Page 694 - Bird R.B. Transport phenomena

674 Chapter 22 Interphase Transport in Nonisothermal Mixtures

In each of these systems, there will be both heat and mass transfer at the interface,
and each of these fluxes will have a molecular (diffusive) and a convective term (here we
have moved the convective term to the left side of the equation):

" x (N A0 + N ) = -[ c® AB ^ (22.1-1)
A0
B0
(22.1-2)

These equations are just Eq. 18.0-1 and Eq. 19.3-6 written at the mass transfer interface
(y = 0). They describe the interphase molar flux of species A and the interphase flux of
energy (excluding the kinetic energy and the contribution from [т • v]). Both N AQ and e 0
are defined as positive for transfer into the local phase except in §22.4 where the fluxes in
each phase are defined as positive for transfer toward the liquid.
In Chapter 14 we defined the heat transfer coefficient in the absence of mass transfer
by Eq. 14.1-1 (Q = hA AT). For surfaces with mass and heat transfer, Eqs. 22.1-1 and 2
suggest that the following definitions are appropriate:

™ло " x (W + W ) = k A Ax (22.1-3)
A0 A0 B0 xA A
E - (Wjl + W H ) = hA AT (22.1-4)
o A0 B0 B0
Here W A0 is the number of moles of species A per unit time going through the transfer
surface at у = 0, and £ 0 is the total amount of energy going through the surface. The
transfer coefficients k xA and h are not defined until the area A and the driving forces Ax A
and AT have been specified. All the comments in Chapter 14 regarding these definitions
may be taken over in this chapter, with the result that a subscript 1, In, a, m, or loc can be
added to make clear the type of driving force that is used. In this chapter, however, we
shall mainly use the local transfer coefficients and occasionally the mean transfer coeffi-
cients. Also, in this chapter, molar fluxes of the species will be used, since in chemical en-
gineering this is traditional. The relations between the mass-transfer expressions in
molar and mass units are summarized in Table 22.2-1.
Local transfer coefficients are defined by writing Eqs. 22.1-3 and 4 for a differential
area. Since dW AQ/dA = N A0 and dE 0/dA = e 0, we get the definitions

N A0 ~ x A0(N A0 + N B0) = k xA>]ocAx A (22.1-5)
e
o ~ (N AOH A0 + N BOH B0) = h locAT (22.1-6)
Next, we note that the left side of Eq. 22.1-5 is /* 0, and that the left side of a similar equa-
tion written for species В is / . However, since ]* = — / and Ax A = — Ax , we find that
B0
B
AQ
B0
KA,\OC = Кв,\оо а п с * therefore we can write both mass transfer coefficients as k xXoc , which
has units of (moles) /(area)(time). Furthermore, if the heat of mixing is zero (as in ideal
gas mixtures), we can replace H A0 by C (T 0 — T°), where T° is an arbitrarily chosen ref-
pA0
erence temperature, as explained in Example 19.3-1. A similar replacement may be made
for H . With these changes we get
B0
N^o " x (N + N ) = k ^x (22.1-7)
A0 A0 B0 x A
e ' (N C + N C, )(T - Г) = h AT (22.1-8)
0 A0 pAt0 B 0 B / 0 0 ]oc
We remind the reader that rapid mass transfer across phase boundaries can distort the
velocity, temperature, and concentration profiles, as we have already seen in §18.2 and
in Example 19.4-1. The correlations provided in §22.2, as well as their analogs in Chap-
ters 6 and 14, are all for small net mass-transfer rates, that is, for situations in which the
convective terms in Eqs. 22.1-7 and 8 are negligible relative to the first term. Such situa-
tions are common, and most correlations in the literature suffer from the same limita-
tion. In §22.8 we consider the deviations associated with high net mass-transfer rates and
decorate the transfer coefficients at these conditions with a superscript "•" (see §22.8).