Falcon 9 V1.1 Reverse Engineering

7/21/2019 Falcon 9 V1.1 Reverse Engineering

http://slidepdf.com/reader/full/falcon-9-v11-reverse-engineering 1/18

Embry-Riddle Aeronautical University

1

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

Multiple performance and operating characteristics of the Falcon 9 two stage rocket were calculated

based on published data of its Merli 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*

= Area, choked m2

M b = Mass, burnout kg

A e = Area, exit of nozzle m2

M l = Mass, payload kg

a = Acceleration m/s2

M p = Mass, propellant kg

c*

Actu al = Velocity, characteristic of actual rocket / M s = Mass, structure kg

c*

Ideal = Velocity, characteristic of idealized rocket / ṁ = Mass flow rate kg/s

C Ƭ,Actual = Thrust coefficient, actual rocket / P = Pressure, static Pa

C Ƭ,Ideal = Thrust coefficient, ideal rocket / P *

= Pressure, choked Pa

d *

= Diameter, choked m P 0 = Pressure, combustion Pa

d e = Diameter, exit m P e = Pressure, exit Pa

ε = Structural coefficient / Ɍ = Mass ratio /

γ = Specific heat ratio / R = Gas constant, specific J/kg*K

g e = Acceleration of gravity, Earth m/s2

R U = Gas constant, universal J/mol*K

h = Altitude m ρ = Density, static kg/m3

h Optimal = Altitude, optimal m Ƭ = Thrust N

I sp = Impulse, specific s T = Temperature, static K

λ = Payload ratio / T C = Temperature, combustion K

M = Mass, molar kg/kmol t b = time, burn s

M e = Mach number, exit of nozzle / u eq = velocity, equivalent of exhaust m/s

M 0 = Mass , total kg v = velocity m/s

I.

Introduction

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 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.

T




2

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 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 engines are aligned in an

‘Octaweb’ pattern which streamlines the manufacturing process by arranging the engines in a simple arrangement of

eight engines in a circle and an ninth in the middle. The second stage is powered by a single Merlin 1D engine

designed with a larger nozzle for optimal performance in the near-vacuum of space.

III.

Parameter Determination

Seven parameters from Falcon 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, 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 1, where M02 is

the total mass of the second stage (109,720 kilograms) and M01 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.

021

01 02

M

M M

Equation 1

The payload ratio for the second stage is defined in Equation 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

02

L

L

M

M M

Equation 2

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 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.

11

01 02

s M

M M

Equation 3

The structural coefficient of the second stage is found using Equation 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.

22

02

s

L

M M M

Equation 4

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

However, the first stage can carry more payload than the second stage. Note that because of the multiple stage

configuration of Falcon 9, the “payload” of stage 1 is the mass of all stages above it.

Using the previous two ratios, a third can now be calculated. The mass Ratio, ( Ɍ ), is defined as the ratio of the

total mass of the stage to the mass at burnout. In Equation 5 below, substitutions were made based on the definitions




3

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.

1 R

Equation 5

B. Mass Flow Rate

The mass flow rate for a rocket determines how much propellant is being expelled over time. This will cause thetotal 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 6, assuming a constant mass flow rate

over the burn time

/ p bm M t 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 is used.

C. Δv

Δv is an important parameter that characterizes a rockets ability to accelerate, neglecting drag and gravity.

Equation 7 defines this parameter, where ueq is the equivalent exhaust velocity and R is the mass ratio that was

defined above.

lneqv u R Equation 7

Before 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.

eq sp eu I g Equation 8

Now, going back to Equation 7 the 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 the rocket is found by adding the Δvs from each stage. Thisgives a total 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 (M0i) 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 9.

0( ) M t M 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. With acceleration (a) now equaling the divisor of thrust (T), assumed to be a constant, and the timedependent mass (Mi(t)), acceleration as a function of time was now found.

( ) / ( )a t T M t Equation 10

To enhance analysis, this acceleration was then divided by Earth’s gravity (g e) to return the acceleration of the

rocket in terms of “G’s,” as shown in Equation 11.

