Look-Up Tables for Adiabatic Systems
For an equilibrium, adiabatic, single-mixture-fraction case, the mean temperature, density, and species fraction are functions of the and only (see Equations 15.2-16 and 15.2-21). Significant computational time can be saved by computing these integrals once, storing them in a look-up table, and retrieving them during the FLUENT simulation.
Figure 15.2.8 illustrates the concept of the look-up tables generated for a single-mixture-fraction system. Given FLUENT's predicted value for and at a point in the flow domain, the mean value of mass fractions, density, or temperature ( ) at that point can be obtained by table interpolation.
The table, Figure 15.2.8, is the mathematical result of the integration of Equation 15.2-16. There is one look-up table of this type for each scalar of interest (species mass fractions, density, and temperature). In adiabatic systems, where the instantaneous enthalpy is a function of only the instantaneous mixture fraction, a two-dimensional look-up table, like that in Figure 15.2.8, is all that is required.
For systems with two mixture fractions, the storage and interpolation costs of look-up tables are too expensive since four-dimensional tables would be necessary. Instead, the instantaneous properties are tabulated as a function of the fuel mixture fraction and the secondary partial fraction (see Equation 15.2-12), and the PDF integrations (see Equation 15.2-14) are performed at run-time. This two-dimensional table is illustrated in Figure 15.2.9.
3D Look-Up Tables for Non-Adiabatic Systems
In non-adiabatic systems, where the enthalpy is not linearly related to the mixture fraction, but depends also on wall heat transfer and/or radiation, a look-up table is required for each possible enthalpy value in the system. The result, for single mixture fraction systems, is a three-dimensional look-up table, as illustrated in Figure 15.2.10, which consists of layers of two-dimensional tables, each one corresponding to a normalized heat loss or gain. The first slice corresponds to the maximum heat loss from the system, the last slice corresponds to the maximum heat gain to the system, and the zero heat loss/gain slice corresponds to the adiabatic table. Slices interpolated between the adiabatic and maximum slices correspond to heat gain, and those interpolated between the adiabatic and minimum slices correspond to heat loss.
The three-dimensional look-up table allows FLUENT to determine the value of each mass fraction, density, and temperature from calculated values of , , and . This three-dimensional table in Figure 15.2.10 is the visual representation of the integral in Equation 15.2-24.
For non-adiabatic, two-mixture-fraction problems, it is unreasonable to tabulate and retrieve Equation 15.2-26 since five-dimensional tables are required. Instead, 3D look-up tables of the instantaneous state relationship given by Equation 15.2-14 are created. The 3D table in Figure 15.2.11 is the visual representation of Equation 15.2-14. The mean density during the FLUENT solution is calculated by integrating the instantaneous density over the fuel and secondary mixture fraction space (see Equation 15.2-26).
| Note that the computation time in
FLUENT for a two-mixture-fraction case will be much greater than for a single-mixture-fraction problem. This expense should be carefully considered before choosing the two-mixture-fraction model. Also, it is usually expedient to start a two-mixture-fraction simulation from a converged single-mixture-fraction solution.