PROBLEM-Chapter 1-5 (Heat Transfer, Yanus A. Cengel)

Transcription

1 POBEM-hape -5 Hea ansfe Yanus. ene - he nne and ue sufaces f a bc wa ae ananed a specfed epeaues. he ae f hea ansfe huh he wa s be deened. ssupns Seady pean cndns es snce he suface epeaues f he wa ean cnsan a he specfed vaues. hea ppees f he wa ae cnsan. Ppees he hea cnducvy f he wa s ven be.9 W/. nayss Unde seady cndns he ae f hea ansfe huh he wa s Q cnd.9w/ 5 5 5W. Bc wa. c 5-7 he hea eneaed n he ccuy n he suface f a -W scn chp s cnduced he ceac subsae. he epeaue dffeence acss he chp n seady pean s be deened. ssupns Seady pean cndns es. hea ppees f he chp ae cnsan. Ppees he hea cnducvy f he scn chp s ven be W/. nayss he epeaue dffeence beween he fn and bac sufaces f he chp s... Q Q W.5 W/... Q eac subsae W hp.5

2 - he cnvecn hea ansfe ceffcen f hea ansfe f an eeccay heaed we a s be deened by easun epeaues when seady pean cndns ae eached and he eecc pwe cnsued. ssupns Seady pean cndns es snce he epeaue eadns d n chane wh e. adan hea ansfe s nebe. nayss In seady pean he ae f hea ss f he we equas he ae f hea enean n he we as a esu f essance hean. ha s Q E eneaed VI V W he suface aea f he we s D. c s π D π Q he Newn's aw f cn f cnvecn hea ansfe s epessed as Q hs s Dseadn any hea ansfe by adan he cnvecn hea ansfe ceffcen s deened be Q W h 7.5 W/. s.88 Dscussn If he epeaue f he suundn sufaces s equa he a epeaue n he he vaue baned abve acuay epesens he cbned cnvecn and adan hea ansfe ceffcen. - We cnsde a hn spheca she eeen f hcness n a sphee see F. -7 n he e.. he densy f he sphee s ρ he specfc hea s and he enh s. he aea f he sphee na he decn f hea ansfe a any can s π whee s he vaue f he adus a ha can. Ne ha he hea ansfe aea depends n n hs case and hus vaes wh can. When hee s n hea enean an eney baance n hs hn spheca she eeen f hcness dun a sa e neva can be epessed as Q Q Eeeen whee Eeeen E E ρ Subsun Q Q ρ whee π Q. Dvdn he equan abve by ves Q ρ an he as and yeds ρ snce f he defnn f he devave and Fue s aw f hea cnducn Q Q Q

3 Nn ha he hea ansfe aea n hs case s π and he hea cnducvy s cnsan he ne-densna ansen hea cnducn equan n a sphee beces α whee α / ρ s he hea dffusvy f he aea. - F a edu n whch he hea cnducn equan s ven n s spes by : α a Hea ansfe s ansen b s ne-densna c hee s n hea enean and d he hea cnducvy s cnsan. -5 F a edu n whch he hea cnducn equan s ven n s spes by d d : d d a Hea ansfe s seady b s ne-densna c hee s hea enean and d he hea cnducvy s vaabe. - F a edu n whch he hea cnducn equan s ven by α a Hea ansfe s ansen b s ne-densna c hee s n hea enean and d he hea cnducvy s cnsan. -7 F a edu n whch he hea cnducn equan s ven n s spes by d d : d d a Hea ansfe s seady b s ne-densna c hee s hea enean and d he hea cnducvy s cnsan. - F a edu n whch he hea cnducn equan s ven by : y α a Hea ansfe s ansen b s w-densna c hee s n hea enean and d he hea cnducvy s cnsan W essance heae we wh a specfed suface epeaue s used b wae. he cene epeaue f he we s be deened. ssupns Hea ansfe s seady snce hee s n chane wh e. Hea ansfe s ne-densna snce hee s hea syey abu he cene ne and n chane n he aa decn. hea cnducvy s cnsan. Hea enean n he heae s unf. Ppees he hea cnducvy s ven be W/. nayss he essance heae cnves eecc eney n hea a a ae f W. he ae f hea enean pe un vue f he we s Q Q en en W 8.55 W/ Vwe π π.5.7 D he cene epeaue f he we s hen deened f Eq. -7 be

