Chapter 12-
-
if Ts > Tsur the net heat transfer rate by radiation qrad, net is from the surface, and the
surface will cool until Ts reaches Tsur.
Emission is due to oscillations and transitions of the many electrons that comprise matter,
which are, in turn, sustained by the thermal energy of the matter
Radiation may also be intercepted and absorbed by matter
Absorption results in heat transfer to the matter and hence an increase in thermal energy
stored by the matter
Emission from a gas or a semitransparent solid or liquid is a volumetric phenomenon.
Emission from an opaque solid or liquid is a surface phenomenon
o For an opaque solid or liquid, emission originates from atoms and molecules
within 1 of the surface. Convection is a surface phenomena
In all cases, radiation is characterized by a wavelength and frequency which are related
through the speed at which radiation propagates in the medium of interest:
𝑐
o 𝜆=𝑣

𝑚
𝐶 = 𝑐0 = 2.998 𝑥 108 ( 𝑠 )
o 𝐸 = ℎ𝑓 =

-
-
ℎ𝑐
𝜆
𝐽
𝜇𝑒𝑉
ℎ = 6.626069𝑥10−34 (𝑠) = 4.13567 𝐺𝐻𝑧
When radiation is incident upon a semitransparent medium, portions of the irradiation
may be reflected, absorbed, and transmitted, as discussed in Section 1.2.3 and illustrated
in Figure 12.5a.
Transmission refers to radiation passing through the medium, as occurs when a layer of
water or a glass plate is irradiated by the sun or artificial lighting.
Absorption occurs when radiation interacts with the medium, causing an increase in the
internal thermal energy of the medium.
Reflection is the process of incident radiation being redirected away from the surface,
with no effect on the medium.
We define reflectivity ρ as the fraction of the irradiation that is reflected, absorptivity α as
the fraction of the irradiation that is absorbed, and transmissivity τ as the fraction of the
irradiation that is transmitted.
-
-
-
-
-
Opaque = no transmission
The radiosity, J (W/m2) of a surface accounts for all the radiant energy leaving the
surface.
For an opaque surface, it includes emission and the reflected portion of the irradiation, as
illustrated in Figure 12.5b.
o 𝐽 = 𝐸 + 𝐺𝑟𝑒𝑓 = 𝐸 + 𝜌𝐺
 12.4
o
Radiosity can also be defined at a surface of a semitransparent medium. In that case, the
radiosity leaving the top surface of Figure 12.5a (not shown) would include radiation
transmitted through the medium from below.
Finally, the net radiative flux from a surface, 𝑞"𝑟𝑎𝑑 (W/m2), is the difference between the
outgoing and incoming radiation
o 𝑞"𝑟𝑎𝑑 = 𝐽 − 𝐺
Opaque surface:
o 𝑞"𝑟𝑎𝑑 = 𝐸 + 𝜌𝐺 − 𝐺 = 𝜀𝜎𝑇𝑆4 − 𝛼𝐺
 12.6
Net radiative heat transfer rate : 𝑞𝑟𝑎𝑑 = 𝑞"𝑟𝑎𝑑 𝐴
Radiation Intensity (12.3)
-
Due to its nature, mathematical treatment of radiation heat transfer involves the extensive
use of the spherical coordinate system. From Figure 12.6a, we recall that the differential
plane angle dα is defined by a region between the rays of a circle and is measured as the
ratio of the arc length dl on the circle to the radius r of the circle. Similarly, from Figure
12.6b, the differential solid angle dω is defined by a region between the rays of a sphere
and is measured as the ratio of the area dAn on the sphere to the sphere's radius squared.
-
When viewed from a point on an opaque surface area element dA1, radiation may be
emitted into any direction defined by a hypothetical hemisphere above the surface. The
solid angle associated with the entire hemisphere may be obtained by
integrating Equation 12.8 over the limits φ = 0 to φ = 2π and θ = 0 to θ = π/2.
o
-
Returning to Figure 12.6c, we now consider the rate at which emission from dA1 passes
through dAn. This quantity may be expressed in terms of the spectral intensity 𝐼𝜆,𝑒 of the
emitted radiation.
