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Drying of Slurries in Spray Dryers
Thesis submitted in partial fulfillment of the requirements for the degree of
“DOCTOR OF PHILOSOPHY”
by
Maksim Mezhericher
Submitted to the Senate of Ben-Gurion University of the Negev
September 2008
Beer-Sheva
Drying of Slurries in Spray Dryers
Thesis submitted in partial fulfillment of the requirements for the degree of
“DOCTOR OF PHILOSOPHY”
by
Maksim Mezhericher
Submitted to the Senate of Ben-Gurion University of the Negev
Approved by the advisors ______________________________
Approved by the Dean of the Kreitman School of Advanced Graduate Studies _______
September 2008
Beer-Sheva
This work was carried out under the supervision of
Prof. Irene Borde
Prof. Avi Levy
In the Department of Mechanical Engineering
Faculty of Engineering Sciences
ACKNOWLEDGEMENTS
I wish to express sincere thanks to my advisors Prof. Irene Borde and
Prof. Avi Levy for their attitude, support, guidance, scientific and
administrative help without which the present study was impossible.
I am especially grateful to Prof. G.P. Verkhivker, Prof. A.V. Korolev
and Prof. M.V. Maksimov (Odessa National Polytechnic University)
for their help and valuable advises. I also thank to A. Khodorkovsky,
V. Preigerzon and I. Bronstein for their generous assistance.
To my wife Ira, my parents Efim and Galina and my sister Anastasiya
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TABLE OF CONTENTS
1. INTRODUCTION .................................................................................................................. 1
2. LITERATURE SURVEY ...................................................................................................... 5 2.1 Drying kinetics of single droplet ...................................................................................... 5 2.2 Summary of literature survey of single droplet drying kinetics ..................................... 11 2.3 Theoretical models of spray drying process ................................................................... 11 2.4 Experimental studies of spray drying process ................................................................ 16 2.5 Summary of literature survey of spray drying process ................................................... 19
3. OBJECTIVES AND TASKS OF THE PRESENT STUDY ................................................ 21
4. THEORETICAL DRYING MODEL OF SINGLE DROPLET CONTAINING INSOLUBLE OR DISSOLVED SOLIDS ............................................................................... 23
4.1 Model development ........................................................................................................ 23 4.2 Numerical solution ......................................................................................................... 31 4.3 Results of model validation and discussion .................................................................... 32 4.4 Conclusions .................................................................................................................... 36
5. HEAT AND MASS TRANSFER AND MODELLING OF WET PARTICLE BREAKAGE IN THE SECOND DRYING STAGE ...................................................................................... 37
5.1 Introduction .................................................................................................................... 37 5.2 Model development ........................................................................................................ 37 5.3 Modelling of wet particle cracking/ breakage ................................................................ 40 5.4 Numerical solution and validation of the model ............................................................ 43 5.5 Results of numerical simulations and discussion ........................................................... 44 5.6 Conclusions .................................................................................................................... 49
6. TWO-DIMENSIONAL MODELLING OF SPRAY DRYING PROCESS ........................ 51 6.1. Introduction ................................................................................................................... 51 6.2 Problem setup ................................................................................................................. 51 6.3 Two-dimensional model of spray drying process .......................................................... 52 6.4 Inlet, outlet and boundary conditions ............................................................................. 54 6.5 Numerical solution using Computational Fluid Dynamics ............................................ 55 6.6 Results of numerical simulations and model validation ................................................. 55 6.7 Conclusions .................................................................................................................... 63
7. INFLUENCE OF DROPLET-DROPLET INTERACTIONS ON TWO-DIMENSIONAL MODELLING OF SPRAY DRYING PROCESS ................................................................... 64
7.1 Theoretical modelling of droplet-droplet interactions .................................................... 64 7.2 Results of numerical simulations and discussion ........................................................... 66 7.3 Conclusions .................................................................................................................... 71
8. THREE-DIMENSIONAL MODELLING OF SPRAY DRYING PROCESS .................... 72 8.1 Introduction .................................................................................................................... 72 8.2 Problem setup, numerical grid, inlet, outlet and boundary conditions ........................... 72 8.3 Three-dimensional model of spray drying process......................................................... 73
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8.4 Numerical solution using Computational Fluid Dynamics ............................................ 73 8.5 Results of numerical calculations and discussion........................................................... 74 8.6 Conclusions .................................................................................................................... 84
9. INFLUENCE OF PARTICLE-PARTICLE INTERACTIONS ON MODELLING OF SPRAY DRYING PROCESS .................................................................................................. 87
9.1 Theoretical modelling of particle-particle interactions .................................................. 87 9.2 Results of numerical simulations and discussion ........................................................... 88 9.3 Conclusions .................................................................................................................... 93
10. COMPARISON BETWEEN THE PREDICTIONS OF AIR FLOW PATTERNS IN SPRAY DRYER USING k ε AND REYNOLDS STRESS TURBULENCE MODELS .... 94
10.1 Introduction .................................................................................................................. 94 10.2 Problem setup, inlet, outlet and boundary conditions .................................................. 94 10.3 Results of numerical simulations and discussion ......................................................... 95 10.4 Conclusions .................................................................................................................. 96
11. SUMMARY AND CONCLUSIONS ................................................................................. 97
CONTRIBUTIONS ................................................................................................................ 101
NOMENCLATURE ............................................................................................................... 102
REFERENCES ....................................................................................................................... 106
APPENDIX A. Numerical Solution for Period of Droplet Evaporation ................................ 112
APPENDIX B. Numerical Solution for Second Stage of Droplet Drying ............................. 117
APPENDIX C. Calculation of Thermophysical Properties for Skim and Whole Milk Droplets ................................................................................................................................................ 126
APPENDIX D. Derivation of Non-Linear Pressure and Vapour Fraction Partial Differential Equations ................................................................................................................................ 128
APPENDIX E. Two-Dimensional Axisymmetric Modelling of Drying Air Flow ................ 130
APPENDIX F. Drag, Added Mass and Buoyancy Forces ..................................................... 132
APPENDIX G. Particle Flow and Statistics Calculated by 2D Axisymmetric Spray Drying Model ...................................................................................................................................... 133
APPENDIX H. Three-Dimensional Modelling of Drying Air Flow ..................................... 135
APPENDIX I. Modelling of Particle-Particle Collisions Using Hard-Sphere Approach ...... 137
APPENDIX J. Equations of Reynolds Stress Turbulence Model .......................................... 142
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ABSTRACT
Transformation of a liquid feed containing a solid fraction into dried particles by
supplying the feed as a spray into a chamber with a hot drying agent is named spray drying.
The sprayed liquid feed can be a slurry, paste, suspension or solution. Granules, agglomerates
or powder represent the final dry product. The process of spray drying is utilized in many
industries, among them are food manufactures, pharmaceutical, chemical and biochemical
industries.
An advanced theoretical model for comprehensive study of spray drying process has
been developed. Since spray drying is a complex process, the present research was divided in
five main tasks.
Heat and mass transfer in spray drying process depends on the drying kinetics of
single droplets forming the spray. For this reason, a comprehensive study of single droplet
drying kinetics has been performed. An advanced theoretical model has been developed and
validated by published experimental and theoretical results. This theoretical model takes into
account a temperature profile in dried wet particle as well as calculates distributions of
pressure and vapour fraction in the particle pores. Moreover, the developed model is able to
predict the appearance of thermal and mechanical stresses, which can lead to the particle
cracking/breakage during drying and in this way affect the quality of the obtained spray
drying product.
In the next stage, drying of spray of droplets has been investigated and a two-
dimensional axisymmetric theoretical model of spray drying process has been developed. The
model, which is based on CFD technique, has been validated by published theoretical and
experimental data under steady-state drying conditions. The influence of droplet-droplet
interactions like coalescence or bouncing on the predicted behaviour of spray droplets and
drying air has been examined using transient mode of calculations.
Based on the two-dimensional study, a three-dimensional theoretical model of spray
drying process has been developed. This 3D model has been validated by published
theoretical and experimental data using steady-state drying conditions. The results of
numerical simulations have also been compared to the data predicted by previously developed
2D axisymmetric model in two different cases: a) steady-state drying conditions; b) transient
mode of calculations including droplet-droplet interactions in the spray.
xii
In order to study the effect of particle-particle interactions on calculated flow fields
and particle trajectories in spray drying process, a model of particle-particle collisions in the
dried spray has been proposed. The influence of both droplet-droplet and particle-particle
interactions on particle trajectories has been examined using 2D and 3D transient spray drying
models.
Finally, the effect of different turbulence models ( k ε and Reynolds stress model) on
predicted 2D and 3D flow patterns has been investigated.
The significance of the research is both fundamental and applied. This comprehensive
study has considerably improved the overall understanding of the spray drying process. On
the other hand, the developed theoretical model and results of the study can be helpful in
design of new spray dryers for different products and facilitate engineering optimal process
conditions that ensure operation of the existing spray drying units with regards to minimize
energy consumption and to achieve the required quality of final product.
Keywords:
Computational Fluid Dynamics, Cracking/Breakage, Drying Kinetics, Flow Pattern,
Heat and Mass Transfer, Mechanical Stress, Mathematical Modelling, Particle, Single
Droplet, Spray Drying, Thermal Stress, Two-Phase Flow, Wet Particle
1
Chapter 1
INTRODUCTION
Transformation of a liquid feed containing a solid fraction into dried particles by
supplying the feed as a spray into a chamber with a hot drying agent is named spray drying.
The sprayed liquid feed can be a slurry, paste, suspension or solution. Granules, agglomerates
or powder are representing the resulting dried product. The process of spray drying is utilized
in many industries and among them are food manufactures, pharmaceutical, chemical and
biochemical industries. Plastics, resins, ceramic material, washing powder, pesticides,
dyestuffs, fertilizers, organic and inorganic chemicals, skim and whole milk, baby foods,
instant coffee and tea, dried fruits and juices, enzymes, vitamins – this list of the spray drying
products is far from being complete.
A typical spray drying chamber consists of an atomizer, transforming a liquid feed
into a spray of droplets, and a contact zone of the chamber, where the spray interacts with a
hot drying agent. As a result of this interaction, moisture content of the drying agent increases
due to liquid evaporation from the droplets. During the evaporation, droplets of the spray are
shrinking and are turned into wet solid particles. The drying process proceeds until the dried
particles with the desired moisture content are obtained. Then the final product is recovered
from the drying chamber. Steam or air under atmospheric pressure is usually utilized as a
drying agent in the process (Strongin and Borde, 1987). The typical stages of the spray drying
process are shown in Fig. 1.
The spray of droplets is obtained from the liquid feed with the help of rotary or nozzle
atomizer. Rotary atomizers are using centrifugal energy: the feed is introduced from a central
inlet and leaves the rotating atomizer through the small periphery orifices, disintegrating into
a spray of droplets. Nozzle atomizers are classified as pressure, two-fluid (or multi-fluid)
pneumatic and sonic (supersonic) nozzles, according to the energy type being used. In
pressure nozzles, potential energy of feed under pressure is converted to kinetic energy and
the feed issues from the nozzle orifice as a high-speed unstable film that disintegrates into a
spray of droplets. In two-fluid nozzles, a high-velocity atomizing gas (usually air) is supplied
separately from liquid feed to the nozzle head and breaks the feed into a spray of droplets.
Sonic/ supersonic nozzles are similar to two-fluid nozzles but utilizing sonic/ supersonic
2
energy to break up the liquid feed. Atomization of a liquid feed by rotary vaned-wheel and
pressure nozzle is illustrated by Fig. 2.
Figure 1. Stages of the spray drying process (Masters, 1972)
(a) (b)
Figure 2. Typical methods of feed atomization (Masters, 1972).
(a) – rotary vaned-wheel atomizer, (b) – pressure nozzle
A proper interaction and mixing between the spray of droplets and drying agent is an
important factor in the spray drying process, which is influenced by the flow type and by the
shape of the drying chamber. The flow of the spray and drying agent can be co-current,
counter-current and mixed (see Fig. 3). The most commonly used shape is cylinder-on-cone
spray dryer with co-current flow of spray and drying agent (hereafter cylinder-on-cone spray
dryer is called simply spray dryer unless stated otherwise). Cylinder-on-cone spray dryers can
be of tall-form and short-form shapes. For tall-form spray dryers, the ratio between the
3
chamber height and diameter is greater than 5:1, while for short-form dryers this ratio is about
2:1 (Langrish, 1996; Langrish and Fletcher, 2001). An advantage of the short-form geometry
lies in significant operating flexibility as compared to the tall-form, since both rotary and
nozzle atomizers can be utilized in the short-form chamber. Unlike this, the usage of rotary
atomization in tall-form dryers is impractical due to substantial deposition of spray droplets
on the walls at top of the chamber. In addition, the spray dryers of a short-form are usually
featured by tangential inlet of drying agent with a non-zero swirl angle, which results in better
interaction and mixing between the spray and drying gas and prolongates the residence time
of droplets in the chamber. In contrast to short-form, the tall-form spray dryers are usually
fitted with a flat inlet of drying agent without swirling. Owing to its benefits, the short-form
shape of spray drying chamber is prevailing in industry nowadays.
(a) (b) (c)
Figure 3. Spray-gas interaction in spray drying chambers (Masters, 1972).
(a) – co-current flow, (b) – counter-current flow, (c) – mixed flow
One of the important parameters that characterises the current state of the dried spray
droplets/ wet particles is their moisture content. The moisture content represents a mass ratio
of liquid to solid content in the dried droplet/wet particle:
w
s
mX
m . (1)
During the drying process, the solid mass in the droplet remains unchanged while its liquid
fraction decreases. This process can be divided into stages according to different rates of the
4
droplet drying. In the first drying stage, the droplet outer surface reaches a saturation state
almost immediately after atomization and a constant-rate evaporation proceeds from the
droplet surface while the droplet diameter decreases. Then, when a critical value of the
moisture content is attained, the drying rate begins to fall. From this point, a solid crust is
being formed from the droplet surface towards its centre and the droplet turns into a wet
particle. This period is referred to as a second stage of drying. When the particle solid crust is
fully formed and a small amount of moisture remains only in particle capillaries, the third,
final, stage of drying is beginning. At this stage, additional resistance to the moisture removal
arises due to capillary and binding forces, which results in supplementary decrease of the
particle drying rate. A theoretical drying curve of three-stage droplet drying is shown in Fig.
4a (Masters, 1972).
drying time
dro
ple
tte
mp
era
ture
0
1 2
3
1-st drying stage 2-nd drying stage
Dp
Di
D0
(a) (b)
Figure 4. Theoretical curve of moisture removal from single droplet containing solids (a) and
two-stage drying process (b)
A negligible amount of moisture and a great resistance to moisture removal demand
extremely large energy supply to continue the particle drying after the completion of the
second drying stage. In addition, in practice the obtained products contain minimum 5-7% of
moisture due to technology requirements. For these reasons, the spray drying process
practically ends in the second drying stage and only a two-stage drying process is considered
in most of the published spray drying models (see Fig. 4b).
5
Chapter 2
LITERATURE SURVEY
2.1 Drying kinetics of single droplet
During the last decades a lot of attention was given to experimental studies and
development of a theoretical drying model of a single droplet, containing solids. The
published theoretical models can be related to one of the following general groups: models
based on semi-empirical approach that utilizes the concept of characteristic drying curve
(CDC) (Kuts et al., 1996; Langrish and Kockel, 2001); analytical drying models, which
describe the process using the continuity, motion, energy conservation and diffusion
equations (Sano and Keey, 1982; Abuaf and Staub, 1986; Nešić, 1990; Nešić and Vodnik,
1991; Borde and Zlotnitsky, 1991; Levi-Hevroni et al., 1995; Elperin and Krasovitov, 1995;
Kuts et al., 1996; Dolinsky, 2001) and drying models based on reaction engineering approach
(REA) (Chen and Lin, 2005; Lin and Chen, 2006; Lin and Chen, 2007). Each group of drying
models has its advantages as well as drawbacks. Thus, semi-empirical CDC models are
usually represented by a small set of simplified equations allowing fast computations, yet the
obtained results often give only a qualitative picture of the drying process and the precision of
the calculated data can be inadequate. The analytical models of single droplet drying describe
the process by a set of differential equations with corresponding initial and boundary
conditions. As a result, the numerical solution of such models is a complex problem and the
simulation of the drying process can demand significant computer recourses and time.
Nevertheless, the behaviour of main drying characteristics can be predicted showing a very
good agreement with experimental data, and this opens up possibilities for both fundamental
and applied studies of single droplet drying by utilizing such analytical models. Finally, the
REA models, which have appeared just in recent years, demonstrate a fine agreement with
experiments as well as fast calculations and reduced demands to the computer resources.
However, at the present time the applications of REA are still limited by narrow range of
materials whose drying behaviour can be modelled. The reason is that the REA approach
utilizes an experimental correlation connecting between the partial vapour density over
droplet surface and the average droplet moisture content. So far such correlations have been
experimentally found for few materials: for skim and whole milk (Chen and Lin, 2005),
lactose (Lin and Chen, 2006), cream and whey protein (Lin and Chen, 2007).
6
Because of space limitations, the detailed survey of published experimental works and
methods of measurement (glass filament, free falling, acoustic levitation etc.) of single droplet
drying is not included in the current work and the reader is referred to the publications of Lin
and Chen (2002) and Chen and Lin (2005) for more information.
The following sections represent the detailed review of published drying models of
single droplet containing insoluble or dissolved solids.
Drying models of single droplet with insoluble solids
A receding interface drying model for slurry droplet of sodium sulphate decahydrate
was proposed by Cheong et al. (1986). In the first drying stage, a uniform droplet temperature
was assumed during periods of droplet initial heating-up and equilibrium evaporation. In the
second drying stage, the density of particle wet core was taken as a constant value and its
temperature distribution was neglected. The temperature and vapour concentration profiles in
the particle crust were unjustifiably assumed to be linear. In addition, the crust heat capacity
was considered to be insignificant. As a result, the authors reported a strong discrepancy
between the predicted and measured droplet temperatures, especially in the second drying
stage. This discrepancy was explained by an arguable suggestion: the measured temperature
was that of the particle crust while the model predicted the actual temperature of the particle
wet core.
A theoretical drying model based on Stefan-type vapour diffusion in the second drying
stage was developed by Abuaf and Staub (1987). The period of droplet initial heating-up was
neglected. In the second drying stage, the vapour flux for a single straight crust capillary was
found from the Stefan-type steady-state diffusion equation and the vapour diffusion
coefficient was considered as independent of temperature. The drying rate of wet particle was
determined by multiplying the vapour flux by the cross-section area of capillary and the void
fraction of the crust. It must be noted that such an assumption is not fully consistent, since the
void fraction is a volumetric ratio whereas the flux must be multiplied by an areal ratio
between the pores and the crust. In addition, temperature profile in particle wet core and the
crust heat capacity were neglected by the authors. The results of the model validation were
not presented.
In the theoretical study of Borde and Zlotnitsky (1991) it was proposed to treat the
droplet with insoluble solids as a pseudo porous body. Thus, the droplet void fraction in both
drying stages was determined with the help of parameter named “transient porosity
coefficient”, which allowed the authors to treat the droplet drying without separate modelling
of the first and second drying stages. The droplet drying rate was calculated with the help of
7
an expression analogous to that developed by Abuaf and Staub (1987) but simplified using the
assumption that partial vapour pressure in the particle crust was small when compared to the
total ambient pressure. However, the above assumption is valid for relatively low
temperatures of the drying agent. The authors also neglected the temperature differences in
particle wet core. It is worth noting that unlike the study of Abuaf and Staub (1987), the
vapour diffusion coefficient was treated as a temperature-dependent and the crust heat
capacity was considered in the model. The authors did not cite the results of the model
validation.
The single droplet drying model published by Abuaf and Staub (1987) was extended
by Levi-Hevroni et al. (1995). A steady-state parallel flow of slurry droplets in hot
atmospheric air was considered. The air temperature was changing along the drying chamber.
The authors proposed to evaluate the crust thermal conductivity with the help of correlation
that took into account the geometry of the dry spherical porous crust. The other model
equations were analogous to those presented in Abuaf and Staub’s study. The results of the
model validation were not reported.
In the mathematical model developed by Elperin and Krasovitov (1995), the solid-to-
liquid volume ratio was utilized as a droplet drying characteristic instead of traditional droplet
moisture content (liquid-to-solid mass ratio). The period of droplet initial heating-up was not
considered. The first drying stage was described by the set of mass and energy conservation
steady-state partial differential equations (PDE) only for drying ambient, so the temperature
gradient inside the droplet was neglected. The PDE set was solved analytically assuming that
the properties of boundary layer around the droplet were independent of the droplet surface
temperature. In the second drying stage, the temperature profile in the particle wet core was
neglected, and the set of mass and energy conservation PDE was developed only for crust
region and drying ambient under steady-state conditions. This set of PDE was reduced to the
set of algebraic equations by assuming that coefficient of vapour diffusion and thermal
conductivities of the drying agent and porous crust were temperature-independent. The
vapour evaporation rate for wet particle was calculated with the help of the equation
developed by Abuaf and Staub (1987). The validation results of the model were not
published.
Drying models of single droplet with dissolved solids
The unsteady character of droplet drying was considered in the semi-empirical model
developed by Furuta et al. (1994). The moisture concentration profile was described by
partial differential equations of species conservation in both drying stages. However, the
8
assumption about uniform temperature of droplet/wet particle throughout the drying process
was unjustified. Experimental measurements were proposed as a method for determining the
vapour mass transfer rate during the wet particle drying. No validation results were reported.
The period of droplet initial heating-up was ignored and the droplet moisture content
was connected to experimentally obtained droplet evaporation rate in the semi-empirical
model of single droplet drying developed by Kuts et al. (1996). The droplet/particle
temperature was assumed to be uniform and Luikov’s experimental correlation between
dimensionless particle temperature and moisture evaporation rate was utilized. The heat
capacity of the particle crust was neglected. The predicted droplet/particle temperatures were
reported to be in deviation of 10-15% from experimentally obtained results for liquid
substrate of entobacterin.
A simplified drying kinetics model developed by Dolinsky (2001) can be utilized for
rapid rough estimation of the droplet drying duration. By neglecting the period of droplet
initial heating-up, the author proposed an analytical expression for uniform droplet
temperature calculation in the first drying stage. In the second stage, the crust temperature
profile was arguably assumed to be linear, the wet core temperature was supposed to be
uniform and the particle surface temperature was calculated from the simplified expression
based on the steady state assumption. In addition, the particle drying rate in the second stage
was calculated using assumption that particle mass decreased under constant rate. Time-
dependent droplet moisture content was treated as uniform in both drying stages. No
validation of the model was presented.
An analytical model of single droplet drying was presented by Kuts et al. (1996) in the
same article in which the authors proposed the semi-empirical model discussed above. The
crust heat capacity was taken into account in the model. It was suggested that average droplet/
particle moisture content can be either linear or exponential function of the drying time. In the
first drying stage, the droplet temperature was assumed to be uniform and the period of
droplet initial heating-up was disregarded. In the second stage, the temperature profiles in the
crust and wet core regions were described by appropriate set of PDE equations of energy and
mass conservation. However, an arguable boundary condition was utilized in the equations
set: the temperature of the crust-wet core interface was assumed to be constant and equal to
the wet bulb temperature of drying air. It should be noted that according to the experimental
studies performed by Cheong et al. (1986) and Ali et al. (1990), the temperature of the crust-
wet core interface is time-dependent and increases considerably over the wet bulb value
during the second drying stage. Based on the assumption of uniform temperature distribution
in the region of particle wet core, Kuts et al. (1996) have obtained several analytical
9
expressions: the temperature inside the particle crust as a function of time and radial
coordinate, and time-dependence of position of the crust-wet core interface during drying.
The authors developed two expressions for calculation of the drying process duration,
depending on the assumption of moisture content behaviour (linear or exponential). The
performed theoretical calculations of estimated drying process duration showed up to 15%
discrepancy from the performed experimental measurements in two case studies of
entobacterin substrate droplet drying.
Sano and Keey (1982) developed a theoretical model which calculated the profile of
moisture concentration inside the dried droplet or wet particle. The moisture distribution was
described using Fick’s transport PDE and corresponding boundary conditions. Since the
droplet temperature was assumed to be uniform in both stages of drying, the particle
temperature was considered to be only time-dependent. The effect of possible wet particle
inflation and rupture during drying was taken into account. Experimental investigations
included the studies of skim milk droplet behaviour during drying in atmospheric air under
different conditions. The agreement between the predicted and measured data of skim milk
drying kinetics was found to be satisfactory. However, the comparison between the model
predictions and experimental data provided by Trommelen and Crosby (1970) showed a
substantial discrepancy, and this inconsistency was not properly explained by the authors.
A theoretical drying model of skim milk and whole milk single droplets, based on
REA approach, was developed by Chen and Lin (2005). A uniform droplet temperature was
assumed. The mass transfer rate was determined as a function of the difference between the
vapour concentration values at the droplet/ wet particle liquid interface and in the bulk of
drying medium. The vapour concentration at the interface was connected to its saturation
value with the help of a fractionality coefficient. In turn, this fractionality coefficient was
expressed by the Arrhenius equation using apparent activation energy (AAE) and droplet
temperature. The values of AAE for skim and whole milk single droplets were found
experimentally as functions of two parameters: a) difference between the average moisture
contents of droplet and bulk air, and b) AAE value of the drying air. In addition, the
droplet/particle uniform temperature was described by an ordinary differential equation of
energy balance. The validation of the model showed a very good agreement of theoretical
predictions with experimental data for all applied conditions. However, applications of the
REA approach for modelling droplet drying kinetics are still limited to a narrow range of
materials due to the need for experimental correlation between the AAE and moisture content
for the specific dried product.
10
Drying models of single droplets with insoluble and dissolved solids
Nešić (1989) and Nešić and Vodnik (1991) developed a theoretical drying model of
single droplet containing insoluble or dissolved solids. The droplet temperature distribution in
both drying stages was neglected. The moisture concentration profile was considered by PDE
of Fick’s diffusion and the effect of Stefan flow in the crust pores was disregarded. The
moisture diffusion coefficient was considered to be a function of local moisture concentration
whereas the temperature influence was ignored. The equation of energy conservation was
developed for quasi steady state conditions. A good agreement between the predicted
temperature and mass time-change with experimentally obtained data was observed for
colloidal silica and sodium sulphate droplets. In the case of skim milk droplet drying, the
mass time-change showed a fine coincidence with the measurements, but the calculated and
experimental droplet temperatures demonstrated a strong discrepancy in the second drying
stage.
A theoretical drying model of single droplet with insoluble/dissolved solids developed
by Farid (2003) was based on the average droplet moisture content. The model took into
account the temperature profile inside the droplet during the period of droplet initial heating-
up. In the period of equilibrium evaporation, the droplet temperature distribution was assumed
to be uniform and equal to the wet bulb temperature of the drying air. In the second drying
stage, a common PDE of energy conservation for both crust and wet core regions of the wet
particle was proposed. The boundary conditions for particle centre and its surface were given
correctly. However, Farid made an unjustified assumption that the temperature over the crust-
wet core interface was constant and equal to the wet bulb temperature during drying (similarly
to Kuts et al., 1996). Furthermore, the crust porosity was not taken into account in the heat
balance boundary condition at the crust-wet core interface. In addition to these improper
boundary conditions, the model lacked an essential equation describing the dependence of
particle drying rate on the crust resistance to the vapour mass transfer. Without such a vital
equation, Farid’s mathematical model of the second drying stage led to unrealistic physical
picture, as though the crust was completely permeable to the diffusion of vapour generated at
crust-wet core interface. It must be remarked in addition, that the thermophysical properties of
droplet and wet particle were assumed to be independent of temperature.
11
2.2 Summary of literature survey of single droplet drying kinetics
The detailed review of published drying models of single droplet containing insoluble
or dissolved solids exposed their different shortcomings, such as: assumption of steady-state
conditions, ignoring the period of droplet initial heating-up, unjustified neglecting of
droplet/wet particle temperature profile, disregarding crust porosity, inaccurate calculation of
mass transfer rate, neglecting crust heat absorption and simplification of droplet
thermophysical properties. Some of the published models lacked validation information and a
number of other studies represented the results of validation showing inappropriate
discrepancy with experimental data. In addition, most of the published models can be used to
describe the drying of single droplet containing either insoluble or solely dissolved solids.
In the light of the exposed shortcomings of published models, the further development
of single droplet drying kinetics is still called for.
2.3 Theoretical models of spray drying process
At the beginning of the 1970s, a number of simplified semi-empirical models of the
spray drying process had been developed (e.g., Place et al., 1959; Paris et al., 1971).
However, such semi-empirical models had many limitations, since each had been developed
for the drying process of a specific product in a specific spray dryer. Therefore, these types of
models could not be generalized to study the effects of different chamber geometries or
different operating parameters. Subsequent progress in theoretical and numerical modelling
together with advancements in the computer industry stimulated the development of
computational methods and commercial Computational Fluid Dynamics (CFD) packages.
Crowe (1980) was the first who used a numerical technique for solving a simple spray dryer
model. He applied an Eulerian approach for the gas phase to solve the time-averaged
conservation equations of continuity, momentum, energy and turbulence. The behaviour of
the discrete phase of droplets was modelled by the Particle-Source-in-Cell (PSI-Cell) method
(Crowe et al., 1977). Crowe’s numerical techniques were further developed by O’Rourke and
Wadt (1982), Goldberg (1987) and Reay (1990).
At the end of the 1980s, Oakley et al. (1990) performed numerical simulations of the
air flow in a co-current cylinder-on-cone spray dryer, 0.76 m in diameter and 1.44 m in
height. They used the CFD package FLOW3DTM, Release 1, non-body fitted version
(rectangular numerical grid utilization). The model predictions were compared to
experimental measurements of the air flow using laser Doppler anemometry. Although that
study did not include the influence of the spray on the air flow field, both measurements and
12
simulations showed the presence of periodic oscillations in the size of the recirculation zones
in the drying chamber. The study also found that predictions of the air flow patterns were
sensitive to the values of turbulence parameters (that were set for the annular air inlet) of the
standard k ε turbulence model. It was concluded that these turbulence parameters could not
normally be obtained from experimental measurements and that they should therefore be
treated by numerical fitting.
