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Introduction to Design
Introduction to Optimum Design
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Several systems can usually accomplish the same task, and some are better than others
Cheng-Liang Chen
PSE
LABORATORY
Department of Chemical Engineering National TAIWAN University
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Introduction to Design ➢
Any problem in which certain parameters need to be determined to satisfy constraints can be formulated as an optimum an optimum design problem
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The design of systems can be formulated as problems of optimization where a measure of performance (eg., utilit uti lityy con consum sumpti ption, on, TAC AC,, profi rofit) t) is to be opt optim imize ized d while satisfying all the constraints (eg., (eg., market demand, feedstock of raw material; MB/EB)
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Iterative Design Process
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Iterative Design Process
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Iterative Design Process
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Iterative Design Process
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Iterative Design Process
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Iterative Design Process
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Iterative Design Process
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Conventional Design Process vs Optimum Design Process An objective function measuring system performance is not identified
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Trend information is not calculated to make design decisions for improving systems
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Most decisions are made based on designer’s experience and intuition
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Conventional Design Process vs Optimum Design Process
Conventional Design: ➢
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Optimum Design: ➢
Optimum design process forces designer to identify explicitly a set of design variables, a cost function to be minimized (a performance function to be optimized), and constraints for the system ( proper math formulation)
⇒
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Using trend information to make decisions
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Aided by designer’s interaction
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Basic Terminology and Notation
Basic Terminology and Notation
US-British SI Units
US-British SI Units
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Basic Terminology and Notation
Basic Terminology and Notation
US-British SI Units
Sets and Points ➢
A point or vector in n-dimensional space
xx x = .. = x 1 2
1
xn
T
x2
···
xn
≡ (x1, x2, ··· , x ) n
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Basic Terminology and Notation
Sets and Points
Notation for Constraints ➢
| (x1 − 4)2 + (x2 − 4)2 ≤ 9
S = x = (x1, x2)
Note: (0, 0)
(3, 3)
∈ S,
The set S defines points within and on the circle of radius 3 centered at (4, 4)
| (x1 − 4)2 + (x2 − 4)2 ≤ 9
S = x = (x1, x2)
∈ S
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Basic Terminology and Notation
Basic Terminology and Notation
Norm/Length of A Vector
Functions
n
(x y ) = xT y =
·
xiyi
i=1
=
x y =
·
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Basic Terminology and Notation Geometrical representation for the set
||x||
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√ x · x
n
√
xT x =
=
x
i
i=1
n
y = Ax =
n
y = a x a x = = x a x a (i)
i
i
i=1
·
ij j
j =1
n
xT Ax = (x Ax)
n
i
i=1
n
n
ij xixj
ij j
j =1
n
gi(x) =
n
g (x) =
2
||x||||y|| cos θ
f (x) = f (x1, x2,
i=1 j =1
··· , x ) g (x1, x2, ··· , x ) g1(x) g2(x) ··· i
gm(x)
T
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Basic Terminology and Notation Continuous and Discontinuous Functions A function f (x) of n variables is called continuous at a point x if ∗
for any > 0, there is a δ > 0 such that f (x) whenever x
|| − x || < δ ∗
|
− f (x )| < ∗
Thank You for Your Attention Questions Are Welcome
