John H. Lienhard IV, John H. Lienhard V. A Heat Transfer Textbook (776116), страница 60
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(The dark lines in the pictures areisotherms.)415416Natural convection in single-phase fluids and during film condensation§8.4The analysis of free convection becomes a far more complicated problem at low Gr’s, since the b.l. equations can no longer be used. We shallnot discuss any of the numerical solutions of the full Navier-Stokes equations that have been carried out in this regime.
We shall instead note thatcorrelations of data using functional equations of the formNu = fn(Ra, Pr)will be the first thing that we resort to in such cases. Indeed, Fig. 8.3 reveals that Churchill and Chu’s equation (8.27) already serves this purposein the case of the vertical isothermal plate, at low values of Ra ≡ Gr Pr.8.4Natural convection in other situationsNatural convection from horizontal isothermal cylindersChurchill and Chu [8.10] provide yet another comprehensive correlationof existing data. For horizontal isothermal cylinders, they find that anequation with the same form as eqn.
(8.27) correlates the data for horizontal cylinders as well. Horizontal cylinder data from a variety ofsources, over about 24 orders of magnitude of the Rayleigh number basedon the diameter, RaD , are shown in Fig. 8.6. The equation that correlatesthem is1/40.518 RaDNuD = 0.36 + !1 + (0.559/Pr)9/16"4/9(8.30)They recommend that eqn. (8.30) be used in the range 10−6 RaD 109 .When RaD is greater than 109 , the flow becomes turbulent.
The following equation is a little more complex, but it gives comparable accuracyover a larger range:⎧⎪⎨⎤1/6 ⎫2⎪⎬RaD⎣⎦NuD = 0.60 + 0.387 !"16/9⎪⎪⎩⎭1 + (0.559/Pr)9/16⎡The recommended range of applicability of eqn. (8.31) is10−6 RaD(8.31)Natural convection in other situations§8.4417Figure 8.6 The data of many investigators for heat transferfrom isothermal horizontal cylinders during natural convection, as correlated by Churchill and Chu [8.10].Example 8.4Space vehicles are subject to a “g-jitter,” or background variation ofacceleration, on the order of 10−6 or 10−5 earth gravities. Brief periods of gravity up to 10−4 or 10−2 earth gravities can be exertedby accelerating the whole vehicle.
A certain line carrying hot oil is½ cm in diameter and it is at 127◦ C. How does Q vary with g-level ifT∞ = 27◦ C in the air around the tube?Solution. The average b.l. temperature is 350 K. We evaluate properties at this temperature and write g as ge × (g-level), where ge is gat the earth’s surface and the g-level is the fraction of ge in the spacevehicle. With β = 1/T∞ for an ideal gas400 − 3009.8(0.005)3 gβ∆T D 3300=RaD =g-levelνα2.062(10)−5 2.92(10)−5 = (678.2) g-levelFrom eqn.
(8.31), with Pr = 0.706, we computeNuD =⎧⎨⎩678.20.6 + 0.387 !"16/91 + (0.559/0.706)9/16=0.952so1/6(g-level)1/6⎫2⎬⎭418Natural convection in single-phase fluids and during film condensationg-levelNuD10−610−510−410−20.4830.5470.6481.086h = NuD2.873.253.856.450.02970.005W/m2 KW/m2 KW/m2 KW/m2 K§8.4Q = π Dh∆T4.515.106.0510.1W/mW/mW/mW/mofofofoftubetubetubetubeThe numbers in the rightmost column are quite low.
Cooling is clearlyinefficient at these low gravities.Natural convection from vertical cylindersThe heat transfer from the wall of a cylinder with its axis running vertically is the same as that from a vertical plate, so long as the thermal b.l. isthin. However, if the b.l. is thick, as is indicated in Fig. 8.7, heat transferwill be enhanced by the curvature of the thermal b.l. This correction wasfirst considered some years ago by Sparrow and Gregg, and the analysiswas subsequently extended with the help of more powerful numericalmethods by Cebeci [8.11].Figure 8.7 includes the corrections to the vertical plate results thatwere calculated for many Pr’s by Cebeci. The left-hand graph gives acorrection that must be multiplied by the local flat-plate Nusselt numberto get the vertical cylinder result.
Notice that the correction increaseswhen the Grashof number decreases. The right-hand curve gives a similarcorrection for the overall Nusselt number on a cylinder of height L. Noticethat in either situation, the correction for all but liquid metals is less than1/410% if (x or L)/R < 0.08 Grx or L .Heat transfer from general submerged bodiesSpheres. The sphere is an interesting case because it has a clearly specifiable value of NuD as RaD → 0. We look first at this limit. When thebuoyancy forces approach zero by virtue of:• low gravity,•very high viscosity,• small diameter,•a very small value of β,then heated fluid will no longer be buoyed away convectively. In that case,only conduction will serve to remove heat.
