Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 40
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4.1a can be classified into oneof three types: two-dimensional (2D), axisymmetric, and three-dimensional (3D). The flow is2D if the body is invariant in cross-sectional shape along a long horizontal axis (e.g., a longhorizontal circular cylinder). An axisymmetric flow takes place near a body (Fig. 4.3b) whoseshape can be generated by revolving a body contour about a vertical line; for example, asphere is generated by rotating a semicircle. If the body meets neither the 2D or axisymmetric requirements, its flow is classified as 3D; this class includes the flow around 2D andaxisymmetric bodies whose axes have been tilted.In addition to this geometric classification, the flow can be classified as fully laminar (i.e.,laminar over the entire body), fully turbulent (turbulent over the entire body), or laminar and4.6CHAPTERFOURturbulent (i.e., laminar over one portion and turbulent over the other).
Laminar flow is confined to a boundary layer, but turbulent flow may be either attached or detached, as discussedin the section on turbulent Nusselt number.Thermal Boundary Conditions. The simplest thermal boundary conditions are Tw and Too,both specified constants. If Too varies, it is assumed to be a function only of the elevation z;also, dTJdz is always positive, since, with 13positive, a situation where dTJdz is negative isalways unstable and cannot be maintained in an extensive fluid. Either T~ or q" may be specified on the body surface.
The difference between the temperature at a given point on thebody and Tooat the same elevation is denoted AT; AT is the area-weighted average value of ATover the surface, and AT0 is an arbitrary reference temperature difference, usually set equal toAT. If AT is positive over the lower part of the body and negative over the upper part, the surface flows over each part are in opposite directions, and these two flows meet and detach fromthe body near the line on the surface where AT = 0.Direction of Heat Transfer (Cooling versus Heating).The relationship of Nu to Ra and Prfor a heated body is precisely the same as when the body is inverted and cooled, except thatTw - Toois replaced by Too- Tw.
This principle implies, for example, that the relationship for theheated triangular cylinder in Fig. 4.4a is identical to that for the inverted cooled triangularcylinder in Fig. 4.4b. In this chapter it is normally assumed that the surface is heated, and theresults for a cooled surface can be inferred from results for a corresponding heated surface byusing the above principle.Xw <TooI g l\'Tw > Too(a)(b)FIGURE 4.4 The Nusselt number for a heated body (a) isthe same if the same body is cooled and inverted (b).Definition of Surface Angle ~.
Let b be the unit vector in the direction of the buoyancyforce as follows. For Tw> Too,b is directed vertically upward as shown in Fig. 4.5a and c, and forTw < Too,b is directed vertically downward (Fig. 4.5b and d). The unit outward normal fromthe surface is fi, and the angle between fi and b is defined as the surface angle ~. For 0 ° < ~ <90 °, as in Fig. 4.5c and d, the buoyancy force has components along and away from the surface. For 90 ° < ~ _< 180 °, Fig.
4.5a and b, the buoyancy force is directed along and into thesurface. Clearly, ~ varies locally over a curved surface.Heat Transfer Correlation MethodOverview. The heat transfer correlations given in subsequent sections draw on a certainparadigm, which is outlined in this section. Strictly, the reader need not be aware of thisNATURAL CONVECTION90 ° < ~ < 180 °r-~4.70 o < ~ < 90 °2,,,,__~_Ab-/Tw > Too(a)Tw< Too(b)Tw> Too(c)Tw< Too(d)FIGURE 4.5 Definitionof surface angle ~ for heated and cooled surfaces. For 90° < ~ < 180°the flow remains attached, while it may detach if 0° < ~ < 90°.paradigm to use the equations.
On the other hand, knowing something of it will assist thereader in interpreting the results and in extending the results to shapes not mentioned.Briefly, the idea is to obtain an equation for the laminar Nusselt number Nue that would bevalid for the hypothetical situation where the flow is always fully laminar over the entire surface, to obtain an equation for the turbulent Nusselt number, Nut, that would be valid for thehypothetical situation where the flow is always fully turbulent at all surface locations, and tocombine the equations for Nue and Nu, to obtain an expression for Nu that is valid for allcases: wholly laminar, wholly turbulent, or partly laminar and partly turbulent.The equation for the laminar Nusselt number Nue is obtained in a two-step procedure.
Inthe first step, not only is the flow idealized as everywhere laminar, but the boundary layer istreated as thin. There results from this idealization the equation for the laminar thin-layerNusselt number Nu T. As already explained, natural convection boundary layers are generallynot thin, so the second step is to correct Nur to account for thick boundary layers. This correction uses the method of Langmuir [175]. The corrected Nusselt number is the laminar Nusselt number Nue.Conduction Nusselt Number NucoNo. It is the aim of the method to give expressions for Nu(defined by Eq.
