FLOW IN PIPES AND CHANNELS 59
D
Rr^3(^4) p
{
1 4 / 3 X 2 X^2 C 4 X^3
5
3
X^4
}
(ii)In the core region
Substituting:sDrcDRY/RrDXrin equation (i) for the core velocityucgives:
ucD1
(^) p
{
R
2 rr^2 X^2 r^2 RYrXr}
D
Rr(^) p
{
1
2
1 X^2 X 1 X
}
Rr(^4) p
f 2 1 X^2 4 XC 4 X^2 g
D
Rr(^4) p
{
2 4 XC 2 X^2
}
Theflowrate through the core is:ucr^2 cDucX^2 r^2 DQc
Thus: QcD
Rr(^4) p
X^2 r^2 f 2 4 XC 2 X^2 g
D
Rr^3 (^4) p
f 2 X^2 4 X^3 C 2 X^4 g
The totalflowrate is:QACQcDQ
and: QD
Rr^3
(^4) p
{
1 ^43 XC^13 X^4
}
Putting RD
Pr
2 lthen :QD
Pr^4
8 l pf 1 ^43 XC^13 X^4 gWhen: PD 6 ð 105 N/m^2 ,lD200 m dD40 mm andrD 0 .02 m.
Then: RDP
r
2 lD 6 ð0. 02
400
ð 105 D30 N/m^2(^) pD 0 .150 Ns/m^2
RYD 14 .35 N/m^2
and: XD
RY
R
D
14. 35
30
D 0. 478
Thus: QD
6 ð 105 0. 02 ^4
8 ð 200 ð 0. 150{
1
4
3
ð 0. 478 C1
3
0. 478 ^3
}
D 0 .000503 m^3 /s