Section 5.3.1: Optimisation

Choices in Process Design

Discrete structure choices
- between alternative process routes
- equipment sequencing possibilities
Continuous parameter choices:
- length of a reactor
- thickness of vessel insulation
- diameter of a pipeline

Continuous parameter trade offs

Example: pipeline for constant flow rate.

Capital cost $\propto$ pipe diameter, D - maybe should be D^0.6 or D^0.9.

Operating cost $\propto$ D^-4.75 - assumes friction factor $\propto$ Re^-0.25.

Typically, capital cost versus operating cost.

The Optimisation problem

f(x)	- the objective function
x	- the decision variable
constraints:	$x \geq x_L$
	$x \leq x_U$

Constraints define feasible region.

Minimise f(x) with respect to (w.r.t.) x, subject to (s.t.) $x \geq x_L$ , $x \leq x_U$

Example: (use D instead of x) minimise f(D) = 2.0 D + 0.4 + 0.5 D^-4.75 (s.t.) $D \geq 0.1$ , $D \leq 2.5$ .

If expression available for derivative..

Solve

$\begin{displaymath}\frac{df}{dx} \equiv f^{'}(x) = 0 \end{displaymath}$

for stationary point(s). Try to get all SPs.

Nature of each SP at x_sp is given by sign of second derivative f^''(x):

+	implies local minimum
-	implies local maximum
0	don't know

Example: $f^{'}(D) = 2.0 - 4.75 \times 0.5 D^{-5.75}$
implies $D_{sp} = \left[ \frac{2.0}{4.75 \times 0.5}\right]^{\frac{-1}{5.75}} = 1.030$ m.
and $f^{''}(D_{sp}) = + 5.75 \times 4.75 \times 0.5 D_{sp}^{-6.75} = \frac{ 5.75 \times 4.75 \times 0.5}{1.030^{6.75}} > 0$

Hence local minimum.

Global minimum

Check

all local minima in the feasible region
all constraints (in 1 dimension, simple bounds)

Beware of discontinuities Simple theory assumes that both f(x) and f^'(x) are continuous. If not, global minimum may exist at or near a discontinuity in f or f^'.

Maximisation

Profit, throughput, reactor conversion,...

`Maximise f(x)' is equivalent to `Minimise -f(x)'.

Without loss of generality, a computer code can treat all problems as minimisations.

Next - Section 5.3.2: Non-gradient Methods

Return to Section 5.3 Index

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Last modified: Tue Aug 25 10:29:53 BST