# 9 FUNDAMENTALS OF THERMAL RADIATION

In this chapter, the basic relations governing heat transfer by thermal radiation are presented. The material properties required for determining radiation exchange among surfaces are given, along with some methods for predicting these properties when measured values are unavailable. Finally, some applications of radiation in actual devices and processes and to contemporary micro- and nanoscale uses are presented.

## 9.1 Electromagnetic Waves and Thermal Radiation

Energy transfer by thermal radiation differs in fundamental ways from conduction and convection. Radiative energy is carried by electromagnetic waves, which require no medium for their propagation. Thus, unlike the other heat transfer modes, radiative energy can be transferred through a vacuum, for example allowing us to receive solar energy through the vacuum of space. Electromagnetic (EM) waves are described mathematically by Maxwell's Equations (Siegel and Howell, 2002; Modest, 2003; Bohren and Huffman, 1983), which formulate the propagation of the perpendicular amplitudes of the electric and magnetic components, E and H, of the waves (Figure 9.1). The energy carried by the wave is proportional to the square of the amplitude of the electrical component E of the wave. These equations can be used to predict the interaction of the waves with interfaces between differing materials, allowing prediction of the radiative properties of various materials in terms of electrical and magnetic properties. More details are in Section 9.5 The EM waves can be generated in various ways. Depending on the source of the EM waves, they may have differing wavelengths, extending from very short (nm) to very long (km) (Fig. 9.2). For radiative heat transfer, we are interested in EM waves originating from microscopic energy transitions that occur because of the internal energy state of a substance, which is in turn dependent on the absolute temperature of a material that is in thermodynamic equilibrium. EM waves originating from such a source are called thermal radiation, and this radiation is emitted by any substance that is above absolute zero temperature. This dependence means that all radiative transfer relations

Figure 9.1 Electromagnetic wave propagating in the x-direction with associated electric (Ey) and magnetic (Hz) components
Figure 9.2 The electromagnetic spectrum

must be in terms of absolute temperature. Thermal radiation is roughly in the range of 0.1 < Failed to parse (unknown function\lamda): \lamda < 1000 μm. The thermal radiation portion of the logarithmic wavelength scale of Fig. 9.2 is thus fairly small, and the visible portion of the spectrum is small indeed. Electromagnetic waves travel without attenuation (loss) through a vacuum and through perfectly transparent materials (ideal dielectrics). Some media will absorb wave energy, converting the radiation into internal energy. Air is generally transparent, although some gases such as carbon dioxide and water vapor can absorb radiation in certain ranges of the infrared portion of the spectrum as it travels. Solids, particularly metals, are very strong absorbers, and can completely absorb radiation over very short distances. Other solids (glass, for example) are quite transparent to radiation over wide ranges of wavelength. To provide the basis for computing energy transfer by thermal radiation, we must connect the temperature of a radiating surface to its rate of electromagnetic energy emission by radiation.

### 9.2 The Blackbody as the Ideal Radiator

To define a benchmark of the ability of a material to emit EM thermal energy, consider a thermodynamic argument. Suppose that a material can absorb EM energy that is incident upon it, converting the EM radiation into internal energy of the absorbing substance. Most real surfaces reflect some of the EM energy that is incident. For example, if radiation from a lecturer's laser pointer is directed onto a blackboard, the spot is easily seen, because the blackboard reflects a portion of the beam energy. If the blackboard surface were a perfect absorber, none of the laser energy would be reflected, and the laser spot would not be visible to the class. A material that absorbs 100 percent of the energy incident on it from all directions and at all wavelengths (i.e., has no reflection at all) is defined as a blackbody. Now, consider such a blackbody element suspended within an evacuated enclosure at uniform temperature T (Fig. 9.3). Let G be the total rate of radiant energy incident on the blackbody that originates by emission from the

Figure 9.3 Radiating evacuated enclosure containing a blackbody.

surface of the enclosure. By definition, the blackbody will absorb all of this energy. As with any mode of heat transfer, this absorbed thermal radiant energy will increase the internal energy (and thus temperature) of the blackbody element. However, this process can only continue until the blackbody reaches the temperature of the enclosure surface T; otherwise the system would violate the Second Law by having energy transfer from a colder to a hotter material. Because the blackbody is in an evacuated enclosure, it cannot have heat transfer by conduction or convection. Thermal radiation emission must therefore balance the absorbed radiation. The blackbody will come to equilibrium at temperature T when the absorbed radiation is balanced by the emitted radiation.

Suppose now that the absorbing element is not a perfect absorber, so that some incident energy is reflected and not absorbed. It follows that to maintain thermal equilibrium at temperature T, the element must emit less thermal energy than the perfectly absorbing blackbody element. Therefore, the perfectly absorbing blackbody must also be the best possible emitter of thermal radiation. It remains to quantify the amount of radiative energy emitted by the blackbody at a given temperature, as well as to prescribe how this energy is distributed among wavelengths.

### 9.2.1 The Planck Distribution and its Consequences

The historical development of the expression for the spectral distribution of blackbody thermal radiation is quite interesting and important, as the search for the correct relation led directly to the development of quantum theory, which must be invoked to explain how the derivation of the blackbody relation can be reconciled with experimental measurements. Max Planck derived the correct relation in 1901, and was forced to invoke quantum arguments to explain the form that agrees with experiment. Planck's fundamental relation for the rate of energy emission (into all directions) from an ideal blackbody is

${E_{\lambda b}}\left( {\lambda ,T} \right) = \frac{{2\pi {C_1}}}{{{\lambda ^5}\left( {{e^{{C_2}/\lambda T}} - 1} \right)}} \qquad \qquad( )$

(9.1)

where Eλb is the spectral emissive power of the blackbody, which is the rate of thermal radiation per unit area and per unit wavelength interval emitted by a blackbody at temperature T and at the wavelength λ. The Eλb has units in SI of

(W / m2 − μm)
.

The subscript b indicates that this expression applies to a blackbody, and the λ subscript indicates that it is wavelength dependent. The constants C1 and C2 are combinations of more fundamental physical constants (Planck's constant h, the Boltzmann constant k, and the speed of light in a vacuum, co), and have values of

${C_1} = hc_o^2 = 0.59552 \times {10^8}({\rm{W - \mu }}{{\rm{m}}^{\rm{4}}}{\rm{/}}{{\rm{m}}^{\rm{2}}})$
;
C2 = hco / k = 14,388(μm − K)
.

Observe that eq. (9.1) is the rate of energy leaving the surface; the blackbody generally will also be absorbing energy from other radiating sources in its surroundings, and these will have to be considered to find the net radiative heat transfer rate to/from the blackbody element. Blackbody spectral emissive power Eλb clearly depends on both temperature and wavelength in a complex way. A plot of Eλb at three temperatures is shown in Fig. 9.4. Some observations from this plot include that as temperature increases, the spectral emissive power at each wavelength increases so that the curves never cross; the wavelength for peak emission decreases with increasing temperature; and the total emission (area under the curves) increases rapidly with temperature. Each of these observations has important consequences in analyzing heat transfer by thermal radiation. A more general relation for Eλb can be formed by dividing eq. (9.1) by T5, resulting in

$\frac{{{E_{\lambda b}}}}{{{T^5}}} = \frac{{2\pi {C_1}}}{{{{\left( {\lambda T} \right)}^5}\left( {{e^{{C_2}/\lambda T}} - 1} \right)}}\qquad \qquad( )$

(9.2)

It is clear that the ratio Eλb / T5 is now a function of only the product λT rather than λ and T separately. Equation (9.2) can then be plotted as a single curve as in Fig. 9.5. From this plot, it is seen that the maximum in the blackbody function occurs at a particular value of the λT product, given by

${\left( {\lambda T} \right)_{\max }} = {C_3} = 2897.8(\mu m \cdot K)\qquad \qquad( )$

(9.3)

This result can be derived analytically by taking the derivative of eq. (9.2), and setting it equal to zero to find the location of the maximum of the function. Equation (9.3) is called Wien's Displacement Law, and provides a quick way to determine the wavelength where most blackbody thermal energy is emitted by a blackbody at temperature T.

Figure 9.4: Planck blackbody spectral emissive power from eq. (9.1)
Figure 9.5 Plot of generalized spectral blackbody emissive power.

For engineering calculation, the total energy emitted by a blackbody surface at all wavelengths is generally of interest, and can be found by integrating eq. (9.1) over all wavelengths. This can be done through a change of variables using ξ = C2 / λT and results in

${E_b} = \int_{\lambda = 0}^\infty {\frac{{2\pi {C_1}}}{{{\lambda ^5}\left( {{e^{{C_2}/\lambda T}} - 1} \right)}}d\lambda } = \frac{{2\pi {C_1}{T^4}}}{{C_2^4}}\int_{\xi = 0}^\infty {\frac{{{\xi ^3}}}{{{e^\xi } - 1}}} d\xi = \sigma {T^4}\qquad \qquad( )$
(9.4)

Where σ is the Stefan-Boltzmann constant, again made up of a combination of the more fundamental constants, $\sigma \equiv 2{C_1}{\pi ^5}/(15C_2^4)$. It has a numerical value in SI units of

$\sigma {\rm{ = 5}}{\rm{.6704}} \times {\rm{1}}{{\rm{0}}^{{\rm{ - 8}}}}{\rm{ W/}}{{\rm{m}}^{\rm{2}}} \cdot {{\rm{K}}^{\rm{4}}}{\rm{.}}$

The Stefan-Boltzmann Law, eq. (9.4), is one of the fundamental relations of thermal radiation heat transfer, showing that total energy emission from a blackbody depends on the fourth power of the absolute temperature of a material. (The word total in thermal radiation is used to indicate integration over all wavelengths.) Again, energy absorbed by the blackbody surface that originates from other radiating materials must be accounted for before radiation heat transfer can be calculated. The fourth power dependence of total blackbody emissive power indicates that radiation will rapidly become important or dominating over other heat transfer modes at higher absolute temperatures. This is certainly the case in power plant boilers, where radiation can contribute up to 90 percent of the energy transfer from combustion gases to the steam tubes.

Figure 9.6 Relation of angle and radius for (a): Planar angle and (b) Solid angle

Figure 9.7 Projected area and intensity

edge-on view is approached and the apparent element size approaches zero. This apparent area dAp = dA cosθ is called the projected area of the element. The energy reaching the observer from the element dA thus varies as cos θ, and this effect is known as Lambert's cosine law. It is valid for a blackbody, but may not be followed for real surfaces. We can now define a quantity that will quantify the radiation emitted from a blackbody in a particular direction. This quantity is called the intensity; it is also wavelength and temperature dependent, and for a blackbody is given the symbol Iλb. The spectral blackbody intensity is defined as the rate of energy deλ emitted per unit solid angle subtended by dAs, per unit projected area dA cosθ, per unit wavelength interval, or

${I_{\lambda b}} = \frac{{d{e_\lambda }}}{{dA\cos \theta d\omega d\lambda }} \qquad \qquad( )$
(9.5)

Suppose the elements dA and dAs in Fig. 9.7 are both blackbodies at the same temperature T. The energy leaving dA in the direction of dAs is, from eq. (9.5)

$d{e_{\lambda ,dA - d{A_s}}} = {I_{\lambda b}}(\theta )dA\cos \theta d\Omega d\lambda = {I_{\lambda b}}\left( \theta \right)dA\cos \theta \left( {\frac{{d{A_s}}}{{{R^2}}}} \right)d\lambda \qquad \qquad( )$

(9.6)

Similarly, the energy from dAs that is incident on dA is

$d{e_{\lambda ,d{A_s} - d{A_{}}}} = {I_{\lambda b}}({\theta _s} = 0)d{A_s}d\Omega d\lambda = {I_{\lambda b}}({\theta _s} = 0)d{A_s}\left( {\frac{{dA \cdot \cos \theta }}{{{R^2}}}} \right)d\lambda \qquad \qquad( )$

(9.7)

The projected area of dA has been used in evaluating the subtended solid angle of dA when viewed from dAs. Also, dAs is normal to R so that cosθs = 1. To avoid violating the second law, the energy exchange between the two elements at the same temperature must be the same, and equating (9.6) and (9.7) results in

${I_{\lambda b}}(\theta ) = {I_{\lambda b}}({\theta _s} = 0) = {I_{\lambda b}} \qquad \qquad( )$

(9.8)

so that the blackbody intensity is independent of angle of emission. The intensity is what the eye interprets as the brightness of an emitter.

Examination of Fig. 9.8 allows an alternate definition of the solid angle dΩ subtended by the area dAs on the surface of the hemisphere in terms of the angles θ and φ as

$d\Omega = \frac{{d{A_s}}}{{{R^2}}} = \frac{{d{S_1} \times d{S_2}}}{{{R^2}}} = \frac{{\left( {R\sin \theta d\phi } \right) \times \left( {Rd\theta } \right)}}{{{R^2}}} = \sin \theta d\theta d\phi \qquad \qquad( )$

(9.9)

where dS1 and dS2 are the arc lengths of the sides of dAs. This allows finding the relation between Iλb and Eλb by integrating the energy deλ that leaves through a particular solid angle over all solid angles in the hemisphere (Figure 9.9). Using eq. (9.6) gives

$\begin{array}{l} {E_{\lambda b}} = \int_{{A_s}}^{} {d{e_{\lambda ,dA - d{A_s}}}} = \int_{\Omega = 2\pi }^{} {{I_{\lambda b}}\cos \theta d\Omega } \\ = {I_{\lambda b}}\int_{\phi = 0}^{2\pi } {\int_{\theta = 0}^{\pi /2} {\cos \theta \sin \theta d\theta d\phi } } = 2\pi {I_{\lambda b}}\int_{\theta = 0}^{\pi /2} {\cos \theta \sin \theta d\theta } = \pi {I_{\lambda b}} \\ \end{array} \qquad \qquad( )$
(9.10)

resulting in a remarkably simple relation between the blackbody spectral emissive power Eλb and the blackbody spectral intensity Iλb of Eλb = πIλb.

