Page 43 - Chemical engineering design

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CHEMICAL ENGINEERING
This will be possible for only a few practical design problems. The technique is illus-
trated in Example 1.1, and in the derivation of the formula for optimum pipe diameter in
Chapter 5. The determination of the economic reflux ratio for a distillation column, which
is discussed in Volume 2, Chapter 11, is an example of the use of a graphical procedure
to find the optimum value.
Example 1.1
The optimum proportions for a cylindrical container. A classical example of the optimi-
sation of a simple function.
The surface area, A, of a closed cylinder is:
2
A D ð D ð L C 2 D
4
where D D vessel diameter
L D vessel length (or height)

This will be the objective function which is to be minimised; simplified:
D 2
f D ð L D D ð L C equation A
2
For a given volume, V, the diameter and length are related by:
2
V D D ð L
4
and
4V
L D equation B
D 2
and the objective function becomes
4V D 2
f D D C
D 2
Setting the differential of this function zero will give the optimum value for D
4V
C D D 0
D 2

3 4V
D D

From equation B, the corresponding length will be:

3 4V
L D

So for a cylindrical container the minimum surface area to enclose a given volume is
obtained when the length is made equal to the diameter.
In practice, when cost is taken as the objective function, the optimum will be nearer
L D 2D; the proportions of the ubiquitous tin can, and oil drum. This is because the cost