-
Spectral Intensity, 𝐼𝜆,𝑒 : the rate at which radiant energy is emitted at the wavelength λ in
the (θ, φ) direction, per unit area of the emitting surface normal to this direction, per unit
solid angle about this direction, and per unit wavelength interval dλ about λ
o Defined by area dA1, perpendicular to the direction of the radiation =
dA1cos(theta)
o 𝑑𝑞"𝜆 = 𝐼𝜆,𝑒 (𝜆, 𝜃, 𝜑) cos 𝜃 sin 𝜃 𝑑𝜃𝑑𝜑
-
Spectral, hemispherical emissive power Eλ (W/m2 · μm) or spectral heat flux associated
with emission into a hypothetical hemisphere above dA1: the rate at which radiation of
wavelength λ is emitted in all directions from a surface per unit wavelength
interval dλ about λ and per unit surface area.
2𝜋
𝜋
o 𝐸𝜆 (𝜆) = 𝑞"𝜆 (𝜆) = ∫0 ∫02 𝐼𝜆,𝑒 (𝜆, 𝜃, 𝜙) cos 𝜃 sin 𝜃 𝑑𝜃𝑑𝜙
-
Total emissive power – over all directions and wavelengths
∞
o 𝐸 = ∫0 𝐸𝜆 (𝜆)𝑑𝜆
-
For a diffuse surface, emission is isotropic (the intensity of the emitted radiation is
independent of direction) and –
o 𝐸𝜆 (𝜆) = 𝜋𝐼𝜆,𝑒 (𝜆)
o 𝐸 = 𝜋𝐼𝑒
 Where Ie is the total intensity of the emitted radiation
 Note the constant is pi not 2pi and has the unit steradians
-
Spectral irradiation (W/m^2 * mu *m) – the rate at which radiation from wavelength 𝜆 is
incident on a surface, per unit area of the surface and per unit wavelength interval 𝑑𝜆
about 𝜆
2𝜋
𝜋
o 𝐺𝜆 (𝜆) = ∫0 ∫02 𝐼𝜆,𝑖 (𝜆, 𝜃, 𝜙) cos 𝜃 sin 𝜃 𝑑𝜃𝑑𝜙
 𝐺𝜆 is a flux based on the actual surface area, whereas 𝐼𝜆,𝑖 is defined in
terms of the projected area
-
Total irradiation (W/m^2) – represents the rate at which radiation is incident per unit area
from all directions and at all wavelengths
∞
o 𝐺 = ∫0 𝐺𝜆 (𝜆)𝑑𝜆
o 𝐺𝜆 (𝜆) = 𝜋𝐼𝜆,𝑖 (𝜆)
o 𝐺 = 𝜋𝐼𝑖
Blackbody radiation
- The blackbody:
o An idealization providing limits on radiation emission and absorption by matter.
 For a prescribed temperature and wavelength, no surface can emit more
radiation than a blackbody: the ideal emitter.
 A blackbody is a diffuse emitter – although the radiation emitted by a
blackbody is a function of wavelength and temperature, it is independent
of direction.
 A blackbody absorbs all incident radiation, regardless of wavelength and
direction: the ideal absorber.
