Numerical Heat TransferChapter3NumericalMethodsforSolvingDiffusionEquationandtheirApplications(2)(Chapter4ofTextbook)1/58
1/58 Numerical Heat Transfer Chapter 3 Numerical Methods for Solving Diffusion Equation and their Applications (2) (Chapter 4 of Textbook)
Contents3.1 1-D Heat Conduction Equation3.2FullyImplicit Scheme ofMulti-dimensionalHeat Conduction Eguation3.3TreatmentsofSourceTermand B.C3.4TDMA&ADIMethodsforSolvingABEs3.5FullyDevelopedHTinCircularTubes3.6FullyDevelopedHTinRectangleDucts2/58
2/58 3.1 1-D Heat Conduction Equation 3.2 Fully Implicit Scheme of Multi-dimensional Heat Conduction Equation 3.3 Treatments of Source Term and B.C. Contents 3.4 TDMA & ADI Methods for Solving ABEs 3.6 Fully Developed HT in Rectangle Ducts 3.5 Fully Developed HT in Circular Tubes
3.4TDMA&ADlMethodsforSolvingAlgebraicEquations3.4.1TDMAalgorithm(算法)for1-Dconductionproblem1.General form of algebraic equations of 1-D conduction problems2.Thomasalgorithm3.Treatmentof1stkindboundarycondition3.4.2ADlmethodforsolvingmultidimensionalproblem1.Introductiontothematrixof2-Dproblem2.ADliterationofPeaceman-Rachford3/58
3/58 3.4 TDMA & ADI Methods for Solving Algebraic Equations 3.4.1 TDMA algorithm (算法) for 1-D conduction problem 3.4.2 ADI method for solving multidimensional problem 1. Introduction to the matrix of 2-D problem 1.General form of algebraic equations of 1-D conduction problems 2. ADI iteration of Peaceman-Rachford 2.Thomas algorithm 3.Treatment of 1st kind boundary condition
3.4TDMA&ADIMethodsforSolvingABEqs3.4.1TDMAalgorithmfor1-Dconductionproblem1.Generalformofalgebraicequationsof1-D conductionproblemsThe ABEqs fora,T, +a,T, +..+a,T, +...+amTm=b (i=1,Ml)steady and unsteady(f >0) problems takethe following forma,T,=a,T+awTw+b(I-N)(I+1)-i=lThe matrix (矩阵)Threeof the coefficients is a tri-unknownsdiagonal(三对角）one4/58
4/58 3.4 TDMA & ADI Methods for Solving ABEqs 3.4.1 TDMA algorithm for 1-D conduction problem 1.General form of algebraic equations of 1-D conduction problems The ABEqs for steady and unsteady (f >0) problems take the following form The matrix (矩阵) of the coefficients is a tridiagonal (三对角) one . P P E E W W a T a T a T b 1 1 2 2 1 1 .+ . = 1, 1) i i M M a T a T a T b i M a T （ Three unknowns i I (I-1) I (I+1)
2.Thomasalgorithm(算法）The numbering method of W-P-E is humanized (人性化)but it can not be accepted by a computer!Rewriteaboveequation:AT, = B,T+ +CT- +D, i= 1,2,....M1(a)End conditions: i=1, C,=C,=0; i=Ml, B,=Bmi=0(1)Elimination（消元)-Reducingtheunknownsateachlinefrom3to2Assuming the eq. afterelimination asT- = P-,T, +Qi-1 (b)Coefficient has been treated to 15/58
5/58 2. Thomas algorithm(算法) Rewrite above equation： End conditions：i=1, Ci=C1 =0; i＝M1, Bi=BM1 =0 (1) Elimination (消元)－Reducing the unknowns at each line from 3 to 2 Assuming the eq. after elimination as 1 1 AT BT CT D i M i i i i i i i , 1,2,. 1 (a) T P T Q i i i i 1 1 1 Coefficient has been treated to 1. (b) The numbering method of W-P-E is humanized (人性化), but it can not be accepted by a computer!