Reversing the Dv equation.

[Legendmir]

I have been watching the below video a few times trying to understand how he finds the fuel he will need to achieve his mission.

The Dv equation is easy enough so iv been trying the equations he has sown in the video and checking my answer with the Dv equation afterwards and no matter how many times i try i don't get the Dv i put in to the equation.

Ill quickly run through what iv been doing. i do hope you can spot what iv done wrong.

e^(Dv/Isp/g) = X

Then i find the fuel mass by doing d x (X-1)/1 + X/9 giving me F at least that's the impression i got but its wrong as when i add F to the mass and adjust for dry mass then punch it all in to the Dv equation i get a total different Dv to what i punched in originally.

Edited January 29, 2014 by Legendmir
Answered

[Tank Buddy]

Here's what my aerospace book says:

Dry mass=full mass*2.71828^-(Desired DV/exhaust velocity) Note the negative exponent.

and

Full mass= Dry mass*2.71828^(Desired DV/exhaust velocity)

2.71828 is some constant.

[numerobis]

2.7 is e.

d x (X-1)/1 + X/9

I'm assuming lower-case x is multiplication, and d is your dry mass?

Dry mass times (X-1) is the propellent mass, so that makes sense.

Then you're dividing it by 1 for some reason, which does nothing. Then you add X/9, which is again quite mysterious. I think you've got some typos in there.

What I worked it out to be some time ago is that the dry mass of the tanks you need is d(X-1) / (1 - X + 8) ; the mass of the fuel in those tanks is 8 times more. So you'd have 9d(X-1) / (9 - X) total mass.

[Rusty6899]

Firstly, choose an appropriate engine for the stage, based on thrust requirements/delta-v requirements etc. This comes with experience, but can be estimated reasonably well.

Secondly, add the mass of all the parts in the rocket (except the fuel tanks, because we don't know how much fuel we need). Call this M_total

Calculate the required m0/m1 = EXP(dv required / (Isp*9.81))

calculate m1...

m1= (8*M_total)/(9-(m0/m1)) -----> This only works for fuel tanks where full/empty mass ratio = 9:1. Replace 8 with 4.35 and 9 with 5.35 for the other two tanks.

Calculate m0....

m0 = m1* (m0/m1)

m0 - m1 is the minimum required mass of fuel for the stage.

You may want to add a factor of safety of 1.1 or 1.2 to ensure you have enough, but I have used this method a fair few times and never came up short.

Edited January 29, 2014 by Rusty6899

[Silverware]

g = 9.81

Wet = Full Mass of stage

Dry = Dry Mass of stage

Isp = Specific Impulse (in seconds)

ln = Natural Log

dV = Isp*ln(Wet/Dry)*g

Isp = (dV/ln(Wet/Dry))/g

Wet = Dry * e^(dV/(Isp*g))

Dry = 1/(e^(dV/(Isp*g))/Wet)

All the equations required to do forwards and backwards calculations.

Know the approximate dry mass and delta V required? Use the wet calculation and throw the Isp of the engine in!

[Rusty6899]

Here's an example.

From a low Munar orbit, I want to land on the Mun and return to Kerbin in one stage.

According to the Delta-V map I have, I will need 2350 m/s. (I will round it to 2500 for a factor of safety).

I choose an LV-909 engine, as I need landing legs to fit on it, it provides sufficient thrust for Mun landing/take off and has low mass with a decent Isp (390) in vacuum.

So far my rocket is designed to have...

Mk 1 Lander Can, 3 Microstruts, 3 OXSTATs, a TR-18A decoupler, a Mk16 Parachute and the LV-909.

M_total = 0.66 + 0.045 + 0.015 + 0.05+ 0.1 + 0.5 = 1.37 tonnes

m0/m1 = EXP(2500/(390*9.81)) = 1.922

m1 = 8*1.37/(9-1.922) = 1.548t

m0 = 1.548 * 1.922 = 2.975t

mass of fuel required = 2.975 - 1.548 = 1.427t

Therefore an FL-T200 and a FL-T100 can be used.

Finally, double check this with the Tsiolkovsky Equation...

dv=390*9.81 * ln(3.0575/1.5575) = 2580m/s --------> The values of m0 and m1 are slightly different because the mass of fuel and fuel tank are slightly higher than the calculated minimums.

[UmbralRaptor]

e^(Dv/Isp/g) = X

Looks good for getting the mass ratio of a given stage, though I'd write it as e^(ÃŽâ€V/(Isp*g0)) in the hopes of extra clarity.

Then i find the fuel mass by doing d x (X-1)/1 + X/9 giving me F at least that's the impression i got but its wrong as when i add F to the mass and adjust for dry mass then punch it all in to the Dv equation i get a total different Dv to what i punched in originally

I'm a bit confused by this as written.

A first approximation would be X == (Full_Tankage + Other_Mass) / (Empty_Tankage + Other_Mass)

Other_Mass being upper stages, the payload, engines, or whatever.

For FL-T and Rockomaxx stock parts you can assume that Empty_Tankage * 9 == Full_Tankage

After a bit of Algebra: Empty_Tankage == (Other_Mass - X*Other_Mass)/(X+9)

Note the assumption that you know how much that Other_Mass is.

That said, I put together some formulas for generating single stages of various performance a while back:

(Though it can stick you with intermediate tank sizes)

Edit: What is it with you people and 9.81? KSP uses 9.82 m/s² for g0...

Edited January 29, 2014 by UmbralRaptor

[Rusty6899]

Edit: What is it with you people and 9.81? KSP uses 9.82 m/s² for g0...

Is that true? Can't say it makes a huge difference to be honest.

[UmbralRaptor]

Is that true? Can't say it makes a huge difference to be honest.

It's a small difference, but rather easy to spot if you check the ingame numbers. eg:

[Legendmir]

g = 9.81

Wet = Full Mass of stage

Dry = Dry Mass of stage

Isp = Specific Impulse (in seconds)

ln = Natural Log

dV = Isp*ln(Wet/Dry)*g

Isp = (dV/ln(Wet/Dry))/g

Wet = Dry * e^(dV/(Isp*g))

Dry = 1/(e^(dV/(Isp*g))/Wet)

All the equations required to do forwards and backwards calculations.

Know the approximate dry mass and delta V required? Use the wet calculation and throw the Isp of the engine in!

Thank you so much. the first one i came across that i could understand was the one above.

I quickly checked it against different dry masses and different Dv's and it worked like a charm.

Looks good for getting the mass ratio of a given stage, though I'd write it as e^(ÃŽâ€V/(Isp*g0)) in the hopes of extra clarity.

Id do that if i had any idea how to do that symbol XD

[numerobis]

Is that true? Can't say it makes a huge difference to be honest.

It is true: g0 is 9.82 in KSP. Makes a bit more than a part in a thousand difference.

[Silverware]

Can't say I have noticed it myself, but then again I don't often look too hard at my calculations, as I leave a large amount of wiggle room.

[James_Eh]

Edit: What is it with you people and 9.81? KSP uses 9.82 m/s² for g0...

Ah-HA!!! So THAT'S why I keep getting to 99,898.167 m and then plummeting to the ground !!!