FUNDAMENTAL MODELING AND CONTROL OF FALLING FILM EVAPORATORS by ZDRAVKO IVANOV STEFANOV, B.S., M.S. A DISSERTATION IN CHEMICAL ENGINEERING Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Approved
Chairperson of the Committee
Accepted
Dean of the Graduate School May, 2004
ACKNOWLEDGEMENTS I am greatly indebted to my research advisor, Dr. Karlene A. Hoo, for her support and advice. During the yeara apent under her guidance, ahe helped me to underatand and appreciate the nature and the beauty of proceaa control and the value of multivariate atatiatics. Dr. Hoo waa an excellent adviaor, and I would like to thank her for the opportunity to work with her. I am also grateful for the advice and the proofreading of my manuscripts and my disaertation. The experience with her was one of the moat important experiences in my life. Alao, I would like to thank Dr. Uzi Mann for being on my committee. Dr. Mann waa very helpful with technical discussiona on the aapecta of the evaporator modeling. I highly appreciate hia support for my laboratory inveatigations. I would like to thank Dr. R. Tock and Dr. W. P. Dayawanaa for their willingnesa to aerve on my comitee. Dr. Tock kindly asaisted me with my laboratory inveatigations. Dr. W. P. Dayawanaa was extremely kind to provide advice. I would like to thank him for the excellent experience I had in hia claaaroom. I am grateful to Dr. D. Chaffin for his advice in the area of advanced computing. I am alao grateful to Mr. M. Champagne and Tembec for providing me with a rewarding graduate internahip at the Tembec Mill in St. Franciaville, LA. A special thanks to Mr. Wayne McAdama, and Mra. Janice Hawley at the Tembec Mill in Crestbrook, CA, for providing industrial data and valuable information used in the validation of the evaporator models.
11
I would like to thank my fellow graduate studenta for their support and for the friendly work environment. I also appreciate the financial support of Petroleum Research Foundation and the Dean's fellowship I received from the CoUege of Engineering in my first year. Ultimately, I am alao indebted to my parents Lilia and Ivan for their deep underatanding and aupport.
Ill
CONTENTS
ACKNOWLEDGEMENTS
ii
ABSTRACT
xi
LIST OF TABLES
xih
LIST OF FIGURES
xv
I BACKGROUND AND INTRODUCTION
1
1.1
Evaporation
1
1.2
Evaporator typea
3
1.3
Modeling the evaporation procesa
4
1.4
Controlling the evaporation process
5
1.5
Evaporation of black liquor
6
1.6
Black liquor origin
6
1.7
Kraft pulp mill recovery cycle
7
1.8
Black liquor properties and the recovery boiler operation
1.9
Falling film evaporators for black liquor concentration
10
1.10
Transport phenomena in falling
11
1.11
Disaertation organization
9
12
II MODELING OF A SINGLE PLATE 2.1
films
...
Physical conditions
14 14
2.1.1 Mass flow rates of the liquor
14
2.1.2 Temperature and preaaure
15
iv
2.2
Hydrodynamica of the falling film at nominal conditions . . . .
15
2.3
Heat tranafer coefiicients of turbulent falling
18
2.4
2.3.1 General description of turbulent flow in an open channel .
18
2.3.2 Heat transfer coefiicient of evaporation
22
2.3.3 Heat transfer coefiicient of condenaation
23
2.3.4 Heat transfer coefficient of senaible heating
24
2.3.5 Calculation of the dimensionless film thickness
24
Heat tranafer coefficients of wavy-laminar 2.4.1 Transition from wavy-laminar to turbulent
2.5
films
films
25
flow
25
2.4.2 Heat transfer coefficient of evaporation
27
2.4.3 Heat tranafer coefficient of senaible heating
32
Modehng of single plate
33
2.5.1 Dimenaionality
33
2.5.2 The physical phenomena
... . .
36
2.5.2.1
Black hquor evaporation
37
2.5.2.2
Black hquor heating
44
2.5.2.3
Steam condenaation
46
2.5.2.4
Heat transfer at the wall
56
2.5.3 Dimensionleaa variables
59
2.5.4 Numerical Solution
60
2.5.4.1
Solution method
60
2.5.4.2
OCFE diacretization
62
v
2.5.4.3
Solver package
62
2.5.4.4
Phyaical parameter correlations
63
2.5.5 Validation of the aingle-plate evaporator model
2.6
63
2.5.5.1
Feed dry hquor concentration
64
2.5.5.2
Feed mass flow rate
66
2.5.5.3
Feed temperature
67
2.5.5.4
Steam pressure
70
2.5.5.5
Summary
71
Nomenclature
74
HI MODELING OF A SINGLE EVAPORATOR
77
3.1
Evaporator deaign
77
3.2
Liquor distributor
79
3.3
Plate atack
80
3.4
Evaporator inventory
82
3.4.1 Masa balance
82
3.4.2 Energy balance
84
3.5
In-line mixer
85
3.6
Splitter
86
3.7
Numerical aolution
86
3.7.1 Initialization
87
3.7.1.1
Caae 1-Evaporator cold startup
87
3.7.1.2
Caae 2-Evaporator hot atartup
87
vi
3.7.2 Calculation loop
87
3.8
ODE solver
88
3.9
Results and discussion
89
3.10
3.9.1 Feed flow rate
90
3.9.2 Feed concentration
91
3.9.3 Feed temperature
91
3.9.4 Steam pressure and vapor pressure
92
3.9.5 Summary
93
Nomenclature
99
IV CONTROL OF A SINGLE EVAPORATOR
101
4.1
Typical disturbancea
103
4.2
Senaor laaues
105
4.3
Decentralized control of a single evaporator
105
4.3.1 On aupply operations
106
4.3.2 Pairing of the manipulated and controlled variablea . . . .
108
4.3.3 Control structure
109
4.3.4 Closed-loop response of the evaporator
113
4.3.5 Limitationa of SISO loop controUera
118
4.4
Summary of decentralized control
119
4.5
Model predictive control (MPC)
120
4.5.1 Linear quadratic regulator
123
4.5.2 Dynamic matrix controller
125
vii
4.5.3 Quadratic dynamic matrix controller 4.6
129
MPC related modeling of single evaporator
132
4.6.1 Lumped model of a single evaporator
133
4.6.1.1
Plate atack
133
4.6.1.2
Bottom inventory
139
4.6.2 Validation of the nonlinear lumped parameter model 4.6.3 A linear MPC deaign
...
139 139
4.6.3.1
Linearization
145
4.6.3.2
System-theoretic analyaia
148
4.7
Cloaed-loop reaults
151
4.8
MPC results
153
4.8.1 Reaulta 4.9
154
Nomenclature
172
V EVAPORATOR PLANT
173
5.1
Description
173
5.2
Modeling
177
5.3
Resulta and discusaion
179
5.3.1 Senaitivity to diaturbancea 5.3.1.1
Mass balance
180
5.3.1.2
Energy balance
189
5.3.2 Model vahdation 5.4
180
191
Nomenclature
192 viii
VI CONTROL OF MULTIPLE EVAPORATOR PLANT 6.1
6.2
Decentrahzed control
193 193
6.1.1 Evaporator inventoriea
193
6.1.2 Selection of controlled and manipulated variablea
196
6.1.3 Reaulta
197
Centralized control
201
6.2.1 Nonlinear ODE model
202
6.2.2 The effect of concentration diaturbancea
206
6.2.3 Linear model
210
6.2.4 Reaulta
211
6.2.4.1
Unconatrained and conatrained MPC controllers . . . 214
6.2.4.2
PI control
218
6.2.4.3
Control of the PDE system
219
VII SUMMARY AND FUTURE WORK
247
7.1
Modeling
247
7.2
Control
248
7.3
Decentrahzed control
249
7.4
Centralized control
250
7.4.1 Single evaporator
250
7.4.2 Multiple evaporator plant
251
7.5
Summary on the control reaulta
252
7.6
Future work
252 ix
7.7
Contributions
253
BIBLIOGRAPHY
255
APPENDICES
261
A: TURBULENT HEAT TRANSFER COEFFICIENTS
261
B: BULK FLOW MOMENTUM EQUATION
267
C: BLACK LIQUOR PHYSICAL PROPERTIES
271
D: COMPUTER PROGRAMS
274
D.l
Matlab S-function for aimulation of aingle evaporator
274
D.2
Matlab S-function for simulation of multiple evaporators
D.3
MAPLE code for linearization of single evaporator nonhnear ODE model
289
MAPLE code for linearization of multiple evaporator nonlinear ODE model
291
D.4
E: OCFE DISCRETIZATION
. . . 278
307
ABSTRACT Evaporatora are a common unit operation that can be found in many industries. The evaporator plant, in the pulp and paper induatry providea a major role of regenerating the proceaa chemicala from the fiber line waste liquor. The effectiveness of the recovery, determinea the overall mill economy. Conaequently, the recovery cycle must be fully operation becauae it ia unacceptable to diacard the waste liquor due to ita highly negative effect on the aurrounding ecosystem. The product of the evaporator plant, the concentrated black liquor aervea aa a fuel to the recovery boiler. The recovery boiler is a combination of a chemical reactor and a power boiler. The dry solida concentration of the black liquor affects the recovery boiler performance not only from an economical point of view but also for aafety reaaona. It is known, that if the dry sohds concentration of the black liquor falls below a lower limit, there ia the poaaibility of an exploaion in the recovery boiler. Evaporation of the waste hquor is usually accomplished in a multiple effect evaporator plant. While there are more than one type of evaporator deaign, the most modern and efficient deaign is the falling film plate evaporator. This design ia characterized with very high heat tranafer ratea at small temperature differences and high reaiatance to acaling due to low reaidence times. This research has two main objectives. The firat ia to develop a rigoroua distributed parameter model of the falling film evaporator using the fundamental principlea of masa, energy, and momentum conservation. The aecond is to synthesize an effective
XI
control atructure for the evaporator and the evaporator plant. A bench-acale experiment has ahown that one-dimenaional distributed model of the evaporator plate is aatisfactory to describe the important transfer proceaaea on the plate accurately. Additionally, it was confirmed by experimentation that two different hydrodynamic regimes (turbulent and wavy-laminar) can exiat in the multiple effect black liquor evaporator plants. Inveatigationa into aimple and advanced control approachea have revealed that the closed-loop performance of a proportional-integral-derivative (PID) controller design in feedback with a aingle evaporator can provide satisfactory compenaation. However, in the case of the entire evaporator plant, the advanced control approach of model-predictive control (MPC) provides better control becauae the MPC centralized controller can addreas multiple interactiona, input and output constraints, and unmeasured diaturbancea. This work preaents the development of the distributed parameter model and the aynthesia of the control atructure; and demonatrates the performance of the cloaedloop aystem to meaaured and unmeasured disturbancea and parameter uncertainty.
Xll
LIST OF TABLES 2.1
Nominal parameters with respect to aingle plate
2.2
Predicted Reynolda numbers at the transition points for black hquor at nominal conditions
17 26
2.3 Experimental conditiona for model dimenaionality experiment
35
2.4
Dimensionleaa Variablea
59
2.5
Operating Conditiona
63
3.1
Evaporator operating conditions
89
3.2
Step changea of evaporator operating condition
90
3.3
Evaporator aenaitivity to disturbances
90
4.1 4.2
Variable Selection and Clasaification Control Loop Parametera
107 112
4.3
Evaporator nominal operating conditiona
113
4.4 4.5 4.6 4.7 4.8
Featurea of the proceaa closed loop reaponaes Correlationa for Turbulent and Wavy-laminar regimea Evaporator nominal operating conditiona Tuning parametera for MPC Tuning parameters for the PI controllers
119 136 149 155 155
5.1 5.2 5.3
Operating conditions of the multiple evaporator plant Senaitivity of the dry aolids concentration Changea in steam economy aa a function of the diaturbance
179 188 190
6.1
Operating conditiona of the multiple evaporator plant
199
6.2
Performance of the single-loop control atructure of the nonlinear multiple evaporator plant
200
The effect of boiling point elevation to a 5% decrease in the feed dry solids concentration
209
Timea to reach la hmits of 0.001 kg/kg of the multiple evaporator plant product dry solids concentration
218
6.3 6.4 6.5
Tuning parameters of the MPC and PI controllers for multiple evapo-
6.6
rator plant Integral abaolute error of the controlled variablea
Xlll
220 221
B.l Dimensionleaa Variables
270
D.l Matrix A, columna 1 to 10 .
300
D.2 Matrix A, columns 11 to 20
301
D.3 Matrix A, columns 21 to 30
302
D.4 Matrix A, columna 31 to 32
303
D.5 Matrix B
304
D.6 Matrix C
305
D.7 Eigenvalues of Matrix A . .
306
XIV
LIST OF FIGURES 1.1
Schematic of a kraft pulp mill recovery cycle
2.1 2.2 2.3
Eddy diffuaivity of momentum for a falling water Heat transfer coefficient of wavy-laminar faUing Experimental deaign
2.4
Experimental aetup to determine flow distribution
47
2.5 2.6
Tracer patterna for water at 24°C, Re = 3557 Tracer patterns for water at 60°C, Re = 6970
48 48
2.7 2.8 2.9
Tracer patterna for SCMC at 24°C, Re = 0.699 Tracer patterns for SCMC at OO^C, Re = 1.298 Preaence or absence of a heating zone
48 48 49
2.10 2.11 2.12 2.13 2.14
Differential volume of the evaporating film Differential volume of the heating film Differential volume of the condensing film Wall differential volume element The responae of the single plate evaporator to a 5% decrease in dry aolida concentration
50 51 52 56 64
2.15 The reaponse of the aingle plate evaporator to a 5% increaae in dry solids concentration
65
2.16 The reaponse of the single plate evaporator to a 5% decreaae in feed masa flow rate
67
2.17 The reaponae of the aingle plate evaporator to a 5% increaae in feed mass flow rate
68
2.18 The reaponae of the single plate evaporator to a 5% decrease in the feed temperature
69
2.19 The reaponse of the single plate evaporator to a 5% increaae in the feed temperature
70
2.20 The response of the aingle plate evaporator to a 5% decrease in the steam preaaure
71
2.21 The reaponae of the single plate evaporator to a 5% increaae in the ateam preaaure
73
3.1
Plate type evaporator aheme
78
3.2
Product dry aohda concentration to changea in the feed masa flow rate.
94
3.3
Product dry aolida concentration to changea in feed dry aohda concentration
95
XV
13 film fllm
22 31 34
3.4
Product dry aohds concentration to changea in the feed temperature.
96
3.5
Product dry aohds concentration to changes in the ateam pressure. . .
97
3.6
Product dry aolida concentration to changea in the aecondary vapor aaturation temperature
98
4.1
Feedback control of the evaporator
Ill
4.2
Closed-loop responae to a 5% increase in the feed flow rate
114
4.3
Closed-loop reaponae to a 5% decreaae in the feed dry sohda concentration
115
4.4
Cloaed-loop responae to a 5% decrease in the feed temperature.
...
116
4.5
Cloaed-loop reaponse to a 10% increaae in the aet point of the dry solids concentration in the product stream
117
4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21
Evaporator reaponae to an increase of 10% in the dry solids concentration aet point 118 Nonlinear ODE model response to a 5% decrease in the feed flow rate. 140 Nonlinear ODE Nonlinear ODE concentration Nonlinear ODE concentration Nonlinear ODE Nonlinear ODE Nonlinear ODE
model response to a 5% increase in the feed flow rate. 140 model response to a 5% increaae in the feed dry aolids 141 model reaponse to a 5% decrease in the feed dry solids 141 model reaponae to a 5% decrease in feed temperature. 142 model responae to a 5% increase in feed temperature. 142 model responae to a 5% increase in vapor preasure. . 143
Nonhnear ODE model reaponse to a 5% decrease in vapor preaaure. . Nonhnear ODE model reaponse to a 5% increase in ateam preasure. . Nonhnear ODE model reaponae to a 5% decrease in steam preaaure. . Responses of the linear (LODE) and nonhnear (NLODE) ODE models to 5% decreaae in the steam pressure Reaponaea of the linear (LODE) and nonlinear (NLODE) ODE models to 5% increase of the ateam preaaure Reaponaes of the linear (LODE) and nonlinear (NLODE) ODE modela to 5% decreaae of the vapor pressure Reaponsea of the linear (LODE) and nonlinear ODE modela to 5% increase of the steam preaaure A feedback block diagram with the MPC controller
4.22 MPC performance in the presence of a 5% increase in the feed maaa flow rate 4.23 MPC performance in the preaence of a 5% decreaae in the dry aohda concentration of the feed XVI
143 144 144 150 151 152 153 154 160 161
4.24 MPC performance in the presence of a 5% decrease in the feed temperature
162
4.25 Constrained MPC performance in the presence of a 5% increaae in the feed mass flow rate
163
4.26 Constrained MPC performance in the presence of a 5% decrease in the feed dry aohds concentration
164
4.27 Constrained MPC performance in the presence of a 5% decrease in the feed temperature
165
4.28 PI performance in the preaence of a 5% increase in the feed maaa flow rate
166
4.29 PI performance in the preaence of a 5% decreaae in the feed dry sohda concentration
167
4.30 PI performance in the preaence of a 5% decrease in the feed temperature. 168 4.31 Conatrained MPC performance in the preaence of a 6.6% decrease in the feed temperature 4.32 PI performance in the presence of a 6.6% decreaae in the feed temperature
170
4.33 Nonlinear PDE evaporator ayatem reaponse
171
5.1 5.2
176
5.3 5.4 5.5 5.6 5.7 5.8 5.9
Multiple falling fllm evaporator plant Open-loop response of the PDE model of multiple evaporator plant to a 5% decrease in the flow rate of the feed Open-loop reaponae of the PDE model of multiple evaporator plant to a 5% increase in the flow rate of the feed Open-loop responae of the PDE model of multiple evaporator plant to a 5% decreaae in the dry aolids concentration of the feed Open-loop reaponse of the PDE model of multiple evaporator plant to a 5% increeise in the dry aolida concentration of the feed Open-loop reaponae of the PDE model of multiple evaporator plant to a 5% decreaae in the temperature of the feed Open-loop reaponae of the PDE model of multiple evaporator model for 5% increase in the feed temperature Open-loop reaponae of the PDE model of multiple evaporator plant to a 10% decreaae in the heat tranafer of the superconcentrator Open-loop reaponse of the PDE multiple evaporator model for 10% decrease in the heat transfer of the superconcentrator
169
181 182 183 184 185 186 187 189
6.1
Syatem of three connected tanka
194
6.2
Decentrahzed feedback control atructure for the multiple evaporator plant
198
xvii
6.3 6.4 6.5 6.6
6.7 6.8
6.9
Closed-loop reaponae of the nonlinear multiple evaporator plant to 5% increase in the flow rate of the feed
200
Cloaed-loop responae of the nonlinear multiple evaporator plant to a 7% decreaae in the temperature of the feed 201 Cloaed-loop reaponae of the nonlinear multiple evaporator plant to a 5% decreaae of the dry aohda concentration of the feed 202 Reaponaea of nonlinear PDE (aolid line) and ODE (dotted fine) models of the multiple evaporator plant to a 5% decrease in the flow rate of the feed 203 Responses of nonlinear PDE (aohd line) and ODE (dotted line) models of the multiple evaporator plant to a 5% increase in flow rate of the feed. 204 Reaponaea of nonlinear PDE (aohd fine) and ODE (dotted line) models of the multiple evaporator plant to a 5% decrease in dry solids concentration of the feed 205 Reaponaea of nonlinear PDE (sohd fine) and ODE (dotted fine) models of the multiple evaporator plant to a 5% increaae in the dry sohda concentration of the feed 206
6.10 Reaponsea of nonlinear PDE (sohd hue) and ODE (dotted line) modela of the multiple evaporator plant to a 5% decreaae in the temperature of the feed 6.11 Reaponsea of nonlinear PDE (sohd line) and ODE (dotted line) modela of the multiple evaporator plant to a 5% increaae in the temperature of the feed 6.12 Reaponsea of the nonhnear (aohd line) and linear (dotted line) ODE models to a 5% decrease in the auperconcentrator steam pressurea.
207
208 212
6.13 Reaponaes of the nonhnear (aohd hue) and linear (dotted line) ODE models to a 5% increaae in the superconcentrator steam preaaurea. . 213 6.14 Reaponsea of the nonlinear (aohd line) and linear (dotted hue) ODE modela to a 5% decreaae in the ateam preaaure of the auperconcentrator evaporator SC-1 6.15 Reaponaea of the nonlinear (aohd line) and linear (dotted hue) ODE models to 5% increaae in the ateam pressure of auperconcentrator evaporator SC-1 ; 6.16 Reaponsea of the nonhnear (sohd line) and hnear (dotted hue) ODE modela to a 5% decrease in the aecondary vapor pressure of evaporator E-5 6.17 Responses of the nonhnear (sohd line) and linear ODE models to a 5% increase in the secondary vapor pressure of evaporator E-5
214
215
216 217
6.18 Closed-loop performance of the unconstrained MPC controller in the presence of a feed maas flow rate disturbance
222
6.19 Unconatrained MPC control actiona in the preaence of a feed maas flow rate disturbance
223
XVlll
6.20 Closed-loop performance of the unconatrained MPC controUer in the preaence of a feed dry aolida concentration diaturbance
224
6.21 Unconstrained MPC control actiona in the presence of a feed dry aolids concentration diaturbance
225
6.22 Closed-loop performance of the unconatrained MPC controller in the preaence of a feed temperature disturbance
226
6.23 Unconatrained MPC control actions in the presence of a feed temperature disturbance
227
6.24 Closed-loop performance of the unconstrained MPC controller in the presence of a heat tranafer diaturbance
228
6.25 Unconatrained MPC control actiona in the presence of a heat transfer disturbance 6.26 Cloaed-loop performance of conatrained MPC controller in the preaence of a feed mass flow rate diaturbance 6.27 Constrained MPC control actions in the presence of a feed mass flow rate diaturbance 6.28 Cloaed-loop performance of constrained MPC controller in the preaence of a feed dry solids concentration disturbance 6.29 Constrained MPC control actiona in the preaence of a feed dry solids concentration diaturbance 6.30 Cloaed-loop performance of conatrained MPC controller in the presence of a feed temperature disturbance 6.31 Constrained MPC control actiona in the presence of a feed temperature diaturbance 6.32 Cloaed-loop performance of constrained MPC controller in the preaence of a heat tranafer diaturbance 6.33 Conatrained MPC control actions in the presence of a heat tranafer disturbance 6.34 Closed-loop performance of the PI controUera in the preaence of a feed mass flow rate disturbance 6.35 PI control actions in the presence of a feed mass flow rate disturbance. 6.36 Closed-loop performance of the PI controUera in the presence of a feed dry solida concentration disturbance 6.37 PI control actiona in the presence of a feed dry sohda concentration diaturbance 6.38 Cloaed-loop performance of the PI controllers in the preaence of a feed temperature disturbance 6.39 PI control actions in the preaence of a feed temperature disturbance. 6.40 Closed-loop performance of the PI controUera in the presence of a heat transfer diaturbance 6.41 PI control actions in the presence of a heat transfer diaturbance. . . . XIX
229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245
6.42 Cloaed-loop reaponae of the PDE model to a feed mass flow rate diaturbance uaing the unconstrained MPC controller actiona
246
A.l A fluid element in the fully developed falling
fllm
261
B.l
Force balance on a fluid element in the faUing
film
267
E.l
OCFE discretization of the spatial domain.
XX
307
CHAPTER 1 BACKGROUND AND INTRODUCTION Evaporation is a fundamental process operation found in many diverse induatriea. In this chapter, the baaics of the evaporation proceaa will be diacuaaed, with some specifics aa they relate to the evaporator units used in the pulp and paper industry. There are more than one type of evaporator, however, this work will be restricted to the falling film plate type evaporator and the process fluid of interest is the black liquor that is the effiuent of the kraft proceaa of the pulp milla [1]. The important tranaport phenomena that dominate the evaporation proceaaea aaaociated with the evaporation of the black hquor process fluid wUl be discussed from both a theoretical, experimental, and practical points of view. Understanding theae phenomena is tantamount to the development of an accurate phenomenological model and stable controller deaign to produce a product with proper quahty (dry aohda concentration-') that is eaaential to the economica of the plant.
1.1
Evaporation
Evaporation ia the process of aolvent removal in the form of a vapor from either solution, suspension or emulsion. The objective can be to concentrate the solution, to regenerate the solvent, or both. In many caaea, however, the objective ia to concentrate the solution. In these cases the solvent vapor is not regarded as product and it may or may not be recovered [2, 3]. Evaporation has been used to recover ^The dry solids are all solids that remain after complete evaporation of the water in the black liquor
1
either diaaolved aohda (salts, organic material) or the pure hquid (deaahnation of sea water) [3]; to concentrate diluted streama and thus to improve their value (milk and juice concentration [3]); or to prepare a atream for future treatment (black liquor concentration [1], Bayer alumina process liquor concentration [4]). There are areas in the induatry, where the evaporation is preferred to other aeparation methoda, for example the radioactive waste treatment [5, 6]. The process of evaporation is similar to other aeparation operationa such aa diatillation, drying, and crystallization. All deacribed processes involve vaporization. The evaporation process differs from the distillation process in that the task of the former is not to separate the vapor components. In the case of a drying process, the evaporation process ia different because the product is alwaya a liquid. Finally, the evaporation process differs from a crystallization process because in the latter the objective is crystal growth while in the former the objective is to concentrate the aolution [2]. Uaually, the evaporative proceaa ia performed in unit operations caUed evaporators that need to accomplish certain tasks [2]. These tasks include: • To achieve a designed extent of separation, or in other words to provide a flnal product with certain solids concentration. • To provide large heat transfer with the smalleat possible heat transfer surface. • To provide efficient energy usage, which is measured by the evaporator's utility (eg. steam) economy. The utility economy is the maaa units of water evaporated
by one mass unit of utility per unit of time. • To be compatible with the processed working fluid (hquor), which in many cases can be corrosive, radioactive, etc.
1.2
Evaporator types
There are several types of evaporators that are commercially available [2, 3, 7]. Evaporators can be classified by the circulation type, design, and direction of the flow. The type can be natural or forced; the construction, vertical or horizontal tube bundles or plate stacks with respect to the heating elements; and the flow can be either in the direction of the gravity or otherwise. Often, devices are installed in the evaporator to improve the distribution of the process fluid. All these design features, when combined, lead to a wide variety of evaporators. The most popular designs however, are the rising fllm long tube vertical (LTV) evaporators and the falling film tube or plate evaporators. The LTV evaporator's heating element ia a long tube bundle that enda with a vapor space at the top and the process fluid inventory at the bottom [1, 2, 3]. The feed is supplied at the bottom of the evaporator and the process fluid rises inside the tubes. The tubes are heated by condensing steam. At a given spatial point, the process fluid starts to boil. The vapors leaving the tubes cause the procesa fluid to riae, climbing on the tube walls in a fashion similar to annual flow. In some cases, a circulation stream is present depending on the amount of liquor that needs to be processed.
The heating element of the faUing film type evaporator can be either a tube bundle or a plate stack. In either case, the process fluid enters the heating element from the top of the evaporator. Usually a distribution device ia neceasary to provide uniform distribution of the process fluid. The process fluid flows by gravitational forces in a manner similar to annual fllm flow or flow past a vertical flat plate. Since the residence time on the plates is small (on the order of tens of seconds), forced circulation ia almost always a necessity for this type of evaporator.
1.3
Modehng the evaporation process
A survey of the open literature produced lumped parameter steady-state models of the evaporation process [2, 3, 7]. Also, there appeared to be a scarcity of dynamic models of the evaporation process but a few dynamic modela of the evaporator system. For example a fundamental lumped parameter dynamic model, published by Niemi et al. [8, 9], was used to represent a multiple evaporator plant in the pulp and paper induatry. Thia model waa extended by Ricker and SeweU [10] and Cardoao [11]. A lumped model of flash type evaporators was pubhshed by To et al. [4]. The lumped parameter modehng has certain advantages, such as small computation times and relative simplicity. However, since the evaporator system is a distributed system, the lumped parameter model may not provide an accurate description of the system phenomena in the spatial direction. For the evaporation process, the most common evaporator types are the LTV and faUing fllm evaporators. Based on their construction, these evaporators have a dominant distributed character. In
the case of the LTV evaporator well-mixedness and therefore a lumped parameter representation can be assumed but only in the evaporator inventory and in the section of the tubes where the process fluid is below its boiling point. However, in the rest of the tube space this assumption is not vahd. It is well known [12, 13, 14] that up to seven different flow types can be observed in the rest of the tube space. A distributed parameter model of a rising fllm evaporators was developed by [12]. Cardoso [11] modeled a falling film concentrator using a lumped parameter approach. However, the physics of the process imphed a distributed character of the system since the process fluid flows down onto the heat transfer surface and wellmixedness cannot be assumed. In the work by Schutte et al. [15], a distributed parameter model of a tube type falling film evaporator was described. However, no model equations or detailed description of the modeling approach were provided.
1.4
Controlhng the evaporation process
Since the evaporation process is an important part of an industry such as the pulp and paper industry, it is important to provide stability of the operation and consistent quahty of the product. In certain cases, the stabihty of the operation may be very significant issue. For example, in the evaporative processes used in radioactive wastes the expectation is that near perfect separation should be achieved. A survey of the open literature shows that the most widely employed control strategy is the decentralized strategy of single loop feedback controllers implemented using a proportional-integral-derivative (PID) control law [2, 3, 16]. In recent years, with
the rapid improvements in the computing industry, advanced control concepts such as model based and model predictive control can be used to control these systems [17, 18, 19]. An example can be found in the work of Gil et al. [20] where a multiple evaporator plant is modeled using a lumped parameter model and controlled by a linear quadratic regulator. An illustrative example of generalized predictive controller can be found in [21] that employs a very simple lumped parameter model to develop the GPC.
1.5
Evaporation of black liquor
The evaporation of the black liquor is a operation that is common in the pulp and paper industry.
1.6
Black liquor origin
The black liquor is a spent hquor that originates from the pulp miU fiber hue. The pulp miU's task is to produce hardwood or softwood pulp used in paper production, yarn production (rayon fibers) or other chemical processes (production of carboxymethylceUulose, smokeless gunpowder production, food additives, etc.). The pulp is produced from wood chips that are cooked in units called digesters. The resulting unbleached pulp (brown stock) is washed from the cooking chemicals and either used for paper production or bleached using a sequence of reactors and then re-used for paper production or in other processes. The black hquor is the spent pulp liquor that is collected from the washing process.
During the washing process, the cooking chemicals are transferred from the brown stock to the black hquor. The color of the hquor is a result of the organic compounds dissolved from the wood during the cooking process. The composition of the black hquor depends on the cooking process type (sulphite, sulphate (kraft process)^, and others), the cooking conditions, and the wood type.
1.7
Kraft pulp mill recovery cycle
The kraft process has several significant advantages when compared to the other pulping processes. The two main advantages are: (i) improved physical properties of the produced pulp and (ii) effective recovery of the cooking chemicals. The kraft pulp process uses a solution of sodium hydroxide and sodium sulphide, called a white liquor. The total alkali content is measured as units of sodium oxide (Na20) and the sulphide content is about 15 to 40 % of the total alkali content. When the wood is pulped and the non-cellulose organic part of the wood (hgnin) is dissolved, the sulphur is transferred from the inorganic part of the liquor to the organic one. The resulting pulp and cooking liquor is washed and the resulting black liquor is supplied to the initial point of the recovery cycle (Figure 1.1), the black hquor evaporators. The black hquor is then concentrated to a dry sohds concentration (from 0.65 to 0.75-0.80 kg/kg) and used as a fuel by the recovery boiler. In the recovery boiler, the organic part of the black liquor is used as an energy source and steam is generated. ^The sulphate process is referred to as a kraft process, since the produced pulp has a higher mechanical strength when compared to the sulphite process. The word kraft has a German origin, which means strong.
The inorganic part faUs to the bottom of the recovery boiler and forms a bed of smelt. In the smelt bed, the important recovery reactions occur in a reduction atmosphere. In the boiler bed, the sodium hydroxide from the black hquor is transformed to sodium carbonate and the sulphur is transformed to sodium sulphide. The resulting smelt is discharged from the boiler to a special disaolving tank. Any ash carryover is separated from the flue gases by the electrostatic precipitator (see Figure 1.1). The smelt is dissolved in the dissolving tank and the resulting solution, called a green liquor is transported to the causticizing process. In the causticizing process, the green liquor is treated with calcium oxide (quick lime). Sodium carbonate is transformed to sodium hydroxide, while calcium oxide is precipitated as calcium carbonate. The suspension is separated and the resulting liquid is the white liquor that is recycled back to the fiber line cooking process; at this point the recovery cycle is closed. The hme mud that results from the separation is concentrated to a paste using rotary drum filters and processed in the lime kiln to regenerate the initial calcium oxide. The above described recovery cycle is very effective with 90 to 95 % efficiencies. However, such efficiencies are achieved when the proper operating conditions are assured. The role of the recovery boiler cannot be underemphasized, it is the main unit in the recovery cycle since it recovers not only the chemicals, but also energy from the black liquor organic compounds. Without question, it is imperative that stability of the recovery boiler operation must be guaranteed.
1.8
Black hquor properties and the recovery boiler operation
Since the black liquor is a fuel source for the recovery boiler, its quality affects the boiler operation to a great extent [1, 22, 23]. The main black hquor characteristic that affects the recovery boiler operation is the dry solids concentration in the hquor. The dry solids concentration determines the amount of water to be evaporated in the boiler combustion chamber before the combustion can begin. It is necessary to achieve black liquor drying and stable firing simultaneously. The black liquor is fired as small droplets produced by swirl or splash type burners. It has been shown that the size distribution of the droplets determines how fast the droplets will be dried. As the droplets are dried a solid (residue) is formed. It is important that only this solid reach the boiler smelt bed. If instead, the wet droplet reaches the bed, the bed temperature decreases which in turn reduces the efficiency of the chemical reduction. In contrast, if the droplets are dried too far from the bed, the residue is carried out of the combustion chamber with the flue gases resulting in chemical losses and fume formation. The latter is extremely unwanted because the fumes accumulate on the boiler heat transfer surfaces thereby reducing the heat tranafer rate. The droplet size distribution has been shown to depend on the black liquor viscosity and consequently on the black hquor dry solids concentration. Therefore, it is important to have a consistent dry solids concentration of the black liquor feed for the recovery boiler.
1.9
FaUing fllm evaporators for black hquor concentration
The use of the falling film evaporators for black liquor concentration has numerous advantages. • FaUing fllm evaporators may be used even when small temperature differences exist [1, 2]. The relatively high heat transfer coefficient makes them a suitable choice and smaller temperature differences mean less energy consumption. • Falling film evaporators can be operated at decreased loads up to 25% from the nominal without experiencing scaling problems [1]. This permits a final dry solids concentrations of 65-70%, which is significantly more, compared to the LTV evaporators, where 50-52% is the usual final product dry solids concentration. The falling film evaporators are known to experience much less scaling compared to LTV evaporators because the liquor resides on the plates for a short amount of time and the evaporators operate at smaller temperature differences. • Falling fllm evaporators provide very small carryover with the condensates [1]. When plate type evaporators are used the carryover is as low as 1 ppm Na20. • Falling film evaporators are suitable for processing viscous liquids [2]. Concentrated black liquor falls in this class of hquids with viscosities up to 3-5 Pa.s. All these advantages determine the falling film evaporators as the most advanced solution in the black liquor evaporation. Therefore this kind of evaporators is being studied in the present work. Additional motivation is the lack of fundamental first
10
principles models in the open literature. Good fundamental model can be used for control, design or optimization purposes. It was therefore decided to develop a first principles distributed parameter model that wiU provide detaUed description of the falling fllm evaporator. The distributed parameter representation was accepted to be unavoidable, as already discussed above.
1.10
Transport phenomena in faUing films
A literature survey shows that the transport phenomena in faUing films have been studied extensively. The work of Kapitza [24] is the first significant work in this area and it is focused on describing the wavy nature of falling films and the important transport phenomena that occur. Kapitza's work has been extended by Levich [25], for example. Since the work of Kapitza focused on wavy-laminar falling films and the majority of the industrial applications involve turbulent falling films, new experimental and theoretical resulta appeared that cover both turbulent and wavy-laminar regimes. Important experimental inveatigationa were contributed by Wilke [26]. His results were used by Limberg [27], Chun and Seban [28, 29] and Seban and Faghri [30] to validate their theoretical work. Chun and Seban provide additional experimental results to improve the understanding of falling fllm phenomena. For instance that the falling film structure is more comphcated [31, 32] than originally assumed [24]. Important developments in the area of turbulent flows past flat surfaces are incorporated in the faUing film theoretical framework [27, 33, 34].
11
Other relevant theoretical investigations to improve the understanding of the faUing film phenomena include [35, 36, 37, 38, 39, 40]. In the case of experimental work, see [41, 42, 43], the range of experimental conditions is extended to cover more viscous liquids, which is the focus of this dissertation.
1.11
Dissertation organization
Thia diasertation ia organized as foUows. Chapter 2 develops a fundamental model of the important processes that occur on a plate in a faUing film plate type evaporator. Chapter 3 builds on the results of Chapter 2 by extending the model to represent one complete falling film plate type evaporator. In both cases, the models are validated using a set of expected changes in the feed parameters and utility stream. Chapter 4 introduces, develops, and demonstrates closed-loop control of a single falling film evaporator. The performance of two control strategies are investigated and compared. The first is a decentralized structure that consists of single-loop feedback controllers, and the second is a more sophisticated centralized control strategy, model predictive control. In Chapter 5, a fundamental model of a multiple effect falling film evaporator plant (seven evaporators) is developed and validated. Subsequently, the control of the multiple evaporator plant is examined in Chapter 6 uaing the same two control strategies tested in Chapter 4. Lastly, Chapter 7 summarizes the work, lists the contributions, and provides directiona for future work.
12
a >> o >, > o o
a, 1—I
c3
a zn
CO
!-<
pi
13
CHAPTER 2 MODELING OF A SINGLE PLATE 2.1
Phyaical conditions
In a multiple effect evaporation system, the term effect is frequently used to represent an evaporator unit that has its own vapor and liquid space and its own heating source.^ The physical conditions of the system are determined by the plant operating parameters, i.e., pressures, temperatures, flow rates etc. Since the values of these parameters change from effect to effect, their limits define the feasible operating regimes. Below, a discussion of the limits and their combined effect on the physical phenomena (eg. heating, evaporation, condensation) is presented.
