hapte, Slutin 5. Ye, thi claim i eanable ince in the abence any heat eatin the ate heat tane thugh a plain wall in teady peatin mut be cntant. But the value thi cntant mut be ze ince ne ide the wall i peectly inulated. heee, thee can be n tempeatue dieence between dieent pat the wall; that i, the tempeatue in a plane wall mut be unim in teady peatin. hapte, Slutin 54. Ye, the tempeatue in a plane wall with cntant themal cnductivity and n heat eatin will vay linealy duing teady ne-dimeninal heat cnductin even when the wall le heat by adiatin m it uace. hi i becaue the teady heat cnductin equatin in a plane wall i d / whe lutin i ( x) x egadle the bunday cnditin. he lutin unctin epeent a taight line whe lpe i. hapte, Slutin 55. Ye, in the cae cntant themal cnductivity and n heat eatin, the tempeatue in a lid cylinical d whe end ae maintained at cntant but dieent tempeatue while the ide uace i peectly inulated will vay linealy duing teady ne-dimeninal heat cnductin. hi i becaue the teady heat cnductin equatin in thi cae i d / whe lutin i ( x) x which epeent a taight line whe lpe i. hapte, Slutin 56. Ye, thi claim i eanable ince n heat i enteing the cylinde and thu thee can be n heat tane m the cylinde in teady peatin. hi cnditin will be atiied nly when thee ae n tempeatue dieence within the cylinde and the ute uace tempeatue the cylinde i the equal t the tempeatue the uunding medium. hapte, Slutin 6. he bae plate a huehld in i ubjected t peciied heat lux n the let uace and t peciied tempeatue n the ight uace. he mathematical mulatin, the vaiatin tempeatue in the plate, and the inne uace tempeatue ae t be detemined teady ne-dimeninal heat tane. Aumptin Heat cnductin i teady and ne-dimeninal ince the uace aea the bae plate i lage elative t it thicne, and the themal cnditin n bth ide
the plate ae unim. hemal cnductivity i cntant. hee i n heat eatin in the plate. 4 Heat l thugh the uppe pat the in i negligible. Ppetie he themal cnductivity i given t be W/m. Analyi (a) Nting that the uppe pat the in i well inulated and thu the entie heat eated in the eitance wie i taneed t the bae plate, the heat lux thugh the inne uace i detemined t be Q& 8 W 5, W/m A 4 bae 6 m aing the diectin nmal t the uace the wall t be the x diectin with x at the let uace, the mathematical mulatin thi pblem can be expeed a d d () and 5, ( L) 85 W/m (b) Integating the dieential equatin twice with epect t x yield d ( x) x whee and ae abitay cntant. Applying the bunday cnditin give x : x L: Subtituting detemined t be ( L) L and L L int the eal lutin, the vaiatin tempeatue i L ( L x) ( x) x (5, W/m )(.6 x)m 85 W/m 5(.6 x) 85 (c) he tempeatue at x (the inne uace the plate) i () 5(.6 ) 85 Nte that the inne uace tempeatue i highe than the exped uace tempeatue, a expected.
hapte, Slutin 6. hilled wate lw in a pipe that i well inulated m utide. he mathematical mulatin and the vaiatin tempeatue in the pipe ae t be detemined teady ne-dimeninal heat tane. Aumptin Heat cnductin i teady and ne-dimeninal ince the pipe i lng elative t it thicne, and thee i themal ymmety abut the cente line. hemal cnductivity i cntant. hee i n heat eatin in the pipe. Analyi (a) Nting that heat tane i ne-dimeninal in the adial diectin, the mathematical mulatin thi pblem can be expeed a d d d ( ) and h[ ( )] d ( ) Wate (b) Integating the dieential equatin nce with epect t give d Inulated Dividing bth ide the equatin abve by t bing it t a eadily integable m and then integating, d ( ) ln whee and ae abitay cntant. Applying the bunday cnditin give : : h[ h( ( ln )] ) Subtituting and int the eal lutin, the vaiatin tempeatue i detemined t be ( ) hi eult i nt upiing ince teady peating cnditin exit. hapte, Slutin 87. L Heat i eated unimly in a lage ba plate. One ide the plate i inulated while the the ide i ubjected t cnvectin. he lcatin and value the highet and the lwet tempeatue in the plate ae t be detemined.
Aumptin Heat tane i teady ince thee i n indicatin any change with time. Heat tane i ne-dimeninal ince the plate i lage elative t it thicne, and thee i themal ymmety abut the cente plane hemal cnductivity i cntant. 4 Heat eatin i unim. Ppetie he themal cnductivity i given t be W/m. Analyi hi inulated plate whe thicne i L i equivalent t ne-hal an uninulated plate whe thicne i L ince the midplane the uninulated plate can be teated a inulated uace. he highet tempeatue will ccu at the inulated uace while the lwet tempeatue will ccu at the uace which i exped t the envinment. Nte that L in the llwing elatin i the ull thicne the given plate ince the inulated ide epeent the cente uace a plate whe thicne i dubled. he deied value ae detemined diectly m 5 L ( W/m )(.5 m) 5 5. h 44 W/m L hapte, Slutin 9. ( W/m )(.5 m) 5. ( W/m ) 5 Inulated 54.6 e L5 cm 5 h44 W/m. Heat i eated unimly in a pheical adiactive mateial with peciied uace tempeatue. he mathematical mulatin, the vaiatin tempeatue in the phee, and the cente tempeatue ae t be detemined teady ne-dimeninal heat tane. Aumptin Heat tane i teady ince thee i n indicatin any change with time. Heat tane i ne-dimeninal ince thee i themal ymmety abut the mid pint. hemal cnductivity i cntant. 4 Heat eatin i unim. Ppetie he themal cnductivity i given t be 5 W/m. Analyi (a) Nting that heat tane i teady and ne-dimeninal in the adial diectin, the mathematical mulatin thi pblem can be expeed a and d ( ) d with cntant 8 (peciied uace tempeatue) d () (themal ymmety abut the mid pint) (b) Multiplying bth ide the dieential equatin by and eaanging give e 8
d d Integating with epect t give d (a) Applying the bunday cnditin at the mid pint, d () B.. at : Dividing bth ide Eq. (a) by t bing it t a eadily integable m and integating, d and ( ) (b) Applying the the bunday cnditin at B.. at : Subtituting thi ( ), elatin int Eq. (b) and eaanging give ( ) which i the deied lutin the tempeatue ditibutin in the wie a a unctin. (c) he tempeatue at the cente the phee ( ) i detemined by ubtituting the nwn quantitie t be () ( ) 7 (4 W/m )(.4 m) 8 6 (5 W/ m ) 79 hu the tempeatue at cente will be abut 7 abve the tempeatue the ute uace the phee.