Falcon 9 V1.1 Reverse Engineering

Reverse Engineering the Falcon 9 Two Stage Rocket, Merlin 1D Engine Shane Bonner, Patrick Cieslak, Kyle Criscenzo, Allyson Smith Embry-Riddle Aeronautical University, Daytona Beach, Florida, 32114 alcon 9 Multiple performance and operating characteristics of the F alcon two stage rocket were calculated based on published data of its M erli n 1D rocket engines designed and manufactured by SpaceX . The parameters calculated relate to mass ratios, performance, dimensions, and ideal to actual relationships of individual stages and the rocket as a whole. In order to evaluate some criterion, perfect expansion of exhaust gasses or sea level altitude assumptions were made where described.

Nomenclature A *

m2

= Area, Area, choked choked

2

A e

= Area, exit exit of nozzl nozzlee

m

a

= Accel Accelerati eration on

m/s

c * Actu al = Veloc elociity, chara charact cter eriistic stic of of actua actuall rock ocket * c Ideal

d e

/

= Mas s , b u rn o ut

kg

M l

= Mas s , p ayload

kg

M p p

= Mas s , p ro p ellant

kg

M s s

= Mas s , s tructu re

kg

ṁ

= Mas s flo w rat e

kg / s

P

= Pres s u re, s t at ic

Pa

*

/

P

= Pres s u re, ch o ked

Pa

= Diameter, ch o ked

m

P 0

= Pr P res s u re, co mb us tion

Pa

= Diameter, exit

m

P e

= Pres s u re, exit

Pa

C Ƭ,Ideal = Th ru s t co efficient , id eal rocket d

/

= Veloci elocity, ty, charac character teriistic of ideal idealiized rocket rocket /

C Ƭ,Actual = Th ru s t co efficient , actu al ro cket *

2

M b

ε

= St S tru ctu ral co efficien t

/

Ɍ

= Mas s ratio

/

γ

= Sp ecific h eat rat io

/

R

= Gas co n s t an t, s pecific

J/ kg * K

g e

= Acceleration Acceleration of gravity, gravity, Earth Earth

m/s

R U

= Gas constant, universal

J/mol*K J/mol*K

h

= A ltit ud e

m

ρ

= Den s ity , s tatic

kg / m3

h Optimal = A ltit ud e, o p timal

m

Ƭ

= Th rus t

N

I sp sp

= Impu ls e, s pecific

s

T

= Te Temp erat ure, s tatic

K

λ

= Payload ratio

/

T C

= Temp erat ure, comb u s t io n

K

M

= M as s , mo lar

kg /kmo l

t b

= time, b u rn

s

M e

= M ach nu mb er, exit of n o zzle

/

u eq

= velo city , eq uivalen t o f exh au s t

m/s

M 0

= Mass , total

kg

v

= velo city

m/s

2

I. Introduction

T

he goal of this project was to apply the learned theories and concepts to an existing rocket design and reverse engineer its performance parameters. The Falcon 9 two stage rocket from SpaceX was chosen. The Falcon 9 utilizes nine Merlin nine Merlin 1D liquid 1D liquid propellant engines during the first stage burn followed by a second stage with just one of the same engine. Several parameters of the rocket had to be found before analysis could begin. The mass properties of both stages were found. This includes the mass of the propellant, payload, structure, and total for both stages. The specific impulse, burn time, expansion ratio, chamber pressure, and thrust of the engines are also known. The propellant used was RP-1 and LOX as the oxidizer.

1 Embry-Riddle Aeronautical University

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II. History The Falcon 9 v1.1 is the second rocket in the Falcon orbital launch vehicle family. Developed by SpaceX in 2011, it is still used in combination with the Dragon capsule to resupply the International Space Station for NASA. It is planned to be used with the Dragon V2 capsule to transport crew to the International Space Station under the Commercial Crew Program in the future. The v1.1 has much better performance than the v1.0, with an increase of 60% in the thrust to weight ratio. The rocket has the thrust capability to lift 13,150 kg to low Earth orbit, or 4,850 kg to geostationary transfer orbit. The first stage is powered by nine Merlin nine Merlin 1D engines, 1D engines, producing 5,885 kN of thrust at sea-level. The stage burns for 180 seconds, with 6,672 kN of thrust at the burn-out altitude. The first stage ’s ’s engines are aligned in an ‘Octaweb’ pattern ‘Octaweb’ pattern which streamlines t he manufacturing process by arr anging the engines in a simple arrangement of eight engines in a circle and an ninth in the middle. The second stage is powered powered by a single Merlin 1D 1D engine designed with a larger nozzle for optimal performance in the near-vacuum of space.

