CM3110 Transport Processes and Unit Operations Ifmorriso/cm310/lectures/2019 fluids lecture… ·...
Transcript of CM3110 Transport Processes and Unit Operations Ifmorriso/cm310/lectures/2019 fluids lecture… ·...
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Lecture 9 F. Morrison CM3110 10/3/2019
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CM3110 Transport Processes and Unit Operations I
© Faith A. Morrison, Michigan Tech U.
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The Weissenberg effect is when a viscoelastic, non‐Newtonian fluid will climb a rotating shaft.
Fluid MechanicsNon‐Newtonian fluids –An Introduction
Photo by
Carlos A
rang
oSabo
gal, U.
Wisc
onsin
, Mad
ison
https://www.youtube.com/watch?v=npZzlgKjs0I
© Faith A. Morrison, Michigan Tech U.
Continuum Modeling—Newtonian
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Intro to Non‐Newtonian Fluid Mechanics
• 𝜇 (viscosity) constant• 𝜌 (density) constant often a good assumption• �̃� 𝜇 𝛻𝑣 𝛻𝑣
—Viscosity differs for different materials—Is a function of T, perhaps P
The form of the function 𝝉 𝒗 is known
This is MAJOR. Predictions seem to be right in a wide
variety of situations (water, oil, air)
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Lecture 9 F. Morrison CM3110 10/3/2019
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© Faith A. Morrison, Michigan Tech U.
Continuum Modeling—Non‐Newtonian
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• 𝜇 (viscosity) NOT constant• 𝜌 (density) constant is often a good assumption• �̃� ⋯ ? The form of the function
𝝉 𝒗 is NOT known
For non‐Newtonian Fluids, we need a new form of �̃� 𝑣 that matches material observations.
Intro to Non‐Newtonian Fluid Mechanics
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Momentum Flux
Momentum (𝔭) = mass * velocity
vectors
top plate has momentum, and it transfers this momentum to the top layer of fluid
V
momentum fluxy
z
0zv
Vvz
H )(yvz
Viscosity determines the magnitude of
momentum flux
𝔭 𝑚𝑣
How do Fluids Behave?
Each fluid layer transfers the momentum downward
Intro to Non‐Newtonian Fluid Mechanics
When did we separate out non-Newtonian fluids?
Lecture 2:
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Lecture 9 F. Morrison CM3110 10/3/2019
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Newtonian
Newtonian Fluids
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1
xv
slope = viscosity
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Newton’s Law of Viscosity
(unidirectional flow)
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�̃� 𝜇 𝜕𝑣𝜕𝑥 �̃�
Intro to Non‐Newtonian Fluid Mechanics
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1
xv
slope = o21
shear‐thinning or pseudoplastic
shear‐thickening or dilatant
Bingham plastic
Newtonian
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Non‐Newtonian Fluids
�̃�
�̃�
Non‐Newtonian behavior (unidirectional flow)
�̃� function of 𝜕𝑣𝜕𝑥
Many different
behaviors are observed.
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Lecture 9 F. Morrison CM3110 10/3/2019
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How can we define viscosity for Non‐Newtonian Fluids?
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1
xv
© Faith A. Morrison, Michigan Tech U.
1x
2x
)( 21 xv
(NOTE on coordinate system: Viscosity definition is written for shear flow in x1direction and gradient in x2 direction)
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𝜂 ≡ �̃�𝛾Non‐Newtonian viscosity (unidirectional flow)
• Perform this:
• Measure 𝝉𝟐𝟏
• Calculate this:
log
log
o
log
log
o
Typical polymeric behavior
© Faith A. Morrison, Michigan Tech U.
Viscosity (Greek letter eta)
zero‐shear viscosity (Newtonian plateau)
shear rate (“gamma dot”)
shear thinning
The changes in viscosity with shear rate are so large they must be plotted log‐log
2
1
xv
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Non‐Newtonian behavior (unidirectional flow)
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Lecture 9 F. Morrison CM3110 10/3/2019
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x1
x2)( 21 xv
In addition, for many polymers there are shear‐induced
NORMAL (perpendicular) forces.
force on 2‐surface in1‐direction
force on 2‐surface in2‐direction
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© Fa
ith A. M
orrison
, Mich
igan
Tech U.
H
22S
p dS�̃� 21
S
dS�̃�
© Faith A. Morrison, Michigan Tech U.10
Newtonian Constitutive Equation
�̃� 𝜇𝛾 𝜇 𝛻𝑣 𝛻𝑣 )
Newton’s law of viscosity is a special case of the Newtonian Constitutive equation.
(Unidirectional flow)
How to deal with this?Recall, for Newtonian fluids:
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Lecture 9 F. Morrison CM3110 10/3/2019
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Newtonian Constitutive Equation
© Faith A. Morrison, Michigan Tech U.11
�̃� �̃� �̃��̃� �̃� �̃��̃� �̃� �̃�
𝜇
2 𝜕𝑣𝜕𝑥𝜕𝑣𝜕𝑦
𝜕𝑣𝜕𝑥
𝜕𝑣𝜕𝑧
𝜕𝑣𝜕𝑥
𝜕𝑣𝜕𝑦
𝜕𝑣𝜕𝑥 2
𝜕𝑣𝜕𝑦
𝜕𝑣𝜕𝑦
𝜕𝑣𝜕𝑧
𝜕𝑣𝜕𝑧
𝜕𝑣𝜕𝑥
𝜕𝑣𝜕𝑦
𝜕𝑣𝜕𝑧 2
𝜕𝑣𝜕𝑧
�̃� 𝜇𝛾 𝜇 𝛻𝑣 𝛻𝑣 )Non‐Newtonian𝜂 𝛾
𝜂 𝛾
Generalized Newtonian Fluid
(non‐Newtonian)𝜂 𝛾 We pick the form of this function that works best
with our data.
