# Absorptivity, Reflectivity, and Transmissivity

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1 cen54261_ch21.qxd 1/25/4 11:32 AM Page where f l1 and f l2 are blackbody functions corresponding to l 1 T and l 2 T. These functions are determined from Table 21 2 to be l 1 T (3 mm)(8 K) 24 mm K f l l 2 T (7 mm)(8 K) 56 mm K f l Note that f l1 f l1 f f l1, since f, and f l2 f f l2 1 f l2, since f 1. Substituting, e ( ).1(1.7146).521 That is, the surface will emit as much energy at 8 K as a g surface having a constant emissivity of e.521. The emissive power of the surface is E est 4.521( W/m 2 K 4 )(8 K) 4 12,1 W/m 2 Discussion Note that the surface emits 12.1 kj of energy per second per m 2 area of the surface., W/m 2 Reflected ρ Semitransparent material Absorbed α Absorptivity, Reflectivity, and Transmissivity Everything around us constantly emits, and the emissivity represents the emission characteristics of those bodies. This means that every body, including our own, is constantly bombarded by coming from all directions over a range of wavelengths. Recall that flux incident on a surface is called ir and is denoted by. When strikes a surface, part of it is absorbed, part of it is reflected, and the remaining part, if any, is transmitted, as illustrated in Fig The fraction of ir absorbed by the surface is called the absorptivity a, the fraction reflected by the surface is called the reflectivity r, and the fraction transmitted is called the transmissivity t. That is, Absorptivity: Absorbed a, abs a 1 (21 37) Reflectivity: Reflected r, ref r 1 (21 38) Transmissivity: Transmitted t tr, t 1 (21 39) Transmitted τ FIURE The absorption, reflection, and transmission of incident by a semitransparent material. where is the energy incident on the surface, and abs, ref, and tr are the absorbed, reflected, and transmitted portions of it, respectively. The first law of thermodynamics requires that the sum of the absorbed, reflected, and transmitted energy be equal to the incident. That is, Dividing each term of this relation by yields abs ref tr (21 4) For opaque surfaces, t, and thus a r t 1 (21 41)

2 cen54261_ch21.qxd 1/25/4 11:32 AM Page CHAPTER 21 a r 1 (21 42) This is an important property relation since it enables us to determine both the absorptivity and reflectivity of an opaque surface by measuring either of these properties. These definitions are for total hemispherical properties, since represents the flux incident on the surface from all directions over the hemispherical space and over all wavelengths. Thus, a, r, and t are the average properties of a medium for all directions and all wavelengths. However, like emissivity, these properties can also be defined for a specific wavelength and/or direction. For example, the spectral directional absorptivity and spectral directional reflectivity of a surface are defined, respectively, as the absorbed and reflected fractions of the intensity of incident at a specified wavelength in a specified direction as a l, u (l, u, f) I l, abs(l, u, f) I l, i (l, u, f) and r l, u (l, u, f) I l, ref(l, u, f) I l, i (l, u, f) (21 43) Likewise, the spectral hemispherical absorptivity and spectral hemispherical reflectivity of a surface are defined as l, abs (l) l, ref (l) a l (l) and r l (l) (21 44) where l is the spectral ir (in W/m 2 mm) incident on the surface, and l, abs and l, ref are the reflected and absorbed portions of it, respectively. Similar quantities can be defined for the transmissivity of semitransparent materials. For example, the spectral hemispherical transmissivity of a medium can be expressed as l, tr (l) t l (l) (21 45) The average absorptivity, reflectivity, and transmissivity of a surface can also be defined in terms of their spectral counterparts as a l l dl r l l dl t l l dl a, r, t (21 46) l dl l dl l dl The reflectivity differs somewhat from the other properties in that it is bidirectional in nature. That is, the value of the reflectivity of a surface depends not only on the direction of the incident but also the direction of reflection. Therefore, the reflected s of a beam incident on a real surface in a specified direction will form an irregular shape, as shown in Fig Such detailed reflectivity data do not exist for most surfaces, and even if they did, they would be of little value in calculations since this would usually add more complication to the analysis than it is worth. In practice, for simplicity, surfaces are assumed to reflect in a perfectly specular or diffuse manner. In specular (or mirrorlike) reflection, the angle of reflection equals the angle of incidence of the beam. In diffuse reflection, is reflected equally in all directions, as shown in Fig. (a) (b) (c) Reflected s Reflected s Reflected FIURE Different types of reflection from a surface: (a) actual or irregular, (b) diffuse, and (c) specular or mirrorlike.