III. Parameter Determination Seven parameters from Falcon from Falcon 9 were 9 were determined using the information that was found on the rocket combined with known relations between them. Since the Falcon 9 is a two stage rocket, each of the seven parameters were found for both stages. The parameters that were found include: Payload Ratios, Structural Coefficients, Mass Ratios, Mass Flow Rates, Δv’s, Δv’s, Throat and Exit Diameters, Optimal Altitudes, Characteristic Velocities (actual and ideal), and Thrust Coefficients (actual and ideal). A. Payload Ratio, Structural Coefficient, and Mass Ratio The payload ratio is used to gauge how much payload the rocket will be able to propel to its final destination. Keeping this in mind, a higher value is desired. The payload ratio for stage 1 is defined in Equation in Equation 1, where 1, where M02 is the total mass of the second stage (109,720 kilograms) and M 01 is the total mass of the first stage (531,020 kilograms). Using these values the payload ratio for the first stage is equal to 0.260.

 1 

M 02

Equation 1

M 01  M 02

The payload ratio for the second stage is defined in Equation 2, 2, where ML is the payload mass (13,150 kilograms). Using these values the payload ratio for the second stage is equal to 0.136.

 2 

M L

Equation 2

M 02  M L

The structural coefficient is a parameter that is used to show how much structure is needed to support the rocket. Less structure leaves more room for propellant and payload, therefore a small value for the structural coefficient is desired. The structural coefficient for the first stage is defined below in Equation in Equation 3, where 3, where Ms1 is the structural mass of the first stage. This value was found to be 25,600 kilograms. Plugging in the known values into the equation gives a structural coefficient of 0.061 for the first stage.

 1 

M s1

Equation 3

M 01  M 02

The structural coefficient of the second stage is found using Equation using Equation 4, where 4, where Ms2 is the structural mass of the second stage (3,900 kilograms). Using this value, along with the payload and total mass of stage 2 gives a structural coefficient of 0.040.

 2 

M s 2

Equation 4

M 02  M L

Quickly comparing the two stages shows that the second stage is more structurally efficient that the first.









of the previous ratios. By using the previously calculated payload ratios and structural coefficients, the mass ratios of the first and second stages were found to be 3.924 and 6.435, respectively. Equation 5 1  

R 

  

B. Mass Flow Rate The mass flow rate for a rocket determines how much propellant is being expelled over time. This will cause the total mass of the rocket to change over time as well. The average mass flow rate for each stage can be estimated by dividing the mass of propellant (M p) by the burn time (t b), as seen in Equation in Equation 6, assuming 6, assuming a constant mass flow rate over the burn time

m  M p / t b

Equation 6

For the first stage the propellant mass is known to 395,700 kilograms with a burn time of 180 seconds, giving a mass flow rate of 2,198.3 kilograms per second for the first stage. However, nine of the same engines make up the first stage, so that 2,198.3 kilograms per second being expelled is of all nine engines. The mass flow rate per engine can be found by dividing the total mass flow by 9, giving 244.25 kilograms per second per engine. For the second stage the propellant mass is known to be 92,760 kilograms burned over 372 seconds, giving a mass flow rate of 249.11 kilograms per second. For the second stage, only 1 Merlin 1D engine 1D engine is used. C.

Δv

Δv Δv is an important parameter that characterizes a rockets ability to accelerate, neglecting drag and gravity. Equation 7 defines this parameter, where u eq is the equivalent exhaust velocity and R is the mass ratio that was defined above. Equation 7 v  u ln R eq

Before Δv Δv can be determined, the equivalent velocity needs to be defined. Equation 8 defines the equivalent velocity to be equal to the specific imp ulse multiplied by the acceleration due to gravity at Earth’s surface (9.81 m/s2). For the first stage the specific impulse is known to be 282 seconds, making the equivalent velocity equal to 27,662.4 m/s. For the second stage the specific impulse is known to be about to be 345 seconds, making the equivalent velocity equal to 3,384.5 m/s.

