MODEST College of Engineering Pennsylvania State University University Park, Pennsylvania 8.1 Fundamentals 8.1.1 Emissive power 8.1.2 Solid angles 8.1.3 Radiative intensity 8.1.4 Radiati
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CHAPTER 8
Thermal Radiation
MICHAEL F MODEST
College of Engineering Pennsylvania State University University Park, Pennsylvania
8.1 Fundamentals 8.1.1 Emissive power 8.1.2 Solid angles 8.1.3 Radiative intensity 8.1.4 Radiative heat flux 8.2 Radiative properties of solids and liquids 8.2.1 Radiative properties of metals Wavelength dependence Directional dependence Hemispherical properties Total properties
Surface temperature effects 8.2.2 Radiative properties of nonconductors Wavelength dependence
Directional dependence Temperature dependence 8.2.3 Effects of surface conditions Surface roughness
Surface layers and oxide films 8.2.4 Semitransparent sheets 8.2.5 Summary
8.3 Radiative exchange between surfaces 8.3.1 View factors
Direct integration Special methods View factoralgebra Crossed-strings method 8.3.2 Radiative exchange between black surfaces 8.3.3 Radiative exchange between diffuse gray surfaces Convex surface exposed to large isothermal enclosure 8.3.4 Radiation shields
8.3.5 Radiative exchange between diffuse nongray surfaces Semigray approximation method
Band approximation method
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8.4 Radiative properties of participating media 8.4.1 Moleculargases
8.4.2 Particle clouds Soot
Pulverized coal and fly ash dispersions Mixtures of molecular gases and particulates 8.5 Radiative exchange within participating media 8.5.1 Mean beam length method
8.5.2 Diffusion approximation 8.5.3 P-1 approximation 8.5.4 OtherRTE solution methods 8.5.5 Weighted sum of gray gases 8.5.6 Otherspectral models Nomenclature
References
Radiative heat transfer or thermal radiation is the science of transferring energy in
the form of electromagnetic waves Unlike heat conduction, electromagnetic waves
do not require a medium for their propagation Therefore, because of their ability to travel across vacuum, thermal radiation becomes the dominant mode of heat trans-fer in low pressure (vacuum) and outer-space applications Another distinguishing characteristic between conduction (and convection, if aided by flow) and thermal ra-diation is their temperature dependence While conductive and convective fluxes are more or less linearly dependent on temperature differences, radiative heat fluxes tend
to be proportional to differences in the fourth power of temperature (or even higher)
For this reason, radiation tends to become the dominant mode of heat transfer in high-temperature applications, such as combustion (fires, furnaces, rocket nozzles), nuclearreactions (solaremission, nuclearweapons), and others
All materials continuously emit and absorb electromagnetic waves, or photons, by changing their internal energy on a molecular level Strength of emission and absorp-tion of radiative energy depend on the temperature of the material, as well as on the wavelengthλ, frequency ν, orwavenumberη, that characterizes the electromagnetic
waves,
where wavelength is usually measured inµm(= 10−6m), while frequency is
mea-sured in hertz= cycles/s), and wavenumbers are given in cm−1 Electromagnetic
waves or photons (which include what is perceived as “light”) travel at the speed of
light, c The speed of light depends on the medium through which the wave travels
and is related to that in vacuum,c0, through the relation
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c = c0
wheren is known as the refractive index of the medium By definition, the refractive
index of vacuum isn ≡ 1 For most gases the refractive index is very close to unity,
and thec in eq (8.1) can be replaced by c0 Each wave or photon carries with it an amount of energy determined from quantum mechanics as
whereh is known as Planck’s constant The frequency of light does not change when
light penetrates from one medium to another because the energy of the photon must be conserved On the other hand, the wavelength does change, depending on the values
of the refractive index for the two media
When an electromagnetic wave strikes an interface between two media, the wave
is either reflected or transmitted Most solid and liquid media absorb all incoming
radiation over a very thin surface layer Such materials are called opaque or opaque
surfaces (even though absorption takes place over a thin layer) An opaque material
that does not reflect any radiation at its surface is called a perfect absorber, black
surface, or blackbody, because such a surface appears black to the human eye, which
recognizes objects by visible radiation reflected off their surfaces
