lectures 1&2 f. morrison cm3110 11/3/2019fmorriso/cm310/lectures/2019 fluids...
TRANSCRIPT
Lectures 1&2 F. Morrison CM3110 11/3/2019
1
© Faith A. Morrison, Michigan Tech U.1
As teachers we can choose between
(a) sentencing students to thoughtless mechanical operations and (b) facilitating their ability to think.
If students' readiness for more involved thought processes is bypassed in favor of jamming more facts and figures into their heads, they will stagnate at the lower levels of thinking. But if students are encouraged to try a variety of thought processes in classes, then they can … develop considerable mental power. Writing is one of the most effective ways to develop thinking.
—Syrene Forsman
Reference: Forsman, S. (1985). "Writing to Learn Means Learning to Think." In A. R. Gere (Ed.), Roots in the sawdust: Writing to learn across the disciplines (pp. 162-174). Urbana, IL: National Council of Teachers of English.Professor Faith A. Morrison
Department of Chemical EngineeringMichigan Technological University
Transport/Unit Operations
© Faith A. Morrison, Michigan Tech U.2
Professor Faith A. Morrison
Department of Chemical EngineeringMichigan Technological University
CM2120—Fundamentals of ChemE 2 (Steady Unit Operations Introduction, MEB)CM3110—Transport/Unit Ops 1 (Momentum & Steady Heat Transport, Unit Operations)CM3120—Transport/Unit Ops 2 (Unsteady Heat Transport, Mass Transport, Unit
Operations
Lectures 1&2 F. Morrison CM3110 11/3/2019
2
© Faith A. Morrison, Michigan Tech U.
Why study transport/unit ops?
3
© Faith A. Morrison, Michigan Tech U.
Why study transport/unit ops?
•Modern engineering systems are complex and often cannot be operated and maintained without analytical understanding
4
•Design of new systems will come from high-tech innovation, which can only come from detailed, analytical understanding of how physics/nature works
Image: wikipedia.org
Imag
e: p
lane
tforw
ard.
ca
Lectures 1&2 F. Morrison CM3110 11/3/2019
3
© Faith A. Morrison, Michigan Tech U.
Where are we now?
5
© Faith A. Morrison, Michigan Tech U.
Where are we now?
6
CM2110
CM2120
Summary
CM21101. Steady mass balances2. Steady energy balances (how to calc. energy)3. MEB‐Mechanical Energy Balance (no friction)
CM2120/CM32151. MEB‐Mechanical Energy Balance (with friction)2. Pumps3. Introduction to Unit Operations4. Staged Unit Operations (distillation, absorption)
Lectures 1&2 F. Morrison CM3110 11/3/2019
4
© Faith A. Morrison, Michigan Tech U.
Where are we now?
7
CM3110Summary
CM31101. Steady momentum balances (macro
and micro)2. Rate‐based heat transfer processes
(Fourier’s law, heat transfer coefficients, radiation)
3. Unit Operations involving heat transfer (Heat Exchangers)
© Faith A. Morrison, Michigan Tech U.
CM3110
Transport Processes and Unit Operations I
Professor Faith Morrison
Department of Chemical EngineeringMichigan Technological University
www.chem.mtu.edu/~fmorriso/cm310/cm310.html
CM3110 - Momentum and Heat Transport
8
Lectures 1&2 F. Morrison CM3110 11/3/2019
5
© Faith A. Morrison, Michigan Tech U.
9
EMERGENCY EVACUATION PROCEDURES
Important: The Michigan Bureau of Fire Services has adopted new rules for colleges and universities effective 2015
1. Only residence halls are required to hold fire and tornado drills.2. In lieu of fire drills in other university buildings all faculty and instructional staff are required to do the following on the first day of class:
- Explain the university fire evacuation procedures to the class (see below).- Explain the locations of the primary and secondary exit routes for your class
location.- Explain your designated safe location where the class will meet after evacuating
the building.3. The class instructor is responsible for directing the class during a building evacuation.
