Page 633 - Bird R.B. Transport phenomena

§20.1 Time Dependent Diffusion 613

Table 20.0-1 Analogies Between Special Forms of the Heat Conduction and Diffusion Equations

Unsteady-state nonflow Steady-state flow Steady-state nonflow

§12.1—Exact solutions §12.2—Exact solutions §12.3—Exact solutions
§12.4—Boundary layer in two dimensions by
& 3
solutions analytic functions
Equations (v • VT) = aV T V T = 0
2
2

rations Heat conduction in Heat conduction in Steady heat conduction
"5. solids laminar incompressible solids
< flow
1 1. к = constant 1. к, р = constants 1. к = constant
2.v = 0 2. No viscous dissipation 2. v = 0
1 3. Steady state 3. Steady state

quations VV VL A J ~L>AB X L A V c A = 0
2
ш
с Diffusion of traces of Diffusion in laminar Steady diffusion in
л<
cati A through В flow (dilute solutions of solids
Vppli AinB)
(Л 1. %b , p = constants 1. %b , p = constants 1. %b , p = constants
AB
AB
AB
0 2. v = 0 2. Steady state 2. Steady state
"i
E
э 3. No chemical reactions 3. No chemical reactions 3. No chemical reactions
<л
(Л
4.v = 0
(A
| OR Equimolar counter-
licat diffusion in low
Appl density gases
ons 1. ЯЬ , с = constants
АВ
mpti 2. v* = 0
1 3. No chemical reactions
§20.1 TIME-DEPENDENT DIFFUSION
In this section we give four examples of time-dependent diffusion. The first deals with
evaporation of a volatile liquid and illustrates the deviations from Fick's second law that
arise at high mass-transfer rates. The second and third examples deal with unsteady-
state diffusion with chemical reactions. In the last example we examine the role of inter-
facial-area changes in diffusion. The method of combination of variables is used in
Examples 20.1-1, 2, and 4, and Laplace transforms are used in Example 20.1-3.

EXAMPLE 20.1-1 We wish to predict the rate at which a volatile liquid A evaporates into pure В in a tube of in-
finite length. The liquid level is maintained at z = 0 at all times. The temperature and pres-
UnsteadyState assumed constant, and the vapors of A and В form an ideal gas mixture. Hence the
s u r e a r e
Evaporation of a Liquid molar density с is constant throughout the gas phase, and 4t may be considered to be con-
AB
(the "Arnold Problem") stant. It is further assumed that species В is insoluble in liquid A, and that the molar average
velocity in the gas phase does not depend on the radial coordinate.