Page 592 - Bird R.B. Transport phenomena

572 Chapter 18 Concentration Distributions in Solids and in Laminar Flow

Fig. 18B.6. Sketch of a two-
Stopcock bulb apparatus for measuring
gas diffusivities. The stirrers in
z = L the two bulbs maintain uni-
form concentration in the
bulbs.
Mole fraction of A in Entire gaseous Mole fraction of A in
left bulb is x A = 1 - xX system is at right bulb is xX (t)
constant p and T

I8B.60 Two-bulb experiment for measuring gas diffusivity—quasi-steady-state analysis 6 (Fig. 18B.6).
One way of measuring gas diffusivities is by means of a two-bulb experiment. The left bulb and
the tube from z = - L t o z = 0 are filled with gas A The right bulb and the tube from z = 0 to
z = +L are filled with gas B. At time t = 0 the stopcock is opened, and diffusion begins; then the
concentrations of A in the two well-stirred bulbs change. One measures xX as a function of time,
and from this deduces ЯЬ . We wish to derive the equations describing the diffusion.
АВ
Since the bulbs are large compared with the tube, xX and x~ change very slowly with time.
A
Hence the diffusion in the tube can be treated as a quasi-steady-state problem, with the
boundary conditions that x A = x A and z = —L, and that x A = x A at z = +L.
(a) Write a molar balance on A over a segment Az of the tube (of cross-sectional area S), and
show that N = Q, a constant.
Az
(b) Show that Eq. 18.0-1 simplifies, for this problem, to
dx
N A? = -c A (18B.6-1)

(c) Integrate this equation, using (a). Call the constant of integration C .
2
(d) Evaluate the constant by requiring that x A = xX at z = +L.
(e) Next set x A = x A (or 1 - xX) at z = -L, and solve for N Az to get finally

(18B.6-2)

(f) Make a mass balance on substance A over the right bulb to obtain

M X A ) - V C (18B.6-3)
2
(g) Integrate the equation in (f) to get an expression for xX which contains ЯЬ :
АВ
In (18B.6-4)
LV
(h) Suggest a method of plotting the experimental data to evaluate ЯЬ .
АВ
18B.7. Diffusion from a suspended droplet (Fig. 18.2-3). A droplet of liquid A, of radius r u is sus-
pended in a stream of gas B. We postulate that there is a spherical stagnant gas film of radius
r surrounding the droplet. The concentration of A in the gas phase is x M at r = r x and x A1 at
2
the outer edge of the film, r = r .
2
(a) By a shell balance, show that for steady-state diffusion r N Ar is a constant within the gas
2
film, and set the constant equal to r\N ArV the value at the droplet surface.
(b) Show that Eq. 18.0-1 and the result in (a) lead to the following equation for x :
A
(18B.7-1)
-х dr
А
1
S. P. S. Andrew, Chem. Eng. Sci., 4, 269-272 (1955).