MASS TRANSFER 275
The rising velocity is given by a force balance:
3 d 1 uD^16 d^31 + 1 + 2
g
or: u 1 D
d^21 g
18
+ 1 + 2
DKd^21 relative to continuous phase
The downward liquid velocity is^12 Kd^21
The upward droplet velocity relative to container is:
Kd^21 ^12 Kd^21 D^12 Kd^21
and the time of contact during rise through heightHis:
tcD
H
1
2 Kd
2
1
. (i)
The mass transfer rate is:D∂CA/∂y
.
Thus:
1
CAS
∂CA
∂y
D
∂
∂y
{
erfc
y
2
p
Dt
}
D
∂
∂y
{
2
p
∫ 1
y/ 2
p
Dt
ey
(^2) / 4 Dt
d
(
y
2
p
Dt
)}
D
∂
∂y
Ð
1
2
p
Dt
2
p
∫ 1
y
ey
(^2) / 4 Dt
dt.
or:
∂CA
∂y
D
CAS
p
Dt
ey
(^2) / 4 Dt
(
∂CA
∂y
)
yD 0
D
CAS
p
Dt
The mass transfer rate at the surface is: (moles/areaðtime).
D
(
CAS
p
Dt
)
D
√
D
t
CAS
The mass transfer in timete 1 is:
√
D
CAS
∫te 1
0
t^1 /^2 dtD 2
√
D
t^1 e 1 /^2 CASDKte^1 / 12
Substituting from equation (i):
Mass transfer in moles per unit area of drop/
p
2 H
p
Kd 1
The mass transfer per drop is proportional to:
p
2
√
H
K
d 11 d^21 /
p
2
√
H
K
d 1
The mass transfer per unit timeDMass transfer per dropðdrops/time
or: proportional to:
p
2
√
H
K
d 1 ð
6 Q 1
d^31
/ 8. 48
√
H
K
Q 1
d^21