Pulse analysis - Principles of Chemical Separations with Environmental Applications - page 291

2 R = d p

Figure C.3

1 Interstitial fluid:

t =−∇· N

∂ c

+

α , ∂

where = rate of mass transfer to particle surface;

c V 0
α

D 0
α

∂ c

N z = −

∂

z

2 c

α ∂

c
dt =−

V 0 ∂

c

D 0 ∂

z 2 + .
Now we need to account for the fraction of the packed column which is solid particles:

z + ∂ ∂

solid volume
total volume =

(number of particles) × (volume / particle)

volume

n 3 π R 3

1 − α =

4 d p /

π z

where n = number of particles in a volume element of column.

R 2 k p ( c n 4 π − c i R )

=−

π d p z / 4

where c i R =

concentration on particle surface

c i =

concentration in intra-particle void space

k p = mass transfer coefficient.

Substituting for n :

(1 − α )( π d 2 / 4 z )

R 2 k p ( c

=− π − c i R )

4

4

3 π

R 3 ( π d 2 / 4 z )

3(1 − α )

=− k p ( c − .

c i R )

R

Substituting for in differential equation:

α ∂ V 0 ∂ D 0 ∂

2 c

− α

c c 3(1 )

t =− z + z 2 −

k p ( c − c i R ) (interstitial fluid equation).

∂ ∂ ∂

R