MODELLING AND SIMULATION OF LIQUID ROCKET ENGINE IGNITION TRANSIENTS

Transcription

1 Faculty of Engineering MODELLING AND SIMULATION OF LIQUID ROCKET ENGINE IGNITION TRANSIENTS a Dissertation submitted to the Doctoral Committee of Tecnologia Aeronautica e Spaziale in partial fulfilment of the requirements for the degree of Doctor of Philosophy Tutor Prof. Marcello Onofri Candidate Francesco Di Matteo Co-tutor Ing. Marco De Rosa Academic Year

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3 Contents Nomenclature x 1. Introduction Motivation: what is the ignition transient and why are we interested in it? Key challenges in Rocket Engine start-up Main objectives Organization of this Thesis State of the Art Engine cycles and their start-up and shut-down transients Gas Generator Engine Expander Engine Staged Combustion Engine Modelling: review of previous works ESPSS: European Space Propulsion System Simulation Fluid Properties Library Perfect Gas properties according to CEA Perfect Gas interpolated properties Simplified Liquid interpolated properties Real Fluids interpolated properties Perfect gas mixtures Real Fluid - Perfect gas mixtures Fluid Flow 1D Library Components Classification Junction/Valve i

4 Contents Capacity/Volume Tubes/Pipes Turbomachinery Library Pump & Generic Pump Turbine & Generic Turbine Combustion Chambers Library Injector Cavity Combustor Equilibrium Combustor rate Nozzle Cooling Jacket components Steady State Library Components Overview Ports The type switch D pipes D components: junctions & valves Combustion Chambers Cooling Channels Turbomachinery Pump Turbine Validation Component validations Subsystem validations Engine cycle designs Transient Modelling Injector Plate model Qualitative behaviour Hot Gas side heat transfer coefficient models Models implemented Validation ii

5 Contents 5.3. Q-2D stratification model for HARCC Model description Numerical validation Experimental validation Integrated Validation: RL-10 design and analysis Overview of the RL-10A-3-3A rocket engine Design procedure Turbomachinery modelling Thrust chamber and cooling jacket modelling Lines, valves and manifolds modelling Subsystem simulation: validation at nominal conditions RL-10 Engine start-up Description of the start-up sequences Start transient RL-10 engine shut-down Description of the shut-down sequence Shut-down transient Dynamic Response Analysis Conclusions 197 A. Implementation of Up-wind Roe Scheme b A.1. Governing equations b A equation subset c A.2. Numerical concepts d A.2.1. Roe s numerical scheme d A.2.2. Approximate Riemann Solver e A.3. Reconstruction method i A.3.1. Higher order accuracy i A.3.2. Preconditioning o A.3.3. Variable cross-section p B. Friction Factor Correlations r iii

6 Contents B.1. Single-Phase Friction Factor Calculation. Function hdc_fric... r B.2. Two-Phase Friction Factor Calculation. Friedel Correlation.... r B.3. Elbow Pressure Loss Function s C. Film Coefficient Calculation v iv

7 List of Figures 2.1. Vulcain 2 schematic [46] Vinci schematic [45] Space Shuttle Main Engine schematic [149] Space Shuttle Main Engine start-up sequence [12] Space Shuttle Main Engine shut-down sequence [12] Components in the fluid_flow_1d library Pipe discretisation Components in the turbo_machinery library Components in the comb_chambers library Cooling jacket wall mesh [43] Simplified Cooling Jacket wall disposition [43] Channel with relevant areas and surfaces for heat flux calculation Longitudinal heat fluxes for a segment i Components in the Steady State library Cooling jacket channels wall mesh [43] Schematic of the Pipeline test case. Purple: steady state components. Cyan: transient components Schematic of Combustion Chamber test case. Purple: steady state components. Cyan: transient components Turbopump test case: HM7B power pack transient schematic Turbopump test case: HM7B power pack steady state schematic Chamber test case: HM7B Combustion Chamber transient schematic Chamber test case: HM7B Combustion Chamber steady state schematic HM7B engine system schematic v

