Lab3 Flow Metering Device PDF

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ME211 FLUID MECHANICS LABORATORY MANUAL Experiment #3 Venturi Flow Meter (Bernoulli Equation)

Revision 5 October 2019 Constantine M. Megaridis Mechanical and Industrial Engineering University of Illinois at Chicago

OBJECTIVE In this experiment, variations of static pressure and fluid flow velocity in a Venturi tube (a.k.a. Venturi metering device) are investigated. The main goal of the experiment is to utilize this device to measure liquid flow rate, and provide comparisons between experiment (practical case) and theory (idealized case) via the Bernoulli equation.

THEORETICAL BACKGROUND Figure 1 illustrates the shape of the horizontally-mounted Venturi tube and marks the axial stations where static (normal to the flow direction) pressure measurements are made. Let us apply Bernoulli’s equation between the inlet (station A or point 1) and any other station (index n) along the center streamline. n can take any value from 2 to 6 (see Fig. 1 and corresponding table underneath the figure). Then p1+ρV12/2+ρgz1= pn+ρVn2/2+ρgzn

(1)

which for a horizontal tube (z1=zn) simplifies to p1 / +V12/2g= pn / +Vn2/2g

(2)

after division by  =ρg, the specific weight of the liquid (water in this case). The mass conservation equation (with ρ = const) between stations (1) and ( n) gives V1 A1=Vn An=Q

(3)

where Q is the volume flow rate (constant through the tube). Since p1= h1 and

pn=  hn

(4)

combining equations (2), (3) and (4), we obtain 𝑉1 = √

2𝑔(ℎ1 −ℎ𝑛 ) 2

𝐴 ( 1 ) −1 𝐴 𝑛

(5)

Equation (5) was obtained using the Bernoulli equation, therefore it is based on the underlying assumptions that the flow is frictionless (ideal). Since friction is always present, use of Eq. (5) to calculate the fluid velocity (from the measured h1  hn ) and the fluid flow rate, produces a value that may not match the one measured in the experiment (Qexp). The theoretical (frictionless) value Qideal is given by 𝑄𝑖𝑑𝑒𝑎𝑙 = 𝑉1 𝐴1 = √

In general,

2𝑔(ℎ1 −ℎ𝑛 ) 2

𝐴 ( 1 ) −1 𝐴 𝑛

𝐴1

(6)

Qexp=C Qideal

Equation (5) can be rearranged as

ℎ1 −ℎ𝑛 𝑉12 /2𝑔

=(

(7) 𝐴1 2 𝐴𝑛

) −1

(8)

As does the original Eq. (5), the Eq. (8) also applies for frictionless (inviscid) flow. If the flow were indeed ideal, then a plot of (h1  hn)/(V12/2g) versus (A1/An)2 1 should be a straight line with slope of 45. But viscous effects would cause deviations from the straight-line behavior, so Eq. (7) would give C≠1.

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Fig. 1: Schematic showing the Venturi metering tube and the stations where static pressure measurements are made in the convergent configuration. More information on these stations is given in the table below. All dimensions in mm. Tapping Position A (point 1) B (point 2) C (point 3) D (point 4) E (point 5) F (point 6)

Manometer Legend

h1 h2 h3 h4 h5 h6

Tube Dia. (mm) 25.0 13.9 11.8 10.7 10.0 25.0

Note: The assumed datum position is at tapping A associated with h1

Fig. 2: Schematic showing the Bernoulli Apparatus used in the lab experiment. The Venturi test section extends between the two unions, one at either end, to facilitate reversal for convergent or divergent testing. The module is mounted on the Hydraulics Bench, which facilitates flow control. The hand pump mounted behind the manometer panel is not used in your experiments.

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EXPERIMENT OUTLINE 1. 2. 3. 4. 5.

Review apparatus description and overall procedure. Measure, calculate, or obtain the quantities listed in the data tables. Plot the data, as instructed. Discuss possible causes for differences between experimental and theoretical results. Follow the lab report write-up format given in this course.

