Space X\'s Merlin 1D engine analysis PDF

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Summary

This is a breakdown of the Merlin 1D engine used by Space X on their Falcon rocket. All available info at the time of making of the document was used. ...

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Little bit more detailed analysis of the SpaceX’s Merlin 1D engine

1. Introduction At the beginning, I would like to say that all below data was obtained based on publicly available data including: Wikipedia [2], photographs, reddit threads, etc.. Calculations were performed based on the rocket propulsion theory presented in available literature, mainly in Rocket Propulsion Elements 8th Edition [1]. I hope that you will enjoy information available in this document and it will be a good added value for the education purposes for all of you.

2. Assumptions and basic abbreviations Basic assumptions must be made to have starting point. It might be confusing for some of the readers why exactly such numbers were used for calculations, however that is the industry, nobody share exact specification of the products if they do not need to. Fact is that there are many contradictory information on the internet regarding Merlin 1D and nobody knows exactly which numbers are correct. unit system: SI (one and only one true and proper!) 2.1 General assumptions - Earth’s gravitational acceleration @ sea level g0=9.80665 [m/s^2] - Used naming convention for the nozzle parameters (taken from [1]):

-

Earth’s atmospheric pressure at sea level: 1atm = 0.101325 [MPa] Earth atmospheric pressure at various altitudes is changing per equation found @ [7]:

-

Ratio between inlet temperature and stagnation temperature (T1/T0) was set to 1 (perfect combustion).

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2.2 Assumed parameters for engine: Merlin 1D - booster Parameters are taken from various sources mentioned in section References: -

constant chamber pressure: nozzle expansion ratio: Thrust @ sea level: Specific Impulse @ sea level: Thrust @ vacuum: Specific Impulse @ vacuum:

p1 = 10.8 [MPa] ϵ = 16 [-] F = 845162.107 [kN] / 190000 [lbf] Isp = 288.5 [s] F = 914109.542 [kN] / 205500 [lbf] Isp = 312 [s]

Additional assumption was set to the nozzle exit diameter. There were number of threads on reddit [4], nasaspaceflight.com [5] forum and spaceflight101.com [6] regarding the nozzle bell end-section diameter. Some of the SpaceX photographs were scaled and nozzle end section measured in order to estimate diameter of the bell end diameter. In general, the range of searched diameters varies from 930 [mm] to 1100 [mm] (as a maximum possible diameter due to size constraints). The difference between the diameters may come from many reasons: -

perspective of the measured photograph, measurement accuracy inaccurate dimensions of the booster stage released to the public (?) – few millimeters can make here a difference during scaling, etc.

Nevertheless, the exact end-section diameter of the nozzle in calculations was not set strictly but was as a variable and found by solver. 2.3 Assumed parameters for engine: Merlin 1D - vacuum Parameters are taken from various sources mentioned in section References: -

constant chamber pressure: nozzle expansion ratio: Thrust at vacuum: Specific Impulse @ vacuum:

p1 = 10.8 [MPa] (for initial iteration) ϵ = 165 [-] F = 934126.539 [kN] / 210000 [lbf] Isp = 348 [s]

2.4 Parameters of the propellant Main parameters were derived from [3] by simple interpolation/measurement of graphs. See Appendix 1 with source. Name:

RP-1 / KEROLOX

combustion pressure heat capacity ratio

pc k

10.8 1.2170

oxidizer / fuel ratio combustion temp. (Stagnation temp) unniversal gas constant gas molar mass

T0 R' ɱ

2.33 3625 8.314459848 21.9

propullsion gas constant acoustic velocity of propellant @T0

R a

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379.66 1294.18

[MPa] [-] [-] [K] [J/(mol*K)] [g/mol] [J/(K*kg] [m/s]

3. Methodology Equations behind the calculations are clearly defined and explained in literature so there is no point, however in some cases simplifications of the computational model must be made: -

propellant mass flow rate assumed to be constant, pressure, velocity and nozzle area at exit are to be calculated, throat area of the vacuum and booster versions are assumed to be same.

