SpaceshipOne Rocket Design

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Spacecraft

Detail

Design

December 2015

Team Lead: Patrick Cieslak

Emmanuel Carretero, Alex Harvey, Jon McMaken, Gabe Silva, Allyson Smith

Embry-Riddle Aeronautical University

Aerospace Engineering Department

600 South Clyde Morris Blvd.

Daytona Beach, FL 32114



ii

1 ABSTRACT

The White Knight and SpaceShipOne are industrial marvels showcasing the ingenuity of man-kinds’

desperation to achieve space flight. The goals of this design was to achieve similar levels of achievement

as those that designed the hybrid rocket motor of SpaceShipOne. The mission goals were to design an

economical and functional hybrid motor, launched from an aircraft at 14 kilometers and fly up to the edge

of space, defined as 100 kilometers. It must be capable of repeating the same flight within two weeks.

The preliminary process in which this hybrid rocket motor was designed includes: in-depth research into

history of hybrid rockets, selection and analysis of our propellant including Isp and oxidizer to fuel ratio,

determination of design system parameters, and finally initial design of the combustion ports and

combustion chamber.

The design parameters for the grain burning was the major design consideration for this portion of the

design of the hybrid rocket motor. Determining the shape in which the fuel burns out as a function of time

was heavily investigated to account for as many variables as possible. The selection of fuel and oxidizer

only affected our size and therefore the mass of the system. These calculations finally led to the

preliminary design of the oxidizer tank and combustion chamber.

After the preliminary analysis, further work was done on the analysis of a hybrid rocket motor and the

different considerations involved during its design. First the fuel design must be finalized. This includes

the geometry and mass of the fuel. The support structure inside the grain needs to be designed to find

the mass. With these consideration in mind, the insulation can be calculated, finding the mass and

material, and thus the final geometry of the motor casing is found. This geometry determines the mass,

material, and stresses inside of the motor casing. The next items considered are the oxidizer tank and

pressurant gas tank. The main design points for these tanks are the pressure inside and therefore the

material and mass needed to function safely. After these components are designed the next consideration

is the injector plate, which includes the material and mass.

The next aspect of design in this paper is the rocket ballistic parameters. This will show many of important

parameters of the motor as functions of time. The burn area of the fuel will be shown along with the

combustion chamber pressure. The exit pressure and velocity are analyzed as functions of time. The

efficiency parameters were analyzed next, the specific impulse and thrust as a function of time. The last

consideration was the propellant flux rate as a function of time was plotted versus time to show that there

is still fuel available.



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2 TABLE OF CONTENTS

1 Abstract ................................................................................................................................................. ii

3 List of symbols ...................................................................................................................................... iv

4 Introduction .......................................................................................................................................... 1

5 Research ................................................................................................................................................ 4

6 Propellant Selection .............................................................................................................................. 8

6.1 Propellant Properties .................................................................................................................... 8

7 Determine Designed System Parameters ........................................................................................... 10

8 Initial Design of Major Components ................................................................................................... 11

8.1 Combustion Ports, Mixing Chamber, and Combustion Chamber .............................................. 11

8.1.1 Fuel Grain Geometry ........................................................................................................... 12

8.1.2 Hoop- Stress Analysis .......................................................................................................... 17

8.2 Nozzle .......................................................................................................................................... 17

8.3 Oxidizer Tank ............................................................................................................................... 19

9 Design Thus Far ................................................................................................................................... 19

10 Design of Major Components ......................................................................................................... 21

10.1 Fuel Grain Shape ......................................................................................................................... 21

10.2 Support Structure ....................................................................................................................... 21

10.3 Insulation .................................................................................................................................... 21

10.3.1 Motor Casing ....................................................................................................................... 22

10.3.2 Oxidizer Tank ....................................................................................................................... 24

10.4 Nozzle .......................................................................................................................................... 24

10.5 Motor Casing ............................................................................................................................... 25

10.6 Oxidizer Tank ............................................................................................................................... 26

10.7 Pressurization System ................................................................................................................. 27

10.7.1 Pressurant Tank Relations ................................................................................................... 28

10.7.2 Analytical Solutions ............................................................................................................. 29

10.7.3 Considerations .................................................................................................................... 30

10.7.4 Calculations ......................................................................................................................... 31

10.8 Injector Plate/Sleeve ................................................................................................................... 33

10.9 Polar Boss and Thrust Skirt ......................................................................................................... 33

11 Rocket Ballistics ............................................................................................................................... 33



iv

11.1 Initial Geometry, Regression Rate, and Web Thickness: ............................................................ 34

11.2 Fuel Grain Cross-Sectional Area .................................................................................................. 40

11.2.1 Acorner (Red): ......................................................................................................................... 40

11.2.2 Aside (Green): ........................................................................................................................ 40

11.2.3 Aarc (Blue): ........................................................................................................................... 40

11.2.4 Acomp (Pink): ......................................................................................................................... 42

11.2.5 Total areas for a single port: ............................................................................................... 43

11.3 Burn Perimeter Calculation ......................................................................................................... 44

11.4 Ballistic Analysis .......................................................................................................................... 45

12 Final Design ..................................................................................................................................... 49

13 Summary and Conclusions .............................................................................................................. 50

14 References ...................................................................................................................................... 52

15 Appendix A ...................................................................................................................................... 53

16 Appendix B ...................................................................................................................................... 55

17 Appendix C ...................................................................................................................................... 57

18 Appendix D ...................................................................................................................................... 59

19 Appendix E: Vehicle Performance ................................................................................................... 65

20 Appendix F: Internal Rocket Ballistics ............................................................................................. 66

21 Appendix G: Mission Progress ........................................................................................................ 67

22 Appendix H: Task2Ballistics MATLAB Code ..................................................................................... 68

23 Appendix I ....................................................................................................................................... 77

3 L

IST OF SYMBOLS

(O/F)ratio Oxidizer to fuel ratio

a(t)Acceleration as a function of

time

Aarc)oArc cross-sectional area (See

Figure 22)

AcompArea unaccounted for from arc

and corner (See Figure 22)

Acorner)oInitial cross-sectional area of

fuel in corner

Acs(t)Cross-sectional area as a

function of time

AcsoInitial cross-sectional area per

port

Ae Exit area

AF Area of grain minus port area



v

AG Area of grain

Ainj Area of injector

alt2(t)

Altitude after burnout as a

function of time

alti Altitude

Apo Initial port area

Aport(t)Single port area as a function

of time

As Area of the sector (Figure 7)

AT

Area of purple triangle (Figure

7)

At Throat area

AtotTotal area of all ports

(including fuel)

c* Characteristic velocity

de Nozzle exit diameter

DH1 Hydraulic port diameter

DH2(t)Hydraulic port diameter from

perimeter calculations

dt Diameter of throat

f H Hoop Stress

g Gravity (9.81 m/s2)

Gf (t)Fuel mass flux as a function of

time

Gfo Initial mass flux of the fuel

gforceAmount of gees the crew will

experience

Go Initial total flux of oxidizer

Go(t)Oxidizer mass flux as a function

of time

Goo Initial mass flux of oxidizer

HTHeight of purple triangle(Figure 7)

Isp Specific Impulse

Ln Length of nozzle

Lp Length of the port

m(t)Mass of SpaceShipOne as a

function of time

mf Mass of fuel

ṁf Mass flow rate of fuel

ṁf (t)Total fuel flow rate as a

function of time

ṁf)av Average fuel mass flow rate

ṁf)port(t)Mass fuel flow rate per port as

a function of time

mf)remaining(t)Mass of fuel remaining as a

function of time

Mg

Molecular weight of pressurant

gas

mg Mass of pressurant gas

mg Gross mass

ṁinjector Mass flow rate through each

injector hole

minsul Mass of insulation around

entire motor case

minsul,combMass of insulation around

combustion chamber



vi

minsul,mixMass of insulation around aft

mixing chamber

mL Payload mass

mmc Mass of motor case

mmotor,noz Mass of motor and nozzle

mnoz Mass of nozzle

ṁo(t)Oxidizer mass flow rate as a

function of time

ṁo)av Average oxidizer mass flow

rate

ṁo)port(t)Oxidizer mass flow rate as a

function of time per port

mo)remaining(t)Mass of oxidizer remaining as a

function of time

mox Mass of oxidizer

ṁox Mass flow rate of oxidizer

moxT Mass of oxidizer tank

mp Mass of propellant

ṁp Mass flow rate of propellant

ms Structural mass

msupp Mass of support structure

ṁt(t)Total mass flow rate as a

function of time

N Number of ports

OF Oxidizer to fuel ratio

Pc Chamber pressure

PD Dynamic pressure

PfgFinal pressure of pressurant

gas

PigInitial pressure of pressurant

gas

PoxT Pressure of the oxidizer tank

Pport(t)Burn perimeter as a function of

time

R Gas constant

r Radius of oxidizer tank

ṙ Regression rate

ṙ(t)Regression rate as a function of

time

ṙav Average regression rate

rcRadius of combustion chamber

to wall

rf Outer radius of fuel

ṙf Final regression rate

RG Radius of grain

Rmc Radius of motor case

ro Radius to edge of port

ṙoInitial regression rate of the

fuel

roxT Radius of oxidizer tank

RS

Radius of the circle containing

the arc in the ports

Rt Radius of throat

T(t) Thrust as a function of time

tb Burn time



vii

Tc Flame temperature

TfgFinal temperature of

pressurant gas

TigInitial temperature ofpressurant gas

tinsul,combThickness of insulation around

combustion chamber

tinsul,mix

Thickness of insulation around

aft mixing chamber

toxT thickness of oxidizer tank

tw

Combustion chamber wall

thickness

ue Exit velocity

V2Oxidizer velocity entering the

combustion chamber

v2(t)Velocity after burnout as a

function of time

Vf Fuel volume

Vf)remaining Volume of fuel remaining

Vfg Final volume of pressurant gas

vi Vertical Velocity

Vig Initial volume of pressurant gas

VoxT Volume of oxidizer

W(t) Weight as a function of time

wf Web thickness

x Length of cylindrical

y Web thickness

α Angle defined in Figure 8

β Angle defined in Figure 8

γ

Specific heat ratio (Isentropic

parameter)

δ Angle defined in Figure 8

ΔPfp Feed system pressure drop

ΔPinj Pressure drop across the

injector

ε Expansion ratio

ηC* Combustion efficiency

θ(t) See Figure 22

Θc Half angle

θc Angle defined in Figure 8

ϴcc Half angle of aft mixing

chamber

θS Angle produced by the port arc

Μ Molecular mass

ρAl Density of aluminum

ρCF Density of carbon fiber

ρf Density of the fuel

ρox Density of the oxidizer

ϕ Angle defined in Error!

eference source not found.

φ(t) See Figure 22

Ψo See Figure 23



1

4 I

NTRODUCTION

In this project a new hybrid rocket motor will be designed for SpaceshipOne. This paper will be the first of

three that will complete the project. This paper specifically will go into the project requirements and

constraints, research of different types of rocket motors, and the preliminary design of major componentsas well as propellant properties.

SpaceshipOne was the first to win the Ansari X Prize design competition in 2004. To win the competition,

a privately funded group had to design a spaceplane that could hold three passengers (or equivalent

weight), reach an altitude of 100 kilometers, and be able to repeat the flight within two weeks [1]. An

altitude of 100 kilometers was chosen because this was the altitude that was defined as the edge of space.

SpaceshipOne completed this competition on October 4, 2004 with the completion of their second flight.

Their design was launched at an altitude of 15 kilometers from their mothership, the White Knight. Once

launched, the aircraft’s hybrid rocket motor fueled with nitrous oxide and HTPB kicked in and brought the

aircraft to a maximum of 112 kilometers [2]. On descent, the aircraft could move its wing into their famous

“feather mode” which brought drag to a maximum and aided in reentry. Once the glide phase started, thewings would move to their third position to help bring the aircraft to the ground [1]. This ability to change

their wing configuration in midflight was a major component to the success of SpaceshipOne.

The goal of this project is to design a new hybrid rocket motor that will fit inside of SpaceshipOne while

still completing the mission requirements of the Ansari X Prize design competition. Of these mission

requirements, the most important one for this project will be obtaining an altitude of at least 100

kilometers. The design of this rocket motor will also be required to have a thrust output of 73.5 kilo-

Newtons, a delta-V of 1.7 kilometers per second and lastly a burn time of 80 seconds. These requirements

were taken from data on SpaceshipOne (given to us by Dr. Crispin). Constraints that will be implemented

on this project are mainly geometrical. The rocket motor that is designed must be able to fit inside of

SpaceshipOne without compromising any of the pre-existing structure, such as the fuselage and cabin.Using the scaled drawings of SpaceshipOne and the vehicle components picture in Appendix A, the

geometrical constraints were estimated and shown in Figure 1. All dimensions in this picture are in

centimeters.

Figure 1. Estimated geometrical constraints for the hybrid rocket motor designed in this project. All dimensions are in centimeters.



2

For reference, SpaceshipOne has a nearly spherical oxidizer tank that is stored in the cylindrical section of

Figure 1, while the solid fuel tank runs through part of the cylindrical section through the trapezoidal

section where it meets the nozzle. It should also be noted that wall for pressurizing the cabin of

SpaceshipOne has not been included in defining the geometrical constraints. Seen in Table 1 is a summary

of the basic requirements for this project.

Table 1. Requirements and Constraints of the project.

Requirements

Thrust output 73,500 N

Delta V 1.7 km/s

Burn Time 80 s

The first design goal of this design stage is the multiple major components of the hybrid rocket motor.

The first component is the fuel grain shape. This component is basically the heart of the engine. This

component determines the thrust of the motor. When finalizing the grain, the final geometry and mass

of the system must be determined. The amount of thrust provided from the configuration needs to be

large enough to propel the spacecraft immediately after ignition. A few common grain geometries are

shown below in Figure 2.

Figure 2: Typical thrust profiles for common grain geometries [3].

Every geometry has its advantages and disadvantages. Take for example the 7-Cylinder Cluster, the simple

cylindrical design makes the ballistics analysis a little simpler, however there is much wasted space in this

design. Compare this to the 4-Port Wagon Wheel, the geometry of the burn area as a function of time will

be more complicated, but it wastes much less space. The grain geometry selected for this hybrid rocket

motor was a four port wagon wheel. This design was selected for its good burn profile and large thrust

available.

The geometry of the fuel is important in determining the thrust history of the rocket. In addition to the

thrust history, the configuration of the solid fuel will determine how the chamber pressure, exit pressure,

exit velocity, and specific impulse of the rocket will change over the duration of the burn. The fuel grain

has to be designed in such a way that it will enable the rocket to meet the minimum altitude requirement

of 100 kilometers.

In addition to the geometry of the fuel, support structure inside the grain is important in order to keep

the fuel from folding in on itself. The motor casing material is selected to withstand the explosive forces



3

of the combustion chamber, to determine the thickness required to prevent the combustion chamber

from failure, a hoop stress analysis will be employed [3].

