This page provides a description and theory of the very powerful mixed-grey-gas model for combustion product emissivity. It is used in ZONE models. See our software page for an example of a mixed-grey-gas model written as a simple Excel spreadsheet to calculate the emissivity of typical natural gas combustion products.

1. Radiant emission and absorption by furnace gases.

The gaseous atmosphere in any fuel-fired furnace or combustion plant comprises mainly carbon dioxide and water vapour. Radiant emission and absorption from these gases occurs at discrete bands or wavelengths in the electromagnetic spectrum. There are also bands in the spectrum (‘windows’) within which there is no radiation absorbed or emitted. Radiation of this nature is highly ‘non-grey’ in character.

2. The mixed grey-gas model

The calculation of thermal radiation exchange in furnaces is simplified if the combustion product atmosphere can be assumed to be a grey gas. However, this assumption is a considerable departure from reality and can lead to errors in calculation. It is possible to represent the real furnace atmosphere by a weighted sum of grey gases plus a clear gas. This is often called ‘the mixed grey-gas model’.  This model retains some of the simplicity of the grey gas assumption but enables real furnace atmospheres to be more accurately modelled. 

3. Formulation of the mixed grey-gas model

The emissivity (e_g) of a grey gas is given by:

e_g = 1 – exp(-k_gpL)                                                        (1)

where, k_g is the absorption coefficient (m^-1atm^-1), p is the partial pressure (atm) of the absorbing gas, and L is the beam length.

The total emissivity of a real gas (e_g’) can be represented mathematically by a mixture of N grey gases of different absorption coefficients k_g,n , so that:

e_g=∑a_g,n[1-exp(-k_g,npL)]  for n=1,N                 (2)    

Where a_g,n are the weighting coefficients for each grey gas.

Since e_g cannot exceed unity (emissivity can never be greater than 1),

∑ a_g,n = 1.0    for n=1,N               (3)

In general, one of the gases is assumed to be clear with an absorption coefficient (k_g) of zero. This term represents the ‘windows’ in the spectrum.

3. How is the mixed grey-gas model derived?

Data are available in the open literature for combustion product emissivity and these are often published as plots of total emissivity against the product pL where p equals the sum of the partial pressures of the radiating gases (eg. water vapour plus carbon dioxide).  A simple grey gas formula (equation 1) does not fit the variation of emissivity with pL very closely, except over a very narrow range of values of pL or temperature. A mixed grey gas model comprising Equation 2 however, can give an acceptable fit even with a small number of gases (N=2 or 3).  The mixed grey gas model is fitted to the data by a least squares minimisation technique to determine values for k_g,n and a_g,n.  For a three term (two grey plus one clear gas model), there will be three values for k_g,n (of which k_g,1 = 0), and three values of a_g,n which must satisfy equation 3.

Variation of emissivity with temperature is taken account of by making the weighting coefficients a_g,n simple functions of gas temperature and maintaining the absorption coefficients k_g,n constant. In many published mixed grey gas models, a_g,n is simply given as:

a_g,n = b_1,n+b_2,n T_g                (4)

where, b_1,n and b_2,n are constants and T_g is the gas temperature in degrees K.

Data for a simple three term and four term model for natural gas combustion products are reproduced from Reference 1 in the Table below:

Example:

In typical natural gas combustion products the partial pressures of water vapour and carbon dioxide are 0.18 and 0.09 respectively. Assuming a gas temperature (T_g) of 1500K, use the three-term model to calculate the total emissivity of natural gas combustion products for a volume of gas of mean beam length equal to 5 metres.

Firstly, use equation 4) to calculate the three a-weighting coefficients as follows:

a₁ = 0.437 + 0.0713x10^-3x1500. = 0.54395

a₂ = 0.390 - 0.0052x10^-3x1500. = 0.3822

a₃ = 0.173 - 0.0661x10^-3x1500. = 0.07385

A simple check confirms that a₁+a₂+a₃ =1.0 .

p= 0.18+0.09 = 0.27 atm

L = 5.0 metres

Substitution into equation 2) gives:

e_g = a₁ (1-exp(-0.0x0.27x5)) + a₂ (1-exp(-1.88x0.27x5)) + a₃ (1-exp(-68.8x0.27x5))

    =  0.54395 (1-1) + 0.3822 (1-0.079) + 0.07385(1-4.6x10^-41)

   e_g  =  0.0 + 0.352 + 0.07385 = 0.42585

Thus, the two-grey plus one-clear gas model gives a total emissivity of 0.43 for these conditions.

