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TextState? TextStyle>@ Times New Roman0,0,128Serial_ParPropDefaultW?Normal>@ Arial Black 0,128,128@W? Heading 1>@Arial 0,128,128@W? Heading 2 >@ Arial@W Heading 3 >@ Arial@W Paragraph >@ Arial@WList >@ Arial@WIndent >@Times New Roman@W Title>@Times New Roman@W Subtitle font_style_listO font_styleP  VariablesArial@P  ConstantsArial@P TextArial@P Greek VariablesSymbol@P User 1Arial@P User 2 Courier New@P User 3Arial@P User 4Times New Roman@P User 5Times New Roman@P User 6Arial@P User 7Times New Roman@P SymbolsSymbol@P Current Selection FontArial@P Undefined Font@P HeaderArial@P FooterArial@P Rotated Math FontArial@ TextRegion* docRegionGshpBoxU<(/ :,%,% CharacterMap-RangeMap;%Transient 1D Conductive Heat Transfer ChrPropMap7% ParPropMap9% RangeElem<% ParPropData: RangeData=@WEmbedMap1<LinkMap/%<%LinkData0@NormalTimes New Roman *@U)cO'8hS&S&-Wayne Pafko 11/19/0179<:@W1</<0@NormalTimes New Roman *@U_yt$p`ii- Background7 9 < :@W1</ < 0@ Heading 2Arial!*@UyWGG_G_-AThis Mathcad document shows how to use an finite difference algorithm to solve an intial value transient heat transfer problem involving conduction in a slab. In this problem, the temperature in the slab is initially uniform (Initial Condition). The edges are then instantly changed to a constant temperature boundary condition (Dirichlet BC). The finite difference algorithm then calculates how the temperature profile in the slab changes over time.79"<#:@W1$</%<&0@NormalTimes New Roman '*@U*O`-Physical Model 79(<):@W1*</+<,0@ Heading 2Arial-*@U G1Q`77-@dConduction of heat in a slab is usually described using a parabolic partial differential equation...7d9d.9<@@t;2=@@;>@@d=x?@@=T@@*@U1D@-'Transient conduction of heat in a slab.7'9'@A<'@B:@W1@C</'@D<'@E0@NormalTimes New Roman @F*@Uatp[`-@SThis partial differential equation can be approximated using finite differences... 7S9S@G@@dA= timestepsA?@@A=\DtimeA@@@A Root EntryaB.dK8Yq@Contents" -->here T1c$ArialdI_/D6P@nArialdI_/D6P@nA@@AK@XA@@A|A*@U1D@`-*4) Input left & right boundary conditions.7*9*A<*A:@W1A</*A<*A0@NormalTimes New Roman A@B@U(Qk^`oA@@ pA@@ AA@@dABC.leftA@@5AA@@tA350A@@AKA@B@UQ,k`A@@ pA@@ AA@@dA BC.rightA@@AA@@tA440A@@AKA*@U`-Problem Solution79A<A:@W1A</A<A0@ Heading 2ArialA*@UZ.`JJ-95) Solve the problem using a finite difference algorithm.7999A<9A:@W1A</9A<9A0@NormalTimes New Roman A*@Uh#,XXLXL-AWWhen you pass a vector containing the temperature at each node to the "updateTemps" function it calculates what the temperature at each node will be at the next time step and returns that vector of updated temperatures. Note that the temperature at the edges are always just set to equal the left and right boundary conditions (Dirichlet BC).7W9WA Root EntryaB.dK8Yq@Contents. T1c$6ArialdI_/D6P@nArialdI_/D6P@nB0@@B,K@XB1@@B)1J1`B2*@U!pG0>``&`&-@The last row contains the temperature profile at the end of the simulation. This may or may not be the steady-state solution depending on how much simulation time elapses.79B3<B4:@W1B5</B6<B70@NormalTimes New Roman B8@B@US_p"B9@@ pB:@@B9B;@@@B:B<@@4@B;B=@@dB<TanswerB>@@B timestepsB@@@B>1BA@@B:BB@@@BABC@@@BBBD@@5BBBE@@<@BDBF B8BEBG< ](CVSOleClientItem  ࡱ> Root EntryaB.dK8Yq@Contents("@M_T1c$6ArialdI_/D6P@nArialdI_/D6P@nBH@@BDK@XBI@@BA1J1`BJ*@UH`qq-6) View the results79BK<BL:@W1BM</BN<BO0@NormalTimes New Roman BP@B@UN%BQ@@ pBR@@ BQBS@@@BRBT@@dBSxBU@@BSiBV@@BRBW@@dBViBX@@BV\DxBY*@Ux$-Position of each node in slab79BZ<B[:@W1B\</B]<B^0@NormalTimes New Roman B_@B@UB B`@@ pBa@@B`Bb@@@BaBc@@@BbBd@@@BcBe@@tBd450Bf@@Bd275Bg@@BcBh@@@BgBi@@BgBj@@ BbBk@@ @BjBl@@ @BkBm@@ @BlBn@@4@BmBo@@dBnTanswerBp@@Bn1Bq@@4BmBr@@dBqTanswerBs@@BqBt@@dBsfloorBu@@pBsBv@@5BuBw@@dBv0.25Bx@@Bv timestepsBy@@4BlBz@@dByTanswerB{@@ByB|@@dB{floorB}@@pB{B~@@5B}B@@dB~0.5B@@B~ timestepsB@@4BkB@@dBTanswerB@@BB@@dBfloorB@@pBB@@5BB@@dB0.75B@@B timestepsB@@4BjB@@dBTanswerB@@BB@@dB timestepsB@@B1B@@BaB@@@BB@@@BB@@vB2.9B@@B0B@@BB@@@BB@@BB@@BB@@dBxB@@BcmB )LPosition in Slab (cm))LTemperature (K)&&&&&&&&&& & & & & &&&B*@U) < ,8 `-"We can also animate the results...7"9"B<"B:@W1B</"B<"B0@NormalTimes New Roman B@B@UH ^ 9X B@@ pB@@ BB@@dBtimeB@@BB@@@BB@@dB10B@@BFRAMEB@@B\DtimeB*@UI \ X -Calculate time at each FRAME79B<B:@W1B</B<B0@NormalTimes New Roman B@B@Ug b B@@ pB@@BB@@@BB@@dB timestepsB@@B10B@@BB@@+@B@XB@@BB*@Uq  - Animate FRAME up to this number.7 9 B< B:@W1B</ B< B0@NormalTimes New Roman B@B@U "  B@@ pB@@BB@@@BB@@@BB@@@BB@@tB450B@@B300B@@BB@@@BB@@BB@@BB@@4@BB@@dBTanswerB@@5BB@@dB10B@@BFRAMEB@@BKB@@BB@@@BB@@@BB@@vB2.9B@@B0B@@BB@@@BB@@BB@@BB@@dBxB@@BcmB )LPosition in Slab (cm))LTemperature (K)&&&&&&&&&& & & & & &&&B@B@U` B@@ pB@@BB@@dBtimeB@@BB@@+@B@XB@@B