Page 313 - Modelling in Transport Phenomena A Conceptual Approach

8.4. MASS TRANSPORT WITHOUT CONVECTION 293

8.4.3 Diffusion in Spherical Coordinates
Consider onedimensional diffusion of species A in the radial direction through a
hollow sphere with inner and outer radii of R1 and Rz, respectively, as shown in
Figure 8.30.

Figure 8.30 Diffusion through a hollow sphere.

Since XA = XA(T), Table C.9 in Appendix C indicates that the only non-zero molar
flux component is NA~ and it is given by
NA- = - CVAB - (8.433)
dXA
dr
For a spherical differential volume element of thickness AT, as shown in Figure
8.30, Eq. (8.41) is expressed in the form

(ANA~)I~ - (ANAr)lr+Ar = O (8.434)
Dividing Eq. (8.434) by Ar and taking the limit as AT + 0 gives

(8.435)

(8.436)
Since flux times area gives the molar transfer rate of species A, iz~, is possible
it
to conclude that
ANA^ = constant = iz~ (8.437)
Note that the area A in Eq. (8.437) is perpendicular to the direction of mass flux,
and is given by
A = 4xr2 (8.438)
Substitution of Eqs. (8,433) and (8.438) into Eq. (8.437) and integration gives

(8.439)