This Demonstration illustrates a method to operate a semibatch reactor to avoid thermal runaways.
Consider the following liquid-phase irreversible exothermic reaction taking place in a semibatch reactor equipped with a cooling jacket: , 100 moles of reactant at a concentration of are placed in the reactor vessel, then a stoichiometric amount of is fed at a concentration of and at a constant volumetric flow rate. Thermal runaway is avoided by maintaining the reactor temperature below the temperature trajectory of an ideal case in which all the added species react instantaneously, .
For each reacting system and reactor configuration, the reactor performance can be defined with two control variables: the molar feed rate of the added reactant and the coolant temperature . If the cooling system is sufficient to absorb the heat generated due to the reaction, three characteristic reactor behaviors can be distinguished:
1. If the reactor temperature is always much lower than , a huge accumulation of unreacted reactants occurs. This process is potentially dangerous because in case of a stirring or cooling system failure, the reactor becomes similar to a batch reactor and thermal runaway can occur.
2. The thermal runaway, where the reactor temperature surpasses the target temperature and approaches much higher temperatures.
3. A case in which the operating temperature is close to the maximum temperature for most of the reacting time. This results in an almost instantaneous conversion and low reactant accumulation and it is the most desirable reactor performance.
where is the rate of reaction, is the rate constant, is the pre-exponential factor, is the activation energy, and are the concentrations of and , and is the temperature.
The reactant and product concentrations are:
, , , , where , , and are the number of moles in the reactor; and are the concentrations of the products; and is the time-dependent volume.
These are the material balances:
Here is the inlet volumetric flow rate, is the concentration of in the feed, is time, and is the initial volume of the semibatch reactor.
The energy balance is:
Here is the heat of reaction, and are the constant temperatures of the heat exchanger and the feed to the reactor, respectively, and stands for the heat capacity of the reactants; for a reactor equipped with a cooling jacket, the cooling capacity increases during the addition period:
, where is the initial heat exchanger cooling capacity.
Initial conditions for these differential equations are:
, , , , with .
In the ideal situation in a semibatch reactor, when the reaction rate is equal to the addition rate, the added reactant reacts away immediately. In that case, the reactor temperature follows a trajectory ; this temperature (see  for a detailed derivation) can be approximated by:
Here is the adiabatic temperature rise, is relative volume increase due to the addition of reactant (in this case ), is the heat capacity of the of the reactant initially in the reactor, is the density of the reacting mixture, is the Westerterp number, the ratio of the cooling capacity to the heat capacity of the reactor content both taken at the beginning of the process, and is the ratio of heat capacities of the added reactant to the reactant originally in the reactor .
Notice that during the dosing period the heat exchange surface area increases, thus the value of decreases, while at the end of the dosing it remains constant.
 K. R. Westerterp, M. Lewak, and E. J. Molga, "Boundary Diagrams Safety Criterion for Liquid Phase Homogeneous Semibatch Reactors," Industrial and Engineering Chemistry Research, 53(14), 2014 pp 5778–5791. doi:10.1021/ie500028u.