Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 59
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Streams that flow through ducts are primary examples of internal flows.Heat exchangers are conglomerates of internal flows. This class of fluid flow and395BOOKCOMP, Inc. — John Wiley & Sons / Page 395 / 2nd Proofs / Heat Transfer Handbook / Bejan396123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: INTERNAL FLOWSconvection phenomena distinguishes itself from the class of an external flow, whichis treated in Chapters 6 (forced-convection external) and 7 (natural convection).
In anexternal flow configuration, a solid object is surrounded by the flow.There are two basic questions for the engineer who contemplates using an internalflow configuration. One is the heat transfer rate, or the thermal resistance between thestream and the confining walls.
The other is the friction between the stream and thewalls. The fluid friction part of the problem is the same as calculation of the pressuredrop experienced by the stream over a finite length in the flow direction. The fluidfriction question is the more basic, because friction is present as soon as there is flow,that is, even in the absence of heat transfer. This is why we begin this chapter with thecalculation of velocity and pressure drop in duct flow. The heat transfer question issupplementary, as the duct flow will convect energy if a temperature difference exitsbetween its inlet and the wall.To calculate the heat transfer rate and the temperature distribution through theflow, one must know the flow, or the velocity distribution.
When the variation oftemperature over the flow field is sufficiently weak so that the fluid density andviscosity are adequately represented by two constants, calculation of the velocity fieldand pressure drop is independent of that of the temperature field. This is the case inall the configurations and results reviewed in this chapter. When this approximationis valid, the velocity field is “not coupled” to the temperature field, although, asalready noted, correct derivation of the temperature field requires the velocity fieldas a preliminary result, that is, as an input.The following presentation is based on the method developed in Bejan (1995).Alternative reviews of internal flow convection are available in Shah and London(1978) and Shah and Bhatti (1978) and are recommended.5.25.2.1LAMINAR FLOW AND PRESSURE DROPFlow Entrance RegionConsider the laminar flow through a two-dimensional duct formed between twoparallel plates, as shown in Fig.
5.1. The spacing between the plates is D. The flowvelocity in the inlet cross section (x = 0) is uniform (U ). Mass conservation meansthat U is also the mean velocity at any position x downstream,1U=u dA(5.1)A Awhere u is the longitudinal velocity component and A is the duct cross-sectional areain general. Boundary layers grow along the walls until they meet at the distancex ≈ X downstream from the entrance.
The length X is called entrance length orflow (hydrodynamic) entrance length, to be distinguished from the thermal entrancelength discussed in Section 5.4.1. In the entrance length region the boundary layerscoexist with a core in which the velocity is uniform (Uc ). Mass conservation and thefact that the fluid slows down in the boundary layers requires that Uc > U .BOOKCOMP, Inc. — John Wiley & Sons / Page 396 / 2nd Proofs / Heat Transfer Handbook / Bejan[396], (2)Lines: 85 to 111———-1.92998pt PgVar———Normal PagePgEnds: TEX[396], (2)LAMINAR FLOW AND PRESSURE DROP123456789101112131415161718192021222324252627282930313233343536373839404142434445397Figure 5.1 Developing flow in the entrance region of the duct formed between two parallelplates.
(From Bejan, 1995.)[397], (3)Lines: 111 to 153The length X divides the duct flow into an entrance region (0 < x ≤ X) and a fullydeveloped flow region (x ≥ X). The flow friction and heat transfer characteristics———of the entrance region are similar to those of boundary layer flows. The features-3.10686pt PgVarof the fully developed region require special analysis, as shown in Section 5.2.2.———The entrance length X is indicated approximately in Fig. 5.1. This is not a preciseNormal Pagedimension, for the same reasons that the thickness of a boundary layer (δ) is known * PgEnds: PageBreakonly as an order-of-magnitude length.
The scale of X can be determined from thescale of δ, which according to the Blasius solution is[397], (3)U x −1/2δ ∼ 5xνThe transition from entrance flow to fully developed flow occurs at x ∼ X andδ ∼ D/2, and therefore it can be concluded thatX/D≈ 10−2ReD(5.2)where ReD = UD/ν. The heat transfer literature also recommends the more precisevalue 0.04 in place of the 10−2 in eq.
