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\journal{Journal of Power Technologies}
\begin{document}
\begin{frontmatter}
\title{Partial Pressures Weighted-Sum-of-Gray-Gases Model for radiation numerical modeling in combusting flows}
\author[wut]{Author1\corref{cor1}}
\ead{Author1@xxx}
\author[wut]{Author2}
\ead{Author2@xxx}
\cortext[cor1]{Corresponding author}
\address[wut]{University, Address}
\begin{abstract}
Commercially available Weighted-Sum-of-Gray-Gases Model for radiation numerical modeling in combustion simulations base on temperature dependent weighing factors. Factors obtained by polynomial fitting of experimental data for only two of reagents (H$_{2}$O and CO$_{2}$) to analytical equation for emissivity. In this article the Partial Pressures Weighted-Sum-of-Gray-Gases concept is developed. Five additional combustion species are added to proposed model: CH$_{4}$, CO, NO, N$_{2}$O and NH$_{3}$. The aim of this model is to improve the initial solution of temperature and species mass fraction profiles in non-premixed combustion of methane numerical modeling. The proposed model is validated on turbulent non-premixed methane combustion with swirling air experimental data. Physical implications of the model performance are discussed.
\end{abstract}
\begin{keyword}
radiation modeling \sep combustion \sep absorption coefficient \sep weighted sum of gray gases
%% keywords here, in the form: keyword \sep keyword
%% PACS codes here, in the form: \PACS code \sep code
%% MSC codes here, in the form: \MSC code \sep code
%% or \MSC[2008] code \sep code (2000 is the default)
\end{keyword}
\end{frontmatter}
%% \linenumbers
%% main text
\section{Introduction}
It has been proven \cite{it1, it2, it3, it4, it5, it6, it7} that, mainly due to high temperatures, radiative heat transfer has a significant impact on combustion parameters. Therefore accurate prediction of temperatures and hence on emissions in combustion numerical modeling is strongly dependent on exact radiation modeling. The problem is particularly challenging in the combustion modeling in turbine engines \cite{it2,it4,it6}. In commercial CFD codes Weighted-Sum-of-Gray-Gases Model based on polynomial regression of experimental data for H$_{2}$O and CO$_{2}$ mixture is implemented, which are main but not the only radiating species for hydrocarbon flames. \par
The aim of this article is to calculate the the absorption coefficient as a weighted sum of absorption coefficients of particular combustion reactants and products based on their partial pressures. Some efforts has been made to describe mean absorption coefficient by the weighted average of gray gases absorption coefficients based on partial pressures. In \cite{it8}, absorption coefficients were calculated using RADCAL software \cite{it9} and only for part of combustion reagents: H$_{2}$O, CO$_{2}$, CH$_{4}$ and CO. It was observed \cite{it10}, that in OPPDIF (opposed-flow diffusion flame) calculations (modified to include radiation) of methane/air flames, inclusion of CH$_{4}$ and CO dropped the peak temperature by 5K and on the rich side of the same flames the maximum observed effect of adding the CH$_{4}$ and CO radiation was an 8K reduction in temperature (from 1280K to 1272K for a particular location) in comparison with model including only H$_{2}$O and CO$_{2}$ mixture. \par
The approach presented in this article assumes use of the Planck-mean absorption coefficient based on high-resolution HITRAN database \cite{it11} for: H$_{2}$O, CO$_{2}$, CH$_{4}$, CO, NO, N$_{2}$O and NH$_{3}$. These seven species have been selected because of their vital role in the combustion modeling. Nowadays studies of emissions of CO and nitrogen-containing species, mainly due to requirements for cleaner combustion systems, require advanced modeling of the above species concentrations and hence their participation in the radiative heat transfer. Proposed model aims to enable more accurate (in comparison with currently used WSGGM model) initial solutions of combusting flow fields. \par
Partial pressures of particular gases are used as an average wide-band absorption coefficient weights. Approach describe in this article is planned to be used in numerical simulation of combustion inside small turbine engine combustion chamber. Proposed, partial pressures WSGG Model is implemented into the ANSYS FLUENT commercial CFD code. Numerical results are validated by comparison with experimental data. The test case is turbulent non-premixed methane combustion with swirling air. Experimental data are obtained from the research conducted at University of Sydney \cite{it12,it13,it14,it15,it16}.
