Mass balance for reactant

:

,

where

is the initial concentration of

,

is the dimensionless variable,

(s) is the residence time of the reactive fluid,

stands for the pre-exponential rate factor

,

is the activation energy

,

is the gas constant

, and

is the temperature of the reactants

.

Thermal balance for reactive fluid within the inner pipe:

,

where

is the average reactants density

,

is the heat capacity

,

is the inner pipe overall heat transfer coefficient

,

is the radius for the inner pipe

,

is the coolant temperature

, and

is the heat of reaction

.

Thermal balance for concurrent cooling fluid in the annulus:

.

The equation for countercurrent cooling differs from the above equation by a negative sign because integration is opposite to the direction of flow. Here

is the coolant density

,

is the coolant heat capacity

,

is the ratio of the velocity of the coolant to reactant fluid,

is the outer pipe overall heat transfer coefficient

,

is the radius of the outer pipe

,

is the ratio

, and

is the ambient temperature

.

This split boundary value problem is solved with:

,

.

with user-selected direction of coolant and values of

,

and

.

[1] L. A. Belfiore,

* Transport Phenomena for Chemical Reactor Design*, Hoboken, NJ: John Wiley & Sons, 2003.