1.0 Introduction
In 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 in every unit,
which can be expensive and impractical, or is a destructive measure that
involves 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 the contents of the ruptured
vessel truly reflect the state of the undamaged vessel. This comes in addition to two further
concerns - both 1) the actual loss of the vessel itself - which in some
low-quantity applications may be unacceptable – and 2) the potential
contamination of the contents of the vessel – which could range from undesired
pollutants in mechanical applications to potentially deadly foreign matter in
biological applications.
If presented with a case of a
valuable container than we cannot sacrifice filled with contents that cannot be
contaminated, the search must continue for other externally measurable
variables that give rise to pressure detection.
As the pressure inside the can rises, 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 Analysis
2.1 Hoop Strain
In treating the can as a
thin-walled pressure vessel (where the thickness of the wall is negligible in
comparison to other significant dimensions) a simple material relation between
pressure and strain can be derived.
Consider the following free body diagram of a half-cross-section of the
can, with pressure p, diameter D, tension T and wall
thickness t:
The net upward pressure force per unit height pD must be balanced by the downward tensile force per unit height 2T, a force that can also be expressed as a stress, σhoop, times area 2t. Equating and solving for σh gives
σh = pD
2t
(1)
Similarly, the
axial stress σaxial can be calculated by dividing the total force on
the end of the can, pA=pπ(D/2)2 by the cross
sectional area of the wall, πDt, giving:
σa = pD
4t
(2)
For a flat sheet in biaxial tension, the strain in
a given direction such as the ‘hoop’ tangential direction is given by the
following constitutive relation - with Young’s modulus E and Poisson’s
ratio ν:
ε h = 1(σh – νσa)
E (3)
Finally,
solving for unknown pressure as a function of hoop strain:
p
= (4Et) ε h
D 2-ν
(4)
2.2 Strain Gages
A strain gage is a variable resistive element that
takes advantage of the geometric and physical basis of resistance. As the strain gage undergoes strain in its
sensitive direction, the diameter of the extremely thin (0.056mm for the strain
gages used in this lab) longitudinal wires changes slightly, giving a change in
overall strain gage resistance.
Resistance of a conductor of length L, cross-sectional area A,
and resistivity ρ is
R = ρL
A
(5)
Consequently,
a small differential change in ΔR/R can be expressed as
ΔR = ΔL
- ΔA + Δρ
R L A ρ (6)
Where ΔL/L is longitudinal
strain ε, and ΔA/A is –2νε where ν is the Poisson’s ratio of the
resistive material. Substitution and
factoring out ε from the right hand side leaves
ΔR = (1+2ν + Δρ ) ε
R
ρε
(7)
Where Δρ/ρε can be
considered nearly constant, and thus the parenthetical term effectively becomes
a single constant, the gage factor, Fg
ΔR = Fg ε
R
(8)
2.3 Wheatstone Bridge
Also known as a Full-Wave Bridge Circuit, the
Wheatstone bridge used in lab is shown below with an overlaid schematic.
Vout
can be readily deduced as V34-V12 and by taking each half
bridge to be a simple voltage divider Vout can be evaluated as:
Vout
= R4 Vo - R2
Vo
R3+R4 R1+R2 (9)
Given
that R1=R3=R4=Ro, and R2 (the strain gage) = Ro
+ ΔR, substituting into equation (9) using equation (8) with algebra
yields:
Vout
= - Fg ε Vo
4
3.0 Experimental Procedure
3.1 Apparatus
3.1.1 Strain Gages
Two Measurements Group, Inc.CEA-13-240UZ-120 constantan
alloy strain gages were used, each with a 120.0 Ohms resistance and a 2.085
gage factor at room temperature (24oC). Labeled ‘A’ in Figure 3, both were mounted
with their sensitive direction along the circumference of the can.
3.1.2 Wheatstone Bridge
A
wheatstone bridge was provided for use in this experiment, labeled ‘B’ in
Figure 3. It is a four-resistor resistive divider, with two static resistors, a
variable potentiometer as a trim resistor, and the strain gage acting as the
fourth resistor.
3.1.3 Laboratory
Amplifier
An amplifier was also provided for this experiment,
during which the precise gain was determined.
The amplifier is labeled ‘C’ in Figure 3.
3.1.4 Digital Multimeter and
Power Supply
A Hewlett Packard 3616A DC Power Supply (‘E’, Figure
3) and a Hewlett-Packard 973A Handheld Digital Multimeter (‘D’, Figure 3) were
respectively used to produce and measure voltages.
