conceptual design of an air- launched multi-stage launch...

IN DEGREE PROJECT ENGINEERING PHYSICS,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2020

Conceptual Design of an Air-launched Multi-stage Launch Vehicle

JOHN SIGVANT

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ENGINEERING SCIENCES

l

DEGREE PROJECT IN PHYSICS, SECOND CYCLE 30.0 HP SH204X 1

Conceptual Design of an Air-launched Multi-stageLaunch Vehicle

John Sigvant, MSc student, KTH Royal Institute of Technology, Stockholm, Sweden

Abstract—In the present thesis, the objective was to find themaximum amount of payload mass that can be put into a 500km polar orbit by a 1400 kg air-launched multi-stage rocketlaunched from a fighter jet platform. To fulfill the objectivean algorithm incorporating several modules was developed. Themodules performed calculations based on theoretical models andliterature values to arrive at optimal design variables. From thedesign the maximum payload mass was able to be derived andit was concluded that a three-stage launch vehicle was able todeliver a 22.0 kg payload to the desired orbit.

Sammanfattning—I den här avhandlingen var syftet att hittaden maximala mängden nyttolastmassa som kan transporterasav en 1400 kg flerstegsraket uppskjuten från luften till en 500km polär bana. För att uppfylla målet utvecklades en algoritmmed flera moduler. Modulerna utförde beräkningar baserade påteoretiska modeller och litteraturvärden för att komma framtill optimala designvariabler. Från konstruktionen kunde denmaximala nyttolastmassan härledas och det konstaterades atten trestegsraket kunde leverera en nyttolast på 22.0 kg till denönskade omloppsbanan.

Index Terms—Air-launched, multi-stage, launch vehicle,rocket, LEO, solid propellant, optimization

I. INTRODUCTION

AS of today, the most common way to put a nano or microsatellite into orbit is to rideshare among other payloads

on a rocket [1]. The rideshare approach implicate severalmission constraints determined by the primary payload. E.g.mission parameters such as orbit, launch schedule and launchdestination etc. An alternative approach to rideshare that hasreceived increasing attention the last decade is dedicated air-launches due to their benefits over rideshares. Dedicated air-launches allows the mission parameters to be determineddirectly by the client instead of the main payload.

Further, the mobility of the launch platform of dedicatedair-launches offers high geographical flexibility and enablesoptimized injection into the desired inclination of the targetorbit.

At about 10 km altitude the density of the atmospherehas decreased to approximately 25 % of the density at sealevel. Thus, as the carrier aircraft serves as the reusable firststage it carries the launch vehicle through the densest partsof the atmosphere and this will drastically reduce the realizedvelocity losses caused by drag on the launch vehicle.

By releasing the launch vehicle at an altitude the launchis less dependent on weather conditions which is the mostcommon reason for delayed launches. The reason for this isthat the launch takes place above the troposphere which iswhere most weather phenomenon takes place [2].

However, dedicated air-launches also includes drawbacks.The main challenges are the limitations of the carrying aircraft,such as the geometrical constraints and the restricted take-offweight of the launch vehicle, and the comparably high launchcosts compared to rideshares [3].

The gradual enhancement in instrument performance in-creases the number of operational applications accessiblesuch as communications, intelligence gathering, early warning,surveillance, different types of observation, etc. Therefore, theemerging market for nano and micro satellites is a candidateto be the potential prime market for air launches [4] to clientswhere the benefits overweigh the disadvantages. A strategicrole attributed to the launch vehicle would be the possibility toquickly launch a small satellite with immediate access whichmight be of interest to defense organizations.

The main objective of the present thesis was to investigatethe amount of payload that can be put into a low Earth orbit(LEO) by an air-launched launch vehicle and to propose a lowfidelity model of the potential launch vehicle. The model foraircraft platform planned to carry the launch vehicle was theSweden produced fighter jet Saab JAS 39 Gripen.

The launch vehicle was supposed to be compatible withthe carrying platform without major modifications. Therefore,the geometry and weight constraints of the launch vehiclewere based on utilized weapon systems. The key characteristicattributes of the launch vehicle aimed to be fast responseability, storability and simplicity. The overall essence was touse hardware based on open standards and commercial off-the-shelf products.

II. OBJECTIVE

The primary objective of the present thesis was to inves-tigate the maximum payload mass that can be carried by anair-launched multistage-stage launch vehicle to a circular polarorbit at an altitude of 500 km utilizing direct ascent. Thecircular polar orbit was investigated because missions to thisorbit results in no velocity gain or loss from Earths rotation[5]. Mainly because Earths rotation is perpendicular to thespecific orbit and cannot thereby contribute. Also, the polarorbit is similar to the Sun synchronous orbit (SSO) which hasgreat usability due to the instant illumination from the Sun.The SSO also enables excellent coverage of the entire planeton a regular basis and is used by Earth observation satellites[6].

Saab JAS 39 Gripen was selected as the potential carrierplatform to launch the launch vehicle at an altitude of 12 kmand with a velocity of Mach number, M “ 1.2. At 12 km


altitude is where the end of the so-called wet troposphere islocated [7]. For launch vehicle integration, the idea was to useexisting pylons and use the similar communication protocolsas for flight proven weapon systems in use today. This willfacilitate no need for modifications to the carrier aircraft andthis aspect would benefit the accessibility of the launch vehicleand allow launches in short notice.

Therefore, the geometry constraints were derived from theTaurus KEPD 350E which is an air-launched cruise missilecurrently compatible with the aircraft platform. The maximumweight of the launch vehicle was set to 1400 kg and themaximum overall length to 5.1 m with a maximum diameterof 1 m according to the Taurus Systems [8].

III. THESIS STRUCTURE

Due to the highly complex and interconnected nature ofthe launch vehicles it was necessary to incorporate severaldifferent disciplines. Since the design parameters of the launchvehicle design are restricted by constraints and influence eachother, an optimization algorithm was essential to implement.The developed optimization process is illustrated in Figure1. The modules of the optimization process are described inindividual sections of the thesis explaining the implementationof the different disciplines.

As each module takes a set of input values the determinationof these were both based on values from literature studiesand from values calculated based on theory. The optimizationprocess is an iterative process and therefore values calculatedby the modules were passed as input values to the nextmodule according to the process flowchart in Figure 1. Themotivations for the input values can be found at the end of eachsection describing a module. Now, a summary of the upcomingsections is presented. Firstly, fundamental equations of rocketdynamics are presented followed by an analysis of the velocityrequirements.

Next, the methods used to optimize the mass distribution ofthe launch vehicle are described, along with details about itsimplementation in the optimization process.

Thirdly, the propulsion system is reviewed including overallvolume budget and nozzle configuration. Then the implemen-tation of the trajectory analysis is explained.

Lastly, different computation implementations will be pro-posed and the most significant results will be presented andcommented.

IV. GENERAL EQUATIONS

This section briefly explains the general equations usedthroughout the report starting with thrust. Thrust is the forceproduced by the rocket propulsion system. It is the reactionforce, acting on the vehicles structure, originated from theejection of propellant at high velocities. The force is given bythe thrust equation [9],

T “ 9mve ` ppe ´ paqAe (1)

where 9m is the mass flow rate at which the fuel mass isexpelled, ve is the exhaust velocity, pe is the pressure at theexit plane, pa is the ambient pressure and Ae is the area of

the exit plane. The first term is known as the momentumthrust and the second term is the pressure thrust. The specificimpulse represents thrust per unit propellant mass flow rate.The specific impulse is defined as,

Isp “T

g0 9m“veg0`ppe ´ paq

g0 9mAe, (2)

where g0 is the gravitational acceleration. In last expression onthe right-hand side the thrust has been substituted by equation(1). The gravitational acceleration, g0, is only introduced toachieve the unit of seconds and this is the unit used throughoutthe report. The specific impulse tells how well the propellantflow rate is converted into thrust.

Another important parameter is the thrust coefficient. Thethrust coefficient CF represents the amplification of thrust dueto the gas expansion in the supersonic nozzle as compared tothe thrust that would be exerted if the chamber pressure wasto act over the throat area only. The coefficient is defined as,

CF “T

Atpc(3)

where At is the throat area and pc is the combustion chamberpressure.

The characteristic velocity c˚ is a function of propellantcharacteristics and combustion chamber properties. It is inde-pendent of the nozzle characteristics and is defined as

c˚ “Atpc

9m. (4)

The characteristic velocity can be used as a figure of meritwhen different propellant combinations are compared for com-bustion chamber performance. The value basically describeshow good the chamber is to produce high pressure.