Making long story short, values mentioned in paragraph 2 were used for establishing two models and later they were compared in order to gain the best convergence of the results. For example, the thrust may be defined in two ways: F= ṁ*v2 + (p2-p3)V2 and F = CF*p1*At. For various conditions, different boundary conditions are taken and at the end missing values are calculated. Frankly speaking the assumption was that the real engine performance was as close to the perfect engine as it is possible, which was probably the goal of the Tom’s Muller design team.

4. Results 4.1 Results accuracy It must be indicated that reality is not a Kerbal Space Program nor the simulation in FE software so theoretical results always will be different from the real rocket engine performance during the flight and different from the tests in laboratory. For sure the engine specifications available are not exact and it must be considered that probably it is done on purpose in order to not spread the intellectual property \ know-how of the company. What’s more, rocket equipment is covered by ITAR (not so much into it, but from my understanding some of the information are confidential by law) and simply must not released for public, thus we should not take available data as accurate. Last but not least we do not know whether the public-released data are given as average for one of the nine Falcon 9 engines or the values are specified for one insulated engine in laboratory environment (or some mix of those conditions). It is not secret knowledge that the engines will work with different efficiency when they operate insulated in laboratory and during flight when flow around the rocket as well as surrounding engines influence on the field of pressures around considered engine (that’s why tests are so important). Such corrections must be taken into account when reading results attached herewith. In general, results order of magnitude and ‘shapes’ of the graphs presented below are (or should be) correct, but relative error in some cases may be confusing. Well, that’s the effect of numerical methods (used here). Relative error may be also interpreted as the losses of some real parameters with respect to the ideal engine (?). In few cases below values are repeated but the tables/graphs are copied directly from the spreadsheet, so they were necessary somehow for efficient work. Legend:

solver variable ‘hard' input vaue calculated error / difference

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4.2 Merlin 1D – booster thrust at sea level

F(sl)

845162.107

[N]

thrust in vacuum specific impulse at sea level specific impulse in vacuum propellant mass flow rate at sea level

F(vac) Isp(sl) Isp(vac) ṁ (sl)

914109.542 288.5 312.0 298.73

[N] [s] [s] [kg/s]

propellant mass flow rate in vacuum propellant mass flow rate (taken for calculations) characteristic velocity altitude of optimum nozzle expansion nozzle expansion ratio

ṁ (vac)

298.76

[kg/s]

298.73 1731.263 2838.875 16.000

[kg/s] [m/s] [m] [-]

heat capacity ratio of the propellant nozzle exit velocity (calibrated) nozzle exit velocity (from thermodynamics/ideal) relative error max. theor. nozzle outlet velocity momentum thrust

k= v2 (F,p2,p3,A2)

1.217 2905.9

[-] [m/s]

v2 (k,R,T1) ε(v2) (v2)max ṁ*v2

3021.0 3.809% 3928.976 868074.291

pressure ratio pressure ratio inlet temp. / stagnation temp.

At*p1= p1/p2 p2/p1 T1/T0

517173.760 150.929 0.007 1.000

[m/s] [-] [m/s] [N] [N]

throat/inlet temperature ratio acoustic velocity at throat

Tt/T1= at

0.902 1229.211

diameter

D

area based on expansion ratio aera calculated from thermodynamics relative error for aera temperature specific volume gas density

A

pressure

p

A ε(A) T V ρ = 1/V

ṁ c* H_opt ϵ = A2/At

@ inlet (1) @ throat @ nozzle exit (2) 980.115 246.923 987.692 N/A 47886.459 754472.746 49782.486 N/A 3.81% 3625.000 3270.185 0.127 0.205 7.847 4.882 10.80

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6.061

766183.348

[-] [-] [-] [-] [m/s]

[mm] [mm^2]

777465.717 [mm^2] 1.45% [-] 1481.894 [K] 7.862 [m^3/kg] 0.127 [kg/m^3] 0.071557 [MPa]

@ H_opt @ sea level @ vacuum 868074.291 845162.107 914109.542

thrust

F

atmospheric pressure pressure thrust searched momentum thrust (thrust - pressure thrust) relative error for momentum thrust thrust coefficient (calibrated) thrust coefficient (from thermodynamics/ideal) thrust check equation (for ideal CF) effective exhaust velocity effective exhaust velocity

p3 (p2-p3)A2

specific impulse

Isp (ca)

