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# Heat Transfer 10th edition Solution Holman MCHE 469

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Heat Transfer 10th edition Solution Holman
COURSE
Heat transfer
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TYPE
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443
WORDS
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Heat Transfer, solution, Holman
KARMA
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This 443 page Bundle was uploaded by Nick C on Thursday January 21, 2016. The Bundle belongs to MCHE 469 at University of Louisiana at Lafayette taught by in Spring 2016. Since its upload, it has received 23 views. For similar materials see Heat transfer in Engineering and Tech at University of Louisiana at Lafayette.

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Date Created: 01/21/16
Chapter 9 9-1 {4- \r/3=o'oo77Re \kr"prF) ro'o 4#1"'=ooorrlw]" n=ff16.oo71rto'|+r<r, Tw)1''t 1 [or'p") t0'6 -r)12t3 (0.w77)tLl+ur, U=ffi= TLWJ kf ( ,, )r/r'8 [ortp," ) (0.077fr fit k zlaee- Tw)ft3 y2ilse N.U_ rzt @ 9-2 Flatplate 4r;, o Rt/ _ tix-3trf d3 l4PrkL(r, -r)1t' It7 ,4 L shfsf2 J aprkUr, -r*)131 R"/ -wLr"l l rP 'skf3tes - T*y3' R./ = J.771r " -ttJ L Ff"hfg- ]"- vvt Chapter 9-3 TurbulenFilm E=ff?0077Re,,0'a 4' Rev=ffi lr'T,4) . _Et(r_r-r; _ELwqs _ T,) hhr, hf, hf, E = LW(T8_ T.) z r0.4 hhr, -o'aonlffi) LW(78 -T*) ( ,r, )r/3 Itrn7') orwo'6(r,- r)kr(pr, d''3ls'3 ^ _lo.wt(+)o L pr''o' l 9-5 ?=rylq=90oC L=1.2 py=964 ttf=3.t5x10{ kf =0.676 hf,=2255 kllkg -l rQ6+)z(g.806)(2.2x t0)(0.6t)31'' y h I3[ (.2X3.15xlo-4xloo- 80) = 6082 _ - - J '',f"c q (6082)(.2)(0.3X10080) 43,790 W m - ^!?'=7??=r=0.019skg/sec- 6s.skg/hr 2.255x l0o 94 L=0 .4m 0- 90 30 60o Tw: 98oC Ts l00oc p- 960 = - - 4 hft 2.25 x 106 k 0.68 lt z.gzxl0 g6q2(9.806 -ll E =1.tr[( )(2.z5sx06X0.6s)3sin6-o -r-.rs2-jy L (0.4X2.82xlOa )d) | ;F { q =(14,152)Q.q)2Q)=4528W it 2.008xt0-3 kg/sec 7.23kg/hr 1r.t- Chapter9 9:7 k- 0.684 ,62 It-3.0x104 P=9 (962)'(g.gx2.255xIo6X0.6gq3 1ua w h- l.l | =lo,93o .--5- { (0.5x3x10-4x100-95) I mt.oC q -(10,930x0.5)2(100-95)- I36,510W m--:0.0606l0 f kg/seQ 217.9kg/hr 2.255x 10o 9-8 w Tf = 4'soF k -0.5g3 p-1.37x10-3 p-99e m-oC U4 h=l.l - 4576 y (.5):lJ7x 0-.,:rs)(;) m'.oC - - l'J=( q-(4s i6Xl.t2(1 ,\e) 8.58xlOa q vn= =0.0361 kb/sec=l30 kg/hr hft 9-9 Ts 38oC Tf= 38 + 3o= 34oC p-s90 v-0.345x10-6 2 hfr-llll kJ/kg k-0.501 , a, -[(590)(9.8xr.1r lo6xo.50l)'lt'o n-- " "L I -s877 y (0.4x0.345xro{x8) J m'.oC q - LALT = (5877X0.4)2(8.0) 7522W 1ix-+ =0.00677kg/seQ = 24.38kg/hr hfr 7a? Chapte9 9-1t Assumelaminar condensatron 4"t =328oF hfs=889 Btu/lbm psat 100psia Iw =280oF Tf = 304"F pp= 57.29 tUm/ft3 Btu '=o.Mg hr.ft ft=0.395 Pr= 1'15 - hr.ft.F Py Pr :. p/pf p)= pp2 r _t E=o- --lUafrt1on.ktIr,= '""55 Btu -Z")l h.'ft2oF A=3-Etat(Tut-7,) n - (1855)z(l)(a8=--'-- lbm hf, hfs L 02XS89) hr.ft e-L2 35uC 95oF ps= 0.8237psia pr =(0.8X0.8237)=0.659psia = 88oF= 3l.loC hfs=lM3 Btu/lbm =2426 kVkg Q ,, =Y= l6.55oc p =99g ttfl.lx l0-3 kf=o.596 Ptf =7.8 Lt4 h =0.7251-egq2(9.gXz.4z6xI06 Xo.596)3 -5425 -J (.lx.l0-'X0.05X3l.l 2)r \IF \ ^ / /mo.ocrl q = (5425)n(0.05X7.5)(31.2)=185,970 ,a= 185'n0 --276kglhr 2.426x10o==0.0767 kg/sec 9-t3 Assumelaminar film andheattransferarealftZ Er: Tf =l54oF pf =61.1 ttf=1.02 kf =0.382 Bt-u EtL=-'- -lA: hr.ft2."F Tf=l77oF pf=ffi.6 hfr=gg2.l ltf=0.86 k/=0.388 Bt-o E-t=8-- - hr.ft2.oF A = EIALT 1.02xlOsBtuThr n,= tqeLT --1.37 tOsntuThr Qz- Qt =0.344 34.4vinarease et tr+ Chapter9 9-r4 psat= 690 kPa hfr= 2-o'l lo6 Jlkg T8- l64oc Tw l38oc Tf =l5loc Pf =915 lty- 1.86x loa kf = 0'683 Sat.Steam Condensate L ASSrm (IlninraIr)/: l4 p-g'h,yf- - E = 0.72',e k a .jt to,45o+ oC Check Rr/ =73.6 lrf'('s- -w) mt. a Lamminar q kg q =Ex (tg- T;) 2z,i3I Wm v|2= =38.1 .m hft sec length 9-15 - k 0.684 It=3.0x104 p=96 2 E = 0.72s[962)'(g.B)(2.25x lo6xo.68q3 -7e62 y (3xl0-4x0.3x5) mt.oC ]"' q -hlrdL(Ts-T; = (7962)rc(A3xl5X5)= 5.628x 105W tix + =0.2496 kg/see= 898 kg/hr hfr 7,.5 Chapte9 g-16 coz hft-153-2kfkg at 20"c d =0. m L- 1.0m T*= l5"C Ts =20oC Tf- l'7.50cp-79s kel^t k-0.0897 m.oc r/0.0935 xl0{ - - p (735X0.093x 10-6):6.87xl0-s Pr 3.78 Iegrz(g.gxl53.z o3xo.og9 rl4 h- a.725l| -t s32 y L (6.g7xl0-tx0.lx20-15) m2."c q= (l532yt(0.1)(1.0X15)= 2406W = hhf,=rir(153.2x 103) rit 0.0157kg/sec=56.5kg/hr g-17 n=20 ?=ry=940c Pf=963.2 =0.25 = d in. 0.00635m Ff =3.06x l0+ kf =0.627 hfs=2255 kryke n =o.tzsf rgor)t(1sxr.rss lt'o r*-'"m2 .oc L(3.06x10+X20X0.00635X100-ss)l' 9-= 1+007n(0.00635X7845X1007.51 t05 w/m . 7.51x105 ^ - kg m-- Z.ZSffi = 0.333kg/sec1200 hr.m g-lg - - d 0.05 Ts 100oC Tw: 98oC [= 1.5m p- 960 k- 0.69 p- 2.92xt0-4 h?r hfr- 2.255x106 U4 = e6tD2(9.xo.6g)(z.zssI06) - y E o sssl (2.92xl0-4rx0.05)(2) 12,117 oc m2. q= (2,117)n(0.05X.5X2) 5710W tix 57rc - L'Jzx l0-3kg/see=g.lz 2.255x106 kg/hr 7cc Chapter '9-19 p -962 l0{ t= 0.68 n= l0 d=0.O254 m lt=3.0x o -- (soz)2qsxz.z ss o6Xo.os)3= n o.t zsl (10)(0.02s4Xx3l0'-+X5)lt' 8264.' 