### Modeling of Chemical Processes

```Chemical Process Control
ChE 462
Theoretical Models of Chemical Processes
Develop understanding of process a mathematical hypothesis of process
mechanisms
Dynamic model is used to predict how process responds to given input
Modeling Types
Variables not a function of
time
Dynamic
Variables are a function of
time
Empirical vs. Mechanistic models
Empirical Models
only limited representation of
the process. Simpler model
forms.
Mechanistic Models
Rely on understanding of a
process.
Derived from fundamental
laws of physics and chemistry.
Useful for exploration of new
conditions.
Linear vs Nonlinear
Linear: simpler model form,
solution has the form ezx
(useful for empirical model) ;
Nonlinear: more complex
and difficult to identify.
Modeling procedure
1. Decide modeling variables and fundamental laws (Mass, Energy and/or
Momentum)
2. Make appropriate assumptions (Simplify, e.g. isothermal, no friction,
incompressible flow, etc,…)
3. Develop the model equations and Check model degree of freedom
do we have more unknowns than equations?
4. Determine unknown constants
e.g. fluid density , temperature, viscosity
5. Solve model equations
6. Validity the model
compare to experimental data/sensitivity analysis
Process Dynamic Model
The balance equations to model chemical
engineering processes.
Mass Conservation
Equations
Thermal Energy Balance Equation
Rate of accumulation 
=

of thermal energy

+
Rate of convective 
heat transfer



entering the system 
Net rate of

heat generation 


by reaction

Mass, Moles, or
Energy Balances
+
−
Rate of convective 
heat transfer



leaving the system 
Net rate of heat transfer 
 through the boundaries



of the system

Rate of

Rate Entering 
 Accumulation =  the

System




Rate Leaving
− 

the
System


Rate of Generation by

+ 

Reaction
within
the
System


Part 1: Process Model Examples
1. Blending systems
2. Series Reaction
3. Stirred Tank and Bioreactor
Part 2: Process optimization
Example 1: Control A Blending Process
X1 and W1
V, ρ
X 2,
W2
Product A
X, W
???: X=Xsp
Overall balance:
Component A balance:
Model Equations
X2, W2
Mixture of A and B
X1 and W1
Product
X, W
???: X=Xsp
Feed forward control Model
AC
Method 1
Mixture of A and B
X1 and W1
AT
X2=1
W2
X1
Product
X, W
t
???: X=Xsp
A simple feedback control algorithm (proportional control)
Method 2
Mixture of A and B
X1 and W1
X1
AC
X2
W2
AT
Product
X, W
t
???: X=Xsp
Unsteady State Blending Process
Considering the density of the liquid, ρ, is a constant
Overall balance: 0≠W1+W2-W
Component A balance: 0≠W1X1 +W2X2-WX
dV
ρ
= w1 + w2 − w
dt
ρd (Vx)
= w1 x1 + w2 x2 − wx
dt
How to find dx/dt???
Equation below can be simplified by expanding the accumulation
term using the “chain rule” for differentiation of a product:
Degree of freedom
NF=NV-NE
NF: Degree of freedom
NV: Number of Process Variables (time is not process variable)
NE: Number of Independent Equation
NFC: the control degree of freedom (independent material/energy streams), e.g.,
Temp, Flow, Comp…
Considering the density of the liquid, ρ, is a constant
For example: freedom = ?
dV 1
= ( w1 + w2 − w)
dt ρ
dx w1
w2
=
( x1 − x) +
( x2 − x )
ρV
dt ρV
Calculation (Example 2.1)
A constant liquid holdup:
a) Steady state value of x
V=2 m3 , ρ = 900kg/ m3
b) W1 change to 400kg/min suddenly
Expression of x(t); V is still constant.
Pure A
X2=0.75
W2 =200 kg/min
Mixture of A and B
X1 =0.4 W1 =500 kg/min
AT
Product
X, W
0 = w1 + w2 − w
0 = w1 x1 + w2 x2 − wx
Solve: x(0)=0.5
W1 change to 400kg/min suddenly, V is constant
0 = w1 + w2 − w
dx
Vρ
= w1 x1 + w2 x2 − wx
dt
Initial condition: x(0)=0.5
Plug w1, x1, w2, x2, w into the equation above.
600
Example Plot
500
400
X(t)=0.517-0.017exp (-t/3)
W
300
W1
200
100
Control 1: FF, good model
Control 2: FF, bad model
Control 3: FB, large Kc
Control 4: FB, small Kc
0.52
0
0
10
15
20
25
30
35
T
no control
Feedforward
0.516
???
