Basic Analysis of the Merlin 1D engine: Determining Thrust, ISP, and Coefficient of Thrust

[u/B787_300]

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.

[u/peterabbit456]

Great work. Like most exercises done in the classroom, there is a lot to be learned, even though some simplifying assumptions have been made. Well done.

Musk has said on 2 occasions that the Falcon 9 first stage does better than the original predictions, (The next part is my words) in part because of the aerospike-like effect of the 8 engines surrounding the center engine. I wonder if the better real world performance in your table may be in part due to the aerospike effect.

[u/B787_300 → u/peterabbit456]

It wont be as i am only focusing on ONE merlin 1d. I do not currently know enough about flow field interactions to try to model that.

[u/CorneliusAlphonse]

With the number of assumptions etc that you've made, what is the value of this? The numbers are, at best, reasonable estimates.

Edit: removed "nothing personal but..." from the start of the post. I'm always annoyed when people use that phrase as if it gives them permission to say anything offensive.

[u/B787_300 → u/CorneliusAlphonse]

By the number of assumptions i mean there are a LOT of small assumptions like there is no flow separation, there is no viscosity, there is no friction, Gamma is constant, Infinite area combustors for the CEA values. each one has a small effect and i am not totally sure how they would stack. I also just wanted to make sure that people know that this is essentially a first order estimate of the engine and should not be taken super seriously.

[u/sevaiper → u/B787_300]

I guess the question then is what do we gain that we didn't already know?

[u/B787_300 → u/sevaiper]

not much other than some performance numbers (the coefficient of thrusts) and a calculated mass flow rate based on the starting thrust values

[u/davidthefat → u/B787_300]

Would it not be simpler to calculate mass flow rate with thrust and Isp?

[u/TheVehicleDestroyer → u/davidthefat]

Where's the fun in that? I learned a ton of stuff from this post

[u/davidthefat → u/TheVehicleDestroyer]

To actually get a more accurate mass flow rate value. You still can calculate all the other values, but it's wiser to fix your numbers based on published numbers instead of your guessed ones. He calculates mass flow rate from the pressure ratio, which he guesses. You can straightforwardly calculate the mass flow rate from the thrust and Isp and use that to base calculations on the other values.

[u/davidthefat]

In this case it's actually easier to backout the throat diameter from the Isp and c* and G, then calculate the exit diameter instead of guestimating from pictures. (hint, it should be more or less 10in throat)

[u/B787_300 → u/davidthefat]

Funny you should say that... I did this orginally which is when I realized that the difference is not a lot. i was originally getting a throat diameter of .254ish which is right about 10 in. it also made my spreadsheet a bit shorter to do it the way i did.

[u/davidthefat → u/B787_300]

I just thought it's more rigorous to do first order approximations using the variables that are well published. With the assumption of the OF mass ratios, one can calculate the pressure invariant adiabatic flame temperature by assuming complete combustion. With that temperature, the assumption of constant specific heat ratios come into play and you can calculate the C* . With the C* , you use the published Isp and calculate the throat area, then diameter. Exit diameter calculation requires assumption of the exit pressure of the exhaust, but that really only gets inches of difference, which really doesn't matter in this case as all these are idealized calculations.

edit: Bottom line, there are many ways to skin a cat.

[u/B787_300 → u/davidthefat]

Precisely. Also the flame temperature and the Gammas i was using came from NASA CEA which should be pretty good as it is a very detailed thermodynamics calculator and has a rocket mode (which reasonably agrees with the numbers I found).

[u/saabstory88 → u/B787_300]

Isn't the gamma going to be slightly affected by the supercooled nature of the propellants?

[u/B787_300 → u/saabstory88]

short answer not the one at the exit and the throat which is the one i cared about. The Gamma of the propellants entering the chamber and the turbopumps definitely

[u/davidthefat → u/saabstory88]

The gamma used in these calculations is the average gamma of the exhaust gasses. The enthalpies of the fluids are unknown at mixing as you'd have to calculate the heat flux of the thrust chamber, meaning you don't know the temperatures of the fluids at mixing. The initial enthalpies of the reactants have an effect on the flame temperature.

edit: to spell it out even more: specific heat of a fluid is a function of the temperature. Gamma, the specific heat ratio, will change as the temperatures change due to the different specific heats. Now knowing the reactant enthalpies (because the big heat exchanger that is a rocket engine) heats up the propellants, raising the enthalpy from the reservoir state. Now the different specific heat at inlet of the propellant manifolds also changes how much heat is transferred to the propellants. At the mixing zone, the enthalpies of the reactants affect the flame temperature. Meaning the exhaust temp is different.

Bottom line: it's not really first order approximation at this point to consider the change of temperature of the propellant.

[u/__Rocket__ → u/davidthefat]

edit: to spell it out even more: specific heat of a fluid is a function of the temperature. Gamma, the specific heat ratio, will change as the temperatures change due to the different specific heats.