ueq  I sp g e

Equation 8

Now, going back to Equation to Equation 7 the Δv for Δv for stage 1 and stage 2 is calculated to be 3,872 meters per second and 6,301 meters per second, respectively. The total Δv of Δv of the rocket is found by adding the Δvs from Δvs from each stage. This gives a total Δv equal Δv equal to 10,083 meters per second. Calculating Δv shows how much speed the rocket will gain over a complete burn time, but it is also very important to analyze what the instantaneous change in velocity is over a short time. This term, also known as acceleration, is crucial to know because as propellant is burned, weight is lost and the rocket has more excess thrust available. This means that as a stage progresses through its burn period, it actually accelerates at a faster rate than initially. Due to this, the acceleration of the rocket as a function of time was found. By taking the initial total mass of each stage (M 0i) and subtracting the product of mass flow rate of fuel ( ṁ) and time (t), mass as a function of time was then found as shown in Equation in Equation 9.

M (t )  M 0  mt

Equation 9

Next, Newton’s 2nd Law of motion was utilized to find the acceleration. Although a high mass of propellant is being used in a short time, the assumption of constant mass was made to simplify the calculations, as shown in Equation 10. 10. With acceleration (a) now equaling the divisor of thrust (T), assumed to be a constant, and the time dependent mass (M i(t)), acceleration as a function of time was now found.

a(t )  T / M (t )

Equation 10









With With Equation 11 now giving the acceleration of the rocket in terms of g’s g’s as a function of time, this could be plotted versus each stage’s respective burn times. In In Figure 1, stage 1, stage 1 was plotted from 0s to t b1 =180s. Stage 2 was plotted from t b1 =180s to t b2=372s. Upon inspection of Figure 1, 1, it can be noticed that as time progresses, the acceleration increases with time as predicted. It can also be identified that the time of peak acceleration is right before stage burnout. To find the initial and final values for accelerations of each stage, the respective time was substituted into Equation 11 for the respective stage. For stage 1, the rocket initially experienced 1.13g’s becoming 4.43 as the stage burned out. The second stage initially accelerated at 1.26 g’s becoming 4.79 as that stage burned out. The accelerations experienced towards the end of the stages’ burn might be outside of the design limitations of the rocket or payload, so the Falcon 9 does reduce throttle of the engines as burn time progresses, a factor not accounted for in this repor t3.

Figure 1. The relationship between burn time and gacceleration for both stages of the Falcon 9 rocket.

D. Throat and Exit Diameters To find the dimensions of the nozzle, the first step was getting the exit diameter. The exit diameter of the first and second stage nozzles was found to be .923m and 2.58m by measuring the known width of the body and comparing it to the relative size of the nozzle. nozzle . Notice on the latter that the nozzle’s exit is much larger to allow for more optimal expansion as back pressure is reduced at higher altitudes. With this diameter, the area of the exit for each nozzle could be calculated. To get the throat dimensions, the known exit-to-choked area ratios of 16 for stage 1 and 117 for stage 2 were used. Lastly, these throat areas were converted to diameters for stage 1 and 2 of .231m and .238m respectively. E. Optimal Altitudes It is known that a rocket provides is most thrust when the exit pressure of the exhaust gasses is perfectly expanded to the ambient pressure, Pe = Pa. Assuming isentropic flow through the nozzle, isentropic relations provide an exit pressure-to-chamber pressure ratio for a certain exit-to-choked area ratio as exit Mach numbers will be the same between the two. For both stage engines, the chamber pressure is 9.7MPa. Multiplying that by a ratio of .0064 for stage 1 and .000497 for stage 2, exhaust pressures for stage 1 and 2 are found to be 62.25kPa and 4.82kPa respectively. These pressures can then simply be compared to standard atmospheric tables which related altitude and pressure to find at which altitude the exhaust and ambient temperatures will equal, and hence perfect expansion. From

Figure 2. Liquid Oxygen and Kerosene combustion co mbustion charts.









F. Engine Performance The performance of the engine is characterized by 2 parameters: the characteristic velocity and coefficient of thrust. These values can be defined for an ideal rocket which is then used to compare to actual rocket performance. The ideal values are based on the chemical and physical properties of the exhaust gasses while the actual values come from measure performance parameters. Specifically, the combustion temperature (T 0), molecular weight (M), and the specific heat ratio (γ) were (γ) were estimated using propellant combustion charts as shown in Figure 24. These values were found based on the known combustion pressure of 9.7 MPa and assuming an ambient pressure of 1atm. The combustion temperature was found to be 3,575K, a molecular weight of 21.85 kg/kmol, and a specific heat ratio of 1.217. These three parameters will be used in the calculation of the characteristic velocity and thrust coefficient. 1) Characteristic Velocity The characteristic velocity is a measure of the combustion chamber’s operation independent of the nozzle. The ideal combustion chamber is shown in Equation 12 as a function of the specific heat ratio, combustion temperature, and molecular weight as previously found, but also the universal gas constant (R u), 8.314 J/mol*K. Using these values, an ideal characteristic velocity of 1789 m/s was found for both engines, as they have identical combustion chambers.

