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 
can be presented as
 | (5–106) |
where 
is the emissivity weighting
factor for the 
th fictitious gray gas,
the bracketed quantity is the 
th fictitious gray gas
emissivity, 
is the absorption coefficient
of the 
th gray gas, 
is the sum of the partial
pressures of all absorbing gases, and 
is the path length. For 
and 
Ansys Fluent uses values obtained from [121] and [611]. These values depend on gas composition, and 
also depend on temperature.
When the total pressure is not equal to 1 atm, scaling rules
for 
are used (see Equation 5–113).
The absorption coefficient for 
is assigned a value
of zero to account for windows in the spectrum between spectral regions
of high absorption (
) and the weighting factor for 
is evaluated from [611]:
 | (5–107) |
The temperature dependence of 
can be approximated
by any function, but the most common approximation is
 | (5–108) |
where 
are the emissivity
gas temperature polynomial coefficients. The coefficients 
and 
are
estimated by fitting Equation 5–106 to the table
of total emissivities, obtained experimentally [121], [138], [611].
The absorptivity 
of the radiation from the wall can be approximated
in a similar way [611], but, to simplify
the problem, it is assumed that 
[446]. This assumption is justified unless the medium is optically thin
and the wall temperature differs considerably from the gas temperature.
Since the coefficients 
and 
are
slowly varying functions of 
and 
, they can be assumed constant
for a wide range of these parameters. In [611] these constant coefficients are presented for different relative
pressures of the 
and 
vapor, assuming that the total pressure 
is 1 atm.
The values of the coefficients shown in [611] are valid for 
atm-m and 
K. For 
K, coefficient values suggested by [121] are used. If 
for all 
, Equation 5–106 simplifies
to
 | (5–109) |
Comparing Equation 5–109 with the gray
gas model with absorption coefficient 
, it can be seen that the change
of the radiation intensity over the distance 
in the WSGGM is exactly the same
as in the gray gas model with the absorption coefficient
 | (5–110) |
which does not depend on 
. In the general case, 
is estimated as
 | (5–111) |
where the emissivity 
for the WSGGM is computed using Equation 5–106. 
as defined by Equation 5–111 depends
on 
, reflecting the non-gray nature of the absorption
of thermal radiation in molecular gases. In Ansys Fluent, Equation 5–110 is used when 
m and Equation 5–111 is used for 
m. Note that for 
m, the values of 
predicted by Equation 5–110 and Equation 5–111 are practically identical (since Equation 5–111 reduces to Equation 5–110 in the limit of small 
).
Ansys Fluent allows you to set 
equal to the mean beam length.
You can specify the mean beam length or have Ansys Fluent compute
it according to the following equation from Siegel [596].
 | (5–112) |
where 
is the fluid volume and 
is the total surface area of the
fluid boundaries (walls, inlets, outlets; not symmetry).
See Inputs for a Composition-Dependent Absorption Coefficient in the User's Guide for details about setting properties for the WSGGM.
Important: The WSGGM is implemented in a gray approach. If the WSGGM is used with a non-gray model,
the absorption coefficient will be the same in all bands. Use
DEFINE_GRAY_BAND_ABS_COEFF to change the absorption coefficient per
band or per gray gas.
The WSGGM, as described above, assumes that 
—the total (static) gas pressure—is equal
to 1 atm. In cases where 
is
not unity (for example, combustion at high temperatures), scaling
rules suggested in [158] are used to introduce
corrections. When 
atm or 
atm, the
values for 
in Equation 5–106 and Equation 5–110 are rescaled:
 | (5–113) |
where 
is a non-dimensional value obtained from [158], which depends on the partial pressures and temperature 
of the absorbing gases,
as well as on 
.
Note that the pressure scaling exponent, , is determined based on as follows:
Table 5.1: Pressure Correction Factor Lookup Values
| (atm) | (atm), for lookup in [158] |
|---|---|
| <0.2 | 0.1 |
| 0.2<<1.65 | 0.3 |
| 1.65<<6.5 | 3 |
| >6.5 | 10 |
When soot formation is computed, Ansys 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:
 | (5–114) |
where 
is the absorption coefficient
of gas without soot (obtained from the WSGGM) and
 | (5–115) |
with
and
is the gas mixture density in 
and 
is the soot mass fraction.
The coefficients 
and 
were obtained [572] by fitting Equation 5–115 to data based on the Taylor-Foster approximation [648] and data based on the Smith et al. approximation [611].
See Radiation Properties and Using the Soot Models in the User's Guide for information about including the soot-radiation interaction effects.
Ansys 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 Setting Material Properties for the Discrete Phase in the User’s Guide for more details on the input of radiation properties for the discrete phase.