Experiment #6

Fluid Meters in Incompressible Flow

Stephen Mirdo

Performed on October 28, 2010

Report due November 4, 2010

Table of Contents Object ………………………………………..………………………….………….…. p. 1 Theory …………………….…………………………………….……….………...pp. 1 - 3 Procedure ………………………………………………...………...…………………. p. 4 Results ………………………………………………………..……...……………pp. 5 - 6 Discussion and Conclusion …………………………………………………….………p. 7 Appendix ……………………………………………………..…….........………pp. 8 – 10

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Object

The object of this experiment was to examine the effects of flow rate on the pressure of a venturi meter and relate the observations to the hydrostatic, Bernoulli and continuity equations.

Theory

Flow meters are used in pipe flow scenarios to adjust particular physical characteristics of the fluid flow and enable for the measurement of the adjustment. One type of flow meter is the venture meter, as seen in Figure 1. An incoming flow enters the meter at the region indicated as 1 on the diagram. As the flow approaches the region indicated as 2 in Figure 1, the cross sectional area of the meter decreases. This region is called the throat of the venturi meter. The fluid entering the throat of the venturi meter experiences an increase in velocity due to the reduction in pipe diameter by means of the continuity equation (Equation 4, below).

Figure 1: Diagram of a venturi meter where 1 indicates the upstream portion of the meter, 2 represents the throat of the meter and Δh indicates the pressure drop of the manometer. (Source: A Manual for the Mechanics of Fluids Laboratory, W.S. Janna, 2008) The air over water manometer of the venturi meter indicates a drop in the pressure of the flow at the throat of the meter. This pressure drop, indicated as Δh in Figure 1, is due to the increased velocity that occurs in the throat as a product of the decrease in the pipe diameter and can be determined by the hydrostatic equation (Equation 1). By Bernoulli’s equation, the cause of the pressure drop is determined to be the increase of velocity of the pipe flow (Equation 2). By aggregating the hydrostatic, Bernoulli’s and continuity equations, the theoretical flow rate passing through the venturi meter can be calculated.

1

2

Bernoulli’s equation is an energy balance equation and is given as:

P1/ρ + V12/2 + gz1 = P2/ρ + V2

2/2 + gz2 (Equation 1) where P1 is the pressure of the fluid flow as it enters the meter, ρ is the density of the flowing fluid, V1 is the upstream velocity of the flow, g is gravitational acceleration, z1 is the height of the fluid as it enters the meter, P2 is the pressure of the fluid at the throat of the meter, V2 is the velocity of the flow at the throat and z2 is the height of the fluid at the throat of the meter. Considering a horizontal application, gravitational potential energy is neglected because there is no change in height of the fluid and Bernoulli’s equation can be rewritten as:

P1/ρ + V12/2 = P2/ρ + V2

2/2 (Equation 2) Bernoulli’s equation can then be rearranged to solve the energy balance in terms of the velocities of the flow at state 1 and state 2.

ΔP/ρ = V22/2 – V1

2/2 (Equation 3)

where ΔP is the pressure difference P1 – P2. Because the pressure drop, ΔP, and the velocities V1 and V2 cannot be measured directly, the hydrostatic equation (Equation 4) and the continuity equation (Equation 5) are employed. The Δh variable of the hydrostatic equation is the difference in height of the air over water manometer due to pressure and is measured directly. A1 of the continuity equation is the cross sectional area of the upstream region of the venturi meter, labeled as region 1 in Figure 1. A2 is cross sectional area of the throat of the venturi meter, labeled as region 2 in Figure 1.

ΔP = ρgΔh (Equation 4)

Qth = V1A1 = V2A2 (Equation 5) Equation 5 is rearranged to solve for V1 and is written as follows:

V1 = V2A2/A1 = V2(D22/D1

2) (Equation 6)

where D2 is the diameter of the throat of the venturi meter and D1 is the diameter of the upstream region of the meter. Still, the velocity at state 2 is unknown and can be solved for by rearranging the continuity equation to be substituted into Equation 3.

