# Eq. (1) - brookelynn vizzerra

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FLOW MEASUREMENT AND
ENERGY LOSS IN PIPES
Lab 1
Brookelynn Vizzerra
Jonathan Zacarias
Carolina Santos
Keith McCabe
University of Idaho
CE322-Hydraulics
Introduction: .................................................................................................................................................... 2
Theory: ........................................................................................................................................................... 3
Approach: ....................................................................................................................................................... 6
Results: .......................................................................................................................................................... 7
Discussions: .................................................................................................................................................. 10
Reference: .................................................................................................................................................... 11
Appendix 1: ................................................................................................................................................... 12
Appendix 2: ................................................................................................................................................... 13
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Introduction:
This lab was designed to better understand pressure and energy differences for
incompressible fluids flowing through pipes by using the Scott Fluid Circuit System, represented
in Figure 1. In doing this we had to accomplish (1) the calibration of the Venturi Flow Meter, (2)
determine the friction factor for a pipe at two Reynolds numbers using the Darcy-Weisbach
diagram, and (3) determine the head loss at three different valve openings and pipe
configurations. The main findings that helped us accomplish the lab were several head loss
measurements using the independent differential manometers and the given max flow
equation, several weight and time measurements to reach the desired one-hundred pounds,
and a temperature measurement. To organize this report, we will be using a variety of tools
such as excel and Mathcad to produce our graphs and results. We will first explain the theory
behind this lab and continue onto the approach we used followed by the results and discussion.
Figure 1: Scott Fluid Circuit System
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Theory:
(1) Calibration of the Venturi Flow Meter
A Venturi Flow Meter is used to compare the head loss between a regular diameter and a
reduced diameter within a pipe. By reducing the area, the velocity will increase to satisfy the
principle of continuity and the pressure will decrease to satisfy the conservation of mechanical
energy. For this lab we used the manometers, measuring the differential head, and the
correction factor to find the flow rate. The flow-rate equation can be derived from using the
Bernoulli equation between the two different diameter points:
π12 π1
π22 π2
+ + π§1 =
+ + π§2 + βπ
2π πΎ
2π πΎ
Eq. (1)
Where p = Pressure
V = Velocity
z = Height
g = Gravitational constant
γ = Specific gravity of water
Because of the short distance and no elevation change, the head loss is now considered to be
negligible and the z1 and z2 can cancel out. In this equation the pressure head (p/γ) plus the
elevation head (z) for each point can be considered respectively as h1 and h2, giving:
π12
π22
+ β1 =
+ β2
2π
2π
Eq. (2)
By noting that A1 and A2 are area, we can now apply the continuity equation and rewrite the
previously reduced Bernoulli equation as:
π22 π΄22
π22
( 2) =
+ (β2 − β1 )
2π π΄1
2π
Eq. (3)
In solving V2 we find that:
π2 =
√2π(ββ)
π΄2
√(1 − 22 )
π΄1
Eq. (4)
By multiplying A2, we can reach the flow-rate as our final goal:
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π=
π΄2 √2π(ββ)
√1 −
Eq. (5)
π΄22
π΄12
The final discharge function given within the lab was:
Q = KA√2π(ββ)
Eq. (6)
Where K is the discharge coefficient:
πΎ=
πΆπ
√1 −
πππ πΆπ =
π΄22
π΄12
π΄ππ‘π’ππ πππ πβππππ
π‘βπππππ‘ππππ πππ πβππππ
Eq. (7)
With these equations we can accurately generate the calibration curve for the Venturi Flow
Meter. The K must be experimentally found and the theoretical values must be corrected for
head loss incurred between the two points measured.
(2) Determination of the Darcy-Weisbach Friction Factor
When a flow is fully developed, steady state, and incompressible within a constant diameter
pipe we can use the Darcy-Weisbach equation. This equation uses the friction factors to
applied to a control volume. This momentum equation states that the sum of the external
forces acting on the object is equal to the rate of momentum change in the control volume plus
the rate at which the momentum flows out of the control volume:
∑∫πΉ =
π
∫ πππ∀ + ∫ πππππ΄
ππ‘
Eq. (8)
Because the velocity and density are constant over time, we should note here that the
momentum equation should be equal to zero:
∑πΉ = 0
Eq. (9)
In studying the momentum equation, there are three forces that have to be considered: (1)
Pressure, (2) Shear, and (3) Weight. The pressure force (Fp) in this circuit system is the
difference of pressure between taps twenty-four and thirty-two (see figure 1) with given pipe
diameter (D):
π
πΉπ = (π1 − π2 ) π· 2
4
Eq. (10)
Shear force (Fs), the effect of the wall on the fluid, varies with respect to a change in length.
