Page 49 - Sustainability in the process industry

26 Cha p te r T h r ee

Minimize (or maximize) F(x, y) Objective function,
performance criterion
where x ∈ R (continuous variables) Continuous domain
n
y ∈ Z (integer variables) Discrete domain
n
subject to h(x, y) = 0 Equality constraints
g(x, y) ≤ 0 Inequality constraints

TABLE 3.1 Generic Optimization Problem

Continuous variables are used to model properties (e.g., flow rates
and chemical concentrations) that vary gradually within the feasible
region. Integer variables are used to model the status (ON versus
OFF) of operating devices as well as the selection/exclusion of options
for operating units in synthesis problems. Using only the problem
formulations, it is possible to generate many combinatorially infeasible
sets of the integer variable values, which are later analyzed by the
optimization solver. Especially for larger problems, it’s a good idea to
eliminate these infeasible combinations from the search space or to
build into the optimization solver a mechanism for avoiding them
(Friedler et al., 1996).
The type of the objective function F dictates which extremum—
the minimum or the maximum—to seek. Common performance
criteria are to minimize the process cost or to maximize the profit.
Because some process subsystems (e.g., water networks) do not usually
generate useful product streams, no revenue is directly realized and
so minimizing the total annualized cost is used instead as an
objective function. For complete production systems and supply
chains the objective is usually to maximize the profit. Thus, additional
variables (reflecting sales and customer behavior) and their
relationships may be added to the formulation.
Equality constraints stem not only from material and energy
balances but also from constitutive relations that normalize the
stream compositions to unity. The balances include those for total
flow rates, balances of the chemical components, and energy balances
of heat exchangers, boilers, and turbines. Inequality constraints stem
from limitations on concentrations, flow rates, temperatures,
pressures, throughput, and so forth. One example of a constitutive
relation is calculation of the fluid heat capacity flow rate from its
mass flow rate and specific heat capacity.

3.4 Main Classes of Optimization Problems

This section discusses methods that can be applied to detect
optimality and solve optimization tasks. Choosing a particular