After discussing the topics of conduction and convection in the previous two sections,
let’s talk about the third and final mode of heat transfer, radiation. Radiation is
the transfer of heat energy between two bodies at different temperatures, that occurs in
the form of electromagnetic waves. Since these electromagnetic waves do not need a medium
to travel, radiation occurs even when there is no solid or fluid material medium present.
If we stand directly under the sun, we feel warm. This is because heat energy from the
sun reaches our body through radiation. As mentioned before, unlike conduction and convection,
radiation does not require any intermediate medium. Hence heat energy from the sun can
travel through the vacuum of space to reach earth. Radiation is the only means of heat
transfer in a vacuum, and hence is important when performing thermal analyses of satellites
and similar objects in space. Another example of radiation in our daily lives is the toaster
oven. Here, heat radiated from the heater coils toasts the slice of bread.
So now let’s look at how this radiation process is modeled mathematically. A simplified
form of the equation describing radiation from one surface to another is given by this
formula. Here, Qij is the radiated heat energy that is transferred from surface i
to surface j. Let’s take a moment to discuss the terms that contribute to radiated
heat. Epsilon is the emissivity of the surface which is a measure of the amount of
thermal radiation emitted by the surface. Fij is known as the view factor between the
surfaces. The view factor is the fraction of the total radiant energy that leaves surface
i, which arrives directly at surface j. We will discuss emissivity and view factor in
more detail shortly. Ti and Tj are the surface temperatures in absolute units. It
is important to note that we always use the absolute scale i.e. Kelvin or degree Rankine
when modeling radiation. For everything else in thermal simulations, say phase change or
conduction or convection, we can use any temperature scale, but for radiation, calculations must
be performed in an absolute scale. Last but not least, sigma is the Stefan-Boltzmann constant
having a value of 5.67e-8 W/m^2-K^4. This is a physical constant formulated in the late
19th century by two physicists from which the constant takes its name, and it can be
thought of as converting the absolute temperature to heat intensity.
All materials having temperature greater than absolute zero, emit thermal radiation. Based
on the radiation equation, we can see that radiant energy exchange is directly proportional
to the fourth order of the absolute temperature of the two surfaces. This has two implications
– not only does the amount of heat radiated between two surfaces depend on the temperature
difference between them, but it also depends on their absolute temperature. Let’s
take a simple example of two objects with a difference of 20⁰C but consider two cases
– one, when the two objects are 20 and 40⁰C and another case when they are at 500 and
520⁰C.
Although the temperature difference is only 20 degrees between the two cases, we see that
the radiated heat between them is close to 20 times different! Previously, we typically
considered only the temperature difference between surfaces. However, in the case of
radiation, we take the absolute temperature to the fourth power, then look at the difference.
Thus, even though the temperature difference is the same, the use of absolute temperature
to the fourth power is an important point concerning radiated heat. As was the case
with convection, we will be modeling radiation as a boundary condition in the conduction-based
solver that we use in this course.
Now let us look at how different bodies or materials behave when subjected to incident
thermal radiation. When a certain amount of thermal radiation is incident on a surface,
part of it will be reflected, part of it will be absorbed and the rest of it may be transmitted
through the body. The fraction of the incident thermal radiation that is transmitted by a
body is known as the transmissivity tau, the fraction that is absorbed by the body is known
as absorptivity alpha, and the fraction that is reflected by the body is known as its reflectivity
rho. Since these three components are fractions of the incident radiation, they always add
up to 1. Based on these three components, the behavior of a body or a material can be
characterized. If a body does not transmit any incident radiation, i.e. the transmissivity
= 0, and the absorptivity and reflectivity are non-zero, then the body is said to be
opaque. Conversely, if a body transmits all the incident radiation i.e. when the transmissivity
is 1 and the other two components are 0, the body is said to be transparent. If a body
absorbs all the incident radiation i.e. its absorptivity is one and the other two components
are zero, then that body is called a black body. Conversely, if a body reflects all the
incident radiation in a uniform manner in all directions, it is called a white body.
Black body and white body are idealized physical states, which are not found in nature. But
mirror-like, metallic surfaces reflect radiation well and are closer to a white body. For example,
Aluminium foil reflects most of the incident radiation and hence is closer to a white body.
