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Università degli Studi di Roma “La Sapienza”
Scuola di Ingegneria Aerospaziale
Dottorato di Ingegneria Aerospaziale
XV Ciclo
MODELING AND NUMERICAL SIMULATION OF
IGNITION TRANSIENT OF LARGE SOLID ROCKET
MOTOR S
Dottorando
Ferruccio Serraglia
Tutore
Prof. Marcello Onofri
Correlatore
Prof. Maurizio Di Giacinto
Correlatore
Prof. Bernardo Favini
Anno Accademico 2002/2003
2
3
Index
INTRODUCTION
1. PHENOMENOLOGY OF THE IGNITION TRANSIENT 10
1.1 GENERAL DESCRIPTION 10
1.2 IGNITION AND COMBUSTION OF SOLID PROPELLANTS 14
IGNITION 14
STEADY-STATE BURNING 19
TRANSIENT AND EROSIVE BURNING 21
1.3 MODELING SRM IGNITION TRANSIENT 23
1D MODELS: STATE OF THE ART 23
GENERAL DISCUSSION 29
2 PHYSICAL AND MATHEMATICAL MODEL 32
2.1 GASDYNAMIC MODEL 32
2.2 IGNITER MODEL AND IMPINGEMENT REGION 36
2.3 PROPELLANT SURFACE HEATING, IGNITION AND REGRESSION 41
IGNITION CRITERION 41
CONVECTIVE HEAT TRANSFER: IMPINGEMENT REGION. 41
CONVECTIVE HEAT TRANSFER: STANDARD REGION 44
RADIATIVE HEAT TRANSFER 45
PROPELLANT HEATING EVALUATION 48
BURNING RATE MODEL 51
SURFACE REGRESSION EVALUATION 52
3 NUMERICAL INTEGRATION TECHNIQUE 54
3.1 DISCRETIZED FLUID DYNAMIC MODEL AND SOLUTION 54
GODUNOV’S METHOD 55
APPLICATION OF THE MODIFIED ENO METHOD TO THE IGNITION TRANSIENT ANALYSIS. 57
RIEMANN PROBLEM 61
3.2 THE ESPRIT SIMULATION CODE 63
4 RESULTS 66
4.1 HEAD-END PRESSURE AND IGNITION SEQUENCE 66
SOLID ROCKET MOTOR # 1 ARIANE 4 SOLID BOOSTER (SRM #1) 66
SOLID ROCKET MOTOR # 2 ARIANE 5 SOLID BOOSTER (SRM #2) 68
SOLID ROCKET MOTOR # 3 VEGA, ZEFIRO SOLID ROCKET STAGE (SRM #3) 71
4.2 OVERPRESSURE PEAK AND EROSIVE BURNING 74
4.3 FLOWFIELD FEATURES 80
4.4 INTERSEGMENT SLOTS AND GAS MIXTURE 87
EFFECTS OF THE SLOTS & SUBMERGENCE MODELING 89
EFFECT OF GAS MIXTURE MODELING 90
CONCLUSIONS 93
ACKNOWLEDGMENTS 95
REFERENCES 95
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5
Nomenclature
Ajet Igniter jet final cross section m2
AP Propellant port area m2
Ath Nozzle throat cross section m2
At ig Igniter nozzle throat cross section m2
cf Friction coefficient m
cp Specific heat at constant pressure per unit mass J kg–1
K-1
cv Specific heat at constant volume per unit mass J kg–1
K-1
Djet Impingement jet diameter on grain surface m
Dh Hydraulic diameter m
Dtig Igniter nozzle throat diameter m
e Total internal energy per unit mass J kg-1
f1 , f2,f3 Erosive burning functions
FRC Flame radiation coefficient
GRC Gas radiation coefficient
Hig Igniter gas total enthalpy per unit mass J kg-1
Hj Distance between igniter nozzle and propellant surface m
Hf Propellant gas enthalpy per unit mass J kg-1
Hs Slot gas flow total enthalpy J kg-1
hc Heat transfer convective coefficient J kg-1
s-1
m-2
hc imp Heat transfer convective coefficient at impingement region J kg-1
s-1
m-2
hf Propellant gas enthalpy per unit mass J kg
-1
hR center Central cell radiative transfer coefficient J kg-1
s-1
m-2
hR right Right-side cell radiative transfer coefficient J kg-1
s-1
m-2
hR left Left-side cell radiative transfer coefficient J kg-1
s-1
m-2
Rh Equivalent radiative transfer coefficient J kg
-1s
-1 m
-2
KMR Impingement region proportionality constant
kpr Propellant thermal conductivity N s-1
K-1
keb Erosive combustion constant m3 K J
-1
Mj Igniter jet mach number
m ig Igniter gas mass flow rate kg s-1
m s Total mass flow rate from slots kg s-1
m s,pr Propellant gas mass flow rate from slots kg s-1
m s,ig Igniter gas mass flow rate from slots kg s-1
m s,in Inert gas mass flow rate from slots kg s-1
Nnz Number of igniter nozzles
Nigz Number of canted igniter nozzles
Naxig Number of axial igniter nozzles
n Combustion index
Pr, Prig Propellant and igniter gas Prandtl numbers
Pb , Pw Combustion and wet perimeters m
p , p0 Static and total pressure N m-2
pc Combustion chamber equilibrium pressure N m-2
pcig Igniter chamber total pressure N m-2
Pref Reference pressure for burning rate N m-2
qC Convective heat flux w m-2
qr center Gas radiative heat flux from central cell w m-2
qr right Gas radiative heat flux from right-side cell w m-2
qr left Gas radiative heat flux from left-side cell w m-2
qt Total heat transfer flux w m-2
6
rightr q Equivalent radiative heat flux from right-side cell w m
-2
leftr q Equivalent radiative heat flux from left-side cell w m
-2
Rej Igniter jet Reynolds number
r, r0ref Burning rate and reference burning rate m s-1
r0 Saint-Robert pressure dependent burning rate m s-1
reb Erosive burning rate m s-1
Sb Grain propellant burning surface m2
T Static temperature K
Tf Flame temperature K
TS0, TS Initial and actual propellant surface temperatures K
t Time s
u Velocity component in the x-direction m s-1
V Igniter control volume m3
vinj Igniter gas velocity m s-1
x Longitudinal coordinate m
Xj Coordinate of igniter jet stagnation point m
Xi Coordinate of the i-th cell center
Z Erosive burning rate exponent
ig Inclination of igniter jet axis Rad
pr Propellant thermal diffusivity m2 s
-1
1 , 2 Function in the exponent of erosive burning rate formula
ii Radial distance from igniter jet stagnation point m
Gas viscosity N s m2
Gas density kg m-3
pr Propellant gas density kg m-3
ig Igniter gas density kg m-3
in Inert gas density kg m-3
p Solid propellant density kg m-3
Stefan-Boltzman constant w-2
K-4
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INTRODUCTION
The material presented in this doctorate thesis represents the synthesis of the research activities
carried out by the author, during last years, at the Department of Mechanics and Aeronautics of the
University of Rome “La Sapienza”; results and proceedings of these activities have also been
presented in several international meetings and conferences 39, 40, 41, 42, 43
.
The aim of this work is to present the formulation of a detailed physical-mathematical model, and
its numerical solution, for the analysis of the ignition transient of large solid rocket motors (SRM);
the result obtained is a complex simulation code able to deal with different rocket engines and
igniters, capable of numerical results that show a very good agreement with available experimental
data.
The operating conditions of the solid rocket motors during the ignition transient are often more
critical than those occurring in the subsequent operative phases; within a fraction of a second in the
combustion chamber extreme events such as pressure jumps of tenths of atmospheres and
temperature peaks of thousands of Kelvin degrees can occur. The ignition transient in fact is an
operative phase of the engine characterized by a very strong unsteady behavior that often causes
net thrust and pressure transients, possible overpressures, possible hang-fires or misfires, transients
of pressure loads on the propellant grain, dynamic loads on the launch vehicle and the ground
structure etc.. All these phenomena can seriously compromise not only the engine performance but
the overall efficiency of the whole system. The capability to predict and evaluate the above
phenomena by means of a numerical simulation model can significantly increase the efficiency of
the preliminary design process with a reduction of both the motor development and operational
costs.
Modeling of this phenomenon has for many years been conducted using zero-dimensional, volume-
filling or one-dimensional analyses, and only recently two-dimensional analyses. Currently, with
the increasing of computational capabilities, three-dimensional fully coupled flow/structural
ignition models are being developed. Although this kind of simulations represent a significant
improvement for the analysis of the internal ballistics and other major issues in SRM operations,
for what strictly concerns ignition transient, unfortunately, the results obtained by such a great
effort often appear insufficient, and do not match with the experimental measurements better than
simpler schematizations1. Also, numerical factors such as insufficient grid resolution, or inadequate
integration schemes has been invoked as a cause of this disagreement, nevertheless inadequacy of
physical modeling is still the real limit of direct simulations2.
The main problem of the simulation of the ignition transient is, substantially, the lack of universally
accepted theoretical models and experimental data about the main physical phenomena that affect
this operative phase of the motor; this is mostly due to the peculiar behavior of each propellant and
the high unsteadiness and the strong interactions that exist among heat transfer, combustion and
fluid-dynamics.
In the open literature, a usual approach for the analysis of the SRM's ignition transient is based on a
simplified 1-D schematization of the motor chamber geometry and for the solution of the chamber
fluid-dynamics (for example, most models completely neglect the presence of the igniter), coupled
with few simplified submodels for the simulation of the main physical phenomena involved3.
8
Simple correlations require the identification of many empirical coefficients that have to be
introduced in the model in order to obtain a good agreement between the experimental and
numerical results in each single motor. These empirical coefficients, that often take the form of
complicated functions of both space and time, relies on "a posteriori" calibrations of empirical
laws, rather than on a reliable mathematical modeling of the physics, and, as a consequence, the
model overtakes the declared physical phenomenology. In fact, whereas these models are well
suited to investigate the influence of slight modifications of the already existing motor
configuration they have been calibrated for, they show a significant decay of their prediction ability
when they are applied to the preliminary design of a new different motor.
However, we deem that an unsteady quasi-one-dimensional approach, providing accurate and
rational physical models, could be still pursued in order to develop an effective simulation tool for
the analysis and prediction of SRM ignition transients. Indeed, this model should be able to
adequately represent the strong wave propagation phenomena that affect the flowfield inside the
combustion chamber and the interactions with the phenomena that characterize the flow-field,
during this operative phase of the motor, such as mass addition from the burning propellant grain,
etc.
Therefore, the major task is to identify and isolate the most significant “ingredients” for the
construction of a simplified model able to represent the solid rocket motor during the ignition
transient. So the gasdynamic model is coupled with several suitable semi-empirical submodels that
take into account the main and most significant phenomena affecting the ignition transient, such as
a detailed igniter model, models for the evaluation of the heat exchange by conduction, convection
and radiation, propellant heating, ignition and combustion, flame spreading, burning rate variations,
etc.
The goal is to substantially reduce the number of ad hoc modeling assumptions; of course,
calibration of the individual submodel is still necessary, but this can be done just once and for all,
and this is very far from the definition of the space and time-dependent functions often found in
open literature.
In summary, the formulation of the simulation model is enriched by the identification of the
physical mechanisms that drive each of the main characterizing phenomena, which are handled by
suitable mathematical models. In this regard, the availability of experimental data obtained by
different motor concepts and configurations is mandatory. The expected accuracy in the simulation
of each single motor is usually lower than in the previous approach, but, generally, the capability of
a blind prediction, without any "a priori" information on the transient behavior, should be better.
The main objective is then to develop an accurate simulation of the ignition transient for very
different engine configurations, in terms of size, performance and shape of the chamber, geometry
and placement of the propellant grain, thereby exhibiting different characteristic times and
sensibilities to each of the main physical phenomena.
In conclusion, in the model development the following basic specifications have been assumed:
comparable accuracy in the prediction capability from motor to motor;
correct sensitivity of the model to the main design options;
low computing cost, thus allowing to carry out through large parametric analyses.
The presentation of the work is organized as follows:
9
Chapter 1 consists in a brief phenomenological description of the ignition transient and the
main physico-chemical phenomena involved, followed by a discussion about the main issues
and the state-of-the-art of modeling SRM ignition.
Chapter 2 is dedicated to the description of the physical and mathematical model developed for
our simulation code.
Chapter 3 deals with the numerical integration technique adopted and a schematic description
of the code.
Chapter 4 is dedicated to the presentation and the discussion of the results obtained by means
of our simulations.
10
1. PHENOMENOLOGY OF THE IGNITION TRANSIENT
1.1 GENERAL DESCRIPTION
The study of the ignition transient of solid rocket engines is fundamental to an accurate prediction
of the time histories of several physical parameters, first of all the chamber pressure, from the very
start-up of the motor until reaching of design operative conditions (quasi-steady phase).
A simple scheme of a generic SRM is shown in Fig.1.
Fig.1 Generic SRM scheme and cross-sections
Although this operative phase is very short compared to the overall combustion time of the engine
(Fig.2), the occurrence of several strongly time-dependent phenomena, such as mass addition from
the igniter, wave propagation, heat and mass exchange between the flowing gas and propellant
grain, flame spreading on propellant surface, chamber filling, etc., contributes to make the ignition
transient a very critical and complex operative phase. Pressure time history, chamber filling delays
and maximum pressure gradients are strongly dependent on the geometrical and operative
characteristics of the engine, the igniter and on the typology of the propellant grain.
Fig.2 Head-end pressure time-history (whole operative life)
11
During the ignition transient, sudden variations in the flow conditions inside the chamber, caused
by violent pressurization and, consequently, strong static and dynamic, structural and thermal loads
on the engine and the launcher system in general, have been the cause for many launch mission
failures (e.g. Challenger, SRMU-PQM1)2.
The ignition transient of a solid propellant rocket is defined as the time interval between the first
electric signal for the motor start-up and the attainment of the steady design operative condition of
the engine; it can be divided into four different phases (Fig.3):
1. electric delay: in between the application of the electric signal to the start-up of the motor
igniter (few milliseconds);
2. induction interval: in between the start-up of the igniter and the very first detection of the
flame on the propellant surface (about 1/5 of the ignition transient).
3. flame spreading: in between the first detection of the flame on the propellant surface and the
instant of the complete ignition of the grain (about 1/5 of the ignition transient).
4. chamber filling: in between the complete ignition of the propellant grain and the reaching of
the “equilibrium condition” into the combustion chamber, corresponding to the attainment of
the design equilibrium pressure characterizing the quasi-steady operative phase of the engine.
During this phase, that takes about 3/5 of the whole ignition transient, the pressure rises by a
factor of 5.
Particular relevance, being one of the main issue in the study of the SRM ignition transient, has the
appearance of an overpressure peak that is considered always as the reference upper limit for the
structural resistance design of the motor case; in the current literature the overpressure peak is
typically considered to be the result of a dynamic coupling phenomenon among the sudden
chamber pressurization, complex transient burning phenomena and the enhancement of the burning
rate due to erosive burning, caused by high flow velocities. This particular behavior is typical of
the HVT (high velocity transient) motors that have a very low area ratio between the chamber port
area and the nozzle throat section, Ap/Ath.
Fig.3 Head-end pressure time-history (ignition transient)
12
Otherwise, as it will be shown later from the results of our simulations, this phenomenon is also
strongly dependent on the igniter operation and on the particular local conditions established in the
impact zone of the igniter jets on the propellant surface (impingement region), even for non-HVT
engines.
The sequence of the main phenomena occurring during the ignition transient can be summarized as
follows.
Induction
After the electric signal application, the igniter (we will consider only pirogen igniters) injects into
the combustion chamber hot gases, liquid and solid particles generating a travelling shock wave
that propagate through the combustion chamber. The igniter jets typically impinge on the
propellant surface: if impingement is strong, it generates an oblique shock that enhances surface
heating; the localization of the impingement region depends on the number, the location and the
inclination of the igniter nozzles, on the penetration capabilities of the jets and the geometry of the
propellant grain. The balance between the main chamber pressure and the pressure inside the
igniter chamber determines the extension and the shape of the impingement region, and the
intensity of the impinging jets.
Fig.4 Hoop strain gages measurements2
Usually, there is a protection diaphragm (or weather seal) located in the nozzle throat section of the
motor.
When the shock wave, generated by the igniter, reaches the bottom-end of the chamber and the seal
in the throat section of the nozzle, it is partly reflected and partly, after the breaking of the seal,
passes through the nozzle. There are several experimental evidences that clearly show the igniter'
shock propagation and its multiple reflections inside the combustion chamber, for example in Fig.4
we can see measurements from hoop strain gages applied on a Space Shuttle SRM, during a static
firing2. The presence of the igniter shock and its successive reflections along the chamber is
evident.
Propellant grain ignites when proper conditions of pressure, temperature and concentration of the
reacting species occur near the surface, are able of activating a self-sustained combustion reaction.
