ME 322: Mechanical Engineering Laboratory II
LABORATORY 6:
HEAD LOSS EXPERIMENT
Objectives
This experiment demonstrates the use of concomitant methods to determine average flow velocity of a
fluid. That is, you will use two different experimental pressure methods to determine the average flow
velocity.
Student Learning Outcomes
Form this lab, you will be expected to be able to:
• Connect and operate several types of manometers
• Use a pitot-static tube to determine dynamic pressure and velocity
• Traverse a flow duct to convert dynamic pressures into flow rates (as per ASME Code)
• Analyze experimental data to compute head loss
Required Reading
The following material should be studied before undertaking this laboratory:
Ref 1
Ref 2
Ref 2
Section 8.4.1-8.4.2
Section 9.2; 9.6
Section 10.4
Viscous Flow in Pipes
Manometers; Pressure in Moving Fluids
Volume Flow Rate (including Example 10.1)
Background
Much of the analysis of fluid flow is centered around Bernoulli’s equation. Since this equation is based on
the analysis of energy in the fluid system, one often sees it with terms that have units of energy or potential
(i.e. pressure). However, by simple algebra one may obtain the form given by Equation (6.1), which is
particularly useful for flow in pipes. Notice that the units of the terms in this equation are displacement, or
“head” of the fluid involved. “Head” and “pressure” are often used interchangeably, for example 6” H20 is
the pressure developed from a 6” head (column) of H20.
p
V2
p1 V12
+
+ z1 = 2 + 2 + z 2 + h L
ρg 2g
ρg 2g
(6.1)
This equation relates the pressure, p, average flow velocity, V, and “altitude”, z, of points 1 and 2 in a
section of pipe. Reality (viscosity) necessitates the inclusion of a “head loss” term, hL, that includes what
amounts to frictional losses in non-ideal (viscous) flow. Note that the specific weight of the fluid, γ, is
sometimes used instead of the product ρg, where ρ is fluid density and g is gravitational acceleration (e.g.
Equation 10.8 of your text). If the diameter of the pipe is constant, V1 and V2 are identical. If the pipe
section is horizontal, z1 and z2 are identical. These conditions leave us with Equation (6.2).
p1 p 2
=
+ hL
(6.2)
γ
γ
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ME322: Mechanical Engineering Laboratory II
Head Loss Experiment
Equations (6.1) and (6.2) form a useful basis for the two methods for the determination of the average flow
velocity and, incidentally, the characterization of pipe quality (surface roughness).
Method 1: First we’ll discuss the use of the head loss term to determine flow velocity. The head loss
accounts for the energy losses between points 1 and 2. For a horizontal pipe section of some diameter, hL is
measured by means of pressure taps at point 1 and 2, as indicated by Equation (6.2). By definition, hL
accounts for the combined effects of what are designated frictional losses, resulting from the viscous
interaction of the fully-developed flow along straight pipe walls, and minor losses, due to changes in the
pipe configuration (bends, diameter changes, valves, obstructions, etc.). To compare head loss in a pipe for
a given average flow velocity, a non-dimensional friction factor, f, may be defined as given by Equation
(6.3).
h L ( friction ) = f
l V2
d 2g
(6.3)
The friction factor depends on certain flow variables (density, ρ, viscosity ,µ, velocity, V, and pipe
diameter, d), which are, fortunately, grouped and non-dimensionalized by the Reynolds Number, Re
(=ρVd/µ) and on the pipe wall relative roughness ε/d. There are engineering tables that list roughness
values ε for different pipes (e.g. see reference [1]). For plastic or drawn copper pipes, a smooth wall may be
assumed.
If one attempts to put friction factors, relative roughness and Reynolds Number together in one chart, one
obtains the well-known Moody Diagram, Figure 6.1 [4]. The Moody Diagram is a valuable tool for
predicting frictional losses, or for determining the relative roughness. But it also gives a means to
determine flow velocity from pressure measurements. At low Re, flow is such that it is called laminar and
has a very predicable head loss (via f), regardless of the quality of the pipe. This phenomena happens to be
the basis for a very simple velocity “measurement” scheme (See Section 10.5 of your text). However,
laminar flow is likely not the case in most engineering situations.