2.1.1
Mass flow rates of the liquor
The total feed and respective product mass flow rates depend on the pulp miU throughput and the evaporation intensity. For a pulp mill with throughput of approximately 250 000 tons/y, it is not unusual to process about 500 tons/hr weak black liquor. The product mass flow rate can be in the range of 50 to 80 tons/hr. However, in the presence of disturbances, or startup and shutdown procedures these values may change. ^In this study, the term effect will be used even if the evaporator unit shares its vapor space with another evaporating unit. The other restrictions still hold.
14
2.1.2
Temperature and pressure
The temperature in the condensation and evaporation zone of each effect is determined mainly by the pressurea in theae zonea. In the case of condenaation, the temperature is the saturation temperature of water vapor at the given pressure. In the case of evaporation, the temperature is also affected by the boihng point rise (BPR). The fresh steam pressure is usually in the range of 0.4-0.5 MPa. The pressure of the secondary vapors of the last effect is in the range of 15 to 30 kPa. The values of the pressures in the intermediate effects are between 15 kPa and 0.5 MPa.
2.2
Hydrodynamics of the falling film at nominal conditions
The hydrodynamics of the falling film is closely related to heat transfer effects. Thus, it is critical to determine the hydrodynamic conditions of each effect to choose the proper correlations to calculate the correct values of the heat transfer coefficients. The hydrodynamics of the falling film can be described using three dimensionless numbers, that describe different relations between the forces acting on the fllm Reynolds number, Prandtl number and Kapitza number [43, 44]. The Reynolds number, given by
4r Re=—
(2.1)
represents the ratio between the inertial and the viscous forces acting on the fluid where F is the mass flow rate per unit width of the channel and /x is the dynamic
15
viscosity. The Prandtl number, given by
Fr = ^
(2.2)
represents the ratio of momentum transfer due to convection and thermal conduction where fi is the kinematic viscosity of the fluid, A is the fluid thermal conductivity and c is the fluid heat capacity. Finally, the Kapitza number
4
Ka = ^
(2.3)
describes the competition between capillary (creation of waves) and viscous forces (attenuation of waves) at the film surface where g is the gravitational acceleration conatant, p ia the fluid denaity, and a is the fluid surface tension. The values of these dimensionless numbers, together with the nominal conditions of the multiple effect evaporator system are given in Table 2.1 where Pyap is the vapor pressure, Tyap is the vapor temperature, Gpi is the black liquor mass flow rate at plates inlet and Xpi is the black liquor dry solids at plates inlet. From Table 2.1, it can be observed that the flow conditions change dramatically from effect 5 to the last section of the auperconcentrator (SCI). The Reynolds number decreases by about two orders of magnitude; the Prandtl number increases by about the same order; and the Kapitza number increaaea by about eight orders of magnitude. Thus, the flow characteristic in effect 5 is turbulent, with the transfer of
16
Table 2.1: Nominal parameters with respect to single plate. Effect
P
Pa
T
K
Gpi,kg/s
Xpi,kg/kg
Re
Pr
Ka
2022 4793 2710 1297 1100 102 25.5
27.01 17.89 21.01 33.19 73.48 605 1697
9.2X-S 1.9X-8 3.9X-8
^ vap •) ^ ^
5 4 3 2 SC3 SC2 SCI
21372 40269 65768 115870 204996 204996 204996
337 351 364 381 402 407 411
2.7 4.3 2.9 2.3 4.5 3.5 2.5
0.2261 0.2152 0.2716 0.3804 0.5144 0.6527 0.7020
2.7x-^ 7.0X-6 3.4X-2
2.0
heat dominated by the eddy momentum transport. In SCI, the flow characteristic is wavy-laminar [28]. The Prandtl number changes dramatically when going from effect 5 to SC3 to SC2, and to SCI. The values of the Kapitza number suggest that the capillary forces are larger than the viscous forces in effect 5, resulting in wavy film characteristics. In SCI, the viscous forces dominate and wave-like behavior of the film is suppressed. It is reasonable to conclude that two flow characteristics-wavy laminar, and turbulent-may exist in a multiple evaporator plant. To account for this different behavior, different correlations to calculate the heat transfer coefficients must be used. The following discusses these three cases and the selected correlations to determine the heat transfer coefficients properly.
17
2.3 2.3.1
Heat transfer coefficients of turbulent faUing films
General description of turbulent flow in an open channel
In turbulent flow, all transport processes, such as momentum, heat and mass transfer, are dominated by the eddy viscosity (e), also known aa eddy viscosity of Boussinesq [44]. The eddy viacoaity is the analog of the dynamic viacoaity term found in Newton'a law of viacoaity. The expression for turbulent flow is given by [44]
_(
tdvj;
dVj;
(2.4)
where r* is the Reynolds stress, p is the density of the fluid and e is the turbulent equivalent of the kinematic viscosity of the fluid. Assuming the mechanics of the motion of the eddiea are analogous to the motion of molecules of a gas, the following expression for the wall shear stress was derived by Prandtl [45]
•yx
= -pP
dvx dvx dy dy '
(2.5)
where I is the Prandtl mixing length.^ Combining Equations (2.4) and (2.5) and transforming to dimensionless variables, gives the eddy diffusivity of momentum as [43] = /+2
dii+
dy^
(2.6)
^The Prandtl mixing length is similar to the average length of the free path of a gas particle.
18
where u+ is the dimensionless film velocity, y+ is the dimensionless distance normal to the waU, and the other variables are as before. Therefore, to model turbulent flow, expressions for the mixing length (l) and the velocity gradient [du-^/dy^) are necessary. The derivation of these expressions ia closely related to the structure of the turbulent flow in an open channel. The flow can be divided into three regions waU boundary layer, turbulent core, and free surface boundary layer [43, 46, 47]. According to Bird [44], the relationship for the mixing length in the wall boundary layer region is linear and can be obtained from the following expression,
l^ = ky+,
(2.7)
where k is the Von Karman constant, which is assigned a value of 0.4. The value of Von Karman's constant is experimentally determined to be in the range of 0.36 < A; < 0.4 based on velocity profiles in round tubes [44]. In the turbulent core region, the dependence deviates from linear behavior, and according to Limberg [27], this deviation can be accounted for by multiplying the right hand side of Equation (2.7) by the following function -CLy+ F = e 5+ ,
(2.8)
where 5"*" is the dimensionless film thickness and c is a constant to be determined. Dampening of the turbulence at the wall boundary layer is accounted for by using
19
the Van Driest damping function [33]
D^ = l-e
^t
,
(2.9)
where A+ is aaaigned a value of 26 that correaponds to the best fit to the experimental data used in [33].^ Thus, the expression for the mixing length in the waU boundary layer and the turbulent core is given by
Z+ = ky+Dy,F.
(2.10)
Combining Equations (2.10) and (2.6), gives an expression for the eddy diffusivity of momentum
- = k'y^'DlF'
dy^
(2.11)
An expression for the velocity gradient is obtained by using the bulk flow momentum equation for fully developed conditions and zero vapor shear, which after integration^ is given by [43]
- ! ^ - ( ^ - ^ ) ^ -
(-^)
^In the same source [33], for another independent set of experimental data, the value of A+ is found to be 27. One may conclude that this parameter is almost constant. "^The integration procedure is provided in Appendix B
20
Near the wall, this expression can be approximated by
l«(l + - ) ^
(2.13)
y+ since close to the wall —
l = _ l + iyrf4^2^T2^p^.
(2.14)
In the free boiindary layer region, Alhusseini et al. [43] developed an expression for the eddy diffusivity of momentum, based on absorption data obtained by Chung and Mills [48], and Won and Mills [49]. The expression is given by
e
1.199 X 10-16 Re^™ / ( ^ + - y +
where Af is an empirical function of the Kapitza number that corrects for the assumption of a smooth surface when the correct representation is that of a wavy surface. The final expression for the eddy diffusivity of momentum in the film is given by,
-^ + \V^ - = {
^
+ {2ky+D^F)^
0
,
,
1 l.mxW-'^Re'f5^-y^\ Ka ^+i V At
21
y+
(2-16)
where y+ is the dimensionless distance from the waU to the starting point of the free interface. A typical distribution of the eddy diffusivity of momentum is shown in Figure 2.1. The experimental research of Alhusseini et al. [43] confirms that the proposed
50
100
150
250
Dimensionless film thickness. 5^
Figure 2.1: Eddy diffusivity of momentum for a falling water film. The values of ejv are determined for a Reynolds number of 9925.
model provides very good agreement with the data for the heat transfer coefficient of evaporation and sensible heating, and mass transfer coefficient of absorption. The precision is in the range of ± 10%.
2.3.2
Heat transfer coefficient of evaporation
Using the turbulent model, the heat transfer coefficient of evaporation is calculated by integrating the energy conservation equation with a constant heat flux at the wall 22
and the free boundary surface [43, 50] (see Appendbc A for detaUs). The expression, in dimensionless variables, is given by
hl =
6+'Pv 5+
(2.17)
1 -'r Pr
Prt
dy' ^u)
where h% is the dimensionless heat transfer coefficient of evaporation and Prt, the turbulent Prandtl number, is defined by [43], -1 Prf.=
(2.18)
In the above equation, Pet = Pr{e/v) is the turbulent Peclet number and Pri = 0.86 and C" = 0.2 are constants determined by Kays and Krawford [51] and
/..i^^^ h% =
2.3.3
(2.19)
Heat transfer coefficient of condensation
The expression to determine the heat transfer coefficient of condensation, according to Stephan [50], is identical to the one developed by Alhusseini et al. [43], i.e.. Equation (2.17). It is derived by integration of the energy equation using the same boundary conditions as stated above.
23
2.3.4
Heat transfer coefficient of sensible heating
The heat transfer coefficient for sensible heating is calculated by integrating the energy conservation equation with the assumptions of zero heat flux at the free boundary surface and a constant heat flux at the wall [43]
5+'Pr
^'^ir^pw^v' Pr ^ Prt Vi/.
"-'"
where u"^(^+) is the dimensionless velocity distribution, defined by [43] to be
u+ = /
2.3.5
r-2/+ 1 _ i i YT^de.
(2.21)
Calculation of the dimensionless film thickness
The dimensionless film thickness, 5~^, is used in the calculation of the heat transfer coefficients. It is determined iteratively using the following equation [43]
r Re ^4-
f^^
=4 / 1^ Jo
u+(iy+
(2.22)
The Reynolds number is calculated using the known parameters, F, the mass flow rate per unit width of the channel and p,, the dynamic viscosity of the liquid. In this study, the dimensionless film thickness is calculated first, followed by the calculation of the relevant heat transfer coefficient variables.
24
2.4 2.4.1
Heat transfer coefficients of wavy-laminar films
Transition from wavy-laminar to turbulent flow
With any change in the hquor properties (e.g., increase in the dry sohds composition) and operating conditions (e.g., decrease in the mass flow rate) the falling film experiences a transition from the turbulent regime to the wavy-laminar and eventuaUy laminar regime. Determination of the transition point has been studied by several researchers. Kapitza [24], in his early work on viscous fluids proposed that laminar flow is described by Reynolds numbers up to 1500. A similar range can be found in Levich [25]. Bird et al. [44] provide a Reynolds number value of about 350. Such approximations, while useful, are not precise enough (the data are for a limited number of liquids such as water and water-ethanol used in the study of Kapitza and Levich) and may not be generalizable to other fluids. A more quantitative approach is taken by Chun and Seban[28]. In their work, two correlations are provided to determine the Reynolds number at the point of transition. One correlation is based on the Prandtl number; the other is based on the Kapitza number. They are given by
Ret = 2460Pr-°-^^
(2.23)
Ret = 0.2l5Ka-'^,
(2.24)
respectively. These correlations were validated by experimental data provided by Chun and Seban [28]. However, in the case of the fluid studied here, the predic25
tions obtained from these correlations, shown in Table 2.2, are different for the same conditions that Chun and Seban obtained their validation. Table 2.2: Predicted Reynolds numbers at the transition points for black liquor at nominal conditions. Effect 5 4 3 2 SC3 SC2 SCI
Pr
Ka
27.01 9.206X-8 17.89 1.939X-S 21.01 3.944X-8 33.19 2.661x-^ 73.48 6.969X-6 605 3.263X-2 1697 2.034
Ret{Pr)
Ret{Ka)
288.7 377.34 339.91 252.51 150.63 38.264 19.57
47.615 80.029 63.163 33.426 11.256 0.673 0.17
The differences in the predictions of the transition Reynolds number by these correlation make the application of these correlations tenuous. One alternative is to use the turbulent model of Alhusseini et al. [43] together with an empirical correlation for the wavy-laminar regime [41]. The proposed expression. Equation (2.25), uses the calculated heat transfer coefficients for both turbulent and wavy-laminar regimes, thus providing a smooth transition
hE,totai = {h%^i + h%/^.
(2.25)
It is shown in [41] that this expression fits the experimental data for all Reynolds numbers between 30 and 15 000, including the transition region. A similar expression
26
for condensation is given by [50]
hE,total = {fh'E,i + h%t)h
(2.26)
where / is a correction factor that has a value of 1.15. Bird et al. [44] propose expressions with the same structure for the combined Nusselt number when accounting for free and forced convection heat transfer in the turbulent boundary layers. Since it is common to determine the heat transfer coefficient by applying a superposition of the combined effects (e.g., Equations [2.25] and [2.26]), it is reasonable to use Equation (2.25), provided by Alhusseini et al., to calculate the heat transfer coefficients for evaporation, condensation, and sensible heating over the entire range of operating conditions.
2.4.2
Heat transfer coefficient of evaporation
The heat transfer of evaporating or condensing films, in the laminar region, has been a well studied subject beginning with the work by Nusselt [44, 52]. In the case of a vertical tube in which a saturated vapor is condensing, Nusselt derived the following equation
where AH^v is the latent heat of vaporization, L is the channel length, and AT is the temperature difference between the wall and the film. It was shown that this expression underpredicts the value of the heat transfer coefficient by as much as 50%
27
[52]. The discrepancy was attributed to the fact that falhng films do not have smooth interfaces, but rather wavy ones. The first significant attempt to describe the motion and heat transfer in wavy falling films was made by Kapitza [24]. In that work, it is assumed that the waves on the film surface are sinusoidal. The mass, momentum, and energy balances are then solved to obtain approximate expressions (second order approximations) for the velocity distribution, film thickness, wave ampHtude, etc. Kapitza also proposed the following equation for the Reynolds number when the flow character changes from laminar to wavy-laminar.
Ret,i = 2.43 ^ J 7 ^ .
(2.28)
Additionally, Kapitza calculated that the heat transfer coefficient of sensible heating should be 1.21 times larger than the value calculated at laminar conditions. An experimental study by Kapitza confirmed the proposed theoretical developments even though large errors (25-50 %) between the calculated and measured values of the phase velocity and wavelength were found. It can be concluded that the theory proposed by Kapitza does not represent the underlying phenomena rigorously. Indeed, the discrepancy between the theory and the experiments can be observed in the photographs, taken by Kapitza, of the water and alcohol falling films. There, large sohtary waves with non-sinusoidal form are visible. Moreover, at higher flow rates, the solitary waves are seen to be foHowed by one or two smaU waves. Other studies have taken a pure experimental approach. Chun and Seban [28] have
28
proposed the following correlation for the local heat transfer coefficient of evaporation
/ ^3 \ 5 / - p X -0-22
hEM = 0-QOG[-J) [-)
(2.29)
Chun and Seban find that this correlation fits their experimental data satisfactory for the evaporation of water and water-glycol mixtures. However, the above correlation (Equation (2.29)) is vahd for a narrow range (1.77 to 5.7) of Prandtl numbers. For a complex Hquid such as black liquor, it has been shown (see Table 2.2) that Chun and Seban experimental correlations may not be appropriate. The rapid advances in computing hardware have made solving very computational intensive calculations possible. Thus, there are a number of accurate numerical solutions to the fundamental momentum, mass and energy balances for wavy films. Alekseenko et al. [35] solve the Navier-Stokes equation for falling films using the Karman-Poulhausen method of integral relations. This method assumes a parabolic velocity profile (for details on the application of Von Karman integral balance equations see Bird et al. [44]) for the falhng film. The results of Alekseenko et al. show close similarity between the calculated and experimentally observed velocity and wave profiles. A similar study by Chang et al. [36] provides another solution to the problem. Here, too, good agreement was found with the experimental data provided by Stainthorp and Allen [53]. A general characteristic of these studies is the occurrence of large irregular (e.g., varying amplitude, asymmetric) waves on the film surface.
29
Research by [54, 55]
conflrmed this observation by direct measurements. Mudawar and Houpt [54] used LASER-Doppler velocimetry to study water and water-glycerol falling films. Jayanti and Hewitt [55] simulated, in great detail, different wave forms and confirmed known experimental results about the increase of the local heat transfer coefficient.
The
reason for the increase in the heat transfer is attributed to the transport of a large part of the fluid in the observed solitary waves. Thus, the average film thickness decreases, hence the increase in the value of the heat transfer coefficient. Many of the studies have a common and significant drawback, they are computationally intensive. For example, the study by Chang et al. [36] required about 500 hours of processing time on a Convex machine to calculate the film evolution on a plate of approximately 2 meters and a duration of about 250 seconds. Such demanding computational loads continue to challenge even today's computing technology. A compromise between the detailed models and the computing burden is to develop more effective engineering correlations. Such an approach is found in the studies of Miyara and Miyara and coworkers. In Miyara and Uehara [38], wavy falling films were solved numerically from the perspective of flow dynamics and heat transfer capabilities. This study confirmed that the heat transfer enhancement is mainly due to a decrease in the average film thickness. In a foUow-up study by Miyara [39] the influence of the Prandtl number on the heat transfer coefficients of condensation and evaporation were investigated. Since the range of Prandtl numbers studied was between 0.1 to 100, the results found is of particular interest to the present study. Miyara concluded that for a Reynolds 30
number of 100, an increase in the Prandtl number in the range cited resuhed in an increase of ^ 56% in the amount of heat transferred (see Figure 2.2).
1.6
o
1.35
20
Miyara's data Power law fit, h*/h = 1.431Pr" 40
60 Prandtl number
80
100
Figure 2.2: Heat transfer coefficient of wavy-laminar falling film. The graph shows the relative increase of the heat transfer coefficient of evaporation/condensation found by Miyara [39].
In Miyara [40], a very useful comparison between the empirical correlations of Kutateladze [56], Chun and Seban [28], and Uehara and Kinoshita [40] is provided. These correlations, however, are functions of only the Reynolds number. However, the Reynolds number neither accounts for the influence of the capillary forces nor the heat conduction. Unfortunately, the study by [40] did not provide a correlation that reflects the influence of the Prandtl or Kapitza numbers on the heat transfer coefficient of evaporation.
31
In the present study, the correlation, obtained experimentally by Alhusseini et al. [41, 42] and given by hsM = 2.65i?e-°-i5«i^a°-°^63
(2.30)
will be used. This choice is dictated by the fact that this correlation accounts for the influence of the Kapitza number even though it does not account for the influence of the Prandtl number. The correlation is shown to be valid by Alhusseini et al. [41] using experimental data in the Prandtl number range of 1.7 to 47. According to Miyara (see Figure 2.2), the influence of the Prandtl number is significant in the range of 0.1 to 10. For higher values, the influence is much smaller as can be seen in Figure 2.2.
2.4.3
Heat transfer coefficient of sensible heating
The information on the sensible heating of wavy-laminar falling films in the open literature is considerable less than the information on the condensation and evaporation. Heating without evaporation of wavy films is not well studied either experimentally or theoretically. One of the few reliable sources is the work of Kapitza [24]. As mentioned in section 2.4.2, Kaptiza determined experimentally and theoretically that the heat transfer coefficient of sensible heating is 1.21 times greater than that for pure laminar conditions. If this information is to be used, it is necessary to calculate the heat transfer coefficient of sensible heating for pure laminar regime. This problem already has been extensively studied and solved initially by Pohlhausen 32
[57]. The analytical solution is based on the solution of the differential energy balance of incompressible flow without heat generation [14, 44, 58]. The general expression to calculate the heat transfer coefficient is given by.
1 „
1
hH,y,i = 0.332Re-^Pr-^
(2.31)
It then follows that to account for wavy interface influence, the final correlation should be, hHM = 0.402Re'^Pr^.
2.5 2.5.1
(2.32)
Modeling of single plate
Dimensionality
The problem of model dimensionality is of importance in any modeling application especially when the system is of the class of distributed parameter systems. Thus, it is necessary to determine how many spatial dimensions should be included in the model to represent satisfactorily the behaviors of interest. The models, describing fahing films usually are two-dimensional, nonlinear, and very complex [35, 36, 40]. However, such systems of equations, while complete often pose large computational overhead. In the modeling of falling film evaporators, there is general agreement on neglecting the details of the film itself
It is enough to
represent the spatial (distributed) behavior of the film but the dimensionality of the its distributed nature (axial or plate direction or both axial and transverse directions)
33
must be established. In this respect, an experiment was performed to determine the number of spatial dimensions. The experiment employed a tracer to determine if there are any significant velocity components in the transverse direction. The experimental device, shown in Figure 2.3 was constructed to introduce the tracer into heated or non-heated falhng films. The device consists of a tin plate with a dimple. The size of the dimple was
Flow distributor
Lip actuators
Temperature controller Rotameter
Yty-
Relay
~ Power, 12 V, 10 A
0 Temperature sensor
Heating element Manual valve
Tin plate
J
Figure 2.3: Experimental design.
34
Centrifugal pump
F
chosen to be close to that found in real industrial evaporators. A flow distributor is placed at the top of the plate to create a uniform falling film. The flow distributor is equipped with a bending hp connected to six actuators for adjustment of the opening of the lip, thus assuring film uniformity The liquid flows down the plate and is collected into a tank where it is recirculated using a centrifugal pump (Teel 1P799A, Dayton Electric Mfg. Inc.). The flow rate is measured using a Brookfield Instruments' rotameter with tube (R-8M127-1) and float (8-RS-14). The tin plate is mounted onto a glass plate, heated from the opposite side by a resistance wire heating element (power of about 1200 W at 12 volts). The temperature of the plate is measured with a temperature sensor that is controlled by an on-off temperature controller. The controller hysteresis was adjusted to assure ± 1 K deviation from the set point. The dimensions of the tin plate and dimple are on Figure 2.4. Table 2.3: Experimental conditions for model dimensionality experiment. Temperature, °C
Fluid
F, kg/s.m
p, mPa.s
Re
24 60 24 60
Water Water SCMC SCMC
0.819 0.819 0.916 0.916
0.921 0.470 5237 2821
3557 6970 0.699 1.298
The experiment was performed at two different temperatures, 24°C and 60°C. Two different liquids were used, water and a water solution of sodium carboxymethylcellulose (SCMC). The solutions of SCMC are very viscous even at low concentrations. The viscosity, depending on the brand of SCMC may reach 10 000 mPa.s for 1-2% 35
solutions^ The objective was to investigate the two limiting conditions observed in the evaporation of a black liquor fluid (see Table 2.1). The experimental conditions for each of the four tests are presented in Table 2.3. The viscosity of the water was calculated fiom tabulated data and the viscosity of the SCMC at room temperature was measured using Cannon-Fenske A919-500 viscometer. The viscosity of SCMC at 60°C was calculated using the data provided in [59]. The results of the experiment are shown in Figures 2.5 to 2.8. It is clear, that the velocity component in the transverse (horizontal) direction is negligible in the case of SCMC; there is no tracer spreading in the horizontal direction. In the case of water, at Re = 3557, a very smaU dispersion of the tracer in the transverse direction can be observed. At Re = 6970, there is a significant dispersion in the transverse direction when the tracer is introduced at the edge of the dimple. However, such high Reynolds numbers are not present in the system studied here (see Table 2.1). Based on the experimental studies, it is concluded that a onedimensional falling film model is satisfactory to represent the behaviors of interest.
2.5.2
The physical phenomena
The two main phenomena that occur on the plate are evaporation of the black liquor at the liquor side and condensation of water vapors at the steam side. Depending on the temperature of the liquor that enters the evaporator, there are two possibilities. If the temperature of the entering liquor is below the boiling point at ^When the water-SCMC solution was used, the flow rate was measured by using a standard volume vessel and a stopwatch
36
the operating pressure, then a heating zone will exist on the plate. In this zone, the liquor is heated until it reaches its boUing temperature. However, if the temperature is greater than the boihng point temperature at the operating pressure conditions, the liquor that enters the evaporator flashes before the temperature decreases to the boiling point temperature. Thus, the hquor enters the plate at the boiling temperature. These two possibilities are shown in Figure 2.9. There are three main physical processes that occur on the plate evaporation and heating of the black liquor and condensation of water. Additionally, heat transfer at the wah cannot be neglected. The fohowing is adopted from Stefanov and Hoo [60].
2.5.2.1
Black liquor evaporation
The film evaporation is modeled using a differential volume approach. Figure 2.10 presents the differential volume in the black liquor film, where Gs is the mass flow rate of the solids, G^ is the mass flow rate of the water, W is the mass flow rate of the evaporated water, Q is the heat entering the differential volume, 5 is the film thickness, a is the film(plate) width and Az is the differential volume length. The following general assumptions have been made. Assumption 2.5.1. The average film velocity in the vertical direction is independent of the horizontal position parallel to the plate. Thus,
^
= 0.
37
(2.33)
Assumption 2.5.2. There is no nucleate boiling. The assumption is reasonable because high temperature differences are not favored in the evaporation of the black liquor. When nucleate boiling is present at high temperatures, it is hypothesized that the accompanying high mixing rate increases collisions between the safl molecules permitting the formation of crystals [61]. This is the primary reason for evaporator scaling, which decreases heat transfer significantly [1, 61, 62]. Assumption 2.5.3. There are no significant changes in the black liquor parameters in the differential volume. Assumption 2.5.4. There is ideal mixing.
Thus, the heat of dilution of the black
liquor can be excluded. This assumption holds for a single plate, as it has been shown that the total error in the evaporation energy requirements in the concentration of softwood black liquor (from 20% to 65-90% dry solids) is about 4-6% [63, 64]. The change in the dry sohds for a single evaporator is about 10%, thus, the influence of the heat of dilution can be neglected. Assumption 2.5.5. The pressure at the vapor and steam side does not change significantly due to friction losses. According to Perry [65] section 11-110, the pressure drop in a falhng film evaporator is negligible; and the pressure over the falling film is essentially the same as the vapor head pressures. 38
Energy balance of the differential volume In the development of the energy balance, the mass flow rate of the black hquor is divided in two parts
mass flow rate of the solids, Gs and the mass flow
rate of the water, Gy, (see Figure 2.10). The overall differential volume element energy balance can be written as dF - ^
=
Q-Ws
+ GsU {hs \z ^\vl + gz) + Gy, I {hy, I +\vl + gz)
-
Gs \z+Az {hs \z+/\z +\vl
+
g{z + Az)) -W{K
+ g{z + A z ) ) - Gyj \^+AZ [K J^+A.^ +\V'^2
+ Ivl, + g{z + Az)) , (2.34)
where /itu is the enthalpy of the water, hg is the enthalpy of the solids, hy is the enthalpy of the water vapors, Ws is the shaft work the system applies to the surroundings, Vy^z is the vapor velocity in z direction and g is the gravitational acceleration. The following simplifying assumptions are made, - The shaft work is negligible. - The kinetic energy terms and the potential energy terms are very small as compared to the internal energy terms. With these assumptions, Equation (2.34) reduces to
——^
=
Q -\- Gs \z hs \z +Gyj \z hw \z —Gs Iz+Az hs \z+Az
dt
(2.35) ~
Gyj
\z+Az
hy,
jz+Az
39
—Why.
The material balance of the differential volume is given by
dMdy _ , - ^ = Gy, I -Gy, |,+A. -W.
(2.36)
Since the solids do not change during the evaporation,
Gs \z— Gs \z+Az
Rearranging the terms in Equation (2.35) gives
dEdy dt
=
n^n \ (h \ h \ \ ^ ^M^y Q + Lrs \z [ris \z -IT'S |z+Azj H -rr-h " ' ^ '^' ' '^ ^ '^^"^^ ' dt w \z
+
Gyi
jz+Az
\hw
\z
—hy,
jz+Az)
~
W^
[hy
— hy,
(2.37) |z) .
It is assumed, that the material accumulation term is small with respect to the other terms in the energy balance. The specific enthalpy is given as
h = Cp{T -Tr),
where for convenience the reference temperature Tr is taken to be 0. Using assumption 2.5.3, Equation (2.37) can be re-written as
——^ = Q -\-Gs \z {Cp,sT \z —Cp,sT (z+Az) + dt Gy)
jz+Az
(Cp.ti;-/
\z
—Cp^yjl
40
Iz+Azj
~
^
{'T'V "
hyj
(2.38) |zj ,
where Cp^s, Cp^w are the specific heat capacities of the sohds and water, respectively in the differential volume. As the phase change occurs, the temperature of the film along the plate does not change significantly. The only change, due to the boiling point rise can be neglected as the boihng point rise along the plate is in on the order of 1 K. Thus, it is reasonable to assume that
T \z= T |z+Az •
The difference, hy - /l^u|z, represents the heat of evaporation, AHey Thus, Equation (2.38) becomes
dEdy = Q- WAHey. dt
(2.39)
The typical film thickness in falling film evaporators is usuahy on the order of 10-^ m. Thus, it is reasonable to conclude that the film reaches equihbrium rapidly. It then follows that
Q = WAHey
(2-40)
where Q = hEaAz{Ty,-Tf).
41
(2-41)
In the above equation, Ty, is the wall temperature, Tf is the film temperature, hs is the local heat transfer coefficient of evaporation, and a is the plate width. • Material balance of the diflFerential volume element Applying the principle of material conservation to a differential volume of size Az (see Figure 2.10) gives
M \t+M -M
\t= G^At-G
Iz+Az At -
WAt,
where M is the mass of the black liquor, G is the mass flow rate, W is the mass flow rate of the evaporated water and At is the time difference. Dividing through by At and taking the limit as At ^ 0 yields
^
= G Iz - G Iz+Az -W.
(2.42)
To derive the mass balance with respect to the black liquor mass flow rate, it is assumed, that the residence time r of the hquor in the differential volume is constant, i.e. r |z = r |Z+AZ- Using this assumption, the residence time can be expressed as _ Az
(2.43)
Vz '
Substitution of the above equation into Equation (2.42) gives
dG AzdG _, , _, . ^j. T^ = — ^ = G z - G z+Az -W, at Vz ot 42
,^ ^^. 2.44
where it is assumed that the residence time does not change significantly with time. Defining W = W/Az and dividing Equation 2.44 by Az gives
1 uG G L —G Uj-Ar - ^ = ^ - W. Vz dt Az
2.45 ^ ^
Taking the limit as Az —> 0 yields
f+-f-^'-
P-^«)
The mass balance with respect to the dry solids is derived in a similar fashion. Thus
Using Equation (2.43), the above equation becomes
G—^+^s—^ Vz
ot
= {GXs) \z -{GXs) Iz+Az .
(2.48)
Vz ot
Dividing Equation (2.48) by Az and taking the limit as Az -^ 0 gives
G9x.^x,sa_t^sx,^^SG:
Vz dt
Vz dt
V dz
dz
^^^^g,
Substituting Equation (2.46) into the above yields
dXs , dXg „,, Xs — +y^^=WVz^. 43
, . (2.50)
To summarize, the mathematical model that describes the evaporation process is given by dG
dG + Vz^r- =
'dt^'^'Tz
-W'vz (2.51)
ax dt 2.5.2.2
dXs .^., Xs dz
G
Black hquor heating
The same approach that uses the differential volume element will be applied to modehng the heating portion of the fluid. Figure 2.11 shows the differential volume element with the important flows and variables. • Energy balance of the differential volume element The general energy balance of the differential volume element is given by
^
=
Q-Ws
+ Gr\z{h\z+\vl
+ gz)
-
Gi Iz+Az f hi Iz+Az +2^2 + 9{z + Az)
where G; is the liquor mass flow rate and hi is the black liquor specific enthalpy. As in the evaporation case, the shaft work is assumed to be negligible; and the kinetic and potential energy terms are neglected. It then follows that
^
= Q + Gi\zhi\z
-Gi Iz+Az hi Iz+Az .
(2-53)
at Only changes in the density of the black liquor are considered during heating because density is a strong function of temperature. However, the density changes
44
are assumed to be small hence
Gi \z= Gi |z+Az= Gi.
It follows that dE - ~
= Q + Giihi\z-hi\z+Az).
(2-54)
Using the specific enthalpy relation
hl=Cp,l{T-Tr),
where as before the reference temperature T^ = 0 and assumption 2.5.3, Equation (2.54) becomes
dEdy dt
Q + GiCp,i{T\z-T\z+Az).
(2-55)
The mass flow rate of the liquor can be expressed as
Gi = aSpvz,
where 6 is the film thickness. The total energy of the differential volume can be assumed to be equal to its internal energy. The internal energy is given by
U = MdyCy^iiT - Tr) = apSAzCyjT. 45
(2.56)
The heat entering the differential volume can be expressed as
Q = hHaAz{Ty,-T).
(2.57)
Using Equations (2.56) and (2.57), the energy balance becomes
ap6AzCy^i— = hnaAziTy,
- T) + a5pCp^iVz(T |, - T |,+AZ).
(2-58)
Taking the limit a Az —>• 0, the energy balance is given by
f + „ . ^ = M ^ ot oz pocp^i
(2.59)
The assumption that Cy^i w Cp^i (recall Cp — Cy = R, the universal gas constant) is used; thus the difference between the heat capacities at constant volume and constant pressure can be neglected.
2.5.2.3
Steam condensation
The model describing the condensation process is similar to that developed for the evaporation process. The assumptions used above are presumed to be valid here. The differential volume with the important flows and variables is shown in Figure 2.12.
46
102.5
344
72
Figure 2.4: Experimental setup to determine flow distribution. The dimensions of the dimpled plate are in mm. The size of the dimple is comparable with the size of the dimples in the real evaporator.
47
;,< I '
i t-
h ^
7
'•
-•1'
Figure 2.5: Tracer patterns for water at 24°C, Re = 3557. .
.
.
.
.
^
),wr
y
v
:>
1^
Figure 2.6: Tracer patterns for water at 60°C, Re = 6970. it'' '
y
|] Figure 2.7: Tracer patterna for SCMC at 24°C, Re = 0.699
J
/
1
Figure 2.8: Tracer patterna for SCMC at 60°C, Re = 1.298. 48
'ledrculation < Tjat
Iredrculation ^
'sat
Vapor Liquor | inlet
13
Condensation zone
:l'
*-
Steam
.^
Flash zone ^
\
Evaporator wail I
Evapqrator wall
Heating zone
I Condensation " ^ zone
Evaporation zone
• - Evaporation zone
Steam
Vapor
Vapor
Plate
Plate
Figure 2.9: Presence or absence of a heating zone.
49
Liquor inlet
X
^ W , Z ) ^ S , Z 5 V2
A
r plate '
AZ~^
W
a
liquor film
Gw,z+Az , G ,z Az 5 V
Figure 2.10: Differential volume of the evaporating film.
50
Tz+Az 5 Vz
Figure 2.11: Differential volume of the heating film.
51
L
condensate film
W.
Gz+Az5 Vz
Figure 2.12: Differential volume of the condensing film. • Energy balance of the differential volume element The overall energy balance of the differential volume is given by
dEdy
dt
=
-Q
-Ws
+ Gc \z (he \z +\vl
+ gz) - Gc |z+Az {he Iz+Az
(2.60)
+
i^z + 9{^ + A^)) + W, {hst + \vlz + g{z + Az))
Neglecting the kinetic and potential terms, the above balance becomes
dEdy
dt
= -Q
+ Gc \z he \z -Ge
52
Iz+Az he |z+Az +Wehst.
(2-61)
The material balance of the differential volume is given by
dMdy - ~
^ , = Ge \z -Ge
Iz+Az + I ^ c .
(2.62)
Rearranging the terms in Equation (2.61) yields
dEdy J,
_ —
^ , dMdy^ 1 , ^ 1 ,u \ u \ \ -~W H 'lf~ '^ 1-^ " ^ ^ c U+Az l/^c \z —n-c Iz+Azj
+
We {hst - /ic Iz) •
(2.63)
It is assumed, that the material accumulation term is small with respect to the other terms in the energy balance. Using assumption 2.5.3 and the fact that the specific enthalpy is given by
h = Cp{T — Tr)
Tr = 0.
Equation 2.63 can be rewritten as
dE. '± = -Q + Gc \z+Az {Cp,eT \z -Cp,cT Iz+Az) + We {hst ' he \z) , dt
(2-64)
where Cp^e is the specific heat capacity of the condensate. As the phase change occurs, the temperature of the film along the plate does not change, thus T |z= T |z+Az •
53
AdditionaUy, the heat of the condensation (evaporation), AHe, is given by the difference hst - hy, \z. It then follows that the energy balance is given by
dEdy
dt
-Q + WeAHe.
(2.65)
As in the solution of the evaporation process, the energy balance is solved at steady state. Thus Q = WeAHe,
(2.66)
where Q = heaAz{T-Ty,,e)-
(2.67)
• Material balance of the differential volume element Applying the principle of material conservation to a differential volume of size, Az, (see Figure 2.12) gives
M \t+At -M
\t= Ge \z At - Ge Iz+Az A t +
WeAt,
where M is the mass of the condensate, Ge is the mass flow rate, W is the mass flow rate of the condensing steam, and At is the time difference. Dividing through by At and taking the limit as At —> 0 yields
^
= Ge Iz -Ge
dt
54
Iz+Az +W^c-
(2.68)
To derive the mass balance with respect to the condensate mass flow rate, it is assumed, that the residence time r of the condensate in the differential volume is constant, that is r | z = T |z+Az •
The residence time then can be expressed as
_ Az
(2.69)
Vz
Substitution in Equation (2.68) gives
dGe
Az dGe
T ^ - = ^ = Ge \z -Ge Iz+Az + W , . dt Vz ot
(2.70)
Defining W', = We/Az
and dividing Equation (2.70) by Az gives
1 dGe Vz dt
Gc \z —Ge \z+Az_ Az
y^f
(^ 71)
Taking the limit as Az -> 0 yields
^
+ , ^ ^
dt
dz
55
= VF>z.
