1. POLITECNICO DI MILANO
School of Industrial and Information Engineering
Master of Science in Space Engineering
DUCTED ROCKET TRAJECTORY OPTIMIZATION:
MODELING AND PROPELLANT INVESTIGATION
Supervisor: Dr. Filippo Maggi
M.Sc. Candidate:
Micol Spitale,
852324
Academic year 2016-2017
2.
3. al mio Nanetto,
“ Dell’autismo si sa molto e non si sa nulla [..]
Sappiamo di piú delle galassie lontane, dei buchi neri,
della struttura piú recondita della materia.
Non é che l’autismo é un problema con troppe variabili?
Una sfida troppo difficile? O non ci interessa abbastanza?”
“Se ti abbraccio non aver paura”. Fulvio Ervas
4.
5. Acknowledgment
Un grazie particolare va a tutte le persone che mi hanno sostenuta e che hanno con-
tribuito alla conclusione di questo percorso di studi. Senza di loro non sarei riuscita a
raggiungere tale traguardo, grazie.
Prima di tutto, un grazie va a tutto il team del SPLab. Non mi sarei mai aspettata
di poter lavorare in un ambiente cosı́ familiare e piacevole. Ovviamente la persona che
devo ringraziare piú di tutte é Filippo, grazie al quale ho imparato davvero tanto e che
mi ha spronato a rendere il mio lavoro di tesi sempre migliore. Grazie alla sua infinita
pazienza, so quanto io possa essere stressante e assillante, ai suoi saggi consigli e sug-
gerimenti e al tempo dedicatomi, nonostante i suoi numerosi impegni. Un immenso
grazie va anche a Stefano e Stefania, che in ogni momento di difficoltá e in ogni situ-
azione problematica hanno saputo aiutarmi e supportarmi con grande pazienza, davvero
grazie. Come dimenticare la persona che mi ha fatto divertire piú di tutte all’interno de
laboratorio, colui che ha coniato il mio fantastico soprannome “mammina” conoscendo
le mie grandi predisposizioni alle mansioni materne. Grazie Christian, hai allietato le
mie giornate di studio regalandomi sempre un grande sorriso. Un grazie va anche al
signor Colombo, sempre pronto a risolvere i problemi di chiunque necessitasse. In-
fine, ringrazio il professor Galfetti di avermi dato la possibilitá di partecipare a una
conferenza di grande importanza come quella dell’EUCASS.
I
6.
7. Abstract
Ducted rocket motors are the combination of solid rockets and ramjets. Thanks to
their relatively high specific impulse and specific thrust, they can be potentially applied
for the atmospheric phase of a space-access mission and used as medium-long range
tactical missile. The improved performances derive from the possibility to exploit the
air as oxidizer, allowing a reduction of the oxidant content of the propellant on board.
The present work deals with the analysis of fuel-rich propellant compositions and
the trajectory optimization of a ducted rocket motor using the investigated formulations.
In order to obtain the aim of the work, a three-fold activity was conceived. As a
first step, a model which simulates the dynamics of the ramjet was implemented in
Modelica and the motor applications were examined varying the initial missile parame-
ters. Secondly, to obtain the control law which derives from the minimum time-to-climb
problem, an optimization procedure was built on Matlab. It was solved with two dif-
ferent numerical methods and the results were further compared, in order to find the
optimum solution. Finally a set of propellant compositions was produced and experi-
mentally tested, deriving the ballistic properties for a realistic simulation. The oxidant-
lean propellants were formulated with aluminum and magnesium addition: the former
can enhance the performances and the latter can improve the ignitability.
The hybrid nature of this study, both numerical and experimental, supports the con-
clusions that such kind of technoligies cannot be applied for the atmospheric phase of
space access because of the low speed reached, while ducted rockets can be used for
long range missile applications.
Keywords: ducted rocket, fuel-rich propellant, magnesium, optimization, trajectory
III
8.
9. Sommario
Il ducted rocket é un motore nato dalla combinazione di un razzo a propellente solido
e un esoreattore di tipo ramjet. Tali motori possono essere applicati sia per una fase
iniziale di accesso allo spazio che per missioni missilistiche di medio-lungo raggio,
grazie al loro elevato impulso specifico ed elevata spinta specifica. Le loro prestazioni
sono dovute alla possibilitá di sfruttare l’aria come ossidante, permettendo una riduzione
del contenuto di ossidante del propellente a bordo.
Il presente lavoro si occupa dell’analisi della composizione dei propellenti ricchi di
combustibile e dell’ottimizzazione di traiettoria dei motori ducted rockets, provvisti del
propellente studiato.
Per raggiungere l’obiettivo di questo studio, sono state pensate tre procedure di la-
voro. Come primo passo, un modello, che simuli la dinamica di un ramjet, é stata
implementata in Modelica e le applicazioni di questo tipo di motore sono state esami-
nate variando i parametri iniziali del missile. Successivamente, per ottenere la legge di
controllo che deriva dal problema di minimizzazione del tempo di salita, é stata costruita
con Matlab una procedura di ottimizzazione. Quest’ultima é stata risolta con due metodi
numerici diversi, i cui risultati sono stati confrontati per trovare la soluzione migliore.
Per concludere il lavoro, due propellenti sono stati prodotti e testati sperimentalmente
in laboratorio, ottenendo cosı́ le proprietá balistiche necessarie per una simulazione piú
realistica. Le formulazioni del propellente povero di ossigeno sono state scelte con
l’aggiunta di alluminio e magnesio: il primo puó migliorare le prestazioni e il secondo
l’infiammabilitá del propellente.
La natura ibrida di questo studio, contemporaneamente numerica e sperimentale,
permette di rafforzare i risultati ottenuti, che concludono che tale tecnologia non puó
essere utilizzato per la fase iniziale di un lanciatore, a causa della bassa velocitá finale;
mentre questo tipo di motore puó essere usato per applicazioni missilistiche di lungo
raggio.
Parole chiave. ducted rocket, propellente ricco di combustibile, magnesio, traiettoria,
ottimizzazione
V
20. LIST OF TABLES LIST OF TABLES
6.9 Parametric analysis of initial velocity . . . . . . . . . . . . . . . . . . 95
6.10 Initial velocity variation: altitude and ground range . . . . . . . . . . . 97
6.11 Parametric analysis of initial altitude . . . . . . . . . . . . . . . . . . 98
6.12 Initial altitude variation: altitude and ground range . . . . . . . . . . . 99
6.13 Parametric analysis of initial mass . . . . . . . . . . . . . . . . . . . . 101
6.14 Initial mass variation: altitude and ground range . . . . . . . . . . . . 102
6.15 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.16 Final time obtained with different optimization methods . . . . . . . . . 110
XVI
21. Chapter 1
Motivation and objectives
1.1 Motivation
For space access, the conventional solid rocket motors are used because of its sim-
plicity and compact design. They grant high volumetric specific impulse, high thrust
to weight ratio and extended storability. However solid rocket motors cannot increase
the performances without ann increment in weight and size (more costs). Furthermore
solid rockets are a very efficient propulsive solution as short ranges tactical missiles are
concerned.
The ramjet propulsion can be a valid solution for space access and tactical missile
applications. One of the advantages of air-breathing propulsion is the higher specific
impulse. The main reason is that the oxidizer is not carried on board but it is taken from
the atmosphere during flight. Thanks to their high specific impulse, ramjet motors allow
missile applications for longer range with respect to the rocket motors.
In this study, a particular type of ramjet is investigated: the ducted rocket, which
combines the advantages of ramjet and rocket propulsion. As for ramjets, the vehicle
shall be accelerated up to a certain speed, from which the engine can start working. A
gas generator combustor contains a fuel-rich solid propellant, which gases combustion
products act as fuel in the ramjet combustion chamber (ramburner), where they react
with the air. The solid propellant is oxidant-lean in order to provide a sustained com-
bustion. So high volumetric loading is obtained, allowing a compact design. Therefore,
ducted rockets are attractive for applications where small weight and volume are re-
quired, as for climber application (useful for first stage launcher of sounding rocket) or
also as sustainer, like medium range air-to-air missile.
In space and missile applications the ducted rocket role has to be clarified. There-
fore, a feasibility study of their use is investigated in order to choose the optimum solu-
tion for each application.
1
22. Thesis objectives 1. Motivation and objectives
1.2 Thesis objectives
The main goal of this thesis study is to optimize the trajectory of a ducted rocket
motor. In order to implement a model, all the components are investigated. In partic-
ular the fuel-rich propellant for airbreathing application is examine in term of burning
rate starting from previous work by Zadra [9]. After a thermochemical analysis, the
formulations have been selected for the experimental investigation.
Once the gas generator fuel-rich propellant has been chosen, the ducted rocket model
can be implemented. In order to find the optimal trajectory for minimum time-to-climb,
a literature survey of optimization methods is presented. Therefore, a comparison be-
tween different methods is performed and the optimum solution is chosen.
1.3 Thesis overview
A brief description of each chapter is reported in order to better understand the work
structure:
- Chapter 2. Ramjets and ducted rocket state of the art is analyzed in depth, then a
description of composite solid propellant focusing on the metal additives combus-
tion process is presented. At last, a literature survey of the trajectory optimization
methods is reported briefly.
- Chapter 3. A theoretical analysis of the performances for different propellant
composition is made. The main objective is to examine the propellant candidates
and compare their propulsive performances.
- Chapter 4. The propellant compositions chosen and the experimental set-up are
analyzed. Also the manufacturing propellant procedure and burning rate experi-
mental line are reported.
