RatePS

Chemical Reactor Design Toolbox Reference Manual

ChemReactorDesign.Basic.Gas.Rates.RatePS

Description

The component determines the molar fluxes due to a heterogeneous chemical reaction (gas-solid)

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

using a power law rate expression

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

with

$\begin{equation*} a_{i} = \frac{\varphi_{i} \, x_{i} \, p}{1 \, bar} \end{equation*}$

since for any solid species it holds

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

Temperature Dependent Parameters

Rate Constant

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

or

$\begin{equation*} k = k_{\infty} \, \exp\left\{-\frac{E_{a}}{R} \left(\frac{1}{T} - \frac{1}{T_{ref}} \right) \right\} \end{equation*}$
Equilibrium Constant

$\begin{equation*} K_{a}(T) = \exp \left\{- \left( \frac{\sum\limits_{i}^{N^G} \nu^{G}_{i} \, \Delta_{f} G^G_{i}(T)}{R \, T} + \frac{\sum\limits_{i}^{N^S} \nu^{S}_{i} \, \Delta_{f} G^S_{i}(T)}{R \, T} \right) \right\} \end{equation*}$

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.
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

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

and

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

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

Variables

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

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

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

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

Ports

Conserving

Gas conserving port
```
Port_B = Gas;  %
```

Input

Physical signal that represents the volume
```
V = {0,'l'}; % V
```
Dependencies: The port is only visible when rateReference is set to Volume.
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 mass
```
m = {0,'l'}; % m
```
Dependencies: The port is only visible when rateReference is set to Mass.

Parameters

Options

Option to select the reversibility of the reaction
```
reversibility = Reversibility.Irreversible;
```
Irreversible | Reversible
Option to select reparametrisation of reaction rate constant
```
reparametrisation = OnOff.On;      
```
Off | On
Option to select the reference frame
```
rateReference = RateReference.Volume; 
```
Volume | Area | Mass
Option to select calculation of the equilibrium constant
```
calculate_Ka = OnOff.Off;   
```
On | Off

Kinetics

Frequency Factor
```
kfinfV = {0,'mol/(l*s)'}; 
```
The parameter is only visible when the option rateReference is set to Volume.
```
kfinfA = {0,'mol/(cm^2*s)'};
```
The parameter is only visible when the option rateReference is set to Area.
```
kfinfm = {0,'mol/(g*s)'};   
```
The parameter is only visible when the option rateReference is set to Mass.
Activation Energy
```
Ea = {0,'kJ/mol'}; 
```

Gas

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.
Stoichiometric Coefficients for Gas Phase
```
nu = {[-1; 2],'1'};   
```
Note Initially for the gas 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.

Solid

Stoichiometric Coefficients for Solid Phase
```
nu_S = {[-1;0],'1'}; 
```
Note 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.

Thermodynamics

Equilibrium Constant
```
Ka0 = {1.0e+30,'1'};
```
The parameter is only visible when the option calculateKa is set to Off.
Reference Temperature
```
Tref = {298.0,'K'}; 
```
The parameter is only visible when the option reparametrisation is set to On.

Nomenclature

$A$	area
$a_{i}$	activity of species A_i
$E_{a}$	activation energy
$F^{G}_{i}$	molar flow rate of species A_i in gas phase
$F^{S}_{i}$	molar flow rate of species A_i in solid phase
$\Delta_{f} H_{i}$	molar enthalpy of species A_i
$k$	reaction rate constant
$K_{a}$	equilibrium constant
$m$	mass
$N$	total number of species in the gas phase
$N^S$	total number of species in the solid phase
$p$	pressure
$r$	reaction rate
$R$	universal gas constant
$\Delta_{f} S_{i}$	molar entropy of species A_i
$T$	temperature
$V$	volume
$x_{i}$	mole fraction of species A_i
$\nu_{i}$	stoichiometric coefficient of species A_i in gas phase
$\nu^{S}_{i}$	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 (forward reaction)
$\for{\lambda}^{S}$	boolean indicator (forward reaction)
$\back{\lambda}^{S}$	boolean indicator (backward reaction)
$\varphi_{i}$	fugacity coefficient of species A_i
$\Phi^{G}$	energy flow rate in gas phase
$\Phi^{S}$	energy flow rate in solid phase