Chapter 5: 8, 16, 25, 27, 28, 35, 36, 49, 52, 57, 72, 82 and 87.

Transcription

1 aper : 8,,, 7, 8,,, 9,, 7, 7, 8 and e fne dfference frulan f sead w-densnal ea cnducn n a edu w ea eneran and cnsan eral cnducv s ven b, n n,, n n, n, n, n, n recanular crdnaes. s relan can be dfed fr e ree-densnal case b spl addn aner nde j e eperaure n e z drecn, and aner dfference er fr e z drecn as, n, j, n, j, n, j, n, j, n, j, n, j, n, j, n, j, n, z, n, j j - A plane wall w n ea eneran s subjeced specfed eperaure a e lef nde and ea flu a e r bundar nde 8. e fne dfference frulan f e bundar ndes and e fne dfference frulan fr e rae f ea ransfer a e lef bundar are be deerned. Assupns Hea ransfer ru e wall s ven be sead, and e eral cnducv be cnsan. Hea ransfer s ne-densnal snce e plae s lare relave s cness. ere s n ea eneran n e edu. Analss Usn e ener balance apprac and an e drecn f all ea ransfers be wards e nde under cnsderan, e fne dfference frulans bece Lef bundar nde: R bundar nde: A q A r 8 Hea ransfer a lef surface: Q lef surface A

2 - A ln ranular fn aaced a surface s cnsdered. e ndal eperaures, e rae f ea ransfer, and e fn effcenc are be deerned nuercall usn equall spaced ndes. Assupns Hea ransfer aln e fn s ven be sead, and e eperaure aln e fn var n e drecn nl s a. eral cnducv s cnsan. Prperes e eral cnducv s ven be 8 W/. e essv f e fn surface s.9. Analss e fn len s ven be L c, and e nuber f ndes s specfed be M. erefre, e ndal spacn s L M.. - e eperaure a nde s ven be, and e eperaures a e reann ndes are be deerned. erefre, we need ave equans deerne e unquel. Ndes,,, and are nerr ndes, and e fne dfference frulan fr a eneral nerr nde s baned b appln an ener balance n e vlue eleen f s nde. Nn a ea ransfer s sead and ere s n ea eneran n e fn and assun ea ransfer be n e edu fr all sdes, e ener balance can be epressed as Q Alef Ar Acnv εσasurface[ surr 7 } all sdes Ne a ea ransfer areas are dfferen fr eac nde n s case, and usn eercal relans, e can be epressed as A A lef r He wd He / w L [ / ] [ / ] w L w / csθ { εσ[ anθ anθ Asurface Len wd w / csθ Subsun, w[ L. ] anθ w[ L. ] anθ Dvdn eac er b wl anθ / ves surr 7 ]} εσ L L Lsnθ Lsnθ surr Subsun, : εσ.. [ surr 7 ] L L Lsnθ Lsnθ : εσ.. [ surr 7 ] L L Lsnθ Lsnθ : εσ.. [ surr 7 ] L L Lsnθ Lsnθ : εσ.. [ surr 7 ] L L Lsnθ Lsnθ / / [ 7 ] An ener balance n e nde ves e equan, : / / an θ εσ [ surr 7 ] csθ csθ

3 Prb. - cnnued Slvn e equans abve sulaneusl fr e unnwn ndal eperaures ves 77., 7., 7., 8., and. b e al rae f ea ransfer fr e fn s spl e su f e ea ransfer fr eac vlue eleen e aben, and fr w s deerned fr Q fn Q eleen, Asurface, εσa surface, [ 7 Nn a e ea ransfer surface area s w /csθ fr e bundar ndes and, and wce as lare fr e nerr ndes,,, and, we ave w Q fn [ ] csθ w εσ {[ 7 surr ] [ 7 surr ] [ 7 surr ] [ 7 surr ] csθ [ W 7 surr ] [ 7 surr ]} surr ]