- Chapter 5. The model of the ducted rocket implemented in Modelica is explained
deeply. Furthermore, the trajectory optimization methods used are examined and
the Matlab code is presented.
- Chapter 6. The results of the experimental burning rate analysis and the simulation
of the optimized trajectory are here collected. Also a parametric analysis of the
variables involved is performed to have a complete overview of the possible ducted
rocket applications.
- Chapter 7. The conclusions and possible future work developments are reported,
making a summary of the results obtained.
2
23. Chapter 2
State of the art
2.1 Ramjet
A ramjet engine provides a simple and light propulsion system for atmospheric high
speed flight. It cannot operate from a standing start; indeed its startup depends on rocket
boosters or some other methods, such as being launched from an aircraft, for being
accelerated to takeover flight speed. At supersonic flight above Mach 2, the ramjet
engine becomes attractive for flight within atmosphere for many reasons. In fact there
are several advantages for using airbreathing engine instead of rockets:
- the oxidant required for the combustion with fuels comes from atmosphere. This is
a great advantage because the oxidant shall not be put on-board, as for rocket, and
the engine mass is minimized in favor of payload mass, which can be maximized;
- the engines produce much higher engine efficiency, in term of relationship be-
tween theoretical energy additives contained in fuels and the real energy exploited,
over a larger portion of the flight and a longer powered range than rockets;
- they have the ability to change efficiently powered flight path and maneuverability;
- there is thrust modulation for efficient cruise and acceleration missions;
- they can be not only refurbishable but also reusable;
- for space access they have short turnaround time with a potential cost reduction of
10-100 times per pound of payload [10].
It is possible to recognize two classes in the ramjet’s family [10] according to the
speed range:
1) Ramjet, subsonic combustion cycle behaviour. There are different configurations:
3
24. Ramjet 2. State of the art
a) the traditional can-type ramjet (CRJ). A tandem booster is required to pro-
vide static and low-speed thrust, which pure ramjets alone cannot provide.
Here the air is diffused to a subsonic speed through a shock before reaching
the combustion. It can be a liquid-fueled (LFRJ) or gaseous-fueled ramjet
(GFRJ) based;
b) the integral rocket ramjet (IRR) which in turn can be liquid-fueled (LFIRR)
or solid-fueled (SFIRR) based. Usually this configuration is preferred with
respect to the previous one because of the simplicity of the fuel supply;
c) the ducted rocket (DR) where a fuel-rich propellant is used to generate a fuel
supply for the subsonic combustor.
2) Scramjet, supersonic combustion cycle behavior. In a traditional scramjet engine
the air enters at supersonic or hypersonic speeds and it is diffused to a lower speed,
still supersonic, toward the combustor. Here, the fuel is injected and mixed, then
it burns with air in a generally diverging area combustor. So combined effects
generate a shock train, which strenght depend on flight conditions.
Ramjets with subsonic combustion and hydrocarbon fuel have an upper speed limit
about Mach 5 [7].
The basic ramjet engine consists of an inlet, diffuser, combustor and exhaust nozzle
represented in Fig.2.1 [10]. Starting from the inlet and diffuser, the ramjet exploits these
fixed components to compress and accelerate intake air by ram effect, as a compressor,
depending on the vehicle speed. The air is delivered to a combustion chamber and it is
mixed with injected fuel. At this point the mixture is ignited and burns with the aid of a
flameholder that stabilizes the flame. The burning fuel imparts thermal energy to the gas,
which expands to high velocity through the nozzle at speeds greater than the entering
air, which produces forward thrust. Thrust is produced by increasing the momentum of
the air as it passes through the ramjet without employing rotating devices.
Relatively high dynamic pressure q is required to provide adequate static pressure
in the combustor for good combustion and to provide sufficient thrust. A typical Mach
number-altitude airbreathing flight corridor is shown in Fig.2.2. As speed increases,
there is less need for mechanical compression. The upper boundary is imposed by a
combustion limit, driven by unburned fuel; while the lower boundary is a region of high
skin temperature and pressure loading thereby establishing design and material limits
[10].
As Curran explains in [1], the airbreathing engine systems operate only in a portion
of Mach number range but features a very high specific impulse, while rockets provide
the necessary thrust-weight ratio to extend the Mach number range but in turn they grant
a very low specific impulse with respect to the ramjets.
4
25. 2. State of the art Ramjet
Figure 2.1: Ramjet configuration
Figure 2.2: Airbreathing flight corridor
Ramjets can be combined with other propulsion technologies to extend the operating
envelope. For instance, a combined rocket and airbreathing propulsion systems can be
used for space-launch applications, as [11] investigate.
In general, the rocket-based combined-cycle systems (RBCC) can operate in different
modes: the Rocket ejection mode, the Ramjet mode, the Scramjet mode and the Rocket-
5
26. Ramjet 2. State of the art
only mode. The main issues for the development of such propulsion system are the
engine selection/integration and the flow-path design for multimodes operations.
2.1.1 Ducted rocket
The ducted rocket consists of a gas-generator containing a gas-generating pyrolant
and a ramburner in which the air continuously flows from the atmosphere through the
air-intake. The configuration of ducted rockets is reported in Fig.2.3. As the figure
shows, there are two combustion processes: the former is the gas generator one in which
a oxidant-lean mixture is burned and expanded in a primary nozzle, then the gaseous
fuel mixes with air flow from atmosphere and burns with it in the ramburner. At last,
the reacted mixture is expanded through the secondary nozzle, producing thrust.
Figure 2.3: Ducted rocket geometry
The thrust of ducted rockets is generated by the momentum difference between the
exhaust gas from the combustor and the air taken in from the atmosphere, depending on
flight speed and altitude [2].
TDR = (ṁa + ṁf)ve − ṁava (2.1)
The same formulation can be written for the liquid-fueled ramjet but what essentially
differs from ducted rocket is the fuel supply. Indeed, in the ramjet systems the fuel is
an atomized liquid that evaporates and mixes with the air for combustion; while in the
ducted rockets the fuel supply consists of high-temperature gaseous products. So the
ignition time and the reaction time of the latter result much shorter then those of the
former fuels. It results in a higher combustion ignitablity and stability. By utilizing
the atmosphere as an oxidizer, the gravimetric impulse and, thus, the total impulse of
the rocket motor are greatly increased, since most of the oxidizer needed for complete
6
27. 2. State of the art Ramjet
combustion of the rocket fuel is not carried within the rocket vehicle itself. This leads to
an increasing payload capability or an improving range, thanks to the weight reduction.
Analyzing the integrated momentum conservation balance, the range computation can
be provided. Eq.2.2 is similar to the Breguet range equation for aircraft, since it presents
the velocity u dependence on the efficiency of the propulsion system represented by the
specific impulse Isp and the masses ratio. The Eq.2.2 can be integrated in order to
compute distances.
u = g[Isp log(
mo
mf
) − t] (2.2)
The Fig.2.4 shows the difference downranges for rockets and for air-breathing sys-
tems. The interval range for the air-breathing engines, which is the lighter region, is
larger than the rocket one, the darker.
Figure 2.4: Comparison between rocket and ducted rocket ranges [1]
The operational conditions of a ducted rocket are dependent on the mass generation
rate of the fuel-rich gas and the airflow rate induced from the atmosphere. As far as
concerned the gas flow, the control system links the ramburner with the gas generator.
Its configuration depends on the needed thrust for each position. In fact, when a required
flight trajectory from high altitude to low altitude or from high speed to low speed is
7
28. Gas generator pyrolant 2. State of the art
specified a control system is needed. There are different configurations [2]:
- Non-chocked fuel-flow system. In this configuration, the combustion pressure of
the gas generator is equal to the ram-pressure, that in turn depends on the shock
wave pressure formed at the air-intake. Obviously, the pressure is affected by
the flight speed and the altitude of the ducted rocket. Since the mass generation
rate of the pyrolant burning is dependent on pressure, it is also varied changing
the altitude and the flight speed. The non-chocked fuel-flow system is a self-
adjustable mass flow system and its mechanical structure is a simple one without
any moving parts.
- Fixed fuel-flow system. This asset provides a constant fuel-flow rate, thanks to a
chocked orifice that is attached at gas generator aft-end. In this way, the mass gen-
eration rate of the fuel-rich gas is therefore independent of the ram-pressure. The
operational applications are restricted due to the dependence on the airflow rate
affected by altitude and flight speed. Indeed, the air-fuel ratio in the combustion
chamber is also altered and in turn also the thrust, restricting the flight envelope.
- Variable fuel-fixed system. This ducted rocket has a control valve attached to the
chocked nozzle which controls the fuel-flow rate according to the air-flow rate
induced into the ramburner, in order to obtain an optimized combustible gas. The
mass generation rate is altered by changing the throat area, thanks to a throattable
valve attached to the end of the gas generator.
With respect to the solid and liquid ramjets, the ducted rockets provide less gravimet-
ric specific impulse, because gas generator pyrolants contain small amount of oxidant
to sustain combustion while the solid and liquid ramjets do not, but in turn higher than
the one provided by solid rocket system. However the use of air as oxygen ensures mass
savings, leading to incremented payload mass fractions.
2.2 Gas generator pyrolant
As already mentioned in the previous section, the choice of solid propellant instead
of liquid is driven by advantages in terms of simplicity and the size reduction, due to
the higher density values. In addition the solid propellant is storable and it has high
reliability.
