compressible flow - zachary warner · 7. record the upstream volumetric flowrate from the...

Compressible Flow

Zach Warner

Lab Partners:

Fletcher Ryan & Phillip Hoff

ME 4031W

Basic Measure Lab

Section: 3

University of Minnesota

October 27, 2015

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Table of Contents

Abstract……………………………………………………………………………………………2

Introduction……………………………………………………………………………………..2-3

Method………………………………………………………………………………………......3-6

Equipment List………………………………………………………………………….3-4

Procedure………………………………………………………………………………..4-6

Case 1: Constant Upstream Pressure, Varying Downstream Pressure……….....4-5

Case 2: Constant Downstream Pressure, Varying Upstream Pressure.………....5-6

Case 3: Continuously Choked Orifice……………………………...…………......6

Results & Discussion……………………………………………………………………..……7-11

Nomenclature Table…………………………………………………………………...…..7

Characterization of Flows Through an Orifice………………………………...………8-11

Determination of the Discharge Coefficient……………….………………………….…11

Conclusion…………………………………………………………………………………….…12

References………………………………………………………………………………………..12

Appendix……………………………………………………………………………………..13-17

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Abstract

In our experiment, flow was studied through a critical orifice. A special apparatus was used (see

Figure 1) to determine the volumetric and mass flow rates versus the pressure ratios. This

allowed for a better understanding of when the orifice becomes choked and how it sets a limit on

the volumetric flowrate. Then a log-log plot of volumetric flowrate and pressure difference was

constructed to determine the power of relationship between these two variables. With a 0.052”

diameter the volumetric flowrate limit was set at 30 ft3/h. Additionally, the discharge coefficient

of this orifice was determined to be 0.87 ± 0.66% at a 95% confidence level.

Introduction

Throughout industry and research critical orifices are used to regulate gas flow rates by setting

an upper limit to the volumetric flow rate. This occurs when the flow becomes sonic and the flow

velocity reaches the speed of sound. Sonic flow can also be defined when the pressure ratio

across the orifice reaches the critical pressure ratio given by equation 1, where k is the ratio of

specific of the gas.

Equation 1: Critical Pressure Ratio

When the flow becomes sonic, the orifice is referred to as choked. A choked orifice has an upper

limit on its volumetric flowrate that is theoretically found using equation 2, where k is the ratio

of specific heats of the gas, R is the specific gas constant, Tup is the temperature upstream of the

orifice, and A* is the cross-sectional area of the orifice.

Equation 2: Volumetric Flowrate Through a Choked Orifice

A good measure of how the experimental flowrate compares to the theoretical flowrate is the

discharge coefficient. This is a dimensionless number that is the ratio of the actual discharge to

the theoretical discharge given by equation 3. Differences from theory is due to separation of

flow from the walls of the orifice.

Equation 3: Discharge Coefficient

This experiment uses three different cases to study the volumetric flowrate through a critical

orifice and to determine the orifice’s discharge coefficient. The first case consists of setting the

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upstream pressure to atmospheric while the downstream pressure is varied up to maximum

vacuum. The second case consists of setting the downstream pressure to atmospheric while the

upstream pressure is varied up to 2 atm. The third case consists of setting the downstream

pressure to maximum vacuum while the upstream pressure is varied from 0 to 2 atm. In the first

two cases, the flow will become choked at some given critical pressure ratio, but in the third case

the orifice will always be choked. Once the discharge coefficient is found experimentally, it is

contrasted with other typical coefficients for cylindrical orifices.

Method

Equipment List (refer to Table 1 and Figure 1):

1. Building Supply Air

2. Speedaire Pressure Regulator

3. Air Filter

4. Dwyer Rotameter, 10 SCFH or 50 SCFH

5. Three-way Control Valve for Rotatameters

6. Ashcroft 6” Grade 3A 0-30 psi Test Gauge

7. Cylindrical Orifice

8. Ashcroft 6” Grade 3A 0-30 inches Hg Test Gage

9. Vacuum Pump

10. Several 3/8” Quick Disconnect Couplings

Figure 1: Apparatus

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Item Manufacturer Model No. Uncertainty

Pressure Regulator Speedaire 4ZL01 -

Rotameter Dwyer RMB-50 3%

Rotameter Dwyer RMB-52 3%

Three-way Control

Valve

Swagelok B-43XF4 -

6” Grade 3A 0-30 psi

Test Gauge

Ashcroft 1082 0.25% accuracy,

resolution of 0.1 psig

6” Grade 3A 0-30

inches Hg Test Gage

Ashcroft 1082 0.25% accuracy,

resolution of 0.1 Hg

Vacuum Pump - TA-0040-V -

Table 1: Equipment and their corresponding information

Procedure:

Case 1: Constant Upstream Pressure, Varying Downstream Pressure

Figure 2: Case 1 Fluid Power Schematic

1. Record the initial atmospheric pressure and temperature. This is used to find the absolute

pressures from the gage pressures.