13
The heat exchange between the hot impinging igniter gases and the propellant surface, mainly due
to convection, increases the temperature of the solid grain and then cause the ignition of the
impingement region.
Flame spreading
After attaining self-sustained reaction, the propellant grain of the impingement region begins to
burn. Combustion reactions at the propellant surface produce additional hot gases; the mass flow of
both hot gases, from the igniter and from the already ignited grain surface, increases the heat flux
(by convection, gas radiation and conduction) towards the unignited propellant surface. In addition,
the flame located at the surface of the ignited grain region radiates towards the surrounding
propellant surfaces. Both phenomena, combustion and radiation, contribute to the spread of the
combustion process over the whole grain surface. In Fig.5 the spatial and temporal characterization
of the flame spreading is shown. This curve has been obtained by means of a numerical simulation
of the ignition of a large motor (SRM#1) described later. An early ignition occurs around the
impingement region, subsequently most part of the motor is ignited with a sudden propagation of
the flame towards the nozzle, while the head-end region is ignited more slowly.
Fig.5 Ignition time sequence
Hot gases generated by the igniter and the already ignited grain region gradually push the inert gas,
originally filling the combustion chamber, toward the diaphragm seal in the nozzle throat. At a
prescribed value of the pressure gap between the combustion chamber and the external
environment, the diaphragm bursts with the onset of a wave system similar to that generated by a
shock tube. The gas inside the combustion chamber is suddenly accelerated towards the nozzle and,
after a short time, the choking condition at the throat section is attained.
Chamber filling
When the igniter is extinguished, the propellant grain surface is completely ignited, the wave
propagation phenomena inside the combustion chamber are significantly damped, and a quasi-
steady operation of the motor is usually attained after a short time.
14
1.2 IGNITION AND COMBUSTION OF SOLID PROPELLANTS
A comprehensive description of solid propellant combustion is beyond the scope of the present
work; we only briefly present a general discussion about the ignition and combustion characteristic
of solid propellants, merely from the point of view of modeling SRM ignition transient.
Many different chemicals have been used in making solid propellants. Different compositions are
chosen to obtain optimized combustion characteristics for different purposes. There are two major
types of propellants which give different combustion characteristics, burning rate, specific impulse,
flame temperature and molecular weight of the combustion products; those in the first class are
called double-base or homogeneous propellants, because oxidizer and fuel are linked chemically in
the same molecule in a homogenous physical structure. Nitrocellulose is a typical example of
homogenous propellant. The second type are called heterogeneous or composite propellants
because their structure is heterogeneous and consist of a physical mixture of oxidizer and fuel.
Typical composite propellants are based on ammonium perchlorate, ammonium nitrate and
potassium perchlorate as oxidizers, polybutadiene or polyuretane that act as fuels and binders.
In general, solid propellants consist of several chemical ingredients such as oxidizer, fuel, binder,
plasticizer, stabilizer, curing agent, crosslinking agent, bonding agent, burning rate catalyst,
antiaging agent, opacifier, flame suppressant, and combustion instability suppressant.
Their complex composition makes solid propellants complicated systems in any aspect of
combustion that may be of interest. Quite naturally, there is a tendency for the investigator of a
particular aspect of solid propellant combustion to simplify the characterization of the process or
the propellant in some manner that retains the essential characteristic of interest and simplifies the
rest. The applicability of the results of this selective characterization depends to a large extent on
the ability of the modeler to make the appropriate selection of essential characteristics, suitable to
his own particular investigation.
Ignition
Ignition of a generic solid propellant is the process occurring in between initial application of some
energetic stimulus to a quiescent chunk of propellant and full-scale combustion. The term
“ignition” is used commonly in two ways:
1. the process of achieving full-scale combustion;
2. the point in time at which sufficiently full-scale combustion is deemed to occur.
Interpretative difficulties are possible, therefore, between what ignition means among practical
systems; even if these difficulties are recognized, whether they are resolved in the various theories
and experiments in a manner satisfactory to practical situations frequently depends on the intent of
the investigator and the degree of involvement with practical engineering.
However, it is possible to say that ignition of the propellant is a transient process with a definite
point of initiation but an endpoint that depends completely upon the definition of that endpoint
itself 4.
Whatever the form of the energetic stimulus, that may be chemical, thermal or even photochemical,
transient thermal processes involving net exothermic chemical reactions, together with transient
diffusional and chemical changes in reactant concentration, are inherent in the ignition process and
are necessary for ignition to be achieved.
15
The total ignition process therefore involves a transition from a non-reactive to a reactive state via
some thermochemical “runaway” in time, followed by an essentially equally rapid transition to full-
scale self-sustained combustion. Practically all studies of solid propellant ignition are concerned
with the time required for the thermal “runaway” to occur and the mechanism, or sequence of them,
responsible of that.
Solid propellant ignition is a very complex phenomenon that involves several chemical and
physical processes (see also Fig.6), for example:
energy transfer by conduction, convection and radiation;
phase transition;
binder fuel pyrolisis;
multi-phase chemical reactions, with energy release;
development of the combustion wave and the flame zone.
In a solid rocket motor the application of the external energy stimulus, necessary for ignition to be
achieved, is obtained typically by the igniter that, as already mentioned, generates hot gas jets that,
whether or not directly impinging on the propellant, increase the surface temperature of the grain,
and thus locally activate the combustion reaction. When the increasing heat feedback from these
chemical reactions reaches the level sufficient to keep the process going on, self-sustained
combustion takes place and the propellant keeps on burning even if the external energy source is
removed.
Fig.6 Solid propellant interface reactions and energy balance
The attainment of the self-sustained combustion condition depends on several different factors: first
of all on the propellant typology (composite or double-base), physical characteristics and
composition, then on the amount and on the characteristics of the external energy stimulation, on
the external pressure and the propellant initial temperature. Other parameters can, of course, be
important, such as oxidant concentration, propellant geometry, orientation and location into the
16
chamber, shape of the port area of the grain and the flow characteristics over the solid propellant
surface, etc.
Fig.7 shows the interesting results obtained from a solid propellant slab ignition experimental
campaign5, investigating the fundamental dependence of the ignition delay on the heat flux
intensity, and in Fig.8 the dependence from the external pressure.
Fig.7 Generalized ignition map: event limits5
Consider a propellant subjected to a given heat flux, in this case by laser radiation, corresponding
to the vertical line in Fig.7. In the beginning the rise in surface temperature will be insufficient to
induce any significant reaction; if the flux is cut off at such times, the propellant will appear to be
macroscopically unchanged. If otherwise the flux is continued, the surface temperature will reach a
level at which reactions leading to propellant gasification begin (point L1a). This sudden change in
behavior defines the “first effect” or gas evolution boundary. It is usually the result of chemical
processes in the condensed phase and, therefore, is pressure independent.
If the flux is maintained past the crossing of L1a, gas evolution continues and the various species
entering the initially cold gas phase begin to undergo further reactions, enhancing the heat
feedback. This event must be detected by a proper sensitive detection system and substantially it is
indicative of the onset of flame development in the gas phase (points L1b, L1c).
Continued irradiation leads to the self-sustained ignition line, denoted L1d. As the name implies,
when this boundary is crossed, sustained burning of the propellant follows removal of the radiation.
Note that, as the external heat flux increases the overall ignition delay decreases. The lower curve
is somewhat insensitive to the pressure, but the upper curve drops downward as p increases. Figure
8, taken from another experimental work6, clearly shows the effect of external pressure, reducing
the ignition delay of a propellant slab subjected to a radiative external heat flux.
17
Fig.8 Measured effect of flux at different pressure levels for PBAN/AC propellant6
This description is a very broad outline of the ignition phenomenon; each class of propellants
(composite or double-base) shows its own particular ignition behavior, with different characteristic
preheating times, different chains of reactions and different response to external stimuli or
environmental conditions. Due to the complexity and the large number of ingredients that
constitute solid propellant grains, the large amount of chemical reactions involved and the
definition of the complex energy balance near the propellant surface, it is very difficult to formulate
a single theory able to adequately represent and completely describe the phenomenon in all its
possible configurations.
However, for particular types of propellants, controlling chemical reactions can be identified
described and analytically solved, thereby enabling to describe ignition and combustion
characteristics with an acceptable accuracy.
The search for a suitable theory of solid propellant ignition process has led to several analytical
models, three of which appear to represents the primary schools of thought:
Solid-phase ignition theory.
Gas-phase ignition theory.
Heterogeneous ignition theory.
The most important differences among the models are the location of the exothermic reaction with
respect to the propellant surface, and the physical state of the reacting ingredients.
The Solid-phase theory assumes that the ignition process is controlled by the thermal balance
between the external heat source and the heat generated by the on-going exothermic reactions,
inside the propellant grain. This model discards any interaction between the solid-state propellant
and the gaseous species diffused into the environment; thus it is not sensible to the external
pressure and the environment composition.
18
The Gas-phase theory assumes that, if subjected to a proper external heat addition, the solid
propellant generates fuel and oxidizer gases by endothermic decomposition. Fuel vapors diffuse
into the hot, oxidizing gas and react exothermically near the propellant surface. These chemical
reactions are all strongly influenced by the composition and the pressure conditions of the external
environment. The hot gas phase products establish a strong heat feedback towards the propellant
surface and therefore an equilibrium condition can be attained, with a steady gasification rate of the
propellant.
The Heterogeneous ignition theory proposes that, following a thermal induction by external
heating, oxidizer starts to decompose and provides hot oxidizing gas that attacks the fuel surface
exothermically, providing the controlling runaway reaction. It is assumed that the oxidizer gas may
emerge either at the exposed oxidizer surface or at oxidizer/binder interfaces near the surface.
Each one of these theories can generate several models that differs in terms of their governing
equations, initial hypothesis on the solid/gas-phase interface conditions, a proper ignition criterion,
etc.
As far as the ignition criterion is concerned, one of the major difficulties arises, again, in the
identification of the physico-chemical controlling mechanism of the phenomenon; for example, the
most common ignition criteria for theoretical models are7 :
Achievement of a critical value of the temperature of the propellant surface.
Achievement of a critical value of the growing rate of the temperature of the propellant surface.
Achievement of a critical value of the temperature in any point of the gas-phase.
Increase of the gas-phase temperature when the external heat source has been removed.
Presence of an inflection point in the surface temperature time-history.
Achievement of a critical value of the maximum reaction rate in the gas-phase.
Achievement of equilibrium condition in the heat balance between gas-phase heat generation
and heat loss to the solid-phase by conduction.
The above theories attempt identifying the controlling set of physico-chemical processes
responsible for thermal runaway under some defined stimulus, calculating the time required to the
runaway to occur, and its dependence upon macroscopic system parameters such as pressure, gas
velocity, temperature, etc. The fact that literature presents a very wide range of possibilities for the
definition of ignition theories and criteria, in theoretical models such as in experimental
observations, originates from the strong dependence of the ignition process, not only on heat
exchange and the environment conditions, but above all on composition and the characteristics of
the propellant adopted.
C. E. Hermance at the end of his “Ignition Theories and Experiment4
” appropriately summarizes
the current situation about solid propellant ignition analysis:
<< if one still decides to ask “what is the mechanism of solid propellant ignition?”, perhaps is
clear that the oversimplified, blunt answer is that we still do not know if there IS one and only one
dominant mechanism, or its nature. But honestly really requires that the questioner specify whether
he is concerned with double-base or composite propellant, whether it is heavily metallized or not,
what kind of igniter is involved, what kind of igniter-propellant geometry is involved, what
propellant environment, what time scale is of (practical) interest, and from what viewpoint these
questions are asked. >>.
19
Back to the problem of SRM, a model for the simulation of the whole ignition transient must be
able not only to describe the ignition of the solid propellant but also to evaluate its combustion
characteristics once full ignition is achieved.
Steady-state Burning
While dealing with combustion we will encounter quite all the same difficulties and uncertainties
we already indicated in the analysis of propellant ignition: strong dependencies on the propellant
class, ingredients, environment pressure, propellant initial temperature, and various other
parameters of aerothermochemistry in rocket motors. The heat feedback, between the gas phase
and the burning surface, retains a dominant role even once full self-sustained combustion
conditions is attained.
Burning rate is defined as the regression distance per unit of time perpendicular to the burning
surface of the propellant grain. It is well known that the pressure in the combustion chamber (thus
the thrust) is strongly dependent on the burning rate of the propellant; in fact the equilibrium
pressure inside the chamber can be written as:
CdA
rSp
th
bP
C
(1)
where r is the burning rate, p is the propellant density, Sb the burning surface of the grain, Ath the
nozzle throat section, Cd is a function of the chamber temperature and the characteristics of the
propellant combustion products.
It is therefore very important to establish the variation with time of the burning rate.
The determination of the burning rate under various operating conditions is the focus of current
propellant combustion research, and the nature of the burning rate must be still completely
explained and related to the physical and chemical properties of propellants.
The burning process is largely dependent on the combustion wave structure. This wave structure is
a function of the propellant composition, pressure and other operating conditions. Propellant
combustion produces heat and high-temperature gas; the heat feedback from the high temperature
gas to unburned propellant brings the latter to the decomposition temperature. As a result, unburned
propellant gasifies and produces heat by an exothermic chemical reaction. This feedback process
makes the propellant burning continuously to sustain steady state.
Figure 9 is a schematic drawing of a simplified combustion wave structure of a generic solid
propellant. The temperature increases rapidly from the initial temperature T0 to the burning surface
temperature Ts just beneath the burning surface. Reactive gases are generated by either
endothermic or exothermic decomposition reactions. This reaction zone is called the solid-phase
reaction or condensed-phase reaction zone. The reactive gases leaving the burning surface generate
a gas-phase region, called gas-phase reaction zone. The temperature increases in the gas-phase
reaction zone and reaches a maximum in the luminous flame zone. In the luminous flame zone, the
exhotermic reaction is completed and equilibrium combustion products are formed. The
combustion products flow away continuously downstream of the combustion zone and the burning
surface regresses in the opposite direction of the combustion product flow.
20
Fig.9 Combustion wave – surface heat balance for a steady burning propellant8
Let’s consider a very simple 1D model8 for the description and the solution of the combustion
wave: if one assumes the propellant burning surface to be stationary, the combustion zones and the
temperature profile remain stationary with time and space. The heat balance at the burning surface
is shown in Fig.9 and is expressed as:
fSP
S
g
S
PIrQ
dx
dT
dx
dT
(2)
where T is the temperature, x the axial distance, r the burning rate, Qs the net heat of reaction at the
burning surface, If the heat flux feedback from the luminous flame zone to the burning surface, is
the density, the thermal conductivity, and the subscripts p means propellant, g gas, s- solid
propellant at the burning surface and s+ gas phase at the burning surface.
Since each term of eq.(2) is dependent on the various physico-chemical parameters in the
combustion zones, such as propellant composition, pressure, and initial propellant temperature, the
burning rate characteristics of the propellant can be determined with a detailed knowledge of the
combustion wave structures.
The conservation equations, for mass, momentum, energy and chemical species in the combustion
wave, neglecting body forces and pressure gradient diffusion, can be written as:
21
0 v
pvv
iip
QTTvc (3)
iii
yDvy
where v is the mass-averaged velocity vector, D is the mass diffusion coefficient, Qi is the heat of
formation per unit mass of species i, cv the specific heat, yi is the mass fraction of species i, and i
is the rate of production of species i.
The detailed combustion wave structure of a solid propellant, including the burning rate, can be
determined by solving these conservation equations.
We see that even this extremely simplified model requires the solution of a complex system of
differential equations; space and time scales for a numerical solution of this problem can be very
different from those that characterize the fluid-dynamics of the combustion chamber during
ignition transient, and, of course, a multidimensional solution is required. Thus could be a very
difficult task to integrate together these different issues for a complete numerical solution of the
problem, computationally not too expensive.
For SRM ignition transient modeling, in order to avoid the complications of the direct solution of
the combustion wave, it is common practice to use simple empirical relations for the burning rate
evaluation, for example, the largely and successfully used expression of Vielle's-de Saint Robert's
law9:
napr (4)
where the parameters n and a in the expression are determined from experiments (strand burner)
and depend upon propellant composition and its initial temperature.
This assumption implies that the rate processes at the propellant surface are quasi-steady in the
sense that their characteristic times are short compared to that of the pressure transient and that the
thickness of the flame zone can be neglected if compared with the characteristic dimensions of the
chamber.
Transient and Erosive Burning
Relation (4), however, represents only the steady-state dependence of the combustion
characteristics on pressure, while it acts on the combustion wave only changing its dimensions, but
retaining its shape and morphology. The instantaneous burning rate of a solid propellant under a
rapidly changing non-steady pressure condition could be significantly different from the steady-
state burning rate measured in strand burner experiments. This means that, under transient
conditions, the instantaneous pressure level alone does not dictate the magnitude of the regression
rate of a solid propellant. Physically, the transient burning effect is introduced during rapid
pressure changes by the finite time interval required for temperature profiles inside the condensed
phase, and possibly the reaction zone, to follow transient pressure variations. Again, the problem
lies in time-scales: the magnitude of the transient burning effect is dependent upon the
characteristic times associated with the thermal diffusion process in the condensed phase, in the
surface region and in the gas phase. These characteristic times must be compared to the time scale
associated with pressure fluctuations: only if the pressure time scale is much larger than all others,
22
then transient burning effects can be reasonably neglected, otherwise the model for the burning rate
evaluation must be suitably modified10
.
