Page 137 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering

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Encyclopedia of Physical Science and Technology EN002G-100 May 19, 2001 18:49

Chemical Process Design, Simulation, Optimization, and Operation 759

Newton’s method and quasi-Newton techniques make or intersection of constraints. The most common solu-
use of second-order derivative information. Newton’s tion technique employed is the Simplex method. In re-
method is computationally expensive because it requires cent years, primal–dual interior-point linear programming
analytical first-and second-order derivative information, algorithms have been introduced that have more favor-
as well as matrix inversion. Quasi-Newton methods rely able solution properties when compared to the Simplex
onapproximatesecond-orderderivativeinformation(Hes- method.
sian) or an approximate Hessian inverse. There are a Quadratic programming involves the optimization of
number of variants of these techniques from various a quadratic function subject to linear constraints. Often-
researchers; most quasi-Newton techniques attempt to times, a quadratic program is solved as a subproblem
find a Hessian matrix that is positive definite and well- while solving general nonlinear programming problems.
conditioned at each iteration. Quasi-Newton methods are There are several techniques for solving nonlinear pro-
recognized as the most powerful unconstrained optimiza- gramming problems, including the generalized reduced
tion methods currently available. gradient (GRG) as well as successive quadratic program-
ming (SQP) approaches.
The application of mixed-integer linear (MILP) and
D. Constrained Optimization nonlinear (MINLP) programming approaches is rapidly
increasing in popularity. These problems involve con-
Constraints in optimization problems often exist in such a
straints that take on integer values. Improvements in pro-
fashion that they cannot be eliminated explicitly—for ex-
cessingpowerandalgorithmicstabilityaswellasthestudy
ample, nonlinear algebraic constraints involving transcen-
of hybrid systems have led to the increasing use of these
dental functions such as exp(x). The Lagrange multiplier
techniques. Global optimization techniques are also in-
method can be used to eliminate constraints explicitly in
creasing in use, particularly for solving steady-state opti-
multivariable optimization problems. Lagrange multipli-
mization problems where locating all solutions is practical
ers are also useful for studying the parametric sensitivity
computationally.
of the solution subject to the constraints.
In general, an optimization problem involving con-
straints has the form:
V. PROCESS DESIGN
min f (x 1 ,..., x n )
(16) Process design is a broad area. At one extreme it can in-
h(x 1 ,..., x n ) = 0 clude, for example, the specification of a replacement heat
exchanger for an existing process. On the other hand, it
where f (x) is a nonlinear function to be minimized, can involve the design of all process-related equipment for
and h(x) is a vector of nonlinear functions denoting the an entirely new (grassroots) plant. A more common situ-
equality and inequality constraints. The necessary condi- ation is a “retrofit” of an existing plant, where significant
tions for a local minimum of a general nonlinear function process equipment modifications are being made.
are given by the Kuhn–Tucker conditions for optimality.
These conditions are often the basis for the design and
A. Process Synthesis
termination criteria for optimization algorithms.
In addition to a wide variety of problem types, there The conceptual development of a typical chemical pro-
are three common types of constrained optimization cess remains somewhat of an art. Usually, experience with
problems that are typically of interest: linear programs similar process plants leads to an initial process flowsheet.
(LPs), quadratic programs (QPs), and nonlinear programs Parameter optimization on that flowsheet can be used to
(NLPs). determine the best economic design. Other flowsheets can
Linear programming is one of the most common opti- be generated by adding/removing unit operations and pro-
mization techniques applied. LPs are commonly used on cess streams and changing the structure of the process
production scheduling and resourcing problems. A lin- flowsheet. A simplified depiction of this synthesis process
ear program is a class of optimization problems where is shown in Fig. 5.
the objective function and constraints are linear. The ob- Early work in process synthesis focused on the solution
jective function and constraints of a linear program are of specific problems, such as the best sequence of distil-
convex; therefore, a local optimum is the global op- lation columns to perform separation of components in
timum. In addition, LPs demonstrate the characteristic feedstreams into product streams. Another early problem
wherein the optimum solutions of LPs lie on a constraint was the synthesis of heat-exchanger networks.