( ) ( ) /e

g t a t g Equation 11




4

With Equation 11 now giving the acceleration of

the rocket in terms of g’s as a function of time, this

could be plotted versus each stage’s respective burn

times. In Figure 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, 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.

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. 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 altitudeand pressure to find at which altitude the

exhaust and ambient temperatures will

equal, and hence perfect expansion. From

this method, it was found that the first stage

will be perfectly expanded at 3,930m and the

second at 21,182m.

Figure 2. Liquid Oxygen and Kerosene combustion charts.

The top left graph shows optimal mixture ratios versus combustion

pressures at P e=1. The top right graph estimates the combustion

chamber temperature while the bottom left graph estimates the

molecular weight of the products. The bottom right graph

estimates the specific heat ratio.

Figure 1. The relationship between burn time and g-

acceleration for both stages of the Falcon 9 rocket.




5

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 (T0), molecular weight (M),

and the specific heat ratio (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.

( 1)/( 1)

* 01 1

2ideal

RT c

M

Equation 12

To find the actual combustion chamber performance, the combustion pressure, throat area (A*

), and massflow rate of propellant are analyzed as shown in Equation 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 .045m2 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.

** 0

actual

P Ac

m

Equation 13

2)

Coefficient of Thrust

Now that the combustion chamber has been analyzed, it is necessary to analyze the nozzle’s operation. The

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, 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.( 1)/( 1)/( 1)2

*

0 0

2 11

1 2ideal

e e a eT

P P P Ac

P P A

Equation 14

To compare the real nozzles’ performance, a similar equation to that of the real characteristic velocity will

be 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.

*

0actual T

T c

P A

Equation 15

IV.

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

efficient transportation device while doing it on just two stages versus the normal three to get into space. Although

these calculations are just approximations based on available information, they all seem to be within reasonable

means, but further verification of the results would be desirable. However, due to the extremely competitive nature

of the rocketry business, most performance specifications can be expected to remain company proprietary for many

years.




6

Figure 3. A Merlin 1D engine

utilizes regenerative cooling as

shown by circumferential tubes on

its nozzle5.

V.

Regenerative Rocket Engine Cooling

The Falcon 9 utilizes nine Merlin 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 incombustion. 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 arise2. Both the

combustion chamber and nozzle must be cooled in order to retain the structural

integrity of the 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 as shown in Figure 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 . 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

fuel absorbs heat energy from the engine which was heated by the exhaust

gases. This conduction and radiation of heat into the liquid coolant adds energy

to it which then increase performance by a small fraction once injected into the

combustion chamber and utilized as a propellant1. This process is shown in Figure 4.

Now that is has been shown that cooling of the engine is necessary in most applications, it must now be identified

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 engineis “detrimental to engine performance” according to Naraghi3, so the channels’ geometry or flow characteristics

must be varied at the throat when compared to

the rest of the engine. Many different cross-

sectional 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

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 toincrease the number of channels4. In essence,

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

going against the flow which will require a more powerful pump, this will aid in adding turbulence in the channel,

bettering heat absorption in the coolant. With a certain combination of the discussed techniques, the required cooling

at the throat and the rest of the engine can be obtained using regenerative cooling in not only the Merlin 1D, but

most liquid rocket engines.

Figure 4. A simplified schematic of a basic regenerative cooling processused in liquid propellant rocket engines. 1




7

Appendix

A. MATLAB Code

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

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

C. Δv ...................................................................................................................................................... 3

D. Throat and Exit Diameters ................................................................................................................ 4

E. Optimal Altitudes .............................................................................................................................. 4

F. Engine Performance .......................................................................................................................... 5

1) Characteristic Velocity ...................................................................................................................... 5

2) Coefficient of Thrust ......................................................................................................................... 5

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

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

C. Constants and Parameters ................................................................................................................ 7

D. 1. Payload, Structural, Mass Ratios ................................................................................................... 9

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

F. 3. Delta v and Acceleration ............................................................................................................. 10

G. 4. Throat and Exit Diameter ............................................................................................................ 13

H. 5. Optimal Altitudes ........................................................................................................................ 14

I. 6. Characteristic Velocity ................................................................................................................ 14

J. 7. Thrust Coefficient ........................................................................................................................ 15

K. Results ............................................................................................................................................. 16

B.

