# Models of an Industrial Evaporator System for Education and Research in Process Control.

код для вставкиСкачатьDev. Chem Eng Mineral Process., lO(i/2),pp.105-127,2002 Models of an Industrial Evaporator System for Education and Research in Process Control Kiew M. Kam”, Prabirkumar Saha’, Moses 0. TadC’* and G. P. Rangaiah2 ‘ Department of Chemical Engineering, Curtin University of Technology, GPO Box UI987, Perth 6845, Western Australia Department of Chemical and Environmental Engineering, National University of Singapore, Singapore I 1 9260 Current affiliation. Chemical & Process Engineering Centre, National University of Singapore, Singapore I I9260 ’ Realistic models of typical industrial processes are essential for effective and practice-oriented education and research in process control This paper presents two mechanistic models of an industrialfwe-eflect evaporator system, and also indicates their potential applications in process control-related research and training Simulation programs for these models in four diflerent platforms (viz, MAPLE, M A TLAB, Simulink and C++) are developed Relative merits and applicationspecific usefulness of these four platforms in process control studies are discussed Introduction The growth of computer-based training during recent years has been phenomenal in chemical engineering education [I]. It has emerged as the tool of choice for most training and development programs due to speed of computation, portability of software, visual graphic utilities and various other user interfaces (e.g. printer, plotter, etc.). Particularly in the field of process control applications, the cause and effect of different control schemes and controller tuning parameters on chemical process systems are readily demonstrated through visual computer simulations rather than just “being taught” fiom the textbooks. Furthermore, use of computer models for case study projects in process control courses allows practical and open-ended problem solving [2] for the training of graduate students. For example, a single-stage forcedcirculation evaporator of Newell and Lee [3] was implemented in Simulink as the computer model for the case study project performed in the undergraduate process control course in the Dept of Chemical Engineering at Curtin University of * Authorfor correspondence (email tadem@che curtin edu au) 105 K M.Kam,P. Saha. M . 0 Tad6 and G.P Rangaiah Technology On completion of the project, students remarked that it has greatly improved their understanding of the concepts of control system design. Further, they understood the importance of process identification for optimal controller tuning, advanced process control concepts (such as feedforward and cascade control) and process interaction compensation techniques (such as relative gain array for pairing the variables and decoupling control). An evaporator was chosen as an exemplary process for the aforementioned process control course, due to its importance in the chemical and mineral processing industries Various mechanistic and black box models have been developed for industrial evaporator systems by researchers in the last decade [e g 4-1 11 Many of these models were developed for only single or double effect bench scale evaporators. Although they are suitable for computer simulation and process control studies, they were not adequate to show a clear picture of an industrial evaporator system, which are mostly multi-effect in nature. Some of the cited works deal with industrial-scale evaporators; however complete details of the models are not reported, possibly due to proprietary reasons. More recently, Kam and Tad6 [12] presented two mechanistic models of a fiveeffect evaporator system, and used them in control studies. Comprising numerous state, input and output variables, and having the severe nonlinearity and difficult dynamic (minimum phase) characteristics, these evaporator models provide a complete base for extensive nonlinear process control studies. The purpose of the present work is three-fold: 1. To present and compare two mechanistic models of the five-effect evaporator for possible application in process control related research and training. 2. To develop simulation programs of these models in four different platforms (viz. MAPLE, MATLAB, Sirnulink and C++) and compare the merits, dements and application-specific usefulness of these four platforms in process control studies. 