September 25, 2026
Whitman's Picture: Two Films and an Interface
When a gas like ammonia dissolves into water, it has to cross from the bulk gas, through the interface, and into the bulk liquid. In 1923 Walter Whitman proposed a simple model of that journey. Each bulk phase is assumed to be well mixed, with uniform composition. Right next to the interface, though, sits a thin stagnant film on each side, one in the gas and one in the liquid, and all the resistance to transfer is concentrated in those two films, where the solute can only move by molecular diffusion.
The model makes one more key assumption: the interface itself offers no resistance, so the gas and liquid compositions right at the interface are in equilibrium with each other. That means the compositions jump at the interface according to the equilibrium relationship, while the concentration falls steadily across each film. The flux through each film is then written with its own coefficient, N = k_y(y − y_i) on the gas side and N = k_x(x_i − x) on the liquid side, and at steady state the two fluxes must be equal.
Adding Resistances: Gas-Film and Liquid-Film Control
The interface compositions are almost impossible to measure, so the two film equations are combined into an overall coefficient based on bulk compositions alone. For a linear equilibrium line of slope m, the result is 1/K_y = 1/k_y + m/k_x. Like thermal resistances in series, the two film resistances simply add, but the liquid-side resistance is weighted by the equilibrium slope.
That weighting tells you which film controls. For a very soluble gas like ammonia in water, m is small, the liquid term shrinks, and the gas film controls the rate, so increasing gas turbulence helps and stirring the liquid barely matters. For a sparingly soluble gas like oxygen or carbon dioxide in water, m is large, the liquid film dominates, and the reverse is true. Lewis and Whitman laid out this framework in their 1924 paper, and it remains the first thing to check when deciding how to speed up an absorber.
Where the Model Is Wrong, and What Replaced It
Film theory predicts that the mass transfer coefficient is proportional to the diffusivity, k = D/δ, because diffusion through a fixed stagnant layer is linear in D. Experiments disagree. Measured liquid-side coefficients in real equipment scale much closer to the square root of diffusivity. The stagnant film is a fiction; real liquid surfaces are constantly being swept away and replaced by fresh liquid from below.
Ralph Higbie's 1935 penetration theory took that seriously. It assumes each small element of liquid sits at the surface for a fixed short time, absorbing gas by unsteady diffusion, before being mixed back in, which gives k = 2√(D/πt) and the square-root dependence. In 1951 Peter Danckwerts generalised this into surface renewal theory, where surface elements are replaced at random with a fractional rate s, giving k = √(Ds). Both match the data better than film theory.
Why Engineers Still Use the Film Model Anyway
Despite being physically wrong in its details, film theory remains the standard framework for design. The reason is that the additive-resistance structure, the overall coefficient built from two film coefficients weighted by the equilibrium slope, holds regardless of which model describes each film. Penetration and surface renewal theories change how you would predict k_x from first principles, not how you combine it with k_y.
In practice, designers rarely predict film coefficients from theory anyway. They use measured volumetric coefficients, k_y·a and k_x·a, from correlations for specific packings or trays, and plug them into the film-theory framework to get transfer units and column height. The model's job is to organise the calculation and point at the controlling resistance, and it does that job very well.
References
- Whitman, W. G. (1923). The two-film theory of gas absorption. Chemical and Metallurgical Engineering, 29(4), 146–148.
- Lewis, W. K., & Whitman, W. G. (1924). Principles of gas absorption. Industrial & Engineering Chemistry, 16(12), 1215–1220. doi:10.1021/ie50180a002
- Higbie, R. (1935). The rate of absorption of a pure gas into a still liquid during short periods of exposure. Transactions of the American Institute of Chemical Engineers, 31, 365–389.
- Danckwerts, P. V. (1951). Significance of liquid-film coefficients in gas absorption. Industrial & Engineering Chemistry, 43(6), 1460–1467. doi:10.1021/ie50498a055
- Treybal, R. E. (1980). Mass-Transfer Operations (3rd ed.), Chapter 5. New York: McGraw-Hill.