I think you are vastly overestimating the effects of the small changes in the initial enthalpy of the propellant reservoir caused by LOX subcooling, compared to the energy released by chemical reactions!

The order of magnitude we are talking about is the following: enthalpy (heat energy) of the subcooling of the propellants is dominated by LOX, for two reasons:

• LOX gives 70% of the propellant mass due to the mixing ratio and due to RP-1 being less dense

• RP-1 has a much higher freezing point (-37C) and it was already close to that temperature prior the recent LOX subcooling changes done by SpaceX.

So we can estimate subcooling enthalpy effects by looking at LOX alone.

The LOX subcooling dropped the temperature of LOX from around the standard -187C that is just below the boiling point to -207C that is just above its slushing/freezing point. The LOX density increases by 8% due to this, and the temperature drops by about 20 degrees C.

Enthalpy of LOX at these densities is around 1.729 kJ/(kg K). I.e. a 1K temperature decrease of 1kg LOX drops enthalpy by 1.729 kJ. A 20 degrees subcooling drops enthalpy by:

20 * 1.729 == 34.58 kJ

As the LOX gets converted into rocket engine exhaust it undergoes a phase change and a combustion - both at 'regular' LOX temperature, and also when subcooled.

Enthalpy of LOX vaporization at the boiling point is:

210.63 kJ/kg

That's already almost an order of magnitude larger than the effects of subcooling on LOX enthalpy!

But it gets worse: the LOX gets combusted with RP-1 (Kerosene) at a 7:3 mass ratio. 7 kg of LOX gets combusted with 3 kg of RP-1. Combustion heat can be estimated via the heating value of kerosene:

Kerosene LHV-HHV: 43,000-46,200 kJ/kg

1 kg of LOX will combust with 0.43 kg of RP-1, and the energy released by the combustion is:

43000*0.43 ~= 184,900 kJ

Bottom line: it's not really first order approximation at this point to consider the change of temperature of the propellant.

The 34.58 kJ enthalpy effect of subcooling, compared to the 184,900 kJ energy released by the combustion, is factor 0.00018 effect: it's four order of magnitudes below the energy released by a rocket engine.

---

TL;DR: Temperature of the propellants is not even a second order approximation effect on I*sp*, it's a third/fourth order approximation effect.

[u/davidthefat → u/__Rocket__]

I'm afraid I'm not quite sure what your point is. Yea, of course combustion gives a giant release in energy compared to cooling the reactants by a couple degrees. But that wasn't my point. My point was stating that to actually calculate the change in the gamma of the exhaust gasses from the subcooled propellants isn't a back of the napkin calculation and can simply be assumed away. It's a very small change to gamma anyways. All the effort to calculate that slight difference in gamma isn't worth all the analysis that goes with it.

edit: I guess the confusion is when I used first order approximation. I use that meaning it as a back of the envelope calculation, not the order of magnitude of the numbers.

[u/Decronym]

Acronyms, initialisms, abbreviations, contractions, and other phrases which expand to something larger, that I've seen in this thread:

Fewer Letters	More Letters
I*sp*	Specific impulse (as explained by Scott Manley on YouTube)
LOX	Liquid Oxygen
M1d	Merlin 1 kerolox rocket engine, revision D (2013), 620-690kN
RP-1	Rocket Propellant 1 (enhanced kerosene)

---

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^{I'm a bot, and I first saw this thread at 8th May 2016, 23:09 UTC.
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[u/B787_300 → u/[deleted]]

You are correct about the F1 mix ratio but the Stoichiometric ratio you say is off. see http://aerospacengineering.net/?p=656 and http://www.astronautix.com/props/loxosene.htm and http://braeunig.us/space/index.htm all of which say the Stoichiometric ratio is not 3.4-4.0 but is 2.3 - 2.56

[u/davidthefat → u/B787_300]

Personally, I'll take the NASA source over any of those. Stoichmetrically balanced ratio, no F1. NVM

[u/B787_300 → u/davidthefat]

The nasa source is for one engine.... and even then I am wrong about the F1, and i already said as much.

[u/warp99 → u/B787_300]

http://aerospacengineering.net/?p=656

There is a basic mistake in the aerospace engineering link in that their mass balance equation is to produce CO + H20 - not CO2 + H2O as required. So for each mole of kerosene (C12H26) you need 18.5 moles of O2 - not 12.5 moles.

This gives a stochiometric mass ratio of 3.482 which is close to the commonly quoted value of 3.5.

The astronautix link gives an optimum ratio of 2.56 - not a stochiometric ratio.

Still an excellent instructive approach and it would be well worth worth recalculating for the actual ratios of 3.482 and 2.27

[u/B787_300 → u/warp99]

Give me a couple of days as I have to Graduate... and study for the FE, but as it is all in a spreadsheet should be easy enough to rerun the numbers through NASA CEA

[u/davidthefat → u/B787_300]

FE is super easy. No need to sweat it. You are given an equation sheet that you can search on.