The oxidizer is another major component that needs an immense amount of inspection to ensure mission

success. The geometry of the oxidizer tank needs to be taken into account to hold the required volume of

the oxidizer. The corresponding pressure from the oxidizer demands the tank to be capable of

withstanding the pressures with regards to the material, which will relate to the total mass of the oxidizer

system. With regards to the oxidizer system, the pressurization system is the system that ensures the

pressure of the oxidizer tank remains the same. There are different ways to maintain pressure. The main

two ways are either pressure fed or pump fed [3]. In pump fed, there are mechanical pumps that increase

pressure. These tend to be more efficient but increase complexity and cost. Pressure fed can be either

blowdown or regulated. In a blowdown system the oxidizer tank is just over-pressurized. In a regulated

system there is a secondary tank of pressurized gas that is later fed into the oxidizer tank [3]. This method

is generally heavier than a blowdown but is more efficient. The pressurization system must also account

for the pipes and plumbing throughout the system for the oxidizer and include a material selection for the

system.

The injector plate also needs to be designed in such a way as to deliver the oxidizer in an efficient manner

to the fuel. This can be considered a part of the pressurization system since there is an inherent drop in

pressure associated with the injector. The injector plate is the housing of the injectors which aerate the

oxidizer to make it more useful for combustion.

Insulation is an important part of many different components. It provides the thermal protection that

materials need to perform their jobs. In the case of the oxidizer, a layer of insulation is required to keep

the liquid oxygen in cryogenic temperatures. Without the insulation, the metal tank would transfer heat

from outside to the interior hull. In the case of the combustion chamber and mixing chamber, the

insulation is inside the chambers protecting the metal casing from the high temperature and harsh

environment inside during the burn. Since the melting point of carbon fiber is around 3500 degrees Celsius

and our combustion temperature is 2770 degrees Celsius, it is highly likely that there could be heat stress

concentrations that would excite the casing enough to cause it to fail if it were to come into contact with

the interior. Therefore the insulation will be designed in such a way that it either reduces or eliminates

the length of time the case is exposed to the combustion process. The nozzle itself is going to be made

out of a pseudo-insulation material. Where it will be made of a stiff plastic that is resistant to heat on its

own. This is because the nozzle is exposed to high heats similar to the combustion chamber. The nozzle is

also exposed to the ambient conditions as well as an oxygen rich environment from combustion. For these

reasons, the nozzle will be made out of a phenolic material.

The rocket ballistics is a considerable component to the analysis of any rocket. This ballistics analysis

investigates the capabilities of the rocket and whether or not it can achieve the desired results. Since

thrust is equal to mass flow rate multiplied by exit velocity, finding these two numbers will provide the

amount of thrust produced which will in turn propel our rocket.

When all of these major components are finalized and designed, the first step of designing a hybrid

rocket motor will have been completed and on its way to further analysis and testing.



4

5 RESEARCH

A hybrid rocket motor is exactly like the name implies, a cross between a liquid rocket motor and a solid

rocket motor. This is achieved by combining the most appealing aspects of both types into a single system

with a liquid oxidizer and a solid fuel. The liquid oxidizer is in a chamber with a system of valves, pipes,

and depending on the system, pumps. The oxidizer tank may either be pressurized externally or if selected,

certain oxidizers are self-pressurizing. However, those certain oxidizers have their own set of design issues.

Pipes connect from the oxidizer tank to a system for injecting the oxidizer into the ports of the fuel grain.

The container that is holding the fuel is also the same location for the combustion. Once the oxidizer has

been injected into the combustion chamber the ignitor ignites the oxidizer that then ignites the fuel.

Alternatively, if using certain propellants, the oxidizer and fuel will combust upon contact, however these

propellants have extreme toxicity characteristics [4]. This combustion process is passed along to the

mixing chamber which is just behind the combustion chamber. This mixing chamber exists to ensure the

exhaust gases are well mixed to produce steady thrust when exiting the nozzle.

Figure 3: Breakdown of individual components of a hybrid rocket [9].

Hybrid rockets are just under a 100 year old technology, first developed in the 1930’s. The first hybrid

rocket was Soviet in origin with liquid oxygen and gellified gasoline producing 60 pound force. It was built

by Tikhonravov and the Korolev group and its first flight was in August 1993. The next big development of

hybrid rockets came during the cold war; specifically this time of research fell between 1960 and 1980.

One large achievement during the 1960’s was the testing done by Chemical Systems Division of UTC. They

tested up to 40 kilo pound force rockets that used Lithium, Lithium Hydrogen, PBAN, and Fluorine with

liquid oxygen motors. The ISP for these models reached upwards of 400 seconds. In recent history, the

Hybrid Propulsion Development Program (HPDP) developed and tested a 250 kilo pound force thrust

utilizing liquid oxygen and HYPB motors. The next large achievement was that of Scaled Composites and

SpaceDev creating SpaceShipOne [8]. The rocket for this vessel developed 20 kilo pound force usingbutylated rubber and nitrous oxide. [Space Propulsion Group]. Currently, the Space Propulsion Group is

testing and analyzing the results of their new design using paraffin wax. This wax has a high regression

rate, which produces the larger thrust required for large scale rockets. This larger thrust and regression

rate is because the wax actually melts when heated so droplets of the wax is released into the air. These

droplets provide a greater area for burning than just a smooth surface could provide [8].



5

A hybrid rocket motor configuration “is mechanically simple and reduces the opportunity for chemical

explosion, both in flight and during ground operations“[5]. The system is simpler because there are less

pumps and pipes to design when compared to a pure liquid rocket motor. The system is safer because the

two different components for combustion are separated unlike in solid rocket motors. In a solid rocket

grain, the two components are mixed together, and while inert, are just a disturbance away from disaster.

Additionally, the design of a hybrid rocket has multiple benefits, and some negatives, when compared toeither a solid rocket or a liquid rocket.

When compared to a solid rocket, the hybrid rocket motor is more complex. But this complexity gives the

system immense amounts of flexibility during missions. This comes from the ability of controlling the

amount of oxidizer flowing into the combustion chamber and even stopping and restarting the

combustion process during midflight [5]. When handling the oxidizer and fuel in a combined mixture in

solid form, there are inherent risks of explosions. Since all of the components of combustion are present,

all the motor needs to combust is heat. In contrast a hybrid rocket motors’ oxidizer and fuel are separated,

and there is almost no chance of this explosion occurring outside of the launch pad. Even when on the

launch pad, the solid rocket motor’s fuel has a chance of having defects and cause an uncontrolled

combustion and explosion, and possibly hurting, maiming, or killing individuals [5]. The likelihood of anuncontrolled combustion is less likely to occur in a hybrid rocket motor since the flow is controlled. One

similarity solid and hybrid rockets have in common is the design of a fuel grain. Liquid rockets do not use

solids so a grain is not a design component for those rockets. For solids and hybrids, a solid grain is well

designed when it can provide the required thrust using the minimal amount of fuel, and leaving minimal

amounts of fuel afterwards. There are many different designs that are utilized; some designs are as simple

as a single cylindrical grain as seen in Figure 4, or as difficult as a star grain. These different designs are

the causes for how much thrust is going to be produced throughout the launch.

Figure 4: Different options for considering fuel grains in solid and hybrid rockets [3].

When compared to a liquid rocket, the hybrid rocket is simpler. This simplicity leads to a factor of safety.

With liquid rocket motors there are twice the number of pipes and pumps required as compared to the

hybrid rocket motors. If a single factor contributes to a significant loss of oxidizer or fuel to the combustion

chamber, the liquid rocket motor would lose thrust and could cause “a massive explosion, destroying the

vehicle and damaging the Launchpad”[5]. Since the fuel is in a solid form in a hybrid rocket, it is also denser

and therefore takes up less space than the liquid fuel in liquid rockets.

In addition to those benefits, there are a few costs that accompany the hybrid rocket motor. One of the

most prominent problems is the ignition to the solid fuel grain because the fuel grain does not support its

own combustion [6]. Also, in order to keep the oxidizer to fuel ratio the same, the amount of oxidizer



6

being introduced to the system has to increase. This is because as the fuel burns away, the available

surface area increases which increases the demand for the oxidizer. One last problem with using a hybrid

rocket motor is the regression rate, the rate at which the solid phase recedes, that accompanies this

selection. Hybrid rocket propellants have a wide range of regression rates, which means that they could

produce either a small thrust for a long period of time or a large thrust over a short time. This balance

requires a large enough thrust to actually propel SpaceShipOne, but not too large that it would injure thecrew. Taking this into consideration, it is crucially important to our design to select the most applicable

propellant.

Propellants come in different compositions, shapes, and carry different levels of risks. A liquid motor is

where both the oxidizer and the fuel are in liquid form. Each liquid is then stored in separate pressurized

tanks or chambers. When investigating liquid rocket motors there are three basic classifications:

petroleum, cryogens, and hypergols. “Petroleum fuels are those refined from crude oil and are a mixture

of complex hydrocarbons” [4]. These types of propellants generally have a lower ISP than cryogenic fuels,

however the petroleum fuels are in liquid form at room temperature. Cryogenic fuels are those fuels that

require low temperatures to remain in the liquid phase. Some of the most common cryogenic fuels are

liquid hydrogen with liquid oxygen for the oxidizer [4]. An example of this combination can be seen on allflights of the Space Shuttle. The U.S. Space Shuttle utilized this propellant in its main engines [7]. One

optional propellant from the cryogen family is a substance called fluorine. However, fluorine is extremely

toxic and usually reacts violently. Despite the drawbacks, in conditions that allow the use of toxic

propellants, fluorine has an impressive engine performance. Since our rocket will be within the

atmosphere, these toxic gases will prevent us from using any kind of toxic waste. Lastly, liquid rockets

have hypergolic propellants. These fuels and oxidizers are unique in that they combust instantaneously

with each other without the need of any ignition source. However, these propellants like fluorine, are

highly toxic and pose different handling difficulties. A similar problem that both liquid and hybrid rockets

encounter is the dilemma between simplicity and inefficiencies to complexity and cost when considering

how to pressurize the oxidizer. There are two methods that either of these rockets could employ; one is

to use pumps to increase the pressure manually and the second method is to select an oxidizer that

pressurizes on its own. While the pumps would reduce weight from the chamber, it would increase

complexity and increase costs and likely hood of mechanical failures. With the self-pressurizing oxidizer,

this reduces the complexity of the system, but increases the weight but decreases efficiencies as seen in

Figure 5. This efficiency drop is from the drop in pressure which relates to slower flow of oxidizer

regardless of the valve state in the pipes.



7

Figure 5: Normalized pressure and temperature data from a small-scale cold flow test. Points in the upper right of each plot

indicate measurement uncertainty. Data has been normalized to start at (0,1) and go to (1,0), corresponding to the potion of

the test where liquid is present within the tank [11].

VaPak was developed to prevent this phenomenon. As stated earlier, there are two options for

pressurizing the oxidizer: pumps, or self-pressurizing. This system falls under the self-pressurizing category.

This system works by using thermodynamic basics to increase the temperature as the fluid is removed for

combustion. This system starts with a combination of saturated liquid and saturated vapor. When the

fluid is removed, the pressure drops which causes cavitation. This creates more vapors which balance the

pressures back towards the initial state.

Figure 6: Initially both phases are equal. Then when the l iquid is removed, the vapor expands. Then the liquid boils and re-

pressurizes vapor. Finally equilibrium is reached with a slightly lower total pressure [10].

The simplest of rocket designs is the solid motor. A solid rocket motor is where the fuel and oxidizer is

mixed together in one mass. When the propellant is ignited, the whole mass is burned from every available

surface until there is nothing left. Even though this is the simplest design, solid motors can still be broken

down into two families: homogeneous and composite. Though there are two families, they both share

some of the same advantages such as their densities, stability, and ease of storage.

There are a few examples of hybrid rocket motors. It is difficult to scale hybrid rocket motors up to the

necessary size to launch large payloads like the Space Shuttle, so most examples are smaller in scale. One

example is the motor tested by the Space Propulsion Group that uses oxygen as the oxidizer and a fuel

called paraffin wax. This wax has a high regression rate, which produces the larger thrust required for

larger scale rockets.



8

6 PROPELLANT SELECTION

Based on the research in the previous section, it was decided that the hybrid rocket should use liquid

oxygen for the oxidizer with HTPB for the fuel. The reason for this oxidizer-fuel combination was that this

propellant combination gives the largest possible ISP while still using a liquid oxidizer.

Using the data from SpaceshipOne, an estimate for the mass of propellant needed can be made. The gross

mass of the vehicle, structural mass, and payload mass are known from previous work with SpaceshipOne

and is shown below in Table 2.

Table 2. Mass estimates of SpaceshipOne.

Gross mass, mo (kg) 3600

Structural mass, ms (kg) 1200

Payload mass, mL (kg) 500

Using Equation 1 the mass of propellant that can be carried by SpaceshipOne is found to be 1900 kg.

Equation 1 3600 1200 500 1900

This propellant mass is the total amount of propellant that is needed, since the propellant is made up of

an oxidizer and fuel, the propellant mass is simply the sum of the oxidizer mass and fuel mass. Equation 2

illustrates this relationship. Where mox is the mass of oxidizer and mf is the mass of fuel.

Equation 2

6.1

P

ROPELLANT

P

ROPERTIES

This was determined using the thermochemical data in Appendix B of Humble’s book [3]. The graphs are

reproduced in Appendix B of this paper. Figure 26 (B) shows the graph that determines the maximum ISP

that is attainable using this oxidizer fuel and the corresponding oxidizer to fuel ratio (OF). Using this graph,

a maximum ISP of about 318 seconds corresponds to an OF ratio of nearly 2. This OF ratio will be used to

determine all of the thermochemical properties for the liquid oxidizer-HTPB configuration. Figure 26 (A)

shows that for an OF ratio of 2, the isentropic parameter at the throat is 1.235; Figure 27 (B) shows that

the molecular mass of the combustion products at the chamber is 22.4 kg/kmol; lastly, Figure 27 (A) shows

that the flame temperature needed is 3543 K. The thermochemical data is summarized in Table 3 below.

Table 3. Thermochemical data for liquid oxygen oxidizer and hydroxy terminated poly butadiene fuel at an OF ratio of 2.0.

OF 2.0

Max ISP (s) 318

γ 1.235

Μ (kg/kmol) 22.4

TC (K) 3543

The thermochemical data obtained above gives the necessary parameters that are needed to solve for

the characteristic velocity, c*. The equation for c* is shown below in Equation 3.



9

∗ ∗ 2 1 +− Equation 3

8314

Μ

Equation 4

Where ηc is the combustion efficiency, γ is the isentropic parameter, R is the gas constant Joules per

kilogram-Kelvin (given by Equation 4), Μ is the molecular weight measured in kilograms per kmol, and Tc

is the flame temperature measured in Kelvin. The thermochemical data in Table 3 assumes the frozen-

flow approximation, therefore the combustion efficiency, ηc, is taken to be 1. Plugging in all of the know

values gives a characteristic velocity of 1750 km/s.

∗ 1750 /

The thermochemical values in Table 3 also allow for the masses of oxidizer and fuel to be found. This

information can be used to find the volume of each needed as well as an approximation for the required

mass flow rate of each. The masses and volumes of the oxidizer and fuel are calculated using the OF ratio,

mp, and the densities of the oxidizer and propellant, defined in Table 4. These values were obtained from

Appendix B in Humble’s book.

