Final Practical: Using a Sharp -edged circular orifice to measure the flow rate PDF

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This practical carried the most mark in the first semester....

Description

Using a Sharp-edged circular orifice to measure the flow rate

P2:

By Azim Ahmed 3 rd December 2013 (Submission deadline- 3 rd December 2013)

1

Contents

Glossary of notation

3

Abstract

3

Introduction

4

Theory

5

Methodology

8

Results and Calculations

9

Errors Analysis

12

Evaluation

13

Discussion

14

Conclusion

15

References

16

2

Glossary of notation Symbol

Denotation

SI Unit

HA

Liquid level

mm

H

Velocity head of liquid jet

mm

g

Gravitational acceleration

m s-2

V

Actual velocity

m s-1

Vi

Ideal/theoretical velocity

ms

Ao

Cross sectional area of orifice

m

Aj

Cross sectional area of jet

m

dj

Diameter of jet stream

m

do

Diameter of orifice opening

mm

ṁ

Mass flow rate

kg/s

Cd

Coefficient of discharge

Cv

Coefficient of velocity

Cc

Coefficient of contraction

-1

2

2

Abstract The aim of the experiment was to understand the fundamentals of fluid mechanics and to improve our knowledge on the different types of measurement flows. In this investigation we had to determine the coefficient of velocity (CV ), coefficient of contraction (Cc) and coefficient of discharge (Cd ) of the orifice and also to determine the use of an orifice plate as a flow meter. Also we had to compare the coefficient of discharge of the venturi meter with the orifice plate. As the water flows through the orifice plate, the velocity head of the jet can be calculated from the height of the water reached in manometer. Hence we can deduce the coefficient of velocity (CV ). By using a sharp small blade we measured the diameter of the water jet and therefore determine the coefficient of contraction (C c). Moreover since graph 1 shows a positive correlation between the actual velocity and the mass flow rate, we can use the gradient of the line to determine the coefficient of discharge (Cd ). It is clear from the results produced from the experiment that both the venturi meter and orifice plate supported Bernoulli’s equation. However the results showed that the coefficient of discharge was different for both the venturi meter and orifice plate.

3

Introduction A circular sharp-edged orifice meter is a device used to measure the volumetric or mass flow rate. In our investigation we used this device to calculate the mass flow rate. Figure 1 below shows a detailed structure of a sharp-edged circular orifice.

Figure 1: Labelled diagram of an orifice As figure 1 indicates an orifice meter is a differential pressure flow meter, which reduces the flow area using an orifice plate. An orifice is a thin flat plate with a hole exactly in the middle of the plate, which is machined to a sharp edge. The orifice plate is inserted between two flanges perpendicularly to the flow; hence the flow passes through the hole with the sharp edge of the orifice pointing to the upstream 1. As the fluid reaches the orifice plate, the fluid must pass through the middle of the hole. This causes a point of convergence, therefore as the fluid flows through the orifice plate the velocity increases, this consequently creates a pressure drop due to the device being a conduit. The pressure continues to drop until the vena contracta is reached and only then the pressure gradually starts to 1 increases . As figure 2 shows the liquid passes through the middle of the hole there is a build up of cavity of air around the orifice plate. As the fluid passes beyond the vena contracta the velocity and pressure change again. Therefore by measuring the difference in pressure we can calculate the mass and volumetric flow rate from Bernoulli’s equation.

Figure 2: Shows the convergence point of the orifice

4

A sharp-edged circular orifice is mostly used in continuous measurement of fluid flow in pipes. It is often used in industry for measurement of single-phase streams and also commonly used in rivers to measure the different flow rates at locations where the river passes through a drain2 . Theory As the liquid flows through the constriction (which rarely occurs) into the available space, this may cause leaks or emergency discharges. Nevertheless, the orifice plate is still used to measure volumetric or mass flow rate. In these situations, the jet characteristics are difficult to comprehend; hence the free jet is a convenient experimental model. A jet that is surrounded by air is referred to as free jet, which is acted upon by gravity3.

H

H

O

O

Liquid Jet Pitot

Figure 3: Flow from the Solvent Tank With reference to Figure 3, the plane O-O, level with the tip of the pitot tube, is taken as a datum. The datum is often described as the reference point. As constriction is placed within the orifice carrying the fluid it will increase in velocity, and hence an increase in kinetic energy at the point of constriction. Therefore using Bernoulli’s equation we can calculate the fluid moving from the liquid surface in the tank to plane O-O through the orifice. Bernoulli’s equation considers 2 points on a perpendicular streamline. P 1 + ½ V12 + gz1 = p 2 + ½ V22 + gz2 Where Z2 =0 V2=V1 following:

Z1=HA P2 =0 therefore rearrange Bernoulli’s equation into the (P 1/ g) +z1 =h

Since P1 =0 due to being open to the atmosphere and the initial fluid speed is negligible (V1=0) per unit mass of fluid, we get equation 1:  



 

(1)

5

Where V1 is the velocity of the ideal fluid based on the cross sectional area of the orifice. Since it is an ideal fluid we assume it must have a steady, incompressible and inviscid flow. The pitot tube with its opening at plane O-O measures the velocity head of the fluid in the jet and this results in the water level C in the manometer tube. Hence:



   

(2)

where V is the actual velocity of the fluid at the centre of the jet. We can rearrange equation 2 to calculate the actual velocity of the fluid. Hence the equation becomes V=√2gh (3)

Figure 4:Theoretical velocity of a perfect orifice plate Hence we can express the coefficient of velocity (Cv ) as the ratio of the actual velocity to the theoretical velocity through the orifice. Due to friction and other losses the actual velocity is always less than the theoretical velocity. Therefore the theoretical velocity is always an over-estimate of the real velocity. We can obtain an equation for C v using equation 1 and 2:         





(4)

Another alteration of the equation 4 can be written as the following: uactual= C vutheoretical

Figure 5: Actual velocity of the orifice plate

6

Figure 6: Detail of jet As shown in figure 6, the vena contra is the point of the fluid stream where the diameter of the stream is at the minimum and hence the velocity of the fluid at its maximum4. From this we can deduce the coefficient of contraction (Cc), which is defined as the ratio of the jet area to the orifice area.   