APPARATUS DESCRIPTION The test section (Fig. 1) is an accurately-machined clear acrylic duct of varying circular cross section. It is provided with a number of side hole pressure tappings, which are connected to the manometers housed on the rig. These tappings allow the measurement of static pressure head simultaneously at each of 6 sections. The dimensions of the test section, the tapping positions and the test section diameters are shown in Fig. 1 and the associated Table. In this schematic, five of the tappings are in the converging portion of the test section. A hypodermic metal tube, the provided total pressure head probe, may be positioned to read the total pressure head (h8) at any section of the duct. This total pressure head probe may be moved after slackening the gland nut; this nut should be re-tightened by hand. To prevent damage, the total pressure head probe should be fully inserted during transport/storage of the apparatus. An additional tapping is provided to facilitate setting up. All eight pressure tappings are connected to a bank of pressurized manometer tubes. Pressurization of the manometers is facilitated by removing the hand pump from its storage location at the rear of the manometer board and connecting its flexible coupling to the inlet valve on the manometer manifold. This is a step that you will not have to do in your experiment. In use, the apparatus is mounted on a base board, which is placed on the work surface of the bench. This base board has feet, which may be adjusted to level the apparatus. A level glass is provided as part of the base. The TA’s would have adjusted the device in advance, so you would not need to level the device yourselves. The inlet pipe terminates in a female coupling, which may be connected directly to the bench supply. A flexible hose is attached to the outlet pipe, which should be directed to the volumetric measuring tank on the hydraulics bench. A flow control valve is incorporated downstream of the test section. Flow rate and pressure in the apparatus may be varied independently by adjusting the flow control valve and the bench supply control valve.

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NOMENCLATURE Column Heading

Units

Nom.

Type

Volume of fluid collected

m3

Vol

Measured

Time to collect

s

t

Measured

Volume flow rate

m3/s

Q

Calculated

Q=Vol/t

hx

Marked

Manometer identification labels

Given

Position of manometer tappings given as distance from the datum at tapping h1. See Fig. 1

Manometer legend

Description Taken from scale on hydraulics bench. The volume collected is measured in liters. Convert to cubic meters for the calculations (divide reading by 1000). Time taken to collect the known volume of water in the hydraulics bench.

Distance into duct

m

Area of duct

m2

An

Known

The area of the duct at each tapping station n. See test section dimensions in Table under Fig. 1

Static head

m

hn

Measured

Measured value from the appropriate manometer. The manometer readings are taken in mm Water. Convert to m Water for calculations.

Velocity

m/s

Vn

Dynamic head

m

Total head (TH)

Distance of TH probe into duct TH Probe reading

m

ho

m

m

h8

Calculated

Velocity of fluid at station n in duct (Q /An)

Calculated

Vn2/g, see theory

Calculated

hn + Vn2/2g, see theory

Measured

Position of the total head probe from the datum at tapping h1

Measured

Head recorded by total head probe

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EXPERIMENTAL PROCEDURE Method We measure flow rates and both static and total pressure heads in a rigid, transparent convergent/divergent tube of known geometry for a range of steady flow rates. Equipment • The F1-10 Hydraulics Bench, which allows us to measure flow by timed volume collection. • The F1-15 Bernoulli Apparatus Test Equipment • Your cell phone timer to determine the flow rate of water. Technical Data The dimensions shown in Fig. 1 and the Table below that figure are used in the appropriate calculations. Theory Equation (5), as derived earlier, can be used to determine the velocity V1 at tapping station 1. The ideal (frictionless) volume flow rate Qideal is calculated using Eq. (6). Notice that you can obtain one Qideal using the values of An and hn for each tapping station. Later on, you will use Eq. (8) to compare Qideal with the experimentally-measured Qexp values. Procedure - Equipment Set up Level the apparatus Set up the Venturi apparatus on the hydraulic bench so that its base is horizontal; this is necessary for accurate height measurement from the manometers. The TA’s would have adjusted the apparatus in advance, so this is something you would not need to do. Note the direction of the test section Ensure that the test-section has the 14° tapered section converging in the direction of flow. Connect the water inlet and outlet Ensure that the rig outflow tube is positioned above the volumetric tank, in order to facilitate timed volume collections. Connect the rig inlet to the bench flow supply; close the bench valve and the apparatus flow control valve, and then start the pump. Gradually open the bench valve to fill the test rig with water. Bleeding the manometers In order to bleed air from pressure tapping points and manometers, close both the bench valve, the rig flow control valve and open the air bleed screw and remove the cap from the adjacent air valve. Connect a length of small bore tubing from the air valve to the volumetric tank. Now, open the bench valve and allow flow through the manometers to purge all air from them; then, tighten

