Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 9
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1.48. Substituting this into Eq. 1.54 we have the following generalrelation:DTDPq,,,p C p - - ~ t - = V , k V T + T [ 3 - ~ + laO +(1.56)For an ideal gas, ~ = l/T, and thenDTDPpCp --~- = V . k V T + ~+ gO + q"(1.57)Note that Cp need not be constant.We could have obtained Eq. 1.57 directly from Eq. 1.54 by noting that for an ideal gas,di = Cp d T where Cp is constant and thusDiDTDt - Cp DtFor an incompressible fluid with specific heat c =cp = cv we go back to Eq. 1.52 (du = c d T ) toobtainDTpc - - ~ = V .
k V T + gO + q"(1.58)Equations 1.52, 1.54, and 1.56 can be easily written in terms of energy (heat) and momentumfluxes using relations for fluxes given in Tables 1.4, 1.6, and 1.8. The energy equation given byEq. 1.58 (with q'"= 0 for simplicity) is given in Table 1.9 in different coordinate systems.For solids, the density may usually be considered constant and we may set V = 0, and Eq.1.58 reduces to/)Tpc - - ~ = V .
k V T + q"(1.59)which is the starting point for most problems in heat conduction.The Energy E q u a t i o n f o r a Mixture. The energy equations in the previous section are applicable for pure fluids. A thermal energy equation valid for a mixture of chemical species isrequired for situations involving simultaneous heat and mass transfer. For a pure fluid, conduction is the only diffusive mechanism of heat flow; hence Fourier's law is used, resulting inthe term V. kVT. More generally this term may be written -Vq', where q" is the diffusive heatflux, i.e., the heat flux relative to the mass average velocity. More specifically, for a mixture, q"is now made from three contributions: (1) ordinary conduction, described by Fourier's law,-kVT, where k is the mixture thermal conductivity; (2) the contribution due to interdiffusionof species, given by ~,i jiig and (3) diffusional conduction (also called the diffusion-thermoeffect or Dufour effect [6, 12]).
The third contribution is of the second order and is usuallynegligible:q" = - k V T + ~i]iii(1.60)BASIC CONCEPTS OF HEAT TRANSFERTABLE 1.91.21The Energy Equation* (for Newtonian Fluids of Constant p and k)Rectangular coordinates (x, y, z):pc~ -ff+~-gx +~Ty +WTz :k[-~x~+ ~Dy+2 --~Z2 ]+2~ ~(0u+~,~/+~---~--z/j+~ ~+ ~x/+ ~+ Ox/+ ~+-~~-y/lCylindrical coordinates (r, 0, z):pep "-~'Jt" Vr'-~r -[---r-~"lr"Vzf['DVr \2~Z = k31-2~.1,[/~) "t-~r r -~r + - ~ - - ~ + Dz2 ])12 IDVzi21 [[Dvo l D v z ) 2 (DVz DVr~2 [1DVrD (-~)] 2}~ D0 "~"vr -1.\ DZ } J -I-~].l/~ 31-7 --~ "t-~ Dr + Dz ] + -~- + r-ffr-r[I(DVoSpherical coordinates (r, 0, 00):(aTpCp ~aT+ Vr ~1 O2T]+ r2sin20 D~2]vo aTv, ~~)[1 O ( a T )1O(aT)"Jr-" kr2+sin 0r ~rsin0 5~--~r-~rr2sin0 D0~"Jr"~f[DVr \2+g(_~DV 0~r)2 ( 1DV~r sin 0 Dc~Vrrv0 cot 0) 2}r{[ D (~e_) 1 DVr]2 [ 1 DVrD (~_)] 2 [Sinr0 _ ~ ( v , )r~r+--r --~-1 + rsin 0 D00 + r ~+~1 Dye]2)+ r sin 0 DO* The terms contained in braces [ }are associatedwithviscousdissipationand may usuallybe neglected except in systemswith large velocity gradients.Here ji is a diffusive mass flux of species i, with units of mass/(area x time), as mentionedbefore.