The burning process becomes even more complicated as the propellant burns in the presence of a
cross flow of combustion gases through the grain port area. Usually, the burning rate of the
propellant increases as the cross flow gas velocity increases. The velocity dependent contribution to
the burning rate of a solid propellant is called erosive burning and it may seriously affect the
performance of solid propellant rocket motors, in particular during ignition transients, when the
flow velocity in chamber can be very high. The erosive burning mechanism is believed to be
caused by the increase in gas-to-solid heat feedback, and by the turbulence-enhanced mixing and
reaction of the oxidizer- or fuel-rich gases pyrolyzed from composite propellants11
.
Occurrence of erosive burning can result in unpredicted pressurization of the chamber, unequal
propellant-web consumption and early exposure of part of the rocket motor casing to the hot
combustion products. If erosive burning effects are not properly taken into account in the design,
rocket chamber failure may occur due to overpressurization immediately after ignition.
Substantial work (see the review presented by Razdan and Kuo11
), both experimental and
theoretical, has been done on this subject in recent years.
Lenoir and Robillard12
, were the first to develop a model based on the heat-transfer theory. Two
mechanism of gas-to-solid heat transfer were proposed:
1. from the primary burning zone, which is independent of the core gas velocity and is a function
only of pressure;
2. from the core of the hot combustion gases, which depends on gas velocity.
Using this approach, total burning rate is expressed as
ebrrr
0 (5)
where r0 is Saint-Robert pressure dependent burning component, and re the erosive burning
component. The erosive burning rate was postulated to be proportional to the convective heat
transfer coefficient including effects of transpiration.
This approach, in his original form or slightly modified13, 14
, has been widely used in rocket motor
performance calculations, and, as detailed in the next chapters, Lenoir-Robillard correlation is also
the base for the burning rate model used in our simulations.
23
1.3 MODELING SRM IGNITION TRANSIENT
1D Models: state of the art
Although ignition transient is an inherently strongly unsteady and non-spatially homogeneous
phenomenon, the literature reports several simulation models based upon quasi-steady or spatial
homogeneity hypotheses for several of the main variables and parameters involved. Historically
this is due to the fact that the very first solid rocket engines featured high Ap/Ath ratio, low
volumetric coefficients, and above all low values of L/D (length-to-diameter ratio), with attendant
low velocity transients. Further, such drastic simplifications matched the necessity of deriving
computationally inexpensive simulation models to be successfully utilized during the preliminary
phase of the design process.
However, modern design of solid rocket motors is clearly oriented toward high L/D and low Ap/Ath
configurations, such as used for the boosters of the main launch systems, then the characteristic
physical parameters cannot be considered spatially uniform along the chamber, and their behavior
must be considered strongly time-dependent.
Models for the prediction of the overall ignition transient can be subdivided into three major
categories:
1. p(t), lumped parameters or volume-filling models;
2. p(x), quasi-steady one-dimensional flow models;
3. p(x,t), unsteady one-dimensional flowfield models;
Early ignition transient modeling, see De Soto and Friedman18
, uses volume-filling methods
whereby the equation for conservation of mass for the rocket chamber is solved to obtain the
history of chamber pressure, by assuming isothermal filling. Propellant ignition is defined by
empirically specifying an ignition delay time and a flame spreading time. Lumped parameters,
volume-filling models assume that pressure is uniform across the combustion chamber, so they are
also known as zero-dimensional models.
Later, more accurate studies, presented by Baker19
and Bradley20
build dynamic p(t) models, by
assuming the system to be adiabatic and homogeneous: they solve two conservation equations, for
mass and energy. The solution of the energy conservation equation allows estimating a time-history
of the gas temperature inside the combustion chamber, and then evaluating propellant heating and
ignition; then, the empirical and inaccurate ignition delay time concept is no longer needed.
However, large boosters with large length-to-diameter (L/D) ratios generate large total and static
pressure drops along the chamber that must be predicted using, at least, one-dimensional models,
such as p(x) models, that assume quasi-steadiness for pressure, temperature and gas velocity within
the bore.
Motors with large L/D, often, have also very low Ap/Ath so that the characteristic velocities inside
the chamber are very high; this kind of motors features strongly time-dependent behavior, and an
accurate prediction, therefore, requires time-dependent modeling, such as p(x,t) models.
Let's briefly consider some example.
24
Zero-dimensional, lumped parameters analysis21
Unsteady case.
Suppose a mass flux min(t) of calorically perfect gas at constant temperature Tin is being pumped
into a volume V that may be changing with time, while a mass flux mout(t) of gas exits from the
volume (see Fig.10). Conservation of mass and energy in the volume provides two equations for
the history of pressure p(t) and temperature T(t) of the gas which is initially at p0=p(0) and
T0=T(0):
Fig.10 Lumped parameters analysis22
dtV
dVp
dtT
dTpmm
V
RT
dt
dpoutin
(6)
vv
outinoutininC
q
dtVC
dVRTTmmTmTm
pV
RT
dt
dT
(7)
where R is the specific gas constant, Cv is the heat capacity of the gas at constant volume, is the
ratio of specific heats of the gas and q is the heat flux lost per unit mass of mass to chamber wall.
These equations can be solved exactly for a number of special cases of inflow mass flux min(t) and
outflow mass flux mout(t) such as pure pressurization problems (mout=0) or adiabatic expansion
through choked nozzles (blowdown, min=0, mout=choked nozzle mass flow rate).
Quasi-steady case
Suppose that an adiabatic cavity is suddenly opened to a gas of constant high pressure pe and
temperature Te; the pressure and temperature in the volume will rise until steady state is reached
when its pressure has risen to pe. However at steady state dT/dt = 0, so that eq.(7) shows for mout=0
that the steady state temperature T = Tin the volume actually exceeds that of the injected gas due
to adiabatic compression. On the other hand, if outflow is occurring, equations show that quasi-
steady state is reached only when min = mout so that eq.(7) shows that T=Tin. Furthermore, if
products are in flowing from solid propellant of density p, surface Sb and burning rate n
apr ,
and outflow is through a choked nozzle of throat area Ath then eq.(6) reduces to the classical
expression:
)1/(1
*
)(
n
th
bp
A
cSatP
(8)
25
It is important to remark that, since eq.(8) represents a mass flow balance at the nozzle throat, the
predicted pressure applies only to the aft-end of the chamber. However, in long motors a significant
pressure drop, caused by mass addition, occurs along the bore such that the head-end pressure can
be as much as 10% higher then the value predicted by the above analysis21
.
Fig.11 Chamber pressure drop down as a function of Mach number and Ap/Ath22
By subdividing the propellant surface into N axial segments, the surface temperature, time of
ignition, and mass efflux on each segment can be solved independently, and provide a simple one-
dimensional model of flame spreading.
One-dimensional, quasi-steady analysis.
The injection of mass flow along the bore of a solid rocket grain causes both static and stagnation
pressure to drop, Fig.11, Fig.12. This effect can be accounted for using Shapiro23
’s influence
coefficients based on one-dimensional compressible flow with mass addition, area change and
friction:
D
dxf
m
dmM
p
dp2
2
0
0
(9)
2
2
12/142
M
M
D
fdxQ
A
dA
m
dmQ
p
dp
(10)
2221/21 M
D
fdxM
A
dA
m
dmM
V
dV
(11)
where 21
2
11 MQ
M is the flow Mach number, D is the hydraulic diameter, f is the friction factor, and the incremental
mass flux injected over the axial distance dx along the bore is PdxrdmP
, where P is the
26
average burning perimeter and r the average burn rate of the propellant of density P along a chunk
of length dx.
Thus stagnation pressure will decrease continuously along a burning bore, especially in axial fins
and past radial slots where mass addition is large; static pressure also drop unless a large port area
increase occurs.
Fig.12 Pressure drop along the bore (Shuttle Solid Booster)
2
One-dimensional, unsteady analysis.
For an accurate analysis of the ignition transient of modern solid boosters, featuring high L/D and
low Ap/Ath ratios, at least a p(x,t), one-dimensional, unsteady model is required.
Most of the characteristic gasdynamic phenomenology of the ignition transient of solid rocket
motors is very similar to those in classical shock tube: the shock formed by firing a pyrogen igniter
propagating along the combustion chamber, the rupturing of a weather seal in the nozzle throat,
sudden venting of an interstage cavity, ignition overpressure in a silo. Thus, pressure-based CFD
codes must be able to accurately deal with these strongly unsteady fluid dynamic phenomena, such
as shock waves, discontinuities and contact surfaces.
For example, consider the combustion chamber of a generic SRM ignited by a classical pyrogen
igniter; assume the chamber is a cylindrical shock tube, and a perfect gas, inviscid one-dimensional
flow. The rapid gas ejection from the igniter acts like a gaseous piston, and propagates compression
waves down the chamber that eventually coalesce into a shock wave.
This fast traveling shock wave is followed by a slower contact surface, that represents the
separation between hot igniter gases and the cold pressurizing gas that filled the bore before the
ignition. At the same time an expansion wave propagates towards the head end of the motor
(Fig.13).
27
Fig.13 Chamber flow-field features (shock-tube behavior)
24
Typically a seal closes the aft-end section of the chamber, before the ignition of the motor.
When this shock reaches the aft closure, it partially begins to reflect back towards the head end,
until the growing difference between chamber and external pressure becomes sufficient to break
the seal. At this point a new Riemann problem originates in this section. The rupture of the seal
generates a new shock wave traveling right, that soon exits from the chamber, while an expansion
wave propagates left towards the inner part of the bore; a contact surface originates between the
last two waves (Fig.14).
Fig.14 Chamber flow-field features (nozzle seal rupture)
It's easy to understand that these two gasdynamic patterns, generated by the two different Riemann
problems, in different times and different locations of the chamber, interact in a very complex and
hardly predictable way.
A suitable solution of such a difficult problem is required for an accurate simulation of the ignition
transient. The flowfield inside the chamber is characterized by strong, time dependent
perturbations, such as temperature jumps of thousands of Kelvin degrees, strong shock waves, local
sonic regions, contact surfaces and discontinuities between different gaseous species; these
28
phenomena, of course, can strongly influence the whole propellant ignition process along the
chamber.
The work of Peretz et alii3 is a milestone in the numerical simulation of the solid rocket motor
ignition transient by means of a quasi-1D approach; since this model represents the starting base
for the formulation of our own simulation model, we briefly summarize its main features.
Peretz's model rests on the following assumptions:
all chemical reactions occur at the propellant surface in a combustion zone, which is so thin
that it can be considered as a plane. The combustion products enter the main stream with zero
axial momentum;
the flow in the chamber port is one-dimensional;
rate processes at propellant surface are quasi-steady in the sense that their characteristic times
are short compared to those associated with the pressure transient;
the propellant and pyrogen igniter combustion products have the same values of Cp, W, ;
the products obey the perfect gas law.
The mathematical formulation is based on mass, momentum and energy conservation equations in
unsteady, quasi-one-dimensional form, plus the equation of state for the gas, with proper initial
conditions (onset of the simulated igniter flow located at the head-end of the chamber), two
boundary conditions at the head-end of the propellant section, representing the source terms for the
igniter mass addition, one boundary condition for the nozzle, for either choked and unchoked flow.
Semi-empirical correlations are used for the convective heat-transfer and friction coefficient, while
the burning rate law for solid propellant includes the effect of initial temperature, pressure and
velocity over the surface (erosive burning). A solid phase heat-up equation is used for the
determination of the propellant surface temperature during the induction interval, and is coupled to
a critical temperature ignition criterion.
The numerical solution of the governing equations is based on central differences in space and
generalized implicit differences in time.
We underline that the igniter is not materially present into the chamber volume but its action is
represented by source terms in the conservation equations. These source terms are located at the
very head-end of the bore and simulate the effect of the igniter jets, as if these were injected purely
axially. This feature strongly simplifies the physics that characterize the behavior of the igniter and
its real action on the propellant surface, as the identification of an impingement region.
Another important issue is the lack of a radiative heat-transfer model.
A large number of works followed adopting similar approaches. Among then, it is worth
mentioning the work of Caveny et alii15
, with an improvement of the basic Peretz's model for
application to large segmented motors, the work of Johnston and Murdock17, 25
of the Aerospace
Corporation, who develop an axisymmetric solution of the fluid motion and a suitable radiative
heat transfer model, and the application to fluid-structural coupling issues.
Lal/Santha/Padmanabhan26
extended the analysis of Peretz to account for canted-port igniters and
Pardue/Han27
to account for two-phase flow.
29
A particular mention deserves the huge and precious material published by Mark Salita2, 21
of
Thiokol -TRW Space and Missiles Systems, concerning ignition transient, as well as other
important issues of solid rocket technology. Several such models were implemented by Rozanski28
,
also at Thiokol, in the ignition simulation code SHARP1-DIT.
General Discussion
Discussing the different driving phenomena involved in SRM ignition transient and their modeling
task, a typical reference motor chamber can be subdivided into three regions (see Fig.15):
Fig.15 Combustion chamber main regions
Standard region. Standard models for the convective heating (i.e. the Dittus-Boelter expression for
the convective heating coefficient) can be adopted where the propellant surface is wetted by a fully
developed flow of hot gases tangential to the surface itself. In this region erosive burning,
occurring after the ignition, can also be evaluated by means of standard formulations, such as the
Lenoir-Robillard correlation seen above. This is the usual flow configuration along most part of the
combustion chamber, involving smooth variations of the cross sectional area and moderate
unsteadiness. Otherwise, close to sharp variations of the cross sectional area, the flowfield could be
affected by strong multidimensionality (flow separation, turbulent structures, vortices, etc.) and
significant unsteadiness. This occurrence could be a critical item, mainly for the simulation of the
propellant heating, when quasi-1D approaches are adopted. The radiative contribution to the
propellant heating, even if usually lower than the convective one, can be also easily accounted for
in this region.
Igniter-head and impingement region. Mass addition from the igniter and, in particular, the
impingement of the igniter jets onto the propellant surface, are the main driving mechanisms of the
motor start-up. Due to the peculiar flow conditions, the strong convective heating of the propellant
surface impinged by the igniter jets has to be evaluated by an "ad hoc" heat transfer model. The
same holds for the evaluation of the local erosive burning, which can be significant, even for
motors with large overall port-to-throat area ratios. In other words, the standard and typical
convective heating and erosive combustion models are not able to take into account the phenomena
induced by the igniter jets and their interactions with the combustion chamber conditions.
Fig.16 shows the results of an investigation15
about the influence of pyrogen igniter characteristics,
such as igniter mass flow, on ignition and flame spreading in SRM. The impingement region where
30
the initial ignition occurs, it’s clearly identified; from it the flame spreading proceeds forward and
backward along the combustion chamber.
Fig. 16 Ignition front propagation as a function of igniter mass flow rate15
Motor head-end, slots (in segmented grains) and submergence regions. In these regions, the flow
velocity is very low and the heating of the propellant surface is mainly due to the radiation from hot
gases and from nearby burning areas and to conduction. Their ignition is delayed, compared to the
standard region, and, once attained, the erosive combustion contribution to the burning rate is
negligible. A reliable radiation model is then required to properly describe ignition of the
propellant surfaces located in these regions. Both the filling of slots and submergence volumes
before their ignition and subsequent consumption, immediately after ignition, affect the overall
ignition transient behavior. The flow has a multidimensional nature but has also a low average
velocity, so convective heat flux is also very low, thus the simplified description of the local
flowfield, obtained by means of quasi-1D modeling, does not compromise too much the prediction
capability of the model in this regions.
An in-depth description and analysis of each of the phenomena involved in SRM ignition transient
is in itself a very difficult task; several problems are still unsolved, different theories or models can
be utilized for the same issue, each of them with its proper limits of application. Furthermore, a
compromise among accurate modeling, numerical accuracy and computational time requisites is
always needed, for an overall simulation of the ignition transient.
Theoretical models adopting sufficiently detailed analyses of heat transfer and fluid dynamics, but
simplified analysis for ignition (i.e., critical surface temperature as ignition criterion), have been
successfully utilized to predict the process of continuous flame spreading over solid propellants
under strong forced convection conditions3,16,17
. The success of such simplified analyses lies in the
fact that when convection is the dominant energy transfer mode, and chemical reaction times are
short in comparison to the propellant preheating time, as long as heat transfer is accurately
predicted, a suitable ignition temperature can adequately correlate experimental data.