c

* ideal



1    1 



(  1) / ( 1)



  2 

RT 0

Equation 12

M

To find the actual combustion chamber performance, the combustion pressure, throat area (A *), and mass flow rate of propellant are analyzed as shown in Equation in Equation 13. Between 13. Between both engines the combustion pressure is the same as shown previously, but the throat areas and mass flow rates are slightly different, being .042m 2 and 244.3kg/s for stage 1 and .045m 2 and 249.1 kg/s for stage 2. As a result, it is found that the actual combustion chamber performance is just slightly lower than ideal, being 1662 m/s for stage 1and 1737 m/s for stage 2.

c

* actual



P0 A*

Equation 13

m

2) Coefficient of Thrust Now that the co mbustion chamber has been analyzed, it is necessary to analyze the nozzle’s operation. The nozzle’s nozzle’s ideal operation is characterized by a thrust coefficient. It is a function of the specific heat ratio, exit pressure (Pe), chamber pressure, and exit-to-choked area ratio. Upon inspection of Equation 14, 14, the second term accounts for non-ideal expansion, however ideal expansion will be assumed in this analysis. For stage 1, the ideal coefficient of thrust is 1.682 and 1.881 for stage 2.

cT ideal

2 2    1       1  2 

(  1) / (  1)

  P ( 1) /  P  P A 1   e    e a e* P0 A   P0  

Equation 14

To compare the real nozzles’ performance, a similar equation to that of the real characteristic velocity will be used. Equation used. Equation 15 shows how the real coefficient of thrust is a factor of thrust, chamber pressure, and throat area. Thrust for the first stage, assuming sea level conditions, will be the per-engine thrust, 645kN, and simply 801kN for the second stage. From this, a very well designed nozzle is found at perfect expansion conditions, with actual values of 1.661 for stage 1 and 1.851 for stage 2.

cT actual  IV.

T

Equation 15

P0 A*

Conclusion

When studying the calculated parameters and comparing them ideal cases or to other rockets where possible, it is evident that the Falcon 9 was and still is a well-designed vehicle. Its structural and payload ratios make it an









V.

Regenerative Rocket Engine Cooling

The Falcon 9 utilizes nine Merlin 1D 1D engines to achieve the required thrust to go into orbit, so it should be no surprise that these rockets operate at high temperatures; high flame temperatures lead to excellent thrust performance. However, this flame temperature is limited in part by the chemical properties of the materials that make up the combustion chamber and nozzle of the engine. Even the most resilient materials that exist today cannot by themselves withstand the temperatures generated when the oxidizer and propellant mix resulting in combustion. In addition to the high operating temperatures, rocket engines also have burn times that can last up to several minutes. These thermal stresses over a long period of time make the need for engine cooling arise aris e 2. Both the combustion chamber and nozzle must be cooled in order to retain the structural integrity of the engine engine4. Although several techniques have been used throughout the evolution of rocketry, one of the most popular options has been that of regenerative cooling for liquid fueled rocket engines. By observing the nozzle on a Merlin 1D engine 1D engine as shown in Figure in Figure 3, metallic 3, metallic tubes can be seen around the circumference. These tubes, as will be discussed, play a pivotal role in keeping the rocket assembly at a manageable operating temperature. The structural weight of any vehicle, rockets inclusive, has a large impact on its overall performance. The heavier the structure, the lower the efficiency of the vehicle. In order to minimize weight of rockets, designers can utilize the regenerative cooling method like used for the Merlin 1D. 1D . Cooling is accomplished by pumping liquid propellant from the storage tank and routing it around channels in the nozzle and combustion chamber on its way to the fuel injectors for burning. During its passage through the channels of the engine, the Figure 3. A Merlin 1D engine utilizes regenerative cooling as fuel absorbs heat energy from the engine which was heated by the exhaust shown by circumferential tubes on gases. This conduction and radiation of heat into the liquid coolant adds energy its nozzle nozzle5. to it which then increase performance by a small fraction once injected into the combustion chamber and utilized as a propellant propellan t1. This process is shown in Figure in Figure 4. Now that is has been shown that cooling of the engine is necessary in most applications, it must now be identifi ed where the highest heat flux occurs along the axial direction of the combustion chamber and nozzle. Although one might assume that this occurs in the combustion chamber, the highest heat flux actually occurs at the throat of the nozzle. This happens due to three factors: Flow compression, combustion, and a minimal radial diamete r 3 . With an increase in the local heat flux rate, the cooling rate must be increased in this area as well. While the throat could be the design condition for a constant geometry cooling channel system, overcooling of the other sections of the engine is “detrimental to engine performance” according to Naraghi Naragh i 3, so the channels’ geometry or flow characteristics must be varied at the throat when compared to the rest of the engine. Many different crosssectional shapes have been used for coolant channel design, but one of the most popular is a high aspect ratio rectangle, or a very tall and skinny shape perpendicular to the engine’s wall wall1. This increases the area of the sidewall around the coolant allowing a higher heat transfer rate from the engine wall to the liquid coolant. Another method to increase this sidewall area around the nozzle is to decrease the cross-sectional area of the channels, but to Figure 4. A simplified schematic of a basic regenerative cooling process 1 increase the number of channels channel s4. In essence, used in liquid propellant rocket engines. taking a few big tubes and separating them into more numerous smaller tubes. Although it can be observed that an increase in wall area will provide more friction











