Page 631 - Bird R.B. Transport phenomena

Problems 611

The diffusion equation of Eq. 19.1-16 with the known velocity components is to be solved
= p A0 at z = 0 at z = °°; and dp A/dr = 0 at r =
under the boundary conditions that: p A = 0; p A
0, oo. Since there can be but one solution to this linear problem, it may be seen that a solution
of the form p A(z) can be found that satisfies the differential equation and all the boundary con-
ditions. Thus, the solution for p A does not depend on the radial coordinate in the region
considered.
(a) Show that at steady-state Eq. 19.1-16 gives

J7 = -£--4r (19D.4-5)
4 be d£ z
(b) Solve Eq. 19D.4-5 to get, for large Schmidt number,

3
тг~ = 1 — " — A — exp(-{aSc£ )d£ (19D.4-6)
HAO T() -> 0
(c) Show that the mass flux at the surface of the disk is 7

JAZ\Z=O = 0-620 A * (19D.4-7)

for large Schmidt number. Clearly, if desired, one could use higher terms in the series expan-
sion for Я and extend the Schmidt-number range. 10 This system has been used for studying
the removal of solid behenic acid from stainless-steel surfaces. 11

10 D. Schuhmann, Physicochemical Hydrodynamics (V. G. Levich Fextschrift), Vol. 1 (D. B. Spalding ed.),
Advance Publications Ltd., London (1977), pp. 445-459; see also K.-T. Liu and W. E. Stewart, Intl. Jnl.
Heat and Mass Trf., 15,187-189 (1972).
11 C. S. Grant, A. T. Perka, W. D. Thomas, and R. Caton, AIChE journal 42,1465-1476 (1996).