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 mul­

tivariate 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 under-

atanding 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 . . . 9

1.9 Falling film evaporators for black liquor concentration 10

1.10 Transport phenomena in falling films 11

1.11 Disaertation organization 12

II MODELING OF A SINGLE PLATE 14

2.1 Physical conditions 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 films 18

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

2.4 Heat tranafer coefficients of wavy-laminar films 25

2.4.1 Transition from wavy-laminar to turbulent flow 25

2.4.2 Heat transfer coefficient of evaporation 27

2.4.3 Heat tranafer coefficient of senaible heating 32

2.5 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 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

2.6 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

3.9.1 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

3.10 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 129

4.6 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 . . . 139

4.6.3 A linear MPC deaign 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 154

4.9 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 180

5.3.1.1 Mass balance 180

5.3.1.2 Energy balance 189

5.3.2 Model vahdation 191

5.4 Nomenclature 192

viii

VI CONTROL OF MULTIPLE EVAPORATOR PLANT 193

6.1 Decentrahzed control 193

6.1.1 Evaporator inventoriea 193

6.1.2 Selection of controlled and manipulated variablea 196

6.1.3 Reaulta 197

6.2 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 . . . 278

D.3 MAPLE code for linearization of single evaporator nonhnear ODE model 289

D.4 MAPLE code for linearization of multiple evaporator nonlinear

ODE model 291

E: OCFE DISCRETIZATION 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 regen­

erating 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 evap­

orator 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 charac­

terized 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 ex­

periment 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. How­

ever, in the case of the entire evaporator plant, the advanced control approach of

model-predictive control (MPC) provides better control becauae the MPC central­

ized 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 cloaed-

loop aystem to meaaured and unmeasured disturbancea and parameter uncertainty.

Xll

LIST OF TABLES

2.1 Nominal parameters with respect to aingle plate 17

2.2 Predicted Reynolda numbers at the transition points for black hquor

at nominal conditions 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 Variable Selection and Clasaification 107

4.2 Control Loop Parametera 112

4.3 Evaporator nominal operating conditiona 113

4.4 Featurea of the proceaa closed loop reaponaes 119

4.5 Correlationa for Turbulent and Wavy-laminar regimea 136

4.6 Evaporator nominal operating conditiona 149

4.7 Tuning parametera for MPC 155

4.8 Tuning parameters for the PI controllers 155

5.1 Operating conditions of the multiple evaporator plant 179

5.2 Senaitivity of the dry aolids concentration 188

5.3 Changea in steam economy aa a function of the diaturbance 190

6.1 Operating conditiona of the multiple evaporator plant 199

6.2 Performance of the single-loop control atructure of the nonlinear mul­

tiple evaporator plant 200

6.3 The effect of boiling point elevation to a 5% decrease in the feed dry

solids concentration 209

6.4 Timea to reach la hmits of 0.001 kg/kg of the multiple evaporator

plant product dry solids concentration 218

6.5 Tuning parameters of the MPC and PI controllers for multiple evapo­

rator plant 220

6.6 Integral abaolute error of the controlled variablea 221

Xll l

B.l Dimensionleaa Variables 270

D.l Matrix A, columna 1 to 10 .

D.2 Matrix A, columns 11 to 20

D.3 Matrix A, columns 21 to 30

D.4 Matrix A, columna 31 to 32

D.5 Matrix B

D.6 Matrix C

D.7 Eigenvalues of Matrix A . .

300

301

302

303

304

305

306

XIV

LIST OF FIGURES

1.1 Schematic of a kraft pulp mill recovery cycle 13

2.1 Eddy diffuaivity of momentum for a falling water film 22

2.2 Heat transfer coefficient of wavy-laminar faUing fllm 31

2.3 Experimental deaign 34

2.4 Experimental aetup to determine flow distribution 47

2.5 Tracer patterna for water at 24°C, Re = 3557 48

2.6 Tracer patterns for water at 60°C, Re = 6970 48

2.7 Tracer patterna for SCMC at 24°C, Re = 0.699 48

2.8 Tracer patterns for SCMC at OO C, Re = 1.298 48

2.9 Preaence or absence of a heating zone 49

2.10 Differential volume of the evaporating film 50

2.11 Differential volume of the heating film 51

2.12 Differential volume of the condensing film 52

2.13 Wall differential volume element 56

2.14 The responae of the single plate evaporator to a 5% decrease in dry aolida concentration 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

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 I l l

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 Evaporator reaponae to an increase of 10% in the dry solids concentration aet point 118

4.7 Nonlinear ODE model response to a 5% decrease in the feed flow rate. 140

4.8 Nonlinear ODE model response to a 5% increase in the feed flow rate. 140

4.9 Nonlinear ODE model response to a 5% increaae in the feed dry aolids concentration 141

4.10 Nonlinear ODE model reaponse to a 5% decrease in the feed dry solids concentration 141

4.11 Nonlinear ODE model reaponae to a 5% decrease in feed temperature. 142

4.12 Nonlinear ODE model responae to a 5% increase in feed temperature. 142

4.13 Nonlinear ODE model responae to a 5% increase in vapor preasure. . 143

4.14 Nonhnear ODE model reaponse to a 5% decrease in vapor preaaure. . 143

4.15 Nonhnear ODE model reaponse to a 5% increase in ateam preasure. . 144

4.16 Nonhnear ODE model reaponae to a 5% decrease in steam preaaure. . 144

4.17 Responses of the linear (LODE) and nonhnear (NLODE) ODE models to 5% decreaae in the steam pressure 150

4.18 Reaponaea of the linear (LODE) and nonlinear (NLODE) ODE models to 5% increase of the ateam preaaure 151

4.19 Reaponaes of the linear (LODE) and nonlinear (NLODE) ODE modela to 5% decreaae of the vapor pressure 152

4.20 Reaponsea of the linear (LODE) and nonlinear ODE modela to 5% increase of the steam preaaure 153

4.21 A feedback block diagram with the MPC controller 154

4.22 MPC performance in the presence of a 5% increase in the feed maaa flow rate 160

4.23 MPC performance in the preaence of a 5% decreaae in the dry aohda concentration of the feed 161

XVI

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 169

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 Multiple falling fllm evaporator plant 176

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

5.3 Open-loop reaponae of the PDE model of multiple evaporator plant to a 5% increase in the flow rate of the feed 182

5.4 Open-loop responae of the PDE model of multiple evaporator plant to a 5% decreaae in the dry aolids concentration of the feed 183

5.5 Open-loop reaponse of the PDE model of multiple evaporator plant to a 5% increeise in the dry aolida concentration of the feed 184

5.6 Open-loop reaponae of the PDE model of multiple evaporator plant to a 5% decreaae in the temperature of the feed 185

5.7 Open-loop reaponae of the PDE model of multiple evaporator model for 5% increase in the feed temperature 186

5.8 Open-loop reaponae of the PDE model of multiple evaporator plant to a 10% decreaae in the heat tranafer of the superconcentrator 187

5.9 Open-loop reaponse of the PDE multiple evaporator model for 10% decrease in the heat transfer of the superconcentrator 189

6.1 Syatem of three connected tanka 194

6.2 Decentrahzed feedback control atructure for the multiple evaporator plant 198

xvii

6.3 Closed-loop reaponae of the nonlinear multiple evaporator plant to 5% increase in the flow rate of the feed 200

6.4 Cloaed-loop responae of the nonlinear multiple evaporator plant to a 7% decreaae in the temperature of the feed 201

6.5 Cloaed-loop reaponae of the nonlinear multiple evaporator plant to a 5% decreaae of the dry aohda concentration of the feed 202

6.6 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

6.7 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

6.8 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

6.9 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 207

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 208

6.12 Reaponsea of the nonhnear (aohd line) and linear (dotted line) ODE models to a 5% decrease in the auperconcentrator steam pressurea. 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-1 214

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 ; 215

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-5 216

6.17 Responses of the nonhnear (sohd line) and linear ODE models to a 5% increase in the secondary vapor pressure of evaporator E-5 217

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

XVll l

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 229

6.26 Cloaed-loop performance of conatrained MPC controller in the preaence of a feed mass flow rate diaturbance 230

6.27 Constrained MPC control actions in the presence of a feed mass flow rate diaturbance 231

6.28 Cloaed-loop performance of constrained MPC controller in the preaence of a feed dry solids concentration disturbance 232

6.29 Constrained MPC control actiona in the preaence of a feed dry solids concentration diaturbance 233

6.30 Cloaed-loop performance of conatrained MPC controller in the presence of a feed temperature disturbance 234

6.31 Constrained MPC control actiona in the presence of a feed temperature diaturbance 235

6.32 Cloaed-loop performance of constrained MPC controller in the preaence of a heat tranafer diaturbance 236

6.33 Conatrained MPC control actions in the presence of a heat tranafer disturbance 237

6.34 Closed-loop performance of the PI controUera in the preaence of a feed mass flow rate disturbance 238

6.35 PI control actions in the presence of a feed mass flow rate disturbance. 239

6.36 Closed-loop performance of the PI controUera in the presence of a feed dry solida concentration disturbance 240

6.37 PI control actiona in the presence of a feed dry sohda concentration diaturbance 241

6.38 Cloaed-loop performance of the PI controllers in the preaence of a feed temperature disturbance 242

6.39 PI control actions in the preaence of a feed temperature disturbance. 243

6.40 Closed-loop performance of the PI controUera in the presence of a heat transfer diaturbance 244

6.41 PI control actions in the presence of a heat transfer diaturbance. . . . 245

XIX

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. 307

XX

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 theoreti­

cal, 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 con­

centrate 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 dia-

tillation, 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 compu­

tation times and relative simplicity. However, since the evaporator system is a dis­

tributed system, the lumped parameter model may not provide an accurate descrip­

tion 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 sec­

tion 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 ap­

proach. 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 well-

mixedness 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 pro­

duction, 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 car­

bonate. 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 vis­

cosity 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 concen­

tration. 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]. Concen­

trated 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 major­

ity of the industrial applications involve turbulent falling films, new experimental and

theoretical resulta appeared that cover both turbulent and wavy-laminar regimes. Im­

portant 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 experimen­

tal 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 rep­

resent 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 pa­

rameters 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 approx­

imately 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 de­

termined 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 num­

ber 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

5 4 3 2

SC3 SC2 SCI

P Pa ^ vap •) ^ ^

21372 40269 65768 115870 204996 204996 204996

T K

337 351 364 381 402 407 411

Gpi,kg/s

2.7 4.3 2.9 2.3 4.5 3.5 2.5

Xpi,kg/kg

0.2261 0.2152 0.2716 0.3804 0.5144 0.6527 0.7020

Re

2022 4793 2710 1297 1100 102 25.5

Pr

27.01 17.89 21.01 33.19 73.48 605 1697

Ka

9.2X-S 1.9X-8 3.9X-8

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 be­

havior, 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 Heat transfer coefficients of turbulent faUing films

2.3.1 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

dy

dvx 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

(2.11) - = k'y^'DlF' dy^

An expression for the velocity gradient is obtained by using the bulk flow mo­

mentum 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 — <C 1. It follows that the expression for the eddy diffusivity

of momentum in the wall boundary layer and the turbulent core is then given by [43]

l = _ l + i y r f 4 ^ 2 ^ T 2 ^ 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 as­

sumption 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 < y + < y +

- = { , , (2-16) ^ 1 l.mxW-'^Re'- f5^-y^\ y+<y+^s+

Ka ^+i V At

21

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 Fig­

ure 2.1. The experimental research of Alhusseini et al. [43] confirms that the proposed

50 100 150 Dimensionless film thickness. 5^

250

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+ 1

(2.17)

-'r Prt ^u)

dy'

Pr

where h% is the dimensionless heat transfer coefficient of evaporation and Prt, the

turbulent Prandtl number, is defined by [43],

Prf.=

- 1

(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% = (2.19)

2.3.3 Heat transfer coefficient of condensation

The expression to determine the heat transfer coefficient of condensation, accord­

ing 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 bound­

ary 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

r-2/+ 1 _ i i u+ = / YT^de. (2.21)

2.3.5 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 f^^ Re ^4- =4 / u+(iy+ (2.22)

1^ Jo

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 Heat transfer coefficients of wavy-laminar films

2.4.1 Transition from wavy-laminar to turbulent flow

With any change in the hquor properties (e.g., increase in the dry sohds compo­

sition) 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 even-

tuaUy 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 predic-

25

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 Pr Ka Ret{Pr) Ret{Ka)

5 27.01 9.206X-8 288.7 47.615 4 17.89 1.939X-S 377.34 80.029 3 21.01 3.944X-8 339.91 63.163 2 33.19 2.661x-^ 252.51 33.426

SC3 73.48 6.969X-6 150.63 11.256 SC2 605 3.263X-2 38.264 0.673 SCI 1697 2.034 19.57 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 expres­

sions 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 develop­

ments 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. In­

deed, 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 / - pX -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 so­

lutions 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 equa­

tions 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 prob­

lem. 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. Research by [54, 55]

29

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 computa­

tionally 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 demand­

ing 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

1.35

o Miyara's data Power law fit, h*/h = 1.431Pr"

20 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 evapo­

ration. Heating without evaporation of wavy films is not well studied either exper­

imentally 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 theoreti­

cally 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^. (2.32)

2.5 Modeling of single plate

2.5.1 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 signifi­

cant 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

Temperature sensor

Lip actuators

Temperature controller

Yty- Relay

~ Power, 12 V, 10 A 0

Heating element

Tin plate

J

Rotameter F

Manual valve

Centrifugal pump

Figure 2.3: Experimental design.

34

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 distribu­

tor 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 carboxymethylcel-

lulose (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 one-

dimensional 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. De­

pending 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 boil­

ing 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. (2.33)

37

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 sig­

nificantly due to friction losses.

According to Perry [65] section 11-110, the pressure drop in a falhng film evapo­

rator 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)) - Gyj \^+AZ [K J^+A.^ +\V'^

+ g{z + Az)) -W{K + Ivl, + g{z + Az)) ,

2

(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, j z + A z —Why.

39

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 n^n \ (h \ h \ \ ^ ^M^y = Q + Lrs \z [ris \z -IT'S |z+Azj H -rr-h w \z dt " ' ^ '^ ' ' ' ^ '^^"^^ ' dt

+ Gyi j z + A z \hw \z —hy, j z + A z ) ~ W^ [hy — hy, | z ) .

(2.37)

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 (2.38)

Gy) j z + A z (Cp . t i ; - / \z —Cp^yjl I z + A z j ~ ^ {'T'V " hyj | z j ,

40

where Cp^s, Cp^w are the specific heat capacities of the sohds and water, respec­

tively 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 dt

= Q- WAHey. (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). (2-41)

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

Vz ' (2.43)

Substitution of the above equation into Equation (2.42) gives

dG AzdG _, , _, . ^j. ,^ ^^. T^ = — ^ = G z - G z+Az -W, 2.44

at Vz ot

42

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. 2.45 Vz dt Az ^ ^

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—^ = {GXs) \z -{GXs) Iz+Az . (2.48) Vz ot Vz ot

Dividing Equation (2.48) by Az and taking the limit as Az -^ 0 gives

G9x.^x,sa_t^sx,^^SG: ^ ^g, Vz dt Vz dt V dz dz

Substituting Equation (2.46) into the above yields

dXs , dXg „,, Xs , . — +y^^=WVz^. (2.50)

43

To summarize, the mathematical model that describes the evaporation

process is given by

+ Vz^r- = -W'vz dG dG 'dt^'^'Tz

(2.51) ax dXs .^., Xs dt dz G

2.5.2.2 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 be­

cause 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, Equa­

tion (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. (2.56)

45

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 ^ (2.59) ot oz pocp^i

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

72

344

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 '

h ^ '•

- • 1 '

i t-

7

Figure 2.5: Tracer patterns for water at 24°C, Re = 3557. . . . . . ^

y 1

v ),wr

: >

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

Condensation zone

Steam

13 :l' Liquor | inlet

Vapor

Flash zone ^ \

Evapqrator wall

Heating zone

Evaporation * - zone

. ^ Vapor

I Condensation " ^ zone

Steam

Liquor inlet

Evaporator wail I

• - Evaporation zone

Vapor

Plate Plate

Figure 2.9: Presence or absence of a heating zone.

