Porous media parameters using Forchheimer plot

Jmaes · June 15, 2023, 16:31

Hello everyone!

I am trying to find the porous media parameters using the Forchheimer plot but I don't know how to calculate the dp/dx term. Does anybody know how? You can find the equation in the attachment.

I tried to plot pressure vs distance graph and fit a second degree polynomial, differentiate it with respect to x and used that final equation to find pressure. Didn't work

fluid23 · June 15, 2023, 17:20

I have never seen this form of the Forchheimer equation, and I think there are some syntax errors. For example, the dot product of a scalar and vector is not a thing, and you somehow end up with a single pressure gradient term out of a 3d velocity vector. I think your equation needs to be fixed, then simplified to 1D. Then you will either use an existing dp/dx vs vf curve to solve for your k and beta values or you will use known k and beta to solve for a pressure gradient (like in a CFD model). Note that in my syntax, bold values denote a tensor of rank 3. If you have isentropic permeability/resistance then those tensors can be a scalar.

DFEq.JPG

LuckyTran · June 15, 2023, 17:43

These are semantic arguments. It is very common to use dots as a symbol for multiplication because x look awful and not using any symbol is very ambiguous. Nobody ever said dots means dot product... And since we're already deep into semantic arguments, there's also more than one dot product. Nobody ever said dot means the inner dot product (versus outer dot product). Darcy's law has roots in 1D and is then generalized to 3D, it shouldn't be mindblowing to see it in a 1D form.

A Darcy/Forchheimer plot is to plot the pressure drop or dp/dx at different flowrates (e.g. velocities). The intercept is related to the kappa and the slope gives you the beta. dp/dx is just pressure drop over the porous media divided by the streamwise length.

fluid23 · June 15, 2023, 18:41

I think it’s reasonable to assume a dot next to a vector means a dot product, however, I take your point. They have used dots to separate scalars and vectors so it’s not clear. Shame on me for assuming.

However, I think my point still stands. The equation was not written correctly since it would require the product of a vector and a scalar to yield a scalar instead of a vector.

That being said, I think you might be inviting confusion with the equating of flow rate and velocity as you did. Maybe I misunderstand.

Just keep in mind that if your charts use flow rate, the constants in the equation are expecting values derived based on vf = Q / A.

Obvious, I know… just a common mistake I see with beginners and worth mentioning.

LuckyTran · June 15, 2023, 19:59

Real Forchheimer plots are plotted against flowrate. It's not inviting confusion when it's part of the answer to the question. It needs to be said.

Most CFD software have massflow rate boundary conditions when there is only a velocity in the navier-stokes equation. Is that also inviting confusion? Maybe. But guess which one is the more practical device used to solve problems by a huge margin?

fluid23 · June 15, 2023, 20:32

EDIT: comment deleted to avoid continued bickering

Jmaes · June 16, 2023, 09:53

hey guys, thank you for the replies. The equation comes from this article: https://luk.staff.ugm.ac.id/jurnal/f...nPractice2.pdf

So, it's just pressure drop vs velocity then? I was just confused because the authors plotted an X vs Y graph (For those of you didn't want to open the article https://files.fm/u/efz4uacjq) and I tried solving for dp/dx but the pressure drop wasn't the same as the data given in Table2 (https://files.fm/u/r3btcuhy2).

LuckyTran · June 16, 2023, 15:23

Part 1 of the paper is also helpful

What you need to do is curve fit the data in Table 2 (pressure drop vs velocity) with a 2nd order polynomial, or equivalently curve Fig 2. You will need to know the measurement distance L=0.9 m. Be careful that there is a factor of 10^-3 in the table.

When you curve fit it, you get something like
dp/dx = av^2+bv+c = 25381*x^2 289180*x+136030

Kappa is 0.000742784/289180 and Beta is 25381*0.000742784/994.49 and that is precisely what is reported in Part 1.

You get slightly different coefficients depending on whether you curve dp vs v or dp/dx vs v or dp vs Q but I trust you to be able to manipulate those parameters. If you give this problem to a lab technician then they plot the measurands dp vs Q and directly give you kappa and beta.

Btw Part 2 spends a lot of time discussing different pressure drops (4-1, 3-1, 2-1, 3-2) and come to the conclusion that the pressure drop over the longest section (4-1) is the most accurate. This result is quite intuitive from practical experience when the accuracy of the distance measurement is more accurate than the differential pressure reading. Curve fitting and using all the available pressure data is the superior approach. It is uncommon knowledge among engineers but widely known among statisticians (because more data must improve confidence!) the Gauss-Markov theorem. Probably the authors and any reviewers of the paper were not aware of this theorem. Or maybe they are fans of the Feynmann approach–reinventing the wheel is a great check that wheels can still be invented.

We can sit here and complain about vector calculus notations used when the problem requires no vectors and no calculus or we can just curve fit the results and calculate the porous media parameters.

Jmaes · June 16, 2023, 15:42

I actually made a mistake, the link was supposed to be for the part 1. And I am aware of everything you said. I did plot the data and in fact I made an excel sheet of all the calculations they did. But like I said, in their X vs Y graph in part 1, I solved the equation for the valu - oh wow I am so stupid. The value of dp/dx I calculated was the approximation and not the experimental value. That's why it was different.

But anyway, I have the pressure vs velocity data from my experiment through which I calculated the permeability and the inertial resistance. Although the fluent simulation gives good approximation at lower velocity, the pressure drops at higher velocities diverge quite a bit. I've tried to reduce mesh size, tried different turbulence models but nothing works, so I've narrowed down the root of the problem to be high amounts of errors in the measurement equipment (highly likely) or lack of knowledge to better estimate the parameters.

