Autothermal Reactor

In this autothermal reactor, an exothermic reaction takes place in a packed bed of catalyst (a plug flow tubular reactor). The feed gas enters an annular region around the packed bed, but flows in the opposite direction from that in the packed bed. Some energy from the reaction in the packed bed is removed by heat transfer through the reactor wall and preheats the gas in the annular region. The feed temperature, Tf, is known, but the temperature at the entrance to the packed bed, To, must be guessed in order to obtain initial conditions for the ODEs. This value is correct when the entrance temperature to the annular region equals Tf; this solution approach is known as a shooting method. In this Demonstration, a slider is used to change the To value (the blue circle in the graph) until the correct Tf value (the red circle in the graph) is obtained and the initial conditions are satisfied. The ODE for the annular region is solved by integrating against the direction of flow, and the signs in the ODE are adjusted accordingly. Two solutions are possible, one of which corresponds to essentially no reaction. Which solution is obtained depends on how the reactor is started up. 

Download the CDF file to view the simulation using the free Wolfram CDF player.




Details
Material balance on packed bed reactor: 
where FA and CA are the molar flow rate (mol/s) and concentration (mol/m3) of A, z is the distance down the reactor (m), r is the cross-sectional radium of the reactor (m), k is the rate constant (1/s), k0 is a pre-exponential factor (1/s), Ea is the activation energy (J/mol), R2 is the ideal gas constant (J/[mol K]), T is temperature (K), v is volumetric flow rate (m3/s), FA,0 is the inlet molar flow rate of A (mol/s), R is the ideal gas constant ([m3 bar]/[mol K]), and P is pressure (bar).

Energy balance on packed bed:

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.