4. Mixed grey-gas models and gas absorptivity

Gas absorptivity is more complex since it depends on the temperature of the source of radiation and its spectral distribution, in addition to the temperature and composition of the gas itself. The total absorptivity of a gas at temperature T_g to grey body radiation from a surface can however be fitted to the same mixed grey-gas formula used to determine emissivity (e.g. using the same coefficients (b_1,n, b_2,n and k_g,n), thus:

a_g=∑a_s,n[1-exp(-k_g,npL)]  for n=1,N                 (5)    

In this case, the a-weighting coefficients are determined using the source temperature T_s as follows:

a_s,n = b_1,n+b_2,n T_s                (4)

This model provides acceptable accuracy for many engineering applications although, in theory, it might be expected that a_s,n should be a function of both T_g and T_s. A more rigorous mixed grey gas models has been published by Smith et al. (reference 2) for natural gas and fuel oil combustion products, where the weighting coefficients are described by third order polynomials in both T_g and T_s.

5. How is the mixed grey-gas model applied?

The mixed grey-gas model can be used as a simple correlation to derive combustion product emissivity or absorptivity as illustrated in the example above. Its most powerful use though is in calculating radiation transfer between a non-grey gas and its surroundings.  If the gas temperature and surrounding surface temperature are known, the heat transfer can be calculated from a simple a-weighted summation of the heat transfer derived assuming a simple grey gas of absorption coefficient k_g,1,k_g,2,k_g,3 etc. The procedure is as follows:

Step1 – Calculate the a_g - weighting coefficients for the given gas temperature.

Step 2 – Using an appropriate formula or model of the combustion plant, calculate the heat transfer from the combustion products assuming a simple grey gas of absorption coefficient k_g,1

Step 3 – Repeat step 2 for the other grey gases k_g,2,k_g,3 etc.

Step 4 – Calculate the a_g-weighted sum of the heat transfer for each of the grey gases; this gives the total heat transfer emitted from the non-grey gas.

The process can be repeated to derive the total heat transfer from the surrounding surface that is absorbed by the combustion products (using the a_s - weighting coefficients and surface temperature T_s).  The net heat transfer (emitted – absorbed) between the gas and surfaces is thus determined.

This procedure forms the basis of the ZONE method of analysis for radiation heat transfer in furnace enclosures comprising non-grey gases. 

Glossary terms

Grey gas - A gas whose emissivity (and absorptivity) do not vary with wavelength.

Emissivity - The emissivity of a real body or surface is defined as the ratio of the radiative flux emitted by the body, to the flux from a blackbody at the same temperature.

Absorption coefficient  - The fractional attenuation of a beam of radiation per unit distance through a gaseous medium per unit pressure (atmosphere) of absorbing gas component. Units are m^-1atm^-1. Term often used to mean Extinction Coefficient.

Partial pressure – The pressure of a component gas in a mixture of gases, that it would exert if it alone occupied the same volume and temperature as the mixture.

Total emissivity – Emissivity integrated over all wavelengths.

Gas Absorptivity - The fraction of radiation passing through a volume of gas that is absorbed.

Spectral distribution – The variation of thermal radiation with wavelength.

Total absorptivity – Absorptivity integrated over all wavelengths.

[1] J.M.Rhine and R.J.Tucker, Modelling of Gas-fired Furnaces and Boilers, McGraw-Hill, 1991.

[2] T.F.Smith, Z.F. Shen and J.N.Friedman, Evaluation of coefficients for the weighted sum of grey gases model, Trans.ASME, J.Heat Transfer, 104, pp602-608, 1982.

© Zerontec Ltd                  (Last updated 04/09/2007)