Figure 7. Inter and intraparticle mass transfer of a single porous spherical bead of radius R. Substrate concentration profiles across the stagnant liquid film and inside the solid particle, S(r). Sf, concentration on the bulk liquid; Ssur, concentration on the solid surface; K, partition coefficient. Source: From Ref. 11.

ternal surface. This external mass transfer is usually modeled by means of a hypothetical liquid film that creates resistance to the transport, and it is characterized by the external mass transfer coefficient. The value of the coefficient, and thus of the external mass transfer velocity, depends on the physical properties of the liquid and the superficial velocity of the liquid around the solid; thus beds with more mixing will present better conditions for external mass transfer. When external mass transfer resistance is negligible, the concentration at the surface of the particle is the same as in the fluid phase (Sf = Ssur, according to Figure 7). An additional possibility is the occurrence of partitioning phenomena in the substrate, between the particles and the liquid, as a result of the material properties. In that case, the ratio between the substrate concentration at both sides of the solid-liquid interface will be given by the partition coefficient, K. Equally, partitioning phenomena can be determined for the product. Once in the solid particle, the substrate will diffuse inside the solid, following Fick's law, and simultaneously the reaction will take place. Therefore the corresponding equations of kinetic reaction and diffusion in the solid have to be solved simultaneously (10,11) in order to obtain the internal concentration profiles. To define this part correctly, the values for the intrinsic kinetic parameters of the reaction (that is, in absence of mass transfer limitations) and the effective dif-fusivities for the substrate and the product of the reaction in the solid have to be known. Moreover, the substrate conversion implies product generation, and this can affect the kinetics (for example, by product inhibition); usually the diffusion and reaction analysis is made simultaneously for both substrate and product. The overall activity of the bio-

catalytic particles will be dictated by the relative velocities of the two phenomena taking place inside them: reaction and diffusion. Systems with low diffusion rates with respect to the reaction rates will be diffusion controlled; on the other hand, systems with high diffusion rates with respect to the reaction rate will be controlled by the reaction. In the first case, the low diffusion will limit the efficiency of the particle because the reaction potential of the immobilized biocatalyst will not be fully used. Usually, this is reflected in terms of the efficiency factor, which is defined as the ratio between the actual reaction rate occurring in the system, and the reaction rate that would occur if no diffusional limitations existed, that is, when all solid particles would present a uniform concentration, equal to that of their surface. The relationship between the effectiveness factor and some moduli, such as the observable modulus, being proportional to the ratio between reaction and diffusion rates, is given in Figure 8. One of the interesting properties of such graphs is that they are very similar for different types of geometries and kinetics. Therefore they enable a direct analysis of the degree of diffusional limitations in a given type of particle in a fluidized bed and also suggest quantitative modifications to be performed to avoid such limitations, for example, changes in the particle size or diffusion conditions. In general terms, fluidized beds are interesting with regard to these aspects because since they require comparatively small-diameter particles for better fluidization, the potential diffusion limitations are reduced.

As mentioned at the beginning of this section, the combination of the different aspects we discussed allows the elaboration of mathematical models describing the behavior of fluidized-bed bioreactors. The reliability of these models depends greatly on the accuracy of the determination of the various parameters involved in the definition of the reactor and the reaction system (flux model, intrinsic kinetic parameters, external mass and heat transfer coefficients, internal effective diffusivities), and they will serve for various purposes such as conceptualization and understanding of the bioreactor itself (an important aspect required for building the model), design of similar bioreac-tors (especially in the case of scaling-up), simulation of the bioreactor operation at different conditions, and bioreactor control. As an example of how the output of a mathematical model can describe the internal concentration profiles in a fluidized-bed bioreactor, the data corresponding to a continuous fermenter for the production of ethanol from glucose by the bacteria Zymomonas mobilis immobilized in carrageenan beads (10) are given in Figure 9.

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