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1.0 IntroductionIn many pressure vessel filling operations, the evaluation of success or failure is

often difficult. Any direct measure of pressure requires either the installation of a gauge inevery unit, which can be expensive and impractical, or is a destructive measure thatinvolves puncturing the vessel in some form. One major downside to any destructive

testing method is the existence of a certain degree of doubt as to whether or not thecontents of the ruptured vessel truly reflect the state of the undamaged vessel. This comesin addition to two further concerns both !" the actual loss of the vessel itself which insome low quantity applications may be unacceptable # and $" the potential contaminationof the contents of the vessel # which could range from undesired pollutants in mechanicalapplications to potentially deadly foreign matter in biological applications.

If presented with a case of a valuable container than we cannot sacrifice filled withcontents that cannot be contaminated, the search must continue for other externallymeasurable variables that give rise to pressure detection. As the pressure inside the canrises, the vessel responds elastically %assuming the vessel is properly designed to not fail",and there is a direct relationship between hoop strain and internal pressure as determined

by simple constitutive strain theory. The experiment is based upon this relationship.

2.0 Theoretical Analysis2.1 Hoop Strain

In treating the can as a thin walled pressure vessel %where the thic&ness ofthe wall is negligible in comparison to other significant dimensions" a simplematerial relation between pressure and strain can be derived. 'onsider thefollowing free body diagram of a half cross section of the can, with pressure p, diameter D, tension T and wall thic&ness t (

)igure !( 'an 'ross *ection )ree +ody iagram

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The net upward pressure force per unit height pD must be balanced by thedownward tensile force per unit height $ T , a force that can also be expressed as astress, - hoop , times area $ t . quating and solving for - h gives

- h / pD

$t %!"*imilarly, the axial stress - axial can be calculated by dividing the total force

on the end of the can, pA=p(D/2) 2 by the cross sectional area of the wall, Dt ,giving(

- a / pD 4t %$"

)or a flat sheet in biaxial tension, the strain in a given direction such as the0hoop1 tangential direction is given by the following constitutive relation with

2oung1s modulus E and 3oisson1s ratio 4( 5 h / !%- h # 4- a"

E %6"

)inally, solving for un&nown pressure as a function of hoop strain(

p / %7 Et" 5 h D $ 4 %7"

2.2 Strain GagesA strain gage is a variable resistive element that ta&es advantage of thegeometric and physical basis of resistance. As the strain gage undergoes strain in itssensitive direction, the diameter of the extremely thin %8.89:mm for the strain gagesused in this lab" longitudinal wires changes slightly, giving a change in overallstrain gage resistance. ;esistance of a conductor of length L, cross sectional area A,and resistivity is

R / L A %9"

'onsequently, a small differential change in < R= R can be expressed as

< R / < L < A > < R L A %:"

$

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?here < L/L is longitudinal strain 5, and < A/A is #$45 where 4 is the 3oisson1s ratioof the resistive material. *ubstitution and factoring out 5 from the right hand sideleaves

< R / %!>$4 > < " 5

R %@"

?here < = can be considered nearly constant, and thus the parenthetical termeffectively becomes a single constant, the gage factor, F g

< R / F g 5 R % "

2.3 Wheatstone Bridge

Also &nown as a )ull ?ave +ridge 'ircuit, the ?heatstone bridge used in

lab is shown below with an overlaid schematic.

)igure $ # ?heatstone +ridge

B out can be readily deduced as B 67 B!$ and by ta&ing each half bridge to be a simplevoltage divider B out can be evaluated as(

V out / R4 V o - R2 V o R3+R4 R1+R2 %C"

Diven that R1=R3=R4=R o, and R2 %the strain gage" / Ro + R , substituting intoequation %C" using equation % " with algebra yields(

V out / Fg V o 7

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3.0 E peri!ental "rocedure

3.1 Apparatus

)igure 6( Top down view of lab setup

3.1.1 Strain GagesTwo Eeasurements Droup, Inc.' A !6 $78FG !$8 constantan

alloy strain gages were used, each with a !$8.8 Ohms resistance and a $.8 9gage factor at room temperature %$7 o'". Habeled 0A1 in )igure 6, both weremounted with their sensitive direction along the circumference of the can.

3.1.2 Wheatstone BridgeA wheatstone bridge was provided for use in this experiment, labeled

0+1 in )igure 6. It is a four resistor resistive divider, with two static resistors,a variable potentiometer as a trim resistor, and the strain gage acting as thefourth resistor.

3.1.3 #a$oratory A!pli%ier

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An amplifier was also provided for this experiment, during which the precise gain was determined. The amplifier is labeled 0'1 in )igure 6.

