[Anna Tian]

Hi,

We can calculate the permeability by the following law,

u=q/Ac=-(k/mu)(dp/dl)

u= fluid velocity
q= flow rate
k= permeability
Ac= cross sectional area
mu= viscosity of the fluid
l= length
dp/dl= pressure gradient in the direction of the flow

May I ask, for u=0 and dp/dl=0, what the permeability value is? Is it infinitely large?

I need that value to estimate(by interpolation) the permeability at low speed.

Thank you very much!

[cdegroot]

Permeability is a constant for low flow rates (i.e. Darcy regime). You can see from the expression that you give that if you plot u vs dp/dl -k/mu is the slope of the line. At zero flow the permeability is still a constant; your equation just reduces to 0=0 since both velocity and dp/dl are zero.

[Anna Tian]

Originally Posted by cdegroot

Permeability is a constant for low flow rates (i.e. Darcy regime). You can see from the expression that you give that if you plot u vs dp/dl -k/mu is the slope of the line. At zero flow the permeability is still a constant; your equation just reduces to 0=0 since both velocity and dp/dl are zero.

Is there a way to calculate the permeability at low speed manually?(We have CFD data for higher flow speed but not low flow speed. We need an estimation of the permeability at low speed.)

How do you define 'low speed'?

What do you mean by 'Darcy regime'? I search for this concept online but didn't find anything.
Could you suggest me a paper about permeability calculation?

[Anna Tian]

Could anyone help me on this question? Thank you very much!

[cdegroot]

Hi, you should look up Darcy's law including the extended form of Forchheimer (http://en.wikipedia.org/wiki/Darcy's_law). The extended form adds a quadratic velocity term. This should be what you need if your flow is not slow (i.e. pressure drop is not linear with velocity). In this case fit a quadratic curve to your pressure vs velocity data and from the coefficients you can find the permeability (and the coefficient on the quadratic term if you need it).

[Anna Tian]

Originally Posted by cdegroot

Hi, you should look up Darcy's law including the extended form of Forchheimer (http://en.wikipedia.org/wiki/Darcy's_law). The extended form adds a quadratic velocity term. This should be what you need if your flow is not slow (i.e. pressure drop is not linear with velocity). In this case fit a quadratic curve to your pressure vs velocity data and from the coefficients you can find the permeability (and the coefficient on the quadratic term if you need it).

Thanks for your answer. I looked at the page. I calculated the permeability with the data for high flow velocity(the data I have) in that way. But now I need an estimation of the permeability for low speed flow. At least I need to know the way to calculate the permeability for zero velocity. These contents are not included in that page.

[cdegroot]

Permeability is a constant. It should work for zero velocity as well. I suggest looking up the book "Principles of Heat Transfer in Porous Media" by Kaviany. It is a really thorough book and I think it will answer your questions.

[Anna Tian]

Originally Posted by cdegroot

Permeability is a constant. It should work for zero velocity as well. I suggest looking up the book "Principles of Heat Transfer in Porous Media" by Kaviany. It is a really thorough book and I think it will answer your questions.

Really? Permeability is the same for all the flow velocity? If it's only for low velocity, is there a way to calculate it manually?

[cdegroot]

In the context of the extended Darcy law yes it can be treated as a constant. If you don't retain the quadratic term and just use Darcy's law then permeability will change with velocity when the velocity gets large. A constant permeability simply means that at low flow rates the pressure drop is linearly related to the flow rate. To put this another way, permeability is a function of pore geometry only. The quadratic term treats influences of inertia.

[lyes43]

Hey you can contact me to send you interesting materials on permeability.

Regards

[Anna Tian]

Originally Posted by cdegroot

In the context of the extended Darcy law yes it can be treated as a constant. If you don't retain the quadratic term and just use Darcy's law then permeability will change with velocity when the velocity gets large. A constant permeability simply means that at low flow rates the pressure drop is linearly related to the flow rate. To put this another way, permeability is a function of pore geometry only. The quadratic term treats influences of inertia.

How do you define 'low speed'? Laminar?

[cdegroot]

By low speed, I mean that the pressure drop is linear with respect to velocity

I don't mean laminar. For porous media flows, usually the Reynolds number based on pore diameter should be less than one to be considered low speed (i.e. Darcy's law is valid), but it depends a bit on what geometry you are looking at. If you plot pressure drop vs. flow velocity and it is not a straight line then you can assume that the flow speed is not low enough.

[Anna Tian]

Originally Posted by cdegroot

By low speed, I mean that the pressure drop is linear with respect to velocity

I don't mean laminar. For porous media flows, usually the Reynolds number based on pore diameter should be less than one to be considered low speed (i.e. Darcy's law is valid), but it depends a bit on what geometry you are looking at. If you plot pressure drop vs. flow velocity and it is not a straight line then you can assume that the flow speed is not low enough.

So at the high velocity, will that slope larger than the slope at low speed? Or smaller?

[cdegroot]

At higher flow rates the slope will be higher, i.e. more pressure loss for the same increase in flow rate.

[Anna Tian]

Originally Posted by cdegroot

At higher flow rates the slope will be higher, i.e. more pressure loss for the same increase in flow rate.

So a reasonable permeability vs. velocity graph should look like the attached picture?

[cdegroot]

Yes if you are just using dp/dl=mu/k*u with no quadratic velocity term the yes that graph could make sense. The slope is practically vertical on the right part of the graph which I don't think would happen. But the trend seems right.

[Anna Tian]

I'd like to include the inertia loss term. May I ask what is the formula to compute the loss coefficient for the nonlinear term? And if the loss term has to be included then the extended Darcy law with the inertia loss term need to be used. So we can't use the simplified Darcy law formula that we discussed before in this post to compute the permeability anymore. Then how to compute the permeability?