Appendix A. MATLAB Code

A.

Payload Ratio, Structural Coefficient, and Mass Ratio ............ ..................... .................. .................. .................. .................. ................... ............ .. 2

B.

Mass Flow Rate ................................. ................ .................................. ................................... ................................... ................................... .................................... .......................... ........ 3

C.

Δv

D.

Throat and Exit Diameters Diamet ers ................................. ................ ................................... ................................... ................................... ................................... .......................... ......... 4

E.

Optimal Optima l Altitudes Altitude s ................................. ................ .................................. ................................... ................................... ................................... .................................... ....................... ..... 4

F.

Engine Performance Perfor mance ................................... .................. .................................. .................................. ................................... .................................... ................................... ................. 5

1)

Characteri Chara cteristic stic Velocity Velocit y ................................. ................ ................................... ................................... ................................... ................................... ................................ ............... 5

2)

Coefficient Coeff icient of Thrust .................................. ................. .................................. .................................. ................................... .................................... ................................... ................. 5

A.

MATLAB Code ................................... .................. .................................. ................................... ................................... ................................... .................................... .......................... ........ 7

B.

Final Project - Falcon 9 Two Stage Rocket, Merlin 1D Engine.......... Engine................... .................. .................. .................. .................. ............. .... 7

C.

Constants Const ants and Parameters Parame ters ................................. ................ ................................... ................................... ................................... ................................... .......................... ......... 7

D.

1. Payload, Structural, Struc tural, Mass Ratios ................................... ................. ................................... ................................... .................................... ............................. ........... 9

E.

2. Mass Flow Rates .................................. ................. .................................. ................................... ................................... ................................... .................................... .................. 10

F.

3. Delta v and Acceleratio Accel eration n ................................... .................. ................................... ................................... ................................... ................................... ..................... .... 10

G.

4. Throat and Exit Diameter Diamete r .................................. ................. ................................... ................................... ................................... ................................... ..................... .... 13

H.

5. Optimal Altitudes Altitude s ................................... .................. .................................. .................................. ................................... .................................... ................................. ............... 14

I.

6. Characterist Charac teristic ic Velocity Veloci ty ................................... .................. ................................... ................................... ................................... ................................... ........................ ....... 14

J.

7. Thrust Coefficient Coeffic ient ................................... .................. .................................. .................................. ................................... .................................... ................................. ............... 15

K.

Results Result s ................................... .................. ................................... ................................... ................................... ................................... ................................... ................................... ................... 16

...................................................................................................................................................... 3

B. Final Project - Falcon 9 Two Stage Rocket, Merlin 1D Engine %{









atmprops 0

= [... 288.15 9.807 101325

1.225

1000 281.65 2000 275.15 3000 268.66

9.804 89880 1.112 9.801 79500 1.007 9.797 70120 0.9093

4000 262.17 5000 255.68

9.794 61660 0.8194 9.791 54050 0.7364

6000 249.19 9.788 47220 0.6601 7000 242.7 9.785 41110 0.59 8000 236.21 9.782 35650 0.5258 9000 229.73 9.779 30800 0.4671 10000 223.25 9.776 26500 0.4135

