DEGREE PROJECT REPORT

Numerical approach of a hybrid rocketengine behaviour

Modelling the liquid oxidizer injection using aLagrangian solver

Gustave SPORSCHILL

Master’s Program in Aerospace Engineering

Examiner: Elena GUTIERREZ FAREWIK

Supervisors: Stefan WALLINJean-Yves LESTRADE

Multi-Physics for Energetics Department

Master’s thesis carried out from 06/02/2017 to 07/07/2017

UNCLASSIFIED

Abstract

To access and operate in space, a wide range of propulsion systems has been developed,from high-thrust chemical propulsion to low-thrust electrical propulsion, and new kind ofsystems are considered, such as solar sails and nuclear propulsion. Recently, interest inhybrid rocket engines has been renewed due to their attractive features (safe, cheap, flexible)and they are now investigated and developed by research laboratories such as ONERA.

This master’s thesis work is in line with their development at ONERA and aims at findinga methodology to study numerically the liquid oxidizer injection using a Lagrangian solverfor the liquid phase. For this reason, it first introduces a model for liquid atomiser developedfor aeronautical applications, the FIMUR model, and then focuses on its application to ahybrid rocket engine configuration.

The FIMUR model and the Sparte solver have proven to work fine with high massflow rates on coarse grids. The rocket engine simulations have pointed out the need ofan initialisation of the flow field. The methodology study has proven that starting with areduced liquid mass flow rate is preferable to a simulation with a reduced relaxation betweenthe coupled solvers. The former could not be brought to conclusion due to lack of time butgives an encouraging path to further investigate.

Key-words: Hybrid propulsion, combustion, numerical simulations, two-phase flow, La-grangian approach

v

Contents

Contents v

List of Figures vi

List of Tables vii

List of Symbols viii

Acknowledgement 1

Introduction 3

1 Context of the Master’s Thesis 51.1 Space propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Hybrid propulsion main concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Hybrid propulsion characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Liquid injection 112.1 Two-phase flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Atomisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Primary atomisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.2 Secondary atomisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.3 Numerical description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4 Objectives of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 General settings 193.1 CEDRE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 FIMUR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.2 Implementation in Sparte . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.3 The atomiser in the simulations . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3 Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 FIMUR test cases 254.1 Test settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5 Simulations on the rocket geometries 295.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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5.2 Initialisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.3 Catalysed gas injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.4 Starting the liquid injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.5 Need of a nozzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.6 Alternative approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Conclusion 39

A Physical properties of H2O and H2O2 43

B The experimental setting atomiser 51

Bibliography 53

List of Figures

1.1 Ariane 5 at launch (Credit: ESA – CNES) . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Liquid propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Solid propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Hybrid propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Studied configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 Diffusion flame inside a hybrid rocket . . . . . . . . . . . . . . . . . . . . . . . . . . 71.7 Relation between Isp and oxidizer-to-fuel ratio O/F [3] . . . . . . . . . . . . . . . . . 9

2.1 Break-up of a jet flow [27] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Primary break-up regimes [22] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Regime domains [24] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Secondary break-up regimes [20] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.5 Jet flow atomisation with DNS [15] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.6 Comparison between ELSA model and DNS (2D cut) [15] . . . . . . . . . . . . . . . 152.7 Particle velocity in a hollow-cone atomisation using the FIMUR model (2D cut) . . 15

3.1 Geometry of the atomiser modelled by the FIMUR model [25, 26] . . . . . . . . . . 203.2 The vertical position shift ∆y of the atomiser . . . . . . . . . . . . . . . . . . . . . . 21

4.1 The test case mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Gas and droplet temperature at t = 0.1 s . . . . . . . . . . . . . . . . . . . . . . . . . 264.3 Evaporation of H2O2 in case 1 without reactions at t = 0.1 s . . . . . . . . . . . . . . 274.4 Mass fraction distribution in case 2 with H2O2 decomposition reaction at t = 0.1 s . 274.5 Profiles in both test cases at x = 0.15 m . . . . . . . . . . . . . . . . . . . . . . . . . 28

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5.1 The rocket engine geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.2 Mesh details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.3 Pressure waves due to combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.4 Gas injection with a turbulence model . . . . . . . . . . . . . . . . . . . . . . . . . . 325.5 Liquid injection with reverse gas flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.6 Liquid injection with the narrow atomiser . . . . . . . . . . . . . . . . . . . . . . . . 355.7 Liquid injection with the narrow atomiser and the nozzle . . . . . . . . . . . . . . . 365.8 Working liquid injection with m = 30 g s−1 . . . . . . . . . . . . . . . . . . . . . . . . 38

A.1 Density for H2O2 and H2O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44A.2 Vapour pressure for H2O2 and H2O . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45A.3 Heat of vaporization for H2O2 and H2O . . . . . . . . . . . . . . . . . . . . . . . . . 46A.4 Heat Capacity for H2O2 and H2O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47A.5 LiquidViscosity for H2O2 and H2O . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48A.6 Thermal conductivity for H2O2 and H2O . . . . . . . . . . . . . . . . . . . . . . . . . 49A.7 Surface tension for H2O2 and H2O . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

B.1 Delavan WDA atomiser specification sheet (extract) . . . . . . . . . . . . . . . . . . 51

List of Tables

1.1 Possible propellants for hybrid propulsion . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1 The atomiser parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.1 Dimensions of the geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

A.1 Constant physical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43A.2 Sparte data for density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44A.3 Sparte data for vapour pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45A.4 Sparte data for heat of vaporization . . . . . . . . . . . . . . . . . . . . . . . . . . . 46A.5 Sparte data for heat capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47A.6 Sparte data for liquid viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48A.7 Sparte data for thermal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . 49A.8 Sparte data for surface tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

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List of Symbols

Variablesd Droplet diameter [m]F Thrust [N]Isp Specific impulse [m s−1] or [s]k Rate of reaction [mol m−3 s−1]m Mass [kg]m Mass flow rate [kg s−1]O/F Oxidizer-to-fuel ratio [–]Oh Ohnesorge number [–]R Radius [m]Rel Liquid Reynolds number [–]T Temperature [K]U Velocity [m s−1]Ue Effective exhaust velocity [m s−1]u, v, w Velocity axial, radial and tangential components [m s−1]We Weber number [–]X Air core ratio [–][X] Molar concentration of species X [mol m−3]YX Mass fraction of species X [–]µ Dynamic viscosity [Pa s]ρ Density [kg m−3]σ Surface Tension [N m−1]θS Half spray-angle [°]

Constantsg0 Earth’s gravitational acceleration at sea level [m2 s−1]R Universal gas constant [J mol−1 K−1]

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Subscripts0 At the atomiser’s nozzleg Gas phasel Liquid phase

AcronymsCEDRE CFD software for Energetics developed by ONERACFD Computational Fluid DynamicsDNS Direct Numerical SimulationDPS Discrete Particle SimulationELSA Eulerian-Lagrangian Spray AtomiserFIMUR Fuel Injection Model by Upstream ReconstructionPIV Particle Image Velocimetry

1

Acknowledgement

First of all, I would like to thank Jérôme Anthoine, head of the Propulsion Laboratory unitat ONERA, for letting me carry out my master’s thesis in his unit and giving me the opportunityto witness the rocket propulsion research environment.

I am grateful to Jean-Yves Lestrade, my supervisor, and Jérôme Messineo, both researchersof the unit, for their feedbacks and suggestions to progress in this thesis work.

I also thank Jouke Hijlkema and Olivier Rouzaud, senior users and former developers ofSparte, for their help with this solver, and more broadly Jean-Mathieu Senoner, head deve-loper of Sparte, and the CEDRE software support team for their unlocking explanations andsolutions.

Finally, I would like to thank Jean-Étienne Durand and Quentin Levard, PhD students, DucMinh Le, postdoc researcher, and everyone else at the Propulsion Laboratory unit for welcomingme among them and the nice atmosphere they provided.