-
The isothermal cavity (Hohlraum) – the closers approximation to a blackbody is a cavity
whose inner surface is at uniform temperature
o After multiple reflections, virtually all radiation entering the cavity is absorbed
o Emission from the aperture is the maximum possible emission achievable for the
temperature associated with the cavity and is diffuse
 The spectral intensity of radiation leaving the cavity is independent of
direction
o Blackbody radiation exists within the cavity irrespective of whether the cavity
surface is highly reflecting or absorbing
-
Diffuse emission:
o
o Constant directional emissivity
-
Spectral intensity of blackbody (Planck):
o 𝐼𝜆,𝑏 (𝜆, 𝑇) =
-
2ℎ𝑐𝑜2
ℎ𝑐𝑜
)−1]
𝜆𝑘𝐵 𝑇
𝜆5 [exp(


ℎ = 6.626 × 10−34 [𝐽 ∙ 𝑠]
𝐽
𝑘𝐵 = 1.381 × 10−23 [𝐾]


𝑐𝑜 = 2.998 × 108 [ 𝑠 ] – speed of light in vacuum
𝑇 = 𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑡𝑒𝑚𝑝 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑙𝑎𝑐𝑘𝑏𝑜𝑑𝑦 [𝐾]
𝑚
Spectral emissive power of blackbody (Planck’s Distribution/Law):
𝐶1
o 𝐸𝜆,𝑏 (𝜆, 𝑇) = 𝜋𝐼𝜆,𝑏 (𝜆, 𝑇) = 5
𝐶2
𝜆 [𝑒𝑥𝑝(

𝜆𝑇
)−1]
𝐶1 = 2𝜋ℎ𝑐𝑜2 = 3.742 × 108 [𝑊 ∙
𝜇𝑚4
𝑚2
]
o
𝐶2 = (

Emitted radiation varies continuously with wavelength
 𝐸𝜆,𝑏 𝑣𝑎𝑟𝑖𝑒𝑠 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠𝑙𝑦 𝑤𝑖𝑡ℎ 𝜆
At any wavelength the magnitude of the emitted radiation increases with
increasing temperature
 𝐸𝜆,𝑏 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑠 𝑤𝑖𝑡ℎ 𝑇
The spectral region in which the radiation is concentrated depends on
temperature, with comparatively more radiation appearing at shorter
wavelengths as the temperature increases
 The fractional amount of total blackbody emission appearing at
lower wavelengths increases with increasing T
A significant fraction of the radiation emitted by the sun, which may be
approximated as a blackbody at 5800 K, is in the visible region of the
spectrum. In contrast, for T < 800 K emission is predominantly in the
infrared region of the spectrum and is not visible to the eye



-
ℎ𝑐𝑜
⁄𝑘 ) = 1.439 × 104 [𝐾 ∙ 𝜇𝑚]
𝐵

Wein’s Displacement Law: the blackbody spectral distribution has a maximum and the
corresponding wavelength X depends on temperature:
o 𝜆𝑚𝑎𝑥 𝑇 = 𝐶3
 𝐶3 = 2898 [𝜇𝑚 ∙ 𝐾]
 (𝑑𝑎𝑠ℎ𝑒𝑑 𝑙𝑖𝑛𝑒 𝑜𝑓 𝑓𝑖𝑔𝑢𝑟𝑒 12.12)
 Maximum spectral emissive power is displaced to shorter wavelengths
with increasing temperature
-
Total emissive power of a blackbody (Stefan-Boltzmann Law):
o 𝐸𝑏 = 𝜎𝑇 4
 𝜎 = 5.670 × 10−8 [𝑊/(𝑚2 ∙ 𝐾 4 )]
-
Total intensity associated with blackbody emission:
𝐸
o 𝐼𝑏 = 𝜋𝑏
-
The fraction of the total blackbody emission in a prescribed wavelength interval or band
is:
𝜆
o 𝐹(𝜆1 −𝜆2 ) = 𝐹(0−𝜆2 ) − 𝐹(0−𝜆1 ) =
𝜆
∫0 𝐸𝜆,𝑏 𝑑𝜆
-
𝜆
∫0 2 𝐸𝜆,𝑏 𝑑𝜆−∫0 1 𝐸𝜆,𝑏 𝑑𝜆
𝜎𝑇 4
o 𝐹(0−𝜆) = 𝜎𝑇 4 = 𝑓(𝜆𝑇) - in general
 Numerical results given in table 12.2 (pg 555)
“Emission from the aperture of any isothermal enclosure will have the characteristics of
blackbody radiation.”
o “Irradiation of any small object inside the enclosure may be approximated as
being equal to emission from a blackbody at the enclosure surface temperature.
