Experiment 12

ME 310 Fluid Mechanics Experiment 12 – The Venturi Meter, The Determination of Discharge from a Pipe ______________________________________________________ This experiment is designed to help you understand the measurement of discharge from a pipe and the effect of viscosity on pressure loss. Objective: • •

To calculate Q and CV at a number of head pressures. To draw a relationship between these properties and head pressure.

Reference: Sections 8-6 through 8-11 in Intro. to Fluid Mechanics; Fox, McDonald, and Pritchard; 6th edition. Description of the Apparatus: The apparatus consist of a flow bench that allows water flow to the venture meter. Inside the flow bench is weighing tank connected to one end of a lever arm. The end of the lever arm protrudes from the side of the flow bench so that the amount of weight on this end of the lever arm may be adjusted (as shown in Figure 12.2). The purpose of the lever arm is to measure the actual mass flow rate of water flowing through the measuring devices. When using the hydraulic bench, placing weight on the lever arm closes the trip valve of the inner tank. When water entering the tank is sufficiently heavy enough to counterbalance the weight on the arm, the arm will rise and the trip valve will open. Dividing the mass of water contained in the tank by the amount of time it takes for the internal tank to fill will yield the actual mass flow rate. Since the adjustable weight end of the lever arm has a three-to-one advantage over the water tank end, the mass of the water in the tank will equal three times the mass added to the lever arm. The weight of the hanger is accounted for in the design of the equipment; therefore, do not add the weight of the hanger to weights placed on the hanger.

Figure 12.1 Venturi Meter and Hydraulic Bench. 12.1

Outer Tank

Flow Measurement Apparatus Exit

Inner Tank Lever Arm

Inner Tank Weight Measured Water Main Tank Valve

Trip Valve Water Supply

Pump Exit – To Flow Measurement Figure 12.2 Hydraulic Bench.

Figure 12.3 Venturi Meter. 12.2

A

B C D

E

F G

H J

K L

7

8 34

22

22

37

54

52

67 82

102

Figure 12.4 Diagram of Venturi Meter (all distances in mm)

Manometer Tube # A(1) B C D(2) E F G H J K L

Table 12.1 Diameter of Cross Section (mm) 26.00 23.20 18.40 16.00 16.80 18.47 20.16 21.84 22.53 25.24 26.00

12.3

Distance From Inlet (mm) 0 20 32 46 61 76 91 106 121 136 156

Experimental Procedure:

1- Make sure the air purge valve on the upper manifold is tightly closed. 2- Set both apparatus flow control and bench supply valve to approximately 1/3 their fully open positions. 3- Switch on bench supply valve and allow water to flow. (Tap manometer tubes in order to remove air bubbles from apparatus.) 4- Close apparatus flow control valve. 5- Release air purge valve to allow water to rise approximately 2/3 the way up the manometer tubes. 6- Open apparatus flow control valve to obtain full flow. (At this condition the pressure difference between the Venturi inlet [A] and throat [D] is approximately 240mm. 7- Make 10 runs, being sure measure and calculate flow rate. Also measure h1 and h2, where h1 is the height of water in manometer tube A (inlet) and h2 is the height of water in manometer tube D (throat). Vary the flow rates so that (h1 – h2) goes from approximately 240mm to 0mm. It is advisable to use enough weight on the arm that the weighing tank takes about 60-90 seconds to fill. 8- Make an additional two runs (at relatively high flow rates) taking pressure readings from all tubes along the length of the Venturi meter.

Theory:

Assumptions: • Steady flow. • Incompressible flow. • Frictionless flow. • Flow along a streamline. In these equations the subscript 1 is for manometer tube section A and the subscript 2 is for manometer tube section D. For flow though the Venturi meter, Bernoulli’s theorem states that: ui2 u2 u2 + h1 = 2 + h2 = n + hn 2g 2g 2g Equation 1

Where g is gravity, un is the velocity and h n is the manometer reading at section n. The continuity equation says: Q = constant = u1a1 = u2 a2 = un an Equation 2

12.4

where Q is the volumetric flow rate or discharge and a is the cross-sectional area. Solving the continuity equation for u1 and substituting it into the Bernoulli equation gives: u22 a2 2 g a1

2

+ h1 =

u22 + h2 2g

Equation 3

Solving for u gives: 1 2

u2 =

Q = u2 a2 and thus:

2 g (h1 − h2 ) a 1− 2 a1

2

Equation 4 1 2

Qcalculated = a2

2 g (h1 − h2 ) a 1− 2 a1

2

Equation 5

The preceding equations are only valid for ideal situations in which viscosity is ignored. Therefore, the values for Q that are measured will be slightly less than the values that are calculated. A constant can be determined experimentally that accounts for the effects of viscosity. 1 2

Qmeasured = Cv a2

2 g (h1 − h2 ) a 1− 2 a1

Equation 6

2

The value for Q that is measured experimentally is equal to the above equation. Once the value for Q has been calculated and measured, the measured value can be divided by the calculated value to determine the value of Cv , the discharge coefficient (typically between 0.90-0.99). Qmeasured = Cv Qcalculated Equation 7

12.5

Lastly, the actual pressure distribution (head pressure) along the convergent-divergent pipe from Bernoulli’s equation is: u12 − un2 hn − h1 = 2g Equation 8

In equation 8, the recorded height values are used (the left side of the equation). One may also use the right hand side since the equality equals the dimensionless actual pressure distribution, but for the purpose of this lab the left side will be used. The ideal pressure distribution can be expressed as a fraction of the velocity head at the throat of the meter (by combining equation 2 with equation 8):

hn − h1 =

a2 a1

2

a − 2 an

2

u 22,ideal 2g

Equation 9

Note from continuity that:

u 2,ideal = Qc a 2 Equation 10

Also note the following equations: Volumetric flow rate Q (m3/s): Q = A * V = Mass flow rate m (kg/s): m =

m

ρ

mass time

Report Requirements:

1- Using the values of h1 and h2 and the measured and calculated values for Q obtained from the 10 runs, calculate Cv (discharge coefficient) for each of the different flow rates. Note that due to errors the Cv may be greater than 1 for some of the calculations, therefore calculate the avg value. 1

2- Graph (h1 − h2 )2 vs. Q for the Venturi meter. 3- Graph Cv vs. Q for the Venturi meter.

12.6

4- Graph the actual and ideal pressure distributions (head pressures) vs. distance from inlet (A) to exit (L) (i.e. for subscript n=b the equation for the actual pressure distribution is hb − h1 and the equation for ideal pressure distribution is

a2 a1

2

a − 2 ab

2

u 22,ideal 2g

) using values obtained from both runs (11 and 12)

Pressure Distribution

where all pressure readings were taken. There will be 4 curves total.

10 20 30 40 Distance from Inlet in mm 5-In the results section, discuss the experimental data, results, and sources of error.

Answer the following question(s) in the conclusion of the report: Should the pressure distribution in the venturi meter at the inlet (A) be the same at the outlet (L), why or why not? The report should include sample calculations; compile collected data and calculated results in tabular form with column headings. The material in this report is used by permission of © TecQuipment Ltd, suppliers of the Venturi Meter.

12.7

Experimental Data and Sample Calculations

12.8