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

Question

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?

Jeffrey J Weimer · Accepted Answer · 2018-09-06T00:51:06.500

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.