Our bioreactor is similar to a traditional packed-bed reactor. In packed bed reactors, porous pellets support a small catalyst pellet at the center. Likewise, in our system, inert alginate beads support E. coli - effectively a catalyst for microcystin breakdown. For a fluid carried through the bioreactor, there are three steps for mass transport and the reaction.
1. Mass transfer through the boundary layer
2. Molecular diffusion down the length of the pore
3. Reaction
We modeled mass transfer through the boundary layer using a simple mass transfer model. The model is based on correlated data for forced convection through boundary layers.
Where 𝚽 is the mass flux and k is the mass transfer coefficient. The mass transfer coefficient was determined by correlation.
Here, D is the diffusion constant for microcystin-LR, d is the pellet diameter, Re is the dimensionless Reynolds number, and Sc is the dimensionless Schmidt number. The Reynolds number is related to the fluid flow regime - a ratio of the inertial to viscous forces in the fluid. The Schmidt number relates the resistance of momentum diffusion to mass diffusion. We found that the diffusion constant D = 1.4 × 10-6 cm2/s according to Zastepa et. al (1).
We anticipated that the boundary layer diffusion would likely not be the limiting factor in our system due to the relatively inviscid behavior of water. Regardless, we aimed to introduce mixing and eddy flow in the reactor to induce turbulent flow, since the mass transfer coefficient is proportional to the Reynolds number (turbulent flows are characterized by high Reynolds numbers). Figure 1 illustrates the dependence of the mass transfer coefficient on the degree of turbulence, or mixing, in the system.
We coupled our modeling for molecular diffusion and the actual reaction - the classical treatment by chemical engineers. The E. coli are encapsulated in porous alginate pellets. The microcystins, once they diffuse through the boundary layer, must diffuse down the length of a pore and then react in the bacteria. We approximated the reaction as a first-order reaction. Michaelis-Menten kinetics, commonly used in biological systems, is approximately a first-order reaction at low substrate (microcystin) concentrations.
To relate the reaction rate to the timescale on which pore diffusion takes place we introduce a new dimensionless group, the Thiele modulus (2).
R is the pellet radius, D is the diffusion constant for microcystin, and k’ is the rate constant divided by the total pellet volume. In a randomly packed reactor, the total pellet volume is approximately 64% of the total reactor volume.
We can relate the Thiele modulus to another group called the effectiveness factor, the ratio of the actual reaction rate to the ideal reaction rate. The effectiveness factor tells us how much mass transfer resistance (on a molecular level) harms the overall reaction rate. Analytically, the effectiveness factor is
Where coth is the hyperbolic cotangent function. Figures 2 and 3 predict how the effectiveness factor varies with the Thiele modulus and pellet diameter, respectively.
Our modeling has informed our decision to minimize the size of the pellet to decrease the characteristic time for molecular diffusion down the length of the pore. We made the pellets as small as we reasonably could (3 mm diameter). We lacked the ability with our available tools to make smaller pellets; however, decreasing the pellet size further should be a trivial matter with more advanced manufacturing techniques and greater control over the process.
1. Zastepa, A., Pick, F. R., & Blais, J. M. (2017). Distribution and flux of microcystin congeners in lake sediments. Lake and Reservoir Management, 33(4), 444-451.
2. Levenspiel, O. (1999). Chemical Reaction Engineering (3rd ed.). John Wiley and Sons.