Table 4. Oxidizer and Fuel densities [3]. ∙ − 950 ∙ − 1142

The calculations for the point design (best OF) are shown below. The oxidizer to fuel ratio is defined as

the ratio of the mass flow rates of oxidizer and fuel. Since the mass flow rate is the mass divided by the

burn time, the OF ratio is also the ratio of the masses of oxidizer to fuel, shown in Equation 5.

// Equation 5

Using Equation 2 and Equation 5, an expression that solves directly for the mass of fuel can be found,

shown by Equation 6. Plugging in the values for the propellant mass and OF ratio, the mass of fuel is found

to be 633.3 kg. Now using Equation 2, the mass of oxidizer is found to be 1266.7 kg

1 Equation 6

633.3 1266.7

Now that the propellant masses and densities are known, the fuel and oxidizer volume are readilyobtained using the definition of density (density equals mass divided by volume). Using the masses of fuel

of oxidizer that were just found, as well as, the densities defined in Table 4, the volume of fuel and oxidizer

needed are found to be 0.667 and 1.11 cubic meters, respectively.

0.667 1.11



10

The OF ratio of 2 that was used to find all the thermochemical data parameters assumes that the rocket

is running at its maximum ISP possible. Since this will not always be the case during mission flights, an OF

ratio will be defined to account for the rocket running at slightly lower specific impulse. Referring back to

Figure 26 (B), the OF range will be defined as the set of oxidizer to fuel ratios that gives a minimum I SP of

315 s. Figure 26 (B) shows that this range is approximately 1.75 to 2.45. This range will allow for more

flexibility in the rocket’s performance without sacrificing too much specific impulse.

7 DETERMINE DESIGNED SYSTEM PARAMETERS

In this section some of the design system parameters will be defined. This will included the combustion

chamber pressure, the required flow rates for the oxidizer and fuel, the feed system pressure drop, the

oxidizer tank pressure, and the dynamic pressure at the injector.

In order to determine all these parameters, a combustion chamber pressure must be assumed. For the

design of this rocket motor, a chamber pressure of 3.7 MPa is assumed. This value is used because it is

similar to the chamber pressure in SpaceshipOne.

The required flow rates of the oxidizer and fuel is simply of the mass of the oxidizer/fuel divided by the

burn time. From the propellant properties section, the mass of oxidizer and fuel needed are 1266.7 and

633.3 kilograms, respectively. Equation 7 shows that the mass flow rate of the oxidizer is 15.83 kg/s.

Equation 7

15.83 /

Equation 8 shows that the mass flow rate of the fuel is 8.075 kg/s. It should be noted that for this

calculation the mass of the fuel was assumed to be a slightly higher value of 646 kilograms. This is to

account for the small amount of fuel that will not be burned due to the rounding of the sharp edges in the

initial port geometry.

Equation 8

8.075 /

For the feed system pressure drop, any change in pressure will cause the oxidizer to flow through the

injector into the combustion chamber. As long as the pressure in the oxidizer tank is higher than the

pressure in the combustion chamber. For an initial design, the feed system pressure drop is assumed to

be 20% of the chamber pressure, this follows the “rule of thumb” defined for unthrottled engines [3]. This

will give a feed pressure drop of 0.74 MPa, as shown in Equation 9.

∆ 0.2 Equation 9

∆ 0.74

The feed system pressure drop is loss in pressure caused by the oxidizer traveling to the combustion

chamber. It is typically in the range of 35,000 to 50,000 Pa, in order to be conservative it is assumed to be

50,000 Pa for the following calculations.

The last parameter that needs to be determined is the dynamic pressure. This is simply the pressure at

which the oxidizer enters the combustion chamber, and is given by Equation 10. Where, ρox is the density



11

of the oxidizer (1142 kg/m3) and V2 is the velocity of oxidizer through the injector (10 m/s). Using these

values, the dynamic pressure comes out to 0.0571 MPa. The velocity is typically 10 m/s and will therefore

be used to calculate the dynamic pressure.

12

Equation 10

0.0571

Now, with all of these pressures known, the pressure of the oxidizer tank can be determined using

Equation 11. Where, Pc is the chamber pressure (3.7 MPa), ∆ is the pressure drop across the injector

(0.74 MPa), ∆ is the feed line pressure drop (0.05 MPa), and PD is the dynamic pressure (0.0571 MPa),

giving an oxidizer tank pressure of 4.55 MPa.

∆ ∆ Equation 11 4.55 Now, knowing the definition of the mass flow rate. The cross sectional area of the injector holes can be

found. Using Equation 12, the area of the injector holes per port is found to be 1.05 cm2, where

is

the mass flow rate of the oxidizer (15.83 kg/s), N is the number of ports (4), K is the head loss coefficientfor radiused inlet (1.2), is the density of the oxidizer (1142 kg/m3), and ∆ is the pressure drop

across the injector plate (0.76 MPa).

2∆ Equation 12

0.000105 1.05

An injector area of 1.05 square centimeters would be the area of a one hole injector per port. For the

design of this hybrid rocket motor, each port is going to have 5 holes for the injector, making the area of

each hole of the 20 total hole’s 0.21 square centimeters.

8 I

NITIAL

D

ESIGN OF

M

AJOR

C

OMPONENTS

In this section the initial design of the major components of the hybrid rocket will be described. Specifically

the combustion chamber, nozzle, and oxidizer tank will be looked at. It is important to design these

parameters carefully to ensure our objective of 100 kilometers is achieved. Not only is functionality a

concern in this design, but the safety is another condition that shall be considered.

8.1 C

OMBUSTION

P

ORTS

,

M

IXING

C

HAMBER

,

AND

C

OMBUSTION

C

HAMBER

The fuel grain for this hybrid rocket motor was decided to take the shape of a four port wagon wheel, as

shown in Figure 4. This design was chosen because it leaves the least amount of un-burned fuel behind,

and is therefore the most efficient in terms of using all of the space. Since this is not a perfect world, some

un-burned fuel will be left behind in the corners. This is due to the initial port area having sharp corners.

As the fuel is burned, the sharp corners will round out. The combustion of the fuel and oxidizer occurs in

the combustion ports, making the combustion chamber and combustion ports the same thing. The mixing



12

chamber is where the oxidizer and fuel mix, which coincidentally enough occurs in the combustion

chamber. Therefore, the design of the combustion chamber is done with the design of fuel grain geometry.

8.1.1 Fuel Grain Geometry

The first step in designing the geometry of the fuel grain is determining the initial area of each port. For

the four port wagon wheel each port is the same size. To determine the initial port area, the following

equation is used. Where Goo is the initial mass flux rate of the oxidizer, is the mass flow rate of the

oxidizer, N is the number of ports, and Apo is the initial port area.

Equation 13

0.0112

The initial mass flux rate of oxidizer is assumed to be 350 kg/m2s because this is usually the upper limit on

oxidizer mass flux per Humble’s book [3]. It is also going to be assumed that the mass flow rate of the

oxidizer is going to remain constant throughout the burn while the mass flow rate of the fuel changes

during the burn. Knowing that N is equal to 4 and

is equal to 15.83 kg/s, the initial port area comes

out to be 0.0112 square meters.

Next the initial mass flux of the fuel can be determined using Equation 14. Where is the initial mass

flow rate of the fuel and can be estimated using the mass flow rate of the fuel calculated in Equation 8.

Equation 14

180.2 /

The port and fuel dimensions are set by the fuel grain shape, volume of fuel, and port area. The fuel grain

shape was selected as explained above, fuel volume was determined in the Propellant Properties section,

and lastly the port area was determined from Equation 13. Due to the symmetry of the fuel grain, thegeometric calculations only needed to be done on a single port with its corresponding fuel.

Figure 7. (a) Blue shaded area is the cross section of a single port, (b) Purple shaded area is the sector defined by the arc of the

circular section of the port, (c) projection of the port over the sector from b, (d) the remaining sector area from c is divided

The geometry was broken up into sections, to simplify calculations. Figure 7 shows the different sections

that were used to evaluate the geometry. The blue sector identifies the port area, the purple sector from

(b) identifies the area that the arc of the circular part of the port makes, and the two purple triangles in

(d) show the portion of the purple sector remaining after the projection of the port onto it. The key here

is to observe that the purple sector is composed of the two triangles and the port.



13

Determining of the dimensions is rather complex, requiring involvement of three subjects: regression

rates, trigonometry, and geometry. The regression rates tie in with the trigonometry problem because of

the web thickness term, they tie into the geometry problem because of the port length term. The

geometry and trigonometry problem are coupled because of the shape.

8 1 1 1

Regression Equations

The initial mass flux of the fuel, Go, can be defined as the sum of the initial mass flux of the oxidizer, Goo,

and initial mass flux of the fuel, Gfo, as shown in Equation 15.

Equation 15

The initial regression rate of the fuel is defined by Equation 16, where Go was defined above, LP is the

length of the port, and a, n, and m are constants that are defined to be 2(10-5), 0.75, and -0.15, respectively.

Equation 16

The average regression rate is defined using the initial and final regression rates, and .

≅ 2 Equation 17

It can be estimated that the final regression rate is one quarter of the initial regression rate, shown by

Equation 18.

14 Equation 18

Finally, the final web thickness can be approximated using Equation 19, where tb is the burn time.

≅

Equation 19

8 1 1 2

Geometric Equations

Since the purple sector is composed of the two triangles and the port, as seen in Figure 7, The area of the

sector is found using Equation 20. Where AS is the area of the sector, AT is the area of one of the purple

triangles from Figure 7 part d, and Apo is the initial area of the port.

2 Equation 20

The area of the sector is simply the fraction of the area of a circle, with radius , as seen in Equation 21.

Where RS is the radius of the circle containing the arc in the ports and is the corresponding angle

produced by the arc. This fraction is simply the sector angle divided by 360°.

360° Equation 21

The general area of a triangle is half the base multiplied by the height given by Equation 22. Where AT is

the area of one of those triangles and HT is the height of said triangle.

12 Equation 22

The area of the grain is simply the area of a circle, AG, with radius , shown by Equation 23.



14

Equation 23

Based on the geometry, the radius of the grain can be defined using Equation 24, where wF is the final

web thickness.

Equation 24

The fuel area, AF, is the area of the grain with the initial area of the four ports removed, leading to Equation

25.

4 Equation 25

The volume of any extruded section is simply the area of the cross section multiplied by the extruded

length, in this case the port length, LP. This is represented in Equation 26.

Equation 26

In order to simplify the geometry, Equation 27 and Equation 28 are defined where N is the number of

ports, which is 4 in this design. Figure 8 shows how the angle γ, α, and θS is defined.

≡ 180° Equation 27

≡ 2 Equation 28

8 1 1 3

Trigonometric Equations

Figure 8. Triangle from Figure 7 part d.

The following equations were obtained from trigonometry and Figure 8. Using the law of sines, the radius

to the port arc, RS, can be defined using Equation 29Figure 8. The angles δ and β are defined in Figure 8.

sinsin Equation 29



15

Similarly, the side A of one of the purple triangles can be defined using Equation 30. This side is the

same as one of the sides of the port, as can be seen from Figure 8.

Equation 30

Because the sum of the angles contained in a triangle is 180 degrees, δ, α, and β are related by Equation

31.

180° Equation 31

δ and γ are also related by Equation 32, which is obtained from Figure 8.

180° Equation 32

Using the basic trigonometric relations, the height of the triangle, HT can be defined by Equation 33.

sin Equation 33

Figure 9. Cross section of one port section from Figure 7 , to show location of B and w F .

Using Figure 9 and noting that the half angle of quarter circle is 45°, B can be defined using Equation 34.

√ 2 Equation 34

Using Equation 15 through Equation 30 the following two equations are made. See Appendix C for how

these 2 equations were derived.

sinsin sinsin () Equation 35

sinsin √ 2 4 58 √ 2/ Equation 36

Equation 35 and Equation 36 represent the combination of the three subjects: regression rates,

trigonometry, and geometry. The derivations of these equations is shown in Appendix C. Note that there

are only two unknowns in these equations, and B. There are as many equations as there are unknowns,



16

meaning that a solution to this system of non-linear equations will solve the problem. The Solver ADD-IN

on Excel was used to solve the set of equations, and the resulting and B were used to obtain the other

dimensions, these steps are shown below. All values required for the proceeding calculations can be found

in the following 8 steps. Note that some of the calculations use values obtained in a previous step.

1.

Find

from Equation 27 and

from Equation 32, where N is 4.

180°

45°

180°

135°

2.

Solve Equation 35 and Equation 36 to obtain and B, where Apo = 0.11 m2, GOO = 350 kg/m2s, Gfo

= 180.2 kg/m2s, a = 2E-5, n = 0.75, m = -0.15, tb = 80 s, φ = 45°, and δ = 135°.

sinsin

sinsin () sinsin √ 2 4 58 √ 2

/

21.72° 0.141

3.

Compute port length using Equation 108, found in Appendix C. The values for this equation are

in the previous step.

58 √ 2

/

1.96

4.

Determine web thickness using Equation 34, where B is 0.141 m. √ 2

0.10

5.

Calculate the length of the side of the port using Equation 111 (from Appendix C), where B, α,

and φ are 0.141 m, 21.72°, and 45°.

0.13

6.

Calculate the radius of the sector using Equation 112, where δ = 135°, B = 0.141 m, α = 21.72°,

and φ = 45°.

sinsin

0.25



17

7.

Obtain the radius of the grain using Equation 24, where wF is 0.10 m and RS is 0.25 m.

0.35

8.

Compute the average regression rate using Equation 105, where the values for the equation are

in found step 2. ≅ 58

r ≅ 0.00125 /

8.1.2 Hoop- Stress Analysis

The Hoop-Stress used in solid rocket motors can be applied to hybrid rockets as well. This stress analysis

focuses on analyzing the maximum stress applied to the thin walls of the combustion chamber. This

analysis ensures that the chamber will not fracture or explode during operation. The following equation

lays the ground work for the analysis:

Equation 37

Equation 37 can be rearranged to find tw, the thickness of the wall necessary to comply with safety

standards. In the equation f H is the stress the wall can experience, which can be substituted with the

ultimate tensile strength of the material chosen, in this case 2219-Aluminum was selected and has a value

of 0.413 GPa. The Pc is the burst pressure which is the combustion chamber pressure, 3.7 MPa, times a

factor of safety of 2. Lastly, the radius, rc, is from the earlier analysis of the rocket grain and is the distance

from the center to the chamber wall, 0.354 meters. Plugging in these values gives a wall thickness of

0.00649 meters, or 6.49 millimeters.

6.32

Even though this value accounts for a factor of safety of 2, it is common practice in other motors such as

liquid rocket motors to use a thickness 1.5 to 2.5 times thicker than this number to account for stress

concentrations and other stresses not modeled with this simple equation. Following this practice, we

selected the median of these and selected 2. So our final number is actually 12.98 millimeters.

12.6

8.2 NOZZLE

The nozzle design includes the throat area (At), exit area (Ae), half angle (θc), and length of nozzle (Ln). For

simplicity, a conical shaped nozzle is going to be used with an assumed half angle of 20°. In addition to

assuming a half angle, an expansion ratio of 40 will be assumed. Using this information, the nozzle can be

fully defined. In order to find the throat area, Equation 38 is used, where Pc is the chamber pressure in

Pascals, At is the throat area in square meters, c* is the characteristic velocity in m/s, and m is the mass

flow rate of the propellant in kg/s.

