dynamic model of a small once-through boiler
TRANSCRIPT
TtKK-tT--55
LAPPEENRANNAN TEKNILLINEN KORKEAKOULU UDK 536.717 :LAPPEENRANTA UNIVERSITY OF TECHNOLOGY 621.18 :
519.876
TIETEELLISIA JULKAISUJA RESEARCH PAPERS
55
TIMO TALONPODCA
Dynamic Model of a Small Once-Through Boiler
Thesis for the degree of Doctor of Technology to be presented with due permission for public examination and criticism in Auditorium 10 at Lappeenranta University of Technology (Lappeenranta, Finland) on the 29th of November, 1996, at noon.
LAPPEENRANTA1996 ttSTRSUBONOrBis DOCUMBiT B WUMJTH)
ISBN 951-764-088-9 ISSN 0356-8210
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Abstract
Lappeenranta University of Technology Research Papers 55
Timo TalonpoikaDynamic Model of a Small Once-Through Boiler Lappeenranta, 1996.
86 pages, 35 figures, 4 tables
ISBN 951-764-088-9, ISSN 0356-8210 UDK 536.717 : 621.18 : 519.876
Key words: dynamic modelling, once-through boiler
In this study, a model for the unsteady dynamic behaviour of a once-through counter flow boiler that uses an organic working fluid is presented. The boiler is a compact waste-heat boiler without a furnace and it has a preheater, a vaporiser and a superheater. The relative lengths of the boiler parts vary with the operating conditions since they are all parts of a single tube. The present research is a part of a study on the unsteady dynamics of an organic Rankine cycle power plant and it will be a part of a dynamic process model. The boiler model is presented using a selected example case that uses toluene as the process fluid and flue gas from natural gas combustion as the heat source. The dynamic behaviour of the boiler means transition from the steady initial state towards another steady state that corresponds to the changed process conditions.
The solution method chosen was to find such a pressure of the process fluid that the mass of the process fluid in the boiler equals the mass calculated using the mass flows into and out of the boiler during a time step, using the finite difference method. A special method of fast calculation of the thermal properties has been used, because most of the calculation time is spent in calculating the fluid properties.
The boiler was divided into elements. The values of the thermodynamic properties and mass flows were calculated in the nodes that connect the elements. Dynamic behaviour was limited to the process fluid and tube wall, and the heat source was regarded as to be steady. The elements that connect the preheater to the vaporiser and the vaporiser to the superheater were treated in a special way that takes into account a flexible change from one part to the other.
The model consists of the calculation of the steady state initial distribution of the variables in the nodes, and the calculation of these nodal values in a dynamic state. The initial state of the boiler was received from a steady process model that is not a part of the boiler model. The known boundary values that may vary during the dynamic calculation were the inlet temperature and mass flow rates of both the heat source fluid and the process fluid.
A brief examination of the oscillation around a steady state, the so-called Ledinegg instability, was done. This examination showed that the pressure drop in the boiler is a third degree polynomial of the mass flow rate, and the stability criterion is a second degree polynomial of the enthalpy change in the preheater. The numerical examination showed that oscillations did not exist in the example case.
The dynamic boiler model was analysed for linear and step changes of the entering fluid temperatures and flow rates. The problem for verifying the correctness of the achieved results was that there was no possibility to compare them with measurements. This is why the only way was to determine whether the obtained results were intuitively reasonable and the results changed logically when the boundary conditions were changed.
The numerical stability was checked in a test run in which there was no change in input values. The differences compared with the initial values were so small that the effects of numerical oscillations were negligible. The heat source side tests showed that the model gives
' results that are logical in the directions of the changes, and the order of magnitude of the time scale of changes is also as expected. The results of the tests on the process fluid side showed that the model gives reasonable results both on temperature changes that cause small alterations in the process state and on mass flow rate changes causing veiy great alterations. The test runs showed that the dynamic model has no problems in calculating cases in which the temperature of the entering heat source suddenly goes below that of the tube wall or the process fluid.
IV
V
ACKNOWLEDGEMENTS
This work is a continuation and a part of the process modelling that has been done in the ORC- project, which is a part of the research in high speed technology in the Department of Energy Technology at Lappeenranta University of Technology. I have participated in the research since 1993 and the present work was actually started in August 1995.
I wish to express my sincere thanks to the research group and all my colleagues and friends who by encouragement and discussions have helped me do this work.
I am deeply grateful to my supervisor and long time superior Prof. Pertti Sarkomaa for his push and encouragement to have this work done, as well as for the guidance in the final stages of this work. During the years that I have worked at the Department of Energy Technology, the many discussions with him have been most fruitful for a young researcher to learn to see the important things and to have the right angle of examination.
Special thanks are also due to Ass. Prof. Jaakko Laijola, who has been the leader of the research group in which this study is made. I apologise that I was not able to make "the simple boiler model" he wanted; instead, it became this thesis.
The contribution of my colleague Juha Honkatukia has been most important. He has made a huge effort in programming the static process models and checked and tested those sometimes wild ideas that have arisen in many long discussions with him. I also wish the best luck to Marko Maunula who is making the dynamic process model that uses this boiler model.
I wish to thank my assistants and colleagues who during this autumn helped me by handling the practical things in the teaching courses, so that I could concentrate on my thesis. Special thanks to Satu Ranta for the work she did with the frustrating manipulation of the many figures in my work.
Thanks also to our children who showed special interest and encouraged me to finish my thesis.
Very special thanks belong to my wife Sinikka who during all these years has supported and encouraged me in so many ways. She has also taken care of our home so that I have been able to concentrate on this work.
Lappeenranta, November 13th, 1996.
Timo Talonpoika
VI
Vll
CONTENTS
AbstractAcknowledgements Contents List of Tables List of Figures Nomenclature
1 Introduction.............................................................................................................................. 11.1 Organic Rankine Cycle..................................................................................................... 11.2 Design of the ORC cycle..................................................................................................31.3 The present study..............................................................................................................4
2 Dynamic modelling .................................................................................................................. 52.1 The present study..............................................................................................................52.2 Governing primary equations............................................................................................8
2.2.1 Mass conservation................................................................................................... 82.2.2 The energy equation.............................................................................................. 102.2.3 The equation of linear momentum ........................................................................11
2.3 Different approaches to dynamic modelling ................................................................. 122.3.1 Power plant modelling...........................................................................................122.3.2 Modular simulation programs............................................................................... 142.3.3 Once-through waste heat boiler model................................................................. 16
3 Boiler model description ........................................................................................................183.1 General............................................................................................................................183.2 Special features of the dynamic boiler model ............................................................... 193.3 Defining the nodes and elements in the boiler...............................................................203.4 Element types and numbering of nodes and elements.................................................. 213.5 General description of unknowns in interior nodes........................................................223.6 Calculation of thermodynamic properties...................................................................... 24
4 Steady state calculation......................................................................................................... 274.1 Input values for the steady state calculation................................................................. 274.2 Pressure loss ...................................................................................................................27
4.2.1 Pressure loss and stability in a once-through boiler............................................274.2.2 Pressure loss coefficient ....................................................................................... 374.2.3 Pressure loss on the heat source passage ............................................................38
4.3 Effect of temperature dependence of fluid properties..................................................394.4 Calculation of the initial state ........................................................................................ 42
4.4.1 General ..................................................................................................................424.4.2 Counter flow heat exchanger................................................................................434.4.3 Calculation of the preheater..................................................................................464.4.4 Calculation of the superheater..............................................................................484.4.5 Calculation of the mixed elements and the vaporiser.......................................... 494.4.6 Calculation of the process fluid mass and the tube wall temperatures...............51
5 Dynamic calculation.............................................................................................................. 535.1 General ........................................................................................................................... 535.2 Analysis of thermodynamic variables and their functional interdependency................545.3 Iteration of the pressure level ........................................................................................ 56
5.3.1 Nodal enthalpies of the process fluid .................................................................. 595.3.2 Nodal temperatures and pressures and process fluid mass in the elements....... 605.3.3 Tube wall temperature.......................................................................................... 635.3.4 Temperature of the heat source fluid.................................................................... 65
6 Calculation program.............................................................................................................. 67
7 Test runs of the dynamic boiler model..................................................................................697.1 Input values from the steady state off-design program ................................................ 697.2 Test runs.........................................................................................................................71
8 Results and discussion........................................................................................................... 728.1 Corrected heat transfer coefficients and calculated constants ..................................... 728.2 Stability of the boiler at the steady state....................................................................... 728.3 Discussion of the results of the test runs...................................................................... 748.4 Discussion of the simplifications....................................................................................80
9 Conclusions ........................................................................................................................... 829.2 Further development of the dynamic model ..................................................................83
References...................................................................................................................................84
viii
Appendix A: Appendix B:
Analysis of thermodynamic variables and their functional interdependency Test runs
LIST OF TABLES
4.1 Temperature distribution (mean values) ..............................................................404.2 Temperature distribution (temperature dependent values) ................................. 407.1 Input values from the steady state off-design program ........................................707.2 Changes of the boundaiy values in the test runs ................................................. 71
LIST OF FIGURES
2.1 Differential tube element ........................................................................................ 82.2 Tube element ...........................................................................................................92.3 Once-through waste heat boiler by Dolezal.......................................................... 16
3.1 Boiler as a part of the process ............................................................................. 183.2 The three parts of the boiler ................................................................................ 183.3 One element of the boiler..................................................................................... 203.4 Definition of nodes and elements in the boiler .................................................... 213.5 Definition of the process fluid pressure in the boiler ...........................................223.6 Specific enthalpy definition of elements in the boiler ..........................................233.7 Validity range of the fast calculation functions ................................................... 25
4.1 A boiler tube ........................................................................................................ 294.2 Possible forms of the pressure loss as a function of the mass flow .................... 354.3 Specific heat capacity of water at 18 MPa and at saturation ..............................394.4 Specific enthalpy of water at 18 MPa and at saturation ..................................... 394.5 Temperatures in the preheater (mean specific heat capacity) .............................404.6 Temperatures in the preheater (specific heat capacity as a function of temp.) .. 404.7 Temperature difference between flue gas and feed water .................................. 414.8 Temperature differences between accurate and mean calculations .................... 414.9 Notation of the input values at the boundaries of the boiler parts ..................... 42
4.10 Vaporiser part parameter definition in the mixed elements ................................ 454.11 One element as a counter flow heat exchanger ................................................... 454.12 Linear interpolation and extrapolation ................................................................484.13 Pressure definition of elements in the boiler ....................................................... 494.14 Specific volume approximation in the preheater side mixed element ................. 51
5.1 Calculation of process fluid nodal values ............................................................. 545.2 Energy flows in an element .................................................................................. 595.3 "Crossing" temperatures as a result of a sudden rise of the process fluid
temperature ..........................................................................................................595.4 Energy balance of the tube wall node .................................................................. 635.5 New temperature guess in the iteration of the heat source temperature ............65
7.1 Organic Rankine cycle used in the test runs in the h, s diagram of toluene .......... 69
8.1 Pressure drop as a function of the mass flow .......................................................738.2 Stability of the boiler as a function of the enthalpy change in the preheater ..... 738.3 Process fluid temperatures in the boiler ..................................................................778.4 Heat source fluid temperatures in the boiler ...........................................................778.5 Process fluid pressures in the boiler ........................................................................78
ix
X
xi
NOMENCLATURE
AAeAdB, B\, Bi bo, b\, b2CCtCi, Ci, C3
Cl, c2, c3CPdEeFfh3kktL1MmmNnPP99mRRuTtAtuVKvwX,YxXlB, X2Bzzz
area, m2outside area of the tube element, m2 cross sectional flow area, m2 coefficients coefficientsheat capacity rate, W/K wall friction factor, (dimensionless) coefficients in the pressure drop equation coefficients in the pressure drop equation specific heat capacity, J/kg K diameter, mtotal number of elements element number force, Nfriction factor, (dimensionless)specific enthalpy, J/kgindex (of a node or in a calculation loop)overall heat transfer coefficient, W/m2Kpressure loss coefficient, m-6length, mheat of vaporisation, J/kg molar mass, kg/mol mass, kg node number number of tubes time counter perimeter, m pressure, Paheat flow rate per tube length, W/m mass flow, kg/sheat capacity rate ratio, (dimensionless) universal gas constant, 8.31451 J/mol K temperature, K (or °C) time, s time step, sspecific internal energy, J/kg volume, m3volume of an element, m3 specific volume, m3/kg flow velocity, m/s co-ordinatesquality, mass fraction of vapour, (dimensionless) vaporiser-part-parameters, (dimensionless) compressibility, (dimensionless) number of transfer units (NTU), (dimensionless) co-ordinate axis
Xll
Greek letters
A difference (of a value)s effectiveness, (dimensionless)
tube bend loss coefficient, (dimensionless)6 temperature difference, K£ inlet and outlet loss coefficient, (dimensionless)p density, kg/m3x viscous stress, N/m2$5 heat flow rate, W
Superscripts
(1), (2),... element number in an array(1), (2) number of successive iteration roundsn time, current timen-1 time, preceding time
Subscripts
1,21, 2, 3, 4 A B Ca, b, cauxcfdeffgfwgiinjj-1mlAm2AmlBm2BmlCm2Cmemmin
the first and the second root of an equationboundaries in the boilerpreheatervaporisersuperheaterdifferent casesauxiliaryprocess fluiddesign valueelementfrictionheat source fluid feed water gravitation inside of the tube into (the boiler) node number, present node node number, previous node first node of the preheater last node of the preheater first node of the vaporiser last node of the vaporiser first node of the superheater last node of the superheater old value for comparison minimum value
xiii
0 outside of the tubeold previous value of the variableout out (of the boiler)P pressureq calculated from flowSum sumsat saturationsi saturation liquidsv saturation vapourtot totaltw tube wallV calculated from specific volumew wallz, z+dz positionX viscous stress
XIV
1
1 Introduction
1.1 Organic. Rankine Cycle
In this study, a model for the unsteady dynamic behaviour of a once-through counter flow boiler
that uses an organic working fluid is presented. Thermal energy for the boiler is provided by hot
gas received from a furnace that is not included in the model. The organic process fluid is chosen
according to the process conditions; toluene, fluorinol, isobutane, and some hydrocarbons are the
most common choices. The model is presented using a selected example case that uses toluene as
the process fluid and flue gaPftofflunatural gas combustion as the heat source. The model is
general in the sense that there is a possibility to change the process fluid or the heat source fluid
and the process parameters. Toluene and flue gas are used only as examples and this is why the
terms process fluid and heat source fluid are used in this study. The dynamic behaviour of the
boiler means transition from the steady initial state towards another steady state that corresponds
to the changed process conditions.
The boiler consists of a set of parallel tubes. Each tube can be considered to contain a preheater
section in the upstream end, in which liquid process fluid is heated to its saturation temperature.
The middle part of the tube is a vaporiser in which the process fluid undergoes a phase transition
from liquid to vapour. The downstream section of the tube is a superheater, and here gaseous
process fluid is heated to a sufficiently high temperature so that it can be used profitably in the
turbine. A liquid separator between the vaporiser and the superheater, which is common in many
large once-through boilers, is not used. The relative lengths of the preheater, the vaporiser and
the superheater vary with the operating conditions since they are all parts of a single tube.
The dynamic model of the boiler, which is presented in this work, is a part of a high speed
research programme in the Department of Energy Technology at Lappeenranta University of
Technology. High speed technology refers to a system in which a turbomachine rotor and an
electric machine have a common shaft rotating faster than the synchronous speed of the electric
network. This arrangement leads to a significant decrease both in the weight and the volume of
the turbogenerator, compressor, and pump (Laijola et.al., 1995; Laijola & Nuutila, 1995).
2
The aim of the high speed research programme is to develop a Rankine cycle that is capable of
using waste heat at a moderate inlet temperature as the heat source. The best efficiency and
highest power output are usually obtained by using a suitable organic fluid instead of water in the
Rankine cycle. To identify this aspect of the power plant, it is said to operate as an Organic
Rankine Cycle, ORC.
The advantage of using organic working fluid arises from the fact that the specific vaporisation
heats of organic fluids are much lower than that of water. Thus the temperature difference
between the heat source to be cooled and the organic working fluid is smaller than when water is
used. Another advantage comes from the fact that the shape of the saturation vapour curve in the
h,s-diagram is backwards-leaning. This leads to an increase of superheat of the vapour that
expands in the turbine, whereas steam gets to the two-phase region. Also, the typically low drop
of specific enthalpy in turbines with organic fluids makes turbine design easy. In most cases a
single-stage turbine may be used instead of a multi-stage one, as required for steam. Especially
when the output of a plant is below 1 to 2 MW, significantly higher output and simpler design are
usually achieved by using an organic fluid instead of water (Laijola, 1995).
The ORC power plant uses a single stage high speed radial or axial turbine. The rotational speed
of the turbine is in the range of 25,000 to 50,000 rpm and it may be varied to achieve the best
efficiency. An electric machine, which can be used as a generator or a motor, runs on the same
shaft as the turbine and the feed pump. A double acting inverter connects the electric machine to
the normal 50 Hz electric power grid.
In the operation of an ORC plant, thermal energy is obtained from a source that has a relatively
low temperature. This may be waste heat from some high temperature process or from direct
burning of wood or heating oil. This limits the temperature of the superheated vapour to a value
that is much lower than in a steam power plant. To obtain a high efficiency, the pressure of the
superheated organic fluid must be quite high and thus close to its critical point. A complication
that arises from the high pressure is the strong variation of thermodynamic properties near the
critical point, which causes the necessity of calculating the initial steady distribution of the
temperatures in the boiler in small parts. Because of the high pressure and low latent heat of
vaporisation, the preheater is the largest part of the boiler, in many cases up to 80 % of the area.
3
The efficient power range of a single-stage turbine is about 50 to 100 % of the nominal power. A
wider power range for the plant is achieved efficiently by using sliding pressure and changing the
number of parallel turbines in steps. The sliding pressure and the small superheater also set a high
standard for the control system of the process, to prevent wet vapour flow into the turbine.
However, because the degree of superheat of the organic vapour increases during the expansion
at the turbine, the regulation of the degree of superheat is not so critical as for steam that gets
wet at the expansion.
1.2 Design of the ORC cycle
The research on high speed technology and the ORC cycle was started in the Department of
Energy Technology at Lappeenranta University of Technology in 1981. A new concept for the
plant was developed in the beginning of this decennium. Starting in 1993, a computer program
was written to analyse the steady operation of the plant. This program, which will henceforth be
called the steady process model, has been used to plan the process in the plant and to choose the
equipment and heat exchangers for it (Honkatukia, 1994). A separate computer program was
developed in 1995 for the prediction of the off-design behaviour of the plant. This will be called
the off-design process model. Both models are based on standard thermodynamic analysis and
provide thermodynamic properties at the inlets and outlets of the preheater, vaporiser, and
superheater. They do not give the details of the property variations along the boiler tube. An
important task in the dynamical simulation of the boiler is to set its initial conditions properly.
The present research is a part of a study of the unsteady dynamics of the ORC power plant and it
will be a part of a dynamic process model. The dynamic process model will be used to simulate
start-ups, change of power, change of turbine connections, and shut-downs. Also the parameters
of the controllers will be calculated using the process model.
4
1.3 The present study
With the initial conditions obtained from the off-design process model, the method for carrying
out the transient calculation needs to be established. The modelling method chosen and the
primary equations governing the flow of fluids in straight tubes are discussed in Chapter 2,
together with a review of other approaches to study unsteady flow in once-through boilers. The
finite difference formalism to solve the governing equations is presented in Chapter 3. The
method makes use of conservation laws applied to control volumes on the finite difference grid.
The important issue of the calculation of the thermodynamic properties is also discussed in
Chapter 3.