Figure 9.8 Solid angle in terms of angles dθ and d
Figure 9.9: Hemisphere for integration of intensity over $0 \le \theta \le \pi /2$ and $0 \le \phi \le 2\pi$

### 9.2.2 The Blackbody Fraction

It is often necessary or useful to find the fraction of blackbody spectral emissive power that lies within a particular range of wavelength. This is most conveniently done through use of eq. (9.2). Integration of that equation over the range $0 \le \left( {\lambda T} \right)* \le \lambda T$ results in the fraction of blackbody emission in that range as

$\begin{array}{l} {F_{0 - \lambda T}} = \int_{\left( {\lambda T} \right)* = 0}^{\lambda T} {\frac{{{E_{\lambda b}}}}{{{T^5}}}} d(\lambda T)*/\int_{\left( {\lambda T} \right)* = 0}^\infty {\frac{{{E_{\lambda b}}}}{{{T^5}}}} d(\lambda T)* \\ = \frac{1}{\sigma }\int_{\left( {\lambda T} \right)* = 0}^{\lambda T} {\frac{{2\pi {C_1}}}{{\left( {\lambda T} \right){*^5}\left( {{e^{{C_2}/\left( {\lambda T} \right)*}} - 1} \right)}}d\left( {\lambda T} \right)*} \\ \end{array} \qquad \qquad( )$

(9.11)

This can be integrated by parts using the same change of variables as for eq. (9.4) Chang and Rhee (1984) to give the analytical relation

${F_{0 - \lambda T}} = \frac{{15}}{{{\pi ^4}}}\sum\limits_{n = 1}^\infty {\left[ {\frac{{{e^{ - n\xi }}}}{n}\left( {{\xi ^3} + \frac{{3{\xi ^2}}}{n} + \frac{{6\xi }}{{{n^2}}} + \frac{6}{{{n^3}}}} \right)} \right]} \qquad \qquad( )$

(9.12)

where$\xi \equiv {C_2}/\lambda T$. Equation (9.12) converges rapidly, generally requiring only a few terms for accurate evaluation. Keeping only five terms in the summation gives results accurate to five significant figures up to F0-λT of at least 0.99812 (at λT = 100,000), and four terms is accurate to five significant figures up to λT = 6,000, where F0-λT = 0.73778. Figure 9.10 is a plot of eq. (9.11) or (9.12). Given eq. (9.11), the blackbody emissive power in any interval (λT)1 to (λT)2 can be found from

$\begin{array}{l} {F_{{{\left( {\lambda T} \right)}_2} - {{\left( {\lambda T} \right)}_1}}} = \frac{1}{\sigma }\int_{\left( {\lambda T} \right)* = {{\left( {\lambda T} \right)}_1}}^{{{\left( {\lambda T} \right)}_2}} {\frac{{2\pi {C_1}}}{{\left( {\lambda T} \right){*^5}\left( {{e^{{C_2}/\left( {\lambda T} \right)*}} - 1} \right)}}d\left( {\lambda T} \right)*} \\ = \frac{1}{\sigma }\left[ {\int_{(\lambda T)* = 0}^{{{(\lambda T)}_2}} {\frac{{2\pi {C_1}}}{{(\lambda T){*^5}[{e^{{C_2}/(\lambda T)*}} - 1]}}d(\lambda T)*} - \int_{(\lambda T)* = 0}^{{{(\lambda T)}_1}} {\frac{{2\pi {C_1}}}{{(\lambda T){*^5}[{e^{{C_2}/(\lambda T)*}} - 1]}}d(\lambda T)*} } \right] \\ = {F_{0 - {{(\lambda T)}_2}}} - {F_{0 - {{(\lambda T)}_1}}} \\ \end{array}$

(9-13)

Figure 9.10 Plot of the blackbody fraction vs. λT product.
Figure 9.11 Blackbody energy in the ranges 0 <λT < (λT) 1 and 0 <λT < (λT) 2

Interpreting eq. (9.13) graphically, Fig. 9.11 shows the difference in the blackbody normalized emissive power (the cross-hatched area) in the range

T)1$\le \lambda T \le$

(λT) 2. If this is divided by the area under the entire curve (which has a value of σ), then the areas are the F0-λT values on the right-hand-side of eq. (9.13).

Example 9.1 Find the fraction of emission from blackbodies at T = 1500 K and at T = 5780 K (the apparent temperature of the Sun) that lie in the range $1 \le \lambda \le 5$μm. Solution: Using eq. (9-12), for T = 1500K, at (λT)1 = 1500 μm-K, F0-1500 = 0.01285, and at (λT) 2 = 7500 μm-K, F0-7500 = 0.83436, so F1500-7500 = F0-7500 - F0-1500 = 0.83436- 0.01285 = 0.82151. Thus, 82.2 percent of the blackbody emission lies in the prescribed range for a 1500K blackbody. For T = 5780 K, F0-5780 = 0.71828 and F0-28900 = 0.99390, so F5780-28900 = 0.99390 - 0.71828 = 0.27562, or 27.6 percent of the solar energy lies in the range $1 \le \lambda \le 5$ μm. This smaller fraction than for the 1500K blackbody reflects the fact that the peak of the blackbody curve at the solar effective temperature is in the shorter wavelength range and thus so is a large fraction of the solar energy.

## 9.3 Properties of Real Surfaces: Definitions, Measurements and Prediction

The characteristics of the blackbody are independent of material properties, because the blackbody is an idealization that can only be approached in reality, and its emission properties depend only on the independent parameters T and λ or their product. Real materials will absorb (and therefore emit) less thermal radiation than predicted for a blackbody. To account for the differences between the ideal blackbody and the performance of real materials, the radiative properties of opaque surfaces are now introduced. These properties show how the real material behaves in its ability to emit, absorb, and reflect radiation relative to a blackbody.

### 9.3.1 Opaque Surface Property Definitions

The radiative properties are usually expressed as surface properties, although on a microscopic basis radiation incident on a surface may actually penetrate for some distance into the bulk material. We found that emitted or absorbed intensity is independent of angle for the blackbody; however, the surface properties of real materials are angularly dependent; i.e., the intensity emitted by a real surface may vary with respect to the angle relative to the surface normal, as does the ability of the surface to absorb radiation. However, there is sparse data for the angular dependence of surface radiative properties, so that the surface properties are usually presented for either properties normal to the surface, or averaged over all directions. The properties may also be wavelength dependent; this dependence leads to important effects that can be exploited in some applications such as the design of solar collectors and the control of surface temperatures. Although directional effects can be important for certain types of surfaces (e.g., grooved, patterned, mirrored), most surfaces used in practical engineering applications such as furnaces, ovens, boilers, etc. have weak directional dependence. Because of the dearth of available data on directional properties, emphasis here will be on spectral (wavelength) dependence of the properties. For experimental determination of radiative properties, it will often be necessary to compare the radiative characteristics of the real surface with those of a blackbody at the same temperature and possibly wavelength. Remembering the relation between the ability of a blackbody to absorb all incident energy, which implies that the blackbody will emit the greatest possible energy, it can be seen that building a device that absorbs all incident energy means that it will also emit as a blackbody. As mentioned before, no real surface has this characteristic. However, it is possible to closely approach a perfect absorbing surface through the use of a heated cavity. Consider a deep hole in a surface which has the surface of the cavity coated with a highly absorbing coating. The radiation absorbing ability of such a device is similar to shining a flashlight into

Figure 9.12 Experimental blackbody cavity for reference.

a narrow cave; very little of the flashlight energy will reflect from the cave. The imagined surface of the mouth of the cave then has radiative characteristics very close to those of a black surface. A laboratory blackbody standard is constructed similarly (Fig. 9.12), with a deep cavity with its internal surface coated with a highly absorbing coating. The cavity is drilled into a good thermally conducting material to minimize temperature gradients, and the cavity itself is insulated on its exterior. If the cavity is heated to uniform temperature, then the radiation leaving the cavity will closely approach the Planck blackbody characteristics. Emissivity: The ability of a real surface to emit radiation at a particular wavelength is expressed by its spectral emissivity, which is the energy emitted by the real surface in a narrow wavelength interval dλ around the wavelength λ divided by the radiation that would be emitted by a blackbody at the same wavelength and temperature. It is given the symbol ε . Emissivity is thus a dimensionless quantity, $0 \le \varepsilon \le 1$. At the direction normal to the surface, the emissivity is

${\varepsilon _\lambda }(\theta = 0,\lambda ,T) \equiv {\varepsilon _{\lambda ,n}}(\lambda ,T) = \frac{{{I_\lambda }\left( {\theta = 0,\lambda ,T} \right)}}{{{I_{\lambda b}}(\lambda ,T)}} = \frac{{{I_{\lambda ,n}}\left( {\lambda ,T} \right)}}{{{I_{\lambda b}}(\lambda ,T)}} \qquad \qquad( )$

(9.14)

The subscript n indicates a quantity evaluated in a direction normal to the surface (θ = 0). The normal spectral emissivity must be measured for any given material. This can be done by heating the material to temperature T, and then using a radiation detector to measure the emitted radiation normal to the surface. Filters or a diffraction grating can be used to provide a signal at the desired wavelength. This measurement gives the numerator of eq. (9.14). The radiating surface is then replaced by the reference blackbody cavity at the same temperature, and the detector will read the value to be placed in the denominator of eq. (9.14); alternatively, the value for denominator may be computed directly from the blackbody relations.

The hemispherical spectral emissivity, λ, is a measure of the energy emitted into all directions by the real surface in a narrow wavelength interval dλ around the wavelength λ relative to that from a blackbody, and is defined by

${\varepsilon _\lambda }(\lambda ,T) = \frac{{{E_\lambda }\left( {\lambda ,T} \right)}}{{{E_{\lambda b}}\left( {\lambda ,T} \right)}} \qquad \qquad( )$

(9.15)

Again, the numerator is found experimentally, and the denominator can be measured experimentally or found from eq. (9.1). To gain some understanding of the meaning of eq. (9.15), consider Fig. 9.13. The top curve is the blackbody emissive power from eq. (9.1) for a blackbody at 1000 K. The ratio of the values of the Eλ curve to the Eλb curve at the same wavelength is the emissivity of the surface. Taking this ratio at each λ and plotting gives the curve of spectral emissivity. The emissivity calculated for the curves in Fig. 9.13 is shown in Fig. 9.14.

Figure 9.13 Curves of emissive power for a blackbody, Eλb, and a real surface, Eλ, both at 1000 K
Figure 9.14 Spectral hemispherical emissivity calculated from curves in Fig. 9.13 using eq. (9.15)

For calculation of heat transfer, it is useful to be able to compute the total (all wavelengths) radiation that is emitted by a surface. The total hemispherical emissivity is given by

$\varepsilon (T) = \frac{E}{{{E_b}}} = \frac{E}{{\sigma {T^4}}} \qquad \qquad( )$

(9.16)

Figure 9.15 Hemispherical total emissivity as the ratio of total emissive power for the real surface (area under the smaller curve) to blackbody emissive power at the same temperature (area under larger curve).

The numerator of eq. (9.16) can be either measured directly by a detector that is sensitive to total radiation, or, if spectral data such as in Fig. 9.13 is available, then that data can be numerically integrated using the definition of eq. (9.15) to give

$\varepsilon (T) = \frac{{E(T)}}{{\sigma {T^4}}} = \frac{{\int_{\lambda = 0}^\infty {{E_\lambda }(\lambda ,T)d\lambda } }}{{\sigma {T^4}}} = \frac{{\int_{\lambda = 0}^\infty {{\varepsilon _\lambda }(\lambda ,T){E_{\lambda b}}(\lambda ,T)d\lambda } }}{{\sigma {T^4}}} \qquad \qquad( )$

(9.17)

The meaning of this relation is illustrated in Fig. 9.15. The ratio of the smaller shaded area to the total larger area (part of which lies beneath the smaller area) is the right-hand term of eq. (9.17), and the ratio of these areas is the total emissivity ε (T). The total normal emissivity follows a parallel derivation and interpretation. For total normal emissivity, the result is

${\varepsilon _n}(T) = \frac{{{I_n}(T)}}{{{I_b}}} = \frac{{\int_{\lambda = 0}^\infty {{I_{\lambda ,n}}(\lambda ,T)d\lambda } }}{{(\sigma {T^4}/\pi )}} = \frac{{\pi \int_{\lambda = 0}^\infty {{\varepsilon _{\lambda ,n}}(\lambda ,T){I_{\lambda b}}(\lambda ,T)d\lambda } }}{{\sigma {T^4}}}\qquad \qquad( )$

(9.18)

Equation (9.14) is substituted to obtain the final term on the right.

