1. Part II. Dimensional Analysis and Experimentation
Chap. 3: Pipe Flow
1. Introduction
• Dimensional analysis: - relationship btw. flow variables and physical
properties in flows
- useful in experimental design
2. Dimensional analysis
• N variables ~ D dimensions (time, length, mass, etc.)
~ combined into indep. dimensionless groups, G=N-D
• Buckingham pi theorem: reduces the number of variables (N  G)
useful in experimental design

2. 3. Smooth pipe flow: Newtonian fluid
Dimensionless groups
• Smooth circular pipe flow: pressure-driven flow (Q & A: drag flow?)
6 variables (N=6)
3 dimensions (D=3)
Indep. dimensionless groups,
G=N-D=3
Aspect ratio, dimensionless
|p| ~ dimensionally same as stress (v/D)
Dimension check?
: ½ v2 ~dimensionally same as stress

3. Reynolds
number
Aspect ratio
• Friction factor:
Unique function of Re for smooth pipe flow of all incompressible Newtonian fluids.

4. • Definition of friction factor:
Fk: force by the fluid motion
A: characteristic area
f: friction factor
k: characteristic kinetic energy
per unit volume
For horizontal pipe flow,
Hagen-Poiseuille Eq.
(Sec. 8.4)

5. Friction factor (f) - Reynolds number (Re) data
• f ~ unique function of Re
• Two distinct regions: Laminar flow (Re < 2,100), f = 16/Re
Transition region (2,100<Re<4,000)
Turbulent flow (chaotic): good for intensive mixing
- Blasius Eq.: ,000<Re<105
- von Karman-Nikuradse Eq.

6. Capillary viscometry
• Viscosity ~ an intrinsic physical property of a Newtonian fluid.
Ex. 3-1)
End effects
• Entry length, Le:
- Important in capillary viscometers for high viscous liquids

7. Example) Pressure buildup at entrance region in capillary viscometer
Pressure distribution

8. Physical meaning of Re
• Ratio of the inertial to viscous forces at work in the fluid
• Friction factor ~ ratio of the net imposed external force
to the inertial force
• Friction factor in laminar flow (no inertia)
Right term is indep. of FI in laminar flow

9. 4. Power
Power input
• Rate at which work is being done on the flowing system
(for the case of incompressible Newtonian liquid through a pipe)
Force at upstream position 1:
Force at downstream position 2:
Net force on fluid:
Work done to move the fluid:
Rate of work:
Dissipation
• Work done on the flowing system ~ increases the system energy

10. • Estimation of max. increase in fluid temp. using pumping power
- Max temp. occurs at adiabatic pipe
Adiabatic increase in temp.
• Lost work or viscous loss
- Viscous dissipation: Power / unit volume for viscous losses,
- Dissipation in laminar pipe flow:
Check Eq.
Ex. 3-2)
Optimal pipe diameter (skip)

11. 5. Commercial pipe Relative roughness
• Previous f-Re  for tube with a very smooth inner surface
• One additional dimensionless group for above geometry
k/D: relative roughness,
Hydraulically smooth
- In laminar flow, f indep. of k/D
- After transition, f indep. of Re
Complete turbulence

12. Pipe roughness
• f-Re-k/D for commercial pipes
Hydraulic smoothness &
complete turbulence regions exist.
• Colebrook formula:
• For 4,000  Re  2107,
Ex. 3-3)
Nominal and real diameter
• Diameter of commercial pipe = nominal size
- Inner pipe D depends on designed strength of pipe by the wall thickness.
(see Table 3-3)

13. 6. Noncircular cross sections
• Finding flow rate and pressure drops in noncircular conduits
 “Equivalent” circular pipe  Use f-Re diagrams
• Diameter for cylinder,
• Hydraulic diameter for any other conduit:
For constant and noncircular area,
e.g., rectangle ?
Transition exists (2,100<Re<4,000)
Comment: Hydraulic radius