The initial state of the flow in the boiler is not obtained in sufficient detail from the off-design
process model. Chapter 4 discusses the calculation of the accurate and consistent initial
conditions for the analysis of the boiler. Special consideration is given to the effect of thermal
properties in the calculation of a heat exchanger. A short theoretical examination of the
connection of pressure loss and instability of the flow in a once-through boiler is done, as well.
In Chapter 5, a detailed dynamic boiler model is presented. The flow diagrams for the
calculations are included in Appendix A and the descriptions of the computer procedures are
given in Chapter 6, both having the aim of furthering the understanding of the details of the
calculation method. Attention is drawn to the importance of good initial guesses in an iterative
calculation.
Chapter 7 describes the test runs performed to test the dynamic model. These include step and
linear changes both in the process fluid and in the heat source inlet temperatures and mass flow
rates. A numerical examination of the instability of the flow is also presented. The results of the
test runs are presented in Appendix B and they are analysed in detail in Chapter 8. The
conclusions drawn from the study are presented in Chapter 9, where suggestions for further work
are also given.
5
2 Dynamic modelling
2.1 The present study
This study presents a dynamic model of a small once-through counter flow boiler that uses
organic process fluid. The purpose of the boiler model is to calculate the pressures and the outlet
temperatures of both the heat source fluid and the process fluid. The boiler model is a part of a
dynamic process model, which determines the calculation environment and the input variables.
The steady state input values for the dynamic process model are received from the off-design
process model described above. Using these values, the nodal values of temperatures, pressures,
specific enthalpies and specific volumes are calculated, and these are the initial conditions for the
unsteady calculation.
The boundary conditions of the boiler model, which vary with time in the dynamic calculation, are
received from the dynamic process model. These variables are the inlet temperatures and the inlet
and outlet mass flows of both the heat source fluid and the process fluid. When testing the boiler
model, these values are given by the main program. The boundary conditions that vary in the
course of time are received from the calling main program. When the boiler model is joined to the
dynamic process model, the boundary conditions will be given by the process model.
A separate dynamic model was made, even though there are several large dynamic simulation
programs available. The reason for this is that the high speed ORC process includes some
components, like a high speed radial turbine and pump, and a double acting inverter that typical
processes do not have. A separately designed model also offered better chances for utilising the
work done while developing the earlier process models. One of the aims was also to produce a
dynamic process model that can be run in a micro computer.
A typical solution method for making a process model is to write the primary differential
equations that govern the case and to solve them. In this case these equations are the one
dimensional forms of the unsteady mass balance, the unsteady momentum equation, and the
unsteady thermal energy balance for the process fluid, the heat source, and the tube wall. Since
the fluid is compressible, acoustic waves will appear in the dynamical simulation. They travel at
6
speeds much faster than the flow velocity and if the aim is to present them accurately in the
simulation, very short time steps are needed. The contribution of weak acoustic waves on the
thermodynamics of the flow is negligible and it would be beneficial to ignore them completely.
This can be done by dropping the unsteady term from the momentum equation. The drawback of
this procedure is that the pressure must now be obtained by iteration. The governing primary
differential equations used in this study, as well as their discretised forms, are presented in
Chapter 2.2.
The solution method chosen was to define the pressure of the boiler so that the mass of the
process fluid in the boiler at every instant, calculated using specific volumes, is equal to the mass
calculated using the known mass flows in and out of the boiler. For the calculation, the boiler is
divided into elements using nodes that connect the elements. The nodal values of temperature,
pressure, specific enthalpy, and specific volume are calculated using a finite difference method
with forward marching Euler approximation. In the elements, temperature and fluid velocity are
approximated as constant in the cross sectional area of the tube. As a result of this, the flow can
be modelled as one-dimensional. Also, an approximation of homogeneous flow is made both in
the one-phase parts of the preheater and the superheater and in the two-phase part of the
vaporiser. The homogeneous model can be used in this study with good accuracy, because it
gives good results for the total pressure gradient with the quotient of liquid and vapour densities
p\ /pv <10 (Whalley, 1996), and in this study pi /pv <5. In the vaporiser the two phases,
vapour and liquid, are approximated to be in a thermodynamic equilibrium, and their velocity is
equal. The dynamic calculation consists of a set of algebraic equations and a set of partial
differential equations that are transformed to finite difference equations. These equations are
solved at each instant of time.
There are five major simplifications made in this study.
1) The effect of gravitation is not considered, which means that the boiler is treated like a set of
horizontal tubes. The error caused by this simplification is small because the hydrostatic pressure
of the process fluid caused by the actual height of the boiler is small compared with the total
pressure drop across the boiler. The input data received from the off-design process model
includes the same simplification.
2) The total pressure drop across the boiler is calculated in a steady state and it is assumed to
be proportional to the square of the mass flow. The total pressure drop received from the off-
7
design process model includes the effect of the friction loss, the minor losses, and the acceleration
of the fluid flow.
3) The effect of the acceleration of the fluid flow is not calculated separately, but it is included
in the total pressure drop. The error caused by this simplification is very small because the effect
of the acceleration is only a few per cents of the total pressure drop across the boiler, and
because it is included in the total pressure drop, the error is caused only by the change in its
relative significance.
4) The heat source fluid is calculated in steady state, which means that the storage of energy in
it and the time constant of the flow are not taken into consideration. The heat source fluid is
assumed to be gas with a low heat capacity, which means that the time constant of the heat
source fluid is much smaller than the time constant of the boiler, and thus the error caused by this
approximation is insignificant.
5) The heat losses in the boiler caused by convection and radiation from the outer surface of the
boiler are approximated to be veiy small because of the small size of the boiler compared with its
heat flow rate. The input data calculated in the off-design process model include the same
simplification.
Specific features of the model are:
- the model consists of well known thermodynamic analysis and calculation methods that are
combined in a new way,
- the relative areas of the boiler parts vary with the dynamic transition, but the total area is fixed,
- the effect of temperature and pressure on fluid properties, especially on specific heat, has been
taken into account both in steady and unsteady calculations,
- very large and quick changes of input temperatures are allowed; e.g. incoming heat source
temperature may drop below superheated vapour temperature,
- a special method for shortening the calculation time of the thermal properties of the process
fluid has been used, because in many cases most of the calculation time is spent in calculating the
fluid properties.
8
2.2 Governing primary equations
The behaviour of a flow system can be described by applying well known primary differential
relations to a special case. These primary equations are adaptable in all systems and their sub
systems. The secondary equations, which are presented in Chapters 4 and 5, are based on these
primary equations, but they include simplifications and in many cases experimental and statistical
correlations. In this study the best available secondary equations and their parameters are chosen
and applied to the calculation of the boiler.
The governing primary equations of an infinitesimal fluid system in a general form are the
differential equation of mass conservation (the continuity equation), the differential equation of
energy, and the differential equation of linear momentum. The differential equation of angular
momentum is neglected in this study.
2.2.1 Mass conservation
Figure 2.1: Differential tube element
For a differential volume element dVc of a tube, presented in Figure 2.1, which has a length of dz,
a cross sectional area Ad, and a cylindrical surface area dAe, and in which the fluid is flowing at a
velocity w, the unsteady differential equation of mass conservation is
dmtQm? <7m,z+dz ~
dt(2.1)
9
Using pAidz = me for the mass, and pw Ai - qm for the mass flow rate, the equation can be
presented as
dp [ 1 d{pA& w) dt Ad dz
(2.2)
Figure 2.2: Tube element
For a finite volume element Ve of the tube in Figure 2.2, Equation (2.2) can be discretised for a
time step (n-l)At to (n)At, and using specific volume v instead of density p, as
(2.3)
For one-phase flow in the preheater and the superheater the mean specific volume in the element
is
and for two-phase flow in the vaporiser the mean specific volume in the element is
v" +xi-i<i-i)+((1-JcjKi +*jXv,j)]
(2.4)
(2.5)
where x is the quality, index si refers to the saturated liquid, and sv to the saturated vapour.
10
2.2.2 The energy equation
In the dynamic analysis of the boiler, the boiler tube is assumed to be straight and horizontal, and
thus the effect of gravitation is neglected. Also, mechanical work is not done on the differential
element. The differential equation of energy for the fluid in the differential element of the tube in
Figure 2.1 is
^ + (Q,m(" + /?v + 2w2))z -(?mO + pv + \w2))^ = («e(z/= +^W*))(2.6)
where dtf> is the differential heat flow rate from the tube wall to the fluid element and u is the
internal energy.
Using the definition of the enthalpy h = u + pv, pAd dz = m6 for the mass, and pw Ad = qm for
the mass flow rate, the equation can be presented as
+ 2W*))+ fw2)) =_L d_Ad dz'
d(j>dK
(2.7)
At low fluid velocities the kinetic energy can be neglected, and solving the internal energy from
the definition of enthalpy Equation (2.7) becomes
-t{ph-p)+±-j-(A'Pwh) = jt (2.8)
Discretising Equation (2.8) for the finite element in Figure 2.2 it becomes
W-f): -(f A-fT' = (& +(?.%' -(%.A)?-')^ (2.9)
The mean values in the element, indexed with e, are calculated as arithmetic means
(Ph ~ P)c = i{(Ph ~ P)U + (Ph - P)j) (2.10)
11
2.2.3 The equation of linear momentum
The differential equation of linear momentum for the differential volume element in Figure 2.1 is
d(mewe)dFg + dF +dFr = (<?mw)z+d2 - (qmw)z +-dt
(2.11)
For a horizontal tube the gravitation force is ignored, dFg = 0. The surface forces, dFp and dFr,
are a result of pressure gradient and viscous stresses on the tube wall r w
-dF -dFr = Addp + dAerw= Ad f£a+JLr 1[dz Ad WJ dz (2.12)
Using pAddz = me for the mass, pwAd = qm for the mass flow rate, Pdz = dAe for the outer
surface of the element, and Addz = dVe for the differential element volume, the equation can be
presented as
dpdz
d(pww) d(pw) dz dtdw
CM?' -j- Wdz
d{pw)dz
dw dp+ P------bW----dt dt
(2.13)
The sum of the second and the fourth term of the right side of Equation (2.13) equal zero
because they form the mass balance as presented in Equation (2.2), and the equation becomes
dpdz
(dw dw \dt+W~dZ; (2-14)
The term in the brackets in the right side of Equation (2.14) is the total differential of fluid
velocity w in respect to time t and the equation becomes
dp dA, dw(2.15)
When the viscous stress on the tube wall is expressed using the wall friction factor Cf
as rw =C(jpw2, the pressure gradient is finally achieved as
dpdz
dAdVc
2Ct^7fM,2+P dw~dt (2.16)
12
Presuming the change of fluid velocity in the element, —, is very small compared with theat
friction loss, the steady state pressure loss for a finite tube element can be written as
- Pi-\ - p-: - 2 Q ~tV v&%m,e (2.17)
where the index e in the mass flow and in the specific volume refers to the mean values in the
element.
In this study the total pressure drop across the boiler is approximated using Equation (2.17). The
effects of acceleration and gravitation are neglected because they are presumed to be small
compared with the effect of pressure loss caused by friction. The effect of this simplification in
the example case is discussed in Section 8.4.
2.3 Different approaches to dynamic modelling
Process models and test conditions similar to the one presented in this thesis are not found in the
literature. Either the boiler type or the boundary conditions in the tests are so different that the
results cannot be compared. In the following, a very brief survey on some dynamic calculation
programs is presented. The presented models are the one-dimensional two-fluid model of
APROS, the model of the once-through boiler oflnkoo power plant by Raiko, and the model of a
once-through waste heat boiler by Dolezal. Before that, a brief review of recent power plant
modelling is given.
2.3.1 Power plant modelling
Concerning process dynamics, boiler—turbine systems are very complex continuous processes
because of the structural complexity, strong nonlinearity, interaction among subsystems,
relevance of distributed parameter phenomena, and considerable uncertainty of process
parameters at macroscopic knowledge level (Maffezzoni, 1992). To tackle the complexity of a
13
detailed model, user interfaces based on a modular description of the process plant are usually
introduced. The modularity allows an engineering approach to the simulation, which is a process
description adherent to the structure of the real process and gives the possibility to represent the
individual components in less detail. The approach to modularity may be quite different in
different models, depending on how the interaction between subsystems and components are
described and treated in the model solution.
The principal scope of modular approach is to reuse the modelling software for different
applications. This leads to the adoption of very elementary modules, to the so called
micromodules approach in which volumes, junctions and heat conductors are the basic building
blocks general enough to fit any real process structure. The main drawback of the micromodule
approach is that the modular structure of any plant model is very complex and the task of
building a module is not far from that of writing equations, since modules are essentially
thermodynamic elementary systems to which a certain set of standard equations corresponds. In
many cases the macromodule approach is most effective. In this approach the modules
correspond to the plant components, and so defining the modular structure of a plant amounts to
drawing its process flow-diagram.
New possibilities for model structuring are provided by the development of software engineering
concepts and technology. Object-oriented databases with their inheritance properties allow
efficient combination of flexible modularization and simplicity of model building procedures
(Maffezzoni, 1992). The novel method of neural networks has also been developed for boiler
simulation. The neural network predictive models are trained using data from the results of a
series of step response tests for an actual once-through boiler (Reinschmidt & Ling, 1994). The
results show that the neural network is a satisfactory plant simulator, capable of accurately
reproducing the measured response of the boiler to a set of control inputs. These kinds of models
need to have data from existing power plants of the same type as the simulated one.
Within the module, the task is to transform the governing partial differential equations to a set of
ordinary algebraic equations. Most often, this is done using the method of finite differences (e.g.
Raiko, 1982). Some solution methods transform the partial differential equations into ordinary
ones using the characteristic curve method (Isomura, 1995). Some of these programs use a
reduced mathematical model in connection with the measurements made at the power plant
14
(Unbehauen & Kocaarslan, 1991). The vast amount of data that must be processed leads easily to
a need of veiy efficient computing capabilities e.g. parallel processing computers (Serman &
Mavracic, 1990).
Many of the first process simulation models came from the process control community. Also
nowadays, there are many simulation programs which are able to adapt a control system to the
calculation (e.g. Jarkovsky et.al., 1989; Serman & Mavracic, 1990; Shikolenko & Bessonov,
1990; Unbehauen & Kocaarslan, 1991).
2.3.2 Modular simulation programs
Most of the existing calculation methods divide the boiler into elements or blocks that are
connected to each other using mass and energy flows. In these models also the fact that the
relative lengths of the preheater, vaporiser and superheater vary with the operating conditions can
be taken into account (Jingrong & Chenye, 1991; Dolezal et.al., 1989; Grobbelaar et.al., 1994).
AFROS is a process simulator program developed by the Technical Research Centre of Finland
and Imatran Voima Oy (Silvennoinen et.al., 1989). The one-dimensional two-fluid model of
AFROS simulates systems containing gas and liquid phases. The system is governed by six
differential equations, from which the pressures, void fractions and phasial velocities, and
enthalpies are solved (Hanninen & Ylijoki, 1992). The phases are coupled with empirical friction
and heat transfer terms that strongly affect the solution. The governing equations are the
conservation equations of mass, momentum and total energy for the liquid and gas phase. The
governing equations are discretized with respect to time and space, and the resulting linear
equation groups are solved by the equation solving system of AFROS. The pressures, void
fractions, and enthalpies are solved in defined calculation points, nodes. The flow velocities in
branches that connect adjacent nodes can be solved directly after the pressure solution. The
model has been tested by calculating several well-known two-phase test cases that show that the
calculated results follow the measured data fairly well in all cases.
15
The division of the heat transfer surface to nodes and to branches between the nodes in APROS
is similar to the one presented in this study. The APROS model pays much attention to the
calculation of interfacial phenomena between the liquid and gas phase, which makes it rather
accurate in the calculation of two-phase flow. As a result of several phenomena concerned, the
discretization of the equations is complicated and the solution of the model needs much
calculation capacity compared to the method presented in this study.
Raiko’s study describes the structures and working principles of the non-linear models that are
used to simulate the transition states of once-through boilers (Raiko, 1982). Four models are
described in detail, and on the basis of these a new model is made in order to simulate the start-up
of the boiler at the Inkoo power plant. The aim of the simulation is to decrease the use of fuel
during a cold start-up. The Inkoo power plant has Benson type sliding pressure once-through
boilers with superheated steam values of 211 kg/s, 18.6 MPa, and 530 °C. Coal is used as fuel,
except for start-ups, where oil is used instead.
Raiko’s solution method is such that iteration is not needed. The furnace is calculated using
steady state equations in nine zones. The flue, too, is divided into nine zones in which the heat
transfer is calculated. The flue gas is calculated in a steady state, which means that there is e.g. no
delay in the mass flow of the flue gas when combustion is increased. To simplify the calculation
of the vaporiser, the pressure losses are calculated using steady state equations, the solution of
the energy equation is explicit, and the mass flow of water is equal in all elements. The extra
water is removed in water separator.
Raiko does not present detailed equations and their discretization. The model is modular in the
sense that in general for example one heat exchanger that is not divided into smaller elements
forms one module. Despite the differences of the power plant, modelling method, and the aim of
the work compared with this study, there are several similar approximations and simplifications
made, as e.g. steady flue gas and steady pressure loss. Anyway, the differences compared with
this study are so great that the results of these two models are difficult to compare.
16
2.3.3 Once-through waste heat boiler model
Dolezal has presented a model of a once-through waste heat boiler, which is capable of
simulating the transition behaviour during large and non-linear changes of state, such as start-up,
shut down, malfunctions, and load changes (Dolezal et.al., 1988). The behaviour of the boiler
during control actions is investigated, as well. The calculation program has been later revalidated
on the operating unit by means of long time transients by comparing the measured transient
curves with simulated ones (Dolezal, 1992).
Figure 2.3 (Dolezal, 1992) shows a boiler that consists
of an economiser EC, an evaporator EV, and a
superheater SH. There is no locally fixed boundary
between the economiser and the evaporator. A water
separator WS between the evaporator and superheater
makes it certain that no water is fed into the
superheater. The temperature of the superheated steam
LS is regulated using water injection IW, which reduces
the steam temperature by 70 K at full load. The values
of the superheated steam are 2.5 kg/s, 40 bar, and 440
°C, which are close to the values in the example case
used in this study.
The dynamic calculation program is based on the
decoupled semi-analytical calculation method, which is
especially convenient for the iteration-free computation
of non-linear and time-variant processes in heat exchangers with distributed parameters. Because
of the modular structure of the model it is possible to reproduce the heat balances of the most
varied plant structures of power plant units without iteration in an uncomplicated and rapid
manner (Dolezal et.al., 1988). To simulate the process dynamics of a steam generator, a
segmentation of its heat exchangers and pipe system is necessary.
The transient behaviour of the once-through waste heat boiler was studied using small and
sudden reductions in boiler feed rate and heating (Dolezal et.al., 1988). The computations were
Figure 2.3: Once-through waste heat boiler by Dolezal
17
carried out for a narrow range of boiler initial conditions including three steady initial
distributions of 100 %, 98 %, and 95 % boiler feed rate, and full gas turbine output in all three
cases. Pressure was supposed to be constant in each simulation. The tests with a reduction of
boiler feeding rate were carried out with a sudden reduction of the working fluid mass flow rate
by 4 %, 7 %, and 10 %, and the gas turbine output was supposed to be the full load. The tests
with a reduction of the boiler heating were carried out using 4 %, 7 %, and 10 % reduction of
flue gas mass flow, and corresponding temperature drops.
In the later tests by Dolezal, the inlet gas temperature, the feed water flow, and the steam
pressure were given as the boundaiy conditions for the boiler model and they were based on a
simulated gas turbine process (Dolezal, 1992). The model calculated for example steam mass
flow at the boiler outlet, steam temperature at the superheater, and steam temperature at the
evaporator outlet. The duration of the input transients was 10 to 45 minutes and the transients
were followed during 120 minutes.
