Details

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:

, .

, , , , , and ,

with user-selected direction of coolant and values of , and .

Reference

[1] L. A. Belfiore, Transport Phenomena for Chemical Reactor Design, Hoboken, NJ: John Wiley & Sons, 2003.