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

Transport processes in biotechnological systems
Scale-up and mass transfer

As pointed out by Oldshue, when the Reactor volume or impeller speed is varied, the different parameters that can affect the k_la product change in different ways*.

* Oldshue J., Biotech. Bioeng., 8, 3 (1966).

1. The turbulent Reynolds number Re_t determines u_rms and, consequently, the bubble mass transfer coefficient k_l:

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2. The impeller tip speed πN_iD_i determines the maximum shear rate y, which in turn affects both the maximum size of stable bubbles and the size of microbial flocs (Sec. 8.3), and can also cause damage to viable Cells (Sec. 8.8.4).

3. The power input per unit volume, P/V, via Ret determines the mass transfer coefficient and the particle size of the dispersed phase. In the laminar and transition aeration regimes, according to Fig. 8.8, b,

In the turbulent regime, the power number is constant, therefore,

Then, assuming the reactor volume V is proportional to D_i³, we obtain

4. The power consumption during aeration is

Thus,

The latter expression allows the Determination of the motor power required to carry out a given microbiological process in a given bioreactor.

5. If the reactor contents are effectively mixed by an internal agitator, the system in the reactor must have a characteristic Circulation time. The liquid recirculation rate F_l in the impeller zone varies proportionally to the cross-sectional area πD_i², and the average velocity in the reactor is proportional to N_iD_i. Thus,

The value of N_i is of interest because it is inversely proportional to the time the liquid spends outside the zone of the homogenizing action of the impeller.

All of these parameters affect the process, and they all depend differently on the mixing conditions. This raises the question: which of these parameters should be used as the basis for scale-up? By scale-up basis, we mean a simple or complex quantitative parameter that, being valid for a smaller reactor, can be transferred without change to larger vessels. For example, if we choose constant power input per unit volume as the scale-up criterion, then for mechanical agitation in the turbulent regime, this means that

Here, subscripts 1 and 2 denote parameters in the small and large reactors, respectively. In early studies on industrial penicillin production, a constant power input per unit volume (about 1 hp per 379 L) was adopted as the scale-up criterion for bioreactors of similar geometry. As shown in Fig. 8.15, a, this scale-up criterion provides a fairly constant penicillin yield in reactors with capacities ranging from 5 to 757 L. At the same time, varying the P/V value causes the penicillin yield to fluctuate depending on the reactor scale.

Another frequently used scale-up criterion is a constant volumetric mass transfer coefficient k_la. Fig. 8.15, b shows the curves of the bacterial metabolite vitamin B₁₂ yield versus the corresponding k_lap values at different scales. Here, the complex parameter k_lap includes the total oxygen pressure, which accounts for the increased driving force of oxygen transfer at elevated pressure, typical of large-scale industrial bioreactors. A satisfactory correlation was achieved for the vitamin B₁₂ yield as a function of k_lap, although yields in small laboratory reactors are usually lower than in large-scale industrial reactors. Note also that, unlike the previously discussed situations where Cell growth was limited by oxygen transfer and higher k_la values were required, the curve of vitamin B₁₂ yield versus k_lap shows a distinct maximum. This is likely because, in reactors of different volumes at the same k_la, other mixing and liquid phase flow parameters (e.g., maximum shear rate or circulation time) may generally differ.

FIG. 8.15. a — penicillin yield as a function of power consumption in reactors of various volumes [after Gaden E., Sci. Rep. Ist. Super. Sanita, 1, 61 (1961)]; b — vitamin B₁₂ yield (μg/g) as a function of the complex mass transfer parameter k_lap (see text) [after Bartholomew W. H., Adv. Appl. Microbiology, 2, 289 (1960).]

It follows from the above that when scaling up under a single constant condition, we must keep in mind that other mixing characteristics and flow patterns will change. This is clearly illustrated by the data of Oldshue, who studied changes in process parameters during the scale-up of a stirred bioreactor from 80 to 10,000 L*. In this scale-up, D_i and V increase by a factor of 5 and 125, respectively. A value of 1.0 in each Column of Table 8.5 indicates that this particular parameter has the same value in both the small and large reactors. The values of each parameter are normalized with respect to the 80-liter reactor; therefore, in the 'small-scale reactor' column, all numerical values of the parameters are 1.0. The data presented in the table show that scale-up at constant P/V is accompanied by a 70% increase in the maximum shear rate (N_iD_i = 1.7) and an approximately threefold increase in circulation time. On the other hand, scale-up at constant circulation time F_l/V requires a 3125-fold increase in power in the large reactor!

Table 8.5. Relationship between various parameters during scale-up^a

^a Oldshue S. Y., Fermentation Mixing Scale-up Techniques, Biotech. Bioeng., 8, 3 (1966).

* Oldshue S. Y., Fermentation Mixing Scale-up Techniques, Biotech. Bioeng., 8, 3 (1966).

The diverse dependencies of key transport phenomena on the impeller design make scale-up in stirred-tank reactors somewhat of an art. In this process, we must attempt to select a parameter as the scale-up criterion that is governing in the given bioprocess. This problem, in turn, can present several difficulties, because transport phenomena, which depend on impeller size and rotational speed, can unpredictably affect shear-sensitive cells.

The overall Oxygen Uptake Rate in the reactor is determined based on k_la and the corresponding reactor design description. If the liquid phase is homogeneous in composition and the bubbles are uniformly distributed throughout the solution volume, then the mass transfer rate is given by a simple expression:

Vk_la(c* — cliq)_e = O₂ uptake rate, mol/s (8.111)

Here, the subscript e indicates that the exit gas composition is referred to in this case (if we assume that the reactor is perfectly mixed, then the COMPOSITION OF THE phase at the reactor outlet does not differ from the composition of this phase inside the reactor; Ch. 9).

If the bubbles in the reactor rise in a plug-flow regime, but the impeller still provides perfect mixing of the liquid phase, then c* varies depending on the bubble position in the reactor. The instantaneous oxygen loss by a bubble as it rises to a height dz is

or

or The rate of oxygen transfer from the gas phase to the liquid phase. Since p_O2 = Mc_l* (where M is the constant in Henry's law equation), then

At a constant bubble rise velocity u_b, dt = dz/u_b, and the dependence of c* on z is described by the equation

or

Consequently, the overall mass transfer rate in the volume Ah is equal to

The interaction between mixing, microbial kinetics, and mass transfer will be discussed in the remaining chapters of this book.