Now the method: If the pipe roughness is known (such as “smooth”), there is an iterative method for
determining the flow velocity. Simply: One begins by estimating (educated guess!) the flow velocity, from
which you calculate Re. Using the Moody diagram and your relative roughness, one then looks up the
friction factor (hint: don’t forget about interpolation to “pick-off” values from the Moody diagram, which
happens to be on a log-log scale). Then using your experimental data for hL and your value for f from your
trail velocity, compute a velocity (i.e. use Equation (6.3)) and repeat the process. As the process continues,
the velocity should converge to a final value, usually within about four iterations. Your TA will help you
on this if you ask.
Method 2: There are other methods to determine flow velocity. In Sections 9.6 and 9.8 of your text, you
see how Bernoulli’s equation (with no head loss) is used as the basis of determining fluid velocity from the
dynamic pressure as measured by a Pitot-static tube. However, this is the velocity of the fluid at the
location of the Pitot-static tube. Real flows in pipes have characteristic velocity profiles—for example,
centerline flow has the highest velocity, flow at the walls is zero, and the rest falls in between. What is
needed is a method for using the Pitot-static tube to find the average flow velocity. Incidentally, the
product of this velocity and the cross-sectional area gives the volume flow rate (then you could use density
to find the mass flow rate). Likewise, given either volume- or mass-flow rate measurements (i.e. via
sensors for measuring either volume- or mass-flow rate), one could calculate the average flow velocity,
which is what is needed to calculate quantities like Re.
While averaging a large number of velocities from measurements across the pipe would work, it is tedious.
However, there are standard engineering methods for using fewer measurements to come up with a
statistically accurate measurement. All of the methods are based on collecting a number of measurements at
systematically selected points and then calculating a weighted average. The actual locations of these points
and the weights used are fixed by the assumptions used to derive the method. Here we are entering into the
very old art of numerical integration: using the fewest number of data points, how do I get the most
accurate integral? We’ll skip the theory and just get to the integration methods.
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ME322: Mechanical Engineering Laboratory II
Head Loss Experiment
Your text documents the first of these in Section 10.4. Example 10.1 does a four-point calculation across
three separate lines of traverse. That example happens to calculate volume flow rate, but note that it’s a
simple matter of algebra to pose it as a average flow velocity calculation. The readings along different
“lines” in this method are to account for any asymmetry in the velocity profile – more accurate but not
always necessary. The example use an approach called the “method of equal-area centroids.” There are
other ways to do this.
The other integration methods are Newton-Cotes, Chebyshef, and Gauss—names you may recognize from
your numerical methods courses. Table 6.1 (from [3]) presents the locations of the measurement point
(assume a pipe radius of ro) and the weighting factors used to compute the average flow velocity as the sum
of the products of the data and the corresponding weights. (Note: In Example 10.1, N = 4 and each radial
station is listed as i and the weights are all ¼.)
It is interesting that all four methods use the same weights for the 2-station averaging—they can’t
all give the same result, can they? But remember there is some detailed mathematics and assumptions going
into each method. Understand that a 4-station measurement will be more accurate than a 2-station
measurement and that the right-most methods are more accurate in general. A four-point Gaussian average
has an error for most any velocity profile that is smaller than you can reasonably determine at this point!
However, some of the methods are easier to implement than others. For example, r/ro=1 corresponds to a
measurement at the wall, for which the dynamic pressure should be 0, though your Pitot-static tube will
never tell you that—it’s too big to work in such close quarters.
Table 6.1: Station Location and Weights for Averaging
Method
No. of
Stations
N
2
Equal-Area
Centroids
r/ro
w
0.5000
0.5
0.8660
0.5
Newton-Cotes
Chebyshef
Gauss
r/ro
0
1
W
0.5
0.5
R/ro
0.4597
0.8881
W
0.5
0.5
r/ro
0.4597
0.8881
W
0.5
0.5
3
0.4082.