'oc L I q -(82(/l00)a(0.0254X2X0.3048Xs)2.0rl0s w =320.9 ,a= hft o.og9lkglsec kg/hr 9-20 =98.5'c P=960 lt=2.82x0{ ft0'68+ ? m."C n =10 d =0.0254m xto6x9'os)3-l n o.tzsl(goo)2(q.sxz'zss =e526-"y= L (10X0.0254)(2.82x10-X3I --'"c q =(g526)(100)z(0.0254X3X0.3048X3)085 105W ,ir=hfg 0.0925kg/sec=332.9kg/hr 9-22 hfr 1135kfkg Tf:ry=86oF-3ooc P=596 V=0.J,9>10-6 k=0.50; n-20 d -0.00635m 1ua =0. ,2-- (5g6)(9.8.135l06Xo.5oT3 :ss04 E 72sl 9xl0-6Xo.oo63sxeo 82)(;) m" 'oc L( J q-(s904X 0063sxo3o48xeo -s2)(;)= 63,87W - -L - m hft- 0.0562g/sec 202kg/hr 711 Chapter 9-23 zr=10,000kg/hr2.778kglsec hf,=130.09kVkg t*-*'O q=hhyr=361,400W ? = =95oF=35oC p=t276 y=o.l93xlo{ k"=0.07+ n=25 d= o.ol2 m'oC rrzzoxs'8xl'?1 f=o.zzsf 1''osLJ'a-w Itzsxo.otzsruo{f)ro.le3)J ;E q=MLT=hnrd,LLT 36l,400(823.4)(625)n(0.0r20/:) L=3.22m \9/ 9.?,4 d =0.012m r, -32'2+-26'=29.45oc p =1295 "2 k=O.M v=0.194x10{ c=9M+ hfs=l33.6skvkg kg'oc =n3.e5 h'fs + (0.6sX0.e84X10)(;)=w.kVkg u4 h 0.5551(1295)(9.8137xl06xo.oT3 -r-7-r\';[[ L (0.194xl0-6X0.012X5.5I T g =hTrdYr-\14- - t rz)(ro)f:303-Wm / \ - v r \ g ) L 9-25 LT,- I07-100=7oC - fr=7.96(7)4 19II kflm2 q- (19,110X0.35734W ?c8 Chapter 9-26 HzOvapor 213"F p - 14.993 SurfaceensionH2O at212"F=58.8dyne/cm - = = tensionrce firzp, p) pressureforce Lwo surface HzO at2l2"F Pt= 14.696 r=-=0.00226tn. Pv-Pt g-29 LTr-119-100-19"C c6=0.013 q- t.zW^2 A q -(t.zlo6xo3)2- loS,ooo g-32 Re r=4r h*1ffi]"- n=ll:h*h ltr +(r?2 snsnn'lt=o h=lr(ffi h' 3[ r4+pfuT 3\",),=L -nrct ) E-LTL-q=hfsT r= - hfs Re1= Re-r/3(m)"'= r(m)''' fr=ffi n=(i)'[ (i)-"Re,-,/r=dl\$''' ffi) r.466Re1-"=r(ffi) 9-33 q LT*- l0oc A -r.9MV^2 7L, tChapt9r 9-34 7", ll0oc Tb^ug=96oC A1,= l6"C d -_1.cm um=1.2 m/sec BrasCrf=0.006forptat xtla wfmz X=r.trl04 #=7.885 2 ,ulr** (7.88"to*{ffi)' = ,.0, ro5flm n l X = (8.02t0syz10.0Iz5)1,494wlm r-lboiling ForcedConvection At 96"C p = 960 =2.96xl0+ t= 0.6g pr=l.g3 (96oXL2N'orl25). lt p"= = 4g,650 2.96x10- ,ft = 73e6 #L(0.01e)(4&oso)0.8(r.83;0.+ # nl = = il"o,,u396)n(0.0125X14)4066Wm 4 =31,494+ 4066=35,560Wm Zltotal 9-3s q (2.3)(2.2xl06; 5.I87xt06yhr =l44W g A=#=20,382 wl^, *=5.56(LTr)a LT,--7.78"c =107.8"C" A T. 9-36 Cu-H2O p=latm d=0.005m T*-Trut=lloC ,Crf 0.0128 Cy(plat) FromFig.9-8 at LT, I loC Q=0.11MW/m2 A Forl=1.0m A=zr(0.005)=0.0157m2 q =(0.lx106X0.01571) t72gW lmlength 31o : Chapter 9'31 AT,= Iloc gl =4 =4000 4 =t161ffi wlmz Alptat Alcu hr'ft' L =(126,160)z(0.0051982Wm L 9-38 A4=10"C p=3MN/^2=435psia d=0.02 L=lm h- 2.54(lgr3t's=r17,56l+- m' .oc q= (17,OnQ.02)(lXl0)=I l,042W h7,at435psia771.3Btu/lbm=1.794xt06 rytg *= ]j'rY?==6.15xl0-3 kglsec22.16kgihr 1.794xl0' 9-39 4=a.zMflm2=200kwfmz d=0.003 L=7.5cm A p=1.6 atm ft 5.56(4Tr)3 Mr, A= 2oo to3 5.56(T)4G.6)0.4 LT*=l3.l4oc = =ll3.6oC '7. atl.l6atm p=23.52psia 4ut 236.5"F = l3.6+13.14=l26.7oC 940 d=2.5cm p= l0l *34 135kPa LTx 4oC Assumeh :orizonh= 5.56(4)3 355.+ m'.oC =htr =(355.8X4)t.423 A kflm2 aboverelationdoesnot apply h=t042(4)tt31654+ .oC 1=htr=6.616 kwfm2 m' A rit08 kg/hr= 0.252kg/sec q =(0.252)(2.ZSt06)=6616A A= 85.96m2 trdL t= 1095m 711 GhapteI 9-41 Ar,=lsoc t,o.m f+') =2xlo54 \4,/\$,31s1 hr.ftr =6.308xt0s w/m' n*=%q =42,050# (f)_"*, fotycerine.83X42,050=34,900 ;fl;" g-43 A?, =30oF Crf=0.008 Forplat = r.* 1gs B\ t = (2.5xt"{ffi)'=,.orr*rou f,|," ;S= 3.38 u3flm2 9-45 LTr=110-100=lOoC FromFigure O., IvrW/m2 *,= q -(1)*t= (0.7x106)zr(0.001xrz) 264w g-46 _944+100 Tf= 2 = 97"C d- 004'm P=962 k-o'68 p-=3xxl-4 ,8),. 0- hft 3 ,55Xl0(6vkg -[, 2)(9E)(2.ssl(00("X68)8)'t4 E=t0.7251ee )(2 :0.6 | = 12,533 Y L( .3:l0rXy)oaxlDO0 94.]J m''oc q_= urd,T=|(2, l3)zr(0,xlx1004 = L s33n('004000 94\,9450 Wm 71L Chapter9 g-47 4 =4*4 Ahotut Alrc Aluoline Lr,-ll0-98=l2oC =2r.6o1 flr_ =4x104 =r.26x105 flm2 # At 100"C k= 0.68 Pr 1.7 n= j<o.ot9x40,000)0'811.7;0'+ +- m'.oC cprut Alurast(r.26lssrf ] =rz.eto5 flm2 ["u*"r J ql 12.8 lo5 /ru flbr*- =lo.6-oJ- " LT ll0-98 mr.oc mo.oC tl}49 :,hfs2.255xt06ryt<g pv=1.5 kel^3 pt=998 kel^3 ' =70 mN/m I = ro6xr.rrl-to.ozlts.s)tssoll''o = Al** 24'.zssx 'L (l.s)' J \r*g998) r.8mdm2' 9-51 =2.255x yt<g = = hfs to6 T, 813r 4"t 373K (a) ,,= = 320oC=593K pt=9ffi pu= 0.365 Y ks=0.042 ttu20.6x 10{ cou2Cf|l0 LTr=440 , n.^f(o.o+z)3(o.rot(g00x9.8x2x106 2000)(0.4x4000)lt/4 (0.012sx20x0ro{x+ao) j W =171.5m' .oC - 4 _ (5.66x t0-8)(0.8X8133734= 43.04 8t3-373 i1t Chapte9 h- (1|.s)4h-rt+\$.04 - /eQ,.Jy m'. oC q - hLTx (204.7)(4=0) A 9006gw I^, 9-52 Tw=28OoC 4at=l00oc v= 3m/sec d:0.0004 m = at25"cflow | Llhr1000xl0{ x996=0.996 kg/hr2.767xt04 kg/s ff=280+100 l90oC 463K hfs 2.ZSSx106 pr = 31= 996 Puy 0.478 cp,=1980 o =58.8mN/m 280 lm\=2.qtzxto6 A.=2.255x10+ogsof-\2) . rykg 312.