General evacuation procedure:- Use the nearest safe exit route to exit the building. The nearest safe exit from room 07-0100 is the back (south) entrance that is close to the Portage Canal and just outside our door, to the right. The secondary exit is the campus (north) exit, that exits near the Husky. The nearest safe exit from Fisher 139 is - Close all doors on the way out to prevent the spread of smoke and fire.- After exiting, immediately proceed to a safe location at least 100 feet from the building. Our designated safe location is in front of the MUB, near the circle drive.- Do not re-enter the building until the all-clear is given by Public Safety or the fire department.
TR Section
© Faith A. Morrison, Michigan Tech U.
10
EMERGENCY EVACUATION PROCEDURES
Important: The Michigan Bureau of Fire Services has adopted new rules for colleges and universities effective 2015
1. Only residence halls are required to hold fire and tornado drills.2. In lieu of fire drills in other university buildings all faculty and instructional staff are required to do the following on the first day of class:
- Explain the university fire evacuation procedures to the class (see below).- Explain the locations of the primary and secondary exit routes for your class
location.- Explain your designated safe location where the class will meet after evacuating
the building.3. The class instructor is responsible for directing the class during a building evacuation.
General evacuation procedure:- Use the nearest safe exit route to exit the building. The nearest safe exit from room 15-139 is the front (south) entrance that is close to highway 41. The secondary exit is the campus (north) exit, that connects to the main path through campus.- Close all doors on the way out to prevent the spread of smoke and fire.- After exiting, immediately proceed to a safe location at least 100 feet from the building. Our designated safe location is east of Fisher, in the parking lot of the Center for Diversity and Inclusion.- Do not re-enter the building until the all-clear is given by Public Safety or the fire department.
MW Section
Lectures 1&2 F. Morrison CM3110 11/3/2019
6
© Faith A. Morrison, Michigan Tech U.
Why study fluid mechanics?
11
© Faith A. Morrison, Michigan Tech U.
Why study fluid mechanics?
•It’s required for my degree
12
Lectures 1&2 F. Morrison CM3110 11/3/2019
7
© Faith A. Morrison, Michigan Tech U.
Why study fluid mechanics?
•It’s required for my degree (too literal)
13
© Faith A. Morrison, Michigan Tech U.
Why study fluid mechanics?
•It’s required for my degree (too literal)
•Fluids are involved in engineered systems
14
Image from: money.cnn.com
Image from: newegg.com
Lectures 1&2 F. Morrison CM3110 11/3/2019
8
© Faith A. Morrison, Michigan Tech U.
Why study fluid mechanics?
•It’s required for my degree (too literal)
•Fluids are involved in engineering systems (many devices that employ fluids can be operated and maintained and sometimes designed without detailed mathematical analysis)
15
© Faith A. Morrison, Michigan Tech U.
Why study fluid mechanics?
•Modern engineering systems are complex and often cannot be operated and maintained without analytical understanding
16
•Design of new systems will come from high-tech innovation, which can only come from detailed, analytical understanding of how physics/nature works
Image: wikipedia.org
Imag
e: p
lane
tforw
ard.
ca
•It’s all part of learning to think and perform like an engineer. We will be intentionally ordering our knowledge and practicing asking relevant questions.
Lectures 1&2 F. Morrison CM3110 11/3/2019
9
© Faith A. Morrison, Michigan Tech U.
O2 heat
exchanger
pump
membrane oxygenator – oxygen goes into blood; carbon dioxide comes out
O2
MO
spent blood
fresh blood
Membrane Oxygenator, MO(Heart-Lung Machine)
Secondary flow drives the O2 mass transfer
17
© F
aith
A.
Mor
rison
, M
ichi
gan
Tech
U.
Artificial Heart
Image: health.howstuffworks.com/medicine/modern/artificial-heart.htm
18
Lectures 1&2 F. Morrison CM3110 11/3/2019
10
Microfluidics – Lab on a Chip
© F
aith
A.
Mor
rison
, M
ichi
gan
Tech
U.
www.nature.com/nmeth/journal/v4/n8/full/nmeth0807-665.html
19
Sensors, diagnostics
And more. . . .
© F
aith
A.
Mor
rison
, M
ichi
gan
Tech
U.
