Page 88 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering

P1: LLL Revised Pages
Encyclopedia of Physical Science and Technology EN002E-49 May 17, 2001 20:13

52 Batch Processing

orthogonal to the functions being integrated. The prob- Because of the accumulation of these errors, in time or
lems found in the use of the second method are in the space, the numerical results which are generated by sim-
inversion of the final transform, which leads to the solu- ulation may be far from the true solution. This problem
tion of the PDE. may be recognized by comparing two solutions of the
same problem using 100 times smaller time or space in-
crements, or both, when it is the case. Solutions passing
D. Evaluation of the Solutions
that test are likely to be correct. But such a test may be too
Numerical evaluation of the solutions to these problems is stringent in a good number of cases since it may require
not easy. Most analytical solutions to PDEs are expressed too much computing time or too much computer memory
in terms of infinite series. To get numerical answers one to be economical to run.
must evaluate series functions, which often are not sharply It is proper to emphasize the fact that there is no math-
convergent; thus, the number of terms to be accounted for ematical way of determining how correct is the numeri-
in the summation may be very large, or the accuracy of cal solution of a nonlinear differential equation, just by
the answer may be largely reduced. Alternatively, when looking at what an algorithm predicts for linear forms of
the solution is obtained by numerical methods, the accu- the same problem. Even if a computer program gener-
racy is affected by the approximations made in replacing ates accurate numerical solutions which are in agreement
partial derivatives by finite difference expressions. Then with analytical solutions of the linear forms of a nonlinear
special consideration must be given to estimating the trun- equation, the solutions in the nonlinear domain may not
cation, roundoff, and generated errors introduced by such be correct.
approximations. Another problem inherent in numerical
solutions is that there is no way of telling whether the
answer obtained is correct. It is customary to solve the IX. SAFETY
problem independently by a different numerical method
and compare the answers to check for errors due to the In batch plants, as in continuous plants, hazardous mate-
other numerical method. However, the accuracy of the rials are stored or handled. The operating conditions of
answers cannot be estimated except by comparison with continuous plants are set to completely avoid fire and ex-
analytical solutions of particular cases. Therefore, in eval- plosions. Risks are taken only in the startup or shutdown
uating the solution, it is important to give consideration to of the plant, when process leaks occur or undesired mate-
the following topics: (1) analysis of errors resulting from rials are released for protection of the equipment. In batch
numerical approximations, (2) comparison of numerical plants the safety risks are more numerous, since equip-
solutions to check the correctness of the software, and (3) ment and materials are brought more often in contact with
solution of analytical approximations to verify the validity ambient conditions or are subjected periodically to tran-
of numerical solutions. sient temperature, pressure, or concentration conditions,
which may constitute safety hazards. Therefore, safety is
much more important in the successful management and
E. How Good is the Model?
operation of batch plants than in that of continuous pro-
A mathematical model of a plant or a section of a plant cessing plants. Safety must be a major consideration in
can be judged only by comparison with actual plant data. the storage and handling of volatile or chemically unsta-
The model may be considered as good when the simu- ble reagents or intermediates. Equipment selection and
lated variables can predict with some level of confidence design must consider the vapor pressure and the thermal
the plant parameters which are important in determining stability of the reagents and protection of the equipment
the cost and quality of the finished product. Failures of the against excessive pressure due to process conditions, to
model are likely to be a result of: (1) oversimplification of runaway reactions, and to emergency situations arising
the equations that constitute the model, (2) inadequacy of in a fire or explosion. Process equipment is designed to
the numerical solution of the equations. withstand internal pressure or vacuum and must meet the
The solution to the first problem is limited by the in- requirements of Section VIII of the ASME (American So-
crease in time or the computer capacity available to solve ciety of Mechanical Engineers) code. Storage equipment
more complete or more advanced equations. The second usually withstands very low internal pressure or vacuum
problem is even more difficult to acknowledge. It may be and is constructed to meet the regulations of the ASME
due to error accumulation through the nonlinear domain. and the API (American Petroleum Institute) codes for stor-
The numerical solution of a differential equation is based age vessels. Safety and emergency relief equipment and
on the approximation of time and, in the case of PDEs, design must fulfill the requirements of the NFPA (National
space partial derivatives, by finite-difference equivalents. Fire Protection Association) codes and procedures. Safety