S15C02LeakRate

Abstract

This experiment is related to the Mu2e experiment, which is an experiment that seeks Charged Lepton Flavor Violation. 25000 Mylar® straws filled with Ar and CO₂ will be used to build the tracker to detect electrons produced by process. Straws’ walls are only about 15 μm thick so CO₂ gas can permeate through the straws, while Ar can hardly leak from the straws. Because it is crucial to make the space outside the straws vacuum, the leak rate of CO₂ from straws must be limited under 33×10^-5 ccm/min. A theory about the leak rate is proposed in this report. In our experiment, the leak rate measurements at different pressures were performed and the CO₂ permeation theory using Fick’s law was tested.

Theory

The speed of CO₂ concentration change is calculated as following:

where is the speed of CO₂ concentration change, C is the concentration of CO₂ and t is time. The change of CO₂ concentration in the chamber need to be tested when there is there is no straw in the chamber (background). The increase rate of CO₂ concentration caused by the straw is:

It is assumed that CO₂ can permeate through the walls of the straws. The permeation process and the quantitatively analysis are described as following.

There are six steps for the gas to penetrate the straw [3]:

1. Adsorption

2. Dissociation

3. Dissolution

4. Diffusion

5. Association

6. Desorption.

However, sometimes there is no dissociation and association happens [4]. Figure 1 shows the general process of the gas permeation. There is usually no dissociation and association process when the wall is only made of nonmetal or the gas is monoatomic or the gas is too stable to dissociate.

Figure 1 The process of gas permeation (Original figure)

Because the diffusion is stable when it reaches equilibrium, our assumption is that the diffusion of the gas follows Fick’s First Law [3], [4]:

where J is the diffusion flux, D is the diffusion coefficient, C(r,t) is the concentration of the CO₂ inside the straw, r is the radius of the straw.

Because the straw is very thin, its inner surface and outer surface are almost equal. In addition, because the diffusion process is stable after the system has reached equilibrium, the flux J is the same everywhere inside the straw. The diffusion flux J can be calculated using [2]:

where C_in and C_out are the concentration of CO₂ in and out of the straw, δ is the thickness of the straw.

According to the laws above, it is reasonable to speculate that the power of P is related to the number of atoms in each gas molecules. Therefore, it is assumed that

where j is a constant for each kind of gas. Because j=1 for monoatomic gases and usually j=2 for diatomic gases when the materials used are metals, it is speculated that j=3 for CO₂ when the materials used have metal surface. The value of j could be determined in the experiment.

The permeation rate of CO₂ is

Where P_in and P_out are partial pressure of CO₂ in and out of the straw, Q is the permeation rate (leak rate), S is the surface area of the straw.

D relates to the diffusion activation energy [4]. K relates to the sorption heat [4]. E_k is the permeate activation energy and does not relate to T. [3]

So we have

where T is temperature, D₀ and K₀ are the values of D and K when 1/T→0.

When temperature is a constant and P_out→0,

where S₁ is a constant. j can be determined by plotting Q vs. P_in and fitting the data into a straight line, then try to find the value of j when the uncertainty of the slope is the smallest with χ²=1 . It is assumed that the total pressure does not affect the leak rate, the leak rate is only determined by the partial pressure of CO₂. However, this theory have limitations. According to some research about H₂ permeation, the value of j predicted by the Henry’s law or Sieverts’ law is valid when the concentration of H₂ is low [1] and Sieverts’ law may be not valid under room temperature [5].

Experimental Methods

I. Apparatus

• 1. The preparation for straws

In order to get the straw connected to CO₂/Ar a gas line with pressure meter and flow meter to be pressurized. The tested straws were glued with end pieces and viton on both end of it. It can be seen on figure 2.

Figure 2. The figure on the left is the end of straw with the end piece and the viton tube glued on (this figure is adapted from a picture in citation [1]).The figure on the right is the photo of the end of straw with end piece and viton tube. (Original figure)

2. The copper chamber and electronic devices

A copper chamber has been built to provide an isolated environment for the straws. This chamber can be seen on figure 3. There is an electronic box at one end of the chamber. The CO₂ sensor EE891 made by AirTest Technologies Incorporated is placed in the box to measure the CO₂ concentration. The accuracy of the sensor is about (50+2% of the reading value) ppm and it can measures in the range of 0~2000ppm. In order to keep the CO₂ inside the chamber evenly distributed, a fan was placed inside the electronics box. A microcontroller board called “Arduino Uno” manufactured by SmartProjects is connected between computer and EE891 in order to transfer data to computer. Figure 3 shows the general experimental setup.

Figure 3 General experimental setup (Original figure)

II. Experimental Procedure

• 1. Background test

A background test is needed to be conducted in order to determine how much the chamber itself leaks. The chamber was flushed with N2 first so the CO2 concentration inside it is lower than the concentration in the air. After running with the valve closed for around 10 hours. A concentration (ppm) vs. time(s) plot can be made from the data to estimate the CO2 leakage of the chamber.

2. Straw leakage test

— Pressurize the straw.

1. Connect the viton tube to a needle tubing which is connected to gas line with a flow meter and a pressure gauge.

2. By controlling the flow meter, straw is flushed by CO2/Ar mixture for several seconds.

3. A small cap was used to seal the end which is not connected to the gas line.

4. Wait until the pressure inside the straw had reached the expected value. Then the viton tube connected to the needle tubing was clamped immediately to stopthe gas flowing into the straw.

