Thank you for this. I'm currently doing research on how a near-Newtonian fluid will flow into paper. I think with this playlist I can get some math going. Is there a textbook on this you would recommend? Thank you
Great video Professor. However, does the effective (or interstitial) velocity is same as "pore velocity" (or physical velocity) in packed beds, as described in many textbooks?
As explained in the video, many textbooks (not all) state that the interstitial velocity v_e equals the apparent velocity divided with the porosity. As explained in the video, that is incorrect. Hence the crossed over equal sign in the video miniature. The tortuosity of the flow channels influences the interstitial velocity, just as explained in the video
Re-reading your question, which I find a bit difficult to understand, perhaps I should clarify: The effective velocity v_e I talk about in this video is the _actual_ velocity in the pores. Many text books on various topics where porous media is dealt with (including books on packed beds) get this wrong and forget to take tortuosity into account. Unfortunately, different text books use different terms for the velocity (which adds to the confusion). Text books that do deal with tortuosity are unfortunately also sometimes confused, confusing the tortuosity factor with tortuosity. The tortuosity factor is the tortuosity squared. So if the first level of understanding is that of these erroneous text books, then the second level is to realize that tortuosity matters. Epsteins article on this matter is probably rather difficult for the novice in the field of porous media, but its clear argumentation proves that these text books simply are wrong. The third level of understanding is to realize that you typically have a distribution of flow velocities in each flow channel, just as text books on fluid dynamics explain regarding (laminar or turbulent) flow in pipes. The effective velocity I talk about in this video is the flux weighted average of the real, physical flow velocity in the flow channels.
@@PLE_LU Thanks for the response. My query is regarding the usage of V_e, not on the tortuosity. Let me put the question in the following way: "A typical representative elementary volume (REV) in a packed bed has different pores. Now, there should exist a velocity component within the pores (voids) which is termed as, say 'pore velocity (or true physical velocity)'. How this pore velocity relates to the interstitial velocity? Can the mean pore velocity be called the interstitial velocity?" I totally agree that the adoption of different characteristic lengths and velocities in the literature adds to the confusion. Thus, leading to ambiguity over such topics.
@@rajendrakumar-bn7ch well, although you claim your question is not about tortuosity, it really is. You see, what some (or perhaps even most) text books on packed beds refer to as the "true velocity" (or some similar name) is not the true velocity at all, but just the value you get when you divide the apparent velocity with the porosity.This fake velocity is always less or equal to the _actual_ true velocity (equal to if tortuosity = 1). To calculate the true (average) velocity you need to multiply with the tortuosity as explained in the video above. Why some text books on packed beds insist on calling a fake velocity the "true velocity" I do not understand, but I think the best way to judge if a text book is wrong about what the velocity in reality is in the pores is to check if they say anything about tortuosity. If they don't, I would say the chance is 95% that the text book is wrong. Funny enough, you might find a text book that doesn't mention tortuosity but uses Kozeny-Carmans equation or Ergun's equation to describe the flow in the pores. Those equations take tortuosity into account, but hide the tortuosity in the factor 150 (or sometimes 200). Epstein's article explains where this factor comes from. As I understand it, the typical tortuosity in a packed bed with spherical particles is the square root of 2, i.e. approximately 1.4. What many text books refer to as the "true" velocity in the pores is thus actually about 1/1.4=0.71 => 71% of what the true velocity in the pores actually is.
Dear sir Nice video. Can you please guide me how to get proof of below equation? P(Rate of change of saturation of water w.r.t time) +(Rate of change of Velocity of water in one dimensional flow through porous media) =0. Where P is the porosity of the medium. It is continuity equation
Do I understand you correctly that you are asking for a proof of the continuity equation? The continuity equation is a way to express as an equation the principle of conservation of mass. That principle has been proven through experiments to be true in most cases, exceptions being things like nuclear fission, nuclear fusion etc. You can't prove the continuity equation through other equation, in the same way that you can not prove through equations that the speed of light is constant.
![[CFD] Porous Zones in CFD](http://i.ytimg.com/vi/sOQMXxoKFQM/mqdefault.jpg)
Thank you for this. I'm currently doing research on how a near-Newtonian fluid will flow into paper. I think with this playlist I can get some math going. Is there a textbook on this you would recommend? Thank you
Convection in Porous Media by Nield & Bejan might help you. But be careful with different terminologies from various texts.
