Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 14
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(3.26) through (3.28). This leads toW_ S ¼ ZZ½R v n dS ¼ Zr ½R vdV ¼ VSr ½ðpI þ sÞ vdVVð3:48Þ_ S can be rewritten asAfter manipulation, WZW_ S ¼ ðr ½pv þ r ½s vÞdVð3:49ÞVIf q_ V represents the rate of heat source or sink within the material volume perunit volume and q_ S the rate of heat transfer per unit area across the surface area ofthe material element, then Q_ V and Q_ S can be written asZZZQ_ V ¼ q_ V dV Q_ S ¼ q_ s n dS ¼ r q_ s dVð3:50ÞVSVApplying the Reynolds transport theorem and substituting the rate of work andheat terms by their equivalent expressions, Eq.
(3.46) becomesdEdtZ MV@ðqeÞ þ r ½qve dV¼@tVZZZZ¼ r q_ s dV þ ðr ½pv þ r ½s vÞdV þ ðf b vÞdV þ q_ V dVVVVVð3:51Þ603 Mathematical Description of Physical PhenomenaCollecting terms together, the above equation is transformed toZ V@ðqeÞ þ r ½qve þ r q_ s þ r ½pv r ½s v f b v q_ V dV ¼ 0@tð3:52ÞFor the volume integral in Eq. (3.52) to be true for any control volume, theintegrand has to be zero.
Thus,@ðqeÞ þ r ½qve ¼ r q_ s r ½pv þ r ½s v þ f b v þ q_ V@tð3:53Þwhich represents the mathematical description of energy conservation or simply theenergy equation written in terms of specific total energy. The energy equation mayalso be written in terms of specific internal energy, specific static enthalpy (orsimply specific enthalpy), specific total enthalpy, and under special conditions interms of temperature.3.6.1 Conservation of Energy in Terms of Specific InternalEnergyTo rewrite the energy equation [Eq. (3.53)] in terms of specific internal energy, thedot product of the momentum equation [Eq. (3.23)] with the velocity vector isperformed resulting in@ðqvÞ þ r fqvvg v ¼ f v@tð3:54ÞAfter some manipulations Eq.
(3.54) becomes@@vðqv vÞ qv þ r ½qðv vÞv qv ½ðv rÞv ¼ f v@t@tð3:55ÞRearranging and collecting terms the following is obtained:@@vðqv vÞ þ r ½qðv vÞv v qþ ðv rÞv ¼ f v@t@t|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}¼fEq: ð3:21Þð3:56ÞNoticing that the third term on the left side is v f and replacing f by itsequivalent expression, an equation for the flow kinetic energy is obtained as3.6 Conservation of Energy61 @11q v v þ r q v v v ¼ v rp þ v ½r s þ f b v@t22ð3:57ÞThis equation can be modified and rewritten in the following form: @11q vv þr q vv v@t22ð3:58Þ¼ r ½pv þ pr v þ r ½s v ðs : rvÞ þ f b vSubtracting Eq. (3.58) from Eq. (3.53), the energy equation with specific internalenergy as its main variable is obtained as@ðq^uÞ þ r ½qv^u ¼ r q_ s pr v þ ðs : rvÞ þ q_ V@tð3:59Þ3.6.2 Conservation of Energy in Terms of Specific EnthalpyRewriting the energy equation in terms of specific enthalpy is straightforward andfollows directly from its definition according to which the specific internal energyand specific enthalpy are related by^u ¼ ^h pqð3:60ÞSubstituting ^h p=q for ^u in Eq.
(3.59) and performing some algebraicmanipulations, the energy equation in terms of specific enthalpy evolves as@ ^Dpqh þ r qv^h ¼ r q_ s þþ ðs : rvÞ þ q_ V@tDtð3:61Þ3.6.3 Conservation of Energy in Terms of Specific TotalEnthalpyThe energy equation in terms of specific total enthalpy can be derived by expressinge in terms of ^h0 to get1p 1pe ¼ ^u þ v v ¼ ^h þ v v ¼ ^h0 2q 2qð3:62Þ623 Mathematical Description of Physical PhenomenaThen by substituting ^h p=q for e in Eq. (3.53) and performing some algebraicmanipulations, the energy equation in terms of specific total enthalpy is obtained as@ ^@pqh0 þ r qv^h0 ¼ r q_ s þþ r ½s v þ f b v þ q_ V@t@tð3:63ÞAll forms of the energy equation presented so far are general and applicable toNewtonian and non-Newtonian fluids.