20
HelicoptersAirplanesQuieter fansFlexible body armorUndersea oil drillingSurgeryFood processingPlastics2D and 3D printingBattery manufactureCelestial explorationVolcanos Biomedical devices (stents, artificial organs, prosthetics)Sensor development…
Ima
ge fr
om
: e
n.w
ikip
edi
a.o
rg
Lectures 1&2 F. Morrison CM3110 11/3/2019
11
© Faith A. Morrison, Michigan Tech U.
Where to start?
21
© Faith A. Morrison, Michigan Tech U.
Where to start?
22
We’ve already started.
Lectures 1&2 F. Morrison CM3110 11/3/2019
12
© Faith A. Morrison, Michigan Tech U.
We’ve already started.
23
1. We’ve learned fluid statics.
www.youtube.com/watch?v=zeNQOqr63cc
DrMorrisonMTU on YouTube: Introduction to Manometers: Two Essential Rules
On 3Sept19 #views >134,000!
© Faith A. Morrison, Michigan Tech U.
We’ve already started.
24
2. There are flow problems that can be addressed with one type of macroscopic energy balance:
2
,
2 2
, ,212 1 2 12 1 21
ΔΔΔ
2
p p(z z )
2
s on
s on
Wvpg z F
vg
m
m
v WF
1. single-input, single output2. Steady state3. Constant density (incompressible fluid)4. Temperature approximately constant5. No phase change, no chemical reaction6. Insignificant amounts of heat transferred
The Mechanical Energy Balance
Assumptions:
𝐹 friction
Lectures 1&2 F. Morrison CM3110 11/3/2019
13
© Faith A. Morrison, Michigan Tech U.
Flow in Pipes
25
For example:
pump
ID=3.0 in ID=2.0 in
75 ft
tank
50 ft 40 ft
20 ft
8 ft
1
2
1. Single-input, single output2. Steady state3. Constant density (incompressible fluid)4. Temperature approximately constant5. No phase change, no chemical reaction6. Insignificant amounts of heat transferred
Mechanical Energy Balance
© Faith A. Morrison, Michigan Tech U.
)( ftH
)(1,2 VH
)(, VH sd
)/( mingalV
valve 50% open
valve full open
pump
system
Centrifugal Pumps
What flow rate does a centrifugal pump produce?
Answer: Depends on how much work it is asked to do.
26
For example:
Calculate with the Mechanical Energy
Balance(CM2110, CM2120,
CM3215)
Lectures 1&2 F. Morrison CM3110 11/3/2019
14
© Faith A. Morrison, Michigan Tech U.
27
We can apply the MEB to many important engineering systems
1. single-input, single output2. Steady state3. Constant density (incompressible fluid)4. Temperature approximately constant5. No phase change, no chemical rxn6. Insignificant amounts of heat
transferred
MEB Assumptions:
Ima
ge fr
om
: w
ww
.ep
a.g
ov
Image from: www.directindustry.com
Image from: www.processindustryforum.com
Image from: www.directindustry,com
Calculate: Work,
pressures, flows
© Faith A. Morrison, Michigan Tech U.
Example 1
28
How many horsepower (hp) will it take to raise water at 4.0 𝑔𝑝𝑚 from a flooded basement to the surface (see figure)? The hose is smooth; as a first calculation we can neglect friction.
37.0𝑚Inner diameter=0.50𝑖𝑛
0.46𝑚
𝑊 , shaft work
Lectures 1&2 F. Morrison CM3110 11/3/2019
15
© Faith A. Morrison, Michigan Tech U.
Example 1
29
ANSWER: 0.12 hp
(our TA has the solution: HW/Example help session Sunday 6:30-7:30, 19-211)
How many horsepower (hp) will it take to raise water at 4.0 𝑔𝑝𝑚 from a flooded basement to the surface (see figure)? The hose is smooth; as a first calculation we can neglect friction.
© Faith A. Morrison, Michigan Tech U.
30pages.mtu.edu/~fmorriso/cm310/convert.pdf
Handy Sheet for Unit Conversions
Lectures 1&2 F. Morrison CM3110 11/3/2019
16
© Faith A. Morrison, Michigan Tech U.
