Chapter 2
Mathematical Modeling of
Chemical Processes
Mathematical Model (Eykhoff, 1974)
“a representation of the essential aspects of an existing
system (or a system to be constructed) which
represents knowledge of that system in a usable form”
Everything should be made as simple as possible, but
no simpler.
General Modeling Principles
• The model equations are at best an approximation to the real
process.
Chapter 2
• Adage: “All models are wrong, but some are useful.”
• Modeling inherently involves a compromise between model
accuracy and complexity on one hand, and the cost and effort
required to develop the model, on the other hand.
• Process modeling is both an art and a science. Creativity is
required to make simplifying assumptions that result in an
appropriate model.
• Dynamic models of chemical processes consist of ordinary
differential equations (ODE) and/or partial differential equations
(PDE), plus related algebraic equations.
Chapter 2
Table 2.1. A Systematic Approach for
Developing Dynamic Models
1. State the modeling objectives and the end use of the model.
They determine the required levels of model detail and model
accuracy.
2. Draw a schematic diagram of the process and label all process
variables.
3. List all of the assumptions that are involved in developing the
model. Try for parsimony; the model should be no more
complicated than necessary to meet the modeling objectives.
4. Determine whether spatial variations of process variables are
important. If so, a partial differential equation model will be
required.
5. Write appropriate conservation equations (mass, component,
energy, and so forth).
Table 2.1. (continued)
Chapter 2
6. Introduce equilibrium relations and other algebraic
equations (from thermodynamics, transport phenomena,
chemical kinetics, equipment geometry, etc.).
7. Perform a degrees of freedom analysis (Section 2.3) to
ensure that the model equations can be solved.
8. Simplify the model. It is often possible to arrange the
equations so that the dependent variables (outputs) appear
on the left side and the independent variables (inputs)
appear on the right side. This model form is convenient
for computer simulation and subsequent analysis.
9. Classify inputs as disturbance variables or as manipulated
variables.
Chapter 2
Modeling Approaches
 Physical/chemical (fundamental, global)
• Model structure by theoretical analysis
 Material/energy balances
 Heat, mass, and momentum transfer
 Thermodynamics, chemical kinetics
 Physical property relationships
• Model complexity must be determined
(assumptions)
•
Can be computationally expensive (not realtime)
•
May be expensive/time-consuming to obtain
•
Good for extrapolation, scale-up
•
Does not require experimental data to obtain
(data required for validation and fitting)
• Conservation Laws
Theoretical models of chemical processes are based on
conservation laws.
Chapter 2
Conservation of Mass
 rate of m ass   rate of m ass   rate of m ass 




in
out
 accum ulation  
 

(2-6)
Conservation of Component i
 rate of com ponent i   rate of com ponent i 



accum ulation
in

 

 rate of com ponent

out

i


 rate of com ponent

produced

i


(2-7)
Conservation of Energy
Chapter 2
The general law of energy conservation is also called the First
Law of Thermodynamics. It can be expressed as:
 rate of en ergy   rate of en ergy in   rate of en ergy o u t 




accu
m
u
lation
b
y
con
vection
b
y
con
vection

 
 

 n et rate of h eat ad d ition 


  to th e sy stem from



th e su rrou n d in gs


n et rate of w ork




 p erform ed on th e sy s tem 
 b y th e su rrou n d in gs 


(2-8)
The total energy of a thermodynamic system, Utot, is the sum of its
internal energy, kinetic energy, and potential energy:
U tot  U int  U K E  U PE
(2-9)
 Black box (empirical)
• Large number of unknown parameters
• Can be obtained quickly (e.g., linear regression)
Chapter 2
• Model structure is subjective
• Dangerous to extrapolate
 Semi-empirical
• Compromise of first two approaches
• Model structure may be simpler
• Typically 2 to 10 physical parameters estimated
(nonlinear regression)
• Good versatility, can be extrapolated
• Can be run in real-time
• linear regression
y  c 0  c1 x  c 2 x
2
• nonlinear regression
Chapter 2
y  K 1  e
 t /