31
In summary, we deem that, in order to obtain an accurate simulation model of the whole
phenomena of ignition transient, suitable submodels must be developed to adequately represent all
processes involved and their interactions. In particular must be defined:
a gasdynamic model to describe the complex strongly unsteady fluid dynamic evolution inside
the combustion chamber, as intense wave propagation phenomena, under variable boundary
conditions and space-time evolution of both mass addition and geometrical configuration, take
place;
an ignition criterion for the solid propellant;
an igniter model, to simulate the presence of the igniter in the bore, its operative
characteristics, and its interactions with the inner environment of the chamber; the igniter
modeling has a strong impact on the predicted timing and location of the very beginning of the
propellant ignition, and, on the pressure time-history of the chamber;
a model to estimate the heat transfer between propellant grain and combustion products, by
conduction, convection and radiation; this model has to include, in an exhaustive and correct
manner, the dependencies of this phenomenon on gas and propellant characteristics, flow
conditions and geometrical configurations;
a model to evaluate the combustion rate; once the propellant grain has been ignited, this model
evaluates the mass addition from the grain burning surface accounting for effects of flow
conditions and propellant characteristics.
a model to estimate the variations of the chamber geometry caused by the surface regression of
the burning propellant.
In this context and from this starting point we developed our own model.
32
2 PHYSICAL AND MATHEMATICAL MODEL
2.1 GASDYNAMIC MODEL
The previous phenomenological description points out once again that many phenomena interact
one another in a very complex way; among them gasdynamics plays a crucial role. The presence of
three different gases inside the combustion chamber has to be taken into account. In fact, the
properties of the pressurizing gas, usually nitrogen, are strongly different from those of the igniter
and the propellant combustion products. Moreover, the igniter gas composition could also be very
different from that of the propellant combustion products. Both the velocity of the propagation
waves and the energy content of the flow may be altered if the mixture nature of the gas is not
considered and mean gas properties are more simply assumed.
The gasdynamic model we adopted is an unsteady quasi-1D Euler flow model with mass addition
and geometry evolution in space and time. The computational domain includes both the converging
and diverging portion of the nozzle. The cross sectional area does not account for possible presence
of grain slots and submergence regions. Their effects are accounted for as mass and energy
exchange terms (see below). The hypothesis of a single phase (gas only), non-reacting mixture of
thermally and calorically perfect gases are also made. The properties of each gas (pressurizing gas,
igniter and propellant combustion products) are properly taken into account and the evolution of
the mixture composition too. The usual assumption that the chemical reactions are very fast and
occur in an ideal thin layer at the propellant surface is made. Mass addition from the propellant
surface, the grain slots and the submergence region is assumed to occur with zero axial momentum.
The overall heat transfer from the gas to the propellant surface and the friction effects are taken
into account until ignition and thereafter neglected. Based on the previous hypotheses, the classical
conservation equations are formulated as follows:
Momentum Conservation
2
21,2
u
fc
V
injavv
PA
igm
x
PA
ppApuxt
PuA
(12)
Energy conservation
V
sHP
Asm
WPtq
V
igH
PA
igm
fhpb
rPx
PuApe
t
PeA
(13)
Mass Conservation
V
PAprsm
prbrP
x
PuApr
t
PApr ,
(14)
V
Am
V
Am
x
uA
t
APigsPigPigPig ,
(15)
33
V
Am
x
uA
t
A PinsPinPin ,
(16)
where k kssmm
, and
k k (17)
with k = in, ig, pr
The equation system is closed by means of the equation of state: RTp
Subscripts in, ig and pr indicate inert pressurizing gas, igniter and propellant combustion products
respectively. Where not diversely stated, all quantities in the above equations indicate averaged
properties of the gas mixture. In order to evaluate mixture properties, the standard relationships for
a mixture of ideal gases have been applied.
Equations (12-16) are a set of hyperbolic partial differential equations, characterized by strongly
non-linear source terms. About these source terms, the following remarks have to be made:
1. the source terms including the quantities ms and ms,k are non-zero just in the flow regions
nearby the grain slots and the submergence region;
2. the source terms due to the igniter (subscript ig) are non-zero only in the igniter control volume
during its operation;
3. the source terms due to the propellant combustion (subscript pr) are non-zero only when and
where the propellant ignition has already occurred;
4. the source terms due to the friction effects (cf) and the total heat transfer (qt) between the gas
and the propellant surface are non-zero only when and where the propellant ignition has not yet
occurred.
Initial and Boundary Conditions.
Initial conditions can be imposed once the initial geometry of the propellant surface, the initial state
of the pressurizing gas inside the combustion chamber and the ambient conditions inside the
diverging portion of the nozzle are known. Wall boundary conditions are initially assumed at both
the head-end of the motor and the nozzle throat. However, the wall condition at the throat is
removed after the diaphragm bursts. Thereafter, the outside pressure value is prescribed at the exit
section of the diverging nozzle, until supersonic flow condition at the nozzle’s exit section are
established. Many simulation models merely prescribe a sonic condition at the nozzle throat.
Actually, flow unsteadiness during the ignition transient induces oscillation of the sonic line about
the throat section (i.e. the passage of the initial contact discontinuity). The boundary condition
adopted here is able to account for this phenomenon and, as a consequence, a slight damping effect
on the pressure waves traveling through the throat has been detected.
Slots & Submergence Region Model.
The evaluation of the evolution of the state inside both the slots and the submergence region is
obtained by means of a single model.
34
The model here proposed is based on a set of two ordinary differential equations, derive by
volume-averaging of the mass and energy conservation equations.
Fig.17 Cavity
For simplicity we can imagine a cavity whose length is exactly equal to one of the cells derived
from the chamber discretization, as depicted in Fig.17
Thus, assuming to know at a certain time t, the state of the fluid inside the cavity, pcav, Tcav, cav, its
volume, Vcav and the mass fractions of the gas species, ccav,pr, ccav,ig, ccav,in, we have:
total gas mass inside the cavity: CAVCAVCAV
Vm
mass of individual gas species in the cavity: CAVkCAVCAV
mcmk
,
with k = in, ig, pr
constant pressure specific heat and gas constant of the gas mixture in the cavity:
inPinCAVigPigCAVprPprCAVPCAVccccccc
,,,,,,
inginCAViggigCAVprgprCAVCAVRcRcRcR
,,,,,,
The state of the fluid in the chamber cell, outside the chamber/cavity interface is pCHAM, TCHAM,
CHAM, and the mass fractions of the gas species, cCHAM,pr, cCHAM,ig, cCHAM,in.
The coupling between the flow evolution along the chamber and the state evolution inside the slots
is ensured by evaluating their mass exchange as a function of the pressure jump at their interface.
Since conservation equations for the slots are averaged on its volume, the amount of the mass
exchange is calibrated in order to obtain a delay corresponding to a finite mass transfer velocity.
Pressure inside the cavity at time t+dt, can then be expressed as:
CHAMCAVCAVCAV
pppp ˆ (18)
where is a calibration coefficient.
Once CAV
p is known the variation of the enthalpy inside the cavity in the interval dt can be
evaluated as:
35
CAVprCAV
HHHH ˆ (19)
where
CAVCAVpCAVCAVTmcH is the enthalpy inside the cavity at time t;
fpbprTdtdxPrH is the enthalpy addition inside the cavity due to propellant combustion
at the non-inhibited surface;
CAV
CAV
CAVpCAVCAVR
VpcH ˆˆ is the enthalpy inside the cavity at time t+dt.;
From the sign of H we obtain the direction of the flow across the cavity/chamber interface;
if 0H we have mass spillage from the chamber to the cavity:
CHAMpCHAM
CAVTc
Hm
is the variation of the total mass inside the cavity;
CAVkCHAMkCAVmcm
,, is the variation of the mass of the single gas specie inside the cavity;
If 0H we have mass spillage from the cavity to the chamber;
CAVpCAV
CAVTc
Hm
is the variation of the total mass inside the cavity;
CAVkCAVkCAVmcm
,, is the variation of the mass of the single gas specie inside the cavity.
The total mass inside the cavity and the species mass fraction can now be updated as:
dtdxPrmmmprbcprCAVprCAVprCAV
om,,,
ˆ , for the propellant gas;
igCAVigCAVigCAVmmm
,,,ˆ for the igniter gas;
inCAVinCAVinCAVmmm
,,,ˆ for the pressurizing gas.
The total mass inside the cavity is: inCAVigCAVprCAVCAV
mmmm,,,
ˆˆˆˆ
The source terms relating to the chamber/cavity interaction, needed for the resolution of the
conservation equations (12-16), (three for the mass conservation and one for the energy
conservation:
dtdx
mm
prCAV
prS
,
, ;
dtdx
mm
igCAV
igS
,
, ;
dtdx
mm
inCAV
inS
,
, (20)
and
dtdx
TcmH
CHAMpCHAMCAV
S
if 0H (21)
dtdx
TcmH
CAVpCAVCAV
S
if 0H (22)
36
2.2 IGNITER MODEL AND IMPINGEMENT REGION
As far as we know, it is a common practice in the simulation models based on a quasi-1D flow
assumption to do not account for the overall dimensions of the igniter, the actual location and
design of its nozzles, and the modeling of their exhausting jets. Usually, the role of the igniter into
the combustion chamber is modeled in terms of its contributions in terms of mass, momentum and
energy, as required by the gasdynamic model (i.e., the igniter provides the boundary conditions or
the source terms just at the head-end of the motor3, 16). Even in more refined simulation models,
based on a multidimensional flow assumption, the behavior of the igniter jets and its fluid-dynamic
interactions with the propellant are often ignored17. An igniter model is proposed here that, at least
in principle, allows identifying the influence of its design parameters and operating conditions on
the motor start-up.
From a geometrical point of view, the following information is required:
1. igniter: external shape, dimensions and location inside the combustion chamber;
2. igniter nozzles: number, shape, dimensions, location and orientation.
As far as the igniter operation is concerned, information is required:
1. the time evolution of the pressure inside the igniter (or any equivalent information)
2. all thermodynamic data referring to the combustion products
Fig.18 Pyrogen igniter
A semi-empirical model has been developed which, on the basis of the above information,
estimates the dimensions and location of the impingement region on the propellant surface, as a
function of time. Pyrogen igniters with both axial and inclined nozzles have been considered. The
model is based on the simplifying hypothesis of an averaged, simplified shape of the propellant
grain and a conical shape of the impinging jets. The size of the jet cone is obtained by imposing
that the jet gas, flowing isentropically like in a fictitious divergent nozzle, extending beyond the
actual igniter nozzle, reaches the motor chamber pressure right on the grain surface. Due to the
pressure evolution inside both the igniter and the combustion chamber, the igniter jet flow regime
in the divergent is initially subsonic, for a short period, then supersonic and, finally, subsonic
37
again. The last phase, as shown in the results, can also take place when the igniter mass flow is at
its maximum and, consequently, its duration can be significant. This occurrence depends on the
interaction between the igniter pressure profile and the chamber pressurization process.
In order to identify the geometry of the impact area of the igniter jets (impingement region), the
time-history of the fluid dynamic variables at the throat section of the igniter nozzles, assumed to
be simply convergent, must be determined.
As the igniter chamber total temperature Tcig and the mass flow rate are assumed to be given, it is
possible to analyze the behavior of the igniter as a function of the motor chamber pressure Pa; if p*
is the critical pressure at a generic instant, and Pcig the pressure inside the igniter chamber, we
have:
1*
2
1
pPc
Ig (23)
Values corresponding to choked nozzle condition are denoted with the subscripts cr. Atig is the
igniter nozzle throat section.
Accordingly the three different operating conditions can be modeled as follows:
adapted unchoked nozzle Pcig<Pcig-cr:
at the igniter start-up Pcig begins increasing while the flow is subsonic throughout, with the exhaust
pressure Pe = Pa.
*pppp
ate (24)
1
*
1
2
Cig
pp (25)
1t
M (26)
12
11
2
Cig
a
Cig
a
cig
tCig
p
p
p
p
RT
Apm (27)
1
11
2
Cig
a
cigetp
pRTvv (28)
adapted choked nozzle Pcig = Pcig-cr:
Pcig attains the critical value, sonic choking condition is established in the throat section; the mass
flow rate reaches its maximum value:
crate
ppppp**
(29)
crCig
ppp*
1*
1
2
(30)
38
1t
M (31)
cig
tigCigtigCig
RT
Ap
c
Apm
* (32)
t
cig
Cig
cigeta
RT
p
pRTvv
1
21
1
2
1
*
(33)
underexpanded choked nozzle Pcig>Pcig-cr:
Pcig keeps rising, the sonic condition at the nozzle exit is maintained but not the adaptation
condition that will occur externally, by the generation of a supersonic conical diverging jet.
atepppp
* (34)
crCig
ppp*
1*
1
2
(35)
1t
M (36)
cig
tigCigtigCig
RT
Ap
c
Apm
* (37)
t
cig
Cig
cigeta
RT
p
pRTvv
1
21
1
2
1
*
(38)
The external pressure Pa (the motor chamber pressure), increases during the ignition transient, and,
eventually modifies the igniter nozzle operative condition driving it back to subsonic flow, until the
igniter burnout (see Figs.20-21).
The time history of the pressure inside the igniter is often not known, from experimental or post-
fire test data, only the time history of the mass flow rate is given. Therefore, the pressure inside the
igniter chamber must be recovered from mass flow rate information, by comparing the minimum
pressure value inside the igniter chamber required to establish a choked nozzle throat:
tig
cigig
tig
ig
cA
RTm
A
cmp
TS
*
(39)
to the value of the total pressure corresponding to the simultaneous condition of adapted and
choked nozzle:
acpp
SON
1
2
1
(40)
39
If Pcson=Pcts , we have a choked adapted nozzle and equations (30–33).
If Pcson<Pcts , we have a choked underexpanded nozzle and the equations (35–38).
Than, in both cases, the final section of the conical igniter jet is :
1
11
2
TSTS c
a
c
a
tig
jet
p
p
p
p
AA
(41)
and the final diameter of the jet
jetjet
AD 2 (42)
See also Fig.19.
If Pcson>Pcts, the igniter nozzle is not choked and equations (25)-(28) hold, thereby:
tigjetAA and
tigtigjetADD 2 ;
In any case, in order to evaluate the source terms for the conservation equations, the outflow
velocity vinj=ve must be projected along the longitudinal symmetry axis, and both axial and inclined
nozzles must be considered: NaxIg is the number of axial nozzles and NzIg is the number of inclined
nozzles.
Accordingly, an averaged value of the velocity vav, inj can be estimated as:
IgIg
IgiginjIginj
injavNzNax
NzvNaxvv
cos
, (43)
Fig.19 Igniter jet geometry
40
0 0.1 0.2 0.3 0.4 0.5 0.6 0 .7 0 .8
T em po (s)
0
0.1
0.2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
1
Ma
ch
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Po
rta
tain
ma
ss
aig
nit
ore
(Kg
/s)
M ach uscita ignitore
P orta ta in m assa ignitore
Fig. 20 Igniter mass flow rate and exhaust Mach number (SRM#1)
0 0.1 0.2 0.3 0.4 0.5 0.6 0 .7 0 .8
T em po (s)
0
0.1
0.2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
1
Ma
ch
0
10
20
30
40
50
60
Pre
ss
ion
ed
iu
sc
ita
ign
ito
re(B
ar)
P ressione di uscita ignitore
M ach uscita ignitore
Fig. 21 Igniter chamber pressure and exhaust Mach number (SRM#1)
41
2.3 PROPELLANT SURFACE HEATING, IGNITION AND REGRESSION
Ignition Criterion
As already mentioned, several theoretical models and mathematical formulations have been
proposed in the past in order to describe the mechanisms of the propellant ignition. In the present
model a simple, widely used ignition criterion is adopted, based on the assumption that the start up
of the propellant combustion begins when the surface temperature reaches an assigned value (the
ignition temperature depending on the specific propellant being considered).
Based upon previous observations about the strong sensibility of the ignition delay on environment
pressure, a simple empirical dependence of the ignition temperature on the local pressure level has
been considered too.
Accordingly, an accurate evaluation of the grain preheating is required for an adequate estimation
of the ignition sequence, and more in general for the correct evaluation of the whole ignition
transient.
During the ignition transient, the grain surface is integrally heated until ignition. Along the
combustion chamber we can have particular situations characterized by local specific phenomena,
for example in the impingement region, and very different flowfield features such as recirculations,
shock waves, discontinuities etc. that can strongly affect heat transfer between the hot gases and the
grain surface. The propellant grain is rapidly heated to ignition by both convective and radiative
heat flux from the hot gas issuing from both the igniter and the upstream segments of the grain
already ignited. The gasdynamic model aims at evaluating all these complex phenomena.
Convective Heat Transfer: impingement region.
Impingement region is delimited by the intersection of the igniter jets with the grain surface
(Fig.22) and is assumed to be circular. Its diameter, Ximp, is proportional to the diameter of the jet
Djet, already evaluated in the igniter model (eq. 42).
