58:080 Experimental Engineering

Title: Isentropic Blow-Down Process and Discharge Coefficient

Steven Cooke

Department of Mechanical EngineeringThe University of Iowa

Iowa City, IA 52242

Submitted to: Professor: Hongtao DingTA: Yuwei Li & Wei Li

Date Submitted: Friday, November 8th, 2013

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Abstract

The objective of the experiment was to study the transient discharge of a rigid pressurized

tank and coefficients for several different diameter orifices as well as a long tube comparing

actual blow down processes with an isentropic process for an ideal gas. Along with these

processes we were able to study the concept of choked flow. This was done through the

comparison of the experimental measurements and numerical model.

An MKS Baratron transducer was calibrated with a DAQ card by measuring the voltage

at different pressure levels. A T-type thermocouple was then calibrated using a thermistor as the

lab standard taking DAQ output readings at ambient room temperature and in an ice point cell.

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Table of Contents PageIntroduction…………………………………………………………………………………… 4

Experimental Considerations……………………………………….................. ................... 4

Results and discussion………………………………………………………… ………….. 7

Conclusions and Recommendations………………………………………………………… 11

References…………………………………………………………………………………… 13

Appendix A:

Appendix B:

Appendix C:

Note: For List of Equipment please refer to experimental engineering lab notebook

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Introduction

The objective of the lab was to study the transient discharge of a rigid pressurized tank

and discharge coefficient (CD) for several different diameter orifices as well as a long tube, then

compare actual blow down processes with an isentropic process for an ideal gas. Also, from

these processes study and understand the concept of choked flow. This was done through the

comparison of the experimental measurements with the numerical model.

Choked flow is a limiting condition, which occurs when the mass flow rate will not

increase with a further decrease in the downstream pressure environment while upstream

pressure is fixed. This can be observed when upstream pressure is constant while downstream

pressure is reduced. Once gas flow doesn’t increase, the flow is considered choked. An

isentropic process is for the most part ideal in that no heat is added to the flow and none is lost

from friction. This means that the process is reversible and can be repeated with the same results

forwards and backwards. This relates to the discharge coefficient, which is the ratio of actual

discharge to the theoretical discharge coefficient, which is witnessed, in isentropic processes.

EXPERIMENTAL CONSIDERATIONS

The experimental setup to acquire the necessary data of the “isentropic” blow down tank

workstation included a pressure vessel, pressure transducers, a pressure gauge, an internal

thermocouple, and a computer assisted data acquisition system (Omega OMB-DAQ-3001 and

LabView 2012 software). Figure 1 shows the main components of the experiment. The Omega

absolute pressure transducer with a digital display was used as a standard to calibrate the MKS

Baratron pressure transducer, which was used to acquire the pressure data. A T-type

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thermocouple and the MKS Baratron transducer were connected to the DAQ to record the data.

The data acquisition is controlled with the software LabView 2012.

Figure 1 Apparatus for “isentropic” blow down process and discharge coefficient measurements

Figure 2 Schematic drawing for the pressure discharge experimental setup

In order to pressurize the tank, close the release valve and open the air line. Adjust the main

control valve until absolute pressure is read. When plugging in the DAQ card make sure to

insert the pressure transducer and thermocouple into channels 0 and 1 respectively. The leads

are plugged into the (+) 4 and (-) 5 to insure proper output.

Barometric pressure of the room was recorded and converted to be 14.77 psi. The

difference between the Omega absolute pressure transducer at atmospheric pressure and

Barometric pressure was recorded to be 0.47 psi. This difference was added to all future results

to “zero” the Omega transducer within its resolution so that it could be used as the lab standard.

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Calibration of an MKS Baratron transducer was carried out by measuring voltage output

from the DAQ card at 10 different pressure levels varying from 14 – 70 psi. The output voltages

for the DAQ card can be seen in Appendix A, Table 1. A 14.4-psi pressure was repeated 10

times in order to allow for repeatability error calculations in the error analysis section with raw

data in Table A.2.

Calibration of a T-type thermocouple was carried out with a thermistor as the standard.

