# Modelling gas desorption from a still solution (e.g. a beer)

The common way to model or estimate gas (e.g. CO2) desorption rate from a liquid solution is by using an equation similar to this one

$J_{CO_2} = -k_L a(H·C^G_{CO_2} - C^L_{CO_2})$

where $J_{CO_2}$ is the molar (or mass) flowrate, $k_L$ is the mass transfer coefficient, $a$ is the specific contact surface, $H$ is one of the many expressions used for Henry's constant, and the G and L scripts stand for gas and liquid phase respectively.

The problem with this approach is that all the correlations I know used to estimate $k_L$, $a$, or most commonly $k_La$ imply that there is a gas stream with a given gas velocity (e.g. aeration or sparging). How could I estimate $k_La$ for a still solution, for instance a glass containing beer where the gas bubbles are stripped slowly? Furthermore, if I wanted to assess the effect of stirring the beer with a spoon, how could I estimate the effect?

• Why not have a glass filled with the beer, and measure the gas desorption over time stirred and not stirred, then calculate the correlations from your results? – Solar Mike Aug 28 '18 at 15:29
• The experiment is truly not as simple to consider. The switch stirring/stagnate changes the meaning of $C^L$ and may also change $H$. In a perfect world, the gas above the liquid would still remain stagnate (meaning the stirring will have to be done internally to the liquid not with a spoon). Any change in whether the gas is or is not stagnate also affects the meaning of $k_L$. At best, the experiment proposed can provide a correlation of $J$ (flux) with different experimental conditions. How all of it maps back to the desire to estimate $k_L a$ in different cases is unclear. – Jeffrey J Weimer Aug 28 '18 at 16:15
• @JeffreyJWeimer if people did not make the effort to do experiments, what theory would we not have.... – Solar Mike Aug 28 '18 at 17:14
• @SolarMike Of course the experiment could be possible, but I don't only care about a beer glass; actually I have encountered this problem more than once when modelling ethanol fermentation tanks, methane stripping from anaerobic digestion, etc. The beer glass is the easiest example and, as you can see from Jeffrey J Weimer's answer, is far from simple – Toulousain Aug 29 '18 at 10:30
• As you noted, the foundations to your post are complex and have applications to a range of separation processes. In the meantime, would you recommend that your question(s) have been addressed? Or are further clarifications warranted? – Jeffrey J Weimer Sep 5 '18 at 21:42

## 1 Answer

In perfectly stagnate gas systems, transport of a component through a gas collapses to diffusion of the component through the gas. In this case, $k_L a \propto (D / \delta)^n$ where $D$ is diffusion coefficient, $\delta$ is a film thickness, and $n$ can be correlated or derived from first principles (typically $n \rightarrow 1$ in film theories). References can be taken from correlations of the Sherwood number to the Grashof and Schmidt numbers.

A perfectly stirred liquid says $C^L$ is uniform throughout. A perfectly stagnate liquid requires that transport through the liquid also be modeled by diffusion. The approaches for diffusion in gas and diffusion in liquid are similar in concept.

The transport models to review are film theory, penetration theory, and two-film theory.

When you really are considering flow of a gas dissolved in a liquid, solubility can be an additional concern. The process of gas bubbles forming from dissolved gas is nucleation + growth. The bubbles become buoyant. This motion is not modeled by diffusion, rather by a process that might be equivalent to free convection. The combined transport of gas bubbles from a dissolved gas in a stagnate liquid must be modeled to include nucleation + growth of bubbles + free convection of the bubbles. I would have to wonder whether the net result (nucleate + grow + rise out of liquid) tends to a homogenized bubble concentration in the container at some point. Considerations made in the absence of gravity will, as will stirring.

## References

Here are some links that I found in a search "stagnate air, mass transfer rate" with or without "stirred liquid".

Methane transfer across the water-air interface in stagnate wooded swamps ...

Rates of evaporation of low-solubility contaminants from water bodies ...

Seasonal variations in air-water exchange of PCBs in Lake Superior

Mass transfer characteristics of solvent extraction ...

Significance of liquid-film coefficients in gas adsorption

Gas-liquid mass transfer coefficient in stirred tanks ...

Gas-liquid mass transfer coefficient in stirred tanks ... eddy structure ..