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
                                                                   1.1
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.
                                                                  1.2
Thus in laminar flow the head loss varies as V and inversely as D2. Comparing equation 1.1 and equation 1.2 it can be shown that
                                                                 1.3
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 1.2 can be solved for the pressure drop as a function of total discharge to obtain
                                                                 1.4

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.
                                                             1.5
For smooth pipes the friction factor is independent of roughness and is given by the smooth pipe law.
                                                            1.6
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.
                                                      1.7

The Colebrook equation can be used to determine the absolute roughness, e, by experimentally measuring the friction factor and Reynolds number. 
                                                        1.8
Alternatively the explicit equation for the friction factor derived by Swamee and Jain can be solved for the absolute roughness.
                                                       1.9
When solving for the roughness it is important to note that the quantity in equation 1.9 that is squared is negative!
                                                      1.10
Equations 1.8 and 1.10 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 1-1). 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.

Table 1-1.        Recommended measurements.
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

• Measure and record the distance between pressure ports.
• Make sure the cold water tap is fully open.
• Open the needle valve slowly and purge all air from the tubing.
• Close the needle valve.
• 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!
• Open the needle valve again to ensure that the test section is full of water.
• Close the needle valve.
• Open the Easy Data software.
• Verify that the data frequency is set to 1 Hz.
• Set the output of the 3 sensors to zero by clicking on  at the top of the Easy Data window.
• Enable logging data  and create a new file in the cee 331 folder.
• Open the needle valve until the head loss as recorded by the 7 kPa pressure sensor is the desired value (see Table 1-1)
• 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”).
• Record data for 30 seconds or until the volumetric detector fills up!
• 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.”)
• 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

• 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?
• Determine the absolute roughness, e, for the brass tubing using equation 1.8.
• 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.
• 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

• 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.
• Plug the 2 7-kPa sensors into the middle row of ports. Plug the 200-kPa sensor into the top row of ports.
• 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.
• 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.
• Use 3/8” tubing to connect the tap at the sink to the pressure regulators.
• 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.

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