3. To indicate different ways of using the two models for education and research in process control. The Evaporator System The selected evaporator system is the first step in the liquor burning process associated with the Bayer process for alumina production at an alumina refinery. It consists of one falling film, three forced-circulation, and a super-concentration evaporators in series. The main components of each stage are a flash tank (FT), a flash pot and a heater (HT). A simplified schematic o f the evaporator system is depicted in Figure 1. Note that the flash pots are not shown in this figure for simplicity of the schematic. Spent liquor, which is recovered after precipitation of the alumina from its solution, is fed to the falling film stage (FT #1). The volatile component, water in this case, is removed under a high recycle rate and the product is further concentrated through the three forced circulation stages (FT #2 to #4). The super-concentration stage (FT #5) is used to remove the residual ‘flashing’ of the concentrated liquor without recycle. In each of the forced-circulation and super-concentration stages, the spent liquor is heated through a shell and tube heat exchanger (heater) and water is removed as Models of an industrial Evaporator System HT#= heater no ; FTIl =flash tank no ;C = Condenser I Figure 1. Simpl@ed schematic of the industrial evaporator system vapour at lower pressure in the flash tank. The vapour given off is used as the heating medium in the heaters downstream. Live steam is used as the heating medium for HT #3, #4 and #5 The flashed vapours from FT #3 and #4 are combined and used in HT #2, while the vapour fiom FT #2 is used in HT ## 1. The steam condensates 6om the heaters are collected in the flash pots. Live steam to HT #3 is set in ratio to the amount of live steam entering HT #4, while the amount of live steam to HT #5 is set depending on the amount of residual ‘flashing’ to be removed. The cooling water flow to the contact condenser (C in Figure 1) is set such that all remaining flashed vapour is condensed. The evaporator system is crucial in the aluminium refinery operation and is difficult to control using classical controllers due to complex characteristics such as strong process interactions and nonlinearities. There are 15 state variables that are of interest. They are the liquor levels in the the liquor densities (pi, i=l,2,. .S),and temperatures of the flash tanks (hi, i=1,2,. -3, product streams leaving each stage (Ti, i=1,2,. ..5). The controlled outputs in the plant are the liquor levels in the flash tanks (hi, i=1,2,,..5), the product liquor density of stage 4 (p4), and the product liquor temperature of the super-concentration stage (Ts). The manipulated inputs are the liquor product flows (Qpi,i=1,2,...5), the steam rate to HT #4 ( hs4 ), and the vapour withdrawal rate from FT #5 ( mv5 ) In the actual plant situation, all these manipulated inputs and the controlled outputs are measured online. Additionally, on-line measurements of the product liquor temperatures of the fvst four stages (TI, T2, T3, Tq) are available. The product liquor densities of the fust three stages (p,, pz, p3), however, are not measured on-line. 107 K M. Kam,P.Saha, M . 0 Tad6 and G.P Rangaiah Models for the Evaporator System Two mathematical models, M1 and M2, were developed for the industrial evaporator system using unsteady-state mass and energy balances around the units in Figure 1. For this modelling only the first four stages of the evaporator system are considered in order to avoid additional complexities. Each of the models consists of 12 ordinary differential equations with 5 input and 5 output variables. The state, input and output variables for the two evaporator models are summarized in Table 1 It should be noted that, due to proprietary issues, the actual values of these variables are multiplied with an arbitrary factor to honour the confidentiality And the reported data do not alter the basic characteristics of the models It should also be noted that values of state variables in this table are only approximate steady-state values; exact values depend on the model(s) selected, and will have to be obtained in each case. All controlled and manipulated variables (y’s and u’s respectively) were selected according to the actual plant situation. The differences between M1 and M2 are due to different assumptions (summarized in Table 2) that were used in their development. These and the model equations presented below and in the Appendix A, have been reported by Kam and Tade [12, 131, and are included here for completeness and convenience. Table 1. State, input and output variables and their nominal values for the evaporation system States T4 XI2 PI P2 P3 1.5 2.25 2.25 2 25 1.54 66 0 90.6 129 135 1.357 1.422 149 Inputs (u ’s) and Outputs (y s) QPI 32.736 u2 QP2 27.713 u3 QP3 23.850 u4 QP~ 2 1.642 u5 ms4 2.3814 UI Y4 hl h2 h3 h4 Ys P4 Y1 Y2 Y3 1.5 2.25 2.25 2.25 1 54 Model M1 The model equations for the second stage of the evaporator system are presented below; model equations for other stages can be found in the Appendix A. Further information on the evaporator model and its derivation are available in 1141. 