## Q Qq2 Qq4 Qq6 Qqb Q1 Q12 Q14 Q16

Superficial gas velocity (m/s)

Figure 11. Some correlations proposed for prediction of gas holdup in the riser of external ALRs (Table 2). The gas holdup is presented as a function of the superficial gas velocity.

Effects of Liquid Rheology. The effect of rheology on the reactor behavior and performance is of great interest because in most biotechnological processes an increase in biomass provokes changes in the rheology of the fluid, especially in the case of mycelial growth. This effect is enhanced when in addition to the biomass growth, a product of the process is released into the medium in appreciable amounts. A good example of this scenario is the biosynthesis of polysaccharides, which cause an increase in the liquid viscosity.

The effect of viscosity on gas holdup in bubble columns has been studied by a number of authors. The main problem to be overcome is that of non-Newtonian flow. If the viscosity is not constant, but changes with changes in the shear rate, then the evaluation of shear rates becomes particularly relevant for the identification of the system. Several authors have confronted this issue. Nishikawa et al. (89,90) analyzed the problem of heat transfer in a bubble column with non-Newtonian liquids. They found a direct proportionality between the superficial gas velocity and the global shear rate:

This global shear rate was then used to calculate a global viscosity. In shear-sensitive cultures, the definition of a global shear rate in itself is of great importance.

A number of researchers, Henzler (91), Kawase and Moo-Young (59), Schumpe and Deckwer (92,93) have followed the approach of Nishikawa et al. (89) but have suggested different proportionality constants relating the global shear rate to the superficial gas velocity. This approach is questionable from the rheological point of view because it will predict the same shear rate for a certain superficial gas velocity, no matter which liquid is used. El-Tamtamy et al. (94) introduced an improvement by calculating the shear rate from the bubble velocity divided by the bubble diameter. However, accurate evaluation of the latter two parameters is difficult. Henzler and Kauling (95) suggested relating the shear rate to power input based on dimensional analysis by expressing the shear rate as a function of the power input per unit volume, (P/[Vqv])1/2. Their analysis gives different shear rates for liquids that are rheologically different.

The above-described relationships predict different shear rates that vary in up to three orders of magnitude. It is thus generally agreed that the correct solution is still to be found. Recently, a more general approach, known as a global approach, has been proposed by Merchuk and Ben-Zvi (Yona) (96). The shear stress in a bubble column was defined as being equal to the acting force, which can be calculated from the power input divided by the sum of the areas of all the bubbles:

where LR is an effective length that represents the mean circulation path of a bubble in the system considered, P is the power input, Sab is the total surface of all of bubbles, and s is the shear stress.

Assuming ideal gas isothermal expansion, the power input P can be calculated. The interfacial area can be evaluated from correlations or can be obtained by direct measurement if available. A correlation taking into account other variables, like sparger configuration, surface tension, etc., will broaden the range of applications of this approach.

If a constitutive equation describing the rheology of the system is available (such as the power law, which has been reported to correspond to many biological systems), equation 17 facilitates the calculation of a global shear force acting on the liquid. The shear rate can be in this case expressed as:

where y is shear rate and k is behavior coefficient, and equation 17 can be now used to express y as:

where the subindexes 1 and 2 represent the two extremes of the section considered.

Equation 19 thus gives a global shear rate that is a function of both fluid dynamics and rheology. This approach has been found to be useful for the presentation of results on mass transfer rates in bubble columns (96).

In contrast to the marked influence of rheology on gas holdup in bubble columns, the data available for ALRs show clearly that the effect of liquid viscosity is less dramatic, but not simpler. Figure 12 (65) illustrates the effect of the addition of glycerol to water in an internal-loop ALR. At low concentrations of glycerol, a moderate increase of the gas holdup is evident, particularly in the downcomer but also in the riser. These increases are caused by the lower free rise velocity of the bubbles, which increase the gas retention due to the longer residence time. In addition, the entrapment of the bubbles is increased, and this is reflected mainly in ud. When the concentration of glycerol becomes too high, a strong decrease of the gas holdup is

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