Page 132 - 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

754 Chemical Process Design, Simulation, Optimization, and Operation

The ij element of the Jacobian represents the partial integration time. These methods often require several nu-
derivativeofequationi withrespecttovariable j.Ifanalyt- merical iterations (usually a nonlinear algebraic equation
ical derivatives are not available, elements of the Jacobian is solved) for each integration step; Gear’s method is such
are obtained by perturbation of the state variable, requir- a procedure.
ing n + 1 function evaluations for an n-equation system of Most commonly used ordinary differential equation
equations. Various quasi-Newton techniques provide ap- (ODE) solvers provide options of several different inte-
proximations to the Jacobian and do not require as many gration techniques. Most solvers also automatically vary
function evaluations, thus reducing computational time. the integration step size during the simulation to allow the
In practice, the Jacobian matrix is not inverted; rather, best trade-off between accuracy and solution time, based
a set of linear algebraic equations is solved for x(k + 1) on user-specified numerical tolerances. There is no single
best integration technique—different methods work better
J(k)(x(k + 1) − x(k)) =− f (x(k)) (8)
for various problems.
Some process models have more than one feasible solu- So-called “stiff” differential equation models are par-
tion. Most numerical methods have local convergence, so ticularly challenging to solve. Stiff models have dynamic
the solution obtained is dependent upon the initial guess behavior that encompasses a wide range of time scales. An
for the solution before the first iteration. There is an ongo- example would be fast kinetics combined with long fluid-
ing effort to develop techniques that have global conver- residence times in a chemical reactor. Gear’s method is
gence or to find all solutions to multisolution problems. perhaps the most commonly used technique for solving
Some chemical process systems may have a single these types of problems.
steady state (single solution to a process model) under Differential algebraic equations commonly arise when
some design or operation conditions and multiple solu- physical property or kinetic expressions must be evaluated
tions under other design conditions. There are automatic in dynamic problems. These systems have the following
techniques to vary a parameter of a system model to deter- form:
mine when these solutions branch from a single solution
dx
to multiple solutions. The FORTRAN code AUTO is per- MxY= M = f (x)
dt
haps the most widely used code for this.
A dynamic bifurcation occurs when the dynamic be- where M is possibly singular. The most commonly used
havior of the solution to a system undergoes a qualitative software code to solve these types of problems is DASSL.
change. For example, a subcritical Hopf bifurcation oc-
curs when a dynamic system changes from a stable node
C. Partial Differential Equations
to a limit cycle. Again, AUTO can be used to determine
parameter changes that cause this bifurcation to occur. A common method for solving partial differential equa-
tions (PDEs) is known as the “method of lines.” Here,
finite difference approximations for spatial derivatives are
B. Ordinary Differential Equations
used to convert a PDE model to a large set of ordinary
Here we consider initial-value, ordinary differential equa- differential equations, which are then solved using any of
tions which often arise when modeling time-dependent the ODE integration techniques discussed earlier.
behavior of perfectly mixed systems. The general form is Typically, the numerical solutions techniques used are
dx very specific to the problem. Particularly challenging

xY= = f (x) problems include “moving front” problems where con-
dt
(9) centration profiles, for example, may vary widely over a
x 0 = x(0)
short distance but may not change much at other spatial
The explicit Euler integration technique involves specify- locations. The spatial discretization must be small close to
ing the integration step size, t: the front for accuracy and numerical stability, but must be
larger at other locations to reduce computation time. Var-
x(k + 1) = x(k) + tf (x(k)) (10)
ious adaptive grid techniques to change the spatial step
This method often requires very small integration step sizes have been developed for these problems. One of the
sizes to obtain a desired level of accuracy. Runge–Kutta more common codes to solve fluid-flow-related problems
integration has a higher level of accuracy than Euler. It is is FLUENT.
also an explicit integration technique, since the state val- In general, the numerical solution of PDEs is much
ues at the next time step are only a function of the previous more difficult to automate than the solution of initial-value
time step. Implicit methods have state variable values that ODEs. The best method to be used is very dependent on
are a function of both the beginning and end of the current the problem being solved.