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Carbon dioxide is a linear molecule and has four vibrational modes, but only two fundamental IR vibration frequencies \cite{Herzberg:1949}. It has five distinct bands that are included in RADCAL, see Table~\ref{Table::CO2}.
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\begin{table}[h!]
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\centering
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\caption{Spectral bands of $\rm CO_2$ included in RADCAL.}
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\caption[Spectral bands of $\rm CO_2$ included in RADCAL]{Spectral bands of $\rm CO_2$ included in RADCAL.}
Methane is a spherical top molecule of tetrahedral shape with the carbon atom occupying the center of the tetrahedron \cite{Herzberg:1949}. It belongs to the point group $T_d$. The methane IR spectrum is the result of the vibration-rotation modes of the $\rm C-H$ groups. It has nine vibrational modes, but due to its symmetry, this translates into only two distinct IR active fundamental vibration frequencies. In RADCAL, the methane IR spectrum is divided into three distinct bands (fundamentals + degenerates), see Table \ref{Table::CH4}.
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\begin{table}[ht]
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\centering
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\caption{Spectral bands of $\rm CH_4$ included in RADCAL.}
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\caption[Spectral bands of $\rm CH_4$ included in RADCAL]{Spectral bands of $\rm CH_4$ included in RADCAL.}
Ethylene is a molecule with a plane symmetrical form and belongs to the point group $D_{2h}$\cite{Herzberg:1949}. The ethylene IR spectrum is the result of the vibration-rotation modes of the $\rm C=C$, $\rm CH$, and $\rm CH_2$ groups. It has 12 vibrational modes. In RADCAL, its IR spectrum is divided into four distinct bands, see Table \ref{Table::C2H4}.
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\begin{table}[ht]
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\centering
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\caption{Spectral bands of $\rm C_2H_4$ included in RADCAL.}
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\caption[Spectral bands of $\rm C_2H_4$ included in RADCAL]{Spectral bands of $\rm C_2H_4$ included in RADCAL.}
Ethane has a three-fold axis of symmetry and belongs to the point group $D_{3d}$\cite{Herzberg:1949}. The ethane IR spectrum is the result of the vibration-rotation modes of the $\rm C-C$, $\rm CH$, and $\rm CH_2$ groups. It has 18 vibrational modes; its IR spectrum is divided into three distinct bands, see Table~\ref{Table::C2H6}.
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\begin{table} [ht]
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\centering
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\caption{Spectral bands of $\rm C_2H_6$ included in RADCAL.}
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\caption[Spectral bands of $\rm C_2H_6$ included in RADCAL]{Spectral bands of $\rm C_2H_6$ included in RADCAL.}
Propylene has only one plane of symmetry and belongs to the point group $C_s$\cite{Herzberg:1949}. The propylene IR spectrum is the result of the vibration-rotation modes of the $\rm C-C$, $\rm C=C$, $\rm CH$, $\rm CH_2$, and $\rm CH_3$ groups. It has 21 vibrational modes; its IR spectrum is divided into three distinct bands, see Table \ref{Table::C3H6}.
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\begin{table}[ht]
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\centering
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\caption{Spectral bands of $\rm C_3H_6$ included in RADCAL.}
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\caption[Spectral bands of $\rm C_3H_6$ included in RADCAL]{Spectral bands of $\rm C_3H_6$ included in RADCAL.}
Propane has two planes of symmetry and two axes of rotation. It belongs to the point group $C_{2v}$\cite{Herzberg:1949}. The propane IR spectrum is the result of the vibration-rotation modes of the $\rm C-C$, $\rm CH_2$, $\rm CH_3$ groups. It has 27 vibrational modes; its IR spectrum is divided into two distinct bands, see Table \ref{Table::C3H8}.
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\begin{table}[ht]
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\centering
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\caption{Spectral bands of $\rm C_3H_8$ included in RADCAL.}
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\caption[Spectral bands of $\rm C_3H_8$ included in RADCAL]{Spectral bands of $\rm C_3H_8$ included in RADCAL.}
Toluene has only one plane of symmetry. It belongs to the point group $C_{s}$\cite{III2011}. The toluene IR spectrum is the result of the vibration-rotation modes of the $\rm C=C$, $\rm CH$, and $\rm CH_3$ groups. It has 39 vibrational modes. For ease of modeling using statistical narrow band models, its IR spectrum has been divided into five distinct bands, see Table \ref{Table::C7H8}.
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\begin{table}[ht]
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\centering
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\caption{Spectral bands of $\rm C_7H_8$ included in RADCAL.}
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\caption[Spectral bands of $\rm C_7H_8$ included in RADCAL]{Spectral bands of $\rm C_7H_8$ included in RADCAL.}
\textit{n}-heptane has two planes of symmetry and two axes of rotation. It belongs to the point group $C_{2v}$\cite{III2011}. The \textit{n}-heptane IR spectrum results from the vibration-rotation modes of the $\rm C-C$, $\rm CH_2$, and $\rm CH_3$ groups. It has 63 vibrational modes. For ease of modeling using statistical narrow band models, its IR spectrum has been divided into two distinct bands, see Table \ref{Table::C7H16}.