Langrish et al. (1992) conducted an experimental and theoretical study of the air flow
in a 1.5-m diameter, 1.9-m-high cylinder-on-cone spray dryer fitted with a vaned-wheel
atomizer. The steady-state flow field characteristics of the gas phase were investigated by
solving numerically Navier-Stokes equations. Air flow was assumed to be axially symmetric,
two-dimensional, incompressible and adiabatic. The influence of the dispersed phase was
neglected, since no liquid was sprayed through the atomizer, and the k ε turbulence model
was applied with the standard constants for general free turbulent flows. Comparison between
the results of the numerical simulations and the experimental data showed that the main
features of the flow were indeed predicted, although many differences were observed between
the experimental and theoretical results. The authors reported that they had difficulty in
obtaining converged numerical solutions and that their finest grid was comparatively coarse.
The influence of different turbulence models on flow patterns was investigated by
Oakley and Bahu (1993). In their study, the geometry of the spray dryer (cylinder-on-cone)
and the vaned disk atomizer were the same as those in the study of Oakley et al. (1990), but a
body-fitted FLOW3DTM package was utilized for numerical simulations. Introduction of a
discrete droplet model for simulating the spray drying process facilitated the prediction of the
gas flow pattern, the spray/gas mixing, and the temperature and moisture histories of the
particles. In that study, more accurate consideration was given to the region of the air inlet to
the drying chamber (where a high swirling was observed) than in the earlier study of Oakley
et al. (1990). The simulations were carried out using assumptions of two-dimensional,
axisymmetric, steady-state flow. One of the most important outcomes of that study was the
comparison between the patterns of air flow obtained experimentally (Oakley et al., 1990) and
those predicted by numerical simulations utilizing k ε and Reynolds stress turbulence
models. It was concluded that the Reynolds stress model was more appropriate than the k ε
model for swirling flows, since it did not assume isotropy of eddy viscosity. However,
utilization of the Reynolds stress model was found to be more resource-consumptive than the
k ε model, and poor convergence and grid-dependence of the numerical solution was
observed.
13
The behaviour of wall deposition rate (mass of wet particles that is deposited on the
wall per unit time) was studied both experimentally and theoretically by Langrish and
Zbicinski (1994) for various air inlet geometries and spray cone angles of a cylinder-on-cone
spray dryer, 0.935 m in diameter and 1.69 m high, fitted with a two-fluid nozzle. Air inlet
velocity and spray atomization parameters were obtained experimentally for spraying of 20%
(mass fraction) sodium chloride solution. In the theoretical model, assumptions of axially
symmetric, incompressible steady-state flow were made. The discrete phase model was based
on the PSI-Cell concept, and the isotropic eddy k ε turbulence model with different sets of
the turbulence constants was utilized. Thus, at the beginning of the numerical simulations, a
set of standard constants was applied, and then, after comparison with the experimental
measurements, fitted constants were used for the subsequent calculations. All particles were
assumed to stick to the walls on collision. The rates of deposition, predicted by the
calculations, showed a strong dependence on the constants used in the k ε turbulence
model, and the discrepancy between calculated and measured data was in the range of 11-57%
(better agreement was achieved after repeated simulations with fitted turbulence constants).
In his theoretical study Zbicinski (1995) developed an original mathematical model,
which can be applied for spray dryers with nozzles. The flow was assumed to be co-current,
and the form of the spray itself was axisymmetric. Equations of motion for droplets/particles
were in steady-state three-dimensional form. Drying kinetics of the droplets was
approximated by a two-stage drying curve analogous to that used previously by Langrish and
Zbicinski (1994). The model of Zbicinski (1995) did not consider the continuous phase; only
discrete phase equations were developed. The data necessary for modelling the behaviour of
the continuous phase were obtained by means of separate calculations of the air flow field
with the use of the CFXTM package. The k ε turbulence model with a set of constants
recommended for recirculating flows was applied. As a result of this approach, Zbicinski’s
model lacked coupling between the discrete and continuous phases. Such uncoupled solution
is one of the main model disadvantages, since the atomized spray can substantially influence
the air flow pattern. Thus, according to Southwell and Langrish (2001), spraying can lead to
changes in the form of the air flow field inside the dryer, and this affects the trajectories of
droplets/particles. Zbicinski (1995) validated the developed model by comparing the
calculation results with experimental data obtained by Langrish and Zbicinski (1994) for a
spray dryer fitted with a two-fluid nozzle. As a result, the simulations predicted only two air
recirculation regions in the spray dryer, whereas the experimental study showed the existence
of three recirculation zones. Such a discrepancy was obtained by Zbicinski (1995) probably
due to the lack of spray-gas coupling in his model.
14
The objectives of the studies of Kieviet et al. (1997) and Kieviet and Kerkhof (1996,
1997) were to develop a theoretical drying model that would predict the quality of the final
product. The study included both measurements and modelling of air pattern, temperature and
humidity flow fields. The flow fields were simulated using the CFD package FLOW3DTM,
body-fitted version 3.3. The spray dryer was a cylinder-on-cone 2.215-m diameter, 3.73-m
high, dryer fitted with a centrifugal pressure nozzle. Two cases were considered: one with and
the other without liquid spray injection. For simulations without spray, swirl was not
considered, and the air flow pattern was compared with that obtained with the measured data.
The swirl angle was set at 5º for the case of droplet spraying, and the experimentally
determined temperature and humidity profiles were compared with the profiles predicted by
simulations. The flow field was considered to be in steady state, and time-averaged Navier-
Stokes equations were solved applying the k ε turbulence model. The droplet drying
kinetics was assumed to be the same as this for drying pure water droplets. Mass transfer was
calculated by using standard mass transfer correlations, and the Antoine equation was used for
estimation of the vapour pressure as a function of temperature. The model took into account
the buoyancy of the air due temperature and humidity differences and heat loss through the
chamber walls (however, no information was given about the value of heat transfer
coefficient). The model was validated by comparing the predicted results with those obtained
by measurements, and a good qualitative agreement was obtained.
A number of studies on spray drying modelling were published by researchers of
NIZO Food Research (Straatsma et al.,1999a, 1999b; Verdurmen, 2002; Verdurmen et al.,
2002). In these studies, two computer packages were utilized: DrySPEC2 and DrySim. The
DrySPEC2 model was developed in order to determine the conditions of drying process that
ensure an optimal exploitation of the existing drying installations with regard to energy
consumption and properties of the obtained product (Verdurmen et al., 2002). A near-
equilibrium state of water vapour pressure between the powder and outlet water was assumed;
this eliminated the need for a detailed description of heat and mass transfer during drying.
One of the substantial drawbacks of the DrySPEC2 was that this model could not predict the
profiles of velocity, temperature and moisture content of the drying agent, as well as the
trajectories, temperature and moisture content of the dried droplets and wet particles. For
these purposes, a two-dimensional model of the spray drying process was developed and
realized in the DrySim package (Straatsma et al., 1999a). In this model, a steady-state gas
flow was described by the time-averaged Navier-Stokes equations in combination with the
standard k ε turbulence model. The spray influence on the gas flow pattern (and vice versa)
was taken into account. The drying kinetics of droplets with dissolved solids was modelled by
15
assuming a moisture distribution and a uniform temperature inside a spherical droplet/wet
particle. The time-changed moisture content of the dried product was described by PDE of
Fick’s diffusion and the coefficient of moisture diffusion was estimated as a function of both
droplet temperature and moisture content. Straatsma et al. (1999b) enabled the calculation of
drying kinetics of droplets with insoluble solids by semi-empirical sub-model added to the
DrySim. The heat and mass transfer between the drying gas and the spray of droplets was
estimate with the help of the classical Ranz-Marshall correlations. All the numerical
simulations were performed using a body-fitted grid and an orthogonal curvilinear coordinate
system. However, it must be noted that the published information about the spray drying
simulation using the DrySim was not complete. Thus, information about the function used to
model the distribution of droplet diameters in the injected spray was not presented. The
applied boundary conditions were not given as well. Furthermore, the validation results of the
DrySim model were not reported. Finally, the presented predictions of flow fields of velocity,
temperature and humidity of the drying air and trajectories of the dried particles for several
industrial spray dryers can not be compared with other reports due to lack of the input data.
Huang et al. (2003a) used a CFD technique for investigating the flow parameters
inside a spray dryer. The spray drying model, incorporated in FLUENT 6.0, was adopted for
simulation of the drying process in a cylinder-on-cone drying chamber fitted with a pressure
nozzle (this spray dryer is similar to that considered in the studies of Kieviet et al., 1997 and
Kieviet and Kerkhof 1996, 1997). The air flow was co-current, and a steady-state, two-
dimensional and axisymmetric flow pattern was assumed. A pressure nozzle located at the top
of the chamber was used to atomize a liquid into a spray. A discrete phase model (DPM) was
used for tracking the particles’ trajectories inside the drying chamber (the DPM incorporated
in FLUENT is similar to the PSI-Cell method). Two-way coupling of heat, mass, and
momentum transfer between the continuous and discrete phases was assumed. All particles
were considered to “escape” from the calculations when the hitting walls of the drying
chamber. The other boundary conditions were taken to be the same as those given in the
works of Kieviet and Kerkhof (1996, 1997) so as to facilitate comparisons of the experimental
results with FLUENT simulations. The spray of droplets was assumed to have a solids content
of 42.5%, and the spray angle was set at 76º. Droplet distribution was modelled using the
Rosin-Rammler distribution, and 20 streams of droplet diameters were tracked. Droplet
drying kinetics was simplified by assuming the physical properties of the feed to be similar to
those of the water, with the exception that the volatile content changed as the drying
proceeded. The effects of turbulence were considered by the standard k ε turbulence model.
The results of the simulations for velocity, temperature and humidity profiles at different
16
chamber levels showed a good agreement with published experimental data (Kieviet et al.,
1997; Kieviet and Kerkhof, 1996; Kieviet and Kerkhof, 1997).
Huang et al. (2003b) and Huang and Mujumdar (2005) utilized the model of Huang et
al. (2003a) to show the power and flexibility of CFD techniques for investigating the flow
characteristics, optimal efficiency and effectiveness of different types of spray dryer chambers
- cylinder-on-cone, conical, hour-glass, and lantern-type shapes, and horizontal spray dryers.
Recently, the above authors enhanced their model to simulate 3D flow fields of both
continuous and discrete phases in spray dryers fitted with nozzles or rotary atomizers, see
Huang et al. (2004, 2005, 2006). However, a drawback of these studies was that the
assumptions of axisymmetric and steady-state flow fields were applied.
A three-dimensional transient model of gas flow field in a spray drying chamber was
developed by Lebarbier et al. (2001). The numerical simulations were performed with the
help of CFD package CFX 5. The validation of the developed model was performed by
comparison of the predicted air flow field with experimentally obtained data. In their
experimental studies the authors performed flow visualization in a spray chamber by using
water instead of hot gas. Since the properties of water and gas (e.g., air) are quite different, it
is obvious that the validation results of this study are suspicious.
Guo et al. (2003) studied numerically the gas flow instability in the spray dryer
chamber of 0.942 m in diameter and 2.8 m in height, with flat air inlet and without liquid
spraying. The gas flow field was simulated using a CFD code CFX 4.4. The transient
Reynolds averaged Navier-Stokes equations were solved and standard k ε model of
turbulence was applied. The calculations predicted the presence of a self-sustained flapping
oscillation in the central core of gas flow in the drying chamber, which made the flow
strongly three-dimensional and time-dependent. It was concluded that the flow instability was
not the result of any interaction between the gas flow and the conical camber walls, but it was
caused by sudden expansion of the gas upstream. The authors suggested that the gas flapping
motion was driven by the pressure difference between the recirculation zones across the
central core of the gas flow.
2.4 Experimental studies of spray drying process
During the literature survey it was found that several papers studied the spray drying
process both theoretically and experimentally. In such cases the experimentally obtained data
were utilized for verification and validation of the developed theoretical model (Kieviet and
17
Kerkhof, 1996, 1997; Kieviet et al., 1997; Langrish et al., 1992; Langrish and Zbicinski,
1994; Lebarbier et al., 2001; Oakley et al., 1990; Zbicinski, 1995 and others). The review of
these studies is given in the above section.
An experimental study on the unsteady air flow patterns was performed by Langrish et
al. (1993). In this work experimental measurements were carried out in a 1.5 m in diameter
and 1.9 m in height cylinder-on-cone spray dryer fitted with rotary wheel atomizer. Flow
parameters were measured in the absence of liquid spray by a hot-wire anemometer when
rotation of the wheel atomizer was co-current and counter-current with respect to inlet swirl
of the drying air. The inlet swirl angles were set to 0º and 30°, since the study of Oakley et al.
(1988) suggested that central recirculation zones, associated with significant low-frequency
velocity oscillations in the flow field, appear when the swirl angle is greater than 25º. In the
presence of the inlet swirl, the performed measurements proved the occurrence of low-
frequency oscillations due to vortex breakdown. The strongest oscillations were found to be in
range of 1-3 Hz. For the case when air swirl angle was set to 30°, the presence of central
recirculation zones was observed, see Fig. 5. The low-frequency oscillations were also
detected for the case with flat air inlet, though their areas were more offset towards the
cylindrical chamber walls of the dryer. These observations agreed with experimental results of
Kieviet and Kerkhof (1997), which showed a time-dependent nature of the air flow inside the
spray dryer for both cases with and without swirl at air inlet. The presence of central
recirculation zones was not noticed for the case of drying air inlet without swirling.
In the work of Southwell and Langrish (2000) the experimental studies were
performed on a pilot-scale of 1.39 m in height and 0.8 m in diameter cylinder-on-cone spray
dryer fitted with a wheel atomizer. The inlet of drying air was realized without swirling. The
characteristics of drying air were measured for two cases with water spray and without
spraying. The performed experimental measurements showed an unsteady behaviour of three-
dimensional air flow field with unstable central vortex in both studied cases. A sketch of the
flow field observed by Southwell and Langrish is shown in Fig. 6.
The effect of air swirl on flow stability was studied experimentally by Southwell and
Langrish (2001). In this paper, 0.8 m diameter and 1.61 m tall co-current cylinder-on-cone
spray dryer fitted with a two-fluid atomizer was utilized. The experimental measurements
were performed using swirl angles in the range of 0° and 45º (5° of increment) with the help
of laser Doppler velocimetry (LDV) techniques. Two case studies of water spray and without
spraying were examined. The observations demonstrated that the spraying changed
significantly the behaviour and stability of flow field for each swirl angle. The authors found
18
that no one of the tested swirl angles resulted in clearly steady behaviour of the flow field in
the dryer.
For swirl angle of 45º the precessing vortex core (PVC) in the flow field was
observed, see Fig. 7. Based on the experimental data showed the time-dependency and non-
asymmetric behaviour of flow patterns in the spray dryer, both with liquid spray and without
spraying, Southwell and Langrish (2001) concluded a limited usefulness of axisymmetric and
steady-state numerical models of the spray drying process. Furthermore, the authors found
that the introduction of the spray had a significant effect on the flow behaviour and, therefore,
air-only studies could not be used to adequately represent the actual flow conditions of
operating spray dryers.
(a) (b)
Figure 5. Experimental frequencies of air flow oscillation (the given values are in Hz). (a) -
without inlet air swirl and (b) - air inlet swirl of 30°. Atomizer wheel co-rotated
with air inlet swirl. (Langrish et al., 1993)
19
Figure 6. Sketch of the flow field in a pilot-plant spray dryer (no air inlet swirling), observed
by Southwell and Langrish (2000)
Figure 7. Sketch of precessing vortex core in a horizontal cut of drying chamber,
(Southwell and Langrish, 2001)
2.5 Summary of literature survey of spray drying process
In the literature survey both theoretical and experimental studies of the spray drying
process were reviewed. The experimental studies of Kieviet and Kerkhof (1996, 1997),
Kieviet et al. (1997), Langrish et al. (1992, 1993) and Southwell and Langrish (2000, 2001)
showed that the main features of the flow fields are three-dimensionality, transience, presence
of periodic oscillations and precessing vortex core. At the same time, most of the existing
spray drying models assumed axisymmetric two-dimensional flow fields (except Guo et al.,
2003) and steady-state process conditions (except Guo et al., 2003 and Oakley et al., 1988).
The fully three-dimensional transient simulations that were performed by Guo et al. (2003)
20
predicted the flow fields of the drying agent in the absence of the liquid spraying. None of the
existing spray drying models took into account the influence of the particle-particle
interactions on the flow fields, which is one of the important parameters affecting trajectories
of the dried particles and their deposition on the walls of drying chamber. The studies of
Huang et al. (2003a, 2003b) uncovered the importance of particle-wall interactions and
Langrish and Zbicinski (1994) assumed that all particles stuck to chamber walls on collision.
The rest of the reviewed spray drying models disregarded the particle-wall interactions. The
drying kinetics was approximated by simplified drying models in the studies of Langrish and
Zbicinski (1994) and Zbicinski (1995). The research groups of NIZO Food Research
(Straatsma et al., 1999a; Verdurmen, 2002; Verdurmen et al., 2002) considered only moisture
diffusion in their model of drying kinetics. Furthermore, the other reviewed studies either
incorporated the drying kinetics models based on evaporation of pure water droplets or
considered only the flow of the drying agent without spray injection. In addition, the influence
on the drying process of heat transfer from the spray chamber walls to the ambient was not
clearly understood.
The review of the published spray drying models shows that none of the models
considers all or even substantial parts of the issues addressed above. Therefore, there is still a
need to develop an advanced spray drying model that will be suitable for comprehensive
analysis of the spray drying process and that will be a tool for the design of new spray dryers
and for improvement and intensification of conventional spray drying systems.
21
Chapter 3
OBJECTIVES AND TASKS OF THE PRESENT STUDY
The main objective of the present work is to develop an advanced theoretical model
for comprehensive study of the spray drying process.
The spray drying is a complex process and thus the study is divided in five major tasks
described below.
The process of heat and mass transfer in the spray drying depends on drying kinetics
of single droplets forming the spray. For this reason, the first task of the research is to develop
an advanced theoretical model of single droplet drying. Stages of the task performance are:
comprehensive study of single droplet drying kinetics and development of the
advanced theoretical model of single droplet drying;
validation of the developed advanced model by published theoretical and
experimental data;
investigation of the process of particle cracking/breakage, which affects the quality
of the obtained spray drying product.
The development of two-dimensional theoretical model of the spray drying process is
the second task of the present study. Stages of the task performance are:
development of two-dimensional axisymmetric model of the spray drying process
based on CFD technique;
numerical simulations and validation of the developed 2D spray drying model by
comparing the predicted operating parameters of a pilot-plant spray dryer with
published theoretical and experimental data;
study of influence of droplet-droplet interactions like coalescence or bouncing (that
are typical in the real spray drying processes) on behaviour of the spray droplets
during drying.
The third task of the research is to develop a three-dimensional theoretical model of
the spray drying process. Stages of the task performance are:
development of three-dimensional model of the spray drying process based on CFD
technique;
22
numerical simulations and validation of the developed 3D spray drying model by
comparing the predicted operating parameters of a pilot-plant spray dryer with
published theoretical and experimental data;
comparison between the results predicted by 3D model with the data obtained using
previously developed 2D axisymmetric model when droplet-droplet interactions in
the spray are included.
The fourth task is modelling of particle-particle collisions in the dried spray and study
of the influence of both droplet-droplet and particle-particle interactions on particle
trajectories using 2D and 3D spray drying models.
Finally, the fifth task of the present work is to investigate the effect of different
turbulence models ( k ε and Reynolds stress model) on the predicted 2D and 3D flow
patterns.
23
Chapter 4
THEORETICAL DRYING MODEL OF SINGLE DROPLET
CONTAINING INSOLUBLE OR DISSOLVED SOLIDS
4.1 Model development
The current chapter describes the development of an advanced theoretical drying
model of a single droplet containing solids.
Unlike the previous published models, the present model of single droplet drying
kinetics takes into account the following process features: fully time-dependent character of
heat transfer in the drying process; droplet initial heating-up in the first drying stage;
temperature profile within the droplet/wet particle; heat absorption by the crust region; crust
resistance to diffusion mass transfer (crust porosity) and temperature dependence of
droplet/particle physical properties.
The developed model is suitable for both types of droplets, containing either insoluble
or dissolved solids.
A two-stage drying process of a motionless spherical single droplet, surrounded by a
flow of atmospheric air, is modelled. The air parameters like temperature, pressure, humidity
and velocity remain constant values during drying. The first drying stage consists of droplet
initial heating and subsequent liquid evaporation from the droplet surface. The second drying
stage begins at the point when droplet moisture content falls down to the critical value and dry
porous solid crust is formed around the particle wet core. As a result, in the second drying
stage the evaporation of liquid takes place inside the wet particle at the receding interface
between the crust and wet core. The vapour, generated over this interface, diffuses through the
crust pores towards the particle outside surface, where it forms a thin boundary layer. From
the particle surface the vapour is taken away by a bulk flow of the drying agent. The drying
process of wet particle is considered to be complete when the desired amount of moisture
content remains in the particle.
24
Droplet initial heating in the first drying stage
The droplet initial heating-up is a quick process that usually continues less then tenths
of a second in real industrial conditions. It means that during this period the drying character
is unsteady and the value of Fourier number, 2d dFo =α t / R , can be expected to be less than
0.1. Therefore, temperature distribution within the droplet must be taken into account.
Consequently, for spherical droplet with isotropic properties the equation of energy
conservation is given by:
2d dd p,d d2
T T1ρ c k r
t r r r
, (2)
and the corresponding boundary conditions are:
d
dg d d d
T0, r 0;
rT
h T T k , r R .r
(3)
During the period of initial heating-up the droplet outer radius, dR , is considered to be
unchangeable due to negligible evaporation. The value of heat transfer coefficient, h , is
determined using a well-known Ranz-Marshall correlation (Ranz and Marshall, 1952):
1 2 1 3d dNu 2 0.6Re Pr . (4)
The droplet specific heat can be evaluated as (Kirillin and Sheindlin, 1956):
tp,d p,w p,s
p
qc c 1 c c c
T
. (5)
For ideal solutions and mixtures, considered here, the heat of mixing tq 0 . The mass
concentration of solids, c , is connected to the droplet moisture content, X , by:
c 1/ 1 X . (6)
The droplet density can be determined as follows:
d d,s d,w d,w d,sρ 1 X ρ ρ / ρ Xρ . (7)
To find the droplet thermal conductivity, either series (8) or parallel (9) conceptions are
applied (Chen and Peng , 2005):
d w sk δk 1 δ k , (8)
d w s1/ k δ / k 1 δ / k , (9)
25
where the droplet void fraction, δ , is given by:
3d,w d d,s d,s dδ V / V 1 6m / πρ d . (10)
Since the droplet density, specific heat and thermal conductivity are functions of
temperature, the solution of equations set (2)-(10) is obtained with the help of numerical
methods.
Droplet evaporation in the first drying stage
During the droplet evaporation in the first drying stage (see Fig. 8), Fourier number
can attain the value greater than 0.1 even at the beginning of the process. In this case when
the value of Biot number is less than 0.1, then the droplet lumped heat capacity can be
assumed and the droplet temperature depends on time only. Otherwise, the droplet
temperature must be treated as a function of both temporal and radial coordinates during the
droplet evaporation. These two approaches for droplet temperature calculation are discussed
below.
Q
vm
drying time
dro
ple
t te
mp
era
ture
0
1 2
1-st drying stageD0
Dp
Figure 8. Scheme of droplet evaporation in the first drying stage.
0-1 initial heating-up, 1-2 equilibrium evaporation
The uniform temperature-approach is based on assumption of constant equilibrium
evaporation temperature of the droplet that is established in droplet evaporation period (see
Fig. 8). This results in the following equation of energy conservation:
fg v g d dh m h T T A . (11)
The rate of mass transfer from the droplet surface is determined as follows:
v D v,s v, dm h ρ ρ A . (12)
Combining (10) and (11), we get:
26
D v,s v, g d fgh ρ ρ h T T / h . (13)
The coefficients of heat and mass transfer are determined by the corresponding
Nusselt and Sherwood numbers, which are evaluated using modified Ranz-Marshall
correlations for evaporating spherical droplets (Levi-Hevroni et al., 1995):
0.71 2 1 3dd d
d
d hNu 2 0.6Re Pr 1 B
k , (14)
0.71 2 1 3d Dd d
v
d hSh 2 0.6Re Sc 1 B
D . (15)
The factor 0.71 B
takes into consideration Stefan flow in the droplet boundary layer and
p,v g d fgB c T T / h is called Spalding number.
Combining eqs. (13), (14), (15) and treating both vapour and drying air as ideal gases,
the following equation is obtained:
1 2 1 3
g d v,s v,d v w1 2 1 3
fg d g d g
T T p p2 0.6Re Sc D M
h 2 0.6Re Pr k T T
. (16)
This expression allows evaluating the droplet equilibrium evaporation temperature.
In order to calculate the droplet shrinkage rate, it is assumed that elementary droplet
mass decrease is proportional to elementary droplet diameter shrinkage, i.e. (Levi-Hevroni et
al., 1995):
d
v2d,w d
d 2m
t ρ πd
d
d . (17)
Combining (12), (17) together with the Reynolds number definition, d g d gRe u d / ν ,
and expressing the mass transfer coefficient, Dh , from eq. (15), the following equation is
obtained:
0.5
d 2 d1
d
d 2 C dC
t d
d
d
, (18)
where
0.7
1 v v,s v, d,wC 2D ρ ρ /ρ 1 B
, (19)
1/ 2 1/32 g gC 0.6 u / ν Sc . (20)
27
For droplets with insoluble solids v,s dρ Tf and therefore in equilibrium evaporation
period v,sρ const since dT const . Hence, for such droplets the expression (18) can be
integrated to give the duration of evaporation period:
0.5
2 d,12 1.5 1.5 0.5 0.5ep 2 2 2d,1 d,cr d,1 d,cr d,1 d,cr4 0.5
1 2 2 d,cr
2 C d1 2τ C 2C d d C d d 8 d d 16 ln3C C 2 C d
. (21)
The droplet diameter in the critical drying point (when the critical moisture content remains in
the droplet and the droplet is turned into a wet particle) can be calculated as:
3d,cr d,0 d,0 cr d,cr 0d d ρ 1 X / ρ 1 X . (22)
For droplet with insoluble solids w,crρ ≈ w,1ρ , and then eq. (22) can be reduced:
1 13
d,cr d,0 d,w cr d,s d,w 0 d,sd d ρ X ρ / ρ X ρ . (23)
The rate of droplet mass change is found by combining eqs. (12), (15), (19) and (20):
0.5d v d,w d 1 2 dm m 0.5ρ πd C 2 C d . (24)
The droplet moisture content can be evaluated as follows:
d 0 d,0X m 1 X / m 1 . (25)
The droplet mass can be calculated by integrating eq. (17):
3 3d d,0 d,w d,0 dm m πρ d d / 6 . (26)
For drying air at atmospheric pressure the coefficient of vapour diffusion can be evaluated as
follows (Grigoriev and Zorin, 1988):
1.7510v d,s gD 3.564 10 T T . (27)
In the case when both lumped heat capacity conditions, Bi 0.1 and Fo 0.1 , are not
satisfied simultaneously for the dried droplet, the temperature profile within the droplet must
be taken into account. Consequently, for droplet evaporation period the energy conservation is
described by eq. (2) applied to the time-dependent spatial domain d0 r R t . The
corresponding boundary conditions for this domain are given by:
d
d vg d d fg d
d
T0, r 0;
rT m
h T T k h , r R t .r A
(28)
28
For the set of equations (2) and (28), the value of the droplet outer radius, dR t , is tracked
by combination of (12) and (17).
In contrast to the approaches suggested in the present study (the first approach
assumes droplet temperature profile during initial heating-up and then uniform droplet
equilibrium temperature during evaporation period; the second approach supposes non-
uniform droplet temperature in both periods of the first drying stage), the literature survey of
published models shows that a typical way is to assume a uniform droplet temperature profile
in both periods of the first drying stage. This point of view is based on estimation that for
droplet diameters smaller than 1 mm the corresponding Biot number usually less than 0.1. As
a result, the equation of energy conservation for this case is as follows (Abuaf and Staub,
1986):
dfg v p,d d g d d
Th m c m h T T A
t
d
d . (29)
Second drying stage
From the moment when droplet moisture content falls down to its critical value, the
process of porous crust formation begins. Consequently, the droplet can no longer be treated
as a droplet with liquid and solids, but as a wet particle consisting of a dry porous crust
surrounding a wet core. Hence, the outer diameter of wet particle can be assumed to be
constant. At the same time, the particle wet core shrinks because of evaporation from its
surface and, as a result, the crust thickness increases. Therefore, the wet particle drying is
classified as a problem with internal moving evaporating interface (see Fig. 9).
Q
vm
vmQ
Figure 9. Scheme of wet particle drying
In the present model the crust region of wet particle is treated as a lump pierced by a
large number of identical straight cylindrical capillaries. It is also suggested to consider the
wet particle as a sphere with isotropic physical properties and temperature-independent crust
29
thermal conductivity. These assumptions lead to the following equation of energy
conservation for the crust region:
2cr cr cr2
T α Tr
t r r r
, i pR t r R , (30)
and the corresponding boundary conditions are:
cr wc vcr wc fg i
i
wc cr i
crg cr cr p
T T mk k h , r R t ;
r r A
T T , r R t ;
Th T T k , r R .
r
(31)
The droplet temperature at the point of critical moisture content is taken as an initial condition
for eq. (30).