Great video Professor. However, does the effective (or interstitial) velocity is same as "pore velocity" (or physical velocity) in packed beds, as described in many textbooks?
As explained in the video, many textbooks (not all) state that the interstitial velocity v_e equals the apparent velocity divided with the porosity. As explained in the video, that is incorrect. Hence the crossed over equal sign in the video miniature.
The tortuosity of the flow channels influences the interstitial velocity, just as explained in the video
Re-reading your question, which I find a bit difficult to understand, perhaps I should clarify: The effective velocity v_e I talk about in this video is the _actual_ velocity in the pores. Many text books on various topics where porous media is dealt with (including books on packed beds) get this wrong and forget to take tortuosity into account. Unfortunately, different text books use different terms for the velocity (which adds to the confusion). Text books that do deal with tortuosity are unfortunately also sometimes confused, confusing the tortuosity factor with tortuosity. The tortuosity factor is the tortuosity squared.
So if the first level of understanding is that of these erroneous text books, then the second level is to realize that tortuosity matters. Epsteins article on this matter is probably rather difficult for the novice in the field of porous media, but its clear argumentation proves that these text books simply are wrong.
The third level of understanding is to realize that you typically have a distribution of flow velocities in each flow channel, just as text books on fluid dynamics explain regarding (laminar or turbulent) flow in pipes. The effective velocity I talk about in this video is the flux weighted average of the real, physical flow velocity in the flow channels.
@@PLE_LU Thanks for the response. My query is regarding the usage of V_e, not on the tortuosity. Let me put the question in the following way:
"A typical representative elementary volume (REV) in a packed bed has different pores. Now, there should exist a velocity component within the pores (voids) which is termed as, say 'pore velocity (or true physical velocity)'. How this pore velocity relates to the interstitial velocity? Can the mean pore velocity be called the interstitial velocity?"
I totally agree that the adoption of different characteristic lengths and velocities in the literature adds to the confusion. Thus, leading to ambiguity over such topics.
@@rajendrakumar-bn7ch well, although you claim your question is not about tortuosity, it really is. You see, what some (or perhaps even most) text books on packed beds refer to as the "true velocity" (or some similar name) is not the true velocity at all, but just the value you get when you divide the apparent velocity with the porosity.This fake velocity is always less or equal to the _actual_ true velocity (equal to if tortuosity = 1). To calculate the true (average) velocity you need to multiply with the tortuosity as explained in the video above.
Why some text books on packed beds insist on calling a fake velocity the "true velocity" I do not understand, but I think the best way to judge if a text book is wrong about what the velocity in reality is in the pores is to check if they say anything about tortuosity. If they don't, I would say the chance is 95% that the text book is wrong.
Funny enough, you might find a text book that doesn't mention tortuosity but uses Kozeny-Carmans equation or Ergun's equation to describe the flow in the pores. Those equations take tortuosity into account, but hide the tortuosity in the factor 150 (or sometimes 200). Epstein's article explains where this factor comes from.
As I understand it, the typical tortuosity in a packed bed with spherical particles is the square root of 2, i.e. approximately 1.4. What many text books refer to as the "true" velocity in the pores is thus actually about 1/1.4=0.71 => 71% of what the true velocity in the pores actually is.
@@PLE_LU
It would be helpful if some literature on the estimation of tortuosity in randomly packed beds, or similar can be provided.
Thanks for sharing!
Dear sir
Nice video.
Can you please guide me how to get proof of below equation?
P(Rate of change of saturation of water w.r.t time) +(Rate of change of Velocity of water in one dimensional flow through porous media) =0.
Where P is the porosity of the medium.
It is continuity equation
Do I understand you correctly that you are asking for a proof of the continuity equation? The continuity equation is a way to express as an equation the principle of conservation of mass. That principle has been proven through experiments to be true in most cases, exceptions being things like nuclear fission, nuclear fusion etc.
You can't prove the continuity equation through other equation, in the same way that you can not prove through equations that the speed of light is constant.
@@PLE_LU
Ok sir. Thank you
Thank-you so much!!
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