4 s.55 W/.5 W/ a If he aea sufaces f he d ae nsuaed he hea ansfe suface aea f he cyndca d s he b he p suface aea f he d s πd /. b If he p and he b sufaces f he d ae nsuaed he hea ansfe aea f he d s he aea suface aea f he d π D. - In seady hea cnducn he ae f hea ansfe n he wa s equa he ae f hea ansfe u f. s he epeaue a any pn n he wa eans cnsan. heefe he eney cnen f he wa des n chane dun seady hea cnducn. Hweve he epeaue an he wa and hus he eney cnen f he wa w chane dun ansen cnducn. - he epeaue dsbun n a pane wa w be a sah ne dun seady and ne densna hea ansfe wh cnsan wa hea cnducvy. - he hea essance f a edu epesens he essance f ha edu aans hea ansfe. -5 he cbned hea ansfe ceffcen epesens he cbned effecs f adan and cnvecn hea ansfes n a suface and s defned as h cbned h cnvecn h adan. I ffes he cnvenence f ncpan he effecs f adan n he cnvecn hea ansfe ceffcen and ne adan n hea ansfe cacuans. - Yes. he cnvecn essance can be defned as he nvese f he cnvecn hea ansfe ceffcen pe un suface aea snce s defned as cnv / h. -7 he cnvecn and he adan essances a a suface ae paae snce bh he cnvecn and adan hea ansfes ccu suaneusy. -8 F a suface f a whch he cnvecn and adan hea ansfe ceffcens ae hcnv and had he sne equvaen hea ansfe ceffcen s heqv hcnv had when he edu and he suundn sufaces ae a he sae epeaue. hen he equvaen hea essance w be / h. eqv eqv -9 he hea essance new asscaed wh a fve-aye cpse wa nvves fve sneaye essances cnneced n sees. -E he nne and ue sufaces f he was f an eeccay heaed huse ean a specfed epeaues dun a wne day. he aun f hea s f he huse ha day and s s cs ae be deened. ssupns Hea ansfe huh he was s seady snce he suface epeaues f he was ean cnsan a he specfed vaues dun he e ped cnsdeed. Hea ansfe s nedensna snce any snfcan epeaue adens w es n he decn f he nds he uds. hea cnducvy f he was s cnsan. Ppees he hea cnducvy f he bc wa s ven be. Bu/h f F. nayss We cnsde hea ss huh he was ny. he a hea ansfe aea s Wa 9 9 f he ae f hea ss dun he daye s Q F day Bu / h. f. F f 5 Bu / h f Q

5 he ae f hea ss dun nhe s Q nh Bu/h.f. F f 8 Bu/h f he aun f hea ss f he huse ha nh w be Q Q Q Q Q Q day nh h 5 Bu / h h 8 Bu / h 95 Bu hen he cs f hs hea ss f ha day beces s 95 / Wh$.9 / Wh $ cpse wa cnsss f sevea hzna and veca ayes. he ef and h sufaces f he wa ae ananed a unf epeaues. he ae f hea ansfe huh he wa he neface epeaues and he epeaue dp acss he secn F ae be deened. ssupns Hea ansfe s seady snce hee s n ndcan f chane wh e. Hea ansfe huh he wa s ne-densna. hea cnducves ae cnsan. hea cnac essances a he nefaces ae dseaded. Ppees he hea cnducves ae ven be F B 8 D 5 E 5 W/. nayss a he epesenave suface aea s... he hea essance new and he ndvdua hea essances ae B D E 7 F d d a 5 Q B D E a F.. /W W/.. d.5. /W W/...5. /W 8 W/.... /W 5 W/ W/.. /W..5 /W W/.. d.... d.5 57 W.9 /W 7 d hen seady ae f hea ansfe huh ene wa beces Q a 57 W.9 W.. /W.5 /W /W f a. secn