(5.2) (Schlichting, 1960), although as shown inFig. 5.2 and 5.3, the transition from entrance flow to fully developed flow is smooth.The friction between fluid and walls is measured as the local shear stress at thewall surface,∂u τx (x) = µ ∂y y=0or the dimensionless local skin friction coefficientτwCf,x = 1ρU 22BOOKCOMP, Inc. — John Wiley & Sons / Page 397 / 2nd Proofs / Heat Transfer Handbook / Bejan(5.3)39810 2Cf, x ReD123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: INTERNAL FLOWS121010⫺410⫺310⫺2[398], (4)x/DReDFigure 5.2 Local skin-friction coefficient in the entrance region of a parallel-plate duct.(From Bejan, 1995.)———-3.02599pt PgVar———Normal Page* PgEnds: Eject100(Cf )0⫺x ReD[398], (4)Cf, x ReD161010⫺3Lines: 153 to 16010⫺2xDReD10⫺1Figure 5.3 Local and average skin friction coefficients in the entrance region of a round tube.(From Bejan, 1995.)Figure 5.2 shows a replotting (Bejan, 1995) of the integral solution (Sparrow, 1955)for Cf,x in the entrance region of a parallel-plate duct.
The dashed-line asymptoteindicates the Cf,x estimate based on the Blasius solution for the laminar boundarylayer between a flat wall and a uniform free stream (U ). If numerical factors of order1 are neglected, the boundary layer asymptote isBOOKCOMP, Inc. — John Wiley & Sons / Page 398 / 2nd Proofs / Heat Transfer Handbook / BejanLAMINAR FLOW AND PRESSURE DROP123456789101112131415161718192021222324252627282930313233343536373839404142434445Cf,x ≈Uxν399−1/2orCf,x · ReD ≈xD · ReD−1/2The solid-line asymptote, Cf,x · ReD = 12, represents the skin friction solution forthe fully developed flow region (Section 5.2.2).The local skin friction coefficient in the entrance region of a round tube is indicatedby the lower curve in Fig.
5.3. This is a replotting (Bejan, 1995) of the solutionreported by Langhaar (1942). The upper curve is for the averaged skin friction coefficient,1 xCf =Cf,ξ (ξ) dξ(5.4a)x 00−xorCf 0−x1xτ̄= 1ρU 22x(5.4b)τw (ξ) dξ0The horizontal asymptote serves both curves,Cf,x = 16 = Cf 0−xand represents the solution for fully developed skin friction in a round tube as shownsubsequently in eq. (5.18).5.2.2Lines: 160 to 223———whereτ̄ =[399], (5)Fully Developed Flow RegionThe key feature of the flow in the region downstream of x ∼ X is that the transversevelocity component (v = 0 in Fig. 5.1) is negligible. In view of the equation for massconservation,∂u ∂v+=0∂x∂ythe vanishing of v is equivalent to ∂u/∂x = 0, that is, a velocity distribution that doesnot change or does not develop further from one x to the next. This is why this flowregion is called fully developed.
It is considered as defined byBOOKCOMP, Inc. — John Wiley & Sons / Page 399 / 2nd Proofs / Heat Transfer Handbook / Bejan0.3832pt PgVar———Normal Page* PgEnds: Eject[399], (5)400123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: INTERNAL FLOWSv=0or∂u=0∂x(5.5)This feature is a consequence of the geometric constraint that downstream of x ∼ X,the boundary layer thickness δ cannot continue to grow. In this region, the lengthscale for changes in the transverse direction is the constant D, not the freely growingδ, and the mass conservation equation requires that U/L ≈ v/D, where L is the flowdimension in the downstream direction.
The v scale is then v ≈ UD/L and this scalevanishes as L increases, that is, as the flow reaches sufficiently far into the duct.Another consequence of the full development of the flow is that the pressure isessentially uniform in each constant-x cross section (∂P /∂y = 0). This feature isderived by substituting v = 0 into the momentum (Navier–Stokes) equation for they direction. With reference to Fig.
5.1, the pressure distribution is P (x), and themomentum equation for the flow direction x becomesdPd 2u=µ 2dxdy(5.6)Both sides of this equation must equal the same constant, because at most, the leftside is a function of x and the right side a function of y. That constant is the pressuredrop per unit length,∆PdP=−LdxThe pressure drop and the flow distribution u(y) are obtained by solving eq.
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