Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 46
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The addition of still more4.34CHAPTER FOURchannels decreases the spacing of each channel, so that the heat transfer coefficient falls, butthe total heat transfer continues to increase because the product of heat transfer coefficientand surface area, or equivalently the heat transfer per unit cross-sectional area of the channelq/Ax, increases. As the plate spacing decreases still further, q/Ax passes through a maximumand then falls. Levy et al. [178] and Raithby and Hollands [223] have shown that the boundary layers begin to interfere at 400 < Ra < 800.
From Eq. 4.53, q/Ax can be shown to passthrough a maximum atRa = Ramax = (24c-Ce2-(1/m)) 4/3 -- 60(4.54)Given R a m a x , the spacing S can be found that corresponds to this maximum q/Ax.In a related problem, Sparrow and Prakash [261] have shown that the heat transfer from aseries of parallel-plate channels can be substantially increased, for the same surface area, bybreaking the continuous vertical plates into a staggered array of discrete vertical plates.Uniform Heat Flux Parallel Plates.
If the heat fluxes are specified as q'l' and q'~, respectively, on the surfaces of the vertical plates (where q'l' > q'~ and both q'l' and q'~ are positive,denoting heat transfer from the plate to the fluid), the Nusselt number for the fully developedregime for air is given by [11]NUld = 0.29(Ra*) v2Ra* ~< 5(4.55)where the_terms are defined in Fig. 4.21.
The temperature difference that appears in Nu is, inthis case, Tw(1/2)- To., where Twtl/2)is the average wall temperature at the channel mid-height.In the laminar boundary layer regime, the Nusselt number relation is of the form of thatfor a vertical flat plateNu = cHe(Ra*)1/5102 ~< Ra* ~< 104(4.56)where He is given by Eq. 4.36a and Table 4.1 and c is given in the following text. Equation 4.56may apply well past the upper limit indicated. The following equation satisfactorily fits theresults of Aung et al.
[12] obtained for Ra* < 104:N U = [(NUfd) m 4-(cne(Ra*)l/5)m]1/mm =-3.5(4.57)The Nu value calculated from Eq. 4.57 provides the average temperature at the mid-height ofthe channel. For air, the analysis suggests c = 1.15, while the data of Sobel et al. [254] yieldc = 1.07; the latter value is recommended. These values may be compared to c = 1.00 for anisolated vertical plate in air (i.e., Eq. 4.36a).Bar-Cohen and Rohsenow [14] provide a relation, similar in form to Eq. 4.57, for the localNusselt number, from which the maximum plate temperature can be calculated.Extensive measurements for mercury (Pr --- 0.022) have been reported by Colwell andWelty [68] and Humphreys and Welty [147].
The data at this Prandtl number do not supportEq. 4.57, and the original references should be consulted.Sobel et al. [254] showed through measurements that the heat transfer from an array ofuniform heat flux parallel plates could be substantially increased by interrupting and staggering the plate surfaces.Isothermal Circular Channels.
Measurements and analyses by Elenbaas [88, 89] and Dyer[77] show that the heat transfer from isothermal cylindrical cooling channels (Fig. 4.20c) canbe represented by[( R a / "- - Ral/4)mll/mNu = [\--i--6-] + (cCem =-1.03(4.58)The nomenclature is defined in Fig. 4.22. The experiments for Pr = 0.71 (air) indicate c -- 1.17,while analysis gives 1.22. The value of Ra for the maximum heat transfer per unit of cross-NATURAL CONVECTIONAT4.35r = 2AlpPSpecified wol I temperatureSpecified wall fluxqr|1HNu =_-,__..Nu = -pH (Tw-Tm) k(Tw- Too) kRe = g B ( T w - T m ) r s r__uaHIRo.
= g,Bq ''r4yakr__HuiritRe-FIGURE 4.22 Naturalconvection through a cylindrical cooling channel.sectional area of the tube (for a given height H and temperature difference) is given by Eq.4.54 with 24 replaced by 16; this yields Ramax = 50 for air.Dyer [77] also presents results for air for the case when an unheated entry length Hi isadded to the tube as shown in Fig. 4.22. For fully developed flow, the 16 in Eqs.
4.54 and 4.58is, in this case, replaced by 16(1 + Hi~H). The heat flow is also reduced in the boundary layerregime by increasing Hi, and if Hi becomes sufficiently large, the throughflow is reduced anda large portion of the wall cooling is provided by a downward flow through the central portion of the top of the tube and a return flow upward along the tube walls (thermosiphonexchange).Uniform Flux Cylindrical Channels. The heat flow results from the analysis of Dyer [76]for air and for vertical circular cooling channels with uniform heat flux at the boundary can berepresented closely byNu ={(~)m}l/m+ (0.67(Ra*)~/5) m0.1 < Ra* < 105(4.59)where m = -1.7, Pr = 0.71 (air), and the remaining symbols are defined in Fig. 4.22. From thedefinition of Nu, the Nu value obtained from Eq.
4.59 provides the average wall temperature.Data from experiments in the range 5 < Ra* < 5 x 103 agree well with Eq. 4.59.Isothermal Channels of Other Shapes.Nu =For isothermal cooling channels of other shapes,r( Ra/m+ (cC, Ral4)mjqUmL\f-~-)_Ra ~< 104m =-1.5(4.60)where the nomenclature is defined in Fig. 4.22, f Re is the friction factor-Reynolds numberproduct in Table 4.4, and c has a value of about 1.20 for air and should decrease toward 1.0with increasing Pr. Equation 4.54 with 24 replaced by f Re yields Ra .