4 -7 A plae s subjeced specfed ea flu n ne sde and specfed eperaure n e er. e fne dfference frulan f s prble s be baned, and e unnwn surface eperaure under sead cndns s be deerned. Assupns Hea ransfer ru e base plae s ven be sead. Hea ransfer s nedensnal snce e plae s lare relave s cness. ere s n ea eneran n e plae. Radan ea ransfer s nelble. e enre ea eneraed b e ressance eaers s ransferred ru e plae. Prperes e eral cnducv s ven be W/. Analss e ndal spacn s ven be. c. en e nuber f ndes M beces L. c M. c e r surface eperaure s ven be 8. s prble nvlves unnwn ndal eperaures, and us we need ave equans deerne e unquel. Ndes,, and are nerr ndes, and us fr e we can use e eneral fne dfference relan epressed as snce, fr and e fne dfference equan fr nde n e lef surface subjeced unfr ea flu s baned b appln an ener balance n e alf vlue eleen abu nde and an e drecn f all ea ransfers be wards e nde under cnsderan: Nde r surface - cnvecn : Ndenerr : Nde nerr : q were. c, W/, 8, and q Q / A 8W /., W/. e sse f equans w unnwn eperaures cnsue e fne dfference frulan f e prble. b e ndal eperaures under sead cndns are deerned b slvn e equans abve sulaneusl w an equan slver be, 9, and 9 Dscussn s prble can be slved analcall b slvn e dfferenal equan as descrbed n ap., and e analcal eac slun can be used cec e accurac f e nuercal slun abve.

5 -8 A plae s subjeced specfed ea flu and specfed eperaure n ne sde, and n cndns n e er. e fne dfference frulan f s prble s be baned, and e eperaure f e er sde under sead cndns s be deerned. Assupns Hea ransfer ru e plae s ven be sead and ne-densnal. ere s n ea eneran n e plae. Prperes e eral cnducv s ven be. W/. Analss e ndal spacn s ven be.. en e nuber f ndes M beces L. M. Ndes,,, and are nerr ndes, and us fr e we can use e eneral fne dfference relan epressed as snce, fr,,, and e fne dfference equan fr nde n e lef surface s baned b appln an ener balance n e alf vlue eleen abu nde and an e drecn f all ea ransfers be wards e nde under cnsderan, 8 q 7 W/. W/.. Oer ndal eperaures are deerned fr e eneral nerr nde relan as fllws: : : : : erefre, e eperaure f e er surface wll be. Dscussn s prble can be slved analcall b slvn e dfferenal equan as descrbed n ap., and e analcal eac slun can be used cec e accurac f e nuercal slun abve.

6 - One sde f a vercal plae s be cled b aacn cpper pn fns. e fne dfference frulan f e prble fr all ndes s be baned, and e ndal eperaures, e rae f ea ransfer fr a snle fn and fr e enre surface f e plae are be deerned. Assupns Hea ransfer aln e fn s ven be sead and ne-densnal. e eral cnducv s cnsan. bned cnvecn and radan ea ransfer ceffcen s cnsan and unfr. Prperes e eral cnducv s ven be 8 W/. Analss e ndal spacn s ven be. c. en e nuber f ndes M beces L c M 7. c e base eperaure a nde s ven be. s prble nvlves unnwn ndal eperaures, and us we need ave equans deerne e unquel. Ndes,,,, and are nerr ndes, and us fr e we can use e eneral fne dfference relan epressed as A A p p / A e fne dfference equan fr nde a e fn p s baned b appln an ener balance n e alf vlue eleen abu a nde. en, : p / A : p / A : p / A : p / A : p / A Nde : A p / A were., 7 W/,,, W/ A πd / π. c /.9c.9 and p πd π..78 b e ndal eperaures under sead cndns are deerned b slvn e equans abve sulaneusl w an equan slver be 98., - 97., 9.7, 9., 9.7, 9. c e rae f ea ransfer fr a snle fn s spl e su f e ea ransfer fr e ndal eleens, Q fn Q eleen, p / A surface, p p / A. W d e nuber f fns n e surface s N.f fns 7,778 fns.. en e rae f ea ranfer fr e fns, e unfnned prn, and e enre fnned surface bece Q Q Q fn, al `unfnned al N. f fns Q A Q fn, al Q unfnned fn unfnned 7,778. W,7 W W/,7 7,78 W - 7, W

