Ketelkantel
Like NatiMatt I did some math.
Suppose we have a cilindrical boiler with a width of 20 cm, so a radius of 10, standing upright. Independent of the degree of filling and the height of the boiler, the surface of the fluid in it will always be the same: πr2 .
When we lay the boiler on its side, the surface is dependent on the degree of filling, as can be seen in this cross section.
The level of the fluid is h. Its length is dependent on the angle φ: h = 2 * r * sin( ⅟₂ φ)
The surface of the remaining part of the circle is: O = {(360-φ)/360} * π * r2
The surface of the triangle formed by h, r and r is: D = sin( ⅟₂ φ) * cos( ⅟₂ φ) * r2
And the surface of the cross section of the fluid is the sum of O and D.
Although the real volume of the filling is dependent of the height of the cilinder, the degree of filling is only dependent of φ. Or rather: φ is the result of a degree of filling.
The resulting surface of the fluid is a rectangle, with one side depending on φ and the other on the height of the cilinder. Given a fixed height, the surface will be at its maximum when the boiler is half filled and so φ is 180°.
Now we can find the surfaces of the fluids in layed down kegs dependent on the degree of filling and dependent on the height of the kegs. I took 3 examples: 1 is as high as it is wide, so r = 10 and H = 20. The second is twice its width and the third is 3 times its width.
The y-axis is the percentage of filling, the x-axis is the surface of the fluid in square whatevers.
The 10 % at the top and at the bottom are hardly relevant, as normally one does not fill his boiler to that levels.
An interesting point is the half-filled square keg, that is 20 wide and 20 high: when it is laid on its side we realise the squaring of the circle! At last we, distillers, solved a millennia old mathemathical problem!!
I was surprised to see how small the gain is for this square keg: when half filled the surface of fluid in a laid down keg is only 27 % larger than in an upright keg.
For higher kegs the surface is more or less proportional to the proportion of the keg: when the keg is 2 times as high as wide, the surface of the fluid will be 2 times as large when laid down compared with upright.
But then I would like to see some opinions of chemical technical engineers on heat and fluid and vapour.
And now for a sphere?