DAQ card temperatures were read from the thermocouple while the thermistor read 24.94 C°.

Measurements were also taken in the ice point cell with standard of 0.21 C° read from the

thermistor. The data for both these readings can be seen in Appendix A, Tables 3 & 4

respectively.

Pressure and temperature decay were recorded during the blow-down process for orifice

restriction sizes of 0.035”, 0.028”, 0.021”, 0.0135” and a long tube of 1 mm opening. Using the

calibrated MKS Baratron transducer and internal T-type thermocouple the outputs were recorded

by the DAQ card by repeatidly pressurizing the tank to 70 psi and waiting for thermal

equilibrium. Once equilibrium was reached data recording began and the valve was opened

allowing pressure to release. Data was recorded until ambient pressure was reached. This was

done 5 times for each orifice making sure thermal equilibrium was reached before releasing the

pressure.

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RESULTS AND DISCUSSION

The calibration data was plotted and obtained the calibration curve shown in Figure 3

Figure 3 Calibration curve for MKS Baratron transducerThis revealed equation Equation 1:

Pressure [ psi ]=19.473 ∙Voltage [V ]+14.773 (1)

The error in the fit was calculated using Equation C1 which gives a final equation with

uncertainty of pressure [ psi ]=19.473 ∙ voltage [V ]+14.773±0.0223. The repeatability error was

calculated by taking 10 readings at 14.4 psi. Using Equation C2 the repeatability error was

found to be 0.0002453 V or 0.0047789 psi. The accuracy of the instrument was then calculated

using Equations 2 & 3.

ε calibration=√εbiasmean2 +ε precision2 +ε resolution

2 +ε fit2 (2)

Accuracy=1−|εcalibration|x true

(3)

Example calculations can be found in Appendix C with tabulated results in Table 1.

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Table 1 Calculated Accuracy and uncertainty for MKS Baratron transducer

Calibration data for the T-type thermocouple can be found in Appendix A, Tables 3 & 4. Using

the thermistor as a standard, the accuracy of the tank thermocouple at room temperature (was

calculated using Equations 2 & 3 above removing error in the fit and using data from Tables A.3

& A.4 for calculations. The ice point and ambient temperature errors for the thermocouple were

calculated using pooled statistics and the RSS.

Table 2 Calculated Accuracy and uncertainty for T-type thermocouple using pooled statics and RSS

Pressure and temperature decay for each orifice were then plotted in Figure 4.

Figure 4 Dimensionless pressure decay for each orifice in relation to time

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Note that the time for the dimensionless pressure to decay to one takes less time for the larger

orifices and more time for the smaller which makes sense logically from the equations. There is

a phenomenon witness by the long tube which has the largest diameter hole yet is between the

second smallest hole and the third smallest which can be explained by friction of the wall and

Equation 4.

τ=μ ∂u∂ y (4)

Since pressure is the driving force for this process, the measured pressure in the isentropic

equation was used to calculate the corresponding ideal temperature for an isentropic process. By

using the measured pressure in the isentropic equation, the temperature for an isentropic process

undergoing the measured pressure change can be calculated. The equation relating the initial

(“1”) and final (“2”) states of an ideal gas undergoing an isentropic process is Equation 5.

T2

T1=( P2

P1 )( k−1) /k

(5)

Using the above equation in excel the comparison between experimental and isentropic processes

for each orifice were calculated and graphed which can be seen in Appendix B Figures B.1-B.5.

These show that there is a large difference in measured and isentropic process. Heat transfer is

very noticeable in the process and is witnessed by the large difference in the measured

temperature compared to the isentropic temperature, which is due to the friction caused by

molecules at the orifice. There is a noticeably larger difference for the Long tube orifice, which

is due to the extended time that the air has time to rub on the walls of the tube compared to

simply the orifice end for the limited amount of distance.

