Download The U-Tube Manometer

Be 301
Low-Pressure Manometer
Spring 2001
FINAL PROJECT
GROUP NUMBER T3
Jehann Biggs
David Kim
Anju Mathew
Naomi Sta. Maria
TITLE
Designing a Low-Pressure Manometer
DATE SUBMITTED
May 9, 2001
ABSTRACT
A U-tube manometer was designed and constructed to measure pressure drops to
precisions of 0.0033 psig for flow in a long, horizontal tubing. The objective was to
improve on the current pressure transducer, which is incapable of measuring pressures
below 0.0033 psig. The design of the manometer was an inverted U-tube filled with a
manometer fluid with density less than water (naptha, = 0.75 g/mL). With our
proposed design, the apparatus could ideally measure pressure differences as low as
0.0004 psig. When Re*f vs. Re was plotted, a y-intercept of 68.99 + 10.67(95%CI) was
achieved, which is more accurate than the value when using the pressure transducer,
98.12 + 17.95 (95%CI). The manometer successfully outperformed the transducer by a
factor of two in measuring pressure drops of near-Poiseullian flow.
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Be 301
Low-Pressure Manometer
Spring 2001
BACKGROUND
The U-Tube Manometer
The simple U-tube manometer has been used to measure pressures for many years
because of its simplicity and accuracy. Manometers measure pressure by balancing the
pressure forces directly against the liquid columni. The ideal U-Tube manometer has a
few characteristics that allows it to make simple, accurate measurements:
1)
2)
3)
4)
It requires no calibration
It is cost efficient
It is useful for measuring slowly varying pressures
It is able to measure low changes in pressure
Another method of measuring pressure in the horizontal tube is using a pressure
transducer. Pressure transducers, similar to the one used in the lab, produces a
quantitative electrical signal in response to rapid changes in pressure of fluid flowing in a
tube. The transducer used in the lab has the ability to only pressures low as 0.0033 psig,
which is not accurate enough to measure the pressures at the low flow rates. The
monometer we have calculated will make up for the transducers shortfall and we have
calculated that our best manometer design will be sensitive pressure measurements as low
as 0.0004 psig.
The Manometer Fluid
A light or heavy density material is used in the U-tube depending on the magnitude of the
pressure difference to be measured. For large pressure differences, a very dense liquid
such as mercury can be used and for smaller pressure differences, fluids whose densities
closely match the flowing fluid can be used. The design and orientation of the manometer
with respect to the tube is a factor in determining the type of fluid to be used. In addition,
the fluids should be immiscible, not readily mixed with other fluids, have good wetting
characteristics and be able to form an observable meniscus.
Application of Fluid Statics
The understanding and application of fluid statics is necessary in the design of the
manometer model. The general equation for fluid statics is shown below.
p1 + gz1 = p2 + gz2
(1)
For a U-shaped glass tube that is partially filled with a liquid and that one end of the tube
is open to the atmosphere (Figure A), while the other is connected to the container whose
pressure is to be measured. Applying equation (1) to the manometer fluid (whose density
is ) between the fluid surfaces at 2 and 1:
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Be 301
Low-Pressure Manometer
Spring 2001
where we have noted that the pressure equals the atmospheric pressure
. The
pressure is not necessarily the same as the pressure of the fluid at the center of the
container if the fluid is a liquid. To account for this difference, we apply equation (1) to
the container fluid (of density ) between points 3 and 2:
If the container fluid is a gas, then its density is much less than that of the manometer
liquid , and and are substantially equal. On the other hand, if the container fluid is
a liquid then the pressure difference
may contribute significantly
to the determination of the container pressure and should not be disregarded.
Figure A: A U-tube manometer used to measure the pressure of a fluid in a container.ii
Hagen-Poiseuille’s Law
The Hagen-Poiseulle’s law (4) is derived from
differential analysis of fluid flow through a
straight, horizontal cylindrical tube using the
conservation of linear momentum. An expression
for the velocity profile for parabolic flow is then
obtained. Poiseulle’s law (5) relates the pressure
drop p to the steady, laminar, fully-developed
flow rate Q for a Newtonian fluid.
