prediction of leachate generation from
TRANSCRIPT
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PREDICTION OF LEACHATE GENERATION
FROM
MINERALS PROCESSING WASTE DEPOSITS
by
Graham Mark Davies
Dissertation submitted to the University. of Cape Town in fulfilment of the requirements for the degree of
Master of Science in Engineering
Department of Chemical Engineering University of Cape Town Rondebosch 7700.
September 1995
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The copyright of this thesis vests in the author. No quotation from it or information derived from it is to be published without full acknowledgement of the source. The thesis is to be used for private study or non-commercial research purposes only.
Published by the University of Cape Town (UCT) in terms of the non-exclusive license granted to UCT by the author.
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Synopsis.
The minerals processing industry in South Africa produces significant tonnages of waste material which are disposed of commonly in dedicated waste depositories. These deposits pose a potential to pollute the environment if leachate is generated within the deposit and released to the surroundings. Leachate generation is generally investigated using laboratory columnar experiments which attempt to mimic the physical and chemical processes which occur in the deposit. These experiments, termed lysimeter experiments,
. are time consuming in that they typically last for at least a few months and can last for up to three years. Lysimeter experiments are also costly to conduct. Because of restrictions such as these, relatively few deposits have been characterised to determine the leachate which they generate and thus the risk which they pose to the environment.
There is an urgent need to be able to estimate the environmental risks associated with existing waste deposits. The first step towards assessing this risk would be an ability to predict leachate generation within a specific deposit. Such an ability could be used to identify which of the existing deposits produce significant leachate and thus pose a potential hazard to the environment. Equally, if leachate generation from new deposits could be estimated as a function of waste material and characteristics of the waste deposit, this information could be used to improve the engineering design of waste deposits.
The work presented in this thesis involved identifying suitable modelling strategies which could be used to determine leachate generation within waste deposits which contain waste material typical of that produced by the minerals processing industry. Two modelling strategies have been investigated. The first modelling strategy involved a
macroscopic model in which all effects such as intrinsic chemical kinetics, intra-particle diffusion, external mass transfer and hydrodynamic considerations are lumped into a single parameter. The result of this approach is an effective reaction rate for the release . of hazardous constituents from a volume element of the waste deposit. The effective reaction rate is determined by fitting the model to experimental data base9 on lysimeter tests. The main advantage of this model is that it eliminates the need for a detailed understanding of the individual factors which contribute to leachate generation. This model was investigated both for its inherent simplicity and for use in cases where insufficient information with respect to the intrinsic chemical reaction rates, intra-particle diffusion, external mass transfer or hydrodynamic aspects exist. The main disadvantage of this model is that it has a limited predictive ability in that the individual significance of any one factor which contributes to leachate generation cannot be determined. For this reason a second, more detailed model, termed the heterogenous columnar model, has also been investigated.
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The heterogenous columnar model describes the release of hazardous constituents at the
single particle level and relates this information to the overall leachate generation within
the deposit. This is achieved by calculating the release of hazardous constituents from
the size distribution of particles to the bulk fluid between these particles. The release
of hazardous constituents from individual particles is determined by making use of a
particle-scale chemical reaction 'model. This particle-scale model is sufficiently detailed
to be able to determine the relative contribution to the overall release of hazardous
constituents from the particles of intrinsic chemical kinetics of the reactions to the effects
of diffusion of the fluid reagent into each particle. The heterogenous columnar model
can also be used to determine whether the effective rate of release of hazardous
constituents from the particles (which include intrinsic kinetic and diffusional
contributions) or the flow of fluid reagent through the deposit limits the release of
hazardous constituents from the deposit. This information can be used to determine the
main factors which affect the release of hazardous constituents from waste deposits and
can thus be used to improve the design of waste deposits.
Probably the most important attribute of the heterogenous columnar model is that
methods have been investigated to determine the model parameters from a simple
continuously stirred tank reactor (CSTR) type experiment.
The ability of the heterogenous columnar. model to predict leaching behaviour has been
investigated using data on precious metal leaching found in the literature. The results
are encouraging in that the model can accurately predict the leaching behaviour of
precious metals. A preliminary investigation into determining suitable particle-scale
model parameters for a sample of waste from a CSTR experiment has been conducted.
This too has yielded encouraging results. However, the application of using the
heterogenous columnar model using these parameters to describe leachate generation
within waste deposits or lysimeter experiments still needs to be demonstrated. Once the
heterogenous columnar model has been verified against data pertaining to leachate
generation from a waste deposit it may start to provide the minerals processing industry
with the information which it so desperately requires in order to dispose of wastes in a
manner which minimises the impact on the environment.
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Acknowledgements.
I would like to thank Dr J. G. Petrie for his guidance and assistance in this work. I believe that no graduate student can learn to do research effectively on their own and I am grateful for what I have learnt from Dr Petrie. In particular Dr Petrie has taught me the level of rigour required in research and the ability to constructively question any results obtained to determine their significance and accuracy. In addition to this, working under Dr Petrie I have become aware of the pressing environmental issues facing the chemical and minerals processing industries in South Africa. I believe that such an awareness will be invaluable to me as a chemical engineer.
I am also grateful for having had the opportunity to work in the 'Greenhouse' in the Department of Chemical Engineering at the University of Cape Town. Working with such a diverse and talented -group of people has been most stimulating and has kept me on my toes. The weekend away at Wilderness, the mini investigation into possible methods to clean up oil on the beaches and the commissioning of the 'fish tank' are only a few of the many things which made working in the group so interesting.
Last, but not least, I wish to gratefully acknowledge the financial assistance of MINTEK. Without this assistance this project would not have been possible.
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Table of Contents.
Synopsis.
Acknowledgements.
Table of Contents.
List of Tables.
List of Figures.
Nomenclature.
1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Statement of the Objectives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Research Methodology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Thesis Layout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2. Relevant Considerations when Modelling Leachate Generation and Mobility. 7
2.1 Particulate and Hydrodynamic Considerations. . . . . . . . . . . . . . . . . 7
2.1.1 Particulate Considerations. 7
2.1.2 Hydrodynamic Considerations. 9
2.2 Review of Existing Leachate Generation and Mobility Models. 14 -
2.2.1 Empirical Models. . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.2 Models which make use of the Continuity Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.3 Models Suitable for Predicting Leachate Generation from Solidified Monolithic Structures. . . . . . . . . . . . . 18
2.2.4 Models Describing Contaminant Migration away from Deposits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Summary of the Strategy Adopted in Precious Metal Heap Leaching Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
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2.4 Modelling the Release of Hazardous Constituents from Partially Saturated Granular Waste Deposits. . . . . . . . . . . . . . . . . . . . . . . . 25
3. A Macroscopic, Lumped Parameter Model to Describe Leachate Generation and Mobility in Granular Waste Deposits. • •...•••••••.•.. ·• • • • • • • • . • • 29
3.1 Development of the Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Expressing the Equations in Dimensionless Form. . . . . . . . . . . . . . 33
3.3 Solution Strategy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4 Introduction _ of a Variable Fluid Velocity into the Solution Strategy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.5 Suitable Computer Routines for the Model. . . . . . . . . . . . . . . . . . . 37
3.6 Verification of the Computer Routines. . . . . . . . . . . . . . . . . . . . . . 38
3. 7 Results and Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3. 7 .1 General Behaviour. . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.7.2 Effect of Competing Reactions. . . . . . . . . . . . . . . . . . 44
3.8 Fitting the Model to Lysimeter Experiments. . . . . . . . . . . . . . . . . . 49
3.8.1 Model Requirements. . . . . . . . . . . . . . . . . . . . . . . . . 49
3.8.2 Fitted parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.9 Limitations of the Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4. A Summary of the Chemical Reaction Model Applicable to Single Particles as Developed by Dixon [1992]. . . . • . . . . . . • . • . . . . • . . . . . . . . . . . . . . . . . 52
4.1 Development of the Equations. . .................... ~ . . . . . 52
4.2 Expressing the Equations in Dimensionless Form. . . . . . . . . . . . . . 55
4.3 Suitability of Dixon's Model to Hazardous Constituent Leaching From Waste Particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4 Solution Strategy. . ....................... : . . . . . . . . . . . . 57
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4.5 Suitable Computer Routines for the Model. . . . . . . . . . . . . . . . . . . 59
4.6 Verification of the Computer Routines. . . . . . . . . . . . . . . . . . . . . . 60
4.7 Application pf the Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5. A Model to Describe Leachate Generation from Granular Wastes in a Continuously Stirred Tank Reactor Experiment. . . . . . • . . . . . • . . . • . . • . 66
5 .1 Development of the Mass Balance Equation for the Bulk Fluid Reagent in a CSTR which Contains Equi-Sized Particles. . . . . . . . . 66
5.2 Model Parameters as a Function of Particle Size. . . . . . . . . . . . . . . 68
5.2.1 Determination of the Model Parameters Applicable to Precious Metal Leaching with Respect to a Reference Size Class of Particles. . . . . . . . . . . . . . . . 68
5.2.2 Determination of the Model Parameters Applicable to Leaching of Hazardous Constituents from Waste Particles with Respect to a Reference Size Class of Particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5. 3 Extension of the Bulk Fluid Mass Balance Equation to Incorporate Fluid Reactant Consumption from a Size Distribution of Particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.4 Solution Strategy: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.5 Suitable Computer Routines for the CSTR Model. . . . . . . . . . . . . . 73
5.6 Verification of the Computer Routines. . . . . . . . . . . . . . . . . . . . . . 74
5. 7 Results and Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5. 7.1 General Behaviour. . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5. 7. 2 Effect of Particle Size Distribution on the Fractional Conversion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5. 7. 3 Effect of the Location of Hazardous Constituents on the Fractional Conversion. . . . . . . . . . . . . . . . . . . . . 83
5.8 Fitting the Model to CSTR Results. . . . . . . . . . . . . . . . . . . . . . . . . 85
5.8.1 Model Requirements. . . . . . . . . . . . . . . . . . . . . . . . . 86
5.8.2 Fitted Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . 86
/
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5.9 Applications and Limitations of the Model. . . . . . . . . . . . . . . . . . . 87
5.9.1 Applications of the model. . . . . . . . . . . . . . . . . . . . . 87
5.9.2 Limitations of the model. . . . . . . . . . . . . . . . . . . . . . 87
6. A Microscopic, Columnar Model to Describe Leachate Generation and Mobility in Granular Waste Deposits. • • • • • • • • • • • • • • • • • • • • • • • • • • • 89
6.1 A Modelling Strategy based on Heap Leaching Models. . . . . . . . . . 90
6.2 A Modelling Strategy based on a Rigorous Mathematical Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.3 Inclusion of a Global Wetting Factor into the Solution Strategy. 96
6.4 Suitable Computer Routines for the Heterogenous Columnar Model. ............................................. 97
6.5 Verification of the Computer Routines. . . . . . . . . . . . . . . . . . . . . . 99
6.6 Determination of the Heterogenous Columnar Model Parameters from Appropriate CSTR type Experiments. . . . . . . . . . . . . . . . . . . 103
6. 7 A Summary of the Experimental Data which is Required to Verify the Applicability of the Heterogenous Columnar Model to describe the Leaching of Hazardous Constituents from Waste Deposits ..... 105
7. Summary of the Applications, Limitations and Extensions of the Heterogenous, Columnar Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7 .1 Comparison of the Heterogenous Columnar Model to the Model of Roman et al. [1974] and the Columnar Model of Dixon [1992, 1993] ............................................... 110
7.1.1 Comparison to the Model of Roman et al. [1974] .... 112
7.1.2 Comparison to the Columnar Model of Dixon [1992, 1993] ..................................... 113
7 .2 A Summary of the Potential Engineering Applications of the Heterogenous Columnar Model. .......................... 114
7.2.1 Improved Deposit Design based on Results from the Heterogenous Columnar Model. . . . . . . . . . . . . . . . 114
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7. 2. 2 Using the Heterogenous Columnar Model to Choose Upstream Processes which would Result in More Stable Wastes. . ............................ 116
7.2.3 . Using the Heterogenous Columnar Model to Asses the Risks and Liabilities Associated with Existing Waste Deposits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7. 3 · Limitations and Possible Extensions of the Heterogenous Columnar Model. . ............................................ 117
7.3.1 Incorporation of External Mass Transfer Resistances into the Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.3.2 Inclusion of Intra-Particle Dissolved Species Transport Resistances into the Model. . . . . . . . . . . . . 119
7.3.3 Inclusion of Matrix Dissolution and Hazardous Constituent Re-Precipitation in the Heterogenous Columnar Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.3.4 Inclusion of More Realistic HydrodynamiC Flow Models into the Heterogenous Columnar Model. . . . . 120
7.4 Statement of the Significance of the Work Presented in this Thesis.
References.
Appendices.
Appendix I.
Appendix IL
Appendix III.
Appendix IV.
. Appendix V.
122
Summary of the Method of Characteristics.
Solution Algorithm and Code for Model4D 1.PAS and Model4D2.PAS.
Solution Algorithm and Code for Model2D2.PAS.
Solution Algorithm and Code for Model5El .PAS and Model5E2. PAS.
Solution Algorithm and Code for Model6Cl.PAS and Model6C2.PAS.
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List of Tables.
2-1. Summary of the Models which can be used to Describe the Release of Hazardous Constituents from Waste Deposits. . . . . . . . . . . . . . . . . . . . . . 15
3-1. Summary of the Parameter Combinations Investigated to Determine the Effect a Buffer Material on Hazardous Constituent Release. . . . . . . . . . . 45
3-2. Summary of Parameters used to Demonstrate Extension of the Model to More than One Reactive Hazardous Constituent. . . . . . . . . . . . . . . . . . . . 48
5-1. Size Distribution and si,k used in the Analysis. . . . . . . . . . . . . . . . . . . . . 76
5-2. Reference Size Class Parameters. (Reference Size Class = Size Class 8.) ............ : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5-3. Summary of the Conditions used to Investigate the Effect of Size Distribution on Fractional Conversion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5-4. Summary of the Conditions used to Investigate the Effect of Hazardous Constituent Location on Fractional Conversion. . . . . . . . . . . . . . . . . . . . . 84
6-1. Physical Properties and Operating Conditions. . .................... 100
6-2. Size Distributions. (% Occurrence.) ............................. 101
6-3. Summary of the Conditions in the CSTR Test on a Waste Sample.. . .... 107
6-4. Values of the Fitted Parameters. . .............................. . 108
7-1. Summary of the Properties of the Heterogenous Columnar Model and the Heap I:eaching Models of Roman [1974] and Dixon [1992,1993] ........ 108
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List of Figures.
2-1. Film, rivulet and filament flow patterns as described by Lutran et al. [1991]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
·2-2. CAT scans used to investigate the flow patterns in a square column packed with equi-sized glass spheres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2-3. Layout of a waste form in a landfill. . . . . . . . . . . . . . . ~ . . . . . . . . . . . . . . 13
2-4. Inference of leaching rates following different leaching scenarios. . . . . . . . 20
2-5. An ore heap conceptually divided into columnar sections and the columnar sections divided into disks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2-6. Summary of the strategy used to determine the breakthrough curve in Heap leaching operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2-7. An example of a two dimensional, porous medium model used to simulate the geometrical characteristics of particles in a trickle bed reactor and a typical flow pattern predicted by using the strategy of Zimmerman et al. [1987]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3-L Schematic of a lysimeter which forms the basis for the macroscopic, lumped parameter model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3-2. Comparison of the predicted flow profile with the analytical profile for a lysimeter in which no chemical reactions take place. . . . . . . . . . . . . . . . . . 38
3-3. Chemical species in the model are interchangeable. . . . . . . . . . . . . . . . . . 40
3-4. Profiles of hazardous constituents which react at the same rate are co-incident. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3-5. Fluid reagent profiles and solid reactant profiles within a lysimeter predicted by Model4Dl for various parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3-6. Conversion curves for a lysimeter predicted by Model4Dl for various parameters.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 43
3-7. Concentration profiles for two competing reactions. . . . . . . . . . . . . . . . . . 46
3-8. Breakthrough curves for two competing reactions. . . . . . . . . . . . . . . . . . . 47
3-9. Sample printout of concentration profiles for more than one hazardous constituent. . . . . . . . . ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
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3-10. Sample printout . of breakthrough curves for more than one hazardous constituent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4-1. Schematic diagram of a porous, spherical particle of radius R and a graph showing the concentration gradients within the particle. . . . . . . . . . . . . . . 53
4-2. Concentration profiles predicted by Model2D2 for the parameters as Indicated in the Figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . 61
4-3. The corresponding concentration profiles to Figure 4-2 presented by Dixon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4-4. Fraction conversion profiles predicted by Model2D2 for the parameters as Indicated in the Figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4-5. The corresponding fractional conversion profiles to Figure 4-4 presented by Dixon. . ................... ·. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4-6. Fraction conversion profiles as a function of the variable order power predicted by Model2D2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4-7. The corresponding fractional conversion profiles as a function of variable order power to Figure 4-6 presented by Dixon. . . . . . . . . . . . . . . . . . . . . . 64
5-1. Schematic of a few equi-sized spherical particles submerged in a well stirred beaker of fluid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5-2. Summary of the Solution Strategy used in the CSTR model. . . . . . . . . . . . 73
5-3. Concentration profiles predicted by Model5El for a single size class of particles with a large excess of bulk fluid reagent. . . . . . . . . . . . . . . . . . . . 75
5-4. Concentration profiles predicted by Model2D2 using the same parameters used in the simulation used to generate Figure 5-3. . . . . . . . . . . . . . . . . . 75
5-5. Size distribution of particles used in analysis . . . . . . . . . . . . . . . . . . . . . . 77
5-6. Hazardous constituent location data used in the analysis. . . . . . . . . . . . . . 78
5-7. Fluid reagent and solid reactant profiles as a function of particle size. . , . 79
5-8. Overall conversion for the system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5-9. Summary of the size distributions used in the simulations. . . . . . . . . . . . . 82
5-10. Fractional conversion for the size distributions investigated. . . . . . . . . . . . 83
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5-11. Summary of the hazardous constituent location data used in the simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5-12. Fractional conversion for surface hazardous constituents concentrations investigated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6-1 Graphical comparison between the macroscopic, lumped parameter model and the heterogenous, columnar model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6-2. Summary of the solution strategy used which was based on Roman's solution strategy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6-3. Solution strategy derived from a rigorous mathematical analysis. . . . . . . . . 96
6-4. Summary of the overall organisation of Program Model6Cl.PAS. . . . . . . . 98
6-5. A typical display produced by Model6Cl.PAS showing the solid and fluid reagent profiles for the smallest, largest and reference size class of particles. . .................................. ; . . . . . . . . . . . . . 99
6-6. Size distribution of the particles in L ysimeter 1. 102
6-7. Size distribution of the particles in Lysimeter 2. 102
6-8. Fitted curve and predicted curve for Model6C 1 compared to the experimental points of Roman [1974]. . ......................... .' 103
6-9. Predicted CSTR conversion versus time curve for O.llof 9.5mm to 13.2mm particles and 1l of 48.8 gp/ of acid ............................... 104
6-10. Concentration of dissolved magnesium in the bulk fluid as a function of timd.07
6-11. Fractional conversion versus dimensionless reaction time. . . . . . . . . . . . . . 108
6-12. Model5El versus CSTR Experimental Data on a Waste Sample. . . . . . . . 109
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Nomenclature.
Roman Letters
DG2. I
DG3. I
E I
~i ~i L M
z
stoichiometry number, kg solid reactant ilkg fluid reagent concentration of reagent A, kg Alm/ concentration of reagent A external to particle, kg Alm[ initial extractable grade of solid reactant i, kg ilkg ore concentration of dissolved species i, kg ilmr 3
bulk concentration of dissolved species i, kg ilm/ grade of solid reactant i within particle, kg i/kg ore initial grade of solid reactant i within particle, kg ilkg ore grade of solid reactant i on particle surface, kg ilkg ore initial grade of solid reactant i on particle surface, kg ilkg ore fraction of the void space within the deposit filled with fluid, dimensionless effective pore diffusivity of reagent A, m[lfI\ s effective pore diffusivity of dissolved species i, m[I~ s ratio of the actual fluid percolation velocity to a reference percolation velocity, dimensionless ratio of the chemical reaction rate i at z=O to the rate of fluid reagent replenishment, dimensionless dimensionless stoichiometric ratio which indicates the fluid reagent strength relative to the solid reactant concentration i within the deposit extraction of dissolved species i, dimensionless rate constant of solid reactant i within particle, kg ilkg ore s[Cp]0 [CA] rate constant of solid reactant ion particle surface, kg i/mP2 s [CJ" [CA] deposit depth, m number of size classes in the size distribution of particles, dimensionless number of solid reactants pellet flow Reynolds number, dimensionless volumetric flow rate of fluid into the deposit, m31s radius, m rate of 'production' of fluid reagent A by reaction with solid species i, kglm2sparticle radius, m time, s reference space time for the deposit, s superficial bulk flow velocity, mi lmh2 s reference fluid velocity (percolation velocity), mis fractional conversion of solid reactant i, dimensionless depth, cm
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•.
"'·
Greek Letters
dimensionless concentration of reagent A dimensionless concentration of reagent A external to particle reagent strength parameter relative to solid reactant i, dimensionless modulus in steady-state model ratio of diffusivity of dissolved species i to reagent A, dimensionless bulk solution volume fraction
€ c01 heap void fraction € 0 ore porosity, cm//cm/ !: dimensionless depth T k ratio of the volume of the particles in size class k to the volume of the
particles in the reference size class, dimensionless
Tl effectiveness factor for solid reactant i, dimensionless e dimensionless flow time Kpi Damkohler II number for solid reactant i within particle, dimensionless Ksi Damkohler II number for solid reactant i on particle surface, dimensionless A.i surface fraction of solid reactant i, dimensionless 11 ratio of volume bulk fluid to fluid in particle pores, dimensionless ~ dimensionless radius Z dimensionless particle radius p0 ore density, g ore/cm3 ore a pi dimensionless grade of solid reactant i within particle asi dimensionless grade of solid reactant i on particle· surface T dimensionless diffusion time T ' dimensionless flow time for the deposit ¢pi reaction ·order for solid reactant i within particle, dimensionless <Psi reaction oredr for solid reactant i on particle surface, dimensionless Xi dimensionless concentration of dissolved species i Xib dimensionless bulk concentration of dissolved species i t ratio of the surface grade of hazardous constituents to the bulk grade of
hazardous constituents within the particle, dimensionless
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Chapter 1. Introduction.
Increasing environmental awareness is challenging both the chemical and minerals processing industries to examine the type and quantity of the waste which they produce, and to critically assess the methods used to dispose of this waste. Presently new technologies are being developed to minimise the overall waste produced. Since the total eradication of wastes is not feasible from both thermodynamic and economic perspectives, industry is still faced with the problem of safely disposing of chemical and mineral wastes in such a manner as to ensure their minimal impact on the environment.
One industry in particular which is being forced to assess the impacts of its wastes on the environment is the minerals processing industry. This industry produces high tonnages of waste streams which include slimes, slags, bag house dusts and leached ores. Slimes consist of solid particles suspended in fluid from the extraction process, while slags, dusts and the leached ores are all granular solid waste particles. All these solid particles contain leachable components in addition to significant quantities of inert material.
These wastes are presently disposed of in various ways. In each case attempts are made to minimise the impact on the environment by containing the waste and any leachate generated within the site of disposal. Slimes are usually disposed of in slimes dams or tailings impoundments. These structures are designed to allow the process fluid to drain from the solid particles and to be returned to the process. The solid particles accumulate in the dams or tailings impoundments until the useful capacity of these structures have been exhausted. Once this stage has been reached, the remaining process fluid is allowed to drain and the dams and tailings impoundments resemble deposits of very fine granular material on closure. There is a possibility for leachate to be generated even after closure when rain water periodically percolates through the system. For this reason these structures not only need to be maintained during their useful lifespan but also for extended periods of time after their closure.
Granular solid wastes in tum are usually disposed of in dedicated waste deposits. These deposits are engineered entities which may or may not have clay or polymer liners to prevent any leachate which may be generated from penetrating the environment. In granular deposits leachate may be generated from rain or ground water which penetrates and percolates through the system. In cases where significant leachate is generated, leachate collection sumps and treatment processes are provided. As in the case with slimes dams and tailings impoundments, these deposits need to be maintained well after they have been filled to capacity with waste. Both slimes dams and dedicated deposits have hazards associated with them. The most obvious hazard is their potential to pollute the environment if they are not sufficiently maintained, especially for the extended periods of time after they have been filled with waste. Further, the liners used in these systems are not perfect and may deteriorate with time, allowing potentially harmful leachate to enter the environment.
A more attractive option which eliminates the need for liners, leachate collection sumps
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and treatment systems is where the waste is physically or chemically pretreated in such a manner to either totally eliminate leachate generation, or to reduce it to release rates which the natural environment can assimilate. Although deposits containing stabilised wastes have the advantages of not requiring maintenance and eliminate the possibility of liner failure, their long term stability needs to be assessed. In this respect, the stability of deposits over at least a few decades needs to determined as well as the stability of deposits over much greater time periods. These time periods are consistent with the time scales being considered in Life Cycle Assessment of waste disposal strategies [Clift 1995].
Since leachate generation and subsequent transport through granular waste deposits represents a significant environmental hazard, an ability to predict these phenomena would be most useful. In the case of slimes dams and dedicated waste deposits, such an ability would enable process engineers to calculate the leachate which would require treating and which would penetrate the environment in the case of a liner failure. This information could then be used to assess the potential environmental hazards of such waste disposal strategies and would form part of a rigorous risk assessment of landfill practice. Since upstream processes directly affect the properties of the waste which needs to be disposed of, an ability to predict the leachate generation as a function of the waste properties would enable process engineers to investigate upstream processing strategies which would produce· wastes which limit the production of leachate. This would enable the processing system to be optimised to limit the impact of its waste on the environment. Further, an ability to predict the leachate generation from deposits which contain stabilised wastes could be used to determine the effectiveness of the stabilisation procedures. In summary, an ability to predict leachate generation and subsequent transport through granular waste deposits would start to provide the minerals processing industry with the tools which it requires in order to determine the most appropriate disposal strategy for its wastes.
In order to predict leachate generation and transport through granular waste deposits, the major physical and chemical processes involved need to be identified. In this work it has been assumed that the leachate is generated by the reaction of a fluid reagent with the granular solid waste particles. These particles usually contain several potentially hazardous and leachable constituents in addition to other components which exhibit buffering capacity within an inert matrix. Here buffering capacity refers to any component which will provide a neutralising capacity to acid which flows through the system. As an example, many wastes contain an alkali silicate matrix which behaves as a buffering component. t::>article characteristics which affect the release of hazardous constituents include particle size, shape and hazardous constituent location. Because of the nature of waste streams, these characteristics vary significantly both between the types of wastes as well as between individual waste particles.
Fluid reagent flow characteristics also play an important role in both the release and subsequent transport of hazardous constituents. The flow patterns are important because they determine the extent to which the fluid is in contact with the individual waste particles. In this work it has been assumed that once hazardous constituents have been released they are transported by the bulk convection of the fluid reagent. Thus the fluid flow patterns significantly affect the mobility of the hazardous constituents. Other physical processes which affect this mobility include adsorption and desorption reactions of the released hazardous constituents onto the surfaces of the solid waste particles.
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Closely associated with fluid flow characteristics is the degree of saturation within the deposit. This is an important consideration because it determines the amount of fluid reagent available for reaction as well as the available wetted surface area of the particles. The influence of the degree of fluid saturation is particularly important in the analysis of South African waste deposits which are rarely saturated with fluid. This is in stark contrast to most North American and European deposits which are usually saturated.
The work presented in this thesis is a start towards determining suitable strategies for the prediction of leachate generation and its subsequent transport through waste deposits which contain granular wastes typical of those produced by the minerals processing industry.
1.1 Statement of the Objectives.
The objectives of the thesis can be summarised as follows:
To derive physical models which describe the generation and subsequent transportation of leachate within waste bodies which contain solid granular material with several leachable constituents.
To represent these models mathematically and to propose . suitable strategies for their solution.
To code these models into suitable computer routines using the appropriate solution strategies.
To identify suitable experimental techniques to quantify any model parameters which are required.
J'o explore methods to verify the models against experimental data which is typically in the form of laboratory column experiments.
1.2 Research Methodolo107.
The literature has been reviewed critically to determine the state of the art with respect to leachate generation and mobility within granular waste deposits. As will be shown in the literature review, none of the existing modelling strategies include a sufficient level of complexity required to either predict the leachate generation or to yield sufficient information about the system which could be used to engineer improved disposal practices. Noting similarities between the leaching of hazardous constituents from waste particles and precious metal heap leaching operations, the modelling strategies used for precious metal heap leaching have also been investigated. As will be shown in the literature review, these too do not contain sufficient detail to adequately model leachate
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generation when dealing with waste particles as opposed to mineral ores. Despite this, it seemed feasible that these modelling strategies could be extended to include the required complexities and this forms the major part of this work.
A second area of existing modelling strategies which has value in attempting to address the complexities associated with leachate generation and mobility in waste bodies is the design of trickle bed reactors. These models too have been reviewed and included in the present development.
The research methodology adopted in this work was to identify or develop models which are sufficiently detailed to predict leachate generation and mobility within granular waste deposits and yet not too complex to preclude implementation. Factors which often limit the usefulness of models include large computational loads, a large number of parameters which need to be determined or when required inputs to the model cannot be determined experimentally. An example of a model input which would be difficult to determine experimentally would be if the model required a full speciation of the components within the waste particles.
The first model investigated was a macroscopic model in which the effects of chemical kinetics, fluid flow characteristics, wetting efficiency and fluid saturation were 'lumped' together into a single parameter. The macroscopic, lumped parameter model describes the leachate concentrations within a granular waste deposit as well as the leachate concentration emanating from the base of a deposit as a function of time. The main reason for investigating this model was its inherent simplicity and its ability to characterise granular deposits as reacting entities. The main disadvantages of this model stem from the lumped parameter used in the model. Because this parameter lumps together the effects of chemical kinetics, fluid flow characteristics, wetting efficiency and fluid saturation on the release of hazardous constituents, it is not possible to isolate the individual contributions of any of these factors. This limits the use of this model for waste deposit design because the main characteristics of a deposit which can be altered include the chemical kinetics and hydrodynamic aspects. Without a knowledge· of which particular aspect is dominant in causing contaminant release it is impossible to engineer better deposits-- other than on a trial and error basis. The lumped parameter is specific to a particular situation and must be determined from an appropriate laboratory column experiment. This .is a severe limitation in that a column experiment, usually referred to as a lysimeter experiment, is required for each different waste and fluid flow scenario envisaged. The disadvantages of lysimeter experiments are the costs involved with such experiments and the long duration of the experiments which typically last for at least a few months and can continue up to two or three years. Because of these limitations it was decided to investigate a model which would be capable of isolating the chemical and hydrodynamic effects on the release of hazardous constituents and which, if possible, obviated the need for lysimeter experiments to determine the model parameters.
This leads to the second model investigated. This model, termed the heterogenous columnar model, also describes the leachate concentrations within a waste deposit as well as the concentration of the leachate emanating from the base of the deposit as a function of time. The main difference is that this model is neither a macroscopic nor a lumped parameter model. Instead, this model calculates the leachate generated as a function of
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the individual particles present in the waste deposit as well as the fluid flow characteristics, wetting efficiency and fluid saturation. The other main advantage of this model is that the required model parameters can be determined from appropriate CSTR type experiments on small samples of the waste material. The time savings incurred by using CSTR type experiments over lysimeter experiments are significant.
The limitations of the heterogenous columnar model as it is presented in this thesis include the fact that only perfect plug flow of fluid through the deposit has been considered, interphase mass transfer limitations or resistances between the bulk fluid reagent and particle surfaces have been neglected, and, adsorption and desorption of released hazardous components has not been considered. It is important to note that more complex flow scenarios, interphase mass transfer and adsorption and desorption reactions can be included into the present model. The model has been specifically coded in a modular manner to allow the incorporation of these aspects.
1.3 Thesis Layout.
The thesis commences with a review of relevant considerations when modelling leachate generation and mobility in granular waste deposits. In this section, more detailed information on particulate and hydrodynamic characteristics of the waste and waste deposit is presented. This is followed by a critical review of modelling strategies with respect to the release of hazardous constituents from granular waste deposits. This review covers existing leachate generation modelling strategies, heap leaching modelling strategies and trickle bed reactor design strategies.
Chapter 3 presents the model development for the macroscopic, lumped parameter model. This chapter includes the derivation of the equations required for this model, an investigation into an appropriate solution strategy and details with respect to computer routines which have been written to implement the solution strategy. Also included in this chapter are details of how to fit the model to lysimeter data. The limitations of the macroscopic, lumped parameter model indicated that a more complete, particle scale model would be required to determine sufficient detail about the system in order to facilitate active engineering of the deposit to limit the release of hazardous constituents from granular waste deposits.
The more complete model, which determines the release of hazardous constituents from individual waste particles and relates this information to the overall waste deposit performance, is developed in Chapters 4 to 6. Chapter 4 is a summary of Dixon's [1992] chemical reaction model which describes the progression of reactions within an individual waste particle. As will be shown, this model has limited applicability on its own. Waste deposits consist of a size distribution of particles and a finite amount of fluid reagent. The performance of this system, rather than that of a single particle, is the desired end product. For this reason the chemical reaction model of Dixon [1992] was extended to form a suitable CSTR model in Chapter 5 and a heterogenous columnar type model in Chapter 6. A CSTR type model has been included due to the fact that the model
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parameters required in the heterogenous columnar model can be determined from a CSTR type experiment. The heterogenous, columnar model has been verified against experimental data presented in a paper by Roman et al. [ 197 4]. The details of this verification have also been included in Chapter 6.
The final chapter presents a summary of the conclusions and recommendations which can be made from this work .
...
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Chapter 2. Relevant Considerations when Modelling Leachate Generation and Mobility.
This chapter discusses important aspects of leachate generation and mobility within granular waste deposits. As discussed briefly in the introduction, both the physical composition of the waste particles and the fluid flow characteristics within the waste deposit affect the manner in which hazardous constituents are released from the waste. The first section of this chapter considers these particulate and hydrodynamic considerations which are important in modelling leachate generation. In this section, information from the literature on trickle bed reactor design has been used to help understand the complex flow patterns of fluid within granular waste deposits.
The existing leachate generation and mobility models which can be found in the literature have been reviewed. This section critically assesses which of these models have any potential to predict leachate generation within deposits which contain granular wastes typical of those produced by the minerals processing industry.
As alluded to in the introduction, heap leaching processes, designed to extract precious· metals from ores, share similarities to leachate generation within waste deposits. In both cases the leachate is generated from the reaction of a fluid reagent with solid particles. Both processes have a similar dependence on the solid reactant location within the ore or waste particles and the fluid flow characteristics. The major difference between the two processes is that the generation of leachate is desired in the case of heap leaching and undesired in the leaching of hazardous constituents. This leads to significant differences between the systems. The fluid reagent used in heap leaching for example is usually chosen in such a manner as to selectively extract one desired component from the ore matrix. This is in contrast to the leaching of hazardous constituents where several contaminants are leached simultaneously by fluid percolating through ·the deposit. Further, heap leaching processes are designed to optimise the wetted area of the particles. When waste deposits are designed, if any consideration is given to particle wetting, it would be to limit the surface area of the particles in contact with the fluid percolating through the deposit. Despite these differences between the two systems, the models for heap leaching of precious metals are a good starting point for developing suitable models to predict leachate generation within waste deposits. For this reason the general strategy behind these models is presented.
The final section of this chapter deals with modelling leachate generation and mobility within unsaturated deposits.
2.1 Particulate and Hydrodynamic Considerations.
2.1.1 Particulate Considerations.
Granular solid wastes produced by the minerals processing industry vary considerably from one waste stream to the next in terms of the number and type of reactive
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components present within the particles. Even within a single waste stream, the spatial distribution of the components may vary between the different sized particles. These differences need to be investigated to determine their significance in the modelling of leachate generation.
Number and t,me of reactive components present within the waste particles.
The most desirable form of waste is a waste in which the constituent particles are totally inert to fluid reagent percolating through the deposit. In this case there would be no reactive components and leachate could not be generated. Although wastes of this nature are the goal of pretreatment and stabilisation processes, they are seldom achieved in reality. The simplest waste type which is capable of producing leachate is the waste which contains a single reactive component. Such wastes are very uncommon. Far more probable however are wastes which contain two or more reactive components which display significant leaching potential.
The reactive components in waste particles can either be hazardous constituents or other components, termed buffering components, which merely consume fluid reagent but do not release any hazardous constituents. Typical hazardous constituents in minerals processing wastes are heavy metals while the buffering capacity is usually due to fluxing agents.
It is difficult to provide a general description of the relative amounts of leachable hazardous constituents, buffering and inert material within waste particles. Leached ores or tailings residues, for example, contain a relatively high fraction of inert material with trace amounts of leachable hazardous constituents and buffering material. This is in contrast to a slag waste stream which has a high buffering material content in addition to leachable hazardous constituents and inert material.
Although there may be several potentially leachable hazardous constituents within waste particles, usually only a few occur in sufficient quantities or are sufficiently reactive to be of concern. -For example, although wastes from the minerals processing industry often contain several heavy metals, usually only one or two of these are in significant concentrations to pose a hazard if leached out of the waste.
Hazardous Constituent location within waste particles.
Hazardous constituents present in waste particles may either be distributed homogenously throughout the particle or be concentrated onto its external surface. These differences in hazardous constituent location are both waste specific and particle ·size specific and are usually a result of the conditions under which the waste was produced. As an example of hazardous constituent location as a function of waste type, compare the hazardous constituent location in a slag waste stream and an electric arc furnace dust. The hazardous constituents present in a slag waste stream would be expected to be distributed reasonably homogenously throughout the particle. This differs
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markedly from an electric arc furnace dust which usually exhibits surface concentrations of heavy metals much higher than the bulk concentrations within the particles [Driesinger et al. 1990].
Turning to the case of hazardous constituent location as a function of particle size, work conducted at the University of Cape Town on specific ferro-alloy waste products indicated that different sized particles exhibit different surface hazardous constituent concentrations [von Blottnitz 1994].
2.1.2 Hydrodynamic Considerations.
Hydrodynamic aspects are concerned with the transport of fluid through waste deposits and the manner in which the fluid and solid waste are contacted. Hydrodynamic aspects, for granular waste deposits as well as for solidified monolithic structures, have been investigated in order to determine their significance on the modelling of leachate generation. Monolithic structures are obtained when granular wastes are mixed with suitable binding agents, cement being one example, in order to physically bind the particles together. Even although the particles are physically bound together, monolithic structure exhibit a continuous pore structure through which fluid can enter the structure. Although monolithic structures are not directly within the brief of this work, the fluid flow patterns associated with these structures can be considered as a limiting case for granular waste deposits which contain very fine, densely packed material.
Fluid flow patterns within granular waste deposits.
Fluid flow patterns through granular waste deposits range from near perfect plug flow to highly irregular flow patterns involving a few preferential flow paths for the fluid reagent. Knowledge of the flow patterns is important because it is related directly to the solid-liquid contacting efficiency.
Trickle bed reactors, common within the process engineering community, are packed bed reactors through which one or more fluid is allowed to flow. These reactors exhibit a _ wide variety of fluid flow regimes which range from strictly trickle flow, in which a single liquid flows downward through the reactor under the influence of gravity, to pulse and foaming flow, caused by high flowrates of gases and liquids through these reactors. Since the physical situation in a trickle bed reactor operating in trickling flow regime is comparable to fluid percolation through granular waste deposits, a knowledge of the flow patterns in these reactors can be used to understand the fluid flow patterns in granular waste deposits.
A review of the literature on trickle bed reactors shows that, traditionally, with respect to hydrodynamic considerations, only two quantities, the liquid holdup ·and wetting efficiency, have been measured and correlated [Columbo et al. 1976; Schwartz et al. 1976; Mills and Dudukovic 1981; Ramachandran et al. 1986; Gianetto and Specchia 1992]. Although these two quantities are useful to describe the overall bulk effects of the
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hydrodynamic interactions, they offer little insight into the flow patterns.
Until fairly recently, relatively little was understood about the flow patterns in trickling flow trickle bed reactors due to the difficulties involved with determining these patterns. A novel use for computer assisted tomography (CAT scans) has eliminated most of these difficulties and enabled Lutran et al. [1991] to investigate the flow patterns in a packed bed of equi-sized glass spheres. At the particle level, they identified two distinct types of fluid flow which they termed 'film' and 'rivulet' flows.· Film flow represents the case where the fluid tends to cover most of the particle surface area, while rivulet flow describes the case where the flow of fluid over a particle is restricted to a narrow band. They also identified pendular structures and liquid pockets at the · microscopic level. Pendular structures reside at the contact points between spheres while liquid pockets fill the pore space between spheres. At the macroscopic level Lutran et al. [1991] identified fluid filaments. A filament was defined to be a stream of fluid flowing down the packed bed. In effect, a filament could be considered as a string of connected liquid pockets. These definitions are graphically depicted in Figure 2-1 (taken from Lutran et al. [1991]).
Figure 2-1. Film, rivulet and filament flow patterns as described by Lutran et al. [1991].
Rivulet
Pendulu StructW'H
Lutran et al. [1991] investigated <the effect of liquid flow rate, the influence of surface conditioning, the influence of particle size, the influence of inlet configurations and the effect of flow history on the flow patterns. A typical result from their work is shown in Figure 2-2 (taken from Lutran et al. [1991]). CAT scans (a) through (t) represent successive vertical planes from the front to the back of a square column packed with equi-sized glass spheres. Filament flow can clearly be seen in these figures as the darker regions moving from the top to the bottom of the column. The experimental conditions under which these CAT scans were taken can be summarised as follows: 3mm glass spheres were used, distilled water was evenly supplied through a uniform inlet distributor
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at a rate of 3.1631/m2 s and the particles were initially dry.
Figure 2-2. CAT scans used to investigate the flow patterns in a square column packed with equi-sized glass spheres. (Taken from Lutran et al. [1991].)
a b c
·'··1· ~-!~~~ ., . . ·.
:~~~~; -:.:.
Although Lutran et al. [1991] do not discuss correlating their findings, it is very unlikely that any correlation would employ the particle Reynolds number as a parameter. The main reason for this is due to the fact that pore size and differences in local porosity seem to play an important role in the flow patterns which are established. As an example, Lutran et al. [1991] have shown that for a constant flowrate, film flow is more prevalent in packed beds containing larger particles (6mm spheres) compared to packed beds containing small particles (3mm spheres) in which filament flow is observed. They attribute this to the fact that packed beds which contain larger particles will have larger
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pores between the particles. They suggest that at the fluid flow rate investigated (3.1631/m2 s), there was insufficient fluid to fill any of the pores between the 6mm spheres and thus establish filament flow. As the fluid flowrate was increased, more filaments were established which supports this hypothesis.
By a similar argument, any regions which exhibit a local decrease in porosity would favour the formation of filaments. Filament formation would also be aided in these areas by the increased number of solid-solid contact points which allows fluid to more easily distribute to these areas due to the lower surface tension forces on the fluid surface [Zimmerman and Ng 1986]. Wastes which contain a size distribution of particles or non-uniformly shaped particles usually exhibit localised regions of decreased porosity. This implies that these deposits would favour filament flow. It is important to note that channelling of fluid, where the fluid flows through deposits in preferential flow paths, is also probably a result of regions of decreased porosity. Channelling results in fluid short circuiting sections of the deposits. Although this is a highly irregular flow pattern which is extremely difficult to model, it is a very desirable flow pattern for fluid in waste deposits because it reduces the number of particles with which the fluid comes into contact as it percolates through . the system.
Fluid redistribution on the other hand is accomplished by capillary and viscous forces. Capillary forces are inversely proportional to the pore size [Ng and Chu 1987]. Thus the capillary pressure is higher in deposits which contain small particles because the pores are relatively small in these deposits. Capillary forces also result in higher void space liquid holdup [Ng and Chu 1987]. This implies that deposits containing smaller particles will contain proportionately more fluid in their pore spaces compared to deposits containing larger particles.
Using the information discussed above, the following heuristics of fluid flow patterns in granular waste deposits are suggested. These are intended to give an idea of the most likely fluid flow patterns under different conditions.
The most likely fluid flow patterns in deposits which contain relatively large particles, such as leached ore particles, would be that of film flow. In this case the fluid would most likely tend 'to cover the surfaces of the particles as it progressed from one particle to the next. As the standard deviation for the size distribution of the particles increases, so too will the likelihood for filament formation and eventual channelling increase. Deposits of this nature are also most likely to exhibit a very low degree of fluid saturation. This implies that the external fluid holdup, that is the holdup of fluid in the pore spaces, would be very low.
As the average particle size decreases, the degree of saturation will increase due to capillary effects. Likely flow patterns include film flow with associated filament flows.
In the case where the deposit contains very fine particles, such as are encountered in slimes dam and tailings impoundment residues, the most likely flow patterns would be filament flow, and the deposit would most probably be almost totally saturated. The size distribution . of particles is unlikely to affect the flow characteristics due to the high capillary forces which would be present. In all probability, the flow patterns in such
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deposits could be described by plug flow models.
Fluid flow patterns within _solidified structures.
Cote and Birdle [1987] have discussed the likely flow scenarios which can be associated with solidified monolithic structUres. In summary they considered the situation shown in Figure 2-3 where the monolithic structure is surrounded by materials of different permeabilities.
Figure 2-3. Layout of a waste form in a landfill. (Taken from Cote and Birdle [1987].)
Ground surface
Material 3
Material 2
Material I
The first case they investigated is when the monolithic structure is in contact with a finite volume of static groundwater. This corresponds to the case when the hydraulic conductivity of the materials (1) and (2) are much smaller than the monolithic structure or material (3). Physically this situation represents the case where a pool of water exists on top of the waste deposit. In this situation, the water penetrates the solidified structure by diffusion.
The second situation addressed was when groundwater flows around the monolithic structure. This is the most common scenario because monolithic structures typically have -hydraulic conductivities which are several orders of magnitude lower than that of the surrounding ground. Because groundwater will follow the path of least resistance, very little, if any, will flow through the structure. Even although there may be no convective flux through the structure, there is still the possibility of a lateral diffusive flux into the structure.
The last case considered is when the groundwater flows through the monolithic structure. This typically occurs when the monolithic structure fails and the hydraulic conductivity increases to a point that it is comparable to that of the surrounding materials. In many respects this situation can be compared to a granular deposit where the fragments of the original structure represent the 'particles' in the deposit.
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2.2 Review of Existin2 Leachate Generation and Mobility Models.
Literature models which address leachate· generation and mobility can be divided into four classes. The first two classes deal with predicting the breakthrough curves from granular waste deposits. The breakthrough curves are the curves which are formed by plotting the dissolved hazardous constituent concentrations which emanate from the base of the deposit as a function of time. The third class applies to modelling hazardous constituent release from solidified monolithic structures. The last group is concerned with hazardous constituent migration away from waste deposits.
All of the literature models are summarised in Table 2-1 which highlights their respective areas of application, advantages and disadvantages.
2.2.1 Empirical Models. -
Purely empirical models fit experimental results obtained from lysimeter studies to an exponentially decaying function of time. These models [Demetracopoulos et al. 1986] were the only models up until the mid 1980's to describe hazardous constituent release from waste deposits. The mathematical form of the model can be summarised as:
where C0
t
C = C e <-Pt> 0 (2-1)
hazardous constituent concentration in the leachate at the time of initial breakthrough (t=O),
time, and,
an empirical constant.
Work carried out at the University of Cape Town [Petersen 1994] has shown that the fitting of lysimeter data to such a function often leads to a poor correlation. Even if the correlation was excellent, such a model could not be used to scale up to full scale deposit proportions because no size dimensions appear in the model. Thus in order to determine the breakthrough curve for leachate from granular deposits, where the fluid flow can often be approximated by one . dimensional flow, a lysimeter of identical height to the envisaged deposit would be required. The only use of such a model would be the extrapolation of data to predict the release of hazardous constituents in the future. Because of the poor correlation and limited predictive ability of this model it is concluded that is has little applicability to modelling the release of hazardous constituents from hazardous waste. deposits.
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. V
I
Tab
le 1
. S
umm
ary
of
Mod
els
whi
ch D
escr
ibe
Con
tam
inan
t R
elea
se
from
Haz
ardo
us
Was
te D
epos
its.
Mod
el:
Not
es:
App
lica
tion
:
Lu
et a
l. (1
981)
. F
irst
mod
el t
o de
scri
be
Non
e.
brea
kthr
ough
cu
rve.
Dem
etra
copo
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(1
986)
. C
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ant
rele
ase
base
d o
n t
he
Dir
ect
extr
apol
atio
n an
d in
itia
l co
ntin
uity
equ
atio
n ap
plie
d to
a
anal
ysis
of
lysi
met
er d
ata.
ly
sim
eter
as
a si
ngle
ent
ity.
Bat
chel
or
(199
0).
Mod
el b
ased
on
sem
i-in
fini
te-m
edia
M
odel
ling
con
tam
inan
t re
leas
e fr
om
diff
usio
n th
eory
. so
lidi
fied
str
uctu
res.
C
heng
and
Bis
hop
(199
0).
Row
e an
d B
ooke
r M
odel
s de
scri
be c
onta
min
ant
Det
erm
inat
ion
of
over
all
I 199
0,19
85a,
1985
bl.
mig
rati
on a
way
fro
ni h
azar
dous
en
viro
nmen
tal
impa
ct o
f ha
zard
ous
was
te d
epos
its.
w
aste
dep
osit
s.
Dan
ce a
nd R
eard
on
( 198
3 (.
Mas
lia
et a
l. (1
9921
.
Sud
icky
et
al.
(198
31 .
Vog
t (1
991)
.
Bru
nn
.. , a
l. [ 1
9741
. M
odel
ling
con
tam
inan
t re
leas
e fr
om
gran
ular
haz
ardo
us w
aste
dep
osit
s.
Rom
an
(197
41.
Sha
fer
(197
9).
Dix
on (
1992
).
Con
tam
inan
t re
leas
e ba
sed
on t
he
Mod
elli
ng c
onta
min
ant
rele
ase
from
co
ntin
uity
equ
atio
n ap
plie
d to
the
gr
anul
ar h
azar
dous
was
te d
epos
its.
in
divi
dual
par
ticl
es i
n th
e de
posi
t.
Mod
elli
ng c
onta
min
ant
rele
ase
from
m
onol
ithi
c ha
zard
ous
was
te
depo
sits
.
Adv
anta
ges:
D
isad
vant
ages
:
Rel
ativ
ely
sim
ple
to u
se.
Lum
ped
para
met
er
anal
ysis
.
Uni
quel
y ch
arac
teri
ses
the
depo
sit
as
Lim
ited
abi
lity
to
pred
ict
perf
orm
ance
a
reac
ting
ent
ity.
un
der
diff
eren
t hy
drod
ynam
ic
cond
itio
ns.
Rel
ativ
ely
sim
ple
to u
se.
Lim
ited
abi
lity
to
pred
ict
perf
orm
ance
un
der
diff
eren
t co
ntam
inan
t an
d bu
ffer
co
ncen
trat
ions
.
.
The
se m
odel
s be
gin
to a
ddre
ss t
he
Ass
umes
tha
t di
ffus
ion
of
the
flui
d in
to
unde
rlyi
ng r
elea
se
mec
hani
sms.
th
e in
divi
dual
par
ticl
es i
s th
e ra
te
lim
itin
g st
ep.
Onl
y co
nsid
ers
a si
ngle
·che
mic
al
reac
tion
tak
ing
plac
e.
Onl
y ap
ply
to s
itua
tion
s in
whi
ch a
ll o
f th
e pa
rtic
les
arc
wet
ted
and
in w
hich
no
pre
forc
ntin
l fl
ow p
aths
, st
agna
nt
zone
s o
r dr
y sp
ots
occu
r.
The
y al
so d
o no
t ad
dres
s cy
clic
wet
ting
and
dry
ing
cycl
es.
Doe
s no
t as
sum
e th
at
diff
usio
n is
the
O
nly
appl
y to
sit
uati
ons
in w
hich
all
of
rate
con
trol
ling
mec
hani
sm.
the
part
icle
s ar
c w
ette
d an
d in
whi
ch
no p
refe
rent
ial
flow
pat
hs,
stag
nant
In
clud
es t
he p
ossi
bili
ty o
f m
ore
than
zo
nes
or
dry
spot
s oc
cur.
T
I1ey
als
o d
o
one
chem
ical
rea
ctio
n ta
king
pla
ce.
not
addr
ess
cycl
ic w
etti
ng a
nd d
ryin
g cy
cles
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2.2.2 Models which make use of the Continuity Eguation.
The second group of models makes use of the fluid continuity equation to describe the contaminant migration through the system. Demetracopoulos et al. [1986] have used such an approach to model leachate generation from domestic waste landfills. They considered the refuse material as a homogenous, partially saturated porous medium. Although domestic waste landfills often contain laminates, which preclude any attempts to approximate the fluid flow as one dimensional, Demetracopoulos et al. [1986] limited their study to cases in which the fluid flow can be approximated as one dimensional. Because of these assumptions, the refuse and flow characteristics are comparable to those found in granular waste deposits.
Demetracopoulos et al. [1986] make use of a hydraulic flow equation to predict the flow of fluid through the system. Demetracopoulos et al [1984] describe the derivation of this equation which was then solved numerically by Korfiatis et al [1984]. These equations can be summarised as:
where e
K(O) q D(O) t z
where I/;
ae + aK(8) _ _E_ [D (S) ae 1 =o at az az az (2-2)
q=K(8) -D(8) ~~ ( 2-3)
moisture content, or the fraction of the control volume occupied by liquid (m3/m3); hydraulic conductivity of the medium, (m/s); volumetric flux per unit bulk area (superficial velocity), (m/s); yapillary diffusivity coefficient, defined by equation (2-4) below, (m2/s); time, (s); and; space coordinate, measured vertically downward (m).
D(8) =-K(8) ~ d8
(2-4)
tension suction head which is defined as the negative capillary pressure potential [Shaw 1994].
The tension suction head versus moisture content relationship which was used can be summarised as:
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where s b
indicates saturation conditions; and; fitted empirical constant.
(2-5)
The release of hazardous waste constituents from a granular waste deposit was modelled using a continuity equation. The resulting equation can be summarised as:
where C E(O) q R
a (6c) + a ( qc) =__£_ rn E(S) ac1 +e R at az az az (2-6)
fluid phase concentration of the hazardous constituents (kg/m3);
longitudinal dispersion coefficient (m2/s); volumetric flux per unit bulk area (superficial velocity) (m/s); and; source or sink term (kg/m3s).
By assuming that the source or sink term R, which corresponds to the rate of release of hazardous constituents, is controlled by the mass transfer between the solid and liquid phases, an equation which describes the generation and transport of non-biodegradable hazardous constituents was obtained and can be summarised as:
where k' s
a(ec) + a(qc> =__£_ [e E(6) ac1 +e k'_§_ (c -c) at az az az S0
st ( 2-7)
rate coefficient for mass transfer (l/s); local solid mass fraction of hazardous constituent available for transfer (kg/m3
);
local solid mass fraction of hazardous constituent available for transfer at time t=O (kg/m3
); and; maximum possible hazardous constituent concentration in the fluid phase (kg/m3
).
Models of this nature are not limited to the case where mass transfer effects control the rate of release of hazardous constituents. By using appropriate expressions for the source or sink term, R, the model can be adapted to describe the situation where chemical kinetic or intra-particle diffusion resistances are rate limiting. In each case, a parameter is being used to quantify the rate limiting mechanism in the release of hazardous constituents.
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Demetracopoulos et al. [ 1986] extended the model to incorporate biological activity. This resulted in two coupled partial differential equations, one which describes the growth and transport of the microorganism population and one which describes the release and transport of hazardous constituents.
The merit of such a model is that it offers a relatively simple method by which to study the release of hazardous constituents from waste deposits.
The disadvantage of this model is that there is no way of predicting the parameters used to describe the rates of release of hazardous constituents. These parameters need to be determined from appropriate lysimeter experiments.
2.2.3 Models Suitable for Predictin2 Leachate Generation from Solidified Monolithic Structures.
Batchelor [1990],Bishop [1990] and Cheng and Bishop [1990] all have developed models which are suitable to predict leachate generation from solidified monolithic structures. From CSTR leach tests on fragments of solidified structures, these investigators found the release of hazardous constituents to be dominated by diffusion internal to the solidified structure. The model considers the monolith as a single entity and determines the rate of release of hazardous constituents using semi-infinite.:.media diffusion theory. The following form of equation is almost always used in this approach: ,
where
(2-8)
hazardous constituent loss during leaching period n (kg); initial amount of the hazardous constituent present in the specimen (kg); volume of the specimen (m3
);
surface area of the specimen (m2);
cumulative time to the end of leaching period n (s); and; effective diffusivity (m2/s).
Equation (2-8) is based on the following semi-infinite-media equation:
where c x
C(X, t) =C0
erf( X ) 2./ (D9 t)
(2-9)
hazardous constituent concentration at position X, distance from the outer surface of the monolith, and,
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t time.
A limitation with respect to the use of equation (2-9) which is not mentioned by any of the investigators is that the following condition must hold before this equation can be applied to finite systems [Welty, Wicks and Wilson, 1984]:
v 2S./Det
> 2 (2-10)
One reason for this omission may be because the assumption of a constant mid-plane hazardous constituent concentration can be considered to be a worst case scenario. This implies that the models would tend to predict a greater loss of hazardous constituents than would occur in finite systems. Another reason for this omission could be because the authors did not consider evaluating their models at times sufficiently large to violate this condition. ·
In general, the leach type models are suitable to extrapolate data in time as well as to predict the release of hazardous constituents from various sized monolithic structures.
The limitation of these models is that they cannot be used to extrapolate the results to different hazardous constituent and buffer concentrations within the solidified matrix. The reason for this is that the effective diffusivity is a fitted parameter which is affected by the concentrations of the hazardous constituents, buffer and inert species within the solidified waste [Cheng and Bishop 1990].
Cote and Birdle [1987] have made use of this class of models to investigate several long term leaching scenarios for solidified waste forms. In particular they investigated the effects on the release and mobilisation of hazardous constituents of the hydraulic regime of the groundwater, chemical characteristics of the groundwater, hydraulic conductivity of the solidified structure and chemical speciation of the hazardous constituents within _ the waste matrix. Figure 2-4 is a summary of the results of their work. This figure can be used to estimate the release rates of hazardous constituents from monolithic structures. As an example, consider the release of hazardous constituents from a fractured monolithic structure. Such a situation would resemble a granular waste deposit and fluid would most likely percolate through the fractured structured as discussed in section 2.1.2. Typical release rates of soluble hazardous constituents in such a situation are shown in region (F) in Figure 2-4. This region indicates that very high leaching rates would be observed ( = 10 000 mmol m-2 day-1
) and that the leaching processes would cease within the first few months due to depletion of the hazardous constituents. Insoluble hazardous constituents could be released by active leaching and typical release rates are shown by region (H) in Figure 2-4. Notice that the leaching rates for insoluble constituents ( =1 mmol m-2 day-1) is much lower than for soluble components as expected.
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Figure 2-4.
I ... 0
"' .. I e 0 e e
• 0 -"' -= .s:; u 0
• ~
Inference of leaching rates following different leaching scenarios. (Taken from Cote and Birdle [1987].)
waste Form Static Aow.ng Grounawatet 10 000
Cattamnant Parn.&tllllty Groiniwatet around the waare t~!llewure
IOw pH neutral pH low pH neutral pH
aoUlle IOw
D I I E B ~ hlgll
1000 inaol&'* low G G il ir180iutll8 hlgll tj
100
10
1.0
0.01 @ "" 'I\ -=- =- ·- :. ... ~
--- Calculated rote
Anticipated trend
0.001
o.0001-l------1''-------r-~-----------r-------------r------------_,_----~------
0.01 0.1 1.0 10.0 I 00.0
Time (years)
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2.2.4 Models Describin2 Contaminant Migration away from Deposits.
The last group of models includes work on contaminant migration away from hazardous waste deposits. These models uncouple the hydrodynamic analysis from the chemical reaction aspects, assuming . that no interaction occurs. The hydrodynamic part, which is a groundwater flow problem, is modelled using a Darcy law approach which incorporates a bulk hydraulic conductivity. A bulk hydraulic conductivity is an average measure of the distance which fluid would move through a porous medium per unit time. This quantity is a function of both the fluid type and fluid saturation within the porous medium [Shaw 1994]. Saturated hydraulic conductivities are usually determined experimentally. Hydraulic conductivities under unsaturated conditions can also be determined experimentally but are much more difficult to determine than saturated hydraulic conductivities [Fourie 1995]. An alternative approach to determine unsaturated hydraulic conductivities is to use appropriate models, prese_nted in groundwater flow texts, which use the saturated hydraulic conductivity as a parameter [Freeze and Cherry 1979]. It is important to note that bulk hydraulic conductivities are quantities which are used to describe the macroscopic flow rates of fluids in porous media. As such, no information with respect to localised flows within the medium can be determined. In cases where this level of detail is required or in cases where local variations in hydraulic conductivities preclude the use of a bulk hydraulic conductivity, the full tensorial form of the hydraulic conductivity would need to be evaluated. Knox et al. [1993] present information with respect to the full tensorial definition of hydraulic conductivities.
The chemical reactions are modelled either by adsorption/desorption isotherms, which are based on experimental results [Rowe and Booker 1990, 1985a, 1985b; Dance and Reardon 1983; Maslia et al. 1992; Sudicky et al. 1983] or by totally predictive thermodynamic analyses [Vogt 1991].
One of the assumptions used in these models is the value assigned to the leachate concentration at the base of the deposit. This is estimated by determining the volume of fluid within the granular deposit and assuming that all of the hazardous components are released into this fluid subject to adsorption and desorption equilibria. Once this leachate concentration has been estimated, the subsequent transport of the leachate through the underlying ground is modelled. Since these models do not address leachate generation aspects they could only be used as suitable mobility models within granular waste deposits. If this strategy could be used as a mobility model it would have the advantage of including the effects of adsorption and desorption on leachate mobility.
Before these models could be used to describe leachate mobility in granular waste deposits, the adsorption/desorption isotherms of the hazardous constituents onto the granular particles would need to be determined. The determination of these isotherms for granular waste particles will be significantly more complex than for inert ground samples. This is due to the fact that it is difficult to determine an adsorption/ desorption isotherm for a material which is itself releasing the same components. Methods to deconvolute the effects of leachate generation from adsorption and desorption processes would need to be determined.
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It is unlikely that thermodynamic analyses could assist to predict the chemical reactions taking place because of the complex chemical compositions of waste particles. In most cases it is impossible, due to physical and financial· constraints, to determine the initial hazardous constituent and buffering material speciation required for thermodynamic analyses.
Although this group of models has limited applicability for the objectives as defined, it will be useful when a complete environmental impact assessment of a hazardous waste deposit is required. Once the release of hazardous constituents from a deposit has been calculated, this information can be used as the input data for the migration models. The subsequent migration through the environment, and thus the health risks and environmental impact, could then be determined.
-2.3 Summary of the Strate1:y Adopted in Precious Metal Heap Leachin1:
Models.
The general strategy used [Roman et al. 1974] in the analysis of heap leaching is to divide the heap conceptually into columnar sections. Each column is then considered as a simple one dimensional, non-catalytic reactor. The progression of the reactions in the individual particles is followed using a suitable chemical reaction model. These reaction models require the concentration of the fluid in contact with the particles. Thus by calculating the fluid concentration in contact with the particles and the progression of the reactions as a function of time, the dissolved hazardous component concentration can be determined. These concentrations can then be used to calculate the breakthrough curve. In order to calculate the fluid concentration in contact with the particles, the columnar sections are further divided into a set of discs which are stacked on top of each other as shown in Figure 2-5. The fluid concentration within each disc is assumed to be constant and is calculated by a simple mass balance. This strategy is summarised in Figure 2-6.
Figure 2-5. An ore heap conceptually divided into columnar sections and the columnar sections divided into discs. (Figure taken from Roman et al. [1974].)
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Figure 2-6. Summary of the strategy used to determine the breakthrough curve in heap leaching operations.
Iterate in time.
End.
Iterate in space.
• Calculate the concentration of fluid in contact with the particles within the disc.
• Calculate the progression of the reaction within the particles using a suitable chemical reaction model.
•Update the dissolved product concentration within each disc.
·Calculate the breakthrough curve.
Braun et al [1974], Roman et al. [1974] and Shafer [1979] have used this approach to model heap leaching operations. In all of these cases very simple hydrodynamic behaviour has been considered. This is evident in that all of the particles within each disk were assumed to be totally· wetted. For the application of heap leaching this is not a bad assumption because totally wetted particles is the objective of heap design and lixiviant spray patterns. Although it was not explicitly stated, these investigators have also assumed that no significant preferential flowpaths or stagnant zones are present.
The chemical reaction model almost always used in this approach is a shrinking core model. This implies that diffusion of fluid reagent into the individual particles is the elementary rate controlling step. This can be a severe limitation because in some cases the chemical reactions are sufficiently slow to result in a homogenous reaction. In these cases, the intrinsic kinetics of the chemical reactions become very important. The simple shrinking_ core model as used by Roman et al. [1974] has a further limitation in that it
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only considers a single reaction taking place.
Dixon [1992] has developed a chemical reaction model which eliminates these shortcomings. Dixon's approach is similar to that of Demetracopoulos et al. [1986] in that it makes use of the continuity equation to model chemical release. The main difference is that Dixon applies the analysis to an individual particle while Demetracopoulos applied it to a lysimeter as a single entity.
Dixon's model includes information on the diffusivity of fluid reactant into the particles and the intrinsic kinetics of the multiple reactions which take place. Although the intrinsic kinetics are not explicitly obtained in Dixon's model, the ratio of the diffusivity over the intrinsic kinetics is determined. In many ways this corresponds to determining the effective reaction kinetics, which include the effects of intra-particle diffusion, of the waste particles. The model is developed in such a manner that once the effective reaction kinetics of a particular sized particle have been determined, the technique can be extended to other sized particles. Dixon's model also makes provision for surface solid reactant concentrations which differ from bulk solid concentrations within the particles.
In summary Dixon's chemical reaction model makes provision for the following:
the diffusivity of the fluid reagent into the particles as well as the intrinsic kinetics of the solid reactant within the particles,
multiple competing reactions, and,
surface solid reactant concentrations which differ from the bulk solid concentrations within the particles.
Dixon made use of this model to describe copper release from a heap leaching operation. He made similar assumptions concerning the hydrodynamic interactions to those of Roman et al. [1974]. Not withstanding the complications of preferential flow _ paths, Dixon's model has an excellent capacity to predict the release of hazardous constituents from hazardous waste deposits which contain granular material.
Dixon's model can also be used to model the release of hazardous constituents from a monolithic structure. The ratio of the effective diffusivity of the fluid into the monolith over the rates of the hazardous constituent release can be determined by conducting a CSTR test on small fragments of the monolith. This information can then be used to predict the release of hazardous constituents from the monolithic structure. This is a more sophisticated method to the one adopted by Batchelor [1990], Bishop [1986] and Cheng and Bishop [1990], because it does not assume that diffusion is the controlling resistance or make use of the semi-infinite-media assumption. This method also has a greater predictive power in that it can also be used to investigate the effect of hazardous constituent and buffer concentration on the rate of release of hazardous constituents which the prior methods could not.
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2.4 Modellin& the Release of Hazardous Constituents from Partially Saturated Granular Waste Deposits.
No models exist in the literature which describe the release and subsequent transport of hazardous constituents under unsaturated flow conditions.
Probably the most comprehens_ive attempt to include the effects of unsaturated conditions on leachate generation are those of Demetracopoulos et al. [1986]. They include these effects by using a hydrodynamic equation for unsaturated flow. These equations, described previously in equations (2-2) to (2-5), are often referred to as groundwater flow equations. Hydrodynamic equations or groundwater flow equations yield information about the local levels of saturation as well as local fluid flows. In the equations presented, which are already in one dimensional format, the saturation of fluid and fluid flow within a waste deposit can be determined as a function of position within the waste deposit and time. Although Demetracopoulos et al. [1986] determine the effect of unsaturated conditions on fluid flow, they do not consider the effect of these conditions on the mass transfer of hazardous constituents. Since they made use of mass transfer to describe the rate of release of hazardous constituents in their model, the influence of this factor should have been investigated. To circumvent the problem, Demetracopoulos et al. define a distribution coefficient, which relates the hazardous constituent concentration in the fluid to the hazardous constituent concentration in the waste particles, specific to the situation being investigated. This distribution coefficient needs to be determined experimentally for each situation. In effect, this coefficient includes the average effects of the degree of saturation on hazardous constituent release.
The only other attempts to include the effects of the level of saturation on leachate generation can be found in the heap leaching models which usually include a saturation parameter [Roman et al. 1974]. The saturation parameter represents the average level of saturation within the lysimeter. All of these models further assume that all of the particles within the heap are totally wetted with fluid. Although this assumption is probably valid for heap leaching, it will almost certainly not hold in unsaturated waste deposits. The reason for this is that fluid which percolates through unsaturated waste deposits usually does so in a fairly random manner.
The level of particle wetting is important in the effective operation of trickle bed reactors and several researchers have investigated means to determine a suitable wetting factor [Columbo et al. 1976; Schwartz et al. 1976; Sicardi et al. 1980; Mills and Dudukovic 1981; Ramachandran 1986]. A wetting factor is a simple factor which describes the average fraction of the particle surface area covered by fluid. It is felt that the inclusion of similar wetting factors into the heap leaching models when they are applied to hazardous constituent leaching would be most beneficial.
Unsaturated flow through deposits drastically affects the flow patterns in deposits. As previously discussed, the flow patterns in waste deposits are extremely complex. All the hydrodynamic equations considered in the literature models are relatively simple one dimensional, plug flow models. More suitable hydrodynamic models need to be
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developed for fluid flow in unsaturated deposits.
The first extension to plug flow models which would begin to incorporate irregular flow behaviour is to include axial dispersion. Levenspiel [1972] describes the use of the 'tanks in series' approach to include axial dispersion. In summary, in a 'tanks in series' approach to a plug flow reactor, the plug flow reactor is modelled as a series of continuously stirred tank reactors. A mathematical analysis of the technique yields the number of tanks required in order to approximate plug flow through the system. Dispersion is included into the model by using fewer tanks than required because fewer tanks tend to erode the plug flow nature by allowing greater degrees of mixing within the system.
The 'tanks in series' model discussed above is a one parameter hydrodynamic model. This means that one parameter is required to account for the non-uniform flow characteristics. More complex, multi-parameter models exist which account for highly irregular flow characteristics. These strategies consider the system to consist of several regions which can be described by plug flow, dispersed plug flow and mixed flow models. Zones which contain stagnant fluid can also be incorporated into these strategies. Levenspiel [1972] discusses the implementation of these techniques.
An alternative approach to dealing with the effect of the complex flow patterns on chemical reactions within trickle bed reactors has been investigated by Funk et al. [1990]; Zimmerman et al. [1987], and, Ng and Chu [1987]. These investigators model the fluid flow at the particle level. To accomplish this they make use of a porous medium model and a fluid distribution model. The porous medium model is used to describe the geometric locations of the particles in the trickle bed reactor. To date, these investigators have only investigated porous media which consist of equi-sized spheres. Using Monte-Carlo techniques, a suitable representation of the geometric locations of the spheres within the trickle bed reactor is obtained. The result of one such simulation is shown in Figure 2-7 (taken from Zimmerman et al. [1987]).