Absorptivity: The ability of an opaque material to absorb incident radiation is described by the property of absorptivity, given the symbol α . This property is the amount of incident energy on the surface that is absorbed (converted into internal energy) relative to that which would be absorbed by a blackbody. Because, by definition, all incident energy is absorbed by a blackbody, it is convenient to use the equivalent definition of absorptivity as the incident energy absorbed divided by the incident energy. Absorptivity is thus in the numerical range of $0 \le \alpha \le 1$. As for emissivity, the absorptivity is often measured for incident radiation normal to the surface (useful, for example, for evaluating solar collector materials) and for radiation incident over all directions (common in analyzing materials inside furnaces and ovens).

If radiation is incident normal to the surface, then the rate of absorbed radiation in wavelength interval dλ is ${d^2}{q_{\lambda ,a}}\left( {\theta = 0,\lambda ,T} \right)d\lambda$ and the incident energy can be expressed in terms of the incident radiation intensity as

${d^2}{q_{\lambda ,i}}\left( {\theta = 0,\lambda ,T} \right)d\lambda = {I_{\lambda ,i}}(\theta = 0,\lambda )\cos \theta dAd\omega d\lambda$
.

Taking the ratio of absorbed to incident radiation for the normal direction gives the normal spectral absorptivity as

${\alpha _\lambda }(\theta = 0,\lambda ,T) \equiv {\alpha _{\lambda ,n}}(\lambda ,T) = \frac{{{d^2}{q_{\lambda ,a}}\left( {\theta = 0,\lambda ,T} \right)}}{{{I_{\lambda ,i}}(\lambda ,\theta = 0)dAd\omega }} \qquad \qquad( )$

(9.19)

Observe that the incident intensity Iλ,i (λ) depends on the temperature and spectral characteristics of the source of the incident radiation. In general, it will not have a blackbody spectral distribution. The T dependence in the equation indicates that the absorbed energy rate by the absorbing surface may depend on the absorbing surface temperature, so the normal spectral absorptivity may depend on the T of the absorbing surface.

Consider the case when radiation is incident upon the absorbing surface from many directions. The absorbed energy is $d{q_{\lambda ,a}}\left( {\lambda ,T} \right)d\lambda$. For convenience, give the symbol Gλ to the spectral energy incident per unit area per wavelength interval dλ from all directions; G is called the irradiation. The hemispherical spectral absorptivity is then

${\alpha _\lambda }(\lambda ,T) \equiv \frac{{d{q_{\lambda ,a}}\left( {\lambda ,T} \right)}}{{{G_\lambda }dA}} \qquad \qquad( )$
(9.20)

For radiative heat transfer calculations, total (integrated over all wavelengths) absorptivities are most useful. The normal total absorptivity α n is then

${\alpha _n}(T) \equiv \frac{{d{q_a}\left( {\theta = 0,T} \right)}}{{{I_i}\left( {\theta = 0} \right)dAd\omega }} = \frac{{\int_{\lambda = 0}^\infty {{\alpha _{\lambda ,n}}{I_{\lambda ,i}}\left( {\theta = 0} \right)dAd\omega d\lambda } }}{{{I_i}\left( {\theta = 0} \right)dAd\omega }} = \frac{{\int_{\lambda = 0}^\infty {{\alpha _{\lambda ,n}}{I_{\lambda ,n}}d\lambda } }}{{{I_n}}} \qquad \qquad( )$

(9.21)

and the hemispherical total absorptivity is

$\alpha (T) \equiv \frac{{\int_{\lambda = 0}^\infty {d{q_{\lambda ,a}}\left( {\lambda ,T} \right)} d\lambda }}{{dA\int_{\lambda = 0}^\infty {{G_\lambda }d\lambda } }} = \frac{{\int_{\lambda = 0}^\infty {{a_\lambda }} {G_\lambda }d\lambda }}{{\int_{\lambda = 0}^\infty {{G_\lambda }d\lambda } }} = \frac{{\int_{\lambda = 0}^\infty {{a_\lambda }} {G_\lambda }d\lambda }}{G} \qquad \qquad( )$
(9.22)
Figure 9.16 Equal temperature normal surfaces exchanging radiation

The final two forms in eq. (9.22) are found by substituting eq. (9.20) to eliminate dqλ,a.

Kirchhoff's Law: Because the blackbody is at once the perfect absorber and best possible emitter of radiative energy, it would seem that there might also be a relationship between the properties of emissivity and absorptivity. This is indeed the case, but care must be taken in applying this relationship. Consider two surfaces at the same temperature T. Surface 2 is a blackbody, and is placed normal to surface 1, which has normal spectral emissivity ε λn and spectral absorptivity α λn at wavelength λ (Fig. 9.16). The radiant energy absorbed in the wavelength range dλ by element 1 from element 2 placed normal to surface 1 is then

${\rm{d}}{e_{\lambda ,a}}{\rm{ = }}{\alpha _{\lambda ,n}}{{\rm{I}}_{\lambda ,i}}{\rm{(}}\theta {\rm{ = 0)d}}{{\rm{A}}_2}{\rm{d}}{\Omega _1}{\rm{d}}\lambda {\rm{ = }}{\alpha _{\lambda ,n}}{{\rm{I}}_{\lambda ,i}}{\rm{(}}\theta {\rm{ = 0)}}d{A_2}\frac{{{\rm{d}}{{\rm{A}}_1}}}{{{S^2}}}{\rm{d}}\lambda \qquad \qquad( )$

(9.23)

The energy emitted by surface 1 that is incident on surface 2 is

${\rm{d}}{{\rm{e}}_{\lambda ,e}}{\rm{ = }}{\varepsilon _{\lambda ,n}}{{\rm{I}}_{\lambda b}}{\rm{(}}\lambda {\rm{,T)d}}\Omega {\rm{d}}{{\rm{A}}_1}{\rm{d}}\lambda {\rm{ = }}{\varepsilon _{\lambda ,n}}{{\rm{I}}_{\lambda b}}{\rm{(}}\lambda {\rm{,T)}}\frac{{d{A_2}}}{{{S^2}}}{\rm{d}}{{\rm{A}}_1}{\rm{d}}\lambda \qquad \qquad( )$

(9.24)

Using the Second Law argument that no energy can be transferred between surfaces at the same temperature, these two equations can be set equal, resulting in the relation

${\varepsilon _{\lambda ,n}} = {\alpha _{\lambda ,n}} \qquad \qquad( )$
(9.25)

This is a very general relation. There are some restrictions, such as an inherent assumption that the surfaces are in thermodynamic equilibrium (in both the macroscopic sense and the microscopic sense of having an equilibrium distribution of energy states described by the single parameter T), but this form of Kirchhoff's Law for spectral properties in a particular direction is taken as correct in engineering situations. We have not treated directional emissivities and absorptivities, but an argument similar to that used in deriving eq. (9.25) shows that it is correct that the spectral directional emissivity of a surface is equal to the spectral directional absorptivity at the same wavelength and for the same direction. Using eq. (9.25) to replace the spectral normal absorptivity in eq. (9.21) gives the total normal absorptivity as

${\alpha _n}(T) = \frac{{\int_{\lambda = 0}^\infty {{\varepsilon _{\lambda ,n}}(T){I_{\lambda ,i}}\left( {\theta = 0} \right)d\lambda } }}{{{I_i}\left( {\theta = 0} \right)}} \qquad \qquad( )$

(9.26)

Comparing eq. (9.26) with eq. (9.18) shows that the total normal emissivity is equal to the total normal absorptivity only in a special case. The normal incoming intensity must come from a blackbody at the same temperature as the absorbing surface. Thus, the simple statement that emissivity equals absorptivity is not valid except for the fundamental spectral-directional properties.

Similarly, eq. (9.22) can be modified using directionally-independent spectral properties to give the total hemispherical absorptivity as,

$\alpha (T) = \frac{{\int_{\lambda = 0}^\infty {{\alpha _\lambda }} {G_\lambda }d\lambda }}{G} = \frac{{\int_{\lambda = 0}^\infty {{\varepsilon _\lambda }} {G_\lambda }d\lambda }}{G} \qquad \qquad( )$

(9.27)

and comparing with eq. (9.17) shows that the total hemispherical properties α (T) and ε (T) are equal only if the irradiation Gλ has the same spectral distribution as a blackbody at the same temperature T as the absorbing surface. In addition, if the surface has significant directional property variations, then the directional distribution incident on the surface must be uniform from all directions; the effect of directional characteristics has been ignored here except for the normal case.

The use of Kirchhoff's Law with the appropriate restrictions allows finding some of the radiative properties for opaque surfaces through measurement of the corollary property.

Reflectivity: The final property of interest for opaque surfaces is the fraction of incident radiation that is reflected from the surface, called the reflectivity and given the symbol ρ . This property in its most fundamental form depends on surface temperature, wavelength, and the direction of incidence of the radiation and the direction of reflection. The most useful of these for engineering calculations are defined here; complete derivations for each of the more detailed reflectivities are defined in detail elsewhere (Siegel and Howell, 2002).

The fraction of the radiation incident from all directions in the wavelength interval dλ that is reflected into all directions in the same wavelength interval is called the hemispherical-hemispherical spectral reflectivity, ρ λ (λ,T), and is defined by

${\rho _\lambda }\left( {\lambda ,T} \right) = \frac{{d{q_{\lambda ,r}}}}{{{G_\lambda }dA}} \qquad \qquad( )$

(9.28)

The total (hemispherical-hemispherical) reflectivity is the fraction of the total incident energy from all directions that is reflected into all directions, and is

$\rho \left( T \right) = \frac{{d{Q_r}}}{{GdA}} = \frac{{\int_{\lambda = 0}^\infty {d{q_{\lambda r}}d\lambda } }}{{dA\int_{\lambda = 0}^\infty {{G_\lambda }} d\lambda }} = \frac{{\int_{\lambda = 0}^\infty {{\rho _\lambda }(\lambda ,T){G_\lambda }d\lambda } }}{G} \qquad \qquad( )$

(9.29)

Because radiation is in the form of a propagating wave, the reflectivity of a surface depends on the orientation of the wave relative to the surface. The wave can be resolved into components that vibrate perpendicular to the surface and parallel to the surface, and each of these components has a different reflectivity. The different reflectivities are discussed in Section 9.3.2. For many engineering surfaces (furnace refractory surfaces, soot-covered and oxidized surfaces) the emitted and reflected radiation is unpolarized, and the differing reflectivities are simply averaged to give a single value. However, for highly polished surfaces, polarization effects are very important, as they are at the nanoscale.

Further property relations: Consider one unit of spectral radiation incident per unit area on a surface; i.e., Gλ = 1. Because for an opaque surface this radiation must either be reflected or absorbed, it follows that the fractions absorbed and reflected must sum to unity, or Fraction absorbed atλ + fraction reflected atλ = 1

${\alpha _\lambda } + {\rho _\lambda } = 1\qquad \qquad( )$

(9.30)

where the properties are spectral hemispherical values. If the surface has minimal directional variation in properties, then Kirchhoff's Law can be invoked to give

${\varepsilon _\lambda } + {\rho _\lambda } = 1 \qquad \qquad( )$

(9.31)

Equations (9.30) and (9.31) show that measuring any one of the three spectral hemispherical properties allows evaluation of the others. If one unit of total energy is incident on the surface (G = 1), then a similar analysis gives Total fraction absorbed + total fraction reflected = 1

$\alpha + \rho = \varepsilon + \rho = 1\qquad \qquad( )$
(9.32)

where the stringent restrictions on applications of Kirchhoff's Law for total properties must be observed for the substitution of ε to be valid. The restrictions can be imposed when evaluation is done experimentally, so again measurement of any one of the three total hemispherical properties is sufficient to evaluate all three.

Idealizations for properties: To simplify radiative heat transfer analysis, surface properties are often idealized. These idealizations make analysis of radiative transfer among multiple surfaces tractable, but can introduce serious errors in some cases. They are commonly made because of the major increase in computational effort that is necessary to treat the complete property variations in wavelength and direction for real surfaces.

Figure 9.17 Comparison of (a) measured directional reflectivity for white Krylon paint for radiation incident at 60o (Zaworski et al., 1996; reproduced with permission from Elsevier) (b) idealized reflectivity of a diffuse surface.

The first simplification is to assume that all of the properties for opaque surfaces are independent of direction. Such a surface is said to be ‘‘diffuse’‘; for a diffuse surface, for example, the normal and hemispherical properties are equal, so that α λn = α λ , and ε λn = ε λ . For a diffuse surface, emitted intensity will be uniform into each direction; reflected energy from any direction is also reflected with equal intensity into each direction. As we shall find in Chapter 10, the characteristics of a diffuse surface allow great simplification in analysis. Figure 9.17 shows the reflected intensity from one real surface (Fig. 9.17a) as compared with that for the idealized diffuse surface (Fig. 9.17b). Note the increased reflection for the real surface in the near-mirrorlike direction. This forward reflection component is typical of many surfaces, and some models of reflectivity combine a single mirror-like spike with a diffuse portion to more nearly model the behavior of real surfaces. The second property idealization is to assume that the properties do not vary with wavelength, so that α λ  α and ε λ  ε . Such a surface is called a gray surface. For a gray surface, integrations of emitted, absorbed or reflected radiation over wavelength are not required, again simplifying radiative heat transfer analysis at the expense of accuracy. Figure 9.18 shows the measured spectral emissivity for platinum at a temperature of 1400 K, with an averaged gray emissivity for this temperature of 0.166 computed using eq. (9.17). At 1400 K, Wien's Law [eq. (9.3)] predicts that the peak blackbody emission will be at

${\lambda _{\max }} = \frac{{{C_3}}}{T} = \frac{{2897.8(um \cdot K)}}{{1400K}} = 2.07\mu m$

so the averaged emissivity is heavily weighted by the values in that region of the spectrum.

