Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 68
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The focus in these solutions is on the determination of generalized transport correlations and on the detaileddescriptions of the thermal and flow fields. For turbulent and transitional flow, reliancemust be placed on experimental data and correlations.6.46.4.1HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOWHigh Reynolds Number Flow over a WedgeIn the absence of mass transfer, the governing equations in Cartesian coordinates(x, y, z) for constant property, incompressible three-dimensional flow with the velocityV = ui + vj + wkaccount for continuity (mass conservation), conservation of momentum, and conservation of energy.
The mass conservation equation is∇ ·V=0(6.2)For the conservation of momentum, the x, y, and z components areDu∂u∂pρ=ρ+ ∇ · (uV) = −+ µ ∇ 2uDt∂t∂xDv∂v∂pρ=ρ+ ∇ · (vV) = −+ µ ∇ 2vDt∂t∂yBOOKCOMP, Inc. — John Wiley & Sons / Page 446 / 2nd Proofs / Heat Transfer Handbook / Bejan(6.3a)(6.3b)[446], (8)Lines: 232 to 268———3.70003pt PgVar———Normal PagePgEnds: TEX[446], (8)HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW123456789101112131415161718192021222324252627282930313233343536373839404142434445ρ∂w∂pDw=ρ+ ∇ · (wV) = −+ µ ∇ 2wDt∂t∂zThe conservation of energy requiresDT∂TDpρcp= ρcp+ ∇ · (T V) = k ∇ 2 T + µΦ + βDt∂tDt447(6.3c)(6.4)The last two terms on the right-hand side of the energy equation, eq.
(6.4), accountfor viscous dissipation and pressure stress effects, where, as indicated in Chapters 1and 5, Φ is given by 2 ∂u 2∂v∂w 2Φ=2++∂x∂y∂z2 ∂v∂u∂w ∂v 2∂u ∂w 2×+++++∂x∂y∂y∂z∂z∂x[447], (9)Lines: 268 to 313———2− (∇ · V)235.68127pt PgVar———Normal PageViscous dissipation effects are important in very viscous fluids or in the presence of* PgEnds: Ejectlarge velocity gradients.Two other orthogonal coordinate systems of frequent interest are the cylindricaland spherical coordinate systems. Governing equations for continuity, the force–[447], (9)momentum balance and the conservation of energy are provided in Chapter 1.Equations (6.2)–(6.4) can be normalized using the variables(x, y, z)L(u, v, w)(u∗ , v ∗ , w∗ ) =Utt∗ =L/Upp∗ =ρU 2(x ∗ , y ∗ , z∗ ) =(6.5a)(6.5b)(6.5c)(6.5d)where L and U are appropriate length and velocity scales, respectively. The normalized form of eq.
(6.3a) is then∗∗∗∂u∗∗ ∂u∗ ∂u∗ ∂u+u+v+w∂x ∗∂y ∗∂z∗∂t ∗ 2 ∗∂p ∗µ∂ 2 u∗∂ 2 u∗∂ u=− ∗ +++∂xULρ ∂x ∗2∂y ∗2∂z∗2BOOKCOMP, Inc. — John Wiley & Sons / Page 447 / 2nd Proofs / Heat Transfer Handbook / Bejan(6.6)448123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: EXTERNAL FLOWSThe normalized forms of eqs. (6.3b) and (6.3c) are similar, and for brevity are notshown. Moreover, when all the variables in eq. (6.2) are replaced by their normalizedversions, the Reynolds number emerges as the key nondimensional solution parameter, relating the inertia forces to the viscous forces:Re =ρU 2 L2ρUL=2µ(U/L)Lµ(6.7)To nondimensionalize the energy equation, a normalized temperature is defined asT∗ =T − T∞Ts − T∞(6.8)where T∞ and Ts are the local ambient and surface temperatures, respectively.
