The partial differential equation describing the steady-state substrate mass conservation in the reactor:
stand for the inner and outer radii,
is the substrate,
are the axial and radial coordinates,
are the diffusivities of the fluids in the inner and outer cylinders,
is the laminar velocity in the inner tube,
is the maximum fluid velocity,
is a first-order catalyzed reaction, where
is the reaction rate constant.
The properties of the fluids in the two cylinders are assumed to be constant, and axial diffusion is neglected.
The integration limits are:
is the length of the reactor, with boundary values
The average concentrations of the substrate in the lumen and in the biocatalyst are:
The equations are solved with the built-in Wolfram Language function NDSolve.
You can use the sliders to vary the values of the fluid velocity, the reaction rate constant and the ratio of the outer to the inner radius to observe their effect on the substrate concentration in the reactor.