The results of the test runs done by Dolezal and the results obtained in this study cannot be
compared, although they have many similarities. The main reasons for this are that in Dolezal's
tests there always exists a minimum fixed area of the superheater, and the pressure of the boiler
has been fixed or defined as a boundary condition. The size of the superheater in Dolezal's tests
may be larger than the fixed superheater area if the steam is superheated in the evaporator before
the water separator, while in this study the relative sizes of all heat transfer surfaces may change
with the only limit of total boiler area. Giving the pressure as a boundaiy condition decreases the
number of the unknowns remarkably, while in this study the main task is to solve the boiler
pressure.
3 Boiler model description18
3.1 General
One may think of the once-through boiler as a set of parallel tubes that go across the boiler. Each
tube can be considered to consist of preheater, vaporiser and superheater sections as shown in
figures 3.1 and 3.2. In the following, a thermal analysis of a single tube is carried out, but the
analysis applies to all the other tubes equally well. In figure 3.2 the flow of the process fluid is
from left to right, and the boiler is taken to be a counter flow type so that the heat source fluid
flows from right to left. The process fluid is fed into the boiler from a feed pump through a
recuperator that is the first preheater, and superheated vapour is taken to a turbine.
Turbine
Supe. heater
Generator
Recuperator
Condenser
Feed pump
Superheaterj Vaporizer
Preheater
heat source fluid
“process
distance along tube
Figure 3.1: Boiler as a part of the process Figure 3.2: The three parts of the boiler
The starting values for the dynamic calculation of the boiler are obtained from the steady state
process model. The known steady state input values are:
— the temperature, pressure and mass flow of the heat source fluid,
— the temperature, pressure and mass flow of the process fluid,
— the overall heat transfer coefficients and the heat transfer coefficients on the heat source side,
— the heat exchanger surface areas of the preheater, vaporiser and superheater,
— the diameters, mass, and volume of boiler tubes.
19
The values of the process fluid and the heat source are known at the inlet and the outlet of the
boiler and at the boundaries between the preheater, vaporiser and superheater. Thus the values
for each fluid are known at four locations.
The known changing input values during the dynamic calculation of the boiler are the
temperature, pressure and mass flow of the entering heat source fluid, the temperature and mass
flow rate of the process fluid as it enters the boiler, and the mass flow rate of the process fluid as
it leaves the boiler. When the boiler model is joined to the dynamic process model, these values
will be given as input to the boiler model by the process model. When testing the boiler model,
they are set as boundary conditions. Input values to the boiler model are set in the process model
so that the feed pump gives the mass flow rate of the process fluid as it enters the preheater,
process calculations give the temperature of the process fluid at the entrance to the preheater, the
turbine gives the mass flow rate of the process fluid as it leaves the superheater, and the
temperature and the mass flow rate of the heat source are set according to the expected running
conditions of the power plant.
The dynamic boiler model will be analysed for linear and step changes of the entering fluid
temperatures and flow rates. These will test the model for conditions expected to be received
from the process model, once the latter is completed. As a part of the process model, the boiler
model will be used to calculate the temperature and pressure of the heat source fluid as it leaves
the boiler, the temperature and pressure of the process fluid as it leaves the superheater, and the
pressure of the process fluid at boiler inlet. In addition, the heat power of the boiler and the
boundaries between the liquid and two-phase regions and the two-phase and the vapour regions
of the process fluid in the boiler will give useful engineering information.
3.2 Special features of the dynamic boiler model
The modelling method chosen was dictated largely by the analyses that had been carried out
earlier in the ORC-project. To fit the modelling environment of the previous steady state models
it was necessary to use the same kind of calculation method. This also made available some ready
procedures, for example the functions that calculate the thermodynamic properties of the fluids.
20
The dynamic boiler model has several special features:
1) The initial steady calculation and the dynamic calculation take into account the temperature
dependence of the process fluid. This is especially significant in the calculation of heat exchange
when the pressure and the temperature of the process fluid are close to the critical point.
2) The model uses an efficient way of calculating fluid properties, which means short
calculation time.
3) The entering fluid temperatures may vary so much that the temperature of the heat source
fluid can be lower than the temperature of the tube wall or the other fluid in some part of the
boiler.
It is necessary to carry out several iterative calculations in the dynamic boiler model. Therefore it
is of utmost importance to cany them out in an effective way and to use accurate initial guesses.
3.3 Defining the nodes and elements in the boiler
Each tube of the boiler is divided into E
elements of equal size using £+1 nodes that
connect the elements. The elements are
numbered in the direction of the process fluid
flow as shown in Figure 3.3. The element e has
node j-1 on its left side and node j on its right
side and the indexing is such that j = e. Each
node has values for heat source (subscript fg),
tube wall (tw) and process fluid (cf).
The nodal values of temperatures, pressures,
specific enthalpies, specific volumes, and mass flow rates are taken as unknowns. When their
average values are needed in the elements, an arithmetic mean is used. Heat flow rates are
calculated for the entire elements.
heat source 9m,feTfgj-i
____tube wall ... I_
1 twj-1 — "*tw ~twjprocess fluid — Tcij-i m=c« - *7m,cf
Figure 3.3: One element of the boiler
21
3.4 Element types and numbering of nodes and elements
The number of nodes is selected so that in the beginning of the dynamic calculation the minimum
number of elements for any part of the boiler is two. In a typical case, the superheater has the
smallest surface area, so that it determines the number of elements in the entire boiler. In the case
that is used in the examples the total area of the boiler in the initial steady state is 105.8 m2: the
preheater area is 82.0 m2, the vaporiser area 17.9 m2 and the superheater area 5.9 m2. Letting the
minimum number of elements be two for the superheater, the minimum number for the boiler is
36. In the calculation examples the boiler is actually divided into 40 elements and the area of one
element is 2.645 m2.
There are four types of elements: preheater elements, vaporiser elements, superheater elements,
and mixed elements. The mixed elements are situated between the preheater and the vaporiser,
and between the vaporiser and the superheater, and they have some special features to simplify
the calculations.
1 preheater dements prehside: :raxed dem
yaporizer yuperkside yuperheater darenls 'nixed dem 'denerisy^ heat source
process fluid
.1__vapoijzer.____i„superticatei;..i.preheater.
Figure 3.4: Definition of nodes and elements in the boiler
The nodes are named so that
— t«ia and t»2a are the first and the last node of the preheater,
— t«ib and m-m are the first and the last node of the vaporiser, and
— /Rig and m2c are the first and the last node of the superheater.
The elements from e = thia+1 to e = m2A are preheater elements, from e = »2IB+1 to e = 7»2B
vaporiser elements, and from e = mic+1 to e = m2c superheater elements. The element e = otjb is
the preheater side mixed element and e = m ic is the superheater side mixed element. These are
shown in schematic form in Figure 3.4.
22
3.5 General description of unknowns in interior nodes
In each node the unknowns are:
— the temperatures of the heat source (7}g), tube wall (Ttvl), and process fluid (7^),
— the pressures of the heat source (pfg) and process fluid (pcr),
— the specific volumes of the heat source (vfg) and process fluid (vcf),
— the vapour fraction of the process fluid in the vaporiser (xCf),
— the mass flow rates of the heat source (gWg) and process fluid (#m,Cf).
preheater elements prekside vaporizer superhside superheater• mixed elem. 'elements pdxedelem. elements
Xpr i cess fluid
.1_.vaporner__ i„superfieater_.!
---- actual ---- simplification
Figure 3.5: Definition of the process fluid pressure in the boiler
The pressure of the process fluid is calculated beginning from the first boiler node m\\ so that in
the preheater, vaporiser, and superheater elements the pressure is changed only by the friction
loss. Acceleration is not taken into account because it has only a minor effect on the pressure. In
the mixed elements the pressure is set to be a constant and equal to the saturation pressure,
as shown qualitatively in Figure 3.5. This causes a small error, but it simplifies calculations
considerably. The temperature of the process fluid in the vaporiser nodes corresponds to the
saturation pressure and thus it sinks slightly in the vaporiser.
The specific enthalpy of the process fluid in the nodes is calculated from the energy balance. In
the mixed elements a vaporiser-part-parameter xm is used to weigh the calculation of the
preheater and vaporiser parts, and x# is used for the vaporiser and superheater part. These
parameters show the relative area of the vaporising part of the mixed element and they are
calculated using saturation enthalpy values and enthalpy values in the nodes. Figure 3.6 shows
qualitatively how the vaporiser-part-parameters are defined using saturation enthalpy values for
liquid and vapour, hcf,si and hcfjSV, and an approximation of linear enthalpy change in the mixed
element. In a dynamic calculation the boundaries of the boiler parts move. First, parameters Xjb
and X2B change within the limits of 0 to 1 and when the saturation values change appreciably, the
previous or the next element becomes a mixed element.
23
preheater elements prekside vaporizer superkside superheater• "puxed elem. elements 'nixed elem. 'elements
process fluid
.preheater_________1__.vaporizer___[..superheater.!
Figure 3.6: Specific enthalpy definition of elements in the boiler
The heat transfer coefficients are based on the outer tube surface area. The overall heat transfer
coefficient includes the convective heat transfer coefficients on both sides of the tube, as well as
the. tube wall heat resistance. The overall and outer heat transfer coefficients are obtained from
the steady state calculation as initial data, and the inner side heat transfer coefficients are
calculated using them. The inner heat transfer coefficient also includes the tube wall heat
resistance.
The saturation values of liquid and vapour are not defined in each node, but only one saturation
liquid and one saturation vapour point is used for the whole boiler. Thus the pressure drop in the
vaporiser does not effect the saturation properties. The saturation liquid temperature, specific
enthalpy, and specific volume are defined in the mixed element e = mm, and saturation vapour
values are defined in the mixed element e = m]C. Vapour fraction and all the other values in the
vaporiser elements and nodes are calculated using these saturation values, which simplifies and
speeds up calculations significantly and causes only a minor error.
24
3.6 Calculation of thermodynamic properties
The thermodynamic properties of the heat source and the process fluid are calculated using the
procedures used in the steady state process model (Honkatukia, 1994; Talonpoika, 1994). For
toluene these functions have two levels of calculation accuracy: an accurate calculation method,
which is quite accurate but very slow, and a rough calculation method, which is fast but
somewhat inaccurate. The rough method is also available for fluids for which the functions of
accurate thermodynamic properties are not known. For the dynamic calculation, a faster
calculation method had to be found to keep the calculation time reasonable.
The main difficulty in the calculation of thermodynamic properties is that in addition to using
explicit formulas for the calculation of saturation pressure p<*i(T), specific enthalpy h(T,p), and
specific volume v(T,p), there is also a need to calculate the saturation temperature Tai(p), and
temperature as a function of pressure and enthalpy T(p,h). Since these are implicit functions, an
iterative solution method would be needed.
A standard way to solve the problem of thermodynamic property calculation of implicit functions
is to calculate all the needed functions by explicit formulas when possible and by implicit formulas
when necessaiy, and to construct tables of these functions. Since for a simple compressible
substance, two independent thermodynamic properties fix the thermodynamic state, the most
convenient tabulated variables can be used as the independent ones. The calculation
procedure then consists of finding the right location in the table and carrying out an interpolation
procedure. Finding the right location is reduced to a root finding procedure. The bisection
method and linear interpolation show a good balance in robustness, computational speed and
accuracy in a table with reasonably small changes in properties. Such considerations as the need
to load tables into computer memory are secondary today since computer memory prices have
decreased greatly during the past two years.
25
A variation of this method was adopted here with the aim of using accurate tabulated values that
are close to the new point to be calculated and using a fast interpolation method. An array that
has four elements is used to store the values of the properties in one node. The temperature,
pressure, specific enthalpy, and specific volume arrays of the process fluid at node j are
The first two elements of the arrays are reserved for the values at the preceding time and the
current time. The values of the second elements are copied to the first elements after the
calculation of each time step. The third and fourth elements are reserved for the estimation of the
properties.
ATI0’ T WO
AT
'T ^ valid range J k (
When a new accurate value of for example
specific enthalpy /z(2) is to be calculated using
known values of temperature 7*2) = T and
pressure pm =p, the third elements are given the
same values 7<3) = 7<2) and A(3) = /z(2). Next,
another enthalpy value /z(4) is calculated using a
slightly higher temperature 7(4) = T+AT and the
pressure p. These values are calculated from the
explicit formulas adopted from the steady state
process model, and there is a possibility to
choose between the accurate but slow calculation procedures and the slightly inaccurate but fast
procedures. Each time when new values of specific enthalpy at elements (3) and (4) are
calculated, the specific volumes are calculated as well, and vice versa.
Figure 3.7: Validity range of the fast calculation junctions
The next time the enthalpy of that node is calculated, the validity of the values A0) and hw is first
checked. The validity range in temperature is from 7*3)-A772 to 7<4)+A772, and for the implicit
properties the validity range in enthalpy is from A(3)-AA/2 to A(4)+AA/2 where Ah = A(4)-AP).
The ideas are illustrated schematically in Figure 3.7.
The pressure dependence of specific enthalpy is weak, and this is why within the validity range, a
new value of the specific enthalpy is calculated using linear interpolation as
When the specific enthalpy is known, the temperature is interpolated correspondingly as
r = ?*> + frw - (3.2)
If the input values are not within the validity range, new values for hQ) and A(4) are calculated as
described above. Also the phase of the fluid and the pressure are checked to guarantee the
validity of the function. The pressure must be in the range of p^-Ap to p^+Ap, where />(3) is the
pressure that is used to calculate the enthalpies A(3) and A(4>. Typically values of AT = 10 K and
Ap = 5 kPa are used.
26
For specific volume the pressure dependence is much stronger than for specific enthalpy and a
slightly more complicated equation is used in the vapour phase. From the real gas equation of
state the specific volume is
M p
In the interpolation the specific volume is calculated as
(3.3)
= „P)v = v + (v(4) _ „(3)
T
P
p(?)-J?)
f(4)„(4)
T’P) (3.4)
This new calculation method is most effective in the calculation of vapour properties. In a test of
a superheater calculation the calculation time of one time step was 0.7 seconds using the new
method, 3.3 seconds using the fast but inaccurate method, and 410 seconds using the accurate
but slow method. Clearly, the time to make the property calculations has been reduced
substantially to a great benefit in both the steady state calculations and for calculating the
unsteady behaviour of the boiler.
27
4 Steady state calculation
4.1 Input values for the steady state calculation
The steady state process model yields the following quantities:
— the temperature, the pressure and the mass flow rate of the heat source fluid,
— the temperature, the pressure and the mass flow of the process fluid,
— the overall heat transfer coefficients and the heat source side heat transfer coefficients,
— the heat exchanger surface areas,
— the diameters, the mass, and the volume of the boiler tubes.
Actually the property values are known only at the boundaries between the preheater, the
vaporiser, and the superheater, as well as at the entrance of the preheater and at the outlet of the
superheater.
The above values are used to calculate the initial steady state distribution of the variables at all
the nodes in the boiler. They include the temperatures, pressures, specific volumes, specific
enthalpies, and fluid masses contained in the elements. In addition to these, the pressure loss
coefficients in the three parts of the boiler and the adjusted heat transfer coefficients are
calculated. The heat transfer coefficients need to be adjusted to achieve consistency. This helps
avoid any truncation errors and a cumulative effect of conservation errors. This calculation also
makes it sure that the temperature dependence of fluid properties is properly taken into account.
4.2 Pressure loss
4.2.1 Pressure loss and stability in a once-through boiler
A once-through boiler consists of several parallel tubes of equal length and size, and they are
expected to be heated equally. The tubes start from an inlet header and they end in an outlet
header. The pressure loss between these headers consists of a friction loss, minor losses from
tube bends, tube inlet and outlet, and a change in pressure caused by acceleration. The main
28
variables in the flow system are mass flow, pressure loss, and vapour fraction, and the connection
between them may cause flow oscillations.
The flow oscillations in the boiler are harmful for three reasons: they generate mechanical
oscillation, they cause control problems, and they affect the local heat transfer characteristics and
the critical heat flux. In the flow system, the connection of the mass flow and the pressure loss is
very important and the thermodynamical connection between them may strengthen flow
oscillations (Sarkomaa, 1973 a).
As a result of flow oscillations, the flow velocity of the boundary layer in a vaporising tube
oscillates periodically. At those moments when the velocity is low, a vapour film may be formed
on the tube surface (Sarkomaa, 1973b). This vapour film may cause an unexpected boiling crisis
or a departure from the nucleate boiling. Owing to the flow oscillations, the pressure decreases
locally when superheating of a fluid at the tube surface increases, resulting in breaking of the
liquid films and a dry-out.
The flow may be unstable for the following reasons
— Instability of the flow may take place if the two-phase flow is in transition from bubble flow
to annular flow. The pressure loss in annular flow is smaller than that in bubble flow. Since the
same pressure loss must exist between the ends of parallel tubes, a change of the flow regime in
one tube may cause extensive disturbance in the mass flow rate between that tube and the other
tubes. Because the change in the mass flow rate does not change the heat transfer coefficient
significantly, the heat flow rate to the fluid stays almost constant, and the flow may change back
to bubble flow and the oscillation continues.
— Ledinegg instability arises from the fact that the pressure loss of a boiler tube that has
preheater, vaporiser and superheater parts is a third degree polynomial in the mass flow rate. The
formulation of this function is presented below. Ledinegg instability is also called pressure drop
oscillation. In general it can be predicted and avoided by choosing the operating conditions
properly.
— Dynamical instability means that the flow in a tube may reach resonance in certain cases. If
the mass flow in the tube diminishes for some reason, the vapour fraction at the tube end
increases. This affects the hydrostatic pressure difference, the pressure difference caused by the
29
acceleration, the friction pressure loss, and the heat transfer. If the geometry of the tube and the
other parameters are inappropriately chosen, oscillations may occur.
— Acoustic instability is also possible. At the film-boiling region there is a thermodynamic
feedback between the flow disturbances and the vapour film. The sound velocity changes
significantly when the vapour fraction changes. The frequencies are proportional to the frequency
of the harmonic oscillation of an open tube.
In this study the dynamic behaviour of a boiler is taken to mean a transition in behaviour during a
large and non-linear change of its state under load changes, shut down, malfunction, etc. Other
kinds of dynamic behaviour are of minor interest. All the instability forms presented above deal
with oscillations at about a steady state. The Ledinegg instability may have a wider range of
applicability with respect to the boiler in question and, therefore, a short theoretical examination
of it is presented below. The present stage of boiler development in the project does not make it
possible to inspect other forms of oscillation.
Effenberger has presented an equation for the pressure loss in a once-through boiler tube that has
preheater and vaporiser parts and in which the fluid at the vaporiser outlet is only partly
vaporised. He found that in a one phase flow, either liquid or vapour, the functional dependency
of the pressure loss and mass flow is a second order polynomial, but in a two-phase flow the
functional dependency is of the third degree (Effenberger, 1987). This form of pressure loss
gives also a possibility for hydrodynamic instability, especially in a heat transfer surface that
consists of several parallel tubes.
Preh eater Vaporizer Superheater
Figure 4.1: A boiler tube
30
In the following text the discussion by Effenberger is extended to a case in which the tube has
preheater (index A), vaporiser (B), and superheater (C) parts as shown in Figure 4.1. The steady
state pressure drop is given by the equation
4P- (fA fA + W + (^1+ fB)
, , (4.1)+ fc (4r^+ Cc + fc) + (2^- %))
a Wm
where /a, fa and /c, are the friction factors, £a, Cb and Cc are the tube bend loss coefficients,
and £c are the inlet and the outlet loss coefficients, and and wout are the inlet and the outlet
flow velocities of the fluid.