(2.72)
2.5.2.4
Heat transfer at the waU
The model of the heat transfer process at the wall is also developed using a differential volume approach (see Figure 2.13). The wah calculations do not involve a
bottom
*Z
Figure 2.13: Wall differential volume element.
material balance, because the wall does not exchange any mass with its surroundings. However, an energy balance is necessary. In the energy balance, the potential and kinetic energy terms and shaft work are neglected. Only the internal energy and heat input/output terms remain. There are four boundaries through which the wall differential volume can exchange heat:
• Heat flux from the condensate film to the element, Qcw-
56
• Heat flux from the element to the evaporating film, Qyj^• Heat flux to the top adjacent element
• Heat flux to the bottom adjacent element.
Because a phase change occurs on both sides of the wall surface, the fluxes to the adjacent elements can be neglected (the temperature gradient in the z direction is neghgible). The heat fluxes that are significant are those from and to the films at both sides of the wah. Thus, the overall energy balance of a differential volume at the wall is given by
In the current study, the wall of the plate is made of stainless steel with a thickness is 1.5 mm. According to [66] the thermal response of thin (1.15 mm) metal (copper) walls is on the order of 0.01 seconds. The temperature differences between the walls of the plate are in the range of 1.17 to 8.66 K. Using this information it is possible to solve the energy balance at the wall assuming that equilibrium is instantaneously achieved. Thus, Equation (2.73) becomes
^ew
— ^we-
V^.' ^J
The surface areas at both wall sides are equal. The following equations describe the condensation/heating and condensation/evaporation processes, respectively at the
57
wall
r
hc{Tf,c - Tw,c) = hH{Tyj^h — Tf^h) = ^^(Tlo.c - r^,/i) hc{Tf,e-Ty,^e) =
hE{Ty,^e-Tf^e)^^{Ty,,e-Ty,^e).
(2.75) (2.76)
These equations can be used to calculate the surface temperatures at the wah. To summarize, the mathematical models that describe heating and evaporation of the black liquor fluid are given by
dT^^ dt
dT ^ hH{Ty, - T) ^ dz pScp^i
^ + V z ^ = W^Vz dt dz
(2.77)
hc{Tf,c — Ty,^c) = hH{Tyj^h — Tf^h) = T—(71o,c — Ty,^h)
and
dG
dG
dt
dz
-^+Vz^
dXs ^
= -W'Vz
dXs . , . ,
Xs
__ + , ^ ^ = Wvz^
(2.78)
dGe
dGe
,,„ c^^ r
hc{Tf,c — Tw,c) = hE\Tw,e — Tf,e) = T-~(-^«'.c ~ -Mi),e))
respectively.
58
2.5.3
Dimensionless variables
The model can be transformed into a dimensionless form, by introducing the following dimensionless variables (Table 2.4). Table 2.4: Dimensionless Variables Transformed Variable
'!'* = rp* we
T To T •'•we
(4* --
Po G
Go Ge G* -Gcfi X *
••
X
Xs,0 z 7." =- — L Vz
K.^'-'c,z «.* Kz = tVffl
Definition
Black hquor temperature Wall temperature at evaporation side
Transformed Variable W' W'* = 7777 W^ W W^* = — -
Wio
Mass flow rate of the black liquor
LW^ r] = —p:^— Go
Mass flow rate of the condensate
77c =
' ^c,0
Dry solids content
TJT
HHL pCp5v*jQ
Space coordinate along the plate Black liquor velocity along the plate Condensate velocity along the plate Time
The substitution of these dimensionless variables in the model equations leads to the following dimensionless systems of PDE's ffP*
dT*
-L^<-^-^T{K-T*) dt* _,+,*^^
dz*
(2.79) = W,VzVc
59
and
dG'
.dG'
An analysis of the dimensionless parameter ??, shows that when Go ^ 0, 77 -^ oo and when Go -^ 00, 77 ^ 0. In general, for very high feed flow rates there will be no evaporation, while at very small feed flow rates the evaporation will be almost instantaneous. This is consistent with the physics of the process. Also, W^ is a weak function of Go. The influence of L, the length of the plate is clear. That is, there will be no evaporation at zero plate length and very high evaporation rates at large values of L. In the case of rje, the mass flow rate at the top of the plate is zero because liquid condensate is not present at the top. In this case Gc,o is taken to be a sufficiently small number (0.001 kg/s) that approximates the zero mass flow rate at the top of the plate.
2.5.4 2.5.4.1
Numerical Solution Solution method
Solving of the mathematical models that describe the physical phenomena that occur on a single plate is a challenging task. The system of equations are nonlinear, partial differential and algebraic equations (PDAEs)with non-constant parameters. For example, at the evaporation side, the film velocity, heat transfer coefficient, den60
sity, and heat capacity are functions of the black liquor mass flow rate, dry sohds concentration, and black liquor feed temperature. Thus, it is very difficult to find an analytical solution to the system of equations. A number of methods exist for solving systems of PDEs with two independent variables
space and time. These methods include finite differences, finite element,
orthogonal collocation, Galerkin methods [67] to name a few. The choice of a suitable method for the problem in this work was based on the considerations of computational efficiency and the high probability of a sharp shock-like solution profile [36]. The efficiency of the method is a consideration because the final goal is to solve a high dimensional system of PDAEs for an entire multiple effect evaporator plant. The computational efficient constraint excluded the usage of the Galerkin method (requires numerical integration at each step) even though this method is capable of finding steep solution profiles [67]. The orthogonal collocation method is the method of choice because it more computationally efficient. Within the class of orthogonal collocation methods, the method of Orthogonal Cohocation on Finite Elements (OCFE) is used because it has been proven to be very suitable for solving problems with steep solutions [67]. This method, however, has the inconvenience of reducing the system of PDEs to a system of differentialalgebraic equations (DAEs) and only a few commercial solvers are capable of solving this type of problem accurately.
61
2.5.4.2
OCFE discretization
The OCFE scheme for this problem includes 10 finite elements with 2 internal collocation points, which gives a total of 31 spatial points. In 21 points, the solution is calculated by integration of 21 ODE's for each PDE in the PDE system. The solution at the rest of the points is calculated by satisfying the algebraic conditions for continuity of the solution's first derivatives at the element boundary points. So, for the case of heating of the black liquor, the DAE system includes 93 equations and for the case of evaporation the DAE system includes 124 equations. Of course, for each spatial point the wall temperatures should be calculated, which adds another 31 algebraic equations to each case. The detailed discretization structure can be found in Appendix E.
2.5.4.3
Solver package
The solver used to solve the discretized systems of PDEs is the LSODI (Livermore Solver for Ordinary Differential equations. Implicit systems) solver, included in the ODEPACK package by Painter and Hindmarsh [68, 69, 70]. The LSODI solver handles well-behaved DAE systems (low index). The system in this study has an index of 1 (c/. ODE systems are of index 0). For more information on DAE properties and the determination of their indices the reader is referred to the work of [71].
62
2.5.4.4
Physical parameter correlations
Accurate solutions of the model equations require reliable correlations for all physical parameters. In the case of black liquor, only a hmited number of correlations exist and the accuracy of their predictions may be difficult to establish against real mill data since the latter themselves are variable as they are functions of the raw material properties, composition, and cooking conditions. The correlations used in this study correspond to the general behavior of kraft black liquor. The correlations can be found in the Appendix C.
2.5.5
Validation of the single-plate evaporator model
The operating conditions for a single-plate evaporator system are given in Table 2.5. Table 2.5: Operating Conditions Parameter
Nominal Value
Liquor feed mass flow rate Liquor feed temperature Steam pressure Vapor pressure Plate mass flow rate
58 kg/s 303.15 K 47000 Pa 21372 Pa 2.53 kg/s
To determine the validity of the single plate evaporator model, the system is subjected to a set of expected and unexpected disturbances. These disturbances include changes in the temperature, dry solids concentration, mass flow rate of the feed and the heating source (steam) inlet temperature. The disturbances are introduced 63
as perfect step functions within the range of ± 5% of their nominal values. The results are shown in Figures 2.14 to 2.21.
2.5.5.1
Feed dry liquor concentration 338
6 Plate length, m
8
Figure 2.14: The response of the single plate evaporator to a 5% decrease in dry sohds concentration. -I-: nominal response and o: non-ideal response. Liquor side variables (first three graphs): plate temperature, exit sohds concentration, and exit mass flow rate. Steam side (bottom graph): condensate mass flow rate.
A change of the dry solids concentration in the feed stream results in changes in
64
6 Plate length, m
8
10
12
Initial state Final state
12 Plate length, m
Figure 2.15: The response of the single plate evaporator to a 5% increase in dry sohds concentration. -I-: nominal response and o: non-ideal response. Liquor side variables (first three graphs): plate temperature, exit sohds concentration, and exit mass flow rate. Steam side (bottom graph): condensate mass flow rate.
the variables of interest (hquor mass flow rate, temperature, condensate mass flow rate) with the largest effect on the dry solids concentration on the plate. A moving fiont can be observed (see Figure 2.14) travelhng from the top to the bottom of the plate. The residence time on the plate is approximately 7 seconds [60]. The esponding change in the exiting dry solids concentration is approximately the corr^ 65
same as the change in the input dry solids concentration. The effect of this disturbance on the other variables is very small. These results are consistent with the physics of the process. The heating zone increases since the amount of water to be heated to the boiling point has increased and water has a larger heat capacity than the black liquor sohds. The decrease in the boihng point rise due to the decrease in the inlet dry solids concentration leads to an overall decrease (small) in the boihng point temperature of the hquor. The almost negligible effect of a change in the feed dry solids concentration on the other variables is attributed to the natural compensation by the system itself
2.5.5.2
Feed mass flow rate
Changes to the feed rate do not appear to affect the variables of interest significantly. The temperature responses (Figures 2.16 and 2.17) show that the heating zone increases for either a positive or negative change in the feed mass flow rate. An explanation for this behavior is related to the properties of the system. If the mass flow rate decreases, the heat transfer coefficients decrease resulting in an increase in the length of the heating zone. In the opposite case, the increase in the amount of liquor to be heated wih require an increase in the length of the heating zone. The effect on the dry solids is very small. This is again attributed to a self-compensating effect by the system. Note, that the single plate evaporator system appears to be more sensitive to an increase in the feed rate as compared to a change of the same size in the the feed dry solids concentration. Another consideration is that the evap-
66
orator recirculation ratio is about six, therefore any changes observed at the exit of the plate may be amplified by the same factor or more.
Plate length, m 0.2 JO
-
0.1 O 0
2
4
1
1
1
6
8
10
1
Plate length, m
Figure 2.16: The response of the single plate evaporator to a 5% decrease in feed mass flow rate. +: nominal response and o: non-ideal response. Liquor side variables (first three graphs): plate temperature, exit solids concentration, and exit mass flow rate. Steam side (bottom graph): condensate mass flow rate.
2.5.5.3
Feed temperature
Changes in the feed temperature show significant effects on the variables of
67
Plate length, m
Figure 2.17: The response of the single plate evaporator to a 5% increase in feed mass flow rate. -I-: nominal response and o: non-ideal response. Liquor side variables (first three graphs): plate temperature, exit sohds concentration, and exit mass flow rate. Steam side (bottom graph): condensate mass flow rate.
interest. A 5% increase in the feed temperature leads to a complete disappearance of the heating zone; flashing occurs before the feed reaches the plate. The corresponding increases in the dry solids concentration and mass flow rate can be observed readily. Note that the changes occur with almost the same magnitude in both the dry sohds concentration and the mass flow rate. 68
When the feed temperature is decreased the variables of interest show more sensitivity as compared to an increase in the feed temperature primarily because the heating zone has increased approximately one meter in length. Equivalently this translates to approximately a 10% decrease in the heat transfer surface available for
Plate length, m
Figure 2.18: The response of the single plate evaporator to a 5% decrease in the feed temperature. +: nominal response and o: non-ideal response. Liquor side variables (first three graphs): plate temperature, exit sohds concentration, and exit mass flow rate. Steam side (bottom graph): condensate mass flow rate.
69
337cp-
4
6 Plate length, m
8
10
12
Plate length, m
Figure 2.19: The response of the single plate evaporator to a 5% increase in the feed temperature. +: nominal response and o: non-ideal response. Liquor side variables (first three graphs): plate temperature, exit solids concentration, and exit mass flow rate. Steam side (bottom graph): condensate mass flow rate.
2.5.5.4
Steam pressure
Steam pressure changes affect the variables of interest in almost the same magnitude as the size of the change. The effects observed are in a direction consistent with the physics of the process. There are no significant spatial effects since the change in
70
the steam pressure is distributed along the entire plate and not locally.
Plate length, m
Figure 2.20: The response of the single plate evaporator to a 5% decrease in the steam pressure. -I-: nominal response and o: non-ideal response. Liquor side variables (first three graphs): plate temperature, exit solids concentration, and exit mass flow rate. Steam side (bottom graph): condensate mass flow rate.
2.5.5.5
Summary
To summarize, based on a set of expected disturbances in the range of ± 5 % of their nominal operating values, the developed model described the single plate evaporator
71
system behavior in a manner consistent with the physics of the system. The most important result observed is that the responses of the single plate evaporator system are stable for all prescribed disturbances.
72
Plate length, m
Figure 2.21: The response of the single plate evaporator to a 5% increase in the steam pressure, -f: nominal response and o: non-ideal response. Liquor side variables (first three graphs): plate temperature, exit solids concentration, and exit mass flow rate. Steam side (bottom graph): condensate mass flow rate.
73
2.6
Nomenclature
Ai
empirical wavy interface correction function
Ay,
constant in Van Driest's damping function
a
plate width, m
C
constant in the turbulent Prandtl number formula
c
specific heat, J/kg
Cp
specific heat at constant pressure, J/kg
Cp^i
liquor specific heat at constant pressure, J/kg
CL
constant in the Limberg's function
Cy^i
liquor specific heat at constant volume, [J/kg
Dy,
Van Driest's damping function
Edy
energy of a differential volume, J
Ey,
energy of a wall differential volume, J
F
Limberg's function
/
correction factor in the total heat transfer formulas
Gc
condensate mass flow rate, kg/s
Gi
liquor mass flow rate, kg/s
Gpi
black liquor mass flow rate at plate inlet, kg/s
Gs
solids mass flow rate, kg/s
Gyj
water mass flow rate, kg/s
g
gravitational acceleration, 9.81 m/s^
he
heat transfer coefficient of condensation, W/m^-K
he
enthalpy of condensate, J/kg
hE
heat transfer coefficient of evaporation, W/m^-K
hE,i
laminar heat transfer coefficient of evaporation, W/m^-K
hE,t
turbulent heat transfer coefficient of evaporation, W/m^-K
hE,wi
wavy-laminar heat transfer coefficient of evaporation, W/m^-K
hE,Nu turbulent heat transfer coefficient of evaporation calculated by Nusseh, W/m^-K hn
heat transfer coefficient of sensible heating, W/m^-K
74
hH,wi
wavy-laminar heat transfer coefficient of sensible heating, W/m^-K
hg
enthalpy of sohds, J/kg
hst
enthalpy of steam, J/kg
hy
enthalpy of water vapors, J/kg
hy,
enthalpy of water, J/kg
AHeyap
latent heat of vaporization, J/kg
AHe
latent heat of condensation, J/kg
Ka
Kapitza number, p'^g/pa^
k
Von Karman's constant
L
plate length, m
/
Prandtl mixing length, m
M
mass of condensate, kg
Mdy
mass of a differential volume, kg
Pyap
vapor pressure. Pa
Pet
turbulent Peclet number
Pr
Prandtl number, Cpp/X
Pri
constant in the turbulent Prandtl number formula
Prt
turbulent Prandtl number
Q
heat transition rate, J/s
Qew
heat flux from condensate to wall element, J/s
Qyje
heat flux from wall element to evaporating film, J/s
Re
film
Reynolds number,
4r/p
Ret
transition film Reynolds number
Ret^i
transition film Reynolds number proposed by Kapitza
Tr
reference temperature, K
Pyap
vapor temperature, K
Tf^e
condensate film temperature, K
Tf^e
evaporating film temperature, K
Tf^h
heating film temperature, K
Tyj^c
wall temperature at condensation side, K
Tyj^e
wall temperature at evaporation side, K
Pw,h
wall temperature at heating side, K
t
time, s
75
AT
temperature difference, K
U
internal energy, J
u
film
velocity, m/s
Vx
film
velocity in x direction, m/s
W
mass flow rate of evaporated water, kg/s
Wc
mass flow rate of condensate, kg/s
Xpi
black hquor dry solids concentration at plate inlet, kg/s
Xs
dry sohds concentration, kg/s
X
spatial coordinate, m
y
spatial coordinate, m
2
spatial coordinate, m
yi
distance from the wall to the film free interface, m Greek Letters
F
black hquor mass flow rate per unit length, [kg/s- m
5
film thickness, m
Sw e
wall thickness, m
V, Vc, VT A
dimensionless variables thermal conductivity, W/m- K
Atu
wall thermal conductivity, W/m- K]
Ai
dynamic viscosity. Pa- s
M* u
turbulent dynamic viscosity. Pa- s kinematic viscosity, m^/s
P
density, kg/m^
T
residence time, s
'yx
Reynolds stress, N/m
e
dummy variable, m
eddy viscosity, m^/s
Note: A bar over any variable represents time averaging and an asterisk or plus sign represents a dimensionless variable.
76
CHAPTER 3 MODELING OF A SINGLE EVAPORATOR 3.1
Evaporator design
Falling film, plate evaporators consist of a shell, heating element, and a vapor space. The position of the vapor space in a general falling film evaporator can be either over [1, 2] or below the heating element [1, 2, 3]. There appears to be no specific criterion to determine the position of the vapor space. In the present work, the position of the vapor space is selected to be over the heating element (the plate stack). This design is iUustrated in Figure 3.1. The feed, the black liquor, enters the evaporator at the suction side of the recirculation pump. The fluid is mixed with liquor that has been already processed in the evaporator. A portion of the liquid product stream is recycled to the evaporator by way of a distributor. The role of the distributor is to assure full usage of the available heat transfer surface. As the fluid flows down the plates it becomes concentrated as a result of heating (evaporation of the liquid in the fluid) by steam that is provided to the opposite side of the plate. The steam itself condenses after the exchange of energy. The released vapors from the fluid travels upwards and exists the plate stack. For preventing of any liquor drops, entrained by the vapor flow to leave the evaporator, a separator is installed. The separator^ is a mechanical device that releases only the vapors out of the evaporator and returns any entrained liquor back to the liquor distributor. The concentrated liquor flows downwards existing the plate stack and •'For more information on separators, see [2, 3].
77
Secondary vapor outlet
Entrainment separator
Liquor distributor
Plate stacic
Steam inlet
Condensate outlet Product outlet
Feed inlet
Liquor inventory
Recirculation pump
Figure 3.1: Plate type evaporator sheme. collecting in the bottom of the evaporator. The evaporator bottoms constitutes the evaporator inventory. From the evaporator design, the following distinct sections can be identified:
78
• Liquor distributor, • Plate stack,
• Evaporator inventory,
• In-hne mbcer before the recirculation pump, and • Splitter after the recirculation pump.
The types of processes that occur in each section are presented below.
3.2
Liquor distributor
The function of the liquor distributor is purely mechanical. The process of interest is flashing by the liquor if its temperature is greater than its boiling point at the operating conditions dictated by the vapor pressure and dry solids concentration. Flashing is assumed to occur instantaneously. Moreover, when flashing occurs, the amount of evaporated water and the changes in the liquor parameters (dry solids, temperature and mass flow rate) should be calculated as they determine the input conditions to the plate stack. The steady state mass balance is given by
G^t = Gin - Wf,
where Gout is the mass flow rate of the liquor out of the distributor, dn recirculation mass flow rate and Wf is the mass flow rate of the water vapors. 79
(3.1)
is the
Assuming that the vapor is saturated, the steady state energy balance is given by
GinCp^inlin
= G outCp^outPboil + Wf{AHeyap
+ Cp^wTboil),
(3-2)
where Cp^in and Cp^out are the heat capacities of the recirculating and flashed liquor, respectively; Tin and Tbou are the recirculating and boiling temperature of the liquor, respectively; AHeyap is the heat of evaporation of the liquor at T^on and Cp^y, is the heat capacity of the water. Heat capacity is a function of the dry sohds and the liquor temperature and the boiling temperature is a function of the dry sohds and the saturation temperature. In this case, determination of the liquor parameters after flashing requires solving a system of nonlinear algebraic equations. However, since the boiling point rise is small enough [1] to be neglected^ the saturation temperature can be used in place of of the boiling temperature. The change in the heat capacity is negligible, thus the steady-state energy balance becomes
GinCpTin = GoutCpTsat + Wf[AHeyap
3.3
+ Cp^wTboil)-
(^-3)
Plate stack
The plate stack is a rectangular metal structure in which the plates are packed to form the liquor and steam spaces, see Figure 3.1. The processes on a single plate were developed and modeled in Chapter 2. Since the stack is a collection of these ^For a 0.1 kg/kg change in the dry solids concentration, the boiling point rise varies from 1 to 3 K.
80
plates, the model of the processes that occur on a single plate can be used repeatedly to represent each member of the stack. This concept is used in this work along with the assumption that the conditions on each plate are equal. The most important detail in the evaporator design is the fact that there are no signiflcant material connections between the plate stack and the evaporator sheU. The plate stack is supported only at the bottom. This is possible since the amount of liquid holdup on the plates is small. In the present case, the evaporator inventory (the bottom of the evaporator) is physically separated from the plate stack. This minimizes the influence of the surroundings to the conditions on the plates that may occur due to heat transfer between the plate stack and the surroundings. Also, every second chamber in the stack is a steam condensation chamber, providing a uniform supply of heat to the entire plate stack. Thus, it is reasonable to assume that the conditions are approximately identical from plate to plate. The response time of the wall is smaller than 1 second (~ 0.01 s) [66]. Thus, it is reasonable to conclude that the thermal inertia of the plate stack can be neglected. The output flow out from the distributor is divided equally among the plates and the operating conditions such as the composition of the dry solids, operating temperature, etc. do not change between the boundary of the plate stack and the distributor.
81
3.4
Evaporator inventory
The evaporator inventory contains the liquor at the bottom of the evaporator and its level is well-controlled to maintain smooth operation of the centrifugal pump located at the outlet. The pump recycles the inventory to the evaporator. A lumped modeling approach is used to develop a model of the evaporator inventory. The bottom of the evaporator is viewed as a tank with a fixed area and volume. The important phenomenon to represent well is the dynamics of the evaporator inventory, since this affects the dynamic response of the evaporator to disturbances.
3.4.1
Mass balance
The overall mass balance is given by N
—jT — /__, Gp^i — Gout,
(3.4)
where M is the mass of the liquor in the tank, Gp^i is the mass flow rate at the output of each plate, G^t is the mass flow rate at the exit of the tank and N is the number of the plates. The exit flow from each plate is assumed to be identical, thus the above balance can be re-written as
^ = NGp,i-Gt,out, dt
where A'' is the number of the plates in the stack.
82
(3.5)
A balance of the tank can be rewritten as
d{LAtp) — — — = TVGp - Gt,out,
(3.6)
where At is the tank cross-sectional area and p is the density of the liquor in the tank. Expanding the time derivative at the left hand side of Equation (3.6) and rearranging the equation gives
wr, ivn --n.Gt,out. dL _ NGp dt
pAt
LT. ^n dp pAt dt
(3.7)
The liquor density is an analytical linear function of the temperature and the dry sohds. Therefore, the calculation of its time derivative does not represent a problem. A solids balance in the liquor is given by
riA/r
^
—J— — 2_^ Gs,p — Gs,out,
(3.8)
where Mg is the mass of the dry solids in the tank, Gs,p is the mass flow rate of the dry solids exiting each plate, and Gs,out is the mass flow rate of the dry solids exiting the tank. By analogy with the overall mass balance, the above can be re-written as
^
= NGs,p - Gs,out.
(3-9)
Let X represent the dry solids concentration in mass fraction. It then follows that
83
Equation 3.9 becomes d{MX) dt
= NGpXp - Gt,outX,
(3.10)
where Xp is the dry solid concentration of the liquor exiting the plates. Expanding the derivative at the left hand side of Equation (3.10) and rearranging give
M ^ = NGpXp - Gt outX - ^X. dt ^ ^ ' dt
(3.11) ^
The mass of the liquor can be defined in terms of its density, M = pAtL. Using this relation and Equation (3.5) an expression for the dynamic behavior of the dry solids is given by
3.4.2
Energy balance
The energy balance of the liquor in the tank is given by
^
= Qin-
Qcmt = NGpCpTp
- Gt,outCpT.
(3.13)
dt
The accumulation term can be rewritten as
dE_ ^ d{pAtCpT) dt dt '
84
,^ ^^.
Expanding the derivative at the right hand side of this equation, and substituting the result in Equation (3.13) give
dT _ NGpTp - Gt,outT _ T^dp _ TdL _ T^dc^p dt pLAt p dt L dt Cp dt
(3.15)
In summary, the equations describing the liquor inventory are
dL _ NGp — Gt,out L dp dt pAt pAt dt
dT dt
NGpTp - Gt,outT pLAt
dt
pAtL^
^
'
Tdp p dt
TdL L dt
Tdc,p Cp dt '
where the time derivatives of the density and the heat capacity are
dp ^ ^ dT ^dX -f- = -0.495-— + 6 0 0 ^ dt dt dt ^ = 3.822X^ + 3 . 8 2 2 r ^ - 3322.67^. dt dt dt dT
Equations (3.16) and (3.16) are functions of the black hquor parameters whose correlations can be found in Appendix C.
3.5
In-line mixer
The mixing that occurs before the recirculation pump is assumed to be instantaneous and ideal (no heat effects due to mixing). Also, the liquor holdup in the in-line
85
mbcer is assumed to be negligible. The overah mass balance for the mixer and its energy balance are
Gm,out = Gfeed + Gt^out
(3.16)
^m^out^l^ m,outCp,m,out = G feed-i feedCpJeed + Gt^outPt,outCp,t,out,
(3-17)
respectively. The solids mass balance is given by
Gm,out-^Tn,out = GfeedXfeed
3.6
+ Gt^outXt^out-
(3.18)
Splitter
Since the holdup is negligible, only the steady state mass balance is necessary to represent the division of the flow at the splitter. Thus
^ spl
3.7
^rec + ^prod-
yo.l.jj
Numerical solution
The first step to solve the non-linear balance equations numerically is to start with a feasible set of initial conditions. Initialization conditions for different cases are discussed below.
86
3.7.1 3.7.1.1
Initialization Case 1-Evaporator cold startup
The first case addresses when the evaporator is started after it has been fihed with weak or feed liquor. The tank is assumed to be at its nominal level initially. The temperatme and the dry solids composition of the feed are assumed to be same in all sections. The recirculation mass flow rate is set to be six times the mass flow rate of the feed. The product mass flow rate is set initially to be equal to the mass flow rate of the feed. The values of the steam and the vapor pressures are determined by the operating conditions.
3.7.1.2
Case 2-Evaporator hot startup
This case addresses operating conditions that are set to values obtained at the last point in time following a cold start. The evaporator is assumed to be at its nominal steady state at this time.
3.7.2
Calculation loop
The first step in the solution strategy is to detect if the conditions are favorable for flashing in the liquor distributor. If the conditions indicate that flashing can occur, the flash calculations are performed and the results are passed to the block that solves the plate equations. Otherwise, the parameters of the recirculation stream are passed to the block. The following describes the procedures for one time step. The model of the plate
87
stack uses the input parameters from the distributor and the steam and vapor pressures to integrate the plate equations. The parameters at the last spatial point in all plates of the stack (at the bottom of the plate stack) are inputs to the evaporator inventory block. The results of the model of the inventory block are used as inputs to solve the model of the in-line mixer calculation block. In turn, the results of the in-line mixer block are inputs to solve the model of the splitter block. The last step involves updating the recirculation stream parameters. The procedure is repeated until a termination condition is achieved. The termination condition can be either a termination time or a user defined condition.
3.8
ODE solver
The ordinary differential equation (ODE) solver comes from the same collection of differential equation (DE) solvers, ODEPACK, used for solving the single plate equations presented in Chapter 2. The specific solver used in this case, is LSODE (Livermore Solver for Ordinary Differential Equations) [68, 69, 70]. It is very convenient to use solvers from the same package, as they share similar formats, subroutines, and control parameters. Both LSODI and LSODE require some basic BLAS (Basic Linear Algebra Subroutines) to operate. It is worth noting that the initial experience with the original BLAS subroutines showed very low integration speed (very long computing times). However, by using the more modern and versatile ATLAS (Automatically Tuned Linear Algebra Subroutines) the integration speed and consequently computing time
88
improved dramatically (A 10 fold decrease in the integration time!).
3.9
Results and discussion
The operating conditions of the evaporator are listed in Table 3.1. The evaporator equations are integrated until a steady state is attained (cold startup). In this manner, the initial dynamics do not influence the responses under non-ideal conditions. As discussed in Chapter 1, the dry solid concentration of the product stream is the most important variable to monitor during the evaporation process. Thus, the sensitivity of this parameter to unmeasured disturbances and changing operating condition are shown in Figures 3.2 to 3.6. Table 3.1: Evaporator operating conditions Variable
Value
Feed mass flow rate, Gf
58 kg/s
Feed dry solids, Xf
0.145 kg/kg
Feed temperature, Tf
303.15 K
Vapor pressure, Pyap
24000 Pa
Steam pressure, Ps
47000 Pa
Table 3.2 summarizes the operating changes apphed to the evaporator process. The sensitivity of the evaporator to the disturbances is defined as
ASS Sensitivity = — 3 - ,
89
(3.20)
Table 3.2: Step changes of evaporator operating condition Variable
Change
Feed mass flow rate, Gf
± 5%
Feed dry solids, Xf
±5%
Feed temperature, Tf
± 5%
Vapor pressure, Pyap
± 9.355%
Steam pressure, Pg
± 5%
where Ad is the percent change in the disturbance and ASS is the percent change in the dry sohds concentration with respect to the steady state value. The numerical values are listed in Table 3.3. Table 3.3: Evaporator sensitivity to disturbances Disturbance
3.9.1
Sensitivity: d> 0
Sensitivity: d < 0
Feed mass flow rate, Gf
-0.444
0.523
Feed dry solids, Xf
1.105
-1.086
Feed temperature, Tf
1.431
-1.188
Vapor pressure, Pyap
-0.945
1.157
Steam pressure, Pg
0.911
-0.750
Feed flow rate
The change in the feed mass flow rate does not affect the performance of the evaporator to the same degree when compared to the other disturbances. The direction 90
of the response is as expected. That is, an increase in the throughput rate results in a decrease in the dry sohds concentration in the product because more feed is being processed with the same amount of energy. The magnitude of the response is similar but not the same for either a decrease or an increase in the mass flow rate. The difference in the magnitude is not unexpected because the evaporator system is nonhnear.
3.9.2
Feed concentration
The evaporator performance, based on the change in the dry solids concentration in the product stream, is affected by changes to the feed dry solids concentration. The dry solids concentration in the product stream follows the same direction as increases and decreases in the feed dry sohd concentration.
3.9.3
Feed temperature
The temperature disturbance seems to affect the evaporator more than all other disturbances, especially increase in the temperature. This behavior can be predicted from the results for single plate, where small changes in the temperature affect significantly the temperature profile on the plate, changing the surface area available for evaporation. The nonlinearity of the system is expressed most clearly, as the changes in the product dry sohds concentration differ by about 30% for positive and negative disturbance.
91
3.9.4
Steam pressure and vapor pressure
A decrease in the steam pressure affects the system slightly more so than an increase in the steam pressure. From the physics of the process an increase in the steam pressure leads to more intensive heat transfer (greater temperature difference across the plate) and therefore more intensive evaporation. This in turn results in an increase in the dry solids concentration of the product stream. A similar logic holds in the case of changes in the vapor pressure because the vapor pressure affects the boihng point of the process stream. For all operating changes, the monitored responses of the evaporator are smooth. The graphs indicate that there are no significant process lags from the time the operating conditions are applied to the time the change affects the product stream. This is because the mixer, pump, and sphtter sections contain no significant dynamics (see Figure 3.1). The long-term behavior (low frequency) reflects the heat and mass transfer phenomena that are affected by the changes in the operating conditions. The difference between the effect of the throughput change and the feed concentration change can be understood by reasoning about how these changes affect the heat transfer coefficient of sensible heating. For the operating conditions provided in Table 3.2, the change in the heat transfer coefficient of sensible heating is approximately doubled (4% as compared to 2%) for the same percentage changes in the feed dry solids concentration and the throughput.
92
3.9.5
Summary
The system is open-loop stable (the integrating effect of the evaporator inventory is regulated by a proportional-integral controller) to the expected set of unmeasured disturbances. Additionally, none of the open-loop responses exhibited oscihations. The response times are in the order of 30 minutes, which is not unusual considering the size of the evaporator (a height of about 12 meters and a diameter of about 6 meters) [1, 3, 65].
93
0.18
10
20
30
40
50
60
0.215
0.195
30
Time, min
Figure 3.2: Product dry sohds concentration to changes in the feed mass flow rate. Top: 5% increase; bottom: 5% decrease.
94
0.215
0.195
30
Time, min
Figure 3.3: Product dry solids concentration to changes in feed dry sohds concentration. Top: 5% increase; bottom: 5% decrease.
95
0.215
^
0.21
I
0.205
•5
0.195
f 0.195 "5) tn
1 o (/)
0.19
D 0.185 0.18
Figure 3.4: Product dry solids concentration to changes in the feed temperature. Top: 5% increase; bottom: 5% decrease.
96
0.215
0.195
I ' 0.195 u> w"
;9 o w
0.19
Q 0.185 0.18
Figure 3.5: Product dry solids concentration to changes in the steam pressure. Top: 5% increase; bottom: 5% decrease.
97
0.195
30
Time, min
Figure 3.6: Product dry solids concentration to changes in the secondary vapor saturation temperature. Top: 9.355% increase; bottom: 9.335% decrease.
98
3.10
Nomenclature
At
evaporator inventory cross section area, m
Cpjeed
feed hquor specific heat, J/kg
Cp^in
liquor specific heat at distributor inlet, J/kg
Cm,out
liquor specific heat at mixer outlet, J/kg
Cp^out
hquor specific heat at distributor outlet, J/kg
Ad
change of disturbance with respect to steady state, %
ASS
change of variable with respect to steady state, %
Gfeed
feed liquor mass flow rate, kg/s
Gin
hquor distributor inlet mass flow rate, kg/s
Gout
liquor distributor outlet mass flow rate, kg/s
Gm,out
mixer outlet mass flow rate, kg/s
Gt,out
evaporator inventory outlet mass flow rate, kg/s
Gp
liquor mass flow rate at the outlet of a plate, kg/s
Gprod
product mass flow rate, kg/s
Gp^i
liquor mass flow rate at the outlet of the i*'' plate, kg/s
Grec
recirculation mass flow rate, kg/s
Ggpi
splitter inlet mass flow rate, kg/s
Gg^out
evaporator inventory solids outlet mass flow rate, kg/s
Gs^p
solids mass flow rate at the outlet of a plate, kg/s
AHeyap
latent heat of vaporization, J/kg
L
tank level, m
M
liquor mass in the evaporator inventory, kg
Mg
solids mass in the evaporator inventory, kg
A''
total number of plates
Wf
mass flow rate of water evaporated by flashing, kg/s
T
evaporator inventory temperature, K
Tfeed
feed liquor temperature, K
Tin
liquor distributor inlet temperature, K
Pboii
liquor boiling temperature, K
99
Tm,(mt
liquor mixer outlet temperature, K
Tp
liquor temperature at the outlet of a plate, K
Tgat
liquor saturation temperature, K
t
time, s
X
evaporator inventory dry solids concentration, kg/kg
Xp
dry solids concentration at the exit of a plate, kg/kg Greek Letters
p
liquor density, kg/m^
100
CHAPTER 4 CONTROL OF A SINGLE EVAPORATOR Falling fllm evaporators play a very important role in the pulp mill chemical recovery cycle (refer to Chapter l,Section 1.7). The product of the evaporator plant, the concentrated (thick) black liquor is not only a fuel for the recovery boiler but also a source from which the cooking chemicals, NaOH and Na2S, are recovered. The quality of the concentrated black liquor is based on its dry solids concentration. The importance of maintaining a constant dry solids concentration can be explained based on the operation of the recovery boiler. A dry solids concentration that is greater than the expected value results in more efficient boiler operation. This is because the total amount of water in the fuel to be evaporated in the recovery boiler is smaller, thus reducing the amount of heat available for steam generation. A related issue to the water content of the black liquor is safe operation of the boiler. According to the Black Liquor Recovery Boiler Advisory Committee (BLRBAC) [1], a hquor whose dry solids concentration is less than 0.58 kg/kg should not be fed to the recovery boiler. A high water content (or a dry solids concentration less than 0.58 kg/kg) in the black liquor feed to the recovery boiler is the second major source of smelt/water explosions in the recovery boilers. In the boiler there is a large amount (about 10-15 metric tons, depending on the boiler size) of smelt residing in the bofler bed. The smelt is a by-product of the minerals that precipitate from the liquor during the recovery boiler process [1]. If the water content
101
in the fuel is high (or a dry solids concentration less than 0.58 kg/kg), there may not be enough residence time to evaporate ah the water in the combustion chamber of the boiler. If liquid water reaches the bed, a smelt/water explosion may occur. The feed to the recovery boiler is heated in the form of small droplets, produced by splash or swirl type of burners. The size of the droplets determines the character of the burn. Since the size of the droplets is a function of the black hquor viscosity, any variability in the dry solids concentration directly affects the efficiency of the recovery boiler operation [22, 23]. If the droplets entering the combustion chamber have very different than the expected size distribution they may enter the bed without being completely dried [22]. As a consequence, the bed temperature decreases resulting in a low recovery boiler efficiency.
In contrast, if the droplets are smaller than expected, they are
carried easily out of the chamber with the flue gases. This leads to (i) chemical losses as the electric precipitator (see Figure 1.1, Chapter 1) cannot process the particles in the flue gas effectively and (ii) an increase in the amount of fly ash that sticks to the heat exchange surface in the top section of the combustion chamber (superheaters) [1]. Blockage of the heat exchange surface leads to poor heat transfer and overheating of the superheater. The extent of the overheating may result in catastrophic events such as meltdown of the superheater tubes and eventual total boiler shutdown or destruction. In some mihs, the variability of the dry solids concentration is mitigated by mixing with on-site stored concentrated black liquor inventories. However, there are 102
safety and ecological incentives to reduce the amount of on-site inventories, thus a dependence on these inventories to dampen the variations in the black hquor viscosity should not be relied upon. Since smooth and efficient operations of the recovery boiler is vital economically to the operation of the mill, there is justification to regulate the product (concentrated black hquor) of the evaporator plant to tight specifications.