The rate of mass transfer from the particle outer surface is determined using the law
(12), but the particle surface area, pA , is utilized instead of the droplet surface area, dA . The
mass of particle crust region is calculated using the expression:
3 3cr cr,s p im 4 3π 1 ε ρ R R . (32)
The coefficient of vapour diffusion at particle surface is determined by equation similar to eq.
(27).
For the region of particle wet core, the equation of energy conservation is as follows:
2wc wcwc p,wc wc2
T T1ρ c k r
t r r r
, i0 r R t . (33)
The corresponding boundary conditions are:
wc
cr wc vcr wc fg i
i
cr wc i
T0, r 0;
rT T m
k k h , r R t ;r r A
T T , r R t .
(34)
The rate of interface receding is given by (Levi-Hevroni et al., 1995):
i
v2wc,w i
R 1m
t ερ 4πR
d
d . (35)
30
In the present model it is assumed that the crust pore diameter is much greater than the
mean free path of vapour molecules (i.e. the corresponding value of Knudsen number,
v cKn λ / d , is smaller than unity). Consequently, the process of vapour diffusion in the crust
pores is treated as independent of pore size and governed only by vapour concentration
gradient. As a result, each crust pore can be considered as a semi open Stefan’s system and
the mass transfer rate from a spherical wet particle can be calculated according to the
following equation (Abuaf and Staub, 1986):
β
v,cr w g p i g v,iv
v,cr,s wc,s p ig v p,s2
w D p g
8πε D M p R R p pm ln
pT T R Rp m T
4πM h R T
. (36)
In the above expression the coefficient of vapour diffusion in the crust pores, v,crD , can be
determined similar to eq. (27). The particle moisture content is given by:
p 0 d,0X m 1 X / m 1 . (37)
The mass of solid component in the particle wet core can be found as follows:
3 3wc,s d,0 0 cr,s p im m / 1 X 4 / 3π 1 ε ρ R R . (38)
The density of particle wet core is evaluated with the help of equation similar to eq. (7). On
the other hand, the wet core mass is equal to:
3wc wc im 4 / 3 πρ R . (39)
Hence, combining (7), (38) and (39), the mass of wet core is obtained:
3wc wc,s wc,w wc,s i wc,s wc,sm m ρ ρ 4 3πR ρ m . (40)
Correspondingly, the mass of liquid fraction in the particle wet core can be estimated:
3wc,w wc,w wc,s i wc,s wc,sm ρ /ρ 4 / 3πR ρ m . (41)
Finally, summarizing (32), (38) and (41), the mass of dried particle can be calculated:
d,0 wc,w 3 3p wc,w i p
0 wc,s
m ρ 4m 1 πρ εR 1 ε R
1 X ρ 3
. (42)
The thermophysical properties of particle crust region are determined by treating it as
a porous medium. In this way, the crust thermal conductivity is given by:
cr cr,s cr,gk k 1 ε εk . (43)
31
And the crust density is evaluated as follows:
cr cr,sρ ρ 1 ε . (44)
Since the values of specific heat of solid fraction, p,sc , and specific heat of drying agent, p,gc ,
are usually of the same order, it can be shown that the crust specific heat is equal to: p,crc ≈ p,sc .
The specific heat and thermal conductivity of the particle wet core region are
determined by (5) and (8) or (9) respectively.
4.2 Numerical solution
Period of droplet initial heating-up
The droplet radius is assumed to remain unchangeable during droplet initial heating-
up and therefore the set of equations (2)-(10) can be solved with the help of standard
numerical methods for parabolic PDE of second order, such as full implicit or Crank-
Nicholson finite differences scheme used in the present study, see (Morton and Mayers,
1994).
Period of evaporation from droplet surface
In the case when a uniform equilibrium evaporation temperature within the droplet is
assumed, eq. (16) is solved numerically as a non-linear equation together with eqs. (12) and
(17). This task is performed using a secant method combined with “Regula Falsi” numerical
algorithm (Dowell and Jarratt, 1971), applied to eq. (16), and a fourth-order Runge-Kutta
method (Isaacson and Keller, 1966) for ordinary differential equation (ODE), applied to eqs.
(12) and (17). In the case when a non-uniform droplet temperature distribution is considered,
the PDE (2) is solved using the boundary conditions eq. (28) and eqs. (12) and (17). The
numerical solution of eq. (2) is complicated by the time-varying spatial domain d0 r R t .
This difficulty is overcome by utilizing a fully implicit finite difference scheme with fixed
time-step and solution procedure described by Moyano and Scarpettini (2000) for moving
boundary problems. Finally, if a uniform droplet temperature profile in both periods of the
first drying stage is assumed (this case is used for comparison with other results), then the set
of ODEs (17) and (29) is solved together with eqs. (12), (14) and (15) using a fourth-order
Runge-Kutta method (Isaacson and Keller, 1966). More details on the numerical solution are
presented in Appendix A.
32
Second drying stage
In the second drying stage particle wet core shrinks while particle outer diameter is
assumed to be constant. So, the wet particle drying can be considered as a problem with
internal moving boundary. The fully implicit finite difference scheme with a fixed time-step,
used in the numerical solution of the droplet evaporation period, is applied for PDEs (30) and
(33) that represent the energy conservation in the crust and wet core regions. Utilizing the
common boundary conditions at the crust-wet core interface (see eqs. (31) and (34)) and
based on algorithm proposed by Moyano and Scarpettini (2000) for moving boundary
problems, an original numerical solution procedure to solve this problem of internal moving
front is developed and applied, see Appendix B for details.
4.3 Results of model validation and discussion
Drying of single droplets containing insoluble solids
The results of comparison between the predicted surface temperature and mass of
dried droplet with the data, obtained experimentally and theoretically by Nešić (1990) and by
Nešić and Vodnik (1991) for slurries of colloidal silica, are shown in Fig. 10 (temperature of
drying air gT = 101 ºC, velocity of drying air gu = 1.73 m·s-1) and in Fig. 11 ( gT = 178 ºC, gu =
1.40 m·s-1). In the calculations, the droplet initial mass and temperature are taken to fit the
experimental data. The values of silica thermophysical properties are assumed according to
the literature information (Perry et al., 1997). This results in initial droplet diameter of dd,0
=2.06 mm for the case when gT = 101 ºC and of dd,0= 2 mm for gT = 178 ºC. The porosity is
set ε 0.3 in both case studies.
In Figs. 10, 11, “model 1” corresponds to assumption of temperature profile within the
droplet during its initial heating-up (see eqs. (2), (3)) and then a uniform droplet equilibrium
temperature in the first drying stage (see eq. (16)). “Model 2” represents the approach that
assumes a non-uniform droplet temperature in both periods of the first drying stage (see eqs.
(2), (3) and (28)). Finally, “model 3” considers only time-dependence of droplet temperature
and represents typical literature approach (Abuaf and Staub, 1986; Borde and Zlotnitsky,
1991) of a uniform droplet temperature during the overall first drying stage (see eq. (29)). The
second drying stage is modelled with the help of eqs. (30)-(44) in all the considered cases.
33
0 10 20 30 40 50 60 70 80
20
30
40
50
60
70
80
90
100 experimental temp. experimental mass Nesic & Vodnik (1991) model 1 (surface temp.) model 2 (surface temp.) model 2 (center temp.) model 3 (surface temp.) wet bulb temperature
time, s
Dro
ple
t t
em
pe
ratu
re,
°C2.0
2.5
3.0
3.5
4.0
4.5
Dro
ple
t mas
s, m
g
Figure 10. Drying history of colloidal silica droplet ( gT = 101 ºC, gu = 1.73 m·s-1)
0 5 10 15 20 25 30 35 40 45
20
30
40
50
60
70
80
90
100
110
120
130
140
150 experimental temp. experimental mass Nesic's model (1989) model 1 (surface temp.) model 2 (surface temp.) model 2 (center temp.) model 3 (surface temp.) wet bulb temperature
time, s
Dro
ple
t t
em
pe
ratu
re, °
C
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Dro
ple
t mas
s, m
g
Figure 11. Drying history of colloidal silica droplet ( gT = 178 ºC, gu = 1.40 m·s-1)
The results of simulations given in Figs. 10, 11, show a good agreement between the
predicted and experimental temperature and mass of silica drying for both “models 1” and
“model 2” under different conditions. The predicted drying curves show inessential difference
in droplet temperature and mass time-change during drying for both proposed approaches
(“models 1” and “model 2”) just as for traditional point of view (“model 3”) in all case
studies. It can be observed that in the case of silica drying under Tg = 178 ºC, the “models 1,
2” predict the duration of the first drying stage to be longer by up to 5% than the calculated
using the “model 3”. Furthermore, the simulations using “model 2” show that there can be a
considerable temperature difference between the outer and inner surfaces of particle crust.
Thus, at the end of the silica drying this difference is found to be up to 3.5 °C when drying air
temperature is 101 °C, and up to 22 °C for Tg= 178 °C.
34
Drying of single droplets containing dissolved solids
In the case of drying of single droplets with dissolved solids, the validation of the
model is carried out by comparing the predicted temperature and mass of droplet with data
obtained experimentally by Lin and Chen (2002) for skim milk and whole droplet drying in
atmospheric air. In order to perform the simulations, the “model 2” with some modifications
is utilized. The partial vapour pressure over the surface of the droplet with dissolved solids
depends not only on temperature but also on the droplet moisture content. Moreover, thermal
conductivity, density and specific heat of the solid fraction of skim and whole milk are
functions of temperature, see Appendix C.
According to the reaction engineering approach developed by Chen and Lin (2005),
the density of water vapour over the surface of skim or whole milk droplet can be found as
follows:
v,s v,sat d,sρ ψρ T . (45)
The fractionality, ψ , and apparent activation energy, vΔE , are calculated by:
v d,sψ exp ΔE / T , (46)
c
v v,g d gΔE a ΔE exp b X X , (47)
where a, b and c are empirical coefficients, dX and gX are corresponding moisture contents
of the droplet and drying ambient. The apparent activation energy for drying agent (air),
v,gΔE , is given by:
v,g g v, v,sat gΔE T ln ρ ρ T . (48)
Assuming that drying air is an ideal gas, it can be obtained:
v,g gΔE T ln g , (49)
where g is the gas relative humidity.
The results of validation of “model 2” extended by eqs. (45)-(49) for skim milk
droplet drying under gT = 67.5 ºC, gu = 0.45 m·s-1 are shown in Fig. 12. The experimental
points were measured by Lin and Chen (2002). In the model, the applied values of the
empirical coefficients in eq. (47) are as follows: a= 0.998, b= 1.405, c= 0.930 (Chen and Lin,
2005). Besides, the initial droplet diameter is set dd,0= 1.46 mm, and the porosity value is
assumed ε 0.3 .
35
Figure 12. Drying history of skim milk droplet ( gT = 67.5 ºC, gu = 0.45 m·s-1)
The model theoretical predictions are also compared to the Lin and Chen (2002)
experimental data of whole milk single droplet drying. The following values of empirical
coefficients are applied in eq. (47): a= 0.957, b= 1.291 and c= 0.934 (Chen and Lin,
2005).The results of validation under different temperatures of drying are given in Fig. 13. To
fit the experimental measurements, the following air velocity, droplet diameter and porosity is
applied in the model: gu = 0.45 m·s-1, dd,0= 1.44 mm and ε 0.3 for gT = 67.5 ºC; gu = 0.42
m·s-1, dd,0= 1.44 mm and ε 0.35 for gT = 87.1 ºC; gu = 0.3 m·s-1, dd,0= 1.43 mm and ε 0.4
for gT = 106.6 ºC.
Analysing the calculated results of skim and whole milk droplet drying, shown in
Figs. 12, 13, it can be concluded that the predicted curves of droplet temperature and droplet
mass are in good agreement with the corresponding experimental data. Thus, for skim milk
drying simulation, the maximum discrepancy between predicted and experimental
temperatures is not exceeding 5%, whereas the differences between corresponding mass
values are lower than 4%. At the same time, the maximum discrepancy between the
calculated and experimental temperatures of whole milk droplet are observed to be lower than
7%, and the differences between the corresponding values of droplet mass are predicted to be
less than 4%. It must be noted that no temperature difference between the particle outer
surface and its centre is observed in simulations of skim and whole milk droplet drying.
0 40 80 120 160 200 240 28010
20
30
40
50
60
70
, experimental data model 2
time, s
Dro
ple
t t
emp
erat
ure
, °C
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Dro
plet m
ass, mg
36
0 40 80 120 160 200 240 280 32020
30
40
50
60
70
80
90
100
110
120
Dro
ple
t t
em
pe
ratu
re,
°C
Time, s
T =87.1 °Cg
T =67.5 °Cg
T =106.6 °Cg
0 40 80 120 160 200 240 280 320
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Dro
ple
t m
as
s, m
g
Time, s
T =106.6 °Cg
T =67.5 °Cg
T =87.1 °Cg
(a) (b)
Figure 13. Temperature (a) and mass (b) of whole milk single droplet during drying.
Dots – experimental data of Lin and Chen (2002), lines – model predictions.
4.4 Conclusions
An advanced theoretical model of drying of single droplet, containing either insoluble
or dissolved solids, has been developed. In the case of droplet with insoluble solids, two
approaches are proposed. In the first approach, temperature profile is considered for droplet
initial heating-up, and it is neglected during equilibrium evaporation period. In the second
approach, temperature distribution is taken into account during both periods of the first drying
stage. Both developed approaches are compared to the typical literature point of view that
assumes a uniform droplet temperature in the entire first drying stage. The second drying
stage is modelled by assuming a non-uniform temperature profile within the dried wet particle
for all cases.
The results of numerical simulations show a good agreement between the predicted
and experimental temperature and mass time-change of silica, skim and whole milk droplet
drying under different conditions. A negligible difference in calculations of the droplet drying
behaviour is found for all used approaches of modelling the first drying stage. Regarding the
second drying stage, it is observed that in some cases the simulations predict a considerable
growing temperature difference between the outer and inner surfaces of the particle crust, e.g.,
reaching up to 20 ºC at the end of silica droplet drying under Tg= 178 ºC. Such calculations of
temperature differences in the particle crust can not be compared directly to experimental
data, since the existing experimental methods allow measurement only of the average
temperature of wet particle (Lin and Chen, 2002).
37
Chapter 5
HEAT AND MASS TRANSFER AND MODELLING OF WET
PARTICLE BREAKAGE IN THE SECOND DRYING STAGE
5.1 Introduction
In the present chapter a comprehensive study of heat and mass transfer and breakage
of wet particle in the transient process of drying is presented. In this theoretical investigation,
previously developed drying model of single droplet is extended to take into account the fully
unsteady character of heat and mass transfer in the second drying stage. Numerical
simulations using this extended drying model allow examining the mechanisms of mass
transfer, concentration and pressure time-change within a wet silica particle. Based on the
theory of elasticity and solid mechanics, a criterion of particle cracking/breakage is proposed
and the studies of silica particle breakage under different conditions are performed.
5.2 Model development
A two-stage drying process of droplet, containing small solid particles, is considered.
The drying agent is atmospheric air. In the first drying stage, the process is controlled by
evaporation of pure liquid. The second drying stage begins when droplet moisture content
falls down to the critical value and a dry porous solid crust is formed at the droplet surface.
From this point onwards the droplet is referred to as a wet particle with permanently growing
crust and shrinking wet core regions. In the second stage, liquid evaporation takes place inside
the wet particle at the receding interface between the crust and wet core. The vapour,
generated over the interface, diffuses through the crust pores towards the particle outer
surface and forms a thin boundary layer. From the particle surface, vapour is removed by the
convection of drying air. The process of particle drying stops when the desired value of
moisture content is obtained.
In the first drying stage, a non-uniform droplet temperature is assumed. Consequently,
the mathematical description of the drying first stage is given by the set of equations (2)-(10),
(12), (14), (15), (17), (23), and (25)-(28). The process of heat transfer from the drying air to
the droplet/wet particle is considered to occur due to both convection and radiation
phenomena. Hence, the corresponding coefficient of heat transfer, h , is evaluated as follows:
38
c rh h h . (50)
The coefficient of convection heat transfer, ch , is determined by the Nusselt number with the
help of expression (13). The coefficient of radiation heat transfer, rh , is given by (Holman,
2002):
4 4r r g d g dh σε T T / T T . (51)
In the first drying stage the droplet has an excess of liquid that forms an envelope over the
droplet surface, and therefore the drying is similar to evaporation of a pure liquid droplet. For
this reason, the emissivity of droplet surface can be assumed to be equal to that of the droplet
liquid fraction, i.e. r r,wε ε .
In the second drying stage, the wet particle is treated as a sphere with isotropic
thermophysical properties. The particle outer diameter remains unchanged as the drying
proceeds whereas the particle wet core shrinks due to evaporation from its surface and the
thickness of the crust region increases. The region of porous crust is assumed to be pierced by
identical straight cylindrical capillaries of a small diameter.
The utilized equations of energy conservation for the crust and wet core regions of wet
particle are the same as eqs. (30), (31) and (33), (34). The crust mass and shrinkage rate of the
crust-wet core interface are determined using eqs. (32) and (35). The coefficient of heat
transfer, h , is calculated in the way described above in this section for the first drying stage
(see eqs. (50) and (51)). However, the difference is that the emissivity of the particle outer
surface, rε , is assumed to be equal to the corresponding value of the emissivity of the particle
solid fraction, i.e. r r,sε ε .
In order to describe the process of liquid evaporation from the crust-wet core interface inside
the wet particle and subsequent vapour flow through the crust pores towards the ambient, let
us consider an isolated capillary pore of the particle crust. A single capillary pore is treated as
a straight cylindrical body. Therefore, it is convenient to apply the mass, the momentum and
the energy conservation laws for control volume (capillary pore) in the cylindrical coordinate
system. The origin of this system is located at the crust-wet core interface and its z-axis
coincides with the axis of pore symmetry (see Fig. 14).
The movement of air-vapour mixture in the particle capillary pore is assumed to occur
along z-axis only. Therefore, the mass conservation law is given by:
z
ρρυ 0
t z
(52)
39
Figure 14. Scheme of capillary pore within the crust of wet particle
The development of momentum conservation equation for air-vapour mixture in the
capillary pore results in the well-known Darcy’s law (Cunningham and Williams, 1980):
totalk kz β
pores
AB Bp pυ
μ A z με z
. (53)
The vapour fraction of air-vapour mixture is defined as follows:
v vω ρ /ρ . (54)
Assuming that the vapour fraction changes only along the z-axis of the capillary pore, the
diffusion process within the pore is described by:
v v vz v
ω ω ωρ υ ρD
t z z z
. (55)
Here the value of vapour diffusion coefficient, vD , is calculated using the following semi-
empirical correlation (Eckert and Drake, 1972):
1.815v 0 0D 2.302 10 p / p T/T , (56)
where p0= 0.98·105 Pa, T0= 256 K and the result is in m2/s.
In the present model it is considered that the temperature of air-vapour mixture
changes negligibly along the capillary pore, because of a short pore length of the pore and fast
diffusive motion of the mixture molecules. As a result, the temperature of air-vapour mixture
in the capillary is assumed to be close to the temperature of the crust-wet core interface. Such
an assumption allows developing the following equation of energy conservation for air-
vapour mixture diffusing within the pores of particle crust:
40
p z
T p pρc υ
t t z
d
d
. (57)
The equation of state for the air-vapour mixture within the capillary pore (by assuming
ideal gas) is as follows:
ρp T
M . (58)
The developed set of equations (52)-(58) can be transformed in order to obtain an
explicit dependence of the pressure and vapour fraction on the time and space coordinates
(see Appendix D for details):
2
a v v kβ 2
a v
M M ω B1 p pM
γ p t M M t με z
dd
d d
, (59)
22
v v a v vv 2
a v
ω ω M M ωD M
t z M M z
d
d
. (60)
The following boundary conditions are developed in order to solve this set of equations:
- when z= 0:
i1-βv iv v wc,w
p
Rω RρD 1 ω ε ρ
z R t
d
d
, (61)
v v,sat wc,sω ρ T /ρ , (62)
vv,sat wc,s
v
Mp p T , when z 0
ω M . (63)
- when z= Lp:
p βvv v z D v v,
i
RωρD ω ρυ ρh ω ω ε
z R
(64)
p
gz Lp p
. (65)
5.3 Modelling of wet particle cracking/ breakage
In the previous chapter it has been shown that increase of drying air temperature can
lead to significant temperature gradients (up to 20 °C) in the crust of colloidal silica particle
(see Figs. 10, 11). Because the crust thickness is on the order of microns at the beginning of
the second drying stage and this value is up to an order of millimetre at the end of drying,
41
such substantial temperature gradients can result in considerable thermal stresses appearing in
the region of particle crust.
The radial and tangential components of thermal stresses are evaluated by assuming
that inner and outer surfaces of the particle crust have free radial strains (Timoshenko and
Goodier, 1951):
p
i i
R r3 3T,cr cr 2 2i
T,r cr cr3 3 3cr R Rp i
2α E r Rσ r,t s T s,t s s T s,t s
1 ν r R Rd d
, (66)
p
i i
R r3 3T,cr cr 2 2 3i
T,θ cr cr cr3 3 3cr p i R R
α E 2r Rσ r,t s T s,t s s T s,t s r T r,t
1 ν r R Rd d
. (67)
The Young’s modulus for porous crust structure can be determined using the correlation given
by Gibson and Ashby (1988):
n
cr 1 s,0 m cr m,sE C E 1 α T / T 1 ε . (68)
In the present model the following empirical constants are utilized: C1= 1, n= 2 and αm= 0.35
(Gibson and Ashby, 1988). The value of silica melting point is taken as Tm,s =1600 °C and the
Young’s modulus is set Es,0= 73 GPa; the crust thermal expansion coefficient and the
Poisson’s ratio are assumed to be equal to αT,cr= 5·10-7 K-1 and ν= 0.17 (Perry et al., 1997).
In addition to the thermal stresses acting in the region of particle crust, mechanical
stresses can occur due to the pressure gradient along the crust capillary pores (see eqs. (59)-
(65)). From the pressure equilibrium requirements it can be concluded that the pressure
established at the crust-wet core interface inside the pores is equal to the pressure that acts on
the region of the porous crust from the side of wet core. Hence, the region of the particle crust
can be treated as pseudo solid spherical container subjected to an internal pressure. The
corresponding mechanical stresses in radial and tangential directions are given by
(Timoshenko and Goodier, 1951):
3 3 3i p
r z 0 3 3 3p i
R R rσ r,t p t
r R R
, (69)
3 3 3i p
θ z 0 3 3 3p i
R 2r Rσ r p t
2r R R
, (70)
where
gz 0 z 0
p t p t p . (71)
42
The radial and tangential components of the total stress in the particle crust are
calculated by adding up the corresponding components of the thermal and mechanical
stresses:
- total radial stress
r,tot T,r rσ r,t σ r,t σ r,t ; (72)
- total tangential stress
θ,tot T,θ θσ r,t σ r,t σ r,t . (73)
In order to address the issue about particle cracking/breakage in the second drying
stage, the predicted total stress must be compared with the particle crust strength using a
certain breakage criterion. This breakage criterion and crust theoretical strength are discussed
below.
The silica slurry droplet consists of water and insoluble small solid primary particles,
which are on the order of microns. In the second stage of drying the temperature of the crust
of silica wet particle can potentially exceed the value of water boiling point under
atmospheric pressure. Therefore, when no special binder is utilized in the dried slurry droplet,
it can be assumed that van der Waals forces predominate as a binding mechanism between the
primary particles in the crust of the wet particle, see (Rumpf, 1962). It should be noted that
such agglomeration mechanism is typical, for example, in ceramics processing (Reed, 1995).
Using the above assumption, it can be suggested that the tensile strength of the particle crust
is equal to the upper bound of agglomerate strength due to van der Waals forces and this
strength strongly depends on the size of primary particles (Rumpf, 1962).
For brittle materials like ceramics, one of well-known failure criteria can be applied:
either maximum normal stress criterion or Mohr’s theory (Beer et al., 2002). In the considered
case, the largest total radial and tangential stresses in the spherical crust coincide with the
principal directions and are expected to be tensile. Consequently, both failure criteria result in
the following condition of particle cracking/ breakage:
r,tot θ,tot tmax σ max σ σor , (74)
where σt is particle tensile strength.
43
5.4 Numerical solution and validation of the model
The equations of the developed theoretical model are solved using numerical methods.
For both drying stages, the solution procedure is based on the algorithm proposed by Moyano
and Scarpettini (2000). Details on the numerical method applied in the first drying stage are
given in the section 4.2. In the second drying stage, a predictor-corrector method (Ames,
1965) is applied to solve the set of non-linear PDE (30), (31), (33), (34) and (59)-(65). As a
result, two sets of algebraic equations are obtained after the discretization. Finally, these two
sets of equations are coupled by utilizing the common boundary conditions at the crust-wet
core interface and then numerically solved using the algorithm proposed by Moyano and
Scarpettini (2000).
The validation of the model is based on comparison of the calculated temperature and
mass of the dried particle with the data predicted by a modification of previously developed
theoretical model of single droplet drying, given in the chapter 4 (because this model has been
successfully validated by published experimental data), see Mezhericher et al. (2006) for
details. The results of comparison in the second drying stage for the case of silica droplet
drying are illustrated in Fig. 15. The following values of drying parameters are utilized:
d,0T 19 °C , d,0d 2 mm , gu 1.4 m/s , ε 0.4 and fX 0.05 .
From Fig. 15a it can be observed that the curves of particle surface temperature,
predicted by two different models, are very close to each other and the maximum discrepancy
is not exceeding 1.32%. The temperatures in the particle centre (see Fig. 15b), calculated by
two different ways, show a little difference (about 1.2% maximum) for drying air
temperatures of gT 150 °C, 200 °C and 400 °C . As it can be found, for gT 600 °C and
750 °C the previous model developed in the chapter 4 predicts that corresponding
temperatures of the particle centre should be smaller than 100 °C at the end of the drying
process. In contrast, the present model calculates that these temperatures can reach the values
above 100 °C. From Figs. 15a, b, the greatest difference in temperatures of particle centre is
2.29% for gT 600 °C and it equals to 2.92% for gT 750 °C . At the same time, the particle
mass time-change shows a very good agreement in all considered cases (see Fig. 15c), and the
largest deviation between the curves of the different models is lower than 0.12%.
The model predictions of particle thermal and mechanical stresses can not be validated
at the present stage since the lack of corresponding published experimental and theoretical
data.
44
0 1 2 3 4 5 6 7 8 9 10
40
60
80
100
120
140
160
180 present model Mezhericher et al. (2006)750 °C
600 °C
400 °C
200 °C
Tem
per
atu
re o
f p
arti
cle
su
rfa
ce,
°C
Time, s
Tg =150 °C
0 1 2 3 4 5 6 7 8 9 10
40
50
60
70
80
90
100
110
Tg =150 °C
750 °C 600 °C
400 °C
200 °C
Tem
per
atu
re o
f p
arti
cle
cen
tre
, °C
Time, s
present model Mezhericher et al. (2006)
(a) (b)
0 1 2 3 4 5 6 7 8 9 100.80
0.85
0.90
0.95
1.00
1.05
1.10 present model Mezhericher et al. (2006)
Tg =150 °C
750 °C 600 °C
400 °C200 °C
Ma
ss o
f p
arti
cle,
mg
Time, s
(c)
Figure 15. Drying history of silica particle in the second drying stage. Model validation.
(a) – temperature of the particle outer surface, (b) – temperature of the particle centre,
(c) – mass of the particle
5.5 Results of numerical simulations and discussion
The numerical calculations of silica slurries drying are performed using the presented
model that accounts for cracking/breakage of wet particles. The dried slurry consists of mono-
dispersed amorphous silica spherical particles (median size of primary particles is 0.272 µm
and their density is 1950 kg/m3), deflocculated in water with initial volume fraction of
primary particles of 0.10 (Minoshima et al., 2002). No specific binder is utilized. The
emissivities of liquid and solid fractions of the slurry are considered as fixed values and are
set r,wε 0.96 and r,sε 0.8 correspondingly (Perry et al., 1997). The studied range of the
drying air temperatures is 150-750 °C, velocity is 1.4 m/s, initial slurry droplet diameter is 2
mm and the particle crust porosity is set 0.4. The characteristics of the droplet/wet particle are
tracked during the simulations. A number of calculation results predicted by the developed
45
model for different temperatures of drying air are illustrated by Figs. 16, 17. In these
simulations, the initial droplet temperature is set d,0T 19 º C and the final moisture content is
equal to fX 0.05 .
0 10 20 30 40 50 60 70 80
20
40
60
80
100
120
140
160
180
water boiling temp. at patm
750 °C
600 °C
400 °C
200 °C
Tg=150 °C
droplet surface droplet centre
Tem
per
atu
re o
f d
rop
let/
par
ticl
e, °
C
Time, s0 10 20 30 40 50 60 70 80
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5 Tg =150 °C
Tg =200 °C
Tg =400 °C
Tg =600 °C
Tg =750 °C
Mas
s o
f d
rop
let/
par
ticl
e, m
gTime, s
(a) (b)
Figure 16. Temperature (a) and mass (b) time-change of silica slurry droplet during drying
0 1 2 3 4 5 6 7 8 9 10 11100
105
110
115
120
125
130
135
Pre
ssu
re, k
Pa
Time, s
Tg =150 °C
Tg =200 °C
Tg =400 °C
Tg =600 °C
Tg =750 °C
0 1 2 3 4 5 6 7 8 9 10 110.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Va
po
ur
fra
ctio
n
Time, s
Tg =150 °C
Tg =200 °C
Tg =400 °C
Tg =600 °C
Tg =750 °C
(a) (b)
Figure 17. Pressure (a) and vapour fraction (b) over the crust-wet core interface during drying
of silica wet particle
For the same drying conditions, Fig. 18 demonstrates the predicted typical
distributions of vapour fraction and pressure of air-vapour mixture within the capillary pores
at the end of the drying process ( fX 0.05 ). A similar behaviour of pressure and vapour
fraction is observed at other moments of the second drying stage.