When the friction loss is considered alone, the pressure drop is
(w P)2 ^ r lb (w P)2 ^ , Lc (w p)2T ^7 d 2 pB* Sq-j- tt"+/c
2 Pc(4.2)
The total length of the tube L, is known and must be kept constant, L = LA + Lb + Lc- If the
Reynolds number is large enough, as is presumed here, the laminar sublayer of the turbulent
velocity profile is so thin that it is masked by the roughness elements of the tube wall. In such a
case of complete turbulence, the friction factor f is dependent only on geometry and it is
independent of the Reynolds number, and thus it is independent of mass flow and viscosity
(Schlichting, 1979). The equation is simplified by taking all the friction factors as equal, thus f=
fA=f& =/c- Also, the heat flow per tube length q is taken to be a constant. The inner cross
sectional area Ad of the tube is constant and the mass flux w p= qJAd is constant in which qm is
the mass flow rate in one tube.
If the assumption is made that subcooled boiling of the fluid does not exist because the heat flow
rate is low, the three parts of the boiler tube can be treated as distinct lengths of liquid flow,
boiling, and vapour flow. The length of the preheater part can be determined from
where AhA is the enthalpy change of the fluid in the preheater.
Aa = 9 m (4.3)
31
The length of the vaporiser part is
LB = ” ?m (4.4)
where / is the heat of vaporisation. The length of the superheater part is then
Lc = L - (Lb + Lc) (4.5)
Since liquids are nearly incompressible, the specific volume in the preheater may be taken to be
that of a saturation liquid vs]
va = vsi (4.6)
In the vaporiser, the outlet specific volume is that of saturation vapour vw. In the vaporiser and
the superheater, the specific volumes are calculated as arithmetic means of the inlet and outlet
specific volumes as
vB=i>si + vsv; (4.7)
+HnIt (4.8)
In the superheater, the vapour obeys the simple real gas law
fv-z(&r)r (4.9)
The pressure changes only slightly owing to reversibilities and can thus be assumed to remain
constant. With a constant specific heat, the specific volume change in the superheater is given by
Thus the mean specific volume in the superheater is
(S,------- 1M- (4.10)
Vr = V„ 2-p.c
Ahc (4.11)
32
The change of the enthalpy in the superheater is given by
(Z,-(Z.A+Z,B))g(4.12)
Combining equations (4.3) to (4.11) and the pressure loss equation (4.2) gives the pressure drop
as
AP = C3?m+ C2 <7m + C1 <7, (4.13)
where
2iizl;
,(S)‘ft + I)2 - - v,J I'M, + i)-p.c
- (4.14)
2<//f;
dvdf. „
vsv---------- =- (Ma + l)cp.c
(4.15)
c, = / (S)4 ^Md cpC
(4.16)
These constants depend on the pressure through their dependence on the saturation liquid
enthalpy A,i, saturation specific volumes v,, and v^, and the latent heat of vaporisation /, and on
the construction of the surface through the tube diameter d, length of the boiler tube L, and the
friction factor f. The heat flow q, and the enthalpy in the boiler inlet hm are also needed to specify
them.
The coefficients of the pressure loss equation need to be developed further in order to relate them
to the area of the heat transfer surface instead of its length. The area of the preheater and that of
the entire boiler is
Aa - ndLAN (4.17)
33
A = ycdLN (4.18)
in which Aa is the area of the preheater, A is the area of the entire boiler, and A/ls the number of
parallel tubes in the boiler. The total mass flow rate is
9m,tot ~Nqn (4.19)
Substituting these, equation (4.2) becomes
d 2 pA /a
2 n d1 A\ N3
. /c 2 nd2 A\ N3
f -A K'H'h’J / ±B Kw HJ V Ac fW
■^A VA 9m,tot +
4: VC 9m,tot
/b
2 ttc?2 4? W34 VB 9m,tot + (4.20)
Defining the pressure loss coefficient 4,A, and 4b and 4c correspondingly, as
Ja4, a - ‘
2^4^#3
gives for the pressure loss the expression
(4.21)
AP = *f.A A vA 9m,tot + 4b4 vB 9m,tot + 4.C 4 9m,tot (4.22)
When the friction coefficients are set to be equal/=_/& =/b = /c, it follows that 4 = 4a = 4b = 4,c and
Ap = 4 f/lA vA + Ab vb + Ac vc) ql Xot
in which the pressure loss coefficient is
k{ 2itd2A\N3
The heat flow rate of the whole boiler is
0 = NL q = ---- qn d
(4.23)
(4.24)
(4.25)
34
so that
q = o
L =
nd~A
AirdN
(4.26)
(4.27)
The total pressure drop across the boiler, given by equation (4.13), written as a function of the
total mass flow rate is
Ap = c3 + c (SsMf + c (Sum.) =^ 3 N 2 N 1 N
- C3 «3j *2 2 a3 9m,tot + ~Jj2 9m,tot + ^ 9m,tot
(4.28)
The definitions
(4.29)
put the pressure loss into the form
A? Qj (7m,tot + Qt (7m,tot + Q (7m,t< (4.30)
where
Q$ “ *f 4<P
\{%)
2 c.-rAA^ + ^ - v„; + 4
V
C2=k{ A
p.c
c7v
v„„ - oUP.C
fdv_
2 c,p.c
(4.31)
(4.32)
(4.33)
35
Analogously with the constants cj, c2, and c3, these constants depend on the pressure, the
construction of the surface, the total heat flow rate, and the inlet enthalpy.
Numerical values for the pressure loss coefficient kf and constants Ci, C2, and C3 can be
calculated from the known pressure loss of the boiler. Such calculations give the qualitative
behaviour shown in Figure 4.2. Depending on the factors, the pressure loss can have one of the
three forms shown. The details of the numerical calculation are presented in Chapter 8.2.
pressure loss
mass flow q,
Figure 4.2: Possible forms of the pressure loss as a function of the mass flow
The function (a) is a monotonically rising one. The flow is stable and it is mainly a function of the
first order term. For the limiting case (b), where the first derivative has a double zero, the same
holds true. Further change in the coefficients and mainly C2, leads to a function with extrema as
shown in function (c), and there are three possible mass flows that correspond to a certain
pressure loss. The first and the third are stable and the one in the middle is unstable.
The graph (c) can be explained as follows. The friction loss is proportional to the product of the
square of the mass flow rate and the specific volume of the fluid. If the mass flow rate increases,
the specific volume of the fluid decreases because superheating decreases. The friction loss
increases when either the mass flow rate increases or when the specific volume increases. In the
unstable middle point, the size of the variables is such that an increase in the mass flow rate
results in a decrease of the pressure drop, which consecutively increases the mass flow rate more.
36
The extrema can be found by differentiating the pressure loss with respect to the mass flow rate
and setting the derivative to zero. The mass flow rates at the extrema are
<?m,2 = -^-±^tV^-3C3C1 (4.34)
This results in complex values if C2 - 3 C3 C, < 0 and there are no extrema and the flow is stable.
CIf C|-3Q Q =0, there is an inflection point at qm = and the flow is stable. If
C3 - 3 Cj Q > 0, there are two extrema and the flow is unstable between them.
The stability criterion is
C2-3C3 C, <0 (4.35)
Substituting Equations (4.31), (4.32), and (4.33), the stability criterion (4.35) becomes
11 -Gr) ^ ~Vsv ~ 3Vs,^a 4 —r(5vs> + 3v. (4.36)
where
(S),B =------- (4.37)
cp.c
Equation (4.36) shows that the heat flow rate <f>, the pressure loss coefficient k(, and the area of
the entire boiler A do not affect the stability. At a certain pressure the stability criterion is a
function of AhA alone, because the specific volumes, the specific volume derivative, the latent
heat, and the specific heat capacity are constants. In Equation (4.36) the coefficient of the second
order term of Ah a is positive, and the coefficient of the first order term and the constant term are
either positive or negative depending on the thermal properties. This means that the boiler may
have a stable range between two values of the enthalpy change of the fluid in the preheater. The
stability can be determined only by numerical calculation, which is presented in Chapter 8.2.
37
It is not possible to get the frequency of the oscillation using only the equations presented above.
For frequency calculation also the linear momentum equation is needed. One possible way to get
the oscillation frequency is the momentum integration method (Sarkomaa, 1973a). This
examination is left for future study.
4.2.2 Pressure loss coefficient
A total pressure drop across the boiler includes the effect of friction loss, the minor losses, the
acceleration of the fluid, and hydrostatic pressure. In this study gravitation, and thus hydrostatic
pressure, are not taken into account, and all the other effects are included in the total pressure
drop as described in Section 2.1. To simplify the calculations, the total pressure drop is expected
to be proportional to the square of mass flow rate. The error made in this approximation is not
significant, because the friction and the minor losses obey this rule and the effect of acceleration
is only some per cents of the total pressure drop.
The steady state input data received from the off-design boiler model gives the total pressure
drop separately for the preheater, vaporiser and superheater. This data is used to solve the
pressure loss coefficients Af,A, hs, and k^c for the three parts of the boiler. These coefficients are
taken to have the same value for every element in each part of the boiler.
The pressure drop across element j is from Equation (2.17)
A CfA*P.=PH-P,^ (4.38)
where the pressure loss coefficient for one element is % and Ac denotes the area of the tube
element. Carrying out the integration by the trapezoidal rule gives the total pressure drop for one
part of the boiler, in the following for the preheater from node miA to node ni2A
(4.39)
38
Finally the pressure loss coefficient is obtained as
£f,A (4.40)
Using Equations (4.38) and (4.40) iteratively together with the steady state heat transfer
calculations presented in Section 4.4 gives the pressure loss coefficients for the preheater,
vaporiser and superheater.
4.2.3 Pressure drop on the heat source passage
The heat source fluid flows outside the boiler tubes in counter flow through passages formed by
the spaces between the tubes. It is presumed to be a one phase gas flow, for example flue gas. In
the model the pressure drop in the heat source fluid does not have any significant effect on the
calculations, the only effect is through the thermodynamic properties, e.g. the specific enthalpy as
a function of temperature and pressure for real gas.
Because the effect of the pressure change of the heat source fluid is so insignificant, it is possible
to use a very simple model for the pressure drop. In fact, it will be modelled in the same way as
the pressure drop in the process fluid side using Equations (4.38) and (4.40).
39
4.3 Effect of temperature dependence of fluid properties
It is clear that in a realistic model of a boiler the temperature dependence of the thermodynamic
properties of the working fluids are taken into account. The largest influence is in high pressure
preheaters, as the liquid approaches its saturation state and the specific heat of the liquid
increases greatly. The increase is particularly large in the vicinity of the critical point. This can be
seen as a curvature when the specific enthalpy is plotted as a function of temperature. This is
shown for water in Figures 4.3 and 4.4. With higher specific heat, the temperature changes less
for a unit change in enthalpy than when the specific heat is small.
Heat exchanger calculations are typically carried out using the mean specific heat of the liquid.
This means that in the colder end of the heat exchanger the specific heat used is too large, and the
temperature change of the liquid is too small. At the hotter end the specific heat is too low, and
the temperature change is too large. The effect of an erroneous specific heat manifests itself in the
temperature difference between the hot and cold fluids and it may lead to an underestimation of
the heat transfer surface area and to a too small heat transfer rate.
14
......
Temperature, °C
Figure 4.3: Specific heat capacity of water at 18 MPa and at saturation
Specific enthalpy ,j MJ/kg
Figure 4.4: Specific enthalpy of waterat 18 MPa and at saturation
As an example we shall look at a counter flow case:
hot fluid: gas 375 °C to 325 °C
cold fluid: water 180 bar, 320 °C to 357 °C (saturation)
The specific heat of the gas is 1.2 kJ/kgK. Table 4.1 and Figure 4.5 show the temperature
distribution in a heat exchanger when the calculations are made with the mean specific heat of
40
water. Table 4.2 and Figure 4.6 show the temperature distribution of a heat exchanger with the
same area, the same inlet temperatures, and the same overall heat transfer coefficient, but
calculated using a specific heat that changes with temperature according to Figure 4.3.
Table 4.1: Temperature distribution: (mean values)
A(m2) %s(°C) 7>w (°C)
0.00 325.00 320.0039.41 327.63 321.9478.83 330.62 324.16
118.24 334.01 326.67157.65 337.87 329.52197.07 342.26 332.77236.48 347.24 336.46275.89 352.91 340.65315.31 359.35 345.42354.72 366.68 350.84394.13 375.00 357.00
temperature, °C
fluegeaj
heat exchanger area, m3
Figure 4.5: Temperatures in the preheater (mean specific heat capacity)
Table 4.2: Temperature distribution:(temperature dependent values)
A(m2) %i,(°C) %w(°C)
0.00 326.84 320.0039.41 330.35 323.3178.83 333.96 326.67
118.24 337.71 330.12157.65 341.61 333.64197.07 345.73 337.23236.48 350.14 340.87275.89 354.99 344.55315.31 360.49 348.30354.72 367.00 352.16394.13 375.00 356.19
temperature, *C
fluegas},
feed water330 "
heat exchanger area, m3
Figure 4.6: Temperatures in the preheater (specific heat capacity is a function of temperature)
Figure 4.7 shows the temperature difference between the hot and the cold fluid, and Figure 4.8
shows differences in the temperatures between the two cases. In the temperature dependent case
41
the heat flow is 3.7 % smaller than designed, which in this case is less than the uncertainties of
heat transfer coefficients, but the effect of using mean values is seen.
temperature jo difference 19 7Vr„ K 18
17 16 15
accurate
3 200 30(heat exchanger area, m3
Figure 4.7: Temperature difference betweenflue gas and feed water
temperaturedifference, K
7emperatvri\— 7. . j'oS'TW": j
’•fine gaj—fitd wait
heat exchanger area, m3
Figure 4.8: Temperature differences between accurate and mean calculations
As a conclusion it can be seen that in some cases using the mean properties leads to an
overestimation of the temperature difference between the hot and the cold fluid, and this leads to
an underestimation of the heat exchanger area. This happens especially in those preheaters where
the temperature difference between the hot and the cold fluid is small.
42
4.4 Calculation of the initial state
4.4.1 General
The off-design process model gives values for the
temperatures, pressures, mass flows, overall heat
transfer coefficients, and all the structural data of
the boiler. The indexing of the input values is
presented in Figure 4.9. Only the temperatures
are shown, but the other variables obey the same
indices. The intermediate nodal values in the
boiler must be calculated because the
temperature and the pressure distributions in the boiler are not received, only the values at the
boundaries of the boiler parts are known. In addition to this, quantities such as the pressure loss
coefficient must be calculated before the dynamical calculation can start.
Another reason for calculating the initial state is the fact that there may be some need for
adjusting the input values that are obtained from the off-design process model in order to make
all the values internally consistent. The input values are perhaps calculated using mean specific
capacity, or using a different number of calculation elements if the temperature dependence is
regarded. The overall heat transfer coefficients are checked to make sure that the boiler is stable
in the beginning of the dynamical calculation, and that there is not any change of state that is a
result of bad starting values.
The boiler is calculated in elements so that each element is considered as a small heat exchanger.
The steady state calculation of the intermediate nodes is started from the inlet of the preheater
towards the vaporiser, and then continued from the outlet of the superheater towards the
vaporiser, and last, the vaporiser is calculated between the last node of the preheater and the first
node of the superheater. In this calculation the overall heat transfer coefficients are checked so
that the boundary temperatures at points two and three in Figure 4.9 are set to the input values.
The pressure loss coefficients are calculated at the same time.
Vaporiser Super! eater
distance along tube
Figure 4.9: Notation of the input values at the boundaries of the boiler parts
43
Each element in the boiler is calculated in a manner typical of a counter flow heat exchanger. The
values of temperature and pressure are calculated for each node. The mass flows are constant in
the steady state calculation, but vary in a dynamic calculation. When the nodal values of all the
thermodynamic properties are calculated, also the pressure loss constant kt is calculated for all
three parts of the boiler. In addition, the overall heat transfer coefficient k is adjusted. This leads
to an iterative calculation.
To begin the iterations, temperature and pressure distributions are taken to vary linearly in each
part of the boiler. As an example for the pressures of the process fluid and the heat source in the
preheater this gives
Ay = Pea + J'm^ (Pct.2 -Pa*) (4.41)m2A ~mlA
Pr&j = Pr&i + -/'^-frfg-2 -Pbi) (4 42)m2A ~ m\A
where j is the loop index that varies from mlA to »72a. The pressure in the last node of the
preheater t«2a is set to the saturation liquid input value at boundary 2, and the pressure at the first
node of the superheater m\c is set to the saturation vapour input value at boundary 3.
The thermodynamic properties T, p, h, and v that are calculated in the steady state calculation are
placed to the element (2) in the variable arrays described in Section 3.6. These arrays are used in
the dynamic calculation. The time counter n is not shown in the equation above or in the
following equations.
4.4.2 Counter flow heat exchanger
The steady state calculation of the boiler is done using the effectiveness-NTU method. In this
method a typical definition of the heat capacity rate ratio R is that the smaller heat capacity flow
is divided with the larger one. This gives the possibility of calculating also cases in which the
temperature of the other flow does not change, which means that the value of the other heat
capacity flow is an infinity. Either of the flows can be taken to be the minimum heat capacity
flow, since in the initial state the temperature difference between the inlet and the outlet of neither
44
fluid is zero. In the vaporiser the change of the temperature of the process fluid is very small and
this is why it is convenient to take the heat source to be the minimum heat capacity flow. The
equations are valid for all values of R smaller or larger than a unity.
The heat capacity flow is defined to be the specific heat multiplied by the mass flow rate, but for
the process fluid in a discrete element it can be calculated as
C -c a - ^cf a - ^cf,j avcf “ Cp,cf c/m,cf - Vm.cf ~ T T 9m,cf
cf rcfj “ rcf,j-l
Correspondingly, the heat source heat capacity rate, Cfg = Cm;n, is
\.2 - fyg.1Qg = 9m,fg Tf.-y - 7f„jfg.2 ■ -'fg.l
The heat capacity rate ratio is
(4.43)
(4.44)
(4-45)
The overall heat transfer
coefficients kA> kB, and kc
for the preheater, vapor
iser, and superheater ele
ments are first received
from the off-design pro
gram. During the steady
state calculation they are
adjusted as explained
below in Sections 4.4.3 to
4.4.5. The overall heatFigure 4.10: Vaporiser-part-parameter definition
in the mixed elements
transfer coefficients in the mixed elements are calculated using the vaporiser-part-parameters xm
and X2B shown in Figure 4.10. The parameters are defined using the specific enthalpies in the
nodes of the mixed elements and the saturation enthalpies ACf,si and /zcf>. For the preheater side
mixed element the vaporiser-part-parameter and the overall heat transfer coefficient are
45
*1B =r,cf.mlB
^cf,mlB - Kt.mZA
*IB = H-^IbMa + %1B*B
(4.46)
(4.47)
For the superheater side mixed element the vaporiser-part-parameter and the overall heat transfer
coefficient are
- A:f,m2B
X2B - 7---------------7------"cOnlC " %,m2B
^2B — X2B^B + (\~x2q)
(4.48)
(4.49)
The number of transfer units (NTU), i.e.
dimensionless conductance, is for an element with
the area Ae for the preheater and, correspondingly,
for the vaporiser and the superheater
kAAc (4.50)
The effectiveness, i.e. degree of recuperation, is
5 = 1-1 -R
ez(1 -R) - R(4.51)
%Element} j
1 eatsounx
-I" <j) l mcessjhtk
'''''''
3-1 3
Figure 4.11: One element as a counter flow heat exchanger
The temperature differences of a counter flow element, shown in Figure 4.11, are the temperature
difference of the inlet temperatures 0O = 7f&j - 7’cf j-_1, the temperature change of the heat source
fluid e 0O = Tfg_j - 7fgj_i, and the temperature change of the process fluid Rs 0„ = Tcfj - Tcfj_,.
