ࡱ> q` bjbjqPqP .::@ @ @ @ \ D| $hbz-" www(  wwwZJw p ̩d@ @w0Dw!Fww4wgD ME 146 Homework 3 Solution 1. R=(Tj-Ta)/Q=(65C-40C)/50W=0.5C/W There are many heat sinks that will work. Check this one out as an example: http://www.aavidthermalloy.com/products/microp/AAVID_038288.pdf 2. 3-116 A hot plate is to be cooled by attaching aluminum pin fins on one side. The rate of heat transfer from the 1 m by 1 m section of the plate and the effectiveness of the fins are to be determined. Assumptions 1 Steady operating conditions exist. 2 The temperature along the fins varies in one direction only (normal to the plate). 3 Heat transfer from the fin tips is negligible. 4 The heat transfer coefficient is constant and uniform over the entire fin surface. 5 The thermal properties of the fins are constant. 6 The heat transfer coefficient accounts for the effect of radiation from the fins. Properties The thermal conductivity of the aluminum plate and fins is given to be k = 237 W/m(C. Analysis Noting that the cross-sectional areas of the fins are constant, the efficiency of the circular fins can be determined to be  EMBED Equation   EMBED Equation  The number of fins, finned and unfinned surface areas, and heat transfer rates from those areas are  EMBED Equation    EMBED Equation  <>FHLNf  " # S U 0 :   + ˼jh"QCJU jh"QCJ h"Q6CJh"Q56CJ h"QCJ h"Q5CJhih"QCJhihiCJ h*)CJ hiCJh*)hiCJh*)h*)h*)H*aJh*)h*)aJ h*)5CJ hi5CJ-6 0  / F ;Vgd"Q$a$gd"Q $xa$gd"Q $<<a$gd"Q $x]a$gd"Q $x]a$gdi+ , - . 0 1 B C D E @ARSϼϡσpdbUj < h"QCJUVUjh"QCJEHU$jI7 h"QCJUVmHnHujh"QCJUmHnHuj/p5 h"QCJEHUj/p5 h"QCJUVj6 h"QCJEHU$j6 h"QCJUVmHnHu h"QCJjh"QCJUj6 h"QCJEHU$j6 h"QCJUVmHnHuThen the total heat transfer from the finned plate becomes  EMBED Equation  The rate of heat transfer if there were no fin attached to the plate would be  EMBED Equation  Then the fin effectiveness becomes  EMBED Equation  3 cm 0.6 cm D=0.25 cm STU㵩㡛㡎hmqh"Q6CJ] h"QCJh"QhDhjP h"QCJEHUj < h"QCJUVj=p5 h"QCJEHU$j=p5 h"QCJUVmHnHu h"QCJjh"QCJUjh"QCJEHUgd"Q ,1h/ =!"#$% Dd  B  S A? 2`I$ef=ImD`!`I$ef=Im )8@@x]hUNv&m&&ͦhEH⃶Z)IhB F>) ł` ")VZ<>D /ZTRjmƹ=w sΜwP!7UŒD-ܤvZ:zLsĹjTt@$/8W9_#C=sjAEO=cIJG?Usu'پlS ̖Qn̖QZJ=KQ֖ n}I_cSWN3ϭ@BJ=H摠2Hu${ǘa&G?dn|dL]S#z&chnb%ɐ/R69TnE# gcb QU8VVB{~8+4h]&!{nPx_b_x]QMcN!2phiӕfKwWvA-}YUTҖˠʖvqliשݜaåPeKU4R'y$T"ޢ9{2y]}, Ug 8/yejWzjyE+-Wx5z^D%d2y≮&XŪ{Ǐ TzљU|bUx1[c2?Fl\p5}c&T"'YI!ST"']sRUT"'¹qέdǟ$ gJ^._. _ /sc㟷Mw΅ەM wە1ݝw,)ř=oRqS!