The fluid distribution model is used to determine the fate of fluid which impinges on a single particle in the assembly of particles. Ng [1986] has developed a wetting criterion for particles. For a given flow of fluid onto a particle, this criterion determines whether the fluid will totally cover the particle, which is comparable to ·film flow defined by Lutran et al. [1991], or whether it will be confined to a specific part of the particle -which is comparable to rivulet flow. This information is then used to determine the flow paths of the fluid as it moves from one particle to the next. In this manner the flow patterns of fluid through the trickle bed can be determined. A typical result of such a calculation strategy is also shown in Figure 2-7 (taken from Zimmerman et al. [1987]).
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Figure 2-7. An example of a two dimensional, porous medium model used to simulate the geometrical characteristics of particles in a trickle bed reactor and a typical flow pattern predicted by using the strategy of Zimmerman et al. [1987].
The densely shaded spheres represent complete wetting of the entire particle surface. The spheres with lighter shading over the entire circle are partially wetted. It indicates a liquid film covering more than 50 % of the sphere surface. All unshaded regions on the spheres represent dry areas. Note that the arrows indicate the flow of an isolated liquid rivulet.
Funk et al. [1990] use this strategy to determine the fluid flow patterns and couple this information to a chemical reaction model for reaction in a catalyst pellet. Catalytic reactions are usually steady state reactions and as such are described by suitable ordinary differential equations. The calculations pertaining to the chemical reactions are simplified by making use of an appropriate steady state effectiveness factor. Using this combined model Funk et al. predict overall performance of the trickle bed reactor.
It is very unlikely that such an approach will be able to be applied to leachate generation in granular waste deposits in the near future because of the following limitations. As pointed out earlier, the porous medium model in its present form can only accommodate equi-sized spheres. Although there is no reason why the model cannot be extended to include a size distribution of spheres, the calculation strategy to merely generate the porous medium model will become far more complex. Further, the reactions which take place in waste particles are non-catalytic in nature. Such reactions need to be described
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using partial differential equations. Since these reactions are not steady state in nature, the calculation strategy cannot be simplified by the use of steady state effectiveness factors. This means that the full partial differential equations describing the reactions need to be solved. The computation involved in solving these equations is significant. The computers which would be required to solve the porous media model, the fluid flow path problem and a partial differential equation for each particle in the system makes this approach unattractive. Before such a method is adopted, it is felt that simpler models need to be investigated to determine whether they can be used to solve the problem. Two such simpler models are investigated in the remainder of this thesis. The first model is a macroscopic, lumped parameter model and the second model is a heterogenous columnar model.
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•
Chapter 3. A Macroscopic, Lumped Parameter Model to Describe Leachate Generation and Mobility in Granular Waste Deposits.
In this work, the first model _investigated, which is capable of predicting leachate generation and mobility within granular waste deposits, is a macroscopic, lumped parameter model. There were two main reasons for investigating a model of this nature. The first reason is due to its inherent simplicity. The macroscopic model considers the deposit as a single reacting entity. As such, it does not attempt to analyze the release of hazardous constituents at the particle level. Instead, it makes use of an overall expression which describes the rate of hazardous constituent release from a small volume element of the deposit. This expression includes the effects of chemical kinetics, hydrodynamic aspects, diffusion and hazardous constituent location on the release of contaminants. This is why the model is termed a lumped parameter model. In effect it assumes that not enough is known about the intrinsic kinetics of the system or about the complex hydrodynamics or even the hazardous constituent location to allow a more rigorous analysis. In many respects, it is this simplicity which makes the model very attractive. In any situation where these aspects cannot be determined or are uncertain, the lumped parameter model can still be used.
The second reason for investigating the macroscopic, lumped parameter approach deals with model accuracy. The lumped parameter model does not require any assumptions with respect to particulate or hydrodynamic aspects. Instead, these effects are included in the experimentally determined model parameter. In effect, the rate of release of hazardous constituents from a waste deposit as a whole is being determined. This is in contrast to more detailed models. More detailed models determine the rate of release of hazardous constituents from the individual particles within a granular waste deposit and then predict the deposit performance. Before the deposit performance can be predicted however, assumptions with respect to fluid flow patterns, fluid saturation and particle wetting need to be made. If any of these assumptions are in error then the accuracy of the more detailed models will be effected. When insufficient information about the deposit is available to make confident choices with respect to these aspects, -the most reliable method may be to make use of the lumped parameter which does not require this information ..
The macroscopic, lumped parameter model was derived by applying a one dimensional fluid continuity equation to the deposit as a whole. The approach adopted is very similar to the one followed by Dixon [1992], the only difference being that Dixon applied the fluid continuity equation to a single particle rather than to an assembly of particles in the form of a waste deposit. The model developed is also very similar to the model of Demetracopoulos et al. [1986] which was discussed in the previous chapter. The main differences between the model developed and that of Demetracopoulos et al. is that the 'rate of hazardous constituent release' term, R, used in the present model was assumed to be described by chemical kinetics rather than by mass transfer considerations. The
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reason for this is that the effects of competing chemical reactions can be investigated with this approach. Realizing that the effects of competing chemical reactions can drastically alter the hazardous constituent release profiles, it was felt that it should be incorporated into the model. The other main difference between the present model and that of Demetracopoulos et al. is that the present model does not make use of hydrodynamic flow equations which describe unsaturated flow. Rather, much simpler 'plug flow' hydrodynamic equations were used. It was realised that the more complete hydrodynamic equations could always be included at a later stage and it seemed more effective to first investigate the potential of the model using the simpler hydrodynamic equations.
The remainder of this chapter ·presents the model derivation. It also discusses an appropriate solution strategy. This is followed by a section which indicates how a time dependent fluid percolation velocity can be incorporated into the solution strategy. The solution strategies have been coded into suitable computer routines and the details of these routines are discussed. Typical results obtained from the model are presented. Some discussion which indicates how the model can be fitted to experimental results is offered, and the limitations of the model are summarised.
3.1 Development of the Equations.
Figure 3-1, a schematic of a lysimeter which is thought to be representative of a real waste deposit, forms the basis of the macroscopic, lumped parameter model. It is assumed that the fluid reagent, A, percolates through the lysimeter and reacts with the solid reactants, Bi, according to the following stoichiometric equation:
n A + I: biBi ... dissolved products
i=l
On a mass basis the stoichiometric coefficient, bi, would represent the mass of solid reactant, Bi, required to react with a unit mass of fluid reagent. This is very often more convenient to use than a conventional molar basis since mass concentrations are more easily determined experimentally, and the 'bi' terms then represent an aggregated elemental behaviour rather than a species balance.
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Figure 3-1. Schematic of a lysimeter which forms the basis for the macroscopic, lumped parameter model.
L
Convection of fluid reagent through lysimeter.
Lysimeter.
The continuity equation for the fluid reactant can be obtained from a statement of conservation of mass. This equation in operator format is:
where NA
(3 -1)
fluid flux through the deposit, including a bulk convective contribution and a diffusive contribution; deposit porosity; deposit saturation; and; defined as the rate of production of fluid reagent A by reaction with species i.
The summation is required to account for the production of fluid reagent A by all the participating reactions.
The equation derived by allowing the rate of production of a solid reactant, Bi, to be described by a kinetic expression which is of variable order with respect to the solid reactant and first order with respect to the fluid reagent is:
( 3 -2)
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where CBi CA ~Bi kBi
mass or moles of the solid reactant per unit volume of solid; mass or moles of the fluid reagent per unit volume of fluid; variable reaction order; and; reaction rate constant.
The reaction rate constant has units, depending on the reaction order term ~Bi' such that the units on the right hand side of the equation are rendered to be mass or moles of solid reactant per unit time per unit volume of solid.
In this analysis, the fluid flux through the column has been assumed to dominated by a convective flux due to the fact that the convective flux usually masks the effects of a diffusive flux. An expression for the convective flux is:
(3 -3)
where u represents the superficial fluid velocity.
Substitution of equations (3-2) and (3-3) into equation (3-1) yields (expressed in one dimensional cylindrical coordinates with axial dependence only):
where t z
time; and; axial position within the deposit.
(3-4)
By assuming that the fluid is incompressible and that the relative void space saturation in the deposit remains constant, which implies that saturation is not a function of fluid velocity through the deposit, the second term on the right hand side of the above equation can be shown to be zero due to the fact that the divergence of the fluid velocity is zero. Thus the equation simplifies to: ·
ac ac n k ccl>sic A A _ ~ Bi Bi A
Col%SatECol at = -u az LJ ( 1-eCol) i=l bi
(3-5)
The initial and boundary conditions which apply are:
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CA ( 0' t) =CAinlBtconc. ( 3 -B)
Note that in equation (3-8) the boundary condition imposed is that the inlet concentration of the fluid reagent to the deposit is constant. It is simple to incorporate the boundary condition which considers the inlet concentration of fluid reagent as a function of time into the model - as long as this function is prescribed. Only the use of a constant boundary condition has been demonstrated in the model development since it is very unlikely for the concentration of fluid reactant entering the column to change dramatically with time.
3.2 Expressin2 the Equations in Dimensionless Form.
It is desirable to express model equations in a dimensionless format. When expressing an equation in dimensionless format, · the original variables are grouped into dimensionless parameters which are less numerous than the original number of variables and which tend to have real physical significance. Reducing the number of variables is advantageous in that it provides results of greater generality, thereby enabling the effects of changing conditions within the deposit to be studied more easily. This is also helpful when attempting to plan experimental work or correlate experimental results [Welty, Wicks, Wilson 1976].
Equations (3-2) and (3-5) to (3-8) can be expressed in dimensionless form by defining the following dimensionless parameters and dimensionless groups:
where
CA a=----c
ArnletConc.
~= z L
( 3 -9 )
(3-11)
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c. CJ .=_!!2:..
B1 C. B10
.._1=.!:_ T
(3-10)
(3 -12)
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L T=-u •
where L deposit length scale.
DGl= ___ u __ _ u •ec,nCol, sat
(3-14)
(3-13)
c Azni.,tcanc.
CB. 1o
k C+s1 Bi Bi0 L
u*b. 1
( 3-16)
(3-15)
u • is a reference fluid velocity (percolation velocity) which has arbitrarily been set at lm per 24 hours. It is important· to note that the definition of the reference fluid velocity is totally arbitrary and only serves as a convenient manner to non-dimensionalise the results. Also note that:
UPercolation = u (3-17)·
Following on, T is the equivalent of a reference space time for the column; DG 1 is the ratio of the fluid percolation velocity to the reference fluid percolation velocity; DG2 is the ratio of the chemical reaction rate at z=O to the rate of fluid reactant replenishment and DG3 is a dimensionless stoichiometric ratio which indicates fluid reagent strength relative to the solid reactant within the deposit. The rate of fluid reactant replenishment is defined to be the rate at which the fluid in a given volume of the deposit is totally replaced by new fluid due to the convective flux of fluid reactant through the lysimeter.
Equations (3-4) and (3-5) in dimensionless form and in cylindrical co-ordinates are summarised as:
aa DG aa ~ 4's1 -=- 1--"" DG2.(XOB· a-r:' a~ i=1 1 1 (3-18)
with
a ( C 0) =O (3-19)
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ex (O, i:1) =1 (3-20)
daBi 4's1 --=-DG2. DG3 ·CXOs· di:' .l .l l
( 3 -21)
with
a Bi ( C 0) =1 (3-22)
Equations (3-18) and (3-21) represent the progression of reaction within the deposit.
3.3 Solution Strateay.
Equation (3-18) is a first order hyperbolic partial differential equation. Simple finite difference methods and finite element numerical techniques must be used with care when solving hyperbolic problems since the discontinuous nature of the solution gives rise to difficulties when these techniques are used. The discontinuous nature of the solution arises due to the fact that a sharp reaction front, corresponding to the fluid reagent front, moves through the deposit Although it may be possible in some cases to obtain a stable method, it will invariably be very inaccurate. This implies that the solution strategy may calculate a solution but that this solution is incorrect. Note that solution strategies can also be accurate but not stable. This implies that although the correct solution is being calculated, the strategy breaks down before the entire solution is generated. Suitable solution strategies need to be both stable and accurate.
The method of characteristics is a suitable method of solution for first and second order hyperbolic partial differential equations. This method converts the partial differential equation into a set of simultaneous ordinary differential equations. The ordinary differential equations can then be solved using the standard numerical techniques of finite differences or finite elements.
Recall equation (3-18):
acx acx ~ <Psi -=-DGl-- ~ DG2.cxaBi ai:' a~ i=l .l
(3-23)
Using the method of characteristics, which is summarised in Appendix I, the following set of ordinary differential equations is obtained:
Only two of these equations are independent. The two equations used in the solution strategy are:
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dr 1 =...Et_=-___ d_a __ 1 DGl n :E DG2 .a;acjlBi
i=l .l
( 3 -24)
dr1 _ 1 , d~ - DGl
(3-25)
and
da; n cjl - =- :E DG2. <XO' Bi dr1 i=1
1 (3-26)
Equation (3-25) represents the relationship between the time increment and the spatial increment. This highlights the essence of the method of characteristics. By imposing this restriction (of interdependence between two of the parameters) the partial differential equation can be reduced to a simple ordinary differential equation.
A simple expliCit finite difference numerical technique was found to be an adequate method to solve equations (3-26) and (3-21). Equation (3-26) in numerical format is:
Equation (3-21) in numerical format is: •
where j represents a time index.
3.4 Introduction of a Variable Fluid Velocity into the Solution Strategy.
At this point the velocity of the fluid percolating through the column has been assumed to be constant with respect to time. However, fluid velocity through granular waste deposits does vary as a function of time due to various reasons of which periodic rainfall is one example. This has been incorporated into the model in the following manner.
The dimensionless group, DGl, has been redefined as a function of dimensionless time:
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DG1 = fn ('t' 1) (3-29)
Thus equation (3-25) becomes:
dr1 _ 1 d~ - DG1 ( 't1)
(3-30)
Where before the time increment was constant, it now needs to be evaluated using equation (3-30) before each iteration in time. In the case of a functional relationship for the dimensionless group DGl being prescribed, equation (3-30) can be used to determine the dimensionless time increment. However, it is more likely that the functional relationship for the velocity will be prescribed in dimensional terms. (Either in terms of a mathematical function or discretely.) In this case, the followirig equation, derived from equation (3-30), can used to determine the dimensionless time increment:
(3-31)
where (tn+i-tn) is obtained from:
where ll.z L
tn+l
J u(t) dt =Ecol ll.z tn
(3-32)
is the length of a spatial increment within the deposit; and; is the total height of the deposit.
3.5 Suitable Computer Routines for the Model.
Program Model4Dl.PAS and Model4D2.PAS are PASCAL codes which solve these equations as a function of position and time. These codes make provision for the fluid velocity to vary as a function of time. The output of the codes include graphs of concentration versus position and time, with breakthrough curves as a function of time. The breakthrough concentration of any dissolved species in Model4Dl .PAS has been calculated as the amount of that species exiting the column in the time increment over the total original leachable amount of that species in the column. Model4D2.PAS
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normalises all the breakthrough concentrations of the dissolved species relative to the total original amount of the first listed solid species in the program codes. This allows comparison between the relative amounts of the different species which appear in the leachate at the bottom of the column. As an example where such a comparison is useful, consider a waste which contains one leachable hazardous constituent and a large excess of buffering material. Although the fraction of the total buffering material released at any one time may be low compared to the hazardous constituent, the dissolved concentration of the buffer could be equal to, or much higher than that of the dissolved hazardous constituent concentration due to the large initial excess of the buffer material.
A copy of the codes as well as a solution algorithm can be found in Appendix IL
3.6 Verification of the Computer Routines.
The code was verified by subjecting it to a set of tests. In the first test no reaction was assumed to take place. This represents the problem of fluid reactant progressing through a deposit in plug flow. Excellent agreement was obtained between the calculated and analytical fluid profile as a function of time and is shown in Figure 3-2. Figure 3-2 represents the fluid reagent profiles within the deposit at successive time steps, beginning at the left hand side and progressing towards the right hand side of the graph.
Figure 3-2. Comparison of the predicted flow profile with the analytical profile for a lysimeter in which no chemical reactions take place.
1.0 ------------------c c ~ • .. o.e c: •
,..._O:Jcu\~~ ~ - ~be..\
cc - hC'\eQf', u c 0
o.o 0.2 0.4 0.6 o.e 1.0
" 0.6 • • ! ]
0.4 • c • • Q
0.2
o.o
- ' ~ !S A ~ c. tt-oetu1s°' l .... .... r .
'" tif'le.
1 I I ; I I
01...nsionl••• L_....th
The next test involved introducing a single reaction and determining its effect on the
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concentration profiles as the rate of reaction increased. The deposit would be expected to react in a homogenous manner for a very slow reaction, proceed through a transitional phase at intermediate reaction rates and finally react in a zone:.wise manner for fast reaction rates. A zone-wise reaction refers to the case where the reaction is restricted to a narrow band within the deposit. This can be clearly seen in the horizontal rows of graphs in Figure 3-5 where the reaction rate increases from left to right.
The code was also checked to ensure that it could calculate the profiles when more than one reaction was occurring. Several other self-checking strategies were used to ensure that the code was operating correctly. These included the following:
The model was checked to ensure that chemical species were interchangeable. This ensures that the order in which the chemical species are defined in the computer . routines is not important. This is demonstrated in Figure 3-3. In Figure 3-3, the profiles of the first solid reactant are depicted by solid lines while the profiles of the second solid reactant are depicted by dotted lines. (In Figure 3:.. 3 these profiles never touch the X-Axis. Unfortunately the fluid reactant profiles are also depicted by solid lines but they can be identified as the curves which touch the X-Axis.)
The model was checked to ensure that the concentration profiles of two solid species reacting at the same rate were co-incident. This is demonstrated in Figure 3-4. In this figure the dotted lines cannot be identified because they are co-incident with the solid lines.
Kinetic aspects of the model were also checked by ensuring that the model predicted the same profiles for a deposit with a single solid reactant compared to a deposit which contained two solid reactants in equal quantities to the solid reactant in the first lysimeter but which each reacted at .half the rate.
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Figure 3-3. Chemical species in the model are interchangeable.
c ~ .. • .. .. c • u c • "' c ~ ~ ! 0
0.6
a.•
a.a
a.a
So\\a tcoc'c.oi'\t Z Ut'te~eO
a.z 0.4
.l.O
c ~ .. • .. .. c • u c • "' • • ! c ~ • c
! 0
a.6 o.z 0~4 a!6
~~~·Of\ ~ tir.-e. Di ....... ianless U:nowth
Figure 3-4. Profiles of hazardous constituents which react at the same rate are coincident.
c • .. • .. .. c I u C-
.L.O i
8 0.6 .
• • I
c ~ . c I t
0
Sohd teoc:.t.ol\t. 1. end. 2. (.c:e\~t..)
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3.7 Results and Discussion.
3.7.1 General Behaviour.
In order to determine the- general behaviour of the model and the sensitivity to the model parameters, the following scenario was investigated:
Deposit of 0.5m in length;
Only one solid reactant present;
First order rate dependence in the solid reactant concentration;
Constant fluid velocity set at lm in 24 hours.
These conditions are typical of those used in laboratory lysimeter experiments. In a similar manner, full scale deposit behaviour could have been investigated by merely using appropriate model parameters.
The computer code calculated the dimensionless group DGJ which under the conditions defined is 0.0084.
Figure 3-5 presents the profiles of fluid reagent A (solid curves) and one solid reactant (dashed curves) for all combinations of DG2 = 0.1,1,10 and 100, and, DG3 =0.1,1,10 and 100. Figure 3-6 presents the breakthrough curve of the solid reactant for the same combinations of parameters. Note that in Figure 3-6 each row of graphs have the same Y axis scaling, however the Y axis scaling. changes between the different rows.
For very low values of DG2 the chemical reaction rate is much slower than the rate of fluid reactant replenishment. Under these conditions, most columns would be expected to react in a homogenous manner. This behaviour can clearly be seen in Figure 3-5 by examining the first column of graphs. Only at very high relative fluid reagent concentrations are any solid reactant concentration gradients established. The release of hazardous constituents in columns with low values for DG2 is controlled only by -kinetic factors.
At moderate values for DG2, the type of kinetics depends on the relative fluid reactant concentration. At DG2= 10, non-transient fluid and solid reactant gradients are established for all values of DG3. When DG3 values are greater than 10, the column reacts in a fluid reactant limiting, zone-wise manner.
At high values of DG3 the column reacts only in a fluid reactant limiting, zone-wise manner.
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n . =
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Univers
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Some interesting features of hazardous constituent release can be observed in Figure 3-6. Notice that the breakthrough curves for all columns which react in a homogenous manner resemble an exponential decay function. This is as expected, since a column which reacts in a homogenous manner could be considered as a continuously stirred tank reactor. The concentration profiles for first order reactions which take place in continuously stirred tank reactors are exponential decay functions [Levenspiel 1972]. In contrast the breakthrough curves for columns which react in a zone-wise manner resemble impulse and Heaviside functions (step up followed by a step down function) which are characteristic of perfect plug flow reactors. When the kinetics of the chemical reactions are sufficiently fast for the column always to react in a zone-wise manner, the width of the Heaviside function is determined by the relative fluid reactant concentration. As the fluid reactant concentration increases, the width of the heaviside
· function decreases. In the limit, the heaviside function approaches a Dirac delta as can be seen in the lower right hand corner of Figure 3-6.
3.7.2 Effect of Competini: Reactions.
As previously discussed in Chapter 2, granular waste deposits very often contain more than one reactive component in the matrix. This has a marked effect on the release of the individual hazardous constituents as the following test scenarios indicate.
The effect of a buffering material on the release of a single solid hazardous constituent.
The scenario investigated can be summarised as:
Deposit of 0.5m in length;
One primary solid hazardous constituent and one buffer material present;
First order rate dependence with respect to the solid concentrations;
Constant fluid velocity set at lm in 24 hours.
The parameters . for the solid hazardous constituent were set at DG2conram. = 10 and DG3eontam. =1. .
The combinations investigated are summarised in Table 3-1.
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Table 3-1. Summary of the parameter combinations investigated to determine the effect a buffer material on hazardous constituent release.
Buffer rate constant 100 Buffer rate constant 10 times slower than times slower than contaminant rate const. · contaminant rate const.
10 times as much buffer · DG2Bujfer = 1.00 DG2Buffer - 10.00 as contaminant. DG3Buffer = 0.10 DG3Bujfer - 0.10
100 times as much buffer DG2Buffer = 10.00 DG2Buffer = 100.00 as contaminant. DG3Bujfer = 0.01 DG3Bujfer - 0.01
Figure 3-7 presents the concentration profiles for the above combinations while Figure 3-8 summarises the breakthrough curves.
The concentration profiles in the top left hand corner of Figure 3-7 closely resemble the concentration profiles when no buffer material was present. The breakthrough curves for the same parameter specification show that in this case the buffer material is not sufficiently reactive to affect the release of the solid hazardous constituents; By merely increasing the relative concentration of the buffer material it begins to play a significant role. Alternatively a more reactive buffer material can be used to limit the release of hazardous constituents.
~ .
The buffering material effectively forces the column to react in a more fluid reactant limited, zone-wise manner. This is because the total rate of acid consumption has effectively increased. Further, due to the competition between the solid hazardous constituent and the buffer material for fluid reactant, the hazardous constituent reacts at a slower rate than before and the hazardous constituent breakthrough curves start to resemble exponential decay Junctions.
Notice in the lower row of breakthrough graphs in Figure 3-8 that the solid hazardous constituent persists to breakthrough at large times. This is typical behaviour observed when competing reactions occur.
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Figure 3-8.
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Extension to more than one reactive hazardous constituent.
Figures 3-9 and 3-10 are sample printouts of typical concentration profiles and breakthrough curves when two hazardous constituents and a buffer material are present. The scenario investigated was a second hazardous constituent of equal amount to the first, but which reacted 10 times slower than the first reactant. These parameters are summarised in Table 3-2:
Table 3-2. Summary of parameters used to demonstrate extension of the model to more than one reactive hazardous constituent.
Contaminant 1. DG2 1 - 10.00 DG3 1 - 1.00 -
Contaminant 2. DG2 2 - 1.00 - DG3 2 - 1.00 -
Buffer Material. DG2 3 - 10.00 DG3 3 - 0.01 -
Figure 3-9. Sample printout of concentration profiles for more than one hazardous constituent.
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J..0
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CJ 0.6 • • ! c .2 0.4 • c • < 0
0.2
DiPMtnSionless Length
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Figure 3-10. Sample printout of breakthrough curves for more than one hazardous constituent.
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3.8 Fittin2 the Model to Lysimeter Experiments.
3.8.1 Model Requirements.
The following information needs to be known in order to fit the model to lysimeter results:
The basic chemical reactions which actually take place within the lysimeter;
the effective concentrations of the participating species;
the reaction orders with respect to the solid reactants;
the percolation velocity of the fluid as a function of time; and;
the voidage of the column and the void space saturation.
The level of detail required in the understanding of the basic chemical reactions is to have identified the fluid reagent and the main solid reactants in the system. For example the fluid reagent is usually a dilute acid stream while the solid reactants are leachable heavy metals and buffering components.
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The effective concentration refers to the total leachable concentration of any species, which in many instances is not equal to the total concentration of that species. This is caused by a certain portion of the species being unavailable to the leaching process. One method to determine this concentration is to conduct a CSTR leach test until equilibrium is reached. The initial effective solid concentrations of the dissolved species can then be determined by back calculation.
Where the solid reactant is not in excess, a solid reactant order of unity is suggested for the following reasons. The solid reactants are known to be located in a size distribution of waste particles within the deposit. If the deposit contained equi-sized spherical waste particles, comparison of the variable order kinetic expression, equation (3-2), to a 'grain' reaction model indicates that the appropriate solid reactant order in this case would be 2/3. Dixon [1992] has intimated that this order increases as the standard deviation of the size distribution of spherical 'grains' increases and that a reaction order of unity is reasonable. The appropriate reaction order for excess solid reactant is usually taken to be zero, which implies that the solid reactant concentration does not have an effect on the reaction kinetics.
Usually the percolation velocity is set at a constant value. As previously discussed the model can accommodate a variable fluid velocity as long as the functional relationship for the velocity is prescribed.
3.8.2 Fitted parameters.
Each dimensionless group, DG2;, contains a lumped reaction rate constant for the chemical reaction corresponding to that group. These reaction rate constants cannot be determined a priori and so the fitted parameters in the model are the DG2; groups. The reason why the reaction rate constant cannot be determined from bench scale tests is because they include hydrodynamic effects. The hydrodynamic situation in bench scale tests is not comparable to the hydrodynamic situation in a waste deposit or its pilot-scale lysimeter equivalent.
The DG2; groups are fitted by comparing the breakthrough curves from the model to experimentally determined breakthrough curves from lysimeters. A good method to optimise the agreement between the model and experimental results is to use a simplex search technique to determine the best values for the fitted parameters.
3.9 Limitations of the Model.
The macroscopic, lumped parameter model does have several limitations due to its inherent simplicity. The first limitation of the model is that it can only be used to extrapolate to deposit proportions and in time for deposits which contain identical wastes and which exhibit identical hydrodynamic characteristics to the ones observed in the
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lysimeter. This implies that a separate lysimeter experiment needs to be conducted for each waste stream and hydrodynamic situation investigated. This is a severe limitation considering the expense and time involved with lysimeter experiments.
The second limitation is that the model can be used only in situations where each solid reactant is released at a single overall rate. Often, particles which have most of the solid reactant concentrated onto the surface of the particle, can also have some of the solid reactant distributed in the bulk of the particle. The solid reactants in the two regions very often exhibit different rates of release.. The first rate of release is due to the release of the solid reactants on the surface. Usually this rate of release is much faster than that of the release from within the bulk of the particle due the added diffusional resistances of the fluid reactant into the particle.
The model cannot be used to determine the individual contribution of the intrinsic chemical kinetics, the hydrodynamic aspects or the hazardous constituent location on the release of hazardous constituents. This is a direct consequence of using the lumped parameter approach. Since one aim of this work is to eventually be able to engineer better waste deposits, it is critical to be able to determine these individual contributions.
For these reasons it was deemed necessary to investigate more complex models which would begin to address the limitations of the macroscopic, lumped parameter model. In effect this implied investigating models which describe the release of hazardous constituents at the particle level. This is the focus of the remainder of this thesis.
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Chapter 4. A Summary of the Chemical Reaction Model Applicable to Single Particles as Developed by Dixon [1992].
The investigation into the macroscopic, lumped parameter model revealed a need to describe the release of hazardous constituents at the particle level. As discussed in Chapter 2, Dixon [1992] has developed a chemical reaction model for investigating the leaching behaviour of precious metals from ore particles. His model is sufficiently detailed to include the effects of diffusive and chemical reaction kinetic resistances, competing reactions and precious metal location within the particle on the leaching behaviour. Since these considerations are similar to the ones encountered in leaching of hazardous constituents from waste deposits, it was felt that Dixon's model should be investigated to determine its applicability to modelling contaminant leaching from waste particles.
This chapter presents the details of the investigation into the applicability of Dixon's model to a single waste particle. This involved determining whether or not the model developed by Dixon could be used without modification. For this reason, this chapter summarises the model development followed by Dixon [1992]. Once it was determined that Dixon's particle scale model could be used, an appropriate solution strategy for the model was investigated and implemented. The computer routines which implement the solution strategy have been rigorously checked against the results obtained by Dixon. These details form the reminder of this chapter.
4.1 Development of the Equations.
Figure 4-1 depicts a porous, spherical particle of radius R which is submerged in fluid reactant and which contains small amounts of solid reactant deposits. Dixon [1992] assumed that these solid reactants, Bi, are dissolved by a single fluid reagent A. This is represented by:
n A + :E biBi .... dissolved products
i=l
As in Chapter 3, the continuity equation for the fluid reactant can be obtained from a statement of conservation of mass. The continuity equation applicable to a single particle is:
(4-1)
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where NA €0
rAi
fluid flux into the particle; particle porosity; and; defined as the rate of production of fluid reagent A by reaction i.
I
As before, the summation sign is required to account for the production of fluid reagent A by all the participating reactions.
Figure 4-1. Schematic diagram of a porous, spherical particle of radius Rand a graph showing the concentration gradients within the particle. (taken from Dixon [1992]).
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Co ~, Cso
.,, c:
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Dixon made explicit provision for different kinetics for those reactions which occur in the bulk of the particle compared to those which occur on its surface. Considering first the reactions which occur within the pores of the particle, Dixon assumed that the reaction kinetics can be described by an expression which is of variable order with respect to the solid reactant and first order with respect to the fluid reagent. This expression. can be summarised as:
where cpi
qCpi = k C<P~1 C dt - pi pi A
(4 -2)
mass or moles of the solid reactant in the pores of the particle per unit mass of solid;
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CA mass or moles of the fluid reagent per unit volume of fluid; and; ~i reaction rate constant.
Here the reaction rate constant has units, depending on the reaction order term ~Bi' such that the units on the right hand side of the equation are rendered to be mass or moles of solid reactant per unit time per unit mass of solid.
In situations where the kinetics of any species on the surface of the particle is different to the kinetics of that species in the bulk of the particle, another kinetic expression will be required. The kinetic expression adopted by Dixon can be summarised as:
where csi
dCsi 3ksic!11cA --=-
dt R Po (l-e0 ) (4-3)
mass or moles of the solid reactant on the surface of the particle per unit mass of solid; mass or moles of the fluid reagent per unit volume of fluid; radius of the particle; ore density; and; reaction rate constant.
The following equation makes it a bit easier to see how this expression was obtained:
Mass of pa+ticle dCsi --k 4's1
Area of particle -cit- sicsi CA (4-4)
Thus the units on the left hand side of the equation are mass or moles of solid reactant per unit time per unit area of panicle. Therefore, the units of the reaction constant, ~i' depending on the reaction order term ~si' are such which render the units on the right hand side of the equation to be mass or moles of solid reactant per unit time per unit area of the panicle.
It is important to note that if the kinetics of the species on .the surface of the particle are not significantly different from the same species within the bulk of the particle, that the requirement for equation (4-3) is obviated.
Substituting equation ( 4-2) into equation ( 4-1) gives:
n '\!· NA - po ( 1 -eo) ~
i=l ( 4 -5)
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Dixon assumed that only a diffusive flux of fluid reactant entered the particles and made use of Fick's law to describe this flux. Substituting Fick's law into equation (4-5) and expressing the equation in spherical co-ordinates with radial dependence only gives:
(4-6)
The initial and boundary conditions which apply are:
CA (I, 0) =0 (4-7)
( 4-8)
ac · a: (0, t) =O (4-9)
It is important to note that equation ( 4-8) effectively implies that no film mass transfer resistances are being considered. This assumption is only valid where the film mass transfer is fast compared to the diffusion of the fluid reactant into the particle and the reaction of the fluid reactant within the particle.
4.2 Expressini: the Equations in Dimensionless Form.
Equations (4-3) and (4-6) to (4-9) can be expressed in dimensionless form by defining the following~ dimensionless parameters and dimensionless groups:
c CA IX=--....:'.!. (4-10) IX - b (4-11) b--c CAo Ao·
c. c. (J .=_.E3_ (4-12) (J . = __!!1:_ (4-13)
pi c. Sl. C. Pl.o Sl.o
I (4 -14) De,.t
( 4-15) ~=- 't'= R e R 2
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(4-16) (4-17)
( 4 -18) ( 4 -19)
.Bi represents a dimensionless stoichiometric ratio which indicates the reagent strength relative to the grade of solid reactant i. In a particle of given porosity which is in contact with a fluid reactant of concentration a= 1, a value of .Bi = 1 would imply that there is sufficient fluid reactant within the pores of the particle to completely react all of the solid reactant.
Ai represents the fraction of solid reactant residing on the surface of the particle.
Kpi and Ksi are ratios of the reaction rate of solid reactant i within the particle pores and on the particle surface, respectively, to the porous diffusion rate of fluid reactant A. In an investigation into the effects of flow, diffusion and heat conduction on reactor performance, Damkohler recognised the importance of four dimensionless groups. The second of these is the ratio of the chemical reaction rate to the rate of diffusion [Aris 1975]. Thus Kpi and Ksi correspond to Damkohler numbers of the second type. These ratios can be identified a bit more easily if the equations are written in the following format:
p (1-e) kpic:1:cA
bi Kpi=~~~~~~~-
DeACA R2
(4 -20) ( 4 -21)
The equations in dimensionless form and in spherical co-ordinates are summarised as:
(4-22)
with
a (CO) =O (4-23)
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( 4 -24)
aa. ~ (0,'t) =O (4-25)
(4-26)
with
(4-27)
d A <l>si a si Ksi t' i a si Cl b --= - --,....----d't )..i
(4-28)
with
asi (0) =1 (4-29)
Equations (4-2~) to (4-29) represent the progression of the reaction within a single particle.
4.3 Suitability of Dixon's Model to Hazardous Constituent Leaching From Waste Particles.
The development of the equations used in Dixon's particle scale model do not include any aspects which are unique to precious metal leaching from ore particles. The same equations would have been determined if the leaching of hazardous constituents from a waste particle had been considered. For this reason, Dixon's equations can· be used without modification to model hazardous constituent release from a waste particle.
4.4 Solution Strategy.
Equation (4-22) is a second order parabolic partial differential equation. A suitable solution strategy, which would be both stable and accurate, was desired to solve the set of equations. The Crank-Nicolson formula, which is an implicit finite difference method, is a suitable solution strategy since it is both unconditionally stable and sufficiently accurate [Crank 1975].
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The Crank-Nicolson approach uses a Taylor series expansion of the concentration function to obtain expressions for the partial derivatives. These are:
where
02N I a. 1 .-2cx . . +a. 1 . .... .. .l+ ,] .l,J .I- ,]
a~2 i,j <.d~)2
i is a spatial index and j is a time index.
(4-30)
(4-31)
Crank and Nicolson [1947] suggested that a better approximation for the above quantities would be the average of the quantities evaluated at times j and j + 1. This yields the Crank-Nicolson formulas for the partial derivatives:
(4-32)
and
::12N 1 a. .-2cx .. +ci. . a. . -2cx. . +a. . _er_ ... I· ... _. < .l+l,J .l,J i-1.1 + .l+l,J+l .l,J+l .i-1.1+1)
a~2 .l,] 2 <.d~)2 (.d~)2 (4-33)
Using these formulas, equations (4-22), (4-26) and (4-28) were converted to numerical equations.
Equation (4-22) in numerical format is:
fori=O
(4-34)
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and for i< >0
N 2 ~ (a~)2 .+ -·-ex ._1 . (-i+l) +ex· · (2i+ ~Ki (a~) CJi j-2i a't ) exi+l,J' ( 1- 1)
.l • J i, J i =l p • (4-35
The reason for two equations being required is due to a singularity which exists at the origin in equation ( 4-22). This problem was overcome by noting that total symmetry exists at the origin and thus equation (4-22) can be expressed in cartesian co-ordinates at this point. Note that the canesian co-ordinates are only valid at the origin. The equation used to derive eq!Jation (4-34) is:
(4-36)
The numerical format of equation (4-26) is:
oiJ+1- 0 iJ KPP (~ +~ , a 't , =- 2 ( 1-A.) o i.Jex i,J o i,J+1ex i,J+1) (4-37)
The numerical format of equation (4-28) is similar to equation (4-37).
4.5 Suitable Computer Routines for the Model. -
Program Model2D2.PAS is a PASCAL code which solves these numerical equations as a function of position and time. The output of this code includes a graph of the solid and fluid reagent profiles within the particle as a function of time and a graph of the fractional conversions as a function of time. The fractional conversion is defined as the fraction of a particular solid reactant species which has been released.
A copy of the code can be found in Appendix III.
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4.6 Verification of the Computer Routines.
The computer routines written needed to be checked to determine whether they operate correctly. One method to check the routines would be to reproduce the results presented in Dixon's thesis. To this end, all of the results presented in Chapter 1 of Dixon's thesis which involve concentration profiles and fractional conversions as a function of time have been reproduced. To demonstr(j.te this, Figure 4-2 is a series of graphs produced by Model2D2. These graphs can be compared to those found in Figure 4-3 which are the graphs presented by Dixon for the same parameter specification.
Further comparisons between the results predicted by Model2D2 and Dixon's work can be seen in Figure 4-4 and 4-5; and; Figure 4-6 and Figure 4-7.
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..... 0 0 II
cQ
-II cQ
8 -II cQ
l Figure~r~~----~----------...._ __ ...., ______ .,.
I~ .::lr=l
1.0 J~!.f [~!!~?!'!~~'~:~~~~::::·····. ·· .. a e S.hcl · .. -.. · .. ·._ ._ ·.. .. ·.. .. · 1-'-• · .. · .. · ............... · .. ' .... ·· .. :<::;<:::::::::·, \
0.4- ........
. ..l.0....1 ~t ~iiiiiiiijfiiiiiiiiiiii~~~~::::~-:~.:.~·;;i~~~:;;:~i2i2~i;;~;;;;
a.oJ 0.2 a.4 a.6 , a.o i.o 0
1.0
o.e
0.6
0.4
o.a
a 0 I • I
o.
I
o.e
a.6
0.4
a.a
a.a oJ
I
Dimensionless Radius, ~
lwnm•••••• .............................. . t:::: ::: :::::::::: ::: ::::: ::: : :.:.:::::: :: .... ~:::: :~.-.· .......... .
··· ... ·· .. ·· .....
..
Dimensionless Radius,~
a.a a.4 0.6
Ar=0.0316
a.a
.. _ .. ··-· .... 1.0
Ar=0.001
o.e
Dimensionless Radius, ~
K =100 p
~ I
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Figure 4-3.
{1 = 0.01
{1 =
p = 100
The corresponding concentration profiles to Figure 4-2 presented Dixon. (Figure 1-3 taken from Dixon's Thesis.)
= 0.01
'JI ,~
6r
b. a.a
0.1
a O.t
O.J
6T • ~.le 'JI ...-_-_ ---""-""·---• ....,_,..,_,.,_..,_.;;;_.;.._ --_,,._,,._,.,_;-_-,_..,_-_-_.,
---------------------·-· ---·--·--·---···------·----------·-----------·------·-------------------:;::~{:_ ~~~:~~:.;~~:~-;_ ~~:-:-:.~:.
0.1
a 0.4
OJI+._. __ _,, ____ ~---------~ '14 C.• 0.1 0.1 •.Q
-----------
'. : ~:,~:::-~~:;;;~~ !u
0.0 I c~ ~..l :.• r:.s e.1 1 .o
Dimensionless rocius, f
= 100
:: g:;;:~~~j ------------------------
0.I ... ::::· ... ·.:::·:.:::· .. ·.:::::·.:-
0.1
C.I
C.& ;;~;;~;~~~~~~~~~~~~~~~~~ C.•
0.2
O.I ·--------. ••••••••••••
------------- ... -- ....... o.• ··--......................................
o.: :iiiiiiiiiiiiiiiiiiiiii~-~ ~.o 4------------i OJI Q.J o.• C.I Cl 1.0
QT' - 0.01 t.0 .... _-_-_ .. _ .. _. .................... -....,---, -.
" C.I ··-·· .
0.1
··-0.4
O.J
o.o ~;;;;;;:=;;;G;;;;;;.;;:::::;.;~~~:J o.o o~ o.• e 1 0.1 ,_o
Dimensionless radius. ~
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Figure 4-4. Fraction conversion profiles predicted by Model2D2 for the parameters as Indicated in the Figure.
1.0
O.B
c 0.6 0 ... Ul I. QI ::> c 0 CJ
0.4
0.2
0.0...f!!:~~~~~~~~+--~~~-t-~~~~r-~~~1
o.o 2.0 4.0 6.0 a.o J.0,0
DiMensionless Reaction TiMe <KappaJ>•Beta•Tau>
Figure 4-5. The corresponding fractional conversion profiles to Figure 4-4 presented by Dixon. (Figure 1-4 taken from Dixon's Thesis.)
1.0 -.---~~~~~~~-:::=-~~~~~~~~~---,
x
c 0.8 0 (.J) L
GJ a 5 > . c 0 u 0.4
0 c 0 0.2
-+-' u 0 LQ.O ~::;:::;:: ........ ......,...,...,.......,....,...,...,...,..,....,...,...,...,..,....,...,...,...,..,....,.."T"T""M'"T"T"T"",...,..,.""T""l"'"T"M-:-TT"M:-r1
Li_ 0 2 4 6
Dimensionless reaction
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Figure 4-6. Fraction conversion profiles as a function of the variable order power predicted by Model2D2.
0.6
0.4
0.2
o.o"""f-~~~--r--~~~-t-~~~~t--~~~-+1 ~~~--1 0.0 6.0 .l<!.O .lB.O <!4.0 30.0
Dil'lenSionless Reaction TiMe <Mith r~t to RXN 1 > CKaPDaPl*Betal*Taul
Figure 4-7. , The corresponding . fractional conversion profiles as a function of variable order power to Figure 4-6 presented by Dixon. (Figure 1-9 taken from Dixon's Thesis.)
-co.a 0 (J) ~
~ 0.6 c 0 .u 0.4
0 c Q0.2 ~
(;.)
0 ~a.a o 10
Dimensionless
Kp = 100 p = 1
20 reaction time,
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Due to the excellent agreement between the results of Model2D2 and those presented by Dixon, the routines of Model2D2 can be used with confidence.
4. 7 Application of the Model.
The chemical reaction model at the particle level is not that useful in itself. The model as it stands only describes the progression of the reactions in a single panicle. It is very unlikely that the progression of the reactions within a single solitary particle will be required. Further, the boundary condition used, equation (4.8) or (4-24), implies that the fluid concentration in contact with the surface of the particle is constant. The only situation where this would arise is where a particle is submersed in an infinite amount of fluid and where mass transfer resistances are negligible.
· More realistic situations are when many particles are associated with a finite volume of fluid. Such situations correspond to a CSTR experiment in which the leaching behaviour of a single size class or a size distribution of particles is investigated. Equally, a volume element of a waste deposit could be considered as an appropriate size distribution· of particles associated with a finite volume of fluid. In this case the fluid would be the fluid in the void spaces between the particles.
In these situations, as the chemical reactions within each of the particles progress, fluid reagent will be consumed which will cause the bulk fluid concentration, that is the concentration of the fluid which surrounds the particles, to decrease. This dropping bulk fluid concentration is the appropriate boundary condition which should be used in particle scale model. In effect, this implies that a suitable boundary condition, which would replace equation (4-8) or (4-24), needs to be developed which will account for the decrease in the bulk fluid reagent as a function of time.
It is important to note that a partial differential equation, corresponding to equation (4-8) or (4-24), will be required for each particle size considered when the leaching behaviour of a size distribution of particles is investigated. The reason for this is that the rates of conversion of different sized particles will not be the same. Suitable dimensionless groups, which are defined in terms of a single reference particle size, need to be investigated. Lastly, appropriate solution strategies for the combined solution of the set of partial differential equations, which is comprised of a partial differential equation for each particle size, and the coupled boundary condition need to be determined. These aspects are considered in the following chapter.
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Chapter 5. A Model to Describe Leachate Generation from Granular Wastes in a Continuously Stirred Tank Reactor Experiment.
The continuously stirred tank reactor (CSTR) model describes the increase in the hazardous constituent concentratic;m and the decrease in the fluid reagent .concentration in the bulk fluid of a CSTR type experiment. This is· achieved by linking the particle scale, chemical reaction model, presented in the previous chapter' to an appropriate mass balance equation for the bulk fluid reagent concentration.
The appropriate mass balance equation is developed initially for equi-sized particles submerged in a finite volume of fluid reactant. In a CSTR experiment which contains a size distribution of particles, the release of hazardous constituents from each size class of particles needs to be determined before the overall release of hazardous constituents for the system can be calculated. This implies that a partial differential equation, corresponding to equation (4-22), will be required for each size class of particles. Before these differential equations can be used, the appropriate parameters for each size class of particles need to be determined. For this reason, a section which discusses model parameters as a function of particle size, has been included. This is followed by the development of a suitable mass balance equation for the consumption of fluid reagent in a CSTR experiment which involves a size distribution of particles.
An appropriate solution strategy for the model has been investigated and implemented. A case study has been used to illustrate the general behaviour of the model and the sensitivity of the model to particle size distributions as well as to hazardous constituent distribution within each solid particle. The following section discusses how the model can be fitted to typical CSTR data. Lastly the applications and limitations of the model are summarised.
5.1 Development of the Mass Balance Equation for the Bulk Fluid Reaa:ent in a CSTR which Contains Equi-Sized Particles.
Figure 5-1 depicts a few equi-sized spherical particles in a beaker of fluid reagent. It is assumed that the fluid reactant diffuses into the particles and reacts with the hazardous constituents which then enter the fluid phase.
The fluid reagent mass balance replaces the boundary conditions used in the model of Dixon [1992]. Referring to equations (4.22) to (4.29), note that equation (4-24) is a boundary condition which sets the bulk fluid concentration in contact with the particle at a constant value. This boundary condition needs to be replaced due to the fact that the bulk fluid reactant concentration drops as the fluid reactant diffuses into the particle.
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Figure 5-1. Schematic· of a few equi-sized spherical particles submerged m a well stirred beaker of fluid.
0 0 0
Equi-sized Particles.
Beaker.
A mass balance equation which relates the consumption of the fluid phase reactant to the bulk fluid reactant concentration is:
VPart. Po ( 1-Eo) f dCsi -D 'VC I 3 VPart. =V . bi i=l dt e A R R Liq.
where V Part.
Vu q.
is the total volume of the particles; and; is the total volume of fluid reactant.
(5-1)
Note that the first term in the above equation represents the consumption of fluid reagent due to chemical reactions which take place on the surface of the particles. The second term represents the fluid reagent diffusing into the particles. This quantity is calculated as the product of the diffusive flux into a single particle (the diffusive flux being defined by Fick's Law), the surface area of a single particle and the number of particles in the system. The number of particles present in the system is determined by dividing the total volume of the particles by the volume of a single particle. The fluid reagent which diffuses into a particle is continually being consumed by the chemical reactions taking place within the pore volume of the particle.
The same equation expressed in dimensionless terms is:
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~ cPsi aa. I aa. bulk - ."-'Ks Osi CX.bulk-3 ai: ~=l =V a
i=l i ~ ~ (5-2)
where
v . V = Liq.
Eo VPazt. ( 5-3)
As previously discussed, the chemical reaction model derived in Chapter 4, applies to a single particle. Equation (5-2) is a suitable boundary condition for equi-sized particles submerged in a finite volume of fluid reagent. This boundary condition, in conjunction with the chemical reaction model for the single parti~le, can be used to simulate both the concentration profiles of the solid reactants and fluid reagent within the particles as well as the bulk fluid phase concentration as a function of time.
5.2 Model Parameters as a Function of Particle Size.
As previously discussed, in order to simulate the progression of reactions in a size, distribution of particles, the chemical reaction model needs to be applied to each individual size class. The overall progression of the reaction for the system is then obtained by integrating the results of the individual size classes over the distribution of particles in the ·system.
Before the chemical reaction model cap. be solved for each size class of particles, the model parameters applicable to each size class need to be determined. Two different approaches to defining the model parameters as a function of particle size can be used. In the first approach, model parameters are fitted to only one size class of particles and the parameters of all the other size classes are related to it. The assumptions behind this approach and the resulting relationships for the model parameters as a function of particle size aie discussed in the next section. In some cases, particles in different size classes exhibit sufficiently different properties to preclude any simple relationships between them. In these cases the model parameters need to be determined for each size class individually.
5.2.1 Determination of the Model Parameters Applicable to Precious Metal Leachin2 with Respect to a Reference Size Class of Particles.
Dixon [1992] defined a set of relationships for the model parameters_ in terms of a reference particle size. The parameters of the reference particle size, which are denoted as barred quantities and which need to be specified, can be summarised as:
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Dixon's work [1992] was specific towards the extraction of precious metals from ores and so made use of information with regard to ore preparation to determine appropriate relationships for the model parameters. In summary, Dixon made the following assumptions with respect to ore preparation:
Only the surface fraction, Ai, is affected by crushing, and not the total extractable grade, CEiO• or any other parameter; .and;
A.i is proportional to the ratio of the particle area to the particle volume.
Using these assumptions, the following relationship was obtained:
where
where Rk R
). . ). . =-2
1.k e k
( 5-4)
( 5 :-5)
radius of particle in size class k (k€{1..M}); and; radius of the reference particle.
ek is a dimensionless total particle radius. It is important to note that the dimensionless total particle radius can be greater than one. This occurs when the radius of the reference particle is smaller than the radius of the particles in size class k.
Dixon proceeds to determine relationships for Kpi• Ksi and T in terms of the corresponding reference class parameters combined with Ai and Ok.
5.2.2 Determination of the Model Parameters Applicable to Leachina= of Hazardous Constituents from Waste Particles with Respect to a Reference Size Class of Particles.
The assumptions which Dixon made with respect to ore preparation to determine the model parameters as a function of particle size do not hold for waste particles. The reason for this is that Dixon's assumptions are based on his previous assumption of the solid reactants being present in the form of discrete inclusions within the porous particle. Further, the only reason that the surface concentration of the solid reactants will increase with decreasing particle size is due to the fact that more inclusions stand a chance of
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falling on the external· surface area of smaller particles.
In contrast, the hazardous constituents in waste particles are usually not restricted to discrete inclusions. Also, the concentration of the hazardous constituents on the surface of particles in known to be enhanced in smaller particles in some cases [von Blottnitz 1994; Van Craen et al., 1983]. (These aspects have been discussed previously in section 2.1.1.) For these 'reasons, appropriate relationships for the model parameters with respect to a reference particle size which would be applicable to waste particles need to be determined.
It has been assumed that the initial hazardous constituent concentration within the particles, Cpio' is the same for all particles. In contrast, the initial surface hazardous constituent concentration, Csio' is known to be a function of particle size [Van Craen et al. 1983]. In summary, it has been assumed that all properties of the particles, except the surface concentration and thus the total extractable concentration, remain fairly constant over the range of particle sizes. In a manner similar to Dixon, the reference size class's parameters have been denoted as barred quantities. The reference size class parameters which need to be defined are identical to. those of Dixon.
It is not yet possible to predict the surface hazardous constituent concentration as a function of particle size from purely theoretical arguments. Instead this information needs to be determined from hazardous constituent location analyses for the particle sizes of interest or estimated from existing hazardous constituent location data. This information must be specified in a parameter which is defined as the ratio of the · hazardous constituent concentration on the surface of the particles to the hazardous constituent concentration within the particles:
( 5-6)
where C hazardous constituent concentration of species i on the surface of the si,O,k particle in size class k; and;
C hazardous constituent concentration of species i in the pores of the particle pi,O,k in size class k.
Once the S-i(ak) values have been specified, sufficient information is known about the system to formulate suitable functional relationships for the model parameters. This i~ demonstrated for the A.ik parameter, which represents the fraction of the solid reactant on the surface of the particles:
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(5-7)
In a similar manner, relationships for the other model parameters can be determined. These relationships can be summarised as:
( 5-8 ) (5-9 )
(5-10) (5-11)
( 5-12)
Dixon [1992] has shown that if it can be assumed that the chemical species within the particle would react to the same extent as the species on the surface of the particle, if both were exposed to the same fluid reactant concentration for the same time, then the surface parameters take on the form:
( 5 -13) ( 5-14)
This information can be used to eliminate the need to define Ksi· Thus equation (5-9) could be written as:
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A, . r1, <l>pi
_- 1 ("1r:) a Ksi -K_pi--=- -=- 'Clk
/I: 1-A.i <;i ( 5 -15)
5.3 Extension of the Bulk Fluid Mass Balance Equation to Incorporate Fluid Reactant Consumption from a Size Distribution of Particles.
Equation (5-1) is a bulk fluid reactant mass balance equation which accounts for the fluid reactant being consumed by equi-sized particles. This equation can be extended to apply to a size distribution of particles by summing the fluid reactant consumed by the different size classes. This can be summarised as:
(5-16)
where V Part,k is the total volume of particles in size class k; and; M number of sizes classes.
This equation re-expressed in dimensionless format is:
~ [- ~ y Part,k o"'~i ,.., - 3Y Part,k ( aa.) I ] - VLiq Ksi,k s1,k ""bulk 2 a~ k ~=1 -
k=l i=l e2k a e v uk o Part.
(5-17) where
v y = Part,k Part,k
VPart.
(5-18)
where V Part,k total volume of particles in size class k; and; V Part . total volume of particles in the reference size class.
Equation (5-17) is a suitable boundary condition which applies to a size distribution of particles. This boundary condition, used in conjunction with a suitable partial differential equation, equation (4-22), for each size class of particles can be used to simulate the concentration profiles within all of the particles in the system and the bulk fluid reagent concentration as a function of time.
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5.4 Solution Strateay:
The system of equations which define this model are identical to equations (4-22) through to (4-29) except that the one boundary condition, equation (4-24), is replaced by equation (5-17). This equation was incorporated into the solution strategy using an implicit finite difference technique.
As it has already been pointed out, in order to simulate chemical release from a size distribution of particles, the chemical reaction model needs to be applied to each and every size class of particles. The solution strategy adopted for the CSTR model makes use of the solution strategy used to solve Dixon's model and is summarised in Figure 5-2.
Figure 5-2. Summary of the Solution Strategy used in the CSTR model.
J Within each time iteration: I ~ J,
Guess the bulk fluid reactant concentration. I ' ~
Apply the solution strategy adopted to solve Dixon's model to calculate the fluid and solid reactant gradients in the particles for each size class 1 to M.
Use the bulk fluid mass balance equation, equation (5-15), to calculate the bulk fluid reactant concentration.
~ Iterate until the calculated bulk fluid reactant concentration approximates the guessed fluid reagent concentration.
I
5.5 Suitable Computer Routines for the CSTR Model.
Programs Model5El.PAS and Model5E2.PAS are suitable computer codes for the CSTR model. Model5El .PAS is a code which assumes a reaction order of unity with respect to the solid reactant while Model5E2 .. PAS can accommodate a variable reaction order.
Copies of the code as well as solution algorithms can be found in Appendix IV.
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5.6 Verification of the Computer Routines.
As before, the computer routines were rigorously checked to ensure that they were operating correctly.
The first test applied was to use Model5El to predict the concentration profiles within a single size class of particles with a large excess of fluid reagent. Because of the large excess of fluid reagent, its concentration would not be expected to drop significantly during the CSTR experiment. Such a constant bulk fluid concentration boundary condition corresponds to the boundary condition used in the development of the chemical reaction model described in the previous chapter. Thus, Model2D2, using the same parameters as used in Mode15El, should predict the same profiles as Model5El for an experiment of this nature. This is demonstrated in Figure 5-3 and Figure 5-4.
Several other self-checking_ strategies were employed to ensure that the routines were operating correctly. The most important strategies used included the following:
The model was checked to ensure that it would predict the same concentration profiles for a given size class of particles irrespective of the order in which the size class of particles was entered into the program.
The program was checked further to ensure that if the average particle size in two 'different' size classes were identical that the code would predict identical concentration profiles for both size classes.
The fluid mass balance equation was checked to ensure that it was operating correctly by defining several 'different' size classes of particles all to contain the same sized particles. As long as the sum of the volume fractions of these size classes remain constant, the overall fluid reagent consumption should remain constant The computer routines predicted this expected behaviour.
The fluid mass balance equation was further checked to ensure that as bulk volume of the fluid was decreased in a series of CSTR experiments that the resulting bulk fluid phase concentration of the reagent would decrease.
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Figure 5-3.
Figure 5-4.
Concentration profiles predicted by Model5El for a single size class of particles with a large excess of bulk fluid reagent
Bet•1 J..ooo; K-0.:,eJ. ioo.ooo: L....mae.1 o.ooo; orcter.l i.oo; GOT o.03.16 Beta2 l..000; K..,.,82 100.000: L..-82 o.:soo; Orcltlr2 1.00
1.o~!!!!!!~::::'~~~~~~~S:::~::::""'""7:-
c o.e a -.. I ~ .. c I O.G u c a t.I
• ; ~ 0.4 c ~ ~
' 0 0.2
a.OJ;~~~~~~~~~~~:;;~~~~~~~~,.,.::~~ o.o o.;;i 0.4 0.6
Dinen.5ionless Radius
Concentration profiles predicted by Model2D2 using the same parameters used in the simulation used to generate Figure 5-3.
-1202 Betel. 1.000; Kanoa.1 100.000; L..mc:ta1 0.000; Or<Mr.1 J..CXJ: GOT 0,0:JJ.6
Bet82 1.000; K..,.,e2 100.000: L....ixta2 O.:SOO: Orcltlr2 J..00
1.0.,,...,...,,,,,,, ..... """,,,.,. .... ~~.,.,~~===,.....-=-~~
c o.e ~ .. • ~ .. c I 0.6 u c a t.I
• • • c 0.4
~ ~ • ~
0 0.2
o.o o.;;i 0.4 0.6 1.0
OittenSionless Radius
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5.7 Results and Discussion.
5.7.1 General Behaviour.
In order to determine the general behaviour of the CSTR model, the following scenario, representative of a typical bench scale CSTR experiment, was investigated:
Volume of fluid reactant = ll;
Total volume of solid particles = O. ll;
Size distribution of the particles used is summarised in Table 5-1 and shown in Figure 5-5;
-
Only one solid reactant present;
First order rate dependence in the solid reactant concentration;
r1 k used is summarised in Table 5-1 and shown in Figure 5-6; ,
Table 5-1. Size Distribution and r1 ,k used in the Analysis.
Size Class Average Particle % r1,k Number. Size in Size Occurrence.
Class. (mm)
1 9.5 0.252 1.2 2 8.5 0.428 1.2 3 7.5 1.045 1.3 4 6.5 2.564 1.3 5 5.5 6.191 1.4 6 - 4.5 14.061 1.4 7 3.5 27.105 1.6 8 2.5 34.247 1.8 9 1.5 13.846 2.3
10 0.5 0.260 5.0
Note that the size distribution of particles included in Table 5-1 is representative of a log-normal size distribution. Also, the r1,k values were obtained by assuming them to be inversely proportional to the radius of the particle. The equation used to generate the r1 k values used in Table 5-1 can be summarised as: ,
( 5-19)
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where Rk is the average radius of the particles in size class k.
This corresponds to the situation in which infinitely large particles exhibit identical surface and bulk 'grades' of hazardous constituents. Also, the ratio of the surface to bulk grade for the particles in size class 10, which have an average radius of 0.5mm, has arbitrarily, for demonstration purposes, been chosen to be 5. In retrospect, a better relationship would have been:
( 5-20)
The reason why this relationship is superior to equation (5-19) is because it corresponds to the case where infinitely large particles exhibit a negligible surface grade of hazardous constituents compared to the bulk grade of hazardous constituents. This situation is far closer to what would be expected in reality. The reason for this is because the external surface area of a particle increases in proportion to the square of the radius while the volume of a particle increases in proportion to the cube of the radius. Thus infinitely large particles will have a negligible surface area compared to the volume of the particle. Hence the surface 'grade' of the hazardous constituents will be negligible. The ratio of ·the surface grade of hazardous constituents to the bulk grade of hazardous constituents for the reference size class of particles in equation (5-20) has once again arbitrarily been chosen for demonstration purposes to be 5.
Figure 5-5. Size Distribution of Particles used in Analysis
50 ·································
40 Q) () c • Q)
'.:i 30 () ()
0
* 20
• 10
o -o 0.002
•
•
0.004
• 0.006
Radius (m)
77
. - - ,
0.008 0.01 0.012
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Figure 5-6. Hazardous Constituent Location Data used in the Analysis.
5
4.5 ---Co 4 a. 0 ex: 3.5 "5 .0
0 3 c 0 2.5 8
\ \
\ \ \
--(j) 2 0
J2 ~ 1.5 en 0 c 0 8
0.5
0 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01
Radius (m)
From Table 5-1 size class 8 can be seen to be dominant. For this reason it was used as the reference size class in the simulation. The parameters used in the simulation for this size class are summarised in Table 5-2. The KP1 parameter corresponds to the chemical reaction rate being 10 times faster than the rate of fluid reagent diffusing into the reference particle.
Table 5-2. Reference size class parameters. (Reference Size Class = Size Class 8.)
B1 1.0 Kp,1 10.0 .llT 0.001
Figure 5-7 presents the profiles of fluid reagent A (solid curves) and one solid reactant (dashed curves) for each size class of panicle used in the simulation. Figure 5-8 shows the overall conversion of the system.
As expected, the smallest particles tend to react in a homogenous manner. These particles are sufficiently small for diffusion not to be rate limiting in any way. Instead, the release of hazardous constituents from these particles is dictated by kinetic considerations. As the particles get larger, they are seen to react in a more zone-wise manner.
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Fig
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Figure 5-7 Con't. Fluid Reagent and Solid Reactant Profiles for the Smallest Size Class in the Simulation.
.. • " .. c • u c 0 u
• = c ~
I 0
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0.4
0.2
············-·················-···········-···············
·····--···-····-··-··· ·····-················· ·-·-·································
····-············ ·················· ···········-·················-············ ·················· .......... . -··-· - . . . ................ ·····-· ............ ···-····-······-······ .. ··-···········-············
·····-···········································-·················-···········-······························-············ ·····--·-··-······ .......... --····· ...... ············-···········-···········-···················"'"'""""""-············ ....... ···········-···· .... ::::::::::::::::: .. :::::::::::::::::::::::: .. = : :·:::··:::::.·=.·.::.::·::: :::::::: ................... -·
0.0-l--~~~--l~~~~-+-~~~~1--~~~-+~~~---j o.o a.a o.q o.& o.s 1.0
DinensionJess Radius
Figure 5-8. Overall conversion for the System .
.1. 0
o.e
c 0 ... • I. • 0.6 ::> c 0
CJ
... • c 0 0.4 ... .. u • L. II.
0.2
O.O-t-~~~~~~-+-~~~~~~-+-~~~~~~-+-~~~~~~4--~~~~~---J
o.oo 0. J.O o.;;io 0.30 0.40 0.:50
DiMensionl-s React ion Ti~ <MAT Reference Particle:>
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Note that there is some numerical inaccuracy in the concentration profiles of the smallest size class shown in Figure 5-7. Although this inaccuracy can be eliminated by reducing the time step between iterations there is a· trade off between the time for computation and numerical accuracy. It is also important to note that the smallest particles will display the largest numerical inaccuracy. This can be seen from the fact that the smallest size classes have the largest relative time steps in the simulation (equation 5-12). Further, although the profiles in the smallest size class are inaccurate, they are unconditionally stable. This unconditional stability is a property of solving second order parabolic partial differential equations using the Crank-Nicolson method [Crank 1975]. Since the smallest size class in Table 5-1 contains such a low occurrence of particles, the numerical inaccuracy in this size class was deemed to be acceptable.
Figure 5-8 exhibits some interesting characteristics. The graph consists of two distinct sections: a straight line section accounting for the· fractional conversion at the beginning of the experiment, and a concave section later on in the experiment. The straight line section is characteristic of a kinetic controlled situation with an excess of fluid reactant. Effectively it represents the release of solid reactant from the small particles which react in a kinetic controlled manner. In contrast, the concave section is characteristic of a diffusion controlled reaction. Roman [1974] also observed these trends in conversion or recovery calculations. (Although Roman's calculations were for a column, a CSTR experiment can be considered as a very short column in which no mass transfer limitations are present.) ·
5.7.2 Effect of Particle Size Distribution on the Fractional Conversion.
The following scenarios, summarised in Table 5-3 and Figure 5-9, were used to investigate the effect of size distribution on the release of hazardous constituents. Note that all 51,k values were set to unity to eliminate their effect on the results. The parameters for the reference particle size wert? as previously defined in Table 5-2.
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Table 5-3. Summary of the conditions used to investigate the effect of Size Distribution on Fractional Conversion.
Particle Base Case Size Distrib. Size Distrib. fi(eJ Size. Size with with (mm) Distribution Predominantly Predominantly
Small. Particles. Large Particles.
9.5 0.252 1.0 1.0 1.0 8.5 0.428 1.0 9.0 1.0 7.8 1.045 1.0 55.0 1.0 6.5 2.564 2.0 9.0 1.0 5.8 6.191 2.0 5.·o 1.0 4.5 14.061 5.0 5.0 1.0 3.5 27.105 16.0 5.0 1.0 2.5 34.247 55.0 5.0 1.0 1.5 13.846 16.0 5.0 1.0 0.5 0.260 1.0 1.0 1.0
Figure 5-9. Summary of the Size Distributions used in the Simulations.
50
Q) 40 () c Q)
:; () ()
0
30 ............................ ..
'* 20
10 ...
··········-····················
~\····· ................................ ~ ...
o~t::-~-,-~~-r-~~=;:::~~~=-~*--~~--! 0 0.002 0.004 0.006 0.008 0.01 0.012
Radius (m)
- Base Case -+- Pred. ·Small Part. - Pred. Large Part.
Figure 5-10 presents the fractional conversion curves for these cases. The size distribution with predominantly smaller particles is similar to the base case size distribution. As such, the fractional conversion for the two cases is very comparable. As expected, the simulation using the size distribution with predominantly smaller particles predicts a higher conversion at all times compared to the base case simulation. Further, the linear, kinetic controlled region is larger for the simulation consisting of smaller particles. The fractional conversion is significantly delayed for the size distribution consisting of larger particles. In effect, the conversion is being delayed by diffusional
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resistances.
Figure 5-10. Fractional Conversion for the Size Distributions Investigated.
11odel:5E'.l Beta1 1.000; K-1 10.000; L-l. O.::SOO; GDT wr-'t ~l'Siz.Cl-• 0.0100 a.ta2 o.ooo; 1<--2 o.ooo; L.arll>d-2 o.ooo; u1...c1.... e
1.0
0.8
c a ... • I. I 0.6 :I c a ti .. ~ 0 0.4 .. .. u • I. II.
0.2
0.0-+-~~~-+-~~~-+~~~~+-~~~-+-~~~-1
o.oo 0.10 o.ao o.3o o.40 o.:io
Di,_ionless Reaction Ti..., <MAT Aefer-ence Particle>
5. 7.3 Effect of the Location of Hazardous Constituents on the Fractional Conversion.
The following scenarios, summarised in Table 5-4 and Figure 5-11, were used to investigate the effect of contaminant location on contaminant release. The first case represents the physical situation in which no hazardous co'nstituents have been concentrated o_nto the surface of the particles. In other words these particles have a surface hazardous constituent concentration equal to their bulk concentration. The second and third cases represent the cases where the surface concentration of the smallest particle is 5 times and 10 times the bulk hazardous constituent concentration respectively. The r1,k values were determined as before, using equations similar to equation (5-19), and the parameters for the reference particle size were defined previously in Table 5-2.
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Table 5-4. Summary of the conditions used to investigate the effect of Hazardous Constituent Location on Fractional Conversion.
Particle Size. Size Case 1-r1,i. Case 2-r1 " Case 3-t1,i. ' (mm) Distribution.
9.5 0.252 1.0 1.21 1.47 8.5 0.428 1.0 1.24 1.53 7.5 - 1.045 1.0 1.27 1.60 6.5 2.564 1.0 1.31 1.69 5.5 6.191 1.0 1.36 1.82 4.5 14.061 1.0 1.44 2.00 3.5 27.105 1.0 1.57 2.29 2.5 34.247 1.0 1.80 2.80 1.5 13.846 1.0 2.33 4.00 0.5 0.260 1.0 5.00 10.00
Figure 5-11. Summary of the Hazardous Constituent Location Data used in the Simulations.
9 - --------- --- -------------
8 --
O+-~---,-~---,--, ~-,.-~---.~~.-~-.---~~,~~~. ~--,--~--1
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 Radius (m)
\ -+-- Case 1 . - Case 2. (Base) - Case 3.
Figure 5-12 shows the conversion curves for the different scenarios. As expected, the cases where hazardous constituents are concentrated onto the surface obtain higher fractional conversions for all times. This is caused by the fact that hazardous constituent deposits on the surface of the particles cannot be retarded by diffusional resistances (although they may be retarded by mass transfer resistances which have not been accounted for in this work). Also note that as the surface concentration increases, so too, the straight line kinetic controlled portion of the conversion graph increases.
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Figure 5-12. Fractional Conversion for Surface Hazardous Constituents Concentrations Investigated.
Hodel~l
e•t•J. i.ooo: K-..J. 10.000: L.-.J. o.:500: GOT wrt A9rs1z.c1-• 0.01.00 B<tt-2 0 .000; 1(-..2 0 .000; L.-.bcl-2 0 .000; Uf....Cl-• 9
c a .. • l.
1.0
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~ 0.6 c a u .. ~ ,g 0.4 .. u • L. II.
0.2
0.0-1-~~~--t~~~~--~~~~1--~~~-t-~~~---1
o.oo O.l.O o.ao 0.30 0.40 O.:KJ
Di.-.sionles5 React ion T itoe <MAT Reference Particle>
5.8 Fittini: the Model to CSTR Results.
The complete CSTR model which predicts the fractional conversion as a function of dimensionless time for a size distribution of particles has many parameters. This is due to the fact that each size class of particles has the following parameters associated with it:
It is not possible to determine all these parameters simultaneously from a single CSTR experiment. Instead, a CSTR experiment which contains only a single size class of particles, termed the reference size class, must be used to determine the parameters for that size class. If the properties of the particles in the remaining size classes are sufficiently similar to the reference size class then the remaining parameters can be determined, through the relationships defined in equations (5-8) to (5-12). Alternatively when the particles in the other size classes exhibit sufficiently different properties to the reference size class, a CSTR experiment for each size class needs to be conducted to determine all the parameters.
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5.8.1 Model Requirements.
The following information needs to be known in order to fit the model parameters for a single size class of particles to CSTR results:
The relative stoichiometric consumption of the fluid reagent for each participating solid reactant;
the effective concentrations of the participating species;
the reaction orders with respect to the solid reactants;
the ratio C5/Cpi' (.\i,k), for each species in the reference size class; and;
the voidage of the particles.