49

X

^ W , Z ) ^ S , Z 5 V2

A

plate

r 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

+ i^z + 9{^ + A^)) + W, {hst + \vlz + g{z + Az))

(2.60)

Neglecting the kinetic and potential terms, the above balance becomes

dEdy

dt = -Q + Gc \z he \z -Ge Iz+Az he |z+Az +Wehst. (2-61)

5 2

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 _ ^ , dMdy^ 1 , ^ 1 ,u \ u \ \ J, — -~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) , (2-64)

dt

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 Iz+Az +W^c- (2.68) dt

54

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 Vz

Substitution in Equation (2.68) gives

dGe Az dGe

Defining

W', = We/Az

and dividing Equation (2.70) by Az gives

Vz dt Az

Taking the limit as Az -> 0 yields

(2.69)

T ^ - = ^ = Ge \z -Ge Iz+Az + W, . (2.70) dt Vz ot

1 dGe Gc \z —Ge \z+Az_ y^f (^ 71)

^ + , ^ ^ = VF>z. (2.72) dt dz

55

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) (2.75)

hc{Tf,e-Ty,^e) = hE{Ty,^e-Tf^e)^^{Ty,,e-Ty,^e). (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 ^ hH{Ty, - T)

dt ^ dz pScp^i

^ + V z ^ = W^Vz (2.77) dt dz

hc{Tf,c — Ty,^c) = hH{Tyj^h — Tf^h) = T—(71o,c — Ty,^h)

and

respectively.

dG dG -^+Vz^ = -W'Vz dt dz

dXs ^ dXs . , . , Xs __ + , ^ ^ = Wvz^

dGe dGe ,,„ c^^

r

hc{Tf,c — Tw,c) = hE\Tw,e — Tf,e) = T-~(- «'.c ~ -Mi),e))

(2.78)

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 Definition Transformed Variable Variable

W' Black hquor temperature W'* = 7777

W^ W

Wall temperature at evaporation side W^* = — -

'!'* =

rp* we

(4* --

G* --

X * ••

7." =

K.^ «.*

T

To T •'•we

Po G

Go Ge

Gcfi

X Xs,0

z - —

L Vz

-'-'c,z

Wio LW^

Mass flow rate of the black liquor r] = —p:^— Go

Mass flow rate of the condensate 77c = ' ^c,0

HHL Dry solids content TJT

pCp5v*jQ

Kz =

tVffl

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 Numerical Solution

2.5.4.1 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, den-

60

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 differential-

algebraic 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 han­

dles 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 phys­

ical 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 58 kg/s Liquor feed temperature 303.15 K Steam pressure 47000 Pa Vapor pressure 21372 Pa Plate mass flow rate 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 8 Plate length, m

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 8 Plate length, m

Plate length, m

10 12

Initial state Final state

12

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 signif­

icantly. 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.

JO

O

0.2

0.1

0 2 4

Plate length, m

1

6 1

8 1

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 sen­

sitivity 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 8 Plate length, m

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 magni­

tude 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

5

Sw

e

V, Vc, VT

A

Atu

Ai

M* u

P T

'yx

e

black hquor mass flow rate per unit length, [kg/s- m

film thickness, m

wall thickness, m

eddy viscosity, m^/s

dimensionless variables

thermal conductivity, W/m- K

wall thermal conductivity, W/m- K]

dynamic viscosity. Pa- s

turbulent dynamic viscosity. Pa- s

kinematic viscosity, m^/s

density, kg/m^

residence time, s

Reynolds stress, N/m

dummy variable, m

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 recir­

culation 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 evapora­

tor, 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

Liquor distributor

Plate stacic

Steam inlet

Condensate outlet

Feed inlet

Liquor inventory

Secondary vapor outlet

Entrainment separator

Product outlet

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, (3.1)

where Gout is the mass flow rate of the liquor out of the distributor, dn is the

recirculation mass flow rate and Wf is the mass flow rate of the water vapors.

79

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 + Cp^wTboil)- (^-3)

3.3 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 inven­

tory. 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 in­

ventory, 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, (3.5) dt

where A'' is the number of the plates in the stack.

82

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. . T. ^n (3.7)

dL _ NGp - Gt,out L dp

dt pAt pAt dt

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. (3.11) dt ^ ^ ' dt ^

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

p dT _ NGpTp - Gt,outT _ T^dp _ TdL _ T^dc^

dt pLAt p dt L dt Cp dt

In summary, the equations describing the liquor inventory are

dL _ NGp — Gt,out L dp dt pAt pAt dt

dt pAtL^ ^ '

dT NGpTp - Gt,outT Tdp TdL Tdc, p

(3.15)

dt pLAt p dt L dt 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 . 8 2 2 X ^ + 3 . 8 2 2 r ^ - 3 3 2 2 . 6 7 ^ . dt dt dt dT

Equations (3.16) and (3.16) are functions of the black hquor parameters whose cor­

relations can be found in Appendix C.

3.5 In-line mixer

The mixing that occurs before the recirculation pump is assumed to be instanta­

neous 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 + Gt^outXt^out- (3.18)

3.6 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 ^rec + ^prod- yo.l.jj

3.7 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 Initialization

3.7.1.1 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 pres­

sures 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 termi­

nation 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 conve­

nient 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 Sub­

routines) 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 - , (3.20)

89

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

Feed mass flow rate, Gf

Feed dry solids, Xf

Feed temperature, Tf

Vapor pressure, Pyap

Steam pressure, Pg

Sensitivity: d> 0

-0.444

1.105

1.431

-0.945

0.911

Sensitivity: d < 0

0.523

-1.086

-1.188

1.157

-0.750

3.9.1 Feed flow rate

The change in the feed mass flow rate does not affect the performance of the evap­

orator 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 signif­

icantly 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 concen­

tration 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 approxi­

mately 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 0.19 o (/)

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 0.19 o w

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 mix­

ing 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 crys-

tahze (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 cali­

brated, 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 con­

figurations 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 in­

ventory. 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 pro­

duction 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, respec­

tively 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 oper­

ation 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 - 0 1\T (4.3)

The matrix K is a matrix of steady state gains between the controlled and manip­

ulated 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.

K Kx/Gp ^L/Gp

Kx/Ps KL/P,

-0.00427 0.000212

0.00903 -0.0000727

(4.4)

108

The corresponding RGA matrix is then given by.

A = -0.1936 1.1936

1.1936 -0.1936 (4.5)

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 feed­

back 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 con­

trol. 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 Tl

(4.6)

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 AT = 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-

I l l

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

FC

PC

LC

CC

Gain {Ke)

0.09 s.%/kg

0.00009 %/Pa

-368 kg/s.m

1250000 Pa

Reset time

3.3 s

3.3 s

100 s

33.3 s

{ri) Valve

VlOl

V102

Time constant (r„ )

45 s

90s

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

. x 1 0

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 re­

sponse 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

. x 1 0

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 prod­

uct 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 re­

jection 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 proportional-

integral-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 in­

tegrated, 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 sim­

ple 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

Performance

Feature

Disturbance

Gf Xf Tf

responses

Servo Control

Xp

Time to reach ±la (min) 3.85 4.81 6.99

Oscillation period (min) 5.25 5.62 6.03 5.45

% 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 un­

measured 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 closed-

loop 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 con­

troller actions were very large in comparison to those used in disturbance compen­

sation. 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 Al­

gorithmic 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 Regu­

lator (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)

xo = x{0), (4.14)

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, (4.15)

where K 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), (4.16) Jo

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 and

x'^{T)Mx{T) 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) (4.17)

y = Cx, (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) = ^x{k) + ru{k)

y{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) + CVu{k-l)

y{k) = C^^x{k-2) + 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 coef­

ficients. For stable systems and for A'' sufficiently large, the impulse response coeffi­

cients 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 aiAu{k + i ~ i) +d{i\k), (4.21) e+i 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 + £) = [yk+i...yk+pf

AU{k) = [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 Ef{k + l)'^Ef{k + 1). (4.26) ALT ^—^

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 opti­

mization problem, the QDMC method improves handling of constraints thus making

the QDMC algorithm a very powerful tool for solving complex multivariable con­

strained 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 = I \{AAU{k) - E{k + l))^Q{AAU{k) - E{k + 1)) AC7(fe) 2 L ( 4 . 2 8 )

+AU{k)'^RAU{k)]

subject to the following linear inequalities

ymin <y < ymax (4-29)

Umvn <U < Umax (4-30)

Ax„,in <AX< 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 hori­

zons 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 fac­

tor. 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 vali­

dated, there are transformations that can be used to transform among different linear

representations. In particular, the transformation from the ODE or state space rep­

resentation 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 ob­

tained 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 concen­

tration, 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 tem­

perature, 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 Variable Correlation R^

X^-G^T'^ + Xd 0.881 0.00140 1.195 -1286.15 0.97

X<^G^T^ + TXd -0.136 0.405 -0.0398 -0.00294 0.96

X°'G^T^ + TXd -0.192 0.427 1.456 -26.212 0.93

X^'G^T^ + TXd -0.126 0.236 1.651 -82.651 0.99

Turbulent

Wavy-

laminar

6

Vz

hs

hn

~5

Vz

hE

hn

XO-G^T^ 5.522 0.333 0.540

XO'G^T^ + Gd -0.242 0.981 0.141

X°-G^T^+ Td -0.688 -0.0200 1.323

X°-G^T^ + TXd 0.508 -0.0140 1.242

0.99

-2.377 0.997

-7.529 0.99

-4.717 0.97

• 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 (4.39)

evap

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 equa­

tion 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, - iZkLlZZf l ) ^ G;,c,(r ,„, - Tfg). (4.43)

Equation (4.43) is solved for Lh

T - GfgCp{Tboii - Tfs) Lh 77p—, ^ N - (4.44)

hHan{Ty, —^—)

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.

W = hEa{L

Lh

dX

~dt

-LI

h

dMst ^

dt ^^f''

- ^f'(Y

-Gout-W

' M

\)n{Ty,aU — Tboil{X, Tsat))

^•^evap

GfsCp{Tboil -Tfs)

.^n.n(T.. ^T^oil+Tfs).

138

4.6.1.2 Bottom inventory

The bottom inventory model is analogous to that described in Chapter 3. Thus,

the model equations are

dT ~dt ~

dL lit '

IS Gpip — GoutT

pLAt

_ NGp — Gout L dp

pAt pAt dt

dX _ NGp dt - pAtL^^' ^^

Tdp TdL T dcp

p dt L dt Cp dt

(4.45)

(4.46)

(4.47)

4.6.2 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 con­

troller, 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

-5

-10

-15 J I L_

10

ODE PDE

15 20 25 3D 35 40 45 50

20 25 3D 35 40 45 50 Time, min

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.

X

<

5

0

-5

-ID

-15

xlD

10 15 20 25 30 35 40

PDE ODE

45 50

20 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 °° *3 0.005

10 15 20 25 3D 35 40

ODE PDE

45 50

20 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

T3

a

<

is: •a o

1-^

<

0.01

0.008

D.OQB

D.004

D.D02

0

0.2

0,15

0.1

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 equa­

tion (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 sys­

tem representation is as follows [82]: Consider the nonlinear state and measurement

equations given by

a;(t) = /(x(t) ,n(t)) x(to) = Xo (4.50)

y{t) = h{x{t),u{t)), (4.51)

where x = —, /(-) is a vector valued function, f : X^ x U^ -^ A", h{-) is the CJij

dx

'di measurement function, h:X^xU^^ F ^ states rr G A" C 3?", outputs yeY^cW

145

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 (4-54) dt

146

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)) (4.56)

B = ^{x{t),u{t)). (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 solu­

tion 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 con­

trollable 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 1-^ rank of the np x n observability matrix given by, O — (j (jj[ ... CA^'^

equals n.

In the linearization of the lumped model it is assumed that the level of the evap­

orator 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

Plate inventory dry solids

Evaporator inventory dry solids

Evaporator inventory temperature

3108 kg/s

0.2248 kg/kg

0.2248 kg/kg

339.26 K

Analytical resuhs obtained for the matrices A G Sf? "" , B G ^'''^ and C G "^""^

are as follows:

A =

-0.10247 502.57 0 0

-0.10491 0.10942

0.01736 -0.021771

0.00005 3.0939 -3.8887 -0.019499

(4.59)

B =

-1.4692 1.6478

0.00011953 -0.00011581

0.016383

C =

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

(4.60)

The test of stability shows that all eigenvalues of the A matrix are real and neg­

ative. 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.

306D

LODE NLODE

10 2D 30 40 50

Time, min

BO 70

20 30 40

Time, mm

0.212

339.05

30 40 50

Time, min

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

2-o

3160

3150

3140

3130

3120

3110

3100

3090

r^' i -•• LODE

NLODE

,

•

•

0.24

10 20 30 40

Time, min

50 60 70

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 O)

^ent

oty.

c

CL

3160

3140

3120

3100

3080

r^ ( • •

•

••-- LODE NLODE

, , 10 20 30 40 50 EO 70

Time, min Time, min

2r E

339.5

33B.5

ra 337,5

10 20 30 50 BO 70

Time, mm Time, min

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:

C = 0 0 1 0

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

3100

3080

3040

3020

1 1 1

;

\ ^

1 1

- - • LODE NLODE .

• - - , ,

2r

10 20 30 40 50 BO 70

Time, min

10 20 30 40 50 BO 70

Time, min

o

10 20 30 40 50 BO

Time, min

70 10 20 30 40 50 60 70

Time, mm

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

and constrained MPC. The actual feedback block diagram is presented on Figure 4.21.

I -

MPC Controller

U

NLPDE

NLODE

LODE

^ d

>\ ^'°^

+ ^ '

"-h

iM ^ -

^ - ( < - ^

r-t-

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 Aui Au2

Weights 50 0.01 0.06 0.06

Minimum -0.001 kg/kg -0.2(1) K -10.96 K -18.23 K Maximum 0.001 kg/kg 0.2(1) K 6.04 K 4.77 K 3 K 3 K

Table 4.8: Tuning parameters for the PI controllers.

Ke Ti

XC loop 2000(1000) K 500(250) s

TC loop 30 300 s

The ±la 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 con­

strained 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 ma­

nipulated 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 compensa­

tion (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 immedi­

ately 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.

159

CM I M CM < N CN

6>|/6>| •''°\

CO

—I o

ro ^ 00

IT) C O

CL CL

< <

E

LO t - LO - ! - O O o • O

O O O

o o o CM

O O O CN

,pojd

>l i LU'|8A9"| Bd'^^d V '^dV

9i -^

at ^

Tl OJ

4 - 1 4 - J

f ) 73 OJ

^ H fl)

+^ c« ^-1

•5 o

q=l

0)

> ni

O

ent

> Pl • •—{

-a ! - i

^ - t J

s " 03

- i d i-i

CU O H

fl CZ3

! - i O . . fl -d

•^ fl o o l O

03

fl CO CO CD

a S - l

o ^ 03

s ^ . fl O) CO

CU

.-fl

(U

fl o

t "" fl ;fl «i;

•-' o -.fl OJ CO -i-^

fl > ^ ^ .s I 0)

fl CO CO

•^ .fl Q;

CM ^ i " CM . CU

^ -§ :S S S .23 fl fl:! . ^ b fc

160

i n CO

LO CO

0.21

0.21

0.21

0.2

s^A •'°'\

334.

.poJd M i Bd ' d V d V

H fl a;

• > T3 fl .'^ ..

fl o

0 ) s-l

fl -1-3 03

03 (u

a S-H

I B I -o ^

^ o o a CO

f-l fl

-a o a; g

. . f l CO

.s § CO o3 03 !^

£ fl a <D

S-H

O a o3 >

-^ -l-i CO

. ^ a; OJ o fl - fl ^ T d

LO T d M l

° >.^ O T3 ^

(P O 03

a o fl t H CO

oj a ^ - ^ -t a r-i "3 ^

i -2 ^ a a ^ s - ^

•-f l l.^ CO !-i O .-H CD ,—I n ,

a ^ fl O <D fl3 PH -^ -^ ^ fl X5

- • Q ;

CO CO L ;

^ 2 :S • a ^ 0 S o 1:3 tn ,

[-iH ; f l 03

161

CM O

CM CD

CM CO ^ CO CM CO O

^ CM O CO CO CO CO CO CO O

LO O O

—

o o o o

o o o LO

O O O o LO

em''°\ .po jd

>i 1 LU '\dAQ-\ .A „,s „

Bd d V d V

-1-3 o3 ^ T3 fl .. CD M CO T 3

(-1 rH

CL> fl fl ro

<D >

>> !H O

+J fl

fl 03 -^ t-i .

