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Unformatted text preview: RADIATION HEAT
TRANSFER I 81 INTRODUCTION Preceding chapters have shown how conduction and convection heat transfer
may be calculated with the aid of both mathematical analysis and empirical
data. We now wish to consider the third mode of heat transfer—thermal ra
diation. Thermal radiation is that electromagnetic radiation emitted by a body
as a result of its temperature. In this chapter, we shall ﬁrst describe the nature
of thermal radiaton, its characteristics, and the properties which are used to
describe materials insofar as the radiation is concerned. Next, the transfer of
radiation through space will be considered. Finally, the overall problem of heat
transfer by thermal radiation will be analyzed, including the influence of the
material properties and the geometric arrangement of the bodies on the total
energy which may be exchanged. I 82 PHYSICAL MECHANISM There are many types of electromagnetic radiation; thermal radiation is only
one. Regardless of the type of radiation, We say that it is propagated at the
speed oflight, 3 X 108 m/s. This speed is equal to the product of the wavelength
and frequency of the radiation, C=/\I/ where c 2 speed of light
A : wavelength
:2 = frequency The unit for A may be centimeters, angstroms (l A : 10‘8 cm), or micrometers
(1 urn : 10‘6 m). A portion of the electromagnetic spectrum is shown in Fig.
81. Thermal radiation lies in the range from about 0.1 to 100 um, while the Thermal i radiation
lulu 1A
logicm i l a
3 2 l 0 —1 —2 —3 74 .5 —5 77 ms 79 40711 71H Ultra+1 .Y “0,5 Infrared violet Visible / Radio
waves Fig. 81 Electromagnetic spectrum. visiblelight portion of the spectrum is ver
to 0.75 ,um. The propagation of thermal radiation takes place in the form of discrete
quanta, each quantum having an energy of y narrow, extending from about 0.35 E : it!) (81)
where n is Planck‘s constant and has the value Ii = 6.625 x 10‘34J ‘ s A very rough physical picture of the radiation
considering each quantum as a particle havin
just as we considered the molecules ofa gas. be thought of as a “photon gas” which may ﬂow from one place to another.
Using the relativistic relation between mass and energy‘ expressions for the
mass and momentum of the “particles” could thus be derived; viz, propagation may be obtained by
g energy, mass, and momentum,
So, in a sense. the radiation might E: me2 = hv [11/ m = — I5.2
Irv hr)
Momentum = 0* : —
C2 0 By considering the radiation as such a gas, the principles of quantum~statistical thermodynamics can be applied to derive an expression for the energy density
of radiation per unit volume and per unit wavelength as 87r/1t‘A’5
“A : gilt/ART _ 1 (82) where l; is Boltzmann’s constant, l.38066 x 10‘23 J/molecule  K. When the
energy density is integrated over all wavelengths. the total energy emitted is
proportional to absolute temperature to the fourth power: Eb 2 0T4 (8—3) Equation (83} is called the StefanBoltzmann law. 5,, is the energy radiated
per unit time and per unit area by the ideal radiator, and cr is the Stefan
Boltzmann constant, which has the value a = 5.669 X 10‘8 W/m2  K4 [0.1714 X 10’8 Btufh  ft2  °R4l where E,, is in watts per square meter and T is in degrees Kelvin. In the
thermodynamic analysis the energy density is related to the energy radiated
from a surface per unit time and per unit area. Thus the heated interior surface
of an enclosure produces a certain energy density of thermal radiation in the
enclosure. We are interested in radiant exchange with surfaces—hence the
reason for the expression ofradiation from a surface in terms ofits temperature.
The subscript b in Eq. (83) denotes that this is the radiation from a blackbody.
We call this bloc/(body radiation because materials which obey this law appear
black to the eye; they appear black because they do not reflect any radiation.
Thus a blackbody is also considered as one which absorbs all radiation incident
upon it. Eb is called the emissive pom'er of a blackbody. It is important to note at this point that the “blackness” of a surfaceto
thermal radiation can be quite deceiving insofar as visual observations are
concerned. A surface coated with lampblack appears black to the eye and turns
out to be black for the thermalradiation spectrum. On the other hand, snow
and ice appear quite bright to the eye but are essentially “black” for long
wavelength thermal radiation. Many white paints are also essentially black for
longwavelength radiation. This point will be discussed further in later sections. 83 RADIATION PROPERTIES When radiant energy strikes a material surface, part ofthe radiation is reflected,
part is absorbed. and part is transmitted. as shown in Fig. 82. We deﬁne the
reflectivity p as the fraction reﬂected, the absorptivity a as the fraction ab
sorbed, and the transmissivity r as the fraction transmitted. Thus p+a+r=l (84) Most solid bodies do not transmit thermal radiation. so that for many applied
problems the transmissivity may be taken as zero. Then p+a=l Two types of reflection phenomena may be observed when radiation strikes
a surface. If the angle of incidence is equal to the angle of reflection. the
reﬂection is called specular. On the other hand. when an incident beam is
distributed uniformly in all directions after reﬂection, the reﬂection is called incident radiation Reﬂection Absorbed
Fig. 82 Sketch showing effects of incident radia
Transmitied tlon. diffitse. These two types of reﬂection are depicted in Fig. 83. Note that a
specular reﬂection presents a mirror image of the source to the observer. No
real surface is either specular or diffuse. An ordinary mirror is quite specular
for visible light. but would not necessarily be specular over the entire wave
length range of thermal radiation. Ordinarily. a rough surface exhibits diffuse
behavior better than a highly polished surface. Similarly, a polished surface is
more specular than a rough surface. The inﬂuence of surface roughness on
thermalradiation properties of materials is a matter of serious concern and
remains a subject for continuing research. The emissive povver of a body E is deﬁned as the energy emitted by the
body per unit area and per unit time. One may perform a thought experiment
to establish a relation between the emissive power ofa body and the material
properties deﬁned above. Assume that a perfectly black enclosure is available.
