This autothermal reactor consists of two concentric tubes. The feed enters an annular region between the tubes at temperature \(T_f\) and flows from right to left. The reactants are at temperature \(T_0\) when they enter the inner tube, which is a packed catalyst bed, where they flow left to right. An exothermic reaction in the catalyst bed raises the gas temperature and heat transfers from the catalyst bed to preheat the gas in the annular region. To solve the ordinary differential equations (ODEs) that model this reactor, a shooting method is used; a value of \(T_0\) (the blue circle) is guessed and then changed until the correct value of \(T_f\) is obtained. This is done by moving the slider for \(T_0\) until the red circle is inside the green circle. Two reactor conditions are possible (two solutions to the mass and energy balances), one of which corresponds to essentially no reaction. Reactor startup determines which condition is obtained.

Material balance on packed bed:

\[ \frac{dF_A}{dz} = -\pi\, r^2\, k\, C_A, \] \[ k = k_0\, e^{-E_a/(R\,T)}, \] \[ C_A = \frac{F_A}{v}, \] \[ v = \frac{F_{A,0}\,R_g\,T}{P}, \]

where \(F_A\) and \(C_A\) are the molar flow rate (mol/s) and concentration (mol/m³) of A, \(z\) is distance down the reactor (m), \(r\) is the cross-sectional radius of the reactor (m), \(k\) is the rate constant (1/s), \(k_0\) is a pre-exponential factor (1/s), \(E_a\) is activation energy (J/mol), \(R\) is the ideal gas constant (J/[mol K]), \(T\) is temperature (K), \(v\) is volumetric flow rate (m³/s), \(F_{A,0}\) is the inlet molar flow rate of A (mol/s), \(R_g\) is the ideal gas constant ([m³ bar]/[mol K]), and \(P\) is pressure (bar).

Energy balance on packed bed:

\[ \frac{dT}{dz} = \frac{-\Delta H\, 2\pi\, r^2\, k\, C_A + Q}{F_{A,0}\, C_p}, \] \[ Q = \frac{2\pi\, r^2\, U\, (T_a - T)}{r}, \]

where \(\Delta H\) is heat of reaction (J/mol), \(Q\) is heat transfer rate (J/s), \(C_p\) is mass heat capacity (J/[g K]), \(U\) is the overall heat transfer coefficient (J/[m² K s]), \(T_a\) is the temperature of the annular region (K), \(T_f\) is the feed temperature to the reactor (K), and \(T_0\) is the feed temperature to the packed bed (K).

Energy balance on annular region:

\[ \frac{dT_a}{dz} = \frac{Q}{F_{A,0}\, C_p}. \]

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.

This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com by John L. Falconer using Claude AI. It is a JavaScript/HTML5 implementation of a Mathematica simulation by Rachael L. Baumann and John L. Falconer. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988) in collaboration with Washington State University. Address any questions or comments to LearnChemE@gmail.com.