ClosedVolume

Chemical Reactor Design Toolbox Reference Manual

ChemReactorDesign.Basic.Gas.Volumes.ClosedVolume

Description

The component represents a closed gas volume of fixed size.

Mass Balance

$\begin{equation*} \frac{dn_{i}}{dt} = F_{i} \end{equation*}$

with initial conditions depending on the selected thermodynamic model

Ideal Gas

$\begin{equation*} n_{i}(t=0) = \frac{x_{i_{0}} \, p_{0} \, V_{0}}{R \, T_{0}} \quad \text{for} \quad i=1,\dots,N \end{equation*}$

Peng Robinson

$\begin{equation*} n_{i}(t=0) = \frac{x_{i_{0}} \, p_{0} \, V_{0}}{z(p_{0},T_{0}) \, R \, T_{0}} \quad \text{for} \quad i=1,\dots,N \end{equation*}$

Energy Balance

$\begin{equation*} \sum_{i}^{N} F_{i} \, {\overline H}_{i}(T) + \Delta H_{res} \, \sum_{i}^{N} F_{i} \, + \left( \sum_{i}^{N} n_{i} \, c_{p_{i}}(T) \right) \, \frac{dT}{dt} = \Phi + \dot{Q} + V \, \frac{dp}{dt} \end{equation*}$

with initial condition

$\begin{equation*} T(t=0) = T_{0} \end{equation*}$

Equation of State

Pressure

Ideal Gas

$\begin{equation*} p = \sum_{i}^{N} n_{i} \, \frac{R \, T}{V} \end{equation*}$
Peng Robinson

$\begin{equation*} p = z({\bf x},p,T) \, \sum_{i}^{N} n_{i} \, \frac{R \, T}{V} \end{equation*}$

with

$\begin{equation*} {\overline V} = \frac{V}{\sum\limits_{i}^{N} n_{i}} \end{equation*}$

Volume

The volume is either given by a component parameter or set by a physical input signal which may even be time variant (see Assumptions).

$\begin{equation*} V = \left\{ \begin{array}{lcl} V_{0} \quad \text{or} \\ V_{in}(t) & & \end{array} \right. \end{equation*}$

Assumptions and Limitations

The time response of the volume input signal is assumed to be much slower than the dynamics of the balance component. Thus the volume is regarded as approximately constant.
In some cases considering the expression work explicitly may slow down the convergence rate.
If the volume is read from the physical input port, it must hold: $V_{0} = V_{in}(t=0)$ .

Ports

Conserving

Gas conserving port
```
Port_A = Gas;  %
```
Gas conserving port
```
Port_B = Gas;  %
```
The port is only visible when the option enable2Port is set to On.
Thermal conserving port
```
Port_C = foundation.thermal.thermal;  %
```
Dependencies: The port is only visible when isothermalOperation is set to Off.

Input

Physical signals that controls the volume.
```
Vin = {0,'l'}; % V
```
Dependencies: The port is only visible when volumeInputOutput is set to Input.

Output

Physical signal that represents the current volume
```
Vout = {0,'l'}; % V
```
Dependencies: The port is only visible when volumeInputOutput is set to Output.

Parameters

Options

Option to select volume input/output
```
volumeInputOutput = InOut.Off; 
```
Input | Output | Off
Option to select thermal behaviour of the volume.
```
isothermalOperation = OnOff.On;  
```
Off | On
Option to select compression work
```
compressionWork = OnOff.On;      
```
Off | On

If the option is set to Off the term $V \, \frac{dp}{dt}$ in the energy balance will be skipped.
Option to enable 2nd Port
```
enable2ndPort = OnOff.Off;
```

Geometry

Initial volume
```
V0 = {1,'l'};       % Volume
```

Operation Conditions

Initial mole fractions
```
x0 = {[0;1],'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.
Initial pressure
```
p0 = {1.0,'bar'};   % Initial Pressure
```

Initial Temperature

T0 = {298.15,'K'};  % Initial Temperature

Nominal Values

Nominal Values for Number of Moles
```
n_nom = {1,'mol'}; 
```

Nomenclature

$c_{p_{i}}$	specific heat for species A_i
$F_{i}$	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
$N$	total number of species
$n_{i}$	number of moles of species A_i
$p$	pressure
$p_{0}$	initial pressure
$Q$	heat flow rate (independent of fluid flow)
$R$	universal gas constant
$t$	time
$T$	temperature
$T_{0}$	initial temperature
$V_{0}$	volume
${\overline V}$	molar volume of mixture
$x_{i}$	mole fraction of species A_i
$x_{i_{0}}$	initial mole fraction of species A_i
$z$	compressibility
$\Phi$	energy flow rate