8.1.1 Emissive Power
Every medium continuously emits electromagnetic radiation randomly into all direc-tions at a rate depending on the local temperature and the properties of the material
The radiative heat flux emitted from a surface is called the emissive power E, and there is a distinction between total and spectral emissive power (heat flux emitted
over the entire spectrum or at a given frequency per unit frequency interval), so that the spectral emissive powerEν is the emitted energy/time/surface area/frequency,
while the total emissive powerE is emitted energy/time/surface area Spectral and
total emissive powers are related by
E(T ) =
∞
0
Eλ(T,λ) dλ =
∞
0
It is easy to show that a black surface is not only a perfect absorber, but it is also a perfect emitter, that is, the emission from such a surface exceeds that of any other
surface at the same temperature (known as Kirchhoff ’s law) The emissive power leaving an opaque black surface, commonly called blackbody emissive power, can be
determined from quantum statistics as
E bλ (T,λ) = 2πhc20
n2λ5(e hc0/nλkT − 1) (n = const) (8.5)
where it is assumed that the black surface is adjacent to a nonabsorbing medium
of constant refractive indexn The constant k = 1.3806 × 10−23J/K is known as
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102
103
104
105
106
107
108
Wavelength λ , mµ
.m)
E b
2 µ
T
5762 K
5000 K
3000 K
2000
K
1000 K
500K
Visible part of spectrum
E T C
b
3
(
= / )
Figure 8.1 Blackbody emissive powerspectrum
Boltzmann’s constant The spectral dependence of the blackbody emissive power into
vacuum (n = 1) is shown fora numberof emittertemperatures in Fig 8.1 It is seen
that emission is zero at both extreme ends of the spectrum with a maximum at some intermediate wavelength The general level of emission rises with temperature, and the important part of the spectrum (the part containing most of the emitted energy) shifts toward shorter wavelengths Because emission from the sun (“solar spectrum”)
is well approximated by blackbody emission at an effective solar temperature of
Tsun = 5762 K, this temperature level is also included in the figure Heat transfer
problems generally involve temperature levels between 300 and, say, 2000 K (plus, perhaps, solar radiation) Therefore, the spectral ranges of interest in heat transfer applications include the ultraviolet (0.1 to 0.4µm), visible radiation (0.4 to 0.7 µm,
as indicated in Figure 8.1 by shading), and the near- and mid-infrared (0.7 to 20µm)
Forquick evaluation, a scaled emissive powercan be written as
E bλ
n3T5 = C1
(nλT )5(e C2/nλT − 1) (n = const) (8.6)
where
C1= 2πhc2
0 = 3.7419 × 10−16W· m2
C2=hc0
k = 14,388 µm · K
Equation (8.6) has its maximum at
(nλT )max= C3= 2898 µm · K (8.7)
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which is known as Wien’s displacement law The constants C1, C2, andC3are known
as the first, second, and third radiation constants, respectively
The total blackbody emissive is found by integrating eq (8.6) over the entire spectrum, resulting in
whereσ = 5.670 × 10−8 W/m2· K4is the Stefan–Boltzmann constant It is often
desirable to calculate fractional emissive powers, that is, the emissive power con-tained overa finite wavelength range, say between wavelengthsλ1andλ2 It is not possible to integrate eq (8.6) between these limits in closed form; instead, one resorts
to tabulations of the fractional emissive power, contained between 0 andnλT ,
f (nλT ) =
λ
0 E bλ dλ
∞
0 E bλ bλ =
nλT
0
E bλ
n3σT5 d(nλT ) (8.9)
so that
λ2
λ 1
E bλ dλ = [f (nλ2T ) − f (nλ1T )]n2σT4 (8.10)
An extensive listing off (nλT ), as well as of the scaled emissive power, eq (8.6),
is given in Table 8.1 Both functions are also shown in Fig 8.2, together with Wien’s
distribution, which is the short-wavelength limit of eq (8.5),
E bλ 2πhc02
n2λ5 e −hc0/nλkT = C1
n2λ5e −C2/nλT hc0
As seen from the figure, Wien’s distribution is actually rather accurate over the entire spectrum, predicting a total emissive power approximately 8% lower than the one given by eq (8.8) Because Wien’s distribution can be integrated analytically over parts of the spectrum, it is sometimes used in heat transfer applications
8.1.2 Solid Angles
Radiation is a directional phenomenon; that is, the radiative flux passing through
a point generally varies with direction, such as the sun shining onto Earth from essentially a single direction Consider an opaque surface elementdA i, as shown in
Fig 8.3 It is customary to describe the direction unit vector ˆs in terms of polar angleθ
(measured from the surface normal ˆn) and azimuthal angleψ (measured in the plane
of the surface, between an arbitrary axis and the projection of ˆs); fora hemisphere
0≤ θ ≤ π/2 and 0 ≤ ψ ≤ 2π.