31pages.mtu.edu/~fmorriso/cm310/MorrisonCoverMatter(c)2011.pdf
Handy Sheet of Fluid Mechanics Equations from inside cover of Morrison
© Faith A. Morrison, Michigan Tech U.
Example 2
32
What is the volumetric flow rate at the drain from a constant-head tank with a fluid level h? You may neglect frictional losses.
Lectures 1&2 F. Morrison CM3110 11/3/2019
17
© Faith A. Morrison, Michigan Tech U.
Example 2
33
What is the volumetric flow rate at the drain from a constant-head tank with a fluid level h? You may neglect frictional losses.
ANSWER: 𝜋𝑅 2ℎ𝑔
For more examples: see CM2110/20 notes; HW1;
Prerequisite review readings
34
Exam 1: Next Tues 6:30-8:00pm, Dow 641The exam and solution from 2017 is on the web. TA help session is Sunday night. Exam topics: vectors, linear algebra, integration, balances, MEB, fluid statics
(Rev
iew
)
© F
aith
A.
Mor
rison
, M
ichi
gan
Tech
U.
HW1 (online and in Canvas)
Lectures 1&2 F. Morrison CM3110 11/3/2019
18
35
© F
aith
A.
Mor
rison
, M
ichi
gan
Tech
U.
• It is limited in application:
• It cannot determine flow patterns
• It does not model momentum exchanges
• It cannot be adapted to systems other than those for which it was designed (see list above)
© Faith A. Morrison, Michigan Tech U.
The Mechanical Energy Balance (MEB) is a macroscopic analysis.
36
1. single-input, single output2. Steady state3. Constant density (incompressible fluid)4. Temperature approximately constant5. No phase change, no chemical rxn6. Insignificant amounts of heat transferred
2 2
, ,212 1 2 12 1 21
p p(z z )
2s onv v W
g Fm
𝐹 friction
Lectures 1&2 F. Morrison CM3110 11/3/2019
19
© Faith A. Morrison, Michigan Tech U.
CM3110
Transport Processes and Unit Operations I
Professor Faith Morrison
Department of Chemical EngineeringMichigan Technological University
www.chem.mtu.edu/~fmorriso/cm310/cm310.html
CM3110 - Momentum and Heat TransportCM3120 – Heat and Mass Transport
37
Part 1:
© Faith A. Morrison, Michigan Tech U.
Energy balances (the MEB) can only take us so far with fluids modeling (due to assumptions).
38
To understand complex flows, we must use
the MOMENTUM balance.
Ima
ge fr
om
: w
ha
tsu
pw
ithth
at.c
om
Ima
ge fr
om
: w
ww
.12
3rf
.co
m
Image from: commons.wikipedia.orgNaruto Whirlpools, Japan
Image from: www-math.mtu.edu
Lectures 1&2 F. Morrison CM3110 11/3/2019
20
© Faith A. Morrison, Michigan Tech U.
Momentum Balance: Newton’s 2nd Law of Motion
39
𝑓 𝑚𝑎
Ima
ge fr
om
: w
ww
.te
xtu
rex.
com
PH 2100: apply momentum
conservation to individual bodies
CM 3110: apply momentum conservation
to a continuum
See also: http://youtu.be/6KKNnjFpGto
© Faith A. Morrison, Michigan Tech U.
Fluid Mechanics
40
• Continuum (density, velocity, stress fields)
• Control volume
• Stress in a fluid at a point (stress tensor)
• Stress and deformation (Newtonian constitutive equation)
• Microscopic and macroscopic momentum balances
• Internal flows – pipes, conduits
• External flows – drag, boundary layers
• Advanced fluid mechanics – complex shapes
𝑣𝑣𝑣𝑣
𝜏𝜏 𝜏 𝜏𝜏 𝜏 𝜏𝜏 𝜏 𝜏
𝜌 𝑥,𝑦, 𝑧
control volume
(calc3, differential equations)
(engineering quantities of
interest)
Lectures 1&2 F. Morrison CM3110 11/3/2019
21
© Faith A. Morrison, Michigan Tech U.
Momentum . . .
