Images from Nicols Garca Rosa (Supaero Ph.D)

Two-phase Flows in Combustion

SystemsGrard Lavergne

SUPAERO, ONERA

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CHAPTER I: INTRODUCTION TO COMBUSTION AND FLAMES IN INDUSTRY............................... 4

I. INTRODUCTION........................................................................................................................................... 5 II. COMBUSTION AND FLAMES IN INDUSTRY AND IN THE NATURE. ...................................................................... 8

CHAPTERII: INTRODUCTION TO TWO PHASE FLOWS....................................................................... 29 CHAPTER III: DEFINITIONS AND CLASSIFICATION OF THE DIFFERENT RGIMES OF TWO PHASE FLOWS.................................................................................................................................................. 33

I : DENSITY AND MASS FRACTION ..................................................................................................................... 34 II RELAXATION TIME......................................................................................................................................... 36 III. DILUTED AND DENSE TWO PHASE FLOWS .................................................................................................... 39

CHAPTER IV: DIFFERENT APPROACHES FOR TWO PHASE FLOWS NUMERICAL SIMULATION .................................................................................................................................................... 41

I. GASEOUS PHASE NUMERICAL SIMULATION : .................................................................................................. 42 I.1 Introduction............................................................................................................................................ 42 I.2 Scales and main characteristics of the turbulence ................................................................................. 42 Kinetic energy of turbulence........................................................................................................................ 44 Spectrum of the length scales of the turbulence .......................................................................................... 44

RANS ............................................................................................................................................................... 46 LES................................................................................................................................................................... 46 DNS.................................................................................................................................................................. 46 II. TWO APPROACHES FOR TWO PHASE FLOW MODELING................................................................................... 47

II.1 Euler/Euler approach ........................................................................................................................... 47 II.2 Euler/Lagrange approach..................................................................................................................... 49

CHAPTER V: SPRAY FORMATION ............................................................................................................. 51 I COMBUSTION CHAMBER AND INJECTION SYSTEM ........................................................................................... 52 SPRAY MEASUREMENT ..................................................................................................................................... 55 II PRIMARY AND SECONDARY LIQUID SHEET BEAK UP....................................................................................... 56

Atomisation.................................................................................................................................................. 56 Secondary break-up..................................................................................................................................... 58

III, DROPLET SIZE,DEFINITIONS, NON DIMENSIONAL NUMBERS, DIFFRENT INJECTION SYSTEMS....................... 61 III, DROPLET SIZE,DEFINITIONS, NON DIMENSIONAL NUMBERS, DIFFRENT INJECTION SYSTEMS....................... 62

Size distributions ......................................................................................................................................... 62 Mean size..................................................................................................................................................... 62 Distribution function: .................................................................................................................................. 62

CHAPTER VI: TURBULENT DISPERSION OF THE LIQUID PHASE.................................................... 64 I. DRAG COEFFICIENT OF A SPHERICAL PARTICLE (OR DROPLET) ....................................................................... 65

Drag coefficient for subsonic compressible flows ....................................................................................... 66 Drag coefficient for supersonic compressive flows ..................................................................................... 66 Drag coefficient in rarefied gases ............................................................................................................... 67

II. TURBULENT PARTICLES (OR DROPLETS) DISPERSION .................................................................................... 69 CHAPTER VII: DROPLET EVAPORATION AND COMBUSTION.......................................................... 74

EVAPORATION MODEL FOR AN ISOLATED DROPLET........................................................................................... 75 Liquid phase calculation/Droplet heating ................................................................................................... 76 I.1 Isolated droplet evaporation without convection ................................................................................... 77 I.2 Variable physical properties .................................................................................................................. 81 I.3 Convection correction ............................................................................................................................ 82

I.4 LIQUID PHASE MODEL .................................................................................................................................. 84 D2 model : .......................................................................................................................................... 84 Infinite Conductivity........................................................................................................................... 84 Limited conduction............................................................................................................................. 85 Effective conduction........................................................................................................................... 85 Validation .................................................................................................................................................... 86 Continuity equation for species in spherical coordinates............................................................................ 86 Combustion rate computation ..................................................................................................................... 89 Flame position............................................................................................................................................. 90 - Temperature profile .................................................................................................................................. 90 - Combustion time........................................................................................................................................ 91

QUASI-STEADY THEORY OF AN ISOLATED DROPLET BURNING ........................................................................... 92 CHAPTER VIII DENSE SPRAYS.................................................................................................................. 102

INTRODUCTION ............................................................................................................................................... 105

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EXPERIMENTAL SETUP .................................................................................................................................... 106 Droplet Generator ..................................................................................................................................... 106 Electrostatic Droplet Deflector ................................................................................................................. 106 Measuring techniques................................................................................................................................ 106 Droplet Size Measurements ....................................................................................................................... 107 Droplet Temperature Measurements......................................................................................................... 107 Droplet Velocity Measurements ................................................................................................................ 107 CARS Thermometry ................................................................................................................................... 108

RESULTS AND DISCUSSION .............................................................................................................................. 108 Drag Coefficient under Non Evaporating Conditions:.............................................................................. 108 Reacting Conditions .................................................................................................................................. 109

CHAPTER IX : DROPLET WALL INTERACTION.................................................................................. 117 DROPLET BEHAVIOR ON A HOT WALL.............................................................................................................. 118 FIRST CLASSIFICATION (PHD C. AMIEL SUPAERO) ...................................................................................... 122 SECOND CLASSIFICATION (P. VILLEDIEU) ....................................................................................................... 122

CHAPTER IX: EULER-LAGRANGE APPROACH, TWO WAY COUPLING ....................................... 126 THE PARTICLE SOURCE IN CELL MODEL (PSICM) FOR GAS DROPLET FLOWS............................................... 128 BASIC CONCEPT .............................................................................................................................................. 128 SOURCE TERMS ............................................................................................................................................... 129

CHAPTER X : EXAMPLES OF STUDIED CONFIGURATIONS............................................................. 133 DROPLET TRAJECTORY IN A TURBULENT FLOW............................................................................................... 134 DUMP COMBUSTOR [11] ................................................................................................................................. 134 LEAN PREMIXER PREVAPORISER MODULE [12]................................................................................................ 135 MODELLING OF THE TWO PHASE FLOW IN SOLID ROCKET MOTORS [13,14] ..................................................... 135

REFERENCES.................................................................................................................................................. 139 ACKNOWLEDGEMENTS.............................................................................................................................. 151 EXERCISES...................................................................................................................................................... 152

EXERCISE 1: LIQUID SHEET DISINTEGRATION .................................................................................................. 153 EXERCISE 2: DROPLET EVAPORATION.............................................................................................................. 156 EX 3 : STABILITY OF A TURBOJET COMBUSTION CHAMBER.............................................................................. 157 EXERCISE 4 : COMBUSTION CHAMBER DESIGN ................................................................................................ 158 EXERCISE 5: DROPLET TRAJECTORIES IN AN ACOUSTIC FIELD......................................................................... 167

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Chapter I: Introduction to Combustion and flames in industry

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I. INTRODUCTION Application domains are very extended and diversified. Energy production:

- Thermal power station - Combustion chambers (motors, reactors, turbojet, domestic and industrial

furnaces)

- Industrial processes:

- Manufacture of materials (steel, glass, ceramic..) - Hydrocarbon extraction - Weapons (gun, explosive) - Environment (pollution, forest fire)

The coupling of many complex phenomena induces difficulties of understanding of combustion phenomena. Various scientific topics

- Thermodynamics - Chemical Kinetics - Fluid mechanics - Heat and mass transfer - Turbulence - Material

First the chemical aspect is a the base of these all phenomena which can be characterized by one or several chemical reactions occuring simultaneously. The combustion induces a high heat release in the thinner zones: (de 0.1 1mm) with temperature gaps across the flame characterized by temperature ratios between burned and fresh gases which can be varied from 5 to 7. The combustion also induces a strong and non linear reaction rate (Arrhnus law). In the most situations, flames are located in gaseous medium and that is explained the presence of a third point : mass transfers. They can be of two types: convection movement of gases composing partially or totally the flame or molecular (or turbulent) diffusion of some species in the medium. To save permanent operating conditions of such a system, we need to bring fresh reactants and also to evacuate combustion products inducing fluid mechanics aspect. To highlight these aspects, consider first a candle flame (figure 1). This classic flame takes into account many complex phenomena (perhaps the most complex flame of all flames). The candle flame occurs just above the wick in a gaseous medium. A candle is composed of stearine made with carbon, hydrogen and oxygen atoms and is solid at ambient conditions. The flame is handled at the above extremity of the wick. The flame zone is composed of two parts : a weak blue zone surrounded by a luminous yellow zone. The combustion process can be explained by the following procedure : the liquid stearine goes up by capillarity effect in the wick, then it vaporizes at the extremity. With the contact of oxygen contains in air the stearine vapor burns. The most intense reaction corresponds to the blue zone weakly luminous. This blue color comes from CH radical radiation which is an intermediate unstable component of the chemical reactions. The intense yellow zone is due to soot radiation. Soot

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are carbon particles, C, which collide. The presence of these particles is due to a non complete combustion. That is means that oxygen is lacking to burn all the available amount of fuel. In contrary soot emits a lot of light, thanks to this particle emission the candle can light everybody.!! The candle light comes from a radiative thermal transfer. The thermal transfers also produces the liquefaction of the stearine at the top of the candle by conduction in the wick and by radiation At the last, the fluid mechanics is necessary to mix vapor and air. The natural convection induced by the heat release entrains fresh air along the flame necessary for the combustion and to evacuate the combustion products (CO 2 , H 2 O, carbon particles).

