Page 594 - Bird R.B. Transport phenomena

574 Chapter 18 Concentration Distributions in Solids and in Laminar Flow

(b) Show that, in the absence of chemical reaction in the liquid phase, the concentration pro-
file is linear.
(c) Show that the rate of leaching is given by
= - C A8 )/8 (18B.9-1)
N Az
18B.10 Constant-evaporating mixtures. Toluene (1) and ethanol (2) are evaporating at z = 0 in a
vertical tube, from a binary liquid mixture of uniform composition х л through stagnant nitro-
gen (3), with pure nitrogen at the top. The unequal diffusivities of toluene and ethanol
through nitrogen shift the relative evaporation rates in favor of ethanol. Analyze this effect
for an isothermal system at 60 F and 760 mm Hg total pressure, if the predicted 8 diffusivities
6
6
at 60° F are c% = 1.53 X 10~ , c% = 2.98 X 10~ , c% = 4.68 X 10~ . 6
2 3 3
(a) Use the Maxwell-Stefan equations to obtain the steady-state vapor-phase mole fraction pro-
files y {z) in terms of the molar fluxes N az in this ternary system. The molar fluxes are known to
a
be constants from the equations of continuity for the three species. Since nitrogen has a negligible
solubility in the liquid at the conditions given, N 3z = 0. As boundary conditions, set y x = y 2 = 0 at
z = L, and let у л = y 10 and y 2 = t/o at z = 0; the latter values remain to be determined. Show that
2
D e ~A(L-z)
y 3 (z) = e~ ^ (18B.10-1)
A-B A-B D
+ N 2z
N l 2
A = • B = C = (18B.10-2)
(b) A constant evaporating liquid mixture is one whose composition is the same as that of the
evaporated material, that is, for which N^ /(N + N ) = x . Use the results of part (a) along
Z
2z
x
]Z
with the equilibrium data in the table below to calculate the constant-evaporating liquid com-
position at a total pressure of 760 mm Hg. In the table, row I gives liquid-phase compositions.
Row II gives vapor-phase compositions in two-component experiments; these are expressed
as nitrogen-free values y\/{y\ + y ) for the ternary system. Row III gives the sum of the partial
2
pressures of toluene and ethanol.
I: x, 0.096 0.155 0.233 0.274 0.375
II: y^/(yi+y ) 0.147 0.198 0.242 0.256 0.277
2
III: р^ + p ( m m Hg) 388 397 397 395 390
2
A suggested strategy for the calculation is as follows: (i) guess a liquid composition x ; (ii) cal-
x
culate y 1/20/ a n c l Узо using lines 2 and 3 of the table; (iii) calculate A from Eq. 18B.10-1, with
10/
z = 0; (iv) use the result of iii to calculate N , B, C, and D, and finally y A (0) for assumed val-
2z
ues of N ; (v) interpolate the results of iv to y x (0) = y 10 to obtain the correct N ]z and N 2z for
lz
the guessed x v Repeat steps i-v with improved guesses for x } until N /(N l2 + N ) is close
l2
2z
enough to x . The final x is the constant evaporating composition.
x x
18B.11. Diffusion with fast second-order reaction (Figs. 18.2-2 and 18B.11). A solid A is dissolving
in a flowing liquid stream S in a steady-state, isothermal flow system. Assume in accordance
with the film model that the surface of A is covered with a stagnant liquid film of thickness 8
and that the liquid outside the film is well mixed (see Fig. 18.2-2).
(a) Develop an expression for the rate of dissolution of A into the liquid if the concentration
of A in the main liquid stream is negligible.
(b) Develop a corresponding expression for the dissolution rate if the liquid contains a sub-
stance B, which, at the plane z = к8, reacts instantaneously and irreversibly with A: A + В —»
P. (An example of such a system is the dissolution of benzoic acid in an aqueous NaOH solu-
tion.) The main liquid stream consists primarily of В and S, with В at a mole fraction of x .
Bx

8 L. Monchick and E. A. Mason, /. Chem. Phys., 35,1676-1697 (1961), with 8 read as 5 ma x in Table IV;
E. A. Mason and L. Monchick, /. Chem. Phys., 36, 2746-2757 (1962); L. S. Tee, S. Gotoh, and W. E. Stewart,
Ind. Eng. Chem. Fundam., 5, 356-362 (1966).