3

Introduction

Nowadays, space has become an important scientific, economic and strategic location. Acrowd of satellites orbits the Earth to monitor its atmosphere and oceans, predict the weather,broadcast data such as global positioning to users all over the world, and observe closely warzones. During the last century, space propulsion has been developed to access these orbits andeven space beyond, through chemical propulsion such as liquid and solid rocket engines andelectrical propulsion. As improvements of the existing systems are investigated, new solutionslike solar sails and nuclear propulsion are considered.

Hybrid chemical propulsion, using both liquid and solid propellants, has recently drawnback the attention on it with Virgin Galactic’s SpaceShipOne successful flight. Especiallyinterested in its safety and flexibility features, aerospace laboratories have decided to speed up itsdevelopment. ONERA, the French aerospace laboratory, studies numerically and experimentallyvarious types of hybrid rocket engines in its Propulsion Laboratory unit, part of the Multi-Physics for Energetics Department on the site of Le Fauga-Mauzac.

Carried out there, this master’s thesis work investigates the methodology to follow in orderto run simulations of a hybrid rocket engine with a liquid oxidizer injection, using a Lagrangiansolver and the FIMUR model for the atomiser. First, the context of space propulsion isintroduced, highlighting the characteristics of the hybrid propulsion system, and the liquidinjection is presented. Then the tools and global models for the simulations are described.Finally, the FIMUR model is tested, followed by the analysis of the methodology and of thedifficulties encountered on the rocket geometry.

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Chapter 1

Context of the Master’s Thesis

1.1 Space propulsion

Figure 1.1: Ariane 5 at launch(Credit: ESA – CNES)

Space propulsion systems basic goal is to accele-rate a spacecraft. Since the spacecraft is isolatedwhen in space, its motion can still be alteredby ejecting mass according to the action-reactionprinciple. Derived from Newton’s law of motion,the thrust vector F applied on the spacecraft is:

F = mdU

dt= −mUe (1.1)

where Ue is the velocity vector of the expelledmass. The propulsion system performances are thenevaluated with the specific impulse Isp:

Isp = F

m≡ Ue [m s−1] (1.2)

It represents the propulsion efficiency and cor-responds to the effective exhaust velocity of thepropellant. A high Isp is desirable, since it meanslow propellant consumption. Another widely useddefinition is given in (1.3), which can be interpretedas the length of time for which one kilogram of propellant produces a thrust equivalent to aone-kilogram mass (i.e. a force of about 9.81 N) in Earth’s gravitational field g0.

Isp = F

m g0[s] (1.3)

Space missions are of various types, such as launching payloads into orbit (e.g. Ariane 5in Figure 1.1) or orienting a spacecraft on its orbit, so the propulsion systems are varied aswell. More particularly, for high-thrust applications, chemical propulsion systems are dividedinto three categories:

– Liquid propulsion (Figure 1.2): the propellants are stored separately as liquid in tanks.They are mixed in the combustion chamber before being accelerated through the nozzle.

6 CHAPTER 1. CONTEXT OF THE MASTER’S THESIS

Liquid rockets are characterized by the highest Isp for chemical propulsion (up to 465 s forthe Vinci engine developed by ASL [7]).

– Solid propulsion (Figure 1.3): both oxidizer and fuel are premixed together in one solidphase propellant, the grain, usually with a metallic additive (e.g. aluminium) to increasethe thrust performances. Producing the highest thrust amongst the chemical rockets(14 · 106 N for the Space Shuttle SRB) but with low Isp, it is mainly used on launcherboosters at take-off. Its design is quite simple but it is impossible to throttle or to shutdown. Moreover, the solid propellant is explosive and highly sensitive to manufacturingimperfections.

– Hybrid propulsion (Figure 1.4): the propellants are stored in two different states, asolid one and a liquid one, generally a solid fuel with a liquid oxidizer. Its theoreticalperformances lie between the solid and liquid propulsion performances, with a vacuum Ispthat can exceed 360 s.

Figure 1.2: Liquid propulsion Figure 1.3: Solid propulsion

Figure 1.4: Hybrid propulsion

Out of these three chemical propulsion systems, the first two have been widely studied anddeveloped due to fast breakthroughs and improvements of their performances. The interestin the last one has been recently renewed thanks to Virgin Galactic’s SpaceShipOne, but alsothanks to its promises: safer, cheaper, more flexible.

1.2 Hybrid propulsion main conceptsThe hybrid propulsion can be designed according to different configurations. The traditional

one, studied here and shown in Figure 1.5, consists in a cylindrical combustion chamber, alsocalled a port, inside a tubular solid fuel, as in solid propulsion systems. To this port are addeda pre-combustion chamber, to insure the heating and the vaporisation of the oxidizer beforecombustion, and a post-combustion chamber, to increase the combustion advancement before

1.2. HYBRID PROPULSION MAIN CONCEPTS 7

exiting the gas through the nozzle. The oxidizer is either injected directly as liquid inside therocket engine, or through a catalyst that heats and vaporises the oxidizer before it enters thepre-chamber.

Pre-combustion

chamber

Fuel grain

Port

Post-combustion

chamber

NozzleLiquid

Oxidizer

Figure 1.5: Studied configuration

One of the main differences between the hybrid propulsion and the two other chemicalpropulsions presented earlier lies in the combustion. Indeed, in solid and liquid propulsions,the fuel and the oxidizer are closely mixed in an almost homogeneous phase in the combustionchamber, as fine solid grains or as small injected droplets. This leads to a premixed flame forthe solid rocket and a short diffusion flame with a large flame area for the liquid rocket, bothwith a controlled oxidizer-to-fuel ratio set to the stoichiometric ratio.

In hybrid propulsion, the gaseous fuel is produced at the surface of the solid fuel by pyrolysis(i.e. decomposition of the solid phase directly into fuel gas due to high heat fluxes), while theoxidizer is injected at the entrance of the combustion chamber. This results in a long diffusionflame inside the boundary layer, represented in Figure 1.6, where the propellants must diffusetowards one another in order to burn. The diffusion flame is less efficient here since the fuelis injected on the whole length of the combustion chamber: the fuel blown from the grain endhas less time to correctly burn, and the mixing area between fuel and oxidizer is smaller thanthe one in the liquid rocket, where the flame appears all around each droplet. Adding the post-combustion chamber enlarges the residence time of the propellants and the flame area, therebyhelping gain in performance. Besides, the oxidizer-to-fuel ratio is not constant and decreasesalong the port due the consumption of the oxidizer.

Figure 1.6: Diffusion flame inside a hybrid rocket

Concerning the choice of propellants, several combinations have been investigated. Table 1.1gives examples of fuels and oxidizers used in hybrid rockets. HTPB, HDPE and PP correspondto common rubbers, while paraffins are liquefiable waxes that will produce a thin liquid layerbetween the solid grain and the gaseous reactants.

8 CHAPTER 1. CONTEXT OF THE MASTER’S THESIS

Table 1.1: Possible propellants for hybrid propulsion

Fuels OxidizersHydroxyl-terminated Polybutadiene (HTPB) Liquid Oxygen (LOx)

High Density Polyethylene (HDPE) Hydrogen Peroxide (H2O2)Polypropylene (PP) Nitrogen Peroxide (N2O4)

Paraffins Nitrous Oxide (N2O)Cellulose (Wood) Nitric Acid (NO3H)

. . . . . .

1.3 Hybrid propulsion characteristicsAs mentioned in Section 1.1, interest in hybrid rockets is related to its many advantages [4]:

(+) Safety: The propellants are separated by distance and by phase, with an inert fuel such asrubber, so that a spark cannot light the engine on. This allows for safe manufacturing, safetransportation, safe assembly and safe storage. The launchpad operations are also saferand the risks of catastrophic failure are reduced, while the hybrid rocket can be shut-down(cf. Flexibility). The working pressures, between 10 bar to 50 bar, are lower than in solidand liquid propulsion, from 50 bar up to 150 bar.

(+) Reliability: The mechanical system is less complex than the ones in liquid propellantrockets, since it uses half of the storage and feed system. In the same time, the solid fuelis insensitive to cracks and imperfections, contrary to solid oxidizer-fuel grains in solidrockets.

(+) Flexibility: The hybrid rocket can be throttled, shut-down and restarted, by adjustingthe oxidizer flow inside the combustion chamber. As seen in Table 1.1, hybrid propulsionalso offers a large choice of propellants. Besides, it can be designed for large boosters aswell as for satellite manoeuvre thrusters.

