VLE

Chemical Reactor Design Toolbox Reference Manual

ChemReactorDesign.Basic.Liquid.Transfer.VLE

Description

The component determines the molar flow rates of all species in the respective liquid and gas phase due to $M$ individual vapor-liquid equilibria (VLE).

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

$\begin{equation*} A_{i}^{L} \rightleftharpoons A_{k}^{G} \qquad \text{für} \qquad j = 1,\cdots,M \end{equation*}$

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

The individual rates are given as

$\begin{equation*} r_{vap_{j}} = \left\{ \begin{array}{ccc} k_{j} \, x_{j}^{L} & \text{if} & p \leq p^{bubble} \\ 0 & \text{otherwise} & \end{array} \right. \end{equation*}$

$\begin{equation*} r_{cond_{j}} = \left\{ \begin{array}{ccc} k_{j} \, \frac{x_{j}^{G}}{K_{j} & \text{if} & p^{dew} \leq p \\ 0 & \text{otherwise} & \end{array} \right. \end{equation*}$

with

$\begin{equation*} K_{j} = \frac{\gamma_{j} \, p_{j}^{vap}}{\varphi \, p} \end{equation*}$

and

$\begin{equation*} p_{bubble} = \lambda \, \sum_{j}^{M} x_{i}^{L} \, \gamma_{j} \, p_{j}^{vap} \end{equation*}$

$\begin{equation*} p_{dew} = \frac{1}{\sum\limits_{j}^{N} \frac{\varphi_{j} \, x_{j}^{G}}{\gamma_{j} \, p_{j}^{vap}} \end{equation*}$

In order to account for a vanishing liquid phase the boolean operator $\lambda$ is defined as

$\begin{equation*} \lambda = \left\{ \begin{array}{ccc} 1 & \text{if} & V > V_{min} \\ 0 & \text{otherwise} & \end{array} \right. \end{equation*}$

Then the $j^th}$ mass transfer rate becomes

$\begin{equation*} r_{j} = \lambda \, r_{vap_{j}}- r_{vap_{j}} \end{equation*}$

The heat of vaporization is given as

$\begin{equation*} \Delta_{v} H_{j} = \Delta_{f} H_{j}^{G} - \Delta_{f} H_{j}^{L} \end{equation*}$

Variables

The molar rates for both domains are given as

$\begin{equation*} F_{i}^{G} = A \, \sum_{j}^{M} \nu_{ij}^{G} \, r_{j} \end{equation*}$

$\begin{equation*} F_{i}^{L} = A \, \sum_{j}^{M} \nu_{ij}^{L} \, r_{j} \end{equation*}$

and the energy flow rates

$\begin{equation*} \Phi^{G} = \sum_{i}^{N} F_{i}^{G} \, {\overline H}_{i}^{G} \end{equation*}$

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

Note The heat of vaporization is solely attributed to the liquid domain. Thus, a convective heat transport block with appropriate parameter values should be added in the model to achieve thermal equilibrium in both phases.

Ports

Conserving

Liquid conserving port
```
Port_B_L = Liquid;  %
```
Gas conserving port
```
Port_B_G = Gas;  %
```

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

Geometry

Surface Area
```
A0 = {0,'cm^2'}; 
```
Dependencies: The parameter is only visible when the option areaInput is set to Off.
Volume Threshold $V_{min}$
```
VinMin = {1.0e-10,'l'}; 
```

Stoichiometry

Stoichiometric coefficients for liquid domain
```
nu_L = {[-1;0],'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.
Stoichiometric coefficients for gas domain
```
nu_G = {[1;0],'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.

Kinetics

Rate constants
```
k = {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.

Nomenclature

$A$	area
$F_{i}$	molar flow rate of species A_i
${\Delta_{f} H}_{i}(T)$	molar enthalpy of species A_i
$\Delta_{v} H$	heat of vaporisation
$K$	equilibrium constant
$k$	rate constant
$N$	total number of species
$M$	number of equlibria
$p$	pressure
$r$	mass transfer rate
$R$	universal gas constant
$T$	temperature
$x_{i}$	mole fraction of species A_i
$\Phi$	energy flow rate
$\gamma$	activity coefficient of species A_i
$\varphi$	fugacity coefficient of species A_i