Fig.22 Igniter jet and impingement region
ig
jetMR
impsin
DKX
where KMR is a calibration coefficient. (44)
In this region the convective heat exchange is evaluated utilizing a formulation derived from a
42
model proposed by Martin29
.
Fig.23 Martin model for impinging jet29
According to this model, if Hjet is the axial distance between the exit section of the igniter nozzle
and the propellant grain surface, the value of the forced convection heat coefficient hc progressively
decreases towards the boundary region of the jet:
i
jet
jet
jet
i
jet
jet
i
pr
igic
r
D
D
H
r
D
Fr
Kxh
61.01
1.11
RePr42.0 (45)
where
200
Re1Re2Re
55.0
5.0 jet
jetjetF (46)
jet
IgIg
tot
ig
jetD
NaxNz
m
4Re (47)
where tot
Igm is the total igniter mass flow rate.
Rej is the jet Reynolds number and ri = xi-xj is the distance of the cell center xi, from the xj
intersection of the jet axis with the propellant surface.
For non-circular grain cross sections, the propellant surface closest to the longitudinal axis of the
motor (i.e., the star tip) intercepts first the axis of the jet originating from the igniter nozzles. Both
the xj abscissa and the Hjet distance are accordingly computed.
The validity range of these expressions are given by:
9030ig
(48)
400000Re2000 jet
(49)
43
5.75.2 jeti
Dr (50)
122 jetjet
DH (51)
The first, the second and the third constraint are usually fulfilled by the rocket motors under
consideration. The width of the impingement region along the longitudinal direction is assumed
equal to KmrDjet (with a standard value of Kmr=5). Since during igniter operation the ratio Djet/De
usually varies from 1 to 2-3, then the upper limit of the fourth constraint is always respected. As far
as the lower limit is considered, a suitable cubic extrapolation of relation (45) is here proposed,
based upon experimental results and theoretical considerations of [29] and [30]. So:
jet
jet
jet
i
jet
i
jet
i
pigicD
Fd
D
rc
D
rb
D
raKxh
)(RePr
23
42.0
(52)
where a, b, c are functions of Hjet/Djet.
The convective heat transfer coefficient hc(xi) for any cell of the impingement region can be
evaluated by equation (45), as a function of the value of ri/Djet.
For the generic cell in the impingement region, hc varies in both axial and radial direction (Fig.24),
then an averaged of the coefficient, hcav, between its maximum and minimum values is considered:
2)(
maxmin icic
iavc
xhxhxh
(53)
Fig. 24 Convective coefficient for impingement region
The igniter has a finite number (NzIg) of inclined nozzles oriented towards the grain surface so the
resulting impact surface of the jets involves only a fraction of the surface of the generic cell of the
44
impingement region; the proposed final expression for the convective heat transfer coefficient, for
the ith cell of impingement region is:
iavc
ib
i
ig
jetMR
Ig
iIMPcxh
xP
rsin
DKNz
xh
2
2
22
(54)
and the specific convective heat flux is:
isiicic
xTxTxhxqIMPIMP
(55)
Relation (54) cannot be considered in any way as the result of a rigorous mathematical modeling of
the impingement of a supersonic jet onto an irregular 3D surface like that of a star-shaped grain. At
any rate, simulations results seem to indicate that this relation gives a rough, but realistic, picture of
the igniter role in different motor and igniter configurations. In addition, it involves the main
design parameters of the igniter and its operating conditions, useful in order to have an effective
design tool.
Convective Heat Transfer: standard region
The evaluation of the convective heat transfer coefficient in the so-called standard region of the
motor is obtained by means of a semi-empirical model proposed by Gnielinsky31 which
incorporates both the transitional region and the region of fully developed flows in pipes and
channels. After an extensive investigation still ongoing, we found that, for the particular conditions
inside the bore during the propellant preheating phase, this formulation is often more accurate than
the classical widely used one proposed by Dittus-Boelter32.
According to the classic Dittus-Boelter formulation:
3.08.0PrRe023.0 Nu (55)
valid for: 104<Re<5*10
5 and 0.7<Pr<120; whereas the Gnielinsky formulation prescribes:
1Pr8
7.121
Pr1000Re8
322
1
f
f
Nu (56)
where
2
64.1Relog82.1
f (57)
valid for: 104<Re<5*10
6 and 0.5<Pr<2300.
Accordingly, a convective heat transfer coefficient, and the related heat flux, can be defined for any
cell in the standard region:
45
1Pr8
7.121
Pr1000Re8
322
1
fD
Kf
xh
h
P
iC
(58)
iSiiCiC
xTxTxhxq (59)
The results that will be presented later claim that a careful calibration of the convective heat
transfer model is required in the event of sharp variations of the motor cross section and/or strong
unsteadiness.
Radiative Heat Transfer
Usually, the radiative contribution to the propellant heating is roughly taken into account by a
moderate increase of the convective heat transfer coefficient3.
Where the flow velocities are very low (like in the head-end region of the motor, the grain slots or
the submergence region), convective heating is negligible and propellant is ignited mainly due to
radiative heat transfer.
As far as radiative heat transfer the hot combustion products in the rocket motor, at temperature TG,
is concerned, by the walls, at temperature TS, the exchanged amount of heat per unit area equals:
44
0 SGradRTTCq (60)
This is a common general expression, used in most of the simulation models that deal with
radiative heat transfer, where the coefficient Crad typically is defined as a constant17
or in more
sophisticated expression is exploited to directly take into account the emissivity of the radiating
materials21
. In our model we defined a Gas Radiative Coefficient GRC, such as:
44
0 SGRTTGRCq (61)
The gas radiation coefficient GRC is a function that takes into account the combined effects of gas
emissivity, gas transmissivity and surface absorptivity; a view factor of 1 is assumed too.
The GRC is defined as a function of the actual values of pressure and temperature of the radiative
gas; the hydraulic diameter of the cell is considered too. The proposed GRC function behavior is
presented in Fig.25, that is qualitatively similar to the gas emissivities tables proposed by Hottel for
its engineering method33
.
46
Fig.25 Gas Radiative Coefficient (GRC) Table
A complete quasi-3D radiation model (Fig.26) has also been developed within the framework of
the present study, estimating, for each computing cell, the total contribution of the radiative flux
emitted from the whole gas volume of the chamber, taking into account the complete geometry of
the igniter jets and the geometry of the chamber, with exact view factors, shadow regions, etc.
Fig.26 Radiative quasi-3D model
However, results showed that the generic portion of the grain surface was substantially heated by
radiation only from the gas occupying a very close region of the chamber volume. The model was
then simplified in order to reduce the computational time; in the present application, the radiative
heat flux is evaluated by means of a "three cells model" that takes into account, for the grain
surface of any cell, the gas radiation emission only from the cell own volume and from the left and
right side cells. To make allowance for surface flame radiation due to already ignited fractions of
the propellant surface, enhancing terms, similar to those proposed in [17], are added.
47
Fig. 27 Three cells radiative model
In this model, the radiative heat fluxes from the gas volume of the i-th cell itself and the two cells
located at its right and left side are evaluated. Radiative heat fluxes are evaluated both originating
from the gas flowing into the chamber and from already ignited fractions of the propellant surface
by means of the following expressions:
Radiative gas flux from the i-th cell volume:
iSiiir
xTxTxGRCxqCENTER
44 (62)
Radiative gas flux from the right side cell volume (not ignited):
iSiiir
xTxTxGRCxqRIGHT
4
1
4
1
(63)
Radiative gas flux from the left side cell volume (not ignited):
iSiiir
xTxTxGRCxqLEFT
4
1
4
1
(64)
As stated before, if the boundary portion of the grain surface of the considered i-th cell is already
ignited, we have to take into account also the radiative contribution of the flame, considering a
view factor of 1/2:
Radiative gas flux and flame flux from the right side cell (ignited):
iSiiiSfir
xTxTxGRCxTTxqRIGHT
4
1
4
01
44
2
1
(65)
Radiative gas flux and flame flux from the left side cell (ignited):
iSiiiSfir
xTxTxGRCxTTxqLEFT
4
1
4
01
44
2
1
(66)
48
The sum of the three fluxes evaluated by means of the present model is the total radiative heat flux
contributing to propellant heating. Once the cell is ignited, the radiative heat flux is not computed
anymore because it is assumed that it does not play any role in the propellant combustion process.
Propellant Heating Evaluation
Time history of the propellant grain surface temperature is evaluated by means of the solution of
the well known Fourier equation, for a semi-infinite slab :
t
T
y
TP
2
2
(67)
where y>0, t>0
together with the boundary and the initial condition:
tqy
Tk
P
(68)
for y=0, t>0, and
0STT (69)
for y, t=0.
Equation (67) can be solved either numerically, by an appropriate technique, or by the exact
solution in the form34
:
t
d
k
qTstT
t
00
)()( (70)
In the former approach, a very fine, radial grid must be used to represent the temperature
distribution inside the solid propellant; this must be done for every longitudinal element of the
main chamber axial grid. Then this method is computationally quite expensive.
In the latter approach, the time integral is subdivided into n time steps; the heat flux after i time
steps is held constant for the next (i+1) time step, then the following expression for the wall
temperature results35
:
1
1
0
02)(
inin
ni
i
ntttt
k
qTstT
(71)
This equation is time-marched in conjunction with the flow solution.
This approach requires that the complete time-histories of the heat fluxes q , for each of the cell of
the main grid mesh, must be stored; accordingly this method can be very expensive in terms of
memory occupation.
A different approach is used in the present study: by coupling the classical unsteady 1D Fourier
equation with the gas/solid heat convection equation (with an appropriate set of initial and
49
boundary conditions), an ordinary differential equation is obtained for the evaluation of the time
history of the propellant surface temperature35
. This is a classical approach widely adopted in the
relevant literature3.
A quantity (t) is defined, called the penetration distance, its properties are such that for x>(t) the
slab, for all practical purposes, is at an equilibrium temperature and there is non heat transferred
beyond this point. The penetration distance is analogous to the boundary layer thickness in fluid
mechanics. Then for y=0 the surface temperature is Ts(t), while for y=(t) the temperature is
supposed to be equal to the initial value of Ts0 (see Fig.28).
In order to describe the temperature profile into the semi-infinite slab that represents the propellant
grain, cubic interpolation is adopted by enforcing four boundary conditions:
0
yy
T; 0
2
2
yy
T ; q
y
T
y
0
; 0S
TyT (72)
So we obtain:
0
3
3
*
STy
zT
where
tzq
z
,
3*
(73)
with z=Ts, z*=Ts-Ts0 and the function q(z,t) is the external heat flux.
Fig.28 Temperature variation inside the propellant36
This formulation has been slightly modified in order to take into account the contribution to the
propellant heating due to both the convective and radiative heat fluxes.
The expressions of these external fluxes must be reorganized in order to be used in expression (73);
in particular in our model the radiative heat flux must be expressed in a fashion formally similar to
the convective heat flux, by means of suitable coefficients:
for the central flux:
iSiirir
xTxTxhxqCENTERCENTER
(74)
where
50
iSiiSiiir
xTxTxTxTxGRCxhCENTER
22 (75)
for the right side flux:
iSiirir
xTxTxhxqRIGHTRIGHT
1
(76)
where
iSiiSiiir
xTxTxTxTxGRCxhRIGHT
2
1
2
11
(77)
for the left side flux:
iSiirir
xTxTxhxqLEFTLEFT
1
(78)
where
iSiiSiiir
xTxTxTxTxGRCxhSLEFT
2
1
2
11
(79)
The possible presence of nearby radiating surface flames, is accounted by adding the following
additional terms:
iSfirir
xTTxhxqRIGHT
ˆˆ (80)
iSfirir
xTTxhxqLEFT
ˆˆ (81)
where
iSfiSfir
xTTxTTxh22
2
1ˆ (82)
Then, the total radiative flux, for the generic ith cell, is the sum of all previous contributions:
iririririritotr
xqxqxqxqxqxqLEFTRIGHTLEFTRIGHTCENTER
ˆˆ (83)
The total heat flux is Crtot
qqqtot
; if eq.(67) is multiplied by dy and integrated from 0 to , the
heat balance integral is obtained:
dt
Td
y
tT
y
tTS
P
0
,0,
(84)
where
t
Tdy
0
(85)
Since there no heat is transferred beyond y=:
0,
t
x
T (86)
51
The temperature will be compelled to satisfy the heat balance integral, but not the original heat
conduction equation. The heat conduction equation will, thereby, be satisfied only on the average.
Substituting conditions (72) in eq.(84):
0
,SP
Tdt
dtzf (87)
and the (73) in eq.(85) we obtain
tzf
TTTTSSSS
,4
3300
(88)
that introduced in eq. (87) gives the following ordinary differential equation to be solved for z=Ts:
tzf
Tz
dt
dtzf
S
P,
,3
42
0 (89)
Under the assumption that during the generic integration time step dt the total heat flux can be
considered constant, the function q(z,t) will depend only on the grain surface temperature z=Ts;
then for the i-th cell:
i
i
i
S
SS
SPTf
TT
dt
dTf
2
0
3
4 (90)
that gives for the temperature transient:
1
23
00
23
4
i
i
iiii
i
S
S
SSSSSSP
S
dT
TfdTTTfTTTf
dt
dT (91)
Burning Rate Model
Propellant combustion model is based on the classical assumption that chemical reactions are very
fast and occur in an ideal thin layer at the surface of the solid propellant. Therefore, the overall
macroscopic result of combustion phenomena is a regression rate of the propellant grain quantified
by the so-called burning rate. The complex chemical and physical processes occurring on the
propellant surface and in the nearby combustion region during the propellant combustion are not
explicitly considered. The total burning rate is assumed to be the sum of a term r0 depending upon
the pressure and a term reb (erosive burning rate) depending upon the gas flow rate. A suitable
function, based on the fitting of different motors, has been used to define the coefficient appearing
into this formulation. As far as the standard region and the motor head-end are considered, the
classical formulation due to Lenoir-Robillard12
, with some modifications due to Lawrence13
, is
adopted:
ebrrr
0 (92)
0
,00 S
Tprr and n
apr 0
(93)
the coefficient a also account for the effect of the initial propellant grain temperature:
52
refSPref TT
n
ref
ep
ra
00
(94)
Erosive burning is described as:
u
r
hg
ebeb
P
eD
u
W
Tkr
2.0
8.0
9.0
12.0
03
2
3
1
1
59 (95)
with
keb=f1 (ρpr, csp, Tf, TbS, TS0 ) and bh
PDf 1,2
(96)
where f1 is a well-known standard function and f2 a suitable function obtained by the fitting of
different motors. Experimental model constants and coefficients, related to the propellant adopted
(like rref, n, Tf, etc.) are needed.
As far as the impingement region is concerned, the peculiar fluid dynamics conditions, determined
by the strong impact of the hot igniter jets on the propellant surface, require adopting a different
model to evaluate the local enhanced erosive burning; the following expression of the erosive
burning rate at impingement is here proposed:
injvig
rP
jet
injig
igig
igig
IMPebIMPebe
D
v
W
Tkr
2
2.0
8.0
9.0
12.0
03
2
3
1
1
59 (97)
with
jetig
g
igb
h
jet
h
Mu
u
P
D
D
Df
1,,
1,,
32
(98)
f3 a suitable function based on the fitting of the transient behavior of different motors with different
igniter design, and kebimp is a calibration coefficient.
Surface Regression Evaluation
In order to evaluate the time evolution of the propellant grain surface, two main information are
required: a detailed description of the initial shape and dimensions of the grain and the
instantaneous value of the burning rate at each propellant region. A grain regression model has
been developed5 to deal with the quasi-2D shapes (cylindrical and conical grains, different types of
quasi-2D star shaped grains). This model, based on standard assumptions, provides at any time, by
means of on-line calculations, the geometrical parameters required by the overall simulation model.
Alternatively, an "ad hoc" preprocessor computes the evolution of the strongly 3D regions as a
function of the burnt propellant. Starting from the detailed drawings of the motor chamber, input
geometrical information is recovered, memorized and elaborated, in order to generate, for every
time step, the values of port area, wet and combustion perimeter for every cell of the main
discretization partition of the chamber36
.
The model is able to recognize different geometrical shapes or regions, characterizing the chamber
section, such as:
53
3D star-shaped section region
2D star-shaped section region
simple conical section region
intersegment slot region
submergence region
Examples of the chamber geometry for SRMs considered in this work are given in the figures
below.
Fig. 29 Chamber scheme for SRM#1 (Ariane 4)36
Fig. 30 Chamber scheme for SRM#3 (Zefiro)36
54
Fig. 31 Chamber segments of SRM#2 (Ariane 5)36
3 NUMERICAL INTEGRATION TECHNIQUE
3.1 DISCRETIZED FLUID DYNAMIC MODEL AND SOLUTION
In order to give a global idea of the numerical solution of the physical-mathematical model
exposed before, we begin with analizing the core of the simulation code, represented by the
gasdynamic solver.