By re-arranging equation (4) it can be seen that the charac-teristic velocity is the transfer function between the mass flowrate and the chamber pressure

9m “1

c˚Atpc (5)

And by combining equation (5) and (3) and solving for thethrust gives,

T “ 9mc˚CF . (6)

Furthermore, by re-arranging and combining with previousexpressions we find that,

T

9m“ c˚CF “ Ispg0. (7)

This means that specific impulse is a value that describes theproduct of pressure generation c˚ and high velocity productionCF .

V. VELOCITY REQUIREMENTS

The velocity requirement for space launch systems, oftencalled delta-v, is the result from the difference between theinitial and the terminal speed. To determine the speed gainedby a rocket in an environment free from aerodynamic andgravitational forces Tsiolkovsky’s rocket equation can be used[10],

∆v “ ve ln

ˆ

m0

mf

˙

, (8)


Nozzle Configuration Module

?

Stage Optimizer Module

Total rocket mass

Structure masses

Required orbital velocity

Velocity losses

Exhaust velocities

Optimal structure masses

Optimal propellant masses

Maximum payload mass

Chamber conditions

Propellant characteristics

Operational altitudes

Initial T/W

Rocket geometry

Exhaust velocities

Stage thrust

Stage burn times

Stage mass flow rates

Start

Initial velocity, altitude and flight path angle

Final orbital velocity, altitude and flight path angle

Velocity losses

Operational altitudes Trajectory Module

No

?

Mass Trade-Off Module

|Required orbital velocity - Velocity| < Tolerance

|Injection altitude - Altitude| < Tolerance

Yes

Pitch Maneuver Module

Yes

Finish

Optimal design

No

?No

|New velocity losses - Old velocity losses| < Tolerance

|New exhaust velocities - Old exhaust velocities| < Tolerance

Yes

Fig. 1. Flowchart describing the optimization process. Each module is individually described in the section with the same name.

where ∆v is the acquired velocity, ve is the effective exhaustvelocity, m0 is the initial mass and mf is the final mass. Thus,the rocket will acquire the highest velocity when the entirestorage of fuel is combusted and exhausted.

However, during the ascent phase the launch vehicle isexposed to counteracting forces. The most significant of theseforces are gravitational and aerodynamic forces [11]. Firstly,the losses from gravity forces were calculated by,

∆vgravity “

ż τ

0

gptq dt, (9)

where gptq is the gravity at time t. The total loss from gravitycan be obtained by evaluating the integral from the initial stateto the terminal state.

The two most significant aerodynamic forces acting on thelaunch vehicle are drag and lift. Out of these the contributionfrom drag effects the rocket the most. If the thrust vector isparallel to the velocity vector of the rocket there is no lift atall. However, if the thrust vector is not parallel the lift exertsa bending moment on the rocket which it must be able towithstand. At the same time the force from lift can be usedin a beneficial manner. At this conceptual stage of the launchvehicle design, the thrust vector was assumed to be paralleland lift was therefore neglected. The velocity losses from dragwas calculated using,

∆vdrag “

ż τ

0

Dptq

mptqdt, (10)

where Dptq is the drag acting on the rocket at time t and mptqis the instantaneous rocket mass. The equation for drag,

Dptq “1

2ρptqv2CDA, (11)

constitutes the local density of the surrounding medium ρptq,the drag coefficient CD, the cross-sectional area of the rocketA and the magnitude of the rocket velocity v relative to thesurrounding medium. The velocity losses from gravity anddrag were added to the combined term,

∆vlosses “ ∆vgravity `∆vdrag. (12)

The terminal velocity in this case was the required orbitalvelocity for the payload to be able to remain in orbit. Thisvelocity was calculated as [12],

vorbit “

d

µ

ˆ

2

r´

1

a

˙

, (13)

where µ is the gravitational parameter, r is the distance fromthe center of the Earth to the payload and a is the semi-majoraxis of the elliptical orbit. The terminal orbit was chosen tobe circular which implies, a “ r. By using a circular orbit thevelocity becomes constant, this is not the case in an ellipticalorbit. Thus, the terminal velocity becomes,

vorbit “

c

µ

r. (14)

The velocity was calculated to vorbit “ 7612.6 m/s for acircular orbit at 500 km altitude where the centrifugal forceprecisely balances the Earth’s gravitational attraction of the


rocket. The required velocity, delta-v, needed to be deliveredby the launch vehicle is given by,

∆vmission “ ∆vlosses ` vorbit ´ va{c, (15)

where va{c is the initial velocity of the rocket transferredfrom the aircraft. The initial velocity was calculated using therelation,

va{c “Mc, (16)

where M corresponds to the Mach number and c is the localsound velocity. According to the Atmospheric Model IX-Athe local sound velocity at 12 km altitude is c “ 295.1 m/s.By using equation (16) the initial velocity of the aircraft wascalculated to va{c “ 354.1 m/s.

VI. STAGE OPTIMIZER MODULE

A. Mass Ratios

Fig. 2. Illustration of the definitions of the masses of a tandem three-stagelaunch vehicle. The sub-indices 0 and f corresponds to initial and finalrespectively. s is the index for structural, p is for propellant and pl is forpayload. Other indices represent stage or sub-rocket index. Not to scale.

This section describes the definitions of the masses makingup the launch vehicle. Rocket stages are either stacked ontop of each other, called serial staging, or attached next toeach other, called parallel staging, and they are numberedas the order of firing. When serial staging is implementedthe stages burn successively in opposite to parallel stagingwhere the stages burn simultaneously instead. There existslaunch vehicles that use both serial and parallel staging. Tokeep the cross section of the rocket to a minimum due to thegeometrical constraints of the aircraft serial staging was thechosen method.

The total mass of the rocket is divided into different partsas can be seen in Figure 2. The structure mass of the ith stage

is denoted ms,i, the propellant mass is denoted mp,i and thesum of these make up the total mass of that specific stagedenoted mi. The initial mass of each stage a.k.a. gross massis denoted m0,i and corresponds to the mass before ignition ofthe stage. After the stage propellant is consumed completelythe remaining mass is called final mass and is denoted mf,i.For a rocket with N stages the mass ratio for the ith stage isgiven by [13],

Ri “m0,i

mf,i»

m0

ms `mpl(17)

The payload ratio is a that describes how much of the weightis payload compared to other masses. For the total rocket thepayload ratio is defined as,

π “mpl

m0,1 ´mpl“

Nź

i“0

πi (18)

The payload ratio for each stage is denoted πi. The payloadfor each stage mpl,i consists of all the masses above that stage.

πi “mpl,i

m0,i ´mpl,i“

m0,i`1

ms,i `mp,i, (19)

and for the last stage m0,i`1 “ mpl.In all equations ms is the structural mass, mp is the

propellant mass, mpl is the payload mass and m0 is the initialmass defined as m0 “ ms `mp `mpl. The structural masscoefficient ε is defined as the fraction of structure mass dividedby the total mass of the stage. Hence, it is a measure of howmuch of the weight is support structure. For a N stage rocketthe structure mass coefficient was calculated as,

εi “ms,i

mp,i `ms,i“ms,i

mi. (20)

By combining the payload ratio and the structural coefficient,the mass ratio for the ith stage can be written as,

Ri “1` πiεi ` πi

. (21)

B. Multi-staging

As the rocket progresses in its trajectory to orbit largevolumes of fuel are being burned and thus the remaining rocketconsists of an increasing amount of empty structure mass.Therefore, much of the energy is spent just by acceleratingthe mass of empty structure. The forfeited energy would servebetter purpose by accelerating the payload instead.

By discarding structure as it gets empty this problem canpartly be avoided. This can be facilitated by dividing therocket into several stages containing their own propellant andstructure and when the propellant is consumed the stage isjettisoned. Launch vehicles used in space applications aremostly multistage rockets composed of two up to four stages.

To visualize the performance benefits of multi-staging,consider the case of N rocket stages with equal structure ratio,payload ratio and exhaust velocity. By inserting the mass ratio(21) into the rocket equation (8) and solving for the payloadratio (19) the variation of the payload ratio as a function ofthe performance parameter ∆v{ve can be plotted. This can beseen in Figure 3.


0 1 2 3 4 5 6 7 8 9 10

v / ve

10-3

10-2

10-1

100

N=1 N=2 N=3 N=4

Fig. 3. Variation of the total payload ratio π of a multi-stage rocket withperformance parameter ∆v{ve. The structure coefficient is set to be εi “ 0.15for all stages.