108 Models of an Industrial Evaporator System [ ;" dt 1 1 C O T ~(v2 - h , A 2 ) [ E2 - m v 2 + Pv2( QPl -QP2 -P W BPL$ = 102.69~2- 128.82 Table 2. Assumptions usedfor development of evaporator models MI and M2 Additional assumptions for Model M2 Assumptions for both Models MI & M2 1. Dynamics of instrumentation and 8. Liquor boiling point elevation in each flash tank remains control valves are very fast and unchanged. negligible when compared to the 9 Vapour pressure-temperature dynamics of flash tanks. relation of flashed vapour is 2 . Liquor density and temperature in linear. each flash tank are respectively the same as those of corresponding 10. Latent heat of vaporization of liquor remains constant. discharged stream since perfect mixing is usually achieved in each 11 No accumulation of vapour in the flash tanks and the steam space in flash tank. the shell side of the heaters Specific heat capacities of all 3. remains unchanged. This means process streams remain constant. amount of vapour leaving a FT 4. Effects of falling film on the heat transfer rate and the dynamics of (liz, ) is equal to the amount of heater discharge temperatures are water vaporisation in that FT (E,), neglected. and it can be completely 5 . Liquor in each flash tank is in condensed in the downstream equilibrium with the corresponding heater as per Figure 1 (without flashed vapour. being limited by the heat transfer 6 . Heat losses t?om the evaporator rate such as equation 14). system are negligible. 7. No flashing of liquor in the heaters. 109 K M. Kam, P. Saha, M . 0 Tadk and G.P Rangaiah Boiling point elevation (BPE) of the liquor is dependent on the liquor density as shown in Equation (5). Equation ( 5 ) was determined from the data obtained fiom the plant. MP2 p v 2 = R(273.1+ T2 - BPE,) [ P2 = 0.133exp A - (7) C + 2 7 3 l+T2-BPE2 The constants M and R are the molecular mass of water and the gas constant, respectively, and A, B, C are constants in the Antoine equation for the vapour pressure-temperaturerelation of water vapour The flashed vapour from FT #2 is used in HT #l.As such, the amount of vapour that is drawn fiom FT #2 (i e. m v 2 ) depends on the amount of condensation due to heat exchange with the liquor in HT #I. The rate of vapour withdrawal t7om FT #2 is given as: where the subscripts HF and S refer to heater feed and steam, respectively. The steam temperature in HT # 1 (i.e. Tsl) is the same as the saturation temperature of the flashed vapour 6om FT #2: The liquor density ( p ~ and ~ ~temperature ) (THFl)in FT # 1 are obtained 6om mass and energy balances at the mixing point as follows: PHFI= 110 4 + Qf QHFI Pf Models of an Industrial EvaporaforSystem The recycle rate R , is set to a constant. The evaporation rate E2 in Equation (4) depends on the amount of steam condensing in HT #2, which is equal to the amount of flashed vapour from FT #3 and + m,, ) and is also governed by the rate of heat transfer in HT FT #4 (i.e. ms2 = kV3 #2 given by: R2~2T2+ QPI PICITI THF2 = QHF2 PHF2 = P HF 2cHF 2 R2 + QPI P I QHF 2 Model M2 The additional assumptions (tabulated in Table 2) are used to simplify model M i . The simplified model is referred to as model M2. The model equations for the second stage of the evaporator system are given below; model equations for other stages can be found in the Appendix A Further information on the evaporator model and its derivation are available in [14]. %=-!---( dt COT^ E2 - 4 2 + pv2 P W where 1 1.58M R(273 1 + T2 - B P E 2 ) - (273.1 +pT2 v 2- BPE2) (21) 111 K M.Kam. P. Saha, M.0 Tad6 and G.P Rangaiah MP2 p v 2 = R(273.1+ T, - BPE2) The value of BPEl is constant for model M 2 (viz , 16). The model equations for the liquor temperature of model M2 in Equations (20) and (21) are considerably simpler than Equations (3) and ( 8 ) of model M1 The simplification is due to the additional assumptions that the BPE is independent of the liquor density and that the vapour pressure-temperature relation of the flashed vapour is linear in Equation (26) This was obtained by local linearization of Equation (7) at the nominal liquor temperature. Another noticeable difference between the model equations of the two models is the steam flow to HT #2 and the vapour withdrawal rate from FT #2 Note that, instead of depending on the heat transfer in the heater, the vapour withdrawal rate in model M2 is equal to the rate of evaporation in the FT, i e Equation (24) Values of constant quantities in the evaporator models are presented in Table 3 Some of the characteristics of the evaporator models are summarised in Appendix B Table 3. Values of constant quantities in the evaporator models. 