Page 702 - Bird R.B. Transport phenomena

682 Chapter 22 Interphase Transport in Nonisothermal Mixtures

The Chilton-Colburn /-factors, one for heat transfer and one for diffusion, are defined as 1

Nu 1 (
7H,1OC — ,1/3 (22.3-23)
RePr pC v x
p
2/3
Sh loc
]D,\oc ~ (22.3-24)
R e S c i/3 cv»
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 \f\ oc is only approximate (see §14.4).

The Chilton-Colburn Analogy
The more widely applicable empirical analogy

JH = )D = 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" v*> has to be replaced by the interstitial velocity or the superficial ve-
locity. Equation 22.3-25 is the usual form of the Chilton-Colburn analogy. It is evident
from Eqs. 22.3-20 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.

EXAMPLE 22.3-1 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 a evaporation from the drop, when the drop surface is at T = 70°F and the air (far from the
o
Freely Falling Drop drop) is at Too = 140°F. The vapor pressure of water at 70°F is 0.0247 atm. Assume quasi-
steady state conditions.

SOLUTION Designate water as species A and air as species B. The solubility of air in water may be ne-
glected, so that W = 0. Then assuming that the evaporation rate is small, we may write Eq.
B0
22.1-3 for the entire spherical surface as
X
W = k A0 (22.3-26)
A0
- x
A
The mean mass transfer coefficient, k , may be predicted from Eq. 22.3-21 in the assumed ab-
xm
sence of internal circulation.
The film conditions needed for estimating the physical properties are obtained as
follows:
7} = l(T 0 + TJ = \(70 + 140) = 105°F (22.3-27)
*Af = l(x + x J = ^(0.0247 + 0) = 0.0124 (22.3-28)
A0 A

1
Т. Н. Chilton and A. P. Colburn, Ind. Eng. Chem., 26,1183-1187 (1934).