Lecture 2: Finite Difference Methods in Heat Transfer

Transcription

1 Lecture 2: Fiite Differece Methods i Heat Trasfer V.Vuorie Aalto Uiversity School of Egieerig Heat ad Mass Trasfer Course, Autum 2016 November 1 st 2017, Otaiemi ville.vuorie@aalto.fi

2 Overview Part 1 ( must kow ) Motivatio by the heat sik problem (HW1 & HW2 + ILO s) Solutio of 1d heat eq via fiite differeces Discretizatio i space ad time Ghost cells Stability Part 2 ( good to kow /very useful i HW2) 2d fiite differece method Computatioal examples usig a 2d FD-code

3 Part 1: 1 dimesioal fiite differece method

4 Recall: Oe of the Key ILO s is the Heat Sik Problem: Coductio + Covective Trasport Fi 1 Fi 2 y z Fi 19 x Heat equatio iside the solid material (alumiium). What boudary coditios? Navier-Stokes equatios i the gas, primary directio of covectio is z. What boudary coditios?

5 Average flow directio

6 HW1: Fid temperature distributio umerically alog oe fi (assume that gaps isulated) Fi 1 Fi 2 2 T x 2 T T =T (x,t) 2 2 y y x z Fi 19 Assumptios: - (100/19) W of heat leaves through each fi. - 1d coductio i alumiium - heat eters from left ad exits oly from right - gaps betwee fis are isulated

7 Discretizatio by Fiite Differece Method T t =α 2 T x 2 Recall Taylor series ad basic discretizatio formulae for derivatives Time derivative i cell i at timestep Secod space derivative i cell i at timestep ( T ) T +1 i T i t i Δ t ( 2 T x 2 )i T i +1 2T i +T i 1 Δ x 2 Kow from previous timestep : Ukow o the +1 timestep: T i T i +1

8 Discretizatio by Fiite Differece Method Cotiuous versio of heat equatio T t =α 2 T x 2 Discrete versio of heat equatio T +1 i T i =α T i+ 1 2 T i +T i 1 Δ t Δ x 2 Courat- Friedrichs- Lewy umber (CFL<0.5 for stability). CFL= αδ t Δ x 2 T +1 i =T i +Δ t α T i +1 2 T i +T i 1 Δ x 2 Now we have a explicit update scheme for T i each discrete grid poit i. This is the explicit Euler scheme (most simple timesteppig).

9 Discrete Represetatio of the Scheme t=t +2 i=1 i=2 i=3 T 1 T 2 T 3 i=n+1 T N+1 i=n+2 T N+2 t=t +1 T 1 T 2 T 3 T N+1 T N+2 t=t T 1 T 2 T 3 T N+1 T N+2 t = Δ t,=0,1,2,... T +1 i =T i +Δ t α T i +1 2 T i +T i 1 Δ x 2

10 Boudary Coditio Types The problem: some umerical value eeds to be assiged to the ghost cells Otherwise: we ca ot calculate secod derivative of T i cells i=2 ad i=n+1 Case 1: Give temperature value (Dirichlet or fixed value ) heat flux through boudary Case 2: Isolated (Neuma or zero-gradiet ) o heat flux through boudary Case 3: Give heat flux heat flux through boudary Ghost cell i=1 i=2 i=3 T 2 T 3 i=n+1 T N+1 Ghost cell i=n+2 Case 1: Case 2: x=0 x=l (T 1 +T 2 )/2=T mi T 1 =T 2 Case 3: k (T 2 T 1 )/Δ x=q L k (T N +2 (T N +1 +T N + 2 T N +1 =T N +2 T N +1 )/2=T max )/Δ x=q R I all the cases a ghost cell value is eeded. Ghost cell: we ca imagie a virtual cell outside the domai where we eter a temperature value so that the desired BC becomes exactly fulfilled.

11 Summary of the Numerical Solutio Scheme for 1d Heat Equatio 1) Set boudary coditios to cells 1 ad N+2 usig T from step. T i (Kow) 2) Update ew temperature at timestep +1 i the iteral cells 2...N+1 T +1 i =T i +Δ t α T i +1 2 T i +T i 1 Δ x 2 T i +1 3) Update time accordig to t = t + dt t +1 =t +Δ t 4) Go back to 1)

12 This Scheme is Extremely Short to Program i Matlab Program: /Example1d/HeatDiffusio.m Executio: >> HeatDiffusio What it does: Solves 1d heat equatio i equispaced grid, fixed T left ad T right. Mai for-loop: for(t=1:k) % set boudary coditios T(1) = 2*Tleft - T(2); T(N+2) = 2*Tright - T(N+1); ed % update temperature i ier poits T(i) = T(i) + (dt*kappa/dx^2)*(t(i+1)-2*t(i)+t(i-1)); Note: I use costatly the trick which makes Matlab-programs ofte very fast. % defie a table which refers to the 'ier poits' i = 2:(N+1); Example for N+1 = 5

