ME 321Measurement Systems

Dr. MayhewSpring 2014–2015

Experimental Determination of an Orifice Discharge Coefficient

04/20/2015

Team 4

Zach LehmanJoseph Arthur

Joseph KaltenthalerAndrew Niemann

Rose-Hulman Institute of TechnologyTerre Haute, IN 47803

Introduction

When discussing junctions or objects that impede flow in piping networks it is often useful to know the factor by which these objects impede the flow. The discharge coefficient essentially describes the relationship between the ideal flow in a system and the actual, restricted flow caused by an obstruction of some kind. Thus the goals of this experiment were to determine the discharge coefficient for an orifice and to develop a reasonable uncertainty approximation that will verify the accuracy of the discharge coefficient. Additionally, it was our expectation to demonstrate that the discharge coefficient was between 0 and 1, based upon evidence from external sources.

To accomplish these goals, the data reduction equation was determined for the discharge coefficient of the orifice and the parameters of the system were altered and measured with the appropriate instrumentation. A manometer was used to obtain the pressure differential of the fluid across the orifice and a rotameter measured the flow rate of the fluid. By utilizing transducer accuracies and resolution values, and by altering the number of data trials collected, the uncertainty in the resultant was determined.

Experimental Facilities and Instrumentation

The experimental system used in this experiment consisted of a water tank, a pump, input and output valves, and the necessary piping to direct the flow of the fluid being pumped, in this case water, to the orifice being tested.

Figure 1. Schematic of the Experimental SystemFigure 1 is a schematic of the entire experimental apparatus which includes a tank, the orifice in question, manometer, rotameter, control valves, and the

appropriate piping. Key in determining the discharge coefficient of the orifice are the rotameter and the manometer.

Figure 2. Schematic of the Manometer

The measurands in this system are obtained using the manometer and the rotameter. The rotameter measures the volumetric flow rate of the water from 2 to 20 liters per minute. The manometer measures the amount of pressure difference across the orifice by displaying water height from of 0 to 440 millimeters as shown in Figure 2.

The manometer and rotameter were utilized in this experiment to measure two fluid heights and the fluid flow rate to determine the discharge coefficient of the orifice. Figure 3 is a cross section view of the orifice displaying flow direction and areas impacting fluid flow.

Figure 3. Orifice Schematic

Of the factors contributing to the discharge coefficient, Cd, variables include:

volumetric flow rate ∀, manometer height before and after the orifice, h1 and h2

respectively. Constant values for this experiment include: the orifice area, A2, the

pipe area, A1, the density of the fluid, ρ, and the gravitational acceleration of the

Earth, g. The mass flow rates and pressures outlined in Figure 3 are related to the manometer heights and volumetric flow rates as described in the following section. The first manometer height recorded, h1 is taken upstream of the orifice and the

second reading, h2 is taken downstream of the orifice. These heights are outputs of the manometer at positions 6 and 7 on Figure 2, while the actual measurand is the pressure before and after the orifice. However, the actual manometer readings can be treated as measurands considering they are used directly in our data reduction equation.

In order to collect the necessary information to determine Cd, the pump was turned on to allow the water to pass through the system. With both valves fully open, the pump was run until all of the air had escaped. Some of the water and air was drained from the manometer so that the fluid levels were in the center of the manometer readout and no bubbles remained. Once all of these criteria were met, the experimental trials were conducted.

By opening or closing the input valve, the flow rate of the water was increased or decreased. To collect data, the valve would be set to a specific position and the flow rate would be observed on the rotameter while the heights were recorded from the manometer. Comparing the alignment of the very top of the float in the rotameter with the measuring markings was crucial to the accuracy of this measurement. Specific to our experiment is the calibration of the rotameter to achieve improved accuracy. This calibration is outlined in Appendix B.

With the flow bench operating, one datum was collected at five unique flow rates in a random order. We randomized the order of data collection minimized the potential effects of hysteresis on random uncertainty.

