Friday 2-5

Lab 3

An investigation of the relationship between internal pressure and degree of

carbonation of soda drinks

Tamanna Islam Urmi

Lab Partner: Jack Greenfield

12/11/14

2.671 Measurement and Instrumentation

Dr. Andrew James Barnabas Milne

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Abstract

Spillage right after opening a soda can is a common occurrence. The sudden effervescence of

fizz from the drink occurs after opening the can due to the difference of pressure inside the can

and in the atmosphere. The drinks with higher internal pressure are therefore more likely to cause

the spillage. The question we explored in our experiment is whether the internal pressure of a can

depend on the amount of CO2 in the drink or not. Using strain gauge the change in strain on the

wall of a Lemonade, less carbonated drink and a Sprite can, highly carbonated drink at opening

the can was measured. The result led us to find the internal pressure. The Lemonade can had an

internal pressure of 72.47 ± 0.94 kPa and the Sprite can had an internal pressure of 352.6 ± 4.11

kPa. By comparing our results with data for internal pressure of Lemonade and soda can from the

last 10 years, we estimated the consistency of our results. We then concluded that the can of the

less carbonated drink has a significantly lower internal pressure than that of a more carbonated

drink. Because of the less internal pressure of the less carbonated drink, it will cause less vigorous

spillage at opening compared to the highly carbonated drink that has a higher internal pressure.

1. Introduction

The question answered in this experiment is: Is the internal pressure of can of lemonade, a

less carbonated drink, different from a regular soda that is more carbonated? Opening the can of

carbonated drink often cause high effervescence and spillage of drink. If we had information about

the internal pressure based on the amount of CO2 in it, we could save some trouble and anticipate

the degree of spillage likely to happen at opening the can. One strain gauge was attached to the

circumference of each can. The change in resistance across the strain gauge was found by

measuring the voltage across the strain gauge using a Wheatstone bridge and amplifier circuit

connected to the strain gauge. The voltage change was then used to find the strain incurred on the

strain gauge; the calculated strain was used to find the internal pressure. This method was the most

suitable for our laboratory conditions and is sufficiently reliable to show us the difference in

internal pressures of the two cans. The result obtained was compared to the historical data on strain

and internal pressure of Sprite and lemonade can from 10 years.

The rest of the paper is divided into four parts. Section 2 gives detailed theory behind each

component of the experiment and how these will allow us to find the internal pressures of the cans.

Section 3 gives a detailed description of the experimental setup, including important information

to make the experiment repeatable. The fourth section discusses the results from the analyzed data.

Lastly, section 5 draws the conclusion of our experiment.

2. Theory of using strain gauge and Wheatstone bridge to get internal pressure

In this section we will find out what the relation between mechanical properties of a can with the

internal pressure is and we will also find out how a Wheatstone bridge can measure voltage

difference that can ultimately be used to calculate strain, hence internal pressure.

2.1 Relationships between stress, strain and internal pressure

The derivation of internal pressure using stress and strain is done by modelling the cylinder using

the approximation that the thickness of the wall is much smaller that the diameter of the can and

the height of the can is more than the diameter. The thickness of both the cans are less than 2% of

the diameter hence the approximation is valid. The hoop stress of the can, 𝜎ℎ, in the container wall

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is given by the tension per unit length divided by wall thickness, 𝑡. The tension per unit length is 𝑝𝐷

2. The free-body diagrams showing the forces acting on the circumferential direction and axial

direction are given below:

Figure 11: Free body diagram to show the force actin on the wall of the can circumferentially

𝜎ℎ =𝑝𝐷

2𝑡 where 𝐷 = 2𝑟

(1)

Figure 21: Free body diagram to show the force acting on the can axially

The axial stress of the can, 𝜎𝑙, is given by2

𝜎𝑙 =𝑝𝐷

4𝑡

(2)

The induced hoop strain along the circumference of the can is2

𝜀ℎ =1

𝐸(𝜎ℎ − 𝜈𝜎𝑙),

(3)

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Combining equations (1)-(3) we get expressions for internal pressure of the can in terms of

stress, σ, strain, 𝜀, Young’s modulus, 𝐸 and Poisson’s ratio, ν.