+tz" tt %orop 1r.83 ro-3xss6)(0.0004) ,oo,(0.478X0.0588)0)T = L J 0.01(6{drop mass=teeo)[]zJto'ooozl3 l0-8 kg ffi #ordro's= ?l-^!ij{5 ,ntli""B2eo drops/sec 3.33810-6kgldrop O^* = (0.056X8290) 129.W 9-53 -'N!92 =96oc r'2 d=o.ol25 n=^l1i6=14 p=961 kel^t F=2.96x10{ &=0.68 hfs=2255 kVkg n=o.tzsl l"o=s088 -ft q= (8088X196)zr(0.0125X2.0)000=9.96x105W = rit(225103) rit=0.44kgls 314 Chapter g_54 l@=*91 rr2 =95.5oC P=961yel^t F=2.97x10a = =2255 ft 0.68 hfc kVkg = +Erqoo-gt> - tsoo 2255xt03(z.gt t04) =33.48,- *to3xo.oa)31t-,o "L JJ,-v' ',d(qote.97xtoaxtoo _qt) J tt4=4.234 Z= 6.85 m - (1s00X2.?xl0{) =0.134 p4 kg/s.mdeprh 9-55 =0.008 plat A7,= 15"C Crf CUr 0.013 4 =2.7x105 Btu-=851 kwf m2 Alpnt hr'ft" Mflm2 11,,=tm{g)'=ros ,9-56 p=5atm=0.507MPa L = 2.253(44)3'e62.253(103'e60,550w I^2 A q wl^' h= #= 2055 9-57 Tw=94"C Tf=97"C p=l atm d=0.3m lt=2.8x104 t=0.68 p=961 hfs=2.26xt06 J/t<g -o.tzslrr6Dtr,.8x3''z6* =77'-- y " v'''Jl (2.xl0-4)(0.3X10094) 1''o m2.oc I 9-= OIOO)n(0.3X10094)= 43,50 Wm L 71{ Chapte9 9-58 ,,=Y= 96.5oC hfs=225lxto3 rytg p =96t kel^3 It=2.96x10{ &=0.6g (g6t)(.g)(zzsxI03 6g)3 U4 h -r.r3[ X-. - 12,605 y (0.2X2.96l0 xl00 e3) mt.oC q- (12,605A.2)200 93) 3530w = h(ZZSsxI03) tix0.00156kg/s f= * toaxo.osxo.zxrqo -r)1'_o L(+xz.(e.sxzzssto3)(s6D2 | 8.6l'-s m 9-59 with h= 5ooom' .oC ,"r,r,! lt" =0.6s4 4r=o.76 \A^ .t (No.offinx) (Areaotube+fin)(5.9x10{X50) =0.0295m2 AverageLT - O.76LZ(base) T*^to l00oC z(base) Ar (avg) f,trice+l q 108 6.I 0.01 106 295W 111 8.4 0.03 106 885W tt7 13 0.15x106 44ZSW 9-60 LTr=104-100=4 4=oQ'?S)z =0.M9m2 =rcaz(tT,)r,= = A 1042(4)rt1654w I^2 q= (1654)(0.049)lW 9{1 LT*=104- 100=4oC wl^z X=rUtO A=T+z(0.05X0.|25)=0.03l9m2q=(|654)(0.03l9)=52.8W 3t, Chapter9 9-62 LTr-I loc Fig.9-8 =0.95x 106wvlmo2 Xlo* ,Csf(plaO=0.013 CsfGeflon) 0.0058 q (r')' 7- tr"J - *= (o.exto6{ffi)' t.olxto7 w l2 q 11 A -l'o7xlo7 =6.3xlos y LT 17 m"'oc l= rdL= n(O.OlXl)0.03 14m2 q- (.07x lo7Xo.o314) 3.3'llos w 9-63 = - - = r\o orubes (10X20) 2-0 d 12 mm l =l5 Tw 86"C Ts l00oc Tf= 93oC hfs=z.26xlo6 kf =0.678 p1'-3.06xlOa Vke #L If' Pt =963 l4 ',8=0.12 p2snrrrr3 = o.7zrleorr(g.s)(2.?o*{1x.6'7tJ' 'l"ltd(-4,)J1''o V"'"1(15x3.06xt0ax0 .12x(00-86) = 6s20 y mt.oC :Af.rter 2Dn(0'0l2xl)-7 '54^2 'q,= - = (6920)(7.5400- 86)-7.3IxI0 hA(Ts T;) . 111 Chapte9 g-64 = = = T, 85oC % l00oc Z 50cm W=25 cm Tf =92.5"C Py=964 ttf3.lxl0{ kf=0.617 hfs=2.26xt06rytg Assumelaminar h -r.nlP'sh*kt 1''ol.,r[gu)'(La(2.26xI06)(0.6)3 LLpQr-r;l L (0.)(.lroT (r00 85) ]"- 77'J y = m'.oC - g5) q EALT= (7723)(0.25X0.5)000 = 14,48w = hhy, t'm=ffi4,480 = o'006kg/sec=23J kg/hr " R.= 4d _ (4xo.oo64)_=330 PFf (0.25X3.1l0-) :LaminarassumptioOK g =l--,rt('r -J : [(+xr. ro*xo.ozzxo.t(roo-ss)lt'o " ;',)1''o l =1.3xlOa m 0.13mm 9-65 rit=l|S kgls d=1.25cm Sp =1.9cm Tw=93"C Ts 100"C n=numberof rows Tf =96.5"C 'hfs=2255 kJlkg p=962 lt=2.96x104 ft=0.68 o = Q oz)z.s) -?s to6 )o.os)1''=76,I86 "-V''LJLl (2.96@ ' q=(1.3)(2.255xt06).932x106 WEA(T.-T*) ir= sideof squaarray 3x= length x=n(0.0125)+(n-1X0.019) 'A = nzrd(3r) 0.0oln3 0.0o224n2 ' = -a.00224n2) ,nts32x106 "?r?!'<o.ut37rn3 'Byiteration=25 x=0.769 3x=2.306 numberof tubes n2 625 776 Chapter9 g-66 m= 600 kefi'= 0,667kg/s q- *hfs (0I667)(2.255 106) q- 3.75x l0:w Tf= 98'5oc p 960 tt-2.82x lOa k- 0.68w n=20 d -0.01I I I m.ooc g6cl(e.8)(2.255 106X0.68)lrt+ i= 0.72zsl_ -tI 10r,23 I_oC L (20x0.01x2.82xI0-4x3) J m". q- hnrdL(Ts -Tw ) 3.7:x ( s=((1,123X400)z(0.01)Z(3) 0.983m 719 Chapter 7 7-3 Show that I fo,anidealas. .= r=lfv\ar Pv=nRrr v =nr (#), =T- )p u=Lry(T)=+ 74 Tf=4O"C=313K P=1.128 L=0.3048m F=l.9o6x l0-s Pr 0.905 p =1.13 k=0.027 =0.00319 Gr=1.558108 f d;= q0.393n-l/2(0.952+prlGr-l/41=.014 u=ttnaxu,.6=t= Uur=0.722 Ns Nn O.50g ptrp.g52+ pr)-ucftt -42.o f =1Nu{ = s.o aw:: (Freconvection) 3 L m'.oC ForcedConvection ReyPttPr4'.o=l3,ooo Laminar tr=0.664Rerttvvt(f) = o.ot \L) +m'.oC fi]Forced).03 h(Free)_ 5.0=r.ZM Conclusion Heattransferis stronglyinfluencedbyyof fluid overtedsurface 7-5 T1:325K Pr=0.7;v: l8.2xlOt6:0.02m Gr* (9.8)(l/3250es0)x.2xt0a)24.455l0ex3 F;q.-20a):0.02 (455xI0n;-t,st 0.71-u2 52+0.7)t/ta ;1 1o.e x:0.9 m Chapter7 74 j' T6\q Tw \ A{tR\ u, (v uxr \e:I,-i)' du--ux L(-;x'-#) .*('-#)'] =o dy [* \$t ffi v- 6 F )=d , = max u occursat | * =uxt = umax tJ[. 3y) nux=z7-t' 7:7 Spacing-26 9L= 3.g3Pr-rt2(a.gsz+)l'oGrr-rt4 ,r, 65+ '-' Ir2 =4--5oC L- 0.3 Pr-3.9 - ui1=_(+.sxtol0xo.l)3(ozs)= l.z5xl0l0 l.g Outsideofaminar ranbutuseaboveequationor estimateof. thickness u ot(ffi)o.nt, + 3.e)r/4(t.25r0)-ta=0.N26 m SpacinshouldbeatleastI cm. ?\$o Chapter 7 7-g Tf = l00oF k -0.63 GraPr- (3.x 10r0x0.02)3(t30?q( = 8.8 106 -\1l -0.53 Nu C(GrPr)n C n=! 4 h- =eoe mt .oC g L- nmlr* -r*)= (eoe)rc(0.02)(r3ru{;) = rso4 Wm 7-9 L - 30cm Tf =57.5oC 330.5 K v - 19.23 l0-6 - Grr=t-'9.8)(,,b=)(t"15X0--'-1, .'108 19.23x l0-o - (0'3X31- D - --:9.01 x l0-2 m-9.01cm (1.84 lo8)rt4 7-10 d 3OO+25 a. EdA .^.r, .^J ,r=T=162.5"c=436K v=30-lxl0{ k=0.036 pr =0.68 Grpr - tq.s{#Xmo-za(l)3ro.osl = 4.64xros (30.1xl0*)' 0'936 a = (0.1')(4.6l4x'l/3 u.o I 1 m"'oc q =hA(T* -L) =(e.OXt)2(300 25)=1656 * 7-rl s -*tt4 6z+=tt.o{ff;" =r-r4in *=(o*)'^ 7+l 7-12 - - : H A.2;W 3.0 Tr:60oC 333K v - 23.3x10t;Pr 0.7; k:0.028; p: 1.06 Gr*- 3.47xhd;Eq. (7-20a)6: 0.0139 Eq.(7-20b); ux: 2.74m/s uma,x (0.148)(2.74):0.4m/s rh- ell6xl.06X o.4lx0.0l3e) 0.003a kgls :2.43x1A7; - - GrPr h (0.02sX0.5910.2)(2.410x7)t/4.8 q -(6 8X(0.2X30X100 20):278W 7-13 Tr:300K; H-0. 