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
1 2 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
3
4 Nomenclature
2 Ajet Igniter jet final cross section m 2 AP Propellant port area m 2 Ath Nozzle throat cross section m 2 At ig Igniter nozzle throat cross section m cf Friction coefficient m –1 -1 cp Specific heat at constant pressure per unit mass J kg K –1 -1 cv Specific heat at constant volume per unit mass J kg K 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 -1 Hig Igniter gas total enthalpy per unit mass J kg Hj Distance between igniter nozzle and propellant surface m -1 Hf Propellant gas enthalpy per unit mass J kg -1 Hs Slot gas flow total enthalpy J kg -1 -1 -2 hc Heat transfer convective coefficient J kg s m -1 -1 -2 hc imp Heat transfer convective coefficient at impingement region J kg s m -1 hf Propellant gas enthalpy per unit mass J kg -1 -1 -2 hR center Central cell radiative transfer coefficient J kg s m -1 -1 -2 hR right Right-side cell radiative transfer coefficient J kg s m -1 -1 -2 hR left Left-side cell radiative transfer coefficient J kg s m -1 -1 -2 ˆ Equivalent radiative transfer coefficient J kg s m h R
KMR Impingement region proportionality constant -1 -1 kpr Propellant thermal conductivity N s K 3 -1 keb Erosive combustion constant m K J Mj Igniter jet mach number -1 m ig Igniter gas mass flow rate kg s -1 s Total mass flow rate from slots kg s -1 s,pr Propellant gas mass flow rate from slots kg s -1 s,ig Igniter gas mass flow rate from slots kg s -1 s,in Inert gas mass flow rate from slots kg s 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 -2 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
5 -2 Equivalent radiative heat flux from right-side cell w m qˆ r right -2 Equivalent radiative heat flux from left-side cell w m qˆ r left Rej Igniter jet Reynolds number -1 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 2 Sb Grain propellant burning surface m 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 -1 vinj Igniter gas velocity m s 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 2 -1 pr Propellant thermal diffusivity m s
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 -3 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 -2 -4 Stefan-Boltzman constant w K
6 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.
7 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:
8 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.
9 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)
10 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)
11
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.
12 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.
13 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.
14 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
15 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.
16
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.
17 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. >>.
18 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:
P S b r p C Cd (1) Ath 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.
19
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: dT dT P g P rQ S I f (2) dx S dx S 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:
20 v 0
v v p
vc T T Q p i i (3)
vy i Dy i i 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: r ap n (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,
21 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
rb r0 re (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.
22 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.
23 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 dp RT p dT p dV m in m out (6) dt V T dt V dt dT RT RT dV q m inTin m out T m in m out T (7) dt pV C vV dt C v 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 n products are in flowing from solid propellant of density p, surface Sb and burning rate r ap , and outflow is through a choked nozzle of throat area Ath then eq.(6) reduces to the classical expression:
1 /(1 n ) * a p S b c P (t ) (8) A th
24
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.
22 Fig.11 Chamber pressure drop down as a function of Mach number and Ap/Ath
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:
0 dp dm dx M 2 2 f (9) 0 p m D
2 dp dm dA fdx M 2Q 4 Q 1 / 2 (10) 2 p m A D 1 M
dV 2 dm dA 2 fdx 2 1 M 2M /1 M (11) V m A D
1 where Q 1 1 M 2 2 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 dm P r Pdx , where P is the
25 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).
26
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
27 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.
28 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
29 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.
30 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.
31
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
uA A m A v P 2 P ig P av ,inj 1 2 u p A p p c u (12) t x x V 2 f
Energy conservation
eA e p uA m A H m A H P P ig P ig s P s rP p h q P (13) t x b f V t W V
Mass Conservation
A uA m A pr P pr P s, pr P rP pr (14) t x b V
ig AP ig uA P m ig AP m s ,ig AP (15) t x V V
32 A uA m A in P in P s ,in P (16) t x V where m m and (17) s k s ,k k k with k = in, ig, pr
The equation system is closed by means of the equation of state: p RT 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.
33 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: m CAV CAV V CAV mass of individual gas species in the cavity: m c m with k = in, ig, pr CAV k CAV ,k CAV constant pressure specific heat and gas constant of the gas mixture in the cavity:
c PCAV c CAV , pr c P , pr c CAV ,ig c P ,ig c CAV ,in c P ,in
R CAV c CAV , pr R g , pr c CAV ,ig R g ,ig c CAV ,in R g ,in
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:
pˆ CAV p CAV p CAV p CHAM (18) where is a calibration coefficient.