46
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0100
105
110
115
120
125
130
135
Pre
ssu
re,
kPa
Dimentionless coordinate z/Lp
Tg =150 °C
Tg =200 °C
Tg =400 °C
Tg =600 °C
Tg =750 °C
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Vap
ou
r fr
acti
on
Dimentionless coordinate z/Lp
Tg =150 °C
Tg =200 °C
Tg =400 °C
Tg =600 °C
Tg =750 °C
(a) (b)
Figure 18. Typical distributions of pressure (a) and vapour fraction (b) within the capillary
pores of silica particle at the end of drying ( fX 0.05 )
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
100
105
110
115
120
125
130
135
Pre
ssu
re in
pa
rtic
le p
ore
s, k
Pa
Dimensionless coordinate z/Lp
=0.25 =0.5 =0.75 =1
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Dimensionless coordinate z/Lp
Vap
ou
r fr
acti
on
in
pa
rtic
le p
ore
s, k
Pa
=0.25 =0.5 =0.75 =1
(a) (b)
Figure 19. Typical distributions of pressure (a) and vapour fraction (b) in the pores of silica
particle during drying.
Figs. 19-22 show the time-dependent profiles of pressure, vapour fraction and stresses
in the second drying stage under air temperature of Tg= 600 °C. The results presented in Fig.
19 demonstrate that both pressure and vapour fraction in the crust pores gradually grow as
drying proceeds. Fig. 20 illustrates that at the crust inner surface the absolute value of radial
mechanical stress increases whereas corresponding value of tangential stress component
decreases during drying. Such a behaviour can be explained by shrinkage of particle wet core
and crust thickening.
47
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0-35
-30
-25
-20
-15
-10
-5
0
Dimensionless coordinate z/Lp
Mec
han
ical
rad
ial
stre
ss
r
, kP
a
=0.25 =0.5 =0.75 =1
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0
10
20
30
40
50
60
70
80
90
100
110
120
Dimensionless coordinate z/Lp
=0.25 =0.5 =0.75 =1
Mec
han
ical
tan
ge
nti
al s
tres
s
, kP
a
(a) (b)
Figure 20. Typical distributions of radial (a) and tangential (b) mechanical stresses in the crust
of silica particle during drying.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0
20
40
60
80
100
120
140
Dimensionless coordinate z/Lp
Th
erm
al r
adia
l str
ess
r
, k
Pa =0.25
=0.5 =0.75 =1
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0-300
-200
-100
0
100
200
300
400
500
600
Dimensionless coordinate z/Lp
=0.25 =0.5 =0.75 =1
Th
erm
al t
ang
enti
al s
tres
s
, k
Pa
(a) (b)
Figure 21. Typical distributions of radial (a) and tangential (b) thermal stresses in the crust of
silica particle during drying.
From Fig. 21 it follows that both radial and tangential components of the thermal
stress are growing within the process. Moreover, the tangential component is substantially
greater than the radial. Comparing the data given in Figs. 20 and 21, it can be found that
absolute values of radial component of both mechanical and thermal stresses are of the same
order during the whole second stage of silica slurry drying. At the same time, the absolute
values of the tangential component of mechanical and thermal stresses are of the same order
only at the beginning of the process, whereas at the end of drying the tangential thermal stress
significantly prevails over the tangential mechanical stress.
48
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0-40
-20
0
20
40
60
80
100
120
140
=0.25 =0.5 =0.75 =1
To
tal
rad
ial
stre
ss
r,to
t ,
kPa
Dimensionless coordinate z/Lp
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0-300
-200
-100
0
100
200
300
400
500
600
700
800
Dimensionless coordinate z/Lp
=0.25 =0.5 =0.75 =1
To
tal
tan
ge
nti
al s
tre
ss
,to
t , k
Pa
t
(a) (b)
Figure 22. Typical distributions of radial (a) and tangential (b) total stresses in the crust of
silica particle during drying.
Analysing the data given in Fig. 22, it can be concluded that in the course of drying
the total tangential stress in the crust of silica wet particle is 5÷10 times greater than
corresponding radial component. The maximum stress value is tensile and located at the inner
surface of the particle crust. Furthermore, it can be observed that near the crust inner surface
the calculated total tangential stress exceeds the particle crust strength. Therefore, according
to the breakage criterion eq. (74), the present theoretical model predicts that the dried particle
will be cracked or broken at the end of the simulated drying process (Tg= 600 °C).
The results of calculations of maximal total radial and tangential stresses in the crust
of silica granules (wet particles with the final moisture content) are shown in Fig. 23 (droplet
initial temperature Td,0= 19 °C, initial moisture content X0= 4.6154, final moisture content
Xf= 0.05, drying air velocity ug= 1.40 m·s-1, granule diameter in the range of dp= 0.002-2.1
mm). In these drying simulations, primary particles with median size of 0.272 µm constitute
the crust of wet particle. The calculations are also performed for slurries with primary
particles of median size in the range of 0.1-0.9 µm (not presented here due to space
limitations) and the maximal discrepancy from the data given in Fig. 23 is found to be smaller
than 0.2%.
Analysing the results presented in Fig. 23, it can be concluded that tangential thermal
stresses in the crust of wet particle are substantially greater than the radial component
(approx. 5 times). On the other hand, the silica granules consisting of smaller primary
particles have lower values of maximal total stresses during drying. Therefore, the slurries
that contain smaller primary particles have to be taken in order to prevent granule breakage
due to tangential stresses in the drying process.
49
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
0 100 200 300 400 500 600 700 800 900 1,000 1,100 1,200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
g750
°C
600
°C
600
°C
500
°C 40
0 °C
30
0 °C
300
°C
400
°C
T =7
50 °
C
T,r
T
t
Dia
me
ter of p
rima
ry p
article, m
Maximal total stress, kPa
Gra
nu
le d
iam
ete
r, m
m
500
°C
Figure 23. Calculated maximal stresses in silica particles at different temperatures of drying
air. The particle tensile strength tσ is assumed to be equal to the upper bound of
agglomerate strength due to van der Waals forces, see Rumpf (1962) for details
how this curve is obtained.
5.6 Conclusions
In the current chapter an advanced theoretical model of heat and mass transfer and
breakage for single particle drying has been presented. This model enables the prediction of
possible pressure rising inside the wet particle at elevated temperatures of drying agent.
Two origins of stress in the particle crust during drying are observed: temperature drop
and pressure differences inside the particle. Consequently, the total tangential and radial
stresses are the sums of corresponding components of thermal and mechanical stresses.
In the present study it is assumed that van der Waals forces predominate as a binding
mechanism between the primary particles in the crust of silica particle. Such an assumption
allows developing the breakage criterion of silica particle, according to which absolute
maximal total stress in the crust in the radial or tangential direction should be greater than
particle strength calculated from the van der Waals forces binding mechanism.
The results of silica particle drying simulations show that the maximal value of
absolute radial mechanical stress is growing during the drying whereas the absolute maximum
of tangential mechanical stress is decreasing (see Fig. 20). Both radial and tangential thermal
stresses are growing in the course of drying (see Fig. 21). Analysing the results presented in
Figs. 20-22, it is concluded that mechanical stresses play a substantial role at the beginning of
50
the second drying stage, but at the end of drying the thermal stresses are much more
considerable. Fig. 23 demonstrates that the greatest stresses in silica granules are predicted for
the largest particle diameters under elevated temperatures of drying air. It is found that
tangential stresses in the crust of the wet particle are predominant over the radial component
(approx. 5 times greater).
The calculated total stresses compared to the breakage criterion of silica particle
demonstrate that they can be the reason of particle cracking/breakage during drying. It is
shown that the breakage of silica particle depends on its diameter, temperature of the drying
agent and diameter of the primary particles. Slurries containing smaller primary particles are
recommended to be used in the drying process in order to prevent the particle cracking or
breakage due to appearing thermal and mechanical stresses.
51
x
y
Chapter 6
TWO-DIMENSIONAL MODELLING OF SPRAY DRYING PROCESS
6.1. Introduction
In the current chapter a theoretical model and computational studies of the spray
drying process in 2D coordinate space have been presented. A two-phase flow theoretical
model of the spray drying process is proposed, which is based on an Eulerian approach for the
continuous phase (drying agent) and on a Lagrangian approach for the discrete phase (spray
of droplets). The numerical solution of the developed model and computational simulations
are performed with the help of CFD package FLUENT 6.3.26.
6.2 Problem setup
(a) (b)
Figure 24. Geometry of spray dryer (Huang et al., 2003) (a) and numerical grid used in CFD
simulations (b)
52
A cylinder-on-cone geometry of a drying chamber with co-current flow of drying air
and spray of droplets is considered, because it is currently the most popular type of the
industrial spray dryer. The dimensions of the chamber and air inlet size are taken as the same
as those in the experimental studies of Kieviet and Kerkhof (1996, 1997) in order to provide
the model validation, see Fig. 24a. The diameter and position of the air outlet pipe, which are
not specified by Kieviet and Kerkhof, are assumed to be equal to the values published by
Huang et al. (2003a), who utilized the same spray dryer geometry. The spray atomization is
performed by a centrifugal pressure nozzle atomizer located at the top of the drying chamber.
The drying air enters the top of the chamber through an annulus, and the angle between the air
inlet vector and the vertical axis is 35°. There is no swirl of the drying air at the inlet. In order
to reduce the computational efforts, the air outlet is assumed to be located at the centre of the
spray dryer and therefore the drying chamber can be considered as 2D axisymmetric (Fig.
24b).
6.3 Two-dimensional model of spray drying process
The continuous phase (drying air) is assumed to be an ideal gas and it is treated by an
Eulerian approach using the k-ε model for turbulence description. The discrete phase of
spherical droplets (spray) is considered in Lagrangian terms. The Newton’s Second Law of
Motion is applied to track the droplets trajectories and energy and species conservation
equations are used to calculate the droplets temperature and mass during the drying process.
For the continuous phase, the equations of continuity, momentum, energy, turbulent
kinetic energy, dissipation rate of turbulence kinetic energy and species conservation in 2D
axisymmetric form are applied, see Appendix E. The relationship between air temperature,
pressure and density is given by the ideal gas law eq. (58).
For the discrete phase, the equation of motion can be written as follows:
p p
p
U Fg
t m
d
d
. (75)
Here pF
is the sum of the forces acting on the considered droplet/particle from the gas
phase, from other droplets/particles and walls of the spray dryer chamber. This sum is given
by:
p D A B CF F F F F
, (76)
53
where DF
is the drag force, AF
is the virtual (added) mass force, BF
is the buoyancy force.
The corresponding equations for evaluation of the above forces are given in Appendix F. The
contact force due to collisions of a droplet/particle with another droplet/particle or with the
walls of the drying chamber, CF
, is discussed below in the chapters devoted to the study of
droplet-droplet and particle-particle interactions.
All liquid droplets dried in spray drying chambers contain a solid fraction in insoluble
or dissolved state. During the drying process, moisture evaporates from the droplet surface,
the droplet diameter shrinks, and at a certain moment in time a thin shell of solid crust is
formed on the droplet surface. From this moment, the droplet becomes a particle consisting of
a wet core surrounded by a dry solid crust. As a result of drying, the particle wet core shrinks,
and the thickness of the crust region increases, while the outer diameter of the particle usually
remains unchanged. As it has been shown in chapters 4 and 5 of the present work, the detailed
modelling of drying kinetics for individual spray droplets is not an easy task, and its
numerical solution can require substantial computational efforts. In the actual spray drying
process, there are millions of droplets in the chamber at the same time. Therefore, the detailed
modelling of drying kinetics for each droplet can significantly complicate the numerical
simulations and can require extreme computer resources and computational time. To simplify
the calculations and due to FLUENT restrictions, which have been found during
implementation of drying kinetics model, it is assumed that all spray droplets have the
physical properties of pure water, even though there is a certain constant fraction of solid
content inside the droplets. Consequently, the drying kinetics of a specific droplet is
considered to be the same as this for the evaporation and boiling of a pure water droplet, and
when the entire moisture content has been removed from the droplet, it is treated as a solid
particle.
For droplet temperatures below the boiling point, the temperature profile within the
droplet is neglected and eq. (29) is utilized to describe the energy conservation for the discrete
phase. The rate of moisture evaporation is found by assuming convective mass transfer and
applying eq. (12) for each of the spray droplets. The coefficients of heat and mass transfer are
determined in terms of the corresponding Nusselt and Sherwood numbers, see eqs. (14) and
(15).
When the droplet temperature attains the value of the boiling point, the rate of droplet
diameter shrinkage is given by (Kuo, 1986):
g p,g dd
dp,g d d fg
4k c T Td1 0.23 Re ln 1
t c d ρ h
d
d
. (77)
54
From a moment when the entire droplet volatile content (all or part of the droplet
moisture content) has been evaporated off and only the solid fraction remains, the droplet
mass does not decrease any further and the droplet temperature is determined as follows:
pp p,p c p p
Tm c h T T A
t
d
d (78)
It should be noted that in the light of small droplet diameters atomized by the pressure
nozzle and moderate temperature of drying air entering the spray dryer (details are given
below), the radiation terms are neglected in all governing equations for the discrete phase, see
Mezhericher et al. (2008).
The distribution of droplet diameters in the liquid spray after spray atomization by the
pressure nozzle is simulated using a Rosin-Rammler distribution function (Masters, 1972):
np pd / d
dY e (79)
where pd is the mean droplet diameter, n is the spread parameter, and dY is the mass fraction
of droplets with diameters > dp.
The influence of the discrete phase on the continuous phase (and vice versa) is
considered by two-way coupling of the energy, mass and momentum transfer equations.
6.4 Inlet, outlet and boundary conditions
In order to validate the present model, the inlet, outlet and boundary conditions are
established according to the studies of Kieviet and Kerkhof (1996, 1997), Kieviet et al. (1997)
and Huang et al. (2003a). The validation is performed in two case studies. In the first case,
drying air flows into the chamber and there is no spray from the nozzle. In the second case,
hot air dries a spray of liquid droplets atomized by the nozzle. The chamber walls are assumed
to be made of 2-mm stainless steel, and the coefficient of heat transfer through the walls is set
at 3.5 W/(m2·K). The ambient temperature outside the drying chamber is assumed to be 300
K. All the droplets hitting the chamber walls are considered as “escaped” from the
calculations. Hot air of temperature 468 K and absolute humidity 0.009 kg H2O/kg dry air
enters the spray dryer chamber from the flat horizontal ceiling through a central round inlet
without swirling (see Fig. 24a), and the angle between the air inlet vector and the vertical axis
is 35°. The pressure of the inflowing drying air is assumed to be atmospheric, and the air
pressure in the outlet pipe is set at -150 Pa. For the case of no spray from nozzle, the inlet
velocity of drying air is set at 7.36 m/s, whereas for the case including spray the above value
55
is 9.08 m/s. In addition, the quantities of turbulence kinetic energy, k, and turbulence energy
dissipation rate, ε, at the air inlet are 0.027 m2/s2 and 0.37 m2/s3, according to the
measurements of Kieviet et al. (1997). For the case in which hot air dries the spray of liquid
droplets atomized by the nozzle, the spray cone angle is assumed to be 76°, the droplet
velocities at the nozzle exit are assigned to be 59 m/s, and the inlet temperature of the
atomized feed is set at 300 K. The distribution of droplet diameters in the spray is
approximated by a Rosin-Rammler function (79), where the value of mean droplet diameter is
assumed to be 70.5 μm, the spread parameter is set at 2.09, and the corresponding minimum
and maximum droplet diameters are taken as 10.0 μm and 138.0 μm, respectively. The
corresponding flow rate of the spray droplets is equal to 0.0139 kg/s (50 kg/hr). Droplet-
droplet and particle-particle interactions are disregarded at the current stage and a steady-state
of the spray drying process is assumed.
6.5 Numerical solution using Computational Fluid Dynamics
The numerical solution of the model equations and computational simulations are
performed with the help of a 2D axisymmetric pressure-based solver incorporated in CFD
package FLUENT 6.3.26. The solver is based on the finite volumes technique and enables a
two-way coupled Euler-Lagrange method for treatment of the continuous and discrete phases.
The geometry of the spray drying chamber is meshed using a triangular scheme with 34,852
grid cells of various sizes. Smaller cells are located in the regions of high concentration of the
discrete phase, where the largest gradients of momentum, heat and mass transfer are expected
(mainly, in the downstream direction of the spray injection). Correspondingly, larger grid
cells are located near the top and side walls of the chamber, see Fig. 24b. The results of the
numerical simulations are verified to be independent of the grid cell sizes.
The convergence of the numerical solution is monitored by means of the residuals of
the governing equations and by the net imbalance between the domain incoming and outgoing
mass flow and heat transfer rates.
6.6 Results of numerical simulations and model validation
The results of model validation for the case of no spray from the nozzle and steady-
state assumptions are shown in Figs. 25-28.
Fig. 25 demonstrates the comparison between measured and calculated magnitudes of
velocity, obtained by Kieviet and Kerkhof (1996, 1997), and the data predicted by the present
model in two cases: non-insulated chamber walls (coefficient of heat transfer 3.5 Wm-2K-1)
56
and fully insulated walls of the spray dryer. Since the velocity scale is identical in Figs. 25a,
b, c, it can be concluded that the current calculations (given in Figs. 25b, c) predict the
velocity magnitudes within the ranges of measured velocity oscillations (given by the bars in
Fig. 25a). An additional comparison between the present velocity predictions and measured
values obtained by Kieviet and Kerkhof (1997) is given below in Figs. 28b, c.
(a) (b) (c)
Figure 25. Magnitudes of air velocity at different levels of the drying chamber (no spray) for:
(a) Kieviet and Kerkhof (1996, 1997) experimental (dots) and simulations (curves)
(vertical lines represent measured range of velocity oscillations), (b) current CFD
simulations for a non-insulated chamber, and (c) current CFD simulations for a fully
insulated chamber.
Analazing the data presented in Fig. 26, it can be found that air velocity vectors
shown in Fig. 26a differ from those given in Fig. 26b in the bottom region of the spray dryer.
This discrepancy can be explained in terms of differences in the modelling of the location of
the air outlet pipe: in the study of Kieviet and Kerkhof (1997) the vertical part of the air outlet
pipe is longer and situated closer to the product outlet than in the study of Huang et al. (2003),
from which the geometry used in the present simulations has been taken.
57
(a) (b) (c)
Figure 26. Flow pattern of air velocity in the drying chamber (no spray) for: (a) Kieviet and
Kerkhof (1997) simulations , (b) current CFD simulations for a non-insulated
chamber, and (c) current CFD simulations for a fully insulated chamber.
The air flow patterns in normalized vectors of the velocity magnitude, illustrated in
Fig. 27, show the influence of boundary conditions on the results of CFD simulations. Thus,
according to Kieviet and Kerkhof (1997), there should be a recirculation zone with the centre
near the imaginary border between cylindrical and conical parts of the drying chamber (Fig.
27a). However, it is unclear from the study of Kieviet and Kerkhof (1997) which boundary
conditions at the chamber walls should be applied. The current numerical simulations
performed for the case of non-insulated chamber walls calculate the existence of two
recirculation regions: one near the walls of the cylindrical part and the other in the conical
part of the chamber (Fig. 27b). The appearance of additional recirculation in the cylindrical
part of the spray dryer may be justified from a physical point of view: as a result of heat
transfer through the walls to the ambient air, hot air in the chamber cools down and its density
increases, and thus a descending flow is established near the inside walls of the chamber. For
fully insulated walls, the flow pattern shown in Fig. 27c is predicted, and the location of the
recirculation region completely coincides with that in Fig. 27a. It should be noted that the air
flow pattern given in Fig. 27b seems to be more realistic than those in Fig. 27a, c, since most
of industrial spray dryers do not have perfectly insulated walls.
58
(a) (b) (c)
Figure 27. Flow pattern of air velocity (normalized vectors) in the drying chamber (no spray)
for: (a) Kieviet and Kerkhof (1997) simulations, (b) current CFD simulations for a
non-insulated chamber, and (c) current CFD simulations for a fully insulated
chamber.
Fig. 28 demonstrates the profiles of the velocity magnitude at different levels,
beginning from the top of the chamber, as calculated by Huang et al. (2003a) and evaluated in
the present study. It can be observed that in the central region of the spray dryer the data
predicted by the present calculations show better agreement with experimental points of
Kieviet and Kerkhof (1997) than the profiles calculated by Huang et al. (2003a). Since the
model utilized for current calculations and the model of Huang et al. (2003a) are the same, the
observed differences may be due to higher mesh resolution used in the present study (34,852
triangular cells in the current simulations vs. 3765 rectangular cells of Huang et al., 2003a).
59
0.0 0.2 0.4 0.6 0.8 1.0 1.20
1
2
3
4
5
6
7
8
Vel
oci
ty m
agn
itu
de,
m/s
Radial position, m
0.3 m, predicted 0.6 m, predicted 1.0 m, predicted 0.3 m, measured 0.6 m, measured 1.0 m, measured
(a) (b)
0.0 0.2 0.4 0.6 0.8 1.0 1.20
1
2
3
4
5
6
7
8
Vel
oci
ty m
agn
itu
de,
m/s
Radial position, m
0.3 m, predicted 0.6 m, predicted 1.0 m, predicted 0.3 m, measured 0.6 m, measured 1.0 m, measured
(c)
Figure 28. Comparison of air velocity profiles at different levels of the drying chamber (no
spray) among: (a) simulations of Huang et al. (2003a), (b) current CFD simulations
for a non-insulated chamber, and (c) current CFD simulations for a fully insulated
chamber. The experimental points are taken from the study of Kieviet and Kerkhof
(1997)
The curves of temperature, vapour mass fraction and magnitude of velocity that given
in Figs. 29-31, are calculated for the case of drying of water spray obtained with the pressure
nozzle (the drying process is assumed to occur at steady state). It can be observed that the
results shown in Figs. 29a, b, Figs. 30a, b and Figs. 31a, b demonstrate some differences in
the central region of the spray dryer between the corresponding profiles predicted in the
present study and the data provided by Huang et al. (2003a). Once again, these discrepancies
are attributed to the different mesh resolutions. Comparison of the profiles illustrated in Figs.
29b, c, Figs. 30b, c and Figs. 31b, c shows that wall boundary conditions mainly affect the
flow patterns of temperature, vapour mass fraction and velocity in the central region of the
spray dryer. Moreover, for the cases of non-insulated and insulated chambers the
corresponding temperature profiles differ in the vicinity of the walls as well. Therefore, the
0.0 0.2 0.4 0.6 0.8 1.0 1.20
1
2
3
4
5
6
7
8
Vel
oci
ty m
agn
itu
de,
m/s
Radial position, m
0.3 m, Huang et al., 1993 0.6 m, Huang et al., 1993 1.0 m, Huang et al., 1993 0.3 m, measured 0.6 m, measured 1.0 m, measured
60
wall boundary conditions have a greater influence on the temperature flow field than on the
patterns of vapour mass fraction and velocity in the spray dryer.
0.0 0.2 0.4 0.6 0.8 1.0 1.2350
360
370
380
390
400
410
420
430
440
450
460
470
0.2 m 0.6 m 1.0 m 1.4 m
Tem
per
atu
re,
K
Radial position, m
0.0 0.2 0.4 0.6 0.8 1.0 1.2
340
360
380
400
420
440
460
480
0.2 m 0.6 m 1.0 m 1.4 m
Tem
per
atu
re,
K
Radial position, m
(a) (b)
0.0 0.2 0.4 0.6 0.8 1.0 1.2
340
360
380
400
420
440
460
480
0.2 m 0.6 m 1.0 m 1.4 m
Tem
per
atu
re,
K
Radial position, m
(с)
Figure 29. Comparison of temperature profiles at different levels of the drying chamber (spray
from nozzle) among: (a) simulations of Huang et al. (2003a), (b) current CFD
simulations for a non-insulated chamber, and (c) current CFD simulations for a fully
insulated chamber.
The vectors of air flow pattern and particle trajectories, calculated for the cases of
non-insulated and insulated chamber walls in the presence of spray from nozzle, are shown in
Fig. 32 (the spray is represented by 20 streams of different droplet diameters). Only one of ten
velocity vectors is depicted for ease of display. From the results presented in Fig. 32, it is
clear that insulation of the chamber walls dramatically affects the particle trajectories, which
mainly follow on the pattern of drying air flow.
61
0.0 0.2 0.4 0.6 0.8 1.0 1.20.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0.2 m 0.6 m 1.0 m 1.4 m
Vap
ou
r m
ass
frac
tio
n
Radial position, m 0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0.045
0.050 0.2 m 0.6 m 1.0 m 1.4 m
Vap
ou
r m
ass
frac
tio
n
Radial position, m
(a) (b)
0.0 0.2 0.4 0.6 0.8 1.0 1.20.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0.045
0.050 0.2 m 0.6 m 1.0 m 1.4 m
Vap
ou
r m
ass
frac
tio
n
Radial position, m
(c)
Figure 30. Comparison of vapour mass fraction profiles at different levels of the drying
chamber (spray from nozzle) among: (a) simulations of Huang et al. (2003a), (b)
current CFD simulations for a non-insulated chamber, and (c) current CFD
simulations for a fully insulated chamber.
62
0.0 0.2 0.4 0.6 0.8 1.0 1.20
2
4
6
8
10
12
0.2 m 0.6 m 1.0 m 1.4 m
Vel
oci
ty m
agn
itu
de,
m/s
Radial position, m 0.0 0.2 0.4 0.6 0.8 1.0 1.2
0
2
4
6
8
10
12
14 0.2 m 0.6 m 1.0 m 1.4 m
Vel
oci
ty m
agn
itu
de,
m/s
Radial position, m
(a) (b)
0.0 0.2 0.4 0.6 0.8 1.0 1.20
2
4
6
8
10
12
14 0.2 m 0.6 m 1.0 m 1.4 m
Vel
oci
ty m
agn
itu
de,
m/s
Radial position, m
(c)
Figure 31. Comparison of velocity profiles at different levels of the drying chamber (spray
from nozzle) among: (a) simulations of Huang et al. (2003a), (b) current CFD
simulations for a non-insulated chamber, and (c) current CFD simulations for a fully
insulated chamber.
63
(a) (b)
Figure 32. Predicted air flow patterns and particle trajectories (spray from nozzle) for: (a)
non-insulated drying chamber and (b) fully insulated drying chamber.
6.7 Conclusions
A 2D axisymmetric model of the spray drying process is presented. The model is
based on a two-phase flow Eulerian-Lagrangian approach. A CFD package FLUENT 6.3.26 is
utilized for numerical simulations of the spray drying process in cylinder-on-cone drying
chamber fitted with a centrifugal pressure nozzle; the chamber is 3.73 m high and 2.215 m in
diameter, and air flow and spray injection are co-current. Steady-state conditions are assumed
and droplet-droplet and particle-particle interactions are disregarded. The model is validated
by comparing the predicted velocity, temperature and humidity patterns with experimental
measurements and CFD simulations of Kieviet and Kerkhof (1996, 1997), Kieviet et al.
(1997) and Huang et al. (2003a). As a result, good agreement and consistency between the
present simulations and literature data are obtained. In addition, two cases of wall boundary
conditions are examined: non-insulated and fully insulated chamber walls. It is established
that thermal insulation of the spray dryer changes the air flow pattern and, consequently,
dramatically affects the trajectories of the dried droplets/ particles.
64
Chapter 7
INFLUENCE OF DROPLET-DROPLET INTERACTIONS ON TWO-
DIMENSIONAL MODELLING OF SPRAY DRYING PROCESS
7.1 Theoretical modelling of droplet-droplet interactions
Collisions between droplets/particles that are present in an actual spray drying process
significantly affect their trajectories and complicate the calculations of particle tracks. As a
result, the computational time required for simulations of the spray drying process can
substantially increase. Nevertheless, it is essential to model droplet-droplet and particle-
particle interactions because of their influence on particle residence time in the chamber,
particle diameter and, in this way, on the quality of the final product. At the present stage,
only influence of droplet-droplet interactions is investigated. In order to simplify the model
and accelerate the calculations, rotational motion of the droplets and plastic deformations of
particles are neglected. Also, electrostatic forces arising from friction between two colliding
droplets are not taken into account. Furthermore, breakages of droplets as result of collisions
are not modelled. Only binary (one to one) collisions are considered.
The probabilistic model developed by O’Rourke (1981) is utilized to address the effect
of droplet-droplet collisions. O’Rourke’s approach is based on the concept of parcels of
droplets, where each parcel represents a group of droplets with the same properties like
diameter, position, velocity, etc., and there are no interactions between the droplets inside the
parcel. When two parcels are located in the same continuous-phase cell, they are likely to
collide. In this case, the parcel of droplets with the larger diameter is designated “collector”
(contains n1 droplets) and the parcel of droplets with the smaller diameter is designated
“contributor” (contains n2 droplets). A mean expected number of collisions that the collector
undergoes is given by:
2
2 1 2 rn (r r ) u Δtn
V
, (80)
where 1r and 2r are radii of the collector and contributor droplets, ru is the relative velocity,
Δt is calculation time step, and V is the cell volume. According to O’Rourke, the probability
distribution of the number of collisions experienced by collector, m, follows a Poisson
distribution:
65
m
n nP(m) e
m! . (81)
The probability of at least one collision between the two droplet parcels, 1P P(m 1) , is
complementary to the probability of no collisions, P(m 0) (Guo et al., 2004):
n1P 1 P(m 0) 1 e . (82)
The interaction between two parcels is considered to occur when the probability of at least
one collision exceeds a critical value, e.g. 1P 0.5 (Guo et al., 2004). In such a case, using eq.