The discretized model is based on an equally spaced finite volume approximation of the
mathematical model. The numerical method adopted here belongs to the family of the second order
accurate ENO37 (Essentially Non Oscillatory) methods and is coupled to an exact Riemann solver,
suitably adapted to a non-reactive gas mixture. The main characteristics of this method are the
robustness and the capability to deal with flowfields affected by propagation phenomena with
strong discontinuities and source terms.
Obviously, the accuracy in describing the shape, strength and propagation velocity of flow
discontinuities increases with the space and time resolution adopted. For motor lengths ranging
from 5 to 40 meters, a reliable simulation can be obtained with a number of cells ranging from 100
to 1000. Provided that the wave propagation velocity inside the combustion chamber ranges from
about 300 m/s up to 1700 m/s, about 25000 time steps are required for the numerical simulation of
a one second interval of the ignition transient. Such a spatial and temporal discretization allows to
accurately describe the unsteady phenomena during the motor start-up and the associated natural
frequencies along the longitudinal axis (5-200 hertz).
In the present study, the original ENO approach has been modified, adapted and then applied38 to a
problem characterized by strong variations of the source terms in both space and time,
modifications are aimed to handle the source terms, boundary conditions and geometrical
discontinuities, e.q., occurring at sharp cross section variations in the igniter region. Other
significant adaptations have been required to account for the gas mixture during both the
reconstruction and evolution phases of the numerical technique. For the former, the set of variables
for the slope evaluation are (k, u, T) instead of (, u, p), since pressure equation for a gas mixture
is more complicated than temperature equation. For the latter, i.e., the solution of the Riemann
problem for the evolution phase, the exact solver has been reformulated by introducing the
propagation of the species concentration along the particle trajectory and jump conditions at the
contact discontinuities for gases with different .
55
The ENO method can be considered as a second order extension and a generalization of the first
order Godunov scheme.
Fundamental characteristics of the Godunov methods is that the discretization of the equations is
obtained directly from the physical behavior of the fluid; this integration technique is developed by
means of the interaction of finite-amplitude waves, rather than from infinitesimal ones as in
common finite-differences schemes. The time evolution of the flowfield is determined by means of
the approximate solution of a finite number of initial and boundary conditions problems, in such a
way that Godunov methods can be considered as an extension of the characteristic-based models
for discontinuities problems.
Godunov’s method
Given an initial-value problem by means of a non-linear equations system, of three equations with
three variables (for the 1D schematization), this can be written as:
0
x
UF
t
U (99)
xUxUo
0, (100)
with ;x e ,0t , where:
euUT
;; and upepuuFT
;;2
We assume that xUo
is a piecewise regular function of x and that the matrix :
n
n
UU
FFUA
,.....,
,.....,
1
1
(101)
has real eigenvalues, so that the problem is hyperbolic.
We define as a weak solution of the problem, a solution obtained by integrating eq.(99) over the
generic domain
,0
2,
2
hhD :
t
hxUFt
hxUF
htxUtxU ,
2
ˆ,2
ˆ,,
(102)
where
2
2
,1
,
h
hdtxU
htxU (103)
is a spatial-averaged value calculated in
2,
2
hx
hx ,
while
0
,2
1,
2
ˆ dth
xUFth
xUF (104)
56
is the time-averaged value of the physical fluxes, calculated in tt , , that satisfies the original
equation inside the region free of discontinuities.
The integration domain of eq.(99) is discretized by subdividing the x range in evenly spaced
intervals of length h and the time axis in evenly spaced intervals of duration .
Assume that, at a given time tn , an approximation of the averaged values of txU , for every cell
unknown (j), then the value for nn
tt1
can be computed, once the values of the interface
fluxes for 1
nn
ttt are known, by using eq.(102) in the form:
2
1
2
1
2
1
2
1
1n
j
n
j
n
j
n
jFFUU , where
h
(105)
Eq. 3.4 represents the exact solution and gives the variations during the time interval of the space-
averaged variables as the balance of the time-averaged values of the fluxes.
The algorithm is composed by these steps:
Reconstruction: it is assumed for every cell of the discretization domain an initial data
distribution, piecewise constant, and equal to the averaged value of the status variables inside
the cell itself (Fig.32) . This operation gives to the method a first order space accuracy.
Evolution: this is the phase during which the Riemann problem for every interface between the
cells of the grid is solved.
Fluxes evaluation: the solution of the time integral of the fluxes is immediate if: n
j
j
J
h
min
Where n
j
K
K
n
jmax and
n
j
K is the k
th eigenvalue of the Jacobian matrix in n
jtx , .
Under such conditions, waves originating from discontinuities at cells interfaces, each of them
propagating at the speed corresponding to the eigenvalue, do not reach the neighboring
interfaces, thus the solution for every 2/1
i
xx is constant during the considered time interval.
Fig.32 Reconstruction
57
Application of the modified ENO method to the ignition transient analysis.
ENO methods, developed by Harten et alii37
, generalize the Godunov scheme as a higher accuracy
orders scheme. The one presented below is the second order scheme used in our code.
A function ntxV , can be defined in
2/12/1,
JJxxx , approximating the exact solution
ntxU , lest h
2 terms:
J
n
J
n
J
nxxsVtxV , (106)
where n
JV is a second order estimation of the averaged value of the exact solution
2/1
2/1
2,,
1 j
j
x
x
n
J
nn
JhOtxUdxtxV
hV (107)
and sn
j is an first-order accurate approximation of the local first derivative of the solution. A
possible choice is:
n
i
n
i
n
i
n
i
n
JVVVV
hs
11,modmin
1 (108)
where the minmod function is defined as:
minmod (x,y) = 0 if 0 yx
minmod (x,y) = yxxsign ,min otherwise
This choice for the local slope of the initial data distribution guarantees that no new local extremes
can be introduced in data reconstruction so to avoid unphysical oscillations.
It is well known, in fact, that passing from first-order schemes to higher-order ones, near
discontinuities or strong gradients, the numerical solution can show unphysical oscillations.
Local solution of the initial values problem, as above defined, can be find by a Cauchy method: in
any cell the solution is given by Taylor’s expansion near (xj, tn)
2, hOtt
t
Vxx
x
VVtxV
n
n
J
j
n
J
n
J
(109)
x-derivatives of this equation are all known from the reconstruction step (103), while time-
derivatives can be evaluated from eq.(99)
x
UF
t
U
(110)
Expression (106) does not define univocally the value n
JtxV ,
2/1 at the interface:
58
22
2/1
2/1
n
J
n
J
n
J
n
Lt
Vh
x
VVV
J (111)
2211
1
2/1
2/1
n
J
n
J
n
J
n
Rt
Vh
x
VVV
J (112)
To solve this issue, and to evaluate the integral of the flux, Harten suggests to solve a Riemann
problem whose left and right-side status is defined by equations (111-112). Thus the flux can be
written:
2/12/1
2/1 2/12/1
,,
n
R
n
L
Rn
j JJ
VVFtxF (113)
where 2/12/1
2/12/1
,
n
R
n
L
R
JJ
VVF is the interface flux given by the solution of a Riemann problem, as a
function of the constant left and right-side states.
Once defined the data reconstruction at the time tn by eq.(109) and once evaluated interface fluxes
by the solution of Riemann problem, defined by eq.(113), the solution at the time tn+1
can be found
by applying eq.(107).
At this juncture, by integrating the conservation equations (12-16), featuring also source terms,
over the generic domain
,0
2,
2
hhD of the xt plane, the following integration scheme
follows:
Mass conservation for the three gaseous species:
n
i
n
ik
n
i
n
ikAA
11 2/1
2/1
2/1
2/1
2/1
2/1
2/1
2/1
2/1
2/1
2/1
2/1
n
i
n
i
n
ik
n
i
n
i
n
ikAuAu
x
t
2/1
1
n
ikS (114)
where k = in, ig, pr .
For the density of the gas mixture inside the single ith cell we have:
prigini
Momentum conservation for the gas mixture:
n
i
n
i
n
i
n
i
n
i
n
iAuAu
111 2/1
2/1
22/1
2/1
2/1
2/1
2/1
2/1
n
i
n
i
n
i
n
iAup
x
t
2/1
2/1
22/1
2/1
2/1
2/1
2/1
2/1
n
i
n
i
n
i
n
iAup
2/1
2
2/1
12
2/1
122/1
n
i
n
i
n
in
iS
x
AApt (115)
Energy conservation for the gas mixture:
59
1111
n
i
n
i
n
i
n
i
b
i
n
iAeAe 2/1
2/1
2/1
2/1
2/1
2/1
2/1
2/1
n
i
n
i
n
i
n
iAuep
x
t
2/1
2/1
2/1
2/1
2/1
2/1
2/1
2/1
n
i
n
i
n
i
n
iAuep
2/1
3
n
iS (116)
In these equations the quantities 2/1
2/1
n
i and
2/1
2/1
n
i represent interface values (respectively at
points 2/1i
x and 2/1i
x ) at time 2/1n
t ; these values are evaluated by the second order ENO scheme
and represent an approximation of the solution for the points 2/1
2/1,
n
itx and 2/1
2/1,
n
itx .
2/1n
ijS represents, the source terms for the mass, momentum and energy equations, respectively,
generated by the igniter operations, propellant combustion and interaction with the intersegment
cavities, as discussed in the gasdynamic model, evaluated at time tn+1/2:
2/1
2/1
,
2/1
,,2/12/12/1
,,1
n
i
n
iP
n
iprS
pr
n
iB
n
i
n
iprV
AmPrS
2/1
2/1
,
2/1
,,
2/1
2/1
,
2/1
,2/1
,,1
n
i
n
iP
n
iigS
n
i
n
iP
n
iign
iigV
Am
V
AmS
2/1
2/1
,
2/1
,,2/1
,,1
n
i
n
iP
n
iinSn
iinV
AmS
(117)
2/12
2/1
2/12/1
,
2/1
,2/1
,22
1
n
in
i
n
inj
n
iP
n
iign
iucf
V
vAmS
2/1
2/1
,
2/1
,
2/1
,,
2/1
2/1
,
2/1
,2/12/12/1
,3
n
i
n
iS
n
iP
n
iigS
n
i
ig
n
iP
n
iig
fpr
n
iB
n
i
n
iV
HAm
V
HAmhPrS
2/1
,
2/1
n
iw
n
iPq
An approximation of the solution is evaluated at any point of 2/1,
n
itx by Taylor’s expansion
from the neighboring n
itx , and n
itx ,
1, supposing that all the functions have the proper
differentiability characteristics. The set of variables for the slope evaluation of the reconstruction
phase are (k, u, T). The subscripts L and R denotes for the left and right side of the interface,
respectively.
Thus:
2
|2
|,|2/1
2/1
t
t
x
xtx
n
i
kn
i
kn
ikL
n
ik
2
|2
|,|2/1
2/1
t
t
ux
x
utxuu
n
i
n
i
n
iL
n
i
(118)
2
|2
|,|2/1
2/1
t
t
Tx
x
TtxTT
n
i
n
i
n
iL
n
i
and
60
2
|2
|,|111
2/1
2/1
t
t
x
xtx
n
i
kn
i
kn
ikR
n
ik
2
|2
|,|111
2/1
2/1
t
t
ux
x
utxuu
n
i
n
i
n
iR
n
i
(119)
2
|2
|,|111
2/1
2/1
t
t
Tx
x
TtxTT
n
i
n
i
n
iR
n
i
All these temporal derivatives can be expressed in terms of spatial derivatives by means of
equation of motion:
kk
kkB
x
u
dx
dAu
Axu
t,1
1
2
1B
x
p
x
uu
t
u
(120)
3
111 B
dx
dAu
AT
x
uT
x
Tu
t
T
The source terms of the conservation equations must be adapted in order to fit the integration
scheme also during the reconstruction phase. Bj represents the same source terms 2/1n
iS , for the
mass, momentum and energy equations, respectively, evaluated at time tn+1/2, rewritten in the
appropriately to be included in the reconstruction of the temporal derivatives. So:
ip
iK
iKA
SB
,
,1
,1 with k = in, ig, pr.
ip
inigprii
i
iA
SSSuSB
,
,1,1,1,2,2
11
(121)
ininigigprpriinigpriiii
iiiP
iScvScvScvTSSSuSuS
cvAB
,1,1,1,1,1,1
2
,2,3
,
,32
11
As far as spatial derivatives are concerned, the following numerical form is adopted
n
i
n
i
n
i
n
i
n
JVVVV
hs
11,modmin
1 (122)
where the minmod function, described above in the text (eq.108), avoids the generation of
numerical oscillations.
It is seen that, by the two Taylor’s expansions, two different values for all the variables at the
interface 2/1i
x are obtained. In order to evaluate the fluxes appearing in the discretized balance
equations, this discrepancy is solved by the solution of a Riemann problem, with initial conditions
represented exactly by the values obtained from the series expansions in n
itx , and n
itx ,
1.
61
Riemann problem
The exact Riemann solver has been reformulated in order to handle discontinuities arising between
adjacent cells containing gases with different thermophysical characteristics, in fact three different
gas species are involved: the inert pressurizing gas filling the combustion chamber before ignition,
the igniter gas and the propellant combustion products. These three gaseous species, mixing
together along the chamber, generate the space and time-changing mixture that possesses own
thermophysical characteristics and constitutes the flowing fluid. Accordingly, during the
reconstruction phase of the integration method, the local species concentration must be defined at
any interface in addition to the other variables, k, u, T, p, in order to allow to completely
evaluating the mixture thermophysical characteristics, such as R, , Cp, Cv.
Assuming that, at a specified time the fluid dynamic field on the left and the right-side of a given
interface between two cells of the discretization grid is known, these two initial conditions can be
denoted as status (1) and status (4), as usual dealing with Riemann problems.
Initial condition:
p1, 1, u1, 1 for the left side;
p4, 4, u4, 4 for the right side;
Now the following quantities can be evaluated:
1
1
121
2auIR
and 4
4
411
2auIVR
(123)
2
1 (124)
4
1
1
4
4
1
1
1
p
p
a
aZ (125)
where a1 and a4 represents the speed of sound for the fluid on the left and the right-side of the
interface, respectively; is the value of the specific heat ratio of the gaseous mixture crossing the
interface, as a solution of the initial values problem itself.
If 41
pp then 1
.
If 41
pp then 4
.
The standard iterative algorithm for the exact solution of the Riemann problem assigned, can now
be advanced adopting as iteration variable the velocity of the intermediate states.
For every iteration the left-side wave and the right side wave are evaluated separately, verifying if
they are shock waves or rarefaction waves, utilizing the Rankine-Hougoniot or the isentropic
relations, until convergence is attained.
The left-side solution gives the status denoted as p2, 2, u2, 2, while the right-side solution gives
the status denoted as p3, 3, u3, 3.
The initial guess value of the velocity for the intermediate states, vel, representing the evolution of
the Riemann problem is:
62
Z
IVRIRZuvel
1
12
0 (126)
Once a solution to the problem is found, thus, the value of vel that satisfies the condition 32
pp ,
the direction of the flow across the considered interface must be evaluated, in order to update the
evolution of the gas mixture inside the considered cells and its thermophysical characteristics.
A correct information of the thermophysical characteristics of the flow crossing the interface is
required for an accurate determination of the interface fluxes needed for the integration of the
conservation equations.
Once the solution of the conservation equations, at the current time, is obtained, the mixture
characteristics in every cell is updated.
In fact, for the ith cell, at the current time step, its fluid-dynamic status is given by:
pr,i, ig,i, in,i, ui, pi, ei
Is known then the mass fraction of the individual gas species.
The density of the mixture is prigini
, and the mass fractions are:
i
ik
ikc
,
, with k = in, ig, pr.
Once the mass fractions are known, all the main thermophysical characteristics of the mixture, such
as, specific heats, molecular weight, gas constant, etc. , can be evaluated.
63
3.2 The ESPRIT simulation code
The present simulation model has been implemented in a computer code, named ESPRIT
(Evaluation of Solid Propellant Ignition Transient), of about 6000 instructions in FORTRAN
language. As a reference figure, the simulation of a one second transient, with a spatial grid of 600
nodes (for the largest engine simulated, SRM#2), requires about 15 minutes computing time on a
Pentium III PC with a 800 Mhz clock.
SO LU TIO N O F EU LER ’S
C O N SER V A TIO N LA W S
EQ U A TIO N S
SO U R C E TER M S PR EPR O C ESSIN G
SPA TIA L D ER IV A TIV ES
EV A LU A TIO N
TIM E D ER IV A TIV ES
EV A LU A TIO N
PR IM ITIV E V A R IA B LES R EC O N STR U C TIO N
W EA TH ER SEA L
R U PTU R E C H EC K
B O U N D A R Y A N D FIELD V A R IA B LES
R EC O N STR U C TIO N
IN TER FA C E V A R IA B LES EV A LU A TIO N
B O U N D A R Y C O N D ITIO N
D EFIN ITIO N
IN TER FA C E FLU X EV A LU A TIO N B Y
R IEM A N N PR O B LEM S SO LU TIO N S
Fig.33 Fluid-dynamics numerical solution
Fig.33 reports a flowchart of the numerical algorithm for the gasdynamical model.