1; w -2.0) k-0.61; Pr:5'S5; p:996 - ZAoc;Grpr -(1.glxlotoxo.)'(20) 3.92x108 AT - Gr* 6.53x107 h- (0.61/0.X0-59X3.82xlO*)t'o503 q: (s03)(0.xl'0X20):1006 w - Eq.(7-20a); 6 0-00292m Eq.(7 -20b); v- 8.63x10-7 _ (0.l4gxg.63xr0-70.x5.17x0.g52+ 5.g5)-tt'(0.53x107)1/2 uma:< Q.AZmls fu -(sl16x(ee6x0.02x0.002e2):0.0033 kg/s Z4t- ChapErT 7-14 rf =-3*30=6t.5oc=334.tz+')lL p=+=2.99x10-3 2 P 334.5 v =l9.l9xl0{ k=O.0289 pr= 0.7 Grpr _(g.g06x2.gx l0-3x93- 30)0.g)(0'7) 2'05 l0ro (19'19 rOffi " h=0'?2:9 (0.1X2.0xl0r0'r/34.394 +- = 1.8 - m"'oc q (4.394)z(0.075X1.8X930) 1l7.4W 7-r5 A=n00-95 =1005 Wl*tat 300K fl=3.33x10-3 y = 15.68 10{ Pr =0.7 k -O.O2624 = (9.806X3.3x l0-X1005X6)4 - Gr* (15.68x t0-6)2.02624) v'59 t0r5 1r=0'02924(0.17X6.59x101s)l/3.7 I 6 m'.oc A? = 1995=l5ooC = =llooC 6.7 ?w.rr 150+ 2o Thisdoesnottakeintoaccouradiationwhichwouldlowethetemperature substantiallv. 7_16 LT =24 20 = 4oC t?=(0.95X41rtt1.508+.' m'.oC q= (1.508)n(o.3)(2.0)(4ll.3 w 24, Chapte7 ,7-17 q*= * =222.2 wl^2 Take propertiat300K (0.3), k=O.02624 v=15.68x10{ h=0.71 F=3.33x10-3 Gr*= Q.Bo6(3.33xrc-3)e22.2!;gJr=5.694x108 att5 cm (0.02624)(15.x to)2 =9.llxl09 Gr* at30cm at15cm h=oi'6)4' ro.6)t(5.6e48;1o.zylrs.sI W + at30cm 1r=0'U.J2a(0.6)t(9.ltxtoe;10.24.:19m'.'oC E -lh-=--(r.2s)(4.796)=6.0y, ,.. m'.oC 7-18 L=0.3 Zra 55oC T*=20oC Tf=37.soc= 310.5K v=16.7x10{ k=O.027 pr = 0.7 = tq.a{n+sXss-zoxo.:)3to.zl Grpr (16.xl0-)' =7.4'xr07 c=0.59 *=! 4 E=W(0.59XT.48 x107)r/44.94 y 0.3 "-' ^2 ."c q 2M(7.- = - 2A) 1.1 W L) Q)@.94)(0.3)2(55 7-19 , ar= =6ooc=333K r2 rt=2.r6xlo-7 Q)\$.01x10s) Y-@ =0.293kgl^3 k- 0.159 Pr o.'l (g.s)(#Xo.zgr)2(toooxo.ot)3(o-z) Grpr = -6.87107 (2.1x l0-)z (0.159X0.59X.87 I07)rt-_lL w h- 0.61 + - mt.oC q hA(T*-T*) =(14X0.61)2(1000\- 417W Lqt Chapter'7 1e0 tf'=-57!= 3o.5oC 303.5 K | =16.lx lo{ L ls 0.0265 Pr 0.7 (e.8)(#) -4)(61)3(0.7) GrPr = 67 - 1.Mx l0l2 (16-lx10-6)2--r' 561/2- 0.825- 0.387Ra1/4 =33.M (11lgxo.o265) Nu 1118 h_ :4.'YY y 6.1 m'2oC q- hA(,T*-T*) =(4.86)(1X1.22)(574) = 1907W -0. UsingC 1and *=! 3 Nu = Q.1(1.041012)rt3 l0rz 9.5Tolower 1-2r 'rr 49+ 2l rf=- =35oc - 3ogK |=16.5x10-6 16 0.0268 P1= 03 (e8)(#)(4s-2 rxt)'0.7) GrPr = -2.27 109 (16-5xtO{)2 0'387Ral/a--- 661/2= e.gzs* - Iz.sg M-158.1 (w)et'^u]o''' [t* (158.1X0.0268) y h- =4.24 -oC mt q- hA(t" -T*)= (4.24)G)2gg-21) =118W UsingQ=0. Iand *-! 3 Nu = (0I)(2.27 10e)l/3l3l ITVolower 7-22 Takepropertieat300K v-15.6gx10-6 k-0.02624 Pr 4.7 1or2= {1st(#)ttolrlrg.zl (5.69x10-u)' L -10.35m t4{ Chdpte7 7-23 Take propertieat15.56'C,t =0.595 &=1.08x1010 ,pk - (l'0xl0t:Xl-q00x0.2s)4 cr-in = 7.o9xl0r0 ^,=Y(0.6)(7."9 x loro)r/s2tl o.25' E=ir.Z1hx=264 -j{- -2 .oc looo =3.79oc MLg,,h 264 Tw + 15 l8.79oc F3.79 it isverynearlisothermal. 7-24 cyrifder: h=n2(+\d'o) verq prat: h=r.42(+\t'o \L) .-"vr:-:- q =l<r.rr{4)"0 *or =2.o73d3t4^rst4 L \u/ nati, q=(r.42t+)t'r =r.ee2d3t4^rst4 to, : Very close : i "16 7-25 Tf= 200 +120 r160oC NaK,22Va Na k -24.4 Pr= 0.019 d -0.02m Ir-0.4x 10-3 [= 0.4m p-830 (m-Em t\ p =6eol 760 -e3 l-_Z.5xt0-4 \ ) h - (9.822..xtO*)xgry)2(200 120x0. =l.28x lOs \Ir4rr ,;;: t^=rt @.+xto-3) - 0.518(GrPr)1: ; 0.36+ q..58(128Ilqs-IYj=4.3L Nu 0.36 L'-(8fi3)src)4' @lsn6)4te-v' n=T-J-v.24.4)=5264 y rn2.oc q=EA(T* -T*) =(5264)n(0.02X0.4X200 120)= 10,580w 7-26 Tr=299K; L=0.15; W=0.5; p=2'2atm; p:036 v = l28xl0512.2:58x105; k:0.149; Pr:0'7; AT: 52oC : I l2gg)(52)(0.4Stxt0t)' I 106 6" 1e.8X GrPr:7x105 h= (0.149/0.X0.59X7 xl051t/a17 q: (17X0.lsxO.sxs2)= 66.2W -Z1b); : (58xI4/0.15X0.I8X 0952+A.Dln 11gs1r(51 7) Eq.(7 urnax :0.23 m/s Eq.7-20a); 6 st';10.t1-t'(0'e52+;r/41106;-ri4 ; : = 0.0016kgls rh 1vt010.23X0.025X0.36) 741 1-21 ,,-70+20=45oc=318K v=0.335x10{ ft=0.485 r2 - 2.0 p- 2.45x10-3 Pr =(g.gx2.45x10-3x70- 20x0.03)Q)_- 5.78 108 GrPr (0.33x to{)2 C -0.53 * -L (t\ rv 3zB Y h-r0.03\w.JJlrw'x1o8)r/4-r *2 .oc q =hIrdL(Tw-T*)= (1328)n(0.03)(1X70A)=6260 W 7-29 .(Lzo-20\l/4 ,r nA w h-(1.32\ffi) -7.e8ffi q- 29,300W -7.98n(0.075)L(12020) L- 155.9m L+€ Chapbr7 7-29 Tr=93!38=65.5oC ft=0.659 "2 Grh,= (7,6xtol0xgl- 3gX0.0004)3 26g.2 From Fig.-S logNu= 0.41 Nu 2.57 h-Q's0.0004.se) 423sm'.oc q= - = 1M35)n(0.0004X0.X93 38) 29.2w 7-30 Laminar: n -r.zz(81-2o')t'= e.oo6J- = \ 0.03) - mz'oc q (9.N6)n(0.03X15X85 20)= 82.6W 7-31 r'2l4o!25=82.5t=356K v=2.54x10{ t=0.031 h =A.697 GrPr= (g.g{,fu40t-25X0.0g)(0.697) ,a , (21Ix l0-) =).54x106 C -0.53 *=! 4 h- 0.031(0.5(2.5I'I0\rt4=8.2 y 0.08 / ^2.oC g L\-hnd(Tw -T*)= (8.2)n(0.08)(140S)=237 Wm 7-32 zso-20)t/4- y h-rsz(\ 0.0125) rs.38m' .oC g =(15.38)n(0125)(2502O)- 138.9Wm L\ L19 Chapter7 7-33 Tf= 150+93 - 121.5"C - - - | 0.I24xl0-4 k 0.135 Pr 175 - p -0.7 xl0-3 GrPr = (9.80.7 l0-'Xt 50 93X0.025)3= 6.96x106 (0.124xt04)2 , --(0.53)(6.g6xlO\rtt4,l- .-- W h- 0 . 0 2 5 \ v . v - l \ v .^ 2 . o C^ v + = \$47)n(0.025X150-93)= 658 Wm :200.6 flft L \ 7 7-34 0.3m squareduct Tw l5.6oc T* =27"C LT - 11.40C o Bottom h-nz(+)'' =(r sD(#)"* -3.28 y m'-oC o o w Side_s h r4z(+) " = r 4D(#)'' -3.53--- oC mt. -1.46 y Top h-ose(T)' ,rr2c n =\netr = (0.3X1)(u.4)t3.