(82), a critical value of mean expected number of collisions can be evaluated by:
crn ln 0.5 0.69315 (83)
Correspondingly, the collisions between two droplet parcels are assumed to occur when mean
expected number of collisions is greater than its critical value: crn n .
After collision between two droplet parcels, they may either bounce or coalesce
(breakage of droplets due to collisions is not considered). If the droplet parcels collide head-
on, the outcome of the collision tends to be coalescence, and when the collision is more
oblique, the result is a grazing collision. To determine whether the collision is head-on or
oblique, a criterion of critical offset between two parcel centres is calculated:
1 2cr 1 2
c
2.4 (r , r )b r r min 1.0,
We
f
. (84)
The function f and the collision Weber number, cWe , are given by:
3 2
1 1 11 2
2 2 2
r r rr , r 2.4 2.7
r r rf
, (85)
2
r 1 2c
u r rWe
. (86)
The value of actual offset between the centres of two droplet parcels is evaluated as follows:
1 2 mb r r R , (87)
where Rm is a random number between 0 and 1. By comparing between the values of actual
offset, b , and critical offset, bcr, the type of collision is established. Thus, when b<bcr, the
result of the collision is coalescence of the parcels, in which each droplet of collector
coalesces with droplet from contributor to form an agglomerate on a one-to-one basis. If there
66
is an excess of droplets in the contributor, they remain in the contributor to be tracked in the
next computational step. The velocities of the agglomerate and of remaining droplets within
the contributor are determined by the law of momentum conservation, and the corresponding
contact forces are found using Newton’s Second Law of Motion. The collector radius
increases after the coalescence according to the volume conservation law:
3 3 31,new 1 2r r r . (88)
For a grazing collision between two parcels (when b>bcr), O’Rourke developed the following
equation for new velocity vector after impact:
p,1 1 p,2 2 p,2 1 2 cr
1, newp,1 p,2 1 2 cr
m u m u m u u b bu
m m r r b
. (89)
In the above equation it is assumed that some fraction of the droplet’s kinetic energy is lost
due to viscous dissipation and the generation of angular momentum. The corresponding
vector of contact force, C,1F
, is given by:
C,1 p,1 1, new p,1F m u u /Δt
. (90)
Owing to the fact that C,2 C,1F F
, a new velocity vector of the second colliding parcel can be
calculated using the Newton’s Second Law.
It must be noted that in spite of several disadvantages of the O’Rourke model for
calculation of droplet-droplet collisions (one of them being mesh dependency under certain
conditions), its choice is still justified by the following benefits: the algorithm is consistent
with the stochastic nature of spray simulations and it is of second order accuracy in space
(Schmidt and Rutland, 2000). In addition, the O’Rourke method is implemented in CFD
package FLUENT 6.3.26 that facilitates the numerical simulations curried out in the present
work.
7.2 Results of numerical simulations and discussion
The two-dimensional axisymmetric model of the spray drying process validated in the
previous chapter by published experimental and theoretical data is utilized. The model is
extended by O’Rourke’s probabilistic algorithm of droplet-droplet collisions (eqs. (80)-(90)).
The utilized geometry of the spray chamber, numerical grid, inlet, outlet and boundary
conditions are the same as those used in the case of 2D axisymmetric simulations without
67
droplet-droplet interactions discussed above in the chapter 6. The walls of the drying chamber
are considered to be without thermal insulation.
The introduction of droplet-droplet binary interactions in numerical simulations
changes the predicted flow fields within the spray dryer and results in time-dependent flow of
both continuous and discrete phases. For this reason, all the simulations that include droplet-
droplet interactions are performed by applying the transient mode of calculations.
The transient spray drying process is simulated in the following way. First, a steady-
state solution of air flow only without spray of droplets is obtained. Thereafter, all necessary
parameters of the spray injection, which is represented by 20 streams of different droplet
diameters, are fixed in the computer code. Then, two-way coupled transient simulations of the
spray drying process are performed. The convergence of the obtained solution is controlled by
means of residual values of the governing equations solved.
0.0 0.2 0.4 0.6 0.8 1.0 1.2340
360
380
400
420
440
460
480
0.5 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 0.5 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
Tem
per
atu
re, K
Radial position, m
x = 0.2 m
0.0 0.2 0.4 0.6 0.8 1.0 1.2
340
350
360
370
380
390
400
410
420
430
440x = 1 m
0.5 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 0.5 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
Tem
pe
ratu
re, K
Radial position, m
0.0 0.2 0.4 0.6 0.8 1.0 1.2
360
370
380
390
400
410
420
430
440
450
Tem
per
atu
re, K
Radial position, m
x = 2 m 0.5 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 0.5 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
0.1 0.2 0.3 0.4 0.5 0.6
370
380
390
400
410
420
430
440
450
x = 3 m
Tem
per
atu
re, K
Radial position, m
0.5 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 0.5 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
Figure 33. Effect of droplet-droplet interactions on time-change of temperature profiles at
different levels of the drying chamber (x – level from the chamber top; case 1 –
droplet-droplet interactions are taken into consideration; case 2 – droplet
collisions are neglected)
68
The predicted time-dependent temperature profiles at various levels of the spray dryer
(beginning from the chamber top) are shown in Fig. 33. These results represent calculations in
two cases: in case 1 the model takes droplet-droplet interactions into consideration, and in
case 2 the model neglects the droplets collisions. The curves given in Fig. 33 show that even
after 30 s from the beginning of the spray drying process, the temperature field still shows
unsteady behaviour. It can be observed that in the case that takes droplet-droplet interactions
into consideration, the temperatures at radial positions > 0.1 m are lower than in the case
neglecting droplet collisions, and this difference increases with time for all the examined
levels of the spray dryer.
0.0 0.2 0.4 0.6 0.8 1.0 1.20.008
0.012
0.016
0.020
0.024
0.028
0.032x = 0.2 m
0.5 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 0.5 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
Vap
ou
r m
ass
frac
tio
n
Radial position, m
0.0 0.2 0.4 0.6 0.8 1.0 1.20.008
0.012
0.016
0.020
0.024
0.028
0.032
0.036
0.040
0.5 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 0.5 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
x = 1 m
Vap
ou
r m
ass
frac
tio
n
Radial position, m
0.0 0.2 0.4 0.6 0.8 1.0 1.20.008
0.012
0.016
0.020
0.024
0.028
0.032
0.036
0.5 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 0.5 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
x = 2 m
Vap
ou
r m
ass
frac
tio
n
Radial position, m 0.1 0.2 0.3 0.4 0.5 0.6
0.008
0.012
0.016
0.020
0.024
0.028
0.032
x = 3 m
Va
po
ur
mas
s fr
acti
on
Radial position, m
0.5 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 0.5 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
Figure 34. Effect of droplet-droplet interactions on time-change of vapour mass fraction
profiles at different levels of the drying chamber (x – level from the chamber top;
case 1 – droplet-droplet interactions are taken into consideration; case 2 – droplet
collisions are neglected)
Fig. 34 demonstrates the unsteady profiles of vapour mass fraction calculated at
different levels of the drying chamber. The time-dependent character of the profiles is still
noticeable after 30 s from the beginning of spray injection at all sampled levels. In contrast to
the observations for the temperature, Fig. 34 shows that the vapour mass fractions are greater
in the case accounting for droplet-droplet interactions than in the case disregarding droplet
69
collisions at radial positions > 0.1 m. The difference between the corresponding values of the
vapour mass fraction for two above-mentioned cases increases with time for all considered
levels of the drying chamber.
Despite the influence of droplet-droplet interactions on the time history of the
temperature and humidity profiles, no substantial influence of collisions of droplets on the air
velocity profiles is calculated, and the pattern of air velocity remains almost time-independent
after 10 s from simulation start. These findings can be explained by a negligible momentum
transfer from the discrete phase to the continuous phase, since the droplet mass flow rate is
not great, being equal to 50 kg/hr, and individual droplets/particles are small, varying in the
range of 10-138 μm at the beginning of the spray injection and 7.5-257.2 μm after 30 s of
spraying (for the case including droplet-droplet interactions). The predicted velocity
distributions at different levels (from the chamber top) of the drying chamber after 10 s from
calculation start are shown in Fig. 35.
0.0 0.2 0.4 0.6 0.8 1.0 1.20
2
4
6
8
10
12
Ve
loci
ty m
agn
itu
de,
m/s
Radial position, m
0.2 m 1.0 m 2.0 m 3.0 m
Figure 35. Predicted velocity profiles at different levels (from the chamber top) of drying
chamber in both cases of considering or neglecting droplet-droplet interactions
(after 10 s of simulation start)
Analysing the results of the simulations given in Figs. 33-35, the following
conclusions may be drawn. Collisions between droplets lead to an increase in the number of
droplets with large diameters within the liquid spray due to coalescence (see eq. (88)) and at
the same time, to reduction in the total number of droplets. The increase of droplet diameter
elongates the time required for evaporation of the droplet volatile content. Therefore, more
droplets with substantial amount of liquid fraction get to peripheral regions of the drying
chamber, where they are evaporated, thereby increasing the vapour mass fraction in the drying
air and reducing the air temperature. This implies that droplet-droplet interactions displace the
region of evaporation of droplet moisture from the central core towards the periphery of the
70
chamber. Hence, the peripheral region of the chamber plays a more significant role in the
spray drying process than it has been concluded in the study of Huang et al. (2003a).
For the case including droplet-droplet interactions, snapshots of the positions of
droplets/particles at 0.5, 10, 20 and 30 s from the beginning of spray injection are shown in
Fig. G.1 of Appendix G. These pictures demonstrate the substantial role of the chamber
periphery in the spray drying process: with the lapse of time, more and more particles move
towards the periphery zones, following the pattern of air flow. The particle data corresponding
to the snapshots in Fig. G.1 are summarized in Table G.1 (see Appendix G), which
demonstrates a noticeable difference between the cases accounting for and neglecting droplet-
droplet collisions in terms of particle diameters and number of particles in the spray dryer.
0.0 0.2 0.4 0.6 0.8 1.0 1.20
2
4
6
8
10
12 0.2 m, case 1 1.0 m, case 1 2.0 m, case 1 3.0 m, case 1 0.2 m, case 2 1.0 m, case 2 2.0 m, case 2 3.0 m, case 2
Vel
oci
ty m
agn
itu
de,
m/s
Radial position, m 0.0 0.2 0.4 0.6 0.8 1.0 1.2
360
380
400
420
440
460
480T
emp
erat
ure
, K
Radial position, m
0.2 m, case 1 1.0 m, case 1 2.0 m, case 1 3.0 m, case 1 0.2 m, case 2 1.0 m, case 2 2.0 m, case 2 3.0 m, case 2
(a) (b)
0.0 0.2 0.4 0.6 0.8 1.0 1.20.008
0.010
0.012
0.014
0.016
0.018
0.020
0.022
0.024
Vap
ou
r m
ass
frac
tio
n
Radial position, m
0.2 m, case 1 1.0 m, case 1 2.0 m, case 1 3.0 m, case 1 0.2 m, case 2 1.0 m, case 2 2.0 m, case 2 3.0 m, case 2
(c)
Figure 36. Effect of insulation of drying chamber on profiles of: (a) velocity, (b) temperature
and (c) vapour mass fraction. Droplet-droplet interactions are taken into
consideration, and flow time is 7 s from the beginning of spray injection (case 1 –
non-insulated drying chamber and case 2 – fully insulated chamber)
71
Fig. 36 shows the calculated velocity, temperature and vapour mass fraction profiles in
two cases, one assuming non-insulated walls and the other considering fully insulated walls of
the drying chamber. The presented profiles are obtained after 7 s from the beginning of spray
injection. As for steady-state simulations without droplet-droplet interactions discussed in the
chapter 6, it is observed that the choice of wall boundary conditions significantly affects the
temperature, whereas the influence of these conditions on velocity flow fields is less
considerable. However, in contrast to steady-state calculations, the transient simulations with
droplet-droplet interactions also show a substantial influence of wall boundary conditions on
patterns of the vapour mass fraction.
7.3 Conclusions
The two-dimensional axisymmetric model of the spray drying process validated in the
chapter 6 by published experimental and theoretical data has been extended by O’Rourke’s
probabilistic algorithm of binary droplet-droplet interactions. The results of unsteady-state
numerical simulations show that droplet collisions have a remarked influence on temperature
and humidity patterns. Thus, in the case taking droplet-droplet interactions into consideration,
the temperatures at radial positions > 0.1 m are lower and the values of vapour mass fractions
are greater than in the case neglecting droplet collisions; this discrepancy increases with time
for all examined levels of the spray dryer. The pattern of air velocity remains almost time-
independent beginning from 10 s of simulation time. This phenomenon can be explained by
negligible momentum transfer from the discrete phase to the continuous phase, since the spray
mass flow rate used is not so great (50 kg/hr) and the sizes of individual droplets/particles are
on the order of microns.
Summarizing the simulation results of velocity, temperature and humidity patterns, it
can be concluded that droplet-droplet interactions displace the region of droplet evaporation,
and, in this way, the region of heat and mass transfer, from the central core towards the
periphery of the drying chamber. Therefore, the peripheral region of the drying chamber plays
a significant role in the process of spray drying.
In addition, the transient simulations with droplet-droplet interactions show that
thermal insulation of the spray dryer substantially affects the patterns of temperature and
vapour mass fraction, whereas the influence on velocity flow fields is less considerable.
72
Chapter 8
THREE-DIMENSIONAL MODELLING OF SPRAY DRYING PROCESS
8.1 Introduction
In the present chapter, the process of droplet drying in the spray chamber is modelled
using 3D two-phase flow Eulerian-Lagrangian approach. Three-dimensional numerical
simulations of heat and mass transfer in the spray dryer are performed with the help of CFD
package FLUENT 6.3.26. The results of calculations are compared with the predictions of the
2D axisymmetric model, described in the previous chapters. Based on the data obtained by
steady-state and unsteady-state computations, either 2D or 3D model is recommended for
practical application in some specific cases.
8.2 Problem setup, numerical grid, inlet, outlet and boundary conditions
The utilized geometry of the spray chamber, inlet, outlet and boundary conditions are
identical to those used in the case of the 2D axisymmetric simulations discussed in the
chapters 6 and 7 of the present work. In the current chapter, the walls of the spray dryer are
assumed to be perfectly thermally-insulated.
The three-dimensional numerical grid, corresponding to the considered spray chamber
geometry (see Fig. 24a), is presented in Fig. 37.
Figure 37. Three-dimensional numerical grid used in the study.
y z
x
73
8.3 Three-dimensional model of spray drying process
Similarly to the 2D axisymmetric model of spray drying process presented in the
above chapters, a two-phase gas-droplet flow is considered. The continuous phase (drying air)
is treated by an Eulerian approach (continuity, motion, energy and species conservation
equations) and the k-ε model is utilized for turbulence description. The equations for the
continuous phase used in the 3D model of the spray drying process are given in Appendix H.
The discrete phase of spherical droplets (spray) is considered in Lagrangian terms by
applying Newton’s Second Law of Motion to track the trajectories of droplets/particles (see
eqs. (6.1), (6.2)). During the droplet travel, moisture evaporates from the droplet surface and
the droplet becomes a particle consisting of a wet core surrounded by a dry solid crust. As it
has been shown in chapters 4 and 5 of the present work, the detailed modelling of drying
kinetics for individual spray droplets is not an easy task, and its numerical solution can
require significant computational efforts. In an actual spray drying process, there are millions
of droplets in the chamber at the same time. Therefore, the detailed modelling of drying
kinetics for each droplet can considerably complicate the numerical simulations and can
require extreme computer resources and computational time. To simplify the calculations and
due to FLUENT restrictions, which have been found during implementation of drying kinetics
model, it is assumed that all spray droplets have the physical properties of pure water, even
though there is certain constant fraction of solid content inside the droplets. Consequently, the
drying kinetics of a specific droplet is considered to be the same as this for the evaporation
(eqs. (12), (14), (15) and (29)) and boiling of a pure water droplet (eq. (77)), and when the
entire moisture content has been removed from the droplet, it is treated as a solid particle (eq.
(78)).
For steady-state simulations, the droplet-droplet interactions are neglected. For
transient computations, droplet-droplet coalescence or bouncing are treated utilizing the
probabilistic approach developed by O’Rourke (1981), see chapter 7 for details. Particle-
particle interactions are disregarded at the present stage.
The influence of the discrete phase on the continuous phase (and vice versa) is taken
in account by two-way coupling of mass, momentum, energy and species conservation
equations.
8.4 Numerical solution using Computational Fluid Dynamics
The numerical solution of the model equations and subsequent computational
simulations are performed by utilizing the 3D pressure-based solver incorporated in CFD
74
package FLUENT 6.3.26. The solution is based on the finite volumes technique and enables a
two-way coupled Euler-Lagrange method for treatment of the continuous and discrete phases.
The geometry of the drying chamber is meshed by 619,849 grid cells of triangular,
quadrilateral and wedge shape with the sizes distribution analogous to the 2D case: smaller
cells are located in the regions of high concentration of the discrete phase in the downstream
direction of the spray injection, where greater gradients of momentum, heat and mass transfer
are expected to appear; correspondingly, larger grid cells are located near the top and side
walls of the dryer.
The convergence of the numerical solution is monitored by means of residuals of the
governing equations and, in the cases of steady-state simulations, also by net imbalances
between the domain incoming and outgoing mass flow and heat transfer rates.
8.5 Results of numerical calculations and discussion
Steady-state simulations
The three-dimensional CFD numerical simulations are performed in two case studies.
In the first case, hot drying air flows into the chamber and there is no spray from the nozzle.
In the second case, hot air interacts with a spray of liquid droplets atomized by the nozzle. In
both cases, the inflow of drying air is flat, without swirl at the chamber inlet. The results of
steady-state calculations for the case of no spray from the nozzle are shown in Figs. 38, 39.
The air flow patterns in normalized vectors of the velocity magnitude, illustrated in
Fig. 38, show different results predicted by 2D axisymmetric and 3D models. Thus, the 2D
model calculates a recirculation zone with the centre near an imaginary border between the
cylindrical and conical parts of the spray chamber (Fig. 38a). At the same time, 3D
simulations demonstrate three-dimensionality, complexity, asymmetry and multi-recirculation
of air flow in the spray chamber. Fig. 38b shows the air downward motion in XZ-plane in
both central core and periphery of the chamber. In contrast, Fig. 38c demonstrates downward
motion in the central core and upward air movement near the chamber walls in YZ-plane. The
presence of transversal air flow in positive X-direction, weak recirculation in the central core
and transversal air flow in negative X-direction near the walls of the chamber is illustrated by
Fig. 38d.
75
Figure 2.
Figure 38. Flow pattern of air velocity (normalized vectors) in the spray chamber for: (a) 2D
axisymmetric simulations, (b)-(d) 3D simulations. (steady-state, no spray, z – level
from the chamber top).
Fig. 39 demonstrates profiles of the velocity magnitude at different dryer levels
(beginning from the chamber top), evaluated for the case of no spray from nozzle. Good
agreement with experimental points of Kieviet et al. (1997) can be observed for 2D
axisymmetric simulations (Fig. 39a). Figs. 39b, c show asymmetric character of velocity
magnitude predicted by 3D calculations. Comparing the results of 3D calculations with the
experimental data, the same behaviour is observed with some discrepancy in the region of the
central core. This divergence can be explained by the fact that experimental points are time-
averaged values of the set of measurements performed with some accuracy. Thus, large
variations of local velocity were found during the experiments, and estimated intervals of
variation reached up to ±70% of the average velocity values presented here as experimental
data, see the study of Kieviet et al. (1997) for details. So, comparing the current calculations
with these estimated intervals (not shown here), it can be concluded that all calculated
velocities are in the range of experimentally measured variations of velocity. On the whole, it
can be observed that both 2D and 3D models predict the same behaviour of the velocity
magnitude along the chamber radius, though there are quantitative differences between the
two models in both central and periphery regions of the dryer.
z
y
z
x
y
x
(a) (b) (c) (d)
5 mz=2.00
76
0.0 0.2 0.4 0.6 0.8 1.0 1.20
1
2
3
4
5
6
7
8
Vel
oci
ty m
ag
nit
ud
e, m
/s
Radial position, m
0.3 m, predicted 0.6 m, predicted 1.0 m, predicted 0.3 m, measured 0.6 m, measured 1.0 m, measured
-1 .2 -1 .0 -0 .8 -0 .6 -0 .4 -0 .2 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 1 .20
1
2
3
4
5
6
7
8 Y Z -p la n e
Vel
oci
ty m
agn
itu
de,
m/s
R a d ia l p o s it io n , m
0 .3 m , p re d ic te d 0 .6 m , p re d ic te d 1 .0 m , p re d ic te d 0 .3 m , m e a s u re d 0 .6 m , m e a s u re d 1 .0 m , m e a s u re d
(c)
Figure 39. Comparison of air velocity profiles at different levels of the spray chamber (steady-
state, no spray) among: (a) 2D axisymmetric simulations; (b), (c) 3D simulations.
The experimental points are taken from the study of Kieviet et al. (1997)
The results of calculations for the case of hot air and spray of water droplets obtained
with the pressure nozzle are given in Figs. 40-44.
The air flow patterns in normalized vectors of the velocity magnitude, illustrated in
Fig. 40, show different results predicted by 2D axisymmetric and 3D models for the case of
spray from nozzle. Similarly to the case of no spray from nozzle, 2D calculations calculate a
recirculation zone with the centre near an imaginary border between the cylindrical and
conical parts of the spray chamber (Fig. 40a). It can be seen that 3D simulations of spray
drying demonstrate three-dimensionality, complexity, asymmetry and multi-recirculation of
air flow in the chamber (Figs. 40b-d). Thus, Fig. 40b shows the air downward motion in XZ-
plane in the central core and periphery region of air outlet pipe location. In the periphery
region opposite to the location of the outlet pipe, there is an upward air motion. Fig. 40c
demonstrates downward air motion in the central core, upward air movement in the periphery
region of conical part and asymmetric upward and downward air motion in the cylindrical
(a) (b)
- 1 .2 -1 .0 -0 .8 -0 .6 - 0 .4 -0 .2 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 1 .20
1
2
3
4
5
6
7
8
Vel
oci
ty m
agn
itu
de,
m/s
R a d ia l p o s it io n , m
0 .3 m , p r e d ic te d 0 .6 m , p r e d ic te d 1 .0 m , p r e d ic te d 0 .3 m , m e a s u r e d 0 .6 m , m e a s u r e d 1 .0 m , m e a s u r e d
X Z -p la n e
77
part of the spray chamber. In contrast to the case of no spray from nozzle (see Fig. 38d), Fig.
40d illustrates the presence of transversal air flow in negative X-direction and flow
recirculation in the central core in XY-plane, at the level z= 2.005m.
Figure 40. Flow pattern of air velocity (normalized vectors) in the spray chamber for: (a) 2D
axisymmetric simulations, (b)-(d) 3D simulations (steady-state, spray from nozzle,
z – level from the chamber top).
The profiles of temperature, vapour mass fraction and magnitude of velocity,
presented in Figs. 41-43, show differences between the results obtained using 2D or 3D
models of heat and mass transfer of droplet drying in spray chamber. Thus, Figs. 41 and 42
demonstrate that main difference between the 2D and 3D patterns of temperature and vapour
mass fraction is in the region of central core when radial position <0.1m; 2D model predicts
lower temperatures (Fig. 41a) and greater values of vapour mass fraction (Fig. 42a) in this
region. At the same time, both XZ and YZ cutting planes show asymmetry of the 3D flow
fields of temperature and vapour mass fraction in the spray chamber and dependency of these
fields on the direction of the cutting plane (see Figs. 41b,c and Figs. 42b, c).
Comparison of the velocity magnitudes calculated by the 2D and 3D models for the
case of spray from pressure nozzle (Fig. 43) shows that 2D simulations predict greater values
in the central core region, than 3D model. At the same time, the 3D model calculates higher
velocity magnitudes in the periphery of the chamber; the latter can be explained by the
presence of the transversal air flow (see Fig. 40d). The asymmetry of the velocity pattern can
be observed from Figs. 43b,c.
z=2.005 m
(a) (b) (c) z
x
z
y
(d)
y
x
78
The results of particle trajectories calculation using the 2D axisymmetric and 3D
models are shown in Fig. 44. It can be found that particles injected by the pressure nozzle
follow the air flow pattern established in the spray chamber. Consequently, 3D simulations
using “escaped” boundary conditions predict asymmetric particle flow in the chamber:
particles downward motion in the central core and upward particles movement with
recirculation zone in the half of the chamber, opposite to the location of the pipe of air outlet
(Figs. 44b-d). In contrast, 2D axisymmetric simulations using “escaped” boundary conditions
calculate particles downward movement in the region of the chamber central core and
particles upward motion and recirculation in the periphery of the spray chamber radius (Fig.
44a).
(a) (b)
-1 .2 -1 .0 -0 .8 -0 .6 -0 .4 -0 .2 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 1 .2
3 4 0
3 6 0
3 8 0
4 0 0
4 2 0
4 4 0
4 6 0
4 8 0
Tem
per
atu
re,
K
R ad ial position , m
0 .2 m 0 .6 m 1 .0 m 1 .4 m
YZ-p lane
(c)
Figure 41. Comparison of temperature profiles at different levels of the spray chamber
(steady-state, spray from nozzle) among: (a) 2D axisymmetric simulations; (b), (c)
3D simulations.
0.0 0.2 0.4 0.6 0.8 1.0 1.2
340
360
380
400
420
440
460
480
0.2 m 0.6 m 1.0 m 1.4 m
Tem
per
atu
re, K
Radial position, m
- 1 .2 - 1 .0 - 0 .8 - 0 .6 - 0 .4 - 0 .2 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 1 .2
3 4 0
3 6 0
3 8 0
4 0 0
4 2 0
4 4 0
4 6 0
4 8 0
Tem
per
atu
re,
K
R a d ia l p o s it io n , m
0 .2 m 0 .6 m 1 .0 m 1 .4 m
X Z -p la n e
79
0.0 0.2 0.4 0.6 0.8 1.0 1.20.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0.045
0.050 0.2 m 0.6 m 1.0 m 1.4 m
Vap
ou
r m
ass
frac
tio
n
Radial position, m
-1 .2 -1 .0 -0 .8 -0 .6 -0 .4 -0 .2 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 1 .20 .0 0 5
0 .0 1 0
0 .0 1 5
0 .0 2 0
0 .0 2 5
0 .0 3 0
0 .0 3 5
0 .0 4 0
0 .0 4 5
0 .0 5 0
Va
po
ur
ma
ss f
ract
ion
R ad ia l p o s itio n , m
0 .2 m 0 .6 m 1 .0 m 1 .4 m
Y Z -p la n e
(c)
Figure 42. Comparison of vapour mass fraction profiles at different levels of the spray chamber (steady-state, spray from nozzle) among: (a) 2D axisymmetric simulations; (b), (c) 3D simulations.
0.0 0.2 0.4 0.6 0.8 1.0 1.20
2
4
6
8
10
12
14 0.2 m 0.6 m 1.0 m 1.4 m
Vel
oci
ty m
agn
itu
de,
m/s
Radial position, m
(a) (b)
- 1 .2 -1 .0 -0 .8 -0 .6 -0 .4 -0 .2 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 1 .20
2
4
6
8
1 0
1 2
1 4
Ve
loci
ty m
agn
itu
de,
m/s
R a d ia l p o s it io n , m
0 .2 m 0 .6 m 1 .0 m 1 .4 m
Y Z -p la n e
(c) Figure 43. Comparison of velocity profiles at different levels of the spray chamber (steady-
state, spray from nozzle) among: (a) 2D axisymmetric simulations; (b), (c) 3D simulations.
-1 .2 -1 .0 -0 .8 -0 .6 -0 .4 -0 .2 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 1 .20 .0 0 5
0 .0 1 0
0 .0 1 5
0 .0 2 0
0 .0 2 5
0 .0 3 0
0 .0 3 5
0 .0 4 0
0 .0 4 5
0 .0 5 0
Vap
ou
r m
ass
frac
tio
n
R a d ia l p o s it io n , m
0 .2 m 0 .6 m 1 .0 m 1 .4 m
X Z -p la n e
(a) (b)
- 1 .2 -1 .0 -0 .8 -0 .6 -0 .4 -0 .2 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 1 .20
2
4
6
8
1 0
1 2
1 4
Ve
loc
ity
mag
nit
ud
e, m
/s
R a d ia l p o s it io n , m
0 .2 m 0 .6 m 1 .0 m 1 .4 m
X Z -p la n e
80
(a) (b) (c) (d)
Figure 44. Predicted particle trajectories in the spray chamber for: (a) 2D axisymmetric
simulations (b)-(d) 3D simulations (steady-state, spray from nozzle, z – level from
the chamber top).
Unsteady-state simulations
In the real process of droplets drying in the spray chambers, accidental droplet-droplet
interactions like coalescence or bouncing substantially complicate the process and can make it
time-dependent. For this reason, all numerical simulations of heat and mass transfer in spray
drying chamber including droplet-droplet interactions are performed by applying the transient
mode of calculations.
The transient process of heat and mass transfer in the spray chamber is simulated as
follows. First, a steady-state solution of air flow without spray of droplets is obtained.
Thereafter, all necessary parameters of the spray injection represented by 20 streams of
different droplet diameters are fixed in the CFD package. Then, two-way coupled transient
simulations of the spray drying process are initiated. The convergence of the obtained
numerical solution is controlled by means of residual values of solved governing equations.
The predicted time-changes of temperature profiles at various levels (beginning from
the chamber top) for 2D and 3D cases are shown in Fig. 45. The calculated curves show that
even after 30 s from the beginning of the of spray injection in the chamber, the temperature
flow field still represents the unsteady behaviour. It can be observed that 2D axisymmetric
and 3D simulations predict temperature profiles which substantially differ in the region of the
central core (when radial position <0.1m). At the same time, in the periphery of the chamber
z
x
z
y y
x
z=2.005 m
81
2D and 3D calculations result in close values of the temperature. The figure also shows that
3D model calculates asymmetric temperature profiles for all considered levels of the chamber.