6 b he a hea essance beween ef suface and he pn whee he secns B D and E ee s..5.5 /W a d hen he epeaue a he pn whee he secns B D and E ee beces Q Q a 57 W.5 /W a c he epeaue dp acss he secn F can be deened f Q Q F 57 W.5 /W F -75E sea ppe cveed wh -n hc fbeass nsuan s subjeced cnvecn n s sufaces. he ae f hea ss f he sea pe un enh and he e nvved n neecn he hea essance f he see ppe n cacuans ae be deened. ssupns Hea ansfe s seady snce hee s n ndcan f any chane wh e. Hea ansfe s ne-densna snce hee s hea syey abu he cene ne and n vaan n he aa decn. hea cnducves ae cnsan. he hea cnac essance a he neface s nebe. Ppees he hea cnducves ae ven be 8.7 Bu/h f F f see and. Bu/h f F f fbeass nsuan. nayss he nne and ue suface aeas f he nsuaed ppe ae πd π./ 5 f f. 9 f πd π8 / f f. 9 f he ndvdua essances ae h a h nsuan ppe Bu/h.f. F.9 f. h F/Bu n / n /.75. h F/Bu π π 8.7 Bu/h.f. Ff n / n / 5.5 h F/Bu π π. Bu/h.f. Ff.9 h F/Bu 5 Bu/h.f. F.9 f h F/Bu hen he seady ae f hea ss f he sea pe f. ppe enh beces Q a 5 55 F 9.9Bu/h 5.5 h F/Bu If he hea essance f he see ppe s neeced he new vaue f a hea essance w be a h F/Bu hen he pecenae e nvved n cacuans beces e% h F/Bu.5% 55. h F/Bu whch s nsnfcan. ppe nsuan

7 - he epeaue f a as sea s be easued by a hecupe. he e aes ese 99 pecen f he na s be deened. ssupns he juncn s spheca n shape wh a daee f D.. he hea ppees f he juncn ae cnsan. he hea ansfe ceffcen s cnsan and unf ve he ene suface. adan effecs ae nebe. 5 he B nube s B <. s ha he uped syse anayss s appcabe hs assupn w be vefed. Ppees he ppees f he juncn ae ven be 5 W /. ρ 85 / and p J /.. nayss he chaacesc enh f he juncn and he B nube ae V πd / D. c. suface πd hc 5 W/.. B. 7 <. 5 W/. Gas Snce B <. he uped syse anayss s appcabe. h hen he e ped f he hecupe ead 99% f Juncn he na epeaue dffeence s deened f D. h h 5 W/. - b 95. s ρv ρ 85 / J /.. p p c b. 95 s - e. e 8.5 s

8 - n e s dpped n bn wae. he cn e f he e s be deened. ssupns he e s spheca n shape wh a adus f.75 c. Hea cnducn n he e s ne-densna because f syey abu he dpn. he hea ppees f he e ae cnsan. he hea ansfe ceffcen s cnsan and unf ve he ene suface. he Fue nube s >. s ha he ne-e appae suns he ansen epeaue chas ae appcabe hs assupn w be vefed. Ppees he hea cnducvy and dffusvy f he es ae ven be. W/. and α. - /s. nayss he B nube f hs pcess s h W/.. 75 B.. W/. he cnsans λ and cespndn hs B nube ae f abe - λ.877 and.999 hen he Fue nube beces 7 97 λ.877 θ sph e.999 e heefe he ne-e appae sun he ansen epeaue chas s appcabe. hen he e equed f he epeaue f he cene f he e each 7 s deened be α /s 8 s 7.8 n Wae 97 E -5 he cene epeaue f paes s be weed dun cn. he cn e and f any pa f he paes w suffe chn njuy dun hs cn pcess ae be deened. ssupns he paes ae spheca n shape wh a adus f c. Hea cnducn n he pa s ne-densna n he ada decn because f he syey abu he dpn. he hea ppees f he pa ae cnsan. he hea ansfe ceffcen s cnsan and unf ve he ene suface. 5 he Fue nube s >. s ha he ne-e appae suns he ansen epeaue chas ae appcabe hs assupn w be vefed. Ppees he hea cnducvy and hea dffusvy f paes ae ven be.5 W/ and α. - /s. nayss Fs we fnd he B nube: h W/ 9.. B..5 W /. /s F abe - we ead f a sphee λ.5 and.. Subsun hese vaues n he ne-e sun ves θ λ 5 e 5. e. 75. whch s eae han. and hus he ne-e sun s appcabe. hen he cn e beces α.75. 5s.5 h - α. / s he wes epeaue dun cn w ccu n he suface / and s deened be λ sn λ / sn λ / sn λ / e θ λ / λ / λ / Subsun sn.5 ad Pa 5

