Page 506 - Modelling in Transport Phenomena A Conceptual Approach

486 CHAPTER 11. UNSTEADY MICROSCOPIC BALANCES WITH GEN.

For a spherical differential volume element of thickness AT, as shown in Figure
11.3, l3q. (11.3-2) is expressed in the form

Dividing Eq. (11.3-3) by 4xAr and taking the limit as AT -+ 0 gives

or.

(11.3-5)
Substitution of Eq. (11.3-1) into Eq. (11.3-5) gives the governing differential
equation for the concentration of species d as

(11.3-6)

The initial and the boundary conditions associated with Eq. (11.3-6) are
at t=O CA=O (1 1.3-7)

(11.3-8)
at T = R CA =cfS (11.3-9)

where cfS is the equilibrium solubility of species d in liquid 23.
Danckwerts (1951) showed that the partial differential equation of the form

(11.3-10)

with the following initial and the boundary conditions
at t=O c=O (1 1.3-1 1)
dC
at r=O -=O (11.3-12)
a?-
at T = R CA =c: (11.3-13)

has the solution
t
c = k 4(r), z) e-kq dr) + 4(t, x) (1 1.3- 14)
where +(t, z) is the solution of Eq. (11.3-10) without the chemical reaction, i.e.,

(11.3-15)