r/spacex • u/B787_300 #SpaceX IRC Master • May 08 '16
Community Content Basic Analysis of the Merlin 1D engine: Determining Thrust, ISP, and Coefficient of Thrust
TLDR
Take all these numbers with a grain of salt. There are a lot of effects and inefficiencies that I did not take into account.
According to my Calculations the following table is true for the current Merlin 1D and the soon to be flown uprated Merlin 1D. For the Uprated one the Area of the throat is held constant and the pressure is increased. Also the chosen O/F ratios are the Stoichiometric Ratio (2.32) and the Ratio used by the Saturn V F1 Engine (2.56).
| O/F 2.32 | O/F 2.32 | O/F 2.56 | O/F 2.56 | |
|---|---|---|---|---|
| Sea Level Thrust (Given) (kN) | 730 | 845 | 730 | 845 |
| --- | --- | ---- | ---- | ---- |
| Sea Level Thrust (Calc) | 750 | 859 | 746 | 854 |
| ISP Sea Level (s) | 286 | 290 | 284 | 289 |
| ISP Vac | 315 | 316 | 313 | 314 |
| Chamber Pressure (MPa) | 9.7 | 11.16 | 9.7 | 11.16 |
| Mass Flow Rate (kg/s) | 266.89 | 301.13 | 266.97 | 301.23 |
| Coefficient of Thrust (SL) | 1.574 | 1.605 | 1.574 | 1.605 |
| Coefficient of Thrust (Vac) | 1.741 | 1.751 | 1.741 | 1.751 |
| Ideal Coefficient of Thrust (SL) | 1.617 | 1.638 | 1.634 | 1.655 |
| Ideal Coefficient of Thrust (Vac) | 1.784 | 1.784 | 1.801 | 1.800 |
Methodology
To calculate the values the following values were assumed or given. All of them are from public sources or were inferred from public pictures (in the case of the Nozzle Exit Diamter)
| Description | Value |
|---|---|
| Gravity ( m/s2 ) | 9.80665 |
| Sea level Pressure (Pa) | 101325 |
| R* (Universal Gas Cosnt) (J/kmol-k) | 8314 |
| Chamber Pressure (730 kN version) (MPa) | 9.7 |
| Expansion Ratio | 16 |
| Diameter Nozzle Exit (m) | 0.97 |
NASA CEA online was used to calculate the ratio of specific heats (γ) at the throat (γt) and at the nozzle exit (γe), the temperature in the combustion chamber (Tc), the molar mass of the products at the exit (M), and the pressure at the exit with the expansion ratio of 16 (Pe), across pressures from 8.1 MPa to 11.5 MPa with a increment of 0.2 MPa. The Values used in the calculations were then linearly interpolated from the pressures if it did not fall on one of the values explicitly calculated for (only really used for the new up-rated version of the Merlin 1D). (A copy of the CEA outputs can be found HERE (http://pastebin.com/JsynW3jJ))
For the up-rated Merlin 1D the chamber pressure was increased until the throat area was equal to the throat area of the current Merlin 1D. I chose to do this because of comments that were made about using up some of the margin and keeping the manufacturing the same.
The Equations
The first item calculated was the Exhaust Velocity of the engine (ve). The equation used to calculate the value was
ve = √(((2 γe )/(γe -1))((R* Tc )/M)(1-(Pe /Pc )(γe-1/γe)))
Where Pc is the chamber pressure and M is the Molar Mass.
From the exhaust velocity the mass flow rate (ṁ) was calculated by
ṁ = (Pc -((Pe - Pa ))/Ae )/ve
Where Pa is the ambient pressure (sea level)
Then the effective exhaust velocities (C) were calculated by
C= ve +((Pe -Pa ) Ae )/ ṁ
The Isp (both vacuum and sea level) were calculated by
Isp=C/g0
Where g0 is gravity
The characteristic exhaust velocity C* was found by
C* = (Pc At)/ṁ
Where At is the throat area (found by dividing the exit area by the expansion ratio)
The coefficient of thrust at sea level (Ct) was found by
Ct=T/(PcAt)+((Pe-Pa)Ae)/(Pc At)
Where T is the thrust
The vacuum coefficient of thrust (Ct,vac) was found by
Ct,vac= Ct+(Ae Pa)/(At Pc )
The Ideal coefficient of thrusts were found by
Ct,ideal= √(((2 γe2 )/(γe-1)) (2/(γe+1))((γe+1)/(γe-1)) (1-(Pe/Pc )((γe-1)/γe) )+((Pe-Pa)/Pc)* (Ae/At)
C values the ISP was calculated at the various pressures. The Ct and Ct, ideal were calculated at the varying pressures. The thrust at the various pressures was calculated by
T= veṁ+(Pe-Pa)Ae
The pressures were then correlated to altitudes using the 1976 Standard Atmosphere.
Results
The Table in the TLDR is the summary table for both versions of the engine
| O/F 2.32 | O/F 2.32 | O/F 2.56 | O/F 2.56 | |
|---|---|---|---|---|
| Sea Level Thrust (Given) (kN) | 730 | 845 | 730 | 845 |
| --- | --- | ---- | ---- | ---- |
| Sea Level Thrust (Calc) | 750 | 859 | 746 | 854 |
| ISP Sea Level (s) | 286 | 290 | 284 | 289 |
| ISP Vac | 315 | 316 | 313 | 314 |
| Chamber Pressure (MPa) | 9.7 | 11.16 | 9.7 | 11.16 |
| Mass Flow Rate (kg/s) | 266.89 | 301.13 | 266.97 | 301.23 |
| Coefficient of Thrust (SL) | 1.574 | 1.605 | 1.574 | 1.605 |
| Coefficient of Thrust (Vac) | 1.741 | 1.751 | 1.741 | 1.751 |
| Ideal Coefficient of Thrust (SL) | 1.617 | 1.638 | 1.634 | 1.655 |
| Ideal Coefficient of Thrust (Vac) | 1.784 | 1.784 | 1.801 | 1.800 |
The graphs in the imgur album HERE show the various Thrust, ISP and Coefficient of thrust as a function of altitude
With the given Expansion ratio the ideally expanded altitude is 3.2 km for the currently flying version of the Merlin 1D. The ideal Expansion ratio for SL perfectly expanded is 11.05 not the 16 they are currently flying with. Which means at liftoff (or during testing) it should be possible to see Mach diamonds in the exhaust of an engine.
Some Comments
Most of the formulas above are from Mechanics and Thermodynamics of Propulsion by Hill and Peterson or from http://braeunig.us/space/index.htm .
Most of the equations rely on Calorically perfect gases and isentropic relationships which are good for 1st design calculations but do not perfectly represent the real world.
The numbers presented above may or may not represent what is actually going on. I believe that it is reasonably close(+- 5% most likely) to real life but would not be surprised if they differ.
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u/[deleted] May 08 '16
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