CHEMICAL ENGINEERING

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