9 whch s abve he epeaue ane f f chn njuy f paes. heefe n pa f he paes w epeence chn njuy dun hs cn pcess. enave sun We cud as sve hs pbe usn ansen epeaue chas as fws:.5w/..877 B h 9W/.. α s. h α. / s heefe he suface epeaue s deened f 877. B h. F. -5b whch ves suface...º F. -5a he sh dffeence beween he w esus s due he eadn e f he chas. -7 h d s dpped n bn wae and epeaue easueens ae aen a cean e nevas. he hea dffusvy and hea cnducvy f he h d and he cnvecn hea ansfe ceffcen ae be deened. ssupns Hea cnducn n he h d s ne-densna snce s n and has hea syey abu he cene ne. he hea ppees f he h d ae cnsan. he hea ansfe ceffcen s cnsan and unf ve he ene suface. he Fue nube s >. s ha he ne-e appae suns he ansen epeaue chas ae appcabe hs assupn w be vefed. Ppees he ppees f h d avaabe ae ven be ρ 98 / and p 9 J/.. nayss a F F. -b we have B h.5 he Fue nube s deened f F. -a be.5 B h 59 9 α.7 9. he hea dffusvy f he h d s deened be.. α α /s s b he hea cnducvy f he h d s deened f 7 αρ.7 /s98 / 9 J/..77 W/. p Wae 9 H d

10 c F pa a we have. 5. hen B h h heefe he hea ansfe ceffcen s h.77 W/..5 h 7 W/.5-5 pesn pus appes n he feeze c he qucy. he cene and suface epeaues f he appes and he aun f hea ansfe f each appe n h ae be deened. ssupns he appes ae spheca n shape wh a daee f 9 c. Hea cnducn n he appes s ne-densna because f syey abu he dpn. he hea ppees f he appes ae cnsan. he hea ansfe ceffcen s cnsan and unf ve he ene suface. 5 he Fue nube s >. s ha he ne-e appae suns he ansen epeaue chas ae appcabe hs assupn w be vefed. Ppees he ppees f he appes ae ven be.8 W/. ρ 8 / p.8 J/. and α. -7 /s. nayss he B nube s h 8 W/..5 B W/. he cnsans λ and cespndn hs B nube ae f abe - λ.7 and.9 he Fue nube s 7 α. /s h s /h. >.. 5 hen he epeaue a he cene f he appes beces 5 λ.7. θ sph e.9 e he epeaue a he suface f he appes s θ sph e λ sn λ λ /.7 he au pssbe hea ansfe s a /.9 e.7.. ρv ρ π 8 /.5. π Q..8 J/. p hen he acua aun f hea ansfe beces. sn.7 ad.7 [ 5 ].7 J ppe.55

11 Q Q a θ Q.Q a sph sn λ λ cs λ sn.7 ad.7 cs.7 ad.79. λ.7..7 J 7. J -7 hc wd sab s epsed h ases f a ped f 5 nues. I s be deened whehe he wd w ne. ssupns he wd sab s eaed as a se-nfne edu subjeced cnvecn a he epsed suface. he hea ppees f he wd sab ae cnsan. he hea ansfe ceffcen s cnsan and unf ve he ene suface. Ppees he hea ppees f he wd ae.7 W/. and α.8-7 /s. nayss he ne-densna ansen epeaue dsbun n he wd can be deened f whee h α 5 W/. h α h α h h α h α efc ep efc α α W/ / s5 s.7 Nn ha a he suface and usn abe - f efc vaues 5 efc ep.8 efc Svn f ves 5 H ases whch s ess han he nn epeaue f 5. heefe he wd w n ne. Wd Sab.