Shear force is acting in the opposite direction of the flow so the value will be negative:
πΉπ  = −πππ·βπΏ
Eq. (11)
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Weight force (Fw), the weight of the fluid in the pipe, changes with respect to length and depth.
As it acts in the direction of flow, we will also have a negative value:
π
βπ§
πΉπ = −πΎ ( π·2 ) βπ ( )
4
βπ
Eq. (12)
Combining the above forces into the momentum equation we produce:
(π1 + πΎπ§1 ) − (π2 + πΎπ§2 ) =
4βππ
π·
Eq. (13)
In order to introduce the head loss, we now need the energy equation:
∝
π12 π1
π22 π2
+ + π§1 + βπ =∝
+ + π§2 + βπ‘ + βπ
2
πΎ
2
πΎ
Eq. (14)
Here, we know α = 1 because the velocity profiles are uniformly distributed. Also, h p and ht are
equal to zero because we are not dealing with any pumps or turbines within the system. We
now have the Bernoulli Equation with head loss, called the energy equation:
π12 π1
π22 π2
+ + π§1 =
+ + π§2 + βπ
2
πΎ
2
πΎ
Eq. (1)
From this, V1 and V2 are equal and therefore drop out of the equation. With some slight
rearranging we now have:
(π1 + πΎπ§1 ) − (π2 + πΎπ§2 ) = βπ πΎ
Eq. (16)
In comparing equations (13) and (16) the left side is the same so we may conclude that:
βπ =
4ππΏ
πΎπ·
Eq. (17)
Taking the specific weight (γ) and the local shear stress (τ) as an empirical value, we can yield
the final Darcy-Weisbach equation:
πΏ π2
2π·π
βπ = π π· 2π where π = βπ πΏπ 2
Eq. (18)
Eq. (19)
The Darcy-Weisbach equation along with the Moody diagram can be used to determine the
friction factor. In doing this we will also need the Reynolds number determined from the flowrate (Q), pipe diameter (D), and the kinematic viscosity (ν):
ππ =
ππ·
4π
=
π
ππ·π
Eq. (20)
(3) Head Loss at Different Valve Openings:
Different valve openings causes head loss to change with respect to velocity. The head
difference between the two sides of the gate valve is measured by the second manometer
while the first manometer is used by the Venturi Flow Meter to obtain the flow rate. In order to
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Eq. (21)
obtain the head loss measurements, the two separate manometers must be connected. To
calculate K, it is evaluated by the following equation:
2
Kπ
οh =
2ππ΄2
By arranging the equation into a more suitable manner, we are able to calculate K:
2
K=
2gπ΄ ββ
π
Eq. (22)
2
The data points that are recorded should cover a wide range of flow rate. With these known
theories, we expect our data to verify the theories. The differences between our theoretical
and actual values can be affected due to the aging of the pipes used.
Approach:
For the first experiment, our goal was to calibrate the Venturi Flow Meter. We began by
obtaining a stopwatch to record time, a thermometer in order to obtain viscosity, and a large
bucket attached to a scale to measure the weight of our discharge. We then had to vent the air
pockets that were trapped in the circuit system by using the finger screws at the top of each
independent differential manometer. The screws were then re-tightened to trap the air as
manometer fluid. The head was sustained at a constant height maintaining a constant level of
water inside the reservoir that would pump water through the system.
We first obtained our max flow rate by fully closing valve fifty-two and fully opening valve fortyfive. We noted the time it took to fill the bucket near capacity. For this particular experiment
we used a capacity of 100 pounds. At the exact time this weight was reached, we obtained the
head loss (βh) given from the manometers. Because we were required ten pairs of readings we
chose to reduce the max flow rate by 10% each time and take new numbers. From this
equation:
Q max ο½ KA 2 gοh
Eq. (6)
Our βh will change. We then had to adjust valve forty-five for the new βh which in turn
changed the flow rate. We re-filled the bucket to capacity noting the new time. These steps
were repeated until the ten measurements were taken and the graph is represented in the
result section below.
For the second experiment, we focused solely on the volumetric flow-rate through pipe three at
two different flow rates. Here, both the fill valve and valve forty-five were fully closed so there
was no discharge and the flow was recirculated through the system. Valve fifty-two was used to
control the flow rates and again the lines were freed of air bubbles. The volumetric flow rates
were determined using the Venturi Flow Meter and the corresponding manometer from the
first experiment. A second flow manometer was attached to determine the total head loss
within the system, which was largely due to the friction in the pipes. This head loss was then
used to find the friction factor value of the pipe using the Darcy-Weisbach equation. This
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experiment was completed at the top of the range tested in task one and then again at half as
large to calculate a better friction factor value. The measurements were taken with the
manometer and the difference between water levels within the manometer.