Calculation of the discharge coefficient was done in excel using Equation 6

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t=[ V tank

(√k ∙R ∙T 0 ) ∙CD ∙ Anozzle ] ∙[ 2k−1 ∙( k+1

2 )k +1

2∙ (k−1 )] ∙[( P0

Ptank )(k−1)2∙ k −1] (6)

where,

t=the time¿ the start of theblow−down process

CD=the dischargecoefficient

V tank=the volumeof the tank(3.769025 L)

R=gas constant for air(287 J /kg ∙K )

k=specific heat ratio for air (1.4)

Anozzle=areaof the nozzle( Diametersare :65→0.035 , #70 → 0.028 ,

¿75→0.021, #80 → 0.0135 ,long tube→1mm) T 0=startingtemperature of process (K )

P0=starting pressure of process

Ptank=tank pressure during process

Pstar=Pa ∙1.892929=99284. ∙1.892929=187938.5Pa (7)

Anozzle=π ∙Dnozzle

2

4=π ∙ 8.89×10− 4

4=6.207×10−7m2 (8)

Tabulation of the calculated data for Equation 6 can be found in Table

Table 3 Tabulated data from excel calculations for each orifice

using the data from above the discharge coefficient can be calculated using Equation

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CD=8.64 ∙V tank ∙[( Po

P star )0.42.8−1]

[t star (√k ∙ R∙T 0∙ Anozzle ) ] (9)

This equation is used along with Table 3 to find the discharge coefficients shown in Table 4

Table 4 Discharge coefficients calculated in Excel

The uncertainty of these calculations can be calculated using Equation 9.

uC D=√( ∂CD

∂ P0∙uP0)

2

+( ∂CD

∂T star∙ uT star)

2

+( ∂C D

∂T 0∙ uT0)

2

(10)

The results are tabulated below in Table 4

Table 5 Tabulated results of total uncertainty for discharge coefficient calculations

CONCLUSION

The experimental process of this lab revealed that the smaller the orifice size the longer

the time to reach choked flow for the blow down process. For an isentropic process of an ideal

gas compared to the actual the temperature distribution is largely different because of friction.

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Figure 5 Resulting discharge coefficients in relation to pressure ratio

In future calculations for this lab I would recommend a better calibration process for the T-type

thermocouple because only two points were taken which was not enough to create a calibration

curve. Instead the uncertainty for these calculations were much larger than needed to be causing

added uncertainty.

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References

1. L.D. Chen, “Experimental Engineering Lecture Notes”, http://www.engineering .uiowa.edu/~expeng/Lecture%20Notes/notes.pdf (1996).

2. Experimental Engineering course website, http://www.engineering.uiowa.edu/~expeng/Labs/Labs_Refs/Lab%201c%20-%20Example%20Calcs.pdf

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APPENDIX A: TABLES

Table A. 1 MKS Baratron calibration data

Table A. 2 Repeatability error data

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Table A. 3 Measured temperature of ambient air with standard of 24.94 C°

Table A. 4 Measured temperature in ice point cell with standard of 0.21 C°

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APPENDIX B: FIGURES

Figure B. 1 Comparison between experimental and isentropic process for 0.035” orifice

Figure B. 2 Comparison between experimental and isentropic process for 0.028” orifice

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Figure B. 3 Comparison between experimental and isentropic process for 0.021” orifice

Figure B. 4 Comparison between experimental and isentropic process for 0.035” orifice

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Figure B. 5 Comparison between experimental and isentropic process for a tube with 0.039” orifice

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APPENDIX C: EXAMPLE CALCULATIONS

Equation C1:

u fit=s yx=√∑i=1

n

( y i− yci )2

ν =√ 0.003998 =0.0223 psi

Equation C2:

urand=t v , 95

Sx√N

=2.228 0.0003482√10

=0.0002453V

Equation C3:

sx=√ 1N∑

i=1

N

¿¿¿

Equation C4:

ubias=|x−x true|=|14.3−14.77|=0.47 psi

Equation C5:

ures=resolution

2=0.1

2=0.05 psi

Equation C6:

uQ=EFSR

2m= 10

224 =0.00059V

Equation C7:

Accuracy=1−|εcalibration|x true

=1−|0.47067|

14.77=96.81 %

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