3.2 Methods
3.2.1
Prepare Can and Bond Gage
The soda can was cleaned, dried, and the desired
surface site was sanded to abrade the surface and give purchase for the bonding
agent. The strain gage was placed on
adhesive tape and held upside down and adjacent to the desired measurement
site. M-Bond 200 catalyst was applied to
the strain gage, and the tape was flipped over and held for one minute. Leads were soldered to the pads of the strain
gage.
3.2.2
Calibrate Laboratory Amplifier
The laboratory amplifier was supplied a known
voltage through a simple voltage divider that approximated the voltage seen
across the Wheatstone bridge. The output
was measured and the precise gain determined.
3.2.3
Preparation and Balancing of Wheatstone Bridge
The strain gage and 10V DC were connected to the
Wheatstone bridge, and the unamplified output voltage was trimmed to within
0.1mV of zero. Subsequently the
amplifier was connected to the output of the bridge, and the amplified output
was trimmed to within 20mV of zero.
3.2.4
Opening of Soda Can
With the bridge balanced and the soda can closed,
the amplified bridge voltage was recorded.
The can was then opened and a new voltage reading due to the change in
strain gage resistance was recorded and used to compute internal pressure.
4.0 Results
4.1
Can Dimensions
Pepsi Can Thickness (mm)
|
Pepsi Can Diameter (mm)
|
0.099
|
66.02
|
0.110
|
66.06
|
0.101
|
66.05
|
0.100
|
|
0.106
|
|
Average: 0.103
mm
|
Average: 66.04mm
|
Diet Pepsi Can Thickness (mm)
|
Diet Pepsi Can Diameter (mm)
|
0.105
|
66.03
|
0.097
|
66.00
|
0.095
|
66.13
|
0.098
|
|
0.096
|
|
Average: 0.098
mm
|
Average: 66.05mm
|
4.2
Amplifier Calibrations
Lab Amplifier Serial Number
|
NA2
|
R1 resistance (Ohms)
|
5.03e6
|
R2 resistance (Ohms)
|
1.20e3
|
Vs voltage (V)
|
4.99
|
Vin voltage (mV)
|
1.185
|
Vout voltage (Volts)
|
0.600
|
Gain
(Vout/Vin)
|
504
|
4.3 Bridge Balancing
Calibration Resistance(Ohms)
|
120.9
|
Measured Amplified Vout (Volts)
|
9.06
|
ΔR (Ohms)
|
0.9
|
Ro (Ohms)
|
120
|
Calculated Vout (Volts)
|
9.44
|
Percent Difference (%)
|
4.19
|
4.4 Gage Readings and Calculated Strain, Pressure
Quantity
|
Pepsi Can
|
Diet Pepsi Can
|
Vo Can Closed (mV)
|
2.05±0.1
|
3.0
|
Vo Can Open (V)
|
3.17
|
3.337
|
“
|
3.27
|
3.337
|
“
|
3.24
|
3.337
|
Vo Open Average:
|
3.23
|
3.337
|
Strain (no units)
|
-0.002189
|
-0.0012718
|
Pressure(kPa)
|
316.0
|
316.2
|
4.5 Pertinent Values
Youngs Modulus- Aluminum (GPa)
|
69
|
Poission’s Ratio – Aluminum (1)
|
0.35
|
Gage Factor
|
2.085
|
5 Discussion and Conclusion
5.1 – Theoretical Comparisons
and conclusion
The laboratory handout gave a theoretical
internal pressure range of 250-700 kPa, which clearly brackets our
results. There is also fairly low
relative uncertainty in our results. The
only measurement taken with some noticeable degree of uncertainty was the
amplified voltage for the open can of regular Pepsi. The fluctuations in this measurement were on
the order of 100mV, which by the theoretical methods laid out corresponds to
only 10kPa. This uncertainty is
approximately 3.16% of the full calculated pressure. This lends credibility to the calculated
pressures, and our very similar results coupled with this small relative
uncertainty leads to the conclusion
there exists no verifiably discernable difference in the internal
pressure of diet and regular Pepsi
5.2
– Discussion of Malfunctioning Wheatstone Bridge
Closer examination of the Wheatstone bridge (Figure
2, and element B of Figure 1) shows that power was not simply hooked up to the
bridge – there is an extra BNC cable with miniclips connected to two elements
on the board as opposed to a banana clip going to the intended terminal. This terminal is the negative side of Vo
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). The only part of the
experiment in which this flaw was not caught was the closed can voltage
readings. A reading of a few millivolts
(<20mV) 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 it was known the bridge was functioning during the balancing of
the bridge to a few millivolts difference.
It is safe to say this malfunction does not negate the results and
conclusions of this experiment.
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