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

%{

Shane Bonner, Patrick Cieslak, Kyle Criscenzo, Ally Smith

December 19, 2015

AE 408 Turbines and Rockets

Dr. Mark Ricklick

%}

clear all

close all

clc

C. Constants and ParametersConstants (assume standard atmosphere at sea level)

ge = 9.81; %[m/s^2]

Rbar = 8.314*1000; %[KJ/mol*K]

%Pressure as a function of altitude




8

atmprops = [...

0 288.15 9.807 101325 1.225

1000 281.65 9.804 89880 1.112

2000 275.15 9.801 79500 1.007

3000 268.66 9.797 70120 0.9093

4000 262.17 9.794 61660 0.8194

5000 255.68 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 216.65 9.761 12110 0.1948

20000 216.65 9.745 5529 0.08891

25000 221.55 9.73 2549 0.04008

30000 226.51 9.715 1197 0.01841

40000 250.35 9.684 287 0.003996

50000 248.15 9.654 79.78 0.001027

60000 247.02 9.624 21.96 0.0003097

70000 219.58 9.594 5.2 0.0000828380000 198.64 9.564 1.1 0.00001846];

% h(m) T(K) g(m/s2) P(Pa) rho(kg/m^3)

% LOX/RP-1

gamma = 1.217; %

Tc = 3575; %[K]

Mbar = 21.85; %[kg/kmol]

P0 = 9.7e6; %[Pa]

R = Rbar/Mbar %[J/Kg*K]

% First Stage

m_prop_s1 = 395700; %[kg] %mass of propellant in first stage

m_pay_s1 = 0; %[kg] %mass of payload in first stage m_struct_s1 = 25600; %[kg] %mass of stucture of first stage

m_total_s1 = 531020; %[kg] %initial total mass of rocket

Isp_s1 = 282; %[s] %specific impulse of first stage

tb_s1 = 180; %[s} %burn time of the first stage

AeoverAstar_s1 = 16; % %exit to throat area ratio

TperE_s1 = 654e3; %[N] %Thrust per engine of the first stage

Ttotal_s1 = TperE_s1*9; %[N] %Total thrust of first stage

Mexit_s1 = 3.668; %http://www.dept.aoe.vt.edu/~devenpor/aoe3114/calc.html

PoverP0_s1 = 0.00641798;

PoverPstar_s1 = 0.01143640;

% Second Stage

m_prop_s2 = 92670; %[kg] %mass of propellant in second stage

m_pay_s2 = 13150; %[kg] %mass of payload in second stage

m_struct_s2 = 3900; %[kg] %mass of stucture of second stage

m_total_s2 = 109720; %[kg] %total mass of rocket when second stage begins

Isp_s2 = 345; %[s] %specific impulse of second stage

tb_s2 = 372; %[s} %burn time of the second stage

AeoverAstar_s2 = 117; % %exit to throat area ratio

T_s2 = 801e3; %[N] %Total thrust of second stage




9

Mexit_s2 = 5.153; %http://www.dept.aoe.vt.edu/~devenpor/aoe3114/calc.html

PoverP0_s2 = 0.00049738;

PoverPstar_s2 = 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)

R_s1 = (1+lambda_s1)/(eps_s1+lambda_s1)

% Second Stage

lambda_s2 = m_pay_s2/(M0_s2-m_pay_s2)

eps_s2 = m_struct_s2/(M0_s2-m_pay_s2)

R_s2 = (1+lambda_s2)/(eps_s2+lambda_s2)

M0_s1 =

531020

M0_s2 =

109720

lambda_s1 =

0.2604

eps_s1 =

0.0608

R_s1 =

3.9242




10

lambda_s2 =

0.1362

eps_s2 =

0.0404

R_s2 =

6.4352

E. 2. Mass Flow RatesFirst Stage

mdottotal_s1 = m_prop_s1/tb_s1

mdotperE_s1 = mdottotal_s1/9

%Second Stage

mdot_s2 = m_prop_s2/tb_s2

mdottotal_s1 =

2.1983e+03

mdotperE_s1 =

244.2593

mdot_s2 =

249.1129

F.