When the temperatures in the colder end of the element are known, the nodal temperatures in
node j are calculated as
T _ s " %j-i vfg.j
s - 1(4.52)
46
T’cf.j = T’ctj.i + Re f^fg.j - ^.jj (4.53)
When the temperatures in the hotter end of the element are known, the nodal temperatures in
node j-1 are calculated as
r _ r=f.j ' R s
Tfg.j-1 = Tfg.j ' 5 fTfgj "
(4.54)
(4.55)
The overall heat transfer coefficient k and the outside heat transfer coefficient k0 are received
from the off-design process model, and the inside heat transfer coefficient k\ is calculated using
them. The outside heat transfer coefficient is based on the outer surface area of the tube and it
includes the outside convection heat transfer and the effect of fouling. It is adjusted after the
adjusting of the overall heat transfer coefficient in the steady iteration loop as
K ~ Ki T (4.56)
where subscript d refers to the design value received from the off-design program.
The calculation of the inside heat transfer coefficient is also based on the outer surface area of the
tube and it includes the inside convection heat transfer, the effect of fouling, and the tube wall
heat resistance. It is calculated as
1 _ 1 1 A, A
(4.57)
4.4.3 Calculation of the preheater
The aim of the preheater calculation is to define the intermediate temperatures so that the
temperature of the process fluid at node /W2a corresponds to the input values from the off-design
process model. This is done in an iteration loop by adjusting the overall heat transfer coefficient
kA.
47
Before the iteration, the input values of point 1 in Figure 4.9 are given to the variables of the
node miA. As the first guess, the nodal pressures of the heat source and the process fluid are set
using a linear distribution according to Equations (4.41) and (4.42). A small part of the preheater
is in the preheater side mixed element. That part is first calculated as a small counter flow heat
exchanger, backwards (counter to the process fluid flow) from the saturation point that connects
the preheater and the vaporiser. This is how the temperature of the last preheater node 7»2a is
achieved. The nodal temperatures of the heat source and the process fluid are calculated using a
linear distribution. The pressure loss coefficients of the heat source and the process fluid are
calculated using Equation (4.40) and the pressures are calculated using Equation (4.38).
The iteration has the following four steps:
1) The values of the node m2A are first calculated as a small counter flow heat exchanger,
backwards from the saturation point, as described above.
2) The temperatures of the intermediate nodes from the node /tzia+1 to the node ttz2a are
calculated element by element as small heat exchangers using Equations (4.43) to (4.45) and
(4.50) to (4.53). The specific enthalpies and specific volumes are calculated as well. This step is
repeated so many times that the temperature of the process fluid T-c^a is stable within 0.001 K.
3) The pressure loss coefficient of the process fluid £f,A is calculated using Equation (4.40), and
the pressures are calculated using Equation (4.38).
4) The temperature of the process fluid obtained in step 2 is compared with the value
obtained in step 1. The iteration is completed when these values differ from each other less than
0.01 K. If the values are not close enough, a new guess for the overall heat transfer coefficient kA
is made using temperatures 71~fm->A from step 2 and the corresponding values of kA. The procedure
is such that when the temperatures are on the same side of the temperature calculated in step 1,
the overall heat transfer coefficient kA is increased or decreased five per cent so that the next
temperature rcf,m2A will be closer to the target value. When the temperatures Tcc^ia are on
different sides of the target, a linear interpolation is used to calculate a new value for kA. Figure
4.12 shows the principle of getting a new guess for the overall heat transfer coefficient kA. In the
figure, the X values are the overall heat transfer coefficients and the Y values are the
temperatures.
48
New values for the outer heat transfer coefficient Zt0,a and the inner heat transfer coefficient k-,,\
are calculated using Equations (4.56) and (4.57) after a proper value for the temperature has been
obtained.
validrange
Figure 4.12 Linear interpolation and extrapolationYo is the target value of the dependent variable and AY is the allowed deviation from it,A® and A® are the values of the independent variable from two previous iteration rounds,1® and I® are the values of the dependent variable from two previous iteration rounds,A® is the new guess for the independent variable.The arrows show the direction of the new guess. In case a both previous values of the dependent variable are on the same side of the target value and the new guess is made forwards, in case b the new guess is made backwards. In case c the previous values of the dependent variable are on different sides of the target and the new guess is made using linear interpolation.
4.4.4 Calculation of the superheater
The aim of the superheater calculation is to define the intermediate temperatures so that the
temperature of the process fluid at node m\c corresponds to the input values from the off-design
process model. This is done in an iteration loop by adjusting the overall heat transfer coefficient
kc.
49
Before the iteration, the input values of point 4 in Figure 4.9 are given to the variables of the
node 7M2c. The first guesses and the iteration follow the procedure described for the preheater,
with the exception that the temperature that is considered is the first nodal value T^ic forwards
from the saturation vapour point, and the nodal values are calculated backwards from the node
OT2c to the node mic. As a result, in addition to the nodal temperatures and enthalpies, the
pressure loss coefficient k[iC, the corrected overall heat transfer coefficient Ac, the outer heat
transfer coefficient Ar0-c, and the inner heat transfer coefficient kuC are achieved.
4.4.5 Calculation of the mixed elements and the vaporiser
The aim of the vaporiser calculation is to define the intermediate temperatures and enthalpies so
that the inlet values at node /»2a and the outlet values at node mIC correspond with the values
calculated in the iteration of the preheater and the superheater. This is done in an iteration loop
by adjusting the overall heat transfer coefficient Ag.
The calculation starts at nodes /tz2a to mm in the
preheater side mixed element, proceeds from node mm
to m2B in the vaporiser, and ends at nodes to miC
in the superheater side mixed element. The mixed
elements are calculated in connection with the
vaporiser elements. The reason for this is that the
mixed elements use the heat transfer coefficients of the
vaporiser.
In the mixed elements the pressure is taken to be uniform, equal to the saturation pressure
obtained from the off-design process model as shown in Figure 4.13. The pressure loss in the
vaporiser elements and the pressure loss constant in the vaporiser are calculated in the same way
as in the preheater. The biggest difference between the vaporiser calculation and those of the
preheater or the superheater is that the temperature of the process fluid is a function of pressure,
that is saturation temperature. Owing to pressure loss in the vaporiser, the process fluid
temperature slightly decreases.
preheater vaporiser superheater
process fluX’d"
____ actual ____
Figure 4.13: Pressure definition of elements in the boiler
50
The temperature of the process fluid changes only slightly, whereas its specific enthalpy changes
significantly. This is why the heat capacity rate of the process fluid is much larger than that of the
heat source fluid. Because the temperature of the process fluid differs in the first and the last node
of an element, the calculation of the heat capacity rate may usually be done in the same way as in
the preheater. If the temperature difference is less than 0.1 K, the value of 0.1 K is used as the
temperature difference. This does not cause a significant error because the heat capacity rate ratio
is very close to zero. As mentioned above, the heat capacity rate of the heat source fluid has been
chosen to be the smaller heat capacity rate.
The change of the process fluid temperature in the vaporiser is so small that it is reasonable to
check the convergence of the iteration from the specific enthalpy of the heat source. This is
calculated from the thermal energy balance of the element, after the enthalpy of the heat source
fluid is achieved, as
~~ (^'m.cf.j-l ^cf.j-1 + (<7m,fg.j fyg.j ~ <7m,fg.j-l fyg.j-l)) (4.58)
The specific enthalpy is used to calculate the quality and the vapour specific volume in a node as
x _ Acfj - &c%l(4.59)ct,j Vsv - Kcsi
vcf,j — (1 ~ ^cfj) Vcf,sl *cf,j Vcf,sv (4.60)
where the subscript si refers to saturation liquid and sv refers to saturation vapour. In the
superheater side mixed element the end node is in the vapour phase, and the temperature and the
specific volume are calculated using the enthalpy and pressure as known values as
rc,j = /(Pc(j,&cCj), and vcf j = f(Tcti,pcfj).
Before the iteration, the nodal enthalpies and pressures of the process fluid are set using a linear
distribution. The nodal temperatures and enthalpies of the heat source are calculated using heat
exchanger equations. The nodal enthalpy of the process fluid is calculated from the heat balance
(4.58), and the quality and the specific volume are calculated using Equations (4.59) and (4.60).
The pressure loss coefficients of the heat source and the process fluid are calculated using
Equation (4.40), and the pressures are calculated using Equation (4.38).
51
The iteration has the following four steps:
1) The temperatures of the heat source fluid at the intermediate nodes from the node twib to the
node 77Z2B are calculated element by element as small heat exchangers using Equations (4.43) to
(4.52). The specific enthalpies and specific volumes are calculated as well.
2) The enthalpy of the process fluid is calculated from the energy balance (4.58) from the node
Wib to the node wic.
3) The pressure loss coefficient of the process fluid Af,A is calculated using Equation (4.40), and
the pressures are calculated using Equation (4.38). After this the saturation nodal temperature of
the process fluid is calculated.
4) The enthalpy of the process fluid Acf,mic obtained in step 2 is compared with the value
obtained in the superheater calculation. The iteration is completed when these values differ less
than 1 J/kg. If the values are not close enough, a new guess for the overall heat transfer
coefficient &b is made using the enthalpies Acf-m2A from step 2 and the corresponding values of &b
by linear interpolation as shown in principle in Figure 4.12.
New values for the outer heat transfer coefficient k0,b and the inner heat transfer coefficient k&
are calculated using Equations (4.56) and (4.57) after the iteration loop has been completed.
4.4.6 Calculation of the process fluid mass and the tube wall temperatures
The mass of the process fluid in an element j is
calculated using the arithmetic mean of the
specific volumes in the nodes as
m=f.j =2F.
Vcf,j-1 + vcf,j(4.61)
where Ve is the volume of the element. The mass
in the mixed element is calculated using an
approximation of two linear specific volume
distributions as shown in Figure 4.14. For the
preheater side mixed element this is
m2A tn IB
Figure 4.14: Specific volume approximation in the preheater side mixed element
52
mcf,j = -- ------------------------ - ----------- 7------------------- (4.62)V"X1b) Vcf,m2A + vc£sl + X1B Vcf,mlB
and for the superheater side mixed element the mass is
2 Vmcf- = ---------------------------------------------------------------- (4.63)
X2B Vcf,m2B + Vcf,sv + 0 " X2b) Vcf,mlCFinally, all the element masses are summed up to get the total mass of the process fluid in the
boiler
m2Cmcf = I>=f.j (4.64)
j=mIA+l
The tube wall temperatures are calculated using the steady heat flow rate from the heat source to
the process fluid in nodes. The heat flow rate per area for a preheater node j is
= (Tfg.j ' Tcf.j) (4.65)
The wall temperature in the node j is obtained from the convective heat flow equation as
tw.j 'f&j KA(4.66)
53
5 Dynamic calculation
5.1 General
When fully functional, the dynamic boiler model will be a part of a dynamic process model. In the
dynamic calculation the inlet temperatures and mass flow rates of the process fluid and the heat
source fluid are obtained from the process model. That is, in the simulation of the boiler, certain
boundary conditions change as a function of time, and as discussed in the previous chapter, the
initial conditions are set by interpolating the results from the steady state process model into all
the interior nodes.
A boiler in its dynamic state is a very complicated system with several dominant equations that
must all be valid at each instant in time. These equations include the heat flow, mass flow rate,
and the thermodynamic properties of the fluids, with their interdependence. In a once-through
boiler an added complexity comes from a lack of fixed heat transfer surface areas for liquid
convection, boiling, and vapour convection. In other words, the boundaries between the
preheater and the vaporiser and between the vaporiser and the superheater move. To find out
how different variables affect each other, a thorough analysis of the relationships between the
thermodynamic properties was made. After this, the calculation order was selected.
The aim of the dynamic boiler model is to calculate the outlet temperature and pressure of the
heat source fluid, the outlet temperature and pressure of the process fluid, and the inlet pressure
of the process fluid. Along with these, boiler heat power and the start and end points of boiling
for the process fluid, that is the positions of the boundaries between the different parts of the
boiler, give useful information.
The main task in the dynamic boiler calculation is to solve the boiler pressure at each instant in
time so that all nodal variables have valid calculated values. The analysis of the thermodynamic
relationships showed that the most important variable that influences the pressure is the total
mass of the process fluid in the boiler. This mass is a function of the specific volumes of the
process fluid at the nodes. Another way to calculate this mass is to add the difference of the mass
54
inflow and outflow rates during a time step to the mass that was in the boiler at the preceding
instant. The mass flow rates are received as time dependent boundary conditions.
The solution method chosen was such that
during each instant in time the pressure was
determined so that the total mass of the process
fluid in the boiler, calculated using the two
methods described above, gave the same result.
To do this, an iterative solution method was
obviously necessary.previousnode
! ) Element Q)\
| _ J
v
-/
presentnode
0)
currenttime(n)
preceding
In the dynamic calculation the notation of
variables for an element j is as presented in
Figure 5.1: subscript j-1 is the previous node, j
is the present node, superscript zz-1 is the
Figure 5.1: Calculation of process fluid nodalvalues from the previous to the present node between the preceding and the current instant in time
preceding instant in time, and n is the current instant. The number of an iteration round is
presented as a superscript in parenthesis, if necessary.
5.2 Analysis of thermodynamic variables and their interdependency
The main aim of the analysis of the interdependency was to find out how strongly temperature,
pressure, specific enthalpy, specific volume, and mass in an element affect each other. Because of
the iterative calculation, it is very important to know the strengths of the dependencies, for some
of them are stronger and some weaker than others. This information is most important for certain
feedback variables that close the dependency circle in the same way as in a control system.
In the following, an example of the functional dependencies for the process fluid in the preheater
is presented. The variables of the current instant in time, which are to be solved at each time step
so that they correspond to each other, are written in italics.
— the heat flow rate from the tube wall to the process fluid is calculated using the thermal
energy balance and the values from the preceding instant,
— change of the specific enthalpy of the process fluid is calculated using the heat flow rate,
55
— the specific enthalpy of the process fluid is calculated using the enthalpy change, the
enthalpy in the inlet of the element, and the preceding mass in the element,
— the temperature of the process fluid is calculated using the enthalpy and the pressure,
— the specific volume of the process fluid is calculated using the pressure and the temperature,
— the mass of the process fluid in the element is calculated using the specific volume and the
known volume of the element,
— the total mass of the process fluid in the boiler is calculated by summing up the masses in the
elements,
— this total mass that is calculated using volumes is compared with the mass calculated using
the preceding mass and the massflow into and out of the element
— a new guess for the new pressure of the process fluid is made using the mass difference
received from the comparison.
This example shows the complexity of the interdependence of the variables and the importance of
careful selection of the calculation order. The example case above was taken from the preheater.
The interdependencies in the superheater are the same as in the preheater. In the mixed elements
and in the vaporiser they are far more complicated due to the influence of the saturation values
that are functions of the pressure. The calculation equations that are the basis of the analysis of
the thermodynamic variables and their interdependency are presented in Section 5.3. This section
also presents the calculation order selected on the basis of the analysis. The results of the analysis
are presented in graphical form in Appendix A.
In the preheater and the superheater, two important feedback variables were found: a strong
interdependence between the total mass difference and the pressure through the pressure guess,
and a weak interdependence between the specific volume and the pressure through the pressure
drop. The mass flow in the superheater changes between the nodes because the vapour is
compressible. If the mass flow rates in the nodes are calculated using the nodal values of the
pressure, one more feedback from the pressure to the heat flow rate exists.
In the vaporiser, there is also feedback from the pressure to the saturation liquid and the
saturation vapour values as presented in Appendix A, Figure 2. The vaporiser-part-parameters
also have an effect on the mass of the process fluid in the mixed elements as presented in
Appendix A, Figure 3.
56
5.3 Iteration of the pressure level
The aim of the dynamic boiler calculation is to solve the pressures for the boiler at each instant in
time so that all thermodynamic properties have proper values in each node. For the node j and the
element j at current time n this includes the following variables: the pressure /?"fj, the temperature
7^j, the vapour fraction in the vaporiser node, x"fj-, and the specific volume v"£j, as well as the
mass of the process fluid in the element, m"fj-.
The principle of the solution is to guess such a value for the pressures that the total mass mcCtV
that is calculated using the volumes, and the mass mC(A calculated using the preceding mass and
the mass flow into and out of the element, are equal. The guess for the pressures is made by
guessing the pressure change in the first node of the preheater wzIA during the time step (n-l)At to
Mr as
'dwA (5-1)
The calculations are made in the flow direction of the process fluid so that the values of the
preceding instant in time n-1 are known both at the previous node j-1 and the present node j, and
the nodal values of the current instant in time n are known at the previous node j-1. The total
mass of the process fluid and the mass in every element are known in the preceding instant, as
well.
The steady state heat transfer calculation in the elements was made using the overall heat transfer
coefficient so that heat transfer from the heat source fluid to the process fluid was calculated
directly. In a dynamic state the effect of the tube wall heat capacity makes it necessary to
calculate the heat transfer in two parts as presented in Figure 5.2: first, the heat flow rate $> from
the heat source fluid to the tube wall, and then the heat flow rate $ from the tube wall to the
process fluid. The outside heat transfer coefficient ka and the inside heat transfer coefficient h that
are calculated in the steady state calculation according to Equations (4.56) and (4.57) are used in
this calculation.
57
Before the iteration of the pressure, the mass of the process fluid in the boiler is calculated using
the previous total mass 7wCf,sum and the mean mass flows into and out of the boiler during the time
step. This is called the biown boiler mass mcf,q
^cf,q — ^cCSum G2 (^m.cf.mlA (Zm.cf.mlA) — ~2 C^m.c£jn2C 9m.cCm2c)) ^
The mass flow of the process fluid in a certain instant is expected to be a constant in the
preheater. If the mass flow at the preheater inlet and the superheater outlet differ, the change
occurs only in the vaporiser and the superheater, and it is taken to be linear with respect to the
area as
nn _ „n , J~m2A n 1Vm.cf.j Vm,cf,m2A (Vm,cf,m2C ”m,cf,m2A-' (5.3)
The new nodal enthalpies of the process fluid are calculated before the iteration of the pressure,
as described in Section 5.3.1.
The iteration loop for determining the boiler pressure consists of the following steps:
1) Calculation of the pressure, temperature, and specific volume of the process fluid in the
nodes, and the process fluid mass in the elements, which is described in detail in Section 5.3.2. At
the first iteration round the pressure is expected to be the same as in the preceding time step, that
is, the pressure change in the first node Ap = 0. Because the calculated variables affect each
other, the calculation of the boiler elements is repeated so many times that the change of the total
mass between successive iteration rounds is less than 0.001 %. Typically two or three iteration
rounds are needed.
2) The pressure change and the total mass values of the current iteration round are set into
memory to make the new pressure guess. There are results from the two best iteration rounds this
far in the memory and the values are updated in the following way. If the old values of the total
mass and are on different sides of the known boiler mass mC[A, then the old value that
is on the same side as the mass calculated in this iteration round mcf,v is replaced with the new
value. If the old values of the total mass are on the same side of the known boiler mass mcf,q then
the old value that is farther away from the known boiler mass is replaced.
|
58
3) The convergation is checked comparing the total mass mCf_v just calculated in step 1 and the
value /McCq calculated before the iteration loop, using Equation (5.2). The iteration is finished if
the masses differ less than 0.01 %. If the difference between the masses is too large, a new guess
for the pressure change Ap is made. During the first calculation round the guess is zero. During
the second iteration round a value received from the previous time step Apmcm, Equation (5.4), is
used. This has proved to be a good practice to lessen the oscillations of the pressure change.
