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Dynamic simulation of a recausticizing plant Wang, Lijun 1993

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DYNAMIC SIMULATION OF A RECAUSTICIZING PLANT by LIJUN WANG B.Eng. Tsinghua University, Beijing, 1988 M.Eng. Tsinghua University, Beijing, 1990  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF CHEMICAL ENGINEERING  We accept this thesis as conforming  to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA  February 1993 © Lijun Wang, 1993  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  (Signature)  Department of  ^Cite^E 4k eeArtiwy .  The University of British Columbia Vancouver, Canada  Date  DE-6 (2/88)  ^Ve.,D  2Lh ,^93  Abstract  The recausticizing plant has been recognized as a very important part of the Kraft pulping process. Because of the multiple objectives involved in the operation of the recausticizing plant, different operating conditions, and the complex interdependencies of the various units of the system, the best operating conditions are not obvious. A dynamic model was developed to study the effect of the different operating conditions and the effect of different disturbances and interconnections between the various units on the overall behaviour of a recausticizing plant. Both the thermodynamic and kinetic behaviour of the causticizing reaction have been investigated. A new approach for describing the equilibrium and kinetics of the causticizing reaction was developed, which takes into consideration the nonideality of the green and white liquor. It was found that both the equilibrium and kinetics of the causticizing reaction can be well represented by this approach. A simplified and accurate method was used for describing the lime mud free settling process. Models for all the units in the plant were developed and presented. The models were based on mass balance equations and the pertinent physical properties. The effects of disturbances and different operating conditions were simulated for the entire plant. Simulations showed the complicated dynamic responses of the system and the importance of the control system to achieve better operations. This work has brought to light some previously less known facts about the recausticizing process, and can be used to develop and compare different control strategies before they are implemented in the mills.  ii  Table of Contents  Abstract ^  ii  List of Tables ^  vi  List of Figures ^  vii  Nomenclature ^  x  Acknowledgments ^  xiv  1 Introduction^  1  1.1 Background ^  1  1.1.1 Purpose of the Kraft Recovery System ^  1  1.1.2 The Recausticizing Plant ^  3  1.1.3 Dynamic Simulation in the Pulp and Paper Industry ^  5  1.2 Research Objectives ^ 2 Mathematical Modeling of the Recausticizing Plant ^  6 7  2.1 Introduction ^  7  2.2 Smelt Dissolving Tank (SDT) ^  10  2.3 Green Liquor Storage Tank (GLST) ^  12  2.4 Green Liquor Clarifier (GLC) ^  14  2.5 Slaking/Causticizing (SC) ^  17  2.5.1 Introduction ^  17  2.5.2 Slaking Reaction ^  18  2.5.3 Modeling of the Causticizing Reaction ^  20  2.5.3.1 Modeling of the Reactive Components ^  20  2.5.3.2 Modeling of the Non-reactive Components ^  23  hi  2.6 White Liquor Clarifier (WLC) ^  23  2.6.1 Modeling of the Chemical Components ^  23  2.6.2 Lime Mud Settling ^  26  2.7 Mud Mix Tank (MMT) ^  29  2.8 Mud Washer (MW) ^  32  2.9 Dregs Washer (DW) ^  35  2.10 Weak Wash Tank (WWT) ^  37  2.11 Summary Remarks ^  39  3 Thermodynamic and Kinetic Studies of the Causticizing Reaction^40  4  3.1 Introduction ^  40  3.2 Methodology ^  43  3.2.1 Thermodynamic Equilibrium Constant InK(T) ^  45  3.2.2 Calculation of Pitzer's Parameters at 100°C ^  47  3.3 The effect of Na2S on the equilibrium ^  49  3.4 Calculation of Causticity ^  49  3.5 Results and Discussion ^  50  3.6 Conclusions about the Thermodynamic Model ^  53  3.7 Kinetic Model of the Causticizing Reaction ^  54  3.7.1^Previous Studies ^  54  3.7.2 Kinetic Model for Causticizing Efficiency ^  55  Simulation Results and Discussions  58  4.1 Introduction ^  58  4.2 Steady State Conditions of the Recausticizing Plant ^  59  4.3 Simulations of the Different Units ^  62  43.1 Responses of the Causticizing Reaction to Different Disturbances ^ 62 4.3.2 Tank Level Control ^  65  4.33 Step Response of Storage Tanks and Clarifiers ^  66  iv  4.4 Dynamic Response of the Recausticizing Plant ^ 4.4.1 Dynamic Responses to a Change in Smelt Rate ^  68 68  4.4.2 Dynamic Responses to a Change in Smelt Composition ^ 72 4.4.3 Dynamic Responses to a Change in Lime Mud Filter Flow Rate ^ 74 4.5 Simulation of Typical Disturbances ^  80  4.6 Summary Remarks ^  83  5 Conclusions^  85  6 Further Work^  87  References^  88  A Chemical and Physical Terminology ^  91  B Tank Level Control^  92  v  List of Tables  2.1 List of Units in the Recausticizing Plant ^  8  2.2 indexes for Chemical Components ^  10  3.3 Thermodynamic Data for Species in Causticizing Reaction — Standard State (25°C) Values ^  46  3.4 Ion Interaction Parameters and Their Derivatives at 25°C ^  47  3.5 Calculated Ion Interaction Parameters at 100°C ^  48  3.6 Comparison of Model Parameters ^  48  4.7 General Structure of Modeling in SIMNON ^  59  4.8 Dimensions of the Units in the Recausticizing Plant ^  60  4.9 Output Mass Flow Rates and Volumetric Flow Rates at Normal Steady State ^ 60 4.10 Smelt Component & Flow Rate ^  61  4.11 Output from Lime Mud Filter ^  61  4.12 Output from Scrubber ^  61  4.13 Fresh Water ^  61  4.14 Steady-State Data of Green Liquor ^  62  A.15 Common Terminology ^  91  vi  List of Figures  1.1 A Typical Kraft Pulping Plant ^  3  2.2 The Recausticizing Plant ^  7  2.3 Input — Output Diagram ^  9  2.4 Smelt Dissolving Tank ^  11  2.5 Green Liquor Storage Tank ^  13  2.6 Green Liquor Clarifier ^  15  2.7 Slaking and Causticizing Process ^  18  2.8 White Liquor Clarifier ^  24  2.9 Settling Rate vs Causticizing Efficiency ^  28  2.10 Mud Mix Tank ^  29  2.11 Mud Washer ^  32  2.12 Generalized Dregs Washer Model ^  35  2.13 Weak Wash Tank ^  37  3.14 Causticizing with Pure Sodium Carbonate Solution ^  51  3.15 Causticization at 15% Sulfidity ^  52  3.16 Causticization at 30% Sulfidity ^  52  3.17 Causticity for Causticization at 15% Sulfidity ^  53  3.18 The Effect of Sulfidity on the Equilibrium ^  53  3.19 Kinetic Model of the Causticizing Reaction ^  57  4.20 [0/1'1 and [CO3 ] During the Course of the Causticizing Reaction ^ 63 -  VII  4.21 Effect of the Green Liquor TTA ^  64  4.22 Effect of Lime Charges ^  65  4.23 Effect of the Residence Time ^  66  4.24 Control System for a Typical Tank ^  66  4.25 Tank Level Control ^  67  4.26 Dynamic Response of a Clarifier ^  67  4.27 Changes in Smelt Rate ^  69  4.28 Volumetric Flow Rate Responses to Change in Smelt Rate ^  69  4.29 TTA Responses to a Step Change in Smelt Rate ^  70  4.30 AA Responses to a Step Change in Smelt Rate (1) ^  71  4.31 AA Responses to a Step Change in Smelt Rate (2) ^  71  4.32 Equilibrium Causticity Responses to Changes in Smelt Rate ^  72  4.33 Volumetric Flow Rate Responses to a Change in Smelt Composition ^ 72 4.34 TTA Responses to a Change in Smelt Composition ^  73  4.35 AA Responses to a Change in Smelt Composition (1) ^  74  4.36 AA Responses to a Change in Smelt Composition (2) ^  74  4.37 Equilibrium Causticity Response to a Change in Smelt Composition ^ 75 4.38 Change in the Mass Rate from Lime Mud Filter ^  76  4.39 Volumetric Flow Rate Responses to Change in the Mass Rate from Lime Mud Filter . . ^ 76 4.40 TTA Responses to a Change in the Mass Rate from Lime Mud Filter ^ 77 4.41 AA Responses to Change In the Mass Rate from Lime Mud Filter (1) ^ 77 4.42 AA Responses to Change In the Mass Rate from Lime Mud Filter (2) ^ 78 4.43 Equilibrium Causticity Responses to Change In the Mass Rate from Lime Mud Filter . . ^ 78  4.44 TTA Responses to a Step Change in Smelt Rate with Make-up Water in the Slaker . . . . 79 4.45 TTA Responses to a 2.5 x 10 -3 min -1 Frequency Disturbance in the Smelt Rate (1) . . 80 4.46 TTA Responses to a 5.0 x 10 -3 min -1 Frequency Disturbance in the Smelt Rate (2) . . 81 4.47 TTA Responses to a 1.0 x 10 -2 min -1 Frequency Disturbance in the Smelt Rate (3) . . 81 4.48 TTA Responses to a 2.0 x 10 -2 min -1 Frequency Disturbance in the Smelt Rate (4) . . 82 4.49 Typical Variations in Smelt Rate ^  82  4.50 TTA Responses to Some Mill Disturbances in the Smelt Rate ^  83  B.51 Block Diagram of PI-Controlled Integrating Process ^  92  ix  Nomenclature  Unless otherwise noted in the text, the symbols used in this thesis are defined in the following list. In certain cases, special symbols are used, but these are immediately defined following the equations in which they appear. Superscripts indicate the unit of the recausticizing plant to which variables belong. Subscripts are used to indicate the chemical component to which the variable refers. For example, Wil" is the mass flow rate of Na 2 S from the green liquor clarifier overflow. In the following, the main symbols are listed first, then the subscripts, and finally the superscripts.  Main Symbols  a^ A^  COT initial concentration in the causticizing reaction (mole/I)  Ao^ AV^  area (m2 ) Debye-Huckel Constant lime availability (%)  b^  OH initial concentration in the causticizing reaction (mole/1)  C^  Pitzer parameter  -  concentration of component i (kg/m 3 ) Caus^ causticity (%) CE^ causticizing efficiency (%) CEO^equilibrium causticizing efficiency (%) Cw^water concentration (kg/L) ACp^change of the reaction heat capacity (caVmol K) ACp °^ standard change of the reaction heat capacity (cal/mol K) DR^ displacement ratio 1/2 partial derivatives of coloumbic force term with ionic strength x  G^  Gibbs free energy (kcal/mol)  AG°^  standard change in Gibbs free energy (kcal/mol)  gi(I), g2(1)^functions of ionic strength h^  tank level (m)  AH^  change of reaction enthalpy (kJ/mol or kcal/mol)  AH°^  standard change of enthalpy (kJ/mol or kcal/mol)  I^  ionic strength (moles/kg)  ki^  constant in all the equations  K(T)^  thermodynamic equilibrium constant  Ka(T, X)^activity constant Kc^  Equilibrium constant based on molar concentrations (mole/L)  Km^Equilibrium constant based on molal concentrations (gmol/kg solvent) m^  mass (kg)  in;^  molality of species i (mol/kg)  MW^  molecular weight  Pss^  percent of solids (%)  Qi^volume flow rate from unit i (m 3/sec) R^ gas constant (cal/mol K)  SW^  solid to water ratio  t^  time (sec)  T^  temperature (K)  TAC^ TTA^  total anion concentration (g as Na20/1-) total titratable alkali (g as Na 2 0/L)  U^  velocity of sedimentation (m/sec)  V^  volume of the tank (m 3 )  W^  mass flow rate (kg/sec)  X^  represents composition of system  x^  moles consumed or produced in the causticizing reaction (mole/L)  Zi^  charge associated with ionic species i  a^  constant parameter  Pmx, Avoc(1)^ion interation parameters for MX pair  xi  7i^  9^  activity coefficient for species i ratio between the hydroxide ion concentration squared divided by the carbonate ion concentration (mole/L)  Pw^  density of water (kg/m 3 ) time constant (sec) dynamic viscosity of liquor (Pa s)  Subscripts  0^ 1^ 2^ 3^ 4^ 5^ 6^ 7^ 8^ 9^  H2O (water) N a 2 S (sodium sulphide) NaOH (sodium hydroxide) Na2CO3 (sodium carbonate) Na2SO4 (sodium sulphate) Na2S03 (sodium sulphite) Suspended Solids CaO (calcium oxide) Ca(OH)2 (calcium hydroxide) CaCO3 (calcium carbonate)  Superscripts  dw^ dwo^ dwu^ glc^ glco^  ecu^ gist^ ks^ lmf^ lm^ mmt^  Dregs Washer Dregs Washer Overflow Dregs Washer Underflow Green Liquor Clarifier Green Liquor Clarifier Overflow Green Liquor Clarifier Underflow Green Liquor Storage Tank Kiln Scrubber Lime Mud Filter lime mud in the slaking/causficizing reaction Mud Mixing Tank  mw^ Mud Washer  xu  mwo^ Mud Washer Overflow mwu^ Mud Washer Underflow sc^ Slaking/Causticizing sci^ variables after slaking reaction scl^ liquor part in the slaking/causticizing area after slaking reaction sdt^ Smelt Dissolving Tank wlc^ White Liquor Clarifier wlco^ White Liquor Clarifier Overflow wlcu^ White Liquor Clarifier Underflow wwst^ Weak Wash Storage Tank  Xlll  Acknowledgments  I am deeply indebted to my supervisors, Dr. Patrick Tessier and Dr. Peter Englezos, for their guidance, kind support, encouragement and valuable discussions throughout the course of this research, without which this work would not have been possible. I am also indebted to my colleagues, Dr. Yu Qian, Mrs. Xiaohong Wang, and Mr. Xingsheng Qian, for many hours of valuable discussions. The consistent computer network support from Mr. Rick Morrison and Mr. Kristinn Kristinsson is greatly acknowledged. Special thanks go to Miss Cathrine Gaarder for her patient assistance in preparing my presentations. I am thankful to the members of the PAPRICAN group at UBC, particularly Ms. Rita Penco for her help, and the staff of the chemical engineering department office for their assistance. I would like to thank Dr. Tessier and Dr. Englezos and the National Sciences and Engineering Research Council of Canada for providing financial support and excellent computer facilities. At last I am most grateful to my wife, Jing, for her continuous support and patience during the process of my study.  xiv  Chapter 1: Introduction  Chapter 1 Introduction  1.1 Background In the pulp industry, the objective is to produce fibers (pulp) from wood chips [1]. This process is carried out by bringing into contact wood chips with chemicals in a reactor called a digester. In the Kraft process, the reacting chemicals (cooking liquor) used are mainly an aqueous mixture of sodium hydroxide (NaOH) and sodium sulfide (Na 2 S). In the digester, wood chips are soaked with white or purified cooking liquor and are heated to elevated temperatures for a specified time to dissolve the lignin that holds the fibres together. The chips and cooking liquor are forced into a low pressure blow tank from the digester. The pulp is then washed. In the washing process, the spent liquor containing the lignin compounds removed from the wood is separated from the pulp and the washed pulp is transferred to the bleachery for further processing. The spent liquor, also called weak black liquor, is transferred to the recovery system. The Kraft recovery cycle is composed of a series of processes to recover chemicals from the spent liquor resulting from the wood pulping digester. The weak black liquor is processed in the recovery system to obtain energy from it by combustion of the lignin compounds and to recover the sodium and sulfur chemicals for producing white liquor chemicals which are recirculated to the digester.  1.1.1 Purpose of the Kraft Recovery System The purpose of the recovery system [2] is to supply high quality white liquor to the pulping digester in sufficient quantities, and in a manner efficient enough to maintain overall pulp mill production, profit and environmental objectives. To accomplish those objectives the recovery system has several basic functions: 1. Separate (for re-use) the spent liquor from the pulp in the washing plant. 2. Evaporate the liquor to a solids concentration that is combustible. 1  Chapter 1: Introduction  3. Bum the concentrated liquor in a furnace designed to generate steam and to recover the inorganic compounds. 4. Withdraw the sodium salts from the furnace in a molten form (smelt) and dissolve the smelt in a weak wash to form green liquor. 5. Causticize green liquor with lime to convert the sodium carbonate in the green liquor to sodium hydroxide, while converting the lime to calcium carbonate. 6. Clarify and separate the resulting white liquor from the calcium carbonate precipitate and the remaining calcium hydroxide. 7. Convert calcium carbonate to calcium oxide for re-use in causticizing. 8. Abate or eliminate water and air pollution. Figure 1.1 illustrates the basic features of the Kraft plant. The spent weak black liquor is concentrated by evaporation from about 15% solids to 65-72% solids by weight in the evaporators. The concentrated black liquor is sent to the recovery furnace where additional evaporation, combustion of organics, and chemical conversion of the inorganic chemicals in the black liquor occurs. The energy from the combustion is used to produce steam. The inorganic chemicals leave the furnace as molten smelt. The smelt is dissolved in weak wash (water with a small amount of Na 2 S, NaOH, etc.) in the smelt dissolving tank. The resulting solution (green liquor) consists largely of sodium carbonate (Na2CO3) and sodium sulfide (Na2S). Make-up chemical in the form of sodium sulfate (salt cake  Na 2 SO 4 ) is added to the system at the recovery unit in order to replenish sodium and sulfur losses from the various parts of the process. The green liquor is sent to the slaking and causticizing tanks where hydrated lime reacts with the sodium carbonate to produce sodium hydroxide (NaOH) for the cooking of wood chips. The resulting raw white liquor contains calcium carbonate precipitate (CaCO 3 ) and some unreacted calcium hydroxide (Ca(OH) 2 ) which is separated from the white liquor solution in the white liquor clarifier. The slurry of calcium carbonate and calcium hydroxide is referred to as lime mud. The lime mud is washed and dewatered and then sent to a lime burning unit (kiln) for re-calcination to 2  Chapter 1: Introduction  wood  steam  chips  White liquor  weak black liquor  cooking and washing  strong black o combustion evaporation Hqt j_r.  smelt  weak wash  green liquor  washed pulp  bleachery  bleached  lime mud  effluent system  lime  lime burning  Pulp  treated effluent  smelt disso ving  solids  mud washing and dewatering  4  slaking and causticizing raw white liquor white liquor clarification while liquor  Figure 1.1: A Typical Kraft Pulping Plant  be converted to lime (CaO). The weak soda wash liquor is recycled to the smelt dissolving tank. Clarified white liquor is returned to the digester. 