3.1.& 'igital (ulti!eter and "o)er SupplyA ewlett 3ac&ard 6:!:A ' 3ower *upply %0 1, )igure 6" and a

ewlett 3ac&ard C@6A andheld igital Eultimeter %0 1, )igure 6" wererespectively used to produce and measure voltages.

3.2 (ethods3.2.1 "repare *an and Bond Gage

The soda can was cleaned, dried, and the desired surface site wassanded to abrade the surface and give purchase for the bonding agent. Thestrain gage was placed on adhesive tape and held upside down and adjacentto the desired measurement site. E +ond $88 catalyst was applied to thestrain gage, and the tape was flipped over and held for one minute. Headswere soldered to the pads of the strain gage.

3.2.2 *ali$rate #a$oratory A!pli%ierThe laboratory amplifier was supplied a &nown voltage through a

simple voltage divider that approximated the voltage seen across the?heatstone bridge. The output was measured and the precise gaindetermined.

3.2.3 "reparation and Balancing o% Wheatstone BridgeThe strain gage and !8B ' were connected to the ?heatstone

bridge, and the unamplified output voltage was trimmed to within 8.!mB ofJero. *ubsequently the amplifier was connected to the output of the bridge,

and the amplified output was trimmed to within $8mB of Jero.

3.2.& +pening o% Soda *an?ith the bridge balanced and the soda can closed, the amplified

bridge voltage was recorded. The can was then opened and a new voltagereading due to the change in strain gage resistance was recorded and used tocompute internal pressure.

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&.0 ,esults&.1 *an 'i!ensions

3epsi 'an Thic&ness %mm" 3epsi 'an iameter %mm"8.8CC ::.8$8.!!8 ::.8:

8.!8! ::.898.!888.!8:Average( 8.!86 mm Average( ::.87mm

iet 3epsi 'an Thic&ness %mm" iet 3epsi 'an iameter %mm"8.!89 ::.868.8C@ ::.888.8C9 ::.!68.8C8.8C:Average( 8.8C mm Average( ::.89mm

&.2 A!pli%ier *ali$rationsHab Amplifier *erial Kumber KA$;! resistance %Ohms" 9.86e:;$ resistance %Ohms" !.$8e6Bs voltage %B" 7.CCBin voltage %mB" !.! 9Bout voltage %Bolts" 8.:88Gain - out/ in 0&

&.3 Bridge Balancing'alibration ;esistance%Ohms" !$8.CEeasured Amplified Bout %Bolts" C.8:

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&. "ertinent alues2oungs Eodulus Aluminum %D3a" :C3oission1s ;atio # Aluminum %!" 8.69Dage )actor $.8 9

'iscussion and *onclusion.1 5 Theoretical *o!parisons and conclusion

The laboratory handout gave a theoretical internal pressure range of $98@88 &3a, which clearly brac&ets our results. There is also fairly low relativeuncertainty in our results. The only measurement ta&en with some noticeabledegree of uncertainty was the amplified voltage for the open can of regular 3epsi.The fluctuations in this measurement were on the order of !88mB, which by thetheoretical methods laid out corresponds to only !8&3a. This uncertainty isapproximately 6.!:L of the full calculated pressure. This lends credibility to thecalculated pressures, and our very similar results coupled with this small relativeuncertainty leads to the conclusion there exists no verifiably discernable differencein the internal pressure of diet and regular 3epsi

.2 5 'iscussion o% (al%unctioning Wheatstone Bridge'loser examination of the ?heatstone bridge %)igure $, and element + of

)igure !" shows that power was not simply hoo&ed up to the bridge # there is anextra +K' cable with miniclips connected to two elements on the board as opposedto a banana clip going to the intended terminal. This terminal is the negative sideof B o for the bridge, and throughout our experiment the output of the amplifier,hence the output of the bridge, would drop to ambient levels %a few millivolts". Theonly part of the experiment in which this flaw was not caught was the closed canvoltage readings. A reading of a few millivolts %P$8mB" was expected, with fewer

being better, so what was actually somewhat amplified ambient voltage noise was perceived as an actual reading. This is not to say the bridge was not balanced, as itwas &nown the bridge was functioning during the balancing of the bridge to a fewmillivolts difference. It is safe to say this malfunction does not negate the resultsand conclusions of this experiment.

,e%erencesunter, I.?. and ughey, +.Q. xperiment R6( stimation of Internal 3ressure

?ithin an Aluminum *oda 'an, Haboratory andout, $.:@! Eeasurement andInstrumentation, EIT, *pring $886

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