03 .;-l S-l r f l Q; ':;3 a „ fl CD

•n <D

-^3 -(-= O

• T d

(U -.fl

CD

Td CD

.'^ ^—I

CD

fl + J 03 s-l CD

a 0

- t J

.fl O ^ ^ • ^ I fl O 03

xj "g a fa\ o O LO 03 D. ce "" a O g 03 03 -^3 CJ o3 fl i-i

03 03 ! - i

a

03 fl

0^ O .Jfl

-tJ 7 d hO

.a o gj CO r X ^

03 . - U CJ >i _ fl ± 1 ' ^ c3 "O fl fl - u 03

0 ° -a Q, 03

03 ! - l

fl CO CO

P H ra

CM CO 03

. f l a

S ^ .a

fc ^ ;fl

162

CO - r t

fl « . f l o3

LO CO

IS) >

CL CL < <

1 1

E

E

CM

CD

CM

O

CM LO

O CO

CO CO

CO CO CO

CM O

LO O O

O o o

o o o CM

O O O CM

iPOJd

>l X ,pojd

>1 1 UJ'|8A81 Bd d V d V

03

T 3 03 03 >

o

03

> 03 fl

- ( . ^ . - H

o3 . . '-' Td & • -O ^

^ ^ . CO 03 CO tH

o3 fl

•Td 03

^ a OJ 0^ .H U

.a fl Td o S-H

03

fl a en

03 T3 fl "

„ O >-'

5 ^ a o3

O 03 S - f l

03 O Cl 03 CO 03 ! - i

a Cl)

.d

ifi S-l

-M

fl 03 CD

rl o o

T3

-^ m

fl 1——1

-*-3

^ _bJJ

fl O 03 - f l CO _sH

03 > , -^^

fe 2 ^ a ^ Si o P H

a a o3 03

C O s - l

'S ^ ^ fl • fl 03

.•fl fl .Jfl o3 S -i-j SH ^ f l

X3 ^ CO M b "^ fl - -H 03 O - U fl D ^ ;fl O

03 .^ CM o3

^ -5 -a

hp a fl

163

0) >. rf l SH

6 /6>, ' "^V iPOJd >l 1 U!UJ'|8Ae-|

|A _ ,S

Bd dV dV

fl

O fl •-^ . f l +J - I 03 . . SH T d

-1-^ SH

« -fl 03 rfl o CJ

CO

SH T3

03

03 S-l fl

- J 03 •i-i CD

a a 03 C3 fl

0) T3 O

03

03 SH

S-, ^ a So

CO

. a ^ o

03 O CO 03 o3 CO 03 SH O 03

-Td

lO

03 SH

a S - l

O

a fl O 03

SH -(.^ fl • • 03 03

CO

o3 o HH fl Afl O O 03 rrH - ' H

S ^ T! CO M 03

a "O Td 03

r ^

4 J

,fl 03 O fl

- t J

o fl

prod

- t J

fl (Xl

ssur

e

Cl) s-l

03 CO a

a - fl •-IH • 03

s-l cr> •i^ 03 - t J CO

a 'g 03 o ; a ^ P H b CO

T3 03 fl

•I—I 03 S H

CO fl O

o

o) .a r f l

r ^ SH 03

CD CM

03 T 3

03 -SH r f l

a +

fl T3 03 - O a 03 fl -(.J

. - H O

fe -!d

u 03 > 03

164

CO -Td

fl « .fl o3

03 >

E CD

E

Bm ' "V ,pojd >1 1 LU |aA3" | Bd d V d V

-^^ O

0) r f l

SH O

fl 03 S-l

o3 b s-l fl 03 • ;g

a ^ 03

T d 03

s-l fl td SH

03 03 •- a 03

r f l a 03

•i^ - ( -= O fl

T3 O 03

SH O 0^ - ^

^ ^ fe? 8 LO 03 03 "

r v 02 • ^ to

03 SH

a SH O

a

03 -^^

fl O 03

O o3 fl ^ 03 03

^CO

03 fl

03 S - l fl to CO

a | - ^

^ :3 ;fl' . f l ' O 03

«3 r f l o:> ^ +=

a t ^ SH -3

a -^ SH O CD SH a a 03

O - ^ OH

• i 1 ^ fc o3

^ ' ^ 03 ^ CO CO

a •'fl s " T-H o;

r f l - ^ - ^ SH

03 CM

o; 03 ••

;: 03 + j

165

T—rr

CM CM CM CM CM O O O CO CD

CO CO

CO CO

LO (O C) ^>

CM O

—

6>)/6>| ,pojd, X

^ ,PO.d^ UJ ' |8A81

1.00

5

40

00

20

00

-200

0

Bd'^'d V '^dV

0) -i^ -^ 53 .4J o3 ^ Td

fl .. 0) rH CD T3

2i I ^ o SH 03 fl fl ^

T3 03

-1-3 -(-3 O

T3 03

r f l H

03

o fl

fl 03 ^

03 T3 !H SH

5 -_ 03~

fl

qfl CO CO o3

Td 03

.0)

03 r ^ -1-3

03 s-l 03

a a 03

4 J

fl -o -- o

s-l

a

03 S H fl CO CO 03 SH M T3 ^

" fl ^ a o SH

O vo O) Li

fe\ to g* lO _p > o3 . y r f l

2 CO

03 O . f l to 03 S H

a

fl O r f l

0) ! y ; f l •5 ' ^ 03 _ CO rfl

.,, S H - O 03 -r) rl p ^ a

s-l T3 O

's-l a 03

a ^ PH r "

(X) CM

^ S fl CO tZ2 03 S H

a i 03

03 rf l

fe +l;fl

166

CO 03 fl

Td fl 03

CD

E i -

CM

O CM

O CM

CD

CN

O CO CO

6x/6^ •'°\

CO CO

,po jd

o o o o o o o o o CO - q - CM

o o o CM

LU I8A81 A „ ,s „ dV dV

Td 03 03 >

-tJ 0)

o Td

03 -Jfl

o -1-3

fl H ^

.2 -d

2 rfl fl r, 0 3 0 3

S-l fl O fl O o3

SH

tZ3 T d

03

a

0 3 S-l fl

^ a O 0) CO +3

SH O

T d T d o 03 !H rJ 03 D , CO

03 T d b

rfl fl a ^ O !H

fl g a - CO a 03 r- > CO fl " S O 03

03 ! H CO

511 lO fl ^

O 4J 03 O r f l

^ ^ .Sf o ;f l —< „.. O CD 8 " fl >. ^ 0^ r r t T3 CO T3 ^ 03 M M - ^ 03

a g 03 T 3 - I fl ""

0 3 S-l fl CO 0 3 0 3 S-l

- ^ a fl QL. o3

fl K) tC

S .t5 03

^ a J SH . f l -tJ a b .-H

I I i-H 03

^ ^ ;f l rb:i

03 T d

03

03 CO 03

bp a 03

(35 CM

03

fe

rfl S H

fl

167

LO CO LO

CO

CD - ^

CM CM

O O

6>|/6>i

212

0.21

o ,pojd

X

336

• ^ CN CO CO CO CO

,pojd M i

330

1.02

E Q)' E i-

Oi |8A8"I

LO O O T —

o o o o o o o o o o O LO LO

" ^ ,A ,S

Bd dV dV

b T-H

^ 03

r f l - ) - : ' - t J

fl 03 CO 03 S-l

i ,c r ^

S H -o:

X3

r f l - M s-l fl

a o 03 SH

nes

T3 03

-t~3 - t J O

-a CD

r f l H

ai SH

atu

! H 03

a a 03

Td 03 03

t + H

03 . • ^

• ^ ^

fl . r - l

03 03

decrea

fe? L O

o3 M H

O

03 CJ fl 03 CO 03 S H

a 03

r ^

4 ^

fl •r—)

03 CJ fl o3

a s-l O

S-l

03

a h — 1

PL,

cii CO

^ a:> SH

fl bn

iZ ,

t + H

Td fl o3

03 > 03

>, S H

O - t^ fl 03

• • — (

TJ

03"" S - l

fl 4 J o3 SH 03

a a 0) 4 J

- ^ J CJ fl

Td o a

Td fl o CJ 03 03

. fl o

• ^

03 SH

- t J

fl 03 C3 fl O o

3 o 03

SH

Td

-^ O fl

-d o s-l

a

03 SH

JH

03

03 SH

fl CO CO 03

a E H O

a t 03

r f l - t J

CO

03

fl ; f l - M

j : ^ bC r H

03 r f l

-o fl

03

fl 03 CO 03 S-t

a a 03 03

CO

03 t^ r f l

a + J

CO

168

L O C O

o CO

03 - O

• — 1

'P, '03 03 ^

o , Td h SH

03 O

" 03 . >

03 fl S H " ^

fl .. +J Td 03 iH S-i . - 3 03 r f l a + fl 03"" 0 3 S-l

-^ fl 1 1

6 /6>1' "V UJ |8A8-| d V d V

^ 0 3

^ a rfl a 4 ^ 0 3

a ^ 03 jzi

S 2 fl r 2 •• 03

CO CD o3

O

a

0 ta 2 '-I 03 fl CO 0 3

fl^^ O 03

03 CJ . 3 O. f^ ^

a o ^ 03 " rfl

.Jfl T3 Sf -UJ .Zi i:ri o; C J

fl o3

SH 03

a O PH

Td 03 fl

•3 S-l

- l - ^ 0 3

fl o

O

o 0 3

C3

fl Td o S H

a 03

0 3

Td fl o3 03 ^-1 fl 03 03 CD SH

a

^ .a

03 03

03

03 rfl

03 r f l

CO

03 S H

fl bjO

• r—I

fa

S H

03 T3

fl

03 4 J S H S H

a fl 03 O S H '^-l

169

6 /6>l • ' ' "V UJ'|8A8-| EcJ ' cJ V ' d V

-^ ^ - ^ 03

03 (H 03 '13 SH rj

03 ^

03 a (H o3

-a <v

o

03 r f l

03 >

rSJ

SH O

-^J fl 03 > fl

03 SH . .

fl Td -I.J s-l

S :fl 0 3 -^•=

a 03

03 SH fl 03

o s-l 03 03

^ a

cz

E '8U

JII

r f l J_3 - ^ J _ , - tH

fl o • - ' fl . 03 Td OJ

decr

eas

nd:

pro

pre

ssui

^ O fH fe? CJ o ^ ^ & CD r- >

of a

at

ion

the

03 H -S le

nc

cen

ine

CQ r H 1 1

03 5 iL O -R

he p

: li

dc

ihg

h

-R O 0^ _ 03 r f l

•a >^:: - , !H Td 03 Td r< CJ S fl ie " 03 H 03 cl fl SH fl Td fl

irfo

r p

ro

ress

2< .. a a i ; "^ r " a PL| r'-' 03

PH 03 -R

CO + j 03

^ a ^ S ^ .22 S ) ^ 03

fa ^ ;fl

170

0) -R CO fl o3 03 03 03

6 /6>1 ' '"^V

s-l CJ 03

Td

03 S - l

a 03 fe? O)

l O 0^

^ ;fl

^ -R

Mrg 03 TJ >-< 03 O r f l ^ H

Td

i s ^ fl

CD SH 03

CO

ion

- R (•;

03

O SH

rt O CJ

era

a a 03 -R

C) fl Td o >H

a

03 R fl ° fl -R . rH .JHl -|H> O

a ^ 03

-fl

03 03 fl O

fl .2 ' - R 03 s-l

fl 03

. O

a fl CO O 03 CJ

aS 03 O

-R CO

>. h 03 !H SH " ^ O - R

- R O o3 ^

^ a 03 . .

W O Q H PH • ' ^ 03

S H SH cS ;3 03 4^ fl 03

Ifl >-<

^ a ^ a is

03 ."S ^ - a ^ ^ -^ ^ b 03 r, '"' Id r f l T l bO -+-" 03

fa . f l 4J

171

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 back­

ward (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 disadvan­

tage 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 super-

concentrators 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 C J s-l CD a fl 03

03 fl o

Td fl o3

• ^

I

fa o

4- i

fa] 0 3 S-l

o -R o3 SH o a 03

> CD

cfl bO fl

fl 3

CD s-l o3 Cl) s-l CD

rfl H - R

fl n1

a s-l O

4 ^ 01 SH O

vap

03

a cfl bC fl

I — { Ol

"HH

03

a 4 ^

^ S i-H

1^3

0 ) SH fl bj()

fa

,^ N

CO 1

(J OO o

4-^

I-H

O CO

ors

4.J 03

por

f > 03

(1) 03 ! H

rfl - R

rfl -R

^ SH

O 4-^ o3 SH

- R

fl 03 CJ

176

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 evap­

oration 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, (5.1) at

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 d{MTs) ~dt ~ MmV dt

Expansion of the right hand side of the above equation gives

(5.3)

dP R (d_M^^^dTg^ ^^^^^

dt MmV \dt " dt

The saturation temperature is a function of pressure thus

,P R ^1M iI.iP^i (5,5) dt MmV \ dt dP dt

178

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 RTg ,

5.3 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-3 410 kPa

t^candenser /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 concen­

tration 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

20 40 60 80 100 120

Time, min

409.5

409

1.02

0 20 40 60 80 100 120

0 20 40 60 80 100 120

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 0.196

Superconcentrator 1

2£ •a o a.

X

^

pro

d'

E

Leve

l

0.194

0.192

0.19

331.8

331.7

3316

331.5

331.4

1.02

1.01

1

0.99

0.98

20 40 60 80 100 120

— I

20 40 60 80 100 120

Time, min

0.66

0.64 •

0.62 •

0.58

409.5

I- 408.5

> CD

20 40 60 80 100 120

100 120

40 60 80

Time, min

100 120

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

O)

o C3.

X

:^

prod

E

Leve

l,

0.2

0.195

0.19

0.185

0.18

331.7

331.65

331.6

331.55

331.5

1.02

1.01

1

0.99

0.98

73 O

a.

X

0 68

0.66

0.64

n RO

Superconcentrator 1

•

G 20 40 60 80 100 120

410

0 20 40 60 80 100 120

Time, min

0 20 40 60 80 100 120

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

0.21 Evaporator 5

O)

T^ ^

D O.

X

^ •a a

H

b 1) > CD

—I

0.205

0.2

0.195

0.19

331.9

331.8

331.7

331.6

331.5

1.02

1.01

1

0.99

0.98 20 40 60 80 100 120

Time, min

0.68

0.66

0.64

0.62

410

409.5 •

409

408.5

Superconcentrator 1

03 > 0)

0 20 40 60 80 100 120

0 20 40 60 80 100 120

20 40 60 80 100 120

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

0.21 Evaporator 5

5"

332

H-

03 > 03

20 40 60 80 100 120

0 20 40 60 80 100 120

0.66

0.64

0.62

410

409.5

Superconcentrator 1

0 20 40 60 80 100 120

409-

408.5

1.02

1.01

1

0.99

0.98

•

0 20 40 60

Time, min

, 80

'

, 100 i : 10

E

Leve

l

1.02

1.01

1

0.99

0.98

100 120

0 20 40 60 80 100 120

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

0.21 Evaporator 5

g/kg

js: T3 O O.

X

^ T3 O a.

h-

E "o > 0) _l

02

(iiq

0.18

335

334

333

332

331

1.02

1.01

1

0.99

0.98

0.68 Superconcentrator 1

20 40 60 80 100 120

100 120

20 40 60 80 100 120

Time, min

E

.eve

l

1 02

1.01

1

0.99

0.98

100 120

100 120

40 60 80 100 120

Time, min

Figure 5.7: Open-loop response of the PDE model of multiple evaporator model for 5% increase in the feed temperature.

186

0 196

0.195 L

0.194

332

Evaporator 5 0.66

Superconcentrator 1

? 331.5

331

0 20 40 60 80 100 120

20 40 60 80 100 120

E

> 03

_ l

1.U2

1.01

1

0.99

0.98

'

0

'

20

' '

40 60

Time, min

'

80

'

, 100 i : 10

Leve

l, m

1.02

1.01

1

0.99

0.98 20 40 60 80 100 120

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

Table 5.2: Sensitivity of the dry solids concentration.