It is clear from the graph that an increase in the numberof stages for a fixed payload mass leads to a higher value ofthe performance parameter and thus a higher velocity. Hence,staging offers a significant increase in performance over thesingle-stage rocket, for an identical total payload ratio.

However, more stages also increases the complexity of therocket and thus costs. It can be seen in the graph that theperformance gain becomes smaller for each increase in numberof stages. A comparison of feasible payload mass between atwo-, a three- and a four-stage rocket with similar features wasconducted in the computations described in Computations XI.

1) Structural Masses: To be able to compute the massratios, equation (21), with the payload mass as the variableit was essential to estimate the structural mass coefficient ofeach stage. It varies typically between 5 and 15 % dependingmainly on the size of the rocket, propellant type and buildingmaterials [3].

For comparison purposes the Pegasus system was examined.Pegasus is an air-launched rocket launched from a heavilymodified aircraft. The Pegasus system is a mature and flightproven launch system that has achieved consistent accuracyand dependable performance. The Pegasus XL is a refinedmodel of the original Pegasus and was developed as anincreased performance design evolution to support NASA andthe USAF performance requirements. Pegasus XL is now thebaseline configuration for all commercial Pegasus launches.Although, the Pegasus XL is larger and heavier than the rocketpresented in this thesis it is still a relatively small air-launchedsystem. Therefore, as an estimate the structural masses werederived from the Pegasus XL rocket.

By using inert and propellant mass values from PegasusXL user guide [14], the structural mass for each stage wascalculated using equation (13). The result for the first stagewas 8.4 %, second stage 9.6 % and third stage 14.1 %. Thesevalues were considered conservative due to the evolution inmaterial design since the Pegasus XL development. Further, acomparison with decreased structural mass values of 10 % to

the original values was performed.

C. Stage Optimization Procedure

The objective of the stage optimization procedure was tofind the optimal mass ratios of the stages, equation (21), thatwould maximize the payload mass for a fixed gross lift-offmass of the launch vehicle. To find the optimal stage massesthe method of Lagrange multiplier [15] was utilized. Themethod of Lagrange multipliers is a strategy used for findingthe local maxima or minima of a function subject to equalityconstraints. From the solution of optimal mass ratios the totalmass, the structural mass and propellant mass for each stagecould be calculated by performing back substitution of themass ratios defined in the section Mass Ratios VI-A.

The optimal mass ratios for an N stage rocket can be foundby maximizing the fraction of payload mass divided by theoverall total rocket mass,

mpl

m0,1. (22)

Maximizing this fraction is essentially the same as to minimizethe inverse equation where the total mass is the numeratorand the payload mass is the denominator instead. The re-arrangement was done for simplicity reasons when the prob-lem was implemented in MATLAB®.

Further, the re-arranged expression can be expressed interms of structure coefficients and mass ratios as,

m0,1

mpl“

Nź

i“1

p1´ εiqRi

1´ εiRi. (23)

The result of the natural logarithm of a monotonic functionis also a monotonic function. Thus, an optimization of thenatural logarithm of a function implies an optimization of thefunction itself [16]. Therefore, to simplify the calculations, thenatural logarithm of both sides of equation (23) and expandingthe right-hand side yields,

ln

ˆ

m0,1

mpl

˙

“

Nÿ

i“1

lnp1´ εiqRi

1´ εiRi

“

Nÿ

i“1

rlnp1´ εiq ` lnpRiq ´ lnp1´ εiRiqs. (24)

Equation (24) constitutes the form of the objective functionto be minimized. The constraint of the optimization was setto the burnout velocity since it is determined by the missionspecifications. The launch vehicle must provide enough delta-v to be able to put the satellite into the desired orbit. The totalvelocity provided by the launch vehicle can be expressed asthe sum of the velocity provided by each stage for a serialstaged rocket. By summing the velocity contributions of allstages described by equation (8) we get,

Nÿ

i“1

ve,i lnRi. (25)


The total delta-v required is given by equation (15) and thusthe constraint equation becomes,

Nÿ

i“1

ve,i lnRi “ ∆vmission ñNÿ

i“1

ve,i lnRi´∆vmission “ 0.

(26)Now, by introducing the Lagrange multiplier λ, the augmentedobjective function was expressed as the linear combinationof the objective function (24), the constraint (26) and theLagrange multiplier according to,

h “Nÿ

i“1

rlnp1´ εiq ` lnpRiq ´ lnp1´ εiRiqs`

λ

˜

Nÿ

i“1

ve,i lnRi ´∆vmission

¸

(27)

The goal was then to find values for Ri when h was stationary.This will extremize equation (24) and hence mpl for the pre-scribed values of ∆vmission, ve,i, εi and m0,1. The stationarycondition was fulfilled when the partial derivatives of h wasequal to zero. Performing the derivative and set equal to zerogives,

Bh

BRi“

1

Ri`

εi1´ εiRi

´ λve,iRi

“ 0, i “ 1, 2, ..., N. (28)

and the derivative with respect to the variable λ gave theconstraint equation. As the primary case studied was a three-stage design, from this point on N “ 3. Solving equation (28)for the mass ratios Ri gives,

Ri “λve,i ` 1

λve,iεi, i “ 1, 2, 3. (29)

Substituting equation (29) into constraint equation (26) gives,3ÿ

i“1

ve,i ln

ˆ

λve,i ` 1

λve,iεi

˙

“ ∆vmission (30)

Equation (30) was solved iteratively for λ by numericalmethods. When λ was obtained, it was back substituted intoequation (29) and the mass ratios for all stages could be solvedfor. By returning to the definition of the mass ratio, equation(21), the masses could be obtained as,

R1 “m0,1

ms `mpl“

m1 `m2 `m3 `mpl

ε1m1 `m2 `m3 `mpl(31)

ñ m1 “R1 ´ 1

1´ ε1R1pm2 `m3 `mplq (32)

In the same manner, the remaining masses were obtained as,

m2 “R2 ´ 1

1´ ε2R2pm3 `mplq (33)

andm3 “

R3 ´ 1

1´ ε3R3pmplq. (34)

From these masses, the structure mass and propellant mass foreach stage were obtained by using,

ms,i “ εimi (35)

andmp,i “ mi ´ms,i. (36)

To verify that the optimization of the objective functioncorresponded to a maximum the partial derivatives of equation(28) were examined. Firstly, simple calculations show that,

B2h

BRiBRj“ 0, i, j “ 1, 2, 3 and pi ‰ jq. (37)

Secondly,

B2h

BR2i

“λve,ipεiRi ´ 1q2 ` 2εiRi ´ 1

pεiRi ´ 1q2R2i

. (38)

If the optimization of the objective function corresponds to aminimum it follows that equation (38) needs to be positive forall i which implies the following condition,

λve,ipεiRiq ´ 1q2 ` 2εiRi ´ 1 ą 0. (39)

The described method of calculations was implemented inMATLAB® and is referred to as the Stage Optimizer Module.Hence, by providing the required initial values to the StageOptimizer Module the optimal masses were returned as outputvalues calculated according to this method.

D. The Module Input Values

The total mass, m0,1 was set to 1400 kg according to thecondition set by the objective.

The initial velocity losses were based on losses due togravity and drag and taken from literature. According to Leyet al. [3] a typical value for gravity losses is 1000-1500 m/sand for drag losses the value is 100-150 m/s for a conventionalground launch to a LEO at 200 km. The arithmetic mean valueof the higher and lower bounds gives ∆vlosses “ 1375 m/s.Although, these values are representative for a ground launch,the target orbit is significantly lower than the objective orbit ofthe present thesis and the mean value was therefore used. Thisvalue was of minor importance since the value got updated bythe Trajectory Module IX every iteration and only served asthe initial value.

The required orbital velocity was calculated using equation(14). The objective states the orbit altitude to be 500 km whichis a conservative value for a LEO which according to [11]typically is below 500 km altitude.

The initial exhaust velocity was assumed to be 2400 m/s andthis value was also of minor importance since it got updatedby the Nozzle Configuration Module VIII-A and only servedas the initial value.

VII. PROPULSION SYSTEM

For the propulsion system only chemical systems wereconsidered. Chemical propulsion systems are based on theprinciple of converting chemical energy contained in the pro-pellant to kinetic energy of the exhaust gases [9]. The energyfrom the exothermal combustion reaction of the propellantchemicals results in a high temperature product of the reactiongases. Chemical propulsion systems can be categorized intotwo types, hot gas and cold gas. Hot gas systems are the most


common type of propulsion systems for space applications[17] and was therefore investigated in this study.