Fig. 46 demonstrates the time-dependent profiles of vapour mass fraction calculated at
different levels of the spray chamber. The unsteady character of the profiles is still noticeable
after 30 s from the beginning of spray injection at all sampled levels. Similar to the
temperature profiles, it can be found that the 2D axisymmetric and 3D simulations predict
curves of vapour mass fraction which are considerably different in the region of the central
core when radial position <0.1m. At the same time, in the periphery of the chamber the 2D
and 3D simulations calculate close values of the vapour mass fraction. The figure also shows
that the 3D model predicts asymmetric profiles of vapour mass fraction for all considered
levels of the spray dryer.
Analysing the results given in both Figs. 45 and 46, it can be found that analogous to
2D axisymmetric study (see chapter 7), 3D calculations predict that droplet-droplet
interactions displace the region of droplet evaporation, and, in this way, the region of heat and
mass transfer, from the central core towards the periphery of the drying chamber.
The performed 2D axisymmetric and 3D simulations show that beginning from t= 10
s, the flow patterns of velocity in the spray chamber demonstrate the behaviour that is close to
steady-state (the time-change in values of velocity magnitude <5%). Fig. 47 illustrates the
velocity magnitudes calculated by the 2D and 3D models at t= 20 s. Similar to 2D case (see
chapter 7), 3D simulations calculate no substantial influence of droplet collisions on the air
velocity profiles and it is explained by a negligible momentum transfer from the discrete
phase to the continuous one. It can be found that the 3D model calculates higher velocity
magnitudes in the periphery of the chamber than the 2D axisymmetric model. As in the case
of steady-state simulations, this difference can be explained by the presence of the transversal
air flow in the 3D simulations. The asymmetry of the velocity flow pattern can be observed
from Figs. 47 b, c.
82
0.0 0.2 0.4 0.6 0.8 1.0 1.2390
400
410
420
430
440
450
460
470
480 0.5 s 10 s 20 s 30 s
Tem
pe
ratu
re,
K
Radial position, m
x = 0.2 m
-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2390
400
410
420
430
440
450
460
470
480
Tem
per
atu
re, K
0.5 s, XZ plane 10 s, XZ plane 20 s, XZ plane 30 s, XZ plane 0.5 s, YZ plane 10 s, YZ plane 20 s, YZ plane 30 s, YZ plane
z = 0.2 m
Radial position, m
0.0 0.2 0.4 0.6 0.8 1.0 1.2390
400
410
420
430
440
450
460
470
480 0.5 s 10 s 20 s 30 s
x = 1 m
Tem
per
atu
re, K
Radial position, m
-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2390
400
410
420
430
440
450
460
470
480
Tem
per
atu
re,
K
z = 1 m
Radial position, m
0.5 s, XZ plane 10 s, XZ plane 20 s, XZ plane 30 s, XZ plane 0.5 s, YZ plane 10 s, YZ plane 20 s, YZ plane 30 s, YZ plane
0.0 0.2 0.4 0.6 0.8 1.0 1.2390
400
410
420
430
440
450
460
470
480 0.5 s 10 s 20 s 30 s
Tem
pe
ratu
re, K
Radial position, m
x = 2 m
-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2390
400
410
420
430
440
450
460
470
480
Tem
per
atu
re, K
z = 2 m
Radial position, m
0.5 s, XZ plane 10 s, XZ plane 20 s, XZ plane 30 s, XZ plane 0.5 s, YZ plane 10 s, YZ plane 20 s, YZ plane 30 s, YZ plane
0.1 0.2 0.3 0.4 0.5 0.6390
400
410
420
430
440
450
460
470
480 0.5 s 10 s 20 s 30 s
x = 3 m
Te
mp
era
ture
, K
Radial position, m-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
390
400
410
420
430
440
450
460
470
480
Tem
per
atu
re, K
z = 3 m 0.5 s, XZ plane 10 s, XZ plane 20 s, XZ plane 30 s, XZ plane 0.5 s, YZ plane 10 s, YZ plane 20 s, YZ plane 30 s, YZ plane
Radial position, m
Figure 45. Profiles of temperature at different levels of the spray chamber (unsteady-state,
spray from nozzle; x, z – level from the chamber top). Left column – 2D
axisymmetric, right column – 3D simulations. Filled dots – droplet-droplet
interactions are considered, empty dots – droplet collisions are neglected.
83
0.0 0.2 0.4 0.6 0.8 1.0 1.20.008
0.012
0.016
0.020
0.024
0.028
0.032
0.036 0.5 s 10 s 20 s 30 s
x = 0.2 m
Vap
ou
r m
ass
frac
tio
n
Radial position, m
-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.20.008
0.012
0.016
0.020
0.024
0.028
0.032
0.036 0.5 s, XZ plane 10 s, XZ plane 20 s, XZ plane 30 s, XZ plane 0.5 s, YZ plane 10 s, YZ plane 20 s, YZ plane 30 s, YZ plane
z = 0.2 m
Radial position, m
Vap
ou
r m
ass
frac
tio
n
0.0 0.2 0.4 0.6 0.8 1.0 1.20.008
0.012
0.016
0.020
0.024
0.028
0.032
0.036 0.5 s 10 s 20 s 30 s
x = 1 m
Vap
ou
r m
ass
frac
tio
n
Radial position, m-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.008
0.012
0.016
0.020
0.024
0.028
0.032
0.036z = 1 m
Vap
ou
r m
ass
fra
ctio
n
Radial position, m
0.5 s, XZ plane 10 s, XZ plane 20 s, XZ plane 30 s, XZ plane 0.5 s, YZ plane 10 s, YZ plane 20 s, YZ plane 30 s, YZ plane
0.0 0.2 0.4 0.6 0.8 1.0 1.20.008
0.012
0.016
0.020
0.024
0.028
0.032
0.036 0.5 s 10 s 20 s 30 s
x = 2 m
Vap
ou
r m
ass
frac
tio
n
Radial position, m
-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.20.008
0.012
0.016
0.020
0.024
0.028
0.032
0.036
z = 2 m
Vap
ou
r m
ass
frac
tio
n
Radial position, m
0.5 s, XZ plane 10 s, XZ plane 20 s, XZ plane 30 s, XZ plane 0.5 s, YZ plane 10 s, YZ plane 20 s, YZ plane 30 s, YZ plane
0.1 0.2 0.3 0.4 0.5 0.60.008
0.012
0.016
0.020
0.024
0.028
0.032
0.036 0.5 s 10 s 20 s 30 s
x = 3 m
Vap
ou
r m
ass
frac
tio
n
Radial position, m -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.008
0.012
0.016
0.020
0.024
0.028
0.032
0.036z = 3 m
0.5 s, XZ plane 10 s, XZ plane 20 s, XZ plane 30 s, XZ plane 0.5 s, YZ plane 10 s, YZ plane 20 s, YZ plane 30 s, YZ plane
Vap
ou
r m
ass
frac
tio
n
Radial position, m
Figure 46. Profiles of vapour mass fraction at different levels of the spray chamber (unsteady-
state, spray from nozzle; x, z – level from the chamber top). Left column – 2D
axisymmetric, right column – 3D simulations. Filled dots – droplet-droplet
interactions are considered, empty dots – droplet collisions are neglected.
84
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0
2
4
6
8
10
12
Vel
oci
ty m
agn
itu
de,
m/s
Radial position, m
0.2 m 1.0 m 2.0 m 3.0 m
-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0
2
4
6
8
10
12XZ-plane
0.2 m 1.0 m 2.0 m 3.0 m
Vel
oc
ity
mag
nit
ud
e, m
/s
Radial position, m
(a) (b)
-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.20
2
4
6
8
10
12 YZ-plane 0.2 m 1.0 m 2.0 m 3.0 m
Vel
oci
ty m
ag
nit
ud
e, m
/s
Radial position, m
(c)
Figure 47. Comparison of velocity profiles at different levels of the spray chamber (unsteady-
state, t= 20 sec, spray from nozzle) among: (a) 2D axisymmetric simulations; (b),
(c) 3D simulations.
8.6 Conclusions
In the present chapter the results of numerical simulations using 3D model of the spray
drying process have been presented. The numerical simulations using CFD technique are
performed for cylinder-on-cone thermally-insulated chamber fitted with pressure nozzle. Two
cases are considered: no spray from nozzle (steady-state only) and spray of water droplets
from the pressure nozzle (both steady-state and unsteady-state calculations). The obtained
results are compared with the data predicted by the 2D axisymmetric model of the spray
drying process. Unlike the 2D model, in both cases the 3D calculations demonstrate three-
dimensionality, complexity, asymmetry and multi-recirculation of air flow in the spray drying
chamber (these conclusions coincide with previously reported by Huang et al., 2006).
Additionally, for the case of no spray from nozzle, the results from the 3D simulations of air
velocity magnitude are compared to the literature experimental data (Kieviet et al., 1997) and
the same behaviour and satisfactory quantitative agreement with experiments are concluded.
85
For the case of water spray from pressure nozzle and steady-state assumption, 2D
axisymmetric and 3D profiles of temperature, vapour mass fraction and velocity magnitudes
at different levels of the spray chamber are compared. The same behaviour of all the above
profiles are observed, though the 2D model predicts lower temperatures and greater values of
vapour mass fraction in the region of central core when radial position <0.1m. At the same
time, 3D simulations show the three-dimensionality and asymmetry of the flow fields of
temperature and vapour mass fraction in the spray chamber. It is also found that the 2D model
predict greater values of air velocity magnitudes in the central core region, whereas the 3D
model calculates higher velocity magnitudes in the periphery of the chamber. The latter can
be explained by the presence of the transversal air flow that can not be calculated by the 2D
axisymmetric model. For both 2D and 3D models, the particle trajectories are observed to
follow the air flow pattern established in the chamber. However, using “escaped” boundary
conditions 3D simulations predict asymmetric particles flow in the spray chamber: particles
downward motion in the central core and upward particles movement with recirculation zone
in the half of the chamber opposite to the location of the air outlet pipe. In contrast, 2D
axisymmetric simulations using “escaped” boundary conditions calculate particles downward
movement in the region of the chamber central core and particles upward motion and
recirculation in the periphery of the chamber radius.
For the unsteady-state simulations with liquid spray drying, profiles of temperature and
vapour mass fraction show the unsteady character even after 30 s from the beginning of the
spray injection. Unlike this, the profiles of velocity magnitude demonstrate the steady-sate
behaviour beginning from t= 10 s for both 2D and 3D cases. Such a phenomenon can be
explained by negligible momentum source in the continuous phase because of low mass flow
rate of the water spray (50 kg/h). It is also found that predictions of 2D axisymmetric and 3D
computations are considerably different in the region of the central core when radial position
<0.1m for profiles of temperature and vapour mass fraction. At the same time, in the
periphery of the chamber the 2D and 3D simulations calculate close values of the temperature
and vapour mass fraction, but 3D model predicts higher velocity magnitudes. As in the case
of steady-state simulations, this difference can be explained by the presence of the transversal
air flow in 3D simulations. Analogous to the 2D axisymmetric study (see chapter 7), 3D
calculations predict that droplet-droplet interactions displace the region of droplet
evaporation, and, in this way, the region of heat and mass transfer, from the central core
towards the periphery of the drying chamber. Finally, unlike the 2D axisymmetric model, the
3D model predicts asymmetry of flow patterns in the spray chamber in unsteady-state case of
simulations.
86
Summarizing the study performed, it can be concluded that the 2D axisymmetric
model is suitable for fast and low-resource consumption numerical calculations and it predicts
values of velocity, temperature and vapour mass fraction in the spray chamber with
reasonable accuracy. However, due to its restrictions, the 2D axisymmetric model fails to
predict asymmetry of flow patterns, presence of the transversal air flow and it can not provide
actual three-dimensional picture of particle trajectories inside the spray chamber. Therefore,
though 3D numerical simulations of droplets drying in spray chambers require much more
computational efforts than 2D, in the case when the above characteristics are important, the
utilization of 3D model is essential.
87
Chapter 9
INFLUENCE OF PARTICLE-PARTICLE INTERACTIONS ON
MODELLING OF SPRAY DRYING PROCESS
9.1 Theoretical modelling of particle-particle interactions
The introduction of droplet-droplet interactions into the model of the spray drying
process demonstrate a remarked influence on transient temperature and humidity patterns in
the chamber as well as on size distribution of the obtained particles. At the same time, the
numerical simulations show that the injected droplet quickly turns into a wet particle and
during the most part of its residence time in the dryer the injected droplet behaves like a wet
or a dry particle, but not a droplet. Therefore, it is essential to study the effect of particle-
particle interactions on the flow patterns and particle trajectories in the spray drying process.
In order to simplify the model and accelerate the calculations, rotational motion of the
particles and plastic deformations of particles are neglected. Also, electrostatic forces arising
from friction between two colliding particles are not taken into account. Furthermore,
breakages of particles as a result of collisions are not modelled. No agglomeration or adhesion
between particles is considered, so only bouncing of colliding particles is assumed. Finally,
droplet-particle interactions are neglected and only binary particle collisions are allowed for.
In the present work, it is proposed to model the particle-particle collisions in the spray
drying process by applying a hard-sphere approach of discrete particle interactions to
determine the parameters after collision (Hoomans, 1999; Goldschmidt, 2001), combined
with a probabilistic algorithm of collision detection used earlier for droplet-droplet
interactions (O’Rourke, 1981).
In the hard-sphere approach of particle-particle collision, the interaction forces are
supposed to be impulsive and thus all other finite forces are considered to be negligible during
collision. The particles are assumed to be spherical, quasi rigid and their shape is retained
after impact. Furthermore, quasi-instantaneous contact between two colliding particles is
considered to occur at a point.
Because of the huge number of particles in the spray drying process, the concept of
particle parcels is utilized. Each particle parcel represents a group of particles with the same
properties like diameter, position, velocity, etc., and there are no interactions between the
particles inside the parcel. When two parcels are located in the same continuous-phase cell,
88
they are likely to collide. In this case, the parcel of particles with the larger diameter is
designated “collector” (contains n1 particles) and the parcel of particles with the smaller
diameter is designated “contributor” (contains n2 particles). Correspondingly, eq. (80) is
applied to calculate the mean expected number of collisions that the collector undergoes. In
the case when this value is greater than the critical value of mean expected number of
collisions, see eq. (83), the collision between two particle parcels is considered to occur. The
calculation of post-collision velocity for each parcel is based on equations obtained by
applying Newton’s Second and Third Laws of Motion and three additional constitutive
relations closing the model equations. These constitutive relations are the theoretical
definitions of experimentally found coefficients of normal restitution, e , dynamic friction, μ ,
and tangential restitution, 0β . The equations of hard-sphere model of particle-particle
collision are given in Appendix I. In the numerical simulations the empirical parameters are
set equal to the values obtained experimentally for glass particles by Hoomans (1999):
0.97e , μ 0.1 and 0β 0.33 .
The presented model of particle-particle interactions is utilized to extend the
developed 2D axisymmetric and 3D CFD models of the spray drying process, discussed in the
previous chapters. For this purpose, the model of particle-particle collisions is realized in
computer code using C programming language and compiled to an objective library. Then, the
CFD package FLUENT 6.3.26 is enabled to calculate the particle-particle collisions (in
addition to built-in option of droplet-droplet collisions calculation) by loading the compiled
library as a user-defined function into the package. All numerical simulations are performed
in an unsteady-state mode of calculations.
9.2 Results of numerical simulations and discussion
The geometry of the spray drying chamber, inlet, outlet and boundary conditions are
taken to be identical to those used in the case of 2D axisymmetric and 3D simulations
discussed in the chapters 6-8 of the present work. In the current chapter, the walls of the spray
dryer are assumed to be perfectly thermally-insulated.
The results of comparison between transient CFD calculations accounting for both
particle-particle and droplet-droplet interactions (DD+PP model) and simulations with
droplet-droplet interactions only (DD model) are shown in Figs. 48-53. The corresponding
particle statistics are given in Tables I.1, I.2 of Appendix I.
89
0.0 0.2 0.4 0.6 0.8 1.0 1.2360
380
400
420
440
460
480 0.5 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 0.5 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
Tem
per
atu
re, K
Radial position, m
x = 0.2 m
0.0 0.2 0.4 0.6 0.8 1.0 1.2
360
380
400
420
440
460
480 0.5 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 0.5 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
x = 1 m
Tem
per
atu
re, K
Radial position, m
0.0 0.2 0.4 0.6 0.8 1.0 1.2360
380
400
420
440
460
480 0.5 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 0.5 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
Tem
pe
ratu
re, K
Radial position, m
x = 2 m
0.1 0.2 0.3 0.4 0.5 0.6
360
380
400
420
440
460
480 0.5 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 0.5 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
x = 3 m
Tem
per
atu
re, K
Radial position, m
Figure 48. Comparison of 2D axisymmetric temperature profiles at different levels of the drying chamber (x – level from the chamber top) among: case 1 – DD+PP model, case 2 – DD model.
0.0 0.2 0.4 0.6 0.8 1.0 1.20.008
0.012
0.016
0.020
0.024
0.028
0.032
0.036
0.040
0.044 0.5 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 0.5 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
x = 0.2 m
Vap
ou
r m
ass
fra
ctio
n
Radial position, m 0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.008
0.012
0.016
0.020
0.024
0.028
0.032
0.036
0.040
0.044 0.5 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 0.5 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
x = 1 m
Va
po
ur
ma
ss f
rac
tio
n
Radial position, m
0.0 0.2 0.4 0.6 0.8 1.0 1.20.008
0.012
0.016
0.020
0.024
0.028
0.032
0.036
0.040
0.044 0.5 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 0.5 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
x = 2 m
Vap
ou
r m
ass
frac
tio
n
Radial position, m
0.1 0.2 0.3 0.4 0.5 0.60.008
0.012
0.016
0.020
0.024
0.028
0.032
0.036
0.040
0.044 0.5 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 0.5 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
x = 3 m
Vap
ou
r m
ass
frac
tio
n
Radial position, m
Figure 49. Comparison of 2D axisymmetric vapour mass fraction profiles at different levels of the drying chamber (x – level from the chamber top) among: case 1 – DD+PP model, case 2 – DD model.
90
-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2390
400
410
420
430
440
450
460
470
480T
emp
erat
ure
, K 1 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 1 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
z = 0.2 m
Radial position, m
-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2390
400
410
420
430
440
450
460
470
480 1 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 1 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
Tem
pe
ratu
re, K
z = 1 m
Radial position, m
-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2390
400
410
420
430
440
450
460
470
480 1 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 1 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
Tem
per
atu
re,
K
z = 2 m
Radial position, m -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
390
400
410
420
430
440
450
460
470
480 1 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 1 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
Tem
per
atu
re, K
z = 3 m
Radial position, m
Figure 50. Comparison of 3D temperature profiles at different levels of the drying chamber in XZ cutting plane (z – level from the chamber top) among: case 1 – DD+PP model, case 2 – DD model.
-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2390
400
410
420
430
440
450
460
470
480 1 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 1 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
Tem
per
atu
re, K
z = 0.2 m
Radial position, m -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
390
400
410
420
430
440
450
460
470
480 1 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 1 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
Tem
per
atu
re, K
z = 1 m
Radial position, m
-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2390
400
410
420
430
440
450
460
470
480 1 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 1 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
Te
mp
erat
ure
, K
z = 2 m
Radial position, m
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6390
400
410
420
430
440
450
460
470
480 1 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 1 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
Tem
per
atu
re, K
z = 3 m
Radial position, m
Figure 51. Comparison of 3D temperature profiles at different levels of the drying chamber in YZ cutting plane (z – level from the chamber top) among: case 1 – DD+PP model, case 2 – DD model.
91
-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.20.008
0.012
0.016
0.020
0.024
0.028
0.032
0.036 1 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 1 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
z = 0.2 m
Radial position, m
Vap
ou
r m
ass
fra
cti
on
-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.008
0.012
0.016
0.020
0.024
0.028
0.032
0.036 1 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 1 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
z = 1 m
Vap
ou
r m
ass
fra
ctio
n
Radial position, m
-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.20.008
0.012
0.016
0.020
0.024
0.028
0.032
0.036 1 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 1 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
z = 2 m
Vap
ou
r m
ass
frac
tio
n
Radial position, m -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.008
0.012
0.016
0.020
0.024
0.028
0.032
0.036 1 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 1 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
z = 3 m
Vap
ou
r m
ass
fra
ctio
n
Radial position, m
Figure 52. Comparison of 3D vapour mass fraction profiles at different levels of the drying chamber in XZ cutting plane (z – level from the chamber top) among: case 1 – DD+PP model, case 2 – DD model.
-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.20.008
0.012
0.016
0.020
0.024
0.028
0.032
0.036 1 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 1 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
z = 0.2 m
Radial position, m
Vap
ou
r m
ass
frac
tio
n
-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.008
0.012
0.016
0.020
0.024
0.028
0.032
0.036 1 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 1 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
z = 1 m
Vap
ou
r m
ass
fra
ctio
n
Radial position, m
-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.20.008
0.012
0.016
0.020
0.024
0.028
0.032
0.036 1 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 1 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
z = 2 m
Vap
ou
r m
ass
frac
tio
n
Radial position, m
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.60.008
0.012
0.016
0.020
0.024
0.028
0.032
0.036 1 s, case 1 10 s, case 1 20 s, case 1 30 s, case 1 1 s, case 2 10 s, case 2 20 s, case 2 30 s, case 2
z = 3 m
Vap
ou
r m
ass
frac
tio
n
Radial position, m
Figure 53. Comparison of 3D vapour mass fraction profiles at different levels of the drying chamber in YZ cutting plane (z – level from the chamber top) among: case 1 – DD+PP model, case 2 – DD model.
92
Analysing the calculated profiles of temperature and vapour mass fraction predicted
by the 2D axisymmetric and 3D spray drying models (see Figs. 48-53), it can be observed that
for radial positions >0.1m the temperatures, predicted by the model with both droplet-droplet
and particle-particle interactions, are greater and the vapour mass fractions are lower than the
corresponding values calculated by the model with droplet-droplet interactions only. This
tendency is preserved in time for all considered levels of the drying chamber.
The particle statistics data given in Tables I.1, I.2 of Appendix I show that in the
calculations using DD+PP model, the total number of particles dwelling in the drying
chamber is substantially greater than in the cases utilizing DD model. At the same time, the
amount of small size particles in the cases of DD+PP model is also greater than in the
calculations with DD model (see the columns of arithmetic mean diameter, dp,10, surface mean
diameter, dp,20, and volume mean diameter, dp,30).
It must be noted that the built-in model of droplet-droplet interactions in FLUENT
6.3.26 (DD model) detects all types of collisions (particle-particle, particle-droplet or droplet-
droplet) and treats them as though they always occur between two droplets. Unlike this, the
developed DD+PP model distinguishes between the types of collision and allows strictly
droplet-droplet coalescences or bounces and particle-particle hard spheres collisions.
Based on the above analysis, it can be concluded that introduction of the DD+PP
model into the theoretical modelling of spray drying process results in greater number of
smaller size droplets and particles to be present in the drying chamber. The decrease of
droplet diameter reduces the time needed for evaporation of the volatile contents of the
droplet. Therefore, more droplets are evaporated and turned into non-evaporating particles in
the central region of the chamber, thereby increasing the vapour mass fraction in the drying
air and reducing the air temperature in this region. Since these particles are of small mass,
they travel along the lines of air flow towards the dryer periphery, being heated by air without
evaporation. For this reason, the periphery region remains with higher temperature and
smaller mass fraction compared to the calculations with DD only model. This implies that
consideration of particle-particle together with droplet-droplet interactions narrows the zone
of simultaneous heat and mass transfer towards the region of chamber central core.
93
9.3 Conclusions
A theoretical model of particle-particle collisions in the spray drying process is
proposed. The model is based on a hard-sphere approach of discrete particle interactions
(Hoomans, 1999; Goldschmidt, 2001), combined with a probabilistic algorithm of collision
detection (O’Rourke, 1981).
The model of particle-particle interactions is utilized to extend the developed 2D
axisymmetric and 3D CFD models of the spray drying process, discussed in the previous
chapters. Using the extended spray drying model, transient CFD simulations of the drying
process are performed. The results of calculations for both 2D axisymmetric and 3D cases
show that for radial positions >0.1m the temperatures, predicted by the model with both
droplet-droplet (DD) and particle-particle (PP) interactions, are higher and the vapour mass
fractions are lower than the corresponding values calculated by the model with droplet-droplet
interactions only. Moreover, the total number of particles dwelling in the drying chamber is
substantially greater in the calculations using DD+PP model than in the cases utilizing DD
model only. At the same time, the amount of small size particles in the cases of DD+PP
model is also larger than in the calculations with DD model.
Based on the results of CFD simulations, it is concluded that in contrast to the spray
drying model with droplet-droplet interactions only discussed in the chapter 7, consideration
of particle-particle together with droplet-droplet interactions narrows the zone of
simultaneous heat and mass transfer towards the region of the chamber central core.
94
Chapter 10
COMPARISON BETWEEN THE PREDICTIONS OF AIR FLOW
PATTERNS IN SPRAY DRYER USING k – ε AND REYNOLDS
STRESS TURBULENCE MODELS
10.1 Introduction
The theoretical model of the spray drying process developed and validated in previous
chapters of the current work utilizes the k ε model for turbulence description. The main
drawback of such approach is an assumption of isotropic eddy viscosity (Zhou, 1993). As it
has been found by Oakley and Bahu (1993) in their experimental and numerical study (using
axisymmetric version of FLOW-3D code) of the spray drying process, in some cases the
application of Reynolds stress model of turbulence (RSM) instead of k ε can provide more
appropriate and realistic numerical results. For this reason, calculations of air flow patterns
using 2D axisymmetric and 3D spray drying models with RSM turbulence description are
performed. The obtained results are compared to the published experimental data and to the
flow fields calculated using the k ε turbulence model.
10.2 Problem setup, inlet, outlet and boundary conditions
The utilized geometry of the spray chamber, inlet, outlet, boundary conditions and
numerical grid are identical to those used for the 2D axisymmetric and 3D simulations
discussed in the chapters 6, 8 of the current work. Besides, the drying air enters the chamber
without swirl and no spray from the nozzle is considered in the numerical simulations. A
steady-state is assumed. Two cases of wall boundary conditions are examined: fully thermally
insulated chamber walls and non-insulated walls when the coefficient of heat transfer is equal
to 3.5 W/(m2·K).
The equations of Reynolds stress turbulence model used in the current study are given
in Appendix J. The procedure of numerical solution is the same as described in chapters 6, 8
for steady-state case.
95
10.3 Results of numerical simulations and discussion
The air flow patterns calculated in 2D axisymmetric numerical simulations using RSM
model for turbulence description are shown in Fig.54.
(a) (b)
Figure 54. Two-dimensional axisymmetric flow pattern of air velocity (normalized vectors, no
spray, z – level from the chamber top) in: (a) fully insulated chamber, (b) non-
insulated chamber
Comparing the predicted 2D axisymmetric air flow patterns of air velocity (RSM
turbulence model is used) with the corresponding flow fields presented in Figs. 27 and 32
( k ε model of turbulence is applied), a similarity can be observed. Thus, the current 2D
axisymmetric calculations using RSM model predict upward, downward and recirculation
zones of air flow to be in the same locations as those calculated in the simulations utilizing
k turbulence description.
Fig. 55 shows the profiles of air velocity at different chamber levels, calculated using
the RSM turbulence model, compared to the experimental data provided by Kieviet and
Kerkhof (1997). It can be found that the 2D axisymmetric calculations demonstrate a good
agreement with experimental points.
Unlike the 2D axisymmetric calculations, the results of the 3D air flow predictions
using a three-dimensional spray drying model with RSM (not presented here due to space
limitations) have shown a discrepancy compared to the published experimental results and
theoretical data, predicted with k turbulence description. Such disagreement can not be
96
explained at the present stage and requires an additional comprehensive investigation beyond
the scope of the present work.
0.0 0.2 0.4 0.6 0.8 1.0 1.20
1
2
3
4
5
6
7
8
Vel
oci
ty m
agn
itu
de,
m/s
Radial position, m
0.3 m, predicted 0.6 m, predicted 1.0 m, predicted 0.3 m, measured 0.6 m, measured 1.0 m, measured
0.0 0.2 0.4 0.6 0.8 1.0 1.20
1
2
3
4
5
6
7
8
Vel
oci
ty m
agn
itu
de,
m/s
Radial position, m
0.3 m, predicted 0.6 m, predicted 1.0 m, predicted 0.3 m, measured 0.6 m, measured 1.0 m, measured
(a) (b)
Figure 55. Comparison of air velocity profiles at different levels (no spray) in: (a) fully
insulated chamber, (b) non-insulated chamber. The experimental points are taken
from the study of Kieviet and Kerkhof (1997)
10.4 Conclusions
A spray drying model with Reynolds stress turbulence description has been utilized
for 2D axisymmetric and 3D simulations of air flow without spray of droplets. Two-
dimensional axisymmetric calculations with RSM demonstrate a good agreement and
consistency of air velocity flow fields with the experimental measurements of Kieviet and
Kerkhof (1997) as well as with previous calculations using k turbulence description.
However, the computations of flow fields with RSM appeared to have poor convergence, to
require much more computer resources and to be time-consumptive compared to simulations
with k turbulence model.
Unlike the 2D numerical simulations, the results of air flow predictions using a three-
dimensional spray drying model with RSM show inapplicability of the Reynolds stress
turbulence model for the considered drying conditions. Thus, compared to the experimental
and theoretical data predicted with k turbulence description, a high discrepancy and
inconsistency is observed. Such disagreement can not be explained at the present stage and
requires an additional comprehensive investigation beyond the scope of the current work.