Note that the discussion on model requirements presented in section 3.8.1 with respect to the chemical reactions, effective concentrations of the species and reaction orders is equally applicable to the CSTR model.
5.8.2 Fitted Parameters.
The fitted parameters in this model are the Kpi and Ksi groups. Notice that there is one Kpi and one Ksi group for every chemical reaction taking place. These groups are fitted by comparing the fractional conversion curves determined by the model to experimentally determined conversion curves.
Before the comparison between the model predictions and the experimental results can be made, the experimental results need to be converted into appropriate dimensionless form. Typical experimental results will be in the form of curves which represent the dissolved concentration of the hazardous constituents in the bulk fluid as a function of time. These curves can be converted into fractional conversion versus time curves. This is a straight forward procedure because the total leachable concentration of each species is known. (The total leachable concentrations of each species can be obtained by conducting a leach test until no further hazardous constituents are released. The bulk fluid concentrations can then be used ·to back calculate the leachable concentrations within the waste particles.) Nondimensionalising the time variable is a more complicated procedure. It is usually not feasible to use the dimensionless diffusion time, 't, defined in equation ( 4-15), to non-dimensionalise the time. The reason for this is that the dimensionless diffusion time includes the effective diffusivity for the particle which is unknown. The dimensionless group defined as:
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k -Ccl>Ri C 'K .A ·'t= pi pi,O A,O t
piPi C. EJ.,0
( 5-21)
and which is termed the dimensionless reaction time can be used to non-dimensionalise the time variable. The reason for this is that the total extractable grade, Cei,O , and the group kpiCpi<\Jsi CA,o , which represents the initial reaction rate, are known. (The initial reaction rate can be determined from the initial slope on the experimental concentration versus time graph). Thus the dimensionless reaction time, equation (5-21) can be used to nondimensionalise the time variable to yield a totally dimensionless experimental curve. By coding the model to predict the conversion versus dimensionless reaction time curves, the model predictions can be compared to the experimental results.
Note that if the hazardous constituents on the surface of the particle can be assumed to react in the same manner as the hazardous constituents within the particle then only the Kpi groups are fitted to the experimental results.
5.9 Applications and Limitations of the Model.
5.9.1 Applications of the model.
As the name implies, the CSTR model is particularly suited to the analysis of CSTR experimental data. Once parameters for a reference size class have been determined, the effects of the following factors on the release of hazardous constituents can be investigated:
particle size distribution;
hazardous constituent concentration;
hazardous constituent location; and;
competing reactions.
The CSTR model is also used to determine the model parameters for the Columnar Model presented in the next chapter.
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5.9.2 Limitations of the model.
The CSTR model has two main limitations. Firstly it cannot be used in cases where there are significant mass transfer resistances because these have not been accounted for in the model. Should the functional relationship between the bulk fluid reactant and the surface fluid reactant concentration be known, it would be easy to incorporate the mass transfer resistances into the model. The second limitation is that the model cannot be
applied to a case where the diffusion of the dissolved solid species is rate limiting. Dixon [1993] has included this aspect into his previous model [1992]. The inclusion of dissolved species transport limitations into the model are discussed in section 7.
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Chapter 6. A Microscopic, Columnar Model to Describe Leachate Generation and Mobility in Granular Waste Deposits.
The macroscopic, lumped parameter model, which was described in Chapter 3, essentially modelled a waste deposit as a single columnar entity. No particulate features within the column were identified in this model. Instead, the contributions of the effective chemical reaction rates, hazardous constituent location and hydrodynamic aspects where lumped together in a parameter which described the effective rate of release of hazardous constituents from the column.
As previously discussed, the main limitations of this model result from using this lumped parameter approach. The most notable limitation is that the individual contributions of the effective chemical reaction rates, hazardous constituent location or hydrodynamic aspects on contaminant release cannot be determined. The overall effective chemical reaction rate for the column is a function of the size distribution of the particles within the column since different sized particles react at different rates. Hazardous constituent location within the individual particles also plays an important role in the release of these constituents. This too in known to be a function of particle size and has been discussed in section 2.1. l. The hydrodynamic aspects, previously discussed in section 2.1.2,are affected by the superficial velocity of the fluid entering the column and by the size distribution and packing of the particles within the column. Since these factors are all dependent on particulate features, a more detailed model which includes particulate information has been investigated.
The heterogenous, columnar model is essentially a columnar, non-catalytic, packed bed reactor type model. The main difference between it and the macroscopic, lumped parameter model is that the particles within the column are included in the model description. Figure 6-1 graphically shows the differences between the two models.
Figure 6-1 Graphical comparison between the macroscopic, lumped parameter model and the heterogenous, columnar model.
No internal structure is considered in
~ the Macroscopic, Lum pee Parameter Model.
89
Lysimeter and internal packing considered in Heterogenous,
__. Columnar Model.
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Two methods to determine an appropriate solution strategy for the heterogenous, columnar model are presented. The first and more simple method is based on the heap leaching modelling strategy of Roman et al. [1974] .. The second method, which is a more rigorous mathematical approach, results in a solution strategy identical to Roman's strategy and has been included merely as a validation of the simpler approach.
This presentation is followed by a section which indicates how a global wetting factor has been incorporated into the solution strategy. Next, the details of suitable computer routines which have been written to implement the solution strategy are discussed. The model has been verified against experimental results which have been presented in a paper by Roman et al. [1974]. The last section in this chapter summarises the experimental data which is required in order to verify the applicability of the heterogenous columnar model to describe the leaching of hazardous constituents from waste deposits.
The following chapter summarises the application of the heterogenous, columnar model to predict the release of hazardous constituents from waste deposits. This chapter also includes a discussion on the advantages of the heterogenous, columnar model over existing models as well as giving details with respect to possible extensions to the model.
6.1 A Modelling Strategy based on Heap Leaching Models.
The strategy adopted in heap leaching models has already been discussed in section 2.4. In summary, the general strategy is to conceptually divide the heap into columnar sections. Each column is then further sub-divided into a series of disks within which the fluid concentration is assumed to be spatially uniform. It is important to note that if the column is divided into too few disks, the assumption that the fluid reagent concentration within each disk being spatially uniform will no longer be valid. The accuracy of the assumption with respect to uniform spatial fluid concentration will increase as the number of disks within the column increase. These sub-divisions of the deposit and columnar sections were shown in Figure 2-5.
The flow of fluid through the column is simulated by allowing the fluid from one disc to replace the fluid in the disk below it at specified time intervals. During each time interval, the fluid reactant is allowed to react with the solid particles resulting in the precious metals, or hazardous constituents, being released.
It is important to note that, within each time interval, each disk in the column consists of an assembly of particles associated with a finite volume of fluid reagent. Since film mass transfer effects have been assumed to be negligible, the decrease in the fluid reagent concentration within each disk and time interval can be described by the CSTR model which was developed in the previous chapter even although the physical characteristics of a CSTR are very different from that of a 'reactor slice' or disk. Since each disk is being described by a CSTR model, the behaviour of the column is being approximated by a number of tanks in series. (The number of tanks corresponds to the
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number of disks in the column.)
The modelling strategy has been summarised in Figure 6-2.
Figure 6-2. Summary of the Solution Strategy used which was based on Roman's Solution Strategy.
j For each time interval:
'\ J,
•Allow the fluid from each disk to replace the fluid in the disk below it. The fluid in the first disk will need to be supplied from a reservoir and the concentration of this fluid forms the boundary condition for the column. The fluid from the bottom disk is discharged from the column and represents the breakthrough fluid.
•Apply the CSTR model to each disk. (Note that the CSTR model can be used because mass transfer resistances from the bulk fluid phase to the liquid/solid interface have been assumed to be negligible.)
Iterate in time.
The equations which describe the release of hazardous constituents within each disk during each time interval, which constitute the CSTR model, have been presented in the preceding chapter.
Effectively the time intervals which elapse between successive fluid replacements represent the time which the fluid would have taken to flow through the disk. Thus for a constant volumetric flowrate, q, the time intervals can be determined from:
where €c01
Col% Sat
A~coI
At= Fluid Volume of Disc q
EcolCOlrg Sat A~col L A
q
column voidage;
(6 -1)
saturation . of the void space within the column; dimensionless length of a disk;
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L A
length of the column; and; area of column.
For a volumetric flowrate which is a function of time, At will need to be determined from (tn+ 1-t1) where (tn+ 1-tJ is obtained from:
Cn+l
J q( t) dt=ecol Col, sac A~col LA (6 -2)
t;n
Since the computer routines which' have been written for the CSTR model are in dimensionless format, it would be convenient to express the time intervals between successive fluid replacements in dimensionless form. These time intervals can be converted to non-dimensional quantities by dividing them by a reference time period. One appropriate reference time period is the space time of the column defined by equation (3-13):
L T=·u •
( 6 -3)
where u· is a reference fluid velocity (percolation velocity) which has arbitrarily been set at lm per 24 hours.
6.2 A Modellini: Strate2y based on a Rii:orous Mathematical Approach.
The starting pQint for this analysis is a mass balance equation which describes the bulk fluid reactant concentration within the column. To obtain this equation an approach very similar to the one adopted in the macroscopic model development can be used. The only difference is that the rate term, rai' which represents the 'production' of fluid reagent due to chemical reaction, is retained in equation (3-5) and not replaced by an overall lumped rate expression such as equation· (3-2). The modified form of equation (3-5) is:
acA acA n E Col col =-u col+~ I
Col % Sat: a LI t az i=l ai ( 6 -4)
The same equation expressed in dimensionless terms is:
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acxcol =-DGl acxcol +----L ____ £I
a I ai: *C C 1 ' A1 't "col U Ao Ecol 0 % Sat .i=l
(6-5)
where DGl is as previously defined in equation (3-14), and has the physical significance of being the ratio of the fluid ·percolation velocity to the reference fluid percolation velocity.
The appropriate initial and boundary conditions for this equation are:
«col (~Col' 0) =O (6-6)
( 6 -8)
Equation (6-5) is a first order hyperbolic partial differential equation. As discussed in Chapter 3, the method of characteristics can be used to convert the partial differential equation into two ordinary differential equations. These equations are:
( 6-9)
and
dabulk L I:ri
d't' CA bulk0 Ecol Col% Sat U *
(6-10)
It is interesting to note that the term on the right hand side of equation ( 6-10) represents a normalised - rate of consumption of fluid reactant over the rate of fluid reactant replenishment. In this regard, this term represents the same ratio as the previously defined dimensionless group DG2.
Before these equations can be solved, suitable expressions for the rate of 'production' of fluid reactant A by the i reactions, rAi, need to be determined.
Rather than redevelop these expressions, recall equation (5-16):
~ [ VPazt kPa ( 1-Eo) ~ dCs1 n I 3 VPazt k] dCA .,. · .,. -D vC • -v k b . -dt e A Rk R - Liq. dt
=1 i i=l k . ( 5-16)
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where V Part,k is the total volume of particles in size class k; M number of sizes classes; and; Rk Radius of the particles in size class k.
This equation expressed in dimensionless format is:
~ r-f iPaxt,k a4>s:1 N - 3iPaxt,k ( aa;) I ] = VLiq Ksi,k si,k "'bulk 2 ~~ k ~=l
k=l i=l 0 k2 ak u.., e v o or o Part.
(5-17 or 6-11)
This equation describes the change in the bulk fluid reactant concentration as a result of the fluid reactant being consumed by chemical reactions within a range of different sized particles. Note that the time variable used in equation (6-11) corresponds to a dimensionless diffusion time corresponding to the reference size class of particles. In contrast, the time variable used in equation (6-10) corresponds to a dimensionless space time of the column. If the column time increment is set equal to the reference size class time increment then equation (6-11) can be used in place of equation (6-10). This implies:
where Eo
VPart.
Vu q.
TPart,k
panicle porosity;
da; bulk da; bulk
en - d-r.'
(6-12)
in this case represents the total volume of the reference size class particles within the spatial increment in the column; in this case represents the volume of fluid within the spatial increment in the column; defined in equation (5-18) ·and which represents the ratio of the Yolume of the particles in size class k to the volume of the particles in the reference size class; and; defined in equation (5-5) and which represents the ratio of the average radius of the particles in size class k to the average radius of the particles in the reference size class.
The overall mathematical solution strategy can be summarised as follows. Firstly set a suitable spatial increment, A~c01 , for the column. This increment corresponds to the length of a disk in the previous heap leaching analysis. Equation (6-9) is then used ·to
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determine the time increment for the column. As shown in Chapter 3, an alternative method to determine the time increment for the column is:
(3-31 or. 6-13)
where (tn+ 1-tJ is obtained from:
tn+l
J u ( t) dt = Ecol Col, Sat t:.Z
=ecol Col, sat A~col L (3-32 or 6-14)
Notice the similarity between equation (6-14) and equation (6-2). In effect these two equations are identical.
Within each time increment and within each spatial increment the change in the bulk fluid reactant concentration can be solved using equation (6-12). Before this equation can be solved however, the fluid reagent profiles within each particle size class need to be known. These can be determined by applying the particle scale model of Dixon to each size class of particles. This approach is summarised in section 5.4.
It is important to note that in reality it is not always feasible to set the column scale time increment equal to the particle scale time increment. The reason for this is that the column time increment is usually sufficiently large to result in the Crank-Nicolson method, used to solve the particle scale concentration profiles, becoming totally inaccurate. This problem is easily overcome by using a much smaller time increment for the particle scale calculations and repeating these calculations until the cumulative time increment for these calculations is equal to the column scale time increment. This can be summarised as:
n A'tParticle scale=A'tcolumn Scale (6-15)
where n is the number of times that the particle scale calculations need to be repeated.
The solution strategy has been summarised in Figure 6-3.
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Figure 6-3. Solution strategy derived from a rigorous mathematical analysis.
Choose an appropriate column increment, Ll~c01 ;
Use equation (6-9) to determine an appropriate time increment, Llr';
Within each time increment do:
•Within each spatial increment use equation (6-12) to determine the fluid reagent concentration. (This effectively involves solving a set of equations identical to the CSTR model equations defined in section 5 .4.)
Iterate in time.
Comparison of Figures (6-2) and (6-3) shows that the solution strategy derived from the rig~rous mathematical analysis is identical to the simpler heap leaching modelling strategy.
6.3 Inclusion of a Global Wetting Factor into the Solution Strategy.
A global wetting factor has been incorporated into the solution strategy to account for the partial wetting of particles. The global wetting factor represents an average fraction of particle surface area wetted by fluid reagent. As an example, a global wetting factor of 0.5 implies that all of the particles are only half covered by fluid reagent. It has further been assumed that sectors within particles which are bordered by dry external surfaces will not release hazardous constituents. In effect, this implies that even with dry zones within a particle, the diffusion will remain one dimensional - in the radial direction only.
From the above discussion, wetting can be seen to influence the release of hazardous constituents in the following manner. Firstly, the degree of wetting determines the solid which is available to react with the fluid reagent. As before, a global wetting factor of 0.5 implies that only half of the particle volume is available to react with the fluid reagent. This in effect also determines the maximum release of hazardous constituents _achievable. A column with a wetting factor of 0.5 will have a maximum contaminant release (conversion) of 50%.
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Using this information the effect of partial wetting was incorporated into the solution strategy as follows. Equation (6-12) has previously been used to determine the bulk fluid reagent concentration within the particles as a function of time for totally wetted particles. This equation can still be used in the case of partially wetted particles if V Part.,k
is defined as the product of the volume of the particles in size class k within a spatial increment within the column and the global wetting factor. This can be summarised as:
v = y v Part, k Part, kActual ( 6 -16)
where V Part,k,ActuaI in this case represents the actual volume of the particles in size class k within the spatial increment in the column;
V Part,k in this case represents the volume of the particles in size class k which is available to react with the fluid reagent within the spatial increment in the column; and; is a global wetting factor.
By using a global wetting factor, all particles have been assumed portion of their external surface area covered by fluid reagent. information with respect to particle wetting as a function of particle could be included into the model as follows:
v = y v Part, k k Pazt, kActual (6-17)
where 'Yk is a size dependent wetting factor.
to have an equal If more detailed
size is available, it
6.4 Suitable Computer Routines for the Heteroi:enous Columnar Model.
Program Model6Cl.PAS and Model6C2.PAS are suitable computer codes for the heterogenous, columnar model. As before, Model6Cl.PAS is a code which assumes a solid reaction order of unity while Model6C2.PAS can accommodate a variable reaction order.
These codes are sufficiently large to justify some explanation as to their organisation. From the discussion on the solution strategy in the previous section it became evident that the codes need to calculate, as a function of time, the reactant profiles within each particle size class within each disk of the column. This information is then related to the fluid reactant and dissolved hazardous constituent profiles within the column itself.
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The particle scale calculations, which are essentially identical to the CSTR model calculations are incorporated in Unit6Cl and Unit6C2. These units are almost identical to previous PASCAL codes Model5El and Model5E2 respectively. All the particle scale parameters are defined and set in these units. The column scale calculations are performed in the main program Model6C 1 or Model6C2. All the column parameters are defined and set in these programs. Figure 6-4 summarises the overall organisation of program Model6Cl.PAS.
The output of these codes is a display of the solid and fluid reactant concentration profiles for the smallest, largest and reference size class of particles as a function of time and position within the column. This information helps to visualise how the different size classes are reacting in different positions of the column at different times. This inf9rmation is most useful because it helps to identify which size classes of particles react in a chemical kinetic controlled manner and which react in a diffusion controlled manner. In effect, this information can be used to determine the relative importance of intrinsic chemical kinetics and diffusive resistances on the overall rate of release of hazardous constituents. A typical display of the solid and fluid reactant concentration profiles is shown in Figure 6-5.
Once the required iterations in time have been performed and the calculations are completed the programs display the conversion and breakthrough curves as a function of dimensionless time. Although the computer routines do not presently display the concentration profiles within the column as a function of time, the data required for these curves is already calculated. Thus this feature can easily be incorporated into the computer routines if required.
Figure 6-4. Summary of the overall organisation of Program Model6Cl.PAS.
Model6C.
Column Scale Calculations.
The main program contains all UoitflQ, the column parameters. Particle Scale
Calls Unit6C Calculations. ... ... to perform ... . particle ' This unit contains
scale calculations. all the particle parameters.
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Figure 6-5. A typical display produced by Model6Cl.PAS showing the solid and fluid reagent profiles for the smallest, largest and reference size class of particles.
BetaJ. Beta2
t,I c 0
LJ
Ill Ill ... :i: Q
LI ,. i3 CJ
~ 11 .. c
i:::
Uni tGC.1 i.ooo; KaDDaJ. 10.000; La,.,baaJ. o.ooo; oraerJ. i.oo; GOT o.ooo:5
·oo; KaP?a2 o .ooo; La ... bda.2 o .ooo; oraer2 i .oo
L • 0 l.~~=~.'.~.~ ....... ~ ... ~~~.~ .......... ~=~~.' .. '.~~ ... ~~:::=~~·~·;;;i~;~.;'.:::.., B
a.a.
0.6
0.4
a.a
DiMensionless Radius
SMallast Siza Fraction L~r-a-st Siz• Fraction
0.2 0.4 J.. 0
Di~ionless Radius DiMenSionless Radius
It is worth p.omtmg out that these programs include numerical step size scaling procedures. As such, the programs have been written to detect numerical stability and to increase the step size used in the calculation strategy whenever appropriate. The main advantage of using such procedures is that the overall calculation time can be greatly reduced.
Copies of the code as well as solution algorithms can be found in Appendix V.
6.5 Verification of the Computer Routines.
Roman et al. (1974] conducted two lysimeter tests which were significantly different in terms of lysimeter dimensions, fluid reagent concentrations and flow rates, and the size distributions of particles, in order to determine the predictive capability of their model.
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In summary, they used one set of data to determine the single unknown parameter in their model and then used the model to predict the second set of data. Although the heterogenous, columnar model presented in this chapter is capable of incorporating far more complexities than the model presented by Roman et al. [1974], its predictive capacity can be tested in a similar manner.
The lysimeter experiments conducted by Roman were concerned with copper extraction. The physical properties, operating conditions and size distributions of the particles used in the two experiments have been reproduced from Roman [1974] and are summarised in Table 6-1 and 6-2, and in Figure 6-6 and 6-7 .
Roman et al. did not include the voidage of the ore particles in their data and this value has been assumed to be 1 % . It is important to note that the Kpi parameter used in the heterogenous columnar model is the ratio of the chemical reaction rate within a particle to the effective rate of diffusion of fluid reagent into the particle. The effective rate of diffusion of fluid reagent into the particle is a function of the particle voidage. The effect of particle voidage on the effective diffusivity has not been considered in the heterogenous columnar model. The particle voidage only affects the heterogenous columnar model through the continuity equation ( 4-1). Equation ( 4-6) is equation ( 4-1) in expanded form. The heterogenous columnar model was determined to be relatively insensitive to small changes in particle voidages in the region of 1 % . For this reason a particle voidage of 1 % has been assumed to be an adequate assumption.
The acid consumption which was determ~ned experimentally has been assumed to include the effects of fluid reagent holdup within the particles. This implies:
Acid Consumption=~e- b l-E copper (6-18)
where bcopper is the mass of copper released per mass acid consumed.
Table 6-1. Physical Properties and Operating Conditions.
Lysimeter 1. Lysimeter 2.
Weight of ore. (kg) 121 74.5 Column height. (cm) 176 305 Column diameter. (cm) 25.4 14.3 Solution flowrate. (l per min per m2) 0.155 0.652 Acid concentration. (gpl) 48.8 69.7 % Voidage. 49.8 42.3 % Saturation. 36.1 37.1 Acid consumption. (g Acid per g Cu.) 3.6 3.6 Copper grade. ( % ) 1.9 1.9 Ore specific gravity. 2.7 2.7
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Table 6-2. Size Distributions. (% Occurrence.)
Particle Size. (mm) Lysimeter 1. Lysimeter 2.
37.5 58.1 -37.5 - 25.0 15.8 -25.0 - 19.0 10.3 -19.0 - 13.2 4.1 29.8 13.2 - 9.5 3.2 22.9 9.5 - 6.7 1.8 15.0 6.7 - 4.75 1.3 6.4 4.75 - 3.35 1.0 5.4 3.35 - 2.36 0.6 3.3 2.36- 1.70 0.5 1.9 1.70- 1.00 0.5 1.9 1.00 - 0.85 0.3 1.6 0.85 - 0.60 0.3 1.6 0.60- 0.425 0.3 1.4
- 0.425 1.9 8.8
The size class used as the reference size class in both simulations was arbitrarily chosen as the 9.5mm to 13.2mm size class.
Since no appreciable amount of copper is concentrated on the surface of the particles the 'surface grade'' csi 0• was assumed to be zero for all particles.
Using this data, the only unknown parameter in the simulation is the Kcopper parameter which represents the ratio of the intrinsic chemical reaction rate within the reference size class of particles to the rate of fluid reagent diffusing into these particles. Using the experimental data for the second lysimeter, this parameter was determined to be 4.5 for the reference size class (9.5mm to 13.2mm). It is worth noting that the model is fairly sensitive to this parameter. For example, running the model with Kcopper=4 and Kcopper=5 resulted in significantly different curves to the experimentally determined curve. Further, the model converges to a Kcopper value of 4.5 irrespective of the whether the initial guess for the Kcopper is larger or smaller than 4. 5.
Using a value of 4.5 for the Kcopper parameter, the model was used to predict the performance of the first lysimeter. The results of both simulations compared to the experimental points are shown in Figure 6-8.
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Figure 6-6. Size distribution of the particles in Lysimeter 1.
60,--~~~~~~~~~~~~~~~~~~~~~~~----,
i I i I
.....• ······························1 50 ······················································································································· ··································
I I 40 ······································································ ···················································································· : ................... 1
§ I I ~ i I g 30 ·-.. ···························································································································----·····:··············· .. ·······························-r-·····---·························I
o I * 20 ................................................................................ : ........................................................... ·····!················· ··1
' / II
10+··········:··············································································································--······································,~-·: ....................................... 1
0+1~~~~:::::;:::::;:~··· ~,·~~.~~.=-•1::=••,:::::::m~~,~~~~~~~1-1
~.,.---,--,--,-..,.1~1_,_j,] 0.1 1 10 100
Average Particle Size (mm)
Figure 6-7. Size distribution of the particles in Lysimeter 2.
50 .................................................................................................................. ·--···-·-·····················································································
~ 40-1 ------·································-··· -----······-················ ···········-·-························1
j 30 i········ --··································· -··························· - -···-··-····-··· 7-····· ·························· -····i
* • 20 ..................................................................... ~---·-···········-·····················--·-····-1- - -- - - -1
I . 1 0 ....... ······· ····ii(······· ........ ················ .................................... ······················-!-············ .. ·········
I ·"--, I '--,
i ~--~=~----------~ 07,~~-r----:---:---;-r.,-.,.....,-:-~~-,--,-~~-..,.-,~ ,,,~~~~~~~-.~.
0.1 10 100 Average Particle Size (mm)
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Figure 6-8. Fitted curve and predicted curve for Model6Cl compared to the experimental points of Roman [1974].
0.9 ······················-·····. ·························································· ........................ ············································ ········ ....... ··········
c 0.8 ..................... ' -··················································---·····-····--··············--···················--···················---···············--····----·---·-·-········--······
i ~:: :::~:- _:J;.~::: : mm mm ::: ::mmm-m•••~::::::·:::::::::::::::: ::-j 0 0 ->.. '-
~ 0.4 0 g 0.3 a: cfl. 0.2
0.1 ................................................................................................................ \
O-='--~-,~~~~~~~~~~~~~~~~~~~~~~I 0 5 10 15 20 25 30 35 40
Time (Days)
\ --- Experimental Points -e- Model; Kappa=4.5
Figure 6-8 shows that Model6Cl is capable of accurately predicting the performance of heap leaching experiments.
6.6 Determination of the Heteroaenous Columnar Model Parameters from Appropriate CSTR type Experiments.
In the previous section, the parameters for the heterogenous, columnar model were determined from a lysimeter experiment. Where external mass transfer resistances are negligible, the model parameters can be determined from an appropriate CSTR type experiment.
The model parameters would be determined as follows. A CSTR experiment would be conducted on a single size class of particles which would then represent the reference size class of particles. The CSTR model would then be fitted to the experimental curve as discussed in section 5.8. These parameters could then be used as the reference size class parameters in the heterogenous columnar model.
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This strategy will be demonstrated in reverse. The model parameters in section 6.5 were determined from fitting the heterogenous columnar model to a lysimeter result. Using the parameters determined, the predicted CSTR conversion versus time curve corresponding to 0.1/ of 9.5mm to 13.2mm particles and 1/ of 48.8 gp/ of acid is shown in Figure 6-9. This curve was predicted using the CSTR model presented in the previous chapter. The significance of Figure 6-9 is that it is the conversion versus dimensionless concentration curve which would have been obtained had a CSTR type leach test been used instead of a lysimeter test. This demonstrates that parameters determined from one type of experiment can be used to predict the performance of other CSTR or lysimeter scenarios.
Figure 6-9. Predicted CSTR conversion versus time curve for O. llof 9.5mm to 13.2mm particles and 1/ of 48.8 gp/ of acid.
S.t•J. 0.26-4; Kat>Q•J. 4.~; L•Ml><l•J. o.ooo; GOT ..,rt Rer-s1z.c1 ... o.~oo
S.t-2 o.ooo: KMX>-2 o.ooo: L...VXS-2 o.ooo; V!.....Cl••• :s
c: c -• I.
1.0
O.B
~ 0.6 c: c (J
0.0 :d.O 4.0 6.0 8.0
OiMen5ionJ~s React ion T i.-.e <MAT Reference Part icJe)
10.0
There are distinct advantages in using a CSTR type experiment to determine suitable model parameters for the heterogenous columnar model. Probably the most important advantage is the short duration of a CSTR experiment. Most heap leaching type models make use of lysimeter data in order to determine the model parameters. (This is exactly the procedure used in the previous section to verify the model.) As previously stated, lysimeter tests typically last for at least a few months and can last up to 3 years. In contrast, a CSTR type leach test takes at most a few days to complete. This means that the potential release of hazardous constituents from a proposed deposit can be determined much more quickly than before. This is a significant advantage in that the mineral processing industry needs to know the likely impacts of certain disposal strategies so that it can determine wether or not the present disposal strategies are adequate. Before, the industry would merely dispose of the wastes and conduct a lysimeter experiment to determine the likely impacts of the disposal strategies. This information would then be used to improve future disposal strategies. In cases where a disposal option was determined to pose a significant risk to the environment, the industry was faced with the expensive rehabilitation or reprocessing of the waste material. The ability
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to predict the likely impacts of the disposal strategies· a priori will help to eliminate the costly rehabilitation or reprocessing procedures.
The financial savings incurred by conducting CSTR type experiments over lysimeter experiments are also significant. CSTR experiments are much cheaper to conduct than lysimeter experiments. This has significance in that existing deposits can be investigated to determine their environmental risks. Previously, suitable lysimeter experiments for all waste deposits were not feasible due to the costs associated with these tests. By making use of the cheaper CSTR tests, samples from these deposits can now be tested to determine their potential to pollute the environment.
6.7 A Summary ofthe Experimental Data which is Required to Verifythe Applicability of the Hetero2enous Columnar Model to describe the Leachin2 of Hazardous Constituents from Waste Deposits.
In order to demonstrate the applicability of the heterogenous columnar model to describe the leaching of hazardous constituents from waste deposits, it. needs to be verified against suitable experimental data. Section 6.5 verified that the computer routines were operating correctly by testing them against heap leaching data for copper. Although this shows that the model can be used to accurately predict the leaching of copper from a heap leaching operation, the ability to predict contaminant leaching from a waste deposit still needs to be demonstrated.
This project has not involved any experimental work. Where experimental data has been required, data which has been reported in the open literature has been used. Unfortunately, suitable data is not reported in the literature which can be used to verify the applicability of the heterogenous columnar model to contaminant leaching from waste deposits. This section summarises the type of data which is required and suitable methods to verify the model using this data.
It may be· worth pointing out some of the short comings of data reported in the open literature so that these can be avoided when suitable experiments are planned and conducted. Firstly, although there is data on leach experiments for waste particles, these leach experiments have always involved a size distribution of particles. The reason for this practice is probably due to the fact that leach tests have been typically used to determine the 'leaching potential' of wastes. To determine this 'leaching potential', a sample of the waste, which inherently involves a size distribution of particles, is leached and the final leachate concentration is taken as a measure of the 'leaching potential'. These tests are usually termed Toxicity Characteristics Leaching Procedure tests (TCLP tests) [US Government Printing Office; 1988, 1980]. These experiments have not been used to model the effective release of hazardous constituents from waste particles. It has been continuously demonstrated in this thesis that the leaching of hazardous constituents from waste particles is dependant on the particle size. Thus to determine the effective
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release of hazardous constituents from waste particles a leach test on o single size class of particles is required.
0
The second factor is that the heterogenous columnar model describes the leaching of hazardous components from wastes. Some hazardous components of wastes are released due to dissolution reactions. It is unlikely that the chemical reaction model used in the heterogenous columnar model can be used model dissolution reactions. The reason for this is due to the fact that dissolution reactions are controlled by solubility constraints which have not been included into the model. Instead the model describes the rate of hazardous constituent release by making use of a kinetic expression corresponding to equation (4-2). Thus care should be taken not to apply the model to instances in which hazardous constituent dissolution is significant.
It goes without saying that a model can only be applied to situations for which it was designed. With this in mind, the heterogenous columnar model has been designed to describe the active leaching of hazardous components from granular waste deposits in which there is no external mass transport resistances. (The external mass transport resistances refer to the transport of components from the bulk fluid phase to the fluid/solid interface and vice versa.) Additionally, the particle characteristics as a function of size have been assumed to be fairly uniform. The only exception to this is that the concentration of hazardous constituents on the surfaces of the particles, and thus the total leachable concentration of hazardous constituents, has been allowed to vary as a function of particle size.
There are two stages to verifying the ability of the heterogenous columnar model to describe the leaching of hazardous constituents from waste deposits. The first stage involves determining whether the model can be fitted to lysimeter data and then used to predict other lysimeter operating conditions. This approach is very similar to the strategy adopted by Roman et al. [1974] and used in section 6.5. This approach would involve setting up two different lysimeter experiments. These experiments could differ in terms of the column heights used, the fluid flow rates through the columns (which in turn would affect column saturation and the wetting efficiency), the fluid reagent concentrations and the particle size distributions. The results of one of the columns could be used to determine the parameters in the heterogenous columnar model. The -model could then be used to predict the results of the second column and the predictions compared to the actual results.
The second stage involves determining whether the heterogenous columnar model parameters could be determined from a suitable CSTR experiment. This would involve conducting a CSTR experiment on a single size class of the waste particles and determining the model parameters as described in section 5. 8. These parameters would then be used in the heterogenous columnar model to predict the two column experiments discussed above. Once the model has been verified in this manner it could be used with confidence to predict the leaching behaviour of hazardous waste constituents from waste.
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An initial investigation into determining the particle-scale model parameters from a CSTR type experiment on a single size class of waste particles has been conducted. The data used in this analysis is from an experimental program currently being conducted at the University of Cape Town. The experimental conditions are summarised in Table 6-3.
Table 6-3. Summary of the Conditions in the CSTR Test on a Waste Sample.
Waste type. Stainless steel electric arc furnace dust.
Volume of fluid reagent. 1.6 l
Mass of waste particles. 0.04 kg
Particle Voidage. 1 %
Solid density. 2900 kg/m3
Average particle size. 3e-6 m
The pH remained fairly constant at a value of 10 during the experiment.
The concentration of dissolved magnesium in the bulk fluid was used as a measure of the leach potential. This is shown as a function of time from one CSTR experiment m Figure 6-10.
Figure 6-10. Concentration of Dissolved Magnesium in the Bulk Fluid as a Function of Time.
-E a. a. -c 0 -ca ,_ -c <Ll (.) c 0 u 0)
2
1a~~~~~~~~~~~~~~~~~~~~~~~~-,I
16 ·····························/·~¥·· I
14
12
10
8
6
4
2
I I
0... I
················±-·/···················· f, • . ·····················-r·············
• I
I .................. '-.. .... ,, .................. .
I I ,
I ............ , .. ··················· ··············································································································
I I ,
·I .......... .!
o~,L-~~~~~~~~~~-.--~~---,~~~--.-,~~~---r-~~----j
0 100 200 300 400 500 600 700 Time (min)
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The point labelled (a) in the diagram has been taken to be an outlier. The curve drawn through the experimental points is the expected increase in the magnesium concentration in the bulk fluid with time based on the data points. This curve was used to back calculate the total extractable grade, CEio' of the magnesium.
These initial reaction rate for the release of the magnesium has been determined using the initial increase in the bulk fluid concentration of magnesium as follows:
( 6 -19)
is the dissolved fluid phase concentration of solid reactant i; is the volume of the fluid in the batch test; and; is the mass of the waste particles in the batch test.
The total extractable grade and the initial reaction rate were used to prepare the fractional conversion versus dimensionless reaction curve as described in section (5-8). This curve is shown in Figure 6-11.
Figure 6-11. Fractional Conversion versus Dimensionless Reaction Time.
1.....-~~~~~~~~~~~~~~~~~~~~~~~--,
0.9
0.8 ··············································· .... c Cl. ·§ 0.7 ···················+·····································
Q) • • > 0.6 c 0 0 0.5
························································· .::::::::.:::.:·.:::::::::·.::.::.:.::::::::·.:::::J
I m c 0 0.4
:;:::; (.)
························ ··················································································································································· ···········································1
m ..... LL
0.3
0.2
0.1 : : : : : :~ : : : : : I
0-""--~~~~~..---~----,~~---.,-,~~-,.-,~~--.-~~-,-~~-.--~------1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Dimensionless Reaction Time
Mode15El was fitted to Figure 6-11 and the results are shown in Figure 6-12. The dashed curve in this figure corresponds to the curve in Figure 6-11. The solid curve is the fractional conversion calculated using Mode15El. The parameters fitted in the model
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were B, which represents the dimensionless reagent strength relative to the concentration of the magnesium in the waste; Kpi - Damkohler II number for the magnesium within the waste particles; and; ~which is the ratio of the surface grade of magnesium to the bulk grade of magnesium. The magnesium on the surface of the particles has also been assumed to react in a similar manner to the magnesium in the bulk of the particle. This is the reason why there is no explicit dependence on the surface kinetics in equation (6-19). The values determined by an initial fit are summarised is Table 6-4. There is excellent agreement between Model5El and the CSTR data from a sample of metallurgical waste.
Table 6-4. Values of the Fitted Parameters.
B1 0.8
Koi 30
~ 1.0
Figure 6-12. Model5El versus CSTR Experimental Data on a Waste Sample.
Hodel:5E1. Betai o.eoo; Ka!X)a1 40.000; Lal'\bda1 o.~oo;· GOT Yrt Ae~SizeClass 0.0020 Beta2 o.ooo; Ka!X)a2 o.ooo; LaMbda2 o.ooo; UteYClass 1.
.1. a
a.a
c: 0.. ,, -0 .. , . ._~~"~ ... ,,. UI , I. • • / II a.6 :> L Oo\\tci CUfUC co<~cb c: 0 ' CJ v lo cu.Na \4\ ':f"~\UI!. c.-1t. ... 19 c: 0 a.4 ... ..
te:s~lh ~ 'ic:de16EI. u
5.ltd CW&>e w ~ I. Li.
o.:a
a.a-P-~~~~-1-~~~~--1-~~~~-+-~~~~--+~~~~----i
o.o .l.a a.a 3.a 4.0 ~.o
Dincnsionless Reaction TiMe <WAT Reference Particle>
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Chapter 7. Summary of the Applications, Limitations and Extensions of the Heterogenous, Columnar Model.
The previous chapter presented the development of the heterogenous columnar model. Conceptually, the modelling strategy involved the division of the waste deposit into columnar sections and then determining the release of hazardous constituents from a single column. The release of hazardous constituents from a single column is determined by making use of a one-dimensional, non-catalytic, packed-bed type reactor model. The reactor model is derived by applying a fluid reagent mass balance to the column. Within the column, fluid reagent is consumed due to chemical reactions which occur both on the surfaces and within the waste particles. This consumption is modelled using a particle scale, chemical reaction model which also determines the release of hazardous constituents into the bulk fluid surrounding the particles. Once the hazardous constituents are released from the particles, they are assumed to be transported out of the column by the bulk fluid flow through the column.
This chapter summarises the applications, limitations and possible extensions of the heterogenous columnar model. The first section compares this model to the model of · Roman et al. [1974] and the columnar model of Dixon [1992, 1993]. As previously discussed, although these last two models were developed as precious metal leaching models, several similarities exist between precious metal leaching from ores and the leaching of hazardous constituents from waste particles. Most importantly, this section highlights the advantages of the heterogenous columnar model, which was specifically designed to model the leaching of hazardous constituents from waste deposits, over the heap leaching models.
The next section discusses potential engineering applications of the heterogenous columnar model. In particular, this section summarises the type of information which can be determined using the model and demonstrates how this information can be used to improve waste deposit design and to choose upstream processing options which result in more stable wastes.
The limitations of the heterogenous columnar model, which are largely due to the simplifying assumptions which have been made, are then reviewed and methods to remove these limitations are presented.
Finally, a statement of the significance of this work is made.
7.1 Comparison of the Hetero&enous Columnar Model to the Model of Roman et al. (19741 and the Columnar Model of Dixon (1992, 19931.
The heterogenous columnar model is comparable to the model presented by Roman et al. [1974] and the columnar model presented by Dixon [1992, 1993]. All of these models
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approximate deposit or heap performance by determining the performance of a single column packed with waste or ore partides. Further, all three models approximate the fluid flow through the deposit or heap as perfect plug flow and include a saturation index. This parameter is included to account for unsaturated conditions in the waste deposit or ore heap. The heterogenous columnar model includes a global wetting parameter, presented in section 6.3, to accommodate the effects of incomplete particle wetting on the release of hazardous constituents. Since the fluid reagent distribution systems in ore heaps are specifically designed to maximise particle wetting, incomplete particle wetting is not considered by Roman et al [1974] or Dixon [1992, 1993].
Table 7 .1 summarises the properties of the heterogenous columnar model and the heap leaching models of Roman et al. [1974] and Dixon [1992, 1993].
Table 7 .1. Summary of the properties of the heterogenous columnar model and the heap leaching models of Roman [1974] and Dixon [1992, 1993].
Fluid flow model:
Chemical reaction model:
-
Heterogenous Columnar Model.
Plug flow model.
Constant or variable fluid flow into deposit.
Includes a saturation index and a global wetting factor.
Particle scale model which includes the effects of intrinsic chemical kinetics, diffusion and hazardous constituent location on contaminant release.
Parameter spec. as a function of particle size is particularly suited to waste particles.
Heap Leaching Model of Roman et al.
Plug flow model.
Constant fluid flow into heap.
Only includes a saturation factor.
Particle scale shrinking core model which assumes that diffusion of the fluid reagent is rate limiting.
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Heap Leaching Model of Dixon.
Plug flow model.
Constant fluid flow into heap.
Only includes a saturation factor.
Particle scale model which includes the effects of intrinsic chemical kinetics, diffusion and precious metal location on metal release.
Parameter spec. as a function of particle size is particularly suited to ore particles.
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7.1.1 Comparison to the Model of Roman et al. (19741.
The main difference between the heterogenous columnar model and the model of Roman et al. [1974] is in the method used to determine the rate of hazardous constituent or precious metal release from the individual particles. Roman et al. assumed that the release of the precious metals was dominated by diffusion of the fluid reagent into the ore particles and thus made use of a diffusion controlled, shrinking core reaction model. As previously discussed in section 2.4, this model cannot include the effects of intrinsic chemical kinetics, precious metal location within the ore particle or competing chemical reactions on the rate of precious metal release.
The heterogenous columnar model developed here employs a chemical reaction model which is based on a fluid continuity equation at the particle scale. This model, which was used by Dixon [1992] and summarised in Chapter 4, is used to determine the release of hazardous constituents from waste particles. (The particle scale model of Dixon is distinct from his columnar model.) This approach is capable of including the effects of intrinsic chemical kinetics, intra-particle diffusion, hazardous constituent location within the particle and competing chemical reactions on the rate of release of the hazardous constituents.
The ability of the particle scale chemical reaction model, which is incorporated into the heterogenous columnar model, to include the effects of competing chemical reactions is particularly important when describing release of hazardous constituents from waste particles. The reason for this is that most waste particles consist of two or more solid reactive species present in significant concentrations which compete for fluid reagent. As long as the voidage of the waste particles remains fairly constant, this competition has the effect of reducing the available fluid reagent for each reactive species. Thus the release of an individual species is slowed down for each additional competing reaction which occurs. In contrast, competing reactions which result in an increase in the voidage of the particles may result in increased leaching rates. The reason for this is due to the fact that an increase in the particle voidage will result in an increased effective diffusivity of the fluid reagent into the waste particles. Only the case of constant particle voidage has been considered in this work. The extension of the model to include a dynamic particle voidage is discussed in the next section.
It is important to note that the particle scale chemical reaction model will reflect diffusion control, chemical kinetic control or an intermediate condition depending on the Kpi parameter specification. The Kappa parameter, Kpi• defined in $ection 4.2,represents the ratio of the chemical reaction rate within a particle to the rate of diffusion of fluid reagent into the particle. As such, the Kpi parameter corresponds to a Damkohler number of the second type [Aris 1975]. A high value for Kpi implies that the reaction is controlled by diffusion of the fluid reagent into the particle. Similarly, a very low value for Kpi implies that the reaction is controlled by the intrinsic chemical kinetics of the reactions. Figure 4-2 showed reagent profiles for several parameter combinations. Note that diffusion controlled reactions result in steep fluid reagent and solid reactant gradients within the particles. In contrast, reactions which are controlled by the intrinsic chemical kinetics exhibit very flat fluid reagent and solid reactant gradients.
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7.1.2 Comparison to the Columnar Model of Dixon [1992, 19931.
Dixon [1992, 1993] has also incorporated the particle scale, chemical reaction model into a column type model to predict the performance of heap leaching operations. Probably the most important difference between his model and the heterogenous columnar model developed here is the manner in which the particle scale model parameters have been determined as a function of particle size. In summary, it is desirable to determine or specify parameters for a single size class of particles, termed a reference size class, within the column. The parameters of all the other size classes of particles are then determined from the reference size class parameters using suitable mathematical relationships. The reason why this approach is desirable is that it eliminates the need to define suitable particle scale model parameters for each size class of particles. This greatly reduces the number of parameters required.
Dixon [1992] presented mathematical relationships for the particle scale model parameters as a function of particle size for ore particles. These relationships, discussed in section 5.2.1,were found not to be applicable to waste particles because they required information about precious metal location in the ore particles as a function of particle size. This information was determined from a knowledge of ore preparation. Hazardous constituent location in waste particles is usually highly size specific and a result of the processing operations which generate the waste. For this reason, additional mathematical relationships, which are specifically applicable to the determination of the model parameters for waste particles as a function of particle size, have been proposed in section 5.2.2. It is worth noting that these relationships are more complex than those defined by Dixon, but, under appropriate conditions, will simply to those of Dixon. This implies that the heterogenous columnar model developed in this work can be applied to both the leaching of hazardous constituents from waste deposits as well as to precious metal leaching from ore heaps. These equations have been included in a bulk fluid reagent mass balance. These aspects and an appropriate solution strategy have been discussed in section 5.3 and 5.4 respectively.
The next main difference between the columnar model of Dixon and the heterogenous columnar mod_el is that the new model includes the possibility for a variable fluid reagent flow into the deposit. The inclusion of a variable fluid velocity into the model is important in order to model the effects of periodic rainfall on the release of hazardous constituents from waste deposits. Further, work at the University of Cape Town [Petersen 1994, 1995] has revealed that fluid flow through lysimeter columns can vary significantly during the experiment. Thus, in order to fit the heterogenous columnar model to lysimeter data, it must be capable of including variable fluid reagent flow into the deposit. The strategy adopted to include a variable fluid reagent flow into the heterogenous columnar model is identical to the strategy adopted to include this effect into the macroscopic, lumped parameter model which was discussed in section 3.4. In summary equation (6-9) is redefined in a manner similar to equation (3-29). This equation is then used to determine the appropriate time interval for each successive iteration in time.
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Another difference between the models is that Dixon [1993] has included a mass balance equation to include the effects of dissolved species transport within the particles. Due to time limitations, this was not included in the heterogenous columnar model. The inclusion of this mass balance equation into the heterogenous columnar model is discussed in a later section in this chapter. The importance of including this aspect into the model is that it will extend the applicability of the model to cases where dissolved species transport within the waste particles is significant.
7 .2 A Summary of the Potential Engineering Applications of the Heterogenous Columnar Model.
The heterogenous, columnar model, described in the preceding sections, has two particular attributes which make it attractive in the engineering design of better waste deposits. The first of these attributes is that it is capable of determining the relative contributions of the intrinsic chemical kinetics and the diffusion of fluid reagent into the particles to the release of hazardous constituents. These rate terms are in tum a function of the hazardous constituent location within the particles, the size distribution of the particles, competing chemical reactions and the bulk fluid reagent flow. The second attribute is that the parameters required for this model can be determined from a simple CSTR type experiment. One particular advantage of being able to determine the model parameters from CSTR type experiments is the short duration of these experiments. This implies that the potential release of hazardous constituents can be determined much more quickly than existing methods which are fitted to lysimeter data. This ability will enable the minerals processing industry to determine a priori the suitability of various wastes for disposal in waste deposits. Another distinct advantage of using CSTR tests to determine the model parameters is that CSTR tests are much cheaper to conduct than lysimeter experiments. This implies that more wastes can be tested prior to disposal and that samples from existing waste deposits, which could not be tested previously due to the expense of lysimeter experiments, can now be tested.
7.2.1 Improved Deposit Desi2n based on Results from the Hetero2enous Columnar Model.
Once the particle scale model parameters for a given waste stream have been determined, the heterogenous columnar model can be used to investigate the behaviour of a typical deposit which would contain this material. The results of the heterogenous columnar model will not only give an indication of the release of hazardous constituents from the waste deposit but can also be used to improve the disposal strategy to minimise the release of hazardous constituents.
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Deposits Which React in a Zone-Wise. Fluid Reagent Limited Manner.
As an example, consider the case where the results of the heterogenous columnar model indicate that the deposit is reacting in a zone-wise,fluid reagent limited manner. The rate of the chemical reactions occurring in the particles of such deposits is relatively fast compared to the rate of fluid flow through the deposit. Even if the breakthrough curves determined by the heterogenous columnar model are acceptable in terms of the concentrations of the hazardous constituents, deposits of this nature present an environmental risk. The reason for this is that the release of hazardous constituents is only limited by the amount of fluid flowing through the deposit. Should this increase, so too would the rate of release of hazardous constituents. It is a high priority therefore to keep deposits which react in a fluid reagent limited manner dry. This could be achieved by using suitable liners to prevent rain or ground water from percolating through the deposit.
As an alternative to keeping the deposit dry to limit the release of hazardous constituents, pretreatment options for the waste could be investigated which would slow down the rate of release of hazardous constituents at the particle level. One likely factor which would result in relatively fast chemical reactions would be if a significant fraction of the hazardous constituents are concentrated onto the surface of the particles. (These reactions would be relatively fast because they do not experience any fluid reagent diffusional resistances.) If significant concentrations of hazardous constituents are known to exist on the surface of the particles, their release could be reduced by removing these components in a pretreatment process. Surface hazardous components could be removed from the particles by making use of an active leach procedure. It may be possible to purify the leachate generated by such a process and return the 'contaminants', which are often heavy metals, to the process which generated the waste. If it is not feasible or economical to pretreat the particles using a leach procedure then alternative methods to stabilise the waste need to be investigated. One alternative method would be to agglomerate the particles into larger particles using suitable binding agents. In this process, much of the external surface area of the original particles would be incorporated into the bulk of the larger agglomerated particle. Before the fluid reagent could react with the hazardous constituents it would need to diffuse through the agglomerated particle to the location of the hazardous constituents. Binding agents usually exhibit a buffering capacity with respect to the fluid reagent. This too, would slow down the release of the hazardous constituents from the particles due to the competition between the binding agent and the hazardous constituents for the fluid reagent. Suitable particle scale model parameters can be determined for such agglomerated particles using a CSTR experiment. These parameters could then be used to determine whether the agglomeration would improve the overall deposit performance.
In the case where the release of hazardous constituents from particles is fast and where there is not a significant concentration of hazardous components on the surfaces of the particles, methods to retard the effective chemical reaction rates need to be investigated. The first way in which this can be achieved would be to enhance the diffusional resistances which limit the fluid transport into the particle. As before, this can be achieved by agglomerating the particles using suitable binding agents. Alternatively, attempts can be made to slow down the intrinsic kinetics within the particle by
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chemically altering the speciation of the hazardous constituents. Usually little is known about the complex chemistry of waste particles and any attempts to alter the intrinsic chemical kinetics will most probably require extensive experimentation. In contrast, note that the only instance in which agglomeration will not result in reduced effective chemical reaction rates will be when the intrinsic chemical kinetics are many orders of magnitude slower than the rate of fluid reagent diffusion into the particle. Such a situation will be easy to identify because the particles will react in a totally homogenous manner. (This situation corresponds to a very low Kappa, Kpi• parameter.)
Deposits Which React in a Homogenous Manner.
In contrast to deposits which react in a zone-wise, fluid reagent limited manner, some deposits react in a homogenous manner. The release of hazardous constituents from these deposits is not limited by the flow of fluid through the deposit but rather by the release of hazardous constituents from the individual waste particles. These deposits present little or no environmental risk if the results of the heterogenous, columnar model indicate that the release of hazardous constituents is acceptable. The reason for this is that the . individual particles within the deposit, release hazardous constituents at a sufficiently slow rate which the natural environment can assimilate. An increase in the fluid reagent flow through the deposit will not increase the rate of hazardous constituent release.
Should the results of the heterogenous columnar model indicate that the release of hazardous constituents from the deposit is unacceptable, then methods to slow down the effective chemical reaction rates similar to those discussed in the previous section need to be investigated.
7.2.2 Usine the Heteroeenous Columnar Model to Choose Upstream Processes which would Result in More Stable Wastes.
The properties of a waste stream are dictated by the upstream processes from which it arose. These properties include the chemical composition of the waste stream, the location of the hazardous constituents within individual particles and the size distribution of the particles within the waste stream.
Using the heterogenous columnar model, the particle characteristics which contribute most to the release of hazardous constituents from waste particles can be identified. This information can then be used to identify the upstream processes which result in the waste particles exhibiting the particular characteristic. Alternative processing options can then be investigated which would result in more stable wastes.
As an example, should the results of the heterogenous columnar model indicate that it is mainly the smaller particles within the size distribution of waste particles which are responsible for the release of hazardous constituents (due to smaller diffusional resistances in small particles), then the upstream processes which produce these particles should be identified and investigated. In this case, alternative processes which still
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achieve the desired process goal but which do not produce the small particles would be preferred.
Equally, the heterogenous columnar model could indicate that significant concentrations of hazardous constituents on the surfaces of particles are responsible for the release of contaminants. In this case, process options which aim at recovering the 'contaminants' from the wastes should be investigated.
7 .2.3 Using the Heterogenous Columnar Model to Asses the Risks and Liabilities Associated with Existing Waste Deposits.
South Africa is not unique in the fact that it has many waste deposits which could pose a significant environmental hazard due to the generation and release of leachate. For various reasons, which include financial constraints of conducting lysimeter experiments and ineffective legislation with respect to the disposal of industrial wastes, the risks associated with many of these deposits is unknown. The heterogenous columnar model can be used to estimate the potential leachate generation of these deposits and thus determine the risks associated with them. This would be achieved by conducting a CSTR test on a single size class of the material in the deposit being examined. As before, this would be used to determine appropriate model parameters for the heterogenous columnar model which would then be used to predict the deposit performance. This information could then be used to estimate the risks associated with these deposits and to identify which of the existing deposits require remediation.
7 .3 Limitations and Possible Extensions of the Hetero2enous Columnar Model.
The heterogenous columnar model has been presented as. a suitable model to describe the release of hazardous constituents from waste deposits. The applicability of this model can be .extended by beginning to remove the simplifying assumptions which were made during the model development. This section summarises the limitations of the model and suggests methods to address these limitations.
7.3.1 Incorporation of External Mass Transfer Resistances into the Model.
The heterogenous columnar model assumes that the mass transfer of the fluid reagent between the bulk fluid, which is the fluid between the particles, and the surfaces of the particles is negligible. Such an assumption is only valid when the effective reaction rate of hazardous constituent release is slow compared to the supply of fluid reagent to the particle surface. In cases where this assumption does not hold, external mass transfer resistances need to be accounted for explicitly.
Equation (5-1) is a mass balance equation which relates the consumption of the fluid
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reagent within the particles to the overall bulk fluid reagent concentration between the particles. This equation is valid only when external mass transfer resistances are negligible and needs to be modified when external mass transfer resistances are important. The modification involves relating the consumption of the fluid reagent within the particles to the rate of fluid reagent trapsfer to the particles. Equation (5-1) as well as the new equation are summarised below:
where V Part.
VLiq.
dCA BUllc
dt
is the total volume of the particles; and; is the total volume of fluid reactant.
(5-1 or 7-1)
This equation is equivalent to:
de A sulk
dt (7 -2)
The new equation is:
where~
~
CA.Bulk
CA,Surface
(7 -3)
is the mass transfer coefficient, determined using correlations for Sherwood numbers or j 0 factors; is the total extemal area of the particles; is the bulk fluid reagent concentration; and; is the concentration of the fluid reagent on the particle surface.
Suitable correlations are required to determine the external mass transfer rate, ~. before the heterogenous columnar model can be applied to cases where external mass transfer resistances are important. Several correlations are presented in the literature [Agarwal 1988; Kawase and Ulbrecht 1985; Wakoa and Funazkri 1978; Nelson and Galloway 1975; Mochizuki and Matsui 1973; Calderbank 1967; Wilson and Geankoplis 1966; Rowe and Claxton 1965; Pfeffer 1964; Ranz 1952; Smith 1981] for Sherwood numbers and j 0 factors as a function of Reynolds and Schmidt numbers. The Sherwood number is a ratio of the rate of convective mass transport to the rate of molecular mass transport at the particle surface and can be used to determine the external mass transfer
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coefficient. Similarly j 0 factors can be used to determine external mass transfer coefficients.
It is important to note that the external mass transfer rates are normally defined in terms of average mass transfer coefficients [Smith 1981]. Thus a single value of the mass transfer coefficient can be -used to describe the rates of transfer between the bulk fluid phase and the particle surface. The error introduced in using an average coefficient is not as serious as might be expected, since the correlations for the mass transfer coefficient, ~'are based on experimental data for packed beds of particles [Smith 1981].
7.3.2 Inclusion of Intra-Particle Dissolved Species Transport Resistances into the Model.
The intra-particle dissolved species transport resistances have been assumed to be negligible compared to the diffusion resistances of the fluid reactant into the particles and the chemical reaction rates within the particles. As discussed in section 7 .1.2,Dixon (1993] has included these effects into his columnar model. The strategy adopted was to apply a mass balance equation for each dissolved species at the particle level. The equations derived are very similar to equation ( 4-6) which represented a mass balance for the fluid reagent at the particle level. Equation (4-6), and the mass balances for the dissolved species are summarised as:
(4-6 or 7 -4)
[ a2ci 2 aci J ( il>pi aci n. --+--- +po 1-eo)k .cpi CA=Eo at 1 ar 2 I ar pi (7 -5)
where Ci dissolved hazardous constituent concentration of species i.
Equation (7-5) can be used to determine the dissolved concentration of hazardous constituents within the particle and at the particle surface. Should external mass transport of the dissolved hazardous constituents from the particle to the bulk fluid be significant, a suitable mass transfer correlation, similar to the correlations described in section 7. 3 .1, would be required to determine the bulk concentration of dissolved hazardous constituents. If external mass transport resistances are negligible, then a simple mass balance can be used to determine the bulk concentration of dissolved hazardous constituents.
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7.3.3 Inclusion of Matrix Dissolution and Hazardous Constituent Re-Precipitation in the Hetero2enous Columnar Model.
Neither matrix dissolution or the re-precipitation of dissolved hazardous constituents have been incorporated into the heterogenous columnar model. Matrix dissolution refers to the process of the waste particles becoming more porous due to significant quantities of the particle being dissolved during the life span of a deposit. It is unlikely that the effects of matrix dissolution will, be easily incorporated into the particle scale chemical reaction model. The reason for this is that the particle scale model makes use of an effective diffusivity De to determine the diffusion of fluid reagent into the particle. As matrix dissolution takes place, the voidage of the particles will increase. A larger voidage within the particles will substantially effect the rate of diffusion of fluid reagent into the particles. Thus before the effects of matrix dissolution can be incorporated in the particle scale model, a suitable model for the effective diffusivity, De, would need to be determined. It is worth noting that simple models for the effective diffusivity such as the parallel pore more model, described by Smith [1981] and summarised in equation (7-6), are inappropriate. The reason for this is that the tortuosity factor used in the models is also a function of matrix dissolution and would be an unknown in the model.
where De D €
0
D =ED e 0 .
effective diffusivity; actual diffusivity; particle voidage; and; tortuosity factor.
(7 -6)
Once the hazardous constituents have been dissolved there is a good possibility for them to reprecipitate either within the particle or onto the surfaces of surrounding the particles. These aspects need to eventually be incorporated into the heterogenous columnar model.
7.3.4 Inclusion of More Realistic Hydrodynamic Flow Models into the Hetero2enous Columnar Model.
The only fluid flow pattern through the deposit which was considered in the heterogenous columnar model was perfect plug flow. As discussed in section 2 .1. 2, the hydrodynamic flow patterns within waste deposits are often far more complex than a simple plug flow pattern. These irregular flow patterns need to be incorporated into the heterogenous columnar model.
The simplest manner in which 'non-ideal' flow patterns can be inco.rpOrated into the heterogenous columnar model would be to make use of a one parameter fluid dispersion
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model. In this approach any back m1xrng and short circu1tmg of fluid reagent is considered as fluid dispersion. A suitable one parameter fluid dispersion model which is applicable to columnar reactors is the 'tanks-in-series' approach described by Levenspiel [1972]. In this approach, the reaction in the column is approximated by a number a discrete, equi-sized continuously stirred tank reactors in series. The single parameter in the approach is the number of tanks-in-series which are required to approximate the fluid flow behaviour within the columnar reactor. A tracer study which yields the residence time distribution of the column can be used to determine this parameter. This technique is described by Levenspiel [1972].
This approach would be easy to implement in the heterogenous columnar model. In section 6.1 a modelling strategy for the heterogenous columnar model which was based on heap leaching models was described. In summary the column was assumed to consist of a number of discrete disks. Within each disk the fluid reagent concentration was assumed to be uniform. It was noted in section 6.1 that the column would need to be divided into a sufficient number of disks in order· to ensure that this assumption would be valid. In effect, this corresponds to the case of 'N-oo' in the tanks-in-series approach where N represents the number of tanks in the chain. In reality, the number of 'tanks' or disks required to model perfect plug flow is finite and is determined as the number of disks beyond which model accuracy does not improve. This number of disks represents the minimum number of disks which are required within the column to adequately simulate perfect plug flow in the column. In effect, the 'inaccuracies' introduced when fewer disks are used can be interpreted as fluid dispersion and can be taken as an indication of the performance of columns which exhibit irregular flow patterns. The performance of a particular column which exhibits irregular flow patterns can be approximated by conducting a tracer experiment on the column and using the residence time distribution obtained to determine a suitable number of tanks, N, which will approximate the flow patterns within that column. By using the heterogenous columnar model with N disks, the performance of the column can then be estimated.
More complex multi-parameter dispersion models will be required as the flow patterns within a deposit become more irregular. These models consider the column to consist of several hydrodynamically distinct regions which can each be described by plug flow, dispersed plug flow and mixed flow models. These models were discussed briefly in section 2.5.
To improve the ability of the heterogenous columnar model to predict the release of hazardous constituents under truly unsaturated conditions, the ground water flow equations used by Demetracopoulos et al. [1986] can be incorporated into the model. These equations, presented in section 2.3.2,determine the saturation and fluid velocity as a function of position within the column and time. These factors are included in the heterogenous columnar model by updating equation (6-4). The updated equation can be summarised as:
ac ac a n E Col Acol =-u Acal -c __!::! + "C1 r
Col % Sat: A ~ at az Col az i=l ai (7 -7)
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where the Col% sat is now a function of position within the column and time. (Col% sat
corresponds to 01 Ec01 , where e is the moisture content used in equation (2-2) and (2-3).)
Note that the second term on the right hand side of the equation is a direct result of the divergence of the velocity field within a volume element in the column no longer being equal to zero. (Compare equation 3-1, 3-4 and 3-5.)
7.4 Statement of the Significance of the Work Presented in this Thesis.
Waste deposits which contain wastes from the minerals processing industry pose environmental hazards due to the possibility of leachate generation and the subsequent · release of this leachate into the environment. The impacts of waste deposits have been traditionally investigated using lysimeter experiments. These experiments are both very time consuming and costly. This and other factors, such as ineffective legislation with respect to waste disposal strategies, have resulted in a limited study of the environmental impacts of disposing of mineral wastes in waste deposits.
Due to increased environmental awareness, the minerals processing industry is being encouraged to investigate the environmental impacts of its disposal strategies. As such, suitable modelling strategies which could predict deposit performance would be most useful. The reason for this is that these models could be used to investigate the potential for waste deposits to pollute the environment. If sufficiently detailed, these models could also be used to identify the major contributing factors to the release of hazardous constituents from waste deposits. This information could then be used to engineer better waste deposits in future. Models could also be used to investigate modifications to the upstream processes with the aim of producing more stable wastes. Equally as important, the risks associated with the many existing waste deposits could be determined and remedial action taken before a catastrophic situation arises.
The work presented in this thesis is an investigation into identifying suitable modelling strategies to describe the release of hazardous constituents from waste deposits. The first model investigated was a macroscopic, lumped parameter model. This model was investigated due to its inherent simplicity and its ability to characterise a deposit as a reacting entity. The investigation into this model revealed that it had limited applicability to identify the main factors which result in the release of hazardous constituents from waste deposits. This implies that this model has limited applicability in the design of new deposits. Further, the parameters required for the macroscopic lumped parameter model are determined from fitting the model to lysimeter data. This too is a limitation due to the fact that lysimeter experiments take a minimum of a few months and can take up to three years to complete. For these reasons a second model, termed the heterogenous columnar model was investigated.
The heterogenous columnar model describes the release of hazardous constituents from the individual particles within a waste deposit and relates this release to the overall
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deposit performance. This model has been developed to apply specifically to the release of hazardous constituents from waste deposits. The heterogenous columnar model has the capability of identifying the main factors which can be associated with the release of hazardous constituents. As an example, the heterogenous columnar model can be used to determine the relative contributions of the intrinsic chemical kinetics of the waste particle and the diffusion of fluid reagent into the particle to the release of hazardous constituents. Equally, the significance of the bulk fluid flow through the deposit compared to the effective rate of hazardous constituent release can be determined. This information could be used in the engineering design of waste deposits and was discussed in section 7.2.1. Further, this information could be used to investigate upstream processing options which would result in more stable wastes. This was discussed in section 7.2.2.
The heterogenous columnar model has the distinct advantage that the parameters which it requires can be determined from CSTR type experiments. Probably the most important advantage of being able to determine the heterogenous columnar model parameters from CSTR experiments is that CSTR type experiments are considerably quicker than lysimeter experiments. This means that the large delay times associated with the traditional lysimeter experiments can be eliminated. Using the heterogenous columnar model, the minerals processing industry_ would be able tp determine the suitability of a particular waste for disposal in a waste deposit. The second significant advantage of being able to determine the model parameters from a CSTR type experiment is the relatively low costs associated with these experiments when compared to lysimeter experiments.
The heterogenous columnar model can also be used to investigate the risks associated with existing waste deposits. This would be achieved by collecting samples from these deposits and conducting a CSTR type experiment on a single size class of the particles. The data from this experiment could be used to determine the heterogenous columnar parameters as discussed in section 5.8 and 6.6. The heterogenous columnar model could then be used to estimate the release of hazardous constituents from these deposits. This information could be used to asses the risks with these deposits and to identify which deposits require remedial attention in order to prevent pollution of the environment.
This work has been a first attempt to identify or develop suitable modelling strategies to determine the release of hazardous constituents from waste deposits. The heterogenous columnar model has been identified as a possible candidate for this purpose. This model has been applied to the leaching of precious metals from ore particles and the capability of the model to describe and predict the leaching behaviour is encouraging. The verification of the model against data from waste deposits or lysimeter experiments still needs to be completed. The type of data required for this purpose has been summarised in section 6. 7. Once the heterogenous columnar model has been verified against waste leaching data it may start to provide the minerals processing industry with the information which it so desperately requires in order to dispose of their wastes in a responsible manner which will not pose a threat to the environment.
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References.
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Bishop, P.L. "Prediction of Heavy Metal Leaching Rates from Stabilised/Solidified Hazardous Wastes." In: Proceedings of 18th Mid-Atlantic Industrial Waste Conference. edited by Boardman, G.D.Lancaster:Technomic,p.236-252, 1986.
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Dixon, D.G. Predicting the Kinetics of Heap Leaching with Unsteady-State Models, PhD Thesis, University of Nevada - Reno, 1992
Dixon, D.G. and Hendrix, J.L. "Theoretical Basis for Variable Order Assumption in the Kinetics of Leaching of Discrete Grains." AI CHE 39(5), 1993.
Dixon, D. G. and Hendrix, J .L. "A General Model for Leaching of One or More Solid Reactants from Porous Ore Particles", Metallurgical Transactions B, 24B: 157-169, 1993.
Dixon, D. G and Hendrix, J .L. "A Mathematical Model for Heap Leaching of One or More Solid Reactants from Porous Ore Pellets.", Metallurgical Transactions B, 24B: 1087-1102, 1987.
Dreisinger, D .B., Peters, E., and Morgan, G. "The Hydrometallurgical treatment of carbon steel electric arc.furnace dusts by the UBC-Chapa"al process." Hydrometallurgy 25:137-152, 1990.
Fourie, A. "Predicting Contaminant Migration in Unsaturated Soil: Some recent Advances.", 1995. (University of the Witwatersrand.)
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Godbee, H.W., Compere, E.L., Joy, D.S., Kibbey, A.H., Moore, J.G., and Nestor, C.W. "Application of Mass Transport Theory to the Leaching of Radionuclides from Waste Solids. ", Nuclear and Chemical Waste Management 1:29-35, 1980.
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Appendix I. Summary of the Method of Characteristics.
This appendix presents a summary of the method of characteristics which is used to solve first and second order hyperbolic partial differential equations. Much of the matter presented in this section has been summarised from Chapter 9 of the book Analysis and Solution of Panial Differential Equations by Robert L. Street [1978].
The solution of first-order or second-order partial differential equations with a single dependent variable and two independent variables can be visualised as a surface in (x,y,z) space. Analytical geometry can be used to enhance an understanding of the solution technique for such equations. This is particularly valid for initial-value problems in which we expect the solution to propagate from the region on which the initial data was specified.
In the case of first-order and hyperbolic second-order equations, information from the initial data propagates over well-defined paths in the surface representing the solution. These propagation paths are called characteristics. A knowledge of the existence of characteristics gives considerable insight into the expected behaviour of a problem's solution, even before the solution is known.
This summary is restricted to first-order equations which involve a single dependant variable and two independent variables.
The general first-order partial differential equation in two independent variables (x,y) is:
where
F(x,y, z,p, q) =O
az p=ax (Al -2)
(Al-l)
(Al-3)
If equation (Al-1) is a quasi-linear equation, it can be written as:
Pp+Qq=R (Al-4)
where P,Q and R are functions of x,y,and z but not of p or q. It is assumed that P,Q and R, together with their first derivatives, are continuous in the region of the problem under consideration. Further, P and Q are assumed not to vanish simultaneously. This implies:
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(Al-5)
First-order equations arise in many physical systems. Equations (3-5) and (3-23) were both first-order hyperbolic partial differential equations. These equations model the bulk flow and chemical reaction of a fluid reagent through a column. These equations, which are so-called Convection Equations, are identical to the quasi-linear equation (Al-4). To demonstrate this, equation (3-23) has been written below in exactly the same form as equation (Al-4):
where
1 =P (Al -7)
aa. +DGl aa. =-DG2a.a418.1
ai' ae 81
DG2=Q (Al-8)
(Al-6)
G 4's1 -D 2a.aBi =R (Al-9)
The solution z(x,y) to the quasi-linear equation (Al-4) can be represented as a surface in (x, y ,z) space shown in Figure A 1.1. This surface is called an integral surface because it represents the solution or 'integral' of the partial differential equation. The integral surface for (Al-4) is represented by the formal identity:
I(x,y,z)=z(x,y)-z (Al-10)
Figure Al.1 An integral surface representing the solution to equation (Al-4).
l(x,y, z) = z(x, y) - z = 0
z
Taking the differential of equation (Al-10) will, from analytical geometry considerations,
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result in the. equation for the tangent plane to the integral surface at any point. The equation for the equation plane is thus:
or
ar dx+ ar d + ar =o ax ay y az
az az -dx+-dy-ldz=O ax ay
(Al-11)
(Al-12)
Equation (Al-12) is nothing more than the total differential and can be written as:
dz=p dx+q dy (Al-13)
Equation (Al-13) is a tangent plane to the solution plane at any point. Accordingly, the vector N=pi+qj~lk; that is, the vector (p,q,-l);is normal to the integral surface, depicted in Figure Al .1, at any point.