Disturbance

Feed mass flow rate, Gf

Feed dry solids, Xf

Feed temperature, Tf

Heat transfer, Pyap

Sensitivity, d > 0

-1.067

0.180

0.64

Sensitivity, d < 0

1.414

-0.197

-0.729

-0.215

In the case of a single evaporator, it was found that the feed temperature distur­

bance 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 con­

centration 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 '

J:WC SC-3

(5.7)

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 Change with respect to kg water/ kg steam base case,%

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 evapo­

rator 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 dy­

namic 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 con­

trolled 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

V

A 1 X2

A

1 X3

V - ^

Figure 6.1: System of three connected tanks

Following the modeling approach by Rugh, the dynamic model of the three con­

nected inventories (see Figure 6.1) is

Xl = Fin - qiV^i - ^2

CX2 = qi\/xi - X2 - q2y/x2 - X3 (6.1)

c i s = q2\/x2 - X3 - F^t,

where x ^ R^ represent the three levels. Fin is the inlet flow rate to the first in­

ventory, 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 devel­

oped, using Taylor series expansion about the nominal states, x,Fout- The resulting

linear-time invariant (LTI) system is expressed by the conventional state space rep­

resentation [82] where for convenience all parameters are assumed to have the value

one,

S — •AcyX + JDcyU (6.2)

where ^ = x — ic is a vector of the deviation variables, u — Fcmt ~ Font is a vector of

manipulated variables and

•Ar-i, —

- 1 1 0

1 - 2 1

0 1 - 1

•'^r.t)

0

0

- 1

The companion measurement equation is given by

y = c^^, (6.3)

where y = ^ and

f-'CV

1 0 0

0 1 0

0 0 1

(6.4)

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 struc­

ture, the multiple evaporator plant PDE model is subjected to changes in the operat­

ing 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 - t J 1:3 ^ S-l

- U OJ

.—H

(5 s-l

- t J

a 0 0

,^ 0 oj

r.Q -D 03

M-l

T l 03 N

CO SH

l-l 03 C3 03

Q

CM

CO

03 SH

Pl br fc

! H

03

n - t J

PI 0 0

i=l 0

• 1—1 •l-J

en 0 a B n r i

0 0 s-T OJ

0 SH

- tJ

rt 0

198

Table 6.1: Operating conditions of the muhiple evaporator plant

Parameter

Gfeed

Xfeed

Tfeed

Psteam ( S C - 1 tO SC-3)

^condenscT

116

Value

kg/s, 50/50 split

0.145 kg/kg

303.15 K

460 kPa

21.4 kPa

The disturbances applied to the system are a 5% decrease in the dry solids con­

centration 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

1 0 , 7 3 ;

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

5% increase of Gfeed

5% decrease of Xfeed

7% decrease of Tfeed

Time to ±1(T

27.8 min

20.6 min

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 (Cl) ) .

200

^ i w 0.735

5

^ 0.73

'

\

1

0 10

fl 650

& S 600

s S. 550

5 500

u 10

'

20

'

, 70

30

'

1

30

'

4 0

'

1

40 Time, mli>

'

50

1

50

1

60

'

60

^

7 0

1 -

7 0

.

8

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 inves­

tigated, 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 evap­

orator 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 vali­

dated. 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 evap­

orator 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

developed in Matlab's®: Simulink ® : environment. A computer program is was

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.

.XlO' Evaporator 5

B)

X

<

0 20 40 60 80 100 120

0.02

0,01

0

-0.01

-0.02 0 20 40 60 80 100 120

Time, min

0.04

i 0.02 o

< 0.01

n

Superconcentrator 1

^ '

-

PDE . ODE ,

1

^ 0.5 •D

< -0.5

-1

0 02

0.01

E

< -0.01

-0,02

0

0 20 40 60 80 100 120

20 40 60 80 100 120

"0 20 40 60 80 100 120

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

i <

0

-2

-4

-6

1

0,5

0

-0.5

-1

0.02

.XlO'" Evaporator 5

\ \ ''' ' r /-" ^

• . . .

PDE " ODE ,

E _ i "

<

0.01 L

0

-0 01

-0 02 0 20

0 20 40 60 80 100 120

r

0 20 40 60 80 100 120

40 60 80 Time, min

100 120

§, -0,02

'•a

X^ -0,04 <

-0,06 0

E -J <

Superconcentrator'

V-,. N. " - - - . . . . .

PDE ODE

\ ^ ^

20 40 60 80 100 120

20 40 60 80 100 120

20 40 60 80 100 120

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 sensi­

tive 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 re­

sponses in some cases (feed flow rates and feed concentration changes). This may be

204

40 60 80

Time, mm

100 120

X <

10

5

0

-5,-

XlO"

E

<

PDE ODE

20 40 60 80 100 120

20 40 60 80 100 120

Time, min

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. Oth­

erwise, 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

X

<

.XlO"" Evaporator 5

E

<

-0 02

/<^"^~' ^^^^""""^

/

• I I I

^LZZlZ^lllZ^lZ,!^

PDE ODE •

•

0 20 40 60 80 100 120

0 20 40 60 80 100 120

Time, min

XlO' Superconcentrator 1

100 120

100 120

"0 20 40 60 80 100 120

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

X <

Evaporator 5 Superconcentrator 1

PDE ODE -i

X <

PDE ODE

0 20 40 60 80 100 120 -0,03

0 20 40 60 80 100 120

0 20 40 60 80 100 120 20 40 60 80 100 120

0,02

0.01

-0 01

-0 02

o f ^

0 20 40 60 80 100 120

Time, min

0.02

0,01

0

-0 01

-0 02 0 20 40 60 80 100 120

Time, mm

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

Ul

^

X <

XlO Evaporator 5 Supeconcentrator 1

/ •

PDE ODE

•

0 20 40 60 80 100 120

kg/k

g

•D O

a. X <

0,03

0,02

0,01

0

-0.01

. ^^-^^^^

• • ' ' '

PDE . ODE

, 20 40 60 80 100 120

y-<

E

<

E

<

20 40 60 80

Time, min

20 40 60 80 100 120

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 Xsci Boihng point rise AXprod/^XPDE,% at SC-1, 2 and 3, kPa coefficient, K

410

390

380

360

410

390

380

360

360

340

320

300

0.6560

0.6330

0.6209

0.5954

0.6747

0.6514

0.6391

0.6129

0.7188

0.6877

0.6533

0.6160

7

7

7

7

6.5

6.5

6.5

6.5

4

4

4

4

18

5

-5

-15

20

8

?»0

-17

-5

-25

-49

-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 recircu­

lating 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

ID

o, -2

o .4

XlO

\ \

1 •

NLODE LODE

' ^—' ^ 50 100

Time, min

150 50 100

Time, min

150

.x10

S

50 100

Time, min

50 100

Time, mm

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 temper-

212

NLODE LODE

50 100

Time, mm

150 50 100 150

Time, mm

XlO

50 100

Time, min

150 50 100 150

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 as­

sessed 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 - y{t)\dt. Jo

(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

XlO

/

,

NLODE LODE

50

Time, min

100 150

0

0,2

04

06

0,8

-1

1 1

XlO •

i •

NLODE LODE -

^ " :

50 100

Time, min

50 100

Time, mm

150

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 con­

trollers 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 discon­

tinuities. 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

, X 1 0

I O

<

100 150

s

<

/\

3

2

1

0

-1

XlO

'

NLODE LODE

. Time, mm

50 100

Time, mm

150

. X l O

50 100 150

Time, mm

50 100

Time, mm

150

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 con­

sumption 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

0,01

0 008

•a 0 006

5^" 0 004 <

0,002

°0

. — . ^ /

/ / • "

,•••'

.

— NLODE LODE

. 50 100

Time, mm

50 100

Time, min

150

150

>r

, x 1 0

/ " :

/ / " "

''•

,

NLODE LODE

50 100 150

Time, min

50 100 150

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

X <

-10

xlO

K —" 1

NLODE LODE •

-

\

50 100

Time, min

150

0

-0,002

•a "° °°''

b -0,006

< -0,008

-0,01 •

-0,012

NLODE LODE

0 50 100 150

Time, min

50 100

Time, mm

150 50 100 150

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 PI

Disturbance Unconstrained Constrained

5% increase of Gfeed 13.74 min

5% decrease of Xfeed

5% decrease of Tfeed 12.97 min

10% decrease of hs 12.78 min

case. This is not unexpected whenever constraints are active.

31.1 min

59.92 min

30.34 min

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 ±2a 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 evapora­

tor 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 ef­

fect 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 regu­

lated 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.

219

a

s « s o

03 Ss

^ o o

o

o3 SH O

a OJ

3

a cn SH a;

a o o

c (33

o O H

a SH 03

bjO

fl

LO

i o

ft faO el

•3 3

d o o

O

O

o O 10

PH S

o

o

s

I

Ss

N O N

II a -H o o .a a

PH

O

o

bO

O

O

r i 4

O O

bO r i 4

O

O

bC

bjO Ai

O

o

ft 3

O

o LO

o o C<l

o o

o o CM

a

s s

<

a S

H

a

s

LO

AJ

bO

a 'a p

o a o o

l-H OH

3000

L

oop

XC

3

T-H

Loo

p T

CI

cS PH j ^

m + 3

a ^ 4 J

1=1 ft

1—1

0 0)

4 J 03 I H

- t J

3 ft

03 PH

-^ cn + j

1 CU bC fl

cha

o o o o o o 1>- CO

ft 0 0

HJ

l -H

0 X

ft 0 0

1-J CN 0 X

220

CQ

a>

iab

3 >

-d

)lle

SH

cont

th

e I

o SH O

err

'o CQ -O o3

egra

]

fl 1—1

CO

CO

OJ

Tab

1—1

a.

o OH

S

ined

tr

a

cn fl O

O

o OH

ned

M

• 1—1

03

cons

fl

• ^

Si

S

C<l 5 5

S

S

CO ; = i

cq S i

55

^

S

C-1 SJ

I-H

S i

CN 1—1

CN ^ H

<d>

^ o CD

I - H

CN

CO

CO

o CN

^ O

CO

CX) 0 0

CO

CN

1-H

O

o CN

CO

o CO

C i

0.8

1.76

T3

^ U

'* 05 6? LO

+

LO CN

CN

o CO

CO CO

CO

LO

o CO

o CO

CO I-H

0.0

CN 1—1

CO

I—1 CO

o CO

CO 1—1

o o

CN ^ H

CO

r-H

CO

t J u <u

, ' ^ >< 6? L O

1

L O I-H I-H

N -r-H

CO

LO o CO

N -CN

o

LO CN CN

CO

0.0

CTi

CO

t^

i-H

o CN CN

LO

o CO

CN

0.9

1.79

•a w <u

- • ^

E-i

L O

1—1

CO I-H

CN I—1

CO

^ o CO

CN oq CO

LO CO CN

CO

0.0

1—1

CN

CO CO CN

CO CO

CO

I - H

I—1

CN

K)

-c; fe? o I-H

1

221

CD O fl 03

SH

OJ ft

6>i/6>i • los sa V 6>i/6>feos'sossav M ' I O S I V tJU'tOS ja^a-i

_ CD fl - ^

+3 4-i CD CJ

^ ^ CD O

S .tl

co" I

o CO -d fl 03

S

T3 CD

U-l

03 CN

°6 a CO o , ^

go

ft.2 fl OJ

—• CJ SH fl (D O

1=1 O

2 ^

CJ CO

o b PH T3

' s o) o fl CJ

. f l CD d M SH

+^ Cl cn >H

1=1 .2 O 4^ O 03 C SH ^ ^ CD <D

^ 8 8:2

S b • cS -c V S ^ O ft ^ ^ ftl'S

-^ ^. ^ CD ' ^ fl cn y^ O) - O >

fl

00 -S

^ g I S) £ fl

. .H . r t _^j M H &-I 03

cn O

O

222

B>|/6>( ,|.DS B>|/6>| •^^ '^d ' " ^ d

,puoo Bed d

s o3 CD

- | J

cn

' f t O

- ^ j

CD - f l 4 J

s o

^ • ^

cn SH

fe ai o fl

s-l fl

- ^ J

cn

^ <D

+ J 03 SH

^ o cfl cn cn ai g

X) <D

. 1 ^ U H

c6 M H

O <D o fl CD cn <D SH

ft CD

- f l

_fl

cn fl

_o +3 CJ 03

rol

ont

c; D PL,

-a CD fl '3

SH

nst

o o fl

^ OT , -H

CO CD s-l

fl bO

fo

LTD

H <+-! o <D SH

fl cn cn 0) SH

ft ; H

O ft 03

> >. s-l 03

X) fl O CJ OJ cn

a SH

^ •IS

.. CO 1

O CO -o fl 03

CN

6 CO t lH G cn CD S H

fl cn 73 CD SH

ft

i CO

-o fl O O CD

cn i-H

1

CO H H

O ID SH

fl cn cn CD SH

ft

223

p

CD

cn CD

.tos, B>l/B>|"-'^SaV B>|/6>| ' "^sa V ' "^sa V M ' l O S l V LU '10SI3A3-1

o3 SH

+ J fl CD

one

o cn

T3

'o cn

413 + j

Co' 1

O CO -a fl o3

CN

6 >> CO SH

- d

- d

fee

03

( 4 H O CD CJ

fl CD cn <U s-l ft <D

Xi -i-j

rt SH

a>

trol

fl o o

OH

S

ined

03 SH

-»-= cn fl O O fl

the

u U-l O CD CJ

B 03

a SH

c2 SH <D

ft ft O o

Td <D cn O

O

ctj CO

QJ SH

fl

fa

^(H

O t/3 fl _o o3 SH

+3 fl CD CJ fl o CJ

cn r2 'o cn >, SH

T3

T3 fl o O <D

cn

fl" _o 03 SH

- IJ fl CD O fl O CJ

cn -o ^ O cn

SH

CJ fl

x) o SH

ft

^ U ^

ft o -t-3

(D ,.fl

a o

^ -f^ cn SH

E CD CJ

fl 03

rO SH

fl cn

1 — ! 1

O CO t4H O > i SH O

- t ^ fl CD

> fl - > — 1

xi S H

fl o ^ H

! H

fl - t J 03 SH (D ft 6 CD

- t J

+ i CJ fl

T3 O s-l

ft

224

Bd'^^^d B>|/6)| '^^^d '^^^d ,puoo

Ed d

cn SH

fa CD CJ fl o3

Xi S H

fl - cn ^ fl

.2 ' • ^

cfi

entr

CJ

fl O o cn S "o cn

>. S H

T3

X) CD (D

^ o3 ^4H

o OJ o fl 0) cn (D S H

ft CD

-fl 4-3

,fl cn fl _o '•w CJ 03

ntro

l

o CJ

O PH

S X3 OJ

fl •3

SH - ^ J

cn fl 0 CJ

fl '^ I-H CN

CO

a; SH

fl , _bfl

Lfi

W • ^ H

0 CD SH

fl cn cn CD SH

ft ! H 0 ft

> SH Cfl

X ) fl 0 0 QJ m

XJ SH

s - ^ J

0 0 " 1

0 CO X3 fl o3

CN

0 CO SJH

0 cn CD S H

fl CO

cn OJ SH

ft

a CD

cn

XJ fl 0 CJ CD cn

I-H"

6 CO <4H

0 CD S H

fl cn cn 0) SH

ft

a o3 OJ

- ^ J

cn

ft;

225

fN O a

o CN o o o

6>l/6>l S a V B)(/B>) •"^sa V '"^sa V >1 ' lOSiV UJ 10S|3A31

cn

^ fa (D CJ

fl 03 JD SH

fl cn

X !

fl)

fl - tJ 03

CD

oT JH

fl en SH

m ft

a CD - M

fl X! o SH ft

X f >H

0'-^ a CD

X5 CD

03 M H f5

QJ

enc

UJ ID SH

ft (11

4 ^

CO

O CO T l

fl cri

CM

o ( / J 1*H

o rn

fl xi .2

CD (D 1=1 CJ O fl SH O

CJ X3

o cn

O PLH

•xs X3 CD fl 03 SH

cn fl O CJ fl fl CD

O CD CJ

a SH

,o

Xi fl O CJ CD cn

_o o3 SH

^ OJ CJ fl o CJ cn

O

ft SH

ft^ o 1

X3 CD cn O

O CM CM

O CO

CJ fl

X) 2 5 ft >,

s-l

O ft o

CO OJ

Xi CD +^

bjO p , f - l

fa

rfl SH fl

.O

226

03 CD

ft O

Bd'^^^d

• f CD (N 1 -•sr ^ ' ^ - ^

Bd '"^d ''""d

^ 2.2 T - CN

,PU03 Bd d

CD

o

CD V LO

^

• I-H

X )

ture

o3

per

S OJ

• « CD

t+H

o3 14H

O <U CJ

sen

CD SH

ft OJ

Xi - M

_fl CC fl

_o +3 CJ o3

O SH

-tJ fl o o

O fa S X3 CD fl

•3 SH

cn

con

fl

CO ( N

CO

QJ SH

fl bO

•1—I

fa

fl CC

pres

s-l 0 ft o3 P SH 03

•X} fl 0 0 CD cn

XJ SH

2 r .