1.1.2 The Recausticizing Plant  The attention paid by management to a given area in a Kraft mill appears to diminish the further the area is from the paper machine [3]. The recausticizing system is the most remote, and as a consequence seems to have suffered considerable neglect considering the overworked and abused conditions prevalent in many North American "caustic rooms". Nevertheless, the mill's economic viability depends on the capability of its recausticizing system to convert recovered chemicals to a proper quality cooking liquor at demand flow to maintain a steady pulp production. The economic pressures of today have increased awareness of recausticizing, particularly in an effort to produce a high quality white liquor and reduce energy usage. Thus, it becomes prudent to examine each of the unit operations and their interactions in this area to determine the most effective control technique to be used in a given situation. It also requires of the operators a thorough 3  Chapter 1: Introduction  understanding of the system and the ability to react to emergency situations to maintain cooking liquor availability. These objectives can be achieved by dynamic simulation of the recausticizing plant and its control system. The results will be a stable high quality white liquor and lower cost of operation. The sequence of unit processes which takes the smelt from the recovery furnace and converts it to white liquor for wood chip cooking is referred to as the recausticizing plant. The objective of recausticizing is to regenerate Kraft cooking liquor (white liquor) for the digester from the recycled inorganic chemicals discharged as a molten smelt from the bottom of the recovery boiler. These chemicals, principally sodium carbonate, are mixed with weak liquor from the recausticizing area to a controlled density to form green liquor. The suspended impurities or dregs which cause the green color must be removed. The clarified liquor reacts under controlled conditions with quick-lime to form sodium hydroxide and calcium carbonate. The calcium carbonate and some unreacted calcium hydroxide, called lime mud, are separated and the clear liquor, called white liquor, is used on demand as the cooking liquor in the digester. The separated lime mud is washed to recover soda values, and the mud is calcined in the lime kiln to form quick-lime, which is recycled to the slaker. The primary functions of the liquor preparation process may be stated as follows: 1. To clarify the green liquor prior to causticizing and to wash the resulting dregs. 2. To convert as much as possible of the recovered sodium carbonate (Na2CO3) into sodium hydroxide (N a0 H). 3. To produce a suitably clarified white liquor of the required concentration and flow rate for the cooking of wood chips. 4. To separate, wash, and dewater the lime mud prior to calcination. 5. To calcine lime mud to a high quality lime. 6. To scrub and recover calcium carbonate and lime from calcining exhaust gases. The recausticizing unit operations appear to many as separately controlled functions, and are often treated as such by mill operators. Yet, the characteristics of rebumed lime and the dregs removal efficiency from green liquor directly affect the washing and dewatering of the lime mud. The interdependency of these operations must not only be recognized at the operator level to allow 4  Chapter 1: Introduction  a proper response to upset conditions, but also must be incorporated in the process at the design stage. Efficient equipment must be selected together with compatible instrumentation and controls to permit the operator to make adjustments before upsets occur. A successful dynamic simulation of the recausticizing plant is very useful in training the operators and in designing an effective control system. This will improve the performance of the whole recausticizing plant operation. 1.1.3 Dynamic Simulation in the Pulp and Paper Industry  There has been a very rapid and substantial increase in the use of computer simulation in the pulp and paper industry in recent years [4]. The most common applications have been in the area of process design using general purpose flowsheet simulations, and in the development of control systems for certain areas of the mills. In the 1980's, there was a general acceptance of simulation as a tool for improving decisions on process design and process operation. Although most of these successful developments have been based on steady-state or quasi-steady-state simulation, the need for robust and dynamic simulation is recognized as a requirement for process optimization and process control. An effective dynamic simulation can be used as a mathematical "pilot-plant" to investigate, predict, explain and modify plant behavior, on the basis of design or measured data. Through effective use of this model, the interactions of all the subsystems within the plant become evident and the response of the whole system to disturbances (such as changing feeds, failure of equipment, or different operating strategies) can be investigated. It is certain that, in the near future, expert systems of various degree of sophistication and ultimately, artificial intelligence systems will be integrated into the control of processes. There is no doubt that process simulation will be an essential component of those systems. Simulators are also very useful in training operators. Simulation allows trainees to explore the effects of operation changes and become familiar with the dynamic behaviour of the plant before dealing with the real plant. Process simulation developments have made process system simulation useful in a wide range of applications, ranging from design and analysis to the development of control, advanced control and expert systems. The integration of process simulation with millwide information and control is nearing reality. 5  Chapter 1: Introduction  1.2 Research Objectives  Because of the multiple objectives involved in the operation of the recausticizing plant, the best operating conditions to fulfill all these different requirements have been far from obvious because of the individual dynamics of each mill operation and the complex interdependencies of the various parts of the recausticizing plant. Jacobi and Williams [5] developed models for the units of a recausticizing plant in early 1970's. Their work was limited by incomplete knowledge for each unit operation, especially in the slaking and causticizing area. Some work has been reported in several specific areas in the recausticizing plant, such as slaking/causticizing, smelt dissolving, and lime mud settling. However, very few papers have been found which were concerned with the modeling and dynamic simulation of the whole recausticizing plant. The present research is thus directed toward developing an unsteady-state mathematical model to describe the dynamics of each unit in the system [6]. As part of the development of a suitable model for the causticizing unit, the thermodynamics and kinetics of the causticizing reaction have been studied. Models of all the units in the system are presented. Based on the modeling for each unit in the system, simulations for both the individual units and the whole plant were carried out. Through these simulations, a better understanding of the dynamics of each unit and the whole recausticizing plant is obtained [7].  6  Chapter 2: Mathematical Modeling of the Recausticizing Plant  Chapter 2 Mathematical Modeling of the Recausticizing Plant  2.1 Introduction  Dynamic modeling is an important tool when control systems are being designed. First, the dynamics of the system can be investigated through dynamic simulation. Second, it facilitates the testing of control schemes by computer simulation of both the process and the proposed control system.  The general approach to modeling in this work was to utilize a modular technique. This provides for a flexible and logical procedure which allows one to develop and first test the models of the individual unit processes before finally putting the entire system model together and testing it. Figure 2.2 shows the unit blocks into which the model has been divided. The sequence of units follows 7  Chapter 2: Mathematical Modeling of the Recausticizing Plant  that of the actual material flow in the system. The models for the individual units are presented in this chapter. However, the slaking/causticizing model is presented in the following chapter because of the importance and complexity of this unit. A glossary of the unit symbols used is given in Table 2.1. Some common pulp and paper terms used in the Kraft recovery system are listed in Appendix A.  Name^  Symbol  Smelt Dissolving Tank^ Green Liquor Storage Tank^ Green Liquor Clarifier^ Dregs Washer^ Slaking / Causticizing^ White Liquor Clarifier^ Mud Mixing Tank^ Mud Washer^ Weak Wash Tank^  sdt glst glc dw sc wlc mmt mw wwt  Table 2.1 List of Units in the Recausticizing Plant  The chemicals dissolved in the green and white liquor are mainly Na2S, NaOH, Na 2 CO 3 , Na2SO4, and Na2S03. In a mill, the typical green liquor contains 8% NaOH, 60% Na 2 CO 3 , 21.5% Na 2 S, 6% Na 2 SO 4 and 3% Na 2 SO 3 [2]. Other chemical components, like K, Mg, Mn, Fe, Si, C1 and S203 , also occur but in negligible quantities. The suspended -  solids are carried with the liquor, the concentration of these solids is based upon the volumetric flow rate of the total mixture. The streams between units in the system are expressed in terms of the mass flow rates of each chemical component. The model of the overall system is based on a mass balance equation for each unit. For the sake of simplicity, the temperatures of the liquors throughout the various unit processes within the liquor 8  Chapter 2: Mathematical Modeling of the Recausticizing Plant  preparation system, with the exception of slaking/causticizing, are assumed to be at an average of 82°C. It is also assumed that solids which dissolve completely in the liquor cause only a negligible increase in the volume of the liquor after dissolution. This assumption means that the volumetric flow rate of the liquor without the insoluble solids is equal to that of water. A further simplifying assumption is that the clarifiers and causticizing tanks are completely full. This assumption is accurate, for that is the way these units are designed to operate. The levels of all the other tanks are controlled by using PI controllers [8]. A detailed analysis for this controller can be found in the Appendix B. Figure 2.3 shows all the inputs and the outputs of the recausticizing plant.  An energy balance could be performed around the slaking/causticizing process to keep track of the liquor temperature. For reasons of simplicity and the small benefit of tracking the causticizing temperature, this energy balance was not included in the overall model. Inclusion of the energy balance would be useful if one wanted to study boilover problems, for example. However boilover at the slaker does not present nearly the explosive hazards that are possible in the recovery furnace, 9  ^  Chapter 2: Mathematical Modeling of the Recausticizing Plant  for example. The temperature of the slaking/causticizing reaction is assumed to be between 95 — 100°C, which is the typical range used in mill operations. 2.2 Smelt Dissolving Tank (SDT)  Figure 2.4 shows the input and output streams of the smelt dissolving tank. In the smelt dissolving tank, the smelt coming out of the furnace is dissolved by the weak wash liquor. The smelt is made up of Na2CO3, Na2S, Na2SO4 and Na2S03. A small amount of insoluble inert matter (K, Mg, Mn, Fe, Al, etc.) is carried along with the smelt. Weak wash liquor from the weak wash storage tank is fed to the dissolving tank to dissolve the smelt. The weak wash contains a small amount of NaOH, Na2CO3, Na2S, Na2SO4 and Na2S03 dissolved in water. The mass flow rates of the  weak liquor components are: 147' 4 , i = 0, ..., 5. The mass flow rates of the smelt components are denoted by Wr n  = 1, ..., 6. The subscripts correspond to specific chemical components,  as shown in Table 2.2.  Index^  Chemical Component  ^0^  1120  1^  Na2S  ^2^ ^3^ ^4^ ^5^ ^6^ Table 2.2  NaOH  Na2CO3 Nat SO4 Na2S03 solids  indexes for Chemical Components  Quantities given without a subscript represent total quantities of that stream. The overall mass balance for the smelt dissolving tank is: d m sdt ^=  Wwwt wsm  dt  10  W sdt  (2.1)  Chapter 2: Mathematical Modeling of the Recausticizing Plant  Figure 2.4: Smelt Dissolving Tank  The mass balance for each component is given by d in sdt  dt  = Wwwt  wrn wisdt  i = 1, ...,6  (2.2)  The mass flow rate of each chemical component, i, in the weak wash stream and output stream can be expressed as wiwwt = cr tQ w wwt  W sdt = c.vdt(dsdt^  (2.3)  i = 1, ...,5  Let Vs & be the volume of liquor in the tank, i.e., vsdt = Asdt hsdt^  where Asck is the tank cross-sectional area and W dt is the liquid level. Substituting equations 2.3 and 2.4 into the component mass balances yields 11  (2.4)  Chapter 2: Mathematical Modeling of the Recausticizing Plant  d(VsdtCi) ^Asdtd(113Cit ^dC. hS,wu^2 + 'pa... C. dhSdt dt^dt^dt^dt  = Qwwt C=wt W9M Crdi Cri  ^  (2.5)  = 1, ...,5 The insoluble solids in the smelt is less than 0.5% of the total mass flow rate and occupies a negligible volume. As mentioned before, if we assume that the solids that dissolve completely in the liquor cause only a negligible increase in the volume of the liquor after dissolution, then dhsdt  A sdi^= Q wwt _ Q sdt  (2.6)  dt  Assuming perfect mixing, then Ci = Grit ,i = 1, ..., 5^  (2.7)  Substituting equations 2.6 and 2.7 into equation 2.5 yields AS* h sdt dqsdt = Q wwt (c wt  _ crit) wism  dt  i = 1, ..., 5^(2.8)  The insoluble suspended solids are also treated like a chemical component. The mass balance for the solids is given below: Asdt hsdt dCet dt = Qwwt cgdt  wr^(2.9)  Equations 2.6, 2.8 and 2.9 represent the mathematical model of the smelt dissolving tank. 2.3 Green Liquor Storage Tank (GLST)  Figure 2.5 shows the input and output streams of the green liquor storage tank. The output of the smelt dissolving tank is directed to this storage tank before it is sent to the green liquor clarifier. The input mass flow rates of each component are W sdt^= 0,^6. i  12  Chapter 2: Mathematical Modeling of the Recausticizing Plant  The overall mass balance for the tank is d rag Ist  wsdt wglst  dt  (2.10)  The mass balance for each component is given by d,„„..glst ^ = wsdt  dt  wglst  i  = 1, ...,6  (2.11)  The mass flow rate of each chemical component, i, in the input stream and output stream can be expressed as wsdt csdt Qsdt  Twist = cgIst QgIst^ i  (2.12)  = 1, ...6  Let Vgis t be the volume of liquor in the tank Vg lst  = Aglst hglst^ 13  (2.13)  Chapter 2: Mathematical Modeling of the Recausticizing Plant  where M ist is the tank cross-sectional area and hg i st is the liquid level. Substituting equations 2.12 and 2.13 into the component mass balances yields d (V giSt Ci)^Ag l s t d(hgiSt Ci)^1^ I dhg ^Ag.s. hg.s. dC^Ag6s. dt^dt^dt^dt 4.  Qsdt csdt Qglst cigist  (2.14)  i = 1, ..., 6  Assuming that the density of the green liquor doesn't change in the green liquor storage tank, the overall mass balance becomes Aglst dhglst^nsdt Qgtst  (2.15)  dt  Assuming perfect mixing, then CZ  = O ist ,^i = 1, ..., 6^  (2.16)  Substituting equations 2.15 and 2.16 into equation 2.14 yields l st  t^Qsdt Aglst hg,st d C?  dt  (crlt cgtst)  i = 1, ..., 6^(2.17)  Equations 2.15 and 2.17 represent the mathematical model of the green liquor storage tank.  2.4 Green Liquor Clarifier (GLC)  Figure 2.6 shows the input and output streams of the green liquor clarifier. Prior to recausticizing, the green liquor is clarified to remove the suspended solids, referred to as dregs. It is assumed that all of the suspended solids are removed in the underflow of the green liquor clarifier. The input to the green liquor clarifier is the output from the green liquor storage tank. The mass flow rate of each component of the input stream is Wr ist , i = 0, ..., 6. 14  Chapter 2: Mathematical Modeling of the Recausticizing Plant  Clarified Green Liquor (water, chemicals)  vglco  Green Liquor (water, chemicals, solids)  V  glcu  Dregs (water, chemicals, solids)  Figure 2.6: Green Liquor Clarifier  Since the green liquor clarifier is completely full, the overflow volumetric flow rate,  Qglco, is:  Qglco = Qglst Qglcu ^  (2.18)  where the underflow volume flow rate, Qg", is an operating condition set by the operators. The overall mass balance for the green liquor clarifier is: d ing ic  = wglst wglco wglcu  dt  (2.19)  The mass balance for each component, i, in the overflow compartment is given by arn ; _ .  olCO  dt  wpisto wow  i = 1, ...,5^(2.20)  where Wr ist° is the mass flow rate of specie i in the input stream associated with the liquor which becomes the output stream. The mass flow rate of each chemical component, i, in the input stream and the overflow stream can be expressed as cr.F IS t (2 g 1 CO  wg Ico  cg lc o (p l c °^  i= 15  (2.21)  Chapter 2: Mathematical Modeling of the Recausticizing Plant  Let Vgl" be the volume of the overflow compartment and substitute equation 2.21 into the overflow mass balance to obtain d(Vgl"Ci)^icodCi = (-plc° clist = Vg^ dt^dt^.