Download 4.4 Heat transfer by radiation 4.4.1 Black body radiation f c λ = c f = 1

4.4 Heat transfer by radiation
The considered above energy transfer mechanisms, heat conduction and heat transfer
by convection (free or forced), imply the existence of an interstitial material medium for
the transfer of heat. For example, for the heat conduction there should exists a
temperature gradient in a substance and for the heat transfer by convection there should
exists a fluid flow to sweep away the heat from a body. The third energy transfer
mechanism is electromagnetic radiation which does not need any material medium.
Electromagnetic energy transfer occurs with velocity of light in vacuum.
Below we will consider several basic notions and laws relevant to the energy transfer
by radiation. We will also consider the sun radiation and its propagation in the earth’s
atmosphere. At first we will consider such notions as absorptivity, emissivity, abledo,
reflection, scattering, and transmission.
4.4.1 Black body radiation
Electromagnetic waves do not need any interstitial material medium and propagate
with velocity
c = λf
(4.42)
where c is light velocity is vacuum, λ wavelength, and f frequency of electromagnetic
radiation. The number
1
λ
=
f
c
is called the wave number and often used as an index of
frequency.
Absorptivity of surface is a fraction of the incident radiation absorbed by the surface
and is expressed as
αf =
q af
q if
,
(4.43)
where q af df and q if df are the absorbed and incident radiation energy per unit area per
unit time in the frequency interval f , f + df. Absorptivity depends on the wave length or
frequency i.e., α (λ ) or α ( f ) . For any real body, usually α < 1 and it varies considerably
with frequency. If α = const < 1 for all frequencies and all temperatures then the body is
called gray. If α = 1 then a body is called black. At the atomic level, physical picture of
the absorption of radiant energy (photons) is that atoms and molecules of the irradiated
body go from a low to a high energy state.
Emissivity of a body ε ( f ) (or ε (λ ) ) is a ratio of the actual radiation emitted by a body
to the amount emitted by a black body
εf =
ε ef
,
ε bfe
(4.44)
where ε ef df and ε bfe df are the radiation emitted by body and black body in the
frequency interval f , f + df, respectively. If ε = const < 1 for all frequencies and all
temperatures then a body is called gray. If ε = 1 then a body is called black. Any body
with T > 0 K emits radiation. Physical atomic picture of the emission of radiant energy is
that atoms and molecules of the body go from a high to a low energy state. Emissivity
increases with temperature for nearly all materials.
The Kirchhoff’s law states that at thermal equilibrium
α ( f ) = ε ( f ) or α (λ ) = ε (λ ) .
(4.44)
Both α and ε depend strongly of the properties of surface. The case when ε ≈ α ≈ 0 is
for a highly polished surface. For example, aluminum foil has small absorptivity and
emissivity, Fig. 4.9. Emissivity and absorptivity are also material and temperature
dependent.
At the typical Earth’s temperatures all bodies emit radiation in the range of 3-100 µm.
Even fresh snow emits radiation as a black body between 3-100 µm. It should be kept in
mind that ‘snow behaves as a black body’ means that it emits its ‘own’ radiation that
should be distinguish form the case when it reflects the solar radiation. The sun radiates
as a black body at 6 000 K (average temperature of the sun’s surface). The earth radiates
as a black body at 300 K (average temperature of the earth’s surface).
Fig. 4.9 Cooling of a bottle with water (125 ml) in still air when it is bare (triangles)
and coated with aluminum foil (circles) (Bohren, 1987).
4.4.2 Stefan-Boltzmann law
It was found experimentally that total radiant energy emitted by a black body is
q f = σT 4 ,
(4.46)
where T is absolute temperature and σ = 5.67 ⋅ 10 −8 W/m2K4 the Stefan-Boltzmann
constant. The expression (4.46) is known as the Stefan-Boltzmann law. For nonblack
(real) bodies at temperature T, the emitted radiant energy is
q f = ε (λ )σT 4 ,
(4.47)
where the emissivity ε (λ ) must be evaluated at temperature T.
