Optimal size for heavy-lift launch vehicle

Question

The primary motivation for the development of large capacity launch vehicles seems to be that of efficiency - a bigger vehicle will be more efficient at delivering a given payload to LEO than multiple smaller vehicles.

However, there seems to be a number of difficulties in scaling up. Factors like chamber pressure and fuel chemistry seem to place a hard limit on the amount of thrust and ISP possible for a given size of engine. Engines only seem to get so big; one factor is combustion instability. Bigger launchers then seem to require larger numbers of engines to achieve a given total thrust. For example, the proposed SpaceX ITS booster would have 42 Raptor engines. That raises questions of reliability; more things that can go wrong increases the prospect that something may go wrong.

My question is: do issues associated with scaling up ultimately place a practical limit on the maximum payload to LEO for a single vehicle launch? Is it a hard limit or one which could increase over time with the further evolution of space technology? What is the current practical limit? What could be the limit in 10 years, 20 years, or 50 years?

Answer 1

Note; larger rockets can definitely be cheaper per unit mass to LEO than smaller rockets, through production methods. The Sea Dragon design was intended to be produced in a dry dock, using 8mm steel, and with simple, large engine designs. The intention was to reduce the component count in the engine by two orders of magnitude, compared to the Saturn V. https://en.m.wikipedia.org/wiki/Sea_Dragon_(rocket)

For the SpaceX ITS, 42 Raptors allows a moderate degree of redundancy in the event of several failing, and further, Elon has stated that the optimisation favoured smaller engines, rather than larger ones. Since there are so many engines, the forces can be balanced by throttling or deactivating engines opposite to the defect. In the event of several engines failing; This wouldn't be as much of an issue, as the payloads then go into a parking orbit. So if several engines failed such that the ascent was severely slowed, the second stage would likely still be able to reach orbit, although at a lower level. As it either has to be refueled, or to deliver fuel, this likely wouldn't be a major issue, as burning more fuel would allow the satellite to orbit as previously planned. This could result in further refuelings to compensate.

I'm not sure if the ITS first stage will be able to land when mostly fueled. I'd hope so, although perhaps it is more difficult landing a taller rocket, with fuel mass too. Although the crew compartment would be able to eject from the first stage and land on Earth, as SpaceX implemented in their Dragon design. (Dragon2?).

Answer 2

Let me present a very simple mechanism that would limit the size of rockets. Consider, at some height the added weight from structural materials will outweigh any benefit gained from further reduction of dynamic pressure in the climb.

Computing this exactly would be quite difficult, since dynamic pressure varies greatly over the trajectory. However, I'm not interested in complex calculations here. I'll do the simplified version. I'll consider a rocket moving straight up, at its max Q.

Fraction of mass contained in structural materials, approx:

$$ \alpha = \frac{ 3 x 3 \rho_{rocket} g h }{ \sigma_{yield} } \frac{\rho_{rocket} }{ \rho_{steel} } $$

The retarding pressure from the weight of the steel would be:

$$ P_{steel} = \alpha \rho_{steel} 3 g $$

The tradeoff between more steel vs. higher aerodynamic efficiency will start to become acute when P_steel is roughly in the same ballpark as the dynamic pressure. This is totally a cop-out to obtain a general ballpark figure. Using the above expressions, I can do:

9*(1 g/cm^3)^2*(9.8 m/s^2)^2*(80 m)^2/((500 MPa))*3

(you can plug this into Google)

What I've done here, is that I manipulated the value of h until the pressure came out to around 33 kPa, which I found from this sound was about the Max Q of the shuttle. The values from steel and other stuff are all just educated guesses. Anyway, my answer is 80 meters. A rocket becomes less efficient if it is taller than that.

The Saturn V was actually 110 meters tall, but I think that was counting a relatively slender cone on the top. So historically large rockets might already be close to the "limit". However, this isn't truly a mass limit, just height. Saturn V was very slender. You could imagine its width growing until it was basically a blocky shape... which probably isn't far from what Orion was proposed for. Beyond that, you'd be building some kind of pancake shape rocket, and intuitively something just sounds wrong with that.

Answer 3

Project Orion claimed to be able to put 6100 tons in LEO in a single launch.

Depending on what you are willing to allow for "the future evolution of space technology", the limit could be pretty large.