CO

6 CO X ! fl o3

CN 1

0 CO M H 0

CO CD SH

fl CC cn CD S H

ft a OJ

- (J CD

XJ fl 0

sec

I-H" 1

0 CO <4H

0 OJ S H

fl cn CO 0) SH

ft

227

6>I/B>I' tOS 'X V B>|/6>|'£os'SOSXV >1 ' lOSiV uu ' l O S |8A31

CO

OJ CJ fl OJ cn OJ SH

ft CD

-fl 4 J

.a SH

,S 'o SH •h^

fl O CJ

O fa S X 3 CD fl

•3 S H

CO

fl 0 CJ fl fl OJ

- f l - t J

C4H 0

CD CJ

fl 03

a S H 0

t4H SH

CD

ft

03

CN 1

0 CO ' J H

0 cn C _o o3 SH

- t J

fl CD CJ fl 0 CJ

cn

73 3 cn

S H

T 3

XJ fl 0 CJ OJ cn

fl""

_o VJ o3 SH

4 ^ fl CD 0

fl 0 CJ

cn

r—t

0 cn

SH

ft X3

-loo

X ! OJ CO 0

0

'^ CM

CO

OJ s-l

fl fcuO Si

CJ

fl T 3 0 SH

ft

' f t 0

- t J

CD

r f l 4^

a 0

- M

1—t

1

CO

^ >. SH

0

fl <D

> _fl

r f l SH

fl 0

228

03 OJ

Bd '^^^d Bd '^^^d PUB ^^^d ,puoo

Bd d

ft o - ^ J

CD r f l - i J

a o

^ - t J cn s-l

fa ai o fl o3

Xi SH

fl - tJ cn

^ s-l

.CD t 4 H cn fl 03 SH

• ^

heat

o3 «4H

O OJ

enc

03 CD S H

ft CD

Xi - u

fl • -^ cn fl .2 '-^3

% •—^ o s-l - t J

fl O CJ

O fa S X3 OJ fl

•3 - f j

cn fl 0 CJ fl U)

LCD

fa u-i 0 OJ

fl cn cn OJ SH

ft S H

0 ft o3 P S H

Cfl

-d fl 0 CJ CD cn

X3 S H

13 -^^ CO~

6 CO

X3 fl a

CN

6 CO ^ H

0

cn CD SH

fl cn cn <D SH

ft

a 03 CD

- u cn

xJ fl 0 CJ OJ CO

^ r H

0 LO CO CM 'JH

CO

0) S H

fl bT)

fa

0

OJ S ^

fl CO GO CD SH

ft

229

Bt/BH' los sa V

O CD 9

B>|/6>|'EOS'20SSaV >l ' lOSiV <JU '1.0S |3A31

CO ! H

fa CD CJ fl o3

r Q S H

fl 4 J CO

^ CD

4J o3 s-l

^

fio

CO

^ a

-Td CD (D

t+H

03

C4H O aj CJ r^ 03

res

ft OJ

r f l 4 J

fl • ^

SH CD

trol

l

fl o CJ

O fa S XJ ID fl

•3

str

fl 0 CJ

t4H 0 CD CJ

fl cS

a s-l 0

<a S H

CD

ft

OJ SH

fl

S H CD ft

a 03

-1-^ 0 fl

X3 0 SH

ft

xi SH

2 4 J . CO

1

U CO X3 fl 03

CN

6 CO u-l 0 1/3 fl

, 0

03 SH

- H fl QJ CJ fl 0 CJ

CD

' 0 CC

>, S H

X!

-id fl 0 CJ OJ cn fl .2 + 3 o3

4 J fl QJ CJ

fl 0 CJ CO

X) ' 1

0 cn

s-l ft X3

-loo

X3 CD CC

0

0

CO ( ^ CO QJ SH

fl bO

• I-H

fa

0 fl

X3 0 S H ft

' f t 0

-4J

QJ

rfl -^J

a 0

l-H 1

CO f 1 1

0

>> SH

0 •g QJ

> .a r f l - * J

SH

fl -2

230

• ^

Bd '^^^d B d ' ^ = ^ d ' " ^ d

,puoo Bd d

i ft o CD

Xi

o , S H

cn S H

QJ CJ L O

-2 ^ 03 SH fl CO CC QJ

CC

o3 S H

o qfl 03

r-i '- '

X) QJ

o3

-o fl O ^ ^

QJ I f l

^ CO"

1^ ft CO

. 3 CN

Q CO

CJ

a ro

l

fl o CJ

O fa ^ X )

ine

03 SH

nst

o

r-rsi CO QJ s-l

fl bC

CJ

re CI) S H

fl cn

res

ft

a 03 03

c/>

X3 fl O o 03 CO

1-H

U CO <4H

o QJ

9 C/J

QJ

fa ft

231

CJ

fl X! O

B>l/B>|''-'''saV B^/B^ '"^SQ V '^^^sa v >I'L0S1V Ul LOS |9A31

fl .2 a

CO

X3

CO

Td fl ai

s-l CO

X) ° QJ cn

.Q:" fl

tjH g

03 *4J

- 2 fl QJ o fl O CJ QJ

r\ cn ft -^ CD ^

-fl O • ^ cn rH > .

fli x i fl o

SH

o CJ

O fa

o CJ QJ cn fl'~

_o '-13 03

Td P! S QJ

o3 O

§;H CJ - Q

^ H CO

a rS ^ r x l 'JH o o o

^ SH . !H ft p QJ ^ ^ SH

ft •• o ft p a o ° s; o fl

-Tl 03 . f l X ) - ^ . . Q "^ ^

^ a O r- ^

p^ O fl

OO -Jf 03"

^ S 1 QJ . S-l

s!i « Q

^ a a fa ,o .5

232

Bd'^^^d B>|/B>| ' " ^ d ' " ^ d ,puoo

Bd d

ft c 4 J

QJ

r f l - ^ J

a o

^ - t ^ 03 S H

fa QJ CJ

fl 03

r O S H

fl - t J CC

^ c O

ati

entr

CJ

fl o CJ 03

rs "o 03

>> S H

X) Td

QJ QJ

• 4 5

o3 t + H

O QJ CJ

fl QJ

cn QJ SH

ft QJ

-fl - t J

fl • r H

cn

ion,

- t i CJ

cS

"o ( H

^ o CJ

U fa S X)

QJ

fl •3

SH

^

1 0

Ei M H

0 QJ SH

fl 03 03 QJ S H

ft S H

0

9-£? > SH

03

X )

fl G C3 QJ cn

Td S H

3 - I J

co" 1

0 CO

X3 fl o3

C N 1

0 CO •4-1

0 03 03 SH fl 03 03 QJ S H

ft

a o3 QJ

4 J 73

Td fl 0 C3 QJ CC

I -H 1

n fl ^ 0

0

ai CN CO

QJ

igur

r -PH

o QJ U

fl cn 03 03 S H

ft

i 03 4 J 03

233

B>i/6>)''-^^sav 6)(/B>i'^^^sa V'^^^sa V M ' l -os iV UJ 'LOS (S'lSI

4 J

SH

fa

n> o fl 03

r Q I H

^ rn

T d

QJ S H

fl

rat

CD U (

a 03 4 J

T3 Q) OJ

t4H

03

M H

O CD CJ Cl Q)

res

o, 03

r f l 4 J

fl ''—1

SH Cl)

lizj

'. S H

fl 4 J Cfl

OJ ft

fl OJ

4 J CJ fl

X3 O S H

ft

ird

r l - t J

., CO

C) rn x i fl 03

CN 1

CO '4-1

o 03

fl o

rat

4 J fl C13 CJ fl C> CJ

c ; S H

4 ^

fl o CJ

O fa S T d 03 fl

•3 ^ 4 J cn fl 0 0

M H

0 QJ CJ fl a a SH

0 <4H SH QJ ft

ft 0

^ 1

XJ OJ 03 0

1 1

0

CO CO CO

CD S-l

fl bfl

fa

cn T d 'Xi 0 03

>, ^ T3

T d fl 0 CJ QJ 03

r .

fl _o '-3 03

SH - t J

fl QJ CJ fl 0 CJ

03

73 0 cn

> v S H

2 ^ fl CO

p 0 S H

ft >. S H

0 CD

-^ > OJ fl

r f l - ^ -t^ . . fl - ^

a t, 2 3 t-B 0

234

o CO

ft o

a o

- f l ft o3 SH

bO

fa P OJ cn O 03

• QJ s-l

ft fl 03

Xl O

cfl SH QJ ft OJ

03 Td fl G o

03 LO

OJ . . ,QJ - f l "^ ft

o3 o3

Q "2 o .fl fl rfl OJ 4 J 03 CD 03 SH OJ

ft ^ QJ M

rfl 03 4J O J

s-l

.a ^ 03 S fl 03 o OJ

o3 CO

G - g Cfl SH

"S CN G f^

" 6 O CO

fa ^ S ft « bO '^ X i

fl 03 SH

Bd'^^^d B d ' ^ ^ ^ d ' ^ ' d Bd'P"°=d

03 fl G

o CO ^

03 CO

QJ SH fl

SH

ft

i fa ^

235

4d ^

P

6>l/6)| • LOS SO V 6>|/6>|'EOS'SOS s a V > i ' l O S i V UJ LOSIS'^BI

QJ O fl 03

r Q

QJ SH fl

- tJ o3 I H

QJ ft

CO u

S H O QJ SH

• J H S 03 ft fl .. g Td fH U

4. .f l I f l ^

03 O ^ H ' ^ G - o QJ fl CJ o3 fl CN QJ V OJ ' w ' !H CO ft 4 H QJ G

^ 03 - t ^ fl

fl .2

r2 -t

R 8 O X3 fa fl^

XI b

o3 - O

03 O

fl ' O '^ r, cn

QJ CJ fl 03

fl _o 4 J o3 SH ^ QJ C3 fl O C3

ft 03

ftS 2 ^ O 03 X) QJ 03 G

o CM CO

CO

> 5 S H

X3 - tJ C3 fl

X) G S H

ft

QJ ft b c) fl -* bO 03

fa S

o CO '^-l

O

236

o3 QJ

O CO

Bd '^^^d Bd '^^^d pue ^^^d ,puoo

Bd d

ft O

r f l ft 03 S H

bjO

OJ CJ fl 03

X SH

fl 03

'-B SH

03 =4H 03 fl Cfl S-l

-IS

-1-^

a QJ r f l

03 <4H

G 03 CJ CJ 03 03 QJ SH

ft 03

Xi • ^

fl

03

fl _o 4 J CJ

cfl

.__, G SH

4 J fl G C3

o fa ^ X ) QJ fl o3 S H

cn fl O

o CO CO

CO

QJ S-l

fl hO

fa

3 cn 03 03 SH

ft S H

G

ft 03 > >. SH 03

X3 fl G CJ QJ 03

LO

fa i5 ft o3 SH

bO

X3 SH

'S 4 J

.. cn 03 SH

fl 03 03 QJ SH

ft

a 03 QJ

- ^ J 03

CO

Td fl 03

CN 1

o CO

r f l ft Cfl SH bO

X I fl G CJ QJ 03

Qj"" SH

fl 03 CO QJ SH

ft

237

B>j/6>|'l'='SsaV B>|/B>| ' " ^ s a '^^^sa V > l 'LOSiV ui'LOSISAsn

the

a o

<4H

+ J C/J

fa

nee

03 r O

71 4 .3 OJ

rth:

fl o M H

QJ SI 1

03

QJ

ft

tern

4H CJ fl

X I O

03 X i

^ o cfl

mas

s

X l QJ Cl)

o3 =4-1 O 03 CJ c"! QJ ro Cl>

ft QJ

r f l

r f l

CO

SC-

X I

fl 03

CN

o CO

'^-l

n cn

fl O

03 SH

fl

G SH

4J fl G CJ

CJ fl O CJ

SH

XJ

fa -o QJ fl

- f l G 4 J CJ •41 ' ^ ^ 03

tu fl" fl .2

o fi <4H QJ

I H CJ

S fi ft O

ft "

§3 Td o 5 cn cn 1^ S ^ ^ o -« 6 .. tJ '^

'^ fl '^ CO Td

« 2 „ OJ ft G

fa 5 .a

SH

238

03 IH fl 03 03 QJ

03 03

Bd '^^^d

^ CN

Bd'"^d'"=d

•sr

2.15

2.1 in CN o

(N

,puoo Bd d

r f l ft Cfl IH

c

£ Q)"

'

QJ ^ CJ ffi y QJ CD SH

r Q ft I H fl >-•

S G 03 ft

^ ^ -^3 i>^

I H CO

^ ^ G G

'fl y OJ O] 03

d l O

a fa Td . . 9. -fi

^ ft ^ cS cfl I H

^ H bO

° X3

ence

th

ir

^ I ft S QJ 03

fl ft

fl o3 O QJ

• S 4-= 4-3 03 CJ 03 CO

I'B - t^ cd

§ ^

" 6 fa^

• ^ - ^

i2 ft CO 03

^ bi) QJ SH fl bO

xd fl

u*J u fa 03

239

6>l/6>|''-^^SaV' B>i/B>i '^^^sa ' " ^ s a V > I 'L0S1V LU 'LOS is '^a i

03 03

• - fl fa S cfl

. S H

QJ CD

fl ^ O QJ u -^ fl 4^

4 J C3 CO ; 3

^ X i

fl 2 o ft o3 SH

• ^

fl 03 CJ fl O C3

03

'o 03

X) tH

Td QJ

. 0 3

O CO

Td c 03

CN

'6 C O

^ 03 QJ h

fl ^ s g 03 3 QJ g SH O

ft CJ

^§ .3 "" ^ T l

r2 ' o Td '-I fl

o " R QJ CJ CC

fa fi ^ G

QJ

QJ O

Ifl tH -^ fl QJ CJ fl G CJ

^ ^ QJ CO

ft >>

^^ ^ fl CO

" ^ ^ H

2 ^ ft >.

.. ^ ^ CO o S CO - ^ >

c d ^ . a QJ -^^

T d 03 03

S a bO p

fa

rfl -t3 (H fl

240

o3 QJ

o CO

Bd'^^^d B d ' " ^ d ' " ' d ,PU03

Bd d

a G

G . IH

rfl ft 03 IH bO

o a

IZ •p

ime,

1

1—

o LCI

t i l

QJ CJ

fl o3

Xi ^ fl

4-3 03

'-B fl .2 4-3 03

ncen

tr

O CJ

03

3 "o 03

SH

Td Td QJ QJ

<4H

03

QJ I H

fl

ess

I H

ft SH

G ft

f SH 03

Xi

seco

n

l O

H

rfl ft 03 SH

bO Td

SH

3 43

CC QJ

44 iH O fl CJ QJ

IH

ft

ft o3 QJ

QJ 4J rfl CO

- ^ CO

.a XI CO 3 fl "

.2 ' 5

1^ 03 CO

-a ft fi 03 G IH O bO

hH r^

fa fl G

CO CC

^ Qj" OJ H iH fl rH 03

to S •— IH

fa ft

241

QJ

-fl 4 3

a o

<i3 4-3 1/1 S H

• rH

fa QJ CJ

fl Cfl

tur

rn

x i QJ S H

fl Cfl SH QJ

ft

a 4-3

x i 03

,CD C4H

r f l I H

fl n <4H

r .

S 4 3

era

ft

a 03 - ^ J

+ 3 C)

fl -d u I H

ft

Td IH

-fl - ^ 3

CO 1

O UJ

X I fl 03

03 <N

B>l/B>| • '-'^sa V 6))/6>| '"SgQ V '^^^sa V >i'^==lV uJ'LOSiaASl

242

fl CO 03 QJ

o3 QJ

Bd'^^^d

'S - CN ^

B d ' " ^ d ' " ^ d

3.8

2.2 T - CN

CN

,puoo Bd d

u CO

ft G

rfl

S-l

o o —

£

'£ Q)' £

1-

o un

o

. fl CD CO CJ cn S QJ 03 ' r v

- Q ^ s-l I H

fl o -^ ft .23 Cfl Td > QJ ^ I H h

fi iS 4 3 T d 03 S

03 8 ft OJ

a " OJ LO -t3 1

Td W QJ . .

