This Demonstration shows how one can control the temperature of a batch fermentor using a proportional-integral feedback controller. The governing equations are the following:

where

and

are the temperature of the fermentor and cooling water,

and

are the specific growth rate and its maximum value,

and

are the yield coefficient and heat yield for substrate,

is the saturation coefficient,

and

are the proportional gain and integral time constant of the controller,

and

are the heat transfer coefficient and area of the cooler,

is the inlet temperature of the cooling water,

is the flow rate of the cooling water,

and

are the volumes of the fermentor and the cooler,

and

are the densities of the fermentation media and cooling water,

is the fermentor's temperature set point (chosen equal to 25°C here),

and

are the biomass and substrate concentrations, and finally

and

are the heat capacities of the fermentation media and cooling water.

For a batch reactor, the substrate will be depleted after a certain time and the biomass will reach a constant plateau. If

and

(i.e., no temperature control), the temperature of the fermentor will rise to a relatively large value (≈ 26.5°C) as can be seen in the snapshot. If control is applied, the fermentor's temperature will stay around 25°C for almost the entire growth period (up to

), then will decrease as the fermentor cools down and the substrate concentration reaches a value very close to zero.