For the last experiment, the goal was to determine the head loss characteristics of the three
different gate valve openings. Again, the water was recirculating, valve forty-five and the fill
valve were fully closed, and valve fifty-two was used to control the flow rates. At the three
different positions; fully open, two turns open, and one turn open, the flow rates were changed
according to our βh numbers gathered in task one. The first two positions required five
readings whereas the last one only required three because the manometer was too short when
the flow was high. Head loss was measured from the manometer and the friction factor,
determined in the last experiment, was used to calculate the K value of the valve.
As with any experiment, error was involved to some extent. The scale used was accurate to the
nearest one pound so we could attribute half pound errors in measurement. Another error
noticed was the manometer readings. When taking the values the water was not stabilized. The
last possible error was the accuracy of the timer. This would account for a fairly small impact
and should not affect the laboratory results.
Results:
(1) Venturi Flow Meter Calibration
In the lab, we calibrated the Venturi Flow Meter based on the data that we calculated by using
the difference in the height of the water column and the equation:
Q = KA√2π(ββ)
Eq. (6)
By recording the weight of the water in the bucket over an amount of elapsed time, we were
able to record the flow rate of the system. This in turn provided us with a theoretical flow rate
and an actual flow rate. Figure 2 shows the curve of the flow rate values that we recorded for
the max value of flow rate, and decreased by ten percent with each new reading.
Figure 2: Flow Rate vs Differential Head
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Figure 3: Flow Rate vs. K-value as a function of Reynolds Number
By decreasing the flow rate, we were able to obtain a more accurate flow rate. The uncertainty
in accuracy of the values was due to human error. The uncertainty in elevation readings and
flow rates are evaluated by the following equation:
ππ
ππ 2
ππ‘ 2
√
= (
) +( )
π
π€
π‘
Eq. (23)
By arranging the equation into a more suitable arrangement:
2
2
πΏπ
πΏπ
√
ππ ≤ (
ππ) + ( ππ)
πΏπ
πΏπ‘
Eq. (24)
Since there is one variable used, the variance in the manometer levels are due to turbulences in
the system.
(2) Darcy-Weisbach Friction Factor Determination
To obtain the Reynolds number, friction factors, and the error of propagation for the second
portion of the lab, where we calculated the max flow rate and half of the max flow rate by
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looking at the elevation change and head loss in manometers, the three following equations
the below were used:
Friction factor:
π=
π·2πβπ π·2πβπ π΄2
=
πΏπ 2
πΏπ 2
Eq. (25)
Reynolds number:
ππ =
4π
ππ·π
Eq. (20)
Error of Propagation:
ππ
ππ
ππ &lt; |
ππ| + | ππ|
ππ
ππ
Eq. (26)
The kinematic viscosity number that we obtained was directly inferred from the temperature of
56 degrees Fahrenheit. Once the kinematic viscosity value was found, we were able to calculate
Reynolds number. By interpolation, we were able to find the value of π to be 1.309*10-5 ft2/s.
The error of propagation for the friction factor of the two different flow rates was evaluated by
the following equation:
ππ
ππ
ππ
ππ ≤ |
πβπ | + | ππΏ| + | ππ|
πβπ
ππΏ
ππ
Eq. (27)
This equation, which is based on the flow rate, length of the pipe, and the difference in head
loss values, gives the error as a result. The error of propagation for the Reynolds number for the
two different flow rates were evaluated by the equation:
πππ = |
πππ
ππ|
ππ
Eq. (28)
This equation, based on the flow rate, gives the error as a result. After these values for the two
separate flow rates were calculated, we were then directed to plot the results on a standard
Moody Diagram which can be seen in Figure 3.
Figure 4: Moody Diagram
In using the Moody Diagram, the uncertainty bounds should be taken into account and change
the values accordingly. From referencing the book, we were able to use a relative roughness for
the Drawn Copper to be 0.0015. Table 1 details the values that we were able to calculate.
(3) Gate Valve Head Loss Characteristics
The final portion of the Lab was to obtain flow rate against head loss at valve settings of fully
open, two turns open, and one turn open. To gain a valid representative sample, we were
instructed to obtain five data points for each valve setting, displayed in Figure 4. For the first
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two valve settings, we were able to obtain our representative five data point sample however,
for the one turn open setting, we were only able to obtain three representative samples due to
the sensitivity of the pressure in the system. Our results proved our theory was correct because
when it’s fully open we have a lower head loss due to a bigger area.
Q vs. K
1.4
1.2
y = -0.0151x2 + 0.2423x
1
0.8
Full
y=
+ 0.0244x
2 turns
K
0.6
-0.0003x2
0.4
1 turn
0.2
0
-10
0
10
20
30
40
50
-0.2
-0.4
Q (cfs)
Figure 5: Flow rate vs. change in height
The results from the 3 different valve settings were obtained and are displayed in Table 2.