8 List of Figures Schematic of the RL-10 engine Schematic illustration of an arbitrary injector head Schematics of the injector plates Temperature profiles from original and new model Heat fluxes and wall temperatures results left: 1-D fluid element and energy balance used for conventional 1- D method; right: control volumes of the Q-2D approach integrated in 3D wall elements Cooling jacket wall mesh Methane bulk variables evolution along channel axis Design of the 4 sector HARCC segment Schematic of the experimental test case Wall and fluid thermal stratification,ar = 9.2, p c = 88 bar Wall and fluid thermal stratification, AR = 9.2, p c = 58 bar Wall and fluid thermal stratification, AR = 30, p c = 88 bar Wall and fluid thermal stratification, AR = 30, p c = 58 bar RL-10A-3-3A engine schematic [115] RL-10A-3-3A engine diagram Pumps performance maps Iterative procedure for determining pump parameters Turbine performance maps from P&W [15] Turbine performance maps RL-10A-3-3A chamber contour [15] and discretisation Cooling jacket channels profiles Venturi nozzle profile RL-10A-3-3A schematic model RL-10A-3-3A Valve schedule for Start-up Simulation [15] Valves opening sequence adopted in the simulation Transient results - part Transient results - part Transient results - part Transient results - part vi

9 List of Figures RL-10A-3-3A Valve schedule for Shut-down Simulation [15] Valves closing sequence adopted in the simulation Shut-down results - part Shut-down results - part Shut-down results - part Shut-down results - part TCV throttle results - part TCV throttle results - part TCV throttle results - part TCV throttle results - part OCV throttle results - part OCV throttle results - part OCV throttle results - part OCV throttle results - part A.1. Piece-wise linear reconstruction l B.1. Elbow pressure loss parameters t vii

10 List of Tables D pipe element D Junction element Combustor element Nozzle element Cooling jacket element Regenerative circuit element Pump element Turbine element Pipeline input data Pipeline output data CC input data CC output data HM7B Turbopump input [44] and initial data HM7B Turbopump output data HM7B CC input [44] and initial data HM7B CC output data HM7B input [44] and initial data HM7B engine system output data RL-10A-3-3A input and initial data RL-10A-3-3A engine system output data Injector plate variables comparison Cooling channels geometries Positioning of themocouples RL-10A-3-3A construction data [15] Venturi geometrical data viii

11 List of Tables 6.3. Fuel line valves parameters Oxidiser line valves parameters RL-10A-3-3A engine system output data Engine dynamic response to TCV ±10% operation Engine dynamic response to OCV ±10% operation A.1. Different values of ω k ix

12 Nomenclature A Cross section area AR Aspect ratio C Capacitive C + Cp D E f r G Reduced torque Specific heat Diameter Internal energy Friction factor Mass flow per unit area h, H Enthalpy h Dimensionless characteristic head h c Isp K k L M M MR MW mh N N k Ns n n s Heat transfer coefficient Specific impulse Loss coefficient for design condition Concentrated load losses Characteristic length Mass Mach number Mixture ratio Molar weight Enthalpy flow Speed coefficient Number of moles Specific speed Reduced speed Pump number of suctions n st P P Pr Q Q + Q q Pump number of stages Pressure Perimeter Prandtl number Volumetric flow rate Mass flow coefficient Heat flux Heat flux per unit area R, r Geometrical radius R Universal gas constant Re Reynolds number S Entropy St Stanton number T Temperature TDH Total dynamic head t Wall thickness u Internal energy u Primitive variables vector V Volume v Velocity W Turbopump power We Weber number x Fluid quality x Horizontal abscissa x nc Z Non-condensable mass fraction Compressibility factor x

13 Nomenclature Greek symbols α Fluid void fraction β Volumetric expansivity β Dimensionless characteristics torque γ Variable isentropic coefficient δ total to static pressure ratio ε Absolute roughness ζ Concentrated pressure drop parameter η Efficiency ϑ Pump dimensionless parameter θ Total to static temperature ratio κ Isothermal compressibility λ Thermal conductivity µ Fluid viscosity ν Specific volume ν Reduced flow parameter ξ Pipe pressure drop coefficient Π Total to total pressure ratio ρ Density σ Stefan-Boltzmann constant τ Mechanical torque τ Time constant τ xy φ φ + ψ + Ω ω Fluid shear stress Interaction parameter Mass flow coefficient Head rise coefficient Acentric factor Rotational speed Subscripts aw Adiabatic wall amb Ambient bu Burned condition c Fluid critical point c, cc Combustion chamber ch cond cap conv cav crit chem eff ext eq fu fr Channel Conductive Capacitive Convective Cavity Critical condition Chemical composition vector Effective External Equilibrium condition Fuel Frozen condition g, gas Gaseous phase gg Gas generator hg Hot gas side i, j, k Spatial indices i Internal wall condition jun Junction l, liq Liquid phase mix Mixing condition nc Non-condensable fraction o Initial condition ox Oxidiser p Pump pw Powder R Rated condition rad Radiative ref Reference point sat Saturated condition sound Sound condition t Turbine t Turbulent xi

