Please help me validate these maths

[Doozler]

where is 9.8?

It's a conversion factor, used to make sure that people don't get confused between units. Putting in the g (the 9.8m/s/s) means that you get the same number for Isp whether you measure it in m/s or feet/ s (or whatever).

Criminally lazy people (like me) leave it out.

[K^2]

It's not about ft/s vs m/s. It's about whether you use Newtons or pounds for thrust and weight.

[KASASpace]

It's a conversion factor, used to make sure that people don't get confused between units. Putting in the g (the 9.8m/s/s) means that you get the same number for Isp whether you measure it in m/s or feet/ s (or whatever).

Criminally lazy people (like me) leave it out.

I was referring to the gravitational constant required for calculating D-v.

Unless, by Isp you mean effective exhaust velocity.

[K^2]

I was referring to the gravitational constant required for calculating D-v.

There is no gravitational constant in these equations. There is Earth's gravity, but that's only there if you are using silly units.

[KASASpace]

So, you basically didn't put it into the equation, but you said it afterwards.

It is in the original Tsiokovsky(probably spelled that wrong) equation, though. Isp is specific impulse, not exhaust velocity.

[Doozler]

Isp is specific impulse, not exhaust velocity.

Yes, you are correct, in the thread I think most people are using specific impulse in terms of Mass, not weight, which is equal to the effective exhaust velocity.

It's probably worth pointing out that KSP uses the weight definition, so to stick in KSP values to the above equations, you should replace Isp(mass) everywhere with Isp(weight)*9.8

[tavert]

Good thread. I haven't updated these so the part data's a few versions old now, but have a look at https://dl.dropboxusercontent.com/u/8244638/KSP%20Design%20Optimizer%20v02.html (need to install the Wolfram CDF Player plugin http://www.wolfram.com/cdf-player/ ), or the OpenOffice version here https://dl.dropboxusercontent.com/u/8244638/KSP%20Design%20Optimizer.ods (read over and enable the macro)

For a single stage using a given single type of engine, and a set of discrete fuel tank choices, solving for the number of engines and number of each type of fuel tank to minimize total mass (or cost, or part count, or some linearly weighted combination thereof) subject to minimum delta-V and minimum TWR constraints is a mixed-integer linear program. You can translate the delta-V constraint into a mass ratio constraint by inverting the rocket equation, since Isp is constant for a single type of engine (at a fixed altitude).

Don't try to use the mass ratio of the fuel tanks to get a continuous answer, since fuel tanks aren't continuous in stock KSP. Unless you're looking at a mod that gives you continuously-sized fuel tanks, you can only use integer numbers of each type of fuel tank in the stock game. With tweakables you can now vary the fuel load almost continuously, but the dry mass of the fuel tanks makes this an inherently discrete integer problem.

If you want to consider mixing separate engine types together and/or multi-stage rockets, it becomes a mixed-integer nonlinear problem and quite a bit more challenging to solve.

Edited April 11, 2014 by tavert

[K^2]

So, you basically didn't put it into the equation, but you said it afterwards.

It is in the original Tsiokovsky(probably spelled that wrong) equation, though. Isp is specific impulse, not exhaust velocity.

It was most certainly not in original Tsiolkovsky Rocket Equation, because he was a Russian, and would have used Metric system. In which Specific Impulse is measured in impulse per mass, which is N*s/kg = m/s. These are units of exhaust velocity, and is, in fact, equal to the mean exhaust velocity.

Americans have defined it as lb*s/lb, using pounds for both force and mass, as American engineers often do, which gave them impulse in seconds and the factor of g in the equation. The equation with g in it is a product of American space program.

[Red Iron Crown]

I thought the idea behind specific impulse in seconds was to have it be the same for either unit system. Enter g in ft/sec^2 and you get dV in ft/sec, enter g in m/s^2 and you get dV in m/s.

[tavert]

"Same in either unit system" is an explanation people like to use here, but it makes no sense. If you're working in metric, there's no physical reason to ever insert an acceleration conversion factor in the equation. Conservation of momentum is expressed in terms of effective exhaust velocity, and "specific impulse" should really mean impulse (= product of force by time) per unit mass of propellant. The acceleration factor is arbitrary (the physics here applies the same on Earth, the moon, deep space, or anywhere else), and comes from the ratio between pounds force and pounds mass. That annoying unit conversion factor is what leads to the common convention of referring to a quantity that should really physically be a velocity, as a time instead. KSP should kill that convention with fire and just specify the engine efficiency in terms of effective exhaust velocity.