After collecting the five different flow rates and corresponding values of h1 and h2, the random uncertainty in the discharge coefficient was determined. By comparing this value with the systematic uncertainty in the discharge coefficient, Cd, one could determine how many trials were necessary to prevent the random uncertainty from being a large contributing factor to the total uncertainty. We determined that the random uncertainty should be of equal or lesser magnitude than the systematic uncertainty. Once the standard deviation in Cd was determined from the five trials, it was used in the random uncertainty equation and was compared to the systematic uncertainty. It was determined that 20 trials were necessary to produce a random uncertainty that was comparable to the systematic uncertainty. Details on the mathematics required to obtain the systematic and random uncertainties are outlined in the following section and sensor information is displayed in Appendix C.

Data Reduction and Uncertainty Analysis

The data reduction equation for the discharge is derived from the conservation of mass at the orifice. The fluid flow before the orifice, m1, is equal to the fluid flow through the orifice, m2, as shown by,

m1=m2 . (1)

From fluid principles, it is known that

m=ρVA=ρ ∀, (2)

The density of the fluid is ρ, V is the velocity of the fluid at the given location, A is the cross sectional area that the fluid passes through, and ∀ is the volumetric flow rate of the fluid. Assuming the density is still constant and utilizing equations 1 and 2, it is determined that,

V 2=V 1

A1

A2(3)

From Bernoulli’s equation,

P1+12ρV 1

2=P2+12ρV 2

2, (4)

we solved for the change in pressure over the orifice;

P1−P2=12ρV 2

2 ¿.(5)

Simplifying this we now have a new expression for V at the orifice,

V 2=√ 2(P1−P2)

ρ[1−( A1

A2)

2

].

(6)

The ideal flowrate through the system is

∀ideal=V 2 A2 (7)

And the discharge coefficient essentially is a ratio between the actual flow rate, which is measured, and the ideal flow rate. This relationship can be described as

Cd=∀actual∀ ideal

=mactualmideal

. (8)

Utilizing equations 6, 7, and 8, we arrive at

m=Cd A2

√1−(A1

A2

)2√2 ρ(P1−P2). (9)

Knowing that the pressure is measured by the manometer, the pressure can be simplified to,

P= ρgh, (10)

where h is the height in the manometer and g is the gravitational acceleration of the Earth. Utilizing equations 2, 9, and 10, we arrive at the data reduction equation;

Cd=∀√1−(

A2

A1

)2

A2 √2g(h1−h2¿)¿

.(11)

To determine the total uncertainty in the discharge coefficient, wCd, the systematic uncertainty and the random uncertainty of the discharge coefficient are root sum squared to produce,

wCd2 =wCd ,sys

2 +wCd , rand2 . (12)

The systematic uncertainty in Cd, wCd , sys, is dependent upon the systematic uncertainties of the measurands as

wCd , sys2 =(

∂Cd∂h2

)2

wh2, sys2 +(

∂Cd∂h1

)2

wh1 , sys2 +(

∂Cd∂ ∀

)2

w ∀ , sys2 ,

(13)

where wh1 ,sys is the systematic uncertainty in the height after the orifice, wh2 ,sys is the systematic uncertainty in the height before the orifice, and w ∀ ,sys is the systematic uncertainty in the volumetric flow rate. The uncertainties of the areas factor into this analysis as well except that the uncertainties were not given, nor was any information that would allow us to determine the uncertainty. It can be assured that the accuracies of these values is exceptional. The sensitivity coefficients of the

orifice area, A2, and the pipe orifice, A1, are roughly 2 x10−3 and 1.35 x10−4 respectively. The sensitivity of the gravitational acceleration g is even smaller, being 2.9 x10−5. When these small sensitivity coefficients are combined with the low values of uncertainty in accuracy we determined that the areas, as well as the gravitational acceleration do not significantly impact uncertainty analysis and are assumed to be absolute values. A better determination of the uncertainty can be determined if the uncertainty in the areas is accounted for.

