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 chemical reaction takes place in the gas phase. The reaction scheme and associated kinetics are as follows:

A \rightarrow B + C,\ \ \ \ \ \ \ r = k_{r}\ c_{A},\ \ \text{with}\ k_{r} = k_{0}\exp\left\lbrack - \frac{E_{a}}{RT} \right\rbrack,\ \ {- r}_{A} = r_{B} = r_{C} = r

(1)

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:

Formulate the stationary (spatial) convection-reaction equations for species $A$ , $B$ and $C$ as well as for the temperature assuming a constant gas velocity and model it using solve_ivp.
Solve the set of ordinary differential equations with a variable-velocity formulation using solve_ivp, while assuming constant gas pressure. (Hint: use the molar fluxes $F_i = c_i v$ as unknowns.)
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.
Solve this set of ordinary differential equations assuming constant pressure using a standard Python IVP solver.
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?
Try solving the constant-velocity and the constant-pressure models using your own backward Euler solver.