97
Chapter 11
SUMMARY AND CONCLUSIONS
In the present thesis a theoretical study of spray drying process has been performed.
First, a literature survey of the published theoretical models and experimental studies on the
subject of research has been conducted. This detailed review allowed to reveal the lacks and
shortcomings of the published spray drying models in droplet drying kinetics and in two- and
three-dimensional modelling of the continuous and dispersed phases. Based on the
conclusions of the literature survey, the current theoretical study has been performed into five
major tasks.
In the first task, an advanced two-stage theoretical model of single droplet drying,
containing insoluble or dissolved solids, has been developed. The results of numerical
simulations have shown a good agreement between the predicted and experimental
temperature and mass time-change of silica, skim and whole milk droplet drying under
different conditions. Also, a considerable growing temperature difference in the crust of dried
particle has been predicted in the case of silica drying under air temperature of 178 ºC. Such
calculations of temperature differences in the particle crust can not be compared directly to
experimental data, since the existing experimental methods allow measurement only of the
average temperature of wet particle. Then, a detailed study of heat and mass transfer and
breakage of wet particle in the second drying stage has been conducted. The early developed
model of single droplet drying has been extended to enable the prediction of possible pressure
rising inside the wet particle at elevated temperatures of drying agent. Two origins of stress in
the crust of silica particle have been observed: temperature and pressure differences inside the
dried particle. These differences lead to thermal and mechanical stresses that can be a reason
of particle cracking or breakage. It has been assumed that van der Waals forces predominate
as a binding mechanism between the primary particles in the crust of silica particle. Such an
assumption has allowed developing the breakage criterion, according to which absolute
maximal total stress in the crust in radial or tangential direction should be greater than particle
strength calculated from the van der Waals forces binding mechanism. The performed
numerical simulations have shown that tangential stresses in the crust of dried silica particle
predominated over the radial component (approx. 5 times greater). The calculated total
stresses compared to the breakage criterion of silica particle have demonstrated that they can
98
be the reason of particle cracking/breakage during drying. It has been shown that the breakage
of silica particle depends on its diameter, temperature of the drying agent and diameter of the
primary particles. Slurries containing smaller primary particles have been recommended to be
used in the drying process in order to prevent the particle cracking/ breakage due to appearing
thermal and mechanical stresses.
A two-dimensional modelling of spray drying process has been conducted in the
second task of the present study. A CFD package FLUENT 6.3.26 has been utilized for
numerical simulations of the spray drying process in cylinder-on-cone chamber fitted with a
centrifugal pressure nozzle and co-current flow of air and spray injection. Steady-state
conditions have been assumed and droplet-droplet and particle-particle interactions were
disregarded. The results of model validation have shown a good agreement and consistency
between the calculations and published experimental and theoretical data. It has been found
that thermal insulation of the spray dryer changes the air flow pattern and significantly affects
the trajectories of the dried droplets/ particles. Next, the influence of droplet-droplet
interactions on two-dimensional modelling of spray drying process has been studied. The
previously developed two-dimensional axisymmetric model of spray drying process has been
extended by probabilistic algorithm of binary droplet-droplet interactions. The results of
unsteady-state numerical simulations have shown a remarked influence of droplet collisions
on temperature and humidity patterns in drying chamber. It has been concluded that droplet-
droplet interactions displace the region of droplet evaporation, and, in this way, the region of
heat and mass transfer, from the chamber central core towards the periphery. Consequently, it
has been deduced that the peripheral region of the drying chamber played a significant role in
the process of spray drying. In addition, the transient simulations with droplet-droplet
interactions have demonstrated that thermal insulation of the spray dryer substantially affected
the patterns of temperature and vapour mass fraction, whereas the influence on velocity flow
fields was less considerable.
In the third task of the research, a three-dimensional modelling of spray drying process
has been conducted. Two cases have been considered: no spray from nozzle (steady-state
only) and spray of water droplets from the pressure nozzle (both steady-state and unsteady-
state calculations). Unlike the 2D model, in both cases the 3D calculations have demonstrated
three-dimensionality, complexity, asymmetry and multi-recirculation of air flow in the spray
drying chamber. For the case of no spray from nozzle, the results of the 3D simulations of air
velocity magnitude have been compared to the published experimental data and the same
behaviour and satisfactory quantitative agreement with experiments have been concluded. For
the case of water spray from pressure nozzle and steady-state assumption, 2D axisymmetric
99
and 3D profiles of temperature, vapour mass fraction and velocity magnitudes at different
levels of the spray chamber have been compared. The same behaviour of all the above
profiles was observed, though the 2D model have predicted lower temperatures and greater
values of vapour mass fraction in the region of central core when radial position <0.1m. It has
also found that the 2D model predicted greater values of air velocity magnitudes in the central
core region, whereas the 3D model calculated higher velocity magnitudes in the periphery of
the chamber. The latter can be explained by the presence of the transversal air flow that can
not be calculated by the 2D axisymmetric model. For both 2D and 3D models, the particle
trajectories have been observed to follow the air flow pattern established in the chamber.
However, in contrast to the 2D model, three-dimensional simulations have predicted
asymmetric particles flow in the spray chamber. For the unsteady-state simulations with
liquid spray drying, profiles of temperature and vapour mass fraction have demonstrated the
unsteady character even after 30 s from the beginning of the spray injection. Unlike this, the
profiles of velocity magnitude have displayed a steady-sate behaviour beginning from t= 10 s
for both 2D and 3D cases. Such a phenomenon can be explained by negligible momentum
source in the continuous phase because of low mass flow rate of the water spray (50 kg/h). It
has been also found that predictions of the 2D axisymmetric and 3D computations were
considerably different in the region of the central core when radial position <0.1m for profiles
of temperature and vapour mass fraction. At the same time, in the periphery of the chamber
the 2D and 3D simulations have calculated close values of the temperature and vapour mass
fraction, but the 3D model has predicted higher velocity magnitudes. As in the case of steady-
state simulations, this difference can be explained by the presence of the transversal air flow
in the 3D simulations. Finally, unlike the 2D axisymmetric model, the 3D model has
predicted asymmetry of flow patterns in the spray chamber in unsteady-state case of
simulations. Therefore, it can be concluded that the 2D axisymmetric model is suitable for
fast and low-resource consumption numerical calculations and it predicts values of velocity,
temperature and vapour mass fraction in the spray chamber with reasonable accuracy.
However, due to its restrictions, the 2D axisymmetric model fails to predict asymmetry of
flow patterns, presence of the transversal air flow and it can not provide actual three-
dimensional picture of particle trajectories inside the spray chamber. Consequently, though
3D numerical simulations of droplets drying in spray chambers require much more
computational efforts than 2D, in the case when the above characteristics are important, the
utilization of 3D model is essential.
An influence of particle-particle interactions on modelling of spray drying process has
been studied in the fourth task of the current work. A theoretical model of particle-particle
100
collisions in the spray drying process has been proposed. The model is based on a hard-sphere
approach of discrete particle interactions combined with a probabilistic algorithm of collision
detection. The proposed model of particle-particle interactions has been utilized to extend the
previously developed 2D axisymmetric and 3D CFD models of the spray drying process. The
results of transient CFD for both 2D axisymmetric and 3D cases have shown that for chamber
central core the temperatures, predicted by the model with both droplet-droplet (DD) and
particle-particle (PP) interactions, were higher and the vapour mass fractions were lower than
the corresponding values calculated by the model with droplet-droplet interactions only.
Moreover, the total number of particles dwelling in the drying chamber was substantially
greater in the calculations using DD+PP model than in the cases utilizing DD model only. At
the same time, the amount of small size particles in the cases of DD+PP model was also larger
than in the calculations with DD model. Based on the results of CFD simulations, it has been
concluded that in contrast to the spray drying model with droplet-droplet interactions only,
consideration of particle-particle together with droplet-droplet interactions has narrowed the
zone of simultaneous heat and mass transfer towards the region of chamber central core.
The final task of the present work was to compare between the predictions of air flow
patterns in spray dryer using k ε and Reynolds stress turbulence models. A spray drying
model with Reynolds stress turbulence description has been utilized for 2D axisymmetric and
3D simulations of air flow without spray of droplets. Two-dimensional axisymmetric
calculations with RSM have demonstrated a good agreement and consistency of air velocity
flow fields with published experimental measurements as like as with previous calculations
using k turbulence description. However, the computations of the 2D flow fields with
RSM appeared to have poor convergence, require much more computer resources and are
time-consumptive compared to the simulations with k turbulence model. The results of
the 3D air flow predictions using a three-dimensional spray drying model with RSM have
shown a discrepancy compared to the published experimental results and theoretical data,
predicted with k turbulence description. Such disagreement can not be explained at the
present stage and requires a comprehensive investigation that can be carried out in future
post-doctoral work.
The significance of the research is both fundamental and applied. This comprehensive
study has considerably improved the overall understanding of spray drying process. On the
other hand, the developed theoretical model and results of the study can be helpful in design
of new spray dryers for different products and facilitate engineering optimal process
conditions that ensure operation of the existing spray drying units with regards to minimize
energy consumption and to achieve the required quality of final product.
101
CONTRIBUTIONS
The results of the present Ph.D. study were published in the following journal articles:
1. Mezhericher, M., Levy, A. and Borde, I. Theoretical drying model of single droplets containing insoluble or dissolved solids. Drying Technology 2007, 25 (6): 1025 – 1032.
2. Mezhericher, M., Levy, A. and Borde, I. Modelling of particle breakage during drying. Chemical Engineering and Processing: Process Intensification 2008, 47 (8): 1404 – 1411.
3. Mezhericher, M., Levy, A. and Borde, I. Heat and mass transfer of single droplet/wet particle drying. Chemical Engineering Science 2008, 63 (1): 12 – 23.
4. Mezhericher, M., Levy, A. and Borde, I. The influence of thermal radiation on drying of single droplet/ wet particle. Drying Technology 2008, 26 (1): 78 – 89.
5. Mezhericher, M., Levy, A. and Borde, I. Droplet-droplet interactions in spray drying using 2D Computational Fluid Dynamics. Drying Technology 2008, 26 (3): 265 – 282.
6. Mezhericher, M., Levy, A. and Borde, I. Modelling of droplet drying in spray chambers using 2D and 3D Computational Fluid Dynamics. Drying Technology 2009, 27 (3): 359-370.
7. Mezhericher, M., Levy, A. and Borde, I. Heat and mass transfer and breakage of particles in drying processes. Drying Technology 2009. Accepted for publication.
The results of the present Ph.D. study were also orally presented at several
international conferences:
1. Mezhericher, M., Levy, A. and Borde, I. Theoretical drying model of single droplets containing insoluble or dissolved solids. In: I. Farkas and A. S. Mujumdar (eds.), Drying 2006 – Proceedings of the 15th International Drying Symposium (IDS 2006), Szent István University Publisher, Gödöllő, Hungary, 2006, Vol. A, 211-218.
2. Mezhericher, M., Levy, A. and Borde, I. Spray drying of wet particles. In: Massacci, P., Bonifazi, G. & Serranti, S. (Eds.), The 5th International Conference for Conveying and Handling of Particulate Solids (CHoPS-05), Sorrento, Italy, 27-31 August 2006, Conference Proceedings CD, ORTRA, Tel Aviv, Israel, 2006.
3. Mezhericher, M., Levy, A. and Borde, I. Heat and mass transfer during dewatering of slurries. In: 9th International Conference on Bulk Materials Storage, Handling and Transportation (ICBMH 2007), University of Newcastle, Australia 9 -11 October 2007, Conference Proceedings CD, 2007.
4. Mezhericher, M., Levy, A. and Borde, I. Heat and mass transfer and breakage of particles in drying processes. In: B.N. Thorat and A. S. Mujumdar (eds.), Drying 2008 – Proceedings of the 16th International Drying Symposium (IDS 2008), Institute of Chemical Technology, University of Mumbai, Mumbai, 2008, Vol. A, 170-178.
5. Mezhericher, M., Levy, A. and Borde, I. Comparison between 2D and 3D modelling of spray drying process. In: B.N. Thorat and A. S. Mujumdar (eds.), Drying 2008 – Proceedings of the 16th International Drying Symposium (IDS 2008), Institute of Chemical Technology, University of Mumbai, Mumbai, 2008, Vol. B, 621-628.
102
NOMENCLATURE
a empirical coefficient
A surface area, m2
Apores mean area of crust cross section, occupied by pores, m2
Atotal total mean area of crust cross section, m2
b empirical coefficient; offset between two droplet centres, m
B Spalding number
Bi Biot number
Bk crust permeability, m2
c empirical coefficient; mass concentration of solid fraction, kg·kg-1
cp specific heat under constant pressure, Jkg-1K-1
cv specific heat under constant volume, Jkg-1K-1
d droplet/ wet particle diameter, m
dc diameter of crust capillary, m
dp diameter of primary solid particles; droplet/ particle diameter, m
dp,10 arithmetic mean diameter, m
dp,20 surface mean diameter, m
dp,30 volume mean diameter, m
pd mean diameter of droplets, m
Dv coefficient of vapour diffusion, m2s-1
e coefficients of normal restitution
E Young’s modulus, Pa
Ec Eckert number
Es,0 Young’s modulus of solid component, Pa
ΔEv apparent activation energy, Jmol-1
F force, N
FA virtual mass force, N
FB buoyancy force, N
FC contact force, N
FD drag force, N
Fo Fourier number
g gravity acceleration, m·s-2
h heat transfer coefficient, Wm-2K-1; specific enthalpy, Jkg-1
103
hc coefficient of convection heat transfer, Wm-2K-1
hD mass transfer coefficient, m·s-1
hfg specific heat of evaporation, J kg-1
hr coefficient of radiation heat transfer, Wm-2K-1
k thermal conductivity, Wm-1K-1; turbulent kinetic energy, m2s-2
Kn Knudsen number
Lp length of crust pore, m
m mass, kg; number of collisions that collector undergoes
vm mass transfer rate, kg·s-1
M molecular weight, kg·mol-1
n empirical coefficient; spread parameter; number of droplets/particles
n mean expected number of collisions
nparcels number of parcels
nparticles number of droplets/particles
Nu Nusselt number
p pressure, Pa; pressure of gas phase, Pa
p0 reference pressure, Pa
P probability distribution
P1 probability of at least one collision
Pr Prandtl number
qt specific heat of mixing, Jkg-1
r radial space coordinate, m
R radius, m
Rm random number between 0 and 1
universal gas constant, Jmol-1K-1
Re Reynolds number
s space coordinate, m
Sc Schmidt number
Sh Sherwood number
t time, s
T temperature, K; temperature of gas phase, K
T0 reference temperature, K
Tm temperature of melting point, K
u velocity of drying agent, m·s-1
104
pU
vector of particle velocity, m·s-1
u
vector of gas velocity, m·s-1; vector of collector/contributor velocity, m·s-1
ur relative velocity between collector and contributor, m·s-1
V volume, m3; cell volume, m3
We Weber number
x space coordinate, m; level from chamber top for 2D simulations, m
X moisture content (dry basis), kg·kg-1
y space coordinate, m
z space coordinate, m; level from chamber top for 3D simulations, m
Yd mass fraction of droplets with diameters greater than diameter d
Greek letters
α thermal diffusivity, m2s-1
αm empirical coefficient
αT coefficient of thermal expansion, K-1
β empirical coefficient
β0 coefficient of tangential restitution
γ ratio of specific heats
δ droplet void fraction
Δt calculation time step, s
ε crust porosity; dissipation rate of turbulent kinetic energy, m2s-3
εr emissivity
θ angular coordinate
φ relative humidity of drying air
λ molecular mean free path, m
μ dynamic viscosity, kg·m-1·s-1; coefficient of dynamic friction
ν Poisson’s ratio; kinematic viscosity, m2s-1
ρ density, kg·m-3; density of gas phase, kg·m-3
σ Stefan-Boltzmann constant, Wm-2K-4; surface tension, N·m-1; mechanical stress, Pa
σt particle tensile strength, Pa
σT thermal stress, Pa
τ duration, s; dimensionless time of particle drying; shear stress, Pa
υ velocity, m·s-1
φ relative humidity
ψ fractionality coefficient
105
ωv vapour fraction
Subscripts
a air, dry air fraction
atm atmospheric
c crust capillary; collision
cr particle crust; critical
d droplet
f final point of drying process
flow forced flow
g drying agent; gas phase
i crust-wet core interface
in inflow
ip initial heating-up period
m air-vapour mixture
max maximal
min minimal
new new
out outflow
p droplet/ particle
pores crust pores
r radial direction
s solid fraction; surface
sat saturation
tot total
v vapour; vapour fraction
w water
wc particle wet core
z axial direction
0 initial point of drying process
1 starting point of droplet evaporation period; collector
2 contributor
∞ bulk of drying agent
θ tangential direction
106
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112
APPENDIX A
Numerical Solution for Period of Droplet Evaporation
In the case a non-uniform droplet temperature is assumed during evaporation period of
the first drying stage, the partial differential equation of parabolic type (2) applied to the time-
dependent spatial domain d0 r R t , must be solved simultaneously with its boundary
conditions eq. (28) and additional equations (12) and (17), which define mass transfer rate and
its connection to the rate of droplet shrinkage. It is obvious that numerical solution of eq. (2)
is complicated by the presence of a variable spatial domain 0 dr R t due to droplet
evaporation. To facilitate the solution, the procedure given by Moyano and Scarpettini (2000)
for moving boundary problems is adopted.
Applying Landau's transformation
,
,d
t
r R
, (A.1)
the droplet moving boundary is fixed in range 0 1 and the following expressions are
obtained:
, ,d dT r t T ;
2 2
2 2 2
1,
1,
.
p
p
p
p
r R
r R
dR
t R d
Using the above formulas and assuming dk f r , eq. (2) can be transformed to give:
2
2 2 2
, , ,2, 0 1d d dd d d
p d d
T T TdR
R d R R
. (A.2)
Restoring t :
113
2
2 2 2
, , ,2, 0 1d d dd d d
d d d
T t T t T tdR t
t R t dt R t R t
. (A.3)
The boundary conditions for eq. (A.3) are obtained from eq. (28):
2v
,0, 0;
4 4 , 1.
d
dg d d d d fg
T t
Th T T R t k R t h m
(A.4)
The initial condition for (A.3) is the temperature profile at the end of the initial period of
droplet heating-up, transformed into spatial domain in accordance with (A.1). Equations
(12) and (17) do not need to be transformed since of time-dependence only.
Let the spatial domain 0 1 be subdivided into N equal intervals of width
each and i i , where 0,1, ,i N . Also let t be the fixed time step for simulation and
n denotes its number, 0,1,2,n .So, ,n
d diT T i n t . Then, the discretization of
(A.3) gives:
1 1 11
1 11 21
1 1 1
1 12 21
2
2
2.
n n n n nn n
d d d d dd di i i i in n
d d
n n n n
d d d di i i i
n
d
T T T TR Ri
t tR i R
T T T
R
(A.5)
Denoting
1212
n
dn ii
n
d
tA
R
, (A.6)
11
11
2
n nn
d dn ii n
d
i R R AB
iR
, (A.7)
and substituting these expressions into eq. (A.5) leads to:
1 1 11 1 1 1 1
1 11 2
n n n nn n n n nd i i d i d i i di i i i
T B A T A T B A T
. (A.8)
Now, the boundary conditions (A.4) should be applied to the left-end and right-end
points of spatial mesh. However, the mesh point 0i is a problematic one since both 0
and 0,
0dT t
. Fortunately, Morton and Mayers (1994) have described the solution
114
procedure for such a problem. First, expanding the function ,dT t in Taylor’s series at
0 and 1t n t gives:
22
2
0,1, 0,
2d
d d
T tT t T t
.
Then, the expansion of 22
1 dT
in Taylor’s series results in:
22
2 2
0,13 dd
T tT
.
Combining the above equations:
22 2
, 0,16 d dd
T t T tT
. (A.9)
Substituting eq. (A.9) into eq. (A.2), a finite difference approximation at 0i is obtained:
1 1 1
0 0 1 02 21
6n n n n n
d d d d di
n
d i
T T T T
t R
. (A.10)
Finally, from eqs. (A.10) and (A.6) it can be found that:
1
1 101 10 1 0
0 0
6 1
1 6 1 6
nn n n
d d dn n
AT T T
A A
. (A.11)
For the right-end of the spatial domain at i N , the discretization yields:
1 1
21 1 11 1 1 11 1v4 4
2
n nn n nd dn n n nN N
g d d d d fgN
T Th T T R k R h m
,
1 11 1 1 v1
11 1 12
2
n nn n n fgn
d d g d nN N N nd d
h mT T Bi T T
k R
. (A.12)
In eq. (A.12) 1nBi is a Biot number for time-step 1n :
111
1
nndn
nd
h RBi
k
. (A.13)
Substituting eq. (A.12) into eq. (A.8), written for the case i N , results in:
115
1 11 v1 1 1 1
11 11
1 1 1 1
2 24
1 2 2
n nn n fgn n n n
d N d N N g nN N nn d d
d N n n n nN N N
h mT A T B A Bi T
k RT
A Bi B A
(A.14)
Now the Gauss elimination algorithm can be implemented (Forsythe and Wasow,
1967). Let 1n
d iT
be written as follows:
1 11 1
1
n nn nd i d ii i
T a T b
. (A.15)
Then, from eq. (A.8):
1 1 11 11 1 1
11 1 1 1 1 1 1 11 11 2 1 2
n n n nn nn n d i i ii i i
d di in n n n n n n ni i i i i i i i
T A B bA BT T
A a A B A a A B
. (A.16)
Comparing eqs. (A.15) and (A.16):
1 1 11 1
11 1
1 1 1 1 1 1 1 11 1
, , 1 11 2 1 2
n n n nn nd i i in ni i i
i in n n n n n n ni i i i i i i i
T A B bA Ba b i N
A a A B A a A B
(A.17)
Comparing eqs. (A.11) and (A.15):
1
1 100 01 1 0
0 0
6 1,
1 6 1 6
nnn n
dn n
Aa b T
A A
. (A.18)
And from eqs. (A.14) and (A.15):
1 1v1 1 1 1 1
1 11
1 1
1 1 1 1 11
2 24
0,1 2 1 2
n nn fgn n n n n
d N N N N g nN nd dn n
N N n n n n nN N N N
h mT A b B A Bi T
k Ra b
A a Bi B A
. (A.19)
In this way, for given time step 1n the values of 1Nia and 1N
ib are calculated
according to eqs. (A.17)-(A.19). After that, using eq. (A.15) the values of 1n
d iT
are
estimated in reverse order, beginning from i N and finishing when 0i .
The numerical expressions for mass transfer and droplet shrinkage rates are found
from discretization of equations (12) and (17):
211 11
v v v 4nn nn s
D dm h R
, (A.20)
1
1
v21 1
1
4
n nnd d
n nwd d
R Rm
t R
. (A.21)
116
An algorithm of the numerical solution for the droplet evaporation period, based on
the above discussion, is given in Fig. A.1. The quantities " Res "and " " in this flowchart are
corresponding to the numerical residual and the required degree of convergence.
start
1n n
n
d
Initial guess
of R
v 1
. .20
n
kCalculation of m
eq A
,
0,1, ,
. .17 .19
n ni iEvaluation of a b
i N
eqs A A
, 1, ,0
. .15
n
d iEvaluation of T
i N N
eq A
1k
1k k
v
. .20
n
kCalculation of m
eq A
?
v v 1
n n
k km m
no
yes
v
. .21
nCalculation of m
eq A
v v
v
Resn n
n kn
m m
m
?
Resn
"
n
d
"Regula falsi or
secant method,
giving new R
no
yes
?
d crX X
yes
stop
no
0n
Figure A.1. Flowchart of numerical solution for period of droplet evaporation.
(droplet temperature profile is considered)
117
APPENDIX B
Numerical Solution for Second Stage of Droplet Drying
In the second drying stage, regions of crust and wet core are present in the dried
particle. For each region there is a partial differential equation of parabolic type, describing
energy conservation, and boundary conditions. Among these boundary conditions there are
equations that common for both particle regions. The temperature at the end of droplet
evaporation period is an initial condition for the second drying stage. Two additional
differential equations are included into this system: mass transfer rate from the crust-wet core
interface to the drying air and equation, establishing connection between the rate of wet core
shrinkage and rate of liquid evaporation.
The numerical solution of the above mentioned set of equations is complicated by the
presence of variable spatial domain for each particle region due to evaporating moving crust-
wet core interface and by unknown temperature of this interface.
Similarly to the droplet evaporation period, applying the Landau's transformation
(Moyano and Scarpettini, 2000) allows fixing the position of crust-wet core interface. For the
region of particle wet core:
1 1
1 1 1
, ;
, .i
t
r R
(B.1)
Consequently:
1, ,wc wcT r t T
1
1,
ir R
2 2
2 2 21
1,
ir R
1
1
.i
i
dR
t R d
For the region of particle crust:
118
2 2
2 2 2
, ;
, .i p i
t
r R R R
(B.2)
As a result:
2, ,cr crT r t T
2
1,
p ir R R
2 2
22 22
1,
p ir R R
2
2
1.i
p i
dR
t R R d
Owing to the performed transformations, the initial domain is subdivided in two fixed-
length regions corresponding to particle wet core ( 1 ) and particle crust ( 2 ), see Fig. B.1.
0 ( )iR t pR r
0 1 0 1
12
Figure B.1. Transformation of initial domain in two fixed-length regions
The transformation of equations (33) and (34) with the help of eq. (B.1) gives:
21 1 11
12 2 21 1 1
, , ,2, 0 1wc wc wci wc wc
i i i
T t T t T tdR t
t R t dt R t R t
(B.3)
11
1
1 2 1
,0, 0;
, , , 1.
wc
wc cr
T t
T t T t
(B.4)
In eq. (B.4) the condition of heat balance at the crust-wet core interface is omitted since its
transformation is discussed separately below.
The transformation of equations (30) and (31) using eq. (B.2) yields:
119
2 22
21
22
222
, ,21
,, 0 1
cr cri crp i
i p i
crcr
p i
T t T tdR tR R t
t dt R t R R t
T t
R R t
(B.5)
2 1 2
22 2
2
, , , 0;
,, , 1.
cr wc
crcrg cr
p i
T t T t
T tkh T T t
R R t
(B.6)
The initial condition for equations (B.3) and (B.6) is temperature profile at the end of
the droplet evaporation period transformed into spatial domains 1 and 2 in accordance with
eqs. (B.1) and (B.2). The equations (35) and (36) determining the relation between mass
transfer and droplet shrinkage rates and mass transfer rate from the crust-wet core interface
surface, do not need to be transformed since of time-dependence only.
The set of equations (B.3)-(B.6), (35) and (36) is solved by applying a fully implicit
finite difference scheme with fixed time-step (Moyano and Scarpettini, 2000). Let the spatial
domain 10 1 be subdivided into N equal intervals of 1 width each and 1, 1i i ,
where 0,1, ,i N . Let the spatial domain 20 1 be subdivided into M equal intervals
of 2 width each and 2, 2j j , where 0,1, ,j M . Also, let t be the fixed time step
for simulation and n denotes its number, 0,1,2,n .Therefore, 1,n
wc wciT T i n t and
2 ,n
cr crjT T j n t .
For region of particle wet core, discretization of eq. (B.3) gives:
1 1 11
1 111 21
11
1 1 1
1 12 21
1
2
2
2.
n n n n nn n
wc wc wc wc wci ii i i i in n
i i
n n n n
wc wc wc wci i i i
n
i
T T T TR Ri
t tR i R
T T T
R
(B.7)
Denoting
1212
1
n
wcn ii
n
i
tE
R
, (B.8)
120
11
11
2
n nn
i in ii n
i
i R R EF
iR
, (B.9)
and substituting these expressions into eq. (B.7), the next equation is derived:
1 1 11 1 1 1 1
1 11 2
n n n nn n n n nwc i i wc i wc i i wci i i i
T F E T E T F E T
, (B.10)
1 1i N .
For the left-end and right-end points of spatial mesh 10 1 , the boundary conditions eq.
(B.6) must be applied. The mesh point 0i is a problematic one because both 1 0 and
1
0,0wcT t
. Using the procedure described by Morton and Mayers (1994) (see Appendix
A), the following expression can be obtained:
1
1 1
1 10 1 0
6 1
1 6 1 6
nn n ni
wc wc wcn ni i
ET T T
A E
. (B.11)
Let the value of temperature at the crust-wet core interface, 1, 0,wc crT t T t , be a guessed
parameter. Then, implementing the Gauss elimination algorithm (Forsythe and Wasow,
1967), the temperature 1n
wc iT
can be written as:
1 11 1
1
n nn nwc i wc ii i
T d T e
. (B.12)
Combining the equations (B.10)-(B.12):
1 1 11 1
11 1
1 1 1 1 1 1 1 11 1
, , 1 11 2 1 2
n n n nn nwc i i in ni i i
i in n n n n n n ni i i i i i i i
T E F eE Fd e i N
E d E F E d E F
(B.13)
1
1 10 01 1 0
6 1,
1 6 1 6
nnn ni
wcn ni i
Ed e T
E E
. (B.14)
In this way, in given time step 1n the values of 1nid and 1n
ie are calculated according to
eqs. (B.13) and (B.14). After that, the values of 1n
wc iT
are estimated using eq. (B.12) in
reverse order, beginning from 1i N and finishing by 0i .
For the particle crust region, discretization of eq. (B.3) yields:
121
1 1 11
1 12 1 1 1
2 2
1 1 1
1 1
2 212
21
2
2.
n n n n nn ncr cr cr cr cri ij j j j j
n n n
i p i p i
n n n n
cr cr cr crj j j j
n
p i
T T T TR Rj
t t R j R R R R
T T T
R R
(B.15)
Denoting
11 nnp is R R
, (B.16)
122 1
2
n
cr jnj
n
tG
s
, (B.17)
11 1
2 2111 1
2 2
1
2
n nn n
i i jnj nn n
i
j R R s GH
s R j s
, (B.18)
and substituting these expressions into (B.15), the next equation is derived:
1 1 11 1 1 1 1
1 11 2
n n n nn n n n ncr j j cr j cr j j crj j j j
T H G T G T H G T
, 1 j M
(B.19)
Because it is assumed that the value of temperature at the crust-wet core interface, 0,crT t , is
a guessed parameter, only boundary condition at the particle outer surface is considered.
