Framework for joint data reconciliation and parameter estimation

FRAMEWORK FOR JOINT DATA RECONCILIATION AND PARAMETER ESTIMATION JOE YEN YEN (B.Eng.(Hons.), NUS) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2004 ACKNOWLEDGMENTS I would like to express my gratitude to my supervisors Dr Arthur Tay and A/Professor Ho Weng Khuen of the NUS ECE Department and Professor Ching Chi Bun of the Institute of Chemical and Engineering Sciences (ICES), for their advice and guidance, and for having confidence in me in carrying out this research The support of ICES in financing the research and providing the research environment is greatly acknowledged The guidance and assistance from Dr Liu Jun of ICES is also appreciated Most of the research work is done in the Process Systems Engineering (PSE) Group of the Department of Chemical Engineering in University of Sydney The guidance of Professor Jose Romagnoli and Dr David Wang is valuable in directing and improving the quality of this work I would also like to thank them for accommodating me with such hospitality that my stay in the group is not only fruitful, but also enjoyable Thanks also go to the other members of the PSE for sharing their research knowledge and experience and for making me feel part of the group My friend Zhao Sumin has seen to my well being in Sydney, and most importantly, has been a like-minded confidante from whom I obtain inspiration in carrying out my research work I would like to dedicate this thesis to her, as with her persistent encouragements, I feel that the effort in completing this thesis is partly hers i TABLE OF CONTENTS ACKNOWLEDGMENTS I TABLE OF CONTENTS II SUMMARY IV LIST OF TABLES VI LIST OF FIGURES VII CHAPTER 1: INTRODUCTION 1.1 MOTIVATION 1.2 CONTRIBUTION .3 1.3 THESIS ORGANIZATION CHAPTER 2: THEORY & LITERATURE REVIEW 2.2 DATA RECONCILIATION (DR) .6 2.3 JOINT DATA RECONCILIATION – PARAMETER ESTIMATION (DRPE) 11 2.4 ROBUST ESTIMATION 15 2.5 PARTIALLY ADAPTIVE ESTIMATION 22 2.6 CONCLUSION 24 CHAPTER 3: JOINT DRPE BASED ON THE GENERALIZED T DISTRIBUTION .26 3.1 INTRODUCTION 26 3.2 THE GENERALIZED T (GT) DISTRIBUTION 26 3.3 ROBUSTNESS OF THE GT DISTRIBUTION .29 3.4 PARTIALLY ADAPTIVE GT-BASED ESTIMATOR 30 3.5 THE GENERAL ALGORITHM 32 3.6 CONCLUSION 35 CHAPTER 4: CASE STUDY .36 4.1 INTRODUCTION 36 4.2 THE GENERAL-PURPOSE CHEMICAL PLANT .36 4.3 VARIABLES AND MEASUREMENTS 40 4.4 SYSTEM DECOMPOSITION: VARIABLE CLASSIFICATION .43 4.4.1 Derivation of Reduced Equations 43 4.4.2 Classification of Measurements 45 4.5 DATA GENERATION 47 4.5.1 Simulation Data 47 4.5.2 Real Data 48 4.6 METHODS COMPARED 48 4.7 PERFORMANCE MEASURES 50 4.8 CONCLUSION 51 CHAPTER 5: RESULTS & DISCUSSION 53 5.1 INTRODUCTION 53 5.2 PARTIAL ADAPTIVENESS & EFFICIENCY .53 5.3 EFFECT OF OUTLIERS ON EFFICIENCY 64 5.4 EFFECT OF DATA SIZE ON EFFICIENCY .67 5.5 ESTIMATION OF ADAPTIVE ESTIMATOR PARAMETERS: PRELIMINARY ESTIMATORS AND ITERATIONS 71 5.6 REAL DATA APPLICATION 76 5.7 CONCLUSION 79 ii CHAPTER 6: CONCLUSION 80 6.1 FINDINGS 80 6.2 FUTURE WORKS 82 REFERENCES 83 AUTHOR’S PUBLICATIONS 85 APPENDIX A 86 APPENDIX B .95 iii SUMMARY Objective knowledge about a process is essential for process monitoring, optimization, identification and other general management planning Since measurement of process states always contain some type of errors, it is necessary to correct these measurement data to obtain more accurate information about the process Data reconciliation is such an error correction procedure that utilizes estimation theory and the conservation laws within the process to improve the accuracy of the measurement data and estimates the values of unmeasured variables such that reliable and complete information about the process is obtained Conventional data reconciliation, and