This preview shows page 1. Sign up to view the full content.
Unformatted text preview: ric flow rate: L PR 2
vz 8 L
PR 4
Q
8 L
2 L v P f D 2 v
v z dA
A
Q A.vz Rearrange equation for average velocity
This gives the pressure drop (energy loss) along the
pipe section due to viscous forces (not accounted
for in Bernoulli) . We use this in the MFEE: f laminar 64
Re hf P
g Laminar Pipe Flow of Power Law fluid
Time independent nonNewtonian Fluid ;
Substitute fluid constitutive model (Power Law),
No slip Boundary condition, i.e. r=R, vz = 0 Average pipe velocity
Volumetric flow rate: Q A.vz r
vz vz 3nn11 1 R n n1 n1 1 P Q 2LK n 3nn1 R
1 3 n1
n Rearrange equation for average velocity to get pressure drop in terms of velocity,
pipe dimensions, etc.. Then obtain frictional losses to use in MFEE (hf );
2 L v P f D 2 Re PL v 2 n D n 4n 8n 1 K 3n 1 n P v 2LK n 3nn1 R n vz
dA
A v dvz dr s K n s K hf f L v2
;
D 2g n f 64 / Re PL Laminar Pipe Flow of Power Law Fluid
n 1 n
Power Law Fluid: s K v z 3n 1 r n 1 vz
n 1 R 2.5 n=2
2.0 v/vaverage vz
vz n=1 n=1 This demonstrates how
‘shear thinning’ (n < 1)
and
‘shear thickening’ (n > 1)
significantly affect the
velocity profile in pipe
flow n=0.5 1.5 n=0.25
n=0.1 n = 0.1 1.0 r/R vs n=0.1
r/R vs n =0.25
r/R vs n=0.5
r/R vs n=1
r/R vs n=2 0.5 0.0
1.0 0.5 0.0 0.5 1.0 r/R
r/R Laminar Pipe Flow : Shear Rate at Wall r 2 v z 2vz 1 R Newtonian
Fluid
Shear rate 2r
dv z 2v z 2
R
dr At Wall, r = R dv z
2 2v z dr
R Generalised Shear rate at
wall in pipe flow
Shear stress at wall in
pipe flow 4Q 8v 3 R D sw PR
2L Laminar Pipe Flow : Shear Rate at Wall r
vz vz 3nn11 1 R n Power Law
Fluid n1 Shear rate dv
3n 1 1 z dr
n R At Wall, r = R n 1
n dv z 3n 1 1 vz dr
n
R Shear rate at wall in pipe
flow for PL fluid r n vz
1 W 3n 1 8v z 4n D 3n 1 W 4n Shear stress at wall in
pipe flow for PL fluid sw PR
2L Determining power law relationship in
Laminar
Other key equations for Power law fluid; Pipe Flow
Shear stress and Shear rate at the wall, r = R P sw s K n s w K w n PR
2L n w 34 n1 . Q 4Q 8v 3 This is the shear rate for a Newtonian fluid R D and Q in terms
hear stress P
v 2LK n 3nn1 R
1 n 1
n Example, Steffe p 143
Determine the rheological properties of bread dough using the data from a ‘capillary
viscometer’ (i.e Q and P from pipe flow using various L & Ds Measured Q & P Link to
excel sheet Example, Steffe p 143
Determine the rheological properties of bread dough using the data from a ‘capillary
viscometer’ (i.e Q and P from pipe flow using various L & Ds Measured Q & P Link to
excel sheet Example, Steffe p 143
Determine the rheological properties of bread dough using the data from a ‘capillary
viscometer’ (i.e Q and P from pipe flow using various L & Ds 4Q 3 R Link to
excel sheet 220
5.4 Wall Shear Stress (
Ln( w) w) 200
5.2
180 n = 0.28
K’=eIntercept
K’ = 27.6
sw=27.6 w0.28 y = 27.6 x0.28 Intercept = 3.32
slope = 0.28
r ² 0.9764522388 5.0
160 4n K K ' 3n 1 140
4.8 n 120 K= 24.07 4.6
100 s K n 4.4
80
60
4.2
0
3.5 s w 24w
4.0 200
4.5 400
5.0
5.5 600
6.0 800
6.5
7.0 0.28 s 24 0.28 1000
7.5 1 Generalised shear rate ( s )
Ln Example, Steffe p 143
Determine the rheological properties of bread dough using the data from a ‘capillary
viscometer’ (i.e Q and P from pipe flow using various L & Ds
n w 34 n1 . My answer: s 24 0.2 8 Link to
excel sheet Reynolds Number: Power Law Fluid
For Power Law Fluid: Kw n 1 Re v D Re PL Still valid, but now
viscosity depends
on shear rate v 2n D n
8n1 K 34nn1 n Q
n
n
V
w 34 n 1 4R 34 n 1 8D...
View
Full
Document
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

Click to edit the document details