Hence G = Eb (T)”
o
Emission from Real Surfaces (12.5)
- Emissivity – the ratio of the radiation emitted by the surface to the radiation emitted by a
blackbody at the same temperature
- Note: the spectral radiation emitted by a real surface differs from the Planck distribution
and the directional distribution may be other than diffuse
- Comparison
o
-
Spectral, directional emissivity – ratio of the intensity of the radiation emitted at the
wavelength 𝜆 and in the direction of 𝜃 and 𝜑 to the intensity of the radiation emitted by a
blackbody at the same values of T and 𝜆:
𝐼 (𝜆,𝜃,𝜑,𝑇)
o 𝜀𝜆,𝜃 (𝜆, 𝜃, 𝜑, 𝑇) = 𝜆,𝑒𝐼 (𝜆,𝑇)
𝜆,𝑏
-
Total, directional emissivity – spectral average of 𝜀𝜆,𝜃
𝐼 (𝜃,𝜑,𝑇)
o 𝜀𝜃 (𝜃, 𝜑, 𝑇) = 𝑒
𝐼𝑏 (𝑇)
-
o
Spectral, hemispherical emissivity – using surface properties that represent directional
averages
𝐸 (𝜆,𝑇)
o 𝜀𝜆,𝜃 (𝜆, 𝑇) = 𝜆,
𝐸𝜆,𝑏 (𝜆,𝑇)
2𝜋
o 𝜀𝜆,𝜃 (𝜆, 𝑇) =
-
2𝜋
𝜋
∫0 ∫02 𝐼𝜆,𝑏 (𝜆,𝑇) cos 𝜃 sin 𝜃𝑑𝜃𝑑𝜙
Total hemispherical emissivity – accounts for emission over all wavelengths and in all
directions. Is the ratio of the total emissive power of a real surface, E(T), to the total
emissive power of a blackbody at the same temperature, Eb(T)
𝐸(𝑇)
o 𝜀(𝑇) = 𝐸
𝑏 (𝑇)
-
𝜋
∫0 ∫02 𝐼𝜆,𝑒 (𝜆,𝜃,𝜙,𝑇) cos 𝜃 sin 𝜃𝑑𝜃𝑑𝜙
∞
=
∫0 𝜀𝜆 (𝜆,𝑇)𝐸𝜆,𝑏 (𝜆,𝑇)𝑑𝜆
𝐸𝑏 (𝑇)
4
(𝑇)
o 𝐸(𝑇) = 𝜀(𝑇)𝐸𝑏
= 𝜀(𝑇)𝜎𝑇
 Can be used to compute the emissive power of the surface at any
temperature if 𝜀(𝑇) is known
If 𝜀𝜆 (𝜆, 𝑇) is known, you can compute the spectral emissive power of the surface at any
wavelength and temperature by combining previous equations:
o 𝐸𝜆 (𝜆, 𝑇) = 𝜀𝜆 (𝜆, 𝑇)𝐸𝜆,𝑏 (𝜆, 𝑇) =
-
𝐶1 𝜀𝜆 (𝜆,𝑇)
𝐶
5
𝜆 [𝑒𝑥𝑝( 2 )−1]
𝜆𝑇
Since the directional emissivity remains nearly constant until reaching 40° for conductors
and 70° for nonconductors, the hemispherical emissivity will not differ largely from the
normal emissivity, corresponding to 𝜃 = 0
o 𝜀 ≈ 𝜀𝑛
o
o Also applies to spectral components
o
 Decr. 𝜀𝜆,𝑛 with incr. 𝜆 for metals and different behavior for nonmetals
o Several assumptions can be made:
 The emissivity of metallic surfaces is generally small, achieving values as
low as 0.02 for highly polished gold and silver.
 The presence of oxide layers may significantly increase the emissivity of
metallic surfaces. From Figure 12.18, contrast the values of 0.3 and 0.7 for
stainless steel at 900 K, depending on whether it is polished or heavily
oxidized.
 The emissivity of nonconductors is comparatively large, generally
exceeding 0.6.
 The emissivity of conductors increases with increasing temperature;
however, depending on the specific material, the emissivity of
nonconductors may either increase or decrease with increasing
temperature.
 Note that the variations of εn with T shown in Figure 12.18 are
consistent with the spectral distributions of ελ, n shown in Figure
12.17. These trends follow from Equation 12.43.


Although the spectral distribution of ελ, n is approximately
independent of temperature, there is proportionately more emission
at lower wavelengths with increasing temperature.
 Hence, if ελ, n increases with decreasing wavelength for a
particular material, εn will increase with increasing temperature for
that material.
