Computational Fluid Mechanics and Heat Transfer Third Edition_12 docx

418 Natural convection in single-phase fluids and during film condensation §8.4 g-level Nu D h = Nu D  0.0297 0.005  Q = πD h∆T 10 −6 0.483 2.87 W/m 2 K4.51 W/m of tube 10 −5 0.547 3.25 W/m 2 K5.10 W/m of tube 10 −4 0.648 3.85 W/m 2 K6.05 W/m of tube 10 −2 1.086 6.45 W/m 2 K10.1 W/m of tube The numbers in the rightmost column are quite low. Cooling is clearly inefficient at these low gravities. Natural convection from vertical cylinders The heat transfer from the wall of a cylinder with its axis running verti- cally is the same as that from a vertical plate, so long as the thermal b.l. is thin. However, if the b.l. is thick, as is indicated in Fig. 8.7, heat transfer will be enhanced by the curvature of the thermal b.l. This correction was first considered some years ago by Sparrow and Gregg, and the analysis was subsequently extended with the help of more powerful numerical methods by Cebeci [8.11]. Figure 8.7 includes the corrections to the vertical plate results that were calculated for many Pr’s by Cebeci. The left-hand graph gives a correction that must be multiplied by the local flat-plate Nusselt number to get the vertical cylinder result. Notice that the correction increases when the Grashof number decreases. The right-hand curve gives a similar correction for the overall Nusselt number on a cylinder of height L. Notice that in either situation, the correction for all but liquid metals is less than 1% if D/(x or L) < 0.02 Gr 1/4 x or L . Heat transfer from general submerged bodies Spheres. The sphere is an interesting case because it has a clearly speci- fiable value of Nu D as Ra D → 0. We look first at this limit. When the buoyancy 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 4 §8.4 Natural convection in other situations 419 Figure 8.7 Corrections for h and h on vertical isother- mal plates to make them apply to vertical isothermal cylin- ders [8.11]. in Table 5.4, we compute in this case lim Ra D →0 Nu D = Q A∆T D k = k∆T(S)D 4π(D/2) 2 ∆Tk = 4π(D/2) 4π(D/4) = 2 (8.32) Every proper correlation of data for heat transfer from spheres there- fore has the lead constant, 2, in it. 5 A typical example is that of Yuge [8.12] for spheres immersed in gases: Nu D = 2 +0.43 Ra 1/4 D , Ra D < 10 5 (8.33) A more complex expression [8.13] encompasses other Prandtl numbers: Nu D = 2 + 0.589 Ra 1/4 D  1 +(0.492/Pr) 9/16  4/9 Ra D < 10 12 (8.34) This result has an estimated uncertainty of 5% in air and an rms error of about 10% at higher Prandtl numbers. 5 It is important to note that while Nu D for spheres approaches a limiting value at small Ra D , no such limit exists for cylinders or vertical surfaces. The constants in eqns. (8.27) and (8.30) are not valid at extremely low values of Ra D . 420 Natural convection in single-phase fluids and during film condensation §8.4 Rough estimate of Nu for other bodies. In 1973 Lienhard [8.14] noted that, for laminar convection in which the b.l. does not separate, the ex- pression Nu τ  0.52 Ra 1/4 τ (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 length of travel, τ, of fluid in the unseparated b.l. In the case of spheres without separation, for example, τ = πD/2, the distance from the bottom to the top around the circumference. Thus, for spheres, eqn. (8.35) becomes hπD 2k = 0.52  gβ∆T(πD/2) 3 να  1/4 or hD k = 0.52  2 π  π 2  3/4  gβ∆TD 3 να  1/4 or Nu D = 0.465 Ra 1/4 D This is within 8% of Yuge’s correlation if Ra D remains fairly large. Laminar heat transfer from inclined and horizontal plates In 1953, Rich [8.15] showed that heat transfer from inclined plates could be predicted by vertical plate formulas if the component of the gravity vector along the surface of the plate was used in the calculation of the Grashof number. Thus, the heat transfer rate decreases as (cos θ) 1/4 , where θ is the angle of inclination measured from the vertical, as shown