An important parameter is depending on solid propellant chemical composition: the
burning rate. Besides chemical features, this value is strongly related to the combustion
environment. In fact, the burning rate dependence on pressure is represented by the
Vieille law:
8
29. 2. State of the art Gas generator pyrolant
rb = aPn
c (2.3)
where r is the linear burning rate, P in the pressure, a is a pressure constant and n
is the pressure exponent. The last two variable depend on the nature of the pyrolant. In
addition as the pressure exponent increases, the r becomes more pressure sensitive as it
is shown in Fig. 2.5.
Figure 2.5: Variable flow range of high and low pressure exponent pyrolant [2]
When a chocked gas generator condition is considered, a low n value is preferred in
order to reduce disturbances and maintain stability. Nevertheless when an unchocked
configuration is chosen, an higher n value leads to a wide range of combustion velocity
being the combustion pressure strongly correlated to the total ambient pressure. In
addition, the temperature sensitivity becomes higher when a high exponent pyrolant is
used.
The variation of the pressure exponent n and coefficient a of the Vieille law, Eq.2.3,
are analyzed, in order to better understand the combustion process in the gas generator.
The main parameters which can contribute to the different solid propellant burning
rate values at constant pressure are [12]:
- Chemical composition
- Trasport processes
- Chemical kinetics
9
30. Gas generator pyrolant 2. State of the art
- Physical geometry
A key roles are the transport processes, in fact the pressure exponent is strongly
dependent on them [12].
n ∝
ThermalDiffusivity
Mass Diffusivity
(2.4)
Taking into consideration this relation, the pressure exponent increases with the ther-
mal diffusivity; however it is quite impossible to change the transport processes without
changing the kinetic mechanism.
From literatury, the combustion is stabilized with an increase amount of heat released
in the condensed phase. For this reason, this implies a lower thermal diffusivity and a
lower exponent, in order to be as stable as possible.
It is possible to increase the thermal diffusivity through some energetic additives, such
as metal fuel. Furthermore, a small amount of metal can affect also the mass diffusivity,
leading to lower the exponent.
Sometimes, when a residues is formed, a negative pressure exponent can exist causing
a decrease of regression rate with the pressure due to lowered heat transfer.
As far as concerned the pressure coefficient a, it depends on the chemical kinetics.
Indeed, it is the term in which the sensitivity to the initial temperature on propellant
burning rate is evident.
As for the variation of initial propellant temperature, also the addition of a catalyst to
the solid can affect the pressure coefficient value.
In some situation when an impulse energy is applied, the pressure can rise excessively
and a transition from deflagration to detonation can occur. This process has to be
avoided.
For this reason, it is important that the burning rate exponent remains under the unity if
deflagration is to be sustained. When n overcome the unity, the transition to detonation
occurs.
In Tab.2.1, the variables affecting the transition from deflagration to detonation are re-
ported, where n is the pressure exponent, ρ is the propellant density, D is the mass
diffusivity, cp is the specific heat at constant pressure, k is the thermal conductivity and
α is the thermal diffusivity.
Temperature affects the rate of chemical reaction and thus the initial temperature of
the propellant grain, as already mentioned, influences the burning rate value.
Usually, the sensitivity is expressed as the form of temperature coefficients, which are
reported in Eq.2.5 and Eq.2.6.
σp = (
δln(r)
δT
)p =
1
r
(
δr
δT
)p (2.5)
10
31. 2. State of the art Gas generator pyrolant
Table 2.1: Variables effect on deflagration and detonation
Variables Deflagration Detonation
n < 1 > 1
ρ high low
D high low
cp high low
k low high
α low high
πk = (
δln(p)
δT
)k =
1
p
(
δp
δT
)k (2.6)
where σp is the temperature sensitivity of burning rate, πk is the temperature sensi-
tivity of pressure and k is a geometric function.
2.2.1 Chemical Composition of gas-generator pyrolants
In the ducted rocket, the composition of the solid propellant has less oxidizer con-
tent (30%-50%) with respect to traditional ones, due to the secondary combustion in
the ramburner. Therefore, the air-fuel ratio depends on the air flow rate and fuel-rich
flow rate, which is the product of the first combustion. Indeed, typically gas-generator
pyrolants consist of [13]:
- a binder, whose function is to ensure the cohesion of propellant materials and to
supply combustible fuel fragments through pyrolysis;
- a oxidizer, which is employed to react with the binder to produce a mixture that is
trasported to the ramburner;
- a catalyst;
- additives, such as metal powder, whose function is to improve the propellant char-
acteristics;
They are energetic materials that burn incompletely by themselves to generate fuel-
rich combustion products, which react with air in the ramburner. Though the chemical
components of the pyrolants are highly fuel-rich, the pyrolants are required to maintain
self-sustaining combustion once ignited. The addition of metals to the fuel can pro-
vide even better energetic performance. There are some criteria for the selection of the
11
32. Gas generator pyrolant 2. State of the art
proper binder: the ease of manufacture and the combustion heat values. The binders
containing more oxygen have also lower heat of combustion, such as polyesters and
polycarbonates. Moreover the latter incorporates a huge amount of fuel-rich solids [13],
making HTPB the proper candidate. For this reason, the choice of hydroxyl-terminated
polybutadiene (HTPB) is preferred with respect to the other binders. A typical oxidizer
used in pyrolants is ammonium perchlorate (AP), which generates gaseous oxidizer
fragments when thermally decomposed. Mixture of AP and HTPB forms fuel-rich
pyrolant, whose burning rate characteristics are dependent on the mixture ratio and the
particle size of the crystalline ammonium perchlorate.
As fuel fillers several elements such as the carbon (C), the zirconium (Zr), the
aluminum (Al), the magnesium (Mg) and the boron (B) can be selected. The beryllium
is excluded because of the toxicity of oxide products. Now each element can be analyzed
in order to identify the strengths and weaknesses [13]. The boron fuel-rich propellants
are highly energetic if the particles ignite and burn in the air within a short period of
time, but this is not so easy to obtain. Boron particles are coated with a layer of boric
oxide, which has a very high boiling temperature that interferes with the combustion.
As far as concerned the aluminum, it has all the proper characteristics, but it cannot
be used with high content because of the tendency to obstruct the nozzle of the gas
generator. The magnesium provides a very high flame temperature, so its combustion
is very rapid. Al and Mg particles are favored metals in the formulation of pyrolants
because of their high potential for ignitability and combustion [2], but in turn product of
Al and Mg particles tend to agglomerate to form relatively large metal oxide particles.
The metal fuels are embedded inside propellants in order to enhance specific impulse
performance, increase density and improve the combustion stability of the propulsion
unit [5]. The metal particles have also a key role in the combustion stability, indeed
they act as flameholders to keep the flame in the ramburner during the combustion be-
tween the fuel-rich gas and the ram air induced through the air-intake. Each metal
particle flows with the combustible gas and becomes a hot metal drops or metal oxide.
Due to the presence of the hot particles in the flow, the combustible gas flow veloc-
ity is decreased. A micro-flame becomes attached to each hot particle and ignites the
surrounding combustible gas, acting as igniter and also flameholder. In this way, the
dispersed hot particles ignite the combustible gas.
2.2.2 Aluminum
Aluminum powders have been added to solid propellants for space applications and
its combustion process has been analyzed deeply. Starting from 1970, the first com-
bustion model was implemented by Gany and Caveny [14]. In 2004 Beckstead has
12
33. 2. State of the art Gas generator pyrolant
summarized the aluminum combustion process into a technical report [3].
First of all, the aluminum particles are ignited and the combustion process proceeds
thanks to the conductive heating of loaded particles. Even if the melting point temper-
ature is reached, the oxide coating Al2O3 shell separates the melt fuel from the envi-
ronment. During heating, the oxide shell is subjected to the cracking phenomenon due
to the higher thermal expansion of the metal with respect to the oxide. In addition, the
melt metal increases its volume of about 6%. Since the metal flows out, the Al com-
bustion starts thanks to the heat feedback to the particles, which lead to the aluminum
vaporization. Now the vaporized aluminum can burn in gas-phase with oxidant at some
distance from particle surface.
In the flame zone, the main products are sub-oxides which condense in order to
create melt aluminum oxide [3]. This liquid can be dissociated by the heat release in that
zone and the dissociation temperature is kept constant until the process is completed.
The oxide particles can behave into two different ways. A fraction of the liquid oxide
goes back and settles on the surface particles forming a “oxide-cap”. Since the oxide
keeps accumulating on the surface with high porosity, the “oxide cap” thickness has the
same size order of the initial particle.
The other part of oxide is going outwards as an “oxide smoke”, which is a white tail
behind the particles. Thanks to these condensed combustion products, both particles
and smoke, the acoustic instabilities can be damped.
The Fig.2.6 shows the aluminum combustion process described in [3].
Figure 2.6: Aluminum particle combustion [3]
If the the aluminum oxide doesn’t melt because the temperature is not reached, the
liquid Al oxidizes to solid Al2O3 allowing the self-suture of the shell fractures. This
process leads to the coalesce of metal particles into clusters that can originate the ag-
glomeration process, which will be analyzed deeply in a next section.
13
34. Gas generator pyrolant 2. State of the art
2.2.3 Magnesium
As many studies have demonstrated, the combustion process of magnesium is mainly
driven by diffusion. In particular in composite propellants, magnesium combustion oc-
curs in vapor-phase mode, due to the lower boiling temperature of metal with respect to
its oxide.
Cassel and Liebman [15] have investigated the combustion of magnesium particles.