41
𝜌𝑣𝜌𝑣𝜌𝑣𝜌𝑣
is a vector
Microscopic momentum balance
gvPvvt
v
2
Macroscopic momentum balance
2# #
1 1
cosˆ ˆ
ii
streams streams
CVi i A
A
A vdv pAn R M g
dt
P
So we need vector math.
Ch 6
Ch 9
𝐶𝑎𝑙𝑐 3
zrz
r
xyzz
y
x
v
v
v
v
v
v
v
v
v
v
1233
2
1
Same vector, different coordinate systems, different components.
magnitudevectorvv
vectorunitv
v
v ˆ
© Faith A. Morrison, Michigan Tech U.
We choose coordinate systems for convenience.
Vectors
42
1x rv v v Note:
(usually)
Lectures 1&2 F. Morrison CM3110 11/3/2019
22
© Faith A. Morrison, Michigan Tech U.
creeping flow (sphere)
0.
01
0.2
0.6
1.2
2.0
3.0
Vector plot of the velocity field in slow flow around a
sphere
The flow is a steady upward flow; the length and direction of the vector indicates the velocity at that location.
43
Fluid velocity is a vector field
𝑣 𝑣 𝑥,𝑦, 𝑧
(calc3)
Vectors – Cartesian coordinate system
© Faith A. Morrison, Michigan Tech U.
•We do algebra with the basis vectors the same way as with other quantities
•The Cartesian basis vectors are constant (do not change with position)
44
(three ways of writing the same thing, the Cartesian basis vectors)
𝑣𝑣𝑣𝑣
𝑣 �̂� 𝑣 �̂� 𝑣 �̂�
𝑣𝑣𝑣𝑣
𝑣 �̂� 𝑣 �̂� 𝑣 �̂�
𝑣 𝚤̂ 𝑣 𝚥̂ 𝑣 𝑘
Lectures 1&2 F. Morrison CM3110 11/3/2019
23
© Faith A. Morrison, Michigan Tech U.
Vectors – Cylindrical coordinate system
•The cylindrical basis vectors are variable (depend on position)
ˆ ˆ ˆcos sin
ˆ ˆ ˆsin cos
ˆ ˆ
r x y
x y
z z
e e e
e e e
e e
cos
sin
x r
y r
z z
P
x
y
z
r
z
45
(see inside back cover of
text; also, supplemental
handouts)
𝑣𝑣𝑣𝑣
𝑣 �̂� 𝑣 �̂� 𝑣 �̂�
© Faith A. Morrison, Michigan Tech U.
Vectors – Spherical coordinate system
•The spherical basis vectors are variable (with position)
ˆ ˆ ˆ ˆsin cos sin sin cos
ˆ ˆ ˆ ˆcos cos cos sin ( sin )
ˆ ˆ ˆsin cos
r x y z
x y z
x y
e e e e
e e e e
e e e
sin cos
sin sin
cos
x r
y r
z r
46
(see inside back cover of
text; also, supplemental
handouts)
𝑣𝑣𝑣𝑣
𝑣 �̂� 𝑣 �̂� 𝑣 �̂�
Note: spherical coordinate system in use by the fluid mechanics community uses 0 𝜃 𝜋
as the angle from the 𝑧=axis to the point.
In your calc 3 class, they probably called this 𝜙and the other one 𝜃
Lectures 1&2 F. Morrison CM3110 11/3/2019
24
© Faith A. Morrison, Michigan Tech U.
creeping flow (sphere)
0.
01
0.2
0.6
1.2
2.0
3.0
47
Fluid Velocity is a Vector Field
Velocity magnitude and direction vary with position
𝑣 𝑣 𝑥,𝑦, 𝑧
Example 3: At positions (1,45o,0) and (1,90o,0) in the 𝑟,𝜃, 𝑧coordinate system, the velocity vector of a fluid is given by
© Faith A. Morrison, Michigan Tech U.
48
What is this vector in the usual 𝑥𝑦𝑧 coordinate system?
𝑣𝑣𝑣𝑣
010
Lectures 1&2 F. Morrison CM3110 11/3/2019
25
© Faith A. Morrison, Michigan Tech U.
