Page 134 - 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

756 Chemical Process Design, Simulation, Optimization, and Operation

approach include: (1) recycle systems are more easily han- optimum insulation thickness. Our discussion will focus
dled, and (2) flowsheet optimization is easier. on understanding the basic structure of optimization prob-
lems, explore some of the common solution techniques,
and examine some of the recent directions/applications of
IV. OPTIMIZATION optimization.

A. Introduction
B. Structure of Optimization Problems
The petrochemical industry has undergone incredible
Optimization problems are by their nature mathematical
changes during the past 25 years, due to increasingly strict
in nature. The first and perhaps the most difficult step is
environmental regulations, increases in raw material and
to determine how to mathematically model the system to
energy costs, as well as intense competition from multi-
be optimized (for example, paint mixing, chemical reac-
national competitors. Improvements in productivity have
tor, national economy, environment). This model consists
been impressive, driven in large part by improved opera-
of an objective function, constraints, and decision vari-
tion strategies and process designs. Optimization is per-
haps the most important tool utilized for these improve- ables. The objective function is often called the merit or
cost function; this is the expression to be optimized that
ments.
is the performance measure. For example, in Fig. 3 the
What is meant by optimization? Optimization is the
objective function would be the total cost. The constraints
field of study associated with finding the best solution
are equations that describe the model of the process (for
to mathematically defined problems. Fields that are im-
example, mass balances) or inequality relationships (in-
pacted by optimization include physics, biology, engineer-
sulation thickness >0 in the above example) among the
ing, economics, and mathematics:
variables. The decision variables constitute the indepen-
dent variables that can be changed to optimize the system.
Engineering and semiconductor design
Science and sequencing of the human genome We can show the basic concepts and structure of
Economics and comparing unemployment rate vs. optimization problems by a examining a least squares
problem. The problem in this case is to determine the
inflation rate
two coefficients (α 0 and α 1 ) such that the error be-
tween the measured output and model predicted output is
Most of the optimization techniques in use today have minimized:
been developed since the end of World War II. Consider-
p
able advances in computer architecture and optimization 2
min f = ε j (11)
algorithms have enabled the complexity of problems that
{α 0 ,α 1 }
j=1
are solvable via optimization to steadily increase. Initial
ˆ
work in the field centered on studying linear optimization ε j = Y j − Y j (12)
problems (linear programming, or LP), which is still used ˆ
Y j = α 0 + α 1 x j (13)
widely today in business planning. Increasingly, nonlinear
optimization problems (nonlinear programming, or NLP) In this example, the objective function to be optimized
have become more and more important, particularly for is given by Eq. (11), while the constraints that constitute
steady-state processes. the model are given in Eqs. (12) and (13). The number of
Optimization can be performed on many different time measurements is p in Eq. (11) ε j is the error between the
scales and levels, from production planning over the next measured and model predicted outputs, x is the indepen-
ˆ
year to determining optimal setpoints for a chemical pro- dent variable, Y j is the actual output, and Y j is the model
cess unit operation every minute. Typical optimization predicted output.
levels in the petrochemical industry include management A number of steps are involved in the solution of opti-
decisions, process design, and plant operations. In these mization problems, including analyzing the system to be
cases, the solution to the optimization problem will be the optimized so that all variables are characterized. Next, the
one that maximizes some measure of profit. An example objective function and constraints are specified in terms of
of optimization applied to process design is determination these variables, noting the independent variables (degrees
of the optimum thickness of insulation for a given steam of freedom). The complexity of the problem may necessi-
pipe installation, as shown in Fig. 3. tate the use of more advanced optimization techniques or
The fixed costs increase as the insulation thickness is problem simplification. The solution should be checked
increased; however, the costs associated with heat losses and the result examined for sensitivity to changes in the
decrease. The total cost goes through a minimum at the model parameters.