Migration

Chemical Reactor Design Toolbox Reference Manual

ChemReactorDesign.Basic.Liquid.Transport.Migration

Description

The component generates the molar fluxes for both diffusion due to concentration gradients.

$\begin{equation*} J^{d}_{i} = - D_{iN} \, \frac{d c_{i}}{dz} \quad \text{for} \quad i=1,\dots,N-1 \end{equation*}$

and migration due to potential gradients

$\begin{equation*} J^{m}_{i} = -c_{i} \, z_{i} \, D_{iN} \, \frac{\cal F}{R \, T} \, \frac{d\phi}{dz} \quad \text{for} \quad i=1,\dots,N-1 \end{equation*}$

The velocity of the solvent is chosen as reference frame

$\begin{equation*} J^{d}_{N} = 0 \end{equation*}$

The solvent A_N carries no charge

$\begin{equation*} J^{m}_{N} = 0 \end{equation*}$

Then the molar flow rates become

$\begin{equation*} F_{i} = A \, \left( J^{d}_{i}+J^{m}_{i} \right) \quad \text{for} \quad i=1,\dots,N \end{equation*}$

The energy flow rate is determined as

$\begin{equation*} \Phi = \sum_{i}^{N} F_{i} \, \left({\overline H}_{i}(T)\right)_{averaged} + F_{tot} \, \Big(H_{res}(T,p)\Big)_{averaged} \end{equation*}$

with

$\begin{equation*} F_{tot} = \sum_{i}^{N} F_{i} \end{equation*}$

The positive direction for the fluxes is from port A to port B.

Assumptions and Limitations

Generally, the species A_N is regarded as solvent.

Ports

Conserving

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

Input

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

Parameters

Options

Option to select area input
```
areaInput = OnOff.Off;
```
Off | On

Geometry

Cross sectional area
```
A0 = {1,'m^2'};
```
Dependencies: The parameter is only visible when the option areaInput is set to Off.
Transport distance
```
delta  = {1.0e-03,'m'};x
```

Mass Transport

Diffusion coefficients
```
D = {1.0e-06*[1,1],'m^2/s'};
```
Note Initially only two species are considered, i.e. $N-1 = 1$ . As the number of species can be changed via the properties dialogue, the size of the array must be adjusted accordingly.

Nomenclature

$c_{i}$	concentration of species A_i
$D_{z}$	dispersion coefficient
$F_{i}$	molar flow rate of species A_i
${\overline H}_{i}(T)$	molar enthalpy of species A_i
$\Delta H_{res}$	departure enthalpy of the mixture
$J_{i}$	diffusional flux of species A_i
$N$	total number of species
$t$	time
$T$	temperature
$x_{i}$	mole fraction of species A_i
$\Phi$	energy flow rate
$phi$	potential