Total, hemispherical emissivity is generally high for rough surfaces
Absorption, Reflection, and Transmission by Real Surfaces
-
Semitransparent medium (such as a layer of water or a glass plate)
o For a spectral component of the irradiation, portions of this radiation may be
reflected, absorbed, and transmimtted
 Reflected: 𝐺𝜆,𝑟𝑒𝑓
 Absorbed: 𝐺𝜆,𝑎𝑏𝑠
 Transmitted: 𝐺𝜆,𝑡𝑟
o Complex – depending on both surface conditions, the wavelength of the radiation,
composition/thickness of the material, and can be strongly influences by
volumetric effects within the medium
o 𝜌𝜆 + 𝛼𝜆 + 𝜏𝜆 = 1
-
Opaque
o Governed by surface phenomenon (no volumetric effects) – irradiation absorbed
and reflected by the surface
 𝑎𝑛𝑑 𝐺𝜆,𝑡𝑟 = 0
 No net effect of the reflection process on the medium, while absorption
has the effect of increasing the internal thermal energy of the medium
o 𝜌𝜆 + 𝛼𝜆 = 1
-
Note: Color is due to selective reflection and absorption of the visible portion of the
irradiation that is incident from the sun or an artificial source of light. Unless it is at such
a high temperature (Ts>1000K) that it is incandescent. Emission is concentrated in the
IR region and is hence imperceptible to the eye.
o For a prescribed irradiation, the “color” of a surface may not indicate its overall
capacity as an absorber or reflector, since much of the irradiation may be in the IR
region. A “white” surface such as snow, for example, is highly reflective to
visible radiation but strongly absorbs IR radiation, thereby approximating
blackbody behavior at long wavelengths.
Total, hemispherical absorptivity – the fraction of the total irradiation absorbed by a
surface
-
o
𝛼=
𝐺𝑎𝑏𝑠
𝐺
∞
=
∫0 𝛼𝜆 (𝜆)𝐺𝜆 (𝜆)𝑑𝜆
∞
∫0 𝐺𝜆 (𝜆)𝑑𝜆
(12.51)
o Approximately independent of surface temperature
-
Reflectivity
o Diffuse reflection occurs – if, regardless of the direction of the incident radiation,
the intensity of the reflected radiation is independent of the reflection angle
o Specular reflection occurs – if all the reflection is in the direction of 𝜃2 , which
equals the incident angle 𝜃1 .
o
o Depends on the direction of both the incident radiation as well as the reflected
radiation
o 𝜌=
-
𝐺
∞
=
∫0 𝜌𝜆 (𝜆)𝐺𝜆 (𝜆)𝑑𝜆
(12.57)
∞
∫0 𝐺𝜆 (𝜆)𝑑𝜆
Total, hemispherical transmissivity
o 𝜏=
-
𝐺𝑟𝑒𝑓
𝐺𝑡𝑟
𝐺
∞
=
∫0 𝐺𝜆,𝑡𝑟 (𝜆)𝑑𝜆
∞
∫0 𝐺𝜆 (𝜆)𝑑𝜆
∞
=
∫0 𝜏𝜆 (𝜆)𝐺𝜆 (𝜆)𝑑𝜆
∞
∫0 𝐺𝜆 (𝜆)𝑑𝜆
o Glass/water are semitransparent at short wavelengths become opaque at longer
wavelengths
-
o Transmissivity of glass affected by its iron content
o Transmissivity of plastics, such as tedlar, is greater than glass in the IR region
Kirchhoff’s Law (12.7)
-
-
-
In Sections 12.7 and 12.8 we consider conditions for which the emissivity and
absorptivity are equal.
o Consider a large, isothermal enclosure of surface temperature Ts, within which
several small bodies are confined (Figure 12.24)
o Since these bodies are small relative to the enclosure, they have a negligible
influence on the radiation field, which is due to the cumulative effect of emission
and reflection by the enclosure surface.
o Recall that, regardless of its radiative properties, such a surface forms a blackbody
cavity.
o Accordingly, regardless of its orientation, the irradiation experienced by any body
in the cavity is diffuse and equal to emission from a blackbody at Ts
Net rate of energy transfer must be zero.