4.1
Typical disturbances
Typical disturbances to the evaporators can be summarized as follows: • Feed mass flow rate Changes to the throughput (feed rate) occur to maintain or empty the weak black liquor inventory of the mill. Maintaining the mill inventory is a normal operating state of the mill. Thus, changes in the weak liquor feed rate to evaporators generally follow fiber line load changes. Emptying the weak liquor inventory usually coincides with maintenance schedules. • Feed composition Feed composition changes occur when the raw material of the fiber line or the cooking conditions are changed. Composition changes also may result from washing inconsistencies in the fiber line.
• Feed temperature Changes in the feed temperature are due to changes in the operating conditions
103
in the washing area. If the objective of the pulp washing process is to remove the residues from the cooking process hot wash water (temperature of ~323 K) must be used. • Steam header pressure Steam header pressure disturbances occur either ff another steam consumer requhements changes rapidly or there is a problem with the boiler (s) providing steam to the header. • Vacuum changes The vacuum problems may result from changes in the cooling water temperature or mechanical problems in the re-circulation pumps of the evaporators. Such problems may cause air leaks and loss of vacuum. • Heat transfer changes Rapid scaling occurs if the operating conditions in the evaporator can promote this. Although, the sequence of events that leads to scaling is quite specific, the frequency at which scaling can occur is quite high [61]. Scaling in falling film evaporators can occur if the evaporators are operated over the critical solids concentration in the black liquor. The critical solids concentration is the concentration at which the salts in the liquor start to crystahze (supersaturation concentration). Below this critical limit, scahng is not expected to occur. The black liquor dry solids concentration in the last evaporator in an evaporator 104
plant usually is over the supersaturation hmit. At the nominal operation state, however, there are always crystals present in the black liquor that assure contact nucleation instead of surface nucleation. Thus, crystallization leads to growth of the crystals in the liquor but does not promote deposits on the equipment surfaces. If the dry solids concentration is less than the supersaturation limit for a period of 1/2-1 hour [61] all crystals in the hquor disappear. If the dry solids concentration returns to its normal value, then crystallization will occur on the equipment surfaces.
4.2
Sensor Issues
Measurement issues are related mainly to determination of the black liquor dry solids concentration. The only known reliable means to measure this variable directly is to employ refractometers [72].-^ From practical experience obtained from a mill operated by Tembec Inc., St. Francisvihe, LA,^ this sensor is considered to be highly reliable, only requiring annual recalibration. Further, when the sensor is well calibrated, operations at the mill report that the measurement matches the laboratory data accurately.
4.3
Decentralized control of a single evaporator
The decentralized control of a single evaporator involves single loop control configurations or single-input single-output (SISO) controhers. The controhed variables ^Vendors include: Liquid Solids Control, Inc., K-Patents, and others. ^Private conversation
105
are the dry solids concentration of the product and the level in the evaporator inventory. Additionally, the temperature and flow rate of the product stream may be controlled variables. The manipulated variables are the feed and product flow rates, the steam pressure, and the condenser cooling water flow stream. It is remarked that the product flow rate may be either a manipulated or controlled variable depending on the whether the evaporator operates on supply or on demand. Operating on supply means that the feed to the evaporator determines the production rate. Thus, the evaporator operates in such a way that it is capable of simultaneously processing the feed rate and satisfying the product quality - dry sohds concentration. Operating on demand means that the evaporator must simultaneously meet a specified production rate and product quality specifications. Most pulp mills operate the evaporator plant on supply, i.e. the evaporators need to process all the weak liquor provided by the fiber line and maintain the weak liquor inventory. The opposite case, operating on demand, occurs if there is a reason to increase the recovery boiler load. In this work, a control strategy is designed for on supply operations. A discussion on the flexibility of the control strategy is provided at the end of this chapter.
4.3.1
On supply operations
The controlled variables are the dry solids concentration of the product stream and the level of the evaporator inventory. The product flow rate is not controUed. This follows from the steady state component balance of the evaporator with respect
106
to the dry solids, GinXin
— GoutXout
(4.1)
where Gin and Ai„ are the inlet mass flow rate and dry solids concentration, respectively and Gout and Xout are the outlet mass flow rate and dry solids concentration, respectively Since Xin cannot be manipulated and Gin is fixed at a specified rate, neither Gemt nor Xout can be controlled independently. Table 4.1: Variable Selection and Classification Controlled variables
Manipulated Variables
Product dry solids concentration, X
Steam pressure, Ps
Evaporator inventory level, L
Product mass flow rate, Gp
The temperature of the product stream does not affect the recovery boiler operation to the same degree as compared to the dry solids concentration of the product stream, because the temperature of the exiting stream from the evaporator plant is heated (direct steam injection) prior to entering the recovery boiler. The main reason for the additional heat is related to the previous discussion on viscosity effects and the efficiency of the recovery boiler operations.lt is reasonable to conclude that the evaporator plant product temperature should not be a tightly controlled variable. The selection of controlled and manipulated variables is given in Table 4.1.
107
4.3.2
Pairing of the manipulated and controhed variables
Part of any controller configuration is to decide on how to pair the available manipulated and controlled variables. One popular technique is to apply a Relative Gain Array (RGA) analysis [73, 74]. The method, ahhough valid for steady-state conditions, can provide a first approximation to pair controlled and manipulated variables. The RGA analysis is by no means correct in every situation and may be erroneous in strongly dynamic situations.. For multivariable problems, the elements of the RGA matrix. A, are calculated as foUows [74],
Xij = kijhij
i = l,...,Ny,
j =
l,...,Ny
(4.2)
where A^^^ and Ny represent the number of controlled and manipulated variables and
H = ( K - 01\T
(4.3)
The matrix K is a matrix of steady state gains between the controlled and manipulated variables. It is important to note that the multiplication in Equation (4.2) corresponds to an element-by-element multiphcation and not the standard matrix product. In the current study, the steady state gain matrix is given by.
Kx/Gp
^L/Gp
-0.00427
Kx/Ps
KL/P,
0.00903
0.000212 (4.4)
K
108
-0.0000727
The corresponding RGA matrix is then given by.
A=
-0.1936
1.1936 (4.5)
1.1936
-0.1936
The RGA guidelines are as follows:
Pair the controlled and manipulated variables so that the corresponding relative gains are positive and as close to one as possible. [74] Thus, the dry solids concentration should be controlled by the pressure of the steam and the evaporator level should be controlled by the product flow rate.
4.3.3
Control structure
The SISO control strategy will be based on feedback control principles. Feedback control involves measuring the control variable, comparing it to a pre-specified value (set point) and applying a control law to determine the controller action that causes the control variable to track the set point. When the set point is constant, the feedback controller is said to provide disturbance compensation or regulation. When the set point is a reference trajectory the feedback controller is said to provide servo control. Feedback control is very common within the chemical industries in comparison to for instance, feedforward control, because the action to compensate comes after measuring the response. Thus, feedback control is often referred to as conservative compensation.
109
The feedback control law that wUl be used in this study is the proportional-integral (PI) law. The velocity form of the PI law is given by [74, 75]
Ck - Ck-i =
Kc
{tk-ek-i)
+
ek
(4.6)
Tl
where Ck is the control action at sample time k, e^ is the error between the set point and the output at time k, TJ is the integral time constant or reset time, Ke is the controller gain, and A T = k - {k - 1) is the sampling interval. Note this controller is a two-parameter (gain and reset time) controller. Additionally, it is assumed that these are fixed parameter controllers, that is, once the gain and the reset time are selected (tuned) they are not adapted or modified. This may limit the performance of the closed-loop system (process and controllers) to respond to any disturbance. The control scheme is shown in Figure 4.1. The scheme shows two cascaded, feedback control loops. The introduction of the cascade structure is necessary to assure proper disturbance rejection. Disturbances in the steam pressure can result from changes in the steam header and disturbances in the product flow rate can result from variations in the re-circulation pump head that result from changes in the level of the evaporator inventory, mechanical problems, etc. Cascaded, feedback control loops are commonly used in many industries; what must be guaranteed for these interconnected loops to function correctly is that the time constant for the inner loop must be much faster that the time constant of the outer loop. Actuators such as control valves do not respond immediately to the controller's
110
Vapor
Mass flow rate Dry solids
Mass flow rate Dry solids
Figure 4.1: Feedback control of the evaporator. There are two cascaded feedback control loops. The composition control (CC) loop uses steam to control the product purity and the level loop (LC) uses the product flow rate to control the evaporator level. command. To account for the delay in the response, the dynamics of the steam and product flow valves are modeled as first order systems
du Ty—
- W=
Cy{t),
(4.7)
where Ty is the valve time constant, Cy is the control command, and u is the actual position of the valve. The controller parameters and the valve constants are provided in Table 4.2. There are many methods to determine the values of the PI controllers tuning parameters. Common ones include the method of Cohen-Coon, Integral of Time-
Ill
weighted Absolute Error (ITAE), Ziegler-Nichols open and closed-loop tuning rules, relay method, and auto-tune [74]. Some of these methods require that the model that approximates the input to the output be a first-order plus dead time (FOPDT) system, given by
r'f^=K{t-e)-y,
(4.8)
where K is the system gain, r is the system time constant, and 9 is the dead time or transport lag. In the present work, the controller's parameters are found using a trial and error procedure, (see [74], Chapter 13, pp. 298-309). Table 4.2: Control Loop Parameters. FC: flow controller, PC: pressure controller, LC: level controller, and CC: composition controller Loop
Gain {Ke)
Reset time {ri)
Valve
Time constant (r„ )
FC
0.09 s.%/kg
3.3 s
VlOl
45 s
PC
0.00009 %/Pa
3.3 s
V102
90s
LC
-368 kg/s.m
100 s
CC
1250000 Pa
33.3 s
Loop
Time constant (r)
FC
58 s
PC
131 s
LC
435 s
CC
722 s
112
4.3.4
Closed-loop response of the evaporator
The closed-loop system of the evaporator and the feedback control strategy are subjected to the following operating changes:
• Feed mass flow rate change of -f-5% fiom the nominal value, • Feed dry solids concentration change of -5% from the nominal value, • Feed temperature change of -6.2% from the nominal value, • Product dry sohds concentration of -1-10% from the nominal value. The nominal values of the process are hsted in Table 4.3. Table 4.3: Evaporator nominal operating conditions Variable
Value
Feed mass flow rate, Gf
58 kg/s
Feed dry solids, Xf
0.145 kg/kg
Feed temperature, Tf
303.15 K
Vapor pressure, Pyap
24000 Pa
Steam pressure, Ps
47000 Pa
The simulated results are shown in Figures 4.2 to 4.5. The criteria for satisfactory closed-loop response include: the length of time it takes the controher to return the value of the controlled variable to within ±lcr of the set point, the oscillation period, the type of response, and the time to settle to the desired value. In this study the
113
acceptable value of a for the dry solids concentration is set to 0.001 kg/kg. This value represents the error of the composition analyzer. Table 4.4 summarizes the responses based on these criteria.
x10
E n S5
Time, min
Figure 4.2: Closed-loop response to a 5% increase in the feed flow rate. Top: the dry solids concentration in the products, bottom: the steam pressure. The dotted lines represent the ±la hmit of 0.001 kg/kg dry sohds.
The most important observation is that in all cases the closed-loop responses are stable to the modifications in the inputs. The responses of the controlled and manipulated variables are smooth and there are no violations of any constraints. The pressure of the steam in each case is adjusted within a 3 to 6 kPa range. The controlled variables are within their ±la limit within ~ 5 minutes (see Table 4.4). 114
.x10
Figure 4.3: Closed-loop response to a 5% decrease in the feed dry solids concentration. Top: the dry solids concentration in the products, bottom: the steam pressure. The dotted lines represent the i l a limit of 0.001 kg/kg dry solids. Comparing the performance of the evaporator among the first three input changes, it can be concluded that the evaporator is approximately three times more sensitive to changes in the feed dry solids concentration and feed temperature as compared to throughput changes. This conclusion is based on the time it takes for the response to be within ±la limit of ±0.001 kg/kg. However, the extent of disturbance compensation is comparable for all disturbances. In the case of the fourth input change, which is an increase in the dry solids concentration set point, the time required to stay within the ±la limit is comparable to the disturbance rejection cases. The peak value of the change in the manipulated variable (steam pressure) is ~ 40 kPa, which is an order of magnitude greater than
115
0.198
.x10 0.
Figure 4.4: Closed-loop response to a 5% decrease in the feed temperature. Top: the dry solids concentration in the products, bottom: the steam pressure. The dotted lines represent the ±1CT limit of 0.001 kg/kg dry sohds. the values obtained in the disturbance rejection cases. Regardless, the values of the manipulated variables do not violate any constraints. The performance of the feedback cascade controller, for servo control (higher product dry solids concentration), is not as satisfactory when compared to the disturbance rejection cases. There are at least two possible reasons for this response. First, the controUer parameters (gain and reset time) were selected based on disturbance rejection and not servo control. It is possible that the controller parameters can be adjusted to accommodate both regulation and servo control. Second, the set point change requested is in the form of a perfect step, which may not be achievable for this or any real, nonhnear, multivariable process.
116
30
40
50
60
xlO
E ni
3
to
Figure 4.5: Closed-loop response to a 10% increase in the set point of the dry sohds concentration in the product stream. Top: the dry solids concentration in the products, bottom: the steam pressure. The dotted lines represent the ±1(T limit of 0.001 kg/kg dry solids. In general, when a system transitions from one set point to the next, the reference trajectory is more likely to be a bounded ramp function rather than a step function. Figure 4.6 shows the responses for the same set point change, an increase of 10% in the dry solids concentration in the product stream, using bounded ramps. The first ramp is to achieve the set point in 20 minutes while the second is to achieve the same increase but in 10 minutes or in one-half the time of the first ramp. Not surprisingly, the second graph shows less aggressive controller action for the slower
117
ramp as compared to the faster one. 0.22
^. 0.21
a
0.19
ro 0.22
0.21
0.19
Figure 4.6: Evaporator response to an increase of 10% in the dry solids concentration set point. The first two graphs, starting from the top, represent the closed-loop response and the controller command, respectively for a set point change that occurs over 20 minutes. The last two graphs represent the closed-loop response and the controller command, respectively for a set point change that occurs over 10 minutes.
4.3.5
Limitations of SISO loop controhers
The single-input single-output (SISO) feedback controllers that employ a proportionalintegral-derivative (PID) control law have several disadvantages. Firstly, the single loops are susceptible to poor performance (offset, sluggish or aggressive responses) when a high degree of interactions is present. This is especially true in highly integrated, nonlinear processes. Secondly, there is no explicit means of incorporating
118
input and output constraints in the calculation of the controller action. This poses the most significant drawback of the SISO PID controllers. There are some simple techniques that can be used to ensure non-violation of constraints. These include simple limit checking and detuning the controller. Lastly, the controller parameters are fixed; this means that if the process dynamics change the fixed parameter controller may not be appropriate to regulate the process in a timely and safe manner. In this study, it has been already shown in the case of set point changes that the two-parameter fixed point controllers exhibit a tendency of applying aggressive changes in the control action. Although physical constraints have not been violated or reached, there are no guarantees. Therefore, there is motivation to seek a different control strategy or control law that may address some of these limitations. Table 4.4: Features of the process closed loop responses
Gf
Feature
Servo Control
Disturbance
Performance
Time to reach ±la (min)
Xf
Tf
Xp
3.85
4.81
6.99 5.45
Oscillation period (min)
5.25
5.62
6.03
% compensalsed
88.63
89.27
89.69
4.4
Summary of decentralized control
A decentralized control strategy that consists of multiple single-input single-output configurations and a feedback control law is able to provide satisfactory disturbance 119
compensation. The disturbance set consisted of known levels of expected but unmeasured disturbances. The controllers selected were the two-parameter family of PI controllers tuned using a trial and error approach. The control strategy employed a feedback control law. In addition to the typical characteristics one seeks in a closedloop response a ± l a limit of 0.001 kg/kg was also imposed. It was found that satisfactory disturbance compensation can be achieved for step changes in known input parameters. However, in the case of servo control, the controller actions were very large in comparison to those used in disturbance compensation. It was concluded that either the PI controller tuning parameters should be adjusted or that the set point change be apphed as a bounded ramp rather than an abrupt step change.
4.5
Model predictive control (MPC)
Some of the more important drawbacks of the PID feedback control law have been discussed in Section 4.3.5. In response to these hmitations, a more advanced control strategy, that of model-based control, will be investigated for regulation and servo control of the evaporator. In particular, the study will focus on the class of model predictive controllers (MPC). MPC is a control strategy that has several advantages. One of the most important is that MPC formulation allows the exphcit handling of constraints on the outputs (controlled variables), the inputs (manipulated variables) and the rate of change of the inputs. Also, MPC address muhivariable systems and their interactions. The technical literature contains many examples of applying
120
MPC successfully to control open-loop stable processes and even non-minimum phase processes [21]. The MPC technology began with the successful implementation of Model Algorithmic Control (IDCOM) by Richalet et al. [17]. A subsequent manuscript by Garcia and Morari explored the concept of internal model control (IMC) [18], which was very similar in formulation to IDCOM. In 1979, a group of control experts at the Sheh Co. implemented a control strategy that they referred to as dynamic matrix control (DMC) [19]. The Sheh group also demonstrated that the theory could be implemented in practice. Some of the shortcomings of IMC and DMC were their inability to control open loop unstable systems and to address constraints explicitly on the process inputs or outputs. In response, a variation of DMC was developed, called Quadratic Dynamic Matrix Control (QDMC) [76]. The QDMC formulation used constraint optimization theory to determine the optimal control action at the current time and into the future. The QDMC formulation appeared to be closely similar to the linear quadratic regulator (LQR) [77]. In 1989, Garcia and Morari unified the area of model-based control with the concept of Model Predictive Control (MPC) [78]. This led to other variations such as General Predictive Control and Generic Model-Based Control to name a few [21, 79, 80]. This work focuses on two forms of MPC, DMC and QDMC to control the multivariable falling film evaporator, which is formulated by non-linear PDE's.
121
Consider a general nonlinear system described by
X =
f{x,u,e)
(4.9)
y = 9{x,tp)
(4.10)
^0 =
x{0),
(4.11)
dx where a; = -—-, /(-) is a vector valued function, / : X " x ?7™ x 0"- ^ A", h{-) is the measurement function, h : X"" x U"^ x ip^'y ^ Y^, states a; G A " C 5?", outputs y eY^
CRP, inputs u e C/"* C 3?™ are piecewise continuous functions and d E 3?""^
and tp e 3?"" are vectors of parameters^. In the general, the objective of the controller is to regulate (disturbance compensation) the process about a given nominal state. The design of a controller based on the nonlinear system model is however difficult because issues such as closed-loop stabflity and robustness for nonlinear controher designs are not established in the general case. An alternative is to employ well established linear control theory to design the controller and to apply the controller at the vicinity of the desired operating conditions. There are many cases in the open literature in which a linear controller is employed to regulate a nonlinear process successfully.
Careful study of these processes reveal that although the process is
nonlinear, the operation of the process is about a single nominal operating state. Hence, if a model-based control strategy is to be employed, a linear representation of the process about the nominal operating state suffices as a representative model of ^In the following discussion the scalars are represented by lowercase symbols, the vectors are bold lowercase symbols and matrices are bold uppercase symbols.
122
the nonlinear system. Thus, hnear control methods can be apphed. The linear model can be in one of two forms, an input-output (convolution) model such as a step or impulse response model; or a state-space model. The linear model can be derived using either system identification methods [81] or by a first-order Taylor series expansion about the nominal state. When system identification is used, the system is excited by a suitable input signal (e.g. step, ramp, pseudo-random binary sequence, etc.) of sufficient magnitude to give a reasonable signal to noise ratio at the process output. Time series analysis is then used to fit a suitable model to the input-and output data. When a nonlinear first principles model of the system is available a linearization technique can be used to arrive at a state-space model [82].
4.5.1
Linear quadratic regulator
The simplest linear model based control algorithm is the Linear Quadratic Regulator (LQR) [77]. In the LQR formulation, a Linear Time Invariant (LTI) state-space model is used. The form of the LTI model is given by
X = Ax + Bu
(4.12)
y
= Cx
(4.13)
= x{0),
(4.14)
xo
123
where A e K"''" is the system dynamic matrix, B e 3?"^'" is the input matrbc, and C eW'P
is the output matrix. The state feedback control law is of the form
u = -Kx,
where K
(4.15)
G SR""^" represents the controller gain matrix and u is that input that
minimizes a performance measure given by
J=
/ {x'^Qx + u^Ru)dt + x'^{T)Mx{T), Jo
(4.16)
where Q and M are typically positive-semidefinite matrices, H is a positive-definite matrix,^ and T is the final time but x{T) is the final state. The terms x^Qx x'^{T)Mx{T)
and
are a measure of control and terminal control accuracy, respectively;
and the term u^Ru
is a measure of control effort.
In practice, not all the states of the system are measurable (i.e. p < n) and in many cases only the measured values of the outputs are available. To address this issue, an observer is designed and used to provide an estimate of the states based on the output measurements. An example of a simple observer is given by
i;
=
Ax + Bu - L{y - xC)
y
= Cx,
(4.17) (4.18)
•^A symmetric matrix Q is said to be positive semi-definite if its corresponding quadratic form x'^Qx > OVx and positive definite if x'^Qx > 0 and x'^Qx = 0 only when a; = 0.
124
where L is a feedback gain matrix. Thus
u = -Kx.
(4.19)
For more discussion on the LQR and the related Linear Quadratic Gaussian (LQG) controller, see [83].
4.5.2
Dynamic matrix controller
In digital control implementation, the output is sampled at some sampling interval, A, and because of the nature of digital-analog converters, the input applied to the system is constant over this interval. This necessarily results in a piecewise constant control input with period A. Thus, the set that u{t) is a member of is the class of piecewise constant control functions, such that u{t) = u[k] for each value oft between successive sampling intervals, kA < t < {k + 1)A. Through simple integration, the behavior of the system in Equation (4.13) at every discrete interval k can be represented by
x{k + l) y{k)
= ^x{k) =
+ ru{k)
Cx{k)
where $
,AAt At
T
= B fe^^^'-^^da 0
125
and $ contains the sampled-system dynamics. By successive substitution, the state vector X can be eliminated, yielding a convolution model
y{k)
= C^x{k-l)
y{k)
= C^^x{k-2)
+
CVu{k-l)
+ C^u{k-2)
+
CTu{k-l)
N-l
y{k)
=
^ C $ ^ - i r i t ( ; c - £ + l ) C $ ^ a ; ( f c - i V + l).
The product of the terms multiplying the input u gives the impulse response coefficients. For stable systems and for A'' sufficiently large, the impulse response coefficients are reasonably approximated by zero. Thus, a finite N is sufficient to describe the system. A model based on the response of the system to a step input is given by
AT
y{i\k) = yo + ^aiAu{k
+ l-i),
(4.20)
1=1
where y{i\k) is the output I sample instances in the future with respect to the k"^ time instance. This model is commonly used in dynamic matrix controller design and can be obtained from the model given in Equation (4.5.2), since the step response coefficient, a^ is the sum of the first i impulse response coefficients, and Au{k) — u{k) — u{k — 1) represents the change in the manipulated variables. Equation (4.20)
126
can be re-written to separate the future and the past control actions past
future
N
i
y{^\k) = yo + Y^ aiAu{k + i-i)-\-J2 e+i
aiAu{k + i ~ i) +d{i\k),
(4.21)
i=i
where d{£\k) represents any unmodeled effects. The first term in this equation is the effect of the past inputs if ah the future control actions are the same as u{k — 1), that is, no additional control moves are made at sample instances /c, /c + 1 ,
The second
term represents the predicted behavior due to future inputs. If the linear convolution model is a perfect representation of the process then d{i\k) represents only external disturbances. Equation (4.21) can be more compactly written as
Y{k + i) = Yo-^ BAUpast{k) + AAU{k)
+ D{k + £),
(4.22)
where Y{k + £) = AU{k)
[yk+i...yk+pf
=
[uk...Uk+M-i]
D{k + e) =
[dk+i...dk+pf rp
Upast{k)
=
[Ufc-1 . . . Wfc-iv] -
Equation (4.22) requires knowledge of future disturbances, which is usuaUy not a prior known. To address this issue, future disturbances are made equivalent to the
127
current disturbance at sample instance k. That is
d{k\k) = d{k + l\k) = ... = d{i\k).
The disturbance estimate at sample instance k is obtained from the difference between the measured output and the model prediction
d{k) = ym{k) - y{k).
(4.23)
Define the reference (set point) trajectory, R to be
R{k + £) = [r{k + t)
..r{k
+ P)f
(4.24)
It then foUows that the feedback error prediction is given by
Ef{k + l) = R{k + l)-Y{k
+ l).
(4.25)
Given Equations (4.22), (4.23), and 4.25), the DMC control problem is defined as finding the M future input moves, AU{k),
so that the sum of the squared deviations
between the projections, Y{k + l), and the reference, R{k + t), are minimized. That is min J = y ALT
Ef{k + l)'^Ef{k + 1).
(4.26)
^—^
It is not difficuh to prove that the solution to the above equation is given by the 128
least-squares solution. That is
AU{k)
=
{A^A)-'A^{R{k)-Yo-BAUpast{k)-D{k)) (4.27)
=
{A^A)-'A''E{k
+ l).
In the DMC paradigm, only the controller action computed for the current instance k is implemented. The computation is repeated at every sampling instance k when a new feedback measurement is obtained and used to update the projected deviations, E{k + 1). Failure to compute a control action at each sampling instance could impair the disturbance handling features of the approach. The parameters P and M are important tuning parameters for stability and speed of response. In general M < P and P = M+N, for stability. Thus, DMC is capable of handling non-minimum phase dynamic characteristics such as an inverse response and dead time. DMC also provides feedforward compensation of measureable disturbances and minimization of constraint violations but only in the least-squares sense. Since the DMC control action is found as the solution to an unconstrained optimization problem, the QDMC method improves handling of constraints thus making the QDMC algorithm a very powerful tool for solving complex multivariable constrained control problems.
4.5.3
Quadratic dynamic matrix controUer
In actual applications, the control actions computed using Equation (4.27) may not be implementable due to process constraint violations. There are several types
129
of process constraints:
• Manipulated variable constraints,
• Controlled variable constraints, and • Associated variable constraints. The latter represent key variables that do not have to be controlled to a fixed value but must be kept within certain bounds. The least-squares solution of the DMC problem can be expressed as the following quadratic minimization problem
min J = AC7(fe)
I \{AAU{k)
- E{k + l))^Q{AAU{k)
2 L
- E{k + 1)) (4.28)
+AU{k)'^RAU{k)]
subject to the following linear inequalities
Ax„,in
ymin
< ymax
(4-29)
Umvn
< Umax
(4-30)
AXmax-
(4-31)
130
Equation (4.28) is the QDMC formulation. Applying the linear inequality constraints gives the following quadratic programming problem (QP)
^ S ) ^ = l^U{k)^HAU{k)
- g{k +
l)^AU{k)
subject to ymin S 1/ S
ymax
"•min ^ ^ Ji f^max
(4.32) ^Xmin
S ^X
<
/XXmaxj
where H = A^QA^
+
R^R
g{k + l) = A'^RE{k
+ l).
Solution of Equation (4.32) by a QP algorithm at each sampling instance k produces an optimal set of control actions AU{k),
which satisfies the constraints. The matrix
Q penalizes any non-zero system output, which forces the optimization algorithm to calculate inputs that drive the system outputs to zero. The matrbc R penalizes excessive control actions. All tuning parameters given for the DMC formulation apply for the constrained QDMC formulation. Additionally in QDMC, quahty is influenced by the selection of the projection interval to be constrained. In practice, only a subset of P projections, £ > 1, are constrained. This set forms a constraint window of future instances in time over which QDMC will prevent constraint violations.
131
A general analysis of stability and robustness of MPC schemes is not possible because of the finite horizon formulation. It has been shown, that stabihty can be guaranteed only ff an infinite prediction horizon is chosen. Therefore, the optimal solution of the control problem does not guarantee closed-loop stability The stability of the MPC controllers depends on the prediction and control horizons and any weight and constraint that are applied to the inputs and the outputs of the system. All these parameters can be adjusted to achieve stability. A general rule of thumb is to chose the prediction horizon sufficiently large. This insures that the system will reach steady state at time k + P, therefore satisfying the stability requirement.
4.6
MPC related modeling of single evaporator
From the above discussion, while many factors play a role in the success of the MPC controller formulation, having a satisfactory model is clearly a major factor. Also, the model used in the MPC formulation must be in a non-distributed (lumped)form. The lumped form represents a system of linear or nonlinear ordinary differential equations (ODEs) where time is the independent variable. In the case of conventional MPC, the ODEs are usually linear. Once such a model is found and validated, there are transformations that can be used to transform among different linear representations. In particular, the transformation from the ODE or state space representation to the input-output (convolution) descriptions of either a step or impulse models will be used in this work.
132
The state-space form is obtained starting from a first principles model that governs the physico-chemical phenomena of the process. Hence, while the spatial variations are neglected the temporal phenomena are retained. This nonlinear formulation is then further approximated by a linear (first order) model by applying a Taylor series expansion about a stable, nominal state. The resulting description is then a linear state space model. Alternatively, an input-output description could have been obtained by step testing the system of nonhnear ODEs and then fitting a model between the inputs and outputs.
4.6.1
Lumped model of a single evaporator
In the development of the lumped description of the evaporator, two main sections are considered, the plate stack and the evaporator liquor inventory (see Chapter 3). In this case the distributed nature of the transport processes (heating and evaporation) are ignored.
4.6.1.1
Plate stack
At normal operating conditions, the entering feed stream is being evaporated. The amount that is evaporated can be characterized by the volume and the density of the liquor in the plate stack. If well mixedness is assumed, the mass that accumulates is a function of the average film thickness, the geometry of the plates and the total number of the plates in the plate stack.
• Overall mass balance
133
The overall mass balance for the plate stack is given by
dMst - ^ = Gfg - G^t - W,
(4.33)
where Mgt is the mass of the liquor in the plate stack, Gfg is the stack feed mass flow rate, Go^t is the mass flow rate of the liquor flowing out of the stack, and W is the mass flow rate of the evaporated water. The feed mass flow rate is calculated in the same manner as described in Chapter 3. That is, the feed mass flow rate is the recirculation mass flow rate if flashing conditions exists, or the feed mass flow rate is the difference between the recirculation mass flow rate and the mass flow rate of the water evaporated as a result of flashing. The mass flow rate exiting the stack is a function of the liquor parameters Gc^t = F^tP = a5vz,
(4.34)
where a is the plate width, 5 is the average film thickness over the plate, and Vz is the average liquor velocity over the plate. In Chapter 2, it was shown that both the average film thickness and the average film velocity are functions of the liquor temperature, dry solids concentration, and mass flow rate. In the distributed parameter case, these parameters are available at each point of the film, however, for the lumped model, the mass flow rate is assumed to be equal to the feed mass flow rate of the liquor and the
134
temperature and the dry solids concentration of the exiting stream. Thus, Gout is given by Gout = Gout{Gfs,Xout,Tboii)-
(4.35)
In this development, the computational burden of the calculations of the average film thickness and average film velocity as functions of the liquor temperature, dry solids concentration and mass flow rate are circumvented by fitting power law correlations using data obtained from the original formulas presented in Chapter 2. Table 4.5 shows two correlations for the parameters of interest - the average film thickness, the average film velocity and the heat transfer coefficients of heating and evaporation, for both the turbulent and wavy-laminar regimes. The last parameter {R^) in the table, is the correlation coefficient or the fit of the model to the data. The fit appears to be satisfactory. Composition balance The composition balance is given as
—r— = GfgXfs — GoutXout-
(4.36)
dt Expanding the time derivative and substituting Equation (4.33) gives
f = ^(^'--^)-^^ 135
'"^'
Table 4.5: Correlations for Turbulent and Wavy-laminar regimes Regime Turbulent
Variable
Correlation
R^
6
X^-G^T'^ + Xd
0.881
Vz
X<^G^T^ + TXd
-0.136
hs
X°'G^T^ + TXd
-0.192
hn
X^'G^T^ + TXd
Wavy-
~5
laminar
0.00140 0.405
1.195
-1286.15
0.97
-0.0398 -0.00294
0.96
0.427
1.456
-26.212
0.93
-0.126
0.236
1.651
-82.651
0.99
XO-G^T^
5.522
0.333
0.540
Vz
XO'G^T^ + Gd
-0.242
0.981
0.141
-2.377
0.997
hE
X°-G^T^+ Td
-0.688
-0.0200
1.323
-7.529
0.99
hn
X°-G^T^ + TXd
0.508
-0.0140
1.242
-4.717
0.97
0.99
• Energy balance The energy balance is assumed to be very fast. Hence, it is given by
0 = Qin- Qout = hEaLn{Ty,all - Tboil{X, Tsat)) - AHeyapW.
(4.38)
From this equation, the amount of water evaporated can be found from
W =
hEaLn{Tyjaii — Tboii{X,Tgat)) AH evap
(4.39)
Balances in the case of heatinglf the feed enters the plate stack at a temperature below the boiling point, it must be heated to its boUing point for evaporation
136
to occur. In this case, the mass and composition balances are unchanged as there is no evaporation; however, the energy balance must be modified since it is used to calculate the length of the plate necessary to heat the hquor to its bofling temperature. The energy balance becomes
0 = Gfghfs - GfshT,^, + Qh,
(4.40)
where hfg is the enthalpy of the black liquor at the plate stack input, hrhoti is the enthalpy of the black liquor at boiling temperature, Gfg is the plate stack feed mass flow rate and Qh is the external heat flux. Equation (4.40) can be re-written as Qh = GfsCp{Tboii-Tfs),
(4.41)
where Tfg is the plate stack feed temperature, under the assumption that changes to the heat capacity are negligible. The left-hand side of this equation can be replaced with
QH = hHanL,{Ty, - i Z k i l ^ ) ,
(4.42)
where hn is the heat transfer coefficient of sensible heating, a is the plate width, n is the number of the plates, Lh is the length of the plate that is necessary to heat the black liquor to its bofling temperature, T^ is the plate waU temperature
137
and Tfs is the plate stack feed temperature. Making this substitution gives
QH = hHanLH{Ty, - i Z k L l Z Z f l ) ^ G;,c,(r,„, - Tfg).
(4.43)
Equation (4.43) is solved for Lh
T Lh
GfgCp{Tboii - Tfs) 77p—, ^ N -
hHan{Ty,
(4.44)
—^—)
In summary, a system of ODEs (lumped description) of the plate stack is given by Equations (4.33), (4.37), (4.39), and (4.44), re-stated here for convenience.
dMst
^
dt
dX ~dt W =
^^f'' ^f'(Y
-Gout-W
'
M
hEa{L -LI \)n{Ty,aU — Tboil{X, Tsat)) ^•^evap
Lh
GfsCp{Tboil h.^n.n(T..
138
-Tfs) ^T^oil+Tfs).
4.6.1.2
Bottom inventory
The bottom inventory model is analogous to that described in Chapter 3. Thus, the model equations are
dL _ NGp — Gout lit ' pAt
L dp pAt dt
dX _ NGp dt - pAtL^^' dT IS Gpip — GoutT ~dt ~ pLAt
4.6.2
Tdp p dt
TdL L dt
^^ T dcp Cp dt
(4.45) (4.46) (4.47)
Validation of the nonlinear lumped parameter model
Figures 4.7 to 4.16 show a comparison between the nonlinear lumped and the nonlinear distributed models. It is observed that the nonlinear ODE model represents the main features of the nonlinear PDE model satisfactorily even though the process temperature profiles show the largest deviations. Additionally, the steady-state gains obtained from a ratio of the change in the outputs to the change in the inputs are equivalent. It is concluded that the nonlinear lumped model is satisfactory for MPC controller development.
4.6.3
A hnear MPC design
Although a nonlinear ODE model is available to design a nonlinear MPC controller, a linear, constrained MPC design is first investigated.
139
<
20
25 3D Time, min
Figure 4.7: Nonlinear ODE model response to a 5% decrease in the feed flow rate. Top: deviations in dry solids concentration in the product, bottom: deviations in the process fluid temperature.
X
<
20
25 30 Time, min
Figure 4.8: Nonlinear ODE model response to a 5% increase in the feed flow rate. Top: deviations in dry solids concentration in the product, bottom: deviations in the process fluid temperature.
140
0.015
20
25 30 Time, min
Figure 4.9: Nonhnear ODE model response to a 5% increase in the feed dry sohds concentration. Top: deviations in dry solids concentration in the product, bottom: deviations in the process fluid temperature. xlO ODE PDE
-5
<
-10
-15
J
I
L_
10
15
20
20
25
3D
35
40
45
50
25 3D Time, min
35
40
45
50
Figure 4.10: Nonlinear ODE model response to a 5% decrease in the feed dry solids concentration. Top: deviations in dry solids concentration in the product, bottom: deviations in the process fluid temperature.
141
0.015
20
25 30 Time, min
Figure 4.11: Nonlinear ODE model response to a 5% decrease in feed temperature. Top: deviations in dry solids concentration in the product, bottom: deviations in the process fluid temperature. 5
xlD PDE ODE
0
-5 X
<
-ID -15
10
15
20
20
25
30
35
40
45
50
25 3D Time, min
Figure 4.12: Nonhnear ODE model response to a 5% increase in feed temperature. Top: deviations in dry solids concentration in the product, bottom: deviations in the process fluid temperature.
142
XlO
X
<
20
25 30 Time, min
Figure 4.13: Nonlinear ODE model response to a 5% increase in vapor pressure. Top: deviations in dry sohds concentration in the product, bottom: deviations in the process fluid temperature. D.025 D.D2 CO
^
0.015
J °°^
ODE PDE
*3 0.005 10
15
20
20
25
3D
35
40
45
50
25 30 Time, min
Figure 4.14: Nonlinear ODE model response to a 5% decrease in vapor pressure. Top: deviations in dry sohds concentration in the product, bottom: deviations in the process fluid temperature.