3. Delta v and AccelerationFirst Stage

%delta v

ueq_s1 = Isp_s1*ge

Mb_s1 = m_total_s1 - m_prop_s1

R_s1 = M0_s1/Mb_s1

dv_s1 = ueq_s1*log(R_s1)

%accel assuming constant mdot and thrust throughout burn

syms t

Mt_s1 = vpa(m_total_s1 - mdottotal_s1*t) %mass of stage as a function of time

at_s1 = Ttotal_s1/Mt_s1 %acceleration of stage as a function of time

gt_s1 = at_s1/ge %"g" acceleration as a function of time %assuming constant g per altitude

gti_s1 = double(subs(gt_s1,0)) %g's at begining of stage




11

gtb_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)

%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)) %g's at begining of stage

gtb_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

Mt_s1 =

531020.0 - 2198.3333333333333333333333333333*t

at_s1 =

-5886000/(2198.3333333333333333333333333333*t - 531020.0)

gt_s1 =

-600000/(2198.3333333333333333333333333333*t - 531020.0)




12

gti_s1 =

1.1299

gtb_s1 =

4.4339

ueq_s2 =

3.3845e+03

Mb_s2 =

17050

R_s2 =

6.4352

dv_s2 =

6.3011e+03

Mt_s2 =

109720.0 - 249.11290322580645161290322580645*t

at_s2 =

-801000/(249.11290322580645161290322580645*t - 109720.0)

gt_s2 =

-8900000/(109*(249.11290322580645161290322580645*t - 109720.0))

gti_s2 =

1.2585

gtb_s2 =

4.7889




13

dv_total =

1.0083e+04

G. 4. Throat and Exit DiameterFirst Stage

Dexit_s1 = 1.8796/570*280 %elon musk comparison

Aexit_s1 = (pi/4)*Dexit_s1^2

Astar_s1 = Aexit_s1/AeoverAstar_s1

Dstar_s1 = sqrt(4*Astar_s1/pi)

%Second Stage

Dexit_s2 = 3.66/115*81Aexit_s2 = pi/4*Dexit_s2^2

Astar_s2 = Aexit_s2/AeoverAstar_s2

Dstar_s2 = sqrt(Astar_s2*4/pi)

Dexit_s1 =

0.9233

Aexit_s1 =

0.6696

Astar_s1 =

0.0418

Dstar_s1 =

0.2308

Dexit_s2 =

2.5779

Aexit_s2 =

5.2195




14

Astar_s2 =

0.0446

Dstar_s2 =

0.2383

H. 5. Optimal AltitudesFirst Stage

Pe_s1 = PoverP0_s1*P0 %Exit pressure of the stage

hea_s1 = interp1(atmprops(:,4),atmprops(:,1),Pe_s1) %Altitude where stage is perf expanded

% Second Stage

Pe_s2 = PoverP0_s2*P0 %Exit pressure of the stage

hea_s2 = interp1(atmprops(:,4),atmprops(:,1),Pe_s2) %Altitude where stage is perf expanded

Pe_s1 =

6.2254e+04

hea_s1 =

3.9297e+03

Pe_s2 =

4.8246e+03

hea_s2 =

2.1182e+04

I.

