I was surfing the forum and decided to read this thread, because my son and I are exploring various ideas for fermentation temp control.
I apologize for jumping into this fray and taking sides, but I think some apologies might be in order: from dbdb if he's offbase or from me if I've misunderstood something from this disagreement.
Quote:
Originally Posted by dbdb
I don't really have time or inclination to correct you math, I pointed out it was wrong I would think you could have worked out your error with a little effort. Are you still at school?

I no longer have my college calculus book handy, but I believe the formula you've given is for the
circumference of a circle, not the
area.
1. The
circumference of a circle is
pi*diameter or
2*pi*radius.
2. The
area of a circle is
pi R squared or
pi*radius squared.
Quote:
Originally Posted by dbdb
Area of a 12" diameter circle is 12pi. Area of a 10 inch diameter circle is 10pi.

No, dbdb, the
circumference of a 12" diameter circle is 12pi; that of a 10" diameter circle is 10pi. The
area of a 12" diameter circle is pi*6squared: 3.1416*36=113 sq in. The
area of a 10" diameter circle is pi*5squared: 3.1415*25 = 78.5 sq in.
Quote:
Originally Posted by dbdb
The area of the gap is 12pi  10pi, i.e. 2pi.

No, you've simply given how much bigger around one circle is than the other, appr 6.25" in this case. The gap is simply 1" all the way around(the 10" circle sitting inside the 12" circle).
The
difference in the areas is (pi * 6squared)  (pi * 5 squared) or 11378.5=34.5 sq in. or 30% less than the larger area or 44% larger than the smaller area. Please note that, when comparing these areas, you can't simply plug the difference into the formula "pi*(65)squared" as you did when you were unwittingly computing circumferences instead of areas.
From your formula, I get 37.7"
circumference for the larger circle (12pi) minus 31.42"
circumference of the smaller circle (10pi) for a difference of 6.29 inches or 16.7% fewer linear inches than the larger circle or 20% larger than the smaller circle.
This difference in circumference (gap as you say) is meaningless by itself in this case.
If you applied your calculations to farm land, you've simply calculated how many more linear feet of fence you'd need or save to go around a pasture that was 1000ft vs 1200ft on a side. Your formula won't tell you how much fertilizer or seed you'd need to cover said fields. For square fields, 20% more linear feet of fence, but 44% more volume of seed. The calculations for round fields are a bit more complicated.
Quote:
Originally Posted by dbdb
Volume of the gap at 10 inches high is 2pi*10, i.e. 20pi.

No, this just gives you the surface area of one side of a tube that's 2" in diameter x 10" long. This is not the volume of the gap between a 10" long 10" diameter tube set inside a 10" long 12" diameter tube.
Even if you use the correct formula for area of a circle, you cannot compute volume of the gap simply by plugging the difference in diameter into the formula. You have to compute the volumes first, then subtract them.
Quote:
Originally Posted by dbdb
Volume of a 10 inch cylinder at 15 inches high is 15*10pi, i.e. 150pi. The volume of the gap divided by the volume of the carboy is 20pi/150pi which is 13.33%. The increase in volume is 13.33%.

No, your results do not give volumes. What you have computed is the surface area (of one side) of a tube that is 10 inches in diameter and 15" long. The rest of your statement makes no sense to me, because it appears that you're starting your calculations with the wrong formula  you're trying to jump from circumference to volume (1D or linear to 3D) instead of from area (2D) to volume (3D)
Also, when comparing circumferences and surface areas of tubes of different diameters, it's OK to simply compute the surface area of the difference between the two. The differences are directly proportional and the formulas don't contain parentheses. IOW, pi*diameter1*height  pi*diameter2*height is the same thing as pi*(diameter1diameter2)*height.
OTOH, when comparing the area of circles and volumes of round things, the proportions are exponential, so you can't simply plug the difference between the radii into the formula. This difference, or gap, is a volume. You have to compute the two volumes first, then subtract them.
For example, when comparing the areas of different circles (and, hence, volumes of tubes of various radii), pi*radius1 squared  pi*radius2 squared is not the same as pi*(radius1radius2)squared. Neither is it the same when you multiply these by height. IOW, the difference between the area of 2 circles of 6" and 5" (your 12" and 10" circles) is the area of the 12" circle minus the area of the 10" circle. You can't simply plug the difference between the diameters or radii into your area formula as you can with the formula for circumference.
1. The outside surface area of a tube is circumference*height and is expressed in square units (square inches, square feet, square whatevers).
2. The volume of a tube is radius squared*pi and expressed in cubic units.
Using the correct formula for area of a circle given earlier in this message, the volume of the 10" high 12" diameter container would be 113 sq in * 10 in high = 1130 cu in. The volume of the 10" high 10" diameter container would be 78.5 sq in * 10 in = 785 cu in. The difference in volume (the volume of the gap/space between) would be 1130 cu in  785 cu in = 345 cu in. IOW, 30.5% less volume compared with the larger vessel or 44% greater volume compared with the smaller vessel.
I ran a few other calculations on paper to double check my math and yours.
1. When comparing the differences in circumferences of circles, you can subtract the diameters before multiplying by pi. IOW, (pi*diameter #1)  (pi*diameter #2) is the same thing as pi*(diameter #1 = diameter #2).
2. When comparing the differences in areas, you have to compute the individual areas first, then subtract them. For example, the
volume of the 7" space between a 5" radius tube inserted into a 12" radius tube is a lot greater than the volume of a a 7" radius tube alone.
If you double the diameter of a circle, you also double the circumference as well as the surface area of one surface of a tube of this diameter.
If you double the radius of a circle, you increase the area by quite a bit more than double.
I think all this is much simpler when dealing with rectangles in 2D and 3D. Using circles to make tubes and spheres makes the calculations more complicated.
Quote:
Originally Posted by dbdb
Anyone just sense checking your claim ought to be able to tell without any math that it would never be 30%. Just visualise it.

Visualizing differences in areas of 2D objects and, to an even greater extent, differences in volumes of 3D objects can be deceiving. A rather small difference in one dimension of a vessel can cause a surprising increase in volume.
Dbdb, surely there's a way to explain your stance without being uncharitable or insulting. There are problems in
your calculations (area vs volume), so you should assume a posture that's a little more humble. I think you need to recheck your formulas, redo the math, then see if you're still correct.
I will publicly apologize to dbdb ahead of time, though, if I've totally missed something here and criticized his calculations undeservedly. My wife says I tend to do that when I butt in to a conversation that's already in progress.
Respectfully,
Keith