Trade-offs in developing larger engines vs using more smaller engines

Question

Looking to explore the inverse point of this question:

What challenges are there for more, smaller liquid fuel engines instead of one large?

As stated in the answer to the above question, fewer, bigger engines reduces parts count and increases reliability at the expense of thrust range (relevant for reusable boosters such as Falcon 9), but what happens when you want to go really big?

The largest single combustion chamber liquid-fueled engine ever successfully flown was Saturn V's F1, last flown 45 years ago. It has no equal today (not considering solid-fuel boosters such as the Shuttle SRBs); even the Space Shuttle main engines develop only about half as much thrust.

The Falcon 9 demonstrates that nine engines are enough to provide the variable range of thrust necessary to land an expended booster. So what trade-offs lead SpaceX to employ large numbers of engines for their proposed big heavy boosters (one proposal called for 42 engines, another calls for 31) rather than develop bigger engines to use in smaller numbers?

I know that one problem faced during development of the F1 was combustion instability. Is this the dominant factor inhibiting the development of bigger engines, or are there others, and if so, what? The modern "benchmark" seems to be the SSME; what is impeding development of anything substantially larger, like, for example, that of the Sea Dragon (again, limiting the question to liquid-fueled engines)?

Answer 1

First, clustering of engines saves a lot of engineering work. You only design one engine, rather than more than one. This is especially valuable in the extremely conservative world of man-rated aerospace. It boils down to modularity.

Second, if you're cost-sensitive, and you're trying to mass produce a ship, clustering lets you mass-produce engines. For nearly every production method except (and even including) 3D printing and artisan hand-building, anything you can do to increase the quantity of identical things built greatly reduces the cost per individual item.

Third, multiple engines lets you adjust thrust by turning them on and off. It should be understood that the Falcon 9 lower stage cannot actually hover; with empty tanks it cannot throttle low enough to hover and instead must "hoverslam".

Fourth, large clusters of engines are more likely to have an individual failure, but won't lose a significant amount of thrust from a single failure. The Saturn V would crash if an F-1 engine failed soon enough, since that would take away 1/5 of the total thrust.

Fifth, large rocket engines are problematic. They are more likely to suffer from "combustion instabilties" of the kind that lead to the rocket just exploding. This was part of the reason for the F-1 taking so long to develop, as mentioned by Uwe, and is part of the reason for Russia's tendency to use clusters. The SSME is considered a very high performance engine with attendant extreme cost. More generally, there are often issues that lead to a "sweet spot" between small and large -- small machines often have poor performance/weight, and large ones run up against the unfavorable components of the square cube law much as they take advantage of the favorable ones -- a particular problem for pressure vessels.