143
0.01 0.008 D.OQB T3
a
<
D.004 D.D02
0
0.2 is: •a o
0,15
0.1
1-^
<
0.05
D 20
25 30 Time, min
Figure 4.15: Nonhnear ODE model response to a 5% increase in steam pressure. Top: deviations in dry sohds concentration in the product, bottom: deviations in the process fluid temperature. XlO
X
<
E -0.05
20
25 30 Time, min
Figure 4.16: Nonlinear ODE model response to a 5% decrease in steam pressure. Top: deviations in dry solids concentration in the product, bottom: deviations in the process fluid temperature.
144
4.6.3.1
Linearization
Consider the following system of equations that represent a linear time-invariant (LTI) state-space model (differential equation) and a companion measurement equation (algebraic) [82]
x = Ax + Bu
(4.48)
y = Gx + Du,
(4.49)
where x is the vector of the system state derivatives, x is the vector of system states, u is the vector of system inputs and y is the vector of system outputs, A G 3?"^", B G 3??"'"", G G a?"''^ and D G 3R'"''''. In the general case. A, B, G, D are firnctions of time. In the case of chemical systems, the matrix D is almost always a zero matrix, i.e. there is no direct feed-through of the inputs to the outputs. The procedure to obtain the above representation starting from a nonlinear system representation is as follows [82]: Consider the nonlinear state and measurement equations given by
a;(t) = /(x(t),n(t))
x(to) = Xo
y{t) = h{x{t),u{t)),
(4.50) (4.51)
dx where x = —, /(-) is a vector valued function, f : X^ x U^ -^ A", h{-) is the
'di CJij
measurement function, h:X^xU^^
F ^ states rr G A " C 3?", outputs
145
yeY^cW
inputs u G {7™ C 3?"*, are piecewise continuous functions, to is a initial time and XQ is the vector of the initial system states. Equation (4.50) can be solved for some nominal input u{t) and initial state x(to) to obtain the nominal trajectory x{t). Consider small perturbations from the nominal initial state and input given by
Xo = io + XQS u{t) =u{t) + us{t).
Assuming that the resulting solution is close to the nominal solution then
X{t)
^X{t)+X5{t).
Thus, Equation (4.50) can be re-written as
| x ( t ) + | x , ( t ) = f{x{t) + xs{t), u{t) + Mt)) Xo = Xo + Xo5-
Applying a Taylor series expansion to the right hand side of this equation about the nominal values and retaining only the first order terms gives
| x ( t ) + | x 5 ( t ) ^ f{x{t),u{t))
+ ^{x,u)xs{t)
+ ^{x,u)us{t).
(4.53)
Since d
i{t) = f{x,u),x{t,)=xo
dt 146
(4-54)
it foUows that the relation between xs and Uj is given by
xs{t) = Axs{t) + Bu5{t)
xs{to) =xo- Xo,
(4.55)
where
df A = -^{x{t),u{t)) B = ^{x{t),u{t)).
(4.56) (4.57)
Applying a similar expansion, the linearization of the nonlinear output equation. Equation (4.51), is given by
C=^{x{t),u{t)).
(4.58)
The linear approximation obtained by the Taylor series expansion can either be obtained analytically or numerically. In this work, both methods are used. The numerical function used is provided by Matlab® (Mathworks, Natick, MA)^. The analytical solution is obtained using Maple'^^ (Waterloo, Canada). The Maple solution is provided in Appendix D. ^The specific function is linmod.
147
4.6.3.2
System-theoretic analysis
This section apphes system-theoretic methods to analyze the properties of the linear time-invariant model [82, 84].
Definition 4.6.1. Stability: A linear time invariant system is said to be exponentially stable if and only if all eigenvalues of the matrix A have negative real parts. Definition 4.6.2. State Controllability: A linear time invariant system is state controllable if the rank of the n x nm controllability matrix given by, C =
B
AB
•--
A^'-^B
equals n.
Definition 4.6.3. Output Controllability: A linear time invariant system is output controllable if the rank of the n x nm output controllability matrix given by, Co = C
CAB
•-•
CA^'-^B
equals the rank of the matrix C .
Definition 4.6.4. Observability :A linear time invariant system is observable if the r rank of the np x n observability matrix given by, O —
1-^ (j
(jj[
...
CA^'^
equals n. In the linearization of the lumped model it is assumed that the level of the evaporator inventory is constant (1 meter). The the linear model has four states: (1) the total mass of the black liquor on the plates, (2) the dry solids concentration in the plate stack, (3) the dry solids concentration in the evaporator inventory, and (4) the temperature in the evaporator inventory. The nominal conditions are given in Table 4.6. The operating conditions are provided in Table 4.3.
148
Table 4.6: Evaporator nominal operating conditions Variable
Value
Plate inventory mass
3108 kg/s
Plate inventory dry solids
0.2248 kg/kg
Evaporator inventory dry solids
0.2248 kg/kg
Evaporator inventory temperature
339.26 K
Analytical resuhs obtained for the matrices A G Sf?^""^, B G ^'''^ and C G "^""^ are as follows:
-0.10247
502.57
0
-0.10491
0.10942
0
A =
(4.59)
0.00005
-1.4692 0.00011953
0.01736
-0.021771
3.0939
-3.8887
-0.019499
10
1.6478
0 0
0 10
-0.00011581
0 (4.60)
C =
B =
0 0
10
0 0 0 1
0.016383
The test of stability shows that all eigenvalues of the A matrix are real and negative. By Definition 4.6.1 the LTI model is open-loop stable. The eigenvalues are {-0.1156, -0.1042, -0.0180, -0.0018}. In Figures 4.17 to 4.20, comparisons are
149
shown between the nonlinear lumped and the LTI model when the same size step changes are applied. It can be concluded that the LTI model represents satisfactory the nonlinear system dynamics as the responses are qualitatively the same and the differences in the steady state gains are small.
LODE NLODE
306D
10
2D
30
40
50
BO
0.212
70
Time, min
20
30
30
40
50
Time, min
339.05
40
Time, mm
Figure 4.17: Responses of the hnear (LODE) and nonlinear (NLODE) ODE models to 5% decrease in the steam pressure. Dashed line: LODE, sohd hne: NLODE.
The controUability matrix given in Definition 4.6.2 has a rank of four (same as the number of states), therefore the LTI model is state controllable. Similarly, the rank of the output controUabihty matrix, given in Definition 4.6.3, is the same as the rank of the C matrix and therefore the LTI model is also output controllable. The rank
150
3160 3150 3140 2o
3130 3120 3110 3100 3090
r^' 0.24
•
i
-••
10
20
30
40
,
50
LODE NLODE 60
•
70
Time, min
iS
339.25
Time, mm
Figure 4.18: Responses of the hnear (LODE) and nonlinear (NLODE) ODE models to 5% increase of the steam pressure. Dashed line: LODE, solid hne: NLODE. of the observabihty matrix (Definition 4.6.4) is also full, therefore the LTI model is observable.
4.7
Closed-loop results
The development of the MPC controller applied to the evaporator is accomplished using the Matlab® and associated toolboxes.^ The Simuhnk® tool provides several built-in functions that designs the model predictive controller using a step response model. ^Specifically, the functions nlcmpc, ss2step in Simulink® and Control® toolboxes.
151
3200 3180
^entoty.
O)
3160 3140
c 3120 CL 3100
r^
••-- LODE NLODE
(••
3080
•
10
20
30
40
,
50
,
EO
70
Time, min
Time, min 339.5
E
2r
33B.5
ra
10
20
30
50
BO
337,5
70
Time, min
Time, mm
Figure 4.19: Responses of the linear (LODE) and nonlinear (NLODE) ODE models to 5% decrease of the vapor pressure. Dashed line: LODE, solid hne: NLODE. The controlled variables are the dry solids concentration and temperature of the product stream. These two controlled variables correspond to the third and the fourth states of the LTI model. The G matrix in the measurement equation has the form:
0 0
10
C = 0 0 0 1
The analysis for the controUed variables shows that the system is both state and output controllable, and observable.
152
3140 3120
1
1
1
1
1
--•
;
LODE NLODE
.
3100 3080
\
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50
60
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Time, min
o
10
20
30
40
50
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10
20
30
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Time, mm
Time, min
Figure 4.20: Responses of the linear (LODE) and nonhnear ODE models to 5% increase of the steam pressure. Dashed line: LODE, sohd line: NLODE. 4.8
MPC results
The design of the MPC controUer and its application to regulate the evaporator are accomphshed using Matlab's® Simulink® tool. The most important process variable to control is the dry solids concentration in the product stream. However, it is possible that in some cases the temperature of the product must also be controUed. To wit, in the MPC design, the temperature and the dry solids concentration in the product stream are both controlled variable. From the analysis provided in Appendix D, the manipulated variables are the temperature of the steam and the saturation
153
temperature of the process fluid. In reality these temperatures translate to set points on pressure because MPC is configured as part of a supervisory control layer rather than as a part of the regulatory layer. The simulated responses of two different formulations of the MPC controUer are compared to the SISO PI controller design. The formulations are unconstrained MPC
+^
>\ ^'°^
NLPDE ^ d
'
and constrained MPC. The actual feedback block diagram is presented on Figure 4.21.
"-h MPC Controller I
U
NLODE
iM ^ -
-
r-t-
LODE
^
b
-(< -^
Figure 4.21: A feedback block diagram with the MPC controller.
4.8.1
Results
The resuhs from the simulations are provided in Figures 4.22 to 4.32. The tuning parameters of the MPC and PI controhers are given in Tables 4.7 and 4.8. In Table 4.7, yi corresponds to the product dry sohds concentration, y2 corresponds to the 154
product temperature, ui corresponds to the steam saturation temperature and U2 corresponds to the secondary vapor saturation temperature. Note that aU values in Table 4.7 are deviations from the nominal conditions (zero) since the linear model is developed using deviation variables. Table 4.7: Tuning parameters for MPC. Prediction horizon, P
15
Control horizon, M
5 yi
y2
Ul
U2
50
0.01
0.06
0.06
-0.001 kg/kg 0.001 kg/kg
-0.2(1) K 0.2(1) K
-10.96 K 6.04 K
-18.23 K 4.77 K
Weights Minimum Maximum
Aui
3K
Au2
3K
Table 4.8: Tuning parameters for the PI controllers. Ke
The ±la
Ti
XC loop
2000(1000) K
500(250) s
TC loop
30
300 s
limits for the dry solids concentration and the product temperature
are ±0.001 kg/kg and ±0.2 K, respectively Both limits are based on expected errors associated with the analyzer or the sensor. The temperature limits may be considered narrow; however, the narrow limits may provide better insights in the controller performance.
155
The MPC controUer is tuned to provide satisfactory performance for disturbance compensation. AU disturbances are modeled as step functions. Disturbances of ± 5 % in the feed dry solids composition, mass flow rate, and temperature are applied to the evaporator process. The responses of unconstrained MPC controUer, constrained MPC controller and the PI controllers are monitored. In each case, the controUer provides satisfactory disturbance compensation.
The
responses are smooth without excessive overshoots or oscillations. The response of the manipulated variables used to regulate the controlled variables do not exhibit aggressive moves for each disturbance. All disturbances are compensated within ~35 minutes after the introduction of the disturbance. However, there are noticeable differences in the performance (closed-loop response) of the MPC and PI controllers. There are no significant differences between the performance of the unconstrained and constrained MPC controller. This is attributed to a combination of the range on the manipulated variables (Au and u) and the size of the disturbances. Small differences in the manipulated variables can be observed in the presence of the feed temperature disturbance (see Figures 4.24 and 4.27). In the constrained case, the MPC controller is a httle more aggressive. In the case of the feed temperature disturbance compensation it is observed that the compensation is less demanding as compared to a feed concentration or flow rate disturbance compensation.
This is because temperature regulation is not as
high a priority (see the weights used in Table 4.7 as compared to the regulation the product dry solids concentration. However, in other instances when tight control of 156
temperature is warranted, the weight on the temperature variable can be adjusted appropriately to reflect its importance. When the PI controhers responses are compared to both unconstrained and constrained MPC controller responses it is found that the PI controllers provide tighter control. This is attributed to the tuning parameters selected for the PI controhers, which are unaltered for each disturbance. It is observed that the change in the manipulated variables are more aggressive in all cases as a prioritization between the controlled variables is not an option. To examine the flexibility of the PI and MPC controllers, the magnitude of the set of disturbances is increased by 1.6% and the performance of the constrained MPC controller and the PI controllers is compared. In these tests, the tuning parameters of the PI controller are changed to assure no violation of the input constraints. The tuning parameters for this case are given in parentheses in Table 4.8. For disturbances in the feed mass flow rate and dry solids concentration, no significant differences were observed. However, in the case of a temperature disturbance, the closed-loop response of constrained MPC is superior to that of the PI controllers. These responses are shown in Figures 4.31 and 4.32. For the additional 1.6% increase in the disturbances, the PI controllers exhibit a decrease in closed-loop performance. For instance, the time for complete compensation (time for which the controlled variable is driven back within ±lcr) is about 4 times longer (20 minutes) in comparison to that achieved by the constrained MPC controher (5 minutes). Figure 4.31 also shows that constrained MPC maintains the 157
most important variable, the product dry solids concentration, within the ±lcr, which is not achieved in the case of the PI controllers. The MPC controller provides this critical performance feature while satisfying all other imposed constraints. If the PI controllers are tuned to achieve the same performance with respect to the outputs, they violate the rate of change limits on the input variables. Since the MPC controller is successful in regulating the nonlinear ODE model, the optimal controUer actions, calculated for a 5% decrease in feed temperature, are apphed to the PDE model of the evaporator. The result of this test is shown in Figure 4.33. From this result, it is clear that the dry solids concentration of the product stream is regulated within ±lcr. The response of the product temperature is not as accurate, the error is ~ 1 K. This test indicates that the approach to design a model-based controller using a linear model obtained from the linearization of a nonhnear, first-principles ODE model is vahd in this specific case. Clearly, no generalization can be made to other systems. The economic benefits of applying a model-predictive controller to regulate the product dry solids concentration is difficult to quantify when only one evaporator is used. In reality, an evaporator plant (more than one evaporator) is used to concentrate the entire black liquor feed to the recovery boiler. If however, control is implemented on the dry solids concentration in the product stream of the last evaporator in a multiple-effect evaporator plant, then it is possible to quantify the economic benefits of maintaining a constant dry solids concentration to the recovery boiler. Information, provided in [85] provides an estimate of the difference in net annual economic loss/gain 158
if a black liquor with a concentration of 0.75 and 0.8 kg/kg dry sohds is produced as compared to a base case with a concentration of 0.48 kg/kg. The difference between the net annual economic loss/gain is approximately 600 000 US dollars per year. If 0.8 kg/kg is assumed to be the desired operating point and the process is maintained within ±lo- limit for 80% of the processing time, but deviates within 0.01 kg/kg for the rest of the processing time, this will result in a loss of 24 000 US doUars per year. However, if the process is 80% of the time out of ±lcr limit, for the same desired operating point, the loss increases up to 96 000 doUars per year or an increase of 400 %. This analysis includes only pure energy balance effects on the recovery boiler. Variations in the recovery boiler operating conditions due to droplet size distribution changes are difficult to quantify as they affect not only the boiler, but the entire recovery cycle. In all the simulated results present, it is assumed that the disturbances immediately affect the outputs. This may lead to aggressive controller actions. To avoid this in the case of a multiple-effect evaporator plant, the feed is not supplied to the first evaporator in the suction line of the recirculation pump, but between the evaporator inventory and the plate stack. However, this design change is valid only for the first evaporator. AU other evaporators have the design shown in Figure 3.1 in Chapter 3.
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4.9
Nomenclature
Gin
evaporator inlet mass flow rate, kg/s
Gp
product mass flow rate, kg/s
Gout
evaporator outlet mass flow rate, kg/s
Kc
controUer gain
L
evaporator inventory level, m
M
control horizon
MPC
model predictive control
P
prediction horizon
Ps
steam pressure. Pa
PID
Proportional-Integral-Differential control law
u
vector of manipulated variables
X
product dry solids concentration, kg/kg
Xin
evaporator inlet dry solids concentration, kg/kg
Xout
evaporator outlet dry solids concentration, kg/kg
X
vector of states
y
vector of outputs Greek Letters
A
RGA matrix
Tj
PID controller reset time
172
CHAPTER 5 EVAPORATOR PLANT 5.1
Description
The pulp mills employ more than one evaporator to concentrate the black liquor (Chapter 1). Such multiple evaporators (MEV) form the evaporator plant. In an MEV plant the evaporators or effects are connected in series with respect to their vapor lines [1, 2]. The vapor (steam), produced in a given evaporator, is used as the heating source for the next evaporator in series. Fresh steam is usually supplied to the first effect. The driving force for the operation of the MEV plant is determined by the overall temperature difference between the steam temperature and that of the heat sink (cooling water) [2]. The overall temperature difference is reduced whenever there is a boiling point rise. The latter occurs due to an increase in the concentration of the dry solids in the process fluid. If each evaporator in the MEV plant is capable of evaporating the same amount of water, then increasing of the number of evaporators should increase the MEV plant efficiency. Clearly, a poorly performing evaporator plant is one that requires higher external utility resources. An important measure of the MEV plant efficiency is a parameter caUed steam economy [1, 2]. This parameter is defined as the mass of water evaporated per unit mass of live steam or in the general case the mass of liquid evaporated per unit of heat supplied. In this work, steam economy (kg water/kg
173
steam) wiU be used to evaluate the performance of the multiple evaporator plant. Steam usage is affected by many different parameters including, evaporator type, process fluid and steam characteristics, condensate flashing, and liquid/vapor flow sequence. There are two basic hquid/vapor sequences, forward (cocurrent) and backward (countercurrent) flow [1, 2]. In the case of forward flow the cold feed is the input to the evaporator with the highest temperature. The advantage of this scheme is that pumps are not needed to move the process fluid from evaporator to evaporator, since the fluid is transported by the natural pressure gradient in the MEV plant. The disadvantage is that significant sensible heating should be applied to the process fluid to reach the boiling temperature of the first evaporator. In the case of backward flow, the feed stream does not require a significant amount of additional heating, however the process fluid has to be pumped from evaporator to evaporator. In the case of falling film evaporator, pumping the process fluid, is not a disadvantage because this type of evaporator design requires recirculation pumps by default. Therefore, the reverse flow scheme is preferred when falling film evaporators are used, which is the case studied here. In the case of black liquor evaporation, it is common practice to include superconcentrators as part of the evaporator plant. In these unit operations, the highest concentrations of the dry solids are achieved. These units are usually two or three falling film evaporators mounted in single body and connected in series with respect to the flow path of the process fluid. Each evaporator in the superconcentrator has its own fresh steam supply but share the same steam header and secondary vapor 174
space. A typical black liquor multiple-effect falling film evaporator plant will have both a sequence of evaporators and at least one superconcentrator.
In the evaporator
plant studied here, there are four falling film evaporators in series followed by one superconcentrator that has three evaporators mounted in single unit. A diagram of the multiple evaporator plant is shown in Figure 5.1. Note that the feed is split equally between evaporators E-4 and E-5. This is normal practice in a black liquor evaporation plant [1] in an attempt to maintain an even distribution of the throughput. Fifty-five pound (55 psig) steam is used to heat the superconcentrator. In the real system, the physical configuration has SC-1 to SC-3 mounted in one body; thus they share a common vapor space. Vacuum conditions are achieved by a surface condenser followed by steam ejector (EJ-1) for removal of the non-condensible gases. The flash units, F-1 to F-4, are auxiliary unit operations that are common in an MEV plant. Their purpose is to improve steam usage by flashing the condensates that come from steam (called secondary vapors) produced by the evaporators. In contrast, the condensate that comes from the fresh steam sources are returned to the boiler because flashing these condensates would result in more boiler feed water and hence higher utilities. Thus, any utility savings achieved in the MEV plant would be lost in the boiler. In this work, all condensates (primary and secondary) are flashed in order to achieve the highest possible steam usage.
175
fl
O CJ s-l CD
afl 03 03
fl
o Td
fl
o3 •
^
I
fa o 4-i
fa] 03 S-l
o -R o3
o a SH
03
> CD
cfl bO
fl
fl
3 CD s-l
o3 Cl) s-l CD
rfl
H -R
fl
n1
,^
N
CO 1
a (J s-l
OO
o
4-^
SH
I-H
vap
O 4^ 01
O
O
CO
fl
por ors
03
I—{
f^
a cfl bC
Ol "HH 03
a
4.J 03
> 03
(1) 03 !H
4^
rfl
^
rfl
i-H
^
S
-R
-R
SH
1^3
0) SH
fl
bj()
fa
176
O
4-^
o3 SH
-R
fl03 CJ
5.2
Modeling
Modeling the MEV plant requires developing models of each evaporator unit, the pressure dynamics and condensate flashing. Models of a single evaporator and condensate flashing were developed and presented in Chapter 3. These models can be used for the MEV plant. What remains to be developed are models that describe the pressure dynamics. A model of the pressure dynamics should describe the pressure variations in each vapor/condensate space shared between two evaporators. This space comprises the vapor space of a given evaporator and the condensate space of the next evaporator in series. Therefore, the pressure in this closed space depends on the rates of evaporation and condensation in the corresponding spaces. For simplicity the following assumptions are made: • Water vapor behaves as an ideal gas. • Pressure losses are negligible during vapor transport between the bodies. • Condensation due to heat losses to the surroundings is ignored. With these assumptions, the mass balance of the system is given by
^ = W-Wc, at
(5.1)
where W is the mass flow rate of the evaporated water in the source evaporator and Wc is the mass flow rate of the condensing vapors in the next evaporator. The mass 177
flow rates of the evaporated water and the condensing vapors are calculated using the evaporator model presented in Chapter 3. Using the ideal gas law, the pressure dynamics can be found as follows:
where Mm is the molecular weight of water (18 g/mol); V is the volume of the system (a design parameter) and is equal to the sum of the volumes of the vapor space of the source evaporator, the volume of the piping between the evaporators, and the volume of the condensate space of the next evaporator. The temperature is assumed to be the saturation temperature (T^) of water vapor at the given pressure. Since the volume of the vapor space is assumed constant, Equation (5.2) becomes
dP R ~dt ~ MmV
d{MTs) dt
(5.3)
Expansion of the right hand side of the above equation gives
dP dt
R (d_M^^^dTg^ MmV \dt " dt
^^^^^
The saturation temperature is a function of pressure thus
,P dt
R ^1M MmV \ dt
178
iI.iP^i dP dt
(5,5)
The change in the saturation temperature is a weak function of pressure, thus the term dTg — is very smaU (10~^) and can be omitted from the above equation. Substituting Equation 5.1 gives an expression for the pressure dynamics as
dP
5.3
RTg ,
Results and discussion
The operating conditions of the multiple evaporator plant are provided in Table 5.1. The diameter of each body varies between 4 and 7 meters and their height is about 12 meters. The number of the plates per body is between 130 and 200 plates.^ Table 5.1: Operating conditions of the multiple evaporator plant Parameter
Value
Gfeed
116 kg/s, 50/50 split
Xfeed
0.145 kg/kg
Tfeed
303.15 K
Psteam at SC-1 to SC-3 t^candenser
410 kPa /i.4
Kra
The levels of E-2 to E-5 and SC-1 are controlled using single loop feedback PI controUers with tuning parameters identical to those used in Chapter 4, Table 4.2. The results for the same set of disturbances used in Chapter 4 are shown in Figures ^The actual parameters were supplied by Tembec, Inc. (Canada) and for confidential reasons cannot be published.
179
5.2 to 5.8. It is observed that the responses are stable and smooth (no excessive oscillations or discontinuities). The levels of all evaporators are regulated to within ±2 centimeters, or ±0.2% from their nominal value of 1 meter. In each case, the multiple evaporator plant reaches a new steady state in approximately one hour after the disturbance is introduced. In the case of El-5 the time to reach a new steady state is longer as compared to the time observed in the responses found for a single evaporator (about 20 minutes, see Chapter 3, Figures 3.2 to 3.6). This is attributed to process feedback by the secondary vapors, which were absent in the single evaporator studies. Due to this feedback, the disturbance propagates through the evaporator plant until it is completely attenuated by the system. The results also show that the effect of the disturbances on the dry solids concentration at SC-1 is about 5 times greater than that observed at E-5. This behavior wiU be explained in the following section (also see Figure 5.9).
5.3.1
Sensitivity to disturbances
The sensitivity of the MEV to the set of disturbances is assessed with respect to the dry solids concentration and to steam economy.
5.3.1.1 Mass balance The sensitivity of the dry solids concentration are provided in Table 5.2. The values are calculated using Equation (3.20) in Chapter 3.
180
Evaporator 5
Superconcentrator 1
409.5 409
0
20
40
60
80
100
120
0
20
40
60
80
100
120
1.02
20
40
60
80
100
120
Time, min
Time, min
Figure 5.2: Open-loop response of the PDE model of multiple evaporator plant to a 5% decrease in the flow rate of the feed.
181
Evaporator 5
Superconcentrator 1
0.196
0.66 0.64 •
0.194 2£
0.62 •
•a
o a.
X
0.192 0.19
0.58
331.8
20
40
60
80
100
120
100
120
100
120
409.5
prod'
331.7
^
3316 I-
331.5 331.4
20
40
60
80
100
408.5
120
1.02
Level
E
1.01 1— I
> CD
0.99 0.98
20
40
60
80
100
40
120
60
80
Time, min
Time, min
Figure 5.3: Open-loop response of the PDE model of multiple evaporator plant to a 5% increase in the flow rate of the feed.
182
Evaporator 5
Superconcentrator 1 0 68
0.2 O)
0.195
0.66
0.19
73 O
o
X
C3.
prod
:^
a.
0.185
•
0.64
X
0.18
n RO
331.7
410
G
20
40
60
80
100
120
0
20
40
60
80
100
120
331.65 331.6 331.55 331.5
E
1.01
Level,
1.02
1 0.99 0.98
0
20
40
60
80
100
120
Time, min
Time, min
Figure 5.4: Open-loop response of the PDE model of muhiple evaporator plant to a 5% decrease in the dry solids concentration of the feed.
183
Evaporator 5 0.21 O)
0.205 0.66
T^
^
Superconcentrator 1
0.68
0.2 D O.
X
0.64
0.195 0.19
0.62
331.9 331.8
H
20
40
60
80
100
120
0
20
40
60
80
100
120
20
40
60
80
100
120
410 409.5 •
^
•a a
0
331.7 409
331.6 331.5
408.5
1.02 b 1)
1.01 1
03
> 0)
>
CD —I
0.99 0.98
20
40
60
80
100
120
Time, min
Time, min
Figure 5.5: Open-loop response of the PDE model of multiple evaporator plant to a 5% increase in the dry solids concentration of the feed.
184
Evaporator 5
Superconcentrator 1
0.21
0.66
5"
0.64
20
40
60
80
100
0.62
120
332
0
20
40
60
80
100
120
100
120
100
120
410 409.5
H-
409-
0
20
40
60
80
1.02
100
408.5
120
1.02
'
1.01 •
> 03
Level
03
1.01 E
1 0.99 0.98
0
20
40
60
,
,
80
100
1 0.99 0.98
i : 10
Time, min
0
20
40
60
80
Time, min
Figure 5.6: Open-loop response of the PDE model of multiple evaporator plant to a 5% decrease in the temperature of the feed.
185
Evaporator 5
Superconcentrator 1
g/kg
0.21
0.68
02
js: T3 O O.
X
(iiq 0.18
20
40
60
80
100
120
100
120
100
120
100
120
100
120
335 334 ^ T3 O a.
h-
333 332 331
1 02
1.02 1.01
E
"o
.evel
E 1
>
0) _l
1 0.99
0.99 0.98
1.01
20
40
60
80
100
0.98
120
40
60
80
Time, min
Time, min
Figure 5.7: Open-loop response of the PDE model of multiple evaporator model for 5% increase in the feed temperature.
186
Evaporator 5
Superconcentrator 1
0 196
0.66
0.195 L
0.194
0
20
40
60
80
100
120
20
40
60
80
100
120
'
'
'
'
'
332
? 331.5
331 1.U2
'
1.02 Level, m
1.01 E
>
03 _l
1 0.99 0.98
, 0
20
40
60
80
100
1.01 1 0.99 0.98
i : 10
20
40
60
80
100
Time, min
Time, min
Figure 5.8: Open-loop response of the PDE model of multiple evaporator plant to a 10% decrease in the heat transfer of the superconcentrator.
187
120
Table 5.2: Sensitivity of the dry solids concentration. Disturbance
Sensitivity, d > 0
Sensitivity, d < 0
Feed mass flow rate, Gf
-1.067
1.414
Feed dry solids, Xf
0.180
-0.197
Feed temperature, Tf
0.64
-0.729
Heat transfer, Pyap
-0.215
In the case of a single evaporator, it was found that the feed temperature disturbance had the strongest effect on dry solids concentration. In the case of the MEV, the largest sensitivity is associated with feed mass flow rate (throughput) changes. One explanation for this behavior is the nonlinear dependence of the dry solids concentration on the amount of water in the process fluid (liquor). Consider the following example where the feed stream has a 0.2 kg/kg dry solids concentration. Figure 5.9 shows the amount of water that must be evaporated to achieve a different dry sohds concentration for two different feed mass flow rates. Observe that this relationship is nonlinear. When the dry solids concentration is in the range of 0.27-0.3 kg/kg, the amount of water to be evaporated for each feed rate is not significantly different. However, if the range of dry solids concentration is between 0.6
0.7 kg/kg then the amount of water to be evaporated is significantly different
between the two feed rates. For a 8.9% difference between the feed rates, there is a 13.25% increase in the amount of water to be evaporated for a 0.6 kg/kg dry sohds concentration. It can be concluded then that a larger final dry solids concentration
188
means a greater sensitivity to throughput changes.
— Laad a 125 hg/unil Hme •--• Load= 137 5 kg/uninim8
The feed dry solids concentration is 0 2 kg/kg in the both cases
40
50 60 70 Amount of evaporated water, kg/unit time
90
100
110
Figure 5.9: Open-loop response of the PDE multiple evaporator model for 10% decrease in the heat transfer of the superconcentrator.
5.3.1.2
Energy balance
The sensitivities are calculated as the differences between the steam economy at nominal (ideal) conditions and the steam economies after the disturbances (non-ideal) are introduced. Steam economy sensitivity is calculated as follows SC-1
E^
SE =
E-5 SC-1
'
(5.7)
J:WC SC-3
where W is the mass flow rate of the evaporated water and Wc is the mass flow rate of the condensate. The sensitivity of steam economy to the set of disturbances are 189
provided in Table 5.3. The MEV energy balance shows the largest sensitivity to changes in the feed temperature. It is observed that a decrease in steam economy occurs when the feed temperature decreases because more steam is consumed to heat the liquor to its boiling point. Also steam economy decreases with an increase in the feed mass flow rate, because there is more throughput to be processed. The disturbance in the heat transfer does not appear to affect steam economy, however it leads to a lower dry solids concentration in the product. Thus, if the same dry solids concentration in the product stream is to be achieved, then steam economy must decrease since more steam is consumed. Table 5.3: Changes in steam economy as a function of the disturbance.
Case
Steam economy kg water/ kg steam
Base case
Change with respect to base case,%
3.929
+5% feed mass flow rate
3.8559
1.36%
-5% feed mass flow rate
4.0008
5.17%
-1-5% feed dry solids concentration
3.9252
3.18%
-5% feed dry sohds concentration
3.9364
3.48%
+5% feed temperature
4.1451
8.96%
-5% feed temperature
3.7342
-1.83%
-10% heat transfer limitation
3.9254
3.19%
Although the changes in steam economy may appear to be small, the analysis 190
must consider the effect over the entire duration of the operation. For example, if the multiple evaporator plant is processing 116 kg/s of process fluid (dry solids feed concentration of 0.145 to a flnal dry sohds concentration of 0.73 kg/kg), the amount of steam is correspondingly 24.92 kg/s when the feed temperature is decreased by 5% with respect to the base case and 22.45 kg/s when the feed temperature is increased. If the cost of steam is assumed to be 5 USD/ton, then the per year difference is ~ 400,000 USD, which is not an insignificant number.
5.3.2
Model validation
A smaU amount of industrial steady-state data for a seven effect faUing film evaporator plant with similar physical characteristics, dimensions, and operating conditions have been provided by the Crestbrook Mill that is owned and operated by Tembec, Inc. (Canada). Based on these parameters, the model converged to a steady-state. The final dry solids concentrations achieved by the model are compared to the values provided by the mill. Surprisingly, they differ by ~ 0.006 kg/kg. This shows that the model assumptions and empirical and semi-empirical bases are valid. As no dynamic data are available for further validation, any further comparison between the mill and model cannot be made. However, the steady-state validation together with the fact that the observed responses follow the physics of the process indicates the potential of the PDE model to be used for other studies such as design modifications, optimization, process monitoring and real-time control.
191
5.4
Nomenclature
E —i
the i-th evaporator
Gfeed
multiple evaporator plant feed mass flow rate, kg/s
SC — i
the i-th superconcentrator evaporating element
M
mass of water vapors, kg
P
pressure. Pa
Pcumdenser
vapor prcssurc at the condenser, Pa
Psteam
steam pressure at superconcentrator. Pa
R
universal gas constant, J/mol- K
T
temperature, K
Tfeed
multiple evaporator plant feed temperature, K
Tg
saturation temperature, K
V
volume, m^
W
mass flow rate of evaporating water, kg/s
We
mass flow rate of condensing water vapors, kg
Xfeed
multiple evaporator plant feed dry solids concentration, kg/kg
192
CHAPTER 6 CONTROL OF MULTIPLE EVAPORATOR PLANT As discussed in Chapter 4 the multiple effect falling film evaporator (MEV) plant represents an important part of the kraft pulp mill chemical recovery cycle. Thus, any instabilities in the operation of the MEV will have significant impact on the overall mill operation and profit. Having a fundamental model that describes well the MEV plant permits an evaluation of the performance (stability, robustness, flexibility) of different control strategies. Two basic control strategies wiU be investigated: a decentralized single-loop feedback proportional-integral-derivative (PID) control and centralized constrained model-predictive control (MPC) control.
6.1
Decentralized control
The first step in designing a decentralized control system is to determine the controlled and manipulated variables and to pair them in a proper fashion (see Chapter 4).
6.1.1
Evaporator inventories
For the MEV plant studied in this work, there are seven evaporators, including the three units mounted in a single body that constitute the superconcentrator. The evaporator plant operates on supply mode. The levels of the liquor inventories of E-5 to E-2 are controlled using the product stream mass flow rate. The feed stream to the superconcentrator (SC-3) cannot be selected to control any of the evaporator units,
193
since the evaporator plant operates on supply. A similar example of three tanks in a parallel conflguration was studied by Rugh [82].
Fin
1
A
1
A
1
X2
V
X3
V
- ^
Figure 6.1: System of three connected tanks
Following the modeling approach by Rugh, the dynamic model of the three connected inventories (see Figure 6.1) is
Xl = Fin - qiV^i
CX2 = qi\/xi
-
^2
- X2 - q2y/x2 - X3
c i s = q2\/x2 - X3 -
(6.1)
F^t,
where x ^ R^ represent the three levels. Fin is the inlet flow rate to the first inventory, Fout is the outlet flow rate and also the selected manipulated variable, c is the tank cross-section area, and qj,j = 1 , . . . , 3 are parameters related to the valve characteristics and the line.
194
To analyze suitable control conflgurations, a hnear model of the system is developed, using Taylor series expansion about the nominal states, x,Fout- The resulting linear-time invariant (LTI) system is expressed by the conventional state space representation [82] where for convenience all parameters are assumed to have the value one, S — •AcyX
(6.2)
+ JDcyU
where ^ = x — ic is a vector of the deviation variables, u — Fcmt ~ Font is a vector of manipulated variables and
•Ar-i,
—
0
-1
1
0
1
-2
1
0
1
-1
•'^r.t)
0 -1
The companion measurement equation is given by
y = c^^,
(6.3)
where y = ^ and 1 0 0 f-'CV
0 1 0
(6.4)
0 0 1 The LTI system is completely state and output controllable. However, controlhng the level in each tank is not necessary because the material balances in tanks two 195
and three are not independent of each other. Thus, if the level in the last tank is controlled using the outflow, then the other two levels have unique values for the given operating conditions. Therefore, only the level in the last tank needs to be controUed. In other words 0 0 1
(6.5)
This also means that even the level transmitters on the first two tanks can be omitted because the observability condition is satisfied. Based on this analysis it is decided to use the product flow rates of evaporators E-5 to E-2 to control their levels and to use the final product flow rate of SC-1 to control the level in the superconcentrator.
6.1.2
Selection of controlled and manipulated variables
The selection of the set of controlled and manipulated variables is first guided by process knowledge provided by the control configurations of existing multiple effect evaporator plants (Tembec Crestbrook Mill, Minton [2], Billet [3], and Bojov [16]), and second by the previous work on control of a single evaporator. By making use of existing control configurations performance comparison can be made to assess improvements. The dry solids concentration in the product stream is an important property and therefore is selected as a controUed variable. With respect to the manipulated variable that can be used to control the dry solids concentration in the product stream there are four possible choices. They include: the steam pressures at the superconcentrator and 196
the secondary vapor pressure at the first evaporator (E-5). The final choice is based on the manipulated variable with the shortest time delay and smallest time constant (proximity rule). Thus, the steam pressure at the last superconcentrator evaporator (SC-1) is selected to control the dry solids concentration in the product stream. In practice, to provide robustness to disturbances in the steam header pressure, the control of the dry solids concentrator involves using two cascaded feedback loops. The inner loop is a pressure controUer that regulates the steam pressure of SC-1. The outer loop is the concentration controller that determines the set point of the inner loop. The other two steam pressures at the superconcentrator are assumed to be constant. This assumption is also applied to the secondary vapor pressure at the initial evaporator {E-5). The proposed control scheme is provided on Figure 6.2.
6.1.3
Results
To validate the performance of the proposed decentralized feedback control structure, the multiple evaporator plant PDE model is subjected to changes in the operating parameters (Table 5.1). The steam pressures at the superconcentrator for these tests (open loop responses) have been decreased from the nominal values to provide a wider range to observe the responses to different disturbances. The range of dry solids concentrations at which the real plant operates is between 0.7 and 0.75 kg/kg. Large deviations outside this range may render the empirical correlations of the black liquor physical properties incorrect and even infeasible. The nominal operating conditions for the multiple evaporator plant are provided in Table 6.1.
