Vacuum limit in Fluent

lenzo · December 13, 2016, 05:54

Hello there!

I am currently working in Fluent designing a Vacuum system.

I have considered that, at a certain point, there must be senseless to keep the simulation because there is no more air, thus, the equations can't be right.

So, how can I calculate the limit of vacuum I can simulate assuming that my results are right?

Thank you all and waiting for your answer

lenzo · December 13, 2016, 06:04

I have googled it and i found this explanation:

Not sure If i can apply this equation for my case

HTML Code:

Flow in a Vacuum (Low Pressure Limit)

Autodesk Simulation CFD provides a numerical solution of the Navier-Stokes (N-S) equations. The N-S equations assume that the fluid can be treated as a continuum. In some vacuum flows or very low pressure flows, the flow is no longer a continuum and individual fluid molecules must be considered. This assumption becomes inaccurate as the characteristic dimension of the flow path drops below 10 times the mean free path of the fluid.
We can characterize the mean free path using, , a non-dimensional value defined as:

The symbols are defined:
Symbol Description
non-dimensional vacuum number
absolute viscosity at STP
p static pressure
L characteristic length (hydraulic diameter)
R gas constant
T temperature
The table below shows the range of for which the Navier-Stokes equations are applicable to the flow:
Range Flow Description
< 0.014 The flow is a continuum and governed by the Navier-Stokes equations
0.014 < <1.0 The flow is slip flow where it slips along surfaces, but can still be approximated by the Navier-Stokes equations.
>1.0 The flow is no longer a continuum and cannot be represented by the Navier-Stokes equations.
Note: There are very few true vacuums in industrial applications. The physical requirements (in terms of the compressor pump and seals) needed to create such a flow environment are extremely demanding, and are simply not practical for most industrial applications. This is why we recommend carefully assessing the situation to understand if a vacuum actually exists or if an approximation will suffice.

flotus1 · December 13, 2016, 06:57

Correct. The continuum assumption that the Navier-Stokes equations are based on breaks down for higher Knudsen numbers. You can extend the applicability of continuum methods to some extent with suitable boundary conditions. But Kn=1 as mentioned in your quote seems a bit of a stretch for slip boundary conditions alone. I would not go beyond ~0.1 with a Navier-Stokes solver.

lenzo · December 14, 2016, 11:03

But then I come up with another question!

According to the formula, I need to use absolute viscosity of air. The number of absolute viscosity is 1.8e-5 Pa·s. But it is at STP!!
If i am not wrong I need to know the limit pressure at where knudsen is lower of 0.1 to calculate the absolute viscosity. Which is an iterative calculation.
But, can't really find a reliable way to do this....

Could you help me?

Thank you very much!

flotus1 · December 14, 2016, 16:37

One definition of the Knudsen number is
$Kn=\frac{\lambda}{l}$
where $\lambda$ is the mean free path of the fluid and

is the characteristic length of the flow (obstacle size, hydraulic diameter,...whatever you would use to calculate a Reynolds number).
For an ideal gas the mean free path is inversely proportional to the density
$\lambda \propto \frac{1}{\rho}$ .
Thus you can estimate the Knudsen number of your flow problem using
$Kn=\frac{\rho_\text{ref} \cdot \lambda_\text{ref}}{\rho \cdot l}$
where $\rho_\text{ref}$ is the density at your reference point (use e.g. 1.205 kg/m³) and $\lambda_\text{ref}$ is the mean free path at this reference density (use ~65 nm for air with said reference density).
There are more elaborate definitions of a mean free path and it can be calculated explicitly, but this is a suitable and quick method to estimate the Knudsen number of your flow problem.

lenzo · December 15, 2016, 02:44

Great, thank you!

Then, if I want to optain the limit pressure where I can still assume NS are right, the order is:

- I accept up to 0.1 of Knudsen
- Taking 1.205 kg/m3 and 65nm of reference. My diameter of my tank of vacuum.
- I calculate my limit density
- With the ideal gas equation, I should be able to obtain my limit pressure.

But then I find another, I have no clue how to calculate the temperature I have in this fluid... Can I assume all this process is isothermal?

-BTW, I have estimated I can assume NS is right up to 99,96% of Vacuum (45 Pa)

December 13, 2016, 05:54	Vacuum limit in Fluent	#1
lenzo Member Lenzo Join Date: Feb 2014 Posts: 33 Rep Power: 12	Hello there! I am currently working in Fluent designing a Vacuum system. I have considered that, at a certain point, there must be senseless to keep the simulation because there is no more air, thus, the equations can't be right. So, how can I calculate the limit of vacuum I can simulate assuming that my results are right? Thank you all and waiting for your answer

December 13, 2016, 06:57		#3
flotus1 Super Moderator Alex Join Date: Jun 2012 Location: Germany Posts: 3,400 Rep Power: 47	Correct. The continuum assumption that the Navier-Stokes equations are based on breaks down for higher Knudsen numbers. You can extend the applicability of continuum methods to some extent with suitable boundary conditions. But Kn=1 as mentioned in your quote seems a bit of a stretch for slip boundary conditions alone. I would not go beyond ~0.1 with a Navier-Stokes solver. lenzo likes this.

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December 14, 2016, 11:03		#4
lenzo Member Lenzo Join Date: Feb 2014 Posts: 33 Rep Power: 12	But then I come up with another question! According to the formula, I need to use absolute viscosity of air. The number of absolute viscosity is 1.8e-5 Pa·s. But it is at STP!! If i am not wrong I need to know the limit pressure at where knudsen is lower of 0.1 to calculate the absolute viscosity. Which is an iterative calculation. But, can't really find a reliable way to do this.... Could you help me? Thank you very much!

December 14, 2016, 16:37		#5
flotus1 Super Moderator Alex Join Date: Jun 2012 Location: Germany Posts: 3,400 Rep Power: 47	One definition of the Knudsen number is $Kn=\frac{\lambda}{l}$ where $\lambda$ is the mean free path of the fluid and is the characteristic length of the flow (obstacle size, hydraulic diameter,...whatever you would use to calculate a Reynolds number). For an ideal gas the mean free path is inversely proportional to the density $\lambda \propto \frac{1}{\rho}$ . Thus you can estimate the Knudsen number of your flow problem using $Kn=\frac{\rho_\text{ref} \cdot \lambda_\text{ref}}{\rho \cdot l}$ where $\rho_\text{ref}$ is the density at your reference point (use e.g. 1.205 kg/m³) and $\lambda_\text{ref}$ is the mean free path at this reference density (use ~65 nm for air with said reference density). There are more elaborate definitions of a mean free path and it can be calculated explicitly, but this is a suitable and quick method to estimate the Knudsen number of your flow problem.

December 15, 2016, 02:44		#6
lenzo Member Lenzo Join Date: Feb 2014 Posts: 33 Rep Power: 12	Great, thank you! Then, if I want to optain the limit pressure where I can still assume NS are right, the order is: - I accept up to 0.1 of Knudsen - Taking 1.205 kg/m3 and 65nm of reference. My diameter of my tank of vacuum. - I calculate my limit density - With the ideal gas equation, I should be able to obtain my limit pressure. But then I find another, I have no clue how to calculate the temperature I have in this fluid... Can I assume all this process is isothermal? -BTW, I have estimated I can assume NS is right up to 99,96% of Vacuum (45 Pa)