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Lecture_9 - 682 Chapter 22 Inter-phase Transport in...

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Unformatted text preview: 682 Chapter 22 Inter-phase Transport in Nonisothermal Mixtures The Chilton—Colburn j-factors, one for heat transfer and one for diffusion, are defined as1 N a 2L3 .09?” The three—way analogy in Eq. 22.3-22 is accurate for Pr and Sc near unity (see Table 12.4-1) within the limitations mentioned after Eq. 22.3-17. For flow around other objects, the friction factor part of the analogy is not valid because of the form drag, and even for flow in circular tubes the analogy with % flu, is only approximate (see §l4.4). Sh k: 2f3 ——‘r°‘—s "°‘( " ) (22.324) The Chilton—Colburn Analogy Evaporation from a Freely Falling Drop SOLUTION The more widely applicable empirical analogy j}, = jg = a function of Re, geometry, and boundary conditions (22.3-25) has proven to be useful for transverse flow around cylinders, flow through packed beds, and flow in tubes at high Reynolds numbers. For flow in ducts and packed beds, the “approach velocity” "0.. has to be replaced by the interstitial velocity or the superficial ve- locity. Equation 22.3-25 is the usual form of the Chilton-Colbert: analogy. it is evident from Eqs. 223—20 and 21, however, that the analogy is valid for flow around spheres only when Nu and Sh are replaced by (Nu - 2) and (Sh — 2). It would be very misleading to leave the impression that all mass transfer coeffi- cients can be obtained from the analogous heat transfer coefficient correlations. For mass transfer we encounter a much wider variety of boundary conditions and other ranges of the relevant variables. Non-analogous behavior is addressed in §§22.5—8. A spherical drop of water, 0.05 cm in diameter, is falling at a velocity of 215 cm/s through dry, still air at 1 atm pressure with no internal circulation. Estimate the instantaneous rate of evaporation from the drop, when the drop surface is at To = '70“F and the air (far from the drop) is at T” = 140?. The vapor pressure of water at 70°F is 0.0247 atm. Assume quasi- steady state conditions. Designate water as species A and air as species B. The solubility of air in water may be ne- glected, so that Wm = 0. Then assuming that the evaporation rate is small, we may write Eq. 22.1-3 for the entire spherical surface as x,“ — an Wm = kmsroz) (223-26) 1—3:.“ The mean mass transfer coefficient, km, may be predicted from Eq. 22.3-21 in the assumed ab- sence of internal circulation. I The film conditions needed for estimating the physical properties are obtained as follows: r, = to, + r...) = £670 + 140) = 105°F (223-27) 32,, = ax“ + x,.,) = gem? + 0) = 0.0124 (223-28) ‘ T. H. Chilton and A. P. Colburn, Ind. Eng. Chem, 26, 1183-1187 (1934}. §22.3 Correlation of Binary Transfer Coefficients in One Phase 681 If we can neglect the heat production by viscous dissipation in Eq.11.5-9 and if there is no production of A by chemical reaction as in Eq. 19.5-11, then the differential equa- tions for heat and mass transport are analogous along with the boundary conditions. It follows then that the dimensionless profiles of temperature and concentration (time smoothed, when necessary) are similar, l” = Hi, 6‘, 2, Re, Pr); 5%,, = F0", 9, 2, Re, Sc) (22.344, 15) with the same form of P in both systems. Thus, to get the concentration profiles from the temperature profiles, one replaces T by 5%,, and Pr by Sc. Finally, inserting the profiles into Eqs. 22.3-5 and 6 and performing the integrations and then time-averaging give for forced convection Nu, = GtRe, Pr, L/D); Sh, = G(Re, Sc, L/D) (22.3-16,17l Here G is the same function in both equations. The same formal expression is obtained for Nu,” Nut“, Nulac as well as for the corresponding Sherwood numbers. This important analogy permits one to write down a mass transfer correlation from the corresponding heat transfer correlation merely by replacing Nu by Sh, and Pr by Sc. The same can be done for any geometry and for both laminar and turbulent flow. Note, however, that to get this analogy one has to assume (i) constant physical properties, (ii) small net mass- transfer rates, (iii) no chemical reactions, (iv) no viscous dissipation heating, (v) no ab— sorption or emission of radiant energy, and (vi) no pressure diffusion, thermal diffusion, or forced diffusion. Some of these effects will be discussed in subsequent sections of this chapter; others will be treated in Chapter 24. For free convection around objects of any given shape, a similar analysis shows that Nu", = HtGr, Pr); Sh," = HfGr,” SC) (223-18, 19) Here H is the same function in both cases, and the Grashof numbers for both processes are defined analogously (see Table 222—1 for a summary of the analogous quantities for heat and mass transfer). To allow for the variation of physical properties in mass transfer systems, we extend the procedures introduced in Chapter ‘14 for heat transfer systems. That is, we generally evaluate the physical properties at some kind of mean film composition and tempera- ture, except for the viscosity ratio M's/Mo- We now give three illustrations of how to “translate” from heat transfer to mass transfer correlations: Forced Convection Around Spheres For forced convection around a solid sphere, Eq. 14.4-5 and its mass-transfer analog are: Num = 2 + 0.60 Re”2 Prm'; Sh", = 2 + 0.60 Re”2 Sc”3 (223-20, 2]) Equations 223-20 and 21 are valid for constant surface temperature and composition, re- spectively, and for small net mass-transfer rates. They may be applied to simultaneous heat and mass transfer under restrictions (il-(vi) given after Eq. 22.3-17. Forced Convection along a Flat Plate As another illustration of the use of analogies, we can cite the extension of Eq. 14.44 for the laminar boundary layer along a flat plate, to include mass transfer: Into: = loioc = ifioc = 0-332 RE; ”2 (223-22) ...
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