abaqus經(jīng)典聲學(xué)分析例題a1-appendix-heat-transfer-eqs

1、Heat Transfer TheoryAppendix 1Copyright 2006 ABAQUS, Inc.A1.2OverviewSummary of Governing Equations for ConductionConstitutive RelationFouriers LawThermal Energy BalanceDifferential FormThermal Energy BalanceEquivalent Variational FormFinite Element ApproximationTransient AnalysisEulerian Formulatio

2、n for ConvectionThermal Radiation FormulationAdiabatic Thermal-Stress AnalysisNonlinear Solution SchemeHeat Transfer and Thermal-Stress Analysis with ABAQUSCopyright 2006 ABAQUS, Inc.Summary of Governing Equations for ConductionCopyright 2006 ABAQUS, Inc.A1.4Summary of Governing Equations for Conduc

3、tionSurface SThe heat balance equation enforces the condition that the surface and volumetric heat flux into a body is balanced by the rate of internal energy generation in the body.Differential formn Volume VqrrU = - q + rxVariational formArbitrary volume controldqdqr=dq qdS +dq rdVUdV - qdVxVVSV T

4、he variational form is used as the basis of the finite element discretization.Heat Transfer and Thermal-Stress Analysis with ABAQUSCopyright 2006 ABAQUS, Inc.The details of the derivations of these equations are discussed shortly.A1.5Summary of Governing Equations for ConductionqHeat flux per unit o

5、f current area crossing surface S from environment to body under consideration (scalar quantity); related to the flux vector q and outward surface normal n throughq = -q n,Heat flux per unit volume generated within body, such as:1. Ohmic heating (electrical resistance).2. Heat caused by dissipation

6、of mechanical energy (plastic work in metals, molecular friction in rubbers or polymers, etc.).Current material mass density.Internal energy per unit mass assumed to be a function of temperature only. ( U = material time rate of change of U.)rrUdq Arbitrary variational temperature field.Heat Transfe

7、r and Thermal-Stress Analysis with ABAQUSCopyright 2006 ABAQUS, Inc.Constitutive RelationFouriers LawCopyright 2006 ABAQUS, Inc.A1.7Constitutive RelationFouriers LawExperimental observations have indicated that the heat flux q is proportional to the temperature gradient.The proportionality factor is

8、 the materials thermal conductivity k . In one dimension:q= -k q .xxThe minus sign indicates heat flows in the direction of decreasing temperature.q1q2Dl(q-q )21qx = -kDlqxHeat Transfer and Thermal-Stress Analysis with ABAQUSCopyright 2006 ABAQUS, Inc.A1.8Constitutive RelationFouriers LawGeneralizin

9、g to multi-dimensions,q = -k (q ) q ,xwhere k is the temperature-dependent anisotropic conductivity matrix. Introducing Fouriers Law into the weak statement of the energy balancegivesdqqVVdqrdV =dq qdS +dq rdV .+ k UdVxxSVHeat Transfer and Thermal-Stress Analysis with ABAQUSCopyright 2006 ABAQUS, In

10、c.Thermal Energy Balance Differential FormCopyright 2006 ABAQUS, Inc.A1.10Thermal Energy BalanceDifferential FormConstruction of the energy balance equation for an arbitrary control volume.Surface SWe consider the control volume attached to the material (Lagrangian formulation) and assume that the m

11、aterial is not undergoing large strain.Volume VArbitrary volume controlHeat Transfer and Thermal-Stress Analysis with ABAQUSCopyright 2006 ABAQUS, Inc.A1.11Thermal Energy BalanceDifferential FormThermal energy balance:net rate ofrate of energy rate ofheat enteringgenerated increase ofacross + = in c

12、ontrol internal energy surface volumeof control volumeStated mathematically,qdS + rdV = rUdVSVVsurface flux term qpositive into the bodyvolumetric flux term (source)storage termHeat Transfer and Thermal-Stress Analysis with ABAQUSCopyright 2006 ABAQUS, Inc.A1.12Thermal Energy BalanceDifferential For

13、mTo convert the energy balance equation to differential form:-q = flux crossing surfacenFirst, introduce a heat flux vector,q, by the geometric relation,q = -q n,qwhere n is the unit outward normal to the surface.Heat flux definitionHeat Transfer and Thermal-Stress Analysis with ABAQUSCopyright 2006