Once pˆ CAV is known the variation of the enthalpy inside the cavity in the interval dt can be evaluated as:
34 ˆ H H CAV H pr H CAV (19) where
H CAV c pCAV m CAV TCAV is the enthalpy inside the cavity at time t;
H pr r Pb p dx dt T f is the enthalpy addition inside the cavity due to propellant combustion at the non-inhibited surface; V Hˆ c pˆ CAV is the enthalpy inside the cavity at time t+dt.; CAV pCAV CAV R CAV From the sign of H we obtain the direction of the flow across the cavity/chamber interface; if H 0 we have mass spillage from the chamber to the cavity: H m CAV is the variation of the total mass inside the cavity; c pCHAM TCHAM
m CAV ,k c CHAM ,k m CAV is the variation of the mass of the single gas specie inside the cavity;
If H 0 we have mass spillage from the cavity to the chamber; H m CAV is the variation of the total mass inside the cavity; c pCAV T CAV
m CAV ,k c CAV ,k m CAV 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: mˆ CAV , pr m CAV , pr m CAV , pr r Pc om b pr dx dt , for the propellant gas; mˆ CAV ,ig m CAV ,ig m CAV ,ig for the igniter gas; mˆ CAV ,in m CAV ,in m CAV ,in for the pressurizing gas.
The total mass inside the cavity is: mˆ CAV mˆ CAV , pr mˆ CAV ,ig mˆ CAV ,in 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:
m CAV , pr m CAV ,ig m CAV ,in m S , pr ; m S ,ig ; m S ,in (20) dx dt dx dt dx dt and
m CAV c pCHAM TCHAM H S if (21) dx dt
m CAV c pCAV TCAV H S if H 0 (22) dx dt
35 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
36 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 1 * (23) Pc Ig p 2
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 2 1 p * p (25) Cig 1 M t 1 (26) 2 1 p Cig At 2 p p m a 1 a (27) RT 1 p p cig Cig Cig 1 2 p a vt v e RT cig 1 (28) 1 p Cig 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: * * p e p t p a p p cr (29) 2 1 p * p p * (30) Cig cr 1 37 M t 1 (31) p Cig Atig p Cig Atig m (32) * c RT cig 1 2 p * 2RT cig (33) vt v e RT cig 1 a t 1 p 1 Cig 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. * p e p t p p a (34) 2 1 p * p p * (35) Cig cr 1 M t 1 (36) p Cig Atig p Cig Atig m (37) * c RT cig 1 2 p * 2RT cig vt v e RT cig 1 a t (38) 1 p 1 Cig 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: * m RT m ig c ig cig p (39) cTS Atig Atig to the value of the total pressure corresponding to the simultaneous condition of adapted and choked nozzle: 1 1 p c p a (40) SON 2 38 If Pcson=Pcts , we have a choked adapted nozzle and equations (30–33). If Pcson Than, in both cases, the final section of the conical igniter jet is : Atig (41) A jet 1 2 p p a 1 a 1 p c p c TS TS and the final diameter of the jet D jet 2 A jet (42) See also Fig.19. If Pcson>Pcts, the igniter nozzle is not choked and equations (25)-(28) hold, thereby: and ; A jet Atig D jet D tig 2 Atig 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: v inj Nax Ig v inj cos ig Nz Ig v av ,inj (43) Nax Ig Nz Ig Fig.19 Igniter jet geometry 39 15 1 14 0.9 13 M ach uscita ignitore P ortata in m assa ignitore 12 0.8 ) s / 11 g K 0.7 ( 10 e r o t 0.6 9 i n g i h 8 c a a 0.5 s s M 7 a 0.4 6 m n i 5 a t a 0.3 t 4 r o P 0.2 3 2 0.1 1 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 T e m po (s) Fig. 20 Igniter mass flow rate and exhaust Mach number (SRM#1) 60 1 0.9 50 ) 0.8 r a B ( 0.7 e 40 r P ressione di uscita ignitore o t M ach uscita ignitore i 0.6 n g i h a t c i a 0.5 30 c s M u i 0.4 d e n 20 o i 0.3 s s e r 0.2 P 10 0.1 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 T e m po (s) Fig. 21 Igniter chamber pressure and exhaust Mach number (SRM#1) 40 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 K D MR jet (44) X imp where KMR is a calibration coefficient. sin ig In this region the convective heat exchange is evaluated utilizing a formulation derived from a 41 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: D jet 1 1.1 K 0.42 pr ri (45) h c x i Pr ig F Re jet r H D i jet jet 1 0.1 6 D r jet i where 0.55 Re 0.5 jet F Re jet 2 Re jet 1 (46) 200 tot 4 m ig Re jet D jet (47) Nz Ig Nax Ig tot where m Ig 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: (48) 30 ig 90 2000 Re jet 400000 (49) 42 (50) 2.5 ri D jet 7.5 (51) 2 H jet D jet 12 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: 3 2 r r r F (Re ) 0.42 i i i jet (52) h c x i Pr ig K p a b c d D D D D jet jet jet jet 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: h x h x cmin i cmax i hc ( x i ) (53) av 2 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 43 impingement region; the proposed final expression for the convective heat transfer coefficient, for the ith cell of impingement region is: 2 K D MR jet 2 2 Nz Ig ri 2 sin ig h x h x (54) cIMP i cav i Pb x i and the specific convective heat flux is: q x h x T x T x (55) c IMP i c IMP i i s i 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: Nu 0.023 Re 0.8 Pr 0.3 (55) valid for: 104 f Re 1000 Pr 8 Nu (56) 1 2 2 f 3 1 12 .7 Pr 1 8 where 2 f 1.82 log Re 1.64 (57) valid for: 104 Accordingly, a convective heat transfer coefficient, and the related heat flux, can be defined for any cell in the standard region: 44 f Re 1000 Pr K 8 P h x (58) C i 1 2 2 f 3 D 1 12 .7 Pr 1 h 8 q C x i hC x i T x i TS x i (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: 4 4 q R 0 C rad TG TS (60) This is a common general expression, used in most of the simulation models that deal with 17 radiative heat transfer, where the coefficient Crad typically is defined as a constant 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: 4 4 q R 0 GRC TG TS (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. 45 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. 46 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: 4 q x GRC x T 4 x T x (62) rCENTER i i i S i Radiative gas flux from the right side cell volume (not ignited): 4 q x GRC x T 4 x T x (63) rRIGHT i i 1 i1 S i Radiative gas flux from the left side cell volume (not ignited): 4 q x GRC x T 4 x T x (64) rLEFT i i 1 i 1 S i 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): 1 4 4 4 4 q r x i T f T S x i GRC x i 1 0 T x i 1 T S x i (65) RIGHT 2 Radiative gas flux and flame flux from the left side cell (ignited): 1 4 4 4 4 q r x i T f T S x i GRC x i 1 0 T x i 1 T S x i (66) LEFT 2 47 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 : 2T T (67) P y 2 t where y>0, t>0 together with the boundary and the initial condition: T (68) k P q t y for y=0, t>0, and T TS 0 (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 q( ) d T (t) Ts 0 (70) 0 k t 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: i n 1 q T (t ) Ts 2 t t t t (71) n 0 n i n i 1 i 0 k 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 48 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: 2 T T T q 0 ; 0 ; ; T y TS 0 (72) y y 2 y y y y 0 So we obtain: z * 3 z * 3 where (73) T y T S 0 3 q z, t 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: q x h x T x T x (74) rCENTER i rCENTER i i S i where 49 2 h x GRC x T x T x T 2 x T x (75) rCENTER i i i S i i S i for the right side flux: q x h x T x T x (76) rRIGHT i rRIGHT i i 1 S i where 2 h x GRC x T x T x T 2 x T x (77) rRIGHT i i 1 i 1 S i i 1 S i for the left side flux: q x h x T x T x (78) rLEFT i rLEFT i i 1 S i where 2 h x GRC x T x T x T 2 x T x (79) rSLEFT i i 1 i 1 S i i 1 S i The possible presence of nearby radiating surface flames, is accounted by adding the following additional terms: qˆ x hˆ x T T x (80) rRIGHT i r i f S i qˆ x hˆ x T T x (81) rLEFT i r i f S i where ˆ 1 2 2 h r x i T f T S x i T f T S x i (82) 2 Then, the total radiative flux, for the generic ith cell, is the sum of all previous contributions: q x q x q x q x qˆ x qˆ x (83) r tot i rCENTER i rRIGHT i rLEFT i rRIGHT i rLEFT i The total heat flux is q q q ; if eq.(67) is multiplied by dy and integrated from 0 to , the tot rtot C heat balance integral is obtained: T ,t T 0,t d T S 0 (84) P y y dt where t Tdy (85) 0 Since there no heat is transferred beyond y=: T , t 0 (86) x 50 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): d P f z,t T S 0 (87) dt and the (73) in eq.