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the air bleed screw and partly open the bench valve and test rig flow control valve. Next, open the air bleed screw slightly to allow air to enter the top of the manometers (you may need to adjust both valves in order to achieve this); re-tighten the screw when the manometer levels reach a convenient height. The maximum volume flow rate will be determined by the need to have the maximum (h1) and minimum (h5) manometer readings both on scale. If required, the manometer levels can be adjusted further by using the air bleed screw. Procedure - Acquiring a Set of Results Readings should be taken at 3 flow rates with the forward facing tube. Subsequently, move on to the second apparatus that has a reversed test section in order to repeat the measurements there. Setting the flow rate Take the first set of readings at the maximum flow rate, then reduce the volume flow rate to obtain h1 - h5 (convergent configuration) head difference near 50 mm. Finally, repeat the whole process for one more flow rate, set to give the h1 - h5 difference approximately half way between that obtained in the above two tests. For the reversed (divergent) configuration, use the h1 – h2 value to set the three different flow rates. Reading the static head Take readings hn of the manometers when the levels have steadied. Ensure that the total pressure probe is retracted from the test-section when you take the static head readings. Timed volume collection You should carry out a timed volume collection, using the volumetric tank, in order to determine the volume flow rate. This is achieved by closing the ball valve and measuring (with a stopwatch) the time taken to accumulate a known volume of fluid in the tank, which is read from the sight glass. You should collect fluid for at least one minute to minimize timing errors. Again, the total pressure probe should be retracted from the test-section during these measurements. Enter the test results into the data entry form, and repeat this measurement twice to check for repeatability. Reading the total pressure head (h8) distribution (Trial 1 only) Measure the total pressure head distribution by traversing the total pressure probe along the length of the test section. Start with the measurement at the cross section of the side hole pressure tapping associated with the manometer measuring h1. Then repeat the measurement by moving the probe to the station corresponding to each tap (2 to 6). Reversed test section (These instructions are for the TA’s who will have the two set ups ready for your use) Ensure that the total pressure probe is fully withdrawn from the test-section (but not pulled out of its guide in the downstream coupling). Unscrew the two couplings, remove the test-section and reverse it, then re-assemble by tightening the coupling. RESULTS The results for each run (trial) should be recorded in the tables found on the next page.

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Data Tables Convergent (Taps 1-5 in converging duct) Trial 1

Total Head (manometer reading h8) =

Water Volume Collected

Time to Collect

Vol (m 3)

t (s)

Volume Flow Rate

Static head

(m) 0.00 0.0603 0.0687

1 2 3 4 5 6

0.0732 0.0811 0.1415

A (m2 ) 490.9 x 10−6 151.7 x 10−6 109.4 x 10−6 89.9 x 10−6 78.5 x 10−6 490.9 x 10−6

h (m)

h8

(m)

Total Head at tap 1: h8 =

Volume Collected

Time to Collect

Vol (m3)

t (s)

Volume Flow Rate

Trial 3

Distance into duct

Area of duct

Static head

Tap

Q (m3/s)

(m) 0.00 0.0603 0.0687 0.0732 0.0811 0.1415

1 2 3 4 5 6

Average Flow Rate

A (m 2) 490.9 x 10−6 151.7 x 10−6 109.4 x 10−6 89.9 x 10−6 78.5 x 10−6 490.9 x 10−6

h (m)

Total Head at tap 1: h8 =

Volume Collected

Time to Collect

Vol (m3)

Average Flow Rate

t (s)

Volume Flow Rate

Total head

Tap

Q (m3 /s)

Average Flow Rate

Trial 2

Distance into duct Area of duct

Distance into duct

Area of duct

Static head

Tap

Q (m3/s)