`°  Qg ico cq ico  i=  1, ..., 5^(2.22)  Assuming perfect mixing, then CZ =  C .91 ' ,^i = 1, ...,5^  (2.23)  Substituting this into equation 2.22 gives the model for the green liquor clarifier overflow: dCq i c°^1^rg ist  cg ico)  (2.24)  dt^Tglco  where the residence time rg l c° is defined as T glco  =  vglco  iQglco  (2.25)  The dynamic behaviour of the liquor underflow compartment is approximated by a series of four well-mixed tanks to represent a plug flow with some backmixing. It was found that this way of modeling the underflow compartment of the clarifier can well represent what has been observed in the mill [5]. This also agrees with the design of the clarifiers. In the actual clarifiers, there are often four compartments in the underflow part. The four underflow tank residence times are assumed to be equal and are defined as Tglcu =  vglcu  4Qgicu  (2.26)  The four well-mixed tanks used to express the dynamics of the green liquor clarifier underflow are then described by the following equations: 16  • •  Chapter 2: Mathematical Modeling of the Recausticizing Plant  1 (cy,g1st^ryglcu) T glcu^2  xl  dt dCf21 ' dt dCf31 ' dt dciF4lcu dt  =  1^cglcu cyglcu) , e  Tg cu (^t2  1  ryglcuryglcu) Tglcu (-1 t2^'i3  1  0 .F leu  (2.27)  _ 0.gicu)  T glcu^t3  i=  where cf41C21 is the output concentration of chemical component i in the green liquor clarifier underflow. 2.5 Slaking/Causticizing (SC) 2.5.1 Introduction  The slaking/causticizing process converts the Na 2 CO 3 in the green liquor to NaOH to produce white liquor. The reaction proceeds in two steps. The first step, known as slaking, involves the addition of quick-lime (Ca0) to the green liquor where it reacts with water to form calcium hydroxide (Ca(OH) 2 ) as follows: Ca0 I H2 O Ca(011) 2^ - -  (2.28)  In the second step, the Ca(OH) 2 , which was formed during the slaking step, reacts with Na2CO3 to produce NaOH and CaCO 3 :  Ca(OH) 2 + Na2CO3 2NaOH d CaCO 3^(2.29) -  Although the reactions are written as two steps, they actually overlap because part of the causticizing occurs almost coincidentally with the slaking. The causticizing reaction is reversible 17  Chapter 2: Mathematical Modeling of the Recausticizing Plant  and may proceed in either direction depending upon the relative concentrations of the reactants and products. Because CaCO 3 is less soluble than Ca(OH) 2 , the causticizing reaction proceeds to the right. However, because of the reversibility, all of the Na 2 CO 3 cannot be converted to NaOH regardless of the amount of lime used. The extent of conversion depends on the concentration of sodium compounds in the green liquor entering the slaker. Lower concentrations produce higher conversion, because high concentrations of NaOH progressively reduce the solubility of Ca(OH) 2 . This continues until there are not enough calcium ions present to exceed the solubility limit of CaCO 3 [9]. 2.5.2 Slaking Reaction  The slaking/causticizing reaction occurs in a series of four stirred tank reactors, as shown in Figure 2.7. The first one is called the slaker. Quick-lime is added to the green liquor in the slaker to form Ca(OH) 2 . The slaker has a retention time of about five to ten minutes. The three causticizing tanks that follow have retention times of 30 to 40 minutes each.  Figure 2.7: Slaking and Causticizing Process It is assumed that the slaking reaction occurs instantaneously relative to the causticizing reaction  and perfect mixing is achieved in each tank. 18  Chapter 2: Mathematical Modeling of the Recausticizing Plant  The equilibrium causticizing condition is solved for using Pitzer's model and the kinetics of the reaction is obtained from an empirical equation. The equilibrium and the kinetics of the reaction will be discussed in detail in the next chapter. There are three inputs to the slaker: lime, green liquor and make-up water. The green liquor is the output from the green liquor clarifier overflow compartment. The mass flow rates are denoted by wrIco, i = 0, ..., 5. The mass flow rate of lime is W7 and its availability is AV, which is a parameter that describes the reactive fraction of the total lime. Also, some make-up water, Nnk , may be added to the slaker. The lime that can be used in the slaking reaction, W77., is W7 r = W7 x AV^  (2.30)  by definition of the availability. Assuming that all of the available lime is slaked, the mass flow rate of Ca(OH) 2 into the causticizing process becomes W8 = (MW8 I MW7) X W7 r^(2.31)  where MW8 is the molecular weight of calcium hydroxide and MW7 is the molecular weight of calcium oxide. The total grits mass rate is the difference between the actual lime flow rate and the available lime flow rate: W4m = (1 — AV) x W7^  (2.32)  The mass rate of water consumed in the slaking reaction, WC", is given by: Wen = (mwo /mw7 ) x nn  ^  (2.33)  where MW0 is the molecular weight of water. The mass rate of water after the slaking reaction is given by: inci = wro + KnIc _ wen^(2.34) 19  Chapter 2: Mathematical Modeling of the Recausticizing Plant  The amount of make-up water may be different from mill to mill. No make-up water was added in the design case used here. However, this can be changed according to the actual situation in the mill. 2.5.3 Modeling of the Causticizing Reaction  After the quick slaking reaction, the solution goes through the three causticizing tanks in series. In the course of the causticizing reaction, the chemical components are divided into two groups, reactive chemical components and non-reactive chemical components. The reactive chemical components include Na 2 CO 3 , NaOH, and Ca(OH) 2 . The non-reactive components are N a 2 S, Na 2 SO 4 , and Na 2 S0 3 . The products of the reaction are CaCO 3 and NaOH. The chemical concentrations after the slaking reaction are:  cr =  wpI CO  ip w  i = 1, ..., 5  (2.35)  i = 1, ..., 5  (2.36)  The molar concentrations are: Cg,! = Cri /MW;  where MIVi is the molecular weight of component i. 2.5.3.1 Modeling of the Reactive Components  The conversion of Na 2 CO 3 to NaOH is a two phase (liquid-solid) heterogeneous reaction. Because there is not any information available on the intrinsic kinetics, a simplified approach is adopted. The following empirical equation is used to describe the dynamics of the reaction: dCE = k i (CEeq — CE) k2 dt  (2.37)  where CE represents the causticizing efficiency and CEeq is the equilibrium causticizing efficiency. The constants, k 1 and k 2 , are kinetic parameters of the reaction and are defined in next chapter. The 20  ^  Chapter 2: Mathematical Modeling of the Recausticizing Plant  causticizing efficiency is defined as [NaOH] — [N a0 14, CE =^ x 100 [IV a0 H] — [N a0 Hi ° + [Na 2 CO 3 ]  (2.38)  where the sodium hydroxide concentration [N a011] is corrected for the initial concentration [IV a0 H] o present in the green liquor. The causticizing efficiency is chosen here because it de-  scribes the extent of the completion of the causticizing reaction [10]. The detailed development of this equation is presented in Chapter 3. The stoichiometry of the causticizing reaction can be determined from alternative reaction schema: Ca(OH) 2 + CO 3= .#. CaCO 3 + 20H ^initial  —  conc.^a^b a f ter reaction ^a — x^b + 2x  (2.39)  After the slaking reaction, the concentrations of the reactive components are [COT] = a and [0H ] = b. The concentrations of each component in the green liquor will determine the equilibrium -  causticizing efficiency, C Eeq , and the residence time will give the corresponding causticizing efficiency, CE at the outlet of the last causticizer. Assuming that x moles/L of COT are consumed in the reaction, then the final concentrations will be [COT] = a - x and [0 H ] = b + 2x. According -  to the definition of the causticizing efficiency, we have: (b + 2x) — b^2x CE =^ (b+2x)—b+a—x a+x  (2.40)  Thus, x can be obtained from CE by solving this equation. The final mass flow rates of each reactive component after the reaction can then be calculated as follows: CaCO3 produced:  I/I/el = x x MW9 x Q" 1^(2.41)  21  Chapter 2: Mathematical Modeling of the Recausticizing Plant  Ca(OH) 2 left:  = Wg x x MW8 x Cr i^(2.42) —  All the solids after reaction: Wsc = wpt^+ Ind  (2.43)  = (C1Z + 2 x x) x MW2 x Q sci  (2.44)  Final NaOH: Wr  Final Na2CO3: W;c = (CLcia — x) x MW3 x Qsc l  (2.45)  where Cfmci is the molar concentration of component i after slaking reaction, as calculated in equation 2.36. Due to the creation of a large amount of solids in the reaction, the volume and density of the solution change. The density of the solution is a nonlinear function of the density of the liquor and the solids content. A standard method of calculating the solution density has been summarized by TAPPI. Data can be found from reference [11] to calculate the density of the solution. A third order polynomial regression [5] for the data yields the following expression for the density of the total slurry mix, given the liquor density and the percent solids: ws e Pss =^x ws c 100%  E wr 6  wsc  i=0 psc = Pt + kioPss + k11Ps2s + k12P8 ,  22  (2.46)  Chapter 2: Mathematical Modeling of the Recausticizing Plant  where p" is the density of mixture, PI is the density of the liquor, Pss , the percent solids content, is given by Wsc, the total flow rate existing the causticizer. Once the density of the mixture is known, the total volumetric flow rate can be calculated from  Q  SC = W3C i p SC^  (2.47)  2.5.3.2 Modeling of the Non-reactive Components  The non-reactive chemical components (Na2S, Na2SO4, Na2SO3) are assumed to go through three perfect mixing tanks in series. The residence time of each tank is: T sc  =  Vse  (2.48)  Q sc  where V" is the volume of each tank. Based on a previous similar development, the equations for each non-reactive chemical component are as follows: ^dCfc l 1^• =^(Ct9 c 2 dt rsc " dCr 2^1 ^c9c2 a (c " ) dt rc dCfc3^1 ( c,c2 C s c3 ) ^T dt^-sc ^2 —  (2.49)  i = 1, 4, 5  where C; c3 is the output concentration of chemical component i after the third tank.  2.6 White Liquor Clarifier (WLC)  2.6.1 Modeling of the Chemical Components  Figure 2.8 shows the input and output streams of the white liquor clarifier . The slurry from the causticizers contains calcium carbonate and unreacted calcium hydroxide (lime mud) which must be 23  Chapter 2: Mathematical Modeling of the Recausticizing Plant  Clarified White Liquor White Liquor  V  (water, chemicals, solids)  (water, chemicals)  wlco  wi c u V Lime Mud (water, chemicals, solids)  Figure 2.8: White Liquor Clarifier  removed from the white liquor before it is sent to the digester. The clarification operation is carried out in the same type of equipment as the one used for green liquor clarification. The input to the white liquor clarifier is from the slaking/causticizing tanks. The mass rates of each component are Wr , i = 0, ..., 6. The total volumetric flow rate as computed before is Q". Since the white liquor clarifier is completely full, the overflow volumetric flow rate, Qw/c0 , i s Q  wlco  = Qsc Qwlcu^  (2.50)  where Qw" is the specified underflow volumetric flow rate. According to the design data used in this study, 20% of the total volumetric flow rate appears as underflow while the rest overflows to the white liquor storage tank. In mill operations, the volumetric flow rate of the underflow may be set constant or manipulated according to production requirements. Since it is assumed that all of the lime mud solids settle out in the white liquor clarifier, all the solids are in the underflow and the overflow volumetric flow rate, Q" , is determined only by the 100  liquor volumetric flow rate. The overall mass balance for the white liquor clarifier is: dmwtc dt  wac _ w wico _ w wieu  24  (2.51)  Chapter 2: Mathematical Modeling of the Recausticizing Plant  The mass balance for each component, i, in the overflow compartment is given by dm 0 " = ^ dt  wisco _ wri co  i = 1, ..., 5^ (2.52)  where W:" is the mass flow rate of specie i in the input stream associated with the liquor which becomes the overflow stream. The mass flow rate of each chemical component, i, in the input stream and the overflow stream can be expressed as wro =  cr Qtvico  wr ico = cr Ico Q wico^  (2.53)  i = 1, ...,5  Let Wy k° be the volume of the overflow compartment and substitute equation 2.53 into the overflow compartment mass balance equation to obtain: d(Vw 1 c° Ci) = _ v wico dC dt^dt ^Q wic°67c  -  (rico  cyJico  i = 1, ..., 5^(2.54)  Assuming perfect mixing, then = Cr;" , i = 1, ..., 5  Cii  (2.55)  Substituting this into equation 2.54 gives the model for the white liquor clarifier overflow compartment, namely dC rico^1 dt  Cric°)  -  1 ,-95  (2.56)  where the residence time 7 - wic° is defined as T W1CO  = VW1CONWICO  25  ^  (2.57)  • •▪  Chapter 2: Mathematical Modeling of the Recausticizing Plant  The white liquor underflow storage is considered to be a series of four well-mixed tanks, to represent a plug flow with some backmixing, similar to the scheme used earlier for the green liquor clarifier underflow. The four underflow tank residence times are each given by vwlcu  r ?Dim =  (2.58)  4Qwlcu  Following a similar development as for the overflow, gives the following equations for the white liquor clarifier underflow tank:  c  dqylcu  1 ( c _ Cw icu) dt Twieu^21 dCT2' 1 ' dt 7w1lcu^Qicu)  dClgicu  (cgcu crIgi cu)  dt^7  dCr4 1 "  1  dt^r wicu ( CN icu i  (2.59)  I  cwlcu i)  = 1, ..., 5  where cri lcu is the output concentration of chemical component i in the white liquor clarifier underflow. The lime mud is treated differently in the underflow compartment, as described below. 2.6.2 Lime Mud Settling  The difference in modeling between the white liquor clarifier and the green liquor clarifier is how the solids in the solution have been dealt with. In the green liquor clarifier, all solids settle out and are treated like any other chemical component in the underflow of the green liquor clarifier. In the white liquor clarifier, the lime mud settles slowly in a plug flow fashion in the underflow compartment. This difference in modeling is due to the different settling quality of these two solids. The input-output diagram, Figure 2.8, depicts how separation of the lime mud from the liquor is accomplished. The operation of the white liquor clarifier and the mud washer is very important to the whole causticizing plant, a fact which is seldom recognized. Mud washing practices determine the TTA (Total Titratable Alkali, as defined in Appendix A) of weak wash, which is used to dissolve the smelt and produce green liquor in the smelt dissolving tank. The weak wash TTA will affect the concentration of the green liquor, and therefore, the causticizing efficiency in the following reaction. 26  Chapter 2: Mathematical Modeling of the Recausticizing Plan!  Millet and Damstrom [12] discussed the control of lime mud washing. They showed the effects that uncontrolled mud washing has on slaker performance using dynamic simulations. The underflow density of the mud washer was found to be the most important factor in mud washer control. The model used for the lime mud washer was not given in their paper. The lime mud free settling rate can be approximated by using Stokes' law. Stokes' Law is stated as U = (p s  —  pi)Dp2 11871^  (2.60)  where U = velocity of sedimentation (m/s)  D p = diameter of sludge particle (m) p s = density of sludge particle (kg/m 3 ) pl =  density of the solution (kg/m 3 )  = dynamic viscosity of liquor (Pa s) As can be seen from equation 2.60, the particle size has a strong influence on the settling rate. Jacobi [5] summarized the two methods of defining the mud particle size, which are based on the conditions in causticizing and in calcining. Both methods are complicated and depend on the parameters of other unit operations in the recausticizing system. This type of model would be satisfactory if one wanted to conduct a more detailed study of lime mud settling, but that was not the objective in the present study. Rather, the goal is to model the overall sodium cycle and to, eventually, assist with the design of optimal control systems. Dorris [13] conducted a comprehensive study about sedimentation of lime mud particles. He reported the influence of different factors on lime mud settling and summarized various models proposed to explain the settling mechanism. The settling rates in white liquor and wash water were compared. It was found that wash water, with its lower alkalinity, viscosity and density, leads to a faster settling mud and to a more concentrated sediment, in agreement with mill experience. However, the application of the model depends on many operating data, such as viscosity, density, lime mud diameter, and so on, which are normally not immediately available in mills. 27  Chapter 2: Mathematical Modeling of the Recausticizing Plant  It was found that, in general, conditions leading to an increase in the rate of causticizing resulted in a reduction in the settling rate of the lime mud produced [10]. Rydin [14] studied the relation between the causticizing efficiency and the settling rate and found that the settling rate is almost uniquely determined by the causticizing efficiency of the white liquor. The relation between settling rate and causticizing efficiency is shown in Figure 2.9. This result allows us to calculate the settling rate as a simple function of causticizing efficiency.  300  E  250  200  E a)  150  100  50  C./) 0 ^ 90 50^55^60^65^70^75^80^85  Causticizing efficiency, Figure 2.9: Settling Rate vs Causticizing Efficiency  The lime mud settling process in the white liquor clarifier and mud washer is modeled as a plug flow. The dead time of the plug flow may change as a result of different causticizing efficiencies. By knowing the height of the tank and the settling rate, the dead time of the plug flow can be easily calculated. Since the mud settling rates are different in the white liquor clarifier and the mud washer, we assume that the settling rate in the mud washer is approximately 1.1 times faster than in the white liquor clarifier based on the work done by Dorris [13]. The solids in the white liquor clarifier underflow is equal to the input to the white liquor clarifier with a time delay. 