(+) Environmental friendliness: The hybrid rocket has an environmentally clean exhaust,without hydrogen chloride or aluminium oxide, which impact the environment. Actually,for LOx/HTPB or H2O2/HDPE rockets, exhausts are only composed of H2O, CO, CO2and of unreacted propellants.

(+) Low cost: The safety of the materials and the simplicity of the systems reduce the cost.

However, improvements have still to be investigated to correct or compensate some draw-backs:

(–) Slow regression rate1: Compared to solid propulsion, the fuel grain regression rateis low, making it difficult to obtain sufficient mass flow rate of pyrolysed fuel vapour inorder to achieve high thrust levels. But recent studies have shown possible improvementsthrough innovative fuels and injection methods [4, 18].

(–) Mixing and combustion inefficiencies: The diffusion flame inside the hybrid rocketcombustion chamber is less efficient than the flames found in the other propulsion systems.Moreover, the fuel is “injected” all along the combustion chamber, thus a part of it mightleave the rocket unburned.

1The regression rate corresponds to the thickness of solid fuel that is vaporised per unit time.

1.3. HYBRID PROPULSION CHARACTERISTICS 9

(–) Low volumetric loading: To enhance the combustion efficiency, a post-combustionchamber is added on hybrid rockets, decreasing further the low volumetric loading due toslow regression rates.

(–) Oxidizer-to-fuel ratio shift: The regression of solid fuel leads to an increasing combus-tion chamber diameter and thus a shift in the mixing ratio over time. Noting that the idealratio is the stoichiometric ratio, this shift directly impacts the performances of the hybridrocket, as shown in Figure 1.7. Therefore, current studies investigate solid fuel geometryoptimisation.

Figure 1.7: Relation between Isp and oxidizer-to-fuel ratio O/F [3]

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Chapter 2

Liquid injection

2.1 Two-phase flow

Liquid injection in a combustion chamber lets two phases interface inside the flow, theliquid droplets and the carrying gas stream. This corresponds to one of the different regimes ofmultiphase flows, classified by Ishii [13] as dispersed flow.

Actually, close to the injector, the considered flow consists in a jet flow, i.e. a separatedflow without mixing of the two phases. Different phenomena occur then: atomisation, whichbreaks up the jet flow into droplets and thus into the dispersed flow, heating and evaporation ofthese droplets, but also collisions between droplets and with the wall, or further fragmentation.Figure 2.1 shows a jet flow going through atomisation, resulting in a dispersed flow.

Figure 2.1: Break-up of a jet flow [27]

2.2 Atomisation

Atomisation is an important process that transforms a liquid jet into a diluted spray ofsmall droplets. Its required characteristics depend on the application, which can be chemistry,agriculture or motor design.

In rocket propulsion, the injector vaporizes the propellant to enable the combustion betweengas. The atomisation phenomenon has to produce the smallest drops possible so that theinterface between the two phases is increased, leading to a fast evaporation and a reducedcombustion chamber in length. For a classical hybrid rocket configuration, the fine atomisation

12 CHAPTER 2. LIQUID INJECTION

influences the length of the pre-combustion chamber. Indeed, in an optimized pre-combustionchamber the residence time and the evaporation time of a droplet are equal.

The process can be split into two steps. The primary atomisation consists in the firstbreak-up. The high differential velocity between the jet and the gas produces an importantshear stress on the phases interface, leading to instabilities that detach fragments from the jet.The primary atomisation depends on the considered liquid streams, since the exact mechanismsmay differ between cases (e.g. liquid jet, liquid sheet, swirl jet,...).

The secondary atomisation corresponds to the second break-up: large drops into a sprayof droplets. Carried downstream by the gas and in transverse directions due to turbulence, thelarge drops of the primary break-up are destabilized and break into smaller drops. This step ismostly controlled by the ratio between the destabilizing aerodynamic forces and the stabilizingsurface tension forces acting on a liquid fragment, which is expressed through the Weber numberWe as:

We = ρg(Ug − Ul)2d

σl(2.1)

The destabilization regimes can also be characterized by other non-dimensional numberstaking in account the properties of the gas, of the liquid and of the geometry. For example, theliquid Reynolds Rel corresponds to the ratio between aerodynamic and viscosity forces in theliquid drop, while the Ohnesorge number Oh is the ratio between viscosity and surface tensionforces acting on a drop:

Rel = ρlUld

µl(2.2)

Oh = µl√ρlσld

(2.3)

2.2.1 Primary atomisation

The primary atomisation is an important process, since it gives the secondary atomisation itsinitial conditions, i.e. the large drops and liquid fragments distribution. Therefore, the choiceof the injector is also important. There are three main types of injectors:

– Pressure atomiser: the liquid is injected at high velocity, through a small nozzle, thanksto a high pressure difference between the flows inside and outside the injector. This is theone used in the studied case, and usually used inside propulsion systems.

– Rotary atomiser: the liquid is injected radially from a spinning surface such as a disc. Thecentrifugal energy helps achieve high relative velocity between the liquid stream and thecarrying gas, leading to primary break-up.

– Twin-fluid atomiser: the liquid is injected at low velocity inside a coaxial high velocitygas stream, usually air. This injector, also called airblast or air-assist atomiser, uses thekinetic energy of the airstream to shatter the liquid stream.

Depending on the flow parameters through the previously given non-dimensional numbersWe, Rel and Oh, experiments have shown distinctions between several break-up regimes, whichhave been classified by Reitz [22] into four main regimes, represented in Figures 2.2 and 2.3:

– the Rayleigh mechanism: at low We and Rel, instabilities in the jet stream are due toaerodynamic forces from the gas relative flow, resulting in drops at least as large as thenozzle exit diameter, far from the nozzle.

2.2. ATOMISATION 13

– the first wind induced break-up: the aerodynamics forces apply a torsion to the liquidjet stream, leading to drop of the size of the injector nozzle diameter, far from it.

– the second wind induced break-up: surface tension instabilities produced by theaerodynamic forces are added to the torsion of the previous case. The drops are smallerthan the nozzle diameter.

– the atomisation: for large We or Rel, the jet stream is shattered as soon as it exits theinjector nozzle into really small droplets compared to the nozzle diameter.

Figure 2.2: Primary break-up regimes [22] Figure 2.3: Regime domains [24]

For the liquid injection in rocket propulsion, as well as in all propulsion applications, thelast regime is the required one. The droplets being smaller will let them evaporate faster andmix more efficiently with the gaseous fuel in the combustion chamber.

2.2.2 Secondary atomisation

Following the primary atomisation, the liquid fragments and drops are broken up into smallerdroplets, due to shear stress resulting amongst others from their velocity differential with respectto the carrying gas flow. The secondary break-up is thus mainly characterised by the Webernumber, based on the primary drops and fragments length scale. Once again, several regimeshave been identified [8, 23], starting from a critical Weber number Wec usually set to 12. Theyare shown in Figure 2.4.

– for Wec ≤ We ≤ 100, the bag break-up regime occurs. In this first regime, the dropflattens and grows hollow near the stagnation point, under the effect of the dynamicpressure. It starts to break up at the bottom of the “bag”. On the second half of theWe-range starts a transitional regime: the bag and stamen break-up.

– for 100 ≤ We ≤ 350, the shear break-up (or sheet stripping) regime occurs. Unlike theprevious regime, the flattened drop is deformed into a ligament and broken up startingfrom the edges. This phenomenon could be explained by destabilizing capillary waves orby the stripping of the drop boundary layer due to shear stress.

– for 350 ≤ We, the catastrophic break-up regime occurs. The large relative velocitybetween the drop and the carrying gas of the last regime leads to small-wavelength

14 CHAPTER 2. LIQUID INJECTION

perturbations at the surface of the drop. The resulting small ligaments on the drop thenbreak up into droplets.

Figure 2.4: Secondary break-up regimes [20]

2.2.3 Numerical description

To model accurately the whole atomisation process is quite difficult. It can be done usinga Direct Numerical Simulation (DNS) as proposes Lebas et al. [15], shown in Figure 2.5.This method corresponds to a numerical experiment and yields detailed results that replaceexperimental data, hard to obtain because the sensors are not adapted to the dense spray ofthe primary break-up. It is however limited to short simulation times due to today’s limitedprocessing performances.

