: substrate concentration (mg/L)

: inhibition constant (mg/L)

: maximum specific growth rate

)

: saturation constant (mg/L)

: oxygen saturation constant (mg/L)

: biomass concentration (mg/L)

: dissolved oxygen concentration (mg/L)

: critical dissolved oxygen concentration (mg/L)

: dissolved oxygen saturation concentration or solubility of oxygen in the broth (mg/L)

: gassing rate (L/min)

: stirrer speed

)

: feed substrate rate (L/h)

: proportional control constant

: inhibitory substrate concentration (mg/L)

: dilution rate

)

Inhibitory substrates at high concentrations reduce

, the specific growth rate, below that predicted by the Monod equation. The empirical inhibition function can be written:

.

If substrate concentrations are low, the term

is smaller than

and

, and the inhibition function is represented by coupled Monod equations [2]. The plots show inhibition of the oxygen uptake and specific growth rates.

The oxygen mass transfer rate

, could be represented by [1]:

.

The transfer coefficient

, varies with

and

according to [2]:

with

.

Proportional control of the feed rate is based on the exit concentration using:

where

is the error, represented by

. If

is zero, then

takes the value

and the process runs out of control [2].

[1] P. M. Doran,

*Bioprocess Engineering Principles*, Boston: Elsevier, 1995.

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

*Biological Reaction Engineering*, 2nd ed., Weinheim, Germany: VCH Verlagsgesellschaft mbH, 2003.