\section{Radiation numerical modeling}
Currently, one of the most commonly used models of radiation for combustion modeling, is Discrete Ordinates (DO) model. Discrete Ordinates model became popular in combustion modeling mainly due to moderate computational cost and modest memory requirements. The DO model transforms radiation transfer equation (RTE) into a transport equation for radiation intensity in the $n$-dimensional coordinates. Model considers RTE in the direction $\vec{s}$ as field equation \cite{it17}:
\begin{eqnarray} \label{eq1}
& \Delta(I(\vec{r},\vec{s})\vec{s})+(\kappa+\sigma_{s})I(\vec{r},\vec{s})= \nonumber \\
& \kappa n^{2}\frac{\sigma T^{4}}{\pi}+\frac{\sigma_{s}}{4\pi}\int_{0}^{4\pi}I(\vec{r},\vec{s}')\Phi(\vec{s},\vec{s}')d\Omega'
\end{eqnarray}
where:
$\vec{r}$ - positional vector,
$\vec{s}$ - directional vetor,
$\vec{s}'$ - scattering direction vector,
$s$ - path length,
$\kappa$ - absorption coefficient,
$n$ - refractive index,
$\sigma_{s}$ - scattering coefficient,
$\sigma$ - Stefan – Boltzmann constant,
$I$ - radiation intensity,
$\Phi$ - phase function,
$\Omega'$ - solid angle.
\par
As can be noticed from \ref{eq1} the DO model requires absorption coefficient as an input. Absorption coefficient can be either constant or dependent on local species concentrations, path length and total pressure. In ANSYS Fluent, so called Weighted-Sum-of-Gray-Gases-Model (WSGGM) is implemented. This is a default choice for modeling radiation in combusting flows. It has been developed as a compromise between the oversimplified gray gas model and a complete model which takes into account particular absorption bands.
\par
WSGGM approach assumes that total emissivity over the distance s can be described by:
\begin{equation} \label{eq2}
\epsilon=\sum_{i=0}^{n} a_{\epsilon,i}(T)(1-e^{-\kappa_{i}ps})
\end{equation}
where:
$a_{\epsilon,i}$ - emissivity weighting factor for $i^{th}$ gas,
$\kappa_{i}$ - absorption coefficient of $i^{th}$ gas,
$p$ - sum of partial pressures of all absorbing gases,
$s$ - path length.
\par
For the $i=0$ absorption coefficient value is set to zero $(k_{0}=0)$, simultaneously the weighing factor for $i=0$ is set as complement to unity:
\begin{equation} \label{eq3}
a_{\epsilon,0}=1-\sum_{i=1}^{n}a_{\epsilon,i}
\end{equation}
In the case of remaining gases, the weighing factor is temperature dependent and is approximated by:
\begin{equation} \label{eq4}
a_{\epsilon,i}=1-\sum_{j=1}^{N}b_{\epsilon,i,j}T^{j-1}
\end{equation}
\par where: $b_{\epsilon,i,j}$ are a gas temperature polynomial coefficients, which like $\kappa_{i}$ is obtained by fitting \ref{eq2} to experimental data.
ANSYS Fluent finally calculates absorption coefficient as:
\begin{equation} \label{eq5}
\kappa=\frac{ln(1-\epsilon)}{s}\mbox{: for }s>10^{-4}\mbox{ m}
\end{equation}
\begin{equation} \label{eq6}
\kappa=\sum_{j=1}^{N}a_{\epsilon,i}\kappa_{i}p\mbox{: for }s\leq10^{-4}\mbox{ m}
\end{equation}
\par Traditional WSGGM utilizes use of two separate formulation depending on the path length. As it was mentioned, ANSYS Fluent uses experimental data form \cite{it18} and \cite{it19} where polynomial functions has been fitted to experimental data only for CO$_{2}$ and H$_{2}$O particular mixtures.
\section{Proposed approach}
Proposed approach assumes that in every computational cell there is a gray gas consisting of combustion reagents mixture. Each combustion reagent occupies its own portion of the cell. Ratio of radiation path lengths of each specie to radiation path lengths of whole cell is equal to partial volume of each specie to total volume of the cell ratio. Based on Amagat’s law of additive volume for ideal gas:
\begin{equation} \label{eq7}
\frac{V_{i}}{V_{tot}}=\frac{x_{i}}{x_{tot}}=\frac{p_{i}}{p_{tot}}
\end{equation}
where:
$V_{i}$ - partial volume of the $i^{th}$ gas mixture component,
$V_{tot}$ - total volume of gas mixture,
$p_{i}$ - partial pressure of the $i^{th}$ gas mixture component,
$p_{tot}$ - total pressure of gas mixture,
$x_{i}$ - amount of moles of the $i^{th}$ gas mixture component,
$x_{tot}$ - total amount of moles in gas mixture.