28  Chapter 2: Mathematical Modeling of the Recausticizing Plant  2.7 Mud Mix Tank (MMT)  Figure 2.10 shows the input and output streams of the mud mix tank. The lime mud from the white liquor clarifier is mixed in the mud mix tank with water and then separated from the liquor in the mud wash filter. The inputs to the mud mix tank are as follows: (1) fresh water: Woft", (2) underflow mud slurry from white liquor clarifier: Ww i ",  i  = 0, ..., 6 with the total  volumetric flow rate Qtolcu, (3) stream from kiln scrubber: Wiks , i = 0, ..., 6 with the total volumetric flow rate (2k.99 and (4) filtrate from lime mud filter: Wilmf ,  i  = 0, ..., 6 with the total volumetric flow rate Q lmf.  Output of Kiln Scrubber  Lime Mud from Clarifier  (water, solids)  (water, chemicals, solids)  Lime Mud from Filter  Fresh Water  (water, chemicals, solids)  (water) ^  h  mmt  Lime Mud (water, chemicals, solids)  Figure 2.10: Mud Mix Tank  The overall mass balance for the mud mix tank is:  dmmmt  = w f w wwlcu wks wlm f wmmt  dt  29  (2.61)  Chapter 2: Mathematical Modeling of the Recausticizing Plant  The mass balance for each chemical component, i, is given by dTienint  —  dt  wr lcu wiks wilmf Knmt  i = 1, ...,6^(2.62)  The mass flow rate of chemical component, i, in each stream can be expressed as  wr lcu crI cu Qwlcu Wks  Wlmf  cts  Qks  clru f Qlmf  (2.63)  W 711Mt CrIMtQMITEt  8  i = 1, ...,6  Let Vn't be the volume of liquor in the tank, vmmt = Arnmt hmmt  (2.64)  where Amm t is the tank cross-sectional area and hmm t is the liquid level. Substituting equations 2.63 and 2.64 into the mass balance equation yields t d(h 17 •^ dhmmt Amrnt hmmt dCi = Ammtci ^+ dt^dt^dt^dt ^t wicu ar icu Q ks Gip + Q tmf ctmf Q mmt  d (VmmtCi _= ^  Q_  (2.65)  i = 1, ...,6  The input stream from the white liquor clarifier underflow is composed of two parts: liquor and solids. The total volumetric flow rate of the white liquor clarifier underflow is simply the sum of the liquor and solids volumetric flow rates, i.e., Q  W1C11 =  01.11C11  30  Q swlcu^  (2.66)  Chapter 2: Mathematical Modeling of the Recausticizing Plant  The volumetric flow rate of the mud mix tank can be expressed in the similar way: Q  mmt  =  Q171Mt  g9MMt^  (2.67)  It is found that the percentage of solids in the streams from both lime mud filter and kiln scrubber is very small and, therefore, the volumetric flow rates of the solids in those two streams can be neglected. If we assume constant density for the water in each stream, then the overall mass balance for the liquor becomes  A'  dmmti h^Qfw (elm Qks Qlmf Qmmt dt  (2.68)  A similar equation can be used for the solids if we assume the density of the lime mud to be constant dVmm tsdhmmts Qwlcu ^ = A' ^ dt^dt  _  w  mt  (2.69)  Combining equations 2.68 and 2.69, then dhmmt A' ^ dt  (2, w  (r  im  + 0.5 Qlmf Qmmt  (2.70)  where hmmt  = h MMtl hmmts  (2.71)  Assuming perfect mixing in the mud mixing tank, then =Cmmt^i = 1, 6^  Substituting equation 2.70 into equation 2.65 then yields 31  (2.72)  Chapter 2: Mathematical Modeling of the Recausticizing Plant  Amm, h mmo dCrnt 4^ 4  Qwlcu (Cwlcu crmt) Qks (GIs crint)  +  dt Qlmf (Clmf — Crm t ) — Qf w  (2.73)  Cr mt  = 1, ...,6 Equations 2.70 and 2.73 represent the mathematical model of the mud mix tank. 2.8 Mud Washer (MW)  Figure 2.11 shows the input and output streams of the lime mud washer . The lime mud settles out in the mud washer and is sent to the lime kiln to be calcined. The clarification operation is carried out in the same type of equipment as the one used for white liquor clarification. The input to the mud washer is from the mud mix tank. The mass flow rates of each component are: Wmm t , i = 0, ...,6 and the total volumetric flow rate, as computed before, is Qmmt. i  Lime Mud From Mud Mix Tank (water, chemicals, solids)  Weak Wash V  (water, chemicals)  mWO  V  mwu  Concentrated Lime Mud (water, chemicals, solids)  Figure 2.11: Mud Washer  Since the mud washer is completely full, the overflow volumetric flow rate, Q  MW 0  = Qm mt Q 32  M WU  Qm"° , is:  (2.74)  Chapter 2: Mathematical Modeling of the Recausticizing Plant  where Qm" is the specified underflow volumetric flow rate. According to the design data used in this study, 20% of the total volumetric flow rate goes into the underflow while the rest goes to the weak wash tank as overflow. As mentioned before, this ratio may differ from mill to mill. Since it is assumed that all of the lime mud solids settle out in the mud washer, all the solids will be in the underflow and the overflow volumetric flow rate, Cr", is determined only by the liquor volumetric flow rate. The overall mass balance for the mud washer is: dm" dt  wmmt  wmwo wmwu  (2.75)  The mass balance for each component, i, in the overflow compartment is given by dm7 1-u° dt  = wrnmto wrwo  i  = 1, ...,5^(2.76)  where Wmmto is the mass flow rate of specie i in the input stream associated with the liquor which becomes the overflow stream. The mass flow rate of each chemical component, i, in the input stream and the overflow stream can be expressed as winnito crmt Qmwo Wm  = crwo  i =  Qmwo^  (2.77)  1, ...,5  Let V"'° be the volume of the overflow compartment and substitute equation 2.77 into the overflow compartment mass balance equation to obtain d(Vinw° C^vmwo dC dt^dt  _ n rnwo crit _ Q mwo cr wo 33  i = 1, ...,5^(2.78)  Chapter 2: Mathematical Modeling of the Recausticizing Plant  Assuming perfect mixing, then ci =  Cr'  ,  i = 1, ..., 5^  (2.79)  Substituting this into equation 2.78 gives the following model equation for the mud washer overflow compartment: dCrw° dt  1  T  mwo (Cr m t — Cr w°)^i = 1, ..., 5  (2.80)  where the residence time T in w ° is defined as 771,WO^V771,WO/Q772,WO  (2.81)  The liquor underflow storage is considered to be a series of four well-mixed tanks to represent a plug flow with some backmixing as discussed before. The four underflow tank residence times are each defined as T  Vmwu  rnwu  4 Qmwu  (2.82)  Following a similar development as was carried for the overflow gives the following equations for the mud washer underflow compartment: dCWwu dt  wu (c  r mt _ cru)  dCr2iwu • rrnwu(Cru — C :721w u ) dt dCr3 "^1 (C72'we' —^u • T mwu dt dCri wu^1 • rinwu (C1. 371 " u — Cri" u dt  (2.83)  i = 1, ...,5 where Cru is the output concentration of chemical component i in the mud washer underflow.  The lime mud settling process is modelled in a similar way as that in the white liquor clarifier as discussed before. The output solids in the mud washer underflow is equal to its input with a time delay. 34  Chapter 2: Mathematical Modeling of the Recausticizing Plant  Fresh Water C2 Washed Stream  Unwashed Stream  (water, chemicals, solids)  (water, chemicals, solids)  CO  Cl  Filtrate Stream (water, chemicals)  Figure 2.12: Generalized Dregs Washer Model  2.9 Dregs Washer (DW) The dregs are pumped out of the green liquor clarifier as a concentrated slurry, mixed with wash water, and are usually separated from the liquor in the rotary drum washer known as the dregs washer. The following model, based on concepts used for brownstock washers [15], is introduced here to describe the dregs washing process. The purpose of the dregs washer is to recover the chemicals associated with the solids. All solids are assumed to be removed into the washed stream. The generalized dregs washer is shown in Figure 2.12. Two parameters are used to describe the operation of the washer. The displacement ratio, DR, is defined as follows: CF,11 ' — Cv iwu DR = ^ riglcu  (2.84)  The displacement ratio is the ratio of the actual reduction of the concentration of dissolved chemicals in the washed stream divided by the maximum possible concentration reduction of the dissolved chemicals. The displacement ratio determines the amount of chemicals recovered from the unwashed solids stream. 35  Chapter 2: Mathematical Modeling of the Recausticizing Plant  The other parameter, the solids to water ratio, SW, is the ratio of the mass flow rate of solids divided by the mass flow rate of water in the washed stream, i.e., wglcu SW =  (2.85)  6  wdwu 0  The solids to water ratio determines the amount of water associated with the washed solids stream. These two parameters are assumed to be constant and can be calculated from normal operating conditions. Given the knowledge of these two parameters, the output of the washer can be easily calculated. The input mass flow rates, which come from the green liquor clarifier underflow, are Wf lcu , 0, ...,6. The input volumetric flow rate is Q 91 '. The chemical concentrations of each component from the green liquor clarifier underflow are C741 '. Another input is fresh water, Wof w . The total mass flow rate of water into the dregs washer is W6/w wog rcu wofw According to the definition of the displacement ratio in equation 2.84, the concentrations of each component in the washed stream can be calculated as  Cflwu = (1  —  DR) x q41 '  ^  (2.86)  where Cli"u is the concentration of specie i in the washed stream of the dregs washer. The water in the underflow is: Wsolid  W dwu  = wg Icu (2.87)  = WelCU/sw  So the mass flow rates of water and each chemical component in the dregs washer underflow are:  Wdwu^ •liwu  x  w(d) wu  pw^1,  5^(2.88)  The mass flow rates of each component in the dregs washer overflow are: 36  Chapter 2: Mathematical Modeling of the Recausticizing Plant  Klwo  W  dwo  = wyw Trolwu  w glcu widwu^= 1 t  ,...,  5  (2.89)  Since the dregs washer has a very short residence time in comparison with the other units of the recausticizing plant, the dynamics of the dregs washer is neglected.  2.10 Weak Wash Tank (WWT)  Figure 2.13 shows the input and output streams of the weak wash tank. All the weak wash liquor with some fresh water is contained together in the weak wash tank before it is sent back to the smelt dissolving tank. There are three inputs to the weak wash tank: (1) fresh water: Wo w (2) filtrate from the dregs washer: W  i = 0, ..., 5  (3) overflow from the mud washer: Wm"', i = 0, ..., 5  Weak Wash from Dregs Washer  (water, chemicals)^Weak  Wash from Mud Washer (water, chemicals)  Fresh Water  wwt  h  Weak Wash (water, chemicals)  Figure 2.13: Weak Wash Tank 37  Chapter 2: Mathematical Modeling of the Recausticizing Plant  The overall mass balance for the weak wash tank is dmw"t w wdwo wmwo wwwt dt — v v o  (2.90)  The mass balance for each chemical component, i, is given by dmrwt = widwo wimwo Wwwt dt  i  = 1, ...,5^(2.91)  The mass flow rates of chemical component, i, in each stream can be expressed as WidWO  c ciwo Q dwo  wr wo = Cr wo Qmwo (2.92)  wiwwi = cr wt Q wwt i=  Let V' be the volume of liquor in the tank, V' t = Aw " hw"^  (2.93)  where A' is the tank cross-sectional area and h' is the liquid level. Substituting equations 2.92 and 2.93 into the component mass balance equation yields d (V wwt C =  Awwt  d (hwwt C^Awwt hwwt dC  Awwt .  dhwwt  dt^dt^dt^dt  = QdWO ClIWO QMWO CTI.WO QWWt Cri^  (2.94)  i=  If we assume the density of the liquor is constant, then dhwwt AA"" ^ = dt  Q f w^Qmwo  38  Q  WWt  (2.95)  Chapter 2: Mathematical Modeling of the Recausticizing Plant  Assuming perfect mixing yields Ci  = Cr t ,^i = 1, ...,5^  (2.96)  Substituting equations 2.95 and 2.96 into equation 2.94 then gives  dCrw t^d^ Pr wt h wwt^= Q w° (Cid w° — Cr t ) Q"° (Crw° —  dt  — QfwCrw t  ,  (2.97)  i = 1, ..., 5  Equations 2.95 and 2.97 represent the mathematical model of the weak wash tank. 2.11 Summary Remarks  All of the equations given in this chapter, except the calculation of the causticizing reaction which is carried out by FORTRAN programming, are solved using the dynamic simulation program SIMNON on a SUN/SPARC station 2. More information on SIMNON and programming can be found in Chapter 4. It should also be noted that the operation of the lime kiln area is not of concern in the present study, although the operations of the lime mud filter, kiln scrubber and lime kiln do have direct impacts on the causticizing plant.  39  Chapter 3: Thermodynamic and Kinetic Studies of the Causticizing Reaction  Chapter 3 Thermodynamic and Kinetic Studies of the Causticizing Reaction In a broad sense, all the operations in the recausticizing plant depend on what takes place in the  slaking and causticizing area. Even if the causticizing reaction is an old method for producing sodium hydroxide, and many investigations have been performed, many questions have still to be answered before this reaction may be fully understood. In order to understand the fundamentals of this reaction, both the thermodynamics and kinetics of the causticizing reaction need to be studied. In this chapter, the thermodynamics of the causticizing reaction is first discussed and modelled. Then, the kinetics of the causticizing reaction is investigated and a kinetic model is presented. The definitions of terms used in this chapter can be found in Appendix A. 3.1 Introduction  The causticizing of green liquor with lime proceeds in two stages: Slaking C a0 (9 ) + H2 O -p C a(0 H)2(3)  ^  (3.98)  Causticizing ^ (3.99) C a(0 H) 2( , ) + N a 2 C 0 3 2NaOH + C aC 0 3(s)  The slaking reaction occurs instantaneously relative to the causticizing reaction. Another main component in the green liquor is N a2 S , which is believed to undergo complete hydrolysis with H2 0 according to the reaction: Na2S +1120 > NaHS + NaOH^ (3.100) —  The thermodynamic equilibrium for the causticizing reaction is important for the efficient operation of the Kraft pulping process. It determines the amount of carbonate dead-load in the 40  Chapter 3: Thermodynamic and Kinetic Studies of the Causticizing Reaction  system, the strength of the white liquor available for use in the digester, and the amount of lime required in the causticizing process. The thermodynamic equilibrium constant of the causticizing reaction can be written as: K (T) = Ka (T, X) X Km  (3.101)  Ka (T, X) = 76117C07  (3.102)  I^° H m—  (3.103)  where  and 771 C0  In the above equations, 7 H - and 7c0 3 are the activity coefficients of the hydroxide and carbonate ions, respectively, and moH- and rri,c07 are their concentrations in molality units (mol/kg 112 0). In solution thermodynamics, molality units are used because molality values are independent  of solution density. Ka (T, X) is called the activity coefficient constant and Km is the ratio of the molar concentrations. Considerable work has been done in studying the equilibrium of the causticizing reaction. Most investigators, who have carried out experimental work on the equilibrium of causticizing, used concentration units (mol/L) and computed the following equilibrium constant:  =  Kc  [0 11 ]2 -  [C071  (3.104)  Obviously, they considered the solution to be ideal and the equilibrium constant, K c , was assumed to be equal to K (T). Lindberg and Ulmgren[16] have studied the effects of different ions on the equilibrium. They found that the presence of N a 2 S had a major effect which was attributed to almost complete hydrolysis in the highly alkaline conditions of the liquor. The conversion of the reaction is suppressed due to the common ion effect. Also, they correlated the equilibrium constant (Ks ) with the cation concentrations: 41  Chapter 3: Thermodynamic and Kinetic Studies of the Causticizing Reaction  log K, = 2.95  -  .62 * ([Nal-F[K+] ) °.5^(3.105)  From their experiments, they found that the causticizing reaction is endothermic at high concentrations and the enthalpy of the reaction, AH, ranges from 6 to 9 kJ/mol. Mondal and Krishnapalan [17] proposed a model that can compute the equilibrium causticizing efficiency as a function of green liquor composition. Based on experimental results, a linear relationship was found between the equilibrium constant (K,) and the hydroxide ion concentration. The equation fits the experimental data fairly well; however, as mentioned by Dorris [18], the model failed to take into account the effect of inert ions. Dorris [18] has reported quite different results from Lindberg and Ulmgren [16]. The causticizing reaction was found to be slightly exothermic with a value for AH equal to —1.2 kJ/mol. As a first approximation, the apparent equilibrium constant could be expressed as a unique function of the total ion concentration. His experimental K, value was correlated as a function of the total anion concentration (TAC) expressed as g Na 2 O/L; i.e., log K, = —0.00421 TAC + 2.300^(3.106)  where TAC is defined as TAC = OH +COT + ST + SO 3= + SO4 +^ (3.107) —  Compared with the results of Dorris and others, the K, values of Lindberg and Ulmgren [16] appear very low. Dorris [18] attributed the differences among the various studies to the accuracy of the chemical analyses performed by the various groups. Dorris also studied the effect of non-process ions on the equilibrium. He found that the presence of these ions has a decidedly adverse effect on the conversion of Na 2 CO3 and numerically showed the effect of the ratio between the Non-Titratable Anions and TTA on the equilibrium. This study is very important to Kraft mill operations because non-process ions (K+, Cr, S2 03 , etc.) are always present. More recently, Daily and Genco [19] have used Pitzer's model for aqueous electrolyte solutions to calculate the ratio of the activity coefficients, Ka (T, X), for the hydroxyl (OH ) and carbonate —  42  Chapter 3: Thermodynamic and Kinetic Studies of the Causticizing Reaction  (COT) ions. They also performed a series of experiments, in which pure Na2CO3was causticized  at 100°C, and used the experimental data to obtain the fitted parameters that were required in their calculations. Since they found that the confidence intervals of those parameters were too large, they reduced the number of parameters from five to two by using principal component analysis. Their experimental results were found to agree well with those Lindberg and Ulmgren. Ransdell and Genco [20] studied the effect of sodium sulfide on the equilibrium of the Kraft causticization reaction by using the same approach. Using Pitzer's model for electrolyte solutions was a significant contribution because the non-ideality of the solution was taken into account. The limitation of the previous work, however, is the use of adjustable parameters. Obtaining these parameters by force fitting to experimental data doesn't facilitate the validation of the computational method. The objective of this chapter was to use Pitzer's thermodynamic model to compute the equilibrium constant of the recausticizing reaction without using any adjustable parameters. Hence, we developed a purely predictive method as opposed to the empirical one of Daily and Genco [19]. It is noted, however, that use of this method to study the effect of N a 2 S on the equilibrium requires one adjustable parameter. The results of our calculations were compared with the experimental data and the results from the previously proposed methods [21].  