Propulsion systems operating with hot gases comprises ofliquid-, solid- and hybrid propellant rockets. For a liquidrocket the fuel and oxidizer are stored in separated thin-walledtanks at low pressure. Before combustion they pass throughturbine-driven pumps and are then injected into the combustionchamber. In the combustion chamber, they are ignited and burnat high pressure.

In contrast, solid propellants consist of a premixed combi-nation of fuel, oxidant and binder called propellant grain. Thegrain is stored within the combustion chamber itself and burnswhen the surface is sufficiently heated.

In both liquid and solid systems, the gases then proceedto be accelerated through a converging-diverging nozzle fromwhich they are ejected at high velocities and thereby impart-ing momentum to the launch vehicle. Converging-divergingnozzles consist of a converging section, a constriction orthroat section, and a conical or bell-shaped diverging section,this is described in the Nozzle Configuration Module VIII-A.The maximum exhaust velocity is achieved when all enthalpycontained in the gas at the inlet of the nozzle is converted intokinetic energy by the expansion inside the nozzle.

The selection of hot gas type was based on merits anddisadvantages of liquid propellant rocket engines and of solidpropellant rocket motors from literature. The selection processof the associated rocket propulsion systems is an integral partof any overall vehicle design effort.

Solid propellant rocket motor consists of fewer parts, inparticular moving parts, compared to a liquid propellant en-gine which results in a simpler design and can therefore beproduced at a lower cost. Solid propellant has higher overalldensity and this facilitates more compact design and smallerlaunch vehicle and thus less drag. The storability of solidpropellant is another advantage, it can be stored for 10 to30 years [11]. Therefore, the most eminent advantages ofsolid propellant considered for the launch vehicle were simpledesign, less overall weight for low total impulse application,storability and higher overall density.

On the downside, solid propellants in general have lessspecific impulse than liquid propellant and the propellantgrain cannot be fully inspected prior to takeoff. Ambienttemperature variations during storage or in repeated flight cancause cumulative grain damage and the launch vehicle may nolonger be safe to ignite. Mainly because a possible crack in thegrain might result in an increasing burning area which wouldcause excessive combustion pressures when ignited. Incautioushandling of the propellant grain can cause inadvertent self-ignition. Another difficulty is that it is not possible to start andstop the rocket motor after ignition. Once the motor is ignitedthe solid propellant grain burn until the fuel is consumed whichmakes launch vehicle control more difficult.

Liquid propellants on the other hand, obtain higher chemicalspecific impulse compared to solid propellant which increasesthe payload carrying ability, launch vehicle velocity incrementand the attainable mission velocity. A liquid propellant drivenengine can be arbitrarly throttled, stopped and restarted whichfacilitates better overall control of the vehicle and its velocity.

One of the main disadvantages using liquid propellant isthe relatively more complex design. The liquid propellantengine has more moving parts and components which mayfail. Cryogenic propellants cannot be stored for longer periodsand the propellant loading must occur just prior to takeoff.Thus, this requires cryogenic propellant storage facilities inthe proximity of the takeoff strip. Spills or leaks of certainliquid propellants can be hazardous, corrosive, toxic and causefires. Liquid engines have more overall engine weight for mostshort-duration, low total-impulse applications. Lower averagepropellant density combined with separate pressurizing gasstorage and the relatively inefficient volume occupation of theengine components require more volume.

Although liquid propellant provides a higher value of thespecific impulse which implies a larger payload mass a solidpropellant system was chosen for the design. This decision wasbased on the benefits of the simplicity of the system, higherdensity of the propellant (and thus a more compact design)and the storability aspect. The benefits of solid propellant wereconsidered to overcome the ones of liquid propellant and suitthe launch vehicle better. Due to the simpler overall design theproduction and maintenance costs of a solid propellant systemwould more likely be less.

A. Solid Propellant

A solid rocket motor roughly consists of a propellant grain,a motor case, an igniter and a nozzle. The grain is the solidbody of the hardened propellant and typically accounts for 82to 94 % of the total rocket motor mass [11]. Thus, the typicalstructure mass of the solid rocket motor is about 6 to 18 %.These values matches well the structural values determined inthe Structural Masses VI-B1 of the Stage Optimization Modulesection.

Solid propellants have historically been grouped into twotypes, double-base or composite type. Double-base propellantswere the first production propellants and subsequently thedevelopment of polymers as binders made the compositepropellant types feasible. Modern processed propellants areclassified into even smaller subgroups of the two types. Todaymost solid rocket motors in production use composite-typepropellants and was therefore considered most suitable for thedesign. Propellants, in general, are often tailored to and clas-sified by specific applications of the launch vehicle. Even forthe same propellant type the fabrication processes are usuallynot identical. There are no single simple generalized processflowsheet or fabrication technique to be used. Most launchvehicles use different manufacturers, motor types, sizes andpropellant formulation. However, summarized characteristicsof commonly used composite type propellant was used as anintroductory step of the propellant analysis conducted in thisthesis.

Composite solid propellants consist of a heterogeneous mixof crystalized oxidizer and powdered fuel held together by asynthetic rubber binder that encases and supports them me-chanically. Ammonium perchlorate (AP) is the most commonoxidizer, constituting 60–86 % of the solid propellant. Thenecessity of carrying an oxidizer within the rocket is since the


rocket must be able to operate outside the atmosphere of theEarth. For fuel powdered spherical aluminum (Al) is the mostcommon. The fuel increases the heat of combustion, the overallpropellant density, the combustion temperature and thus thespecific impulse. The concentration of Al fuel in composites isusually about 5–20 %. Finally, the polymeric binder constitutes12–15 % of the composite and competes with the Al forthe oxidizer. Hydroxyl-terminated polybutadiene (HTPB) asthe polymer binder allows a higher solids fraction (i.e. morealuminum) and relatively good physical properties at thetemperature limits.

The characteristic model of the propellant used was ahigh performance, high specific impulse composite containingabout 70 % AP, 20 % Al and 10 % HTPB as polymer binder.The density of this type of composite propellant combinationis approximately 1800 kg/m3 and this value was used for thepropellant density in all computations. The molecular weightof the formulation was assumed to be 29.9 kg/mol accordingto table values found in [11]. Huggett et al. [18] states thatmost hot rocket gases have a specific heat ratio γ near 1.2and that value was used. The influence of variations in thespecific heat ratio on the specific impulse is not as largeas those from changes in chamber temperature or molecularmass. Once the solid propellant ignites, the AP and Al reactexothermically and produce smaller hydrocarbon particles thatreact with chlorine dioxide species in the gas-phase and thiscause a temperature rise to approximately 2800-3500 K [19].Due to the rich concentration of Al, 3300 K was used as theconstant chamber temperature for all computations.

B. Internal Ballistics

This part describes the propellants combustion characteris-tics such as burning rate, burning surface, and grain geometry.When the solid propellant grain is ignited, the burning surfacerecedes perpendicular to the ignited surface. The burningrate describes how much the burning surface decreases inthe perpendicular direction per unit time. The burning areais proportional to the chamber pressure and thus the thrustproduced [9]. Therefore, the geometry of the cavity of thegrain where the surface is ignited has a considerable impacton the thrust profile. If the burning surface remains constant aneutral thrust profile is achieved. If the burning area increasesthe thrust increases and the thrust profile becomes progressiveand if it decreases the thrust profile becomes regressive.

If the grain does not have a cavity it is called end-burninggrain which produces a neutral thrust profile with ratherlow thrust since the cross-sectional diameter often is smallcompared to the length of the grain. Using a cavity oftenresults in a larger burning area than for an end-burning grainand thus creates a larger value of thrust.

The produced thrust largely depends on the geometry of thecavity and thereby a progressive, regressive or neutral thrustprofile can be achieved by design. By designing the cavity inshape of an eight-point star the propellant grain burning areais approximately constant during operation which means thata neutral thrust profile is achieved. In general, a star shapedcavity exposes a larger burning area compared to a cylindrical

cavity when occupying an equal amount of volume. This givesthe eight-point star shaped cavity the advantage of highervolumetric loading fraction of the propellant grain inside themotor case. In this study constant burning rate and constantmass flow was implemented and this feasibly corresponds toan eight-point star shape of the cavity. Based on characteristicdata from literature [11] the cavity of the solid propellant grainvaries from 7.4 % to 11.3 % and 10 % was used in the presentthesis.

