Page 674 - Bird R.B. Transport phenomena

654 Chapter 20 Concentration Distributions with More Than One Independent Variable

(c) Show further that the total moles absorbed across area A up to time t is

1 \ exp(-
M = (20C.3-4)
A
TT J
(d) Show that, for large values of k"'t, the expression in (c) reduces asymptotically to
1
= Ac A0 V°b AB k"'[ t + ^ (20C.3-5)
M A
This result 6 is good within 2% for values of k"'t greater than 4.
20C.4. Design of fluid control circuits. It is desired to control a reactor via continuous analysis of a
side stream. Calculate the maximum frequency of concentration changes that can be detected
as a function of the volumetric withdrawal rate, if the stream is drawn through a 10 cm length
of tubing with an internal diameter of 0.5 mm. Suggestion: Use as a criterion that the standard
deviation of a pulse duration be no more than 5% of the cycle time t 0 = 2тг/о), where o> is the
frequency it is desired to detect.
20C.5. Dissociation of a gas caused by a temperature gradient. A dissociating gas (for example,
Na 2 <=* 2Na) is enclosed in a tube, sealed at both ends, and the two ends are maintained at dif-
ferent temperatures. Because of the temperature gradient established, there will be a continu-
ous flow of Na molecules from the cold end to the hot end, where they dissociate into Na
2
atoms, which in turn flow from the hot end to the cold end. Set up the equations to find the
concentration profiles. Check your results against those of Dirac. 7
20D.1. Two-bulb experiment for measuring gas diffusivities—analytical solution (Fig. 18B.6).
This experiment, described in Problem 18B.6, is analyzed there by a quasi-steady-state
method. The method of separation of variables gives the exact solution 8 for the compositions
in the two bulbs as

(20D.1-1)

in which y n is the nth root of у tan у = N, and N = SL/V. Here the ± sign corresponds to the
reservoirs attached at ±L. Make a numerical comparison between Eq. 20D.1-1 and the experi-
mental measurements of Andrew. 9 Also compare Eq. 20D.1-1 with the simpler result in Eq.
18B.6-4.
20D.2. Unsteady-state interphase diffusion. Two immiscible solvents I and II are in contact at the
plane z = 0. At time t = 0 the concentration of A is c = cf in phase I and c = cf, in phase II.
M
x
For t > 0 diffusion takes place across the liquid-liquid interface. It is to be assumed that the
solute is present only in small concentration in both phases, so that Fick's second law of diffu-
sion is applicable. We therefore have to solve the equations
да д с.
2
-oo < z < 0 (20D.2-1)
л„
£-• О < Z < +00 (20D.2-2)

6
R. A. T. O. Nijsing, Absorptie van gassen in vloeistoffen, zonder en met chemische reactie, Academisch
Proefschrift, Technische Universiteit Delft (1957).
7
P. A. M. Dirac, Proc. Camb. Phil. Soc, 22, Part II, 132-137 (1924). This was Dirac's first publication,
written while he was a graduate student.
R. B. Bird, Advances in Chemical Engineering, Vol. 1, Academic Press, New York (1956), pp. 156-239;
8
errata, Vol. 2 (1958), p. 325. The result at the bottom of p. 207 is in error, since the factor of (-1)" is
+ 1
missing. See also H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd edition, Oxford
University Press (1959), p. 129.
4
S. P. S. Andrew, Chem. Eng. Sci., 4, 269-272 (1955).