Discussions:
For the first part of the experiment we were to generate the calibration curve for the Venturi
Flow Meter. We used the relationship between Q and βh to find values for K through this
equation:
π = πΎπ΄√2πββ
Eq. (6)
To solve for ten different values of Q to find values of βh. With Q and βh known we were able
to plot them in excel and compare the graph obtained to that of the theoretical equation. We
found that our graph looked like a square root function which fit our theoretical equation. Even
though our graph looks correct we know that there are factors of uncertainties involved while
obtaining these values. One of these uncertainties would be the person with the stop watch
recording the time required to get to one-hundred pounds with different valve openings.
Another uncertainty would be that at one measurement we over filled the reservoir and water
poured out of the tube which might have caused a Q higher than the other tests.
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For the second part of the lab we used the values Qmax and Q50% to solve for the DarcyWeisbach friction factors and Reynolds numbers. By using the temperature, length, and given
diameter we were then able to calculate for these values. After our calculations were made our
values for ‘f’ ended up being higher than the theoretical value for drawn copper tubing which is
0.0015. This could be due to the fact that the pipes in the Scott Fluid Circuit System are aged
and differ from the pipes used for the theoretical friction factor.
The purpose of the last part of our lab was to see if our data was relevant to theory when
dealing with gate valves. In theory the bigger the area the flow is going through the lower the
head loss will be. In our situation we first started with a fully open gate valve and we had a max
head loss of 8.25 inches. As we reduce the flow, our head loss reduced with it. When we did
one rotation on the gate valve we then had a head loss of 38.25 inches which proved that going
from a fully open valve to closing it by one rotation reduces the area thus causing velocity to
increase producing a higher head loss. We then did two full rotations on the gate valve thus
increasing head loss even more, proving our theory that different valve openings cause head
loss to vary with respect to velocity.
Reference:
Fundamentals of Hydraulic Engineering
Engineering Fluid mechanics, 10th edition
Lab handout
http://parra.sdsu.edu/
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Appendix 1- Raw Data Sheets:
Attached
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Appendix 2-Sample Calculation:
Attached Graph Paper
Flow Rate
Q value
(cfs)
Q value
(in^3/s)
Qmax
0.034
58.031
0.5*Qmax
0.017
28.885
Table 1: Values for Q, f, Re, and errors
Friction
Factor
0.025
0.028
+/- Error of
Propagation
Friction
Factor
0.001043
0.001217
Fully Open
2 Turns Open
Q (cfs)
Q (cfs)
(in)
(in)
0.0335
8.25
0.0168
45
0.0259
5.5
0.0141
28
0.0196
3
0.0133
14.25
0.0129
1.5
0.0072
7
0.0063
0.25
0.0032
1.75
Table 2: Results of Gate Valve Characteristics
Reynolds
Number
4.995*104
2.486*104
+/- Error of
Propagation
Reynolds
Number
1,481
689.2
1 Turn Open
Q (cfs)
(in)
0.00167
38.25
0.00142
29
0.00114
19.5
Excel Calculations
1st
Q
Max
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
2nd
Q
1
0.5
Time for 100 lbs
47.72
52.97
61.66
68.38
81.82
95.87
124.22
162.18
255.65
476.43
Elevation Change
Left
36.5
10.5
Elevation
Change
Q (cfs)
Q (in^3/s)
58.031
52.279
44.911
40.498
33.845
28.885
22.293
17.075
10.832
5.812
0.034
0.030
0.026
0.023
0.020
0.017
0.013
0.010
0.006
0.003
46
37.5
29
22.5
16.5
11.5
7
4
1.8
0.5
Elevation Change Right
46
11.5
Q (in^3/s)
K (actual)
58.031 0.043
28.885 0.040
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3rd
Q
1
0.8
0.6
0.4
0.2
Full
Q
2 turns
1 turn
Diameter
Area
Velocity 1
Velocity 2
Viscosity
Friction factor
1
Friction factor
2
Reynolds 1
Reynolds 2
k (copper)
0.5
0.4
0.3
0.2
0.1
0
0
0
0.785
0.483982
119.903
59.68258
1.31E-05
Elevation Change
Left
8.25
5.5
3
1.5
0.25
Elevation Change
Left
45
28
14.25
7
1.75
38.25
29
19.5
in
in^2
in/s
in/s
ft^2/s
Elevation Change Right
46
29
16.5
7
1.8
kvalues
0.090
0.088
0.089
0.084
0.103
Elevation Change Right
11.5
7
4
1.8
0.5
0.5
0.25
0.125
0.019
0.019
0.020
0.019
0.019
0
0
0
0.065
ft
9.992
4.974
ft/s
ft/s
0.0253
0.0278
49953.2
24864.57
0.0015
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