Surface area in a towel?

So I'm looking at this ordinary bath towel, maybe a meter long and half meter wide and the tufts are about a millimeter apart and a millimeter high on both sides the tweed (little in joke there🙂, what would the total surface area be? I assume there would be an additional area from each thread that makes up the individual tufts, but don't know how much that represents. Any ideas what the total surface area is?

I would think that the tufts are woven along lines in one direction. Imagining each tuft as a small cilinder along these lines, I would guess the following:
- number of tufts: a line every two (?) milimeters or 250 lines and 1000 in each row (just one mm next to the other) gives 25000 tufts
- surface of one tuft: pi x 1mm = 3.15 square mms
- total added surface 3.15 x 12500 square mms approx 80 000 square mms or 0.08
square meters
- double this for both sides = 0.16 square meters added to the 2x0.5 square meters gives a total surface of 1.16 square meters

just a rough estimate to shoot at 😕

Originally posted by sonhouse
So I'm looking at this ordinary bath towel, maybe a meter long and half meter wide and the tufts are about a millimeter apart and a millimeter high on both sides the tweed (little in joke there🙂, what would the total surface area be? I assume there would be an additional area from each thread that makes up the individual tufts, but don't know how much that represents. Any ideas what the total surface area is?

Almost infinite.

Take a look at the fascinating world of fractal geometry!

http://www.calresco.org/fractal.htm

Originally posted by wolfgang59
Almost infinite.

Take a look at the fascinating world of fractal geometry!

http://www.calresco.org/fractal.htmn

Infinitesimal doesn't mean infinitely long. If it has to have a meaning, the level of detail should be in line with the scale that is relevant. A 1 x 0.5 m towell measurement has no meaning at microscopical level. The Mandelbrott analysis gives an estimate of a 1.25 factor for the coast of Britain. At microscopic level it would be many orders of magnitute higher.

Originally posted by Mephisto2
Infinitesimal doesn't mean infinitely long. If it has to have a meaning, the level of detail should be in line with the scale that is relevant. A 1 x 0.5 m towell measurement has no meaning at microscopical level. The Mandelbrott analysis gives an estimate of a 1.25 factor for the coast of Britain. At microscopic level it would be many orders of magnitute higher.

I see the price of bread has gone up again.

Originally posted by wolfgang59
I see the price of bread has gone up again.

How insightful.

Originally posted by sonhouse
So I'm looking at this ordinary bath towel, maybe a meter long and half meter wide and the tufts are about a millimeter apart and a millimeter high on both sides the tweed (little in joke there🙂, what would the total surface area be? I assume there would be an additional area from each thread that makes up the individual tufts, but don't know how much that represents. Any ideas what the total surface area is?

Imagine that the tufts are in a square grid, and each is a cylinder with radius 0.5mm, then:

tufts per towel side = 1000 * 500
surface area of side of each tuft (assuming tufts are smooth) = 2 * pi * 0.0005 * 0.001

the area of the top of each tuft, plus the portion of the towel uncovered by tufts will be equal to the total area of the towel face untufted, so:

total area of tofty towel = untufty_area + tufts_per_side * n_sides * surface_area_of_a_tuft
total_area_of_tufty_towel = 1 + 1000 * 500 * 2 * 2 * pi * 0.0005 * 0.001
= 4.14 m^2

To see what a fractal towel is like, assume that each tuft is itself like a minature tufty towel, then the area of the tufts will increase by a factor of approximately 4.14

As we go down each level of fractality, because we increase the total area by a factor of more than 1, the series will not converge and an infinitely tufty towel will have an infinite area.

Originally posted by Mephisto2
How insightful.

More insightful than:
"Infinitesimal doesn't mean infinitely long"

Originally posted by wolfgang59
More insightful than:
[b]"Infinitesimal doesn't mean infinitely long"[/b]

Really? You may need a checkup on that statement.

Originally posted by iamatiger
Imagine that the tufts are in a square grid, and each is a cylinder with radius 0.5mm, then:

tufts per towel side = 1000 * 500
surface area of side of each tuft (assuming tufts are smooth) = 2 * pi * 0.0005 * 0.001

the area of the top of each tuft, plus the portion of the towel uncovered by tufts will be equal to the total area of the towel face unt ...[text shortened]... e than 1, the series will not converge and an infinitely tufty towel will have an infinite area.

So the question is how much does water react to surface area? If the surfaces were smooth one fiber tufts, there would be X amount of water that towel can hold determined by experiment, say how much water is left by weight after it stops dripping when immersed in water.

Then compare that to a towel that has multiple threads on each tuft. That would seem to me an experimental verification of the fractility of such a towel. I would think if say, there were 4 tufts per thread then you might correlate that to a fractal ratio of about 4 because it holds 4 times the water.

I think the rate that a towel can take up water depends on its surface area, but the total amount of water it can hold depends on its volume.

As we have seen, the surface area, even with smooth fibres, is quite big, but what about its volume?

To calculate the volume of a non-tufty towel we have to estimate its thickness. Let's say it is 2.5mm thick

then, without tufts, its volume is 0.5*1*0.0025 = 0.00125 m^3

tufts per towel side = 1000 * 500
volume of each tuft (assuming tufts are smooth) = pi * 0.0005^2 * 0.001
volume of all tufts = 0.000785398163 ^3

So making the towel tufty has increased its volume by about 63%, it can absorb a bit more water, although not loads more, but it can absorb this water much faster than if it was non-tufty, which is what makes it a good towel.

Possibly needs a WARNING "BAD" WORDS ETC

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