4.8, where L is any convenient length scale) that cover the entire range in Rafrom zero to infinity, so one needs to consider the Nusselt number in the limit where Ra --> 0.For Ra --->0, there is no fluid motion, and (with the possible exception of radiation) heat transfer is by conduction only. The Nusselt number that applies in this limit is therefore called theconduction Nusselt number NUCOND.NUCONDis zero for 2D flows and nonzero for 3D flows Fora sphere of radius D, for example, NUCOND= 2L/D. Except for simple body shapesmfor whichvalues of NUCONDmay be available in the literature (see, for example, Ref.
292)ma detailedconduction analysis may be required to determine NUCOND. On the other hand, the work ofChow and Yovanovich [296] and, later, of Yovanovich [292] has shown that the quantity(V~/L) NUCoNDis highly insensitive to the body shape, ranging from 3.391 to 3.609 over a,_y_verywide range of shapes, including cubes, lenses, ellipsoids, and toroids; for an ellipsoid, (X/A/L)NUCONDranges from 3.55 to 3.19 as the ratio of major to minor axis ranges from 1 to 104. In theabsence of a conduction analysis for a particular body shape, the Yovanovich method providesa quick method for estimating NUCOND.Laminar Thin-Layer Nusselt Number Nu T.
Raithby and Hollands [223] derived the following equation (similar to Eq. 4.10) as a close approximation to the full solution of the thin-layerequations:4.8CHAPTER FOURNut=mGCe R a TM(4.11)g~ATog3Ra = ~where(4.12)v~and L is any convenient reference length (the dependence on this length cancels out of theequations). G is a ratio of integrals (both of which can often be evaluated in closed_ form) thatdepend on the shape of the body and on thermal boundary conditions, and Ce is an approximately universal function of Prandtl number Pr given by Churchill and Usagi [54]0.671DC e - [1 + (0.492/Pr)9/16] 4/9(4.13)Ce is tabulated in Table 4.1.
For the special case of a heated body with constantG ratio is given for both 2D and axisymmetric flows bya =sTw andZ TM IfO~r 4i/3 sin 1/3 ~ d x ]3/4Too,the(4.14)o ri d xwhere index i = 0 if the flow is 2D and i = 1 if the flow is axisymmetric (see the section on classification of flow). The body coordinates r and x in Eq. (4.14) are defined in Fig. 4.3, x beingthe distance measured along the surface of the body from the lowest point on the body, and Sis the value of x at the highest point on the body.
The angle ~ in the integral is the surface angleexplained previously and in Fig. 4.5. Raithby and Hollands [225-227] and (for the special caseof Pr ~ oo) Stewart [264] have given more general expressions covering the cases of 3D flowsand nonuniform T~ and Too.Hassani and Hollands [126, 127] gave the following approximateequation for G for 3D flows: G = (LIL~2/A2) TM, where Lyis the vertical distance from the lowest point on the body to the highest, and @ is the vertically averaged perimeter of the bodydefined by1 fo'y~(z) dz(4.15)where ~(z) is the local perimeter of the body at elevation z above the lowest point on thebody.
[A horizontal plane at height z will intersect the surface of the body at a flat curve; ~(z)is the circumference of this curve.] For 2D flow, this equation for G reduces to G =(4LIL/p2)1/4, where P/is the perimeter of the cylinder's cross section.To further characterize the laminar thin-layer heat transfer, it is helpful to introduce theidea of the local and average conduction layer thicknesses, Ax and A, respectively: Ax is definedin Fig. 4.3d as the local distance from the wall at which the linear T profile intersects T~, itbeing understood that the T profile is a hypothetical linear temperature profile that has thesame value at the surface and the same slope at the surface as the true temperature profile.Stated differently, Ax is the local thickness of stationary fluid that would offer the same therTABLE 4.1Prandtl Number Dependence of Various CoefficientsMercuryGasesWaterOilsPr0.0220.711.02.04.06.0501002000Ce0.2870.055*0.14"0.3970.5150.1030.1400.6240.5340.1060.1400.6410.5680.1120.1400.6710.5950.1130.1400.6940.6080.1130.1410.7040.6500.0970.1430.7380.6560.0910.1450.7440.6680.0640.1490.754CtvC~vHe* No experimental validation available.NATURAL CONVECTION4.9mal resistance as the thin boundary layer.
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