2. Use Figure 2 to aid in the construction of the apparatus. Each number represents the

corresponding equipment from the equipment list. Upstream pressure should remain at a constant

atmospheric pressure which is why it is left unconnected to the building air supply. In our case, a

0.052” orifice was used.

3. Ensure that the rotameter measures upstream volumetric flow rate and that the hoses are

configured in a way that upstream pressure is measured directly before the orifice while the

downstream pressure is measured directly after the orifice.

4. Zero both gages.

5. Use the three way control valve to make sure the 10 SCFH rotameter is used initially.

6. Close the valve on the vacuum pump by turning it clockwise.

7. Turn on the vacuum pump.

8. Slowly open the valve until the downstream pressure is 0.1” Hg.

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9. Record the flowrate indicated on the rotameter. A proper flowrate value can be found by

reading from the center of the circular indicator.

10. Use the three way control valve to make sure the 10 SCFH rotameter is used initially and

then switch the valve to use the 50 SCFH rotameter when the smaller rotameter becomes

insufficient.

11. Repeat steps 8-10 up to a downstream pressure of 21” Hg. From 0.0-1.0” Hg use increments

of 0.1”, from 1.0-4.0” Hg use increments of 0.2”, and from 4.0-21.0” Hg use increments of 1.0”.

12. Turn off the vacuum pump when finished.

Case 2: Constant Downstream Pressure, Varying Upstream Pressure



corresponding equipment from the equipment list. In our case, a 0.052” orifice was used.

2. Ensure that the upstream pressure is measured directly before the orifice and the downstream

pressure is measured directly after the orifice. The downstream pressure should remain at a

constant atmospheric pressure, which is why it remains unconnected in the diagram.

3. Ensure that the rotameter reads upstream volumetric flowrate.


5. Turn on the building supply air to the pressure regulator and rotate the regulator knob counter-

clockwise until the upstream pressure is set to zero.

6. Adjust the pressure regulator until the upstream pressure is 0.1 psi.

7. Record the upstream volumetric flowrate from the rotameter. A proper flowrate value can be

found by reading from the center of the circular indicator.

8. Use the three way control valve to make sure the 10 SCFH rotameter is used initially and then

switch the valve to use the 50 SCFH rotameter when the smaller rotameter becomes insufficient.

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9. Repeat steps 6-8 for upstream pressures up to 25.0 psi. From 0.0-1.0 psi use increments of 0.1

psi, from 1.0-3.0 psi use increments of 0.2 psi, from 3.0-10.0 psi use increments of 0.5 psi, and

from 10.0-25.0 psi use increments of 1.0 psi.

10. Close the valve on the pressure regulator when finished.

Case 3: Continuously Choked Orifice



corresponding equipment from the equipment list. In our experiment, a 0.052” diameter orifice

was used. In this case, nothing should be left unconnected.

2. Ensure that the upstream pressure is measured directly before the orifice and the downstream

pressure is measured directly after the orifice. Additionally, ensure that the rotameter is

measuring upstream volumetric flowrate.


4. Close the valve on the vacuum and turn it on.

5. Open the valve on the vacuum pump while simultaneously increasing the upstream pressure to

balance out the upstream pressure to zero.

6. Repeat step 5 until the valve on the vacuum pump is completely open.

7. Record the upstream and downstream pressures in addition to the volumetric flowrate from the

rotameter.

8. Increment the upstream pressure by 1 psi and repeat step 7.

9. Repeat steps 7-8 until you reach 25 psi.

10. When data is done being collected, slowly close the pressure regulator valve and the vacuum

pump valve at the same time. Ensure that the pressures always remain within their respective

ranges, this is extremely important.

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Results & Discussion

All of the various variables and subscripts that are used in this report can be found in the

nomenclature table below (Table 2).

Variable Symbol

Pressure P

Absolute Downstream Pressure Pdown

Absolute Upstream Pressure Pup

Absolute Pressure Pabsolute

Gage Pressure Pgage

Atmospheric Pressure Patmospheric

Vacuum Gage Pressure Pvacuum

Ratio of Specific Heats k

Cross-Sectional Area of Orifice A*

Upstream Cross-Sectional Area Aup

Cross-Sectional Area at Orifice Throat Athroat

Diameter of Orifice D

Specific Gas Constant for Air Ra

Specific Gas Constant R

Temperature T

Upstream Temperature Tup

Maximum Upstream Volumetric Flowrate Qup

Maximum Actual Volumetric Flowrate

Through Orifice

Qmax,actual

Maximum Theoretical Volumetric Flowrate

Through Orifice

Qmax,theoretical

Volumetric Flowrate Indicated on Rotameter Qindicated on rotameter

Corrected Volumetric Flowrate Qactual

Discharge Coefficient Cd

Density ρ

Density During Factory Calibration ρduring factory calibration

Density During Experimental Use ρduring actual use

Upstream Density ρup

Density at Orifice Throat ρthroat

Upstream Velocity Vup

Velocity at Orifice Throat Vthroat

Mass Flowrate �̇�

Table 2: Nomenclature Table

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Characterization of Flows Through an Orifice

The pressures recorded in this experiment are actually different than their absolute pressures.