%

15000 20000

216.65 216.65

9.761 12110 0.1948 9.745 5529 0.08891

25000 30000

221.55 226.51

9.73 2549 9.715 1197

40000 50000

250.35 248.15

9.684 287 0.003996 9.654 79.78 0.001027

60000 70000

247.02 219.58

9.624 21.96 0.0003097 9.594 5.2 0.00008283

80000 h(m)

198.64 9.564 1.1 T(K) g(m/s2) P(Pa)

0.04008 0.01841

0.00001846]; rho(kg/m^3)

% LOX/RP-1 gamma

= 1.217;

%

Tc Mbar

= 3575; = 21.85;

%[K] %[kg/kmol]

P0 R

= 9.7e6; = Rbar/Mbar

%[Pa] %[J/Kg*K]

% First Stage m_prop_s1

= 395700;

%[kg] %mass of propellant in first stage

m_pay_s1 m_struct_s1

= 0; = 25600;

%[kg] %mass of payload in first stage %[kg] %mass of stucture of first stage

m_total_s1 Isp_s1

= 531020; = 282;

%[kg] %initial total mass of rocket %[s] %specific impulse of first stage

tb_s1 = 180; AeoverAstar_s1 AeoverAstar_s1 = 16;

%[s} %

%burn time of the first stage %exit to throat area ratio









Mexit_s2 PoverP0_s2

= 5.153; = 0.00049738;

PoverPstar_s2

=

%http://www.dept.aoe.vt. %http://www .dept.aoe.vt.edu/~devenpo edu/~devenpor/aoe3114/ca r/aoe3114/calc.html lc.html

0.00088630;

R = 380.5034

D. 1. Payload, Structural, Mass Ratios M0_s1 = m_total_s1 M0_s2 = m_total_s2 % First Stage lambda_s1 = M0_s2/(M0_s1-M0_s2) eps_s1 = m_struct_s1/(M0_s1-M0_s2) m_struct_s1/(M0_s1-M0_s2) R_s1 = (1+lambda_s1)/(eps_s1+la (1+lambda_s1)/(eps_s1+lambda_s1) mbda_s1) % Second Stage lambda_s2 = m_pay_s2/(M0_s2-m_pay_s2 m_pay_s2/(M0_s2-m_pay_s2) ) eps_s2 = m_struct_s2/(M0_s2-m_pay_ m_struct_s2/(M0_s2-m_pay_s2) s2) R_s2 = (1+lambda_s2)/(eps_s2+la (1+lambda_s2)/(eps_s2+lambda_s2) mbda_s2)

M0_s1 = 531020

M0_s2 =









lambda_s2 = 0.1362 eps_s2 = 0.0404 R_s2 = 6.4352

E. 2. Mass Flow Rates First Stage mdottotal_s1 mdottotal_s1 = m_prop_s1/tb_s1 m_prop_s1/tb_s1 mdotperE_s1 = mdottotal_s1/9 mdottotal_s1/9 %Second Stage mdot_s2 = m_prop_s2/tb_s2

mdottotal_s1 mdottotal_s 1 = 2.1983e+03

mdotperE_s1 = 244.2593

mdot_s2 = 249.1129









gtb_s1 = double(subs(gt_s1,tb_s1)) double(subs(gt_s1,tb_s1)) %g's at end of stage % Second Stage %delta v ueq_s2 = Isp_s2*ge Mb_s2 = m_total_s2 - m_prop_s2 R_s2 = M0_s2/Mb_s2 dv_s2 = ueq_s2*log(R_s2) ueq_s2*log(R_s2) %accel assuming constant mdot throughout burn Mt_s2 = vpa(m_total_s2 - mdot_s2*t) %mass of stage as a function of time at_s2 = T_s2/Mt_s2 %acceleration of stage as a function of time gt_s2 = at_s2/ge %"g" acceleration as a function of time gti_s2 = double(subs(gt_s2,tb_s1)) double(subs(gt_s2,tb_s1)) %g's at begining of stage gtb_s2 = double(subs(gt_s2,tb_s2)) double(subs(gt_s2,tb_s2)) %g's at end of stage % Total dv_total = dv_s1+dv_s2

