Height of liquid in cylindrical tank with hemispherical or cone ends.

• Nov 15th 2011, 01:21 AM
RichardGB
Height of liquid in cylindrical tank with hemispherical or cone ends.
I posted a similar question here yesterday and was extremely impressed with the extremely quick and accurate response from e^(i*pi)

I have a similar but slightly more complicated question.

Given a horizontal cylindrical tank (i.e. where the circular end is in the vertical plane), with two hemispherical ends, a known diameter d, the length of the cylinder and containing a known volume of liquid, what is the height of the liquid h from the base of the tank?

See attached picture where being hemispherical ends the distance s = 1/2 the tank diameter d.

How should the formula be modified for non hemispherical ends where the distance s < tank radius, (i.e. an elliptical end), and secondly where the ends are conical?

• Nov 15th 2011, 06:51 PM
bjhopper
Re: Height of liquid in cylindrical tank with hemispherical or cone ends.
Hi RichardGB,
This is a complicated calculation. It involves calculation of volume of a spherical segment as a function of height and reduced radius and the same calculation for the straight cylinder. One method would be to make calculations in terms of h to make a table showing h and volume. Given volume the h would come from the table
• Nov 16th 2011, 01:55 AM
RichardGB
Re: Height of liquid in cylindrical tank with hemispherical or cone ends.
Hi,

Thanks for the input. I had a similar thought but wanted to avoid a table if at all possible since there would always be small errors between the table records unless the granularity was extremely fine and certainly finer than the inputs required.

Am I right in thinking that the complexity is primarily because two different 'shapes' are concerned? viz. the cylinder and the spherical ends, rather than the reduced radius per se? Since if there is a solution for a regular cylinder/sphere, the reduced 'sphere' would just substitute the smaller 'radius' and rather than have a r^3 term we'd have a r^2 * r(small) term
• Nov 16th 2011, 05:54 AM
bjhopper
Re: Height of liquid in cylindrical tank with hemispherical or cone ends.
Hi Richard,
Go to handy math.com. The site has an on line calculator for horizontal cylinder and for a sphere.Formulas are also available relating volume to the height of the liquid surface of each shape
• Nov 17th 2011, 09:37 AM
RichardGB
Re: Height of liquid in cylindrical tank with hemispherical or cone ends.
Hi,

Thanks for the link. Unfortunately this (along with other similar sites I've found), does not quite do what I want.

My problem is that I have a horizontal tank consisting of different sections. e.g. two elliptical ends connected by a cylinder. Calculating volumes is no problem since the formulae are well known, but working back from a known volume of liquid and trying to work out the height within the composite tank is seemingly non trivial.

Regards
• Nov 17th 2011, 07:04 PM
bjhopper
Re: Height of liquid in cylindrical tank with hemispherical or cone ends.
Quote:

Originally Posted by RichardGB
Hi,

Thanks for the link. Unfortunately this (along with other similar sites I've found), does not quite do what I want.

My problem is that I have a horizontal tank consisting of different sections. e.g. two elliptical ends connected by a cylinder. Calculating volumes is no problem since the formulae are well known, but working back from a known volume of liquid and trying to work out the height within the composite tank is seemingly non trivial.

Regards

I think what you seek is impossible
• Nov 18th 2011, 12:43 AM
RichardGB
Re: Height of liquid in cylindrical tank with hemispherical or cone ends.
Quote:

Originally Posted by bjhopper
I think what you seek is impossible

Hi,

I'll be somewhat surprised if it is totally impossible. Every dimension/quantity needed is I believe known, and I suspect the answer involves calculus. My thoughts in support of this are along the following lines.

Think of each section in it's own right. From e^(i*pi) 's response to my other thread http://www.mathhelpforum.com/math-he...tml#post694930
it is possible to calculate the height from a known volume. So think of the liquid only going into the cylindrical part of the composite tank. Clearly the height can be calculated. But in reality some of this liquid goes into the end sections, thus lowering the height in the overall tank, and the volume in the cylindrical section.

Now assuming we can calculate the reduced volume in the cylindrical section we can again calculate the height. etc. to infinity.

Obviously that assumption of mine is the key and the difficult part of this process.
Regards
• Nov 18th 2011, 01:43 PM
bjhopper
Re: Height of liquid in cylindrical tank with hemispherical or cone ends.
Quote:

Originally Posted by RichardGB
Hi,

I'll be somewhat surprised if it is totally impossible. Every dimension/quantity needed is I believe known, and I suspect the answer involves calculus. My thoughts in support of this are along the following lines.

Think of each section in it's own right. From e^(i*pi) 's response to my other thread http://www.mathhelpforum.com/math-he...tml#post694930
it is possible to calculate the height from a known volume. So think of the liquid only going into the cylindrical part of the composite tank. Clearly the height can be calculated. But in reality some of this liquid goes into the end sections, thus lowering the height in the overall tank, and the volume in the cylindrical section.

Now assuming we can calculate the reduced volume in the cylindrical section we can again calculate the height. etc. to infinity.

Obviously that assumption of mine is the key and the difficult part of this process.
Regards

eipi's reply shows a V and an apex angle.What other info is supplied in the current problem other than length of cylinder and radii of both cyl and sphere
• Nov 18th 2011, 02:03 PM
RichardGB
Re: Height of liquid in cylindrical tank with hemispherical or cone ends.
Quote:

Originally Posted by bjhopper
eipi's reply shows a V and an apex angle.What other info is supplied in the current problem other than length of cylinder and radii of both cyl and sphere

Hi,

Apart from the volume of course, non.

Regards