in Fig. 8.8. Subsequent studies have shown that Rich’s result is substantially cor- rect for the lower surface of a heated plate or the upper surface of a cooled plate. For the upper surface of a heated plate or the lower surface of a cooled plate, the boundary layer becomes unstable and separates at a relatively low value of Gr. Experimental observations of such instabil- ity have been reported by Fujii and Imura [8.16], Vliet [8.17], Pera and Gebhart [8.18], and Al-Arabi and El-Riedy [8.19], among others. §8.4 Natural convection in other situations 421 Figure 8.8 Natural convection b.l.’s on some inclined and hor- izontal surfaces. The b.l. separation, shown here for the unsta- ble cases in (a) and (b), occurs only at sufficiently large values of Gr. In the limit θ = 90 ◦ — a horizontal plate — the fluid flow above a hot plate or below a cold plate must form one or more plumes, as shown in Fig. 8.8c and d. In such cases, the b.l. is unstable for all but small Rayleigh numbers, and even then a plume must leave the center of the plate. The unstable cases can only be represented with empirical correlations. Theoretical considerations, and experiments, show that the Nusselt number for laminar b.l.s on horizontal and slightly inclined plates varies as Ra 1/5 [8.20, 8.21]. For the unstable cases, when the Rayleigh number exceeds 10 4 or so, the experimental variation is as Ra 1/4 , and once the flow is fully turbulent, for Rayleigh numbers above about 10 7 , experi- 422 Natural convection in single-phase fluids and during film condensation §8.4 ments show a Ra 1/3 variation of the Nusselt number [8.22, 8.23]. In the latter case, both Nu L and Ra 1/3 L are proportional to L, so that the heat transfer coefficient is independent of L. Moreover, the flow field in these situations is driven mainly by the component of gravity normal to the plate. Unstable Cases: For the lower side of cold plates and the upper side of hot plates, the boundary layer becomes increasingly unstable as Ra is increased. • For inclinations θ  45 ◦ and 10 5  Ra L  10 9 , replace g with g cos θ in eqn. (8.27). • For horizontal plates with Rayleigh numbers above 10 7 , nearly iden- tical results have been obtained by many investigators. From these results, Raithby and Hollands propose [8.13]: Nu L = 0.14 Ra 1/3 L  1 +0.0107 Pr 1 +0.01 Pr  , 0.024  Pr  2000 (8.36) This formula is consistent with available data up to Ra L = 2 ×10 11 , and probably goes higher. As noted before, the choice of length- scale L is immaterial. Fujii and Imura’s results support using the above for 60 ◦ θ90 ◦ with g in the Rayleigh number. For high Ra in gases, temperature differences and variable proper- ties effects can be large. From experiments on upward facing plates, Clausing and Berton [8.23] suggest evaluating all gas properties at a reference temperature, in kelvin, of T ref = T w −0.83 ( T w −T ∞ ) for 1 T w /T ∞  3. • For horizontal plates of area A and perimeter P at lower Rayleigh numbers, Raithby and Hollands suggest [8.13] Nu L ∗ = 0.560 Ra 1/4 L ∗  1 +(0.492/Pr) 9/16  4/9 (8.37a) where, following Lloyd and Moran [8.22], a characteristic length- scale L ∗ = A/P, is used in the Rayleigh and Nusselt numbers. If §8.4 Natural convection in other situations 423 Nu L ∗  10, the b.l.s will be thick, and they suggest correcting the result to Nu corrected = 1.4 ln  1 +1.4  Nu L ∗  (8.37b) These equations are recommended 6 for 1 < Ra L ∗ < 10 7 . • In general, for inclined plates in the unstable cases, Raithby and Hollands [8.13] recommend that the heat flow be computed first using the formula for a vertical plate with g cosθ and then using the formula for a horizontal plate with g sinθ (i.e., the component of gravity normal to the plate) and that the larger value of the heat flow be taken. Stable Cases: For the upper side of cold plates and the lower side of hot plates, the flow is generally stable. The following results assume that