Heat transfer – lecture 4
THERMAL RADIATION
Thermal radiation differs from heat conduction and convective heat transfer in
its fundamental laws. Heat transfer by radiation does not require the presence of
matter; electromagnetic waves also transfer energy in empty space. Temperature
gradients or differences are not decisive for the transferred flow of heat, rather the
difference in the fourth power of the thermodynamic (absolute) temperatures of the
bodies between which heat is to be transferred by radiation is definitive. In addition,
the energy radiated by a body is distributed differently over the single regions of the
spectrum. This wavelength dependence of the radiation must be taken as much into
account as the distribution over the different directions in space.
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Fundamentals
All the considerations that follow are only valid for radiation that is stimulated
thermally. Radiation is released from all bodies and is dependent on their material
properties and temperature. This is known as heat or thermal radiation. Two theories
are available for the description of the emission, transfer and absorption of radiative
energy: the classical theory of electromagnetic waves and the quantum theory of
photons. These theories are not exclusive of each other but instead supplement each
other by the fact that each describes individual aspects of thermal radiation very
well.
According to quantum theory, radiation consists of photons (= light particles),
that move at the velocity of light and have no rest mass. They transfer energy,
whereby each photon transports the energy quantum
Here, h = (6.626 068 76±0.000 000 52)·10-34 Js is the Planck constant, also
known as Planck’s action quantum; ν is the frequency of the photons. Quantum
theory is required to calculate the spectral distribution of the energy emitted by a
body. Other aspects of heat transfer can, in contrast, be covered by classical theory,
according to which the radiation is described as the emission and propagation of
electromagnetic waves.
Electromagnetic waves are transverse waves that oscillate perpendicular to
the direction of propagation. They spread out in a straight line and in a vacuum at
the velocity of light c0 = 299 792 458 m/s. Their velocity c in a medium is lower than
c0, whilst their frequency ν remains unchanged; the ratio n := c0/c > 1 is the
refractive index of the medium. The wavelength λ is linked to the frequency ν by
The energy transported by the electromagnetic waves depends on λ. This also
has to be considered for heat transfer.
Fig. 4.1 shows the electromagnetic spectrum that extends from λ = 0 to very
large wave lengths (λ → ∞). At small wave lengths (λ < 0.01 µm) we have gammarays and x-rays, neither of which are thermally stimulated and so therefore do not
belong to thermal radiation. The same is true for the region of large wavelengths, (λ
> 103 µm), that is determined by the oscillations of electronic switching networks
(radar, television and radio waves). Neither region has any meaning for thermal
radiation. The thermal radiation region is the middle of the range of wavelengths
between around 0.1 µm and 1000 µm. Within this region bodies, whose
temperatures lie between a few Kelvin and 2·104 K, radiate. This includes the visible
light region between 0.38 µm (violet) and 0.78 µm (red). The designation of this
radiation as light has no physical reason, but instead is based on the peculiarity that
the human eye can “see” in this wavelength range.
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Figure 4.1 Electromagnetic wave spectrum.
The wavelength interval 0.01 µm ≤ λ ≤ 0.38 µm is the range of ultraviolet
(UV) radiation. Between 0.78 µm and 1000 µm lies infrared (IR) radiation. This is the
wavelength range in which most earthly bodies radiate.
Thermal radiation is not only dependent on the wavelength; in numerous
problems, particularly in radiative exchange between different bodies, its distribution
in space must also be considered. This holds for the emission of radiative energy in
the same way as for reflection and absorption of radiation incident on a body.
Radiative heat transfer or thermal radiation is the science of transferring
energy in the form of electromagnetic waves. Unlike heat conduction,
electromagnetic waves do not require a medium for their propagation. Therefore,
because of their ability to travel across vacuum, thermal radiation becomes the
dominant mode of heat transfer in low pressure (vacuum) and outer-space
applications. Another distinguishing characteristic between conduction (and
convection, if aided by flow) and thermal radiation is their temperature dependence.
While conductive and convective fluxes are more or less linearly dependent on
temperature differences, radiative heat fluxes tend to be proportional to differences
in the fourth power of temperature (or even higher).
All materials continuously emit and absorb electromagnetic waves, or photons,
by changing their internal energy on a molecular level. Strength of emission and
absorption of radiative energy depend on the temperature of the material, as well as
on the wavelength λ, frequency ν.
When an electromagnetic wave strikes an interface between two media, the
wave is either reflected or transmitted. Most solid and liquid media absorb all
incoming radiation over a very thin surface layer. Such materials are called opaque or
opaque surfaces (even though absorption takes place over a thin layer). An opaque
material that does not reflect any radiation at its surface is called a perfect absorber,
black surface, or blackbody, because such a surface appears black to the human eye,
which recognizes objects by visible radiation reflected off their surfaces.
