Diffusion and gas–liquid mass transfer

Diffusion: why gas moves slowly through still water

In still water, dissolved gas spreads only by diffusion, the random motion of molecules, and diffusion is slow. Fick's law describes it: the flow of a dissolved gas is proportional to how steeply its concentration changes with distance Fick, 1855. The proportionality constant is the diffusion coefficient, D.

For oxygen in clean water at 20 °C, D was measured at 1.98 × 10⁻⁹ m²/s Jamnongwong et al., 2010. Gases diffuse about 10,000 times faster in air Mommer & Visser, 2005. Salt, sugar and surfactants each lowered D further in the same laboratory tests Jamnongwong et al., 2010.

The time diffusion needs grows with the square of the distance, roughly t = x²/2D Einstein, 1905. With D = 2 × 10⁻⁹ m²/s:

Across the last fraction of a millimeter, at a bubble wall, a root or a gill, diffusion is fast enough. Across centimeters of still water it is not; flow and mixing must do that work, as the next lesson shows.

The two-film model: where the resistance sits

Gas crossing into water meets resistance in two thin films, one on each side of the surface, and for oxygen the water-side film is the bottleneck. The two-film model Lewis & Whitman, 1924 treats gas and water as being in equilibrium only at the interface itself; on each side, gas crosses a thin, poorly mixed film by diffusion Kraakman et al., 2011.

Because oxygen is poorly soluble, the gas-side film offers no significant resistance Kraakman et al., 2011. Nitrogen and hydrogen are less soluble still (see How gases dissolve in water), so the same reasoning applies to them.

Transfer therefore speeds up when something thins or renews the water film (mixing, turbulence, flow), when there is more interface area, or when the gap between saturation and the actual concentration is larger Kraakman et al., 2011. Plants show the film at work: submerged plants in stirred water kept their internal oxygen stable, while in unstirred water it fell quickly Mommer & Visser, 2005.

kLa: the number engineers measure

Transfer rate = kLa × (saturation concentration − actual concentration) Kraakman et al., 2011. kLa combines two things measured together: kL, how fast gas crosses the water film, and a, the interface area per volume of water Jamnongwong et al., 2010. It is usually reported per hour or per minute.

The bracket is the driving force, and it explains two practical rules. First, transfer slows as water approaches saturation, so the last few percent come slowest. Second, anything that raises the saturation value, such as pure oxygen instead of air, widens the gap and speeds transfer.

Clean-water tests give a standard kLa and a standard oxygen transfer efficiency (SOTE), the share of supplied oxygen that dissolves. Real water transfers differently, and engineers correct with an α-factor, the ratio of kLa in process water to kLa in clean water Jamnongwong et al., 2010.

Bubble size, rise speed and contact time

Smaller bubbles offer more surface per liter of gas and rise more slowly, so they give gas more area and more time. A sphere's surface per unit volume is 6 divided by its diameter.

The 3.9 mm bubbles rose at 0.29 m/s with a kL of 4.6 × 10⁻⁴ m/s Jamnongwong et al., 2010. The smaller speeds are computed with Stokes' law for a rigid sphere, the slow limit for small bubbles Manor & Chan, 2009. Measured rise speeds of 20 to 1,000 µm bubbles fell between the rigid-surface and fluid-surface predictions Detsch, 1991, and trace surfactants shift them Manor & Chan, 2009. Small bubbles also shrink as their gas dissolves.

Water chemistry changes bubble size. In a laboratory column with membrane diffusers, salt up to 15 g/L stopped bubbles from merging, and the smaller bubbles more than doubled kLa compared with tap water Behnisch et al., 2018.

At bench scale, bubbles of 100 nm to 2 µm, at 10⁸ to 10⁹ per mL, reached a kLa of 0.4204 min⁻¹ at 0.5 L/min of air in 25 L of water, with SOTE up to 54.33%, about twice the diffuser's at the best airflow Kizhisseri et al., 2025.

Why smaller is not always better

• Surfactants. Surface-active substances collect on bubbles and reduce transfer, more for fine-bubble aerators than for others Rosso et al., 2006. They act two ways: they keep bubbles small, which adds area, but stiffen the surface, which lowers kL Jamnongwong et al., 2010.
• Driving force. No bubble size escapes the bracket: near saturation, transfer slows whatever the bubbles.
• Energy. Aeration accounts for the largest share of energy costs in wastewater treatment plants Rosso et al., 2006. SOTE counts oxygen per oxygen supplied, not per kilowatt-hour. The bench generator above ran a pump at 3.5 to 4.5 bar, and the study did not report energy per kilogram of oxygen Kizhisseri et al., 2025. More turbulence can offset contamination, at the expense of efficiency Rosso et al., 2006.

How cavitation devices break gas into fine bubbles is covered in Cavitation, gas dissolution and fine bubbles.

Transfer runs both ways

The same film physics removes gas. Recirculating fish farms strip carbon dioxide by trickling water through air. In full-scale cascade columns, CO₂ removal was 21–24% at air-to-water ratios of 2.2:1 to 3.4:1 and rose to 35.8–37.2% at 9.5:1 to 9.9:1 Summerfelt et al., 2003. Supersaturated water losing its excess gas is the same process, covered in Water as a gas carrier.