Although the total energy emitted by a black body is proportional to T4, the exchange
of energy between the two black bodies at temperatures T1 and T2 becomes nearly
proportional to the temperature difference T1 - T2, when the difference is small, what is a
common case in the natural environmental conditions. Indeed, writing T1 − T2 ≡ δT , the
difference between the fluxes of radiant energy of the two bodies is expressed as
∆q f = σ {(T1 + δT ) 4 − T14 } ≈ 4σT13δT ,
with an error given by 1.5
δT
T
(4.48)
. For the temperatures ≈ 298 K, the error is only 0.5 % per
Kelvin.
4.4.3 Planck’s radiation law
Historically, classic theory could not explain radiation of a black body. For
explanation Planck introduced a notion of quantum or photon: atom behaves as harmonic
oscillator which can absorb or emit radiant energy only in a discrete amount that is
proportional to its frequency i.e.,
E = hf ,
(4.49)
where h = 6.625 ⋅ 10 −34 Js is the Planck’s constant and f frequency of an atom oscillator.
Amount of energy emitted by an atom is equal to a decrease in potential energy of its
constituents (electrons and nucleus). The expression (4.49) explains the observed line
spectra of atoms which are due to the changes in the energy state of outer electrons in the
atom. In complex molecules composed of several atoms, the radiant energy is derived
from vibration and rotation of individual atoms. Therefore the emitted or absorbed
radiation is in the broad range of frequencies.
The Planck’s radiation law gives the radiant energy distribution of black body
radiation,
qb =
2πhc 2
λe
5
hc
−1
kλT
.
(4.50)
Here the subscript “b” means black body. The Plank’s law (4.50) can also be written in
the frequency form i.e., qb = qb ( f ) . The two forms are related according to
c
qb (λ )dλ = qb ( f )df , where dλ = − 2 df . Physically a value of qbλ dλ means a radiated
f
energy flux from a black surface in the wave length interval λ, λ + dλ. Dimension of the
W
W
energy flux qb is 2 = 3 i.e., the energy flux per unit area per unit length (wave
m m m
length in µm !). Example of radiant energy distribution of black, grey, and real bodies is
shown in Fig. 4.10.
Fig. 4.10 Radiant energy distribution of black, grey, and real bodies as a function
of wave length (Bird, 1960).
The Stefan-Boltzmann law (4.46) can be obtained by the integration of the Planck’s
radiation law (4.49). The Stefan-Boltzmann law describes the total radiation from a black
body in all direction. It is useful to know how the radiant energy is distributed with
respect to angle. This is described by the Lambert’s law
qb ,dθ =
qb
π
cos θ =
(σT 4 )
π
cos θ ,
(4.51)
where qb ,dθ is the energy emitted per unit area per unit time per unit solid angle in the
direction θ. In (4.51), cosθ is due to the fact that projection of the unit area on the
direction θ is dA cos θ.
Differentiation of the Plank’s law (4.49) and then setting the result to zero and solving
for λ gives the Wien’s law for a black body,
λmax =
2.897 ⋅ 10 6
,
T
(4.52)
where λmax is the wavelength at which the radiation energy E(λ) reaches maximum value.
The λ is in nm (nm = 10-9 m) and T in K. The Wien’s law indicates that as temperature of
a black body increases then the apparent color of radiation shifts from red (long
wavelength) to blue (short wavelength). From the Wien’s law one can calculate the
maximum wavelength at which the sun and earth emit maximum energy. For
approximately 6000 K (the sun) and 300 K (the earth) they are 0.48 and 9.7 µm,
respectively.
4.4.4 Change of radiant energy between two nonblack bodies
Let us imagine a nearly black body 1 at temperature T1. This body is totally surrounded
by another a nearly black body 2 at temperature T2. At T2 > T1, the total radiant energy
coming to body 1 is (see (4.47)),
Q = σA1 (α 1T24 − ε 1T14 ) ,
(4.53)
where it is assumed that α 1 = ε 1 (T2 ) . A nearly black body means that only one or two
reflections need to be considered. But if we have a system with highly reflecting surfaces,
the analysis is much more complicated because one has to take into account a complicate
distribution of the emitted, reflected, and transmitted radiation.