It was assumed that the entire grain area was ignited in-stantaneously and the transient behavior to the constant steady-state chamber pressure was negligible. The mass of the igniterwas also assumed to be small and therefore negligible. Further,the chamber was assumed to be an adiabatic system with auniform total pressure throughout the cavity and the differencebetween the static- and total- pressure was negligible. Asreference chamber operating pressure the commonly used1000 psia or 6.895 MPa was used for all computations.

C. Motor Configuration

The solid propellant grain can be either case-bonded orcartridge-loaded to the motor case of the launch vehicle [20]. Ifthe grain is case-bonded the grain is usually cast into the casewith adhesion between the grain and the case. The primaryadvantage of case-bonded design is that the propellant grainact as a thermal insulator between the combustion chamber andthe case and thus keep the case relatively cool during operation[9]. This facilitates lighter case structures since thinner wallsare permissible. Cartridge-loaded is when the grain is formedseparately and then assembled into the motor case. The mainadvantage with this approach is that the freestanding grainscan be replaced when the propellant grain has aged exces-sively. Cartridge-loaded grains is often cheaper and are easierto inspect. However, case-bonded grains give slightly betterperformance, less inert mass and better volumetric loadingfraction. But, case-bonded grains are exposed to a higher levelof stress and often more expensive to manufacture. Based onthis a case-bonded grain would be preferred in the design ofthe present rocket mainly due to the better performance aspect.

VIII. NOZZLE CONFIGURATION MODULE

A. Nozzle Configuration

Most modern rockets which employ hot gas combustionutilizes de Laval nozzles and therefore a supersonic nozzle ofde Laval type was implemented to the design. The equationsrepresent an idealization and simplification of full three-dimensional equations of real aerothermal-chemical behav-ior. The nozzle analysis was done with the assumption ofquasi one-dimensional gas flow. To compensate for the one-dimensional assumption a single correction factor was used.The correction factor was multiplied with the momentumthrust term of thrust equation (1) to compensate for the non-axial-outlet. The correction factor is given by Sutton et al. [11]as,

λ “1

2p1` cosαq, (40)


where α is the half angle of the divergent part of the nozzleillustrated in Figure 4. For a visualization of the impact ofthe correction factor on the momentum thrust, see Figure 5.The length of the convergent and divergent part of the conicalnozzle was calculated using,

L “re ´ rttanα

(41)

where re is the radius of the exit plane and rt is the throatradius. The combustion gas was assumed to behave as an idealgas and the gas flow was assumed to be isentropic. Whenthe propellant was ignited the gas flow was assumed to besteady during full operation of the stage. Due to the high flowvelocities, the gas flow was assumed to be compressible. Withthese concepts and assumptions, the thrust produced by therocket is given by [11],

T “

Atpc

$

&

%

Γ

g

f

f

e

2γ

γ ´ 1

«

1´

ˆ

pepc

˙

γ´1γ

ff

`ppe ´ paq

pc

AeAt

,

.

-

,

(42)where At is the nozzle throat area, Ae is the nozzle exit area,pc is the nozzle entry total pressure, pe is the exit pressureand pa is the ambient pressure of the surrounding medium.The Γ corresponds to a function of the specific heat ratio γand is defined,

Γ “ γ

g

f

f

e

ˆ

2

γ ` 1

˙

γ`1γ´1

. (43)

The ratio between the exit area and the throat area in thesecond term of equation (42) is called the expansion ratio andis denoted,

ε “AeAt. (44)

According to Sutton et al. [11] the inverse expansion ratio canbe expressed as,

1

ε“

ˆ

γ ` 1

2

˙1{pγ´1qˆpepc

˙1{γ

g

f

f

e

γ ` 1

γ ´ 1

«

1´

ˆ

pepc

˙γ´1{γff

.

(45)By combining equation (44) and (45) the throat area can beexpressed as,

At “

Ae

»

–

ˆ

γ ` 1

2

˙1

γ´1ˆ

pepc

˙1{γ

g

f

f

e

γ ` 1

γ ´ 1

«

1´

ˆ

pepc

˙

γ´1γ

ff

fi

fl .

(46)Based on expression (46) the optimum throat area could bedetermined for a chosen value of the exit pressure pe by usingfixed values for the specific heat ratio and chamber pressuredetermined in the Propulsion System VII section. However,since the thrust (42) is incrementally proportional to the throatarea it must also be sufficiently large to produce the requiredthrust.

From continuity 9m is always equal to the mass flow atany cross section within the nozzle for steady flow. Therefore,

when the gas velocity reaches sonic conditions and becomeschoked, which occurs in a supersonic nozzle, the mass flow isgiven by,

9m “ Atpcγ

d

r2{γ ` 1spγ`1q{pγ´1q

γRTc. (47)

where R is the universal gas constant and Tc is the chambertemperature. The value used for the chamber temperaturewas based on literature values described in the PropulsionSystem VII section. The mass flow through the nozzle is thusproportional to the throat area At and the chamber stagnationpressure pc.

The burnout time of each stage was calculated, based on thepropellant masses calculated by the Stage Optimizer ModuleVI, as,

tb,i “mp,i

9mi(48)

where mp,i is the propellant mass and 9mi is the mass flow ofthe ith stage.

a

re

L

rt

Fig. 4. The figure illustrates the cross section in the axial length direction ofa divergent-convergent nozzle. In the figure the length of the divergent partof the nozzle is denoted L, the exit plane radius re, the throat radius rt andthe half angle α.

B. Under- and Over-expanded Nozzles

For optimum gas expansion inside the nozzle it is desirableto match the exhaust pressure pe with the ambient pressurepa. However, during the ascent phase the launch vehicleexperience a range of altitudes. With an increasing altitude theambient pressure is decreasing and therefore the nozzle canonly perform optimally at a certain specific altitude, calledthe design altitude of the nozzle. This might be counterintuitive when examining equation (1) since an increasedvalue of pe would increase the thrust. However, the nozzleexit pressure and velocity are closely and adversely coupledthrough the amount of nozzle expansion [21]. Because theflow is supersonic the exit velocity will increase and the exitpressure decrease as the expansion ratio Ae{At is increasedand vice versa is true as the expansion ratio is decreased.


0 5 10 15 20 25 30

Half angle [degrees]

0

10

20

30

40

50

60

70

80

90

100

L/(

r e -

rt)

93

94

95

96

97

98

99

100

Effic

iency [%

]

Length fraction

Efficiency

Fig. 5. The left side axis represents the length of the divergent part of thenozzle divided by the difference between the exit plane radius and the throatradius as a function of half angle. The axis on the right side represents theefficiency of the nozzle as a function of half angle due to the assumption ofisentropic one-dimensional flow.

In an under-expanded nozzle, the gases are discharged at anexit pressure greater than the ambient pressure (pe ą pa) thisoccur because the nozzle exit area is too small for an optimumexpansion. Gas expansion is therefore incomplete within thenozzle, and further expansion will take place outside of thenozzle exit. This phenomenon will result in a decrease in CFand Isp compared to optimum expansion. The opposite to anunder-expanded nozzle is an over-expanded nozzle.

In an over-expanded nozzle, the gases exits at lower pressurethan the ambient pressure (pe ă pa) because the exit area istoo large for optimum expansion. The gases are fully expandedinside the nozzle and this also leads to lower values of CF andIsp than for an optimum expanded nozzle. When a nozzle isin an over-expanded state flow separation can take place. Flowseparation or free shock separation is when the flow is fullyseparated from the wall and ensuing instationary, asymmetricforces called side-loads might occur. These dynamic side-loads can damage the nozzle and cause serious failure to thelaunch vehicle and are highly undesired. An illustration of thedifferent scenarios are presented in Figure 6.

Because the ambient pressure change with increasing alti-tude a nozzle with a fixed area ratio is always in a under-or over-expanded state except for the design altitude for thenozzle where pe “ pa. There are several methods to investigateat what value flow separation begin to take place.

The model used to describe at what pressure the flowseparation phenomena takes place was the commonly usedcriteria proposed by Summerfield et al. [22]. It is an empiricalformula based on extensive tests in conical nozzles and itstates,

p{pa « 0.4. (49)

In other words when the exit pressure is less than 40 % ofthe ambient pressure the undesired side-loads might occur.An obvious way to improve the payload mass of launchersis hence to increase the area ratio of the first stage nozzle,

Over-expanded nozzle

Optimum expanded nozzle

Under-expanded nozzle

Fig. 6. Illustrations of exhaust gas behavior under different pressure con-ditions. Over-expansion occurs when the ambient pressure is larger than theexit pressure and the gases are fully expanded inside the nozzle. At optimumexpansion, the ambient pressure is equal to the exit pressure and the gasesare fully expanded at the nozzle exit plane. Under-expansion occurs when theambient pressure is less than the exit pressure and the exhaust gases continueto expand outside of the nozzle.

however, this will at the same time reduce the nozzle exitpressure. In order to prevent flow separation and side-loadslaunch vehicles use area ratios far below optimum, but ensurefull-flowing and thus safe operation. Hence, by allowing flowseparation to some extent would considerably improve thelaunchers payload [21].

