Chapter 2 Developing Models for Optimization
Chapter 2
Chapter 2
Developing Models for
Optimization
1
Chapter 2
2
Chapter 2
3
Chapter 2
4
Chapter 2
5
Chapter 2
Everything should be made as simple
as possible, but no simpler
6
Chapter 2
7
Chapter 2
8
Chapter 2
9
Chapter 2
10
Chapter 2
TERMINOLOGY OF MATHEMATICAL MODELS
There are many additional ways to classify mathematical
models besides those used in Chapter 2. For our
purposes it is most satisfactory to first consider grouping
the models into opposite pairs:
deterministic
linear
steady state
lumped parameter
black box
vs.
vs.
vs.
vs.
vs.
probabilistic
nonlinear
nonsteady state
distributed parameter
fundamental (physical)
11
Common Sense in Modeling
Chapter 2
What simplifications can be made?
How are they justified?
Types of Simplifications
(1)Omitting Interactions
(2)Aggregating Variables
(3)Eliminating Variables
(4)Replace Random Variables with Expected Values
(5)Reduce Detail of Mathematical Description
12
Chapter 2
13
Chapter 2
14
Precautions in Model Building
Chapter 2
(1) Limits on availability of data and accuracy of data
Examples: Kinetic coefficients
Mass transfer coefficients
(2) Unknown factors present or not present in scale up
Examples: Impurities in plant streams
Wall effects
(3) Poor measures of deviation between ideal and actual
models
Examples: Stage efficiency
15
Chapter 2
(4) Models used for one purpose used improperly for
another purpose
Example: Invalidity of kinetic models
(5) Extrapolation – using the model outside of the regions
Where it has been validated
16
Chapter 2
17
Chapter 2
18
2. Empirical Models
y a 0 a1 x1 a 2 x2 ...
Chapter 2
y a 0 a11 x12 a12 x1 x2 a 21 x2 x1 a 22 x22
1
G s
a 0 a1s a 2 s 2
Re a (Pr)b ( Sc )c
3. Probabilistic concepts applied to small
physical subdivisions of the process
Not often used
19
Chapter 2
20
Chapter 2
21
Chapter 2
22
Chapter 2
23
Chapter 2
24
Chapter 2
FIGURE E2.3a
Variation of overall heat transfer
coefficient with shell-side flow rate
ws = 80,000.
FIGURE E2.3b
Variation of overall heat transfer
coefficient with tube-side flow rate
Wt for ws = 4000.
25
Semi-empirical Model Fitting
Heat exchanger data, p. 54
Chapter 2
1
1
1
1
=
+
+
u
hs
ht
hf
(a)
curve D in Eqn (3), Figure 2.6
x
= + βx
y
1
=
+β
y
x
(b)
(c)
1
1
1
1
1
=
+
=
+
U
hsf
ht
hsf
k t w t 0.8
hsf k t w t 0.8
U=
k t w t 0.8 +hsf
26
Quadratic Curve Fitting
y = β1 + β2 x + β3 x2 (x1=1, x2 = x1 x3 = x2 )
Chapter 2
Least squares analysis leads to 3 linear equations
in 3 unknowns (n data points)
n
T
x x xi
xi2
xi
xi2
x i3
yi
T
x y= x i y i
x 2y
i i
xi2
x i3
xi 4
What about
y β1e x β 2 e x ?
y β1 β 2 sin x
?
