Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 39
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4.3 has assumed that changes in density are small compared to density itself. For liquids Eq. 4.2 is approximated as follows:(poo(z) - p)g-- pg{exp[-13o(T= - 7)1 - 1} = pgl3o(T- Too)= pogl3o(r- T=)(4.4)where both 130and P0 are evaluated at Tr. Thus, except for the reference temperature at which13is evaluated, the final expression for the buoyancy force is the same for liquids and gases. Anextra degree of approximation was, however, required in Eq. 4.4 as compared to Eq.
4.3, andthe temperature range over which Eq. 4.4 is valid will be considerably less than that for Eq.4.3. For water near 4°C (40°F), the range of validity is so small that, for practical problems,Eq. 4.4 must be replaced by a nonlinear relation between p and T [110].Thus the appropriate equations of motion for natural convection [107] are produced byintroducing Eq. 4.1 for P into the full equations of motion and then introducing Eq. 4.3 or 4.4for the term (poo- p)g. These equations are then further simplified by taking each of the properties p, la, cp, and k as constant at their respective values, P0, go, Cpo,and k0, evaluated at TI.The resulting equations are nondimensionalized by substituting x = x ' L , y - y ' L , z - z'L,T = Too+ OAT, u = you* v = VoV* w = VoW* Pd = 1/2pov2pff, and t = (L/vo)t*, where L is a characteristic dimension of the body, AT = Tw - Too, and v0 = V'g131 ATL/(1 + Vo/~), with 131= [3ooforgases and [30 for liquids.
With the assumption that Too(z) and Tw are constant, the resultingequations for mass, z momentum, and energy become3u*3u*3w*3x* + 3y* + 3z* = 0IIIIII(4.5)IVDw*3P~ , / P r (1 + Pr) V,2w * + (1 + Pr)ODt* - - - z- ORa~ + ¥IDOII/I+Pr- VR--R-~rIIIV,20_ fSTgLcpoAT + ~lTEcIV(4.6)VDP~EcF(3w*/21Dt* + ~ L~,~z * ] + "'"(4.7)where, with ~ = ko/poCpo and v0 = go/P0, Ra (the Rayleigh number) = g~l ATL3/voo~o, Pr (thePrandtl number) = Vo/~, Re (the Reynolds number) = Lvo/Vo, and Ec (the Eckert number) =v2/(CpoAT). See the end of the nomenclature list for the definition of the operators D/Dt* andV .2. The nondimensional x momentum equation is identical to Eq. 4.6 except that u* replacesw*, 3P~/3x* replaces 3P~/3z*, and the term in 0 does not occur; an analogous descriptionapplies to the nondimensional y momentum equation. The subscripts 0 and 1 on the properties will, for simplicity, be dropped in the rest of this chapter.Terms III-V in Eq.
4.7 often have only minor influence and can usually be dropped. TermIII reflects the fact that the pressure of the ambient fluid decreases with elevation so that, ifthe fluid moves upward, it does work on expanding at the expense of internal energy and thisproduces a cooling effect; since fSTg/Cp (the adiabatic lapse rate) is usually small (0.01°C/m or0.006°F/ft for gases) compared to AT/L, this term can be safely dropped. Term IV representsa similar effect, this time due to dynamic pressure changes, and can likewise be neglected.Term V represents viscous heating and has a negligible effect.With terms III-V in Eq.
4.7 deleted, Eqs. 4.5-4.7, together with the x and y m o m e n t u mequations, constitute the "simplified equations of motion" appropriate to natural convectionproblems. For constant Tw and Too,the boundary conditions on these equations are 0 = 1 andu* = v* = w* = 0 on the body and 0 = 0 far from the body. Steady-state laminar solutions tothese equations are those that are obtained after setting the time partials (i.e., terms containing partial derivatives with respect to t*) in the equations equal to zero. Steady-state turbulent4.4CHAPTER FOURsolutions are those that are obtained after setting time partials of mean quantities equal tozero but that incorporate the effect of the unsteady (turbulent) motion.Important Dimensionless Groups. The average Nusselt number for steady-state heat transfer from the body shown in Fig.
