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

Transport phenomena in biotechnological systems
Mass transfer involving freely rising and freely falling bodies
Mass transfer coefficients for gas bubbles and bubble streams

The mass transfer coefficient for a gas bubble is the proportionality constant between the overall flux density and the overall mass transfer driving force (с_l* — с_l). The local flux density at the gas–liquid interface is —D_O2(dc/dz)_z=0 (valid at low mass transfer rates), where z is the coordinate measured into the liquid phase. Thus,

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or in terms of dimensionless parameters:

The relative (dimensionless) concentration c near the interface can be found by solving the transport equations:

Using this expression to evaluate the derivative in Equation (8.28), the desired mass transfer coefficient k_l can, in principle, be expressed in terms of the Sherwood number:

Thus, the dimensionless mass transfer coefficient Sh is a function of only two parameters: Sc and Gr. Here, D is the characteristic bubble diameter.

The literature also describes other expressions for mass transfer coefficients in the case of falling or rising bubbles, droplets, or solid particles, which use different similarity criteria, such as the Reynolds number (Re = р_lDu/μ_l) or the Péclet number (Pe = uD/D_O2). In these dimensionless parameters, u is the velocity of the gas bubbles relative to the liquid velocity. In both cases, expressing the velocity u in terms of the density difference ∆р = (р_l - p_g) leads, in accordance with Equation (8.30), to a dependence of the mass transfer coefficient solely on Gr and Sc.

Mass transfer from an isolated sphere with a rigid interface (this model describes reasonably well a system consisting of small bubbles in a culture broth in the presence of Surfactants) can be determined theoretically for the case of Re ≡ p_lDu/μ_l ≪ 1 and Pe ≡ uD/D_O2 ≫ 1. (Thus, uD/D_O2 ≫ 1 ≫ p_lDu/μ_l, from which it follows that μ_l/р_lD_O2 = Sс ≫ 1. Can the opposite inequality hold?) In aqueous media, where the kinematic viscosity (v = μ_l/р_l) is approximately 10^-2 cm²/s and D_O2 is about 10^-5 cm²/s, the Schmidt number is typically of the order of 10³. Consequently, for Re = 10^-1—10^-2, the analytically derived equation is applicable:

For low Reynolds numbers, where Equation (8.31) is applicable, the terminal velocity u_t of the sphere is given by the equation

Substituting u_t from Equation (8.32) into Equation (8.31) yields

[The expression (D³∆pg)μ_lD_O2 is also known as the Rayleigh number, Ra.] Note that here, as expected, Sh = f(Gr, Sc).

For high Reynolds numbers, another equation has been proposed for a single non-circulating spherical bubble in the laminar regime:

Interestingly, here Sh varies proportionally to the square (rather than cube) ROOT of the relative velocity parameter; this indicates A change in the hydrodynamic regime. Again, substituting the expression for the corresponding terminal velocity instead of u will lead to Sh = f(Gr, Sc).

In many industrial aerated reactors (Fig. 8.2), air bubbles cluster into swarms where they are in such close contact that the flow in the dispersed system and the mass transfer near the gas–liquid interface can no longer be adequately described by a model based on the hydrodynamic characteristics and mass transfer of an isolated bubble. According to Calderbank and Moo-Young, the absorption of sparingly soluble gases in liquids reacting with these gases can be described using only two equations [10]. There are two fundamentally different mass transfer regimes involving bubble swarms; the boundary between these regimes is determined by a critical bubble diameter D_c. In the absence of surfactants, D_c ≈ 2.5 mm. Larger bubbles typically occur in stirred reactors with pure Water and in sieve-plate columns. Smaller bubbles are characteristic of sintered-plate columns, as well as stirred vessels containing aqueous solutions of hydrophilic substances.

For D < D_c = 2.5 mm

For D > D_c = 2.5 mm

Equations (8.33) and (8.35) imply that in a bubble swarm, at the same Sc and Gr values, the mass transfer coefficient is approximately 20% lower compared to isolated bubbles with a rigid surface. It has been shown that equation (8.36) is also valid for airlift systems (Fig. 8.2, a) if a coefficient of 0.50 is used instead of 0.42.

The change in the exponent of the Schmidt number in equation (8.36) [compared to equation (8.35)] indicates a change in the hydrodynamic regime. In the case of Newtonian fluids, whose viscosity is constant and independent of the shear rate (determined by the mixing rate, bubble velocity, and other factors), the transition from the region D < D_c to the region D > D_C is accompanied by a change in bubble shape from nearly spherical (small bubbles) to hemispherical and mushroom-like. The hydrodynamics of bubble flows are discussed in more detail in [10]. The value of D_c is affected by surfactants; it has been reported that in some media, D_c reaches 7.0 mm. In some Non-Newtonian fluids, which we will consider later, the transition from one mass transfer regime to another occurs more smoothly; such systems do not exhibit the sharp jump characteristic of Newtonian fluids.

Results of mass transfer studies for small particles show that as the density difference ∆p decreases, the Sherwood number approaches a lower limit of 2.0. For Suspensions of single Cells, Cell clusters, flocs, etc., as well as for gas oil or other hydrocarbon emulsions, the Sherwood number is more accurately determined by the equation

or

It follows that the relative contribution of the purely diffusional effect (∆p = 0, k_l = 2.0D_O2/D) decreases as the particle size increases. For an isolated cell, 2D_O2/D is approximately 0.1 cm/s, while the second term, derived from the Rayleigh number, is 0.01 cm/s; therefore, mass transfer near The Cell surface is close to that of a sphere in a relatively stagnant medium. The larger sizes of flocs, films, etc., lead to a significant increase in the relative contribution of the second term.