# 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

Example: pipeline for constant flow rate.

Capital cost pipe diameter, D - maybe should be D0.6 or D0.9.

Operating cost D-4.75 - assumes friction factor Re-0.25.

Typically, capital cost versus operating cost.

### The Optimisation problem

 f(x) - the objective function x - the decision variable constraints:

Constraints define feasible region.

Minimise f(x) with respect to (w.r.t.) x, subject to (s.t.) ,

Example: (use D instead of x) minimise f(D) = 2.0 D + 0.4 + 0.5 D-4.75 (s.t.) , .

If expression available for derivative..

Solve

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

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

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

Example:
implies m.
and

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.

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