Multicomponent convection-reaction with first-order chemical kinetics

In this exercise we will consider the unsteady convection-reaction equation for three components:

where the index $i$ refers to one of the 3 components $A$, $B$ or $C$. All species move with the (same constant) velocity. The same kinetics as in exercise 1.2 will be used:

For unsteady convection-reaction consider a first-order temporal discretization where the convection term is evaluated explicitly (using e.g. a TVD scheme) and the reaction term implicitly, i.e. an Implicit-Explicit scheme:

Here the bold-vector notation is used to indicate column vectors of 3 long where the 3 elements correspond to the 3 species. The subscript $j$ indicates the spatial cell and the superscript $n$ the time step.

Suggested approach:

• Use a two dimensional array $c[i,j]$ where the first index, $i=0,..., N-1$, refers to the spatial positions and the second one, $j=0,1$ and 2 , the index of the species.

• Compute the convection term using the concentration on the old time-step

• Next loop over all grid cells $j=1...N$ and in each cell solve (with $\mathbf{r} = \mathbf{J\ c}$):

• Please, notice how close this is to the problem solved in the second part of exercise 1.2.2). Reuse the part of your code developed there.

Questions:

1. Implement the unsteady convection-reaction equations.

2. Perform simulations using the same kinetics as in exercise 1.2. Consider an initially empty column of length $L = 1~\mathrm{m}$. At the inlet feed pure $A$ with $c_{A,in} = 1~\mathrm{mol\cdot m^{-3}}$. Vary the residence time by changing the velocity $v$.

3. Plot concentration profiles in the column for the three species at different times.