Figure 9.18: Comparison of measured normal spectral emissivity for platinum at 1400 K compared with properties using the gray assumption. Experimental values from Harrison et al. (1963).

A diffuse-gray surface automatically obeys Kirchhoff’s Law, so that equations for radiative heat transfer among such surfaces can be written in terms of a single radiative property for each measured total properties for a number of opaque materials are given in Appendix F.

### 9.3.2 EM Theory Predictions of Properties

Although the radiative properties for many materials have been measured, there are still many others for which no data are available. For some classes of opaque materials, it is possible to predict the spectral, directional, total and hemispherical properties through the use of electromagnetic theory. These predictions have limitations in accuracy and applicability, but serve in the absence of other data. Maxwell's Equations can be used to predict the amplitude of an electromagnetic wave that is reflected when it interacts with a surface. Because the energy carried by the wave is proportional to the square of the wave amplitude, this allows prediction of the reflectivity of a surface. Invoking Kirchhoff's Law then allows prediction of the other properties as well. Electromagnetic theory predictions depend on certain approximations. There are terms in the mathematically exact solutions that decay rapidly with distance (a few wavelengths), and these are often neglected to simplify the solutions. Neglect of these "near-field" terms to obtain simplified solutions result in "far-field" solutions, and are used in the results and predictions presented in this section. When nanoscale interactions between EM waves and materials are considered (Section 9.5), the additional terms must be retained, and these "near-field" solutions are more complex. Accurate far-field solutions require that the features of the reflecting material be much smaller than the wavelength of the incident EM wave. Such surfaces are called optically smooth, and have the directional characteristics of a mirror. These surfaces in radiative transfer are often called specular surfaces, not to be confused with spectral (wavelength-dependent) surfaces. The radiative properties of optically smooth surfaces as derived from EM theory are presented here. Again, emphasis is on the wavelength dependence of the materials rather than their directional dependence, although some directional characteristics that lead to useful applications are discussed. Derivation of the results and more complete treatments including directional effects are in Siegel and Howell (2002), Modest (2003), and others.

Properties of Dielectric Materials. Dielectric materials as treated by EM theory are considered to be perfect insulators, so that no coupling occurs between the EM field and electrons in the material. In this case, EM theory predicts that the EM wave is not attenuated in the material, and that the reflective properties can be found in terms of other measurable properties of the material, including the simple index of refraction, n, the dielectric constant, or the electrical permittivity. An EM wave propagating through dielectric material 1 and incident on an interface with dielectric material 2 is shown in Figure 9.19. In most engineering applications, radiation is incident through air or a vacuum; for either of these cases, the refractive index n1 is taken as unity. The EM wave will reflect at an angle of reflection equal to the angle of incidence, θ. A portion of the wave from material 1 will pass through the

Figure 9.19 Interaction of EM wave with a dielectric surface.

interface and be refracted at angle  in material 2. The EM theory predicts that the angle of refraction is given by the familiar Snell's Law:

$\sin \chi = \frac{{{n_1}}}{{{n_2}}}\sin \theta \qquad \qquad( )$

(9.33)

The refractive indices can be directly measured or can be computed from other properties of a dielectric. In terms of the dielectric constant K (also called the electric permittivity ) the relations are

$\frac{{{n_1}}}{{{n_2}}} = \frac{{\sqrt {{K_1}} }}{{\sqrt {{K_2}} }} = \frac{{\sqrt {{\gamma _1}} }}{{\sqrt {{\gamma _2}} }} \qquad \qquad( )$

(9.34)

The reflectivity for unpolarized incident radiation is

${\rho _\lambda }\left( {\lambda ,\theta } \right) = \frac{1}{2}\frac{{{{\sin }^2}\left( {\theta - \chi } \right)}}{{{{\sin }^2}\left( {\theta + \chi } \right)}}\left[ {1 + \frac{{{{\cos }^2}\left( {\theta + \chi } \right)}}{{{{\cos }^2}\left( {\theta - \chi } \right)}}} \right] \qquad \qquad( )$

(9.35)

If the radiation is polarized, the reflectivity for the components of the electric field E (Fig. 9.1) that are parallel and perpendicular to the surface are, after substituting eq. (9.33) to eliminate ,

${\rho _\lambda }_{,\parallel }\left( {\lambda ,\theta } \right) = {\left\{ {\frac{{{{\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}^2}\cos \theta - {{\left[ {{{\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}^2} - {{\sin }^2}\theta } \right]}^{1/2}}}}{{{{\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}^2}\cos \theta + {{\left[ {{{\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}^2} - {{\sin }^2}\theta } \right]}^{1/2}}}}} \right\}^2}\qquad \qquad( )$

(9.36a)

${\rho _{\lambda , \bot }}\left( {\lambda ,\theta } \right) = {\left\{ {\frac{{{{\left[ {{{\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}^2} - {{\sin }^2}\theta } \right]}^{1/2}} - \cos \theta }}{{{{\left[ {{{\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}^2} - {{\sin }^2}\theta } \right]}^{1/2}} + \cos \theta }}} \right\}^2}\qquad \qquad( )$
(9.36b)

For unpolarized incident radiation, the reflectivity is the average of the reflectivities for the two components, or

$\begin{array}{l} {\rho _\lambda }\left( {\lambda ,\theta } \right) = \frac{1}{2}\left[ {{\rho _\lambda },\parallel \left( {\lambda ,\theta } \right) + {\rho _{\lambda , \bot }}\left( {\lambda ,\theta } \right)} \right] \\ = \frac{1}{2}\left( {{{\left\{ {\frac{{{{\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}^2}\cos \theta - {{\left[ {{{\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}^2} - {{\sin }^2}\theta } \right]}^{1/2}}}}{{{{\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}^2}\cos \theta + {{\left[ {{{\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}^2} - {{\sin }^2}\theta } \right]}^{1/2}}}}} \right\}}^2} + {{\left\{ {\frac{{{{\left[ {{{\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}^2} - {{\sin }^2}\theta } \right]}^{1/2}} - \cos \theta }}{{{{\left[ {{{\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}^2} - {{\sin }^2}\theta } \right]}^{1/2}} + \cos \theta }}} \right\}}^2}} \right) \\ \end{array} \qquad \qquad( )$
(9.37)

In eq. (9.36a), the reflectivity for the parallel component becomes equal to zero at the particular incident (and therefore reflected) angle of

$\theta = {\tan ^{ - 1}}\left( {{n_2}/{n_1}} \right)$
.

This angle is called Brewster's angle. Reflected energy at this angle from a mirror-like surface is thus all perpendicularly polarized, and this fact is the basis for polarized glasses. Lenses in polarized glasses filter out the perpendicular component of reflected radiation, eliminating glare from highly reflecting surfaces. For the special case of θ = 0 (normal incidence), eq. (9.37) reduces to

${\rho _{\lambda ,n}}(\lambda ) = {\left( {\frac{{{n_2} - {n_1}}}{{{n_2} + {n_1}}}} \right)^2}\qquad \qquad( )$

(9.38)

The spectral dependence for reflectivity enters because of the wavelength dependence of the refractive index n.

Table 9.1 Optical Property Values and Normal Spectral Reflectivity of Various Dielectric Materials at T = 300 K and λ = 0.589 μm; Data from Lide (2008).

PROPERTY	n	K	 ρ λ,n
Refractive

index Dielectric constant eq. (9.38) MATERIAL

SiO2 (glass)	1.458	4.42	0.035

SiO2 (fused quartz) 1.544 3.75 0.046

NaCl	1.5441	5.9	0.046
KCl	1.4902	4.86	0.039
H2O (liquid)	1.332	77.78	0.020
H2O (ice, 0oC)	1.309	91.6	0.018
vacuum	1.000	1.000	0.000

Kirchhoff's Law may now be invoked to find the emissivity of dielectrics. Using eqs. (9.30) and (9.38), the normal spectral emissivity is

${\varepsilon _{\lambda ,n}}(\lambda ) = {\alpha _{\lambda ,n}}(\lambda ) = 1 - {\rho _{\lambda ,n}}(\lambda ) = 1 - {\left( {\frac{{{n_2} - {n_1}}}{{{n_2} + {n_1}}}} \right)^2} = \frac{{4\left( {\frac{{{n_2}}}{{{n_1}}}} \right)}}{{{{\left[ {\left( {\frac{{{n_2}}}{{{n_1}}}} \right) + 1} \right]}^2}}} = \frac{{4n}}{{{{\left( {n + 1} \right)}^2}}}\qquad \qquad( )$
(9.39)

where n = n2 / n1 has been substituted. Because most engineering properties are used for emission into air, for which n1 ≈ 1, the n in this and following equations is taken as equal to the refractive index of the emitting material, n2. The hemispherical spectral emissivity${\varepsilon _\lambda }(\lambda )$ is found by using eq. (9.37) and Kirchhoff's Law, and then integrating the result over all angles of the hemisphere. The final form is

$\begin{array}{l} {\varepsilon _\lambda }(\lambda ) = \frac{1}{2} - \frac{{\left( {3n + 1} \right)\left( {n - 1} \right)}}{{6{{\left( {n + 1} \right)}^2}}} - \frac{{{n^2}{{\left( {{n^2} - 1} \right)}^2}}}{{{{\left( {{n^2} + 1} \right)}^3}}}\ln \left( {\frac{{n - 1}}{{n + 1}}} \right) \\ {\rm{ }} + \frac{{2{n^3}\left( {{n^2} + 2n - 1} \right)}}{{\left( {{n^2} + 1} \right)\left( {{n^4} - 1} \right)}} - \frac{{8{n^4}\left( {{n^4} + 1} \right)}}{{\left( {{n^2} + 1} \right){{\left( {{n^4} - 1} \right)}^2}}}\ln n \\ \end{array}\qquad \qquad( )$

(9.40)

Properties of Electrical Conductors. Metals are characterized by having a high concentration of electrons, resulting in large electrical and thermal conductivities. A radiative EM field can interact with the electrons, causing rapid attenuation of the EM wave. In this case, solutions of Maxwell's Equations depend on the complex refractive index $\bar n$ of the absorbing material, which is defined here as

$\bar n = n - i\kappa \qquad \qquad( )$
(9.41)

where n is the simple refractive index, i is the imaginary $i = \sqrt { - 1}$, and  is the dimensionless extinction coefficient for the wave. (Some references such as Bohren and Huffman (1983) define the complex index as $\bar n = n + i\kappa$. In Lide (2008), the definition varies from table-to-table.) For the particular case of metals,  is very large, and the polarized components of spectral reflectivity are (assuming incidence is through a nonabsorbing medium with n1 = 1 and ${\bar n_2} = n - i\kappa$)

${\rho _\lambda }_{,\parallel }\left( {\lambda ,\theta } \right) = \frac{{{{\left( {n\cos \theta - 1} \right)}^2} + {{\left( {\kappa \cos \theta } \right)}^2}}}{{{{\left( {n\cos \theta + 1} \right)}^2} + {{\left( {\kappa \cos \theta } \right)}^2}}}\qquad \qquad( )$

(9.42a)

${\rho _{\lambda , \bot }}\left( {\lambda ,\theta } \right) = \frac{{{{\left( {n - \cos \theta } \right)}^2} + {\kappa ^2}}}{{{{\left( {n + \cos \theta } \right)}^2} + {\kappa ^2}}}\qquad \qquad( )$
(9.42b)

${\rho _\lambda }\left( {\lambda ,\theta } \right) = \frac{1}{2}\left[ {\frac{{{{\left( {n\cos \theta - 1} \right)}^2} + {{\left( {\kappa \cos \theta } \right)}^2}}}{{{{\left( {n\cos \theta + 1} \right)}^2} + {{\left( {\kappa \cos \theta } \right)}^2}}} + \frac{{{{\left( {n - \cos \theta } \right)}^2} + {\kappa ^2}}}{{{{\left( {n + \cos \theta } \right)}^2} + {\kappa ^2}}}} \right]\qquad \qquad( )$
(9.43)

For radiation incident at θ = 0, this reduces to

${\rho _\lambda }\left( {\lambda ,\theta = 0} \right) = {\rho _{\lambda ,n}}\left( \lambda \right) = \frac{{{{\left( {n - 1} \right)}^2} + {\kappa ^2}}}{{{{\left( {n + 1} \right)}^2} + {\kappa ^2}}}\qquad \qquad( )$

(9.44)

Invoking Kirchhoff's Law and using eqs. (9.30) and (9.44), the normal emissivity for metals is predicted to be

${\varepsilon _{\lambda ,n}} = {\alpha _{\lambda ,n}} = 1 - {\rho _{\lambda ,n}}\left( \lambda \right) = \frac{{4n}}{{{{\left( {n + 1} \right)}^2} + {\kappa ^2}}}\qquad \qquad( )$

(9.45)

Using eq. (9.43) with Kirchhoff's Law to find ε λ(λ,θ) and then integrating over all angles of a hemisphere and applying some approximations gives the hemispherical spectral emissivity as

$\begin{array}{l} {\varepsilon _\lambda }\left( \lambda \right) = 4n - 4{n^2}\ln \left( {\frac{{1 + 2n + {n^2} + {\kappa ^2}}}{{{n^2} + {\kappa ^2}}}} \right) + \frac{{4n\left( {{n^2} - {\kappa ^2}} \right)}}{\kappa }{\tan ^{ - 1}}\left( {\frac{\kappa }{{n + {n^2} + {\kappa ^2}}}} \right) \\ {\rm{ + }}\frac{{4n}}{{{n^2} + {\kappa ^2}}} - \frac{{4{n^2}}}{{{{\left( {{n^2} + {\kappa ^2}} \right)}^2}}}\ln \left( {1 + 2n + {n^2} + {\kappa ^2}} \right) - \frac{{4n\left( {{\kappa ^2} - {n^2}} \right)}}{{\kappa {{\left( {{n^2} + {\kappa ^2}} \right)}^2}}}{\tan ^{ - 1}}\left( {\frac{\kappa }{{1 + n}}} \right) \\ \end{array}\qquad \qquad( )$

(9.46)

As before, the spectral dependence of emissivity enters through the spectral dependence of n and . Equation (9.46) is said to be accurate within a few percent for most highly polished pure metals (Siegel and Howell, 2002). At long wavelengths (λ> about 5 μm), Maxwell's Equations predict that n and  become equal for metals, and in turn are related to the electrical resistivity of the metal, re, through the relation

$n = \kappa = \left( {\frac{{0.003{\lambda _o}}}{{{r_e}}}} \right)\qquad \qquad( )$

(9.47)

In eq. (9.47), λo is the wavelength in a vacuum in μm, and re is in ohm-cm. The relation is the Hagen-Rubens Equation.