With experimental procedures, they have confirmed the studies conducted by Glass-
man [16]. Basically, the combustion occurs in vapor phase at a certain distance from
the metal surface. In fact during the reaction, the magnesium drops pass through the
oxide layer because it is pulled into pores by capillary forces; then the drops evaporate
being supported by heat transfer from the flame in order to react with atmosphere oxy-
gen [4]. As the experiments provide in [15], the metal drops burn completely while the
oxide shell remains unburned and grows due to agglomeration of magnesia particles.
The Fig.2.7 shows the drawing of a magnesium particle combustion.
Figure 2.7: Magnesium particle combustion [4]
Pisarenko and Kozitskii have shown in [4] that the conglomerate magnesium parti-
cles burn in vapor-phase as usual, however if the conglomerates size is large enough,
the combustion can proceed in steps. The test results show that the combustion time
of conglomerates is quite long with respect to the time for solitary particle of the same
14
35. 2. State of the art Gas generator pyrolant
mass. The main reason is the low thermal conductivity of conglomerate: the heat trans-
ferred to conglomerate by conduction is not sufficient to melt it.
It has been observed that the dispersity of the powder drives the vapor-phase reaction
regime for the conglomerates.
For the composite propellants applications, the fact that the ignition temperature for
Mg is much lower than for Al leads to an increasing of the condensed phase heat re-
lease. Mg droplets are easily ignited on the surface and they undergo rapid transition to
vigorous detached flame burning. Thus their residence time on the surface is less than
other metals such as aluminum and boron. The net result is an increase in burning rate
for magnesium propellant which is bigger than for aluminum but smaller than that for
boron.
Mg-based compositions have efficient combustion characteristics in the ramburner as
studied by King [17] and [18]. The vapors, produced in exhaust streams of the pri-
mary motor, react violently with oxygen, producing oxygen atom, thereby promoting
vigorous ignition and combustion. In [17] it is also reported an increase in theoretical
volumetric heating value on increasing Mg loading.
2.2.4 Agglomeration of metal particles
In solid propellants, the presence of metal fuel complicates the burning process,
because they form condensed combustion products (CCPs). In the combustion process,
the agglomeration phenomenon is the coagulation of the CCPs on the burning propellant
surface and then the burning surface supplies these products to the gas phase [19]. This
leads to specific impulse loss because of the two-phases nozzle flow expansion and
consequent performance degradation.
Regarding the propellant composition with AP, HTPB as binder and micrometric
aluminum powders, the microstructure of this heterogenous propellant can be analyzed.
Starting form the chemical structure, the Fig.2.8 shows the coarse particles of oxidizer
placed randomly while the other smaller particles of metals, fine oxidizer and the binder
fill the gaps. In this way, it is possible to recognize strips of binder which connect
pockets, called interpocket bridges [5].
Once the particles reach the burning surface, they are warmed up by the heat feed-
back from the flame and nearby particles start melting together forming a structure. As
the hottest regions of the flame get in contact with the irregular structure, the metal
temperature rises and the aggregates collapse into a fused drop.
The sequences of the agglomeration process are [20]:
- initial distribution of the aluminum in the propellant matrix;
15
36. Trajectory optimization 2. State of the art
Figure 2.8: Microstructure of propellant with oxidizer and metal particles [5]
- the aluminum tendency to adhere and concentrate on the burning surface without
immediate ignition;
- the sintering of the concentrated particles into an aggregate that will coalesce into
a drop rather than disintegrate when further heated;
- ignition of the agglomerate.
Obviously, the features of the agglomeration process depend on the propellant com-
position and burning conditions. The particles size distribution is another important
parameter to take into account: higher dispersion corresponds to smaller particle sizes
[19].
It is possible also to make a distinction between agglomeration and aggregation [21]:
the former is reserved to the formation of the spherical drops of liquid metal and oxide
in the combustion; the latter is reserved to the formation of the partially oxidized flakes
of irregular shape seen at burning surface, so aggregates can be seen as agglomeration
precursors.
2.3 Trajectory optimization
In this work, the constrained control optimization problem is solved through the
Euler-Lagrange theorem because of its simplicity and reliability. Let’s focus on the
general classification of optimization problem in order to better understand the types
16
37. 2. State of the art Trajectory optimization
of trajectory optimization implemented. For each of them, some examples have been
reported.
2.3.1 Trajectory optimization methods classification
In literature, there are several works done in order to optimize a desired trajec-
tory. Since digital computer have been available, the trajectory optimization was im-
plemented for solving real problem. In Russia, Pontryagin develops studies based on
the calculus of variation, one of the main result was the Pontryagin’s maximum prin-
ciple. From here, modern researchers call this kind of approach as “indirect method”,
which are explained deeply in the next paragraph.
In order to better understand the trajectory problem, it is important to underline the
difference between the initial value and the two point boundary value problems. Typ-
ically,the initial value problem is expressed through linear ordinary differential equa-
tions:
ẋ = f(t, x) (2.7)
where x is N vectors and thus a N + 1 number of initial conditions are needed
in order to solve univocally the problem. The basic idea is to propagate the solution
forward in time starting from initial conditions, exploiting numerical methods.
On the other hand when a two boundary value problem is considered, a final condition
shall be satisfied starting from a set of initial conditions.
Usually, the trajectory optimization is a two boundary value problem, where the initial
conditions are set and a certain final conditions shall be reached.
In literature, there are two main way of optimization: the parameter optimization and
the optimal control theory. The former is referred to a problem where the parameters
that shall be minimized are not function of time; while the latter includes parameters
functions of time and a functional shall be minimized [22].
For handling terminal boundary conditions problem, the most widely used methods
can be divided into direct and indirect [23]:
a) direct method, discretizes the control problem and applies non-linear program-
ming technique to the finite-dimensional optimization problem. The main issue is
that the solution is only approximated.
b) indirect method, finds a solution to the necessary conditions of optimality. It needs
to guess also the optimal solution structure;
Discretizing the problem in different parts, the direct method generates a solution
through a parameter optimization problem. On the other hand, the indirect method
17
38. Trajectory optimization 2. State of the art
sets the problem according to the Potryagin maximum principle and then it is solved
numerically. In addition, the direct methods are easier to solve, but they are not so
accurate as indirect methods are. This is due to the fact that the indirect methods need
the explicit construction of the adjoint equations and gradients. However, Citron [24]
demonstrate in its book that both approaches are valid and equivalent and the user shall
determine which is the best for its particular purpose.
The main disadvantage of the indirect methods, such as indirect shooting method, is to
find an appropriate initial guess for the adjoint variables (which will be explained better
in the next section) while direct methods, such as direct collocation, have lower accuracy
and their iteration can terminate with a non-optimal solution [25]. So in order to handle
the indirect method, the user shall have a deep knowledge of the physical nature of the
problem to find properly the initial guess conditions. This problem came from the limit
in the convergence domain of the Newton method, from which the shooting method is
built. There are other methods such as the gradient method which use the minimum
principle directly for optimization.
As the author [25] has mentioned, the direct approach transforms the problem into a
non-linear programming problem by parameterizing the control variables. The basic
idea is to integrate numerically the equation of motion and choose the control variables
from a finite dimensional space. If in the same way the variables are parameterized, the
equation of motion are satisfied on the collocation points prescribed.
The best solution can be found approximating the solution with direct approach and
then refining it with indirect method.
When in the optimization problem, the objective function is constrained with respect
to variables, a constrained optimization problem can be set. If there aren’t constraints
on objective function, the optimization problem is called unconstrained.
2.3.2 Indirect method - variational approach
Since a solution of the optimal control problem is needed, the indirect approach
is based on the generalization of the calculus of variations. Firstly it is important to
formulate an appropriate two boundary value problem and then solve it numerically. In
particular, a two boundary value problem is made up of states and co-states equations
with its own initial and final conditions.
The Euler-Lagrange theorem is often applied to optimal control case in which a
control input is involved.
So consider the dynamic system:
ẋ = f(x, u), x(0) = x0 (2.8)
18
39. 2. State of the art Trajectory optimization
with the cost function with associated performance index:
J(x0, u) = ψ(x(T)) +
Z T
0
L(x(t), u(t))dt (2.9)
and final conditions constrained:
φ(x(T)) = 0 (2.10)
where J is the cost function, u is the control variable, ψ(x(T)) is the end constrained
penalty and L(x, u) is the running cost. Now a Hamiltonian function can be introduced
in order to set the problem properly:
H(x, u, λ) = L(x, u) + λf(x, u) (2.11)
The λ variables are denoted as Lagrange multipliers, or adjoint variables. The opti-
mal control problem finds the control function that minimizes the cost function, and in
addition the final state conditions and system of equations are satisfied [26].
So, the solution of optimal control problem solves a set of differential equations:
ẋ = Hλ = f(x) (2.12)
λ̇ = −Hx = −
∂L
∂x
− λ
∂f
∂x
(2.13)
Hu =
∂L
∂u
+ λ
∂f
∂u
= 0 (2.14)
The Eq.2.12 is referred to the state equations, Eq.2.13 is the set of costate equations
and the Eq.2.14 is the optimal condition from which the control law can be obtained.
In the next paragraph, an example of the indirect methods application is reported:
gradient-restoration method.
Sequential gradient-restoration algorithm
Miele was one of the pioneer of the gradient restoration method. His studies are
focusing on this type of algorithm implementation.
In [27] the author shows an efficient method for solving the optimization problem. In
his work Miele deals with the optimization of ascent trajectory employing the sequential
gradient-restoration algorithm for optimal control problems.
