FLOW IN PIPES AND CHANNELS 55
At the free surface whereyDS:
usD
(g
k
) 1 /n( n
nC 1
)
S
nC 1
n (i)
The volumetricflowrateQis given by:
QD
∫s
0
wdy
(g
k
) 1 /n( n
nC 1
)(
S
nC 1
n sy
nC 1
n
)
Dw
(g
k
) 1 /n( n
2 nC 1
)
S
2 nC 1
n (ii)
For the non-Newtonianfluid,kD 0. 1 ,nD 0 .5 and equation (ii), when expressed in S.I.
units, becomes:
Dw
(
g
0. 1
) 2
ð 0 .25 s^4 D 25 w
(
gs^2
) 2
(iii)
For the Newtonianfluid,nD1andkD and substituting in equation (ii):
QDw
(
g
)
ð 0. 33 S^3 N (iv)
whereSNis the thickness of the Newtonianfilm.
For equalflowrates, from equations (iii) and (iv):
25 w
(
gs^2
) 2
D 0. 33 w
(
g
)
S^3 N
or: S^3 ND 75
(
g
)
S^4 (v)
For equal surface velocities, the termg/Kin equation (i) can be substituted from
equation (v) and:
For the non-Newtonianfluid:usD
(
g
0. 1
) 2
ð 0. 33 S^3
D 100
(
S^3 N
75 S^4
) 2
ð 0. 33 S^3
D 0. 00592 S^6 N/S^5
For the Newtonianfluid:usD
(
g
)
ð 0. 5 S^2 N
D
(
S^3 N
75 S^4
)
ð 0. 5 S^2 ND 0. 0067 S^5 N/S^4
and: SN/SD 0. 00667 / 0. 00592 D 1. 126