Asphalt on the other hand absorbs most of the incident radiation and is hence closer
to a black body.
Let’s take a moment to discuss the concept of a black body in more detail, as it can
be useful to understand emissivity, which we saw appear in the radiation equation earlier.
A black body absorbs all incident radiation, while reflecting or transmitting none – hence,
it is the perfect absorber. A close idealization of a black body could be a cavity with opaque
walls which has a small hole in it. In this case, once an incident wave enters the cavity
through the small hole, it is either absorbed or reflected. Any reflected component of the
radiation hits another surface of the cavity and part of it get absorbed. This process
of absorption and reflection continues, trapping the incident radiation inside the body till
it is completely absorbed since the chance that a fraction of the reflected radiated
heat will leave through the same opening is small. For thermal equilibrium, i.e. if the
black body has to stay at a constant temperature, it has to emit thermal radiation at the same
rate that it absorbs it. Hence, a black body is also a perfect emitter. While we haven’t
proven this point yet, the emissivity is the same as the absorptivity for a black body,
which is 1.0
Now, real surfaces in nature fall somewhere in between black and white bodies. Such surfaces
are called grey bodies. Grey bodies may transmit, absorb and reflect the radiation incident
on them i.e. they may have non-zero transmissivity, absorptivity and reflectivity which together
add up to 1. It is important to consider the absorptivity, reflectivity and transmissivity
of different materials in the design process. For example, when designing a solar thermal
conversion system, it is critical to choose collector materials which have high absorptivity
and hence can absorb most of the incident radiation.
As noted earlier, any body with temperature greater than absolute 0 or 0 K, will emit
radiation. The amount of thermal radiation emitted by a body depends on its surface property
called emissivity. Emissivity is defined as the ratio of the energy radiated from a material’s
surface to the energy radiated from a black body’s surface, under the same conditions
i.e. at the same wavelength, temperature and viewing conditions. Emissivity is a dimensionless
quantity. For a body to be in thermal equilibrium, the emissivity of a body is equal to its absorptivity.
Thus, surfaces which are good absorbers of thermal radiation are also good emitters.
At first glance, it may seem strange to think of emissivity and absorptivity to be the same
for a body in thermal equilibrium, but the concept of a black body introduced earlier
has helped us understand this point. Thus, while we only saw emissivity in the radiative
heat flow rate equation shown in the beginning of this lesson, we can now understand emissivity
and absorptivity to be the same for our purposes.
Emissivity depends on a number of factors such as the material and nature of the surface,
its temperature, the wavelength and direction of the radiation. For example, a polished,
mirror-like surface has a lower emissivity as compared to a rough surface, since a polished
surface reflects most of the radiation instead of absorbing it. From this table here, we
see that different materials have different emissivity. But apart from bare, polished
metals, the general color or appearance of a surface is not a good way to estimate its
emissivity. For example, pure water absorbs very little visible light and hence may appear
transparent, but it is a very good absorber of infrared radiation. Hence, water has a
high emissivity despite its transparent appearance.
In addition, emissivity may also depend on the direction. If a surface emits the same
amount of radiation in different directions, then the surface is known as a diffuse surface.
On the other hand, if the radiation emitted by a surface changes with direction, then
the surface is known as specular surface. Another factor affecting emissivity is the
wavelength of the radiation. In case of opaque surfaces, the emitted radiation, and hence
emissivity, may vary with wavelength. A spectral distribution plot shows the variation of emitted
radiation with wavelength. Lastly, emissivity may also depend on temperature. For example,
the emissivity of certain Aluminium alloy was found to reduce from 0.3 to 0.1 when its
temperature was increased from 600 K to 800 K.
So now we know how to differentiate between opaque and transparent radiation, between
diffuse and specular radiation, and we also learned what the term black body, white body
and gray body mean. In this course, we will talk about opaque, gray, diffuse radiation
i.e. the bodies considered here cannot transmit any radiation, and their surfaces are diffuse
surfaces which emit or reflect radiation in all directions in a similar manner. As discussed
before, since we are using a conduction-based solver in this course, radiation will be modeled
as a boundary condition.