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Be 301
V 
P 
Low-Pressure Manometer
2
dP R 2   r  
1    
dx 4    R  
Spring 2001
(4)
8 L
Q
R 4
(5)
Friction Factor and Head Loss
Pressure losses occurring in internal flows are results of friction. The HL term actually
represents the decrease (loss) in mechanical energy between points 1 and 2, and in
general contains both major and minor losses. In Equation (2) we see that pressure
changes result from velocity changes, elevation changes and friction losses. For
incompressible case of flow though a straight, horizontal tube of constant diameter,
equation (2) becomes HL = (p1-p2)/g.
Knowing pressure change is known to depend on pipe diameter D, average velocity U,
length L, viscosity , density , and wall roughness, these parameters have been
combined to empirically define the friction factor from the HL, as shown below (6).
2
p
 L U
 HL   
f
g
 D  2g
(6)
Analytically the friction factor f for laminar flow is f = 64/Re, this is derived from
Poiseulle’s Law from the following derivation from Lab experiment 2.
f 
64
Re
(7)
where Q = Vave A = UR2,  = , D=2a
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Be 301
Low-Pressure Manometer
Spring 2001
DESIGN
Figure 1: Manometer Design.
The above figure describes the setup of the manometer. The yellow section represents
naptha, which is less dense than water. The calculation for the difference in pressure
between Pa and Pb shows that ∆P=(ρH20 – ρnaptha)gh, so the height difference can be
maximized by decreasing the difference in density between water and the manometer
fluid.
Figure 2: Experiment Setup
The above figure shows the manometer connected to the long
horizontal tube as in the experiment. The image on the right
shows actual prototype, using lighter fluid (napthalene) as manometer fluid.
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Be 301
Low-Pressure Manometer
Spring 2001
MATERIALS
 Tygon ID tubing diameters (¼”and 3/16”)
 Flow cutoff valve
 U-shaped manometer
o Moving Fluid: Water  = 1000 kg/m3
o Manometer Fluid: Naptha  = 750 kg/m3
 Water tank
 Flow restrictor
 Various connectors, stoppers, and Teflon® tape
METHODS
The actual manometer prototype was constructed using ¼” ID tubing attached in a
U-shape around a meter stick. The top of the tubing had an open connector which
allowed for the easy pouring of naphthalene and also allowed for the easy removal of air
bubbles. After naphthalene had been added and the air bubbles removed, the connector
was closed with a rubber stopper. The ends of the manometer were connected to the
horizontal tubing via connectors and sealed with Teflon® tape. The manometer was
supported by a ring stand.
Measurement of the pressure-flow relationship in a straight, horizontal tube was
made using an apparatus that consists of a reservoir fluid tank with supports elevated
connected to Tygon ID tubing connected to a flow cutoff valve then to a U-shaped
manometer then continuing on to a Tygon ID tubing cut to a determined length with a
flow restrictor, to a collection bucket. In developing a protocol, parameters such as flow
rates, density of moving fluid, density of manometer fluid, and U-shaped manometer setup were tested for greatest efficacy.
In this determination of the pressure-flow relationship, we used naphthalene as the
manometer fluid and water as the moving fluid. Various sizes (diameter) of Tygon tubing
were used as another variable, such as 1/4in and 3/16in. The pressure differential was
directly controlled through flow rates, and we conducted at least 10 trials for each size
tubing.
First, the h in the manometer at zero PSIG was noted (static pressure). The
needle valve was opened with exit blocked, and the height of the water above the
manometer was measured and converted to pressure, with the relationship that water
exerts 1 PSIG per 27.68 inches. The U-shaped manometer was connected above the flow
tube. For a schematic diagram, please refer to the Design section. The manometer, which
looks like an inverted U, contained manometer fluid with density close to but slightly less
than the density of the moving fluid. Ten trials were conducted and average h (pressure
height) and 95% confidence intervals were determined.