ueq_s1 = 2.7664e+03

Mb_s1 = 135320

R_s1 = 3.9242

dv_s1 = 3.7821e+03











gti_s1 = 1.1299

gtb_s1 = 4.4339

ueq_s2 = 3.3845e+03

Mb_s2 = 17050

R_s2 = 6.4352

dv_s2 =









dv_total = 1.0083e+04

G. 4. Throat and Exit Diameter First Stage Dexit_s1 = 1.8796/570*280 1.8796/570*280 %elon musk comparison Aexit_s1 = (pi/4)*Dexit_s1^2 (pi/4)*Dexit_s1^2 Astar_s1 = Aexit_s1/AeoverAstar_s1 Aexit_s1/AeoverAstar_s1 Dstar_s1 = sqrt(4*Astar_s1/pi) sqrt(4*Astar_s1/pi) %Second Stage Dexit_s2 = 3.66/115*81 Aexit_s2 = pi/4*Dexit_s2^2 pi/4*Dexit_s2^2 Astar_s2 = Aexit_s2/AeoverAstar_s2 Aexit_s2/AeoverAstar_s2 Dstar_s2 = sqrt(Astar_s2*4/pi) sqrt(Astar_s2*4/pi)

Dexit_s1 =









Astar_s2 = 0.0446 Dstar_s2 = 0.2383

H. 5. Optimal Altitudes First Stage Pe_s1 = PoverP0_s1*P0 %Exit pressure of the stage hea_s1 = interp1(atmprops(:,4),atm interp1(atmprops(:,4),atmprops(:,1),P props(:,1),Pe_s1) e_s1) %Altitude where stage is perf expanded % Second Stage Pe_s2 = PoverP0_s2*P0 %Exit pressure of the stage hea_s2 = interp1(atmprops(:,4),atm interp1(atmprops(:,4),atmprops(:,1),P props(:,1),Pe_s2) e_s2) %Altitude where stage is perf expanded

Pe_s1 = 6.2254e+04

hea_s1 =









1.6618e+03

CstarIdeal_s1 CstarIdeal_ s1 = 1.7893e+03

CstarActual_s2 CstarActual _s2 = 1.7371e+03

CstarIdeal_s2 CstarIdeal_ s2 = 1.7893e+03

J. 7. Thrust Coefficient First Stage CTActual_s1 = TperE_s1/(P0*Astar_s1) TperE_s1/(P0*Astar_s1) CTIdeal_s1 = sqrt((2*gamma^2/(gamma-1) sqrt((2*gamma^2/(gamma-1))*(2/(gamma+1 )*(2/(gamma+1))^((gamma+ ))^((gamma+1)/ 1)/... ... (gamma-1))*(1-(Pe_s1/P0) (gamma-1))*(1-(Pe_s1/P0)^((gamma-1)/g ^((gamma-1)/gamma)))+ amma)))+... ... (Pe_s1-interp1(atmprops( (Pe_s1-interp1(atmprops(:,1),atmprop :,1),atmprops(:,4),hea_s s(:,4),hea_s1))/P0*Aeov 1))/P0*AeoverAstar_s1 erAstar_s1









K. Results disp('%%%%%%%%%%% disp('%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%' %%%%%%%%%%') ) disp('----------disp('------------------------------------------------------------------------------------------------------------------------------------------' ----------') ) fprintf('1. fprintf( '1. Mass ratio () : Stage 1 = %3.3f Stage 2 = %3.3f\n' %3.3f\n',R_s1,R_s2) ,R_s1,R_s2) fprintf(' fprintf( ' Payload ratio %3.3f\n',lambda_s1,lambda_s2) %3.3f\n' ,lambda_s1,lambda_s2) fprintf(' fprintf( ' Structural coeff.

()

: Stage 1 = %3.3f

Stage 2 =

()

: Stage 1 = %3.3f

Stage 2 = %3.3f\n' %3.3f\n',eps_s1,eps_s2) ,eps_s1,eps_s2)

disp('----------disp('------------------------------------------------------------------------------------------------------------------------------------------' ----------') ) fprintf('2. fprintf( '2. Delta v (m/s) : Stage 1 = %4.0f Stage 2 = %4.0f Total = %4.0f\n' ,dv_s1,dv_s2,dv_total) %4.0f\n',dv_s1,dv_s2,dv_total) fprintf(' fprintf( ' Initial Acceleration (Gs)

