In general the radiative properties of the medium vary with the local composition and wave-length.
The products of combustion, such as CO2 and H2O, are strong selective adsorbers and emitters; but they do not scatter radiation significantly.
Particulates on the other hand, such as pulverised coal, ash and soot, scatter strongly.
The absorptivities of N2, O2 and H2 are so small that these gases are almost completely transparent to radiation.
However, a detailed modelling of the radiative properties is not warranted for the differential radiation models considered here. Rather these models require specification of the mean absorption coefficient, a, in m-1, and the mean scattering coefficient, s, in m-1, which may be interpreted as 'gray' values that are representative of the entire spectrum. These coefficients are equal to the proportion of incident radiation absorption/scattered per unit metre of its passage through the medium, and so they are like reciprocal mean free paths.
It should be noted that the absorption coefficient a is not to be confused with the conventional coefficient which appears in the Radiative Transfer Equation.
The specification of these coefficients has been discussed in some detail by Gosman and Lockwood [1973a, 1973b], although many workers simply use constant values or values dependent only on the local composition.
For example, Hoffmann and Markatos [1988] used a=0.1 m^-1 and s=0.01 m^-1 for hydrocarbon combustion, whereas Khalil et al [1975] used a=0.2*mfu+0.1*mpr where mfu and mpr are the mass fractions of fuel and product species, respectively.
Another approach is to use the "mixed gray and clear" gas formulation of Hottel [1954] ( see also Hottel and Sarofim [1967]) to determine the total gas emissivity εg, which is dimensionless and represents the ratio of the energy flux emitted relative to the black body value. The value of the absorption coefficient a is then obtained from the "pseudo-gray" approximation:
εg = 1 - exp (-a*L) Eqn 9.1
where L is the effective mean path length for radiation, which can be taken as some fraction of the furnace/chamber dimension, or estimated from the simple formula L=3.5*V/A, where V is the total volume of gas, and A is the total surface area of the furnace/chamber ( see Ozisik [1989] ). The effective path length differs from the geometrical distance travelled because the radiation is scattered or absorbed when passing through the medium.
There is much uncertainty in specifying an empirical expression for the absorption coefficient, and many CFD simulations simply use constant values estimated from the foregoing gray-gas method (see also Wang [2014]). Some worked examples are given by Mills [1992]). More sophisticated methods exist.
Further details can be found in the following references: Abbas et al [1984], Gosman and Lockwood [1973b], Khalil [1982], Kjaldman [1993], Siegel and Howell [1992], Taylor and Foster [1974] and Viskanta and Menguc [1987].
A.S.Abbas, F.C.Lockwood and A.P.Salooja, 'The prediction of the combustion and heat transfer performance of a refinery heater', Combustion & Flame, Vol.58, p91, (1984).
A.D.Gosman and F.C.Lockwood, 'Incorporation of a flux model for radiation into a finite-difference procedure for furnace calculations', 14th Symp. Combustion, p661, (1973a).
A.D.Gosman and F.C.Lockwood, 'Prediction of the influence of turbulent fluctuations on flow and heat transfer in furnaces', Mech. Eng. Dept. Report HTS/73/52, Imperial College, London, (1973b).
N.Hoffmann and N.C.Markatos, 'Thermal radiation on fires in enclosures', Appl. Math. Modelling, Vol.12, p129, (1988).
H.C.Hottel, 'Radiant heat transmission', in W.H.McAdams (Ed.), Heat Transmission, McGraw Hill, New York, (1954).
H.C.Hottel and A.F.Sarofim, 'Radiative transfer', McGraw Hill, New York, (1967).
E.E.Khalil, 'Modelling of furnaces and combustors', Abacus Press, (1982).
E.E.Khalil, D.B.Spalding and J.H.Whitelaw, 'The calculation of local flow properties in two-dimensional furnaces', Int. J.Heat Mass Transfer, Vol.18, p775, (1975).
L.Kjaldman, 'Numerical simulation of combustion and nitrogen pollutants in furnaces', TRC Finland, VTT Publications 159, (1993).
F.Liu and J.Swithenbank, 'Modelling radiative heat transfer in pulverised coal-fired furnaces', In Heat Transfer in Radiating and Combusting Systems, Ed. M.G.Carvalho, F.C.Lockwood and J.Taine, p358, Springer Verlag, (1991).
A.F. Mills 'Heat Transfer', pg557-565, Irwin, (1992).
M.N.Ozisik,'Heat Transfer. A Basic Approach', p691, McGraw Hill, (1989)
R.Siegel and J.R.Howell, 'Thermal Radiation Heat Transfer', 3rd Edition, Hemisphere Publishing Corpo2ation, (1992).
P.B.Taylor and P.J.Foster, 'The total emissivities of luminous and non-luminous flames', Int.J.Heat Mass Transfer, Vol.17, p1591, (1974).
P.Wang, F.Fan and Q.Li, "Accuracy evaluation of the gray-gas radiation model in CFD simulation", Case Studies in Thermal Engineering, Vol.3, p51-58, (2014).
R.Viskanta and M.P.Menguc, 'Radiation heat transfer in combustion systems', Prog. Energy Combust. Sci., Vol.13, p97, (1987).