Homogeneous Reaction Parameters

The parameters of homogeneous/bulk reactions can be provided through echemfem.EchemSolver.homog_params, a list containing one dictionary for each reaction. Below is a list of different keys that should appear in each dictionary

• Key:

  "stoichiometry"

  Type:

  dict of int

  Description:

  This entry defines the stoichiometry of the reaction. Each key in this dictionary should be the name of a species (as defined in the "name" key values of conc_params). The corresponding value is the stoichemetry coefficient for the species in this reaction. It should be an integer: negative for a reactant, positive for a product.

• Key:

  "forward rate constant"

  Type:

  float or Firedrake expression

  Description:

  Rate constant for the forward reaction.

• Key:

  "backward rate constant" or "equilibrium constant" (optional)

  Type:

  float or Firedrake expression

  Description:

  Rate constant for the backward reaction or equilibrium constant of the reaction. For a reversible reaction, only of one these is required. For an irreversible reaction, leave unset.

• Key:

  "reference concentration" (optional)

  Type:

  float or Firedrake expression

  Description:

  Value of 1M in the appropriate units. By setting this, the rate constants are assumed to have the same units (typically 1/s), and the equilibrium constant to be nondimensional.

• Key:

  "reaction order" (optional)

  Type:

  dict of int

  Description:

  This entry can be used to enforce reaction order. Each key in this dictionary should be the name of a species (as defined in the "name" key values of conc_params). The corresponding value (positive integer) is reaction order of the species in this reaction. If unset, the absolute value of the stoichiometry coeffcient is the reaction order.

Assuming an arbitrary homogeneous reaction system where the forward and backward rate constants for reaction \(i\) are \(k_i^f\) and \(k_i^b\), respectively, and the stoichiometry coefficient of species \(j\) in reaction \(i\) is \(s_{j,i}\), then, according to the law of mass action, the reaction for species \(j\) is given by

\[R_j = \sum_i s_{j, i} \left( k_i^f \prod_{n, s_{n, i} < 0} c_n^{-s_{n, i}} - k_i^b \prod_{n, s_{n, i} > 0} c_n^{s_{n, i}} \right),\]

where \(c_n\) is the concentration of species \(n\). If the equilibrium constant \(K_i\) is provided instead of the backward rate constant, then it is automatically recovered via \(k_i^b = \dfrac{k_i^f}{K_i}\). The above formula assumes single-step reactions so that the reaction orders are given by the stoichiometry coefficients. If that is not the case, the exponents can be replaced by the "reaction order" inputs.

Note that here the rate constants can have different units, depending on the number of reactants or products. It is also common to write the reactions in terms of dimensionless “activities” instead of concentrations. Then, the rate constants have the same units (typically 1/s) and the equilibrium constants are dimensionless. With this definition of rate constants, we instead write the reaction for species \(j\) as

\[R_j = \sum_i s_{j, i} c_\mathrm{ref} \left( k_i^f \prod_{n, s_{n, i} < 0} a_n^{-s_{n, i}} - k_i^b \prod_{n, s_{n, i} > 0} a_n^{s_{n, i}} \right),\]

where \(a_n = c_n / c_\mathrm{ref}\) is the activity of species \(n\) and \(c_\mathrm{ref} = 1\text{M}\) is a reference concentration, which needs to be provided in the correct units to homog_params.

Here is an example of a reaction system that can be implement via homog_params. Simplified CO2 - bicarbonate homogeneous reactions:

\[ \begin{align}\begin{aligned}\mathrm{CO}_2 + \mathrm{OH}^- \xrightleftharpoons[k_1^b]{k_1^f} \mathrm{HCO}_3^-\\\mathrm{HCO}_3^- + \mathrm{OH}^- \xrightleftharpoons[k_2^b]{k_2^f} \mathrm{CO}_3^{2-} + \mathrm{H}_2\mathrm{O}\end{aligned}\end{align} \]

Here, we assume the dilute solution case where \(\mathrm{H}_2\mathrm{O}\) concentration is not tracked. In conc_params, we will have entries for the other species in this reactions, with "name" key values: "CO2", "OH", "HCO3", and "CO3". We get the following homog_params list:

[{"stoichiometry": {"CO": -1,
                    "OH": -1,
                    "HCO3": 1},
  "forward rate constant": k1f,
  "backward rate constant": k1b
 },
 {"stoichiometry": {"HCO3": -1,
                    "OH": -1,
                    "CO3": 1},
  "forward rate constant": k2f,
  "backward rate constant": k2b
 }]

Now here is another bicarbonate approximation with a high pH approximation, where the reaction system is simplified into a single irreversible reaction:

\[\mathrm{CO}_2 + 2\mathrm{OH}^- \xrightarrow[]{k_1^f} \mathrm{CO}_3^{2-} + \mathrm{H}_2\mathrm{O}\]

Since this is not actually a single-step reaction, we need to specify the reaction order for \(\mathrm{OH}^-\). We can use the following homog_params list:

[{"stoichiometry": {"CO2": -1,
                   "OH": -2,
                   "CO3": 1},
 "reaction order": {"OH": 1},
 "forward rate constant": k1f
 }]

As an alternative to the homog_params interface, the reactions can be written directly and passed as a function in physical_params["bulk reaction"] (Physical Parameters). See examples/cylindrical_pore.py for the direct implementation of the high pH approximation of the bicarbonate bulk reactions. See examples/carbonate.py and examples/carbonate_homog_params.py for two equivalent implementations of the full bicarbonate bulk reactions.