The scalar product of N and the vector (P, Q,R) is zero because:
(P, Q, R) · (p, q, -1) = (Pi+Qj+Rk) · (pi+qj-lk)
= Pp+Qq-R = 0
(Al-14)
from equation (Al-4). Thus (P,Q,R) is perpendicular to N. Consequently, (P,Q,R) is tangent to the integral solution surface at every point and lies in the plane of the tangent equation (Al-13). This means that the first-order equation (Al-4) can be regarded as a geometric requirement that any integral or solution z(x,y)through the point (x,y ,z)must be tangent to the vector (P, Q,R). In fact, by beginning at some point on the integral surface, one line which lies in the integral surface can be determined from the known tangent vector (P,Q,R). This line is termed a characteristic. By determining sets of characteristics, the solution surface can be generated.·
Stated in another way, the vector (P,Q,R) has been determined to be tangent to the solution surface at all points. A single line in the integral surface can be determined by starting on a point on the integral surface, moving in the direction of (P, Q,R) and determining the curve which is tangent to (P,Q,R).
Suppose tha~ the position vector r of a point on a characteristic curve 1s:
r=xi+yj+zk (Al-15)
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This vector can be expressed parametrically in terms of distance s along a particular curve:
r(s)=x(s)i+y(s)j+z(s)k (Al-16)
and the tangent vector to the curve at a particular point is:
ar = ax i + ay . + az k as as as 1 as
(Al-17)
The vector (dxlds;dy/ds;dzlds) is tangent to the characteristic curve and the solution or integral surface. (P, Q,R) is also tangent to the solution or integral surface thus the two vectors are co-incident and their components must thus be proportional. This implies:
dx/ ds = dy/ ds =dz/ ds (Al-18)
or
p Q R
dx= dy =dz p Q R
(Al-19)
The system of equations represented in (Al-19) must be satisfied by any characteristic in the solution surface. Thus the partial differential equation (Al-4) can be converted into a system of ordinary differential equations represented by (Al-19). Equivalent systems of equations (Al-19) are:
dy= Q dz R (Al-20) ---dx p dx p
or
dx p dz R (Al-21) --- ---dy Q dy Q
Equations (Al-20) or (Al-21) are ordinary differential equations which can be solved by conventional means.
F: I WP51 \THESIS\GDTH2AP1 .1
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Appendix II. Solution Algorithm and Code for Model4Dl.PAS and Model4D2.PAS.
Model4 represents the computer routines for the Macroscopic, Lumped Parameter Model Presented in Chapter 3.
Solution Algorithm for Model4D.
Start. I Set/Initialise Variables.
Iterate in Time.
Determine Corresponding Time Increment.
Calculate the appropriate Alpha and sigma vector values within the column.
Increment Fluid Vectors.
I
Graph Breakthrough Curves
I End.
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seg
rap
h,i
nte
gra
t;
var Su
pVel
File
,Exp
Dat
a
SupV
elD
ata
Dat
a
:tex
t;
:arr
ay[1
.• 2
,1 .
• 2]
of d
oubl
e;
:arr
ay[0
.. 2
56,1
.• 3
] of
dou
ble;
ART
,ART
P1,S
RT
1,SR
T_2
,SR
T_3
,Sol
SRT
_1,S
olSR
T_2
,Sol
SRT
_3
Hol
dSRT
1,
Hol
aSR
T 2,
Hol
dSR
T 3
Dat
aSet
X,D
ataS
etY
;lnt
erG
_1,T
nter
G_2
,lnte
rG_3
C
onv
1,C
onv
2,C
onv
3 BT
CA~T
,BTC
S~T_
1,BT
CSRT
_2,B
TCSR
T_3,
Time
Vect
or
lntl
eng
th,l
ntl
eng
thC
rit,
Del
ta t
,Del
taT
,Arb
Val
ue
Cum
Tim
e,E
rror
Sllll
-
-DG
2 1,
DG
2 2,
DG
2 3,
DG
3 1,
DG
3 2,
DG
3 3
,0rd
1,0
rd 2
,0rd
3
Voi
aage
,Len
gth,
usta
r,D
elta
E-
--
--
GfM
ax_X
,GfM
ax_Y
Arb
lnde
x,S
ubln
terv
alln
dex,
Con
tinu
e,P
rint
Fil
terl
ndex
V
ecto
rlnd
ex,H
ardC
opy,
erro
r,N
o of
Dat
aPoi
nts,
Pos
itio
n N
,Nll
llS
ubln
t,lt
erat
ions
,Pri
ntF
ilte
r N
o_of
_P
oint
s
Tim
eCou
nter
NS,
Len
gthS
DG
2 1S
,DG
2 2S
,DG
2 3S
,DG
3 1S
,DG
3 2S
,DG
3 3S
G
Lao
el2,
GL
abel
4,G
[abe
l5 -
--
Sho
rtV
ecto
r;
Sho
rtV
ecto
r;
Ver
ylon
gVec
tor
Ver
yLon
gVec
tor
Ver
yLon
gVec
tor
:Dou
ble;
:D
oubl
e;
:Dou
ble;
:D
oubl
e;
:Rea
l;
: In
teg
er;
: In
teg
er;
: In
teg
er;
: In
teg
er;
:Lon
glnt
;
Str
ng
Str
ng
Str
ng
(---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re D
efin
e_V
aria
bles
;
begi
n Del
taE
:=
0.0
2·
N
:= S
O;
'
uSta
r :=
1/8
6400
;
(Dim
ensi
onle
ss s
pace
in
crem
ent.
C
Nlll
i>er
of
no
des.
(1
/Del
taE
) } }
(Ref
eren
ce f
luid
vel
oci
ty.
Equ
ival
ent
to
}
07[2
8[19
95
13:5
2 F
ilen
ame:
M
ODEL
4D1.P
AS
Page
C1m
in 2
4 ho
urs.
}
Len
gth
:= 0
.5·
(Len
gth
of
the
coll.
llV'I.
(m
) }
Voi
dage
:=
0.4
2;
CV
oida
ge o
f th
e co
ll.llV
'I.
}
Ord
1
:= 1
.0;
(Rea
ctio
n or
der
of
firs
t re
acti
on
. }
Orc
l2
:= 1
.0;
(Rea
ctio
n or
der
of s
econ
d re
acti
on
. }
Orc
0
:=
1.0
; (R
eact
ion
orde
r of
th
ird
rea
ctio
n.
}
DG2
1 :=
10
.0;
(Def
ined
dim
ensi
onle
ss g
roup
. }
DG
2-2
:= 0
.0;
(Def
ined
dim
ensi
onle
ss g
roup
. }
DG
(:3
:=
0.0
; (D
efin
ed d
imen
sion
less
gro
up.
}
DG3
1 :=
1
.0;
(Def
ined
dim
ensi
onle
ss g
roup
. }
DG
r2
:= 0
.0;
(Def
ined
dim
ensi
onle
ss g
roup
. }
DG
3:J
:= 0
.0;
(Def
ined
dim
ensi
onle
ss g
roup
. }
GfM
ax_x
:=
2
.0;
CMax
iffill
ll x·
valu
e fo
r co
nver
sion
gra
ph.
} ((
Dim
ensi
onle
ss T
ime.
) }
GfM
ax_Y
:=
1
.0;
CMax
iffill
Tl y
-val
ue f
or
brea
kthr
ough
gra
ph.
} ((
Con
cent
rati
on -
kg/m
3 fl
uid
.)
}
Iter
atio
ns
= 1
00;
(Ite
rati
on
s in
tim
e.
} P
rin
tFil
ter
= 1
; CG~ p
rin
tin
g f
ilte
r.
} N
umSu
blnt
=
10;
- (
N
r of
su
bin
terv
als
wit
hin
each
sp
atia
l }
(inc
rem
ent.
}
end;
(---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re
lnit
iali
se_
Su
per
fici
al_
Vel
oci
ty_
Fil
e;
begi
n assi
gn(S
upV
elF
ile,
'Sup
Vel
Fi.
Dat
');
rese
t(S
upV
elF
ile)
; su
pVel
Dat
a[1,
1l:=
O;
SupV
elD
ata[
112J
:=O
; re
adC
Sup
Vel
F1l
e,S
upV
elD
ata[
2,1l
);
read
(Sup
Vel
Fil
e,S
upV
elQ
ata[
2,2l
);
lntl
engt
hCri
t:=
Voi
dage
*Del
taE
*Len
gth;
en
d;
(---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re T
imeO
ART
an
d SR
Ts;
(Thi
s p
roce
du
re-i
nit
iali
ses
the
alph
a an
d si
gma
vec
tors
. }
begi
n ART
[Ol
:=1;
SR
T_1
[0]
:=1;
SR
T_2[
0]
:=1;
SR
T 3[
0]
:=1;
So
lSR
T_1
[0J:
=O;
SolS
RT
2[0]
:=0;
SolSRT~ [0
] :=
O;
Hol
dSR
'r 1[
0]
=1
Hol
dSR
T-2
[OJ
=1
HoldSRT~[O]
=1
for
Arb
Tnd
ex
=1-
to N
do
begi
n
2
(j
0 Q.
-~ t-C -· Cll -· = ~
"'1 ~
0 Q.
~
~
t:; """ . ~ r:r
.i .
Univers
ity of
Cap
e Tow
n
07/2
8/19
95
13:5
2 Fi
lena
me:
M
ODEL
4D1.P
AS
Page
3
AR
TCA
rbln
dexJ
:=O
; SR
T_1
CA
rbln
dexJ
:=1;
SR
T_2
[Arb
lnde
xJ:=
1;
SRT
_3[A
rbln
dexJ
:=1;
So
lSR
T_1C
Arb
lnde
xJ :=
O;
SolS
RT_
2CA
rbln
dexJ
:=O
; So
lSR
T 3C
Arb
lnde
xJ :=
O;
Hol
dSRT
1C
Arb
lnde
xJ
=1
Hol
dSR
T-2C
Arb
lnde
xJ
=1
Hol
dSR
T-3C
Arb
lnde
xJ
=1
end·
-
end;
'
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
-·}
Pro
cedu
re D
eter
min
e C
orre
spon
ding
Del
taT
; {T
his
proc
edur
e us
es t
he m
etho
d oT
ch
arac
teri
stic
s to
det
erm
ine
the
} {p
rogr
essi
on i
n tim
e fo
r a
give
n C
harD
elta
E.
It a
llow
s fo
r un
stea
dy
} {
stat
e fl
ow.
}
begi
n Intl
engt
h:=
O;
Del
ta t
:=O
; ·
whi
le-I
ntle
ngth
<>
lntL
engt
hCri
t do
beg
in
if E
OF(
SupV
elFi
le)
then
beg
in
clos
eseg
raph
ics;
w
rite
lnC
'ln
suff
icie
nt
Dat
a in
the
Sup
erfi
cial
V
eloc
ity
Dat
a F
ile
to')
; w
rite
ln(•
exec
ute
the
requ
ired
num
ber
of
itera
tio
ns.
');
wri
teln
('P
rese
nt
num
ber
of
iter
atio
ns:
=
1,T
imeC
ount
er);
C
ontin
ue:=
O;
Intl
eng
th:=
Intl
eng
thC
rit;
re
adln
; en
d;
Intl
engt
h:=
Intl
engt
h+C
Sup
Vel
Dat
aC2,
1J-
Sup
Vel
Dat
a[1,
1J)
*Sup
VeL
Dat
a[2,
2J;
Del
ta t
:=D
elta
t+
(Sup
Vel
Dat
a[2,
1J-
SupV
elO
ata[
1, 1
J );
If
In
tlen
gth>
Intl
engt
hCri
t th
en b
egin
A
rbV
alue
:=In
tlen
gth-
IntL
engt
hCri
t;
Del
ta t
:=D
elta
t-A
rbV
alue
/Sup
Vel
Dat
a[2,
2J;
IntL
engt
h:=
lnt[
engt
hCr1
t;
SupV
elD
ataC
1,1J
:=
Sup
Vel
Dat
a[2,
1J-A
rbV
alue
/Sup
Vel
Dat
a[2,
2l;
SupV
elD
ataC
1,2J
:=Su
pVel
Dat
aC2,
2J;
end
else
beg
in
SupV
elD
ataC
1,1J
:=Su
pVel
Dat
a[2,
1J;
SupV
elD
ataC
112J
:=S
upV
elD
ata[
2,2J
; re
ad(S
upV
elF
1le,
S
upV
elD
ata[
2,1J
);
read
CSu
pVel
File
, Su
pVel
Dat
a[2,
2J>
end;
en
d·
Del
taT
:=D
elta
t*u
Sta
r/L
engt
h;
end;
-
{------------------------------~------------------------------------------}
Pro
cedu
re
Iter
ate
wit
hin
Del
taE
s;
{Thi
s pr
oced
ure
uPcl
ates
th
e al
pha
and
sigm
a ve
ctor
s du
ring
th
e tim
e it
{t
akes
for
th
e fl
uid
to
mov
e th
roug
h a
spat
ial
incr
emen
t.
begi
n for
Arb
lnde
x:=O
to
N d
o be
gin
for
Sub
lnte
rval
lnde
x:=
1 to
Num
Subl
nt
do b
egin
AR
TP1
CA
rbln
dexl
:=
AR
TCA
rbln
dexJ
+ O
elta
T/N
umSu
blnt
*(-O
G2_
1*A
RT
[Arb
lnde
xJ
} }
07/2
8/19
95
13:5
2 Fi
lena
me:
M
ODEL
4D1.P
AS
Page
4
SRT _
1 [A
rb In
dex]
SRT _
2 [A
rb! n
dexl
SRT_
3 [A
rbln
dexJ
ART
[Arb
lnde
xJ
end·
en
d·
' en
d;
'
*exp
(Ord
i*l
nCSR
T 1
CA
rbln
dexJ
)))
+
Del
taT
/Num
Subl
nt*(
-DG
2 2*
AR
TCA
rbln
dexJ
*e
xp(O
rd 2
*ln(
SRT
2C
ArE
lnde
xJ))
)+
Del
tat/N
umSu
blnt
*(-D
G2
3*A
RTC
Arb
lnde
xJ
*exp
(Ord
3*l
n(SR
T_3
CA
r6In
dexJ
)));
:=
SRT
1CA
r6In
dexJ
+Oel
taT
/Num
Subl
nt*
(-DG
2 1*
DG
3 1*
AR
T[A
rbln
dexJ
*e
xpC
Ord
1*T
nCSR
T_1C
Arb
lnde
xJ>>
>;
:=SR
T 2C
ArE
inde
xJ+D
elta
T/N
umSu
blnt
* C-
DG2
2*D
G3
2*A
RTC
Arb
lnde
xJ
*exp
(Ord
2*T
nCSR
T_2
CA
rbln
dexJ
)));
:=
SRT
3CA
r6In
dexl
+Del
taT
/Num
Subl
nt*
C-DG
2 3*
DG
3 3*
AR
TCA
rbln
dexJ
*e
xp(O
rd 3
*TnC
SRT
_3C
Arb
lnde
xJ))
);
:=A
RTP1
CA
r6In
dexJ
;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re
Incr
emen
t F
luid
Vec
tors
; {T
his
proc
edur
e up
date
s tn
e al
pha
vect
or o
nce
the
flu
id h
as p
rogr
esse
d }
{thr
ough
a s
pat
ial
incr
emen
t.
}
begi
n for
Arb
lnde
x:=1
to
N d
o be
gin
SolS
RT
1CA
rbln
dexJ
:=
SolS
RT
1CA
rbln
dexJ
+(H
oldS
RT
1[
Arb
lnde
xJ-
-SR
T 1C
Arb
lnde
xJ)/
N;
-So
lSR
T 2C
Arb
lnde
xJ
:=So
lSR
T 2C
Arb
lnde
xJ+(
Hol
dSR
T
2CA
rbln
dexJ
--
SRT
2CA
rbln
dexJ
)/N
; -
SolS
RT
3CA
rbln
dexJ
:=
SolS
RT
3CA
rbln
dexJ
+(H
oldS
RT
3[A
rbln
dexJ
--
SRT
3CA
rbln
dexJ
)/N
;· -
Hol
dSRT
1[
Arb
lnde
x]
=SR
T-1
CA
rbln
dexJ
; H
oldS
RT=
2CA
rbln
dexJ
=S
RT
=2C
Arb
lnde
xJ;
1
Hol
dSRT
3C
Arb
lnde
xJ
=SRT
3C
Arb
lnde
xJ;
end;
-
-
for
Arb
lnde
x:=N
do
wnt
o 1
do b
egin
A
RTC
Arb
lnde
xJ
:=A
RT
CA
rbln
dex-
1J;
SolS
RT
1CA
rbln
dexJ
:=S
olSR
T 1C
Arb
lnde
x-1l
So
lSR
T-2
CA
rbln
dexJ
:=S
olSR
T-2
CA
rbln
dex-
1J
SolS
RT
-3C
Arb
lnde
xJ:=
SolS
RT
-3[A
rbln
dex-
1J
end;
-
-
ART
[OJ
=1
SolS
RT
1COJ
=O
So
lSR
T-2C
OJ
=O
SolS
RT-
3CO
J =O
en
d;
-
{***
**
{***
**
{***
**
{***
**
Impo
sed
boun
dary
con
d t
on.
Impo
sed
boun
dary
con
d t
on.
Impo
sed
boun
dary
con
d t
on.
Impo
sed
boun
dary
con
d t
on.
} } } }
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re U
pdat
e_B
reak
Thr
ough
Vec
tors
;
begi
n Tim
eVec
tor[
Vec
torl
ndex
J B
TCA
RTC
Vec
torln
dexJ
BT
CSRT
1C
Vec
torl
ndex
J B
TC
SRT
-2[V
ecto
rlnde
xJ
BTCSRT~CVectorlndexJ
end;
-
=Cum
Tim
e· =A
RT [N
J; I
=Sol
SRT
1
[NJ
=Sol
SR
T-2
[NJ
=Sol
SRT=
3 CN
l
Univers
ity of
Cap
e Tow
n
07/2
8/19
95
13:5
2 Fi
lena
me:
M
ODEL
4D1.P
AS
Page
5
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re L
abel
_Gra
ph;
begi
n Str
(N:3
,NS
);
Str
(Len
gth:
4:2,
Len
gthS
);
Str(
OG
2 1:
8:4,
0G2
1S);
St
r(O
G2
2:8:
4,D
G2
2S);
St
r(O
G2=
3:8:
4,0G
2=3S
>;
Str(
OG
3=1:
8:4,
0G3=
1S);
St
r(O
G3_
2:8:
4,0G
3_2S
);
Str
(OG
3_3:
8:4,
0G3_
3S);
G
Lab
el2:
=C
onca
t('
L '
,Len
gth
s,•
Nod
es
',NS
);
Gla
bel4
:=C
onca
t('
DG2_
1 ',O
G2_
1S,'
OG
2_2
',DG
2_2S
,' G
labe
l5:=
Con
cat(
' OG
3 1
',DG
3_1S
,' O
G3_
2 ',D
G3_
2S,'
Labe
lGra
phWi
ndow
(150
,~05
,GLa
bel2
,0,0
) L
abel
Gra
phW
indo
wC
150,
875,
GL
abel
4,0,
0)
Lab
elG
raph
Win
dow
(150
,845
,GL
abel
5,0,
0)
end;
OG
23
',DG
23S
);
OG3=
3 I ,O
G3=
3S);
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re G
raph
1_In
itia
lise
;
begi
n;
init
SE
Gra
phic
s(1f:
\tp
61);
SetC
urre
ntW
indo
w(2
);
Bor
derC
urre
ntW
indo
w(1
);
Set
Axe
sTyp
e(0,
0);
Sca
leP
lotA
rea(
0.0,
0.0,
1, 1
.2);
S
etX
Yin
terc
epts
(0.0
,0.0
);
Set
Col
or(2
);
Dra
wX
Axi
s(0.
2, 1
>;
Dra
wY
Axi
s(0.
2,1)
; L
abel
XA
xis(
1,0)
; L
abel
YA
xisC
1,0)
; T
itle
XA
xis(
'Oim
ensi
onle
ss L
engt
h');
T
itle
YA
xisC
'Dim
ensi
onle
ss C
on
cen
trat
ion
');
Titl
eWin
dow
(1M
odel
4D1'
)· L
abel
Gra
phW
indo
wC
375,
93S,
'C
once
ntra
tion
Pro
file
s',0
,0);
L
abel
_Gra
ph;
for
Arb
lnde
x:=O
to
N d
o O
atas
etX
[Arb
lnde
xl :
=A
rbln
dex*
Del
taE
; en
d;
{---
----
----
----
······
··---
----
----
----
----
······
··---
----
----
----
----
---·
} P
roce
dure
Gra
ph1_
Res
ults
;
begi
n for
Arb
lnde
x:=O
to
N d
o O
ataS
etY
[Arb
lnde
xl :=
AR
T[A
rbln
dexl
; L
ineP
lotD
ata(
Dat
aSet
X,D
ataS
etY
,N+
1,3,
0);
for
Arb
lnde
x:=O
to
N d
o D
ataS
etY
[Arb
lnde
xJ:=
SRT
1
[Arb
lnde
xl;
Lin
ePlo
tOat
aCD
ataS
etX
,Dat
aSet
Y,N
+1,
5,2)
; -
for
Arb
lnde
x:=O
to
N d
o D
ataS
etY
CA
rbln
dexl
:=SR
T 2[
Arb
inde
xl;
Lin
ePlo
tDat
a(D
ataS
etX
,Dat
aSet
Y,N
+1,
4,1)
; -
for
Arb
lnde
x:=O
to
N d
o D
ataS
etY
[Arb
inde
xl:=
SRT
3[A
rbln
dexl
; L
ineP
lotO
ata(
Dat
aSet
X,D
ataS
etY
,N+
1,7,
1);
-en
d;
{·-
-···
···-
····
····
····
-··-
····
···-
----
----
----
---·
··--
----
--·-
····
-·--
···}
P
roce
dure
Gra
ph_B
reak
Thr
ough
Cur
ves;
begi
n clos
eseg
raph
ics;
in
itse
gra
ph
ics(
'f:\
tp6
'>;
07/2
8/19
95
13:5
2 F
ilen
ame:
M
OD
EL40
1.PA
S Pa
ge
6
SetC
urre
ntW
indo
wC
2);
Bor
derC
urre
ntW
indo
w(1
);
SetA
xesT
ypeC
0,0)
; S
cale
Plo
tAre
a(0.
0,0.
0,G
fMax
_X,G
fMax
_Y);
S
etX
Yin
terc
epts
(0.0
,0.0
);
Set
Col
or(2
);
Ora
wX
Axi
s((G
fMax
X
/5),
1);
O
raw
YA
xisC
CG
fMax
-Y/5
),1);
Lab
elX
Axi
sC1,
0>;-
Lab
elY
Axi
sC1,
0);
Tit
leX
Axi
sC'D
imen
sion
less
Tim
e');
T
itle
YA
xisC
'Fra
ctio
n of
T
otal
O
rigi
nal
Con
tam
inan
t R
emov
ed');
T
itleW
indo
wC
'Mod
el40
1 '>
; L
abel
Gra
phW
indo
wC
380,
935,
'Bre
akth
roug
h C
urv
es',0
,0);
L
abel
G
raph
; fo
r A
rbln
dex:
=O
to V
ecto
rlnd
ex d
o D
ataS
etX
[Arb
lnde
xJ:=
Tim
eVec
tor[
Arb
inde
xl;
(*
for
Arb
lnde
x:=O
to
Vec
torl
ndex
do
Dat
aSet
Y[A
rbln
dexJ
:=B
TC
AR
TC
Arb
lnde
xl;
Lin
ePlo
tOat
a(O
ataS
etX
,Dat
aSet
Y,C
Vec
torl
ndex
+1)
,3,0
);
*)
for
Arb
lnde
x:=O
to
Vec
torl
ndex
do
Dat
aSet
Y[A
rbln
dexl
:=BT
CSRT
1C
Arb
lnde
x];
Lin
ePlo
tDat
a(D
ataS
etX
,Dat
aSet
Y,(
Vec
torl
ndex
+1)
,5,1
);
-fo
r A
rbin
dex:
=O.
to V
ecto
rlnd
ex d
o D
ataS
etY
CA
rbln
dexl
:=B
TC
SRT
2[
Arb
lnde
xl;
Lin
ePlo
tOat
a(D
ataS
etX
,Oat
aSet
Y,(
Vec
torl
ndex
+1)
,4,1
);
-fo
r A
rbln
dex:
=O
to V
ecto
rlnd
ex d
o D
ataS
etY
[Arb
lnde
xJ:=
BT
CSR
T
3[A
rbln
dexl
; L
ineP
lotD
ata(
Dat
aSet
X,O
ataS
etY
,(V
ecto
rlnd
ex+
1),7
,1);
-
end;
{---
----
----
----
----
----
---·
-··-
··---
--···
·---
----
----
----
--···
-····
·····-
} {M
AIN
PROG
RAM
}
begi
n
Con
tinu
e:=
1;
Cum
Tim
e :=
O;
Pri
ntF
ilte
rln
dex
:=O
; T
imec
ount
er:=
O;
Vec
torl
ndex
:=O
; T
imeV
ecto
r[V
ecto
rlnd
exJ:
=C
umT
ime;
BT
CA
RT
[Vec
torln
dexl
:=
O;
BTCS
RT_1
[V
ecto
rlnd
ex]
:=O
; BT
CSRT
2[
Vec
torl
ndex
l :=
O;
BTCS
RT=3
[V
ecto
r I n
dexl
: =
O;
Con
v 1[
Vec
torl
ndex
] :=
O;
Con
v=2
[Vec
tor I
ndex
] : =
O;
Con
v_3
[Vec
tor I
ndex
] : =
O;
Def
ine
Var
iabl
es;
Tim
eO A
RT
and
SRTs
; In
itia
lise
Su
per
fici
al V
eloc
ity
Fil
e;
Gra
ph
1_
Init
iali
se;
--
whi
le
(Con
tinu
e=1)
an
d C
Tim
eCou
nter
<It
erat
ions
) do
be
gin Tim
eCou
nter
:=T
imeC
ount
er+1
; P
rin
tFil
terl
nd
ex:=
Pri
ntF
ilte
rln
dex
+1
; O
eter
min
e_C
orre
spon
ding
_Del
taT
;
Univers
ity of
Cap
e Tow
n
07/2
8/19
95
13:5
2 Fi
lena
me:
M
ODEL
4D1
.PA
S Pa
ge
7
Cum
Tim
e:=C
umTi
me+
Del
taT;
It
erat
e w
ithi
n D
elta
Es;
if
Pri
ntF
ilte
rTn
dex
=P
rin
tFil
ter
then
be
gin Vec
torl
ndex
:=V
ecto
rlnd
ex+
1;
Pri
ntF
ilte
rlnd
ex:=
O;
Upd
ate_
Bre
akT
hrou
ghV
ecto
rs;
Gra
ph1
Res
ults
; en
d·
-Inc~
emen
t F
luid
Vec
tors
; en
d;
--
read
ln(H
ardC
opy)
;. if
har
dcop
y=1
then
Scr
eenD
umpC
3,0,
2,1.
0, 1
.0,0
, 1,0
,err
or)
;
Gra
ph_B
reak
Thr
ough
Cur
ves;
re
adln
(Har
dcop
y);
if h
ardc
opy=
1 th
en S
cree
nDum
pC3,
0,2,
1.0
;1.0
,0, 1
,0,e
rro
r);
clos
eseg
raph
ics;
end.
{======================================~==================================}
Univers
ity of
Cap
e Tow
n
07/2
8/19
95
13:5
5 Fi
lena
me:
M
ODEL
4D2.P
AS
Page
Prog
ram
Mod
el4D
1;
{Mod
el4D
2 {
Thi
s pr
ogra
m
is a
cod
e fo
r th
e co
nvec
tive
flu
x m
odel
}
whi
ch
inco
rpor
ates
uns
tead
y st
ate
flow
. }
{ { { { {
Dea
ls w
ith v
aria
ble
so
lid
rea
ctio
n o
rder
s.
} N
ote
that
th
is c
ode
uses
the
NEW
DIM
ENSI
ONLE
SS
GROU
PS
} de
fine
d on
18
Jan
uary
199
5.
} T
his
vers
ion
pres
ents
bre
akth
roug
h cu
rves
rel
ativ
e to
so
lid
}
reac
tatn
1.
}
{Cod
ed:
Gra
ham
Dav
ies.
}
{ D
epar
tmen
t of
Che
mic
al
Eng
inee
ring
. }
<
Uni
vers
ity
of C
ape
Tow
n.
} <
19
Jan
uary
199
5.
} {
10 M
ay
1995
-
Upd
ated
. }
<===
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
==}
{Dec
lara
tion
s:
}
uses
cr
t,st
dh
dr,
gra
ph
,wo
rld
dr,
seg
rap
h,i
nte
gra
t;
var Su
pVel
File
,Exp
Dat
a
SupV
elD
ata
Dat
a
:tex
t;
:arr
ay[1
•• 2
,1 •
• 2J
of d
oubl
e;
:arr
ay[0
.. 2
56,1
.• 3
) of
dou
ble;
. ART
,ART
P1,S
RT
1,SR
T 2,
SRT
3,So
lSR
T
1,So
lSR
T
2,So
lSR
T 3
H
oldS
RT
1,H
olaS
RT
2;H
oldS
RT
3 -
--
Dat
aSet
X,D
ataS
etY
;Int
erG
1,T
nter
G 2
,lnt
erG
3
Conv
1,C
onv
2,C
onv
3 ·
--
-B
TCA
RT,
BTC
SRT_
1,B
TfSR
T_2,
BTC
SRT_
3,Ti
meV
ecto
r
lntL
eng
th,l
ntl
eng
thC
rit,
Del
ta t
,Del
taT
,Arb
Val
ue
Cun
Tim
e,E
rror
Sun
-.
DG2
1,D
G2
2,D
G2
3,D
G3
1[D
G3
2,D
G3
3,0r
d 1,
0rd
2,0r
d 3
Voiaage,Length,uStar,~e
taE
--
--
-
GfM
ax_X
,GfM
ax_Y
Arb
lnde
x,S
ubln
terv
alln
dex,
Con
tinu
e,P
rint
Fil
terl
ndex
V
ecto
rlnd
ex,H
ardC
opy,
err.
or,N
o of
D
ataP
oint
s,P
osit
ion
N,N
umS
ubln
t,It
erat
ions
,Pri
ntF
Tlt
er
No_
of _
Poi
nts
Tim
eCou
nter
NS,
Len
gthS
DG
2 1S
,DG
2_2S
(DG
2_3S
,DG
3_1S
,DG
3_2S
,DG
3_3S
G
Lao
el2,
GL
abe
4,G
Lab
el5
Sho
rtV
ecto
r;
Sho
rtV
ecto
r;
Ver
ylon
gVec
tor
Ver
ylon
gVec
tor
Ver
ylon
gVec
tor
:Dou
ble;
:D
oubl
e;
:Dou
ble;
:D
oubl
e;
:Rea
l;
Inte
ger;
In
tege
r;
Inte
ger
; In
teg
er;
:Lon
glnt
;
Str
ing
S
trin
g
Str
ing
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re D
efin
e_V
aria
bles
;
begi
n Del
taE
N
:=
0.
02;
:= 5
0;
{Dim
ensi
onle
ss s
pace
inc
rem
ent.
<N
l.lllb
er
of
node
s.
(1/D
elta
E)
} }
07/2
8/19
95
13:5
5 Fi
lena
me:
M
OOEL
4D2.P
AS
Page
2
us ta
r :=
1/
8640
0;
{Ref
eren
ce f
luid
vel
oci
ty.
Equ
ival
ent
to
} {1
m
in 2
4 ho
urs.
}
Len
gth
:= 0
.5·
{Len
gth
of
the
colu
nn.
(m)
} V
oida
ge
:= 0
.4~;
{Voi
dage
of
the
colll
ll"I.
}
Ord
1
:= 1
.0;
{Rea
ctio
n or
der
of f
irst
re
acti
on
. }
ora
2
:= 1
.0;
{Rea
ctio
n or
der
of s
econ
d re
acti
on
. }
orc:
Q
:=
1.0
; {R
eact
ion
orde
r of
th
ird
rea
ctio
n.
}
DG2
1 :=
10.
0;
{Def
ined
dim
ensi
onle
ss g
roup
. }
DG
2-2
:=
1.0
; {D
efin
ed d
imen
sion
less
gro
up.
} D
G2:
J :=
10
.0;
{Def
ined
dim
ensi
onle
ss g
roup
. }
DG3
1 :=
1
.0;
{Def
ined
dim
ensi
onle
ss g
roup
. }
DG
3-2
:=
1.0
; {D
efin
ed d
imen
sion
less
gro
up.
} D
G3:
J :=
0.0
1;
{Def
ined
dim
ensi
onle
ss g
roup
. }
GfM
ax_x
:=
4.0
; {M
axin
x.rn
x-v
alue
for
con
vers
ion
grap
h.
} {(
Dim
ensi
onle
ss T
ime.
) }
GfM
ax_Y
:=
0.0
2;
{Max
inx.
rn y
-val
ue f
or b
reak
thro
ugh
grap
h.
} {(
Con
cent
rati
on -
kg/m
3 fl
uid
.)
}
Iter
atio
ns
= 1
00;
{It
erat
ion
s in
tim
e.
} P
rin
tFil
ter
= 1
0;
{G~ p
rin
tin
g f
ilte
r.
} N
umSu
blnt
=
10;
{N
r
of s
ub
inte
rval
s w
ithi
n ea
ch s
pat
ial
} {i
ncre
men
t.
}
end;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re
lnit
iali
se_
Su
per
fici
al_
Vel
oci
ty_
Fil
e;
begi
n assi
gn(S
upV
elF
ile,
•su
pVel
Fi.
Dat
');
rese
t(S
upV
elF
ile)
; S
upV
elD
ata[
1,1J
:=O
; S
upV
elD
ata[
1,2J
:=O
; re
ad(S
upV
elF
ile,
Sup
Vel
Dat
a[2,
1J);
re
ad(S
upV
elF
ile,
Sup
Vel
Dat
a[2,
2J);
ln
tlen
gthC
rit:
=V
oida
ge*D
elta
E*L
engt
h;
end;
{---
----
----
----
----
----
·---
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re T
imeO
ART
an
d SR
Ts;
· {T
his
pro
ced
ure
-in
itia
lise
s th
e al
pha
and
sigm
a v
ecto
rs.
}
begi
n ARH
OJ
:=1;
SR
T_1
[0J:
=1;
SRT
2 [O
J :=
1;
SRT
:J [0
] : =
1;
SolS
RT
_1[0
J:=O
; So
lSR
T 2
[OJ
:=O
; SolSRT~COJ :=
O;
Hol
dSR
T 1C
OJ:
=1;
Hol
dSR
(::2[
0] :
=1;
n 0 Q.. ~
~ -· tll """" :r ~
"'S ~
0 Q.. ~ -~ ~ N . ~ 00
.
Univers
ity of
Cap
e Tow
n
07/2
8/19
95
13:5
5 Fi
lena
me:
M
ODEL
4D2.P
AS
Page
3
Hol
dSRT
3[0
]:=
1;
for
Arb
Tnd
ex:=
1 to
N d
o be
gin
AR
TC
Arb
inde
xl:=
O;
SRT
_1C
Arb
inde
xl:=
1;
SRT
2CA
rbin
dexJ
:=
1;
SRT~CArbindex] :
=1;
So
lSR
T_1
CA
rbin
dex]
:=O
; So
lSR
T 2C
Arb
inde
xl:=
O;
SolSRT~CArbindex] :=
O;
Hol
dSRT
1C
Arb
inde
xl:=
1;
Hol
dSR
T-2
CA
rbln
dexJ
:=1;
H
oldS
RT
=3C
Arb
inde
x]:=
1;
end;
fo
r A
rbin
dex:
=O
to m
axv
do b
egin
B
TCA
RTC
Arb
inde
xl
=O
BTCS
RT
1CA
rbin
dexl
=O
B
TCSR
T-2C
Arb
inde
xl
=O
BTC
SRT-
3CA
rbin
dexl
=O
en
d;
-en
d;
{·--
------
----
----
----
----
----
----
-~--
----
----
----
-------;·---------------}
Pro
cedu
re D
eter
min
e_C
orre
spon
ding
Del
taT
; {T
his
proc
edur
e us
es t
he m
etho
d o
r ch
arac
teri
stic
s to
det
erm
ine
rhe
} {p
rogr
essi
on i
n ti
me
for
a gi
ven
Cha
rDel
taE
. It
all
ows
for
unst
eady
}
{st
ate
flow
. }
begi
n lntl
engt
h:=
O;
Del
ta t
:=O
; w
hile
-lnt
Len
gth<
>In
tlen
gthC
rit
do
begi
n if
EO
F(Su
pVel
File
) th
en b
egin
cl
oses
egra
phic
s;
wri
teln
('In
suff
icie
nt
Dat
a in
the
Su
per
fici
al V
eloc
ity
Dat
a F
ile
to')
; w
rite
ln(•
exec
ute
the
requ
ired
num
ber
of
iter
atio
ns.
•);
·wri
teln
('P
rese
nt
num
ber
of
iter
atio
ns:
=
',Tim
eCou
nter
);
Con
tinue
:=O
; In
tlen
gth:
=ln
tlen
gthC
r1t;
re
adln
; en
d·
lntl
engt
h:=
Intl
engt
h+(S
upV
elD
ata[
2,1J
-Sup
Vel
Dat
a[1,
1J)*
Sup
Vel
Dat
a[2,
2l;
Del
ta t
:=D
elta
t+
(Sup
Vel
Dat
aC2,
1J-S
upV
elD
ataC
1,1J
);
If
IntL
engt
h>In
tlen
gthC
rit
then
beg
in
Arb
Val
ue:=
Intl
engt
h-In
tLen
gthC
rit;
D
elta
t:=
Del
ta t
-Arb
Val
ue/S
upV
elD
ata[
2,2l
; ln
tlen
gth
:=ln
t[en
gth
Cri
t;
SupV
elD
ata[
1,1J
:=Su
pVel
Dat
a[2,
1l-
Arb
Val
ue/S
upV
elD
ata[
2,2l
; Su
pVel
Dat
a[1,
2l :
=Su
pVel
Dat
aC2,
2l;
end
else
beg
in
SupV
elD
ataC
1,1l
:=Su
pVel
Dat
aC2,
1l;
SupV
elD
ataC
112l
:=
Sup
Vel
Dat
a[2,
2l;
read
(Sup
Vel
F1l
e,
SupV
elD
ataC
2, 1
1);
read
CSu
pVel
File
, S
upV
elD
ata[
2,2l
) en
d·
~d•
I
Del
taT
:=D
elta
_t*u
Sta
r/L
engt
h;
end;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re
Iter
ate_
wit
hin_
Del
taE
s;
07/2
8/19
95
13:5
5 F
ilen
ame:
M
ODEL
4D2.P
AS
Page
4
{Thi
s pr
oced
ure
upda
tes
the
alph
a an
d si
gma
vect
ors
duri
ng t
he
tim
e it
{t
akes
for
th
e fl
uid
to
mov
e th
roug
h a
spat
ial
incr
emen
t.
begi
n for
Arb
lnde
x:=O
to
N d
o be
gin
for
Sub
lnte
rval
lnde
x:=
1 to
Num
Subl
nt
do
begi
n A
RT
P1[A
rb!n
dex]
:=
AR
T[A
rbln
dex]
+ D
elta
T/N
umSu
blnt
*(-D
G2
1*A
RT
[Arb
lnde
xl
*exp
(Ord
1*l
n(SR
T
1CA
r6In
dex]
)))+
D
elta
T/N
umSu
blnt
*(-D
G2
2*A
RTC
Arb
lnde
xl
*exp
(Ord
2*l
n(SR
T
2CA
r6In
dex]
)))+
D
elta
T/N
Lim
Subl
nt*(
·DG
2 3*
AR
TCA
rbln
dex]
*e
xpC
Ord
3*l
nCSR
T_3
CA
roln
dexJ
)));
SR
T 1[
Arb
lnde
x]
:=SR
T 1C
Ar6
Inde
xl+D
elta
T/N
umSu
blnt
* -
(·DG
2 1*
DG
3 1*
AR
T[A
rbln
dexl
*e
xp(O
rd 1
*TnC
SRT
_1C
Arb
lnde
x]))
);
SRT
2[A
rbln
dex]
:=
SRT
2CA
r61n
dexJ
+Del
taT
/Num
Subl
nt*
-(-D
G2
2*D
G3
2*A
RT
[Arb
lnde
x]
*exp
(Ord
2*T
nCSR
T 2C
Arb
lnde
xJ))
);
SRT
3[A
rbln
dex]
:=
SRT
3[A
r6In
dex]
+Del
taT
/Nl.l
llSU
blnt
* -
(·DG
2 3*
DG
3 3*
AR
TCA
rbln
dex]
ART
[Arb
lnde
xl
end·
en
d·
'
*exp
(Ord
3*T
nCSR
T 3C
Arb
inde
x]))
);
:=A
RT
P1C
Ar6
Inde
xl;
-
end;
'
} }
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re
Incr
emen
t F
luid
Vec
tors
; {T
his
proc
edur
e up
date
s tn
e al
pha
vec
tor
once
the
flu
id h
as
prog
ress
ed
} {t
hrou
gh a
sp
atia
l in
crem
ent.
}
begi
n for
Arb
lnde
x:=1
to
N d
o be
gin
1
SolS
RT
1[A
rb!n
dex]
:=
SolS
RT
1CA
rbin
dex]
+(H
oldS
RT
1
[Arb
lnde
x]-
-SR
T 1C
Arb
lnde
x])/
N;
-So
lSR
T 2C
Arb
lnde
x]
:=So
lSR
T 2C
Arb
lnde
x]+(
Hol
dSR
T
2[A
rbln
dex]
--
SRT
2CA
rbln
dex]
)/N
; -
SolS
RT
3CA
rbln
dexl
:=
SolS
RT
3CA
rbln
dexJ
+(H
oldS
RT
3[A
rbin
dexJ
--
SRT
_3C
Arb
inde
xJ)/
N;
-H
oldS
RT
1[A
rbln
dexl
=S
RT
1CA
rbln
dexl
; H
oldS
RT
=2[A
rbln
dex]
=S
RT
=2[A
rbln
dexl
; H
oldS
RT
_3[A
rbln
dex]
=S
RT
_3[A
rbln
dexl
; en
d;
for
Arb
lnde
x:=N
do
wnt
o 1
do b
egin
A
RTC
Arb
lnde
xl
:=A
RT
[Arb
!nde
x-1]
; So
lSR
T 1C
Arb
lnde
xJ:=
SolS
RT
1C
Arb
inde
x-1J
So
lSR
T-2
[Arb
lnde
xl :
=Sol
SRT
-2C
Arb
inde
x-1J
So
lSR
T-3
[Arb
inde
x]:=
SolS
RT
-3C
Arb
lnde
x-1J
~d;
--
ARTC
OJ
=1
Sol S
RT
1 [0
] =O
So
lSR
C2C
0l
=O
Sol
SR
C3[
0]
=0 .
en
d;
-
{***
**
<***
**
{***
**
{***
**
Impo
sed
boun
dary
con
dit
on.
l~sed
boun
dary
con
dit
on.
Impo
sed
boun
dary
con
dit
on.
Impo
sed
boun
dary
con
dit
on.
} } } }
{~--
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re U
pdat
e_B
reak
Thr
ough
Vec
tors
;
Univers
ity of
Cap
e Tow
n
07/2
8/19
95
13:5
5 Fi
lena
me:
M
ODEL
4D2.P
AS
Page
5
begi
n Tim
eVec
tor[
Vec
torl
ndex
l =C
umTi
me;
B
TC
AR
T[V
ecto
rlnde
x]
=ART
[NJ;
BTCS
RT
1[V
ecto
rlnd
exJ
=Sol
SRT
1[N
l;
if D
G3-
2<>0
th
en B
TCSR
T 2[
Vec
torl
ndex
l if
DG3~<>0
then
BT
CSR
T-3
[Vec
torln
dexJ
en
d;
-·
-
:=So
lSR
T_2C
Nl*
DG
3_1/
DG
3_2;
:=
SolS
RT
_3[N
l*D
G3_
1/D
G3_
3;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re L
abel
_Gra
ph;
begi
n Str
(N:3
,NS
);
Str
(Len
gth:
4:2,
Len
gthS
);
Str(
DG
2 1:
8:4,
DG
2_1S
);
Str(
DG
2_2:
8:4,
DG
2_2S
);
Str(DG2~:8:4,DG2
3S);
St
r(D
G3
1:8:
4,D
G3
1S);
St
r(D
G3=
2:8:
4,D
G3=
2S);
St
r(D
G3=
3:8:
4,D
G3=
3S);
G
Lab
el2:
=C
onca
t('
L '
,Len
gth
s,'
Nod
es
',NS
);
GL
abel
4:=
Con
cat(
' DG
2_1
',DG
2_1S
,' D
G2_
2 ',D
G2_
2S,'
GL
abel
5:=
Con
cat(
' DG
3 1
',DG
3_1S
,' D
G3_
2 ',D
G3_
2s,'
Lab
elG
raph
Yin
dow
C15
0,90
5,G
Lab
el2,
0,0)
L
abel
Gra
phY
indo
w(1
50,8
75,G
Lab
el4,
0,0)
L
abel
Gra
phY
indo
w(1
50,8
45,G
Lab
el5,
0,0)
en
d;
DG2
3 ',D
G2
3S);
OG
3=3
I ,D
G3=
3S);
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re G
raph
1_1n
itia
lise
;
begi
n;
in1t
SE
Gra
phic
s('f
:\tp
61);
Set
Cur
rent
Yin
dow
(2);
B
orde
rCur
rent
Yin
dow
(1);
S
etA
xesT
ype(
0,0)
· S
cale
Plo
tAre
a(0.
0,0.
0, 1
, 1.2
>;
Set
XY
lnte
rcep
ts(0
.0,0
.0);
S
etC
olor
(2);
D
raw
XA
xis(
0.2,
1);
Dra
wY
Axi
s(o.
2 61);
L
abel
XA
xis(
1,
>;
Lab
elY
Axi
s(1,
0);
Tit
leX
Axi
sC'D
imen
sion
less
Len
gth'
);
Tit
leY
Axi
s('D
imen
sion
less
Co
nce
ntr
atio
n')
; T
itleY
indo
wC
'Mod
el4D
2');
L
abel
Gra
phY
indo
w(3
75,9
35,
'Con
cent
rati
on P
rofi
les'
,0,0
);
Lab
el_G
raph
; fo
r A
rbln
dex:
=O
to N
do
Dat
aSet
X[A
rbln
dexJ
:=A
rbln
dex*
Del
taE
; en
d; _
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re G
raph
1_R
esul
ts;
begi
n for
Arb
lnde
x:=O
to
N d
o D
ataS
etY
CA
rbln
dexJ
:=A
RT
CA
rbln
dexJ
; L
ineP
lotD
ata(
Dat
aSet
X,D
ataS
etY
,N+
1,3,
0);
for
Arb
lnde
x:=O
to
N d
o D
ataS
etY
CA
rbln
dexJ
:=S
RT
1CA
rbln
dexJ
; L
ineP
lotD
ata(
Dat
aSet
X,D
ataS
etY
,N+
1,5,
2);
-fo
r A
rbln
dex:
=O
to N
do
Dat
aSet
YC
Arb
lnde
xJ :=
SRT
2CA
rbln
dexJ
; L
ineP
lotD
ata(
Dat
aSet
X,D
ataS
etY
,N+
1,4,
1);
.-
for,
Arb
lnde
x:=
O
to N
do
Dat
aSet
YC
Arb
lnde
xJ :=
SRT
3CA
rbln
dexJ
; L
ineP
lotD
ata(
Dat
aSet
X,D
ataS
etY
,N+
1,7,
1);
-
. ------------
-
07/2
8/19
95
13:5
5 F
ilen
ame:
M
OD
EL4D
2.PA
S Pa
ge
6
end;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re G
raph
_Bre
akT
hrou
ghC
urve
s;
begi
n clo
sese
gra
ph
ics;
in
itse
gra
ph
ics(
1f:
\tp
6')
; S
etC
urre
ntY
indo
w(2
);
Bor
derc
urre
ntY
indo
w(1
);
SetA
xesT
ypeC
0,0)
; S
cale
Plo
tAre
a(0.
0,0.
0,G
fMax
_X,G
fMax
_Y);
S
etX
Yln
terc
epts
(0.0
,0.0
);
Set
Col
or(2
);
Dra
wX
Axi
sCC
GfM
ax
X/5
),1
);
Dra
wY
Axi
sCC
GfM
ax-Y
/5),1
); L
abel
XA
xis(
1,0)
;-L
abel
YA
xisC
1,0>
; T
itle
XA
xisC
'Dim
ensi
onle
ss T
ime'
);
Tit
leY
Axi
s('B
reak
thro
ugh
Con
e.
Rel
. to
Ori
gina
l Q
uant
ity
Sol
. 1
.');
T
itleY
indo
w('M
odel
4D2
');
' L
abel
Gra
phY
indo
w(3
80,9
35,'B
reak
thro
ugh
Cu
rves
',0,0
);
Lab
el
Gra
ph;
· fo
r A
rbln
dex:
=O
to V
ecto
rlnd
ex d
o D
ataS
etX
CA
rbln
dexJ
:=
Tim
eVec
tor[
Arb
!nde
xl;
(*
for
Arb
lnde
x:=O
to
Vec
torl
ndex
do
Dat
aSet
YC
Arb
!nde
xJ :
=BTC
AR
TCA
rb!n
dexJ
; L
ineP
lotD
ata(
Dat
aSet
X,D
ataS
etY
,(V
ecto
rlnd
ex+
1),3
,0);
*)
fo
r A
rbln
dex:
=O
to V
ecto
rlnd
ex d
o D
ataS
etY
CA
rbln
dexJ
:=B
TC
SRT
1[
Arb
lnde
xJ;
Lin
ePlo
tDat
a(O
ataS
etX
,Dat
aSet
Y,(
Vec
torl
ndex
+1)
,5, 1
);
-.
for
Arb
lnde
x:=O
to
Vec
torl
ndex
do
Dat
aSet
Y[A
rbln
dexJ
:=BT
CSRT
2C
Arb
lnde
xJ;
Lin
ePlo
tDat
a(D
ataS
etX
,Dat
aSet
Y,C
Vec
torl
ndex
+1)
,4, 1
);
-fo
r A
rbln
dex:
=O
to V
ecto
rlnd
ex d
o D
ataS
etY
CA
rbln
dexJ
:=B
TC
SRT
3C
Arb
lnde
xJ;
Lin
ePlo
tDat
a(D
ataS
etX
,Dat
aSet
Y,(
Vec
torl
ndex
+1)
,7,1
);
-
end;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
{MAI
N PR
OGRA
M
)
begi
n
Con
tinu
e:=
1;
Cum
Tim
e :=
O;
Pri
ntF
ilte
rln
dex
:=O
; T
imeC
ount
er:=
O;
Vec
torl
ndex
:=O
; T
imeV
ecto
r[V
ecto
rlnd
exJ:
=C
umT
ime;
BT
CA
RT
[Vec
torln
dexJ
:=
O;
BTCS
RT_1
[V
ecto
rlnd
exJ
:=O
; BT
CSRT
_2 [
Vec
tor I
ndex
] :=
O;
BT
CSR
T_3
[Vec
torln
dex]
:=
0;
Con
v 1[
Vec
torl
ndex
J :=
O;
Con
v-2C
Vec
torl
ndex
J :=
O;
Con
v:3c
Vec
torl
ndex
J :=
0;
Def
ine
Var
iabl
es;
Tim
eO A
RT
and
SRTs
; In
itia
lise
su
per
fici
al V
eloc
ity
Fil
e;
Gra
ph
1_
1n
itia
lise
; -
-
•
Univers
ity of
Cap
e Tow
n
07/2
8/19
95
13:5
5 Fi
lena
me:
M
ODEL
4D2.P
AS
Page
7
whi
le
(Con
tinu
e=1)
an
d (T
imeC
ount
er<
lter
atio
ns)
do
begi
n Tim
eCou
nter
:=T
imeC
ount
er+1
; P
rin
tFil
terl
nd
ex:=
Pri
ntF
ilte
rln
dex
+1
; D
eter
min
e C
orre
spon
ding
Del
taT
; CumTime:=~umTime+DeltaTT
Iter
ate
wit
hin
Del
taE
s;
if P
rin
tFil
terT
nd
ex=
Pri
ntF
ilte
r th
en
begi
n Vec
torl
ndex
:=V
ecto
rlnd
ex+
1;
Pri
ntf
ilte
rln
dex
:=O
; U
pdat
e_B
reak
Thr
ough
Vec
tors
; G
raph
1_R
esul
ts;
en
d·
lnc~ement
Flu
id V
ecto
rs;
end;
-
-
read
ln(H
ardC
opy)
; if
har
dcop
y=1
then
Scr
eenD
ump(
3,0,
2, 1
.0, 1
.0,0
, 1,0
,err
or)
;
Gra
ph_B
reak
lhro
ughC
urve
s;
read
ln(H
ardc
opy)
; if
har
dcop
y=1
then
Scr
eenD
umpC
3,0,
2, 1
.0, 1
.0,0
, 1-,
0,er
ror)
;
clos
eseg
raph
ics;
end.
{=
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
}
Univers
ity of
Cap
e Tow
n
Appendix ID. Solution Algorithm and Code for Model2D.PAS.
Model2 essentially represents suitable computer routines for Dixon's [1992] particle scale model discussed in Chaper 4.
Solution Algorithm for Model2D.
Start. I Set/I nitialis.e Variables.
Iterate in Time.
Calculate the A-Matrix and Y-Vector required for the Crank-Nicolson Method.
l Use the A-Matrix and Y-Vector to calculate the Alpha and Sigma vectors within the Particle.
I Graph profiles.
I
I Graph conversion versus time.
End.
Univers
ity of
Cap
e Tow
n
07/3
0/19
95
15:Q
1 Fi
lena
me:
M
ODEL
2D2.P
AS
-----~--
P;ig
e
Prog
ram
Mod
el2D
2; ·
{Mod
el2D
2.
{ T
his
prog
ram
cal
cula
tes
the
con
cen
trat
ion
pro
file
of
flu
id
} an
d so
lid
rea
ctan
ts w
ithi
n a
par
ticl
e us
ing
the
equa
tion
s as}
{ { { { { { { { { { { { { { { { {
deve
lope
d by
Dix
on.
} In
clud
es t
he p
oss
ibil
ity
of
a se
cond
so
lid
rea
ctan
t.
} In
clud
es t
he p
oss
ibil
ity
of·
a s
urf
ace
conc
entr
atio
n }
dif
fere
nt
to t
he b
ulk
con
cen
trat
ion
. }
Allo
ws
for
a· r
eact
ion
ord
er
in t
erm
s of
th
e so
lid
.rea
ctan
t }
othe
r th
an 1
. }
Thi
s pr
ogra
m a
lso
cal
cula
tes
the
frac
tio
nal
co
nver
sion
. }
Num
eric
al
stra
teg
y u
sed
is a
for
m o
f C
rank
-Nic
olso
n Im
pli
cit
}
fin
ite d
iffe
ren
ce.
}
Ass
umpt
ions
in
th
is m
odel
in
clud
e:
} Th
e so
lid
rea
ctan
t d
epo
sits
wit
hin
the
par
ticl
e }
rese
mbl
e th
ose
on
the
surf
ace.
}
The
dep
osi
ts w
ould
bot
h re
act
to t
he s
ame
exte
nt
} if
eac
h w
ere
expo
sed
to t
he s
ame
acid
co
nce
ntr
atio
n
} fo
r th
e sa
me
tim
e.
} S
urfa
ce a
cid
co
nce
ntr
atio
n i
s sti
ll
assu
med
to
be
} co
nsta
nt
-ie
alw
ays
1.
}
{Cod
ed:
Gra
ham
Dav
ies.
}
<
Dep
artm
ent
of
Che
mic
al
Eng
inee
ring
. }
{ U
nive
rsit
y of
C
ape
Tow
n.
} <
5
Aug
ust
1994
. }
{ 28
Ju
ly
1995
-
Upd
ated
. }
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
{Dec
lara
tion
s:
}
uses
crt
,std
hdr,
gj,g
raph
,wor
lddr
,seg
raph
,int
egra
t,M
2C1B
rent
Roo
ts;
cans
t Bet
a1
=1.
0;
Kap
paP1
=
10.0
;
Lam
bda1
=
0.0;
Ord
SP1
=1.
0;
Bet
a2
=0.
0;
Ka~P2
=0.
0;
La
a2
=0.
0;
Ord
SP2
=O.O
;
N
=24;
Del
taT
=
0.00
05;
Del
taE
=1
/CN
+1);
Nor
m_C
rit
=1e
-10;
Con
v_C
rit ~1
e-8;
sc
one
Tol
=1e
-5·
-,
{Dim
ensi
onle
ss S
toic
hiom
etri
c ra
tio
def
ined
pg
11
} {R
atio
of
reat
ion
rat
e of
so
lid
rea
ctan
t re
sid
ing
}
{wit
hin
the
par
ticl
e to
por
ous
dif
fusi
on
of
flu
id
} {i
nto
the
par
ticl
e.
Def
ined
pg
11
Dix
on.
} {
frac
tio
n o
f th
e te
acha
ble
mat
eria
l on
th
e su
rfac
e.}
{Lam
bda1
=1
impl
ies
all
th
e m
ater
ial
is o
n th
e }
{su
rfac
e.
} {R
eact
ion
orde
r of
th
e so
lid
in
the
por
es.
}
{Hal
f th
e nu
mbe
r of
in
teri
or
po
ints
. r=
O
and
r=R
} {n
ot
incl
uded
. }
{Tim
e In
crem
ent
} {S
pace
In
crem
ent.
C
alcu
late
d fr
om
1/(N
+1)
. (S
ince
}
<
R=i
*dE
and
R i
s at
poi
nt N
+1.)
} {C
onve
rgen
ce c
rite
ria b
ased
on
the
norm
of
vec
tor
} {*
**
} {C
onve
rgen
ce c
rite
ria f
or
the
Bre
nt
Rou
tine
. }
<D
imen
sion
less
con
cent
rati
on o
f so
lid
bel
ow
whi
ch
} {
it
is a
ssum
ed
to b
e n
egli
gib
le.
}
07/3
0/19
95
15:0
1 F
ilen
ame:
M
OD
EL2D
2.PA
S Pa
ge
2
var M
axlt
er
=100
;
lter
atio
ns=
60
0;
Pri
nt
Cri
t=10
; G
f2_M
axX
=3
0;
<Max
imum
it
erat
ion
s fo
r th
e B
rent
R
outi
ne.
{It
erat
ion
s in
tim
e.
{Det
erm
ined
fro
m d
esir
ed P
rin
t D
elta
T/D
elta
T
{Gra
ph
2 m
axim
um d
imen
sio
nle
ss-r
eact
ion
time~
AR
T,A
RTP
1,SR
T1,S
RT1
P1,S
RT1
P1C
SR
T2,
SRT
2P1,
SRT
2P1C
,YV
ecto
r D
ataS
etX
,Dat
aSet
Y,l
nter
G1,
lnte
rG2
Con
v1,C
onv2
A
Mat
rix,
Aln
vers
e A
Mat
Det
,lnte
gVal
,XA
xisM
ax
Nor
m1,
Nor
m2,
A,A
P1,S
,Roo
t,Val
ueA
tRoo
t
Row
s,C
ols,
Arb
lnde
x,D
ataP
oint
s,R
epea
ts
erro
r,P
lot
Var
,Con
v V
ar,N
ewtl
ters
H
ardC
opy1
,Har
dCop
y2-
Bet
a1S,
Kap
paP1
S,la
mbd
a1S,
GD
TS,
GL
abel
1 B
eta2
S,K
appa
P2S,
Lam
bda2
S,G
Lab
el2
Ord
SP1S
,Ord
SP2S
{ART
A
lpha
fn
(r)
at
tim
e T
CA
RTP1
A
lpha
fn
(r)
at
tim
e T+
1 {S
RT1
Sigm
a fn
(r)
at
tim
e T
(S
RTs
P1
Sigm
a fn
(r)
at
tim
e T+
1 {S
RTs
P1C
Si
gma
fn(r
) at
ti
me
T+1
·sh
ort
Vec
tor;
"S
hort
Vec
tor;
·v
eryl
ongV
ecto
r;
·ver
yLon
gVec
tor;
·s
qrM
at;
real
; re
al;
inte
ger
; in
teg
er;
inte
ger
;
str
ng
str
ng
str
ng
(gue
ssed
) (c
alcu
late
d)
rang
e 0 .
. N
rang
e 0 .
. N
rang
e 0 .
. N
rang
e 0 .
. N
rang
e 0 .
. N
{YV
ect
'Con
st'
vec
tor
in C
rank
-Nic
olso
n m
etho
d ra
nge
0 .. N
C
AM
atrix
M
atri
x of
Cra
nk-N
icol
son
coef
ftci
ents
ra
nge
N*
N
{Dat
aSet
X
X v
ecto
r us
ed
in t
he g
raph
ing
r9u
tin
e ra
nge
0 .. N
+1
} } } } } } } } } } } }
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Fun
ctio
n P
ower
(Bas
e,P
ow:r
eal)
:Ext
ende
d;
begi
n if P
ow=O
th
en P
ower
:=1
else
if
Bas
e=O
th
en P
ower
:=O
el
se
Pow
er:=
exp(
Pow
*ln(
base
));
end;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re T
imeO
AR
T an
d SR
Ts;
{Thi
s pr
oced
ure-
uses
th
e in
itia
l co
nd
itio
ns
to s
et t
he
alph
a an
d si
gma
{vec
tors
. } }
surf
ace}
{N
ote
that
ART
[N
+1J,
ARTP
1 tN
+1l,
SRT1
[N
+1J
and
SRT1
P1 [
N+1
J ar
e th
e {c
on
cen
trat
ion
s of
th
e li
qu
id a
nd s
oli
d r
eact
ants
. be
gin fo
r A
rbln
dex:
=O
to N
do
begi
n A
RT
"[A
rbln
dexJ
:=O
; SR
T1"
CA
rbln
dexJ
:=1;
SR
T2"
[Arb
lnde
xJ:=
1;
end;
A
RT. [
N+1
J :=
1;
SRT1
. [N
+1J
:=1;
}
n 0 c..
~
t""4 -· Cl'.> !'
"to,
-· :s ~
""S :: 0 c..
~ N ~
N . ~ 00
.
Univers
ity of
Cap
e Tow
n
07/3
0/19
95
15:0
1 F
ilen
ame:
M
OD
EL2D
2.PA
S Pa
ge
3
• SR
T2"
[N+1
J:=1
; A
RTP1
. [N
+ll
:=1;
(I
mpo
sed
boun
dary
con
diti
on o
f th
e co
nce
ntr
atio
n o
f (t
he
bulk
flu
id
rem
aini
ng c
onst
ant
to t
he o
rig
inal
(v
alu
e.
if K
appa
P2=0
th
en f
or A
rbln
dex:
=O
to m
axc
do
begi
n SRT2
" [A
rbln
dexJ
=O
SR
T2P1
. [A
rbln
dexJ
=O
SR
T2P1
C. [
Arb
lnde
xJ
=O
end·
en
d;
'
} } }
(---
-. --
----
---
----
-----
----
----
----
·; --
----
----
----
----
----
----
----
----
--}
Pro
cedu
re G
uess
SR
TsP
l; {T
his
proc
edur
e-pr
ovid
es t
he in
itia
l "g
uess
" fo
r th
e it
era
tio
n.
It u
ses
} (t
he
prev
ious
tim
e in
terv
al's
val
ues
as
the
gues
s.
} be
gin fo
r A
rbln
dex:
=O
to C
N+1
) do
be
gin
SRT
1P1"
[Arb
lnde
xl:=
SRT
1"[A
rbln
dexJ
; SR
T2P
1"(A
rbin
dex]
:=SR
T2"
[Arb
lnde
xJ;
end·
en
d;
'
{--
----
----
--·-
----
----
----
---·
-·--
----
·---
----
----
----
----
----
----
--·-
---}
P
roce
dure
ReG
uess
SR
TsP1
; {T
his
proc
edur
e pr
ovid
es a
n up
date
d "g
uess
" fo
r th
e ne
xt
itera
tio
n.
It
{use
s th
e SR
T1P1
C ve
ctor
as
the
upda
ted
gues
s.
begi
n for
Arb
lnde
x:=O
to
CN
+1)
do
begi
n SR
T1P1
" [A
rbln
dex]
:=S
RT
1P1C
"[A
rbln
dexJ
; SR
T2P
1"[A
rbln
dexJ
:=SR
T2P
1C"[
Arb
lnde
xl;
end·
en
d;
'
} }
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re C
ale
AM
atri
x an
d Y
vec
tor·
{T
his
proc
edur
e ca
lcu
late
s-tn
e C
rank
-Nic
olso
n co
effi
cien
t m
atri
x.
} be
gin fo
r R
ows:=
O
to (
N+1
) do
be
gin fo
r C
ols:
=O
to (
N+1
) do
A
Mat
rix·
[R
ows,
Col
sJ :
=O;
end;
AM
atri
x"[0
,0J:
=·6
·Del
taE
*Del
taE
*CK
appa
P1*
Pow
er(S
RT
1P1"
[0J
,Ord
SP1)
+K
appa
P2*P
ower
CSR
T2P
1"[0
],0rd
SP2)
) -2
*Del
taE
*Del
taE
/Del
taT
; A
Mat
rix"
CO,
1)
:=6;
Y
Vec
tor"
[OJ
:=A
RT
"[0J
*C6+
Del
taE
*Del
taE
*CK
appa
P1*P
ower
(SR
T1.
[0J,
Ord
SP1)
+K
appa
P2*P
ower
(SR
T2"
[0J,
Ord
SP2)
) ·2
*Del
taE
*Del
taE
/Del
taT
)·A
RT
.[1]
*6;
07/3
0/19
95
15:0
1 F
ilen
ame:
M
OD
EL2D
2.PA
S Pa
ge
4
for
Row
s:=1
to
N-1
do
be
gin AM
atr i
x·
[Row
s, R
ows-
11 :
=R
ows-
1;
AM
atri
x-[R
ows,
Row
sl
:=-2
*Row
s-R
ows*
Del
taE
*Del
taE
*(
Kap
paP1
*Pow
er(S
RT
1P1"
[Row
s],O
rdSP
1)+
Kap
paP2
*Pow
er(S
RT
2P1"
[Row
sJ,O
rdSP
2))
-2*R
ows*
Del
taE
*Del
taE
/Del
taT
; A
Mat
rix"
[Row
s,R
ows+
1J:=
Row
s+1;
Y
Vec
tor"
[Row
sJ
:=A
RT
"[R
ows·
1J*C
·Row
s+1)
+AR
T"[
Row
sl*C
2*R
ows+
Row
s*
end;
Del
taE
*Del
taE
*C
Kap
paP1
*Pow
er(S
RT
1"[R
owsJ
,Ord
SP1)
+K
appa
P2*P
ower
CSR
T2"
CR
owsJ
,Ord
SP2)
) -2
*Row
s*D
elta
E*D
elta
E/D
elta
T)
+AR
T"[
Row
s+1]
*(·R
ows-
1);
AM a
t r ix
. [N
, N
-11
: =N
-1;
AM
atri
x"(N
,NJ
:=·2
*N-N
*Del
taE
*Del
taE
*(K
appa
P1*P
ower
(SR
T1P
1" [
NJ,
Ord
SP1)
+K
appa
P2*P
ower
(SR
T2P
1"[N
],Ord
SP2)
) ·2
*N*D
elta
E*D
elta
E/D
elta
T;
YV
ecto
r"[N
J :=
AR
T"[
N-1
l*C
·N+1
)+A
RT
"[N
J*C
2*N
+N*D
elta
E*D
elta
E*
(Kap
paP1
*Pow
er(S
RT
1"[N
],O
rdSP
1)
+Kap
paP2
*Pow
er(S
RT
2"[N
J,O
rdSP
2))
-2*N
*Del
taE
*Del
taE
/Del
taT
)+
AR
T"[
N+1
J*C
-N-1
)·AR
TP1
"[N
+1]*
(N+1
);
end;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re C
alcS
RT
sP1C
; {T
his
proc
edur
e is
a n
on
-lin
ear
root
fi
nd
ing
pro
cedu
re.
It
uses
Bre
nt'
s }
(Met
hod
to s
olv
e fo
r th
e ro
ot.
}
begi
n if
X]
ex]
(Kap
paP1
<>0)
th
en b
egin
if
Ord
SP1=
1 th
en b
egin
end SR
T1P
1C"[
Arb
lnde
x] :
=SR
T1"
[Arb
lnde
xl*C
1-D
elta
T*K
appa
P1*B
eta1
*AR
T.[
Arb
lnde
/C2*
C1-
Lam
bda1
)))/
(1+
Del
taT
*Kap
paP
1*B
eta1
* A
RT
P1"[
Arb
lnde
xJ/C
2*C
1-L
ambd
a1))
);
else
if
SR
T1P
1"[A
rbln
dexl
<SC
onc_
Tol
th
en S
RT
1P1C
"[A
rbln
dex]
:=S
RT
1P1.
CA
rbln
d
else
beg
in
A
:=A
RT
"[A
rbln
dexl
; A
P1:=
AR
TP1
"[A
rbln
dexJ
; s
:=SR
T1"
CA
rbln
dexJ
; S
RT
1P1C
"[A
rb!n
dexJ
:=B
rent
Roo
ts(0
.0,1
.0,D
elta
T,K
appa
P1,
Bet
a1,L
ambd
a1,0
rdS
P1
,A,A
P1,
X]
S, 1
e-8,
100,
Val
ueA
tRoo
t,er
ror)
; en
d·
end·
'
if
CK
appa
P2<>
0)
then
beg
in
if O
rdSP
2=1
then
beg
in
SRT
2P1C
"[A
rb!n
dexJ
:=SR
T2"
[Arb
lnde
x]*(
1-D
elta
T*K
appa
P2*B
eta2
*AR
T"[
Arb
lnde
/C2*
C1·
Lam
bda2
)))/
(1+
0elt
aT*K
appa
P2*
Bet
a2*
AR
TP1
"[A
rbln
dexl
/C2*
C1-
Lam
bda2
)));
Univers
ity of
Cap
e Tow
n
07/3
0/19
95
15:0
1 F
ilen
ame:
M
ODEL
2D2.P
AS
Page
5
ex]
end
else
if
SRT
2P1"
[Arb
lnde
xJ<S
Con
c_T
ol
then
SR
T2P1
c· [
Arb
lnde
xl :
=SR
T2P
1"[A
rbln
d
else
beg
in
A
:=A
RT"
[Arb
lnde
xl;
AP1
:=A
RT
P1"[
Arb
lnde
xJ;
s :=
SRT
2"[A
rbln
dexl
; SR
T2P
1C"[
Arb
lnde
xl :
=B
rent
Roo
ts(O
.O, 1
.0,D
elta
T,K
appa
P2,
Bet
a2,L
ambd
a2,0
rdS
P2
,A,A
P1,
end·
en
d·
' en
d;
'
S, 1
e-8,
100
,Val
ueA
tRoo
t,er
ror)
;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re C
heck
C
onve
rgen
ce;
{Thi
s pr
oced
ure-
chec
ks w
heth
er
or n
ot
the
solu
tio
n h
as
conv
erge
d by
}
{com
pari
ng t
he g
uess
ed v
alue
of
SRT1
P1
with
a
calc
ula
ted
val
ue o
f SR
T1P1
. }
begi
n Nor
m1:
=0;
{Usi
ng
a no
rm**
* }
Nor
m2:
=0;
for
Arb
lnde
x:=O
to
CN
+1)
do
begi
n C
alcS
RTs
P1C
; if
(B
eta1
<>0)
an
a C
Kap
paP1
<>0)
th
en b
egin
N
orm
1:=C
SRT
1P1C
"[A
rbln
dex]
-SR
T1P
1"[A
rbln
dexJ
)*(S
RT
1P1C
"[A
rbln
dexJ
SR
T1P
1"[A
rbln
dexl
);
end
else
Nor
m1:
=0;
if
(Bet
a2<>
0)
and
CK
appa
P2<>
0)
then
beg
in
Nor
m2:
=(SR
T2P
1C"[
Arb
lnde
x]-S
RT
2P1"
[Arb
lnde
x])*
(SR
T2P
1C"[
Arb
lnde
x]
SRT
2P1"
[Arb
lnde
x] )
; en
d el
se N
orm
2:=0
; en
d·
end;
'
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re U
pdat
e AR
T an
d SR
Ts;
{Thi
s pr
oced
ure
upda
tes
the
alph
a an
d si
gma
vect
ors
for
the
next
it
erat
ion
}
Cby
repl
acin
g th
eir
com
pone
nts
wit
h th
e al
phaT
+1
and
sigm
aT+1
v
ecto
rs.
