Report

CHE412 Process Dynamics and Control BSc (Engg) Chemical Engineering (7th Semester) Week 3 (contd.) Mathematical Modeling (Contd.) Luyben (1996) Chapter 3 Stephanopoulos (1984) Chapter 5 Dr Waheed Afzal Associate Professor of Chemical Engineering Institute of Chemical Engineering and Technology University of the Punjab, Lahore [email protected] 1 Modeling CSTRs in Series constant holdup, isothermal Basis and Assumptions A → B (first order reaction) Compositions are molar and flow rates are volumetric Constant V, ρ, T Overall Mass Balance ρ = ρ0 − ρ1 = 0 i.e. at constant V, F3 =F2 =F1 =F0 ≡ F So overall mass balance is not required! F0 V1 K1 T1 Luyben (1996) F1 CA1 V2 K2 T2 F2 CA2 V3 K3 T3 F3 CA3 2 Modeling CSTRs in Series constant holdup, isothermal Component A mass balance on each tank (A is chosen arbitrarily) 1 = 0 − 1 − 11 1 2 = 1 − 2 − 22 2 3 = 2 − 3 − 33 3 kn depends upon temperature kn = k0 e-E/RTn where n = 1, 2, 3 Apply degree of freedom analysis! Parameters/ Constants (to be known): V1, V2, V3, k1, k2, k3 Specified variables (or forcing functions): F and CA0 (known but not constant) . Unknown variables are 3 (CA1, CA2, CA3) for 3 ODEs Simplify the above ODEs for constant V, T and putting τ = V/F 3 Modeling CSTRs in Series constant holdup, isothermal If throughput F, temperature T and holdup V are same in all tanks, then for τ = V/F (note its dimension is time) 1 1 1 + 1 + = 0 τ τ 2 1 1 + 2 + = 1 τ τ 3 1 1 + 3 + = 2 τ τ In this way, only forcing function (variable to be specified) is CA0. 4 Modeling CSTRs in Series Variable Holdups, nth order Mass Balances (Reactor 1) 1 = 0 − 1 (11 ) = 00 − 11 −11(1)n Mass Balances (Reactor 2) 2 = 1 − 2 (22) = 11 − 22 −22(2)n Mass Balances (Reactor 1) 3 = 2 − 3 (33) Changes from previous case: V of reactors (and F) varies with time, reaction is nth order Parameters to be known: k1, k2, k3, n Disturbances to be specified: CA0, F0 Unknown variables: CA1, CA2, CA3, V1, V2, V3, F1, F2, F3 CV Include MV Controller eqns V1 (or h1) F1 F1 = f(V1) = 22 − 33 −33(3)n V2 (or h2) F2 F2 = f(V2) V3 (or h3) F3 F3 = f(V3) 5 Modeling a Mixing Process Basis and Assumptions F (volumetric), CA (molar); Well Stirred Stephanopoulos (1984) Feed (1, 2) consists of components A and B Enthalpy of mixing is significant Process includes heating/ cooling H H ρ is constant 2 1 Overall Mass Balance (ℎ) = 11 + 22 − (ℎ) = (1 + 2 ) − 3 Q 33 in or out Component Mass Balance ( ) = (1 1 + 2 2) − 3 3 H3 6 Modeling a Mixing Process Conservation of energy (recall first law of thermodynamics) ∆ = ∆ + ∆ + ∆ ± + − ∆ H2 H1 ∆ ≅ ∆ (for constant ρ/ liquid systems ∆ is zero) Energy Balance H3 enthalpy balance (h is energy/mass) () = (1 + 2 ) − 3 ± We were familiar with energy ∆; how to characterize h (specific enthalpy) into familiar quantities (T, CA, parameters, …) H is enthalpy, h is specific enthalpy; CP is heat capacity, cP is specific heat capacity …. 7 Modeling a Mixing Process () = (1 + 2 ) − 3 ± Since enthalpy depends upon temperature so lets replace h with h(T) ℎ1 1 = 0 + 1 1 − 0 ℎ2 2 = (0) + 2 2 − 0 ℎ3 3 = (0) + 3 3 − 0 enthalpy associated with ΔT was easy to obtain, how to obtain h(T0) 0 = 1 + 1 + ∆1(0) 0 = 2 + 2 + ∆2(0) 0 = 3 + 3 + ∆3(0) and are molar enthalpy of component A and B and ∆ is heat of solution for stream i at T0. Put values of h in overall energy balance 8 Modeling a Mixing Process Re-arranging (and using component mass balance equations) 3 3 = 11 ∆ 1 − ∆ 3 + 22 ∆ 2 − ∆ 3 +1 1 1 − 0 − 3 3 − 0 + 2[2 2 − 0 − 3 3 − 0 ] ± If we assume cP1 = cP2 = cP3 = cP 3 = 11 ∆ 1 − ∆ 3 + 22 ∆ 2 − ∆ 3 +1(1 − 3) + cp2(2 − 3) ± If heats of solutions are strong functions of concentrations then ∆ 1 − ∆ 3 and ∆ 2 − ∆ 3 are significant Mixing process is generally kept isothermal (how?) 9 Tips For Assessment (Exam) Introduction + Modeling (week 1-3) In exam, you may be asked short descriptive questions to check your understanding of process control and to prepare a mathematical model for a chemical process or processes and to make the system exactly specified (i.e. Nf = 0) 1. Consult your class notes, board proofs, discussions 2. Stephanopoulos (1984) chapters 1-5, examples and end-chapter problems 3. Luyben (1996) chapter 3 page 40 to 74. Practice examples and end-chapter problems for chapter 3. 10 Week 3 Weekly Take-Home Assignment 1. Follow all the example modeling exercises in Luyben (1996) chapter 3 page 40 to 74. Practice these example processes. 2. Solve at least 10 end-chapter problems from Luyben (1996) chapter 3 (Compulsory) Submit before Friday (Feb 7) Curriculum and handouts are posted at: http://faculty.waheed-afzal1.pu.edu.pk/ 11