1. Chapter 12: Fundamentals of Thermal Radiation
Yoav Peles
Department of Mechanical, Aerospace and Nuclear Engineering
Rensselaer Polytechnic Institute

2. Objectives
When you finish studying this chapter, you should be able to:
Classify electromagnetic radiation, and identify thermal radiation,
Understand the idealized blackbody, and calculate the total and spectral blackbody emissive power,
Calculate the fraction of radiation emitted in a specified wavelength band using the blackbody radiation functions,
Understand the concept of radiation intensity, and define spectral directional quantities using intensity,
Develop a clear understanding of the properties emissivity, absorptivity, relflectivity, and transmissivity on spectral, directional, and total basis,
Apply Kirchhoff law’s law to determine the absorptivity of a surface when its emissivity is known,
Model the atmospheric radiation by the use of an effective sky temperature, and appreciate the importance of greenhouse effect.

3. Introduction
Unlike conduction and convection, radiation does not require the presence of a material medium to take place.
Electromagnetic waves or electromagnetic radiation ─ represent the energy emitted by matter as a result of the changes in the electronic configurations of the atoms or molecules.
Electromagnetic waves are characterized by their frequency n or wavelength l
c ─ the speed of propagation of a wave in that medium.
(12-1)

4. Thermal Radiation
Engineering application concerning electromagnetic radiation covers a wide range of wavelengths.
Of particular interest in the study of heat transfer is the thermal radiation emitted as a result of energy transitions of molecules, atoms, and electrons of a substance.
Temperature is a measure of the strength of these activities at the microscopic level.
Thermal radiation is defined as the spectrum that extends from about 0.1 to 100 mm.
Radiation is a volumetric phenomenon. However, frequently it is more convenient to treat it as a surface phenomenon.

5. Blackbody Radiation A body at a thermodynamic (or absolute)
temperature above zero emits radiation in
all directions over a wide range of
wavelengths.
The amount of radiation energy emitted
from a surface at a given wavelength
depends on:
the material of the body and the condition of its surface,
the surface temperature.
A blackbody ─ the maximum amount of radiation that can be emitted by a surface at a given temperature.
At a specified temperature and wavelength, no surface can emit more energy than a blackbody.
A blackbody absorbs all incident radiation, regardless of wavelength and direction.
A blackbody emits radiation energy uniformly in all directions per unit area normal to direction of emission.

6. Examples of approximate blackbody:
The radiation energy emitted by a blackbody per unit time and per unit surface area (Stefan–Boltzmann law)
s=5.67 X 10-8 W/m2·K4.
Examples of approximate blackbody:
snow,
white paint,
a large cavity with a small opening.
The spectral blackbody emissive power
(12-3)
(12-4)

7. Several observations can be made from this figure:
The variation of the spectral blackbody emissive power with wavelength is plotted in Fig. 12–9.
Several observations can be made
from this figure:
at any specified temperature a
maximum exists,
at any wavelength, the amount of
emitted radiation increases with
increasing temperature,
as temperature increases, the curves
shift to the shorter wavelength,
the radiation emitted by the sun
(5780 K) is in the visible spectrum.
The wavelength at which the peak occurs is given by Wien’s displacement law as
(12-5)

8. We are often interested in the amount of
radiation emitted over some wavelength
band.
The radiation energy emitted by a
blackbody per unit area over a
wavelength band from l=0 to l= l1 is
determined from
This integration does not have a simple closed-form solution. Therefore a dimensionless quantity fl called the blackbody radiation function is defined:
The values of fl are listed in Table 12–2.
(12-7)
(12-8)

9. Table 12-2 ─ Blackbody Radiation Functions fl
(12-8)
(12-9)

10. Radiation Intensity The direction of radiation passing
through a point is best described in
spherical coordinates in terms of the
zenith angle q and the azimuth angle f.
Radiation intensity is used to describe how the emitted radiation varies with the zenith and azimuth angles.
A differentially small surface in space dAn, through which this radiation passes, subtends a solid angle dw when viewed from a point on dA.
dAn

11. Radiation intensity ─ the
The differential solid angle dw subtended by a differential area dS on a sphere of radius r can be expressed as
Radiation intensity ─ the
rate at which radiation energy
is emitted in the (q,f) direction
per unit area normal to this
direction and per unit solid angle about this direction.
(12-11)
(12-13)