3290 BPE3 BPE4 112 1 24.6 I 30.6 kJ/k "C OC Ih I 3 hV4 I 2243600 1 2243600 Mkg kJkg Models of an Industrial Evaporator Sysrem Implementation of the Evaporator Models During the course of our research on model-based control of the evaporator system, models M1 and M2 have been implemented on four computing platforms. Maple, Matlab, Simulink and C++ These simulation programs for the industrial evaporator system can be obtained upon request f?om the corresponding author. Comparative merits of the different computing platforms based on our experience for the evaporator model simulation are summarized in Table 4, and briefly discussed below. These are helpful to educators and researchers in choosing a correct programming platform for their process control studies. Implementation in MAPLE MAPLE allows mixed symbolic manipulation and numerical computation that can achieve greater accuracy and speed in engineering problem solving. The use of MAPLE in control engineering has received significant attention in the literature [ 1, 151. The evaporator model was implemented easily in Maple as it has useful built-in mathematical functionality, such as the linulg package that has a collection of functions for matrix manipulations and evaluations, and the dsolve fknction and DEtools package for solving a system of ordinary differential equations. The linalg package is particularly useful in developing design procedures for an analytical nonlinear controller by using an input-output linearization technique [ 151. While MAPLE is efficient in symbolic designs in terms of less programming effort, it is limited for numerical simulation of closed-loop control systems. This is because the required execution (or clock) time is substantially higher when compared to other numerical simulation packages such as Simulink. The large execution time seems to be due to poor use of memory when computing in MAPLE, and the execution time grows with closed-loop simulation time. For example, memory used for executing a MAPLE statement is not “reclaimed” for other subsequent executions. Therefore, as the number of executed statements increases (i.e. the do-while loop in closed-loop simulation code), the amount of free memory that is available decreases. This causes the execution time to increase due to less available memory. For this reason, numerical simulation involving repetitive and iterative computations in Maple is not viable. Implementation in Matlab The use of MATLAB in process control application has received significant attention in the literature [16]. It allows extensive matrix manipulation and numerical computation in an “interpreter” form Development of programs for simulating evaporator models was found to be easy in MATLAB as it has a group of built-in mathematical functionalities (called toolboxes) such as funfun toolbox that has the ode45 function for solving a system of ordinary differential equations. For control applications, there are lmictrl and lmilab (LMIControl Toolbox), &t (QFT Control Design Toolbox), fizzy (Fuzzy Logic Toolbox), rnpccmds (Model Predictive Control Toolbox), ncd (Nonlinear Control Design Blockset), m e t (Neural Network Toolbox), and robust (Robust Control Toolbox). MATLAB is equally efficient in analytical derivatiodsimplification of fust principle models through its symbolic (Symbolic Math Toolbox). This is particularly useful in the calculation of Jacobian matrices. All 113 K.M. Karn, P. Saha. M.0 Tade‘and G.P Rangaiah these toolboxes are quite useful in developing design procedures for linear and nonlinear controllers The programming in MATLAB is easy but not so transparent if the toolboxes are used extensively For this same reason, applications are sometimes restricted due to limited choice of features in the functional subroutines. The debugging of a program is quite easy if the problem is not due to the limitation of built-in subroutines such as those in toolboxes. Also there are graphic utility toolboxes, such as gruphkd (two dimensional graphs), gruphjld (three dimensional graphs), specgraph (specialized graphs), graphics (Handle Graphics) and uitools (graphical user interface tools). These are convenient and sufficient to have a quick online visualization of simulation results. There are two major demerits of MATLAB simulation First, the program is always platform dependent Although MATLAB compilers are being used to convert the code to C/C++ and compile it to form a stand-alone application file, the use is quite limited. One of the reasons for this is that MATLAB compiler cannot convert a few commands such as “eval” and “plot”, which are considered to be key commands in most of the application programs Second, MATLAB simulation is very slow because it considers all its variables as matrices