About optimal staging and structure fractions

Question

I have recently came across an assignment where I have to do optimal staging of rockets using Lagrange multipliers. I am suppose to optimize the mass of each stage of a well-known rocket(I chose Saturn v), by using their known values of their final speed, payload mass, specific impulses of each stage, and their structural factor. The part that bothers me is the structure fraction, which is given as $$\dfrac{m_{structure}}{m_{propellant}+m_{structure}}$$

I have two questions here.

First of all, I have to optimize an already-launched-rocket, which would mean that the rocket stages will already be optimized at maximum. However, I have no clue how to only find the structure fraction of each stage directly, not by plugging in each $m_{propellant}$ and $m_{structure}$, which is all I can get from my poor searching skills. I doubt that there will be any sites that only give us the structure fraction, as the mass will be decided first, and then the structure fraction.

Secondly, even if I were to just plug in each mass factors, Wikipedia gives me empty mass and gross mass. From what I have learnt, I suspect that the structural mass is the empty mass, and the propellant mass is the difference between gross mass and empty mass for each stage. However, this gives me a structure fraction of $0.057$ for the first stage, which is highly unlikely from what I have learnt. (I have learnt that around 10% is what we can make at maximum, and such low structure fraction is impossible at current technology.) Where am I wrong? How should I interpret empty mass and gross mass?

I really am a newbie to the rocket equation, so I have no idea what's going on. Can somebody help me?

Answer 1

First of all, I have to optimize an already-launched-rocket, which would mean that the rocket stages will already be optimized at maximum

Your assumption here is incorrect. Whenever optimization is discussed, your first question should be "optimized for what?"

Real rockets are not usually optimized for launch mass to payload mass ratio. They are most often intended to be optimized for some combination of development and operational cost and time factors, and often for political factors as well, and problems that crop up during development usually mean they wind up not optimal for much of anything.

Launcher designs may reuse existing components which are not optimal, but have the enormous advantage of already existing. The requirements that were optimized for at the start of the project may change after portions of the design have already been frozen, meaning that only the parts that have not yet been designed can be changed to accommodate the new requirements.

You will certainly find that the Saturn V is not quite optimal from a mass staging perspective. The point of your exercise is finding out exactly how non-optimal it is.

Secondly, even if I were to just plug in each mass factors, Wikipedia gives me empty mass and gross mass. From what I have learnt, I suspect that the structural mass is the empty mass, and the propellant mass is the difference between gross mass and empty mass for each stage. However, this gives me a structure fraction of 0.057 for the first stage, which is highly unlikely from what I have learnt. (I have learnt that around 10% is what we can make at maximum, and such low structure fraction is impossible at current technology.) Where am I wrong? How should I interpret empty mass and gross mass?

Your interpretation of gross and empty mass is essentially correct. As Organic Marble and PcMan have noted in comments, it's your 10% limit that's in error.

The fact sheet that Organic Marble links gives a dry mass fraction of 6.2%, while the Wikipedia numbers give a slightly better 5.7%; this is because modifications to the first stage after Apollo 8 dropped the first stage mass by about 5000kg, and propellant loads were increased. Those details are available in Orloff's Apollo By The Numbers and the statistical reference derived from that book.

Answer 2

$$m_{empty} = m_{structure}$$

$$\frac{m_{structure}}{m_{propellant} + m_{structure}} = \frac{m_{empty}}{m_{gross}}$$

You are unlikely to find structure fractions listed directly, and would indeed have to calculate this yourself in the way you describe, which is correct.

There are however sometimes subtleties, like the mass of coolant, ablative heat shields, detachable equipment, inter-stage structure, tank pressurisation gas, etc. For a launcher those are usually small, but you have to think their classification through regardless of whether you use structure fractions or not.

I have learnt that around 10% is what we can make at maximum, and such low structure fraction is impossible at current technology.

Sanity checks are good. But in this case rocket stages frequently do have structure fractions well below 10%, the S-IC being no exception.