radiation - RADIATION HEAT TRANSFER I 8-1 INTRODUCTION...

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 10
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: RADIATION HEAT TRANSFER I 8-1 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 8-2 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. 8-1. 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 7-4 -.5 —5 77 ms 79 40711 71H Ultra-+1 .Y “0,5 Infrared violet Visible / Radio waves Fig. 8-1 Electromagnetic spectrum. visible-light 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!) (8-1) 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 (8-2) 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 (8-3} is called the Stefan-Boltzmann 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. (8-3) 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 surface-to- 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 thermal-radiation 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 long-wavelength radiation. This point will be discussed further in later sections. 8-3 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. 8-2. 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 (8-4) 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. 8-2 Sketch showing effects of incident radia- Transmitied tlon. diffitse. These two types of reflection are depicted in Fig. 8-3. 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 thermal-radiation 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. 8-4. 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. 8-4 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: (8-5) 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) (8-6) since the absorptivity of a blackbody is unity. If Eq. {85) is divided by Eq. (8-6), 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 {8-8} 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 monochromatic-emissive 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 (8-9) where Em is the emissive power of a blackbody per unit wavelength. If the gray-body condition is imposed, that is, 5,, : constant, Eq. (8—9) reduces to (8-10) 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. (8-2) by the Em : ‘4— (3-11) CIA—5 01' Em = (8-12) 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] (8-13) 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 8-5!) 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 dark-red 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 8-4 RADIATION SHAPE FACTOR Consider two black surfaces A, and A;. as shown in Fig. 8-8. 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'firt-mr. angle 1mm; and r'tn-i/igriruti'mrfactor. The energy leaving surface 1 and arriving at surface 2 is EMA It‘ll anal-13 . . =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, (8-23} = constant. “5 law [Eq. (8-24) 5 the result wavelength ity and ab- zy must be not alw‘dyS gray-body eel-surface gray—body 10. ll. and h gray and Relations between shape factors 393 - EXAMPLE 8-2 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. 8-12 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 8-5 RELATIONS BETWEEN SHAPE FACTORS Some useful relations between shape factors may be obtained by considering the system showu in Fig. 8-19. 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 (8-25) i.e.. the total shape factor is the sum of its parts. We could also write Eq. (8-25) as (8—26] AsFa—m =A3F3—1+A3F3—2 A|.1F1.2—3 :AlFl—3 “42573 Fig. 8-19 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. (8-18) The net heat exchange is therefore 1-2 = A.F.3(E;,! — E“) = A3F21(E,,, — EM) (8-19) 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. 8-8. The angles d). and (ii; are measured between a normal to the surface and the line drawn between the area elements r. ' LO , ., @5454! II-EZZ, £1.44.- IIW'W' aa- ‘01 £44414 _,4 4..- _m%fifllflIIm-m—u-IIII_ mari'llil-flm-Illllu- Vémmm‘pfian, 44- I. . l/JZUIMI‘IIfl-_-!mun MEMZ’IIEII-E’i-II ‘ _ magnum-Inn ; UQHI’IIIII-_IIIIII AI’IIAIIII-II IIII 0.] 0.15 0,: 0.3 0.5 1.0 l5 1 3 30 Ratio XtD 0‘05 \ FiQ- 3-12 Radiation shape factor for radiation between parallel rectangles Ran-o do- FlQ. 8-13 Radiation shape factor for radiation between parallel disks. ...
View Full Document

Page1 / 10

radiation - RADIATION HEAT TRANSFER I 8-1 INTRODUCTION...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online