In [28], the author explains the sequential method and its transformations.
The algorithm is composed by sequential cycles, each of them involves a gradient
and a restoration phase. The procedure involves different steps that can be summarized
[29]: initiation phase, gradient phase, a restoration phase and a gradient-restoration
19
40. Trajectory optimization 2. State of the art
cycle. The algorithm is characterized by two scalar quantities: the constraint error P
and the optimality condition error Q.
1) In the initiation phase, the constraint error P shall be computed for the state and
control functions. If this value doesn’t overcome the tolerance threshold, it is
possible to go on to the gradient phase, otherwise the restoration phase has to be
invoked.
2) In the gradient phase, the linear multi-point boundary value problem shall be
solved and the optimality condition Q error shall be computed. If it is smaller
than a given tolerance value, the algorithm is terminated because either the feasi-
bility and optimality conditions are satisfied. If it is larger than a tolerance value,
another gradient step is performed updating all the state, control and parameters.
Once the gradient phase is completed the inequality of P error must be checked,
in order to determine whether go to the restoration phase or not.
3) The restoration phase starts if P is larger than a given value, and then the system
of equation has to be solved. Also in this phase, all the variables are updated for
each iteration step. In the restoration phase, multiple iterations are performed in
order to obtain P lower than a given value. When this step ends, the computation
is switched to the gradient phase.
4) Finally, the gradient-restoration cycle can starts if the constraints of Q and P are
satisfied. The same procedure iteration must be done in order to compare the per-
formance index of the present cycle with the same index computed at the previous
cycle. It must be provided that the performance index at each cycle is higher than
the one of the previous cycle.
For each cycle iteration, the solution of an auxiliary minimization problem is re-
quired. There are two different kinds of sequential gradient-restoration algorithm: the
primal and dual formulations.
In the former the auxiliary minimization problem is solved according to the state, con-
trol and parameter variations, while the latter solves the auxiliary minimization problem
with the help of the Lagrange multipliers.
2.3.3 Direct method - non-linear programming approach
As already mentioned, the basic idea is to discretize the control problem and then
apply non-linear programming (NLP) technique in order to solve the optimization prob-
lem.
First of all it is necessary to subdivide the optimization time [t0, tf ] into ns control
20
41. 2. State of the art Trajectory optimization
stages, such as t0 < t1 < ... < tns = tf . Then in each subintervals [tk−1, tk], the control
variable shall be approximate as:
u(t) = Uk
(t, ωk
) (2.15)
In this way it is possible to define the Lagrange interpolating polynomials of degree
M:
uj = Uk
j =
M
X
i=0
ωk
i,jφ
(M)
i (
t − tk−1
tk − tk−1
); tk−1 ≤ t ≤ tk (2.16)
where φ can be defined as:
φ
(M)
i (τ) =
1 ifM = 0
QM
q=0
q6=i
τ−τq
τi−τq
ifM ≥ 1
(2.17)
with collocation point 0 ≤ τ0 ≤ τ1 ≤ τM ≤ 1. These variables are depicted in the
Fig.2.9.
Figure 2.9: Collocation points graph
In the next section, a direct method example is reported: the Pseudospectral method.
21
42. Trajectory optimization 2. State of the art
Pseudospectral method
Pseudospectral methods are direct collocation where the optimal control is trans-
formed into a NLP technique by parameterizing the state and control. Some authors
call this approach as “orthogonal collocation”, but essentially the two terms refer to the
same concept. The parameterizations is made using polynomial and collocating the dif-
ferential algebraic equations (DAEs) using nodes [30]. The nodes are computed from a
Gaussian quadrature, in order to better approximate the integral values.
The authors in [30] cite three different methods which differ from each other for the
different form of Gaussian quadrature. These methods used different sets of collocation
points as Fig.2.10 shows:
1) Legendre-Gauss (LG), which includes neither of the endpoints;
2) Legendre-Gauss-Radau (LGR), which includes one of the endpoints;
3) Legendre-Gauss-Lobatto (LGL), which includes both of the endpoints.
Figure 2.10: Collocation points for the pseudospectral method
In literature, the widely studied methods are the Lobotta pseudospectral method and
the Gauss pseudospectral method.
The Gauss pseudospectral method consists in discretizing the state with Lagrange
interpolating polynomials and the Legendre Gauss points are used for orthogonality
collocate the dynamics [31], [32].
So instead of the common method, the dynamics are not collocated at the boundary
points.
22
43. 2. State of the art Trajectory optimization
In this method, the approximation of states is made using a N Lagrange interpolating
polynomials:
Li(τ) =
N
Y
j=0
j6=i
τ − τj
τi − τj
; i = 0, .., N (2.18)
So the state components can be approximated in this manner:
xN
j =
N
X
i=0
xijLi(τ) (2.19)
It is possible to differentiate the series and evaluate is at collocation points τk:
˙
xN
j (τk) =
N
X
i=0
L̇i(τk) =
N
X
i=0
Dkixij = (DX)ij (2.20)
where the DX is N x n matrix. In the same way, the state and control components
can be discretized and defined at the LG points. So collocating the dynamics at the N
LG points:
DX = F(XLG
, ULG
) (2.21)
Finally, the cost function is approximated with Gauss quadrature.
Also the path constraint is collocated at Legendre-Gauss points.
The state equations, the cost function, the boundary and path constraints allow to solve
the non linear problem.
In order to minimize the discontinuities in state and control, the trajectory can be di-
vided into phases which are connected through phase interface constraints [32].
23
45. Chapter 3
Preliminary performance analysis
3.1 Thermochemical computations
A deep analysis in term of thermochemical properties is necessary in order to support
the experimental studies mainly looking at flame temperature and combustion products.
The present thesis uses the CEA (Chemical Equilibrium with Application) code, a ther-
mochemical solver developed by NASA.
3.1.1 What is the CEA code?
CEA software computes chemical equilibrium starting from a set of reactants and
initial conditions. The outputs are the thermodynamic, chemical and transport properties
of the product mixture.
The equilibrium conditions are computed through the minimization of the free Gibbs
energy, which is state function applied at constant pressure and temperature.
In the minimization problem, also the mass constraint of the conservation of the atoms
for each species is applied. Generally, an iterative solver is required in order to obtain
the results for the non-linear equations [33].
The thermochemical computations are based on a fundamental assumption: the Hess
law. The last specified that the total thermal effect of a chemical transformation depends
only on the extreme states, the initial configuration and the final one. In particular, the
enthalpy change is the same if the reaction takes place in one or more steps. Since the
enthalpy is a state variable, the energy conservation can be applied also in the chemical
systems.
However, the heat change modifies the entropy. Even if the entropy is a state function,
the way its change is defined depends on the process. The variation ∆S = S(B)−S(A)
doesn’t depend on the path but only on the initial and end states, but this expression
equates the heat integral if the transformation is reversible. So entropy change, not the
25
46. Thermochemical computations 3. Preliminary performance analysis
absolute entropy, is independent on the path taken. In the chemical process since the
variation of entropy is equal to the difference between absolute product and reactants
entropies, the Hess law can be extended also to this state variable.
Thus, it is not necessary to study specifically the system evolution. Indeed, in this
work, only the reaction in the gas generator, initial state, and in the ramburner, final
state, are considered. This leads to analyze the reaction without separating the self-
sustained combustion of the propellants and the reaction in the ramburner involving the
intake air.
3.1.2 Definitions and formula
The computation of performances has to be done starting from the evaluation of
some propulsive parameters. From the momentum conservation, the uninstalled thrust
for an airbreathing system can be calculate. Some assumptions are done for the balance:
- inviscid flow, because the pressure contributions dominate over the stresses;
- steady state conditions;
- no body forces;
- 1D problem, only engine axis direction.
The uninstalled thrust is the difference between the engine thrust and the loss of mo-
mentum required to stagnate the intake air, which is called the ram drag. This term
(ṁinv∞) is increasing with the flight speed.
TAB = (ṁin + ṁf )ve + (pe − p∞)Ae − ṁinv∞ (3.1)
where ṁin and ṁf are respectively the inlet air mass flow rate and the injected fuel
mass flow rate; ve is the exit flow velocity and v∞ is the flight speed; pe and p∞ are the
pressure at the exit nozzle and the ambient pressure and Ae is the exit area.
As the Eq.3.1 shows, the thrust is composed by two contributions: a dynamic and a
static terms. The former is related to the momentum of the exit flow and the ram drag,
while the latter is the contribution dependent on the pressure.
If a optimum expansion design condition is considered, the exit pressure is equal to the
ambient one and so the static term disappears. This means that a complete expansion of
the combustion products is occurred.
As regards the rocket propulsion, the ṁin doesn’t exist, because no air mass is en-
tering the engine. For this reason the last term of the ram drag disappears. The thrust
expression is simplified as follows:
26
47. 3. Preliminary performance analysis Thermochemical computations
TR = ṁpve + (pe − p∞)Ae (3.2)
where all the terms are already mentioned with the exception of the ṁp which is the
mass flow rate of the propellant expelled by the nozzle. Also in this formulation it is
possible to distinguish the static and the dynamic (without the ram drag) terms.
In the airbreathing system case, the static term vanishes if an optimum expansion is
considered. In the formulation, all the terms suggest that the thrust is strongly dependent
on the choice of the nozzle, in particular on its geometry.
Linked to the thrust, also the thrust coefficient can be defined as:
cf =
TAB
1
2
ρiv2
∞A
(3.3)
where ρi is the inlet air density and A is the maximum section of the engine.
