RateE

Chemical Reactor Design Toolbox Reference Manual

ChemReactorDesign.Basic.Liquid.Rates.RateE

Description

The component determines the molar fluxes and the respective current due to a area based electrochemical reaction using a power law rate expression

$\begin{equation*} r = k(T) \, \left( \exp\left\{-\alpha \, n \, f \, \eta(T) \right\} \, \prod_{i}^{N} a_{i}^{\for{\kappa_{i}}} - \exp\left\{(1-\alpha) \, n \, f \, \eta(T) \right\} \, \prod_{i}^{N} a_{i}^{\back{\kappa_{i}}} \right) \end{equation*}$

formulated in terms of activities

$\begin{equation*} a_{i} = \frac{\gamma_{i} \, c_{i}}{1 \, \frac{mol}{l}} \end{equation*}$

with

$\begin{equation*} f = \frac{\cal F}{R \, T} \end{equation*}$

and

$\begin{equation*} n = \lvert \sum_{i}^{N} \nu_{i} \, z_{i} \lvert \end{equation*}$

The concentrations are obtained from the mole fractions and the temperature dependent molar volumes (c.f. getConc).

Temperature Dependent Parameters

Rate Constant

$\begin{equation*} k(T) = k_{\infty} \, \exp\left\{-\frac{E_{a}}{R \, T} \right\} \end{equation*}$
Open Loop Potential

$\begin{equation*} U_{0}(T) = - \frac{\sum\limits_{i}^{N} \nu_{i} \, \Delta G_{f_{i}}(T)}{n \, {\cal F}} \end{equation*}$
Overpotential

$\begin{equation*} \eta(T) = U-U_{0}(T) \end{equation*}$

Comments

Generally, the reaction is regarded to be reversible. Thus, the individual orders of reaction are calculated from the provided stoichiometric coefficients to ensure equivalence between thermodynamics and kinetics.

Variables

The molar fluxes are obtained as

$\begin{equation*} F_{i} = \nu_{i} \, A \, r \qquad \text{for} \quad i=1,\cdots,N \end{equation*}$

Since the heat of reaction, i.e. the energy change resulting from the change in composition, is implicitly accounted for in the balance equation of the respective volume component, it holds

$\begin{equation*} \Phi = 0 \end{equation*}$

The associated currrent is obtained as

$\begin{equation*} I = - {\cal F} \, \sum_{i}^{N} z_{i} \, F_{i} \end{equation*}$

Ports

Conserving

Liquid conserving port
```
Port_B = Liquid;  %
```

Electrical conserving ports

Port_p = Electrical;  %

Port_n = Electrical;  %

Input

Physical signal that represents the area
```
A = {0,'l'}; % A
```
Dependencies: The port is only visible when rateReference is set to Area.

Parameters

Options

Option to select calculation of the open loop potential
```
calculate_U0 = OnOff.Off;   
```
On | Off

Stoichiometry

Stoichiometric Coefficients
```
nu = {[-1; 2],'1'};   
```
Note Initially only two species are considered. As the number of species can be changed via the properties dialogue, the size of the array must be adjusted accordingly.

Thermodynamics

Equilibrium Potential
```
U0 = {0,'V'};
```
The parameter is only visible when the option calculate_U0 is set to Off.

Kinetics

Frequency Factor
```
kfinfA = {0,'mol/(cm^2*s)'};
```
Activation Energy
```
Ea = {0,'kJ/mol'}; 
```
Factor of Symmetry
```
alpha = {0.5,'1'};
```

Nomenclature

$A$	area
$a_{i}$	activity of species A_i
$c_i$	concentration of species A_i
$E_{a}$	activation energy
$F_{i}$	molar flow rate of species A_i
$\Delta G_{f_{i}}$	Gibbs free energy of species A_i
$I$	current
$k$	reaction rate constant
$n$	number of transferred electrons
$N$	total number of species
$r$	reaction rate
$R$	universal gas constant
$T$	temperature
$U$	potential
$U_{0}$	potential
$x_{i}$	mole fraction of species A_i
$z_{i}$	charge of species A_i
$\cal F$	Faraday constant
$\alpha$	symmetry factor
$\eta$	overpotential
$\nu_{i}$	stoichiometric coefficient of species A_i
$\for{\kappa}_{i}$	order of reaction of species A_i (forward reaction)
$\back{\kappa}_{i}$	order of reaction of species A_i (forward reaction)
$\gamma_{i}$	activity coefficient of species A_i
$\Phi$	energy flow rate