John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 62
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It follows that we look for9 − 5 = 4 pi-groups. The ones we choose areΠ1 = NuL ≡Π3 = Ja ≡hLkcp (Tsat − Tw )hfgΠ2 = Pr ≡Π4 ≡ναρf (ρf − ρg )ghfg L3µk(Tsat − Tw )Two of these groups are new to us. The group Π3 is called the Jakobnumber, Ja, to honor Max Jakob’s important pioneering work during the1930s on problems of phase change. It compares the maximum sensibleheat absorbed by the liquid to the latent heat absorbed.
The group Π4does not normally bear anyone’s name, but, if it is multiplied by Ja, itmay be regarded as a Rayleigh number for the condensate film.Notice that if we condensed water at 1 atm on a wall 10◦ C belowTsat , then Ja would equal 4.174(10/2257) = 0.0185. Although 10◦ C is afairly large temperature difference in a condensation process, it gives amaximum sensible heat that is less than 2% of the latent heat. The Jakobnumber is accordingly small in most cases of practical interest, and itturns out that sensible heat can often be neglected.
(There are important7Note that, throughout this section, k, µ, cp , and Pr refer to properties of the liquid,rather than the vapor.Film condensation§8.5429exceptions to this.) The same is true of the role of the Prandtl number.Therefore, during film condensation⎛⎞ρf (ρf − ρg )ghfg L3NuL = fn ⎝, Pr, Ja ⎠(8.46) µk(Tsat − Tw )secondary independentvariablesprimary independent variable, Π4Equation (8.46) is not restricted to any geometrical configuration,since the same variables govern h during film condensation on any body.Figure 8.10, for example, shows laminar film condensation data givenfor spheres by Dhir8 [8.32].
They have been correlated according toeqn. (8.12). The data are for only one value of Pr but for a range ofΠ4 and Ja. They generally correlate well within ±10%, despite a broadvariation of the not-very-influential variable, Ja. A predictive curve [8.32]is included in Fig. 8.10 for future reference.Laminar film condensation on a vertical plateConsider the following feature of film condensation. The latent heat ofa liquid is normally a very large number. Therefore, even a high rate ofheat transfer will typically result in only very thin films. These films moverelatively slowly, so it is safe to ignore the inertia terms in the momentumequation (8.4):ρg∂u∂2u∂u= 1−+vg+νu∂x∂yρf∂y 2 2udy 2d0This result will give u = u(y, δ) (where δ is the local b.l. thickness)when it is integrated.
We recognize that δ = δ(x), so that u is not strictlydependent on y alone. However, the y-dependence is predominant, andit is reasonable to use the approximate momentum equationρf − ρg gd2 u=−2dyνρf8(8.47)Professor Dhir very kindly recalculated his data into the form shown in Fig. 8.10for use here.430Natural convection in single-phase fluids and during film condensation§8.5Figure 8.10 Correlation of the data of Dhir [8.32] for laminarfilm condensation on spheres at one value of Pr and for a rangeof Π4 and Ja, with properties evaluated at (Tsat + Tw )/2. Analytical prediction from [8.33].This simplification was made by Nusselt in 1916 when he set down theoriginal analysis of film condensation [8.34].
He also eliminated the convective terms from the energy equation (6.40):∂T∂2T∂T+v=αu∂x∂y∂y 20Film condensation§8.5431on the same basis. The integration of eqn. (8.47) subject to the b.c.’s ∂u u y =0 =0and=0∂y y=δgives the parabolic velocity profile:u=(ρf − ρg )gδ22µ yy 22−δδ(8.48)And integration of the energy equation subject to the b.c.’s T y = 0 = Twand T y = δ = Tsatgives the linear temperature profile:T = Tw + (Tsat − Tw )yδ(8.49)To complete the analysis, we must calculate δ. This can be done intwo steps.
First, we express the mass flow rate per unit width of film, ṁ,in terms of δ, with the help of eqn. (8.48):δṁ =0ρf u dy =ρf (ρf − ρg )3µgδ3(8.50)Second, we neglect the sensible heat absorbed by that part of the filmcooled below Tsat and express the local heat flux in terms of the rate ofchange of ṁ (see Fig. 8.11): Tsat − Twdṁ∂Tq = k= hfg(8.51)=k∂y y=0δdxSubstituting eqn.
(8.50) in eqn. (8.51), we obtain a first-order differential equation for δ:khfg ρf (ρf − ρg )dδTsat − Tw=gδ2δµdx(8.52)This can be integrated directly, subject to the b.c., δ(x = 0) = 0. Theresult is1/44k(Tsat − Tw )µx(8.53)δ=ρf (ρf − ρg )ghfg432Natural convection in single-phase fluids and during film condensation§8.5Figure 8.11 Heat and mass flow in an element of a condensing film.Both Nusselt and, subsequently, Rohsenow [8.35] showed how to correct the film thickness calculation for the sensible heat that is needed tocool the inner parts of the film below Tsat .
Rohsenow’s calculation was, inpart, an assessment of Nusselt’s linear-temperature-profile assumption,and it led to a corrected latent heat—designated hfg —which accountedfor subcooling in the liquid film when Pr is large. Rohsenow’s result,which we show below to be strictly true only for large Pr, was⎡⎤c(T−T)psatw⎦(8.54)hfg = hfg ⎣ 1 + 0.68hfg≡ Ja, Jakob numberThus, we simply replace hfg with hfg wherever it appears explicitly inthe analysis, beginning with eqn.