. . . the Rayleigh numberfor maximum heat transfer per unit of cross-sectional area for a channel of given length H andgiven Tw - T~. The relationship of the fully developed Nusselt number to f Re was originallypointed out by Elenbaas [88, 89].TABLE 4.4 Valuesof f Re for Internal Flow in Ducts of Various ShapesDuct shape.........fRe24016A013.314.225........60 °b4b2br'---lb',bb15.5518.70O15.05--I B IC/B = 0.918.234.36CHAPTER FOURExtendedSurfacesHeat transfer from each of the extended surfaces (fins), shown in Fig. 4.23, is now discussed.The prediction of the heat transfer requires a solution for complex 3D motion, so few analyses are as yet available.
Experimental data have been obtained exclusively using air.In many practical applications there is a significant temperature drop between the base ofthe fin and its tip, and this affects both the natural convection flow and heat transfer. Very little information is available on the coupling between fin conduction and fluid convection, sothat attention is restricted to isothermal fins. As a first approximation, the heat transfer coefficient for isothermal fins can be used for the case where the fins are not isothermal, becausethere is such a weak dependence on the temperature difference. Property values are to beevaluated at 0.5(Tw + T~) unless otherwise indicated.Rectangular Isothermal Fins on Vertical Surfaces.
Vertical rectangular fins, such as shownin Fig. 4.23a, are often used as heat sinks. If W/S _>5, Aihara [1] has shown that the heat transfer coefficient is essentially the same as for the parallel-plate channel (see the section on parallel isothermal plates). Also, as W/S ~ O, the heat transfer should approach that for a verticalfiat plate. Van De Pol and Tierney [270] proposed the following modification to the Elenbaasequation [88, 89] to fit the data of Welling and Wooldridge [283] in the range 0.6 < Ra < 100,Pr = 0.71, 0.33 < W/S < 4.0, and 42 < H/S < 10.6:Ra{F / 0 5 ~3/4]}Nu = - - ~ - - 1 - e x p [ - ~ ~ a a ) ](4.61a)24(1 0.483e -°'17/~*)W = {(1 + ix*/2)[1 + (1 - e-°83~*)(9.14~e 3 - 0.61)]} 3-whereIVt~p--IV iVIV IV/-'~T.4sl*- \,-,4sl-,--,-I ~ s j w2WSr = 2W + S " a * = S/Wq"SNu = ~( T .
- T ~ ) kq"rNu : (Tw-Too) kRo -IVq"LNu = ( T . - T = )g.B(Tw-Too) S 3rvaL(a)kg/3(T.-Too) L 3vaRe :g/3 (Tw-Too) r 3(4.61b)vaRe :(c)(b)/-,-Is ~s.I,'-'4s F-(T.-Too) kRo = g,B (T.-T~)S 3va(d)HNu --q"S(T,-Too) kRe =(e)gB (T.-Too)S3vaS--HNu =-~S(Tw-Tm) k'~FIGURE 4.23 Flow configurations and nomenclature for various open cavity problems.d= ~ . Re =g# (T.-T=) S3 Sva(f)DN A T U R A L CONVECTION,0,..37mand where S is dimensionless and equal to -4.65S (for S in cm) or-11.8S (for S in inches).Other nomenclature is defined in Fig. 4.23a.For a given base plate area there are two fin spacings, $1 and $2, of particular interest.
If thefin spacing is decreased, starting from a large value, the heat transfer coefficient remains relatively constant until the spacing $1 (which corresponds to Ra = R a l ) is reached, at which itbegins to fall rapidly because of fin interference. As the spacing is decreased within a specified volume, more fins are added to the base plate, thereby increasing the total surface areafor heat transfer. Since total heat flow is proportional to the product of heat transfer coefficient and surface area, decreasing the spacing below S1 still improves the total heat transferuntil the spacing $2 (which corresponds to Ra = Ramax) is reached; below $2, the total heat transfer falls.
For long fins (0~* << 1), S1 and $2 will fall to roughly the same values as for parallelplate channels: Ral --- 600, Ramax = 60. For short fins, these spacings can be established fromEq. 4.61.Rectangular Isothermal Fins on Horizontal Surfaces. The heat transfer from rectangularfinned surfaces such as shown in Fig.
4.23b (upward-facing for Tw > T= or downward-facing forTw < T=) has been measured by Jones and Smith [150], Starner and McManus [263], and Harahap and McManus [120]. For a given fin width, W = 0.254 m (0.833 ft), Jones and Smith wereable to correlate their measured heat transfer to within about +_25 percent on an Nu-Ra plot.The following equation closely represents this correlation over the data range 2 x 102 < Ra <6 x 105, Pr = 0.71, 0.026 < H/W < 0.19, and 0.016 < S/W < 0.20:Nu =[(Ra)m]500+(0.081 Ra°39)mql/mm = -2(4.62)This simple equation ignores the effect of the geometric parameters H/S and H/W. While H/Sdoes not appear to play a strong role, H/W is known to have significant effect.A parametric study using Eq. 4.62 shows that, for a given base area and temperature difference, the curve of total heat transfer versus fin spacing displays a sharp maximum for highfins (large H) and a less well-defined peak for short fins.
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