Conduction Heat Transfer
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- Silas Alexander Cameron
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1 Cnductn Heat Transfer Practce prblems A steel ppe f cnductvty 5 W/m-K has nsde and utsde surface temperature f C and 6 C respectvely Fnd the heat flw rate per unt ppe length and flux per unt nsde and per unt utsde area The ID and OD f the ppe are 4 mm and 5 mm respectvely [Ans: 673 W/m] Cnsder a cylndrcal element f nsde and utsde dameter ' a ' and ' b ' wth the nsde and utsde temperature levels T and T Fr a lnear varatn f ' ' wth T fnd the average value m whch satsfes T, m T T q ln ba [Ans: ( ( ))] 3 A plane wall f a cmpste f tw materal A and B The wall f materal A has 6 3 q 5 W/m, 75 W/m-K and thcness L 5 mm generatn wth A A The materal B has n energy 5 W/m-K and thcness L mm The nner surface f A s well B nsulated, whle the uter surface f B s cled by a water stream h W/m -K a) Setch the temperature dstrbutn f the cmpste under steady state cndtn, B T 3 C and b) Determne the temperature T f the nsulated surface and temperature T f the cled surface [Ans: T=4:C, T=5:C] 4 A slcn chp s encapsulated such that, under steady state cndtns, all the pwer t dsspates s transferred by cnvectn t a flud stream fr whch h W/m -K and T 5 C The chp s separated frm the flud by a mm thc alumnum cver plate and the cntact resstance f the clp/alumnum surface s 4 5 m -K/W If the chp area s mm and ts maxmum allwable temperature s 85 C, what s the maxmum allwable pwer dsspatn n the chp? [Ans: 567W]
2 5 Fnd the temperature dstrbutn n a D rectangular plate (at steady state) f sdes ab, subject t T, y, Tx,, T a, y and T x, b Ax ( ) ( ) ( ) ( ) 6 An exterr wall f a huse may be apprxmated by a 4 n layer f cmmn brc 7 W/m-K fllwed by a 5 n layer f gypsum plaster 48 W/m-K What thcness f rc-wd nsulatn havng frm the wall by 8 %? 65 W/m-K shuld be added t reduce the heat lss (r gan) [Ans: 5834mm] 7 An alumnum W/m-K prtrudes frm a wall It s 3 mm thc and 75 cm lng The bac s mantaned at 3 C and the ambent temperature s 5 C wth the heat lss per unt depth f the fn h W/m -K Calculate [Ans: 866W/m] 8 A certan materal has a thcness f 3 cm and a thermal cnductvty f 4 W/m-K At a partcular nstant n tme, the temperature dstrbutn wth x, the dstance frm the left face s, where x s n meters Calculate the heat flw rates at x and x 3 cm Is the T 5x 3x sld heatng up r clng dwn? 9 Derve a relatn fr the crtcal radus f nsulatn fr a sphere A plane wall 6 cm thc generates heat nternally at the rate f [Ans: W/m, 435W/m ] [Ans: /h] 3 3 MW/m One sde f the wall s nsulated and the ther sde s expsed t an envrnment at 93 C The ' h ' between the wall and the envrnment s maxmum temperature n the wall 57 W/m -K The thermal cnductvty f the wall s W/m-K Calculate the [Ans: 58:C] Derve an expressn fr the temperature dstrbutn n a hllw cylnder wth heat surce whch vary accrdng t the lnear relatn q a br
