帕坦卡 传热数值计算 课件

1、Computational fluid dynamics with heat and mass transferE4465Fundamentals of transfer processesConservation of massConservation of momentumConservation of energyConservation of chargep-v-t relationsturbulenceboundary layer approximationsNumerical techniques for transfer processesDiscretization metho

2、ds for:conductionconvective diffusionBasic equationsGridNonlinearityBoundary conditionsTransientsRelaxationCalculation of the:flow fieldtemperature distributionconcentration fieldVehicle - CFDComputational Fluid Dynamics (CFD) is a discipline that solves a set of equations governing the fluid flow o

3、ver any geometrical configuration. The equations can represent steady or unsteady, compressible or incompressible, and inviscid or viscous flows, including nonideal and reacting fluid behavior. The particular form chosen depends on the intended application. The state of the art is characterized by t

4、he complexity of the geometry, the flow physics, and the computer time required to obtain a solution.Applications in Power IndustryHeat flux distribution in a boilerHeat transfer and pressure loss in a heat exchangerFlow, heat transfer and erosion in a condenserPerformance of a gas turbine combustio

5、n chamber and flow through a gas turbine test cellFlow through a high pressure steam turbine valveHazard analysisCarsFour views of cylinder intake flow. Data courtesy Ford Motor Co.Side and top views of fuel spray in a diesel engine.Sequence from a car in a wind tunnel animation. Data courtesy Fluen

6、t, Inc.Animation of flow over a car in a wind tunnel. Data courtesy Ford Motor Co.Two views of a car crash.Civil and EnvironmentalEnvironmental Impact Assessment Flood Analysis and Dam Breach Simulation Hydro-Power Systems Wind Interaction with Structures Pollutant Dispersal and Air Quality Fire and

7、 Smoke Spread Heating and Air Conditioning Waste Water Treatment and Water Quality TransportationAerodynamics of Airplanes and trains Cars and trucksMotorcyclesHydrodynamics of ships and submarinesDesign of car componentsexhaustvalveModeling of supersonic/hypersonic flowPassenger safety analysis Fue

8、l sloshing analysisOther applicationsFlow through porous mediaMixing vessel designCyclone designMetal casting Turbine blade designPolymerization reactor vessel designCoating processes Ink jet design Cooling of electronic componentsNumerical Transfer ProcessesE4386Modeling vs. experimentationAdvantag

9、es of modeling:cheapermore complete informationcan handle any degree of complexity as long as Disadvantages of modeling:deals with a mathematical description not with realitymathematical description can be inadequatemultiple solutions can exist and the winner is Conservation principle leads to a gen

10、eral differential equation:Mathematical description of transfer processesDiscretization methodsanalytical solution gives a function (continuous)numerical solution gives a set of numbers (discrete)TxExample: steady 1D heat conductionxSIntegrationPWExweDx(dx)w(dx)eDiscretizationTxPWEweTpTwTEDiscretiza

11、tion equationconductancesheat generationSource term linearizationFour basic rulesRule 1. Flux consistency at control volume faces“when a face is common to two adjacent control volumes, the flux across it must be represented by the same expression in the discretization equations for the two control v

12、olumes”Rule 2. Positive coefficients“all coefficients a must always be positive ”Four basic rulesRule 3. Negative slope linearization of the source term“When the source term is linearized as S=SC+SPTP, the coefficient SP must always be less than or equal to zero. ”Rule 4. Sum of the neighbour coeffi

13、cients“it is required that aP= anb for situations where the differential equation continues to remain satisfied after a constant is added to the dependent variable”exampleBack to: steady 1D heat conductionPWExweDx(dx)w(dx)eDiscretizationTxPWEweTpTwTEDiscretization equationconductancesheat generation

14、Source term linearizationfor example when S=f(T)Nonlinearityfor example when k=f(T)the answer is iterationInterpolation ofinterfacial conductivityPExwe(dx)e(dx)e+(dx)e-First guess:Interpolation ofinterfacial conductivityremember all we need is the fluxInterpolation ofinterfacial conductivityremember

15、 the first guess:for: which one is correct?Grid spacingUniformeasy to set-upcomputationally expensiveNon-uniformharder to set-upneeds iterationcomputationally economicalBlock grids?Multi-grids?Boundary conditions1. Given boundary temperaturesimple2. Given boundary heat fluxa) as a constant b) specif

