John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 58
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Thiswill bring (Tsat − Tw ) into the solution of the momentum equation.Recall that the boundary layer on a flat surface, during forced convection, was easy to analyze because the momentum equation could besolved completely before any consideration of the energy equation wasattempted. We do not have that advantage in predicting natural convection or film condensation.8.3Laminar natural convection on a verticalisothermal surfaceDimensional analysis and experimental dataBefore we attempt a dimensional analysis of the naturalconvection problem, let us simplify the buoyancy term, (ρ − ρ∞ )g ρ, in the momentumequation (8.3).
The equation was derived for incompressible flow, but wemodified it by admitting a small variation of density with temperature inthis term only. Now we wish to eliminate (ρ − ρ∞ ) in favor of (T − T∞ )with the help of the coefficient of thermal expansion, β: 1 − ρ∞ ρ1 ∂ρ 1 ρ − ρ∞1 ∂v =−−=−(8.6)β≡v ∂T pρ ∂T pρ T − T∞T − T∞where v designates the specific volume here, not a velocity component.Figure 8.2 shows natural convection from a vertical surface that ishotter than its surroundings. In either this case or on the cold plateshown in Fig. 8.1, we replace (1 − ρ∞ /ρ)g with −gβ(T − T∞ ).
The sign(see Fig. 8.2) is the same in either case. Thenu∂u∂2u∂u+v= −gβ(T − T∞ ) + ν∂x∂y∂y 2(8.7)401402Natural convection in single-phase fluids and during film condensation§8.3Figure 8.2 Natural convection from avertical heated plate.where the minus sign corresponds to plate orientation in Fig. 8.1a. Thisconveniently removes ρ from the equation and makes the coupling ofthe momentum and energy equations very clear.The functional equation for the heat transfer coefficient, h, in naturalconvection is therefore (cf. Section 6.4) h or h = fn k, |Tw − T∞ | , x or L, ν, α, g, βwhere L is a length that must be specified for a given problem. Notice thatwhile h was assumed to be independent of ∆T in the forced convectionproblem (Section 6.4), the explicit appearance of (T − T∞ ) in eqn.
(8.7)suggests that we cannot make that assumption here. There are thus eightvariables in W, m, s, and ◦ C (where we again regard J as a unit independentof N and m); so we look for 8−4 = 4 pi-groups. For h and a characteristiclength, L, the groups may be chosen asNuL ≡hL,kPr ≡ν,αΠ3 ≡L3 g ,2νΠ4 ≡ β |Tw − T∞ | = β ∆Twhere we set ∆T ≡ |Tw − T∞ |. Two of these groups are new to us:• Π3 ≡ gL3 /ν 2 : This characterizes the importance of buoyant forcesrelative to viscous forces.23Note that gL is dimensionally the same as a velocity squared—say, u2 . Then Π3can be interpreted as a Reynolds number: uL/ν. In a laminar b.l. we recall that Nu ∝1/4Re1/2 ; so here we expect that Nu ∝ Π3 .2§8.3Laminar natural convection on a vertical isothermal surface• Π4 ≡ β∆T : This characterizes the thermal expansion of the fluid.For an ideal gas,1 ∂β=v ∂TRTp=p1T∞where R is the gas constant.
Therefore, for ideal gasesβ ∆T =∆TT∞(8.8)It turns out that Π3 and Π4 (which do not bear the names of famouspeople) usually appear as a product. This product is called the Grashof(pronounced Gráhs-hoff) number,3 GrL , where the subscript designatesthe length on which it is based:Π3 Π4 ≡ GrL =gβ∆T L3ν2(8.9)Two exceptions in which Π3 and Π4 appear independently are rotatingsystems (where Coriolis forces are part of the body force) and situationsin which β∆T is no longer 1 but instead approaches unity.
We therefore expect to correlate data in most other situations with functionalequations of the formNu = fn(Gr, Pr)(8.10)Another attribute of the dimensionless functional equation is that theprimary independent variable is usually the product of Gr and Pr. Thisis called the Rayleigh number, RaL , where the subscript designates thelength on which it is based:RaL ≡ GrL Pr =gβ∆T L3αν(8.11)3Nu, Pr, Π3 , Π4 , and Gr were all suggested by Nusselt in his pioneering paper onconvective heat transfer [8.1]. Grashof was a notable nineteenth-century mechanicalengineering professor who was simply given the honor of having a dimensionless groupnamed after him posthumously (see, e.g., [8.2]).