To avoid the sidal forces from under-expansion but to getmaximum performance the value of pe “ 0.4pa was chosenas the exit pressure to match for the first stage. This meansthat the nozzle will start to perform in an over-expanded statefor the first portion of its path, then arrive at optimum ex-pansion followed by an over-expanded state for the remainingoperation.

For the second and third stage, the nozzles performs innear vacuum, this made it impossible for under-expansionto occur or to match the ambient pressure since it was notpossible to expand the exhaust gases to a pressure that low.To obtain optimum nozzle performance in near vacuum eitherthe nozzle get too big or the thrust produced is not enough for


the correct expansion ratio. This fact left the nozzles of thesecond and third stage in an over-expanded state during theirentire operation.

1) Truncated Contour Nozzle: The length of the propulsionnozzle is often a significant contributor to the overall length ofthe launch vehicle and thereby the weight. The nozzle designcan be further improved by using a contour nozzle instead ofa straight cone. By using a contour nozzle, also known as bellnozzle, the length of the diverging part of the nozzle can bereduced to about 80 % for a 15° half angle conical nozzleand still have the same or even slightly improved efficiency[11]. The performance enhancements resulting from the useof bell-shaped nozzles with high correction factor values aresomewhat lower in solid rocket motors with solid particles inthe exhaust impinging the nozzle walls. Therefore the slightlyimproved efficiency was neglected.

C. The Nozzle Configuration Procedure

To estimate the size of the different parts of the launchvehicle a simple volume budget was performed. The analysiswas done primarily to not exceed the geometrical constraintsof the carrier aircraft defined in the Objective II section. Thevolume budget calculations were implemented and executedas the first step in the present module. The calculations werebased on a cylindrical geometry. From the maximum lengthand radius defined in the Objective the total volume budgetwas calculated as,

Vtot “ LtotAe. (50)

The volume occupied by the propellant was calculated basedon the propellant mass obtained from the Stage OptimizerModule VI. The propellant mass was then divided by thepropellant density determined in the Propulsion System VIIsection. Thus,

Vprop,i “mp,i

ρprop. (51)

From the Internal Ballistics VII-B the volume of the cavitywas determined to occupy 10 % of the volume intended forthe propellant. Thus, the volume of the cavity was calculatedusing,

Vcavity,i “ 0.1 ¨ Vprop,i. (52)

The volume of the payload fairing and the avionics bay wasassumed to occupy a length of 0.6 m and thereby a volumeof,

Vpayload “ 0.6 ¨Ae. (53)

The payload volume was assumed to fit at least a 4U CubeSatincluding its dispenser system. The dispenser system wouldprovide the attachment to the launch vehicle, protection of thepayload during launch and the release mechanism responsiblefor the satellite release into space. The realized payload fairingwould most likely have sections composed of a cylinder anda ogive part to optimize the volume space and minimize drag.

At this point, the rocket consists of the volumes describedup to this point and in addition a volume dedicated for thenozzles was included. Since the nozzles does not necessarilyfill the dedicated volume space only the value for maximum

allowed nozzle length was of interest. The maximum lengthdedicated for the nozzles was calculated as,

Lnozzles “Vtot ´ Vpayload ´

ř3i“1 pVprop,i ` Vcavity,iq

Ae(54)

Theoretically, an ideal nozzle is extremely long and con-sequently is not practically feasible for rocket applications.The huge length is necessary to be able to produce a one-dimensional exhaust profile [21]. The maximum nozzle lengthwas therefore divided equally between the stages to get anestimate of what performance to be achieved. To achieve amore accurate model, the building material and thus weightof the nozzles needs to be considered and this lies outside ofthe scope of the thesis which treats a conceptual level design.From the volume budget the maximum radius of the rocketwas determined to be 0.5 m and this was set as the constrainton maximum radius for the nozzles. For each stage nozzle aiterative loop was performed. The loop was iterated over inthe range of the radius of the nozzle exit plane. The rangewas set to 0.05 ď re ď 0.5 with an increase of 0.1 cm periteration. First, the nozzle exit plane and throat dimension weredetermined. The nozzle exit plane area was calculated using,

Ae “ πr2e . (55)

The throat area was determined from equation (46) wherethe exit pressure was set according to equation (49) with theambient pressure obtained from the Atmospheric Model IX-Asection and from the result the throat radius was calculatedby using equation (55). The expansion ratio could then becalculated using equation (44). Secondly, based on the totalnozzle length the convergent and divergent part of the nozzlewas calculated using equation (41). The half angle for theconvergent section was set to the commonly used value of30°. Thus,

Lconvergent “re ´ rt

tanpπ{6q. (56)

The part left for the divergent section could then be achievedfrom,

Ldivergent “Lnozzles

# of stages´ Lconvergent. (57)

The half angle for the divergent part could now be calculatedby solving equation (41) for α as,

α “ arctan

ˆ

re ´ rtLdivergent

˙

. (58)

The half angle was then used to calculate the correction factorfor non-axial-outlet flow in equation (40). In the next step thelength of the divergent part of the nozzle was reduced to 80 %of the total initial length regardless half angle. The motivationfor the reduction was based on implementation of a contournozzle instead of a straight cone nozzle explained in sectionTruncated Contour Nozzle VIII-B1. Thirdly, the ideal thrustT and thrust coefficient CF were calculated using equation(42) and (3) respectively. At this stage the momentum thrustterm was reduced by the correction factor λ. Next the overallthrust was reduced 0.5 % for the use of jet vanes according


to Sutton et al. [11]. The thrust values were re-calculated forevery increase of 1 km of the operational altitudes due tothe change in atmospheric pressure. By doing this the thrustperformance over the whole operation was analyzed. Next c˚

was calculated from equation (4) and by multiplying withthe thrust coefficient the specific impulse for the operationalaltitudes were calculated. For all loop iterations over the exitplane radii the minimum and maximum value of the specificimpulse as well as the initial and final thrust, the initial andfinal thrust-to-weight ratios and the mean value of thrust weresaved for subsequent comparison. The mass flow rate wascalculated by using equation (47) and the burnout time of thestage was calculated using equation (48). The iterative processfor changes in the exit plane radii was invoked for all stagesseparately based on the operational altitudes of that specificstage.

D. The Module Input Values

The optimal structure masses, optimal propellant masses andoptimal payload mass were calculated by the Stage OptimizerModule VI.

The values for chamber conditions were set based onthe analysis done in Internal Ballistics VII-B. The chamberpressure was set to 6.895 Mpa and the chamber temperaturewas set to 3300 K. The characteristic propellant values weredetermined in section VII-A. The propellant density was setto 1800 kg/m3, the molecular weight was set to 29.9 kg/moland the specific heat ratio was set to 1.2.

The initial operational altitudes were assumed to be 12km, 90 km and 260 km for the first, second and third stagerespectively. These values were of minor importance (thesevalues only effected the nozzle design in the first iteration)since these values were updated by the Trajectory Module IXbased on the calculated trajectory.

IX. TRAJECTORY MODULE

A. Atmospheric Model

To model the atmosphere a 21-layer standard model [23]for the Earth was used, ranging from sea level to a geometricaltitude of 700 km. The model was used in all computationsof atmospheric and transatmospheric trajectories in the subse-quent sections. The atmospheric model used is based on the1976 U.S. Standard Atmosphere for an altitude 0 ď H ď 86km and 1962 U.S. Standard Atmosphere for an altitude above86 km. The two models have a close agreement in the altituderange 0 ď H ď 86 km but there is a large difference in the twomodels in the exospheric region, which begins at an altitudeof 500 km. However, since the injection altitude was chosento 500 km this was accepted.