0.512
X
5
0.508
0.504
Feedback
0.5
0.496
0
5
10
15
Xsp=0.5
20
25
30
35
Example 2: Model for Product Composition for
CSTR with a Series Reaction
r1
r2
→ B 
→ C
A 
F: mass/time
CAO , F, ρ, Vr
r1 =
k1 ⋅ CA2
r2 =
k 2 ⋅ CB
FC
Component A:
FT
Feed
CA, CB, CC
Product
Component B:
AT
Equations for Component
Compositions.
Vr is constant.
Component C:
Example 3: Stirred tank heater
Chapter 2: 2-3
Stirred-Tank Heating Process
Assumptions:
1. Perfect mixing; thus, the exit
temperature T is also the temperature
of the tank contents.
2. The liquid holdup V is constant
because the inlet and outlet flow rates
are equal.
3. The density ρ and heat capacity C of
the liquid are assumed to be constant.
Thus, their temperature dependence is
neglected.
4. Heat losses are negligible.
Figure 2.3
Model Development - I
For a pure liquid at low or moderate pressures, the internal energy
is approximately equal to the enthalpy, Uint ≈ H , and H depends
only on temperature. Consequently, in the subsequent
development,
^
^
we assume that Uint = H and Uint = H where the caret (^) means
per unit mass. As shown in Appendix B, a differential change in
temperature, dT, produces a corresponding change in the internal
energy per unit mass,
^
^
d U int = d H = CdT
(2-29)
where C is the constant pressure heat capacity (assumed to be
constant). The total internal energy of the liquid in the tank is:
U int = ρVUˆ int
(2-30)
Model Development - II
An expression for the rate of internal energy accumulation can be
derived from Eqs. (2-29) and (2-30):
dU int
dT
(2-31)
= ρVC
dt
dt
Note that this term appears in the general energy balance
Suppose that the liquid in the tank is at a temperature T and has an
enthalpy, Hˆ . Integrating Eq. 2-29 from a reference temperature
Tref to T gives,
^
^
(2-32)
H − H ref = C (T − T )
ref
where Hˆ ref is the value of Hˆ at Tref. Without loss of generality, we
assume that Hˆ ref = 0 (see Appendix B). Thus, (2-32) can be
written as:
=
Hˆ C T − T
(2-33)
(
ref
)
Model Development - III
For the inlet stream
Substituting (2-33) and (2-34) into the convection term of (2-10)
gives:
(2-10)
Finally, substitution of (2-31) and (2-35) into (2-10)
Degrees of Freedom Analysis for the Constant
Volume Stirred-Tank Model
parameters:
dT
V ρ C = wC (Ti − T ) + Q
dt
(2-36)
variables:
equation:
The process variables are classified as:
w
Models of Bioreactor Operations
Model of Fed-Batch Bioreactor
Growth Rate
rg = µ X
(2-93)
Monod Equation
µ = µmax
S
Ks + S
µ is Specific growth rate
Fed-batch reactor for a
bioreaction.
(2-94)
Monod Equation
Reaction rate
0 order
1/2 umax
µ = µmax
S
Ks + S
(2-94)
Ks
1st order
Substrate Concentration
Modeling Assumptions
1.
2.
3.
4.
The fed-batch reactor is perfectly mixed.
Isothermal reactor operation can be assumed.
The liquid density is constant.
The broth in the bioreactor can be approximated as a homogenous
liquid.
5. The rate of product formation per unit volume rp can be expressed as
rp = YP / X rg
(2-95)
where the product yield coefficient YP/X is defined as:
YP / X =
mass of product formed
mass of new cells formed
(2-96)
• General Form of Each Balance
{Rate of
accumulation
=
}
{rate in} + {rate of
formation}
(2-97)
Model of Fed-Batch
• Individual Component Balances for Fed Batch
• Cells:
• Product:
• Substrate:
• Overall Mass Balance
• Mass:
Part 2: Process Control and Optimization
• Control has to do with adjusting flow rates to
maintain the controlled variables of the process at
specified set-points.
• Optimization chooses the values for key set-points
such that the process operates at the “best”
economic conditions.
Optimization and Control of a CSTR
Optimizer
RSP
What is optimal
operation temperature?
TC
RSP
Feed
FC
FV
CA0
FT
Steam
A→B→C
TT
Product
CA,CB, CC
Optimization Example (steady state)
Economic Objective Function
Product
Substrate
Φ = Q C A VA + Q C B VB + Q CC VC − Q C A0 VAF
•
•
•
•
VB > VC, VA, or VAF
V is the chemical values.
At low T, little formation of B
At high T, too much of B reacts to form C
Therefore, the exits an optimum reactor
temperature, T*
Graphical Solution of Optimum
Reactor Temperature, T*
Process Design and Economic Analysis
1. Develop a process model
2. Convert process model to financial model
3. Optimization Algorithm
1. Select initial guess (within certain ranges) for reactor
temperature
2. Evaluate CA, CB, and CC
3. Evaluate Φ
4. Choose new reactor temperature and return to 2 until T*
identified.
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