other procedures that involve estimation, have relied on the assumption that observation errors are normally distributed The inevitable presence of gross errors and outliers violates this assumption In addition, the actual underlying distribution is not known exactly and may not be normal Various robust approaches such as the M-estimators have been proposed, but most assumed, in a priori, yet other forms of distribution, although with thicker tails than that of normal distribution in order to suppress gross errors / outliers To address the issue of the suitability of the actual distribution to the assumed one, posteriori estimation of the actual distribution, based on non-parametric methods such as kernel, wavelet and elliptical basis function, is then proposed However, these fully adaptive methods are complex and computationally demanding An alternative is to strike a balance between the simplicity of the parametric approach and the flexibility of the non-parametric approach, i.e by adopting a iv generalized objective function that covers a wide variety of distributions The parameters of the generalized distribution can be estimated posteriori to ensure its suitability to the data This thesis proposes the use of a generalized distribution, namely the Generalized T (GT) distribution in the joint estimation of process states and model parameters The desirable properties of the GT-based estimator are its robustness, simplicity, flexibility and efficiency for the wide range of commonly encountered distributions (including Box-Tiao and t-distributions) that belong to the GT distribution family To achieve estimation efficiency, the parameters of the GT distribution are adapted from the data through preliminary estimation The strategy is applied to data from both the virtual version and a trial run of a chemical engineering pilot plant The results confirm the robustness and efficiency of the estimator v LIST OF TABLES Table 5.1 MSE of Measurements 57 Table 5.2 Estimated parameters of the partially adaptive estimators used to generate Figure 5.5 63 Table 5.3 MSE of Measurements with Outliers 64 Table 5.4 Reconciled Data and Estimated Parameter Values Using Different DRPE Methods 78 vi LIST OF FIGURES Figure 2.1 Plots of Influence Function for Weighted Least Square Estimator (dashed line) and the Robust Estimator based on Bivariate Normal Distribution (solid line)21 Figure 2.2 Partially Adaptive Estimation Scheme 24 Figure 3.1 Plot of GT density functions for various settings of distribution parameters p and q 27 Figure 3.2 GT Distribution Family Tree, Depicting the Relationships among Some Special Cases of the GT Distribution 28 Figure 3.3 Plots of Influence Function for GT-based Estimator with different parameter settings 30 Figure 3.4 General Algorithm for Joint DRPE using partially adaptive GT-based estimator 33 Figure 4.1 Flow Diagram of the General Purpose Plant for Application Case Study 37 Figure 4.2 Simulink Model of the General Purpose Plant in Figure 4.1 38 Figure 4.3 Configuration of the General Purpose Plant for Trial Run 39 Figure 4.4 Reactor Configuration Details and Measurements 39 Figure 4.5 Reactor Configuration Details and Measurements 40 Figure 5.1 MSE Comparison of GT-based with Weighted Least Squares and Contaminated Normal Estimators 57 Figure 5.2 Percentage of Relative MSE 58 Figure 5.3 Comparison of the Accuracy of Estimates for the overall heat transfer coefficient of Reactor cooling coil 60 Figure 5.4 Comparison of the Accuracy of Estimates for the overall heat transfer coefficient of Reactor cooling coil 61 Figure 5.5 Adaptation to Data: Fitting the Relative Frequency of Residuals with GT, Contaminated Normal, and Normal distributions 62 Figure 5.6 MSE of Variable Estimates for Data with