#/gƎe2G=kHVWs-nh>CjR'O-?jŖĒ>-Fz^ŋR]˪0 ]yV3"[)Y voOM0**mԋ|}^mrbb27ȉ㈟lF0?/J$x>8tq6bKHEJޏxKTuU@] ^|OW˿+"ox2<{ ox[Wy<^O^Ss?@;ȭdNjGy[F@m`qOO[b|Q y_Wx=].D^r 1.w- ';^qCb}mu`}oro_{u0f\; IUi7q"w3uuy~ rJP)gҁJ̿I5OJFO1.C'NOFk]b4<9F!9_Fz#@7|xڕK/QϽ}h訠SnTөJW$(*F*J;B7† ;`#b/1w2Efwɮc7MtZxe~ڌtN[ӧTY#PVZ#!keJqϞ W[| Pim~)}G ~/ZY?cx {dvX A}f0m  != "#$'%&(*)+-,./<;123456789:@>S?AEBCDFIGHJKLPMNOQRTWUVXYZ[]^_`abcdefgRoot Entry@ FpýdFData WordDocument?.ObjectPoolB ̩dpýd_919456256 F̩d̩dOle PIC LMETA (  !"#%()*+,-13456789:;<=>?@ABCDEFGILMNOSUVWXYZ[\]^_`abcdegjkoqtuvwxyz{|}~LN*N*.>   .1  ``&& & & MathType-MMFn -n-7MxM --__ MRM  -  - 99M]M--QDQDQ  Times New Roman- 2 6a 2 whp 2 7kA 2 h 2  D 2 k 2 D 2 *h 2 rkD Times New Roman- 2 xccSymbol- 2 3= 2 = 2 n = 2 y= 2  2  2 o =Symbol- 2 : p 2 &p Times New Roman- 2 % 2pTimes New Roman+- 2 ; 4 2 4 2 4 2 35 2 237 2 0 2 0025 2 \!15 2 "37 2  /Y 2 (k 2 )k 2 s(k 2 )(kk 2 m.P 2 n)k 2 ".P 2 6 WP' 2 /Y 2 m 2 '.P 2 'C 2  WP' 2 /Y 2 1m.P 2 'C 2 ) mP 2 $ mP Times New Roman- 2 A2p 2 f%-1Jp & "System- b c a FMathType Equation Equation Equation a= hpkA c = CompObj $PObjInfo&Ole10Native 'Ole10ItemName.hpDkpD 2 /4  = 4hkD  = 4(35 W/m 2 .C)(237 W/m.C)(0.0025 m)  =15.37 m -1All_919456255 F̩d̩dOle /PIC  0LMETA 2HL"X"3   .1  &@g & MathType-  [  6 SymbolN- 2 `(h Times New Roman- 2 finI>p 2 s-1Jp 2 Z-1JpTimes New Roman- 2   mP 2 m mP 2 u  mP 2 uT mPSymbol- 2 `w= 2 `= 2 x 2 u_ 2 `=Times New Roman- 2 tanhY2 tanh(Yk 2  .P 2 .P 2 )k 2 uw .P 2 u.P 2 `%.PTimes New Roman- 2 aL 2 YaLTimes New Roman- 2 i 15 2  37 2 `0 2 203 2 uP 15 2 u 37 2 uG0 2 u03 2 `0 2 `f935 & "System-    CompObjHPObjInfoJOle10NativeK$Ole10ItemNameP FMathType Equation Equation Equation  h fin =tanhaLaL=tanh(15.37 m -1 0.03 m)15.37 m -1 0.03 m=0.935AllLE lEv  K  .1  @& & MathType- 4 '  Times New Roman-_903114799. F̩d̩dOle QPIC RLMETA TH 2 `6nSymbol- 2 `)= 2 ` =Times New Roman- 2 1 2 Y0 2 Y006 2 Y0 2 Yu006 2 ` 27 2 `777 2 w mP 2 Yt m)Pi 2 YZ  m)Pi Times New Roman*- 2 2pTimes New Roman- 2 YJ(i 2 YN.P 2 Y0(i 2 Y4.P 2 `,PK&MathTypeUU n=1 m 2 (0.006 m)(0.006 m)=27,777 & "System-P 2  FMathType Equation Equation EquationCompObjfPObjInfohOle10NativeiOle10ItemNamel n=1 m 2 (0.006 m)(0.006 m)=27,777AllL-e _934648393% F̩d̩dOle mPIC nLMETA 0-e> 8   .1  )&@); & MathType - .