the flow is not obstructed at the edges of the plate; a surrounding adiabatic surface, for example, will lower h [8.24, 8.25]. • For θ<88 ◦ and 10 5  Ra L  10 11 , eqn. (8.27) is still valid for the upper side of cold plates and the lower side of hot plates when g is replaced with g cosθ in the Rayleigh number [8.16]. • For downward-facing hot plates and upward-facing cold plates of width L with very slight inclinations, Fujii and Imura give: Nu L = 0.58 Ra 1/5 L (8.38) This is valid for 10 6 < Ra L < 10 9 if 87 ◦ θ90 ◦ and for 10 9  Ra L < 10 11 if 89 ◦ θ90 ◦ .Ra L is based on g (not g cosθ). Fujii and Imura’s results are for two-dimensional plates—ones in which infinite breadth has been approximated by suppression of end effects. For circular plates of diameter D in the stable horizontal configu- rations, the data of Kadambi and Drake [8.26] suggest that Nu D = 0.82 Ra 1/5 D Pr 0.034 (8.39) 6 Raithby and Hollands also suggest using a blending formula for 1 < Ra L ∗ < 10 10 Nu blended,L ∗ =   Nu corrected  10 +  Nu turb  10  1/10 (8.37c) in which Nu turb is calculated from eqn. (8.36) using L ∗ . The formula is useful for numerical progamming, but its effect on h is usually small. 424 Natural convection in single-phase fluids and during film condensation §8.4 Natural convection with uniform heat flux When q w is specified instead of ∆T ≡ (T w − T ∞ ), ∆T becomes the un- known dependent variable. Because h ≡ q w /∆T , the dependent variable appears in the Nusselt number; however, for natural convection, it also appears in the Rayleigh number. Thus, the situation is more complicated than in forced convection. Since Nu often varies as Ra 1/4 , we may write Nu x = q w ∆T x k ∝ Ra 1/4 x ∝ ∆T 1/4 x 3/4 The relationship between x and ∆T is then ∆T = Cx 1/5 (8.40) where the constant of proportionality C involves q w and the relevant physical properties. The average of ∆T over a heater of length L is ∆T = 1 L  L 0 Cx 1/5 dx = 5 6 C (8.41) We plot ∆T/C against x/L in Fig. 8.9. Here, ∆T and ∆T ( x/L = ½ ) are within 4% of each other. This suggests that, if we are interested in average values of ∆T, we can use ∆T evaluated at the midpoint of the plate in both the Rayleigh number, Ra L , and the average Nusselt number, Nu L = q w L/k∆T . Churchill and Chu, for example, show that their vertical plate correlation, eqn. (8.27), represents q w = constant data exceptionally well in the range Ra L > 1 when Ra L is based on ∆T at the middle of the plate. This approach eliminates the variation of ∆T with x from the calculation, but the temperature difference at the middle of the plate must still be found by iteration. To avoid iterating, we need to eliminate ∆T from the Rayleigh number. We can do this by introducing a modified Rayleigh number, Ra ∗ x , defined as Ra ∗ x ≡ Ra x Nu x ≡ gβ∆Tx 3 να q w x ∆Tk = gβq w x 4 kνα (8.42) For example, in eqn. (8.27), we replace Ra L with Ra ∗ L  Nu L . The result is Nu L = 0.68 +0.67  Ra ∗ L  1/4  Nu 1/4 L  1 +  0.492 Pr  9/16  4/9 §8.4 Natural convection in other situations 425 Figure 8.9 The mean value of ∆T ≡ T w − T ∞ during natural convection. which may be rearranged as Nu 1/4 L  Nu L −0.68  = 0.67  Ra ∗ L  1/4  1 +(0.492/Pr) 9/16  4/9 (8.43a) When Nu L  5, the term 0.68 may be neglected, with the result Nu L = 0.73  Ra ∗ L  1/5  1 +(0.492/Pr) 9/16  16/45 (8.43b) Raithby and Hollands [8.13] give the following, somewhat simpler corre- lations for laminar natural convection from vertical plates with a uniform wall heat flux: Nu x = 0.630  Ra ∗ x Pr 4 +9 √ Pr +10 Pr  1/5 (8.44a) Nu L = 6 5  Ra ∗ L Pr 4 +9 √ Pr +10 Pr  1/5 (8.44b) These equations apply for all Pr and for Nu  5 (equations for lower Nu or Ra ∗ are given in [8.13]). 