12 5-8 pane wa wh vaabe hea enean and cnsan hea cnducvy s subjeced nsuan a he ef nde and adan a he h bunday nde 5. he fne dffeence fuan f he bunday ndes s be deened. ssupns Hea ansfe huh he wa s ven be seady and ne-densna and he hea cnducvy be cnsan. nvecn hea ansfe s nebe. adan Insuaed nayss Usn he eney baance appach and an he decn f a hea ansfes be wads he nde unde su ε cnsdean he fne dffeence fuans bece ef bunday nde: 5 / 5 h bunday nde: εσ su 5 5 /

13 5- uanu pae s subjeced nsuan n ne sde and cnvecn n he he. he fne dffeence fuan f hs pbe s be baned and he nda epeaues unde seady cndns ae be deened. ssupns Hea ansfe huh he wa s seady snce hee s n ndcan f chane wh e. Hea ansfe s ne-densna snce he pae s ae eave s hcness. hea cnducvy s cnsan. adan hea ansfe s nebe. Ppees he hea cnducvy s ven be 8 W/. nayss he nube f ndes s specfed be M. hen he nda spacn beces M.5. - hs pbe nvves unnwn nda epeaues and hus we need have equans deene he unquey. Nde s n nsuaed bunday and hus we can ea as an ne ne by usn he ae cncep. Ndes and ae ne ndes and hus f he we can use he enea fne dffeence ean epessed as f and Fnay he fne dffeence equan f nde 5 n he h suface subjeced cnvecn s baned by appyn an eney baance n he haf vue eeen abu nde 5 and an he decn f a hea ansfes be wads he nde unde cnsdean: Nde ef suface -nsuaed : Ndene : Nde ne : Nde ne : Nde ne : 5 5 Nde 5 h suface - cnvecn : h 5 5 Insuaed h 5 / whee. W/ 8 W/ h W/ and. hs syse f equans wh s unnwn epeaues cnsue he fne dffeence fuan f he pbe. b he nda epeaues unde seady cndns ae deened by svn he equans abve suaneusy wh an equan sve be and 5 5. Dscussn hs pbe can be sved anaycay by svn he dffeena equan as descbed n hap. and he anayca eac sun can be used chec he accuacy f he nueca sun abve.

14 5-9 pae s subjeced specfed hea fu and specfed epeaue n ne sde and n cndns n he he. he fne dffeence fuan f hs pbe s be baned and he epeaue f he he sde unde seady cndns s be deened. ssupns Hea ansfe huh he pae s ven be seady and ne-densna. hee s n hea enean n he pae. Ppees he hea cnducvy s ven be.5 W/. nayss he nda spacn s ven be.. hen he nube f ndes M beces. M. Ndes and ae ne ndes and hus f he q 5 we can use he enea fne dffeence ean epessed as snce f and he fne dffeence equan f nde n he ef suface s baned by appyn an eney baance n he haf vue eeen abu nde and an he decn f a hea ansfes be wads he nde unde cnsdean q 7 W/.5 W/.. Ohe nda epeaues ae deened f he enea ne nde ean as fws: : : : : heefe he epeaue f he he suface w be Dscussn hs pbe can be sved anaycay by svn he dffeena equan as descbed n hap. and he anayca eac sun can be used chec he accuacy f he nueca sun abve. 5-8 uanu pae nay a a unf epeaue s subjeced nsuan n ne sde and cnvecn n he he. he ansen fne dffeence fuan f hs pbe s be baned and he nda epeaues afe 5 n and unde seady cndns ae be deened. ssupns Hea ansfe s ne-densna snce he pae s ae eave s hcness. hea cnducvy s cnsan. adan hea ansfe s nebe. Ppees he cnducvy and dffusvy ae ven be 8 W/ and α. 5 /s. nayss he nda spacn s ven be.. hen he nube f ndes beces M /.8/. 5. hs pbe nvves 5 unnwn nda epeaues and hus we need have 5 equans. Nde s n nsuaed bunday and hus we can ea as an ne ne by usn he ae cncep. Ndes