ṁ*v2 [desired]

0.000 [MPa] 54825.773 [N]

868074.291 867969.861 859283.769

[N]

ε(ṁ*v2)

0.00%

0.01%

1.02%

[-]

CF (F,p2,At)

1.678

1.634

1.768

[-]

CF (k,p1,p2,p3,At,A2)

1.678

1.634

1.785

[-]

868074.291 845266.537 922900.064

[N]

F ca (v2 (k,R,T1))

3020.976

2944.626

3204.507 [m/s]

cb (v2 (F,p2,p3,A2))

2905.918

2829.568

3089.450 [m/s]

308.054

300.268

326.769

[s]

296.321

288.536

315.036

[s]

N/A

288.500

312.000

[s]

0.000%

0.012%

0.952%

[-]

ε(Isp)

N/A

0.012%

0.973%

[-]

Δ(Isp) = ε(Isp) *Isp(prov.)

N/A

0.036

3.036

[s]

b

specific impulse specific impulse (provided)

Isp (c )

relative error for thrust relative error for specific impulse absolute error for specific impulse

ε(F)

Thousands

0.071557 0.101325 0.000 -22807.754

[N]

Isp(prov.)

Merlin 1D:booster - thrust characteristics

1000.0

320.0 922.9 315.0

800.0

Thrust [N]

305.0

Pressure Thrust (p2-p3)*A2 [N] 400.0

Total Thrust F(h) [N] 300.0

Specific Impulse Isp [s] 200.0

295.0 0.0 -10.00

0.00

10.00

20.00

30.00

40.00

50.00

60.00

290.0 70.00

Thousands 285.0

-200.0

Altitude [m]

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Specific Impulse [s]

310.0 600.0

Merlin 1D:booster - nozlle characteristics: velocity & specific volume 10

3500

specific volume [m^3/kg]

3000

velocity [m/s]

2500 2000 1 1500 1000 500 0.1

0 11.0

10.0

9.0

8.0

7.0

6.0

5.0

4.0

3.0

2.0

1.0

0.0

Pressure through nozzle [MPa] velocity through nozzle vi [m/s]

throat section

specific volume Vi [m^3/kg]

Merlin 1D: booster - nozlle characteristics: nozzle area & velocity/spevcific volume ratio

vi/Vi ratio [kg/(s*m^2)]

6000 5000 4000 0.1 3000 2000 1000 0 11.0

0.01 10.0

9.0

8.0

7.0

6.0

5.0

4.0

3.0

2.0

1.0

0.0

Pressure through nozzle [MPa] ratio vi/Vi [kg/(s*m^2)]

throat section

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cross sectional area Ai [m^2]

nozzle cross section area [m^2]

1

7000

7000

4

6000

3.5 3

5000

2.5

4000

2 3000

1.5

2000

1

1000 0 11.0

Mach number [-]

temperature [K]

Merlin 1D: booster - nozlle characteristics: temperature & Mach number

0.5 0 10.0

9.0

8.0

7.0

6.0

5.0

4.0

3.0

2.0

1.0

0.0

Pressure through nozzle [MPa] temperature Ti [K]

throat section

Mach number M [-]

Merlin 1D: booster- nozlle characteristics: specific volume vs nozzle radius 600

radius [mm]

500 400 300 200 nozzle radius r [mm] 100

throat section

0 0

1

2

3

4

5

spercific volume [m^3/kg]

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6

7

8

9

4.3 Merlin 1D – vacuum thrust at sea level

F(sl)

N/A

thrust in vacuum specific impulse at sea level specific impulse in vacuum propellant mass flow rate at sea level

F(vac) Isp(sl) Isp(vac) ṁ (sl)

propellant mass flow rate in vacuum propellant mass flow rate (taken for calculations) characteristic velocity altitude of optimum nozzle expansion nozzle expansion ratio

ṁ (vac)

heat capacity ratio of the propellant nozzle exit velocity (calibrated) nozzle exit velocity (from thermodynamics/ideal) relative error max. theor. nozzle outlet velocity momentum thrust

k= v2 (F,p2,p3,A2)

[N] 934127 0 348

[N] [s] [s] [kg/s]

273.720

[kg/s]

273.720 1742.947 21343.813 165.000

[kg/s] [m/s] [m] [-]

1.217 3320.372

[-] [m/s]

v2 (k,R,T1) ε(v2) (v2)max ṁ*v2

3428.698 3.159% 3928.976 908850.659

pressure ratio pressure ratio inlet temp. / stagnation temp.