} be
gin fo
r A
rbln
dex:
=O
to (
N+1
) do
be
gin
AR
T"[
Arb
lnde
xl :
=AR
TP1
"[A
rbin
dexl
; SR
T1"
[Arb
lnde
xJ:=
SRT
1P1"
[Arb
inde
xl;
SRT
2"[A
rbln
dexl
:=
SRT
2P1"
[Arb
lnde
xl;
end·
en
d;
'
{---
----
----
----
----
----
----
·---
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re C
ale
Con
vers
ion;
{T
his
procedur~ ca
lcu
late
s th
e fr
acti
on
al
conv
ersi
on o
f th
e p
art
icle
fo
r }
{th
e ti
me
inte
rval
. It
use
s th
e fo
rmul
a on
pg
. 13
Dix
on.
The
inte
gra
tor
} {
is t
he Q
uinn
-Cur
tis
vect
or
inte
gra
tor.
}
begi
n
07/3
0/19
95
15:0
1 F
ilen
ame:
M
OD
EL2D
2.PA
S Pa
ge
6
if K
appa
P1<>
0 th
en b
egin
fo
r A
rbln
dex:
=O
to C
N+1
) do
ln
terG
1"[A
rbln
dexJ
:=
C1-
SR
T1"
[Arb
lnde
xJ)*
Arb
lnde
x*D
elta
E*A
rbln
dex*
Del
taE
;
lnte
grat
eVec
torC
lnte
rG1"
,Del
taE
,0,(
N+
1)
Inte
gVal
);
Con
v1·c
conv
_var
J:=3
*<1-
Lam
bda1
)*1n
tegV
a[+L
ambd
a1*C
1-SR
T1"
[N+1
J>;
end·
if
Kap
paP2
<>0
then
beg
in
for
Arb
lnde
x:=
O
to C
N+1
) do
ln
terG
2"[A
rbln
dexJ
:=C
1-S
RT
2"[A
rbln
dexJ
)*A
rbln
dex*
Oel
taE
*Arb
lnde
x*D
elta
E;
Inte
gra
teV
ecto
r(ln
terG
2",
Del
taE
,0,(
N+
1),
lnte
gV
al);
C
onv2
" [C
onv
Var
] :=
3*C
1-L
ambd
a2)*
1nte
gVal
+Lam
bda2
*C1-
SRT
2"[N
+1J)
; en
d;
-
end;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re G
raph
1 In
itia
lise
; '
begi
n -
init
SE
Gra
phic
s(1F
:\tp
61);
Set
Cur
rent
Win
dow
(2);
B
orde
rCur
rent
Win
dow
C1)
; S
etA
xesT
ype(
0,0)
; S
cale
Plo
tAre
a(0
.0,0
.0, 1
.0, 1
.0);
S
etX
Yln
terc
epts
(0.0
,0.0
);
Set
Col
or(2
);
Dra
wX
Axi
s(0.
2,1)
; D
raw
YA
xis(
0.2,
1);
Lab
elX
Axi
s(1,
0);
Lab
el Y
Axi
sC1
,0);
T
itle
XA
xisC
'Dim
ensi
onle
ss R
adiu
s');
T
itle
YA
xis(
'Dim
ensi
onle
ss C
on
cen
trat
ion
');
Tit
leW
indo
w('M
odel
2D2'
);
Str
(Bet
a1:6
:3,B
eta1
S);
S
tr(K
appa
P1:
6:3,
Kap
paP
1S);
S
tr((
Del
taT
*P
rin
t C
rit)
:6:4
,GD
TS
);
StrC
Lam
bda1
:5:3
,Lam
bda1
S);
Str
(Ord
SP
1:5:
2,0r
dSP
1S);
S
tr(B
eta2
:6:3
,Bet
a2S
);
Str
(Kap
paP
2:6:
3,K
appa
P2S
);
StrC
Lam
bda2
:5:3
,Lam
bda2
S);
Str
(Ord
SP
2:5:
2,0r
dSP
2S);
G
Labe
l 1:=
Con
cat(
1 B
eta1
',B
eta1
S, ';
K
appa
1 ',K
appa
P1S
, ';
La
mbd
a1
1 ,L
ambd
a1S
•·
Ord
er1
'Ord
SP
1S
•·
GOT
' G
DTS
)· G
Lab
el2:
=C
onca
tC'
Bet
a2
1,B
etaZ
S,1
; Kapp~2
',Kap
paP
2S,:
; La
mbd
a2
',Lam
bda2
S
. '·
Ord
er2
',Ord
SP
2S);
L
abel
Gra
phW
indo
wC
1,93
0,G
Lab
el1,
0,0)
; L
abel
Gra
phW
indo
w(1
,900
,Gla
bel2
,0,0
);
for
Arb
lnde
x:=O
to
N+1
do
Dat
aSet
X"[
Arb
lnde
xJ:=
Arb
lnde
x*D
elta
E;
Dat
aPoi
nts:
=N
+2;
en
d;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Univers
ity of
Cap
e Tow
n
07/3
0/19
95
15:0
1 F
ilen
ame:
M
ODEL
2D2.P
AS
Pag
e·
7
Pro
cedu
re G
raph
1 R
esul
ts;
begi
n -
if K
appa
P1<>
0 th
en b
egin
fo
r A
rbln
dex:
=O
to (
N+1
) do
Dat
aSet
Y"[
Arb
lnde
xJ :=
AR
TP1.
[A
rbln
dexl
; L
ineP
lotD
ata(
Dat
aSet
x·,o
ataS
etY
",D
ataP
oint
s,3,
0);
for
Arb
lnde
x:=O
to
(N
+1)
do D
ataS
etY
"[A
rbln
dex]
:=S
RT
1P1"
[Arb
lnde
xJ;
Lin
ePlo
tDat
a(D
ataS
etX
",D
ataS
etY
",D
ataP
oint
s,5,
0);
end·
if
Kap
paP2
<>0
then
beg
in
for
Arb
lnde
x:=O
to
(N
+1)
do D
ataS
etY
" [A
rbln
dexl
:=A
RTP1
" [A
rbln
dexl
; L
ineP
lotD
ata(
Dat
aSet
x·,o
ataS
etY
" ,D
ataP
oin
ts,3
,0);
fo
r A
rbln
dex:
=O
to (
N+1
) do
Dat
aSet
Y"[
Arb
lnde
xJ:=
SRT
2P1"
[A
rbln
dexl
; ·
Lin
ePlo
tDat
a(D
ataS
etX
",D
ataS
etY
" ,D
ataP
oin
ts,4
,2);
en
d;
end;
(---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re G
raph
2_In
itia
lise
_and
_Dra
w;
begi
n Set
Cur
rent
Win
dow
(2);
B
orde
rCur
rent
Win
dow
(1);
S
etA
xesT
ype(
0,0)
; X
Axi
sMax
:=K
appa
P1*B
eta1
*Del
taT
*Rep
eats
; S
cale
Plo
tAre
aCO
.O,O
.O,G
f2M
axx,
1.0)
; S
etX
Yin
terc
epts
(0.0
,0.0
);-
Set
Col
or(2
);
Dra
wX
Axi
sCC
Gf2
Max
X/5
),1)
; D
raw
YA
xis(
0.2,
l);
Lab
elX
Axi
sC1,
0);
Lab
elY
Axi
sC1,
0>;
Tit
leX
Axi
sC'D
imen
sion
less
Rea
ctio
n Ti
me
(With
re
spec
t to
RXN
1
) [K
appa
P1*B
eta
1*T
aul'>
; T
itle
YA
xisC
'Fra
ctio
nal
Con
vers
ion'
);
Tit
leW
indo
w('M
odel
2D2'
);
Lab
elG
raph
Win
dow
(1,9
30,G
Lab
el1,
0,0)
; L
abel
Gra
phW
indo
wC
1,90
01G
lab
el2
,0,0
);
if K
appa
P1<>
0 th
en b
egin
fo
r A
rbln
dex:
=O
to C
onv
Var
do
be
gin
Dat
aSet
X"[
Arb
lnde
x]:=
Arb
lnde
x*P
rint
Cri
t*D
elta
T*K
appa
P1*
Bet
a1;
Dat
aSet
Y"[
Arb
lnde
xJ:=
Con
v1"[
Arb
lnde
xl;
end·
Dat~
SetY
. [0
]: =
O;
Dat
aPoi
nts:
=C
onv
Var
; L
ineP
lotD
ata(
Dat
iSet
X",
Dat
aSet
Y",
Dat
aPoi
nts,
5,0)
; en
d·
if K
appa
P2<>
0 th
en b
egin
fo
r A
rbln
dex:
=O
to C
onv
Var
do
be
gin
Dat
aSet
X"[
Arb
lnde
x]::
Arb
lnde
x*P
rint
_Cri
t*D
elta
T*K
appa
P1*
Bet
a1;
Dat
aSet
Y"[
Arb
lnde
xJ:=
Con
v2"[
Arb
lnde
xl;
end·
Dat~
SetY
" [Q
] : =
O;
Dat
aPoi
nts:
=C
onv
Var
; L
ineP
lotD
ata(
Dat
iSet
x·,o
ataS
etY
" ,D
ataP
oin
ts,4
,2);
en
d;
end;
07/3
0/19
95
15:0
1 F
ilen
ame:
M
OD
EL2D
2.PA
S Pa
ge
8
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
{MAI
N PR
OGRA
M:
}
begi
n
new
(AR
T);
new
(SR
T1)
; ne
w(S
RT
2);
new
(YV
Ect
or>;
ne
w(
I nte
rG1)
; ne
w(C
onv1
);
new
(AM
atri
x);
new
(AR
TP1
); ne
w(S
RT
1P1)
; ne
w(S
RT
2P1)
; ne
w(D
ataS
etX
);
new
(lnt
erG
2);
new
(Con
v2);
ne
w(A
lnve
rse)
;
cl r
scr;
G
rap
hl_
Init
iali
se;
repe
ats:
=O
; N
orm
1:=1
; N
orm
2:=1
; P
lot
Var
:=O
; C
onv-
Var
:=O
;
new
(SR
T1P
1C);
new
(SR
T2P
1C);
new
(Dat
aSet
Y);
Tim
eU A
RT
and
SRTs
; G
uess
-SR
TsP
1;-
whi
le-
rep
eats
<I
tera
tio
ns
do
begi
n w
hile
(N
orm
l>N
orm
Cri
t) o
r (N
orm
2>N
orm
Cri
t) d
o be
gin
Cal
e A
Mat
rix
and
Y V
ecto
r;
-G
auss
Jord
anC
AM
atri
x",Y
Vec
tor·
,cN
+1)
,AR
TP
1-,A
inve
rse·
,AM
atD
et);
C
heck
C
onve
rgen
ce;
ReG
uess
SR
TsP1
; en
d;
-
Plo
t_V
ar:=
Plo
t_V
ar+
1;
if P
lot
Var
=P
rint
C
rit
then
beg
in
Gra
pn1_
Res
ults
; P
lot
Var
:=O
; C
onv-
var:
=C
onv
Var
+1;
Cal
e-C
onve
rsio
n;
end;
-
Upd
ate_
AR
T_a
nd_S
RT
s;
Gue
ss
SRTs
P1;
Nor
m1:
=1;
Nor
m2:
=1;
rep
eats
:=re
pea
ts+
1;
end;
read
ln(H
ardC
opy1
);
if
Har
dCop
y1=1
th
en b
egin
Sc
reen
Dum
pC3,
0,2,
1.5
,1.5
,0, 1
,0,e
rro
r);
end;
Cle
arW
indo
w;
Gra
ph2_
Init
iali
se_a
nd_D
raw
;
read
lnC
Har
dCop
y2);
Univers
ity of
Cap
e Tow
n
QJ
"' "-
Vl c.: "-N 0 N _, UJ 0 0 :i::
QJ E
"' c: QJ
.....
CJ
"'
"' °' °' 0 ,...., ,..._ CJ
~
'-0 '-'-QJ
CJ•
CJ•
"' c:~
"'"' QJ .o~
C:N QJ
.t:CJ ... ,....,· 7."0. NE >- :::i a.o 0 c: u QJ -0 QJ '- '-"'u ~(/') ....
-0 -0 .... c: c: ' QJ QJ .....
Univers
ity of
Cap
e Tow
n
Sta
rt.
Se
t/In
itia
lise
V
ari
ab
les.
De
term
ine
mo
de
l p
ara
me
ters
fn
(siz
e).
Ite
rate
in
Tim
e.
Ca
lcu
late
th
e n
ext
Alp
ha
and
S
igm
a V
ecto
rs w
ithin
th
e p
art
icle
.
See
ne
xt f
low
she
et f
or d
eta
ils.
Gra
ph
pro
file
s.
Gra
ph
co
nve
rsio
n v
ers
us
time.
En
d.
Unt
il co
nve
rge
nce
occ
urs
: ~
I G
uess
a b
ulk
flu
id r
ea
ge
nt
con
cen
tra
tio
n.
I I
I Ite
rate
in
siz
e cl
ass
es.
~
Gu
ess
a S
igm
a v
ect
or
for
the
pa
rtic
le.
Ca
lcu
late
th
e A
-Ma
trix
an
d Y
-Vec
tor.
Ca
lcu
late
th
e A
lph
a V
ecto
r.
Ca
lcu
late
th
e b
ulk
flu
id r
ea
ge
nt
con
cen
tra
tio
n. I
Ca
lcu
late
th
e S
igm
a V
ect
ors
usi
ng
th
e c
alc
ula
ted
Alp
ha
Vec
tor.
I
Ite
rate
un
til b
ulk
flu
id r
ea
ge
nt
an
d
all
sig
ma
ve
cto
rs c
on
verg
e.
oo
~
I>
0 0
-0
. c:
C'D
'C".
u;
0 '"1
=
C'D
~
] g -tll 9- C'D
(")
~
0
'""'
3 ~
~
g 0
~
c..
1(!)
'"1
-0
Ul
c:: •
'trj
e.
~
~ >
~
~
00
Q
= Q
-=
""1 ::r
...
...
C'D
Q,.
.....
(") =--~
~ ~ =
~
Q
3 Q
,.
~ ~
~ o.
KS
c,.
~-.
r;; ~
tll
,...
... n
8. 00
Q
s·
• c,
. ("
) ~
::r
.§ &
I~
Vi
""1
Univers
ity of
Cap
e Tow
n
06/1
5/19
95
09:4
8 F
ilen
ame:
M
OD
EL5E
1.PA
S Pa
ge
Prog
ram
Mod
el5E
1;
{Mod
el5E
1.
{ *
Thi
s pr
ogra
m
is a
too
l fo
r *
The
prog
ram
cal
cula
tes
the
and
soli
d r
eact
ants
wit
hin
de
velo
ped
by D
ixon
.
anal
yzin
g TC
LP
dat
a.
} co
nce
ntr
atio
n p
rofi
le o
f fl
uid
}
{ { { { { { { { {
* T
his
prog
ram
onl
y pr
ovid
es a
part
icle
usi
ng
the
equa
tion
s as
) }
for
soli
d r
eact
ant
ord
ers
of
1.
}
*.A
ssum
ptio
ns
in
this
mod
el
incl
ude:
}
-Th
e so
lid
rea
ctan
t d
epo
sits
wit
hin
the
part
icle
}
rese
mbl
e th
ose
on
the
surf
ace.
}
The
dep
osi
ts
wou
ld b
oth
reac
t to
the
sam
e ex
ten
t }
if e
ach
wer
e ex
pose
d to
the
sam
e ac
id c
on
cen
trat
ion
}
for
the
sam
e ti
me.
}
{Cod
ed:
Gra
ham
Dav
ies.
}
{ D
epar
tmen
t of
Che
mic
al
Eng
inee
ring
. }
{ U
nive
rsit
y of
C
ape
Tow
n.
} {
28
Feb
ruar
y 19
95.
} {
29 M
ay
1995
Upd
ated
(G
MO
). }
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
{Dec
lara
tion
s:
}
uses
cr
t,st
dh
dr,
gj,
gra
ph
,wo
rld
dr,
seg
rap
h,i
nte
gra
t,M
2C
1B
ren
tRo
ots
;
cons
t
{***
The
se a
re t
he p
aram
eter
s ap
pli
cab
le t
o t
he
refe
ren
ce s
ize
clas
s.*
**
}
Bet
a1
Kap
paP1
Bet
a2
Kap
paP2
=10
.0;
=0.
12;
-O
O·
:o:o
; V
oida
ge
=0.
01;
{Dim
ensi
onle
ss S
toic
hio
met
ric
rati
o d
efin
ed p
g 11
}
{Rat
io o
f re
atio
n r
ate
of s
oli
d r
eact
ant
resi
din
g
} {w
ithi
n th
e p
art
icle
to
por
ous
dif
fusi
on
of
flu
id
} {i
nto
the
part
icle
. D
efin
ed p
g 11
D
ixon
. }
{Voi
dage
of
the
soli
d p
arti
lces
.(P
oro
sity
) }
{***
The
se a
re p
aram
eter
s w
ith
rep
ect
to t
he C
STR
expe
rim
ent.
****
****
***}
Vol
liq
=0.
159;
{V
olum
e of
L
iqui
d L
ixiv
ian
t.
Cm3)
Tot
VoL
Part
=0.
577;
{T
otal
vo
lum
e o
f th
e so
lid
part
ilces.
M
=2;
{Num
ber
of s
ize c
lass
es.
Ref
Size
CL
=1
; {D
efin
es t
he
refe
ren
ce s
ize
clas
s.
}
} } }
{***
The
se a
re n
umer
ical
m
etho
d pa
ram
eter
s.**
****
****
****
****
****
****
***}
N
=19;
{H
alf
the
num
ber
of
inte
rio
r p
oin
ts.
r=O
an
d r=
R }
{not
in
clud
ed.
} D
el ta
E
=1/C
N+1
); {S
pace
In
crem
ent.
C
alcu
late
d f
rom
1/
(N+
1).
(Sin
ce}
{
R=i
*dE
and
R i
s at
po
int
N+
1.)
}
Del
taT
=
0.00
1;
Clim
e In
crem
ent.
}
Nor
m_C
rit
=1e
·6;
{Con
verg
ence
cri
teri
a b
ased
on
the
norm
of
vec
tor
}
06/1
5/19
95
09:4
8 F
ilen
ame:
M
OD
EL5E
1.PA
S Pa
ge
2
Con
v_C
rit
=1e
·8;
SCon
c To
l =
1e-4
· -
. M
axit
er
=100
;
Iter
atio
ns=
40
0;
Pri
nt
Cri
t=10
; V
iew
S1ze
CL
=2;
{***
}
{Con
verg
ence
cri
teri
a f
or
the
Bre
nt R
outi
ne.
} {D
imen
sion
less
co
nce
ntr
atio
n o
f so
lid
bel
ow
whi
ch
} {
it
is a
ssum
ed
to b
e n
egli
gib
le.
} {M
axim
um
itera
tio
ns
for
the
Bre
nt R
outi
ne.
}
{It
era
tio
ns
in t
ime.
}
{Det
erm
ined
fro
m d
esir
ed P
rin
t D
elta
T/D
elta
T
} {T
he c
on
cen
trat
ion
pro
file
s o
f-th
is s
izecla
ss a
re
} {g
raph
ed
in g
raph
1
. CN
B V
iew
Size
CL
<=
M
.) }
{***
Oth
er
info
rmat
ion
required.*****************************~**********}
Val
idD
ir
=1F
:\T
P6
1;{
Val
id d
irec
tory
fo
r g
rap
hic
s d
riv
ers.
Ord
SP1
Ord
SP2
=1.
0;
=1.
0;
{Rea
ctio
n o
rder
of
the
soli
d
in t
he
po
res.
{F
or
this
pro
gram
th
ese
need
to
be
set
at u
nit
y.
{See
Mod
el5D
2 fo
r v
aria
ble
ord
er r
eact
ion
ord
ers.
} } } }
type
var ar
ray1
= A
rray
[1 .
. 50,
1 ..
2]
of D
oubl
e;
Arb
ind
ex,S
izel
nd
ex,e
rro
r R
epea
ts,P
lot_
Var
,Con
v_V
ar,F
lag1
,Har
dCop
y C
onv1
,Con
v2,D
ataS
etX
,Dat
aSet
Y
: In
teg
er;
Bet
a1
k,B
eta2
k,K
appa
P1
k,K
appa
P2
k,K
appa
S1
k,K
appa
S2
k La
mbd
a1
k,L
amO
da2_
k,C
ontR
atio
V1,
Con
tRat
ioV
2--
: In
teg
er;
:"V
eryL
ongV
ecto
r;
:"S
ho
rtV
ecto
r;
Del
taT
K
A
RT
,AR
fP1,
SRT
1,SR
T2,
SRT
1P1,
SRT
2P1,
SRT
1P1C
,SR
T2P
1C
AM
atri
x,A
lnve
rse
AR
TP1
CV
al,S
um5i
zeD
ata_
2,N
uSta
r L
ambd
a1,L
ambd
a2,K
appa
S1,K
appa
S2
GL
abel
1,G
labe
l2
Siz
eDat
a :
Arr
ay[1
. .5
0, 1
. .2]
o
f R
eal;
CART
A
lpha
fn
(r)
at
tim
e T
{A
RTP1
A
lpha
fn
(r)
at
tim
e T+
1 {S
RT1
Sigm
a fn
(r)
at
tim
e T
{S
RTs
P1
Sigm
a fn
(r)
at
tim
e T+
1 (g
uess
ed)
: "S
hort
Vec
tor;
:"
Arr
ay1;
:·
sqrM
at;
:·sq
rMat
; :E
xten
ded;
:E
xten
ded;
:S
trin
g;
rang
e O
.. M
0
.• N
+1}
rang
e O
•. M
0 ..
N+1
} ra
nge
O ••
M
0 •.
N+1
} ra
nge
O ••
M
0 ..
N+1
} {S
RTs
P1C
Si
gma
fn(r
) at
ti
me
T+1
(cal
cula
ted
) ra
nge
O .. M
0 ..
N+1
} C
YV
ect
'Co
nst
' v
ecto
r in
Cra
nk-N
icol
son
met
hod
rang
e O
•• N
+1
} {A
Mat
rix
Mat
rix
of C
rank
-Nic
olso
n co
effi
cien
ts
rang
e N*
N
}
{Oat
aSet
X
X v
ecto
r us
ed
in t
he
grap
hing
ro
uti
ne
rang
e O
•• N
+1
}
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re S
ize_
Dis
trib
uti
on
_In
itia
lisa
tio
n;
{Thi
s pr
oced
ure
sets
up
the
size
dis
trib
uti
on
dat
a ar
ray
. In
itia
lly
th
e }
{Siz
eDat
a ar
ray
co
nta
ins
rad
ius
info
rmat
ion
and
fr
acti
on
al
volu
me
} {i
nfor
mat
ion
Cie
R a
nd V
p/V
tot)
. O
n o
utp
ut
it c
on
tain
s re
lati
ve r
adiu
s }
{inf
orm
atio
n an
d re
lati
ve v
olum
e in
form
atio
n (
ie R
/R
ref
and
Vp/
Vp
ref)
. }
{Id
eall
y t
his
in
form
atio
n w
ould
be
re
ad
in f
rom
a
dat
a fi
le.
-}
var R
efR
adiu
s,V
oLR
efP
art
:Dou
ble;
(1
0 Q.. ~
t""' -· C
l) ~ -· = O' ..,. ~
0 Q..
. ~ Ul" ~ ~ . ~
00
.
Univers
ity of
Cap
e Tow
n
06/1
5/19
95
09:4
8 Fi
lena
me:
M
ODEL
5E1.P
AS
Page
3
begi
n Siz
eDat
a[1,
1J
=11
.35e
-3;
S ze
Dat
a [1
, 21
= 1
/100
Si
zeD
ata
[2, 1
l =
11.3
5e-3
; S
zeD
ata
[2, 2
1 =9
9/10
0 Si
zeD
ata
[3, 1
1 =
O.O
e-3;
S
zeD
ata
[3, 2
1 =
0/1
00
Si z
eDat
a [4
, 11
= O
.Oe-
3;
S ze
Dat
a[4,
2l
= 0
/100
S
izeD
ata[
5, 11
=
O.O
e-3;
S
zeD
ata
[5, 2
1 =
0/1
00
Siz
eDat
a[6,
1l
= O
.Oe-
3;
s ze
Dat
a[6,
2l
= 0
/100
Si
zeD
ata
[7, 1
l =
O.O
e-3;
s
zeD
ata[
7,2l
=
0/1
00
Siz
eDat
a[8,
1l
= O
.Oe-
3;
s ze
Dat
a [8
, 21
= 0
/100
S
zeD
ata
[9, 2
1 =
0/1
00
Siz
eDat
a[9,
1l
= O
.Oe-
3;
Siz
eDat
a[10
, 11
= O
.Oe-
3;
S ze
Dat
a[10
,2l
= 0
/100
Vol
Ref
Par
t:=
Tot
Vol
Par
t*S
izeD
ata[
Ref
Siz
eCl,
21;
Ref
Rad
ius
:=S
izeD
ata[
Ref
Siz
eCl,
11;
Sum
Size
Dat
a_2:
=0;
for
Siz
elnd
ex:=
1 to
M do
be
gin Siz
eDat
a[S
izel
ndex
, 11:
=S
izeD
ata[
Siz
elnd
ex,1
1/R
efR
adiu
s;
Siz
eDat
a[S
izel
ndex
,2l:
=S
izeD
ata[
Siz
elnd
ex,2
l*T
otV
olP
art/
Vol
Ref
Par
t;
Sum
Siz
eDat
a_2:
=S
umS
izeD
ata_
2+S
izeD
ata[
Siz
elnd
ex,2
l;
end;
·
NuS
tar:
=V
olL
•qtC
Voi
dage
*Vol
Ref
Part
);
{Rat
io o
f vo
lum
e of
bu
lk f
luid
to
flu
id
in p
arti
cle
pore
s.
}
end;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re C
onta
min
ant_
Loc
atio
n_ln
itia
lisa
tion
;
{Thi
s pr
oced
ure
is u
sed
to d
efin
e th
e co
ntam
inan
t ra
tio
vec
tor
(rati
o o
f }
{th
e su
rfac
e co
ntam
inan
t co
ncen
trat
ion
to b
ulk
cont
amin
ant
con
cen
trat
ion
).}
{NO
TE:
If a
ny s
ize
clas
s of
p
arti
cles
hav
e a
surf
ace
conc
entr
atio
n of
}
{ co
ntam
inan
t,
then
so
too
mus
t th
e re
fere
nce
size
cla
ss.
}
begi
n Con
tRat
oV
1"[1
l =O
: C
ontR
at
oV1
-C2l
=O;
Con
tRat
oV
1 -[
3l
=O;
Con
tRat
oV
1"[4
l =O
; C
ontR
at
oV1"
[5l
=O;
Con
tRat
oV
1 -[
6l
=O;
Con
tRat
oV
1 -[
7l
=O;
Con
tRat
oV
1 -[
8l
=O;
Con
tRat
oV
1 -[
9l
=O;
Con
tRat
oV
1"[1
0l
=O;
end;
Con
tRat
ov
2" [
1l
=O
Con
tRat
ov
2"[2
l =O
C
ontR
at
ov2"
[3l
=O
Con
tRat
ov
2"[4
l =O
C
ontR
at
ov2"
[5l·
=O
C
ontR
at
ov2"
[6l
=O
Con
tRat
ov
2"[7
l =O
C
ontR
at
ov2"
[8l
=O
Con
tRat
ov
2"[9
l =O
C
ontR
at
ov2"
[10l
=O
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re D
eter
min
e_M
odel
_Par
amet
ers_
fn_S
ize;
{Thi
s pr
oced
ure
dete
rmin
es
the
mod
el
para
met
ers
each
siz
e cl
as.s
of
{p
arti
cles
. } }
06/1
5/19
95
09:4
8 F
ilen
ame:
M
OD
EL5E
1.PA
S Pa
ge
4
var Fl
ag1,
Flag
.2
:In
teg
er;
begi
n F
lag1
:=0;
F
lag2
:=0;
fo
r S
izel
ndex
:=1
to M
do
begi
n Lam
bda1
k.
[S
izel
ndex
l :=
Con
tRat
ioV
1"[S
izel
ndex
l -
/(1+
Con
tRat
ioV
1"[S
izel
ndex
l >;
La
mbd
a2
k"[S
izel
ndex
l:=
Con
tRat
ioV
2"[S
izel
ndex
l -
/(1+
Con
tRat
ioV
2"[S
izel
ndex
l);
f C
ontR
atio
V1"
[Siz
elnd
exl<
>O
th
en F
lag1
:=F
lag1
+1;
f
Con
tRat
ioV
2"[S
izel
ndex
l<>
O
then
F
lag2
:=F
lag2
+1;
en
d
if
CC
CFl
ag1=
1>
and
(Con
tRat
ioV
1"[R
efS
izeC
ll=
0))
or
((F
lag2
=1)
an
d (C
ontR
atio
v2"[
Ref
Siz
eCll
=0)
))
then
beg
in
clo
sese
gra
ph
ics;
w
rite
lnC
'Con
tam
inan
t L
ocat
ion
Vio
loat
ion
.');
re
adln
; en
d·
Lam
bda1
=L
ambd
a1
k"[R
efS
izeC
ll;
Lam
bda2
=
Lam
bda2
-k"[
Ref
Siz
eCll
; K
appa
S1
=Lam
bda1
*Kap
paP1
/(1-L
ambd
a1);
Kap
paS2
=L
ambd
a2*K
appa
P2/(
1-L
ambd
a2);
for
Siz
elnd
ex:=
1 to
M do
be
gin Del
taT
_k-[
Siz
elnd
ex,1
1 :=
Del
taT
/Siz
eDat
a[S
izel
ndex
,11
/Siz
eDat
a[S
izel
ndex
,11;
B
eta1
k"
[Siz
elnd
exl
:=B
eta1
*(1+
Con
tRat
ioV
1"[R
efS
izeC
ll)
-/(
1+C
ontR
atio
V1"
[Siz
elnd
exl)
; B
eta2
k"
[Siz
elnd
exl
:=B
eta2
*(1+
Con
tRat
ioV
2"[R
efS
izeC
ll)
-/(
1+C
ontR
atio
V2"
[Siz
elnd
exl)
; K
appa
P1
k"[S
izel
ndex
l :=
Kap
paP
1*S
izeD
ata[
Siz
elnd
ex,1
l -
*Siz
eDat
a[S
izel
ndex
, 11;
K
appa
P2
k" [
Siz
elnd
exl
:=K
appa
P2*
Siz
eDat
a[S
izel
ndex
, 11
-*S
izeD
ata[
Siz
elnd
ex,1
1;
{***
**N
ote:
Uni
ty p
ower
as
sum
ptio
n in
volv
ed i
n ne
xt
few
li
nes*
****
}
if C
ontR
atio
V1"
[Siz
elnd
exl=
O
then
Kap
paS1
k"
[Siz
elnd
exl
:=O
el
se
Kap
paS1
k"
[Siz
elnd
exl:
=K
appa
S1*
Con
tRat
ToV
1"[S
izel
ndex
l -
/Con
tRat
ioV
1"[R
efS
izeC
ll*S
izeD
ata[
Siz
elnd
ex, 1
1;
if C
ontR
atio
V2"
[Siz
elnd
exl=
O
then
Kap
paS2
k"
[Siz
elnd
exl
:=O
el
se
Kap
paS2
k"
[Siz
elnd
exl:
=K
appa
S2*
Con
tRat
ToV
2"[S
izel
ndex
l -
/Con
tRat
ioV
2"[R
efS
izeC
ll*S
izeD
ata[
Siz
elnd
ex, 1
1;
end·
en
d·
' {--~----------------------
------------------------------------------------}
Pro
cedu
re T
imeO
_AR
T_an
d_SR
Ts;
{Thi
s pr
oced
ure
uses
th
e in
itia
l co
ndit
ions
to
set
the
alp
ha a
nd s
igm
a }
{vec
tors
. }
{Not
e th
at A
RT[
N+1
l, A
RTP
1[N
+1l,
SRT1
[N
+1l
and
SRT1
P1[N
+1l
are
the
surf
ace}
{c
once
ntra
tion
s of
th
e li
qu
id a
nd s
oli
d r
eact
ants
. }
begi
n for
Siz
elnd
ex:=
1 to
M do
Univers
ity of
Cap
e Tow
n
06/1
5/19
95
09:4
8 Fi
lena
me:
M
ODEL
5E1.P
AS
Page
5
begi
n for
Arb
lnde
x:=O
to
N d
o be
gin AR
T"[
Size
lnde
x,A
rbln
dex]
=O
S
RT
1"[S
izel
ndex
,Arb
lnde
xl
=1
SR
T2"
[Siz
elnd
ex,A
rbln
dex]
=1
en
d·
AR
Tt[
Size
lnde
x,N
+1l
=1
SRT
1"[S
izei
ndex
,N+1
l =1
SR
T2"
CSi
zeln
dex,
N+1
J =1
en
d·
for'
arb
lnde
x:=
O
to m
axv
do
begi
n C
onv1
"[A
rbln
dexJ
:=O
; C
onv2
"[A
rbln
dexl
:=O
; en
d·
end;
'
{---
----
----
----
----
----
----
·---
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re G
uess
_SR
TsP1
;
{Thi
s pr
oced
ure
prov
ides
the
in
itia
l "g
uess
" fo
r th
e it
erat
ion
. It
use
s }
{the
pre
viou
s ti
me
inte
rval
's v
alue
s as
the
gue
ss.
}
begi
n for
Siz
eind
ex:=
1 to
M do
be
gin
' fo
r A
rbin
dex:
=O
to C
N+1
) do
be
gin SR
T1P
1"[S
izei
ndex
,Arb
lnde
xl:=
SR
T1"
CS
izel
ndex
,Arb
lnde
xl;
SR
T2P
1"[S
izel
ndex
,Arb
lnde
xJ:=
SR
T2"
[Siz
eind
ex,A
rbln
dexJ
; en
d·
ARTP
1 · C
Size
inde
x,N
+1]
:=A
RT"
CSi
zeir
dex,
N+1
J ;.
end·
en
d;
'
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re R
eGue
ss_S
RTs
P1;
{Thi
s pr
oced
ure
prov
ides
an
upda
ted
"gue
ss"
for
the
next
it
erat
ion
. It
{u
ses
the
SRT1
P1C
vect
or a
s th
e up
date
d gu
ess.
begi
n for
Siz
eind
ex:=
1 to
M do
be
gin fo
r A
rbln
dex:
=O
to C
N+1
) do
be
gin SR
T1P
1"[S
izei
ndex
,Arb
lnde
xJ :
=SR
T1P
1C"[
Size
lnde
x,A
rbln
dexJ
; SR
T2P
1"[S
izel
ndex
,Arb
lnde
x] :
=SR
T2P
1C"[
Size
inde
x,A
rbln
dexJ
; en
d·
AR
TP1
"[Si
zeln
dex,
N+1
J:=A
RT
P1C
Val
; en
d;
end;
} }
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re C
alc_
Cra
nk_N
icol
son_
Mat
rix(
var
YV
ecto
r :S
hort
Vec
tor)
;
{***
**N
ote:
Uni
,ty p
ower
as
sum
ptio
n in
volv
ed i
n th
is p
roce
dure
****
*
var R
ows,
Col
s : I
nteg
er;
}
06/1
5/19
95
09:4
8 F
ilen
ame:
M
OD
EL5E
1.PA
S Pa
ge
6
begi
n for
Row
s:=O
to
(N
+1)
do
begi
n for
Col
s:=O
to
{N
+1)
do A
Mat
rix"
[Row
s,C
ols]
:=O
; en
d;
AM
atri
x"[O
,OJ:
=C
-6-D
elta
E*D
elta
E*S
izeD
ata[
Siz
elnd
ex,1
J*S
izeD
ata[
Siz
elnd
ex,1
J *(
Kap
paP1
k"
[Siz
elnd
exJ*
SR
T1P
1"C
Siz
eind
ex,O
J +K
appa
P2 k
"[S
izei
ndex
J*S
RT
2P1"
[Siz
eind
ex,0
J)-2
*Del
taE
*Del
taE
*S
izeD
ataC
Siz
eind
ex, 1
J*S
izeD
ata[
Siz
elnd
ex,1
J /O
elta
T
k"[
Siz
ein
dex
,1J)
; A
Mat
rix·
CO,
1]
:=6;
-
YV
ecto
r[O
J :=
AR
T"[
Siz
elnd
ex,0
]*(6
+D
elta
E*D
elta
E*S
izeO
ata[
Siz
elnd
ex, 1
] *S
izeD
ata[
Siz
elnd
ex, 1
l*C
Kap
paP
1_k.
[Siz
eind
ex]
*SR
T1"
[Siz
elnd
ex,O
J+K
appa
P2
k"[S
izei
ndex
J *S
RT
2"[S
izel
ndex
,0])
-2*D
elta
E*O
elta
E*S
izeD
ata[
Siz
eind
ex, 1
1 *S
izeD
ata[
Siz
elnd
ex,1
1/D
elta
T k
"[S
izei
ndex
, 1])
-A
RT
"[S
izei
ndex
,1]*
6;
-
for
Row
s:=1
to N
-1
do
begi
n AM
atri
x"[R
ows,
Row
s·1J
:=R
ows·
1;
AM
atri
x"[R
ows,
Row
s]
:=-2
*Row
s-R
ows*
Del
taE
*Del
taE
*Siz
eDat
a[S
izel
ndex
,1]
*Siz
eDat
a[S
izei
ndex
,1l*
CK
appa
P1_
k·cs
izei
ndex
]*
SRT
1P1"
CSi
zeln
dex,
Row
s]+K
appa
P2
k"[S
izei
ndex
]*
SR
T2P
1"[S
izel
ndex
,Row
s])-
2*R
ows•
oelt
aE
*Del
taE
*Siz
eDat
a[S
izei
ndex
,1]
*Siz
eDat
aCS
izei
ndex
,1l/
Del
taT
_k·c
size
inde
x, 1
1;
AM
atri
x"[R
ows,
Row
s+1]
:=R
ows+
1;
YV
ecto
r[R
owsl
:=
AR
T"[
Size
inde
x,R
ows-
1J*(
-Row
s+1)
+
AR
T"[
Size
inde
x,R
owsl
*C2*
Row
s+R
ows*
Del
taE
*Oel
taE
*S
izeD
ata[
Siz
eind
ex,1
J*S
izeD
ata[
Siz
eind
ex,1
l *C
Kap
paP1
k"
[Siz
eind
exJ*
SR
T1"
CS
izei
ndex
,Row
sJ
+Kap
paP2
k"[
Siz
elri
dexJ
*SR
T2"
[Siz
elnd
ex,R
owsl
) -2
*Row
s*D
elta
E*D
elta
E*S
izeD
ata[
Siz
eind
ex,1
l *S
izeD
ataC
Siz
elnd
ex,1
1/D
elta
T_k
·csi
zeln
dex,
1])
+AR
T"[
Size
lnde
x,R
ows+
1l*C
-Row
s-1)
; en
d;
AM
atri
x · C
N, N
-1 l :
=N-1
; A
Mat
rix"
[N,N
l :=
-2*N
-N*D
elta
E*D
elta
E*S
izeD
ata[
Siz
eind
ex,1
l *S
izeD
ata[
Siz
elnd
ex,1
l*C
Kap
paP
1 k"
CS
izei
ndex
l *S
RT
1P1"
[Siz
eind
ex,N
J+K
appa
P2 k
~[Sizeindexl
*SR
T2P
1"[S
izel
ndex
,Nl)
-2*N
*DeT
taE
*Del
taE
*S
izeD
ataC
Siz
eind
ex, 1
l*S
izeD
ata[
Siz
eind
ex, 1
J .
/Del
taT
k
"[S
izel
nd
ex,1
l;
YV
ecto
rCN
l :=
AR
T"[
Size
lnde
x,N
-1l*
C-N
+1)+
AR
T"[
Size
inde
x,N
l*C
2*N
+ N
*Del
taE
*Del
taE
*Siz
eDat
a[S
izei
ndex
,1]
*Siz
eDat
a[S
izei
ndex
,1l*
CK
appa
P1
k"[S
izel
ndex
l *S
RT
1"[S
izei
ndex
,NJ+
Kap
paP2
k"[
Siz
eind
ex]
*SR
T2"
[Siz
elnd
ex,N
J)-2
*N*D
eTta
E*D
elta
E
*Siz
eDat
a[S
izei
ndex
,1J*
Siz
eDat
a[S
izei
ndex
,1J
/Del
taT
k"
CS
izei
ndex
,1l)
+A
RT
"[S
izei
ndex
,N+
1l*C
-N-1
) -A
RT
P1"1
Size
lnde
x,N
+1J*
{N+1
);
end;
{·--
----
----
----
----
----
----
----
----
----
----
----
-···
··-·
----
----
----
----
--}
Pro
cedu
re C
alc_
AR
TP1_
and_
SRTs
P1C
;
Univers
ity of
Cap
e Tow
n
06/1
5/19
95
09:4
8 Fi
lena
me:
M
ODEL
5E1.P
AS
Page
7
{As
it s
tand
s th
is p
roce
dure
can
cop
e w
ith
a v
aria
ble
rea
ctio
n o
rder
. T
his}
{
is d
ue
to t
he
incl
usio
n of
th
e B
rent
Rou
tine
. }
var M
assB
alV
al,A
Mat
Det
V
alue
AtR
oot
Out
putV
ecto
r,Y
Vec
tor
Dou
ble;
R
eal;
S
hort
Vec
tor;
begi
n {F
irst
cal
cula
te t
he A
RTP1
v
ecto
rs.
for
Siz
elnd
ex:=
1 to
M do
be
gin Cal
e C
rank
N
icol
son
Mat
rixC
YV
ecto
r);
GaussJordanCAMatrix~,YVector,CN+1),0utputVector,Alnverse·,AMatDet);
}
for
Arb
lnde
x:=O
to
N d
o A
RT
P1"[
Size
lnde
x,A
rbln
dex]
:=
Out
putV
ecto
r[A
rbln
dexJ
;
end;
{The
Gau
ssJo
rdan
pro
cedu
re c
alcu
late
s va
lues
for
th
e AR
T ve
ctor
fr
om
{poi
nt 0
to
poin
t N
(al
thou
gh it
mak
es
use
of.
the
N+1
th p
oin
t).
To
{det
erm
ine
the
N+1
th p
oint
val
ue,
mak
e us
e of
th
e m
ass
bala
nce
of
{fl
uid
rea
ctan
t in
-th
e CS
TR.
Mas
sBal
Val
:=O
;
for
Siz
elnd
ex:=
1 to
M do
be
gin
} } } )
Mas
sBal
Val
:=M
assB
alV
al-S
izeD
ata[
Size
lnde
x,2J
/2*C
Kap
paS1
k"
[Siz
elnd
ex]
*SR
T1"
[Siz
elnd
ex,(
N+
1)J*
AR
T'[S
izel
ndex
,(N
+1)
J
end;
+Kap
paS1
k"
[Siz
elnd
exJ*
SR
T1P
1"[S
izel
ndex
,CN
+1)
l *A
RT
P1"[
Size
lnde
x,(N
+1)J
+Kap
paS2
k"[
Siz
elnd
exJ
*SR
T2"
[Siz
elnd
ex,(
N+
1)J*
AR
T"[
Siz
elnd
ex,(
N+
1)l
+K
appa
S2_k
"[Si
zeln
dexJ
*SR
T2P
1"[S
izel
ndex
,(N
+1)
l *A
RT
P1"
[Siz
elnd
ex,(
N+
1)J)
-3*S
izeD
ata[
Siz
elnd
ex,2
l /C
Siz
eDat
a[S
i.:e
lnde
x, 1
J*S
izeD
ata[
Siz
elnd
ex,1
J)/2
*C
CA
RT
"[Si
zeln
dex,
(N+1
)J+A
RT
P1"[
Size
lnde
x,C
N+1
)J)
·CA
RT
"[S
izel
ndex
,NJ+
AR
TP
1"[S
izel
ndex
,NJ)
)/D
elta
E;
AR
TP1
CV
al:=
AR
T"[
1,(N
+1)J
+Del
taT
/NuS
tar*
Mas
sBal
Val
;
{Cal
cula
te t
he S
RTsP
1C
vec
tors
. C
ode
mak
es
use
of
Bre
nt'
s M
etho
d }
CCA
non-
line
ar
root
fi
ndin
g pr
oced
ure)
to
sol
ve f
or
the
roo
t.
)
for
Siz
elnd
ex:=
1 to
M do
be
gin fo
r A
rbln
dex:
=O
to.
CN+1
) do
be
gin
if (
Bet
a1_k
"[S
izel
ndex
l<>
O)
and
(Kap
paP1
_k"[
Size
lnde
xJ<
>0)
th
en
begi
n if O
rdSP
1=1
then
be
gin SR
T1P
1C"[
Siz
elnd
ex,A
rbln
dexJ
:=S
RT
1"[S
izel
ndex
,Arb
lnde
xJ
*(1-
Del
taT
k"[
Siz
elnd
ex,1
J*K
appa
P1
k"[S
izel
ndex
l *B
eta1
k"
[Siz
elnd
exJ*
AR
T"[
Siz
elnd
ex,A
rbln
dexl
/C
2*C
FLam
bda1
k"
[S
izel
ndex
l ))
)/(1
+D
el ta
T
k" [
Siz
elnd
ex, 1
] *K
appa
P1_
k"[S
izel
ndex
l*B
eta1
_k· [
Siz
elnd
exT
.
06/1
5/19
95
09:4
8 F
ilen
ame:
M
OD
EL5E
1.PA
S Pa
ge
8
ex, 1
J,
*AR
TP
1"[S
izel
ndex
,Arb
lnde
xl/C
2*C
1-L
ambd
a1_k
.[S
izel
ndex
J)))
end
else
if
SR
T1P
1"[S
izel
ndex
,Arb
lnde
xJ<
SCon
c To
l th
en
SR
T1P
1C"(
Siz
elnd
ex,A
rbln
dexJ
:=S
RT
1P1"
CS
izel
ndex
,Arb
lnde
xJ
else
be
gin SR
T1P
1C"C
Siz
elnd
ex,A
rbln
dexJ
:=B
rent
Roo
ts(0
.0,1
.0,D
elta
T_k
-CS
izel
nd
Kap
paP1
k"
CS
izel
ndex
J,B
eta1
k"
[Siz
elnd
ex],
L
ambd
a1-k
"(S
izel
ndex
J,O
rdS
PT
,AR
T"[
Siz
elnd
ex,A
rbln
dex]
, A
RT
P1"C
Size
lnde
x,A
rbln
dexl
,SR
T1"
[S
izel
ndex
,Arb
lnde
x],
1e·8
, 100
,Val
ueA
tRoo
t,er
ror)
; en
d·
end·
'
if
CB
eta2
k"
[Siz
elnd
exJ<
>O
) an
d (K
appa
P2
k"[S
izel
ndex
]<>
O)
then
be
gin
--
if O
rdSP
2=1
then
be
gin SR
T2P
1C"[
Siz
elnd
ex,A
rbln
dexJ
:=S
RT
2"[S
izel
ndex
,Arb
lnde
xl
end
*(1-
Del
taT
k"
CS
izel
ndex
, 1l*
Kap
paP2
k"
[S
izel
ndex
] *B
eta2
k"
Csi
zeln
dexl
*AR
T"[
Siz
elnd
ex,A
rbln
dexl
/(2*C1~Lambda2
k"[S
izel
ndex
])))
/(1+
Del
taT
k"
[Siz
elnd
ex, 1
1 *K
appa
P2
k"C
Siz
elnd
exl*
Bet
a2_k
·csi
zeln
dex]
*A
RT
P1"
Csi
zeln
dex,
Arb
lnde
xl/C
2*C
1-L
ambd
a2_k
·csi
zeln
dex]
)))
else
if
SRT
2P1"
CSi
zeln
dex,
Arb
lnde
xJ<S
Con
c To
l th
en
SR
T1P
1C"[
Siz
elnd
ex,A
rbln
dexl
:=S
RT
2P1"
[Siz
elnd
ex,A
rbln
dexJ
el
se
begi
n SR
T2P
1C"[
Siz
elnd
ex,A
rbln
dexJ
:=B
rent
Roo
ts(0
.0, 1
.0,
Del
taT
k"
CS
izel
ndex
,1l,
Kap
paP
2 k"
[Siz
elnd
ex],
B
eta2
K0
[Siz
elnd
ex],
Lam
bda2
_k"l
Siz
elnd
ex],
Ord
SP
2,
AR
T"[
Siz
elnd
ex,A
rbln
dexJ
,AR
TP
1"[S
izel
ndex
,Arb
lnde
x],
SR
T2"
CS
izel
ndex
,Arb
lnde
xJ,1
e·8,
100,
Val
ueA
tRoo
t,er
ror)
; en
d·
end·
'
end·
'
end·
'
end;
'
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re C
heck
_Con
verg
ence
;
{Thi
s pr
oced
ure
chec
ks w
heth
er o
r no
t th
e so
luti
on
has
co
nver
ged
by
} {c
ompa
ring
th
e gu
esse
d va
lue
of S
RT1P
1 w
ith
a ca
lcu
late
d v
alue
of
SRT1
P1.
} {A
lso
com
pare
s th
e ca
lcu
late
d v
alue
of
ARTP
1C
wit
h gu
esse
d va
lue
of
} {A
RTP
1.
)
var N
orm
1,N
orm
2,N
orm
3
begi
n Nor
m1
=O;
{Usi
ng a
nor
m *
**
Nor
m2
=O;
Nor
m3
=O;
for
S ze
lnde
x:=
1 to
M do
be
gin
:Dou
ble;
}
Univers
ity of
Cap
e Tow
n
_,.
06/1
5/19
95
09:4
8 Fi
lena
me:
M
OO
EL5E
1.PA
S Pa
ge
9
for
Arb
lnde
x:=O
to
(N
+1)
do
begi
n·
if (
Bet
a1_k
.[S
izel
ndex
l<>
0) a
nd
(Kap
paP
l_k.
[Siz
elnd
exJ<
>O
) th
en b
egin
N
orm
l:=N
orm
l+(S
RT
1Plc
· [S
izel
ndex
,Arb
lnde
xl
·SR
T1P
l"[S
izel
ndex
,Arb
lnde
xJ)*
(SR
T1P
1C. [
Siz
elnd
ex,A
rbin
dexJ
-S
RT1
P1. [
Siz
elnd
ex,A
rbln
dexl
);
end
else
Nor
ml:=
O;
if (
Bet
a2
k"[S
izel
ndex
]<>
O)
and
(Kap
paP2
k"
[Siz
eind
ex]<
>O
) th
en b
egin
N
orm
2:=N
orm
2+(S
RT
2Plc
·csi
zeln
dex,
Arb
lnde
xJ
-SR
T2P1
" [S
izel
ndex
,Arb
lnde
xl )
*(SR
T1P
lc·
[Siz
elnd
ex,A
rbln
dexl
-S
RT2
P1. [
Siz
elnd
ex,A
rbln
dexl
>;
end
else
Nor
m2:
=0;
end·
en
d·
' Nor~
:=(A
RTP1
CVal
-ART
P1.
[1,N
+ll
)*(A
RT
PlC
Val
-AR
TPl
. [1,
N+
ll )
; if
(N
orm
l<N
orm
Cri
t) a
nd
(Nor
m2<
Nor
m C
rit)
and
(N
orm
3<N
orm
Cri
t)
then
Fla
gl:
=l;
-
-en
d;
{---
----
----
----
----
-:·-
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re U
pdat
e_A
RT_
and_
SRTs
;
{Thi
s pr
oced
ure
upda
tes
the
alph
a an
d si
gma
vect
ors
for
the
next
it
erat
ion
}
{by
repl
acin
g th
eir
com
pone
nts
with
th
e al
phaT
+l
and
sigm
aT+l
v
ecto
rs.
}
begi
n for
Siz
elnd
ex:=
l to
M do
be
gin fo
r A
rbln
dex:
=O
to (
N+l
) do
be
gin ART"[Sizelndex,Arblnd~x]
=A
RT
Pl"
[Siz
elnd
ex,A
rbin
dexl
; S
RT
l"[S
izel
ndex
,Arb
lnde
xl
=S
RT
1P1"
[Siz
elnd
ex,A
rbln
dexJ
; S
RT
2"[S
izel
ndex
,Arb
lnde
xl
=S
RT
2P1"
[Siz
elnd
ex,A
rbln
dexJ
; en
d·
end·
'
end;
'
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re C
alc_
Con
vers
ion;
{Thi
s pr
oced
ure ~a
lculat
es
the
frac
tio
nal
co
nver
sion
of
the
par
ticl
e fo
r }
{th
e ti
me
inte
rval
. It
use
s th
e fo
rmul
a on
pg.
13
Oix
on.
The
inte
gra
tor}
{
is t
he Q
uinn
-Cur
tis
vect
or
inte
gra
tor.
}
var In
terG
1 ln
terG
2 ln
tegv
al,c
onve
rsio
n :V
eryL
ongV
ecto
r;
:Rea
l;
begi
n for
Siz
elnd
ex:=
1 to
M do
be
gin if K
appa
P1
k"[S
izel
ndex
J<>
O
then
be
gin
-·
for
Arb
!nde
x:=O
to
(N
+1)
do
lnte
rG1[
Arb
lnde
xJ:=
(1-S
RT
1"[S
izel
ndex
,Arb
lnde
xJ)*
Arb
lnde
x*O
elta
E
*Arb
lnde
x*D
elta
E;
lnte
grat
eVec
tor(
lnte
rG1,
Del
taE
,0,(
N+
1),l
nteg
Val
);
06/1
5/19
95
09:4
8 fi
lena
me:
M
OO
EL5E
1.PA
S Pa
ge
10
Con
vers
ion
:=3*
(1-L
ambd
a1
k"[S
izel
ndex
l)*l
nteg
Val
+L
ambd
a1
k"[S
izel
ndex
J*(1
-SR
T1"
[Siz
elnd
ex,N
+1l
>;
Con
v1"[
Con
v V
ar]
:=C
onv1
·cco
nv V
ar]+
Con
vers
ion*
Siz
e0at
a[S
izel
ndex
,2J
-/S
umSi
zeO
ata_
2;
end;
if K
appa
P2_k
·csi
zeln
dexJ
<>
O
then
be
gin fo
r A
rbln
dex:
=O
to
(N+1
) do
Jn
terG
2[A
rbln
dexJ
:=(1
-SR
T2"
[Siz
elnd
ex,A
rbln
dexJ
)*A
rbln
dex*
Oel
taE
*A
rbln
dex*
Oel
taE
; ln
teg
rate
Vec
tor(
lnte
rG2
,Del
taE
,0,(
N+
1),
lnte
gV
al);
C
onve
rsio
n :=3*C1-Lambda2_k.[SizelndexJ)~lntegVal
+L
ambd
a2_k
.[S
izel
ndex
J*(1
-SR
T2"
[Siz
elnd
ex,N
+ll
);
Con
v2·c
conv
Var
) :=
Con
v2"[
Con
v V
arJ+
Con
vers
ion*
Siz
eOat
a[S
izel
ndex
,2J
-/S
umSi
zeO
ata_
2;
end·
en
d·
' en
d;
'
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re G
raph
1_1n
itia
lise
;
var B
eta1
S,K
appa
P1S,
Lam
bda1
S,G
OT
S B
eta2
S,K
appa
P2S,
Lam
bda2
S .
Vie
wSi
zeC
lS
stri
ng
st
rin
g
stri
ng
begi
n init
SE
Gra
phic
s(V
alid
Oir
);
Set
Cur
rent
Win
dow
(2);
B
orde
rCur
rent
Win
dow
(1);
S
etA
xesT
ype(
0,0)
; S
cale
Plo
tAre
a(0
.0,0
.0,1
.0,1
.0);
S
etX
Yln
terc
epts
(0.0
,0.0
);
SetC
olor
C2>
; D
raw
XA
xisC
0.2,
1>;
Dra
wY
Axi
s(0.
2,1)
; L
abel
XA
xisC
1,0)
; L
abel
YA
xis(
1,0)
; T
itle
XA
xis(
'Dim
ensi
onle
ss R
adiu
s');
T
itle
YA
xis(
'Dim
ensi
onle
ss C
on
cen
trat
ion
');
Tit
leW
indo
wC
'Mod
el5E
1');
S
tr(B
eta1
k"
[Vie
wS
izeC
l]:6
:3,B
eta1
S);
Str(Kappa~1
k"[V
iew
Size
Cll
:6:
3,K
appa
P1S
);
Str
((D
elta
r•P
rin
t C
rit)
:6:4
,GD
TS
);
Str(
Lam
bda1
k"
[Vie
wSi
zeC
lJ :
5:3,
Lam
bda1
S);
S
tr(B
eta2
k~[ViewSizeClJ:6:3,Beta2S>;
Str(Kappa~2
k"[V
iew
Siz
eCll
:6:
3,K
appa
P2S
);
Str
(Lam
bda2
=k·
cvie
wS
izeC
ll :
5:3,
Lam
bda2
S);
S
trC
Vie
wS
izeC
l:2,
Vie
wS
izeC
lS);
G
labe
l1:=
Con
cat(
' B
eta1
',
Bet
a1S
,';
Kap
pa1
',Kap
paP
1S,';
La
mbd
a1
Lam
bda1
S,';
GOT
wrt
R
efS
izeC
lass
',G
DT
S);
GL
abel
2:=
Con
cat(
' B
eta2
',
Bet
a2S
,';
Kap
pa2
',Kap
paP
2S,';
La
mbd
a2
Lam
bda2
S,';
Vie
wC
lass
',V
iew
Siz
eClS
);
Lab
elG
raph
Win
dow
(1,9
30,G
Lab
el1,
0,0)
; L
abel
Gra
phW
indo
w(1
,900
,Gla
bel2
,0,0
);
end;
Univers
ity of
Cap
e Tow
n
06/1
5/19
95
09:4
8 Fi
lena
me:
M
ODEL
5E1.P
AS
Page
11
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re G
raph
1_R
esul
ts;
begi
n if V
iew
Size
Cl>
M
then
be
gin Clo
sese
grap
hics
; w
rite
lnC
'Gra
ph
1 D
raw
ER
RO
R');
w
rite
ln('
You
hav
e in
stu
cted
the
Gra
phin
g R
outi
ne
to g
raph
th
e co
nv
ersi
on
');
wri
teln
('cu
rves
of
a si
ze c
lass
whi
ch
does
no
t ex
ist.
');
wri
teln
('R
esp
ecif
y =
Vie
wSi
zeC
lass
in
Dec
lara
tion
s se
ctio
n'>
; re
adln
; en
d·
Siz~lndex:=ViewSiz
eCl;
for
Arb
lnde
x:=O
to
(N
+1)
do D
ataS
etx·
[A
rbln
dexl
:=
Arb
lnde
x*D
elta
E;
if K
appa
P1
k"[S
izel
ndex
]<>
O
then
beg
in
for
Arb
Tnde
x:=O
to
(N
+1)
do
Dat
aSet
Y"[
Arb
lnde
xJ:=
AR
TP
1"[S
izel
ndex
,Arb
lnde
xJ;
Lin
ePlo
tDat
a(D
ataS
etX
" ,D
ataS
etY
" ,(N+2~,3,0);
for
Arb
lnde
x:=O
to
CN
+1)
do
Dat
aSet
Y"C
Arb
lnde
xJ:=
SRT
1P1"
CSi
zeln
dex,
Arb
lnde
x];
Lin
ePlo
tDat
a(D
ataS
etX
",D
ataS
etY
" ,(
N+
2),5
, 1);
en
d·
if K
appa
P2
k"[S
izel
ndex
]<>
O
then
beg
in
for
Arb
Tnde
x:=O
to
(N
+1)
do
Dat
aSet
Y"[
Arb
lnde
xl :=
ART
P1"
[Siz
elnd
ex,A
rbln
dexJ
; L
ineP
lotD
ata(
Dat
aSet
x-,D
ataS
etY
" ,(
N+
2),3
, 1);
fo
r A
rbln
dex:
=O .
to (
N+1
) do
D
ataS
etY
"[A
rbln
dex]
:=
SR
T2P
1"[S
izel
ndex
,Arb
lnde
xJ;
Lin
ePlo
tDat
a(D
ataS
etX
",D
ataS
etv·
,(N
+2)
,4,2
);
end;
end;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re G
raph
2_1n
itia
lise
_and
_Dra
w;
var XA
x i sM
ax
:Rea
l;
begi
n SetC
urre
ntW
indo
w(2
);
Bor
derC
urre
ntW
indo
w(1
);
SetA
xesT
ypeC
0,0)
; X
Axi
sMax
:=5;
C
****
*Del
taT
*Rep
eats
;***
*)
Sca
leP
lotA
rea(
0.0,
0.0,
XA
xisM
ax, 1
.0);
S
etX
Yln
terc
epts
(O.O
,O.O
>;
Set
Col
orC
2l;
Dra
wX
Axi
sCC
XA
xisM
ax/5
), 1
);
Dra
wY
Axi
s(0.
2, 1
);
Lab
elX
Axi
s(1,
0);
Lab
elY
Axi
sC1,
0);
Tit
leX
Axi
sC'D
imen
sion
less
Rea
ctio
n Ti
me
(WRT
R
efer
ence
Part
icle
)');
T
itle
YA
xis(
'Fra
ctio
nal
Con
vers
ion'
);
Titl
ewin
dow
('Mod
el5E
1 ')
;
06/1
5/19
95
09:4
8 F
ilen
ame:
M
OD
EL5E
1.PA
S Pa
ge
12
Lab
elG
raph
Win
dow
(1,9
30,G
Lab
el1,
0,0)
; L
abel
Gra
phW
indo
wC
1,90
0,G
labe
l2,0
,0);
if
Kap
paP1
k"
[Siz
elnd
ex]<
>O
th
en b
egin
fo
r A
rbT
ndex
:=O
to
Con
v V
ar
do
begi
n D
ataS
etX
"[A
rbln
dexJ
:;A
rbln
dex*
Pri
nt C
rit*
Del
taT
; D
ataS
etY
"CA
rbln
dexJ
:=C
onv1
"CA
rbln
dexJ
; en
d·
Dat~SetY"[OJ :=
O;
Lin
ePlo
tDat
a(D
ataS
etX
" ,D
ataS
etY
",C
onv
Var
,5,0
>;
end·
-
if K
appa
P2
k"[S
izel
ndex
]<>
O
then
beg
in
for
Arb
Tnd
ex:=
O
to C
onv
Var
do
be
gin
Dat
aSet
X"[
Arb
lnde
x] :
;Arb
lnde
x*P
rint
_Cri
t*D
elta
T;
Dat
aSet
Y"C
Arb
lnde
x] :
=C
onv2
"[A
rbln
dexl
; en
d·
Dat~
SetY
"[OJ
:=O;
L
ineP
lotD
ata(
Dat
aSet
X",
Dat
aSet
v· ,C
onv
Var
,4,0
);
end;
-
end;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
begi
n new
(Bet
a1
k);
new
(Kap
paP1
_k>;
ne
w(K
appa
S1_k
);
new
(Lam
bda1
k)
; ne
w(C
ontR
atT
oV1)
; ne
w(A
RT)
; ne
w(S
RT
1);
new
(SR
T2)
; ne
w(C
onv1
);
new
(Dat
aSet
X);
ne
w(A
Mat
rix)
;
cl r
scr;
new
(Bet
a2
k);
new
(Kap
paP2
_k);
ne
w(K
appa
S2_k
);
new
(Lam
bda2
k)
; ne
w(C
ontR
atT
oV2)
; ne
w(A
RTP
1);
new
(SR
T1P
1);
new
(SR
T2P
1);
new
(Con
v2);
ne
w(D
ataS
etY
);
new
(Aln
vers
e);
Siz
e D
istr
ibu
tio
n I
nit
iali
sati
on
; C
onta
min
ant
Loc
atio
n In
itia
lisa
tio
n;
Det
erm
ine
Mod
el
Par
amet
ers
fn S
ize;
G
raph
1_In
Tti
alis
e;
--
Rep
eats
:=O
; Fl
ag1
:=0;
P
lot
Var
:=O
; C
onv:
:::va
r:=O
;
Tim
eO A
RT
and
SRTs
; G
uess
::::s
RT
sP1;
-
whi
le
Rep
eats
<It
erat
ion
s do
be
gin whi
le (
Fla
g1=
0) d
o be
gin Cal
e AR
TP1
and
SRTs
P1C
; C
hecK
con
verg
ence
; R
eGue
ss_S
RT
sP1;
en
d;
Plo
t_V
ar:=
Plo
t_V
ar+
1;
new
(Del
taT
_k);
new
(SR
T1P1
C);
new
(SR
T2P1
C);
Univers
ity of
Cap
e Tow
n
06/1
5/19
95
09:4
8 F
ilen
ame:
M
OD
EL5E
1.PA
S Pa
ge
13
end;
if P
lot
Var
=P
rint
C
rit
then
be
gin
--
Plo
t V
ar:=
O;
Con
v-V
ar:=
Con
v V
ar+1
; G
rapn
1 R
esu
lts;
C
ale
Con
vers
ion;
en
d;
-
Upd
ate_
AR
T_a
nd_S
RT
s;
Gue
ss
SRTs
P1;
Fla
g1:=
0;
Rep
eats
:=R
epea
ts+
1;
read
ln(H
ardC
opy)
; if
H
ardC
opy=
1 th
en S
cree
nDum
p(3,
0,2,
1.5
,1.5
,0, 1
,0,e
rro
r);
Cle
arlJ
indo
w;
Gra
ph2_
Init
iali
se_a
nd_D
raw
; re
adln
(Har
dCop
y);
if
Har
dCop
y=1
then
Scr
eenD
umpC
3,0,
2, 1
.5,1
.5,0
,1,0
,err
or)
;
clos
eseg
raph
i cs;
end.
Univers
ity of
Cap
e Tow
n
07/3
0/19
95
16:2
7 Fi
lena
me:
M
OD
EL5E
2.PA
S Pa
ge
Prog
ram
Mod
el5E
2;
(Mod
el5E
2.
( *
Thi
s pr
ogra
m
is a
too
l fo
r *
The
prog
ram
cal
cula
tes
the
and
soli
d r
eact
ants
wit
hin
deve
lope
d by
Dix
on.
anal
yzin
g TC
LP d
ata.
}
conc
entr
atio
n p
rofi
le o
f fl
uid
}
( ( ( ( ( ( ( ( ( (
* T
his
prog
ram
onl
y pr
ovid
es
orde
rs.
a p
arti
cle
usin
g th
e eq
uati
ons
as
} } fo
r v
aria
ble
so
lid
rea
ctan
t } }
* A
ssum
ptio
ns
in
this
mod
el
incl
ude:
}
·Th
e s
oli
d r
eact
ant
dep
osi
ts w
ithi
n th
e p
arti
cle
} re
sem
ble
thos
e on
th
e su
rfac
e.
} ·
The
dep
osi
ts w
ould
bot
h re
act
to t
he s
ame
exte
nt
} if
eac
h w
ere
expo
sed
to t
he s
ame
acid
co
nce
ntr
atio
n
} fo
r th
e sa
me
tim
e.
}
(Cod
ed:
Gra
ham
Dav
ies.
}
( D
epar
tmen
t of
Che
mic
al
Eng
inee
ring
. }
( U
nive
rsit
y of
C
ape
Tow
n.
} (
28
Feb
ruar
y 19
95.
} (
29 M
ay
1995
Upd
ated
(G
MO
). }
{···············--········-···········-········--···-~---······-·-····-···}
(Dec
lara
tion
s:
}
uses
crt
,std
hdr,
gj,g
raph
,wor
lddr
,seg
raph
,int
egra
t,M
2C1B
rent
Roo
ts;
cans
t (***
The
se a
re t
he p
aram
eter
s ap
pli
cab
le t
o t
he r
efer
ence
siz
e cl
ass.
**
*}
Bet
a1
Kap
paP1
Ord
SP1
Bet
a2
Kap
paP2
O
rdSP
2
Voi
dage
=10
.0;
=0.
12;
=1.
0;
=0.
0;
=0.
0;
=1.
0;
=0.
01;
(Dim
ensi
onle
ss S
toic
hiom
etri
c ra
tio
def
ined
pg
11
} (R
atio
of
reat
ion
rat
e of
so
lid
rea
ctan
t re
sid
ing
}
(wit
hin
the
par
ticl
e to
por
ous
dif
fusi
on
of
flu
id
} (i
nto
the
par
ticl
e. D
efin
ed p
g 11
D
ixon
. }
{Rea
ctio
n or
der
of
the
soli
d r
eact
ants
wit
hin
the
} ~o~s.
· }
(Voi
dage
of
the
soli
d p
arti
lces
.(P
oro
sity
) }
<***
The
se a
re p
aram
eter
s w
ith
repe
ct
to t
he C
STR
expe
rim
ent.*
****
****
**}
Vol
Liq
=
0.15
9;
(Vol
ume
of
liq
uid
lix
ivia
nt.
(m
3)
Tot
Vol
Par
t=0.
577;
{T
otal
vo
lum
e of
the
so
lid
par
tilc
es.
M
=2;
(Num
ber
of s
ize
clas
ses.
Ref
Si z
eCl
=1;
(Def
ines
th
e re
fere
nce
size
cla
ss.
}
} } }
{***
The
se a
re n
umer
ical
m
etho
d pa
ram
eter
s.**
****
****
****
****
****
****
***}
N
Del
taE
=19;
<H
alf
the
nunb
er o
f in
teri
or
po
ints
. r=
O
and
r=R
} (n
ot
incl
uded
. =
1/(N
+l)
; (S
pace
In
crem
ent.
}
Cal
cula
ted
from
1/
(N+
1).
(Sin
ce}
07/3
0/19
95
16:2
7 F
ilen
ame:
M
OD
EL5E
2.PA
S Pa
ge
2
Del
taT
=
0.00
1;
Nor
m C
rit
=1e
·6;
Con
v C
rit
=1e
·8·
SCon
c To
i =
1e·4
;
Max
lter
=1
00;
lter
atio
ns=
10
0;
Pri
nt
Cri
t=10
; V
iew
S1ze
Cl=
2;
{ R
=i*d
E an
d R
is
at
poin
t N
+1.)
}
{Tim
e In
crem
ent.
{C
onve
rgen
ce c
rite
ria b
ased
on
the
norm
of
ve
ctor
{*
**
{Con
verg
ence
cri
teri
a f
or
the
Bre
nt R
outi
ne.