#a an aar Times New Roman- 2 lA 2 l DL 2  D 2 ^A 2  D 2 Q 2 Q 2 }hA 2 mT 2 T 2 H6Q 2 HhA 2 H T 2 H T Times New Roman+- 2 E bp 2 \ bp Times New Roman- 2 BfinI>p 2 (2p2 unfinnedppI>ppcp 2 $2p2 E finnedI>ppcp 2 E finI>p 2 E pfin,I>p8 2 E maxcp 2 E s finI>p 2 E finI>p 2 d 2p 2 2p2 unfinnedppI>ppcp2 unfinnedppI>ppcp 2 J2pTimes New Roman+- 2 eQ 4 2 l' mP 2 c mP 2 @   WP' 2 @ /Y 2 @ a m 2 @ C 2 @  mP 2 @ C 2   WP' 2 H{)k 2 H WP' 2 H/Y 2 HGm 2 HCSymbol- 2 l= 2 l +Fences- 2 mLx 2 Nx 2 Mx 2 Mx 2 mAOx 2 AQx 2 APx 2 APxSymbol- 2 l= 2 l+Fences- 2 mALx 2 ANx 2 AMx 2 AMx 2 m#Ox 2 #Qx 2 #Px 2 #PxSymbol- 2 lh$= 2 = 2 ]-Fences- 2 i Fx 2 i Hx 2 !i Gx 2 v Ix 2 v Kx 2 !v JxSymbol- 2 c= 2  -Fences- 2 %Lx 2 %Nx 2 %Mx 2 %Mx 2 Ox 2 Qx 2 Px 2 PxSymbol- 2 U= 2 = 2 = 2 - 2 @ = 2 @ T 2 @ - 2 @  2 = 2 H= 2 H. - 2 H<= 2 H: Symbol- 2 E M 2  Times New RomanSy-2 l27777 2 e 42 l27777 2 l0 2 l0025 2 l0 2 l03 2 0 2 0025 2 li%6 2 l;&68 2 12 F27777 2  4 2 P12 27777 2 0 2 0025 2 >4 2 V0 2 86 2 @ 0 2 @ 935 2 @ 35 2 @ 6 2 @ }68 2 @ {100 2 @ u30 2 15 2 300 2 H35 Times New Roman+- 2 2p 2 "2p 2 U 2p 2 U2pSymbol- 2 lU p 2  p 2 lp 2 p 2  p 2 p 2 h 2 hTimes New Roman+- 2 l(k 2 l.P 2 l@)(kk 2 l.P 2 l!)k 2 (k 2 .P 2 W")k 2 l%.P 2 (k 2 .P 2 D)k 2 .PMT Extra- 2 p&k 2 &kTimes New Roman+- 2 (k 2 )k 2 @ E.P 2 @ W(k 2 @ .P 2 @ )(kk 2 @ <.P 2 @ )(kk 2 @ )k 2 ,PMT Extra- 2 &kTimes New Roman+- 2 H[ (k 2 H=(k 2 H.P 2 H)(kk 2 H0 2 HY86 2 HW100 2 HQ 30 2 (2107 2 H".P 2 H)(kk 2 H!)k 2 H mP 2 H"C 2 (C WP' Times New RomanMT- 2 2pSymbol- 2 Hh- 2 H! 2 (= & "System-angesTab Stop ChangesMatrix Matrix ChangeLineCompObj"pfObjInforOle10Native!#sOle10ItemName$ FMicrosoft Equation 3.0 DS Equation Equation.39q A fin =27777pDL+pD 2 4[]=27777p(0.0025)(0.03)+p(0.0025) 2 4[]=6.68 m 2 A unfinned =1-27777pD 2 4()=1-27777p(0.0025) 2 4[]=0.86 m 2 2Q finned =h fin 2Q fin,max =h fin hA fin (T b -T  )=0.935(35 W/m 2 .C)(6.68 m 2 )(100-30)C=15,300 W2Q unfinned =hA unfinned (T b -T  )=(35 W/m 2 .C)(0.86 m 2 )(100-30)C=2107 WAlllaI8I A fin Equation Native _10213157336)F̩d̩dOle PIC &(L=27777DL+D 2 4[]=27777(0.0025)(0.03)+(0.0025) 2 4[]=6.68 m 2 A unfinned =1"27777D 2 4()=1"27777(0.0025) 2 4[]=0.86 m 2 2Q finned = fin 2Q fin,max = fin hA fin (T b "T " )=0.935(35 W/m 2 .C)(6.68 m 2 )(100"30)C=15,300 W2Q unfinned =hA unfinned (T b "T " )=(35 W/m 2 .C)(0.86 m 2 )(100"30)C=2107 WLWTW>   .1   & & MathType` MT ExtraN- 2 &k 2 &k 2 " &kTimes New Roman- 2 `c,PTimes New Roman- 2 `6Q 2 `YQ 2 ` QMETA HCompObj'+fObjInfoOle10Native*, Times New Roman-2 total,>p>c>8 2 finI>p2 ;finnedI>ppcp2  unfinnedppI>ppcpSymbol- 2 `X= 2 `+ 2 `E= 2 `+ 2 `|=Times New Roman- 2 `215 2 `300 2 `2107Times New Roman-2 `s17,407 WPP@ & "System- FMicrosoft Equation 3.0 DS Equation Equation.39q 2Q total,fin =2Q finned +2Q unfinned =15,300+2107=Times New Roman17,407 WAll?