RatePS

Chemical Reactor Design Toolbox Reference Manual

ChemReactorDesign.Basic.Liquid.Rates.RatePS

Description

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

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

using a power law rate expression

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

with

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

since for any solid species it holds

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

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

Temperature Dependent Parameters

Rate Constant

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

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

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

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

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

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

Variables

The molar fluxes for the liquid phase are obtained as

$\begin{equation*} F_{i} = \nu_{i} \, \big[ \, V \, | \, A \, | \, m \, \big] \, r \qquad \text{for} \quad i=1,\cdots,N \end{equation*}$

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

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

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

Ports

Conserving

Liquid conserving port
```
Port_B_L = Liquid;  %
```
Solid conserving port
```
Port_B_ = Solid;  %
```

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 calculation of the equilibrium constant
```
calculate_Ka = OnOff.Off;   
```
On | Off
Option to select the reference frame
```
rateReference = RateReference.Volume; 
```
Volume | Area | Mass

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'}; 
```

Liquid

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

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.

Thermodynamics

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

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
$\Delta_{f} H_{i}$	molar enthalpy of species A_i
$k$	reaction rate constant
$K_{a}$	equilibrium constant
$m$	mass
$N$	total number of species in the liquid phase
$N^S$	total number of species in the solid phase
$r$	reaction rate
$R$	universal liquid constant
$G_{i}$	Gibbs energy 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 liquid 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)
$\gamma_{i}$	activity coefficient of species A_i
$\Phi^{L}$	energy flow rate in liquid phase
$\Phi^{S}$	energy flow rate in solid phase