Problem finding fuel tank size (reverse delta-v calculation)

[FortyCaliber]

I am using PartGen from Lando to create custom tank sizes.

With a given craft of a known weight (minus the fuel and fuel tank) I want to calculate the require tank+fuel size.

I have the following:

1430 m/s = (9.81 x 350) x ln(m1/m2)

craft mass = 5.035

I have the delta-v and the Isp and I even have the craft weight... except this formula assumes that the "empty" weight includes the empty tank mass too.

If I use the formula using this mass, I get a number that tells me the mass of fuel used for that burn. However, since the tank mass is missing from the craft mass, when I test the numbers, I always come in under 1430.

Here's the problem:

I can find the tank mass easily by knowing the amount of fuel (fuel mass/8 = tank mass.) However, I can't calculate the fuel necessary for the 1430 m/s burn without knowing the empty tank mass and I can't calculate the empty tank mass without knowing the fuel. I put this in Excel and got what I expected: a circular reference warning.

I can get close to my number running multiple layers of calculations by adding the tank mass to the craft mass and re-calculating the fuel and then using the average of the first calculation and second calculation of tank mass as my assumed tank mass and using the second fuel calculation as my fuel. This isn't ideal as it's not precise and I don't like it.

Does anyone with a more practiced math mind have a solution for this problem; is there a way to do this with pure math?

Any help is appreciated.

[agentKmurph]

So your only using a single stage right? Because if so you can use the properties of logarithms to solve for the empty weight.

ln( x / y) is equivalent to ln(x) - ln(y)

[numerobis]

Start with m1 = m0 e^{dV/Isp.g0}.

Break down m1 = mpayload + mtanks + mfuel, whereas m0 = mpayload + mtanks. If you're using the normal tanks (not the oscar-B and round-8) it turns out that mfuel = 8 mtanks. Substitute it all in and solve.

Tanks come in integer quantities, so round up.

Edited December 16, 2013 by numerobis

[I_Killed_Jeb]

1430 m/s = (9.81 x 350) x ln(m1/m2)

craft mass = 5.035

1430 = (9.81*350) x ln [ (5.035+ 1.125x)/(5.035 + 0.125x) ]

Solve for x

1.125x is the fuel+tank mass you want

[numerobis]

By the way, g0 is 9.82 in KSP (not 9.81) for reasons unclear.

[FortyCaliber]

"For Reason Unclear"

It's just a Kerbal thing.

Thanks for the help everyone... I'm going to plug some numbers in Excel.

I had dinner and wine so I'm ready for maths.

[I_Killed_Jeb]

It should really only require a calculator at this point..

[FortyCaliber]

Then I guess I should dig my casio watch out.

[I_Killed_Jeb]

your casio watch can do exponents? very impressive

[Imran60]

1430 = (9.81*350) x ln [ (5.035+ 1.125x)/(5.035 + 0.125x) ]

Solve for x

1.125x is the fuel+tank mass you want

Thanks, It is the best method to find the size of fuel tank.