Ergun

Chemical Reactor Design Toolbox Reference Manual

ChemReactorDesign.Basic.Gas.Transport.Ergun

Description

The component generates a volumetric flow rate due to a pressure difference between the ports for flow in a porous medium (after Ergun). To allow for a change in sign upon reversal of flow direction and to eliminate singularities due to flow reversal a modified relation is used

$\begin{equation*} \Delta p = 150 \frac{(1-\overline{\varepsilon})^2} {\overline{\varepsilon}^3} \frac{\mu}{d_p^2} \, \frac{L}{A} + 1.75 \frac{(1-\overline{\varepsilon})} {\overline{\varepsilon}^3} \frac{\rho}{d_p} \, \frac{L}{A^{2}} \, q \, \sqrt{q^{2} + q_{small}^{2}} \end{equation*}$

Then the molar flow rates become

$\begin{equation*} F_{i} = q \, \left(\frac{x_{i} \, p}{z \, R \ T} \right)_{upstream} \end{equation*}$

The energy flow rate is determined as

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

with

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

The positive flow direction is from port A to port B.

Assumptions and Limitations

In some cases setting ${\overline \varepsilon} = 1$ may lead to poor convergence of the numerical solution algorithm.

Remedy Choose ${\overline \varepsilon} = 1-\epsilon$ instead

The parameter $q_{small}$ is internally set to a value of $1.0 \times 10^-6 \, \frac{l}{s}$ .

Ports

Conserving

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

Input

Physical signal that represents the cross sectional area
```
Ain = {0,'m^2'};
```
Dependencies: The port is only visible when areaInput is set to On.
Physical signal that represents the voidage
```
epsbIn = {1,'1'};
```
Dependencies: The port is only visible when inputPorosity is set to On.

Output

Physical signal that represents the volumetric flow rate at upstream conditions
```
qout = {1,'l/s'};
```
Dependencies: The port is only visible when flowOutput is set to On.

Parameters

Options

Option to select porosity input
```
porosityInput = OnOff.Off; 
```
On | Off
Option to select area input
```
areaInput = OnOff.Off; 
```
On | Off
Option to select flow output
```
flowOutput = OnOff.Off; 
```
On | Off

Geometry

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

Mass Transport

Voidage fraction
```
epsb = {1,'1'}; 
```
Dependencies: The parameter is only visible when inputPorosity is set to Off.
Particle diameter
```
dp = {1.0e-06,'m'}; 
```
Threshold for laminar flow
```
qThreshold = {1.0e-10,'l/s'}; 
```

Nomenclature

$A$	cross sectional area
$d_{p}$	particle diameter
$F_{i}$	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
$L$	length
$N$	total number of species
$p$	pressure
$q$	volumetric flow rate
$t$	time
$T$	temperature
$x_{i}$	mole fraction of species A_i
$z$	compressibility
$\overline \varepsilon$	voidage (porosity)
$\rho$	density
$\mu$	viscosity
$\Phi$	energy flow rate