Figure 1 [1] Different physical aspects in a candle flame

Figure 2 [2] Candle flame structure Without gravity the candle can switch off due to the presence of burned gases around the flame (no convection). The three main aspects (chemical, physical and mechanical) of the candle combustion are associated to secondary phenomena:

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liquefaction, evaporation, nucleation and collision of soot particles. The heat conduction in the porous wick and the ascension of the liquid stearine along the wick are also some physical processes participating to the flame. The figure 2 shows a simplified scheme of the flame structure. The zone where the chemical reactions occur shares a gaseous medium where gases are oxidizer (outside) and a gaseous medium (reductor, inside). This kind of flame is called diffusion flame (mass transfer by molecular diffusion) or non-premixed flames.

- Premixed laminar flames

Two main combustion regimes can be encountered: the diffusion flame and the premixed flame. In this last case the oxidizer and the fuel are mix upstream the combustion zone.. Example of premixed flame A 1 liter bottle containing air is filled with fuel gas of a lighter during about 30s to get an appropriated equivalence ratio to ignite the combustion (figure 3). The mixed gases are ignited with the lighter at the nozzle of the bottle.

Figure 3 [1] Flame propagation in a gaseous mixture: a experiment being done with a bottle but caustiously The perpendicular flame propagation can be observed from the nozzle to the other side of the bottle. The propagation velocity of the flame can be. Computed. This experiment is not at all dangerous. The energy provided by the combustion is weak. The blue color is due to CH radical emission, the intensity of the yellow color depends of the equivalence ratio inside the bottle The heat transfers play an important role to heat the fresh premixing. The flame surface shares the fresh gases and the burnt gases The hot gases and the flame heat a small portion of fresh gas which ignites and heat again another part of fresh gas which ignites again and so on.., by this way the flame propagates in the bottle (figure 4).

Mass transfer plays also an important role in this flame. On one part the burnt gases (CO 2 , H 2 O) diffuse towards the fresh gases and on the other hand the high volume necessary for the heated gases expansion induced a high exit velocity of these gases at the nozzle of the bottle (be careful, dont put your hands!!).

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Figure 4 [2] Premixed laminar flame strucure

II. COMBUSTION AND FLAMES IN INDUSTRY AND IN THE NATURE. The synoptic presents a classification of different practical systems of combustion following the type reactant injection types (diffusion or premixed).

Figure 5: Different types of flame

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- motors

Two kinds of motors

Airbreathing using oxygen of the air. no airbreathing (rocket motors for example), oxygen comes from others

sources. Different types of airbreathing motors: Piston engine ignites by a spark plug or diesel The physical processes are so complex that is very difficult to predict the combustion chamber performance from a numerical computation. However the numerical simulation is used to reduce the number of experimental tests and to reduce the cost. Many empirical approaches are yet used for combustion chamber development. For example in France, the Renault society has developed only three motors between 1960 and 1980. The actual motivation in the domain of research is to improve the understanding of the physics of this hind of reacting two phase flows, to develop new models to get some predictive tools for performances computation. From 1960, the car consumption has been divided by two, the volume of the cylinder has been reduced from some liters to 1000 cm3, the power has increased and the weight reduced.

Piston engine, controlled ignition

Figure 6 [3]

Gases (a mixing of air and vapor petrol). are pressured in the cylinder before the valve closure.This mixing is then compressed, heated and then ignited by a spark plug and burns during a certain time. Then the piston go down, the mixing (which has changed its chemical nature and being heated up

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to 2500 or 2700K) is then expanded and cooled and finally exhaust when the exhaust valves are open and the piston go up. The ignition appears for a crank angle about of 20 before the high dead point and the end of combustion of the total volume of gas contained in the cylinder corresponds to about a crank angle of 20 after the high dead point.

Example : If the rotation velocity of a motor is 3000t/mn, the combustion delay is about : (40/360)*(60/3000) seconds, about 2 milliseconds ; this delay is enough long for the chemical reactions. What happens during the 2 milliseconds? it is the same event than the propagation of the flame in the bottle. The mixing enclosed in the cylinder is a mixing of air and fuel, almost perfectly homogeneous if the fuel droplets are well spatially distributed and evaporated and if the spark plug induces the premixed flame propagation in the gaseous mixture. If the gas is weakly disturbed the premixed flame propagates in any direction and get a hemispheric shape (figure 7), that happens for low rotation velocity motors. For high rotation velocity motors the gas inlet by the valve and the ascent of the piston induced movements and recirculations of the mixing and high turbulence level before the generation of the spark plug. The flame propagates in a turbulent medium, the size of the recirculation is higher than the flame thickness, we can observe corrugated flames figure 3. This flame is called premixed turbulent flame.

Figure 7 [1] Flame propagation in a motor with controlled ignition, idealized case without turbulence

Figure 8 [1] Flame propagation in a motor with controlled ignition, case with turbulence

This coupling phenomenum between flame and turbulence is preponderant for the

normal operating conditions of the motors. A laminar flame propagates at about 1 m/s, if the distance for the piston is 4cm, the delay for the propagation of the flame is 40 ms. For this conditions the rotation velocity limit should be 750 cycles per minute (80 milliseconds per cycle). The rotation velocity of actual motors is very higher (up to 12000 cycles per minute) thanks to the turbulent propagation of the flame!!

Diesel motor:

The diesel motor uses the diffusion flame. The gas (air only) enclosed in the cylinder is compressed by the piston and for a crank angle of about 20 before the high dead point a fuel spray is injected in the cylinder (that is the case for direct injection). This jet is disintegrated in fine dense droplets which disperse in the air. Droplets evaporate

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thanks to hot air and the generated vapor burnt with air. The spray core is very dense and plays the role of the wick for the candle. This kind of flame is a diffusion flame.

Figure 9 [4] The liquid jet ignites some fractions of milliseconds after the injection, the liquid injection continue during 1 or 2 milliseconds and the combustion phase stops some fractions of milliseconds after (figure 9). Then the piston goes down, the exhaust valves are open.

Figure 10 [1] Scheme of a flame in a diesel engine with direct injection

The combustion period is not instantaneous, the delay corresponds to 40 of crank angle. The flame structure is those corresponding to a diffusion flame. In fact this type of combustion is spray combustion, if globally the combustion regime seems to be a diffusion flame, many regimes of combustion occur in a spray: isolated droplet combustion, package burning, diffusion flame depending of the effect of turbulence on the spatial repartition of the droplets. The main result of this kind of combustion is an high amount of soot produced by the diesel motor in comparison with the petrol motor, but the thermodynamic efficiency is higher.

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The ramjet: From 60 years others motors have been developed, the simplest is the ramjet which can be used in the future for aircraft propulsion if the velocity reached by other tool is equal to a Mach number about 2. It is called flying stove tube. A scheme of principle is shown below (figure 11). The first ramjet has been designed by Leduc after the second world war.

Figure 11 [1] Scheme of cylindrical ramjet combustion chamber with only one flame holder To improve the performances and to reduce the discretion, different geometry of ramjet have been developed: 1 inlet, 2 inlets or 4 inlets.(figure 12 )

Figure 12: Example of ramjet geometry and physical phenomena occuring in the two phase flow processes (liquid fuel). Gases are compressed, burnt, and then expanded. The compression takes place in the air inlet and the expansion in the nozzle. The combustion occurs in the flame tube situated between inlets and nozzle. In the case of a subsonic regime, the inlet velocity is about 150m/s. Kerosene injection system is located either in the inlets (internal and external radius) or in connection with the dome.