3 wth q s the heat generated per unt vlume at r r The nsde and utsde temperature are T T at r r and T T at r r [Ans: b a T r r r Cln r C 3 3 T ar r br r 36 T 9 4 C r 36 lg r C,, where 3 3 r T r ar r br 36T ln r 9ar ln r 4br ln r 36ln 9ln 4ln r 36 lg r ] A very lng thn cpper rd 64 mm n dameter s expsed t an envrnment at C The base temperature s 5 C and ' h ' between the rd and surrundng s 4 W/m -K Calculate heat lss frm the rd [Ans: W assumng = 39W/m-K] 3 Fr a strp f nfnte y -dmensn and length L n x -dmensn, prve that the slpe f the sthermals s zer at x L fr all respectvely Here T T and T L, y T ; and zer and nfnte at x and L, fr and and the bundary cndtns are T x, T ; T, y T 4 Cnsder a lng metal rd f square crss-sectn L L The upper and lwer sde faces are at T The left face s nductn heated; assume that the effect s a unfrm flux nput at the surface The rght face s at T a Fnd temperature dstrbutn n the rd-materal b Obtan a slutn f the left face s adabatc
4 5 A steel ball Cp 46 J/g-K, 35 W/m-K temperature f mantaned at, 5cm n dameter and ntally at a unfrm 45 C s suddenly placed n a cntrlled envrnment where the temperature s C The cnvectve heat transfer c-effcent s W/m -K Calculate the tme requred fr the ball t attan 5 C Assume specfc gravty f steel t be A large blc f steel 45 W/m-K, 4 m /s f 35 C The surface s expsed t a heat flux a by suddenly rasng the surface temperature t b thrugh a cnstant surface flux f [Ans: 6 hrs], s ntally at a unfrm temperature 5 3 W/m 5 C Calculate the temperature at a depth f 5cm after a tme f 5mn n bth cases 5 7 A large plate f alumnum 5 W/m-K; 84 m /s [Ans: a) 856:C b) 7934:C] 5cm thc and ntally at C s suddenly expsed t a cnvectn surface envrnment f 7 C wth h 55W/m -K Calculate the temperature at a depth f 5cm frm ne f the faces ne mnute after the plate has been expsed t the envrnment Hw much energy per unt area has been remved per unt area f the plate durng ths tme [Ans: 357 :C] 8 A plate f thcness ntally has a snusdal temperature dstrbutn varyng frm T at x t T m at x t T at x If the surfaces f the plate are held at T fr subsequent tmes, fnd the temperature dstrbutn as a functn f tme 9 Cnsder tw sem-nfnte slds, ne ntally at Ans: x, t sn x e C and the ther at, whch are suddenly at placed n ntmate cntact at ther surfaces Assumng n cntact resstance, calculate the subsequent cntact surface temperature Qualtatvely shw the temperature plts n the tw slds at dfferent tme nstants t [Ans: ] A large slab f cpper havng a thcness f 3cm s ntally at t a cnvectn envrnment n the tp surface at 3 C It s suddenly expsed 8 C whle the bttm surface s nsulated In
5 6mnutes, the surface temperature drps t 4 C Calculate the value f 'h' Assume fr cpper, 4 4 W/m-K; m /s A steel sphere cm n dameter s suddenly mmersed n a tan f l at temperature f the sphere s sphere t cl dwn t C, h C? Fr steel, 5W/m -K [Ans: 383 W/m -K] C The ntal Hw lng wll t tae the centre f the 5 45W/m-K; 4 m /s A very thc lw carbn steel plate 3496 BTu/hr-ft F heat treatment bath at ntally at [Ans: 68 s] 6 F, s placed n a 86 F The surface prcess s characterzed by a cnvectn ceffcent f 5Btu/ft -hr-:f Fnd the surface temperature varatn wth tme fr shrt tmes Assume 688ft /hr [Ans: ( ) ( ( ) ( )) where t s n hrs] 3 Steel plates beng thnned by rllng must be perdcally reheated Hw lng must plates 3n thc, whch are at F, be ept n a furnace surrundng at 6 F, rder t reach a mnmum metal temperature f BTu/hr-ft F; =37ft /hr h 3 BTu/hr-ft F, n 3 F? Assume [Ans: 5 s] 4 A sem-nfnte sld ntally at T thrughut s n cntact at ts face wth an envrnment at T e A surface ceffcent 'h' may be used at the expsed surface If T e s a functn f tme, determne f there s any such functn r functns, whch wll permt a smlarty type f slutn, that s n terms f x [Ans: ( ) ( )
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