Even a scalar is taken as a 1x1 matrix in the memory space, which eventually slows down the computations Implementation in Sirnulink The use of Simulink in simulation jobs has found wide acceptance due to its user friendly and menu-driven programming technique It allows the user to draw a flow diagram of the model calculations, with the help of a few pre-assigned blocks, sirnulink (Simulink) and blocks (Simulink block library) are the toolboxes containing various block modules required by the software. Simulink does use the features of MATLAB functions and subroutines or toolboxes, but in a manner, which is not at all transparent to the user. It is definitely easy to develop a program for the model in Simulink, and associate it with available control system utility blocks to generate correct control strategy. However, they have their own limitations in terms of the specific need of a particular control algorithm In those cases, the use of S-function facilitates for users developing their own blocks for control and other applications. The debugging of Simulink programs is the easiest if the problem is not due to the limitation of subroutine blocks. Switching between two control strategies requires nothing more than changing the specific controller block in the main program There are two major demerits of Simulink simulation. First, like MATLAB, the Sirnulink program is always platform dependent. Unlike MATLAB, compilers are not proved to be useful to form any stand-alone application file. Second, Sirnulink does not allow handling of two independent variables in the program. As a result, the control application requiring two independent variables (such as Model Predictive Control for nonlinear processes, where a second time variable is needed for calculation of future prediction of model outputs for optimization purpose) are quite difficult to program. Implementation in C++ The use of C++ in process simulation and control has become quite popular because of its general-purpose structured or object-oriented programming technique that is powerful, efficient and compact. C++ combines the features of a high level language with the elements of the assembler (in “compiler” form) and thus is close to both man I14 bl 4 4 Analysis of program Program I Programming Integration step size Open-loop simulation Feature Needs graph utilities that are easy to use Requires platform Quick, simple and transaarent. Tedious requlrement mcreases and computational speed decays exponentially. Need to be specified. Accuracy depends upon correct choice of + I Easy on-line plotting Requires platform Easy on-line plotting Requires platform Easiest Easy but least transparent Takes internally, not so transparent. But at least 10 times faster than MATLAB Need to be specified. Equipped wlth facility for variable Easy but not transnarent Easier Fastest Slowest Faster than Matlab 1 Needs graph utilities; graphics programming m C++ IS difficult and time consuming Stand-alone EXE file Depends on subprograms developedhsed for numerical integration. Tedious and difficult but transnarent Most tedious 1 Table 4. Comparisons of different computing platforms for evaporator model simulation. I K M . Kam, P Saha, M . 0 Tad6 and G.P Rangaiah and machine. It has now been implemented on virtually every type of computer, fi-om micro to macro. Unlike MATLAB/Simulink, developing the program for simulation of an evaporator model in C++ has been found to be quite time consuming and tedious, It does not have any built-in mathematical functionality for linear algebra or numerical computations. Thus for open-loop simulation as well as for control applications, one needs to develop functional subroutines for everything from ODE solver to graphic utility. However, the programming in C++ is most transparent as it is entirely done by the user. Nevertheless one can take the help of readily available codes for a specific application [ 171; in fact, there are often several choices of functional subroutines and the user has to select the most appropriate code. The debugging of programs is the most tedious out of the four platforms. The greatest advantage of C++ code is that it is compiled to generate a stand-alone executable file for future application. Thus the program is platform independent. If needed, it is possible to maintain the confidentiality of the code. 60 -g - 50 40 30 20 lo- .