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The liquid fuel is generated in this case directly in the flame tube.The fuel is disintegrated then evaporated at the entry of the combustion zone. The combustion chamber performances are depending on disintegration quality but also of the position of the injection devices which influences the spatial repartition of the equivalence ratio. These geometries are defined in order to generate several recirculating zones to attach the flame. We can distinguish on the previous figure three main zones, one between the inlets called dome which is a very mixed zone piloting the stability of the combustion chamber, the other called lateral recirculating zone which is a non well mixed zone and then the jet without recirculation. In the frame of a global approach a network of elementary reactors is considered, the two first zones are modeled as well stirred reactors and the last as piston reactor. The main characteristics of these reactors (volume, residence time, airflow rate, fuel flow rate) have a determinant influence on the ramjet performances. To improve the turbulence level inside the recirculating zones some obstacles are placed in the inlet. Heat and mass transfer occur inside these zones, and the combustion in maintained if the residence time is enough high or if the air velocity is low. The flame regime observed is not diffusion or premixed flame. It is called combustion zone.These zones produced permanently hot gases in the main flow and then the combustion propagates between these different zones.(figure 13).

Figure 13 [1] Some details on flame stabilization from recirculating kernels behind the flame holder In the flame tube the mean air velocity is about 50m/s. One example of numerical simulation of the reactive unsteady two phase flows is shown below for a simplified ramjet configuration.

Figure 14: Example of numerical simulation of reacting two phase flow inside a dump configuration[5] The ramjet propulsion type is often uses in the military domain, for missile propulsion but actually some recent researches are lead for plane propulsion for the cruise flight (scramjet).

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The turbojet: In the domain of aeronautics, the performances of turbojets has increased from the second world war (see the following table) :

[2]

Three generations of motors can be identified : ATAR developed just after the second war, equipped the Mystre II and now Mirage F1, the M53 for Mirage 2000 and the M88 for Rafale. The chamber volume has been divided by 10 and the reaction rate per volume unit Multipliesby2.4

[2] The size reduction of the combustion chamber induced pollutant emissions. Effectively, this reduction decreases the residence time, increases the temperature enhancing NOx emissions. Some prototypes are actually developed. Lean Premixer Prevaporizer modules are introduced upstream the primary zone (combustion zone) of the combustion chamber to improve the air-fuel mixing and the droplets evaporation rate before entering the combustion chamber.

The combustion chamber is located between the compressor and the turbine (figure15). Combustion occurs in the primary zone of the main chamber and also for military motors in the post combustion zone.

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Figure 15: Scheme of an aeronautic reactor

An actual combustion chamber configuration is presented below:

Figure 16: Scheme of a combustion chamber sector

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Figure 17: Scheme of a combustion chamber sector for helicopter application The general shape is annular, the total chamber is composed of several identical sectors (from 12 to 20 equipped each one of the same injection system. Three zones composed the main chamber: the primary zone, the intermediate zone and the dilution zone used to cool burnt gases before the turbine. The air coming from the compressor is divided following three or four directions:

Injection system feeding, the air crossing the injection system is used to atomize, to mix fuel and air in the injection zone and also to participate to the combustion.

Film cooling feeding, this air fraction is only used to cool the walls of the primary zone. The cooling can be achieved by air film generated close to the wall or by multi-perforated walls (TURBOMECA technique)

Primary holes feeding, one part of air feeds the primary holes, the primary jets impinge in the middle of the chamber, one part recirculates towards the primary zone, the other part flowing downstream. The primary zone is composed of different recirculating zones used to stabilize the flame (figure 18)

Figure 18: Example of isothermal flow inside a combustion chamber (hydraulic simulation)

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Dilution holes feeding : the last part of air is injected in the dilution holes to reduce the burnt gases temperature before the turbine.

The operating conditions are ranged depending of the altitude and flight conditions (take off, cruise, taxi).

- 0.2 bar < at compressor exit

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Figure 19: Example of fuel repartition in a combustion chamber of turbojet (numerical simulation [6]) Post-combustion zone

Figure 20 [2] Scheme of post combustion chamber The post combustion zone (PC) or reheat zone is a second combustion chamber located downstream the turbine. It exists only on military aircraft, except on Concorde which uses post combustion to get supersonic regime. This zone is feed by burnt gases coming from the main chamber, expanded in the turbine and by the secondary air flux coming directly from the compressor. These two fluxes must be optimized to get the best possible combustion. The flame is stabilized by the flame holder composed of concentric rings. The kerosene is injected by very simple injector. The operating conditions are the following : 0.3 bar < pressure < 6 bar 800 K < upstream mean temperature

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Combustion instabilities are often encountered in the post combustion zone linked to the presence of flow instabilities in the recirculating zone behind the flame holder. A passive control is generally applies to reduce these instabilities. The ignition of the post combustion zone must be performed just at the starting. Noise and discretion are also some actual research topics. Rockets : Rocket propulsion do not use air for combustion for two main reasons :

- absence of oxygen at high altitude - best performances by using products more oxidizer than oxygen

Rockets were invented at the beginning of the last century in Russia, France, USA and Germany. Different combinations between oxidizer and a reductor product are possible : oxygen/kerosene (figure21), N 2 O 4 -UDMH (Unsymetrical Dimethyl Hydrazine, figure 22) or oxygen/hydrogen ; solid products : ammonium or potassium perchlorate and plastic materials (polyurethan), hydrazine can be also used. In rocket motors reactants are called ergols.

Figure 21[1] Principle of bi-liquid rocket motors Figure 22 [1] Principle of solid

propellant rocket motor For the presentation of the different type of propulsion, we will describe the different stages of Ariane V. (figure 23).

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Figure 23: Scheme of the different stages of Ariane V Ariane V Rocket is composed of three propulsion stages

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Figure 24

- The solid propellant P230 also called Booster propels the rocket during 125s of the launch. It is composed of cylindrical blocs of about 25 meters length located both sides of the main stage (cryotechnic stage) figure 24). This propellant contains a solid propellant called butalane composed of ammonium perchlorate and of 18% of aluminium particles The particles size is about 35 m. This bloc is composed of

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three segments S1, S2 et S3 separated by thermal protections. The igniter is located at the top of segment S3, the nozzle is an integrated nozzle in order to point it up to 7. The combustion takes place at the surface of the grain. The regression surface velocity is about a few millimeters per second and the gas flow rate is proportional to the burnt surface. The gas flow rate being controlled, the nozzle section being fixed, the chamber pressure and the thrust are controlled too. The internal pressure is of 50 bar and the temperature about 3500 K

- This propellant behavior is like a close cavity. It is composed of two waterproof walls and a sonic nozzle. Instability problems can occur from the coupling between acoustic modes of the chamber and aerodynamics instabilities coming from thermal protection. Some aluminium particles contained in the grain agglomerate after their fusion and continue their burning in the burnt gases. The aluminium particle combustion is not yet well known and the modeling is very difficult. Studies linked in laboratories show a deposition of alumina slag on the aluminium particle giving after combustion a alumina residue size about 60 m. After 60S of the launch the zone around the nozzle do not contain any grain and a recirculating zone appears and freeze some alumina droplets. At the end of the launch 2 or 3 tons of alumina is deposited around the nozzle. This aspect reduces the performances of the launcher. These two phenomena: instability and slag in the solid propellant are well studied in France and in USA.. One example of computation on scale 1is presented below :

Figure 25 [7] Example of computation of reactive flow inside the booster The second stage: the cryotechnic stage The geometry of this kind of chamber is simple. Upstream the chamber is equipped of coaxial injectors, liquid oxygen is injected at the center and gaseous hydrogen at the periphery. Hydrogen is generated with a velocity about 200m/s and disintegrates liquid oxygen to create a spray and a igniter generates the combustion. Downstream, the nozzle accelerates the burnt gases to provide the thrust of the motor.

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Figure 26 (Vulcain motor)

In this type of motor appear some times high frequency instabilities, this phenomena is also studied in the research centers.

Bi-ergols motors: the third stage of Ariane V (figure 24), the aestus motor works with this bi-ergols type which have the properties to ignite themselves by contact each other. They are called hyperbolic ergols. The liquids used are mainly N 2 O 4 et UDMH (hydrazin)

At the chamber inlet, ergols are liquid, injectors used are doublet or quintuplet types (figures 27,28). A scheme of doublet type is presented on figure 22. The figure 23 shows spray combustion. We can notice that the flame is not a diffusion flame or premixed flame but different regimes can be observed (isolated burning droplet, package burning).