Figure 9.20 Measured spectral normal emissivity of polished platinum at 1400K (Harrison et al., 1961) compared with the prediction of eq. (9.48) using re (T = 1400K) = 47.3x10-6 Ohm-cm.

Substituting n =  into eq. (9.45), expanding the result in a series, and then substituting eq. (9.47) after retaining only two terms in the series (and making a small adjustment to account for the truncation) results in the ‘‘Davisson-Weeks emissivity equation as modified by’‘ Schmidt and Eckert (1935).

${\varepsilon _{\lambda ,n}}\left( \lambda \right) = 36.5{\left( {\frac{{{r_e}}}{{{\lambda _o}}}} \right)^{1/2}} - 464\frac{{{r_e}}}{{{\lambda _o}}}\qquad \qquad( )$
(9.48)

Table 9.2: Optical Property Values and Normal Spectral Reflectivity of Various Metals at T = 300 K and λ = 0.589 and 10μm PROPERTY Refractive index1 n Extinction coefficient1 Electrical resistivity1

re	Normal spectral reflectivity
ρ λ,n
(Ohm-cm)	eq. (9.44)	eq. (9.48)	Measured2

MATERIAL Aluminum (0.589 μm) 1.15 7.147 2.73 x10-6 0.917 0.923 0.91 (10 μm) 25.4 90.0 " 0.988 0.981 0.988 Chromium (0.589 μm) 3.33 4.39 12.7 x10-6 0.650 0.841 0.57 (10 μm) 13.9 27.7 " 0.944 0.960 0.946 Copper (0.589 μm) 0.47 2.81 1.72x10-6 0.813 0.939 0.76 (10 μm) 28 68 " 0.980 0.985 0.985

Gold (0.589 μm)	0.18	2.84	2.27x10-6	0.924	0.930	0.85

(10 μm) 6.7 73 " 0.995 0.983 0.994

Iron (0.589 μm)	2.80	3.34	9.98x10-6	0.562	0.858	0.517

(10 μm) 6.34 28.2 " 0.970 0.964 0.953

Nickel (0.589 μm)	1.85	3.48	7.20x10-6	0.634	0.878	0.655

(10 μm) 7.59 38.5 " 0.980 0.969 0.965 Platinum (0.589 μm) 2.23 3.92 10.8 x10-6 0.654 0.852 0.673 (10 μm) 10.8 38.2 " 0.973 0.963 0.970

Silver (0.589 μm)	0.26	3.96	1.63 x10-6	0.940	0.941	0.95

(10 μm) 8.22 79 " 0.995 0.985 0.995 Titanium (0.589 μm) 2.01 2.77 39 x10-6 0.520 0.734 0.490 (10 μm) 4.1 19.7 " 0.960 0.930 0.970

Tungsten (0.589 μm)	3.54	2.84	5.44 x10-6	0.506	0.893	0.51

(10 μm) 11.6 48.4 " 0.981 0.973 0.97 1Lide (2008); 2Measured data is the largest reflectivity value from multiple references cited in Touloukian (1970) Using the resistivity of platinum at 1400K, ρ e = 47.3x10-6 Ohm-cm (extrapolated from data for pure platinum), the spectral normal emissivity of polished platinum can be predicted using eq. (9.48). The result is compared with measured values in Fig. 9.20. The predictions at shorter wavelengths usually become worse, and are often in greater error for other metals, especially for surfaces that are not highly polished or are oxidized. However, if no measured data is available, the EM theory predictions provide a useful resource, and can also be used to extend limited data. The total properties from EM theory are found by integrating the spectral properties, which incorporates some further approximations. The results are

${\varepsilon _n}(T) = 0.578{({r_e}T)^{1/2}} - 0.178{r_e}T + 0.0584{({r_e}T)^{3/2}}\qquad \qquad( )$

(9.49)

$\varepsilon (T) = 0.766{({r_e}T)^{1/2}} - \left[ {0.309 - 0.0889\ln \left( {{r_e}T} \right)} \right]{r_e}T - 0.0175{({r_e}T)^{3/2}}\qquad \qquad( )$
(9.50)

Comparisons of predictions of these two equations with experimental data for ten or more metals show good agreement (Siegel and Howell, 2002). Table 9.2 gives data for the various optical properties at T = 300K for a variety of metals along with comparisons of the spectral normal reflectivity, ρ λ,n, calculated from the EM theory relations in this section with measured values. Note the relatively poor prediction of normal spectral reflectivity at the short wavelength (0.589 μm) using eq. (9.48), which is based on the electrical resistivity re, , and the much improved predictions at 10 μm. The assumptions used in the derivation of eq. (9.48) are clearly better satisfied at the longer wavelength. However, the assumption that n ≈  is not met even at 10 μm. The predictions of eq. (9.44), based on the components of the complex refractive index n and, are better at both wavelengths. However, these properties (especially ) are often unavailable, whereas the electrical resistivity is often in handbooks or easily measured.

## 9.4 Application and Exploitation of Radiative

Properties

The fact that real materials are not gray can be used as an advantage. Some materials, such as white paint, naturally have major changes in absorptivity and emissivity with wavelength. It is also possible to tailor materials to enhance the wavelength dependence. This is commonly done by placing thin oxide coatings on a highly polished metallic substrate. At short wavelengths, the oxide layer is highly absorbing, while at longer wavelengths, the radiation penetrates the layer, and the absorptivity takes on the low absorptivity of the metallic substrate. The transition from high to low absorptivity for such surfaces can be quite abrupt; the wavelength at which the transition occurs is called the cutoff wavelength.

### 9.4.1 Spacecraft Thermal Design

Because spacecraft operate in the vacuum of space, convection and conduction exchanges with the environment are not possible, and radiative exchange is the only available mechanism for energy transfer. Spacecraft are exposed to solar energy except when in the Earth's shadow, so they generally absorb solar energy through part of their orbit unless they are actively oriented to minimize solar absorption. Controlling the temperature of a satellite requires a careful balance between absorbed and emitted radiation.

Example 9.2 Consider the flat plate shown in Earth orbit around the sun. One face of the plate is directly facing the Sun; the back of the plate is insulated. The exposed face has the idealized diffuse spectral emissivity shown in the figure. Assuming that the incident solar flux on the plate is given by the solar constant of 1368 W/m2 and has a spectral distribution similar to a blackbody at the solar radiating temperature of 5780K, find the equilibrium temperature of the plate when the value of λc (the cutoff wavelength) is varied from 0.5 to 2.0 μm. Also, find the limiting temperatures when λc is set atλ =0 andλ= ∞.

Solution: The solar absorptivity of the surface can be found using eq. (9.26), noting that for a diffuse surface α λλ = α λnλ,

$\begin{array}{l} {\alpha _n}(T) = \frac{{\int_{\lambda = 0}^\infty {{\varepsilon _{\lambda ,n}}(T){I_{\lambda ,i}}\left( {\theta = 0} \right)d\lambda } }}{{{I_i}\left( {\theta = 0} \right)}} = \frac{{\int_{\lambda = 0}^\infty {{\varepsilon _{\lambda ,n}}(T){I_{\lambda b}}\left( {T = 5680K} \right)d\lambda } }}{{{I_b}\left( {T = 5680K} \right)}} \\ = \frac{{0.9\int_{\lambda = 0}^{{\lambda _c}} {{I_{\lambda b}}\left( {{T_s} = 5680K} \right)d\lambda } }}{{{I_b}\left( {{T_s} = 5680K} \right)}} + \frac{{0.1\int_{\lambda = {\lambda _c}}^\infty {{I_{\lambda b}}\left( {{T_s} = 5680K} \right)d\lambda } }}{{{I_b}\left( {{T_s} = 5680K} \right)}} \\ = 0.9{F_{0 - {\lambda _c}{T_s}}} + 0.1{F_{{\lambda _c}{T_s} - \infty }} = 0.9{F_{0 - {\lambda _c}{T_s}}} + 0.1\left( {1 - {F_{0 - {\lambda _c}{T_s}}}} \right) = 0.1 + 0.8{F_{0 - {\lambda _c}{T_s}}} \\ \end{array}$

At λc = 0, the value of α n will approach 0.1, and at λc → ∞, α n → 0.9. The emissivity of the surface is found in a similar way. Using eq. (9.17),

$\begin{array}{l} \varepsilon (T) = \frac{{\int_{\lambda = 0}^\infty {{\varepsilon _\lambda }\left( {\lambda ,T} \right){E_{\lambda b}}\left( {\lambda ,T} \right)d\lambda } }}{{\sigma {T^4}}} \\ = \frac{{0.9\int_{\lambda = 0}^{{\lambda _c}} {{E_{\lambda b}}\left( {\lambda ,T} \right)d\lambda } }}{{\sigma {T^4}}} + \frac{{0.1\int_{\lambda = {\lambda _c}}^\infty {{E_{\lambda b}}\left( {\lambda ,T} \right)d\lambda } }}{{\sigma {T^4}}} \\ {\rm{ }} = 0.9{F_{0 - {\lambda _c}T}} + 0.1{F_{{\lambda _c}T - \infty }} \\ = 0.9{F_{0 - {\lambda _c}T}} + 0.1\left( {1 - {F_{0 - {\lambda _c}T}}} \right) = 0.1 + 0.8{F_{0 - {\lambda _c}T}} \\ \end{array}$
Figure 9.21 Spectral emissivity ε λλ
Figure 9.22 Equilibrium temperatures vs. cutoff wavelength

Unlike the case for solar absorptivity where the spectral distribution at the solar temperature is known, however, the temperature of the emitting surface is unknown, so that F0-λT usually must be found by an iterative solution. This can be done by equating the absorbed and emitted energy at the required surface temperature to give

$\begin{array}{l} {\rm{Absorbed energy rate }} = {\rm{ }}{\alpha _n}{{q''}_{solar}}A \\ = {\rm{emitted energy rate }} = \varepsilon \sigma {T^4}A \\ \end{array}$

Or

${\rm{ }}T = {\rm{ }}{\left[ {\left( {\frac{{{\alpha _n}}}{\varepsilon }} \right)\frac{{{{q''}_{solar}}}}{\sigma }} \right]^{1/4}}$

Because the solar constant qs and σ are fixed, it is clear that the equilibrium temperature is dependent on the ratio of solar normal absorptivity to surface emissivity. The minimum value for ε must be 0.1 in the present case, and this can be used to find an initial guess for T. This T is then used to determine F0-λT, which is inserted into the equation for emissivity. This is repeated until the ε and T values converge. For the present problem, this must be done for each value of λc . Using the value of α n = 0.8517 for λc = 2 μm,

$\begin{array}{l} {\rm{ }}T = {\rm{ }}{\left[ {\left( {\frac{{{\alpha _n}}}{\varepsilon }} \right)\frac{{{{q''}_{solar}}}}{\sigma }} \right]^{1/4}} \\ = {\left[ {\left( {\frac{{0.8517}}{{0.1}}} \right)\frac{{1368(W/{m^2})}}{{5.6704 \times {{10}^{ - 8}}(W/{m^2} \cdot {K^4})}}} \right]^{1/4}} \\ = \underline {673.3K} \\ \end{array}$

giving F0 − 2x671.4= 0.005626 and ε = .10045. The value of λc = 2 μm gives the largest value of α n , and therefore the largest values for T and F0-λT, and therefore ε . The value of ε = 0.100 thus is a reasonably accurate guess for the whole range of λc. The resulting equilibrium temperatures are in Fig. 9.22. For a gray surface, no matter what the value of α = ε , the equation for equilibrium temperature shows that

$\begin{array}{l} {\rm{ }}{T_{gray}} = {\rm{ }}{\left( {\frac{{{{q''}_{solar}}}}{\sigma }} \right)^{1/4}} \\ = {\left( {\frac{{1368(W/{m^2})}}{{5.6704 \times {{10}^{ - 8}}(W/{m^2} \cdot {K^4})}}} \right)^{1/4}} = 394.1K \\ \end{array}$

This will be the equilibrium temperature when λc is set atλ =0 orλ= ∞.