3 cen54261_ch21.qxd 1/25/4 11:32 AM Page Absorptivity, α White fireclay 2. Asbestos 3. Cork 4. Wood 5. Porcelain 6. Concrete 7. Roof shingles 8. Aluminum 9. raphite Source temperature, K FIURE Variation of absorptivity with the temperature of the source of ir for various common materials at room temperature Reflection from smooth and polished surfaces approximates specular reflection, whereas reflection from rough surfaces approximates diffuse reflection. In analysis, smoothness is defined relative to wavelength. A surface is said to be smooth if the height of the surface roughness is much smaller than the wavelength of the incident. Unlike emissivity, the absorptivity of a material is practically independent of surface temperature. However, the absorptivity depends strongly on the temperature of the source at which the incident is originating. This is also evident from Fig , which shows the absorptivities of various materials at room temperature as functions of the temperature of the source. For example, the absorptivity of the concrete roof of a house is about.6 for solar (source temperature: 578 K) and.9 for originating from the surrounding trees and buildings (source temperature: 3 K), as illustrated in Fig Notice that the absorptivity of aluminum increases with the source temperature, a characteristic for metals, and the absorptivity of electric nonconductors, in general, decreases with temperature. This decrease is most pronounced for surfaces that appear white to the eye. For example, the absorptivity of a white painted surface is low for solar, but it is rather high for infrared. α =.9 Sun α =.6 Kirchhoff s Law Consider a small body of surface area A s, emissivity e, and absorptivity a at temperature T contained in a large isothermal enclosure at the same temperature, as shown in Fig Recall that a large isothermal enclosure forms a blackbody cavity regardless of the radiative properties of the enclosure surface, and the body in the enclosure is too small to interfere with the blackbody nature of the cavity. Therefore, the incident on any part of the surface of the small body is equal to the emitted by a blackbody at temperature T. That is, E b (T ) st 4, and the absorbed by the small body per unit of its surface area is abs a ast 4 The emitted by the small body is FIURE The absorptivity of a material may be quite different for originating from sources at different temperatures. E emit est 4 Considering that the small body is in thermal equilibrium with the enclosure, the net rate of heat transfer to the body must be zero. Therefore, the emitted by the body must be equal to the absorbed by it: A s est 4 A s ast 4 Thus, we conclude that e(t ) a(t ) (21 47) That is, the total hemispherical emissivity of a surface at temperature T is equal to its total hemispherical absorptivity for coming from a blackbody at the same temperature. This relation, which greatly simplifies the analysis, was first developed by ustav Kirchhoff in 186 and is now called Kirchhoff s law. Note that this relation is derived under the condition

4 cen54261_ch21.qxd 1/25/4 11:32 AM Page 973 that the surface temperature is equal to the temperature of the source of ir, and the reader is cautioned against using it when considerable difference (more than a few hundred degrees) exists between the surface temperature and the temperature of the source of ir. The derivation above can also be repeated for at a specified wavelength to obtain the spectral form of Kirchhoff s law: e l (T ) a l (T ) (21 48) This relation is valid when the ir or the emitted is independent of direction. The form of Kirchhoff s law that involves no restrictions is the spectral directional form expressed as e l, u (T ) a l, u (T ). That is, the emissivity of a surface at a specified wavelength, direction, and temperature is always equal to its absorptivity at the same wavelength, direction, and temperature. It is very tempting to use Kirchhoff s law in analysis since the relation e a together with r 1 a enables us to determine all three properties of an opaque surface from a knowledge of only one property. Although Eq gives acceptable results in most cases, in practice, care should be exercised when there is considerable difference between the surface temperature and the temperature of the source of incident. The reenhouse Effect You have probably noticed that when you leave your car under direct sunlight on a sunny day, the interior of the car gets much warmer than the air outside, and you may have wondered why the car acts like a heat trap. The answer lies in the spectral transmissivity curve of the glass, which resembles an inverted U, as shown in Fig We observe from this figure that glass at thicknesses encountered in practice transmits over 9 percent of in the visible range and is practically opaque (nontransparent) to in the longer-wavelength infrared regions of the electromagnetic spectrum (roughly l 3 mm). Therefore, glass has a transparent window in the wavelength range.3 mm l 3 mm in which over 9 percent of solar is emitted. On the other hand, the entire emitted by surfaces at room temperature falls in the infrared region. Consequently, glass allows the solar to enter but does not allow the infrared from the interior surfaces to escape. This causes a rise in the interior temperature as a result of the energy buildup in the car. This heating effect, which is due to the nong characteristic of glass (or clear plastics), is known as the greenhouse effect, since it is utilized primarily in greenhouses (Fig ). The greenhouse effect is also experienced on a larger scale on earth. The surface of the earth, which warms up during the day as a result of the absorption of solar energy, cools down at night by radiating its energy into deep space as infrared. The combustion gases such as CO 2 and water vapor in the atmosphere transmit the bulk of the solar but absorb the infrared emitted by the surface of the earth. Thus, there is concern that the energy trapped on earth will eventually cause global warming and thus drastic changes in weather patterns. In humid places such as coastal areas, there is not a large change between the daytime and nighttime temperatures, because the humidity acts as a barrier 973 CHAPTER 21 T A s, εα, FIURE The small body contained in a large isothermal enclosure used in the development of Kirchhoff s law τ λ.4.2 Visible T E emit Thickness.38 cm.318 cm.635 cm Wavelength λ, µ m FIURE The spectral transmissivity of low-iron glass at room temperature for different thicknesses. Solar reenhouse Infrared FIURE A greenhouse traps energy by allowing the solar to come in but not allowing the infrared to go out.