Outliers 66 Figure 5.7 Comparison of MSE with and without outliers for GT and Contaminated Normal Estimators 66 Figure 5.8 MSE Results of WLS, Contaminated Normal and GT-based estoimators for Different Data Sizes 68 Figure 5.9 Improvement in MSE Efficiency when Data Size is increased 70 Figure 5.10 Iterative Joint DRPE with Preliminary Estimation 72 Figure 5.11 Final MSE Comparison for GT-based DRPE method Using GT, Median and WLS as preliminary estimators 74 Figure 5.12 MSE throughout iterations 75 Figure 5.13 Scaled Histogram of Data and Density Plots of GT, Contaminated Normal and Normal Distributions 77 vii CHAPTER 1: INTRODUCTION 1.1 Motivation The continuously increasing demand for higher product quality and stricter compliance to environmental and safety regulations requires the performance of a process to be continuously improved through process modifications (Romagnoli and Sanchez, 2000) Decision making associated with these process modifications requires accurate and objective knowledge of the process state This knowledge of process state is obtained from interpretation of data generated by the process control systems The modern-day Distributed Control System (DCS) is capable of high-frequency sampling, resulting in vast amount of data to be interpreted, be it for the purpose of process monitoring, optimization or other general management planning Since measurement data always contain some type of error, it is necessary to correct their values in order to obtain accurate information about the process Data reconciliation (DR) is such an error-correction procedure that improves the accuracy of measurement data, and estimates the values of unmeasured variables, such that reliable and complete information about the process is obtained It makes use of conservation equations and other system/model equations to correct the measurement data, i.e by adjusting the measurements such that the adjusted data are consistent with respect to the equations The conventional data reconciliation approach is the least squares minimization, whereby the (square of) adjustments to the measurements are minimized, while at the same time subjecting the measurements to satisfy the system/model equations The least squares method is simple and reasonably efficient; in fact, it is the best linear unbiased, the most efficient in terms of minimum variance, and also the maximum likelihood estimator when the measurement errors are distributed according to the Normal (Gaussian) distribution However, measurement error is made up of random and gross error Gross errors are often present in the measurements and these large deviations are not accounted for in the normal distribution In this case, the least squares method can produce heavily biased estimates Attempts to deal with gross errors can be grouped in two classes The first includes methods that still keep the least squares approach, but incorporate additional statistical tests to the residuals of either the constraints (which can be done prereconciliation) or the measurements (which must be done post-reconciliation) The drawback of these approaches is that there is a need for separate gross-error processing step Also, most importantly, normality is still assumed for the data, while the data may not be best represented by the Normal distribution Furthermore, the statistical tests are theoretically valid only for linear system/model equations, which is a rather constricting restriction in chemical processes where most relationships are nonlinear The second class of gross-error handling approaches comprises the more recent approaches to suppress gross error, i.e by making use of the so-called robust estimators These estimators can suppress gross error while performing reconciliation, so there