Ideal Gas Law & Its Effects on Leak Testing

Pressure, volume, moles, and temperature

During leak testing, changes in pressure in the test part are of primary interest. When a test part is filled with air (or any other gas), it initially expands inside the part to occupy its volume. When the part finally reaches the test pressure, the air inside contracts. This rapid expansion and contraction of air changes its temperature and volume. The test part also undergoes slight changes in temperature and volume because of changes in temperature and volume of the air. 

In order to detect the leakage, the changes in pressure must be taken into consideration. The changes in pressure can occur due to changes in temperature, volume, and number of moles of the gas. Since parameters like pressure, volume, and amount of air are involved in this process, it is governed by the ideal gas law. Mathematically, the law is expressed as:

P = Pressure of the air (or gas) enclosed in the container

V = Volume of the container occupied by the gas

n = Number of moles of the gas

T = Absolute temperature of the gas

R = Ideal gas constant with a value of 0.0821 dm3 atm K-1 mol-1

According to equation (1), the product of the pressure and volume of any quantity of an ideal gas is equal to the product of the number of moles, ideal gas constant, and the absolute temperature of the gas. It can be seen from equation (1) that changes in pressure, temperature, and volume are related and a change in any single parameter can affect other parameters. 

According to the Kinetic Molecular Theory (KMT) of gases, temperature is proportional to the average kinetic energy of a given sample of gas. This can be expressed as: 

Equation (1) relates the actual experimental behavior of gases while equation (2) relates the results of experimental analysis using KMT. Studying equations (1) and (2), we can see how pressure, temperature, and volume are inter-related during leak testing. This means that changes in temperature, test part volume, and number of moles of a gas can affect the part’s pressure during leak testing. 
 

Effect of temperature changes on the test part’s pressure

When the temperature of air increases, the average kinetic energy and velocity of the particles increases. This is expressed in the equation (2) above. These particles now have a greater velocity, and they hit the walls of the part with greater force. The pressure exerted by these particles is in essence the force exerted per unit area of the walls of the part. So the pressure increases with an increase in temperature. Pressure is directly related to the temperature when the volume of the container and number of moles of gas are constant. 

Effect of volume changes on the test part’s pressure

When the test part is pressurized, the walls of the part undergo expansion. This causes a change in volume of the part. When the part expands and its volume increases, the pressure inside drops. This happens because the increase in volume increases the area of the walls. So the force per unit areas (pressure) drops because the force is constant. At constant temperature, the particles have the same average kinetic energy (or velocity) and they exert the same force on the walls. With an increase in area, the overall pressure (force per unit area) drops. Similarly, a decrease in volume causes an increase in pressure if the temperature and number of moles are constant. 
 

Effect of number of moles on the test part’s pressure

An increase in the number of moles of the gas increases the pressure inside the part when the temperature and volume are kept constant. Pressure is generated inside the part when the particles exert force on the walls of the part. With an increase in the number of particles (number of moles) increases, this exerted force per unit area (pressure) also increases. 

Accounting for the leak rates caused by pressure changes

In order for detecting the leak rates correctly, the fluctuations in pressure caused by the factors explained above should be allowed to settle. Oftentimes, this is not the case and in practice, leak testing is carried out without a proper fill and settle time. In order to account for these pressure changes, an adequate relation is needed that accounts for the pressure losses. 

Deriving the relation of pressure loss to leak rate 

Using the ideal gas law, a suitable relation can be derived that allows determination of leak rates through pressure loss in the part. A leak rate can be seen as the volume of gas that escapes through the part per second. This relation can then later be modified to account for the effects of changes in pressure, temperature, volume, and number of moles.

From the ideal gas law, the number of moles in the test part can be expressed by: 

If,

N_L = Number of moles of gas lost 

P = P_ATM = Atmospheric pressure

V = L.R = Leak rate or volume of gas escaping per second

Then equation (3) becomes: 


Therefore, the remaining number of moles NR will be:


Using ideal gas law, at a constant temperature, the remaining pressure after time (t) can be given by:

, we get:

To accommodate these pressure changes, a master part or ‘non-leak part’ is put under test. Despite no real leakage, some pressure change occurs in this non-leak part because of the reasons described above. This pressure loss is called the ‘zero offset’ factor. Equation (7) can then be modified after considering the ‘zero offset’ factor. The modified equation can be used to determine leak rates in test parts. The zero offset factor or the pressure loss of the non-leak part is subtracted from the pressure change of the test part in consideration. By doing this, the pressure changes due to temperature and volume are accommodated for and correct leak rates can be determined. The modified equation is:

Where,

L.R (t)       Leak rate (scc/s)

t                 Time (sec)

ΔP test          Pressure loss in the test part during the test (psi)

ΔP non-leak    Pressure loss in the non-leak part (psi)

V                Volume (cubic cm)
 

Process repeatability

If the difference between ΔP test and ΔP non-leak is not great enough, the repeatability of the process will be compromised by environmental factors influencing the part under test. The selection of Fill and Settle times during the test will directly affect the test outcome. The greatest contributor to successful outcomes will be from the Fill step. Parts that have more compliance (capability to experience expansion) will need to reach a point of equilibrium. At higher test pressures, a longer fill step will allow the thermodynamic process also to reach equilibrium. Reducing or nullifying these two factors will cause a greater difference in the ΔP, thus improving the test.