i.e.._ one which absorbs all the incident radiation falling upon it. as shown
schematically in Fig. 84. This enclosure will also emit radiation according to
the 1"4 law. Let the radiant ﬂux arriving at some area in the enclosure be
171, W/mz. Now suppose that a body is placed inside the enclosure and allowed
to come into temperature equilibrium with it. At equilibrium the energy ab—
sorbed by the body must be equal to the energy emitted: otherwise there would Sour 'c
Source .. L Reﬂected
rays ‘V/ Mirror image
\ .
of source tut U'u Fig. 8~3 (a) Specular (gm : dig} and (b) diffuse reflection. Black
enclosure Fig. 84 Sketch showing model used for derive
lng Kirchhott’s law. be an energy ﬂow into or out of the body which would raise or lower its
temperature. At equilibrium we may write EA = than: (85) If we now replace the body in the enclosure with a blackbody of the same size
and shape and allow it to come to equilibrium with the enclosure at the same
temperature, Em = q,rA(1) (86) since the absorptivity of a blackbody is unity. If Eq. {85) is divided by Eq.
(86), E
‘ : a
El: and we ﬁnd that the ratio of the emissive power of a body to the emissive
power of a blackbody at the same temperature is equal to the absorptivity of
the body. This ratio is deﬁned as the emissivity e of the body. E
: _ 7
E,, [8 ) 50 that e : or {88} Equation (88) is called Kirchhoff‘s identity. At this point we note that the
emissivities and absorptivities which have been discussed are the total prop
erties of the particular material; i.e.. they represent the integrated behavior of
the material over all wavelengths. Real substances emit less radiation than ideal
black surfaces as measured by the emissivity of the material. In reality. the
emissivity of a material varies with temperature and the wavelength of the
radiation. g The Gray Body A gray body is deﬁned such that the monochromatic emissivity e), of the body
is independent of wavelength. The monochromatic emissivity is deﬁned as the
ratio of the monochromaticemissive power of the body to the monochromatic
emissive power of a blackbody at the same wavelength and temperature. Thus a5
“We... The total emissivity of the body may be related to the monochromatic emissivity
by noting that E = 6, EM (M and Eb = I: am (a = or
E J; EA EbA SO that 6 I E; = T (89) where Em is the emissive power of a blackbody per unit wavelength. If the
graybody condition is imposed, that is, 5,, : constant, Eq. (8—9) reduces to (810) 6.1 The emissivities of various substances vary widely with wavelength, tem
perature, and surface condition. Some typical values of the total emissivity of
various surfaces are given in Appendix A. A very complete survey of radiation
properties is given in Ref. 14. The functional relation for EM was derived by Planck by introducing the
quantum concept for electromagnetic energy. The derivation is now usually
performed by methods of statistical thermodynamics. and Em is Shown to be
related to the energy density of Eq. (82) by the Em : ‘4— (311)
CIA—5
01' Em = (812)
where )t I wavelength, urn
T = temperature, K
C. 2 3.743 X 10E W  nm‘sz [1.187 x 108 Btu  ,Ltm‘Vh  ftz]
C3 : 1.4387 x 104 ,um  K [2.5896 X 10“ am  °R] A plot of EM as a function of temperature and wavelength is given in Fig. 8
5a. Notice that the peak of the curve is shifted to the shorter wavelengths for
the higher temperatures. These maximum points in the radiation curves are
related by Wien’s displacement law, AmxT : 2897.6 om  K {5215.6 am A °R] (813) 350— I;
 3“: 10
300 i g
5
:50 a
_ rs
i ‘3
~'_ 300 — .5
5 a 6
3 9
x 3
 a
130 — i; 4
U
100% g
a
E
z 2
50v— ‘2’
3
2
0L
0
Wavelength Ruum
(a)
1:
350
r a
1 ID
300 ~— +
g
zsol— § 3
E x
31 Vz<
"'2 :00 k “j.
5: ‘J 6
2 L :5; 51 = e = 0.6 [Gray body}
HO 5 ‘ 4 L
IDOL ' Real surface
a 1‘
L E 3 Ix VI
50 g ' v‘.
