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Transport processes
7. Semester
Chemical Engineering
Civil Engineering
Course plan
1. Elementary Fluid Dynamics2. Fluid Kinematics3. Finite Control Volume Analysis4. Differential Analysis of Fluid Flow5. Viscous Flow and Turbulence6. Turbulent Boundary Layer Flow7. Principles of Heat Transfer8. Internal Forces Convection9. Unsteady Heat Transfer10. Boiling and Condensation11. Mass Transfer12. Porous Media Flow13. Non-Newtonian Flow
Today's lecture
• Non-Newtonian Fluids
Newtonian fluids
• Newton’s law of fluids
dudy
τ µ=
Shear stressRate of shear strain
Constant of proportionality
Non-newtonian
nduKdy
τ
=
Shear stressRate of shear strain
Constant of proportionality
Flow behavior index
Newtonian vs NN fluids
• Which fluids are Newtonian?– All gasses
– Most common liquids incl. water and oil
• Which fluids are Non-Newtonian (NN)?– Many Slurry flow (eg. Water-sand mixture /quicksand)
– Many “foods” and like (i.e. Toothpaste, mayonnaise )
– Many Polymers (long chained molecules)
Liquids with a more complex molecular structure
Fluids with a simple molecular structure
Non-Newtonian fluids
• Time independent NN fluids– Bingham plastic
– Shear thinning (Pseudoplastic )
– Shear thickening (Dillatant)
• Time dependent NN fluids– Thicxotropic
• i.e. Ketchup
– Rheopectic• i.e. Whipped cream
Bingham plastic
• Same as Newtonian fluid but the curve does not go through origin
• Examples: drilling mud, toothpaste, paper pulp, sewage sludge
when >
when 0
o o o
o
dudydudy
τ τ µ τ τ
τ τ
= +
< =
Shear thinning
• Most NN fluids are in this class
• Represented by the power law (Ostwald-de Waeleequation)
• Examples: Polymer solutions or melts, greases, mayonnaise, paints, …
1
Apparent viscosity
1
1
decreases with increasing shear rate
a
n
n
n
a
duK ndy
du duKdy dy
duKdy
µ
τ
τ
µ
−
−
= <
=
= ⇒
Shear thickening
• Also represented by the power law (Ostwald-deWaele equation)
• Examples: corn flour-sugar solutions, solutions with high concentration of powders in water
1
Apparent viscosity
1
1
increases with increasing shear rate
a
n
n
n
a
duK ndy
du duKdy dy
duKdy
µ
τ
τ
µ
−
−
= >
=
= ⇒
Properties of NN fluids (time-independent)
• Measurement of Δp vs. flowrate V…
• ..in tube of length L and diameter D, then:
capillary-tube viscometer
4wD p
Lτ ∆
= 8
r R
du Vdr D=
=
Intermezzo: determination of τw
• From force balance on a cylindrical element:
Simplifying gives:
At the wall we have:
Rearranging gives:
• Laminar velocity profile:
• thus
Intermezzo: determination of du/dy
2
max 2
2 2 4 81r R r R r R
du d r V V VV rdr dr R R R D= = =
⋅ = − = = =
2
max( ) 1 ru r VR
= −
Properties of NN fluids
• For a power law fluid:
• Coefficients K’ and n’ can be found from curve fitting
• A generalized viscosity can be defined as:
• If the fluid properties are given as K and n than the following holds:
'
w8'
4
nD p VKL D
τ ∆ = = ⋅
' 1'8nKγ −=
'3 ' 1' ; '4 '
nnn n K Kn+ = =
Some values of K, n’ and γ
NN-fluid flow in pipes
• For pipe flow, what are we interested in determining?– Laminar or turbulent flow
– Pressure loss due to friction
– Pressure loss in components
– Velocity profiles
– …
Similar methods to that of Newtonian fluids Theoretic approach for Laminar flow
