Steady state 1D fixed bed reactor model: first order exothermal reaction

In a fixed bed catalyst tube of $0.5~\mathrm{m}$ long a heterogeneously catalyzed exothermic (gas phase) chemical reaction takes place. The reaction scheme and associated kinetics are as follows:

where $k_{0} = 1.0 \times 10^{9}~\mathrm{s^{-1}}$, $E_{a} = 50~\mathrm{kJ \cdot mol^{-1}}$. The reaction is exothermic with $\Delta H_{r} = 15~\mathrm{kJ \cdot mol^{-1}}$ at the reference temperature of $293~\mathrm{K}$. The molar heat capacities of the gases equal $100, 60$ and $40~\mathrm{J \cdot (mol \cdot K)^{-1}}$, for $A$, $B$ and $C$, respectively.

Here we will model the reactor as adiabatic. (Of course the reactor should be cooled but this will be considered in a later tutorial). The inlet gas stream consists of pure $A$ and is at a pressure of $1~\mathrm{bar}$ and a temperature $293~\mathrm{K}$. The interstitial inlet gas velocity $v$ equals $2.0~\mathrm{m \cdot s^{-1}}$.

Questions:

1. Formulate the stationary convection-reaction equations for species $A$, $B$ and $C$ as well as for the temperature assuming a constant gas velocity.

2. Solve the set of ordinary differential equations assuming constant gas velocity using a standard Python IVP solver.

3. Formulate the stationary convection-reaction equations for species $A$, $B$ and $C$ as well as for the temperature assuming a constant pressure throughout the reactor. Hint: be cautious about placing the velocity inside or outside of the spatial derivative.

4. Solve this set of ordinary differential equations assuming constant pressure using a standard Python IVP solver.

5. Are the results according to your expectations? Is the temperature increase consistent with the adiabatic temperature rise? Is the increase in velocity in accordance with expectations?

6. Try solving both sets of ODE’s using your own Euler backward solver.