1. CHE412 Process Dynamics and Control BSc (Engg) Chemical Engineering (7 th Semester) Week 2/3 Mathematical Modeling Luyben (1996) Chapter 2-3 Stephanopoulos (1984) Chapter 4, 5 Seborg et al (2006) Chapter 2 Dr Waheed Afzal Associate Professor of Chemical Engineering Institute of Chemical Engineering and Technology University of the Punjab, Lahore wa.icet@pu.edu.pk 1

2. Test yourself (and Define): Dynamics (of openloop and closedloop) systems Manipulated Variables Controlled/ Uncontrolled Variables Load/Disturbances Feedback, Feedforward and Inferential controls Error Offset (steady-state value of error) Set-point 2 Hint: Consult recommended books (and google!) Luyben (1996), Coughanower and LeBlanc (2008) Stability Block diagram Transducer Final control element Mathematical model Input-out model, transfer function Deterministic and stochastic models Optimization Types of Feedback Controllers (P, PI, PID)

3. 3 Types of Feedback Controllers (Stephanopoulos, 1984)

4. Mathematical representation of a process (chemical or physical) intended to promote qualitative and quantitative understanding Set of equations Steady state, unsteady state (transient) behavior Model should be in good agreement with experiments 4 Mathematical Modeling Experimental Setup Set of Equations (process model) InputsOutputs Compare

5. 1.Determine objectives, end-use, required details and accuracy 2.Draw schematic diagram and label all variables, parameters 3.Develop basis and list all assumptions; simplicity Vs reality 4.If spatial variables are important (partial or ordinary DEs) 5.Write conservation equations, introduce auxiliary equations 6.Never forget dimensional analysis while developing equations 7.Perform degree of freedom analysis to ensure solution 8.Simplify model by re-arranging equations 9.Classify variables (disturbances, controlled and manipulated variables, etc.) 5 Systematic Approach for Modelling (Seborg et al 2004)

6. To understand the transient behavior, how inputs influence outputs, effects of recycles, bottlenecks To train the operating personnel (what will happen if…, ‘emergency situations’, no/smaller than required reflux in distillation column, pump is not providing feed, etc.) Selection of control pairs (controlled v. / manipulated v.) and control configurations (process-based models) To troubleshoot Optimizing process conditions (most profitable scenarios) 6 Need of a Mathematical Model

7. Theoretical Models based on principal of conservation- mass, energy, momentum and auxiliary relationships, ρ, enthalpy, c p, phase equilibria, Arrhenius equation, etc) Empirical model based on large quantity of experimental data) Semi-empirical model (combination of theoretical and empirical models) Any available combination of theoretical principles and empirical correlations 7 Classification of Process Models based on how they are developed

8.  Theoretical Models  Physical insight into the process  Applicable over a wide range of conditions  Time consuming (actual models consist of large number of equations)  Availability of model parameters e.g. reaction rate coefficient, over-all heart transfer coefficient, etc.  Empirical model  Easier to develop but needs experimental data  Applicable to narrow range of conditions 8 Advantages of Different Models

9.  State variables describe natural state of a process  Fundamental quantities (mass, energy, momentum) are readily measurable in a process are described by measurable variables (T, P, x, F, V)  State equations are derived from conservation principle (relates state variables with other variables) (Rate of accumulation) = (rate of input) – (rate of output) + (rate of generation) - (rate of consumption) 9 State Variables and State Equations

10.  Basis Flow rates are volumetric Compositions are molar A → B, exothermic, first order  Assumptions Perfect mixing ρ, c P are constant Perfect insulation Coolant is perfectly mixed No thermal resistance of jacket 10 Modeling Examples Jacketed CSTR Coolant F i, C Ai, T i F, C A, T

11.  Overall Mass Balance (Rate of accumulation) = (rate of input) – (rate of output)  Component Mass Balance (Rate of accumulation of A) = (rate of input of A) – (rate of output of A) + (rate of generation of A) – (rate of consumption of A) 11 Modeling of a Jacketed CSTR (Contd.) Coolant Fc i,Tc i VCATVCAT F i, C Ai, T i F, C A, T Coolant Fc o,Tc o  Energy Balance (Rate of energy accumulation) = (rate of energy input) – (rate of energy output) - (rate of energy removal by coolant) + (rate of energy added by the exothermic reaction)

12. 12 Modeling of a Jacketed CSTR (Contd.) Coolant Fc i,Tc i VCATVCAT F i, C Ai, T i F, C A, T Coolant Fc o,Tc o Input variables: C Ai, F i, T i, Q, (F) Output variables: V, C A, T

13. N f = N v - N E Case (1): N f = 0 i.e. N v = N E (exactly specified system) We can solve the model without difficulty Case (2): f > 0 i.e. N v > N E (under specified system), infinite number of solutions because N f process variables can be fixed arbitrarily. either specify variables (by measuring disturbances) or add controller equation/s Case (3): N f < 0 i.e. N v < N E (over specified system) set of equations has no solution remove N f equation/s We must achieve N f = 0 in order to simulate (solve) the model 13 Degrees of Freedom (N f ) Analysis

