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MAE 322

Thermal and Fluids Lab

Instructor

Dr. Gall

Flow Measurement using the Bernoulli Principle

Submitted by

April 3, 2013

1. Introduction



2. Experimental Apparatus and Procedure

2.1 Apparatus

2.2 Procedure

Pitot-Static Tube and Turbine Meter

1. First, the local air temperature and barometric pressure are recorded.

2. The manometer is set to zero with no flow in the duct.

3. The pitot-static tube is aligned with the axis of the duct.

4. The blower to drive the air flow in the duct is turned on.

5. Next, the deflection in the manometer is recorded.

6. The time for the flow meter to complete 10 revolutions is recorded.

7. The pitot tube is then moved in quarter inch increments towards the far wall of the duct.

8. Steps 5-7 are repeated until the tube is at the desired position.

Venturi Meter and Rotameter

1. First, the water pump is turned on and the desired flow rate is set on the rotameter.

2. The valve on the outlet of the venture meter is set, so the water levels in the manometer

are within the graduated scale.

3. The values for each tube are recorded.

4. The flow rate is increased by the 1 gallon per minute.

5. Steps 3 and 4 are repeated for the desired values until the flow rate can no longer be

increased.

2.3 Data Reduction



First, the atmospheric pressure will be converted from millimeters of mercury to pounds per

square foot. Next the density will be calculated will be calculated using the determined

atmospheric pressure and the room temperature. The ideal gas law is used

and is given by the following equation.

After determining the pressure and density, the change in pressure will be determined using the

density, gravity, and the height. The equation is given by

The velocity of the pipe will be determined using a simplified version of the Bernoulli equation.

Next, numerical integration will be used to determine the mass flow rate. Since the

measurements were taken in quarter inch increments, the velocity will be taken for quarter inch

spacing. The results will be tabulated in a table. The equation for mass flow and the integral over

the radius are given by the following equations

The integral will be taken from 0 to the total radius, but since the integration will be calculated

numerically, a summation will be used. The equation is given by

The average velocity will be calculated using the total mass flow rate, area, and density. The

equation is given by



Next, the flow rate will be determined for the pitot tube and the flowmeter. The flow rate can be

calculated using a simple formula using the velocity and the cross sectional area of the pipe.

First, the velocity of the fan will be determined by taking the change in distance of the change in

time, and the determined velocity of the fan will be used to determine the velocity of the pipe.

Then, the flow rates can be determined using the following equation

Next, the calculations will be determined for the venturi meter. First, the it will be

assumed that the actual flow rate and the flow rate for the rotometer are equal. The equation for

the actual flow rate is given by the following formula

where A2 is the throat area and A1 is the inlet area. The equation can be rearranged to determined

the value for C. The equation is given by

The change in pressure will be calculated by using the difference between the highest and lowest

reading values on the venturi meter. The equation is given by

Then the height of H2O will be converted to a change in pressure given in pounds per foot

squared. Lastly, the Reynolds number will be calculated using the following formula.



3. Theory

The theory of this lab is simple overall. Measurements were taken using various devices.

This lab focused on the application of the Bernoulli equation. In the first part of the experiment,

the velocity was determined using a simplified portion of Bernoulli’s equation. The portion used

is given by

Also, the bernouli principle was used to measure the flow rate of the fluid using the venturi

meter. The flow in a venturi meter is calculated in a very similar way to the flow calculated in

the orifice meter. For the pitot-static tube, the flow velocity is accurately measured as long as the

flow behaves normally. Basically, the Bernoulli principle is applied in nearly all fluid flow

applications. The principle states that for incrompressible flow, the summation of the dynamic

pressure, static pressure, and pressure head are equal to a constant.

4. Discussion and Results

The first part of the lab was done in the Pitot Tube where it was moved a quarter of an inch

down from the top after each recording. These results are found in Table 1: Pitot Tube Data. This data

was then used to determine the velocity of the air being blown through the pipe. The closer the pitot

tube got to the center of the pipe the higher the velocity got. In Figure 4.1, a plot of the radial position

versus the velocity can be seen. It can be seen that when the pitot tube passes the center of the pipe the

velocity will begin to decrease.



Table 1: Pitot Tube Data

r Δh Tamb Pamb ρamb ΔP V

in in R lb/ft2 sl/ft3 lb/ft2 ft/s

2.375 0.125 534 2039 0.002225147 0.650708333 24.18403928

2.125 0.375 534 2039 0.002225147 1.952125 41.88798476

1.875 0.625 534 2039 0.002225147 3.253541667 54.07715579

1.625 0.875 534 2039 0.002225147 4.554958333 63.98495362

1.375 1.125 534 2039 0.002225147 5.856375 72.55211783

1.125 1.375 534 2039 0.002225147 7.157791667 80.2093842

0.875 1.625 534 2039 0.002225147 8.459208333 87.19679366

0.625 1.875 534 2039 0.002225147 9.760625 93.66438136

0.375 2.125 534 2039 0.002225147 11.06204167 99.71334839

0.125 2.375 534 2039 0.002225147 12.36345833 105.4157833



Figure 4.1: Plot of the radial position versus the velocity of the air in the pipe creating a velocity

profile.

