CHEMICAL ENGINEERING

MASS TRANSFER 237

PROBLEM 10.19

A crystal is suspended in fresh solvent and 5% of the crystal dissolves in 300 s. How
long will it take before 10% of the crystal has dissolved? Assume that the solvent can
be regarded as infinite in extent, that the mass transfer in the solvent is governed by
Fick’s second law of diffusion and may be represented as a unidirectional process, and
that changes in the surface area of the crystal may be neglected. Start your derivations
using Fick’s second law.

Solution

The mass transfer process is governed by Fick’s second law:

∂CA

∂t

DD

∂^2 CA

∂y^2

(equation 10.66)

and discussed in Section 10.5.2
The boundary conditions for the crystal dissolving are:

WhentD 00 <y< 1 CAD 0

t> 0 yD1 CAD 0

t> 0 yD 0 CADCAsthe saturation value

These boundary conditions allow the solution of equation 10.66 using Laplace trans-
forms as the most convenient method:

∂CA

dt

D

∫ 1

0

ept

∂CA

∂t

dt (equation 10.102)

D

[

eptCA

] 1

0 Cp

∫ 1

0

eptCAdtD 0 CpCNA (equation 10.103)

Taking Laplace transforms of both sides of equation 10.66:

pCNADD

∂^2 CNA

∂y^2

∴

∂^2 CNA

∂y^2



p

D

CNAD 0

and: CNADAe

p

p/D

yCBe

p

p/D

y (equation 10.105)

WhenyD1,CAD 0 ∴ CNAD0andAD 0

WhenyD 0 ,CADCAs and CNADCAs/po,BDCAs/p

∴ CNAD

CAs

p

e

p

p/D

y

Inverting: CADCAserfcy/ 2

p

Dt

 (See Volume 1, Appendix Table 13)