Page 203 - Design and Operation of Heat Exchangers and their Networks

192 Design and operation of heat exchangers and their networks

and databanks, designers are able to use more accurate models and more
powerful optimization algorithms to solve their design problems, no matter
how complicated they would be.

5.1 Design criteria and constraints

Before discussing the optimization methods, let us take a look of design criteria
and constraints. Those design criteria and constraints included minimum
investment costs, minimum operating costs, minimum weight or material,
minimum volume or heat transfer surface area, and minimum labor costs.
When a single performance measure has been defined quantitatively and is
to be minimized or maximized, it is called “objective function” in a design
optimization. A particular design may also be subjected to certain customer
requirement such as required heat transfer; allowable pressure drop; and lim-
itations on height, width, and length of the exchanger. These requirements are
the constraints in a design optimization.
The basic optimization problem may be expressed mathematically as
T

½
min f xðÞ x ¼ x 1 , x 2 , …, x n Š (5.1)

T
sb: gx ðÞ 0 g ¼ g 1 , g 2 , …, g p (5.2)

T
hxðÞ ¼ 0 h ¼ h 1 , h 2 , …, h q (5.3)
The vector x contains n unknowns, which are to be adjusted (or optimized) to
minimize the objective function f(x). Any solution must also satisfy the con-
ditions defined by the p inequality constraints g i (x) 0 and the q equality
constraints h j (x)¼0. A maximization problem may be described by
Eqs. (5.1)–(5.3) by making f(x) the negative of the function of interest.
The functions f(x), g(x), and h(x) may be linear or nonlinear algebraic func-
tions of the design variables. However, in most actual problems, one or more
of these functions requires substantial computational efforts for its evaluation.
If the function f(x) can be expressed algebraically and if no constraints are
imposed, setting its first derivative to zero will deliver the optimum. How-
ever, this optimum might be one of the local minimums. Furthermore, most
functions in practical problems are either more complex in nature or involve
more than one independent variable with constraints imposed on some vari-
ables. Therefore, the optimization is a complex task.
Numerical nonlinear programming techniques are one of the powerful
tools for solving nonlinear optimization problems. There is a good match