I couldn’t find any analysis of how much of a difference cross-feed fueling makes, so I tried to make my own.

The Falcon Heavy is America’s next heavy-lift launcher. It’s going to lift double the amount of the Space Shuttle. Powered by the future Merlin 1-D, it’s got some impressive power. What’s even more impressive is their approach to staging.

If you just want the numbers, click here to scroll down to the bottom of the article.

The Falcon Heavy is built by strapping three Falcon 9 first stages together and topping it with a Falcon 9 second stage with the payload on top.

Fig. Falcon 9 and Falcon Heavy, from www.spacex.com/falcon_heavy.php

Normally when a rocket launches it’ll consume a stage, drop off the stage, and light the next one as it moves into space. This is done to drop off dead weight that gets hauled up.

Fig. Staging of a Saturn 1-B Rocket, image from www.real-world-physics-problems.com

So the intuitive thing would be “Why not make rockets 30 stages and be super-efficient?” The short story is diminishing returns and additional points of failure. And it’s expensive. Long story short, you want to be sitting around 2-3 stages if you can help it. Single stage to orbit is a topic I’ll cover in another post.

Then how can we squeeze out extra performance?

The Falcon Heavy answers this challenge by introducing cross-feed fuel transfer. What this means is that when the rocket first launches, the three Falcon 9 first stage cores (hereafter referred to as ‘F9Cs’) fire at full power at the same time using fuel only from the two F9Cs on the sides. The center core burns with fuel cross-fed from the side cores. When the side cores are out of fuel, they separate, leaving a fully fueled center core already at altitude and speed, ready to continue to orbit.

The closest comparison I could make to say the ‘normal’ way this is done is the Delta IV Heavy. The Delta IV Heavy has three Common Booster Cores (the first stage of the Delta IV is named Common Booster Core, or ‘CBC’) that fire as the first stage of this rocket. The two side CBCs fire at full power, while the center CBC is throttled lower to conserve fuel until the side boosters run out and separate. Once the boosters are gone, the center core is throttled up until it’s expended its fuel and then separates, allowing the rest of the rocket to continue.

There are two things that are most important in your rocket performance: your fuel to structure ratio for each stage, and your exhaust gas velocity. Your fuel-to-structure ratio is the weight of your stage when it’s fully fueled vs. the weight of your stage when it’s empty. Given that everything else stays the same, the higher that ratio is, the better your rocket will perform. Your exhaust gas velocity is a function of your engine chamber pressure, which is tied to your mass flow rate of fuel. Cross-feed fueling deals with the first. There are other things that can leap ahead your rocket’s performance such as fuel composition, reaction design in your engines, and other things, but aren’t currently being pursued and aren’t important in this article.

The short, short answer:

Cross-feeding fuel is the difference between success and failure for the Falcon Heavy. It gives the vehicle the additional performance it needs to lift up everything as it says on the website and have the cushion room for propellant to return and land each stage as they hope to do. This configuration outperforms having all three stages firing at the same time, or having the center stage throttled low until the side boosters separate. (This is expected, because I doubt any company would develop a new technology with no payoff). I just wanted to see how much of a difference it makes. The answer is 0.5 km/s of an available speed boost if you just kept pushing your payload and didn’t bother with landing. This is the easiest metric I could use to express this bonus, even though this isn’t the way it’s going to be used.

The short answer:

I challenged myself to see if I could use the available data on SpaceX’s website to calculate some of the parameters of the Falcon 9, and then use that information to guess at the performance of the Falcon Heavy. Part of the challenge was using information only available from SpaceX explicitly, and filling in the holes with some of my older stuff in Aero. I’d love to compare notes/find out how close I got, but really the goal was to see the difference cross-feeding makes. Whether my numbers are close or not doesn’t really matter because changing how the fuel is used through cross-feeding will have the same effect on indicating improvement.

I used the 200 km circular orbit parameter with the F9 fully loaded launching at Cape Canaveral’s inclination at 28.5 degrees, basically the optimum launch parameters to get the most payload into orbit. My target total delta-V (rocket velocity) was 9.4 km/s to achieve orbit – 7.8 km/s for a 200km circular orbit + at least 1.6 km/s loss in velocity due to gravity and drag. The velocity and drag loss usually ranges from 1.4-2.0 km/s.

With that said, I estimate that compared to firing all three F9C cores at full power and separating them at once, the cross-fed fuel configuration will add between 0.5 and 0.7 km/s of delta-V which is a significant improvement on flight performance (around 5%). 0.5 km/s might not seem like that much, but it’s 1118 miles/hour. Especially with the goal of making rockets fully reusable, this can make the difference of having the room to recover all your stages.