(85) we obtain 3T 3T T T S S 0 S S 0 (88) 4 f z, t that introduced in eq. (87) gives the following ordinary differential equation to be solved for z=Ts: 2 4 d z T f z,t S 0 (89) P 3 dt f z,t 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: 2 4 d TS TS f T i 0 (90) P S i 3 dt f TS i that gives for the temperature transient: 1 dT d f T S i 4 3 2 S i P f T S 2T S T S f T S T S T S (91) dt 3 i i 0 i i 0 dT S i 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: r r0 reb (92) r r p,T and r ap n (93) 0 0 S 0 0 the coefficient a also account for the effect of the initial propellant grain temperature: 51 r 0 ref T T a e P S 0 ref (94) n p ref Erosive burning is described as: 1 2 r 3 0.12 0.8 P 3 9 5 T0 u u r k e (95) eb eb 0.9 0.2 1 W g D h with keb=f1 (ρpr, csp, Tf, TbS, TS0 ) and f 2 D h ,1 Pb (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: 1 2 2 P r 0.12 0.8 v ig 3 9 ig 53 T0 ig v inj ig inj r k e (97) eb IMP eb IMP 0.9 0.2 ig 1W ig D jet with u D h D h 1 g 1 2 f 3 , , , , (98) D P u M jet b ig ig jet 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: 52 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 53 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 . 54 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: U F U 0 (99) t x U x,0 U o x (100) with x ; e t 0, , where: U T ; u; e and F T u; u 2 p ; e p u We assume that U o x is a piecewise regular function of x and that the matrix : F ,....., F AU 1 n (101) U 1 ,....., U n 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 h h : D , 0, 2 2 ˆ h ˆ h U x,t U x,t F U x ,t F U x ,t (102) h 2 2 where h 1 2 U x, t h U x , t d (103) h 2 h h is a spatial-averaged value calculated in x , x , 2 2 while ˆ h 1 h F U x , t F U x , t d (104) 2 0 2 55 is the time-averaged value of the physical fluxes, calculated in t, t , 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 U x, t for every cell unknown (j), then the value for t n 1 t n can be computed, once the values of the interface fluxes for t n t t n 1 are known, by using eq.(102) in the form: 1 1 n n n 1 n 2 2 U j U j F 1 F 1 , where (105) j j h 2 2 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. h min j Fluxes evaluation: the solution of the time integral of the fluxes is immediate if: n J j n K n K n th n Where j max K j and j is the k eigenvalue of the Jacobian matrix in x j , t . 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 x xi1 / 2 is constant during the considered time interval. Fig.32 Reconstruction 56 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. n A function V x, t can be defined in x x J 1/ 2 , x J 1/ 2 , approximating the exact solution U x, t n lest h2 terms: n n n V x, t V J s J x x J (106) n where V J is a second order estimation of the averaged value of the exact solution x n 1 j 1 / 2 n n 2 V J V x, t dx U x J , t O h (107) x h j 1 / 2 n and s j is an first-order accurate approximation of the local first derivative of the solution. A possible choice is: n 1 n n n n s J min mod Vi Vi1 ,Vi1 Vi (108) h where the minmod function is defined as: minmod (x,y) = 0 if x y 0 minmod (x,y) = sign x min x , y 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 n any cell the solution is given by Taylor’s expansion near (xj, t ) n n n V V n 2 V x, t V J x x j t t O h (109) x J t J x-derivatives of this equation are all known from the reconstruction step (103), while time- derivatives can be evaluated from eq.