Substituting Eq. 1.60 in, for example, Eq. 1.54, we obtain the energy equation for amixture:DiDPP Dt - Dt + V • k V T -V •(~i)jiii + gO + q"(1.61)For a nonreacting mixture the term V • (~'.i jiii) is often of minor importance. But whenendothermic or exothermic reactions occur, this term can play a dominant role. For reactingmixtures the species enthalpiesii = i °i +f;o Cp'i d Tmust be written with a consistent set of heats of formation i,°. at T O[13].T h e C o n s e r v a t i o n E q u a t i o n for S p e c i e sFor a stationary control volume, the conservation equation for species is~)Ci=-V. (CiV)V .
ji+ri"i)trate of storagenet rate ofnet rate of diffusion productionrateof species i per convectionof speciesof species i perof species i perunit volumei per unit volumeunit volumeunit volume(1.62)1.22CHAPTERONEUsing the mass conservation equation, the above equation can be rearranged to obtainDm----L= - V . ji + r ~"P Dt(1.63)where mi is mass fraction of species i, i.e., where m i = C i / p , where p is the density of the mixture, ~ i Ci = P, and Ci is a partial density of species i (i.e., a mass concentration of species i).The conservation equation for species can also be written in terms of mole concentrationand mole fractions, as shown in Refs. 10, 12, and 13.
The mole concentration of species i isci = C~/Mo where M~ is the molecular weight of the species. The mole fraction of species iis defined as X i -" Ci/C, where c = ~i Ci" As is obvious, ~-~-im~ = 1 and ~ i Xi -~ 1.Equations 1.62 and 1.63 written in different coordinate systems are given in Ref. 10.Use of Conservation Equations to Set Up ProblemsFor a problem involving fluid flow and simultaneous heat and mass transfer, equations of continuity, momentum, energy, and chemical species (Eqs.
1.41, 1.44, 1.54, and 1.63) are aformidable set of partial differential equations. There are four i n d e p e n d e n t variables: threespace coordinates (say, x, y, z) and a time coordinate t.If we consider a pure fluid, there are five equations: the continuity equation, three momentum equations, and the energy equation. The five accompanying d e p e n d e n t variables are pressure, three components of velocity, and temperature. Also, a thermodynamic equation ofstate serves to relate density to the pressure, temperature, and composition. (Notice that fornatural convection flows the momentum and energy equations are coupled.)For a mixture of n chemical species, there are n species conservation equations, but one isredundant, as the sum of mass fractions is equal to unity.A complete mathematical statement of a problem requires specification of boundary andinitial conditions.
Boundary conditions are based on a physical statement or principle (forexample: for viscous flow the component of velocity parallel to a stationary surface is zero atthe wall; for an insulated wall the derivative of temperature normal to the wall is zero; etc.).A general solution, even by numerical methods, of the full equations in the four independent variables is difficult to obtain. Fortunately, however, many problems of engineeringinterest are adequately described by simplified forms of the full conservation equations, andthese forms can often be solved easily. The governing equations for simplified problems areobtained by deleting superfluous terms in the full conservation equations. This appliesdirectly to laminar flows only.
In the case of turbulent flows, some caution must be exercised.For example, on an average basis a flow may be two-dimensional and steady, but if it is unstable and as a result turbulent, fluctuations in the three components of velocity may be occurring with respect to time and the three spatial coordinates. Then the remarks about droppingterms apply only to the time-averaged equations [7, 12].When simplifying the conservation equation given in a full form, we have to rely on physical intuition or on experimental evidence to judge which terms are negligibly small. Typicalresulting classes of simplified problems are:Constant transport propertiesConstant densityTimewise steady flow (or quasi-steady flow)Two-dimensional flowOne-dimensional flowFully developed flow (no dependence on the streamwise coordinate)Stagnant fluid or rigid bodyBASIC CONCEPTS OF HEAT TRANSFER1.23Terms may also be shown to be negligibly small by order-of-magnitude estimates [7, 12].Some classes of flow that result are:Creeping flows: inertia terms are negligible.Forced flows: gravity forces are negligible.Natural convection: gravity forces predominate.Low-speed gas flows: viscous dissipation and compressibility terms are negligible.Boundary-layer flows: streamwise diffusion terms are negligible.DIMENSIONLESS GROUPS AND SIMILARITY IN HEAT TRANSFERModern engineering practice in the field of heat transfer is based on a combination of theoretical analysis and experimental data.
Often the engineer is faced with the necessity ofobtaining practical results in situations where, for various reasons, physical phenomena cannot be described mathematically or the differential equations describing the problem are toodifficult to solve. An experimental program must be considered in such cases.
However, incarrying the experimental program the engineer should know how to relate the experimentaldata (i.e., data obtained on the model under consideration) to the actual, usually larger, system (prototype). A determination of the relevant dimensionless parameters (groups) provides a powerful tool for that purpose.The generation of such dimensionless groups in heat transfer (known generally as dimensional analysis) is basically done (1) by using differential equations and their boundary conditions (this method is sometimes called a differential similarity) and (2) by applying thedimensional analysis in the form of the Buckingham pi theorem.The first method (differential similarity) is used when the governing equations and theirboundary conditions describing the problem are known.
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