The Engineer’s Answer:

Using the given information from the website, I calculated:

Merlin 1-C’s performance (per engine):

Throttle Setting: Locked at 85% of capable maximum

Exhaust Gas Velocity at Sea Level: 2,501 m/s

In vacuum: 2, 981 m/s

Fuel Flow at SL (Takeoff) = 222 kg/s

In vacuum = 204 kg/s

I used an exponential atmospheric pressure decay model to estimate the flight average values. This is way less complicated than it sounds. It’s one equation saying that the launch vehicle is moving such that the atmosphere’s pressure that it feels exponentially decays per time as opposed to a linear decay. I could have also written a code to more exactly calculate how the rocket would perform, but this is for back-of-the-envelope and should be good enough for the purpose of this article. Using the exponential averaging:

Flight average thrust for F9 first stage = 5,384,016 Newtons

Flight average ISP for first stage = 294 seconds

Exhaust gas velocity: 2,886 m/s

Fuel Flow F9: 1866 kg/s

Merlin 1-C Vac (from website)

Thrust: 445,000N

Isp 342 seconds

Then I calculated

Exhaust gas velocity = 3,354 m/s

Fuel Flow = 133 kg/s

Now, maximum payload to orbit is 10,454 kg for the F9, and we need enough fuel to get the payload to orbit AND return each stage back to earth. We also have the burn times of each stage which can give us the amount of fuel consumed per stage. We have the takeoff thrust setting, and we know the delta-V we have to achieve, and from photos and the ratio of fuel on board each stage I estimated the ratio of sizes between the first and second stage of the Falcon 9.

With that, the fully loaded F9 is probably around 500 metric tons. If all available fuel on board was consumed (not used to return the stages to Earth), the maximum available delta-V is then around 10.8 km/s. <- This number needs to be what the Falcon Heavy needs to be around to accomplish all mission objectives. If the F9 can do it, the FH should be able to do it, too.

I’ll throw up a spreadsheet with all the stats later in down below.

Turning to the Falcon Heavy, the website indicated the structure to fuel ratio is greater than 30, so I’ll give them room and say it’s slightly above 30. The total mass of the Heavy fully loaded with payload will be 1.4 million metric tons, with a maximum payload carrying capacity of 53 metric tons to orbit. I know the booster burn times are going to be a little more than 2/3s the burn time of a Falcon 9 first stage because each booster is pumping a third of its own fuel to the central core and they will be travelling more slowly and lower than the end of a Falcon 9 first stage at its end of mission, so they need less fuel to return. Similarly, the central core needs to have a bit more fuel and a slightly shorter burn time so it can make the return trip. And the final stage of the Falcon Heavy I kept the same as the F9 second stage. Based on available data for the FH, the Merlin 1-D engines are throttled at 92.5% maximum rating to achieve the launch thrust desired. Same simplifications as the F9 are made for performance variation through the atmosphere.

The engines I replaced with the available information for the Merlin 1D. I also had to fill in the blank for the Merlin 1D Vac, since there is no data on it. (Last minute update, the Merlin 1-D vac was just tested at 80 tons of thrust, which by my calculation is around 20% of its probable maximum performance). Based on the calculation as I was doing it, I sacrificed thrust for additional specific impulse, as that was what was necessary to reach the delta-V necessary for mission success. I didn’t do anything in depth, just enough to make logical sense for a probable direction of design. Again, that’s something a program could work out, but this was for the sake of qualitatively determining the additional performance.

With that, the delta-V difference between a Falcon Heavy with all 27 engines firing full blast for the whole first stage and the cross-fed Falcon Heavy came out to be over 0.5 km/s. It doesn’t seem like much, but when your target velocity is 9.4 km/s, and you come in at 8.9 km/s, you aren’t going to make it. Also if you don’t think that’s a lot, 0.5 km/s is an additional 1,118 miles per hour. It should be noted that in the event of a problem, the Falcon Heavy still has the capacity to get its payload to orbit without cross-feed fuel, but won’t have enough fuel to return its stages safely to the ground.

Now, the raw data – I’d love to hear/work with an engineer that was on this project to see how close I got. I know all information isn’t public for good reason but this was just a challenge to me to stay in shape with rockets. It should (hopefully) be within the ballpark for any enthusiasts who are curious. There’s just no analysis online of how much of a difference the cross-feeding system actually makes, and I was curious to see for myself.



With that, here’s a cliffs notes of my spreadsheet. All the numbers take into account that the rockets need sufficient fuel to slow down and land each stage, but are tabulated as if the rocket is going full burn, all engines, full power. I also added fudge-factors for additional weight for extra pumps and equipment for x-feed fuel while there are weight reductions in other areas of the rocket hardware (like the F9C boosters, since they won’t have all the equipment of a full stage and the website explicitly states their mass ratios will be different. I calculated the Falcon 9 characteristics first and used them to predict how the Falcon Heavy would be. Anything that I found on the website is marked “Given”, anything else should be assumed calculated. Disclaimer: Error bars on this tabulation are gigantic and nothing here should be taken as official, confirmed, demonstrated, or actual.

