February 14 Homework Solutions


 Solomon Lambert
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1 llege f Engineeing and pute Science Mechanical Engineeing Depatent Mechanical Engineeing 75 Heat ansfe Sping 7 Nube 769 Instuct: Lay aett Febuay Hew Slutins.8 nside an aluinu pan used t c stew n tp f an electic ange. he btt sectin f the pan is L.5 c thic and has a diaete f D 8 c. he electic heating unit n the ange tp cnsues 9 f pwe duing cing, and 9 pecent f the heat eated in the heating eleent is tansfeed t the pan. Duing steady peatin, the tepeatue f the inne suface f the pan is easued t be 8. Assuing tepeatuedependent theal cnductivity and nediensinal heat tansfe, expess the atheatical fulatin (the diffeential equatin and the bunday cnditins) f this heat cnductin pble duing steady peatin. D nt slve (Figue f Çengel, Heat and Mass ansfe) e can stat with the eal heat cnductin equatin. ρ c t x x y y z z p e e ae tld that the pcess is steady and nediensinal s we can igne the tie, y and z deivative tes. e ae als tld that thee is a heat input f 9 f an electic heate f which 9% entes the btt f the pan. his is a bunday cnditin heat flux f (9%)(9 )/[(p/)(.8 ) ],8 /. hee is n heat eatin with the btt f the pan s we can set the heat eatin te t ze. ith these assuptins, the diffeential equatin beces d d dx dx d dx nst he bunday cnditins ae the heat flux fund in the pevius paagaph at x and a tepeatue f 8 at x L. e can wite these as fllws. d at x and 8 dx at x L.79 A  esistance heate wie with theal cnductivity f /, a diaete f D, and a length f L.9 is used t bil wate. If the ute suface tepeatue f the esistance wie is s, deteine the tepeatue at the cente f the wie. e can use the equatin f the tepeatue pfile in a slid cylinde and set t get the tepeatue at the cente f the wie. apply this equatin we ust fist cpute the heat suce te, which is the heat eated pe Jacaanda (Engineeing) Mail de Phne: Eail: 88 Fax:
2 Febuay hew slutins ME 75, L. S. aett, Sping 7 Page unit vlue. e ae given the ttal heat eatin f and we divide by the vlue f the wie t get the heat eatin te. Q V R L.768x (. ) (.9 ) e nw apply the equatin f the tepeatue in a slid cylinde with heat eatin using t get the cente tepeatue x [(. ) ] R ute suface nside a lng slid cylinde f adius c and theal cnductivity 5 /. Heat is eated in the cylinde unifly at a ate f 5/c. he side suface f the cylinde is aintained at a cnstant tepeatue f s 8. he vaiatin f tepeatue in the cylinde is given by s Based n this elatin, deteine (a) if the heat cnductin is steady tansient, (b) if it is ne, tw, theediensinal, and (c) the value f heat flux n the side suface f the cylinde at. (a and b) he equatin f tepeatue has a single independent vaiable,. hee is n tie in the equatin. Hence we cnclude that the pcess is (a) steady and (b) nediensinal. aing the fist deivative f this equatin gives the heat flux. 8 q d d 5 c.5x e c (. ) 5.87 nside a lage 5cthic bass plate ( / ) in which heat is eated unifly at a ate f x 5 /. One side f the plate is insulated while the the side is expsed t an envinent at 5 with a heat tansfe cefficient f /. Explain whee in the plate the highest and the lwest tepeatues will ccu, and deteine thei value. (Figue at ight taen f Çengel, Heat and Mass ansfe.) Since n heat leaves f the plate at x we expect the axiu tepeatue t ccu thee. he lwest tepeatue shuld ccu at the the side f the plate. he ttal heat eatin is the pduct f the heat eatin ties the vlue, AL. Since all this heat is cnvected away f the plate at x L, we can use the heat balance at that lcatin t deteine the suface tepeatue, L at x L.
3 Febuay hew slutins ME 75, L. S. aett, Sping 7 Page Q L 5 h x (.5 ) ha e AL cnv ( L ) L 5. e can use the equatin f the heat flux and apply the esult that the heat flux is at x t get an equatin f the tepeatue at this pint. Substituting the given data pvides the desied esult. ( ) L L q x L x x x x L q x L 5 x (.5 ).5 x L. hen the theal cnductivity f a ediu vaies linealy with tepeatue, is the aveage theal cnductivity always equivalent t the cnductivity value at the aveage tepeatue? A linea vaiatin has the f a b. Using the aveage tepeatue gives a idpint theal cnductivity value f a b( )/. Using the definitin f aveage theal cnductivity with a b gives the fllwing esult. ( ) b( ) a / d ( a b ) d a b hus the aveage theal cnductivity is always the sae as the theal cnductivity at the aveage tepeatue if the theal cnductivity vaies linealy with tepeatue.. nside a cylindical shell f length L, inne adius, and ute adius whse theal cnductivity vaies linealy in a specified tepeatue ange as () ( ) whee and ae tw specified cnstants. he inne suface f the shell is aintained at a cnstant tepeatue f, while the ute suface is aintained at. Assuing steady nediensinal heat tansfe, btain a elatin f (a) the heat tansfe ate thugh the wall and (b) the tepeatue distibutin () in the shell. In this pble we can fllw the basic pcess in the ntes n heat eatin t slve the basic diffeential equatin f nediensinal heat tansfe in a cylindical shell with heat eatin. hat diffeential equatin is.
4 Febuay hew slutins ME 75, L. S. aett, Sping 7 Page d d d d Multiplying by and ding an indefinite integatin ne tie gives d d d [] d d n d d [] If the heat eatin is nt a functin f distance the heat eatin te siply beces, and a secnd integatin (afte ultiplying by d/) gives / ( ) d d [] d e can evaluate the tw cnstants f integatin by applying the bunday cnditins that the tepeatues at and ae and, espectively. e Subtacting equatin [] f equatin [5] gives e ( ) e can slve this equatin f. ( ) ( ) Substituting this esult int equatin [5] [6] gives the value f. Once the cnstants ae nwn, the tepeatue pfile is given by the fllwing dificatin f equatin []. his quadatic equatin in tepeatue can be slved f f any given value f. ± he heat tansfe at any value f can be fund f equatin [] and the definitin f heat tansfe. [] [5] [6] [7] [8] [9]
5 Febuay hew slutins ME 75, L. S. aett, Sping 7 Page 5 e L d d L Aq Q e d d [] e can apply this equatin f any value f between and,.using the value f given by equatin [7] If the heat eatin te is ze, u equatin f beces [] Substituting this value f int equatin [] with the heat eatin te set t ze gives the heat flw as L Q [] Substituting the value f f equatin [] int equatin [] with the heat eatin te set t ze gives [] he tepeatue f n heat eatin is fund by substituting equatins [] and [] int equatin [9] ± [] ± [5]
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