For any surface in the enclosure, the total, hemispherical emissivity of the surface is
equal to its total, hemispherical absorptivity if isothermal conditions exist and no net
radiation heat transfer occurs at any of the surfaces
o 𝜀=𝛼
 If the irradiation corresponds to emission from a blackbody at the surface
temperature, T, in which case 𝐺𝜆 (𝜆) = 𝐸𝜆,𝑏 (𝜆, 𝑇) 𝑎𝑛𝑑 𝐺 = 𝐸𝑏 (𝑇)
 Or the surface is gray (𝜀𝜆 𝑎𝑛𝑑 𝛼𝜆 𝑎𝑟𝑒 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑜𝑓𝜆)
o 𝜀𝜆 = 𝛼𝜆
 Applicable if the irradiation is diffuse (𝐼𝜆,𝑖 𝑖𝑠 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑜𝑓 𝜃 𝑎𝑛𝑑 𝜑)
or if surface is diffuse (𝜀𝜆,𝜃 𝑎𝑛𝑑 𝛼𝜆,𝜃 𝑎𝑟𝑒 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑜𝑓 𝜃 𝑎𝑛𝑑 𝜑
o 𝜀𝜆,𝜃 = 𝛼𝜆,𝜃
 No restrictions. Always applicable since they are inherent surface
properties.
 Independent of the spectral and directional distributions of the emitted and
incident radiation
The Gray Surface (12.8)
- Because the total absorptivity of a surface depends on the spectral distribution of the
irradiation, it cannot be stated unequivocally that α = ε. For example, a particular surface
may be highly absorbing to radiation in one spectral region and virtually nonabsorbing in
another region (Figure 12.25a).
-
Accordingly, for the two possible irradiation fields Gλ, 1(λ) and Gλ, 2(λ) of Figure 12.25b,
the values of α will differ drastically. In contrast, the value of ε is independent of the
irradiation. Hence there is no basis for stating that α is always equal to ε.
-
Gray Surface - one for which αλ and ελ are independent of λ over the spectral regions of
the irradiation and the surface emission (wavelength independence).
o Irradiation and surface emissions concentrated in a region for which the spectral
properties of the surface are approximately constant
-
o
Diffuse, gray surface - A surface for which 𝛼𝜆,𝜃 and 𝜀𝜆,𝜃 are independent of θ and λ
(diffuse because of the directional independence and gray because of the wavelength
independence).
Environmental Radiation
Solar Radiation
- Sun: 1.39 × 109 m in diameter and is located 1.50 × 1011 m from the earth.
o Emits approximately as a blackbody at 5800 K
- At the outer edge of the earth's atmosphere, the flux of solar energy has decreased by a
factor of (rs/rd)2, where rs is the radius of the sun and rd is the distance from the sun to
the earth
- The solar constant,3 Sc, is defined as the flux of solar energy incident on a surface
oriented normal to the sun’s rays, at the outer edge of the earth's atmosphere, when the
earth is at its mean distance from the sun (Figure 12.27). It has a value of 1368 ± 0.65
W/m2
- For a horizontal surface (that is, parallel to the earth's surface), solar radiation appears as
a beam of nearly parallel rays that form an angle θ, the zenith angle, relative to the
surface normal.
- The extraterrestrial solar irradiation, GS, o, defined for a horizontal surface, depends on
the geographic latitude, as well as the time of day and year.
o 𝐺𝑆,𝑜 = 𝑆𝑐 ∙ 𝑓 ∙ cos 𝜃
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o
However, as the solar radiation propagates through the earth's atmosphere, its magnitude
and both its spectral and directional distributions experience substantial modification.
This change is due to absorption and scattering of the radiation by atmospheric
constituents.
The sun is essentially a nuclear reactor, with temperatures as high as 40,000,000 K in the
core region. The sun emits radiation energy at a rate of 3.8 x 1026 W. Less than a
billionth of this energy (about 1.7 x 1017 W) strikes the earth.
Chapter 13
Blackbody Radiation Exchange (13.2)
- Matters are simplified for surfaces that may be approximated as blackbodies, since there
is no reflection. Hence energy leaves only as a result of emission, and all incident
radiation is absorbed.
- Consider radiation exchange between two black surfaces of arbitrary shape (Figure 13.8).
Defining qi → j as the rate at which radiation leaves surface i and is intercepted by
surface j, it follows that
o Net rate at which radiation leaves a surface I as a result of its interaction with j,
which is equal to the net rate at which j gains radiation due to its interaction with i
o