Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 60
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(5.6)subject to u = 0 at the walls (y ± D/2), where y = 0 represents the center plane ofthe parallel plate duct: 3y 2u(y) = U 1 −(5.7)2D/2withU=D2dP−12µdx(5.8)In general, for a duct of arbitrary cross section, eq. (5.6) is replaced bydP= µ ∇ 2 u = constantdxwhere the Laplacian operator ∇ 2 accounts only for curvatures in the cross section,∇2 =∂2∂2+ 22∂y∂zBOOKCOMP, Inc. — John Wiley & Sons / Page 400 / 2nd Proofs / Heat Transfer Handbook / Bejan[400], (6)Lines: 223 to 271———0.44623pt PgVar———Normal Page* PgEnds: Eject[400], (6)LAMINAR FLOW AND PRESSURE DROP123456789101112131415161718192021222324252627282930313233343536373839404142434445401that is, ∂ 2 /∂x 2 = 0. The boundary conditions are u = 0 on the perimeter of the crosssection. For example, the solution for fully developed laminar flow in a round tubeof radius r0 is 2 r(5.9)u = 2U 1 −r0withU=r02dP−8µdx(5.10)This solution was first reported by Hagen (1839) and Poiseuille (1840), which iswhy the fully developed laminar flow regime is also called Hagen–Poiseuille flow orPoiseuille flow.5.2.3Hydraulic Diameter and Pressure Drop[401], (7)Lines: 271 to 331———Equations (5.8) and (5.10) show that in fully developed laminar flow the mean veloc-0.74576ptity U (or the mass flow rate ṁ = ρAU ) is proportional to the longitudinal pressure———gradient P /L.
In general, and especially in turbulent flow, the relationship betweenNormal Pageṁ and ∆P is nonlinear. Fluid friction results for fully developed flow in ducts are* PgEnds: Ejectreported as friction factors:τwf = 1(5.11)2ρU[401], (7)2where τw is the shear stress at the wall. Equation (5.11) is the same as eq. (5.3), withthe observation that in fully developed flow, τw and f are x-independent.The shear stress τw is proportional to ∆P /L.
This proportionality follows from thelongitudinal force balance on a flow control volume of cross section A and length L,A ∆P = τw pL(5.12)where p is the perimeter of the cross section. Equation (5.12) is general and isindependent of the flow regime. Combined with eq. (5.11), it yields the pressure droprelationshippL 1 2∆P = f(5.13)ρUA 2where A/p represents the transversal length scale of the duct:rh =Aphydraulic radius(5.14)Dh =4Aphydraulic diameter(5.15)BOOKCOMP, Inc.
— John Wiley & Sons / Page 401 / 2nd Proofs / Heat Transfer Handbook / BejanPgVar402123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: INTERNAL FLOWSTABLE 5.1 Scale Drawing of Five Different Ducts That Have the SameHydraulic DiameterCross SectionDiagramCircularSquare[402], (8)Lines: 331 to 362Equilateral triangle———2.89604pt PgVar———Normal Page* PgEnds: EjectRectangular (4:1)Infinite parallel platesSource: Bejan (1995).Table 5.1 shows five duct cross sections that have the same hydraulic diameter.The hydraulic diameter of the round tube coincides with the tube diameter.
Thehydraulic diameter of the channel formed between two parallel plates is twice thespacing between the plates. For cross sections shaped as regular polygons, Dh is thediameter of the circle inscribed inside the polygon. In the case of highly asymmetriccross sections, Dh scales with the smaller of the two dimensions of the cross section.The general pressure drop relationship (5.2) is most often written in terms ofhydraulic diameter,BOOKCOMP, Inc. — John Wiley & Sons / Page 402 / 2nd Proofs / Heat Transfer Handbook / Bejan[402], (8)403LAMINAR FLOW AND PRESSURE DROP123456789101112131415161718192021222324252627282930313233343536373839404142434445∆P = f4LDh1 2ρU2(5.16)To calculate ∆P , the friction factor f must be known and it can be derived from theflow solution.
