Chapter 12: Radiation Heat Transfer

Transcription

1 Chapter : adaton Heat Transfer adaton dffers from Conducton and Convecton heat t transfer mechansms, n the sense that t does not requre the presence of a materal medum to occur. nergy transfer by radaton occurs at the speed of lght and suffers no attenuaton n vacuum. adaton can occur between two bodes separated by a medum colder than both bodes. Accordng to Maxwell theory, energy transfer takes place va electromagnetc waves n radaton. lectromagnetc waves transport energy lke other waves and travel at the speed of lght. lectromagnetc waves are characterzed by ther frequency ν (Hz) and wavelength (µm), where: c / ν where c s the speed of lght n that medum; n a vacuum c 0.99 x 0 8 m / s. Note that the frequency and wavelength are nversely proportonal. The speed of lght n a medum s related to the speed of lght n a vacuum, c c 0 / n where n s the ndex of refracton of the medum, n for ar and n.5 for water. Note that the frequency of an electromagnetc wave depends only on the source and s ndependent of the medum. The frequency of an electromagnetc wave can range from a few cycles to mllons of cycles and hgher per second. nsten postulated another theory for electromagnetc radaton. Based on ths theory, electromagnetc radaton s the propagaton of a collecton of dscrete packets of energy called photons. In ths vew, each photon of frequency ν s consdered to have energy of e hν hc / where h 6.65 x 0 -.s s the Planck s constant. Note that n nsten s theory h and c are constants, thus the energy of a photon s nversely proportonal to ts wavelength. Therefore, shorter wavelength radaton possesses more powerful photon energes (X-ray and gamma rays are hghly destructve). Chapter, &C 09, Sprng 005. Majd Bahram

2 Fg. -: lectromagnetc spectrum. lectromagnetc radaton covers a wde range of wavelength, from 0-0 µm for cosmc rays to 0 0 µm for electrcal power waves. As shown n Fg. -, thermal radaton wave s a narrow band on the electromagnetc wave spectrum. Thermal radaton emsson s a drect result of vbratonal and rotatonal motons of molecules, atoms, and electrons of a substance. Temperature s a measure of these actvtes. Thus, the rate of thermal radaton emsson ncreases wth ncreasng temperature. What we call lght s the vsble porton of the electromagnetc spectrum whch les wthn the thermal radaton band. Thermal radaton s a volumetrc phenomenon. However, for opaque solds such as metals, radaton s consdered to be a surface phenomenon, snce the radaton emtted by the nteror regon never reach the surface. Note that the radaton characterstcs of surfaces can be changed completely by applyng thn layers of coatngs on them. Blackbody adaton A blackbody s defned as a perfect emtter and absorber of radaton. At a specfed temperature and wavelength, no surface can emt more energy than a blackbody. A blackbody s a dffuse emtter whch means t emts radaton unformly n all drecton. Also a blackbody absorbs all ncdent radaton regardless of wavelength and drecton. Chapter, &C 09, Sprng 005. Majd Bahram

3 The radaton energy emtted by a blackbody per unt tme and per unt surface area can be determned from the Stefan-Boltzmann Law: b σt where σ ( W / m ) 8 W m K where T s the absolute temperature of the surface n K and b s called the blackbody emssve power. A large cavty wth a small openng closely resembles a blackbody. Fg. -: Varaton of blackbody emssve power wth wavelength Spectral blackbody emssve power s the amount of radaton energy emtted by a blackbody at an absolute temperature T per unt tme, per unt surface area, and per unt wavelength. b C C ( T ) πhc hc k [ exp( C / T ) ].7 0 / k.9 C 8 W m. µ m ( W. µ m / m ) 0 ( µ m. K ) ( / K ) Boltzmann's constant Ths s called Plank s dstrbuton law and s vald for a surface n a vacuum or gas. For other medums, t needs to be modfed by replacng C by C /n, where n s the ndex of refracton of the medum, Chapter, &C 09, Sprng 005. Majd Bahram

