ES 202
Fluid and Thermal Systems
Lecture 10:
Pipe Flow (Major Losses)
(1/6/2003)
Assignments
• Reading:
– Cengel & Turner Section 12-5, 12-6
• Homework:
– 12-25, 12-35, 12-42 in Cengel & Turner
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Road Map of Lecture 10
• Announcements
• Comments on Lab 1
• Recap from Lecture 9
– Modified Bernoulli’s equation
• Concept of viscosity
• Pipe friction
–
–
–
–
–
friction factor
significance of Reynolds number
laminar versus turbulent
Moody diagram
flow chart to determine friction factor
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Announcements
• Lab 2 this week
– dress casually
– you may get wet
– formation of lab group of 2-3 students (need to split into 2 groups, 1.5
periods per group)
– report your lab group to me by the end of today via email, otherwise I
will assign you
• Extra evening office hours this week (8 pm to 10 pm)
• (From tutor) Review package for Exam 1 available at the
Learning Center and the new residence hall
• Review session for Exam 1 on Saturday evening 8 pm to 10 pm
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Comments on Lab 1
•
Write your memorandum as if your project manager will not read your attachments:
•
Fundamental rules of plotting
– list your p groups
– state the functional relationship between p groups
– always label your axes clearly
– always plot the dependent variable against the independent variable!
•
Unit conversion is a good practice but NOT necessary in forming p groups (one of
the many advantages of p group)
•
What is a log-log plot for? (Not necessary if a linear functional relationship exists!)
•
Invariance of p term does NOT imply invariance of dependent variable
•
Marking scheme (total 10 points):
–
–
–
–
correct p group formulation: 4 points
correct plotting of data: 4 points
correct conclusion of data: 2 points
coherence (adjustment up to 2 points)
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Summary of Lab 1 Write-up
• Upon dimensional analysis, the relevant p terms are
found to be
Part a:
p1 
Part b:
P

, p2 
h
D
p1 
Vavg
D
, p2 
h
D
• Data analysis reveals
Part a:
P
Part b:
D
 constant  

h
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2
Vavg
D
ES 202 Fluid & Thermal Systems
 constant
6
Recap from Lecture 9
• The Torricelli experiment (A2<< A1)
Area = A1
H
V
Area = A2
V
2 gH
 A2 
1   
 A1 
2
• The “Bent” Torricelli experiment
H
Lecture 10
V
ES 202 Fluid & Thermal Systems
hmax  H
7
“Modified” Bernoulli’s Equation
• What if fluid friction causes some losses in the
system, can I still apply the Bernoulli’s equation?
• Recall the “conservation of energy” concept from
which we approach the Bernoulli’s equation
• Remedy: introduce a “head loss” factor
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One Major Reason for the Losses
• Fluid friction
– also termed “Viscosity”
– basketball-tennis-ball demonstration
– exchange of momentum at the molecular scales (nature prefers
“average”)
– no-slip conditions at the solid surface (imagine thin layers of fluid
moving relative to one another) generates velocity gradients
– the two-train analogy
– stress-strain relation in a Newtonian fluid
stress  viscosity  strain rate
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du
 
dy
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Frictional Pipe Flow Analysis
• Recall Modified Bernoulli’s equation
• How does the head loss manifest itself?
– flow velocity is constant along the pipe (which physical principle
concludes this point?)
– pressure is the “sacrificial lamb” in frictional pipe flow analysis
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Pressure Drop in Pipe Flow
• Recall supplementary problem on dimensional analysis of pipe
flow
– In dimensional representation (7 variables)
P  f  ,V , D,  , , l 
– In dimensionless representation (4 p groups)


P
VD

 g
,
2

V

 
 Reynoldsnumber
Lecture 10



l 
,
D
D


relative roughness

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Significance of Reynolds Number
• Definition of Reynolds number:
V D
Re 

• The Reynolds number can be interpreted as the ratio of inertial
to viscous effects (one of many interpretations)
• At low Reynolds number,
– viscous effect is comparable to inertial effect
– flow behaves in orderly manner (laminar flow)
• At high Reynolds number,
– viscous effect is insignificant compared with inertial effect.
– flow pattern is irregular, unsteady and random (turbulent flow)
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Introducing the Friction Factor
• Recall results from dimensional analysis of pipe flow
 VD  l 
P
 g1 
, , 
2
V
D D
 
• From hindsight, cast the above equation as
 VD  
P
l
 g 2 
, 
2
V
D  
D


P  f
V 2 l
2 D
friction factor / 2
• The friction factor (as defined) only depends
– Reynolds number
– relative roughness
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How to find the friction factor?
• Since the friction factor only depends on two independent p
groups, it is simple to represent its variation with multiple
contour lines on a 2D plane
• Show the Moody diagram
– representation of two p groups
– partition of different flow regimes
• The whole problem of finding the pressure drop across piping
system is reduced to finding the friction factor on the Moody
diagram
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Flow Chart
Find Reynolds number
• fluid properties (, )
• geometry (D)
• flow speed (V)
Laminar
(Re < 2300)
Re 
V D

Turbulent
(Re > 2300)
Find relative roughness
64
f 
Re
Look up Moody diagram
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