4.1a depends on the dimensionless groups that arise in thenondimensionalized equations of motion and their boundary conditions [78]. With Tw and To.constant, the only dimensionless groups that appear in the boundary conditions are thoseassociated with the body shape. Provided the simplified equations (Eqs. 4.5--4.7) are valid, theonly other dimensionless groups are the Rayleigh and Prandtl numbers. Thus, for a givenbody shape,qLN u - AATk- f ( R a , Pr)(4.8)where A is the surface area transferring the total heat rate q.The role of the Prandtl number and its relative importance can be surmised by inspectingEqs.
4.6 and 4.7. For Pr >> 1, the buoyancy force that drives the motion (term IV in Eq. 4.6)can be seen to be balanced almost exclusively by the viscous force (term III). With thisapproximation, and because, for Pr >> 1, Pr disappears from Eq. 4.7, Ra is the only remainingdimensionless group in the simplified equations. For Pr << 1 the viscous force is unimportant,and the buoyancy force must be balanced by the dynamic pressure and inertial terms; in thiscase the product Ra Pr remains as the only dimensionless group.
It is therefore concluded thatNu = f(Ra)for Pr >> 1(4.9a)Nu = f(Ra Pr)for Pr << 1(4.9b)and the Prandtl number governs the relative importance of viscous and inertial forces in theproblem.Validation of the Simplified Equations. Steady-state laminar solutions to the simplifiedequations for flow around a long horizontal circular cylinder immersed in air with both T.. andTw constant have been obtained by Kuehn and Goldstein [165] and Fujii et al. [101], usingnumerical methods.
Their results, in terms of the Nusselt number, are plotted in Fig. 4.2,IiII0~/;V/Nu =_-~ ~0.39~RGI/4Savil le ~ ChurchiPl [240]_-I~. . . . . . . . .I0 -2i0 0I0 2• Kuehn ~ Goldstein [1651• Fujii, Fujii & Ma~sunaga [101]Approximate range ofexperimental dataI.....I....J104106108Ro= 185xI0 4Pr = 071Ro(a)(b)FIGURE 4.2 Measurements and predictions for a circular cylinder (a). (b) shows isotherms fornatural convection around a horizontal cylinder. (Courtesy o f E R G Eckert.)NATURAL CONVECTION4.5xTwzz +dziT(a)(b)(c)Ax(d)=YFIGURE 4.3 Surfacecoordinate systems for two-dimensional (a), axisymmetric (b), and three-dimensional(c) natural convection flows; (d) is a schematicof the temperature distribution near the boundary.together with the band in which the experimental data for this problem lie.
The agreement isgood; the differences for Ra > 106 are believed to be due to turbulence, which is not includedin their steady-state laminar solution. This comparison lends support to the approximationsinherent in the simplified equations of motion.Thin-Layer Approximation. Laminar analyses often make the further approximation thatthe boundary layer is so thin that when the simplified equations of motion are rewritten interms of local surface coordinates, i.e., in terms of the x and y of Fig. 4.3a, several terms normally associated with curvature effects can be dropped.
The Nusselt number equation, basedon solutions to such laminar "thin-layer" equation sets, always takes the formNu = c Ra TM(4.10)where c is independent of Ra but does depend on geometry and Prandtl number. For example, c - 0.393 for the horizontal circular cylinder in air (Pr = 0.71) [240]. Plotted in Fig. 4.2, Eq.4.10 with this value of c falls consistently below the experimental data; also (since no singlepower law can fit the data) its functional form is incorrect. At high Ra it approaches Kuehnand Goldstein's laminar solution to the complete simplified equations of motion, but, asalready discussed, the presence of turbulence has rendered laminar solution invalid in thisrange.The thin-layer approximation fails because natural convective boundary layers are notthin. From the interferometric fringes in Fig.
4.2b (which are essentially isotherms), the thermal boundary layer around a circular cylinder is seen to be nearly 30 percent of the cylinderdiameter. For such thick boundary layers, curvature effects are important. Despite this failure, thin-layer solutions provide an important foundation for the development of correlationequations, as explained in the section on heat transfer correlation method.Problem ClassificationClassification of Flow. The flow near bodies like that in Fig.
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