From: vince@offshore.ai (Vincent Cate)
Newsgroups: sci.space.tech
Subject: Pressure Tank Mass (was Re: Basci question about rocket shapes.
NNTP-Posting-Host: 207.42.133.230
I could not reply to my May 25 post with my code for pressure tank
mass calculations. I found the error in my code. The:
double SphereVolume = 4/3 * PI * Rcubed;
Needs to be:
double SphereVolume = 4.0/3.0 * PI * Rcubed;
Because in Java or C a simple 4/3 is equal to 1.
The book "Ballistic Missile and Space Vehicle Systems" by Seifert
and Brown has a derivation for tank mass which I will summarize here.
These are from formulas 10.30 to 10.34 on pages 227 and 228 in my book.
Strength will mean tensile strength property of the tank material.
Density will mean the density of the tank material.
The thickness of sphere = Ts = Pressure * Radius / (2 * Strength)
The thickness of cylinder = Tc = Pressure * Radius / Strength
So Tc = 2 * Ts, so cylinder wall is twice as thick as sphere.
Mass of sphere = Ms = 4 * Pi * R^2 * Ts * Density
Substituting for Ts and then Volume for 4/3 * Pi * R^3
you can get:
Ms = 3/2 * Pressure * Volume * Density / Strength
Mass of tank section = Mt = 2 * Pi * R * Tc * Length * Density
Substituting for Tc and then Volume for Pi * R^2 * Length
you can get:
Mt = 2 * Pressure * Volume * Density / Strength
There are 2 very interesting points.
1) The ratio of Mass/Volume does not depend on size. So
2 small spheres theoretically weigh as much as one with
twice the volume. Same with cylinder sections. There will be
minimum sizes for materials, of course.
2) The cylinder section is 2/1.5 = 1 1/3 the mass of a sphere
of the same volume. Since any sized sphere has the same
mass/volume ratio, there is 1/3 more mass for tank sections.
This is why my the 4/3 in the code for the sphere volume
being converted to 1 made the sphere and tank seem equal. :-)
There were several errors in the derivation in the book.
They had a Pi^2 when they should have had a Pi*R^2.
They don't include the length of the cylinder.
They once put the symbol for Pressure when they mean Density.
And when all done, they said "Thus a cylindrical section of
a tank weighs 50 per cent more than an equivalent spherical
tank of the same volume." The error here is that it is
33.33% more by their results (2 vs 1.5).
In practice, a cylindrical tank that is twice as long as it
is wide will be about 25% more mass than a sphere of
equal volume, because part of this tank is a sphere (the
two ends).
My corrected applet, with tank mass samples at the end of the
sample inputs, is at:
http://spacetethers.com/rocketequation.html
Fixed source code is on the site too.
-- Vince
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Vincent Cate Space Tether Enthusiast
vince@offshore.ai http://spacetethers.com/
Anguilla, East Caribbean http://offshore.ai/vince
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You have to take life as it happens, but you should try to make it
happen the way you want to take it. - German Proverb