\par
For such mixture, the radiative heat flux loss per unit volume can be calculate as:
\begin{equation} \label{eq8}
q_{loss}=\sum_{i=1}^{N}p_{i}\kappa_{i}\sigma(T^4-T_{b}^4)
\end{equation}
where:
$\kappa_{i}$ - Planck mean absorption coefficient for $i^{th}$ species $\mbox{ (m Pa)}^{-1}$,
$T$- cell temperature.
Putting \ref{eq7} to \ref{eq8}:
\begin{equation} \label{eq9}
q_{loss}=\sum_{i=1}^{N}\frac{x_{i}}{x_{tot}}p_{tot}\kappa_{i}\sigma(T^4-T_{b}^4)
\end{equation}
Marking mixture weighted absorption coefficient as $\kappa_{mix} \mbox{ (m)}^{-1}$, based on \ref{eq9} we can write:
\begin{equation} \label{eq10}
\kappa_{mix}=\sum_{i=1}^{N}\frac{x_{i}}{x_{tot}}p_{tot}\kappa_{i}
\end{equation}
Planck mean absorption coefficients for individual gas species are calculated based on data from HITRAN \cite{it11} high-resolution database as \cite{it20}:
\begin{equation} \label{eq11}
\kappa_{i}=\frac{\pi}{\sigma T^{4}}\int_{0}^{\infty}I_{b\eta}\sum_{j}\kappa_{\eta j}d\eta=\sum_{j}\left(\frac{\pi I_{b\eta 0}}{\sigma T^{4}}\right)S_{j}
\end{equation}
where:
$\eta$ - wavenumber $\mbox{ (m)}^{-1}$,
$S_{j}$ - $j^{th}$ line integrated absorption coefficient,
$I_{b\eta}$ - blackbody radiative intensity for given wave number ($I_{b\eta 0}$ is evaluated at the center of each spectral line),
$\kappa_{\eta j}$ - absorption coefficient for given wavenumber.
\par Individual functions were fitted to the experimental data. Function were fitted using Curve Fitting Tool (cftool) from Matlab. A number of different types of fit have been tried: Exponential, Fourier, Gaussian, Polynomial, Power and Sum of Sin function. Best fit was judged based on sum of squared errors of prediction – $SSE$ (\ref{eq12}) and coefficient of determination - $R^{2}$ (\ref{eq13}):
\begin{equation} \label{eq12}
SSE=\sum_{i=1}^{n}\left(y_{i}-f(x_{i})\right)^2
\end{equation}
\begin{equation} \label{eq13}
R^{2}=1-\frac{\sum_{i}\left(y_{i}-f(x_{i})\right)^2}{\sum_{i}\left(y_{i}-\overline{y}\right)^2}
\end{equation}
where:
$y_{i}$ - $i^{th}$ value of variable,
$x_{i}$ - $i^{th}$ value of the explanatory variable,
$f(x_{i})$ - predicted value of $y_{i}$,
$\overline{y}$ - mean of the observed data:
\begin{equation} \label{eq14}
\overline{y}=\frac{1}{n}\sum_{i=1}^{n}y_{i}
\end{equation}
where $n$ is the number of observations.
\par Table \ref{table:tab1} contains the $SSE$ and $R^2$ for each of for each of the considered fit functions. Shaded cells indicate chosen fit function. Fitting function coefficients are presented in Table \ref{table:tab2} for individual species.
