The CRC Handbook of Mechanical Engineering. Chapter 4. Heat and Mass Transfer (776127), страница 12
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With uniformheat flux, the surface and fluid temperatures vary linearly except in the entrance region where the higherheat transfer coefficient leads to a smaller difference between the surface and fluid temperatures. Thevariation of the fluid temperature in the two cases is shown in Figure 4.2.24.Convective Heat Transfer in Noncircular TubesLaminar Flows: The Nusselt numbers for laminar flows have been analytically determined for differentnoncircular ducts.
Some of them can be found in Kakac et al. (1987), Kays and Crawford (1993), and© 1999 by CRC Press LLC4-51Heat and Mass TransferFIGURE 4.2.23 Variation of h with Tb in 1-, 2-, and 4-cm-diameter tubes with water flow rates of 0.2 kg/sec and0.4 kg/sec with uniform surface temperature.FIGURE 4.2.24 Variation of fluid temperature in a tube with (a) uniform surface temperature and (b) uniform heatflux.Burmeister (1993). A few of the results are given below. The characteristic length for forming theReynolds number and Nusselt number is the hydraulic mean diameter defined asdh =4 cross-sectional areawetted perimeterInfinite parallel plates: a = spacing between plates, dh = 2aBoth plates maintained at uniform and equal temperatures: Nu = 7.54Both plates with imposed uniform and equal heat fluxes: Nu = 8.24Rectangular ducts: a = longer side, b = shorter side, dh = 2ab/(a + b)b/aUniform surface temperatureUniform heat flux*10.70.50.250.1252.983.613.083.733.394.124.445.335.66.49Equilateral triangle: dh = a/31/2, a = length of each sideUniform surface temperature: Nu = 2.35Uniform surface heat flux:* Nu = 3.0Coaxial tubes: With coaxial tubes many different cases arise — each tube maintained at uniform butdifferent temperatures, each tube subjected to uniform but different heat fluxes (an insulated*Uniform axial heat flux but circumferentially uniform surface temperature.© 1999 by CRC Press LLC4-52Section 4surface is a special case of imposed heat flux being zero), or a combinations of uniform surfacetemperature of one tube and heat flux on the other.
The manner in which the heat transfercoefficient is determined for uniform but different heat fluxes on the two tubes is described below.Define:dh = 2(ro - ri )qi¢¢= hi (Ti - Tb )qo¢¢ = 0Nu i =Nu ii =hi dhkhi dhkr * = ri roqo¢¢ = ho (To - Tb )Nu o =Nu oo =and qi¢¢= 0ho dhkho dhkThenNu i =TABLE 4.2.6Nu iiq ¢¢1 - o q*iqi¢¢and Nu o =Nu ooq ¢¢1 - i q*oqo¢¢(4.2.106)Values for Use with Equation (4.2.106)*NuiiNuooqi*qo*0.050.10.20.40.60.81.017.8111.918.4996.5835.9125.585.3854.7924.8344.8834.9795.0995.245.3852.181.3830.9050.6030.4730.4010.3460.02940.05620.10410.18230.24550.2990.346rSome of the values needed for the computations of Nui and Nuo (taken from Kays and Crawford,1993) are given in the Table 4.2.6.For a more detailed information on heat transfer and friction factors for laminar flows in noncirculartubes refer to Kakac et al.
(1987).Turbulent Flows: For noncircular tubes, estimates of the convective heat transfer coefficient can beobtained by employing equations for circular tubes with dh replacing d in the computations of theReynolds and Nusselt numbers. To determine the heat transfer coefficients in developing regions andfor more-accurate values with turbulent flows in noncircular tubes refer to Kakac et al.
(1987) and thereferences in that book.Mixed ConvectionIf the fluid velocity is low, the effect of natural convection becomes significant and the heat transfer ratemay be increased or decreased by natural convection. From a review of experimental results, Metaisand Eckert (1964) developed maps to delineate the different regimes where one or the other mode isdominant and where both are significant. Figures 4.2.25 and 4.2.26 show the relative significance ofnatural and forced convection in vertical and horizontal tubes. The maps are applicable for 10–2 < Pr(d/L)< 1 where d and L are the diameter and the axial length of the tube.
The maps show the limits of forcedand natural convection regimes. The limits are delineated “in such a way that the actual heat flux underthe combined influence of the forces does not deviate by more than 10 percent from the heat flux that© 1999 by CRC Press LLCHeat and Mass Transfer4-53FIGURE 4.2.25 Map delineating forced, mixed, and natural convection — vertical tubes.FIGURE 4.2.26 Map delineating forced, mixed, and natural convection — horizontal tubes.would be caused by the external forces alone or by the body forces alone.” The Grashof number is basedon the diameter of the tube.For flows in horizontal tubes, correlations were developed for the mixed convection regime inisothermal tubes by Depew and August (1971) and for uniform heat flux by Morcos and Bergles (1975).Uniform Surface Temperature.