7 - w cas rn sea ppes are cnneced eac er ru w -c c flanes, and ea s ls fr e flanes b cnvecn and radan. e fne dfference frulan f e prble fr all ndes s be baned, and e eperaure f e p f e flane as well as e rae f ea ransfer fr e epsed surfaces f e flane are be deerned. Assupns Hea ransfer ru e flane s saed be sead and ne-densnal. e eral cnducv and essv are cnsans. nvecn ea ransfer ceffcen s cnsan and unfr. Prperes e eral cnducv and essv are ven be W/ and ε.8. Analss e dsance beween ndes and s e cness f e ppe,. c.. e ndal spacn aln e flane s ven be c.. en e nuber f ndes M beces L c M 7 c s prble nvlves 7 unnwn ndal eperaures, and us we need ave 7 equans deerne e unquel. Nn a e al cness f e flane s., e ea cnducn area a an lcan aln e flane s Acnd πr were e values f rad a e ndes and beween e ndes e d pns are r., r., r., r.7, r.8, r.9, r. r.8, r., r., r.7, r.8, r.9 en e fne dfference equans fr eac nde are baned fr e ener balance be as fllws: Nde : π r πr Nde : π r πr [π r r / ] / { εσ[ surr 7 ]} Nde : π r πr πr { εσ[ surr 7 ]} Nde : π r πr πr { εσ[ surr 7 ]} Nde : π r πr πr { εσ[ surr 7 ]} Nde : π r πr πr { εσ[ surr 7 ]} Prb. - cnnued Nde : π r [π / r r / πr ]{ εσ[ surr 7 ]} were.,., W/, ε.8, 8, n, 9 K and surr -8, 8 W/,.7 W/ K. W/ σ e sse f 7 equans w 7 unnwns cnsues e fne dfference frulan f e prble. b e ndal eperaures under sead cndns are deerned b slvn e 7 equans abve sulaneusl w an equan slver be.,., 9., 8.9, 8.7, 8., and 8.,

8 c Knwn e nner surface eperaure, e rae f ea ransfer fr e flane under sead cndns s spl e rae f ea ransfer fr e sea e ppe a flane secn Q fn Q eleen, Asurface, εσa were A surface, are as ven abve fr dfferen ndes. surface, [ 7 surr ]. W

9 -9 A ln sld bd s subjeced sead w-densnal ea ransfer. e unnwn ndal eperaures are be deerned. Assupns Hea ransfer ru e bd s ven be sead and w-densnal. ere s n ea eneran n e bd. Prperes e eral cnducv s ven be W/. Analss e ndal spacn s ven be l., and e eneral fne dfference fr f an nerr nde fr sead w-densnal ea cnducn fr e case f n ea eneran s epressed as ndel lef p r b nde nde lef p r b / a ere s ser abu e nsulaed surfaces as well as abu e danal lne. erefre, and,, and are e nl unnwn ndal eperaures. us we need nl equans deerne e unquel. Als, we can replace e ser lnes b nsulan and ulze e rrr-ae cncep wen wrn e fne dfference equans fr e nerr ndes. Ndenerr : Nde nerr : Nde nerr : 8 8 / / Slvn e equans abve sulaneusl ves 8 9 / b ere s ser abu e nsulaed surface as well as e danal lne. Replacn e ser lnes b nsulan, and ulzn e rrr-ae cncep, e fne dfference equans fr e nerr ndes can be wren as Ndenerr : Nde nerr : Nde nerr : Nde nerr : / / / Slvn e equans abve sulaneusl ves.9 8. / Dscussn Ne a an advanae f ser splfed e prble real.