426 Natural convection in single-phase fluids and during film condensation §8.4 Some other natural convection problems There are many natural convection situations that are beyond the scope of this book but which arise in practice. Natural convection in enclosures. When a natural convection process occurs within a confined space, the heated fluid buoys up and then fol- lows the contours of the container, releasing heat and in some way re- turning to the heater. This recirculation process normally enhances heat transfer beyond that which would occur by conduction through the sta- tionary fluid. These processes are of importance to energy conserva- tion processes in buildings (as in multiply glazed windows, uninsulated walls, and attics), to crystal growth and solidification processes, to hot or cold liquid storage systems, and to countless other configurations. Survey articles on natural convection in enclosures have been written by Yang [8.27], Raithby and Hollands [8.13], and Catton [8.28]. Combined natural and forced convection. When forced convection along, say, a vertical wall occurs at a relatively low velocity but at a relatively high heating rate, the resulting density changes can give rise to a super- imposed natural convection process. We saw in footnote 2 on page 402 that Gr 1/2 L plays the role of of a natural convection Reynolds number, it follows that we can estimate of the relative importance of natural and forced convection can be gained by considering the ratio Gr L Re 2 L = strength of natural convection flow strength of forced convection flow (8.45) where Re L is for the forced convection along the wall. If this ratio is small compared to one, the flow is essentially that due to forced convection, whereas if it is large compared to one, we have natural convection. When Gr L  Re 2 L is on the order of one, we have a mixed convection process. It should be clear that the relative orientation of the forced flow and the natural convection flow matters. For example, compare cool air flow- ing downward past a hot wall to cool air flowing upward along a hot wall. The former situation is called opposing flow and the latter is called as- sisting flow. Opposing flow may lead to boundary layer separation and degraded heat transfer. Churchill [8.29] has provided an extensive discussion of both the con- ditions that give rise to mixed convection and the prediction of heat trans- §8.4 Natural convection in other situations 427 fer for it. Review articles on the subject have been written by Chen and Armaly [8.30] and by Aung [8.31]. Example 8.5 A horizontal circular disk heater of diameter 0.17 m faces downward in air at 27 ◦ C. If it delivers 15 W, estimate its average surface temper- ature. Solution. We have no formula for this situation, so the problem calls for some judicious guesswork. Following the lead of Churchill and Chu, we replace Ra D with Ra ∗ D /Nu D in eqn. (8.39):  Nu D  6/5 =  q w D ∆Tk  6/5 = 0.82  Ra ∗ D  1/5 Pr 0.034 so ∆T = 1.18 q w D  k  gβq w D 4 kνα  1/6 Pr 0.028 = 1.18  15 π(0.085) 2  0.17 0.02614  9.8[15/π(0.085) 2 ]0.17 4 300(0.02164)(1.566)(2.203)10 −10  1/6 (0.711) 0.028 = 140 K In the preceding computation, all properties were evaluated at T ∞ . Now we must return the calculation, reevaluating all properties except β at 27 +(140/2) = 97 ◦ C: ∆T corrected = 1.18 661(0.17)/0.03104  9.8[15/π(0.085) 2 ]0.17 4 300(0.03104)(3.231)(2.277)10 −10  1/6 (0.99) = 142 K so the surface temperature is 27 + 142 = 169 ◦ C. That is rather hot. Obviously, the cooling process is quite ineffec- tive in this case. [...]