MASS TRANSFER 245
Then:
∂CN^0
∂t
D
∫ 1
0
ept
∂C^0
∂t
dt (equation 10.102)
D[eptC^0 ]^10 Cp
∫ 1
0
eptC^0 dtDpCN^0 (equation 10.103)
Since the Laplace transform operation is independent ofy,
∂^2 C^0
∂y^2
D
∂^2 C^0
∂y^2
(equation 10.104)
Taking Laplace transforms of both sides of equation 10.100:
pCN^0 DD
∂^2 CN^0
∂y^2
∂^2 CN^0
∂y^2
p
D
CN^0 D 0
From which: CN^0 DAe
p
p/D
yCBe
p
p/D
y (equation 10.105)
When yD1, CN^0 D0andAD 0
When yD 0 , CN^0 DC^0 i/p and BDC^0 i/p
∴ CN^0 D
C^0 i
p
e
p
p/D
y
dCN^0
dy
D
C^0 i
p
pD
e
p
p/D
y
Inverting:
∂C^0
∂y
D
C^0 i
p
D
ð
1
p
t
ey
(^2) / 4 Dt
(See Volume 1, Appendix Table 12)
The mass transfer rate at the surface,NA (^) tDD
(
∂C^0
∂y
)
yD 0
DC^0 i
√
D
t
at timet
The average rate of mass transfer in timet:
1
t
∫t
0
C^0 i
√
D
t
dtD 2 C^0 i
√
D
t
Taking 1 m^2 of surface, the area disrupted by the bubbles per second is:
120 ð 15 / 10000 D 0. 18 /s
∴Average surface age durationD 1 / 0. 18
D 5 .55 s
C^0 iD 0. 16 0. 03
/ 100 D 0 .0013 kg O 2 /kg steelD 0. 0013 / 32
ð 7100
D 0 .2885 kmol/m^3