1. Radiation: Overview Radiation - Emission Radiation - Absorption
thermal radiation is the emission of electromagnetic waves when matter is at an absolute temperature greater than 0 K
emission is due to the oscillations and transitions of the many electrons that comprise the matter
the oscillations and transitions are sustained by the thermal energy of the matter
emission corresponds to heat transfer from the matter and hence to a reduction in the thermal energy stored in the matter
Radiation - Absorption
radiation may also be absorbed by matter
absorption results in heat transfer to the matter and hence to an increase in the thermal energy stored in the matter

2. Radiation: Overview Emission Dual Nature
emission from a gas or semi-transparent solid or liquid is a volumetric phenomenon
emission from an opaque solid or liquid is a surface phenomenon
emission originates from atoms & molecules within 1 μm of the surface
Dual Nature
in some cases, the physical manifestations of radiation may be explained by viewing it as particles (A.K.A. photons or quanta); in other cases, radiation behaves as an electromagnetic wave
radiation is characterized by a wavelength λ and frequency ν which are related through the speed at which radiation propagates in the medium of interest (solid, liquid, gas, vacuum)
in a vacuum

3. Radiation: Spectral Considerations
Electromagnetic Spectrum
the range of all possible radiation frequencies
thermal radiation is confined to the infrared, visible, and ultraviolet regions of the spectrum
Spectral Distribution
radiation emitted by an opaque surface varies with wavelength
spectral distribution describes the radiation over all wavelengths
monochromatic/spectral components are associated with particular wavelengths

4. Radiation: Directional Considerations
Emission
Radiation emitted by a surface will be in all directions associated with a hypothetical hemisphere about the surface and is characterized by a directional distribution
Direction may be represented in a spherical coordinate system characterized by the zenith or polar angle θ and the azimuthal angle ϕ.
- The amount of radiation emitted from a surface, dAn, and propagating in a particular direction (θ,ϕ) is quantified in terms of a differential solid angle associated with the direction, dω.
dAn  unit element of surface on a hypothetical sphere and normal to the (θ,ϕ) direction

5. Radiation: Directional Considerations
Solid Angle
the solid angle ω has units of steradians (sr)
the solid angle ωhemi associated with a complete hemisphere

6. Radiation: Spectral Intensity
Spectral Intensity, Iλ,e
a quantity used to specify the radiant heat flux (W/m2) within a unit solid angle about a prescribed direction (W/m2-sr) and within a unit wavelength interval about a prescribed wavelength (W/m2-sr-μm)
associated with emission from a surface element dA1 in the solid angle dω about θ, ϕ and the wavelength interval dλ about λ and is defined as:
the rational for defining the radiation flux in terms of the projected area (dA1cosθ) stems from the existence of surfaces for which, to a good approximation, Iλ,e is independent of direction. Such surfaces are termed diffuse, and the radiation is said to be isotropic.
the projected area is how dA1 appears along θ, ϕ
[W/m2-sr-μm]

7. Radiation: Heat Flux
The spectral heat rate (heat rate per unit wavelength of radiation) associated with emission
The spectral heat flux (heat flux per unit wavelength of radiation) associated with emission
The integration of the spectral heat flux is called the spectral emissive power
spectral emission (heat flux) over all possible directions

8. Radiation: Heat Flux
The total heat flux from the surface due to radiation is emission over all wavelengths and directions  total emissive power
If the emission is the same in all directions, then the surface is diffuse and the emission is isotropic

9. Radiation: Irradiation
electromagnetic waves incident on a surface is called irradiation
irradiation can be either absorbed or reflected
Spectral Intensity, Iλ,i
a quantity used to specify the incident radiant heat flux (W/m2) within a unit solid angle about the direction of incidence (W/m2-sr) and within a unit wavelength interval about a prescribed wavelength (W/m2-sr-μm) and the projected area of the receiving surface (dA1cosθ)

10. Radiation: Irradiation Heat Flux
The integration of the spectral heat flux is called the spectral irradiation
spectral irradiation (heat flux) over all possible directions
The total heat flux to the surface due to irradiation over all wavelengths and directions  total irradiative power

11. Radiation: Radiosity Radiosity Spectral Intensity, Iλ,e+r
for opaque surfaces
accounts for all radiation leaving a surface
emission
reflection
Spectral Intensity, Iλ,e+r
a quantity used to specify emitted and reflected radiation intensity
The integration of the spectral heat flux is called the spectral radiosity
spectral emission+reflection (heat flux) over all possible directions
The total heat flux from the surface due to irradiation over all wavelengths and directions  total radiosity

12. Radiation: Black Body Black Body
an idealization providing limits on radiation emission and absorption by matter
for a prescribed temperature and wavelength, no surface can emit more than a black body  ideal emitter
a black body absorbs all incident radiation (no reflection)  ideal absorber
a black body is defined as a diffuse emitter
Isothermal Cavity – Approximation of Black Body
after multiple reflections, virtually all radiation entering the cavity is absorbed
emission from the aperture is the maximum possible emission for the temperature of cavity and the emission is diffuse
cumulative effect of emission and reflection off the cavity wall is to provide diffuse irradiation corresponding to emission from a black body

