Helpful rewording of the rocket equation

[GoSlash27]

The standard form of the rocket equation

ÃŽâ€V= 9.81 x Isp x in( (M+F)/(M))

is useful for predicting what a rocket can do, but we rarely need to know that. What would be truly useful would be an equation to tell you exactly *how much* rocket you need to do a given job. Knowing this ahead of time helps efficiency by avoiding over-engineering your stages (thus making the entire rocket bigger and heavier than it needs to be) or under- engineering (thus wasting time with failed missions or needless math).

This form of the equation assumes that you know:

-How much payload /assorted crap you need to move (in metric tons)

-How much delta-vee you need to generate (in m/sec)

-what kind of rockets you intend to use (mass in metric tons and Isp in seconds) and how many (N)

and

-the ratio of tank mass to fuel in the tanks you'll be using. This ratio is generally constant from one size tank to another for the same type and manufacturer.

The equation will spit out the mass of fuel you need to accomplish the goal and (by extension) the mass of tankage needed to carry it.

I will break it down here into smaller chunks to keep it manageable and because the entire equation looks like crap in text.

First step is to add up all the weight you will be lifting that's not directly related to holding your fuel or spitting it out. This would be your payload, engines, ladders, decouplers, science, etc. It does *not* include tanks or fuel (because you don't know that yet)

M= P+Sr+E

where

P = payload and E=total mass of your engines and Sr = random stuff (in metric tons)

and

Rwd= e^( ÃŽâ€V/9.81x Isp)

where e=the natural logarithm base (2.718)

ÃŽâ€V= your desired change in velocity in M/Sec

and

Isp is the specific impulse of your chosen engine type in seconds.

With these established, we can express the entire equation

F=M(Rwd-1)/(1+Rtf)

where

F= the required mass of fuel and oxidizer (in metric tons)

M= defined above (in metric tons)

Rwd= defined above (ratio)

and

Rtf= the ratio of the mass of your chosen tank type to the mass of the fuel it holds. (ratio)

Having defined F, we can figure out how much mass our tanks will cost us using the equation

Mt= F*Rtf

where Mt = tank mass in metric tons.

And so our stage mass is known.

Ms=F+E+Sr

Add this to the payload, and you know how much mass the preceding stage will have to accelerate.

Edited June 2, 2014 by GoSlash27
typo in the equation...

[Vanamonde]

I made a version of this for myself, but yours is tidier.

(Also, thread moved to the how-to section.)

[UmbralRaptor]

Yeah, reversing the rocket equation to generate stages is a good idea. I should probably update my notes to include ARM tanks, though.

[davidpsummers]

I came up with a negative number (1-Rwd and Rwd-1 can't both be positive?). An example of this calculation would be great.

[GoSlash27]

You're right! There was a typo in the equation. I'll fix it after I post this.

As a very simple example, say we want to design a rocket to deliver a 10 ton payload directly from LKO to the mun. We will use the LV-909 engine due to it's efficient Isp.

our required delta v is 1070, but we'll add 10% as a safety margin to cover pilot error and rendezvous. call it 1200.

Our engine makes 390 seconds of ISP in vacuum and weighs .5t.

We will be docking, so we will need to add RCS and 4 thrusters as well as a brain and reaction wheel. 1.55 tons. Plus a docking port to decouple the load, so 1.6 tons plus the mass of our engine is 2.1 tons. Add our payload and it's 12.1 tons.

Our Rwd is e^(deltaV/(9.81*Isp)

2.72^(1200/(9.81*390)

0r 1.368

Almost all liquid fuel rocket tanks have a ratio of .125 for tank to fuel, so Rtf is .125

So now we would fill in the blanks.

F=M(Rwd-1)/(1+Rtf)

F=(12.6)(1.368-1)/(1+.125)

F=(12.6)(.368)/(1.125)

F= 4.12

We need to carry 4.12 tons of fuel and F(Rtf) or (4.12) (.125) = .515 tons of tank.

Closest we can get to that is an FL- T800 plus another 120k of fuel. That's a round 8 + an Oscar B.

So that's 4.5 + .136 + .079 = 4.715t loaded and .5+ .025 + .015 =.054t dry

Our mass including the fuel is 12.6 + 4.715 = 17.315 and the mass with this stage dry is 12.6+.054 = 12.654.

plugging these numbers into the original rocket equation as a sanity check

deltaV= 9.81*390* ln(17.315/12.654) = 1,199.8 deltaV, as expected. So we can use these numbers to plan the preceding stage.

best,

-Slashy