Where fit functions are described by following functions \cite{it21}: \\
Exponential:
\par-one-term:
\begin{equation} \label{eq15}
\kappa(T)= \alpha e^{\beta T}
\end{equation}
\par-two-term:
\begin{equation} \label{eq16}
\kappa(T)=\alpha e^{\beta T}+\gamma e^{\delta T}
\end{equation}
Fourier $n^{th}$ degree:
\begin{equation} \label{eq17}
\kappa(T)=\sum_{i=0}^{n}\left[\alpha_{i} \cos(i \omega T)+\beta_{i} \sin(i \omega T)\right]
\end{equation}
Gaussian n-th degree:
\begin{equation} \label{eq18}
\kappa(T)=\sum_{i=0}^{n} \alpha_{i} e^{-\left(\frac{T-\beta_{i}}{\gamma_{i}}\right)^{2}}
\end{equation}
Polynomial n-th degree:
\begin{equation} \label{eq19}
\kappa(T)=\sum_{i=0}^{n} p_{i}x^{n+1-i}
\end{equation}
Power:
\par -one-term:
\begin{equation} \label{eq20}
\kappa(T)=\alpha T^{\beta}
\end{equation}
\par -two-term:
\begin{equation} \label{eq21}
\kappa(T)=\alpha + \beta T^{\gamma}
\end{equation}
Sum of Sines:
\begin{equation} \label{eq22}
\kappa(T)=\sum_{i=0}^{n} \alpha_{i}sin\left(\beta_{i}T+\gamma_{i}\right)
\end{equation}
Obtained Planck mean absorption coefficients are presented in the form of points in the Figure \ref{fig:fig1} and Figure \ref{fig:fig2} for temperature range 300-2500K.
\begin{figure}[h]
\centering
\captionsetup{justification=centering}
\includegraphics[width=7cm]{Figure1.png}
\caption{Planck mean absorption coefficient fitting functions against experimental data for H$_{2}$O, CO$_{2}$, N$_{2}$O and NH$_{3}$.}
\label{fig:fig1}
\end{figure}
\begin{figure}[h]
\centering
\captionsetup{justification=centering}
\includegraphics[width=7cm]{Figure2.png}
\caption{Planck mean absorption coefficient fitting functions against experimental data for CO, NO and CH$_{4}$.}
\label{fig:fig2}
\end{figure}
Equation \ref{eq10} is valid for all of combustion species. Since in this article seven combustion species are considered, direct usage of the same equation assign absorption coefficient of zero for the remaining ingredients. In reality the remaining species have nonzero absorption coefficient. Based on HITRAN database review \cite{it11} combustion species other than H$_{2}$O, CO$_{2}$, N$_{2}$O, NH$_{3}$ and CO, NO and CH$_{4}$ (important due to its participation in pollutant formation) have negligible Planck mean absorption coefficient. Base on above, assumption that the absorption coefficients of remaining species are equal to zero, is an appropriate approximation of reality. Therefore absorption coefficient is calculated by \ref{eq10}. Proposed approach has been successfully programmed in C language and implemented into commercial ANSYS FLUENT code.
\begin{table*}[t]
\captionsetup{justification=centering}
\caption{Sum of squared errors of prediction and coefficient of determination for individual fitting function to experimental data.}
\label{table:tab1}
\centering
\begin{tiny}
\begin{tabular}{cccccccccccccc}
\noalign{\hrule height 1pt}
\multicolumn{2}{c}{\textbf{}} & \multicolumn{2}{c}{\textbf{Exponential}} & \multicolumn{2}{c}{\textbf{Fourier}} & \multicolumn{2}{c}{\textbf{Gaussian}} & \multicolumn{2}{c}{\textbf{Polynomial}} & \multicolumn{2}{c}{\textbf{Power}} & \multicolumn{2}{c}{\textbf{Sum of Sin fun}} \\