(m) 0.00 0.0603 0.0687

1 2 3 4

0.0732 0.0811 0.1415

5 6

8

A (m 2) 490.9 x 10−6 151.7 x 10−6 109.4 x 10−6 89.9 x 10−6 78.5 x 10−6 490.9 x 10−6

h (m)

Divergent (Taps 2-6 in diverging duct) Trial 4

Total Head at tap 1: h8 =

Volume Collected

Time to Collect

Vol (m3)

t (s)

Volume Flow Rate

Area of duct

Static head

Tap

Q (m3/s)

(m) 0.00 0.0605 0.0684 0.0729 0.0813 0.1415

1 2 3 4 5 6

Average Flow Rate

Trial 5

Distance into duct

A (m 2) 490.9 x 10−6 78.5 x 10−6 89.9 x 10−6 109.4 x 10−6 151.7 x 10−6 490.9 x 10−6

h (m)

Total Head at tap 1: h8 =

Volume Collected

Time to Collect

Vol (m3)

t (s)

Volume Flow Rate

Distance into duct

Q (m3/s)

(m) 0.00 0.0605 0.0684 0.0729

1 2 3 4 5 6

Average Flow Rate

Trial 6

Area of duct

Static head

Tap

0.0813 0.1415

A (m 2) 490.9 x 10−6 78.5 x 10−6 89.9x 10−6

h (m)

109.4 x 10−6 151.7 x 10−6 490.9 x 10−6

Total Head at tap 1: h8 =

Volume Collected

Time to Collect

Vol (m3)

Average Flow Rate

t (s)

Volume Flow Rate

Distance into duct

Area of duct

Static head

Tap

Q (m3/s)

(m) 0.00 0.0605 0.0684 0.0729 0.0813 0.1415

1 2 3 4 5 6

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A (m 2) 490.9 x 10−6 78.5 x 10−6 89.9x 10−6 109.4 x 10−6 151.7 x 10−6 490.9 x 10−6

h (m)

DATA ANALYSIS AND CONCLUSIONS 1.

Plot the total head distribution h8 along the axis of the Venturi meter (Trial 1 only) For steady, incompressible, frictionless flow, we would expect this value to be constant. Is it, according to your measurements? Explain why, or why not.

2.

For station n = 5 (nozzle throat in convergent configuration), plot the measured values of (h1-h5)1/2 versus the flow rate Q (three values of Qexp and Qideal for each trial). According to Eq. (6), which is valid for frictionless flow, Qideal is proportional to (h1 hn)1/2. Consequently, the (h1-h5)1/2 versus Qideal curve would be a straight line through the origin in this plot. However, the points in that plot, as represented by the pairs of (h1-h5)1/2 and Qexp, may not lie on a straight line due to the inevitable viscous (friction) effects.

3.

Repeat the above for station n=2 in the divergent configuration, where the throat is during trials 4-6.

4.

For each of the two configurations (convergent, divergent), plot the ratio C=Qexp/Qideal versus Qexp (there are three values of Q, one for each trial). The ratio C quantifies the agreement between Qexp and Qideal (perfect agreement when C = 1). It is important to note that the value of C is not constant as Q varies. Does the value of C get closer to unity as Q (or likewise, velocity) gets larger? If so, explain why.

5.

For each of the two configurations (convergent, divergent): For the maximum and minimum flow rates, and using the given cross sectional areas, find the velocity value V1=Qexp/A1 and plot the quantity 2g (h1  hn)/V12 versus the area factor (A12/An2 1) for both ideal flow (show as a continuous curve; Eq. 8), as well as the actual flow (show as points). Comment on the agreement between the two data sets.

QUESTIONS FOR FURTHER DISCUSSION 1.

Why are stations #1 and 5 best suited for the calculation of Q and V1? What would be the effect of using any other pair on the accuracy of the result?

2.

Ideally (if no friction), in the convergent configuration (Fig. 1), h6 should be equal to h1. Are these two quantities equal in the experiment? If not, provide an explanation why they are different.

3.

Assuming that a fluid particle moves from 1 to 6 along the centerline of the Venturi tube (Fig. 1). Can you estimate using the experimental data what is the energy loss of the particle during that period?

4.

Would a different Venturi meter give the same values of C?

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