∗ Equation 38



18

The mass flow rate of the propellant is calculated using Equation 39, where mp is the propellant mass in

kilograms and tb is the burn time in seconds. As defined above in the propellant section, the mass of

propellant is 1900 kg and the burn time is 80 seconds. Equation 39 gives a mass flow rate of 23.75 kg/s.

Equation 39

23.75 /

Now that the mass flow rate of the propellant is known, Equation 38 can be rearranged to solve for the

throat area. Knowing that chamber pressure is 3.7 MPa, c* is 1750 m/s, and the mass flow rate is 23.75

kg/s, the throat area is calculated to be 0.112 square meters.

0.0112

Using the area of a circle, the diameter of the throat is found to be 12 centimeters, as shown by Equation

40.

14

Equation 40

12

Now using the definition of the expansion ratio (ϵ), the exit area of the nozzle is found to be 0.45 square

meters. Equation 41 shows how the exit area was found using the expansion ratio of 40 and the throat

area of 0.0112 square meters.

ϵ Equation 41

0.448

Similarly, the equation for the area of a circle is used to find the diameter of the nozzle exit. This is shownby Equation 42, where the diameter at the exit is found to be 75.5 centimeters.

14

Equation 42

75.5

The last parameter needed to fully define the nozzle is the length. This value is found using Equation 43,

where de and dt is the diameter of the nozzle exit and throat in meters, respectively, and θc is the half

angle. The length of the nozzle was found to be 0.875 meters.

2tan

Equation 43

0.872

Now the preliminary sizing of the nozzle is done. A more finalized design concept will come in Task 2

where the material properties will be defined.



19

8.3

OXIDIZER TANK

From the propellant properties section the volume of oxidizer needed for the mission was determined to

be 1.11 cubic meters. Using this volume, the geometrical shape of the oxidizer tank can be estimated. Due

to space constraints in SpaceshipOne, the oxidizer tank will be designed as a capsule shape. The spherical

ends of the tank will allow for pressure to be distributed evenly in the tank, eliminating the possibility of

stress concentrations.

Using the geometry of such a shape, the volume inside the tank can be found using Equation 44. Where x

is the length of the cylindrical section and r is the radius of both the spherical and cylindrical section (see

Figure 10).

43 Equation 44

Figure 10. Schematic of a capsule tank showing the definition of x and r. Note this is NOT to scale.

Because SpaceshipOne has a maximum diameter of about 1.53 meters in the section that contains the

rocket, a radius of 0.5 meters will be assumed. This gives an extra 0.53 meters around the volume of the

oxidizer to account for the thickness of the tank and any insulation or supporting material needed. Using

a radius of 0.5 meters, and a volume of 1.11 cubic meters, the length of the cylindrical section is found to

be 0.7466 meters.

0.7466

Now the preliminary sizing of the oxidizer tank is done. A more finalized design concept will come in Task

2 where the material properties and stress analysis will be completed.

9 DESIGN THUS FAR

Throughout this report, the initial design of a hybrid rocket motor for SpaceShipOne was considered. The

initial design shown in Figure 11 and Figure 12 were created using the values calculated in this report.

Using this design, the rocket will be able to easily fit within the geometrical constraints defined by Figure

1. There is ample room around the oxidizer tank (roughly 50 centimeters) to allow for support structure

or a larger oxidizer tank if needed. It should be noted that the rocket fits everything up to the throat of

the nozzle.



20

Figure 11. Initial design concept for the hybrid motor rocket designed in this report.

The injector plate and sleeve were made not using anything calculated in this paper, other than the

injector area. The placement of the injector holes were made such that they were directly over the center

of the initial port area. The injector sleeve was made to be the same radius as the combustion chamber

casing. It was also made to be thick enough so that part of the oxidizer tank rests inside of it. In Task 2 the

injector plate, injector sleeve, and ignitor (which will be inside of the injector sleeve) will be designed

more thoroughly. See Appendix D for detailed drawings of the rocket motor.

Figure 12. Exploded view of the initial design.



21

10 DESIGN OF MAJOR COMPONENTS

In this section, the design of the major components of the hybrid rocket motor will be considered.

Specifically the finalized fuel grain geometry, support structure, insulation, nozzle design, motor casing,

oxidizer tank, pressurization system, injector plate, polar boss, and thrust skirt will all be considered here.

Note that the design of some of these components are estimated due to time constraints, unavailability

of a proper stress analysis, or the use of empirical formulas.

10.1

FUEL GRAIN SHAPE

Throughout the design of this hybrid rocket motor, many things might have changed. The geometry of the

fuel grain however is not one of them. The numbers from Task 1 are the same numbers used for the rest

of the calculations of Task 2. These numbers are summed in Table 5.

Table 5. Summary of major parameters of the fuel grain.

RG 35.25 cm

ṙavg 0.00125 m/s

wf 0.10 m

Lp 1.96 m

Vf 0.68 m3

It should be noted, again, that the volume of the fuel is slightly higher than what was required from the

calculations in Task 1. This is to account for the small amount of fuel that will not be burned in the corners

due to the geometry of the fuel grain.

10.2 SUPPORT STRUCTURE

The support structure is assumed to be 10% of the inert mass (or structural mass). The inert mass of

SpaceShipOne is known to be 1200 kilograms from Task 1. This will give a structural mass of 120 kilograms. 0.10 Equation 45 120

This amount of mass will be used to mount the hybrid rocket motor inside of SpaceShipOne. Other support

structures include a support that is placed inside of the fuel grain to prevent it from collapsing on itself. If

the fuel grain were to collapse on itself then unburned fuel could be expelled out of SpaceShipOne and

could cause the spacecraft not to reach its destination of 100 kilometers. The fuel support structure will

take the shape of a cross and will physically break up the grain into 4 equally sized pieces. The cross hair

will be made out of a slow burning plastic and will be assumed to be 1 centimeter thick. This will bring the

radius of the grain from 35.25 centimeters to 35.75 centimeters.

35.75

10.3 INSULATION

Insulation, as stated in the introduction, protects materials from extreme temperatures. There are two

different categories, or uses, for insulation. The first is where there exist a structure exposed to extremely



22

high temperatures, such as the combustion and mixing chambers. In this case the insulation needs to be

able to withstand the high temperatures and protect the casing from melting or failing due to thermal

stresses. The second category is where there exists a structure exposed to extremely low temperatures,

such as the oxidizer tank. In this case the low temperatures of the oxidizer need to be sheltered from any

heat transfer from outside the tank. This means there needs to be a layer of insulation outside the tank

preventing heat transfer.

10.3.1 Motor Casing

In addition to the physical stresses and strains from explosive reactions dictated by Newton, there are

thermal stresses that the engine is subjected to. With this in mind, insulation is added to the system in

order to protect the structural integrity of the motor. It makes sense that when considering insulation

that the higher temperature areas with a large flow of hot gases are going to be the main concern of this

segment of the design. The design of insulation will be broken down into two major components:

combustion chamber and mixing chamber. Insulation is needed for the protection of the motor casing

inside the combustion chamber. Without insulation, the motor casing would be exposed to higher

temperatures than it can withstand and would fail rapidly and dangerously. For this reason, the insulation

is placed between the motor casing and the fuel grain itself.

Since the combust chamber and mixing chamber are one piece, it is only logical to use one insulation

material. Table 6 lists common insulators used in modern rockets. From this table and the

recommendations found in Humble’s text, polycrystalline graphite was selected as the insulator inside the

combustion chamber and mixing chamber.

Table 6: Materials used for solid rocket motor insulation.

Material Density [kg/m3] ė [mm/s]

Pyrolytic Graphite 2200 0.05

Polycrystalline Graphite 1700 0.10

2-D Carbon/Carbon 1400 -

3-D Carbon/Carbon 1900 0.10

Carbon/Phenolic 1400 0.18

Graphite/Phenolic 1400 0.28

Silica/Phenolic 1700 1.3

Glass/Phenolic 1900 1.5

Paper/Phenolic 1200 1.9

The selected material has the following properties: a density of 1700 kg/m3, and an erosion rate of .1

millimeters per second. Taking the erosion rate found in this table, it is a simple calculation to find the

thickness of insulation required. Taking into consideration the time the insulation is exposed to the hot

gases multiplied by their erosion rates including a factor of safety, a simple equation is shown in Humble’s

text.

∗ ∗ Equation 46

It is common practice to use a factor of safety equal to 2 in this design. The first variable, texp is the length

of time the insulation is exposed to the volatile temperatures. The grain protects the combustion chamber

for majority of the burn so it will only experience minimal heat. Taking this into account it can be reduced



23

to about 10 seconds of exposure. Plugging in these values gives a thickness of insulation inside the

combustion chamber of 2 millimeters.

, 2

The mixing chamber however is exposed to the hot fumes during the entire burn of the motor. This means

the thickness will increase by a large margin. Changing the value of 10 seconds to 80 gives a thickness of

16 millimeters.

, 16

Now that thicknesses of the insulation has been found, it is necessary to find the mass of all the insulation.

The mass of the insulation can be found by multiplying the volume by the density. The volume of the

insulation in the combustion chamber can be found simply by finding the volume of the cylinder using the

grain radius plus the thickness of insulation minus the volume of the cylinder using just the grain radius.

This can be seen in Equation 47.

, ( ,) Equation 47

The radius of the grain is 35.75 centimeters. Plugging in values gives a mass of insulation inside the

combustion chamber of 15.03 kilograms.

, 15.03

Finding the volume of the mixing chamber is a little more complex. It is assumed that the mixing chamber

insulation takes the form of a truncated cone. This has some overlap with the combustion chamber, but

it is negligible and can be considered as an extra factor of safety. The volume of a truncated is given in

Equation 48.

Figure 13 Representation of a truncated cone for reference

13 ℎ Equation 48

R1 and R2 are the radii of the nozzle throat and the fuel grain respectively. The h in this equation is the

length of the mixing chamber in the axial direction. This is done using Pythagoreans theorem and comes

to a value of 34.86 centimeters. The same principle of finding the volume of the insulation in the



24

combustion chamber applies here. Add the thickness of the insulation to the radius of the casing and

subtract the volume of just the casing. This can be seen below.

13 ( , , , ,)ℎ 13 ℎ

Equation 49

After plugging in these values, the volume equals 0.007726 cubic meters.

0.007726

Then finding the mass is simply the volume multiplied by the density and gives a mass of the insulation in

the mixing chamber of 13.13 kilograms.

, , 13.13

The total mass of insulation is the sum of the two and equals 28.16 kilograms. 28.16

It is noted here that it would probably be easier to have the same thickness in the combustion chamber

as the mixing chamber, however this would increase the mass of the insulation in the combustion

chamber by 700%. This mass savings seemed too high of a cost for a small concession in simplicity.

10.3.2 Oxidizer Tank

Insulation is required for the oxidizer tank to prevent heat transfer to the cryogenic temperatures inside

it. One challenge with designing insulation for the tank is from its geometry. With rounded edges it

becomes impossible to use the same methods of design as with the combustion chamber and mixingchamber. Not only that, but this insulation will not be burning away. So the thickness of the insulation is

not determined by an erosion rate. It is determined by the amount of allowable heat transfer [12].

An excellent strategy for insulation is to use multiple layers of aluminum foil and fiber-glass. The aluminum

rejects radiative heat acting as a reflector, while the space in between is created to be a vacuum and

prevents conductive heat transfer. There are some risks that there could be a crack or leak which would

cause losses in insulation effectiveness [12]. Insulation for this type of design would be roughly 2.0574

centimeters [12].

10.4

N

OZZLE

The nozzle was designed in Task 1 and was found to have a throat area of 0.0112 square meters, giving a

throat radius of 6 centimeters. Assuming an expansion ratio of 40, the exit radius was found to be 37.75

centimeters. In addition, assuming the half angle of the nozzle is 20°, the length of the nozzle was found

to 0.872 meters. Hybrid rocket motor nozzles are typically made out of phenolic materials because they

do not oxidize as easily as materials such as graphite. An empirically derived expression for the nozzle

mass (Equation 50) will be used to estimate the mass of the nozzle for this design, where mp is the mass



25

of propellant (1900 kg) and is the expansion ratio of the nozzle (40). This will give a nozzle mass of 88.1

kilograms.

125

5400

10

Equation 50

88.1

The nozzle that will be used for the hybrid rocket motor will be thicker around the throat to reinforce the

throat and insulate it a little better (since phenolic materials are a common insulation material anyways).

For simplicity, the thickness of the nozzle is assumed to be the same thickness of the aft mixing chamber

(16 mm). Please see the drawings of the final design to see how the nozzle is constructed in Appendix I.

10.5 MOTOR CASING

The motor casing houses the combustion chamber. In Task 1, Aluminum 2219 was the material chosen

for the combustion chamber. This material has since been changed. Aluminum 2219 has a much too low

melting point and would most likely melt when subjected to combustion chamber temperatures in the

3000°C range. So, it has been decided that the motor case will be made out of a carbon fiber/epoxy

composite material due to its higher strength to weight ratio and ability to withstand higher temperatures.

In the design of this hybrid rocket motor, the motor case, throat, and nozzle will all be one piece. Since

the motor casing is made of a composite material, construction of more complicated shapes can easily be

accomplished. The nozzle and combustion chamber will be wrapped with a carbon fiber/epoxy. Essentially

the motor casing will extend all the way to the nozzle, making it one fluid piece.

Table 7. Mechanical properties of carbon fiber/epoxy resin [13]..

Ultimate Tensile Strength (MPa) 600

Density (kg/m3) 1600

The mechanical properties of standard carbon fiber fabric with epoxy resin are summarized in Table 7.

Using these material properties, the wall thickness of the motor casing can be re-evaluated using the hoop

stress analysis described in Task 1, shown by Equation 51. This equation takes the original hoop stress

equation and solves for the wall thickness (tw) as well as including the factor of safety of 4 into the equation.

A factor of safety of 2 accounts for stress concentrations and other stresses not modeled by the equation.

The remaining factor of safety of 2 accounts for the burst pressure being estimated by 2 times the

combustion chamber pressure. In this case, the pressure used will be the chamber pressure (3.7 MPa),

and the ultimate tensile strength (600 MPa) found in Table 7. The radius used will be the grain radius

(0.3575 m) plus the thickness of insulation around the combustion chamber (0.002 m). With all of this

taken into account, the thickness of the combustion chamber comes out to be 8.87 millimeters.

4 Equation 51

0.00887 8.87

The shape of the combustion chamber has also been slightly changed in this rendition of the design. An

aft mixing chamber was included to ensure that all the fuel is burned. The aft mixing chamber typically

has a length-to-diameter ratio between 0.5 and 1.0. Instead of keeping the aft mixing chamber cylindrical



26

in shape, it is going to be deigned as the converging part of the nozzle, with the end of the mixing chamber

lining up with the throat of the nozzle.

Figure 14 shows the geometry of the combustion tank, with scaled dimensions. Using this geometry, the

length of the aft mixing chamber comes out to be about 34.86 centimeters. This gives an L/D ratio of 0.5,

on the low end of the range of typical values.