14、 ABAQUS, Inc.A1.13Thermal Energy BalanceDifferential FormSubstituting the heat flux vector rUdV = -q ndS + rdVVSVand applying the divergence theorem SVq ndS = qdVxyields V qdV + rdV .VrUdV = -xVHeat Transfer and Thermal-Stress Analysis with ABAQUSCopyright 2006 ABAQUS, Inc.A1.14Thermal Energy Balanc

15、eDifferential FormSince the volume is arbitrary, we obtain the differential form of the heat balance equation:rU = - q + r.xThis “strong” form of thermal equilibrium holds pointwise in a body.Heat Transfer and Thermal-Stress Analysis with ABAQUSCopyright 2006 ABAQUS, Inc.Thermal Energy BalanceEquiva

16、lent Variational FormCopyright 2006 ABAQUS, Inc.A1.16Thermal Energy BalanceEquivalent Variational FormWe cannot enforce the “strong” form of thermal equilibrium numerically.We need to relax the equation and enforce thermal equilibrium in an average sense over finite volumes (finite elements).To do t

17、his, we multiply the differential thermal energy balance equation by an arbitrary variational temperature field dq and integrate over the volume: V qdV + dq rdV .VdqrUdV = -dqVxThe chain rule allows us to rewrite the middle term: (qdq ) + q dq dV +dq rdV . - VVdqrUdV =xxVHeat Transfer and Thermal-St

18、ress Analysis with ABAQUSCopyright 2006 ABAQUS, Inc.A1.17Thermal Energy BalanceEquivalent Variational FormWe again apply the divergence theorem, this time to convert a volume integral to a surface integral, then convert our heat flux vector back to its scalar form: (qdq )dV = -n qdq dS = qdq dS.-xVS

19、SThis produces the “weak” form of the thermal energy balance equation,which is used as the basis of the finite element discretization:dqVVdqr qdV =dq qdS +dq rdV .UdV -xSVHeat Transfer and Thermal-Stress Analysis with ABAQUSCopyright 2006 ABAQUS, Inc.Finite Element ApproximationCopyright 2006 ABAQUS

20、, Inc.A1.19Finite Element ApproximationWe discretize the volume geometrically using finite elements and interpolate the temperatures within the elements from the element nodaltemperatures, qN ,byq = N NqN ,( x )where N Nare the interpolation functions for the element(summation is implied over the re

21、peated superscripts).Using the Galerkin approach, the variational temperature field dq is interpolated with the same functions:dq = N NdqN .( x )In ABAQUS the functions N Nare linear or parabolic polynomialproduct forms for one-, two-, and three-dimensional elements, as well as for axisymmetric elem

22、ents.Heat Transfer and Thermal-Stress Analysis with ABAQUSCopyright 2006 ABAQUS, Inc.A1.20Finite Element ApproximationIntroducing this interpolation into the heat balance equation we obtain the approximationNNM MNdqNrUdV +NdVq = dqNNN k NrdV +NqdS .xxVVVSSince the variational quantities dqNare indep

23、endent, this providesN NN M M NrUdV + = NrdV + NqdSNNNdVq k xxVVVSThis set of equations is the basis for uncoupled, conductive heat transfer in ABAQUS.Under steady-state conditions U 0,hand side disappears.For transient problems the equation must be integrated in time.and so the first term on the le

24、ft-Heat Transfer and Thermal-Stress Analysis with ABAQUSCopyright 2006 ABAQUS, Inc.A1.21Finite Element ApproximationWe can writedUdqq = cq ,U =where c (q )is the specific heat of the material. The approximate heatbalance can be writtenC + K = Q,where= rcN N NM dV is the heat capacitance matrix,CNMVN

25、 NN M= NM k KdV is the conductivity matrix,xxV= N N qdS + N N rdV is the external (applied) flux vector, andQNSV is the vector of nodal temperature degrees of freedom, qHeat Transfer and Thermal-Stress Analysis with ABAQUSCopyright 2006 ABAQUS, Inc.N .Transient AnalysisCopyright 2006 ABAQUS, Inc.A1.

