Capillary Action and Paper Towel Physics

[yz2275]

Hey guys,

I am a little confused with the concept of Capillary actions with specific regards to paper towels. I understand the mechanics behind a single capillary tube and the height that water reaches in relation to surface tension. However, my professor claims that paper towels absorb as if a lot of capillary tubes are bunched up together, and function based on capillary action.

I sort of get this, but I can't seem to grasp how the equation for a single capillary tube would transfer over to a paper towel. I.E how to find the height of water that a paper towel will absorb based on the capillary equation.

Would someone mind helping me see this?

 

[haruspex]

Think of the paper towel as a mass of filaments. Take a horizontal section through this. You would see a circular or elliptical section of each filament. Add together the circumferential lengths of these and you have the total meniscus length. Multiply that by the surface tension and you have the total uplift force. The water will stop rising when this equals the weight of water lifted.
How you figure out the total meniscus length is an interesting question. Crudely, you could suppose it's all circles packed in a hexagonal matrix to fill the area, but that will be an overestimate.