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13.3.8 Radiation in Combusting Flows

The Weighted-Sum-of-Gray-Gases Model

The weighted-sum-of-gray-gases model (WSGGM) is a reasonable compromise between the oversimplified gray gas model and a complete model which takes into account particular absorption bands. The basic assumption of the WSGGM is that the total emissivity over the distance $s$ can be presented as

 \epsilon = \sum_{i=0}^I a_{\epsilon,i}(T)(1 - e^{-\kappa_i ps}) (13.3-81)

where $a_{\epsilon,i}$ are the emissivity weighting factors for the $i$th fictitious gray gas, the bracketed quantity is the $i$th fictitious gray gas emissivity, $\kappa_i$ is the absorption coefficient of the $i$th gray gas, $p$ is the sum of the partial pressures of all absorbing gases, and $s$ is the path length. For $a_{\epsilon,i}$ and $\kappa_i$ FLUENT uses values obtained from [ 67] and [ 344]. These values depend on gas composition, and $a_{\epsilon,i}$ also depend on temperature. When the total pressure is not equal to 1 atm, scaling rules for $\kappa_i$ are used (see Equation  13.3-87).

The absorption coefficient for $i=0$ is assigned a value of zero to account for windows in the spectrum between spectral regions of high absorption ( $\sum_{i=1}^I{a_{\epsilon,i} < 1}$) and the weighting factor for $i=0$ is evaluated from [ 344]:

 a_{\epsilon,0} = 1 - \sum_{i=1}^I{a_{\epsilon,i}} (13.3-82)

The temperature dependence of $a_{\epsilon,i}$ can be approximated by any function, but the most common approximation is

 a_{\epsilon,i} = \sum_{j=1}^J {b_{\epsilon,i,j}T^{j-1}} (13.3-83)

where $b_{\epsilon,i,j}$ are the emissivity gas temperature polynomial coefficients. The coefficients $b_{\epsilon,i,j}$ and $\kappa_i$ are estimated by fitting Equation  13.3-81 to the table of total emissivities, obtained experimentally [ 67, 79, 344].

The absorptivity $\alpha$ of the radiation from the wall can be approximated in a similar way [ 344], but, to simplify the problem, it is assumed that $\epsilon = \alpha$ [ 247]. This assumption is justified unless the medium is optically thin and the wall temperature differs considerably from the gas temperature.

Since the coefficients $b_{\epsilon,i,j}$ and $\kappa_i$ are slowly varying functions of $ps$ and $T$, they can be assumed constant for a wide range of these parameters. In [ 344] these constant coefficients are presented for different relative pressures of the CO $_2$ and H $_2$O vapor, assuming that the total pressure $p_T$ is 1 atm. The values of the coefficients shown in [ 344] are valid for $0.001 \leq ps \leq 10.0$ atm-m and $600 \leq T \leq 2400$ K. For $T > 2400$ K, coefficient values suggested by [ 67] are used. If $\kappa_i ps \ll 1$ for all $i$, Equation  13.3-81 simplifies to

 \epsilon = \sum_{i=0}^I {a_{\epsilon,i}\kappa_i p s} (13.3-84)

Comparing Equation  13.3-84 with the gray gas model with absorption coefficient $a$, it can be seen that the change of the radiation intensity over the distance $s$ in the WSGGM is exactly the same as in the gray gas model with the absorption coefficient

 a = \sum_{i=0}^I {a_{\epsilon,i} \kappa_i p} (13.3-85)

which does not depend on $s$. In the general case, $a$ is estimated as

 a = - \frac{\ln(1-\epsilon)}{s} (13.3-86)

where the emissivity $\epsilon$ for the WSGGM is computed using Equation  13.3-81. $a$ as defined by Equation  13.3-86 depends on $s$, reflecting the non-gray nature of the absorption of thermal radiation in molecular gases. In FLUENT, Equation  13.3-85 is used when $s \leq 10^{-4}$ m and Equation  13.3-86 is used for $s > 10^{-4}$ m. Note that for $s \approx 10^{-4}$ m, the values of $a$ predicted by Equations  13.3-85 and  13.3-86 are practically identical (since Equation  13.3-86 reduces to Equation  13.3-85 in the limit of small $s$).

FLUENT allows you to specify $s$ as the mean beam length or the characteristic cell size. The model based on the mean beam length is the recommended approach, especially when you have a nearly homogeneous medium and you are interested in the radiation exchange between the walls of the enclosure. You can specify the mean beam length or have FLUENT compute it. If you do decide to use the WSGGM based on the characteristic cell size, note that the predicted values of $a$ will be grid dependent (this is a known limitation of the model). See Section  8.8.1 for details about setting properties for the WSGGM.


The WSGGM cannot be used to specify the absorption coefficient in each band when using the non-gray DO model. If the WSGGM is used with the non-gray DO model, the absorption coefficient will be the same in all bands.

When $p_{\rm tot} \neq 1$ atm

The WSGGM, as described above, assumes that $p_{\rm tot}$--the total (static) gas pressure--is equal to 1 atm. In cases where $p_{\rm tot}$ is not unity (e.g., combustion at high temperatures), scaling rules suggested in [ 94] are used to introduce corrections. When $p_{\rm tot} < 0.9$ atm or $p_{\rm tot} > 1.1$ atm, the values for $\kappa_i$ in Equations  13.3-81 and  13.3-85 are rescaled:

 \kappa_i \rightarrow \kappa_i p_{\rm tot}^m (13.3-87)

where $m$ is a non-dimensional value obtained from [ 94], which depends on the partial pressures and temperature $T$ of the absorbing gases, as well as on $p_{\rm tot}$.

The Effect of Soot on the Absorption Coefficient

When soot formation is computed, FLUENT can include the effect of the soot concentration on the radiation absorption coefficient. The generalized soot model estimates the effect of the soot on radiative heat transfer by determining an effective absorption coefficient for soot. The absorption coefficient of a mixture of soot and an absorbing (radiating) gas is then calculated as the sum of the absorption coefficients of pure gas and pure soot:

 a_{s+g} = a_g + a_s (13.3-88)

where $a_g$ is the absorption coefficient of gas without soot (obtained from the WSGGM) and

 a_s = b_1 \rho_m [1 + b_T (T - 2000)] (13.3-89)


b_1 = 1232.4 \; \mbox{m}^2/\mbox{kg} \; \; \mbox{and} \; \; b_T \approx 4.8 \times 10^{-4} \; \mbox{K}^{-1}

$\rho_m$ is the soot density in kg/m $^3$.

The coefficients $b_1$ and $b_T$ were obtained [ 318] by fitting Equation  13.3-89 to data based on the Taylor-Foster approximation  [ 370] and data based on the Smith et al. approximation [ 344].

See Sections  8.8 and 20.3.3 for information about including the soot-radiation interaction effects.

The Effect of Particles on the Absorption Coefficient

FLUENT can also include the effect of discrete phase particles on the radiation absorption coefficient, provided that you are using either the P-1 or the DO model. When the P-1 or DO model is active, radiation absorption by particles can be enabled. The particle emissivity, reflectivity, and scattering effects are then included in the calculation of the radiative heat transfer. See Section  22.14 for more details on the input of radiation properties for the discrete phase.

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