10 - A ln sld bd s subjeced sead w-densnal ea ransfer. e unnwn ndal eperaures are be deerned. Assupns Hea ransfer ru e bd s ven be sead and w-densnal. ere s n ea eneran n e bd. Prperes e eral cnducv s ven be W/. Analss e ndal spacn s ven be l., and e eneral fne dfference fr f an nerr nde fr sead w-densnal ea cnducn fr e case f n ea eneran s epressed as ndel lef p r b nde lef p r b nde a ere s ser abu a vercal lne passn ru e ndes and. erefre,,, and,,, and are e nl unnwn ndal eperaures, and us we need nl equans deerne e unquel. Als, we can replace e ser lnes b nsulan and ulze e rrr-ae cncep wen wrn e fne dfference equans fr e nerr ndes. Nde nerr : Nde nerr : Nde nerr : Nde nerr : Slvn e equans abve sulaneusl ves 7 b ere s ser abu a vercal lne passn ru e ddle. erefre, and. Replacn e ser lnes b nsulan and ulzn e rrr-ae cncep, e fne dfference equans fr e nerr ndes and are deerned be Ndenerr : Nde nerr : Slvn e equans abve sulaneusl ves Dscussn Ne a an advanae f ser splfed e prble real.

11 -7 e epsed surface f a ln cncree dan f ranular crss-secn s subjeced slar ea flu and cnvecn and radan ea ransfer. e vercal secn f e dan s subjeced cnvecn w waer. e eperaures a e p, ddle, and b f e epsed surface f e dan are be deerned. Assupns Hea ransfer ru e dan s ven be sead and w-densnal. ere s n ea eneran wn e dan. Hea ransfer ru e base s nelble. eral prperes and ea ransfer ceffcens are cnsan. Prperes e eral cnducv and slar absrpv are ven be. W/ and α s.7. Analss e ndal spacn s ven be l, and all ndes are bundar ndes. Nde n e nsulaed bundar can be reaed as an nerr nde fr wc lef p r b nde. Usn e ener balance apprac and an e drecn f all ea ransfer be wards e nde, e fne dfference equans fr e ndes are baned be as fllws: l l l / l sn s s Nde : l l l l l l l l l l s qs l l sn Nde : l l l l l Nde : Nde : l l / [ sqs ] l sn Nde : [ α q ] Nde : [ ] were l,. W/, W/,, W/,, α s.7, and q s 8 W/. e sse f equans w unnwns cnsues e fne dfference frulan f e prble. e ndal eperaures under sead cndns are deerned b slvn e equans abve sulaneusl w an equan slver be p.,., ddle.,.,., b. Dscussn Ne a e es eperaure ccurs a a lcan fures awa fr e waer, as epeced.

12 -7 A plane wall w varable ea eneran and cnsan eral cnducv s subjeced unfr ea flu q a e lef nde and cnvecn a e r bundar nde. e eplc ransen fne dfference frulan f e bundar ndes s be deerned. Assupns Hea ransfer ru e wall s ven be ransen, and e eral cnducv be cnsan. Hea ransfer s ne-densnal snce e plae s lare relave s cness. Radan ea ransfer s nelble. Analss Usn e ener balance apprac and an e drecn f all ea ransfers be wards e nde under cnsderan, e eplc fne dfference frulans bece Lef bundar nde: A A A q A / R bundar nde: A A A A /