Oxydant

Fuel

Figure 27 Scheme of doublet injector type

Figure 28 Visualization of ergols combustion[ ]

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Figure 29

The burners Current development of industrial or domestic furnaces, boilermust be take into account different criteria : efficiency, reduction of pollutant emissions and noise. We can distinguish gas boiler and the boiler using solid fuel (coal for example). The lighter is the simple boiler figure 30, the fuel is methane. The methane jet entrains a certain amount of airflow rate. After ignition we can see a diffusion flame as in candle case.

Figure 30 [1] Scheme of a lighter flame Bunsen burner used in laboratories generates a premixed flame. It is composed of premixing tube in which is located an injector surrounded by an air flux

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Figure 31 [1] Scheme of a bunsen burner flame At the exit the mixing is quite homogeneous, a conic premixed flame is attached at the tube exit. The flame length is shorter than a diffusion flame one. Close to the wall, the gas velocity is weak, the flame is attached. In the others zones, the gas velocity being higher than the flame velocity, the flame structure has a conic shape. Sometimes the flame can propagate upstream towards the injector and deteriorates it in the case of abnormal working. For these reasons most of the systems work with a diffusion flame. However some systems work with a premixed flame (gas stove..) but in this case the boiler injector is composed of small injection holes in order to avoid the upstream propagation of the flame (quenching distance of the flame). In the case of industrial burner, the air injection is better controlled as the injection in a turbojet combustion chamber. The turbulence level is high in the chamber, in this case the combustion regime is called combustion zone (figure 32).

Figure 32 [1] Gas burner with two air inlets with contra rotary swirls The fuel oil burner (figure 33 ) supplies a diffusion flame regime. A spray of fine droplets is injected at a certain temperature (150C) with a swirl effect to create recirculating zone in order to attach the flame. The recirculating zone close to the head of injection feed with air and fuel permits to stabilize a hot gases kernel like in the case of flame holder. From this kernel a diffusion flame is developed.

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Overall air and fuel flow rate is injected upstream in the burner. The different combustion regimes are also in the case present at the droplet scale.

Figure 33 [1] Scheme of fuel oil burner in an industrial furnace Fires Combustion plays an important role in fire, explosion and detonation. First consider an explosion creates by a leak of gas in a room, a spark can induce a premixed flame propagation which is called explosion. The first results came from Mallard et Le Chatelier works in the domain of firedamp explosion 1881. In stagnant air, the explosion induces pressure oscillations which can sometimes couple with the flame propagation to lead to a detonation propagating at high velocity level (1000m/s). This phenomena has been studied for liquid or solid explosives. The scramjet propulsion has also many common points with detonation propagation (weak detonation). The vertical wall fire creates also a diffusion flame between air of the room and vapor coming from wall material(figure 34). Diffusion flame can be also observed on the sea surface resulting of a liquid sheet combustion, in this case too, convection plays an important role(figure 35).

Figure 34 [1] Flame propagating close to an open polymer vertical wall

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Figure 35 [1] Scheme of a flame above of liquid fluid reservoir Forest fires. At high scale the fire can be seen as a premixed flame, but we can observe a diffusion flame around each tree or around a tree package and also isolated flame around a branch for example. In fact we can observe the same combustion regime as in spray. Wind and turbulence have also an important effect on the flame shape (wake flame).

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Refere

Figure 36 : Combustion systems classification : FAETH

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ChapterII: Introduction to two phase flows

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Multiphase Flows can be divided into four types: - gas-liquid - gas-solid - liquid-solid - Liquid-solid-gas

Gas-liquid Bubbly flows

Stratified flows Gas-droplets flows

Gas-solid gas-particles flows pneumatic Transport Fluidizided beds

Liquid-solid Particles transport in a liquid Hydraulic transport Sediment Transport

Liquid-solid-gas Droplets-particles in a gaseous flow

Classification and multiphase flows examples

Many applications are concern by sprays, formation and droplet transport. The transformation of a liquid jet in spray involved many complex phenomena (primary and secondary atomization, droplet turbulent dispersion, droplet evaporation, droplets collision, spray-wall interaction .. and has many industrial applications;

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INDUSTRY

- Spray evaporation and combustion - Cooling by evaporation - Powder materials - Painting - .

AGRICULTURE -Culture treatment ENVIRONNEMENT

- Humidification - Pollutant transport - Fire

MEDECINE

- Aerosols -

GAZ EXPLORATION

- Two phase flows metering (Venturi measurement for example)

PROPULSION

- Gas turbine - Rocket motor - Diesel motor - Burner - Furnace

EXAMPLES

Injection system studiesSpray studies

Figure 1 : Example of turbojet reactor

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InjectionPrimary disintegration

EvaporationCombustion

Droplet-droplet interactionZone dense

Secondary break up

Turbulent dispersion

Droplets-wall interaction

This support is oriented to multiphase flows encountered in the domain of propulsion and particularly towards dispersed two phase flows. The continuous phase will be represented by air for airbreathing propulsion (ramjet, turbojet, piston engine..) and the dispersed phase by fuel droplet, but the proposed modeling can be also applied to many others applications such as water ingestion in aero-engine inlet for example. For rocket propulsion, four propulsion types are used: - Cryotechnic propulsion (H2/O2), liquid O2 (droplets: dispersed phase) and gaseous H2, ( continuous phase) VULCAIN propellant of Ariane V. - Bi-liquids propulsion: two sprays injection (fuel and oxydizer) without continuous

phase except the vapor coming from evaporation ( AESTUS motor of Ariane V) - Solid propulsion: powder bloc containing fuel and oxydizer + aluminium particle (

Booster, P230 motor of Ariane V) - Hybrid propulsion: atomization of liquid oxydizer on a solid bloc (N2O, PBHT for

example)

Figure 2 : Physical phenomena in a combustion chamber

Figure 3 : Injection in a piston engine

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Chapter III: Definitions and classification of the different rgimes of two phase flows

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The objective of this chapter is the introduction of some definitions for two phase dispersed flows. I : DENSITY AND MASS FRACTION The continuous phase will be the gaseous phase The dispersed phase will be the particulate phase (droplets or solid particles) The volumetric fraction of the dispersed phase will be represented by the volume of particles per unit of volume..

V

VNi

pii

P

=

with: iN particle number of the class i having a volume:

Pipi DV 6=

PiD is the equivalent diameter of the particle. Then: F )1( P= The bulk density of the dispersed phase is the particles mass per unit of volume:

PPPbP c ==

The gaseous density is given by

FPbF )1( =

The mixing density is given by:

PbP

bFm =+= 1( PPF +)

The droplets concentration is expressed by:

VNn PP = ( particles number per unit of volume)

The loading factor is written by : FFP

PPP

UU

)1( = =dispersed phase mass flux

/continuous phase mass flux (mainly applied in gas-solid two phase flows) Droplet spacing influence on the classification of the two phase flows regimes The classification of the two phase flows regimes in terms of diluted or dense is depending on the droplet spacing parameter. This parameter represents the ratio between the mean distance between the droplets and the droplet mean size.

35

For a cubic arrangement, the mean distance between the particles is given by:

31

)6

(PPD

L= with 3

3

6LD

P =

A volume fraction of 1% , the spacing parameter is 3.74 and for 10% is 1.74. Works carried out on monosized droplet stream show that the droplet drag coefficient, the evaporation and the burning rates are highly affected for values of the spacing parameter lower than 80 for the drag coefficient and 20 for the burning rate. (see chapter VIII). Close to injection systems, the droplet concentration is very high and the spacing parameter is often lower than 10. The classical models developed for an isolated droplet must be corrected to take into account the influence of droplets interactions. The following diagram (figure 1) shows the classification of the different regimes. For low volume fractions 5.10-4 droplet-droplet interaction must be added to the Two Way coupling method. (Four Way Coupling).

Figure 1 : Two phase flows classification

36

Figure 2: Other classification given by J. Bellan : If Ri and Rd are respectively the mean radius of the sphere of influence around the droplet and the droplet radius, the proposed classification is the following: At low pressure:

- Ri/Rd

37

The particle Reynolds number is defined by: =epRc

c vuD

Dividing by the particle mass, the equation can be written by:

)(24

182 vu

RCDdt

dv ePDP

c =

(2)

c is the viscosity of the continuous phase

For low Reynolds number ( ePR = 1, Stokes flow), the factor Cd=Rep/24 Dynamical relaxation time: Introducing the dynamic relaxation time in the equation (2):

c

Pv

D

18

2

=

Then the equation (2) can be written as: )(1 vudtdv

v

= The solution of this equation, for a constant gas velocity u and for a droplet velocity v=0, is:

)1( vt

euv =

v represents the time for a droplet to get 63% ( )1ee of the gaseous phase velocity. This result is only valid for Stokes flows (Rep 1 ). For example for a water droplet of 100 m size moving in air .30msv = For higher Reynolds numbers:

c

Pv

D

18

2

=f1

with f=24

epD Rc , DC corresponding in this case to the drag coefficient related to the Reynolds

number. (see chapter VI). Thermal relaxation time: The thermal balance equation of a particle can be written (neglecting the radiative flux):

)( PccPP TTDNudtdTmc = Nu is the Nusselt number, Pc the specific heat of the particle

and cK the thermal conductivity of the gaseous phase.

