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15.2.3 Non-Adiabatic Extensions of the Non-Premixed Model

Many reacting systems involve heat transfer through wall boundaries, droplets, and/or particles. In such flows the local thermochemical state is no longer related only to $f$, but also to the enthalpy, $H$. The system enthalpy impacts the chemical equilibrium calculation and the temperature and species of the reacting flow. Consequently, changes in enthalpy due to heat loss must be considered when computing scalars from the mixture fraction, as in Equation  15.2-13.

In such non-adiabatic systems, turbulent fluctuations should be accounted for by means of a joint PDF, $p(f,H)$. The computation of $p(f,H)$, however, is not practical for most engineering applications. The problem can be simplified significantly by assuming that the enthalpy fluctuations are independent of the enthalpy level (i.e., heat losses do not significantly impact the turbulent enthalpy fluctuations). With this assumption, $p(f,H) = p(f) \delta(H-\overline{H})$ and mean scalars are calculated as

 \overline{\phi_i} = \int^1_0 \phi_i (f,\overline{H}) p(f) df (15.2-24)

Determination of $\overline{\phi_i}$ in the non-adiabatic system thus requires solution of the modeled transport equation for mean enthalpy:

 \frac{\partial}{\partial t} (\rho \overline{H}) + \nabla \cd... ...\left( \frac{ k_t }{ c_p } \nabla \overline{H} \right) + S_{h} (15.2-25)

where $S_{h}$ accounts for source terms due to radiation, heat transfer to wall boundaries, and heat exchange with the dispersed phase.

Figure  15.2.6 depicts the logical dependence of mean scalar values (species mass fraction, density, and temperature) on FLUENT's prediction of $\overline{f}$, $\overline{f^{'2}}$, and $\overline{H}$ in non-adiabatic single-mixture-fraction systems.

Figure 15.2.6: Logical Dependence of Averaged Scalars $\overline{\phi_i}$ on $\overline{f}$, $\overline{f^{'2}}$, $\overline{H}$, and the Chemistry Model (Non-Adiabatic, Single-Mixture-Fraction Systems)

When a secondary stream is included, the mean values are calculated from

 \overline{\phi_i} = \int^1_0 \int^1_0 \phi_i (f_{\rm fuel},p... ... p_1(f_{\rm fuel}) p_2(p_{\rm sec}) df_{\rm fuel} dp_{\rm sec} (15.2-26)

As noted above, the non-adiabatic extensions to the PDF model are required in systems involving heat transfer to walls and in systems with radiation included. In addition, the non-adiabatic model is required in systems that include multiple fuel or oxidizer inlets with different inlet temperatures. Finally, the non-adiabatic model is required in particle-laden flows (e.g., liquid fuel systems or coal combustion systems) when such flows include heat transfer to the dispersed phase. Figure  15.2.7 illustrates several systems that must include the non-adiabatic form of the PDF model.

Figure 15.2.7: Reacting Systems Requiring Non-Adiabatic Non-Premixed Model Approach

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