• number of parameters affects accuracy of model,
but confidence limits on the parameters fitted must
be evaluated
• objective function for data fitting – minimize sum of
squares of errors between data points and model
predictions (use optimization code to fit
parameters)
• nonlinear models such as neural nets are
becoming popular (automatic modeling)
Chapter 2
Number of
births (West
Germany)
Number of sightings of storks
Uses of Mathematical Modeling
•
to improve understanding of the process
•
to optimize process design/operating conditions
•
to design a control strategy for the process
•
to train operating personnel
•Development of Dynamic Models
Chapter 2
•Illustrative Example: A Blending Process
An unsteady-state mass balance for the blending system:
 rate of accum ulation   rate of 



 of m ass in the tank   m ass in 
 rate of 


 m ass out 
(2-1)
or
d V ρ 
dt
 w1  w 2  w
(2-2)
Chapter 2
where w1, w2, and w are mass flow rates.
• The unsteady-state component balance is:
d V ρ x 
dt
 w1 x1  w 2 x 2  w x
(2-3)
The corresponding steady-state model was derived in Ch. 1 (cf.
Eqs. 1-1 and 1-2).
0  w1  w 2  w
(2-4)
0  w1 x1  w 2 x 2  w x
(2-5)
Chapter 2
The Blending Process Revisited
For constant  , Eqs. 2-2 and 2-3 become:

dV
dt
 w1  w 2  w
 d V x 
dt
 w1 x1  w 2 x 2  w x
(2-12)
(2-13)
Equation 2-13 can be simplified by expanding the accumulation
term using the “chain rule” for differentiation of a product:

d V x 
 V
dt
dx
 x
dt
dV
(2-14)
dt
Chapter 2
Substitution of (2-14) into (2-13) gives:
V
dx
 x
dV
dt
dt
 w1 x1  w 2 x 2  w x
(2-15)
Substitution of the mass balance in (2-12) for  dV / dt in (2-15)
gives:
V
dx
dt
 x  w1  w 2  w   w1 x1  w 2 x 2  w x
(2-16)
After canceling common terms and rearranging (2-12) and (2-16),
a more convenient model form is obtained:
dV

1
dt

dx
w1
dt

 w1  w 2  w 
V
 x1  x  
w2
V
(2-17)
 x2  x 
(2-18)
Chapter 2
Chapter 2
Stirred-Tank Heating Process
Figure 2.3 Stirred-tank heating process with constant holdup, V.
Stirred-Tank Heating Process (cont’d.)
Chapter 2
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  and heat capacity C of the liquid are assumed to
be constant. Thus, their temperature dependence is neglected.
4. Heat losses are negligible.
Chapter 2
For the processes and examples considered in this book, it
is appropriate to make two assumptions:
1. Changes in potential energy and kinetic energy can be
neglected because they are small in comparison with changes
in internal energy.
2. The net rate of work can be neglected because it is small
compared to the rates of heat transfer and convection.
For these reasonable assumptions, the energy balance in
Eq. 2-8 can be written as
dU int


  w H  Q
dt
  denotes the difference
betw een outlet and inlet
conditions of the flow ing
stream s; therefore
U int  the internal energy of
the system
H  enthalpy per unit m ass
w  m ass flow rate
(2-10)


-Δ w H = rate of enthalpy of the inlet
Q  rate of heat transfer to the system
stream (s) - the enthalpy
of the outlet stream (s)
Chapter 2
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 Uˆ int  Hˆ 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, dUˆ int ,
dUˆ int  dHˆ  C dT
(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   V Uˆ 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):
Chapter 2
dU int
 V C
dt
dT
(2-31)
dt
Note that this term appears in the general energy balance of Eq. 210.
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,

