热流科学与工程西步文源大学E教育部重点实验室NumericalHeatTransferChapter3NumericalMethodsforSolvingDiffusionEquationandtheirApplications(1)(Chapter4ofTextbook)roInstructorTao,Wen-QuanKeyLaboratoryofThermo-FluidScience&EngineeringInt.JointResearchLaboratoryofThermalScience&EngineeringXi'an Jiaotong UniversityInnovativeHarborofWestChina,Xian2022-Sept-20CFD-NHT-EHTG1/55CENTER
Instructor Tao, Wen-Quan Key Laboratory of Thermo-Fluid Science & Engineering Int. Joint Research Laboratory of Thermal Science & Engineering Xi’an Jiaotong University Innovative Harbor of West China, Xian 2022-Sept-20 Numerical Heat Transfer Chapter 3 Numerical Methods for Solving Diffusion Equation and their Applications (1) (Chapter 4 of Textbook) 1/55
热流科学与工程西步文源大堂G教育部重点实验室Contents(Chapter4of Textbook)Remarks: Chapter3 in the textbook will be studiedlaterfor the students'convenience of understanding3.11-D Heat Conduction Equation3.2Fully Implicit Scheme of Multi-dimensionalHeat ConductionEguation3.3TreatmentsofSourceTermandB.C3.4TDMA&ADIMethodsforSolvingABEs3.5FullyDevelopedHTinCircularTubes3.6*FullyDevelopedHTinRectangleDuctsCFD-NHT-EHTΦ2/55CENTER
2/55 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 (Chapter 4 of Textbook) 3.4 TDMA & ADI Methods for Solving ABEs 3.6* Fully Developed HT in Rectangle Ducts 3.5 Fully Developed HT in Circular Tubes Remarks: Chapter 3 in the textbook will be studied later for the students’ convenience of understanding
热流科学与工程西步文源大堂G教育部重点实验室3.1 1-D Heat Conduction Equation3.1.1 General equation of 1-D steadyheatconduction3.1.2 Discretization of G.G.E. by CV method3.1.3 Determination of interface thermalconductivity3.1.4Discretizationof 1-Dunsteadyheatconductionequation3.1.5Mathematical stability can't guaranteesolutionphysicallymeaningful（有意义的）中CFD-NHT-EHT3/55CENTER
3/55 3.1 1-D Heat Conduction Equation 3.1.1 General equation of 1-D steady heat conduction 3.1.3 Determination of interface thermal conductivity 3.1.4 Discretization of 1-D unsteady heat conduction equation 3.1.2 Discretization of G.G.E. by CV method 3.1.5 Mathematical stability can’t guarantee solution physically meaningful (有意义的)
热流科学与工程西步文源大堂E教育部重点实验室3.11-D HeatConduction Equation3.1.1G.E.of 1-D steadyheat conduction1.Two ways of codingfor solving engineeringproblemsSpecial code(专用程序)：FLOWTHERN,POLYFLOW.....Having some generality within itsapplicationrange.General code(通用程序):HT,FF,CombustionMT, Reaction, Thermal radiation, etc.; PHOENICS.FLUENT, CFX. STAR-CDDifferent codes tempt to have some generality(通用性)Generality includes: Coordinates; G.E.; B.Ctreatment; Source term treatment; Geometry...CFD-NHT-EHTΦ4/55CENTER
4/55 3.1 1-D Heat Conduction Equation 1. Two ways of coding for solving engineering problems Special code(专用程序): FLOWTHERN， POLYFLOW.Having some generality within its application range. Different codes tempt to have some generality(通用性）. Generality includes：Coordinates；G.E.；B.C. treatment；Source term treatment；Geometry. General code(通用程序): HT, FF, Combustion, MT, Reaction, Thermal radiation, etc.；PHOENICS, FLUENT, CFX, STAR-CD , . 3.1.1 G.E. of 1-D steady heat conduction
热流科学与工程西步文源大堂G教育部重点实验室2.General governing equations of 1-D steadyheatconductionproblemdT1dI+S=0[aA(x)dxA(x) dxT----Temperature,x----Independent spacevariable（独立空间变量）normal to cross section;A(x)----Area factor, normal to heat conductiondirection;a----Thermal conductivity;S---- Source term, may be a function of both x and T.ΦCFD-NHT-EHT5/55CENTER
5/55 2. General governing equations of 1-D steady heat conduction problem 1 [ ( ) ] 0 ( ) d dT A x S A x dx dx x-Independent space variable (独立空间变量), normal to cross section; A(x)-Area factor, normal to heat conduction direction; -Thermal conductivity; S- Source term, may be a function of both x and T. T-Temperature;