During the third iteration round and after that the guess is made with the latest two pressure
guesses Ap(1) and Ap(2) and calculated masses and using the linear extrapolation or
interpolation in the same way as described in Section 4.4.3, Figure 4.12. If the latest two values
of the boiler mass are both either smaller or greater than the known boiler mass, a new guess for
Ap is made changing the latest value with 27 per cent of the difference between Apm and Ap(2).
After the iteration loop, a weighed new guess for the pressure change for the next time step is
made using the old guess of the pressure change with a weight of 2/3 and the latest pressure
change with a weight of 1/3 as
APmem = f2 ^Pmem.old + Ap) / 3 (5.4)
The reason for this is that it lessens the oscillation of the pressure change in the iteration, because
usually the next pressure change is close to the latest one.
The temperatures of the tube wall are calculated as described in section 5.3.3, and the calculation
of the temperatures of the heat source fluid is described in section 5.3.4. After the calculations of
the current instant in time have been finished, the nodal values of the preceding time are replaced
with the latest values so that the first elements of the arrays described in section 3.6 are replaced
with new values.
59
5.3.1 Nodal enthalpies of the process fluid
New nodal values for the enthalpies of the
process fluid are calculated using the
temperatures and masses of the preceding
instant in time. The changing boundary
conditions are taken into account so that the
current time values are used for the mass flow
and the inlet enthalpy. The new nodal enthalpies
are used to calculate the enthalpy of the next
node.
Element j
.tubem II
Figure 5.2: Energy flows in an element
To start the dynamic calculation, the inside heat transfer coefficients k-^, k\$, and k-,tC are chosen
according to the element type. For the mixed elements they are calculated using the vaporiser-
part-parameters Xib and X2B as described for the overall heat transfer coefficient in Section 4.4.2,
Equations (4.46) and (4.48). The heat flow rate from the tube wall to the process fluid in an
element shown in figure 5.2 is calculated using the logarithmic mean temperature difference as
/"rn-l 7H-I 1 /T'n-I '7'n—1 ij.n = j. A (Aw.j-1 - Af.j-1/ - (Aw,j - Af.j ) ri,j *i "e Tn-1 Tn-1
1 Aw.j-1 - -'cf.j-1 7H-I 7-n-lAw.j -Ar.j
(5.5)
If the temperature difference between the tube
wall and the process fluid is very close to
constant or the temperatures are crossed as
shown in Figure 5.3, the logarithmic mean
temperature difference is not available and the
mean arithmetic temperature difference is used
instead. This gives the possibility to calculate
cases where the temperature of the process fluid
entering the element suddenly changes to a valueFigure 5.3: "Crossing" temperatures as a
result of a sudden rise of the process fluid temperature
60
greater than the temperature of the tube wall.
- w. 4% + (5.6)
The energy balance of an element was expressed in discretised form in Section 2.2.2 as
w-p): -(p =(&+(?.&)# - wor')# (^)'c
Some further simplification of this equation is needed to carry out the pressure iteration
successfully. When the difference forward is made using the values from the previous instant of
time for the mass in the element me = pVc and the pressure p, and approximating the change of
the specific enthalpy in successive nodes to be equal, the specific enthalpy change of the process
fluid in node j, may be solved from the following thermal energy balance
This energy balance uses the current time values of the mass flow and the inlet enthalpy. The
specific enthalpy of the process fluid in node j is thus
Acfj = Acnfj + Qyj At (5.8)
5.3.2 Nodal temperatures and pressures and process fluid mass in the
elements
The calculation of the temperature, pressure, and specific volume of the process fluid in the
nodes, and the mass in the elements is presented in this section. The nodal enthalpies of the
process fluid are calculated before this as described in Section 5.3.1.
The enthalpy of the first node of the preheater in current time is known as a boundary value.
Using the pressure change guess Ap, the new pressure and specific volume in this node are
calculated according to Equation (5.1) as
61
Pct,m 1A = Pcf.ml A + AP @.9)
and the specific volume of the first preheater node is calculated as a function of temperature and
pressure. The calculation then proceeds element by element from node t«ia+1 to node m2c in the
direction of the process fluid flow: the preheater elements are calculated first, then the preheater
side mixed element, after that the vaporiser elements, then the superheater side mixed element,
and finally the superheater elements.
In a preheater element, the saturation liquid temperature Tcf,aiK and enthalpy /zecaux that correspond
to the pressure of the previous node are calculated first. If the enthalpy of the present node, Acny,
is less than the saturation enthalpy Ac&mx, then the element is a preheater element and the node j is
a preheater node, otherwise the element is a preheater side mixed element and the node j is a
vaporiser node.
For the preheater node, the pressure is calculated using the pressure drop equation (4.38). A
geometric mean is used for the square of the mass flow and an arithmetic mean is used for the
specific volume as
Pet,] = Plf,]A - 2 &CA <7m,cf,j-l tfm.cfj (vcf,j-l + vcf.j^ (5.10)
Next, the temperature rcy is calculated as a function of pressure and specific enthalpy, and the
specific volume v"f j- is calculated as a function of temperature and pressure. The mass in the
element, m"g, is calculated using the nodal specific volumes and the known volume of the
element as described in Section 4.4.6.
In the preheater side mixed element, when calculating node j, the number of the last node of the
preheater and the number of first node of the vaporiser are set as /M2a -j-1 and m1B =j. The node
t»2a is a preheater node and mm is a vaporiser node. The saturation liquid temperature and
enthalpy are set to the saturation auxiliary values 7cW = rcf,aux and hzw = Acr,aux calculated as
presented above, and the saturation specific volume is calculated. The pressure is set to be the
same in both nodes of the mixed element because of the simplification that there is no pressure
loss in the mixed element, = pCf j.,. The temperature of the node mm is set to the saturation
value that corresponds to the pressure, and the vapour fraction x”fj, the specific volume v”f j,
62
the vaporiser-part-parameter xlB, and the mass of the process fluid in the mixed element, m"f j,
are calculated as described in Section 4.4 using Equations (4.59), (4.60), (4.46), and (4.62).
In a vaporiser element, the saturation vapour enthalpy hcf,aux that corresponds to the pressure of
the previous node is calculated first. If the enthalpy of the present node, , is less than the
saturation enthalpy hcf-aux, then the element is a vaporiser element and the node j is a vaporiser
node, otherwise the element is a superheater side mixed element and the node j is a superheater
node.
For a vaporiser node, the pressure is calculated using the pressure drop equation (4.38), and the
temperature is set to the saturation value that corresponds to the pressure. The vapour fraction
and the specific volume are calculated as in the preheater side mixed element described above,
and the mass in the element is calculated in the same way as in the preheater element.
In the superheater side mixed element, when calculating the node j, the number of the last node of
the vaporiser and the number of the first node of the superheater are set as /W;>b = j-1 and mic = j.
The node m& is a vaporiser node and m1C is a superheater node. The pressure is set to the same
value as in the previous node because there is no pressure loss in the mixed element,
Pcii = pcf.j-i • The saturation vapour temperature and enthalpy are set to the saturation auxiliary
values Tcfsv = and hcf4] = Z?cf,aux that correspond to the pressure, and the saturation specific
volume is calculated. If there is no superheater, the saturation vapour values are set in the
pressure of the last node of the vaporiser.
In the superheater node mlc, the temperature is calculated when the pressure and the specific
enthalpy are known, the specific volume is calculated as a function of temperature and pressure,
and the vaporiser-part-parameter of the superheater side mixed element X2B is calculated using
Equation (4.48). The mass in the mixed element is calculated using an approximation of two
linear specific volume distributions with Equation (4.63).
In a superheater element, the pressure is calculated using the pressure drop equation, the
temperature is calculated when the pressure and the specific enthalpy are known, the specific
63
volume is calculated as a function of temperature and pressure, and the mass in the element is
calculated as described above.
Finally, all the masses of the elements are summed up to get the total mass of the process fluid in
the boiler as
m=r.v =2Xf.jj
(5.11)
5.3.3 Tube wall temperature
Because of the calculation method used, the current time values of the tube wall or the heat
source temperature do not affect the process fluid values. So, new values for these are calculated
after the iteration of the boiler pressure. The temperature of the tube wall in a dynamic state is
calculated using an energy balance which includes the outside heat flow from the heat source to
the tube wall <fa and the inside heat flow from the tube wall to the process fluid <j\. The inside heat
transfer coefficient k\ includes the thermal resistance of the tube wall. The inside heat flows are
calculated at the same time as the nodal enthalpies of the process fluid using Equation (5.5) or
(5.6).
The boiler is divided to E elements and it has
2?+l nodes. So there is one node and also one
tube wall temperature more than there are heat
flows. This is why the tube wall temperatures
are calculated so that the thermal energy balance
of the tube wall temperature 7tWj includes half of
the heat flows of the previous element <f>0j.i and
$j.i and half of the heat flows of the next
element <j>0j and as shown in Figure 5.4. At
the first node of the preheater only half of the
heat flow of the first element is used, and at the last node of the superheater only half of the heat
flow of the last element is used. In this way there are E+\ energy balances, that is, one energy
balance for every node.
Elemental \ Elementj
heat source
tube v all
process fluid
Figure 5.4: Energy balance of the tube wall node
64
The outside heat transfer coefficient in the element is chosen according to the element type in the
same way as described for the inside heat transfer coefficients. The outside heat flow rate is
calculated using the logarithmic mean temperature difference as
/7TI-1 7111-1 \ 7111—1 \(-'fgj.l - AwJ-1/ " ~Aw,j )
nr'n-l o^n-lKi = 4 4In-
7»n—1 710—14&j -■‘tw.j
(5.12)
If the temperature difference of the heat source fluid and the tube wall is very close to constant or
if the temperature of the heat source fluid in one end of the element is less and in the other end
greater than the temperature of the tube wall, a logarithmic mean temperature difference is not
available, and a mean arithmetic temperature difference is used instead as
- 44 (5.13)
ATThe change of the temperature of the tube wall, —, is calculated from the thermal energy
balance as
A7;w _ \(4,h + - \(fx,y\ + Ki)At ^■p.tw^e.tw
(5.14)
For the first node of the preheater, when calculating element j = OTia+1, it is
A7L _ +At ^p.tw^e.tw
(5.15)
and for the last node of the superheater, when j = m2c, it is
A7^ At <'p,tw^e.tw
(5.16)
The new temperature of the tube wall is calculated as
(5.17)
65
5.3.4 Temperature of the heat source fluid
The mass of the heat source fluid in the boiler is expected to be so small that energy storage in it
has no significant effect, and it is assumed to be at steady state. The temperature of the heat
source is calculated in the direction of the flow of the heat source fluid from the last to the first
node in the boiler.
There are two ways to calculate the heat flow rate from the heat source fluid to the tube wall.
The first one is to use the temperature difference between the fluid and the tube wall, and
calculate the heat flow rate $,j using Equation (5.12) or (5.13). The second way is to use the
equation for the cooling of the fluid as
This equation shows that the heat flow rate $,,m is a function of the unknown temperature rf”j_,
only. In the steady state the heat flows <j>0j and <j>0fa are equal.
The greatest possible value of the heat source
temperature, T^_u is the inlet temperature
2}|j, and the smallest possible value is the tube
wall temperature j_,. This is why the heat
source temperature Tf&j-i must lie on the line
that goes through the point $,,m( T’tw.j-i) and the
zero point %) in the heat flow rate
temperature diagram. The solution for T^_x is
rF
Figure 5.5: New temperature guess in the iteration of the heat source temperature
found by iteration on this line in the point where
the heat flow rates <j>0j and $,,m are equal as shown in Figure 5.5.
66
Having two successive guesses and 7^2) for the unknown heat source temperature, and the
heat flow rates <j> <f> o.L calculated using them, the next guess for the temperature is
calculated as
%K,=B\ +B2
(5.19)
where
Bi
B2
TP) _ TO) Jfg ifg
(1)
(5.20)
(5.21)
The iteration is completed when the difference between the heat flow rates <f>0j and <f>0fa is less
than 0.01 %. After the calculation of the nodal temperatures, the specific volume of the heat
source fluid is calculated, and the new nodal pressures of the heat source fluid are calculated as
described in Section 4.2.3.
67
6 Calculation program
The calculation equations presented in Chapters 4 and 5 have been written to a Pascal program
using the Turbo Pascal 7.0 (© Borland International Inc.) programming language. The program
consists of a main program and three program units DYNSTAT.PAS, DYNCALC.PAS, and
DYNFUNC.PAS. In addition to these, the program uses routines that were made during the
development of the steady state process model and the off-design process model.
The size of the main program is 780 lines and 25 kBytes, the size of DYNSTAT.PAS is 1330 lines
and 42 kBytes, the size of DYNCALC.PAS is 890 lines and 28 kBytes, and the size of
DYNFUNC.PAS is 500 lines and 16 kBytes. The size of the compiled program is 175 kBytes.
The main program first sets the initial data received from the off-design process model to the
variables of the dynamic calculation and asks the user for time step and test case, then calls the
initial steady calculation procedure, and after that starts the time loop in which it calls the
dynamic calculation procedure. In the time loop the main program also changes the boundary
values in the course of time. The input data is given as constants in the main program. At the
present stage of program development the input data at the boundaries of the heat transfer areas
in the boiler can be changed only by making the changes into the program.
The main program calculates the smallest number of elements in the boiler, so that the smallest
part of the boiler has at least two elements, and then suggests this value while asking the user to
give the number of elements in the boiler. Then the program asks for the time step and the names
of the two output files. The first of the output files is for saving the selected nodal values at
selected instants in time. The second of the output files is for saving the temperature and pressure
distribution at selected instants in time. Typically, the data is written into the first file at time steps
of 0.1 seconds from 0 seconds, which is the beginning of the simulation, to 10 seconds; at time
steps of 1 second up to the running time 100 seconds, and at time steps of 10 seconds after that.
Into the second file, the data is written at time steps of ten seconds. During the program run, the
program writes selected nodal values on the display so that the user is able to follow the progress
of the calculation. After the program run, the output data is imported into EXCEL (© Microsoft
Corp.) worksheets for presenting the results as can be seen in Appendix B.
68
The procedures in the unit DYNSTAT.PAS control the calculation of the steady nodal values in
the boiler. There are subroutines for calculating the steady nodal values of the preheater, the
vaporiser, and the superheater.
The procedures in the unit DYNCALC.PAS control the calculation of the dynamic nodal values in
the boiler. There are subroutines for doing the pressure iteration loop in which the subroutines for
calculating the nodal specific enthalpy of the process fluid, the nodal values of pressure,
temperature, specific volume, mass in the elements, and the total mass in the boiler, as well as the
nodal temperatures of the tube wall and the heat source are called.
The program unit DYNFUNC.PAS has several writing routines and functions that calculate the
thermodynamic properties of the fluids using the fast method. The subroutines in DYNFUNC.PAS
call the thermodynamic properties procedures that were made during the development of the
steady process model and the off-design process model.
69
7 Test runs of the dynamic boiler model
7.1 Input values from the steady state off-design program
This study presents a dynamic model of a small once-through counter flow boiler, which is a part
of a dynamic process model. The boundary and initial values for the boiler model are received
from the off-design process model at the boundaries of the three parts of the boiler, as described
in Chapters 1, 2 and 3. The purpose of the boiler model is to calculate the pressures and the
outlet temperatures of both the heat source fluid and the process fluid.
-4000-1
^'‘^satu'ration line
condenserKecuflerMoE.
specific entropy (kJ/kgK)
Figure 7.1: Organic Rankine cycle used in the test runs in the h,s diagram of toluene
The process in the selected test case consists of a boiler, a high speed turbine, a condenser, a feed
pump, and a recuperator, as shown in Figure 3.1 on page 18. Toluene is used as the organic
process fluid, and flue gas from natural gas combustion with the excess air ratio of 2.2 is used as
the heat source fluid. The known input values are the temperatures and pressures of both the heat
source and the process fluid, as well as the mass flows of both fluids. These values are presented
numerically in Table 7.1 and in the h,s diagram in Figure 7.1. Also, the overall heat transfer
coefficients, the heat source side heat transfer coefficients, the heat exchanger surface areas, and
70
the diameter, mass and volume of boiler tubes are known. The test case is selected so that the
process corresponds to typical values being planned. Unfortunately there is no possibility to
compare the results of the test runs to measured values in a real process, because these kinds of
power plants have not been built yet.
Table 7.1: Input values from the steady state off-design program
point fluid temperature pressure
(°Q (kPa)
preheater inlet heat source 190.9 99.470
process fluid 166.7 3314.7
vaporiser inlet heat source 324.5 99.852
process fluid 299.1 3234.9
superheater inlet heat source 382.6 99.948
process fluid 297.9 3188.2
superheater outlet heat source 420.0 100.000
process fluid 335.7 3150.0
mass flows heat source 5.90 kg/s
process fluid 2.53 kg/s
The heat transfer areas of the boiler are for the preheater 82.0 m2, the vaporiser 17.9 m2, and the
superheater 5.9 m2, ant the total for the entire boiler is 105.8 m2. The overall heat transfer
coefficients and the heat source side heat transfer coefficients are for the preheater 428.2 W/m2K
and 623.8 W/m2K, for the vaporiser 438.2 W/m2K and 654.3 W/m2K, and for the superheater
504.9 W/m2K and 669.0 W/m2K.
71
7.2 Test runs
The boundary values that change in the course of time during the dynamic calculation of the
boiler are the temperature, the pressure and the mass flow of the heat source fluid entering the
superheater, and the temperature and the mass flow of the process fluid entering the preheater.
The mass flow of the process fluid at the boiler outlet changes in the course of time depending on
the pressure and the specific volume of the process fluid, because a simplified turbine model has
been used to characterize the outflow from the boiler.
The boiler model was tested using step changes and linear changes of the boundary values in nine
dynamic test runs as presented in Table 7.2. Changes of the boundary values were made at
running time 0 seconds. A time step of 0.05 seconds was used in the calculation and all test runs
used the new, fast calculation method for thermodynamic properties. The results were printed
every 0.1 seconds from start to 1 second, every second from 1 to 100 seconds, and every 10
seconds after 100 seconds. Graphs for the results of the test runs are presented in Appendix B.
Table 7.2: Changes of the boundary values in the test runs
step change linear change
heat source
temperature Test run 1:
-50 °C
Test run 2:
-10 °C/s for 10 s
fluid mass flow Test run 3:
-20 %
Test run 4:
-10 %/s for 9 s
process
temperature Test run 5:
-20 °C
Test run 6:
-0.5 °C/s for 100 s
fluid mass flow Test run 7:
-20 %
Test run 8:
-1 %/s for 90 s
steady case Test run 9:
no change of input values
72
8 Results and discussion
8.1 Corrected heat transfer coefficients and calculated constants
As a result of the steady state calculation the corrected heat transfer coefficients (in W/m2 K) are
obtained as
part of the boiler overall heal source side
— preheater 496.0 722.6
— vaporiser 438.4 654.6
— superheater 503.5 667.1
The great difference in the heat transfer coefficients of the preheater between the input value
428.2 W/m2 K and the corrected value 496.0 W/m2 K is a result of the fact that the steady state
off-design model calculated the preheater using a mean specific heat capacity of the process fluid.
The calculated pressure loss coefficients (in m"6) are
part of the boiler process fluid side heat source side
— preheater 240,300 0.2264
— vaporiser 318,400 0.3020
— superheater 269,500 0.3776
8.2 Stability of the boiler at steady state
The stability of the boiler at the steady state was examined using the Ledinegg criterion in Section
4.2.1. The pressure loss coefficient can be solved from Equation (4.23) as
kt =Ap
(aava + 4svb +(8.1)
The areas in Equation (8.1) must be solved using the presumption of constant heat flow rate per
area. Using the numerical values received from the off-design model, the areas of the boiler parts
are
73
4\ / A <7m,tot 61.08 m9
(8.2)
Ab = — Aq jot = 27.08 m29
(8.3)
Aq = A - (Aa + Ab) = 17.64 m2 (8.4)
The areas are somewhat different compared with the real areas, 82.0 m2, 17.9 m2, and 5.9 m2.