The only limitation is that they are applicable to a fixed control volume.3.6.4 Conservation of Energy in Terms of TemperatureTo be able to write the energy equation with temperature as the main variable someconstraints have to be imposed. Assuming ^h to be a function of p and T, the fluid isexpected to be Newtonian. Therefore the derivations to follow are applicable toNewtonian fluids only.
If ^h ¼ ^hðp; T Þ, then d ^h can be written as!!^h^@@hd ^h ¼dT þdpð3:64Þ@T@ppTUsing the following ordinary equilibrium thermodynamics relation:! ^@ ^h@V^¼V T@p@T pð3:65ÞT^ is the specific volume, the expression for d ^h can be modified towhere V" #^^ T @Vd ^h ¼ cp dT þ Vdp@T pð3:66ÞThe left side of the specific enthalpy [Eq. (3.61)], with d ^h given by Eq. (3.66),can be rewritten in terms of T as" #^@ ^D^hDT@VDP^^qh þ r qvh ¼ q¼ qcpþq V T@tDtDt@T p Dt" #DT1@ ð1=qÞDP¼ qcpþq TDtq@Tp Dt" #DT@ ðLnqÞDPþ 1þ¼ qcpDt@ ðLnT Þ p Dtð3:67Þ3.6 Conservation of Energy63Substituting Eq.
(3.67) into Eq. (3.61) gives the energy equation with T as itsmain variable asqcpDT@ ðLnqÞ Dp¼ r q_ s þ ðs : rvÞ þ q_ VDt@ ðLnT Þ p Dtð3:68ÞThe above equation is equivalently given by@@ ðLnqÞ DpðqT Þ þ r ½qvT ¼ r q_ s þ ðs : rvÞ þ q_ Vcp@t@ ðLnT Þ p Dtð3:69ÞThe heat flux q_ S appearing in all forms of the energy equation represents heattransfer by diffusion, which is a phenomenon occurring at the molecular level and isgoverned by Fourier’s law according toq_ s ¼ ½krT ð3:70Þwhere k is the thermal conductivity of the substance. The above equation states thatheat flows in the direction of temperature gradient and assumes that the material hasno preferred direction for heat transfer with the same thermal conductivity in alldirections, i.e., the medium is isotropic.
However some solids are anisotropic forwhich Eq. (3.70) is replaced byq_ s ¼ ½j rT ð3:71Þwhere j is a second order symmetric tensor called the thermal conductivity tensor.Consequently, the heat flux in anisotropic medium is not in the direction of thetemperature gradient. In the derivations to follow the medium is assumed to beisotropic and Eq. (3.70) is applicable. Replacing q_ s using Fourier’s law, the energyequation, Eq. (3.69), becomescp@@ ðLnqÞ DpðqT Þ þ r ½qvT ¼ r ½krT þ ðs : rvÞ þ q_ V ð3:72Þ@t@ ðLnT Þ p DtThe expression for ðs : rvÞ in terms of the flow variables in a three-dimensionalCartesian coordinate system is given by10 2 2 2@u@v@w2þ2þ2CB2@u @v @w@y@zCB @xþ þðs : rvÞ ¼ kþ lB 2 2 2 C@@x @y @z@u @v@u @w@v @w Aþþþþþþ@y @x@z @x@z @yð3:73Þ643 Mathematical Description of Physical PhenomenaDefining W and U as@u @v @w 2þ þW¼ð3:74Þ@x @y @z" # @u 2@v 2@w 2@u @v 2@u @w 2@v @w 2þþþU¼2þþþþþ@x@y@z@y @x@z @x@z @yð3:75ÞThe energy equation in terms of temperature reduces to@@ ðLn qÞ DpðqT Þ þ r ½qvT ¼ r ½krT þ kW þ lU þ q_ Vcp@t@ ðLn T Þ p Dtð3:76ÞFor later reference the energy equation is expanded to@qcp T þ r qcp vT ¼ r ½krT @tDcp@ ðLn qÞ Dpþ kW þ lU þ q_ Vþ qT@ ðLn T Þ p DtDt|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}QTð3:77Þand rewritten as@qcp T þ r qcp vT ¼ r ½krT þ QT@tð3:78ÞThe energy equation is rarely solved in its full form and depending on thephysical situation several simplified versions can be developed.