The friction factors derived from the Hagen–Poiseuille flows describedby eqs. (5.7) and (5.9) aref = 24 ReDh = 2Dparallel plates (D = spacing)(5.17) 16ReDhDh = Dround tube (D = diameter)(5.18)DhEquations (5.17) and (5.18) hold for laminar flow (ReDh ≤ 2000). Friction factors forother cross-sectional shapes are reported in Tables 5.2 and 5.3. Additional results canbe found in Shah and London (1978). All Hagen–Poiseuille flows are characterized by[403], (9)Lines: 362 to 436TABLE 5.2Duct Flow———Effect of Cross-Sectional Shape on f and Nu in Fully DevelopedCross Section*f/ReDhB=πDh2 /4ANu = hDh /kUniform q Uniform T013.30.60532.3514.20.7853.632.891614.3643.6618.31.265.354.65241.578.2357.54241.575.3854.86Source: Bejan (1995).BOOKCOMP, Inc. — John Wiley & Sons / Page 403 / 2nd Proofs / Heat Transfer Handbook / Bejan18.75684pt PgVar———Normal PagePgEnds: TEX[403], (9)404123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: INTERNAL FLOWSTABLE 5.3 Friction Factors and Nusselt Numbers for Heat Transfer to Laminar Flowthrough Ducts with Regular Polygonal Cross SectionsNu = hDh /kCross SectionSquareHexagonOctagonCircleUniform Heat FluxIsothermal Wallf · ReDhFullyDevelopedFlowFullyDevelopedFlowSlugFlowFullyDevelopedFlowSlugFlow14.16715.06515.381163.6144.0214.2074.3647.0837.5337.6907.9622.9803.3533.4673.664.9265.3805.5265.769Source: Data from Asako et al.
(1988).[404], (10)CReDhf =(5.19)where the constant C depends on the shape of the duct cross section. It was shown inBejan (1995) that the duct shape is represented by the dimensionless groupB=πDh /4Aduct(5.20)and that f · ReDh (or C) increases almost proportionally with B. This correlation isillustrated in Fig. 5.4 for the duct shapes documented in Table 5.2.5.35.3.1HEAT TRANSFER IN FULLY DEVELOPED FLOWMean TemperatureConsider the stream shown in Fig. 5.5, and assume that the duct cross section A isnot specified. According to the thermodynamics of open systems, the first law for thecontrol volume of length dx is q p = ṁ dh/dx, where h is the bulk enthalpy of thestream.
When the fluid is an ideal gas (dh = cp dTm ) or an incompressible liquidwith negligible pressure changes (dh = c dTm ), the first law becomesdTmq p=dxṁcp(5.21)In heat transfer, the bulk temperature Tm is known as mean temperature. It isrelated to the bundle of ministreams of enthalpy (ρucp T dA) that make up the bulkenthalpy stream (h) shown in the upper left corner of Fig.
5.5. From this observationit follows that the definition of Tm involves a u-weighted average of the temperaturedistribution over the cross section,BOOKCOMP, Inc. — John Wiley & Sons / Page 404 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 436 to 477———0.39122pt PgVar———Normal PagePgEnds: TEX[404], (10)HEAT TRANSFER IN FULLY DEVELOPED FLOW123456789101112131415161718192021222324252627282930313233343536373839404142434445405[405], (11)Lines: 477 to 484———0.23903pt PgVar———Normal PagePgEnds: TEX[405], (11)Figure 5.4 Effect of the cross-sectional shape (the number B) on fully developed frictionand heat transfer in a straight duct.
(From Bejan, 1995.)Tm =1UAAuT dA(5.22)In internal convection, the heat transfer coefficient h = q /∆T is based on thedifference between the wall temperature (T0 ) and the mean temperature of the stream:namely, h = q /(T0 − Tm ).5.3.2 Thermally Fully Developed FlowBy analogy with the developing velocity profile described in connection with Fig.