The reason for this is the higher heat flow rate per area at the vaporiser and the superheater in the
real case.
The pressure loss coefficient is calculated to be
kf= 55988 m"^
and the coefficients of the stability criterion
(4.35) are
C3 = 5559.8 Pa/(kg/s)3
C2 = -49469 Pa/(kg/s)2
Ci = 154672 Pa/(kg/s)
The stability criterion is calculated to be
pressure drop 350 - (kPa) 3oo-
boiler massflow design value
inversion point
mass flow (kg/s)
Figure 8.1: Pressure drop as a function of the mass flow
C22-3C,C, = -132.6-106 < 0 (8.5)
which means that the boiler is stable in the steady
state. The pressure drop of the boiler as a
function of the process fluid mass flow is
presented in Figure 8.1.
Equation (4.36) on page 36 gives the stability
criterion as a function of the enthalpy change in
the preheater AAa as
b2Ah2A + blAhA +b0< 0 (8.6)
stability
too 200 300 400 Ma (kJ/kg)
Figure 8.2 Stability of the boiler as afunction of the enthalpy change in the preheater
where the coefficient of the second order term is
74
b2 = k}A2[^ = 0.010533 (ml kg)2(J/ kg)"2(8.7)
in which the coefficient B is defined in Equation (4.37). The coefficient of the first order term is
h = k2A21 (Bl - Vsv -3vsI) = -5915.1 (ml kg)2(J/ kg)"1 (8.8)
and the constant term is
2 \/2 = 655.98 • 106 (ml kg)2 (8.9)(5^ + 3vJ/ + [^
The roots of Equation (8.6) are A/?a,i = 152 kJ/kg and AAa.2 = 410 kJ/kg, which means that the
boiler is instable at small enthalpy changes of the fluid in the preheater, stable between the
enthalpy changes from 152 kJ/kg to 410 kJ/kg, and unstable at large enthalpy changes. This is
shown in Figure 8.2. The design value of the enthalpy change is 343 kJ/kg, which shows that the
design state is well inside the stable range.
8.3 Discussion of the results of the test runs
The problem with verifying the correctness of the results achieved using the dynamic boiler model
is that there is no possibility to compare them with measurements made in a real boiler plant. This
is why the only way is to try to determine whether the obtained results are intuitively reasonable,
and the results change logically when the boundaiy conditions are changed. The tests were
chosen in order to find the responses to step and linear changes, which is the typical way of
describing the operation of the boiler in a dynamic state.
The changes made in the boundary conditions were chosen to be decreases in temperatures and
mass flow rates because the initial state of the boiler was that of full power. As a result of this,
many tests end when the vapour in the boiler outlet contains a notable amount of liquid. It is
necessary to model this kind of operation of the boiler for the simulation of the turbine by-pass
valve, as well.
75
The results of the test runs are presented as graphs in Appendix B. The figures show the
temperatures (Figure 1), the pressures (Figure 2), the mass flows (Figure 3), the boiler mass
(Figure 4), and the relative areas of the preheater and vaporiser parts, and the vapour fraction at
the superheater outlet (Figure 5) in the boiler. Temperatures, pressures, and mass flow rates are
plotted at the boiler inlet and outlet.
Steady state test
Steady state test 9 was made to test the numerical instabilities of the calculation. The figures on
pages 9.1 and 9.2 in Appendix B show the values of the variables compared with the initial
values. The oscillations of the temperatures are less than 0.5 °C except for the tube wall
temperature at the boiler outlet, which shows an increase of2.3 °C in 3000 seconds. This maybe
a result of the somewhat different calculation method in the steady state calculation that uses the
heat transfer coefficients, and in the dynamic calculation that uses the inside and outside heat flow
rates. The change of the tube wall temperature seems to have no effect on other temperatures,
which shows that the initial state of the process fluid and the heat source fluid is stable.
The small oscillation of the outlet mass flow rate is a result of the numerical instability of the
turbine model. This causes also the pressure oscillations which are small, mostly less than 1 kPa.
The oscillation of the total mass is very small, less than 0.02 kg compared to the total mass of
120 kg.
As a conclusion, the differences compared with the initial values are so small that the effects of
numerical oscillations are negligible in the dynamic calculation.
Changes of the temperature and the mass flow of the heat source
In test runs 1 to 4 the effects of changes in the temperature and mass flow rate of the heat source
fluid were tested. The mass flow rate and the temperature of the process fluid at the boiler inlet
were constant in these cases.
76
Decreasing the temperature or the mass flow rate of the heat source fluid causes a decrease of the
heat power in the boiler. As a result of this, the superheat of the process fluid decreases, and in
cases 1, 2 and 4 the superheater part of the boiler vanishes and the vapour flowing out of the
boiler contains liquid. The relative areas of the preheater and the vaporiser are increased and as a
result of this, the total mass in the boiler increases. In these cases, calculation was stopped when
the superheater part of the boiler vanished. Case 3 shows how the boiler closes to a steady state
that corresponds to the new boundary values. The steady state is reached in about 20 minutes.
The pressure of the process fluid is decreased in all these cases. It is mainly a result of the
decrease of the superheater part of the boiler. The pressure decreases when vapour is replaced
with liquid, and the total mass of the fluid increases when the mass outflow of the boiler is almost
constant.
The boiler model is also able to calculate cases in which the temperature of the heat source
decreases below the temperature of the tube wall at the boiler outlet, as can be seen in case 2.
After a short time the tube wall loses its thermal energy and its temperature gets lower than the
temperature of the heat source.
In the following three figures the temperature profiles of the process fluid and the heat source, as
well as the pressure of the process fluid in the boiler, are shown in test run 4 at times 0, 10, 20,
40, and 60 seconds. The temperature drop in the heat source is large after 9 seconds because the
mass flow is only 10 % of the initial value. Figure 8.3 shows that the superheater has vanished in
40 seconds, and due to the small mass flow rate of the heat source and the increased temperature
difference between the heat source and the process fluid, the temperature drop in the heat source
is large. In Figure 8.5 the position of the superheater side mixed element can be seen to move
from 100 m2 to the boiler outlet. The position of the preheater side mixed element is all the time
about 80 m2. Test run 4 was stopped when the quality at the boiler outlet fell below 80 %.
As a conclusion of the heat source side tests, the model gives results that are logical in the
direction of the changes, and the order of magnitude in the time scale of the changes is also as
expected. Comparisons with future measured values will give more precise information.
77
Figure 8.3 Process fluid temperatures in the boiler, initial heat source temperature is also shown
heal source fluidtemperature <8o (°C) 4 60
-40 s
boiler area (m2)
Figure 8.4 Heat source fluid temperatures in the boiler
78
process fluid pressure 3.50 (MPa)
3.40
3.30
3.20
3.10
3.00
2.90
2.80
2.70
2.60
2.50
2.40
boiler area (°C)
Figure 8.5 Process fluid pressures in the boiler
Changes of the process fluid temperature and pressure
In test runs 5 to 8 the effects of changes in the temperature and the mass flow rate of the process
fluid were tested. The mass flow rate and the temperature of the heat source fluid at the boiler
inlet were constant in these test runs.
A decreasing in the temperature of the process fluid causes an increase in the temperature
difference between the heat source and the process fluid, which increases the heat power of the
boiler. At the same time the enthalpy change needed to heat the liquid to saturation increases, and
as a result the relative size of the preheater and the total mass of the process fluid are increased,
as can be seen in the temperature drop in cases 5 and 6. In cases 5, 6, and 7 the boiler closes to a
steady state that is reached in about 20 minutes.
The changes of the heat source temperature follow the changes in the inlet temperature of the
process fluid as shown on pages 5.1 and 6.1 in Appendix B. The outlet temperature of the
process fluid, as well as the outlet temperature of the heat source fluid is increased, as can be
79
expected, when the inlet mass flow rate of the process fluid is decreased, which is shown in cases
7 and 8.
The changes of boiler pressure are small in cases 5 and 6 when the inlet temperature of the
process fluid is decreased. In cases 7 and 8 when the inlet mass flow rate of the process fluid
decreases, the decrease of the pressure is noticeable. As a result of the step change in the mass
flow of the inlet process fluid in case 7, the pressure decreases very quickly. This causes a short
decrease in the process fluid outlet temperature and an increase in the vaporiser area when the
vaporisation is increased quickly. After the pressure has been stabilised, the superheater and
vaporiser areas start to increase and the total mass is decreased. The outlet mass flow calculated
by the turbine model reaches the inlet mass flow in less than 100 seconds.
The total mass in the boiler is increased in the temperature drop cases as a result of the increase
of the relative preheater and vaporiser areas. Vice versa, the total mass is decreased when the
relative area of the superheater is increased in the cases where the inlet mass flow rate of the
process fluid is decreased.
In case 8 the final mass flow rate of the process fluid is only 10 % of the initial value. The results
on pages 8.1 and 8.2 show how the total mass in the boiler is decreased when the mass flow of
the process fluid into the boiler is decreased to a very small value. As a result of a large pressure
decrease, also the mass flow at the outlet of the boiler decreases, though it is greater than the
inflow all the time. As a result, the outlet temperature of the process fluid increases close to the
heat source inlet temperature, the heat source outlet temperature rises, the relative superheater
area reaches almost the entire boiler, and the total mass of the process fluid goes almost to zero.
The run was stopped at 230 seconds when the pressure of the process fluid reached the saturation
pressure corresponding to the inlet temperature.
As a conclusion of the tests on the process fluid side, the model shows reasonable results both on
temperature changes that cause small alterations in the process state and on mass flow rate
changes causing very large alterations.
80
8.4 Discussion of the simplifications
The effects of the simplifications made in the pressure drop calculation in Section 4.2 need some
further examination. The total pressure drop received from the off-design model includes the
effects of friction loss and the minor losses of one-phase flow in the preheater and the
superheater. For the two-phase flow in the vaporiser, also the effect of acceleration is included in
the pressure drop. The effect of the hydrostatic pressure drop is not taken into account in any
part of the boiler.
The question is whether the pressure loss coefficient kf includes all the components of the
pressure drop in such a precision that the inaccuracy of the calculation is negligible.
A calculation based on the input data obtained from the off-design model was made to check the
effect of the acceleration on pressure drop in different parts of the boiler. The results show that in
the individual parts of the boiler the relative extent of the acceleration on pressure drop is 0.9 %
in the preheater, 14 % in the vaporiser, 8 % in the superheater, and 6 % in the entire boiler. In
addition, 64 % of the entire effect of the acceleration takes place in the vaporiser.
As presented above, the effect of acceleration is included in the pressure loss coefficient of the
vaporiser. The error caused by the simplification is a result of the change in the relative
magnitudes of the different parts of the pressure drop, and thus it is not determined as a quotient
of the acceleration pressure drop and the total pressure drop. In addition, the uncertainty in one-
phase flow is often 25 % (Wallis, 1969). The uncertainty in two-phase flow is much greater than
this. Usually the total pressure drop includes four terms, the friction loss, the effect of phase
change, the effect of acceleration, and the effect of gravity. Some researchers also add the effect
of the change of the velocity profile in the linear momentum equation (Sarkomaa, 1970). In a
boiler that has several tube bends the effect of this may also be considerable.
As a conclusion, the inaccuracy caused by the simplifications made in the calculation of the
pressure drop may be regarded as negligible.
81
The simplification of the steady heat source can be judged by comparing the time that the heat
source flows through the boiler to the 20 minutes the boiler needs to reach the new steady state
that corresponds to the changed boundary values. The flowing time of the heat source is only a
few seconds and thus the simplification does not cause a significant error.
l
82
9 Conclusions
The results of the tests show very reasonable and logical behaviour of temperatures, pressures,
mass flow rates, the total mass of the process fluid, and the relative areas of the boiler parts. The
test cases were chosen so that many of the test runs were stopped when the heat flow rate in the
boiler became so small that the vapour was not superheated and the superheater disappeared.
After this the process fluid at the boiler exit was wet and there was no reason to continue
calculation, because the fluid as it enters the turbine must be superheated. In general, the dynamic
state of the boiler approaches the new steady state that corresponds to the changed input values,
though in many cases it was not possible to reach this state.
The tube wall temperature follows very well the temperatures of the heat source and the process
fluid, and there seems to be no problems caused by the increase of the tube wall temperature,
which was noticed in the steady state test run. The model has no difficulties in calculating cases in
which the temperature of the entering heat source fluid suddenly drops below the tube wall
temperature.
The results show that with this dynamic model of a boiler, calculations on a very large variety of
different cases can be carried out. The case in which the boiler dries owing to a decrease of the
inlet mass flow of the process fluid shows that the model is able to simulate very large changes.
The shortcoming of this study is that it was not possible to compare the results obtained using the
model with measured values of a real plant. The only way of verifying the correctness of the
results achieved using the dynamic boiler model was to determine whether the results change
logically when the boundary conditions are changed and the obtained results are intuitively
reasonable.
83
9.1 Further development of the dynamic model
The most important test of the dynamic model developed in this thesis will be the comparison of
the calculated results to those obtained from tests of the pilot plant. This is left for future studies.
The dynamic boiler model that is presented in this study is a part of a dynamic process model of
an ORC power plant. During the autumn 1996 the boiler model will be connected to the processmodel that will be used to examine the behaviour of the process when large transitions of state |
take place. It will also be used to trim the controllers of the plant.
Several things that need further research have arisen during this work. Some of them are already
under examination and some of them have to be left for the future. The most important future
work is a revision of the many simplifications made in the model. The homogeneous model that is
used for the two-phase flow in the vaporiser could be replaced with a more precise model. The
consideration of the compressibility of the fluid in the vaporiser and in the superheater using a
more precise method would give a chance to determine the pressure drop and the mass flow in
calculation elements more precisely.
The dependence of the overall heat transfer coefficient on the mass flow has recently been
adapted to the off-design process model. Later it will be connected to the dynamic boiler model,
as well. This is still a simplification, which does not take the effects of the thermal and transport
properties, fouling, and the Reynolds number properly into account. The separation of the tube
wall heat resistance from the inside heat transfer coefficient should be put under consideration.
The pressure drop model used in this thesis has several simplifications. The effects of
acceleration, tube bends, and hydrostatic pressure drop should be included in the pressure drop
model. The simplification made with pressure and mass in the discretion of the energy balance of
an element needs checking, as well.
The oscillations of the flow, both in one tube and in parallel tubes; should be paid more attention
to and more a profound examination of the linear momentum equation should be done at the
element balances. The effect of the compressibility of two-phase flow and the various two-phase
models on pressure drop in the vaporising is left for further research.
: %
84
References
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Isomura, S., 1995, Supercritical Sliding Pressure Boiler Steam Temperature Dynamics. In English and Japanese. Transactions of the Japan Society of Mechanical Engineers, Part C, Vol. 61, No. 581, January 1995. p. 49-56.
Jarkovsky, J. & Fessl, J. & Medulova, V., 1989, A Steam Generator Dynamic Mathematical Modelling and its Using for Adaptive Control Systems Testing. Power Systems Modelling and Control Applications — Selected papers from the IFAC Symposium, Brussels, September 5-8, 1988. IFAC Proceeding Series No. 9. Pergamon Press, Elmsford, USA. p. 167-174.
Jingrong, Z. & Chenyue, Z., 1991, Real-Time Mathematical Model for the Heat Transfer in the Furnace and the Evaporating Condition of a Once-Through Boiler. In Chinese. Proceedings of the Chinese Society of Electrical Engineering, Vol. 11, No. 1, January 1991., p. 63-72.
Laijola, Jaakko, 1995, Electricity from industrial waste heat using high-speed organic Rankine cycle (ORC). Int. Journal of Production Economics, Vol. 41, 1995. p. 227-235.
Laijola, Jaakko & Nuutila, Matti, 1995, District heating plant converted to produce also electric power. 27th UNICHAL Congress Stockholm, June 12-14, 1995. Report 228 E. 14 p.
85
Laijola, Jaakko & Backman, Jan & Sallinen, Petri & Sahlberg, Pekka & Esa, Hannu & Seppanen, Jukka & Pitkanen, Harri, 1995, Suumopeustekniikan perns-ja soveltava tutkimus Lappeenrannan teknillisen korkeakoulun Energiatekniikan osastolla 1/1991 - 5/1995, Loppuraportti. (The Basic and Applied Research of High Speed Technology at the Department of Energy Technology of Lappeenranta University of Technology from 1/1991 to 5/1995, Final Report. In Finnish.) Lappeenranta University of Technology, Department of Energy Technology, Publication EN D- 38. Lappeenranta. 30 p. ISBN 951-763-972-4, ISSN 0785-8256.
Maffezzoni, C., 1992, Issues in modelling and simulation of power plants. I FAC Symposium on Control of Power Plants cmd Power Systems, Munich, 1992. IF AC Symposia Series No. 9, 1992. Pergamon Press, USA. p. 15-23.
Raiko, Markku, 1982, Lapivirtauskattiloiden epalineaaristen simulointimallien rakenne ja kaytto. (The structure and use of non-linear simulation models of once-through boilers. In Finnish.) Licentiate thesis. Helsinki University of Technology.
Reinschmidt, K. F. & Ling, B., 1994, Neural networks for plant simulation and control. Proceedings of the 56th Annual American Power Conference, Vol. 56, Part 1, 1994. Illinois Inst. ofTechnology, Chicago, USA. p. 796-801.
Sarkomaa, Pertti, 1970, Ydinreaktorin lammon siirtymisesta ja kaksifaasivirtauksesta.Raportti TKK-F-B5.
Sarkomaa, Pertti, 1973a, Kevytvesireaktoreiden jaahdytteen hydrodynamiikka. TKY/Opetusmonisteet B-4. Otaniemi. ISBN 951-763-002-6.
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Schlichting, H., 1979, Boundary-Layer Theory. 7th Edition. McGraw-Hill, New York.ISBN 0-07-055334-3.
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86
Talonpoika, Timo, 1994, Termodynaamisen kiertoprosessin valiaineen aineominaisuuksien mallitus. (Modelling the properties of fluid in thermodynamic cycle. In Finnish.) Lappeenranta University of Technology, Department of Energy Technology, EN B-82. Lappeenranta. 23 p. ISBN 951-763-847-7, ISSN 0787-0043.
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1
Analysis of thermodynamic variables and their functional interdependency
The following three diagrams show the results of the analysis of thermodynamic variables and their functional interdependency for the calculation of the process fluid at the present boiler node at the current time instant (j,n). The values of the present node at the preceding time (j,n-l), and those of the previous node both at preceding time (j-l,n-l) and at current time (j-l,ri) are known, and they are not included in the diagrams. The discussion of the analysis is presented in Section 5.2.
The graphs show those variables that change during the pressure iteration that is described in Section 5.3. Along with these, the calculation of the nodal enthalpy is presented. The arrows in the graphs show the direction of the interdependence, for example the nodal specific volume is a function of temperature and pressure, v(T,p).
The mass of the process fluid in the boiler m^fflow) is calculated using the mass obtained from the previous instant in time and the mass flow rates into and out of the boiler <7m,e£miA and <7m,c£m2c that are received as boundary conditions. Thus the mass of the process fluid in the boiler does not vary during the pressure iteration. The mass flow rates into and out of the elements do not change during the iteration, either, because the mass flow rate in the preheater is equal in all nodes and has the boundary value <7m,cCmiA as the liquid is incompressible, and the mass flow rate in the vaporiser and the superheater is calculated using a linear distribution and is thus a function of the boundary values. The masses of the process fluid which are used to calculate the energy balances in the elements, are the values from the previous instant in time, and so they do not change during the pressure iteration. In this way, the heat flow rate into the element the change of the nodal enthalpy (Ah/At)c, and the nodal specific enthalpy Acy are constants during the pressure iteration.
