Steady Heat ansfe ebuay, 7 Steady Heat ansfe wit Cnductin and Cnvectin ay Caett Mecanical Engineeing 375 Heat ansfe ebuay, 7 Outline eview last lectue Equivalent cicuit analyses eview basic cncept pplicatin t seies cicuits wit cnductin and cnvectin pplicatin t cmpsite mateials pplicatin t te gemeties w-dimensinal sape facts eview Steady, -D, ectangula Q & & Speical sell ( π / / ( q e& gen Cylindical sell π ( ( is an aveage temal cnductivity ( a cnstant value if is cnstant, tempeatues at x,;, tempeatues at inne ( and ute( adii eview Heat Geneatin Vaius penmena in slids can geneate eat Define e& gen as te eat geneated pe unit vlume & pe unit time igue - fm Çengel, Heat and Mass ansfe e gen I ρ I I ρ V 3 eview Heat Geneatin II empeatue and eat flux equatins.8.6 Plt f ( - /( - f Heat Geneatin in a Slab e& genx e& ( x ( gen & e& genx & & ( Q x Egen Qx x empeatuediffeence ati...8.6.. & gen e H (...3..5.6.7.8.9 H H. H. H H H 5 H x/ 5 6 ME 375 Heat ansfe
Steady Heat ansfe ebuay, 7 Steady Heat ansfe Definitin In steady eat tansfe te tempeatue and eat flux at any cdinate pint d nt cange wit time Bt tempeatue and eat tansfe can cange wit spatial lcatins, but nt wit time Steady enegy balance (fist law f temdynamics means tat eat in plus eat geneated equals eat ut 7 ectangula Steady Cnductin igue 3- fm Çengel, Heat and Mass ansfe e eat tansfe is cnstant in tis D ectangle f bt cnstant & vaiable d dx igue -63 fm Çengel, Heat and Mass ansfe 8 emal esistance Heat flw analgus t cuent empeatue diffeence analgus t ptential diffeence Bt fllw Om s law wit apppiate esistance tem 9 emal esistance II Cnductin ( cnd Cnvectin s f ( s f cnv adiatin ad 3 3 σ cnd cnv ( ad Wee Des te Heat G? Enegy cnsevatin equies tat cnductin eat tug wall equals te eat leaving te wall by cnvectin and adiatin Q & & & Q Q3 Wee Des te Heat G? II igue -8 fm Çengel, Heat and Mass ansfe igue -8 fm Mass ansfe igue 3-5 fm Çengel, Heat and Mass ansfe ME 375 Heat ansfe
Steady Heat ansfe ebuay, 7 Paallel esistances ( su Cmbined Mdes igue 3-5 fm Çengel, Heat and Mass ansfe su ttal ttal s s ttal cnv cnv ad s 3 ad Define ttal eat tansfe cefficient, ttal ttal cnv ad ( Cnvectin cnvectin plus adiatin igue 3-6 fm Mass ansfe Cnductin ( ll values ae te same Cnvectin cnvectin plus adiatin ( Cmbined Mdes II Cmbined Mdes III is aea nmal t eat flw igue 3-6 fm Mass ansfe is aea nmal t eat flw igue 3-6 fm Mass ansfe ttal cnv, wall cnv, Seies esistance mula 5 & Q If yu nw,,,,, and, but yu d nt nw and, can yu find te eat flux? Once yu fund te eat flux fm te infmatin give, can yu find and? 6 Pblem use as a in tic bic wall wit.6 Btu/ ft. e intei tempeatue is 7 and te extei tempeatue is. e inside and utside cnvectin plus adiatin cefficients ae 3 Btu/ ft and Btu/ ft, espectively. ind te eat flux tug te wall. Given: Wall wit in / ft and.6 Btu/ ft as cnvectin n tw sides. 7,, 3 Btu/ ft and Btu/ ft. ind: 7 is aea nmal t eat flw Slutin igue 3-6 fm Mass ansfe Q 6.5 Btu & ft 7 ft ft ft ft 3 Btu.6 Btu Btu ind values f and. Can yu cec tese values? 8 ME 375 Heat ansfe 3
