CHEMICAL ENGINEERING

222 CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS

Solution

If the particle has a radiusr, and is surrounded by a spherical shell of radiussthen,
Moles per unit time diffusing through the shell,Mis given by:

MD 4 s^2

(

D

dCA

ds

)

At steady state,Mis constant and:

M

∫s 2

s 1

ds

s^2

D 4 D

∫CA

2

CA 1

dCA

M

(

1

s 1



1

s 2

)

D 4 DCA 1 CA 2

IfCA 1 is the concentration ats 1 DrandCA 2 is the concentration ats 2 D1, then:

M/rD 4 DCA

The mass transfer coefficient:hdD

M

ACA

D

M

4 r^2 CA

hD 4 r^2 CA

/rD 4 DCA

hDDD/rD 2 D/d

hDd/DDShD 2

PROBLEM 10.7

Show that the concentration profile for unsteady-state diffusion into a bounded medium
of thicknessL, when the concentration at the interface is suddenly raised to a constant
valueCAiand kept constant at the initial value ofCAoat the other boundary is:

CACAo

CAiCAo

D 1 

z

L



2



[nD1

∑

nD 1

1

n

expn^2 ^2 Dt/L^2 sinnz/L

]

.

Assume the solution to be the sum of the solution for infinite time (steady-state part) and
the solution of a second unsteady-state part, which simplifies the boundary conditions for
the second part.

Solution

The system is shown in Fig. 10a.
The boundary conditions are:
At time, tD 0 CADCAo 0 <y<L
t> 0 CADCAi yD 0
t> 0 CADCAo yDL