RateH

Chemical Reactor Design Toolbox Reference Manual

ChemReactorDesign.Basic.Liquid.Rates.RateH

Description

The component determines the molar fluxes due to a chemical reaction using a hyperbolic rate expression

$\begin{equation*} r = k(T) \, \frac{\left( \for{\lambda} \, \prod\limits_{i}^{N} a_{i}^{\for{\kappa_{i}}} - \back{\lambda} \, \frac{1}{K_{a}(T)} \prod\limits_{i}^{N} a_{i}^{\back{\kappa_{i}}} \right)} {\left[ \beta_{0} + \sum\limits_{k}^{L} \left( \beta_{\infty_{k}} \, \exp\left\{ -\frac{E_{k}}{R \, T} \right\} \, \prod\limits_{i}^{N} a_{i}^{{\tilde \kappa}_{ik}} \right) \right]^{n}} \end{equation*}$

formulated in terms of activities

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

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

Temperature Dependent Parameters

Rate Constant

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

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

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

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.
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

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

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

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

Ports

Conserving

Liquid conserving port
```
Port_B = Liquid;  %
```

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

Model Parameters

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.
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.
Activation Energy
```
Ea = {0,'kJ/mol'}; 
```
Equilibrium Constant
```
Ka0 = {1.0e+30,'1'};
```
The parameter is only visible when the option calculateKa is set to Off.
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.
Constant Term ( $\beta_{0}$ Denominator)
```
beta0 = {1,'1'}; 
```
Multipliers ( $\beta_{\infty_{k}}$ Denominator)
```
betainf = {0,'1'}; 
```
The size of this array must be equal to $[L,1]$ .
Activation Energies ( $E_{k}$ Denominator)
```
Es = {0,'kJ/mol'}; 
```
The size of this array must be equal to $[L,1]$ .
Orders of Rection ( ${\tilde \kappa}_{ik}$ Denominator)
```
kappak = {[0; 0],'1'}; 
```
The size of this array must be equal to $[N,L]$ .
Number of Active Sites ( $n$ Denominator)
```
n  = {1,'1'};
```

Nomenclature

$A$	area
$a_{i}$	activity of species A_i
$c_{i}$	concentration of species A_i
$E_{a}$	activation energy
$E_{k}$	activation energy (denominator)
$F_{i}$	molar flow rate of species A_i
$\Delta H_{i}$	molar enthalpy of species A_i
$k$	reaction rate constant
$K_{a}$	equilibrium constant
$L$	number of terms (denominator)
$m$	mass
$N$	total number of species
$n$	number of active sites (denominator)
$r$	reaction rate
$R$	universal gas constant
$G_{i}^{o}$	Gibbs energy of species A_i
$T$	temperature
$V$	volume
$x_{i}$	mole fraction of species A_i
$\beta_{0}$	constant term (denominator)
$\beta_{\infty_{k}}$	multipliers (denominator)
$\nu_{i}$	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)
$\tilde{\kappa}_{ik}$	order of reaction of species A_i (denominator)
$\for{\lambda}$	boolean indicator (forward reaction)
$\back{\lambda}$	boolean indicator (backward reaction)
$\gamma_{i}$	activity coefficient of species A_i
$\Phi$	energy flow rate