Another important parameter to be evaluated is the specific impulse. The last is con-
nected to the combustion reaction inside the chamber, indeed it is dependent on the
reactants and product compositions characteristics.
For an airbreathing system, it is expressed as the ratio between the uninstalled thrust
and the weight fuel flow rate:
IspAB
=
TAB
ṁf g0
(3.4)
which measured unit is [s].
Analogously, the specific impulse for the rocket can be defined:
IspR
=
TR
ṁpg0
(3.5)
Another important parameter is the volumetric specific impulse which is affected by
the density, thus the higher is the metal additives density, the higher is the volumetric
specific impulse:
Iv = Ispρ (3.6)
As already mentioned, the specific impulse depends on the energetic of the propel-
lant for the rocket, and on the mixture ratio for the ducted rocket. The mixture ratio is
the ratio between the fuel flow rate coming from the gas generator and the airflow rate
coming from the intake [2].
The CEA code provides propulsive performance results only for rocket propulsion
even when the ramjet problem is imposed. In this respect, the software can obtain the
27
48. Thermochemical computations 3. Preliminary performance analysis
combustion properties in the ramburner, according to the peculiar oxidizer-to-fuel ratio.
For this reason a relation between the airbreathing and the rocket systems performance
data is necessary.
For the rocket the expelled mass is equal to the propellant mass flow rate ṁe = ṁp; on
the other hand for the airbreathing systems the expelled mass results to be equal to the
sum of the inlet and fuel mass flow rates: ṁe = ṁi + ṁf . Since the basic goal is to
find a correlation between the expression of rocket and air-breathing system to compute
performances, the following relation can be obtained:
TR = TAR + ṁiv∞ (3.7)
Starting from this expression, also the relation for the specific impulse can be com-
puted.
Firstly, the ratio of air to fuel mass flow rate has to be introduced:
φ =
ṁa
ṁf
(3.8)
Now with some adjustments, the expression can be computed:
IspAB
= IspR
(1 + φ) −
v∞φ
g0
(3.9)
This expression of the specific impulse is valid for the ramjet, considering the rocket
propellant as the sum of the fuel and the air introduced by the intake.
It is possible to rewrite this relation according to the thermochemical properties obtained
from CEA calculations. Indeed, from the code the molar mass Mm, the flame tempera-
ture, which is the chamber temperature, Tc and the specific heat ratio γ are obtained.
For all these evaluations, some assumptions are selected such as, calorically perfect gas
mixture, steady-state isentropic flow and frozen chemistry during expansion. The last
implies that the time for reaction is higher than the time of the fluidynamic phenomena.
All the properties in the combustion chamber are now defined and the gravimetric spe-
cific impulse can be rewrite for the rocket as follows:
IspR
=
1
g0
s
2γ
γ − 1
R
Tc
Mm
(1 − (
pe
pc
)
γ−1
γ ) (3.10)
As regards the airbreathing system, the expression for the specific impulse becomes:
IspRJ
=
1
g0
s
2γ
γ − 1
R
Tc
Mm
(1 − (
pe
pc
)
γ−1
γ )(1 + φ) −
v∞φ
g0
(3.11)
28
49. 3. Preliminary performance analysis Performances analysis
Thus, as already mentioned, the combustion chamber is ruled by the specific impulse
parameter (∝
q
R Tc
Mm
) while the nozzle is dominated by the thrust value.
In this work, the optimum expansion nozzle design condition is chosen, with the exit
nozzle pressure equal to the ambient one, in order to avoid the static term in the per-
formance parameters expressions for CEA calculations. In addition, for all the thermo-
chemical computations, the chemically frozen expansion is assumed because it allows
a close algebraic formula. This choice delivers a conservative evaluation of the specific
impulse.
3.2 Performances analysis
In this section, the analyses made are reported. The thermochemical properties are
obtained by the CEA code, while the performances evaluation are computed using the
expression mentioned in the previous section.
The aim is to recognize which expressions can optimize the combustion chamber in
term of performances and how the metal additives, such as the magnesium, act during
the combustion process.
3.2.1 Selection of propellant compositions and parameters model
Starting from the selection of propellant, it is important to underline that this analysis
is based on the Zadra [9] and Colciago [34] previous theses.
In fact, the family of AP50 propellant, which are pyrolants with 50 % of ammonium
perchlorate, is already investigated by the previous works. This study is focusing on
the family of AP40 propellants, which are investigated according to their gravimetric
specific impulse. This is the figure of merit for the selection of the formulation analyzed.
In addition, another important parameter is the volumetric impulse since the propellant
is solid and it has to be stored in a compact way.
The decreasing of oxidant percentage can increase the specific impulse, this is the reason
why this family is investigated.
In Tab.3.1 the thermochemical properties for the ingredients considered are reported.
Besides the study of aluminum as metal additive, the magnesium behavior is investi-
gated in order to better understand its role in the combustion process.
First of all, a detailed knowledge of the thermal decomposition characteristic of
polybutadine binders is essential [35]. The composition of binder HTPB used in this
work is based on [2] as:
C7.075H10.65O0.223N0.063 (3.12)
29
50. Performances analysis 3. Preliminary performance analysis
Table 3.1: Thermochemical data of ingredients
Ingredients Density [kg/m3] ∆H f [kJ/mol] Reference tempereature [K]
AP 1950 -295.77 300
HTPB 920 -58 300
Al 2700 0 300
Mg 1738 0 300
AlH3 1490 -11.4 300
MgH2 1450 -75.2 300
As far as concerned the chemical composition of the ammonium perchorate, it is
formulated as follows: NH4ClO4.
The oxidizer chemical composition fits the nominal air gases percentage for all the
cases examined:
Table 3.2: Chemical composition of air, as oxidizer
Oxidizer
Chemical composition N2 O2 Ar CO2
Percentage [%] 78.09 20.9 0.93 0.01
The analysis starts from not considering metal content passing through the possible
propellant compositions with different metal additives concentrations.
For the propellant analysis, the chamber pressure has an key role therefore a pressure
model shall be implemented. The total pressure above the shock waves at the inlet,
which for assumption is equal to the combustion pressure, is computed by the relation:
πd = 1 − 0.075(Ma − 1)1.35
; 1 < Ma < 5 (3.13)
πd =
800
(Ma4 + 935)
; Ma > 5 (3.14)
The πd is the pressure ratio referred to the pressure losses through the shock across
the inlet.
30
51. 3. Preliminary performance analysis Performances analysis
The atmospheric standard model is used to compute the total pressure at the entry ac-
cording to the altitude (this model is explained deeply in the Chapter 5).
3.2.2 AP40 with no metal
As a first guess, the composition with no metal content is analyzed. In order to
accomplish this goal, the performances at different altitudes and flight speeds are con-
sidered. In Tab.3.3 the mission parameters considered for this analysis are collected.
Table 3.3: Mission profile parameters for AP40 with no metal content, assuming combustion
chamber stagnation conditions
z [km] M Ttot [K] Ptot [bar] Pc [bar]
10 1.5 323.8 0.9728 0.9442
10 2 401.9 2.0735 1.9180
10 2.5 502.4 4.5278 3.9407
10 3 625.2 9.7342 7.8732
10 3.5 770.4 20.2122 14.9895
10 4 937.9 40.2363 26.9381
10 4.5 1127.7 76.6948 45.4830
20 1.5 314.1 0.203 0.1970
20 2 389.9 0.4326 0.4002
20 2.5 487.3 0.9447 0.8222
20 3 606.5 2.0310 1.6427
20 3.5 747,3 4.2171 3.1274
20 4 909.7 8.3950 5.6204
20 4.5 1093.8 16.0017 9.4896
30 1.5 328.4 0.0439 0.0426
30 2 407.7 0.0937 0.0866
30 2.5 509.6 0.2045 0.1780
30 3 634.2 0.4397 0.3556
30 3.5 781.4 0.9130 0.6771
30 4 951.3 1.8175 1.2168
30 4.5 1143.8 13.4643 2.0545
40 1.5 363.1 0.0105 0.0102
40 2 450.7 0.0225 0.0208
40 2.5 563.4 0.0490 0.0427
40 3 701.1 0.1054 0.0853
40 3.5 863.9 0.2189 0.1623
40 4 1051.7 0.4358 0.2917
40 4.5 1264.5 0.8306 0.4926
31
52. Performances analysis 3. Preliminary performance analysis
Another important aspect is the chemical composition of the propellant selected for
the performance computations, which is reported in the Tab. 3.6.
Table 3.4: Chemical composition of propellant for AP40
Label AP [%] HTPB [%]
AP40 40 60
As the Tab.3.6 and 3.2 shows, the oxidizer is made by the air composition while the
fuel is the propellant composition in the gas generator. In this case, a lower specific
impulse is expected with respect to the case in which the metal is present.
Figure 3.1: Performances at M=2.5, for AP40 with no metal content
From Fig.3.1 to Fig.3.5, all the performances as a function of altitude and Mach
number are reported. In each plot, the variation of the chosen figure of merit, the specific
impulse, with the altitude is shown at constant flight speed, or Mach number.
Since the specific impulse is determined by mixture ratio (OF) of the fuel from gas
generator and the airflow rate from the intake, the specific impulse is not only driven by
the energetics of gas pyrolants but also by the air-intake from atmosphere.