QJ Xi ' ^ ft o3 o3

SH

^ bO

OJ ' 2 CJ . f l fl rfl QJ 43 cn

pre

res,

r f l 03 4 J 03 _ I H

.3 ^ fi 1 O QJ

4 3 - t^ -g cn 03 CO

"o -^ ti fl ^ ' O CN

fa ^ ,:..• r f l

s ft CO 3^ ^ bb

Figu

re

seco

nd

243

> ft O fl

QJ

6>l/6>1 ' LOS s a V B v B V E O S ' S O S S a V > l ' L O S i V iiJ'L0S|3Aa-|

o -^ ih IH

c i fl G

4 3 <44 cn

• ti CD fa ^ QJ CJ fl o3

rO SH

fl

Td

o3

QJ

ft

CJ

fl Td G QJ

03 ft fl

03 -^^ OJ

r f l CO • ^ I

a O •44 CO G r ^ 03 fl CJ 03 fl 03 cn r\

fel ^ ft "44 QJ G

r f l cn 43 fl

fl -2

CN

03 03 s-l

O 8 j3 fi o " O 03

HH : 2

QJ tn r f l > . 4 3 SH

G ..

fl § g QJ a to

t l .2 03 4 3

ft g

& fi ° 8

fl G C3 03

-fl O 03

Xl QJ 03

_o

^ SH

fl-5 o bO o ^

•rH j _ | ^_ |

fa ft G

244

B d ' l O S d Bd EOSd 'SOSd Bd 'puood

X! fl G C3 QJ 03

Qj"

03 03 03

o3 03

O CO

ft O

O , S H

r f l ft Cfl S-l

bJ3

fa *-^ QJ

• SH OJ fl CJ 03 fl cn

3 S SH O ft

>> SH 03

Td fl O O QJ 03 Cfl

QJ r f l

cfl fa H H

QJ -5 tJ S* fi S

ft .a ^ 5 - ^ 03" H QJ

2 S ! H 03 G QJ

• •g ft r^ a O 03 SH 03

4 3 4 3

fl tn

8 ^ '-' "2 fa CN •^ 6 1 ^ CO

QJ . .

fi ft bO ifl

fa bO

245

o L O

x L

6>|/6>|'''°\v .POJd — >\ i V LU |8A91

O S H

o CJ

O fa

xi QJ

fl

03

fl G O fl fl 03 <4H

-fl O • ^ cn bC fl fl O

CJ

fl Td O s-l

a XJ iH

-^3

co" I

O CO X)

I

O CO

03

fl QJ CJ fl o3

r Q

cfl S H

^ OJ C3 fl O C3

Td 03

o3

O 03

Cfl ' d 03 03 03

Td QJ

,QJ

QJ X3 G

fl QJ CJ fl O o

fl 03

Q ^ fa " QJ b

rfl Td 2 O

03 X l 03 o

CO

fl G ft 03 03

O

. . S H

- ^ G ft -S o fi I ft 43

r2 -fl 1 4 J

o § b r2 ±J fl

6i 2 ^ fa CO ^

QJ 52 S H fl

fl o QJ ft

bO -43 a rT' 2 tD M H o3 4 3

246

CHAPTER 7

SUMMARY AND FUTURE WORK

7.1 Modeling

This work developed a fundamental model of a multiple effect falling film evap­

orator 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 hydrody­

namic 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 us­

ing 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 inte­

grated 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 (proportionally-

integral 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 Centralized control

7.4.1 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 concen­

tration 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 sig­

nificant 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 for­

mulations 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 con­

centrations 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 lu 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 ro­

bustness. 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 crystahiza-

tion 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 concentra­

tions 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

BIBLIOGRAPHY

[1] G. Hough. Chemical Recovery in the Alkaline Pulping Processes. TAPPI Press, Atlanta, GA, 1st edition, 1985.

[2] P. E. Minton. Handbook of Evaporator Technology. Noyes Publications, New Jersey, NJ, first edition, 1986.

[3] R. Billet. Evaporation Technology. VCH Pubhshers, New York, NY, first edition, 1974.

[4] L. C. To, M.O. Tade, and M. Kraetzl. Robust Nonlinear Control of Industrial Evaporation Systems. World Scientific, New Jersey, NJ, first edition, 1999.

[5] Y. Yamamoto, N. Mitsushi, and S. Kadoya. Design and operation of evaporators for radioactive wastes. Technical Report 87, IAEA, Vienna, Austrua, 1968.

[6] K. Vijayaraghavan, A. Nikolov, and T. D. Wasan. Radioactive waste evaporation: Current methodologies employed for the development , design and operation of waste evaporators at the Savannah River site and Hanford waste treatment plajit. The 9th International Conference on Radioactive Waste Management and Environmental Remedation,Examination School, Oxford, England, icemOS-4515(WSRC-MS-2003-00050, Rev.O), 2003.

[7] W. L. McCabe, J. C. Smith, and P. Harriot. Unit Operations of Chemical Engi­neering. McGraw-Hih, Inc., New York, NY, fifth edition, 1993.

[8] A. Niemi and R. Koistinen. Dynamic Modeling of Multiple-Effect Evaporator Plant. Paper och Tra, 54(2):297-306, 1972.

[9] A. Niemi, E. 0 . Hartikainen, and R. Koistinen. Dynamics and Control of Multiple-Effect Evaporator and Drum Washer Plants. Proceedings of 4ih IFAC/IFIP International Conference on Digital Computer Applications to Pro­cess Control,Zurich, pages 503-519, 1974.

[10] L. N. Ricker and T. Seweh. Dynamics of Multiple-Effect Evaporators: Computer Simulation and Plant Tests. AIChE Symposium Series: Applications of Chemical Engineering in the Forest Products Industry, 80:87-94, 1984.

[11] A. J. L. Cardoso. Modelizagdo e Controlo Robusto de uma Cadeia de Evaporagdo. Master of science, Universidade de Coimbra, Coimbra, Portugal, 1995.

[12] M. G. Kelly and L. L. Edwards. Two-Phase Flow and Heat Transfer in Rising Film Evaporators: A Distributed Parameter Model. AIChE Symposium Series: Applications of Chemical Engineering in the Forest Products Industry, 80:95-109, 1984.

[13] F. P. Incropera and D. P. DeWitt. Introduction to Heat Transfer. John Wiley & Sons, Inc., New York, NY, third edition, 1996.

255

[14] M. Kaviany. Principles of Heat Transfer. John Wiley k Sons, Inc., New York, NY, first edition, 2002.

[15] P. Schutte, A. Gorak, and L. Jeromin. Stationare und dynamische Simulation eines Fallfilmverdampfers. Chem.Ing.Tech., 64(5):468-469, 1992.

[16] I. S. Bojov. Basics of Automatic Control. Technika, Sofia, Bulgaria, third edition, 1992.

[17] J. A. Richalet, A. Rault, J. D. Testud, and J. Papon. Model predictive heuristic control. Automatica, 14:412, 1978.

[18] E. C. Garcia and M. Morari. Internal model control: a unifying review and some new results. Industrial Engineering Chemical Process Design and Development, 21:308-323, 1982.

[19] C.R. Cutler and B. L. Ramaker. Dynamic matrix control a computer control algorithm. AIChE 86th Annual Meeting, Paper WP5-B, 1979.

[20] P. Gil, H. Ramos-Duarte, and A. Dourado. Dynamic Modelling and Optimal Hierarchical Control of a Multiple-Effect Evaporator-Superconcentrator Plant. 17th IFIP Conference on System Modeling and Optimization, Prague, Czech Republic, 1995.

[21] E.F. Camacho and Bordons C. Model Predicitve Control in the Process Industry. Springer Verlag, New York, first edition, 1985.

[22] W. J. Frederick, T. Noopha, and M. Hupa. Combustion behavior of black liquor at high sohd firing. TAPPI Journal, 74(12):163-170, 1991.

[23] R. Karvinen, P- Hyoty, and P. Siiskonen. The effect of dry solids content on recovery bofler furnace behavior. TAPPI Journal, 74(12): 171-177, 1991.

[24] P.L. Kapitza. Collected Papers of Kapitza. Pergamon Press, Oxford, United Kingdom, 1 edition, 1965.

[25] V. G. Levich. Physicochemical Hydrodynamics. Prentice Hah Inc., Englewood Cliffs, NJ, 1962.

[26] W Wilke. Warmeiibergang an rieselfilme. VDI Forsh.Heft, 490, 1962.

[27] Limberg. H. Warmeiibergang an turbulente und laminare rieselfilme. Int. J.Heat Mass Transfer, 16:1691-1702, 1973.

[28] K. R. Chun and R. A. Seban. Heat transfer to evaporating hquid films. J. Heat Transfer, 93:391, 1971.

[29] K. R. Chun and R. A. Seban. Performance prediction of falhng-film evaporators. J. Heat Transfer, 94:432, 1972.

[30] R. A. Seban and A. Faghri. Evaporation and heating with turbulent falling liquid films. J. Heat Transfer, 98:315-318, 1976.

256

[31] H. Brauer. Stromung und warmeiibergagn bei rieselfilmen. VDI Forsh.Heft, 457 1956.

[32] A. S. Telles. Liquid Film Characteristics in Vertical Two-Phase Flow. Doctor of phylosophy. University of Houston, Texas, USA, 1968.

[33] E. R. vanDriest. On turbulent flow near a wah. J.Aeronaut.Sci., 23(11):1007-1011, 1956.

[34] D. K. Chung and A. F. Mflls. Heat Transfer Across Turbulent Falling Films. Int. J. Heat and Mass Transfer, 16:694-696, 1976.

[35] S. V. Alekseenko, V Ye. Nakoryakov, and B. G. Pokusaev. Wave formation on a vertical falhng liquid film. AIChE Journal, 31:1446-1460, 1985.

[36] H. Chang, E. A. Demekhin, and E. Kaladin. Simulation of noise driven wave dynamics on a fahing film. AIChE Journal, 42:1553-1568, 1996.

[37] E. Stuhltrager, A. Miyara, and H. Uehara. Flow dynamicas and heat transfer of a condensate film on a verical wall-ii. Flow dynamics and heat transfer. Ind.J.Heat Mass Transfer, 38(15):2715-2722, 1995.

[38] A. Miyara and H. Uehara. Flow dynamics and heat transfer of a condensate film on a vertical wall - ii. flow dynamics and heat transfer. Int. J.Heat.Mass. Transfer, 38(15):2715-2722, 1995.

[39] A. Miyara. Numerical analysis on flow dynamics and heat transfer of falling liquid films with interfacial waves. Heat and Mass Transfer, 35:298-306, 1999.

[40] A. Miyara. Flow dynamics and heat transfer of wavy condensate film. Transac­tions of ASME, 123:492-500, 2001.

[41] A. A. Alhusseini. Heat and Mass Transfer of Falling Film Evaporation of Viscous Liquids. Doctor of phylosophy, Lehigh University, Bethlehem, Pennsylvania, 1994.

[42] A. A. Alhusseini, K. Tuzla, and J. C. Chen. Falhng film evaporation of single component liquids. Int. J .Heat Mass Transfer, 41:1623-1632, 1997.

[43] A. A. Alhusseini and J. C. Chen. Transport phenomena in turbulent falling films. Ind.Eng.Chem.Res., 39:2091-2100, 2000.

[44] R. B. Bird, W. E. Stewart, and E. N. Lightfoot. Transport Phenomena. John Wiley & Sons, New York, NY, 2 edition, 2000.

[45] L Prandtl. Zeits.f.angew.Math.u.Mech, 5:136-139, 1925.

[46] F. Blangetti. Lokaler Warmeiibergang bei der condensation mit iiberlagerter Zwangskonvection im vertikalen Rohr. Doctor of phylosophy. University of Karl­sruhe, Karlsruhe, Germany, 1979.

257

[47] G.L. Hubbard, A.F. Mflls, and D.K. Chung. Heat transfer across turbulent falling film with concurrent vapor flow. J. Heat Transfer, 98:319, 1976.

[48] D. K. Chung and A. F. Mills. Experimental study of gas absorbtion into turbu­lent falling films of water and ethylene glycol mixtures. Int. J. Heat and Mass Transfer, 19:51, 1976.

[49] Y. S. Won and A. F. Mills. Correlation of the effects of viscosity and surface tension on gas absorbtion rates into freely falling turbulent liquid films. Int. J. Heat and Mass Transfer, 25:223, 1982.

[50] K. Stephan. Heat Transfer in Condensation and Boiling. Springer-Verlag, Berlin, Germany, 1 edition, 1992.

[51] W. M. Kays and M. E. Crawford. Convective Heat and Mass Transfer. McGraw-Hfll, New York, NY, 2nd edition, 1980.

[52] W.L. Badger and W.L. McCabe. Elements of chemical engineering. McGraw Hill Book Company, Inc., New York, London, second edition, 1936.

[53] P.P. Stainthorp and J.M. Allen. The development of ripples on the surface of hquid films flowing inside a vertical tube. Trans. Inst. Chem. Eng., 43:85, 1965.

[54] I. Mudawar and R.A. Houpt. Measurement of mass and momentum transport in wavy-laminar falling liquid fllms. Int. J.Heat.Mass.Transfer, 36(17):4151-4162, 1993.

[55] S. Jayanti and G.F. Hewitt. Hydrodynamics and heat transfer of wavy thin film flow. Int.J.Heat.Mass.Transfer, 40(1):179-190, 1997.

[56] S.S. Kutateladze. Fundamentals of Heat Transfer Academic Press Inc., New York, NY, translated from second edition, 1963.

[57] E. Pohlhausen. Z.angew.Math.u.Mech, 1:115, 1921.

[58] C O . Bennett and J.E. Myers. Momentum, Heat and Mass Transfer. McGraw Hfll, Inc., New York, NY, second edition, 1974.

[59] A.A. Khahd, S.R. Hosahahi, G. Doyon, and C. Toupin. Effects of concentra­tion and temperature on carboxymetylcehulose rheology. Int. J.Food Sci.&Tech., 29:243-253, 1994.

[60] Z. I. Stefanov and K. A. Hoo. A Distributed-Parameter Model of Black Liquor Falling Film Evaporators. Part I. Modeling of a Single Plate. Ind.Eng. Chem.Res, 42:1925-1937, 2003.

[61] T. N. Adams. Sodium salt scaling in black liquor evaporators and concentrators. TAPPI Journal Peer Rewieved Paper, 84(6): 1-18, 2001.

[62] S. J. Severtson, R Y. Duggirala, P. W. Carter, and P. E. Reed. Mechanism and chemical control of calcium carbonate scaling in the kraft process. TAPPI Journal, 82(6): 167-174, 1999.

258

[63] A.L. Pricke and M.A. Stoy. Development of a method for measuring the heat of dilution of kraft black hquor and water. TAPPI Journal, 77(8): 169-174, 1994.

[64] A.L. Fricke and M.A. Stoy. Enthalpy concentration relations for black hquor. TAPPI Journal, 77(9):103-110, 1994.

[65] R. H. Perry, Green D. W., and Maloney J. 0 . Perry's Chemical Engineer's Handbook. McGraw-Hill, New York, NY, 7th edition, 1997.

[66] J.C. Patterson, T. Graham, W. Schopf, and S.W. Armfield. Boundary layer development on a semi-infinite suddenly heated vertical plate. J.Fluid Mech, 453:39-55, 2001.

[67] B. A. Finlayson. Nonlinear analysis in chemical engineering. McGraw-Hih, 1980.

[68] A.C. Hindmarsh. Odepack, a systemized collection of ode solvers. Scientific Computing, I of IMACS Transactionson Scientific Computing:55-64, 1982.

[69] A.C. Hindmarsh and L.R. Petzold. Algorithms and software for ordinary differen­tial equations and differential/algebraic equations-part i. Computers in Physics, 9(1):34-41, 1995.

[70] A.C. Hindmarsh and L.R. Petzold. Algorithms and software for ordinary dif­ferential equations and differential/algebraic equations-part ii. Computers in Physics, 9(2): 148-155, 1995.

[71] C. Tischendorf. Solution of index-2 differential algebraic equations and it's ap­plication in cirquit simulation. Doctor of phylosophy, Humboldt-Universitat, Berhn, Germany, 1996.

[72] Black Liquor Recovery Boiler Advisory Comitee. Safe firing of black liquor in black liquor recovery boilers, www.blrbac.org, 10 2003. Freely available to the public.

[73] E. H. Bristol. On a measure of interactions for multivariable process control. IEEE Trans. Auto. Control, AC-11:133, 1966.

[74] D. E. Seborg, T. F. Edgar, and D. A. Mellichamp. Process Dynamics and Control. John Wiley and Sons, New York, first edition, 1989.

[75] G. Stephanopoulos. Chemical Process Control. Prentice-Hall, NJ, fhrst edition, 1984.

[76] C. E. Garcia. Quadratic dynamic matrix control (QDMC) of nonlinear pro­cesses: An apphcation to a batch reaction process. AIChE National Meeting, San Francisco, CA, November 1984.

[77] G. F. Franklin, J. D. Powel, and M. L. Workman. Digital Control of Dynamic Systems. Addison Wesley, Boston, MA, 1994.

[78] E. C. Garcia, D. M. Prett, and M. Morari. Model predicitve control: Theory and practice a survey. Automatica, 25(3):335-348, 1989.

259

[79] D. W. Clarke, C. Mohtadi, and P. S. Tuffs. Generalized predictive controkl.the basic algorithm. Automatica, 23(2): 137, 1987.

[80] D. W. Clarke, C. Mohtadi, and P. S. Tuffs. Generalized predictive con-trokll.extensions and interpretations. Automatica, 23(2):149, 1987.

[81] L. Ljung. System Identification. Prentice Hall Inc., Englewood Cliffs, NJ, 1962.

[82] W. J. Rugh. Linear System Theory. Prentice Hall, Upper Saddle River, NJ, second edition, 1996.

[83] P. Dorato, V. Cerone, and C. Abdaflah. Linear Quadratic Control: An Intro­duction. Prentice Hah, Englewood Cliff, NJ, 1995.

[84] P J. Antsaklis and A. N. Michel. Linear System. McGraw-Hih, New York, NY, 1997.

[85] J. L. Clement and J. W. Blake. Economics of high solids black hquor combustion. TAPPI Engineering Conference, pages 243-252, 1996.

