Lesson 4 - Gas Transfer

The previous discussion of Henry’s law and the worked example described the solubility of a single gas in a liquid and we used the parameter of partial pressure, for a single gas, and total gas pressure inter-changeably. Of course, we have also considered Dalton’s law of partial pressures in detail, in which a gas mixture can be considered in terms of its respective partial pressures. Let’s now consider how we numerically tackle a mixed gas dissolution situation by way of a worked example.

A mixed gas stream (75% by volume of nitrogen and 25% by volume of carbon dioxide) is used to pressurise and dispense a keg of stout. If the stout temperature was 5°C and the gas pressure was 2.5 bar g, what mass of dissolved N₂ and CO₂ per litre of stout would result? (Atmospheric pressure is 100 kPa).

The Henry’s law constants for CO₂ and N₂ in stout are 900 bar a/mol fr and 54,000 bar a/mol fr respectively, both at 5°C. You may assume that the molecular mass for stout is the same as water and that stout’s density is 1,000 kg/m³.

OK, this may appear a bit daunting, but if you’ve followed the concept behind the previous worked example, you’ll be able to follow this! We’ll take it slowly, so let’s get on with ‘dispensing’ with this example…

The approach that we’ll use will be to consider each gas in turn, from its behaviour in the headspace via Dalton’s law to use of Henry’s law to determine the quantity of gas at saturation. Let’s start with nitrogen:

So, what do we know about the behaviour of nitrogen in the gaseous headspace? Well, we know the total headspace pressure is 2.5 bar g (3.5 bar a) and that the composition of the gas is 75% by volume of nitrogen and 25% by volume of carbon dioxide. So we can apply Dalton’s law and determine the partial pressure of nitrogen:

We can express Dalton’s law as the following:

And introducing the molar fraction of N₂:

Remember that the molar fraction is equivalent to the volume fraction so x_N2 = 0.75:

Recall we converted the gas pressures from gauge into absolute pressure.

Considering Henry’s law for N₂ and rearranging for the molar fraction of N₂ in solution x: (Again, don’t get confused with the molar fraction of N₂ in the gas phase – Dalton’s law – and dissolved in the liquid phase – Henry’s law):

We were given the Henry’s law constant for N₂ 54,000 bar a/mol fr:

Noting that the molecular masses of N₂ and H₂O (stout) are 28 and 18 kg/kmole, respectively, gives the mass ratio:

We are assuming that the density of stout is 1,000 kg/m³, so 1 kg is equivalent to 1 litre. Hence, the saturation concentration of nitrogen is 7.6x10^-5 kg/L, or 0.076 g/L.

We can conduct the same approach for CO₂. Below are the relevant intermediary results of calculations for you to work towards.

Hence the relative dissolved concentrations of N₂ and CO₂ are 0.076 and 2.4 g/L, respectively.

As we’ve just been dealing with nitrogenated and carbonated stout, why don’t you do a bit of further reading on Michael Ash (1927-2016) and his role in nitrogenating one of the world’s most well-known stouts, Guinness™.