Sorption Sorption

Chemical Reactor Design Toolbox Reference Manual

ChemReactorDesign.Basic.Liquid.Transfer.Sorption

Description

The component generates the molar flow rates for all species present in the respective liquid and interphase domain due to $M$ individual sorption equilibria according to Langmuir.

Since for every equilibrium under consideration only one species per domain is involved, only one respective stoichiometric coefficient in the $j^{th}$ sorption reaction is different from zero. Using this criterion the relevant data are extracted and used in calculating the sorption rate.

The $j^{th}$ sorption is modelled as reversible reaction between the respective species in both domains.

$\begin{equation*} A_{j}^{L} + \Theta_{0}^{\nu_{j}^{I}} \rightleftharpoons \nu_{j}^{I} \, A_{j}^{I} \qquad \text{for} \qquad j = 1,\cdots,M \end{equation*}$

The sorption rates are given in terms of the selected model.

Langmuir

$\begin{equation*} r_{j} = k_{ads_{j}} \, \left( {\tilde a}_{j} \, \Theta_{0}^{n_{j}} - \frac{{\tilde \Theta}_{j}^{n_{j}}}{K_{j}} \right) \quad \text{for} \quad j=1,\cdots,M \end{equation*}$
Frumkin

$\begin{equation*} r_{j} = k_{ads_{j}} \, \left( {\tilde a}_{j} \, \Theta_{0}^{n_{j}} - \frac{{\tilde \Theta}_{j}^{n_{j}}}{K^{*}_{j}} \right) \quad \text{for} \quad j=1,\cdots,M \end{equation*}$

mit

$\begin{equation*} K_{j}^{*} = K_{j} \, \exp\left\{g \, \sum\limits_{i}^{N^{I}} {\tilde \Theta}_{i} \right\} \end{equation*}$
Tempkin

$\begin{equation*} r_{j} = k_{ads_{j}} \, \left( {\tilde a}_{j} - \frac{{\tilde \Theta}_{j}^{n_{j}}}{K^{*}_{j}} \right) \quad \text{for} \quad j=1,\cdots,M \end{equation*}$

mit

$\begin{equation*} K_{j}^{*} = K_{j} \, \exp\left\{g \, \sum\limits_{i}^{N^{I}} {\tilde \Theta}_{i} \right\} \end{equation*}$
Linear Sorption

$\begin{equation*} r_{j} = k_{ads_{j}} \, \left( {\tilde a}_{j} - \frac{{\tilde \Theta}_{j}}{K_{j}} \right) \quad \text{for} \quad j=1,\cdots,M \end{equation*}$

with

$\begin{equation*} \Theta_{0} = 1- \sum_{i}^{M} {\tilde \Theta}_{i} \end{equation*}$

and

$\begin{equation*} {\tilde a}_{j} = \frac{\gamma_{j} \, c_{j}}{1 \, M} \end{equation*}$

as well as

$\begin{equation*} n_{j} = \sum_{i}^{M} {\tilde \nu}_{ij} \end{equation*}$

The equilibrium constant is regarded as temperature dependent, as

$\begin{equation*} K_{j}(T) = K_{j_{0}} \, \exp\left\{ -\Delta H_{ads_{j}} \, \left(\frac{1}{T} - \frac{1}{298.15 \, K} \right) \right\} \end{equation*}$

with $\Delta H_{ads_{j}} \neq \Delta H_{ads_{j}}(T)$ .

Then the molar flow rates for both domains become

$\begin{align*} F_{i}^{L} & = A \, \sum_{j}^{M} \nu^{L}_{ij} \, r_{j} \\ F_{i}^{I} & = A \, \sum_{j}^{M} \nu^{I}_{ij} \, r_{j} \end{align*}$

The energy flow rate for the gas domain is given as

$\begin{equation*} \Phi^{L} = \sum_{i}^{N} F_{i}^{L} \, {\overline H}_{i}^{L} + A \, \sum_{j}^{M} r_{j} \, \Delta H_{ads_{j}} \end{equation*}$

Ports

Conserving

Liquid conserving port
```
Port_B_L = Liquid;  %
```
Interface conserving port
```
Port_B_I = Interface;  %
```

Input

Physical signal that represents the surface area
```
Ain = {0,'m^2'};
```
Dependencies: The port is only visible when areaInput is set to On.

Parameters

Options

Option to select area input
```
areaInput = OnOff.Off;
```
Off | On
Option to select sorption type
```
select = SorptionMode.Langmuir;
```
Langmuir | Frumkin | Tempkin | Linear

Geometry

Surface Area
```
A0 = {0,'cm^2'}; 
```
Dependencies: The parameter is only visible when the option areaInput is set to Off.

Thermodynamics

Heat of Adsorption
```
dHads = {0,'kJ/mol'};
```
Note Initially only one equilibrium is considered. When the number of individual equilibria is increased, the size of the array must be adjusted accordingly.

Stoichiometry

Stoichiometric coefficients for liquid domain
```
nu_L = {[-1;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.
- Stoichiometric coefficients for interface domain
```
nu_I = {[1;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.

Kinetics

Sorption rate constants
```
kads = {0,'mol/(m^2*s)'}; 
```
Note Initially only one equilibrium is considered. When the number of individual equilibria is increased, the size of the array must be adjusted accordingly.
Equilibrium constant at standard conditions
```
K0 = {1,'1'};           
```
Note Initially only one equilibrium is considered. When the number of individual equilibria is increased, the size of the array must be adjusted accordingly.
Interaction parameter
```
g = {0,'1'};    
```
Dependencies: The parameter is only visible when the option select is set to Frumkin or Tempkin, respectively.

Nomenclature

$a_{i}$	activity of species A_i
$A$	area
$F_{i}$	molar flow rate of species A_i
${\overline H}_{i}(T)$	molar enthalpy of species A_i
$g$	Frumkin/Tempkin interaction parameter
$\Delta H_{ads}$	Heat of adsorption
$K$	equilibrium constant
$k$	sorption rate constant
$N$	total number of species
$M$	number of sorption equlibria
$p$	pressure
$r$	sorption rate
$R$	universal gas constant
$T$	temperature
$x_{i}$	mole fraction of species A_i
$z$	compressibility
$\Phi$	energy flow rate
$\gamma$	activity coefficient of species A_i
$\Theta$	surface coverage of species A_i