7. Heat Transfer by Radiation
Radiation Exchange with Emitting and Absorbing Gases
Introduction
Heat transfer by radiation needs no medium to be occurred (complete vacuum), but radiation can
be happened in case of presence of fluids such that water liquid or gases. The participation of
such medium depends on whether this medium absorbs, emits, or scatters radiation. If this
medium is completely transparent, it has no effect on radiation passing through it, mediums such
that is called nonparticipating medium. Air at ordinary temperatures and pressures is
nonparticipating medium. Gases that of monatomic molecules such as Ar and He and symmetric
diatomic molecules such as N2 and O2 are transparent to radiation, except at extremely high
temperatures.
Gases with asymmetric molecules such as H2O, CO2, CO, and SO2, and hydrocarbons HmCn may
participate in the radiation process by absorption at moderate temperatures, and by absorption
and emission at high temperature such as those encountered in combustion chambers.
The propagation of radiation through a medium can be complicated further by presence of
aerosols such as dust, ice particles, liquid droplets, and soot (unburned carbon) that scatter
radiation. Scattering refers to the change of direction of radiation due to reflection, refraction,
and diffraction.
Radiation Properties of a Participating Medium
Consider a participating medium of thickness L, as shown in
figure (1). A spectral radiation beam of intensity Iλ,0 is
incident on the medium, which is attenuated as it propagates
due to absorption. The decrease in the intensity of radiation as
it passes through a layer of thickness dx is proportional to the
intensity itself and the thickness dx. According to Beer’s law,
one can write:
𝑑𝐼𝜆 (𝑥) = −𝑘𝐴 𝐼𝜆 (𝑥) 𝑑𝑥
Where the constant of proportionality 𝑘𝐴 is the spectral
absorption coefficient of the medium, whose unit is m-1.
Separating the variables and integrating from x = 0 to x = L
gives:
Figure (1) The attenuation of a radiation
beam while passing through an absorbing
medium of thickness L
𝐼𝜆,𝐿
= 𝑒 −𝑘𝜆𝐿
𝐼𝜆,0
Where the absorptivity of the medium is assumed to be independent of x. The spectral
transmissivity (τλ) of a medium can be defined as:
𝐼𝜆,𝐿
= 𝑒 −𝑘𝜆𝐿
𝐼𝜆,0
The spectral transmissivity of a medium represents the fraction of radiation transmitted by the
medium at a given wavelength. For nonscattering medium (nonreflection) medium, radiation is
τ𝜆 =
1
either absorbed or transmitted. Therefore αλ + τ𝜆 = 1, and the spectral absorptivity of a
medium of thickness L is:
𝛼𝜆 = 1 − 𝜏𝜆 = 1 − 𝑒 −𝑘𝜆𝐿
According to Kirchoff’s law, the spectral emissivity of the medium is:
𝜀𝜆 = 𝛼𝜆 = 1 − 𝜏𝜆 = 1 − 𝑒 −𝑘𝜆𝐿
Note that the spectral absorptivity, transmissivity, and emissivity of a medium are dimensionless
quantities, with values less than or equal to 1. The spectral absorption coefficient of a medium
(and thus 𝜀𝜆 , 𝛼𝜆 , 𝑎𝑛𝑑 𝜏𝜆 ), in general, vary with wavelength, temperature, pressure, and
composition.
Emissivity and Absorptivity of Gases and Gas Mixtures
The spectral absorptivity of, for example, CO2
is given in figure (2) as a function of
wavelength. It is clear the band nature of
absorption
and
the
strong
nongray
characteristics. The shape and the width of this
absorption bands vary with temperature and
pressure, but the magnitude of absorptivity
also varies with the thickness of the gas layer.
Therefore, the absorptivity values without
specified thickness and pressure are
meaningless.
Figure (2) Spectral absorptivity of CO2 at 830 K and 10 atm for a
In spite of the non-gray nature of absorptivity path length of 38.8 cm.
of the gas, satisfactory results can be obtained by assuming the gas to be gray, and using an
effective total absorptivity and emissivity determined by some averaging process.
Radiation properties of an absorbing and emitting gas are usually reported for a mixture of the
gas with non-participating gases rather than the pure gas. The emissivity and absorpitivity of a
gas component in a mixture depends primarily on its density, which is a function of temperature
and partial pressure of the gas.
For single participating gas in a mixture, the emissivity of such gas is function of its partial
pressure (Pw for H2O, Pc for CO2), temperature (Tg) and the mean distance traveled by the
radiation beam (L). Figure (3-a) and Figure (3-b) show charts of the emissivity of water vapor in
air and Carbon dioxide (CO2) in air at total pressure equals to 1 atm and at different temperature
(Tg) ; respectively.
