ver-1gc.md

ver-1gc

Series Chemical Reactions

Problem set up

This verification problem is taken from !cite and builds on the capabilities verified in ver-1g for simple chemical reactions. This case is simulated in [/ver-1gc.i].

This problem models a set of chemical reactions in series with three species: $A$, $B$, and $C$. The system was configured so that the enclosure initially contained only species $A$. At time $t \geq 0$ s, the reactions were allowed to proceed. The reactions that were modeled are

\begin{equation} \label{eq:chemical_reaction} A \xrightarrow{\mathit{k_1}} B \xrightarrow{\mathit{k_2}} C, \end{equation} with $k_1$ and $k_2$ the reaction rates.

The concentration of each species is therefore described as \begin{equation} \label{eq:chemical_reaction_reaction_A} \frac{dc_A}{dt} = - k_1 c_A, \end{equation} \begin{equation} \label{eq:chemical_reaction_reaction_B} \frac{dc_B}{dt} = k_1 c_A - k_2 c_B, \end{equation} \begin{equation} \label{eq:chemical_reaction_reaction_C} \frac{dc_C}{dt} = k_2 c_B, \end{equation} with $c_i$ the concentration of species $i$.

Analytical solution

!cite provides the analytical equations for the time evolution of the concentrations of $A$ and $B$ as \begin{equation} \label{eq:chemical_reaction_solution_A} c_A(t) = c_{A0} \exp\left( -k_1 t\right), \end{equation} and \begin{equation} \label{eq:chemical_reaction_solution_B} c_B(t) = k_1 c_{A0} \frac{\exp\left( -k_1 t\right) - \exp\left( -k_2 t\right)}{k_2-k_1}, \end{equation} where $t$ is the time in s, $c_{A0} = 2.415 \times 10^{14}$ atoms/m$^3$ is the initial concentration of species $A$, $k_1 = 0.0125$ s$^{-1}$, and $k_2 = 0.0025$ s$^{-1}$.

The concentration of $C$ was found by applying a mass balance over the system in !cite. From the stoichiometry of this reaction it was found that \begin{equation} \label{eq:chemical_reaction_solution_C} c_C(t) = c_{A0} - c_A(t) - c_B(t). \end{equation} The time evolution of the species concentrations from the analytical solution is provided in [ver-1gc_comparison_diff_conc].

Results and comparison against analytical solution

The comparison of TMAP8 results against the analytical solution is shown in [ver-1gc_comparison_diff_conc]. The match between TMAP8's predictions and the analytical solution is satisfactory, with root mean square percentage errors (RMSPE) of RMSPE = 4.78 % for species $A$, RMSPE = 3.20 % for species $B$, and RMSPE = 0.38 % for species $C$. The larger values of RMSPE for species $A$ and $B$ are due to the small average values of these concentrations over time, and do not reflect on the accuracy of the TMAP8 solve itself.

!media comparison_ver-1gc.py image_name=ver-1gc_comparison_diff_conc.png style=width:50%;margin-bottom:2%;margin-left:auto;margin-right:auto id=ver-1gc_comparison_diff_conc caption=Comparison of partial pressures of species in a series reaction predicted by TMAP8 and provided by the analytical solution. The RMSPE is the root mean square percent error between the analytical solution and TMAP8 predictions.

Input files

!style halign=left The input file for this case can be found at [/ver-1gc.i], which is also used as test in TMAP8 at [/ver-1gc/tests].

!bibtex bibliography