The strong and weak connection between the thermodynamic properties are defined using their well-known nature. For the function h(T,p) the dependency between the enthalpy and the temperature is strong and the dependency between the enthalpy and the pressure is weak, because the pressure affects the enthalpy of a vapour only slightly and the enthalpy of a liquid almost not at all. For the function v(T,p) both the temperature and the pressure affect the vapour noticeably, but for the liquid only the temperature has a strong effect. For the pressure drop in the elements, the guess of the pressure change Ap that is used to calculate the pressure of the first node of the boiler pcyniA, has much stronger effect than the nodal specific volumes, and thus the feedback (2) from the specific volume is weak.
In the vaporiser, the saturation properties of the liquid 7cf,si, /?cW, and vcfy and of the vapour 7cf>, Aetsv, and v^, are functions of the saturation pressure in the elements Wib and Otic. This is why the temperature and the pressure of an individual vaporiser node do not affect the saturation properties or the mass in the element. The effect of the pressure comes through the mixed elements, which is shown by the weak feedback (3). In the mixed elements this feedback is strong, because the pressure determines the saturation properties.
In the two mixed elements, the calculation of the exit nodes is different, because the exit node of the preheater side mixed element mm is a vaporiser node, and the exit node of the superheater
Appendix A
2
side mixed element m\C is a superheater node, as shown in Figure 3.4 on page 21. The feedback (2) from the specific volume to the pressure does not exist, because the pressure is uniform in the mixed element. The heat flow rate into the element is calculated using the values from the previous instant in time, and this is why the heat transfer coefficients that are functions of the vaporiser-part-parameters xiB and x2B do not affect the calculation.
The notation used in the diagrams is as follows
Dpm guess of the pressure change in the process fluid at iteration round number #,Pd pressure of the process fluid,m^fflow) mass of the process fluid in the boiler calculated from the mass flow rate, mtffvol) mass of the process fluid in the boiler calculated from the specific volume,7* temperature of the process fluid,m mass of the process fluid in one element,vcf specific volume of the process fluid,Acf specific enthalpy of the process fluid,dh/dt change in the specific enthalpy of the process fluid per time,fa heat flow into the element,vc£si saturation liquid specific volume of the process fluid,Vctsv saturation vapour specific volume of the process fluid,/?cf.si saturation liquid specific enthalpy of the process fluid,Actsv saturation vapour specific enthalpy of the process fluid,Tcfy i saturation liquid temperature of the process fluid,rc£*v saturation vapour temperature of the process fluid,Xm vaporiser-part-parameter in the preheater side mixed element,X2B vaporiser-part-parameter in the superheater side mixed element,
Superscriptsn current time,
Subscriptsj node number,e element number,ml A first preheater node,mlB first vaporiser node,m2B last vaporiser node,
Lines= strong dependence,- weak dependence.
Other(1),... number of feedback.
3
Calculation of a preheater or a superheater element
dh, n
guess
4
Calculation of a vaporiser element
dh. ndt'e
guess
(3)
5
Calculation of a mixed element
dh. ndty e
guess
(p) in the preheater side mixed element (s) in the superheater side mixed element
Appendix B
Test runs
The results of the nine test runs described in Chapter 7 are presented as figures in this appendix. In each case, Figure 1 shows the temperatures, Figure 2 the pressures, Figure 3 the mass flow rates, Figure 4 the process fluid mass in the boiler, and Figure 5 the relative areas of the preheater and vaporiser and the quality of the process fluid at the boiler outlet. Toluene is used as the organic process fluid, and flue gas from natural gas combustion with excess air ratio of 2.2 is used as the heat source fluid. The detailed input values are given in Table 7.1 on page 70.
The following test runs are presented:
Heat source temperature changes:Test run 1: -50 °C step decrease of inlet temperatureTest run 2: -10 °C/s linear decrease of inlet temperature for 10 seconds
Heat source mass flow changes:Test run 3: -20 % step decrease of inlet mass flowTest run 4: -10 %/s linear decrease of inlet mass flow for 9 seconds
Process fluid temperature changes:Test run 5: -20 °C step decrease of inlet temperatureTest run 6: -0.5 °C/s linear decrease of inlet temperature for 100 seconds
Process fluid mass flow changes:Test run 7: -20 % step decrease of inlet mass flowTest run 8: -1 %/s linear decrease of inlet mass flow for 90 seconds
Steady state test:Test run 9: Response to steady input values.
The notation used in the figures is as followsTcf process fluid (toluene) temperature,Ttw tube wall temperature,% heat source (flue gas) temperature,P<f process fluid (toluene) pressure,qmcf process fluid (toluene) mass flow,qmfg heat source (flue gas) mass flow,mc/Stim total mass of the process fluid (toluene) in the boiler,mlA first preheater node (boiler inlet),m2C last superheater node (boiler outlet).
Appendix B, Page 1.1
Test run: 1 Datafile: runl.outDate: 1996-11-07
Case: Step change of incoming heat source temperature.-50 K
Step: 0.05 seconds
Other: Stopped at 500 seconds.Fast thermal properties calculation is used.
= 250
100 150 200 250 300 350 400 450 500
---------TcflmlA]...............Ttw[mlA]--------- Tfg[mlA]Tcf[m2C].............Ttw[m2C]----------Tfg[m2C] time (s)
Figure 1: Temperatures in the boiler
-50 0 50 100 150 200 250 300 350 400■pcflmlA]---------pcfIm2C]
450 500time (s)
Figure 2: Pressures in the boiler
\ORC\DYNAMICRUNlJOJS
Appendix B, Page 1.2
£1 ' ' ' ' 1 : 1 ! i i 1 1 !! i 1 1
W A _
1 ii i
1!
6 41 3 -
11
I 2- 11 1
1
_____ _____
j
-50 0 50 100 150 200 250 300 350 400 450 500
----------qmcflmlA]--------- qmcf[m2C]-----------qmfg[m2C] time (s)
Figure 3: Mass flows in the boiler
Figure 4: Process fluid mass in the boiler
1
§ 0 8- T i i
---- > 1 11ii
!3 U-B
1^6 .
i. —
1 !it
S 0 4-
ii
!i
!
j
5 02.i!
!
!
1
!
n . _______
; ; i I
i i!
1i
!
-50 0 50 100 150 200 250 300 350 400 450 500
------------ preheater--------------vaporiser--------------- quality time (s)
Figure 5: Relative areas of the preheater, vaporiser and superheaterand quality at boiler outlet
\0RCDYNAMICRUN1J3JS
Appendix B, Page 2.1
Testnin: 2 Datafile: run2.outDate: 1996-11-06
Case: Linear change of incoming heat source temperature.-10 K/s for 10 seconds.
Step: 0.05 secondsOther: Stopped at 500 seconds.
Fast thermal properties calculation is used.
• — Tcf[mlA]............ TtwfmlA]---------- Tfg[mlA]—Tcf[m2C]............ Ttw[m2C]---------- Tfg[m2C] time (s)
Figure 1: Temperatures in the boiler
-prflmlA]---------pcf[m2C] time (s)
Figure 2: Pressures in the boiler
\ORC\DYNAMIC\RUN2JOS
Appendix B, Page 2.2
A
3A .6 4
1 3 - _____!i
S3 jI 2- .. i i l i
| „| |! i
A! ! ii i I
-50 0 50 100 150 200 250 300 350 400 450 500
----------qmcflmlA]--------- qmcf[m2C]-----------qmfg[m2C] time (s)
Figure 3: Mass flows in the boiler
nn .!1
E '
.o lO/C .ns azo2
; i22. I /T 1 1
! 1 /Z i
no .
TN-Z i1
-50 0 50 100 150 200 250 300 350 400 450 500
--------mcfSum time (s)
Figure 4: Process fluid mass in the boiler
1
$ 08 -7T !
I1
c3 U.5§><5 £?a f. _
r-+~ii
S 01-3 02.
A -
1!i _____
!1 _____ ____
-100 -50 0 50 100 150 200 250 300 350 400 450 500
--------- preheater----------vaporiser----------- quality time (s)
Figure 5: Relative areas of the preheater, vaporiser and superheaterand quality at boiler outlet
\ORODYNAMIORUN2JXLS
Appendix B, Page 3.1
Test run: 3 Datafile: runSn.outDate: 1996-11-12
Case: Step change of incoming heat source mass flow rate.-20 %
Step: 0.05 secondsOther: Stopped at 1900 seconds.
Fast thermal properties calculation is used.
200 400 600 800 1000 1200 1400 1600 1800 2000---------TcflmlA]..............Ttw[mlA]---------- Tfg[mlA]— —TcfIm2C] ............Ttw[m2C]---------- Tfg[m2C] time (s)
Figure 1: Temperatures in the boiler
ii
|I
|
i\ 1 i ii\
i \i i ! i
j \ ill!
&3ioo -
t N_____ A
vi'ii
ft3050 -NJ i n
i I\
--------3000 -
2950 -\! ! |
2900 -r-'--J i
oocn . i i! ! ! ~~ --
-200 0 20 0 400 600 800 1000 1200 1400 1600 1800 2000pcffmlA] pcf[m2C] time (s)
Figure 2: Pressures in the boiler
\ORODYNAMIORUN3NJOJS
Appendix B, Page 3.2
-200____ 0_ 200 400 600 800 1000 1200 1400 1600 1800 2000time (s)-qmcf[mlA]---------qmcfIm2CJ-----------qmfg[m2C]
Figure 3: Mass flows in the boiler
200 400 600 800 1000 1200 1400 1600 1800 2000time (s)mcfSum
Figure 4: Process fluid mass in the boiler
1
(3 0 8-i l 1 i i i
f !i J
i t— -
03 °-8
C2 ,§*n e. .
1 i i i i ii i i i l i
i i
i i
11?? 0 4-
! !! i
1i
i!
5
0 2 -
i ■
' ii i
iii
0 -! 1 i
1
-2()0 0 200 400 600 800 1000 1200 1400 1600 1800 2000---------preheater----------vaporiser----------- quality time (s)
Figure 5: Relative areas of the preheater, vaporiser and superheaterand quality at boiler outlet
\ORCDYNAMIC\RUN3NJ{LS
Appendix B, Page 4.1
Test run: 4 Datafile: run4.outDate: 1996-11-05
Case: Linear change of incoming heat source fluid mass flow rate.-10 %/s for 9 seconds.
Step: 0.05 secondsOther: Stopped at 60 seconds.
Fast thermal properties calculation is used.
a 300
£ 250
15 20 25 30 35 40 45 50 55 60
Tcf[mlA]............Ttw[mlA]---------- Tfg[mlA]-TcfIm2C]............Ttw[m2C]---------- Tfg[m2C] time (s)
Figure 1: Temperatures in the boiler
'lACiCi
3300 -1 1 ' ' 1 ! 1
—:----- ill'll! !
1 ! lit! 1 i
- N\ 1 ' -
r^3100 - Ki ;i
jg oonn .i !
i____ i
!
2700 -I
i—
o(500 -i i i
v~s! S ! i 1
i \ ii 1 i i 1 i i
-5 0 5 10 1 5 20 25 30 35 40 45 50 55 60
pcfJmlA] pcf!m2C] time (s)
Figure 2: Pressures in the boiler
\ORCDYNAMICRUN4JHS
Appendix B, Page 4.2
\
S'ti) a .
\\
6 4
1 3 -X\
q- j! 2.
i\ T i i
V\
1 i ! i i i
li
X. u.-L. i . L. . ____ L- . L. .i!1 1 i
0 5 10 15 20 25 30 35 40 45 50 55 60
----------qmcf[mlA]--------- qmcf[m2C]-----------qmfg[m2C] time (s)
Figure 3: Mass flows in the boiler
15 20 25 30 35 40 45 50 55 60
time (s)mcfSum
Figure 4: Process fluid mass in the boiler
-5 0 10 15 20 25 30 35 40 45 50 55 60time (s)--------- preheater----------vaporiser ■ -quality
Figure 5: Relative areas of the preheater, vaporiser and superheaterand quality at boiler outlet
\ORODYNAMIC\RUN4JCLS
Appendix B, Page 5.1
Test run: 5 Datafile: runS.outDate: 1996-11-07
Case: Step change of incoming process fluid temperature.-20 K
Step: 0.05 secondsOther: Stopped at 1800 seconds.
Fast thermal properties calculation is used.
u
1
TcffmlA] - - —Ttw[mlA] ---------TfgfmlA]----------TcfIm2C] -- - - -Ttw[m2C] ---------Tfg[m2C] time (s)
Figure 1: Temperatures in the boiler
3320
3300
3280 |3260
2 3240 13220
O,3200
3180
3160
3140
3120-200 0 200 400 600 800 1000 1200 1400 1600 1800
time (s)
Figure 2: Pressures in the boiler
pcf[mlA]---------pcf[m2C]
f i1 ^—!__ ! i !i ! ! i i
|
! - ! 1 I I
ii
lii!
i ! 1 i 1 i1 1 i I I I |
i i i ; i 1i i ! i i |
\s-----! I I l l !; , > i •! 1 i ! i
\OROiDYNAMJC\R UN5JOS
Appendix B, Page 5.2
&
"! "1
5g> A .
1i
6 41 3-
1i
1 2- ! i
! i 11 1
i!
!
i I i i
-200 0 200 400 600 800 1000 1200 1400 1600 1800
qmcf[mIA] qmcf[m2C] - - qmfg[m2C] time (s)
Figure 3: Mass flows in the boiler
-mcfSum time (s)
Figure 4: Process fluid mass in the boiler
-r OR-
1 I .1 1!1i
8 u,°C9O
f) (Z -
1
!g 5>0-6 ^04-
! ! 1
! ! i
I i1!c v.z
0 ■ 1 i i-2()0 0 200 400 600 800 1000 1200 1400 1600 1800
- preheater vaporiser quality time (s)
Figure 5: Relative areas of the preheater, vaporiser and superheaterand quality at boiler outlet
XORCDYNAMIORUNSJCLS
Appendix B, Page 6.1
Test run: 6 Data file:Date: 1996-11-07
Case: Linear change of incoming process fluid temperature. -0.5 K/s for 100 seconds.
Step: 0.05 secondsOther: Stopped at 1800 seconds.
Fast thermal properties calculation is used.
---------Tcf[mlA] - - —Ttw[mlA]--------- Tfg[mlA]----------Tcf[m2C] -- — Ttw[m2C] - Tfg[m2C] time (s)
Figure 1: Temperatures in the boiler
1—4 i
f
3280 -i i --------- |
1 1 jf^3°60 -
i 11 i
1______
§300Q . i i 1
1 1 !, i
i
3160 -i
-- ! i| 1
-200 0 200 40 0 600 800 1000 1200 1400 1600 1800
pcf[mlA] pcfj[m2C] time (s)
Figure 2: Pressures in the boiler
\ORC\DYNAM/C\RUN6JCLS
Appendix B, Page 6.2
6
< .
g 4-
1 3 ■
1 2-1 1 1 1
1 1 1
A .1
-200 0 200 400 600 800 1000 1200 1400 1600 1800
qmcf[mlA] qmcfIm2C] - - qmfg[m2C] time (s)
Figure 3: Mass flows in the boiler
800 1000 1200 1400 1600 1800
time (s)------ mcfSum
Figure 4: Process fluid mass in the boiler
1
flf$4 A 0 .
__11
c3 u-6
|*06.
11
$ 01 -
t0 2 -
A .
-200 0 200 400 600 800 1000 1200 1400 1600 1800
------------- preheater---------------vaporiser-----------------quality time (s)
Figure 5: Relative areas of the preheater, vaporiser and superheaterand quality at boiler outlet
\ORCDYNAMICRUN6J{LS
Appendix B, Page 7.1
Testmn: 7 Datafile: run7m.outDate: 1996-11-11
Case: Step change of incoming process fluid mass flow rate.-20 %
Step: 0.05 secondsOther: Stopped at 1500 seconds.
Fast thermal properties calculation is used.
S 250
---------Tcf[mlA]...............TtwfmlA]--------- TfgfmlA]Tcf[m2C].............Ttw[m2C]----------Tfg[m2C] time (s)
Figure 1: Temperatures in the boiler
g 3100
a 3000
1000 1200 1400 1600
pcflmlA]---------pcftm2C] time (s)
Figure 2: Pressures in the boiler
\ORCOYNAMICRUN7MJCLS
Appendix B, Page 7.2
a
ts
6
5
4
3
2
1
0-200 0 200 400 600 800 1000 1200 1400 1600
11 i
!i
\
qmcf[mlA]---------qmcf[m2C]-----------qmfg[m2C] time (s)
Figure 3: Mass flows in the boiler
Figure 4: Process fluid mass in the boiler
1 08 -
| V------
3 04 .!i
5 !1;
0 ■
!ii
-2()0 0 200 400 600 800 1000 1200 1400 1600
---------preheater----------vaporiser----------- quality time (s)
Figure 5: Relative areas of the preheater, vaporiser and superheaterand quality at boiler outlet
\ORC\DmAMIORUN7MJCLS
Appendix B, Page 8.1
Test run: 8 Datafile: run8m.outDate: 1996-11-11
Case: Linear change of incoming process fluid mass flow rate.-1 %/s for 90 seconds.
Step: 0.05 secondsOther: Stopped at 230 seconds.
Fast thermal properties calculation is used.
-------Tcf[mlA]--------Tcf[m2C]
-Ttw[mlA]--------- Tfg[mlA]-Ttw[m2C]--------- Tfg[m2C] time (s)
Figure 1: Temperatures in the boiler
g 2000
o, 1500
100 125 150 175 200 225 250
pcflmlA]---------pcflm2C] time (s)
Figure 2: Pressures in the boiler
\ORC\DYNAMIC\RUNSKfJiLS
Appendix B, Page 8.2
-25 25 50 75 100 125 150 175 200 225 250
time (s)-qmcflmlA]---------- qmcf[m2C]------------ qmfg[m2C]
Figure 3: Mass flows in the boiler
mcfSum time (s)
Figure 4: Process fluid mass in the boiler
---------- preheater----------- vaporiser ■ -quality time (s)
Figure 5: Relative areas of the preheater, vaporiser and superheaterand quality at boiler outlet
\ORCDYNAMICRUN8MJCLS
Appendix B, Page 9.1
Test run: 9 Datafile: run9.outDate: 1996-11-04
Case: Differences compared with the initial state.No change of input values.
Step: 0.05 seconds
Other: Stopped at 3000 seconds.
Fast thermal properties calculation is used.
O
I£=2
----------TcflmlA].................Ttw[mlA]-----------Tfg[mlA]
TciIm2C] ...............Ttw[m2C]----------- Tfg[m2C] time (s)
Figure 1: Change of temperatures in the boiler
time (s)pcfjmlA]------ — pcf[m2C]
Figure 3: Change of pressures in the boiler
\ORC\DYNAMIC\RUN9JaS
Appendix B, Page 9.2
-qmcfJmlA]---------qmcf[m2C]-----------qmfg[m2C] time (s)
Figure 2: Change of mass flows in the boiler
= 0.01
time (s)------ mcfSum
Figure 4: Change of process fluid mass in the boiler
0.3
0.25
S' 0.2
4:o.i5 1 0.1
I0.05
0§-0.05
-0.1
1j
1
\
!j
i i
-500 0 500
---------preheater----------vaporiser
1000 1500 2000 2500 3000time (s)
-whole boiler
Figure 5: Change of areas of preheater and vaporiser
\ORC\DYNAKfIORUN9XLS