12. The radiation flux for emitted radiation is the emissive power E
The emissive power from the surface into the hemisphere surrounding it can be determined by
For a diffusely emitting surface, the intensity of the emitted radiation is independent of direction and thus Ie=constant:
(12-14)
(12-15)
(12-15)
(12-16)

13. For a blackbody, which is a diffuse emitter, Eq
For a blackbody, which is a diffuse emitter, Eq. 12–16 can be expressed as
where Eb=sT4 is the blackbody emissive power. Therefore, the intensity of the radiation emitted by a blackbody at absolute temperature T is
(12-17)
(12-18)

14. Intensity of incident radiation
Ii(q,f) ─ the rate at which radiation
energy dG is incident from the (q,f)
direction per unit area of the
receiving surface normal to this
direction and per unit solid angle
about this direction.
The radiation flux incident on a surface from all directions is called irradiation G
When the incident radiation is diffuse:
(12-19)
(12-20)

15. Radiosity (J )─ the rate at which radiation energy leaves
a unit area of a surface in all
directions:
For a surface that is both a diffuse emitter and a diffuse reflector, Ie+r≠f(q,f):
(12-21)
(12-22)

16. Spectral Quantities ─ the variation of radiation with wavelength.
The spectral radiation
intensity Il(l,q,f), for
example, is simply the total radiation intensity I(q,f) per unit wavelength interval about l.
The spectral intensity for emitted radiation Il,e(l,q,f)
Then the spectral emissive power becomes
(12-23)
(12-24)

17. Then the spectral blackbody emissive power is
The spectral intensity of radiation emitted by a blackbody at a thermodynamic temperature T at a wavelength l has been determined by Max Planck, and is expressed as
Then the spectral blackbody emissive power is
(12-28)
(12-29)

18. Radiative Properties
Many materials encountered in practice, such as metals, wood, and bricks, are opaque to thermal radiation, and radiation is considered to be a surface phenomenon for such materials.
In these materials thermal radiation is emitted or absorbed within the first few microns of the surface.
Some materials like glass and water exhibit different behavior at different wavelengths:
Visible spectrum ─ semitransparent,
Infrared spectrum ─ opaque.

19. Emissivity
Emissivity of a surface ─ the ratio of the radiation emitted by the surface at a given temperature to the radiation emitted by a blackbody at the same temperature.
The emissivity of a surface is denoted by e, and it varies between zero and one, 0≤e ≤1.
The emissivity of real surfaces varies with:
the temperature of the surface,
the wavelength, and
the direction of the emitted radiation.
Spectral directional emissivity ─ the most elemental emissivity of a surface at a given temperature.

20. Spectral directional emissivity
The subscripts l and q are used to designate spectral and directional quantities, respectively.
The total directional emissivity (intensities integrated over all wavelengths)
The spectral hemispherical emissivity
(12-30)
(12-31)
(12-32)

21. The total hemispherical emissivity
Since Eb(T)=sT4 the total hemispherical emissivity can also be expressed as
To perform this integration, we need to know the variation of spectral emissivity with wavelength at the specified temperature.
(12-33)
(12-34)

22. Gray and Diffuse Surfaces
Diffuse surface ─ a surface which properties are independent of direction.
Gray surface ─ surface properties are independent of wavelength.

24. Absorptivity, Reflectivity, and Transmissivity
When radiation strikes a surface, part of it:
is absorbed (absorptivity, a),
is reflected (reflectivity, r),
and the remaining part, if any, is transmitted (transmissivity, t).
Absorptivity:
Reflectivity:
Transmissivity:
(12-37)
(12-38)
(12-39)

25. Dividing each term of this relation by G yields
The first law of thermodynamics requires that the sum of the absorbed, reflected, and transmitted radiation be equal to the incident radiation.
Dividing each term of this relation by G yields
For opaque surfaces, t=0, and thus
These definitions are for total hemispherical properties.
(12-40)
(12-41)
(12-42)

26. Spectral directional absorptivity
Like emissivity, these properties can also be defined for a specific wavelength and/or direction.
Spectral directional absorptivity
Spectral directional reflectivity
(12-43)
(12-43)

