11.3.3 Particle Reaction

The particle composition vector is represented as

where $Y_k$ is the $k$th specie mass fraction, $T$ is the temperature and $p$ the pressure.

For the reaction fractional step, the reaction source term is integrated as

where $S$ is the chemical source term. Most realistic chemical mechanisms consist of tens of species and hundreds of reactions. Typically, a reaction does not occur until an ignition temperature is reached, but then proceeds very quickly until reactants are consumed. Hence, some reactions have very fast time scales, in the order of $10^{-10}$ s, while others have much slower time scales, on the order of 1 s. This time-scale disparity results in numerical stiffness, which means that extensive computational work is required to integrate the chemical source term in Equation  11.3-8. In ANSYS FLUENT, the reaction step (i.e., the calculation of $\phi^1$) can be performed either by Direct Integration or by In-Situ Adaptive Tabulation (ISAT), as described in the following paragraphs.

A typical steady-state PDF transport simulation in ANSYS FLUENT may have 50000 cells, with 20 particles per cell, and requires 1000 iterations to converge. Hence, at least $10^9$ stiff ODE integrations are required. Since each integration typically takes tens or hundreds of milliseconds, on average, the direct integration of the chemistry is extremely CPU-demanding.

For a given reaction mechanism, Equation  11.3-8 may be considered as a mapping. With an initial composition vector $\phi^0$, the final state $\phi^1$ depends only on $\phi^0$ and the mapping time $\Delta t$. In theory, if a table could be built before the simulation, covering all realizable $\phi^0$ states and time steps, the integrations could be avoided by table look-ups. In practice, this a priori tabulation is not feasible since a full table in $N+3$ dimensions ( $N$ species, temperature, pressure and time-step) is required. To illustrate this, consider a structured table with $M$ points in each dimension. The required table size is $M^{N+3}$, and for a conservative estimate of $M=10$ discretization points and $N=7$ species, the table would contain $10^{10}$ entries.

On closer examination, the full storage of the entire realizable space is very wasteful because most regions are never accessed. For example, it would be unrealistic to find a composition of $Y_{\rm OH}=1$ and $T=300K$ in a real combustor. In fact, for steady-state, 3D laminar simulations, the chemistry can be parameterized by the spatial position vector. Thus, mappings must lie on a three dimensional manifold within the $N+3$ dimensional composition space. It is, hence, sufficient to tabulate only this accessed region of the composition space.

The accessed region, however, depends on the particular chemical mechanism, molecular transport properties, flow geometry, and boundary conditions. For this reason, the accessed region is not known before the simulation and the table cannot be preprocessed. Instead, the table must be built during the simulation, and this is referred to as in-situ tabulation. ANSYS FLUENT employs ISAT [ 277] to dynamically tabulate the chemistry mappings and accelerate the time to solution. ISAT is a method to tabulate the accessed composition space region "on the fly" (in-situ) with error control (adaptive tabulation). When ISAT is used correctly, accelerations of two to three orders of magnitude are typical. However, it is important to understand how ISAT works in order to use it optimally.