38

Dividing by Pcm. :

)(12

2 2 dcPPcP TTDc

KNudt

dT = For low Reynolds numbers, the ratio Nu/2 is close to 1. The other factor is thermal relaxation time:

c

PPT K

Dc12

2 = Then :

)(1 dcT

P TTdt

dT = For the previous example (water droplet), the thermal relaxation time is T =145 ms. Link between the two characteristic times :

Pr1

32

32

3212

18 22

P

c

Pcd

c

PP

cP

T

v

cc

ccK

cK

DcKD ====

with Pr the Prandtl number and cC the

specific heat at constant pressure. Stokes number The Stokes number is defined by:

F

vvSt

= where F is the characteristic time of the gaseous phase. For turbulent flow the characteristic time will be the temporal integral scale of the turbulence. For a periodic unsteady flow (coherent structures) this time will be represented by the period of the vortices. For a Venturi flow for example, the characteristic time will be represented by the ratio between the diameter of the Venturi and the flow velocity. If vSt 1 The particles are not affected by the gaseous phase. If vSt 1 In the case of unsteady flow, the particles will be located at the periphery of the vortices (mixing layer configuration for example), some of them will be centrifuged. In this case the expansion will be maximum.

39

III. DILUTED AND DENSE TWO PHASE FLOWS A two phase flow is considered as a diluted flow if the motion of the particles is controlled by the fluid forces (drag and lift forces).

That is quantified by : c

v

1< where c is the mean collision time between particles.

If : c

v

>1 The particle has not enough time to respond to the fluid forces before the next

collision. In this case the flow is dense. Collision frequency estimation (figure 3) Considering a group of particles with an uniform diameter, a particle crosses the particle group with a relative velocity rv . During t , this particle will intercept all the particles in a tube of 2D diameter, with a length of rv t . The particle number in the tube is: trvDnN 2= with n the particle number. The frequency is then given by: rc vDnf

2=

Then the collision time can be expressed by: rc

c vDnf 211

==

c

v

is expressed by : c

v

c

rP vDn

18

4

= =c

rP Dv

3

=Co

P represents the density of the dispersed phase; if all the particles, in the same volume, have

the same mass m, we can write:

P =nm

P + mc =

PPccm += If Co

40

rP

c

vD

3<

rc

c

vZD

3< with c

dMMZ = mass flow rates ratio between the dispersed phase and the gaseous

phase. In the case where the relative velocity is only represented by the fluctuating velocity of the

gas, this relative velocity can be related to the standard deviation which is equal to 2'u .

In this case

c

c

ZD

33.1<

Figure 4: Example of two colliding monodisperse droplet streams

41

Chapter IV: Different approaches for two phase flows numerical simulation

42

I. GASEOUS PHASE NUMERICAL SIMULATION : I.1 Introduction The statistic modeling of a turbulent flow, based on RANS (Reynolds Average Navier Stokes equations) is devoted to turbulent flows statistically steady or to the flows where the time evolution of the physical properties is very low. With this statistic approach, all the scales of the turbulence are modeled, but the models used are not universal and they are adjusted with some constants. However the statistic approach permits to simulation complex configurations because the mean values do not need very precise spatial and temporal discretization in comparison with unsteady flow field presenting high gradients. In parallel some deterministic approaches are now used. The Direct Numerical Simulation consists in solving all the spatial and temporal scales of the turbulence from the energetic largest one to the dissipative scales (Kolmogorov scale). This approach is not yet used to simulate very complex geometries such as a combustion chamber of a turbojet. An intermediate approach which is now more and more used is the LES technique (Large Eddy Simulation). In this approach, the largest scales (the more energetic one) are computed and the smaller one are modeled (Sub Grid Model). This technique is well adapted for the simulation of reacting two phase flows in combustion chambers. The presence of jets, mixing layers, recirculation zones induces the formation of large vortices piloting the fuel-air mixing an the turbulent transfers. I.2 Scales and main characteristics of the turbulence Turbulence intensity : u ),( tx j , v

),( tx j , w ),( tx j are the fluctuating velocities

The following quantity is called turbulence intensity :

UuIu

2'

= for 1D flow

222

2'2'2'

WVUwvuIe ++

++= for 3D flow whereU is the mean value of the component in x direction of the velocity vector Micro and macro Taylor scales : The one point correlation function is defined by :

2'2'

''

'

vu

vuR vu =

43

For Ctettt === 0' , respectively 'xx = Ctex == 0 , this definition leads to the temporal respectively spatial auto-correlation function. For homogeneous turbulent fields and for unsteady state, these functions are written :

1),(.),(

),(),(),(

02'

02'

0'

0'

0 ++=

trxutxu

trxutxutrR or 1

),(.),(

),(),(),(0

2'0

2'

0'

0'

0 ++=

trxutxu

ttxutxutxR

R

1

0 r ou

The macro temporal and spatial Taylor scales are written :

=0

),()( dtrRt and =0

),()( drrRr These values give an order of time or distance step where the fluctuating velocity ),(' txu is itself correlated The micro-scales (temporal and spatial) are defined by :

)(2)),(( 202

2

trtrR

r =

= and )(

2)),(( 2022

rrR

=

=

These micro-scales provided us an estimation of the amount of dissipation, they do not represent scales of vortices. Now we will introduce the Kolgomorov scales characterising the dissipation of the structure. Kolgomorov scales We associate to the smallest scales of the flow the different scales respectively: the length

( ) , and the velocity =v

These smallest scales correspond to a balance between the inertial and viscosity forces:

1v (Reynolds number)

44

Following Kolgomorov : 21

)(

is the dissipation rate such as : 22'

30 u=

in the case of isotropic and homogeneous turbulence Then we can deduced the two scales of kolgomorov (the smallest scales in the flow), depending of the dissipation rate and the viscosity ,

41

)(3

== kl et 4

1

).( =v Kinetic energy of turbulence

)''(21 22'2 wvuk ++=

Spectrum of the length scales of the turbulence The kinetic energy of the turbulence is distributed on different wave numbers from 0 to

= 0 )( dnnEk , n : wave number

Ln(tklnE )( )

5 Slope =5/3 3 Ln(nlt) Figure 1: Kolgomorov spectrum Lt : spatial integral scale of the turbulence = r

21'

k

ltL = = temporal integral scale of the turbulence = t , where cte='

45

E(K) RANS (modeling) DNS LES computation LES modeling

Kc K

Figure 2: Different approaches for flow computation

U DNS LES RANS

t

Figure 3: Example of flow velocity computations from different approaches.

46

Comparison DNS/LES/RANS Approaches Advantages

Disadvantages

RANS Low computation cost and low resolution

Mean aerodynamic fields only

LES Unsteady computation producing a real behavior of the large scales of the turbulence. Application to the computation of the unsteady reactive flow inside a combustion chamber

Only one part of the spectrum is computed, high computation cost

DNS No model Very high computation cost, application only on simple geometries. dcoulements

(K, ) turbulence model

tD l

kC23

= whereCD=cte= 0.09 43

+

=

+ 2)()(

yv

yk

Syykv

xkv x

ce

tyx

kC

yv

kC

ySyyv

xv xt

c

tyx

2

22

1 )()(

+

=

+

where yx vandv are the longitudinal and transversal mean velocities cce SSCC ,,, 21 are constants respectively equal to : 1.45, 1.95, 1, 1.3. t is the turbulent viscosity coefficient defined by :

2kCt = where C =0.09

47

II. TWO APPROACHES FOR TWO PHASE FLOW MODELING - Euler (continuous phase), Euler for the dispersed phase - Euler (continuous phase), Lagrange for the dispersed phase The first approach will be presented very briefly, the second will be improved in the frame of these lectures. II.1 Euler/Euler approach Two methods are distinguished: deterministic and statistics Deterministic eulerian approach (dilute flow or two fluid model):

- hypothesis: - H1 the particle phase is treated as a continuous field - H2 the volume of the particles is negligible and the Stokes number computed

on the collision characteristic time is lower than1. - H3 the gas specific heat is constant, the gas is considered perfect and

chemically frozen. - H4 the particle specific heat is constant, and there is no temperature gradient

inside the particles. - H5 the particles are spherical with no roughness - H6 the density of tha particle is very higher than the gas one - H7 the Brownian motion of the particles is negligible. - H8 the trajectory of the particle is computed using a deterministic method.

