Mass Transfer Effects: Immobilized and Heterogeneous Reaction Systems - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 859

17.7. External and Internal Surface
Effects

17.8.2. Time Required to Completely
Dissolve a Porous Sphere Full
of Fast-Reactive Materials

899

903

17.8. The Shrinking Core Model 900

17.8.1. Time Required to Completely
Dissolve a Porous Slab Full
of Fast-Reactive Materials

17.9. Summary 904

Problems 908

902

With the exception of oxygen transfer, we have so far implicitly treated the bioreactions as
homogeneous. In the case of multiphase systems, we have assumed that the phases are well
mixed and there is no transport limitation between phases. Therefore, effectively, mass trans-
fer effects have not been considered in our analyses other than oxygen transfer.
Ordinarily two phases are involved: gas/liquid, gas/solid, or liquid/solid. Bioreactions
frequently involve three phases: gas/liquid/solid. For example, oxygen is supplied from
gas phase, CO 2 produced is released into gas phase. The media are commonly liquid, or
suspensions. Fermenting organisms may be viewed as solid (to be suspended into the media).
Heterogeneous catalytic reactions by nature involve a serial transport/reaction rate
process, since the reaction species (substrates) must be transported to and removed from
the surface site where the chemical transformation occurs. Therefore, mass transfer plays
an important role in the reaction kinetics. In this chapter, we examine the effect of mass trans-
fer on the overall rate of a reaction system.

17.1. M OLECULAR DIFFUSION AND MASS TRANSFER RATE

Mass transfer usually refers to any process in which diffusion of species plays a role. Diffu-
sion of species is the spontaneous intermingling or mixing of atoms or molecules by random
“thermal” motion. It gives rise to motion of the species relative to motion of the mixture as
a whole. In the absence of other gradients (such as temperature, electric potential, pressure,
or gravitational potential), molecules of a given species within a single phase will always
diffuse from regions of higher concentrations to regions of lower concentrations due to the
random motion of the molecules. This concentration gradient results in a molar flux of the
species, which is described in general by the Fick's law,
J A ¼D A CVx A

(17.1)
where J A is the diffusional flux of species A, D A is the molecular diffusivity (against another
species B: D AB as in binary diffusion), C is the total concentration, and x A is the volume frac-
tion of species A in the mixture. Commonly, the volume fraction of A is understood as the
same as the mole fraction, which hold true in gaseous phase.
Table 17.1 shows the magnitude of diffusivity in gaseous phase, liquid phase, and solid
phase, and their relationships with respect to temperature and pressure. One can observe
that the diffusivity is lowest in solid phase and highest in gas phase.
When flow-induced diffusion or dispersion is dominant, mass transfer is much faster than
the molecular diffusion alone. Thermal randommotion is limited by the system temperature.