### CHE412 Process Dynamics and Control BSc (Engg)

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]
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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
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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
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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.
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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
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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 ….
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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
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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?)
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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.
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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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