Consider an irreversible first-order reaction
taking place in a batch reactor. The production of
proceeds in cycles, each cycle being composed of a reaction period (time
) and a dead time
. The dead time entails preparation of the reaction mixture before the reaction begins (filling the reactor and heating the mixture to the desired temperature, etc.) and preparation of the reactor for the next reaction period after the reaction time
(cooling of the reaction mixture to stop the reaction, emptying the reactor, and cleaning the reactor).
This Demonstration shows plots of the conversion, production, and cost curves as functions of time. The green dot corresponds to the maximum production option, while the cyan dot represents the maximum profit (i.e., the minimum cost function). The cyan and green dots are obtained by making the orange and red lines tangent to the conversion versus time curve (i.e., the blue curve). The cost function,
, is given by:
are the cost coefficients for the reaction and dead time and
is the cost of miscellaneous expenses. The dead time
is taken to be equal to 1 hour. It is clear that the two possible operating points (maximum profit and maximum production) do not coincide. Depending on the value of the parameters
, we conclude that it is often preferable to minimize costs even if this leads to smaller production, as can be seen in the cost versus time curve.