To determine the uncertainties in the heights, the accuracy and readability of the manometer, are utilized in the following way, and can be applied to both h1 and h2,

wh , sys2 =wmanometer , accuracy

2 +wmanometer ,readability2 . (14)

The flow rate uncertainty was determined from the rotameter calibration as described in Appendix B. The data reduction equation for this calibration is

∀= ∆∀∆ t

, (15)

where ∆ ∀ is the volume change in the tank and ∆ t is the change in time during which the volume change occurred. From equation 15, the systematic uncertainty of the flow rate becomes

w ∀ ,sys2 =( ∂ ∀

∂∆ ∀)

2

(w¿¿ sight glass , acc2+w sight glass ,read2 )+( ∂ ∀

∂∆ t )2

(w stopwatch ,acc2 +w stopwatch ,read

2 )¿

.

(16)

The subscripts sight glass and stopwatch in equation 16 refer to properties of the sight glass on the tank and stopwatch respectively. Their accuracies and readabilities determine the new uncertainty in volumetric flow rate since they were used to calibrate the readings from the rotameter.

The random uncertainty in Cd is determined from the number of trials, n, the

related t-distribution value, t n−1, and the sample standard deviation of the Cd values

for the given number of trials, SCd using the equation,

wCd , rand=t n−1SCd

√n.

(17)

The uncertainties for the accuracy and readabilities of the transducers are given in Appendix C and the resulting uncertainties as well as the partial derivatives of the data reduction equation in equation 11 are given in Table 1.

Table 1. Variable impact on the overall Cd uncertainty

Parameter

Units Typical Value (mean) Partial Sensitivity Coeff i Term i

∀ mm3 6 ∂ ∀∂∀

3.521×10−2 3.88×107

∆ t sec 29.3 ∂ ∀∂∆ t

−7.33×103 3.36×106

h2

mm 141∂Cd∂h1

-3.06×10-4 1.17×10−6

h1

mm 194∂Cd∂h2

3.06×10-4 1.17×10−6

W ∀ , sysmm3

sec1.25×104

W h , sysmm 1.12×102

WCd , rand 0.001851

WCd , total 0.020

Rotameter calibration was essential as its uncertainty caused W ∀ , sys to be

1.179 x105 mm3

sec, which is an order of magnitude greater than the calibrated

uncertainty in flow rate and three orders of magnitude greater than the uncertainty caused by the heights. For these reasons, the rotameter was calibrated with the available tank and stopwatch. Uncertainties for areas and gravity are not

provided as they were deemed to have a negligible effect on the overall uncertainty.

Results and Discussion

The mean result for the orifice discharge coefficient with associated uncertainty is

Cd=0.564±0.020

Figure 4 shows the relationship between the discharge coefficient and the fluid volumetric flow rate as well as the uncertainty associated with each value using the rotameter calibration outlined in Appendix B.

9 10 11 12 13 14 150.5

0.51

0.52

0.53

0.54

0.55

0.56

0.57

0.58

0.59

0.6

Volumetric Flow Rate (L/min)

Dis

ch

arg

e C

oeff

icie

nt

(-)

Figure 4. Discharge coefficient with uncertainty of an orifice at varying volumetric flow rates

Figure 4 reflects the mean of four measurements at each of the flow rates measured. Complete data is available in Appendix D.

When discussing fluid dynamics it is essential to mention Reynolds number, which is a non-dimensional term that characterizes the flow of the fluid. The relationship

Note: uncertainty is larger at lower flow rates

Relative uncertainty

Experimental data

between Reynolds number and discharge coefficient can be seen in Figure 5 to exhibit a near-parabolic behavior when transitioning into higher Reynolds numbers from the 10 to the 2000 range. Figure X compares our measured values of discharge coefficient and Reynolds numbers to those of an orifice plate with a

similar A2

A1 ratio. Our data reflects a small range of Reynolds numbers as compared

to the reference data.

10 100 1000 100000.5

0.55

0.6

0.65

0.7

0.75

Reynolds Number (-)Dis

ch

arg

e C

oeff

icie

nt

(-)

Reference data fit

Experimen-tal data

Note the measured values represent a small segment of the reference data

Figure 5. This relationship is based on tests done on a similar orifice plate to the one used in our experimentation. Source: Discharge Coefficient Performance of Venturi, Standard Concentric Orifice Plate, V-Cone and Wedge Flow Meters at

Small Reynolds Numbers, Authored by Colter L Hollingshead, Utah State University.