64
C O M B U ST IO N C H A M B E R
FLU ID -D Y N A M IC FIE LD
SO L U T IO N
FLU ID D Y N A M IC S E Q U A T IO N S
SO U R C E T ER M S U PD A T E
IG N IT ER
O PE R A T IO N
E V A LU A T IO N
C O N V EC T IV E H E A T
T R A N SFER
E V A LU A T IO N
R A D IA TIV E H EA T
T R A N SFER
E V A LU A T IO N
G R A IN SU R FA C E
T EM PE R A T U R E
U PD A T E
&
IG N IT IO N C H E C K S
G EO M E TR IC A L PR EPR O C E SSO R
&
C O D E IN IT IA L IZA TIO N
C H A M B ER G EO M E TR Y U PD A TE
D A T A IN PU T
T IM E STE P
L O O P
G A SD Y N A M IC A L
M O D E L
Fig.34 ESPRIT simulation code
The gasdynamic model represents the core of our simulation code; Fig.34 reports a simplified flow
chart of whole simulation code. The gasdynamic model, is coupled and associated with the main
physical submodels such as the igniter, the evaluation of the heat fluxes, the grain surface update,
the ignition criterion checks, grain combustion and regression.
Flowcharts of some of the main submodels adopted is reported in Fig. 35-36:
65
IG N IT ER PE R FO R M A N C E
&
IM PIN G EM EN T R EG IO N
D E FIN ITIO N
Igniter m ass flow rate
from data tabs
Igniter m ass flow rate.
Ign iter cham ber pressure.
Ign iter jets diam eter , tem perature and
im pact velocity
Im pingem ent region boundaries
C ham ber flu id-dynam ic status:
p(t),T (t), (t)
Fig.35 Igniter submodel implementation
PR O PE LL A N T SU R FA C E
H E A T E X C H A N G E
E V A LU A T IO N A N D
IG N IT IO N C H EC K
Surface T em perature:
T s(j,t)
G as radiative
coefficient: G R C (j,t)
C onvective heat transfer
coefficient: hconv(j,t)
Surface T em perature:
T s(j,t+ t)
R adiative gas flux:
q radg(j,t+ t)
R adiative flam e flux:
q radf(j,t+ t)
C onvective heat flux:
q conv(j,t+ t)
G as tem perature:
T (j,t)
F lam e tem perature:
T f
Fig.36 Surface temperature evaluation model (ith
cell)
66
4 RESULTS
Numerical results presented in this section refer to three different solid rocket motor families.
These motors are prototypes or tentative configurations developed and built up at FIAT-AVIO in
the past. All the motor data (e.g. shapes, sizes, dimensions, propellant and grain characteristics,
igniter designs, etc.) have been kindly made available by FIAT-AVIO in the framework of a
cooperation program with our institution. Experimental data, obtained by FIAT-AVIO during static
tests of these motors, are utilized to evaluate the accuracy and reliability of the simulation model.
The comparison is carried out in terms of pressure evolution at the head-end of the motor, the only
useful experimental data available. Sometimes, pressures dimensions and times can be reported in
the graphics as non-dimensional quantities.
4.1 HEAD-END PRESSURE AND IGNITION SEQUENCE
Solid Rocket Motor # 1 Ariane 4 Solid Booster (SRM #1)
Motor size: medium size engine (less than ten tons of total mass, combustion chamber length
about 8.5 m) with a L/D ratio of about 8.5 and an average mass flow rate of about 300 kg/s.
Grain design: single segment grain of composite propellant, star shaped port area with an initial
Ap/Ath ratio of about 2.5, volumetric loading fraction of about 0.75, Sb/Ath ratio of about 240.
Igniter: pyrogenic igniter at the head-end of the motor providing a peak mass flow rate of about
15 kg/s, a total temperature of the combustion products of about 2200 K, with one central and
six lateral nozzles.
Fig. 37 Combustion chamber scheme
The solid black line in Fig.38 shows the time history of pressure measurements at the head-end of
the SRM #1 during a static test of the motor. The overall dimensions of this motor, the propellant
grain and the igniter designs lead to a short ignition transient characterized by a steep pressure rise.
The pressure gradient reaches a peak of about 550 atm/sec and the overall average value, during the
entire transient, is about 280 atm/sec.
The steep rise of the pressure is mainly due to the low initial volume of the chamber void and the
high value of the burning surface.
The orange curve in Fig.38 represents the numerical results obtained by the present simulation
model with a spatial discretization of 300 cells. In the whole, the comparison shows a very good
qualitative and quantitative agreement.
Fig.39 shows the ignition time history of the cells along the whole motor chamber.
67
Fig. 38 Head-end pressure comparison
The start of the pressure rise is well-timed as well as the pressure gradient variation due to the
flame spreading start (t=0.06s). The maximum slopes of both curves (very similar) indicate a
correct evaluation of the flame spreading rate too. The fitting of the curves at the quasi-steady
conditions has been obtained applying a scale factor of 1.033 to the pressure dependent term of the
burning rate. The percentage error in the pressure evaluation indicated an averaged departure of
about 3%.
Fig.39 Ignition time sequence
68
The ignition of the impingement region is strongly anticipated with respect to the remaining of the
combustion chamber. This is due to the strong convective heat flux generated by the igniter jets
impinging onto the grain surface. Fig.40 shows the evolution of the convective heat flux close to
the impingement region during the early ignition transient. The port section of this motor is almost
uniform along the chamber, so the heating of the propellant surface in the regions outside the
impingement is quite uniform too and the critical ignition temperature is achieved almost
simultaneously along the bore. Therefore, the flame spreads in the standard region at a very fast
rate. After the flame spreading start-up, significant gas velocities are attained in the combustion
chamber that is affected by strong unsteady gasdynamic phenomena.
T IM E
P
0 0.1 0 .2 0 .3 0.4 0.5
10
20
30
40
50
N on-dim ensiona l C ham ber Length
No
n-d
ime
ns
ion
al
Tim
e
0 0.05 0.1 0 .15 0.2 0 .250
0.1
0 .2
0 .3
0 .4
im pingem ent region ignition
flam e spreading start
head-end full ignition
ignite r's head
N on-dim ensiona l C ham ber Length
Co
nv
ec
tiv
eH
F(k
W/m
2)
0
500
1000
1500
2000
2500
3000
3500
0.1 0 .20
Fig. 40 Convective heat flux growing near the impingement
Solid Rocket Motor # 2 Ariane 5 Solid Booster (SRM #2)
Motor size: large size engine (total mass about 270 tons, combustion chamber length about 24
m) with a L/D ratio of about 8 and an average mass flow rate of about 2000 Kg/s.
Grain design: segmented grain of composite propellant, one short segment with star shaped
cross section at the head-end of the motor followed by two cylindrical segments, low initial
value of the Ap/Ath ratio at the motor end (about 2), volumetric loading fraction of about 0.78,
initial Sb/Ath ratio of about 230.
Igniter: large size and high energy pyrogenic igniter at the head-end of the motor with a peak
value of about 140 kg/s of mass flow rate and a total temperature of about 3300 K.
The large dimensions of this motor lead to a relatively long ignition transient (two times longer
than SRM#1). In spite of its size, the design concept is similar to that of the SRM#1. Due to the
volume of the combustion chamber void (20 times greater than SRM#1) this motor is characterized
by a slow pressure rise with peak values of about 250 atm/sec and an overall average value of about
110 atm/sec. As consequence of a low Ap/At ratio, significant gas velocities are reached in the
combustion chamber.
69
Fig. 41 Combustion chamber scheme
For this family of boosters we tested four slightly different configurations, that present different
geometrical details and variations in the composition of the propellant grain mixture.
The results of these simulation have been obtained by the default un-tuned response of the code.
The experimental curves of Fig.42 represents two ground fire tests and show the time history of the
pressure (about 100 measurements) at the head-end of the SRM#2. The other couple of figures, Fig.
43, represents the simulation comparisons of two experimental data sets obtained from in-flight
response of slightly different configurations. As general remark, the comparison shows a good
agreement between all the experimental and numerical pressure diagrams during the ignition
transient.
Fig. 42 Ground fire test simulations
Fig.44 represents the ignition sequence and the chamber section of SRM#2 and shows the delay in
the start-up of the flame spreading at the initial section of the conical region, before the full ignition
of the star shaped region.
The ignition sequence for this motor is much more complicated than for SRM#1, this is due to the
complex geometrical configuration of the motor chamber.
This is a long segmented engine, with major morphological bore section variations along the
chamber: a star-shaped zone that comprehends the impingement region, followed by two long
conical port section segments. The port area in the transition zone between the first two grain
segments (near slot#1) is very small and it acts as a sort of throat section. Therefore the local flow
velocity is very high and the associated strong convective heat flux quickly ignites the grain.
70
Fig. 43 Comparisons with in-flight data
Fig. 44 Ignition time sequence
The quite simultaneous starting of flame spreading along the star-shaped zone and the first conical
segment generates a sudden important mass addition that, coupled to the travelling compression
waves originated by the igniter start-up, is the cause of the “knee” apparent in the pressure diagram.
The adoption of a quasi-1D gasdynamic model seems to lead to a rough approximation of the
flow/surface heat exchange, where the port area shows sudden variations, with an overestimation of
the local convective heat flux. Parametric analysis suggests that an accurate validation of
convective heat transfer modeling must be still carried on. Fig.44 shows also that the simulation
model detects the ignition delay of the slots between the different segments and of the submergence
zone. The fitting of the all the curves at the quasi-steady conditions has been obtained by applying
a scale factor of 1.03.
71
Solid Rocket Motor # 3 Vega, Zefiro Solid Rocket Stage (SRM #3)
Motor size: medium size engine (total mass about 17 tons, combustion chamber length about 4
m) with a low L/D ratio (about 2.5) and an average mass flow rate of about 200 kg/s.
Grain design: single grain of slow burning composite propellant, cylindrical shape of the port
area with an enlarged star shaped section at the end of the motor, high ratio Ap/Ath (initial value
of 10 at the motor end), high volumetric load fraction (about 0.85), Sb/Ath ratio of about 210.
Igniter: highly energetic igniter at the head end of the motor with a peak value of about 15 kg/s
of mass flow rate and a total temperature of about 3000 K.
Although the class of this motor is similar to that of the SRM#1, the design concept is different.
The SRM#3 family is characterized by a slower pressure rise with peak values of the pressure
gradient of about 380 atm/s and an overall average value of about 160 atm/s The main reasons of
this behavior are: i) the slow burning propellant; ii) the low value of the burning surface. The
highly energetic igniter lead to a short delay of the flame spreading start-up.
Fig. 45 Chamber port section
Fig. 46 Head-end pressure comparison
72
SRM#3, as the SRM#2 family, presents a major port section variation along the bore. Also in this
case, the un-tuned application of the simulation model can bring to a remarkable overestimation
and displacement of the pressure curve with respect to the experimental data. The problem, as
pointed out for SRM#2, lies again in the coupling of the convective heat model to the simple quasi-
1D simulation of the flow inside the chamber: the overstimation of the convective heat occurs at
the sudden variation of the port area between the uniform conical region and the diverging star-
shaped zone. 1D simulation is not able to describe the flow recirculation that probably occurs in
this region of the motor chamber. By artificially reducing the convective heat transfer in the star
shaped region of the grain, the local flame spreading is slowed down and the associated pressure
shows a good agreement with the experimental data, Fig.46.
Fig. 47 Head-end pressure comparison (second prototype)
For this motor, the limits of a quasi-1D gasdynamic model are more tangible than in the previous
motor. A new formulation of the convective heat transfer model is under development in order to
overcome at least in part, these limits. Fig.47 presents the simulation code application, with no
more further tuning, to the experimental data of a second prototype of the SRM#3 family, with a
slightly different grain geometry and propellant characteristics. Good results confirms an overall
accuracy of the code.
The complex ignition behavior of SRM#3 is evidenced by the ignition sequence represented in Fig.
48.
As final remark, the short motor length induces high frequency pressure oscillations of small
amplitude at the very beginning of the ignition transient. These oscillations are properly noticed by
the experimental curve, since the available number of pressure measurements, contrary to the
SRM#1 test, is now appropriate for their detection. The numerical simulation is also able to detect
these pressure oscillations.
The fitting of the curves at the quasi-steady conditions has been obtained by applying a scale factor
of 1.04.
73
Fig. 48 Ignition time sequence
74
4.2 OVERPRESSURE PEAK AND EROSIVE BURNING
Figure 49 shows chamber pressure and mass balance time history of SRM#1: mass flow rates from
igniter, impingement region, standard region together with the mass flow exiting from the nozzle
are depicted. The plot evidences the coincidence of the pressure peak with the instantaneous
balance between the mass flow through the motor nozzle and the total mass flow originating from
both the propellant and the igniter combustion. The contribution of the impingement region to the
occurrence of the overpressure peak is evidenced in Fig.50.
Fig. 49 Chamber mass flow balance (SRM#1)
Fig. 50 Effect of erosive burning at impingement (SRM#1)
75
The rationale of the simulation model has been altered by artificially imposing the evaluation of the
erosive burning rate at the impingement region by means of the standard convective heat transfer
model. Due to the negligible mass flow at the head-end side of the motor, the local erosive burning
at the impingement region becomes very low. The reduced mass addition from the propellant
surface of the impingement region leads to a decrease of the chamber pressurization rate. The
corresponding pressure curve in Fig.50 is labeled as “numerical pressure without erosive burning”.
The comparison of the mass flow from the impingement region obtained by the modified model
with respect to that obtained by the nominal numerical simulation (with the proper erosive burning
model at impingement) shows a remarkable difference. The most interesting consequence is that
overpressure peak is missing.
Fig.51 Erosive burning time history along the chamber (SRM#1)
Fig.52 Total burning rate time history along the chamber (SRM#1)
Figure 51 shows erosive burning rate time history along the combustion chamber of SRM#1: the
strong local erosive burning effect of the igniter jets in the impingement region is evidenced by the
pike near the head-end of the chamber; erosive burning is very small along the standard region of
the bore but raises again downstream near the end of the combustion chamber, where high flow
76
velocity is attained.
Figure 52 shows the time history of the total burning rate obtained by the sum of the erosive
burning rate and the pressure contribution terms (see eq.92): comparing Fig 51 and 52, the pressure
contribution seems to be quite uniform along the chamber.
Figure 53 shows the total propellant mass addition rate time history along the combustion chamber:
mass addition from the burning propellant does not exactly follow the behavior of the total burning
rate because it depends also on the local combustion perimeter (i.e. the contribution of the
submergence region near the end of the chamber is evidenced near cell 250).
Fig.53 Mass addition from propellant grain combustion (SRM#1)
Figures 54 and 55 show the chamber pressure and mass balance time history during the ignition
transient in the combustion chambers of SRM#2 and SRM#3, respectively. These motors have
different dimensions, geometrical configuration and overall ballistic behavior, but overpressure
peaks, as SRM#1, again characterize the chamber filling phase at the end of the flame spreading;
our simulation model is able to reproduce them, and to explain their origin. Again the overpressure
peaks take place exactly at the time when the total added mass flow equals the mass flow exiting
from the motor nozzle, as observed for SRM#1. This occurrence, verified in motors with different
values of L/D, in addition to the similarity between the pressure and nozzle mass flow evolution,
clearly indicates the absence of strong propagation phenomena inside the combustion chamber. In
other words, the peak is not originated by unbalancing due to “dynamic phenomena”. The absence
of a peak in the curve of the grain surface mass flow (that does not include the impingement
region) demonstrates that the overpressure is not a consequence of possible temporary increase of
the erosive burning in the standard region of the chamber during the transient.
77
Fig. 54 Chamber mass flow balance (SRM#2)
On the other hand, it is seen that the decreasing mass flow from the igniter and, consequently, from
the propellant grain (mainly from the impingement region) lead to a lower value of the actual
“equilibrium pressure”. Further the shape of the pressure curve near the peak, is very similar to that
of igniter mass flow rate, for all the simulated motors.
In conclusion, the impingement of the igniter jets plays a significant role, not only in local
propellant heating, but also in the strong erosive burning induced at the impingement region.
Fig. 55 Chamber mass flow balance (SRM#3)
During the steady-state operative phase, it is well known that erosive burning can be a significant
78
effect, due to the high velocities of the steady flow in the combustion chamber, for motors with
high L/D and low Ap/Ath ratios (port area/throat area). It is common practice to suppose that erosive
burning occurs only for this typology of motors even during the ignition transient. But, as it has
been evidenced before, erosive burning at the impingement region, together with igniter mass flow
rate, clearly influences the ignition transient even in a motor with a very low L/D ratio, such as
SRM#3. So it could be improper to refer to a global behavior of erosive burning only on the basis
of the averaged geometrical characteristics of the motor chamber. Peculiar flow conditions (like at
the impingement region), unsteady phenomena (high velocity transients) and/or a non-uniform port
section of the combustion chamber, discourage adopting this simplified approach.
