Merge pull request #13883 from mcgratta/master

FDS Tech Guide: Minor note about radiation

Manuals/FDS_Technical_Reference_Guide/Radiation_Chapter.tex

@@ -337,9 +337,9 @@ \subsection{Correction of the Emission Source Term}
 
 In calculations of limited spatial resolution, the source term, $I_{\rm b}$, defined in Eq.~(\ref{emission_source_term}) requires special treatment in the flaming region of the fire. Typical FDS calculations use grid cells that are tens of centimeters in size, and consequently the computed temperatures constitute a bulk average for a given grid cell and are considerably lower than the maximum temperature in a diffusion flame. Because of its fourth-power dependence on the temperature, the source term must be modeled in those grid cells where combustion occurs. Elsewhere, the computed temperature is used directly to compute the source term. It is assumed that this ``flaming region'' is where the local, nominal radiative loss is greater than a specified lower bound, $\chi_{\rm r} \dq'''>10$~kW/m$^3$. In this region, the global radiative fraction model is used. The emission source term is multiplied by a corrective factor, $C$:
 \be I_{\rm b,f}(\bx) = C \, \frac{\sigma \, T(\bx)^4}{\pi}   \quad ; \quad
-    C = \min \left( 100 \; , \; \max \left[ 1 \; , \; \frac{\sum_{\chi_{\rm r}\dq'''_{ijk}>10} \left( \chi_{\rm r} \, \dq'''_{ijk} + \kappa_{ijk} \, U_{ijk} \right) \, V_{ijk}}{\sum_{\chi_{\rm r} \dq'''_{ijk}>10} \left( 4 \, \kappa_{ijk} \, \sigma \, T^4_{ijk} \right) \, V_{ijk}} \right] \right) \label{corrected_emission_source_term}
+    C = \min \left( 100 \; , \; \max \left[0.1 \; , \; \frac{\sum_{\chi_{\rm r}\dq'''_{ijk}>10} \left( \chi_{\rm r} \, \dq'''_{ijk} + \kappa_{ijk} \, U_{ijk} \right) \, V_{ijk}}{\sum_{\chi_{\rm r} \dq'''_{ijk}>10} \left( 4 \, \kappa_{ijk} \, \sigma \, T^4_{ijk} \right) \, V_{ijk}} \right] \right) \label{corrected_emission_source_term}
 \ee
-When the source term defined in Eq.~(\ref{corrected_emission_source_term}) is substituted into Eq.~(\ref{net_emission}), the net radiative emission from the flaming region becomes the desired fraction of the total heat release rate. Note that this correction factor is bounded below by 1 and above by 100. The upper bound is arbitrary, meant to prevent spurious behavior at the start of a simulation.
+When the source term defined in Eq.~(\ref{corrected_emission_source_term}) is substituted into Eq.~(\ref{net_emission}), the net radiative emission from the flaming region becomes the desired fraction of the total heat release rate. Note that this correction factor is bounded below by 0.1 and above by 100. These bounds are somewhat arbitrary, meant to prevent spurious behavior at the start of a simulation.
 
 The radiative fraction, $\chi_{\rm r}$, is a useful quantity in fire science. It is the nominal fraction of the combustion energy that is emitted as thermal radiation. For most combustibles, $\chi_{\rm r}$ is between 0.3 and 0.4~\cite{Beyler2:SFPE}. However, in Eq.~(\ref{corrected_emission_source_term}), $\chi_{\rm r}$ is interpreted as the fraction of energy radiated from the flaming region.  For a small fire with a base diameter less than approximately 1~m, the local $\chi_{\rm r}$ is approximately equal to its global counterpart. However, as the fire increases in size, the global value will typically decrease due to a net re-absorption of the thermal radiation by the increasing smoke mantle~\cite{Takahashi:1}.