Figure 5 shows that our data increases for the three lower Reynolds number trials. When compared to the reference data for a similar orifice plate, this trend does not agree. We have no justification as to why this is. This would suggest that for our trials, the discharge coefficient is not constant. Our results however, indicate that for our trials, discharge coefficient is constant.

75 95 115 135 155 175 195 21530

32

34

36

38

40

42

44

46

Volumetric Flow Rate2 (L2/min2)

Tu

be H

eig

ht

Diff

ere

nce (

mm

) Experimental Data

Note the strong linear correla-tion

Figure 6. The volumetric flow rate squares with the difference in manometer tube height

Figure 6 shows how the manometer tube height difference changes with the square of the volumetric flow rate. If equation 11 is squared and rearranged, one arrives at

∀2=Cd2 A2

22 g(h1−h2)

(1−( A2

A1)

2

),

(18)

which describes the relationship in Figure 6. This relationship, which is linear, strongly suggests that the discharge coefficient Cd, is constant. In other words, since the volumetric flow rate squared and the manometer height difference vary proportionately to one another, then every other element in this equation must be constant. This evidence supports our resultant, Cd, being a single value at all flow rates and is more conclusive than the Reynolds number comparison.

Conclusions

This experiment measured the resultant discharge coefficient and the related uncertainty of an orifice plate, and successfully demonstrated that this resultant was between 0 and 1, as we expected due to previous experimentation and research conducted on the topic. We successfully proved this within and uncertainty of ±0.020 .

It is recommended that similar experiments use an input range for the volumetric flow rate between 4 and 20 liters per minute. The measurement of volumetric flow rate was the largest source of uncertainty in our analysis. It is recommended to improve sensor accuracy by calibrating the rotameter prior to use. Once the calibration is successfully executed using the water tank and stopwatch, the uncertainty dramatically decreases.

Appendix A - Principles of Operation - Orifice Flowmeter

The manometer consists of a set of glass tubes that follow hydrostatic principles to provide a readout of pressure, displayed in water column height. This allows the user determine the pressure difference following equation 10. The tubes are connected at different points in the flowstream, shown in Figure 7, in order to determine the pressure difference across the orifice plate.

Figure 7. Assembly photograph

When power is inputted, the pump sends water out of the holding tank at a constant rate through the system. Air accumulates inside the system because some of the pipes and valves in the setup are exposed to the air. The excess air is released from the system by adjusting the manometer valve marked on Figure 4. Once there is no excess air remaining in the system, the system is stabilized and accurate readings can be made on the manometer and the rotameter. The meniscus of the water level is read directly from the height gage attached to the manometer. In determining the discharge coefficient, tubes numbered 6 and 7 were used for analysis because they are connected immediately before and after the orifice, as shown in Figure 7. The pressure difference across the orifice is proportional to the flow rate. The rotameter contains a metallic float that, once the system is stabilized, remains at a constant level within the device. This device is pre-calibrated in the system to display an output of liters per minute. The rotameter is the second sensor used in experimentation. It produces the final measurand, flow rate, used for the original DRE, equation 11.

Appendix B - Principles of Operation – Rotameter Calibration

The rotameter was calibrated as it was the highest source of uncertainty in the measurement system. This calibration allowed for a more precise discharge coefficient to be reported. This calibration consisted of the water tank, stopwatch, and rotameter. The sight glass that displays the water tank volume is outlined in Figure 8.

Figure 8. Tank sight glass used in rotameter calibration

To begin, an initial flow rate was recorded from the output of the rotameter. Then the input valve was closed so that the tank would fill. An initial fluid level was recorded and after a specific amount of time elapsed the final volume was recorded. The difference in these times and the difference in volume measurements was used in equation 15 to determine the calibrated flow rate. The calibrated flow rates were used in the systematic uncertainty calculation of the volumetric flow rate.

