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Suppose I have an approximately rectangular prism composed entirely of folded paper. If I place 600lbs on top of these rectangular sheets of paper, the paper should compress. How does this affect water permeability across the sheet? Is the rate across which the fluid flows across the membrane slowed down?

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  • $\begingroup$ Is your water permeating vertically (across all the sheets) or horizontally (along each sheet)? By capillary action or under pressure? Any particular type of paper? $\endgroup$
    – rob
    Dec 11 '14 at 0:22
  • $\begingroup$ It can't permeate vertically as the 600lb weight is on top of the sheet. The bottom is also compressed by the floor. We assume capillary action of flow of fluid, as any type of pressure may alter composition of the paper. The paper is similar to paper towels, except the sheets are folded. $\endgroup$
    – Wise
    Dec 11 '14 at 15:41
  • $\begingroup$ A bit of context regarding the intent of running water through permeable sheets of paper would be very helpful in providing the kind of answer that the OP is looking for. $\endgroup$ Nov 19 '18 at 21:06
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This is outside of my area of expertise, but I figure I'll take a stab at it.

You elaborate in a comment:

We assume capillary action of flow of fluid, as any type of pressure may alter composition of the paper. The paper is similar to paper towels, except the sheets are folded.

The height $h$ of a column in a capillary with radius $r$ is given by $$ h = \frac{2\gamma \cos\theta}{\rho g r} $$ where $\gamma$ is the surface tension, $\theta$ is the contact angle, $\rho$ is the liquid density and $g$ the local acceleration due to gravity. You could presumably measure the ratio $\cos\theta/r$ for your paper towels by putting one in water and measuring how high the wetness climbs; you can get $\gamma,\rho,g$ from other sources.

If you model your paper towels as an array of narrow capillary tubes, you might instead think of the product $$ \rho g h = \frac{2\gamma\cos\theta}{r} $$ as a sort of "capillary pressure," which pulls the liquid up into the capillary gaps in the paper. You might then use this capillary pressure in the laminar flow equations to estimate the the flow rate across each paper towel.

If you can successfully model the behavior of an uncompressed paper towel you can start to consider the compressed ones. There will be several competing effects. Most notably you'll be changing the distance between adjacent layers of paper (surely you've noticed how a folded or two-ply paper towel dries better than a single layer). You'll also crush the paper somewhat and change the structure of the internal capillaries.

You'll want to be careful what you mean by "rate of flow." I think it's plausible that a compressed block of paper would see the dry end get damp faster than an uncompressed block, but to find that the uncompressed block of paper has a higher volume flow rate for a given pressure head on the wet side.

This might be a case where it's simpler just to do the experiment.

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