With these hypothesis, it is possible to identify the mass concentration of gas and its density.. Considering m classes of particles, the equations of conservation of gas can be written:

=

=+ m

jjjPNudivt 1

,)()( &r

{ } =+ =m

jjPjjP uNuudivut 1

,,)(r&rrr j

m

jjP FNr

=1,

{ } jmj

jP

m

jjPjjP

m

jjP

jPjPjjP QNuFNh

uuNquuEdivE

t ===

+=++

1

,1

,,1

,,,

, 2.

.)( rrrr

&rrr

jFr

: gas force on a particle of class j (mainly the drag force)

jQ : heat transfer from the gas to a particle of class j (convection heat)

j& : mass transfer between gas and particle of class j (evaporation, condensation..) To these equations, a turbulence model must be added. The more used is a two

equations model ,K , respectively the turbulent kinetic energy and the viscous dissipation of the turbulent kinetic energy (Jones et Launder)

48

)()(32""

i

j

j

it

K

Ktijji x

uxu

xukuu

+

+= A gradient diffusion model is used to compute the scalar flux:

jt

tj x

u =

"" with t the turbulent Prandtl number for ????? The turbulent viscosity is given by:

2kCt = with 09.0=C

The turbulent kinetic energy k= 2/"" ii uu can be obtained from the following transport equation:

+

+

=

ii

t

j

iji

jk

t

jjj x

Pxx

uuuxk

xxku 2

"")(rr with k the turbulent

Prandtl number for k. The turbulent dissipation rate can be obtained from the following transport equation:

kC

xP

xxuuu

kC

xxxu

ii

t

j

iji

j

t

jjj

2

22""

1 )~

()(

+

+

=

The recommended values for the constants are:

09.0=C 44.11 =C 92.12 =C 0.1=k 30.1= 7.0=t

In the case of the model k, lm (mixing length), these three parameters are linked by the

relation:

23

43 kClm =

The turbulent viscosity is then computed:

kCt 41

= ml The equations of conservation for the aerodynamic parameters fv YE ,,, are close

after having derived the two transport equations for ,k or lm. Advantages and disadvantages of this approach Advantages:

49

- Easy elaboration of the code, the computations for the two phases are identical - The volume occupied by the dispersed phase is taken into account in the equations. - The action of the dispersed phase on the gas phase . (Two Way Coupling) is

naturally taken into account.

Disadvantages:

- The integration of the physical models due to the presence of the dispersed phase is very difficult: droplet evaporation, condensation, atomisation, droplet-wall interaction, secondary break up, collision.

- Difficulties to considered a polydisperse size distribution for the dispersed phase, that is a main disadvantage for this method, in different burners at the exit of injection devices the spray is polydisperse.

- The cost can become high by considering a polydisperse size distribution. - Conclusion : The Euler approach is mainly devoted to treat dense two phase flows

in non reactive regime and for a low droplet size dispersion. Some researches are now lead to couple the two approaches (Euler-Euler and Euler-Lagrange) to solve very complex flows presenting dense and dilute zones.

II.2 Euler/Lagrange approach The continuous gaseous phase is always computed with an eulerian approach (same

method), except the coupling between the two phases. The chapter IX will be devoted to this approach.

Lagrangian approach for the dispersed phase: - Simplified approach (limited to a steady computation): Individual trajectory is

computed and each particle represents a certain percentage of the total mass of the dispersed phase.

- General approach (valid for steady and unsteady flows): particles or droplets are considered as packages. These packages are injected simultaneously or with different injection frequencies. The particles velocity and temperature in the package have the same value as an individual droplet.

On For droplet trajectories computation, two methods can be used: the deterministic approach (no effect of the turbulence on the droplet trajectory)and the stochastic approach (influence of the turbulence on the droplet dispersion). - The coupling with the continuous phase can be done for each time step or for some time step depending of the application and the importance of source terms by computing and introducing the source terms in the equations of conservation.

Advantages and disadvantages: Advantages

- The using is very simple (some problems can be encountered for the Two Way coupling depending of the importance of sources terms.

50

- The integration of the physical models is very easy, it is for this reason that this approach is often used to simulate the reactive two phase flows inside a combustion chamber of air-breathing or rocket engines

- Different injection points can be chosen with for each point different size classes (example : to compute a spray, 10 injection points are generally chosen with 5 size per point, each class representing a droplet package. Each package can be injectedwith the one frequency. The droplet size, velocity, temperature and frequency are provided by experiment by using optical techniques such as: Malvern, PDPA (for the droplet size and velocity), LDA (for the aerodynamic field), rainbow and LIF (for the droplet temperature)

Disadvantages: - The computation cost can become high

The volume occupied by the particles is not taken into account, inducing some problems for dense two flow computations. However, to consider a four way coupling some empirical correlation can be used to treat the droplet-droplet interactions. Some correlation have been derived by ONERA to correct the evolution of the drag coefficient, the evaporation rate and the burning rates with the droplet spacing (ratio between the mean distance between the drplets and the mean size of the droplets.

Figure 4 : Droplets evaporation in a backward facing step configuration

51

Chapter V: Spray formation

52

I COMBUSTION CHAMBER AND INJECTION SYSTEM

Figure 1: CFM 56 reactor

Figure 2: Scheme of combustion chamber

53

Figure 3: Main combustion phenomena

Figure 4: View of one sector of the combustion chamber

54

Monosized injector for research

Piezoelectric

Vg

30m< Dg 2 < C = Sg / Dg < 7

Orifice Disk

Liqui

+-

~

Thermocoupl

Turbojet Airblast

Fue

Ai

Pressure

Fue

Rocket

Oxidiser Hypergol

Fuel

Cryogenic

Gas H2

Liquid

55

SPRAY MEASUREMENT

Figure 5: Main injection devices for airbreathing and rocket engines

Figure 6: Spray visualization and drop size measurements

56

II PRIMARY AND SECONDARY LIQUID SHEET BEAK UP Atomisation The initial conditions for the dispersed phase in the most hard problem to be solved in two phase flows modelling. Some injection devices are shown on the figure 5. Atomisation regime is not yet well known and usually the droplet characteristics (size, velocity and fuel mass flow rate) close to the injection point are coming from measurements by mean PDPA, MALVERN techniques (figure 6). However, Some researches are developed in the domain of atomisation. As example [3] we will present an approach to improve the knowledge of liquid sheet disintegration by aerodynamic forces (airblast atomizer). A basic experiment was designed to study the break up of a planar liquid sheet induced by a high velocity air stream. A liquid sheet is generated from the central duct with a speed up to 9 m/s (figure 7). The liquid of simulation is water. The liquid film is 300 m thickness and 18 mm width. The injector is located at the exit of the air duct. The flow velocity can be greater than 100m/s. This experiment allows a parametric study about the evolution of the break up with air velocity, liquid velocity, the turbulence level (air and air), liquid sheet thickness and liquid properties by adding tracers to modify the surface tension and the viscosity. Visualisations of the disintegration is carried out by using Video camera and stroboscopic back lighting technique ([4], figure 7). Longitudinal waves ( called primary instability) appear first on the planar sheet by a primary instability mechanism. After these waves are perturbed and become unstable and produce the 3D waves. This phenomena is called the secondary instability. Actually we think that these 3D waves are produced not by the primary instability but by the instabilities generated by the two co-flowing air streams. From these Waves are produced ligaments which then give large droplets and after small droplets by droplet secondary break up. The characteristics of the droplets produced far from the exit of the injector are extremely dependant of wavelength of the secondary oscillation. The wavelength of the instability is provided by a post processing of the images recorded. The expression obtained is the following :

Mf 1.0* = Where *f

lutf= is the non dimensional frequency, M is the momentum ratio of the two

fluxes, t the liquid sheet thickness, U l the liquid velocity and f the frequency of the global oscillation of the sheet. The wavelength of the secondary oscillation [5] is expressed by a direct relation of the frequency of the global oscillation of the waves (figure 7).

+= 548.0)(sec mm )(5.479

Hzf g

Recent works on annular liquid sheet (figure 7 ) give the same results as planar sheet if the ratio between the liquid sheet thickness ant the curvature radius is small. The next step is the break up of the ligaments in large droplets. Rayleigh theory is often used to compute the droplet size :

Dp = 1.9 Dl

57

with Dl is the size of ligament

These large droplets are after break up by air, this phenomena will be describe in the next section.