Figure 2.5: Jet flow atomisation with DNS [15]

Another way is to use a complex model developed for the atomisation process, such as theELSA model (Eulerian-Lagrangian Spray atomisation) based on the work of Vallet and Borghi[28]. It describes the dense zone using an eulerian approach of a two-phase fluid, i.e. the fluidhas a variable density and two species. The model also calculates the mean area of the liquid-gas interface, in order to switch to Lagrangian approach for the liquid phase (see below) in thedilute zone. The good accuracy in the near nozzle region comes with some drawbacks, amongstwhich the limitation to a 3D mesh for a correct description [12]. Figure 2.6 compares the fieldscalculated with the ELSA model to DNS.

2.2. ATOMISATION 15

(a) Liquid volume fraction (b) Liquid-gas surface density

Figure 2.6: Comparison between ELSA model and DNS (2D cut) [15]

In the present study, to avoid this still heavy modelling, the FIMUR model (Fuel InjectionModel by Upstream Reconstruction) introduced by Sanjosé [25, 26, 2, 11] will be used. Thismodel, developed for spray simulations in aircraft motors, directly produces the droplets dis-tribution of a swirl pressure atomiser after the primary and secondary break-ups, based on theatomiser and the spray characteristics. As shown in Figure 2.7, the resulting spray forms ahollow cone, due to the swirl motion given to the injected liquid. The model is detailed inSection 3.2.

Figure 2.7: Particle velocity in a hollow-cone atomisation using the FIMUR model (2D cut)

Finally, to model the dispersed phase, the large number of particles (liquid in the currentcase) makes it difficult to reproduce them numerically. Therefore, three different formulationsfor CFD software have been developed [19]:

– DPS (Discrete Particle Simulation): this is the direct approach, where all the physicalparticles are tracked individually. The same constraints as for the DNS apply: calculationsare extremely heavy and can only be carried out on small parts during short simulationtimes.

– Lagrangian approach: simplification of the DPS, it solves for numerical particles,also called parcels, containing several “real” particles with the same characteristics (size,velocity,...) and located by their center of gravity. Quite accurate, it still can be heavy

16 CHAPTER 2. LIQUID INJECTION

in calculation depending on the number of parcels to compute and is difficult to speed upthrough parallel processing. It has to be coupled with an eulerian solver for the gas flow.The spray distribution in Figure 2.7 illustrates the Lagrangian approach.

– Eulerian approach: similar to gas solvers, it solves the flow fields based on conservativeequations, generally on a finite volume grid. More suitable for implicit algorithms andparallel processing, it is confronted to numerical diffusion and precision issues since thedispersed phase is modelled by average per cell. Moreover, it cannot explicitly take intoaccount some particle phenomena, such as collisions.

2.3 Evaporation

In the considered hybrid rocket configuration, the oxidizer is injected as liquid dropletsat storage temperature (approximatively 300 K) inside the combustion chamber. There, thedroplets are heated and evaporated.

While the liquid droplet is cold, only little vapour is present at its surface, which leads tolow evaporation rate and mass transfer. Therefore, the received heat is mainly used to increasethe droplet temperature. This increase produces vapour, decreasing the heat flux at the liquidsurface and slowing down the temperature increase. The temperature inside the droplet becomesuniform and the evaporation is accelerated.

The evaporation rate depends on the carrying gas (pressure, temperature and properties),on the liquid phase (temperature, diameter and properties) and on the relative velocity of thedroplet with respect to the carrying flow.

The evaporation is generally not modelled globally for the whole dispersed phase, butindependently for each droplet. Three main models have been developed [1]:

– the D2 model: the simplest model, the temperature is supposed uniform in the dropletand constant over time. All the received heat is used to evaporate, with a law proportionalto the square of the drop diameter.

– the infinite conductivity: the conductivity of the droplet is supposed infinite, so thetemperature is uniform in the liquid, but it varies with respect to time.

– the effective conductivity: in this model, the heating of the droplet is more detailed,leading to a non-uniform temperature inside the droplet.

Using one of these models implies some simplification hypotheses, among which:

– the process is quasi-stationary– the considered drop is spherical and isolated– the liquid only contains one species– the phase change is way faster than the vapour transport in the ambient air (the vapour isproduced at the surface temperature of the drop and its corresponding vapour pressure)

– the heat transfer by radiation is negligible

2.4 Objectives of the thesis

The objectives of the current master’s thesis is to find a methodology to get simulations ofliquid H2O2 injection inside a hybrid rocket to run, using a Lagrangian description.

2.4. OBJECTIVES OF THE THESIS 17

Such simulations have already been carried out, especially by Lazzarin et al. [14] withN2O. At ONERA, simulations of hybrid rocket engines have been first investigated in 2009 byLestrade [16], and highlighted some difficulties with the Lagrangian solver and its evaporationmodels. Until now, these simulations have therefore been performed with catalysed injection,i.e. the oxidizer enters the pre-chamber already as a hot gas. This work focuses back on theliquid injection, as the solvers and their models have been further developed and improved.

When Lestrade introduced the liquid spray inside the rocket engine, no atomiser modelswere implemented in the software. The droplets were thus injected from a large distributionof point injectors, each having unvarying characteristics, to produce the injection spray basedon experimental data. Since then, the FIMUR model was developed and should simplify theinjection settings while yielding a better droplets distribution in the spray. As the atomisermodel has been implemented for aeronautic propulsion systems, the thesis investigates itsapplication with the Lagrangian solver to rocket propulsion systems with much larger massflow rates (100 g s−1 instead of usually around 2 g s−1).

19

Chapter 3

General settings

3.1 CEDRE

All simulations have been carried out on CEDRE1, the CFD software developed by ONERA[21]. It is dedicated to numerical simulations for energetics and propulsion, and is redistributedinternally but also externally to aerospace industrials. Contrary to commercial CFD codes,CEDRE is meant for research applications and focuses therefore more on the correctness of thephysics and less on the robustness, by including regularly new models developed by researchersand PhD students.

The software is divided into several solvers interacting with one another in order to considerthe different physical phenomena occurring in this kind of flow, for example reactive compressibleflow, but also dispersed flows, thermal conduction in the walls or heat radiation.

In the present study, only the Lagrangian solver for dispersed flow Sparte is used in additionto Charme, the main solver for reactive compressible flows. The gas phase is modelled as athermal perfect gas while the heat radiation is neglected, though it can play a significant rolein rocket combustion chambers, depending on the chemical species involved and on the gasconditions (temperature, pressure). The input grid is designed with Gmsh, an open source 3Dfinite element grid generator [9].

Both solvers Charme and Sparte can be used in steady or unsteady flow. In the case ofCharme, a steady simulation corresponds to a simulation using local time step size, which canbe specified through several models. When no local time step model is chosen, the simulationis unsteady. For Sparte, the steady simulation consists in finding the trajectory of an injectedparticle until either it evaporates or exits the domain. However, this kind of simulation isreserved for dilute sprays where the particles do not have important influence on the carryinggas. Therefore, when introducing droplets, both solvers are set to unsteady simulation andthe convergence is assessed by looking to the variation of parameters such as the pressure, thetemperature and the mass fractions.

Concerning evaporation, the Sparte solver proposes only two of the presented models inSection 2.3: infinite conductivity and effective conductivity. To keep the model simple and fastto solve, the first one is used in all simulations.

The interaction between both solvers is managed by CEDRE as follow:

– One-Way coupling: Only Charme affects Sparte. The gas is solved as if no particleswere injected, while the particles follow the gas flow and feed on its energy. It corresponds

1Calcul d’Écoulements Diphasiques Réactifs pour l’Énergétique: Two-phase Reactive Flow Calculation forEnergetics, version 6.1.1

20 CHAPTER 3. GENERAL SETTINGS

to a correct approximation for highly dilute dispersed flows in terms of volume fraction,where the influence of the particles on the carrying gas is negligible (e.g. particles in PIV).

