Page 44 - Chemical engineering design

INTRODUCTION TO DESIGN
will include that of forming the vessel and making the joints, in addition to cost of the
material (the surface area); see Wells (1973). 27
If the vessel is a pressure vessel the optimum length to diameter ratio will be even
greater, as the thickness of plate required is a direct function of the diameter; see
Chapter 13. Urbaniec (1986) gives procedures for the optimisation of tanks and vessel,
and other process equipment.

1.10.3. Multiple variable problems

The general optimisation problem can be represented mathematically as:

f D f v 1 , v 2 , v 3 ,. .., v n 1.2
where f is the objective function and v 1 , v 2 , v 3 ,... , v n are the variables.
In a design situation there will be constraints on the possible values of the objective
function, arising from constraints on the variables; such as, minimum flow-rates, maximum
allowable concentrations, and preferred sizes and standards.
Some may be equality constraints, expressed by equations of the form:

 m D  m v 1 , v 2 , v 3 ,. .., v n D 0 1.3
Others as inequality constraints:

 p D  p v 1 , v 2 , v 3 ,.. . , v n P p 1.4
The problem is to find values for the variables v 1 to v n that optimise the objective function:
that give the maximum or minimum value, within the constraints.

Analytical methods
If the objective function can be expressed as a mathematical function the classical methods
of calculus can be used to find the maximum or minimum. Setting the partial derivatives
to zero will produce a set of simultaneous equations that can be solved to find the optimum
values. For the general, unconstrained, objective function, the derivatives will give the
critical points; which may be maximum or minimum, or ridges or valleys. As with single
variable functions, the nature of the first derivative can be found by taking the second
derivative. For most practical design problems the range of values that the variables
can take will be subject to constraints (equations 1.3 and 1.4), and the optimum of the
constrained objective function will not necessarily occur where the partial derivatives
of the objective function are zero. This situation is illustrated in Figure 1.16 for a two-
dimensional problem. For this problem, the optimum will lie on the boundary defined by
the constraint y D a.
The method of Lagrange’s undetermined multipliers is a useful analytical technique for
dealing with problems that have equality constraints (fixed design values). Examples of
the use of this technique for simple design problems are given by Stoecker (1989), Peters
and Timmerhaus (1991) and Boas (1963a).