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Pressure Tank Mass (was Basci question about rocket shapes.
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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~ Vincent Cate Space Tether Enthusiast http://spacetethers.com/ Anguilla, East Caribbean http://offshore.ai/vince ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~ 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 |
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