<I,|I 2Q total,fin =2Q finned +2Q unfinned =15,300+2107=1.74Ole10ItemName-Equation Native X_9031148130 F̩d̩dOle 10 4  W=Times New Roman~1~7~.~4~ ~k~WL4)\4)> X  .1  ``%& %  & MathType Times PIC /2LMETA CompObj14PObjInfoNew Roman- 2 ^A 2 6Q 2 hA 2 T 2  T Times New Roman- 2 8 bp Times New Roman-2 no finpp8I>p 2 2p2 no finpp8I>p2 no finpp8I>p 2 %&2p 2 %=2pTimes New Roman- 2  mP 2 2 mP 2   mP 2  WP' 2 /Y 2 #m 2 C 2  mP 2 C 2 # WP'Symbol- 2 = 2 8 = 2 = 2  - 2  = 2  2 - 2 E 2  = Symbol- 2  Times New Roman- 2 (k 2 )(kk 2 w )kMT Extra- 2 ]&kTimes New Roman- 2 7(k 2 W )k 2 (k 2 .P 2 )(kk 2 )(kk 2 )k 2 1 2 1 2 % 1 2 35 2 Y1 2 100 2 30 2  !2450 & "System- L M N O      FMathType Equation Equation Equation A no fin =(1 m)(1 m)=1 m 2 2Q no fin =hA no fin (T b -T  )=(35 W/m 2 Ole10Native35Ole10ItemName_1021315792:F̩d̩dOle .C)(1 m 2 )(100-30)C=2450 WAllLE lE4   PIC 79LMETA HCompObj8<fObjInfo   .1  &@ & MathType-l  SymbolN- 2 @2e Times New Roman+- 2 finI>p 2 finI>p2 dno finpp8I>pSymbol- 2 @a= 2 @ = 2 @ =MT Extra- 2 h&k 2 &kTimes New Roman- 2 q_ ,PTimes New Roman+- 2 mQ 2 =QTimes New Roman- 2 q17 2 q 407 2 92450Times New Roman+- 2 @ 7.10P & "System- FMicrosoft Equation 3.0 DS Equation Equation.39q e fin =2Q fin 2Q no fin =17,4072450=Times New Roman7.10Ole10Native;=Ole10ItemName>Equation Native  1Table\All?DII  fin =2Q fin 2Q no fin =17,4002450=Times New Roman~7~.~1~0Oh+'0  < H T`hpxSummaryInformation(A DocumentSummaryInformation88CompObjqME 146 Homework 3 SolutionMAE310CNormalMAE310C2Microsoft Office Word@F#@'d@'d՜.+,0 hp  91  ME 146 Homework 3 Solution Title  FMicrosoft Office Word Document MSWordDocWord.Document.89q<@< "QNormalCJ_HmH sH tH DA@D Default Paragraph FontRi@R  Table Normal4 l4a (k@(No List C0/F60000000000000000000000I000I000I00@0@0@00`CI00K00K00I00K00 + S  +-0BD 24:::::::8+,@P+(  b "be+)  #" `  c $sss?"y#)|` &Lc* # z#p& T  C v @&Lc*T  C v @&c*hT  C v @&c*t &Lc* # #" z#!&,%T   C v @&Lc*T   C v @&c*hT   C v @&c*t &Lc*  # #" z#%&i)T   C v @&Lc*T  C v @&c*hT  C v @&c*  B1111g ?.$bJ& rB  6D$$E  B1111g ?%:'a   B1111g ?-'Re+ B S  ?uI"t"$em "3333DhX*)irL"Qmq@m3 (@0@UnknownG: Times New Roman5Symbol3& : Arial"qhcBcB  !24 2QHX)?"Q2ME 146 Homework 3 SolutionMAE310CMAE310C