26、23Transient AnalysisRecall the finite element approximation to the thermal energy balance equation,N NN M M NVrUdV + V= NVrdV + NqdS.NNNdVq k xxSThe finite elements we introduced discretize the problem in space. For transient problems this equation must also be integrated through time.The time integ

27、ration operator used in ABAQUS/Standard for transient solid body heat transfer is the backward difference algorithm:= (Ut+Dt-Ut ).Ut+DtDtHeat Transfer and Thermal-Stress Analysis with ABAQUSCopyright 2006 ABAQUS, Inc.A1.24Transient AnalysisIntroducing the time integration operator into the thermal e

28、nergy balance equation results in the following system of equations:N NN M M1V-Ut )dV + r (Ut+DtNdVqt +Dt kt+DtNDxxtV- N N rt+Dt dV - N N qt+DtdS = 0.VSHeat Transfer and Thermal-Stress Analysis with ABAQUSCopyright 2006 ABAQUS, Inc.Eulerian Formulation for ConvectionCopyright 2006 ABAQUS, Inc.A1.26E

29、ulerian Formulation for ConvectionRecall the Lagrangian form of the energy balance for an arbitrary control volume: qdV + rdV .rUdV = -xVVVThe divergence theorem has been applied to the heat flux term.An Eulerian formulation is used for forced convection problems, so expansion of the material time r

30、ate of change of internal energy, U, required:is ( rU ) + u ( rU ) dV = - V qdV +rdV .VxtxVHeat Transfer and Thermal-Stress Analysis with ABAQUSCopyright 2006 ABAQUS, Inc.A1.27Eulerian Formulation for ConvectionqUdUdqsinceU = U (q )= c x ,c =Noting thatand,xand using the chain rule we can writeqtV()

31、rUdV =rcVdVtandq ()rUdV =rcu Vu r ).dV(assumingconstantxxVApplying these results to an arbitrarily small volume, the differential equation of the thermal energy balance becomesrcq + rcu q = - q + r.xxHeat Transfer and Thermal-Stress Analysis with ABAQUSCopyright 2006 ABAQUS, Inc.A1.28Eulerian Formul

32、ation for ConvectionUpwindingPetrov-Galerkin finite elements are used to model systems with high Peclet numbers accurately. These elements use nonsymmetricweighting functions that are different from those used as shape functions.That is, we interpolate the variational field asdq ( x ) = W N ( x )dqN

33、and the temperature field asq ( x ) = N N ( x )qwhere N N ( x ) W N ( x ).The weighting functions are of the formN ,N NW= W( x,y ) = N+y NNN.xHeat Transfer and Thermal-Stress Analysis with ABAQUSCopyright 2006 ABAQUS, Inc.A1.29Eulerian Formulation for ConvectionUpwinding formHeat Transfer and Therma

34、l-Stress Analysis with ABAQUSCopyright 2006 ABAQUS, Inc.A1.30Eulerian Formulation for ConvectionThese weighting functions are biased, or “upwinded,” with the extent of the upwinding defined by y on an element-by-element basis.The upwinding functions act to control numerical diffusion and dispersion

35、and, thus, to stabilize results.The upwinding term is partly a function of the element Peclet number, g.The full expression for the weighting functions is given in the ABAQUS Theory Manual.Heat Transfer and Thermal-Stress Analysis with ABAQUSCopyright 2006 ABAQUS, Inc.Thermal Radiation FormulationCo

36、pyright 2006 ABAQUS, Inc.A1.32Thermal Radiation FormulationSeveral assumptions are made to calculate the matrix of geometric viewfactors, radiation fluxes, and resultant temperatures for a cavity.Geometric assumptionsCalculation of Fij requires that each cavity be divided into a series of individual

37、 surfaces.In ABAQUS/Standard these surfaces are further subdivided into a series of individual facets, which is easily accomplished in the context of finite elements.For the purpose of radiation calculations, each of these facets is considered to be at a uniform temperature (isothermal) and to posse

38、ss a single value of emissivity over the facet (isoemissive).Heat Transfer and Thermal-Stress Analysis with ABAQUSCopyright 2006 ABAQUS, Inc.A1.33Thermal Radiation FormulationRadiative surface property assumptionsThe radiative characteristics of the surfaces are assumed to possess certain characteri

39、stics:The surfaces are assumed to be “gray” bodies.Thermal radiation is transported via electromagnetic waves. The thermal radiation emitted by a real surface encompasses a range of wavelengths. The amount of radiation can vary with the length of the propagating waves. This variance is best quantifi