Prior to the start of the experiment, the needle valve was opened again to
eliminate air bubbles out of the tubing by obtaining a slow flow at first. Then, ten series
of pressure-flow measurements were conducted by opening the cutoff clamp a given
amount, recording the pressure output as h (+0.01cm) and the flow rate from recording
the volume differences and dividing by the time length of accumulation. Q = (Vfinal –
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Be 301
Low-Pressure Manometer
Spring 2001
Vinitial)/t. The height of the water in the tank was kept constant by continuously
refilling the tank with the emptying fluid into the collection-graduated cylinder after each
trial. For very low flow rates, the graduated cylinder was placed so that the lip of the
cylinder touched the flowing water droplets thereby ensuring a constant flow. The water
flow from the tank is driven by gravity and controlled by a needle valve. Bubbles were
minimized in the tube since they increase the Darcy friction factor and other losses. The
same protocol for pressure-flow measurements were used for the all other parameters
stated initially.
RESULTS
Figure 3. Plot of Friction vs. Reynold’s Number
f vs. Re
2.5
2.0
1/4 in ID tube
1.0
3/16 in ID tube
f
1.5
0.5
0.0
0
500
1000
1500
2000
Re
Figure 3 shows the Darcy friction factor versus Reynold’s number plot. The graph shows
laminar flow and that friction is approximately inversely proportional to the Reynold’s
Number.
Figure 4a. Plot of Reynold’s Number * Friction vs. Reynold’s Number
1/4 in ID tube
Re*f
Re*f vs. Re
3/16 in ID tube
200
180
160
140
120
100
80
60
40
20
0
0
500
1000
1500
2000
Re
Using the calculated values of friction and Reynold’s Number, the product of the two
variables was plotted against the Reynold’s Number. For Re<500, the product appears to
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Be 301
Low-Pressure Manometer
Spring 2001
be constant, while for Re>500, the product increases. Extrapolating a regression of the
data to the y-intercept for Re<2000 yields an intercept of 9818 (95%CI), as shown
below in Figure 4b.
Figure 4b. Plot of Reynold’s Number * Friction vs. Reynold’s Number for
Reynold’s number less than 500
Re*f vs. Re
100
Re*f
80
60
y = 0.0795x + 68.999
R2 = 0.3822
40
20
0
0
50
100
150
200
250
300
Re
Figure 5. Results using one pressure transducer from T3 experiment 2
Re*f vs. Re
250
water, 5/16"
Re*f (unity)
200
water, 1/4"
150
water, 3/16"
sucrose, 5/16"
100
sucrose, 1/4"
sucrose, 3/16"
50
0
0
500
1000
1500
2000
2500
3000
3500
4000
Re (unity)
Figure 5 shows results using one pressure transducer with a 98.12 ± 17.95 (95%CI) yintercept to which the results from figures 4a and 4b were compared.
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Low-Pressure Manometer
Spring 2001
Table 1. Sample Trial Data for ¼” ID Tubing
Pressure drop values (in yellow highlight) show precision to 0.004 psig.
Table 2. Sample Entrance Length Effects Data from ¼” ID Trials
Trial
1
2
3
4
5
6
7
8
9
10
3
Q m /s
6.7E-07
1.01E-06
2.67E-06
4.93E-06
6.25E-06
2.02E-07
3.35E-07
6.27E-07
7.84E-07
0.000
V m/s
0.021
0.032
0.084
0.156
0.197
0.006
0.011
0.020
0.025
0.038
XL
0.000
0.039
0.102
0.188
0.239
0.008
0.013
0.024
0.030
0.046
The laminar development length XL (m) for the flow to become fully developed is given
by Boussinesq XL = 0.03ReD. Table 1 shows a sample data set for resulting entrance
lengths due to different flows.