14 Nomenclature th Throat v, vap Vapour w Wall wet Wet xii

15 1. Introduction In this introductory chapter the problem of the ignition start-up and shut-down transients for liquid rocket engine is discussed. The object of this thesis is not limited to the modelling and simulation of liquid rocket engines start-up but it aims to the creation of a tool able to model the dynamic behaviour and to obtain the functional design of liquid rocket engine systems. Once the motivation and the interest of this problem are shown, the attention will be focused on the critical aspects that characterize liquid rocket engine propulsion system during these particular phases. Finally, the main objectives of this dissertation as well as a brief overview of its structure will be shown Motivation: what is the ignition transient and why are we interested in it? In the past, until the end of the 70s, the starting process development of liquidpropellant engines was usually achieved empirically by testing different schemes and start cyclogrammes directly during test firing. It required a big amount of resources and was time consuming. Start-up calculation methods at that time did not reflect the main factors affecting the process and could not serve as reliable means for start development. More than 30% of the engine failures occurred during start-up [68]. Recently, with more powerful and complex engines, the need of more rationale and reliable methods of the starting process development appeared. The prediction of the start-up characteristics of liquid propellant rocket engines is important to the engine configuration and control system design processes. Despite that, engine start systems have received secondary considerations since high-power performance was the chief design objective. 1

16 1. Introduction A careful synchronization of control actions with transient start processes is required to deliver the smooth and reliable thrust build-up characteristics desired. For turbopump-fed engines, the ability to power turbomachinery prior to combustion chamber ignition is an important design concern [7]. Indeed, thrust build-up can be delayed or inhibited if turbine power is insufficient to accelerate propellant pumps. Many difficulties associated with engine start predictions stem from the nonlinear mass flow and heat transfer characteristics associated with filling unconditioned engine systems with cryogenic propellants. The use of cryogenic fluids imposes additional problems during start-up when they flow into a system with ambient wall temperatures. Because of strong evaporation inside tubes, cryogenic fuels may lead to severe over-pressures and even Thermal choking during the cool-down processes. Accurate predictions are especially important in engine cycles where this initial propellant flow provides all of the available turbine power for starting, e.g., expander cycles. During the start of liquid-propellant engine the main role belongs to the hydrodynamic processes (mass flow variations, pressure surges, phase change, etc.). Hydrodynamic processes are practically the only ones that can be used to act upon all other phases of the physical-chemical process forming the start-up of the engine Key challenges in Rocket Engine start-up Ever since the first liquid rocket engines were developed, performance analyses have been implemented to examine their operation under various conditions. Steady state assessments for design conditions are wide-spread and useful in pre-design phases [17, 47, 61, 94, 100]. Further testing of the nominal operational modes are performed during initial testing of new liquid rocket engines when various tests are run in order to verify that the engine has been constructed according to the original design and performance requirements. This leads to engines that are extensively reliable during their nominal operation. Nominal conditions however do not illustrate the extreme conditions in which most engine components are required to work during transient phases, such as 2