4 The wavelength at whch the peak emssve power occurs for a gven temperature can be obtaned from Wen s dsplacement law: ( T ) power m. K max µ It can be shown that ntegraton of the spectral blackbody emssve power b over the entre wavelength spectrum gves the total blackbody emssve power b : b ( T ) ( T ) d σt ( W / m ) 0 b The Stefan-Boltzmann law gves the total radaton emtted by a blackbody at all wavelengths from 0 to nfnty. But, we are often nterested n the amount of radaton emtted over some wavelength band. To avod numercal ntegraton of the Planck s equaton, a non-dmensonal quantty f s defned whch s called the blackbody radaton functon as f ( T ) 0 The functon f represents the fracton of radaton emtted from a blackbody at temperature T n the wavelength band from 0 to. Table - n Cengel book lsts f as a functon of T. Therefore, one can wrte: f f b σt ( T ) d ( T ) f ( T ) f ( T ) ( T ) f ( T ) b b (T) Fg. -: Fracton of radaton emtted n the wavelength between and Chapter, &C 09, Sprng 005. Majd Bahram

5 xample - The temperature of the flament of a lght bulb s 500 K. Assumng the flament to be a blackbody, determne the fracton of the radant energy emtted by the flament that falls n the vsble range. Also determne the wavelength at whch the emsson of radaton from the flament peaks. Soluton The vsble range of the electromagnetc spectrum extends from 0. to 0.76 mcro meter. Usng Table -: T T f 0.µ m 0.76µ f ( 500K ) m( 500K ) µ m. K f 900µ m. K f whch means only about 5% of the radaton emtted by the flament of the lght bulb falls n the vsble range. The remanng 95% appears n the nfrared regon or the nvsble lght. adaton Propertes A blackbody can serve as a convenent reference n descrbng the emsson and absorpton characterstcs of real surfaces. mssvty The emssvty of a surface s defned as the rato of the radaton emtted by the surface to the radaton emtted by a blackbody at the same temperature. Thus, 0 ε mssvty s a measure of how closely a surface approxmate a blackbody, ε blackbody. The emssvty of a surface s not a constant; t s a functon of temperature of the surface and wavelength and the drecton of the emtted radaton, ε ε (T,, θ) where θ s the angle between the drecton and the normal of the surface. The total emssvty of a surface s the average emssvty of a surface over all drecton and wavelengths: ( T ) ( T ) ( T ) ε ( T ) b σ T Spectral emssvty s defned n a smlar manner: ( T ) ε ( T ) ε ( T ) σ T where (T) s the spectral emssve power of the real surface. As shown, the radaton emsson from a real surface dffers from the Planck s dstrbuton. b ( T ) ( T ) Chapter, &C 09, Sprng Majd Bahram

6 Fg. -: Comparson of the emssve power of a real surface and a blackbody. To make the radaton calculatons easer, we defne the followng approxmatons: Dffuse surface: s a surface whch ts propertes are ndependent of drecton. Gray surface: s a surface whch ts propertes are ndependent from wavelength. Therefore, the emssvty of a gray, dffuse surface s the total hemsphercal (or smply the total) emssvty of that surface. A gray surface should emt as much as radaton as the real surface t represents at the same temperature: ( T ) ε 0 ( T ) ( T ) ε σ T mssvty s a strong functon of temperature, see Fg. -0 Cengel book. b d Absorptvty, eflectvty, and Transmssvty The radaton energy ncdent on a surface per unt area per unt tme s called rradaton, G. Absorptvty α: s the fracton of rradaton absorbed by the surface. eflectvty ρ: s the fracton of rradaton reflected by the surface. Transmssvty τ: s the fracton of rradaton transmtted through the surface. adosty : total radaton energy streamng from a surface, per unt area per unt tme. It s the summaton of the reflected and the emtted radaton. Chapter, &C 09, Sprng Majd Bahram

7 absorptvty : reflectvty : transmssvty : absorbed radaton Gabs α ncdent radaton G reflected radaton Gref ρ ncdent radaton G transmtted radaton Gtr τ ncdent radaton G 0 α 0 ρ 0 τ Applyng the frst law of thermodynamcs, the sum of the absorbed, reflected, and the transmtted radaton radatons must be equal to the ncdent radaton: G abs + G ref + G tr G Dvde by G: α + ρ + τ Incdent radaton G, W/m eflected ρg adosty, (eflected + mtted radaton) mtted radaton ε b Absorbed αg Sem-transparent materal Transmtted τg Fg. -5: The absorpton, reflecton, and transmsson of rradaton by a semtransparent materal. For opaque surfaces τ 0 and thus: α + ρ. The above defntons are for total hem-sphercal propertes (over all drecton and all frequences). We can also defne these propertes n terms of ther spectral counterparts: Chapter, &C 09, Sprng Majd Bahram