Example of visualisation of liquid sheet disintegration

0,00

0,50

1,00

1,50

2,00

2,50

3,00

0 200 400 600 800 1000 1200

Global Oscillation Frequency (Hz)

Liga

men

t spa

cing

(mm

)

Planar

Annular

Correlation

Example of visualization of liquid sheet disintegration

Figure 7: Ligament spacing evolution with frequency of the global sheet oscillation

58

Secondary break-up Secondary break up model has been validated from basic experiment (figure 8) by introducing monodispersed droplets in a mixing layer produced by two co flowing air streams. A good classification [5] of the different break up regimes is given by Pilch and Erdman (figure 9). This experiment [6] has permitted qualitative and quantitative visualisations of the break up phenomena (figure10). It appears that break up is not so idealist than those proposed in the literature. The break up is a continuous phenomena and simulations like the one used give some interesting information but not completely the reality. According those remarks on the droplet size repartition a new model is going to be implemented. First the duration of the fragmentation process is considered as a random variable following a Poisson's law. This mean that during a time step t , the break up probability for a given numerical drop is calculated as :

=bupP

max0

max)exp(

rrif

rrifrrK

rmax is the maximum size of fragments and is given by the following experimental correlation :

061.0

max 1.2 rWer=

where 0r is the initial radius. The parameter r is calculated in order to respect a correlation on the mass median size ( meanr ) of fragments. According to Pilch and Ederman [6] we have :

drerKr

rr

max0

3 = 30r

59

For numerical purposes, the exponential law has to be discretised [7]. This current model gives results in a good agreement with experimental results (figure 10). Others models of primary and secondary break up will be proposed in one exrercise. References: [1] FAETH, G.M, current status of droplet and liquid combustion. Progress. Energy Comb. Sciences, Vol3, pp191-224, 1997 [2] BORGHI, R, LOISON, S, 24th Symp (Int) on combustion, pp 1541-1547, 1992. [3] CARENTZ,H, Etude de la dsintgration dune nappe liquide mince PHD Thesis, April 2000, university Paris VI. [4] BERTHOUMIEU, P, CARENTZ, H, LAVERGNE, G Study of planar liquid sheet disintegration. ILASS 97 [5] BERTHOUMIEU, P, CARENTZ. Experimental Study of a Thin Planar Liquid Sheet Disintegration. Paper submitted to ICLASS 2000. [6] PILCH, M, ERDMANN, C, A, Use of break-up time data and velocity history data to predict the maximum size of stable fragments for acceleration induced break-up of liquid drop Int Journal Multiphase Flow 13 (6), pp741-757 [7] BERTHOUMIEU, P, CARENTZ, H, LAVERGNE, G, VILLEDIEU P Contribution to droplet break-up analysis Int Journal of Heat and Fluid Flow 20(1999), pp 492-498

60

Figure 5

Secondary break-up

A

Orifice fordropletsinjection

Air inlet

Monosizedinjector

4 mm

20 m/s 80 m/s

20 mm

Initial droplet size : 320 mm

Example of break-up

Figure 8 : Basic experiment on droplet secondary break up, visualisation

61

Different secondary break up regimes

Experimental evolution of drop size with downstream distance form injection point

Various secondary break-up regimes (Pilch et Erdman)

Numerical evolution

20 60

100

140

180

220

260

300

340

380

420

0-2

mm

8-1

0 m

m

16-

18 m

m

0-2 mm

2-4 mm

4-6 mm

6-8 mm

8-10 mm

10-12 mm

12-14 mm

14-16 mm

16-18 mm

18-20 mm

10 40 70

100

130

160

190

220

250

280

310

340

370

400

430

0-2 mm

14-16 mm

0-2 mm

2-4 mm

4-6 mm

6-8 mm

8-10 mm

10-12 mm

12-14 mm

14-16 mm

16-18 mm

18-20 mm

Figure 9 : Model of Pilch and Erdman

Figure 10 : Comparison experiment/model

62

III, DROPLET SIZE,DEFINITIONS, NON DIMENSIONAL NUMBERS, DIFFRENT INJECTION SYSTEMS Size distributions The objective of this chapter is to give the main statistic characteristics of the droplet size distributions. If the size dispersion is lower than 10%, the distribution is considered as monodisperse. A distribution function f is introduced. F is the probability for a droplet to have a size ranging between D and D+dD in the spatial domain x et x+dx with a velocity vet v+dv. We consider that all the droplets of the same size have the same velocity.

),( Dxff = We can defined local parameters as: - The spray density dDDxfDx l ),()(

36=

- The spray momentum dDDxfDxvDxv ls ),(),()(3

6 =

- l is the liquid density (for a droplet) If ),( Dxm is the evaporation rate of a droplet of a diameter D, the evaporation rate by unit of volume is:

=00

0

),(),()()( dDDxfDxmxnxvap with n the total number of droplets per unit of volume. Mean size The different used diameters are expressed by the following formula and the table designs the different diameters.

jK

K

j

jk

dDxDfD

dDxDfDxD

=

),(

),()(

0

0

j K name Constant value 1 0 linear Droplet number 2 0 surfacic Total surface 3 0 cubic Total Mass 3 2 Sauter Mean

Diameter Ratio between the mass and the surface

In energetic, propulsionthe DMS is currently used. Distribution function: The histograms of sizes obtained by PDPA or Malvern techniques can be described by classic analytical functions:

63

- Normal law 2)(log)( DD eDf

= - Rosin Ramler law DeDDf = 1)(

- Nukiyama Tanasawa law:

DeDDf = 5)(

,, are constants evaluated from experiments.

64

Chapter VI: Turbulent dispersion of the liquid phase

65

I. DRAG COEFFICIENT OF A SPHERICAL PARTICLE (OR DROPLET) Equation of motion of a particle in a gaseous field: Odar and Hamilton equation:

Main hypothesis:

- Droplets are inert, spherical - No rotation of the droplets - The droplet density is very higher than the gas one.

gVVVVDC

dtVd

PgPgPP

dgP rrrrrr

+= )(43

(I) PP V

dtXd rr

=

PVr

et gVr

respectively the instantaneous velocities of the particle and the gas

P et g respectively the densities of the particle and the gas =dC drag coefficient of the particle (or the droplet) =PD particle diameter (or the droplet)

g

PgPgP

DVVRe

rr = Reynolds number In the case of a numerical simulation, at each time step, the values of

the previous parameters must be known. Others equations for evaporation and combustion must be introduced and solve simultaneously.

PPPd D

KnDLCMRefC === ,,,(

With M : Mach number, C : spacing parameter, Kn : Knudsen number

For incompressible subsonic dilute flows

66

If 1PRe P

dst ReC 24= (Stokes flow)

If 1000PRe )15.01(24 687.0PP

d ReReC +=

If 1000fPRe 437.0=dC

For compressible subsonic, transonic and supersonic diluted flows

),( MRefC Pd =

Drag coefficient for subsonic compressible flows

[ ]S

ReM

MMReRe

ReReRe

M

SRe

TTTT

SReC

P

PP

PP

P

P

g

P

g

P

Pd

6.0)exp(1

2.01.048.003.01

4803.0(38.05.4)5.0exp(

)247;0exp()353.01

53.165.3(33.4(24

82

1

+

++++

++

+

++=

M et PRe are computed with the relative velocity, S is the ratio of the

molecular velocity Drag coefficient S=M2

Drag coefficient for supersonic compressive flows

5.0

45.0

25.0

2

)(86.11

1)(058.122)(86.134.09.0

+

++++

=ReM

STT

SSReM

MC g

P

d

In the transonic regime 1

67

Drag coefficient in rarefied gases The previous expressions must be corrected in function of Knudsen

number.

DKn = ratio between the free path line of the particles and the particle

diameter The gas velocity is proportional to gc (c : sound velocity) The Knudsen number can be written: :

DKn =

P

P

Pg

g

ReM

cD=

4 regimes are identified : - Free Molecule flow:

Flow treated as individual droplet motion - Transitional flow: the collision between particles appears - Slip flow : no adherence of the flow to the wall - Continuum flow Some studies show (Crowe et al) that a particle in a rocket nozzle

crosses these four regimes.

Continuum flow Kn

68

This expression is finally used for low Mach numbers. The expression often used in solid rocket propulsion is the following:

)2

exp()()2(2)(07.3

MRe

MkMheCC PRe

MRegK

dstdP

P ++= With K the ratio of specific heats and g and h, 2 functions:

P

PPP Re

ReReReg278.111

)548.0278.12(1)( +++= and

g

P

TT

MMh 7.1

16.5)( ++=

PT et gT the particle and gas temperatures.

