Optimal shared-engine TSTO configuration

[sevenperforce]

One of the ways that SpaceX keeps costs down is using the same fuel and engine on the second stage that it uses on the first stage. Only having a single engine design for the entire launch vehicle is a really good idea. 

I was wondering, though: how close to optimal is the 9:1 configuration? SpaceX started development of a 5:1 configuration so there is obviously some room for variance. I don't know what other TSTO launchers use matching engines so I'm not sure there is anything else to compare it to.

The second stage engine has a higher specific impulse and thrust due to the extended nozzle, but that extended nozzle also weighs more, so that may or may not have an impact. I assume the primary driver here is the relative masses of the two stages: you want your launch vehicle to have a T/W ratio of nearly 2:1 at launch, but you want your second stage to be roughly one fifth the mass of your first stage and have closer to a 1:1 T/W ratio at separation, suggesting a nearly 10:1 thrust ratio...pretty close to Falcon 9.

Any other considerations? What about altitude-compensating nozzles?

[StrandedonEarth]

Well, the Saturn V used the same J-2 engines for the second (5) and third (1) stages...

[sevenperforce]

30 minutes ago, StrandedonEarth said:

Well, the Saturn V used the same J-2 engines for the second (5) and third (1) stages...

Oh, good example. Of course, one difference there is that the second stage is already in motion, post-gravity-turn, and thus has less stringent thrust requirements than with a static launch. 

[wumpus]

I think 5:1 is much closer to "ideal", but the way you are stating ignores recovery situations.  Typically a good approximation has the delta-v of each stage the same.  Another issue is that with a TWR of 2:1, you will likely get a max pressure that the rocket can't handle (unless it throttles down).  Typically rocket scientists use every bit of thrust they can get, and only give it up rather late (look at how late in the launch Saturn V shut down the center engine).

Space-x can recover the first stage.  They don't have a hope of recovering the upper stage.  That means using a 9:1 configuration allows 90% recovery (if successful).  Another recovery issue is that they can "only" throttle to 60% (most engines can't throttle at all, and this probably wider than anything but Blue Origins) so that the 9-engine falcon is performing a suicide burn at TWR>1.  With a 5-engine falcon that TWR would be nearly twice as high and closer to the doomed "punch a hole in the barge" falcon attempting to land on three engines (they normally use one).

Even KSP has a ton of considerations that make "ideal considerations" pretty moot.  Often the cheapest source of delta-v at launch are SRBs.  Even if you want to use liquid first stages, the specialized upper stages are so vastly more efficient it would be silly not to use them.  Example (basic mark 1 stage, mark 1 parachute, full heat shield to orbit)

non-theory rocket: 1 thumper (dialed to 90% for TWR=2.0 at launch), 1 FL-T400 fuel tank, 1 terrier: ~3,800 vacuum delta-v. 3,862 kredits.  11.7 tons, ~300 m/s delta-v left in orbit.

TSTO: 2 reliant engines (first stage with FL-T400&FL-T800, second stage with FL-T400)  ~3,400 vacuum delta-v 6,122 credits. 12.8 tons.  Barely got to orbit (I'm less familiar with this type of rocket).

same fuel? first stage reliant+FLT-800 and FLT-200, second stage spark + FLT-200.  ~3,700 vacuum delta-v 4,722 kredits, 9.4 tons.  Unstable, couldn't get to orbit (note that the spark is much later in the tech tree than everything else.  Also note that heat shield really needs to go.  A mark 1 capsule is on the edge of things the spark can deal with).

improved liquid rocket (first stage with FL-T800 & reliant, second stage with FL-T400 and terrier) ~3,800 vacuum delta, 4,912 kredits, 9.8 tons.  Didn't fly, but I'm sure it doesn't have issues (I've used it enough before).

Judging by the ISP, I'd guess that the reliant and spark assumes RP1/LOX while the terrier assumes LH/LOX.  Using the same fuel presumably is great for the logistics of rocket launch (remember that the cost of the rocket is 60% or so of a falcon launch.  The logistics are expensive).  Note that for big NASA jobs like the Saturn V this is pretty moot: each stage was generally designed/built by different contractors, and that coordinating them together would probably be more expensive than letting each fuel their own stage.