5 I
2 V"
G _ .1 I! I
0 1 2 3 4 5 6 Wavelengih 7M ,um
(h)
Fig. 8—5 (a) Blackbody emisswe power as a functron of wave‘ength and temperature: (0) companson of emissive power of IdeaF bfackbodies and
gray bodies with that of a real surface. Figure 85!) indicates the relative radiation spectra from a blackbody at 3000°F
and a corresponding ideal gray body with emissivity equal to 0.6. Also shown
15 a curve indicating an approximate behavior for a real surface, which may
differ considerably from that of either an ideal blackbody or an ideal gray body.
For analysis purposes surfaces are usually considered as gray bodies, with
emissivities taken as the integrated average value. The shift in the maximum point of the radiation curve explains the change
in color of a body as it is heated. Since the band of wavelengths visible to the
eye lies between about 0.3 and 0.7 am. only a very small portion of the radiant
energy spectrum at low temperatures is detected by the eye. As the body is
heated, the maximum intensity is shifted to the shorter wavelengths, and the
ﬁrst visible sign of the increase in temperature of the body is a darkred color.
With further increase in temperature, the color appears as a bright red, then
bright yellow, and ﬁnally white. The material also appears much brighter at
higher temperatures because a larger portion of the total radiation falls within
the visible range. I 84 RADIATION SHAPE FACTOR
Consider two black surfaces A, and A;. as shown in Fig. 88. We wish to obtain
a general expression for the energy exchange between these surfaces when
they are maintained at different temperatures. The problem becomes essentially
one of determining the amount ofenergy which leaves one surface and reaches
the other. To solve this problem the radiation s/tripefucrors are deﬁned as Fl, 3 : fraction of energy leaving surface 1 which reaches surface 2
F34 : fraction of energy leaving surface 2 which reaches surface l
Fm,” : fraction of energy leaving surface or which reaches surface it Other names for the radiation shape factor are vicu'firtmr. angle 1mm; and
r'tni/igriruti'mrfactor. The energy leaving surface 1 and arriving at surface 2 is EMA It‘ll anal13 . .
=u0§¢ti c056); H1 (1;! "1:3! dql _ 1 —nct Fig. 8—5 Sketch snowing area elements used in deriving radiation
shape factor. appearance
mductor: (b) ’aces are a
1gs. These
ncident ra—
Nith wave
nd surface
igs. Let us
‘ll"‘"T area.
i as the
he surface, (823} = constant.
“5 law [Eq. (824) 5 the result
wavelength
ity and ab
zy must be
not alw‘dyS
graybody
eelsurface
gray—body
10. ll. and
h gray and Relations between shape factors 393  EXAMPLE 82 Heat transfer between btack surfaces Two parallel black plates 0.5 by 1.0 m are spaced 0.5 m apart. One plate is maintained at 1000°C and the other at 500°C. What is the net radiant heat exchange between the
two plates? Soiuticm The ratios for use with Fig. 812 are r 0.5 X
—:—~=I.0 —
D D 0.5 so that FI2 = 0.285. The heat transfer is calculated from (1" = AiFt1(Em * Em) 2 UAIFIZ(TI4 * T14] = (5.669 X 10’“)(0.5)(0.285)(1273“‘ — 7734)
: t8.33 kW [62.540 Btu/h] I 85 RELATIONS BETWEEN SHAPE FACTORS Some useful relations between shape factors may be obtained by considering the system showu in Fig. 819. Suppose that the shape factor for radiation from A3 to the combined area AM is desired. This shape factor must be given very
simply as F3—.2 = F37! 'l’ Fs—z (825) i.e.. the total shape factor is the sum of its parts. We could also write Eq.
(825) as (8—26] AsFa—m =A3F3—1+A3F3—2
A.1F1.2—3 :AlFl—3 “42573 Fig. 819 Sketch showing some relations be
tween shape factors. and that energy leaving surface 2 and arriving at surface I is
E112A2F21 Since the surfaces are black, all the incident radiation will be absorbed, and
the net energy exchange is EaiAIFiz ’ EhZAZFZl = Qli'l If both surfaces are at the same temperature, there can be no heat exchange,
that is, Q,_3 = 0. Also Eh] I Era:
so that AlFlg = AZFE. (818)
The net heat exchange is therefore
12 = A.F.3(E;,! — E“) = A3F21(E,,, — EM) (819) Equation (8—18) is known as a reciprocity relation, and it applies in a general
way for any two surfaces m and n: AUJ'FHHI = AHFIHH Although the relation is derived for black surfaccs. it holds for other surfaces
also as long as diffuse radiation is involved. We now wish to determine a general relation for F]; (or Fgl). To do this, we
consider the elements of area (M. and dA; in Fig. 88. The angles d). and (ii; are measured between a normal to the surface and the line drawn between the
area elements r. ' LO , ., @5454!
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UQHI’IIIII_IIIIII AI’IIAIIIIII IIII 0.] 0.15 0,: 0.3 0.5 1.0 l5 1 3 30
Ratio XtD 0‘05 \ FiQ 312 Radiation shape factor for radiation between parallel rectangles Rano do FlQ. 813 Radiation shape factor for radiation between parallel
disks. ...
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 Fall '08
 Schetz,J

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