B. Gravity-turn Trajectory

A gravity-turn or zero-lift turn is a maneuver used whenlaunching a spacecraft into an orbit. It is a trajectory opti-mization that uses gravity to steer the vehicle onto its desiredtrajectory. It offers two main advantages over a trajectorycontrolled solely through the vehicle’s own thrust. First, the

T

D

V

g

mg

Local horizon

Fig. 7. The forces acting through the center of gravity on the rocket in thelocal horizon frame. It was assumed that the thrust force, T , and the drag,D, always were acting axis-symmetric to the velocity vector, V , of the rocketwhich would result in no lift force. The γ corresponds to the flight path angleand mg to the gravitational force. The forces are not to scale.

X

H

R

h1

h2

h3

Fig. 8. The right-handed vector set h is located at the center of the Earth.The vector h1 is in the downrange direction and h2 is always pointing at therocket. RC corresponds to the Earth radius, H is the height over ground andX is the arc length.

thrust is not used to steer the launch vehicle and therefore moreof the thrust can be used to accelerate the launch vehicle intoorbit. More importantly, the vehicle can maintain low or evenzero angle of attack which minimizes transverse aerodynamicstress and allowing a smaller structure coefficient. The gravity-turn is not the optimal way to minimize the overall ascentvelocity losses [24] but was considered sufficient to get aninsight of the velocity losses.

To obtain an estimate of the velocity losses, mainly fromdrag and gravity, exerted on the launch vehicle a two-dimensional flight path trajectory was implemented. The tra-jectory analysis was performed by a module of the opti-mization algorithm. The development of a fully optimizedtrajectory would demand a mature launch vehicle contourdesign which was outside of the scoop of this thesis. Therefore,the gravity-turn trajectory was considered adequate for this


low fidelity conceptual design phase. The realized equationsof motion were derived in the local vertical local horizontalframe. The equations constitute a system of ordinary differen-tial equations (ODE) which were solved simultaneously. Thesystem of equations states [16],

9V “T

m´D

m´

˜

g ´9X2

RC `H

¸

sin γ (59)

9γ “ ´1

V

˜

g ´9X2

RC `H

¸

cos γ (60)

9X “ V cos γ (61)9H “ V sin γ (62)9m “ ´β (63)

where V being the velocity of the launch vehicle, γ is theflight path angle, T is the thrust and D is drag. An illustrationof the defined directions of the variables can be seen in Figure7. The value of T was obtained from the Nozzle ConfigurationModule VIII-A, more specifically equation (42) combined withthe correction factor (40) applied to the momentum thrustalong with the efficiency reduction of 0.5 % for the use of jetvanes. The thrust was assumed to be constant during operationand the mean value of the thrust produced in the atmosphericinterval was chosen. The drag D was evaluated at each point ofintegration based on the speed and density at the instantaneousaltitude by implementing equation (11) into (59). Further, Xthe is the arc length at height H over the ground and RC isthe Earths radius which are illustrated in Figure 8. Finally, βis the mass flow rate assumed to be constant given by equation(47) which implies a neutral burning grain.

The dot represents the time derivative of the respective vari-able. Since the launch vehicle mass decreases with time andthe drag D varies with speed there are no analytic solutionsand the system of ODEs had to be solved numerically.

At a certain percentage of burnt fuel of the last stage themodel switched to a tangent steering law [25] in which theflight path angle γ is prescribed as,

tan γptq “ tan γ0

ˆ

1´t

tBurnout

˙

(64)

where γ0 is the initial flight path angle at which the switchto the steering law and tBurnout the time at which all fuel hasbeen burnt. This was implemented for the launch vehicle toarrive in horizontal position at burnout. The trajectory analyzedescribed was implemented in MATLAB®.

C. Mass Trade-Off Module

At the point of injection into orbit the launch vehicle musthave gained the required orbital velocity and be oriented inhorizontal position relative to the local horizon frame. Theburnout velocity largely depends on the amount of payload andfuel carried by the launch vehicle. The trade-off ratio betweenthe payload and propellant is one-by-one for the last rocketstage. Thus, one or the other can be increased or decreased tomatch the desired burnout velocity.

If the final velocity V was not sufficient for the requiredorbital velocity after the system of ODEs (59-63) was solvedthe Mass Trade-Off Module was invoked. By reducing thepayload mass by a small amount and increasing the propellantmass with the equivalent a higher value of final velocity couldbe achieved. If, on the other hand, the final velocity was higherthan the required orbital velocity the opposite conversion wasexecuted. The tolerance, i.e. the maximum allowed differencebetween final and required velocity, was set to 100 m/s.

D. Pitch Maneuver Module

If the final altitude was not within the tolerance of 100 mto the required altitude the Pitch Maneuver Module added orsubtracted a small quantity to the value of the nudge parameter.The nudge force created a net torque on the launch vehicle andthereby changed the previously vertical flight path. This wasdone to obtain the right nudge value for dynamics to positionthe launch vehicle in a almost horizontal position at the desiredaltitude for the tangent steering law to take over.

E. The Module Input Values

Integrations of the first stage were started with the initialmass m0 “ 1400 kg, the initial velocity calculated by equation(16) to va{c “ 354.1 m/s, H “ 12 km and γ “ π{2. Forthe sequential stages the initial values were provided fromcalculations by the preceding stage integration. The exhaustvelocities, thrust of the stages, burn times and mass flow rateswere obtained from the Nozzle Configuration Module VIII. Forthe drag coefficient CD a value of 0.25 was used based on [26].The final orbital velocity was determined by equation (14),which is the required velocity to maintain a circular orbit atthe desired altitude. The final altitude was set in the ObjectiveII section to be 500 km for all computations.

X. COMPUTATION TOOL ARCHITECTURE

The flowchart of optimization process implemented in thecomputation tool was presented in Figure 1 and the moduleswere described in individual sections throughout the report. Tocontrol the overall flow according to the flowchart a controllerscript was developed in MATLAB®. The controller script wasresponsible for calling the customized modules provided withthe required input parameters according to the optimizationprocess. The control script also determined if the velocitylosses, required orbital velocity and final altitude were withintolerances. The tolerances were set based on trial and error tovalues where the computation results could be obtained withina reasonable time limit and the final values was consideredclose enough to the desired values.

The tolerance for the velocity losses was set to 10 m/s whichcorresponds to 0.7 % of the initial guess value of 1400 m/s.The tolerance for the exhaust velocity of the nozzles was setto 10 m/s from the value calculated in the previous iteration.This tolerance value corresponds to 0.4 % of the initial exhaustvelocity of 2400 m/s. Hence, if one of the values were notwithin the tolerances the control script passed the most recentvalue of the velocity losses to the Stage Optimizer Module


VI and the operational altitudes to the Nozzle ConfigurationModule VIII-A and invoked the next iteration.

For the required orbital velocity the tolerance was set to100 m/s which is 1.3 % of the required orbital velocity andthe tolerance for the altitude was set to 100 m and thiscorresponds to 0.02 % of the final altitude. If the value of therequired orbital velocity was not within the tolerance the MassTrade-Off Module IX-C was invoked and converted a smallamount of payload mass into propellant mass able to furtheraccelerate the slightly lighter rocket into a higher velocity. Thefinal tolerance check to pass was the altitude tolerance. If thefinal altitude was too far from the desired altitude the PitchManeuver Module IX-D was invoked and adjusted the nudgeforce to a value closer to the value that nudged the rocketjust the right amount for gravity to turn the rocket to a finalhorizontal position. When all values were within the tolerancesthe optimization process was considered finished.

The algorithm of the optimization process was developedmainly for evaluation of a three-stage rocket design. Thisdecision was made based on the analysis performed in theMulti-staging VI-B section where it was concluded that anincrease in the number of stages was theoretically beneficialbut practically disadvantageous. Furthermore, as the numberof stages increases the improvement from the previous numberof stages is asymptotically less.

XI. COMPUTATIONS

The thrust-to-weight ratio expresses the acceleration as afraction between the thrust T and the weight W of the launchvehicle, denoted T {W . The initial T {W values were variedbetween above 1 and 4 based on values from [28]. For the mostbeneficial three-stage design, obtained from the optimizationalgorithm, a comparison between two and four stages withsimilar characteristics was performed.

For the comparison the Stage Optimizer Module VI wasmodified to handle two and four stages instead of three.The structure mass and specific impulse of the first and laststage of the three-stage design was used in the modifiedStage Optimizer Module to obtain the maximum payload massallowed for a two-stage rocket. For the four-stage comparison,the structure mass and specific impulse of the second stage ofthe three-stage design was duplicated to a new stage. In both(the two- and four-stage) cases the velocity losses obtainedfrom the Trajectory Module IX were assumed to be identicalas to the three-stage launch vehicle.

