Material balance on packed bed:

,

,

,

,

where

and

are the molar flow rate (mol/s) and concentration (

) of

,

is distance down the reactor (m),

is the cross-sectional radium of the reactor (m),

is the rate constant (1/s),

is a pre-exponential factor (1/s),

is activation energy (J/mol),

is the ideal gas constant (J/[mol K]),

is temperature (K),

is volumetric flow rate (

),

is the inlet molar flow rate of

(mol/s),

is the ideal gas constant (

), and

is pressure (bar).

Energy balance on packed bed:

,

,

where

is heat of reaction (J/mol),

is heat transfer rate (J/s),

is mass heat capacity (J/[g K]),

is the overall heat transfer coefficient (

),

is the temperature of the annular region (K),

is the feed temperature to the reactor (K), and

is the feed temperature to the packed bed (K).

Energy balance on annular region:

.

The sign of the heat transfer term in the annular region is negative because integration is opposite to the direction of flow.

For the solution in which the reaction rate is nonzero, the feed temperature in the annular region increases as the feed flows to the left and is heated by the packed bed. The temperature increases in the packed bed due to the exothermic reaction and some of the energy transfers to the feed gas in the annular region. As the reactant is used up, the temperature in the packed bed reaches a maximum and decreases as heat is transferred to the feed gas.

The screencast video at [2] shows how to use this Demonstration.

[1] J. G. Ekerdt and J. B. Rawlings,

*Chemical Reactor Analysis and Design Fundamentals*, Madison, WI: Nob Hill Publishing, 2002 pp. 326–332.