The ideal semi-batch reactor

In actual practice pure batch reactors are rarely used, but semi-batch reactors find very wide application. They have the obvious advantage that the reaction rate can be controlled by the feed rate of one reactant. 

F is the volume flow containing reactant B with concentration c.  

This is a set of coupled differential equations. They can be solved with the help of an additional mass balance that describes the relation between the reactant concentrations c, and c at the time t  

By eliminating c from the first and c from the second differential equation, and 

Obviously the solution of these differential equations is not simple, but a numerical calculation is always possible if all the constants are known. Using a fitting procedure one can find the rate constant from measurements of one of the concentrations with time. 

---