Figure 14. Geometry of the combustion chamber including aft mixing chamber, all dimensions are in centimeters.

Now with the geometry and material of the combustion chamber determined, the mass of the chamber

can be determined using Equation 37. Where is the density of the carbon fiber composite (1600

kg/m3), tw is the thickness of the combustion chamber (0.00887 m), Lp is the length of propellant (1.9625

m), Rmc is the radius of the grain plus the thickness of insulation (0.3595 m), R t is the radius of the throat

(0.06 m), and is the half angle of the aft mixing chamber (40°), giving a mass of 69.6 kilograms.

Note that this mass is for the combustion chamber only. While the motor case, throat, and nozzle will all

be one piece, their masses will be kept separate. Therefore, the mass of the one large piece will be the

sum of the mass of the motor case and nozzle, which roughly equals 157.7 kg. Note that this mass is a

rough estimate since an empirical formula was used to estimate the mass of the nozzle.

10.6 OXIDIZER TANK

The shape of the oxidizer tank will also be changing in this design phase. Leaving an extra half meter

around the oxidizer tank will make it more difficult to mount inside of SpaceShipOne. It could also makeit more susceptible to moving around during flight and possibly breaking the support structure holding it

up. This would, of course, be catastrophic and could lead to a fail in the mission. To remedy this, the

oxidizer tank will have an increased diameter. However, it has also been decided to incorporate the

pressurant volume within the oxidizer tank, separating the two by a layer. This will maintain the shape of

the oxidizer tank as a pill bottle, but with different dimensions. The 1.11 cubic meter volume of oxidizer

as well as the radius of the tank are the two constraints in determining the geometry of the tank. Also to

2 Equation 52 69.6

71.5

196.25



27

ensure that the rocket can fit within the SpaceShipOne, there is a length constraint on the oxidizer tank.

Using this length, radius, and volume of oxidizer, the volume of pressurant needed can be determined.

The maximum radius that is available is 76.7 centimeters. To allow for room for insulation and connection

of the motor to the fuselage, a radius of 72 centimeters is assumed. From the length of the combustion

chamber, and the two semispherical halves of the oxidizer tank, about 20 centimeters is left to be used as

the cylindrical length of the oxidizer tank. Plugging in these values for r and x into Equation 44, the total

volume of the oxidizer tank comes out to be 1.88 cubic meters. Subtracting the 1.11 cubic meters for the

oxidizer leaves a volume of 0.78 cubic centimeters for the pressurant.

2 43 3 Equation 53

1.88

The oxidizer tank thickness will be found using the hoop stress analysis outlined in the Motor Casing

section of this paper. In this case, the pressure will be taken to be the oxidizer tank pressure (as opposed

to the combustion chamber pressure). Similar to the motor casing, the oxidizer tank will also be made out

of a carbon fiber composite. Fiber composites are usually incompatible with liquid propellants, due to thepermeability of fiber composites over time. To remedy this, an aluminum liner is typically used between

the composite tank and liquid propellant, this will allow for a sealing barrier to prevent the permeability

of the tank. Due to small thickness and low mass of the aluminum liner, the increase in mass is assumed

to be negligible. The wall thickness of the oxidizer tank was found to be 1.95 centimeters, using PoxT of

4.55 MPa, rc of 0.72 m, and f H of 600 MPa.

4 Equation 54

0.022 2.2

Now with the thickness and material, the mass of the oxidizer tank can be found. The mass of the pillshaped oxidizer tank will be the sum of the cylindrical section and 2 hemispherical sections (or 1 sphere).

This is described by Equation 55, where roxT is the radius of the spherical tank (.72 m), tc is the thickness of

the oxidizer tank (0.022 m), x is the length of the cylindrical section (.2 m), and is the density of the

carbon fiber composite (1600 kg/m3). The extra term at the end of the equation account for the disc of

material that separates the pressurant volume from the oxidizer volume. It is assumed that the pressurant

will require the same thickness as the oxidizer, as an approximation. This gives an oxidizer tank mass of

318.5 kilograms.

2 4 Equation 55

318.5

It should be noted that this mass of 318.5 kilograms includes both the mass from the pressurant ‘tank’

and oxidizer tank.

10.7 PRESSURIZATION SYSTEM

The regulated pressurization system will help maintain a constant pressure in the oxidizer tank as the

oxidizer is forced into the combustion chamber, allowing the oxidizer mass flow rate to remain constant

throughout the launch. The purpose of this system is to maintain the mass flow rate of the oxidizer into



28

the combustion chamber. The system relies on a gas pressurant tank that is connected to the oxidizer tank

via a regulator valve; the pressure is controlled by changing opening and closing the regulator valve. The

generic setup of a pressure-fed regulated system is shown is Figure 15. The gas that was used as the

pressurant was helium, due to it being inert.

Figure 15. General regulated pressure fed system over the duration of the burn.

10.7.1

Pressurant Tank Relations

The initial condition of the pressurant tank and the oxidizer tank follows the ideal gas equation shown as

Equation 56 below. The initial pressure of the pressurant tank (Pig) depends on the mass (mg), thepressurant volume (Vfg), and the initial temperature of the pressurant (Tig). The molecular weight of the

pressurant gas (Mg) and universal gas constant (R) are constants, whose values are found in the

Calculations section. The initial condition of the system is such that both tanks are full before flow is

started. Due the pressurant tank having a higher pressure than the oxidizer tank, the flow is forced one

way in the direction of the oxidizer tank. The initial condition is seen in Figure 15, above.

() Equation 56

The final condition of the pressurization system is shown in Figure 15; here one can see that the pressurant

has filled the oxidizer tank. The ideal gas equation is used to relate the final pressure (Pfg), mass, and finaltemperature (Tfg), seen in Equation 57. The pressure in the oxidizer tank remains constant over time.

The final volume of the pressurant (Vig) is the volume of the initial pressurant and the volume of the

oxidizer tank (Vox), as shown in Equation 58. As the oxidizer is ejected into the combustion chamber, the

pressurant is used to uphold the same pressure in the oxidizer tank in order to maintain the mass flow

rate of the oxidizer flow.



29

() Equation 57

Equation 58

The process of venting the gas to the oxidizer tank is assumed to be an isentropic process. This process

does not account for loss in pressure in the feed lines or across the regulator which controls the flow ofthe pressurant into the oxidizer tank. Also, it is assumed that there is no heat transfer with the walls or

the oxidizer. Equation 59 compares the initial and final pressures to the initial and final temperatures of

the isentropic process to this pressurization system.

− Equation 59

10.7.2

Analytical Solutions

The above equations are a set of four independent equations. As part of the design process, the known

quantities are: initial temperature of the pressurant gas, volume of the oxidizer tank, molecular weight of

the pressurant gas, universal gas constant, initial pressure of the pressurant gas, final pressure of the

pressurant gas. The values for these will be discussed in the Pressurant Tank Relations section. The

unknown variables are: final temperature of the gas, mass of pressurant gas required, final volume

occupied by the pressurant gas, and volume of pressurant tank. Since the number of independent

equations is equal to the number of unknowns, the four variables can be determined. A closed form

solution was obtained through use of substitution, shown in the steps below.

The temperature ratio is related to the pressure ratio by Equation 59. Equation 56 and Equation 57 relate

the temperature to the pressure of the gas; thus, another relation for the temperature and pressure ratio

can be obtained by dividing Equation 57 by Equation 56 to make Equation 60. In addition, the terms R,

Mg, and mg are conveniently cancelled out.

Equation 60

The expression for temperature ratio, Equation 59, is then substituted into Equation 60 in order to develop

an equation containing only the volume ratio and the pressure ratio. The result is simply rearranged

algebraically so that the unknowns are on the left of the equal sign and the knowns are on the right.

Equation 61

Equation 58 is substituted into Equation 61, in order to make an expression that contains a single unknownquantity, Vig.

Equation 62

This expression is rearranged algebraically so that Vig is on the left of the equal sign and the other

quantities are on the right. Notice that one of the unknowns has just been solved.



30

1

Equation 63

The mass of the gas can now be solved for through use of Equation 56, since it is now the only unknown

in this equation. First, the expression is rearranged to give Equation 64.

Equation 64

Equation 63 is substituted into Equation 64 to give the mass as a function of the original known quantities.

A second unknown quantity has just been obtained.

[

1]

Equation 65

The remaining unknowns can be readily obtained. Equation 59 was rearranged to solve for the final

temperature (this could have been done at the beginning).

− Equation 66

10.7.3

Considerations

The values for Tfg, mg, Vfg, and Vig may now be obtained, but first necessary considerations are made. The

primary considerations arise from the structural capability of the material from which the pressurant tank

is made, temperature limitations of the pressurant and oxidizer, the space available for the pressurant

tank, and the mass of the pressurant structure and gas. The method taken to solve these limitations was

to meet the temperature and volume constraints, while minimizing the mass of the gas used. Limited

structural considerations were made for the pressurant tank; these are discussed in the Oxidizer Tank

section. It should be noted that this approach does not necessarily result in the lightest overall system,

since structural mas is determined after the optimization.

Recalling the assumption made that there is no heat transfer between the pressurant gas and the liquid

oxidizer, it is apparent that this would give misleading results if any of the oxidizer is evaporated. The

boiling temperature of liquid oxygen is 90.19 K; however, the temperature of the pressurant gas can be

much higher, since there is little time for heat transfer (80 s burn time). Additionally, the coefficient for

heat convection of a gas is much higher than that of a liquid. The actual case would involve some of the

oxidizer evaporating, but this is neglected here. The upper limit of the temperature would be the initial

temperature, which is assumed to be 273.15 K.

The lower temperature limit arises from the fact that if the temperature is too low, the pressurant gas will

undergo condensation. The boiling point of helium is 4.2 K, which is far below any temperature the system

could experience. Still, this limit is kept in mind in the design.



31

Due to the size limitations, the resulting oxidizer-pressurant tank shape takes the form of a pill shape (as

seen in Appendix ); the specific dimensions were such as to achieve the minimum mass of pressurant gas,

as governed by the relationships in the following section.

10.7.4 Calculations

Since the temperature is decreasing over time, the initial temperature will be the highest. Therefore, the

Tig is set to the upper temperature limit: Tig = 273.15 K. The volume of the oxidizer tank is obtained from

section Oxidizer Tank to be 1.11 m3. The molecular weight of helium is Mg = 4.003 kg/kmol, and the

universal gas constant is R = 8314 J/kmol-K. The final pressure will be the lowest pressure achieved by the

pressurant gas. This must be the same as the pressure of the oxidizer Pfg = 4.55 MPa, obtained from section

Oxidizer Tank. The ratio of specific heats for helium is 1.66. Any value of P ig which satisfies all the

previously mentioned constraints can be selected, but it is better to use Equation 65 to minimize the mass

of the pressurant gas. The effects the initial pressure has on the initial volume, mass, and final

temperature of the pressurant gas are shown in Figure 16, Figure 17, and Figure 18.

Figure 16. Relationship between initial volume of pressurant gas required and the initial pressure of the gas

Figure 17. Relationship between the mass of pressurant gas required and the initial pressure of the gas



32

Figure 18. Relationship between the final temperature of pressurant gas and the initial pressure of the gas

It can be observed from Figure 17 that the mass of pressurant gas required first decreases with pressure,

but there is a transition around 21 MPa where the mass is minimum and then begins to increase with

pressure. It would be desirable to design the system at the conditions for minimum mass, and fortunatelyenough the amount of space available for the pressure tank is enough to satisfy the volume requirement

for this condition. The pressurant mass, initial volume, and final temperature are determined from

Equation 63, Equation 65, and Equation 66, shown below. In Equation 63, the VoxT, γ, Pfg, and Vig are known

to be 1.11 m3, 1.66, 4.55 MPa, and 0.78 m3, respectively. Solving the equation gives the initial pressure in

the pressurant tank to be 19.8 MPa.

1

Equation 63

19.8

The initial pressure of the pressurant tank which minimizes the mass of the gas required is 21 MPa, the

19.8 MPa that was just solved for is fairly close to this value. The corresponding volume, mass, and final

temperature of the pressurant gas are determined below.

[

1]

Equation 65

27.2

−

Equation 66

113



33

10.8 I

NJECTOR

P

LATE

/S

LEEVE

The injector is used to bring the liquid oxidizer to the solid fuel so that combustion may occur. The injector

is used to vaporize the liquid oxidizer before it comes into contact with the solid fuel. This allows for

ignition of the oxidizer-fuel mix to occur more easily compared to having to ignite a liquid-solid mix. From

Task 1, it was decided that each port will have 5 small injector holes. This will ensure that the proper mass

flow rate of oxidizer is being injected into the combustion chamber. Each port will have to have the samemass flow rate so that the flow characteristics are the same. If the flow characteristics are not the same

in each port then the thrust history of the rocket could be off. Each port will have a mass flow rate equal

to one fourth the oxidizer mass flow rate (3.96 kg/s), meaning each of the 5 injector holes will have a mass

flow rate of 0.7915 kg/s, or one-twentieth of the oxidizer mass flow rate. Each of the 20 holes will have

an area of 0.21 square centimeters, corresponding to a radius of about 0.26 centimeters.

An injector plate will be connected to the top face of the combustion chamber, containing all 20 injector

holes as well as 4 ignitor holes (one for each port). An injector sleeve will be connected to the injector

plate and will connect the oxidizer tank to the combustion tank. The injector sleeve will serve to funnel

oxidizer from the oxidizer tank to the injector plate. The ignitor will also be housed within the injector

sleeve and will be used to ignite the oxidizer as it enters the combustion chamber. The injector sleeve, in

addition to funneling the oxidizer to the combustion chamber, serves the structural purpose of

transferring the loads from the combustion chamber/nozzle assembly to the oxidizer tank. Since this is a

structurally important piece, it will be made out of titanium to ensure structural integrity.

10.9

POLAR BOSS AND THRUST SKIRT

The polar boss and thrust skirt are both structural elements that bring the whole design together with the

body of the spaceplane. Specifically, the polar boss is used to connect the motor case to the nozzle,

therefore it must be effective in transferring the loads from the nozzle to the motor case. Since it was

decided to create a single structure consisting of both the motor case and nozzle, there will be no need

for a polar boss. Instead, there will be an increase in thickness around the throat as a means to supportthe throat loads. This will then all be encased in the single motor case-nozzle piece.

The thrust skirt connects the motor case to the vehicle itself. In the case of this hybrid rocket motor, the

motor casing is going to be cantilevered from the oxidizer tank. The oxidizer tank will have its own thrust

skirt, so-to-speak, and this will be the connection point of the rocket to the vehicle. Similar to

SpaceShipOne, the oxidizer tank will be the main point of connection for the hybrid rocket motor to the

vehicle, and will use an elastomeric compound to bind the fuselage to the oxidizer tank. Elastomer is a

rubbery material that is capable of being deformed while returning to its original shape, allowing the

material to absorb any loads that are being transferred from the fuselage to the oxidizer tank (and vice

versa).

11 ROCKET BALLISTICS

The rocket ballistics analysis is the center of the design process. This is the stage where it is determined

whether the design thus far will be successful or if it needs to be changed or scraped altogether. The most



34

critical components of this analysis will show the thrust, velocity, and altitude as functions of time. This is

done by finding the geometry of the fuel and how it burns. Then finding the burning perimeter calculations.

