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

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

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

(OP)
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?

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

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  

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

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

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

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

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

(OP)
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?

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

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

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

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

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

(OP)
Hi zdas04,

Read my original question again winky smile

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?  

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

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

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

(OP)
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!

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

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
 

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

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

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

Need more data:

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

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

(OP)
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

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

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

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

(OP)
Finite difference model based on the Reynolds equation!!

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

"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.

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


What is the Reynold number do you calculate?

Matt

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

"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

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

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.

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

Matt, I doubt dr. ir. is calculating Reynolds number.  The Reynolds equation is a statement of several conservation principles. It is derived by solving the equation of continuity simultaneously with the Navier-Stokes Equations.  It can be used as a model for thin film lubricant flow between two parallel surfaces.

Is the density being calculated within each finite difference?  Or, is the density calculated as boundary conditions or limits?  The correct answer depends on which one.

So, is this an undergraduate or postgraduate assignment?

Good luck,
Latexman

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

Sorry, reading in a hurry. I was trying to follow up a hunch that Darcy doesn't apply.  6 bar to 3 bar seems rather high for the incompressible assumption, but I wanted to check applicability before we go down the route of sectioning the bed for roughly constant density.  I'm just curious whether turbulence has been checked for.  If it is higher, we have a chance of offering something with extra loss terms.

Matt

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

From the WIKI, "Darcy's law is a simple proportional relationship between the instantaneous discharge rate through a porous medium, the viscosity of the fluid and the pressure drop over a given distance."

In original form it was for developed for simple incompressible fluids, but as is also stated there, if you didn't know already, is that it is used to describe flow of water, oil and gas through petroleum resevoirs, so it certainly can be made applicable to complex compressible flows.  You just have to be able to somehow calculate the density and find the appropriate relationships of density to shear force of your fluid, viscosity.  You only need the equation of state for your fluid, relating density to it's state of pressure and temperature, and viscosity to that density and temperature.

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

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

When I read drir's original post and subsequent comments I away with the impression that s/he is only interested in the answer to the question, "Is it right to use the mean pressure of the input and output pressure in order to calculate the density of air?"

S/He doesn't appear to be interested in knowing if s/he's using the right formula or method.  S/He also doesn't appear willing to share other information about the problem -- though a lot of exclamation marks appear which gave the impression that drir thought all the people trying to read his/her mind were idiots for not providing him/her with the obviously simple YES answer s/he wanted.

I would suggest that drir go find a Crane Technical Paper 410, "Flow of Fluids Through Valves, Fittings and Pipes."  That way s/he can come to a conclusion based on facts rather than wishful thinking or mind reading.  Note I recognize that Crane might not be a perfect fit, but it has a lot of information about when Darcy's is applicable and provides other formulae for when it's not.  It also provides the answer to the question about whether the mean pressure should be used (by the way, it's not (P1 + P2)/2.  Crane also has a different weighted average than that given by BigInch -- but it applies to more situations than gas pipelines.)

Patricia Lougheed

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

vpl,

I sensed that too, but I'm practicing self-control this week.

Good luck,
Latexman

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

Patricia, Answering the OP's original question, "Is it right the use the mean pressure of the input and output pressure in order to calculate the density of air?" would not give an answer in the total context in which it was asked.  As we have found, the density can be calculated at any pressure and temperature, irrespective of velocity and ultimately mass rate, which is what was really required.

As we have also found, the arithmatic mean pressure would only apply to his/her situation, if the pressure drop was linear and that average mass rate through the system were to be some known function of the observed pressure drop.  Those can be easily related in the context of limited scenarios, such as gas through a pipe, but is actually quite a stretch, if there are no constraints applied, which consequently resulted in some controversy moving along the rocky road to the "answer" we have gotten to so far.

 

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

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

If you chose not to apply it to compressible flow, you can do that if you wish.

If you wish, and when the fluid is other than water at standard conditions, the conductivity is replaced by the permeability of the media. The two properties are related by,

K = kρ g / μ = kg / ν

ρ = density

http://bioen.okstate.edu/Darcy/LaLoi/basics.htm

Is this really important now?

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

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

Darcy's Law ≠ Darcy-Weisbach Equation

Katmar Software - Engineering & Risk Analysis Software
http://katmarsoftware.com

"An undefined problem has an infinite number of solutions"

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

Is it a big step to get there?

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

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

(OP)
Hi guys,

Here I'm back with the solution I was looking for. I made the equation independent of the 'unknown' density. The mass flow rate through an porous media for an isothermal fluid is equal to

[itex]\beta\beta[/itex]

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

(OP)
Hi guys,

Here I'm back with the solution I was looking for. I made the equation independent of the 'unknown' density. The mass flow rate through an porous media for an isothermal fluid is equal to
 
1/2 * rho_s/p_s * k/viscosity * (p_s^2 - p_a^2)/length * surface

k = permeability
p = pressure
rho = density
s = supply
a = ambient

Thanks everyone for the helpful feedback!

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

I see a lot of similarities between your equation and a simplification (pure viscous flow) of the equation in my Perry's Handbook for "the isothermal flow of an ideal gas through porous media".  The only confusing part to me is the "surface".  Is "surface" an area (m2)?

Good luck,
Latexman

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

(OP)
Yes with the surface I mean the traversing area in m²!

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

Then the two equations are dimensionally equal.

Good luck,
Latexman

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