Calculate wine volume in a horizontal barrel using a dipstick

Question

I suck at math, but still need a way to mark a dipstick to measure the volume of wine in a barrel.

This question has been asked, but the only answer is to cryptic for me to understand!

My barrel has a Height of 430 mm, Small radius of 136 mm and large radius of 175mm.

Could someone show me how to calculate the volume in this barrel at Depth of 5, 10, 15 mm....?

A spreadsheet would be nice (I don't know how to do Integrations!)

Someone pointed to this as a possible solution, but the French and the math is above my grade!

Answer 1

Assuming you want the solution to the practical (rather than the mathematical) problem, and since you don't specify the exact shape of the barrel, the easy solution would be to pour in known quantities of water and mark your dipstick after each addition.

Answer 2

Assuming your French mathematical find is useful to you, and the last line calculates the amount you're interested in (my French isn't good):

Make yourself a spreadsheet.

Place your values for the following variables into the given cells:

d -> B1
D -> B2
L -> B3
h -> B4

Then paste this equation into any other cell:

=3.14159*(4*POWER(B1-B2,2)/(5*POWER(B3,4))*(POWER(B3/2,5)-POWER(B3/2-B4,5))+2*B2*(B1-B2)/(3*POWER(B3,2))*(POWER(B3/2,3)-POWER(B3/2-B4,3))+B4*POWER(B2/2,2))

And it should show you the result of the calculation (which is given in the last line of the equations that starts $V = \pi ...$)

Note that I've not tested this equation much!

There's a downloadable xlsx version here:

https://www.dropbox.com/s/k83jozysjaxuj74/barrelCalculation.xlsx?dl=0

Note that I originally tried to make a Wolfram Alpha link to solve this, but the site complained about the equation being too big!

Answer 3

The French text gives four different equations, depending on the geometry. There are three equations for a barrel lying on its side ("Pour un tonneau couché") and one for a standing barrel ("Pour un tonneau debout"). The first three equations cover three cases:

• $h \le \frac{D-d}{2}$ ("Si $h \le \frac{D-d}{2}$, alors");
• $\frac{D-d}{2} \le h \le \frac{D+d}{2}$ ("Si $\frac{D-d}{2} \le h \le \frac{D+d}{2}$, alors"); and
• $h \ge \frac{D+d}{2}$ ("Si $h \ge \frac{D+d}{2}$, alors").

It should be clear why three cases are needed: the geometry of the surface changes as $h$ increases. In particular, in the second case, it is bounded by the ends of the barrel.

Unfortunately, the text doesn't give explicit formulae for these three cases. Instead, it gives integrals, for example $\int 2\pi y^2 dx$. You didn't tell us the shape of your barrel (how the radius varies along the axis of the barrel), but that is the $y$ that you must use in the integral.

So...not very helpful, I would say.