(99) U F U (110) t x n Expression (106) does not define univocally the value V x J 1 / 2 , t at the interface: 57 n n V h V V n 1 / 2 V n (111) L J 1 / 2 J x J 2 t J 2 n n V h V V n 1 / 2 V n (112) R J 1 / 2 J 1 x J 1 2 t J 1 2 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: F x , t n F R V n 1 / 2 ,V n 1 / 2 (113) j 1 / 2 L J 1 / 2 R J 1 / 2 where F R V n 1 / 2 ,V n 1 / 2 is the interface flux given by the solution of a Riemann problem, as a L J 1 / 2 R J 1 / 2 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 h h of the xt plane, the following integration scheme D , 0, 2 2 follows: Mass conservation for the three gaseous species: n 1 n 1 n n t n 1 / 2 n 1 / 2 n 1 / 2 n 1 / 2 n 1 / 2 n 1 / 2 n 1 / 2 k Ai k Ai k u i 1 / 2 Ai 1 / 2 k u i 1 / 2 Ai 1 / 2 S 1 (114) i i x i 1 / 2 i 1 / 2 k i where k = in, ig, pr . th For the density of the gas mixture inside the single i cell we have: i in ig pr Momentum conservation for the gas mixture: n 1 n 1 n 1 n n n t n 1 / 2 n 1 / 2 n 1 / 2 2 n 1 / 2 i u i Ai i u i Ai p i1 / 2 i1 / 2 u i1 / 2 Ai1 / 2 x n 1 / 2 n 1 / 2 n 1 / 2 n 1 / 2 n 1 / 2 2 n 1 / 2 n 1 / 2 Ai 12 Ai 12 n 1 / 2 p i 1 / 2 i 1 / 2 u i 1 / 2 Ai 1 / 2 t p i S 2 (115) x i Energy conservation for the gas mixture: 58 n 1 b 1 n 1 n n n 1 t n 1 / 2 n 1 / 2 n 1 / 2 n 1 / 2 i ei Ai i ei Ai p i1 / 2 ei1 / 2 u i1 / 2 Ai1 / 2 x n 1 / 2 n1 / 2 n1 / 2 n1 / 2 n 1 / 2 p i1 / 2 ei1 / 2 u i1 / 2 Ai1 / 2 S 3 i (116) n 1 / 2 n 1 / 2 In these equations the quantities i 1 / 2 and i 1 / 2 represent interface values (respectively at n 1 / 2 points xi1 / 2 and xi1 / 2 ) at time t ; these values are evaluated by the second order ENO scheme n 1 / 2 n 1 / 2 and represent an approximation of the solution for the points x i1 / 2 , t and x i1 / 2 , t . n 1 / 2 S represents, the source terms for the mass, momentum and energy equations, respectively, j i 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: n 1 / 2 n 1 / 2 n 1 / 2 n 1 / 2 m S , pr ,i A P ,i S r n 1 / 2 P 1, pr ,i i B i pr n 1 / 2 V i n 1 / 2 n 1 / 2 n 1 / 2 n 1 / 2 n 1 / 2 m ig ,i A P ,i m S ,ig ,i A P ,i S 1,ig ,i n 1 / 2 n 1 / 2 V i V i n 1 / 2 n 1 / 2 n 1 / 2 m A S S ,in ,i P ,i (117) 1,in ,i n 1 / 2 V i n 1 / 2 n 1 / 2 n 1 / 2 n 1 / 2 m ig ,i A P ,i v inj 1 n 1 / 2 S cf u 2 2 ,i n 1 / 2 i V i 2 n 1 / 2 n 1 / 2 n 1 / 2 n 1 / 2 n 1 / 2 n 1 / 2 n 1 / 2 m ig ,i A P ,i H ig m S ,ig ,i A P ,i H S ,i S r n 1 / 2 P h q n 1 / 2 P n 1 / 2 3,i i B i pr f n 1 / 2 n 1 / 2 i w ,i V i V i n 1 / 2 An approximation of the solution is evaluated at any point of xi , t by Taylor’s expansion n n from the neighboring x i , t and xi1 , t , 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: n 1 / 2 n k n x k n t k | L k x i , t |i |i i 1 / 2 x 2 t 2 n 1 / 2 n u n x u n t u i1 / 2 |L u xi , t |i |i (118) x 2 t 2 n 1 / 2 n T n x T n t Ti 1 / 2 | L T x i , t |i |i x 2 t 2 and 59 n 1 / 2 n k n x k n t k | R k x i 1 , t |i 1 |i 1 i 1 / 2 x 2 t 2 n 1 / 2 n u n x u n t u i1 / 2 | R u xi1 , t |i1 |i1 (119) x 2 t 2 n 1 / 2 n T n x T n t Ti 1 / 2 | R T x i 1 , t |i 1 |i 1 x 2 t 2 All these temporal derivatives can be expressed in terms of spatial derivatives by means of equation of motion: k k 1 u dA u u k B1,k t x A dx x u u 1 p u B 2 (120) t x x T T u 1 u dA u 1T 1T B 3 t x x A dx The source terms of the conservation equations must be adapted in order to fit the integration n 1 / 2 scheme also during the reconstruction phase. Bj represents the same source terms S i , 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: S 1K ,i B1K ,i with k = in, ig, pr. A p ,i 1 1 B 2 ,i S 2 ,i u i S 1, pr S 1,ig S 1,in (121) i A p ,i 1 1 2 B 3,i S 3,i u i S 2,i u i S 1, pr S 1,ig S 1,in Ti cv pr S 1, pr cv ig S 1,ig cv in S 1,in A P ,i cv i i 2 As far as spatial derivatives are concerned, the following numerical form is adopted n 1 n n n n s J min mod Vi Vi1 ,Vi1 Vi (122) h 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 xi1 / 2 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 n n represented exactly by the values obtained from the series expansions in xi , t and xi1 , t . 60 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: 2 2 R I u a and R IV u a (123) 2 1 1 1 4 4 1 1 4 1 1 (124) 2 1 a p 1 4 1 (125) Z 4 1 a1 p 4 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 p1 p 4 then 1 . If p1 p 4 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: 61 Z R 2 I R1 IV vel u 0 (126) 1 Z Once a solution to the problem is found, thus, the value of vel that satisfies the condition p 2 p 3 , 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 i in ig pr , and the mass fractions are: k ,i c k ,i with k = in, ig, pr. i 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. 