12.1 Fundamental Concepts
• Radiation can be transferred in medium and vacuum.
• Conduction and convection can only be transferred in medium.
• All matters at non-zero Kelvin temperature emit radiation. At T = 0 K there will be no radiation.
• Examples around us: furniture, walls, ground, buildings, atmosphere and sun.
• Radiation is viewed as electromagnetic waves.
That explains the transport phenomena in vacuum.
• Electromagnetic waves are expressed in frequency () and wavelength () where = c / .
c = speed of light = 2.998e+8 m/s, units of is s-1 and units of is usually m
12.2 Radiation Intensity
• Radiation HT involves spherical coordinate system, figure 12.6.
Recall plane angle = arc length / radius ( = l / r).
Similarly solid angle = sphere area / radius square ( = A / r2)
• Consider emission from dA in particular direction.
• Direction is specified in terms of the zenith and azimuthal angles, and therefore d = sin d d
If all directions are needed over an opaque surface, then a hemisphere is obtained.
• Radiation from dA1 through dAn can be expressed in terms of spectral intensity (I,e)
Spectral Intensity (I,e) is rate at which radiant energy is emitted in (, ) direction, per unit area normal to direction,
per unit solid angle, per unit wavelength .This means that I,e = f (, , )
• To calculate the spectral hemispherical emissive power E() we must integrate over the zenith and azimuthal
angles, and .
𝐸𝜆 (𝜆) = ∬ 𝐼𝜆,𝑒 𝑑𝜃 𝑑𝜙
• E = {W/m2} = Total hemispherical emissive power
= rate at which radiant energy is emitted per unit area in all directions at all wavelengths
It is integrated over wavelength, , as well. 𝐸 = ∭ 𝐼𝜆,𝑒 𝑑𝜃 𝑑𝜙𝑑𝜆
• The integrations above are just to give an idea how complex radiation could be.
Definitions:
• Emissive power, E (W/m2) is radiation rate emitted from a surface per unit area. 𝐸 = 𝜀𝜎𝑇𝑠4
• Irradiation, G(W/m2) is radiation rateincident upon a surface per unit area
• Radiosity, J(W/m2) is radiation leaves a surface per unit area 𝐽 = 𝐸 + 𝜌 𝐺
Figure 12.6
Figure 12.5
12.3 Blackbody Radiation
• Blackbody absorbs all incident radiation (regardless of wavelength and direction) ( = 1)
• For prescribed temperature and wavelength, no surface can emit more energy than a blackbody
• Radiation emitted by blackbody is independent of direction (Blackbody is diffuse emitter)
12.4 Real Surfaces
• It is convenient to use blackbody as a reference to describe emission from real surfaces
• Define emissivity (ε) as ratio of radiation emitted by a surface to radiation emitted by blackbody (at the same
temperature)
• emissivity (ε) for blackbody = 1.0
• emissivity (ε) for real surface is 0 < ε < 1.0
• We will assume emissivity (ε) is constant even though (ε) = f(, , , T)
12.5 Absorption, Reflection, and Transmission by real surfaces
• When irradiation G is received by glass, some will be reflected, some will be absorbed, andsome will be transmitted.
See figure 12.5
• Opaque surface don’t transmit any energy ( = 0)
• Color we observe is due to selective reflection and absorption but not due emission (unless Ts> 1000 K)
• Absorptivity () is a property that determines the fraction of irradiation (G) absorbed by a surface.
• Reflectivity () is a property that determines the fraction of irradiation (G) reflected by a surface.
• Transmissivity () is a property that determines the fraction of irradiation (G) transmitted by a surface.
G=G +G +G
++=1
12.55
For opaque surfaces ( = 0)
+=1
12.57
Figure 12.5
12.6 Kirchhoff’s Law
Consider large isothermal enclosure of surface temperature ofTs. Also consider within several small bodies are confined.
Since these bodies are small relative to enclosure, they have negligible influence on the radiation field.
Regardless of the cavity’sradiative properties it forms a blackbody cavity.This means that any body in the cavity is diffuse
and equal to emission from a blackbody at Ts.
This means that
* G 1 – E1 = 0
or
This means
𝐸1
𝛼1
𝐸
= 𝛼2 = ⋯ = 𝐸𝑏
E ≤ Eb
2
and
(known as Kirchhoff’s Law)
12.59
=
12.60
12.7 The Gray Surface
Gray surface is one for which and are independent of . This means = .
Diffuse means independent of direction
Gray surface means independent of wavelength
0