He did not work with natural convection.403404Natural convection in single-phase fluids and during film condensation§8.3Figure 8.3 The correlation of h data for vertical isothermalsurfaces by Churchill and Chu [8.3], using NuL = fn(RaL , Pr).(Applies to full range of Pr.)Thus, most (but not all) analyses and correlations of natural convectionyield PrNu = fn Ra , (8.12)secondary parameterprimary (or most important)independent variableFigure 8.3 is a careful selection of the best data available for naturalconvection from vertical isothermal surfaces. These data were organizedby Churchill and Chu [8.3] and they span 13 orders of magnitude of theRayleigh number.
The correlation of these data in the coordinates ofFig. 8.2 is exactly in the form of eqn. (8.12), and it brings to light thedominant influence of RaL , while any influence of Pr is small.Thedata correlate on these coordinates within a few percent up toRaL [1+(0.492/Pr9/16 )]16/9 108 . That is about where the b.l. starts exhibiting turbulent behavior. Beyond that point, the overall Nusselt number, NuL , rises more sharply, and the data scatter increases somewhatbecause the heat transfer mechanisms change.Laminar natural convection on a vertical isothermal surface§8.3Prediction of h in natural convection on a vertical surfaceThe analysis of natural convection using an integral method was doneindependently by Squire (see [8.4]) and by Eckert [8.5] in the 1930s.
Weshall refer to this important development as the Squire-Eckert formulation.The analysis begins with the integrated momentum and energy equations. We assume δ = δt and integrate both equations to the same valueof δ:ddx δ02u − uu∞ δ∂u dy = −ν+ gβ (T − T∞ ) dy∂y y=00(8.13)= 0, sinceu∞ = 0and [eqn. (6.47)]ddxδ0∂T qwu (T − T∞ ) dy == −αρcp∂y y=0The integrated momentum equation is the same as eqn. (6.24) exceptthat it includes the buoyancy term, which was added to the differentialmomentum equation in eqn. (8.7).We now must estimate the temperature and velocity profiles for use ineqns. (8.13) and (6.47).
This is done here in much the same way as it wasdone in Sections 6.2 and 6.3 for forced convection. We write down a setof known facts about the profiles and then use these things to evaluatethe constants in power-series expressions for u and T .Since the temperature profile has a fairly simple shape, a simple quadratic expression can be used: 2 yyT − T∞+c=a+bTw − T∞δδ(8.14)Notice that the thermal boundary layer thickness, δt , is assumed equal toδ in eqn.
(8.14). This would seemingly limit the results to Prandtl numbers not too much larger than unity. Actually, the analysis will also proveuseful for large Pr’s because the velocity profile exerts diminishing influence on the temperature profile as Pr increases. We require the following405406Natural convection in single-phase fluids and during film condensation§8.3things to be true of this profile: T y = 0 = Twor • T y = δ = T∞or••∂T =0∂y y=δorT − T∞ =1=aTw − T∞ y/δ=0T − T∞ =0=1+b+cTw − T∞ y/δ=1dd(y/δ)T − T∞Tw − T∞y/δ=1= 0 = b + 2cso a = 1, b = −2, and c = 1.
This gives the following dimensionlesstemperature profile: 2 T − T∞yyy 2+=1−2= 1−(8.15)Tw − T∞δδδWe anticipate a somewhat complicated velocity profile (recall Fig. 8.1)and seek to represent it with a cubic function: 2 3 yyy+c+d(8.16)u = uc (x)δδδwhere, since there is no obvious characteristic velocity in the problem,we write uc as an as-yet-unknown function. (uc will have to increase withx, since u must increase with x.) We know three things about u:we have already satisfied this condition by• u(y = 0) = 0writing eqn. (8.16) with no lead constant• u(y = δ) = 0or∂u =0∂y y=δor•u= 0 = (1 + c + d)uc∂u = 0 = (1 + 2c + 3d) uc∂(y/δ) y/δ=1These give c = −2 and d = 1, souy=uc (x)δ1−yδ2(8.17)We could also have written the momentum equation (8.7) at the wall,where u = v = 0, and created a fourth condition:gβ (Tw − T∞ )∂2u =−∂y 2 y=0ν§8.3Laminar natural convection on a vertical isothermal surfaceFigure 8.4 The temperature and velocity profiles for air (Pr =0.7) in a laminar convection b.l.and then we could have evaluated uc (x) as βg|Tw − T∞ |δ2 4ν.
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