{Dim
ensi
onle
ss c
on
cen
trat
ion
of
soli
d b
elow
whi
ch
{it
is
ass
umed
to
be
neg
lig
ible
. {M
axim
um
iter
atio
ns
for
the
Bre
nt
Rou
tine
.
{It
erat
ion
s in
ti
me.
{D
eter
min
ed f
rom
des
ired
Pri
nt
Del
ta T
/Del
ta T
{T
he c
on
cen
trat
ion
pro
file
s o
f-th
is s
izec
lass
are
{g
raph
ed
in g
raph
1.
(N
B V
iew
Size
Cl
<= M
.)
} } } } } } } } } } }
(***
Oth
er
info
rmat
ion
requ
ired
.***
****
****
****
****
****
****
****
****
****
*}
Val
idD
ir
=1F:
\TP6
1;{
Val
id d
irec
tory
fo
r gr
aphi
cs d
riv
ers.
}
type
var ar
ray1
= A
rray
[l .•
50,
1 .•
2]
of
Dou
ble;
Arb
lnde
x,S
izel
ndex
,err
or
Rep
eats
,Plo
t_V
ar,C
onv
Var
,Fla
g1,H
ardC
opy
Con
v1,C
onv2
,Dat
aSet
X,D
ataS
etY
,
:In
teg
er;.
Bet
a1
k,B
eta2
k,
Kap
paP1
k,
Kap
pap2
k,
Kap
paS1
k,
Kap
paS2
k
Lam
bda1
k,
Lam
6da2
_k,C
ontR
atio
V1,
Con
tRat
ioV
2'
-
: In
teg
er;
:·ve
ryL
ongV
ecto
r;
:"S
hort
Vec
tor;
Oel
taT
K
AR
T,A
RfP
1,SR
T1,S
RT2
,SR
T1P1
,SR
T2P1
,SR
T1P1
C,S
RT2
P1C
A
Mat
rix,
Aln
vers
e A
RT
P1C
Val
,Sum
Size
Dat
a 2,
NuS
tar
Lam
bda1
,Lam
bda2
,Kap
paS1
,Kap
paS2
G
Lab
el1,
Gla
bel2
Siz
eDat
a:
Arr
ay[1
•• 5
0,1
•• 2
l of
Rea
l;
CART
A
lpha
fn
(r)
at
tim
e T
CA
RTP1
A
lpha
fn
(r)
at
tim
e T+
1 {S
RT1
Sigm
a fn
(r)
at
tim
e T
{S
RTs
P1
Sigm
a fn
(r)
at
tim
e T+
1 (g
uess
ed)
: ·sh
ort
Vec
tor;
:"
Arr
ay1;
:·
sqrM
at;
:·sq
rMat
; :E
xten
ded;
:E
xten
ded;
:S
trin
g;
rang
e O
.. M
0
.• N
+1}
rang
e O
.. M
0
.• N
+1}
rang
e O
•. M
O
.. N
+1}
rang
e 0 .
. M
0 ..
N+1
} {S
RTs
P1C
Si
gma
fn(r
) at
ti
me
T+1
(cal
cula
ted
) ra
nge
O .. M
O
•. N
+1}
{YV
ect
'Con
st'
vec
tor
in C
rank
-Nic
olso
n m
etho
d ra
nge
0 .•
N+1
}
CA
Mat
rix
Mat
rix
of C
rank
-Nic
olso
n co
effi
cien
ts
rang
e N*
N
}
{Dat
aSet
X
X v
ecto
r us
ed
in t
he
grap
hing
ro
uti
ne
rang
e o •
. N+1
}
{·--
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Fun
ctio
n P
ower
(Bas
e,P
ow:r
eal)
:Ext
ende
d;
begi
n if P
ow=O
th
en P
ower
:=1
else
if
Bas
e=O
th
en P
ower
:=O
el
se
.Pow
er:=
expC
Pow
*ln(
base
));
(j
0 Q..
~
~ -· (I} -s· ~
-0 .., ~
0 a.
·~· -· Ul ttj·
N
- .. ~
U:i
•·
Univers
ity of
Cap
e Tow
n
07/3
0/19
95
16:2
7 F
ilen
ame:
M
OD
EL5E
2.PA
S Pa
ge
3
end;
{---
----
----
----
----
----
----
----
----
----
----
-·--
----
----
----
----
----
----
--}
Pro
cedu
re S
ize_
Dis
trib
uti
on
_In
itia
lisa
tio
n;
{Thi
s pr
oced
ure
sets
up
the
size
dis
trib
uti
on
dat
a ar
ray
. In
itia
lly
the
}
{Siz
eDat
a ar
ray
con
tain
s ra
dius
in
form
atio
n an
d fr
acti
on
al
volu
me
} {i
nfor
mat
ion
(ie
R a
nd V
p/V
tot)
. On
ou
tput
it
con
tain
s re
lati
ve
radi
us
} {i
nfor
mat
ion
and
rela
tiv
e vo
lum
e in
form
atio
n C
ie R
/R
ref
and
Vp/
Vp
ref)
. }
{Id
eall
y t
his
in
form
atio
n w
ould
be
re
ad
in f
rom
a d
ata
file
. -
}
var R
efR
adiu
s,V
olR
efP
art
:Dou
ble;
begi
n Siz
eDat
a[1,
1J
=11
.35e
-3;
S ze
Dat
a C1
, 21
= 1
/100
S
izeD
ata[
2, 1
l =
11.3
5e-3
; S
zeD
ata
[2, 2
1 =9
9/10
0 S
izeD
ata[
3, 1
l =
O.O
e-3;
S
zeD
ata
[3, 2
1 =
0/1
00
Siz
eDat
a[4,
1l
= O
.Oe-
3;
S ze
Dat
a [4
, 21
= 0
/100
S
izeD
ata[
5,1l
=
O.O
e-3;
, S
zeD
ata
[5, 2
1 =
0/1
00
Siz
eDat
a[6,
1l
= O
.Oe-
3;
S ze
Dat
a [6
, 21
= 0
/100
S
izeD
ata[
7, 1
l =
O.O
e-3;
S
zeD
ata
[7, 2
1 =
0/1
00
Siz
eDat
a[8,
1l
= O
.Oe-
3;
S ze
Dat
a[8,
2l
= 0
/100
=
O.O
e-3;
S
zeD
ata[
9,2l
=
0/1
00
Siz
eDat
a[9,
1l
Siz
eDat
a[10
, 1J
= O
.Oe-
3;
S ze
Dat
a[10
,2l
= 0
/100
Vol
Ref
Par
t:=
Tot
Vol
Par
t*S
izeD
ata[
Ref
Siz
eCl,
21;
Ref
Rad
ius
:=S
izeD
ata[
Ref
Siz
eCl,
11;
Sum
Size
Dat
a_2:
=0;
for
Siz
elnd
ex:=
1 to
M do
be
gin Siz
eDat
a[S
izel
ndex
,1l:
=S
izeD
ataC
Siz
elnd
ex, 1
1/R
efR
adiu
s;
Siz
eDat
a[S
izel
ndex
,2l:
=S
izeD
ata[
Siz
elnd
ex,2
l*T
otV
olP
art/
Vol
Ref
Par
t;
Sum
Siz
eDat
a_2:
=S
umS
izeD
ata_
2+S
izeD
ataC
Siz
elnd
ex,2
l;
end;
NuS
tar:
=V
olL
iq/C
Voi
dage
*Vol
Ref
Par
t);·
{Rat
io o
f vo
lum
e of
bu
lk f
luid
to
flu
id
in p
arti
cle
pore
s.
end;
}
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re C
onta
min
ant_
Loc
atio
n_In
itia
lisa
tion
;
{Thi
s pr
oced
ure
is u
sed
to d
efin
e th
e co
ntam
inan
t ra
tio
vec
tor
(rati
o o
f }
{th
e su
rfac
e co
ntam
inan
t co
nce
ntr
atio
n t
o b
ulk
cont
amin
ant
con
cen
trat
ion
).}
{NO
TE:
If a
ny s
ize
clas
s of
p
arti
cles
hav
e a
surf
ace
con
cen
trat
ion
of
} {
cont
amin
ant,
th
en s
o to
o m
ust
the
refe
renc
e si
ze c
lass
. }
begi
n Con
tRat
ov
1 · C
1l =O
C
ontR
at
ov2·
[11
=O
Con
tRat
oV
1 ·
C2l
=O
Con
tRat
ov
2· [
21
=O
Con
tRat
oV
1 · [
3]
=O
Con
tRat
ov
2 · [
3]
=O
Con
tRat
oV
1 · C
4l =O
C
ontR
at
ov2
· C4l
=O
07/3
0/19
95
16:2
7 F
ilen
ame:
M
OD
EL5E
2.PA
S Pa
ge
4
Con
tRat
ov
1·cs
1 =O
C
ontR
at
oV1"
[6l
=O
Con
tRat
oV
1"[7
] =O
C
ontR
at
oV1"
[8J
=O
Con
tRat
oV
1"[9
] =O
C
ontR
at
oV1"
[10J
=O
Con
tRat
ov
2"[5
] =O
C
ontR
at
ov2"
[6]
=O
Con
tRat
ov
2"[7
] =O
C
ontR
at
ov2"
[8]
=O
Con
tRat
ov
2"[9
J =O
C
ontR
at
ov2"
[10J
=O
en
d;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re D
eter
min
e_M
odel
_Par
amet
ers_
fn_S
ize;
{Thi
s pr
oced
ure
dete
rmin
es
the
mod
el
para
met
ers
each
si
ze c
lass
of
} {
par
ticl
es.
}
var F
lag1
,Fla
g2
:In
teg
er;
begi
n F
lag1
:=0;
F
lag2
:=0;
fo
r S
izel
ndex
:=1
to M
do
begi
n Lam
bda1
k"
CS
izel
ndex
J:=
Con
tRat
ioV
1"C
Siz
elnd
exJ
-/(
1+C
ontR
atio
v1·c
size
lnde
xJ);
La
mbd
a2
k"C
Siz
elnd
exJ:
=C
ontR
atio
V2"
CS
izel
ndex
l -
/(1+
Con
tRat
ioV
2"[S
izel
ndex
J);
if C
ontR
atio
v1·c
size
lnde
xJ<
>O
the
n F
lag1
:=F
lag1
+1;
if
Con
tRat
iov2
·csi
zeln
dexJ
<>
O t
hen
Fla
g2:=
Fla
g2+
1;
end;
if
CC
CFl
ag1=
1)
and
(Con
tRat
ioV
1"[R
efS
izeC
ll=
0))
or
((F
lag2
=1)
an
d (C
ontR
atio
v2"[
Ref
Siz
eCll
=0)
))
then
beg
in
clo
sese
gra
ph
ics;
w
rite
lnC
'Con
tam
inan
t L
ocat
ion
Vio
loat
ion
.');
re
adln
; en
d·
Lam
bda1
=L
ambd
a1
k"[R
efS
izeC
lJ;
Lam
bda2
=
Lam
bda2
-k"[
Ref
Siz
eCll
; K
appa
S1
=Lambda1~KappaP1/C1-Lambda1);
Kap
paS2
=L
ambd
a2*K
appa
P2/C
1-L
ambd
a2);
for
Siz
elnd
ex:=
1 to
M do
be
gin Del
taT
_k-C
Siz
eind
ex, 1
] :=
Del
taT
/Siz
eDat
a[S
izel
ndex
, 1J
/Siz
eDat
a[S
izel
nd
ex,1
J;
Bet
a1
k"[S
izel
ndex
J :=
Bet
a1*C
1+C
ontR
atio
V1"
CR
efSi
zeC
lJ)
-/C
1+C
ontR
atio
V1"
[Siz
elnd
exJ)
; B
eta2
k"[
Siz
elnd
exJ
:=B
eta2
*(1+
Con
tRat
ioV
2"[R
efS
izeC
l])
-/C
1+C
ontR
atio
V2"
[Siz
elnd
ex])
; ·K
appa
P1
k" [
Siz
elnd
exJ
:=K
appa
P1*
Siz
eDat
a[S
izel
ndex
, 11
-*S
izeO
ata[
Siz
elnd
ex,1
J;
Kap
paP2
k"
CS
izel
ndex
J:=
Kap
paP
2*S
izeD
ataC
Siz
elnd
ex,1
J -
*Siz
eDat
a[S
izei
ndex
,11;
if
Con
tRat
ioV
1"C
Size
lnde
xJ=
O
then
Kap
paS
1_k"
[Siz
elnd
exJ
:=O
el
se
Kap
paS1
k"
[Siz
elnd
ex]
:=K
appa
S1*P
ower
((C
ontR
atio
V1"
CSi
zeln
dexJ
-
/Con
tRat
ioV
1"[R
efS
izeC
ll),
Ord
SP
1)
*Siz
eDat
a[S
izel
ndex
,1J;
if
Con
tRat
iov2
· C
Size
lnde
xJ=O
th
en K
appa
S2
k"[S
izel
ndex
J :=
O
else
K
appa
S2_
k"[S
izel
ndex
J :=
Kap
paS
2*P
ower
((C
ontR
atio
V2"
[Siz
elnd
exJ
Univers
ity of
Cap
e Tow
n
07/3
0/19
95
16:2
7 Fi
lena
me:
M
OD
EL5E
2.PA
S Pa
ge
5
/Con
tRat
ioV
2"[R
efS
izeC
ll )
,Ord
SP2)
*S
izeD
ata[
Siz
elnd
ex, 1
J;
end·
en
d·
' {-
-'.·
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
-}
Pro
cedu
re T
imeO
_AR
T_an
d_SR
Ts;
{Thi
s pr
oced
ure
uses
th
e in
itia
l co
ndit
ions
to
set
th
e al
pha
and
sigm
a } }
surf
ace}
{v
ecto
rs.
{Not
e th
at A
RT[
N+1
J, A
RTP
1[N
+1],
SRT1
CN+1
J an
d SR
T1P1
[N+1
] ar
e th
e {c
once
ntra
tion
s of
th
e li
qu
id a
nd
soli
d r
eact
ants
.
begi
n for
Siz
elnd
ex:=
1 to
M do
be
gin fo
r A
rbln
dex:
=O
to N
do
begi
n AR
T"[
Size
inde
x,A
rbin
dexJ
=O
S
RT
1"[S
izei
ndex
,Arb
inde
xJ
=1
SR
T2"
[Siz
eind
ex,A
rbin
dexJ
=1
en
d·
AR
T![
Size
inde
x,N
+1J
=1
SRT
1"[S
izei
ndex
,N+
1l
=1
SRT
2"[S
izei
ndex
,N+1
J =1
en
d·
for'
arbi
ndex
:=O
to
max
v do
be
gin·
C
onv1
"[A
rbin
dexJ
:=O
; C
onv2
"[A
rbin
dexJ
:=O
; en
d·
end;
'
}
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re G
uess
_SR
TsP
1;
{Thi
s pr
oced
ure
prov
ides
the
in
itia
l "g
uess
" fo
r th
e it
erat
ion
. It
use
s }
{the
pre
viou
s ti
me
inte
rval
's v
alue
s as
th
e gu
ess.
}
begi
n for
Siz
eind
ex:=
1 to
M do
be
gin fo
r A
rbin
dex:
=O
to (
N+1
) do
be
gin SR
T1P
1"[S
izei
ndex
,Arb
inde
xJ:=
SR
T1"
CS
izei
ndex
,Arb
inde
xJ;
SR
T2P
1"[S
izei
ndex
,Arb
inde
xJ:=
SR
T2"
[Siz
eind
ex,A
rbin
dexl
; en
d·
. A
RT
P1"[
Size
inde
x,N
+1J
:=A
RT
"[Si
zein
dex,
N+
1J;
end·
en
d;
'
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re R
eGue
ss_S
RTs
P1;
{Thi
s pr
oced
ure
prov
ides
an
upda
ted
"gue
ss"
for
the
next
it
erat
ion
. It
{u
ses
the
SRT1
P1C
vect
or a
s th
e up
date
d gu
ess.
begi
n for
Siz
eind
ex:=
1 to
M do
be
gin fo
r A
rbln
dex:
=O
to (
N+1
) do
} }
07/3
0/19
95
16:2
7 F
ilen
ame:
M
OD
EL5E
2.PA
S Pa
ge
6
begi
n SR
T1P
1"[S
izel
ndex
,Arb
lnde
xJ :
=S
RT
1P1C
"[S
izel
ndex
,Arb
lnde
xJ;
SR
T2P
1"[S
izei
ndex
,Arb
inde
xJ :
=S
RT
2P1C
"[S
izei
ndex
,Arb
lnde
xJ;
end·
AR
TP1
· [S
izel
ndex
,N+
1J :=
AR
TP1C
Val
; en
d·
end·
'
{--'
.---
----
----
--J-
----
----
----
----
----
----
----
----
----
----
----
----
----
---}
P
roce
dure
Cal
c_C
rank
_Nic
olso
n_M
atri
x(va
r Y
Vec
tor
:Sho
rtV
ecto
r);
var R
ows,
Col
s : I
nte
ger
;
begi
n for
Row
s:=O
to
(N
+1)
do
begi
n for
Col
s:=O
to
(N
+1)
do
AM
atri
x· [
Row
s,C
olsJ
:=O
; en
d;
AM
atri
x"[0
,0J:
=C
-6-D
elta
E*D
elta
E*S
izeD
ata[
Siz
eind
ex,1
J*S
izeD
ata[
Siz
elnd
ex,1
J *(
Kap
paP1
k"
[Siz
elnd
exJ*
Pow
er(S
RT
1P1"
[Siz
eind
ex,O
J,O
rdS
P1)
+
Kap
paP
2_K
-[S
izel
ndex
]*P
ower
(SR
T2P
1"C
Siz
eind
ex,O
J,O
rdS
P2)
) -2
*Del
taE
*Del
taE
*Siz
eDat
a[S
izel
ndex
, 1J*
Siz
eDat
a[S
izel
ndex
,1J
/Del
taT
_k
-[S
izei
nd
ex,1
J);
AM
atri
x" [
0, 1
] :=
6;
YV
ecto
r[O
J :=
AR
T"[
Siz
eind
ex,O
J*(6
+D
elta
E*D
elta
E*S
izeD
ata[
Siz
eind
ex,1
J *S
izeD
ata[
Siz
eind
ex,1
J*(K
appa
P1
k"[S
izei
ndex
J *P
ower
(SR
T1"
[Siz
elnd
ex,O
J,O
rdS
Pl)
+K
appa
P2
k"[S
izei
ndex
l *P
ower
(SR
T2"
[Siz
elnd
ex,O
J,O
rdS
P2)
) -
-2*D
elta
E*D
elta
E*S
izeD
ata[
Siz
elnd
ex, 1
J *S
izeD
ata[
Siz
elnd
ex,1
]/D
elta
T k
"[S
izei
nd
ex,1
])
-ART
" [S
izel
ndex
, 1]*
6;
-
for
Row
s:=1
to N
-1
do
begi
n AM
atri
x"[R
ows,
Row
s-1]
:=R
ows-
1;
AM
atri
x"[R
ows,
Row
s]
:=-2
*Row
s-R
ows*
Del
taE
*Del
taE
*Siz
eDat
a[S
izei
ndex
, 1]
*Siz
eDat
a[S
izel
ndex
,1J*
(Kap
paP
1 k"
[Siz
eind
exl
*Pow
er(S
RT
1P1"
[Siz
eind
ex,R
owsJ
,Ord
SP
1)
+Kap
paP2
k"
[Siz
elnd
ex]
*Pow
er(S
RT
2P1"
[Siz
eind
ex,R
owsJ
,Ord
SP
2))
-2*R
ows*
Del
taE
*Del
taE
*Siz
eDat
a[S
izel
ndex
, 1l
*Siz
eDat
a[S
izel
ndex
,1]/
Del
taT
k"[
Siz
eind
ex, 1
l;
AM
atri
x"[R
ows,
Row
s+1l
:=R
ows+
1;
-Y
Vec
tor[
Row
sl
:=A
RT
"[Si
zein
dex,
Row
s-1J
*(-R
ows+
1)+
end;
AM
atri
x"[N
,N-1
J:=
N-1
;
AR
T"[
Size
inde
x,R
owsJ
*C2*
Row
s+R
ows*
Del
taE
*Del
taE
*S
izeD
ata[
Siz
elnd
ex,1
J*S
izeD
ata[
Siz
elnd
ex,1
J *C
Kap
paP1
k"
[Siz
elnd
exl
*Pow
er(S
RT
1"[S
izei
ndex
,Row
sJ,O
rdS
P1)
+K
appa
P2
k"[S
izel
ndex
l *P
ower
CSR
T2"
[Siz
elnd
ex,R
owsJ
,Ord
SP2>
> -2
*Row
s*D
elta
E*D
elta
E*S
izeD
ata[
Siz
elnd
ex,1
] *S
izeD
ata[
Siz
elnd
ex,1
J/D
elta
T k
"[S
izel
nd
ex,1
l)
+AR
T"[
Size
lnde
x,R
ows+
1J*C
-Row
s-1)
;
AM
atri
x"[N
,Nl
:=-2
*N-N
*Del
taE
*Del
taE
*Siz
eDat
a[S
izel
ndex
, 1J
*Siz
eDat
a[S
izei
ndex
,1J*
CK
appa
P1
k"[S
izei
ndex
l *P
ower
(SR
T1P
1"[S
izel
ndex
,NJ,
Ord
SP
1)+
Kap
paP
2_k-
[Siz
eind
exl
Univers
ity of
Cap
e Tow
n
07/3
0/19
95
16:2
7 Fi
lena
me:
M
ODEL
5E2.P
AS
Page
7
YV
ecto
r[N
J
*Pow
er(S
RT
2P1-
[Siz
eJnd
ex,N
J,O
rdSP
2))-
2*N
*Del
taE
*Del
taE
*S
izeD
ata[
Siz
elnd
ex,1
J*S
izeD
ata[
Siz
elnd
ex,1
l /D
elta
T
k·cs
izel
ndex
, 1l;
:=
AR
T-[
Size
lnde
x,N
-1J*
(-N
+1)+
AR
T-[
Size
lnde
x,N
l*C
2*N
+ N
*Del
taE
*Del
taE
*Siz
eDat
a[S
izel
ndex
, 1l
*Siz
eDat
a[S
izel
ndex
, 1J*
CK
appa
P1
k-[S
izel
ndex
l *P
ower
CSR
T1-
[Siz
elnd
ex,N
J,O
rdSP
l)+K
appa
P2
k·cs
izel
ndex
J *P
ower
(SR
T2-
[Siz
elnd
ex,N
l ,O
rdSP
2))-
2*N
*DeT
taE
*Del
taE
*S
izeD
ata[
Siz
elnd
ex,1
l*S
izeD
ata[
Siz
elnd
ex,1
l /D
elta
T
k·cs
izel
ndex
, 1l )
+AR
T-[
Size
lnde
x,N
+ll*
C-N
-1>
-AR
TP1
.[Siz
eJnd
ex,N
+1J*
CN
+1);
end;
{--
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
---}
P
roce
dure
Cal
c_A
RTP
1_an
d_SR
TsP1
C;
var M
assB
alV
al,A
Mat
Det
V
alue
AtR
oot
Out
putV
ecto
r,Y
Vec
tor
Dou
ble;
R
eal;
S
hort
Vec
tor;
begi
n .
{F
irst
cal
cula
te t
he A
RTP1
v
ecto
rs.
}
for
Siz
elnd
ex:=
1 to
M do
be
gin Cal
e C
rank
N
icol
son
Mat
rixC
YV
ecto
r);
Gau
ssJo
rdan
(AM
atri
xT,Y
Vec
tor,
CN
+1)
,0ut
putV
ecto
r,A
lnve
rse·
,AM
atD
et);
fo
r A
rbln
dex:
=O
to N
do
ART
P1. [
Siz
elnd
ex,A
rbln
dexl
:=
Out
putV
ecto
r[A
rbJn
dexl
;
end;
{The
Gau
ssJo
rdan
pro
cedu
re c
alcu
late
s va
lues
fo
r th
e AR
T ve
ctor
fro
m
{poi
nt 0
to
poin
t N
(al
thou
gh
it m
akes
us
e of
th
e N+
1 th
po
int)
. To
{d
eter
min
e th
e N+
1 th
po
int
valu
e,
mak
e us
e of
th
e m
ass
bala
nce
of
(flu
id r
eact
ant
in-t
he
CSTR
.
Mas
sBal
Val
:=O
;
for
Siz
elnd
ex:=
1 to
M do
be
gin
} } } }
Mas
sBal
Val
:=M
assB
alV
al-S
izeD
ata[
Siz
elnd
ex,2
J/2*
(Kap
paS
1_k-
[Siz
elnd
exl
*SR
T1-
[Siz
elnd
ex,(
N+
1)J*
AR
T.[
Size
lnde
x,C
N+
1)J
end;
+Kap
paS1
k·
csiz
elnd
exl*
SR
T1P
1.[S
izel
ndex
,CN
+1)
J *A
RT
P1-[
Size
lnde
x,C
N+1
)J+K
appa
S2
k·cs
izel
ndex
l *S
RT
2.[S
izel
ndex
,CN
+1)
J*A
RT
·csi
zeln
dex,
(N+
1)l
+Kap
paS2
k·
csiz
elnd
exJ*
SR
T2P
1.[S
izel
ndex
,CN
+1)
l *A
RTP1
-[S
izel
ndex
,CN
+1)
J)-3
*Siz
eDat
a[S
izel
ndex
,2l
/(S
izeD
ata[
Siz
elnd
ex,1
J*S
izeD
ata[
Siz
elnd
ex, 1
J)/2
*C
CA
RT
-[Si
zeln
dex,
CN
+1)J
+AR
TP1
·csi
zeln
dex,
CN
+1)J
) -C
AR
T.[
Siz
elnd
ex,N
J+A
RT
P1.
[Siz
elnd
ex,N
J))/
Del
taE
;
AR
TP1
CV
al:=
AR
T.[1
,CN
+1)J
+Del
taT
/NuS
tar*
Mas
sBal
Val
;
{Cal
cula
te t
he S
RTsP
1C
vec
tors
. C
ode
mak
es
use
of
Bre
nt'
s M
etho
d <C
A no
n-li
near
roo
t fi
ndin
g pr
oced
ure)
to
sol
ve f
or
the
roo
t.
} }
07/3
0/19
95
16:2
7 F
ilen
ame:
M
OD
EL5E
2.PA
S Pa
ge
8
for
Siz
elnd
ex:=
1 to
M do
be
gin fo
r A
rbln
dex:
=O
to C
N+1
) do
be
gin
ex, 1
J,
if
(Kap
paP
1_k.
[Siz
elnd
exJ=
0)
then
SR
T1P
1C-[
Siz
elnd
ex,A
rbln
dex]
:=O
els
e be
gin if O
rdSP
1=1
then
be
gin
end SR
T1P1
c· [
Siz
elnd
ex,A
rbln
dexJ
:=S
RT
1.[S
izel
ndex
,Arb
lnde
xJ
*(1-
Del
taT
k·
csiz
elnd
ex,1
J*K
appa
P1
k·c
size
lnd
exl
*Bet
a1
k·cs
izel
ndex
J*A
RT
.[S
izel
ndex
,Arb
lnde
xl
/C2*
C1
7La
mbd
a1
k·c
size
lnd
exJ)
))/(
1+
Del
taT
k
·csi
zeln
dex
, 1J
*Kap
paP
1_k.
[Siz
elnd
exJ*
Bet
a1_k
.[S
izel
ndex
] *A
RT
P1.
[Siz
elnd
ex,A
rbln
dexl
/C2*
C1-
Lam
bda1
_k.[
Siz
elnd
ex])
))
else
if
s'R
T 1P
1 ·[
Si z
el n
dex,
Arb
l nde
xl <
SCon
c_To
l th
en
SR
T1P
1c·c
size
lnde
x,A
rbln
dexJ
:=S
RT
1P1·
csiz
elnd
ex,A
rbln
dexJ
el
se
begi
n ,
SR
T1P
1C.[
Siz
elnd
ex,A
rbln
dexJ
:=B
rent
Roo
ts(0
.0,1
.0,D
:lta
T_k
-[S
izel
nd
Kap
paP1
k·
csiz
elnd
exJ,
Bet
a1
k·c
size
lnd
exJ,
L
ambd
a1-k
·csi
zeln
dexJ
,Ord
SP
l,A
RT
.[S
izel
ndex
,Arb
lnde
xJ,
AR
TP
1·cs
izel
ndex
,Arb
Jnde
xJ,S
RT
1.[S
izel
ndex
,Arb
lnde
xl,
le-8
,100
,Val
ueA
tRoo
t,er
ror)
; en
d·
end·
'
if ~K
appa
P2_k
.[Si
zeln
dexJ
=0)
then
SR
T2P
1C-[
Siz
elnd
ex,A
rbln
dexJ
:=O
els
e be
gin if O
rdSP
2=1
then
be
gin
end S
RT
2P1c
·csi
zeln
dex,
Arb
lnde
xJ:=
SR
T2.
[Siz
elnd
ex,A
rbln
dexJ
*C
1-D
elta
T
k·cs
izel
ndex
,1J*
Kap
paP
2 k·
csiz
elnd
exl
*Bet
a2 k
·csi
zeln
dexl
*AR
T·c
size
lnde
x,A
rbln
dexl
/C
2*C
17
Lam
bda2
k
·csi
zeln
dex
J)))
/(1
+D
elta
T k
·csi
zeln
dex
, 1J
*Kap
paP
2_k.
[Siz
elnd
exJ*
Bet
a2_k
·csi
zeln
dex]
*A
RT
P1-
[Siz
elnd
ex,A
rbln
dex]
/(2*
C1-
Lam
bda2
_k.[
Siz
elnd
exJ)
))
else
if
SR
T2P
1-[S
izel
ndex
,Arb
lnde
x]<
SC
onc
Tol
th
en
SR
T1P
1c·c
size
lnde
x,A
rbln
dexJ
:=S
RT
2P1·
csiz
elnd
ex,A
rbln
dexJ
el
se
begi
n SR
T2P
1C.[
Siz
elnd
ex,A
rbln
dexJ
:=B
rent
Roo
ts(O
.O, 1
.0,
Del
taT
k·
csiz
elnd
ex,1
J,K
appa
P2
k·c
size
lnd
exJ,
B
eta2
K·c
size
lnde
x],L
ambd
a2 k
·csi
zeln
dexl
,Ord
SP
2,
AR
T.[
Siz
elnd
ex,A
rbln
dexl
,AR
iP1.
[Siz
elnd
ex,A
rbln
dexJ
,
end·
en
d·
' en
d·
' en
d·
' en
d;
'
SR
T2.
[Siz
elnd
ex,A
rbln
dexJ
,1e-
8,10
0,V
alue
AtR
oot,
erro
r);
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re C
heck
_Con
verg
ence
;
{Thi
s pr
oced
ure
chec
ks w
heth
er o
r no
t th
e so
luti
on
has
con
verg
ed b
y }
Univers
ity of
Cap
e Tow
n
07/3
0/19
95
16:2
7 F
ilen
ame:
M
OD
EL5E
2.PA
S Pa
ge
9
{com
pari
ng
the
gues
sed
valu
e of
SR
T1P1
w
ith
a ca
lcu
late
d v
alue
of
SRT1
P1.
} {A
lso
com
pare
s th
e ca
lcu
late
d v
alue
of
ARTP
1C
with
gu
esse
d va
lue
of
} {A
RTP
1.
}
var N
orm
1,N
orm
2,N
orm
3
begi
n
:Dou
ble;
Nor
m1
=O;
{Usi
ng a
nor
m *
**
} N
orm
2 =O
· N
orm
3 =O
'. fo
r S
zein
dex:
=1
to M
do
begi
n for
Arb
lnde
x:=O
to
(N
+1)
do
begi
n if (
Bet
a1
k"[S
izel
ndex
]<>
O)
and
(Kap
paP1
. k"
[Siz
elnd
exl<
>O
) th
en b
egin
N
orm
1::N
orm
1+C
SRT
1P1C
"[Si
zeln
dex,
Arb
inde
xl
·SR
T1P
1"[S
izel
ndex
,Arb
lnde
x])*
(SR
T1P
1C.[
Siz
elnd
ex,A
rbln
dexl
·S
RT
1P1"
[Siz
elnd
ex,A
rbln
dexJ
);
end
,els
e N
orm
1 :=
O;
if (
Bet
a2_k
"[S
izel
ndex
l<>
0)
and
(Kap
paP2
k"
[Siz
elnd
ex]<
>O
) th
en b
egin
N
orm
2:=N
orm
2+(S
RT
2P1C
"[Si
zeln
dex,
Arb
lnde
xl
-SR
T2P
1"[S
izel
ndex
,Arb
lnde
x])*
(SR
T1P
1C"
[Siz
elnd
ex,A
rbln
dexl
-S
RT
2P1"
[Siz
elnd
ex,A
rbln
dexl
);
end
else
Nor
m2:
=0;
end·
en
d·
' Nor~:=(ARTP1CVal·ART
P1"(
1,N+
1])*
(ART
P1CV
al·A
RTP1
"[1,
N+1l
);
if
(Nor
m1<
Nor
m C
rit)
and
(N
orm
2<N
orm
Cri
t) a
nd
(Nor
m3<
Nor
m C
rit)
th
en F
lag1
::1;
-
-en
d;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re U
pdat
e_A
RT_
and_
SRTs
;
{Thi
s procedu~e
upda
tes
the
alph
a an
d si
gma
vect
ors
for
the
next
it
erat
ion
}
{by
repl
acin
g th
eir
com
pone
nts
wit
h th
e al
phaT
+1
and
sigm
aT+1
v
ecto
rs.
}
begi
n for
Siz
elnd
ex:=
1 to
M do
be
gin fo
r A
rbln
dex:
=O
to C
N+1
) do
be
gin AR
T"[
Siz
elnd
ex,A
rbln
dexl
=
AR
TP
1"[S
izel
ndex
,Arb
inde
xl;
SR
T1"
[Siz
elnd
ex,A
rbln
dexl
=
SR
T1P
1"[S
izel
ndex
,Arb
lnde
xl;
SR
T2"
[Siz
elnd
ex,A
rbln
dexl
=
SR
T2P
1"[S
izel
ndex
,Arb
inde
xl;
end·
~d·
, ~d;
,
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re C
alc_
Con
vers
ion;
{Thi
s pr
oced
ure
calc
ula
tes
the
frac
tio
nal
co
nver
sion
of
the
par
ticl
e fo
r }
{th
e ti
me
inte
rval
. It
use
s th
e fo
rmul
a on
pg.
13
Dix
on.
The
inte
gra
tor
} {
is t
he Q
uinn
-Cur
tis
vect
or
inte
gra
tor.
}
07/3
0/19
95
16:2
7 F
ilen
ame:
M
OD
EL5E
2.PA
S Pa
ge
10
var ln
terG
1,ln
terG
2 ln
tegV
al,C
onve
rsio
n :V
eryL
ongV
ecto
r;
:Rea
l;
begi
n for
Siz
elnd
ex:=
1 to
M do
be
gin if K
appa
P1
k"[S
izel
ndex
l<>
O
then
be
gin
-fo
r A
rbln
dex:
=O
to C
N+1
) do
ln
terG
1[A
rbln
dex]
:=
(1-S
RT
1"[S
izel
ndex
,Arb
lnde
xJ)*
Arb
lnde
x*D
elta
E
*Arb
lnde
x*D
elta
E;
lnte
gra
teV
ecto
r(Jn
terG
1,D
elta
E,0
,(N
+1
),ln
teg
Val
);
Con
vers
ion
:=3*
(1-L
ambd
a1
k"[S
izel
ndex
J)*l
nteg
Val
+
Lam
bda1
_k-[
Siz
elnd
ex]*
(1-S
RT
1" [
Siz
elnd
ex,N
+1l
);
Con
v1"(
Con
v V
ar]
:=C
onv1
"[C
onv
Var
J+C
onve
rsio
n*S
izeD
ata[
Siz
elnd
ex,2
] -
/Sum
Size
Dat
a_2;
. e
nd;
if K
appa
P2
k"[S
izel
ndex
l<>
O
then
be
gin
-;
for
Arb
lnde
x:=O
to
(N
+1)
do
lnte
rG2C
Arb
lnde
xJ:=
(1·S
RT
2"[S
izel
ndex
,Arb
lnde
xJ)*
Arb
lnde
x*D
elta
E
*Arb
lnde
x*D
elta
E;
lnte
gra
teV
ecto
r(ln
terG
2,D
elta
E,0
,(N
+1
),ln
teg
Val
);
Con
vers
ion
:=3*
(1-L
ambd
a2
k"[S
izel
ndex
J)*l
nteg
Val
+L
ambd
a2
k"[S
izel
ndex
]*(1
-SR
T2"
[Siz
elnd
ex,N
+1l
);
Con
v2"[
Con
v_V
arJ:
=C
onv2
"[C
onv_
Var
J+C
onve
rsio
n*S
izeD
ata[
Siz
elnd
ex,2
J
end·
en
d·
' en
d;
'
/Sun
Siz
eDat
a_2;
{----------------------------------------------~--------------------------}
Pro
cedu
re G
rap
h1
_1
nit
iali
se;
var B
eta1
S,K
appa
P1S,
Lam
bda1
S,G
DT
S B
eta2
S,K
appa
P2S,
Lam
bda2
S V
iew
Size
ClS
begi
n init
SE
Gra
phic
s(V
alid
Dir
);
Set
Cur
rent
Win
dow
(2);
B
orde
rCur
rent
Win
dow
(1);
S
etA
xesT
ype(
0,0)
; S
cale
Plo
tAre
a(0
.0,0
.0,1
.0,1
.0);
S
etX
Yln
terc
epts
(0.0
,0.0
);
Set
Col
or(2
);
Dra
wX
Axi
s(0.
2,1>
; D
raw
YA
xis(
o.2 61>
; L
abel
XA
xis(
1,
);
Lab
elY
Axi
s(1,
0);
Tit
leX
Axi
s('D
imen
sion
less
Rad
ius'
);
Tit
leY
Axi
s('D
imen
sion
less
Con
cent
rati
on'>
; T
itle
Win
dow
('Mod
el5E
2'>
; S
tr(B
eta1
k"
CV
iew
size
Cll
:6:3
,Bet
a1S
);
Str
(Kap
paP
1_k"
[Vie
wS
izeC
lJ:6
:3,K
appa
P1S
);
stri
ng
st
rin
g
stri
ng
Univers
ity of
Cap
e Tow
n
07/3
0/19
95
16:2
7 F
ilen
ame:
M
OD
EL5E
2.PA
S Pa
ge
11
Str
((D
elta
T*P
rint
Cri
t):6
:4,G
DT
S);
St
r(L
ambd
a1
k"C
Vie
wSi
zeC
lJ:5
:3,L
ambd
a1S)
; S
tr(B
eta2
k~CViewSizeClJ:6:3,Beta2S);
Str
(Kap
paP
2_k"
CV
iew
Siz
eClJ
:6:3
,Kap
paP
2S);
St
r(L
ambd
a2
k"C
Vie
wS
izeC
ll:5
:3,L
ambd
a2S
);
Str
(Vie
wS
izeC
l:2,
Vie
wS
izeC
lS);
G
labe
l1:=
Con
cat(
' B
eta1
',
Bet
a1S
,';
Kap
pa1
',Kap
paP
1S,';
La
mbd
a1
Lam
bda1
S,';
GOT
wrt
R
efS
izeC
lass
',G
DT
S);
Gla
bel2
:=C
onca
t('
Bet
a2
',Bet
a2S
, ';
K
appa
2 ',K
appa
P2S
,';
Lam
bda2
L
ambd
a2S,
'; V
iew
Cla
ss
',Vie
wS
izeC
lS);
L
abel
Gra
phW
indo
wC
1,93
0,G
labe
l1,0
,0);
L
abel
Gra
phW
indo
wC
1,90
0,G
labe
l2,0
,0);
en
d;
{····
-·--
----
----
----
----
----
----
·---
----
----
----
----
----
----
----
----
----
-}
Pro
cedu
re G
raph
1_R
esul
ts;
begi
n if V
iew
Size
Cl>
M
then
be
gin Clo
sese
grap
hics
; w
rite
ln('
Gra
ph 1
Dra
w
ERR
OR
');
wri
teln
C'Y
ou h
ave
inst
uct
ed t
he G
raph
ing
Rou
tine
to
gra
ph
the
con
ver
sio
n')
;
wri
teln
C'c
urve
s of
a s
ize
clas
s w
hich
do
es
not
ex
ist.
');
wri
teln
C'R
espe
cify
= V
iew
Siz
eCla
ss
in D
ecla
rati
ons
sect
ion
'>;
read
ln;
end·
Siz~Index:=ViewSizeCl;
for
Arb
inde
x:=O
to
CN
+1)
do D
ataS
etX
"[A
rbln
dexl
:=
Arb
inde
x*D
elta
E;
if K
appa
P1
k"[S
izel
ndex
l<>
O
then
beg
in
for
Arb
Tnde
x:=O
to
CN
+1)
do
Dat
aSet
Y"[
Arb
lnde
xl :=
AR
TP1-
[Siz
elnd
ex,A
rbln
dexl
; L
ineP
lotD
ata(
Oat
aSet
X",
Dat
aSet
Y-,
CN
+2)
,3,0
);
for
Arb
lnde
x:=O
to
(N
+1)
do
Dat
aSet
Y"[
Arb
lnde
xl :
=S
RT
1P1"
[Siz
elnd
ex,A
rbln
dexl
; L
ineP
lotD
ata(
Dat
aSet
X"
,Dat
aSet
Y"
,(N
+2)
,5, 1
);
end·
if
K
appa
P2
k"C
Size
lnde
xJ<>
O
then
beg
in
for
Arb
Tnde
x:=O
to
CN
+1)
do
Dat
aSet
Y"C
Arb
inde
xJ:=
AR
TP1
"CSi
zeln
dex,
Arb
lnde
xJ;
Lin
ePlo
tDat
a(O
ataS
etX
" ,D
ataS
etv·
,CN
+2)
,3,1
);
for
Arb
lnde
x:=O
to
CN
+1)
do
Dat
aSet
Y"[
Arb
lnde
xJ:=
SR
T2P
1"C
Siz
elnd
ex,A
rbln
dexl
; L
ineP
lotD
ata(
Oat
aSet
X",
Dat
aSet
Y",
CN
+2)
,4,2
);
end;
end;
{---
------
----
----
:·--
-~--
----
----
----
----
----
----
------------------------}
Pro
cedu
re G
raph
2_In
itia
lise
_and
_Dra
w;
var X
Axi
sMax
begi
n SetC
urre
ntW
indo
wC
2);
:Rea
l;
07/3
0/19
95
16:2
7 F
ilen
ame:
M
OD
EL5E
2.PA
S Pa
ge
12
Bor
derC
urre
ntW
indo
w(1
);
Set
Axe
sTyp
eC0,
0);
XA
xisM
ax:=
5;
(**
**
*o
r D
elta
T*R
epea
ts;*
***)
S
cale
Plo
tAre
a(0.
0,0.
0,X
Axi
sMax
, 1.0
);
Set
XY
lnte
rcep
ts(0
.0,0
.0);
S
etC
olor
C2)
; D
raw
XA
xisC
CX
Axi
sMax
/5),
1>;
·ora
wY
Axi
s(0.
2, 1
>;
Lab
elX
Axi
sC1,
0);
Lab
elY
Axi
sC1,
0);
Tit
leX
Axi
s('D
imen
sion
less
Rea
ctio
n Ti
me
CWRT
R
efer
ence
Part
icle
)');
T
itle
YA
xisC
'Fra
ctio
nal
Co
nv
ersi
on
');
Tit
leW
indo
w('
Mod
el5E
2');
L
abel
Gra
phW
indo
wC
1,93
0,G
Lab
el1,
0,0)
; L
abel
Gra
phW
indo
wC
1,90
0,G
labe
l2,0
,0);
if
Kap
paP1
k"
[Siz
elnd
exl<
>O
the
n be
gin
for
Arb
Tnd
ex:=
O
to C
onv
Var
do
be
gin
Dat
aSet
X"[
Arb
lnde
xl:=
Arb
lnde
x*P
rint
_Cri
t*D
elta
T;
Dat
aSet
Y"C
Arb
lnde
xl:=
Con
v1"C
Arb
lnde
xJ;
end·
Da
t~Se
tY"[
Ol :=
O;
Lin
ePlo
tDat
a(D
ataS
etX
",D
ataS
etY
-,C
onv_
Var
,5,0
);
end·
if
Kap
paP2
k"
[Siz
elnd
exJ<
>O
th
en b
egin
fo
r A
rbT
ndex
:=O
to
Con
v V
ar d
o be
gin
Dat
aSet
X"[
Arb
lnde
xl:=
Arb
lnde
x*P
rint
C
rit*
Del
taT
; D
ataS
etY
"CA
rbin
dexJ
:=C
onv2
"[A
rbin
dexJ
; en
d·
Dat~SetY"COl :=
O;
Lin
ePlo
tDat
a(D
ataS
etX
",D
ataS
etv·
,con
v_V
ar,4
,0);
en
d;
end;
{··:
·---
----
----
----
----
----
----
----
----
----
----
----
----
-be
gin
----
----
----
----
-}
new
(Bet
a1
k);
new
(Kap
paP1
_k>;
ne
w(K
appa
S1_k
>;
new
(lam
bda1
k
);
new
(Con
tRat
ToV
1);
new
CA
RT)
; ne
w(S
RT
1);
new
(SR
T2)
; ne
w(C
onv1
);
new
(Dat
aSet
X);
ne
w(A
Mat
rix)
;
cl r
scr;
new
(Bet
a2
k);
ne
w(K
appa
P2_k
);
new
(Kap
paS2
_k);
ne
w(L
ambd
a2
k);
new
(Con
tRat
ToV
2);
new
(AR
TP1)
; ne
w(S
RT
1P1)
; ne
w(S
RT
2P1)
; ne
w(C
onv2
);
new
(Dat
aSet
Y);
ne
wC
Ain
vers
e);
Siz
e D
str
ibu
tio
n I
nit
iali
sati
on
; C
onta
m n
ant
Loc
atio
n In
itia
lisa
tio
n;
Det
erm
ne
Mod
el
Par
amet
ers
fn S
ize;
G
raph
1_In
Tti
alis
e;
--
Rep
eats
:=O
; Fl
ag1
:=O
; P
lot
Var
:=O
; Co
n.v:
::va r
: =O
;
new
(Del
taT
_k);
new
(SR
T1P
1C);
new
(SR
T2P
1C);
Univers
ity of
Cap
e Tow
n
07/3
0/19
95
16:2
7 F
ilen
ame:
M
OD
EL5E
2.PA
S Pa
ge
13
Tim
eO A
RT
and
SRTs
; G
uess
:::sR
TsP
1;-
whi
le
Rep
eats
<Jt
erat
ions
do
begi
n whi
le (
Fla
g1=
0)
do
begi
n Cal
e AR
TP1
and
SRTs
P1C
; C
hecK
Con
verg
ence
; R
eGiJe
ss S
RTs
P1;
end;
-
Plo
t_V
ar:=
Plo
t_V
ar+
1;
if P
lot
Var
=P
rint
C
rit
then
be
gin
--
Plo
t V
ar:=
O;
Con
v-V
ar:=
Con
v V
ar+1
; G
rapl
l1
Res
ult
s;
Cal
e co
nver
sion
; en
d;
-
Upd
ate
ART
and
SRTs
; G
uess
SR
TsP1
; -
Fla
g1:=
0;
· R
epea
ts:=
Rep
eats
+1;
end;
read
ln(H
ardC
opy)
; if
Har
dCop
y=1
then
Scr
eenD
ump(
3,0,
2, 1
.5,1
.5,0
,1,0
,err
or)
;
Cle
arW
indo
w;
Gra
ph2_
Init
iali
se_a
nd_D
raw
; re
adln
CH
ardC
opy)
; if
H
ardC
opy=
1 th
en S
cree
nD
um
p(3
,0,2
,1.5
,1.5
,0,1
,0,e
rro
r);
cl o
sese
grap
h i c
s;
end.
Univers
ity of
Cap
e Tow
n
Sta
rt.
Initi
alis
e va
rious
var
iabl
es.
Initi
alis
e th
e S
ize
Dis
trib
utio
n.
Initi
alis
e C
onta
min
ant
Loca
tion.
Initi
alis
e S
uper
ficia
l V
eloc
ity F
ile.
Initi
alis
e C
olum
n C
ondi
tions
. In
itial
ise
Par
ticle
Con
ditio
ns.
Gen
erat
e U
nit
Hea
p F
iles.
Det
erm
ine
appr
opri
ate
step
siz
es f
n(si
ze).
Itera
te i
n Ti
me.
Det
erm
ine
the
appr
opria
te D
elta
T.
Itera
te i
n U
nit
Hea
ps.
Rea
d U
nit
Hea
p V
ecto
rs f
rom
Uni
t H
eap
File
.
Cho
ose
ap
pro
pri
ate
ste
p sr
ze t
or
unit
heap
cal
cula
tions
.
Det
erm
ine
Del
ta T
fn(
size
).
Exe
cute
Uni
t6.
(Equ
iv.
CS
TR
mod
el.)
~ ~~
I>
-("'
) 0
-=
c:
g. U
pd
ate
Hea
p C
onve
rsio
n, H
eap
Vec
tors
1=-
• ~
(3\
~
and
Writ
e U
nit
Hea
p R
esul
ts t
o F
ile.
§ 8..
>;
=
-·
(1)
~
'h..
_
:::3
'"O
""'"
>
; •
Incr
em
en
t H
ea
p V
ect
ors
ana-
1 6
9 ~
Bre
akt
hro
ug
h V
ect
ors
. 15;
.§
g
<
~
0 V
l •
::r
>;
·Wri
te c
on
vers
ion
s to
a f
ile. I
1; ?
' g-
s= ~
0 ("'
)• Q
..
. ""
l 0
c.. =
:;;
: 3
~ ~
Gra
ph
Bre
akt
hro
ug
h C
urv
es.
I
' lo
~
~ c"
~
~ C
l=
0:::
>;
I-lo
E
nd
. I
1("1
g
~
s· >
2:::
Dl (I
J
O'
~ ~
>;
= -·
...... c..
~
g == ~
~ o
a ~
c..
8 ~
fllO
I (J
Q
........
(1)
~ =
g
Cl
Q..
~ N
•
Q
~Cl
Z'
(IJ
Q
3 •
Q..
~ ~
~ :s.
I=' g.
"'1
Univers
ity of
Cap
e Tow
n
07/3
1/19
95
08:2
7 F
ilen
ame:
M
OD
EL6C
1.PA
S Pa
ge
1
Prog
ram
Mod
el6C
1;
(Mod
el6C
1.
( ( ( { (
*Thi
s pr
ogra
m
is a
cod
e fo
r he
ap
leac
hing
an
aly
sis.
*I
ncor
pora
tes
unst
eady
sta
te f
low
. *U
ses
Uni
t6C
1 w
hich
is
a m
odi.t
ied
form
of
th
e ch
emic
al
CSTR
m
odel
. *M
odel
6C1
val
id f
or
firs
t or
der
kin
etic
s,
use
Mod
el6C
2 fo
r v
aria
ble
ord
er
kin
etic
s.
} } } } } )
(Cod
ed:
Gra
ham
Dav
i es.
)
( D
epar
tmen
t of
C
hem
ical
E
ngin
eeri
ng.
) (
Uni
vers
ity
of
Cap
e To
wn.
}
( 6
Mar
ch
1995
. }
{ 6
June
19
95.
Upd
ated
. (G
MO)
}
(===
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
==}
{Dec
lara
tion
s:
}
uses
crt
,std
hdr,
gj,g
raph
,wor
lddr
,seg
raph
,int
egra
t,M
2C1B
rent
Roo
ts,U
nit6
C1;
cons
t
var {*
**T
hese
are
the
par
amet
ers
app
liab
le t
o th
e co
lum
n.**
****
****
****
****
}
Col
Len
gth
=1.
76;
(Col
umn
leng
th.
(m)
} C
olV
oida
ge =
0.49
8;
(Voi
dage
of
the
colu
mn.
}
Sat
Fra
c ~0.361;
(Fra
ctio
n o
f th
e vo
id s
pace
fil
led
wit
h fl
uid
. }
(Ass
umed
to
be
cons
tant
in
ti
me.
ie
Doe
s no
t ch
ange
} {w
ith
a ch
ange
in
the
f lo
wra
te.
} G
lobU
etFa
c =
1.0;
{G
loba
l w
etti
ng f
acto
rs o
f th
e p
arti
cles
. }
{***
The
se a
re c
olum
n nu
mer
ical
m
etho
d pa
ram
eter
s.**
****
****
****
****
****
}
NoU
ni tH
eaps
=5;
{Num
ber
of
un
it h
eaps
. }
Col
Del
taE
=1
/NoU
nitH
eaps
;
uSta
r =
1/86
400;
(Ref
eren
ce s
up
erfi
cial
v
elo
city
. E
quiv
alen
t to
1m
· }
{in
24
hour
s.
}
Spe
cCol
lter
s=1;
GfM
ax X
Gf
Hax:
::Y
=1.
0;
=1.
0;
{Max
imum
x-v
alue
fo
r co
nver
sion
gra
phs.
{H
axin
um y
-val
ue f
or
conv
ersi
on g
raph
s.
} }
{***
Oth
er
info
rmat
ion
requ
ired
.***
****
****
****
****
****
****
****
****
****
)
Val
ici'.
>ir
='F:
\TP6
\HO
DE
L6
1;C
Val
id d
irec
tory
fo
r gr
aphi
cs d
riv
ers.
}
{***
****
****
****
****
****
****
****
****
****
****
****
****
****
****
****
****
**}
supV
elfi
le
SupV
elD
ata
11.a
tchS
i zeD
T
Tex
t;
arra
y[1
.. 2
,1 .
• 2]
of D
oubl
e;
arra
y[1
•. m
axcJ
of
In
teg
er;
lntl
engt
hCri
t,In
tLen
gth,
Col
Del
ta_t
,Col
Del
taT
,Arb
Val
ue
Hol
dCon
v1,H
oldC
onv2
N
umSu
bint
,Tim
eCou
nter
Dou
ble;
D
oubl
e;
Inte
ger
;
07/3
1/19
95
08:2
7 F
ilen
ame:
M
ODEL
6C1.P
AS
Page
Co
nti
nu
e,C
oli
ters
AR
TBTC
,SR
T1B
TC,S
RT2
BTC
,Tot
Hea
pCon
v1,T
otH
eapC
onv2
T
imeV
ecto
r T
otH
eapA
RT
Vec
,Tot
Hea
pSol
SRT
1Vec
,Tot
Hea
pSol
SRT
2Vec
OT
fo
r W
atch
Size
co
nvR
esul
ts
:In
teg
er;
"Ver
yLon
gVec
tor;
·v
eryL
ongV
ecto
r;
"Sho
rtve
ctor
;·
"Sqr
Mat
; T
ext;
<===
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
==}
Pro
cedu
re
Init
iali
se_
Var
iou
s_V
aria
ble
s;
begi
n Vol
Liq
T
otV
olP
art
:=C
olD
elta
E*C
olV
oida
ge*C
olL
engt
h*S
atF
rac;
:=
Col
Oel
taE
*(1-
Col
Voi
dage
)*C
olle
ngth
*Glo
bWet
fac;
Cum
Tim
e =O
; U
nitH
eapC
onv1
=O
; U
nitH
eapC
onv2
=O
; C
oli
ters
=O
; T
imeV
ecto
r"[O
J =O
; en
d;
(---
----
----
----
----
----
----
----
·-··
···-
··--
····
·---
-···
-···
····
····
····
··}
P
roce
dure
ln
itia
lise
_S
up
erfi
cial
_V
elo
city
_F
ile;
begi
n assi
gn(S
upV
elF
ile,
'Sup
Vel
Fi.
Dat
');
rese
t(S
upV
elF
ile)
; Su
pVel
Dat
aC1,
1l:=
O;
SupV
elD
ata[
112J
:=O
; re
ad(S
upV
elF
1le,
Sup
Vel
Dat
aC2,
1J>
; re
ad(S
upV
elF
ile,
Sup
Vel
Dat
a[2,
2J>
; In
tlen
gthC
rit:
=C
olV
oida
ge*C
olD
elta
E*C
olle
ngth
; en
d;
(---
----
-·--
-·-·
----
---·
·-·-
·-···
····-
----
----
·---
----
----
-·-·
----
---·
----
) P
roce
dure
Co
l_In
itia
l_C
on
dit
ion
s;
begi
n for
Arb
lnde
x:=O
to
max
c do
be
gin Tot
Hea
pAR
TV
ec"[
Arb
lnde
xJ
=O
Tot
Hea
pSol
SRT
1Vec
"[A
rbin
dexJ
=O
T
otH
eapS
olSR
T2V
ec"[
Arb
inde
xJ
=O
end;
Tot
Hea
pAR
TV
ec"[
1J:=
1;
for
Arb
inde
x:=O
to
max
v do
be
gin AR
TBTC
" [A
rbin
dexJ
:=
O;
SRT1
BTC
"[A
rbin
dexJ
:=O
; SR
T2B
TC
"[A
rbin
dexJ
:=O
; T
otH
eapC
onv1
"CA
rbin
dexJ
:=O
; T
otH
eapC
onv2
"[A
rbin
dexJ
:=O
; en
d·
end;
'
{***
Impo
sed
BC
for
heap
. }
2
n Q
Q.. ~
~ -· .tll ~ s· Q' .., :: Q
Q.. ~
,;\ n ~ ~ 0
0 .
Univers
ity of
Cap
e Tow
n
07/3
1/19
95
08:2
7 Fi
lena
me:
M
OO
EL6C
1.PA
S Pa
ge
3
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re G
ener
ate_
Uni
tHea
pDat
aFil
es;
{Thi
s pr
oced
ure
is u
sed
to g
ener
ate
the
Uni
tHea
pDat
aFil
es.
The
file
s are
}
{nam
ed M
6C1H
1 th
roug
h to
M6C
1HX.
)
var U
nitH
eapN
oS,F
ileN
ame
Hea
pFil
e
begi
n for
Uni
tHea
plnd
ex:=
1 to
NoU
nitH
eaps
do
be
gin Str
(Uni
tHea
plnd
ex,U
nitH
eapN
oS);
Fi
leN
ame:
=Con
catC
'M6C
1H',U
nitH
eapN
oS,'
.Oat
');
assi
gn(H
eapF
ile,
Fil
eNam
e);
rew
rite
(Hea
pFil
e);
for
Siz
elnd
ex:=
1 to
M do
be
gin fo
r A
rbln
dex:
=O
to N
+1
do
begi
n
:Str
ing
; :T
ext;
wri
te(H
eapF
ile,
AR
T"[
Siz
elnd
ex,A
rb!n
dexJ
,' ')
; w
rite
(Hea
pFil
e,S
RT
1"[S
ize!
ndex
,Arb
!nde
xl, •
')
; w
rite
lnC
Hea
pFil
e,S
RT
2"C
Siz
e!nd
ex,A
rbln
dexJ
>;
end·
w
rite
lnC
Hea
pFil
e);
end;
',Uni
tHea
pCon
v2);
w
rite
lnC
Hea
pFil
e,U
nitH
eapC
onv1
, •
wri
teln
(Hea
pFil
e);
wri
teln
(Hea
pFil
e,'C
umT
ime
1,C
umT
ime)
; cl
ose(
Hea
pFil
e);
end·
en
d;
'
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re R
ead_
AR
T_an
d_SR
Ts;
var U
nitH
eapN
oS,F
ileN
ame
Hea
pFil
e
begi
n Str
(Uni
tHea
plnd
ex,U
nitH
eapN
oS);
Fi
leN
ame:
=Con
cat(
'M6C
1H',U
nitH
eapN
oS, '.
Oat'
);
assi
gn(H
eapF
ile,
Fil
eNam
e);
rese
tCH
eapF
ile)
; fo
r S
izel
ndex
:=1
to M
do
begi
n for
Arb
lnde
x:=O
to
N d
o be
gin read
(Hea
pFil
e,A
RT
"[S
izel
ndex
,Arb
Jnde
xl )
; re
ad(H
eapF
ile,
SR
T1"
[Siz
elnd
ex,A
rbln
dexl
);
read
ln(H
eapF
ile,
SR
T2"
CS
izeJ
ndex
,Arb
lnde
xl);
en
d·
read
(Hea
pFil
e,A
rbV
alue
);
AR
T"[
Size
lnde
x,N
+1l
:=T
otH
eapA
RT
Vec
"[U
nitH
eapl
ndex
l; re
ad(H
eapf
ile
SRT
1"[S
izel
ndex
,N+1
J >;
re
adln
CH
eapF
i(e,
SR
T2"
[Siz
elnd
ex,N
+1J
);
:Str
ing
; :T
ext;
07/3
1/19
95
08:2
7 F
ilen
ame:
M
OO
EL6C
1.PA
S Pa
ge
4
end·
re
adln
(Hea
pFil
e,H
oldC
onv1
,Hol
dCon
v2);
cl
ose(
Hea
pFi
le);
end;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--)
Pro
cedu
re Y
rite_
AR
T_a
nd_S
RT
s;
var U
nitH
eapN
oS,F
ileN
ame
Hea
pF i
le
begi
n Str
(Uni
tHea
plnd
ex,U
nitH
eapN
oS);
F
ileN
ame:
=C
onca
t(1M
6C1H
1,U
nitH
eapN
oS,
1.0
at1);
assi
gn(H
eapF
ile,
Fil
eNam
e);
rew
rite
(Hea
pF
ile)
; fo
r S
ize!
ndex
:=1
to M
do
begi
n for
Arb
lnde
x:=O
to
N+1
do
be
gin if A
RT-
CS
izel
ndex
,Arb
inde
xl>
FC
onc
Tol
th
en
wri
te(H
eapF
ile,
AR
T"[
Siz
elnd
ex,A
rbJn
dexl
,' el
se wri
te(H
eapF
ile,
0.00
0,0
');
if S
RT
1"[S
izel
ndex
,Arb
lnde
xJ>
SC
onc_
Tol
th
en
wri
te(H
eapF
ile,
SR
T1"
[Siz
elnd
ex,A
rbln
dexJ
,' el
se wri
te(H
eapF
i le
,0.0
00
, •
• );
if S
RT
2"[S
izel
ndex
,Arb
lnde
x]>
SC
onc_
Tol
th
en
wri
teln
(Hea
pFil
e,S
RT
2"[S
izel
ndex
,Arb
lnde
xl)
else
wri
teln
(Hea
pF
ile,
0.0
00
);
I )
:Str
ing
; :T
ext;
I )
end;
w
rite
ln(H
eap
Fil
e);
end·
w
rite
ln(H
eapF
ile,
Uni
tHea
pCon
v1,
• ',U
nitH
eapC
onv2
);
wri
teln
(Hea
pF
ile)
; w
rite
ln(H
eapF
ile,
•cum
Tim
e 1,C
umtim
e);
clos
e(H
eapF
i le
);
Uni
tHea
pCon
v1:=
0;
Uni
tHea
pCon
v2:=
0;
end;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re O
eter
min
e_M
axim
um_O
elta
T_f
n_Si
ze;
begi
n for
Uni
tHea
plnd
ex:=
1 to
NoU
nitH
eaps
do
be
gin fo
r S
izel
ndex
:=1
to M
do
end;
OT
for
Yat
chS
ize"
[Uni
tHea
plnd
ex,S
izel
ndex
J :=
Max
Oel
taT
-
*S
ize0
ata[
Siz
eln
dex
,1l*
Siz
e0at
a[S
izel
nd
ex,1
J /S
ize0
ata[
M,1
J/S
ize0
ata[
M,1
l;
for
Uni
tHea
plnd
ex:=
1 to
NoU
nitH
eaps
do
Y
atch
Siz
eOT
[Uni
tHea
plnd
exl:
=M
;
end;
Univers
ity of
Cap
e Tow
n
07/3
1/19
95
08:2
7 Fi
lena
me:
M
ODEL
6C1.P
AS
Page
5
(-·-
----
----
----
----
--·-
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re D
eter
min
e_C
orre
spon
ding
_Col
Oel
taT
;
(Thi
s pr
oced
ure
uses
th
e m
etho
d of
ch
arac
teri
stic
s to
det
erm
ine
the
} {p
rogr
essi
on i
n tim
e fo
r a
give
n C
harC
olO
elta
E.
It a
llow
s fo
r un
stea
dy
} {
stat
e fl
ow.
}
begi
n Intl
eng
th
:=O
; C
olD
elta
t:=
O;
whi
le
IntL
engt
h<>
lntL
engt
hCri
t do
be
gin
if E
OF(
SupV
elFi
le)
then
beg
in
clos
eseg
raph
ics;
w
rite
ln('
Insu
ffic
ien
t D
ata
in t
he S
up
erfi
cial
Vel
ocit
y D
ata
Fil
e to
');
wri
teln
('ex
ecu
te t
he r
equi
red
num
ber
of
itera
tio
ns.
');
wri
teln
('P
rese
nt
num
ber
of
iter
atio
ns:
=
1,T
imeC
ount
er);
C
ontin
ue:=
O·
IntL
engt
h:=
intL
engt
hCri
t;
read
ln;
end·
In
tlen
gth:
=ln
tLen
gth+
(Sup
Vel
Dat
a[2,
1]-S
upV
elD
ata[
1, 1
J)*S
upV
elD
ata[
2,2J
; C
olD
elta
t:=
Col
Del
ta_t
+(S
upV
elD
ata[
2,1]
-Sup
Vel
Dat
a[1,
1])
; If
In
tlen
gth>
lntL
engt
hCri
t th
en b
egin
A
rbV
alue
:=In
tlen
gth-
Intl
engt
hCri
t;
Col
Del
ta t
:=C
olD
elta
t-
Arb
Val
ue/S
upV
elD
ata[
2,2J
; In
tLen
gtfi
:=In
tLen
gthC
rit;
S
upV
elD
ata[
1,1J
:=S
upV
elD
ata[
2,1J
-Arb
Val
ue/S
upV
elD
ata[
2,2J
; su
pVel
Dat
a[1,
2J:=
Sup
Vel
Dat
a[2,
2J;
end
else
beg
in
SupV
eLD
ata[
1, 1
1:=
Sup
Vel
Dat
a[2,
1J;
SupV
elD
ata[
1,2]
:=
Sup
Vel
Dat
a[2,
2J;
read
(Sup
Vel
Fi l
e,
SupV
elD
ata
[2.,
1]
);
read
(Sup
Vel
Fil
e,
Sup
Vel
Dat
a[2,
2])
end·
en
d·
' C
olD
elta
T:=
Col
Del
ta t
*uS
tar/
Col
Len
gth;
C
umTi
me:
=Cum
Tim
e+C
oTD
elta
T;
end;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re D
eter
min
e_A
ppro
pria
te_R
ef_S
ize_
Cla
ss_D
elta
T;
begi
n Siz
eind
ex:=
Wat
chS
izeD
T[U
nitH
eapl
ndex
];
Iter
atio
ns:=
INT
(CoL
Del
taT
/CD
T
for
Wat
chS
ize"
[Uni
tHea
plnd
ex,S
izei
ndex
J))
+1 ·
--
Del
taT
:=C
olD
elta
T/I
tera
tion
s;
wri
teln
(Siz
eind
ex);
en
d;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re C
heck
_Whe
ther
_Wat
chS
ize_
Sti
ll_V
alid
;
var F
lag
Inte
rG1
Inte
gVal
Inte
ger;
V
eryL
ongV
ecto
r;
Rea
l;
07/3
1/19
95
08:2
7 F
ilen
ame:
M
OD
EL6C
1.PA
S Pa
ge
6
begi
n Siz
eind
ex:=
Wat
chS
izeD
T[U
nitH
eapi
ndex
];
Flag
:=O
; if
AR
T"[
Siz
eind
ex,N
]<0.
5*A
RT
"[S
izei
ndex
,(N
+1)
] be
gin fo
r A
rbin
dex:
=O
to N
do
then
begi
n . i
f (A
RT
"[S
izei
ndex
,Arb
inde
xJ<
0.8*
AR
T"[
Siz
eind
ex,C
Arb
inde
x+1)
l)
or
(ART
0
[Siz
elnd
ex,A
rbin
dex]
>1.
2*A
RT
"[S
izei
ndex
,(A
rbin
dex+
1)])
th
en F
lag:
=F
lag+
1;
end·
~d•
I
if F
lag=
O
then
be
gin fo
r A
rbln
dex:
=O
to N
+1
do
Inte
rG1
[Arb
inde
xJ:=
AR
T"[
Siz
eind
ex,A
rbln
dexJ
; In
tegr
ateV
ecto
r(In
terG
1,D
elta
E,0
,(N
+1)
,lnt
egV
al);
fo
r A
rbln
dex:
=O
to N
+1
do
if
(AR
T"[
Siz
elnd
ex,A
rbin
dex]
<(0
.9*I
nteg
val)
) or
(A
RT
"[S
izei
ndex
,Arb
inde
xl>
(1.1
*1nt
egva
l))
then
Fla
g:=
1;
if
(Fla
g=O
) an
d (W
atch
Size
DT
[Uni
tHea
pind
exl>
1)
then
W
atch
Size
DT
[Uni
tHea
pind
exJ:
=W
atch
Size
DT
[Uni
tHea
pind
exJ-
1;
end
else
if
AR
T"(
Size
inde
x,N
t1l>
FCon
c_T
ol
then
DT
_for
_Wat
chSi
ze"[
Uni
tHea
plnd
ex
,Siz
eind
exl
:=2*
DT
_for
_Wat
chS
ize"
[Uni
tHea
plnd
ex,S
izei
ndex
l;
·
end;
(---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re U
pDat
e_T
otH
eapC
onv;
begi
n Tot
Hea
pCon
v1"[
Col
lter
sJ:=
Tot
Hea
pCon
v1"(
Col
iter
sJ
+Glo
bWet
Fac*
Uni
tKea
pCon
v1/N
oUni
tHea
ps;
Tot
Hea
pCon
v2·c
coli
ters
] :=
Tot
Hea
pCon
vz·c
coll
ters
] +G
lobW
etFa
c*U
nitH
eapC
onv2
/NoU
nitH
eaps
; en
d;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re U
pDat
e_T
otH
eapV
ecto
r;
begi
n Tot
Hea
pAR
TV
ec"[
Uni
tHea
plnd
ex]
:=A
RT
"C1,
N+1
l; T
otH
eapS
olSR
T1V
ec"[
Uni
tHea
plnd
exl:=
Tot
Hea
pSol
SRT
1Vec
"[U
nitH
eapi
ndex
J+
(Uni
tHea
pcon
v1-H
oldC
onv1
);
Tot
Hea
pSol
SRT
2Vec
"[U
nitH
eapi
ndex
l:=T
otH
eapS
olSR
T2V
ec"[
Uni
tHea
plnd
ex]+
(U
nitH
eapC
onv2
-Hol
dCon
v2);
en
d;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re
Incr
emen
t_T
otH
eapV
ecto
rs_a
nd_B
TC
Vec
tors
;
begi
n AR
TB
TC
"[C
olite
rs +
1]
:=T
otH
eapA
RT
Vec
"[N
oUni
tHea
psl;
SRT
1BT
C"[
Col
lters
+1J
:=G
lobW
etFa
c*(T
otH
eapS
olSR
T1V
ec·[
NoU
nitH
eaps
J);
SRT
2BT
C"[
Col
iters
+1J
:=G
lobW
etFa
c*(T
otH
eapS
olSR
T2V
ec"[
NoU
nitH
eaps
]);
for
Arb
lnde
x:=N
oUni
tHea
ps d
ownt
o 2
do
begi
n Tot
Hea
pArt
Vec
"[A
rbin
dexJ
:=
Tot
Hea
pArt
Vec
"[A
rbln
dex-
11;
Tot
Hea
pSol
SRT
1Vec
"[A
rbln
dexJ
:=T
otH
eapS
olSR
T1V
ec"[
Arb
lnde
x-1J
;
Univers
ity of
Cap
e Tow
n
07/3
1/19
95
08:2
7 Fi
lena
me:
M
ODEL
6C1.P
AS
Page
7
Tot
Hea
pSol
SRT
2Vec
·[Arb
lnde
x] :
=T
otH
eapS
olSR
T2V
ec·[
Arb
lnde
x-1J
; en
d·
Tot
Hea
pArt
Vec
"[1]
:=1;
T
otH
eapS
olSR
T1V
ec"[
1]:=
0;
Tot
Hea
pSol
SRT
2Vec
·[1]
:=O
;
end;
{---
----
----
----
----
----
----
----
--·-
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re G
raph
_Con
vers
ion_
and_
BT
C_C
urve
s;
begi
n init
SE
Gra
phic
s(V
alid
Dir
);
SetC
urre
ntW
indo
w(2
);
Bor
derC
urre
ntW
indo
w(1
);
Set
Axe
sTyp
e(0,
0);
Sca
leP
lotA
rea(
0.0,
0.0,
(Tim
eVec
tor·
[col
lter
sl),
GfM
ax Y
); (*
**}
Set
XY
lnte
rcep
ts(0
.0,0
.0);
-
Set
Col
or(2
);
Dra
wX
Axi
s((T
imeV
ecto
r·cc
ollt
ersJ
/5),
1);
(*
**}
Dra
wY
Axi
s((G
fMax
Y
/5),
1);
L
abel
XA
xisC
1,0)
;-L
abel
YA
xis(
1,0)
; T
itle
XA
xis(
'Dm
lss
Tim
e (D
mls
s T
ime=
(tim
e *
uS
tar)
/Co
llen
gth
');
Tit
leY
Axi
s('F
rac.