Empirical basis for turbulent flow
Laminar flow of NN-fluids
• Reynolds number:
Laminar : Re<2000
Turbulent: Re>10.000
' 2 ' ' 2 ' 2
gen ' 11
Re'8 3 18
4
n n n n n n
n nn
D V D V D VK nK
n
ρ ρ ργ
− − −
−−
= = =+
Laminar flow of NN-fluids
• Pressure drop in pipe due to friction:– Method 1: From derivation
– Method 2: using friction factors (fanning) • Friction factor relation from Newtonian fluids but using the generalized
Reynolds number for NN-fluids
'' 4 8 nK L VpD D
∆ =
2
42
L Vp fD
ρ∆ =Re, gen
16fN
=
19
20
21
Flow of NN-fluids
• Mechanical energy balance:
2 21 2
1 1 2 22 2 lossV VP g z P g z Pρ ρρ ρα α
+ + = + + + ∆
Coefficient due to the viscous effects
Laminar Turbulent
Newtonian
Non-Newtonian
0.5α = 1.0α ≈
1.0α ≈( )( )( )2
2 1 5 33 3 1n n
nα
+ +=
+
Flow of NN-fluids
• Pressure loss in components:– Losses in contractions and fittings
same as for Newtonian fluids
– Losses in sudden expansion in Laminar flow
– Losses in sudden expansion in Turbulent flow flow
same as for Newtonian fluids
( )( )( ) [ ]
4 22 1 1
12 2
3 3 13 1 3 /2 1 2 5 3 2 5 3ex
nD Dn nh V J kgn n D D n
+ + + = − + + + +
Turbulent flow of NN-fluids
• Use the friction factor approach (empirical):– Empirical coefficients for smooth pipes (no data for rough pipes!)
– Note: Fanning friction factor used in Geankoplis
2
42
L Vp fD
ρ∆ =
25
26
Laminar Velocity profile of NN fluids
Laminar Velocity profile of NN fluids
• Found using same procedure as for Newtonian fluids:
1 11 1
maxv 1 =v 11 2
n nnn n n
nn p r rRn LK R R
+ ++ ∆ = − − +
Laminar Velocity profile of NN fluids
• Flow rate:
• Average velocity:
• Velocity ratio:
1 1
maxv1 2
nnnn p R
n LK
+∆ = +
1 1
av 2v3 1 2
nnnQ n p R
n LKRπ
+∆ = = +
av
max
v 1v 3 1
nn+
=+
( )1 3 1
3 1 2
nnnn pQ u r dA R
n LKπ +∆ = = + ∫
Bingham plastic fluids
• Stress relationship
when >o o odudy
τ τ µ τ τ
= +
Yield stress
Laminar stress(same as for Newtonian fluids)
when 0odudy
τ τ< =
Bingham plastic fluids• Velocity profile
• Relationship between r0 and τ0:
( )2
2maxu 1 0
4o
centerline orpr u u R for r r
K Rµ ∆ = = = − < <
2o op rL
τ ∆=
( )22
0u 1 116
opD r rr R for r rL R R
τµ µ
∆ = − − − >
Bingham plastic fluids• Flow rate
• Buckingham-Reiner equation:
44 4 118 3 3
o o
w w
pRQL
τ τπµ τ τ
∆ = − +
( ) ( ) ( )0
002 2
r R
rQ u r dA u r rdr u r rdrπ π= = +∫ ∫ ∫
2wpRwhereL
τ ∆=
33
34
35
Bingham plastic fluids
• There exist friction factors to calculate the pressure loss for both laminar and turbulent flow for Bingham plastic fluids (not shown in Geankoplis)
2
42
L Vp fD
ρ∆ =
Laminar Bingham Plastic Flow
( )
−+= 73
4
Re3Re61
Re16
BPBPBP fHeHef
20
2
∞
=µτρDHe
∞
=µρVD
BPRe
Hedstrom Number
(Non-linear)
Turbulent Bingham Plastic Flow
( )Hex
BPa
ea
f5109.2
193.0
146.01378.1
Re10−−
−
+−=
=
Excercises
• Exercises, solutions for this lecture: On the web
• Examination questions will be available 2 weeks prior to the exam. They will be submitted by email and I will submit them on the web as well!
(k7c-1-e10@bio.aau.dk); (k7c-2-e10@bio.aau.dk); (k7c-3-e10@bio.aau.dk); (k7c-4-e10@bio.aau.dk); (k7c-5-e10@bio.aau.dk); (k9-4-e10@bio.aau.dk); (k9-6-e10@bio.aau.dk); (k9-7-e10@bio.aau.dk)
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