June 15, 2023, 20:32	Porous media parameters using Forchheimer plot	#6
fluid23 Senior Member Matt Join Date: Aug 2014 Posts: 947 Rep Power: 18	EDIT: comment deleted to avoid continued bickering

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June 15, 2023, 17:20		#2
fluid23 Senior Member Matt Join Date: Aug 2014 Posts: 947 Rep Power: 18	I have never seen this form of the Forchheimer equation, and I think there are some syntax errors. For example, the dot product of a scalar and vector is not a thing, and you somehow end up with a single pressure gradient term out of a 3d velocity vector. I think your equation needs to be fixed, then simplified to 1D. Then you will either use an existing dp/dx vs vf curve to solve for your k and beta values or you will use known k and beta to solve for a pressure gradient (like in a CFD model). Note that in my syntax, bold values denote a tensor of rank 3. If you have isentropic permeability/resistance then those tensors can be a scalar. DFEq.JPG

June 15, 2023, 17:43		#3
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,761 Rep Power: 66	These are semantic arguments. It is very common to use dots as a symbol for multiplication because x look awful and not using any symbol is very ambiguous. Nobody ever said dots means dot product... And since we're already deep into semantic arguments, there's also more than one dot product. Nobody ever said dot means the inner dot product (versus outer dot product). Darcy's law has roots in 1D and is then generalized to 3D, it shouldn't be mindblowing to see it in a 1D form. A Darcy/Forchheimer plot is to plot the pressure drop or dp/dx at different flowrates (e.g. velocities). The intercept is related to the kappa and the slope gives you the beta. dp/dx is just pressure drop over the porous media divided by the streamwise length.

June 15, 2023, 18:41		#4
fluid23 Senior Member Matt Join Date: Aug 2014 Posts: 947 Rep Power: 18	I think it’s reasonable to assume a dot next to a vector means a dot product, however, I take your point. They have used dots to separate scalars and vectors so it’s not clear. Shame on me for assuming. However, I think my point still stands. The equation was not written correctly since it would require the product of a vector and a scalar to yield a scalar instead of a vector. That being said, I think you might be inviting confusion with the equating of flow rate and velocity as you did. Maybe I misunderstand. Just keep in mind that if your charts use flow rate, the constants in the equation are expecting values derived based on vf = Q / A. Obvious, I know… just a common mistake I see with beginners and worth mentioning.

June 15, 2023, 19:59		#5
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,761 Rep Power: 66	Real Forchheimer plots are plotted against flowrate. It's not inviting confusion when it's part of the answer to the question. It needs to be said. Most CFD software have massflow rate boundary conditions when there is only a velocity in the navier-stokes equation. Is that also inviting confusion? Maybe. But guess which one is the more practical device used to solve problems by a huge margin?

June 16, 2023, 09:53		#7
Jmaes New Member James Ranawadia Join Date: Apr 2023 Posts: 15 Rep Power: 3	hey guys, thank you for the replies. The equation comes from this article: https://luk.staff.ugm.ac.id/jurnal/f...nPractice2.pdf So, it's just pressure drop vs velocity then? I was just confused because the authors plotted an X vs Y graph (For those of you didn't want to open the article https://files.fm/u/efz4uacjq) and I tried solving for dp/dx but the pressure drop wasn't the same as the data given in Table2 (https://files.fm/u/r3btcuhy2).

June 16, 2023, 15:23		#8
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,761 Rep Power: 66	Part 1 of the paper is also helpful What you need to do is curve fit the data in Table 2 (pressure drop vs velocity) with a 2nd order polynomial, or equivalently curve Fig 2. You will need to know the measurement distance L=0.9 m. Be careful that there is a factor of 10^-3 in the table. When you curve fit it, you get something like dp/dx = av^2+bv+c = 25381x^2 289180x+136030 Kappa is 0.000742784/289180 and Beta is 25381*0.000742784/994.49 and that is precisely what is reported in Part 1. You get slightly different coefficients depending on whether you curve dp vs v or dp/dx vs v or dp vs Q but I trust you to be able to manipulate those parameters. If you give this problem to a lab technician then they plot the measurands dp vs Q and directly give you kappa and beta. Btw Part 2 spends a lot of time discussing different pressure drops (4-1, 3-1, 2-1, 3-2) and come to the conclusion that the pressure drop over the longest section (4-1) is the most accurate. This result is quite intuitive from practical experience when the accuracy of the distance measurement is more accurate than the differential pressure reading. Curve fitting and using all the available pressure data is the superior approach. It is uncommon knowledge among engineers but widely known among statisticians (because more data must improve confidence!) the Gauss-Markov theorem. Probably the authors and any reviewers of the paper were not aware of this theorem. Or maybe they are fans of the Feynmann approach–reinventing the wheel is a great check that wheels can still be invented. We can sit here and complain about vector calculus notations used when the problem requires no vectors and no calculus or we can just curve fit the results and calculate the porous media parameters.

June 16, 2023, 15:42		#9
Jmaes New Member James Ranawadia Join Date: Apr 2023 Posts: 15 Rep Power: 3	I actually made a mistake, the link was supposed to be for the part 1. And I am aware of everything you said. I did plot the data and in fact I made an excel sheet of all the calculations they did. But like I said, in their X vs Y graph in part 1, I solved the equation for the valu - oh wow I am so stupid. The value of dp/dx I calculated was the approximation and not the experimental value. That's why it was different. But anyway, I have the pressure vs velocity data from my experiment through which I calculated the permeability and the inertial resistance. Although the fluent simulation gives good approximation at lower velocity, the pressure drops at higher velocities diverge quite a bit. I've tried to reduce mesh size, tried different turbulence models but nothing works, so I've narrowed down the root of the problem to be high amounts of errors in the measurement equipment (highly likely) or lack of knowledge to better estimate the parameters.