is no need for a separate procedure to remove the gross errors Most of these approaches are physical model of the process or system is available, but also with other empirical and approximation models The usual precaution that the model inaccuracy is negligible compared to the measurement inaccuracies holds, of course; but this also holds for other estimators and is not a restriction specific to the GT-based estimator 6.2 Future Works In this work the GT-based DRPE strategy has been applied successfully to various data profiles, both simulation and real data These various applications have covered quite general situations, which make the GT-based strategy a viable choice to deal with a wide range of situations However, recalling that the GT is a symmetric distribution, its limitation may be that in the case when the error is actually not distributed symmetrically, the adapted GT may not be efficient It is possible to further generalize the GT distribution to include non-symmetric distributions Another useful extension would be to explore the application of the GT-based strategy in the dynamic operation of the system This poses challenges in the efficient optimization of the differential algebraic model, which is not encountered in the steady-state case 82 REFERENCES Albuquerque, J S., Biegler, L.T (1996) Data Reconciliation and Gross-Error Detection for Dynamic Systems AIChE J., Vol 42, No 10, pp 2841-2856 Arora, N., Biegler, L.T (2001) Redescending Estimators for Data Reconciliation and Parameter Estimation Comp Chem Eng., Vol 25, pp 1585-1599 Bickel, P.J (1982) On adaptive estimation, Annals of Statistics 10, 647-671 Butler, R.J., McDonald, J.B., Nelson, R.D., White, S.B (1990) Robust and Partially Adaptive Estimation of Regression Models Rev Econ Stat., Vol 72, No 2, pp.321327 Crowe, C.M (1989) Observability and redundancy of process data for steady state reconciliation Chemical Engineering Science 44, 2909-2917 Crowe, C.M (1986) Reconciliation of process flow rates by matrix projection Part II: The non-linear case AIChE.J 32, 616-623 Gill, P., Murray, W.A., Saunders, M.A., and Wright, M.H (1986) User Guide for NPSOL (Version 4.0); a FORTRAN program for Nonlinear Programming, Technical Report SOL 86-2 Stanford University, Department of Operation Research, Stanford, CA Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J., Stahel, W.A (1986) Robust Statistics: The Approach Based on Influence Function Wiley, New York Huber, P.J (1981) Robust Statistics Wiley, New York Huber, P.J., (1964) Robust Estimation of a location parameter Annals of Mathematical Statistics, 64, 73-101 Joris, P., and Kalitventzeff, B (1987) Process measurements analysis and validation Proc CEF ’87: Use Comput Chem Eng., Italy, pp.41-46 Kim, I.W., Liebman, M J., and Edgar, T.F (1990) Robust error-in-variable estimation using non-linear programming techniques AIChE J., Vol 36, 985-993 McDonald, J.B., Newey, W.K (1988) Partially Adaptive Estimation of Regression Models via the Generalized T Distribution Econometric Theory, Vol 4, pp.428-457 Narasimhan , S.; Mah, R.S.H Generalized Likelihood Ratio Method for Gross Errors Identification AIChE J 1987, 33, 1514 83 Potscher, B.M and Prucha, I.R (1986) A Class of Partially Adaptive One-Step MEstimators for the Non-linear Regression Model with Dependent Observations Journal of Econometrics, 32, 219-251 Reilly, P.M., and Patino-Leal, H (1981) A Bayesian study of the error-in-variable models Technometrics, Vol 23, 221-231 Romagnoli, J and Sanchez, M (2000) Data Processing and Reconciliation for Chemical Process Operations Academic Press Rogmanoli, J., and Stephanopoulos, G (1980) On the rectification of measurement errors for complex chemical plants Chemical Engineering Science, 35, 1067-1081 