16、ied via a heat transfer coefficient and the temperature of the surrounding fluidBoundary conditionsPWEIBiIBDx(dx)iiqBqiSBoundary conditionsIBDx(dx)iiqBqiSBoundary conditionsif qB specified as a constant then:Boundary conditionsif qB specified via a heat transfer h coefficient and the temperature of

17、the surrounding fluid Tf : Solution of the linear algebraic equationsPossibilitiesIteration Gaussian eliminationTri-Diagonal-Matrix-Algorithm TDMAPWE1i-1ii+1Nchange of notationTDMAPWE1i-1ii+1Nwhat happens when a boundary temperature is given ?TDMAPWE1i-1ii+1NTDMAPWE1i-1ii+1NTDMA algorithmCalculate:P

18、1=b1/a1 and Q1=d1/a1 Use recurrence relations to get Pi and Qi for i=2,3N.Set TN =QN Use recurrence relations to getTi =Pi Ti+1 +Qi for i=N-1, N-23, 2, 1 to obtainTN-1, TN-2, T3, T2, T1.Unsteady1D heat conductionsteady:unsteady:(transient)PWExweDx(dx)w(dx)etIntegrationTxPWEweTpTwTEintegrate with res

19、pect to x:integrate with respect to t:Time integrationrepeat for points E and W:tTExplicit vs implicitf=0 explicit schemef=0.5 semi-implicit (Crank-Nicolson) schemef=1 fully implicit schemeTpoldtTpnewt+Dtf=0f=1f=0.5Explicit schemefor:in order to give realistic solutionsCrank-Nicolson schemecan give

20、unrealistic solutionsImplicit schemealways gives realistic solutionsUnsteady2-D heat conductionxyz=1PWEweDxNSnsDyDiscretized unsteady 2-D heat conduction equationUnsteady 3-D heat conduction equationDiscretized unsteady 3-D heat conduction equationSolution of the equationsdirect methodGauss-Seidel m

21、ethodline-by-line methodalternating-direction implicitstrongly implicit procedureRelaxationoverrelaxationunderrelaxationRelaxationrelaxation through inertia:Convection and diffusionassume a known flow fieldgeneralize T and k as F and Gcontinuity and momentum equations:Steady1D convection and diffusi

22、onconvective-diffusionequation:continuityPWExweDx(dx)w(dx)eFxPWEweFpFwFEDiscretization using central differencescentral differencescheme:Discretization using central differencesDiscretization using upwindingcentral differencescheme:upwind scheme:Discretization using upwindingdefine a newoperator:mea

23、ning:the greater of A and Bthen:Discretization using upwindingExact solution1D convective-diffusion equation:has an exact solution:given simple boundary conditionswhere the Peclet number is:Exact solution and the Peclet numberxFFLFoLP1P=1P=-1P=0Calculation of the flow fieldmass conservation orcontin

24、uity equation is obtained for f =1:general conservation equation:Momentum equationsor for i=1 (x direction):momentum equation obtained for f=ui:similar equations for i=2 and 3Momentum equationsor for i=2 (y direction):Momentum equationsor for i=3 (z direction):Where is the problem?equations are tran

25、sient (time dependent)?we do not have as many equations as we have unknowns?equations look ugly?equations are complicated?equations are nonlinear?eqautions are coupled?the pressure field is unknown?there is no obvious equation to obtain the pressure field?Some more problemswith pressureintegratedpre

26、ssureterm:PWExweDx(dx)w(dx)eSome more problemswith pressurepxPWEwepppwpEcentral differencescheme: so what? wavy fields so what?Similar problems with velocityintegratecontunuityequation:PWExweDx(dx)w(dx)eRemedy:the staggered gridxyz=1PWEweNSnsbenefit: no interpolationno wavy fieldspressure “driving”

27、the flowprice: more bookeepingx-momentum equationor for i=1 (x direction) ui=u1=u:general momentum equation :x-momentum equation discretizedin the x direction:PWEweNSnseenesexyy-momentum equationor for i=2 (y direction) ui=u2=v:general momentum equation :y-momentum equation discretizedin the y direction:xySPWEweNnsnnnenwz- momentum equationor for i=3 (z direction) ui=u3=w:general momentum equation :z-momentum equation disc