Merlin 1-C (per engine)

Reported Maximum Sea Level Thrust = 653,889 N

Reported Maximum Vacuum Thrust = 716,164 N

Given Isp SL = 255 s

Given Isp Vac = 304 s

Exhaust Gas Velocity SL = 2,501 m/s

Exhaust Gas Velocity Vac = 2,981 m/s

Sea Level Fuel Flow = 222 kg/s

Vacuum Fuel Flow = 204 kg/s

Merlin 1-C Vac

Given Thrust = 445,000 N

Given Isp = 342s

Exhaust gas velocity = 3,354 m/s

Fuel Flow = 133 kg/s

Falcon 9

Payload Mass (Given) = 10,454 kg

Total Structural Mass = 16,537 kg

Total Loaded Propellant = 462,009 kg

Predicted Fully Loaded Launch Mass = 489,000 kg

Given Launch Thrust = 5,000,000 Newtons

Calculated Maximum Throttle Setting = 85% Engine Rating

Calculated Launch Thrust = 5,002,251 Newtons

Exponential Averaging Calculation (1st stage only)

Flight Average in-flight thrust = 5,384,016 N

Flight Average Isp = 294s

Flight Average Exhaust Gas Velocity = 2,886 m/s

Flight Average Fuel Flow = 1,866 kg/s

F9 Stage 1

Given Burn Time= 171 s

Total Loaded Fuel (including for re-entry and landing) = 410,039 kg

Structural Coefficient = 0.035

Payload Ratio = 0.151

Structural Mass = 14689 kg

Mass Ratio Z = 6.193

Total Available dV (including for re-entry and landing) = 5,262 m/s

F9 Stage 2

Given Burn Time = 354 s (Conflicts in Manual/Website, but I’ve calculated that 354s is most likely)

Total Loaded Fuel = 51,970 kg

Structural Coefficient = 0.034

Payload Ratio = 0.194

Structural Mass = 1,848 kg

Mass Ratio Z = 5.225

dV2 = 5,545 m/s

Total F9 dV Available including reentry and landing fully loaded all engines full power = 10.8 km/s

dV necessary to achieve orbit including gravity and drag losses 9.2 – 9.9 km/s (so this configuration works)

Falcon Heavy, assuming improvements, extra pumps, and I guessed the direction the Merlin 1-D vac’s design would be to lower thrust in exchange for higher specific impulse (at least that’s what I would do if i had to choose).

Falcon Heavy

Given Maximum Payload = 53,000 kg

Structural Mass = 44,016 kg

Total loaded propellant mass = 1,303,030 kg

Given total launch mass = 1,400,000 kg

Total calculated launch mass = 1,400,046 kg

FH Stage 1 (the two side boosters combined)

Burn Time if All Engines On, including reentry and landing for both boosters = 159s per booster

(Burn time increases after mission burn time by 1/3 because the center core engines are gone)

Mission Burn Time 114+ seconds (they may have more time because the boosters will be lower and slower than a normal Falcon 9 stage at the end of their cycle)

Loaded Fuel for both boosters combined = 841,078 kg <- Had to be similar to F9 stage 1, maybe a little extra fuel on board if boosters are slightly modified from normal stage.

Structural Coefficient = 0.032(SpaceX’s website says that the boosters will have a structural ratio >30, this coefficient gives a ratio of 31.6)

Payload Ratio = 0.61

Empty dual booster mass = 27,478 kg

Mass Ratio Z = 2.5

dV1 (including return and landing, not accounting for staging weight shift) = 2.7 km/s

FH Stage 2 (Center Falcon 9 Stage 1 Core)

All Engines On available burn time= 217s

Loaded Fuel = 410,039 kg <- Had to be similar to F9 Stage 1

Structural Coefficient = 0.035 (Structure Ratio 26.23)

Payload Ratio = 0.251

Stage Mass (including extra pumps and hardware for x-feed and booster attachment) = 14,689 kg <- had to be close to F9 Stage 1 mass

Mass Ratio Z = 4.38

dV2 (including return and landing) = 4.5 km/s

FH Stage 3

Total Available Burn Time = 560s

Total Loaded Fuel = 51,913 kg <- Matches F9 Stage 2

Structural Coefficient = 0.034 (Structure Ratio 29.1)

Payload Ratio = 0.986

Mass = 1,848 kg <- Matches F9 stage 2

Mass Ratio Z = 1.946

dv3 (including return and landing) = 2.9 km/s

Since there’s no information on the Merlin 1D Vac yet, I had to guess based on the relationship between the Merlin 1C and the 1C vac and taking into account the needs of the mission

Merlin 1D Vac

Thrust: 400,000 N

Isp: 440 s

Exhaust Gas Velocity 4,315 m/s

Fuel Flow 93 kg/s

With these parameters, this theoretical Falcon Heavy would achieve orbit with some cushion room to return its stages back to Earth. I didn’t confirm that it would be enough to complete a landing, but there’s certainly enough cushion room where it’s possible.

I can also show that a similar Falcon Heavy that doesn’t perform Cross-Feed Fuel propulsion does not have that cushion room and like I said, loses 0.5 km/s dV of potential velocity.

And that’s all! Hope it helps in case anyone was curious about the difference cross-feeding fuel makes. Like I said, I would love to compare notes with a SpaceX engineer on this, or at least know how close I got just from working backwards just for fun. In any case, I hope people enjoyed finding out this new approach to staging makes a big difference!