Sanchez, M., Bandoni, A., and Romagnoli, J (1992) PLADAT – A package for process variable classification and plant data reconciliation Computers and Chemical Engineering, S16, 499-506 Serth, R.; Heenan, W Gross Error and Data Reconciliation in Steam Measuring Systems AIChE J 1986 32, 733 Stein, C (1956) Efficient nonparametric testing and estimation In: Proceedings of the third Berkeley symposium on mathematical statistics and probability, Vol I (University of California Press, Berkeley, CA) Swartz, C.L.E (1989) Data reconciliation for generalized flowsheet applications 197th National Meeting, American Chemical Society Dallas, TX Tjoa, I., and Biegler, L (1991) Simultaneous strategies for data reconciliation and gross error detection of nonlinear systems Computer and Chemical Engineering, 15, 679 Valko, P., Vadja, S (1987), An extended Marquardt-type Procedure for Fitting Error-inVariable Models Comp Chem Eng., Vol 11, pp 37-43 Wang, D., Romagnoli, J.A (2003) A Framework for Robust Data Reconciliation Based on a Generalized Objective Function Ind.Eng.Chem.Res.,Vol.42, No.13, pp 30753084 84 AUTHOR’S PUBLICATIONS • Joe, Y.Y., Wang, D., Romagnoli, J and Tay, A (2004) Robust and Efficient Joint Data Reconciliation and Parameter Estimation Using a Generalized Objective Function Proceedings of the 7th Dynamics and Control of Process Systems (DYCOPS7) Boston, USA Accepted • Joe, Y.Y., Wang, D., Tay, A., Ho, W.K., Ching, C.B., and Romagnoli, J (2004) A Robust Strategy for Joint Data Reconciliation and Parameter Estimation Proceedings of the 14th European Symposium on Computer-Aided Process Engineering Elsevier, Lisbon, Portugal • Joe, Y.Y., Xu, H., Dong, Z.Y., Ng, H.H., and Tay, A (2004) Searching Oligo Sets of Human Chromosome 12 using Evolutionary Strategies, International Journal of Systems Sciences, Accepted • Joe, Y.Y., Xu, H., Dong, Z.Y., Ng, H.H., and Tay, A (2003) Searching Oligo Sets of Human Chromosome 12 using Evolutionary Strategies, Proceedings of the 2003 Congress on Evolutionary Computation, IEEE Press, Canberra, Australia 85 APPENDIX A Description of the Matlab/ Simulink Model of the General-Purpose Pilot Plant GENERAL DESCRIPTION The plant is a pilot-scale setting containing two CSTRs, a mixer, a feedtank and a number of heat exchangers Material feed from the feed tank is heated before being fed to the first reactor and the mixer The effluent from first reactor is then mixed with the material feed in the mixer, and then fed to the second reactor The effluent from the second reactor is, in turn, fed back to the feed tank and the cycle continues RUNNING THE MODEL The plant model is in twocstr3.mdl (excluding pumps) UNITS NOTE: Operating values (initial conditions, parameters) used in the current model right now are still the ones taken from the references, not the values from the pilot plant REACTOR Tc Cooling water OUT Fin, Tin, Cain V, T, Ca Ps F, T, Ca Product OUT Reaction inside the reactor: A → B 1.1 Model I/O and Parameters Matlab S-function: s_reactor1.m 86 Fin,Tin,Cain F,T,Ca Tcin V Fsin Tc Fcin Ts F REACTOR1 1.1.1 Input: Fin : volume flowrate of input stream (vol/time) Tin : temperature of input stream Cain : concentration of component A in the input stream (mole/vol) Fcin : volume flowrate of incoming cooling stream Tcin : temperature of incoming cooling stream Fsin : volume flowrate of incoming heating stream F : volume flowrate of output stream (as controlled by output flow controller) 1.1.2 Output: F : volume flowrate of output stream T : temperature of output stream = (assumed) temperature inside reactor vessel Ca : concentration of A in the output stream = (assumed) inside reactor vessel V : volume of content of reaction vessel Tc: Temperature of outgoing cooling stream Ts: Temperature of steam jacket