Hˆ  Hˆ ref  C T  Tref

(2-32)
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  Tref

(2-33)
Model Development - III
For the inlet stream

Chapter 2
Hˆ i  C Ti  Tref

(2-34)
Substituting (2-33) and (2-34) into the convection term of (2-10)
gives:

 w Hˆ
  w  C  Ti  Tref    w  C  T  Tref  
(2-35)
Finally, substitution of (2-31) and (2-35) into (2-10)
V C
dT
dt
 w C  Ti  T   Q
(2-36)
steam-heating:
V C
dT
dt
Q  ws  H v
 w C (Ti  T )  w s  H v (1)
0  w C (Ti  T )  w s  H v (2)
subtract (2) from (1)
V C
dT
dt
 w C (T  T )  ( w s  w s )  H v
divide by wC
V  dT
w
dt
 T T 
H v
wC
( ws  w s )
Define deviation variables (from set point)
y  T T
T is desired operating point
Chapter 2
u  ws  w s
 V dy
w s (T ) from steady state
 y
w dt
note w hen
H v
1
dt
note that
wC
dy
0
dt
dy
u
H v
wC
 K p and
V
w
 1
y  K pu
  y  K pu
G eneral linear ordinary differential equ ation solution: sum of exponential(s)
S uppose u  1 (unit step response)
t



y (t )  K p  1  e 1






Chapter 2
Example 1:
Ti = 40 C, T = 90 C, Ti = 0 C
o
o
s.s. balance:
o
w C ( T - Ti ) = w s  H v
w s = 0.83  10 g hr
Chapter 2
6
 H v = 600 cal g
o
C = l cal g C
4
w = 10 kg hr
 = 10 kg m
3
V = 20 m
3
3
 V  2  10 kg
4
 
V
w
2
dy
dt
2  10 kg
4

4
 2hr
10 kg hr
= -y + 6  10 u
y  T  T
u  ws  ws
-5
dynamic model
Step 1: t=0 double ws
T (0) = T
y(0) = 0
u = +0.83  10 g hr
6
Chapter 2
2
dy
dt
= -y + 50
y = 5 0 l - e
final
-0 .5 t
o
T = y ss + T = 50 + 90 = 140 C
Step 2: maintain
Step 3: then set
2
dy
dt

o
T = 140 C / 24 hr
u = 0, w s = 833 kg hr
= -y + 6  10 u, y(0) = 50
-5
Solve for u = 0
y = 50e
-0.5t
t  
y  0
(self-regulating, but slow)
how long to reach y = 0.5 ?
Chapter 2
Step 4: How can we speed up the return from 140°C to 90°C?
ws = 0 vs. ws = 0.83106 g/hr
at s.s ws =0
y  -50°C T  40°C
Process Dynamics
Process control is inherently concerned with unsteady
state behavior (i.e., "transient response", "process
dynamics")
Stirred tank heater: assume a "lag" between heating
element temperature Te, and process fluid temp, T.
Chapter 2
heat transfer limitation = heA(Te – T)
Energy balances
w C Ti +h e A (Te -T )-w C T =m C
Tank:
Chest:
Q - h e A(T
dT
At s.s.
dt
 0,
e
dT e
- T) = m e C e
dT
dt
dT e
dt
 0
dt
Specify Q  calc. T, Te
2 first order equations  1 second order equation in T
Relate T to Q (Te is an intermediate variable)
y=T-T
u=Q-Q
mm e C e d 2 y
 m eC e
m eC e
m  dy
1