Power‐Law Model
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1 121
2 2
ndv dvmdx dx
m or K = consistency index (m = for Newtonian)n = power‐law index (n = 1 for Newtonian)
rateshearxv
2
1
© Faith A. Morrison, Michigan Tech U.
(does notmodel normal stresses)
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�̃� =
Non‐Newtonian behavior (unidirectional flow)
�̃� function of 𝜕𝑣𝜕𝑥
𝜂 𝛾 𝑚𝛾
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Lecture 9 F. Morrison CM3110 10/3/2019
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© Faith A. Morrison, Michigan Tech U.
What does the power‐law model predict for viscosity?
1
21 21 1
21
2
ndvmdxdv
dx
1
1 121
2 2
ndv dvmdx dx
On a log‐log plot, this would give a straight line:
2
1log1loglogdxdvnm
Y B M X13
�̃� �̃� �̃�
Power‐Law Fluid
2
1
xv
© Faith A. Morrison, Michigan Tech U.
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log
Newtonian
shear thickening
shear thinning
Non‐Newtonian viscosity
1,1 nm n
1,1 nm n
1,1 nm n
log14
�̃�
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Lecture 9 F. Morrison CM3110 10/3/2019
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© Faith A. Morrison, Michigan 15
www.chem.mtu.edu/~fmorriso/cm310/Navier.pdf
vvt
The one with “𝝉” is for non‐
Newtonian fluids
Where do we use the power‐law expression?
16© Faith A. Morrison, Michigan Tech U.
http://www.chem.mtu.edu/~fmorriso/cm310/stpl.pdf
We have a handout for that:
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Lecture 9 F. Morrison CM3110 10/3/2019
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Where do we use the power‐law expression?
rL
PPgL Lorz
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e.g., Poiseuille flow in a tube:
zrz
dvdr
Newtonian
1nz z
rzdv dvmdr dr
non‐Newtonian, power‐law
solve for 𝒗𝒛 𝒓
© Faith A. Morrison, Michigan Tech U.
1‐direction = r2‐direction = z 17
�̃�
�̃�
�̃�
EXAMPLE III: Pressure‐driven flow of a Power‐law fluid in a tube
•steady state•incompressible•well developed•long tube
g
z cross-section A:
A
r
r
z
L vz(r)
R fluid
Calculate velocity and stress profiles
© Faith A. Morrison, Michigan Tech U.18
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Lecture 9 F. Morrison CM3110 10/3/2019
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Calculate the velocity field forPressure‐driven flow of a power‐law (PL) fluid:
© Faith A. Morrison, Michigan Tech U.
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no Lz z
rz
L g P Pdv dvm rdr dr L
rrvz
1n nz z zdv dv dvm m r
dr dr dr
Solve for vz(r)19
�̃�
Non‐Newtonian behavior (unidirectional flow)
�̃� PL function of 𝜕𝑣𝜕𝑥
Boundary Conditions:
© Faith A. Morrison, Michigan Tech U.
?
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(same as before in the Newtonian case)
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Lecture 9 F. Morrison CM3110 10/3/2019
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Velocity fieldPoiseuille flow of a power‐law fluid:
1 1 1
12
20 0
112 1
( )1 ( )1 3 2
n no Lz
R no L
z
R L g P P R rv rLm R
n
R P Pnv v r rdrd RR n mL
© Faith A. Morrison, Michigan Tech U.21
Non‐Newtonian behavior (unidirectional flow)
�̃� PL function of 𝜕𝑣𝜕𝑥
Solution to Poiseuille flow in a tubeincompressible, power‐law fluid
0.0
0.5
1.0
1.5
2.0
0 0.2 0.4 0.6 0.8 1 1.2r / R
vz/v
z,av
0.0
0.2
0.4
0.6
n=1.0
0.8
© Faith A. Morrison, Michigan Tech U.22
Non‐Newtonian behavior (unidirectional flow)
�̃� PL function of 𝜕𝑣𝜕𝑥
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Lecture 9 F. Morrison CM3110 10/3/2019
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© Faith A. Morrison, Michigan Tech U.
•Processing (design, costs, production rates)
Rheology affects:
•End use (food texture, product pour, motor-oil function)
•Product quality (surface distortions, anisotropy, strength, structure development)
www.math.utwente.nl/ mpcm/aamp/examples.html
www.corrugatorman.com/ pic/akron%20extruder.JPG
Pomar et al. JNNFM 54 143 1994
Rheology (Non-Newtonian Fluid Mechanics)
Non‐Newtonian behavior (all flows)�̃� nonlinear function of 𝛻𝑣
© Faith A. Morrison, Michigan Tech U.24
At Michigan Tech:CM4650 Polymer Rheology (Even years spring)www.chem.mtu.edu/~fmorriso/cm4650/cm4650.html
Rheology (Non-Newtonian Fluid Mechanics)
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Next:
© Faith A. Morrison, Michigan Tech U.