Fig. 56 Effect of erosive burning along the chamber (SRM#1)
Figures 56, 57, 58 show for the three motors, respectively, the difference between the nominal
numerical pressure time history and the pressure curve obtained by artificially neglecting the
contribution of the erosive burning along the combustion chamber, except at the impingement
region. The strong dynamic phenomena that characterize the initial ignition transient appear to be
ineffective as far as erosive burning in the “standard regions” of the chamber is concerned,
although very high transient flow velocities can be temporarily attained. The difference between
the curves becomes evident when the mass flow is significant and quasi-steady flow conditions are
approached.
SRM#1 and SRM#2 show a quite similar behavior evidencing that erosive burning strongly affects
these motors in the same qualitative and also quantitative way. On the other hand, Figure 58
confirms that the erosive burning for SRM#3 is not effective for the standard regions of the
chamber, while, as shown before, the erosive burning at impingement plays a decisive role for the
attainment of the overpressure peak.
79
Fig. 57 Effect of erosive burning along the chamber (SRM#2)
Fig. 58 Effect of erosive burning along the chamber (SRM#3)
In conclusion the erosive burning model has to be able to reproduce even local or temporary
effects, such as in the impingement region or in other particular locations of the chamber
characterized by peculiar geometrical configurations or gasdynamic patterns.
80
4.3 FLOWFIELD FEATURES
The influence on the overall simulation capability of the introduction of some important features
and characteristics of the gasdynamic model adopted is evidenced by means of proper numerical
experiments; the numerical results are also exploited in order to analyze in deeper detail some
aspects of the gasdynamic behavior of the motors.
By means of results obtained from the simulations, it is possible to have a clear insight of the main
fluidynamic events characterizing the ignition transient, by analyzing the time history of the
flowfield inside the combustion chamber.
For this task, results obtained for SRM#1, characterized by a monolithic propellant grain with a
simple port geometry, are analyzed first.
Fig.59 Chamber pressure (SRM#1)
Fig.60 Chamber pressure (SRM#1)
81
Figures 59 and 60 show the evolution of the pressure profile inside the combustion chamber during
the initial part of the ignition transient (0-0.05s); each line is taken at a different time, as shown in
the contour legend. Accordingly it is possible to understand in detail the fluidynamic behavior of
the system during this phase, and its similarity to the shock-tube test study, as already discussed in
the first part of this work (chapter 1). Figures 61-62, 63-64, show the detail of the two phases in
which the considered time-interval can be divided.
Fig.61 Pressure profiles (phase 1) (SRM#1)
Fig.62 Temperature profiles (phase 1) (SRM#1)
82
As the igniter is started-up, a series of compression waves begins to propagate along the
combustion chamber (Figures 61-62, zone 1). At the same time a contact surface is generated, and
propagates slowly, behind the compression waves, and represents the interface between the cold
pressurizing gas, residing in the chamber before ignition, and the hot gases generated by the igniter
and by the ignited regions of the grain (Figures 61-62, zone 2). In the pressure profiles a small
spike localized on the position of the contact surface is noted; this presence is unphysical and it is
due only to a small numerical error of the scheme. We also notice the presence of the protection
seal at the nozzle throat, still unbroken, although the pressure difference between the chamber and
the environment is growing (Figures 61-62 zone 3). Near the head-end of the chamber we can see
red scatters that represent the first cells of the impingement region that begin to ignite.
Fig.63 Pressure profiles (phase 2) (SRM#1)
Fig.64 Temperature profiles (phase 2) (SRM#1)
83
Figures 63 and 64 show pressure and temperature profiles, respectively, during the immediate
following instants: while the contact surface reaches and crosses the middle of the chamber (Figure
63 zone 1), pressure waves reach the nozzle seal and begin to reflect (Figure 63 zone 2).
When the pressure difference across the seal reaches the critical design value, the seal breaks
(Figure 63 zone 3). Meanwhile, the reflected shock reaches the contact surface inside the chamber
and interacts with it, accelerating towards the head-end of the chamber (Figure 63 zone 4), where
the ignited region keeps expanding (Figure 63 zone 5).
Fig.65 Pressure profiles (seal rupture) (SRM#1)
Fig.66 Temperature profiles (seal rupture) (SRM#1)
84
Figures 65 and 66 show in detail the pressure and temperature profiles near the nozzle seal, during
its rupture: values at rupture are evidenced in black. The seal rupture generates a shock wave and a
contact surface that propagate outside through the diverging part of the nozzle, while an expansion
wave propagates backward towards the head-end, located immediately behind the shock previously
reflected by the still unbroken seal, see also Fig.67.
Fig.67 Gasdynamic patterns (seal rupture) (SRM#1)
The fluid-dynamic behavior of SRM#2 is also analyzed.
The geometry of the combustion chamber is much more complicated than SRM#1; Figure 68-
bottom shows the chamber section plotting the equivalent cylindrical cross section radius. Sudden
port variations and the presence of the intersegment slots lead to complex fluid-dynamic
phenomena that develop inside the motor, as shown in Figure 65-top. Compression waves,
generated by the igniter start-up, propagate forward through the chamber (zone 1); the presence of
the intersegment slot (mid of the chamber) generates a first reflected shock (zone 2). Compression
waves arrive in the nozzle zone and start reflecting (zone 3) upon convergent walls and from the
throat seal, still unbroken. The reflected shock wave is very intense and starts propagating
backwards through the chamber until it arrives near the first grain segment and starts interacting
with the on-going flame spreading, indicated by the red scatters (zone4). Finally, the reflected
shock arrives at the head-end of the motor (zone 5).
Figure 69 shows the detail of the interaction among the reflected waves coming from the nozzle, a
first shock (zones 1-2), an expansion and a second shock (zone 3) and the flame spreading. Flame
is propagating starting from two distinct points of the chamber, the impingement region with a star-
shaped cross-section (A) and the conical-shaped cross-section region located immediately after the
slot between the first and the second segments of the propellant grain (B). In this region, in fact, the
port area is very small and velocity is high, thus a locally high convective heat flux quickly
increases the propellant surface temperature, up to the critical ignition value.
85
Fig.68 Pressure profiles and chamber scheme (SRM#2)
The complex interaction between the waves and the flame spreading is the cause of the appearance
of the first knee clearly visible in the head-end pressure curve (see Figure 42 or 71): this
phenomenon also generates a slow, quasi-steady pressure oscillation along the chamber,
corresponding to the first natural acoustic frequency of the combustion chamber (about 20 Hz).
This oscillation is clearly visible in the head-end pressure curve (Figure 42 or 71) as in Fig.70 (red
lines) that shows pressure profile in chamber during the first part of the ignition transient.
86
Fig.69 Pressure profiles and flame spreading (SRM#2)
Fig.70 Pressure profiles and oscillations (SRM#2)
87
4.4 INTERSEGMENT SLOTS AND GAS MIXTURE
Numerical results presented in this section refers to the simulation of the ignition transient of the
solid booster of the Ariane 5 prototype, SRM#3 family. A specific calibration of the convective
heat transfer coefficient in the standard region has been performed in order to have the best
agreement with experimental data. This calibration is constituted by a single value of a single
coefficient, that multiplies the convective heat transfer coefficient and is based on an a posteriori
comparison with the available experimental data; no further specific calibrations have been applied
(Figure 71).
Fig.71 Head-end pressure SRM#2 (nominal)
The first ignition occurs a t 0.05 s, while the beginning of flame spreading, represented by a
sudden change of the curve slope, occurs a t 0.15 s. A further significant change of the curve
slope is located a t 0.20 s. Less pronounced inflexion points can be observed with a period of
about 0.05 s. This behavior, as discussed above, is due to the first longitudinal frequency of the
motor (about 20 hertz). Even if the low value of the Ap/Ath ratio leads to significant gas velocities,
the overpressure peak at about t 0.55 s is mainly due to the igniter operation6. The large
dimensions of the motor lead to a relatively long ignition transient (about 0.8 s) characterized by a
slow pressure rise with peak values of about 750 atm/s and an overall average value of about 240
atm/s.
88
Fig.72 Comparison of head-end pressure histories
Fig.73 Head-end pressure histories (detailed view)
The results of two different numerical simulations are presented in both Figs.69 and 70, based on
the following assumptions:
The properties of both the igniter gases and the pressurizing gas are assumed equal to those of
the propellant combustion products (no-mixture solution).
The presence of the void volumes due to the grain slots and the submergence region are
neglected, while their combustion surfaces are correctly accounted for (no-slots solution).
Two different overall behaviors are evidenced in the figures. The no-slot numerical solution shows
that pressure is overestimated up to overpressure peak, while the quasi-steady condition is correctly
89
attained thereafter. Further, the qualitative agreement of the curve is not satisfactory. On the other
hand, the no-mixture numerical solution is sufficiently in agreement, from the qualitative point of
view, with the experimental curve, but the pressure profile is significantly underestimated for a
large period of the ignition transient. These results, and their motivations, will be discussed in more
details in the following paragraphs.
Effects of the Slots & Submergence Modeling
Figure 74 shows the evolution of significant properties and quantities inside the submergence
region during the ignition transient. The figure presents the results obtained by means of the
reference numerical solution. Similar behaviors could be obtained if the grain slots were
considered.
Fig.74 Gas state inside the submergence volume
As the pressure inside the combustion chamber increases, the pressure inside the submergence
volume (red curve in the figure) increases in a similar way. Therefore, the total mass of gas inside
the submergence volume (black line in the figure) increases from 2 kg until about 9 kg. Oscillations
of the mass curve near to the peak value are due to the passage of significant pressure waves inside
the combustion chamber. Until t 0.17 s, the submergence volume is filled with pressurizing gas
only. In fact, the pressurization of the submergence region is due to pressure waves, while the hot
gases produced by both the igniter and the propellant combustion have not yet reached into the
submergence region. Thereafter, a slight amount of igniter gases and propellant combustion
products enters the submergence volume (green line and blue line respectively). This is due to the
arrival of the flame spreading, which is so fast, that it reaches the submergence region basically ate
the same time with the hot gases produced at the impingement region and by the igniter. The total
mass of gas is increased; at t 0.20 s the combustion surface inside the submergence volume starts
to ignite. This ignition is the consequence of both the surface preheating due to the radiation from
the gas mixture inside the volume (very moderate) and the radiation from the flames on the
propellant surfaces nearby the submergence region (more significant). The high-density
90
pressurizing gas is then expelled, and the volume starts to be filled by the low-density gas produced
by the propellant combustion products. Therefore, the total mass inside the volume is greatly
reduced, even if its pressure is further increased. The evolution of the gas mixture temperature,
orange curve, is in agreement with the previous description of the phenomenon.
Fig.75 Ignition time sequence
The no-slot simulation model is not able to take into account the filling of the slots and
submergence volumes. Provided that their total volume is about 5% of the total volume of the
chamber, the underestimation of the chamber volume leads to a more rapid pressurization, as
shown in Figure 73. In addition, the relevant combustion surface is supposed to ignite as soon as
flame spreading reaches the submergence region (i.e. without any delay). The local significant
discrepancy between the numerical and the experimental curves at t = 0.18-0.20 s (as evidenced in
Figure 73) is mainly due to this latter occurrence. This kind of qualitative discrepancy could be
very significant when the accurate prediction of the pressure gradient is required. Obviously, larger
volumes and combustion surfaces of these portions of the propellant grain involve more severe
limits in the applicability of the no-slot model.
Effect of Gas Mixture Modeling
As mentioned before, the pressure history obtained by the no-mixture numerical simulation is
largely shifted towards later times. The delay appears already at an early stage of the transient,
during the ignition of the impingement region, and increases until the onset of the flame spreading.
The maximum delay is attained approximately at the end of this phase, around t 0.22 s, when it
stands at about 0.02 s. This value remains approximately constant up to the pressure peak, when it
begins decreasing, and eventually disappears at quasi-steady condition.
91
The plots of the ignition time sequence along the combustion chamber for the two cases of gas
mixture and no-mixture (Figure 75), show that the different pressure histories are generated by
different ignition times. A small delay already appears at the beginning of the ignition of the
impingement region. In fact, even if the energy source term due to the igniter is correctly accounted
for, when the igniter gas properties are replaced by the propellant gas properties, small differences
in the cps could introduce significant differences in the gas temperature evaluation.
The no-mixture model also replaces the pressurizing gas properties by the propellant gas properties.
Because the two gases exhibit different values of R and , the propagation speed of the pressure
waves is significantly altered. A higher value of the speed of sound for the mixture model brings to
a faster waves propagation and therefore to a stronger energy exchange between the hot gas and the
propellant grain.
Snapshots taken at early stages of the ignition transient history shed light on this process (Figure
76). Effects of the hot gas addition from the igniter are similar to a shock-tube: as a matter of fact it
is evidenced from the velocity, pressure and temperature distributions along the bore at t=0.097 s.
A compression wave occurs near cell number 400, followed by a contact surface near cell 110;
other strong variations near cell 80 and cell 310 are induced by the port area geometry. The step
appearing near cell 530 corresponds to the diaphragm located at the nozzle throat, still closed at
that time. By comparing the same variable distributions along the bore for the no-mixture case, it
appears clearly that the gas mixture model exhibit an enhanced wave speed of propagation.
92
A
Fig.76 Gas properties along the chamber
(t=0.097s):
A) reference numerical solution
B) no-mixture model
B
93
CONCLUSIONS
An unsteady quasi-1D numerical simulation model is developed in order to predict the behavior of
large solid motors during the ignition transient. In particular, this model is finalized to be used as a
numerical tool during the preliminary design phase, when no information about the future behavior
of the motor is available. An Euler flow model is adopted coupled to suitable semi-empirical
models that take into account the main phenomena affecting the ignition transient. Non-reacting
gas mixture is considered and the original numerical integration technique, a second order ENO
scheme, accordingly modified. Special attention is devoted to simulate the effects of the
impingement of the igniter jets on the grain propellant surface (heating, ignition and combustion of
the impingement region). A radiation model is also proposed. Further, the behavior of the slots and
the submergence region has been considered by a simple integral model. Their geometry is taken
into account and their ignition accordingly evaluated. The simulation model is extensively tested
and numerical results compared to experimental results obtained for widely different motor
concepts and configurations. Significant insight about the role of some phenomena affecting the
ignition transient is drawn from a critical discussion of these results.
On the overall, results show a satisfactory agreement between the numerical calculations and the
experimental data. For the rocket motors considered here, this agreement is quite uniform, in spite
of the different size and driving concepts of the relevant designs. More significant discrepancies
between numerical and experimental results appear only in cases where the 1D gasdynamic model
does not account for significant multidimensional effects. The level of these results is due to an
accurate and detailed analysis of the phenomena involved the ignition transient and in the
modeling. In particular:
the proper timing of ignition at the impingement region, obtained by the igniter model and the
associated heat transfer model;
the proper timing of the ignition at the head-end region, mainly thanks to the radiative model
and the ignition criterion;
the proper evaluation of erosive combustion at the impingement region, thanks to the igniter
model and the relevant combustion model;
the gas mixture model allows proper evaluation of propagation phenomena and energetic
processes during the flow transient;
besides a proper filling of the combustion chamber, the slot and submergence model take into
account their delayed ignition and the subsequent expulsion of inert gas from their volumes.
The role of these improvements is demonstrated by the discussion of presented results.
A specific calibration (convective heat flux at sharp cross section variations) has been required just
when and where the 1D gasdynamic model ignores significant multidimensional effects.
The following further modeling improvements are foreseen at the moment: i) development of a
two-phase (gas/particle) gasdynamic model; ii) local analysis of some particular regions of the
combustion chamber (sudden change in the cross section shape and dimensions, igniter contraction,
etc.) by means of more sophisticated 2D or 3D flow models. The overlapping and linking, without
loosing the handiness of our approach, among the basic 1D model and the local multidimensional
models could be a critical point. More refined burning rate relationships (especially for unsteady
conditions) and propellant ignition criteria can lead to a further improvement of the overall
simulation model.
94
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ACKNOWLEDGMENTS
The author would like to thank first of all Professor Marcello Onofri and Professor Carlo
Buongiorno, Tutor and Chief Coordinator of the doctorate course, respectively. Special thanks, of
course, to Professor Maurizio Di Giacinto and Professor Bernardo Favini for the patience, the
precious cooperation and the “human support” granted during all these years.
The author would like to thank also Professors M. Valorani, D. Lentini and F. Nasuti for their
cooperation, useful suggestions and technical support.
Thanks also to Dr. R. De Amicis, Dr. A. Annovazzi and Dr. A, D’Acunzo of FIAT-AVIO
(Colleferro factory) which have kindly granted the motor data and the experimental results.
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