1 Chapter 2 Developing Models for Optimization Chapter 2
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Chapter 2
Developing Models for
Optimization
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as possible, but no simpler
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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 vs. probabilistic
linear vs. nonlinear
steady state vs. nonsteady state
lumped parameter vs. distributed parameter
black box vs. fundamental (physical)
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Common Sense in Modeling
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
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Precautions in Model Building
(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
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(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
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2. Empirical Models
0 1 1 2 2
2 2
0 11 1 12 1 2 21 2 1 22 2
2
0 1 2
...
1
Re (Pr) ( )b c
y a a x a x
y a a x a x x a x x a x
G sa a s a s
a Sc
3. Probabilistic concepts applied to small
physical subdivisions of the process
Not often used
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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.
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Semi-empirical Model Fitting
s t f
1 1 1 1 = + + (a)
u h h h
x = + βx (b)
y
1 = + β (c)
y x
Heat exchanger data, p. 54
curve D in Eqn (3), Figure 2.6
0.8
sf t sf t t
1 1 1 1 1 = + = +
U h h h k w
0.8
sf t t
0.8
t t sf
h k wU =
k w +h
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Quadratic Curve Fitting 2 2
1 2 3 1 2 1 3y = β + β x + β x (x =1, x = x x = x )
Least squares analysis leads to 3 linear equations
in 3 unknowns (n data points) 2
i i
T 2 3
i i i
2 3 4
i i i
n x x
x x x x x
x x x
i
T
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2
i i
y
x y= x y
x y
What about
(coefficients must appear linearly)
? x sinββy
? eβeβy
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x
2
x
1
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Factorial Design and Least Squares Fitting
2 3 4 13 variables: x , x , x (x = 1)
o
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t 220x = ( C)
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p-3x = (atm)
2
m-200x = (kg/h)
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for data matrix on p. 65 (see Fig. E2.6)
T
11 0 0 0diagonal!
0 8 0 0x x easy to invert,
0 0 8 0well-conditioned
0 0 0 8
2 3 4y = 58.810 + 12.124 x + 11.402 x + 0.689 x
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