From this same data the velocity profile can be used for numerical integration. This integration

can be used to find the mass flow rate of the air in the pipe. In Table 2: Numerical Integration of Velocity

Profile, these values can be seen. As the radial position gets closer to the center the mass flow rate

decreased.

-2.500

-2.000

-1.500

-1.000

-0.500

0.000

0.500

1.000

1.500

2.000

2.500

0 20 40 60 80 100 120

R a d i a l P o s i t i o n

Velocity (ft/s)



Table 2: Numerical Integration of Velocity Profile

Location r r from CL dr ΔP V ṁ

in in in in H20 ft/s sl/s

1 0.125 2.375 0.25 0.09 20.50967 0.00118173

2 0.375 2.125 0.25 0.13 24.64955 0.00127076

3 0.625 1.875 0.25 0.15 26.47786 0.00120443

4 0.875 1.625 0.25 0.16 27.34622 0.00107807

5 1.125 1.375 0.25 0.17 28.18784 0.00094029

6 1.375 1.125 0.25 0.175 28.59936 0.00078056

7 1.625 0.875 0.25 0.18 29.00505 0.00061571

8 1.875 0.625 0.25 0.18 29.00505 0.00043979

9 2.125 0.375 0.25 0.18 29.00505 0.00026388

10 2.375 0.125 0.25 0.175 28.59936 8.6729E-05

Table 3 shows a comparison of the mass flow rate, velocity and the volumetric flow rate of the

pitot tube and fan flowmeter. These values were all greater at the pitot tube than at the fan flowmeter.

This could be because the pitot tube was closer to the air source than the fan flowmeter which was at

the end of the tube.

Table 3

ṁ Vavg Q

sl/s ft/s CFM

Pitot Tube 0.007862 25.9253 211.5505

Fan Flowmeter 0.000212 0.693 5.67



In the second part of the lab pressure differences were looked at through a venture meter using

different flow rates. There were a total of eleven different tubes that were filled with different H2O

heights which were due to the pressure at that point in the pipe. These values and they’re pressure in

lb/ft2 can be seen in Table 4. These pressure results were then graphed on Figure 4.2 and it can be seen

that as the flow rate decreases the less the pressure differential there is between the largest pressure

and smallest pressure poin

Figure 4.2: Plot of the pressure differential versus the position of the tube for each flow rate.

0

5

10

15

20

25

30

35

40

45

50

0 2 4 6 8 10 12

Δ P

x

5.7 GPM

4.7 GPM

3.7 GPM

2.7 GPM

1.7 GPM

Δh ΔP Δh ΔP Δh ΔP Δh ΔP Δh ΔP Δh ΔP Δh ΔP Δh ΔP Δh ΔP Δh ΔP Δh Δ

GPM in H2O lb/ft2

in H2O lb/ft2

in H2O lb/ft2

in H2O lb/ft2

in H2O lb/ft2

in H2O lb/ft2

in H2O lb/ft2

in H2O lb/ft2

in H2O lb/ft2

in H2O lb/ft2

in H2O lb/5.7 213 43.65382 207 42.42413 151 30.94707 69 14.14138 88 18.03538 132 27.05307 155 31.76686 172 35.25097 184 37.71034 192 39.34992 196 40.1

4.7 231 47.34287 227 46.52308 186 38.12024 128 26.23328 139 28.4877 174 35.66087 190 38.94003 203 41.60434 209 42.83403 211 43.24392 220 45.0

3.7 209 42.83403 205 42.01424 180 36.89055 144 29.51244 150 30.74213 175 35.86581 184 37.71034 192 39.34992 196 40.16971 200 40.9895 202 41.

2.7 197 40.37466 194 39.75982 179 36.6856 159 32.58665 161 32.99655 174 35.66087 179 36.6856 184 37.71034 185 37.91529 188 38.53013 189 38.7

1.7 201 41.19445 200 40.9895 189 38.73508 176 36.07076 178 36.48066 184 37.71034 186 38.12024 189 38.73508 190 38.94003 191 39.14497 193 39.5

Table 4

Tube F Tube G Tube H Tube J Tube K Tube LFlow Rate

Tube A Tube B Tube C Tube D Tube E



Table 5 shows the comparison of the discharge coefficient and the Reynolds Number for each

flow rate. This comparison was then plotted on Figure 4.3.

Table 5

FlowrateC

ΔP Re

GPM lb/ft2

5.7 0.947698 29.48 17387.88

4.7 0.927592 21.09 14394.88

3.7 0.913297 13.31 11259.36

2.7 0.877025 7.78 8266.368

1.7 0.671029 5.12 5130.849

Figure 4.3: Plot of discharge coefficient versus the Reynolds Number.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5000 10000 15000 20000

C

Re



5. Conclusions and Recommendations

In conclusion this lab gave greater detail on different ways of measuring pressure in pipes of

different fluids. The pitot-static tube is useful in determining the pressure at different internal radiuses

of a pipe, which then can be used to determine velocity profile in the pipe. This can be helpful in

knowing the type of flow that is being produced in the pipe. In the venturi meter the measured pressure

drop of the fluid flowing in the pipe is very helpful when the pipe may only be able to hold a certain

pressure drop. By changing the flow rate of the fluid entering the pipe can change how the pressure

drop is affected.

References

MAE 322 “Impact of a Jet.” Experiment #1 handout.



Appendices

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