49
𝑣
0
𝑣1
00
ANSWERS:
ℎ𝑖𝑛𝑡: 010
�̂�
Example 3: At positions (1,45o,0) and (1,90o,0) in the 𝑟,𝜃, 𝑧coordinate system, the velocity vector of a fluid is given by
What is this vector in the usual 𝑥𝑦𝑧 coordinate system?
𝑣𝑣𝑣𝑣
010
© Faith A. Morrison, Michigan Tech U.
We use Calculus in Fluid Mechanics to:
1. Calculate flow rate, 𝑄
2. Calculate average velocity, ⟨𝑣⟩
3. Express forces on surfaces due to fluids (vectors)
4. Express torques on surfaces due to fluids (vectors)
50
These are quantities of interest.These items are what we are learning to calculate.
Lectures 1&2 F. Morrison CM3110 11/3/2019
26
© Faith A. Morrison, Michigan Tech U.
2
0 0
ˆ( ) ( )
( )
area
R
z
Q v n d area
Q v r rdrd
1. Calculate Flow rate: or VQ
is the component of v in the direction normal to the area
ˆv n
General:
Tube flow:
51
© Faith A. Morrison, Michigan Tech U.
Common surface shapes in the standard coordinate systems:
2
: ( )
: ( )
: ( )
: ( ) ( ) sin sin
rectangular d area dxdy
circular d area r drd
surface of cylinder d area Rd dz
spherical d area rd r d r d d
52
(see inside back cover of
text; also, supplemental
handouts)
Lectures 1&2 F. Morrison CM3110 11/3/2019
27
© Faith A. Morrison, Michigan Tech U.
Example 4: Calculate the flow rate in flow down an incline plane of width W.
53
22
2
)cos()( xH
gxvz
HMomentum balance calculation gives:
(we will learn how to get this equation for 𝑣 𝑥 ; here it is given)
© Faith A. Morrison, Michigan Tech U.
Example 4: Calculate the flow rate in flow down an incline plane of width W.
54
22
2
)cos()( xH
gxvz
HMomentum balance calculation gives:
ANSWER:
𝑄𝐻 𝜌𝑔𝑐𝑜𝑠𝛽
3𝜇
(we will learn how to get this equation for 𝑣 𝑥 ; here it is given)
Lectures 1&2 F. Morrison CM3110 11/3/2019
28
© Faith A. Morrison, Michigan Tech U.
2. Calculate Average velocity:
2
Qv
area
Qv
R
v
“area” is the cross-sectional area normal to flow
General:
Tube flow:
55
© Faith A. Morrison, Michigan Tech U.
Example 5: The shape of the velocity profile for a steady flow in a tube is found to be given by the function below. Over the range 0 <r<10 mm, (R=10mm), what is the average value of the velocity?
56
Lectures 1&2 F. Morrison CM3110 11/3/2019
29
© Faith A. Morrison, Michigan Tech U.
57
ANSWER: 𝑣2
Example 5: The shape of the velocity profile for a steady flow in a tube is found to be given by the function below. Over the range 0 <r<10 mm, (R=10mm), what is the average value of the velocity?
© Faith A. Morrison, Michigan Tech U.
3. Express forces on surfaces due to fluids
58
𝐹 𝑛 ⋅ Π 𝑑𝑆Total fluid force on a
surface
Π ≡ 𝜏 𝑝�̳� Total stress tensor(calc3)
Lectures 1&2 F. Morrison CM3110 11/3/2019
30
© Faith A. Morrison, Michigan Tech U.
Example 6: In a liquid of density , what is the net fluid force on a submerged sphere (a ball or a balloon)? What is the direction of the force and how does the magnitude of the fluid force vary with fluid density?
59
(p81)
H0f
air
x
z
© Faith A. Morrison, Michigan Tech U.
Solution: We will be able to do this in this course (Ch4, p257).
From expression for force due to fluid, obtain: (in spherical coordinates)
We can do the math from here. (Calc 3)
60
𝐹 𝑛 ⋅ Π 𝑑𝑆Total fluid force on a
surface
𝐹 𝜌𝑔𝑅 𝐻 𝑅𝑐𝑜𝑠𝜃 �̂� sin𝜃 𝑑𝜃𝑑𝜙
Lectures 1&2 F. Morrison CM3110 11/3/2019
31
© Faith A. Morrison, Michigan Tech U.
