Biochemical Engineering Fundamentals Part 1 - Bailey J., Ollis D. 1989

Transport Processes in Biotechnological Systems
Mass Transfer Across Free Surfaces

Gas transfer across free surfaces of gas-liquid systems (Fig. 8.10) plays a major role in supplying oxygen to Animal Cell Cultures and removing CO₂ from them. Surface mass transfer is also important for culture growth in shaken vessels and small stirred bioreactors designed for microbial cultivation. Gas transfer across a free liquid surface is essential for the reaeration of natural Water streams and for the Respiration of aerobic organisms in the near-surface layers of marine and lake biocenoses. Mass transfer across free liquid surfaces plays a significant role in various microbiological processes in trickling-flow reactors, which are used, in particular, in vinegar production and wastewater Treatment. In these cases, the depth of oxygen penetration depends on the scale of eddies near the liquid surface. Much work has been devoted to studying mass transfer involving a falling liquid film, although not all of the results obtained are applicable to microbiological processes. We will first consider mass transfer in falling liquid films.

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FIG. 8.10. Various modes of mass transfer across free surfaces: a — stagnant surface (calm lake); b — stirred reactors; c — falling film; d — turbulent water stream.

For a falling liquid film of thickness h, length L, and width W, at a zero initial concentration of dissolved gas, the area-integrated gas absorption rate is expressed by the equation

where u_max is the free surface velocity. In deriving this equation, it was assumed that the concentration of dissolved gas near the solid surface is always zero; in other words, during the film flow time, the solute does not have enough time to penetrate the entire thickness of the film [5].

In this case, the Reynolds number is defined as

where The Role of the length scale is played by the hydraulic radius R_h:

From the Definition of the mass transfer coefficient, it follows:

Total absorption rate = k_l(c_l* → c_l)WL      (8.84)

If c_l is much less than c_l* (c_l ≈ 0), then equations (8.83) and (8.84) can be written in the form Sh = f (Sc, Re):

Here, h is the length scale for the Sherwood number, and b = (L/h)^1/2. Thus, Sh is proportional to Re^1/2.

Livansky et al. studied the absorption of CO₂ by liquid films falling down an inclined surface of known area; water, algal Suspensions, and nutrient media were used as absorbing liquids [18]. For all three systems, the value of kl at Re = (7—8) ∙ 10³ was identical; only algal suspensions, for which turbulent flow is most likely, were studied at various Reynolds numbers. The results obtained can be described by the equation

k_l = 4 ∙ 10^-5Re^2/3 for 2000 ≤ Re ≤ 8000

In turbulent water streams, the scale of circulating eddies becomes highly important, as it determines the thickness of the layer into which water transports fresh portions of nearly saturated gas solution from the surface into the liquid phase. If we imagine the circulating eddy of length and depth A shown in Fig. 8.11, the average Sherwood number for turbulent mass transfer can be defined, by analogy with equation (8.28), as:

where are dimensionless coordinates expressed on the scale of , the coefficient value averaged over the eddy length.

FIG. 8.11. Schematic of eddies circulating near a free liquid surface.

The mass transfer rate ultimately depends locally on DO₂, as well as on the rate at which the liquid near the surface is renewed due to Circulation. According to Higbie's equation, which was one of the first proposed, the mass transfer coefficient at The surface of a gas-liquid system, assuming equal residence time of liquid elements, is*

This equation was modified by Danckwerts to make it applicable to various residence time distributions of liquid elements at the surface; it turned out that in this case, too, k_l is proportional to (D_O2)^1/2. It has been suggested that the renewal time τ of a turbulent flow is equal to The ratio of its depth h to the average velocity <u_w>:

It follows that

If the stream width is W, then the interfacial area per unit volume of the stream is

* Higbie R., Trans. AIChE, 35, 365 (1935).

Thus,

The latter equation has been used quite successfully to describe reaeration rates in oxygen-depleted lakes and rivers. Another equation accounts for the variation of k_l depending on the position w within the eddy:

Here, u_rms is the ROOT-mean-square circulation velocity of the eddy. In general, there is every reason to assume that ⋀ and u_rms are proportional to the mean depth and mean velocity of the stream, respectively, with the same proportionality constant:

It is well known that gas transfer is strongly influenced by wave action at the air-water interface. Unlike the cases discussed above, the key factors here are the gas flow parameters, such as the mean and turbulent Components of the wind speed, which determine The Nature of ocean waves. This topic, however, is too complex for this textbook; readers interested in this subject can find more details in the literature listed at the end of the chapter.