3.2 Methodology Equation 3.101 can be written as follows: lnKm = in K(T)  —  in Ka (T,  X)  (3.108)  This is a general equation. Any thermodynamic model that is capable of describing a strong aqueous multicomponent electrolyte solution could be used. Zamaitis et al. [22] present a review of all the currently available models. In this work, we use Pitzer's model [22] because it is well validated and has been extensively used in a variety of applications. In Pitzer's method, the activity coefficients are expressed in terms 43  Chapter 3: Thermodynamic and Kinetic Studies of the Causticizing Reaction  of interaction parameters for ions in solutions. Pitzer [23][24] related the excess Gibbs free energy (G E ) of an electrolyte solution to the number of moles of ions present with the equation:  E^E Oijk ninink  GE^11 Mw f AT:  RT  (3.109)  ij^w ijk  ^where M,, is the number of kilograms of solvent and ^etc. are the number of moles of the ionic species i, j, etc. The function f represents, in essentially the Debye-Huckel manner, the effects of long range electrostatic forces. The Aij and cbijk are second and third virial coefficients. These coefficients account for the effects of two and third ion interactions. By applying Pitzer's model into causticizing reaction, equation 3.108 becomes: In  ^—  7/1 [  2^  °H  = InK(T) + 2p  MCOT^  —  *(0)^r-v*^,,,2 ONa+ In Na + — ‘ '1Va+ 1 "Na+ -  P1(■2a +01.1- [2 mNa+ ( 291 (/) — moil- 92 (-0)]  ^(3.110)  +4a) +cci3 [ 2 mNa+ (91 (I) + inc0 3 92(I))]  where^  Em  p =^  [  i  2 () 1 n (1 + 1 .2 PM] 1 + 1.2/ 1 / 2^1.2 /1/2  gi(I) = ,t21 [1 ^+ ar1 / 2 )exp(—aIll 2 )1  1 .4_ (j. a/ 1 / 2 — 2 i)exp(—a./ 112 )] 92(I) = a2/2^ 2a  At a temperature of 100°C, A, is equal to 0.4603 and a is equal to 2.0. Equation 3.110 is linear in the thermodynamic equilibrium constant lnK(T) and two of the four parameters, *(o) and C * N a+  Na+'  which are combinations of Pitzer's model parameters, are given as PNa+ 4 02oH 2 0tco3^ CA4 `r a + = 4 CNaOH 2 CNa 2 CO 3  44  (3.112)  Chapter 3: Thermodynamic and Kinetic Studies of the Causticizing Reaction  Daily and Genco [19] considered the equilibrium constant, lnK(T), as an adjustable parameter. They obtained lnK(T) and the above four parameters  1) a*(o)^*^a(i)^( ri  - and 13N +COr fitting experimental data. In the present work, we will compute the equilibrium constant and these l`^ Wa+ 9  -1  /Va+ 9 l'IVa+OH '  four parameters from thermodynamic considerations, forgoing any reliance on experimental data. 3.2.1 Thermodynamic Equilibrium Constant lnK(T)  In a typical Kraft pulp mill, the causticizing reaction takes place at a temperature around 90 — 100°C. The equilibrium constant at this temperature will be calculated from standard data at 25°C as follows. From thermodynamics, it is known that the equilibrium constant, lnK(T), is related to the standard Gibbs free energy by  (3.113)  lnK(T) = AG° RT  Differentiation of the above equation gives the well known Van't Hoff's equation  R  dInK(T)^d(AG°(T)IT) OH dT^dT T2  (3.114)  The enthalpy difference, D H, can be expressed as a function of temperature in terms of the constant pressure heat capacity, C p , as follows: All = H°^ACp(T) dT  ^  (3.115)  T°  By assuming ACp constant ACT) , the enthalpy difference, zH, becomes  OH = H° ACT, (T — T°)^ 45  (3.116)  Chapter 3: Thermodynamic and Kinetic Studies of the Causticizing Reaction  Finally, equations 3.114 and 3.116 may be combined to give R  d ln K AH°^(1 T°) dT^T2^L'P T T2  (3.117)  Data for AG°, AH° and AC; at 25°C are tabulated in Table 3.3. Because the heat capacities for the two ions in the reaction change less than 3% over the temperature range from 25°C to 100°C, and the changes of heat capacities for the two solid components basically cancel each other out, the assumption of constant ACp is very reasonable and fairly accurate for this reaction.  Species OH CO3= -  Ca(OH)2  CaCO3  State aqueous aqueous solid solid  All° Kcal/mol -54.970 -161.84 -235.68 -288.51  AG° Kcal/mol -37.594 -126.17 -214.76 -269.55  Cp°cal/mol K -57 -151  20.91 19.42  Table 3.3 Thermodynamic Data for Species in Causticizing Reaction — Standard State (25°C) Values [25][26]  Equation 3.117 can be integrated and then solved for lnK(T). The calculated value at 100°C was found to be 6.533. It is noted that the calculated value at 25°C is 6.430. It can be seen that lnK(T) is only a weak function of temperature. We also found that OH is equal to 7.533 1d/mol  at 100°C and –3.891 kJ/mol at 25°C, which means the reaction is exothermic at 25°C and becomes endothermic at 100°C. Since the enthalpy of the reaction is a small number in either case, the reaction is only slightly exothermic or endothermic. This result compares well with the results obtained from Lindberg and Ulmgren's experiments at 100°C [16]. 46  Chapter 3: Thermodynamic and Kinetic Studies of the Causticizing Reaction  on  Na2CO3 0.0362  NaOH 0.0864  /3(1)  1.51  0.253  C 103 0,0V0T  0.0052 1.79  0.0044 0.7  103 00 (1)MT  2.05  0.134  103 0C/OT 105 02 0 (°)/0T2  0.0  -0.1894  -4.22  -2.00  105020(i)/er2  -16.8  -2.1  105 02 C/OT 2  0.0  0.29  Table 3.4 Ion Interaction Parameters and Their Derivatives at 25°C  3.2.2 Calculation of Pitzer's Parameters at 100°C  As shown in equation 3.110, four parameters are needed. These parameters are the following: a (1)^(4(1 4aol- CNa+ PIVa+OH ' an'A l'Ara +C07. -  )  We have seen that these parameters are related to Pitzer's  model parameters which are temperature dependent. Pitzer [22] noted that there was very little change in these parameters over the range of temperatures from 25 to 300°C. Nevertheless, he gave the first temperature derivatives of the parameters for various species. Later, Peiper and Pitzer [27] studied experimentally a system which involved Na2CO3 and NaOH and presented the ion interaction parameters at 25°C together with the first and second derivatives of the parameters. We used the temperature derivatives in a Taylor series expansion in order to compute Pitzer's model parameters at 100°C from the values at 25°C. The calculation was carried out as follows, using the parameter Pl cio)0,, c, as an example. AT2( 0200) ) _,^ "(0)^ ( ^s 01 °0 )0. C = )3 V ). C + AT ^OT  ^+ 2^0T2  (3.118)  25°C^25°C  The ion interaction parameters at 25°C and the first and second derivatives are shown in Table 3.4. 47  Chapter 3: Thermodynamic and Kinetic Studies of the Causticizing Reaction  Na2CO3  NaOH  on  0.05176  0.08265  00)  1.19191  0.20399  C  0.0052  -0.001649  Table 3.5  Parameter Ln K(T) P;siT+  C7,  +  /3(11)oli /3(/%1/a 2C0 3  Calculated Ion Interaction Parameters at 100°C  Results of This Study 6.533 0.227  Results of Daily and Genco 7.23 0.460  -0.017 0.204  -0.0272 -1.18  1.192  -2.54  )  Table 3.6  Comparison of Model Parameters  In Table 3.5, the calculated value of the parameters at 100°C are given. Finally, using the above calculated values of Pitzer's model parameters, we obtain the following values needed in equation 3.110:  °) = 4/3 ( cria) 0H — 23C,?LCO3 = 0.227  i3;j a +  A  Iv  Ck a + 4 CNaOH 2 CNa 2 CO 3 13(Alia) +OH _ =  0.204  =  —  0.017  (3.119)  13(.I\r) +CO3=^1.192  Table 3.6 is the comparison of model parameters obtained from this study with that of Daily and Genco. There is a big difference between those two results. 48  Chapter 3: Thermodynamic and Kinetic Studies of the Causticizing Reaction  With all Pitzer's parameters and lnK at 100°C, the molal compositions of moll- and mcoz- at equilibrium can be easily calculated using an iterative procedure. 3.3 The effect of N a2 S on the equilibrium  In a Kraft mill, the solution in the causticizer does not consist of Na 2 CO 3 alone. Among all the other components which are present, Na 2 S, which exists as the HS ion, has the most significant -  effect on the equilibrium. The method proposed here can also be used to calculate the equilibrium composition of the reaction in the presence of Na2 S. In this case, additional ion interactions have to be considered. The resulting equation for Km is given below.  In  [  in  °H- = InK(T)-F 2p — p.Atin iva + — ck a +m2,va , mcoT - 13(.1■11a) +OH- [ 27nNa+ ( 2g1 CO — MOH - 92( I ))]  (3.120)  + 13(1%lia) +CO3= [2mN a + (m. (I) + inc0722(0)] i_ a o.) -  ri-wa-f-Hs- [2MNa+MHS- 92( 1)]  Unfortunately, there are no independent thermodynamic data available to compute a value for 131‘( 1.a+Hs _. Hence, Ransdell and Genco's experimental data [20] at 30% sulfidity are used to estimate a (r) (1) +Hs _ . We considered only the operating range in the industry, and obtained fjNa^ Na +1-1S- = 0.0253. It should be noted that all the other Pitzer's parameters and thermodynamic equilibrium constant K(T) are the same as before.  3.4 Calculation of Causticity  The causticity has been widely used in industry to describe the quality of the white liquor and as a parameter for slaker control. Our proposed methodology can be easily used to determine the causticity at equilibrium. The sodium molality can be calculated from the total titratable alkali (TTA). The TTA is expressed here in g as Na 2 0/kg and can be changed to g as Na 2 0/L provided the water 49  Chapter 3: Thermodynamic and Kinetic Studies of the Causticizing Reaction  concentration (Cw ) is available. Cw is used in converting concentrations to molalities since raj = Ci/Cw^  (3.121)  and is calculated from the solution density p by the equation: Cw = p  —  Eci x mwi^(3.122)  By knowing TTA, the sodium molality can be calculated as follows ^  mAra+ = TTA X 2 X MWNa+/MW. Na 2 o  (3.123)  and further = (mNa+ x sul fidity)12^  (3.124)  The rest of the iterative calculation procedure will be the same as before. In mills, TTA is always expressed by g as Na 2 O/L. Thus to estimate TTA in molality, data for the water concentration (or solution density) are required. From the mill data, TTA, water concentration, and sulfidity, then the causticity at equilibrium can be easily calculated. 3.5 Results and Discussion  The results of causticizing pure sodium carbonate solutions are shown in Figure 3.14. In this figure, we plot lnif in versus the ionic strength of the solution. In the same graph, the results of Daily and Genco [19] are also plotted. It can be seen that the proposed method shows the same general dependence of the equilibrium constant upon the ionic strength of the solution. Over the typical operating range in a Kraft mill [16], our values are slightly lower than the results of Daily and Genco. Our results were also compared with Lindberg and Ulmgren's results. Due to the different units used in two studies, the two figures can not be plotted together directly. There is only a small 50  Chapter 3: Thermodynamic and Kinetic Studies of the Causticizing Reaction  1.5^2^2.5^3^3.5^4^4.5^5  ^  5.5  ^  6  IONIC STRENGTH (MOLES/KG)  Figure 3.14: Causticizing with Pure Sodium Carbonate Solution  difference between K, and Km over the mill ionic range. The constants, K, and Km , can be converted to each other provided information is given about the water concentration, which is almost 1 (kg/L) in the ionic range of interest. The constant K., has been converted to Km by using the available water concentrations [20]. Figure 3.15 shows the results of causticization at 15% sulfidity. At a sodium concentration greater than 3 mol/kg all three investigations show quite similar results. At lower concentrations, however, the results given by Ransdell and Genco give consistently higher values when compared with other results. This suggests that our approach is valid to describe the causticizing equilibrium. Since the approach is independent of the reaction itself, it validates the application of Pitzer's method to the causticizing reaction. The results of causticizing at a sulfidity of 30% are shown in Figure 3.16. The standard deviation of the differences between our model predicted values and the experimental data was found to be 0.0225. It can be seen that the proposed method yields a similar accuracy compared with the prediction of Ransdell and Genco's model over the mill operating range. The same approach can also be used to study the effect of other ions on the causticizing equilibrium provided the model parameters are known from independent (not related to the causticizing reaction) thermodynamic data. In Figure 3.17, a comparison of the causticities at 15% sulfidity between this work and Ransdell and Genco's is shown. Over the operating range in mills (around 110-135 g as Na20/kg), the two 51  Chapter 3: Thermodynamic and Kinetic Studies of the Causticizing Reaction  Experimental Data (Lindberg & Ulmgren) 5.5 -  Ransdell and Genco ^ This Work^ +  5 4.5-  -  Mill Range 4-  3.5 -  1.5^2^2.5^3^3.5^4^4.5  5.5^6  TOTAL SODIUM (MOLES/KG)  Figure 3.15: Causticization at 15% Sulfidity  Experimental Data 5.5  Ransdell and Genco ^ This Work  5  . Mill Range .  4.5 E"  3.5  1.5^2^2.5^3^3.5^4^4.5^5^5.5 IONIC STRENGTH (MOLES/KG)  Figure 3.16: Causticization at 30% Sulfidity  results agree very well. In Figure 3.18, the effect of sulfidity on the causticity is shown. It can be seen that the reaction is somewhat suppressed with the increase of sulfidity. This agrees with the analysis discussed earlier. It also shows that there is a large gap between the equilibrium and the operating causticity.  52  Chapter 3: Thermodynamic and Kinetic Studies of the Causticizing Reaction  100  95  90  85  80  75  40^60^80^100^120^140^160 TOTAL TITRATABLE ALKALI as g Na 20/kg  Figure 3.17: Causticity for Causticization at 15% Sulfidity 95  90  85  80  75 80^90^100^110^120^130^140^150^160  170  TOTAL TITRATABLE ALKALI as g Na 20/kg  Figure 3.18: The Effect of Sulfidity on the Equilibrium  3.6 Conclusions about the Thermodynamic Model  This research presents a methodology for calculating the equilibrium constant and the equilibrium composition of the recausticizing reaction in a kraft pulp mill. The methodology is based on Pitzer's model for aqueous multicomponent electrolyte solutions. Although Pitzer's model was previously applied in describing the causticizing reaction, adjustable parameters were employed. Our methodology, however, does not require any adjustable parameters for the causticizing of pure 53  Chapter 3: Thermodynamic and Kinetic Studies of the Causticizing Reaction  sodium carbonate solutions. The effect of the presence of Na 2 S was also calculated. The results of the calculation were found to be in excellent agreement with experimental data over the mill operating range. The effect of other components can also be studied using the same approach depending upon the availability of the Pitzer's parameters. For the effect of Na2S, one adjustable parameter was necessary because ,4 . +Hs _ is not available from independent thermodynamic data. It was noticed )  that the recausticizing reaction shifts from exothermic to endothermic when the temperature increases from 25°C to 100°C. It should be noted that although the calculation were done at 100°C, results at any other temperatures could be obtained in a similar manner.  3.7 Kinetic Model of the Causticizing Reaction  3.7.1 Previous Studies  Considerable work has been done to study the equilibrium of the causticizing reaction. However, the actual degree of causticizing that can be obtained in a mill is also dependent on the reaction kinetics. Comparatively, very few studies related to the kinetics of the causticizing reaction have been found. A preliminary review of mathematical modeling the slaking/causticizing reaction was conducted by Jacobi and Williams [5]. They assumed that the slaking reaction occurred instantaneously relative to the causticizing reaction, and that the causticizing reaction was diffusion limited. The equilibrium causticizing condition is solved for, using a simple steady state approach. Due to the limited knowledge about the thermodynamics and kinetics of the causticizing reaction available at that time, this approach is rather empirical and not accurate for different operating conditions. Rydin [28] proposed two kinetic models assuming pseudo-homogeneous reaction conditions. Rydin set up both 3 and 5 parameter models starting from lime as the calcium reactant. Rydin estimated the parameters from experimental kinetic data. However, when the models were evaluated under equilibrium conditions by assuming long reaction times, neither model could successfully predict the equilibrium composition data found in the literature. Activity coefficients were not used in either of the Rydin models. 