As an example let us consider the total Earth’s albedo R. Under the Earth’s albedo we
I
will understand a fraction of incoming solar radiation reflected to space i.e., R = λR ,
Iλ
where I λ and I λR are total incoming solar radiation and the reflected radiation,
respectively. Incoming solar radiation partly reflects from a homogeneous cloud layer
and partly transmits to the Earth’s surface, Fig. 4.11. Then again, a part of the radiation
reflects from the Earths surface, transmits the cloud layer, and reflects from the cloud
layer back to the surface and so on. In this case the total Earth’s albedo is calculated
according to a formula
Rtot =
I 0 Rc + I 0T 2 R g + I 0T 2 R g2 Rc + I 0T 3 R g3 Rc2 + ...
I0
= Rc +
= Rc + T 2 R g (1 + R g Rc + R g2 Rc2T + ..
T 2 Rg
1 − R g Rc
where I0 is the total incoming solar radiation, T =
,
(4.54)
I γT
a fraction of radiation passed
Iλ
through the cloud layer i.e., transmission, and Rc and Rg are the cloud and surface albedo,
respectively.
Fig. 4.11 Multiple reflection of solar radiation in a cloud-surface system. External solar
radiation is I0. Rc is a fraction of short wave radiation reflected from a cloud layer, Rg
from the earth’s surface. T = 1 - Rc is a fraction of solar radiation transmitted through the
cloud layer.
If a body is not totally surrounded by the other one then for calculation we should take
into account a direction (angle) at which the bodies ‘see’ each other. This is so called
view-factor, which is tabulated for the simplest geometry. If two bodies are far from each
other then the first can be considered as a point source of energy and the second one as a
disk, as it is in the case of the sun and the earth.
4.4.5 Radiant energy transport in material medium
Above we have considered the processes in which it has been implicitly assumed that
absorption and reflection occurred only on surfaces and radiation passed through the
interstitial material medium (in our case the atmosphere) undisturbed. But during the
transport of radiant energy through gas or liquid (material medium) some fraction of the
radiant energy is absorbed and reflected by the molecules and the remaining fraction is
transmitted. Sum of these fractions equals to unity i.e.,
α λ + Rλ + Tλ = 1 ,
(4.55)
where α λ , Rλ and Tλ are absorbivity, albedo, and transmitivity, respectively.
When electromagnetic radiation passes through the gas or liquid medium, a part of
photons strike the molecules and aerosol particles. As a result, these photons are either
absorbed or scattered. The absorption and scattering lead to attenuation of the initial
radiation. The absorption removes radiant energy so that the material medium is heated.
In contrast to absorption, scattering only changes the direction and polarization of
radiation. Scattering is a process at which an atom or molecule, at first, absorb a photon
to raise to ‘excited state’ and then return spontaneously to lower initial energy state when
emitting a photon. Scattering depends strongly on wavelength and size and structure of
scattering atoms and molecules. Since in the atmosphere, scattering is uniform in all
directions it also leads to attenuation of the initial radiation beam. For example, scattering
of the blue light (λ = 400 nm) is responsible for the blue color of the sky and the apparent
redness of the sun’s disk is an evidence of that blue light has been preferentially removed
from the initial incoming solar radiation. A second example of the light scattering is a
laser beam which we observe. Without the scattering medium (gas molecules) we would
not see it, as it is, for example, in space. It should be emphasized that all bodies emit
(thermal) radiation even if they had not been irradiated by the external radiation.
In geometrical optics, notions of refraction and reflection are used. These phenomena
can be considered as coherent scattering. There are two types of reflection: mirror
reflection and diffusion reflection. In the former case, the reflection occurs according to
the law of geometrical optics. In the case of diffusion refraction a light beam disperses in
all directions. Reflection strongly depends on surface properties. When light angle
increases then mirror reflection increases and adsorption decreases.
It should be noticed that a photon can be scattered many times. Such systems, in which
a photon is scattered many times, are called optically thick. Multiple scattering is
important in the atmospheric science. For example, optical thickness of cirrus clouds
varies from 0.02 for tropical thin high altitude cirrus to about 0.5 for lower mid-latitude
cirrus. In cirrus clouds, light is scattered mainly on ice crystals. Another example of the
optical thick system is milk. In milk, a fat spherical particle strongly scatters blue light
and therefore color of milk should be bluish. But because of that milk is optically thick
the incoming white light is scattered so many times that the out-coming scattered light is
white too.