– Two-Way coupling: Charme and Sparte affect each other. The Lagrangian solveradds source terms in the gas flow, corresponding to the exchanges in mass, momentumand energy between the particles and the gas. To avoid large gradients when starting theliquid injection, a relaxation parameter moderates the interaction of Sparte on Charmeand can be set between 0 (i.e. One-Way coupling) and 1 (i.e. 100% Two-Way coupling).

The software do not include the Four-Way coupling, which also solves the interactionsbetween each particle in very dense dispersed flows.

3.2 FIMUR model

3.2.1 General description

As indicated in Section 2.2.3, the FIMUR model defines the distribution of the dispersedphase after the atomisation of a pressure swirl atomiser (shown in Figure 3.1), to avoid its com-plex modelling. It also greatly simplifies previous “manual” technique described in Section 2.4.

Figure 3.1: Geometry of the atomiser modelled by the FIMUR model [25, 26]

Given the input parameters for the atomiser and the spray, the model yields the dropletdistribution for position, velocity and diameter downstream the atomisation process, and re-constructs the path of the droplets upstream to the atomiser nozzle to insure continuity of theinjection. The atomiser is characterized by the mass flow-rate of the injected liquid m, the orificeradius R0 and the air core radius Ra related to the air core ratioX, defined in Equation (3.1), andthe spray by its half spray-angle θS and the characteristic diameter of the droplet distributiond after atomisation.

X = AaA0

=(RaR0

)2= sin2 θS

1 + cos2 θS(3.1)

3.2.2 Implementation in SPARTE

In addition to the parameters mentioned above, the Lagrangian solver requires the position,the orientation and the injection period of the atomiser, which determines the number of newnumerical particles injected per time step.

3.2. FIMUR MODEL 21

The position of the atomiser and the time step for the Sparte solver are constraints by theinputs parameters. Indeed, for each time step, the atomiser model places the new numericalparticles inside a cone, determined by the axial velocity of the particles and the half spray-angle,as shown in Figure 3.2. If this cone goes over the limits of the mesh, some particles (in red inFigure 3.2a) will be statistically placed outside the mesh and lead to errors.

θS

Δy = ul Δt tan θS

ul Δt

(a) Bad placement

θS

Δy = ul Δt tan θS

ul Δt

(b) Correct placement

Figure 3.2: The vertical position shift ∆y of the atomiser

Due to that, the atomiser cannot be placed on the symmetry axis in a 2D axisymmetricmesh but have to be slightly shifted inside, and the closer from the axis it is, the smaller thetime step must be, thus increasing the processing time.

To avoid the shift, one could think of inclining the injection cone of θS , while reducing thehalf spray-angle to the spray half thickness angle δθS , to reproduce the same injection. However,the velocity distribution (3.2) inside the spray is defined according to the injection axis: thiswill therefore lead to a wrong distribution.

u(x0, r, φ) = m

ρlπR20(1−X)

v(x0, r, φ) = 0

w(x0, r, φ) = m

ρlAp

R0 +Ra2r

(3.2)

3.2.3 The atomiser in the simulations

Using Sparte and FIMUR, the modelled injector is a swirl pressure atomiser, which param-eters are given in Table 3.1. These parameters are based on the Delavan WDA nozzle, used inthe experimental settings and presented in Appendix B.

This corresponds to a dense hollow cone injection, with a distribution of the particles within±5° of the half spray-angle. The initial temperature of the droplets is set to 300 K. The dropletdiameter distribution is given by a log-normal distribution, defined by its probability densityfunction p(d) in (3.3):

p(d) = Log-N (ln(dm), σ2) = 1√2πdσ

exp(−(ln(d)− ln(dm))2

2σ2

)(3.3)

where σ = 0.4 is the standard deviation.

22 CHAPTER 3. GENERAL SETTINGS

Table 3.1: The atomiser parameters

Parameter ValueMass flow rate m 100 g s−1

Orifice radius R0 0.89 mmAir core ratio X 0.14

Half spray-angle θS 30°Droplet mean diameter dm 50 µm

According to Equation (3.2), the droplets at the injector nozzle have an axial velocity of32.4 m s−1. This leads to a vertical position shift of the atomiser of about 5 · 10−5 m using atime step ∆t = 10−6 s (as it will be used in the following chapters). Considering a margin andnoting that the first cell height at the injection point is of 2 · 10−4 m in the rocket grids (seeSection 5.1), the chosen shift is ∆y = 10−4 s. This offset works fine and no particles are lost.

3.3 ChemistryIn the considered hybrid rocket design, the propellants are stored as liquid hydrogen peroxide

H2O2 and solid HDPE (high-density polyethelene). In the combustion chamber, due to hightemperature and heat fluxes, the H2O2 evaporates and decomposes itself into O2 and H2Ofollowing Reaction (3.4), while the HDPE yields mainly C2H4 by pyrolysis.

H2O2ka H2O + 1

2 O2 (3.4)

No catalyst is used, so that the H2O2 decomposition is thermally driven. The kinetics is givenby a first order reaction rate according to Giguère and Liu [10], modelled with an Arrheniusequation:

ka = Aa exp(−Ea,aRT

)[H2O2] (3.5)

where R = 8.314 J mol−1 K−1 is the universal gas constant and

Aa = 1 · 1013 s−1 Ea,a = 200 832 J mol−1

The hydrogen peroxide is naturally unstable and is difficult to sustain above a concentrationof 99% in an aqueous solution. Practical and economical considerations therefore limit itsconcentration to 98% in industrial applications [5]. This leads to a two-step evaporation, thewater evaporating at a lower temperature than H2O2 for a given pressure (cf. Appendix A.2).Although the Sparte solver manages multi-species particles, it cannot yet evaporate each speciesindependently. For that reason, and to keep the model simple, the hydrogen peroxide is assumedto be pure in the following simulations. For simplification still, the fuel pyrolysis is assumed toyield only gaseous C2H4.

The combustion occurs between the gaseous fuel C2H4 and the decomposition product O2,which is the real oxidizer of the combustion. It is described according to the two-step reactionmodel (3.6), using the coefficients established by Westbrook and Dryer [29].

C2H4 + 2O2kb 2CO + 2H2O

CO + 12 O2

kck−c

CO2

(3.6)

3.3. CHEMISTRY 23

The rates of reaction are modelled with the following Arrhenius equations:

kb = Ab exp(−Ea,bRT

)[C2H4]0.1[O2]1.65 (3.7)

kc = Ac exp(−Ea,cRT

)[CO][H2O]0.5[O2]0.25 (3.8)

k−c = A−c exp(−Ea,−cRT

)[CO2] (3.9)

where

Ab = 7.589 · 107 m2.25 mol−0.75 s−1 Ea,b = 125 520 J mol−1

Ac = 1.259 · 1010 m2.25 mol−0.75 s−1 Ea,c = 167 360 J mol−1

A−c = 5 · 108 s−1 Ea,−c = 167 360 J mol−1

Finally, the propellants are considered to be in stoichiometric proportions, corresponding tothe optimum oxidizer-to-fuel ratio O/F = 2.5 for the couple O2/C2H4 [4]. The H2O2 mass flowrate is set to 100 g s−1, i.e. an injected O2 mass flow rate of 47 g s−1, this leads to a fuel massflow rate of 18.8 g s−1.

25

Chapter 4

FIMUR test cases

4.1 Test settingsSeveral test cases have been carried out to investigate the general behaviour of the model

with high mass flow rate and evaporation.For these quick simulations, a simple 2D-axisymmetric coarse grid (0.2 × 0.15 m) has been

used with the time step ∆t = 10−6 s. It is shown in Figure 4.1:

X

Y

Z

AXIS

WALL

INLET

OUTLET

ATOMISER

Figure 4.1: The test case mesh

It contains 2000 elements and its boundary conditions are defined as follows:

– on the left stands an adiabatic wall,– below is the symmetry axis,– on the top, the boundary condition is set to a pressure condition, acting as inlet as default,– on the right, an outlet with fixed pressure,– the atomiser is placed in the refined lower left corner.

After some test cases with the default species, i.e. air as a species for the gas phase andwater for the liquid phase, an input file for the Sparte solver, defining the liquid hydrogenperoxide properties, has been written (cf. Appendix A) and two cases have been tested:

26 CHAPTER 4. FIMUR TEST CASES

1. The liquid H2O2 simply evaporates into the gas phase without any chemical reactions,2. The liquid H2O2 evaporates into the gas phase and the gaseous H2O2 decomposes according

to Reaction (3.4).