8 G where ρ G τ τ ρ ρ ( T, ) ( T, ) ( T, ) α α thus + τ G ρ + τ + α + α G spectral reflectvty spectral absorptvty spectral transmssvty Note that the absorptvty α s almost ndependent of surface temperature and t strongly depends on the temperature of the source at whch the ncdent radaton s orgnatng. For example α of the concrete roof s about 0.6 for solar radaton (source temperature 576 K) and 0.9 for radaton orgnatng from the surroundngs (source temperature 00 K). Krchhoff s Law Consder an sothermal cavty and a surface at the same temperature T. At the steady state (equlbrum) thermal condton G abs α G α σ T and radaton emtted emt ε σ T Snce the small body s n thermal equlbrum, G abs emt ε(t) α(t) The total hemsphercal emssvty of a surface at temperature T s equal to ts total hem-sphercal absorptvty for radaton comng from a blackbody at the same temperature T. Ths s called the Krchhoff s law. T G T A, ε, α emt Fg. -6: Small body contaned n a large sothermal cavty. The Krchhoff s law can be wrtten n the spectral form: Chapter, &C 09, Sprng Majd Bahram

9 and n the spectral drectonal form ( T ) α ( T ) ε ( T ) α ( T ) ε, θ, θ The Krchhoff s law makes the radaton analyss easer (ε α), especally for opaque surfaces where ρ α. Note that Krchhoff s law cannot be used when there s a large temperature dfference (more than 00 K) between the surface and the source temperature. Solar adaton The solar energy reachng the edge of the earth s atmosphere s called the solar constant: G s 5 W / m Owng to the ellptcty of the earth s orbt, the actual solar constant changes throughout the year wthn +/-.%. Ths varaton s relatvely small; thus G s s assumed to be a constant. The effectve surface temperature of the sun can be estmated from the solar constant (by treatng the sun as a blackbody). The solar radaton undergoes consderable attenuaton as t passes through the atmosphere as a result of absorpton and scatterng: Absorpton by the oxygen occurs n a narrow band about 0.76 µm. The ozone layer absorbs ultravolet radaton at wavelengths below 0. µm almost completely and radaton n the range of µm consderably. Absorpton n the nfrared regon s domnated by water vapor and carbon doxde. Dust/pollutant partcles also absorb radaton at varous wavelengths. As a result the solar radaton reachng the earth s surface s about 950 W/m on a clear day and much less on a cloudy day, n the wavelength band 0. to.5 µm. Scatterng and reflecton by ar molecules (and other partcles) are other mechansms that attenuate the solar radaton. Oxygen and ntrogen molecules scatter radaton at short wavelengths (correspondng to volet and blue colors). That s the reason the sky seems blue! The gas molecules (mostly CO and H O) and the suspended partcles n the atmosphere emt radaton as well as absorbng t. It s convenent to consder the atmosphere (sky) as a blackbody at some lower temperature. Ths fcttous temperature s called the effectve sky temperature T sky. G sky σ T sky T sky 0 K for cold clear sky Chapter, &C 09, Sprng Majd Bahram

10 T sky 85 K for warm cloudy sky Usng Krchhoff s law we can wrte α ε snce the temperature of the sky s on the order of the room temperature. The Vew Factor adaton heat transfer between surfaces depends on the orentaton of the surfaces relatve to each other as well as ther radaton propertes and temperatures. Vew factor (or shape factor) s a purely geometrcal parameter that accounts for the effects of orentaton on radaton between surfaces. In vew factor calculatons, we assume unform radaton n all drectons throughout the surface,.e., surfaces are sothermal and dffuse. Also the medum between two surfaces does not absorb, emt, or scatter radaton. F j or F j the fracton of the radaton leavng surface that strkes surface j drectly. Note the followng: The vew factor ranges between zero and one. F j 0 ndcates that two surfaces do not see each other drectly. F j ndcates that the surface j completely surrounds surface. The radaton that strkes a surface does not need to be absorbed by that surface. F s the fracton of radaton leavng surface that strkes tself drectly. F 0 for plane or convex surfaces, and F 0 for concave surfaces. Plane surface, F 0 Convex surface, F 0 Concave surface, F 0 Fg. -7: Vew factor between surface and tself. Calculatng vew factors between surfaces are usually very complex and dffcult to perform. Vew factors for selected geometres are gven n Table - and!-5 and Fgs. - to - n Cengel book. Chapter, &C 09, Sprng Majd Bahram