-- . . * 25 20 I5 10 0 5 0 g N c o -5 -10 -I5 -20 -25 -30 -35 32 28 24 20 16 12 - - 8 5 m 4 g -4 cv o -8 0 1 2 3 time (hr) 4 5 0 1 2 3 time (hr) 4 5 12 10 8 6 4 2 o -2 -4 -6 -12 -16 -20 -24 -28 -a -10 -12 -14 0 1 2 3 time (hr) 4 5 Legend for all the figures +20% change in m,4 for M1 model -20% change in ms4 for MI model +20% change in m,4 for M 2 model -20% change in ms4 for M2 model Figure 2. Open-loop responses of evaporator models A41 and M2 for a change in from 2 3814 to 2 8577 liquor level in the four flash tanks 116 Models of an Industrial Evaporator System Comparison and Use of Evaporator Models Open loop simulation of the two evaporator models subjected to a S O % increase in the steam flow to HT #4 was performed in each of the four platforms. The results obtained in all the platforms are identical The open loop responses of the evaporator models obtained from Simulink are given in Figures 2 to 4. In these figures, outputs and other state variables are expressed in terms of percentage deviations 60m their respective nominal steady-state conditions (in Table 1). The responses of the two models differ significantly from each other In Figure 3, the responses of the liquor temperatures are in the opposite directions (e.g. T3 of M1 increases while T3 of M2 decreases). In the case of the liquor densities, the ultimate responses of the M1 and M2 are different indicating uncertainty in the model gains. The effect of steam flow on the liquor densities of model M2 is larger than those of model MI. Structural model uncertainties are evident from the difference in the transient responses of the liquor levels, densities and temperatures. Model mismatch between M1 and M2 was also shown using closed-loop simulation [12]. All these differences are due to the additional assumptions for model M2 stated previously . . I 50 40 . I I -4 -6 I -20- . 0 1 2 3 4 5 0 1 time (hr) 4 , -4 -3 - -6 4 10 1 8 2 3 time (hr) 4 5 6 - & n b 4 2 o -2 0 1 2 3 time (hr) 4 5 0 I I I I I 2 3 time (hr) 4 5 Legend for all the figurer +20% change in ms4 for MI model -20% change In mS4 for MI model +20% change in m,4 for M2 model -20% change in mS4 for M2 model Figure 3. Open-loop responses of evaporator models MI and M2 for a change in ms4fiom 2 3814 to 2 8577 liquor temperature in the fourfrash tanks 117 K M.Kam, P,Saha. M . 0 Tad6 and G.P Rangaiah 3 , 10 05 E- oo a -05 -lo -15 I ...... o.... 1 - 0 1 2 3 time (hr) 4 5 0 5 1 2 3 time (hr) 4 5 1 0 -3 -4 0 1 2 3 time (hr) 4 5 *....* I I I I I 0 1 2 3 4 5 time (hr) Legend for all the figures +20% change in m,4 for MI model -20% change in ms4 for MI model 4 +20% change in ms4 for M2 model -20% change in ms4 for M 2 model Figure 4. Open-loop responses of evaporator models MI and M2 for a change in ms4from 2 3814 to 2 8577 liquor density in the fourflash tanks The challenge involved in control studies using models M1 and M2 can easily be understood by the “process characterisation cube” described by Ogunnaike and Ray [18]. The three axes of this cube correspond to three process characteristics, viz degree of dynamic complexity, degree of nonlinearity and degree of interaction. Having severe nonlinearity and structural complexity (as shown by many equations above, and open-loop responses in Figures 2 to 4), models M1 and M2 belong to the most difficult “Category VIII Processes”. Model M 1 is considered relatively more realistic for the industrial evaporator system than model M2 due to fewer assumptions. However, fiom a controller design point of view, model M 2 is relatively viable for nonlinear model-based controller design due t o its simplicity. Furthermore, 118 Models of an Industrial Evaporator System the control-nonaffine nature of model M1 (i.e. temperature state equations are nonlinear with respect to the manipulated inputs) and the exponential terms make it computationally less attractive for nonlinear control system design Nevertheless, the availability of two models for industrial evaporator systems allows more rigorous and realistic nonlinear control studies through computer simulation [ 12, 13, 18, 191 These studies have generally employed model M 1 as the process and model M2 for controller design. The suggested control techniques for this kind of system would be a full-scale multivariable, nonlinear, model based control law with capabilities for handling difficult dynamics, such as Generic Model Control and Nonlinear Model Predictive Control [30].Apart from this use of models M1 and M2, they can be used in several other ways in process control research and education. A few possible ways are: 1. Process identification of either model M1 or M2, and the identified model can then be used for model-based controller design for the simulated process 2 Either model M1 or M2 can be used as the process for training in classical control techniques at undergraduate and/or post-graduate level 3 Various evaporator configurations (such as one or a few stages) are possible for process flexibility and controllability studies This would require the users to set up relevant model equations from either models MI or M2, and select the input and output variables. While