C
onv.
an
d BT
C C
urve
s');
T
itleW
indo
wC
'Mod
el6C
1');
L
abel
Gra
phW
indo
w(1
,930
,GL
abel
1,0,
0);
Lab
elG
raph
Win
dow
(1,9
00,G
labe
l2,0
,0);
fo
r A
rbln
dex:
=O
to C
oll
ters
do
Dat
aSet
X"[
Arb
lnde
xJ
=T
imeV
ecto
r-[A
rbln
dexJ
; fo
r A
rbln
dex:
=O
to C
oll
ters
do
Dat
aSet
Y"[
Arb
lnde
xJ
=Tot
Hea
pCon
v1"[
Arb
lnde
xJ;
Lin
ePlo
tDat
a(D
ataS
etx·
,oat
aSet
Y·,
ccol
lter
s +
1),
5,0
) fo
r A
rbln
dex:
=O
to C
oll
ters
do
Dat
aSet
Y"[
Arb
lnde
x]
=Tot
Hea
pCon
v2"[
Arb
lnde
xJ;
Lin
ePlo
tDat
a(D
ataS
etx·
,Dat
aSet
Y",
(Col
lter
s +
1),
4,0
) fo
r A
rbln
dex:
=O
to C
oll
ters
do
Dat
aSet
Y-[
Arb
lnde
xJ
=AR
TB
TC
-[A
rbln
dexl
; L
ineP
lotD
ata(
Dat
aSet
x· ,D
ataS
etY
",(C
ollt
ers
+1)
,3, 1
) fo
r A
rbln
dex:
=O
to C
oll
ters
do
Dat
aSet
v· [
Arb
lnde
xl
=SR
T1B
TC
"[A
rbln
dexJ
; L
ineP
lotD
ata(
Dat
aSet
x",D
ataS
etv·
,cco
llte
rs +
1),
5,1
) fo
r A
rbln
dex:
=O
to C
oll
ters
do
Dat
aSet
Y"[
Arb
lnde
x]
=SR
T2B
TC
"[A
rbln
dexJ
; L
ineP
lotD
ata(
Dat
aSet
x",D
ataS
etY
",(C
ollt
ers
+1
),4
,1)
end;
(===
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
==}
{MAI
N PR
OGRA
M:
} {=
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
} be
gin ne
w(B
eta1
_k);
ne
w(K
appa
P1_k
); ne
w(K
appa
S1_k
); ne
w(L
ambd
a1
k);
new
(Con
tRat
loV
1);
new
( AR
T);
new
(SR
T1)
; ne
w(S
RT
2);
new
(Bet
a2
k);
new
(Kap
paP2
_k);
new
(Kap
paS2
_k);
new
( Lam
bda2
k)
; ne
w(C
ontR
atlo
V2>
; ne
wC
AR
TP1)
; ne
w(S
RT
1P1)
; ne
w(S
RT
2P1)
;
new
(Del
taT
_k);
new
(SR
T1P1
C);
new
(SR
T2P1
C);
07/3
1/19
95
08:2
7 F
ilen
ame:
M
ODEL
6C1
.PA
S Pa
ge
8
new
(Dat
aSet
X);
ne
w(A
Mat
rix)
; ne
w(A
RTB
TC);
new
(Tot
Hea
pAR
TV
ec);
new
(Tot
Hea
pCon
v1);
ne
w(D
T_f
or_W
atch
Size
);
new
(Dat
aSet
Y);
ne
w(A
lnve
rse)
; ne
w(S
RT1
BTC
); ne
w(T
otH
eapS
olSR
T1V
ec);
ne
w(T
otH
eapC
onv2
);
assi
gn(C
onvR
esul
ts,'C
onvR
esul
ts.D
at')
; re
wri
te(C
onvR
esul
ts);
cl
ose(
Con
vRes
ults
);
lnit
iali
se_
Var
iou
s_V
aria
ble
s;
Siz
e D
istr
ibu
tio
n I
nit
iali
sati
on
; C
onta
min
ant
Loc
atio
n In
itia
lisa
tio
n;
Oet
erm
ine_
Mod
el_P
aram
eter
s_fn
_Siz
e;
lnit
iali
se_
Su
per
fici
al_
Vel
oci
ty_
Fil
e;
, C
ol
Init
ial
Con
diti
ons;
P
arti
cle
Init
ial
Con
diti
ons;
G
ener
ate=
Uni
tHea
poat
aFil
es;
Det
erm
ine_
Max
imum
_Del
taT
_fn_
Size
;
whi
le C
ollt
ers<
Spe
cCol
lter
s do
be
gin Co
llte
rs:=
Co
llte
rs+
1;
Det
erm
ine
Cor
resp
ondi
ng C
olD
elta
T;
Tim
eVec
tor·
ccol
lter
sJ:=
Cum
Tim
e;
for
Uni
tHea
plnd
ex:=
1 to
NoU
nitH
eaps
do
begi
n Rea
d AR
T an
d SR
Ts;
if A
RT-
cT,N
+TJ>
FCon
c To
i th
en
begi
n -
Clo
sese
grap
hics
;
new
(SR
T2B
TC);
new
(Tot
Hea
pSol
SRT
2Vec
);
new
(T im
eVec
tor)
;
Det
erm
ine
App
ropr
iate
Ref
S
ize
Cla
ss D
elta
T;
Det
erm
ine-
Del
taT
fn
Siz
e;-
--
Exe
cute
Uni
t6C
1;-
-C
heck
_Wne
ther
_Wat
chS
ize_
Sti
ll_V
alid
; en
d·
upD
ate_
Tot
Hea
pcon
v;
UpD
ate
Tot
Hea
pVec
tor;
W
rite
ART
an
d SR
Ts;
end;
-
--
lncr
emen
t_T
otH
eapV
ecto
rs_a
nd_B
TC
Vec
tors
;
Ass
ign(
Con
vRes
ults
,'Con
vRes
ults
.Dat
');
App
end(
Con
vRes
ults
);
Wri
teln
(Con
vRes
ults
,Tim
eVec
tor-
[Col
lter
sJ:8
:4,'
,Tot
Hea
pCon
v1"[
Col
lter
sJ:8
:4);
C
lose
(Con
vRes
ults
);
end;
Gra
ph_C
onve
rsio
n_an
d_B
TC
_Cur
ves;
read
ln(H
ardC
opy)
;
Univers
ity of
Cap
e Tow
n
~
~
ro ~ ~ II
II II II II II II II II II II
" II II
" ~ " ~ II 0 II ~ " ~ " ~ " 0 " " II
II
0 II
" ~ II ~ II ~ II ~ II
~ II II
u II ~ II ~ N " w " 0 0 " a " ~ " ~ II
~ II ~ II E E II ro ~ II c 0 II ~ c II
~ II ~ II ~ II u II ~ II
II c II ~ II ~ .. II ~ 00 II
~ u II N II
II ~ II ~ ~ ~ ~ a ~ ro " 0 ~ " u ~ " ~ ~ ~ " ~ ~ 00 " ~ ro ~ " ~ 00 " 0 " ~ " ~ u
~ " " ~ c II 0 ~ ~
Univers
ity of
Cap
e Tow
n
06/1
6/19
95
15:5
1 F
ilen
ame:
U
NIT
6C1.
PAS
Page
1
Uni
t U
ni t.
6C1;
{Uni
t6C
1.
{ *
Th
is p
rogr
am
is s
imil
ar t
o
* Th
e pr
ogra
m c
alcu
late
s th
e an
d so
lid
rea
ctan
ts w
ithi
n de
velo
ped
by D
ixon
.
Mod
el5E
1.PA
S.
} co
nce
ntr
atio
n p
rofi
le o
f fl
uid
}
{ { { { { { { { {
* T
his
prog
ram
onl
y pr
ovid
es a
par
ticl
e us
ing
the
equa
tion
s as
} }
for
soli
d r
eact
ant
ord
ers
of
1.
}
* A
ssum
ptio
ns
in t
his
mod
el
incl
ude:
}
-Th
e so
lid
rea
ctan
t d
epo
sits
wit
hin
the
part
icle
}
rese
mbl
e th
ose
on
the
surf
ace.
}
-Th
e d
epo
sits
wou
ld
both
re
act
to t
he
sam
e ex
ten
t }
if e
ach
wer
e ex
pose
d to
the
sam
e ac
id c
on
cen
trat
ion
}
for
the
sam
e ti
me.
}
{Cod
ed:
Gra
ham
Dav
ies.
}
{ D
epar
tmen
t of
C
hem
ical
E
ngin
eeri
ng.
} {
Uni
vers
ity
of
Cap
e To
wn.
}
{ 28
F
ebru
ary
1995
. }
{ 06
Ju
ne
1995
U
pdat
ed
(GM
O).
} {=
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
} In
terf
ace
{===
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
==}
{Dec
lara
tio
ns:
}
uses
crt
,std
hd
r,g
j,g
rap
h,w
orl
dd
r,se
gra
ph
,in
teg
rat,
M2
C1
Bre
ntR
oo
ts;
cons
t
{***
The
se a
re t
he p
aram
eter
s ap
pli
cab
le t
o t
he r
efer
ence
siz
e cl
ass.
**
*}
Bet
a1
Kap
paP1
0.
264;
4
.5;
Bet
a2
= 0
.0;
Kap
paP2
=
0.0
;
Voi
dage
0.
01;
M
=1
0·
Ref
Siz
eCl
=5;
' V
iew
Size
Cl=
5;
{Dim
ensi
onle
ss S
toic
hiom
etri
c ra
tio
def
ined
pg
11
} {R
atio
of
reat
ion
rat
e of
so
lid
rea
ctan
t re
sid
ing
}
{wit
hin
the
par
ticl
e to
por
ous
dif
fusi
on
of
flu
id
} {
into
th
e p
art
icle
. D
efin
ed p
g 11
D
ixon
. }
{Voi
dage
of
the
soli
d p
arti
lces
.(P
oro
sity
) }
{Num
ber
of s
ize
clas
ses.
}
{Def
ines
the
ref
eren
ce s
ize
clas
s.
} {T
he
con
cen
trat
ion
P.r
ofil
es o
f th
is s
izec
lass
are
}
{gra
phed
in
gra
ph
1:
(NB
Vie
wSi
zeC
l <=
M
.) }
{***
The
se a
re n
umer
ical
m
etho
d pa
ram
eter
s.**
****
****
****
****
****
****
***}
N
=19;
Del
taE
=
1/(N
+1)
;
Max
Del
taT
=
0.00
1;
Nor
m C
rit
=le
-6·
-,
{Hal
f th
e nu
mbe
r of
in
teri
or
po
ints
. r=
O
and
r=R
} {n
ot
incl
uded
. }
{Spa
ce
Incr
emen
t.
Cal
cula
ted
from
1/
CN
+l)
. (S
ince
}
{ R
=i*d
E an
d R
is
at p
oint
N+1
.) }
{Max
imum
per
mit
ted
tim
e in
crem
ent
for
part
icle
s.
}
{Con
verg
ence
cri
teri
a b
ased
on
the
norm
of
vec
tor
} {*
**
}
06/1
6/19
95
15 :5
1 F
ilen
ame:
U
NIT
6C1.
PAS
Page
Con
v C
rit
=1e
-8
SCon
c To
l =1
e-4
FC
on
()o
l =1
e-4
Max
lter
=1
00;
Pri
nt_
Cri
t=1
;
{Con
verg
ence
cri
teri
a f
or
the
Bre
nt R
outi
ne.
} {D
imen
sion
less
co
nce
ntr
atio
n o
f so
lid
bel
ow w
hich
}
{Dim
ensi
onle
ss c
on
cen
trat
ion
of
flu
id
reac
tan
t }
{bel
ow w
hich
it
is
ass
umed
to
be
neg
igib
le.
} {
it
is a
ssum
ed
to b
e n
egli
gib
le.
} {M
axim
um
itera
tio
ns
for
the
Bre
nt R
outi
ne.
}
{***
Oth
er
info
rmat
ion
requ
ired
.***
****
****
****
****
****
****
****
****
****
*}
Val
idD
ir
='F
:\T
P6
\';{
Val
id d
irec
tory
fo
r g
rap
hic
s d
riv
ers.
}
Ord
SP1
Ord
SP2
=1.
0;
=1.
0;
{Rea
ctio
n or
der
of
the
soli
d
in t
he
po
res.
{F
or
this
pro
gram
th
ese
need
to
be
set
at u
nit
y.
{See
Uni
t6C
2 fo
r v
aria
ble
ord
er
reac
tio
n o
rder
s.
} } }
{***
****
****
****
****
****
****
****
****
****
****
****
****
****
****
****
****
***}
type
var ar
ray
l= A
rray
[1 .
. 50,
1 ..
2J
of D
oubl
e;
Tot
Vol
Par
t,V
olL
iq,D
elta
T,I
tera
tion
s,C
umT
ime
Uni
tHea
pCon
v1,U
nitH
eapC
onv2
A
rbin
dex,
Siz
elnd
ex,e
rror
,Uni
tHea
pind
ex
Rep
eats
,Plo
t_V
ar,F
lag1
,Har
dCop
y D
ataS
etX
,Dat
aSet
Y
.
:Dou
ble;
:D
oubl
e;
: In
teg
er;
Bet
a1_k
,Bet
a2 k,KappaP1_k,K~ppaP2_k,KappaS1_k,KappaS2_k
Lam
bda1
k,
Lam
£ida
2_k,
Con
tRat
10V
1,C
ontR
atio
V2
Del
taT
K
AR
T,A
RTP
1,SR
T1,S
RT2
,SR
T1P1
,SR
T2P1
,SR
T1P1
C,S
RT2
P1C
A
Mat
rix,
Ain
vers
e
: In
teg
er;
:·ve
ryL
ongV
ecto
r;
: "S
hort
Vec
tor;
: ·
sho
rtV
ecto
r;
AR
TP1
CV
al,S
umSi
zeD
ata
2,N
uSta
r L
ambd
a1,L
ambd
a2,K
appa
S1,K
appa
S2
GL
abel
1,G
labe
l2,G
Lab
el3
Siz
eDat
a :
Arr
ay[1
.•
50, 1
•• 2
] of
Rea
l;
{ART
A
lpha
fn
(r)
at
tim
e T
{A
RTP1
A
lpha
fn
(r)
at
tim
e T+
1 {S
RT1
Sigm
a fn
(r)
at
tim
e T
{S
RTs
P1
Sigm
a fn
(r)
at
tim
e T+
1 (g
uess
ed)
: "A
rray
1;
:"S
qrM
at;
:·sq
rMat
; :E
xten
ded;
:E
xten
ded;
:S
trin
g;
rang
e o· •
. M
0 •. N
+1}
rang
e O
.• M
0
.• N
+1}
rang
e 0
•. M
0 ..
N+1
} ra
nge
O •.
M
0 •. N
+1}
{SR
TsP1
C
Sigm
a fn
(r)
at
tim
e T+
1 (c
alcu
late
d)
rang
e O
•• M
0
•. N
+1}
{YV
ect
'Co
nst
' v
ecto
r in
Crank~Nicolson
met
hod
rang
e O
•. N
+1
} C
AM
atrix
M
atri
x of
Cra
nk-N
icol
son
coef
fici
ents
ra
nge
N*
N
} {D
ataS
etX
X
vec
tor
used
in
the
gra
phin
g ro
uti
ne
rang
e 0
.• N
+1
}
{***
****
****
****
****
****
****
****
****
****
****
****
****
****
****
****
****
***}
Pro
cedu
re S
ize
Dis
trib
uti
on
In
itia
lisa
tio
n;
Pro
cedu
re C
onta
min
ant
Loc
atT
on
Init
iali
sati
on
; P
roce
dure
Det
erm
ine
Mod
el
Par
amet
ers
fn S
ize;
P
roce
dure
Det
erm
ine-
Del
taT
fn
Siz
e; -
-P
roce
dure
Par
ticl
e_T
nit
ial=
Co
nd
itio
ns;
P
roce
dure
Gue
ss
SRTs
P1;
Pro
cedu
re R
eGue
ss
SRTs
P1;
Pro
cedu
re C
ale
Cra
nk
Nic
olso
n M
atri
x(va
r Y
Vec
tor
:Sh
ort
Vec
tor)
; P
roce
dure
Cal
c=A
RTP
1=an
d_SR
TsP1
C;
2
Univers
ity of
Cap
e Tow
n
"
06/1
6/19
95
15:5
1 Fi
lena
me:
U
NIT
6C1.
PAS
Page
3
Pro
cedu
re C
heck
Con
verg
ence
; P
roce
dure
Upd
ate_
AR
T_an
d_SR
Ts;.
Pro
cedu
re C
ale
Con
vers
ion;
P
roce
dure
Gra
pn1_
Init
iali
se;
Pro
cedu
re G
raph
1 R
esul
ts;
Pro
cedu
re E
xecu
te_U
nit6
C1;
f ~~1;
~;~~
~~i~
~===
====
====
====
====
====
====
====
====
====
====
====
====
====
====
=}
{===
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
==}
Pro
cedu
re S
i ze_
Di s
trib
uti
on_!
ni. ti
al i
sat
ion;
{Thi
s pr
oced
ure
sets
up
the
size
dis
trib
uti
on
dat
a ar
ray
. In
itia
lly
the
}
{Siz
eDat
a ar
ray
con
tain
s ra
dius
in
form
atio
n an
d fr
acti
on
al
volu
me
} {i
nfor
mat
ion
(ie
R a
nd V
p/V
tot)
. On
out
put
it c
onta
ins
rela
tiv
e ra
dius
}
{inf
orm
atio
n an
d re
lati
ve
volu
me
info
rmat
ion
Cie
R/
R re
f an
d V
p/V
p re
f).
} {
Idea
lly
th
is
info
rmat
ion
wou
ld.b
e re
ad
in f
rom
a
data
fil
e.
-}
var R
efR
adiu
s,V
olR
efP
art
:Dou
ble;
begi
n Size
Dat
aC.1
, 11
= 3
7.00
e-3/
2 s
zeD
ata
[1, 2
1 =
58.
1/1
00
s i z
eDat
a [2
, 11
= 3
1.25
e-3/
2 S
zeD
ata[
2,21
=
15.
8/10
0 Si
zeD
ataC
3, 1
1 =
22.
00e-
3/2
S ze
Dat
a[3,
21
= 1
0.3/
100
Si z
eDat
a [4
, 11
= 1
6.10
e-3/
2 S
zeD
ata[
4,2l
=
4.
1/10
0 Si
zeD
ata
[5, 1
1 =
11.
35e-
3/2
S ze
Dat
a [5
, 21
=
3.2/
100
Siz
eDat
a[6,
11
=
8.10
e-3/
2 s
zeD
ata
[6, 2
1 =
1.
8/10
0 S
izeD
ata[
7, 11
=
5.
73e-
3/2
S ze
Dat
a[7,
2l
=
1.3/
100
Siz
eDat
a[8,
11
=
4.05
e-3/
2 S
zeD
ata
[8, 2
1 =
1.
0/10
0 S
izeD
ata[
9, 11
=
2.
86e-
3/2
S ze
Dat
a[9,
21
=
0.6/
100
Siz
eDat
aC10
,1l
=
1.18
e-3/
2 S
zeD
ata[
10,2
l =
3.
8/10
0
Vol
Ref
Par
t:=
Tot
Vol
Par
t*S
izeD
ata[
Ref
Siz
eCl,
21;
Ref
Rad
ius
:=S
izeD
ata[
Ref
Siz
eCl,
11;
Sum
Size
Dat
a_2:
=0;
for
Siz
elnd
ex:=
1 to
M do
be
gin Siz
eDat
a[S
izel
ndex
, 11:
=S
izeD
ata[
Siz
elnd
ex,1
1/R
efR
adiu
s;
Siz
eDat
a[S
izel
ndex
,2J:
=S
izeD
ataC
Siz
elnd
ex,2
J*T
otV
olP
art/
Vol
Ref
Par
t;
Sum
Size
Dat
a 2:
=Sum
Size
Dat
a 2+
Siz
eDat
a[S
izel
ndex
,2l;
en
d;
--
NuS
tar:
=V
olL
iq/(
Voi
dage
*Vol
Ref
Par
t);
{Rat
io o
f vo
lum
e of
bu
lk
flu
id t
o fl
uid
in
par
ticl
e po
res.
end;
}
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re C
onta
min
ant_
Loc
atio
n_ln
itia
lisa
tion
;
{Thi
s pr
oced
ure
is u
sed
to d
efin
e th
e co
ntam
inan
t ra
tio
vec
tor
(rati
o o
f }
{th
e su
rfac
e co
ntam
inan
t co
ncen
trat
ion
to b
ulk
cont
amin
ant
con
cen
trat
ion
).}
{NO
TE:
If a
ny s
ize
clas
s of
p
arti
cles
hav
e a
surf
ace
conc
entr
atio
n of
}
{ co
ntam
inan
t,
then
so
too
mus
t th
e re
fere
nce
size
cla
ss.
}
06/1
6/19
95
15:5
1 F
ilen
ame:
U
NJT
6C1.
PAS
Page
4
begi
n Con
tRat
oV
1"[1
l =
0.00
C
ontR
at
oV1"
C21
=0.
00
Con
tRat
oV
1"[3
1 =
0.00
C
ontR
at
oV1"
C4l
=0
.00
Con
tRat
oV
1"[5
] =
0.00
C
ontR
at
oV1"
[6]
=0.
00
Con
tRat
oV
1"[7
] =
0.00
C
ontR
at
oV1"
[8]
=0.
00
Con
tRat
oV
1"[9
] =
0.00
C
ontR
at
ov1·
c101
=
0.00
en
d;
Con
tRat
ov
2"[1
J =O
C
ontR
at
ov2"
[2J
=O
Con
tRat
ov2
"[3]
=O
C
ontR
at
ov2"
[4l
=O
Con
tRat
ov
2"[5
] =O
C
ontR
at
ov2"
[6J
=O
Con
tRat
ov
2"[7
J =O
C
ontR
at
ov2"
[8J
=O
Con
tRat
ov
2"[9
J =O
C
ontR
at
ov2"
C10
J =O
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re D
eter
min
e_M
odel
_Par
amet
ers_
fn_S
ize;
{Thi
s pr
oced
ure
dete
rmin
es
the
mod
el
para
met
ers
each
si
ze c
lass
of
} {
par
ticl
es.
Not
e th
at
the
Del
taT
is
def
ined
sep
arat
ely
in
anot
her
} {p
roce
dure
. ·
}
var F
lag1
,Fla
g2
:In
teg
er;
begi
n F
lag1
:=0;
F
lag2
:=0;
fo
r S
izel
ndex
:=1
to M
do
begi
n Lam
bda1
k"
[Siz
elnd
exl
:=C
ontR
atio
V1"
[Siz
elnd
exl
-/C
1+C
ontR
atio
V1"
[Siz
elnd
exJ)
; La
mbd
a2
k"[S
izel
ndex
] :=
Con
tRat
ioV
2"[S
izel
ndex
] -
/C1+
Con
tRat
ioV
2"C
Siz
elnd
exJ)
; if
Con
tRat
ioV
1"[S
izel
ndex
J<>
O
then
Fla
g1:=
Fla
g1+
1;
if C
ontR
atio
V2"
[Siz
elnd
exJ<
>O
th
en
Fla
g2:=
Fla
g2+
1;
end;
if
CC
CFl
ag1=
1)
and
(Con
tRat
ioV
1"[R
efS
izeC
lJ=
0))
or
CC
Flag
2=1)
an
d (C
ontR
atio
v2"C
Ref
Siz
eClJ
=0)
))
then
beg
in
clo
sese
gra
ph
ics;
w
rite
ln('
Con
tam
inan
t L
ocat
ion
Vio
loat
ion
.');
re
adln
; en
d;
Lam
bda1
:=
Lam
bda1
_k-C
Ref
Size
ClJ
; La
mbd
a2
:=La
mbd
a2
k"[R
efS
izeC
ll;
Kap
paS1
:=
Lam
bda1
wK
appa
P1/C
1-L
ambd
a1);
Kap
paS2
:=
Lam
bda2
*Kap
paP2
/C1-
Lam
bda2
);
for
Siz
elnd
ex:=
1 to
M do
be
gin Bet
a1
k"[S
izel
ndex
] :=
Bet
a1*(
1+C
ontR
atio
V1"
[Ref
Siz
eCl]
) -
/C1+
Con
tRat
ioV
1"[S
izel
ndex
J);
Bet
a2 k
"[S
izel
ndex
J :=
Bet
a2*C
1+C
ontR
atio
V2"
[Ref
Size
CL
J)
-/C
1+C
ontR
atio
V2"
[Siz
eJnd
exJ)
; K
appa
P1_k
"CSi
zeln
dexJ
:=KaepaP1*Siz~Data[Sizelndex,1J
*S1z
eDat
aCS
1zel
ndex
,1J;
K
appa
P2_k
-CSi
zeln
dexl
:=KaepaP2*Siz~Data[Sizelndex,1J
*S1z
eDat
a[S
1zel
ndex
,1J;
{*
****
Not
e:U
nity
pow
er
assu
mpt
ion
invo
lved
in
next
fe
w
line
s***
**
}
Univers
ity of
Cap
e Tow
n
06/1
6/19
95
15:5
1 Fi
lena
me:
U
NIT
6C1.
PAS
Page
5
if C
ontR
atio
V1"
[Siz
elnd
exJ=
O
then
Kap
paS1
k"
[Siz
elnd
exJ
:=O
el
se
Kap
paS1
_k-
[Siz
elnd
exJ:
=K
appa
S1*C
ontR
atT
oV1"
[Siz
elnd
exJ
/Con
tRat
ioV
1"[R
efS
izeC
LJ*
Siz
eDat
a[S
izel
ndex
,1J;
if
Con
tRat
ioV
2"[S
izel
ndex
J=O
th
en K
appa
S2
k"[S
izel
ndex
J:=
O e
lse
Kap
paS2
_k-
CSi
zeln
dexJ
:=K
appa
S2*C
ontR
atT
oV2"
CSi
zeln
dexJ
/C
ontR
atio
V2"
[Ref
Siz
eCL
J*S
izeD
ata[
Siz
elnd
ex, 1
J;
end
~d·
, {-
-'.-
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
-}
Pro
cedu
re D
eter
min
e_D
elta
T_f
n_S
ize;
begi
n for
Siz
elnd
ex:=
1 to
M do
D
elta
T
k"[S
izel
ndex
,1J
:=D
elta
T/S
izeD
ata[
Siz
elnd
ex,1
J -
/Siz
eDat
a[S
izel
ndex
,1J;
end;
{---
----
----
----
----
----
----
----
---C
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re P
arti
cle_
Init
ial_
Co
nd
itio
ns;
{Thi
s pr
oced
ure
uses
th
e in
itia
l co
ndit
ions
to
set
th
e al
pha
and
sigm
a } }
surf
ace}
{v
ecto
rs.
{Not
e th
at A
RTCN
+1],
AR
TP1[
N+1
J, SR
T1CN
+1J
and
SRT1
P1[N
+1J
are
the
{con
cent
rati
ons
of
the
Liq
uid
and
soli
d r
eact
ants
.
begi
n for
Siz
elnd
ex:=
1 to
M do
be
gin fo
r A
rbln
dex:
=O
to N
do
begi
n ART"
[S
izel
ndex
,Arb
inde
xJ
:=O
S
RT
1"[S
izel
ndex
,Arb
lnde
xl:=
1 SR
T2"
CSi
zeln
dex,
Arb
lnde
xJ :=
1 SR
T1P1
C" [
Siz
elnd
ex,A
rbin
dex]
=O
;. SR
T2P
1C"[
Size
lnde
x,A
rbin
dexJ
=O
; en
d·
AR
Tt[
Size
lnde
x,N
+1J
:=1;
SR
T1"
[Siz
elnd
ex,N
+1J
:=
1;
SRT2
" [S
izel
ndex
,N+
1J :
=1;
SR
T1P
1C"[
Size
lnde
x,A
rbln
dexl
:=O
; SR
T2P
1C"[
Size
lnde
x,A
rbln
dexJ
:=O
; en
d;
end;
}
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re G
uess
_SR
TsP1
;
{Thi
s pr
oced
ure
prov
ides
the
in
itia
l ''g
uess
" fo
r th
e it
erat
ion
. It
us
es
} {t
he p
revi
ous
tim
e in
terv
al's
val
ues
as
the
gues
s.
}
begi
n for
Siz
elnd
ex:=
1 to
M do
be
gin fo
r A
rbln
dex:
=O ·
to C
N+1
) do
be
gin SR
T1P
1·cs
izel
ndex
,Arb
lnde
xJ:=
SR
T1"
CS
izel
ndex
,Arb
Jnde
xJ;
SRT
2P1"
CSi
zeln
dex,
Arb
lnde
xJ :
=S
RT
2"[S
izel
ndex
,Arb
Jnde
xJ;
end;
06/1
6/19
95
15:5
1 F
ilen
ame:
U
NJT
6C1.
PAS
Page
6
AR
TP
1"[S
izel
ndex
,N+
1J:=
AR
T"[
Siz
eind
ex,N
+1J
; en
d·
end;
'
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re R
eGue
ss_S
RT
sP1;
{Thi
s pr
oced
ure
prov
ides
an
upda
ted
"gue
ss"
for
the
next
it
era
tio
n.
It
} {u
ses
the
SRT1
P1C
vec
tor
as t
he
upda
ted
gues
s.
}
begi
n for
Siz
elnd
ex:=
1 to
M do
be
gin fo
r A
rbln
dex:
=O
to (
N+1
) do
be
gin SR
T1P
1"[S
izel
ndex
,Arb
Jnde
xJ :
=S
RT
1P1C
"[S
izel
ndex
,Arb
lnde
xJ;
SR
T2P
1"[S
izel
ndex
,Arb
Jnde
xJ :
=S
RT
2P1c
·csi
zeln
dex,
Arb
inde
xJ;
end·
A
RT
P1"C
Size
lnde
x,N
+1J:
=AR
TP1
CV
al;
end·
en
d;
' {-
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
} P
roce
dure
Cal
c_C
rank
_Nic
olso
n_M
atri
x(va
r Y
Vec
tor
:Sho
rtV
ecto
r);
{***
**N
ote:
Uni
ty p
ower
as
sum
ptio
n in
volv
ed
in
this
pro
cedu
re**
***
}
var R
ows,
Col
s : I
nte
ger
;
begi
n for
Row
s:=O
to
(N
+1)
do
begi
n for
Col
s:=O
to
CN
+1)
do A
Mat
rix"
CR
ows,
Col
sJ:=
O;
end;
AM
atri
x·co
,OJ:
=(-
6-D
elta
E*D
elta
E*S
izeD
ata[
Siz
eJnd
ex, 1
J*S
ize0
ata[
Siz
elnd
ex, 1
J *C
Kap
paP1
k"
[Siz
elnd
exJ*
SR
T1P
1"[S
izel
ndex
,OJ
+Kap
paP2
f"[
Siz
elnd
exJ*
SR
T2P
1"[S
izel
ndex
,0J)
-2*D
elta
E*D
elta
E
*Siz
eDat
a[S
izel
ndex
,1J*
Siz
eDat
a[S
izel
ndex
,1J
/Del
taT
k
"[S
izel
nd
ex,1
J);
AM
atrix
" CO
, 1J
:=6;
-
YV
ecto
rCO
J :=
AR
T"[
Siz
elnd
ex,O
J*C
6+D
elta
E*D
elta
E*S
izeD
ata[
Siz
elnd
ex,1
J *S
izeD
ata[
Siz
elnd
ex,1
J*C
Kap
paP
1_k-
[Siz
elnd
exJ
*SR
T1"
[Siz
eind
ex,O
J+K
appa
P2_
k-[S
izel
ndex
J *S
RT
2"[S
izel
ndex
,0J)
-2*D
elta
E*D
elta
E*S
izeD
ata[
Siz
elnd
ex,1
J *S
izeD
ata[
Siz
elnd
ex,1
]/D
elta
T k
"[S
izel
nd
ex,1
])
-AR
T"[
Siz
elnd
ex,1
J*6;
-
for
Row
s:=1
to N
-1
do
begi
n AM
atri
x"[R
ows,
Row
s-1J
:=R
ows-
1;
AM
atrix
"[R
ows,
Row
sJ
:=-2
*Row
s-R
ows*
Del
taE
*Del
taE
*Siz
eDat
a[S
izel
ndex
,1J
*Siz
eDat
a[S
izel
ndex
,1J*
(Kap
paP
1_k-
CS
izel
ndex
J*
SRT
1P1"
[Siz
elnd
ex,R
owsJ
+Kap
paP2
k·
csiz
elnd
exJ*
SR
T2P
1.[S
izel
ndex
,Row
sJ)-
2*R
owsw
oeL
taE
*D
elta
E*S
izeD
ata[
Siz
elnd
ex,1
J *S
izeD
ataC
Siz
elnd
ex,1
J/D
elta
T_k
·csi
zeln
dex,
1J;
A
Mat
rix·
[R
ows,R
ows+
1] :
=Row
s+1;
Y
Vec
torC
Row
s]
· :=
AR
T"[
Size
lnde
x,R
ows-
1J*C
-Row
s+1)
+ A
RT
.[Siz
elnd
ex,R
owsJ
*C2*
Row
s+R
ows*
Del
taE
*Del
taE
Univers
ity of
Cap
e Tow
n
06/1
6/19
95
15:5
1 Fi
lena
me:
U
NIT
6C1.
PAS
Page
7
*Siz
eDat
a[S
izel
ndex
,1J*
Siz
eDat
a[S
izel
ndex
, 13
*(K
appa
P1
k"[S
izel
ndex
J*S
RT
1"[S
izel
ndex
,Row
sJ
+Kap
paP2
K0
[Siz
e!nd
exJ*
SR
T2"
[Siz
elnd
ex,R
owsJ
) ·2
*Row
s*D
elta
E*D
elta
E*S
izeD
ata[
Siz
elnd
ex, 1
J *S
izeD
ata[
Siz
elnd
ex,1
3/D
elta
T k
"(S
izel
ndex
,1])
+A
RT
"[Si
zeln
dex,
Row
s+1J
*C-R
ows-
1);
end;
AM
atri
x"[N
,N-1
J:=
N-1
; A
Mat
rix"
[N,N
l :=
-2*N
-N*D
elta
E*D
elta
E*S
izeD
ata[
Siz
elnd
ex,1
l *S
izeD
ata[
Siz
elnd
ex,1
J*C
Kap
paP
1 k"
[Siz
elnd
ex]
*SR
T1P
1"[S
izel
ndex
,NJ+
Kap
paP2
k~[Sizelndexl
*SR
T2P
1"[S
izel
ndex
,NJ
)-2*
N*D
eTta
E*D
elta
E
*Siz
eDat
a[S
izel
ndex
,1J*
Siz
eDat
a[S
izel
ndex
, 1]
/Del
taT
k"
[Siz
elnd
ex, 1
3;
YV
ecto
r[N
J :=
AR
T"[
Size
lnde
x,N
-1J*
(-N
+1)+
AR
T"[
Size
lnde
x,N
J*C
2*N
+
end;
N*D
elta
E*D
elta
E*S
izeD
ata[
Siz
elnd
ex, 1
J *S
izeD
ata[
Size
lnde
x,1J
*CK
appa
P1
k"[S
izel
ndex
J *S
RT
1"[S
izel
ndex
,NJ+
Kap
paP2
k"
[Siz
elnd
exJ
*SR
T2"
[Siz
elnd
ex,N
J)-2
*N*D
eTta
E*D
elta
E
*Siz
eDat
a[S
izel
ndex
, 1J*
Siz
eDat
a[S
izel
ndex
, 1J
/Del
taT
k"
[S
izel
ndex
, 1J)
+AR
T"[
Size
lnde
x,N
+1J*
C-N
-1)
-AR
TP1
"[Si
ze!n
dex,
N+1
J*(N
+1);
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re C
alc_
AR
TP1_
and_
SRTs
P1C
;
{As
it s
tand
s th
is p
roce
dure
can
cop
e w
ith a
var
iab
le r
eact
ion
ord
er.
Thi
s}
{is
due
to
the
in
clus
ion
of
the
Bre
nt R
outi
ne.
}
var M
assB
alV
al,A
Mat
Det
V
alue
AtR
oot
· O
utpu
tVec
tor,
YV
ecto
r
begi
n {F
irst
cal
cula
te t
he A
RTP1
v
ecto
rs.
for
Siz
elnd
ex:=
1 to
M do
be
gin
Dou
ble;
R
eal;
S
hort
Vec
tor;
Cal
e C
rank
Nic
olso
n M
atri
x(Y
Vec
tor)
; GaussJordanCAMatrix~,YVector,(N+1),0utputVector,Alnverse·,AMatDet);
for
Arb
lnde
x:=O
to
N d
o
end;
if O
utPu
tVec
tor[
Arb
lnde
xJ>F
Con
c To
i th
en
AR
TP1
"(Si
zeln
dex,
Arb
lnde
xJ :
=O
utpu
tVec
tor[
Arb
lnde
xl
else
·
AR
TP1
"(Si
zeln
dex,
Arb
!nde
xl :
=O;
{The
Gau
ssJo
rdan
pro
cedu
re c
alcu
late
s va
lues
for
th
e AR
T ve
ctor
fr
om
{poi
nt 0
to
poin
t N
(al
thou
gh it
mak
es
use
of
the
N+1
th p
oin
t).
To
{det
erm
ine
the
N+1
th p
oint
val
ue,
mak
e us
e of
th
e m
ass
bala
nce
of
{fl
uid
rea
ctan
t in
-th
e CS
TR.
Mas
sBal
Val
:=O
;
} } } } }
06/1
6/19
95
15:5
1 F
ilen
ame:
U
NIT
6C1.
PAS
Page
8
for
Siz
elnd
ex:=
1 to
M do
be
gin Mas
sBal
Val
:=M
assB
alV
al-S
izeD
ata[
Siz
eind
ex,2
J/2*
CK
appa
S1
k"[S
izel
ndex
l *S
RT
1"[S
ize!
ndex
,CN
+1)
J*A
RT
"[Si
zeln
dex,
CN
+1)
J +
Kap
paS
1_k"
[Siz
elnd
exJ*
SR
T1P
1"[S
izel
ndex
,(N
+1)
J *A
RT
P1"
[Siz
elnd
ex,(
N+
1)J+
Kap
paS
2_k"
[Siz
elnd
exJ
*SR
T2"
[Siz
elnd
ex,C
N+
1)J*
AR
T"[
Siz
elnd
ex,(
N+
1)J
+Kap
paS2
k"
[Siz
elnd
ex]*
SR
T2P
1"[S
izel
ndex
,CN
+1)
J *A
RT
P1"
[Siz
e!nd
ex,(
N+
1)J)
-3*S
izeD
ata[
Siz
elnd
ex,2
J /C
Siz
eDat
a[S
izel
ndex
,1J*
Siz
eDat
a[S
izel
ndex
,1J)
/2
*((A
RT"
(S
ize!
ndex
,(N
+1)
J+A
RT
P1"
[Siz
elnd
ex,(
N+
1)])
-C
AR
T"[
Siz
elnd
ex,N
J+A
RT
P1"
[Siz
elnd
ex,N
J))/
0elt
aE;
end;
AR
TP1
CV
al:=
AR
T"[
1,(N
+1)J
+Del
taT
/NuS
tar*
Mas
sBal
Val
;
{Cal
cula
te t
he S
RTsP
1C
vec
tors
. C
ode
mak
es
use
of
Bre
nt'
s M
etho
d }
{(A
n
on
-lin
ear
root
fi
ndin
g pr
oced
ure)
to
so
lve
for
the
roo
t.
}
for
Siz
elnd
ex:=
1 to
M do
be
gin fo
r A
rbln
dex:
=O
to (
N+1
) do
be
gin
ex, 1
J,
if
(Bet
a1_k
-[S
izel
ndex
J<>
O)
and
(Kap
paP
1_k"
[Siz
elnd
exJ<
>O
) th
en
begi
n if O
rdSP
1=1
then
.
begi
n SR
T1P
1C"[
Siz
elnd
ex,A
rbln
dexJ
:=S
RT
1"[S
izel
ndex
,Arb
lnde
xJ
end
*(1-
Del
taT
k"[
Siz
elnd
ex,1
J*K
appa
P1
k"[S
izel
ndex
] *B
eta1
_k"[
Siz
elnd
exJ*
AR
T"[
Siz
elnd
ex,A
rbln
dex]
/C
2*C
1-La
mbd
a1
k"[S
izel
ndex
])))
/(1+
Del
taT
k"
[Siz
elnd
ex,1
] *K
appa
P1_
k·cs
izel
ndex
J*B
eta1
k
"[S
izel
nd
exr
*AR
TP1
"[Si
zeln
dex,
Arb
!nde
xJ/(
2*C
1-L
ambd
a1
k"[
Siz
eln
dex
])))
.
I -
else
if
SRT
1P1"
[Siz
elnd
ex,A
rbln
dexl
<SC
onc_
Tol
th
en
SR
T1P
1C"[
Siz
elnd
ex,A
rbln
dex]
:=
SR
T1P
1"[S
izel
ndex
,Arb
lnde
xJ
else
be
gin SR
T1P
1C"[
Siz
elnd
ex,A
rbln
dexJ
:=B
rent
Roo
ts(0
.0,1
.0,D
elta
T_k
·csi
zeln
d
Kap
paP1
k"
[Siz
elnd
exJ,
Bet
a1
k"[S
izel
ndex
J,
Lam
bda1
-k"C
Siz
elnd
exJ,
Ord
SP
T,A
RT
"[S
izel
ndex
,Arb
lnde
xJ,
AR
TP
1"[S
izel
ndex
,Arb
lnde
xJ,S
RT
1"[S
izel
ndex
,Arb
lnde
xl,
1e-8
,100
,Val
ueA
tRoo
t,er
ror)
; en
d·
end·
'
if
CB
eta2
_k-[
Siz
elnd
exJ<
>O
) an
d (K
appa
P2_
k"[S
izel
ndex
J<>
0)
then
be
gin ff o
rdSP
2=1
then
be
gin
end S
RT
2P1C
"[S
izel
ndex
,Arb
lnde
xJ:=
SR
T2"
[Siz
elnd
ex,A
rbln
dexJ
*C
1-D
elta
T
k"[S
izel
ndex
,1J*
Kap
paP
2 k"
[Siz
elnd
ex]
*Bet
a2 k
"[S
izel
ndex
l*A
RT
"[S
izel
ndex
,Arb
lnde
xl
/C2*
C17
Lam
bda2
k"
[Siz
elnd
exJ)
))/(
1+D
elta
T k
"[S
izel
ndex
,11
*Kap
paP2
k"
[Siz
elnd
exJ*
Bet
a2_k
"[S
ize!
ndex
r *A
RT
P1"
[Siz
elnd
ex,A
rbin
dexJ
/C2*
C1-
Lam
bda2
_k"[
Siz
elnd
exJ)
))
else
if
SRT
2P1"
[Siz
elnd
ex,A
rb!n
dexl
<SC
onc_
Tol
th
en
Univers
ity of
Cap
e Tow
n
06/1
6/19
95
15:5
1 Fi
lena
me:
U
NIT
6C1.
PAS
Page
9
SR
T1P
1C"[
Siz
elnd
ex,A
rbln
dexJ
:=S
RT
2P1"
[Siz
elnd
ex,A
rbln
dex]
el
se
begi
n SRT2
P1C"
[S
izel
ndex
,Arb
lnde
xJ:=
Bre
ntR
oots
(O.O
, 1.0
, D
elta
T
k"[S
izel
ndex
,1J,
Kap
paP
2 k"
CS
izel
ndex
l,
Bet
a2 f
"[S
izel
ndex
],L
ambd
a2
k"1S
izel
ndex
J,O
rdS
P2,
A
RT
"[S
izel
ndex
,Arb
inde
xJ,A
RiP
1"[S
izel
ndex
,Arb
lnde
xJ,
SR
T2"
CS
izel
ndex
,Arb
lnde
xl, 1
e-8,
100
,Val
ueA
tRoo
t,er
ror)
; en
d·
end·
'
end·
·'
en
d·
' en
d;
'
{---
---·
----
-·--
----
----
······
······
·---
----
----
----
----
----
----
----
----
--}
Pro
cedu
re C
heck
_Con
verg
ence
;
{Thi
s pr
oced
ure
chec
ks w
heth
er
or n
ot
the
solu
tio
n h
as
conv
erge
d by
}
{com
pari
ng
the
gues
sed
valu
e of
SR
T1P1
w
ith
a ca
lcu
late
d v
alue
of
SR
T1P
1.}
<Als
o co
mpa
res
the
calc
ula
ted
val
ue o
f AR
TP1C
w
ith
gues
sed
valu
e of
}
{AR
TP1.
}
var N
orm
1,N
orm
2,N
orm
3
begi
n
:Dou
ble;
Norm
1 =O
; {U
sing
a
norm
***
}
Nor
m2
=O·
· N
orm
3 =O
! fo
r S
zein
dex:
=1
to M
do
begi
n for
Arb
lnde
x:=O
to
(N
+1)
do
begi
n ·
if (
8eta
1 k"
[Siz
elnd
ex]<
>O
) an
d CK
appa
P1
k"[S
izel
ndex
l<>
O)
then
beg
in
Nor
m1:
=Nor
m1+
(SR
T1P
1C"[
Size
lnde
x,A
rbln
dexJ
-S
RT
1P1"
[Siz
elnd
ex,A
rbln
dex]
)*C
SR
T1P
1C"[
Siz
elnd
ex,A
rbln
dex]
-S
RT
1P1"
[Siz
elnd
ex,A
rbln
dexl
);
end
else
Nor
m1:
=0;
. if
(B
eta2
_k"[
Siz
elnd
exJ<
>O
) an
d (K
appa
P2_k
"[Si
zeln
dexJ
<>
0)
then
beg
in
· N
orm
2:=N
orm
2+C
SRT
2P1C
"[Si
zeln
dex,
Arb
lnde
xl
-SR
T2P
1"[S
izel
ndex
,Arb
lnde
xJ)*
(SR
T1P
1C"[
Siz
elnd
ex,A
rbln
dexl
-S
RT
2P1"
[Siz
elnd
ex,A
rbln
dexl
);
end
else
Nor
m2:
=0;
end·
en
d·
' Nor~:=(ARTP1CVal-ARTP1.
C1,N
+1J
)*(A
RTP
1CV
al-A
RTP
1. C
1,N
+1l
);
if (
Nor
m1<
Nor
m C
rit)
and
(N
orm
2<N
orm
Cri
t) a
nd
(Nor
m3<
Nor
m C
rit)
th
en
Fla
g1::
1;
--
end;
{---
----
----
----
----
----
·---
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re U
pdat
e_A
RT_
and_
SRTs
;
{Thi
s pr
oced
ure
upda
tes
the
alph
a an
d si
gma
vect
ors
for
the
next
it
erat
ion
}
{by
repl
acin
g th
eir
com
pone
nts
with
th
e al
phaT
+1
and
sigm
aT+1
v
ecto
rs.
}
begi
n
06/1
6/19
95
15:5
1 F
ilen
ame:
U
NIT
6C1.
PAS
Page
10
for
Siz
elnd
ex:=
1 to
M do
be
gin fo
r A
rbln
dex:
=O
to (
N+1
) do
be
gin AR
T"[
Siz
elnd
ex,A
rbln
dexl
=A
RTP1
" [S
izel
ndex
,Arb
inde
xl;
SR
T1"
[Siz
elnd
ex,A
rbln
dexl
=
SR
T1P
1"[S
izel
ndex
,Arb
lnde
xJ;
SR
T2"
[Siz
elnd
ex,A
rbln
dexl
=
SR
T2P
1"[S
izel
ndex
,Arb
lnde
xJ;
end;
en
d·
end;
'
{---···················-····--·-··---·············-·····-·-·-···-·········}
P
roce
dure
Cal
c_C
onve
rsio
n;
{Thi
s pr
oced
ure
calc
ula
tes
the
frac
tio
nal
co
nver
sion
of
the
par
ticl
e fo
r }
{the
tim
e in
terv
al.
It u
ses
the
form
ula
on
pg.
13
Dix
on.
The
inte
gra
tor}
{
is t
he Q
uinn
-Cur
tis
vec
tor
inte
gra
tor.
}
var ln
terG
1,ln
terG
2 ln
tegV
al,C
onve
rsio
n :V
eryl
ongV
ecto
r;
:Rea
l;
begi
n for
Siz
elnd
ex:=
1 to
M do
be
gin if K
appa
P1
k"[S
izel
ndex
l<>
O
then
be
gin
-fo
r A
rbln
dex:
=O
to (
N+1
) do
ln
terG
1 C
Arb
lnde
xl:=
(1·S
RT
1"[S
izel
ndex
,Arb
lnde
x])*
Arb
lnde
x*D
elta
E
*Arb
lnde
x*D
elta
E;
lnte
grat
eVec
tor(
lnte
rG1,
Del
taE
,01(N
+1)
,lnt
egV
al);
C
onve
rsio
n :=
3*(1
·Lam
bda1
k
[Siz
elnd
exJ)
*lnt
egV
al
+Lam
bda1
k"
[Siz
elnd
exJ*
(1-S
RT
1"[S
izel
ndex
,N+
1J )
; U
nitH
eapC
onv1
:=U
nitH
eapC
onv1
+C
onve
rsio
n*S
izeD
ata[
Siz
elnd
ex,2
J /S
umSi
zeD
ata_
2;
end;
if K
appa
P2
k"[S
izel
ndex
]<>
O
then
be
gin
-fo
r A
rbln
dex:
=O
to (
N+1
) do
ln
terG
2CA
rbln
dexJ
:=(1
·SR
T2"
[Siz
elnd
ex,A
rbln
dexJ
)*A
rbln
dex*
Del
taE
*A
rbln
dex*
Del
taE
; ln
teg
rate
Vec
tor(
lnte
rG2
,Del
taE
,0,(
N+
1),
lnte
gV
al);
C
onve
rsio
n :=
3*(1
-Lam
bda2
k"
[Siz
elnd
ex])
*lnt
egV
al
+Lam
bda2
k"
[Siz
elnd
exJ*
(1-S
RT
2"[S
izel
ndex
,N+
1J);
U
nitH
eapC
onv2
:=U
nitH
eapC
onv2
+C
onve
rsio
n*S
izeD
ata[
Siz
elnd
ex,2
J /S
umSi
zeD
ata
2;
end;
-
end·
en
d;
'
{-·-
-··
-··
·--··
-··
--··
·-··
·--·-
-··
-··
····
·-··
····
-··
-·-
·-··
·--·-
·-·-
····
··}
P
roce
dure
Gra
ph
l_ln
itia
lise
;
var G
raph
Typ
e B
eta1
S,K
appa
P1S,
Lam
bda1
S,G
DT
S B
eta2
S,K
appa
P2S,
Lam
bda2
S O
rdSP
1S,O
rdSP
2S
Inte
ger
; S
trin
g;
Str
ing
; S
trin
g;
Univers
ity of
Cap
e Tow
n
06/1
6/19
95
15:5
1 Fi
lena
me:
U
NIT
6C1.
PAS
Page
11
Cum
Tim
eS,U
nitH
eapN
oS,V
iew
Size
ClS
begi
n init
SE
Gra
phic
s(V
alid
Dir
);
SetC
urre
ntW
indo
w(3
);
Bor
derC
urre
ntW
indo
w(1
);
SetA
xesT
ype(
O,O
>;
Sca
leP
lotA
rea(
0.0,
0.0,
1.0
, 1.2
);
Set
XY
inte
rcep
ts(0
.0,0
.0);
S
etC
olor
(2);
D
raw
XA
xis(
0.2,
1);
D
raw
YA
xis(
0.2,
1);
L
abel
XA
xisC
1,0)
; L
abel
YA
xis(
1,0)
; T
itle
XA
xisC
'Dim
ensi
onle
ss R
adiu
s');
T
itle
YA
xis(
'Dm
lss
Con
e');
T
itleW
indo
wC
'Uni
t6C
1'>
; S
tr(B
eta1
:6:3
,Bet
a1s)
; S
tr(K
appa
P1:
6:3,
Kap
paP
1S);
S
tr((
Del
taT
*Pri
nt C
rit)
:6:4
,GD
TS
);
Str(
Lam
bda1
:5:3
,Lam
bda1
S);
Str
(Ord
SP
1:5:
2,0r
dSP
1S);
S
tr(B
eta2
:6:3
,Bet
a2S
);
Str
(Kap
paP
2:6:
3,K
appa
P2S
);
Str(
Lam
bda2
:5:3
,Lam
bda2
S);
Str
(Ord
SP
2:5:
2,0r
dSP
2S);
St
r(C
umT
ime:
8:4,
Cum
Tim
eS);
S
tr(U
nitH
eapl
ndex
:2,U
nitH
eapN
oS);
S
tr(V
iew
Siz
eCl:
2,V
iew
Siz
eClS
);
:Str
ing
;
Gla
bel1
:=C
onca
t('
Bet
a1
',B
eta1
S,'
; K
appa
1 ',K
appa
P1S
,';
Lam
bda1
L
ambd
a1S,
'; O
rder
1 ',O
rdS
P1S
,';
GOT
',GD
TS)
; G
Lab
el2:
=Con
catC
' B
eta2
',B
eta2
S, '
; K
appa
2 ',K
appa
P2S,
';
Lam
bda2
L
ambd
a2S,
'; O
rder
2 ',O
rdS
P2S
);
Gla
bel3
:=C
onca
t('
Cum
Tim
e 1,C
umTi
meS
,1 •
Pro
file
s of
u
nit
hea
p '
Uni
tHea
pNoS
,' an
d si
zecl
ass
',Vie
wS
izeC
lS);
L
abel
Gra
phW
indo
wC
1,90
0,G
Lab
el1,
0,0)
; L
abel
Gra
phW
indo
wC
1,85
0,"G
Lab
el2,
0,0)
; L
abel
Gra
phW
indo
wC
160,
800,
GL
abel
3,0,
0);
for
Arb
lnde
x:=1
to
2 d
o be
gin if A
rbln
dex=
1 th
en G
raph
Typ
e:=9
els
e G
raph
Typ
e:=1
0;
SetC
urre
ntW
indo
w(G
raph
Typ
e);
Bor
derc
urre
ntW
indo
wC
1>;
Set
Axe
sTyp
e(0,
0);
Sca
leP
lotA
rea(
0.0,
0.0,
1.0
, 1.2
);
Set
XY
inte
rcep
ts(0
.0,0
.0);
S
etC
olor
(2);
D
raw
XA
xis(
0.2,
1);
D
raw
YA
xis(
0.2,
1);
L
abel
XA
xis(
1,0)
; L
abel
YA
xisC
1,0)
; T
itle
XA
xis(
1D
imen
sion
less
Rad
ius'
);
Titl
eYA
xisC
'Dm
lss
Con
e');
T
itle
Win
dow
('Uni
t6C
1');
if
Arb
lnde
x=1
then
Lab
elG
raph
Win
dow
(200
,900
,'Sm
alle
st
Siz
e F
ract
ion
•,0
,0)
else
Lab
elG
raph
Win
dow
(200
,900
,1L
arge
st S
ize
Fra
ctio
n',
0,0
);
end;
·
end;
06/1
6/19
95
15:5
1 F
ilen
ame:
U
NIT
6C1.
PAS
Page
12
(·········································································}
P
roce
dure
Gra
ph1_
Res
ults
;
var G
raph
Inde
x : I
nte
ger
;
begi
n if V
iew
Size
Cl>
M
then
be
gin Clo
sese
grap
hics
; w
rite
ln('
Gra
ph
1 D
raw
ER
RO
R');
w
rite
lnC
'You
hav
e in
stu
cted
the
Gra
phin
g R
outi
ne
to g
raph
th
e co
nv
ersi
on
');
wri
teln
('cu
rves
of
a si
ze c
lass
whi
ch
does
no
t ex
ist.
');
wri
teln
C'R
espe
cify
= V
iew
size
Cla
ss
in D
ecla
rati
on
s se
cti
on
');
read
ln;
end·
fo
r1
Gra
phln
dex:
=1
to 3
do
begi
n if G
raph
lnde
x=1
then
be
gin Siz
elnd
ex:=
Vie
wS
izeC
l;
Set
Cur
rent
Win
dow
(3);
en
d el
se i
f G
raph
lnde
x=2
then
be
gin Siz
elnd
ex:=
M;
Set
Cur
rent
Win
dow
(9);
en
d el
se
begi
n Siz
elnd
ex:=
1;
Set
Cur
rent
Win
dow
(10)
; en
d·
for1
Arb
lnde
x:=O
to
(N
+1)
do D
ataS
etX
"[A
rbln
dexJ
:=A
rbin
dex*
Del
taE
; if
Kap
paP1
<>0
then
beg
in
1
for
Arb
inde
x:=O
to
(N
+1)
do
Dat
aSet
Y"[
Arb
lnde
xJ:=
AR
T"[
Siz
elnd
ex,A
rbln
dexJ
; L
ineP
lotD
ata(
Dat
aSet
X",
Dat
aSet
Y",
(N+
2),3
,0);
fo
r A
rbin
dex:
=O
to (
N+1
) do
D
ataS
etY
"[A
rbln
dexJ
:=S
RT
1"[S
izel
ndex
tArb
lnde
xJ;
Lin
ePlo
tDat
a(D
ataS
etX
",D
ataS
etY
",(N
+2)
,),0
);
end·
if
Kap
paP2
<>0
then
beg
in
for
Arb
inde
x:=O
to
(N
+1)
do
Dat
aSet
Y"[
Arb
lnde
xJ:=
AR
T"[
Siz
elnd
ex,A
rbin
dexJ
; L
ineP
lotD
ata(
Dat
aSet
X",
Dat
aSet
Y",
(N+
2),3
,0);
fo
r A
rbin
dex:
=O
to C
N+1
) do
D
ataS
etY
"[A
rbln
dexJ
:=
SR
T2"
[Siz
eind
ex,A
rbln
dexJ
; L
ineP
lotD
ata(
Dat
aSet
x·,o
ataS
etY
",(N
+2)
,4,2
);
end·
en
d·
' en
d;
'
(===
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
==}
Pro
cedu
re E
xecu
te_U
nit6
C1;
begi
n clrs
cr;
Univers
ity of
Cap
e Tow
n
06/1
6/19
95
15:5
1 Fi
lena
me:
U
NIT
6C1.
PAS
Page
13
Gra
ph1_
1nit
iali
se;
Rep
eats
:=O
; Fl
ag1
:=O
; Pl
ot_V
ar:=
O;
Gue
ss_S
RT
sP1;
Gra
ph1_
Res
ults
;
whi
le (
Rep
eats
<lt
erat
ions
) an
d C
AR
T.[1
,N+1
J>FC
onc_
Tol
) do
be
gin whi
le
CFl
ag1=
0)
do
begi
n Cal
e AR
TP1
and
SRTs
P1C
; Ch
ecK
Con
verg
ence
; .R
eGue
ss
SRTs
P1;
end;
-
Plo
t_V
ar:=
Plo
t_V
ar+
1;
if P
lot
Var
=P
rint
Cri
t th
en
begi
n· -
-P
lot
Var
:=O
; G
rapF
i1_R
esul
ts;
end;
Upd
ate_
AR
T_an
d_SR
Ts;
Gue
ss
SRTs
P1;
Flag
1T=O
; R
epea
ts:=
Rep
eats
+1;
end;
Gra
ph 1
Res
ults
; C
alc_
Con
vers
ion;
end;
end.
{===
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
==}
Univers
ity of
Cap
e Tow
n
07/3
1/19
95
09:0
5 Fi
lena
me:
M
ODEL
6C2.P
AS
Page
Prog
ram
Mod
el6C
2;
{Mod
el6C
2.
{ *T
his
prog
ram
is
a c
ode
for
heap
l~aching
anal
ysi
s.
*Inc
orpo
rate
s un
stea
dy s
tate
flo
w.
} } } } }
{ { {
*Use
s U
nit6
C2
whi
ch
is a
mod
ifie
d fo
rm
of
the
chem
ical
CS
TR
mod
el.
*Mod
el6C
2 v
alid
for
var
iab
le o
rder
kin
etic
s.
{Cod
ed:
Gra
ham
Dav
ies.
}
{ D
epar
tmen
t of
C
hem
ical
E
ngin
eeri
ng.
} {
Uni
vers
ity
of
Cap
e To
wn.
}
{ 6
Mar
ch
1995
. }
{ 6
June
19
95.
Upd
ated
. (G
MO)
}
{===
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
==}
{Dec
lara
tion
s:
}
uses
crt
,std
hdr,
gj,g
raph
,wor
lddr
,seg
raph
,int
egra
t,M
2C1B
rent
Roo
ts,U
nit6
C2;
cons
t
var {*
**T
hese
are
the
par
amet
ers
app
liab
le t
o th
e co
lum
n.**
****
****
****
****
}
Col
leng
th
=1.
76;
Col
Voi
dage
=D
.498
; S
atF
rac
=0.
361;
Glo
blJe
tFac
=
1.0;
{Col
umn
leng
th.
(m)
} {V
oida
ge o
f th
e co
lum
n.
} {F
ract
ion
of
the
void
spa
ce f
ille
d w
ith
flu
id.
} {A
ssum
ed
to b
e co
nsta
nt
in t
ime.
ie
Doe
s no
t ch
ange
} {w
ith a
cha
nge
in t
he f
low
rate
. }
{Glo
bal
wet
ting
fa
cto
rs o
f th
e p
art
icle
s.
}
{***
The
se a
re c
olum
n nu
mer
ical
m
etho
d pa
ram
eter
s.**
****
****
****
****
****
}
NoU
ni tH
eaps
=5;
{N1.1
nber
of
un
it h
eaps
. }
Col
Del
taE
=1
/NoU
nitH
eaps
;
uSta
r =
1/86
400;
{Ref
eren
ce s
up
erfi
cial
v
elo
city
. E
quiv
alen
t to
1m
} {i
n 2
4 ho
urs.
}
Spe
cCol
lter
s=1;
GfM
ax X
Gf
Max
::::v
=1.
0;
=1. O
'; {M
axim
um x
-val
ue f
or
conv
ersi
on g
raph
s.
{Max
imum
y-v
alue
fo
r co
nver
sion
gra
phs.
} }
{***
Oth
er
info
rmat
ion
requ
ired
.***
****
****
****
****
****
****
****
****
****
}
Val
idO
ir
=1F:
\TP6
\MO
DEL
61;{
Val
id d
irec
tory
for
gra
phic
s d
riv
ers.
}
{***
****
****
****
****
****
****
****
****
****
****
****
****
****
****
****
****
**}
Sup
Vel
Fil
e Su
pVel
Oat
a IJ
atch
Size
OT
Tex
t;
arra
y[1
•. 2
,1 •
• 2]
of D
oubl
e;
arra
yC1
.. m
axcl
of
In
teg
er;
lntL
engt
hCri
t,ln
tLen
gth,
Col
Del
ta t
,Col
Del
taT
,Arb
Val
ue
Hol
dCon
v1,H
oldC
onv2
-
. N
umsu
bint
,Tim
eCou
nter
C
onti
nue,
Col
lter
s
Dou
ble;
D
oubl
e;
Inte
ger
; In
teg
er;
07/3
1/19
95
09:0
5 F
ilen
ame:
M
OD
EL6C
2.PA
S Pa
ge
2
AR
TBTC
,SR
T1B
TC,S
RT2
BTC
,Tot
Hea
pCon
v1,T
otH
eapC
onv2
T
imeV
ecto
r T
otH
eapA
RT
Vec
,Tot
Hea
pSol
SRT
1Vec
, Tot
Hea
pSol
SRT
2Vec
OT
fo
r IJ
atch
Siz
e co
nvR
esul
ts
·ver
yLon
gVec
tor;
·v
eryL
ongV
ecto
r;
"Sho
rtV
ecto
r;
"Sqr
Mat
; T
ext;
{===
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
==}
Pro
cedu
re
lnit
iali
se_
Var
iou
s_V
aria
ble
s;
begi
n Vo
lliq
T
otV
olP
art
:=C
olD
elta
E*C
olV
oida
ge*C
olL
engt
h*S
atF
rac;
:=
Col
Del
taE
*(1-
Col
Voi
dage
)*C
olle
ngth
*Glo
blJe
tFac
;
Cum
Tim
e =O
; U
nitH
eapC
onv1
=O
; U
nitH
eapC
onv2
=O
; C
oll
ters
=O
; T
imeV
ecto
r"[O
] =O
; en
d;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--)
Pro
cedu
re
Init
iali
se_
Su
per
fici
al_
Vel
oci
ty_
Fil
e;
begi
n assi
gn(S
upV
elF
ile,
'Sup
Vel
Fi.
Dat
');
rese
t(S
upV
elF
ile)
; S
upV
elD
ata[
1, 1
1:=
0;
SupV
elD
ataC
112l
:=O
; re
ad(S
upV
elF
1le,
Sup
Vel
Dat
aC2,
1l);
re
ad(S
upV
elF
ile,
Sup
Vel
Dat
a[2,
2]);
1
Intl
engt
hCri
t:=
Col
Voi
dage
*Col
Del
taE
*Col
Len
gth;
en
d;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re C
ol_l
niti
al_C
ondi
tion
s;
,
begi
n for
Arb
lnde
x:=O
to
max
c do
be
gin Tot
Hea
pAR
TV
ec"[
Arb
lnde
x]
=O
Tot
Hea
pSol
SRT
1Vec
"[A
rbln
dex]
=O
T
otH
eapS
olSR
T2V
ec"[
Arb
lnde
x]
=O
end;
Tot
Hea
pAR
TV
ec"[
1]:=
1;
for
Arb
lnde
x:=O
to
max
v do
be
gin AR
TBTC
"[A
rbin
dex]
:=
O;
SRT
1BT
C"[
Arb
inde
xl:=
O;
SRT
2BT
C"[
Arb
lnde
xl:=
O;
Tot
Hea
pcon
v1"C
Arb
lnde
xJ :=
O;
Tot
Hea
pCon
v2"[
Arb
lnde
xJ:=
O;
end;
en
d;
{***
Impo
sed
BC
for
heap
. }
{---
---·
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
---}
Q
Q..
(1) ~ -· ~ ~ :;·
8' .., ~ Q
.. (1
) - °' (1 N ~ r:n .
Univers
ity of
Cap
e Tow
n
07/3
1/19
95
09:0
5 Fi
lena
me:
M
ODEL
6C2.P
AS
Page
3
Pro
cedu
re G
ener
ate_
Uni
tHea
pDat
aFil
es;
{Thi
s pr
oced
ure
is u
sed
to g
ener
ate
the
Uni
tHea
pDat
aFil
es.
{nam
ed M
6C1H
1 th
roug
h to
M6C
1HX.
var U
nitH
eapN
oS,F
ileN
ame
Hea
pFil
e
begi
n for
Uni
tHea
plnd
ex:=
1 to
NoU
nitH
eaps
do
be
gin Str
(Uni
tHea
plnd
ex,U
nitH
eapN
oS);
Fi
leN
ame:
=Con
cat(
1M
6ClH
1,U
nitH
eapN
oS,1
.Dat
');
assi
gn(H
eapF
ile,
Fil
eNam
e);
rew
rite
(Hea
pFil
e);
for
Siz
elnd
ex:=
l to
M do
be
gin fo
r A
rbln
dex:
=O
to N
+1
do
begi
n
The
fi L
es
are
} }
:Str
ing
; :T
ext;
wri
te(H
eapF
ile,
AR
T"[
Siz
elnd
ex,A
rbln
dexJ
,' ')
; w
rite
(Hea
pFil
e,S
RT
l"[S
izel
ndex
,Arb
lnde
xl,'
');
wri
teln
(Hea
pFil
e,S
RT
2"C
Siz
elnd
ex,A
rbln
dexl
);
end·
w
rite
ln(H
eapF
ile)
; en
d;
',Uni
tHea
pCon
v2);
w
rite
ln(H
eapF
ile,
Uni
tHea
pCon
vl,'
wri
teln
(Hea
pFil
e);
wri
teln
(Hea
pFil
e,'C
umT
ime
1,C
umTi
me)
; cl
ose(
Hea
pFil
e);
end·
en
d;
'
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re R
ead_
AR
T_an
d_SR
Ts;
var U
nitH
eapN
oS,F
ileN
ame
Hea
pFil
e
begi
n Str
(Uni
tHea
plnd
ex,U
nitH
eapN
oS);
Fi
leN
ame:
=Con
cat(
1M
6ClH
1,U
nitH
eapN
oS,1
.Dat
1);
assi
gn(H
eapF
ile,
Fil
eNam
e);
rese
t(H
eapF
ile)
; fo
r S
izel
ndex
:=l
to M
do
begi
n for
Arb
lnde
x:=O
to
N d
o be
gin read
(Hea
pFil
e,A
RT
"CS
izel
ndex
,Arb
lnde
xJ);
re
ad(H
eapF
ile
SR
T1"
[Siz
elnd
ex,A
rbln
dexl
);
read
ln(H
eapF
ile,
SR
T2"
[Siz
elnd
ex,A
rbln
dexJ
);
end;
re
ad(H
eapF
ile,
Arb
Val
ue);
A
RT
"CSi
zeln
dex,
N+l
J:=T
otH
eapA
RT
Vec
·cun
itHea
plnd
exJ;
re
ad(H
eapF
ile
SRT
1"C
Size
lnde
x,N
+1J)
; re
adln
(Hea
pFil
e,S
RT
2"[S
izel
ndex
,N+
1J);
en
d;
:Str
ing
; :T
ext;
07/3
1/19
95
09:0
5 F
ilen
ame:
M
OD
EL6C
2.PA
S Pa
ge
4
read
ln(H
eapF
ile,
Hol
dCon
v1;H
oldC
onv2
);
clos
e(H
eapF
ile)
;
end;
{-------------------------~-----------------------------------------------}
Pro
cedu
re W
rite_
AR
T_a
nd_S
RT
s;
var U
nitH
eapN
oS,F
ileN
ame
Hea
pFil
e
begi
n Str
(Uni
tHea
plnd
ex,U
nitH
eapN
oS);
Fi
leN
ame:
=Con
cat('
M6C
1H1,U
nitH
eapN
oS,1
.Dat
1);
assi
gn(H
eapF
ile,
Fil
eNam
e);
rew
rite
(Hea
pFil
e);
for
Siz
elnd
ex:=
l to
M do
be
gin fo
r A
rbln
dex:
=O
to N
+1
do
begi
n if A
RT
"[Si
zeln
dex,
Arb
lnde
xJ>
FCon
c T
ol
then
w
rite
(Hea
pFil
e,A
RT
"CS
izel
ndex
,Arb
inde
xl,'
else
wri
te(H
eap
Fil
e,0
.00
0,'
');
if S
RT
l"[S
izel
ndex
,Arb
lnde
xJ>
SC
onc
Tol
then
w
rite
(Hea
pFil
e,S
RT
1"[S
izel
ndex
,Arb
lnde
xl,'
else
wri
te(H
eap
Fil
e,0
.00
0,'
'>;
if S
RT
2"[S
izel
ndex
,Arb
lnde
xJ>
SCon
c To
l th
en
wri
teln
(Hea
pFil
e,S
RT
2"[S
izel
ndex
,Arb
lnde
xJ)
else
wri
teln
(Hea
pFil
e,0.
