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Unformatted text preview: Average Velocity v v Recall from last lecture for
shell (annulus) of fluid: vz
dA
A dA= 2 r dr A = R2 n1
2 n1
vz
constant
dA 2 R n r r n dr
A
R2 R n1 r 2 r n 1 R n 2 2 nn1 1 0
2 n1 L v P
2LK P
2LK 1
n 1
n n
n 1 2
R2 n1 R 2 R n 1 R n 2 2 nn1 1 2 n1 n
n (3n 1) R 2nR (3n 1)
3n 1 n1 n
n 1 n 1 (3n 1) 2n (3n 1) 1
n1 n
P 2LK n R n 3n 1 P 2LK nn1 R
1
n n1
n Laminar Pipe Flow of Power Law Fluid 4Q 3 R n
w 34 n1 s K n s w K w n This puts the wall shear stress and shear rate as a direct function
of the measurable pressure drop and flow rate.
4Q For a Newtonian fluid, i.e. n=1, this reduces to Newtonian 3 R Therefore, to simplify equations for Power law fluid, this shear
rate is used as a reference, i.e. apparent shear rate is : 4Q 8v 3 R D n w 34 n1 . Laminar Pipe Flow of Power Law Fluid
Average Velocity for Power Law Fluid:
P v 2LK n 3nn1 R
1 n1
n Volumetric Flow rate , Q: P Q v.A R 2 2LK n 3nn1 R
1 n1
n P
Q 2LK n 3nn1 R
1 3 n1
n Note, wall shear stress is directly proportional to the pressure drop: sw PR
n K w 2L Power law model and the wall shear rate can be determined by subbing into equation for Q:
n 4Q w 34 n1 3 R 1 s n w w K Laminar Pipe Flow of NNF: General Approach
General expression can be used for any fluid model :
(see derivation in Steffe) 1
Q
3
R 3 s w tw 0 s 2 f (s )dt Use Leibnitz’ rule to change variables to differentiate as function of sw and solve for
shear rate at the wall, gives: d ds w w f (s w ) 3 1 s w 4
4 x dy d (ln y ) y dx d (ln x) d (ln ) w 3 1 4
4 d (ln s w ) d (ln )
1 3n'1 w n' d (ln s w ) 4n' NB : 4Q 8v 3 R D RabinowitschMooney
equation n’ can be obtained directly from
P and Q data !
For Power Law, n’ = n...
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This note was uploaded on 04/17/2013 for the course CHEE 2003 taught by Professor Jasonstokes during the One '12 term at Queensland.
 One '12
 JasonStokes

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