197
03 03 03
o (X 03 r—1
'o o o
13
> 03
o
-u cd
^ o a (Tl > 0^ 03
Cu -;j
;=!
S 03
^-u SH
n
!-l-l 03 !-l
Pl -tJ
1:3
^
S-l -U OJ .—H
(5
s-l -tJ
a 0 0
!H
03
n ,^ 0 oj
r.Q
-D
03
M-l
Tl 03 N CO SH
l-l
03 C3
-tJ
PI 0 0
i=l
0 • 1—1
•l-J
en 0
a
B n ri
03
Q 0 0 CM CO
OJ
03
0
Pl
-tJ
SH
br
fc
198
s-T SH
rt 0
Table 6.1: Operating conditions of the muhiple evaporator plant
Parameter
Value 116
Gfeed
kg/s, 50/50 split
Xfeed
0.145 kg/kg
Tfeed
303.15 K
Psteam ( S C - 1 tO S C - 3 )
460 kPa
^condenscT
21.4 kPa
The disturbances applied to the system are a 5% decrease in the dry solids concentration in the feed stream, a 5% increase in the mass flow rate of the feed stream, and a 7% decrease in the temperature of the feed stream. The tuning parameters of the single-loop controllers are the same values as those used in the feedback control of a single evaporator (see Table 4.2). The closed-loop responses of the nonlinear multiple evaporator plant (nonlinear system of partial differential equations) to the change in the operating conditions are shown in Figures 6.3 to 6.5. The results show that in all cases the disturbances are compensated within 40 to 50 minutes. The responses of both the controlled and manipulated variables are smooth and the size of the control action does not imply excessive changes. The length of time it takes to bring the dry solids concentration of the product to within the iblcr limit of 0.001 kg/kg are listed in Table 6.2. In Figure 6.5, to compensate for the decrease in the dry sohds concentration the steam pressure is decreased. This is in contrast to the results of the the closed
199
10,73;
Figure 6.3: Closed-loop response of the nonlinear multiple evaporator plant to 5% increase in the flow rate of the feed. Top: dry solids concentration in the product stream, bottom: steam pressure of SC-1. The dotted lines represent the ±lcr limit of 0.001 kg/kg. Table 6.2: Performance of the single-loop control structure of the nonlinear multiple evaporator plant Disturbance
Time to ±1(T
5% increase of Gfeed
27.8 min
5% decrease of Xfeed
20.6 min
7% decrease of Tfeed
20.4 min
loop tests provided in Chapter 4 where the pressure was observed to increase as the dry solids concentration in the product stream decreased. It is also observed that initially after the disturbance in the dry solids concentration is applied, the dry solids concentration of the product stream increases. This behavior while unexpected can be explained by the physics of the process. That is, when the evaporator plant produces a product with dry solids concentration in the range of 0.7 to 0.75 kg/kg, the boiling point elevation increases significantly (see Appendix C, Equation ( C l ) ) . 200
'
^
i
w 0.735
'
'
'
1
^ .
\
5 ^
0.73
1
0 fl
10
20
30
40
650
& S 600
'
'
'
50
1
60
70
'
1
60
70
8
-
s S. 550 5
500
u 10
,
70
1
1
30
40 Time, mli>
50
BO
Figure 6.4: Closed-loop response of the nonhnear muhiple evaporator plant to a 7% decrease in the temperature of the feed. Top: dry solids concentration in the product stream, bottom: steam pressure of SC-1. The dotted lines represent the ±lcr limit of 0.001 kg/kg. For such operating conditions, when the dry sohds concentration in the feed stream is increased, any consequent increase in the dry solids concentration of the product stream is eliminated by the increase in the boiling point elevation. The increased boiling point elevation decreases the total available temperature difference (driving force) of the multiple evaporator plant. Thus, the total effect of the increase in the dry solids concentration of the feed is a decrease of the dry solids concentration of the product. More details on this behavior of the system is provided in Section 6.2.2.
6.2
Centralized control
In this control strategy, a model predictive control (MPC) formulation is investigated, see Figure 4.21 in Chapter 4. The performance of the MPC controller is demonstrated on the nonlinear ODE and PDE representations of the multiple evaporator plant. The strategy reUes on a model of the system to determine the control
201
^
0.7421-
«• "•'* " • 0.738 § 0,736 o ^
«
0 734 -
480
Figure 6.5: Closed-loop response of the nonlinear multiple evaporator plant to a 5% decrease of the dry solids concentration of the feed. Top: dry solids concentration in the product stream, bottom: steam pressure of SC-1. The dotted lines represent the ±la limit of 0.001 kg/kg. actions now and in the future. As discussed in Chapter 4, this formulation uses a linear-time invariant (LTI) model (ordinary differential equations) or an input-output (convolution) model. To generate the LTI model, first a nonlinear ordinary differential equation (ODE) description of the multiple evaporator plant is developed and validated. Second, the validated nonlinear ODE model is used to develop the LTI model. Third, the LTI model is transformed to an input-output (step response) model, which is used to develop the MPC controller.
6.2.1
Nonlinear ODE model
The development of the nonlinear ODE or lumped model of the multiple evaporator plant follows the development of the lumped model of a single evaporator discussed in Chapter 4 (see Equations (4.33), (4.37) and (4.44) to 4.47)). The model was
developed in Matlab's®: Simulink ® : environment. A computer program is
202
provided in Appendix D. To validate the lumped ODE model, the responses of the distributed PDE model and the lumped ODE model are compared for the same changes in the operating conditions. The responses are shown in Figures 6.6 to 6.11. Evaporator 5
.XlO'
Superconcentrator 1 0.04
^'
B) i
0.02
-
0.01
PDE . ODE
o
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<
<
n 0
,
20
40
60
80
100
120
20
40
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1 ^
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120
0
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0,01 E 0 <
-0.01
-0.01 -0.02 0
20
40
60
80
100
-0,02 "0
120
Time, min
Time, min
Figure 6.6: Responses of nonlinear PDE (solid line) and ODE (dotted line) models of the multiple evaporator plant to a 5% decrease in the flow rate of the feed. Left column: Evaporator E-5, right column: superconcentrator SC-1. Top: dry sohds concentration of the product stream {AXprod), middle: temperature (ATj^od), and bottom: inventory (AL).
The responses show that the nonlinear ODE and PDE models have similar trends. The profile of the responses of the most important variable, the dry sohds concen-
203
Evaporator 5
.XlO'" 0
Superconcentrator'
\
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40
60
80
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20
40
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X^ -0,04 < -0,06 0
PDE ODE
20
40
60
80
100
120
20
40
60
80
100
120
20
40
60
80
100
120
1 0,5 0
r
-0.5 -1
0
0.02 0.01 L E _i"
E 0
-J
<
< -0 01 -0 02 0
20
40
60
80
100
120
Time, min
Figure 6.7: Responses of nonlinear PDE (solid line) and ODE (dotted line) models of the multiple evaporator plant to a 5% increase in flow rate of the feed. Top: dry sohds concentration of the product stream {AXprod), middle: temperature {ATprod), and bottom: inventory (AL). tration are similar and the steady state gains are on the same order. The responses to feed temperature disturbances show that the nonlinear ODE model is less sensitive to this particular disturbance, especially in the case of the first evaporator, E-5. The temperature disturbances effect on the superconcentrator evaporator SC-1 are represented with satisfactory accuracy. There are noticeable differences in the profiles of the temperature and level responses in some cases (feed flow rates and feed concentration changes). This may be
204
10
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5 X
0
PDE ODE
< -5,-
20
40
20
40
60
80
100
120
60
80
100
120
E
<
40
60
80
100
120
Time, min
Time, mm
Figure 6.8: Responses of nonlinear PDE (solid line) and ODE (dotted line) models of the multiple evaporator plant to a 5% decrease in dry solids concentration of the feed. Top: dry solids concentration of the product stream {AXprod), middle: temperature (ATprod), and bottom: inventory (AL). amplified because the controller parameters of the level controllers are different. Otherwise, it is always possible to tune the level controllers in the both cases to achieve a closer correspondence. This was not done to demonstrate the effect of level on the performance of the models. In can be concluded, that the nonlinear ODE (lumped) model is suitable to develop a centrahzed control strategy for the multiple evaporator plant.
205
Evaporator 5 .XlO""
Superconcentrator 1
XlO'
/<^"^~'^^^^^""""^
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PDE ODE •
/ X
•
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0
20
I
40
I
60
I
80
100
120
100
120
100
120
100
120
E
< -0 02 0
20
40
60
80
100
"0
120
Time, min
20
40
60
80
Time, min
Figure 6.9: Responses of nonlinear PDE (sohd line) and ODE (dotted line) models of the multiple evaporator plant to a 5% increase in the dry solids concentration of the feed. Top: dry sohds concentration of the product stream {AXprod), middle: temperature {ATprod), and bottom: inventory (AL). 6.2.2
The effect of concentration disturbances
The nonlinear ODE model can be used to study the operation of the multiple evaporator plant. It also has an advantage in that the ODE model can be integrated in about 20-30 seconds on a Pentium''''^III 1.33 GHz machine). In comparison, the nonhnear PDE model reqmres 24 hours on 7 1.2 GHz AMD™Athlon® nodes of a Rocks cluster^ for the same duration. ^ Rocks is an open source high performance Linux® cluster solution. For more information, see www. rocksclusters .org
206
Evaporator 5
Superconcentrator 1 PDE ODE
PDE ODE
-i
X
X
<
< 0
20
40
60
80
100
120
0
20
40
60
80
100
120
-0,03
0,02
0.02
0.01
0,01
of^
20
40
60
80
100
120
20
40
60
80
100
120
20
40
60
80
100
120
0 -0 01
-0 01 -0 02 0
0
20
40
60
80
100
-0 02 0
120
Time, mm
Time, min
Figure 6.10: Responses of nonlinear PDE (sohd line) and ODE (dotted line) models of the multiple evaporator plant to a 5% decrease in the temperature of the feed. Top: dry solids concentration of the product stream {AXprod), middle: temperature (ATprod), and bottom: inventory (AL). The performance of the nonlinear ODE model of the multiple evaporator plant is evaluated at different operating conditions and different boiling point elevation coefficients {K). The results are summarized in Table 6.3 and are given as deviations from the final dry solids concentration values in the product stream. The results show that in the range 6.5 < A" < 7 that is associated with the calculation of the boiling point rise, depending on the value of the steam pressure, there is a point at which the effect of increases to the feed dry sohds concentration
207
Evaporator 5
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Supeconcentrator 1 kg/kg
0,03
Ul
^
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PDE ODE
•
•
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X
<
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40
60
80
100
0,01
.
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PDE . ODE
0 20
40
20
40
,
60
80
100
120
60
80
100
120
y-
<
E
E
<
<
20
40
60
80
Time, min
Time, min
Figure 6.11: Responses of nonlinear PDE (solid line) and ODE (dotted hne) models of the multiple evaporator plant to a 5% increase in the temperature of the feed. Top: dry sohds concentration of the product stream (AXprod), middle: temperature (ATprod), and bottom: inventory (AL). on the product dry solids concentration is reversed. That is, for different increases in the feed dry solids concentration, the product dry solids concentration increases at some operating conditions but decreases at others (see Table 6.3). At low dry solids concentrations (< 0.63 kg/kg), the response of the muhiple evaporator plant to a decrease in the dry solids concentration if the feed stream leads to a decrease in the dry solids concentration in the product stream. This resuh was observed in the open loop tests of the PDE model (see Figures 5.4 and 5.5). However, ff the
208
Table 6.3: The effect of boiling point elevation to a 5% decrease in the feed dry sohds concentration. Steam pressure at SC-1, 2 and 3, kPa
Xsci
Boihng point rise coefficient, K
AXprod/^XPDE,%
410
0.6560
7
18
390
0.6330
7
5
380
0.6209
7
-5
360
0.5954
7
-15
410
0.6747
6.5
20
390
0.6514
6.5
8
380
0.6391
6.5
?»0
360
0.6129
6.5
-17
360
0.7188
4
-5
340
0.6877
4
-25
320
0.6533
4
-49
300
0.6160
4
-75
boihng point elevation is less sensitive to changes in the dry solids concentration (e.g., a boiling point rise coefficient of i<' = 4) then a dry solids concentrations > 0.7 kg/kg in the product stream does not result in a reverse response, i.e., for a decrease in the feed dry solids concentration there is a decrease in the product dry solids concentration. The reduced pressures in the case of K = 4 are necessary to keep the dry sohds concentration in the product within feasible limits so that the correlations used to calculate the physical properties of the process fluid remain valid. It suffices to conclude that the boiling point elevation of the process fluid determines to a large
209
extent the response of the evaporator plant to changes in the dry sohds concentration of the feed
6.2.3
Linear model
The linear-time invariant state-space model of the multiple evaporator plant has been calculated analytically, using Maple'^'^ The procedure is similar to the one used in Chapter 4; the Maple code is presented in Appendix D. The resulting state-space representation matrices A,B,
and C also are given in Appendix D.
In the development of this linear model, the equation that models the heating zone length is omitted (Equation (4.44 in Chapter 4)). AU levels are assumed constant with a value of 1 meter. The manipulated variables are the same as in the case of the decentralized control - the steam pressures at the superconcentrator (SC-1 to SC-3) and the secondary vapor pressure at E-5. The real part of all thirty-two eigenvalues of the dynamic constant coefficient matrix A are negative; hence open-loop stability is assured [82, 84]. Two of the eigenvalues are complex conjugates, which would imply some oscihations in the open loop responses. However, the magnitude of the imaginary parts of these eigenvalues is very small (order 10""* when compared to the largest eigenvalue of the system (-2.1465).
Thus, the oscihations do not dominate the response.
In general, the
system has a dominant mode that corresponds to the largest eigenvalue. All the other eigenvalues are much smaller between one to three orders of magnitude, see Appendix D).
210
The controUability analysis shows that the system is not state controllable [84]; the rank of the controUabflity matrix is four but the number of states is thhty-two. Since only four of the states are assumed measurable, output controUability analysis can be used to establish the controUabihty Suppose that the measured variables are the dry solids concentrations of E-5, SC-3, SC-2 and SC-1 in their respective output streams. Controllability analysis shows that the system is not output controllable (the rank of the output controUabflity matrix is 3). Another potential set of control variables include ah three superconcentrator dry solids concentrations in the recirculating streams and the multiple evaporator plant product (SC-3) temperature. For this set of four controlled variables, the output controUabihty condition is satisfied, the rank of the output controllability matrix is four. The responses of the linear model to changes in the manipulated variables are shown in Figures 6.12 to 6.17. The responses show that the linear model represents the nonlinear model well. The steady state gains are comparable. Further analysis shows that there are smaller differences between the response of the linear and nonlinear models of the evaporator that is closer to where the change is implemented.
6.2.4
Results
The closed-loop performance of the centralized unconstrained and unconstrained MPC controllers, and the decentralized single-loop (PI) controllers are compared for disturbance rejection. In the PI control scheme, the dry solids concentrations at each
211
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\
o,
-2
o
.4
•
NLODE LODE
1 \
'^^—'^^ 50
100
150
50
Time, min
100
150
Time, min
.x10
S
50
100
50
100 Time, mm
Time, min
Figure 6.12: Responses of the nonlinear (sohd hne) and linear (dotted line) ODE models to a 5% decrease in the superconcentrator steam pressures. superconcentrator unit is controlled by the corresponding steam pressure; the product temperature is controlled by the secondary vapor pressure at E-5. The parameters of the MPC and PI controller are listed in Table 6.5. In this table, yi is the dry solids concentration at SC-3, 2/2 is the dry solids concentration at SC-2, ys is the dry solids concentration at SC-1, and 2/4 is the temperature of SC-1. Also, Ui,U2 and U3 are the steam pressures at SC-3, SC-2, and SC-1, respectively; and u^ is the secondary vapor pressure at E-5. Note that all values in Table 6.5 are deviations from the nominal conditions since the linear model is developed using deviation variables. The responses of the controlled variables to the different disturbances are shown in Figures 6.18 to 6.41. The mass flow rate, dry solids concentration, and the temper212
NLODE LODE 50
100
150
50
Time, mm
100
150
100
150
Time, mm
XlO
50
100
150
50
Time, min
Time, mm
Figure 6.13: Responses of the nonlinear (solid line) and linear (dotted line) ODE models to a 5% increase in the superconcentrator steam pressures. ature of the feed have been decreased by 5% each and the heat transfer coefficients of the evaporators have been decreased by 10%. The performance in all cases is assessed by the length of time each response reaches ±lcr limits of 0.001 kg/kg, and the integral of the absolute deviation between the set point (y^p) and the measured value {y{t)).
I A E = / \ysp Jo
y{t)\dt.
(6.6)
The time to reach ±la limit is for each disturbance is given in Table 6.4. The values of the lAE are given in Table 6.6.
213
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NLODE LODE -
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NLODE LODE
, 50
100
^
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:
-1 1 1
150
50
Time, min
100
150
Time, min
50
100 Time, mm
Figure 6.14: Responses of the nonlinear (solid line) and linear (dotted line) ODE models to a 5% decrease in the steam pressure of the superconcentrator evaporator SC-1. 6.2.4.1
Unconstrained and constrained MPC controUers
All the closed-loop responses of the unconstrained and constrained MPC controllers are stable. There are no offsets in the responses of the controUed variables and the approach to set point is continuous without excessive oscillations or discontinuities. Similar descriptions can be stated for the controller actions (manipulated variables). The closed-loop responses show a greater sensitivity to throughput (feed mass flow rate change) changes. This is reflected in the length of time the response takes to reach ± l a limit of 0.001 kg/kg (see Table 6.4) and the values of the lAE (see Table
214
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NLODE LODE
-1
150
.
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Time, mm
100
150
100
150
Time, mm
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50
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150
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Time, mm
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Figure 6.15: Responses of the nonlinear (sohd hne) and linear (dotted line) ODE models to 5% increase in the steam pressure of superconcentrator evaporator SC-1. 6.6). The ± l a limit is used as a constraint in the MPC formulation of only the final product stream of SC-1. Constraints on the other two dry solids concentration limits are not as stringent since only the final dry solids concentration should be maintained in a narrow range. The temperature of the product stream of SC-1 is constrained to within ± 3 K from the desired set point. The limits on the manipulated variables are selected to satisfy reasonable process limits such as maximum on the steam consumption or a minimum on the secondary vapor pressure at E-5. The rate of change for the manipulated variables is at most 30 kPa/min. The closed-loop responses to changes in the temperature of the feed and the heat transfer parameters disturbances are similar. The response with respect to feed dry 215
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,
150
50
Time, mm
50
100
150
100
150
Time, min
100
150
50 Time, min
Time, min
Figure 6.16: Responses of the nonhnear(solid line) and linear (dotted hne) ODE models to a 5% decrease in the secondary vapor pressure of evaporator E-5. solids concentration is less sensitive when compared to the responses for the other disturbances. The dry solids concentration in the product stream does not violate the ±la limit for either MPC formulations. It takes approximately 13 minutes for the closed-loop responses to reach ±lcr limit for all disturbances, except in the case of the change in the feed dry sohds concentration where the product dry solids concentration never leaves these limits. Also, it is observed that the effect of a change in the dry solids concentration at SC-3 is more significant than at SC-2 and SC-1 based on the larger control actions (steam pressure Pscs) acting on SC-3. Data from a pulp mfll operated by Tembec, Inc. (St. Francisvihe) show that
216
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NLODE LODE
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50
100
-0,012 0
150
50
Time, min
50
100
150
100
150
Time, min
100
150
50
Time, mm
Time, min
Figure 6.17: Responses of the nonlinear (sohd line) and linear ODE models to a 5% increase in the secondary vapor pressure of evaporator E-5. variations with amplitudes as large as the ±2cr limit do not affect the stability of the recovery boiler. The effect of deviations in this dry sohds concentration range can manifest themselves only in the longer term operation of the boiler where changes in the burning characteristics of the black liquor can be detected. The results show no constraint violation on the inputs, outputs, and rate of change of the inputs in all cases. The controller actions (maximum of rate of change of 30 kPa/min is applied in 1 minute or 15 kPa every 30 seconds, which is the sample time) are more aggressive to compensate for feed mass flow rate, feed temperature, and heat transfer disturbances on SC-1 (see Figures 6.27, 6.31, and 6.33). The lAEs calculated for the constrained MPC controller are larger when compared to the unconstrained
217
Table 6.4: Times to reach la limits of 0.001 kg/kg of the multiple evaporator plant product dry solids concentration. MPC Disturbance 5% increase of Gfeed
Unconstrained
PI Constrained
13.74 min
31.1 min
5% decrease of Tfeed
12.97 min
59.92 min
10% decrease of hs
12.78 min
30.34 min
5% decrease of Xfeed
case. This is not unexpected whenever constraints are active.
6.2.4.2
PI control
In the case of the decentralized strategy that employs single PI loops, the responses are oscillatory with a period of approximately 12.5 minutes. The length of time to reach ±la
limit in the range of 30 to 60 minutes. This is longer when compared
to the performance of the both MPC controUers. The ± 2 a limit is violated in the case of temperature and heat transfer disturbances. This duration of the violation is not excessive, however this shows that the unconstrained MPC controller achieved better results as a function of the selected tuning parameters. The sensitivities to all disturbances are similar to those observed in the case of MPC control. The most important comparison between the MPC and PI controllers is in the values of the lAEs. The lAE values of the closed-loop response under PI Control show that the dry solids concentrations at SC-3 and SC-2 are more tightly controlled 218
when compared to the values obtained in the case of the MPC controllers. This can be advantageous if tight control at SC-3 and SC-2 is necessary However, since the most important variable to be controlled is the dry sohds concentrations of the product stream at SC-1, and since it is not as tightly controlled by the PI controher when compared to the responses of both MPC controllers, it can be concluded that the performance of both MPC controllers is better than the performance of the PI controher.
6.2.4.3
Control of the PDE system
Closed-loop control of the PDE system that describes the multiple effect evaporator plant is carried out in the same manner as in Chapter 4. That is, the manipulated variables found from the control of the nonlinear ODE description of the multiple effect evaporator plant are implemented on the PDE system when disturbance in the feed mass flow rate are present. The closed-loop response of the PDE system is shown in Figure 6.42. The response shows that the MPC controller actions provide excellent compensation. The dry solids concentration in the product stream of SC-1 is regulated to within ±la limit in about 20 minutes. Also the temperature of the product stream is maintained at the desired set point. It can be concluded that nonlinear ODE representation and the subsequent linear ODE derived from it is satisfactory to develop an effective MPC controller for the actual multiple effect evaporator plant.
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CHAPTER 7 SUMMARY AND FUTURE WORK 7.1
Modeling
This work developed a fundamental model of a multiple effect falling film evaporator plant that processes a black liquor feed stream normally found in pulp and paper mills. The model strived to represent the important transport phenomena of evaporation and sensible heating of the falling film, as well as water film condensation were modeled in detail. It also accounted for turbulent and wavy-laminar hydrodynamic regimes found in these processes. No attempt was made to describe the wavy nature of the falling films or other small scale phenomena. Using a simple laboratory experiment, laboratory tracer experiment on the falling film past a vertical dimpled plate, it was shown that a one spatial dimension model adequately described the phenomena of interest at the necessary scale. The resulting system of partial differential equations was subsequently solved using orthogonal cohocation on finite element (OCFE). The model was validated by examining the response to a set of expected disturbances in the feed stream and changes in the utility stream (steam) pressure. The results showed that feed mass flow rate had the least eflFect on the dry solids concentration in the product while feed temperature disturbances had the largest effect. The single evaporator PDE model was used to develop the models for the multiple effect evaporator (MEV) plant. The multiple evaporator plant consisted of seven
247
single evaporators interconnected in a countercurrent design. The pressure dynamics were modeled by system of four ordinary differential equations. Also the condensate flashing was modeled using a steady state assumption on the flashing process. The resulting model of the MEV was integrated using parallel processing, in this case MPI (Message Parsing Interface), where each evaporator's equations are integrated on a separate processing node. The model was validated using an expected set of disturbances to the feed stream and steam utilities. The responses showed that the system physics are well represented. To further validate the MEV model, steady state data from an industrial plant were used. The results showed excellent agreement - steady state values of the product dry solids concentration showed very small diflterences of 0.006 kg/kg or about 0.8%.
7.2
Control
To study control, a nonlinear model in the form of a system of ordinary differential equation (ODE) was developed. Validation of the ODE and PDE models showed that for the same set of disturbances the ODE model satisfactorily represented the distributed parameter system. The nonlinear ODE model was then approximated by a linear ODE model which permitted linear control theory to be applied to develop a suitable controller. Two sets of linear controllers were applied, decentralized single loop feedback control law (proportional-integral controller) and a centralized model predictive controller.
248
7.3
Decentralized control
A decentralized control scheme was tested on the single evaporator represented by the PDE model. The control scheme includes two cascaded feedback (proportionallyintegral or PI) loops controhing the level in the evaporator inventory with the flow rate of the evaporator product mass flow rate and the dry sohds concentration of the product with the steam pressure. The evaporator was controlled for on-supply operations. The results showed that although not optimally tuned, the PI controllers provided satisfactory performance for disturbance compensation. The control action was more aggressive in the case of set point changes (servo control), however if the changes were applied as ramps for one or half of the system time constant aggressive control actions may not occur. A similar decentralized control scheme was tested on the multiple evaporator plant represented by the PDE model. The control scheme included six cascaded feedback (proportionally-integral or PI ) loops controlling the level in five of the evaporator inventories with the flow rate of the corresponding evaporators product streams. The dry solids concentration of the product of the last evaporator was controlled with the steam pressure supplied to the evaporator. The results showed satisfactory disturbance compensation in the presence of feed mass flow rate, feed temperature, and feed dry solids concentration disturbances. The response time in which the product dry solids concentration was within ±lcr limits were satisfactory.
249
7.4 7.4.1
Centralized control
Single evaporator
Unconstrained and constrained MPC formulations were implemented using the nonlinear ODE models. In this case the controlled variables were the product dry solids concentration and the product temperature. The evaporator level inventory was controhed using a cascaded single loop feedback (PI) controller. The closed loop system was subjected to different disturbances and the performance of the system was evaluated. The unconstrained MPC controller was able to provide successful disturbance compensation in the presence of a feed mass flow rate, dry solids concentration and temperature changes and to provide regulation of the product dry solids concentration within ±lcr hmit in less than flve minutes. The constrained MPC controller showed improved performance, however, no significant difference between the unconstrained and constrained implementations can be observed. The results of both unconstrained and constrained MPC controllers were compared to equivalent decentralized control scheme. The performances were found to be similar for disturbance compensation. However, when the magnitude of the disturbances are increased, the performance of the constrained MPC controller is superior to that of the decentralized scheme. This demonstrates that the MPC controller is a more robust controller. The final validation of the MPC controller was the application of the controller actions found for the nonhnear ODE model to regulate the distributed parameter model (PDE model).
The results showed that the most important variable, the 250
product dry solids concentration was well controlled and regulated within ±la limit in approximately seven minutes.
7.4.2
Multiple evaporator plant
Similar to control of a single evaporator, unconstrained and constrained MPC formulations were tested using the nonlinear ODE model. The controlled variables were the last three evaporators (SC-3, SC-2 and SC-1) product streams' dry solids concentrations and the last evaporator (SC-1) temperature. The manipulated variables were the steam pressures at the last three evaporators (SC-3, SC-2 and SC-1) and the secondary vapor pressure at the first evaporator (E-5). The closed-loop system was subjected to feed mass flow rate, feed dry solids concentration, feed temperature and heat transfer disturbances. The feed dry solids concentration disturbance was well compensated by both constrained and unconstrained MPC. In the presence of the other disturbances, the performance of constrained MPC was found to be superior to that of unconstrained MPC. For unconstrained MPC, the response of the product dry solids concentration remained within the r t l u limit within ten to thirteen minutes after the introduction of the disturbance, or about one fifth of the open-loop system time constant. When compared to the equivalent decentralized control scheme, the advantages of the MPC formulation were more obvious than in the case of single evaporator. The responses of the decentralized scheme showed oscillatory character with the response of the product dry solids concentration remaining within ±a limits in about twenty
251
to thirty minutes. The final validation of the MPC controller involved the apphcation of the controher actions found for the nonlinear ODE model to the distributed parameter model. The results showed that the most important variable, the product dry solids concentration was well controlled as in the case of single evaporator.
7.5
Summary on the control results.
The results of the application of the different control strategies showed that both decentralized and centralized control can be used to control the single evaporator and multiple evaporator plant successfully. However, constrained MPC provided robustness. Therefore, the control strategy applied should be determined by evaluating the quality requirements on the controlled variables. For tight constraints on the controlled and manipulated variables the centralized control strategy should be used.
7.6
Future work
The following are areas in which future development may be very useful. First, it is necessary to build at least a single laboratory scale falling film evaporator and to refine the PDE model using dynamic data.
Laboratory scale experiments are
preferable as there will be less uncertainty in the measured parameters. The model should be validated using real plant data if available. It is possible to use different working fluids. Suitable choice can be a carboxymethylceUulose solution, which is non-corrosive and its dry sohds concentration can be measured easily through the
252
solution conductivity. Second, there is room for improvements in the numerical techniques used to solve the PDE model. The lengthy calculation times found in this work obstructed more detailed investigation of the PDE model. Lastly, additional models that describe the scaling process or possible crystahization in the evaporators can be developed and incorporated in the PDE model. If the computation times are reduced by optimization of the numerical solution method a long term operation of the evaporators can be simulated in order to predict how scaling affects the performance of falhng film evaporators. This knowledge can be used to optimize the scheduling of the washing cycles that are common when high dry solids concentration is to be achieved in the final evaporators/concentrators.
7.7
Contributions
The contributions of this work are fohows: • A fundamental first principles distributed parameter model of a falling film plate type evaporator was developed and • Two control strategies were developed to regulate the dry solids concentration in the product stream of evaporator, which is essential to provide a stable feed to the recovery bofler. • A fundamental model of a multiple evaporator plant (seven evaporators) was developed and validated.
253
• An effective MPC strategy was developed to regulate the dry sohds concentrations in three evaporators and the temperature of the final product stream. • Stefanov, Z.I. and K.A. Hoo (2003) "A Distributed Parameter Model of Black Liquor Falling Fflm Evaporators. Part I. Modehng of a Single Plate," Industrial & Engineering Chemistry Research, 42, pp. 1925-1937.
254
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260
APPENDIX A TURBULENT HEAT TRANSFER COEFFICIENTS
The derivations of the formulas for the heat transfer coefficients are taken from [86]. Consider a vertical wah with a fluid flowing down the length of the wah and that the flow is fully developed (see Figure A.l). The side of the wall opposite the
Figure A.l: A fluid element in the fully developed falhng film.
flowing fluid is heated uniformly. Suppose that the transport of energy by convection is neghgible in comparison to the energy transport due to conduction. Then, the heat flux is given by q = -{X + Xt)
dT dy'
(A.1)
where A is the fluid thermal conductivity. At is the turbulent thermal conductivity, T is the fluid temperature, and y is the spatial coordinate normal to the wall. Accordingly, 261
the turbulent thermal diffusivity {at) is defined as
at = - ^ , pcp
(A.2)
where p and Cp are the density and the heat capacity of the fluid, respectively. By analogy with the molecular Prandtl number, the turbulent Prandtl number is defined ag Prt = - , at
(A.3)
where e is the eddy viscosity. Using the quantities defined by Equations (A.3) and (A.2), Equation (A.l) can be re-written as
^ .
Pr e\ dT
, , ^,
It is useful to make the heat flux equation dimensionless using the following change of variables listed in Table A.l. It then follows that the dimensionless heat flux equation is given by
1 /
Pr e\ dT+
262
For a completely developed non-evaporating film, the temperature profile in the direction of the flow {x) is uniform, thus
d ( Tnw - T dx \ Ty, - TM
= 0
(A.6)
j^udy where TM is the mean film temperature. Table A.l: Dimensionless Variables Transformed Variable pCpU*{Ty,-T)
T^ = — y+ = u'y u+ = 6+ "^Z.
U
Definition
T.
., ,
qw
Spatial coordinate normal to the wall
u*
Liquid velocity
u
Fflm thickness
u* = ^/g5
,
Liquid temperature
Friction velocity
For constant flux at the wah, Qy,, the term {Ty,—TM) is uniform in the x direction. Therefore dT _ dTyj _ dTM dx dx dx
263
/ . o\
The energy equation for a Newtonian fluid with constant thermal conductivity and density is given by [87]
pCp
dT g,+^^^ , ,'d^T dx'^ +/i
\dy
dT
dT + ^y
d^T dy'^
dT dz
dy
d^T\ dz"^
dx J
dux dx
{dz
dx J
+
dUy
dUy
dx
dx
\dz
dy J
+ PQ,
(A.9)
where Q is the external radiant energy transmission rate per unit mass and equals zero in the studied case. As the velocity distribution in the studied case is assumed to be Ux =
(A.IO)
Ux{y)
and the axial conduction is assumed to be negligible, using Fourier's law [87]
A— dy
(A.ll)
the energy balance equation (A.9) reduces to the form
dT dx
pCp u-
264
dy
(A.12)
Using Equations (A.5), (A.7) and (A.12) the dimensionless form of the heat transport equations can be obtained as
/
u'^dy'^ (A.13)
Re
T q T+
=
(A. 14) Pr /
Prt V
T+u+dy+
Jo
(A.15)
Re
T The integral in Equation (A.15) can be further simplified using integration by parts. Thus, the dimensionless heat transfer coefficient of sensible heating is given by
qw'''
S+V3pr
{Ty,-TM)\g"^
T+
2/3
h*H
5+V^Pr (
u'^dy'^
\
-dy^.
(A.16)
1 - Jo Re
T
&+ Pr
/
Prt V
In the case of an evaporating or condensing film, the energy supplied through the wall almost equals the energy release at the film surface as the heat of evaporation. The wall heat flux is then preserved across the film {q = g^) and the temperature
265
distribution is obtained by direct integration of Equation (A.5). That is
y"*"
I
r-=/^ X 7 X I ' Pr
'^•'"
Prt u
The heat transfer coefficient of evaporation is then given by
h-E = —.
(5+'Pr - ^
/ X-X7I\*^ Pr Prt Vi/
266
•
{A.18)
APPENDIX B BULK FLOW MOMENTUM EQUATION
Consider a vertical wall with a film of fluid flowing down the length of the wall and that the flow is fufly developed (see Figure B.l).
5
\
I dz
-^1
7
J^
xdz 1
''
pg(5-•y)dz
Figure B.l: Force balance on a fluid element in the falling film.
The following holds for the time averaged velocity components:
Vx =0
(B.l)
v„ = 0
(B.2)
Vz = Vz{y).
(B.3)
267
For incompressible, steady flow, and the conditions of Equations (B.l) to (B.3) the momentum balance (Navier-Stokes equations) for turbulent flow is given by
dP d , , cl^vz • ^ = -P^(..t;.) + / . ^ ,
,^ , (B.4)
where P is the global pressure acting on the film and the term lyOJ represents the fluctuations of the velocity components about the average velocities in the y and z directions. The force balance over a differential volume of thickness y, width A, and length dz is given by Ay dP = -TA
dz,
(B.5)
where r is the shear stress. Using Equation (B.5), the integration of Equation (B.4) in the y direction gives dvz /
-dy = -p
Jo y
d{u^) Jo
•+ p, Jo
yd
dy
or T = -pVyVz
268
, dVz + p - ^ .
,.r> a\ (B.6)
Here the term, -pVyV^, is the Reynolds stress, rt (see for example Bird et al. [44]). The Reynolds stress can also be expressed as
Tt = pe--. dy
(B.7
The shear stress in the film is a function of y an^d (see [58]) is given by
= r. (l - I) ,
(B.8)
where Ty, = p6g is the shear stress at the wall and 6 is the film thickness. Combining Equations (B.6) to (B.8) results in the momentum balance
piy \
oJ
\
uJ dy
or ^9 (^ y\ fs I ^\ ^'"^ ?(-D-('-;)fdy
The dimensionless form of Equation (B.9) is given by
e\ dvt
using the variable transformations given in Table B.l.
269
i^-^'
Table B.l: Dimensionless Variables Transformed Variable
y+ =
Definition
Spatial coordinate normal to the wall
(J+ =
^
Fflm thickness
f;+ = —p=
Liquid velocity
Vz
Tg
270
APPENDIX C BLACK LIQUOR PHYSICAL PROPERTIES
For the properties below all units are in SI system.
Density [1]
p = 1007 - 0.495(r - 273.15) + 6X 303K
kg/kg,
where T is the black liquor temperature and X is the dry solids concentration. Kinematic viscosity [88]
V
exp(6.1373 x 10^ - 42.178 x WX + 335.12 x lO^X^ - 349.23 x lO^X^) (r3 - 2.4273 + 9.1578X - 56.723X2 + 72.666X3)0.0000035 2,mK
271
Thermal conductivity [89]
A - 0.63 -F 0.001(r - 311) - 0.2737X
303i^ < r < 450i^ 0 kg/kg < X < 85 kg/kg
The original correlation was derived for temperature range from 0 to 100 centigrade, however the in this study the correlation is extrapolated to the temperature range indicated above. Heat capacity [1]
=(0.98 - (-9.122 X 10~^r + 0.793)X)4190 2,Q2K
Surface tension The information available in Hough [1] and Guihchsen [88] shows, that the surface tension of the black liquor does not change significantly with changes in the dry solids concentration and temperature. In this work it has been assumed to be an average value of 0.033 N/m.
272
Boiling point rise [1]
BPR = K-
0 kg/kg < X < 80
^ 1-X kg/kg,
where ii' is a constant that varies from 6.5 to 7.5 K. In this work an average value of 7 is used.
