Page 85 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering

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

Batch Processing 49

This is done in simulating complex reactions such as poly- space change of certain variables is important, the descrip-
merization. The zeroth, first, and second moments of the tion of these process variables must be in terms of partial
molecular weight distribution are used to characterize the differential equations (PDEs). The changes of these vari-
kinetics of the reaction. ables may be interpreted only by distributed parameter
models (DPMs). An illustration is encountered in a system
of large viscosity where the temperature changes along the
B. Stochastic Modeling
direction of heat flow are as important as the time changes
Stochastic modeling is used when a measurable output is in determining product quality.
available but the inputs or causes are unknown or cannot Algebraic equations (AEs) describe relations among
be described in a simple fashion. The “black-box” ap- variables of the system which are independent of time
proach is used. The model is determined from past input or space changes. They represent variables not related by
and output data. An example is the description of incom- material energy or momentum balances. They may char-
plete mixing in a stirred tank reactor, which is done in acterize physicochemical or other type of relationships
terms of contributions of dead zones and short circuiting. between physically independent portions of the system,
In these cases, a sequence of output called a time series as the vapor and liquid spaces of a reaction vessel.
is known, but the inputs or causes are numerous and not
known; in addition, they may be unobservable. Though the
D. Single-Stage Modeling
causes for the response of the system are unknown, the de-
velopment of a model is important to gain understanding The model for a single unit or a train of equipment which
of the process, which may be used for future planning. is a part of a plant may consist of one or two PDEs, com-
Two modeling methods are current. The first is deter- bined with a few ODEs and some AEs. The mathematical
mining the parameters of a model of the form: solution of a model having more than two PDEs may be
beyond the budget of many industrial organizations. The
y(k) = f (k) + w(k), k = 1, 2,..., N (2)
PDEs stand for a temperature and a concentration or an-
where f (k) is a given function and w(k) represents a ran- othervariablewhosevariationsinspaceareasimportantas
dom noise sequence. The objective is to select the param- their changes in time for the proper characterization of the
eters in such a manner that w(k) is a zero-mean white system (i.e., for determining product quality). The ODEs
noise sequence of minimum possible variance. The com- represent the kinetic or dynamic changes of the unit with
mon forms of f (k) are a finite power series in k or a simple time, and the AEs indicate the equilibrium relationships
exponential. The second modeling method involves con- between the various phases present.
sidering that the time series is a linear transformation of Although the model is only for a single unit, it is by
a zero-mean white noise sequence. In simple cases, this no means simple. Its solution may be pictured as a plot
may be regarded as the output of a linear, time-invariant, whichshowsthechangesofthemeanconcentrationsofthe
discrete-time system subject to white noise input, and the reactants, some intermediates, and the products with time
object of the modeling is to estimate the parameters of during a batch operation. It will also indicate the space
the transfer function or the difference equation of the sys- distribution at various times of the temperature and the
tem. These are recognized as stationary time series mod- particular concentration which characterizes the product
els. When a model of this type is not adequate, it may be quality.
necessary to fit a difference equation model to the nth suc- Asingle-stagemodelisthesimplestmodelwhichcanbe
cessive differences of the given time series; this is called proposedtosimulateasingleunitoronetrainofequipment
a nonstationary time series model. that is part of a plant. The model represents the operation
of units or pieces of equipment which must run for the
same length of time.
C. System Modeling
Mathematical modeling of systems for which character- E. Multistage Modeling
istic variables are time-dependent only and not space-
dependent is done by ordinary differential equations Since a model of just a single unit of a plant is by itself
(ODEs). The situation is found in a nearly well-mixed complex, modeling of a complete plant, no matter how
batch reactor. There one may find differences in temper- simple it is, cannot be achieved unless the plant is split
ature or concentrations from one site to another due to into sections which are relatively independent from each
imperfect mixing. When space changes are not important other and small enough to be represented by single-stage
to the model, the process variables can be approximated models.Thus,plantmodelingisconvertedintoacollection
by means of lumped parameter models (LPMs). When the of single-stage models of which several may be handled