Dependence of flow rate on pore radius
I think some of us have been confusing each other in our discussions of the dependence of flow rate on pore radius. Jim thinks in terms of a single pore, and I agree that there is an r^4 dependence. Therefore, if you double the pore radius, flow would increase 16X with everything else being equal. However, when I talk about this, I usually think in terms of the membrane, with a constant porosity (probably more relevant in the engineering world). If the porosity is kept constant, there is an r^2 dependence, where doubling the radius results in a 4X increase in flow. So, in effect, the resistive term is really just an r^2 dependence, with pore area providing the extra r^2 in single-pore modeling. Perhaps I was the only one confused by this, but I think it is worth pointing out.
Further, for ultrathin membranes, something interesting appears to happen – the r^2 resistive term weakens a bit. Based on the Tong paper, for a constant porosity, and a 15nm thick membrane, doubling the pore radius from 5nm to 10nm, increases predicted flow by only 3.1X. Doubling again increases flow by only 2.8X. IF you kick the thickness up to 1 micron, and do the same thing, you get the expected 4X dependence. I have probably come at this backword – the big deal is that as you approach smaller pore size, a higher than expected flow is maintained. So there must be a pore aspect-ratio dependence that may be worth investigating a bit. Has anyone else noticed this, because I have not been thinking about it this way…
Chris,
I’m a bit lost. I understand that the flow rate through a single pore will have a r^4 dependence as stated in the Tong paper and Jim’s post. But where is the derivation for the r^2 dependence for an entire membrane area? Am I missing something obvious?
Also, are you talking about fluid or gas flow? Does this hold true for both types?
I was just thinking about fluid flow, but it should be a similar argument for gas. It’s just geometry. Even if there was no such thing as friction or viscosity, you would expect that a larger pore could flow more water, because there is more open area. This change in area is obviously an r^2 factor for a circle. Then, when you consider the fluid dynamics invloved when you shrink a pore and increase the surface/volume ratio, this is apparently an additional r^2 factor that Jim and Tom can explain better than I can. Combining both effects gives the r^4 dependence.
Now if you choose to keep porosity constant, as you decrease the area of a single pore, you add more pores to keep the open area constant. This effectively cancels out the area factor, so I believe we have an r^2 dependence under the constant-porosity condition.
I’m not sure if the physics is 100% correct, but this reasoning seems to lead to reasonable intuition…