the strict validity of both models MI and M2 to the real evaporator system on-site has not been established, model M2 is considered to satisfactorily represent the actual plant owing to the success of the implementation trial of geometric nonlinear control on the evaporator system simulator on-site [21] In the implementation trial, model M2 was utilised for the geometric nonlinear controller design. Should model M2 fail to represent the evaporator system satisfactorily, such a success would not have been possible Being more realistic than model M2 from the physical point of view, it is understood that model M1 also represents a valid model to the actual evaporator system. Several measured andor unmeasured load disturbances may occur during the operation of the industrial evaporator system. Common disturbances are due to changes in the feed (such as rate, density and temperature in F in Figure 1) to the evaporator system, andor changes in the heat transfer coefficient of heaters For control studies on the evaporator models by researchers and students, the following disturbances are suggested. 1 . Change in flow rate of feed, F from the nominal value of 37 7 to 39.7 or 35 7. 2. Change in heat transfer coefficient, UA in heaters 1 and 2 from the nominal value of 1999780 and 965390 to 1699810 and 820580 respectively. 3. Change in density of feed, F from the nominal values of 1.3 10 to 1 245 or 1 376 4. Change in temperature of feed, F from the nominal values of 60 to 54 or 66 5. Change in the set point of product liquor density from FT#4 from the nominal value of 1.54 to 1.46 or 1 62 Based on our experience, the authors recommend the use of MATLAB and/or Simulink platforms for performing control studies owing to the availability of various control system toolboxes. However, the Maple platform is better for easy analysis and design of analytical nonlinear controllers with its iinarg package C++ platform will 119 K M. Karn, P. Saha, M . 0 Tadk and G.P Rangaiah be recommended to those who want absolute transparency in the entire simulation code. Conclusions Two different mechanistic models of a complex multistage industrial evaporator system are presented. The models are significantly different from each other, and they are useful for training and research in process control at the undergraduate and/or postgraduate levels. The models are simulated in four different platforms of programming language/package viz. MAPLE, MATLAB, Simulink and C++. The merits, demerits and usefulness of these codes are critically analyzed, which will enable the reader to choose the appropriate programming technique for their application-specific need. Nomenclature Abbreviations BPE boiling point elevation CW cooling water D underflow F liquor feed FT flashtank HD heater discharge HF heater feed HT heater HW hot water P product flow S steam flow V vapour flow Symbols C h Q T U X Y m P specific heat capacities flash tank level volumetric flow liquor temperature manipulated input variable state variable controlled output variable mass flow liquor density Subscripts P liquor product v vapour S steam References Mackenzie, J G and Allen, M 1998 Mathematical power tools Chem Eng Educ ,32(2), 156 Bequette, B W , Schott, K D , Prasad, V ,Natarajan, V and Rao, R R 1998 Case study projects in an undergraduate process control course Chem Eng Educ ,32(3), 214 3 Newell, R B and Lee, P L 1989 Applied Process Control A Case Study, Prentice-Hall, New York 4 Montano, A , Silva, G and Hernandez, V 1991 Nonlinear control of a double effect evaporator In Advanced Control of Chemical Processes, Pergamon Press, New York, 167 5 Allen, R M and Young, B R 1994 Gain scheduled lumped parameter multi-input multi-output models of a pilot plant climbing film evaporator Control Engineering Practice, 2(2), 219 6 Quaak, P , Vanwijck, M P C M and Vanharan, 3 J 1994 Comparison of process identification and physical modeling for falling film evaporators Food Control, 5(2), 73 7 Vanwijck, M P C M , Quaak, P and Vanharan, J J 1994 Multivariable supervisoly control of a 4effect falling film evaporator Food Control, 5(2), 83 8 Wang, F Y and Cameron, I T 1994 Control studies on a model evaporation process constrained state driving with conventional and higher relative degree system Journal of Process Control, 4(2), 59 9 To, L C ,Tadt, M 0 , Kraetzl, M and Le Page, G P 1995 Nonlinear control of a simulated industrial evaporation process Journal of Process Control, 5(3), 173 10 Young, B R and Allen, R M 1995 MlMO identification of a pilot plant climbing film evaporator Control Engineering Practice, 3(8), 1067 1 2 - 120 Models of an Industrial Evaporaior System 15 16 17 18 19 20 21 To, L C , Tade, M 0 and Le Page, G P 1997 Implementation of input output linearization