Another application of radiative heat transfer for spacecraft is in the design of radiators to reject waste heat. The net rate of heat rejection per unit area, q", is the difference between the emitted and the absorbed radiative fluxes. To minimize the absorbed flux, the radiator can be oriented edge-on to the sun, so that the energy absorbed comes only from the Earth and from space. Space itself has an apparent background temperature of 4K, and radiation from this source can be neglected. The heat rejection rate from the space radiator is

$q = q''A = \left( {2\varepsilon \sigma T_{rad}^4 - {{q''}_{Earth}}} \right)A\qquad \qquad( )$
(9.51)

Note that the factor of two enters the equation because both sides of the radiator emit energy, but only one side is exposed to the Earth's radiation. If the spacecraft is far from Earth or other planets so that q''Earth = 0, then the radiator area required to reject a certain rate of energy is

$A = \frac{q}{{2\varepsilon \sigma T_{rad}^4}}\qquad \qquad( )$

(9.52)

For a given radiator temperature T, choosing a radiator coating with the largest possible value of emissivity, ε , will allow the smallest possible radiator area. Clearly, a higher heat rejection temperature can greatly reduce the required area for heat rejection because of the fourth-power relation.

Example 9.3 A nuclear-powered Carnot engine is to produce 100 kW of work output to drive an electrical generator for a proposed small space station. The temperature of the energy input to the cycle is limited to 1000 K. Find the heat rejection temperature from the cycle (the average temperature of a heat rejection radiator) that will minimize the area of a radiator that rejects the cycle energy.

Solution: The energy output of the cycle is given by the Carnot efficiency relations

$w = {q_{in}}(1 - \frac{{{T_{rej}}}}{{{T_{in}}}}) = (w + {q_{rej}})(1 - \frac{{{T_{rej}}}}{{{T_{in}}}})$

Or

$w = {q_{rej}}\left( {\frac{{{T_{in}}}}{{{T_{rej}}}} - 1} \right)$

Substituting eq. (9.52) to eliminate qrej gives, for an isothermal radiator with ε = 1,

$w = 2\sigma T_{rej}^4A\left( {\frac{{{T_{in}}}}{{{T_{rej}}}} - 1} \right)$

or

<center>$A = \frac{w}{{2\sigma T_{rej}^4\left( {\frac{{{T_{in}}}}{{{T_{rej}}}} - 1} \right)}}$

To determine the minimum area for this case, treat this as a max-min problem, and take the derivative dA / dTrej, and set it equal to zero and solve for the resulting value of Trej that will minimize the area. This gives

$A = \frac{w}{{2\sigma }}{\left[ {T_{rej}^4\left( {\frac{{{T_{in}}}}{{{T_{rej}}}} - 1} \right)} \right]^{ - 1}}$
$\begin{array}{l} \frac{{dA}}{{d{T_{rej}}}} = - \frac{w}{{2\sigma }}\frac{{\left[ {4T_{rej}^3\left( {\frac{{{T_{in}}}}{{{T_{rej}}}} - 1} \right) - \frac{{T_{rej}^4{T_{in}}}}{{T_{rej}^2}}} \right]}}{{{{\left[ {T_{rej}^4\left( {\frac{{{T_{in}}}}{{{T_{rej}}}} - 1} \right)} \right]}^2}}} \\ = \frac{w}{{2\sigma }}\frac{{\left( {4T_{rej}^3 - 3T_{rej}^2{T_{in}}} \right)}}{{{{\left[ {T_{rej}^3\left( {{T_{in}} - {T_{rej}}} \right)} \right]}^2}}} = 0 \\ \end{array}$

The numerator $\left( {4T_{rej}^3 - 3T_{rej}^2{T_{in}}} \right)$ must equal to zero, so $T_{rej}^{} = \frac{3}{4}{T_{in}} = 750K$, giving the required radiator area as

$\begin{array}{l} A = \frac{w}{{2\sigma T_{rej}^4\left( {\frac{{{T_{in}}}}{{{T_{rej}}}} - 1} \right)}} \\ = \frac{{100(kW) \times 1000\left( {W/kW} \right)}}{{2 \times 5.6704 \times {{10}^{ - 8}}(W/{m^2} \cdot {K^4}) \times {{\left( {750K} \right)}^4}\left( {\frac{{1000}}{{750}} - 1} \right)}} = 8.36({m^2}) \\ \end{array}$

Observe that to optimize collector size, the heat rejection temperature from the Carnot cycle should be set at $T_{rej}^{} = 3{T_{in}}/4$ regardless of both the power requirement and maximum temperature set for the cycle. This results in a Carnot efficiency for the cycle of ηCarnot = 1 − Trej / Tin = 1 − 3 / 4 = 0.25 or 25%. The collector area is seen to be directly proportional to the work output of the cycle.

### 9.4.2 Solar Thermal Energy Collectors

Solar collector analysis is similar to that for spacecraft, with the added complications that the collector surface usually has convective losses in addition to radiation; the collector is exposed to not only solar but also environmental radiation; and the collector is not isothermal (see Fig. 9.23).

So called flat plate collectors are used to gather thermal energy, such as those employed for swimming pool heating, domestic hot water production, residential and commercial building heating, and possibly for use with thermally-driven cooling cycles. The collector is made up of an absorber plate, which absorbs the solar radiation, and transfers it to a fluid flowing through channels in the plate. These are often of fin-tube design. For low temperatures such as for swimming pool heaters, the absorber surface is often uncovered. For intermediate to higher temperatures, convective losses to the environment become significant, and a transparent cover plate may be placed above the absorber plate to add additional resistance to losses. The back of the collector is insulated. The instantaneous rate of net energy collected by the working fluid with mass flow rate $\dot m$ and specific heat cp is given by a First Law energy balance as

$q = \dot m\left( {{H_{out}} - {H_{in}}} \right) = \dot m{c_p}\left( {{T_{out}} - {T_{in}}} \right)\qquad \qquad( )$

(9.53)

Alternately, a heat transfer balance for the collector gives

$q = {\alpha _{eff}}{q''_{solar}}A\cos \theta - \bar hA(\overline {{T_s}} - {T_\infty }) - \varepsilon \sigma A(\overline {T_s^4} - T_\infty ^4)\qquad \qquad( )$
(9.54)

where q"solar is the solar flux on a surface normal to the sun and θ is the angle of the sun relative to the normal of the solar collector. This form includes the following assumptions: First, the value α eff is the effective solar absorptivity of the collector, which is also assumed to be a diffuse surface property. As eq. (9.54) is written, it is inherently assumed that the solar energy incident on the collector comes only from the direction of the sun, and is what is known as direct solar energy. The direct component is the major solar input on a clear day. However, when days are hazy or cloudy, a so-called diffuse component may be present, which is incident from many other directions than θ onto the collector. Additionally, a tilted collector may gather energy reflected from surrounding structures or the ground. The latter two components are omitted in the simplified treatment given here. The α eff in eq. (9.54) includes not only the absorptivity of the solar collector plate itself, but also any effects of cover glasses on the overall absorption of solar energy.

Figure 9.23 Typical construction of solar thermal collector

The final two terms in eq. (9.54) are the heat losses between the collector and the surroundings. The $\overline {{T_s}}$ is the average solar collector plate surface temperature, taken as the average of the fluid inlet and outlet temperatures,

$\overline {{T_s}} = ({T_{in}} + {T_{out}})/2$
.

Because most flat plate collectors have a fairly small temperature change in the working fluid between the collector inlet and outlet, this is a valid approximation to use in determining a temperature difference for use in the convective and radiative loss terms. Often, a sky radiating temperature that is lower than the environmental (air) temperature is used as the radiative sink temperature in the final term.

The final term in eq. (9.54) comes from approximating the radiative loss as an exchange between the collector surface and the surroundings, both of which are near enough in absolute temperature that the IR emissivity and absorptivity are close to the same value. If cover glasses are present, the radiative loss calculation becomes more complex, as most “transparent” materials in cover glasses actually become opaque at wavelengths longer than about 2 μm, although some materials have transparent “windows” in the infrared spectrum. The rate of energy collection is time dependent. Even on a clear day, the angle of the sun relative to the collector, θ, will vary with time of day and day of the year. In addition, weather conditions can change the value of the direct solar radiation q"solar over short time periods. The convective heat transfer coefficient $\bar h$ is an overall value, accounting for natural convection between the collector plate and any cover glasses, conduction through the cover glasses, and forced convection from the outer cover glass surface and the environment due to average wind conditions. Its determination is challenging, because it can involve free convective transfer between parallel surfaces (absorber plate and cover glass, and possibly between multiple parallel cover glasses.) If the plate is installed on a slanted rooftop, for example, then the free convective flows are dependent on the slant angle of the collector. The free convection patterns undergo a transition from multiple cells similar to Bénard cells for the horizontal collectors to a single large rotating cell found in a vertical collector (Fig. 9.24), although the figure is quite simplified (the flow patterns are three-dimensional because of the presence of corners in the closed rectangular geometry). The transition occurs at some tilt angle that depends on the ratio of length L to the plate-to-cover spacing height H, L / H. For L / H > 12 (typical of solar collectors,) the critical angle according to Hollands et al. (1976) is about 70o. Analysis of the effect of tilt and the transition angle on solar collector configurations has been carried out by Catton et al. (1974), Hollands et al. (1976), Randall et al. (1979) and Torrance and Catton (1980). Randall et al. report that the average Nusselt number in the laminar flow regime based on plate spacing H is adequately correlated (within ± 8%) for tilt angles  from 45 to 90o by

Figure 9.24 Natural convection patterns in parallel enclosures as found in solar collectors.
$\overline {N{u_H}} = 0.118{\left[ {G{r_H}\Pr {{\cos }^2}\left( {\phi - 45} \right)} \right]^{0.29}}\qquad \qquad( )$

(9.55a)

for $4 \times {10^3} \le G{r_H} \le 3.1 \times {10^5}$ and aspect ratios $9 \le L/H \le {\rm{36}}$. Note that within this range of L / H, the predicted Nusselt number is independent of aspect ratio. For smaller aspect ratios, other investigators have observed a dependence on aspect ratio. For angles less than the critical angle, Hollands et al. (1976) suggest the correlation

$\overline {N{u_H}} = 1 + 1.44\left[ {1 - \frac{{1708}}{{R{a_H}\cos \phi }}} \right]\left[ {1 - \frac{{1708{{\left( {\sin 1.8\phi } \right)}^{1.6}}}}{{R{a_H}\cos \phi }}} \right] + \left[ {{{\left( {\frac{{R{a_H}\cos \phi }}{{5830}}} \right)}^{1/3}} - 1} \right]\qquad \qquad( )$

(9.55b)

which applies for $L/H \ge 12,\phi \le {70^o}$. If any term is negative, it is set equal to zero. The efficiency of a solar collector is defined as the amount of energy transferred to the working fluid passing through the collector over the energy incident from the sun, or

$\begin{array}{l} {\eta _{coll}} = \frac{q}{{{{q''}_{solar}}A\cos \theta }} = {\alpha _{eff}} - \frac{{\bar h{T_\infty }}}{{{{q''}_{solar}}\cos \theta }}\left( {\overline {{\Theta _s}} - 1} \right) - \frac{{\varepsilon \sigma T_\infty ^4}}{{{{q''}_{solar}}\cos \theta }}\left( {\overline {\Theta _s^4} - 1} \right) \\ {\rm{ }} = {\alpha _{eff}} - \bar H\left( {\overline {{\Theta _s}} - 1} \right) - \bar S\left( {\overline {\Theta _s^4} - 1} \right) \\ \end{array}\qquad \qquad( )$

(9.56)

where $\overline {{\Theta _s}} {\rm{ }} = {T_s}/{T_\infty }$ and the two dimensionless groups describing the relative importance of convection and radiation losses to incident solar flux are

$\bar H = \frac{{\bar h{T_\infty }}}{{{{q''}_{solar}}\cos \theta }}{\rm{ and }}\bar S = \frac{{\varepsilon \sigma T_\infty ^4}}{{{{q''}_{solar}}\cos \theta }}$
.

