Fundamentals of Thermal
Radiation 2
Radiation Intensity
• The direction of radiation passing
dA
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.
n
• The differential solid angle dw subtended by a differential area
dS on a sphere of radius r can be expressed as
dS
d w  2  sin q dq df
r
(12-11)
• 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.
dQe
dQe
 W  (12-13)
I e q , f  



dA cos q  dw dA cos q sin q dq df  m2  sr 
• The radiation flux for emitted radiation is the
emissive power E
dQe
dE 
 I e q , f  cos q sin q dq df
dA
(12-14)
• The emissive power from the surface into the
hemisphere surrounding it can be determined by

E
dE  
2

 /2
f 0 q 0
I e q , f  cos q sin q dq df
2
W
m

(12-15)


hemisphere
• For a diffusely emitting surface, the intensity of
the emitted radiation is independent of direction
and thus Ie=constant:
E

hemisphere
dE  I e 
2

 /2
f 0 q 0
cos q sin q dq df   I e

(12-16)
• For a blackbody, which is a diffuse emitter, Eq.
12–16 can be expressed as
Eb   I b
(12-17)
• where Eb=sT4 is the blackbody emissive
power. Therefore, the intensity of the radiation
emitted by a blackbody at absolute temperature
T is
sT 4
I b T  



Eb

W m2 ×sr

(12-18)
• 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
G

dG  
2

 /2
f 0 q 0
I i q , f  cos q sin q dq df
(12-19)
W m 
2
hemisphere
• When the incident radiation is diffuse: G   Ii
(12-20)
• Radiosity (J )─ the rate at
which radiation energy leaves
a unit area of a surface in all
directions:
J 
2

 /2
f 0 q 0
I e r q , f  cos q sin q dq df
W m 
2
(12-21)
• For a surface that is both a diffuse emitter and a
diffuse reflector, Ie+r≠f(q,f):
J   I e r ( W m 2 )
(12-22)
• 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)


dQe
W
I l ,e  l , q , f  
 2

dA cos q  dw  d l  m  sr  μm 
(12-23)
• Then the spectral emissive power becomes
El  
2
f 0
 /2
q
0
I l ,e  l ,q , f  cos q sin q dq df
(12-24)
• 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
2hc02
Ibl  l , T   5
l exp  hc0 l kT   1


W/m2  sr  μm (12-28)
• Then the spectral blackbody emissive power is
Ebl  l , T    Ibl  l , T 
(12-29)
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.
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.
• Spectral directional emissivity
e l ,q  l ,q , f , T  
I l ,e  l , q , f , T 
Ibl  l , T 
(12-30)
• The subscripts l and q are used to designate
spectral and directional quantities, respectively.
• The total directional emissivity (intensities
integrated over all wavelengths)
eq q , f , T  
I e q , f , T 
I b T 
• The spectral hemispherical emissivity
El  l , T 
el l,T  
Ebl  l , T 
(12-31)
(12-32)
• The total hemispherical emissivity
e T  
E T 
(12-33)
Eb T 
• Since Eb(T)=sT4 the total hemispherical
emissivity can also be expressed as
e T  
E T 
Eb T 



0
e l  l , T  Ebl  l , T  d l
sT
4
(12-34)
• To perform this integration, we need to know
the variation of spectral emissivity with
wavelength at the specified temperature.
Gray and Diffuse Surfaces
• Diffuse surface ─ a surface
which properties are
independent of direction.
• Gray surface ─ surface
properties are independent of
wavelength.
Absorption, Reflection, and Transmission
• 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).
Absorbed radiation Gabs
a

Incident radiation
G
Gref
Reflected
radiation
• Reflectivity:
r

Incident radiation
G
Transmitted radiation Gtr

• Transmissivity: t 
Incident radiation
G
• Absorptivity:
• The first law of thermodynamics requires that
the sum of the absorbed, reflected, and
transmitted radiation be equal to the incident
radiation.
G

G
G
G
ref 
tr 
Gabs

G

G

G
abs
ref
tr
a  r t  1
(for a semi-transparent medium)
• For opaque surfaces, t =0, and thus
a  r 1
• These definitions are for total, hemispherical
properties.
• Spectral, hemispherical absorptivity: al  l  
• Total, hemispherical absorptivity: a  

0


Gl d l
rl  l  

• Total, hemispherical reflectivity: r 

0

Gl  l 
a l Gl d l
0
• Spectral, hemispherical reflectivity:
Gl ,abs  l 
Gl ,ref  l 
Gl  l 
rl Gl d l

0
Gl d l
• Spectral, hemispherical transmissivity: t l  l  

• Total, hemispherical transmissivity: t 

0

Gl ,tr  l 
Gl  l 
t l Gl d l

0
Gl d l
• 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.
Specular
Diffuse
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.
• The radiation absorbed by the small body per
unit of its surface area is
Gabs  a G  as T 4
• The radiation emitted by the small body is
Eemit  es T 4
• Considering that the small body is in thermal
equilibrium with the enclosure, the net rate of
heat transfer to the body must be zero.
Ases T 4  Asas T 4
• Thus, we conclude that
e T   a T 
(12-47)
• The restrictive conditions inherent in the derivation of
Eq. 12-47 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.
e l T   a l T 
(12-48)
• The form of Kirchhoff’s law that involves no
restrictions is the spectral directional form
e l ,q T   al ,q T 
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.
• The solar energy reaching the earth’s atmosphere is called the
total solar irradiance Gs, whose value is
Gs  1373 W/m 2
(12-49)
• 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
 4 L  G   4 r s T
2
2
s
4
sun
(12-50)
• 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.
• 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.
• 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:
Gsolar  GD cos q  Gd  W/m 2 
(12-49)