Bothtemperatures could, in general, vary with location and/or time. The resulting normalized energy equation isDT ∗1Dp ∗2 ∗(6.9)=T+2Ec·Pr·Φ+2βT·Re·Pr·Ec∇Dt ∗Re · PrDt ∗where β is the coefficient of volumetric thermal expansion and Pr is the Prandtlnumber:Pr =cp uν=αk(6.10a)a measure of the ratio of diffusivity of momentum to diffusivity of heat. The EckertnumberEc =V22cp (Ts − T∞ )(6.10b)which expresses the magnitude of the kinetic energy of the flow relative to the enthalpy difference. In eq.
(6.9), βT = 1 for ideal gases, and typically, βT 1 forliquids. The pressure stress term is thus negligible in forced convection whenever theviscous dissipation is small.The wall values of shear stress and heat flux are, respectively,∂u µU ∂u∗ τs = µ =(6.11a)∂y sL ∂y ∗ y ∗ =0and∗−T)∂Tk(T∂Ts∞ =qs = −k− ∗ ∂y sL∂y y ∗ =0(6.11b)From a practical perspective, the friction drag and the heat transfer rate are the mostimportant quantities. These are determined from the friction coefficient, Cf :BOOKCOMP, Inc.
— John Wiley & Sons / Page 448 / 2nd Proofs / Heat Transfer Handbook / Bejan[448], (10)Lines: 313 to 378———1.50526pt PgVar———Long Page* PgEnds: Eject[448], (10)HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW1234567891011121314151617181920212223242526272829303132333435363738394041424344452τs2Cf ==2ρURe∂u∗ ∂y ∗ y ∗ =0449(6.12a)and the Nusselt number Nu, defined asqs L∂T ∗ Nu ==− ∗k(Ts − T∞ )∂y y ∗ =0(6.12b)The boundary layer development for two-dimensional flow in the x- and y-coordinate directions over the wedge shown in Fig. 6.6 is based on the governing equationsrepresenting continuity, the x and y components of momentum, and the conservationof energy,[449], (11)∂u ∂v+=0∂x∂y(6.13) 2∂u∂u∂uX 1 ∂p∂ u ∂ 2u+ 2+u+v=−+ν∂t∂x∂yρρ ∂x∂x 2∂y 2∂v∂ 2v∂v∂vY1 ∂p∂ v+ 2+u+v= −+ν∂t∂x∂yρρ ∂y∂x 2∂y 22 ∂T∂T∂Tµ∂ TβT Dpq ∂ T+u+v++Φ+=α+∂t∂x∂y∂x 2∂y 2ρcpρcp Dtρcp(6.14a)∂pmX 1 ∂p=−∂xρρ ∂xand−(6.14b)(6.15)∂pmY1 ∂p= −∂yρρ ∂ywhere the motion pressure pm is the difference between the local static and localhydrostatic pressures.On each face of the wedge shown in Fig.
6.6, the x direction is measured along thesurface from the point of contact (the leading edge) and the y direction is measurednormal to the surface. The free stream velocity, which is designated by U (x) at aFigure 6.6 Boundary layer development for high-Reynolds-number flow over a wedge.BOOKCOMP, Inc. — John Wiley & Sons / Page 449 / 2nd Proofs / Heat Transfer Handbook / Bejan———2.23122pt PgVarIn eqs. (6.14), X and Y are the body forces in the x and y directions.
The bodyforce and the static pressure gradient terms on the right-hand sides of eqs. (6.14) canbe combined to yield−Lines: 378 to 418———Long PagePgEnds: TEX[449], (11)450123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: EXTERNAL FLOWSgiven x, remains unchanged for large values of y. The velocity varies from zero at thesurface [u(y = 0) = 0] to the free stream velocity within a fluid layer, the thicknessof which increases with x.The region close to the surface where the velocity approaches a value close tothe local free stream level defines the hydrodynamic boundary layer, which has thethickness δ(x). When heat transfer occurs, the temperature at the surface, T (y = 0) =Ts , is not equal to the free stream temperature T∞ , and a thermal boundary layer alsoexists. In the thermal boundary, the temperature adjusts from the wall value to nearthe free stream level.