[86] I. A. Mudawar and M. A. El-Masri. Momentum and heat transfer across freely-falling turbulent liquid films. Int.J.Heat.Mass.Transfer, 12(5):771-790, 1986.

[87] J. C. Slattery. Advanced Transport Phenomena. Cambridge University Press, New York, NY, first edition, 1999.

[88] J. Guihchsen, J. Guihchsen, and H. Paulapuro. Chemical Pulping. Fapet Oy, Atlanta, GA, 1st edition, 1999.

[89] P. Ramamurthy, A. R. P. van Heiningen, and G. J. Kubes. Viscosity and thermal conductivity of black liquor. TAPPI Journal, 76:175-179, 1993.

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 = - ^ , (A.2) pcp

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 = - , (A.3) at

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 di­

rection of the flow {x) is uniform, thus

d ( Tn - T w

dx \ Ty, - T M = 0 (A.6)

j^udy

where TM is the mean film temperature.

Table A.l: Dimensionless Variables

Transformed Definition Variable

pCpU*{Ty,-T) T. . , , , T^ = — Liquid temperature

qw

Spatial coordinate normal to the wall

Liquid velocity

Fflm thickness

u* = ^/g5 Friction velocity

For constant flux at the wah, Qy,, the term {Ty,—TM) is uniform in the x direction.

Therefore

dT _ dTyj _ dTM / . o\ dx dx dx

y+

u+

6+

=

=

"^Z.

u'y

U

u*

u

263

The energy equation for a Newtonian fluid with constant thermal conductivity and

density is given by [87]

pCp dT dT dT dT g,+^^^ + ^y dy dz

, ,'d^T d^T d^T\

dx'^ dy'^ dz"^ dux

dx + dUy

dx

dUy

dx

+/i \dy dx J {dz dx J \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 = Ux{y) (A.IO)

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 pCp u-

dx dy (A.12)

264

Using Equations (A.5), (A.7) and (A.12) the dimensionless form of the heat transport

equations can be obtained as

T+ =

/ u'^dy'^

Re

T q

Pr Prt V

/ T+u+dy+ Jo

Re

T

(A.13)

(A. 14)

(A.15)

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

h*H qw'''

2/3 S+V3pr

{Ty,-TM)\g"^ T+

5+V^Pr

(

&+

1 - Jo u'^dy'^

\ -dy^. (A.16)

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

(5+'Pr h-E = —. - ^ • {A.18)

/ X-X7I\*^ Pr Prt Vi/

266

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).

\

I dz

5

- 1 J

xdz 1

pg(5-

^

•y)dz

'

7

'

Figure B.l: Force balance on a fluid element in the falling film.

The following holds for the time averaged velocity components:

Vx =0

v„ = 0

Vz = Vz{y).

(B.l)

(B.2)

(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 d{u^) •+ p, yd Jo y Jo Jo dy

or

, dVz ,.r> a\ T = -pVyVz + p - ^ . (B.6)

268

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--. (B.7 dy

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-('-;)f- i - ' dy

The dimensionless form of Equation (B.9) is given by

e\ dvt

using the variable transformations given in Table B.l.

269

Table B.l: Dimensionless Variables

Transformed Definition Variable

y+ = Spatial coordinate normal to the wall

(J+ = ^ Fflm thickness

Vz

f;+ = —p= Liquid velocity 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 <T < 450K

0 kg/kg < X < 85 kg/kg,

where T is the black liquor temperature and X is the dry solids concentration.

Kinematic viscosity [88]

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 <T < 450K

0 kg/kg < X < 85 kg/kg

V

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 centi­

grade, however the in this study the correlation is extrapolated to the temper­

ature range indicated above.

Heat capacity [1]

=(0.98 - (-9.122 X 10~^r + 0.793)X)4190

2,Q2K <T < 450K

0 kg/kg < X < 85 kg/kg

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- ^ 1 - X

0 kg/kg < X < 80 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

C O M P U T E R P R O G R A M S

D. l Mat lab 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); %Cal-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

%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 con-centration% 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

Twal l in= (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; dT l = Twallout-Tboil; dT =(dT2-dTl) / log(dT2/dTl) ; 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 declarations%

a = *; % 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> ts tep %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 derivatives% 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 outputs%

case 2 , 4 , 9 %Unusedflags% sys = [];

otherwise

278

error (['Unhandled flag = ',num2str(flag)]); %Error handling% end %End of csfunc%

%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 ) = eqth2o(u(6)); Tsat(2) = eqth2o(y(ll)); Tsat(3) =eqth2o(y(17)); Tsat (4) = eqth2o(y(23)); Tsat(5) = eqth2o(y(29)); Tsat(6) = Tsat(5); Tsat(7) = Tsat(5);

%>Recirculation flows no flash%

Grecnf(l) = Gpump(l) u( l ) Grecnf(2) = Gpump(2) u(2) Grecnf(3) = Gpump(3) u(3) Grecnf(4) = Gpump(4) u(4) Grecnf(5) = Gpump(5); Grecnf(6) = Gpump(6); Grecnf(7) = Gpump(7) u(5);

%iRecirculation compositions no flash, E-5 to E-2'%

Xrecnf(l) = y(4); Xrecnf(2) = (y(4)*u(l)-fy(9)*(Gpump(2)-u(l)))/Gpump(2); Xrecnf(3) = (Xrecnf(2)*u(2)+y(15)*(Gpump(3)-u(2)))/Gpump(3); Xrecnf(4) = (Xrecnf(3)*u(3)+y(21)*(Gpump(4)-u(3)))/Gpump(4);

%)Recirculation temperatures no flash, E-5 to E-2%)

lYecnf(l) = y(5); IVecnf(2) = (y(10)*(Gpump(2)-u(l))+y(5)*u(l))/Gpump(2); 'IVecnf(3) = (y(16)*(Gpump(3)-u(2))+y(10)*u(2))/Gpump(3); 'IVecnf(4) = (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)<y(32))

Gbout(5) = -Gbout(5); %// inverse flow% end;

else

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)<y(37))

Gbout(6) = -Gbout(6); %// inverse flow%o end;

else Gbout(6) = 0; 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)4-y(28)*Gbout(5))/Gmixout(6);

end;

%}Recirculation composition and temperature SC-2%o

Xrecnf(6) = Xmixout(6); Trecnf(6) = Tmixout(6);

%)Recirculation composition and temperature SC-l%o

Xrecnf(7) = (y(33)*Gbout(6)+y(38)*(Gpump(7)-Gbout(6)))/Gpump(7); lVecnf(7) = (y(39)*(Gpump(7)-Gbout(6))+y(34)*Gbout(6))/Gpump(7);

%)Flash detection%o

for i = 1:7 if Ti-ecnf(i) > 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) = Ket(y(2),Gin(l)/pl(l),Tbl(y(2),Tsat(l))); Ketb(2) = Ket(y(7),Gin(2)/pl(2),TbI(y(7),Tsat(2))); Ketb(3) = Ket(y(13),Gin(3)/pl(3),Tbl(y(13),Tsat(3))) Ketb(4) = Ket(y(19),Gin(4)/pl(4),Tbl(y(19),Tsat(4))) Ketb(5) = Ket(y(25),Gin(5)/pl(5),Tbl(y(25),Tsat(5))) Ketb(6) = Ket(y(31),Gin(6)/pl(6),Tbl(y(31),Tsat(6))) Ketb(7) = 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) = y(l)*ut(y(2),Gin(l)/pl(l),Tbl(y(2),Tsat(l)))/L; Gout(2) = y(6)*ut(y(7),Gin(2)/pI(2),Tbl(y(7),Tsat(2)))/L; Gout(3) = y(12)*ut(y(13),Gin(3)/pl(3),Tbl(y(13),Tsat(3)))/L Gout(4) = y(18)*ut(y(19),Gin(4)/pl(4),Tbl(y(19),Tsat(4)))/L Gout(5) = y(24)*ut(y(25),Gin(5)/pl(5),Tbl(y(25),Tsat(5)))/L Gout(6) = y(30)*ut(y(31),Gin(6)/pl(6),Tbl(y(31),Tsat(6)))/L Gout(7) = 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

((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)));

W(3) = (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)));

W(4) = (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)));

W(5) = (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)));

W(6) = (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)));

W(7) = (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) + (y(2)/y(l))*W(l); dy(7) = Gin(2)*(Xin(2)-y(7))/y(6) + (y(7)/y(6))*W(2); dy(13) = Gin(3)*(Xin(3)-y(13))/y(12) + (y(13)/y(12))*W(3) dy(19) = Gin(4)*(Xin(4)-y(19))/y(18) + (y(19)/y(18))*W(4) dy(25) = Gin(5)*(Xin(5)-y(25))/y(24) -f (y(25)/y(24))*W(5) dy(31) = Gin(6)*(Xin(6)-y(31))/y(30) + (y(31)/y(30))*W(6) dy(36) = Gin(7)*(Xin(7)-y(36))/y(35) + (y(36)/y(35))*W(7)

%Evaporator inventory 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) = (Gout(l)*y(2)+Gfeed*Xfeed)/Gtin(l); Xtin(2) = (Gout(2)*y(7)-hGfeedl*Xfeedl)/Gtin(2); Xtin(3) = y(13); Xtin(4) = y(19) Xtin(5) = y(25) Xtin(6) = y(31) Xtin(7) = y(36)

%olnventory inlet temperatures%o

Ttin( l ) = (Gfeed*Tfeed + Gout(l)*Tbl(y(2),Tsat(l)))/Gtin(l); Ttin(2) = (Gfeedl*Tfeedl + Gout(2)*Tbl(y(7),Tsat(2)))/Gtin(2); Ttin(3) = Tbl(y(13),Tsat(3)); Ttin(4) = Tbl(y(19),Tsat(4)) Ttin(5) = Tbl(y(25),Tsat(5)) Ttin(6) = Tbl(y(31),Tsat(6)) Ttin(7) = 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

u(i) ; u(2); u(3); u(4) + Gbout(5); Gbout(5) + Gbout(6); Gbout(6);

%Inventory mass, dry solids concentration and temperature derivativesYo

%oDry solids concentration derivativesYo

dy(4) = dy(9) = dy(15) = dy(21) = dy(27) = dy(33) = dy(38) =

(Gtin(l)* (Gtin(2)* : (Gtin(3) -• (Gtin(4) •• (Gtin(5) •• (Gtin(6) : (Gtin(7)

(Xtin(l) (Xtin(2) *(Xtin(3) *(Xtin(4) *(Xtin(5) *(Xtin(6) *(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) - (y(3)/rhot(l))*(-0.495*dy(5) + 600*dy(4)); dy(8) = (Gtin(2) - Gtout(2))/(rhot(2)*Ac) - (y(8)/rhot(2))*(-0.495*dy(10) + 600*dy(9)); dy(14) = (Gtin(3) - Gtout(3))/(rhot(3)*Ac) - (y(14)/rhot(3))*(-0.495*dy(16) + 600*dy(15)) dy(20) = (Gtin(4) - Gtout(4))/(rhot(4)*Ac) - (y(20)/rhot(4))*(-0.495*dy(22) + 600*dy(21)) dy(26) = (Gtin(5) - Gtout(5))/(rhot(5)*Ac) - (y(26)/rhot(5))*(-0.495*dy(28) + 600*dy(27)) dy(32) = (Gtin(6) - Gtout(6))/(rhot(6)*Ac) - (y(32)/rhot(6))*(-0.495*dy(34) + 600*dy(33)) dy(37) = (Gtin(7) - Gtout(7))/(rhot(7)*Ac) - (y(37)/rhot(7))*(-0.495*dy(39) + 600*dy(38))

VoPressure derivatives%o

VoCondensate mass flow rates%o

Wc(l) = W(l)*hevap(Tbl(y(2),Tsat(l)))/hevap(Tsat(2)); Wc(2) = W(2)*hevap(Tbl(y(7),Tsat(2)))/hevap(Tsat(3)); Wc(3) = W(3)*hevap(Tbl(y(13),Tsat(3)))/hevap(Tsat(4)); Wc(4) = W(4)*hevap(Tbl(y(19),Tsat(4)))/hevap(Tsat(5)); Wc(5) = W(5)*hevap(Tbl(y(25),Tsat(5)))/hevap(eqth2o(u(7))) Wc(6) = W(6)*hevap(Tbl(y(31),Tsat(6)))/hevap(eqth2o(u(8))) Wc(7) = W(7)*hevap(Tbl(y(36),Tsat(7)))/hevap(eqth2o(u(9)))

dy( l l ) = (W(2)-Wc(l))*(R*(Tbl(y(7),Tsat(2)))+Tsat(l))/(2*M*V(2)); dy(17) = (W(3)-Wc(2))*(R*(Tbl(y(13),Tsat(3)))+Tsat(2))/(2*M*V(3)); dy(23) = (W(4)-Wc(3))*(R*(Tbl(y(19),Tsat(4)))-fTsat(3))/(2*M*V(4)); dy(29) = (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 := - .0105965767- | -3 .644585759 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 : -3500 ;

> Kcl ; - 20720: Kc2:- 9616: Kc3:-7778:

> Kc4:-9464: Kc6;-3583: Kc6 : -

> 26431: Kc7 : - 2891:

> p l l : - 136: pl2:-140: pl3;-13B: pl4:-160:

> pl5:-64: p l6 : -64:

> p l7 : -64:

> VI;- 291: V2:-186: V3:-211: V4:-277: V6:-352;

> cpl;- 3665; cp2:" 3675: cp3:- 3532:

> cp4;-3302: cp5:" 3082: cp6:-

> 2882: cp7:- 2823;

L:" 10.96: M:- 0.018: R:- 8.

Gfeedinl:- 58:

67: G£eedin4:-

GfeedinB:- 41:

GfQQdin2:" 98:

- 59:

GfeedinS:- 32:

314:

Gfeedin3:

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 :- 30 + 273.15:Ac ;- 16:

> yln;-3111; y2n;=0.2266: y3n:-l:

> y4n:-0.2149: ySn:=332.2; y6n:-4213: y7n:-0.2221; y8n;-l;

> y9n;-0.2152: yl0n;-346.1: ylln:-38910: yl2n;=3791: yl3n;-0.2972;

> yl4n;-l: yl5n:-0.2972; yl6n:-364.6: yl7n;-66090: yl8n;-5113;

> yl9n;-0.4322; y20n;-l: y21n;-0.4322: y22n;-382.6: y23n:-116700;

> y24n:-34S5; y25n;-0.5745: y26n:-1.018: y27ii;-0.5745; y28n:.403:

> y29n;-201300; y30n;-4494: y31n;-0.6963: y32n:-1.007: y33n;-0.6963; > y34u;"409.6; y35n;"5930; y36n;-0.7365:

> y37n:-l; y38n:=0.7355: y39n;"413.1:

> uln ;- 21732: u2n ;- 410000: u3n :- 410000: u4n :- 410000;

> Tsat l ;-3.9487 4

> 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 ;

Gbout1 ' T8at5:

-39.45; Tsat7 :- Tsat5:

0bout2:-78.73; Gbout3:

Gpump4 Gbout4: Gin5

> 59.96; Gbout4:-41.3; GboutS:-

> 30.48; Cbout6;- 24.79: Gbout7:- 23.05;

> Ginl :- Gpumpl - Gboutl: Gin2 :- Gpump2

> Gbout2: Gin3 :- Gpump3 - Gbout3: Gin4

> Gpump5:Gin6 ;- GpumpG; Gin7

> :- Gpump7 - Gbout7; > Xinl :- y4; Xin2 :-

> (y4»Gboutl4y9.(Gpump2-Gboutl))/Gpump2; Xin3 :-

> {Xin2»0bout2+yl5»(Gpump3-Gbout2))/Gpump3:

> Xin4 :- (Xin3*Gbout34y21.(Gpump4-Gbout3))/Gpump4:

> Tinl ;- y5; Tin2 :-

> (ylO"(Gpump2-Gboutl)+y4*Oboutl)/Gpump2: Tin3 :-

> (yl6»(Gpump3-Gbout2)4yl0«Gbout2)/Gpunp3:

> 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; Tin5 :- Tmixout5:

* Gmiiin6 ;- GpumpS 4 Gbout6 - GboutS:

• Gmixout6 ;= Gpump6 4 Gbout6:

• XmiioutG := (y33«Gmixin6 4 Gbout5*Xin5)/Gmixout6:

Tmiioute :- (y34*Gmixin64Tin5*Gbout5)/Gmixout6:

Xin6 := Xmixout6; Tin6 ;= Tmixout6: Xin7 ;.