32
53. 3. Preliminary performance analysis Performances analysis
Figure 3.2: Performances at M=3, for AP40 with no metal content
Figure 3.3: Performances at M=3.5, for AP40 with no metal content
33
54. Performances analysis 3. Preliminary performance analysis
Figure 3.4: Performances at M=4, for AP40 with no metal content
Figure 3.5: Performances at M=4.5, for AP40 with no metal content
In all plots, it is highlighted that the airflow rate decreases with the increasing flight
34
55. 3. Preliminary performance analysis Performances analysis
altitude because of the lower air density at the same flight velocity. So the specific
impulse decreases with altitude as a function of mixture ratio.
The graphs show the same specific impulse behavior for different flight altitude and
flight velocity with the exception of the higher altitude (40 km). The performances
increases normally with mixture ratio up to a certain value then it starts to increase
smoother due to the variation of the thermochemical parameters.
Since the specific impulse is strongly related to the gas generator pyrolant charac-
teristics (∝
q
Tc
Mm
), a graph of the Tc
Mm
has been reported for different altitude at Mach
equal to 2.5, (Fig.3.6), in order to better understand this particular path.
Figure 3.6: The Tc
Mm ratio as a function of mixture ratio for different altitudes at Ma = 2.5
The Fig.3.6 provides that at 40 km of altitude the ratio path is flatter with respect to
the other altitudes. For this particular case, this is the reason why the specific impulse
presents such a kind of behavior.
35
56. Performances analysis 3. Preliminary performance analysis
3.2.3 AP40 and Al
In this section, the case with the same amount of oxidizer and different percentage
of metal content is considered. The metal is the aluminum (Al). As in the previous case,
the mission profile is presented in Tab.3.5. For this situation, not all the Mach numbers
previously considered are analyzed, but only the situation with Ma = 3.5.
Table 3.5: Mission profile parameters for AP40 with Al
z [km] M Ttot [K] Ptot [bar] Pc [bar]
10 3.5 770.4 20.2122 14.9895
20 3.5 747,3 4.2171 3.1274
30 3.5 781.4 0.9130 0.6771
40 3.5 863.9 0.2189 0.1623
In Tab.3.5 the parameters of pressure inserted in the CEA code are reported, while
in Tab.3.6 the chemical compositions for the reactants are collected.
Table 3.6: Chemical compositions of propellant for AP40 with Al
Label AP [%] HTPB [%] Al [%]
AP40Al5 40 55 5
AP40Al10 40 50 10
AP40Al15 40 45 15
AP40Al20 40 40 20
Tab.3.6 shows that different percentages of Al metal content are considered, starting
from 5% (reducing the HTPB content and keeping the AP content constant) up to
20%. Thus as regards the performance, the Fig.3.7,3.8, 3.9 and 3.10 report the behavior
of the specific impulse as a function of the mixture ratio (φ = OF) with the variation
of metal content percentage at the different altitude. As already mentioned, the Mach
number is considered the same for all the cases (Ma = 3.5). As it is expected, in this
case the gravimetric specific impulse reached is higher with respect to the one computed
for the composition with no metal content.
Each plot represents the contributes of metal additives addition at different altitudes.
It is important to notice that the specific impulse decreases as the altitude increases for
the same mixture ratio value; however the increasing in metal concentration doesn’t give
36
57. 3. Preliminary performance analysis Performances analysis
Figure 3.7: Performances at h=10 km, for AP40Al
Figure 3.8: Performances at h=20 km, for AP40Al
37
58. Performances analysis 3. Preliminary performance analysis
Figure 3.9: Performances at h=30 km, for AP40Al
Figure 3.10: Performances at h=40 km, for AP40Al
many benefits to the specific impulse enhancement. Only for mixture ratio values quite
low, the metal addition can bring some improvement. Nevertheless, the metal presence
contributes in enhancement of ballistic properties and energy-density.
38
59. 3. Preliminary performance analysis Performances analysis
3.2.4 AP40 and Mg
This section is focused on the performance analysis of the propellant with the ad-
dition of magnesium (Mg). Just as the previous case, the flight conditions consider a
constant Mach number of 3.5 and a variable altitude.
Table 3.7: Mission profile parameters for AP40 with Mg
z [km] M Ttot [K] Ptot [bar] Pc [bar]
10 3.5 770.4 20.2122 14.9895
20 3.5 747,3 4.2171 3.1274
30 3.5 781.4 0.9130 0.6771
40 3.5 863.9 0.2189 0.1623
In Tab.3.7, the flight conditions are reported for the case studied. The chemical
composition of the gas generator pyrolant is collected in Tab.3.10, where the Mg content
is varying from 5% to 20% maintaining the AP concentration constant.
Table 3.8: Chemical compositions of propellant for AP40 with Mg
Label AP [%] HTPB [%] Mg [%]
AP40Mg5 40 55 5
AP40Mg10 40 50 10
AP40Mg15 40 45 15
AP40Mg20 40 40 20
The Fig.3.11, 3.12, 3.13, 3.14 show the performance with the Mg metal content.
The specific impulse is computed as a function of the mixture ratio varying the metal
content concentration at different altitudes.
As opposed to Al case, the curves of specific impulse increase slower with the mix-
ture ratio in the case of high metal content, while the composition with less Mg at high
mixture ratio behaves better.
For example as the first plot at 10 km of altitude is concerned, the behavior changes
at a 8 mixture ratio value. If the ratio Tc
Mm
is considered (Fig.3.15), there is smoother
variation of the flame temperature for the propellant composition with lower magnesium
content.
This temperature flame change can be attributed to the peculiar burning process of
39
60. Performances analysis 3. Preliminary performance analysis
Figure 3.11: Performances at h=10 km, for AP40Mg
Figure 3.12: Performances at h=20 km, for AP40Mg
the magnesium. With respect to the Fig.3.6, the highly magnesium content ratio de-
creases monotonically with the mixture ratio increasing; on the other hand, the propel-
40
61. 3. Preliminary performance analysis Performances analysis
Figure 3.13: Performances at h=30 km, for AP40Mg
Figure 3.14: Performances at h=40 km, for AP40Mg
lant with lower magnesium content behaves as expected.
41
62. Performances analysis 3. Preliminary performance analysis
Figure 3.15: Tc
Mm ratio at h=10 km, for AP40Mg
42
63. 3. Preliminary performance analysis Performances analysis
3.2.5 AP40 and AlH3
Another important energetic additives which can improve the gravimetric specific
impulse are the hydrides. Thanks to the presence of hydrogen (H) the products mo-
lar mass is expected to be lower than the case with only metal element. As already
explained, with the decreasing of the molar mass the specific impulse will increase
(Isp ∝
q
R Tc
Mm
).
For this reason this section is focusing on the aluminum hydride (AlH3).
The mission profile parameters have the same assumptions of the other cases. The input
pressure for the chemical equilibrium computations are collected in Tab.3.9.
Table 3.9: Mission profile parameters for AP40 with AlH3
z [km] M Ttot [K] Ptot [bar] Pc [bar]
10 3.5 770.4 20.2122 14.9895
20 3.5 747,3 4.2171 3.1274
30 3.5 781.4 0.9130 0.6771
40 3.5 863.9 0.2189 0.1623
The percentage of energetic additives is considered 5%-20% as shown in Tab.3.10.
Table 3.10: Chemical compositions of propellant for AP40 with AlH3
Label AP [%] HTPB [%] AlH3 [%]
AP40AlH35 40 55 5
AP40AlH310 40 50 10
AP40AlH315 40 45 15
AP40AlH320 40 40 20
For this situation, as previously the specific impulse increases with the mixture ratio
as the hydride content increases, as confirmed in Fig.3.16, 3.17, 3.18 and 3.19.
But let’s focus on the enhancement of the performances due to the presence of the
hydrogen which allow a decreasing in products molar mass. With respect to the case
with only aluminum inside, the hydride products molar mass present a lower value as
shown in Fig.3.21 and Fig.3.20: passes from about 29 g/mol to less than 28 g/mol for
the highest concentration case. In particular as the hydride concentration increases, the
molar mass decreases.
43
64. Performances analysis 3. Preliminary performance analysis
Figure 3.16: Performances at h=10 km, for AP40AlH3
Figure 3.17: Performances at h=20 km, for AP40AlH3
It can be also observed that the variation of additives concentration is more evident
for the Al case with respect to the hydride one; so the aluminum hydride content doesn’t
44
65. 3. Preliminary performance analysis Performances analysis
Figure 3.18: Performances at h=30 km, for AP40AlH3
Figure 3.19: Performances at h=40 km, for AP40AlH3
affect very much the performance behavior. In fact, for the aluminum case the molar
mass variation is about 2 g/mol from the lowest metal content up to the higher one;
whereas for the hydride case, this variation is only about 1 g/mol.
For the Tc
Mm
ratio, the hydride aluminum and the aluminum plots do not differ so
45
66. Performances analysis 3. Preliminary performance analysis
much. In fact for the AlH3 case, the temperature flame is lower then the Al because of
the less energy content. As a result the overall ratio is not so different for the two cases.
The hydride doesn’t enhance much the performances with respect to the Al case.
Figure 3.20: Products molar mass at h=10 km, for AP40AlH3
Figure 3.21: Products molar mass at h=10 km, for AP40Al
46
67. 3. Preliminary performance analysis Performances analysis
3.2.6 AP40 and MgH2
The other hydride analyzed in this work is the magnesium hydride (MgH2). As for
the aluminum hydride case, the products lowered molar mass allows an increasing of
the gravimetric specific impulse.
The mission profile respects the already mentioned assumptions, and the parameters are
recap in Tab.3.11.