5 cen54261_ch21.qxd 1/25/4 11:32 AM Page Earth s surface = s cos Earth s atmosphere n^ Sun s s s, W/m 2 FIURE Solar reaching the earth s atmosphere and the total solar irradiance. on the path of the infrared coming from the earth, and thus slows down the cooling process at night. In areas with clear skies such as deserts, there is a large swing between the daytime and nighttime temperatures because of the absence of such barriers for infrared ATMOSPHERIC AND SOLAR RADIATION The sun is our primary source of energy. The energy coming off the sun, called solar energy, reaches us in the form of electromagnetic waves after experiencing considerable interactions with the atmosphere. The energy emitted or reflected by the constituents of the atmosphere form the atmospheric. Here we give an overview of the solar and atmospheric because of their importance and relevance to daily life. Also, our familiarity with solar energy makes it an effective tool in developing a better understanding for some of the new concepts introduced earlier. Detailed treatment of this exciting subject can be found in numerous books devoted to this topic. The sun is a nearly spherical body that has a diameter of D m and a mass of m kg and is located at a mean distance of L m from the earth. It emits energy continuously at a rate of E sun W. Less than a billionth of this energy (about W) strikes the earth, which is sufficient to keep the earth warm and to maintain life through the photosynthesis process. The energy of the sun is due to the continuous fusion reaction during which two hydrogen atoms fuse to form one atom of helium. Therefore, the sun is essentially a nuclear reactor, with temperatures as high as 4,, K in its core region. The temperature drops to about 58 K in the outer region of the sun, called the convective zone, as a result of the dissipation of this energy by. The solar energy reaching the earth s atmosphere is called the total solar irradiance s, whose value is s 1373 W/m 2 (21 49) Sun 1m 2 r E b = σt 4 sun s 1m 2 (4 πr 2 )E b Earth FIURE The total solar energy passing through concentric spheres remains constant, but the energy falling per unit area decreases with increasing radius. L (4 πl 2 ) s The total solar irradiance (also called the solar constant) represents the rate at which solar energy is incident on a surface normal to the sun s s at the outer edge of the atmosphere when the earth is at its mean distance from the sun (Fig ). The value of the total solar irradiance can be used to estimate the effective surface temperature of the sun from the requirement that (4pL 2 ) s (4pr 2 ) st 4 sun (21 5) where L is the mean distance between the sun s center and the earth and r is the radius of the sun. The left-hand side of this equation represents the total solar energy passing through a spherical surface whose radius is the mean earth sun distance, and the right-hand side represents the total energy that leaves the sun s outer surface. The conservation of energy principle requires that these two quantities be equal to each other, since the solar energy experiences no attenuation (or enhancement) on its way through the vacuum (Fig ). The effective surface temperature of the sun is determined from Eq to be T sun 578 K. That is, the sun can be treated as a

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