13. Radiation: Black Body Planck Distribution
the spectral emission intensity of a black body
determined theoretically and confirmed experimentally
spectral emissive power

14. Radiation: Black Body Planck Distribution
emitted radiation varies continuously with wavelength
at any wavelength, the magnitude of the emitted power increases with temperature
the spectral region where the emission is concentrated depends on temperature
comparatively more radiation at shorter wave lengths
sun approximated by 5800 K black body
The maximum emission power, Eλ,b, occurs at λmax which is determined by Wien’s displacement law

15. Radiation: Black Body Stefan-Boltzmann Law
the total emissive power of a black body is found by integrating the Planck distribution
the fraction of the total emissive power within a wavelength band
(λ1 < λ < λ2) is
Stefan-Boltzmann Law
this can be rewritten as
the following function is tabulated

16. Radiation: Black Body

17. Example: Radiation
According to its directional distribution, solar radiation incident on the earth’s surface consists of two components that may be approximated as being diffusely distributed with the angle of the sun θ. Consider clear sky conditions with incident radiation at an angle of 30° with a total heat flux (if the radiation were angled normal to the surface) of 1000 W/m2 and the total intensity of the diffuse radiation is Idif = 70 W/m2-sr. What is the total irradiation on the earth’s surface?

18. Example: Radiation
The human eye, as well as the light-sensitive chemicals on color photographic film, respond differently to lighting sources with different spectral distributions. Daylight lighting corresponds to the spectral distribution of a solar disk (approximated as a blackbody at 5800 K) and incandescent lighting from the usual household lamp (approximated as a blackbody at 2900 K).
Calculate the band emission fractions for the visible region for each light source.
Calculate the wavelength corresponding to the maximum spectral intensity for each light source.

19. Radiation: Surface Properties
Real surfaces do not behave like ideal black bodies
non-ideal surfaces are characterized by factors (< 1) which are the ratio of the non-ideal performance to the ideal black body performance
these factors can be a function of wavelength (spectral dependence) and direction (angular dependence)
Non-Ideal Radiation Factor
emissivity, ε
Non-Ideal Irradiation
absorptivity, α
reflectivity, ρ
transmissivity, τ

20. Radiation: Emissivity
characterizes the emission of a real body to the ideal emission of a black body and can be defined in three manners
a function of wavelength (spectral dependence) and direction (angular dependence)
a function of wavelength (spectral dependence) averaged over all directions
a function of direction (angular dependence) averaged over all wavelengths
Spectral, Directional Emissivity
Spectral, Hemispherical Emissivity (directional average)
Total, Directional Emissivity (spectral average)

21. Radiation: Emissivity
Total, Hemispherical Emissivity (directional average)
to a reasonable approximation, the total, hemispherical emissivity is equal to the total, normal emissivity
which can be simplified to

22. Radiation: Emissivity
Representative spectral variations
Representative temperature variations

23. Radiation: Absorption/Reflection/Transmission
Three responses of semi-transparent medium to irradiation, Gλ
absorption within medium, Gλ,abs
reflection from the medium, Gλ,ref
transmission through the medium, Gλ,tr
Total irradiation balance
An opaque material only has a surface response – there is no transmission (volumetric effect)
The semi-transparency or opaqueness of a medium is governed by both the nature of the material and the wavelength of the incident radiation
the color of an opaque material is based on the spectral dependence of reflection in the visible spectrum

24. Radiation: Absorptivity
Spectral, Directional Absorptivity
assuming negligible temperature dependence
Spectral, Hemispherical Absorptivity (directional average)
Total, Hemispherical Absorptivity

25. Radiation: Reflectivity
Spectral, Directional Reflectivity
assuming negligible temperature dependence
Spectral, Hemispherical Reflectivity (spectral average)
Total, Hemispherical Reflectivity
specular – polished surfaces
diffuse – rough surfaces

26. Radiation: Reflectivity
Representative spectral variations

27. Radiation: Transmissivity
Spectral, Hemispherical Reflectivity
assuming negligible temperature dependence
Total, Hemispherical Transmissivity
Representative spectral variations

28. Radiation: Irradiation Balance
Semi-Transparent Materials
Opaque Materials
and
and

29. Radiation: Kirchhoff’s Law
spectral, directional surface properties are equal
Kirchhoff’s Law (spectral)
spectral, hemispherical surface properties are equal
for diffuse surfaces or diffuse irradiation
Kirchhoff’s Law (blackbodies)
total, hemispherical properties are equal
when the irradiation is from a blackbody at the same temperature as the emitting surface

30. Radiation: Kirchhoff’s Law
Kirchhoff’s Law (spectral)
true if irradiation is diffuse
true if surface is diffuse
Kirchhoff’s Law (blackbody)
true if irradiation is from a blackbody at the same temperature as the emitting surface
true if the surface is gray ? ?

