
LookUp Tables for Adiabatic Systems
For an equilibrium, adiabatic, singlemixturefraction case, the mean temperature, density, and species fraction are functions of the and only (see Equations 15.216 and 15.221). Significant computational time can be saved by computing these integrals once, storing them in a lookup table, and retrieving them during the FLUENT simulation.
Figure 15.2.8 illustrates the concept of the lookup tables generated for a singlemixturefraction 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 15.216. There is one lookup 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 twodimensional lookup 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 lookup tables are too expensive since fourdimensional 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 15.212), and the PDF integrations (see Equation 15.214) are performed at runtime. This twodimensional table is illustrated in Figure 15.2.9.
3D LookUp Tables for NonAdiabatic Systems
In nonadiabatic systems, where the enthalpy is not linearly related to the mixture fraction, but depends also on wall heat transfer and/or radiation, a lookup table is required for each possible enthalpy value in the system. The result, for single mixture fraction systems, is a threedimensional lookup table, as illustrated in Figure 15.2.10, which consists of layers of twodimensional 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 threedimensional lookup table allows FLUENT to determine the value of each mass fraction, density, and temperature from calculated values of , , and . This threedimensional table in Figure 15.2.10 is the visual representation of the integral in Equation 15.224.
For nonadiabatic, twomixturefraction problems, it is unreasonable to tabulate and retrieve Equation 15.226 since fivedimensional tables are required. Instead, 3D lookup tables of the instantaneous state relationship given by Equation 15.214 are created. The 3D table in Figure 15.2.11 is the visual representation of Equation 15.214. The mean density during the FLUENT solution is calculated by integrating the instantaneous density over the fuel and secondary mixture fraction space (see Equation 15.226).

Note that the computation time in
FLUENT for a twomixturefraction case will be much greater than for a singlemixturefraction problem. This expense should be carefully considered before choosing the twomixturefraction model. Also, it is usually expedient to start a twomixturefraction simulation from a converged singlemixturefraction solution.