0.7071
0.9129
0.3333
0.3333
0.3333
0
0.7071
1
0.1667
0.6667
0.1667
0.3827
0.7071
0.9239
0.3333
0.3333
0.3333
0.3357
0.7071
0.9420
0.2778
0.4444
0.2778
4
0.3536
0.6124
0.7906
0.9354
0.2500
0.2500
0.2500
0.2500
0
0.5774
0.8165
1
0.1250
0.3750
0.3750
0.1250
0.3203
0.6382
0.7699
0.9473
0.2500
0.2500
0.2500
0.2500
0.2635
0.5745
0.8185
0.9647
0.1739
0.3261
0.3261
0.1739
A final comment. Whenever losses or pressures are quantified in terms of a head, a fluid is always
specified. For example, a head loss of 2 inches means nothing unless the fluid is known. A head loss of
two inches of water is much less than a head loss of two inches of mercury. In this laboratory, pressure is
always measured using manometers in inches of water. But, the exercises at hand deal with losses for air
flowing in a pipe. Regardless of whether the head loss is reported in inches of air or inches of water, the
pressure change (p1-p2) is the same. After some manipulation of Equation (6.2), this relationship is
illustrated in Equation (6.4).
∆P = h L ,air γ air = h L,H 2 0 γ H 2 0
(6.4)
Then, from Equation (6.4),
h L,air =
γ H2 0
γ air
h L, H 2 0 =
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ρ H2 0
ρ air
h L,H 2 0
(6.5)
ME322: Mechanical Engineering Laboratory II
Head Loss Experiment
Figure 6.1: Moody Diagram
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ME322: Mechanical Engineering Laboratory II
Head Loss Experiment
From Equation (6.5) it can be seen that the conversion from head loss in terms of water to a head loss in
terms of air is accomplished by multiplying by the density ratio. Values of density for appropriate
temperatures are in most fluids or thermodynamics texts or handbooks.
Procedure
The experimental apparatus to be used today was designed to investigate several fluid flow phenomena. We
will be using only some of its features. It consists of a blower, powered by a permanent magnet DC motor,
at the inlet of a short “pipeline”. Side-wall taps are installed at fixed locations to measure the static pressure
in the pipe. Equipment for measuring these pressures includes a 0-1” and a 0-6” H20 slant-tube manometers
and a 0-2.5” H20 digital manometer. A pitot-static tube is available for traversing the pipe and measuring
dynamic pressure in the flow.
Your TA will explain the operation of the apparatus, particularly the motor controller. For your work today,
bear in mind the following:
1.
2.
3.
Do not exceed at motor controller setting of “9”. This controller was designed for a larger
motor than that which it is currently controlling.
The slant-tube manometer has a pronounced dynamic response. You must decide as group
how long you will wait before taking each reading. Note that 30 seconds is not long enough –
have patience and with a few observations you will know how long to wait!
While pressure tap readings at the highest motor speeds will require the use of the liquid
manometer, at very low speeds you can also use the digital manometer if it is available.
However, it is critical that you do not exceed 2.5” H20 head on this unit! Take a moment to
verify that the digital unit is correctly calibrated—don’t take it for granted. Reading the
pressure tap data is straight forward but tedious.
Assignment
You are asked to determine the average flow velocity using the two methods described above: the iterative
head-loss method on a section of smooth pipe and the weighted averaging of velocities from Pitot-static
tube readings. You are to do this at two motor speeds, one of which must be “9”. Selection of method(s) of
integration is left to you and your group, though your instructor may give some guidelines. These are
concomitant methods – likely you will not get exactly the same answer! But the answers should be
reasonably close. However, both methods are accepted engineering methods to determine flow velocity.
Compare and discuss your results. Think about which of the methods you feel more comfortable with and
discuss why. Each method contains a measure of uncertainty due to the nature of the technique, information
used, measurements taken, and your experience at using each. So discuss your confidence in your
experimental measurements and the effect of errors on your results. Don’t forget to include your final value
of the Reynolds number in your reporting of results.
Of course, this is all to be done in the standard report format.
References
1.
2.
3.
4.
Munson, B.R, Young, D., Okiishi, T., Fundamentals of Fluid Mechanics, 3rd Ed., Wiley, 1998.
Beasley, D. and Figliola, R., Theory and Design for Mechanical Measurements, John Wiley and Sons,
2nd edition, 1995..
Fluid Meters, Theory and Applications, 6th ed., American Society of Mechanical Engineers, New
York, 1971.
Flow of Fluids Through Valves, Fittings, and Pipe, Technical Paper No. 410, Crane Company, New
York, 1982.
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