Using shape factor number 4Natural convection in other situations§8.4Figure 8.7 Corrections for h and h on vertical isothermal plates to make them apply to vertical isothermal cylinders [8.11].in Table 5.4, we compute in this caselim NuD =RaD →0Q Dk∆T (S)D4π (D/2)===22A∆T k4π (D/2) ∆T k4π (D/4)(8.32)Every proper correlation of data for heat transfer from spheres therefore has the lead constant, 2, in it.5 A typical example is that of Yuge [8.12]for spheres immersed in gases:1/4NuD = 2 + 0.43 RaD ,RaD < 105(8.33)A more complex expression [8.13] encompasses other Prandtl numbers:1/4NuD = 2 + !0.589 RaD"4/91 + (0.492/Pr)9/16RaD < 1012(8.34)This result has an estimated uncertainty of 5% in air and an rms error ofabout 10% at higher Prandtl numbers.5It is important to note that while NuD for spheres approaches a limiting value atsmall RaD , no such limit exists for cylinders or vertical surfaces.
The constants ineqns. (8.27) and (8.30) are not valid at extremely low values of RaD .419420Natural convection in single-phase fluids and during film condensation§8.4Rough estimate of Nu for other bodies. In 1973 Lienhard [8.14] notedthat, for laminar convection in which the b.l. does not separate, the expression1/4Nuτ 0.52 Raτ(8.35)would predict heat transfer from any submerged body within about 10%if Pr is not 1. The characteristic dimension in eqn.
(8.35) is the lengthof travel, τ, of fluid in the unseparated b.l.In the case of spheres without separation, for example, τ = π D/2, thedistance from the bottom to the top around the circumference. Thus, forspheres, eqn. (8.35) becomes1/4gβ∆T (π D/2)3hπ D= 0.522kναor1/4 3/4 2πhDgβ∆T D 3= 0.52kπ2ναor1/4NuD = 0.465 RaDThis is within 8% of Yuge’s correlation if RaD remains fairly large.Laminar heat transfer from inclined and horizontal platesIn 1953, Rich [8.15] showed that heat transfer from inclined plates couldbe predicted by vertical plate formulas if the component of the gravityvector along the surface of the plate was used in the calculation of theGrashof number.
Thus, g is replaced by g cos θ, where θ is the angle ofinclination measured from the vertical, as shown in Fig. 8.8. The heattransfer rate decreases as (cos θ)1/4 .Subsequent studies have shown that Rich’s result is substantially correct for the lower surface of a heated plate or the upper surface of acooled plate. For the upper surface of a heated plate or the lower surfaceof a cooled plate, the boundary layer becomes unstable and separates ata relatively low value of Gr. Experimental observations of such instability have been reported by Fujii and Imura [8.16], Vliet [8.17], Pera andGebhart [8.18], and Al-Arabi and El-Riedy [8.19], among others.§8.4Natural convection in other situationsFigure 8.8 Natural convection b.l.’s on some inclined and horizontal surfaces.
The b.l. separation, shown here for the unstable cases in (a) and (b), occurs only at sufficiently large valuesof Gr.In the limit θ = 90◦ — a horizontal plate — the fluid flow above a hotplate or below a cold plate must form one or more plumes, as shown inFig. 8.8c and d.
In such cases, the b.l. is unstable for all but small Rayleighnumbers, and even then a plume must leave the center of the plate. Theunstable cases can only be represented with empirical correlations.Theoretical considerations, and experiments, show that the Nusseltnumber for laminar b.l.s on horizontal and slightly inclined plates variesas Ra1/5 [8.20, 8.21]. For the unstable cases, when the Rayleigh numberexceeds 104 or so, the experimental variation is as Ra1/4 , and once theflow is fully turbulent, for Rayleigh numbers above about 107 , experi-421422Natural convection in single-phase fluids and during film condensation§8.4ments show a Ra1/3 variation of the Nusselt number [8.22, 8.23].
In the1/3latter case, both NuL and RaL are proportional to L, so that the heattransfer coefficient is independent of L. Moreover, the flow field in thesesituations is driven mainly by the component of gravity normal to theplate.Unstable Cases: For the lower side of cold plates and the upper sideof hot plates, the boundary layer becomes increasingly unstable as Ra isincreased.• For inclinations θ 45◦ and 105 RaL 109 , replace g with g cos θin eqn.
(8.27).• For horizontal plates with Rayleigh numbers above 107 , nearly identical results have been obtained by many investigators. From theseresults, Raithby and Hollands propose [8.13]:1 + 0.0107 Pr1/3, 0.024 Pr 2000 (8.36)NuL = 0.14 RaL1 + 0.01 PrThis formula is consistent with available data up to RaL = 2 × 1011 ,and probably goes higher. As noted before, the choice of lengthscale L is immaterial. Fujii and Imura’s results support using theabove for 60◦ θ 90◦ with g in the Rayleigh number.For high Ra in gases, temperature differences and variable properties effects can be large. From experiments on upward facing plates,Clausing and Berton [8.23] suggest evaluating all gas properties ata reference temperature, in kelvin, ofTref = Tw − 0.83 (Tw − T∞ )for1 Tw /T∞ 3.• For horizontal plates of area A and perimeter P at lower Rayleighnumbers, Raithby and Hollands suggest [8.13]1/40.560 RaL∗NuL∗ = !1 + (0.492/Pr)9/16"4/9(8.37a)where, following Lloyd and Moran [8.22], a characteristic lengthscale L∗ = A/P , is used in the Rayleigh and Nusselt numbers.
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