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
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, σ h, in the container wall Tamanna Islam Urmi, 12/11/20142014 1
is given by the tension per unit length divided by wall thickness, t. The tension per unit length is pd. The free-body diagrams showing the forces acting on the circumferential direction and axial 2 direction are given below: Figure 1 1 : Free body diagram to show the force actin on the wall of the can circumferentially σ h = pd 2t where D = 2r (1) Figure 2 1 : Free body diagram to show the force acting on the can axially The axial stress of the can, σ l, is given by 2 σ l = pd 4t The induced hoop strain along the circumference of the can is 2 ε h = 1 E (σ h νσ l ), (2) (3) Tamanna Islam Urmi, 12/11/20142014 2
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 = (4Et D ) ε h 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 is 2, R R = F gε (5) F g, 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 will be in the order of R 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 Tamanna Islam Urmi, 12/11/20142014 3
V out is the sum of the differential voltages across R 1 and R 2 in series and R 3 and R 4 in series. When R 1 = R 3, the Wheatstone bridge is balanced and V R 2 R out = 0. Any change in the resistances 4 will result in a corresponding change in V out. We will use this circuit with the strain gauge as R 2. 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 R 2, we will find corresponding change in V out using this equation 2 : V out = R 4R 0 V 0 The amplifier connected to the wheatstone bridge will amplify the reading by a factor of G. V meas = GV out = GV 0 R = K R 4R 0 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 V meas 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. (6) Tamanna Islam Urmi, 12/11/20142014 4
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. Amplifier Wheatstone Bridge Digital Micrometer 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 Tamanna Islam Urmi, 12/11/20142014 5
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 V out was measured using the DMM. The final R 2 was found by R 2,final = ( 1 1 1 + ) R gauge R parallel Here R gauge is the resistance of the strain gauge initially and R parallel is the resistance of the resistor added in parallel. Then we will have R R = R 2,final R gauge = ( 1 1 1 + ) R R gauge R gauge parallel (8) 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 V meas varied with R. The slope of this line is the calibration constant, K. The slope gives 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 1 mm and Tamanna Islam Urmi, 12/11/20142014 6
(1.09 ± 0.30) 10 1 mm 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 1 mm thick and the wall of Sprite can was (1.09 ± 0.30) 10 1 mm 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 1 mm 0.735 ± 0.002V Sprite 66.10 ± 0.119mm (1.09 ± 0.30) 10 1 mm 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. Tamanna Islam Urmi, 12/11/20142014 7
Fraction of sample 0.25 0.2 Lemonade Sprite 0.15 0.1 0.05 0 50 70 90 110 130 150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450 470 Internal pressure (kpa) 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. Tamanna Islam Urmi, 12/11/20142014 8
Internal Pressure (kpa) 500 450 400 350 Sprite My Sprite My Lemonade Lemonade 300 250 200 150 100 50 0 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 Strain 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. Tamanna Islam Urmi, 12/11/20142014 9
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) Tamanna Islam Urmi, 12/11/20142014 10