+(2X3.53+1.46140.4 Wm .length 7-35 + lo Tr2=240 - 125oc=398K v- 6.76x10-s k- 0.78 - Pr 0.7 I (e.8)(#)t40 ro)(2.54o-s )3 GrPr = (0.71) 1.42x10-s (6.76xlo-s)2 (o.5l8Xl.42xlo-s)rt4 Nu 0.36 + = Q.384 F.(%P)e/ru)o'' h- (0.384X0.178)-269r 2.54 10-J LL=Q6gt)n(2.54x 10-s)(Z+O-10) 4g.4 Wm ?50 Chapter7 736 Heatlatm L-2m d -0.1m Tw = 93oC= 366 K T* =-l8oc - 255K = Tf 3lo'5K v - 134.6x10-6 ^2 k-0. I 524 w = It m.oC Pr A.7 (e8)(nts) - GrPr = e66 zssxo.l)37) 1.356x 105 (134.6x l0-) C -0.53 * -! 4 h-= 0.I124(0.53)(---x6los 4- y 0.1 \ '\ tU L5.5m z . o c q= neg* -T*) = (15.5)z(0. = lX2)(366-255) 10gt W 7-37 Tf= 135"C -408K v-26.83x10-6 - - k 0.0342 Pr 0.688 A* (e.8)(#)1250-20X3)3(0.688) GrPr = 1.43x l0lI 1l/6 | 1' Nul/2= 0.6+0.382] | =2378 [l,.(B#ffi Nu =565.3 h- (565.3X0.0342)=6.44 3.0 g=(6.M)rT(3.0X250- L\/ 20)- |3.g7kflm 7-38 Tf = 40oC= Jl3 K v -0.A0022 k- 0.296 p-0.5x10-3 - Pr 2.45 (9.gx0.x 10-3x6020x0.g21 Grpr = .45)='19.37 (o.ooo2D2 _ Q.5lg)(7g3Trt: - Nu o.36+ I.677 L t,6 I J h-_(\.- .X0.296-=23.98 0.02 q -(23.98)n(0.02X0.6X60-20D=36.15 W aft ChapErT 7-39 fr=soo wl^' p=* 1=(1500)zr(0.035)=165vm Propertiat350 K AI =l00t v = 2O.76l0{ ;, 0.03003 Pr=0.7 0=65" Taket=lm Grpr _tg.s{#Irooxl.o)3 (o'7)5'4xloe (20-76Xr;f ll ;+g(sin 01t'rt.32 - Nu1 t0.6(0.488)(s65.)1.03J(x.t0e;0.122M.9 ft (206'9x0'03003= 6.21 ftiinsensititorbecauseof,.32exponent. -)t)oa Ar=156.2r With properties - at400K and LT =242 v 25.9x10-6 k- 0.03365 Nur = 237.6 l5:oo GrPr-8.32x109 h-9.0 LT= - rg7.5oc T; = 20+I87.J ZA7.soC= 8 = 4g0.5K Tf 386 K closenough 7-40 25+20 Tf= =22.5oC= 295.5K v-15.3x10-6 k -0.0259 - q'8)("risQs- 2o)(o3)0.7) Pr 0.7 GrPr= - I.34x107 c -0.53 m--I 4 Wm a{L Cha4erT 7-41 r2=260:20=l40oC=413K d=o.l25m v=27.2x10{ ft 0.035 = Pr 0.687 (9.8h)e60 - 20X-:---l-'--'l-<A,3xlo7 GrPr= (27.2x10-) c=0.53 ^=! 4 a = 99?lt0.s3xr.03ro7yrr+8.41-J- 0.125 -2 .oC q =3'7,0Wt6= 18.71)n(0.t25)11260 L= 46.7m Therewill abesubstantiradiatiheating. 7-42 r,.=180-+60=r20oF k=0.644 s\p'", =4.g9x1010 '2pk = - GrPr (4.8xtoroxo.os)31rso*f'\9/ 4.075108 c=0.53 *=! 4 p=Y10.53X4.07s x_lo8yrr+,," y 0.05._.__, ^2.oc q =hrdL(T* T*)- (e7o)n(o.05x3x180 :3.o47x uo'\9/ 104 w zy, Chapter7 7-43 +27 J2=77 = Szoc - 3zsK r/=12.83x10-6 k-0.0281 - Pr 0.7 GrPr - = 2.5i4xl0lo (18.23xl0{)2 t'u I r{_rr. l ^61/2=0.6+o.3g7J Gttt ,= | -r7eer F.. =323.9 llt.tff1 Nu n_Q23.9X0.0291) =4.55 w F= q- LMUT; - T*)- (4.55)n(2X20X77 -27) - 2.g6x 104w Using C -0.13 and*=! 3 Nu = (013)(2.54x l0l0)l/3 gVolower 293.7=) 7-44 ',==9o+2o - tf = 55oc 32gK p -3.049x lo-3 r/=1g.52x lo-6 k- 0.0294 - Pr 0.'l GrPr = (18.52x l0-u)' (0.7)-1.153x108 h,- -.---+L2+0.43-\-153x-108'rtJ- 4.407 y 0.3 m".oC q - (4.407)tc(4)I0. - Z0)= g7 ,2(90 .23W 7-45 Ts-35+10 =z2.5oc - r2 k 0.604 = - - Grpr (l.46xl0l0x35 l0x0.0zr3 5.703x106 ft- 0.604 _\___ 0.0251 / t Je_ mZ.oC -(556)4 - q rc(0.0t2r2(35 t0) 27.3 75+ Chapter 146 Tf=-25"C=248K '=;(287)(248)1.967x10a k=O.72 Ir=1.488x10-5 F=4.o32x10-3 k=o.o223 (9.SX4.0x 10-3X1.967I Grpr= L0-2X50X2.4)Q.72)=3.44xrAl (1.48xlO-s)2 n= 0'973e +o.43(3.44r071rt| 0.32s+- 2-4 m'.oC q=(0.325)rc(4)(1.2)2(294'vv ForcedConvection: Re_ (1.96x lo-2x0.1)(2=952 1.488 10-' = 7r=o2.423tr'3Trl52\o'6.2llm'-oC = q= (o.2ID@n)(r.2)'(SOl9l 747 = 38+15 =26.5oc k=0.6r4 7r2 GrPr= (1.91tolox:S-15X0.04)3 2.81x107 n=0;6!!-tz+0.43(2.81x7;r/-51tl += 0.04 m'.oC q= (5lD{@irc)(O.02)215) 59.1ry zf, Chapter7 7-49 r -Td I L- 2 Q =0.52 m-- 4 M=0.52[.onr 4)'l k v'v-L-v'r*("z.) J 1v4 r(I\e| I rou Pr)tt4 0.464(cu Prft+ (a) +-Nud:(os J Compare to Nu -2+0.43(GraPr)l/a (b) GrPr Nu (a) (b) Nu 104 4.64 6.3 10s 8.25 9.64 14.67 15.6 106 7-49 Flatplate h -!(o.rexcr Pr)tt I.42(g)1/4 L I " . J - , , \ \ f T ) L , Eq.(7-50) h ,:+o.43*rorprft+ t4 o.43 = (.42)ffi (f)t -1 ot(+)''o *ro,Pr;r/+ {)r/4 h-zkd+l .\$( \d) ?-fc, Chapte7 zs0 ?F ZM+93 rf =T -148.5oC- 42L5K He at3 atm rt=-2L5x 10-6=73.gx10-6 k_0.1g3 pr_ O.7l 0.3)3(0.71) GrPr_ =9'08x106 (7 c -0.53 m--I 4 E=W19.08x torrt4=33.5+ q=Eir(r, -r-)= (33.5)z(0 ,rilrrr?,*e3)= 36,43w 7-51 ', =Y= 70oc=343K v=2o.0sx - l0{ k=o.0295 Pr 0.7 (e8)(#)t20 -2oxo.3;310.7; GrPr= l.34xlo8<loe (20.0xlO{) c -0.53 lll, 4 t=@(0.53X1.34 x w lo8)r/45.61;z,C q=EA1T,- T*)=(5.61)a(0.3X100X12 o_20)= s2,84w 7-52 r,=ff= 85oc=358K F= z.79zx10-3 = k - - v 21.5810{ 0.0306 Pr 0.7 GrPr = - 20)(0.15)3 0.7)-1.806x107 w - ffi q (9.03)78(0.0712_120)_lg.W ?.5r? Chapter 7'53 69*20=40oC rf=_ z v=0.0N24 k=0.1M Pr=2870 F=O.7xl0-3 _ (9.8X0x l0-3X60 29X0.3)3(28=0) crpr (o.Cffizqz 3.69 108 h=o'l\$ ro.tt(3.69x108)l/3.6 Y 0.3 m'.oC q =(51.6X0.3)2(60)=,185.w 7_54 laminar: L=6mm AT=500-20 q=2kW rop: h=r.:2(#)"o = 22.2 #* y== Botrom: o=o.ut(_A]t" = 16.2^' ."C [(0.006f) q=2W=(16.23+22.2)(O.OO6X{480) L=18.07m i-ss ,,-25+28 =26.5oc=299.5K v--16.84x10{ k=o.o2624 r2 Pr 0.71 ts.s{#)(gxro3to.zrl Grpr _ =2.42xl'u (16.8x l0{)' t -0'o?tge.E)(2.42x101)1/=2.454 t0 q =(2.454)(10)2(736W Lrt Chapte7 ,7-56 TJ2=ry =32.5oc=305.5 K | = l6.ZSx10-6 - Pr- 