control technique on an industrial evaporative system CHEMECA 1997, Rotorua, New Zealand, PC2d Kam, K M and Tade, M 0 2000 Simulated control studies of five-effect evaporator models Comput Chem Eng,23(11-12), 1795 Kam, K M and Tade, M 0 1999 Nonlinear control of a simulated industrial evaporation system using an input-output linearization technique with a reduced-order observer Ind Eng Chem Res ,38,2995 Kam, K M and Tad6, M 0 1997 Technical Report 1/97 Dynamic Modeling and Differential Geometric Analysis of an Industrial Multi-stage Evaporation Unit, School of Chemical Engineering, Curtin University of Technology, Western Australia Kam, K M , Tade, M 0 , To, L C and Le Page, G P 1999 Implementation of MAPLEprocedures for simulating an industrial multi-stage evaporator MapleTech, 5(2&3), 27 Henson, M A and Seborg, D E 1997 Nonlinear Process Control, Prentice-Hall Inc ,New Jersey Press, W H , Teukolsky, S A , Vellerling, W T and Flannery, B P 1997 Numerical Recipes in C The Art of Scientific Computation, Cambridge University Press Kam, K M Tadt5, M 0 Rangaiah. G P and Tian. Y C 2001 Strategies for enhancing geometric nonlinear control of an industrial evaporator system Ind Eng Chem Res ,40,656 Rangaiah, G P , Saha, P and Tadt, M 0 2001 Nonlinear model predictive control of an industrial four-stage evaporator system Chem Eng J . accepted - in press Ogunnaike, B A and Ray, W H 1994 Process Dynamics, Modelling, and Control, Oxford University Press, New York Kam, K M , Implementation and Simulation of Nonlinear Control Systems for Mineral Processes, Ph D Thesis, Chapter 8, Curtin University of Technology, Western Australia (2000) . . Received 26 January 2001; Accepted after revision. 19 March 2001 Appendix A Equations for stages 1, 3 and 4 for both models MI and M2 are presented in this appendix. Further details of the derivation of these equations can be found in [ 141 Model MI Stage 1 121 K M.Kam. P. Saha, M . 0 Tade' and G.P Rangaiah where "' M4 = R(273.1+ T, - BPE,) P,=0.133ex d A- C+273 l+TI-BPEI Stape 3 (A.12) ( A 13) %= dt -[ 1 1 C O T ~(P'3 -hy43) (E3 - m ~ 3+ P V ~ Q( P -~ Q P ~ --PE3 w -q -&[ - COT3 C0RhO3[ E3( - 1) - QP2 P P 2 ( I)] (A.14) - I))]] where (A. 15) 122 Models of an Industrial Evaporator System hs3 = (5.76 / 2.38)m,4 (A.16) BPE3 = 102.69~3-128.82 (A.17) MP3 p v 3 = R(273 1 + T3 - BPE,) d P3=0 133ex A - " 1 (A.19) C + 273.1 + T3 - BPE3 CoRh03 = -102 69pv3 1 B (C+273.1+T3-BPE3)2 (273 1+T3-BP&) (A.21) (A.22) mv3 = 1 Stage 4 - * L[ dt = h4A4 E4[ (A.23) -1) -Q P ~ P P ~ [ -I ) ] (A.24) (A.25) where ' 3Q P ~~ 4 ~ 4 7 '+4 ms44s4 E4 = Q P ~~ 3 ~ 3 7 Av4 K M. Kam, P. Saha, M . 0 Tad6 and C P Rangaiah BPE4 = 1 0 2 . 6 9 ~ 4- 128.82 (A.27) (A.28) P4 = O 133ex CoRh04 d A- (A.29) C + 2 7 3 l+Tq-BPE4 = - 102 69pv4 B 1 (A.3 1) (C+273.1+T4-BPE4)2 (273 1+T4 -BpE4) Model M2 Stage 1 where (A.36) 124 Models of an Industrial Evaporator System (A 37) (A.38) MPI p v l = R(273 1 + T, - BPE,) COT,= (Vl - Alh, PVl 0.75M R(273 1 + T, - BPE1)- (273.1 + , '7 - BPEl) Stape 3 -=-[ dh3 dt 1 A3 QPZ - Q P ~ -P W dt = -!--[ h3A3 E3( - I)- Qpz P P ~ ( 5 = L( E3 - k v 3+ py3( Qf.2 dt CoT3 -Qp3 -I)] -Pw where (A.45) kv3 (A.46) = E3 ms3 = (5.76 l2.38)k,, (A.47) MP3 p v 3 = R(273.1+ T3 - BPE3) (A.48) P3 = 4.09T3 - 410.38 COT, = (V3 - A3h3 (A.49) 4.09M Pv3 R(273.1+ T3 - BPE3)- (273.1 + T3 - BPE3) 125 K M. Kam, P. Saha. M.0.Tad6 and G.P Rangaiah Stage 4 -[ dh4 - 1 -- dt A4 -"I - Q P ~ Q P ~ (AS 1) PW * 1[ (2 dt = h4A4 - 1 ) - Qp3 pp3 E4 (2 - (A.52) I)] (A.53) where (A.54) (A 5 5 ) pV4 = MP4 R(273.1+ T4 - B P E 4 ) P4 = 4.09q - 434.95 (A 57) 4 09M Pv4 R(273 1 + T 4 - B P E 4 ) - ( 2 7 3 1 + T 4 - B P E 4 ) Appendix B Linear models for M1 and M2 models were obtained by performing local linearization about the steady-state operating conditions given in Table 1. They are expressed in state space form as: QP~,ks4IT and d = [QJ, pfi TAT.The matrices A, Y and r are the Jacobian matrices for state, input and disturbance variables respectively, and were found by local linearization of the nonlinear evaporator models where x = [hi, hz, h3, h4, p4, p i , pz, p3, 7'1, T2,7'3, 126 7'4IT, u = [QPL Qp2, Qp3, Models of an Industrial Evaporator System It can be shown that the Eigen values of the linearized models LM 1and LM2 are. - 8.1757 - 2.4121 k 0.58431' %hi, -1.7389 - 1.3465 - 0.5397 = 0.23 17 - 0.1133 - 0 01 12 f 0.0042i - 0.0025 - 2.5283 k 0.33731- - 0.9957 k 0.66861' 0.1033 -I 0.0184i - 0 0169 k 0 0652i 0 1040 x lo-* 0.4095 x 0.2297 x - 0.3782 x lo-' 0 0015 Some of the Eigen values of both MI and M2 models are positive which imply that the evaporator models are open loop unstable There are also noticeable differences between the two sets of Eigen values indicating dissimilarity between the evaporator models M1 and M2. 127

1/--страниц