CHE412 Process

Dynamics

and Control

BSc (Engg) Chemical Engineering (7 th 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

𝑑ρ𝑉 𝑑𝑡

= ρ𝐹

0

− ρ𝐹

1

= 0

i.e. at constant V, F So overall mass balance is not required! 3 =F 2 =F 1 =F 0 ≡ F

F 0

Luyben (1996) V 1 K 1 T 1

F 1 C A1 V 2 K 2 T 2 F 2 C A2 V 3 K 3 T 3 F 3 C A3

2

Modeling CSTRs in Series constant holdup, isothermal

Component A mass balance on each tank

(A is chosen arbitrarily) 𝑑𝐶 𝐴1 𝑑𝑡 𝑑𝐶 𝐴2 𝑑𝑡 𝑑𝐶 𝐴3 𝑑𝑡 = = = 𝐹 𝑉 1 𝐹 𝑉 2 𝐹 𝑉 3 𝐶 𝐴0 − 𝐶 𝐴1 𝐶 𝐴2 𝐶 𝐴1 − 𝐶𝐴 2 − 𝐶𝐴 3 − 𝑘 1 𝐶 𝐴1 − 𝑘 2 𝐶 𝐴2 − 𝑘 3 𝐶 𝐴3

k n

depends upon temperature k

n = k 0 e -E/RTn where n = 1, 2, 3 Apply degree of freedom analysis!

Parameters/ Constants (to be known): V

1 , V 2 , V 3 , k 1 , k 2 , k 3

Specified variables (or forcing functions): F and C

A0

constant) . Unknown variables are

3

(C A1 , C A2 , C A3 (known but not ) 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)

𝑑𝐶

𝐴1

𝑑𝑡 + 𝐶

𝐴1

𝑘 + 1

τ

= 1

τ

𝐶

𝐴0

𝑑𝐶

𝐴2

𝑑𝑡 + 𝐶𝐴

2

𝑑𝐶 𝑑𝑡

𝐴3

+ 𝐶𝐴

3

𝑘 + 𝑘 + 1

τ

1

τ

= = 1

τ

𝐶

𝐴1

1

τ

𝐶

𝐴2 In this way, only forcing function (variable to be specified) is C

A0

. 4

Mass Balances (Reactor 1)

𝑑𝑉 𝑑𝑡 1 = 𝐹 0 − 𝐹 1 𝑑(𝑉 1 𝐶 𝐴1 ) 𝑑𝑡 = 𝐹 0 𝐶 𝐴0

Modeling CSTRs in Series

Variable Holdups, n th order − 𝐹 1 𝐶 𝐴1 −𝑉 1 𝑘 1 (𝐶𝐴 1)

n

Changes from previous case: V of reactors (and F) varies with time, reaction is n th order Parameters to be known:

Mass Balances (Reactor 2)

𝑑𝑉 𝑑𝑡 2 = 𝐹 1 − 𝐹 2 𝑑(𝑉 2 𝐶 𝐴2 ) 𝑑𝑡 = 𝐹 1 𝐶 𝐴1 − 𝐹 2 𝐶 𝐴2 −𝑉 2 𝑘 2 (𝐶 𝐴2 )

n

k 1 , k 2 , k 3 , n

Disturbances to be specified:

C A0

, F

0

Unknown variables:

C A1 , C A2 , C A3 , V 1 , V 2 , V 3 , F 1 , F 2 , F 3

Mass Balances (Reactor 1)

𝑑𝑉 𝑑𝑡 3 = 𝐹 2 − 𝐹 3 𝑑(𝑉 3 𝐶 𝐴3 ) 𝑑𝑡 = 𝐹 2 𝐶 𝐴2 − 𝐹 3 𝐶 𝐴3 −𝑉 3 𝑘 3 (𝐶 𝐴3 )

n

V 1 V 2

CV

(or h

1

) (or h

2

)

MV

F 1 F 2

Include

Controller eqns

F 1 F 2 = f(V 1 ) = f(V 2 ) V 3

(or h

3

)

F 3 F 3 = f(V 3 )

5

Modeling a Mixing Process

Basis and Assumptions

F (volumetric), C

A

(molar); Well Stirred Feed (1, 2) consists of components A and B

Stephanopoulos (1984)

