Page 670 - Bird R.B. Transport phenomena

650 Chapter 20 Concentration Distributions with More Than One Independent Variable

with a constant, provided that H o = H (p, T) and C pA = C fl8 = constant. It is further assumed
a
that c, k, and p are essentially constant. Here the dissipation term (T:VV) and the work term
2 (ja * ga) a r e appropriately neglected. {Hint: Use the species equation of continuity of Eq.
a
19.1-10.)
(b) Show that the solution of Eq. 20B.2-1, with the initial condition that T = T. at t = 0, and
A
the boundary conditions that T = T at z = 0 and T = Т. at z = <*, is
o
л
f ) (20B.2-2)
T,-T o
with
and <p = v* Г- (20В.2-3)
T

(c) Show that the interfacial mass and energy fluxes are related to T and Т. by
л
o
N
— _ A0 + Nb0 — = V^r(l + erf <р )<р ехр ^ (20В.2-4)
т
(
[q /C (T - TJ]
o p o
so that N /q 0 and N /q 0 are constant for f > 0. This nifty result arises because there is no
A0
B0
characteristic length or time in the mathematical model of the system.
20B.3. Stoichiometric boundary condition for rapid irreversible reaction. The reactant fluxes in
Example 20.1-2 must satisfy the stoichiometric relation
at z - z (t), c {v - v ) - ~ c (v - v ) (20B.3-1)
R A lA R P zB K
in which v R = dz /dt. Show that this relation leads to Eq. 20.1-31 when use is made of Fick's
R
first law, with the assumptions of constant с and instantaneous irreversible reaction.
20B.4. Taylor dispersion in slit flow (Fig. 2B.3). Show that, for laminar flow in a plane slit of width
IB and length L, the Taylor dispersion coefficient is

20B.5. Diffusion from an instantaneous point source. At time t = 0, a mass m A of species Л is in-
jected into a large body of fluid B. Take the point of injection to be the origin of coordinates.
The material A diffuses radially in all directions. The solution may be found in Carslaw and
Jaeger: 2

О (20В.5-1)
л н
(a) Verify that Eq. 20B.5-1 satisfies Fick's second law.
(b) Verify that Eq. 20B.5-1 satisfies the boundary conditions at r = <».
(c) Show that Eq. 20B.5-1, when integrated over all space, gives m , as required.
A
(d) What happens to Eq. 20B.5-1 when t -> 0?
20B.6. Unsteady diffusion with first-order chemical reaction. Use Eq. 20.1-43 to obtain the concen-
tration profile for the following situations:
(a) The stationary semi-infinite system of Problem 20C.3.

2
H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd edition, Oxford University Press
(1959), p. 257.