11 Vew Factor elatons adaton analyss of an enclosure consstng of N surfaces requres the calculatons of N vew factors. However, all of these calculatons are not necessary. Once a suffcent number of vew factors are avalable, the rest of them can be found usng the followng relatons for vew factors. The ecprocty ule The vew factor F j s not equal to F j unless the areas of the two surfaces are equal. It can be shown that: A F j A j F j The Summaton ule In radaton analyss, we usually form an enclosure. The conservaton of energy prncple requres that the entre radaton leavng any surface of an enclosure be ntercepted by the surfaces of enclosure. Therefore, N j F j The summaton rule can be appled to each surface of an enclosure by varyng from to N (number of surfaces). Thus the summaton rule gves N equatons. Also recprocty rule gves 0.5 N (N-) addtonal equatons. Therefore, the total number of vew factors that need to be evaluated drectly for an N-surface enclosure becomes N N + N ( N ) N( N ) xample - Determne the vew factors F and F for the followng geometres: D L D A L A A L A A A A A ) Sphere of dameter D nsde a cubcal box of length L D. Chapter, &C 09, Sprng 005. Majd Bahram

12 ) Dagonal partton wthn a long square duct. ) nd and sde of a crcular tube of equal length and dameter, L D. Assumptons: Dffuse surfaces. Soluton: ) sphere wthn a cube: By nspecton, F By recprocty and summaton: F F A A + F F πd 6L F π 6 π 6 ) Partton wthn a square duct: From summaton rule, F + F + F where F 0 By symmetry F F Thus, F 0.5. From recprocty: A L F F A L ) Crcular tube: from Fg. -, wth r / L 0.5 and L / r, F 0.7. From summaton rule, F + F + F wth F 0, F - F 0.8 From recprocty, The Superposton ule A πd / F F A πdl The vew factor from a surface to a surface j s equal to the sum of the vew factors from surface to the parts of surface j. Chapter, &C 09, Sprng 005. Majd Bahram

13 + Fg. -8: The superposton rule for vew factors. F (,) F + F The Symmetry ule Two (or more) surfaces that possess symmetry about a thrd surface wll have dentcal vew factors from that surface. xample: - Fnd the vew factor from the base of a pyramd to each of ts four sdes. The base s a square and ts sde surfaces are sosceles trangles. From symmetry rule, we have: F F F F 5 Also, the summaton rule yelds: F + F + F + F + F 5 Snce, F 0 (flat surface), we fnd; F F F F Pyramd n example -. Chapter, &C 09, Sprng 005. Majd Bahram

14 The Crossed-Strngs Method Geometres such as channels and ducts that are very long n one drecton can be consdered two-dmensonal (snce radaton through end surfaces can be neglected). The vew factor between ther surfaces can be determned by crossstrng method developed by H. C. Hottel, as follows: F j ( crossed strngs) ( uncrossed strngs) ( strng on surface ) D L C L L L 5 L 6 L L 5 L 6 L L L B L A Chapter, &C 09, Sprng 005. Majd Bahram Fg. -9: Cross-strng method. F ( L + L ) ( L + L ) Note that the surfaces do not need to be flat. adaton Heat Transfer 5 The analyss of radaton exchange between surfaces s complcated because of reflecton. Ths can be smplfed when surfaces are assumed to be black surfaces. The net radaton between two surfaces can be expressed as A F b b ( W ) 6 L radaton leavng surface radaton leavng surface that drectly stkes surface that drectly stkes surface A F Applyng recprocty A F A F yelds A Fσ ( T T ) ( W ) Consder an enclosure consstng of N black surfaces mantaned at specfed temperatures. For each surface, we can wrte

15 N N j A Fjσ j j ( T T ) ( W ) Usng the sgn conventon, a negatve heat transfer rate ndcates that the radaton heat transfer s to surface (heat gan). Now, we can extend ths analyss to non-black surfaces. It s common to assume that the surfaces are opaque, dffuse, and gray. Also, surfaces are consdered to be sothermal. Also the flud nsde the cavty s not partcpatng n the radaton. adosty s the total radaton energy streamng from a surface, per unt area per unt tme. It s the summaton of the reflected and the emtted radaton. For a surface that s gray and opaque (ε α and α + ρ ), the adosty can be expressed as ε ε ε b b b + ρ G + ( ε ) G ( W / m ) σt j ( for a blackbody) Note that the radosty of a blackbody s equal to ts emssve power. Usng an energy balance, the net rate of radaton heat transfer from a surface of surface area A can be expressed as A A ( G ) ( W ) ε ε b Aε ε ( ) In electrcal analogy to Ohm s law, a thermal resstance can be defned as b ε A ε where s called the surface resstance to radaton. b Surface b Chapter, &C 09, Sprng Majd Bahram Fg. -0: Surface resstance to radaton.

