RateES

Chemical Reactor Design Toolbox Reference Manual

ChemReactorDesign.Basic.Liquid.Rates.RateES

Description

The component determines the molar fluxes due to a heterogeneous electrochemical reaction involving species in the solid and liquid phase

$\begin{equation*} \sum_{i}^{N^{S}} \nu_{i}^{S} \, A^{S}_{i} + \sum_{i}^{N^{L}}} \nu^{L}_{k} \, A^{L}_{k} + n \, e^{-} = 0 \end{equation*}$

using a power law rate expression

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

with

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

and

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

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

since for any solid species it holds

$\begin{equation*} a_{i}^{S} = 1 \quad \text{and} \quad \kappa_{i}^{S} = 0 \end{equation*}$

The reference frame is set to the area of the electrode. 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.
As already mentioned, the reaction is of 0^th order for any solid species involved. Thus, the rate should become zero if the amount of the respective solid species approaches zero. Therefore, respective boolean indicators are defined

$\begin{equation*} {\for \lambda}^{S} = \left\{ \begin{array}{lcl} 0 & \text{if} & \sum\limits_{i}^{N} \left(x_{i}^{S} \leq 0 \; \& \; \nu_{i}^{S} < 0 \right) > 0 \\ 1 & \text{else} & \end{array} \right. \end{equation*}$

$\begin{equation*} {\back \lambda}^{S} = \left\{ \begin{array}{lcl} 0 & \text{if} & \sum\limits_{i}^{N} \left(x_{i}^{S} \leq 0 \; \& \; \nu_{i}^{S} > 0 \right) > 0 \\ 1 & \text{else} & \end{array} \right. \end{equation*}$

and incorporated in the rate expression for the forward and the backward reaction.

Variables

The molar fluxes for the liquid phase are obtained a

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

and for the solid phase

$\begin{equation*} F^{S}_{i} = \nu^{S}_{i} \, A \, r \qquad \text{for} \quad i=1,\cdots,N^{S} \end{equation*}$

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

$\begin{equation*} \Phi^{L} = 0 \qquad \text{and} \qquad \Phi^{S} = 0 \end{equation*}$

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_L = Liquid;  %
```
Solid conserving port
```
Port_B_S = Solid;  %
```

Electrical conserving ports

Port_p = Electrical;  %

Port_n = Electrical;  %

Input

Physical signal that represents the area
```
A = {0,'m^2'}; % A
```

Parameters

Options

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

Kinetics

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

Thermodynamics

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

Liquid

Stoichiometric Coefficients
```
nu = {[-1; 0],'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.

Solid

Stoichiometric Coefficients for Solid Phase
```
nu_S = {[1;0],'1'}; 
```
Notes Initially for the solid phase 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.

Nomenclature

$A$	area
$a_{i}$	activity of species A_i
$c_i$	concentration of species A_i
$E_{a}$	activation energy
$F^{L}_{i}$	molar flow rate of species A_i in liquid phase
$F^{S}_{i}$	molar flow rate of species A_i in solid phase
$\Delta G_{f_{i}}$	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
$\nu^{S}_{i}$	stoichiometric coefficient of species A_i in solid phase
$\for{\kappa}_{i}$	order of reaction of species A_i (forward reaction)
$\back{\kappa}_{i}$	order of reaction of species A_i (backward reaction)
$\gamma_{i}$	activity coefficient of species A_i
$\Phi^{L}$	energy flow rate in liquid phase
$\Phi^{S}$	energy flow rate in solid phase