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Department of Materials Science and Engineering Massachusetts Institute of Technology 3.044 Materials Processing- Spring 2013 Exam I Wednesday, March 13, 2013 The Rules: I) No books allowed; no computers allowed; etc. 2) A simple calculator is allowed 3) One hand written 3x5 index card may be used as a crutch 4) Complete 5 out of the 6 problems. If you do more than 5 problems, I will grade the first 5 that are not crossed out. 5) Make sure that you READ THE QUESTIONS CAREFULLY 6) Reasonable assumptions are permitted, but please state them clearly. 7) Supplementary materials are attached to the end of the test (eqns., etc.) 8) WRITE YOUR NAME HERE: Problem #1: Why Not Start at the Beginning, with a Heat Balance Problem? Let's derive the heat conduction equation, but with some bells and whistles added. Let's add anisotropy! dy Consider a small square (2D) patch as shown; we will work the problem in 2D and assume a unit thickness on the Z axis, which we will call Lz. Unfortunately, this material is anisotropic and has two relevant conductivities, kx and ky on the two axes. ( dx } ) \ IC.l.l ·r· 1\C.C t)fV' \JLAT cD Part B: Replace the words with mathematical symbols, but do not rearrange yet. Part C: Simplify to the greatest extent possible, to provide a differential equation that is the I ooodro(equati:f;;2(-D\anisotropicplat(e!aT ( Kx ~ x) xrdY. + k't I k." fC..f ~_}\ ~ l'J+dy Jy Jx + - ;)y \ )T y J :::;;: yCf el-i. 'C{T -d'I'- T) Part D: Show that you can reproduce the standard, isotropic form of the 2D conduction equation in the limiting case where kx = ky ( '2 "1. k-.::: k..:><- =Ly I Tllf:N dI _h. d t - fCp d T + 0----z:. d X.7.. '2 c<. 7 T cl '1 Problem #2: Hot and Bothered, Cooled and Bored Take a small block (L = 0.5 em) of a refractory material like Ah0 3, and consider how it cools from a very high temperature (1500° C) while sitting in air. Part A) On its way from high temperatures to low, we have a competition between radiation, convection, and conduction. At what temperatures is each of these dominant? 0- ::::. ~ .==:. 0 OQ g :9 No [iltADt £.N"l S IN S6l-l £) f...\/6fC 01 1 _ , ., ' CoNDV&iiSN' Nelle~<.. t--IMI-r!Nt.J · ~ RJ"-'()ID.Ii<JN \/<,. " q /1 v !"'. :=SoN I....X, 'tt..J>.r:> '. -:d K. "'"' 1..0 ""/ ('/'. coNVECIIil'N~ "n(T 5 -T.) h ?::::: :> . \ t_tr(is*-T~) Jlf:..tr .c.T ,__, oS f:>'l J::: n /V"\. 5 I u;-) :;=- 0.10 . (!<.AD ••• .[::6MINAN1) r61"H '-'<. 1':\ :: Q. 3 5 r,l:: \Otx::>o; '-" - 1 \I':,_"Tif:fl._ r:;-r r::.,l ::- >)ad'' Q - , ~ \ c,T ~~ =so1; Q-::;; 13 (cON..J Do·MIN~T) t;T Fll'R... ~l-z_03 Part B) On the axes below, draw a graph showing schematically how the temperature at the surface of the block drops with time. Explain in one sentence what you have drawn. Part C) Now a small bore-hole.is made through the center of the block along one axis. Draw a second curve on your graph, showing how this hole affects the cooling of the block. Again, explain in one sentence or so what your thoughts are that led you to this drawing. ~Dl ~>.<toN '' I'Jt.,.,lioNIAN c.oct_II'.J (:1 1 \j-11..\fl'!-lSr?