13 The Followig Folders ad Files for Matlab Programs Provided (Week 1) /Example0d/ cool0d.m /Examples1d/ HeatDiffusio.m CovectioDiffusio.m /HowToPlot/ DrawigSurface.m PlottigFigure.m SurfaceAimatio.m /Examples2d/ CaseDefiitio.m computedt.m GradX.m GradY.m HeatDiffusio2d.m project.m solvetemperature.m circle.m DivDiv.m GradXskew.m GradYskew.m Laplacia.m settbcs.m visualizeresults.m Execute by >> cool0d Execute by e.g. >> HeatDiffusio Demos o plottig Figures. Execute by e.g.: >> DrawigSurface 2d heat trasfer code. Execute by: >> HeatDiffusio2d

14 Part 2: 2 dimesioal fiite differece methods

15 Overview Previously you leared about 0d ad 1d heat trasfer problems ad their umerical solutio Here we exted thigs ito 2d (3d) cases which is straightforward We cosider the simple case of a square domai Real geometries: Welcome to CFD (sprig) & CFD-modelig (autum) courses Real geometries: fiite volume methods or fiite elemet methods

16 Thikig task How would you model heat trasfer across a brick wall? What could be boudary coditios for brick heat trasfer? How could you model the holes? Why are there holes? How ca you explai the holes by heat coductivity or thermal resistace

17 Discretizatio by Fiite Differece Method Geeral form of heat equatio T t = α T Time derivative i cell (i, j) at timestep ( T ) T +1 T t Δ t X: ( 2 Terms opeed i 2d T t = x α T x + y α T y Secod space derivatives at at cell (i,j) T x 2 ) T i +1, j 2 T +T i 1, j Δ x 2 Y: ( 2 T y 2 ) T +1 2 T Δ y 2 +T 1

18 Discretizatio by Fiite Differece Method T +1 =T Δ T Explicit Euler timesteppig for 2d heat equatio: +Δ t α T i +1, j Which is equal to the delta form: =Δ t α T i +1, j Where: Δ T =T +1 T 2 T +T i 1, j +Δ t α T +1 Δ x 2 2 T +T i 1, j +Δ t α T + 1 Δ x 2 2 T +T 1 Δ y 2 CFL= αδ t Δ x 2 2 T +T 1 Δ y 2

19 Visualizatio of 2d Cartesia Grid Ghost cells where y=ly boudary coditios give. For example: Zero gradiet: T ghost =T i 1, j T ghost x=0 x=lx T i 1, j T +1 T T i+1, j This corer cell would be idle /useless Ghost cells where boudary coditios give Fixed temperature: T ghost =2 T target T i 1, j y=0 T 1 Δ y Cell face ceter Cell ceter Δ x

20 Summary of the Numerical Solutio Scheme for 2d Heat Equatio 1) Set boudary coditios (BC's) to the ghost cells usig T from step. T (Kow ad hece BC update possible) 2) Update ew temperature at timestep +1 i the iteral cells 3) Update time accordig to t = t + dt T +1 =T +Δ t α T i +1, j 2T +T i 1, j +Δ t α T +1 Δ x 2 +1 T t +1 =t +Δ t 2T +T 1 Δ y 2 4) Go back to 1)

21 Also This Scheme is Extremely Short to Program i Matlab Program: /Examples2d/HeatDiffusio2d.m Executio: >> HeatDiffusio2d What it does: Solves 2d heat equatio i Cartesia grid, various BC's possible. It is also possible to simulate materials with variable heat diffusivity to simulate coductio i e.g. layered materials. The heart of this 2d code is the computatio of dt i the.m file computedt.m dt=dt*divdiv(t,u,east,west,orth,south,ix,iy,dx,dy); - I short the cryptic fuctio DivDiv.m is eeded to evaluate divergece of the diffusive flux (coservatively) i.e. α T - The cotets of the fuctio DivDiv.m looks legthy but it is so oly to accout for the fact that the code assumes diffusivity (x,y) - depedet. The fuctio solvetemperature.m applies the 4 th order Ruge-Kutta method (for better stability tha Euler) to evaluate dt several times ad fially accumulates them together to ed up i essetially the same outcome as Euler method. T ew =T old +Δ T

22 T left =500K T right =293K T left =T top =500K T right =300K Bottom ad top walls are zero gradiet keepig solutio 1d T left =500K T right =293K Top & Bottom zero gradiet Bottom wall is zero temperature gradiet i.e. thermally isulated Coductig Isulated pocket (k=0) Nocoductig Zoom: ote that isolies meet the surface perpedicular for zero-gradiet

23 Thak you for your attetio!