Page 570 - Bird R.B. Transport phenomena

550 Chapter 18 Concentration Distributions in Solids and in Laminar Flow

Then from Eq. 18.2-14 we get

c\n(x /x )
B]
B2
(N \ ._ )(z -z,)RT
A z
z
2
9
(7.26 X 1(Г )(17Л)(82.06)(273)
" (755/760X2.303 lo (755/722))
gl0
= 0.0636 cm /s (18.2-21)
2
This method of determining gas-phase diffusivities suffers from several defects: the cooling
of the liquid by evaporation, the concentration of nonvolatile impurities at the interface, the
climbing of the liquid up the walls of the tube, and the curvature of the meniscus.

EXAMPLE 18.2-3 (a) Derive expressions for diffusion through a spherical shell that are analogous to Eq. 18.2-11
(concentration profile) and Eq. 18.2-14 (molar flux). The system under consideration is shown
Diffusion through a in Fig. 18.2-4.
Nonisothermal (b) Extend these results to describe the diffusion in a nonisothermal film in which the tem-
Spherical Film
perature varies radially according to
T _ { r
(18.2-22)

where T, is the temperature at r = r,. Assume as a rough approximation that % varies as the
AB
§-power of the temperature:
3/2
»AB
(18.2-23)
AB,\
V
in which ЯЬ is the diffusivity at T = T . Problems of this kind arise in connection with dry-
}
АВА
ing of droplets and diffusion through gas films near spherical catalyst pellets.
The temperature distribution in Eq. 18.2-22 has been chosen solely for mathematical sim-
plicity. This example is included to emphasize that, in nonisothermal systems, Eq. 18.0-1 is
the correct starting point rather than N Az = -% (dcJdz) + x {N Az + N ), as has been given
AB
A
Bz
in some textbooks.
SOLUTION (a) A steady-state mass balance on a spherical shell leads to
2
f(rN ) = 0 (18.2-24)
Ar
r

s*^ Temperature T = Tj
2
-Temperature T]
i \ I < \ 1 \ \

\ у у у
Gas film ^^ ~~ Fig. 18.2-4. Diffusion through a hypotheti-
cal spherical stagnant gas film surrounding
^ r 2 ^
a droplet of liquid A.