54  Chapter 3: Thermodynamic and Kinetic Studies of the Causticizing Reaction  The following simple rate equation was proposed by Lindberg and Ulmgren [29] to describe the reaction kinetics: d0 = kri (K — Or dt  (3.125)  e^[01-1-]2 [Cal]  (3.126)  where 0 is defined as  At equilibrium, 0 is equal to the stoichiometric equilibrium constant K. They also found that a value of 2 for the exponent n yielded a good fit for most of the data. The constant kr, is a measure of the overall rate of reaction and is useful when evaluating how different factors affect the rate of reaction. This model structure is simple and correlates very well with their experimental data. The influence of different factors in the reaction can easily be shown by using this model. However, the stoichiometric equilibrium constant used in this model was obtained in an empirical way and no theoretical basis was given for this method. Another limitation of their work is that analytical-grade lime was used in all their experiments. Their results may serve as a good reference when discussing process conditions. However, since analytical-grade lime is, of course, nonexistent in a Kraft mill, their results do not necessarily reflect the situation in the mill, particularly as to the quality of lime. Dorris [10] has conducted a large number of experiments on the causticizing reaction. The effects of lime-to-liquor ratio, mixing during slaking, lime and liquor temperatures on causticizing and lime mud settling have been studied. Also the effect of the physical structure of the lime on the causticizing reaction has been reported. This work is very important in understanding the nature of the causticizing reaction. Due to the complexities of the causticizing reaction itself, no mathematical model was given in Dorris's paper. 3.7.2 Kinetic Model for Causticizing Efficiency  The causticizing reaction is a two phase (liquid-solid) heterogeneous reaction. Much more work needs to be done before the mechanism of this reaction will be fully understood. The development 55  Chapter 3: Thermodynamic and Kinetic Studies of the Causticizing Reaction  of a real kinetic model based on physical and chemical principles of the reaction is difficult at the moment. Hence a simple, yet accurate, approach was employed in our current research. The slaking reaction is assumed to occur instantaneously relative to the causticizing reaction and to go to completion. The causticizing reaction is an equilibrium reaction. The kinetics data are presented using CE, the causticizing efficiency, as a function of time. At equilibrium, CE is equal to the equilibrium causticizing efficiency CE eq , which can be calculated from our thermodynamic equilibrium model. This way of presenting the data simplifies the evaluation of how different factors affect the rate of reaction and makes it possible to use the simple reaction rate equation: dCE = k i (CEeq — C E) k3 dt  (3.127)  The constant k1 and k2 are measures of the overall rate of reaction. They are functions of temperature and lime quality. This model structure is quite similar to the one given by Lindberg and Ulmgren. However, it gives the final result of causticizing efficiency directly, which is really what concerns the mill operators. At the same time, a real reburned lime of average reactivity was used in the reaction to determine the empirical constants k1 and k2 in the equation. The integration of equation 3.127 will give CE as a function of time and constant k1 and k2. The following function was minimized by using the least squares criterion:  F(k1, k2) = ti x (cE./(ki,  k2, i) - cEesp(i))2  (3.128)  The difference between the model predictions and experimental results was time weighted to account for the fact that the latter part of the curve is more important than the initial reaction. The values which yielded the optimal fit to Dorris's data [10] were found to be 2.1 x 10 -4 min -1 for k1 and 2.86 for k2. The standard deviation is equal to 10.05. The fitted reaction rate curve is compared with experimental results in Figure 3.19. The lime samples used by Dorris were collected in 10 56  Chapter 3: Thermodynamic and Kinetic Studies of the Causticizing Reaction  80  e  70  g'  6°  a> 1:5  w ta) N 1.5 ..17; M CO C.)  so  +  40  +  30  Experimental Data Model Prediction  20 10  20^40^60  ^  80^100^120  Time (min)  Figure 3.19: Kinetic Model of the Causticizing Reaction  Canadian Kraft mills. The experimental data used in our fitting are for the reaction with a lime of moderate reactivity. This method of modeling the kinetics ensures that the causticizing efficiency will converge at CEeq as time approaches infinity. Because mill data were used in the fitting, a better representation for the causticizing reaction in the mill can be expected. What is more important is that the causticizing efficiency is given directly by this model. This is what the mill operators really need to know.  57  Chapter 4: Simulation Results and Discussions  Chapter 4 Simulation Results and Discussions  4.1 Introduction  Several computer simulation runs were made with typical disturbances. The steady state data for the simulation were taken from the design data of a 1000 ton pulp/day mill [30]. Simulators such as GEMS, MASSBAL, GEMCS, MAPPS, and PAPMOD, are all based on steady-state mass and energy balance and mainly used for process design and analysis. Dynamic simulators are necessary for the analysis of process dynamics. There are a few dynamic simulators available now for the pulp and paper industry. Among them are PAPDYN, DYNSYS, DYSCO. These simulators have built in models for some processes (mixers, tanks, etc.). However, models for some nonlinear and complex units, such as the slaker/causticizer, are not provided in those simulators. SIMNON, a general purpose dynamic simulation program, though not developed for pulp and paper system, can be used to simulate the more complex systems encountered in the pulp and paper industry. All the models were programmed by using the SIMNON dynamic simulation package combined with FORTRAN on a Sun/SPARC station 2. SIMNON is a program that can solve sets of simultaneous differential and difference equations. It can also be used to simulate dynamic systems that are composed of many subsystems. Such systems are common, e.g., in control engineering. SIMNON has evolved over more than a decade. It has been used for education and research in such diverse disciplines as automatic control, biology, chemical engineering, economics, electrical engineering, etc. at many universities. SIMNON has also been used worldwide in industry for simulation of control systems [31]. The new version of SIMNON for UNIX operating systems supports the feature of linking externally written subroutines into SIMNON. These subroutines may be written in Fortran, C, or any language linkable to Fortran and C routines. This feature facilitates the computation of complicated algorithms. The general structure of a unit model in SIMNON is shown in Table 4.7. The system description starts with CONTINUOUS SYSTEM <identifier>. It is terminated by a line which contains END. 58  Chapter 4: Simulation Results and Discussions  CONTINUOUS SYSTEM <identifier> INPUT <identifier> ... <identifier> OUTPUT <identifier> ... <identifier> STATE <identifier> ... <identifier> DER <identifier> ... <identifier> TIME <identifier>  computation of auxiliary variables computation of derivatives parameter assignment initial value assignment END Table 4.7 General Structure of Modeling in SIMNON  The first part of the program is the declaration part. The inputs, outputs, state variables, and their derivatives are declared. Then, the body of the system contains the calculation and the definitions of all the parameters and the initial values. Each unit is connected to the others by their input and output variables. In solving the differential equations, 4th-order Runge-Kutta method is used. The simulation step size is chosen automatically by SIMNON to ensure the desired accuracy. An iterative procedure is used to solve the equilibrium of the causticizing reaction. The kinetic model for the causticizing efficiency is solved by using a 4th-order Runge-Kutta method. Both the equilibrium and the kinetics of the causticizing reaction are programmed as an additional unit in FORTRAN and are connected to the slaking/causticizing unit in SIMNON. 4.2 Steady State Conditions of the Recausticizing Plant  First of all, the steady state conditions of the whole plant are established based on the design data of a 1000 ton pulp/day mill [30]. The dimensions of each unit in the plant are shown in Table 4.8. Table 4.9 shows the output mass flow rates as well as the output total volumetric flow rates of each individual unit at normal steady state. All mass flow rates are in kg/min. 59  Chapter 4: Simulation Results and Discussions  Units  Total Volume (m 3 )  Operating Volume (m 3 )  Area (m 2 )  Height (m)  sdt  93.0  64.7  19.6  3.3  gist  376.0  265.2  44.2  6.0  glco  74.5  74.5  /  glcu  74.5  74.5  /  sc  103 x 3  103x3  /  / / /  wlco  346.0  346.0  /  /  wlcu  346.0  346.0  /  mmt  50.0  35.3  mwo  477.5  477.5  / 12.6 /  mwu  500.0  500.0  /  wwt  2164.0  1514.7  / 142.9  Table 4.8  2.8 / 10.6  Dimensions of the Units in the Recausticizing Plant  Units  water  N a2 S  NaOH  Na 2 CO3  N a2 SO 4  N a2 SO3  Solids  Q (m 3/min)  sdt  2791.4  109.4  48.4  354.0  39.8  11.0  2.8  2.791  gstl  2791.4  109.4  48.4  354.0  39.8  11.0  2.8  2.791  glco  2733.4  107.1  47.4  346.6  39.0  10.8  0.0  2.733  glcu  58.0  2.3  1.0  7.4  0.8  0.2  2.8  0.058  sc  2671.2  107.1  276.5  43.1  39.0  10.8  352.1  2.722  wlco  2177.9  87.3  225.4  35.1  31.8  8.8  0.0  2.178  wlcu  493.3  19.8  51.1  8.0  7.2  2.0  352.1  0.544  mmt  3247.9  22.2  58.3  9.2  8.1  3.3  376.8  3.299  mwu  608.7  4.2  10.9  1.7  1.5  0.6  376.8  0.660  mwo  2639.2  18.0  47.4  7.5  6.6  2.7  0.0  2.639  dwo  106.1  2.3  1.0  7.3  0.8  0.2  0.0  0.106  dwu  1.2  0.0  0.0  0.1  0.0  0.0  2.8  0.001  wwt  2791.4  20.2  48.4  14.8  7.4  2.9  0.0  2.791  Table 4.9  Output Mass Flow Rates and Volumetric Flow Rates at Normal Steady State  Tables 4.10 to 4.12 show the inputs to the recausticizing plant. The mass flow rates of each 60  Chapter 4: Simulation Results and Discussions  component at steady state in the smelt are shown in Table 4.10. The mass flow rates of each component from the lime mud filter and the scrubber are shown in Table 4.11 and Table 4.12, respectively. A large amount of fresh water is added to the plant. Table 4.13 shows the locations and flow rates of that input water.  Components  Na2S  Na2CO3  Na2SO4  Na2S O3  Solids  kg/min  89.1  339.2  32.4  8.1  2.8  Table 4.10  Smelt Component & Flow Rate  Components  Water  Na2S  NaOH  Na2CO3  Na2S O4  Na2S O3  Solids  kg/min  1192.1  2.4  7.2  1.3  0.9  1.3  10.5  Table 4.11 Output  from Lime Mud Filter  Components  Water  Solids  kg/min  1152.8  14.3  Table 4.12 Output from Scrubber  Positions  Dregs Washer  kg/min  49.3  Mud Mixing Tank _ 409.7 Table 4.13 Fresh Water  61  Weak Wash Tank _ 46.3  Chapter 4: Simulation Results and Discussions  4.3 Simulations of the Different Units  First, the dynamic responses of the most important units in the system are shown. Then the simulation of the complete plant is shown and the results are discussed. 4.3.1 Responses of the Causticizing Reaction to Different Disturbances  Green liquor and lime are the two inputs to the slaker. They are the main disturbances. Some make-up water may also be added to compensate for the water consumption in the slaking reaction in mills, but for the time being we assume that no water is added in the slaker. The effect of make-up water in the slaker will be discussed later in this chapter. Initial steady state mass flow rates of each component of the green liquor are shown in Table 4.14. The mass flow rate of lime at steady state is 215.4 kg/min and the availability of the lime is 90%. In each simulation, the amount of lime charged is based on the calculated amount required to reach equilibrium, which is equal to a charge of 1.02. This way of calculating the charge gives a value lower than that obtained if the calculation is based on the stoichiometric amount required.  Component Na2S  NaOH  Na2CO3  Na2S 04  N a2 S 03  kg/min  47.4  346.6  39.0  10.8  107.1  Table 4.14  water 2733.4  Steady-State Data of Green Liquor  In the slaking and causticizing reaction, the concentrations of the two ions, OH and COT, —  undergo a tremendous change due to the causticizing reaction as discussed in Chapter 3. The dynamic responses of the two major ions are shown in Figure 4.20. The concentration of the OH ion increases —  from 0.15 mole/L to 2.51 mole/L and the concentration of CO3 ion decreases from 1.32 mole/L to —  0.14 mole/L. It can be seen that about 90% of the equilibrium concentration is reacted in about 30 minutes. After 30 minutes, the reaction proceeds slowly. It is very important to know the responses of the causticizing reaction to different disturbances and under different operating conditions. This can help us understand the nonlinearity of the reaction 62  Chapter 4: Simulation Results and Discussions  3  [OW]  ;71  2.5  0  E  2  1.5  2[CO ]  0.5  3  o  o  10^20^30^40^50  ^  60  ^  70  ^  80  ^  90  Time (min)  Figure 4.20: [OH ] and [CO3 ] During the Course of the Causticizing Reaction -  -  and its effect on the whole plant. In the following simulations, the kinetics data are presented using the variable 9, defined as  e  =  [0 H 1 2 -  [c(4  -  (4.129)  ]  This way of presenting the data shows, more conveniently, the differences between each operating condition. A change in the composition of the green liquor is an important disturbance to the slaking and causticizing reaction. This change may come from variations in smelt rate and composition or from variations in composition of the weak wash liquor. A change in the composition of the green liquor can be viewed as a change of TTA. The effect of different green liquor TTA values is shown in Figure 4.21. The TTAs simulated are 91.8, 115.3, and 138.9 g as Na2O/L. Correspondingly, the values of 0 at equilibrium for each case are 65.1, 51.5, and 42.3 mole/L respectively. The causticities decrease to 93.5%, 90.2% and 86.6%, respectively. Due to the limited reaction time (90 minutes), the reaction conversion is equal to about 90% of the equilibrium value in all three cases. Obviously, the low TTA will facilitate the reaction to approach completion. However, the 'PTA is dictated by 63  Chapter 4: Simulation Results and Discussions  the requirement of white liquor density and the burden in the evaporator. The TTA of the solution has no significant effect on the rate of reaction. The lime charge is 1.02 in all cases. As discussed earlier, the TTA determines the maximum possible extent of reaction.  70  0  50  E  .7,  40  /  CV -^/1 , - -- - -- -0 30 ^ CV-^/ _.^ .°' ' / --------2^/^ 0 20 - / /  I  to 4 /  '  20  ^  20% increase in TTA (138.9 g/L) normal TTA (115.3 g/L) 20% decrease in TTA (91.8 g/L) _  40^60^80^100^120  Time (min)  Figure 4.21: Effect of the Green Liquor TTA  It is often of practical interest to know in advance how a change in the lime feed will affect the final quality of the white liquor. The change in the lime charge can be attributed to a change in the amount of lime, the availability of the lime, and the reactivity of the lime. The responses of 0 to different lime charges are shown in Figure 4.22. The differences between the curves are explained by the fact that an insufficient amount of lime is charged. It should also be noticed that an excess charge does not increase the rate of reaction. When the lime charge is equal or greater than 1.0, the equilibrium 0 is 51.5 mole/L. The causticity in this case is 87.6%. The real output of 0 in this case (residence time 90 minutes) is 46.5 mole/L, which is equivalent to 90% of the equilibrium value. Another possible disturbance to the slaking and causticizing reaction is the volumetric flow rate of the green liquor. Since the causticizing tanks are completely full, a different flow rate of green liquor will result in a different residence time in the slaker/causticizers. The lower the flow rate, 64  Chapter 4: Simulation Results and Discussions  60  Equilibrium 50  40 ..........................  •••  Charge 0.95 0.97 1.02  20  10  0^  20^40^60  ^  80  ^  100  ^  120  Time (min)  Figure 4.22: Effect of Lime Charges  the longer the residence time and, therefore, the higher conversion of the causticizing reaction. This is shown in Figure 4.21. The change in the residence time will also change the dynamic response of the other components in the slaking/causticizing unit. For example, Figure 4.23 shows the dynamic response of Na 2 S in a unit having two different residence time to a 10% increase in the input concentration of the green liquor. The shape of the response is typical of three perfect CSTRs in series. The dynamic responses of the other non-reactive components are similar in shape to that of Na 2 S. 4.3.2 Tank Level Control  Figure 4.24 shows the typical PI feedback control system used for tank level control. Figure 4.25 shows the responses of the level of the smelt dissolving tank and its output flow rate to a step change in the weak wash flow rate Q1. The output volumetric rate of the smelt dissolving tank is denoted by Q2. The tank level is represented by h. The mathematical development for this control system can be found in Appendix B. The single tuning parameter r is related to the desired response speed. A value of 1 min for  7  gives the desired second-order response with a damping factor of 1  [8]. It can be seen that the output flow rate of the smelt dissolving tank follows the input change very quickly and the tank level is quickly stabilized at its initial steady state value. The tuning parameter 65  Chapter 4: Simulation Results and Discussions  40.5  cl  ,q, 39.5 --  coN  39  Z • C 0  38.5  //  37  36.5 0  /  /  /  /  /  //  / //  //  , //  ,  ...•  .......  ^ Residence Time 90 min .  ^ Residence Time 120 min . /^  50^100^150^200^250^300  Time (min) Figure 4.23: Effect of the Residence Time  Q1  Q2 manipulated variable Figure 4.24: Control System for a Typical Tank  r can be changed according to the desired response. All the other tanks show a similar dynamic  response except that they may have different response times. 