5. This end of the straw was sealed by inserting another cap into the viton tube.

— Flush the chamber and begin testing

1. Place the pressurized tube inside the chamber.

2. Flushed the chamber with N2 until the CO2 sensor gives readings around 40ppm. Then close the valve.

3. Start the program on the computer and Arduino to take data for 2 hours.

Results

Figure 4. The leak rate of CO₂ from straw No. 00135 measured at different CO₂ partial pressures on several days. (Original figure)

Figure 4 includes all results measured using straw 00135. It was found that the black viton began to release CO₂ after 1.5 hours. The black viton needed more than 19 hours to relax. Therefore, only one measurement could be made on each day. According to these results, after the CO₂ partial pressure had been increased to 17.35 psi and 12.35 psi, the CO₂ leak rate measured at 7.35 psi CO₂ partial pressure increased a lot compared to its previous measurement and then decreased to the value that was close to its previous measurement. The CO₂ leak rate measured at 12.35 psi CO₂ partial pressure did not changed a lot after some measurements had been done at 17.35 psi and 7.35 psi CO₂ partial pressures. However, the leak rate obviously increased after the straw had been measured at 22.35 psi CO₂ partial pressure.

Figure 5. The leak rate of CO₂ from straw No. 00139 measured at different CO₂ partial pressures on several days. (Original figure)

Again, as shown in figure 5, the leak rates at 12.35 CO₂ partial increased after the straw had been measured at 14.85 psi, 24.35 psi and 17.35 psi CO₂ partial pressure.

Figure 6. The concentration of CO₂ in the chamber vs. Time when the straw 00135 was inserted without injecting CO₂ and was sealed with air in it. (Original figure)

The chamber was closed after flushing N₂. However, as shown in figure 6, the CO₂ concentration in the chamber still increased at a rate of about 8ppm/hour, which was significantly above the background increase 0.5ppm/hour. One day before, the straw 00135 was filled with CO₂ and measured.

According to figure 6, the straws were still releasing CO₂ over about 24 hours after they had been filled with CO₂. The black viton attached to the straw was tested and the result showed that the black viton which had ever filled with CO₂ stopped releasing CO₂ 19 hours after it each measurement. Therefore, it indicates that there was still a lot of CO₂ left in the wall of the straw after 24 hours. A guess of the explanation of the phenomenon mentioned above is that the CO₂ left in the straw may cause the increase of the leak rate. Several days later, the leak rate came back to its original value after the CO₂ concentration inside the wall of the straws had decreased under some value.

Figure 8. The leak rate of CO₂ from straw No. 00139 at different CO₂ partial pressures (All of the data included except the bad data that caused by the black viton) (Original figure)

According to figure 7, the leak rate increased very fast when the partial pressure of CO₂ increased. This data can hardly be fit into a straight line. If fitting these data into a curve that can be described using a power function, the power would be about 8, which means j=1/8. There is too less data to see any trend in figure 8.(No more data because straw00139 was broken).

Some paper says the value of j is not a constant and can be decreasing constantly, but still larger than 1 [5]. So the result for j may be incorrect. There are three possibilities for this result: 1. Our measurements are not accurate enough. 2. j is smaller than one. 3. Fick’s law may not apply in this case because the diffusion coefficient is related to concentration or it is incorrect to assume total pressure does not affect the leak rate.

Conclusions

1. It seems some CO₂ is left inside the wall of the straw after the measurements and it may cause the leak rate to increase when measured at a lower CO₂ partial pressure. However, more experimental methods needed to be used to test this hypothesis.

2. The permeation theory using Fick’s law may not apply in our experiment.

References

1. Rivera, David, Marjorie Corcoran, and Aseet Mukherjee. "Straw Assembly for CO2 Leak Testing", Rice University, 2013.

2. Basile, Angelo. InAdvanced Membrane Science and Technology for Sustainable Energy and Environmental Applications (Cambridge: Woodhead, 2011), 218-219.

3. Vadrucci, Monia et al. "Hydrogen permeation through Pd–Ag membranes: Surface effects and Sieverts' law." International Journal of Hydrogen Energy 38(10) (2013): 4144-4152. Accessed April 8, 2015. doi:10.1016/j.ijhydene.2013.01.091

4. Valentina Siracusa, “Food Packaging Permeability Behaviour: A Report,” International Journal of Polymer Science, vol. 2012, Article ID 302029, 11 pages, 2012. doi:10.1155/2012/302029

5. "H2 Permeation through Pd Films." H2 Permeation through Pd Films. Accessed May 10, 2015. https://www.wpi.edu/Pubs/ETD/Available/etd-011006-123013/unrestricted/Guazzone2.pdf

6. "EE891 Module." EE891 Module. Accessed April 8, 2015. http://www.airtest.com/oem/ee891module.htm.

7. "Arduino - ArduinoBoardUno." Arduino - ArduinoBoardUno. Accessed April 8, 2015. http://arduino.cc/en/Main/ArduinoBoardUno.

Acknowledgements

In our experiment, professor Clement Pryke, professor Dan Cronin-Hennessy, Mr. Hajime Muramatsu, Mr. Daniel Ambrose, Andrew Void, Yuguang Chen provided a lot of assistances and helpful suggestions. We would like to thank to all of them for their assistances.