Solution: We will be able to do this in this course (Ch4, p257).
From expression for force due to fluid, obtain: (in spherical coordinates)
61
ANSWER: (see p83)
𝑓
00
4𝜋𝑅 𝜌𝑔3
𝐹 𝑛 ⋅ Π 𝑑𝑆Total fluid force on a
surface
𝐹 𝜌𝑔𝑅 𝐻 𝑅𝑐𝑜𝑠𝜃 �̂� sin𝜃 𝑑𝜃𝑑𝜙
© Faith A. Morrison, Michigan Tech U.
4. Express torques on surfaces due to fluids
ˆS at surface
total fluid torqueR n dS
on a surfaceT
R lever arm
We will learn to write the stress tensor for our systems; then we can calculate stresses, torques.
62
pI total stress tensor
(Points from axis of rotation to position where torque is applied)
Lectures 1&2 F. Morrison CM3110 11/3/2019
32
Example 7, Torque in Couette Flow: A cup-and-bob apparatus is widely used to measure viscosities for fluids. For the apparatus below, what is the torque needed to turn the inner cylinder (called the bob) at an angular speed of ?
© Faith A. Morrison, Michigan Tech U.
63
© Faith A. Morrison, Michigan Tech U.
Torque in Couette FlowSolution:
1. Solve for velocity field (microscopic momentum balance)
2. Calculate stress tensor3. Formulate equation for torque (an integral)4. Integrate5. Apply boundary conditions
64
Lectures 1&2 F. Morrison CM3110 11/3/2019
33
© Faith A. Morrison, Michigan Tech U.
Torque in Couette FlowSolution:
65
See problem 6.22 p487
2
2
0
1
0r z
R r Rv
R r
Velocity solution:
Tv v
pI
ˆS at surface
total fluid torqueR n dS
on a surfaceT
What is lever arm, R?
etc…
𝜏𝜏 𝜏 𝜏𝜏 𝜏 𝜏𝜏 𝜏 𝜏
© Faith A. Morrison, Michigan Tech U.
Torque in Couette FlowSolution:
66
See problem 6.22 p487
ˆS at surface
total fluid torqueR n dS
on a surfaceT
ANSWER: (see p308)
𝑇4𝜋𝑅 𝜅 𝐿𝜇Ω𝜅 1
001
Lectures 1&2 F. Morrison CM3110 11/3/2019
34
© Faith A. Morrison, Michigan Tech U.
Summary of Quick Start
A: Mechanical Energy Balance
B): Use Calculus in Fluid Mechanics to
1. Calculate flow rate 2. Calculate average velocity3. Express forces on surfaces due to fluids4. Express torques on surfaces due to fluids
67
1. SI-SO, steady, incompressible, no rxn, no Δ𝑇, no 𝑄2. Macroscopic, based on energy (not momentum)3. Choose points 1 and 2 wisely4. Solve for 𝐹 or 𝑊 , or 𝑝, velocity, elevation
𝐹 friction2
,
2 2
, ,212 1 2 12 1 21
ΔΔΔ
2
p p(z z )
2
s on
s on
Wvpg z F
vg
m
m
v WF
© Faith A. Morrison, Michigan Tech U.
68
Summary of Quick Start
A: Mechanical Energy Balance
B): Use Calculus in Fluid Mechanics to
1. Calculate flow rate 2. Calculate average velocity3. Express forces on surfaces due to fluids4. Express torques on surfaces due to fluids
1. SI-SO, steady, incompressible, no rxn, no , no 2. Macroscopic3. Choose points 1 and 2 wisely4. Solve for or or , velocity, elevation
End of Quick Start.
We have reviewed:• MEB (an energy balance)• Math tools
Now, on to Fluid Mechanics, i.e. momentum transport.
Lectures 1&2 F. Morrison CM3110 11/3/2019
35
© Faith A. Morrison, Michigan Tech U.
69