V22 = (Qth/A2)2 (Equation 7)

where Qth is the theoretical flow rate.

2

3

Substituting Equation 4 in for ΔP and Equation 6 in for V1, the Bernoulli equation (Equation 3) becomes:

gΔh = [V22/2 - V2(D2

2/D12)] (Equation 8)

Tidying up Equation 8 yields the following:

2gΔh = V22 [1 - (D2

2/D12)] (Equation 9)

Substituting Equation 7 into Equation 9 yields the following:

2gΔh = Qth2/A2

2 [1 - (D22/D1

2)] (Equation 10) Rearranging to solve for the theoretical flow rate Qth yields the following: __________________

Qth = A2√(2gΔh)/ [1 - (D22/D1

2)] (Equation 11) The Reynolds number of the pipe flow can be calculated using the following equation:

Re = V2D2 / ν (Equation 12) where ν is the kinematic viscosity of the fluid. The coefficient of discharge, Cv, can be calculated using the following equation

Cv = Qact / Qth (Equation 13) where Qth is the theoretical flow rate and Qact is the indicated flow rate of the testing apparatus.

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Procedure Equipment

Fluid Meters Apparatus Experiment

1) Ensure that the sump tank has water and that all valves of the testing apparatus are in the open position.

2) Engage the pump, as seen in the lower left of the apparatus in Figure 2. 3) Control the flow rate with the valve nearest the pump. Use the turbine-type

flow meter to establish a desired flow rate. This flow rate should be low, for example, 5 LPM.

4) Record the change in height, Δh, from the manometer specific to the venturi meter from the manometer board.

5) Increase the flow rate with the valve nearest the pump. Again, record the change in height from the manometer specific to the venturi meter.

6) Repeat step 5 for at least five more data points.

Figure 2: Diagram of the fluid meters apparatus, including the turbine type meter used to record actual flow rates and the subject of the experiment, the venturi meter. (Source: A Manual for the Mechanics of Fluids Laboratory, W.S. Janna, 2008)

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Results Table 1: Dimensions of the upstream and throat portions of the venturi meter and the kinematic viscosity of water at room temperature required to solve for the theoretical flow rate of the venturi meter.

D1 Upstream Diameter (m) 0.026035 D2 Throat Diameter (m) 0.015875 Kinematic Viscosity of Water ν (m2/s) 9.14E-07

Table 2: Theoretical and experimental flow rates of the venturi meter with the calculated coefficient of discharge and the Reynolds number for each actual flow rate.

Q actual (m3/s) Δh (m) Q theoretical

(m3/s)

Velocity2 at Throat (m/s)

Coefficient of Discharge

Cv Re

8.4168E-05 0.0127 1.0643E-04 0.4252 0.7908 7390 1.7167E-04 0.0413 1.9187E-04 0.8673 0.8947 15072 2.5351E-04 0.0873 2.7907E-04 1.2808 0.9084 22257 3.3667E-04 0.1524 3.6869E-04 1.7009 0.9132 29559 4.2468E-04 0.2350 4.5778E-04 2.1456 0.9277 37286 5.1001E-04 0.3350 5.4665E-04 2.5767 0.9330 44778 5.8701E-04 0.4351 6.2297E-04 2.9657 0.9423 51539 6.6835E-04 0.5588 7.0599E-04 3.3766 0.9467 58680

Table 3: Percent error calculation between theoretical and indicated flow rates of the testing apparatus.

Q actual (m^3/s)

Q theoretical (m^3/s) % Error

8.4168E-05 1.0643E-04 20.92% 1.7167E-04 1.9187E-04 10.53% 2.5351E-04 2.7907E-04 9.16% 3.3667E-04 3.6869E-04 8.68% 4.2468E-04 4.5778E-04 7.23% 5.1001E-04 5.4665E-04 6.70% 5.8701E-04 6.2297E-04 5.77% 6.6835E-04 7.0599E-04 5.33%

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0.0000E+00

1.0000E-04

2.0000E-04

3.0000E-04

4.0000E-04

5.0000E-04

6.0000E-04

7.0000E-04

8.0000E-04

0.0000 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000

Δh (m)

Flow

Rat

e (m̂

3/s)

Experimental Theoretical

Figure 3: Graph of theoretical and actual volume flow rate through a venturi meter as a function of head loss.

