### 8.6.3. Weighted Sum of Gray Gases

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.

#### 8.6.3.1. Weighted Sum of Gray Gases Model Parameters

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.

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