13 -8 A uranu plae nall a a unfr eperaure s subjeced nsulan n ne sde and cnvecn n e er. e ransen fne dfference frulan f s prble s be baned, and e ndal eperaures afer n and under sead cndns are be deerned. Assupns Hea ransfer s ne-densnal snce e plae s lare relave s cness. eral cnducv s cnsan. Radan ea ransfer s nelble. Prperes e cnducv and dffusv are ven be 8 W/ and α. /s. Analss e ndal spacn s ven be.. en e nuber f ndes beces / L M.8/.. s prble nvlves unnwn ndal eperaures, and us we need ave equans. Nde s n nsulaed bundar, and us we can rea as an nerr ne b usn e rrr ae cncep. Ndes,, and are nerr ndes, and us fr e we can use e eneral eplc fne dfference relan epressed as e fne dfference equan fr nde n e r surface subjeced cnvecn s baned b appln an ener balance n e alf vlue eleen abu nde and an e drecn f all ea ransfers be wards e nde under cnsderan: : Nde cnvecn Nde nerr : Nde nerr : Ndenerr : : Nde nsulaed r were, W/, 8 W/, W/.,, and. α /s. e upper l f e e sep s deerned fr e sabl crera a requres all prar ceffcens be reaer an r equal zer. e ceffcen f s saller n s case, and us e sabl crera fr s prble can be epressed as α / / snce α /. Subsun e ven quanes, e au allwable e e sep beces. s ] /8 W/... W/ /s[.. erefre, an e sep less an. s can be used slve s prble. Fr cnvenence, le us cse e e sep be s. en e es Furer nuber beces α / s s Subsun s value f and er ven quanes, e ndal eperaures afer / e seps n are deerned be Afer n: 8.9, 8.,.8,., and 9.9

14 b e e needed fr ransen peran be esablsed s deerned b ncreasn e nuber f e seps unl e ndal eperaures n lner cane. In s case sead peran s esablsed n ---- n, and e ndal eperaures under sead cndns are deerned be,, 9,, and Dscussn e sead slun can be ceced ndependenl b bann e sead fne dfference frulan, and slvn e resuln equans sulaneusl.

15 -87 e fran f f n e lass surfaces f a car s be prevened b aacn elecrc ressance eaers e nner surfaces. e eperaure dsrbun ruu e lass n afer e srp eaers are urned n and als wen sead cndns are reaced are be deerned usn e eplc ed. Assupns Hea ransfer ru e lass s ven be ransen and w-densnal. eral cnducv s cnsan. ere s ea eneran nl a e nner surface, wc wll be reaed as prescrbed ea flu. Prperes e cnducv and dffusv are ven be.8 W/ and α 9. /s. Analss e ndal spacn s ven be. c and c. e eplc fne dfference equans are deerned n e bass f e ener balance fr e ransen case epressed as Q G V All sdes eleen eleen We cnsder nl 9 ndes because f ser. Ne a we d n ave a square es n s case, and us we wll ave rel n ener balances ban e fne dfference equans. Usn ener balances, e fne dfference equans fr eac f e 9 ndes are baned as fllws: Nde : Nde : Nde : Nde : 7 Nde : 8 Nde : 9 Nde 7: W Nde 8: Nde 9: were.8 W/., α /. 9 /s, - W/., W/.,., and.. e upper l f e e sep s deerned fr e sabl crera a requres e ceffcen f n e epressn e prar ceffcen be reaer an r equal zer fr all ndes. e salles prar ceffcen n e 9 equans abve s e ceffcen f 9 n e epressn snce s epsed s cnvecn per un vlue s can be verfed. e equan fr nde can be rearraned as 9 α α erefre, e sabl crera fr s prble can be epressed as

16 α α Subsun e ven quanes, e au allwable value f e e sep s deerned be r, s W/ W/ /.9 s erefre, an e sep less an.8 s can be used slve s prble. Fr cnvenence, we cse e e sep be s. en e eperaure dsrbun ruu e lass n afer e srp eaers are urned n and wen sead cndns are reaced are deerned be fr e EES sluns ds n: -., -., -., -.8, -., -.7, 7., 8.7, 9 9. Sead-sae: -., -., -., -.8, -., -.7, 7., 8.7, 9 9. Dscussn Sead peran cndns are reaced n abu 8 n.