40、ed by plotting emissivity as a function of wavelength for a typical engineering surface. In some materials variance of emissivity with wavelength can be quite strong.A gray body has an emissivity that is independent of wavelength for all levels of radiation.Heat Transfer and Thermal-Stress Analysis

41、with ABAQUSCopyright 2006 ABAQUS, Inc.A1.34Thermal Radiation Formulation“range” of thermal radiation1.5gray bodyreal surface010 11021031011Wavelength (m m)Typical emissivity vs. wavelength Thus, by assuming that surfaces are gray, an average emissivity for the spectrum of electromagnetic energy invo

42、lved in thermal radiation is used.Heat Transfer and Thermal-Stress Analysis with ABAQUSCopyright 2006 ABAQUS, Inc.EmissivityA1.35Thermal Radiation FormulationThe surfaces are assumed to be “opaque.”Thermal radiation is assumed to be a surface effect rather than a volumetric effect. At a surface inci

43、dent radiation can be reflected and absorbed but not transmitted (see figure). Kirchhoffs Law states that the ability of the surface to absorb is equal to its ability to emit (relative to a blackbody).incident radiationreflected radiationtransmitted radiation = 0Opaque surface behaviorHeat Transfer

44、and Thermal-Stress Analysis with ABAQUSCopyright 2006 ABAQUS, Inc.surfaceabsorbed radiationA1.36Thermal Radiation FormulationThe surfaces are assumed to be “diffuse.” Incident radiation, reflected radiation, and emitted radiation at the surface are assumed to have no directional tendencies; they are

45、 diffuse. This assumption is reasonable for most surfaces. However, all surfaces exhibit some departure from diffuse behavior. Highly polished surfaces, for example, may tend to invalidate this assumption.Radiative transfer occurs within cavities that are filled with a nonparticipating medium.Any ga

46、s or other medium within the confines of a cavity has no influence on the thermal radiation. The radiation exchange involves only the surfaces of the cavity. Energy exchange by radiation in an absorbing, emitting medium (such as molten glass) is beyond the scope of these notes.Heat Transfer and Ther

47、mal-Stress Analysis with ABAQUSCopyright 2006 ABAQUS, Inc.A1.37Thermal Radiation FormulationSurface emissivity is often a function of surface temperature (see figure) and other surface conditions (e.g., oxidation). Emissivity may depend on temperature or predefined field variables.1.0stainless steel

48、0.5aluminum0500Temperature (C)Typical emissivity vs. temperature1000Heat Transfer and Thermal-Stress Analysis with ABAQUSCopyright 2006 ABAQUS, Inc.EmissivityA1.38Thermal Radiation FormulationRadiation fluxThe ABAQUS/Standard implementation of cavity radiation is based on the definition of one or mo

49、re cavities in the model.Each cavity is a collection of distinct surfaces. ABAQUS/Standard automatically creates the geometric viewfactor matrix for a cavity, including treatment of symmetry conditions (which are discussed later).Each surface is composed of facets.A facet is a side of an element in

50、axisymmetric and two- dimensional cases and is a face of a solid or shell element in three-dimensional cases.Each facet is assumed to be isothermal and to have a uniform emissivity.Radiative fluxes are imposed on each of these facets.Heat Transfer and Thermal-Stress Analysis with ABAQUSCopyright 200

51、6 ABAQUS, Inc.A1.39Thermal Radiation Formulation The radiation flux per unit area into cavity facet i can be written as(T- Ti4 ),jk-qi = see14FCijikkjjwhere= dkj - (1- ei ) FkjCkjis the reflection matrix.Heat Transfer and Thermal-Stress Analysis with ABAQUSCopyright 2006 ABAQUS, Inc.A1.40Thermal Rad

52、iation FormulationIn these expressions:ei, ej are the emissivities of facets i, j;o is the Stefan-Boltzmann constant;Fijis the geometrical viewfactor matrix;Ti, Tj are the absolute temperatures of facets i, j; anddijis the Kronecker delta function.In the special case of blackbody radiation, e = 1, no reflection takes place: Ckj = dkj.Heat Transfer and Thermal-Stress Analysis with ABAQUSCopyright 2006 ABAQUS, Inc.A1.41Thermal