Fully developed velocity profile, developing temperature profile:L/d < 28.425 < Gz < 7120.7 ´ 105 < Gr < 9.9 ´ 105ms = dynamic viscosity, evaluated at the wall temperatureAll other properties at the average bulk temperature of the fluid© 1999 by CRC Press LLC4-54Section 4Gz =[ṁC pGr = gbDT d 3 n 2kL(Nu d = 1.75 Gz + 0.12 GzGr 1 3 Pr 0.36)0.88 1 3](mbms )0.14(4.2.107)Uniform Heat Flux. Properties at (Ts + Tb)/2: 3 ´ 104 < Ra < 106, 4 < Pr < 175, 2 < hd2/(kwt) < 66, kw =tube wall thermal conductivity, t = tube wall thickness.( )Grd* = gbd 4 qw¢¢ n 2 kPw = kd (kw t )Ra d = gbDT d 3 Pr n 20.265 2 üìéù ïæ Grd* Pr 1.35 öï2ú ýNu d = í4.36 + ê0.145ç÷0.25Pwêú ïèøïëû þî0.5(4.2.108)In Equation (4.2.107) and (4.2.108) evaluate fluid properties at the arithmetic mean of the bulk and walltemperatures.NomenclatureAsddhfhkLeLe,thNudNuiiNuooPrq²qi¢¢qo¢¢RedTbTsvmmsr— surface area— diameter— hydraulic mean diameter— friction factor— convective heat transfer coefficient— fluid thermal conductivity— hydrodynamic entrance length— thermal entrance length— Nusselt number— Nusselt number with only inner tube heated— Nusselt number with only outer tube heated— Prandtl number— heat flux— heat flux on the inner tube surface— heat flux on the outer tube surface— Reynolds number (rvd/m)— bulk temperature— surface temperature— average fluid velocity— dynamic viscosity— dynamic viscosity at surface temperature— fluid density© 1999 by CRC Press LLCHeat and Mass Transfer4-55ReferencesBurmeister, L.C.
1993. Convective Heat Transfer, 2nd ed., Wiley-Interscience, New York.Depew, C.A. and August, S.E. 1971. Heat transfer due to combined free and forced convection in ahorizontal and isothermal tube, Trans. ASME 93C, 380.Dittus, F.W. and Boelter, L.M.K. 1930. Heat transfer in automobile radiators of the tubular type, Univ.Calif. Pub. Eng., 13, 443.Gnielinsky, V. 1976. New equations for heat and mass transfer in turbulent pipe channel flow, Int. Chem.Eng., 16, 359.Gnielinsky, V.
1990. Forced convection in ducts, in Handbook of Heat Exchanger Design, Hewitt, G.F.,Ed., Begell House/Hemisphere, New York.Kakac, S., Shah, R.K., and Win Aung, Eds. 1987. Handbook of Single-Phase Convective Heat Transfer,Wiley-Interscience, New York.Kays, W.M. and Crawford, M.E. 1993.
Convective Heat and Mass Transfer, 3rd ed., McGraw-Hill, NewYork.Metais, B. and Eckert, E.R.G. 1964. Forced, mixed, and free convection regimes, Trans. ASME 86C, 295.Morcos, S.M. and Bergles, A.E. 1975. Experimental investigation of combined forced and free laminarconvection in a horizontal tube, Trans. ASME 97C, 212.Sieder, E.N. and Tate, C.E. 1936. Heat transfer and pressure drop of liquids in tubes, Ind. Eng. Chem.,28, 1429.Sleicher, C.A. and Rouse, M.W. 1976. A convenient correlation for heat transfer to constant and variableproperty fluids in turbulent pipe flow, Int. J.
Heat Mass Transfer, 18, 677.© 1999 by CRC Press LLC4-56Section 44.3 Radiation*Michael F. ModestNature of Thermal RadiationAll materials continuously emit and absorb radiative energy by lowering or raising their molecular energylevels. This thermal radiative energy may be viewed as consisting of electromagnetic waves or of masslessenergy parcels, called photons. Electromagnetic waves travel through any medium at the speed of lightc, which is c0 = 2.998 ´ 108 m/sec in vacuum and approximately the same in most gases such as air andcombustion products.
These are characterized by their wavelength l or frequency n, which are related by(4.3.1)n=c lThe strength and wavelengths of emission and absorption depend on the temperature and nature of thematerial.The ability of photons to travel unimpeded through vacuum and gases makes thermal radiation thedominant mode of heat transfer in vacuum, low-pressure environments, and outer space applications(due to the near absence of conduction and convection). Its temperature dependence [as given by Equation(4.3.3) below] on the other hand, guarantees that radiative heat transfer is of utmost importance in hightemperature applications (including solar radiation: with the sun being a high-temperature heat sourceat an effective temperature of Tsun = 5762 K).When an electromagnetic wave traveling through a gas (or vacuum) strikes the surface of a medium,the wave may be partly or totally reflected, and any nonreflected part will penetrate into the medium.If a wave passes through a medium without any attenuation, the material is called transparent.
A bodywith partial attenuation is known as semitransparent, and a body through which none of the incomingradiation penetrates is called opaque. Most gases are rather transparent to radiation (except for narrowspectral regions, called absorption bands), while most solids tend to be strong absorbers for mostwavelengths, making them opaque over a distance of a few nanometers (electrical conductors, i.e., metals)to a few micrometers (ceramics, semiconductors), or more (dielectrics).Blackbody RadiationThe total amount of radiative energy emitted from a surface into all directions above it is termed emissivepower; we distinguish between spectral (at a given wavelength l, per unit wavelength) and total(encompassing all wavelengths) emissive power.
The magnitude of emissive power depends on wavelength l, temperature T, and a surface property, called emissivity e, which relates the ability of a surfaceto emit radiative energy to that of an ideal surface, which emits the maximum possible energy (at agiven wavelength and temperature). Such an ideal surface is known as a “blackbody” or “black surface,”since it absorbs all incoming radiation; i.e., it reflects no radiation and is, therefore, invisible (“black”)to the human eye.
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