0.7 k 0.0267 _(e.8)(30*)(s- I5X4)3(0.7) Grpr - 1.x l0ll Upper: lsxl.exlor w 4=ry(O. \rt3-5 .7s Lower: -TC ht=ry(0.27)(l .9xlot\rt4-l.tg w - ;rc q (5.75+I.tg)(q2(50l5)= 3gg6W -57 7 Tf= =350K r/=2.075x l0{ k -0.03003 Pr- 0.697 --+4=I=_= L 4s X- P3L66 =_= =7.SCm Grpr= Xo.o7t3(0.697- (2'075xt0{ I .gl108 )2- h=ry3 l5Xl.glx108)rt3 34.5g w 0.07i(0. ffi q -(34.58)(0'4-t2 '2\ (400-300)- 350w 7-5e Tw =40"C T* =20"C T,=40-(0.25)(40_20)=35"g k=0.626 GrPr (2.89 xtOl0;10.2(4o_ 20)= 4.624x - lOe e 300 E=Y-(0.56) [(4.624x10e)cos3gof/- 440 w - ffi q (440)(0.2)2(-0ZO)= 352.w 7-59 0 = -30o = M" (0'14)t(4.624xl}s1r-(zx loe)l/+'Y.56L(v.24xrPs)u3gss39o//4 Mr = 56.85+140.87 =197.72 " h -(0.626X197.72- 618.e y - mt.oC q (168.g)(O.Z(+O-ZO)= 495.1W as0 Chapter7 7-60 AssumeT-20"C inside Tm =20* =35oC = p -3.247 2 30gK x l0-3 v-l6.4gx10-6 - - k 0.0268 Pr 0.7 GraPr = (9.gx3.24x l0-3x30x0.0125\3.Ar\ (0.7) 4g03 ke =16 q - (0'0268X'8Xl.2X30)-13g.9w 0.0125 7-61 160+ 40 Tm= -100oc-373K p-2.69x10-3 k-0.0317 l'olx lo4 Pr= 0.69 '-_ (287)(373)0.0946 Ir- z.rTzxlo{ (g.gx2.6gx 10-3x0.g46)(160-40xq.wlr(0.6g) Gr5Pr 2.I1x 104 (2.i e-20" C 0.212 ,=! 4 k\(o.ztz)[(z.l1x104)cos 2golta-2.516 - q (2.516X0.017)Q)2(160-40)- g.57w 7-62 - ew 700Wl^' L-l m e -60o at30"C v-15.98x10-6 Pr-0.7 k-0.027 p-3.3x10-3 Gr*_ (9-gX3.3xl0-3x700)(g4_3 (0.027X15.9x l0-u)- .zg6xl0l2 h-ry(o. \ 17/ L \g6xlotr/ ' l J "5.6sv ry I ^ Z . O C LT =ry- 5.65 I z3.8oc Tw= | z3.g+30=l53.goc 1-63 otA(T*4 -Tr4) hA(Ts -To) (5.G6gx 4 - t0-txo.5x30g 3a34)= (6.5X303- To) Ta =300.5K -27.5oC ?SCt. Chapterz 7-64 ?r=80-fttg-20)=65oC=338K | v=t9.54x10{ '= 308 k=O.0291 Pr 0.7 Downwardfacing 0=45o -zolto.l)3to.zt crh _tg.s{#)(eo =3.5 106 (19.5x 10-)' =o'2'?'(0.s6)(3.5 a = o' 0.1 x106x 0.707)tt-.4-m'.oc Upwardfacing G%=1.05x10e %Pr=7.35x108 crl <Gr.sothathu=#,0.56X3.5x106 xo.707)rta.46 ,ft q =(2)(6.46Xt)2(80 20) 752w 1-65 I Te=5o-i<to -20)--42.5"c p=99o lt--6.2x10{ ft 0.635 Pr 0.41 0 30" cosO 0.866 - Grpr_ (9.8X2.0x l0{X50 20X9.05)3(4.1X99=7.95x 107 (6.2xI0-)' o,= =@7.8 ffi(0.56x2.9sx107x0.866)r/4 ,,Y- SinceGt <Gt \=ha = - q (2)(&7.8X0.05)2(520) 97.2W 746 4 =38"C=3ll K Tn=54OoC Z=30cm = = qO d 6.5mm 4d fullydeveloped Airu^=30 rVs v=16.6x10{ k=0.O28 Pr=0.7 Re- =(lo)(o'qq)=lL75o turbulent 16.6 lo{ Nu 0.023Re0'8pr0'=(0.023Xll,zso)0.8(o-D35.95 _(35.esx0.028) y '. 0.0065 = 155m' .oC 76o Chapter 7-67 It2100-+15=55oC=328K y=l8.Zxl0{ &=0.02E5 Pr- LJl L-Q{r'\2)= 7.scm=0.0m 75 pr _(9.8X328X100 I5X0:97t3(0._) Gr (1 2.5x 106 c =o.52 ^ =L 4 I =0-o?E @.sz)(2.tsx06)l/47.56 y 0.075 m' ."C e =(Z.S)2'+ @)(2.5X5)62.5cmz - - q =M(7, T*)=(7.56)(62.tO4 ;1t00I5) 4.O2W 7-68 = = e I2')'sin60o 692.8cm2 p= (3X40) 120cm +25 2= L--%--5i77 *, gt =55 = 4soc==313'K P' 12o.' 2 p=3.195x10-3 v=17x10{ k=0.o272 Pr=0.7 (9.SX3.19x l0-3X55.--25X0.0577)3 Grpr = o.7)=4.37 xt05 ltztto-[t' h = ros --6.s46 ffi e.s4)(4.37 )r/4 # q =(6.546)(0.0693X525) 13.6W 749 Tf =N"C= 313K F =3.195x10-3 v=17 xl0{ k =O.O272 - 4= ! =0.02 Characteristiceth P4 m (9.SX3.195 pr= 0.7 Grpr = x10-3X50:-10X0.02)3(0="t.2t10a (tzxt0{)2 =O'O?J2 =7.71 1, 0.02 @.54\0.21x04)l/4 m''oc = - = q (7.71)E(0.04)2(30) 0.775W z(.I Chapter7 7 -70 .r, 400 +27 -213.5oC=486.5K rf= 2 r/ 36.23xt0-6 k- 0.03949 Pr-0.6g1 p- I I I I L-5.2Tcm 4g6.j L=i3.*g A ts.s{a#)(400 -27)(0.0s2)3(0.681) urrr - _5.54x105 h 0.0522x\ ' t'-t'\^v,r4- lt.' g y - m..oc A (2)(rrz+(4)(lsx8)=e30 q -(t2.39X930x l0-4X400_ 27) 429 W 7-71 111 * L- 0.075m - 7= or5 o.rs v 17xl0-6 k- 0.a27 Pr- 0.7 8)(#)Ar(o.o GrPr =(e. 7r3(0.7)- 3.34x lOaLT A- 6(0.rc)z Q.135 ^2 h- y3=(0.6X 3.34x lga LTt/4 0.075 )rt4 Z.92LTI/4 50 (2.92)(0l35XATft+ Tw= 48.2+20 = LT=4g .ZoC 68,2oC 7-72 os(Tt4-T;o)=h(To-7,) (5.669xt0-tX0.95X3034 -zg3\ = 5(To_303) Ta=325 K- 52"C f.i+ Chapte7 7-73 L -30 + 15 45cm Tr - 45+20 = 32.5oC=306 K r2 v-16.5x10-6 k-0.026g Pr-0.7 = (e8)(#)fos- 2oxo.4s)30.7)- GrPr 1.88 108 C -0.52 ^ -! 4 tu=ry(0.52)(r.88xlos - - ' o v ? 0 . 4 5v ' v - / \ )rt4\ ' v^ 2 , O C' ' + = h+(T* -T*) - (3.62)(2x0+30.r5x45-20)- 8r.5Wm LL 7-74 T _ 38+60 =.- - = 2 k 0.644 Pr 3.64 -(4.Bgx -38X0.0 - Gr6Pr 10^'Xoo rzr3 2.1x106 - +k =\0.4./\z.r106/rt\(3.,4)0'+)'' 6.26 \1.25) q -(6.26)(0.644X60 3gxo3\2 -638.5 oi W 7-75 rm--80+20=500c-323K k-0.156 143.3 10-6- - v- 1.LOZx10-4 Pr 0.7 1.3 (e.8)(#)fro- 2oxo.02)3(0.7) GtaPr = (l.102 104tZ = 839 ) ke-k (0.156)(.4)2(3020) q- =74.88W 0.02 a(t Chapter7 7-76 Tm =30t k- 0.62 - GtdPr (2.2s xl0l0x40- zo)(0.02)33.6x 106 C 0.40 n-0.ZO ? =(0.40)(3.x 10)o-2=g.l9l _20) q- x40 =2860 W 1:11 lm=30-10 -10"C=283K , J -0.05X1.x l0s) 2 ' (257X293) =O.O624 k -0.0249 p=3.53x10-3 p-l.gl4xl0-s Pr- 0.71 G6Pr= r, (l.r4x (o,1)=1454 k - 7 =(a.229)(145q0.226I.lg2 q- =ll0.7W 7-78 Tm- 400!l4o =27ooF = l32.2oC 405K | = 26.2x10-6 k-0.034 p=2.47x10-3 pr-0.7 (e.8)(2.xI0-,tt+oo I40{;){o.os1r Gta Pr= _ 0.7) 9.63xlOa (26.2xr0-)" c - 0.197 n-- I m I 4 9 35,-lt9 ? =(o.rs7)(.63xIO+;r+( =2.64 \3I =(2-64X0.03?!100- r40)(;) x = 432w I2 26 + Chapter7 7'79 r oo rm=#=0O l45oC=4lB K v-26.97 x10-6 k- 0.0349 ) p - 2.39 lo-3 Pr 0.685 - g0x0.025)3(0.6g5) GtOPr -(g.gx2.3gxl0-3x200 = I.7gxlOa (26.9x t0{)2 c=0.r97 ,=L 49 ^=-L t'n kr= 1o.tst)(3.7o+;r+(?o=)- = 2.086 k \2.5) q=(2.o86)(0.034ex.zf('!