The hoop strain is

𝑝 =(

4𝐸𝑡𝐷 ) 𝜀ℎ

2 − 𝜈

(4)

For our experiment, we will only measure strain induced circumferentially on the can to find the

hoop strain. The cans are made of Aluminum. It has a Young modulus of 69.0 GPa and a

poisson’s ratio of 0.35. These values along with the strain measured by the strain gauge will be

used to find the internal pressure of the can.

2.2 Finding strain using Wheatstone Bridge and strain gauge

A strain gauge experiences change in resistance as it experiences a mechanical strain. The

proportion of change in resistance and the initial resistance of the gauge is proportional to the

strain the gauge is experiencing. That is2,

∆𝑅

𝑅= 𝐹𝑔𝜀

(5)

𝐹𝑔, the proportionality constant, is the gauge factor which is 2.1±0.5%. We used the gauge factor

value from the specifications sheet of the strain gauge provided by the manufacturer. We have to

address two issues for measuring 𝜀 correctly. Firstly, the ratio ∆𝑅

𝑅 will be in the order of

miliohms. That is too small to precisely measure using a digital multimeter. Secondly, to make

our results accurate we need to know the scaling factor of the amplification. So, we will use a

Wheatstone Bridge circuit connected to an amplifier to get amplified value of resistance change.

The scaling factor of magnification will be found by changing the resistance by a known value

and calculate the magnification.

Figure 3: Wheatstone bridge circuit

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𝑉𝑜𝑢𝑡 is the sum of the differential voltages across 𝑅1 and 𝑅2 in series and 𝑅3 and 𝑅4 in series.

When 𝑅1

𝑅2=

𝑅3

𝑅4, the Wheatstone bridge is balanced and 𝑉𝑜𝑢𝑡 = 0. Any change in the resistances

will result in a corresponding change in 𝑉𝑜𝑢𝑡.

We will use this circuit with the strain gauge as 𝑅2. The circuit diagram for the modified version

of Wheatstone bridge that we will use is given below:

Figure 4: Wheatstone bridge with the strain gauge and amplifier connected to it. This is the circuit used in our

experiment

By changing 𝑅2, we will find corresponding change in 𝑉𝑜𝑢𝑡 using this equation2:

𝑉𝑜𝑢𝑡 = −∆𝑅

4𝑅0𝑉0

(6)

The amplifier connected to the wheatstone bridge will amplify the reading by a factor of 𝐺.

𝑉𝑚𝑒𝑎𝑠 = 𝐺𝑉𝑜𝑢𝑡 = −𝐺𝑉0∆𝑅

4𝑅0= −𝐾∆𝑅

The constant 𝐾 is called calibration constant that takes into account the ratio of change in

resistance with initial resistance and the scaling factor. K can be defined as

𝐾 ≡−𝑉𝑚𝑒𝑎𝑠

∆𝑅

(7)

To cause the change in resistance we had connected known resistance in parallel with the strain

gauge. Before measuring the change in voltage due to opening the can, the calibration constant

will have to be determined using the concept discussed above. Once we have the calibration

constant, the change in voltage due to opening the can be found and the result can be used to get

the corresponding ∆𝑅. Then we can use equation (5) to find the ɛ. In the following section, the

experimental procedure and the application of the theory discussed in this section will be

described.

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3. Setup of experiment

In this section, the apparatus used to carry the experiment out will be described in section 3.1. In

section 3.2, the complete method of the experiment is explained. In the second section it is

explained how the theories mentioned in the previous section is used in the experiment.

3.1 Apparatus used

We used the apparatus from the laboratory of course 2.671 at MIT. The amplifier serial number

was 2671-09. We used HP 973A digital micrometer, digital calipers, digital micrometers, HP

power supply, a can of Sprite, a can of Lemonade, strain gauge made of constantan with a gauge

factor of 2.10±0.5%, Wheatstone Bridge, BNC cables, resistors of 147kΩ, 33.3 kΩ, 100 kΩ, 49.9

kΩ, 68 kΩ resistances, soldering equipment, cable and wires.