Converting between meters and feet is much much easier than having to explain every 5 minutes that the conversion factor in the rocket equation doesn't change with local gravity.

Edited April 11, 2014 by tavert

[KASASpace]

It was most certainly not in original Tsiolkovsky Rocket Equation, because he was a Russian, and would have used Metric system. In which Specific Impulse is measured in impulse per mass, which is N*s/kg = m/s. These are units of exhaust velocity, and is, in fact, equal to the mean exhaust velocity.

Americans have defined it as lb*s/lb, using pounds for both force and mass, as American engineers often do, which gave them impulse in seconds and the factor of g in the equation. The equation with g in it is a product of American space program.

Tsiokovsky's equation had effective exhaust velocity, which is equal to:

Isp (weight definition) multiplied by the gravitational constant of Earth.

Interesting to note, that Tsiokovsky got his famous equation from the thrust equation.

[K^2]

Tsiokovsky's equation had effective exhaust velocity, which is equal to:

Isp (weight definition) multiplied by the gravitational constant of Earth.

Tsiolkovsky equation was stated with specific impulse. And "specific" means per unit mass to everyone but the U.S. engineers.

Tsiolkovsky woudl not use exhaust velocity, because there is no exact value for that. You can say that there is an effective mean exhaust velocity, and that it is equal to specific impulse, but you still have to define specific impulse first. So there is no reason to talk about exhaust velocity. Specific impulse is an unambiguous value, and that's what Tsiolkovsky would have used.

Edited April 13, 2014 by K^2

[KASASpace]

Tsiolkovsky equation was stated with specific impulse. And "specific" means per unit mass to everyone but the U.S. engineers.

Tsiolkovsky woudl not use exhaust velocity, because there is no exact value for that. You can say that there is an effective mean exhaust velocity, and that it is equal to specific impulse, but you still have to define specific impulse first. So there is no reason to talk about exhaust velocity. Specific impulse is an unambiguous value, and that's what Tsiolkovsky would have used.

Notice how I said effective exhaust velocity.

Plus, you misspelled "would"

[tavert]

There's a perfectly valid derivation (http://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation#Derivation) where you can look at it as a stream of propellant leaving the rocket with mass flow rate mdot and effective velocity relative to the rocket of ve. Applying conservation of momentum, you get that the acceleration of the rocket is effective exhaust velocity times mdot divided by the remaining mass of the rocket (subject to your sign convention on mdot). A bit simplistic, but nowhere was impulse or specific impulse absolutely required.

[K^2]

That is a high school derivation used only for being very intuitive. While Tsiolkovsky did, as it turns out, use something similar, it ignores a number of problems. The foremost being variations in velocity, direction, and composition of the exhaust gas flow. Indeed, these things are going to vary even with respect to position relative to the nozzle of the rocket. Not to mention interactions with atmosphere during ascent. Worse, this derivation takes a rather primitive look at what the effects of acceleration and loss of momentum to gas flow actually are. They do end up canceling each other out in case of momentum, but using same analysis on energy leads to errors.

Correct derivation is frame-independent, and is written in the following manner. I'm going to use primed terms, such as p', to denote time derivatives: p' = dp/dt.

First, velocity depends only on mass flow. Velocity cannot be "carried away" by any part of the rocket leaving, so the equation for v' is simple.

v' = ∂v/∂m m'

In contrast, momentum is carried away by the mass flow. So we have to use an expression familiar from fluid dynamics.

p' = ∂p/∂t + ∂p/∂m m'

It's easy to see that momentum lost to the flow is equal to the fraction of mass being lost.

∂p/∂t = p m'/m

Finally, we know that velocity and momentum are always related.

v = p/m

So we differentiate with respect to time using the chain rule.

v' = (d/dt)(p/m) = p'/m - p m'/m²

Substituting result for p' and v' we get the following.

∂v/∂m m' = (∂p/∂t + ∂p/∂m m')/m - p m'/m² = (p m'/m + ∂p/∂m m')/m - p m'/m²

The p m'/m² terms cancel on the right side, and we can cancel the common m' factor on both sides to leave us with what we expect.

∂v/∂m = ∂p/∂m (1/m)

If we assume that ∂p/∂m = I_SP is a constant, we can integrate the above to yield Tsiolkovsky equation.

v₁ = v₀ + ∂p/∂m ln(m₀/m₁)

Note that specific impulse, defined as the impulse gained per unit mass, shows up explicitly. Exhaust velocity has not yet been considered. It is, however, easy to see that the effective velocity has to be defined as V_e = ∂p/∂m = I_SP. This is a consequence of the rocket formula, and not one of its assumptions.