[sevenperforce]

One of the hidden advantages of the SRB is that it can be "throttled" by shaping the burn surface without essentially adding dead weight. In contrast, downthrottling a liquid-fueled engine renders those engines essentially dead weight; any thrust capacity you aren't using is useless dry mass and cuts directly into payload. An all-liquid launcher needs quite a bit of engine to hit that 2:1 T/W ratio at launch to minimize gravity drag, but instantly becomes overpowered as it loses fuel weight. Of course, the faster it launches, the faster drag builds up, so the extra thrust isn't completely wasted...but SRBs really have a big advantage here.

Still hate them because they can't be restarted or refueled, but the usefulness is hard to deny.

SpaceX seems to have sidestepped this issue by building ridiculously lightweight and powerful engines, so that the high T/W ratio mitigates engine downthrottle penalty.

I like air augmentation for its easily-attainable positive effects on specific impulse, but it only meagerly augments static thrust, which means you still need either a supercharger or a larger engine cluster, both of which add to dry weight and thus reduce payload. Thrust-augmented nozzles and dry mass injection both help with vertical takeoff, while a rolling takeoff is also possible (not to build up aerodynamic lift but to build up airflow for air augmentation of thrust).

Where does the 5:1 ratio come from?

[A Fuzzy Velociraptor]

Short answer: It depends

Long answer: It depends on a large number of factors which can't easily and fully be broken down into an explanation over the course of a post in a thread. One of the first factors being the vehicle mission. A vehicle designed to be reusable will have different parameters than one that will be entirely disposable. For instance a reusable vehicle that intends to propulsively land will require an amount of delta V to perform the landing that is no longer used towards the orbit system. Different missions will also have different trajectories that may require more or less thrust on the upper stage compared to lower stage. Different feed systems will vary the mass ratios as will different geometries. Upper stage engines may also have extended nozzles and even without them will have substantially different effective velocities than those used in the low atmosphere. There are a lot of different other factors as well. You can compare what various companies use but ultimately it will be a case by case basis what will work best and it may simply come down to what is convenient or good enough rather than what is optimal.

At least for a disposable system you can estimate stage mass ratios with the following.

x=[1:100];
y=zeros(1,100);
c=1;
% Mass Fractions
nuL=0.07;
nuU=0.07;
% Effective Velocity
EVL=3400;
EVU=3550;
% Total Rocket Mass
Mt = 1920;
% Payload Mass
Mp = 80;
for c=1:100
    % Lower Stage Mass
    Ml=Mt*x(c)/100;
    % Upper Stage Mass
    Mu=Mt*(100-x(c))/100;
    % Delta U estimation
    delU = EVL*log((Ml+Mu+Mp)/(nuL*Ml+Mu+Mp))+EVU*log((Mu+Mp)/(Mu*nuU+Mp));
    y(c) = delU;
end
plot(x,y)
title('Delta U Estimation');
xlabel('Percentage First Stage of Total Vehicle Mass');
ylabel('Delta U')
grid on

[wumpus]

21 hours ago, sevenperforce said:

Where does the 5:1 ratio come from?

It was an earlier design for a Falcon that was eventually scrapped.  Generally 3:1 seems closer to the ideal (and KSP seems to agree with building a test rocket out of a Mark 1-2 capsule, Rokomax-16 fuel tanks, and reliant engines (rocket wasn't flown due to iffy overall construction, but had >4000 m/s delta-v).

This leads to the question of would a three-stage "falcon [ultra]-heavy" work better with a 9 (or presumably 12, but 9 is obviously available):3:1 stage setting.   Presumably the chance of recovering the middle stage is too low to bother with (and designing the current "heavy" is also easier than shoving a middle stage in).  Without cross-feeding, I'm less sure (but I have even less confidence of landing a higher stage.  Personally, I'd strap some COTS (commercial off the shelf) SRBs to the [unrecoverable] second stage for more delta-v.