273
APPENDIX D COMPUTER PROGRAMS
D.l
M a t l a b S-function for simulation of single evaporator
function [sys,yO,str,ts] = csfunc(t,y,u,flag) a = *; %Plate width, m% Kw = *; %Wall heat transfer coefficient, W/m^.K% Kc = *; %Heat transfer coefficient of condensation, W/m^KVo pi = *; %Number of plates% L = *; %Plate length, m% Gpump = 58*7; if t > tstep %Introducing of step disturbance% Gfeed = 60.9; else Gfeed = 58; end; Xfeed = 0.145; Gfeed = 58; cp = 3600; %Liquor heat capacity, J/kg.K% Ac = *; %Inventory crossection area, va'^% %Dispatch the flag% switch flag, case 0 [sys,yO,str,ts]=mdllnitiaIizeSizes; %Initialization% case 1 sys = mdlDerivatives(t,y,u,a,Kw,Kc,pl,L,Gpump,Gfeed,Xfeed,Tfeed,cp,Ac); culate derivatives% case 3 sys — mdlOutputs(t,y,u); %Calculate outputs% case 2, 4, 9 % Unused flags% sys = []; otherwise error(['Unhandled flag = ',num2str(flag)]); %Error handling% end %End of csfunc%
%mdlInitializeSizes%
274
%Cal-
%Retums the sizes, initial conditions, and sample times for the% %S-function.%
function [sys,yO,str,ts] = mdllnitializeSizes % Call simsizes for a sizes structure, fill it in and convert it% %to a sizes array.Vo sizes = simsizes; sizes.NumContStates = 5; sizes.NumDiscStates — 0; sizes.NumOutputs = 5; sizes.Numlnputs = 3; sizes.DirPeedthrough — 0; %Matrix D is empty% sizes.NumSampleTimes = 1; sys = sinisizes(sizes); %Initialize the initial conditions.% yO = [* * * ...]; VoEnter the nominal steady state values here.% YoFor initialization start with close, feasible values.% %str is an empty matrix.% str = []; %Initialize the array of sample times; in this case the sample% %time is continuous, so set ts to 0 and its offset to 0.% ts = [0 0]; %End of mdllnitializeSizes.% %^===============================^^====% %mdlDerivatives% %Retums the derivatives for the continuous states.% % = ^ = = = = = = = = = = = ^ = = = = = = = = = ™ = = = = = = = = = = = = %
function sys =
mdlDerivatives(t,y,u,a,Kw,Kc,pI,L,Gpump,Gfeed,Xfeed,Tfeed,cp,Ac)
Gin = Gpump - u(3); %,Recirculation flow% Xplatein = (y(4)*(Gpump-Gfeed)-(-Xfeed*Gfeed)/Gpump; %Stack inlet dry solids concentration% Tplatein = (y(5)*(Gpump-Gfeed)-|-Tfeed*Gfeed)/Gpump; %Steck inlet temperature% Tavgh = 0.5*Tplatein -I- Tbl(Xplatein,u(2))*0.5; %Average heating zone temperature% Khtb = Kht(Xplatein,Gin/pl,Tavgh); %Sensible heating heat transfer coefficient% rhoe = dnslblq(y(2),Tbl(y(2),u(2))); %Liquor density on plates% TboU = Tbl(Xplatein,u(2)); %Boiling temperature% Ketb = Ket(y(2),Gin/pl,Tbl(y(2),u(2))); %Heat transfer coefficient of evaporation% Gout = y(l)*ut(y(2),Gin/pl,Tbl(y(2),u(2)))/L; %Plate stack outlet flow% 275
%Calculation of the log average temperature difference for heating zone and% %heating zone lengthVo T w a l l i n = (Kw*Kc*u(l)/(Kw+Kc)+Khtb*Tplatein)/(Khtb+Kw-Kw^2/(Kw+Kc)); Twaflout = (Kw*Kc*u( 1)/ (Kw-fKc)+Khtb*Tboil)/ (Khtb+Kw-Kw^2/ (Kw+Kc)); dT2 = Twallin-Tplatein; d T l = Twallout-Tboil; dT = ( d T 2 - d T l ) / l o g ( d T 2 / d T l ) ; Lh = -Gin*(Tplatein-Tbon)*cp/(Khtb*aV*(dT)); %Plate stack mass and dry solids concentration
derivatives%
dy(l) = Gin- Gout (Ketb*(L-Lh)*a*pl/hevap(Tbl(y(2),u(2))))*... ((Kw*Kc*u(l)/(Kw+Kc)+Ketb*Tbl(y(2),u(2)))/... (Ketb-f Kw-Kw^ 2/ (Kw+Kc) )-Tbl(y(2) ,u(2))); dy(2) = Gin*(Xplatein-y(2))/y(l) + (y(2)/y(l))*pI*(Ketb*(L-Lh)*a/hevap(Tbl(y(2),u(2))))*. ((Kw*Kc*u(l)/(Kw+Kc)+Ketb*Tbl(y(2),u(2)))/... (Ketb+Kw-Kw^2/(Kw-fKc))-Tbl(y(2),u(2))); %Inventory
parameters%
Gtout = Gpump-Gfeed; Gtankin = Gout; Xtankin = y(2); Ttankin = Tbl(y(2),u(2)); dG — Gtankin Gtout; rhotO = dnsIblq(Xtankin, Ttankin); rhot =dnslblq(y(4),y(5)); hctO = cp; hct = cp; %Inventory
mass, dry solids concentration and temperature
derivatives%
dy(4) = (Gtankin*(Xtankin - y(4))/(y(3)*Ac*rhot)); dy(5) = ((Gtankin*Ttankin*hctO -Gtout*y(5)*hct)/(Ac*rhot*hct*y(3))- ... (y(5)*dG/(Ac*rhot*y(3))) ((y(5))/hct)* ... (3.822*y(5)-3322.67)*(Gtankin*(Xtankin-y(4))/(y(3)*Ac*rhot)))*... (hct/(hct + 3.822*y(5)*y(4))); dy(3) = (dG)/(rhot*Ac) (y(3)/rhot)*(-0.495*dy(5) + 600*dy(4)); sys = dy; %End of
mdlDerivatives.%
276
%Retum the block outputs.%
function sys = mdlOutputs(t,y,u) sys = y; %End of mdlOutputs.%
277
D.2 Matlab S-function for simulation of multiple evaporators function [sys,yO,str,ts] = csfunc(t,y,u,flag) %Parameter a = *;
declarations% % Plate width, m%
Kw = *; % Wall heat transfer coefficient, W/m^.K% Kc=[* ***** *]; %Heat transfer coefficients of condensation, W/m^K% Pl = [ * * * * * * *]; %Number of plates per evaporator% V = [ * * * * *]; %Vapor spaces volumes, m^% cp = [ * * * * * *]; VoLiquor heat capacities, J/kg.K% L = *; YoPlate length, m% M = 0.018; %Water molecular mass, kg/mole% R = 8.314; %Universal gas constant, J/mol.K% Gfeedin = [58 98 67 59 41 32 26]; %Nominal stack feed flow rates, kg/s% Gpump = Gfeedin*6 -1- Gfeedin; if t > t s t e p %Introducing of step disturbance% Xfeed = 0.145-1-0.05*0.145; Xfeedl = 0.145-^0.05*0.145; else Xfeed = 0.145; Xfeedl = 0.145; end; Gfeed = 58; %Feed flow rate of E-5, kg/s% Gfeedl = 58; %Feed flow rate of E-4, kg/s% Tfeed = 30-1- 273.15; 7oFeed temperature of E-5, K% Tfeedl = 30 + 273.15; %Feed temperature of E-4, K% Ac = *; %Inventories crossection area, rn^% % Dispatch the flag% switch flag, case 0 [sys,yO,str,ts]=mdllnitializeSizes; %Initialization% case 1 sys — mdlDerivatives(t,y,u,a,Kw,Kc,pl,L,Gpump,Gfeed,Xfeed,... Gfeedl,Xfeedl,Tfeed,Tfeedl,Ac,V,M,R,cp); %Calculate case 3 sys = mdlOutputs(t,y,u,a,Kw,Kc,pl,L,Gpump,Gfeed,Xfeed,... Gfeedl,Xfeedl,Tfeed,Tfeedl,Ac,V,M,R,cp); %Calculate case 2 , 4 , 9 sys = []; otherwise
%Unusedflags%
278
derivatives%
outputs%
error (['Unhandled flag = ',num2str(flag)]); end %End of csfunc%
%Error handling%
%mdlInitializeSizes% %Retums the sizes, initial conditions, and sample times for the% %S-function.% %=================^=============.==^====% function [sys,yO,str,ts] = mdUnitializeSizes % Call simsizes for a sizes structure, fill it in and convert it% %to a sizes array.% sizes = simsizes; sizes.NumContStates = 39; sizes.NumDiscStates = 0; sizes.NumOutputs = 11; sizes.Numlnputs = 10; sizes.DirFeedthrough = 0; %Matrix D is empty% sizes.NumSampleTimes = 1; sys = simsizes(sizes); %Initialize the initial conditions.% yO = [* * * ...]; %Enter the nominal steady state values here.% %For initialization start with close, feasible values.%) %)Str is an empty matrix.%o
str = 0; %)Initialize the array of sample times; in this case the sample%i %)time is continuous, so set ts to 0 and its offset to 0.%, ts = [0 0]; %)End of mdllnitializeSizes.%) %omdlDerivatives% %Retums the derivatives for the continuous states.Vo function sys= mdlDerivatives(t,y,u,a,Kw,Kc,pl,L,Gpump,Gfeed,Xfeed,... Gfeedl,Xfeedl,Tfeed,Tfeedl,Ac,V,M,R,cp) dy = zeros(l,39); Xmixout = zeros(7,l); Tmixout = zeros(7,l); Gmixin = zeros(7,l); Gmixout = zeros(7,l);
279
%)Saturation temperatures at the vapor spaces%o Tsat(l) Tsat(2) Tsat(3) Tsat (4) Tsat(5) Tsat(6) Tsat(7)
= eqth2o(u(6)); = eqth2o(y(ll)); =eqth2o(y(17)); = eqth2o(y(23)); = eqth2o(y(29)); = Tsat(5); = Tsat(5);
%>Recirculation flows no flash% Grecnf(l) Grecnf(2) Grecnf(3) Grecnf(4) Grecnf(5) Grecnf(6) Grecnf(7)
= = = = = = =
Gpump(l) Gpump(2) Gpump(3) Gpump(4) Gpump(5); Gpump(6); Gpump(7)
u(l) u(2) u(3) u(4)
u(5);
%iRecirculation compositions no flash, E-5 to E-2'% Xrecnf(l) Xrecnf(2) Xrecnf(3) Xrecnf(4)
= = = =
y(4); (y(4)*u(l)-fy(9)*(Gpump(2)-u(l)))/Gpump(2); (Xrecnf(2)*u(2)+y(15)*(Gpump(3)-u(2)))/Gpump(3); (Xrecnf(3)*u(3)+y(21)*(Gpump(4)-u(3)))/Gpump(4);
%)Recirculation temperatures no flash, E-5 to E-2%) lYecnf(l) IVecnf(2) 'IVecnf(3) 'IVecnf(4)
= = = =
y(5); (y(10)*(Gpump(2)-u(l))+y(5)*u(l))/Gpump(2); (y(16)*(Gpump(3)-u(2))+y(10)*u(2))/Gpump(3); (y(22)*(Gpump(4)-u(3))+y(16)*u(3))/Gpump(4);
%,Outlet flow SC-3%, ify(26)~=y(32) Gbout(5) =0.0506*dnslblq(y(27),y(28))*sqrt(2*9.81*abs(y(26)-y(32))); Gmixin(5) = Gpump(5) u(4); Gmixout(5) = Gpump(5); Xmixout(5) = (y(27)*Gmixin(5) + u(4)*Xrecnf(4))/Gpump(5); Tmixout(5) = (y(28)*Gmixin(5) + Ti:ecnf(4)*u(4))/Gmixout(5); if (y(26)
280
Gbout(5) = 0; Gmixin(5) = Gpump(5) u(4); Gmixout(5) = Gpump(5); Xmixout(5) = (y(27)*Gmixin(5) + u(4)*Xrecnf(4))/Gmixout(5); Tmixout(5) = (y(28)*Gmixin(5)+'IVecnf(4)*u(4))/Gmixout(5); end; %iRecirculation composition and temperature SC-3%} Xrecnf(5) = Xmixout (5); Trecnf(5) = Tmixout(5); %oOutletflow SC-2%0 ify(32)~=y(37) Gbout(6) =0.0506*dnslblq(y(33),y(34))*sqrt(2*9.81*abs(y(32)-y(37))); Gmixin(6) = Gpump(6) Gbout(5); Gmixout(6) = Gpump(6); Xmixout(6) = (y(33)*Gmixin(6) + Gbout(5)*y(27))/Gmixout(6); Tmixout (6) = (y(34)*Gmixin(6)-Fy(28)*Gbout(5))/Gmixout(6); if (y(32) Tbl(Xrecnf(i),Tsat(i)) flash(i) = 1; 281
else flash(i) = 0; end; end; %oStack inlet flows calculation%o for i = 1:7 if flash(i) = = 0 Gin(i) = Grecnf(i); Xin(i) = Xrecnf(i); Tin(i) = Trecnf(i); else Wfl(i) =
Grecnf(i)*hcblq(Xrecnf(i),'IVecnf(i))*(Trecnf(i)-Tbl(Xrecnf(i),Tsat(i)))/... hevap(Tbl(Xrecnf(i),Tsat(i))); Gin(i) = Grecnf(i)-Wfl(i); Xin(i) = Grecnf(i)*Xrecnf(i)/(Gin(i)); Tin(i) = Tbl(Xin(i),Tsat(i)); end; end; %jffeai transfer coefficients of evaporation%o Ketb(l) Ketb(2) Ketb(3) Ketb(4) Ketb(5) Ketb(6) Ketb(7)
= = = = = = =
Ket(y(2),Gin(l)/pl(l),Tbl(y(2),Tsat(l))); Ket(y(7),Gin(2)/pl(2),TbI(y(7),Tsat(2))); Ket(y(13),Gin(3)/pl(3),Tbl(y(13),Tsat(3))) Ket(y(19),Gin(4)/pl(4),Tbl(y(19),Tsat(4))) Ket(y(25),Gin(5)/pl(5),Tbl(y(25),Tsat(5))) Ket(y(31),Gin(6)/pl(6),Tbl(y(31),Tsat(6))) Ket(y(36),Gin(7)/pl(7),Tbl(y(36),Tsat(7)))
VoHeating zone length calculation%o for i = 1:7 if flash(i) = = 0 Tavgh(i) = (Tin(i) + Tbl(Xin(i),Tsat(i)))/2; Khtb(i) = Kht(Xin(i),Gin(i)/pl(i),Tavgh(i)); ifi < 5 Twallin(i) = (Kw*Kc(i)*Tsat(i+l)/(Kw+Kc(i))+Khtb(i)*Tin(i))/... (Khtb(i)+Kw-Kw^2/(Kw+Kc(i))); Twallout(i) = (Kw*Kc(i)*Tsat(i+l)/(Kw+Kc(i))-f... Khtb(i)*Tbl(Xrecnf(i),Tsat(i)))/(Khtb(i)-hKw-Kw^2/(Kw+Kc(i))); dT2(i) = Twallin(i)-Tin(i); dTl(i) = Twallout(i)-Tbl(Xrecnf(i),Tsat(i)); dT(i) = (dT2(i)-dTl(i))/log(dT2(i)/dTl(i)); Lh(i) = -Gin(i)*(Tin(i)-Tbl(Xrecnf(i),Tsat(i)))*cp(i)/(Khtb(i)*a*pl(i)*(dT(i))); 282
else Twallin(i) = (Kw*Kc(i)*eqth2o(u(i+2))/(Kw+Kc(i))+Khtb(i)*Tin(i))/.. (Khtb(i)-hKw-Kw'^2/(Kw-hKc(i))); Twallout(i) = (Kw*Kc(i)*eqth2o(u(i+2))/(Kw-hKc(i))+... Khtb(i)*Tbl(Xrecnf(i),Tsat(i)))/(Khtb(i)+Kw-Kw^2/(Kw+Kc(i))); dT2(i) = Twallin(i)-Tin(i); dTl(i) = Twallout(i)-Tbl(Xrecnf(i),Tsat(i)); dT(i) = (dT2(i)-dTl(i))/log(dT2(i)/dTl(i)); Lh(i) = -Gin(i)*(Tin(i)-Tbl(Xrecnf(i),Tsat(i)))*cp(i)/(Khtb(i)*a*pl(i)*(dT(i))); end; else Lh(i) = 0; end; end; VcEvaporation calculation%o %oDensities%o
rhoe(l) = dnslblq(y(2),Tbl(y(2),Tsat(l))); rhoe(2) = dnslblq(y(7),Tbl(y(7),Tsat(2))); rhoe(3) = dnslblq(y(13),Tbl(y(13),Tsat(3))) rhoe(4) = dnsIblq(y(19),Tbl(y(19),Tsat(4))) rhoe(5) = dnslblq(y(25),Tbl(y(25),Tsat(5))) rhoe(6) = dnslblq(y(31),Tbl(y(31),Tsat(6))) rhoe(7) = dnslblq(y(36),Tbl(y(36),Tsat(7))) VoPlate stack outlet flows%o Gout(l) Gout(2) Gout(3) Gout(4) Gout(5) Gout(6) Gout(7)
= = = = = = =
y(l)*ut(y(2),Gin(l)/pl(l),Tbl(y(2),Tsat(l)))/L; y(6)*ut(y(7),Gin(2)/pI(2),Tbl(y(7),Tsat(2)))/L; y(12)*ut(y(13),Gin(3)/pl(3),Tbl(y(13),Tsat(3)))/L y(18)*ut(y(19),Gin(4)/pl(4),Tbl(y(19),Tsat(4)))/L y(24)*ut(y(25),Gin(5)/pl(5),Tbl(y(25),Tsat(5)))/L y(30)*ut(y(31),Gin(6)/pl(6),Tbl(y(31),Tsat(6)))/L y(35)*ut(y(36),Gin(7)/pl(7),Tbl(y(36),Tsat(7)))/L
%)Derivatives for M and X on the platesVo VoEvaporated water flow rates%o W ( l ) = (Ketb(l)*(L-Lh(l))*a*pl(l)/hevap(Tbl(y(2),Tsat(l))))*... ((Kw*Kc(l)*Tsat(2)/(Kw+Kc(l))+Ketb(l)*TbI(y(2),Tsat(l)))/. (Ketb(l)+Kw-Kw^2/(Kw+Kc(l)))-Tbl(y(2),Tsat(l))); W(2) = (Ketb(2)*(L-Lh(2))*a*pl(2)/hevap(Tbl(y(7),Tsat(2))))*... 283
W(3)
W(4)
W(5)
W(6)
W(7)
((Kw*Kc(2)*Tsat(3)/(Kw+Kc(2))+Ketb(2)*Tbl(y(7),Tsat(2)))/... (Ketb(2)+Kw-Kw^2/(Kw-hKc(2)))-Tbl(y(7),Tsat(2))); = (Ketb(3)*(L-Lh(3))*a*pl(3)/hevap(Tbl(y(13),Tsat(3))))*... ((Kw*Kc(3)*Tsat(4)/(Kw+Kc(3))+Ketb(3)*Tbl(y(13),Tsat(3)))/... (Ketb(3)+Kw-Kw^2/(Kw+Kc(3)))-Tbl(y(13),Tsat(3))); = (Ketb(4)*(L-Lh(4))*a*pl(4)/hevap(Tbl(y(19),Tsat(4))))*... ((Kw*Kc(4)*Tsat(5)/(Kw+Kc(4))+Ketb(4)*Tbl(y(19),Tsat(4)))/... (Ketb(4)+Kw-Kw^2/(Kw+Kc(4)))-Tbl(y(19),Tsat(4))); = (Ketb(5)*(L-Lh(5))*a*pl(5)/hevap(Tbl(y(25),Tsat(5))))*... ((Kw*Kc(5)*eqth2o(u(7))/(Kw+Kc(5))+Ketb(5)*Tbl(y(25),Tsat(5)))/. (Ketb(5)+Kw-Kw^2/(Kw+Kc(5)))-Tbl(y(25),Tsat(5))); = (Ketb(6)*(L-Lh(6))*a*pl(6)/hevap(Tbl(y(31),Tsat(6))))*... ((Kw*Kc(6)*eqth2o(u(8))/(Kw+Kc(6))+Ketb(6)*Tbl(y(31),Tsat(6)))/. (Ketb(6)-FKw-Kw^2/(Kw-|-Kc(6)))-TbI(y(31),Tsat(6))); = (Ketb(7)*(L-Lh(7))*a*pl(7)/hevap(Tbl(y(36),Tsat(7))))*... ((Kw*Kc(7)*eqth2o(u(9))/(Kw+Kc(7))-fKetb(7)*Tbl(y(36),Tsat(7)))/. (Ketb(7)+Kw-Kw^2/(Kw+Kc(7)))-Tbl(y(36),Tsat(7)));
%)Liquor mass derivatives%o dy(l) = Gin(l)-Gout(l)-W(l); dy(6) = Gin(2)-Gout(2)-W(2); dy(12) = Gin(3)-Gout(3)-W(3) dy(18) = Gin(4)-Gout(4)-W(4) dy(24) = Gin(5)-Gout(5)-W(5) dy(30) = Gin(6)-Gout(6)-W(6) dy(35) = Gin(7)-Gout(7)-W(7) %}Liquor dry solids concentrations
derivativesVo
dy(2) = Gin(l)*(Xin(l)-y(2))/y(l) + dy(7) = Gin(2)*(Xin(2)-y(7))/y(6) + dy(13) = Gin(3)*(Xin(3)-y(13))/y(12) dy(19) = Gin(4)*(Xin(4)-y(19))/y(18) dy(25) = Gin(5)*(Xin(5)-y(25))/y(24) dy(31) = Gin(6)*(Xin(6)-y(31))/y(30) dy(36) = Gin(7)*(Xin(7)-y(36))/y(35) %Evaporator inventory
(y(2)/y(l))*W(l); (y(7)/y(6))*W(2); + (y(13)/y(12))*W(3) + (y(19)/y(18))*W(4) -f (y(25)/y(24))*W(5) + (y(31)/y(30))*W(6) + (y(36)/y(35))*W(7)
calculationsVo
%,Inventory inlet flows%} Gtin(l) = Gout(l) -f- Gfeed; Gtin(2) = Gout(2) -t- Gfeedl; for i = 3:7 Gtin(i) = Gout(i); 284
end; Volnventory inlet compositionsVo Xtin(l) Xtin(2) Xtin(3) Xtin(4) Xtin(5) Xtin(6) Xtin(7)
= = = = = = =
(Gout(l)*y(2)+Gfeed*Xfeed)/Gtin(l); (Gout(2)*y(7)-hGfeedl*Xfeedl)/Gtin(2); y(13); y(19) y(25) y(31) y(36)
%olnventory inlet temperatures%o Ttin(l) Ttin(2) Ttin(3) Ttin(4) Ttin(5) Ttin(6) Ttin(7)
= = = = = = =
(Gfeed*Tfeed + Gout(l)*Tbl(y(2),Tsat(l)))/Gtin(l); (Gfeedl*Tfeedl + Gout(2)*Tbl(y(7),Tsat(2)))/Gtin(2); Tbl(y(13),Tsat(3)); Tbl(y(19),Tsat(4)) Tbl(y(25),Tsat(5)) Tbl(y(31),Tsat(6)) Tbl(y(36),Tsat(7))
%olnventory inlet and bulk densities, heat capacities%) for i = : L:7 rhotO(i) = dnslblq(Xtin(i), Ttin(i)); hctO(i) = hcblq(Xtin(i), Ttin(i)); end; rhot(l) rhot(2) rhot(3) rhot (4) rhot(5) rhot(6) rhot(7) hct(l) hct(2) hct(3) hct(4) hct (5) hct(6) hct(7)
= = = = = = =
dnslblq(y(4),y(5)); dnslblq(y(9),y(10)); dnslblq(y(15),y(16)) dnslblq(y(21),y(22)) dnslblq(y(27),y(28)) dnslblq(y(33),y(34)) dnslblq(y(38),y(39))
=. hcblq(y(4),y(5)); == hcblq(y(9),y(10)); == hcblq(y(15),y(16)) == hcblq(y(21),y(22)) == hcblq(y(27),y(28)) =-- hcblq(y(33),y(34)) == hcblq(y(38),y(39))
%olnventory outlet flow rates%) 285
Gtout(l) Gtout(2) Gtout (3) Gtout(4) Gtout(5) Gtout(6) Gtout (7)
Gpump(l Gpump(2 Gpump(3 Gpump (4 Gpump(5 Gpump(6 Gpump(7
%Inventory
mass, dry solids concentration and temperature derivativesYo
u(i); u(2); u(3); u(4) + Gbout(5); Gbout(5) + Gbout(6); Gbout(6);
%oDry solids concentration
derivativesYo
dy(4) = (Gtin(l)* (Xtin(l) dy(9) = (Gtin(2)* (Xtin(2) dy(15) = : (Gtin(3) *(Xtin(3) dy(21) = -• (Gtin(4)*(Xtin(4) dy(27) = •• (Gtin(5)*(Xtin(5) dy(33) = •• (Gtin(6)*(Xtin(6) dy(38) = : (Gtin(7) *(Xtin(7)
y(4))/(y(3)*Ac*rhot(l))); y(9))/(y(8)*Ac*rhot(2))); y(15))/(y(14)*Ac*rhot(3))) - y(21))/(y(20)*Ac*rhot(4))) - y(27))/(y(26)*Ac*rhot(5))) - y(33))/(y(32)*Ac*rhot(6))) y(38))/(y(37)*Ac*rhot(7)))
%o Temperatures derivatives%o dy(5) = ((Gtin(l)*Ttin(l)*hctO(l) Gtout(l)*y(5)*hct(l))/(Ac*rhot(l)*hct(l)*y(3)) (y(5)*(Gtin(l) Gtout(l))/(Ac*rhot(l)*y(3))) ((y(5))/hct(l))* ... (3.822*y(5)-3322.67)*(Gtin(l)*(Xtin(l)-y(4))/(y(3)*Ac*rhot(l))))*... (hct(l)/(hct(l) + 3.822*y(5)*y(4))); dy(lO) = ((Gtin(2)*Ttin(2)*hctO(2)
...
Gtout(2)*y(10)*hct(2))/(Ac*rhot(2)*hct(2)*y(8))
(y(10)*(Gtin(2) Gtout(2))/(Ac*rhot(2)*y(8))) ((y(10))/hct(2))* ... (3.822*y(10)-3322.67)*(Gtin(2)*(Xtin(2)-y(9))/(y(8)*Ac*rhot(2))))*... (hct(2)/(hct(2) + 3.822*y(10)*y(9))); dy(16) = ((Gtin(3)*Ttin(3)*hctO(3)
Gtout(3)*y(16)*hct(3))/(Ac*rhot(3)*hct(3)*y(14))
(y(16)*(Gtin(3) Gtout(3))/(Ac*rhot(3)*y(14))) - ((y(16))/hct(3))* ... (3.822*y(16)-3322.67)*(Gtin(3)*(Xtin(3)-y(15))/(y(14)*Ac*rhot(3))))*... (hct(3)/(hct(3) + 3.822*y(16)*y(15))); dy(22) = ((Gtin(4)*Ttin(4)*hctO(4)
Gtout(4)*y(22)*hct(4))/(Ac*rhot(4)*hct(4)*y(20))
(y(22)*(Gtin(4) Gtout(4))/(Ac*rhot(4)*y(20))) - ((y(22))/hct(4))* ... (3.822*y(22)-3322.67)*(Gtin(4)*(Xtin(4)-y(21))/(y(20)*Ac*rhot(4))))*... (hct(4)/(hct(4) + 3.822*y(22)*y(21)));
286
dy(28) = ((Gtin(5)*Ttin(5)*hctO(5)
Gtout(5)*y(28)*hct(5))/(Ac*rhot(5)*hct(5)*y(26))
(y(28)*(Gtin(5) Gtout(5))/(Ac*rhot(5)*y(26))) ((y(28))/hct(5))* ... (3.822*y(28)-3322.67)*(Gtin(5)*(Xtin(5) y(27))/(y(26)*Ac*rhot(5))))*... (hct(5)/(hct(5) + 3.822*y(28)*y(27))); dy(34) = ((Gtin(6)*Ttin(6)*hctO(6)
Gtout(6)*y(34)*hct(6))/(Ac*rhot(6)*hct(6)*y(32))
(y(34)*(Gtin(6) Gtout(6))/(Ac*rhot(6)*y(32))) ((y(34))/hct(6))* ... (3.822*y(34)-3322.67)*(Gtin(6)*(Xtin(6)-y(33))/(y(32)*Ac*rhot(6))))*... (hct(6)/(hct(6) + 3.822*y(34)*y(33))); dy(39) = ((Gtin(7)*Ttin(7)*hctO(7)
Gtout(7)*y(39)*hct(7))/(Ac*rhot(7)*hct(7)*y(37))
(y(39)*(Gtin(7) Gtout(7))/(Ac*rhot(7)*y(37))) ((y(39))/hct(7))* ... (3.822*y(39)-3322.67)*(Gtin(7)*(Xtin(7)-y(38))/(y(37)*Ac*rhot(7))))*... (hct(7)/(hct(7) + 3.822*y(39)*y(38))); %iMass derivatives (levels)%o dy(3) = (Gtin(l) - Gtout(l))/(rhot(l)*Ac) dy(8) = (Gtin(2) - Gtout(2))/(rhot(2)*Ac) dy(14) = (Gtin(3) - Gtout(3))/(rhot(3)*Ac) dy(20) = (Gtin(4) - Gtout(4))/(rhot(4)*Ac) dy(26) = (Gtin(5) - Gtout(5))/(rhot(5)*Ac) dy(32) = (Gtin(6) - Gtout(6))/(rhot(6)*Ac) dy(37) = (Gtin(7) - Gtout(7))/(rhot(7)*Ac)
- (y(3)/rhot(l))*(-0.495*dy(5) + 600*dy(4)); - (y(8)/rhot(2))*(-0.495*dy(10) + 600*dy(9)); - (y(14)/rhot(3))*(-0.495*dy(16) + 600*dy(15)) - (y(20)/rhot(4))*(-0.495*dy(22) + 600*dy(21)) - (y(26)/rhot(5))*(-0.495*dy(28) + 600*dy(27)) - (y(32)/rhot(6))*(-0.495*dy(34) + 600*dy(33)) - (y(37)/rhot(7))*(-0.495*dy(39) + 600*dy(38))
VoPressure derivatives%o VoCondensate mass flow rates%o Wc(l) Wc(2) Wc(3) Wc(4) Wc(5) Wc(6) Wc(7)
= = = = = = =
W(l)*hevap(Tbl(y(2),Tsat(l)))/hevap(Tsat(2)); W(2)*hevap(Tbl(y(7),Tsat(2)))/hevap(Tsat(3)); W(3)*hevap(Tbl(y(13),Tsat(3)))/hevap(Tsat(4)); W(4)*hevap(Tbl(y(19),Tsat(4)))/hevap(Tsat(5)); W(5)*hevap(Tbl(y(25),Tsat(5)))/hevap(eqth2o(u(7))) W(6)*hevap(Tbl(y(31),Tsat(6)))/hevap(eqth2o(u(8))) W(7)*hevap(Tbl(y(36),Tsat(7)))/hevap(eqth2o(u(9)))
dy(ll) dy(17) dy(23) dy(29)
= (W(2)-Wc(l))*(R*(Tbl(y(7),Tsat(2)))+Tsat(l))/(2*M*V(2)); = (W(3)-Wc(2))*(R*(Tbl(y(13),Tsat(3)))+Tsat(2))/(2*M*V(3)); = (W(4)-Wc(3))*(R*(Tbl(y(19),Tsat(4)))-fTsat(3))/(2*M*V(4)); = (W(5)+W(6)+W(7)-Wc(4))*(R*(Tbl(y(25),Tsat(5)))+Tsat(4))/(2*M*V(5));
sys = dy; 287
VoEnd of mdlDerivatives.%0 %===^==================^^^===^^^^^^^===% %Retum the block outputs.%o
function sys = mdlOutputs(t,y,u,a,Kw,Kc,pl,L,Gpump,Gfeed,Xfeed... ,Gfeedl,Xfeedl,Tfeed,Tfeedl,Ac,V,M,R) sys = [y(3) y(8) y(14) y(20) y(26) y(32) y(37) y(27) y(33) y(38) y(39)]; %oEnd of mdlOutputs.%0
288
D.3
MAPLE code for linearization of single evaporator nonlinear ODE model
>
restart:
>
yln ;-3108:
>
y2n :- 0.2248527:
>
y3n :- 0.2248527:
>
y4n :- 339.2606:
>
Tsatn :- 337.23:
>
Tsn :- 352.96:
>
Gfeedn :- 58:
>
Xfeedn :- 0.145:
>
Tfeedn :- 30 + 273.15:
>
Tbl :- 7«y2/(l-y2) 4 Taat:
>
L :- 10.96:
>
pli- 136:
>
a:»1.219:
>
cp:=3600:
>
Gpump :» 406:
>
K»:-3500:
>
Kc:"10000:
>
GiD :- 316:
>
Xplatein :• Cy4*CGpump-Gfeedn)+Xfeedn*Gfeedii)/Gpump;
>
Tplatein :- (y4*(GpuiDp-Gfeedn)+Tfeedn*Gfeedn)/Gpump:
>
Tblh :» 7«Xplatein/(l-Xplatoin) 4 Tsat:
>
Taigh :- (Tplatein + Tblh)/2:
>
Kh :-(Xplatein-C-0.126))»((Gin/pl)-0.236)»(Tavgh-(1.661))+Tavgh*Xplatein»(-B2.6S):
>
uev : = (y2"(-0.136))•((0in/pl)-0.404606)»CTbl"(-0.0398414))+Tbl«y2«(-0.0029444):
>
Kev :-(y2-(-0.1915))«C(Giii/pl)-0.4272)»(Tbl-(l.4557))4Tbl«y2»(-26.2117):
>
Gout :- yl*uev/L;
>
Tvallin :- CK»»Kc»Ta/(Ku+Kc)41Qi«Tplatein)/(Kii4Kw-Kv-2/CKy4Kc:)) :
>
Twallout :- CKw«Kc»Ts/CKw+Kc)4Kh»Tbl)/(lUi+Kw-Kw-2/(Ky+Kc)):
>
dT2 := Twallin-Tplatein:
>
dTl :- Tuallout-Tbl:
>
dT :- (dT2-dTl)/log(dT2/dTl):
>
Lh :- -(Gin)«(Tplatein-Tbl)»cp/(iQi«a*pl*(dT)):
>
hevap :- 2812200 - 349.98*Tbl - 3.034»(Tbl-2):
>
dyl :"Gin-Oout-CKe»*(L-Lh)*a»pl/hevap)«((Kw'KoTs/(Kv+Kc)+Kov«Tbl)/(Kev+Kv-K»-2/(Kw+Kc) )-Tbl) :
>
dyllin :»mtaylor(dyl, [yl=yln,y2=y2n,y3=y3n,y4-y4n,Ts-Tsn,T8at-Tsatn] .2) : dyllin
: = 119.8944559-I-551.3177104 j/2 4- 1.911766548 Taat - .1041463166 i/J - 1.461569548 Ta - .2221735804 ]/.(
289
>
dy2:-Gin*(Xplatein-y2)/yl + (y2/yl)*pl*(Kev*(L-Lh)*a/hevap)*C(Kw*Kc*Ts/{Kw+Kc)+Kev*Tbl)/{Kev+Kw-Kw-2/(Kw+Kc))-Tbl):
>
dy211ii:-iDtaylorCdy2,[yl-yln,y2-y2n,y4-y4n,T8-TBn,TBat-Tflatii] , 2 ) ;
dy2lin := 29.54739342 + .08716444634 y4 - .09790570141 y2 - .009506629896 yl + .0001057393369 Ts - .0001214356797 Tsat >
Gtout :- 290:
>
Gtankin .• Gout:
>
Xtankin :- y2:
>
Ttankin :- Tbl:
>
dG :- Gtankin - Gtout:
>
rhot0:-1007.4 - 0.496*(Ttankin - 273.15) + 600*Xtankin:
>
rhot:-1007.4 - 0.495«Cy4 - 273.15) + 600*y3:
>
Ac:-15:
>
dy4 : - Gtankin*CXtankin - y3)/Cl*Ac*rhot) :
>
dy41in:-nitaylorCdy4,[yl-yln,y2-y2n,y3-y3n,y4=y4n,TBat-T8atn],2);
>
dy4lin : = .019447884011/2 - .019447884017/5 dy5:-CCGtankiii*Ttankin*cp-Gout*y4*cp)/(Ac*rhot*cp)-Cy4*(Gtankin-Gout)/(Ac*rhot))-
>
C (y4) /cp)•(3.822«y4-3322.67)*(Gtankin*(Xtankin-y3)/(Ac*rhot)))*(cp/(cp + 3.822*y4*y3)):
>
dy51in:"iiitaylDr(dy5, [yl-"yln,y2-y2n,y3-y3n,y4«y4n,Tsat"TBatn,Tfesd=Tf8edn] ,2) ;
dySlin := - . 0 1 0 5 9 6 5 7 6 7 - | - 3 . 6 4 4 5 8 5 7 5 9 i/S-b .01799084545 Tsat - .3381179629 1 0 " ^ yJ - .01799084492 y^ - 3.434988058 i/5
290
D.4
MAPLE code for linearization of multiple evaporator nonhnear ODE model
>
a:-
1.219;
>
Kw : - 3 5 0 0 ;
>
Kcl ; - 20720:
>
Kc4:-9464:
>
26431:
>
p l l : - 136:
>
pl5:-64:
Kc2:- 9616:
Kc6;-3583:
Kc3:-7778:
Kc6 : -
Kc7 : - 2891: pl2:-140:
pl3;-13B:
pl4:-160:
pl6:-64:
>
pl7:-64:
>
V I ; - 291:
>
cpl;- 3665;
>
cp4;-3302:
>
2882:
V2:-186:
V3:-211:
cp2:" 3675:
V4:-277:
V6:-352;
cp3:- 3532:
cp5:" 3082: cp6:-
cp7:- 2823;
314: L:" 10.96: M:- 0.018: R:- 8. Gfeedinl:- 58: GfQQdin2:" 98: Gfeedin3: 67:
G£eedin4:-- 59:
GfeedinB:- 41:
GfeedinS:- 32:
Gfeedin?
