110 Basic Radiation

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11Basic Relations of Radiation11.1 Therm al R adiation

Thermal radiation is the energy emitted by bodies because oftheir tem perature level. Other types of radiation include gamm a rays, x-rays, fo r instance. Radiation is often treated as electromagnetic wavesthat propagate according to Maxwell's classic electromagnetic theory.Radiation may also be treated as photons as prescribed in Max Planck'sconcept of the qua ntum of energy. The electromagnetic theory has beenused to predict the radiant properties of materials, while the quantumtheory h as been used to p redict the am oun t of radiant energy emitted by abody because of its level of temp erature.

Infrared-0.7- 1000urn Visib le -0.4 - 0.7 urn

^ wi

^ w Ultraviolet~0.4 - 10"2um

10" 1012 1013 1014 1015 1016 10 ' 1018 Frequency, s"1

10 4 10' 102 10' 10" lO'1 10'2 10'3 10"4 Wavelength, urn

\Solar radiation -0.1-3urnThermal rad.- 0.1 - 100 urn

Figure 11.1 Electromagnetic wave spectrum.

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In Fig. 11.1, is shown a large range of the electromagnetic-wavespectrum. In theory, electromagnet ic waves of zero to inf ini tywavelengths have thermal radiant energy. In practice, a big portion ofthe thermal radia t ion l ies in the range from about 0.1 to 100 u,m. Thisportion is labeled as such in the figure. The visib le range is from 0.4 to0.7 urn; i t is im po rtan t to the extent that i t tells the scholars of heattransfer to use their eyes to obtain insight into the thermal radiationphenom enon . W hen r ad ia t ion is considered an electromagnetic wave, it stransport in a medium takes place with the speed of light, c. Thewavelength X and the f r equency / are related to the speed of l ight by c= / . When thermal radiat ion t ravels in a vacuum, for instance, formost of the distance between the sun and the earth, the speed of l ight is2.9979 x 108 m /s. Th is speed is attenuated by the atm ospheresu r round ing the earth.11.2 Rad ia t ion In tens i ty and Blackbody

dA

Figure 11.2 Notation for radiation intensity.

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contained in a solid angle over an entire hemisphere is obtained by theintegration of Eq. (1 1.2) asq= /cos6k/Q (11.3)where th e symbol indicates the integration with respect to a solidangle over an ent i re hemisphere . A s shown in Fig. 11.2, 0 is the polarangle and 9 is the azim uth al ang le. Since dQ . = s i n G d O d c p , Eq. (1 1.3) m aybe written as

(11.4)The d ime n s io n s of q are energy per unit t im e, per unit area of the surface(e.g., kJ/h.m 2 .)

There is a ma x imu m a mo u nt of radiant energy emitted by a bodyat a given absolute temperature T at a wavelength A, . This maximumamount of radiant emission is the spectral blackbody radiation intensityUb(T); th e emitter of such radiation is named a blackbody. This spectralblackbody radiation intensity is independent of direction. For ablackbody at an absolute temperature T and emitting radiative energyinto a v a c u u m, Ixb(T) is calculated from the relation given by Planck,1 959 [1], in the form

where h ( = 6.6256 x 10'34 J.s) and k ( = 1.38054 x 10'23 J.K) are thePlanck and Boltzmann constants, respectively, T is the absolutetemperature and c is the speed of light in a vacuum.

For engineering practice, the spectral blackbody emissive fluxq^b(T) at a surface is defined as

(11.6)

A s Ixb(T) is independent of direction,

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(11.7)The quantity in Eq. (1 1.7) is the a m oun t of radiative en ergy e m itted by ablackbody at tem perature T per un it of i ts surface, per unit t im e, per unitwavelength in all directions in the hem ispherical space. Substituting Eq.( 1 1 . 5 ) i n t o E q . ( 1 1 . 7 ) ,

(11-8)

where q^,(T} is the spectral blackbody emissive flux as the surface(W/m 2 .um),

d = 27ihc2 = 3.743 x 108 W .u m 4/m 2c 2 = hc/k= 1.4387 x!04um.K.Figure 1 1 .3 is a plot of the spectral b lackb od y em issive flux as afunction of wa velength at various tempe ratures. From this figure, it isclear that at any given wavelength, the radiative energy emitted by ablackbody increases as the absolute temperature of the body increases.