\textbf{Species} & \textbf{Indicator} & \textbf{1-term} & \textbf{2-term} & \textbf{4-deg} & \textbf{5-deg} & \textbf{4-deg} & \textbf{5-deg} & \textbf{4deg} & \textbf{5deg} & \textbf{1-term} & \textbf{2-term} & \textbf{4-deg} & \textbf{5-deg} \\
\noalign{\hrule height 1pt}
& \textbf{$SSE$} & 1525 & 116.7 & 7.858 & 1.855 & 1.964 & \cellcolor[HTML]{C0C0C0}0.310 & 87.35 & 75.48 & 2881 & 994.1 & 17.16 & 7.999 \\
\multirow{-2}{*}{\textbf{CO$_{2}$}} & \textbf{$R^2$} & 0.761 & 0.982 & 0.999 & 1.000 & 1.000 & \cellcolor[HTML]{C0C0C0}1.000 & 0.986 & 0.988 & 0.549 & 0.844 & 0.997 & 0.999 \\
\hline
& \textbf{$SSE$} & 0.967 & 0.999 & 0.999 & 0.473 & 0.591 & \cellcolor[HTML]{C0C0C0}0.003 & 147.2 & 58.88 & 5.945 & 5.847 & 75.13 & 39.37 \\
\multirow{-2}{*}{\textbf{H$_{2}$O}} & \textbf{$R^2$} & 165.3 & 3.287 & 3.527 & 1.000 & 1.000 & \cellcolor[HTML]{C0C0C0}1.000 & 0.979 & 0.988 & 0.999 & 0.999 & 0.985 & 0.992 \\
\hline
& \textbf{$SSE$} & 23.83 & 0.321 & 0.052 & 0.023 & 0.051 & \cellcolor[HTML]{C0C0C0}0.012 & 1.848 & 0.181 & 37.26 & 17.65 & 0.052 & 0.035 \\
\multirow{-2}{*}{\textbf{CO}} & \textbf{$R^2$} & 0.621 & 0.995 & 0.999 & 1.000 & 0.999 & \cellcolor[HTML]{C0C0C0}1.000 & 0.971 & 0.997 & 0.407 & 0.719 & 0.999 & 0.999 \\
\hline
& \textbf{$SSE$} & 0.588 & 0.024 & 0.013 & 0.001 & \cellcolor[HTML]{C0C0C0}0.000 & 0.000 & 0.000 & 0.000 & 10.6 & 0.478 & 0.007 & 0.016 \\
\multirow{-2}{*}{\textbf{CH$_{4}$}} & \textbf{$R^2$} & 0.994 & 1.000 & 1.000 & 1.000 & \cellcolor[HTML]{C0C0C0}1.000 & 1.000 & 1.000 & 1.000 & 0.892 & 0.995 & 1.000 & 1.000 \\
\hline
& \textbf{$SSE$} & 10.14 & 9.052 & \cellcolor[HTML]{C0C0C0}1.415 & 3.420 & 38.230 & 31.69 & 32.91 & 6.438 & 175 & 96.49 & 4.408 & 9.124 \\
\multirow{-2}{*}{\textbf{NH$_{3}$}} & \textbf{$R^2$} & 0.998 & 0.999 & \cellcolor[HTML]{C0C0C0}1.000 & 1.000 & 0.994 & 0.995 & 0.995 & 0.999 & 0.972 & 0.985 & 0.999 & 0.999 \\
\hline
& \textbf{$SSE$} & 7.412 & 0.106 & 0.021 & \cellcolor[HTML]{C0C0C0}0.012 & 0.078 & 0.041 & 1.300 & 0.230 & 12.88 & 6.237 & 0.068 & 0.016 \\
\multirow{-2}{*}{\textbf{NO}} & \textbf{$R^2$} & 0.728 & 0.996 & 0.999 & \cellcolor[HTML]{C0C0C0}1.000 & 0.997 & 0.999 & 0.952 & 0.991 & 0.527 & 0.771 & 0.998 & 0.999 \\
\hline
& \textbf{$SSE$} & 677 & 19.870 & 1.481 & 1.164 & 2.949 & \cellcolor[HTML]{C0C0C0}0.514 & 88.06 & 8.400 & 1362 & 605.5 & 1.507 & 1.424 \\
\multirow{-2}{*}{\textbf{N$_{2}$O}} & \textbf{$R^2$} & 0.819 & 0.995 & 1.000 & 1.000 & 0.999 & \cellcolor[HTML]{C0C0C0}1.000 & 0.976 & 0.998 & 0.635 & 0.838 & 1.000 & 1.000 \\
\noalign{\hrule height 1pt}
\end{tabular}
\end{tiny}
\end{table*}
\begin{table}
\captionsetup{justification=centering}
\caption{Fitting function coefficients.}
\label{table:tab2}
\centering
\begin{tiny}
\begin{tabular}{@{}lccccccc@{}}
\toprule
\noalign{\hrule height 1pt}
\multicolumn{1}{c}{\textbf{}} & \textbf{CO$_{2}$} & \textbf{H$_{2}$O} & \textbf{CO} & \textbf{CH$_{4}$} & \textbf{NH$_{3}$} & \textbf{NO} & \textbf{N$_{2}$O} \\ \midrule
\noalign{\hrule height 1pt}