>
Gpumpl :- Gfeedinl*6 + Gfeedinl;
>
Gpump2 :- Gfeedin2*6 + Gfeedin2:
>
Gpump3 ;• Gfeedin3*6 + GfBedin3:
>
Gpump4 := Gfeedin4*6 + Gfeedin4:
>
Gpump5 := Gfeedin5«6 + GfeedinS:
>
Gpump6 :- Gfeedin6.6 + Gfeedin6:
>
Gpump7 := Gfeedin7*6 + Gfeedin7;
>
Xfeed :- 0.145;Xfeedl := 0.145:Gfeed :=
>
58:Gfeedl := 58:Tfeed ;= 30 +
>
273.15:Tfeedl
>
yln;-3111;
:- 30 + 273.15:Ac ;- 16:
>
y4n:-0.2149:
ySn:=332.2;
>
y9n;-0.2152:
yl0n;-346.1:
>
yl4n;-l:
y2n;=0.2266:
yl5n:-0.2972; y20n;-l:
y3n:-l: y6n:-4213:
y7n:-0.2221;
ylln:-38910:
yl6n:-364.6:
yl7n;-66090:
>
yl9n;-0.4322;
>
y24n:-34S5;
> >
y29n;-201300; y30n;-4494: y31n;-0.6963: y34u;"409.6; y35n;"5930; y36n;-0.7365:
>
y37n:-l;
y38n:=0.7355:
>
uln ;- 21732:
>
T s a t l ;-3.9487 4
> > > > > >
y21n;-0.4322:
y25n;-0.5745:
y8n;-l;
yl2n;=3791:
y22n;-382.6:
y26n:-1.018:
yl3n;-0.2972; yl8n;-5113; y23n:-116700;
y27ii;-0.5745; y32n:-1.007:
y28n:.403: y33n;-0.6963;
y39n;"413.1:
u2n ;- 410000:
u3n :- 410000:
u4n :- 410000;
l7.711.1n(ul/1000)-0.24263«ln(ul/1000)-2+0.19674nn(ul/1000)-3+273.16; TBat2 ;-3.9487 + i7.7ll.ln(yll/10O0)-0.24263.1n(yll/1000)-2+0.19674.1n(yll/lO0O)-34273. 15; Tsat3 ;=3.94a7 + i7.711.1n(yl7/1000)-0.24263.1n(yl7/1000)-2+0.19674-ln(yl7/1000)-34273.
291
>
T»at4 :-3.9487 4
>
'^•'"•l"(y=3/1000)-0.24263.1n(y23/1000)-240.19674.1n(y23/1000)-34273
>
15;
>
TsatS :-3.9487 +
>
"•"»*l"(y29/1000)-0.24263.1n(y29/1000)-240.I9674.1n(y29/1000)-34273
>
15; TsatS ; ' T8at5: Tsat7 :- Tsat5: Gbout1 -39.45; 0bout2:-78.73; Gbout3:
>
59.96;
Gbout4:-41.3;
>
30.48;
Cbout6;- 24.79:
GboutS:-
>
Ginl :- Gpumpl - Gboutl:
>
Gbout2:
>
Gpump5:Gin6 ;- GpumpG;
> >
:- Gpump7 - Gbout7; Xinl :- y4; Xin2 :-
>
(y4»Gboutl4y9.(Gpump2-Gboutl))/Gpump2;
>
{Xin2»0bout2+yl5»(Gpump3-Gbout2))/Gpump3:
Gbout7:- 23.05; Gin2 :- Gpump2
Gin3 :- Gpump3 - Gbout3:
Gin4
Gpump4
Gin5
Xin3 :-
> >
Xin4 :- (Xin3*Gbout34y21.(Gpump4-Gbout3))/Gpump4: Tinl ;- y5; Tin2 :-
>
(ylO"(Gpump2-Gboutl)+y4*Oboutl)/Gpump2:
>
(yl6»(Gpump3-Gbout2)4yl0«Gbout2)/Gpunp3:
Tin3 :-
> *
Tin4 :- (y22«(Gpump4-Gbout3)+yl6»Gbout3)/Gpu«ip4: Gmiiin5 :- Gpuinp5 + GboutS - Gbout4:
*
GmixoutS :- Gpump5 + Gbout5:
'
XmiioutS :- (y27»Gmiiin5 4 Gbout4»Xin4)/Gmixout6;
•
Tliiiiout5 := (y28»Gmiiin54Tin4«Gbout4)/GmiioutS:
•
Xin5 :- Xmixout5;
*
Gmiiin6 ;- GpumpS 4 Gbout6 - GboutS:
•
Gmixout6 ;= Gpump6 4 Gbout6:
•
Gbout4:
Gin7
Tin5 :- Tmixout5:
XmiioutG := (y33«Gmixin6 4 Gbout5*Xin5)/Gmixout6: Tmiioute :- (y34*Gmixin64Tin5*Gbout5)/Gmixout6: Xin6 := Xmixout6; Xin7 ;.
Tin6 ;= Tmixout6:
(Xffliiout6*Gbout6+y38« (Gpump7-Gbout6)) /Gpump7;
Tin7
(y39» (Gpump7-Gbout6)4Tmixout6*Gbout6) /Gpump7: Tbll :- 7»y2/(l-y2) 4 Tsatl: 7»y7/(l-y7) 4 Tsat2:
Tbl2 :-
Tbl3 :-
7«yl3/(l-yl3) + T3at3: Tbl4 ;» 7«yl9/{l-yl9) + T8at4: 7«y25/(l-y25) + TsatS:
Tbl6
7>y31/(l-y31) + Tsat6;
Tbl7
Tbl5 ;•
7«y36/(l-y36) + T3at7:
Kevl :((y2)-(-0.1915))«((Ginl/pll)-0.4272)»(Tbll-(l.4567))4Tbll»y2»(-25.2117
): KBV2
:=
( (y7) - (-0.1915) ) • C (Gin2/pl2)-0.4272)* (Tbl2-(1.4557) )+Tbl2«y7» (-26.2117
): Kev3 :((yl3)-(-0.1915))«({Gin3/pl3)-0.4272)«(Tbl3-(1.4657))4Tbl3«yl3«(-26.21 17): Kev4 ;((yl9)-(-0.1915))»((Gin4/pl4)-0.4272)«(Tbl4-(1.4557))+Tbl4»yl9«(-26.21 17):
292
>
Kev5
>
( (y25) • (-0.1915) ) • ((Gin5/pl5) "0.4272) • (TblB" (1.4657)) +Tbl5«y25. (-26.21
>
17);
>
Kev6 ;-
>
((y31)-(-0.1915))>((Gin6/pl6)-0.4272)«(Tbl6-(1.4657))+Tbl6«y31.(-26.21
>
17):
>
Kav7 :-
>
((y36)-(-0.1915))*((ain7/pl7)-0.4272)»(Tbl7-(1.4657))+Tbl7«y36«(-26.21
>
17):
>
uevl :-
>
((y2)-(-0.136))«((Ginl/pll)-0.404606).(Tbll-(-0.0398414))4Tbll.y2<(-0.
>
0029444) :
>
uev2 :-
>
((y7)-(-0.136)) •((Gin2/pl2)-0.404606) •(Tbl2-(-0.0398414))+Tbl2
>
0029444);
^
uev3 :-
>
((yl3)-(-0.136))«((Gin3/pl3)-0.404606)»(Tbl3-(-0.0398414))+Tbl3.yl3«(-
>
0.0029444):
>
uev4 ;-
>
((yl9)-(-0.136))«((Gin4/pl4)-0.404606). (Tbl4-(-0.0398414) )+Tbl4«yl9»(-
>
0.0029444):
>
uev5 ;-
>
((y25)-(-0.136))»((Gin5/pl5)-0.404606)«(Tbl5-(-0.0398414))+Tbl6»y25»(-
>
0.0029444):
>
uev6 :=
>
((y31)-(-0.136)).((Gin6/pl6)-0.40460S)»(Tbl6-(-0.0398414))+Tbl6-y31»(-
>
0.0029444);
>
uev7 :"
>
((y36) - (-0.136) )» ( (Gin7/pl7) "0.404506) • (TbU" (-0.0398414)) +Tbl7.y36- (-
>
0.0029444);
>
Ooutl :- yl«uevl/L: Gout2 :- y6»uov2/L;
>
Gout3 :» yl2*uev3/L; Gout4 :-
>
yl8*uev4/L; Gout5 :° y24*uev5/L:
>
Gout6 :- y30.uev6/l.; Gout7 :- y35«uev7/L:
>
hevapl :- 2812200 - 349.98«Tbll -
>
3.034.(Tbl 1-2):
>
349.98»Tbl2 - 3.034«(Tbl2-2);
>
hevap3 :- 2812200 - 349.98»Tbl3 -
>
3.034«(Tbl3-2):
>
349.9B*Tbl4 - 3.034«(Tbl4-2):
>
bevapS ;- 2812200 - 349.98»Tbl5 -
>
3.034-(Tbl5-2):
>
349.98»Tbl6 - 3.034»(Tbl6-2);
>
bevapT :- 2812200 - 349.98«Tbl7 - 3.034.(Tbl7-2);
>
dyl ;- (Ginl- Goutl
>
lievap2 :- 2812200 -
hevap4 :- 2812200 -
hevap6 ;- 2812200 -
(Ke,l.(L).a.pll/hevapl).((Kv-Kcl.Tsat2/(Kv4Kcl)4Kevl.Tbll)/(Kevl4Kw-Kw
>
-2/(Kv*Kcl))-Tbll)):
>
dyllin ;-mtaylor(dyl.tyl-yl».y2-y2n.yll=y"°.<'l-»l°]'2''
iylHu
; = 219.6171890-F 562.0571158 v2 + .001868925221.1 - .1117241480 yj
- .001022022882 v J l > >
dy6 := (Gin2- Gout2 (Ke.2.(L).a.pl2/bevap2).((Kw.Kc2.Tsat3/(K.4Kc2)4Kev2.Tbl2)/(Kev24K»-K.
>
-2/(Kw4Kc2))-Tbl2));
>
dyGlin
:.mtayl,>r(dy6,[y6-y6n.y7.y7n,yll-yllb.yl>ynn].2);
293
dySlin
:= 398.6807444 + 840.1964808 !/7-I- .0012009334961/11 - .1396122867 yfi
- .0006570382728 yl7 >
dyl2 ;- (Gin3- Gout3 -
>
(Kev3»(L)"a'plS/bevapS)•((Kw»Kc3»TBat4/(Kw4Kc3)4Kev3«Tbl3)/(Kev34Kw-Kw
^
-2/(Kw4Kc3))-Tbl3));
>
dyl21in ; - mtaylor(dyl2,tyl2-yl2n,yl3-yl3n,yl7-yl7n,y23-y23n].2); dylSlin
: = 197.2718227 -I- 635.1460892 yJS -I- .0007138376571 yl? - .1029428143 yJS
- .0003685287372 yS3 >
dyl8 ; - (Gin4- Gout4 -
>
(Kov4<(L)«a"pl4/hevap4)»((K»"Kc4.Tsat6/(Kw4Kc4)+Kev4»Tbl4)/(Kov4+Kw-Kw
J
-2/(Kv4Kc4))-Tbl4)):
>
d y i a i i n ; - mtaylor(dyl8. [yl8-yl8n.yl9-yl9n,y23-y23n,y29-y29n].2); dylShn
: = 6.76041234 -I- 772.9072428 yl9 + .0005564371608 ySS - .06908486165 yl8
- .0002607352908 v29 >
dy24 :- (Gin5- GoutS -
>
(Kev5»(L)>a'pl6/hevap6)«((Kw.Kc5"(3.9487 4
>
17.711«ln(u2/1000)-0.24263»ln(u2/1000)-2+0.19674»ln(u2/1000)-3+273.16)
>
/(Kw4Kc5)4Kev5"Tbl5)/(Kev5+Kv-Kw-2/(Kw+Kc5))-Tbl5)):
>
dy241in :- iDtaylor(dy24, [y24-y24n,y26-y25n,y29-y29n, u2-u2n].2): dy24Un : = - 5 1 . 4 7 7 1 3 2 4 2 -I- 546.6003755 yS5 + .0002012946815^29 - .07996422167 y2.J - .00006356448214 u2
>
dy30 :- (Gin6- Gout6 -
>
(Kev6.(L).a*pl6/hevap6)*((KM*Kc6*(3.9487 4
>
17.711'ln(u3/1000)-0.24263»ln(u3/1000)-2+0.19674«ln(u3/1000)-3+273.16)
>
/(Kw4Kc6)4Kev6»Tbl6)/(Kev6+Kw-Kw-2/(Kw4Kc6))-Tbl6)):
>
dy301in :- mtaylor(dy30,[y29-y29n,y30-y30n,y31-y31n. u3-u3n],2); dySOlin := - 3 1 6 . 0 1 1 7 9 6 1 + 731.9940856 y5i + .0002482335077 y29 - .04857655176 ySO - .00006167511763 u 3
>
dy36 := (Gin7- Gout7 -
>
(Kev7.(L)«a.pl7/heTap7).((Kw»Kc7«(3.9437 *
>
17,711.1n(u4/1000)-0.24263.1n(u4/1000)-2+0.19674.1n(u4/1000)-34273.15)
>
/(Kw+Kc7)+Kev7.Tbl7)/(Kev74Kw-Kw-2/(Kw4Kc7))-Tbl7)):
>
dy351in := mtaylor(dy35,ty29.y29n,y35-y35n,y36-y36n. u4-u4n].2); dySSlin : = - 5 7 1 . 2 3 3 3 7 0 6 4- 939.7440874 y36 + .0002498576954 y29 - .02651654546 ySS - .00003161463720 W
> >
dy2 ; - (Ginl«(Xinl-y2)/(yl) + (y2/yl)«pll»(Ke7l.{L)«a/hevapl)»((Kw»Kcl*Tsat2/(Kw+Kcl)+Kevl«Tbll)/(Ke
>
vl4Ku-Kw-2/(Kw4Kcl))-Tbll));
>
dy21in ; .
mtaylor(dy2,[yl-yln.y2-y2n,y4.y4n,yll-ylln,ul=uln],2)l dy2lin
•= .000173775148 4- .1178238508 y4 - .1140341851 y2 4- .47743879 l O " " yj
+ .7444242529 lO"'' yll > > >
- .1187297698 IQ-" ul
dy21in :.173775148e-34.1178238508.y4-.2498576954e-3.y24.47743879e-8.yl4.744424 2529e-7»yll-.1187297698e-6«ul: dy2hn
:= .000173775148 -f .1178238508 W " .0002498576954 y2 + .47743879 l O " " yl
-I- .7444242529 lO"'' yll
- .1187297698 1 0 " ^ ul
294
> >
dy7 :- (01n2»(Xin2-y7)/(y6) t (y7/y6)•pl2»(Kev2»(L)•a/hevap2)•((Kv*Kc2.Taat3/(Kw+Kc2)+Kev2«Tbl2)/(Ke
> >
v24Kw-Ku-2/(Kw4Kc2))-Tbl2)): dyTlin ;-
>
mtaylor(dy7,ty4-y4n,y6-y6n,y7-y7n,y9-y9n,yll-ylln.yl7-yl7n],2); dylhn
> > > >
:= .000050860734 + .008289212242 y.j + .1358527294 y9 - .1410675806 y7
+ .29571838 lO"'' y6 + .3463759801 lO"'' yl7 - .5270434982 1 0 " ^ yli dyl3 :- (Gin3«(Xin3-yl3)/(yl2) • (yl3/yl2)«pl3.(Kev3»(L)"a/hevapS)•((Kw*Kc3*Tsat4/(Kw+Kc3)+Kov3»Tbl3)/( Kev34Kw-Kw-2/(Kw4Kc3))-Tbl3)): dyl31in ;-
>
mtaylor(dyl3.[y4-y4n.y9-y9n,y12-yl2n,yl3-yl3n,yl5-yl5n,yl7-yl7n,y23.y2
>
3n:,2);
dylSlin
:= .000282617581 -I- .001041628642 y4 H- .01707135610 y9 -I- .08978730541 ylS
- .1052644402 yl.J + .22142658 1 0 " ^ yJ2 + .2889125316 1 0 ~ ^ y2J - .4492075590 1 0 - ' yJ7 >
dyl9 :- (0in4"(Xin4-yl9)/(yl8) 4
>
(yl9/yl8).pl4.(Kev4.(L)•a/hevap4)>((Kv.Kc4«Tsat6/(Kw4Kc4)+Kev4.Tbl4)/(
> > >
Kev44Kw-Kw-2/(Ku+Kc4))-Tbl4)): dyl91in : mtaylor(dyl9, [y4-y4n.y9-y9n,yl5=yl5n,yl8-yl8n,yl9-yl9n,y21-y21n,y23-y2
>
3n.y29-y29n],2); dyiglin
: = .001305700549 -|- .0001018699762 y.( -f .001669557240 yS
4- .008781085988 yJ5 + .06214453356 y2J - .07338375398 yJ9 + .21994151 1 0 - * yl8 •\- .2203985775 1 0 - ' y29 - . 3 3 4 0 3 1 1 3 9 1 1 0 - ^ y2S >
dy25 :- (Gin5«(Xin5-y25)/(y24) +
>
(y25/y24)-pl5.(Kev5«(L)«a/bevap5).({Kw«Kc5«(3.9487 +
>
17.711»ln(u2/1000)-0.24263«ln(u2/1000)-2+0.19674»ln(u2/1000)-34273.15)
>
/(Kw4Kc5)+Kev5«Tbl5)/(Kev5+Kw-Kw-2/(Kw+Kc5))-Tbl5));
>
dy251in :-
> >
mtaylor(dy25,[y4-y4n,y9=y9n,yl5-yl5n,y21=y21n,y24-y24n,y25-y25n,y27=y2 7n,y29=y29n,u2-u2n].2); dy25lin
:= .002446007754 -I- .07226195361 y27 -I- .00001514247835 y4
+ .0002481715938 y9 + .001305265883 yl5 4- .009237483790 »2J - .08545026195 y25 -F .3872921410-° y24 4- .1056954992 1 0 " ' u2 - .1821474478 1 0 - ' y29 >
dy31 ;- (Gin6.(Xin6-y31)/(y30) 4
>
(y31/y30)*pl6*(Kev6*(L)*a/hevap6)*((Kw*Kc6*(3.9487 4
>
17.711*ln(u3/1000)-0.24263*ln(u3/1000)-240.19674*ln(u3/1000)-3+273.15)
> > >
/(Kv4Kc6)4Kev6«Tbl6)/(Kev6+Kw-Kw-2/(K»+Kc6))-Tbl6)): dySllin : • mtaylor(dy31.[y4-y4n.y9-y9n,yl5-yl5n,y21-y21n.y27=y27n,y29«y29n,y30=73
>
0n,y31-y31n,y33=y33n,u3=u3n],2): dySlhn
:= .005847498477+ .04373767164 ySS -I- .005312180785 y27
+ .1113166451 10-° y4 + .00001824379643 y9 -I- .00009595379026 yJ5 4- .0006790735844 y2J - .05781927408 y5J - .335133 1 0 " ' " ySO -I- .9555937782 10-° uS - .1658290203 1 0 " ^ y29 >
dy36 ;= (0in7»(Xin7-y36)/(y35) +
>
(y36/y36)»pl7»(Kev7<(L)*a/bevap7)«({Kw«Kc7»(3.94a7 +
> >
l7,711.1n(u4/1000)-0.24263»ln(u4/1000)-2+0.19674«ln(u4/1000)-34273.15) /(Kw4Kc7)4Kev7»Tbl7)/{Kev7+Kw-Kw-2/(Kw+Kc7))-Tbl7)):
295
>
dy361in ;-
>
mtaylor(dy36, [y4-y4n,y9-y9n.yl5-yl5n.y21-y21n,y27-y27n,y29-y29n,y30-y3
>
0n,y31-y31n,y33-y33n,y35-y35n,y36-y36n,y38-y38n,u4-u4n] ,2) ;
dySelin
:= .003395351142 + .003203698862 y5S + .0003891068478 y 2 7
4- .8153726437 lO-'^ y^ + .133632239010-'' y9 -t- .7028427378 1 0 - ^ ylB 4- .00004974081130 y2J + .02315339168 yS« - .03144812364 y56 + .5857079 1 0 - ' ° ySS Gtini
- Goutl 4 Gfeed:
Gfeedl Gtin4 Xtinl
+ .3921174648 10-** u4 - .6861010248 1 0 - * y29
Gtin2 :- Gout2 +
Gtin3 :- Gout3: - Gout4;
Gtin5 :- GoutS:
Gtin6 :- Gout6:
Gtin7 :- Gout7:
- (Goutl*y24Gfeed*Xfeed)/Gtinl: - (Gout2«y7+0feedl»Xfeedl)/0tin2: >
yl9;
>
y25:
Xtin3 :- yl3:
Xtin4 :-
Xtin5 ;-
>
Xtin6 ;- y31:
>
Ttinl ;- (Gfeed*Tfeed +
Xtin7 ;- y36:
>
Goutl"Tbll)/Gtinl:
>
Gout2«Tbl2)/Gtin2:
>
Ttin3 ;- Tbl3:
>
Tbl5;
Ttin2 :- (Gfeedl'Tfeedl 4
Ttin4 :- Tbl4:
Ttin6 :• Tbl6;
Ttin5 ;-
Ttin7
>
;- Tbl7:
>
rbot01:=1007.4 - 0.495"(Tinl - 273.15) •
>
600«Xtinl;
>
0.496«(Tin2 - 273.15) + 600.Xtin2:
>
rbot03:-1007.4 - 0.495*CTin3 - 273.15) +
rhot02:-1007.4 -
>
600»Xtin3;
>
0.495*(Tin4 - 273.15) 4 600*Xtin4:
rbot04:-1007.4 -
>
rhot05;-1007.4 - 0.495«(Tin5 - 273.15) +
>
600*Xtin5:
> >
0.495*(Tin6 - 273.15) + 600*Xtin6: rhot07:-1007.4 - 0.495*(Tin7 - 273.15) + 600*Xtin7:
>
rhotl;-1007.4 - 0.495*(y5 - 273.15) +
rhot06:-1007.4 -
>
600*y4:
>
0.495»(yl0 - 273.15) 4 600«y9;
rhot2:-1007.4 -
>
rbot3:-1007.4 - 0.495»(yl6 - 273.15) *
>
600*yl5:
>
0.495»(y22 - 273.15) 4 600»y21:
>
rhot5:-1007.4 - 0.495»(y28 - 273.16) +
>
600*y27;
> >
0.495*(y34 - 273.15) + 600«y33: rhot7:.1007.4 - 0.495«(y39 - 273.15) + 600.y38:
>
Gtoutl :- Gpumpl:
>
Gboutl;
Gtout3 :- Gpump3 -
>
Gbout2;
Gt0Tlt4 :- Gpump4 - Gbout3:
>
GtoutS ;= Gpump5 - Gbout4 + GboutS;
rhot4:-1007.4 -
rhot6:-1007.4 -
Gtout2 :- Gpump2 -
>
Gtoute ;= Gpump6 - GboutS 4
> >
Gbout6: Gtout7 ;- Gpump7 - Gbout6; dy4 ;- (Gtinl«(Xtinl - y4)/(Ac»rliotl)):
>
dy41in ;. mtaylor(dy4,[yl-yln,y2-y2n,y4=y4u,y6-y5n,ul-uln].2);
dy4lin
:= .0003500256197 + .02055535709 y2 -
.1682995620 1 0 " " ul
+ . 7 8 7 1 3 7 8 1 8 0 1 0 - ^ yl - .02442277403 y.( + .3342213811 1 0 " ^ yS >
dy9 :- (Gtin2«(Xtin2 - y9)/(Ac»rhot2));
>
dy91in ;- „taylor{dy9, [y6-y6n.y7.y7n.y9.y9n.yl0-fl0n,yll-ylln] .2);
296
dy91in
:= .0003425465190 + .03529394737 y 7 - . 8 4 1 0 1 9 2 7 6 7 1 0 - ' "
yll
-H .5836158006 1 0 " ^ yfi - .03914780618 y9 - .3573504048 1 0 " ' ylO
> >
dyl6 : - (Gtin3»(Xtin3 - yl6)/(Ac"rhot3)); dylSlln :-
>
mtaylor(dyl6,[yl2-yl2n,yl3-yl3n,yl5-yl6n,yl6-yl6n,yl7-yl7n],2); dylSHn
:= .02281295040yI5 -
> >
dy21 :- (0tin4«(Xtin4 - y21)/(Ac«rhot4)): dy211in : -
^
mtaylDr(dy21.Cyl8-yl8n,yl9-yl9n,y21-y21n,y22-y22n,y23-y23n] .2) ;
> >
dy27 ;- (Gtin5»(XtinB - y27)/(Ac»rhotS)): dy271in :-
>
mtaylor(dy27, [y24-y24n,y25-y25n,y27-y27n.y28-y2an,y29-y29n] ,2) ;
dySlUn
.02281295040yJS
;= 01942016058 y i 9 - .01942016058 y2J
dy27lin := .01430197150 y25 - .01430197150 y27 > >
dy33 :- (GtinS-(XtinS - y33)/{Ac»rhot6)); dy331in :-
>
mtaylor(dy33. [y29-y29n,y30-y30n.y31-y31n,y33-y33n,y34-y34n] ,2) : dySSlin .= .01071975221 yJJ - .01071975221 y i S
> > >
dy38 : - (Gtin7»(Xtin7 - y38)/(Ac*rhot7)): dy381in : mtaylor(dy38, [y29-y29n,y35-y35n,y36-y36n,y38=y38n,y39-y39n] .2) ;
>
dy5 :- ((Gtinl»Ttinl*cpl -
>
Gtoutl*y5*cpl)/(Ac*rhotl*cpl)
dy38Un
:= .007599453546 ySS - .007599453546 yj«
(y5*(Gtinl - Gtoutl)/(Ac*rhotl))
>
((y5)/cpl)*(3.822»y5-3322.67)»(Gtinl«{Xtinl
>
y4)/(Ac»rhotl)))«(cpl/(cpl 4 3.822*y6»y4)):
>
dySlin :- mtaylor(dy5, Cyl-yln,y2=y2n,y4=y4n,yS-ySn,ul-uln] ,2) :
dyShn
:= 7.075366564 + 3.641466247 y2 + .00001945594909 ul + .00004443758676 yl
- .02272964074 y5 - 4.231102010 y^
>
dylO : - ((Gtin2*Ttin2*cp2 -
>
Gtout2*ylO*cp2)/(Ac»rbot2*cp2)
>
(ylO*(Gtin2 - Gtout2)/(Ac.rliot2))
((yl0)/cp2)»(3.822»yl0-3322.67)»(Gtin2«(Xtin2 -
> >
y9)/(Ac»rhot2)))»(cp2/(cp2 + 3.822«yl0»y9)); dylOlin ; -
>
mtaylor(dylO, [y6-y6n,y7-y7n,y9=y9n,yl0-yl0n,yll=ylln] .2) : dylOlin : = 11.66642519 + 6.357635266 y7 + .00002020590792 yll + .00004352515937 y6 - .03633391475 yJD - 6.843347540 y9
>
dyl6 ;= ((Gtin3*Ttin3*cp3 -
>
Gtout3»yl6»cp3)/(Ac»rhot3«cp3) - (yl6»(Gtin3 - Gtout3)/{Ac»rhot3))
>
((yl6)/cp3)*(3.822*yl6-3322.67)»(Gtin3*(Xtin3 -
>
yl5)/(Ac»rhot3)))•(cp3/(cp3 + 3.822«yl6*yl5));
> >
dylSlin ; mtaylor(dyl6,[yl2-yl2n,yl3-yl3n,yl5=yl6n,yl6-yl6n.yl7-yl7n],2)i dyielin
: = 6.825559911 + 4.358579553 y l 5 + .8047148055 1 0 - ^ y J 7
- . 5 0 0 8 2 6 0 8 5 0 1 0 - ^ y l 2 - .02041900728 vJ6 - 4.064588546 y J 5 >
dy22 ;= ((0tin4«Ttin4»cp4 -
>
Gtout4-y22»cp4)/(Ac»rhot4.cp4)
>
- ((y22)/cp4)»{3.822«y22-3322.67)«(Gtin4«(Xtin4 -
(y22.(Gtin4 - 0tout4)/(Ac.rhot4))
>
y21)/(Ac.rhot4)))«(cp4/(cp4 + 3.322»y22.y21));
297
> >
dy221in ; mtaylor(dy22.[yl8-yl8n,yl9-yl9n.y21-y21n,y22-y22n,y23-y23n],2); dy22lin := 5.611870850 -I- 3.869533132 yJ9 + .4019402614 1 0 - ° y23 - .1879287265 1 0 - ° yl8 - ,01630095061 y22 - 3.512803682 y21
>
dy2S :- ((Gtin5.Ttin5*cp5 -
>
GtoutS*y28.cp5)/(Ac*rhot5*cp5) - (y28*(Gtin5 ~ GtoutB)/(Ac*rhotS))
> > > >
((y28)/cpS)«(3.822«y28-3322.67).(Gtin6«(Xtin5 y27)/(Ac»rhotS)))»(cp5/(cp5 + 3.a22.y28»y27)): dy281in : mtaylor(dy28,[y24-y24n,y26-y2Sn,y27-y27n,y28-y28n.y29-y29n].2)i
dy28lin
:= 3.879282745-F 3.018434417 y2S 4- .1752203488 1 0 - ^ y29
+ .1447790319 1 0 - * y2.< - .01111175042 y2S - 2.590176612 y 2 7
>
dy34 ; - ((Gtin6*Ttin6*cp6 -
>
Gtout6*y34«cp6)/(Ac*rhot6*cp6) - (y34*(Gtin6 - Gtout6)/(Ac*rhot6))
>
((y34)/cp6)»(3.a22*y34-3322.67)»(Gtin6>(Xtln6 -
> >
y33)/(Ac.rhot6)))*(cp6/(cp6 + 3.B22.y34"y33)); dy341in : -
>
mtaylor(dy34,[y29-y29n,y30-y30n,y31-y31n,y33-y33n.y34-y34n],2); dyS4lin : = 2.528888357 + 2.531704678 yW -I- . 1 2 2 6 4 0 5 4 3 7 1 0 - * y29 4- .7415241923 1 0 " ' ySO - .007778028536 jJ.J - 1.942710398 y55
>
dy39 ; - ((Gtin7.Ttin7»cp7 -
>
Gtout7.y39*cp7)/(Ac.rhot7«cp7)
> > > >
(y39*(Gtln7 - Gtout7)/(Ac"rhot7))
((y39)/cp7)«(3.822.y39-3322.67)»(Gtin7*(Xtin7 y38)/(Ac.rhot7))).(cp7/(cp7 4 3.822«y39.y3a)): dy391in ; mtaylorCdy39,[y29=y29n.y35-y35n,y36-y36n,y38-y38n.y39=y39n],2): dy39Un := 1.655990567+ 1.914027680 yJ6 + .8494399618 1 0 - * y29 - .3740020638 1 0 - " ' y S 5 - .005384428758 y59 - 1.373817363 y5«
>
>
Ul
:-
(Kevl'(L).a«pll/hevapl)«((K»»Kcl«T3at2/(Kw4Kcl)+Kevl«Tbll)/(Kevl*Kw-Kw
>
-2/(Kw+Kcl))-Tbll):
>
H2
>
:-
(Kev2*(L)*a«pl2/hevap2)*((Kw*Kc2*Tsat3/(Kw4Kc2)4Kev2*Tbl2)/CKev24Kw-Kw
>
-2/(Kw4Kc2))-Tbl2):
>
U3 ;-
>
(Kev3»(L)•a»pl3/hevap3)»((Kw»Kc3"Tsat4/(Kw+Kc3)+Kev3»Tbl3)/(Kev3+Kv-Kw
>
-2/(Kw4Kc3))-Tbl3):
>
W4 :-
>
(Kev4»(L)"a.pl4/hevap4)*((Kw»Kc4«Tsat5/{Kw4Kc4)4Kev4«Tbl4)/(Kev4+Kw-Kw
>
-2/(Kw4Kc4))-Tbl4):
>
US :-
>
(KevS»(L)»a'plS/hevap6)»((Kw»Kc5.(3.9487 +
>
17.711.1n(u2/1000)-0.24263*ln(u2/1000)-240.19674«ln(u2/1000)-34273.15)
>
/(Kw4Kc5)4Kev5«Tbl5)/(KevB4Kw-Kv-2/(Kw4Kc6))-Tbl5):
>
U6 ;-
>
(Kev6«(L)»a»pl6/hevap6)«((Kv»Kc6*(3.9487 +
>
i7.7ll.ln(u3/1000)-0.24263«ln(u3/1000)-240.19674»ln(u3/1000)-3+273.15)
>
/(K«4Kc6)4Kev6.Tbl6)/(Kev64Ky-Kw-2/(Kv4Kc6))-Tbl6);
>
U7 :-
>
(Kev7.(L).a-pl7/bevap7)»((Kw.Kc7.(3.94a7 4
>
17.711.1n(u4/1000)-0.24263»ln(u4/1000)-2+0.19574»ln(u4/1000)-34273.16)
>
/(Kw4Kc7)4Kev7«Tbl7)/(Kev7+Kw-Kw-2/(Kw4Kc7))-Tbl7);
>
Ucl ;- Ul.hevapl/(2812200 - 349.98.Tsat2 - 3.034.(Tsat2-2)):
298
>
Uc2 ;- H2«bovap2/(2812200 - 349.98.Taat3 - 3.034«(Tsat3-2));
>
Uc3 :- W3»hevap3/(2812200 - 349.98.TBat4 - 3.034«(Tsat4-2));
>
Uc4 :- W4>hoiap4/(2812200 - 349.98.Tsat5 • 3.034»(Tsat6-2)):
>
WcS ;- W5«hevap6/(2812200 -
>
349.98.(3.9487 +
> > >
17.711.1n(u2/1000)-0.24263»ln(u2/1000)-2+0.19674»ln(u2/1000)-3) 3.034*((3.9487 4 17.711.1n(u2/100O)-0.24263«ln(u2/100O)-240.19674«ln(u2/100O)-34273.15)
>
-2));
>
Uc6 :- U6-bovap6/(2812200 -
>
349.98»(3.9487 •
> > >
17.711«ln(u3/1000)-0.24263»ln(u3/1000)-2+0.19674»ln(u3/1000)-3) 3.034»({3.94S7 + 17.711.1n(u3/1000)-0.24263«ln(u3/1000)-240.19674»ln(u3/1000)-34273.16)
>
-2)):
>
Uc7 :- U7«hBvap7/(2812200 -
>
349.98* (3.9487 +
> > >
17.711»ln(u4/1000)-0.24263»ln(u4/1000)-240.19674.1n(u4/1000)-3) 3.034.((3.9487 4 17.711.1n(u4/1000)-0.24263«ln{u4/1000)-2+0.19674»ln(u4/1000)-3+273.I5)
>
-2));
>
dyll :- (W2-Wcl)»R*(Tbl24Tsatl)/(2"H.V2):
>
dylllin :-
>
mtaylor(dyll, [y2-y2n,y7=y7n,yll=ylln,yl7-yl7n,ul-uln] ,2) ;
dylllin
:= - 1 4 1 1 . 5 9 5 1 2 1 - 21939.40106 y 7 - 1.741495870 y l l + 1.419688293 ul
> >
+ .5591625114 y J 7 + 26534.18253 y2 dyl7 :- (U3-Uc2)«R«(Tbl34Taat2)/(2»H»V3): dyl71in : -
>
oitaylor(dyl7, [y7-y7n,yll-ylln,yl3=yl3n,yl7=yl7n,y23-y23n] ,2) ; dynhn
: = 1912 218789 - 23234.82822 yJS - .9725000415 yJ7 + .7992333911 y l l
+ .2875389213 y23 + 20540.03877 y7 >
dy23 := (U4-Uc3)»R«(Tbl44T6at3)/(2.H.V4):
>
dy231in : -
>
mtaylor(dy23.Cyl3-yl3n.yl7-yl7n,yl9=yl9n,y23=y23n,y29-y29n].2); dy23hn
:= 6940.794186 - 31514.67300 yJ9 - .4806113365 y2J + .3656276542 y l 7
+ .1617310647 v29 + 18928.87074
>
dy29 :- (U54U64U7-Uc4)»R«(Tbl5+Tsat4)/(2«H.V5):
>
dy291in ; -
ylS
>
mtaylor(dy29.[yl3-yl3n,yl7«rl7n,yl9-yl9n,y23-y23n.y25-y25n,y31-y31n,y3
>
6=y36n,y29=y29n,u2=u2n,u3=u3n,u4=u4n],2); dy29hn
:= 32387.26637 - 16771.66175 y2S - .2764099638 y29 + .2077223744 y23
- 30493.38398 y j l + .03156874297 u J + .03253582442 u2 + 26599.38652 yJ9 - 20324.32384 v5e + .01618212326 u.i
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APPENDIX E OCFE DISCRETIZATION
The discretization scheme is presented on Figure E.l. The plate length is normalized to value of one and is divided in 10 finite elements. Each finite element has two internal collocation points.
0,0211 I—0
0.D768 o—)—0
osZU 1—0
0—1
0.978B o—I
Figure E.l: OCFE discretization of the spatial domain.
As all partial differential equations have the same form, here only the discretization of the black liquor mass flow rate equation is provided
^ + y*^ = -W'*v:V' dt* ^ dz* ^'
(E.l) ^ '
The transformation from partial differential equation to ordinary differential equations in OCFE method is done by expanding the spatial derivatives as polynomials at each internal collocation point for every element. For K elements with equal length hK, and N internal collocation points, according to [67], on every A;*'' element, for i =
307
2 to N-M
^^k,i
_
Vzk,i V ^
,
^ ,
,
dt*
(E.2)
In the studied case N = 2, therefore we have
— V z,k,2
N+2
/
dt*
IK
dGh
-V. z,k,3
dt*
hK
. ^2jGfc 2
K:2v:,k,2 (E.3)
N-{-2
E^3.G':,3-H^C3
where A is a matrix of coefficients, see Finlayson [67]. To assure continuity of the solution, on all adjacent elements: N+2 1 / , ^N+2jGj
i=i
J
•Ar-1-2
= E^ii^^=
i=i
k-l
where A; is the index of the element. Therefore each element gives rise to two ordinary differential equations and each two adjacent elements give rise to one algebraic equation. For 10 finite elements this is 20 -h 9 = 29 equations. The initial and boundary conditions determine the equations at the initial and the last point of the interval. In the studied case we have
dt* dGlo^ dt*
=
0 —V 2,10,4
(E.4)
N+2 10,4 "
HK
308
^10,4^z,10,4)
which leads to total number of 31 algebraic and differential equations per partial differential equation.
309