Last is the actual ballistic analysis where the graphs can be produced.

11.1

INITIAL GEOMETRY, REGRESSION RATE, AND WEB THICKNESS:

Using the initial parameters obtained during Task 1, an approximation for regression rate r will be

obtained. Initial regression rate ro is obtained from the r equation,

r aGoLp Equation 67

Where, 2 5 0.75 0.15 1.96

Go, the initial total flux through all ports, is the sum of the initial oxidizer and fuel fluxes, Goo and Gfo,

respectively. Goo is estimated to be 350∙, and Gfo is calculated according to

Where, m fis the average fuel mass flow rate, N is the number of ports, and Apo is the initial area of a

single port. For this estimation, the average fuel mass flow rate is assumed close to the initial rate, and is

therefore used for calculating the initial flux. The average value is estimated by dividing the total fuel mass

mf by the total burn time tb.

Initial port area Apo is calculated by rearranging the initial mass flux equation for the oxidizer:

⇒

m o is calculated similarly to m f: total oxidizer mass is divided by total burn time. Goo having been

already estimated, all variables are now known and can be plugged into the regression rate equation to

obtain an approximate initial value:



35

2 5[350

∙ 63380

4 1,266.67804 ∙ 3 5 0 ∙ ]

1.983

Final regression rate rf is estimated as a quarter of the initial regression rate.

0.25

0.496

A linear regression rate is assumed between the initial and final points, allowing regression rate to be

plotted as a function of time as an equation of the form rt kt b, where k is the slope of the line and

b is ro. Since regression rate is assumed linear, the slope is calculated by subtracting the ro from rf and

dividing by the burn time:

0.496 1.983

80

0.0186

An initial estimation for rt has been obtained, and is plotted below in Figure 19:

rt 0.0186t1.983 mms Equation 68



36

Figure 19: Estimated Regression Rate vs. Time

An average regression rate is obtained for use in calculating the final web thickness, which is an essential

parameter to the initial grain geometry. Due to the linearity of the assumed regression rate,

2

1.983 0.496 2

1.239

Final web thickness wf is calculated by multiplying the average regression rate by the total burn time:

1.241 ∙80

99.1

Next, the cross-sectional area of the solid fuel (per port) Acso is derived by dividing the total fuel volume

Vf by the port length Lp and the number of ports N:



37

⇒

0.667

1.94 ∙4

.0859

Initial grain geometry parameters are defined below, and described in Figure 20: y wf Equation 69

ro N(Aco Apo)π y Equation 70

z ro

y

y Equation 71

Figure 20: Initial Grain Geometry

wf

r o

y

wfy

y

wf

wf

wf

wf

z y

z



38

From the figure, it is clear that the parameter y is equal to the final web thickness w f , satisfying Equation

69.

The total area of all four ports (including the fuel) Atot can be derived from the outer radius rf and the

equation for the area of a circle. The outer radius is easily described in terms of ro and y. The total area

for all four ports is derived from multiplying the sum of the port area and the cross sectional area by the

number of ports:

Combining the above equations and solving for ro yields Equation 70.

Finally, Equation 71 can be derived by applying the Pythagorean Theorem and solving for z:

The entire initial geometry is now defined by Equation 69, Equation 70, and Equation 71 in terms of only

y, z, and ro in Figure 20. Values are listed in Table 8 below. Outer radius rf is also included in the table for

later reference.

Table 8: Initial Grain Geometry Parameters

Parameter Length [m]

y 0.0993

z 0.133

ro 0.253

rf 0.352

Grain geometry can now be described as a function of time based on the initial geometry and the

regression rate in Equation 68. The corners are assumed not to burn fully. The remaining fuel in each port

is divided into four sections, color coded in Figure 21. In red, the three corners are assumed to burn from

the center of the port outwards, behaving as quarter circles of increasing perimeter throughout the burn.

The green rectangles are assumed to burn uniformly towards the walls of the port, and will remain

rectangular. The blue arc will be treated as a percentage of a circle based on the ratio of arc angle θo to

2π radians. The angle θo is pictured in Figure 21. Although θ increases as ro increases throughout the burn,

the arc area is calculated only based on the initial angle θo. Therefore, θ(t) will not be considered until the

perimeter calculations are made later on. Since the time variation of θ is not accounted for here, the areas

between the arc and the corners (pink) must be accounted for separately, and are approximated as right

triangles burning outward from the port. The thin dashed line in the figure represents the web progression

at time t during the burn.



39

Figure 21: Grain Geometry as a Function of Time

Before the time varying geometry of the grain can be analyzed, the web thickness must now be derivedas a function of time. Regression rate rt is already known. Stated another way, regression rate is the

change in web thickness as a function of time. Therefore, the web thickness at a given time is found by

integrating the regression rate equation as a function of time:

∫

This integration is approximated in MATLAB using the Riemann Sum

∑ − ∗

= −

Where, time t varies from 0 to 80 seconds in 1000 increments (n = 1000). Initial web thickness wi=1 is zero

by definition. Web distance as a function of time is plotted in Appendix F: Internal Rocket Ballistics.

w(t)

w(t)

w(t)

w(t)

w(t)

w(t)

w(t)

w(t)

θo



40

11.2

FUEL GRAIN CROSS-SECTIONAL AREA

Now, that w(t) has been obtained, each of the four color coded sections above can be broken down and

analyzed separately.

11.2.1

A

corner

Red):

The initial cross-sectional area of the fuel in each corner can be described using the equation for the areaof a quarter circle:

14

where the final web thickness wf is used as the radius of the circle. As fuel is expended, this cross sectional

area will decrease. The magnitude of the decrease can be found by subtracting the area of fuel burned at

time t from the initial area. Area burned as a function of time depends on the same quarter circle equation

used for the initial area, this time substituting web thickness at time t for the burn radius. The equation

for Acorner as a function of time is therefore:

14 14

Acoet 14 wf wt Equation 72

11.2.2

A

side

Green):

The initial fuel cross-sectional area for each green piece is simply its length times its width, or y·z. The area

as a function of time is obtained by subtracting the width times the web thickness at time t from the initial

area:

Adet zy wt Equation 73

11.2.3

A

arc

Blue):

In order to derive the arc area as a function of time, the angle θo must first be known as a function of time.

In order to facilitate this, a new angle φ is defined. Both will be derived as a function of time in this section,

although only φo and θo are needed for the area calculation. Refer to Figure 22 below.



41

Figure 22: Arc Area Derivation

From the figure, θ(t) is not easy to obtain, but φ(t) can be easily calculated from the right triangle formed

by the arc’s inner burn radius r(t), the side thickness y, and the sum of y, z, and w(t):

−

Where,

Now that φ is known, a relation between φ and θ must be obtained. Consider the angle [ φt

], which

will be defined as ψ(t). This is the angle between r(t) and the vertical left hand side of the port depicted.From the figure, it is clear that subtracting twice this angle from will give θ(t) for this geometry:

2 2

2 2 2

φ(t)

θ(t)

w(t)

r(t)

w(t)

y

z

y



42

θt 2φt π2 Equation 74

Now that θ(t) is known, θo can be obtained from Equation 74 and used to calculate the angle ratio of the

arc to a circle. Values are listed below in Table 9.

Table 9: Initial θ Values

θo (rad) θo (deg) 2

0.763 43.7 0.122

The angle ratio is now applied to find the initial and time varying arc cross-sectional areas. The initial area

is calculated by applying the angle ratio to the area equation for a circle with a concentric circular hole.

The outer radius rf and the inner radius ro are used in this calculation:

2 (

)

The arc cross-sectional area as a function of time is found by simply replacing the ro term with r(t).

Act θo2π πrf rt Equation 75

11.2.4

A

comp

Pink):

Acomp is necessary to compensate for the neglect of θ variance which was used to simplify the arc area

calculation. Since Acomp is small, it can be approximated as a right triangle with hypotenuse wf . (Note that

this approximation is reflected in Figure 21. The two base corners of each triangle are not flush with the

outer radius of the grain, and the center of the base intersects the outer arc at its midpoint. Because the

area is small, the grain area lost on one side of the triangle is roughly equal to the area gained on theother.) Figure 23 below shows a close-up of Acomp.



43

Figure 23: Compensated Area Calculation

The initial area of the triangle can be calculated using the triangle area formula, where the base is

wf ·sin(ψo) and the height is approximated as wf . The burned area as a function of time is calculated using

w(t) to determine a new base and height at time t (blue in Figure 23). This area is then subtracted from

the initial.

12 sin 12 sin

Acopt 12 sinψo wf wt Equation 76

11.2.5

Total areas for a single port:

Looking at Figure 21, it is clear that there are three red sections, two green, one blue, and two pink.

Combining Equations 6, 7, 9, and 10 yields a comprehensive expression for fuel cross-sectional area as a

function of time for a single port:

Act 3Acoet2Adet Act 2Acopt Equation 77

The single port area as a function of time is easily obtained by subtracting the cross-sectional area from

the total area of the quarter fuel grain.

Apot 14 πrf Act Equation 78

≈w(t)

wf

φo

≈wf

wf sin(ψo)

ψo



44

Finally, the hydraulic port diameter is calculated. Hydraulic port diameter is defined as the diameter of a

circular port of identical area to the actual port diameter, making it a useful tool for comparison. Hydraulic

port diameter is calculated according to Equation 79.

DHt 4Apotπ Equation 79

11.3 B

URN

P

ERIMETER

C

ALCULATION

Now that area has been obtained, the burn perimeter is now calculated for each port. Once again, the

port is broken up into sections, shown in Figure 24. Only three sections will be used for perimeter, as the

time variation of θ is easily accounted for and the compensation section used in the area calculation is

unnecessary.

Figure 24: Perimeter Calculation

The burn perimeter of the arc as a function of time is calculated from the arc length formula incorporating

the r(t) and θ(t) equations. The contribution of the green sections remains fixed at z. Finally, the perimeter

in the corners can be calculated using the arc length formula at radians and rcorner(t), where rcorner(t) =

w(t). Adding the three sections yields the total inner burn perimeter as a function of time for a single port.



45

Ppot θtrt 2 z 3 π2 wt Equation 80

Hydraulic diameter is calculated again from the perimeter to check the area expression in Equation 77 for

accuracy.

DHt Ppotπ Equation 81

Both hydraulic diameter calculations are plotted in Appendix F: Internal Rocket Ballistics. Since the two

match closely, the area equation is confirmed accurate in spite of its approximation of Acomp and use of a

Riemann sum for web thickness.

Next, the port length Lp is recalculated using the first data point from the Acs(t) equation instead of the

Acso calculated initially.

0

Because the corners are assumed not to burn in the derived Acs(t) model, the effective cross-sectional

area of usable fuel is reduced. Therefore, in order to facilitate the same burnable fuel volume Vf , the port

length Lp must be increased to 2.18 m.

The final grain geometry parameters are listed below in Table 10. The entire solid rocket grain can be

constructed using only these values.

Table 10: Final Grain Geometry Parameters

Parameter Length [m]

y 0.0993

z 0.133

ro 0.253Lp 2.18

11.4 BALLISTIC ANALYSIS

Equation 77 and Equation 79, for area Acs and perimeter Pport, respectively, can now be used to perform

ballistic analysis on the rocket. First, the volume of fuel remaining as a function of time is derived to test

the accuracy of the assumptions in the area equation. If the equation is accurate, Acs and Vf)remaining should

be very close to zero at burnout. Using Equation 82 below, the remaining fuel volume at tb is calculated

to be 0.0004 m3, once again confirming the accuracy of the Acs equation.

Vfet ActLpN Equation 82

Next, mass flow rates for both the fuel and the oxidizer are calculated. For a single port, fuel mass flow

rate is calculated by multiplying the perimeter Pport, length Lp, and regression rate r, yielding a volume of

fuel burned per second. This quantity is then multiplied by the density of the fuel ρ to obtain mass flow

rate:

m fpot PpotLprtρf Equation 83



46

Total fuel mass flow rate can be is obtained by multiplying Equation 83 by the number of ports N.

m f t m fpotN Equation 84

Oxidizer mass flow rate remains constant, and was already obtained by dividing the total oxidizer mass by

the burn time. The oxidizer mass flow rate for a single port is obtained by dividing the total rate (constant,

equal to the average rate) by the number of ports.

m ot m o Equation 85

m opot m otN m oN Equation 86

Total mass flow rate can now be found by adding the oxidizer and fuel rates and multiplying by the number

of ports.

m t m fpot m opot N Equation 87

Fuel and oxidizer mass fluxes are obtained by dividing the port mass flow rates by the port area Aport from

Equation 78. Total mass flux is obtained by dividing the total mass flow rate by the port area times the

number of ports. Graphs for fuel, oxidizer, and total mass flux are included in Appendix F: Internal Rocket

Ballistics.

Gf t m fpotNApotN m fpotApot Equation 88

Got m opotApot Equation 89

Gt m tApotN Equation 90

Oxidizer to fuel ratio can be calculated as a function of time using Equation 83 and Equation 86, and is

included in Appendix E: Vehicle Performance.

OFo t

m opotm fpot

Equation 91

Multiplying the total mass flow rate from Equation 87 by the exit velocity ue yields thrust as a function of

time (Appendix E: Vehicle Performance).

Tt m tue Equation 92

Fuel mass remaining as a function of time is calculated by multiplying the remaining fuel volume in

Equation 82 by the fuel density. Remaining oxidizer mass is calculated by subtracting the product of the



47

port oxidizer mass flow rate, the number of ports, and the time from the initial oxidizer mass mo

(distinguished from gross mass m0). Total vehicle mass as a function of time is found by adding the

remaining fuel and oxidizer masses to the structural and payload masses, ms and ml, respectively. A plot

of vehicle mass vs time can be found in Appendix E: Vehicle Performance.

mfet Vfetρf Equation 93

moet mo m opotNt Equation 94

mt m m mfet moet Equation 95

Equation 93 can be divided by the total solid fuel mass mf and multiplied by 100% to obtain the percentage

of fuel remaining as a function of time. This curve is plotted in Appendix G: Mission Progress.

Next, another pair of checks is performed to ensure the accuracy of the model, this time ensuring that all

the fuel is being burned. In the first check, Equation 93 is used to calculate mf)remaining at burnout. The valueobtained is 0.382 kg, close to the 0 kg expected. As a percentage of the initial fuel weight mf , this value is

six hundredths of a percent of the total fuel, demonstrating the ability of the model to reach a reasonable

final state. For the second check, the total amount of fuel burned is checked by integrating the fuel mass

flow rate using another Riemann sum, where mcheck)i represents the total amount of fuel burned at a given

time (therefore, mcheck)1 = 0). Time t varies from 0 to tb = 80 in n = 1000 increments.

∑ − ∗= −

The integration approximation yields 635 kg of solid fuel burned by tb, a deviation of only three tenths of

a percent from the expected value mf = 633 kg. The accuracy of the above equations is again confirmed.

Next, rt is recalculated according to Equation 67, this time using the new Lp value and the time-varying

Go(t) obtained in Equation 89.