4.3.3 Step Response of Storage Tanks and Clarifiers  One of the characteristics of the recausticizing plant is its slow dynamics. This is due to the fact that there are many storage tanks in the system. The storage tanks and storage capacities of the settling tanks, to a large extent, dictate the dynamics of the system. The smelt dissolving tank has the 66  • Chapter 4: Simulation Results and Discussions  C  E  3.7  -___ Q2  3.6  ----------  3.5 CC •  O LL  01  3.4  0 3.3  -  ^ Input Flow Rate 01 ^ Output Flow Rate Q2 Tank Level h  3  2.9 0  2^4^6^8^10^12  ^  14  ^  16  ^  18  20  Time (min) Figure 4.25: Tank Level Control  42  Overflow of the Green Liquor Clarifier  41  Underflow of the Green Liquor Clarifier rn  40  _NC  C  .2 39 7:3  I:8  0  38  •  C O  co  37  81"\I Z 36 35  0  500  1000  1500^2000  2500  3000  3500  4000  Time (min)  Figure 4.26: Dynamic Response of a Clarifier  fastest storage dynamics of the order of 1/2 hours. The slowest dynamics are those of the underflow parts of the clarifiers, which have retention times of the order of 10 to 20 hours. Figure 4.26 shows the response to a 10% increase in the input concentration of Na 2 S in the green liquor clarifier. From the graph, it can be seen that the concentration response of the overflow is much faster than that of the underflow. Due to the perfect mixing assumption in the overflow, the 67  Chapter 4: Simulation Results and Discussions  response is a typical first order response. On the other hand, the underflow behaves like a plug flow with back-mixing and is characterized by a much slower response. At steady state, the concentrations in both the overflow and underflow have the same value. The dynamic responses for other chemical components in the clarifiers are quite similar to that of Na 2 S. 4.4 Dynamic Response of the Recausticizing Plant  After simulating some typical units of the system, all the units were connected together to simulate the whole recausticizing plant. Based on the established steady state conditions, some typical disturbances were then introduced to study their effects on the whole plant behaviour. Here, three disturbances, smelt rate, smelt composition and lime mud filter rate, are chosen as examples. In these dynamic simulations, the most important output variables of the plant are volumetric flow rate, total titratable alkaline (TrA), active alkaline (AA), and equilibrium causticity. The responses of these variables to the disturbances at different locations in the plant are presented and discussed. Since the time for dissolved chemicals to go through the plant is of the order of 24 hours, all the simulation runs carried out in this study are of 72 hours (4320 minutes) real time duration. A simulation of the whole plant takes about an hour on a Sun/SPARC station. PI level control is applied to all the incompletely filled tanks in these simulations. There are not any other types of controllers in operation. 4.4.1 Dynamic Responses to a Change in Smelt Rate  The change in smelt rate is shown in Figure 4.27. At 90 minutes, the total smelt rate is increased by 20% from its steady state value (471.6 kg/min to 565.9 kg/min). The mass fraction of each component in the smelt is kept constant during this change. For each simulation, enough lime was charged in the slaker for the slaking/causticizing reaction. Figure 4.28 shows the responses of the green liquor and white liquor volumetric flow rates to the disturbance. About three hours after the disturbance, the white liquor volumetric flow rate increases slightly. This is because the total mass flow rate has increased as a result of the disturbance. Therefore, a small increase in the volumetric flow rate out of the white liquor clarifier can be seen in the response. Due to the increase in the total volumetric flow rate out of the causticizer, more 68  Chapter 4: Simulation Results and Discussions  580  -E- 560  y; 540 T  E  co 520  0 a) 11 cc  500  0 0 4 80 (.6  0,  2  46°o  500  1000  1500^2000^2500  3000  3500  4000  Time (min)  Figure 4.27:  Changes in Smelt Rate  2.8  E  2.7  Green Liquor a) 2.5  White Liquor  01 E 2.3  O 2.2  2.1  0  500  1000  ^  1500^2000^2500  ^  3000  3500  4000  4500  Time (min)  Figure 4.28: Volumetric Flow Rate Responses to Change in Smelt Rate  weak wash is sent back from the plant cycle. It can be seen that there is also a very small increase in the green liquor volumetric flow rate. Figure 4.29 shows the responses of the TTA to the disturbance. Since we are looking at an open loop simulation, the TTA in the smelt dissolving tank follows the change in smelt rate very quickly. The TTA response is slower in the green liquor clarifier. The TTA of the white liquor 69  • -  Chapter 4: Simulation Results and Discussions  clarifier overflow will feel the effect of the disturbance about 3 hours later. The response time may be different from mill to mill due to the different number and volume of the intermediate tanks in each mill. It is noticed that the average TTA of the white liquor is higher than that of the green liquor. This results from the consumption of water in the slaking reaction. The TTA in the smelt dissolving tank first reaches its temporary steady state after 200 minutes. However, it begins to rise again after 1000 minutes because the disturbance has brought back its effect onto the smelt dissolving tank through the recausticizing cycle. This is also true for the TTA in the green liquor clarifier. The TTAs in both the causticizer and the white liquor clarifier begin to experience the second rise without even reaching a temporary steady state. The following responses are very slow due to the long retention time of the plant. It takes about three days for the system to reach a new steady state. It can be seen that dynamic behaviour of the plant as a whole is characterized by a quick response at the beginning followed by a slow one.  145  140 v". --- 135-^/' * 0 CP^I /  I  130  •  Smelt Dissolving Tank  1  125 1 -  Green Liquor Clarifier  I.  Causticizer White Liquor Clarifier  120^  115  0  500^1000^1500^2000^2500^3000^3500^4000^4500  Time (min)  Figure 4.29: TTA Responses to a Step Change in Smelt Rate  Figures 4.30 and 4.31 show the dynamic responses of the green liquor and white liquor AA composition to the disturbance. The shape of these responses are quite similar to that of the TTA composition. The same remarks apply here. It can be seen that the white liquor AA composition 70  Chapter 4: Simulation Results and Discussions  53 52 51  Smelt Dissolving Tank 4  Green Liquor Clarifier  46 45  500^1000^1500^2000^2500^3000^3500  4000^4500  Time (min)  Figure 4.30: AA Responses to a Step Change in Smelt Rate (1)  1000  ^  1500^2000^2500  ^  3000  ^  3500  ^  4000  Time (min)  Figure 4.31: AA Responses to a Step Change in Smelt Rate (2)  (in the causticizers and the white liquor clarifier) is much higher than that of the green liquor (in the smelt dissolving tank and the green liquor clarifier). This results from the causticizing reaction. A large amount of sodium hydroxide (N a0 H) has been created in the reaction, which contributes significantly to the white liquor AA composition. Figure 4.32 shows the response of the equilibrium causticity of the causticizing reaction to the disturbance. The shape of the dynamic response is similar to that of the TTA composition, but in the 71  Chapter 4: Simulation Results and Discussions  Figure 4.32:  Equilibrium Causticity Responses to Changes in Smelt Rate  28 2.7  2.6  lC  Weak Wash  0  Green Liquor  CE 2.5  0 2.4  White Liquor E 2.3  O 2.2  2.1  0  500  1000  ^  1500  ^  2000^2500  ^  3000  3500  4000  4500  Time (min)  Figure 4.33: Volumetric Flow Rate Responses to a Change in Smelt Composition opposite direction. As explained before, the equilibrium causticity has decreased as a result of higher green liquor TTA. The residence time for the reaction is roughly the same as that of the previous steady state case because the increase in the volumetric flow rate can be neglected. 4.4.2 Dynamic Responses to a Change in Smelt Composition The dynamic response to a disturbance due to a change in smelt composition is quite similar  72  Chapter 4: Simulation Results and Discussions  to what has just been shown for the change in smelt rate. The change in smelt composition has a very small effect on the volumetric flow rates in the plant. However, it affects the liquor TTA, and therefore, the causticizing reaction. The effect of a change in Na 2 S in smelt on the recausticizing plant is shown below. At 90 minutes, the mass flow rate of Na 2 S in smelt is increased by 20% from its steady state value (89.1 kg/min to 106.9 kg/min). The mass flow rates of the other components in the smelt are kept constant during this change. Figure 4.33 shows the responses of the weak wash, green liquor and white liquor volumetric flow rates to the disturbance. As discussed before, there are not appreciable changes in any of those volumetric flow rates. Figures 4.34, 4.35, 4.36 and 4.37 show the dynamic responses of the TTA, green liquor AA, white liquor AA and equilibrium causticity to the disturbance respectively. All the responses show the similar dynamic behaviour as those to the disturbance in smelt rate. However, the new steady state values for all the responses are different from what have seen before as a result of different compositions in smelt and the nonlinearity of the causticizing reaction.  128 • Z^  127 ------^  z • •••-  126 125  I^/ I  I^I  0 124  (o)  , If;^  123  ftl  Smelt Dissolving Tank  122  Green Liquor Clarifier  •:( 121  Causticizer 120—  White Liquor Clarifier 119 —  -J  118  0^SOO^1000^1500^2000^2500  3000^3500^4000^4500  Time (min)  Figure 4.34: TTA Responses to a Change in Smelt Composition  73  ^  Chapter 4: Simulation Results and Discussions  51  50  /  _I 49  O  I  z  a  48  g  Dl * 47  ; ;  a a  Smelt Dissolving Tank  I  Green Liquor Clarifier  46  45  44 ^ 0^500^1000^1500^2000^2500^3000^3500^4000^4500  Time (min)  Figure 4.35: AA Responses to a Change in Smelt Composition (1)  118  117  Causticizer  i  ;  a  I I  Ca  1  White Liquor Clarifier  a  114  I  a a  113  i 1120^  500  ^  1000  ^  1500  ^  2000^2500  ^  3000  ^  3500  ^  4000  ^  4500  Time (min)  Figure 4.36: AA Responses to a Change in Smelt Composition (2)  4.4.3 Dynamic Responses to a Change in Lime Mud Filter Flow Rate  The importance of a stable green liquor control on the slaking/causticizing operation has been widely recognized [32]. In addition to a disturbance from the smelt rate change, the green liquor concentration can also be affected by the weak wash liquor. The change in the concentration of the 74  Chapter 4: Simulation Results and Discussions  86.2  86  E 42 85.4 85.2  85  0  500  1000  1500  2000^2500  Time (min)  3000  3500  4000  4500  Figure 4.37: Equilibrium Causticity Response to a Change in Smelt Composition  weak wash may come from different sources including changes in the fresh water flow rate, output  from the lime mud filter, output from the kiln scrubber, and the operation of the mud washer. The effect of the change in the mass flow rate from the lime mud filter has been taken as an example and is studied below. The step change in the mass flow rate from lime mud filter is shown in Figure 4.38. The system starts from steady state. At 90 minutes, the mass flow rate from the lime mud filter is increased by 20% from 1216 kg/min to 1459 kg/min. The mass fraction of each component from the lime mud filter is kept constant during this change. Figure 4.39 shows various volumetric flow rate responses to the disturbance. As about 98% of the output from the lime mud filter is water, the disturbance from this change manifests itself mainly as an increase in the volumetric flow rate. Due to the tank level control system for all the non-filled tanks, the volumetric flow rates of weak wash liquor, green liquor, and white liquor are increased almost immediately as a result of the disturbance. An overshoot can be seen in each response as a result of the tank level control system illustrated in Figure 4.25. Figure 4.40 shows the 'PTA dynamic responses in different units to the disturbance shown in Figure 4.38. The effect of the disturbance first appears in the smelt dissolving tank and then in the 75  Chapter 4: Simulation Results and Discussions  Figure 4.38: Change in the Mass Rate from Lime Mud Filter  Figure 4.39: Volumetric Flow Rate Responses to Change in the Mass Rate from Lime Mud Filter  green liquor clarifier. Since the major component in the output from the lime mud filter is water, the  TTA is decreased as a result of a general lowering of concentration in the liquor. The effect of the disturbance is felt in the causticizer and white liquor clarifier about three hours later. As discussed before, the average TTA of the white liquor is higher than that of the green liquor due to the water consumption in the slaking reaction. Notice the small overshoot in the TTA of the smelt dissolving tank. This results from the tank level control system which produces an overshoot in the response 76  Chapter 4: Simulation Results and Discussions  122  120  118  c5  a  116  1 1^Smelt Dissolving Tank t Green Liquor Clarifier 1 1^ t %, Causticizer 1; t White Liquor Clarifier . \ t  114 ................. a  112  0^500^1000^1500^2000^2500^3000^3500^4000^4500  Time (min)  Figure 4.40: TTA Responses to a Change in the Mass Rate from Lime Mud Filter  Figure 4.41: AA Responses to Change In the Mass Rate from Lime Mud Filter (1)  of the volumetric flow rates. Figures 4.41 and 4.42 show the dynamic responses of the green liquor and white liquor AA composition to the disturbance. These responses are quite similar in shape to those of TTA composition. The analysis given for the TTA responses with respect to the time the disturbance reaches the different units can also be applied here. Notice the overshoot in the green liquor AA 77  Chapter 4: Simulation Results and Discussions  1500^2000^2500  ^  3000  3500  4000  4500  Time (min) Figure 4.42: AA Responses to Change In the Mass Rate from Lime Mud Filter (2) 87.8 87.6 87.4 Z.* 87.2 * CS * foi 87  86.4 86.2  860^  500^1000^1500^2000^2500^3000^3500^4000^4500  Time (min)  Figure 4.43: Equilibrium Causticity Responses to Change In the Mass Rate from Lime Mud Filter composition response. There is, however, no appreciable overshoot in the white liquor composition  thanks to the green liquor storage tank and the green liquor clarifier that damped this temporary overshoot created by the tank level control system. Figure 4.43 shows the response of the equilibrium causticity of the causticizing reaction to the same disturbance. The dynamics of this response is similar to that of the TTA composition, but in the opposite direction. The curve shows a quick response at the beginning followed by a very slow 78  Chapter 4: Simulation Results and Discussions  increase. As discussed before, lower TTA facilitates the reaction to approach completion. This result agrees with the analysis done before in Chapter 3. It should be noticed that make-up water may be added in the slaker to compensate for the water consumption in the slaking reaction. The dynamic responses are similar to those that have just been shown. The TTA will be lower than in the case where no water is added. If the water consumed in the reaction can be exactly compensated, the TTA of the white liquor will be the same as that of the green liquor. Figure 4.44 shows the responses of the TTA to a step change in smelt rate with make-up water in the slaker. The disturbance is the same as what has been shown before. At 90 minutes, the total smelt rate is increased by 20% from its steady state value. The dynamics of the TTA responses is similar to that of TTA responses without make-up water. However, the steady state values are slightly different due to the make-up water. It can be seen that the TTA of the white liquor is equal to the TTA of the green liquor in the simulation.  135  U1  cd 130  rn  1. 2. 3. 4.  1 25  1000  1500  Smelt Dissolving Tank Green Liquor Clarifier Causticizer White Liquor Clarifier  2000^2500  Time (min)  3000  3500  4000  4500  Figure 4.44: TTA Responses to a Step Change in Smelt Rate with Make-up Water in the Slaker  79  Chapter 4: Simulation Results and Discussions  4.5 Simulation of Typical Disturbances  All the simulations performed in the previous sections are the responses of the recausticizing plant to step changes in individual, selected inputs. However, more irregular variations are present in the mill. The plant responses to this kind of disturbances can also be predicted with our simulator. Figures 4.45 to 4.48 show the TTA responses to the disturbances in smelt rate. The disturbances are introduced in the smelt rate. The smelt rate fluctuates around its steady state value (471.6 kg/min) with a magnitude of 5% in a sine wave form. The make-up water is added in the slaker in all the simulations. From the following graphs, it can be seen that the TTA of the smelt dissolving tank can follow the disturbance closely and its magnitude is not depressed very much. The TTAs of the following units have different time delay and different magnitude. The higher the frequency of the disturbance, the smaller the magnitude of the TTA responses. Disturbances with high frequencies, as shown in Figure 4.48 have very little effect on the TTA of the white liquor.  