Rates

Chemical Reactor Design Toolbox Reference Manual

ChemReactorDesign.Basic.Interface.Rates.RateP

Description

The component determines the molar fluxes due to an area based chemical reaction using a power law rate expression

$\begin{equation*} r = k(T) \, \left( \for{\lambda} \, \prod_{i}^{N} a_{i}^{\for{\kappa_{i}}} - \back{\lambda} \, \frac{1}{K_{a}(T)} \prod_{i}^{N} a_{i}^{\back{\kappa_{i}}} \right) \end{equation*}$

formulated in terms of activities

$\begin{equation*} a_{i} = \Theta_{i} \end{equation*}$

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

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

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

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

Variables

The molar fluxes are obtained as

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

Ports

Conserving

Interface conserving port
```
Port_B = Interface;  %
```

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

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.

Kinetics

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

Thermodynamics

Isothermal Temperature
```
T0 = {298.15,'K'};   
```
The parameter is only visible when the option temperatureInput is set to On.
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 Temperature 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.
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.

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
$\Theta_{i}$	surface coverage of species A_i