\textbf{$\alpha_{0}$} & --- & --- & --- & --- & 27.57 & 0.04736 & --- \\
\textbf{$\alpha_{1}$} & 2.129 & 8.05E+06 & 1.332 & 1.695 & 44.05 & -1.021 & 9.123 \\
\textbf{$\beta_{1}$} & 634.7 & -1053 & 542.4 & 370.9 & -9.732 & -1.098 & 415.8 \\
\textbf{$\gamma_{1}$} & 82.51 & 370 & 154.2 & 193.7 & --- & --- & 193.2 \\
\textbf{$\alpha_{2}$} & 4.428 & 2952 & 0.7574 & 3.147 & 22.34 & -1.228 & 5.47 \\
\textbf{$\beta_{2}$} & 673.9 & -1277 & 424.5 & 553.6 & -12.64 & -0.6306 & 675.5 \\
\textbf{$\gamma_{2}$} & 242.7 & 712.4 & 103 & 440.2 & --- & --- & 209.7 \\
\textbf{$\alpha_{3}$} & 3.393 & 44.61 & 1.681 & 1.086 & 6.366 & -0.7685 & 17.23 \\
\textbf{$\beta_{3}$} & 522.8 & -879.1 & 706.6 & 741.1 & -7.845 & -0.2624 & 625.4 \\
\textbf{$\gamma_{3}$} & 80.4 & 1127 & 256.4 & 772.1 & --- & --- & 556.2 \\
\textbf{$\alpha_{4}$} & 8.35E+05 & 0 & 1.256 & 0 & 0.5755 & -0.3156 & 1.398 \\
\textbf{$\beta_{4}$} & -1.24E+04 & 738.6 & 929.9 & 220.4 & -2.236 & -0.09312 & 1642 \\
\textbf{$\gamma_{4}$} & 4182 & 1.44 & 419.1 & 10.41 & --- & --- & 653.3 \\
\textbf{$\alpha_{5}$} & -58.47 & 3.815 & 0.7134 & --- & --- & -0.08977 & -3.901 \\
\textbf{$\beta_{5}$} & 190.8 & -158.7 & 1296 & --- & --- & -0.02962 & 330.1 \\
\textbf{$\gamma_{5}$} & 392.6 & 1980 & 838.5 & --- & --- & --- & 114.6 \\
\textbf{$\omega$} & --- & --- & --- & --- & 0.001738 & -0.001958 & --- \\ \bottomrule
\noalign{\hrule height 1pt}
\end{tabular}
\end{tiny}
\end{table}
\section{Results and discission}
Comparison and evaluation of the proposed Partial Pressures Weighted-Sum-of-Gray-Gases Model was carried out on the basis of Swirling Turbulent Non-premixed Flames of Methane experiment \cite{it12,it13,it14,it15,it16} which is official validation test case \cite{it22} for ANSYS Fluid Dynamics software. In the experiment methane and swirling jet air are supplied from separate inlets of the burner. Around the swirling air jet, the non-swirling co-flow of air is present.
Geometry and boundary conditions for numerical simulations are the same as for experiment. The methane inlet has a diameter of 3.6 mm, the swirling air inlet an inner diameter 50 mm and an outer diameter of 60 mm. Area of co flow was limited to cylinder of 310 mm diameter and length of 950 mm. Sine 2D axisymmetric swirl numerical calculations are performed, the computational domain is modeled as half of cylinder slice (Figure~\ref{fig:fig3}).
\begin{figure}
\centering
\captionsetup{justification=centering}
\includegraphics[width=7cm]{Figure3.png}
\caption{Swirling Turbulent Non-premixed Flames of Methane experiment.}
\label{fig:fig3}
\end{figure}
\begin{figure}
\centering
\captionsetup{justification=centering}
\includegraphics[width=7cm]{Figure4.png}
\caption{Computational grid for numerical calculations in the methane and swirling air inlets region (inlet area).}
\label{fig:fig4}
\end{figure}
\begin{figure*}[p!]
\centering
\captionsetup{justification=centering}
\includegraphics[width=15cm]{Figure5.png}
\caption{Temperature profiles at several downstream locations. \\ a) x = 20 mm, b) x = 40 mm, c) x = 55 mm, d) x = 75 mm.}
\label{fig:fig5}
\end{figure*}
\begin{figure*}[p!]