(8.51).Finally, the heat transfer coefficient is obtained fromk1k(Tsat − Tw )q=(8.55)=h≡Tsat − TwTsat − TwδδsoNux =xhx=kδ(8.56)Thus, with the help of eqn. (8.54), we substitute eqn. (8.53) in eqn. (8.56)Film condensation§8.5433and get⎡Nux = 0.707 ⎣ρf (ρf − ρg )ghfg x 3µk(Tsat − Tw )⎤1/4⎦(8.57)This equation carries out the functional dependence that we anticipated in eqn. (8.46):Nux = fnΠ4 , Ja , Pr eliminated in so far as weneglected convective termsin the energy equationthis is carried implicitly in hfgthis is clearly the dominant variableThe liquid properties in Π4 , Ja, and Pr (with the exception of hfg ) areto be evaluated at the mean film temperature.
However, if Tsat − Tw issmall—and it often is—one might approximate them at Tsat .At this point we should ask just how great the missing influence ofPr is and what degree of approximation is involved in representing theinfluence of Ja with the use of hfg . Sparrow and Gregg [8.36] answeredthese questions with a complete b.l.
analysis of film condensation. Theydid not introduce Ja in a corrected latent heat but instead showed itsinfluence directly.Figure 8.12 displays two figures from the Sparrow and Gregg paper.The first shows heat transfer results plotted in the formNux3= fn (Ja, Pr) → constant as Ja → 04Π4(8.58)Notice that the calculation approaches Nusselt’s simple result for allPr as Ja → 0.
It also approaches Nusselt’s result, even for fairly largevalues of Ja, if Pr is not small. The second figure shows how the temperature deviates from the linear profile that we assumed to exist in thefilm in developing eqn. (8.49). If we remember that a Jakob number of0.02 is about as large as we normally find in laminar condensation, it isclear that the linear temperature profile is a very sound assumption fornonmetallic liquids.434Natural convection in single-phase fluids and during film condensation§8.5Figure 8.12 Results of the exact b.l.
analysis of laminar filmcondensation on a vertical plate [8.36].Sadasivan and Lienhard [8.37] have shown that the Sparrow-Gregg formulation can be expressed with high accuracy, for Pr 0.6, by includingPr in the latent heat correction. Thus they wrote! "(8.59)hfg = hfg 1 + 0.683 − 0.228 Pr Jawhich includes eqn. (8.54) for Pr → ∞ as we anticipated.Film condensation§8.5435The Sparrow and Gregg analysis proves that Nusselt’s analysis is quiteaccurate for all Prandtl numbers above the liquid-metal range. The veryhigh Ja flows, for which Nusselt’s theory requires some correction, usually result in thicker films, which become turbulent so the exact analysisno longer applies.The average heat transfer coefficient is calculated in the usual way forTwall = constant:1 Lh=h(x) dx = 43 h(L)L 0so⎡NuL = 0.9428 ⎣ρf (ρf − ρg )ghfg L3µk(Tsat − Tw )⎤1/4⎦(8.60)Example 8.6Water at atmospheric pressure condenses on a strip 30 cm in heightthat is held at 90◦ C.
Calculate the overall heat transfer per meter, thefilm thickness at the bottom, and the mass rate of condensation permeter.Solution.⎤1/4−T)µx4k(Tsatw⎦δ=⎣ρf (ρf − ρg )ghfg⎡where we have replaced hfg with hfg :0.228 4.211(10)hfg = 2257 1 + 0.683 −= 2281 kJ/kg1.862257so1/44(0.677)(10)(2.99 × 10−4 ) x= 0.000141 x 1/4δ=961.9(961.9 − 0.6)(9.806)(2281 × 103 )Thenδ(L) = 0.000104 m = 0.104 mm436Natural convection in single-phase fluids and during film condensation§8.5Notice how thin the film is. Finally, we use eqns. (8.56) and (8.59) tocomputeNuL =4(0.3)4 L== 38463δ3(0.000104)soq=NuL k∆T3846(0.677)(10)== 8.68 × 104 W/m2L0.3(This would correspond to a heat flow of 86.8 kW on an area about halfthe size of a desk top. That is very high for such a small temperaturedifference.) ThenQ = (8.68 × 104 )(0.3) = 26, 040 W/m = 26.0 kW/mThe rate of condensate flow, ṁ isṁ =26.0Q= 0.0114 kg/m·s =hfg2281Condensation on other bodiesNusselt himself extended his prediction to certain other bodies but wasrestricted by the lack of a digital computer from evaluating as many casesas he might have.
In 1971 Dhir and Lienhard [8.33] showed how Nusselt’smethod could be readily extended to a large class of problems. Theyshowed that one need only to replace the gravity, g, with an effectivegravity, geff :geff ≡ x x gRg1/3R4/3(8.61)4/3dx0in eqns. (8.53) and (8.57), to predict δ and Nux for a variety of bodies.The terms in eqn.
(8.61) are (see Fig. 8.13):• x is the distance along the liquid film measured from the upperstagnation point.• g = g(x), the component of gravity (or other body force) along x;g can vary from point to point as it does in Fig. 8.13b and c.Figure 8.13 Condensation on various bodies.
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