where L is the characteristic length. When the Bo number tends to infinity, the mixing conditions are similar to those of a plug-flow reactor, and the reactor can be considered as well-mixed for low Bo numbers.

The alternative approach of Buffham and Mason states that the mixing characteristics of a piece of equipment should be expressed as the variance a2 of the distribution obtained by injection of a pulse of tracer without adopting any mechanistic model (116). The relationship between Bo and a2 depends on the reactor configuration (117). The approach of Buffham and Mason facilitates the presentation of mixing characteristics free of any modeling assumptions. The variance a2 is the second moment of the distribution and carries information on the spread of the distribution around the mean value (first moment). Nevertheless, most of the data on mixing in bioreactors are presented either as tm or as overall Bo numbers, which can be obtained by relatively simple experiments of pulse injection.

Single-pass mixing in the ALR is due to mixing in the individual and interrelated sections of the reactor—riser, separator, downcomer, and bottom. Repeated passage mixing is the sum of the mixing in the subsequent passages. The latter is usually reported as the mixing time (tm), the former as Bo or a2. Indeed, these parameters are interrelated, and knowledge of Bo or tm is sufficient for calculating, theoretically, the mixing time (108,118) based on the deviation of the envelope of the maxima in the response curve to a pulse, which is a measure of the degree of in-homogeneity. Verlaan et al. (80) and Lin et al. (119) correlated their results as follows:

where Mis a constant equal to 0.093 (80) or to 0.089 (119). The coefficient Mgiven by Verlaan et al. is in exact agreement with the theoretical relationship derived by Murakami et al. (118) for Bo > 50 and a degree of inhomogeneity, I = 0.05.

Equation 28 shows that the circulation path, which enters in the definition of Bo, has a linear effect on the mixing time. If the mean circulation time and the axial dispersion coefficients are known, it is possible to theoretically estimate the mixing time using equation 28. Experimental details must, however, be carefully planned to avoid complications. Note that in order to simplify data processing it is important to inject the signal and to measure the response at exactly the same point (120) (often the position of the injection point is not specified despite its effect on the mixing time). In a study of the effect of the injection point on the dynamics of the mixing time, Schugerl et al. (12) concluded that the gas-liquid separator is the best choice for tracer injection for short mixing times. Fields and Slater (114) reported a marked dependence of the respiratory quotients upon the injection point of methanol during unlimited fed-batch growth of Methylophilus methylotropus in a concentric-tube ALR. The lines in Figure 19 show experimental data for Bo (as overall values) reported for different types of ALR (in which the reactor is considered as a single unit). The dimensions of the reactors are given in Table 4. Because of the definition of the overall Bo, the values are specific to the reactor for which they were obtained and can be used only as indication of trends and orders of magnitude.

As explained above, the ALR is, in fact, a combination of several regions having quite different fluid dynamic characteristics. The overall mixing is the result of the contributions of each of them, and the overall Bo represents tm _ MBo

Table 4. Bodenstein Number as a Function of the Superficial Gas Velocity (Key for Fig. 19)

Curve Ref.

Table 4. Bodenstein Number as a Function of the Superficial Gas Velocity (Key for Fig. 19)

Curve Ref.







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