: Stage 1 = %2.1f

Stage 2 = %2.1f\n' ,gti_s1,gti_s2)

fprintf(' fprintf( ' Final Acceleration (Gs) : Stage 1 = %2.1f Stage 2 = %2.1f\n' ,gtb_s1,gtb_s2) disp('----------disp('------------------------------------------------------------------------------------------------------------------------------------------' ----------') ) fprintf('3. fprintf( '3. Mass Flow Rate \n',mdotperE_s1,mdot_s2) \n' ,mdotperE_s1,mdot_s2)

(kg/s) : Stage 1 = %3.1f

Stage 2 = %3.1f

disp(' disp(' - Average per Engine' Engine') ) disp('----------disp('------------------------------------------------------------------------------------------------------------------------------------------' ----------') ) fprintf('4. fprintf( '4. Throat Diameter (m) : Stage 1 = %3.3f Stage 2 = %3.3f\n' ,Dstar_s1,Dstar_s2) %3.3f\n',Dstar_s1,Dstar_s2) fprintf(' fprintf( ' Exit Diameter

(m)

: Stage 1 = %3.3f

Stage 2 =

%3.3f\n',Dexit_s1,Dexit_s2) %3.3f\n' ,Dexit_s1,Dexit_s2) disp('----------disp('------------------------------------------------------------------------------------------------------------------------------------------' ----------') ) fprintf('5. fprintf( '5. Optimal Altitude

(m)

%3.0f

%3.0f\n' ,hea_s1,hea_s2)











%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% %%%%%%%%% ------------------------------------------------------------------------------------------------------------------------------------------------------------1. Mass ratio Payload ratio Structural coeff.

() () ()

: Stage 1 = 3.924 : Stage 1 = 0.260 : Stage 1 = 0.061

Stage 2 = 6.435 Stage 2 = 0.136 Stage 2 = 0.040

-------------------------------------------------------------------------------------------------------------------------------------------------------------2. Delta v (m/s) : Stage 1 = 3782 Stage 2 = 6301 Total = 10083 Initial Acceleration (Gs) Final Acceleration (Gs)

: Stage 1 = 1.1 : Stage 1 = 4.4

Stage 2 = 1.3 Stage 2 = 4.8

-------------------------------------------------------------------------------------------------------------------------------------------------------------3. Mass Flow Rate (kg/s) : Stage 1 = 244.3 Stage 2 = 249.1 - Average per Engine -------------------------------------------------------------------------------------------------------------------------------------------------------------4. Throat Diameter (m) : Stage 1 = 0.231 Stage 2 = 0.238 Exit Diameter (m) : Stage 1 = 0.923 Stage 2 = 2.578 -------------------------------------------------------------------------------------------------------------------------------------------------------------









Published with MATLAB® R2014b

References Presentations and Theses 1 Boysan, Mustafa E., “Analysis of Regenerative Cooling in Liquid Propellant Rocket Engines,” [Thesis], URL: https://etd.lib.metu.edu.tr/upload/12610190/index.pdf https://etd.lib.metu.edu.tr/upload/12610190/index.pdf [cited 19 November 2015]. 2 “Heat Transfer and Cooling,” [Presentation], Massachusetts [Presentation], Massachusetts Institute Institute of Technology. URL: Technology. URL: http://ocw.mit.edu/courses/aeronautics-and-ast http://ocw.mit.edu/courses/aeronautics-and-astronautics/16-50-introduction-to-propulsion-system ronautics/16-50-introduction-to-propulsion-systems-spring-2012/lectures-spring-2012/lecturenotes/MIT16_50S12_lec14.pdf. [cited notes/MIT16_50S12_lec14.pdf. [cited 7 December 2015]. 3 Naraghi, Mohammad Mohammad H., “Thermal Analysis Analysis of Liquid Rocket Engines,” Engines,” [Presentation], [Presentation], Manhattan Manhattan College, Department Department of Mechanical Engineering. URL: home.manhattan.edu%2F~mohammad.naraghi%2Frte% home.manhattan.edu%2F~m ohammad.naraghi%2Frte%2Fnotes%2Flp%2520heat%2520transf 2Fnotes%2Flp%2520heat%2520transfer%2520notes.ppt. er%2520notes.ppt. [cited 30 November 2015]. Web Sites 4 Braeunig, Robert A., “Rocket Propulsion,” [webpage], UR L: L: www.braeunig.us [cited 19 November 2015]. 5 “New Merlin Engine Firing,” SpaceX . [webpage], URL: http://www.spacex.com/media-galle http://www.spacex.com/media-gallery/detail/1661/172 ry/detail/1661/172 [cited 4 December 2015].

Falcon 9 V1.1 Reverse Engineering

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