If s is not too far from unity (see Example 10.1 for a quantitative discussion), the final term in eq. (9.56) can be linearized by the transformation

$\begin{array}{l} \left( {\overline {\Theta _s^4} - 1} \right) = \left( {\overline {\Theta _s^2} + 1} \right)\left( {\overline {\Theta _s^2} - 1} \right) = \left( {\overline {\Theta _s^2} + 1} \right)\left( {\overline {\Theta _s^{}} + 1} \right)\left( {\overline {\Theta _s^{}} - 1} \right) \\ {\rm{ = }}\left( {\overline {\Theta _s^2} + 1} \right)\left( {\overline {\Theta _s^{}} + 1} \right)\left( {\overline {\Theta _s^{}} - 1} \right) \approx 4\left( {\overline {\Theta _s^{}} - 1} \right) \\ \end{array}\qquad \qquad( )$

(9.57)

For example, if the collector average surface temperature is $\overline {{T_s}}$ = 67oC = 340K and T = 27oC = 300 K, then Θs = 340/300 = 1.13. The error in the approximation in eq. (9.57) is then

$\frac{{\left( {\overline {\Theta _s^2} + 1} \right)\left( {\overline {\Theta _s^{}} + 1} \right) - 4}}{{\left( {\overline {\Theta _s^2} + 1} \right)\left( {\overline {\Theta _s^{}} + 1} \right)}} = \frac{{\left( {{{1.13}^2} + 1} \right)\left( {1.13 + 1} \right) - 4}}{{\left( {{{1.13}^2} + 1} \right)\left( {1.13 + 1} \right)}} = 0.18$

or 18%. If this level of approximation is acceptable (which depends on the magnitude of $4\overline S$ relative to $\bar H$), then eq. (9.55) can be rewritten using eq. (9.57) as

$\begin{array}{l} {\eta _{coll}} = {\alpha _{eff}} - \bar H\left( {\overline {{\Theta _s}} - 1} \right) - \bar S\left( {\overline {\Theta _s^4} - 1} \right) = {\alpha _{eff}} - \bar H\left( {\overline {{\Theta _s}} - 1} \right) - 4\bar S\left( {\overline {{\Theta _s}} - 1} \right) \\ {\rm{ }} = {\alpha _{eff}} - \left( {\bar H + 4\bar S} \right)\left( {\overline {{\Theta _s}} - 1} \right) = {\alpha _{eff}} - \left( {\bar H*} \right)\left( {\overline {{\Theta _s}} - 1} \right) \\ \end{array}\qquad \qquad( )$

(9.58)

The $\bar H*$ = ($\bar H$ + $4\overline S$) is a modified dimensionless heat transfer coefficient that now accounts for both convective and (linearized) radiative losses to the surroundings. Equation (9.58) shows that after this linearization, the collector efficiency is directly dependent on the temperature difference (Θs -1). Examining eq. (9.58), it is clear that collector efficiency is affected by the radiative surface properties. Choosing a collector plate coating with large α eff is a benefit; however, a spectrally selective coating that has large absorptivity for solar energy and a small emissivity at large wavelength (as idealized in the figure in Example 9.2) will not only have a large value of α eff, but also a small value of hemispherical emissivity at the plate temperature, which will make the value of $\overline S$ small and thus help to reduce radiative loss to the environment. At some high collector plate average temperature, the absorbed energy is balanced by the heat transfer to the environment, so no useful energy is collected and the efficiency equals to zero. This can occur if flow is lost to the collector. At this condition, the collector reaches its maximum or stagnation temperature given by

${\Theta _{stag}} = 1 + \frac{{{\alpha _{eff}}}}{{\left( {\bar H*} \right)}}{\rm{ or }}{T_{stag}} = {T_\infty } + \frac{{{\alpha _{eff}}{T_\infty }}}{{\bar H + 4\bar S}} = {T_\infty } + \frac{{{\alpha _{eff}}{{q''}_{solar}}\cos \theta }}{{\bar h + 4\varepsilon \sigma T_\infty ^3}}\qquad \qquad( )$
(9.59)

Example 9.4 A solar thermal collector is to be used as a swimming pool heater. For this use, no cover plate is used, and to maximize solar absorption, a carbon black-filled plastic is used for the absorber plate material. The effective solar absorptivity α eff is very nearly that of a blackbody. On a clear midsummer day the solar flux on a surface normal to the sun is measured to be 1000 W/m2 at solar noon. At the latitude of the pool, at solar noon (when the sun is at its smallest angle θto the collector), the sun is at an angle of θ = 15o to the normal to the horizontal pool collector. A pump forces the pool water at 27oC into the collector. The convective heat transfer coefficient between the collector surface and the air at 20oC is 6 W/m2-K. (a) If the temperature rise of the pool water through the collector is 6K, what is the collector efficiency at solar noon? (b) The melting point of the plastic collector is 120oC. If flow is lost to the collector, will the collector experience melting? (c) What mass flow rate per unit of collector area. $(\dot m/A)(kg/s \cdot {m^2})$

of pool water passing through the collector?

Solution: The temperature rise through the collector is 6 K, so the average temperature $\overline {{T_s}}$ ==27+3=30oC and Θs = (30+273)/(20+273) = 1.03. The parameter $\bar H*$ = ($\bar H$ + $4\overline S$) is then $\bar H*$ =($\bar H$ + $4\overline S$) =$\frac{{\bar h{T_\infty }}}{{{{q''}_{solar}}\cos \theta }}{\rm{ }} + \frac{{4\varepsilon \sigma T_\infty ^4}}{{{{q''}_{solar}}\cos \theta }}$

$\begin{array}{l} {\rm{ = }}\frac{{6(W/{m^2} \cdot K) \times (20 + 273)(K)}}{{1000(W/{m^2}) \times \cos \left( {{{15}^o}} \right)}}{\rm{ }} \\ {\rm{ }} + \frac{{4 \times 1 \times 5.6704 \times {{10}^{ - 8}}(W/{m^2} \cdot {K^4}) \times \left( {20 + 273} \right)_{}^4({K^4})}}{{1000(W/{m^2}) \times \cos \left( {{{15}^o}} \right)}} \\ {\rm{ = 1}}{\rm{.82 + 1}}{\rm{.73 = 3}}{\rm{.55}} \\ \end{array}$

From eq. (9.56), the collector efficiency is then

${\eta _{collector}} = {\alpha _{eff}} - \left( {\bar H*} \right)\left( {\overline {{\Theta _s}} - 1} \right) = 1 - 3.55 \times \left( {1.03 - 1} \right) = 1.00 - 0.11 = 0.89$

so the collector converts 89 percent of the incident radiation (called the insolation) into increased enthalpy of the fluid flowing through the collector. The stagnation temperature from eq. (9.59) is

$\begin{array}{l} {T_{stag}} = {T_\infty } + \frac{{{\alpha _{eff}}{{q''}_{solar}}\cos \theta }}{{\bar h + 4\varepsilon \sigma T_\infty ^3}} \\ = {20^o}C + \frac{{1 \times 1000(W/{m^2}) \times \cos {{15}^o}}}{{6(W/{m^2} \cdot K) + 4 \times 1 \times 5.6704 \times {{10}^{ - 8}}\left( {W/{m^2} \cdot {K^4}} \right) \times {{(293)}^3}({K^3})}} \\ \end{array}$
$= {20^o}C + {82.5^o}C = \underline {{{102.5}^o}C}$

so the collector is safe under the given conditions.

Using eq. (9.53),

This result can be used to determine the necessary collector size to provide the necessary rate of energy to balance heat loss from the pool.

Because the sun traverses the sky during the day, and the path traversed varies with time of year, and the radiation reaching the collector depends on local weather conditions, it is very difficult to accurately predict the annual integrated energy that can be produced by a given collector. Some methods that have been developed to account for these factors (Duffie and Beckman, 2006). Year-to-year variations can also be significant. It is possible to use the collector to reject energy from the working fluid. During the hours of night, the collector energy balance, eq. (9.54), can be applied with q"solar = 0 to give

$q = - \bar hA\left( {\overline {{T_s}} - {T_\infty }} \right) - \varepsilon \sigma A\left( {\overline {T_s^4} - T_{sky}^4} \right)\qquad \qquad( )$
(9.60)

The effective night sky temperature Tsky has been substituted for T in the radiative loss term, since the sky temperature, especially on a clear night in a dry climate, may be much lower than the air temperature that governs convective loss. Both loss terms are negative as long as the plate temperature is above the environment and sky temperatures, so energy will be lost from the collector, and the temperature of the fluid passing therough the collector will be lowered as given by eq. (9.53). There is no particular advantage to linearizing eq. (9.60) because the sink temperatures of the two heat transfer modes are different, so the linearized form can't be combined with the convection term.

Example 9.5 An environmentally neutral cooler takes advantage of the cold night sky to provide cooling in hot climates. Consider a shallow pan of water that is outdoors on a clear night with no wind. The air temperature is 25oC, and the night sky temperature is 250K. The free convective heat transfer coefficient for a cool surface facing upwards is small, and in this case is about 5 W/m2-oC. What will be the equilibrium temperature of the water during the night, assuming no evaporation occurs, and the pan is situated on an insulating surface? Note that water at long wavelengths acts very much as a blackbody Solution: In this case, the net heat transfer to the surface in thermal equilibrium is zero, and eq. (9.60) gives

$- \bar hA\left( {{T_s} - {T_\infty }} \right) = \varepsilon \sigma A\left( {T_s^4 - T_{sky}^4} \right).$

This can be written as

$\begin{array}{l} {T_s} = {T_\infty } - \frac{{\varepsilon \sigma }}{{\bar h}}\left( {T_s^4 - T_{sky}^4} \right) \\ = {25^o}C - \frac{{5.6704 \times {{10}^{ - 8}}(W/{m^2} \cdot {K^4})}}{{5(W/{m^2} \cdot K)}}\left[ {T_s^4 - {{250}^4}\left( {{K^4}} \right)} \right] \\ \end{array}$

The value of the water temperature can be solved by guessing Ts, substituting into the RHS to generate a new guess, continuing until convergence, giving Ts= 286.5 K, or 13.5oC, compared with the air temperature of 25oC. It is possible to produce ice using this phenomenon even though the air temperature is well above freezing.

To increase the rate of energy collection per unit area by a solar collector, the incident solar flux can be multiplied by using concentrators. In this case, eq. (9.53) is modified to give

$q = C{\alpha _{eff}}{q''_{solar}}A - \bar hA\left( {\overline {{T_s}} - {T_\infty }} \right) - \varepsilon \sigma A\left( {\overline {T_s^4} - T_\infty ^4} \right)\qquad \qquad( )$
(9.61)

In this form, C is the concentration ratio, or the number of effective sun fluxes incident on the collector due to multiple concentrators. Some concepts for solar collection have concentration ratios of from C = 2-4 (flat plates with mirrored grooves), 30-60 (parabolic mirrored troughs), to over C = 600 for solar tower collectors with many heliostats to direct sunlight onto the collector surface. Solar towers can operate at very high temperatures, and linearization of the radiative loss term is inappropriate.

### 9.4.3 Other Property Choices for Radiation/Surface Interactions

In certain applications, it is desired to minimize solar absorption while maximizing radiative loss from a surface. This is the case in hot climates, where rooftops may be painted white to minimize solar absorption. The effect can be augmented by the use of a selective surface with low spectral absorptivity up to a cutoff wavelength followed by a high absorptivity (and thus emissivity) at the longer wavelengths where most surface emission occurs. Such a surface will radiate strongly to the night sky to provide overnight cooling, while minimizing solar gain during the day. Coatings with these characteristics are particularly useful to minimize boil-off from large storage tanks for cryogenics or hydrocarbons. Many materials that appear to be white to the eye are actually good selective surfaces. Common bright white paint uses titanium dioxide as the pigment, and has a solar absorptivity of about 0.12, yet its room temperature emissivity is above 0.92. Using these values for the plate in orbit in Example 9.2 gives an equilibrium temperature of

$\begin{array}{l} {\rm{ }}T = {\rm{ }}{\left[ {\left( {\frac{{{\alpha _n}}}{\varepsilon }} \right)\frac{{{q_{solar}}}}{\sigma }} \right]^{1/4}} = {\left[ {\left( {\frac{{0.12}}{{0.92}}} \right)\frac{{1368(W/{m^2})}}{{5.6704 \times {{10}^{ - 8}}(W/{m^2} \cdot {K^4})}}} \right]^{1/4}} \\ = 237K = - {36^o}C \\ \end{array}\qquad \qquad( )$
(9.62)

which can be compared with the equilibrium temperature of 673 K = 400oC found for the high ( α n/ ε ) selective surface used in Example 9.2 for the insulated flat plate in orbit. Of course, for the case of objects on Earth, the effect of convective transfer must be included in determining the equilibrium temperature as was shown in Example 9.5. 9.5 High-energy Radiation-Surface Interactions When the features of a surface or the size of particles is of the same order as the wavelength of the incident radiation, then some effects differ from those discussed for traditional engineering applications of radiative heat transfer. Radiation can generally be treated as an electromagnetic wave until the wavelength becomes of the order of the size of the atoms in a solid interacting with the radiation. Thus, gamma rays (Fig. 9.1) have effective wavelengths that are of the order of atomic sizes, and gamma rays do not undergo surface refraction as governed by Maxwell's equations. Radiation in the range of wavelengths common in engineering heat transfer can generally be treated by electromagnetic theory. As discussed in Section 9.1, the propagation of thermal radiation can be treated in terms of electromagnetic waves. Maxwell's equations relating the electric field vector E and the magnetic field vector H are

(9.63)

For plane waves propagating in an unbounded medium, the form of the electric field Ey (Fig. 9.1) that satisfies Maxwell's equations is (Bohren and Huffman, 1983)

${E_y} = {E_{y,o}}\exp \left\{ {i\omega \left[ {t - (n - i\kappa )\frac{x}{{{c_o}}}} \right]} \right\} = {E_{y,o}}\exp ( - \frac{{2\pi \kappa x}}{\lambda })\exp (i\omega t - i\frac{{2\pi nx}}{\lambda })\qquad \qquad( )$

(9.64)

where Eo is the maximum wave amplitude,  is the angular frequency ω = 2πco / λ, n is the simple refractive index,  is the imaginary component of the complex refractive index, $\bar n = n - i\kappa$, and λ is the wavelength in a vacuum.

