Page 415 - Modelling in Transport Phenomena A Conceptual Approach

9.5. MASS TRANSFER WITH CONVECTION 395

The boundary conditions associated with Eq. (9.560) are

(9.561)

at <=l 0=1 (9.5-62)
The use of the substitution
u=1-8 (9.563)
reduces Eqs. (9.5-60)-(9.562) to

(9.5-64)

(9.5-65)

at [=l u=O (9.5-66)
Equation (9.3-61) can be solved for Sh by the method of Stodola and Vianello w
explained in Section B.3.4.1 in Appendix B.
A reasonable fist guess for u which satisfies the boundary conditions is

UI = 1 - (2 (9.567)
Substitution of Eq. (9.5-67) into the left-side of Eq. (9.5-64) gives

-$ ((g) =-2Sh((-2s3+s5) (9.568)

The solution of Eq. (9.5-68) is
(11 - 18J2 + 9c4 - 2E6
u = Sh (9.5-69)
36
fli0
Therefore, the first approximation to the Sherwood number is
s
I’ (1 - e2)2.fi(E) de
Sh(l) = (9.5-70)
I’ s (1 - s2)m 4

Substitution of fi(<) from Eq. (9.5-69) into Eq. (9.570) and evaluation of the
integrals gives
Sh = 3.663 (9.5-71)
On the other hand, the value of the Sherwood number, as calculated by Graetz
(1883, 1885) and Nusselt (1910), is 3.66. Therefore, for a fully developed concen-
tration profile in a circular pipe with a constant wall concentration Sh = 3.66 for
all practical purposes.