DISCUSSION
Poiseulle’s law states that for laminar flow, the Darcy friction factor f varies inversely
with Re. Figure 3 exhibits an near-inverse relationship and therefore is misleading. A
much steeper decrease should be seen if it were truly inversely related. Thus, Poisuellian
flow was not observed.
Theoretically, for a long, horizontal tube the product Re*f must be a constant value of 64
for Re<2000 and increase with Re>2000. However, our data, as shown in Figure 4a,
indicate an increase immediately after Re>500 which reconfirms non-Poiseullian flow.
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Be 301
Low-Pressure Manometer
Spring 2001
In Figure 4b, an extrapolated regression of the graph fRe versus Re to the y-intercept for
Re<500 yields an intercept of 68.99 + 10.67(95%CI). As the flow approaches zero, the
data exhibits Poiseullian flow, which is theoretical, limiting flow. Equation 5.24 in H&B
yields an intercept of 64, which in well within our confidence intervals.
Results from Experiment 2 using one pressure transducer, as shown in Figure 5, yielded a
y-intercept of 98.12 ± 17.95 (95%CI). The confident limits do not include the theoretical
number, 64, for Poiseullian flow. The electronic transducer does not measure accurate
pressure difference for low pressure flows. In addition, Table 1 shows the apparatus
measuring pressure differences as low as 0.0004 psig. Thus, the manometer is a better
apparatus for measuring low pressure flows.
Recommendations:
The design and construction of a manometer may not be the time efficient for future
laboratory experiments. However, certain design aspects of the manometer may be
applied to Experiment 2. Instead of using just one pressure transducer located at the base
of the water tank, the protocol may be improved by using two pressure transducers. The
placement of the pressure transducers will be critical due to the effects of entrance and
exit effect. The first transducer should be placed after a maximum calculated entrance
length such as shown in Table 2. The second transducer is needed because the
assumption of zero gage pressure at the outlet may be inaccurate due to exit effects,
especially if a restrictor has been used. The second transducer should be placed near, but
not at, the exit to avoid these effects.
In addition, when LabView software is used to monitor pressure when using electronic
transducers. If the flow is very low, the fluid exits in small constant droplets. This
produces a “noisy” signal because the pressure is similar to static pressure between
droplets. We accommodated this by placing the lip of the graduating cylinder touching
the droplet, thereby breaking surface tension and producing a constant flow.
According to Bernoulli’s Equation along the streamline region, the pressure decreases as
the core accelerates. So as the flow becomes fully developed, the core accelerates thereby
decreasing the pressure drop p. For this experiment, the flow was not fully developed
for the entire length of the tube as shown in Table 2. This causes the p to increase,
thereby increasing the Darcy friction factor and fRe, as shown in the Boussinesq
relationship(XL = 0.03ReD). Compared to the tube length of 2 meters, the resulting XL
values in Table 2 of approximately 20cms definitely affects the p and should have been
taken into account.
Ideally, the designed manometer can also be used for faster flows if the ends of the
manometer were positioned closer to each other along the tube. The measured pressure
drop would be small in comparison to the pressure drop if the ends were farther apart.
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Be 301
Low-Pressure Manometer
Spring 2001
CONCLUSIONS




i
ii
A U-tube manometer successfully measured pressure differences as low as 0.0004
psig.
An ideal manometer fluid should have density close to that of the moving fluid to
produce the greatest h
A plot of Re*f vs. Re yielded a y-intercept of 68.99 + 10.67(95%CI) which is
more accurate than the value when using the pressure transducer, 98.12 + 17.95
(95%CI) when compared to the theoretical value of 64.
The manometer successfully outperformed the transducer by a factor of two in
measuring pressure drops of near-Poiseullian flow.
Miller, Richard, A. Flow Measurement Engineering Handbook, 3rd ed., McGraw-Hill, New York 1996
Fluid Statics: http://www.mas.ncl.ac.uk/~sbrooks/book/nish.mit.edu/2006/Textbook/Nodes/chap02/eqhez
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