^^2-o)= " \ 0.02s ) 28.s3 7-80 165 T- =X)+ =172.5oC=445.5K Pr 0.684 2 F =2.245x l0-3 v =28.59x10-6 ft 0.0368 pr -(g.axz.zfxto-3xtqt_(o.00ro)3 -^r" lo{;2 (0.6s4)12.45 Q8'5gx k'=k q -(o'oJ99xtos=37e5 wl^, A 0.0016 7-E1 atl72.5oC k--0.669 Gr6pr= (10.1xtOl0)(t6SXO.OOtO)3 Gtak=6.83x104 C=0.13 n=0.3 rn=0 &.z=(0.13)(6.x lo4)o'33.67 k =2.453 k,=(3.67)(0.669) i=ffi=213xtos flm2 265 Chapte7 7-83 T^ =ry= 35t p =g4 =7.2.xr0a It k=0.626 Pr 4.82 rflP" =2'89x loro f &a Pt =(2.89xtOl0y1O.O+)3(t=5.55x 107 L Y=(O.M6X5.55 xlOT)rt3 17.55 - q (r7.ss)(0.626x0.5)r(s0'zO)=rouo * 7& 'Y- (0.057)(5.xlo7 - K )rt3 21.74 - q (2r.74)(.626)(s)2(ryJ = 2ss2w \ 0.04 ) 1-87 Assume2O"C inside Tm=20*+=285oC =302 K p -3.32x10-3 Pr- A.7l 2 r/=15.6gx10-6 k=0.0262 (2:gl.32x10-rxt7x0.1)3(0.t) GtaPr= -l..598x106 (5.68X )4)2 C 0.073 n It ml-=1 tt 9 k.e '1.51 6 ( Z \-U9 k :(0.07ril 98 I0')rl 6.1lg q 6.11g 0.026 ,') .O.J,, xl 2X1,7 7.25 wl^^2 A 0.I -?,cG Chapter7 7-88 T^=350K ,-20'76xl}a ft 0.030O3 p =2 atm Pr= 0.7 T pr- (e'8{#X400 - 300x0.06)3(o.t)(z)2 Gr5 ffi=3'e3xlo6 C=0.073 ^=l't '= -6I = (0.03003X0.073X ,;r{ \-lle w e" 3.93xroe -? =o'224 \0^06) ,n* q= k,A+ = e.224)(31z{1L19)= rrn * 7-sl f^=T=60"C=333K v=l9.O4xl0{ k=0.O2g7 Pr 0.7 (e.8)(#Xtoo-2oxo.o8)3(o.z) GtaPr= 2.33x106 C-0.073 I n-!3 m--g ne- ' ' '^ / -(0.073)(2.33lo6)l/3fI )-ll-7'304 k \o^og,l q- (0.0287)(7.3M)(l),( _209.6W 2C1 Chapter7 7-gl Takepropertieat300K rl 15.69x 10-6 k- 0.02624 Pr -0.7 (s'a)(#)(Eoxo'01)3(o'z) c6 pr - - 2787 (1.6gx l0{)2 Forhorizontalplate C-0.059 n-0.4 ne- - (0.05)(z7g1o-+- I.41 k\ _ (t.4t)(0.02624)(03(30) q = 9.g8W For3ocmur*,?*tot.r" - = -0.59 I GrPr (278T'\'I) 7.52x 107 C m=- 4 0'02624 n- - (0'59(7 '52xa7 'rt4 {-806 y 0.3 m'2.C q- hAg* - T*)(4.g06X03)2(30)- l2g.gw the9.98W isareduction of 93Vo 7-g2 rm t2oP =7ooc- - 2 343K 'r-2.04x10-5 k 0.0295 Pr- o.'1 f =lcm - Gra 1700 _ p2(s.s)(#)(rzo z0x0.or)3(o.z) (2.04x 10-s)2 p2 =0.3537 p=0.595 At Iatm p-1.A23 0.595 Thusat 6-lcm '- -- = 0.582atm (58.96kPa) I.023 For theothespacings: 6 (cm) p2 p (atm,kPa) 2 0.4M2 0.2103 0.2056,20.93 5 0.00293 0.0532 0.0525.27 , r0 3.537 l0-3 0.0189 0.01841 ,.86 7ts Chapter7 7-94 q- hndLLT ft= 1.,(+)'' Q- LT5l4 Qss-( s3 27)s/4 r7s [ss -27) Qgs= 4l6W 7-95 ,rf=-)4ot?o -lo5oF 4o.6oc = 3r4K - tyt-17.llx10-6^2l* k-0.0273 Pr- 0.7 - Grpr - roe- tq.a)(#)tr+o z-o{i)"3 (17'lx1fl(0'7) f,- 0.701m - y h %0.701)(roe \ 'rt4=4.09 mz.oc g=h/lr=(4.\ext.\ -7q-.\g)= l5e wl^, A u-umaxaty-3= A.l48u* ux =CFll' C1 =4.43 = - ux (4.43)(0.70rtz 3 | thnax (0.148)(.7l) 0.55m/s (, 7, Chapter7 7-et d=0.025 m Tf=50"C=323K v=18.02xt0{ m2/s k=0.02798 Pr 0.7 - gort.025)3(0.7) Grapr= tq.g)(#)tzo =40'878, (18'O2x10f t4-3.014 +k\= (o.ztz)(40,g7g)l ke=0.0843 -+ oC m. =r3s +A(o.o84r(#J0.025 ) wl^z Ford-0.01 m Gr5Pr=2616 + =0.059(zio;o'+I.374 k g-e.374xo.o27sq(Z.pJ = A \ '/\- -',\o.ol) \$4 'wl^z 7-97 ')t GrsPr=1700 -+O,e'/.3( p=1 \latm/ rin.' RT F=0---4atnr 7-98 lzoi 80 T'2= =loooF 37.goc = 3l l K (a) At v=16.81x10{ k=0.027O7 pr=0.7 (e:s)(#)(z0 -80)(;){o.ozs)3(0.7) GrPr = =27,100 -' w hJ 0.025[2+'0.43(z'\t)\'Jr+Ol'4 ;Z"C q - hAAr= (8.r4)4n(0.r2s)2(40)[;)0.36w '2,1 a Chap|€ir7 ^) SltPlco (b) water = 3.3x l0l0 t =0.630 ttk Grpr =(3.3xroroxo.o2s)3lzo-*)f*l= 1.14x 6107 \9'i o= #r2 +o.5o(l.46x tollrtol783 -ft q= h,tLr- (783)4n(o.0r2sGr{;) : 34.2w 7_99 Tf =35O K v =20.76x l0{ /c=0.03003 pr= 0.7 ro=0.05 6=0.01 4.=0.04 pr -(s's)(#)l+00- :00x0'01)3(0'7) Gr5 (20'76*ro 6F--- = 4548 lt=p.zzt)(4548)o.22= 1.53 k ft" (0.03003X1.5=30.0459 _ az(0.0a5.9X0.0aX0.05X100) n 0.05 0.04 =11.54W 7-100 d-L-0.08m c-0.77s ftr=0.21 Tf=zg7.5K v- 14.6x10-6 k- a.}zsz pr- 0.7r tq.g{#s)troo- zzsl(0.08)3(0.71) Grpr_ -I.'45x106 (14'6xl0{f h-ry*(o.77sxr.4s - y 0.08 \ z\ x106'/.2r 4.8r mZ.oc hA= (4.8 - l)l2n(a.04)+ rc(0.08x0.08)0.l45z pcv- (gggx4l80)z(0.04)2O.OA) rclg h4--0'1452-g.65xro-s pcv 1679 290- 275 _ expt-8.6x l0-scJ 3-0 n5- - r 5905sec 1 .64hr 'lt.1 Chapter7 7-101 L = , - = - = _ p 4rrd 4 - l2o-+80= = 7r2 loooF 37.8t ^t SltPl_cp=3.3xl0lo ft=0.63 pk 0'95)'1no crpr= (3.3 rotof - 80-\9/=1.43106 " \4)' Uppersurface C=0.54 *=L 4 n= Ofi (0.54X1.43x106)=-/42-i{- 0.0514 ' '-'^2 .oc q=(e42)n(o.o2r2(n{;)= 4r.rw Lowersurface C=0.27 ^=! 4 h=47r m'.oC q=20.6w 7-ffiz 0=45" Z=0.1m t=350r Te=4OO0.25(469300) = 375K v=23.33x 10{ ft 0.0318 pr= 0.69 CrLPrcosO= 1s.s{#X+oo-rooXo.rfto.zlto.zoz=) 2.54x106 w.Q40'e119(0.56)(2.s4 1= x106)r/47"'^ y 0.1 m2."C q = bAAT= (7.llxo.l)2(40300)=7.1Iw 7-1A3 Upward facingsurface BecauseGq < Gq lst termof Q46) dropsoutandthresultisthesameas in Prob.7-98). L1 7- Chapter7 7-104 3s 2, L- 50cm-0.5m Tf=350K TtW v -20.76x10-6 k -0.03003 Pr A3 (e8)(#X4oo - Gry= 3ooxo.s)3-8.lzx108 d^in= (0.5X35X8.1 4 Z xl08)-rt= e104m 7-l0g ConstheatfluxTw 65oC - 338K T*= Z1"C= 293K L- 0.5m TJ265!20 = 4z5oc -316 K v-!7.ix10-6 k-0.7 k-o.oz7| Forlaminar h -2h--, 4 rtLr (e. - GrPr= 8)(fr.)tu52oxo.5)3(0.