Steady-state 2D fixed bed reactor model: first order exothermal reaction

This is the 2D version of exercise 1.3. In a fixed-bed catalyst-tube of 0.5 m long a heterogeneously catalyzed exothermic (gas phase) chemical reaction takes place. The reaction scheme and associated kinetics are as follows:

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

(1)

where $k_{0} = 1.0 \times 10^{9}\ s^{- 1}$ , E~a~=50 kJ/mol. The reaction is exothermic with --∆H~r~=15 kJ/mol at the reference temperature of 293 K. The molar heat capacities of the gases equal 100, 60 and 40 J/(mol K), for $A$ , $B$ and $C$ , respectively. The (constant) inlet gas stream consists of pure A and is at a pressure of 1 bar and a temperature 293 K. The (constant and uniform) interstitial inlet gas velocity $v$ equals 2.0 m/s.

Here we will model the reactor in a 2D fashion. The radius of the tubular reactor is 0.01 m. The reaction is highly exothermic and therefore the catalyst bed needs to be cooled at the wall of the tube. The imposed wall temperature equals 293.0 K. Effective thermal conduction and dispersion are modelled in the radial direction with values $\lambda_{e,r} = 0.1\ W/(m \cdot K)$ and $D_{e,r} = 10^{- 3}\ m^{2}/s$ , respectively.

Questions:

Make a 2D Python reactor model. Note that, because dispersion effects in the axial directions can be neglected, you can use the methods of lines by performing the discretization in the spatial direction and get a set of coupled ODE’s in the axial direction.
Calculate the adiabatic temperature rise.
Compute the (radially averaged) axial concentration and temperature profiles. Compare the results with the 1D profiles.
Compute the exit concentration and temperature profiles. Explain the observed radial temperature profile and explain how the maximum temperature can exceed the inlet temperature augmented with the adiabatic temperature rise.