They are relative to the local value of the atmospheric pressure and are called gage pressures.

There are two different equations that can be used to calculate the actual absolute pressures. For

pressures that are higher than atmospheric pressure (upstream pressures in this case) use equation

4.

Equation 4

For pressures that are lower than atmospheric pressure (downstream pressures in this case) use

equation 5.

Equation 5

Since this experiment was performed under conditions that are different than the conditions

carried out during the calibration of the rotameters, a correction was carried out to ensure that the

most accurate values of the flowrates are determined. The manufacturers recommend to use

equation 6 to make this adjustment. To find the density values, the ideal gas law (Equation 7) can

be used based on the ambient conditions. The factory calibration was at a pressure of 12.7 psia

and a temperature of 70°F. The atmospheric conditions of this experiment were at a pressure of

14.35 psi and a temperature of 71°F. The variable R in the ideal gas law is the specific gas

constant for the gas. The specific gas constant for air is 0.287 kJ/kg-k.

Equation 6: Dwyer Rotameter Volumetric Flowrate Correction

Equation 7: The Ideal Gas Law

A good way to study how the air flows through the orifice is to look at how the orifice’s

volumetric flowrate is affected by the pressure ratio from downstream to upstream. This allows

you to calculate the critical pressure ratio using equation 1 and make conclusions on when the

orifice becomes choked. The experimental results of how the volumetric flowrate was affected

by the pressure ratio for case 1 and case 2 (from the procedure) are shown in Figure 5.

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Figure 5: Displays the relationship between the orifice volumetric flowrate and the

pressure ratio at the orifice.

The dotted line on the graph is placed to show where the critical pressure ratio is. It can be seen

in both cases that the orifice volumetric flowrate remains relatively constant before this critical

ratio. After this ratio the volumetric flowrate decays towards zero as the downstream pressure

gets closer to the upstream pressure. The orifice creates an upper limit on the volumetric flowrate

of around 30 ft3/h.

In additional method of analyzing the flow through an orifice is to determine how the mass

flowrate is affected by the pressure ratio. This allows us to see how the mass flowrate changes

before and after it becomes choked. To calculate the mass flowrate we can use the continuity

equation shown below as equation 8.

Equation 8: Continuity Equation

The density at the rotameter can be calculated by once again using equation 7, evaluating using

the atmospheric temperature, the specific gas constant for air, and the upstream pressures

indicated on the test gauge. The mass flowrate can then simply be calculated by multiplying the

actual volumetric flowrate values by the density at the rotameter. The experimental results of

how the volumetric flowrate was affected by the pressure ratio for case 1 and case 2 (from the

procedure) are shown in Figure 6.

0

5

10

15

20

25

30

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Qac

tual

[ft3

/ho

ur]

Pdown / Pup

Orifice Volumetric Flowrate vs. Pressure Ratio

Case 1

Case 2

Critical Pressure Ratio

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Figure 6: Displays the relationship between the orifice mass flowrate and the pressure ratio

at the orifice.

Once again the critical pressure ratio is shown with a dotted line. In case 1, the mass flowrate

behaves much like that of the volumetric flowrate. Below the critical pressure ratio, the mass

flowrate remained constant at about 35 kg/hour, but after it decayed to zero as the downstream

pressure reached the upstream pressure. In case 2, the orifice mass flowrate doesn’t seem to be

affected by the orifice being choked. This is due to the fact that in this case the density of the air

at the orifice doesn’t stay constant.

The logarithmic actual volumetric flowrate was then graphed as a function of logarithmic

pressure difference (Figure 7). Additionally, Figure 7 displays linear fits and 95% confidence

intervals for when both cases show volumetric flowrate increasing as well as when both cases

show constant volumetric flowrate. This graph allows for a better understanding of how these

variables affect each other because on a log-log plot the slope of the best fit line displays the

power of the relationship. It can be seen that when the logarithmic pressure difference becomes

around 0.75 the volumetric flowrate hits a constant limit, indicating when the orifice is choked.

This demonstrates that these variables have no relationship when log ∆P > 0.75 or in other words

when the pressure difference gets too high. Prior to this limit, both cases show a relationship

between these two variables. In case 1, the relationship between actual volumetric flowrate and

pressure difference has a power of 0.4789. In case 2, the relationship between actual volumetric

flowrate and pressure difference has a power of 0.4383. Now that there is a better understanding

of the flow conditions and how the orifice controls the flowrate, the orifice’s discharge

coefficient will be calculated.