27. Spectral hemispherical absorptivity
Spectral hemispherical reflectivity
Spectral hemispherical transmissivity
(12-44)
(12-44)
(12-44)

28. The average absorptivity, reflectivity, and transmissivity of a surface can also be defined in terms of their spectral counterparts as
The reflectivity differs somewhat from the other properties in that it is bidirectional in nature.
For simplicity, surfaces are assumed to reflect in a perfectly specular or diffuse manner.
(12-46)

29. Kirchhoff’s Law Consider a small body of surface area
As, emissivity e, and absorptivity a at
temperature T contained in a large
isothermal enclosure at the same
temperature.
Recall that a large isothermal enclosure forms a blackbody cavity regardless of the radiative properties of the enclosure surface.
The body in the enclosure is too small to interfere with the blackbody nature of the cavity.
Therefore, the radiation incident on any part of the surface of the small body is equal to the radiation emitted by a blackbody at temperature T.
G=Eb(T)=sT4.

30. The radiation emitted by the small body is
The radiation absorbed by the small body per unit of its surface area is
The radiation emitted by the small body is
Considering that the small body is in thermal equilibrium with the enclosure, the net rate of heat transfer to the body must be zero.
Thus, we conclude that
(12-47)

31. The restrictive conditions inherent in the derivation of Eq
The restrictive conditions inherent in the derivation of Eq should be remembered:
the surface irradiation correspond to emission from a blackbody,
Surface temperature is equal to the temperature of the source of irradiation,
Steady state.
The derivation above can also be repeated for radiation at a specified wavelength to obtain the spectral form of Kirchhoff’s law:
This relation is valid when the irradiation or the emitted radiation is independent of direction.
The form of Kirchhoff’s law that involves no restrictions is the spectral directional form
(12-48)

32. Atmospheric and Solar Radiation
The energy coming off the sun, called solar energy, reaches us in the form of electromagnetic waves after experiencing considerable interactions with the atmosphere.
The sun:
is a nearly spherical body.
diameter of D≈1.39X109 m,
mass of m≈2X1030 kg,
mean distance of L=1.5X1011 m from the earth,
emits radiation energy continuously at a rate of Esun≈3.8X1026W,
about 1.7X1017 W of this energy strikes the earth,
the temperature of the outer region of the sun is about 5800 K.

33. The solar energy reaching the earth’s atmosphere is called the total solar irradiance Gs, whose value is
The total solar irradiance (the solar constant) represents the rate at which solar energy is incident on a surface normal to the
sun’s rays at the outer edge
of the atmosphere when the
earth is at its mean distance
from the sun.
The value of the total solar irradiance can be used to estimate the effective surface temperature of the sun
from the requirement that
(12-49)
(12-50)

34. The several dips on the spectral distribution of radiation on the
The solar radiation undergoes considerable attenuation as it passes through the atmosphere as a result of absorption and scattering.
The several dips on the spectral
distribution of radiation on the
earth’s surface are due to
absorption by various gases:
oxygen (O2) at about l=0.76 mm,
ozone (O3)
below 0.3 mm almost completely,
in the range 0.3–0.4 mm considerably,
some in the visible range,
water vapor (H2O) and carbon dioxide (CO2) in the infrared region,
dust particles and other pollutants in the atmosphere at various wavelengths.

35. The solar energy reaching the earth’s surface is weakened considerably by the atmosphere and to about 950 W/m2 on a clear day and much less on cloudy or smoggy days.
Practically all of the solar radiation reaching the earth’s surface falls in the wavelength band from 0.3 to 2.5 mm.
Another mechanism that attenuates solar radiation as it passes through the atmosphere is scattering or reflection by air molecules and other particles such as dust, smog, and water droplets suspended in the atmosphere.

36. Diffuse solar radiation Gd: the scattered radiation is
The solar energy incident on a surface on earth is considered to consist of direct and diffuse parts.
Direct solar radiation GD: the part of solar radiation that reaches the earth’s surface without being scattered or absorbed by the atmosphere.
Diffuse solar radiation Gd:
the scattered radiation is
assumed to reach the earth’s
surface uniformly from all
directions.
Then the total solar energy incident on the unit area of a horizontal surface on the ground is:
(12-49)