Figure 5: Experimental setup showing how the soda cans, wheatstone bridge, digital micrometer and amplifier

were connected

3.2 Methods

In this section the detailed procedure we followed to measure the internal pressure of the soda cans

will be discussed. At first we cleaned the surface of the cans and attached the strain gauge along

the circumference. Piece of wires were soldered to the strain gauge. The loose ends of the wire

were connected to the digital multimeter to find the initial resistance of the strain gauge. The

resistance of the strain gauge on Lemonade was 120.421 ± 0.022𝛺 and that on the Sprite can read

120.061 ± 0.022Ω. The circuit of power supply, soda can, Wheatstone bridge, amplifier and

digital micrometer was set up. A supply of 10V was provided from the power supply. The setup

Amplifier

Wheatstone Bridge

Digital Micrometer

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was then used to calibrate the circuitry. A resistor of known resistance was connected in parallel

to the soda can with strain gauge and the change in 𝑉𝑜𝑢𝑡 was measured using the DMM.

The final 𝑅2 was found by

𝑅2,𝑓𝑖𝑛𝑎𝑙 = (1

𝑅𝑔𝑎𝑢𝑔𝑒+

1

𝑅𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙)

−1

(8)

Here 𝑅𝑔𝑎𝑢𝑔𝑒 is the resistance of the strain gauge initially and 𝑅𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 is the resistance of the

resistor added in parallel. Then we will have ∆𝑅

∆𝑅 = 𝑅2,𝑓𝑖𝑛𝑎𝑙 − 𝑅𝑔𝑎𝑢𝑔𝑒 = (1

𝑅𝑔𝑎𝑢𝑔𝑒+

1

𝑅𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙)

−1

− 𝑅𝑔𝑎𝑢𝑔𝑒

We were using a HP 973A digital multi-meter. Before each measurement the Wheatstone bridge

is balanced by rotating the potentiometer in it until the voltage output is within 0.1mV of zero. An

amplifier is used in this process because the change in voltage is too small to detect. Amplifying

the output allowed us to amplify the error and get a higher accuracy in both balancing the

Wheatstone bridge and measuring the change in output Voltage. This was repeated four times for

four different known resistors of known resistance. Using the results, we were able to find five

values of K, the mean of which gave us a calibration constant, interval K of 10.648 ± 0.114A

with 95% confidence interval. The plot below shows how 𝑉𝑚𝑒𝑎𝑠 varied with ∆𝑅. The slope of this

line is the calibration constant, K.

Figure 1: Plot of voltage measured as the resistance of one leg of the Wheatstone bridge was varied using

resistors of known resistance

We then used digital calipers to measure the diameters of the cans. The tip of the calipers were

very sharp so we had to be careful to avoid puncturing the can. Four measurements were taken for

each can. The diameters of Lemonade and Sprite can were 65.84 ± 0.119𝑚𝑚 and 66.10 ±0.119𝑚𝑚 respectively and the thicknesses of the cans were (1.05 ± 0.30) × 10−1𝑚𝑚 and

The slope

gives K

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(1.09 ± 0.30) × 10−1𝑚𝑚 for Lemonade and Sprite can respectively. There measurements were

done with 95% confidence interval.

Next, we had to find the voltage change due to opening the can to use the value to find the

resistance change which can then give us the strain acting on the strain gauge. After balancing the

circuit each can is carefully opened while still connected to the circuit. The change in voltage as

shown in the multi-meter was recorded. The Lemonade can was opened with the strain gauge

connected to the circuit and the digital multi-meter displayed a voltage right after opening the can.

This is the change in voltage due to opening the can. It gave a voltage of 0.735 ± 0.002V. The

Sprite can gave a voltage of 3.483 ± 0.004V.

After this step we emptied the can and cut a small section of the cans to measure thickness. Using

digital micrometers, the thickness of the can wall at four different locations were measured. The

mean of the four measurements were found for each of the can. The wall of Lemonade can was (1.05 ± 0.30) × 10−1mm thick and the wall of Sprite can was (1.09 ± 0.30) × 10−1mm thick.

All these measurements have 95% confidence interval.