Emissive Power
Every medium continuously emits electromagnetic radiation randomly into all
directions at a rate depending on the local temperature and the properties of the
material. The radiative heat flux emitted from a surface is called the emissive power
E, and there is a distinction between total and spectral emissive power (heat flux
emitted over the entire spectrum or at a given frequency per unit frequency interval),
so that the spectral emissive power Eν is the emitted energy/time/surface
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area/frequency, while the total emissive power E is emitted energy/time/surface
area. Spectral and total emissive powers are related by
It is easy to show that a black surface is not only a perfect absorber, but it is
also a perfect emitter, that is, the emission from such a surface exceeds that of any
other surface at the same temperature (known as Kirchhoff ’s law). The emissive
power leaving an opaque black surface, commonly called blackbody emissive power,
can be determined from quantum statistics as
where it is assumed that the black surface is adjacent to a nonabsorbing medium of
constant refractive index n. The constant k = 1.3806×10-23 J/K is known as
Boltzmann’s constant.
Figure 4.2 Blackbody emissive power spectrum.
The spectral dependence of the blackbody emissive power into vacuum (n =
1) is shown fora numberof emittertemper atures in Fig. 4.2. It is seen that emission
is zero at both extreme ends of the spectrum with a maximum at some intermediate
wavelength. The general level of emission rises with temperature, and the important
part of the spectrum (the part containing most of the emitted energy) shifts toward
shorter wavelengths. Because emission from the sun (“solar spectrum”) is well
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approximated by blackbody emission at an effective solar temperature of Tsun = 5762
K, this temperature level is also included in the figure. Heat transfer problems
generally involve temperature levels between 300 and, say, 2000 K (plus, perhaps,
solar radiation). Therefore, the spectral ranges of interest in heat transfer
applications include the ultraviolet (0.1 to 0.4 µm), visible radiation (0.4 to 0.7 µm,
as indicated in Figure 4.2 by shading), and the near- and mid-infrared (0.7 to 20
µm). For quick evaluation, a scaled emissive power can be written as
where
Equation above has its maximum at
which is known as Wien’s displacement law. The constants C1,C2, and C3 are known
as the first, second, and third radiation constants, respectively.
The total blackbody emissive is found by integrating over the entire spectrum,
resulting in
where
is the Stefan–Boltzmann constant.
Radiative properties of solids and liquids
Because radiative energy arriving at a given point in space can originate from
a point far away, without interacting with the medium in between, a conservation of
energy balance must be performed on an enclosure bounded by opaque walls (i.e., a
medium thick enough that no electromagnetic waves can penetrate through it).
Strictly speaking, the surface of an enclosure wall can only reflect radiative energy or
allow a part of it to penetrate into the substrate. A surface cannot absorb or emit
photons: Attenuation takes place inside the solid, as does emission of radiative
energy (and some of the emitted energy escapes through the surface into the
enclosure). In practical systems the thickness of the surface layer over which
absorption of irradiation from inside the enclosure occurs is very small compared
with the overall dimensions of an enclosure—usually, a few angstroms for metals and
a few micrometers for most nonmetals. The same may be said about emission from
within the walls that escapes into the enclosure.
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Thus, in the case of opaque walls it is customary to speak of absorption by
and emission from a “surface,” although a thin surface layer is implied. If radiation
impinging on a solid or liquid layer is considered, a fraction of the energy will be
reflected (reflectance ρ, often also referred to as reflectivity), another fraction will be
absorbed (absorptance α, often also referred to as absorptivity), and if the layer is
thin enough, a fraction may be transmitted (transmittance τ, often also referred to as
transmissivity). Because all radiation must be either reflected, absorbed, or
transmitted,
If the medium is sufficiently thick to be opaque, then τ = 0 and
All surfaces also emit thermal radiation (or, rather, radiative energy is emitted
within the medium, some of which escapes from the surface). The emittance ε is
defined as the ratio of energy emitted by a surface as compared to that of a black
surface at the same temperature (the theoretical maximum).
All of these four properties may vary in magnitude between the values 0 and
1; for a black surface, which absorbs all incoming radiation and emits the maximum
possible,
They may also be functions of temperature as well as wavelength and
direction (incoming and/or outgoing). One distinguishes between spectral and total
properties (an average value over the spectrum) and also between directional and
hemispherical properties (an average value over all directions).
Figure 4.3 Normal, spectral emittances for selected materials. (From White, 1984.)
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Radiation shields
If it is desired to minimize radiative heat transfer between two surfaces, it is
common practice to place one or more radiation shields between them (usually, thin
metallic sheets of low emittance). In these situations any two shields Ai and Aj often
enclose one another, or are very close together. The radiative heat transfer between
two diffusely reflecting plates is then
where Rij is termed the radiative resistance.
Figure 4.4 Arrangement of parallel or concentric radiation shields
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Summary
Unlike conduction and convection, radiative heat transfer between two
surfaces or between a surface and its surroundings is not linearly dependent on the
temperature difference and is expressed instead as
where F includes the effects of surface properties and geometry and σ is the Stefan–
Boltzmann constant
For modest temperature differences, this equation can be linearized to the
form
where hr is the effective “radiation” heat transfer coefficient,
and for small
is approximately equal to
It is of interest to note that for temperature differences on the order of 10 K,
the radiative heat transfer coefficient hr for an ideal (or “black”) surface in an
absorbing environment is approximately equal to the heat transfer coefficient in
natural convection of air. The radiation thermal resistance, analogous to the
convective resistance, is seen to equal
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