Density of air for calculating the mass flow rate from Darcy's law 3

[drir]

Hi,

I use Darcy's law to calculate first the volumetric flow rate of air through a porous material. From this I want to calculate the mass flow rate namely the product of the volumetric flow rate and the density. However the density is function of the pressure. Is it right the use the mean pressure of the input and output pressure in order to calculate the density of air?

 

[GannetAM]

I would use the output pressure to calculate air density and volumetric flow rate, because this would nullified the pressure drop across the porous material.
Furthermore, the output condition is probably a better indicator of the porous material performance

 

[zdas04]

On this kind of problem I always work the solution three ways. Density based on upstream. Density based on Downstream. Density based on the "pipeline average"(see GPSA Field Data Book). Then I look hard at the answer. If I get a material difference I will dig further (and almost always end up with the pipeline average, which is a bit higher than the mean). Most of the time the difference is not within the accuracy of the calculation and it just doesn't matter.

David

 

[BigInch]

You should use the weighted average. Because pressure drop along the length of flow is not linear, so the average is around 2/3 of the difference, with a little twist

Pavg = 2/3 * (P1+P2 - P1*P2/(P1+P2))

Assuming that your porous medium has a pressure drop similar to a gas pipeline.

We will design everything from now on using only S.I. units ... except for the pipe diameter. Unk. British engineer

 

[drir]

The pressure drop is around 3 bar! The application is an porous air bearing! Supply pressure is around 6 bar! Which of the three (different) answers are closest to my application?

 

[Latexman]

I suggest you contact technical support of the porous air bearing supplier for their recommended methodology and data ([α] and [β] if they use Green and Duwez's article for the isothermal flow of an ideal gas).

Good luck,
Latexman

 

[zdas04]

I just read the Wiki article on Darcy's Law and there is no density term in any of the versions of the equations that they reference.

David

 

[drir]

Hi zdas04,

Read my original question again ;-)

I use Darcy's law to calculate first the volumetric flow rate of air through a porous material. From this I want to calculate the mass flow rate namely the product of the volumetric flow rate and the density. However the density is function of the pressure. Is it right the use the mean pressure of the input and output pressure in order to calculate the density of air?

 

[Latexman]

What is the basis (T & P) for the volumetric flow rate you calculate? That is the basis for the density you need.

Good luck,
Latexman

 

[drir]

If you look to darcy's law, it is the difference between the input and output pressure that you need to calculate the volumetric flow rate!!!!

You can't use a pressure difference for calculating the density! Therefore my question which 'absolute' pressure value which I have to use to calculate the density!

 

[mbt22]

Darcy's law is an empirical rule for estimating pressure drop for viscous laminar flow of an incompressible fluid. Have you checked that the Reynolds number is less than 10-ish? I would be surprised if it was for the pressure drop you are talking about.

The equation is only really valid for constant density, which is why you can not find anything to tell you which density to use.

You might get away with using it for slow flow of gas with a small pressure drop.

Matt

 

[Latexman]

A volumetric flow rate of a gas is meaningless without a T & P reference. Ditto for a volumetric flow rate equation of a gas without a T & P reference. Show the equation you are using. What reference are you using? This information may help get an answer.

Good luck,
Latexman

 

[GannetAM]

Need more data:

Have you determine whether the expansion of the gases is either of the following:
isothermal - p/? = constant
isentropic or adiabatic - p /?^k = constant
polytropic - pV^n = constant

Is the input area ands the output area the same? If not, what are their areas.

Do you have a manufacturer and catalog number for this porous air bearing?

 

[drir]

Hi,

I'm making a numerical model for porous air bearings, so I don't have a manufacturer.

In my model I assume isothermal behaviour of the gas, thus p/rho = cst!

You can assume that the input and output area is the same!

The equation of Darcy I use is like the first equation on Wikipedia:

http://en.wikipedia.org/wiki/Darcy's_law

 

[Latexman]

Please describe "numerical model". Is it a differential model (it sums up very thin sections of media)? Or, is it an integrtated model (calculates the entire thickness of the media in one step)?

Good luck,
Latexman

 

[drir]

Finite difference model based on the Reynolds equation!!

 

[btrueblood]

"which 'absolute' pressure value which I have to use to calculate the density! "

Either pressure, at either end of the pressure drop, presuming you have the temperature value there as well. The mass flow in and out should be the same.

 

[mbt22]

What is the Reynold number do you calculate?

Matt

 

[BigInch]

"I want to calculate the mass flow rate namely the product of the volumetric flow rate and the density."

Calculating the density at any pressure will give you the mass/unit volume, but it will not give you the mass flow rate. Even if you use the average pressure, you will only get the average density.

The different pressures at various points will all have different respective densities, and instantaneous pressure drops at those points as well, which will integrate into the overall pressure drop across the medium, but he's not interested in finding that. Problem is that density at any point cannot be related to mass flow rate. You need to use a pressure drop to flow rate relationship, one either for dP vs volumetric flow rate, or dP vs mass flow rate and solve for mass flow rate to find out what the mass flowrate is.

To do that, the easiest way would be to use Inlet Pressure and the Outlet Pressure, calculate the differential pressure, relate that to flowrate. If you used a dP to mass flowrate relationship, you're finished, otherwise you'd have to continue by calculating the density (presumedly at the overall average pressure) and relate that density to your volumetric flowrate to get the mass flowrate as the final answer.

Got it?



We will design everything from now on using only S.I. units ... except for the pipe diameter. Unk. British engineer

 

[GannetAM]

Personally, I am back to my original answer. Your design end point is to support some weight that requires a certain mass flow rate. I would work up that require output mass flow rate and work backwards to determine your input requirements.