The "dynamics and feedback control" plot shows the startup of the bioreactor, initially under batch growth conditions (between hour 0 and hour 3.5). Feeding of the substrate starts after hour 3.5. Between hour 3.5 and hour 15, the setpoint for cellular concentration is 2.8 g/L. After hour 15 the setpoint is

, trying to control

near the maximum of biomass concentration. This is done to observe the response of the controlled reactor to a step change in

.

The "rates" plot shows the steady state, at which the specific growth rate becomes equal to the dilution rate

[1].

: substrate concentration (g/L) (orange line)

: specific growth rate (1/h) (yellow line)

: maximum specific growth rate (1/h)

: saturation constant (g/L)

: biomass concentration (g/L) (blue line)

: maximum biomass concentration (g/L) (green dashed line)

: proportional controller constant (

)

: integral control time constant (h)

: setpoint for cellular concentration (g/L) (red dashed line)

: dilution rate (1/h) (black line)

: biomass-substrate yield (kg/kg)

: feed substrate rate (L/h)

The specific growth rate

predicted by the Monod equation has the following form [1]:

.

Consult either reference for the model for a continuous bioreactor.

The turbidometer is modeled by a proportional-integral control of the feed rate of substrate concentration and can be set by:

where

is the error and is represented by

. If

takes the value zero, then

takes the value

and the process runs out of control [2].

Snapshot 1: turn off the controller (

). Note that

is influenced by the

at steady state.

Snapshot 2: turn on the controller (

). Note that

begins to approach

.

Snapshot 3: vary the integral control time constant to see its effect

Snapshot 4: vary the flow rate to see its effect

[1] P. M. Doran,

*Bioprocess Engineering Principles*, 2nd ed., Boston: Academic Press, 2013.

[2] I. J. Dunn, E. Heinzle, J. Ingham and J. E. Prvenosil,

*Biological Reaction Engineering: Dynamic Modelling Fundamentals with Simulation Examples*, 2nd, completely rev. ed., Weinheim, Germany: VCH Verlagsgesellschaft mbH, 2003.