The thermal boundary layer thickness δT (x) characterizes thethermal boundary layer.When R 1, thin hydrodynamic and thermal boundary layers of thickness δ(x)and δT (x), respectively, develop along an object in general and on the surfaces of thewedge in Fig. 6.6 in particular. Under this condition, δ/L 1 and δT /L 1 andthe following scaled variables can be defined:tt0xx∗ =Lyy∗ =δt∗ =uU (x)vv∗ =Vsu∗ =∗pm=pm − p∞ρU 2T − T∞Ts − T ∞yyT∗ =δTT∗ =(6.16a)Lines: 418 to 454(6.16b)10.15335pt PgVar——————Normal Page(6.16c)* PgEnds: Eject(6.16d)[450], (12)(6.16e)(6.16f)(6.16g)(6.16h)where Vs is the fluid velocity normal to the surface imposed by induced or forcedfluid motion, U (x) the velocity field in the potential flow region outside the boundarylayers, t0 a time reference, and T∞ the temperature of the environment.The corresponding continuity equation is∂u∗Vs L ∂v ∗+=0∗∂xU δ ∂y ∗(6.17)and in order for the two terms to be of the same order of magnitude, Vs must be ofthe order of δU/L:BOOKCOMP, Inc.
— John Wiley & Sons / Page 450 / 2nd Proofs / Heat Transfer Handbook / Bejan[450], (12)HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW123456789101112131415161718192021222324252627282930313233343536373839404142434445Vs = OδUL451Then the normalized momentum equations become ∗L ∂u∗δ 2 ∂ 2 u∗∂u∗∂u∗∂pmνL∂ 2 u∗+u ∗ +v ∗ =− ∗ ++ ∗2U t0 ∂t ∗∂x∂y∂xL ∂x ∗2∂yU δ2(6.18)and as Re 1,νL= O(1)U δ2δ11=O1/2LReLso that[451], (13)and therefore, 2 ∗∗δL ∂v ∗∗ ∂v∗ ∂v+u+vLU to ∂t ∗∂x ∗∂y ∗ ∗∂pm1∂ 2 v∗δ 2 ∂ 2 v∗=− ∗ ++ ∗2∂yReL ∂x ∗2∂yLines: 454 to 509———6.25131pt PgVar(6.19)As Re → ∞, eq. (6.19) becomes0=−∗∂pm+0∂y ∗[451], (13)which implies that the pressure inside the boundary layer is a function of x and canbe evaluated outside the boundary layer in the potential flow via the solution to theEuler equation−1 dpmdU=Uρ dxdxThe energy equation is∗L ∂T ∗δ ∗ ∂T ∗k∗ ∂T+u+v=U t0 ∂t ∗∂x ∗δT ∂yT∗ρcp U L+∂ 2T ∗+∂x ∗2LδT2µULβT U L Dp ∗q L∗Φ++cp ∆T t0 Dt ∗ρcp U ∆Tρcp ∆T δ2∂ 2T ∗∂yT∗2(6.20)Here the transient term is important if L/t0 U = O(1) and steady conditions areapproached for L/t0 U 1.
In addition, the pressure stress term vpy upx and thenormalized equation resulting from letting Re = O(L2 /δ2 ) isBOOKCOMP, Inc. — John Wiley & Sons / Page 451 / 2nd Proofs / Heat Transfer Handbook / Bejan———Normal Page* PgEnds: Eject452123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: EXTERNAL FLOWSδ ∗ ∂T ∗1u+v∗ =∗∂xδT ∂yTPr∗ ∂T∗δδT2∂ 2T ∗∂yT∗2+ 2Ec · Φ∗ + 2βT · Ec · u∗dp ∗q L+dx ∗ρcp U ∆T(6.21)The corresponding dimensional form of eq. (6.21) isu∂TβT dp∂T∂ 2Tµu+v=α 2 ++∂x∂y∂yρcp dxρcp∂u∂y2+q ρcp(6.22)Equation (6.20) can be used to demonstrate that when Pr 1,δT1=OδPr1/2[452], (14)and when Pr 1,δT=Oδ1Pr1/3Lines: 509 to 566———-2.12363pt PgVar———Long PagePgEnds: TEX6.4.2 Similarity Transformation Technique for Laminar BoundaryLayer FlowFollowing the simplifications of eqs.
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