Page 669 - Bird R.B. Transport phenomena

Problems 649

20A.4. Effect of bubble size on interfacial composition (Fig. 20A.2). Here we examine the assump-
tion of time-independent interfacial composition, (o , for the system in Fig. 20A.2. We note
A0
that, because of the interfacial tension, the gas pressure p depends on the instantaneous
A
bubble radius . The equilibrium expression
r
s
Рл = p + у 1 (20А.4-1)
x
is adequate unless drjdt is very large. Here p is the ambient liquid pressure at the mean ele-
x
vation of the bubble, and a is the interfacial tension.
For a sparingly soluble solute, the interfacial liquid composition ш depends on p ac-
А0 A
cording to Henry's law
(20A.4-2)
<*>AO = Hp A
in which the Henry's law constant, H, depends on the two species and on the liquid tempera-
ture and pressure. This expression may be combined with Eq. 20A.4-1 to obtain the depen-
dence Of U) ОП r .
A0 s
For a gas bubble dissolving in liquid water to T = 25°C and p x = 1 atm, how small must
the bubble be in order to obtain a 10% increase in OJ above the value for a very large bubble?
AO
Assume a = 72 dyn/cm over the relevant composition range.
Answer: 1.4 microns
20A.5. Absorption with rapid second-order reaction (Fig. 20.1-2). Make the following calculations
for the reacting system depicted in the figure:
(a) Verify the location of the reaction zone, using Eq. 20.1-38.
(b) Calculate N A0 at t = 2.5 s.

20A.6. Rapid forced-convection mass transfer into a laminar boundary layer. Calculate the evapo-
ration rate n (x) for the system described under Eq. 20.2-52, given that a) = 0.9, Q> — 0.1,
A0
A0
AK
and Sc = 2.0. Begin by determining, by trial and error, the values of К and П'(0, Sc, K) consis-
tent with Table 20.2-2 and Eqs. 20.2-51 and 52. Or use the plots given in §22.8.
Answer: n (x) = 0.33Vpv^fx/x
A0
20A.7. Slow forced convection mass transfer into a laminar boundary layer. This problem illus-
trates the use of Eqs. 20.2-55 and 57 and tests their accuracy.
(a) Estimate the local evaporation rate, n , as a function of x for the drying of a porous
A0
water-saturated slab, shaped as in Fig. 20.2-2. The slab is being dried in a rapid current of air,
under conditions such that ш А0 = 0.05, o> = 0.01, and Sc = 0.6. Use Eq. 20.2-55 for the calcu-
Ax
lation.
(b) Make an alternate calculation of n A0 using Eq. 20.2-57.
(c) For comparison with the preceding approximate results, calculate n A0 from Eq. 20.2-47
and Table 20.2-2. The К values found in (a) will be sufficiently accurate for looking up П'(0,
Sc, K).
Answers: (a) n (x) = 0Ш88Л/pv^fi/x; (b) n (x) = 0.0196Vpv fi/x)
A0 A0 x
(c) n AQ (x) = 0.0188VpUcc/x/*
20B.1. Extension of the Arnold problem to account for interphase transfer of both species. Show
how to obtain Eqs. 20.1-23, 24, and 25 starting with the equations of continuity for species A
and В (in molar units) and the appropriate initial and boundary conditions.
20B.2. Extension of the Arnold problem to nonisothermal diffusion. In the situation described in
Problem 20B.1, find the analogous result for the temperature distribution T(z, t).
(a) Show that the energy equation [Eq. (H) of Table 19.2-4] reduces to

^
§ + ? = «^T (20B.2-1)
dt dz s 1