\centering
\captionsetup{justification=centering}
\includegraphics[width=15cm]{Figure6.png}
\caption{CO, CO$_{2}$ and H$_{2}$O Mass fraction profiles at several downstream locations. \\ a) x = 20 mm, b) x = 40 mm, c) x = 55 mm, d) x = 75 mm.}
\label{fig:fig6}
\end{figure*}
\par Methane inlet velocity: 32.7 m/s, swirling air axial velocity: 38.2 m/s, swirl velocity: 19.1 m/s. Co-flowing air velocity taken as 20 m/s. Non-premixed equilibrium model with non-adiabatic energy treatment and probability density function was used. Turbulence was modeled with realizable k-ε model and standard wall function. Walls are treated as adiabatic.
\par A mesh convergence test with the use of adaptation tool in ICEM CFD was performed. Solutions were obtained with: 68163, 85152, 135635, 170901 cells grid resulting respectively in the following discrepancy of peak temperature 20 mm above air co-flow inlet between each mesh: 5.04\%, 1.83\%, 0.92\%. Since the discrepancy between results obtained with the use of last two grids was lower than 1\%, last grid is used in further calculations.
\par Uniform velocity profiles at each inlet were assumed. This is proper assumption since swirling air and methane inlets’ diameters used in the experimental burner are couple of times smaller than their length, thus providing proper distance to unify the flow. Moreover the grid for inlets was extended in order to capture the impact of the boundary layer. Turbulence intensity of co-flow air is assumed to be 2\% according to experiment data.
\par Boundary conditions at 20 mm radial distance above air co-flow inlet where verified against experiment \cite{it12}. Maximum calculated discrepancy of radial and swirl velocity in relation to experiment was 4.3\%. Based on this assumed boundary condition are in good agreement with experiment data.
\par In Figure~\ref{fig:fig5}, a comparison between commercially standard WSGGM, Partial Pressures WSGGM and experimental data is shown. Results are analyzed on four selected planes, located at x = 20 mm, 40 mm, 55 mm and 75 mm. Based on this comparison it can be observed that in every location the peak temperature from numerical calculations using proposed model is closer to the experimental data. Maximum calculated relative discrepancy for both models is up to 18\% (for peak temperature on location x = 75 mm). In every location maximum discrepancy occurs for peak temperature. The reason why the proposed partial pressures WSGGM model results adapts better to the experimental data, is a direct link between absorption coefficient and mole fraction of individual species in the reagents mixture. Taking into account more species had also a positive effect on the quality of the results. Presented temperature profiles show that radiative heat transfer has a large influence on the results when used with equilibrium chemistry non-premixed combustion. It is suspected that incorporating larger quantities of species and utilization of more complex chemistry model (such as GRI-Mech 3.0) during the numerical calculation can improve results towards the experimental data.
\par Discrepancy between the results of numerical calculations for the standard WSGGM and partial pressures WSGGM and experimental data is partially caused by the fuel mixture used in the experiment. Numerical calculations presented in this paper intercepted air as a mixture of the 79\% mass fraction of nitrogen and 21\% of oxygen. When it comes to the fuel, pure methane was assumed in calculations. Also large variation occurs in the experiment, but presented experimental results are only Reynolds averaged mass fractions. To give an example of experimental data variation, the standard deviation of peak combustion temperature on location x=20 mm is equal to 42\% of mean value. This shows how unstabilized, therefore hard to simulate with RANS equations, are combustion processes. It is extremely difficult to recreate conditions during the experiment in numerical calculations and usually it can be done only to a certain degree of approximation. Adopted partial pressures WSGGM model determinates the absorption coefficients of non-modeled reagents as equal to zero, which in the general case introduces errors. Finally, as mention in the introduction, developed model is planned to be used in order to obtain only approximate, initial solution. Further calculation are going to be performed with the use of more sophisticated radiation models together with complex reaction mechanisms.
\section{Conculsion}
The new Partial Pressures Weighted-Sum-of-Gray-Gases Model has been developed and successfully implemented into the ANSYS FLUENT commercial CFD code. The proposed Partial Pressures WSGGM has been validated with the use of turbulent non-premixed methane combustion with swirling air experimental data. The new, partial pressures model shows slightly better results (closer to experimental data) than the currently available standard WSGGM used with equilibrium chemistry non-premixed combustion model. Proposed model strengths can be therefore noticed in rapid (coarse) combustion calculations in which significant benefits are observed with insignificant calculation time increase.
\par Proposed Partial Pressures Weighted-Sum-of-Gray-Gases Model for radiation numerical modeling in combustion simulations can be used in a wide range of numerical codes, adjusted to individual needs. It can be extended to any number of combustion species and any reaction mechanisms.
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