62 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. SOURCE TERMS PREPROCESSING WEATHER SEAL RUPTURE CHECK BOUNDARY AND FIELD VARIABLES RECONSTRUCTION SPATIAL DERIVATIVES TIME DERIVATIVES EVALUATION EVALUATION INTERFACE VARIABLES EVALUATION BOUNDARY CONDITION DEFINITION INTERFACE FLUX EVALUATION BY RIEMANN PROBLEMS SOLUTIONS SOLUTION OF EULER’S CONSERVATION LAWS EQUATIONS PRIMITIVE VARIABLES RECONSTRUCTION Fig.33 Fluid-dynamics numerical solution Fig.33 reports a flowchart of the numerical algorithm for the gasdynamical model. 63 D AT A IN PU T GEOMETRICAL PREPROCESSOR & CODE INITIALIZATION IG NIT ER CONVECTIVE HEAT RADIATIVE HEAT OPERATION TRANSFER TRANSFER EVALUATION EVALUATION EVALUATION GRAIN SURFACE TEMPERATURE UPDATE & IGNITION CHECKS T IM E STE P LOOP FLUID DYNAMICS EQUATIONS SOURCE TERMS UPDATE GASDYNAMICAL MODEL COMBUSTION CHAMBER FLUID-DYNAMIC FIELD SOLUTION CHAMBER GEOMETRY UPDATE 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: 64 Chamber fluid-dynam ic status: p(t),T(t), (t) Igniter m ass flow rate from data tabs IGNITER PERFORMANCE & IMPINGEMENT REGION DEFINITION Igniter m ass flow rate. Igniter chamber pressure. Igniter jets diameter, temperature and impact velocity Impingement region boundaries Fig.35 Igniter submodel implementation Convective heat transfer coefficient: hconv(j,t) Gas radiative coefficient: GRC(j,t) Gas temperature: T (j,t) PROPELLANT SURFACE Surface Temperature: HEAT EXCHANGE Flame temperature: T s(j,t) EVALUATION AND T f IGNITION CHECK Surface Temperature: T s(j,t+ t) Radiative gas flux: q radg(j,t+ t) Radiative flame flux: q radf(j,t+ t) Convective heat flux: q conv(j,t+ t) Fig.36 Surface temperature evaluation model (ith cell) 65 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. 66 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 67 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. 35005.40 ignite r's hea d 3000 40 hea d-end full ignition ) 0.3 2 2e 500 m m / i T W l k a ( 30 2n 000 F o i H s P 0.2 n e e v flam e spre ading start i t 1500 m i c 20 d e - v n n o o 1000 N C 0.1 10 500 im pinge m ent region ignition 0 00 00..015 00..21 0.1 0..135 00.4.2 00..52.25 N on-dim e nsionTaIMl CEha m be r Le ngth 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. 68 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. 69 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. 70 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 71 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. 72 Fig. 48 Ignition time sequence 73 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) 74 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 75 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. 76 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 77 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. 78 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. 79 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) 80 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) 81 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) 82 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) 83 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. 84 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. 85 Fig.69 Pressure profiles and flame spreading (SRM#2) Fig.70 Pressure profiles and oscillations (SRM#2) 86 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. 87 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 88 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 89 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. 90 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. 91 A B Fig.76 Gas properties along the chamber (t=0.097s): A) reference numerical solution B) no-mixture model 92 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. 93 94 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. 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