The grid is initialized with water vapour at 30 bar, 2000 K. The atomiser is placed on theaxis and injects liquid hydrogen peroxide at 300 K with the mass flow rate m = 100 g s−1.

4.2 Results and discussionIn the steady state, the injection cone is deformed by the flow it has created and tends

to spread toward the axis. This flow also cools down the inner part of the cone to 300 K byconvecting downstream the heat transfer with the droplets exiting the atomiser (see Figure 4.2).

(a) Case 1 (b) Case 2

Figure 4.2: Gas and droplet temperature at t = 0.1 s

As it can be seen in Figure 4.3a for case 1, the smallest droplets are driven closest to theaxis due to their low inertia and end up in the cold region Therefore they cannot evaporate.The heaviest droplets are little affected by the flow on the grid length scale and go through thehot gas. They need more time to heat, so they evaporate little in the grid. Finally, the gaseousH2O2 forms in-between these two groups (Figure 4.3b).

In case2, the H2O2 forms a line between the cold, where the droplets do not evaporate, andthe hot regions, where it decomposes quickly and completely (see Figure 4.4). Reaction (3.4) isexothermic and thus releases heat, which affects the temperature profile and causes the H2O2to decompose faster. Consequently, a sharp increase in temperature parallel to a sharp dropin peroxide mass fraction can be observed in Figure 4.5. Also, this sustained high temperaturein the spray increases the evaporation rate of the droplet compared with the first case : inFigure 4.2b, the droplets almost disappear inside the light green zone, whereas they can be seendown to the cold border in Figure 4.2a.

4.2. RESULTS AND DISCUSSION 27

(a) Droplets diameter and streamlines (b) Gas phase H2O2 mass fraction

Figure 4.3: Evaporation of H2O2 in case 1 without reactions at t = 0.1 s

(a) Gas phase H2O2 (b) O2

Figure 4.4: Mass fraction distribution in case 2 with H2O2 decomposition reaction at t = 0.1 s

At this point, no specific issues are encountered with the FIMUR model, or more broadlywith Sparte, despite the dense spray and the importance of the mass flow rate, and cantherefore be applied to a more complex geometry. However, the resulting spray will have tobe compared to coming experimental measurements to insure the accuracy of the model. Ifnecessary, the FIMUR model could be fitted to the experimental observations in the range ofthe rocket propulsion conditions (high mass flow rate, high pressure, high temperature).

28 CHAPTER 4. FIMUR TEST CASES

(a) Temperature (b) Mass fractions

Figure 4.5: Profiles in both test cases at x = 0.15 m

29

Chapter 5

Simulations on the rocket geometries

5.1 Geometry

The simulations are processed on two different 2D-axisymmetric geometries. The first one(Figure 5.1a) represents only the pre-chamber and the port. It is a simplified case of the hybridrocket, containing only the parts of interest. Indeed, the focus is put on the atomiser and onthe region close to it.

1

3

2

5

4

x

y

Lprechamber Lport

Rprechamber

Rport

(a) Without nozzle

1

3

2

5

x

y

Lprechamber Lport

Rprechamber

Rport

Lnozzle

2

4

(b) With nozzle

Figure 5.1: The rocket engine geometries

The second geometry (Figure 5.1b) adds a short post-chamber and a nozzle, to represent thewhole internal flow from the injection to the nozzle throat. This sets more correctly the outletcondition and solves some issues explained in the following sections, but it also adds complexityand new difficulties. The throat radius R∗ has been calculated using (5.1), derived from themass flow rate definition with the isentropic relations.

A∗ = π(R∗)2 = m

P0

√RT0Mgasγ

(γ + 1

2

)(γ+1)/2(γ−1)

(5.1)

where P0 and T0 are the pressure and temperature conditions inside the combustion chamber.Mgas and γ are the molecular mass and the heat capacities ratio for the mixed gas, and are

30 CHAPTER 5. SIMULATIONS ON THE ROCKET GEOMETRIES

based on the exit gas of simulations on the first geometry.The dimensions are given in Table 5.1. They are chosen approximatively according to usual

dimensions and do not represent an experimental set-up. Therefore they are not optimized yetfor perfect working conditions.

Table 5.1: Dimensions of the geometries

Part Length Radius[mm] [mm]

Pre-chamber 50 37.5Port 230 12.5Nozzle 75 3.81

On these geometries, fives boundary conditions are defined:

1. The inlet (in green on the figures). During initialisation (cf. Section 5.2) and until theinjection is fully liquid, this boundary injects decomposed H2O2, as if catalysed upstream.Hence it injects O2 and H2O at 1225 K, with the mass fraction YO2 = 0.47. The radiusof the inlet is chosen at the beginning of the simulation, but can be changed between twodifferent cases. Once the atomiser injects the whole oxidizer in liquid phase, the inlet isturned off and becomes an adiabatic wall.

2. Adiabatic walls (in black).

3. The fuel pyrolysis (in red). It corresponds to a wall that injects C2H4 at mfuel =18.8 g s−1 when its temperature exceeds 900 K.

4. The outlet (in blue). In the first geometry, it is a simple subsonic outlet set to 30 bar.In the second geometry, it is set to supersonic outlet once the nozzle throat is choked. Inorder to reach that condition, the outlet is first set to a low pressure condition (≈ 1 bar).

5. The symmetry axis (in yellow).

Due to the simplicity of the first geometry, the mesh can be regularly structured withrectangle elements and refinement close to the boundaries. The corner at the port entry is alsorefined, to better capture the gradients where the injection flow separates into a recirculatingflow inside the pre-chamber and the port flow (Figure 5.2a).

The second geometry takes the same structured grid in the pre-chamber and the port.However, to avoid too many cells at the nozzle throat, the post-chamber is partially unstructured,as shown in Figure 5.2b. The post-chamber and nozzle regions are not of interest in this studyand are only used here to force the exit condition. Therefore, the slight irregularities at theborders between the structured and the unstructured regions are not important here.

The resulting grids consist in between 30 000 and 40 000 elements. These numbers are basedon previous works at ONERA on a similar geometry to obtain a sufficient precision, enough hereto define a methodology. Nevertheless, a grid refinement study should be carried out to insurethe accuracy of the processed result fields.

1This corresponds to the throat radius, the pre-chamber and post-chamber radii being equal.

5.2. INITIALISATION 31

X

Y

Z(a) The injection and port entrance mesh

XZ

Y

(b) The nozzle mesh

Figure 5.2: Mesh details

5.2 Initialisation

Before introducing liquid H2O2 droplets inside the grid, an initialisation of the flow must becarried out.

Indeed, a direct simulation with the particles leads to divergence in less than 20 iterations.The cause for that is the sudden and important gradient in momentum at the injection point ofthe pre-chamber. Another downside with this method, the cold droplets will not be able to startthe combustion. In a real case, the combustion would actually be started with a pyrotechnicigniter.

Therefore, the rocket is first ignited with a gas phase injection, corresponding to the case of ahybrid rocket with catalysed injection. The objective here is to obtain a converged velocity field,with a stable flame. Once this stage is reached, the injection is to be progressively transferredto liquid droplets.

The time step size is driven by the chemistry. The chemical characteristic time is hard toassess, because it depends on many local variables, such as the local concentration or the localtemperature. The time step has been reduced until the disappearance of the numerical instabil-ities due to the combustion, which provoked amplifying pressure waves with high gradients atthe entrance edge of the port (Figure 5.3). The resulting value is ∆t = 10−6 s, corresponding toa maximum CFL number below 0.5 in the mesh.

Figure 5.3: Pressure waves due to combustion

32 CHAPTER 5. SIMULATIONS ON THE ROCKET GEOMETRIES

5.3 Catalysed gas injection

The injected gas density being three order of magnitude lower than the liquid phase, the inletdiameter is chosen equal to the port diameter. The resulting injection velocity for m = 100 g s−1,a bit less than 30 m s−1, is acceptable regarding to the liquid injection velocity.