... Ing., 38:872, 1915 References [8.2] C J Sanders and J P Holman Franz Grashof and the Grashof Number Int J Heat Mass Transfer, 15:562–563, 1972 [8.3] S W Churchill and H H S Chu Correlating equations for laminar and turbulent free convection from a vertical plate Int J Heat Mass Transfer, 18:1323–1329, 1975 [8.4] S Goldstein, editor Modern Developments in Fluid Mechanics, volume 2, chapter 14 Oxford University... horizontal cylinder Int J Heat Mass Transfer, 18:1049–1053, 1975 [8.11] T Cebeci Laminar-free-convective -heat transfer from the outer surface of a vertical slender circular cylinder In Proc Fifth Intl Heat Transfer Conf., volume 3, pages 15–19 Tokyo, September 3–7 1974 [8.12] T Yuge Experiments on heat transfer from spheres including combined forced and natural convection J Heat Transfer, Trans ASME, Ser... Eckert and R M Drake, Jr Analysis of Heat and Mass Transfer Hemisphere Publishing Corp., Washington, D.C., 1987 [8.6] A Bejan and J L Lage The Prandtl number effect on the transition in natural convection along a vertical surface J Heat Transfer, Trans ASME, 112:787–790, 1990 [8.7] E M Sparrow and J L Gregg The variable fluid-property problem in free convection In J P Hartnett, editor, Recent Advances in Heat. .. single-phase fluids and during film condensation heat transfer cofficient of the tube b The tube is cooled by cold water flowing through it and the thin wall of the copper tube offers negligible thermal resistance If the bulk temperature of the water is 275 K at a location where the outside surface of the tube is at 290 K, what is the heat transfer coefficient inside the tube? c Using the heat transfer coefficients... constant-wall-temperature heater in calculating the convective part of the heat transfer? The surroundings are at 20◦ C and the surrounding room is virtually black 8.44 A vertical plate, 11.6 m long, condenses saturated steam at 1 atm We want to be sure that the film stays laminar What is the lowest allowable plate temperature, and what is q at this temperature? 8.45 A straight horizontal fin exchanges heat by laminar... heat that is less than 2% of the latent heat The Jakob number is accordingly small in most cases of practical interest, and it turns out that sensible heat can often be neglected (There are important 7 Note that, throughout this section, k, µ, cp , and Pr refer to properties of the liquid, rather than the vapor Film condensation §8.5 429 exceptions to this.) The same is true of the role of the Prandtl... single-phase fluids and during film condensation Figure 8.11 §8.5 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 to cool the inner parts of the film below Tsat Rohsenow’s calculation was, in part, an assessment of Nusselt’s linear-temperature-profile assumption, and it led... been asked to design a vertical wall panel heater, 1.5 m high, for a dwelling What should the heat flux be if no part of the wall should exceed 33◦ C? How much heat will be added to the room if the panel is 7 m in width? 8.23 A 14 cm high vertical surface is heated by condensing steam at 1 atm If the wall is kept at 30◦ C, how would the average Problems 447 heat transfer coefficient change if ammonia, R22,... acetone were used instead of steam to heat it? How would the heat flux change? (Data for methanol and acetone must be obtained from sources outside this book.) 8.24 A 1 cm diameter tube extends 27 cm horizontally through a region of saturated steam at 1 atm The outside of the tube can be maintained at any temperature between 50◦ C and 150◦ C Plot the total heat transfer as a function of tube temperature... heat transfer from spheres including combined forced and natural convection J Heat Transfer, Trans ASME, Ser C, 82(1):214, 1960 [8.13] G D Raithby and K G T Hollands Natural convection In W M Rohsenow, J P Hartnett, and Y I Cho, editors, Handbook of Heat Transfer, chapter 4 McGraw-Hill, New York, 3rd edition, 1998 453 . space, the heated fluid buoys up and then fol- lows the contours of the container, releasing heat and in some way re- turning to the heater. This recirculation process normally enhances heat transfer. Yuge’s correlation if Ra D remains fairly large. Laminar heat transfer from inclined and horizontal plates In 1953, Rich [8.15] showed that heat transfer from inclined plates could be predicted by. introduce Ja in a corrected latent heat but instead showed its influence directly. Figure 8 .12 displays two figures from the Sparrow and Gregg paper. The first shows heat transfer results plotted in the