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.