(coefficients must appear linearly)
27
Chapter 2
28
Chapter 2
Factorial Design and Least Squares Fitting
3 variables: x2, x3 , x4 (x1 = 1)
t 220 o
x2 =
( C)
20
p-3
x3 =
(atm)
2
m-200
x4 =
(kg/h)
50
for data matrix on p. 65 (see Fig. E2.6)
11
0
T
x x
0
0
0
8
0
0
0
0
8
0
0
0
0
8
diagonal!
easy to invert,
well-conditioned
yˆ = 58.810 + 12.124 x2 + 11.402 x3 + 0.689 x4
29
Chapter 2
Developing Models for
Optimization
1
Chapter 2
2
Chapter 2
3
Chapter 2
4
Chapter 2
5
Chapter 2
Everything should be made as simple
as possible, but no simpler
6
Chapter 2
7
Chapter 2
8
Chapter 2
9
Chapter 2
10
Chapter 2
TERMINOLOGY OF MATHEMATICAL MODELS
There are many additional ways to classify mathematical
models besides those used in Chapter 2. For our
purposes it is most satisfactory to first consider grouping
the models into opposite pairs:
deterministic
linear
steady state
lumped parameter
black box
vs.
vs.
vs.
vs.
vs.
probabilistic
nonlinear
nonsteady state
distributed parameter
fundamental (physical)
11
Common Sense in Modeling
Chapter 2
What simplifications can be made?
How are they justified?
Types of Simplifications
(1)Omitting Interactions
(2)Aggregating Variables
(3)Eliminating Variables
(4)Replace Random Variables with Expected Values
(5)Reduce Detail of Mathematical Description
12
Chapter 2
13
Chapter 2
14
Precautions in Model Building
Chapter 2
(1) Limits on availability of data and accuracy of data
Examples: Kinetic coefficients
Mass transfer coefficients
(2) Unknown factors present or not present in scale up
Examples: Impurities in plant streams
Wall effects
(3) Poor measures of deviation between ideal and actual
models
Examples: Stage efficiency
15
Chapter 2
(4) Models used for one purpose used improperly for
another purpose
Example: Invalidity of kinetic models
(5) Extrapolation – using the model outside of the regions
Where it has been validated
16
Chapter 2
17
Chapter 2
18
2. Empirical Models
y a 0 a1 x1 a 2 x2 ...
Chapter 2
y a 0 a11 x12 a12 x1 x2 a 21 x2 x1 a 22 x22
1
G s
a 0 a1s a 2 s 2
Re a (Pr)b ( Sc )c
3. Probabilistic concepts applied to small
physical subdivisions of the process
Not often used
19
Chapter 2
20
Chapter 2
21
Chapter 2
22
Chapter 2
23
Chapter 2
24
Chapter 2
FIGURE E2.3a
Variation of overall heat transfer
coefficient with shell-side flow rate
ws = 80,000.
FIGURE E2.3b
Variation of overall heat transfer
coefficient with tube-side flow rate
Wt for ws = 4000.
25
Semi-empirical Model Fitting
Heat exchanger data, p. 54
Chapter 2
1
1
1
1
=
+
+
u
hs
ht
hf
(a)
curve D in Eqn (3), Figure 2.6
x
= + βx
y
1
=
+β
y
x
(b)
(c)
1
1
1
1
1
=
+
=
+
U
hsf
ht
hsf
k t w t 0.8
hsf k t w t 0.8
U=
k t w t 0.8 +hsf
26
Quadratic Curve Fitting
y = β1 + β2 x + β3 x2 (x1=1, x2 = x1 x3 = x2 )
Chapter 2
Least squares analysis leads to 3 linear equations
in 3 unknowns (n data points)
n
T
x x xi
xi2
xi
xi2
x i3
yi
T
x y= x i y i
x 2y
i i
xi2
x i3
xi 4
What about
y β1e x β 2 e x ?
y β1 β 2 sin x
?
(coefficients must appear linearly)
27
Chapter 2
28
Chapter 2
Factorial Design and Least Squares Fitting
3 variables: x2, x3 , x4 (x1 = 1)
t 220 o
x2 =
( C)
20
p-3
x3 =
(atm)
2
m-200
x4 =
(kg/h)
50
for data matrix on p. 65 (see Fig. E2.6)
11
0
T
x x
0
0
0
8
0
0
0
0
8
0
0
0
0
8
diagonal!
easy to invert,
well-conditioned
yˆ = 58.810 + 12.124 x2 + 11.402 x3 + 0.689 x4
29