15 and ae ne ndes and hus f he we can use he enea epc fne dffeence ean epessed as he fne dffeence equan f nde n he h suface subjeced cnvecn s baned by appyn an eney baance n he haf vue eeen abu nde and an he decn f a hea ansfes be wads he nde unde cnsdean: h : Nde cnvecn Nde ne : Nde ne : Ndene : : Nde nsuaed ρ h h whee 5 W/ 8 W/ W/. h and.5 α /s. he uppe f he e sep s deened f he saby cea ha eques a pay ceffcens be eae han equa ze. he ceffcen f s sae n hs case and hus he saby cea f hs pbe can be epessed as α h h h / / snce α /. Subsun he ven quanes he au awabe he e sep beces 5. s ] /8 W/... 5 W/ /s[.5. heefe any e sep ess han 5.5 s can be used sve hs pbe. F cnvenence e us chse he e sep be 5 s. hen he esh Fue nube beces α / s5 s Subsun hs vaue f and he ven quanes he nda epeaues afe 5 /5 e seps 5 n ae deened be h Insuae

16 fe 5 n: and 9.9 b he e needed f ansen pean be esabshed s deened by nceasn he nube f e seps un he nda epeaues n ne chane. In hs case seady pean s esabshed n ---- n and he nda epeaues unde seady cndns ae deened be 9 5 and Dscussn he seady sun can be checed ndependeny by bann he seady fne dffeence fuan and svn he esun equans suaneusy.

17 5-87 Hea cnducn huh a n -shaped sd ba wh specfed bunday cndns s cnsdeed. he epeaue a he p cne nde # f he bdy afe 5 and n s be deened wh he ansen epc fne dffeence ehd. ssupns Hea ansfe huh he bdy s ven be ansen and wdensna. hea cnducvy s cnsan. Hea enean s unf. Ppees he cnducvy and dffusvy ae ven be 5 W/ and α. /s. nayss he nda spacn s ven be.5. he epc fne dffeence equans ae deened n he bass f he eney baance f he ansen case epessed as Q G V sdes eeen eeen ρ he quanes h q and d n chane wh e and hus we d n need use he supescp f he. s he eney baance epessns can be spfed usn he defnns f hea dffusvy α ρ / and he densness esh Fue nube α / whee y. We ne ha a ndes ae bunday ndes ecep nde 5 ha s an ne nde. heefe we w have ey n eney baances ban he fne dffeence equans. Usn eney baances he fne dffeence equans f each f he 8 ndes ae baned as fws: Nde : h q ρ Nde : h 5 ρ Nde : h ρ I can be eaaned as h h Nde : q 5 ρ Nde 5 ne: 5 5 Nde : h 7 5 ρ Nde 7: h ρ Nde 8: h ρ 5 7 q h Insuae

18 7 whee W/ q 8 W/.5 5 W/ h 8 W/ and 5. he uppe f he e sep s deened f he saby cea ha eques he ceffcen f n he epessn he pay ceffcen be eae han equa ze f a ndes. he saes pay ceffcen n he 8 equans abve s he ceffcen f n he epessn snce s epsed s cnvecn pe un vue hs can be vefed and hus he saby cea f hs pbe can be epessed as h h/ α h / snce α /. Subsun he ven quanes he au awabe vaue f he e sep s deened be..5.s /s[ 8 W/..5 /5 W/. ] heefe any e sep ess han. s can be used sve hs pbe. F cnvenence we chse he e sep be 5 s. hen he esh Fue nube beces α. /s5 s..5 f 5 s Usn he specfed na cndn as he sun a e f sweepn huh he 9 equans abve w ve he sun a nevas f 5 s. Usn a cpue he sun a he uppe cne nde nde s deened be 5 and 59 a 5 and n especvey. I can be shwn ha he seady sae sun a nde s San wh an eney baance n a vue eeen he hee-densna ansen epc fne dffeence equan f a enea ne nde n ecanua cdnaes f y z f he case f cnsan hea cnducvy and n hea enean s be baned. nayss We cnsde a ecanua en n whch hea cnducn s snfcan n he and y decns. n hee s n hea enean n he edu and he hea cnducvy f he edu s cnsan. Nw we dvde he -y-z en n a esh f nda pns whch ae spaced y and z apa n he y and z decns especvey and cnsde a enea z ne nde n whse cdnaes ae y ny ae z z. Nn ha he vue eeen ceneed abu he enea ne nde n nvves hea cnducn f s sdes h ef - y fn ea p and b and epessn he a pevus e sep he ansen epc fne n dffeence fuan f a enea ne nde can be epessed as

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