the Real Gas model
The Ideal Gas assumptions define a direct proportionality between the pressure and the amount of gas: double pressure, double the moles.
Unfortunately, at high pressures and thus high densities, this is no longer true, especially with Helium: the intermolecular distances become quite short, the attractive forces between molecules become significant and the volume of the molecules themselves becomes significant relative to the size of the container.
These facts have effect on the gas properties, of which the pressure.
In case of mixtures, this issue becomes even more complex, especially when Helium is part of the mixture: the interactions between the dissimilar molecules of the mixture create new forces which are not present in either of the pure components. (more in the section “What is different in mixtures”).
What is the order of magnitude of these differences? Are they negligible ?
Even considering the best case, Helium as pure gas, at 200 bar gauge and 20 °C, it has an actual concentration of 7.53082 Mol/L, compared to 8.24708 calculated according to the Ideal Gas formula.
That is a deviation of 9.51%: no comment, the order of magnitude of the error cannot be considered negligible !
Moreover, the partial pressure blending is by far the most common blending method in the technical community, and it uses the pressure to determine the quantity of gas. The pressure, however, is only a property of gases, easy to understand and practical to measure but it is not reliable with the Ideal Gas model.
For this reason the Real Gas models have been adopted to predict gas behavior.
The blender’s interest is the relationship pressure - moles per liter (mol/L). This is the key point, easy to understand, less to implement, and that is exactly the task of the application: relieve the user from these complications and to make the number of moles predictable by the pressure.
The Real Gas models
A Real Gas model is an equation used for predicting the behavior of gas, and it integrates the Ideal Gas model with a corrective factor Z:
P V = Z n R T
The corrective Z is called Z factor, a complex function of pressure and temperature which varies not only for each gas, but also for each gas composition of the mixture.
In thermodynamics, the Z factor is determined experimentally for a certain range of pressure and temperature at regular intervals; these experimental data , in addition to being used in tables, are the basis for implementing the equations of state and, in the case of mixtures, the binary interaction parameters.
Looking at the real gas formula, we can see that by keeping “p”, “v” and “t” stable , increasing the value of Z, there is a decrease in the amount of gas “n” .
There are a large number of equations of Real Gas, dedicated to different industries, with different complexity. The majority is focused on specific sectors; they correctly reproduce certain gas behavior in a certain range of temperatures and pressures. This accuracy is not necessarily the same in other scenarios; for example , regardless of the quality of the model, no one would think of applying an equation intended for hydrocarbons to trimix blending...
Discussions about Real Gas are frequently referred to the van der Waals equation, the first to take into account the non-zero size of molecules and the attraction between them, introducing two correction factors to the Ideal Gas model: ( P – n2 a/ V2 )(V - nb) = nRT
However, the van der Waals equation, while representing a step in the right direction, is not sufficient to accurately represent the behavior of gases of our interest, especially at the pressures we use: the result with helium at 200 bar and 20 °C becomes with van der Waals 6.94651 mol/L, with a deviation from the experimental data of 7.56%, even in this case very considerable.
It should be also noted that the van der Waals equation can be used only to estimate the single gas, not the mixtures of which the gas is a fraction.
Some may think that the compressibility of a gas has to do only with a difference of breathing gas available and perhaps with a difference in cost; it actually has a direct impact on the blending procedure by partial pressures, with an impact sometimes negligible in case of Nitrox, not with Trimix, especially with high fractions of Helium.
The Z factor is commonly associated only with the concept of the compressibility of gas, but it is important to note that the issue is not limited to the behavior of the individual gases at different pressures and temperatures, but it extends to their mixtures, where the molecular interactions that take place are not present when the gases are alone.