Overhead

Chemical Reactor Design Toolbox Reference Manual

ChemReactorDesign.Basic.Solid.Volumes.Overhead

Description

The component represents an variable solid volume with gas overhead. The total volume of gas and solid is constant.

Mass Balance

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

initial conditions

$\begin{equation*} n_{i}(t=0) = \frac{m_{i_{0}}}{M_{i}} \quad \text{for} \quad i=1,\dots,N \end{equation*}$ 1

Energy Balance

$\begin{equation*} \sum_{i}^{N} F_{i} \, {\overline H}_{i}(T) + \left( \sum_{i}^{N} n_{i} \, c_{p_{i}}(T) + n_{0}^{G} \, c_{p}^{G} \right) \, \frac{dT}{dt} = \Phi + \dot{Q} + \left( V_{max}-V \right) \, \frac{dp}{dt} \end{equation*}$

with

$\begin{equation*} n_{0}^{G} = p_{0} \, \frac{V_{max}-V_{0}}{R \, T_{0}} \end{equation*}$

with initial condition

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

Equation of State

Pressure

The gas pressure exerted on the solid is given as

$\begin{equation*} p = p_{0} \frac{\left(V_{max}-V_{0}\right)}{\left(V_{max}-V\right)} \end{equation*}$

The pressure at the port is the sum of internal and potential pressure

$\begin{equation*} p_{A} = p + \rho \, g \, \left( \frac{V}{A_{0}}+h_{0} \right) \end{equation*}$

with

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

Volume

$\begin{equation*} \frac{dV}{dt} = \sum_{i}^{N} {\overline V}_{i}(T) \, F_{i} \end{equation*}$

with

$\begin{equation*} V(t=0) = V_{0} \end{equation*}$

Variables

To adjust the nominal values for the component variables use the Nominal Values tab in the dialogue box.

Assumptions and Limitations

The heat capacity of the gas phase is regarded as independent of temperature.
The gas is modelled as an ideal gas phase.

Ports

Conserving

Solid conserving port
```
Port_A = Solid;  %
```
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.
```
pin = {1,'bar'}; % p
```
Dependencies: The port is only visible when pressureInput is set to On.

Output

Physical signal that represents the current volume
```
Vout = {0,'l'}; % V
```
Dependencies: The port is only visible when volumeOutput is set to On.
Physical signal that represents the current solid mass
```
mout = {zeros(length(selectSpecies),1),'g'}; % m
```
The individual masses are obtained as

$\begin{equation*} m_{i} = n_{i} \, M_{i} \end{equation*}$

Dependencies: The port is only visible when massOutput is set to On. Arbitrary species can be selected by providing the index array selectSpecies accordingly.

Parameters

Options

Option to select thermal behaviour of the volume.
```
isothermalOperation = OnOff.On;  
```
Off | On
Option to select volume output
```
volumeOutput = OnOff.Off; 
```
On | Off
Option to select mass output
```
massOutput = OnOff.Off; 
```
On | Off

Geometry

Total Volume
```
Vmax = {1,'l'};  % Total Volume
```
Initial Solid Volume
```
V0 = {1,'l'};       % Volume
```
Cross Sectional Area
```
A0 = {10,'cm^2'};
```
Geodetic Height
```
h0 = {0,'m'}; 
```

Operation Conditions

Option to select mass output
```
massOutput = OnOff.Off; 
```
On | Off 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

Gas Properties

Specific Heat

cpGas = {1.0e-03,'kJ/(mol*K)'}; % Specific Heat Gasphase

Nominal Values

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

Nomenclature

$A_{0}$	cross sectional area
$c_{i}$	molar concentration of species A_i
$c_{p_{i}}$	specific heat for species A_i
$c_{p}^{G}$	specific heat of gas phase
$F_{i}$	molar flow rate of species A_i
$h_{0}$	geodetic height
${\overline H}_{i}(T)$	molar enthalpy of species A_i
$m_{i}$	solid mass of species A_i
$M_{i}$	molar weight of species A_i
$N$	total number of species
$n_{i}$	number of moles of species A_i
$n_{0}^{G}$	number of moles in the gas phase
$p$	pressure
$p_{0}$	initial pressure
$p_{ext}$	external pressure
$Q$	heat flow rate (independent of fluid flow)
$R$	universal gas constant
$t$	time
$T$	temperature
$T_{0}$	initial temperature
$V_{0}$	initial solid volume
$V_{max}$	total volume
${\overline V}_{i}$	molar volume of species A_i
$x_{i}$	mole fraction of species A_i
$x_{i_{0}}$	initial mole fraction of species A_i
$\Phi$	energy flow rate