Discretization of the second equation in the set (B.6) gives:
1 1111 1 1
122
n nnn cr crcrn M M
g cr nM
T Tkh T T
s
,
1 1 1 1
21 12
n n n n
cr cr cr g crM M MT T Bi T T
. (B.20)
Here 1n
crBi
is Biot number of the crust particle crust for time-step 1n :
1 11
1
n nn
cr n
cr
h sBi
k
. (B.21)
Substituting eq. (B.20) into eq. (B.19), written for j M , yields:
1 11 1 11 21
11 1 12
2 2
1 2
n n nn n nn cr M cr cr M M gM M
cr M nn n nM cr M M
T G T Bi H G TT
G Bi H G
. (B.22)
122
After applying the Gauss elimination algorithm (Forsythe and Wasow, 1967), 1n
cr iT
is
written as:
1 11 1
1
n nn ncr j wc jj j
T f T g
(B.23)
Then, from (B.19), (B.22) and (B.23) it follows that:
1 1 11 1
11 1
1 1 1 1 1 1 1 11 1
, , 2 11 2 1 2
n n n nn ncr j j jj j jn n
j jn n n n n n n nj j j j j j j j
T G H gG Hf g j M
G f G H G f G H
,
(B.24)
11 11 11 11 1 1 01 1
1 11 11 1
, ,1 2 1 2
n nn nn ncr crn n
n n
T G H TG Hf g
G G
(B.25)
1 1 1 1 12 11 1
11 1 1 1 12 1
2 20,
1 2 2
n n n n n ncr cr M M g M Mn n M
M M nn n n n nM cr M M M M
T Bi H G T G gf g
G Bi H G G f
. (B.26)
In this way, at given time step 1n the values of 1njf and 1n
jg are calculated according to
eqs. (B.24)-(B.25). After that, the values of 1n
cr jT
are estimated using eq. (B.23) in reverse
order, beginning from j M and finishing by 1j .
There is also a supplementary condition, which must be satisfied for the both crust and
wet core regions of the particle. This condition is the energy balance at the crust-wet core
interface. Such balance is given by the first equation in boundary conditions (31) and by the
second expression in eq. (34):
2 2v4 4 ,cr wc
cr i wc i fg i
T Tk R k R h m r R t
r r
. (B.27)
The above equation can be rewritten:
v20 0
, , , ,lim lim
4cr i cr i wc i wc i
cr wc fgr r
i
T R t T R r t T R r t T R t mk k h
r r R
.
(B.28)
Discretization of eq. (B.28) gives:
1 11 1* *1 1 v2 10
1 2* 10 * 12 1
, 1 1 , 1
4
n nn nn n fgcr cr wc wcN
cr wc nn N ni i
h mT n t T T T n tk k
s R R
,
(B.29)
123
where *1 and *
2 are dimensionless distances from the crust-wet core interface. These
parameters are connected by the following relationship:
1* * 11 2
n niR s . (B.30)
The above expression is obtained from the next demand: in eq. (B.29) the positions
corresponding to the temperature values *11 , 1wcT n t and *
2 , 1crT n t must
be equidistant from the both sides of the crust-wet core interface in the space domain r (see
Fig. B.2).
1n
iR
1
1 1
n
ir R 12 2
nr s
r
1
1N N0 1
2
*1
*1 2r r
M
pR
0
0
Figure B.2. Determination of *1
As a result, to check the satisfaction of the condition (B.29) by taking
1*2 1, 1
n
cr crT n t T , it must be assumed *2 2 in (B.30) and then the value of
*1 can be calculated:
1*
1 2 1
n
n
i
s
R
. (B.31)
After that, the value of *11 , 1wcT n t must be obtained with the help of interpolation
in the domain 10 1 .At the next step, the residual, Res, can be estimated using eq. (B.29):
1 11 *1 v1
1 2* 11
11*
1 2 0* 102
1 , 1
4Res 1
, 1
n nnn fgwc wcN
wc nN ni in
nn cr cr
cr n
h mT T n tk
R R
T n t Tk
s
. (B.32)
When the obtained residual is smaller than the required degree of accuracy, then the necessary
calculation precision is considered to be attained.
124
Similarly, taking 1*1 1
1 , 1n
wc wc NT n t T
leads to *
1 1 . In this case
from eq. (B.30):
1
*2 1 1
n
in
R
s
. (B.33)
Then, the value of *2 , 1crT n t must be obtained with the help of interpolation in the
domain 20 1 , and after that the residual value can be estimated from eq. (B.32). The
interpolation can be performed using one of well-known interpolation methods, e.g. linear,
quadratic cubic spline or cubic piecewise.
Numerical expressions for evaluation of mass transfer and wet core shrinkage rates are
obtained by discretization of equations (35) and (36):
1
1
v21 1
1
4
n nni i
n nwwc i
R Rm
t R
, (B.34)
1 111 ,
v 1 111 v
v1 2
4ln
4
n nneffn g v iv w g p i
n nNn
g crn Mgw D p
p pD M p R Rm
s pT p m TTM h R
. (B.35)
The algorithm of numerical simulation of droplet drying in the second stage is
presented in Fig. B.3.
125
start
1n n
n
i
Initial guess
of R
v 1
. .34
n
kCalculation of m
eq B
1k
0n
, , ,
0,1, , 1
1, 2, ,
. .13 , .14 ,
.24 .26
n n n ni i j j
Evaluation of
d e f g
i N
j M
eqs B B
B B
. .35
n
wc NCalculation of T
eq B
1, ,0
, ,1
. .12 , .23
n
wc i
n
cr j
Evaluation of T
i N and
T j M
eqs B B
1k k
v
. .34
n
kCalculation of m
eq B
?
v v 1
n n
k km m
no
yes
Res
. .32
nCalculation of
eq B
?
Resn
"
n
i
"Regula falsi or
secant method,
giving new R
no
yes
?
p fX X
yes
stop
no
Figure B.3. Flowchart of numerical solution for droplet drying in second drying stage.
126
APPENDIX C
Calculation of Thermophysical Properties for Skim and Whole Milk
Droplets
a) Thermal conductivity
Thermal conductivity of the milk droplet is equal to (Chen and Peng, 2005):
d serial parallelk 0.5 k k , (C.1)
where kserial and kparallel are thermal conductivities, calculated with the help of serial and
parallel models. Correspondingly,
serial w w p p f f c c a ak ε k ε k ε k ε k ε k , (C.2)
pw c afparallel
w p f c a
εε ε εεk
k k k k k . (C.3)
In the above equations, is a volume fraction of component and superscripts "w", "p", "f",
"c" and "a" are corresponding to water, protein, fat, carbohydrate and ash components of milk
droplet. The thermal conductivities of milk components are calculated as given by Chen and
Peng (2005):
6 2w
6 2p
7 2f
6 2c
6 2a
k 0.57109 0.0017625 T 6.7036 10 T
k 0.17881 0.0011958 T 2.7178 10 T
k 0.18071 0.0027604 T 1.7749 10 T
k 0.20141 0.0013874 T 4.3312 10 T
k 0.32962 0.0014011 T 2.9069 10 T
(C.4)
Here T is temperature of milk droplet in Celsius degrees.
b) Density
Density of milk droplet can be estimated as follows (Chen and Peng, 2005):
dpw c af
w p f c a
1ρ
cc c cc
ρ ρ ρ ρ ρ
, (C.5)
127
where c is mass fraction of component. The densities of the components are given by (Chen
and Peng, 2005):
w
p
f
c
a
ρ 997.18 0.0031439 T 0.0037574 T
ρ 1329.9 0.5185 T
ρ 925.59 0.41757 T
ρ 1599.1 0.31046 T
ρ 2423.8 0.28063 T
(C.6)
c) Specific heat
Specific heat of milk droplet is evaluated according to Chong and Chen (1999):
6 2p,smc 1.09 0.00352 T 3.54 10 T , (C.7)
6 2p,wmc 1.21 0.00306 T 2.74 10 T (C.8)
The information about composition of skim and whole milk powders, used in
experimental studies of Lin and Chen (2002) and utilized for the validation of droplet drying
kinetics model in the present work, is given in Table C.1.
Table C.1. Gross composition of milk powders, used in experiments of Lin and Chen (2002)
Component Mass fraction, %
Skim milk Whole milk
Fat 0.6 26.5 Total protein 36.5 28.0 Lactose 49.8 36.8 Mineral 9.3 5.9 Moisture 3.8 2.8
128
APPENDIX D
Derivation of Non-Linear Pressure and Vapour Fraction Partial
Differential Equations
In order to obtain the explicit dependences of the pressure and vapour fraction on time
and space coordinates, the set of equations (5.3)-(5.9) is subjected to transformations given
below.
The equation of mass conservation (5.3) can be rewritten as:
zυρρ 0
t z
d
d
. (D.1)
Substituting the equation of state (5.9) into eq. (D.1), yields:
zυp M pM TM p pM 0
t t T t z
d d d
d d d
; (D.2)
Performing the same operation on eq. (5.8) results in:
zp
T T p pυ
t Mc p t z
d
d
, (D.3)
or
p
T T p
t Mc p t
d d
d d
. (D.4)
Differentiation of eq. (5.4) by z and disregarding changes of permeability, viscosity and
porosity along the crust pore:
2z k
β 2
υ B p
z με z
. (D.5)
Substituting eqs. (D.4) and (D.5) into eq. (D.2) gives:
2kβ 2
p
B Mp M p pM p p 0
t t c t με z
d d d
d d d
. (D.6)
For air-vapour mixture it can be shown that
a v
v v v a
M MM
M 1 ω ω M
, (D.7)
129
and therefore
a v a v v
2
v v v a
M M M M ωM
t tM 1 ω ω M
dd
d d
, (D.8)
2a v v
a v
M M ωMM
t M M t
dd
d d
. (D.9)
Similarly, it can be derived that:
2a v v
a v
M M ωMM
z M M z
. (D.10)
Substitution of eq. (D.10) into eq. (D.6) yields:
2a v v k
β 2a v
M M ω B1 p pM
γ p t M M t με z
dd
d d
, (D.11)
where p vγ c / c is ratio of specific heats.
The equation (D.11) establishes dependence of both vapour fraction and pressure of
air-vapour mixture on time and space coordinates. In order to solve this equation, a
supplementary relationship between p and ωv is needed. For this purpose, the equation of
mass diffusion (5.6) is rewritten as follows:
v vv
ω ωρ ρD
t z z
d
d
. (D.12)
Then, by utilizing the equation of state (5.9), it can be found that:
v vv
ω ωpM pM D
t z z
d
d
. (D.13)
Differentiation of (5.7) gives:
1.815v
0 20
D T 1 p2.302 10 p
z T p z
,
or
vv
D 1 pD
z p z
. (D.14)
Then, using eqs. (D.10) and (D.14), the equation (D.13) can be finally rewritten:
22v v a v v
v 2a v
ω ω M M ωD M
t z M M z
d
d
. (D.15)
130
APPENDIX E
Two-Dimensional Axisymmetric Modelling of Drying Air Flow
The continuous phase (drying air) is treated using Eulerian approach and k-ε
turbulence model. The corresponding conservation equations for two-dimensional
axisymmetric air flow in cylindrical coordinates are as follows (Zhou, 1993):
- continuity
c
1U r V S
t x r r
(E.1)
- axial momentum
2e e e
e c gp,x
1 p U 1 U UU U r UV r
t x r r x x r r r r x x1 V
r g US ΣFr r x
(E.2)
- radial momentum
2e e e
ee c gp,r2
1 p V 1 V UV UV r V r
t x r r r x r r r r x r2 V1 V
r VS ΣFr r r r
(E.3)
- energy conservation
e er h
h h
1 h 1 hh Uh r Vh r q S
t x r r x r r r r
(E.4)
- turbulent kinetic energy
e ek b
k k
1 k 1 kk Uk r Vk r G G
t x r r x r r r r
(E.5)
- dissipation rate of turbulence kinetic energy
e e1 k 2
1 1U r V r C G C
t x r r x r r r r k
(E.6)
131
- species conservation
e v e vv v v c
Y Y
Y Y1 1Y UY r VY r S
t x r r x r r r r
. (E.7)
In eqs. (E.5) and (E.6) the production of turbulence kinetic energy due to mean velocity
gradients is equal to:
2 2 2
k T
U V U VG 2 2
x r r x
. (E.8)
The production of turbulence kinetic energy due to buoyancy is given by:
TbG
T
Tg
x
, (E.9)
where is coefficient of thermal expansion:
1
pT
, (E.10)
The universal standard constants are: 1C 1.44 , 2C 1.92 , k h Y Tσ 0.9 and
1.3 . The effective viscosity, e , is found as:
e T , (E.10)
where T is turbulent viscosity
2
T
kC
. (E.11)
In the above expression C 0.09 .
132
APPENDIX F
Drag, Added Mass and Buoyancy Forces
The drag force acting on droplet/particle is found as follows (Fan and Zhu, 1998):
2D D p p pF C U U U U d
8
, (F.1)
where the drag coefficient, DC , is evaluated according to empirical correlations (Zhou, 1993):
p p
2/3D p p
p
p
24 / Re , Re 1
24 1C 1 Re , 1 Re 1000
Re 6
0.44, Re 1000
(F.2)
Here the Reynolds number, pRe , for a droplet/particle is expressed by:
p p
p
d U URe
μ
. (F.3)
The components of added mass (virtual-mass) force, required to accelerate the gas
surrounding the droplet/particle, are determined as follows (Crowe et al., 1998):
3p p
A
d UUF ρ
12 t t
dD
D d
(F.4)
where vmC is the virtual-mass coefficient and i =1,2,3 denotes the axial, radial and tangential
directions in cylindrical coordinate system.
The buoyancy force is given by:
3p
B
dF g
6
. (F.5)
133
APPENDIX G
Particle Flow and Statistics Calculated by 2D Axisymmetric Spray Drying
Model
(a) (b) (c) (d)
Figure G.1. Snapshots of droplet/particle positions (shown by dots) for flow times of:
(a) 0.5 s; (b) 10 s; (c) 20 s; and (d) 30 s. Droplet-droplet interactions are
considered
The particle statistics corresponding to the particle flow shown in Fig. G.1 are summarized in
Table G.1 on the next page.
134
Table G.1. Summary of predicted particle parameters at different times of transient spray
drying (case 1 – droplet-droplet interactions are taken into consideration, case 2 –
droplet collisions are neglected)
Flow time Case dp,min, μm dp,max, μm nparcels nparticles mp,tot, kg
0.5 s 1 7.52 204.32 8595 9.739·107 3.259·10-3
2 7.52 138.00 9965 5.772·108 3.380·10-3
10 s 1 7.52 234.54 40324 2.455·108 8.508·10-3
2 7.52 138.00 61154 4.063·109 1.836·10-2
20 s 1 7.52 276.33 43700 2.617·108 9.093·10-3
2 7.52 138.00 68429 5.496·109 2.134·10-2
30 s 1 7.52 257.20 44461 2.715·108 9.354·10-3
2 7.52 138.00 72087 6.210·109 2.250·10-2
135
APPENDIX H
Three-Dimensional Modelling of Drying Air Flow
The continuous phase (drying air) is treated using Eulerian approach and k-ε turbulence
model. The conservation equations for three-dimensional air flow in tensor form are as
follows (Zhou, 1993):
- continuity
j cj
u St x
(H.1)
- momentum
jii j i e i pi c gp
j i j j i
uupu u u g U S F
t x x x x x
(H.2)
- energy conservation
ej r h
j j h j
hh u h q S
t x x x
(H.3)
- species conservation
e vv j v c
j j Y j
YY u Y S
t x x x
. (H.4)
- turbulent kinetic energy
ej k b
j j k j
kk u k G G
t x x x
(H.5)
- dissipation rate of turbulence kinetic energy
ej 1 k 2
j j j
u C G Ct x x x k
(H.6)
The production of turbulence kinetic energy due to mean velocity gradients is equal to:
k TG ji i
j i j
uu u
x x x
. (H.7)
The production of turbulence kinetic energy due to buoyancy is given by:
TbG j
T j
Tg
x
, (H.8)
136
where is coefficient of thermal expansion:
1
pT
, (H.9)
The universal standard constants are: 1C 1.44 , 2C 1.92 , k h Y T 0.9 and
1.3 . The effective viscosity, e , is found as:
e T , (H.10)
where T is turbulent viscosity
2
T
kC
. (H.11)
In the above expression C 0.09 .
137
APPENDIX I
Modelling of Particle-Particle Collisions Using Hard-Sphere Approach
Hard-spheres approach considers binary (one-on-one) collisions between spherical
particles and assumes that only impulsive interaction forces are important while other finite
forces are negligible during the collision. In the present study the model of Hoomans (1999) is
adopted.
The normal unit vector between two colliding particles a and b (see Fig. I.1) is given
by:
a b
a b
r rn
r r
(I.1)
Figure I.1 Scheme of binary particle collision (Hoomans, 1999).
The point of origin of normal vector is the contact point. Before the collision, the spheres with
radii Ra and Rb and masses ma and mb have translation velocity vectors a,0v
and b,0v
. Rotation
of particles is disregarded in the current model. Using the Newton's Second and Third Laws
of Motion, the following set of equations can be derived:
a a a,0
b b b,0
m v v J
m v v J
. (I.2)
138
The impulse vector J
is given by:
ct
ab
0
J F td
, (I.3)
where tc is collision duration.
The relative velocity at the contact point is defined as follows:
ab a,c b,cv v v
. (I.4)
Since the particle rotation is neglected:
ab a bv v v
. (I.5)
Then, the tangential unit vector between two colliding particles can be derived as follows:
ab,0 ab,0
ab,0 ab,0
v n v nt
v n v n
(I.6)
Using the set of equations (I.2) and eq. (I.5) it can be found that:
ab ab,0a b
1 1v v J
m m
(I.7)
Denoting
2a b
1 1B
m m , (I.8)
then
ab ab,0 2v v B J
. (I.9)
To close the model equations, constitutive definitions of three parameters are introduced:
- coefficient of normal restitution ( 0 1e ):
ab ab,0v n v ne
; (I.10)
- coefficient of dynamic friction ( 0 );
n J n J
. (I.11)
- coefficient of tangential restitution ( 00 1 ):
ab 0 ab,0n v n v
. (I.12)
Combining (I.9) and (I.10) yields:
139
ab,0n
2
v nJ J n 1
Be
(I.13)
Regarding the tangential component of impulse vector, two types of collisions can be
distinguished: sticking and sliding. When tangential component of relative velocity is
sufficiently high compared to the coefficients of friction and tangential restitution (so that
gross sliding occurs throughout the whole duration of the contact), the collision is of the
sliding type. Otherwise, there is a sticking-type collision.
The following criterion can be used to determine the type of collision:
- sliding collision
0 ab,0
n 1
1 v t<
J B
, (I.14)
- sticking collision
0 ab,0
n 1
1 v t
J B
, (I.15)
where 1 2
7B B
2 .
Then, for sticking collision the tangential component of impulse vector is given by:
ab,0 ab,0t 0 0
1 1
n v v tJ J t n J 1 1
B B
. (I.16)
In the case when collision of sliding type occurs:
t nJ J . (I.17)
Now, the total impulse vector is calculated as follows:
n tJ J n J t
, (I.18)
and the post-collision velocities can be determined from the set of equations (I.2).
The energy dissipation during particle-particle collision can be obtained as follows:
c ct t
diss,tot diss,n diss,t ab,n n ab,t t
0 0
E E E v J v Jd d
. (I.19)
The energy dissipation in normal direction is given by:
140
2ab,n,0 2
diss,n2
vE 1
2Be . (I.20)
For tangential direction, the energy dissipation is determined as follows:
- sticking collision
2ab,t,0 2
diss,t 01
vE 1
2B . (I.21)
- sliding collision
t n ab,0 1 n
1J J v t J
2B
. (I.22)
Table I.1. Summary of calculated particle statistics of transient 2D axisymmetric spray drying
simulations (case 1 – DD+PP model, case 2 – DD model)
time case nparcels nparticles mp,tot,
kg
dp,max,
μm
dp,min,
μm
dp,10,
μm
dp,20,
μm
dp,30,
μm
0.5
s
1 8892 4.54·108 3.11·10-3 642.05 7.52 14.02 18.08 23.56
2 8532 0.92·108 3.08·10-3 200.90 7.52 24.81 32.03 40.02
10 s 1 49000 3.03·109 1.46·10-2 312.30 7.52 13.08 16.43 20.97
2 38496 0.25·109 0.784·10-2 204.82 7.52 23.76 31.18 38.95
20 s 1 58653 4.35·109 1.80·10-2 464.76 7.52 12.67 15.71 19.93
2 43438 0.27·109 0.846·10-2 207.58 7.52 23.67 31.22 39.21
30 s 1 62280 4.97·109 1.93·10-2 309.78 7.52 12.53 15.44 19.51
2 44823 0.26·109 0.866·10-2 203.53 7.52 23.85 31.53 39.75
141
Table I.2. Summary of calculated particle statistics of transient 3D spray drying simulations
(case 1 – DD+PP model, case 2 – DD model)
time case nparcels nparticles mp,tot,
kg
dp,max,
μm
dp,min,
μm
dp,10,
μm
dp,20,
μm
dp,30,
μm
1 s 1 3168 5.83·108 4.88·10-3 292.98 7.52 15.94 20.12 25.20
2 2147 0.44·108 3.53·10-3 334.08 7.52 39.18 45.42 53.40
10 s 1 10348 2.36·109 1.63·10-2 294.23 7.52 15.68 19.31 23.63
2 6020 0.16·109 0.698·10-2 331.62 7.52 34.42 38.55 43.53
20 s 1 13190 3.40·109 2.14·10-2 306.92 7.52 15.24 18.71 22.90
2 8275 0.22·109 0.926·10-2 429.94 7.52 34.35 38.17 42.89
30 s 1 14292 3.78·109 2.36·10-2 277.23 7.52 15.32 18.76 22.87
2 9823 0.27·109 1.12·10-2 360.31 7.52 33.99 37.83 42.90
The statistical mean diameters, dp,10, dp,20 and dp,30 shown in Tables I.1-I.2, are
determined as follows (Fan and Zhu, 1998):
- arithmetic mean diameter:
N
0p,10
N
0
b b b
d
b b
f d
f d
(I.23)
- surface mean diameter:
2N
2 0p,20
N
0
b b b
d
b b
f d
f d
(I.24)
- volume mean diameter:
3N
3 0p,30
N
0
b b b
d
b b
f d
f d
(I.25)
In the eqs. (I.23)-(I.25) N bf is number density function of particles with size b.
142
APPENDIX J
Equations of Reynolds Stress Turbulence Model
The closed system of Reynolds stress transport equations in tensor form can be written as
(Zhou, 1993):
/ / / / / / / /
1
2 3
2
3
2 2 2;
3 3 3
i j k i j s k l i j i j ijk k l
ij ij k ij ij b ij ij ij
ku u u u u C u u u u C u u k
t x x x k
C P G C G G P G
(J.1)
/ / / / / / / / / /
/ / / / / /2 2
1 2 3 ;
ii k i sT k l i i j k
k k l k k
ii T i T k T i
k
uk Tu T u u T C u u u T u u u T
t x x x x x
ug T C u T C u T C g T
k x
(J.2)
/ / / / / / / /2 2 2 212 ;k T k l k
k k l k
k TT u T C u u T u T T
t x x x x R k
(J.3)
/ / ;k s k l k bk k l
k kk u k C u u G G
t x x x
(J.4)
/ /
2
1 2 31 .k k l k b fk k l
ku C u u C G G C R C
t x x x k k
(J.5)
Here the stress production is given by:
/ / / /j iij i k j k
k k
u uP u u u u
x x
. (J.6)
The buoyancy production is calculated as follows:
/ / / /
ij i j j iG g u T g u T , (J.7)
The production of turbulence kinetic energy due to mean velocity gradients is equal to:
143
k TG ji i
j i j
uu u
x x x
. (J.8)
The production of turbulence kinetic energy due to buoyancy is determined by:
TbG j
T j
Tg
x
, (J.9)
where T 0.9 .
The ratio of turbulence kinetic energy production due to swirl to total production is equal to:
2
2 2f
k rR
r r
(J.10)
The model constants are given in Table J.1
Table J.1. Empirical constants used in the Reynolds stress turbulence model
Cs C1 C2 C3 C1T C2T C3T R CsT CT C Cε1 Cε2 Cε3
0.24 2.2 0.55 .55 3.0 0.5 0.5 0.8 0.11 0.13 0.15 1.44 1.92 0.8
תקציר
ייבוש טיפות המכילות פאזה מוצקה או מומסת עד להפיכתן לחלקיקי מוצק קרוי יבוש תרסיסים. התהליך
להיות בוש יכול החומר המוזרם לתא היי הזורם לתא עם מדיה חמה ויבשה.\הטיפות\מתבצע כתוצאה מהזנת התרסיס
התוצר הסופי המתקבל בסוף התהליך הינו או תמיסה. תרחיף,רסק, )slurry( תערובת דלילה של מים וחומר מוצק
תהליך זה של יבוש תרסיסים נפוץ מאוד ונמצא בשימוש רחב בתעשיות .גרגרים, אגלומרטים או אבקה בצורת
ביוכימית. בעבודה זו פותח מודל תיאורטי מתקדם המית וכיה התעשייהפרמצבטיקה, הביניהם תעשיית המזון, , רבות
לאפיון מקיף של יבוש תרסיסים, תוך חלוקת המחקר לחמישה תתי נושאים נרחבים.
המצויה המעבר חום וחומר בתהליך יבוש תרסיסים תלוי בקינטיקת תהליך הייבוש של טיפה בודד
אשר תאם לתוצאות הטיפה בודדשל שהייבו תהליך לתיאורפותח מודל תיאורטי נרחב לצורך כך. בתרסיס
ומחקרים תיאורטיים בפרסומים מדעיים. מודל זה לוקח בחשבון את פרופיל הטמפרטורה של חלקיק מיובש ומחשב
המודל יכול לחזות את הופעתם של מאמצים ,את פילוג הלחץ ופרקציית האדים בחללים בין החלקיקים. בנוסף
הוביל לסדק או שבר בזמן תהליך הייבוש ובכך לפגום במוצר המוגמר.תרמיים ומכאניים אשר יכולים ל
. מודל זה מבוסס ממדית-בשלב הבא נלמד ונחקר תהליך הייבוש של התרסיס במערכת אקסיסימטרית דו
מול פרסומים ועבודות שונות בספרות. השפעת האינטראקציה בין וואושש ונבדק. תחזיות המודל CFDעל טכניקת
בזמן כתלות על ההתנהגות החזויה של התרסיס והאויר היבש, נבדקה והחזרותהתחברות - גון הצמדותטיפה לטיפה כ
)transient. (
ובמצב מתמיד נבדקתחזיות המודל . מדימ-תלת יתיאורטממדית, פותח מודל -בהתבסס על האנליזה הדו
בנוסף בוצעה השוואה בין תחזיות הסימולציות תוצאות ניסוייות ופרסומים שונים בספרות. השוואתם לע"י וואושש
בהתחשב באינטראקציות בין טיפה לטיפה.ועבור זמן מעבר והן עבור מצב מתמיד הנומריות הדו והתלת ממדיות הן
תנועת החלקיקים ועלזרימה ה השד עלבכדי ללמוד את השפעת האינטראקציה בין חלקיק לחלקיק
תנגשויות בין חלקיק למשנהו בתרסיס היבש. השפעתה של האינטראקציה בתהליך זה, הוצע מודל המתבסס על הה
הן בין טיפה לטיפה והן בין חלקיק לחלקיק בתהליך הייבוש בזמן מעבר על מסלול התנועה של החלקיק נבדקה גם
.יותלת ממד ידו ממדבמודלים הכן
and Reynolds stress model לבסוף נחקרה השפעתם של מודלים שונים של טורבולנציה ( k ε על (
.יתלת ממדוהן עבור ה ידו ממדהאופני הזרימה הן עבור
תרם משמעותית להבנה זה שימושית. מחקר מבחינה תיאורטית והן מבחינה כפולה, הן המחקר חשיבות
תקני ייבוש תרסיסים יכולים לשמש לעזר רב בתכנון של ממחד והמודלים שפותחו תרסיסיםשל ייבוש השל תהליך
מאידך. שימוש במודלים יכול להקל בפיתוח מתקני יבוש ומציאת חדשים עבור מוצרים שונים בתעשיות שונות
חסכון באנרגיה וקבלת מוצר סופילתוך התייחסות התזה קיימים יייבשלפעולה אמינה של מ םהתנאים האופטימאליי
. האיכות הנדרשת עם
: מילות מפתח
שבירה, קינטיקת ייבוש, תבנית זרימה, מעבר חום וחומר, מאמץ מכני, מודל /ימה חישובית, סדיקהזר תדינאמיק
פאזית, חלקיק רטוב-מתמטי, חלקיק, טיפה בודדה, ייבוש תרסיסים, זרימה דו
העבודה נעשתה בהדרכת
פרופ' אירנה בורדה פרופ' אבי לוי
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September 2008 אלול תשס''ח
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ייבוש בוצות במייבשי התזה
מחקר לשם מילוי חלקי של הדרישות לקבלת תואר "דוקטור לפילוסופיה" מאת
מז'ריצ'ר מקסים
הוגש לסינאט אוניברסיטת בן גוריון בנגב
September 2008 אלול תשס''ח
באר שבע