The results are shown in tabulated form below:

Table 1: Results obtained for the diameter, thickness and voltage at opening for the lemonade and Sprite cans

Diameter Thickness Voltage at opening

Lemonade 65.84 ± 0.119𝑚𝑚 (1.05 ± 0.30) × 10−1𝑚𝑚 0.735 ± 0.002V

Sprite 66.10 ± 0.119𝑚𝑚 (1.09 ± 0.30) × 10−1𝑚𝑚 3.483 ± 0.004V

4. Results and Discussion

Using equation (4) the internal pressures of the Sprite can had significantly high internal pressure

than the Lemonade can. The table below shows the values for Internal Pressure and Strain of the

two cans:

Table 2: The calculated value for internal pressure and strain of the two cans

Internal Pressure Strain

Lemonade 72.47 ± 0.94 kPa (2.730 ± 0.03) × 10−4

Sprite 352.6 ± 4.11 kPa (1.297 ± 0.02) × 10−3

The internal pressure inside the Sprite can is about 5 times higher than the Lemonade can. Our

data shows the more carbonated a drink, the higher the internal pressure of the can is. In our

experiment, we used two cans of the same size, took measurements at the same environmental

conditions (for example, temperature, humidity and atmospheric pressure). Only one variable was

changed in between the two cans which is the amount of CO2 in the drink. We have compared our

results with the data from previous years. The graph below shows a histogram of internal pressures

of Lemonade can and Sprite can.

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Figure 2: Histogram of data on the internal pressure of Lemonade and Sprite can from the last 10 years

measured in 2.671 laboratory

The peak for Lemonade and Sprite are very distinctly apart. This supports our data as

Lemonade has a lower internal pressure than Sprite according to this histogram. However,

although the pressure of the Sprite can that we measured fall within the range that has highest

frequency, i.e. 350.1-370.0, the pressure of the Lemonade does not. The pressure of the lemonade

is within a range, 70.01-90.00 that is one of least frequents. We have also examined the relationship

between strain measured by the gauge and the internal pressure of the can to check the validity of

our data analysis and find out our position in the historical data. The diagram below shows the plot

of internal pressure against the strain.

0

0.05

0.1

0.15

0.2

0.25

50 70 90 110 130 150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450 470

Fra

ctio

n o

f sa

mp

le

Internal pressure (kPa)

Lemonade

Sprite

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Figure 3: Plot of the internal pressure of Lemonade and Sprite can against the strain measured by the strain

gauge. The data is taken from the 10 years of measurements done in 2.671 laboratory. The graph also shows the

position of the result of our experiment

This graph also shows that the strain and internal pressure that we measured for Lemonade can is

lower than what most people found but the strain and internal pressure for Sprite is in the crowded

region of points for Sprite, meaning that both the results are close to what most people found.

However, both of the points fall on the line of best fit of this plot which indicates that the method

of data analysis that we chose gave was consistent with other scientists.

5. Conclusions

The goal of our experiment was to find the difference in internal pressure of can of carbonated

soda and can of less carbonated soda. By measuring the internal pressure of a Lemonade can and

a soda can we found that can of less carbonated soda has a lower internal pressure. The internal

pressure of the Lemonade can was 72.47 ± 0.94 kPa and the internal pressure of Sprite was

352.6 ± 4.11 kPa. Highly carbonated drinks, like Sprite are more likely to cause spillage due to

pressure change as the can is opened than less carbonated drinks, like Lemonade.

In future, the correlation between the CO2 concentration and internal pressure can be found.

The amount of spillage should be quantified in future research so that the relationship between

concentration of CO2, internal pressure and the spillage it causes be found more accurately.

0

50

100

150

200

250

300

350

400

450

500

0.00E+00 2.00E-04 4.00E-04 6.00E-04 8.00E-04 1.00E-03 1.20E-03 1.40E-03 1.60E-03 1.80E-03

Inte

rnal

Pre

ssu

re (

kP

a)

Strain

Sprite

My Sprite

My Lemonade

Lemonade

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Acknowledgments

The author would like to thank Professor Ian Hunter and Dr. Barbara Hughey for compiling

all the information for us to carry out the experiment, Dr. Susan Brenda Swithenbank and Dr.

Andrew James Barnabas Milne for providing us with useful insight and guidance for the

experiment.

References

1. Efunda, “Thin Walled Pressure Vessels”,

http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/pressure_vessel.cfm

2. B.J. Hughey and I.W. Hunter, “Experiment 3: Estimation of internal pressure within an

aluminum soda can,” 2.671 Laboratory Instructions, MIT, Fall 2014 (unpublished)

3. B.J. Hughey, “Soda can pressure database,” 2.671 Laboratory Data, MIT, Fall 2014

(unpublished)