To get an understanding of the influence of the initial massof the launch vehicle the Stage Optimizer Module was invokedwith all parameters obtained from the final three-stage launchvehicle design, but with the initial mass varied between 1200kg to 1600 kg.

Further, the effect of an increase in the overall specificimpulse by 5 % was investigated along with a separatecomputation with a 10 % decrease in the structural mass ofall three stages. Initial values used for all computations canbe seen in Table I.

TABLE IINITIAL FLIGHT PATH ANGLE , INITIAL ROCKET VELOCITY ACHIEVED

FROM AIRCRAFT, LAUNCH ALTITUDE & INITIAL MASS

γ [rad] Va{c [m/s] H [km] m0 [kg]π{2 354 12 1400

XII. RESULTS

As the initial T {W value was increased the gravity lossesdecreased and the drag losses increased. An increase in T {Wresulted in lower values of the total velocity losses for allperformed computations. The results for T {W = 4 for a three-stage launch vehicle was concluded most suitable and aretherefore presented. The design parameters provided by themodules of the optimization process are shown in Table 2-3. Table 3 reveals the maximum payload mass of 22.0 kg.Figure 9-11 shows graphs from the ascent phase obtained bythe Trajectory Module IX.

The comparison between different number of stages forsimilar performance characteristics showed that a two-stagelaunch vehicle could deliver 8.28 kg and a four-stage 26.9 kginto the objective orbit.

An increase of the specific impulse with 5 % resulted in apayload mass of 26.7 kg instead of 22.0 kg and the reductionof structural mass with 10 % resulted in a payload mass of23.2 kg.

Finally, Figure 12 shows the variation of the payloadmass as a function of the initial mass with the performancecharacteristics of the final design. This gives an indication ofallowed payload mass if the constraint for the initial mass ofthe launch vehicle was subject to change.

XIII. DISCUSSION

The positive result emerging from the conducted study isthat it is most likely possible to put a small size satellite inthe 20 kg range into a 500 km polar orbit by a three-stageair-launched launch vehicle. As the three-stage launch vehicledesign was concluded to be the best choice for the rocket typeexamined.

The values of the specific impulse seems reasonable com-paring with values delivered by other solid propellant drivenrockets from literature [11]. Although liquid propellant wouldincrease the performance of the rocket it would also increasethe complexity of the design and thereby the overall cost. Aliquid propelled rocket would further reduce the storabilityaspect. Further, the preparations before launch would beexpanded which would affect the availability of the launchvehicle in a negative way.

The Trajectory Module IX was merely used to achievevalues associated with the drag and gravity losses duringthe ascent phase. Another study [27] suggests a releaseangle where the rocket benefits from lift and thus a moresophisticated trajectory analysis needs to be done for thisconcept to be analyzed. The gravity-turn used in the presentthesis is a near optimum trajectory and thus gives a morepessimistic value of the payload weight. To implement a moresophisticated optimized trajectory rather mature rocket design


0 20 40 60 80 100 120 140

Time [s]

0

1000

2000

3000

4000

5000

6000

7000

8000

Velo

city [m

/s]

Fig. 9. Velocity vs. time from the final iteration of the computation of thethree-stage rocket calculated by the Trajectory Module described in sectionIX.

0 20 40 60 80 100 120 140

Time [s]

0

1

2

3

4

5

6

Altitude [m

]

105

Fig. 10. Altitude vs. time from the final iteration of the three-stage compu-tation performed by the Trajectory Module section IX.

is required. By using a mature design, it would be possibleto achieve accurate values of the drag and lift coefficientsduring operation and thereby accurate values of the relatedforces. The use of an optimized trajectory was outside thescope of this thesis and therefore the gravity-turn trajectorywas implemented.

The computations showed that the velocity losses decreasedwith an increase in the initial T {W . This is reasonable sincethe gravity losses constitute the major part of the velocitylosses. A lower initial T {W value implies a longer time spentin the atmosphere and thus a higher value of the velocitylosses from gravity. In contrast, the drag is proportional tothe velocity squared and therefore a higher value of the initialT {W would increase the velocity and the drag. The increasein drag were of minor importance since the density of theatmosphere at the launch was relatively low. The contributionfrom drag to the velocity losses did not significantly effect

0 20 40 60 80 100 120 140

Time [s]

0

200

400

600

800

1000

1200

1400

Mass [kg]

Fig. 11. Mass vs. time from the final iteration of the three-stage computationperformed by the Trajectory Module section IX.

1200 1250 1300 1350 1400 1450 1500 1550 1600

Initial launch vehicle mass [kg]

18

19

20

21

22

23

24

25

26

Paylo

ad m

ass [kg]

Fig. 12. This plot presents the linear correlation between the initial massof the launch vehicle and the allowed payload mass. The payload masscalculated by the Stage Optimizer Module VI is plotted against the initiallaunch vehicle mass. These calculations were based on the injection altitude,exhaust velocities and velocity losses attained from the three-stage launchvehicle design presented in Table II and III.

the total velocity losses and thereby the benefits of a highervalue of initial T {W . As the present study was conductedat conceptual level only initial T {W according to [28] wastested.

In addition, a higher value of the initial T {W increases thedynamic loads acting on the launch vehicle which must beaccounted for in the structural design. Of course, the dynamicloads must be within the constraints set by the payload.

Because the nozzle mass was not included in the nozzleanalysis, the design is not completely optimized but rathergives an insight of what performance values is to be expected.The evaluation of the operational altitudes of the nozzles wasnot a necessary procedure, because at the operational altitudesof the second and third stage the ambient pressure was so lowthat it was impossible to create a nozzle expansion to match the


ambient pressure and at the same time provide enough thrust.For the motors of the upper stages the minimum exit pressurewas constrained to 5 kPa, which is the minimum exit pressurefor solid rocket motors operating in close to vacuum conditionsaccording to [29]. For the first stage the ambient pressure wasthe same for all iterations since the launch altitude did notchange. Furthermore, it would be advantageous to individuallydetermine the nozzle length of each stage to optimize theoverall design instead of assigning an equal nozzle lengthto all stages as in this study. It would also be interesting toinvestigate multiple nozzles on a single stage.

TABLE IICHARACTERISTICS OF THE LAUNCH VEHICLE STAGES

Stage # 1 2 3ms,i [kg] 85.9 29.2 7.36mp,i [kg] 936.6 275.3 43.61mi [kg] 1022.5 304.6 50.97tb [s] 41.0 51.9 42.0Isp [s] 246.1 284.6 289.5T [kN] 60.2 14.8 2.95T {W [-] 4.0 4.0 4.0T {W [-] (Final) 13.4 14.7 9.84Ae [m2] 0.312 0.128 0.0320At [m2] 4.79ˆ 10´3 1.11ˆ 10´3 2.18ˆ 10´4

ε [-] 65.0 115.1 147.0α [deg] 21.7 12.1 5.24

TABLE IIIPAYLOAD MASS, FINAL VELOCITY, VELOCITY LOSSES & FINAL ALTITUDE

mpl [kg] V (Final) [m/s] Vlosses [m/s] H (Final) [km]22.0 7657 1381 500

XIV. CONCLUSION

With the final design of a three-stage launch vehicle thestudy showed that it was possible to put a payload of 22.0kg into a 500 km polar orbit launched from the Saab JAS 39Gripen. The conceptual launch vehicle design was able to meetthe geometrical and weight constraints as they were derivedfrom the flight proven Taurus KEPD 350 system.

The study serves the purpose of a low fidelity model of alaunch vehicle and it further gives important insights in whattypical payload mass to be expected based on the presentedconstraints.

The modularity of the optimization process facilitates theability to interchange individual modules. For future work twoobvious steps would be to refine the Nozzle ConfigurationModule VIII and the Trajectory Module IX. In the NozzleConfiguration Module, conduct a more thorough nozzles anal-ysis including material design and thereby weight. For theTrajectory Module the definition of the geometrical design ofthe launch vehicle would support an extended aerodynamicanalysis. With a geometrical design in place a more sophisti-cated trajectory optimization could be performed with moreaccurate values of lift and drag coefficients based on thedesign.

By investigating different manufacturing processes and ma-terials, e.g. additive manufacturing, it is probably possible to

reduce the structure mass further and improve the performanceof the launch vehicle. In addition, an updated and morethorough survey of propellant types for the propulsion systemwould probably reveal the existence of more efficient fueltypes to further improve the launch vehicle design.

APPENDIX AACKNOWLEDGMENT

I would like to thank my supervisor professor ChristerFuglesang for the support throughout the degree project andfor the motivation to complete the work.

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