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Unformatted text preview: 682 Chapter 22 Interphase Transport in Nonisothermal Mixtures The Chilton—Colburn jfactors, one for heat transfer and one for diffusion, are deﬁned as1 N a 2L3 .09?” The three—way analogy in Eq. 22.322 is accurate for Pr and Sc near unity (see Table
12.41) within the limitations mentioned after Eq. 22.317. 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 (22.325) 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 22.325 is the usual form of the ChiltonColbert: 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. Nonanalogous 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.
22.13 for the entire spherical surface as x,“ — an Wm = kmsroz) (22326) 1—3:.“ The mean mass transfer coefﬁcient, km, may be predicted from Eq. 22.321 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 (22327)
32,, = ax“ + x,.,) = gem? + 0) = 0.0124 (22328) ‘ T. H. Chilton and A. P. Colburn, Ind. Eng. Chem, 26, 11831187 (1934}. §22.3 Correlation of Binary Transfer Coefﬁcients in One Phase 681 If we can neglect the heat production by viscous dissipation in Eq.11.59 and if there
is no production of A by chemical reaction as in Eq. 19.511, 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. 22.35 and 6 and performing the integrations
and then timeaveraging give for forced convection Nu, = GtRe, Pr, L/D); Sh, = G(Re, Sc, L/D) (22.316,17l 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) (22318, 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. 14.45 and its masstransfer analog are:
Num = 2 + 0.60 Re”2 Prm'; Sh", = 2 + 0.60 Re”2 Sc”3 (22320, 2]) Equations 22320 and 21 are valid for constant surface temperature and composition, re
spectively, and for small net masstransfer rates. They may be applied to simultaneous
heat and mass transfer under restrictions (il(vi) given after Eq. 22.317. 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 = 0332 RE; ”2 (22322) ...
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