17 1. Introduction start-up, shut-down, and throttling. Transient phenomena range from combustion high frequency instabilities to water hammer effects in feed lines. High pressure and temperature peaks, inherent to transient phases, may lead to failures in part of the engine system. The potentially resulting system failures may cause loss of payload, serious damage to the ground segment, if not loss of human life. Anomalies in the past, such as the RL-10 anomaly of the Atlas-Centaur flight AC-71 in 1992 [15], or the Aestus anomaly of Ariane 5G flight 142 in 2001 [74, 82] have demonstrated that a hard transient may indeed lead to significant system failures. In order to have a better understanding of the main problems that may occur during start-up and shut-down, a description of the main failures for the Japanese LE-7 engine and for the American SSME engine are here described. The Space Shuttle Main Engine (SSME) is the high performance LOX/LH 2 engine which was used in the Space Shuttle, producing a thrust of about 1700 kn by means of a staged combustion cycle. For this very complex engine 5 years of analysis were necessary to model the transient behaviour of the propellants and of the hardware during start-up and shut-down. The SSME engine was sensitive to small changes to propellant conditions and valve tuning was critical, a 2% error in the valve position or a 0.1 s timing error could lead to significant damage to the engine. A step-by-step approach was necessary to explore the start-up sequence with small time increments: that required 19 tests, 23 weeks, 8 turbopump replacements to reach 2 s into 5 s of start-up. Additional 18 tests, 12 weeks and 5 turbo-pump replacements were necessary to touch the minimum power level [12]. The LE-7 is a high performance LOX/LH 2 engine employed in the H-II rocket, which produces a thrust of about 1000 kn by means of a staged combustion cycle similar to that of the SSME. During H-II Flight 8 in 1999 a failure occurred in the LE-7 engine. The failure was determined by mechanical vibration problems into the fuel pump that caused high cycle fatigue and so the premature engine failure [129]. During development tests of the LE-7 other issues can be addressed to phenomena occurring during transient phases [50]: 3

18 1. Introduction High sensitivity to heat transfer from hardware to fluids and to ignition timing reproducibility of both combustion devices Functional instability to inlet pump vibration of pressure Problem of rotating cavitation on oxidiser and fuel pumps Over power during start-up caused damages to hardware Explosion at Main Oxidiser Valve opening 1.3. Main objectives Given the increased power of today s computers, more frequent use is made of CFD codes to perform detailed assessments of the flow behaviour in single engine components. The goal is to understand observed variations in fluid properties and to find the source of unexpected behaviours. Such CFD methods however require extensive computational times, making speedy parametric analyses impossible. Additionally, most require long input preparation times. Although the qualities of a thorough and in full-depth analysis may be desirable in the study of specific single components and scenarios, for complex, multi-component systems, the long computational times become insurmountable. An intelligent simplification of the underlying processes allows the reduction of the 3-D governing partial differential equations to one-dimensional or quasi 1-D differential equations which no longer require complex solution methods thus allowing much faster computational times. Results obtained from such studies rely significantly on imposed initial and boundary conditions. Simplifications introduced in these models often lead to an incorrect transient behaviour which does not correspond to the measured and observed physics. This is, amongst other reasons, due to the ignored interaction between downstream and upstream lying components. Tools concentrating on one component only are thus not sufficient as they do not help in understanding how components affect each other during transient phases, what their impact on system frequencies is, and how this interaction may lead to a major component or system failure. 4

19 1. Introduction A system approach is then necessary to take into account all the interactions between all the components of an engine. This choice is fundamental if a detailed estimation of the engine transient behaviour is the task. To this purpose the present work was initiated with the aim of studying and modelling by numerical tool the ignition transient of liquid rocket engines, improving and implementing more complex and accurate models in system modelling tools for transient analysis. This aim is achieved in three steps: A suitable physical and mathematical collection of models able to design and analyse the steady state behaviour of liquid rocket engine systems is developed and implemented in a numerical code. The model library has been successfully validated with respect to open literature data, performing design simulations and off-design analyses of actual rocket engines. Basic and simple models used to simulate the injector plate of the engine, the hot-gas-side heat transfer correlations in the combustion chamber and the regenerative circuit are exchanged with improved, more sophisticated and more physical models. These models are able to describe accurately the convective and radiative heat transfer at the injector plate face, the hot-gas-side heat transfer coefficient, and to describe also the influence of thermal stratification in high aspect ratio cooling channels. At last, the validation of a design procedure and the models developed are achieved by the design, the start-up, the shut-down and the dynamic response simulation of a real engine, the RL-10A-3-3A Organization of this Thesis The work performed is here presented in six chapters as described below. Chapter 1 provides a brief introduction of the motivations which led this work, followed by a description of the main issues related to transient phases in liquid rocket engines. Issues presented practical consequence as described for the case of LE-7 engine failures or the SSME start-up campaign. 5

20 1. Introduction In addition, a description of the main objectives of this work have been provided explaining which approach has been adopted for the modelling of the components behaviour of a liquid rocket engine, and what is the final aim of such a Ph.D. dissertation. Chapter 2 is dedicated to a summary of the state of the art in the field of liquid rocket engine transient simulation. