Measuring the Friction Factor
in Small Pipes
Objective
To study the variation in friction factor, f, used in the Darcy
Formula with the Reynolds number in both laminar and turbulent flow. The
friction factor will be measured as a function of Reynolds number and the
roughness will be calculated using the Colebrook equation.
Theory
The loss of head resulting from the flow of a fluid through a
pipeline is expressed by the Darcy Formula
hf =flv^2/2Dg
where hf is the loss of head (units of length) and the
average velocity is V. The friction factor, f, varies with Reynolds number and
a roughness factor.
Laminar flow
The Hagen-Poiseuille equation for laminar flow indicates that
the head loss is independent of surface roughness.
Thus in laminar flow the head loss varies as V and inversely as
D2. Comparing equation and equation it can be shown that
indicating that the friction factor is proportional to viscosity
and inversely proportional to the velocity, pipe diameter, and fluid density
under laminar flow conditions. The friction factor is independent of pipe
roughness in laminar flow because the disturbances caused by surface roughness
are quickly damped by viscosity.
Equation can be solved for the
pressure drop as a function of total discharge to obtain
Turbulent flow
When the flow is turbulent the relationship becomes more
complex and is best shown by means of a graph since the friction factor is a
function of both Reynolds number and roughness. Nikuradse showed the dependence
on roughness by using pipes artificially roughened by fixing a coating of
uniform sand grains to the pipe walls. The degree of roughness was designated
as the ratio of the sand grain diameter to the pipe diameter (e/D).
The relationship between the friction factor and Reynolds
number can be determined for every relative roughness. From these
relationships, it is apparent that for rough pipes the roughness is more
important than the Reynolds number in determining the magnitude of the friction
factor. At high Reynolds numbers (complete turbulence, rough pipes) the
friction factor depends entirely on roughness and the friction factor can be
obtained from the rough pipe law.
For smooth pipes the friction factor is independent of roughness
and is given by the smooth pipe law.
The smooth and the rough pipe laws were developed by von Karman
in 1930.
Many pipe flow problems are in the regime designated
“transition zone” that is between the smooth and rough pipe laws. In the
transition zone head loss is a function of both Reynolds number and roughness.
Colebrook developed an empirical transition function for commercial pipes. The
Moody diagram is based on the Colebrook equation in the turbulent regime.
The Colebrook equation can be used to determine the absolute
roughness, e, by experimentally
measuring the friction factor and Reynolds number.
Alternatively the explicit equation for the friction factor
derived by Swamee and Jain can be solved for the absolute roughness.
When solving for the roughness it is important to note that
the quantity in equation that is squared is negative!
Equations and are not equivalent and will
yield slightly different results with the error a function of the Reynolds
number.
Experimental Apparatus
The experimental apparatus consists of a pressure reducing
valve, shutoff valve, flow control valve, a test section of tubing with
pressure taps and a pressure transducer (Figure ). The pressure-reducing valve
is used to minimize the effects of pressure fluctuations in the tap water
supply. A 10-cm-diameter volumetric detector will be used to measure the flow
rates. A section of 3/8” OD tubing should be installed between the flow
control valve and the test section so the flow can become laminar.
Experimental Methods
The experiment consists of measuring the head loss in a length
of tubing as a function of discharge. Head loss will be measured in small
diameter brass pipe using pressure transducers. Discharge will be obtained by
measuring the volume of discharge over a time interval using a volumetric
detector. An 85 cm section of tubing with an inside diameter of 3.4 mm will be
used as the test section.
Pressure transducer
(for head loss)
|
Desired head loss (cm)
|
7 kPa
|
1
|
7 kPa
|
2
|
7 kPa
|
4
|
7 kPa
|
8
|
7 kPa
|
16
|
7 kPa
|
32
|
200 kPa
|
64
|
200 kPa
|
120
|
200 kPa
|
250
|
200 kPa
|
500
|
200 kPa
|
1000
|
200 kPa
|
max
|
1)
Measure and record the distance between pressure ports.
2)
Make sure the cold water tap is fully open.
3)
Open the needle valve slowly and purge all air from the
tubing.
4)
Close the needle valve.
5)
Verify that the tubes connecting the pressure sensors
to the ports contain water and if necessary purge the air by carefully removing
the pressure sensor while clamping the tubing between your finger and thumb.
After the pressure sensor is removed allow a small amount of water to discharge
into a sponge and then reclamp and reconnect the pressure sensor. Be very
careful to not get the outside of the pressure sensor wet!
6)
Open the needle valve again to ensure that the test
section is full of water.
7)
Close the needle valve.
8)
Open the Easy Data software.
9)
Verify that the data frequency is set to 1 Hz.
10) Set
the output of the 3 sensors to zero by clicking on 
at the
top of the Easy Data window.
at the
top of the Easy Data window.
11) Enable
logging data 
and
create a new file in the cee 331 folder.
and
create a new file in the cee 331 folder.
12) Open
the needle valve until the head loss as recorded by the 7 kPa pressure sensor
is the desired value (see Table )
13) Use
the ability to write notes in the data file 
to
record that you are acquiring data at a stable flow rate (As an example, type
in “begin 1 cm head loss”).
to
record that you are acquiring data at a stable flow rate (As an example, type
in “begin 1 cm head loss”).
14) Record
data for 30 seconds or until the volumetric detector fills up!
15) Record
a note in the data file indicating the end of the good data before you change
the flow rate! (Type “end 1 cm head loss.”)
16) Repeat
steps 12-15 until you have acquired data for all the desired flow rates,
remembering to change the pressure transducer as needed. (Note that when you
change the pressure transducer you should stop acquiring data and then restart
by repeating steps 4-11)
Data Analysis
1)
What is the advantage of expressing the friction factor
as a function of the Reynolds number rather than as a function of the flow rate?
2)
Determine the absolute roughness, e, for the brass tubing using equation .
3)
Create a diagram similar to the one created by Moody
showing the friction factor as a function of Reynolds number (log-log plot).
Clearly indicate the laminar and turbulent regions. In addition to your data,
plot the equation obtained by Hagen-Poiseuille in the laminar region and the
Swamee-Jain equation in the turbulent region using your best estimate of the
roughness of the brass tubing. Make sure that when plotting equations that you
plot sufficient points to create smooth curves and that you don’t show any
“data” points.
4)
Why are two different pressure transducers used to
measure the head loss? (The answer isn't explicitly in the lab manual!)
Lab Prep Notes
Setup
1)
Configure the top row of ports to have a maximum
voltage of 100 mV. The middle row of ports should have a maximum voltage of 20
mV.
2)
Plug the 2 7-kPa sensors into the middle row of ports.
Plug the 200-kPa sensor into the top row of ports.
3)
Set up the physical apparatus and create configuration
files for the Easy data software. The pressure sensors used to measure head
loss should be configured to measure head loss in cm. The pressure sensor for
the volumetric detector should be configured to measure volume in L. Although
only one of the head loss sensors will be used at a time, configure the
software to monitor both of them so the same configuration can be used for the
entire experiment.
4)
Install a 3/8” valve and 3/8” tubing from the port near
the bottom of the volumetric detector so that it can easily be drained.
5)
Use 3/8” tubing to connect the tap at the sink to the
pressure regulators.
6)
Set the screw on the pressure regulators so that the
effluent (regulated) pressure is reduced by approximately 10 kPa from the
pressure of the cold water tap.
Table . Equipment list.
Description
|
Supplier
|
Catalog number
|
Pressure transducer
|
Omega
|
PX26-001DV
|
Pressure transducer
|
Omega
|
PX26-030DV
|
Nupro angled 3/8 swage valve
|
Rochester Valve & Fitting Co., INC.
|
B-6JNA
|
3/8" OD tubing
|
Cole-Parmer
|
H-06490-15
|
Pressure reducer
|
ID Booth
|
FB-38
|
Volumetric detector
|
CEE shop
|
|
Pipe test section
|
