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?

 

[Latexman]

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

 

[mbt22]

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

 

[BigInch]

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

 

[vpl]

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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[Latexman]

vpl,

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

Good luck,
Latexman

 

[BigInch]

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

 

[ione]

As already noted above (mbt22) the Darcy’s law applies to incompressible flow. Check page 1-40 at the link below.

http://home.shirazu.ac.ir/~eor/pdf/Heinemann_-_Fluid_Flow_in_Porous_Media.pdf

For compressible flow through porous media

http://www.intechopen.com/source/pd...y_state_compressible_flow_in_porous_media.pdf

 

[BigInch]

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 = k[ρ] 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

 

[katmar]

Darcy's Law [≠] Darcy-Weisbach Equation

Katmar Software - Engineering & Risk Analysis Software

http://katmarsoftware.com

"An undefined problem has an infinite number of solutions"

 

[BigInch]

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

 

[drir]

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]

 

[drir]

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!

 

[Latexman]

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

 

[drir]

Yes with the surface I mean the traversing area in m²!

 

[Latexman]

Then the two equations are dimensionally equal.

Good luck,
Latexman