The flow inside the rocket engine is turbulent, so a turbulence model is added to yieldresults closer to reality. However, the input turbulent parameters are hard to estimate, as nomeasurements inside the combustion chamber are available. The chosen input values are thustheoretical and let the numerical convergence correct them.

The turbulence is modelled with the two-equation k-ω SST model, based on Messineo’swork [17]. In the input file, CEDRE allows the user to enter the two scalars needed as turbulencelevel Tu and turbulent length scale `, and converts them automatically to k and ω. For acylindrical combustion chamber, these values are usually given by (5.2):

Tu ≈ 0.05 and ` ≈ 0.07D = 0.0018 m (5.2)

The gas-initialised solution displayed in Figure 5.4 does not exactly correspond to the steadystate. The O2 is still convected inside the pre-chamber recirculation zone and will carry on untilthe mass fraction is homogeneous inside of it, but its field in the port is converged. Moreover, thevelocity field has converged and the diffusion flame is well defined and stable, which correspondsto the goals of this initialisation. Liquid injection can therefore start progressively.

(a) Mass fraction of O2 and streamlines

(b) Gas temperature, showing the diffusion flame

Figure 5.4: Gas injection with a turbulence model

5.4. STARTING THE LIQUID INJECTION 33

5.4 Starting the liquid injectionThe liquid particles are added in Two-Way coupling, using the FIMUR atomiser described

in Section 3.2. Starting directly with a full coupling leads to fast divergence, due to the largesource terms in the dense region. Especially two source terms are in cause.

The first one is the momentum source term. In this case, the calculation states that there isa cell with negative density or negative pressure in the near region of the atomiser. Due to thetransfer of momentum, the gas is highly accelerated in the first cells of the mesh and might endup empty of gas. This is actually one of the drawbacks of the FIMUR model: the CFD codedoes not take into account the separate phase flow in the vicinity of the atomiser.

The second source term causing divergence is the energy exchange. This can be explainedby the coupling mechanism. Each time step is divided into two sub-steps for each solver. First,the Charme solver processes the gas phase based on the previous time step solution. At theend of this sub-step, each cell possesses a determined amount of energy in form of heat. Thenthe Sparte solver moves the particles inside the gas and calculates the new source terms tosend to the gas solver. Especially, if the gas temperature in a cell is higher than the temperatureof the particles that it contains, heat is transferred from the cell to the particles. However,each particle absorbs an amount of heat as if it was alone inside the cell, while the gas of thecell is not refreshed and keeps its heat until the end of the time step. Indeed, the couplingdoes not take into account interactions between particles, whatever their proximity, and theevaporation model considers the droplets as isolated (Section 2.3). Also, note that the particlesare located with their center of gravity, without consideration of their spatial extend: a dropletastride several cells only produces source terms in one of them. As a result, in the case of highevaporation rate of a dense spray, more heat is absorbed by the particles than available insidethe cell, causing its absolute temperature to become negative. Since such a behaviour is notphysical, the calculation stops.

Three approaches were considered to solve this issue. The first one consists in using largercells, and thereby reducing the accuracy of the simulation, so that each cell contains more gasand thus more heat to exchange for a given time step. The coupling actually works fine on acoarser grid, as it has been demonstrated with the FIMUR test cases (Chapter 4), but in the longrun the objectives are to obtain a good representation of the flow inside the combustion chamber,so the cells will have to be kept fine. Besides, having coarse cells close to the injector and fineelsewhere affects the grid quality as well as the accuracy of the flow field in the pre-chamber.

The time step can also be reduced to let the evaporation occurs more progressively. Thetime step was therefore set to 10−7 s, because reducing further the time step increases harshlythe computation time.

The last method consists in reducing the influence of the particles on the gas by reducing therelaxation parameter. This parameter needs then to be progressively increased as the flow fieldconverges towards its steady solution. To get the energy issue corrected and the simulation run,the relaxation parameter had to be reduced below 5% on the whole grid, in a first phase. Tothat, a mask in the vicinity of the atomiser was added. This mask allows to choose a differentrelaxation parameter inside a rectangular area, but only one can be defined and its limits aresharp, leading to a jump in the relaxation parameter value. In the current cases, it was setbelow 0.1%, i.e. almost One-Way coupling, to hide the atomisation process region.

34 CHAPTER 5. SIMULATIONS ON THE ROCKET GEOMETRIES

5.5 Need of a nozzle

When adding the droplets, a new strange effect leading to divergence occurs: the flow in theport reverses, as illustrated in Figure 5.5a.

(a) Droplets and streamlines

(b) Mass fraction of gaseous H2O2

(c) Gas temperature field

Figure 5.5: Liquid injection with reverse gas flow

It can be explained by the vortices generated at the front of the jet flow, which amplify therecirculation inside the pre-chamber and create a depression. This flow inversion might alsoamplify the recirculation of the droplets, as it pushes them inside the pre-chamber. Also, dueto their inertia, a great part of droplets accumulates and evaporates on the fuel wall inside thepre-chamber (Figure 5.5b). Although it is expected, according to the burning deformation ofthe fuel block observed in experimental set-ups, the phenomenon is quite difficult to model here.

5.5. NEED OF A NOZZLE 35

On one hand, it is impossible to observe the flame behaviour on this side of the fuel, so there isno approximated values for the regression rate and the fuel mass flow rate. On the other hand,the accumulation of droplets in a few cells will lead to numerical divergence, their evaporationat the wall causing then too large source terms per cell.

Therefore, the injection angle is changed and the FIMUR model is set for a narrower atomiserwith θS = 10°±5°. The new atomiser does not correspond to the experimental set-up, butsimplifies the case to further investigate the methods regarding the liquid injection inside thecombustion chamber. The accumulation of droplets on the pre-chamber wall has vanished, yetthe vortices still reverse the flow, as shown in Figure 5.6.

(a) Droplets and streamlines

(b) Mass fraction of gaseous H2O2

(c) Gas temperature field

Figure 5.6: Liquid injection with the narrow atomiser

36 CHAPTER 5. SIMULATIONS ON THE ROCKET GEOMETRIES

To counter that effect, the nozzle is added and the calculations are processed on the secondgeometry. Once choked, it forces the mass flow rate at the throat. With that outlet condition,the flow does not reverse any more and the simulation can run further (Figure 5.7). Note thatthe sudden start of the evaporation (x = 0.02 m) is due to the lower coupling mask.

(a) Droplets and streamlines

(b) Mass fraction of gaseous H2O2

(c) Gas temperature field

Figure 5.7: Liquid injection with the narrow atomiser and the nozzle

Nevertheless, though all the corrections and new parameters chosen to run the simulation,the calculation still breaks, but without clearly stated divergence reasons.

5.6. ALTERNATIVE APPROACH 37

5.6 Alternative approachAnother method has been investigated: instead of starting with a low relaxation coefficient

at full mass flow rate, the simulation is performed with full Two-Way coupling and progressivelyincreasing mass flow rate. The mask in the atomiser vicinity is still active to prevent the issuesalready identified. To keep the injection velocity constant despite the changing mass flow rate,the atomiser radius R0 was artificially reduced in the input parameters based on (3.2). Theadvantage of this method is that the density limitation for the spray with the Sparte solvercan be tested.

Figure 5.8 shows the converged simulation with a mass flow rate of 30 g s−1, representing30% of the intended mass flow rate. In Figure 5.8d, a cold axis region can be observed. Thisis due to the same reasons as for the FIMUR test cases, and strongly reduces the evaporationand decomposition rates of H2O2. As a result, the droplets do not evaporate completely in therocket engine. Note that the mass fraction of O2 in the still injected gas has been reduced to 0.3to highlight the decomposition zone. This zone, in red in Figure 5.8c, is unsurprisingly near theflame zone. The reduction in injected O2 also affects the flame, which seems to go out upstreamin the port.

Unfortunately, the simulation could not be investigated further, due to lack of time. Indeed,this simulation is particularly slow, due to the great number of particles (only 1% is actuallydisplayed in 5.8a). Also, the mass flow rate has to be increased with small steps and let eachstep converge properly to prevent divergence due to transitory behaviours.

From the behaviour observed until the results shown in Figure 5.8, this methodology stillseems to have some issues, especially due to the high cooling of the axis. It might be difficult,through increase in liquid injection mass flow rate, to heat this zone and to increase theevaporation rate. Moreover, the sustainability of the flame without gas injection is unsure.