Fs: Flowrate of input steam to steam jacket 1.1.3 Parameters [Luyben, 1989]: NOTE: • The parameters are currently coded at the S-function; so to simulate two reactors with different parameters, two of this S-function are needed, each with different set of parameters Symbol Description Nominal Values Used in Simulation Mass density of stream 50 lbm/ft3 rho, ρ Specific heat of stream 0.75 Btu/lbm R Cp, C p alpha, α E, E R, R Temperature constant for Arrhenius expression Activation energy for reaction Molar gas constant 7.08E10 /h = 1.9667e+007 /s 30,000 Btu/lb.mol 1.99 Btu/lb.mol R 87 lambda, λ Uc, U c Ac, Ac Reaction energy heat transfer coefficient of reactor cooling coil area of reactor cooling coil -30,000 Btu/lb.mol 150 btu/h ft2 R = 0.0417 btu/s ft2 R 250 ft2 rhoc, ρ c mass density of cooling stream 62.5 lbm/ft3 Cpc, C pc specific heat of cooling stream Btu/lbm R Vc, Vc volume of reactor cooling coil 3.85 ft3 hos, hos Aos, Aos heat transfer coefficient of reactor heating jacket area of reactor heating jacket Ms, M s molar mass of heating stream Avp, Avp Constant for steam jacket pressure -8744.4 R Bvp, Bvp Constant for steam jacket pressure 15.70 Hs_hc, H s − hc (Vapor_enthalpycondensation_enthalpy) 1000 btu/h ft2 R = 0.2778 btu/s ft2 R 56.5 ft2 18 lbm/lb.mol 939 Btu/lbm 1.2 Model Equations [Luyben, 1989] 1.2.1 Mass Balance dV = Fin − F dt 1.2.2 Component Balance d (VC a ) E = Fin C ain − FC a − α exp(− )VC a dt RT Cb = ( ρ − M a C a ) / M b 1.2.3 Energy Balance a Reaction Vessel λα dVT E = FinTin − FT − exp(− )VC a − Qc − Q j ρC p dt RT Qc = U c Ac (T − Tc ) Q j = − hos Aos (Ts − T ) b Cooling Coil dTc Fc Qc = (Tcin − Tc ) + ρ c C pcVc dt Vc Qc = U c Ac (T − Tc ) 88 c Steam Jacket dρ V j s = ws − wc dt MPj ρs = RTs Pj = exp( Avp Ts Total Continuity Perfect Gas Law + Bvp ) Simple vapor-pressure equation ws = C vs X s Pjin − Pj Q j = −hos Aos (Ts − T ) wc = − Qj H s − hc Energy equation, ignoring internal energy change & assuming steady-state, i.e ws = wc HEAT EXCHANGER Matlab S-function: s_heatex_btu.m In1 Out1 In2 Out2 HEAT_EX1 In1: [Fin1, Tin1, Cain1] T In2: [Fin2, Tin2, Cain2] T Out1: [F1, T1, Ca1] T Out2: [F2, T2, Ca2] T 2.1 Model I/O and Parameters 2.1.1 Input: Fin1/2: Volume flowrate of input stream 1/2 Tin1/2: Temperature of input stream 1/2 Cain1/2: Concentration of component A in input stream 1/2 2.1.2 Output: F1/2: Volume flowrate of output stream 1/2 T1/2: Temperature of output stream 1/2 Ca1/2: Concentration of component A in output stream 1/2 2.1.3 Parameters: [Luyben, 1989; Malleswararao, 1992; Holman, 1972] Symbol Description Nominal Values Used in Simulation 89 ρ1 ρ2 V1 V2 Cp1 Cp U A Mass density of stream Mass density of stream Volume of stream in the heat exchanger Volume of stream in the heat exchanger Specific heat of stream Specific heat of stream Heat transfer coefficient Area of heat exchange 50 lbm/ft3 50 lbm/ft3 3.85 ft3 3.85 ft3 0.75 btu/lbm R 0.75 btu/lbm R 1000 btu/h R 20 ft2 2.2 Model Equations 2.2.1 Mass Balance F1 = Fin1 F2 = Fin 2.2.2 Component Balance C a1 = C ain1 C a = C ain 2.2.3 Energy Balance [Holman, 1972; ASHRAE, 1993] dT1 [F ρ (T −T ) − Qt /C p1] = dt ρ1V1 1 in1 dT2 = [ F ρ (T −T ) + Qt /C p2] dt ρ2V2 2 in2 Qt =εQmax where: ⎧ 1− exp(− N (1− C )) ⎪ ;C ≠1 ε = ⎪⎨1− C exp(− N (1− C )) N ;C =1 ⎪ ⎪ N +1 ⎩ N =UAh /Cmin C = Cmin /Cmax Cmin = min{F1ρ1Cp1,F2 ρ2Cp2} Cmax = max{F1ρ1Cp1,F2 ρ2Cp2} Qmax = Cmin (Th −Tc ) Th = max{Tin1,Tin2 } Tc = min{Tin1,Tin2} 90 MIXER / FEED TANK Matlab S-function: s_mixer_btu_cl.m F1, T1, Ca1 F, T, Ca F2, T2, Ca2 V F MIXER 3.1 Model I/O and Parameters 3.1.1 Input: Fin1/2: Volume flowrate of input stream 1/2 Tin1/2: Temperature of input stream 1/2 Cain1/2: Concentration of component A in input stream ½ F : Volume flowrate of output stream (as