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
wa.icet@pu.edu.pk
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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
𝑑ρ𝑉
𝑑𝑡
= ρ𝐹0 − ρ𝐹1 = 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)
𝑑𝐶𝐴1 𝐹
=
𝐶𝐴0 − 𝐶𝐴1 − 𝑘1𝐶𝐴1
𝑑𝑡
𝑉1
𝑑𝐶𝐴2 𝐹
=
𝐶𝐴1 − 𝐶𝐴2 − 𝑘2𝐶𝐴2
𝑑𝑡
𝑉2
𝑑𝐶𝐴3 𝐹
=
𝐶𝐴2 − 𝐶𝐴3 − 𝑘3𝐶𝐴3
𝑑𝑡
𝑉3
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)
𝑑𝐶𝐴1
1
1
+ 𝐶𝐴1 𝑘 +
= 𝐶𝐴0
𝑑𝑡
τ
τ
𝑑𝐶𝐴2
1
1
+ 𝐶𝐴2 𝑘 +
= 𝐶𝐴1
𝑑𝑡
τ
τ
𝑑𝐶𝐴3
1
1
+ 𝐶𝐴3 𝑘 +
= 𝐶𝐴2
𝑑𝑡
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)
𝑑𝑉1
= 𝐹0 − 𝐹1
𝑑𝑡
𝑑(𝑉1𝐶𝐴1
)
𝑑𝑡
= 𝐹0𝐶𝐴0 − 𝐹1𝐶𝐴1 −𝑉1𝑘1(𝐶𝐴1)n
Mass Balances (Reactor 2)
𝑑𝑉2
= 𝐹1 − 𝐹2
𝑑𝑡
𝑑(𝑉2𝐶𝐴2)
𝑑𝑡
= 𝐹1𝐶𝐴1 − 𝐹2𝐶𝐴2 −𝑉2𝑘2(𝐶𝐴2)n
Mass Balances (Reactor 1)
𝑑𝑉3
= 𝐹2 − 𝐹3
𝑑𝑡
𝑑(𝑉3𝐶𝐴3)
𝑑𝑡
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)
= 𝐹2𝐶𝐴2 − 𝐹3𝐶𝐴3 −𝑉3𝑘3(𝐶𝐴3)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
𝑑(𝜌𝐴ℎ)
= 𝜌1𝐹1 + 𝜌2𝐹2 −
𝑑𝑡
𝑑(ℎ)
𝐴
= (𝐹1 + 𝐹2 ) − 𝐹3
𝑑𝑡
Q
𝜌3𝐹3
in or out
Component Mass Balance
𝑑(𝑐𝐴 𝑉)
𝐴
𝑑𝑡
= (𝐹1 𝑐𝐴1 + 𝐹2 𝑐𝐴2) − 𝐹3 𝑐𝐴3
H3
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Modeling a Mixing Process
Conservation of energy
(recall first law of thermodynamics)
∆𝐸 = ∆𝑈 + ∆𝑘𝑒 + ∆𝑝𝑒 ± 𝑄 + 𝑊𝑠𝑤 − 𝑝∆𝑣
H2
H1
∆𝑈 ≅ ∆𝐻 (for constant ρ/ liquid systems 𝑝∆𝑣 is zero)
Energy Balance
H3
enthalpy balance (h is energy/mass)
𝑑(𝜌𝑉𝒉𝟑)
= 𝜌(𝐹1 𝒉𝟏 + 𝐹2 𝒉𝟐) − 𝜌𝐹3 𝒉𝟑 ± 𝑄
𝑑𝑡
We were familiar with energy 𝑚𝐶𝑃 ∆𝑇; 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
𝑑(𝜌𝑉𝒉𝟑)
= 𝜌(𝐹1 𝒉𝟏 + 𝐹2 𝒉𝟐) − 𝜌𝐹3 𝒉𝟑 ± 𝑄
𝑑𝑡
Since enthalpy depends upon temperature
so lets replace h with h(T)
ℎ1 𝑇1 = 𝒉𝟏 𝑇0 + 𝑐𝑃1 𝑇1 − 𝑇0
ℎ2 𝑇2 = 𝒉𝟐(𝑇0) + 𝑐𝑃2 𝑇2 − 𝑇0
ℎ3 𝑇3 = 𝒉𝟑(𝑇0) + 𝑐𝑃3 𝑇3 − 𝑇0
enthalpy associated with ΔT was easy to obtain, how to obtain h(T0)
𝜌𝒉𝟏 𝑇0 = 𝑐𝐴1𝐻𝐴 + 𝑐𝐵1𝐻𝐵 + ∆𝐻𝑆1(𝑇0)
𝜌𝒉𝟐 𝑇0 = 𝑐𝐴2𝐻𝐴 + 𝑐𝐵2𝐻𝐵 + ∆𝐻𝑆2(𝑇0)
𝜌𝒉𝟑 𝑇0 = 𝑐𝐴3𝐻𝐴 + 𝑐𝐵3𝐻𝐵 + ∆𝐻𝑆3(𝑇0)
𝐻𝐴 and 𝐻𝐵 are molar enthalpy of component A and B and ∆𝐻𝑆𝑖 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)
𝑑𝑇3
𝜌𝑐𝑃3 𝑉
= 𝑐𝐴1𝐹1 ∆𝐻 𝑠1 − ∆𝐻 𝑠3 + 𝑐𝐴2𝐹2 ∆𝐻 𝑠2 − ∆𝐻 𝑠3
𝑑𝑡
+𝜌𝐹1 𝑐𝑝1 𝑇1 − 𝑇0 − 𝑐𝑝3 𝑇3 − 𝑇0 +
𝜌𝐹2[𝑐𝑝2 𝑇2 − 𝑇0 − 𝑐𝑝3 𝑇3 − 𝑇0 ] ± 𝑄
If we assume cP1 = cP2 = cP3 = cP
𝑑𝑇3
𝜌𝑐𝑝 𝑉
= 𝑐𝐴1𝐹1 ∆𝐻 𝑠1 − ∆𝐻 𝑠3 + 𝑐𝐴2𝐹2 ∆𝐻 𝑠2 − ∆𝐻 𝑠3
𝑑𝑡
+𝑐𝑝𝜌𝐹1(𝑇1 − 𝑇3) + cp𝜌𝐹2(𝑇2 − 𝑇3) ± 𝑄
 If heats of solutions are strong functions of concentrations
then ∆𝐻 𝑠1 − ∆𝐻 𝑠3 and ∆𝐻 𝑠2 − ∆𝐻 𝑠3 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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