Steady Heat ansfe ebuay, 7 Slutin II Slutin III is aea nmal t eat flw igue 3-6 fm Mass ansfe is aea nmal t eat flw igue 3-6 fm Mass ansfe 6.5 Btu ft 7 9.5 3 Btu ft 6.5 Btu ft 5. Btu ft 9 Hw can we cec esults belw fund fm analysis f veall pblem and cnvectin pcesses? 6.5 Btu 9.5 5. ft nalyze cnductin step f cnsistency..6 Btu 9.5 5. 6.5 Btu ( ft ft ft ( Cmpsite Mateials Cmpsite Mateials II Hw wuld yu analyze tis pblem? igue 3-9 fm Mass ansfe igue - 5 fm Çengel, Heat and Mass ansfe eview Cylindical Sell & cnstant π π 3 π ( Q Cylindical Sell wit Cnvectin π π igue 3-5 fm Çengel, Heat and Mass ansfe cnv, cnd cnv, π π π π cnv, π cnv, ME 375 Heat ansfe
Steady Heat ansfe ebuay, 7 Cylinde plus Cnvectin esult igue 3-5 fm Çengel, Heat and Mass ansfe π π π We can eaange tis equatin as swn belw ( π 5 Pblem t-wate pipe ( 35 Btu/ ft in a use, made f ¾ inc scedule pipe (OD.5 in; ID.8 in is ft g and cntains wate at. e ai aund te pipe is at 6. e eat tansfe cefficients inside and utside te pipe ae, espectively, and 3 Btu/ ft. Detemine te eat lss fm te pipe. π( 6 igue 3-5 fm Mass ansfe Slutin Given: 6,, ID/. in, OD/.55 in, 35 Btu/ ft, ft, Btu/ ft, 3 Btu/ ft ind: Q & π( 7 Slutin II Given: 6, ft, ID/. in, OD/.55 in, 35 Btu Btu/ ft, ft, Btu/ ft, 3 Btu/ ft ft 35 Btu,9 Btu ft 3 Btu. ind: Q &.55 π( π( 6 ( ft ft (.6.7 7.69 Btu in in ft.55in. in in in ft 8 Cmpsite Cylindical Sell Cmpsite Cylindical Sell II igue 3-6 fm Mass ansfe π igue 3-6 fm Çengel, Heat and Mass ansfe 9 π 3 3 3 3 ME 375 Heat ansfe 5
Steady Heat ansfe ebuay, 7 Cmpsite Cylindical Sell III igue 3-6 fm Çengel, Heat and Mass ansfe 3 3 π π π π 3 π 3 nte Pblem Insulatin wit. Btu/ ft is t be added t te pipe in te pevius example pblem. Detemine te eat tansfe if te insulatin is ne inc tic. 3 π π π π3 ( π 3 3 Knw all tems fm pevius example except tese tw 3 nte Pblem II Uncanged esistances fm pevius example ft.53 Btu New and mdified esistances ft in ft 3.55 5. 33 Btu in..55 Btu ft 3 Btu 3.55 in.63 ft in ft Btu 33 nte Pblem III ( π 3 3 π( 6 ( ft 86 Btu ft (.6.7 5.33.63 Btu Insulatin and ute cnvectin esistances ae lagest Inne cnvectin and pipe cnductin negligible Oute cnvectin esistance less wit insulatin 3 5 5 Heat lss (Btu/ 5 Effect f Insulatin icness 3 5 6 7 8 9 icness (in Insulatin Inceases Q &? Wy des initial amunt f insulatin incease eat tansfe? adeff f tw esistances dded insulatin adds cnductin esistance dded insulatin als inceases ute adius wic deceases te ute cnvectin esistance /( ute ute /( ute π ute 35 36 ME 375 Heat ansfe 6
Steady Heat ansfe ebuay, 7 8 esistance 6 ( ft /Btu esistances f Pipe Insulatin.5.5.5 3 Insulatin icness (in Insultatin esistance Cnvectin esistance tal esistance 37 te adius f Maximum Q & ( π i maximum Q & : d Q & d π ute adius temal cnductivity f ute laye i ( te / f maximum Q & 38 adius f Maximum Q & / f maximum Q & In te example pblem 3 Btu/ ft, and. Btu/ ft s Q &.667 ft.8 in f maximum Pipe adius was.55 in;.8 in gives an insulatin ticness f.75 in Nte tat / des nt depend n i and is usually lage tan i Speical Sell wit Cnvectin π cnv, sp ee is n adius f minimum Q & 39 igue 3-5 fm Çengel, Heat π and Mass ansfe π sp cnv, cnv, sp cnv, π π π Speical Sell esult π igue 3-5 fm Çengel, Heat and Mass ansfe π π π sp π( Cnductin Sape acts Simplified analysis f multidimensinal gemeties wit eac suface at a unifm tempeatue Use sape fact, S, wse equatin is fund fm tables lie Çengel able 3-7 Basic equatin: Q & S( S must ave dimensins f lengt Equatins f S depend n paametes in te diffeent gemeties ME 375 Heat ansfe 7
Steady Heat ansfe ebuay, 7 Example Sape act m able 7- in Çengel, Heat and Mass ansfe 3 S/ 3.5 3..5..5..5. Buied Pipe Sape act 3 5 6 7 8 9 z/d ME 375 Heat ansfe 8