The solid angle with which a surfaceA j is seen from a certain pointP (or dA iin
Fig 8.3) is defined as the projection of the surface onto a plane normal to the direction vector, divided by the distance squared, as also shown in Fig 8.3 for an infinitesimal
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TABLE 8.1 Blackbody Emissive Power
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TABLE 8.1 Blackbody Emissive Power (Continued)
elementdA j If the surface is projected onto a unit sphere above pointP , the solid
angle becomes equal to the projected area, or
Ω =
A jp
dA jp
S2 =
A j
cosθ0dA j
whereS is the distance between P and dA j Thus, an infinitesimal solid angle is simply an infinitesimal area on a unit sphere, or
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dΩ = dA jp = (1 × sin θ dψ)(1 × dθ) = sin θ dθ dψ (8.13) Integrating over all possible directions yields
2 π ψ=0
π/2
Figure 8.2 Normalized blackbody emissive power spectrum
n
dAi
sin d
d
dAjp
dAj
n0
1
dA j⬙
P
d
Figure 8.3 Definitions of direction vectors and solid angles
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for the total solid angle above the surface If a point inside a medium removed from the surface is considered, radiation passing through that point can strike any point
of an imaginary unit sphere surrounding it; that is, the total solid angle here is 4π,
with 0 ≤ θ ≤ π, 0 ≤ ψ ≤ 2π Similarly, at a surface one can talk about an upper
hemisphere (outgoing directions, 0≤ θ < π/2), and a lowerhemisphere (incoming
directions,π/2 < θ ≤ π).
8.1.3 Radiative Intensity
The directional behavior of radiative energy traveling through a medium is charac-terized by the radiative intensityI, which is defined as
I ≡ radiative energy flow/time/area normal to the rays/solid angle
Like emissive power, intensity is defined on both spectral and total bases, related by
I (ˆs) =
∞
0
However, unlike emissive power, which depends only on position (and wavelength),
the radiative intensity depends, in addition, on the direction vector ˆs Emissive power
can be related to emitted intensity by integrating this intensity over the 2π solid angles
above a surface, and then realizing that the projection ofdA normal to the rays is
dA cos θ Thus,
E(r) =
2π
0
π/2
0
I (r, θ, ψ) cos θ sin θ dθ dψ =
2 π I (r, ˆs)ˆn · ˆs dΩ (8.16) which is, of course, also valid on a spectral basis For a black surface it is readily shown, through a variation of Kirchhoff’s law, thatI bλis independent of direction, or
Using this relation in eq (8.16), it is observed that the intensity leaving a blackbody
(or any surface whose outgoing intensity is independent of direction, or diffuse) may
be evaluated from the blackbody emissive power(oroutgoing heat flux) as
I bλ (r,λ) = E bλ (r,λ)
In the literature the spectral blackbody intensity is sometimes referred to as the Planck
function.
8.1.4 Radiative Heat Flux
Emissive power is the total radiative energy streaming away from a surface due to emission Therefore, it is a radiative flux, but not the net radiative flux at the surface,
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because it only accounts foremission and not forincoming radiation and reflected radiation Extending the definition of eq (8.16) gives
(qλ)out=
cosθ>0 Iλ(ˆs) cos θ dΩ ≥ 0 (8.19) whereIλ(θ) is now outgoing intensity (due to emission plus reflection) Similarly, for
incoming directions (π/2 < θ ≤ π),
(qλ)in=
cosθ<0 Iλ(ˆs) cos θ dΩ < 0 (8.20) Combining the incoming and outgoing contributions, the net radiative flux at a surface is
(qλ)net= qλ· ˆn = (qλ)in+ (qλ)out=
4 π Iλ(ˆs) cos θ dΩ (8.21) The total radiative flux, finally, is obtained by integrating eq (8.21) over the entire spectrum, or
q · ˆn =
∞
0
qλ· ˆn dλ =
∞
0
4 π Iλ(ˆs)ˆn · ˆs dΩ dλ (8.22)
Of course, the surface described by the unit vector ˆn may be an imaginary one (located somewhere inside a radiating medium) Thus, removing the ˆn from eq (8.22) gives
the definition of the radiative heat flux vector inside a participating medium:
q=
∞
0
qλdλ =
∞
0
4 π Iλ(ˆs)ˆs dΩ dλ (8.23)
Because radiative energy arriving at a given point in space can originate from a point
faraway, without interacting with the medium in between, a conservation of energy balance must be performed on an enclosure bounded by opaque walls (i.e., a medium
thick enough that no electromagnetic waves can penetrate through it) Strictly speak-ing, the surface of an enclosure wall can only reflect radiative energy or allow a part
of it to penetrate into the substrate A surface cannot absorb or emit photons: Atten-uation takes place inside the solid, as does emission of radiative energy (and some
of the emitted energy escapes through the surface into the enclosure) In practical systems the thickness of the surface layer over which absorption of irradiation from inside the enclosure occurs is very small compared with the overall dimensions of an enclosure—usually, a few angstroms for metals and a few micrometers for most non-metals The same may be said about emission from within the walls that escapes into