To confirm this behaviour, the simulation has to be carried out further, possibly with adifferent gas injection strategy (to prevent the flame to go out). Also, additional physicalphenomena could be taken into account, such as the radiation, which might help heating upprogressively the droplets and the gas upstream. This new setting would however need animportant work, as the radiation solvers in CEDRE only work on 3D meshes and are not yetcoupled with the droplets, not to mention the incompatibility with the evaporation models(Section 2.3) leading to further developments.

38 CHAPTER 5. SIMULATIONS ON THE ROCKET GEOMETRIES

(a) Droplets and streamlines

(b) Mass fraction of gaseous H2O2

(c) Mass fraction of gaseous O2

(d) Gas temperature field

Figure 5.8: Working liquid injection with m = 30 g s−1

39

Conclusion

This master’s thesis aimed to model the liquid oxidizer injection inside a hybrid rocket engineusing a Lagrangian solver. Consisting in a challenging simulation, especially due to the largemass flow rate of the atomiser, it has already been briefly investigated at ONERA a few yearsago. Since then, the CFD code has been further developed, while a new model to simplify theinjection has been implemented.

The current work have shown some difficulties to perform the simulation but yields in theend encouraging results. First the FIMUR model was discovered and tested on coarse grid,and the solvers did work well despite the extreme conditions of rocket combustion chamber interms of mass flow rate, pressure and temperature. The accuracy of the model and potentialadjustments will be assessed through coming experiments. The cases were then progressivelymade more complex, using hybrid rocket engine geometries.

The thesis highlighted the need of an initialisation, based on catalysed gas injection, andof a mask in the atomiser vicinity, to hide the liquid phase atomisation region. Two methodswere then investigated. The first one applied the intended mass flow rate at a reduce relaxationcoefficient of the dispersed flow source terms toward the carrying gas solver, but could not run toconvergence. The second one increased gradually the liquid mass flow rate without changing thecoupling between the solvers. Although its results seems promising, they could not be broughtto a conclusion due to lack of time, the simulation time becoming extremely long. This lastmethod will therefore have to be further investigated.

41

Appendix

43

Appendix A

Physical properties of H2O and H2O2

This appendix presents the physical properties of the injected liquid oxidizer H2O2 and thoseof the default chemical species, H2O, used by the Sparte solver to accurately model the droplets.

The Sparte solver reads an external data file to get the physical properties of the chemicalspecies that compose the modelled particles.

The data file includes three constant values: the molar mass, the critical temperature andthe critical pressure (Table A.1).

The rest of the data is supposed function of the temperature only. The density, the heatcapacity and the thermal conductivity are given for the liquid and the solid phase, in case ofsolidification. The liquid viscosity and the surface tension are used for further fragmentationor coalescence of the droplets. These five functions are interpolated as 6th degree polynomials,with a chosen reference temperature T0 of 300 K. The heating and evaporation models also needthe vapour pressure and the heat of vaporization, interpolated as non-polynomials. The boilingpoint, required as well in the calculations, can be directly deduced from the vapour pressure(Antoine equation) since they are reciprocal functions.

The other parameters, such as the heat of solidification, are not to be used and their datacannot be found in the reference data compilation [6], so they are let to their default valueswithout impact on the simulations.

The interpolated values are presented in the following sections. The Table A.(x+ 1) gathersthe interpolation coefficients read by the Sparte solver according to Equation (A.x), and theresulting curves are drawn in Figure A.x.

Table A.1: Constant physical properties

Species Molecular mass Critical Temperature Critical Pressure[kg mol−1] [K] [Pa]

H2O 18 · 10−3 647.30 2.2048e7H2O2 34.015 · 10−3 730.15 2.1684e7

44 APPENDIX A. PHYSICAL PROPERTIES OF H2O AND H2O2

Density

ρp(T ) =N∑i=0

aρ,i

(T

T0

)i(A.1)

Table A.2: Sparte data for density

Index i H2O coefficients H2O2 coefficients0 −2.0671 · 104 −6.8268 · 103

1 9.5758 · 104 3.5697 · 104

2 −1.7386 · 105 −6.1706 · 104

3 1.6617 · 105 5.5343 · 104

4 −8.8202 · 104 −2.7434 · 104

5 2.4628 · 104 7.129 16· 103

6 −2.8276 · 103 −7.6037 · 102

Figure A.1: Density for H2O2 and H2O

45

Vapour pressure

ps,p(T ) = exp(ap −

bpT + cp

)(A.2)

Table A.3: Sparte data for vapour pressure

Coefficient H2O coefficients H2O2 coefficientsap 2.3196 · 101 2.3296 · 101

bp 3.8164 · 103 4.3871 · 103

cp −4.6130 · 101 −5.0703 · 101

Figure A.2: Vapour pressure for H2O2 and H2O

46 APPENDIX A. PHYSICAL PROPERTIES OF H2O AND H2O2

Heat of vaporization

Lv,p(T ) = aL

(1− T

Tc

)bL(A.3)

Table A.4: Sparte data for heat of vaporization

Coefficient H2O coefficients H2O2 coefficientsaL 3.1253 · 106 1.8198 · 106

bL 3.8000 · 10−1 3.2450 · 10−1

Tc 6.4730 · 102 7.3015 · 102

Figure A.3: Heat of vaporization for H2O2 and H2O

47

Heat capacity

cp,p(T ) =N∑i=0

acp,i

(T

T0

)i(A.4)

Table A.5: Sparte data for heat capacity

Index i H2O coefficients H2O2 coefficients0 1.5341 · 104 1.8771 · 103

1 −3.4806 · 104 6.4119 · 102

2 4.0591 · 104

3 −2.1156 · 104

4 4.2130 · 103

Figure A.4: Heat Capacity for H2O2 and H2O

48 APPENDIX A. PHYSICAL PROPERTIES OF H2O AND H2O2

Liquid viscosity

µp(T ) =N∑i=0

aµ,i

(T

T0

)i(A.5)

Table A.6: Sparte data for liquid viscosity

Index i H2O coefficients H2O2 coefficients0 2.2543 · 10−1 3.7253 · 10−1

1 −8.4848 · 10−1 −1.66842 1.3284 3.17673 −1.1039 −3.27084 5.1276 · 10−1 1.91295 −1.2611 · 10−1 −6.0093 · 10−1

6 1.2824 · 10−2 7.9058 · 10−2

Figure A.5: LiquidViscosity for H2O2 and H2O

49

Thermal conductivity

λp(T ) =N∑i=0

aλ,i

(T

T0

)i(A.6)

Table A.7: Sparte data for thermal conductivity

Index i H2O coefficients H2O2 coefficients0 −4.2670 · 10−1 4.9360 · 10−1

1 1.7071 −1.4085 · 10−1

2 −7.2059 · 10−1

3 4.9005 · 10−2

Figure A.6: Thermal conductivity for H2O2 and H2O

50 APPENDIX A. PHYSICAL PROPERTIES OF H2O AND H2O2

Surface tension

σp(T ) =N∑i=0

aσ,i

(T

T0

)i(A.7)

Table A.8: Sparte data for surface tension

Index i H2O coefficients H2O2 coefficients0 3.3111 · 10−1 2.0919 · 10−1

1 −8.8277 · 10−1 −3.4727 · 10−1

2 1.4199 4.6993 · 10−1

3 −1.3203 −4.1876 · 10−1

4 7.0312 · 10−1 2.0891 · 10−1

5 −2.0343 · 10−1 −5.4640 · 10−2

6 2.4770 · 10−2 5.8946 · 10−3

Figure A.7: Surface tension for H2O2 and H2O

51

Appendix B

The experimental setting atomiser

The atomiser used in the experimental settings and modelled through the FIMUR modelcorresponds to the Delavan WDA atomiser, nozzle number 35.0 with a 1.78 mm diameter.Converting the indicated flow rate at the maximum recommended pressure in Figure B.1, i.e.267 L h−1, into mass flow rate, the atomiser can deliver up to:

m = ρH2O2V = 0.106 kg s−1 (B.1)

Figure B.1: Delavan WDA atomiser specification sheet (extract)

53

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