0

10

20

30

40

50

60

70

80

90

100

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Mas

s Fl

ow

rate

[kg

/ho

ur)

Pdown / Pup

Orifice Mass Flowrate vs. Pressure Ratio

Critical Pressure Ratio

Case 1

Case 2

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Figure 7: Displays the power of relationship between the change in pressure and the

volumetric flowrate at the orifice.

Determination of the Discharge Coefficient

In this section, the discharge coefficient will be calculated for the orifice used in this experiment.

Using equation 2, the theoretical volumetric flow rate for each of the data points can be

calculated. The orifice area can be calculated using the orifice’s diameter along with equation 9.

𝐴∗ =𝜋

4𝐷2

Equation 9: Area of Circular Cross-Section

For our experiment, Ra = 0.287 kJ/kg-k, D = 0.052” = 0.00132 m, Tup = 71°F = 294.82 K, k = 1.4

(for air). The theoretical volumetric flowrate for this experiment was calculated to be 2.73 x 10-4

m3/s or 34.7 ft3/h. Once this is done equation 3 can be used to calculate a discharge coefficient

for each data point. From these calculations an average was found and error analysis was used to

determine the orifice’s discharge coefficient to a 95% confidence level. The experimental

discharge coefficient for the 0.052” orifice was calculated to be 0.87 ± 0.66%. The error in the

value was concluded to be largely due to the sample mean precision error. A study done by A. J.

Ward-Smith pursued to determine the characteristics of cylindrical orifices. He concluded that

the discharge coefficient fell between the range of 0.81-0.86. This is agrees with the discharge

coefficient that was determined in this experiment. The difference in discharge coefficient was

concluded to be due to variations in the sharpness of the leading edge of the orifice as well as to

the slight error in the measurements of orifice diameter. Errors in orifice diameter can lead to

errors in flowrate calculations.

y = 0.4789x + 1.1377R² = 0.9913

y = 0.4383x + 1.1299R² = 0.9884

y = -6E-14x + 1.4828R² = #N/A

y = 0.0127x + 1.471R² = 0.1678

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

-1.5 -1 -0.5 0 0.5 1 1.5

log

Qac

tual

log ∆P

log Qactual vs. log ∆P

Case 1

Case 2

Case 1 Confidence Interval

Case 2 Confidence Interval

Linear (Case 1 Linear)

Linear (Case 2 Linear)

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Conclusion

This experiment analyzed the flowrate through a 0.052” diameter orifice in three different

conditions: upstream pressure set to atmosphere and downstream varied to maximum vacuum,

upstream pressure varied up to 2 atm and downstream set to atmospheric, and upstream pressure

varied up to 2 atm while downstream is held at maximum vacuum. The first two cases allowed

for us to study how the orifice flowrates react to differences in upstream and downstream

pressures. It was determined that this particular orifice set the volumetric flowrate limit to 30

ft3/h when it was choked. It was also concluded that since upstream density varies in the second

case, the mass flowrate wasn’t held to a constant value. The third experimental case was used to

determine the overall discharge coefficient, a good measurement of how the flowrate differs

from the theoretical flowrate when using a particular orifice. The experimental cylindrical orifice

yielded a discharge coefficient of 0.87 ± 0.66%, which agreed with past experimental values. To

add certainty to this experiment it should be attempted to reduce any possible human error in

measuring pressure values. Additionally, further testing should be done with cylindrical orifices

of different diameters and lengths. This would allow for a better understanding of how variations

in orifice dimensions affect the overall orifice discharge coefficient.

References

Dahl, Scott. Statistical Analysis of Experimental Data. Twin Cities: University of Minnesota,

n.d. PDF.

Lab 4: Compressible Flow. Twin Cities: University of Minnesota, n.d. PDF.

McMurry, Peter. Fluid Flow Rate Measurement. Twin Cities: University of Minnesota, 5 Aug.

2015. PDF.

McMurry, Peter. Regression Analysis. Twin Cities: University of Minnesota, 5 Aug. 2015. PDF.

McMurry, Peter. Uncertainty Analysis, Statistical Analysis, and Error Propagation. Twin

Cities: University of Minnesota, 5 Aug. 2015. PDF.

Ward-Smith, A. J. "Critical Flowmetering: The Characteristics of Cylindrical Nozzles with

Sharp Upstream Edges." International Journal of Heat and Fluid Flow 1.3 (1979): 123-

32. UMN Libraries. Web. 5 Nov. 2015.

Appendix

Lab Notebook Pages

compressible flow - zachary warner · 7. record the upstream volumetric flowrate from the...

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