14. Basis/ Assumptions  Perfectly mixed, Perfectly insulated  ρ, c P are constant 14 Stirred Tank Heater: Modeling and Degree of Freedom Analysis Steam Fst A  Degree of Freedom Analysis  Independent Equations: 2 Variables: 6 (h, F i, F, T i, T, Q)  N f = 6-2 (= 4) Underspecified

15. N f = 4  Specify load variables (or disturbance) Measure F i, T i (N f = 4 - 2 = 2)  Include controller equations (not studied yet); specify CV-MV pairs: 15 Stirred Tank Heater: Modeling and Degree of Freedom Analysis Steam Fst A CVMV hF TQ Can you draw these control loops?

16. FZFZ mBmB mDmD VBVB DxDDxD BxBBxB R x D Reboiler Condenser Reflux Drum (Stephanopoulos, 1984) 16 Basis/ Assumptions 1.Saturated feed 2.Perfect insulation of column 3.Trays are ideal 4.Vapor hold-up is negligible 5.Molar heats of vaporization of A and B are similar 6.Perfect mixing on each tray 7.Relative volatility ( α ) is constant 8.Liquid holdup follows Francis weir formulae 9.Condenser and Reboiler dynamics are neglected 10.Total 20 trays, feed at 10 2, 4, 5 → V 1 = V 2 = V 3 = … V N (not valid for high-pressure columns) Modeling an Ideal Binary Distillation Column = 20

17. 17 V 20 DxDDxD RxDRxD Reflux Drum N = 20 V 20 R V 19 L 20 Top Tray mDmD Modeling Distillation Column

18. 18 N th Stage mNmN vNvN L N+1 LNLN v N-1 Modeling Distillation Column Feed Stage (10 th ) mNmN v 10 L 11 L 10 v9v9 FZFZ

19. 19 VBVB L1L1 V1V1 L2L2 1 st Stage Modeling Distillation Column VBVB L1L1 Column Base mBmB B VBVB

20. 20 Modeling Distillation Column Equilibrium relationships (to determine y)

21. 21 Modeling Distillation Column Hydraulic relationships (to determine L)

22. 22 Modeling Distillation Column Degree of Freedom Analysis Total number of independent equations:  Equilibrium relationships (y 1, y 2, …y N, y B ) → N+1 (21)  Hydraulic relationships (L 1, L 2, …L N ) →N (20) (does not work for liquid flow rates D and B)  Total mass balances (1 for each tray, reflux drum and column base) →N+2 (22)  Total component mass balances (1 for each tray, reflux drum and column base) →N+2 (22)  Total Number of equations N E = 4N + 5 (85) 44 differential and 41 algebraic equations Note the size of model even for a ‘simple’ system with several simplifying assumptions!

23. 23 Modeling Distillation Column Degree of Freedom Analysis Total number of independent variables:  Liquid composition (x 1, x 2, …x N, x D, x B ) → N+2  Liquid holdup (m 1, m 2, …m N, m D, m B ) → N+2  Vapor composition (y 1, y 2, …y N, y B ) → N+1  Liquid flow rates (L 1, L 2, …L N ) → N  Additional variables →6 (Feed: F, Z; Reflux: D, R; Bottom: B, V B )  Total Number of independent variables N V = 4N + 11 Degree of Freedom = (4N + 11) – (4N + 5) = 6 System is underspecified

24. 24 Modeling Distillation Column Degree of Freedom Analysis (4N + 11) – (4N + 5) = 6  Specify disturbances: F, Z (N f = 6-2 = 4)  Include controller equations (Recall our discussion on types of feedback controllers )  General form, of P-Controller c(t) = c s + K c Є(t) Controlled Variable Manipulated Variable xDxD R xBxB VBVB mDmD D mBmB B R = K c (x s - x D ) + R s V B = K c (x Bs -x B ) + V Bs D = K c (m Ds -m D ) + D s B = K c (m Bs -m B ) + B s N f = 4 - 4 = 0 Controller Equation (Proportional Controller) Can you draw these four feedback control loops?

25. 25 Feedback Control on a Binary Distillation Column CVMVloop xDxD R1 xBxB VBVB 2 mDmD D3 mBmB B4 R (Stephanopoulos, 1984)

26. Week 2/3 Weekly Take-Home Assignment 1.Define all the terms on slide 2 with examples whenever possible. 2.Prepare short answers to ‘things to think about’ (Stephanopoulos, 1984) page 33-35 3.Prepare short answers to ‘things to think about’ (Stephanopoulos, 1984) page 78-79 4.Solve the following problems (Chapter 4 and 5 of Stephanopoulos, 1984): II.1 to II.14, II.22, II.23 (Compulsory) 26 Submit before Friday Curriculum and handouts are posted at: http://faculty.waheed-afzal1.pu.edu.pk/