Each curve displays a peak, and the peaks shift toward smallerwavelengths as the temperature rises. The locus of the peaks calculatedanalytically by Wien's displacement rule isWmax = 0.28976 cm .K = 28997.6 /fln.K (11.9)The blackbody radiation intensity Ib(T) is found by the integration ofIM (T ) over the wavelengths ranging from 0 to oo.

( i i . i o )Put Eq .(l 1 .5) into Eq .(l 1.10) and integrate,

Ib(T ) = (11 .11)awh ere the Stefan-Boltzm ann constant cr = 5.6697 x 10"8 W/(m 2 .K 4).

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nThe generalized idea of a blackbody is one that possesses the

characteristic of allowing all incident radiation to enter th e mediumwithout surface reflection and without allowing it to leave th e mediumagain. A blackbody absorbs all incident radiation from all directions atall frequen cies without reflecting, transmitting, or scattering it outwards.The blackbody emits as much radiative energy as it absorbs, if it is atthermal equilibrium with th e enclosure walls. For practical purposes, acavity such as a hollow sphere whose interior surfaces are kept at auniform temperature T can be used to approximate a blackbody. If avery tiny hole (compared to the cavity) is made, any radiation enteringthe cavity through th e hole is almost entirely absorbed since it has verylittle possibility to escape through the hole. Such a cavity is consideredan approximate blackbody. By a similar argument, radiation leaving th ecavity through th e hole is considered almost a blackbody radiation attemperature T.11.3 Reflectivity, Absorptivity, Emissivity and TransmissivityReal Surfaces

Consider a beam of radiant energy incident on a real surface.Part of this radiation is reflected, part of it is absorbed and the rest istransmitted. Let l% be the spectral radiation intensity incident on thesurface. The spectral radiant heat flux incident on the surface can beexpressed asq ^ = [ /^ cos 9* dQ ? energy/( time x area x wavelength ) (11.14)w here 0' is the polar an gle between the direction of the incident radiationand the normal to the surface. The spectral h em ispherical reflectivity pxis defined as

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_ radiant energy reflected/( time x area x wavelength )P x ~ V * . (11 .15 )The spectral he m isp he rical abso rptivity ax is defined as

radiant energy absorbed/( tim e x area x wavelength )a^ =-.^ A (11.16)

For an opaque surface, the relationship between the spectralhemispher ica l reflectivity and the spectral hem isphe rical absorptivity isP x + a x = l . (11 .17)For m u c h of engineering practice, the reflectivity and the absorptivity,averaged over th e entire wavelengths, is of relevance. W hen this is done,the resu lt ing hem isph erical reflect ivi ty p and the hem isphericalabsorpt ivi ty a are defined as fo l lows:-

( 1 U8 )

(11.19)

For an op aque surface,p + a = l . (1 1.2 0 )

The radiant energy emit ted by a real surface at an absolutetemperature T is always less than that emitted by a blackbody surface atthe same temperature . Let qx(T) be the spectral emissive flux from a realsurface at an ab solute tem perature T and q^b(T) be the spectral blackbody

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em issive heat flux for a blackbody surface at the same temperature. Thehem ispherical emissivity ex of the surface is defined as

= (11.21)The hemispherical emissivity e over the entire range of wavelengths isfound by

where q(T) and qb(T) are the emissive fluxes from the real surface attemp erature T , and the blackbody at tem perature T, respectively.

Incidentradiation Reflected

Absorbed

TransmittedFigure 1 1 .4. Inciden t radiation on a translucent body.When radiation is incident on a translucent body, part of theincident radiation is reflected, part is absorbed, and the remainder istransm itted through the translucent body (Fig. 11.4). A n exam ple of atranslucent body is a pane of glass. The relationsh ip between the spectralreflectivity px, the spectral absorptivity ax and the spectral transmissivity

T X of the translucent body is(11.23)

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When these radiative properties are averaged over all wavelengths, w egetp + a + x = l . (1 1.2 4)

