# What is wrong with this burn time calculation?

Length of Burn

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ΔT = [ (ML * EV) / F ] * [1 – EXP[-(ΔV / EV)]

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ΔT = Length of burn in seconds

ML: Total mass of the rocket at the beginning of the burn.

EV = Exhaust Velocity in meters/second.

F: Thrust of the rocket in Newtons.

ΔV = Delta-V of burn in meters/second.


The burn time for the Falcon 9 first stage is 162 seconds. In order to get 162 seconds from this calculation, it requires a velocity of 4176 m/sec / 15033.6 km/h which is absurd. The latest launch, Telstar 18 reached 8162 km/h at 'meco' nearly HALF. I know that they throttle down for max-q.

Using estimated values for the Falcon 9 FT

Velocity Change (ΔV) -

Mass: 549,054 kg

ISP: 282s seconds

Thrust: 7607000 Newtons

[ (549,054 * 2766.42) / 7607000 ] * [1 – exp[-(4176 / 2766.42) ] = 162.001814183


TL:DR

Does anyone know how to calculate burn time based on weight?

• Is the missing element gravity losses? I'm not seeing them in the equation. Any thrust directed down has to fight gravity as well as accelerate the rocket. Subtract 9.8 m/s per second of thrust -- but this only applies to the vertical component, not the horizontal. Sep 14, 2018 at 15:42
• "The burn rate for the Falcon 9 first stage is 162 seconds" This makes no sense. Burn rates are not measured in seconds. Sep 14, 2018 at 16:10
• The title of 162 second FT is "Burn time". I am looking to find out how they calculate the 162 second amount from the first stage Sep 14, 2018 at 16:16

The equation works for a straight-line burn with no gravity losses, but the first-stage burn of an orbital launcher arcs from vertical to nearly horizontal, and loses velocity to gravity.

To compute burn time, just divide total propellant mass by the mass flow rate; flow rate can be calculated from thrust divided by exhaust velocity (or specific impulse in seconds * 9.81 m/s^2): 845 kN / 2.77 km/s = 305.1 kg/s per Merlin 1D. Note that full-throttle flow-rate, unlike thrust and specific impulse, doesn't change significantly with altitude, which is helpful.

That does leave the throttle schedule as an unknown; if you're still trying to figure out the "actual" launch mass of the Falcon 9, this is going to be another dead end for you.

Assuming 410900 kg of propellant in the first stage, full-throttle burn would empty the stage in 410900 / (9 x 305.1) = 149.6 seconds. The throttle-down appears to extend the burn time by 12 seconds.

The "expended velocity" figure you computed of 4176 m/s is reasonable, incidentally. Circular LEO velocity is about 7800 m/s, but ascent from Earth's surface to LEO takes roughly 9300-9700 m/s of expended ∆v (varying with the ascent trajectory and the size of the launcher) because of gravity, arc, and drag losses. The first stage pays for the vast majority of the 1500-1900 m/s difference. So you'd expect the 4176 m/s of expended ∆v to yield a linear velocity in the ballpark of 2500 m/s, or 9000 kph.

• "Exhaust Velocity", no? Sep 14, 2018 at 17:56
• (waves hands wildly) Sep 14, 2018 at 18:00
• Excellent answer. Through MAX-Q, I've read that they throttle down the 9 engines down to 70%, but with everything, it is up in the air. Sep 14, 2018 at 18:10
• There’s likely also throttle down towards the end of the first stage burn to limit the maximum g-force. Sep 14, 2018 at 19:57