000)
; en
d·
wri
teln
(Hea
pF
ile)
;
I )
:Str
ing
; :T
ext;
I)
end·
w
rite
ln(H
eapF
ile,
Uni
tHea
pCon
v1,'
wri
teln
(Hea
pFil
e);
wri
teln
(Hea
pFil
e,'C
umT
ime
clos
e(H
eapF
i le)
; U
nitH
eapC
onv1
:=0;
U
nitH
eapC
onv2
:=0;
',Uni
tHea
pCon
v2);
• ,C
umtim
e);
end;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re D
eter
min
e_M
axim
um_D
elta
T_f
n_Si
ze;
begi
n for
Uni
tHea
plnd
ex:=
1 to
NoU
nitH
eaps
do
begi
n for
Siz
elnd
ex:=
1 to
M do
end;
DT
for
wat
chS
ize·
cuni
tHea
plnd
ex,S
izei
ndex
J:=
Max
Del
taT
-
*Siz
eDat
a[S
izel
ndex
,1J*
Siz
eDat
a[S
izel
ndex
,1J
/Siz
eDat
a[M
,1l/
Siz
eDat
a[M
,1l;
for
Uni
tHea
plnd
ex:=
1 to
NoU
nitH
eaps
do
Wat
chSi
zeD
T[U
nitH
eapl
ndex
J :=
M;
end;
Univers
ity of
Cap
e Tow
n
07/3
1/19
95
09:0
5 Fi
lena
me:
M
ODEL
6C2.P
AS
Page
5
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re D
eter
min
e_C
orre
spon
ding
_Col
Del
taT
;
{Thi
s pr
oced
ure
uses
th
e m
etho
d of
ch
arac
teri
stic
s fo
de
term
ine
the
} {p
rogr
essi
on i
n tim
e fo
r a
give
n C
harC
olD
elta
E.
It a
llow
s fo
r un
stea
dy
} {
stat
e fl
ow.
}
begi
n lntl
eng
th
:=O
; C
olD
elta
t:=
O;
whi
le
Intl
engt
h<>
Intl
engt
hCri
t do
beg
in
if E
OFC
SupV
elFi
le)
then
beg
in
clos
eseg
raph
ics;
w
rite
ln('
Insu
ffic
ien
t D
ata
in t
he S
up
erfi
cial
Vel
ocit
y D
ata
Fil
e to
');
wri
teln
(•ex
ecut
e th
e re
quir
ed n
umbe
r of
it
era
tio
ns.
');
wri
teln
('P
rese
nt
num
ber
of
iter
atio
ns:
=
',Tim
eCou
nter
);
Con
tinue
:=O
; In
tlen
gth
:=In
tlen
gth
Cri
t;
read
ln;
' en
d·
Intl
engt
h:=
Intl
engt
h+(S
upV
elD
ata[
2, 1
J-S
upV
elD
ataC
1,1J
)*S
upV
elD
ataC
2,2l
; C
olD
elta
t:
=C
olD
elta
t+
(Sup
Vel
Dat
a[2,
1]-
Sup
Vel
Dat
a[1,
1J);
If
In
tlen
gth>
Intl
engt
hCri
t th
en b
egin
A
rbV
alue
:=In
tlen
gth-
Intl
engt
hCri
t;
Col
Del
ta t
:=C
olD
elta
t-
Arb
Val
ue/S
upV
elD
ataC
2,2l
; In
tLen
gtfi
:=In
iLen
gthe
rit;
S
upV
elD
ata[
1,1J
:=S
upV
elD
ataC
2,1J
-Arb
Val
ue/S
upV
elD
ata[
2,2l
; S
upV
elD
ata[
1,2J
:=S
upV
elD
ata[
2,2l
; en
d el
se b
egin
·
SupV
elD
ata[
1, 1
] :=
SupV
elD
ataC
2, 1
l;
SupV
elD
ata[
112J
:=
Sup
Vel
Dat
a[2,
2J;
read
(Sup
Vel
F1l
e,
SupV
elD
ataC
2, 1
J);
read
(Sup
Vel
Fil
e,
Sup
Vel
Dat
a[2,
2J)
end·
en
d·
' C
olD
elta
T:=
Col
Del
ta t
*uS
tar/
Col
leng
th;
Cum
Tim
e:=C
umTi
me+
CoT
Del
taT;
en
d;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re D
eter
min
e_A
ppro
pria
te_R
ef_S
ize_
Cla
ss_D
elta
T;
begi
n Siz
eind
ex:=
Wat
chS
izeD
T[U
nitH
eapi
ndex
J;
Iter
atio
ns:=
INT
CC
olD
elta
T/C
DT
fo
r W
atch
Siz
e·cu
nitH
eapi
ndex
,Siz
eind
exJ)
) +1
•
--
Del
taT
:=C
olD
elta
T/I
tera
tion
s;
wri
teln
CS
izei
ndex
);
end;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re C
heck
_Whe
ther
_Wat
chS
ize_
Sti
ll_V
alid
;
var Fl
ag
Inte
rG1
Inte
gVal
begi
n
Inte
ger
; ve
rrL
ongV
ecto
r;
Rea
;
07/3
1/19
95
09:0
5 F
ilen
ame:
M
OD
EL6C
2.PA
S Pa
ge
6
Siz
eind
ex:=
Wat
chS
izeD
TC
Uni
tHea
pind
exJ;
Fl
ag:=
O;
if A
RT
-[S
izel
ndex
,NJ<
0.5*
AR
T-C
Siz
eind
ex,(
N+
1)]
then
be
gin fo
r A
rbln
dex:
=O
to N
do
begi
n if
CA
RT
-[S
izel
ndex
,Arb
lnde
xl<
0.8*
AR
T-[
Siz
elnd
ex,C
Arb
lnde
x+1)
])
or
CA
RT
·csi
zeln
dex,
Arb
lnde
xl>
1.2*
AR
T-[
Siz
elnd
ex,(
Arb
inde
x+1)
l)
then
Fla
g:=
Fla
g+1;
en
d·
end·
'
if
Flag
=O
then
be
gin fo
r A
rbln
dex:
=O
to N
+1
do
lnte
rG1[
Arb
lnde
x]:=
AR
T-C
Siz
elnd
ex,A
rbin
dex]
; ln
teg
rate
Vec
tor(
lnte
rG1
,Del
taE
,0,(
N+
1),
lnte
gV
al);
fo
r A
rbln
dex:
=O
to N
+1
do
if
(AR
T-[
Siz
elnd
ex,A
rbin
dex]
<(0
.9*1
nteg
val)
) or
CA
RT
-CS
izel
ndex
,Arb
inde
xl>
C1.
1*1n
tegv
al))
th
en F
lag:
=1;
if
C
Flag
=O)
and
(Wat
chSi
zeD
TC
Uni
tHea
pind
exJ>
1)
then
W
atch
Size
DT
CU
nitH
eapi
ndex
l:=W
atch
Size
DT
CU
nitH
eapi
ndex
J-1;
en
d el
se i
f A
RT
-[Si
zein
dex,
N+1
J>FC
onc_
Tol
th
en D
T_f
or_W
atch
Siz
e·cu
nitH
eapl
ndex
,S
izel
ndex
] :=
2*D
T_f
or_W
atch
Siz
e"[U
nitH
eapl
ndex
,Siz
elnd
exl;
end;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re U
pOat
e_T
otH
eapC
onv;
begi
n Tot
Hea
pCon
v1·c
coL
iter
sl :
=T
otH
eapC
onv1
·cco
lite
rsJ
+Glo
bWet
Fac*
Uni
tHea
pCon
v1/N
oUni
tHea
ps;
Tot
Hea
pCon
v2·c
coL
iter
sl :=TotHeapConvz·ccolit~rsl
+Glo
bWet
Fac*
Uni
tHea
pCon
v2/N
oUni
tHea
ps;
end;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re U
pOat
e_T
otH
eapV
ecto
r;
begi
n Tot
Hea
pAR
TV
ec·c
unitH
eapi
ndex
] :=
AR
T-[
1,N
+1l;
Tot
Hea
pSol
SRT
1Vec
·cun
itHea
plnd
exJ:
=T
otH
eapS
olSR
T1V
ec·c
unitH
eapl
ndex
J+
(Uni
tHea
pCon
v1-H
oldC
onv1
); ·
T
otH
eapS
olSR
T2V
ec·c
unitH
eapl
ndex
l :=
Tot
Hea
pSol
SRT
2Vec
·cun
itHea
pind
ex]+
(U
nitH
eapC
onv2
-Hol
dCon
v2);
en
d;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re
Incr
emen
t_T
otH
eapV
ecto
rs_a
nd_B
TC
Vec
tors
;
begi
n AR
TB
TC
-[C
ollte
rs +
1]
:=T
otH
eapA
RT
Vec
·[NoU
nitH
eaps
l; S
RT
1Brc
·cco
Lit
ers
+1J:
=Glo
bWet
Fac*
CT
otH
eapS
olSR
T1V
ec·c
NoU
nitH
eaps
]);
SR
T2B
rc·c
coll
ters
+1l
:=G
lobW
etFa
c*(T
otH
eapS
olSR
T2V
ec·c
NoU
nitH
eaps
l);
for
Arb
inde
x:=N
oUni
tHea
ps d
ownt
o 2
do
begi
n Tot
Hea
pArt
Vec
·[A
rbln
dexl
:=T
otH
eapA
rtV
ec·[
Arb
inde
x-1]
; T
otH
eapS
olSR
T1V
ec·[
Arb
lnde
x]:=
Tot
Hea
pSol
SRT
1Vec
·cA
rbln
dex-
1l;
Tot
Hea
pSol
SRT
2Vec
·[A
rbin
dexJ
:=T
otH
eapS
olSR
T2V
ec·c
Arb
lnde
x-1J
;
Univers
ity of
Cap
e Tow
n
07/3
1/19
95
09:0
5 Fi
lena
me:
M
ODEL
6C2.P
AS
Page
7
end·
T
otH
eapA
rtV
ec"[
1] :
=1;
T
otH
eapS
olSR
T1V
ec"[
1J:=
O;
Tot
Hea
pSol
SRT
2Vec
·[1J:
=O;
end;
(---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re G
raph
_Con
vers
ion_
and_
BT
C_C
urve
s;
begi
n init
SE
Gra
phic
s(V
alid
Dir
);
Set
Cur
rent
Yin
dow
(2);
B
orde
rCur
rent
Yin
dow
C1>
; S
etA
xesT
ype(
0,0)
; .
Sca
leP
lotA
rea(
0.0,
0.0,
CT
imeV
ecto
r·cc
ollt
ers]
),G
fMax
Y);
{**
*}
Set
XY
lnte
rcep
ts(0
.0,0
.0);
-
Set
Col
or(2
);
Dra
wX
Axi
sCC
Tim
eVec
tor"
[Col
lter
s]/5
), 1
);
{*
**
}
Dra
wY
Axi
s((G
fMax
Y/5
), 1
);
Lab
elX
Axi
sC1,
0);-
Lab
elY
Axi
sC1,
0);
Titl
eXA
xisC
'Dm
lss
Tim
e CD
mlss
Ti
me=
Ctim
e *
uS
tar)
/Co
llen
gth
');
Tit
leY
Axi
sC'F
rac.
C
onv.
an
d BT
C C
urve
s');
T
itleY
indo
w('M
odel
6C2
1);
Lab
elG
raph
Yin
dow
C1,
930,
GL
abel
1,0,
0);
Lab
elG
raph
Yin
dciw
C1,
900,
Gla
bel2
,0,0
);
for
Arb
inde
x:=O
to
Co
llte
rs
do D
ataS
etX
"[A
rbln
dexl
=T
imeV
ecto
r· [
Arb
lnde
xl;
for
Arb
lnde
x:=O
to
Co
llte
rs
do D
ataS
etY
"[A
rbln
dex]
=
Tot
Hea
pCon
v1"[
Arb
lnde
xl;
Lin
ePlo
tDat
a(D
ataS
etX
",D
ataS
etY
-,C
Col
lter
s +
1),
5,0
) fo
r A
rbln
dex:
=O
to C
oll
ters
do
Dat
aSet
Y"[
Arb
lnde
xl
=T
otH
eapC
onv2
"[A
rbln
dexl
; L
ineP
lotD
ata(
Dat
aSet
x·,o
ataS
etY
-,(C
oll
ters
+1
),4
,0)
for
Arb
lnde
x:=O
to
Co
llte
rs
do D
ataS
etY
"[A
rbln
dexl
=A
RT
BT
C"[
Arb
lnde
xl;
Lin
ePlo
tDat
a(D
ataS
etx"
,Dat
aSet
Y-,
(Col
lter
s +
1),
3,1
) fo
r A
rbln
dex:
=O
to C
oll
ters
do
Dat
aSet
Y"[
Arb
lnde
x]
=SR
T1B
TC
"[A
rbln
dexJ
; L
ineP
lotD
ata(
Dat
aSet
x" ,D
ataS
etY
",(C
ollt
ers
+1
),5
,1)
for
Arb
lnde
x:=O
to
Co
llte
rs
do D
ataS
etY
"[A
rbln
dexl
=S
RT
2BT
C"[
Arb
lnde
xl;
Lin
ePlo
tDat
a(D
ataS
etx·
,oat
aSet
Y-,
CC
ollt
ers
+1)
,4, 1
)
end;
{===
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
==}
(MAI
N PR
OGRA
M:
} (=
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
} be
gin
~
new
(Bet
a1
k);
new
CK
appa
P1_k
); ne
w(K
appa
s1_k
);
new
(Lai
nbda
1 k)
; ne
w(C
ontR
atio
V1)
; ne
w(A
RT)
; ne
w(S
RT
1);
new
CSR
T2);
new
(Dat
aSet
X);
new
(Bet
a2
k);
new
CK
appa
P2_k
); ne
w(K
appa
S2_k
); ne
w(L
ambd
a2
k>;
new
CC
ontR
atio
V2)
; ne
w(A
RTP
1);
new
CSR
T1P1
); ne
w(S
RT2
P1 )
; ne
wC
Dat
aSet
Y);
new
CD
elta
T_k
);
new
(SR
T1P1
C);
new
CSR
T2P1
C);
07/3
1/19
95
09:0
5 F
ilen
ame:
M
OD
EL6C
2.PA
S Pa
ge
8
new
CA
Mat
rix);
new
(AR
TBTC
); ne
wC
TotH
eapA
RTV
ec);
new
(Tot
Hea
pCon
v1);
ne
wC
DT
_for
_Yat
chSi
ze);
new
(Aln
vers
e);
new
(SR
T1B
TC);
new
(Tot
Hea
pSol
SRT
1Vec
);
new
(Tot
Hea
pCon
v2>;
assi
gn(C
onvR
esul
ts,'C
onvR
esul
ts.D
at')
; re
wri
te(C
onvR
esul
ts);
•
clos
e(C
onvR
esul
ts);
lnit
iali
se_
Var
iou
s_V
aria
ble
s;
Siz
e D
istr
ibu
tio
n I
nit
iali
sati
on
; C
onta
min
ant
Loc
atio
n In
itia
lisa
tio
n;
Det
erm
ine_
Mod
el_P
aram
eter
s_fn
_siz
e;
lnit
iali
se_
Su
per
fici
al_
Vel
oci
ty_
Fil
e;
Col
In
itia
l C
ondi
tion
s;
Par
ticl
e In
itia
l C
ondi
tion
s;
Gen
erat
e=U
nitH
eapo
ataF
iles
;
Det
erm
ine_
Max
imum
_Del
taT
_fn_
Size
;
whi
le C
ollt
ers<
Spe
cCol
lter
s do
be
gin Co
llte
rs:=
Co
llte
rs+
1;
Det
erm
ine_
Cor
resp
ondi
ng_C
olD
elta
T;
Tim
eVec
tor"
[Col
lter
s] :=
Cum
Tim
e;
for
Uni
tHea
plnd
ex:=
1 to
NoU
nitH
eaps
do
be
gin Rea
d AR
T an
d,SR
Ts;
if
AR
T"[
l,N+l
l>FC
onc
Toi
then
be
gin
-C
lose
segr
aphi
cs;
new
(SR
T2B
TC);
new
(Tot
Hea
pSol
SRT
2Vec
);
new
CT
imeV
ecto
r);
Det
erm
ine
App
ropr
iate
Ref
S
ize
Cla
ss D
elta
T;
Det
erm
ine-
Del
taT
fn
Siz
e;-
--
Exe
cute
Uni
t6C
2;-
-C
heck
_Yne
ther
_Yat
chS
ize_
Sti
ll_V
alid
; en
d·
UpD~te_TotHeapConv;
UpD
ate
Tot
Hea
pVec
tor;
Y
rite_
AR
T_a
nd_S
RT
s;
end;
lncr
emen
t_T
otH
eapV
ecto
rs_a
nd_B
TC
Vec
tors
;
Ass
ign(
Con
vRes
ults
,'Con
vRes
ults
.Dat
');
App
end(
Con
vRes
ults
);
Yri
teln
(Con
vRes
ults
,Tim
eVec
tor"
[Col
lter
sl :
8:4
,1
,Tot
Hea
pCon
v1"[
Col
lter
s]:8
:4);
C
lose
(Con
vRes
ults
);
end;
Gra
ph_C
onve
rsio
n_an
d_B
TC
_Cur
ves;
read
lnC
Har
dCop
y);
if H
ardC
opy=
1 th
en S
cree
nD
um
p(3
,0,2
,1.5
,1.5
,0,1
,0,e
rro
r);
Univers
ity of
Cap
e Tow
n
~
w ro ~ ~ II
II II II II II II II II II II II
w < ~ II N II u II ~ I ~ w 0 0 ~
w E ro c w
~ ~ u 0
~ ~ ~ I 0 ro II
~ II m II ~ w II ~ ~ II ~ w II
~ II 0 II
~ II ~ u
~ II II
~ c II 0 w ~
Univers
ity of
Cap
e Tow
n
07/3
1/19
95
09:0
4 F
ilen
ame:
U
NIT
6C2.
PAS
Page
1
Uni
t U
nit6
C2;
{Uni
t6C
2.
*T
his
pro
gram
is
sim
ilar
to
*
The
prog
ram
cal
cula
tes
the
and
soli
d r
eact
ants
wit
hin
Mod
el5E
2.PA
S.
} { { { { { { { { { {
conc
entr
atio
n p
rofi
le o
f fl
uid
}
a p
arti
cle
usin
g th
e eq
uati
ons
as )
de
velo
ped
by D
ixon
. *
Th
is p
rogr
am p
rovi
des
for
var
iab
le s
oli
d r
eact
ant
ord
ers.
} )
* A
ssum
ptio
ns
in t
his
mod
el
incl
ude:
}
· ·
The
soli
d r
eact
ant
dep
osi
ts w
ithi
n th
e p
arti
cle
} re
sem
ble
thos
e on
th
e su
rfac
e.
) -
The
dep
osi
ts w
ould
bo
th
reac
t to
the
sam
e ex
ten
t }
if e
ach
wer
e ex
pose
d to
the
sam
e ac
id c
on
cen
trat
ion
}
for
the
sam
e ti
me.
}
{Cod
ed:
Gra
ham
Dav
ies.
}
{ D
epar
tmen
t of
C
hem
ical
E
ngin
eeri
ng.
} {
Uni
vers
ity
of
Cap
e To
wn.
}
{ 28
F
ebru
ary
1995
. }
{ 06
Ju
ne
1995
Upd
ated
(G
MO
). }
<===
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
==)
Inte
rfac
e
<===
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
==)
{Dec
lara
tion
s:
}
uses
crt
,std
hdr,
gj,g
raph
,wor
lddr
,seg
raph
,int
egra
t,M
2C1B
rent
Roo
ts;
cans
t <***
The
se a
re t
he p
aram
eter
s ap
pli
cab
le t
o t
he r
efer
ence
siz
e cl
ass.
**
*)
Bet
a1
= 0
.264
; K
appa
P1
= 4
.5;
Ord
SP1
= 1
.0;
Bet
a2
= 0
.0;
Kap
paP2
=
0.0
; O
rdSP
2 =
0.0
;
Voi
dage
=
0.0
1;
M
=10·
R
efSi
zeC
l =
5;'
Vie
wSi
zeC
L=S
;
{Dim
ensi
onle
ss S
toic
hiom
etri
c ra
tio
def
ined
pg
11
) {R
atio
of
reat
ion
rat
e of
so
lid
rea
ctan
t re
sid
ing
}
{wit
hin
the
part
icle
to
por
ous
dif
fusi
on
of
flu
id
} {
into
th
e p
art
icle
. D
efin
ed p
g 11
D
ixon
. }
{Rea
ctio
n or
der
of
the
soli
d i
n t
he
pore
s.
}
{Voi
dage
of
the
soli
d p
arti
lces
.(P
oro
sity
)
{Num
ber
of
size
cla
sses
. {D
efin
es t
he
refe
renc
e si
ze c
lass
. {T
he c
on
cen
trat
ion
pro
file
s of
th
is s
izec
lass
{g
raph
ed i
n gr
aph
1.
(NB
Vie
wSi
zeC
l <=
M
.)
} } } ar
e ) }
<***
The
se a
re n
umer
ical
met
hod
para
met
ers.
****
****
****
****
****
****
****
*}
N
=19;
Del
taE
=1
/(N
+1);
Max
Del
taT
=0.
001;
{Hal
f th
e nu
mbe
r of
in
teri
or
po
ints
. r=
O
and
r=R
} {n
ot
incl
uded
. {S
pace
Inc
rem
ent.
{
} C
alcu
late
d fr
om
1/(N
+1)
. (S
ince
}
R=i
*dE
and
R i
s at
po
int
N+1
.) }
{Max
imum
per
mit
ted
tim
e in
crem
ent
for
part
icle
s.
}
07/3
1/19
95
09:0
4 F
ilen
ame:
U
NIT
6C2.
PAS
Page
2
Nor
m_C
rit
=1e
-6;
Con
v C
rit
=1e
-8
scon
e To
i =
1e-4
FC
onC
:)ol
=1
e-4
Max
lter
. =
100;
Pri
nt_
Cri
t=1
;
{Con
verg
ence
cri
teri
a b
ased
on
the
norm
of
vec
tor
} <*
**
} {C
onve
rgen
ce c
rite
ria f
or
the
Bre
nt R
outi
ne.
) {D
imen
sion
less
co
nce
ntr
atio
n o
f so
lid
bel
ow w
hich
}
{Dim
ensi
onle
ss c
on
cen
trat
ion
of
flu
id r
eact
ant
} {b
elow
whi
ch
it
is a
ssum
ed
to b
e n
egig
ible
. }
{it
is
ass
umed
to
be
neg
lig
ible
. }
{Max
imum
it
erat
ion
s fo
r th
e B
rent
Rou
tine
. }
{***
Oth
er
info
rmat
ion
requ
ired
.***
****
****
****
****
****
****
****
****
****
*}
Val
idD
ir
=1F
:\T
P6\
1;{
Val
id d
irec
tory
for
gr
aphi
cs d
riv
ers.
}
{***
****
****
****
****
****
****
****
****
****
****
****
****
****
****
****
****
***}
type
var ar
ray1
= A
rray
[1 .. 5
0,1
.. 2J
of
Dou
ble;
Tot
Vol
Par
t,V
olL
iq,D
elta
T,I
tera
tion
s,C
umT
ime
Uni
tHea
pCon
v1,U
nitH
eapC
onv2
A
rbin
dex,
Siz
elnd
ex,e
rror
,Uni
tHea
plnd
ex
Rep
eats
,Plo
t_V
ar,F
lag1
,Har
dCop
y D
ataS
etX
,Dat
aSet
Y
:Dou
ble;
:D
oubl
e;
: In
teg
er;
Bet
a1
k,B
eta2
k,
Kap
paP1
k,
Kap
paP2
k,
Kap
paS1
k,
Kap
paS2
k
Lam
bda1
k,
Lam
6da2
k,
Con
tRat
ioV
1,C
ontR
atio
V2-
-
: In
teg
er;·
:"
Ver
yLon
gVec
tor;
: "
Sho
rtV
ecto
r;
:"S
hort
Vec
tor;
:"
Arr
ay1;
:"
Sqr
Mat
; :"
Sqr
Mat
; :E
xten
ded;
:E
xten
ded;
:S
trin
g;
Del
taT
K
-A
RT,
AR
fP1,
SRT1
,SR
T2,S
RT1
P1,S
RT2
P1,S
RT1
P1C
,SR
T2P1
C
AM
atri
x,A
lnve
rse
AR
TP1
CV
al,S
umSi
zeD
ata_
2,N
uSta
r L
ambd
a1,L
ambd
a2,K
appa
S1,K
appa
S2
GL
abel
1,G
labe
l2,G
Lab
el3
Siz
eDat
a :
Arr
ay[1
. .5
0, 1
. .2]
of
Rea
l;
CART
A
lpha
fn
(r)
at
tim
e T
{A
RTP1
A
lpha
fn
(r)
at
tim
e T+
1 {S
RT1
Sigm
a fn
(r)
at
tim
e T
{S
RTs
P1
Sigm
a fn
(r)
at
tim
e T+
1 (g
uess
ed)
rang
e 0
.• M
ra
nge
O •.
M
rang
e 0 ..
M
rang
e O
.• M
{S
RTs
P1C
Si
gma
fn(r
) at
ti
me
T+1
(cal
cula
ted
) ra
nge
O .. M
C
YV
ect
'Co
nst
' v
ecto
r' i
n C
rank
-Nic
olso
n m
etho
d ra
nge
O .. N
+1
{AM
atri
x M
atri
x of
Cra
nk-N
icol
son
coef
fici
ents
ra
nge
N*
N
{Dat
aSet
X
X v
ecto
r us
ed
in t
he
grap
hing
ro
uti
ne
rang
e 0 ..
N+1
0 ..
N+1
} 0 ..
N+1
} 0 ..
N+1
} 0 ..
N+1
} 0 ..
N+1
} } } }
{***
****
****
****
****
****
****
****
****
****
****
****
****
****
****
****
****
***}
Pro
cedu
re S
ize
Dis
trib
uti
on
ln
itia
lisa
tio
ni
Pro
cedu
re C
onta
min
ant
Loc
atio
n In
itia
lisa
tio
n;
Pro
cedu
re D
eter
min
e M
odel
·P
aram
eter
s fn
Siz
e;
Pro
cedu
re D
eter
min
e-D
elta
T f
n S
ize;
--
Pro
cedu
re P
arti
cle
Init
ial-
Co
nd
itio
ns;
P
roce
dure
Gue
ss
SRTs
P1;
-P
roce
dure
ReG
uess
SR
TsP1
; P
roce
dure
Cal
e C
rank
Nic
olso
n M
atri
xCva
r Y
Vec
tor
:Sho
rtV
ecto
r);
Pro
cedu
re C
alc-
AR
TP1
-and
SR
TsP1
C;
Pro
cedu
re C
hecK
_Con
verg
ence
;
Univers
ity of
Cap
e Tow
n
07/3
1/19
95
09:0
4 Fi
lena
me:
U
NIT
6C2.
PAS
Page
3
Pro
cedu
re U
pdat
e AR
T an
d SR
Ts;
Pro
cedu
re C
ale
Con
vers
ion;
Pr
oced
ure.
Gra
pn1
Init
iali
se;
Pro
cedu
re G
raph
1-R
esul
ts;
Pro
cedu
re E
xecu
te_U
nit6
C2;
f ;~[~;~~;:;1~~============================================================}
(===
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
==}
Fun
ctio
n P
ower
(Bas
e,P
ow:r
eal)
:Ext
ende
d;
begi
n if P
ow=O
th
en P
ower
:=1
else
if
Bas
e=O
th
en P
ower
:=O
el
se
Pow
er:=
exp(
Pow
*ln(
base
));
end;
(---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re S
ize_
Dis
trib
uti
on
_In
itia
lisa
tio
n;
(Thi
s pr
oced
ure
sets
up
the
size
dis
trib
uti
on
dat
a ar
ray
. In
itia
lly
the
}
(Siz
eOat
a ar
ray
cont
ains
ra
dius
in
form
atio
n an
d fr
acti
on
al
volu
me
} (i
nfor
mat
ion
(ie
R a
nd V
p/V
tot)
. On
out
put
it c
onta
ins
rela
tiv
e ra
dius
}
(inf
orm
atio
n an
d re
lati
ve
volu
me
info
rmat
ion
(ie R
/R
ref
and
Vp/
Vp
ref)
. }
(Id
eall
y t
his
in
form
atio
n w
ould
be
read
in
fro
m a
dat
a fi
le.
-}
var R
efR
adiu
s,V
olR
efP
art
:Dou
ble;
begi
n Si z
eDat
a [1
, 1J
= 3
7.00
e-3/
2 S
zeD
ata[
1,2l
=
58.
1/10
0 Si
zeD
ata
[2, 1
1 =
31.
25e-
3/2
S ze
Dat
a C2
, 21
= 1
5.8/
100
Si z
eDat
a C3
, 1J
= 2
2.00
e-3/
2 S
zeD
ata
C3, 2
1 =
10.
3/10
0 Si
zeD
ataC
4, 11
=
16.
10e-
3/2
S ze
Dat
a C4
, 21
=
4.1/
100
Si z
eDat
a CS
, 1J
= 1
1.3S
e-3/
2 S
zeD
ata
CS, 2
1 =
3.
2/10
0 S
izeD
ataC
6, 11
=
8.10e~3/2
s ze
Dat
a C6
, 21
=
1.8/
100
Siz
eDat
a[7,
11
=
S. 7
3e-3
/'2
S ze
Dat
a C7
, 21
=
1.3/
100
Si z
eDat
a [8
, 11
=
4.0S
e-3/
2 S
zeD
ataC
8,2l
=
1.
0/10
0 Si
zeD
ata
[9, 1
J =
2.
86e-
3/2
S ze
Dat
a C9
, 21
=
0.6/
100
Siz
eDat
a[10
, 11
=
1.18
e-3/
2 s
zeD
ata[
10,2
l =
3.
8/10
0
Vol
Ref
Par
t:=
Tot
Vol
Par
t*S
izeD
ataC
Ref
Siz
eCl,
21;
Ref
Rad
ius
:=S
izeD
ata[
Ref
Siz
ecl,
11;
Sum
Size
Dat
a_2:
=0;
for
Siz
elnd
ex:=
1 to
M do
be
gin Siz
eDat
a[S
'izel
ndex
, 11:
=S
izeD
ataC
Siz
elnd
ex,1
1/R
efR
adiu
s;
Siz
eDat
a[S
izel
ndex
,21
:=S
izeD
ata[
Siz
elnd
ex,2
l*T
otV
olP
art/
Vol
Ref
Par
t;
Sum
Size
Dat
a 2:
=Sum
Size
Dat
a 2+
Siz
eDat
aCS
izel
ndex
,21;
en
d;
--
NuS
tar:
=V
olL
iq/(
Voi
dage
*Vol
Ref
Par
t);
(Rat
io o
f vo
lum
e of
bul
k fl
uid
to
flu
id
in p
arti
cle
pore
s.
}
07/3
1/19
95
09:0
4 F
ilen
ame:
U
NIT
6C2.
PAS
Page
4
end;
(---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re C
on
tam
inan
t_L
oca
tio
n_
lnit
iali
sati
on
;
{Thi
s pr
oced
ure
is u
sed
to d
efin
e th
e co
ntam
inan
t ra
tio
vec
tor
(rati
o o
f }
{th
e su
rfac
e co
ntam
inan
t co
nce
ntr
atio
n t
o b
ulk
cont
amin
ant
con
cen
trat
ion
).}
{NO
TE:
If a
ny s
ize
clas
s of
p
arti
cles
hav
e a
surf
ace
con
cen
trat
ion
of
} (
cont
amin
ant,
th
en s
o to
o m
ust
the
refe
ren
ce s
ize
clas
s.
}
begi
n Con
tRat
oV
1"[1
1 =0
.00
Con
tRat
oV
1"C2
1 =0
.00
Con
tRat
oV
1"C
3l
=0.0
0 C
ontR
at
oV1"
C4l
=0
.00
cont
Rat
ov1
·cs1
=0
.00
Con
tRat
oV
1"[6
l =0
.00
Con
tRat
oV
1"C
7l
=0.0
0 C
ontR
at
oV1"
[8l
=0.0
0 C
ontR
at
oV1"
C9l
=0
.00
Con
tRat
ov
1·c1
01
=0.
00
Con
tRat
ov
2"C
1l
=O
Con
tRat
ov
2"C
2l
=O
Con
tRat
ov
2"C
3l
=O
Con
tRat
ov
2"C
41
=O
Con
tRat
ov
2"[S
l =O
C
ontR
at
ov2"
C6l
=O
C
ontR
at o
v2"C
71
=O
Con
tRat
ov
2"C
81
=O
Con
tRat
ov
2"[9
l =O
C
ontR
at
ov2"
C10
l =O
en
d;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re D
eter
min
e_M
odel
_Par
amet
ers_
fn_S
ize;
{Thi
s pr
oced
ure
dete
rmin
es
the
mod
el
para
met
ers
each
siz
e cl
ass
of
} {
par
ticl
es.
Not
e th
at
the
Del
taT
is
def
ined
sep
arat
ely
in
anot
her
} {p
roce
dure
. }
var F
lag1
,Fla
g2
:In
teg
er;
begi
n F
lag1
:=0;
F
lag2
:=0;
fo
r S
izel
ndex
:=1
to M
do
begi
n Lam
bda1
k"
CS
izel
ndex
l:=
Con
tRat
iov1
·csi
zeln
dexl
-
/(1+
Con
tRat
iov1
·csi
zeln
dexl
>;
Lam
bda2
k"
CS
izel
ndex
l:=
Con
tRat
iov2
-csi
zeln
dexl
-
/(1+
Con
tRat
ioV
2"[S
izel
ndex
l);
if C
ontR
atio
V1"
[Siz
elnd
exl<
>O
the
n F
lag1
:=F
lag1
+1;
if
Con
tRat
iov2
·csi
zeln
dexl
<>
O t
hen
Fla
g2:=
Fla
g2+
1;
end;
if (
((F
lag1
=1)
an
d (C
ontR
atio
v1-C
Ref
Siz
eCll
=0)
) or
((
flag
2=1)
and
(C
ontR
atio
v2"C
Ref
Siz
eCll
=0)
))
then
beg
in
clo
sese
gra
ph
ics;
w
rite
ln('
Con
tam
inan
t L
ocat
ion
Vio
loat
ion.
1);
read
ln;
end·
La
mbd
a1
=Lam
bda1
k-
[Ref
Siz
eCll
; La
mbd
a2
=Lam
bda2
-k"C
Ref
Size
Cll;
K
appa
S1
=Lam
bda1
•Kap
paP1
/(1-
Lam
bda1
);
Kap
paS2
=L
ambd
a2*K
appa
P2/C
1-L
anbd
a2);
Univers
ity of
Cap
e Tow
n
07/3
1/19
95
09:0
4 Fi
lena
me:
U
NIT
6C2.
PAS
Page
5
for
Siz
elnd
ex:=
1 to
M do
be
gin Bet
a1_k
· [S
izel
ndex
] :=
Bet
a1*(
1+C
ontR
atio
V1"
[Ref
Siz
eClJ
) /(
1+C
ontR
atio
V1"
[Siz
elnd
exJ)
; B
eta2
k"
[Siz
elnd
exJ
:=B
eta2
*(1+
Con
tRat
ioV
2"[R
efS
izeC
lJ)
-/(
1+C
ontR
atio
vz· C
Size
lnde
xJ >
; K
appa
P1
k"[S
izel
ndex
J:=
Kap
paP
1*S
izeD
ata[
Siz
elnd
ex,1
J -
*Siz
eDat
aCS
izel
ndex
,1J;
K
appa
P2
k"[S
izel
ndex
J :=
Kap
paP
2*S
izeD
ata[
Siz
elnd
ex, 1
J -
*Siz
eDat
a[S
izel
ndex
,1J;
if
Con
tRat
iov1
·csi
zeln
dexJ
=O
the
n K
appa
S1_
k.[S
izel
ndex
J :=
O e
lse
Kap
paS1
k"
[Siz
elnd
ex]
:=K
appa
S1*
Pow
er((
Con
tRat
iov1
·csi
zeln
dexJ
-
/Con
tRat
ioV
1"[R
efS
izeC
LJ)
,Ord
SP
1)
*Siz
eDat
a[S
izel
ndex
,1J;
if
Con
tRat
ioV
2"[S
izel
ndex
J=O
the
n K
appa
S2_k
"[Si
zeln
dexJ
:=O
els
e K
appa
S2
k"C
Size
lnde
xJ :
=K
appa
S2*
Pow
er((
Con
tRat
iovz
·csi
zeln
dexJ
-
/Con
tRat
ioV
2"[R
efS
izeC
lJ),
Ord
SP
2)
*Siz
eDat
a[S
izel
ndex
, 1J;
end·
~d·
, {-
-~--
----
----
----
----
----
----
----
----
----
----
----
------------------------}
Pro
cedu
re D
eter
min
e_D
elta
T_f
n_S
ize;
begi
n for
Siz
elnd
ex:=
1 to
M do
D
elta
T
k"[S
izel
ndex
, 1J
:=D
elta
T/S
izeD
ata[
Siz
elnd
ex, 1
J -
/Siz
eDat
a[S
izel
ndex
, 1J;
end;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re P
arti
cle_
lnit
ial_
Co
nd
itio
ns;
{Thi
s pr
oced
ure
uses
th
e in
itia
l co
ndit
ions
to
set
the
alph
a an
d si
gma
{vec
tors
. '
} } ar
e th
e su
rfac
e}
{Not
e th
at A
RT[
N+1
], A
RTP1
CN+1
J, SR
T1CN
+1]
and
SRT1
P1CN
+1]
{con
cent
rati
ons
of
the
liq
uid
and
so
lid
rea
ctan
ts.
begi
n for
Siz
elnd
ex:=
1 to
M do
be
gin fo
r A
rbln
dex:
=O t
o N
do
begi
n AR
T"C
Size
Jnde
x,A
rbln
dexJ
:=
O
SR
T1"
[Siz
elnd
ex,A
rbln
dexJ
:=1
SRT
2"C
Size
lnde
x,A
rbln
dexJ
:=1
SRT
1P1C
"[Si
zeln
dex,
Arb
lnde
x]
=O;
SR
T2P
1c·r
size
lnde
x,A
rbln
dexJ
=O
; en
d·
ART
t CSi
zeln
dex,
N+1
J :=
1;
SRT1
. [Si
zel
nde
x,N
+1 J
: =1;
SR
T2"
[Siz
elnd
ex,N
+1J
:=
1;
SRT
1P1C
"[Si
zeln
dex,
Arb
lnde
xJ:=
O;
SRT
2P1C
"[Si
zeln
dex,
Arb
lnde
xl :
=0;
en
d;
end;
}
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re G
uess
_SR
TsP
1;
07/3
1/19
95
09:0
4 F
ilen
ame:
U
NIT
6C2.
PAS
Page
6
{Thi
s pr
oced
ure
prov
ides
the
in
itia
l "g
uess
" fo
r th
e it
erat
ion
. It
use
s }
{the
pre
viou
s ti
me
inte
rval
's v
alue
s as
th
e gu
ess.
}
begi
n for
Siz
elnd
ex:=
1 to
M do
be
gin fo
r A
rbln
dex:
=O
to (
N+1
) do
be
gin SR
T1P
1"[S
izel
ndex
,Arb
lnde
xJ :
=S
RT
1"C
size
lnde
x,A
rbln
dexJ
; S
RT
2P1"
[Siz
elnd
ex,A
rbln
dex]
:=
SR
T2"
[Siz
elnd
ex,A
rbln
dexJ
; en
d·
AR
TP
1"[S
izel
ndex
,N+
1J:=
AR
T.[
Siz
elnd
ex,N
+1J
; en
d·
· en
d;
'
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re R
eGue
ss_S
RT
sP1;
{Thi
s pr
oced
ure
prov
ides
an
upda
ted
"gue
ss"
for
the
next
it
erat
ion
. .I
t }
{use
s th
e SR
T1P1
C v
ecto
r as
the
upd
ated
gue
ss.
}
begi
n ~
for
Siz
elnd
ex:=
1 to
M do
be
gin fo
r A
rbln
dex:
=O
to (
N+1
) do
be
gin SR
T1P
1"[S
izel
ndex
,Arb
lnde
xJ:=
SR
T1P
1C"[
Siz
elnd
ex,A
rbln
dexJ
; S
RT
2P1"
CS
izel
ndex
,Arb
lnde
xJ:=
SR
T2P
1C"C
Siz
elnd
ex,A
rbln
dexJ
; en
d·
AR
TP1
"[Si
zeln
dex,
N+1
J:=A
RT
P1C
Val
; en
d;
end·
{-
-~--
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
} P
roce
dure
Cal
c_C
rank
_Nic
olso
n_M
atri
x(va
r Y
Vec
tor
:Sho
rtV
ecto
r);
var R
ows,
Col
s
begi
n
: In
teg
er;
for
Row
s:=O
to
(N
+1)
do
begi
n for
Col
s:=O
to
(N
+1)
do A
Mat
rix"
CR
ows,
Col
sJ:=
O;
end;
AM
atri
x"[0
,0J:
=(-
6·D
elta
E*D
elta
E*S
izeD
ata[
Siz
elnd
ex,1
J*S
izeD
ata[
Siz
elnd
ex,1
J *(
Kap
paP1
k·
csiz
elnd
exJ*
Pow
er(S
RT
1P1"
[Siz
elnd
ex,O
J,O
rdS
P1)
+K
appa
P2 K
·csi
zeln
dexJ
*Pow
er(S
RT
2P1·
csiz
elnd
ex,O
J1or
dSP2
>Y
·2*D
elta
t*D
elta
E*S
izeD
ata[
Siz
elnd
ex,1
J*S
izeD
ata[
S1z
eind
ex,1
J /D
elta
T
k·c
size
ind
ex,1
J);
AM
atri
x·co
, 1J:
=6;
-
YV
ecto
rCO
J :=
AR
T"C
Siz
elnd
ex,O
J*C
6+D
elta
E*D
elta
E*S
izeD
ata[
Siz
elnd
ex,1
J *S
izeD
ata[
Siz
elnd
ex,1
J*C
Kap
paP
1 k·
csiz
elnd
exJ
*Pow
er(S
RT
1"C
Size
inde
x,O
J,O
rdSP
T)+
Kap
paP2
k·c
size
lnde
x]
*Pow
er(S
RT
2"C
Size
inde
x,O
J,O
rdSP
2))
-·2
*Del
taE
*Del
taE
*Siz
eDat
a[S
izel
ndex
,1l
*Siz
eDat
aCS
izel
ndex
,11/
Del
taT
k·c
size
lnd
ex,1
1)
·ART
" [S
izel
ndex
,1J*
6;
-
for
Row
s:=1
to
N-1
do
Univers
ity of
Cap
e Tow
n
07/3
1/19
95
09:0
4 Fi
lena
me:
U
NIT
6C2.
PAS
Pase
7
begi
n AM
atri
x.[R
ows,
Row
s·1]
:=R
ows·
1;
·AM
atrix
.CR
ows,R
ows]
:=
·Z*R
ows·
Row
s*O
elta
E*O
elta
E*S
izeO
ata[
Size
lnde
x, 1
) *S
izeD
ata[
Siz
elnd
ex, 1
]*(K
appa
P1_
k.[S
izel
ndex
l *P
ower
(SR
T1P
1.[S
izel
ndex
,Row
sJ,O
rdSP
1)
+Kap
paPZ
k"
[Siz
elnd
exl
*Pow
er(S
RT
2P1·
csiz
elnd
ex,R
owsJ
,Ord
SP2)
) ·2
*Row
s*D
elta
E*O
elta
E*S
izeO
ata[
Siz
elnd
ex, 1
) *S
izeD
ata[
Siz
elnd
ex, 1
]/D
elta
T k
"CS
izel
ndex
, 11;
A
Mat
rix.[R
ows,
Row
s+1J
:=R
ows+
1;
-Y
Vec
torC
Row
sl :=
AR
T.C
Size
lnde
x,R
ows·
1l*C
·Row
s+1)
+
end;
AR
T.C
Size
lnde
x,R
owsl
*C2*
Row
s+R
ows*
Del
taE
*Oel
taE
*S
izeD
ataC
Si'z
elnd
ex, 1
J*S
ize0
ata[
Siz
elnd
ex, 1
1 *C
Kap
paP1
k·
csiz
elnd
exl
*Pow
er(S
RT
1·cs
izel
ndex
,Row
sJ,O
rdSP
1)
+Kap
paP2
k"
[Siz
elnd
exl
*Pow
erC
SRT
2"[S
izel
ndex
,Row
s],O
rdSP
2))
·2*R
ows*
Del
taE
*Del
taE
*Si z
eOat
a [S
i zel
ndex
, 1'J
*Siz
eDat
a[S
izel
ndex
,1l/
Del
taT
k·c
size
lnd
ex,1
])
+AR
T"[
Size
lnde
x,R
ows+
1J*C
·Row
s·1)
;
AM
atri
x·cN
,N·1
]:=
N·1
; A
Mat
rix.C
N,N
l :=
·2*N
·N*O
elta
E*D
elta
E*S
izeO
ata[
Siz
elnd
ex,1
l *S
izeD
ata[
Size
lnde
x,1l
*CK
appa
P1
k·cs
izei
ndex
l *P
ower
(SR
T1P
1"[S
izel
ndex
,NJ,
Ord
SP1)
+Kap
paP2
k·c
size
lnde
xl
*Pow
er(S
RT
2P1.
CSi
zeJn
dex
1N
J,O
rdSP
2))·2
*N*D
eTta
E*O
elta
E
*Siz
eOat
a[S
izel
ndex
, 1l*
S1z
eDat
a[S
izel
ndex
, 1)
/Del
taT
k·
csiz
elnd
ex, 1
1;
YV
ecto
rCN
l :=
AR
T.[S
izeJ
ndex
,N·1
l*C
·N+1
)+A
RT
·csi
zeln
dex,
Nl*
C2*
N+
N*O
elta
E*D
elta
E*S
izeO
ata[
Siz
elnd
ex,1
] *S
izeD
ata[
Size
lnde
x,1l
*CK
appa
P1
k·cs
izel
ndex
l *P
ower
CSR
T1.
[Siz
elnd
ex,N
l,Ord
SPl)
+Kap
paP2
k·c
size
lnde
xl
*Power(SRTz·csizelndex~Nl,OrdSP2))·2*N*OeTtaE*DeltaE
*Siz
eDat
a[S
izel
ndex
,11
Siz
eOat
a[S
izel
ndex
,11
/Del
taT
k"
[Siz
elnd
ex,1
J)+
AR
T·c
size
lnde
x,N
+1l
*C·N
·1)
·AR
TP1
.[Si
zeln
dex,
N+
1]*(
N+
1);
end;
{··
····
····
····
····
····
····
····
····
····
····
····
····
····
····
····
····
····
···}
P
roce
dure
Cal
c_A
RTP
1_an
d_SR
TsP1
C;
{As
it s
tand
s th
is p
roce
dure
can
cop
e w
ith
a v
aria
ble
rea
ctio
n o
rder
. T
his}
{
is d
ue
to t
he i
nclu
sion
of
the
Bre
nt R
outi
ne.
}
var M
assB
alV
al,A
Mat
Det
V
alue
AtR
oot
Out
putV
ecto
r,Y
Vec
tor
Dou
ble;
R
eal;
S
hort
Vec
tor;
begi
n {F
irst
cal
cula
te t
he A
RTP1
v
ecto
rs.
for
Siz
elnd
ex:=
1 to
M do
be
gin Cal
e C
rank
Nic
olso
n M
atri
xCY
Vec
tor)
; GaussJordanCAMatrix~,YVector,CN+1),0utputVector,Alnverse·,AMatDet);
for
Arb
lnde
x:=O
to
N d
o
}
07/3
1/19
95
09:0
4 F
ilen
ame:
U
NIT
6C2.
PAS
Page
8
end;
if O
utPu
tVec
torC
Arb
lnde
xl>F
Con
c To
l th
en
AR
TP
1·cs
izel
ndex
,Arb
lnde
xJ:=
Out
putV
ecto
r[A
rbln
dexl
el
se AR
TP
1.[S
izel
ndex
,Arb
lnde
x]:=
O;
{The
Gau
ssJo
rdan
pro
cedu
re c
alcu
late
s va
lues
for
th
e AR
T v
ecto
r fr
om
} {p
oint
0 t
o p
oint
N (
alth
ough
it
mak
es
use
of
the
N+1
th p
oin
t).
To
} {d
eter
min
e th
e N+
1 th
po
int
valu
e,
mak
e us
e of
th
e m
ass
bala
nce
of
} {
flu
id r
eact
ant
in-t
he
CSTR
. }
Mas
sBal
Val
:=O
;
for
Siz
elnd
ex:=
1 to
M do
be
gin Mas
sBal
Val
:=M
assB
alV
al·S
izeD
ataC
Size
lnde
x,2l
/2*C
Kap
paS1
k·
csiz
elnd
exl
*SRT
1 · [
Si z
el n
dex,
(N+1
)] *
ART
· CS i
zel
nde
x, (N
+1)]
+K
appa
S1
k·cs
izel
ndex
]*S
RT
1P1·
csiz
elnd
ex,C
N+
1)l
*AR
TP
1·cs
izei
ndex
,CN
+1)
l+K
appa
S2_
k·cs
izel
ndex
l *S
RT
2·cs
izei
ndex
,CN
+1)
l*A
RT
·csi
zeln
dex,
(N+
1)J
+Kap
paS2
k·
csiz
elnd
exl*
SR
T2P
1·cs
izel
ndex
,(N
+1)
] *A
RT
P1.
[Siz
elnd
ex,(
N+
1)])
·3*S
ize0
ata[
Siz
elnp
ex,2
l /(
Siz
e0at
a[S
izel
nd
ex,1
J*S
ize0
ata[
Siz
eln
dex
,1])
/2
*CC
AR
T.C
Size
lnde
x,(N
+1)
J+A
RT
P1·c
size
lnde
x,(N
+1)
l)
·CA
RT
.CSi
zeln
dex,
Nl+
AR
TP1
.CSi
zeln
dex,
NJ)
)/D
elta
E;
end;
AR
TP1
CV
al:=
AR
T.[1
,(N+1
)]+0
elta
T/N
uSta
r*M
assB
alV
al;
{Cal
cula
te t
he S
RTsP
1C v
ecto
rs.
Cod
e m
akes
,use
of
Bre
nt'
s M
etho
d }
{(A
no
n-l
inea
r ro
ot
find
ing
proc
edur
e)
to s
ofve
for
th
e ro
ot.
}
for
Siz
elnd
ex:=
1 to
M do
be
gin fo
r A
rbln
dex:
=O
to (
N+1
) do
beg
in
ex, 1
1,
if
(Bet
a1
k·cs
izel
ndex
l<>
O)
and
CKap
paP1
k·
csiz
elnd
exl<
>O
) th
en
begi
n -
-if
Ord
SP1=
1 th
en
begi
n SR
T1P
1c·c
size
lnde
x,A
rbln
dexl
:=
SR
T1·
csiz
elnd
ex,A
rbln
dexl
end
*C1·
Del
taT
k·
csiz
elnd
ex,1
l*K
appa
P1
k·cs
izel
ndex
J *B
eta1
k"
[Siz
elnd
exJ*
AR
T·c
size
inde
x,A
rbin
dexl
/C
2*C
17
Lam
bda1
k·
csiz
elnd
exJ)
))/(
1+D
elta
T k
·csi
zeln
dex,
11
*Kap
paP1
_k.C
Size
lnde
xl*B
eta1
k
·csi
zeln
dex
f *A
RT
P1"
[Siz
elnd
ex,A
rbln
dexl
/C2*
C1·
Lam
bda1
_k·c
size
lnde
x]))
)
else
if
SRT
1P1·
csiz
elnd
ex,A
rbln
dex]
<SC
onc
Tol
th
en
SR
T1P
1c·c
size
lnde
x,A
rbln
dexl
:=S
RT
1P1·
csiz
elnd
ex,A
rbln
dexl
el
se
begi
n SR
T1P
1c·c
size
lnde
x,A
rbln
dexl
:=B
rent
Roo
ts(0
.0,1
.0,D
elta
T_k
·csi
zeln
d
end;
Kap
paP1
k·
csiz
elnd
exJ,
Bet
a1
k·c
size
lnd
exl,
L
ambd
a1-k
·csi
zeln
dex]
,Ord
SP
l,A
RT
·csi
zeln
dex,
Arb
inde
xl,
AR
TP
1"[S
izel
ndex
,Arb
lnde
xl,S
RT
1.[S
izel
ndex
,Arb
lnde
xl,
1e·8
,100
,Val
ueA
tRoo
t,er
ror)
;
Univers
ity of
Cap
e Tow
n
07/3
1/19
95
09:0
4 Fi
lena
me:
U
NIT
6C2.
PAS
Page
9
end·
if
~Be
ta2_
k"CSiz
elnd
exJ<
>0)
and
(Kap
paP2
_k.C
Size
lnde
xJ<>
0)
then
be
gin if O
rdSP
2=1
then
be
gin SR
T2P
1C"[
Size
lnde
x,A
rbln
dexJ
:=S
RT
2"C
Size
lnde
x,A
rbln
dexJ
end
*C1·
Del
taT
k"
CSi
zeln
dex,
1J*
Kap
paP2
k"
CS'
izel
ndex
J *B
eta2
k"
CSi
zeln
dexJ
*AR
T"[
Size
lnde
x,A
rbln
dexJ
/(
2*(1
7Lam
bda2
k"
CS
izel
ndex
J)))
/(1+
Del
taT
k"[
Siz
elnd
ex,1
J *K
appa
P2
k"C
Size
lnde
xJ*B
eta2
k"[
Siz
elnd
exr
*AR
TP
1"C
size
lnde
x,A
rbln
dexJ
/(2*
C1·
Lam
bda2
_k·c
size
lnde
xJ))
)
else
if
SRT
2P1"
CSi
zeln
dex,
Arb
lnde
xJ<S
Con
c To
i th
en
SRT
1P1C
"[Si
zeln
dex,
Arb
lnde
xJ :
=SR
T2P
1"C
Size
lnde
x,A
rbln
dexJ
el
se
begi
n SRT
2P1C
"[Si
zeln
dex,
Arb
lnde
xJ :=
Bre
ntR
oots
(O.O
, 1.0
, D
elta
T
k"C
Size
lnde
x,1J
,Kap
paP2
k"
CS
izel
ndex
J,
Bet
a2 f
"[S
izel
ndex
],L
ambd
a2 k
"1S
izel
ndex
J,O
rdS
P2,
A
RT
"CSi
zeln
dex,
Arb
lnde
xJ,A
RT
P1"C
Size
lnde
x,A
rbln
dexJ
, SR
T2"
CSi
zeln
dex,
Arb
lnde
xl, 1
e·8,
100
,Val
ueA
tRoo
t,er
ror>
; en
d·
end·
'
end·
'
end·
'
end;
'
{··
····
·-··
···-
-··-
····
·---
----
---·
-··-
----
--·-
·-··
··--
----
---·
----
----
---}
P
roce
dure
Che
ck_C
onve
rgen
ce;
{Thi
s pr
oced
ure
chec
ks w
heth
er
or n
ot
the
solu
tio
n h
as c
onve
rged
by
} {c
ompa
ring
the
gue
ssed
val
ue o
f SR
T1P1
w
ith
a ca
lcu
late
d v
alue
of
SRT1
P1.
} {A
lso
com
pare
s th
e ca
lcu
late
d v
alue
of
ARTP
1C
with
gu
esse
d va
lue
of
} CA
RTP1
. }
var N
orm
1,N
orm
2,N
orm
3 :D
oubl
e;
begi
n Norm
1 =O
; {U
sing
a n
orm
***
}
Nor
m2
=O;
Nor
m3
=O;
for
S ze
lnde
x:=
1 to
M do
be
gin fo
r A
rbln
dex:
=O
to (
N+1
) do
be
gin if (
Bet
a1
k"[S
izel
ndex
l<>
O)
and
CKap
paP1
k"
CSi
zeln
dex]
<>O
) th
en b
egin
N
orm
1:=N
orm
1+C
SRT
1P1C
"CSi
zeln
dex,
Arb
lnde
xl
-SR
T1P
1"C
Size
lnde
x,A
rbln
dexJ
)*(S
RT
1P1C
"[Si
zeln
dex,
Arb
lnde
xl
-SR
T1P
1"C
Size
lnde
x,A
rbln
dexJ
);
end
else
Nor
m1:
=0;
if C
Bet
a2
k"C
Size
lnde
xJ<>
O)
and
(Kap
paP2
k"
CSi
zeln
dex]
<>O
) th
en b
egin
No
rm2:
:Nor
m2+(
SRT2
P1C"
[Siz
elnd
e~,A
rbln
dexJ
-S
RT
2P1"
CSi
ze!n
dex,
Arb
lnde
xJ)*
CSR
T1P
1C"[
Size
lnde
x,A
rbln
dexl
-S
RT
2P1"
CSi
zeln
dex,
Arb
lnde
xJ);
en
d el
se N
orm
2:=0
;
07/3
1/19
95
09:0
4 F
ilen
ame:
U
NIT
6C2.
PAS
Page
10
end·
en
d·
' N
oriri
3:=(
AR
TP1C
Val
-AR
TP1"
[1,
N+1
])*C
AR
TP1C
Val
-AR
TP1"
C1
N+
1l);
if
(N
orm
1<N
orm
Cri
t) a
nd
CNor
m2<
Nor
m C
rit)
an
d (N
or~<
Norm
Cri
t)
then
F
lag1
::1;
-
-en
d;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re U
pdat
e_A
RT_
and_
SRTs
;
{Thi
s pr
oced
ure
upda
tes
the
alph
a an
d si
gma
vec
tors
for
th
e ne
xt
iter
atio
n}
{b
y re
plac
ing
thei
r co
mpo
nent
s w
ith
the
alph
aT+1
an
d si
gmaT
+1
vec
tors
. }
begi
n for
Siz
elnd
ex:=
1 to
M do
be
gin fo
r A
rbln
dex:
=O
to (
N+1
) do
be
gin AR
T"[
Size
lnde
x,A
rbln
dexJ
=A
RT
P1"C
Size
lnde
x,A
rbin
dexJ
; S
RT
1"[S
izel
ndex
,Arb
lnde
xl
=S
RT
1P1"
[Siz
elnd
ex,A
rb!n
dexJ
; S
RT
2"[S
izel
ndex
,Arb
lnde
xJ
=S
RT
2P1"
[Siz
elnd
ex,A
rbln
dex]
; en
d·
end·
'
end;
'
{---
-·--
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re C
alc_
Con
vers
ion;
{Thi
s pr
oced
ure
calc
ula
tes
the
frac
tio
nal
co
nver
sion
of
the
par
ticl
e fo
r }
{the
tim
e in
terv
al.
It u
ses
the
form
ula
on
pg.
13 D
ixon
. Th
e in
teg
rato
r}
{is
the
Qui
nn-C
urti
s ve
ctor
in
teg
rato
r.
}
var I n
terG
1 I n
terG
2 In
tegv
al,c
onve
rsio
n :V
eryL
ongV
ecto
r;
:Rea
l;
begi
n for
Siz
elnd
ex:=
1 to
M do
be
gin if K
appa
P1
k"C
Size
lnde
xJ<>
O
then
be
gin
-fo
r A
rbin
dex:
=O
to C
N+1
) do
.
Inte
rG1[
Arb
lnde
xJ:=
C1-
SR
T1"
CS
izel
ndex
,Arb
inde
xJ)*
Arb
lnde
x*D
elta
E
*Arb
lnde
x*D
elta
E;
lnte
grat
eVec
tor(
Inte
rG1,
Del
taE
,0t(
N+
1),l
nteg
Val
);
Con
vers
ion
:=3*
C1-
Lam
bda1
k
[Siz
elnd
exJ)
*lnt
egV
al
+Lam
bda1
k"
[Siz
elnd
exJ*
C1-
SR
T1"
[Siz
elnd
ex,N
+1J
);
Uni
tHea
pCon
v1:=
Uni
tHea
pCon
v1+
Con
vers
ion*
Size
Dat
a[Si
zeln
dex,
2]
/Sum
Size
Dat
a_2;
en
d;
if K
appa
P2
k"[S
izel
ndex
]<>
O
then
be
gin
-fo
r A
rbln
dex:
=O
to C
N+1
) do
ln
terG
2CA
rbln
dexJ
:=(1
-SR
T2"
[Siz
elnd
ex,A
rbln
dexJ
)*A
rbln
dex*
Del
taE
*A
rbln
dex*
Del
taE
; ln
tegr
ateV
ecto
r(ln
terG
2,D
elta
E,0
t(N
+1)
,lnt
egV
al);
C
onve
rsio
n :=
3*C
1-L
ambd
a2
k C
Siz
elnd
exJ)
*Int
egV
al
+Lam
bda2
_k"[
Size
lnde
xJ*C
1-SR
T2"
CSi
zeln
dex,
N+1
J);
Univers
ity of
Cap
e Tow
n
07/3
1/19
95
09:0
4 Fi
lena
me:
U
NIT
6C2.
PAS
Page
11
Uni
tHea
pcon
v2:=
Uni
tHea
pCon
v2+
Con
vers
ion*
Size
Oat
a[Si
zeln
dex,
2l
/Sum
Size
Dat
a_2;
en
d·
end·
'
end;
'
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re G
raph
1_1n
itia
lise
;
var G
raph
Typ
e B
eta1
S,K
appa
P1S,
Lam
bda1
S,G
DT
S B
eta2
S,K
appa
P2S,
Lam
bda2
S O
rdSP
1S,O
rdSP
2S
Cum
Tim
eS,U
nitH
eapN
oS,V
iew
Size
ClS
Inte
ger
; st
rin
g;
Str
ing
; S
trin
g;
Str
ing
;
begi
n init
SE
Gra
phic
s(V
alid
Dir
);
SetC
urre
ntW
indo
w(3
);
Bor
derC
urre
ntW
indo
w(1
);
Set
Axe
sTyp
e(0,
0);
Sca
leP
lotA
rea(
0.0,
0.0,
1.0
, 1.2
);
Set
XY
lnte
rcep
ts(0
.0,0
.0);
S
etC
olor
(2);
D
raw
XA
xis(
0.2,
1);
D
raw
YA
xis(
0.2,
1);
Lab
elX
Axi
s(1,
0);
Lab
elY
Axi
s(1,
0);
Tit
leX
Axi
s('D
imen
sion
less
Rad
ius'
);
Tit
leY
Axi
s('D
mls
s C
one'
);
Tit
leW
indo
w('U
nit6
C2'
);
Str
(Bet
a1:6
:3,B
eta1
S);
S
tr(K
appa
P1:
6:3,
Kap
paP
1S);
S
tr((
Del
taT
*Pri
nt_C
rit)
:6:4
,GD
TS
);
Str(
Lam
bda1
:5:3
,Lam
bda1
S);
Str
(Ord
SP
1:5:
2,0r
dSP
1S);
S
tr(B
eta2
:6:3
,Bet
a2S
);
Str
(Kap
paP
2:6:
3,K
appa
P2S
);
Str(
Lam
bda2
:5:3
,Lam
bda2
S>;
Str
(Ord
SP
2:5:
2,0r
dSP
2S);
St
r(C
llllT
ime:
8:4,
Cum
Tim
eS);
S
tr(U
nitH
eapl
ndex
:2,U
nitH
eapN
oS);
S
tr(V
iew
Siz
eCl:
2,V
iew
Siz
eClS
);
Gla
bel1
:=C
onca
t('
Bet
a1
',B
eta1
S,'
; K
appa
1 1,K
appa
P1S
,';
Lam
bda1
La
mbd
a1S
•· O
rder
1 '
Ord
SP1S
•·
GO
T '
GD
TS)·
GL
abel
2:=C
onca
tC'
Bet
a2 •:s~ta2S,'i
K~ppa2
•:K~
ppaP2s,•;
Lam
bda2
L
ambd
a2S,
'; Order~
',Ord
SP
2S);
G
Lab
el3:
=C
onca
t('
CllT
ITim
e ',C
umTi
meS
1'
Pro
file
s of
un
it h
eap
' U
nitH
eapN
oS['
and
s1ze
clas
s ',V
iew
size
ClS
);
Lab
elG
raph
Win
dow
(1,9
00,G
Lab
e 1
,0,0
);
Lab
elG
raph
Win
dow
C1,
850,
GL
abel
2 0
,0)·
L
abel
Gra
phW
indo
wC
160,
800,
GL
abe[
3,0,
0);
for
Arb
lnde
x:=1
to
2 d
o be
gin if A
rbln
dex=
1 th
en G
raph
Typ
e:=9
els
e G
raph
Typ
e:=1
0;
SetC
urre
ntW
indo
w(G
raph
Typ
e>;
Bor
derC
urre
ntW
indo
w(1
);
SetA
xesT
ypeC
0,0)
· S
cale
Plo
tAre
a(0
.0,0
.0,1
.0,1
.2);
S
etX
Yln
terc
epts
(0.0
,0.0
);
07/3
1/19
95
09:0
4 F
ilen
ame:
U
NIT
6C2.
PAS
Page
12
Set
Col
or(2
);
Dra
wX
Axi
s(0.
2,1)
; D
raw
YA
xis(
0.2 01>
; L
abel
XA
xis(
1,
>;
Lab
elY
Axi
s(1,
0>;
Tit
leX
Axi
s('D
imen
sion
less
Rad
ius'
);
Tit
leY
Axi
sC'D
mls
s C
one'
>;
Tit
leW
indo
w('U
nit6
C2'
>;
if A
rbln
dex=
1 th
en L
abel
Gra
phW
indo
w(2
00,9
00,
1S
mal
lest
Siz
e F
ract
ion
',0
,0)
else
Lab
elG
raph
Win
dow
(200
,900
,'Lar
gest
Siz
e F
ract
ion
',0
,0);
en
d;
end;
{---
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
----
--}
Pro
cedu
re G
raph
1_R
esul
ts;
var G
raph
Inde
x : I
nte
ger
;
begi
n if V
iew
Size
Cl>
M
then
be
gin Clo
sese
grap
hics
; w
rite
ln('
Gra
ph 1
Dra
w
ERR
OR
');
wri
teln
('Y
ou h
ave
inst
uct
ed t
he G
raph
ing
Rou
tine
to
gra
ph
the
con
ver
sio
n')
;
wri
teln
('cu
rves
of
a si
ze c
lass
whi
ch
does
not
ex
ist.
');
wri
teln
('R
esp
ecif
y =
Vie
wS
izeC
lass
in
Dec
lara
tio
ns
sect
ion
'>;
read
ln;
end·
fo
r1
Gra
phln
dex:
=1
to 3
do
begi
n if G
raph
lnde
x=1
then
be
gin Siz
elnd
ex:=
Vie
wS
izeC
l;
Set
Cur
rent
Win
dow
(3);
en
d el
se i
f G
raph
lnde
x=2
then
be
gin Siz
elnd
ex:=
M;
Set
Cur
rent
Win
dow
(9);
en
d el
se
begi
n Siz
elnd
ex:=
1;
Set
curr
entW
indo
w(1
0);
end·
fo
r1
Arb
lnde
x:=O
to
(N
+1)
do D
ataS
etX
"[A
rbln
dex]
:=
Arb
lnde
x*D
elta
E;
if K
appa
P1<>
0 th
en b
egin
fo
r A
rbln
dex:
=O
to (
N+1
) do
D
ataS
etY
0
[Arb
lnde
x]:=
AR
T0
[Siz
elnd
ex,A
rbln
dex]
; L
ineP
lotD
ata(
Dat
aSet
X0
,Dat
aSet
Y0
,(N
+2
),3
,0);
fo
r A
rbln
dex:
=O
to (
N+1
) do
D
ataS
etY
"[A
rbln
dex]
:=S
RT
1"[S
izel
ndex
iArb
lnde
xl;
Lin
ePlo
tDat
a(D
ataS
etX
0
,Dat
aSet
Y",
(N+
2),,
,0);
en
d·
if K
appa
P2<>
0 th
en b
egin
fo
r A
rbln
dex:
=O
to (
N+1
) do
D
ataS
etY
°CA
rbln
dexl
:=A
RT
0
[Siz
elnd
ex,A
rbln
dex]
; L
ineP
lotD
ata(
Dat
aSet
X0
,Dat
aSet
Y0
,(N
+2
),3
,0);
fo
r A
rbln
dex:
=O
to (
N+1
) do
Univers
ity of
Cap
e Tow
n
07/3
1/19
95
09:0
4 Fi
lena
me:
U
NIT
6C2.
PAS
Page
13
Dat
aSet
Y"[
Arb
lnde
x] :
=S
RT
2.[S
izel
ndex
,Arb
lnde
xl;
Lin
ePlo
tDat
a(D
ataS
etx·
,oat
aSet
Y.,
(N+
2),4
,2>
; en
d·
· en
d·
' ~d;
•
<===
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
==}
Pro
cedu
re E
xecu
te_U
nit6
C2;
begi
n cl r
scr;
Gra
ph1_
Init
ial i
se;
Rep
eats
:=O
; Fl
ag1
:=O
; Pl
ot_V
ar:=
O;
Gue
ss_S
RTs
P1;
Gra
ph1_
R.e
sul t
s;
whi
le
(Rep
eats
<lt
erat
ions
) an
d C
AR
T.C
1,N
+1l>
FCon
c T
ol)
do
begi
n ·
-w
hile
(Fl
ag1=
0) d
o be
gin Cal
e AR
TP1
and
SRTs
P1C
; Ch
ecl<
Con
verg
ence
; .
· R
eGue
ss_S
RTs
P1;
end;
Plot~Var:=Plot_Var+1;
if P
lot
Var
=P
rint
Cri
t th
en
begi
n -
-P
lot
Var
:=O
· G
rapl
i1_R
esul
ts;
end;
Upd
ate_
AR
T_an
d_SR
Ts;
Gue
ss_S
RTs
P1;
· Fl
ag1
:=0;
Repeats:=Repeat~+1;
end;
Gra
ph1
Res
ults
; ca
lc_c
onve
rsio
n;
end;
end.
<===
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
====
==}
l... "
r--
----..
,_ __ "