113)  1300  ^  1400  Causticizer  Smelt Dissolving Tank  Green Liquor Clarifier ---------- White Liquor Clarifier  Figure 4.45: TTA Responses to a 2.5 x 10 -3 min  80  -  1  Frequency Disturbance in the Smelt Rate (1)  ^ ^  Chapter 4: Simulation Results and Discussions  ct  - -,  120  tN 03 Z 119 01 118  117  1100^1200  1400  1500  Time (min) Smelt Dissolving Tank ^  causticizer  Green Liquor Clarifier -----------White Liquor Clarifier Figure 4.46:  TTA Responses to a 5.0 x 10 -3  min -1 Frequency Disturbance in the Smelt Rate (2)  122  900^1000^1100^1200  ^  1300  ^  1400  ^  1500  Time (min)  Figure 4.47: TTA Responses to a 1.0 x 10 -2 min -1 Frequency Disturbance in the Smelt Rate (3)  Figure 4.49 shows typical operational variations of the smelt rate. The steady state mass flow rate of the smelt is at 472 kg/min. Figure 4.50 shows the TTA responses in different locations in the plant to these variations. No make-up water is added in the slaker in this simulation. The TTA in the smelt dissolving tank follows 81  Chapter 4: Simulation Results and Discussions  120  119.5  --I 119  117.5 -  117  116.5 ^ 300  900^1000^1100^1200^1300^1400^1500  Time (min)  Figure 4.48: TTA Responses to a 2.0 x 10 -2 min -1 Frequency Disturbance in the Smelt Rate (4) 510  _  C 500  490  (/)  0  480  470  E2 460 0  LT_ 450 440  430^  100^200^300^400^500^600^700^800  ^  Time (min)  900^1000  Figure 4.49: Typical Variations in Smelt Rate  the variations fairly closely. However, it can be seen that the high frequency content of the input variations has been suppressed by the smelt dissolving tank. Only the low frequency content affects the TTA responses in the following units, and the magnitude of these variations are also damped out. Since the magnitude of the response variations in the white liquor clarifier is very small, the effect of the chemicals coming back from the plant cycle on the smelt dissolving tank is negligible. 82  Chapter 4: Simulation Results and Discussions  122  •  „.  „  121  —1 120  O  117  116 -  115 0  100^200^300^400^500^600^700^800^900^1000  Time (min) ^ Smelt Dissolving Tank ^ Causticizer ^ Green Liquor Clarifier ----------- White Liquor Clarifier  Figure 4.50: TTA Responses to Some Mill Disturbances in the Smelt Rate  The study of such realistic disturbances is crucial when the control system is being designed. The high frequency variations can be neglected in the control system design as they will be damped out by the storage tank.  4.6 Summary Remarks  The simulations presented here were based on the design data for a 1000 ton pulp/per day mill. The response of the plant to step changes and typical input variations have been presented. The dynamic simulations following other disturbances and for different operating conditions can be performed in a similar way. The transport lag in the plant piping is not considered in this work. However, transport lags can be included, if mill data are available. In a Kraft mill, the flowsheet for the recausticizing plant may be different from the one we have considered here, and the dimensions of the different equipment may change; however, these new 83  Chapter 4: Simulation Results and Discussions  values can be easily changed in the code to suit different designs. This simulation package can easily incorporate any new development in each individual area in the recausticizing plant. As discussed before, the recausticizing plant shows slow dynamics and long delay responses to different disturbances. The responses are complicated by the cyclic nature of the plant. The nonlinear behaviour of the causticizing reaction also increases the complexity of the dynamics of the whole plant. An effective control system must be developed to achieve stable operation of the recausticizing plant to under the influence of many different disturbances. The production of high quality white liquor is the primary purpose of the recausticizing plant. However, it is hard to achieve this objective by only dealing with the slaking/causticizing area. To overcome the slow dynamics and long delays, feed-forward control should be used. It can be seen that having a green liquor source with stable properties is vital to the whole plant. The parameters used to characterize the green and white liquor, including TTA, AA and causticity, can not be measured directly. The conductivity sensors may be used to relate liquor conductivities to those important parameters [33].  84  Chapter 5: Conclusions  Chapter 5 Conclusions  The importance of the recausticizing plant in the Kraft recovery system has been much more recognized in recent years. In order to obtain a better understanding of the dynamics of the recausticizing plant, a dynamic mathematical model has been developed and the dynamics of individual units in the system as well as their inter-relationships have been studied. First of all, every unit in the system has been modelled based on mass balance equations. Units such as the mixing tank, clarifier and dregs washer are represented by models of quite different structure that appropriately reflect their typical dynamic behaviour. The thermodynamics and kinetics of the slaking and causticizing reaction have also been studied. A new approach, based on Pitzer's model for strong electrolyte solutions, has been proposed to describe the equilibrium conditions of the causticizing reaction. This approach aims at predicting the equilibrium constant of the causticizing reaction without the need for fitting experimental data. The results of this predictive approach show excellent agreement with experimental results over the entire range of the ionic strength of the solution. The kinetics of the causticizing reaction has been described by a simple model which utilizes the better understanding of equilibrium conditions developed in this study. The parameters of the kinetic model were fitted to experimental data obtained using mill lime. The model can yield directly the value of the causticizing efficiency, which is the most important variable in a mill. Simulations of the different units and the entire plant have been carried out using typical input disturbances. The tanks in the system show a typical first order response to disturbances. The underflow compartment of the different clarifiers are represented in terms of plug flow with some backmixing. Careful studies of the slaking/causticizing reaction has yielded a number of new conclusions that can be summarized as follows: The reaction between the inputs — green liquor and lime — features considerable non-linearity; the higher the green liquor concentration (TTA), the lower the causticizing efficiency. A proper 85  Chapter 5: Conclusions  amount of lime charged is crucial to the reaction; lower charge will reduce the equilibrium causticity while higher charge will have an adverse effect on the following operations. Simulations of the whole plant have shown complex responses of the recausticizing plant to step disturbances. Responses to changes in smelt rate and in the volumetric flow rate show a fast response at the beginning, followed by very slow dynamics which may take days to reach a new steady state. Due to interactions between the plant and its tank level control system, overshoots may appear in some units with different magnitudes. Disturbances propagate to different units at different times. It has been noticed that the effect of high frequency disturbances is reduced by intermediate tanks, and their magnitudes are also damped out by tanks before they can affect the outputs. The effects of other types of disturbances can be studied in a similar manner with our simulator.  The overall establishment of dynamic models, particularly the model for the plant as a whole, has opened the door for further research and development in enhancing the performance of the recausticizing plant in the kraft recovery system. The present simulations and interpretation of these studies on the results have enriched current knowledge of the dynamics of the recausticizing plant. They may lead to efficient ways of training mill operators as one of the immediate applications. Moreover, much of the research not only helps achieve a better understanding of the process, but also provides beneficial guidelines for mills to improve quality of their products. It also can be seen from the dynamic simulation results that it is necessary to design a suitable control system for the recausticizing plant in order to overcome the disturbances promptly or even in advance. The control system may employ various control techniques, such as feed forward control, ratio control, cascade control, and model-based control. The dynamic modelling and simulation presented in this work has paved the way for control engineers to fully investigate for best control strategies.  86  Chapter 6: Further Work  Chapter 6 Further Work  Extensions of the research work discussed herein can be made in several ways: (1) The modeling of the lime kiln, lime mud filter, and kiln scrubber could be one extension of this work since the operation of the lime kiln affects the operation of the recausticizing plant, and vise versa. However, most of the papers published on the modeling and control of the lime kiln stop at obtaining the temperature profile of the lime kiln rather than using the temperature profile to help determine lime quality. Lime quality, in terms of its availability and reactivity, has a direct impact on the causticizing reaction. Thus it is considered to be necessary and important to do more work in this area. (2) Further investigations could be made in modeling the solids in the solution. In this work, the solids are assumed to join in the underflow of the clarifiers according to the design data available. Based on achievements in this work, it will be of no difficulty to obtain simulation results based on the assumption that a certain percentage of solids go into the overflow of the clarifiers, provided that the operational data of the solids in each stream at normal operating conditions are available. However, this is still a steady state description for the solids. A dynamic description for the solids requires the information of the diameter distribution of the lime mud, density of the lime particle, and the properties of the white liquor. The whole dynamic simulation can be expected to be improved if a more detailed model for the solids could be included. (3) The immediate application of this research is in designing a control system for the recausticizing plant. Based on mill data, different control strategies can be designed and tested by dynamic simulation of the control system under different disturbances, operating conditions, and design requirements of the mill. A more stable and improved operation of the entire recausticizing plant can be achieved by an effective control system.  87  References [1]  G. SMOOK. Handbook for Pulp & Paper Technologists. TAPPI and CPPA, Second Edition, 1992.  [2]  M. KOCUREK. Pulp and Paper Manufacture, Volume 5 Alkaline Pulping. TAPPI and CPPA, 1989.  [3]  R. GREEN and G. HOUGH. Chemical Recovery in the Alkaline Pulping Processes. TAPPI, Atlanta, Third Edition, 1992.  [4]  A. ROCHE and D. BOUCHARD. Process simulation in the pulp and paper industry. Technical report, PAPRICAN, Miscellaneous Reports 22, 1982.  [5]  E. JACOBI and T. WILLIAMS. A dynamic model and advanced direct digital control system for a kraft mill liquor preparation system. Technical report, Report No.56, Purdue Laboratory for Applied Industrial Control, 1973.  [6]  L. WANG, P. ENGLEZOS, and P. TESSIER. Modelling, simulation and control of a recausticizing plant. In 42nd Canadian Chemical Engineering Conference Proceeding, 1992.  [7]  L. WANG, P. ENGLEZOS, and P. TESSIER. Dynamic simulation of a recausticizing plant. Submitted for Publication to Computers & Chemical Engineering, 1993.  [8]  P. TURNER, B. ALLISON, and J. OEI. Brown stock washer control Part II: filtrate tank level control. In 78th Annual Meeting, Technical Section, CPPA, 1992.  [9]  B. BLACKWELL. Increasing white liquor causticity by addressing the diffusion limitation. Pulp & Paper Canada, 1987.  [10] G. DORRIS and L. ALLEN. Operating variables affecting the causticizing of green liquors with rebumed limes. Journal of Pulp and Paper Science, 1987. [11] TAPPI. Technical information sheets. Technical report, No. 102.02 TAPPI, Atlanta, Ga., 1965. [12]  G. MITTET and G. DAMSTROM. Lime mud washing control — a key to improve slaker performance. In International Chemical Recovery Conference Proceedings, 1992. 88  [13] G. DORRIS. The physical characterization of hydrated rebumed lime and lime mud particles. In International Chemical Recovery Conference Proceedings, 1992. [14]  S. RYDIN, P. HAGLUMND, and E. MATTSSON. Causticizing of technical green liquors with various lime qualities. Svensk Papperstidning, 1977.  [15] R. CROTOGINO, N. POIRIER, and D. TRINH. The principles of pulp washing. Tappi Journal, 1987. [16] H. LINDBERG and P. ULMGREN. Equilibrium studies of white liquor preparation in kraft mills. J. Pulp & Paper Sci., 1983. [17]  D. MONDAL and A. KRISHNAGOPALAN. Predicting equilibrium causticizing efficiencies by a model based on ionic equilibrium. Tappi Journal, 1989.  [18]  G. DORRIS. Equilibrium of the causticizing reaction and its effect on the slaker control strategies. In Pulping Conference, TAPPI Press, 1990.  [19]  C. DAILY and J. GENCO. Thermodynamic model of the kraft causticizing reaction. J. Pulp & Paper Sci., Jan. 1992.  [20]  J. RANSDELL and J. GENCO. The effect of sodium sulfide on the equilibrium of the kraft causticization reaction. Tappi Journal, Aug. 1991.  [21]  L. WANG, P. TESSIER, and P. ENGLEZOS. A thermodynamic study of the kraft causticizing reaction equilibrium. Submitted for Publication to Journal of Pulp and Paper Science, 1992.  [22]  J. ZEMAITIS, et al. Handbook of Aqueous Electrolyte Thermodynamics. AIChE, 1986.  [23]  K. PI1 LER. Thermodynamics of electrolytes. I. theoretical basis and general equations. Journal of Physical Chemistry, 1973.  [24]  K. PILLER and G. MAYORGA. Thermodynamics of electrolytes. II. activity and osmotic coefficients for strong electrolytes with one or both ions univalent. Journal of Physical Chemistry, 1973.  [25] D. WAGMAN, et al. Selected Values of Chemical Thermodynamic Properties. National Bureau of Standards, Washington, DC., 1973. 89  [26]  C. M. CRISS and J. W. COBBLE. The thermodynamic properties of high temperature aqueous solutions. v. the calculation of ionic heat capacities up to 200°. entropies and heat capacities above 200°. J. Am. Chem. Soc., 1964.  [27] J. PEIPER and K. Pl't LER. Thermodynamics of aqueous carbonate solutions including  mixtures of sodium carbonate, bicarbonate, and chloride. J. Chem. Thermodynamics, 1982. [28]  S. RYDIN. The kinetics of the causticizing reaction. Svensk Papperstidning, 1978.  [29] H. LINDBERG and P. ULMGREN. The chemistry of the causticizing reaction — effects on the operation of the causticizing department in a kraft mill. Tappi Journal, 1986. [30] P. TESSIER. Design of a recausticizing plant for a 1000 ton/day kraft pulp mill. Confidential Report, 1989. [31] Simnon User's Guide for UNIX Systems, Version 3.1, May 1991.  [32]  W. MUSOW. White liquor quality control starts out at the smelt dissolving tank. In International Chemical Recovery Conference, Seattle, 1992.  [33]  G. DORRIS. Conductivity sensors for slaker control Part II: Probe calibration and performance at a mill site. Pulp & Paper Canada, 1990.  90  Appendix A Chemical and Physical Terminology  The active cooking chemicals in the pulping liquor (white liquor) are sodium hydroxide and sodium sulfide; these are termed active alkali (AA). The amount and concentration of these chemicals are determined primarily by the types of wood, degree of cooking, the desired Kappa number, and to a lesser extent by the manner the chemical recovery area is operated. Since sodium is the common element, general North American practice is to express the concentration of these chemicals in terms of equivalent Na 2 0 even if this component doesn't exist. In mills, it is common practice to use the following terms (Table A.15) to characterize the cooking liquor. All values are based on equivalent Na2 0.  Total titratable alkali (TTA) Active alkali (AA) Effective alkali  + Na2CO3 + Na2S + 1/2Na2S03 NaOH + Na2 S NaOH + 1/2Na 2 S NaOH  Causticizing efficiency % (CE)  NaOH (less NaOH in green ligtior)x100 NaOH (less NaOH in green liquor)+Na,CO,  Causticity °'  Na0Hx100 Na0H-1-Na 9 CO,  Sulfidity % Total chemical  Na2sx100  Na0H+Na9S  All sodium salts Table A.15  Common Terminology  91  Appendix B Tank Level Control  All the tanks in the recausticizing plant are integrating processes. Figure B.51 shows an integrator subject to a load disturbance and regulated by a PI controller. The process gain is Kp and the load gain is Ki. The process gain Kp is the change in output (AY) per unit time (At) caused by a one unit change in the manipulated variable (AU). K p may be obtained empirically from an open-loop step response (Kp = AY/At/AU), or analytically, as in this case is the 1/A.  Disturbance Load (L)  Controller T s+1 Kc r  (U)  Tr s  .41  Figure B.51: Block Diagram of PI-Controlled Integrating Process  The closed-loop transfer function relating the process output to the load disturbance for the system in Figure B.51 is given by Y(s) L(s)^1 + Kc 1.0±1. 1S.E. Tr s KITr Kcifp[ Rins 2 Tr s + 11  Gil=  (B.130)  which is the impulse response of a second order system with standard form -  Gd(s) = K 7 2s2 + 2ers +1 -  (B.131)  where K is the gain, e is the damping factor and r is the natural frequency . The gain K is a function of the load gain K1 which is not necessarily known, nor required to design the controller. 92  By equating coefficients of s in the above equations, the second-order time constant and damping factor are given by T  2  Tr Ka i = Tr  =  (B.132)  1r  A damping factor equal to 1 will insure a non-oscillatory behaviour and simplify the above equations to Tr = 27 2 K= c K pr  (B.133)  which is a tuning rule where the desired natural frequency r is the only user specified tuning parameter and is related to the desired speed of response. The settling time of a critically damped second order system is T  67.  93  

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