69

II. TURBULENT PARTICLES (OR DROPLETS) DISPERSION For combustion applications the droplet motion is computed using a classical way. The equation of motion of the droplet is derived from the linear theory of stokes, and the work of Odar and Hamilton [8]. The only forces which are considered in our simulations are the drag force, and the gravity force. This approximation is motivated by the studied configurations, in which the other terms are negligible. The drag term is the one expressed by Clift and al [9]. This expression is only valid for a very diluted two phase flow. We will see in the following parts of this paper that the drag coefficient is very affected in the case of high droplet concentration. The temporal integration is performed with a fourth order Runge-Kutta method, in the case of non-evaporating droplets, but with a second order Runge Kutta method in the case of evaporating droplets [10]. This difference in the integration method is explained by the fact that when droplet evaporates, the diameter variation is calculated with a second order accuracy, making a fourth order method unappropriated. The value of the aerodynamic quantities are interpolated at the exact position of the droplet, with a trilinear method, which has been chosen to ensure continuity of the gas fields, and as the most suitable on in 3D curvilinear meshes. Droplets are dispersed by the gas flow, according to the stochastic model developed by Gossman [11]. This model is a single particle one, which means that only the droplet is followed in its motion (not also a fluid particle, as in two particles models).

Fluctuating velocities are randomly sampled, according to the local turbulence intensity, and remain till decorrelation scale is elapsed. This decorrelation scale is separated in two scales, a spatial one, and a temporal one (figure 7). The first one is evaluated according to the local turbulent integral spatial scale, whereas the second one is the minimum between the turbulent integral time scale and the interaction time of a droplet and a turbulent structure. This model provides a good agreement with experimental data (figure 7) [12].

The different aerodynamic parameters appearing in the droplet equation of motion are

unsteady values.

To compute the particles transport, three ways are possible: Deterministic approach in a mean gaseous field, the computed trajectories will be

deterministic (no influence of the turbulence).

Deterministic approach in an unsteady gaseous field computed by LES or DNS

techniques. In this configuration at each time step the unsteady gas velocity is known. The droplet dispersion is produced by the unsteadiness of the gas phase, but the trajectories are always deterministic.

gVVVVDC

dtVd

PgPgPP

dgP rrrrrr

+= )(43

(I) PP V

dtXd rr

=

70

Stochastic or steady:

Numerical simulation of the gaseous fiels by RANS approach and a turbulence model using two equations: K et for example. This kind of computation gives for any mesh the mean flow field and the mean velocity fluctuations thanks to the knowledge of the turbulent kinetic energy K and the dissipative rate et la dissipation turbulente .

)wvu(21K

2'22' ++=

xu2K

32u t2

=

xv2K

32v t2

=

xw2K

32w t2

=

)xv

yu(vu t

+

=

The fluctuating velocities wvu ,, must be modeled. An unsteady turbulent gaseous flow must be simulate from a stochastic model. The fluctuating velocity 'u will be randomly sampled such as:

utzyxUtzyxu += ),,(),,( 0,000,00 idem for v and w Then the new position of the particle corresponding to the time 1t is computed with the droplet equation of motion (I) using the previous equation. The computation continue up to 0tti < (interaction time), the new positions of the particle ( iii zyx ,, ) are computed with the same fluctuating gaseous velocity. Then a new fluctuating gaseous velocity is randomly sampled knowing the mean fluctuating velocities (Reynolds tensor) at the last position of the particle.

Droplet dispersion model

71

Hypothesis : Isotropic homogeneous turbulence 222 wvu == K32=

Fluctuating velocities de-correlated following 3 directions Fluctuating velocities following a gaussian distribution To get the fluctuating velocity, a randomly sample is done respecting:

K32'u 2 =

The fluctuating velocity will stay constant during a certain delay. This delay must be evaluated. Temporal integral scale of the turbulence

KC

K

lml

43

23

32

== , It is the first temporal scale of de-correlation, it is

called Eddy Life time model, this scale represents the lifetime of the turbulent structure. The turbulent structure is characterized by its size ml and l its lifetime. with 09.0=C and 2u2

3K = , l and ml can be written:

=

5.12

mu3.0l and

=2

lu3.0

The transit time of the particle in the turbulent structure can be expressed by considering only the Stokes drag force:

gPP

mPt

VVl rr = 1ln(

with P the relaxation of the particle : g

PPP

D

18

2

= The interaction time between the particle and the turbulent structure will be the minimum time between the transit time and the temporal integral scale of the turbulence:

),( tlMin = Remark: Computation of P anf t Analytical solution of the equation of droplet motion (without gravity term and with Stokes hypothesis)

72

).......(36

3

PgPgPP

P VVDdtVdD rrr

= (I) Hyp : we suppose CteVg =

r

New variable: PP VVV

rrr =

)18

()0()( 2PP

g

Dt

EXPVtV = rr solution of (I)

Let: 18

2PP

PD=

P

t

gPgP eVVVtV+= ).)0(()( rrrr

Droplet transit time to cross the spatial integral scale of the turbulence: mL :

)1()0()(0

== Pt

t eVdttVL PPm

))0(

1ln(V

L

P

mPt =

References: [8] ODAR, F, and HAMILTON, N , S, Forces on a sphere acceleration in a viscous fluid J; Fluid Mech, vol 18, pp302-314, 1964 [9] CLIFT, R, GRACE, J, R, and WEBER, M, E, Bubbles, Drops and Particles Academic Press, New York, 1978 [10] BEARD, P, BISCOS, Y, BISSIERES, D, LAVERGNE G Experimental and numerical studies of droplet dispersion in basic configurations of two phase flows. 7th Workshop on two phase flow prediction - Erlangen 1994 [11] GOSMAN, A, D, IOANNIDES, E Aspects of computer simulation of liquid fuelled combustors AIAA 81-0323, 19th Aerospace Sciences Meeting, Saint Louis, 1981. [12] BEARD, P, BISSIERES, D, LAVERGNE, G, ROMPTEAUX, A, Experimental and numerical studies of droplet turbulent dispersion in two phase flows.

73

ASME fluids Engineering Division Summer Meeting, FED Vol185, pp15-22, Lake Tahoe, NEVADA

Droplet dispersion Comparison model - experiment

74

Chapter VII: Droplet evaporation and combustion

75

EVAPORATION MODEL FOR AN ISOLATED DROPLET We first consider the vaporisation of a motionless, cold, droplet after it is placed in a hot, stagnant, gravity-free environment of infinite extent. Assuming the system pressure is much less than the critical pressure of the liquid, critical phenomena are not important. The lack of forced or natural convection then implies the vaporisation of a spherical droplet as illustrated in figure 8 . Since the droplet temperature, in particular its surface value, is lower than that of the ambience, heat is transferred towards the droplet through conduction. At the surface, a part of this heat further transferred to the droplet interior causing the droplet to heat up. The rest is used to evaporate the liquid such that a high concentration of fuel vapour, generally at its saturation value, exists at the droplet surface. When the fuel vapour concentration in the environment is lower than that at the surface, a concentration gradient exists through which the fuel vapour is transported outward. The depletion of the fuel vapour at the surface gives possible further evaporation. Thus, through the above mechanism, a liquid mass can be continuously converted to vapour and eventually dispersed to the ambience i.e. droplet vaporisation is effected. The initial vaporisation rate is low because the droplet is cold. As the liquid temperature rises, this rate will increase as a result of higher fuel vapour concentration at the droplet surface. This has two effects, an increasing portion of the energy reaching the droplet surface must supply the heat of vaporisation of the evaporating fuel, and the outward flow of fuel vapour reduces the rate of heat transfer to the droplet. This slows the rate of increase of the liquid surface temperature and later in the process temperatures become more uniform in the liquid phase. For a pure liquid, droplet heating is mostly over in the early part of the droplet lifetime such that the subsequent rate of its surface area, or equivalently its diameter squared, remains constant with time, as predicted by the d2 law (see the following section). When a droplet is surrounded by a hot oxidising medium, it can ignite, giving rise to a reaction zone in its immediate vicinity. The resulting spherically- symmetric burning, shown in figure 8 , is of the diffusion flame type in which the outwardly-diffusing fuel vapour and the inwardly-diffusing oxidiser gas approach a reaction zone in approximately stochiometric proportion. The ensuing reaction is rapid and intense, implying the reaction zone is thin and very little reactants can leak through the flame. The heat generated is transported both outward to the ambience and inward for droplet heating and evaporation. Similar to the vaporisation case, for a pure fuel, much of the droplet heating is rapidly over and the droplet surface area then regresses at a constant rate. In fact droplet burning bears many similarities with droplet vaporisation and apart from the gas-phase reactions, the detailed transport mechanisms within the d