Page 45 - Chemical engineering design

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Feasible region
f(v) CHEMICAL ENGINEERING

y = a
Minimum of
function

v
Figure 1.16. Effect of constraints on optimum of a function

Search methods
The nature of the relationships and constraints in most design problems is such that
the use of analytical methods is not feasible. In these circumstances search methods,
that require only that the objective function can be computed from arbitrary values of
the independent variables, are used. For single variable problems, where the objective
function is unimodal, the simplest approach is to calculate the value of the objective
function at uniformly spaced values of the variable until a maximum (or minimum) value
is obtained. Though this method is not the most efficient, it will not require excessive
computing time for simple problems. Several more efficient search techniques have been
developed, such as the method of the golden section; see Boas (1963b) and Edgar and
Himmelblau (2001).
Efficient search methods will be needed for multi-dimensional problems, as the number
of calculations required and the computer time necessary will be greatly increased,
compared with single variable problems; see Himmelblau (1963), Stoecker (1971),
Beveridge and Schechter (1970), and Baasel (1974).
Two variable problems can be plotted as shown in Figure 1.17. The values of the
objective function are shown as contour lines, as on a map, which are slices through the
three-dimensional model of the function. Seeking the optimum of such a function can be

75%
Yield contours
80%
85%
90%
Pressure

Temperature
Figure 1.17. Yield as a function of reactor temperature and pressure