6. Characteristic VelocityFirst Stage

CstarActual_s1 = P0*Astar_s1/mdotperE_s1

CstarIdeal_s1 = sqrt((1/gamma)*((gamma+1)/2)^((gamma+1)/(gamma-1))*(Rbar*Tc/Mbar))

% Second Stage

CstarActual_s2 = P0*Astar_s2/mdot_s2

CstarIdeal_s2 = sqrt((1/gamma)*((gamma+1)/2)^((gamma+1)/(gamma-1))*(Rbar*Tc/Mbar))

CstarActual_s1 =




15

1.6618e+03

CstarIdeal_s1 =

1.7893e+03

CstarActual_s2 =

1.7371e+03

CstarIdeal_s2 =

1.7893e+03

J.

7. Thrust CoefficientFirst Stage

CTActual_s1 = TperE_s1/(P0*Astar_s1)

CTIdeal_s1 = sqrt((2*gamma 2/(gamma-1))*(2/(gamma+1))^((gamma+1)/...

(gamma-1))*(1-(Pe_s1/P0) ((gamma-1)/gamma)))+...

(Pe_s1-interp1(atmprops(:,1),atmprops(:,4),hea_s1))/P0*AeoverAstar_s1

% Second Stage

CTActual_s2 = T_s2/(P0*Astar_s2)

CTIdeal_s2 = sqrt((2*gamma 2/(gamma-1))*(2/(gamma+1))^((gamma+1)/...

(gamma-1))*(1-(Pe_s2/P0) ((gamma-1)/gamma)))+...

(Pe_s2-interp1(atmprops(:,1),atmprops(:,4),hea_s2))/P0*AeoverAstar_s2

CTActual_s1 =

1.6112

CTIdeal_s1 =

1.6818

CTActual_s2 =

1.8511

CTIdeal_s2 =

1.8809




16

K. Results

disp('%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%')

disp('----------------------------------------------------------------------------------')fprintf('1. Mass ratio () : Stage 1 = %3.3f Stage 2 = %3.3f\n',R_s1,R_s2)

fprintf(' Payload ratio () : Stage 1 = %3.3f Stage 2 =

%3.3f\n',lambda_s1,lambda_s2)

fprintf(' Structural coeff. () : Stage 1 = %3.3f Stage 2 = %3.3f\n',eps_s1,eps_s2)

disp('----------------------------------------------------------------------------------')

fprintf('2. Delta v (m/s) : Stage 1 = %4.0f Stage 2 = %4.0f Total =

%4.0f\n',dv_s1,dv_s2,dv_total)

fprintf(' Initial Acceleration (Gs) : Stage 1 = %2.1f Stage 2 = %2.1f\n',gti_s1,gti_s2)

fprintf(' Final Acceleration (Gs) : Stage 1 = %2.1f Stage 2 = %2.1f\n',gtb_s1,gtb_s2)

disp('----------------------------------------------------------------------------------')

fprintf('3. Mass Flow Rate (kg/s) : Stage 1 = %3.1f Stage 2 = %3.1f

\n',mdotperE_s1,mdot_s2)

disp(' - Average per Engine')

disp('----------------------------------------------------------------------------------')

fprintf('4. Throat Diameter (m) : Stage 1 = %3.3f Stage 2 =

%3.3f\n',Dstar_s1,Dstar_s2)

fprintf(' Exit Diameter (m) : Stage 1 = %3.3f Stage 2 =

%3.3f\n',Dexit_s1,Dexit_s2)

disp('----------------------------------------------------------------------------------')

fprintf('5. Optimal Altitude (m) : Stage 1 = %3.0f Stage 2 = %3.0f\n',hea_s1,hea_s2)

disp(' - Pe = Pa')

disp('----------------------------------------------------------------------------------')

fprintf('6. C* (Actual) (m/s) : Stage 1 = %3.0f Stage 2 =

%3.0f\n',CstarActual_s1,CstarActual_s2)

fprintf(' C* (Ideal) (m/s) : Stage 1 = %3.0f Stage 2 =

%3.0f\n',CstarIdeal_s1,CstarIdeal_s2)

disp('----------------------------------------------------------------------------------')

fprintf('7. C_T (Actual) () : Stage 1 = %3.3f Stage 2 =

%3.3f\n',CTActual_s1,CTActual_s2)

fprintf(' C_T (Ideal) () : Stage 1 = %3.3f Stage 2 =

%3.3f\n',CTIdeal_s1,CTIdeal_s2)

disp(' - at respective optimal altitudes')

disp('----------------------------------------------------------------------------------')

disp('%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%')