The magnitude of the energy carried by this plane wave $\left| {\mathbf{S}} \right|$ , is

$\left| {\mathbf{S}} \right| = \frac{{\bar n}}{{\mu {c_o}}}E_y^2\qquad \qquad( )$

(9.65)

where μ is the magnetic permeability of the medium and S is the Poynting vector, E×H. The value of μ in a vacuum, μo, is 4x10-7N/A2. The radiative intensity I is directly proportional to $\left| {\mathbf{S}} \right|$.

The interaction of a plane wave propagating through one medium and interacting at the smooth interface (relative to the wavelength) with a second material with differing properties can be analyzed using Equations (9.63)-(9.65), and is the basis for the property predictions in Section 9.3. When a plane wave traveling through space interacts with a spherical dielectric particle of differing refractive index, part of the wave is refracted into the particle and the complex patterns of interference and reflection within the particle give a highly lobed structure for the radiation reflected from the particle. The exact form of the radiated pattern depends on the complex refractive index as well as the ratio of particle diameter to wavelength. Gustav Mie (1908), is usually given credit for the analytical solution for this case. The complete analytical solution includes terms that decay rapidly with distance from the particle (within a very few wavelengths), and these decaying fields are called evanescent waves. These near-field terms are usually neglected in the so-called Mie scattering solutions, since the particles in most suspensions (interstellar dust, ash particles in combustion products, atmospheric aerosols,

Figure 9.25 Near-field intensity enhancement of laser radiation caused by interaction with a dielectric (silica) sphere (Heltzel, 2006)

etc.) are quite far apart on a wavelength scale. Neglect of the terms is unimportant in these cases, and the resulting solutions are called ‘‘far-field’‘ solutions. However, for particles that are close to another object or surface, the evanescent waves greatly change the energy transfer between the particle and object, and their effect must be included. Figure 9.25 shows the effect of interaction of a continuous-wave laser with a spherical silica sphere. Very near the sphere (in the near-field), the laser energy is greatly enhanced in the forward direction, resulting in a 15-fold increase in the peak forward intensity just past the sphere itself. This effect of magnifying the laser intensity over a very small area can be used as a means to focus intense laser energy onto a substrate at nanometer scales.

### 9.5.1 Nanoscale surface modification for tailoring properties

Much contemporary research in radiative transfer is aimed at exploiting near-field effects. If a sphere is placed on or very near (within a few wavelengths) of a substrate, then the enhancement shown in Fig. 9.25 can be used to focus incident laser energy onto a very small area (smaller than the nanosphere itself.) Theppakuttai (2006) used this effect to imprint surface features on a silicon substrate (Fig. 9.26). Heltzel et al. (2005) and Heltzel (2006) compared the experimental results of Theppakuttai (2006) with predictions from using solutions of the near-field equations to provide boundary conditions for solving the conventional heat equation within the substrate. Predictions of the melt zone diameter agreed well with the observed values (Fig. 9.27). No damage was predicted or observed below a laser fluence of 50 mJ/cm2.

Figure 9.26 Scanning electron microscope (SEM) of surface features on a silicon substrate produced by irradiating 1.76 mm SiO2 spheres on the silicon surface with 532 nm nanosecond laser pulse at different laser fluencies (Theppakuttai, 2006).
Figure 9.27 Comparison of predicted and measured feature diameters on a silicon substrate produced by irradiating 1.76 mm SiO2 spheres on the silicon surface with 532 nm nanosecond laser pulse at different laser fluences (Heltzel et al., 2005)
Figure 9.28 Directional spectral emissivity from a grating surface on SiC. Grating lines are separated by approximqately 6.25 μm, and are about 1 μm in depth. Larger amplitude line: theory (300K). Smaller amplitude line: measurement (800K) (Courtesy of Prof. J.-J. Greffet)

Greffet et al. (2002) showed that nano-contoured surfaces can produce highly directionally and wavelength dependent surface properties. Figure 9.28 shows an example of the emissivity distribution that can be achieved.

### 9.5.2 Macroscale laser-surface interactions

High-power lasers are used in many manufacturing processes, including cutting, welding, brazing, surface texturing and surface cleaning. The thermal analysis of these processes requires treatment of the laser-surface radiative energy transfer coupled to a transient conduction analysis within the work piece. These interactions are treatable without near-field effects in many cases. For welding and brazing, the analysis is complicated by the presence of the moving boundary between the molten and solid material. In addition, a layer of metal vapors or ionized gases near the hot surface, called the Knudsen layer, may interfere with the incident laser energy, requiring analysis of radiative transfer through a participating medium. Rosenthal (1946) derived the first simplified analysis of the welding process, and reviews of recent modeling improvements are in Frewin and Scott (1999), who used finite element modeling with variable material properties for ultrafast laser welding. In these analyses, it is usually assumed that the laser beam for a continuous wave laser has a Gaussian shape for normal incidence of

${q_{absorbed}}(r) = C(1 - {\rho _\lambda }){q_{laser}}\exp ( - {r^2}/R_{laser}^2)\qquad \qquad( )$

(9.66)

where C accounts for interaction with the the Knudsen layer, ρ λ is the reflectivity of the surface, r is the radial beam position, Rlaser is the laser beam radius, and qlaser is the incident laser power. The value of C depends on the particular material being welded, the weld velocity v, and the laser power q laser . For a pulsed laser, the shape in space and time is given by

${q_{absorbed}}(r,t) = C(1 - {\rho _\lambda })F(t){q_{laser}}\exp ( - {r^2}/R_{laser}^2)\qquad \qquad( )$
(9.67)

where F(t) is the temporal pulse shape. The laser is generally not positioned at normal incidence for these processes to avoid reflections into the beam optics, so the shape of the incident beam is generally ellipsoidal rather than circular (Fig. 9.29). Rosenthal (1946) used an analytical solution to show the three-dimensional temperature profile present around a point source moving at velocity v on the surface of a thick material with thermal conductivity k and thermal diffusivity α . The temperature profile approaches a steady solution with the respect to the source position. The result is

$T(\xi ,y,z) = {T_o} + \frac{q}{{4\pi kR}}\exp \left( { - \frac{{v\xi }}{{2\alpha }}} \right)\exp \left( { - \frac{{vR}}{{2\alpha }}} \right)\qquad \qquad( )$

(9.68)

where

$R \equiv {\left( {{\xi ^2} + {y^2} + {z^2}} \right)^{1/2}}$

and  = vt.

Figure 9.29 Coordinate system for laser welding.

This quasi-steady solution is invalid near the x-boundaries of the welded material. Additional information on numerical modeling is in Rohsenow et al. (1998) and Han et al. (2005).

## 9.6 Light Pipes and Fiber Optics

Snell's Law as given by eq. (9.33) gives the angle of refraction for radiation traveling through medium 1 with refractive index n1 and encountering an interface with a second material with refractive index n2. Suppose that the radiation is traveling in glass or other transparent material and encounters a surface such as air with n2 < n1. In this case, Snell's Law predicts

$\sin \chi = (\frac{{{n_1}}}{{{n_2}}} > 1)\sin \theta \qquad \qquad( )$

(9.69)

Thus, unless the angle of incidence on the interface θ < sin − 1(n2 / n1) , the angle of refraction is predicted to be  > /2. Practically, whenever θ > sin − 1(n2 / n1), the radiation within the higher refractive index material will undergo 100 percent reflection at the interface. This means that once radiation enters a perfectly transmitting material (a perfect dielectric) such as an optical fiber in which no radiation absorption occurs, the radiation will propagate along the fiber with the perfect wall reflectivity not allowing any losses by refraction through the fiber surface. This phenomenon is observed in a planar geometry by swimmers (and fish), who can only see through the water surface above them within a cone of angles described by

$\theta < {\sin ^{ - 1}}\left( {\frac{{{n_{air}}}}{{{n_{water}}}}} \right) = {\sin ^{ - 1}}\left( {\frac{1}{{1.33}}} \right) = {\sin ^{ - 1}}\left( {0.75} \right) = {48.6^o}$

.

At greater angles, the water surface appears to be a mirror (see Fig. 9.30). For a light-pipe or fiber optic, radiation enters the flat end of a long circular cylinder. If the refractive index ratio n2 / n1 exceeds $\sqrt 2$, all radiation entering the light-pipe end will encounter the cylindrical surface at greater than the critical angle (Qu et al., 2007b). An example of the pattern of radiation transmitted through a light-pipe is shown in Fig. 9.31. Because of the increasing reflectivity of dielectrics with angle [eqs. (9.36)], the radiation actually entering the lightipipe end is from near-normal angles. Qu et al. (2007a, 2007b) have analyzed the errors that arise because of various signal loss mechanisms that can occur in a light-pipe, such as the presence of very minor surface imperfections, effects of blocking and shadowing of reflected energy from the surface being measured, changing the temperature of the measured object by radiative transfer to the probe itself, and any environmental radiation that may enter the light-pipe walls and enter the

Figure 9.30 Critical angle for view through the surface of a material with refractive index greater than surroundings.

(a) Experimental observation (b) Predicted pattern

internal signal path by encountering surface imperfections or scattering centers within the light-pipe. The signal entering the detector at the end of an LPRT or optical fiber consists of emitted plus reflected energy from the object being viewed. If the environment is cold relative to the viewed object so that reflected energy can be neglected, and the detector is only sensitive in a small wavelength range around a particular wavelengthλ, then the detected emission is, from eq. (9.15),

${E_\lambda }(\lambda ,{T_{act}}) = {\varepsilon _\lambda }{E_\lambda }(\lambda ,{T_{act}}) = {\varepsilon _\lambda }\frac{{2\pi {C_1}}}{{{\lambda ^5}\left[ {\exp \left( {\frac{{{C_2}}}{{\lambda {T_{act}}}}} \right) - 1} \right]}} \qquad \qquad( )$

(9.70)

However, rather than the actual temperature Tact , the detector reads an apparent temperature Tapp that would appear to originate from a blackbody,

${E_\lambda }(\lambda ,{T_{app}}) = \frac{{2\pi {C_1}}}{{{\lambda ^5}\left[ {\exp \left( {\frac{{{C_2}}}{{\lambda {T_{app}}}}} \right) - 1} \right]}} \qquad \qquad( )$
(9.71)

Equating the two emitted energy rates relates the actual and apparent temperatures as

${\varepsilon _\lambda }\frac{{2\pi {C_1}}}{{{\lambda ^5}\left[ {\exp \left( {\frac{{{C_2}}}{{\lambda {T_{act}}}}} \right) - 1} \right]}} = \frac{{2\pi {C_1}}}{{{\lambda ^5}\left[ {\exp \left( {\frac{{{C_2}}}{{\lambda {T_{app}}}}} \right) - 1} \right]}} \qquad \qquad( )$

(9.72)

For most engineering conditions, the factor of (-1) in the denominator can be neglected relative to the exponential term, and the actual temperature in terms of the apparent absolute temperature then becomes

${T_{act}} = \frac{{{T_{app}}}}{{\left[ {1 + \frac{{\lambda {T_{app}}}}{{{C_2}}}\ln {\varepsilon _\lambda }} \right]}} \qquad \qquad( )$

(9.73)

As ε λ → 1, the apparent and actual temperatures approach one another. eq. (9.75) can be used for many spectrally-based temperature measurements, remembering the proviso that the environmental temperature must be low in order get an accurate temperature measurement from an optical pyrometer or light-pipe radiation thermometer.

## 9.7 Infrared Sensing, Cameras and Photography

Some digital video sensors and cameras are constructed so that the charge-coupled detectors (CCDs) are sensitive at infrared wavelength ranges well outside the visible. Such equipment must have optical components that are highly transmitting in the infrared; most common glasses are opaque at these wavelengths. Cameras then detect signals that can be calibrated to be proportional to the temperature of locations on a viewed surface (if it is black; otherwise compensation must be made for reflected radiation from the surroundings), allowing scanning for hot spots in electronic equipment, heat leaks from structures to aid in energy conservation, and scanning of power lines to find current leakage. Military night-vision equipment is similar, but uses signal amplification to enhance the weak IR signals emitted by warm objects. Conventional film cameras can make use of film that is sensitive to radiation in the near-IR region. Because the reflectivity of many materials, especially foliage, varies strongly between the visible and near-IR region, interesting artistic effects can be obtained using IR film. The two photographs

shown in Fig. 9.32 of the same scene, one using conventional film and the other using IR film, illustrate how reflectivity can vary. 9.8 Other Contemporary Applications and Research Research on the radiative properties of materials and how they can be modified, tailored for a specific application, or measured for use in new applications is a continuing research field. At the macroscopic level, there is much interest in biomedical applications such as skin absorption, thermal imaging for cancer detection, monitoring of burn severity by observing changes in tissue reflectivity and others. Highly coupled problems involving radiation include spacecraft reentry and ablation determination; radiation from incandescent solid and porous surfaces in combustion systems; use of "ground truth" known properties to calibrate infrared satellite sensors used for gathering temperature data; the use of high-energy radiation to preheat materials during machining, particularly for ceramics; and the continuing need to gather and predict radiative properties in the design and analysis of infrared ovens, dryers, utility boilers, chemical processing equipment, and for high temperature manufacturing processes such as cutting, grinding, turning, joining and others. At the micro- and nanoscale level, research is ongoing in the use of near-field radiation phenomena in measuring various properties through the use of near-field radiative energy interactions, sub-wavelength microscopy, and micro-manufacturing of surface features using near-field energy enhancement to tailor selective directional and spectral radiative properties