The radiative absorption and emission from a gas can be characterized
by the emissivity as a function of temperature and pL, that is the product of the partial pressure and the path length.
In the context of typical combustion systems, the dominant emitters
of radiation are carbon dioxide and water vapor (although hydrocarbons,
CO and SO2 also make a minor contribution).
Hottel and Sarofim [48] have published emissivity charts for CO2 and H2O that have been obtained by a combination
of measurement and extrapolation. These plots show that emissivity
is strongly dependent on 
and also has a weaker
dependence on the gas temperature. This functional dependence can
be accurately correlated by assuming that the emissivity arises as
the result of independent emission from a sufficient number of gray
gases:
 | (8–40) |
Because emissivity must be proportional to absorptivity by Kirchoffs’
law, it follows that 
must approach
unity as 
. This imposes a constraint on the gray
gas weights or amplitudes:
 | (8–41) |
Also the requirement that 
is a monotonically
increasing function of 
is satisfied if
all the 
are
positive.
If the number of gray gases, 
,
is large, then 
may
be thought of as the fraction of the energy spectrum, relative to
the blackbody energy, for which the absorption coefficient is approximately 
. Then, the methodology described for the Multiband
model can be used directly.
Hadvig [49] has published charts of emissivity of combined CO2-H2O mixtures, for mixtures with different
relative proportions of CO2 and H2O. For the case of natural gas combustion, it can be
shown that the proportions of water vapor and carbon dioxide in the
products of combustion is such that partial pressure ratio, 
/ 
is approximately equal
to 2. Similarly, this ratio is 1 for oils and other fuels with the
empirical formula, (CH2)x. Most other hydrocarbon fuels have combustion products with a 
/ 
ratio lying between
1 and 2. Starting from the charts of Hottel and Sarofim (1967) [48] for CO2 and H2O and applying their correction
factor for mixtures, Hadvig has evaluated the emissivity of a gas
mixture with 
/ 
= 1
and 2 and presented the results as a function of 
and 
. Leckner [50] has also published
emissivity data, based on integrating the measured spectral data for
CO2 and H2O, which is
in reasonable agreement with the Hottel charts where the charts are
based on measured data.
Taylor and Foster (1974) [51] have integrated the spectral data and constructed a multigray gas representation:
 | (8–42) |
where the 
are
represented as linear functions of 
:
 | (8–43) |
As well as CO2 and H2O, the model developed by Beer, Foster and Siddall [52] takes into account the contributions of CO and unburned hydrocarbons (for example, CH4), which are also significant emitters of radiation. These authors generalize the parameterization of the absorption coefficients as follows:
 | (8–44) |
where 
is
the partial pressure of CO and 
is
the total partial pressure of all hydrocarbon species.
The values of 
, 
[K-1], 
[m-1 atm-1] and 
[m-1 atm-1] are given in Table 8.1: Gray gas emissivity parameters for a carbon dioxide / water
vapor / hydrocarbon mixture., together with a similar correlation for 
= 3, derived by Beer, Foster and
Siddall [52],
and suitable defaults for 
= 2
or 1 (single gray gas) representations.
Table 8.1: Gray gas emissivity parameters for a carbon dioxide / water vapor / hydrocarbon mixture.
| Ng | i | Gaseous Fuels pH2O/pCO2 = 2 | Oils pH2O/pCO2 = 1 | ||||||
|---|---|---|---|---|---|---|---|---|---|
| b1i | b2i | ki | kHCi | b1i | b2i | ki | kHCi | ||
| 1 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
| 2 | 1 | 0.437 | 7.13 | 0 | 3.85 | 0.486 | 8.97 | 0 | 3.41 |
| 2 | 0.563 | -7.13 | 1.88 | 0 | 0.514 | -8.97 | 2.5 | 0 | |
| 3 | 1 | 0.437 | 7.13 | 0 | 3.85 | 0.486 | 8.97 | 0 | 3.41 |
| 2 | 0.390 | -0.52 | 1.88 | 0 | 0.381 | -3.96 | 2.5 | 0 | |
| 3 | 0.173 | -6.61 | 68.83 | 0 | 0.133 | -5.01 | 109 | 0 | |
| 4 | 1 | 0.364 | 4.74 | 0 | 3.85 | 0.4092 | 7.53 | 0 | 3.41 |
| 2 | 0.266 | 7.19 | 0.69 | 0 | 0.284 | 2.58 | 0.91 | 0 | |
| 3 | 0.252 | -7.41 | 7.4 | 0 | 0.211 | -6.54 | 9.4 | 0 | |
| 4 | 0.118 | -4.52 | 80 | 0 | 0.0958 | -3.57 | 130 | 0 | |
Note: To satisfy the requirement that the 
factors
sum to unity, the 
factors must sum to 1.0 and the 
factors must sum to
0.