The mass balance for the reactant is:

with boundary conditions:

,

,

,

.

The velocity of the fluid in the annulus is [1]:

and the maximum velocity is

,

.

Here

stands for the reactant specific density,

is the concentration of reactant

,

is the reactant diffusivity,

is the pressure difference,

is the radius of the outer cylinder,

is the length of the reactor,

is the ratio of the UV lamp radius to the radius of the outer cylinder,

is the fluid viscosity, and the axial and radial variables are

and

, respectively.

The average (cup) concentration of the reactant is:

.

This system of equations is solved with the built-in Wolfram Language function

NDSolve and plots of the velocity profile, contours of the reactant concentration and the average (cup) conversion of the reactant are shown.

[1] R. B. Bird, W. E. Stewart and E. N. Lightfoot,

*Transport Phenomena*, 2nd ed., New York: J. Wiley, 2002.