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

Transport Processes in Biotechnological Systems
Mass Transfer by Forced Convection
Equations for Determining Mass Transfer Coefficients and Interfacial Area

The gas mass transfer coefficient depends primarily on the hydrodynamic Properties of the liquid film surrounding the bubble, which, for most of the bubble's residence time, are in turn determined mainly by buoyancy (natural convection) and turbulence, i.e., the Reynolds number. Therefore, equations describing The behavior of freely rising bubbles most often include the Reynolds number in the form of Equation (8.62c).

In sufficiently large reactors equipped with baffles to ensure maximum mixing rates in the continuous phase (Ch. 9), METABOLISM/18.html">The Influence of free surface effects (the Froude number) becomes negligible. Gas exchange at the free surface can play a major role in small laboratory bioreactors, but with increasing Reactor volume, this effect practically disappears. In the absence of surface influence, dimensionless solutions for the velocity and concentration fields yield

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Therefore, the Sherwood number depends only on Re and Sc:

Calderbank proposed the equation [11]

Sh (turbulent aeration) = 0.13Sc^1/3Re^3/4      (8.67)

Using Equation (8.52), it can be shown that the dependence of k_l(Sh) on the power consumed per unit volume of the reactor, P/V, is given by the expression

Thus,

In the turbulent regime, when Equation (8.69) becomes applicable, an increase in P rapidly reduces the influence of this parameter on the dependence of k_l on P:

Having familiarized ourselves with bubble breakup processes, we can expect to find mathematical expressions for the dependence of the maximum stable bubble diameter D_c on σ and the variables determining the dynamic stress. Since this value is in many cases very close to the characteristic mean true diameter D_sm, it is not surprising that the equations for D_sm have a similar form. The inclusion of additional terms in these equations to account for the gas holdup H and the viscosity of the dispersed phase μd indicates important but non-governing contributions of other factors. The previously mentioned requirement for similar coalescence properties in both the experimental system used to derive the empirical equations and the system under study remains valid for the equations presented in Example 8.2.

Example 8.2. Equations for determining the maximum (D_c) and Sauter mean (D_sm) diameters of bubbles or droplets.

Experimentally determined values of D_c for freely rising bubbles are described by the equation*

As for stirred reactors, the following experimental results on the dependence of D_sm on the power consumed per unit volume of the reactor can be presented:

Liquid–liquid system**:

Gas–liquid electrolyte system**:

Gas–aqueous alcohol system***:

Gas–viscous liquid system****:

* Hu S., Kintner R. C., The Fall of Single Liquid Drops Through Water, AIChE J., 1, 42 (1955).

** Calderbank P. H., Trans. Inst. Chem. Eng., 36, 443 (1958).

*** McDonough J. A., Tomme W. J., Holland C. D., Formulation of Interfacial Areas in Immiscible Liquids by Orifice Mixers, AIChE J., 6, 615 (1960).

**** Bhavaraju S. M., Russell T. W. F., Blanch H. W., Design of Gas Sparged Devices for Viscous Liquid Systems, AIChE J., 24, 454 (1978).

For turbulent flow in a pipe [11]

For flow through an orifice in a pipe (determined at a point 0.3 m from the pipe outlet)*

Here, We_pipe and Re_pipe are defined by equations (8A2.9).

For given D_c or D_sm and gas holdup H, the value of a can be calculated using equation (8.60). In complex cases, such as in most of the agitated two-phase reactors shown in Fig. 8.2, H must be measured directly in the reactor or determined from empirical equations developed specifically for a reactor of this design. Several such equations were presented in Example 8.1.

* McDonough J. A., Tomme W. J., Holland C. D., Formulation of Interfacial Areas in Immiscible Liquids by Orifice Mixers, AIChE J., 6, 615 (1960).