31. Radiation: Gray Surfaces
a surface where αλ and ελ are independent of λ over the spectral regions of the irradiation and emission
Gray approximation only valid for:

32. Radiation: Example
The spectral, hemispherical emissivity absorptivity of an opaque surface is shown below.
What is the solar absorptivity?
If Kirchhoff’s Law (spectral) is assumed and the surface temperature is 340 K, what is the total hemispherical emissivity?

33. Radiation: Example
A vertical flat plate, 2 m in height, is insulated on its edges and backside is suspended in atmospheric air at 300 K. The exposed surface is painted with a special diffuse coating having the prescribed absorptivity distribution and is irradiated by solar-simulation lamps that provide spectral irradiation characteristic of the solar spectrum. Under steady conditions the plate is at 400 K.
(a) Find the plate absorptivity, emissivity, free convection coefficient, and irradiation. (b) Estimate the plate temperature if if the irradiation was doubled.

34. Radiation: Exchange Between Surfaces
Overview
Enclosures consist of two or more surfaces that envelop a region of space (typically gas-filled) and between which there is radiation transfer.
Virtual, as well as real, surfaces may be introduced to form an enclosure.
A nonparticipating medium within the enclosure neither emits, absorbs, nor scatters radiation and hence has no effect on radiation exchange between the surfaces.
Each surface of the enclosure is assumed to be isothermal, opaque, diffuse and gray, and to be characterized by uniform radiosity and irradiation.

35. Radiation: View Factor (Shape Factor)
View Factor, Fij
geometrical quantity corresponding to the fraction of the radiation leaving surface i that is intercepted by surface j
General expression
consider radiation from the differential area dAi to the differential area dAj
the rate of radiosity (emission + reflection) intercepted by dAj
The view factor is the ratio of the intercepted radiosity to the total radiosity
the view factor is based entirely on geometry

36. Radiation: View Factor Relations
Reciprocity
Summation
from conservation of radiation (energy), for an enclosure

37. Radiation: View Factors
2-D Geometries

38. Radiation: View Factors
3-D Geometries

39. Radiation: Blackbody Radiation Exchange
For a blackbody there is no reflection (perfect absorber)
Net radiation exchange (heat rate) between two “blackbodies”
net rate at which radiation leaves surface i due to its interaction with j OR
net rate at which surface j gains radiation due to its interaction with i
Net radiation (heat) transfer from surface i due to exchange with all (N) surfaces of an enclosure
(heat loss from Ai)

40. Radiation: Gray Radiation Exchange
General assumption for opaque, diffuse, gray surfaces
Equivalent expressions for the net radiation (heat) transfer from surface i
thus for gray bodies the resistance at the surface is
and the driving potential is

41. Radiation: Gray Radiation Exchange
Net radiation (heat) transfer from surface i due to exchange with all (N) surfaces of an enclosure
thus for gray bodies the resistance between two bodies (space or geometrical resistance)
and the driving potential is
Radiation energy balance on surface i :
net energy leaving = energy exchange with other surfaces

42. Radiation: Gray Radiation Exchange
The equivalent circuit for a radiation network consists of two resistances
resistance at the surface
resistances between all bodies

43. Radiation: Gray Radiation Exchange
Methodology of an enclosure analysis
apply the following equation for each surface where the net radiation heat rate qi is known
apply the following equation for each remaining surface where the temperature Ti (and thus Ebi) is known
determine all the view factors
solve the system of N equations for the unknown radiosities J1, J2, …, JN
apply the following equation to determine the radiation heat rate qi for each surface of known Ti and Ti for each surface of known qi

44. Radiation: Gray Radiation Exchange
Special Case
enclosure with an opening (aperture) of area Ai through which the interior surface exchange radiation with large surroundings at temperature Tsur
Tsur Ai
Treat the aperture as a virtual blackbody surface with area Ai,
Ti = Tsur and

45. Radiation: Two Surface Enclosures
Simplest enclosure for which radiation exchange is exclusively between two surfaces and a single expression for the rate of radiation transfer may be inferred from a network representation of the exchange

46. Radiation: Two Surface Enclosures
Special Cases

47. Radiation: Reradiating Surface
idealization for which GR = JR hence qR = 0 and JR = Eb,R
approximated by surfaces that are well insulated on one side and for which convection is negligible on the opposite (radiating) side
Three-surface enclosure with a reradiating surface

48. Radiation: Reradiating Surface
The temperature of the reradiating surface TR may be determined from knowledge of its radiosity JR. With qR = 0 a radiation balance on the surface yields

49. Radiation: Multimode Effects
In an enclosure with conduction and convection heat transfer to/from one or more surface, the foregoing treatments of the radiation exchange may be combined with surface energy balances to determine thermal conditions
Consider a general surface condition for which there is external heat addition (e.g., electrically) as well as conduction, convection and radiation
appropriate analysis for N-surface, two-surface, etc. enclosure

50. Example: Radiation Exchange
A cylindrical furnace for heat treating materials in a spacecraft environment has a 90-mm diameter and an overall length of 180 mm. Heating elements in the 135 mm long section maintain a refractory lining at 800 °C and ε = the other linings are insulated but made of the same material. The surroundings are at 23 °C. Determine the power required to maintain the furnace operating conditions.