controlled by output flow controller) 3.1.2 Output F: Volume flowrate of output stream T: Temperature of output stream Ca: Concentration of component A in output stream V: Volume of tank 3.1.3 Parameter None 3.2 Model Equations 3.2.1 Mass Balance dV = Fin1 + Fin − F dt 3.2.2 Component Balance dVCa = F1Ca1 + F2 Ca2 − FCa dt 3.2.3 Energy Balance 91 dVT = F T + F T − FT in1 in2 dt SPLITTER Matlab S-function: not implemented in s-function Out1 In Out2 splitter In1: [Fin, Tin, Cain, Cbin] T Out1: [F1, T1, Ca1, Cb1] T Out2: [F2, T2, Ca2, Cb2] T Parameters: [eta] T 4.1 Model I/O and Parameters 4.1.1 Input: Fin: Volume flowrate of input stream Tin: Temperature of input stream Cain: Concentration of component A in input stream Cbin: Concentration of component B in input stream 4.1.2 Output F1/2: Volume flowrate of output stream 1/2 T1/2: Temperature of output stream 1/2 Ca1/2: Concentration of component A in output stream ½ 4.1.3 Parameters Symbol eta, η Description Split fraction ratio (F1/Fin) Nominal Values Used in Simulation 0.4 4.2 Model Equations 4.2.1 Mass Balance F1 = ηFin F2 = Fin − F1 4.2.2 Component Balance Ca1= Ca2 = Cain 92 4.2.3 Energy Balance T1 = T2 = Tin MIXING JUNCTION Matlab S-function: threeway.m In1 Out In2 junction In1: [Fin1, Tin1, Cain1] T In2: [Fin2, Tin2, Cain2] T Out1: [F, T, Ca] T 5.1 Model I/O 5.1.1 Input: Fin1/2: Volume flowrate of input stream 1/2 Tin1/2: Temperature of input stream 1/2 Cain1/2: Concentration of component A in input stream 1/2 5.1.2 Output F: Volume flowrate of output stream T: Temperature of output stream Ca: Concentration of component A in output stream 5.2 Model Equations 5.2.1 Mass Balance F = Fin1 + Fin 5.2.2 Component Balance F1Ca1 + F2 Ca2 − FCa = 5.2.3 Energy Balance F1Tin1 + F2Tin − FT = REFERENCES ASHRAE (1993) Toolkit for HVAC System Energy Calculations, 1993 De Nevers, Noel (1991) Fluid Mechanics for Chemical Engineers McGraw-Hill Holman, J P (1972) Heat Transfer McGraw-Hill 93 Luyben, William, L (1989) Process modeling, simulation, and control for chemical engineers McGraw-Hill Malleswararao, Y S N, Chidambaram, M (1992) Non-linear controllers for a heatexchanger In J Proc Cont, Vol 2, No 1, p17-21 94 APPENDIX B Steady-State Equations of the General Purpose Pilot Plant Reactor Mass Balance: F2 − F5 = U c1 Ac1 (T − T ) = ρC p rx1 c1 U A Cooling Coil Energy Balance: Fc1 (Tc ,in − Tc1 ) + c1 c1 (Trx1 − Tc1 ) = ρ c C pc Energy Balance: F2T2 − F5T5 − Mixer Mass Balance: F5 + F7 − F9 = Energy Balance: F5T5 + F7T7 − F9T9 = Reactor Mass Balance: F9 − F12 = U c Ac (T − T ) = ρC p rx c U A Cooling Coil Energy Balance: Fc (Tc ,in − Tc ) + c c (Trx − Tc ) = ρ c C pc Energy Balance: F9T9 − F12T12 − Feed Tank Mass Balance: F12 − F13 = Energy Balance: F12T12 − F13T13 = Heat Exchanger Energy Balance: F ρ (T14 − T7 ) − Qt / C p = F ρ (T6 − T15 ) + Qt / C =0 hx phx where Qt is a function of T7, T14, T6 and T15 Heat Exchanger Energy Balance: F2 ρ (T16 − T2 ) − Qt / C p = F1 ρ hx (T1 − T17 ) + Qt / C phx =0 95 where Qt is a function of T2, T16, T1 and T17 Heat Exchanger Energy Balance: Fc (Tc ,in − Tc ) + F13 (T13 − T13 a ) − U c Ac (T − T ) = ρ c C pc 13a c U c Ac (T − T ) = ρ c C pc 13a c T 96 ... estimator for a joint data reconciliation – parameter estimation strategy is formulated The algorithm consists of three main steps: the preliminary estimation and the estimation of the GT parameters... steps corresponding to data reconciliation and parameter estimation are merged into a single joint data reconciliation – parameter estimation (DRPE) step 11 The problem formulation, taking the weighted... simulation cases A comprehensive literature review of the data reconciliation and joint data reconciliation – parameter estimation, and the technical aspects associated with them is conducted