Got m opotApot Equation 89

rt aGotLp Equation 67

The recalculated

r curve is displayed in Appendix F: Internal Rocket Ballistics alongside the linear curve

that was assumed initially.

Finally, the vehicle’s expected performance is checked against the mission requirements to ensure the

design is feasible. Because this analysis is not a requirement for this report, it is greatly simplified and

intended only as a “sanity check” to screen for major discrepancies between requirements and

performance. The spacecraft is assumed to be in a vertical orientation (nose up), with all forces acting

along its major axis, from launch at 14 km until burnout. Pull-up time after launch is ignored, as is air

resistance (and therefore drag).



48

First, the vehicle weight, W, as a function of time is obtained by multiplying the time varying mass in

Equation 95 by the Earth’s gravity constant g=9.81.

Wt mtg Equation 96

Acceleration is then obtained by subtracting the weight from the thrust:

∑

Rearranging,

at Tt Wtmt Equation 97

Equation 97 is plotted in Appendix G: Mission Progress.

G-force is calculated to ensure the spacecraft crew will not be subject to excessive accelerations during

the burn. In this case, the actual g-forces experienced by the crew (excluding the pull-up maneuver) will

be less due to drag, making the zero air resistance estimate the worst-case scenario.

gfocet atg Equation 98

From the g-force plot in Appendix E: Vehicle Performance, it is clear that the spacecraft and its crew never

experience more than 3 g’s.

Velocity during the burn is calculated by integrating the acceleration curve with a Riemann sum. Initial

velocity v1 is assumed 0 (since this is the y-component), and the integration takes place in n = 1000

intervals from t = 0 to tb = 80 s. Altitude is calculated similarly, this time integrating the velocity curve witha Riemann sum and assuming initial altitude alt1 to be 14 km.

v ∑t t− ∗= a v− Equation 99

alt ∑t t− ∗= v a l t− Equation 100

Velocity and altitude after the burn are calculated by assuming constant acceleration due to gravity. Time

t is reset to 0 for use in these equations, with v1000 and alt1000 (velocity and altitude at burnout from

Equation 99 and Equation 100) representing initial velocity and altitude at time t = 0 in the equations

below.

vt gt v Equation 101



49

altt 22 gt vt alt Equation 102

Equation 101 and Equation 102 are added onto the ends of Equation 99 and Equation 100 to obtain the

altitude and velocity equations for the entire mission. Altitude and velocity curves are presented in

Appendix C. From the graphs, the maximum ΔV attainable under ideal circumstances is equal to 1540 ,

just shy of the initial 1700 goal for this design. Under engine power, an altitude of 66.6 km is attained

at burnout. However, the vehicle is still able to coast to a maximum altitude of 187.6 km, almost twice the

mission requirement of 100 km. Therefore, even with the ΔV lower than expected, the altitude

requirement is met with room to spare. Even with air resistance factored in, 100 km should be easily

attainable due to the low air density at mission altitudes.

Relevant graphs are included in Appendix E: Vehicle Performance – Appendix G: Mission Progress. The

MATLAB code used to obtain the data and graphs for this section is included in Appendix H: Task2Ballistics

MATLAB Code.

12 FINAL DESIGN

Throughout this report, the initial design of a hybrid rocket motor for SpaceShipOne was modified as some

of the major components were designed. The results of this rendition of the design, shown in Figure 25,

was created using the calculations performed in this report. Again, the rocket was designed so that it can

easily fit within the geometrical constraints defined by the fuselage of SpaceShipOne.

Figure 25. Expanded view and assembled view of the current design of the hybrid rocket motor.

In this version of the design, the oxidizer tank was modified to be a spherical shape. This allows the tank

to be closer to the inside of the fuselage to allow for easier mounting. With this design, there is about 11

centimeters between the oxidizer tank and the fuselage. This leaves enough room to fit insulation around

the outside of the tank as well as enough room to mount the rocket assembly using an elastomeric

compound. The injector sleeve was modified to hold ignitors for each of the four ports. The injector plate

now features holes for each of the ignitors to allow for combustion of the oxidizer and fuel.



50

The injector sleeve and oxidizer tank have been fitted with a flange to allow for them to be connected to

each other. The other side of the injector sleeve and the combustion chamber have also been fit with the

same flange to connect them. Now the entire hybrid rocket can be constructed into one assembly. It

should be noted that the injector plate and sleeve were not designed using anything that was previously

calculated, other than the injector area. Other major changes to the design include the addition of an aft

mixing chamber and combining the combustion chamber and nozzle into one piece.

See for Appendix I detailed drawings of each part.

13 S

UMMARY AND

C

ONCLUSIONS

The objective stated for this mission was to design a hybrid rocket motor capable of launching

SpaceShipOne to an altitude of 100 kilometers carrying three passengers and 600 kilograms of payload.

This preliminary design of our hybrid rocket motor utilizes a combination of liquid oxygen and a solid fuel

of HTPB. This propellant was selected for its beneficial characteristics in safety and simplicity, along withits performance record. The initial sizing of our propellant gives mass and volume for our oxidizer and fuel

as 1266.7 kilograms and 1.11 cubic meters, and 633.3 kilograms and 0.667 cubic meters respectively. The

port design is that of a four port wagon wheel where the apex of each port is 0.142 meters out radially.

The angle of each port is 90 degrees. The length of each side of the port is 0.13 meters. The port length

was found to be 1.94 meters with an outer radius of 0.35 meters. The arc connecting the two legs is of

the same curvature as the outer wall of the chamber. The oxidizer tank is a capsule in shape with half

sphere ends on either side with radii of 50 centimeters and a length of 74.66 centimeters. There are four

injectors, one for each port, with a total injector area of 17.2 square centimeters. The throat of the nozzle

was calculated to be 0.112 square meters, and is expanded to an exit area of 0.45 square meters. The

nozzle length is designed with a length of 0.875 meters. A summary of these values is shown below. It

should be noted that these are all preliminary calculations and are subject to change throughout the final

design of the motor.

Table 11. Summary of initial design parameters.

mf 646 kg At 0.0112 m2

mox 1266.7 kg dt 12 cm

Vf 0.68 m3 Ae 0.448 m2

Vox 1.11 m3 de 75.7 cm

ṁox 15.83 kg/s Ln 0.872 m

ṁf 8.075 kg/s Lox 0.7466 m

PoxT 4.44 MPa rox 0.50 m

Ainj 3.85 cm2 Lp 1.94 mA 0.13 m Rs 0.25 m

RG 0.35 m ṙav 0.00125 m/s

B 0.142 m ϕ 45°

tw 12.98 mm

wF 0.10 m



51

The next thing that was looked into was the detailed design of the major components. The major

components were the fuel grain, support structure, insulation, nozzle, motor casing, oxidizer tank,

pressurization system, injector plate/sleeve, and polar boss and thrust skirt. The fuel grain design did not

change from Task 1. The radius of the fuel is 35.25 centimeters, but including the support structure it

becomes 35.75 centimeters. The average regression rate is 0.00125 meters per second. The web thickness

is 10 centimeters. The length of the fuel is 1.96 meters, and lastly the volume of the fuel is 0.68 cubicmeters. The support structure inside the fuel grain is to just hold it in place. The support structure outside

of the casing will hold the motor in place. Both of these supports has a mass of approximately 120

kilograms. The insulation thickness of the combustion chamber, mixing chamber, and oxidizer are 2

centimeters, 16 centimeters, and 2.06 centimeters respectively. The mass of insulation only accounts for

the mass in the motor casing which is 28.16 kilograms. The nozzle has a length of 87.2 centimeters and a

mass of 88.1 kilograms. The motor casing is made of a carbon fiber/epoxy and has a thickness of 8.87

millimeters. The length of the combustion chamber is 196.25 centimeters with a diameter of 71.5

centimeters. The half angle in the mixing chamber is 40 degrees. The throat has a diameter of 12

centimeters. The mass of the motor case is 69.6 kilograms; combined with the nozzle the mass of the

motor case nozzle assembly is 157.7 kilograms. The volume required of the oxidizer is 1.11 cubic meters.

The thickness needed to withhold that pressure is 1.95 centimeters. The mass of the oxidizer tank is

161.75 kilograms. The injector plate/sleeve will have 4 sets of 5 holes, 1 set for each port. The mass flow

through each hole will be 0.79 kilograms per second with a radius of 0.26 centimeters. The polar boss and

thrust skirt were able to become obsolete because the motor casing and nozzle became one piece that

was cantilevered at the oxidizer tank.

Table 12: Summary of final design parameters.

RG 0.3575 m

ṙavg 0.00125 m/s

wf 0.10 m

Lp 1.96 m

Vf 0.68 m3

msupp 120 kg

tinsul,comb 0.02 m

tinsul,mix 0.16 m

tinsul,oxT 0.0206 m

minsul 28.16 kg

mnoz 88.1 kg

tw 8.87 mm

θcc 40°

Rt 6 cm

mmc 69.6 kgVoxT 1.11 m3

toxT 1.95 cm

moxT 318.5 kg

ṁinjector 0.79 kg/s

TOTAL MASS 624.36 kg

The next final numbers were the numbers from the ballistics analysis. These numbers are the performance

details of the hybrid rocket motor. The goal “delta-V” for this mission was 1700 meters per second. The

ideal circumstances of our motor is equal to 1540 meters per second. Though the engine is only under

power up to 66.6 kilometers, it is estimated that SpaceShipOne will coast to 187.6 kilometers. Howeverthis does not take into consideration the effects of air drag. While the “delta-V” for this maneuver is lower,

it appears that it should be capable of reaching the required altitude.

Table 13: Summary of final performance parameters.

ΔV 1540 m/s

Max Altitude 187.6 km



52

14 REFERENCES

[1]

Rutan, Burt. "SpaceShipOne (SS1) | Spacecraft." Encyclopedia Britannica Online. Encyclopedia

Britannica, 31 Oct. 2010. Web. 15 Sept. 2015.

[2]

Sharp, Tim. "SpaceShipOne: The First Private Spacecraft | The Most Amazing Flying Machines Ever |

Space.com."Www.Space.com. N.p., 2 Oct. 2014.

[3]

Humble, W. Ronald, Henry, N. Gary, Larson, J. Wiley. Space Propulsion Analysis And Design. New York:

McGraw-Hill, 1995. Print.

[4]

Braeunig.us, 'Basics Of Space Flight: Rocket Propellants'. N.p., 2008. Web. 22 Sept. 2015.

[5]

Cantwell, Brian. 'Wax Fuel Gives Hybrid Rockets More Oomph'. Spectrum.ieee.org. N.p., 2009.

[6]

'Hybrid Rocket Motor Design'. Space Safety Magazine. N.p., 2014. Web.

[7]

Rocket.com, 'RS-25 Engine'. N.p., 2015. Web. 22 Sept. 2015.

[8]

Spg-corp.com, 'Space Propulsion Group |SPG | Propulsion Technologies |Hybrid Rocket Propulsion

Systems'. N.p., 2012.

[9]

Murc.ws,. 'Spacedev Flies Prototype Hybrid Rocket Lunar Lander'. N.p., 2007. Web. 24 Sept. 2015.[10]

Ewig, Ralph. Vapor Pressurization (Vapak) Systems History, Concepts, And Applications . 1st ed.

Renton: Holder Consulting Group, 2009. Print.

[11]

Zimmerman, Jonah E., Benjamin S. Waxman, Brian Cantwell, and Greg Zilliac. "Comparison Of Nitrous

Oxide And Carbon Dioxide With Applications To Self-Pressurizing Propellant Tank Expulsion

Dynamics." (2013). Print.

[12]

United States of America. National Aeronautics and Space Administration. Office of Technology

Utilization. Design of Liquid Propellant Rocket Engines. By Dieter K. Huzel and David H. Huang.

Washington, D.C.: n.p., 1967. Web. 12 Nov. 2015.

[13]

"Mechanical Properties of Carbon Fiber Composite Materials." (n.d.): n. pag. ACP Composites. ACP

Composites, Dec. 2014. Web.



53

15 APPENDIX A



54



55

16 APPENDIX B

Figure 26. Plots taken from Appendix B in Humble’s textbook, page 712 [3]. The lines show how important values were

determined for this project.

A)

B)



56

Figure 27. Plots taken from Appendix B in Humble’s textbook, page 711 [3]. The lines show how important values were

determined for this project.

A)

B)



57

17 APPENDIX C

Combined Equations:

Equation 15 IN Equation 16: Equation 103

Equation 103


58 Equation 104


≅ 58 Equation 105


≅ 58 Equation 106


√ 2 Equation 107

Equation 107 REARANGED: Equation 108

58 √ 2

/ Equation 108

Equation 21 & Equation 20, IN Equation 22: Equation 109

2 Equation 109

Equation 32 IN Equation 31, AND REARAGNED: Equation 110

Equation 110


Equation 111


sinsin Equation 112

Equation 29 & Equation 33 & Equation 110 & Equation 112, IN Equation 109: Equation 35



58

sinsin sinsin Bsin Equation 35


4 Equation 113

Equation 113 IN Equation 26, AND REARANGED: Equation 114

4 Equation 114


4 Equation 115


sinsin

4 Equation 116

Equation 110 & Equation 34 & Equation 108, IN Equation 116: Equation 36

sinsin √ 2 4 58 √ 2/

Equation 36



59

18 APPENDIX D



60



61



62



63



64



65

19 APPENDIX E: VEHICLE PERFORMANCE



66

20 APPENDIX F: INTERNAL ROCKET BALLISTICS



67

21 APPENDIX G: MISSION PROGRESS



68

22 APPENDIX H: TASK2BALLISTICS MATLAB CODE

% Task 2 Ballistics

close all % closes figures from previous runs of the program

clear

clc

format('long') % increases significant figures tracked

%%%%%%%%%%%%%%%%%%%%%%% INPUTS %%%%%%%%%%%%%%%%%%%%%%%

deltaV = 1700;

tb = 80; % burn time

F = 73500; % Thrust

m0 = 3600; % gross mass

ms = 1200; % inert mass

ml = 500; % payload mass

mp = 1900; % propellant mass

g = 9.81; % gravity

gamma = 1.235; % an initial ratio of specific heats is assumed

eta = 0.98; % combustion efficiency

slantyR = 8314; %

slantyM = 22.4; % molecular weight

Tc = 3543;% flame temperature

Pc = 3.7*(10^6); % chamber pressure

OFratio = 2; % oxydizer to fuel ratio (initial assumption)

Isp = 318.5; % specific impulse

epsilon = 25; % nozzle expansion ratio

rhof = 950; % fuel density

rhoo = 1142; % oxydizer density

N = 4; % number of ports

%%%%%%%%%%%%%%%%%%%%%%% OUTPUTS %%%%%%%%%%%%%%%%%%%%%%%

t = linspace(0,tb,1000); % time

TWratio = F/(m0*g); % thrust to weight ratio

R = slantyR/slantyM;

ac = sqrt(gamma*R*Tc);% characteristic speed of sound

fgamma = gamma*((2/(gamma + 1))^((gamma + 1)/(2*gamma - 2)));

cstar = (eta*ac)/fgamma; % characteristic speed

mf = mp/(1 + OFratio); % fuel mass

mo = mp - mf; % oxydizer mass

mdotav = mp/tb; % average mass flow rate

ue = F/mdotav; % nozzle exit speed

At = (mdotav*cstar)/Pc; % throat area

dt = sqrt((4/pi)*At); % throat diameter

Ae = epsilon*At; % exit area

de = sqrt((4/pi)*Ae);% exit diameter

Vf = mf/rhof; % fuel volume

Vo = mo/rhoo; % oxydizer volume

Vfp = Vf/N; % fuel volume per port

rhomixture = Pc/(R*Tc); % density after combustion, assuming ideal gas

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TASK 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