5.1,there is a thermal entrance region of length XT . In this region the thermal boundaryBOOKCOMP, Inc. — John Wiley & Sons / Page 405 / 2nd Proofs / Heat Transfer Handbook / Bejan406123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: INTERNAL FLOWSq⬙u dA.mrvuxq⬙Tm ⫹ dTmTm[406], (12)x x + dxFigure 5.5 Nomenclature for energy conservation in a duct segment. (From Bejan, 1995.)layers grow and effect changes in the distribution of temperature over the duct crosssection.
Estimates for XT are given in Section 5.4.1. Downstream from x ∼ XT thethermal boundary layers have merged and the shape of the temperature profile acrossthe duct no longer varies. For a round tube of radius r0 , this definition of a fullydeveloped temperature profile is rT0 (x) − T (r, x)(5.23)=φT0 (x) − Tm (x)r0The function φ(r/r0 ) represents the r-dependent shape (profile) that does not dependon the downstream position x.The alternative to the definition in eq. (5.23) is the scale analysis of the sameregime (Bejan, 1995), which shows that the heat transfer coefficient must be constant (x-independent) and of order k/D.
The dimensionless version of this seconddefinition is the statement that the Nusselt number is a constant of order 1:∂T /∂r r=r0hD=D= O(1)(5.24)Nu =kT0 − T mThe second part of the definition refers to a tube of radius r0 . The Nu values compiledin Tables 5.2 and 5.3 confirm the constancy and order of magnitude associated withthermally fully developed flow.
The Nu values also exhibit the approximate proportionality with the B number that characterizes the shape of the cross section (Fig.5.4). In Table 5.3, slug flow means that the velocity is distributed uniformly over theBOOKCOMP, Inc. — John Wiley & Sons / Page 406 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 484 to 500———0.05212pt PgVar———Normal PagePgEnds: TEX[406], (12)407HEAT TRANSFER IN DEVELOPING FLOW123456789101112131415161718192021222324252627282930313233343536373839404142434445cross section, u = U .
Noteworthy are the Nu values for a round tube with uniformwall heat fluxNu =48= 4.36411uniform wall heat fluxand a round tube with isothermal wallNu = 3.665.4isothermal wallHEAT TRANSFER IN DEVELOPING FLOW5.4.1 Thermal Entrance Region[407], (13)The heat transfer results listed in Tables 5.2 and 5.3 apply to laminar flow regionswhere the velocity and temperature profiles are fully developed.
They are valid in thedownstream section x, where x > max(X, XT ). The flow development length X isgiven by eq. (5.2). The thermal length XT is determined from a similar scale analysisby estimating the distance where the thermal boundary layers merge, as shown inFig. 5.6. When Pr 1, the thermal boundary layers are thicker than the velocityboundary layers, and consequently, XT X. The Prandtl number Pr is the ratio ofthe molecular momentum and thermal diffusivities, ν/α. When Pr 1, the thermalboundary layers are thinner, and XT is considerably greater than X. The scale analysisof this problem shows that for both Pr 1 and Pr 1, the relationship between XTand X is (Bejan, 1995)XT≈ PrX(5.25)Pr 10XT ⬃ Pr XXxPr 10XXT ⬃ Pr XFigure 5.6 Prandtl number effect on the flow entrance length X and the thermal entrancelength XT .
(From Bejan, 1995.)BOOKCOMP, Inc. — John Wiley & Sons / Page 407 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 500 to 535———7.09703pt PgVar———Normal PagePgEnds: TEX[407], (13)408Equations (5.25) and (5.2) yield the scaleXT ≈ 10−2 Pr · Dh · ReDh(5.26)which is valid over the entire Pr range.5.4.2 Thermally Developing Hagen–Poiseuille FlowWhen Pr 1, there is a significant length of the duct (X < x < XT ) over whichthe velocity profile is fully developed, whereas the temperature profile is not.
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