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\begin{table}[ht]
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\centering
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\caption{Spectral bands of $\rm C_7H_{16}$ included in RADCAL.}
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\caption[Spectral bands of $\rm C_7H_{16}$ included in RADCAL]{Spectral bands of $\rm C_7H_{16}$ included in RADCAL.}
Methanol has only one plane of symmetry. It belongs to the point group $C_{s}$\cite{Herzberg:1949}. The methanol IR spectrum results from the vibration-rotation modes of the $\rm C-O$, $\rm OH$, and $\rm CH_3$ groups. It has 12 vibrational modes. For ease of modeling using statistical narrow band models, its IR spectrum has been divided into four distinct bands, see Table \ref{Table::CH3OH}.
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\begin{table}[ht]
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\centering
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\caption{Spectral bands of $\rm CH_3OH$ included in RADCAL.}
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\caption[Spectral bands of $\rm CH_3OH$ included in RADCAL]{Spectral bands of $\rm CH_3OH$ included in RADCAL.}
Methyl Methacrylate or MMA has the most complex IR spectrum of all the fuels presented above. With 15 atoms, it has 39 vibrational modes \cite{III2011}. The MMA IR spectrum results from the vibration-rotation modes of the $\rm C-O$, $\rm C=O$, $\rm C=C$, $\rm CH_2$, and $\rm CH_3$ groups. For ease of modeling using statistical narrow band models, its IR spectrum has been divided into six distinct bands, see Table \ref{Table::C5H8O2}.
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\begin{table}[ht]
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\centering
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\caption{Spectral bands of $\rm C_5H_8O_2$ included in RADCAL.}
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\caption[Spectral bands of $\rm C_5H_8O_2$ included in RADCAL]{Spectral bands of $\rm C_5H_8O_2$ included in RADCAL.}
\caption[FDS geometry used for developing the cone reference flux.]{\label{fig:cone_ref_geom} FDS geometry used for developing the cone reference flux. Red is the cone heater and green is the sample surface. The domain is clipped at the plane y=0~m.}
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\caption[FDS geometry used for developing the cone reference flux]{\label{fig:cone_ref_geom} FDS geometry used for developing the cone reference flux. Red is the cone heater and green is the sample surface. The domain is clipped at the plane y=0~m.}
\caption{Extinction criteria for the \ct{'EXTINCTION 1'} model.}
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\caption[Extinction criteria for the {\small\tt'EXTINCTION 1'} model]{Extinction criteria for the \ct{'EXTINCTION 1'} model.}
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\label{extinction_1_sketch}
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\end{figure}
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If $X_{\OTWO,ijk}<X_{\OTWO,\lim}$, local extinction is assumed and $\dot{m}_\alpha'''=0$ and $\dot{q}'''=0$ for that grid cell at that time step. At an ambient temperature of 20~$^\circ$C, the default limiting oxygen volume fraction is 0.135. This value is consistent with the measurements of Morehart et al.~\cite{Morehart:1991}, who measured the oxygen concentration near self-extinguishing flames. They found that flames self-extinguished at oxygen volume fractions of 12.4~\% to 14.3~\%. Note that their results are expressed as volume, not mass, fractions. Beyler's chapter in the SFPE Handbook references other researchers who measured oxygen concentrations at extinction ranging from 12~\% to 15~\%.
\caption[Illustration of the fan curve, system curve, and operation point for an HVAC fan.]{\label{hvac_curves} Illustration of the fan curve, system curve, and operation point for an HVAC fan.}
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\caption[Illustration of the fan curve, system curve, and operation point for an HVAC fan]{\label{hvac_curves} Illustration of the fan curve, system curve, and operation point for an HVAC fan.}
\caption[Illustration of determining duct runs.]{\label{HVAC_ductrun} Illustration determining duct runs. Left side has a single duct run. Right side has two duct runs.}
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\caption[Illustration of determining duct runs]{\label{HVAC_ductrun} Illustration determining duct runs. Left side has a single duct run. Right side has two duct runs.}
This is the default mode of FDS. For optically thin flames, however, where the yield of soot is small compared to the yields of $\rm CO_2$ and water vapor, the gray gas assumption can lead to an over-prediction of the emitted radiation. From a series of numerical experiments using methane as the fuel, it has been found that six bands ($N=6$) provide an accurate representation of the most important radiation bands of the fuel, $\rm CO_2$, and water vapor~\cite{Hostikka:3}. Table~\ref{banditos} through Table~\ref{band_MMA} list the band limits for various fuel species. The location of the bands have been adjusted to accommodate most of the features of the fuels spectra. If the absorption of the fuel is known to be important, separate bands can be reserved for fuel, increasing the total number of bands, $N$. The number of additional bands depends on the fuel, as discussed in Appendix~\ref{absorption_coefficients}.
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\begin{table}[p]
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\caption{Limits of the spectral bands for methane (CH$_4$).}
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\caption[Limits of the spectral bands for methane (CH$_4$)]{Limits of the spectral bands for methane (CH$_4$).}
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