Here

,

and

are the species concentrations;

and

are the temperatures of the reactor and the heat exchange fluid, respectively;

is the reactant fluid density;

is the maximum laminar parabolic velocity in the reactor;

is the heat capacity of the reactants;

and

are the axial and radial coordinates, while

stands for the reactor radius;

and

are the pre-exponential factors, and

and

are the activation energies of the two consecutive reactions;

is the gas constant;

is the concentration of reactant

entering the reactor;

is the diffusion coefficient;

is the thermal conductivity;

is the temperature of the fluids entering both the reactor and the heat exchanger;

is the product of the heat transfer coefficient and the heat exchanger surface area; and the suffixes in the ordinary differential equation refer to properties of the fluids flowing in the heat exchanger

and the reactor

.