%"g" plot

t1_vec = [0:1:tb_s1];

t2_vec = [tb_s1:1:tb_s2];g_s1 = subs(gt_s1,t1_vec);

g_s2 = subs(gt_s2,t2_vec);

figure (1)

plot(t1_vec,g_s1,t2_vec,g_s2)

title('G load per Time')

xlabel('t (s)')

ylabel('Gs (1/ge)')

legend('Stage 1','Stage 2','location','northwest')




17

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

----------------------------------------------------------------------------------

1. Mass ratio () : Stage 1 = 3.924 Stage 2 = 6.435

Payload ratio () : Stage 1 = 0.260 Stage 2 = 0.136

Structural coeff. () : Stage 1 = 0.061 Stage 2 = 0.040

----------------------------------------------------------------------------------

2. Delta v (m/s) : Stage 1 = 3782 Stage 2 = 6301 Total = 10083

Initial Acceleration (Gs) : Stage 1 = 1.1 Stage 2 = 1.3

Final Acceleration (Gs) : Stage 1 = 4.4 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

----------------------------------------------------------------------------------

5. Optimal Altitude (m) : Stage 1 = 3930 Stage 2 = 21182

- Pe = Pa

----------------------------------------------------------------------------------

6. C* (Actual) (m/s) : Stage 1 = 1662 Stage 2 = 1737C* (Ideal) (m/s) : Stage 1 = 1789 Stage 2 = 1789

----------------------------------------------------------------------------------

7. C_T (Actual) () : Stage 1 = 1.611 Stage 2 = 1.851

C_T (Ideal) () : Stage 1 = 1.682 Stage 2 = 1.881

- at respective optimal altitudes

----------------------------------------------------------------------------------

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




18

Published with MATLAB® R2014b

References

Presentations and Theses1 Boysan, Mustafa E., “Analysis of Regenerative Cooling in Liquid Propellant Rocket Engines,” [Thesis], URL:

https://etd.lib.metu.edu.tr/upload/12610190/index.pdf [cited 19 November 2015].2 “Heat Transfer and Cooling,” [Presentation], Massachusetts Institute of Technology. URL:

http://ocw.mit.edu/courses/aeronautics-and-astronautics/16-50-introduction-to-propulsion-systems-spring-2012/lecture-notes/MIT16_50S12_lec14.pdf. [cited 7 December 2015].

3 Naraghi, Mohammad H., “Thermal Analysis of Liquid Rocket Engines,” [Presentation], Manhattan College, Department of

Mechanical Engineering. URL:home.manhattan.edu%2F~mohammad.naraghi%2Frte%2Fnotes%2Flp%2520heat%2520transfer%2520notes.ppt. [cited30 November 2015].

Web Sites4 Braeunig, Robert A., “Rocket Propulsion,” [webpage], UR L: www.braeunig.us [cited 19 November 2015].

5 “New Merlin Engine Firing,” SpaceX . [webpage], URL: http://www.spacex.com/media-gallery/detail/1661/172 [cited 4

December 2015].

http://www.mathworks.com/products/matlab


http://ocw.mit.edu/courses/aeronautics-and-astronautics/16-50-introduction-to-propulsion-systems-spring-2012/lecture-notes/MIT16_50S12_lec14.pdf


http://www.spacex.com/media-gallery/detail/1661/172

http://www.spacex.com/media-gallery/detail/1661/172




Falcon 9 V1.1 Reverse Engineering

Documents

Transcript of Falcon 9 V1.1 Reverse Engineering