The reflectivity, absorptivity and t ransmissivi ty of a t ranslucentbody depend in large part on the surface conditions, th e wavelength ofthe radiation, the composit ion of the mater ia l and the thickness of thebody. Since the a ttenuation of radiat ion w ithin a body should beanalyzed as a bulk process, th e evaluation of the reflectivity andtransmissivity of a translucent object is more involved.Graybody

For simplici ty , the graybody assumption is used in manyapplications. The radiative properties px, c^ , e^ and T X are assumed to beuniform over the entire wavelength spectrum. In other words,graybo dies have radia tive properties p, a , e and T that are ind epen den t ofwavelength .11.4 Kirchhoff s Law of Radiat ion

The absorptivity and the emissivi ty of a body can be related byKirchhoff s law of radiation, Planck, 1959 [1]. Consider a body inside ablack, closed container whose walls are kept at a uniform absolutetemperature T and has reached thermal equilibrium with the walls of thecontainer. If flux q^ (T) is the spectral radiative heat flux from the wallsat temperature T incident on the body and a^(T) is the spectralabsorptivity of the body, then the spectral radiative heat flux q^(T)absorbed by the body at the wavelength A , isqi(T) = a,(T)q\(T). (11-25)Since the body is in radiative equi l ibr ium, qx(T) also expresses th espectral radiative flux emitted by the body at the wav elength X. Theincident radiation q\(T) comes from th e black walls of the enclosure attemperature T, and the emission by the walls is not influenced by thebody regardless if it is a blackbody or not. Let qxb(T) be the spectralblackbody emissive flux at temperature T. Then,

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(11.26)From Eqs. (1 1.25) and (1 1.26),

(11.27)

The spectral emissivity ex(T) of the body for radiation at temperature T isdefined as the ratio of the spectral emissive flux qx(T) of the body to thespectral blackbody emissive flux qxb(T) at the same temperature.Expressed m athem atical ly,

(11.28)

From Eqs. (1 1 .27) and (1 1 .28), it can be deduced thatex(T) = ax(T). (11.29)Equation (1 1 .29) is Ki rchhof f s law of radiat ion. The law states that th espectral emissivi ty for the emiss ion of radiat ion at temperature T is equalto the spectral absorptivity for radiat ion from a blackbody at the sametemp erature T. The relat ione(T) = a(T) (11.30)holds only if the inciden t and em itted radiation have the sam e spectraldistribution or when th e body is gray. This later characteristic is onewhe re the radiat ive propert ies are independe nt of wavelength.PROBLEMS11.1. The average internal tem perature of an oven is 1500C, and theemissivi ty of the internal surface is e = 0.9 at this temperature.Calculate the radiant energy coming from the oven through anopening 10 cm by 10 cm.

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11.2. A blackbody enclosure at 1000C has a small aperture into theenvironment . Determine (i) the blackbody radiation intensityemerging from the aperture, and ( i i ) the blackbody radiation heatflux from the blackbody.11.3. The surface of an outer space station receives solar radiation at arate of 1.2 k W / m 2 . The surface has an absorptivity of a = 0.75fo r solar radiation and an emissivi ty of e = 0.86. There are no

heat losses into the space station. How ever, heat is dissipated bythermal radiat ion into the space at absolute zero. Determine theequi l ib r ium temperature of the surface.11.4. A solar collector surface receives solar radiation at 1 kW /m 2 , and

its other side is insula ted . The absorptivity of the surface to solarradiat ion is a = 0.8 whi l e its emiss iv i ty is e = 0.6 . A ssum ing thesurface loses heat by radiation into a clear sky at an effectivetempera ture of 10C, calculate the temperature of the surface.R E FE R E N CE S

1. M Planck. The Theory of Heat Radiat ion. New York: DoverPublications , 1959.

Blackbody and GraybodyA blackbod y absorbs all incide nt radiat ionAt a l l f requencies and from all different directionsNo ph en om en a of ref lec ting, t ransm i t t ing or scat ter ing

It is emi t t i ng as m u c h as it is absorbing.A t a n y con di t ion s , graybody has uni form propert iesThey are no t dep en den t on other propert iesR adia t ive propert ies are uni form over all wavelengthsGraybody has proper t ies independent of wavelength .

K . V . W o n g

110 Basic Radiation

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