The dissipation termΦ has negligible values except for large velocity gradients at supersonic speeds.Moreover, for incompressible fluids the continuity equation implies that W ¼ 0 andbecause the density is constant it follows that ð@ ðLnqÞ=@ ðLnT ÞÞ ¼ 0. Therefore theenergy equation [Eq. (3.77)] for incompressible fluid flow is simplified toDcp@qcp T þ r qcp vT ¼ r ½krT þ q_ V þ qT@tDtffl}|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflð3:79ÞQTEquation (3.79) is also applicable for a fluid flowing in a constant pressuresystem. For the case of a solid, the density is constant, the velocity is zero, and ifchanges in temperature are not large then the thermal conductivity may be considered constant, in which case the energy equation becomes3.6 Conservation of Energy65qcp@T¼ kr2 T þ q_ V@tð3:80ÞFor ideal gases ð@ ðLnqÞ=@ ðLnT ÞÞ ¼ 1 and the energy equation for compressible flow of ideal gases reduces to@DpðqT Þ þ r ½qvT ¼ r ½krT þþ kW þ lU þ q_ Vcp@tDtð3:81ÞIf viscosity is neglected (i.e.
the flow is inviscid), Eq. (3.81) is further simplifiedtocp@DpðqT Þ þ r ½qvT ¼ r ½krT þþ q_ V@tDtð3:82Þ3.7 General Conservation EquationFrom the above, the governing equations describing the conservation of mass,momentum, and energy are written in terms of specific quantities or intensiveproperties, i.e., quantities expressed on a per unit mass basis. The momentumequation, for example, expressed the principle of conservation of linear momentumin terms of the momentum per unit mass, i.e., velocity.
The same type of conservation equation may be applied to any intensive property /, e.g., concentration ofsalt in a solution or the mass fraction of a chemical species. The variation of / inthe control volume over time can be expressed as a balance equation of the formTerm ITerm IITerm IIIFor the fixed control volume shown in Fig. 3.9, the change of / over time withinthe material volume can be written using the Reynolds transport theorem as01ZZ d@@Aðq/Þ þ r ðqv/Þ dVðq/ÞdV ¼Term I ¼dt@tMVð3:83ÞVwhere q is the fluid density and V the volume of the control volume of surface areaS. The term qv/ represents the transport of / by the flow field and is denoted by theconvective flux, i.e.,663 Mathematical Description of Physical PhenomenaFig. 3.9 Arbitrary fixedcontrol volumedSVVJ/convection ¼ qv/ð3:84ÞThe second term represents variation of / due to physical phenomena occurringacross the control volume surface.
For the physical phenomena of interest in thisbook, the mechanism causing the influx/out flux of / is due to diffusion, which isproduced by molecular collision and is designated by J/diffusion . Denoting the diffusion coefficient of / by C/ , the diffusion flux may be written asJ/diffusion ¼ C/ r/ð3:85Þand Term II becomesZTerm II ¼ SJ/diffusion n dS ¼ Zr J/diffusion dV ¼VZr C/ r/ dVVð3:86Þwhere n is the outward unit vector normal to the surface and the negative sign is dueto the adopted sign convention (i.e., inward flux is positive). Term III can be writtenasZð3:87ÞTerm III ¼ Q/ dVVwhere Q/ is the generation/destruction of / within the control volume per unitvolume, which is also called the source term.
Thus the conservation equation can beexpressed asZ VZZ@ðq/Þ þ r ðq/vÞ dV ¼ r C/ r/ dV þ Q/ dV@tVVð3:88Þ3.7 General Conservation Equation67which can be rearranged intoZ V@ðq/Þ þ r ðqv/Þ r C/ r/ Q/ dV ¼ 0@tð3:89ÞFor the integral to be zero for any control volume, the integrand has to be zerogiving the conservation equation in differential form as@ðq/Þ þ r ðqv/Þ r C/ r/ Q/ ¼ 0@tð3:90ÞFor later reference the above equation may be rewritten as@ðq/Þ þ r J/ Q/ ¼ 0@tð3:91Þwhere the total flux J/ is the sum of the convective and diffusive fluxes given byJ/ ¼ J/;C þ J/;D ¼ qv/ C/ r/ð3:92ÞThe final form of the general conservation equation, Eq. (3.90), for the transportof a property / is expressed as@ðq/Þ þ r ðqv/Þ ¼ r C/ r/ þ Q/|{z}|fflfflfflfflfflffl{zfflfflfflfflfflffl}@t|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}|fflfflffl{zfflfflffl}source termconvection termunsteady termð3:93Þdiffusion termBy comparing Eq.
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