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15.2.4 Chemistry Tabulation

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 $\overline{f}$ and $\overline{f'^2}$ 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 $\overline{f}$ and $\overline{f^{'2}}$ at a point in the flow domain, the mean value of mass fractions, density, or temperature ( $\overline{\phi_i}$) 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.

Figure 15.2.8: Visual Representation of a Look-Up Table for the Scalar $\overline{\phi_i}$ as a Function of $\overline{f}$ and $\overline{f^{'2}}$ in Adiabatic Single-Mixture-Fraction Systems

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 $\phi_i$ are tabulated as a function of the fuel mixture fraction $f_{\rm fuel}$ and the secondary partial fraction $p_{\rm sec}$ (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.

Figure 15.2.9: Visual Representation of a Look-Up Table for the Scalar $\phi_i$ as a Function of $f_{\rm fuel}$ and $p_{\rm sec}$ in Adiabatic Two-Mixture-Fraction Systems

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 $\overline{f}$, $\overline{f^{'2}}$, and $\overline{H}$. This three-dimensional table in Figure  15.2.10 is the visual representation of the integral in Equation  15.2-24.

Figure 15.2.10: Visual Representation of a Look-Up Table for the Scalar $\overline{\phi_i}$ as a Function of $\overline{f}$ and $\overline{f^{'2}}$ and Normalized Heat Loss/Gain in Non-Adiabatic Single-Mixture-Fraction Systems

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.

Figure 15.2.11: Visual Representation of a Look-Up Table for the Scalar $\phi_i$ as a Function of $f_{\rm fuel}$, $p_{\rm sec}$, and Normalized Heat Loss/Gain in Non-Adiabatic Two-Mixture-Fraction Systems

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