-. '1 1'0 "-f\J()-i'.j i\~1':.; l \ ~vs-r S\.14\.JLO 1411'.\\:) dF 1)'C C...~EP.SIN~ PaN ft.... L.Aif'i IIJt Nq) (a><. -----"!> b\ --r 1"' ') [:;, '1 I I ' Au..'-! I IS '50 \?cdc. To< t 11 CooL..\ f\1 C4 15e-rW'E:.t:N f.l '~I" \J .... \ -y ( DofSN o,. c.t...EA 'LL"' r= M PC f.-A.TIJI~:f.. .DIU> Ps r..-r 11--l f: {'..II df\1 aN L-'l ' \..l. CS'N f:-\1 (:_((_/ fi..Jsi 0 "D I !-"'"' 1 ' M.AT'IER.. 6f SOW\ E D1c..1 A--ft: 0, E.. \\.1 £ ,, l.l I! "' '--'(, L)N\ BE:() ,, G,:l. AND \0 'i"'oR. ;-J3tx:/''~ .t:.T<= 1Soo 1 Bo\\4 R.f.::.DI P.TtaN f!x USNVEC-ridtJ C-{S'NTR..\e,0\E 1 AcC6LEI2A'IJI\J~ fliE (...Oc.ll....II\J(.;), ~01<... AI<::.. )C0°C1 ~Df~\10('..! 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Let's try to formalize this a bit more. ~lk/:" -Fi';(.ED Part A) Write down the proper form of the conduction equation for this geometry, and also write down the assumed form for a 1-D Cartesian problem. u.::,.e_:T ESIAN "d\ --;: ;;; r.X. ~-c. Part B) Compare the two forms of the diffusion equation, and identify the terms that need to be ~jLT neglected in order for the problem to become 1-D Cartesian r\ "d ( ~ d-z_l a-.?.. I~ IS dl\ (" -;(-) ::- rI > > _L ( dT ~ EJ I (> ( p..cc:E.PIA'BLE.~ + r l:J: \ __ ~ ·-r .---2 ar 7.) - THEN / T .En w.1£ "THE r d (' d '(" AI'~XtMAit ~ CA12-I£SIA.N MUST -rl-1 E NE:qLfC..\ _l. fl I d y ..,--_eft\ IE • I Part C) Write down a mathematical condition under which you would consider it is OK to neglect the cylindrical geometry in favor of the 1-D Cartesian one. If there is any assumption here, please state it expJicitly,. d'l.."f > IO ;;,.,-"- -r\ ""-r ~ CAN \t.lE: Bf u.-no AS MP (2,1>;<. i\'S (IZ -y J / ::::to J - . 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Part A: Qualitatively explain the flaw in Rudiger's logic. Part B: Write down a proper set of equations that define this problem, including the equation we must solve, and the boundary conditions for steady state. (Don't solve it, just write the equations) A L)Ni F012_fV\ 'LS Tfi'V\PERfrrv P-£ T\.l ~ STEAD 'i \)N 1 <r oQ.Y'I\. SN 'fl P-..rl'-1 N\fN7, 1N 1\ b6\)ND M.)-j CaNDi'"(i 6'NS, 0 _ r- J ;) y-2 d-f' ( 2 Ji\ r () r ) • v-fiJEe...f: No L}R-..ADI(i\fiS, lfl \"i -t cASt:. I rL.Sllli'f d d¢ S"'f AIE +" 6R.. A Sf>\.1 £: 12_E, oofS 1\JCJ r\A\1£ \.)tJIFcJR.JI'\ (siN~ ~\ d ~ . ) : Problem #5: Enlighten Me Each of the five statements below is false (or at best, is not always true!). Make a simple but nontrivial change (for example, deleting a few words, adding a few words, or both) to each sentence to make it generally true. Write 1 sentence of explanation for each to back up your changes. 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I T Y"f>l & ·'<-r _,.., t:,D, "'(\0!2.. .Jv _, =0 c_ 'l -r 'i sl!l2f-19JIU£ J • Problem #6: Wherein a Block of Steel Meets a Fate Common to Lobsters A cubic centimeter block of steel is at room temperature, and suddenly immersed in I 0 cm 3 of water at I 00 C. 2Cf'c Part A: What is the steady state for this system? 2. 2 II-IF lj \<lI S \...o V'lf ST \.N'r.~1 MIT OpenCourseWare http://ocw.mit.edu 3.044 Materials Processing Spring 2013 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.