(341c) Comparison of a Fully Coupled and a Decoupled Solver for the Simulation of Fluid Catalytic Cracking

The ongoing research effort presented here aims at finding the most
time-effective way to solve reactive flow problems. The target is to be able to
perform steady-state simulations of important chemical processes involving
multiphase flow and detailed kinetics with minimal CPU time requirements.

We focus on the Fluid Catalytic Cracking process, as a benchmark to assess the
performance of our steady-state solvers. Fluid Catalytic Cracking (or
FCC) is a major conversion process in the petrochemical industry,
used extensively to convert heavy crude oil fractions to lighter, commercially
more valuable, transport fuels. It is a prime example of a multiphase process,
involving gas, solid and liquid phases. It is characterized by high feed
concentrations and important density variations, a feature which sets it apart
from combustion, where the heat effect, rather than the volumetric expansion
effect, of the reactions is of major concern for the numerical scheme.

The FCC process was chosen for several reasons. The research group
has built up experience with this process, both in the area of modelling and
simulation. It is a numerically challenging process. The density variations
induced by the reactions are found to have an important impact on the
robustness of the numerical scheme. Furthermore, the numerical behaviour of
this process is sufficiently different from problems in combustion, an area of
active research, to warrant separate attention. And, not in the least, it is a
process of relevance to chemical engineering.

Finding the least time-consuming way to solve the equations of reactive flow is
the long-term target of this research. In this presentation, two approaches
which have led to concrete results will be discussed. The first approach is a
fully coupled solver in line with traditional Navier-Stokes solvers for
compressible, nonreactive flow. The second approach is a generalization of the
post-processing approach, frequently used for the a posteriori
imposition of chemical reactions on a previously determined flow field.

Fully coupled solver

In the first approach, the equations for reactive flow are all treated alike.
This means that the Navier-Stokes equations and the continuity equations are
discretized and integrated simultaneously. The ultimate result of the
discretization of the original partial differential equations, is a set of
coupled linear algebraic equations. Each of these algebraic equations expresses
the changes in the state variables due to convection, diffusion, reaction and
other source terms. Because the state variables are linked to one another, a
matrix equation has to be solved to obtain the changes for the individual state
variables. Obviously there are as many equations as there are state variables,
and, hence, the coefficient matrix has size n-by-n, n
representing the number of state variables. It is to be noted that the
computational work for solving a matrix equation with an n-by-n
matrix is at least of the order n². This implies that
the work is more than proportional to the number of equations, a point that
will become of importance as the number of reactive components increases.

Simulations of the FCC process with the fully coupled solver indicate
that it is a stable and robust solver. By using an implicit treatment of the
source terms, the solver copes well with the numerical stiffness caused by the
chemical reactions, and the iteration count is nearly identical to the
iteration count of a solver for nonreactive flow.

The main advantage of the fully coupled approach is that it is a logical
extension of a Navier-Stokes equation solver, in which the additional
continuity equations are discretized and integrated in exactly the same way as
the Navier-Stokes equations. The solver apparently copes with issues such as
the density variations, to be discussed subsequently, without any problems. Its
most serious disadvantage is that for large numbers of reactive components,
i.e. for complex kinetic schemes, the coefficient matrix becomes too large.
Since the work involved in solving the matrix equation is at least proportional
to n², it follows that any reduction in the size of this
matrix is rewarded with a more-than-proportional decrease in execution time.
This is the major motivation for the algebraically decoupled approach discussed
next.

Algebraically decoupled solver

The algebraically decoupled approach differs from the fully coupled approach,
in that the Navier-Stokes equations and the continuity equations are decoupled
at the algebraic level. Consider the coefficient matrix of the fully coupled
solver. If certain elements of the coefficient matrix are set to zero, the
equations decouple algebraically. The matrix equation with the modified
coefficient matrix can be split in two matrix equations, each with a smaller
coefficient matrix. Evidently, the total number of algebraic equations remains
the same. However, the computational work involved is reduced
significantly.

The algebraically decoupled approach is a generalization of the post-processing
approach which is often used for the `superposition' of chemical reactions on a
previously determined flow field. Often, a flow field is determined using a CFD
package, excluding chemical reactions, producing density, pressure, temperature
and velocity fields that are then used as an input to another package that
takes care of the chemical reactions. Provided the reactions do not influence
the flow field significantly, this chemical post-processing is then the last
stage of a reactive flow simulation. The generalization consists in
transferring control from the flow part to the reaction part (and vice versa)
several times, instead of just once. The post-processing approach is a limiting
case of this, consisting of one pass for the Navier-Stokes equations, followed
by one pass for the continuity equations.

Simulations of the FCC process with the algebraically decoupled
solver are found not be successful. The experience gained with the fully
coupled solver is now of value to understand the cause. It is found that some
elements in the coefficient matrix, set to zero in the algebraically decoupled
solver, in fact do not have negligible values. It is exactly this which causes
the failure of the algebraically decoupled solver for the FCC
process. However, the value of those elements can be quantified, and linked to
operating conditions and/or physical properties. Hence, we can derive an
objective criterion which can be calculated a priori, to determine the
success of the application of the algebraically decoupled solver.

The advantage of the algebraically decoupled approach is that it is relatively
straightforward from a conceptual and technical point of view, compared to
other methods of decoupling. Since the continuity equations are of the same
nature as the Navier-Stokes equations, viz. of
convection-diffusion-source type, no different discretization or integration
methods need to be used. Also, the same boundary conditions as in the fully
coupled solver can be applied. A disadvantage is that the algebraically
decoupled solver loses its competitive edge over the fully coupled solver if
the number of reactive components grows large. Another disadvantage is that
this decoupled solver does not succeed unconditionally. However, the conditions
for successful application can be quantified, and linked to physical properties
and/or operating conditions, quantities which are known a priori. Since the
post-processing approach is a limiting case of this algebraically decoupled
approach, the same conditions apply to it, making the results obtained here
more generally useful for assessing the feasibility of chemical post-processing
on given flow fields.