6.1 Introduction to Radiation
Any matter with temperature above absolute zero (0 K) emits electromagnetic radiation. In a simplified picture, radiation comes from the constantly changing electromagnetic fields of the oscillating atoms. Electromagnetic radiation can be visualized as waves traveling at the speed of light. The two prominent characters of the wave are the wavelength (λ) and frequency (ν). The wavelength is the distance between crest to crest on the wave. The frequency is related to wavelength by the following:
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(Eq. 6.1)
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where c is the speed of light, approximately equal to 3×108 m/s in vacuum. The wavelength is measured in units of length, and the frequency is given in cycles per second (hertz, Hz).
The amount of radiation emitted by a body depends on its temperature, and is proportional to T4. This relation shows that as the temperature of the object increases, the amount of radiation emitted increases very rapidly. The emitted radiation will travel at the speed of light until it is absorbed by another body. The absorbing medium can be gas, liquid, or solid. Radiation does not require a medium to pass through. This is demonstrated by solar radiation which pass through interplanetary space to reach the earth.
6.2 Electromagnetic Spectrum
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Electromagnetic radiation is categorized into types by their wavelengths. The types of radiation and the respective wavelength ranges are shown in Figure 6.1. Radiation with shorter wavelengths are more energetic, evident by the harmful gamma and x-rays on the shorter end of the spectrum. Radio waves, which are used to carry radio and TV signals, are much less energetic; however, they can pass through walls with no difficulty due to their long wavelengths.
The type of radiation emitted by a body depends on its temperature. In general, the hotter the object is, the shorter the wavelengths of emitted radiation, and the greater the amount. Most of the radiation we see from a cooking standpoint is in the infrared range (a 500 K body, for example, emits the most radiation at ~2×10-5 m). A much hotter body, such as the sun (~5800 K), emits the most radiation in the visible range.
The total energy emitted by a body, regardless of the wavelengths, is given by:
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(Eq. 6.2)
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Figure 6.1 The electromagnetic spectrum |
where ε is the emissivity of the body, A is the surface area, T is the temperature, and σ is the Stefan-Boltzmann constant, equal to 5.67×10-8 W/m2K4. Emissivity is a material property, ranging from 0 to 1, which measures how much energy a surface can emit with respect to an ideal emitter (ε = 1) at the same temperature.
6.3 Radiative Properties
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When radiation strikes a surface, a portion of it is reflected, and the rest enters the surface. Of the portion that enters the surface, some are absorbed by the material, and the remaining radiation is transmitted through. This is shown in Figure 6.2.
The ratio of reflected energy to the incident energy is called reflectivity, ρ. Similarly, transmissivity (τ) and absorptivity (α) are defined as the fraction of the incident energy that is transmitted through or absorbed by the object, respectively. The three radiative properties all have values between zero and 1. Furthermore, since the reflected, transmitted, and absorbed radiation must add up to equal the incident energy, the following can be said about the three properties:
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Figure 6.2 Radiative interaction
at a surface |
Generally, an object with high reflectivity has low absorptivity and emissivity. Those with low reflectivities tend to have high absorptivities and emissivities.
6.4 Radiative Heat Transfer
Consider the heat transfer between two surfaces, as shown in Figure 6.3. What is the rate of heat transfer into Surface B? To find this, we will first look at the emission from A to B. Surface A emits radiation as described in Eqn. 6.2:
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(Eq. 6.4)
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This radiation is emitted in all directions, and only a fraction of it will actually strike Surface B. This fraction is called the shape factor, F. The amount of radiation striking Surface B is therefore:
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(Eq. 6.5)
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Figure 6.3 Radiation exchange between two surfaces |
The only portion of the incident radiation contributing to heating Surface B is the absorbed portion, given by the absorptivity αB:
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(Eq. 6.6)
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Equation 6.6 is the amount of radiation going into Surface B from Surface A. To find the net heat transfer rate at B, we must now subtract the amount of radiation emitted by B:
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(Eq. 6.7)
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The net radiative heat transfer rate at Surface B is Eqn. 6.6 minus Eqn. 6.7:
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(Eq. 6.8)
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6.5 Shape Factors
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Shape factor, F, is a geometrical factor which is determined by the shapes and relative locations of two surfaces. Figure 6.4 illustrates this for a simple case of cylindrical source and planar surface. Both the cylinder and the plate are infinite in length. In this case, it is easy to see that the shape factor is reduced as the distance between the source and plane increases. The shape factor for this simple geometry is simply the cone angle (θ) divided by 2π.
Shape factors for other simple geometries are available in heat transfer texts. However, for more complicated geometries, the following two rules must be applied to find shape factors based on simple geometries.
The first is the summation rule. This rule says that the shape factor from a surface (1) to another (2) can be expressed as a sum of the shape factors from (1) to (2a), and (1) to (2b). Using this rule allows you to break up complicated geometry into smaller pieces for which the individual shape factors can be found.
The second rule is the reciprocity rule, which relates the shape factors from (1) to (2) and that from (2) to (1) as follows:
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(Eq. 6.9)
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Thus, if the shape factor from (1) to (2) is known, then the shape factor from (2) to (1) can be found by:
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(Eq. 6.10)
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Figure 6.4 Effect of distance on the shape factor
Figure 6.5 Summation rule
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