 



y

u

wC
w  dt
wC
 h eA e
Chapter 2
wh e A e dt
2
T i fixed
Note Ce  0 yields 1st order ODE (simpler model)
Fig. 2.2
Rv
q =
1
R
h
v
Rv: line resistance
A
dh
Chapter 2
dt
 qi 
1
h
P  p   gh
(2 - 57)
Rv
linear ODE
P  p   gh
If
q = Cv
A
dh
dt
*
P - Pa
 qi  C v
Pa : ambient
*
nonlinear ODE
ρgh  q i  C v h
pressure
(2-61)
Chapter 2
Chapter 2
Table 2.2. Degrees of Freedom Analysis
1. List all quantities in the model that are known constants (or
parameters that can be specified) on the basis of equipment
dimensions, known physical properties, etc.
2. Determine the number of equations NE and the number of
process variables, NV. Note that time t is not considered to be a
process variable because it is neither a process input nor a
process output.
3. Calculate the number of degrees of freedom, NF = NV - NE.
4. Identify the NE output variables that will be obtained by solving
the process model.
5. Identify the NF input variables that must be specified as either
disturbance variables or manipulated variables, in order to
utilize the NF degrees of freedom.
Chapter 2
Degrees of Freedom Analysis for the Stirred-Tank
Model:
3 parameters:
V ,  ,C
4 variables:
T , Ti , w , Q
1 equation:
Eq. 2-36
Thus the degrees of freedom are NF = 4 – 1 = 3. The process
variables are classified as:
1 output variable:
T
3 input variables:
Ti, w, Q
For temperature control purposes, it is reasonable to classify the
three inputs as:
2 disturbance variables:
Ti, w
1 manipulated variable:
Q
Biological Reactions
Chapter 2
• Biological reactions that involve micro-organisms and enzyme catalysts
are pervasive and play a crucial role in the natural world.
• Without such bioreactions, plant and animal life, as we know it, simply
could not exist.
• Bioreactions also provide the basis for production of a wide variety of
pharmaceuticals and healthcare and food products.
• Important industrial processes that involve bioreactions include
fermentation and wastewater treatment.
• Chemical engineers are heavily involved with biochemical and
biomedical processes.
Bioreactions
• Are typically performed in a batch or fed-batch reactor.
Chapter 2
• Fed-batch is a synonym for semi-batch.
• Fed-batch reactors are widely used in the pharmaceutical
and other process industries.
• Bioreactions:
cells
substrate  m ore cells + products
(2-90)
• Yield Coefficients:
YX
/S

YP / S 
m ass of new cells form ed
(2-91)
m ass of substrate consum ed to form new cells
m ass of product form ed
m ass of substrate consum ed to form product
(2-92)
Chapter 2
Fed-Batch Bioreactor
Monod Equation
rg   X
(2-93)
Specific Growth Rate
   m ax
Figure 2.11. Fed-batch reactor
for a bioreaction.
S
Ks  S
(2-94)
Chapter 2
• Modeling Assumptions
1. The exponential cell growth stage is of interest.
2. The fed-batch reactor is perfectly mixed.
3. Heat effects are small so that isothermal reactor operation can
be assumed.
4. The liquid density is constant.
5. The broth in the bioreactor consists of liquid plus solid
material, the mass of cells. This heterogenous mixture can be
approximated as a homogenous liquid.
6. The rate of cell growth rg is given by the Monod equation in (293) and (2-94).
• Modeling Assumptions (continued)
7. The rate of product formation per unit volume rp can be
expressed as
Chapter 2
r p  Y P / X rg
(2-95)
where the product yield coefficient YP/X is defined as:
YP / X 
m ass of product form ed
(2-96)
m ass of new cells form ed
8. The feed stream is sterile and thus contains no cells.
• General Form of Each Balance
 R ate of
accum ulation    rate in    rate of form ation 
(2-97)
• Individual Component Balances
• Cells:
d ( XV )
dt
Chapter 2
• Product:
 V rg
d  PV
dt
• Substrate:
d ( SV )
dt
 F Sf
• Overall Mass Balance

(2-98)
 V rp
1

YX
/S
(2-99)
V rg

1
YP / S
V rP
(2-100)
• Mass:
d (V )
dt
 F
(2-101)
Chapter 2
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