24 CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
Noting that: RADRB
AtsDa: usAD 0 usBD 0
On the centre plane: AtsD 0
usADusB
AD
P
(^) Al
a^2
2
C
RA
(^) A
a
BD
P
(^) Bl
a^2
2
C
RB
(^) B
a
Thus: usAD
P
(^2) Al
fa^2 s^2 g
RA
(^) A
as
and: usBD
P
(^2) Bl
fa^2 s^2 gC
RA
(^) B
as sinceRADRB
Centre line velocity, uACLD
Pa^2
(^2) Al
C
RAa
(^) A
and: uBCLD
Pa^2
(^2) Bl
C
RAa
(^) B
Equating:
P^2 a
2 l
{
1
(^) A
1
(^) B
}
DaRA
{
1
(^) A
C
1
(^) B
}
∴ RAD
Pa
2 l
{
(^) B (^) A
(^) BC (^) A
}
Substituting:
usAD
P
(^2) Al
a^2 s^2
Pa
(^2) Al
as
(^) B (^) A
(^) BC (^) A
D
P
(^2) Al
{
a^2 s^2 C
(^) A (^) B
(^) AC (^) B
aas
}
Totalflowrate ofADQAis given by:
QAD
∫a
0
busdsD
Pb
(^2) Al
{∫a
0
[
a^2 s^2 C
(
(^) A (^) B
(^) AC (^) B
)
a^2 as
]
ds
}
D
Pb
(^2) Al
[(
a^2 s
s^3
3
)
C
[
(^) A (^) B
(^) AC (^) B
](
a^2 sa
s^2
2
)]a
0
D
Pb
(^2) Al
[
2 a^3
3
C
[
(^) A (^) B
(^) AC (^) B
]
a^3
2
]
D
Pba^3
(^2) Al
[
2
3
C
(^) A (^) B
2 AC (^) B