Figure 9 is the comparison of the rotameter flow rates and the more accurately determined calibrated flow rates found from the tank filling calibration.

10 11 12 13 14 15 169

10

11

12

13

14

15

Rotameter Flow Rate (L/min)

Cali

bra

ted F

low

Rate

(L

/min

)

Experimental data

Figure 9. Rotameter Calibration curve

The calibration equation for the rotameter is

Calibrated Flow Rate=0.9426∗Rotameter Flow Rate+0.413, (19)

which is used to generate more accurate flow rates from the rotameter flow rates and ultimately leads to more accurate discharge coefficients. The uncertainty in

the volumetric flow rates for the calibrated flow rates is equation 16 instead of the rotameter equivalent of equation 14.

Suggestions for Future Experimentation EffortsCalculating values for discharge coefficients can be simplified and expedited by using the rotameter on the orifice bench, and not timing tank fillings. For simplicity a calibration of the rotameter has been completed in this appendix yielding the calibration equation 19. Future experiments can be conducted using this calibration, but it is suggested that flow rates above 4 liters per minute are measured. Our results are reported in Figure 10.

9 10 11 12 13 14 150.5

0.51

0.52

0.53

0.54

0.55

0.56

0.57

0.58

0.59

0.6

Rotameter Volumetric Flow Rate (L/min)

Dis

ch

arg

e C

oeff

icie

nt

(-)

Figure 10. Discharge coefficient and uncertainty before calibrating the rotameter

The larger uncertainty at lower flow rate is due to the greater systematic uncertainty of the transducers in their lower operation ranges. It is suggested that flow rates greater than 4 liters per minute are used to mitigate this issue.

Uncertainty is larger at lower flow rates

Experimental data

Appendix C - Systematic Uncertainties for the Measurands

Table 2 provides a list of all sources of sensor uncertainties apparent in our experimentation.The resolutions for the manometer and rotameter were taken directly from their readouts and their readabilities and accuracies are half of their resolutions. The tank was found to have an accuracy of an eighth of a liter an and equal readability. The readability of the stopwatch was half of the resolution but the accuracy was made a quarter from judgement based on the difficulty in operating the stopwatch at the correct time.

Table 2. Sensor Accuracies and Readabilities

Sensor Resolution Readability Accuracy

Manometer (mm) 5 2.5 2.5

Rotameter (L/min) 1 0.5 0.5

Tank Volume (L) 0.25 0.125 0.125

Stopwatch Time (s) 0.01 0.005 0.25

Appendix D – Experimental Data

Table 3 contains the measurands and the resultant Cd calculated from equation 11 for the four trials at five separate volumetric flow rates. The flow rates displayed are the rotameter readings post calibration using equation 19. The mean of the twenty values of Cd is the value reported for this orifice.

Table 3. System measurands and calculated resultant for randomized trials

V [ Lmin ] h1 [mm ] h2 [mm ] ∆ h [mm ] Cd ¿

10 165 203 38 0.54482115 148 225 77 0.56592312 157 207 50 0.56589511 161 204 43 0.56119314 150 217 67 0.56741113 154 211 57 0.57259210 165 203 38 0.54482115 150 226 76 0.56963412 158 208 50 0.56589511 161 205 44 0.5547814 152 219 67 0.56741113 154 211 57 0.57259210 166 202 36 0.5597515 149 225 76 0.56963412 157 207 50 0.56589511 162 204 42 0.56783514 152 218 66 0.57169413 153 211 58 0.56763410 165 201 36 0.5597515 150 226 76 0.56963412 158 208 50 0.56589511 160 204 44 0.5547814 153 219 66 0.571694

13 153 211 58 0.567634

To utilize the values above and equation 11 to determine the discharge coefficients, the areas, density, and gravitational acceleration must be known. These values are presented below in Table 4.

Table 4. Experimental variables that are presumed constant

Constant

Orifice Area, A2(mm2) 792

Pipe Area, A1(mm2) 314

Density of Water, ρ(kg

m3) 1000

Graviational Acceleration,

g( m2

sec)

9.81