  





  





(5)

 

Like equation 4, there is an alternation to equation 5 which is the following: A actual=C CAorifice Using equation 4 and 5 we can derive the Coefficient of Discharge (Cd), which is described as the ratio of actual mass flow rate to the theoretical mass flow rate. The theoretical mass flow rate would occur if the ideal velocity (Vi) existed through the full area of the orifice:   



  



  

   

m is the measured mass flow rate and



(5)

is the density of the liquid (1000 kg/m3

for water). Alternatively C d is also determined in terms of the coefficients of velocity and contraction:   

 

  

    (6)

The Cd can also be expressed in another form as shown below: Q=uA

where Q=Volumetric flow rate

Qactual=Aactualu actual

A=Area

=C cCvAorificeu theoretical

u=velocity

=C dA orificeutheoretical =CdA orifice√2gh

7

Method 1. Control the flow to the tank to obtain a steady water level of about 35cm. 2. Swing the pitot into the centre of the jet and ensure there is no trapped air around its nose. The nose should be positioned as high as possible. 3. Read the tank and pitot manometers to obtain HA and H. This will enable you to calculate C v from equation (3). 4. Insert the small sharp-edged blade into the pitot tube and align the blade edge at right angles to the axis of the traverse screw – a 1mm pitch singlestart thread. 5. Using the micrometre wheel to traverse the blade, measure the diameter of the jet. Position the blade at a distance of about half the orifice diameter below the orifice and ensure that the blade and not the pitot tube itself pass through the jet. In measuring, always bring the blade from the air towards the water to avoid any attachment effect. Each member of the group should make three independent measurements of the jet diameter, (i.e. do not let the others see your readings until all are done). 6. Record all results in Table 1. Taking a mean of your δj values will give a good value of the jet diameter from which CC can be calculated using equation (4). 7. Using the hydraulic bench unit, determine the actual mass flow rate for eight steady values of HA within the range 35cm down to 12cm. Wait for the level to stabilise after any flow adjustment before starting to measure the mass flow rate, ṁ. Ensure the pitot tube is out of the jet. 8. Observe HA continuously during a water collection interval and record one representative value only. Record the results in Table 2. You will be able to determine Cd from the results as shown below.

8

Results and Calculations Table 1: Jet Diameter Measurement Reading Number

Jet Diameter

Deviation

("#)

J

mm

Standard deviation

("# − "&)

("# − "&) 

mm

mm2

1

44-31.7= 12.3

-0.2

0.04

2

43.2-30.5=12.7

0.2

0.04

3

47.6-30.4=17.2

4.7

22.09

4

48.1-34.9= 13.2

0.7

0.49

5

47.3-34.9= 12.4

-0.1

0.01

6

46.1-37.5=8.6

3.9

15.21

7

47.2-34.6=12.6

0.1

0.01

8

48.9-33.8=15.1

2.6

0.76

9

48.6-35.6=13.0

0.5

0.25

10

43.7-33.4=10.3

-2.2

4.84

11

41.7-31.3=10.4

-2.1

4.41

12

44.8-32.2=12.6

0.1

0.01

' "#  No. of readings, N =

150.4

'("# − "& )  

12

Mean value / mm

54.16 12

Variance / mm2

' "# /)  "& 

150.4/12 = 12.5

*   '("# − "&) / ) 

Standard error/mm

2.12/√12=0.612

Standard deviation / √4.51 =2.12 mm /  √ *

+ 

,

√.

54.12/12=4.51

9

Table 2: Flowrate Measurements HA

H A0.5

cm

m0.5

+/- 0.1

Mass of water collected

Time for collection

Kg

s

+/- 0.1kg

+/- 1.0

+/- 0.001

0.592

15/92.5= 15.0

92.5

0.162 15/97.5=

32/100= 0.320.5 32

0.566

15.0

97.5

0.154 15/106.5=

28/100= 0.280.5 28

0.529

15.0

106.5

0.141 15/109=

25/100= 0.250.5 25

0.500

15.0

109

0.138

22/100= 0.220.5 22

0.469

15/114= 15.0

114

0.132

18/100= 0.180.5 18

0.424

15/126= 15.0

126

0.119 15/139.5=

15/100= 0.150.5 15

0.387

15.0

139.5

0.108

12/100= 0.120.5 12

0.346

Coefficient of velocity (CV ):

0 kg/s +/- 0.1kg/s

35/100= 0.350.5 35

Mass Flow Rate

15/163= 15.0

163 

0.092

121

        0.993  123 

As stated in the theory Cv is the ratio of actual velocity to the theoretical velocity. As you can see the value of 0.993 is very close to 1. Hence our value of Cv is extremely close to the theoretical velocity. Coefficient of contraction (CC ):

8  



9



; 

....