Enthalpy of mixing is significant Process includes heating/ cooling

H 2

ρ is constant

H 1

Overall Mass Balance

𝑑(𝜌𝐴ℎ) 𝑑𝑡 𝑑(ℎ) = 𝜌 1 𝐹 1 + 𝜌 2 𝐹 2 − 𝜌 3 𝐹 3 𝐴 𝑑𝑡 = (𝐹 1 + 𝐹 2 ) − 𝐹 3

Component Mass Balance

𝑑(𝑐𝐴 𝑉) 𝐴 𝑑𝑡 = (𝐹 1 𝑐 𝐴1 + 𝐹 2 𝑐 𝐴2 ) − 𝐹 3 𝑐 𝐴 3

H 3

Q in or out

6

Modeling a Mixing Process

Conservation of energy

H 1

(recall first law of thermodynamics) ∆𝐸 = ∆𝑈 + ∆𝑘𝑒 + ∆𝑝𝑒 ± 𝑄 + 𝑊 𝑠𝑤 − 𝑝∆𝑣 ∆𝑈 ≅ ∆𝐻 (for constant ρ/ liquid systems 𝑝∆𝑣 is zero)

H 2

Energy Balance

H 3

enthalpy balance (h is energy/mass) 𝑑(𝜌𝑉𝒉 𝟑 ) 𝑑𝑡 = 𝜌(𝐹 1 𝒉 𝟏 + 𝐹 2 𝒉 𝟐 ) − 𝜌𝐹 3 𝒉 𝟑 ± 𝑄 We were familiar with energy 𝑚𝐶 𝑃 ∆𝑇 ; how to characterize h (specific enthalpy) into familiar quantities (T, C

A

, parameters, …) H is enthalpy, h is specific enthalpy; C P heat capacity ….

is heat capacity, c P is specific 7

Modeling a Mixing Process

𝑑(𝜌𝑉𝒉 𝑑𝑡 𝟑 ) = 𝜌(𝐹 1 𝒉 𝟏 + 𝐹 2 𝒉 𝟐 Since enthalpy depends upon temperature ) − 𝜌𝐹 3 𝒉 𝟑 so lets replace h with h(T) ± 𝑄 ℎ 1 ℎ 2 𝑇 𝑇 1 2 = 𝒉 𝟏 𝑇 0 + 𝑐𝑃 1 = 𝒉 𝟐 (𝑇 0 ) + 𝑐 𝑃2 𝑇 1 − 𝑇 𝑇 2 − 𝑇 0 0 ℎ 3 𝑇 3 = 𝒉 𝟑 (𝑇 0 ) + 𝑐 𝑃3 𝑇 3 − 𝑇 0 enthalpy associated with ΔT was easy to obtain, how to obtain

h(T 0 )

𝜌𝒉 𝟏 𝑇 0 = 𝑐𝐴 1 𝐴 + 𝑐𝐵 1 𝐵 + ∆ (𝑇 0 ) 𝜌𝒉 𝜌𝒉 𝟐 𝟑 𝑇 𝑇 0 0 = 𝑐𝐴 2 𝐴 = 𝑐 𝐴3 𝐴 + 𝑐𝐵 2 + 𝑐𝐵 3 𝐵 𝐵 + ∆ (𝑇 0 + ∆ (𝑇 0 ) ) 𝐴 and 𝐵 are molar enthalpy of component A and B and solution for stream

i

at

T 0

. ∆ is heat of Put values of

h

in overall energy balance 8

Modeling a Mixing Process

Re-arranging (and using component mass balance equations) 𝜌𝑐𝑃 3 𝑉 𝑑𝑇 3 𝑑𝑡 = 𝑐𝐴 1 𝐹 1 ∆ 𝑠1 − ∆ 𝐻 𝑠3 + 𝑐 𝐴2 +𝜌𝐹 1 𝑐 𝑝1 𝜌𝐹 2 [𝑐𝑝 2 𝑇 1 − 𝑇 𝑇 2 − 𝑇 0 0 − 𝑐𝑝 − 𝑐𝑝 3 3 𝑇 3 − 𝑇 𝑇 3 − 𝑇 0 0 + ] ± 𝑄 𝐹 2 ∆ 𝑠2 − ∆ 𝑠3 If we assume c P1 𝜌𝑐𝑝 𝑉 𝑑𝑇 3 𝑑𝑡 = 𝑐𝐴 1 = c 𝐹 1 P2 = c ∆ P3 = c 𝑠1 − ∆ 𝐻 P 𝑠3 + 𝑐 𝐴2 𝐹 2 ∆ 𝑠2 − ∆ 𝑠3 +𝑐𝑝𝜌𝐹 1 (𝑇 1 − 𝑇 3 ) + c p 𝜌𝐹 2 (𝑇 2 − 𝑇 3 ) ± 𝑄   If heats of solutions are strong functions of concentrations then ∆ 𝑠1 − ∆ 𝐻 𝑠3 and ∆ 𝑠2 − ∆ 𝐻 𝑠3 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. N f = 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.

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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/

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