69

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

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

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

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

%%%%%%%%%%%%%%%%%%%%%%% SOLID FUEL DESIGN %%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%% INPUTS %%%%%%%%%%%%%%%%%%%%%%%

a = 2*(10^(-5));

n = 0.75;

m = -0.15;

Goo = 350; % initial oxidizer flux (assumed 350 in initial program)

mdotoav = mo/tb; % average oxidizer mass flow rate (constant over time)

Apo = mdotoav/(Goo*N); % initial port cross sectional area

mdotfav = mf/tb; % average fuel mass flow rate (only valid on first iteration, where linear mdot

is assumed)

Gfo = mdotfav/(N*Apo); % initial fuel flux (mdotfo is assumed to be close to mdotfav)

Lp = 1.94; % chosen value for port length (constraint from Task 1)

Acso = Vf/(Lp*N); % initial cross sectional fuel area (single port) Go = Goo + Gfo; % initial total flux

rdoto = a*(Go^n)*(Lp^m); % initial regression rate

rdotf = 0.25*rdoto; % final regression rate

rdotav = 0.5*(rdoto + rdotf); % average regression rate

wf = rdotav*tb; % final web thickness

% initial grain geometry and rdot

y = wf; % grain geometry parameter

ro = sqrt(((Acso + Apo)*N)/pi) - y; % grain geometry parameter (total engine area > rf > rf - y =

ro

z = sqrt((ro^2)-(y^2)) - y; % grain geometry parameter

% parameters for rdot estimation below, based on y = mx + b (in this case, rdot = kt + b) format,assuming linear rdot between estimations for rdoto and rdotf

k = (rdotf - rdoto)/(tb);

b = rdoto; % rearranging the equation format, b = rdot(t) - k*t...initial condition at t = 0 is

known to be rdoto, so b = rdoto

% rdot estimation

for time = 1:1000

% rdot = kt + b

rdot(time) = k*t(time) + b; % rdot as a function of time, based on linear change in rdot

between initial and final conditions

rdotmm(time) = rdot(time)*1000;

end

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

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

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

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

%%%%%%%%%%%%%%%%%%%%% GRAIN GEOMETRY AS f(t) %%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%% Needed (P) %%%%%%%%%%%%%%%%%%%%%



70

% rdot(t)

% y, z (geometry parameters)

% thetao, phio

% ro

%%%%%%%%%%%%%%%%%%%%% Needed (CS) %%%%%%%%%%%%%%%%%%%%%

% rdot(t)

% y, z

%%%%%%%%%%%%%%%%%%%%% Equations %%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%% Needed for both CS and P Sections %%%%%%%%%%%%%%%%%%%%%

w(1) = 0;

rcorner(1) = 0;

for time = 2:1000

w(time) = (t(time) - t(time - 1))*rdot(time) + w(time - 1); % web as a funtion of time

(integrated using a Riemann sum: rdot = dw/dt --> int(dw)=int(rdot*dt))

wmm(time) = w(time)*1000; % web thickness converted to millimeters

rcorner(time) = w(time); % inner burn radius of each corner as a function of time

end

wf2 = w(1000); % accurate final web thickness (doesn't assume linear regression rate)

for time = 1:1000

r(time) = ro + w(time); % inner burn radius as a funtion of time

phi(time) = acos(y/r(time)); % angle from horizontal to top arc corner as a function of time

theta(time) = phi(time) - ((pi/2) - phi(time)); % angle from origin to major arc corners as a

function of time

end

ro2 = r(1);

rf = r(1000); % final radius from origin (fuel depleted)

thetao = theta(1); % initial theta

thetaodeg = (thetao*180)/pi;

thetaf = theta(1000); % final theta

thetafdeg = (thetaf*180)/pi;

%%%%%%%%%%%%%%%%%%%%% Fuel Cross Sectional Area %%%%%%%%%%%%%%%%%%%%%

phio = phi(1); % initial phi

phiodeg = (phio*180)/pi;

Cf = 2*pi*rf;

Afpie = (Cf^2)/(4*pi); % cross sectional area of entire engine as it relates to circumference

(circumference was used instead of radius in case the ratio of arc length to circumference was

also needed later), used in the next equation

Afslice = (thetao/(2*pi))*Afpie; % total area between origin and inner burn arc, using ratio of

thetao to total radians in a circle to obtain area proportion

for time = 1:1000

Cpie(time) = 2*pi*r(time); % compare Cpie(1000) with Cf, they should be the same

Apie(time) = ((Cpie(time))^2)/(4*pi);

Aslice(time) = (thetao/(2*pi))*Apie(time);

Aarc(time) = Afslice - Aslice(time); % area of fuel remaining in major arc as a function of

time

Aside(time) = (y*z) - (w(time)*z); % area of sides (spokes of the wheel) as a funtion of time

Acorner(time) = (0.25*pi*(y^2)) - (0.25*pi*((rcorner(time))^2)); % corner area as a funtion



71

of time

Acomp(time) = (0.5*y*(wf2*sin((pi/2)-phio))) - (0.5*((wf2*sin((pi/2)-

phio))/wf2)*((w(time))^2)); % initial Acomp minus the burned portion of Acomp (using triangle

geometry) as a function of time

Acs(time) = Aarc(time) + (2*(Acomp(time))) + (3*(Acorner(time))) + (2*(Aside(time))); % cross

sectional area of fuel as a function of time (single port)

Aport(time) = (0.25*pi*(rf^2)) - Acs(time); % port area as a function of time, used with mdot

to calculate flux

DH1(time) = sqrt((4*Aport(time))/pi); % hydraulic diameter (equivalent diameter for a

circular port)

end

%%%%%%%%%%%%%%%%%%%%% Burn Perimeter %%%%%%%%%%%%%%%%%%%%%

for time = 1:1000

s(time) = (theta(time))*(r(time)); % arclenth of inner burn radius of major arc as a function

of time

Pmajor(time) = (2*z) + s(time); % inner perimeter of the sides and arc as a function of time

scorner(time) = 2*pi*rcorner(time)*0.25; % arclength of inner burn radius of each corner as a

function of time

Pminor(time) = 3*scorner(time); % inner perimeter of the corners as a function of time Ptot(time) = Pminor(time) + Pmajor(time); % total burn (inner) perimeter as a function of

time (single port)

DH2(time) = Ptot(time)/pi; % hydraulic diameter, calculated from perimeter instead of area

end

%%%%%%%%%%%%%%%%%%%%% Outputs %%%%%%%%%%%%%%%%%%%%%

Lp2 = Vf/(Acs(1)*N); % length of propellant tube

for time = 1:1000

Vfpremaining(time) = Acs(time)*Lp2; % volume of fuel remaining per port as a function of time

(should be very close to 0 at tb, as a check of this method's accuracy)

Vfremaining(time) = N*Vfpremaining(time); % total fuel volume remaining as a function of time

(all ports) end

for time = 1:1000

mdotfp(time) = Ptot(time)*Lp2*rdot(time)*rhof; % fuel mass flow rate as a function of time

(per port)

mdotf(time) = mdotfp(time)*N; % fuel mass flow rate as a function of time

fuelflux(time) = mdotfp(time)/(Aport(time)); % fuel mass flux as a function of time

mdotop(time) = mdotoav/N;

oxidizerflux(time) = mdotop(time)/Aport(time); % oxidizer mass flux as a function of time

OFratiovarying(time) = mdotop(time)/mdotfp(time);

Tp(time) = (mdotfp(time) + mdotop(time))*ue; % Thrust as a function of time (per port)

mdot(time) = N*(mdotfp(time) + mdotop(time)); % Total masas flow rate as a function of time

(all ports)

massflux(time) = mdot(time)/(Aport(time)*N);

T(time) = mdot(time)*ue; % Total thrust as a function of time (all ports)

TkN(time) = T(time)/1000;

massfremaining(time) = Vfremaining(time)*rhof; % fuel mass as a function of time

percentfuelremaining(time) = (massfremaining(time)/mf)*100; % percent of fuel remaining

massoremaining(time) = mo - mdotop(time)*t(time)*N; % oxidizer mass as a function of time

(true as long as mdotop is constant; an integral is needed if oxidizer flow rate is varied)

mass(time) = ms + ml + massoremaining(time) + massfremaining(time); % total mass of the



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spacecraft

end

masscheck(1) = 0; % masscheck integrates mdotf from 0 to tb to make sure all the fuel is being

burned

for time = 2:1000

masscheck(time) = (t(time) - t(time - 1))*mdotf(time) + masscheck(time - 1);

end

% now that values have been obtained from the initial approximation of

% rdot, rdot can be expressed more precisely. A new average rdot is also derived.

rdottotal2 = 0;

for time = 1:1000

rdot2(time) = a*(massflux(time)^n)*(Lp2^m); % new, accurate rdot based on the data above

rdot2mm(time) = rdot2(time)*1000; % rdot in millimeters per second

rdottotal2 = rdot2(time) + rdottotal2; % in preparation for taking a new average rdot, below

end

rdotav2 = rdottotal2/1000; % accurate rdot average (without assuming linear)

% altitude while boosting

altitude0 = 14000; % launch altitude (m) velocity(1) = 0; % y-component of launch velocity

altitude(1) = 14000; % launch altitude(m)

for time = 1:1000 % acceleration vs. time (assumes constant g, ignores air resistance)

weight(time) = mass(time)*g; % total weight of the spacecraft as a function of time

acceleration(time) = (T(time) - weight(time))/mass(time);

gforce(time) = acceleration(time)/g;

end

for time = 2:1000

velocity(time) = (t(time) - t(time - 1))*acceleration(time) + velocity(time - 1); % integral

of the acceleration equation (Riemann sum); all velocity is assumed to be in the y-direction

altitude(time) = (t(time) - t(time - 1))*velocity(time) + altitude(time - 1); % integral of

the velocity equation (Riemann sum)

altitudekm(time) = altitude(time)/1000;end

altitudekm(1) = altitude(1)/1000;

% altitude after boost

t2 = linspace(0,240,3000); % time after burnout

for time = 1:3000

velocity2(time) = ((-g)*t2(time)) + velocity(1000);

altitude2(time) = (0.5*(-g)*((t2(time))^2)) + (velocity(1000)*t2(time)) + altitude(1000);

altitudekm2(time) = altitude2(time)/1000;

end

% velocitymission and altitudemission are created to combine the velocity and altitude before and

after burn

for time = 1:1000 % values during burn are added to new variables

velocitymission(time) = velocity(time);

altitudemission(time) = altitude(time);

end

for time = 1:3000 % values after burn are added to new variables

velocitymission(time + 1000) = velocity2(time);

altitudemission(time + 1000) = altitude2(time);

end

for time = 1:4000



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altitudekmmission(time) = altitudemission(time)/1000;

end

tmission = linspace(0,320,4000); % creates a time variable with dimensions to match

velocitymission and altitudemission, allowing for plots to be extended to after burnout

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

% Plotting each Time Varying Value

% 1. Vehicle Performance

figure('Name','Vehicle Performance','NumberTitle','off')

% Thrust

subplot(2,2,1)

plot(t,TkN)

title('Thrust vs. Time')

xlabel('Time [s]')

ylabel('Thrust [kN]')

% O/F Ratio

subplot(2,2,2)

plot(t,OFratiovarying)

title('O/F Ratio vs. Time')

xlabel('Time [s]')

ylabel('O/F Ratio')

% Vehicle Gross Mass

subplot(2,2,3)

plot(t,mass)

title('Vehicle Gross Mass vs. Time')

xlabel('Time [s]')

ylabel('Vehicle Gross Mass [kg]')

% G Forces Experienced

subplot(2,2,4)

plot(t,gforce)

title('G Force vs. Time')

xlabel('Time [s]')

ylabel('G Force')

% 2. Internal Ballistics

figure('Name','Internal Rocket Ballistics','NumberTitle','off')

% Total Mass Flux

subplot(2,3,1)

plot(t,massflux)

title('Total Mass Flux vs. Time')

xlabel('Time [s]')

ylabel('Mass Flux [kg/(s*m^2)]')

% Oxidizer Mass Flux

subplot(2,3,2)

plot(t,oxidizerflux)



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title('Oxidizer Mass Flux vs. Time')

xlabel('Time [s]')

ylabel('Oxidizer Mass Flux [kg/(s*m^2)]')

% Fuel Mass Flux

subplot(2,3,3)

plot(t,fuelflux)

title('Fuel Mass Flux vs. Time')

xlabel('Time [s]')

ylabel('Fuel Mass Flux [kg/(s*m^2)]')

% Regression Rate

subplot(2,3,4)

plot(t,rdotmm,'r',t,rdot2mm,'b')

legend('assumed rdot', 'calculated rdot')

title('Regression Rate vs. Time')

xlabel('Time [s]')

ylabel('Regression Rate [mm/s]')

% Web Distance subplot(2,3,5)

plot(t,wmm)

title('Web Distance vs. Time')

xlabel('Time [s]')

ylabel('Web Distance [mm]')

% Hydraulic Port Diameter

subplot(2,3,6)

plot(t,DH1,'r',t,DH2,'b')

legend('DH (Area)', 'DH (Perimeter)') % shows DH calculated from both perimeter and area

geometries...they correspond well, indicating similarly accurate approximations

title('Hydraulic Port Diameter vs. Time')

xlabel('Time [s]')ylabel('Hydraulic Diameter [m]')

% 3. Estimated Mission Progress

figure('Name','Mission Progress','NumberTitle','off')

% Altitude

subplot(2,2,1)

plot(tmission,altitudekmmission)

title('Altitude vs. Time')

xlabel('Time [s]')

ylabel('Altitude [km]')

% Velocity

subplot(2,2,2)

plot(tmission,velocitymission)

title('Vertical Velocity Component vs. Time')

xlabel('Time [s]')

ylabel('Vertical Velocity [m/s]')

% Acceleration

subplot(2,2,3)



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plot(t,acceleration)

title('Accleleration vs. Time')

xlabel('Time [s]')

ylabel('Acceleration [m/(s^2)]')

% Percent of Fuel Remaining

subplot(2,2,4)

plot(t,percentfuelremaining)

title('Percent Fuel Remaining vs. Time')

xlabel('Time [s]')

ylabel('Remaining Fuel Percentage')

% displaying useful properties for CATIA modeling

fprintf('Final Web Thickness = %f\n', wf2)

fprintf('Initial Inner Radius = %f\n', ro2)

fprintf('Outer Radius = %f\n', rf)

fprintf('Port Length = %f\n', Lp2)

Final Web Thickness = 0.099240

Initial Inner Radius = 0.252530

Outer Radius = 0.351771

Port Length = 2.182509



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Published with MATLAB® R2014b

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





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SpaceshipOne Rocket Design