16 Note that the surface resstance to radaton for a blackbody s zero. For nsulated or adabatc surfaces, the net heat transfer through them s zero. In ths cases, the surface s called reradatng surface. There s no net heat transfer to a reradatng surface. Net adaton between Two Surfaces Consder two dffuse, gray, and opaque surfaces of arbtrary shape mantaned at unform temperatures. The net rate of radaton heat transfer from surface to surface j can be expressed j A j A Fj F j A Applyng recprocty F ( ) ( W ) j j j j ( W ) In analogy wth Ohm s law, a resstance can be defned as j j j A F where j s called the space resstance to radaton. j j b j j bj j j Surface Surface j Fg. -: lectrcal network, surface and space resstances. In an N-surface enclosure, the conservaton of energy prncple requres that the net heat transfer from surface to be equal to the sum of the net heat transfers from to each of the N surfaces of the enclosure. N N j j j j We have already derved a relatonshp for the net radaton from a surface j ( W ) Chapter, &C 09, Sprng Majd Bahram

17 Combnng these two relatonshps gves: b b N j j j ( W ) ( W ) Method of Solvng adaton Problem In radaton problems, ether the temperature or the net rate of heat transfer must be gven for each of the surfaces to obtan a unque soluton for the unknown surface temperature and heat transfer rates. We use the network method whch s based on the electrcal network analogy. The followng steps should be taken:. Form an enclosure; consder fcttous surface(s) for openngs, room, etc.. Draw a surface resstance assocated wth each surface of the enclosure. Connect the surface resstances wth space resstances. Solve the radatons problem (radostes) by treatng t as an electrcal network problem. Note that ths method s not practcal for enclosures wth more than surfaces. xample -: Hot Plates n oom Two parallel plates 0.5 by.0 m are spaced 0.5 m apart. One plate s mantaned at 000 C and the other at 500 C. The emssvtes of the plates are 0. and 0.5, respectvely. The plates are located n a very large room, the walls of whch are mantaned at 7 C. The plates exchange heat wth each other and wth the room, but only the plate surfaces facng each other are to be consdered n the analyss. Fnd the net heat transfer rate to each plate and the room; neglect other modes of heat transfer,.e., conducton and convecton. Assumptons: Dffuse, gray, and opaque surfaces and steady-state heat transfer. Soluton: Ths s a three-body problem, the two plates and room. The radaton network s shown below. Chapter, &C 09, Sprng Majd Bahram

18 T 000 C 0.5 m 0.5 m room at 7 C T 000 C.0 m Fg. -: Schematc for Problem - where, T 000 C 7 K A A 0.5 m T 500 C 77 K T 7 C 00 K ε 0. ε 0.5 b b b σt Fg. -: Thermal network for Problem -. We can assume that the room s a blackbody, snce ts surface resstance s neglgble: ε A >> A ε 0 From Fg. - n Cengel book, the shape factor F 0.85 Usng recprocty and A A, F F 0.85 Applyng summaton rule F + F + F Chapter, &C 09, Sprng Majd Bahram

19 Snce F 0 (flat plate), F Fnally, from symmetry F F 0.75 The surface resstances are Space resstances are ε A ε ε A ε A F A F A F 0. ( 0.)( 0.5m ) 0.5 ( 0.5)( 0.5m ) 0.85 ( )( 0.5m ) 0.75 ( )( 0.5m ) 0.75 ( )( 0.5m ) We need to fnd the radosty for surface and only, snce surface s a blackbody, b σt For node : For node : where b b b b b σt σt σt + + kw / m 0.59 kw / m 0 0 kw / m Substtutng values and solvng two equatons, one fnds:.69 kw/m and 5.05 kw/m The total heat loss by plate s: b b.5.59 kw kw Chapter, &C 09, Sprng Majd Bahram

20 The total radaton receved by the room s Note that from an energy balance, we must have: kw + Chapter, &C 09, Sprng Majd Bahram