Table 3.11: Mission profile parameters for AP40 with MgH2
z [km] M Ttot [K] Ptot [bar] Pc [bar]
10 3.5 770.4 20.2122 14.9895
20 3.5 747,3 4.2171 3.1274
30 3.5 781.4 0.9130 0.6771
40 3.5 863.9 0.2189 0.1623
The chemical compositions are presented in Tab.3.12 and the performances are re-
ported in Fig.3.22, 3.23, 3.24 and 3.25.
Table 3.12: Chemical compositions of propellant for AP40MgH2
Label AP [%] HTPB [%] MgH2 [%]
AP40MgH25 40 55 5
AP40MgH210 40 50 10
AP40MgH215 40 45 15
AP40MgH220 40 40 20
As in the Mg case, the specific impulse increases with hydride content until a certain
mixture ratio value and then it decreases with the increasing of the hydride content. The
temperature flame behavior drives this particular path. With respect to the Mg solution,
the magnesium hydride is characterized by this transition for lower mixture ratio values.
Also for the hydride magnesium case, it is expected to obtain an increasing of the
specific impulse with respect to the Mg case due to lower products molar mass. How-
ever on the other hand, the decreasing of flame temperature has an important role in
the performances enhancement. Basically the MgH2 acts as the aluminum hydride: the
final result is a Tc
Mm
ratio comparable to the Mg case.
47
68. Performances analysis 3. Preliminary performance analysis
Figure 3.22: Performances at h=10 km, for AP40MgH2
Figure 3.23: Performances at h=20 km, for AP40MgH2
48
69. 3. Preliminary performance analysis Performances analysis
Figure 3.24: Performances at h=30 km, for AP40MgH2
Figure 3.25: Performances at h=40 km, for AP40MgH2
49
70. Final considerations 3. Preliminary performance analysis
3.3 Final considerations
In order to support this theoretical analysis, the ducted rocket OF range of appli-
cation is an important investigation aspect. In [36], the author analyzes the equiva-
lence ratio, which is the ratio between stoichiometric mixture ratio and current one
(Φ = OFstoic
OF
). He focuses on the equivalence ratio effects on combustion in a ducted
rocket. Since above the unity equivalence ratio the particles burn incompletely, the typ-
ical operational range of a ducted rocket is within 0.5 and 1 values of Φ.
In this study, the propellant is chosen as a compromise between the different com-
positions analyzed. In fact a fuel-rich propellant composition with aluminum and mag-
nesium as metal additives is selected. As the plots reveals, the addition of magnesium
doesn’t affect significantly the specific impulse value, but thanks to its chemical prop-
erties, it can improve the ignitability of the propellant. Since the stoichiometric mix-
ture ratio value is about 4, the operational range of the selected composition does not
overcome the 8-10 mixture ratio values. So the performances just examined should be
contextualized for the real ducted rocket operational envelope.
50
71. Chapter 4
Materials and experimental set-up
4.1 Materials
In order to compute the burning rate for different propellant compositions, it is nec-
essary to produce them. Basically, in this work the solid propellants are prepared to be
tested.
The composition choice is driven by the theoretical analysis and previous thesis work.
Indeed, the propellant investigated in this thesis are:
Table 4.1: Propellants composition
AP [%] FAP [%] Binder [%] Al (type II) [%] Al (type III)[%] Mg [%]
34 6 24 28.8 0 7.2
34 6 24 0 28.8 7.2
Let’s focus on each chemical components of the oxidant-lean propellant, starting
from the binder up to the metal additives.
4.1.1 Binder
Binders are used in composite propellant in order to held together all the solid ingre-
dients providing a structure. It acts as a fuel which reacts with oxidant in the combustion
process.
Polyurethane reaction
The polyurethane is produced by an exothermic reaction between an isocyanate (R−
(N = C = O)n) and a polyol containing hydroxyl groups (R0
− (OH)n) as Fig.4.1
51
72. Materials 4. Materials and experimental set-up
shows.
Figure 4.1: Polyurethane reaction
As in [37] is explained, the reaction of isocyanate with polyol can be performed with
a catalyst or not. The two kind of reactions are quite similar but differ in the type of
applications.
The most important parameter which imposes the polyurethane chemical characteristics
is the NCO
OH
ratio. If the ratio is equal to one, it means that both isocyanate and polyol
groups are bound together, while if it is far away from identity, some vacant will be
remained.
Since this ratio affects the chemical composition, also the mechanical properties are
driven by this parameter.
HTPB
HTPB is long-chain, cross-linked and high molecular weight polymer. Its heating
rate affects strongly the pyrolisis of HTPB [38].
If a low heating rate is considered, the pyrolisis occurs into two different phases. The
first phase is depolymerization, cyclization and crosslinking of material [35], forming
monomer butadiene, cyclopentene, 1.3-cyclohezadiene and 4-vynilcyclohexene as the
main gaseous products [38]. The second stage consists in the decomposition of the
residue yielded in the first stage.
If higher heating rate is considered, the first stage prevails, with depolymerization as
the main degradation process.
In ducted rocket engine, since HTPB is exposed at very high temperature (above 2000
K), pressure and heating rate, there is no time for exothermic cross linking and cycliza-
tion to occur. For this reason, the dominant process is the depolymerization one.
Hydroxyl-terminated polybutadiene (HTPB) is an oligomer of butadiene terminating
at each end with hydroxyl functional group. HTPB can react with isocyanates in order
to create polyurethane polymers. In solid propellant applications, it binds the oxidant
and the metal additives into a solid and elastic composition mass.
There are many advantages in using HTPB [38]: excellent hydrolytic stability, low wa-
ter absorption, excellent low temperature flexibility, high compatibility with fillers and
extenders, and formulation flexibility. In Fig.4.2 the chemical structure is represented
while in Tab.4.2 the properties are collected.
52
73. 4. Materials and experimental set-up Materials
Figure 4.2: HTPB chemical structure
Table 4.2: HTPB chemical properties
Chemical properties
Molecular weight [ g
mol ] 2800
Density [ kg
m3 ] 920
Hydroxyl functionality 2.464
Hydroxyl value [meq
g ] 0.88
IPDI and TIN in DOA
IPDI is the isophorone diisocyanate, which has the role of the curing agent. This
particular kind of isocyanate differs in its reactivity with respect to the other type of the
same family, due to the difference in the location of −NCO group.
The ratio NCO
OH
drives the curing level.
In SPLab, for the production of lean-oxidant propellant with AP40 a curing level of 1 is
chosen in order to provide the required mechanical properties. In fact, if a curing level
lower than one is considered, the mechanical properties are poorer with respect a case
in which a unity level is chosen.
The chemical formulation of the IPDI is the following and the chemical structure is
reported in Fig.4.3:
C12H18N2O2 (4.1)
As regards the chemical properties, the Tab.4.3 collects all the chemical details.
As far as concerned the DOA, dioctyl adipate, it acts as a plasticizer. The TIN in
DOA used in this procedure has 0, 01% of concentration ( TIN
DOA
= 0.01%). The TIN has
a role of reaction catalyst.
53
74. Materials 4. Materials and experimental set-up
Figure 4.3: IPDI chemical structure
Table 4.3: IPDI chemical properties
Chemical properties
Molecular weight [ g
mol ] 222.3
Density [ kg
m3 ] 1062
Melting point [◦C] -60
Boiling point [◦C] 140
The chemical properties of DOA are reported in Tab.4.4 and its formulation is the fol-
lowing:
C22H42O4 (4.2)
Table 4.4: DOA chemical properties
Chemical properties
Molecular weight [ g
mol ] 370
Density [ kg
m3 ] 920
Melting point [◦C] -67.8
Boiling point [◦C] 158
For the composition of the propellant produced, the binder has the chemical percent-
age reported in Tab.4.5:
54
75. 4. Materials and experimental set-up Materials
Table 4.5: Binder chemical composition
Binder chemical composition
HTPB R-45 79.21 %
IPDI 7.68 %
TIN in DOA 13.11%
4.1.2 Metal additives
As far as concerned the energetic additives, metal powders are used in order to in-
crease performances. The metal selected came from the results of Zadra [9] thesis work.
His results show how can be increased the ignitability with the addition of the magne-
sium, and how differently the propellant behaves with the different aluminum type.
In this work, the composition with both material is studied deeply.
For the magnesium a 325 mesh 44 µm size is chosen, while for the aluminum different
type are analyzed. Indeed a type III (30 µm) and type II (15 µm) aluminum are chosen
for the experimental analysis.
Aluminum and magnesium powders
The literature has no investigated deeply the effect of Al-Mg powders additives in
the solid propellant compositions. In [39], the author explains the burning procedure of
these metal additives. Also in [40], experiments are reported in order to better under-
stand the behavior of the alloy.
From these studies, the burning process can be analyzed. It consists of two stages: the
magnesium burns in the first step while the aluminum in the second one.
The main idea is that the magnesium burning is dominant in first step, in fact when
the magnesium is exhausted the burning aluminum rate increases. In the first stage, the
flames forms an inhomogeneous luminescence zone of products which surrounds the
particles [39], in addition the particles dimension and flame zone remain constants dur-
ing burning process.
When this step is concluded, the second stage can start. This is characterized by the
aluminum combustion since it has great similarity with the well-know aluminum alone
combustion.
If the magnesium content is greater than 30 %, the transition between the first and the
second step is continuous, while if it is less than 30 % a discontinuity characterizes the
transition process.
Other important conclusions are reported in [39]: the dimension of stationary burning
55
