Rates

Chemical Reactor Design Toolbox Reference Manual

ChemReactorDesign.Basic.Interface.Rates.RatePG

Description

The component determines the molar fluxes due to a heterogeneous area based chemical reaction (interface-gas) using a power law rate expression

$\begin{eqnarray*} r & = & k(T) \\ & & \left( \left[ \for{\lambda} \, \prod_{i}^{N} a_{i}^{\for{\kappa_{i}}} \right]_{G} \, \left[ \for{\lambda} \, \prod_{i}^{N} a_{i}^{\for{\kappa_{i}}} \right]_{I} - \right. \\ & & \left. \frac{1}{K_{a}(T)} \, \left[ \back{\lambda} \, \prod_{i}^{N} a_{i}^{\back{\kappa_{i}}} \right]_G \, \left[ \back{\lambda} \, \prod_{i}^{N} a_{i}^{\back{\kappa_{i}}} \right]_I \, \right) \end{eqnarray*}$

formulated in terms of activities

$\begin{equation*} a^{I}_{i} = \Theta_{i} \qquad \text{and} \qquad a_{i}^{G} = \frac{\varphi_{i} \, x_{i} \, p}{1 \, bar} \end{equation*}$

Temperature Dependent Parameters

Rate Constant

$\begin{equation*} k(T) = k_{\infty} \, \exp\left\{-\frac{E_{a}}{R \, T} \right\} \end{equation*}$
The equilibrium constant $K_{a}$ is either regarded as constant or is retrieved for the given temperature from a lookup table provided by the user.

Comments

For an irreversible reaction the individual orders of reaction for the reactands can be arbitrarily chosen. For a reversible reaction, however, the individual orders of reaction are calculated from the provided stoichiometric coefficients to ensure equivalence between thermodynamics and kinetics.
In order to account that for a 0^th order reaction the rate should become zero if the amount of the stoichiometric limiting species approaches zero. Therefore, respective boolean indicators are defined for both domains

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

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

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

Variables

The molar fluxes are obtained as

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

and

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

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

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

Ports

Conserving

Interface conserving port
```
Port_B_I = Interface;  %
```
Gas conserving port
```
Port_B_G = Gas;  %
```

Input

Physical signal that represents the area
```
A = {0,'l'}; % A
```
Dependencies: The port is only visible when rateReference is set to Area.
Physical signal that represents the temperature
```
Tin = {298.15,'K'}; 
```
Dependencies: The port is only visible when temperatureInput is set to On.

Parameters

Options

Option to select the reversibility of the reaction
```
reversibility = Reversibility.Irreversible;
```
Irreversible | Reversible
Option to select calculation of the equilibrium constant
```
calculate_Ka = OnOff.Off;   
```
On | Off

The option is only visible when the option reversibility is set Reversible.
Option to select temperature input
```
temperatureInput = OnOff.Off;   
```
On | Off

Thermodynamics

Isothermal Temperature
```
T0 = {298.15,'K'};   
```
The parameters are only visible when the option temperatureInput is set to On.
Equilibrium Constant
```
Ka0 = {1.0e+30,'1'};
```
The parameter is only visible when the option reversibility is set to Reversible as well as the option calculateKa is set to Off.
Temperature Table
```
table_T = {[300,400],'K'}; % Temperatur Table
```
The parameter is only visible when the option reversibility is set to Reversible as well as the option calculateKa is set to On.
Equilibrium Constant Table
```
table_Ka  = {[1,2],'1'};   % Equilibrium Constant Table
```
The parameter is only visible when the option reversibility is set to Reversible as well as the option calculateKa is set to On.

Kinetics

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

Gas

Stoichiometric Coefficients for gas domain
```
nu_G = {[0 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.
Reaction Orders for Forward Reaction for Gas Domain
```
kappaf_G  = {[0;0],'1'};
```
The parameter is only visible when the option reversibility is set to Irreversible.

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.

Interface

Stoichiometric Coefficients for interface domain
```
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.
Reaction Orders for Forward Reaction for Interface Domain
```
kappaf = {[0; 0],'1'};  
```
The parameter is only visible when the option reversibility is set to Irreversible.

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.

Nomenclature

$A$	area
$a_{i}$	activity of species A_i
$E_{a}$	activation energy
$F_{i}$	molar flow rate of species A_i
$k$	reaction rate constant
$k_{\infty}$	frequency factor
$K_{a}$	equilibrium constant
$N$	total number of species
$r$	reaction rate
$R$	universal gas constant
$T$	temperature
$\nu_{i}$	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)
$\for{\lambda}$	boolean indicator (forward reaction)
$\back{\lambda}$	boolean indicator (backward reaction)
$\Phi$	energy flow rate
$\varphi$	fugacity coefficient of species A_i
$\Theta_{i}$	surface coverage of species A_i