热流科学与工程西步文源大堂G教育部重点实验室Od+S=02LA(r)A(x) dxdxAreaCoordi-Indep.IllustrationModevariablefactor（图示）natea11(unit)Cartesianx2Cylin-rr（arc弧度Shaded rediondricalarea)（阴影区）r2rSpherical(spherical3surface)VariableA(x),xcrossI HeatPerpendicu-4sectionlar to sectionconductionA(x)direction中CFD-NHT-EHT6/55CENTER
6/55 Mode Coordi - nate Indep. variable Area factor Illustration （图示） 1 Cartesian x 1(unit) 2 Cylin - drical r r (arc弧度 area) 3 Spherical r r 2 (spherical surface) 4 Variable cross section x Perpendicu - lar to section A(x), Heat conduction direction 1 [ ( ) ] 0 ( ) d dT A x S A x dx dx A(x)
热流科学与工程西步文源大堂E教育部重点实验室3.1.2Discretizationof Gener.Govern.Eq.byCVMMultiplying two sides by A(x)d福1ddtAA(x)I+S.A(x)=0]+S=0LACxA(x) dxdxdxdxLinearizing(线性化) source term : S(x,T)= S_+ S,TSand Sp are constant in the CV.(&z)w(ar)Adopting piecewise linear profile)(i+1for temperature;i+1Integrating over control volume PWEyielding(得)[A(x) 1 -[aA(x)-1. + [(Se + S,T,)A(x)dx =0ΦCFD-NHT-EHT7/55CENTER
7/55 3.1.2 Discretization of Gener. Govern .Eq. by CVM [ ( ) ] ( ) 0 d dT A x S A x dx dx Multiplying two sides by A x( ) Linearizing (线性化) source term ： ( , ) C P P S x T S S T Adopting piecewise linear profile for temperature; [ ( ) ] [ ( ) ] ( ) ( ) 0 e w C P P dT dT A x A x S S T A x dx dx dx Integrating over control volume P 1 [ ( ) ] 0 ( ) d dT A x S A x dx dx yielding(得) Sc and SP are constant in the CV
热流科学与工程西步文源大堂E教育部重点实验室Using the piecewise linear profile for temperature:-T++(Se +S,T,)·Ap(x)x =0F-1Y(Sx)e(8x),Moving terms with T, to left side while those with Te, Twto right sideA,(x)2T,[A(0)2+ 4.(0)2-SpA,(x)Ax)=T,[A(0)+ ScA,(x)Ax(8x).(8x)(8x).(8x)eWe adopt followingapT,=aT+awTw+bwell-accepted formfor discretized eqs.:1.A(x)e2A(x), b= ScAp(x)Ax= ScAVae(Sx)(Sx)eap=ae+aw-Sp△VΦCFD-NHT-EHT8/55CENTER
8/55 ( ) ( ) ( ) ( ) 0 ( ) ( ) E P P W e e w w C P P P e w T T T T A x A x S S T A x x x x Moving terms with to left side while those with to right side TP , T T E W ( ) ( ) ( ) ( ) [ ( ) ] [ ] [ ] ( ) ( ) ( ) ( ) ( ) e e w w e e w w P P P E W C P e w e w A x A x A x A x T S A x x T T S A x x x x x x We adopt following well-accepted form for discretized eqs.： P P E E W W a T a T a T b ( ) ( ) , , ( ) ( ) ( ) e e w w E W C P C e w A x A x a a b S A x x S V x x P E W P a a a S V Using the piecewise linear profile for temperature:
热流科学与工程西步文源大堂G教育部重点实验室Physical meaning of coefficients ae,aw1aeThermalresistancebetweenPandE(Sx)。 /[2,A(x)]a, is the reciprocal(倒数)of thermal conductionresistance between Points P and E. It represents the effectof the temperature of point E on point P, and is calledinfluencingcoefficient(影响系数）---Physicalmeaning！3.1.3Determinationof interface thermal conductivity1.Arithmeticmean（算术平均法）or(Sx)(8x)(8x)e(8x)eap+AUniform grid(a)(or),t2中CFD-NHT-EHT9/55CENTER
9/55 3.1.3 Determination of interface thermal conductivity Physical meaning of coefficients , E W a a 1 ( ) /[ ( ) ] E e e e a x A x Thermal resistance betwe 1 en P and E 1. Arithmetic mean (算术平均法) ( ) ( ) ( ) ( ) e e e P E e e x x x x Uniform grid 2 P E e is the reciprocal(倒数) of thermal conduction resistance between Points P and E. It represents the effect of the temperature of point E on point P, and is called influencing coefficient(影响系数) E a -Physical meaning!
瓶科字与工程西步文通大堂G教育部重点实验室2.Harmonicmean（调和平均法）Assuming that conductivities of P, E are differentaccording to the continuum requirement of heat flux（热流密度的连续性要求）atinterfaceedr)Te-T,T-T.T,-Tp(8x)e(8x)et(8x)(8x)。MEpAe2AlgebraicLeft sideRight side(8x),-(ox),+operationruleT, -T,T, -T,(8x)(8x)(8x)(Sx)et(Sx)。(Sx)eΛpreME1MEΛpInterface conductivityHarmonic mean中CFD-NHT-EHT10/55CENTER
10/55 Right side 2. Harmonic mean (调和平均法) Assuming that conductivities of P，E are different, according to the continuum requirement of heat flux (热流密度的连续性要求) at interface e ( ) ( ) E e e P e e E P T T T T x x ( ) ( ) ( ) E P E P e e e E P e T T T T x x x Algebraic operation rule Left side Interface conductivity ( ) ( ) ( ) e e e e E P x x x ( ) ( ) E P e e E P T T x x Harmonic mean