0.1000

1.0000

10.00001000 10000 100000

Reynolds Number

Coe

ffici

ent o

f Dis

char

ge

Figure 4: Log-Log graph of the calculated coefficient of discharge for the venturi meter as a function of the Reynolds number

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Discussion & Conclusion

Unlike the venturi meter, where the pressure drop is recorded at the area of constriction of the flow, the orifice meter’s pressure drop is indicated downstream from the obstruction. This is due to the fact that the flow that has passed through the orifice has not yet regained its flow characteristics it had prior to passing through the obstruction. The pressure of the fluid is more or less the same at its indicated point as it was as it passed through the orifice. This method would not work with the venturi meter. To account for this discrepancy, the formulation of the theoretical flow rate equation uses gravitational potential indicated by the air over water manometer instead of the pressure difference over the density of the fluid. Both methods of calculation yield an energy equation.

Of the flow meters discussed, the best type will depend on the application. For example, a venturi meter is best suited for applications where pressure differences of a small magnitude are desired. In situations where the pressure difference is desired to be of a higher magnitude than the venturi is capable of, an orifice meter is better suited. An application where an orifice meter would be used is in a large scale industrial scenario, such as that of a dam where the desired pressure differences would be large in magnitude. For smaller scale applications, such as residential plumbing where pressure differences a fraction of those needed for a dam, the venturi meter is better suited.

The venturi meter experiences the smallest pressure loss of all the aforementioned flow meters. That does not imply that this is the best type of meter to be used for all applications. The amount of pressure drop desired is entirely dependent on the application in which the flow meter will be used. Because of the smaller pressure losses experienced by the venture meter, it is the more accurate of the meters discussed. The orifice meter may be able to produce a higher level of precision, but it is inaccurate at lower flow rates. Accuracy is the tendency of measurements to center around an accepted or actual value of some measured quantity. Precision is the tendency of measurements to center around a particular value, whether it is the expected value or not.

There were sources of error in this experiment. One source of error was due to

the measurement of the head loss, Δh, from the manometer board. Due to nonsteady flow in the testing apparatus, the air over water manometer did not give a steady reading. In order to compensate for this discrepancy, the lowest value the fluctuating fluid took was the recorded value. Another source of error was present due to the neglecting of friction in the theoretical flow rate calculation. This discrepancy explains the difference of values plotted in Figure 3. Because friction was neglected, the theoretical values of the flow rate appear to be higher than the experimental values. In reality, the viscous forces of the fluid and the pipe cause the flow rate to be lower than the calculated values.

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Appendix Data Usage Sample calculation of the theoretical flow rate at a head loss of 0.0127 m: _________________________________________________

π/4 * 0.015875m2 * √(2*9.81 m/s2 * 0.0127 m) / [1 – (0.015875m)4/(0.026035m)4] = 1.0643E-04 m3/s

Sample calculation of the velocity at region 2 (referenced from Figure 1) at flow rate of 8.4168E-05 m3/s:

8.4168E-05 m3/s / [(π/4)*(0.015875m)2] = 0.4252 m/s Sample calculation of the percent error in theoretical and experimental flow rates at head loss of 0.0127 m:

|1.0643E-04 m3/s - 8.4168E-05 m3/s | / 1.0643E-04 m3/s * 100 = 20.92%

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Bibliography

Introduction to Fluid Mechanics, 3rd Edition William S. Janna (1993)

A Manual for the Mechanics of Fluid Laboratory

William S. Janna (2008)

The Engineering Toolbox – Types of Flow Meters http://www.engineeringtoolbox.com/flow-meters-d_493.html