At*p1= p1/p2 p2/p1 T1/T0

477078.757 3102.867 0.000 1.000

[m/s] [-] [m/s] [N] [N]

throat/inlet temperature ratio acoustic velocity at throat

Tt/T1= at

0.902 1229.211

[-] [m/s]

diameter

D

@ inlet (1) @ throat @ nozzle exit (2) 980.115 246.923 3171.782

[mm]

area based on expansion ratio aera calculated from thermodynamics relative error for aera temperature specific volume

A

gas density pressure

ρ = 1/V p

N/A

ṁ c* H_opt ϵ = A2/At

A ε(A) T V

N/A 47886.459 754472.746 49448.743 N/A 3.16% 3625.000 3270.185 0.138 0.222 7.239 9.963

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4.503 5.591

7901265.776

[-] [-] [-]

[mm^2]

8159309.084 [mm^2] 3.16% [-] 1481.894 [K] 102.206 [m^3/kg] 0.010 [kg/m^3] 0.0032108 [MPa]

@ H_opt @ sea level @ vacuum 908850.659 133624.341 934126.539

thrust

F

atmospheric pressure pressure thrust searched momentum thrust (thrust - pressure thrust) relative error for momentum thrust thrust coefficient (calibrated) thrust coefficient (from thermodynamics/ ideal) thrust check equation (for ideal CF) effective exhaust velocity effective exhaust velocity

p3 (p2-p3)A2

specific impulse

Isp (ca)

0.003 0.000

ṁ*v2 [desired]

908850.659

0.1013250 -775226.317

[N]

0.000 [MPa] 25369.438 [N]

908850.659 908757.101

[N]

ε(ṁ*v2)

0.00%

0.00%

0.01%

[-]

CF (F,p2,At)

1.905

0.280

1.958

[-]

CF (k,p1,p2,p3,At,A2)

1.905

0.280

1.958

[-]

133624.341 934220.096

[N]

F

908850.659

ca (v2 (k,R,T1))

3428.698

596.506

3521.382 [m/s]

cb (v2 (F,p2,p3,A2))

3320.372

488.180

3413.056 [m/s]

349.630

60.827

359.081

[s]

338.584

49.780

348.035

[s]

b

specific impulse specific impulse (provided)

Isp (c ) Isp(prov.)

N/A

N/A

348.000

[s]

relative error for thrust relative error for specific impulse absolute error for specific impulse

ε(F)

N/A

N/A

0.010%

[-]

ε(Isp) Δ(Isp) = ε(Isp) *Isp(prov.)

N/A

N/A

0.010%

[-]

N/A

N/A

0.035

[s]

Merlin 1D:vacuum - thrust characteristics

350.0

1000.0

934.2

300.0

800.0 600.0

250.0

Thrust [N]

400.0 200.0

200.0 0.0

-10.00

0.00 -200.0

10.00

20.00

30.00

40.00

50.00

60.00

70.00

Thousands

-400.0

150.0 100.0

Pressure Thrust (p2-p3)*A2 [N]

-600.0

Total Thrust F(h) [N]

-800.0

Specific Impulse Isp [s]

-1000.0

50.0 0.0

Altitude [m]

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Specific Impulse [s]

Thousands

1200.0

Merlin 1D:vacuum- nozlle characteristics: velocity & specific volume 1000

3500

100

velocity [m/s]

2500 2000

10 1500 1000

1

specific volume [m^3/kg]

3000

500 0.1

0 10.0

9.0

8.0

7.0

6.0

5.0

4.0

3.0

2.0

1.0

0.0

Pressure through nozzle [MPa] velocity through nozzle vi [m/s]

throat section

specific volume Vi [m^3/kg]

Merlin 1D: vacuum - nozlle characteristics: nozzle area & velocity/spevcific volume ratio

vi/Vi ratio [kg/(s*m^2)]

5000 1

...