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Figure (3) Emissivities of H2O and CO2 gases in a mixture of non participating gases at a total pressure of 1 atm for a mean
beam length of L (1m. atm = 3.28 ft.atm)
Emissivity at a total pressure P other than P = 1 atm is determined by multiplying the emissivity
value at 1 atm by a pressure correction factor Cw for water vapor and Cc for CO2, (as shown in
figure (4) and accordingly:
𝜀𝑤 = 𝐶𝑤 𝜀𝑤,1𝑎𝑡𝑚
and
𝜀𝐶 = 𝐶𝐶 𝜀𝐶,1𝑎𝑡𝑚
Figure (4) Correction factors for the emissivities of H2O and CO2 gases at pressures other than 1 atm for use in the relations
𝜀𝑤 = 𝐶𝑤 𝜀𝑤,1𝑎𝑡𝑚 and
𝜀𝐶 = 𝐶𝐶 𝜀𝐶,1𝑎𝑡𝑚
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In case of the existence of water vapor and CO2 together in the gas mixture, an equivalent
emissivity of the gas is given by:
𝜀𝑔 = 𝜀𝐶 + 𝜀𝑤 − Δ𝜀
Where Δ𝜀 is the emissivity correction factor, which accounts for the overlap of emission bands.
For a gas mixture that contains both CO2 and H2O gases, Δ𝜀 is plotted in figure (5).
Figure (5) Emissivity correction Δε for use in 𝜀𝑔 = 𝜀𝐶 + 𝜀𝑤 − Δ𝜀
The emissivity of a gas also depends on the mean length an emitted radiation beam travels in a
gas before reaching a boundary surface, and thus the shape and size of the gas body involved.
The mean beam length (L) for various gas volume shapes are listed in table (1).
Table (1) Mean beam length L for various gas volume shapes
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The absorptivity of a gas containing CO2 and H2O gases for radiation emitted by a source at
temperature Ts can be determined similarly from:
𝛼𝑔 = 𝛼𝐶 + 𝛼𝑤 − Δα
Where Δα = Δε and is determined from figure (5) at the source temperature Ts. The
absorptivity of CO2 and H2O can be determined from the emissivity charts [figure (3) and
(4)] as:
𝑇𝑔 0.65
For CO2:
𝛼𝐶 = 𝐶𝐶 × ( 𝑇 )
For H2O:
𝛼𝑤 = 𝐶𝑤 × ( 𝑇 )
𝑠
𝑇𝑔 0.45
𝑠
𝑇
× 𝜀𝐶 (𝑇𝑠 , 𝑃𝐶 𝐿 𝑇𝑠 )
𝑔
𝑇
× 𝜀𝑤 (𝑇𝑠 , 𝑃𝑤 𝐿 𝑇𝑠 )
𝑔
As it is clear the emissivities should be evaluated using Ts instead of Tg, PcLTs/Tg instead of
Pc L and PwLTs/Tg instead of Pw L. Also it is noted that the absorpitivity of the gas depends
on source temperature Ts as well as the gas temperature Tg. The pressure correction
factors Cc and Cw are evaluated using PcL and PwL as in emissivity calculation.
When the total emissivity of a gas εg at temperature Tg is known, the emissive power of the
gas (radiation emitted by the gas per unit surface area) can be expressed as: 𝐸𝑔 = 𝜀𝑔 𝜎 𝑇𝑔 4 .
Then the rate of radiation energy emitted by a gas to a boundary surface of area As
becomes:
𝑄̇𝑔,𝑒 = 𝜀𝑔 𝐴𝑠 𝜎 𝑇𝑔 4
If the boundary surface is black at temperature Ts, the surface will emit radiation to the gas at a
rate of 𝐴𝑠 𝜎 𝑇𝑠 4 without reflection any, and the gas will absorb this radiation at a rate of
𝛼𝑔 𝐴𝑠 𝜎 𝑇𝑠 4 , where 𝛼𝑔 is the absorpitivity of the gas. Then the net rate of radiation heat transfer
between the gas and a black surface surrounding it becomes:
For black enclosure:
𝑄̇𝑛𝑒𝑡 = 𝐴𝑠 𝜎(𝜀𝑔 𝑇𝑔 4 − 𝛼𝑔 𝑇𝑠 4 )
If the surface is not black, a correction is made for the case of 𝜀𝑠 > 0.7, and the net rate of
radiation heat transfer becomes:
𝑄̇𝑛𝑒𝑡,𝑔𝑟𝑎𝑦 =
𝜀𝑠 + 1
𝜀 +1
̇ 𝑛𝑒𝑡,𝑏𝑙𝑎𝑐𝑘 = 𝑠
×𝑄
𝐴𝑠 𝜎(𝜀𝑔 𝑇𝑔 4 − 𝛼𝑔 𝑇𝑠 4 )
2
2
The emissivity of wall surfaces of furnaces and combustion chambers are typically greater than
0.7, and thus the previous relation is suitable to be applied in such cases.
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