=4.08x 108 Nu,= = 0.6(GrNu, Prll/s75.05 i-ny-L=T5.05X0.0^7=)4.13--6__- m t . o C /{\ w E =[+k+.rrt =5. lr \z+,/ FC 4conv= Ee1r,-L) =(5.16x2)(o.s)2(oso)=l16.lw = oeAlT*4 T*o = (5.66g yzy1o.s)2 _ erad ) xto-8 (l:sa zsl1= l6l W Qolrrt 16.+ 161 277.W 7-109 + l0 L-0.2m T.=30oC Tw -lgoc Tr230 = 2AoC - r/ 0.00118 -t k 0.296 Pr- 12.5x103 p-0.5x10-3 -gP\I: (9.8x0.xt0-3x30- t0x0.2)3(t2xt03) Grpr pr= - 7.038 106 v' (0.00119)2 Nu = C(GrPr)n (0.59)(7.0x 10)rt =30.39 T_ (30.39x0.296) .^ ., w h=T=43.46;FC -Eeg* q -T*) - (43.46)(2x0.2)r(l0)= 69.53w a7, Chapte7 ?'110 Tw= 43.3"C L = l0"C Tf =26.65oC L--d= 0.3m altn-?= pr ryItx l.9lxl0lo = 5.85 k=0.614 GryPr=(1.91xtor0xo.lyz1+zl0) t.72xt}ro A l.72xl0to Grtr=ff- =2.94xloe= Gra -J*= o.l5 ! =r.o> o.l5 GrL"* L Treatasverticalplate Sides C=0.1 ^=! 3 =o'!lo = E 0.3 ro.De.l2xl0r0)r/3528*2."C q =EA(r,- =(528)z(0.3X0.3X43.3 = L) l0) 4974w ToB andBottom L: + =4 =0.075m P4 l.72xl0ro Ufrtsf=------ =259x1O8 (4)' Tsp C=0.15 *=! 3 Bottom C =O.27 *=! A=4 05F10. 44 /r(Top) rc)e.6gx10')r/392 + 0.075 m'.oc lr(Bottom) z7)(z.6108)r/4283 0.075. mt'oc q(Top+Bottom)=(7g2*rrl@-@3.3 -10)=2539 ry q(Total) 4974+253A= 7504W 7.-l+ Chapter7 SecondMethod Treasolid asverticalwith characterdimensionL+ ''\2)= 0.6 m = = GrPr (r.72rorS(;f)' ,.r, rorr C 0.1 ^=! = 1r1u(0.IXI.3x10ll)l/3 516 ft_(sl6xo.6r4=528 y 0.6 m' .oC (9!)1-]= A(totar)rdL ,(+)= of,o:uo3) + o.oro^, \4/ L 2 J q(total)4o.grt(T**)= (528X0.424)(43.3-107457w Goodagreemenbetweenthetwo methods 7-ttl ?=3SOr u=20'8:(-10-6 02 k=0.0307 =0.7025 Pr pr= tg.{#)t+00 l00l!0.1)to.zozt Gr (20.x 10-6)2 = ro,z22 c=0.53 ^=! 4 E =of?9' (0.s3)(lo,222ytr+6 J- 0'01 *2'oC q EA(TW -L) = 116.36'1n(0.01Xr.25X400 64.25W ?,1{ Chapter 7 1-lLz --*zcm F L=2 cm=0.O2 m= L 4=400t< tr=0.1 =300 f ez=O.15 d1= 91, Q.) & 0.0300, 1= pr=0.7 I-1''- m.oC ^r=2 - aZ!L- h -2^r =[o.s^z.+/\2Y400 - 3002^r) o( y+tprT o.z/[ o.o2 ) = 40.15t =/'(r-T'_ 2LT 400-300- (2x4015) fl.""o L ) =(o.o3o{3 0.02 =29.59Wl^z 4 _o{rro:ha) _ (s.o6xto-8x+004_1004)_ Aaa. Tl,"o=F=ffi=63'3lwl^z 4Alrotat9.59+63.3r=92.89wl^' 7-tt3 Tn=!0011 T*=20"C=293K d=0.4m tt =--=3473 K v=20.5x10{ t=0.03 h=0.7 (e.8)(#)@oo-zs3)(0.4)3(0.7) GrPr= -J.23xl08 Nu =2+O.S(Grpr)rt42+ (0.5X3.x 108)l/469.0 _(6exo.o3) y t O.4 = 5.18 m'.oc q= EA(T*- T-)=(5I8)42o.Dzg.N- zg3) 279w erad oeAlT*4 T*4)=(5.66x to-8)(o.g)@n\(0.2)21+zgza)_467w a7c ChapbrT 7-ll4 L=O.2m 4=350f Tz,=4OOK d=0.02m 7- 350+4oo=375K 2 Heliumatp=2atm l8l_>110=90.5x10{ pr=0.71 ,=- z k=o.l7r &ak = = tg's{#)!gDt-15-%9;0zl3to=ze062000 ryo, Therefore k=rc q=k,A{d(0.171-" -'>r(o0.020))=lz.rw ?-115 Tw=125"C=398K T*=25"C=298K Tf=3+8K v =20.7xl0{ & 0.03 pr=0.7 =4 =L =! = CharacteristP4L4 0.075m - crpr_ ts.s)(#Xrzs zsl(0.075)3(=r'e4xlo6 (2o'7x'o-uff =0.54 Iap C *=I 4 Bottom C=O.27 ^=! 4 Isp D'=910.54X1.94 x106)r/48.06 I.oC 0.075 m' Bottom 6=9-'2--7,'t4.03 y 0.54' mt'oC q(total)8.0+4.03X0.31ltzSZS)=108.w -?.77 Chapter7 7-116 T*=134"C=407K T*=2OoC=293K d=0.001m pr=0.697 ?=350f k=0.03003 v=20.76x10-6 tg.s{#Irr+- zoxo.00l)3to.oszl Grpr= =5.16 (20.7x 10-)' C=I.02 m=0.148 Nu C(GrPr)- =(1.02X5.16)0'11.3 _ (1.3x0.03003) y [ 0.001 =3e.05m' .oC q =EA(Tn- L =(3e.05)x(0.00lx34 zo) r5.2w r =qv = 4)t ) ' '\4 . 05.21x4) ) q-#=l.94xtO7 flm3 z(0.001)'(l) -T*=#=W=o.o76oc From chap.2 T, To=134.076'C rLbc= _L (70x10{)(100x4= n, P"i= 0.89Cl ,'2 "('ii- q.R=15.21W E =[(15.21X0.89)=l3.68Volts 1-ll7 L=0.3m d=0.105m L=l5oC=288K Zr=l00oc=373K ?=330.5f v=!8.7x10{ k =0.O29 Pr=0.7 tg.s{#Xroo --rslto.l)3 GrP= = l.e5 lOs (18.xl0*)' d - 0'10=0.35 35 1 <0'35 treaasverticalplate L 0.3 W=O'296 Gr1Pr= (1.9xIO8XO.Z) 1.365 108 C=0.59 *=i Nu=C(GrPr)'=(0.59X1.365x108;l/a63.77 E _63.77)(9.o2=)6-16 y 0.3 m' 'oC q=EA(T* -T_) =(6.16)z(0.105X0.3X115) 51.91y' a7g ChapErT 7-lt8 Tw=865K L=300K d=0.025mm Tf=582.5K v = 48.9xl0{ & 0.0455 pr= 0.6g3 p, - Grpr =spvrd3 _tg-s{#s)tsos 300x0.02510-')'(0.68-4.24x10-s C=O.675 n =0.058 Nu=C(Grpr). =(0.675)(4.2+xt0-s)0.0586 (0.376X0.045_s) ft- = 685 y.oC O.O2x l0-r m' econvEa1r*-L) = (6g5)z.025xto-3;1aOS300)3o.4 Wm. length qrad oAe(T*aT*4)=(5.66x to-8)z(0.05to-3xo.gxg65a goo4) =2.2I length Wm' 7-tlg Tn=l6g L=300K ?=350f v=20.76x10{ k=0.03003 n r(. l\ pr=o.6e7 A=2r)€r) =8,] L=o.2m L (2 ) 4 : characteridimension4 =t'? = L= =0.144L=0.0288m P 3L 443 tq.s)(#X+oo r00lto.ozse)3to.oszl Grpr_ =l.0e 105 . (20.7x l0-)' C =O.54 *=t Nu= C(GrPr)' (0.54X1.x l0s)l/9.81 ; , _(e.81)(o.o3o=ro.23+ 0.0288 m' .oC q =EA(T,-T*)= (10.23.z>f eoo- 300)r7.7w a1e Chapter7 7-120 \ilator d= l0 cm Tw 49"C L =l0oC Tf=29.5"C characterisdim= L = !- o.ol5m n ) . P4 9FP',c=2.Zxlyro ltk &=0.61g GrPr= (2.2tol0;1o.o2r3gg-l0)=1.34 107 C= 0.15 *=t Nu =C(GrPr)' =(0.15X1.3x107)l/3 35.6 ; _ (3s.6)(0.6=8) y [ 0.025 881m' .t q=EA(T* - = = L) (881)z.os)2(49-l0) z7ow 7-125 t,= ttlto=2.5oC--216K r2 v=13.5x10{ m2/s k=o.024:3 h=0.71 d=0.025m Z=1.0m (e.8{#X0.02s)3(0.71)(r5 Gr.spr_ +10) - (lm=5'1'231 C=0.197 "=j ^=-; 1 1-l/9 k

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