(Xffliiout6*Gbout6+y38« (Gpump7-Gbout6)) /Gpump7; Tin7

(y39» (Gpump7-Gbout6)4Tmixout6*Gbout6) /Gpump7:

Tbll :- 7»y2/(l-y2) 4 Tsatl: Tbl2 :-

7»y7/(l-y7) 4 Tsat2: Tbl3 :-

7«yl3/(l-yl3) + T3at3:

Tbl4 ;» 7«yl9/{l-yl9) + T8at4: Tbl5 ;•

7«y25/(l-y25) + TsatS: Tbl6

7>y31/(l-y31) + Tsat6; Tbl7

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):

7«y36/(l-y36) + T3at7:

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<y7.(-0.

> 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): lievap2 :- 2812200 -

> 349.98»Tbl2 - 3.034«(Tbl2-2);

> hevap3 :- 2812200 - 349.98»Tbl3 -

> 3.034«(Tbl3-2): hevap4 :- 2812200 -

> 349.9B*Tbl4 - 3.034«(Tbl4-2):

> bevapS ;- 2812200 - 349.98»Tbl5 -

> 3.034-(Tbl5-2): hevap6 ;- 2812200 -

> 349.98»Tbl6 - 3.034»(Tbl6-2);

> bevapT :- 2812200 - 349.98«Tbl7 - 3.034.(Tbl7-2);

> dyl ;- (Ginl- Goutl

> (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 : .mtay l ,>r (dy6 , [y6-y6n .y7 .y7n ,y l l -y l lb .y l>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)):

> dyia i in ; - 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 := -51 .47713242 -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 := -316 .0117961 + 731.9940856 y5i + .0002482335077 y29

- .04857655176 ySO - .00006167511763 u3

> 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 := -571 .2333706 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,yl l-yl ln,ul=uln],2)l

dy2lin •= .000173775148 4- .1178238508 y4 - .1140341851 y2 4- .47743879 lO"" 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 lO"" yl

-I- .7444242529 lO"'' yll - .1187297698 10"^ 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 10"^ 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 10"^ yJ2 + .2889125316 10~^ y2J

- .4492075590 10-' 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 - .334031139110-^ 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 10"' u2

- .1821474478 10-' 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 10"'" ySO

-I- .9555937782 10-° uS - .1658290203 10"^ 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) ;

Gtini

Gfeedl

Gtin4

Xtinl

dySelin := .003395351142 + .003203698862 y5S + .0003891068478 y27

4- .8153726437 lO-'^ y^ + .133632239010-'' y9 -t- .7028427378 1 0 - ^ ylB

4- .00004974081130 y2J + .02315339168 yS« - .03144812364 y56

+ .5857079 10-'° ySS + .3921174648 10-** u4 - .6861010248 1 0 - * y29

- Goutl 4 Gfeed: Gtin2 :- Gout2 +

Gtin3 :- Gout3:

- Gout4; Gtin5 :- GoutS: Gtin6 :- Gout6: Gtin7 :- Gout7:

- (Goutl*y24Gfeed*Xfeed)/Gtinl:

- (Gout2«y7+0feedl»Xfeedl)/0tin2: Xtin3 :- yl3: Xtin4 :-

> yl9; Xtin5 ;-

> y25:

> Xtin6 ;- y31: Xtin7 ;- y36:

> Ttinl ;- (Gfeed*Tfeed +

> Goutl"Tbll)/Gtinl: Ttin2 :- (Gfeedl'Tfeedl 4

> Gout2«Tbl2)/Gtin2:

> Ttin3 ;- Tbl3: Ttin4 :- Tbl4: Ttin5 ;-

> Tbl5; Ttin6 :• Tbl6; Ttin7

> ;- Tbl7:

> rbot01:=1007.4 - 0.495"(Tinl - 273.15) •

> 600«Xtinl; rhot02:-1007.4 -

> 0.496«(Tin2 - 273.15) + 600.Xtin2:

> rbot03:-1007.4 - 0.495*CTin3 - 273.15) +

> 600»Xtin3; rbot04:-1007.4 -

> 0.495*(Tin4 - 273.15) 4 600*Xtin4:

> rhot05;-1007.4 - 0.495«(Tin5 - 273.15) +

> 600*Xtin5: rhot06:-1007.4 -

> 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) +

> 600*y4: rhot2:-1007.4 -

> 0.495»(yl0 - 273.15) 4 600«y9;

> rbot3:-1007.4 - 0.495»(yl6 - 273.15) *

> 600*yl5: rhot4:-1007.4 -

> 0.495»(y22 - 273.15) 4 600»y21:

> rhot5:-1007.4 - 0.495»(y28 - 273.16) +

> 600*y27; rhot6:-1007.4 -

> 0.495*(y34 - 273.15) + 600«y33: > rhot7:.1007.4 - 0.495«(y39 - 273.15) + 600.y38:

> Gtoutl :- Gpumpl: Gtout2 :- Gpump2 -

> Gboutl; Gtout3 :- Gpump3 -

> Gbout2; Gt0Tlt4 :- Gpump4 - Gbout3:

> GtoutS ;= Gpump5 - Gbout4 + GboutS;

> 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

+ .787137818010-^ 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 y7 - .841019276710-'" 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 - .02281295040yJS

> dy21 :- (0tin4«(Xtin4 - y21)/(Ac«rhot4)): > dy211in : -

^ mtaylDr(dy21.Cyl8-yl8n,yl9-yl9n,y21-y21n,y22-y22n,y23-y23n] .2) ;

dySlUn ;= 01942016058 y i9 - .01942016058 y2J

> dy27 ;- (Gtin5»(XtinB - y27)/(Ac»rhotS)):

> dy271in :-

> mtaylor(dy27, [y24-y24n,y25-y25n,y27-y27n.y28-y2an,y29-y29n] ,2) ;

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 yiS

> dy38 : - (Gtin7»(Xtin7 - y38)/(Ac*rhot7)): > dy381in : -> mtaylor(dy38, [y29-y29n,y35-y35n,y36-y36n,y38=y38n,y39-y39n] .2) ;

dy38Un := .007599453546 ySS - .007599453546 yj«

> dy5 :- ((Gtinl»Ttinl*cpl -

> Gtoutl*y5*cpl)/(Ac*rhotl*cpl) (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 - ^ yJ7

- .500826085010-^ y l2 - .02041900728 vJ6 - 4.064588546 yJ5

> dy22 ;= ((0tin4«Ttin4»cp4 -

> Gtout4-y22»cp4)/(Ac»rhot4.cp4) (y22.(Gtin4 - 0tout4)/(Ac.rhot4))

> - ((y22)/cp4)»{3.822«y22-3322.67)«(Gtin4«(Xtin4 -

> 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 y27

> 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- .122640543710-* 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 := -1411 .595121 - 21939.40106 y7 - 1.741495870 y l l + 1.419688293 ul

+ .5591625114 yJ7 + 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 yl7

+ .1617310647 v29 + 18928.87074 ylS

> dy29 :- (U54U64U7-Uc4)»R«(Tbl5+Tsat4)/(2«H.V5):

> dy291in ; -

> 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

299

O O O O O O O - ^ O J 05 O

t ^ o o o o o o o o o o o o o o o o o o o o ti

C M O O O O r t O O - H i o l O O O O O - 4 O --I

O T}< O CO ' ^ CM •

,-; ^ o , o V l o 00

O O O O C O O O O O O O O O O O O O O O O O O O CO CM cn CXl

CM

O O O O O O O C O - H CO

;o o CO o

o o o o o o o o o o o o o o o o o o o o o o o o

o l-H

o

o o o o o CO lO 00 lO CO T—(

lO r-l C5 CO C3 f—i

lO CO CO

^ 00 en

O O I-H

h-O t-rH O

o o o o r to T-H

o p (-

O O O O 00 ^ CM o o o

o o o o o w CM 0 0

0 0 ( 0 0 0

o Tt< CO

cn

o o

s-l

03

iO CO 05 1—1

o Tf OO

T-H

<o Ol CO .—1

o '

~ o --I

Tfl I-H

o '

CTl CD

M CD OT

CM

Tf O) CM lO CO

o o

OO

o 1 Tf 00

lO

lO CO CO N-in CO

CO

lO

o W lO CO

Tji

Tf CO Oi ^H c '

o

o o o o o o o o o o o o o o o o o o o o o o Tf

o o Tf lO

o CM

o o o o o o o o o o o o o o o o o o o o o o o

Q

E3

o o o o o o o o o o o o o o o o o o o o o o o

Tf CM 00 t~-^H

^4

M Tf CO

CO

CM Tf Tf CM

p f—i

CM O CD

' I-H T ^

CO CM

1

O cn 00 CM 00 o o

O O O C M O O O O C M O O O T f O O r-l rt O

<=> e O O

d d

W

O O O CD O O

o 00

o w lO

T - H C O i O C O O O O O

l O O CM CO lO

i O CD i n Tf O r-l CM T f p p d CO

O O O O O O O O O O O O O O O O O O O O O O O

CM OJ 00 i n

1 - o o o ^

o o o o o o o o o o o o o o o o o o o o o o o o o o

t _ t ^ Tf t ~ 00 Tf Tf t-^ Tji

300

o o o o o o o o o o o o o o o o o o o CD CD 03 o

o o o o o o o o o 0 0 o

0 0

o o 03 CD 0 0 CO in -in in t^ o 00 o CM O

d d

o (r<i

o 44>

O O O O O O O O O O O O O O O

O O O O O O O O O O O O O O O

0 0 O

^ CO

CO 1

o

i n T f .—1 CM CO o

o

o

CM T f O) 1-4

P r-3

CD O

(^ CM

4.0

-0.0

163

0 0 CM irt i n CO

I-H CO O 0 0 T f

d

O

o

St i o t^ d °° 1 CO

o o

o o

O N-co CM O l o o

o [i in Tf

o o o o o o o o o

' C M O O O O O O O O CM 1~-O CM

O O O O O O O O O O O

0 0 0 0 03

t o o o o

in o

03

cn

O o

< X

..—I s-l

4 J 133

O O O O O O O O O O O O O O C M O O C M C O t ^ 0 0 0 0 0 3 0 0 0 0 0 0 0 0

o o o o o o o o o o o o o o

1 - CO o CO C73 1^ Csi °. t- o 1^ 1

OO 03

o o CO W O CM

Tf CO 03 i n

0.01

.8

69

CO

O 1^ o M 0 3

T f

i n rH CO

' o o o o o

CO

03 03 i n CO CM

o o o o o o o o o

Q o o o o CD 0 0

CO O

S^ O CD

CO CO irt 03 i n i n

^ I-H

O

o o

0 0

o ti i n T-H

o CO

o f^ i n

o

i n CM

0 3

d

o o o o o o o o o o o o o o o o o

O O O O O O O O O O O O C M O O O O O O O O O O O O O O O O O O O

O O O O O O O O O O t ^ rt C33 OO 00 in t - CM T f 03 CM CO 00 O o

° d Tf O I I

O r^ 00 t ^ OO o o

o o o i n o o o o i n o o CO

o o CO

CD

o CO

o

o o o o o o o o o rt CD CO 00 00 CO CM - H in _-; Tf in 00 00 2 , rt o CM in S

1 - g ^ ?? CO O H T f CM CO I O '

o o o o 0 0

0 0 CM CJ3 0 0

o o o o o o o o o o o o o

301

O O O O O O O O O O O O O O O O O O O O O O O C O O O O O r t T f CM CO o CM

T f TJI

CO 03

in 03 00 T f 03 Cq rt in o CO t ^ TJH O O rt

d P °? I O i-l

O O O O O O O O O O O O O O O O O O O O O O O O O O O O CM I-H i n rH ^ r'•^ CM W O

d

O 00

o I

H

o o o o o o o o o o o o o o o o o o o o o o o o o O O OO o o o o r-o o

o CO

CD

fl

o o

o o o o o o o o o o o o o o o o o o o o o o o o o 00 CM rt CO t^ r-t- O CM 00 rt Tf Tf O 03

°. d r^ O I I

O Tf o o o CM CO o o

O O O O O O O O O O O O O O O O O O O O O O O T f ^ H C ^ I C M i n O O O O „ Tf 00 t^ g CD t^ o 2 CJ3 in rt S -; p =? CO CO o o

1 t ^ 1

O O O O O O O O O O O O O O O O O O O O O O O O O O r t O 00

o o o o o

ci

CO

Q CD

E2

o o o o o o o o o o o o c o o o C<I o

S w p CM d CM

I

o o rt rt 00 o o o W O OO O _ ;

O CD rt 00 OO O T f T f O ^1 CD cq ^1

H t - o H i n CM o b-<>: c6 <=>. .A rt i O I

o CD in CD o i^ o w CD T f 00

O O O O O O O O O O O O O O O O O O O O O O r t O O O O O O O O O

^H

o d

O O O O O O O O O O O O O O O O O O O O C M C O O O CO T f —'

Tf O o CO in TH

CM CM

o d

in Tf in 00 p (-3

i H

o d

CM O CO Tf i H

O

o 03

2.5

1

Tf CO Tf 00 r-l O

O O CM O O O ^H CO in o o d

03 o o 00 CO o o o d

t ^ o o o o o o o o Xl t-t-co

302

CM CO

tn fl

O

o

< X.

o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o CO in o o d

I

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O C O C O C M i n 1^ OO rt O CO CO o t^

d I

Ttl

Q

303

CQ

03

O O O O O O O O O O O O O O O O O O O O C M O O O O i n 0 3 0 0 o o o t i CO I—f O l

LO

Q r r t

3 03 H

o

CO

o W 00 i r t

O

t ~

o

O

o I - H

f i c i CM T f

0 0 CD

o o o o o o o o o o o o o o o o o o o o c M i n c n o o o o o o o o o (^ th (^ CO 1 - CD ^H i H LO CO CO CD

1

O O O O O O O O O O O O O O O O i n o O O O C M O O O O O O O O O O o fh iil Ci CO CO i n CO p CM CD i-i CO

i n O O O O C M O O O O O O O O O O O O O O O O O O O O O O O O CO

i n 1 ^ 03 CO

CM rt rt rt

304

o X SH

4 J

03

CX>

Q

i

O O O rt

O O rt o

o o o o

o o o o

o o o o

O rt o o

o o o o

o o o o

o o o o

o o o o

rt o o o

o o o o

o o o o

o o o o

o o o o

o o o o

o o o o

o o o o

o o o o

o o o o

o o o o

CZl <Z> CZl CD

o o o o

o o o o

o o o o

o o o o

o o o o

o o o o

o o o o

o o o o

o o o o

o o o o

305

<: X

cn

fl CU h C

• l-H

w

Q

t^ o o 03 ^H Tf o CM o

.—1 in 03 o o CO CD Irt

o 1.01

1111

75 CD

CM O OO I--t^ r-o o

03 CM TS< Tf 00 CO in o o

00 i^ o in CO Tf ^H r^

CM 03 t^ 00 CO 00 CO 00

in CO CD in t^ 03 00 CO

CM

o 00 in in Tf t--^H

in ^H CD CM 03 o Tl< rH

t^ CD I^ 00 Tf Tf CO -H

r-o CO CM CD CO CM ' — i

03 CM t^ CM Tf in i H rH

L142

465 in CM

rH CD OO CM O Irt

O O O O O O O

CO i H in Tf T f o CO CO I-H LO CO 00 CD CD OO t - -

o o

irt I r t CO m T f in CM m 00 in - ^ CM OO 00 0 0 rH ^H r ^ CO I r t CM r H t ^ CO 0 0 o rt i n CO CO CD CD CO

o o o o o o o

00 CD CO Tl< CO CO o o o d d d

CD CO CO lO LO CD CM 03 rt CM lO t^ CD CM CM CM o o d d

CO CO

o o T f T f

CO CD 2 ° 2 <=> o o

+

in

. 03 o o

CD in

CM CM CO I r t - H O T f TJI O

S S 9 o o --I r H

O o o o d d

+ CO

o CO 0 0 I r t CM O O

306

APPENDIX E

OCFE DISCRETIZATION

The discretization scheme is presented on Figure E.l. The plate length is normal­

ized to value of one and is divided in 10 finite elements. Each finite element has two

internal collocation points.

0,0211 0.D768 osZU 0.978B I—0 o—)—0 0—1 1—0 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' (E.l) dt* ^ dz* ^ ' ^ '

The transformation from partial differential equation to ordinary differential equa­

tions 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*

In the studied case N = 2, therefore we have

(E.2)

dt*

dGh dt*

— V z,k,2 N+2

IK / . ^2 jGfc 2 K:2v:,k,2

-V. z,k,3 N-{-2

h K E ^ 3 . G ' : , 3 - H ^ C 3 < M . i = i

(E.3)

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 =

k-l

•Ar-1-2

= E ii -i = i

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 equa­

tion. 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 N+2 (E.4)

2,10,4

HK 10,4 " ^10,4^z,10,4)

308

which leads to total number of 31 algebraic and differential equations per partial

differential equation.

309