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CHE412 Process Dynamics and ControlBSc (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 TechnologyUniversity of the Punjab, Lahore

wa.icet@pu.edu.pk

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Modeling CSTRs in Seriesconstant holdup, isothermal

F0

F1

CA1

F2

CA2 F3

CA3

V1

K1

T1

V2

K2

T2

V3

K3

T3

Basis and AssumptionsA → B (first order reaction)Compositions are molar and flow rates are volumetricConstant V, ρ, T Overall Mass Balance i.e. at constant V, F3 =F2 =F1 =F0 ≡ FSo overall mass balance is not required!

Luyben (1996)

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Modeling CSTRs in Seriesconstant holdup, isothermal

Component A mass balance on each tank (A is chosen arbitrarily)

kn depends upon temperature kn = k0 e-E/RTn where n = 1, 2, 3Apply 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

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Modeling CSTRs in Seriesconstant 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)

In this way, only forcing function (variable to be specified) is CA0.

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Modeling CSTRs in Series Variable Holdups, nth order

Mass Balances (Reactor 1)

n

Mass Balances (Reactor 2)

n Mass Balances (Reactor 1)

)n

Changes from previous case: V of reactors (and F) varies with time, reaction is nth order Parameters to be known: k1, k2, k3, nDisturbances to be specified:CA0, F0 Unknown variables: CA1, CA2, CA3, V1, V2, V3, F1, F2, F3

CV MVInclude

Controller eqnsV1 (or h1) F1 F1 = f(V1)

V2 (or h2) F2 F2 = f(V2)

V3 (or h3) F3 F3 = f(V3)

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H1

H2

H3

Qin or out

Modeling a Mixing Process

Overall Mass Balance Component Mass Balance 3

Basis and AssumptionsF (volumetric), CA (molar); Well StirredFeed (1, 2) consists of components A and B Enthalpy of mixing is significant Process includes heating/ coolingρ is constant

Stephanopoulos (1984)

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H1

H2

H3

Modeling a Mixing ProcessConservation of energy

(recall first law of thermodynamics)

(for constant ρ/ liquid systems is zero)

Energy Balance

enthalpy balance (h is energy/mass)

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 ….

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Modeling a Mixing Process

Since enthalpy depends upon temperature so lets replace h with h(T)

enthalpy associated with ΔT was easy to obtain, how to obtain h(T0)

and are molar enthalpy of component A and B and is heat of solution for stream i at T0.

𝑑 (𝜌𝑉 𝒉𝟑)𝑑𝑡 =𝜌(𝐹 ¿¿1𝒉𝟏+𝐹2𝒉𝟐)−𝜌 𝐹 3𝒉𝟑±𝑄 ¿

Put values of h in overall energy balance

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Modeling a Mixing ProcessRe-arranging (and using component mass balance equations)

If we assume cP1 = cP2 = cP3 = cP

cp If heats of solutions are strong functions of concentrations

then and are significant Mixing process is generally kept isothermal (how?)

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Tips For Assessment (Exam)Introduction + Modeling (week 1-3)

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.

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)

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Week 3Weekly 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/