M7.1 Background

Biochemical reactors are used in a wide variety of processes, from waste treatment to alcohol fermentation. Biomass (cells) consume substrate (sugar or waste chemicals) and produce more cells. A typical control and instrumentation diagram, with biomass concentration as the measured output, is shown in Figure M7-1.

Model

The modeling equations for a bioreactor are

Equation M7.1

where the state variables are x₁ = biomass (cell) concentration = mass of cells/volume, and x₂ = substrate concentration = mass of substrate/volume.

The manipulated input is D = dilution rate = F/V = volumetric flow rate/reactor volume, and the disturbance input is x_2f = substrate feed concentration.

Two possible expressions for the specific growth rate are monod and substrate inhibition kinetics, which include

Notice that the monod-specific growth rate model is a subset of the substrate inhibition model (k₁ = 0).

Scale Up

One motivation for working with the dilution rate as the manipulated input is that the resulting dynamic model is independent of scale. A reactor volume of 1 liter with a flow rate of 0.3 liters/hour has the same dynamic behavior as a reactor volume of 1000 liters with a flow rate of 300 liters/hour. Thus, a small-scale (laboratory or pilot plant) reactor can be used to predict the behavior of a production-scale reactor.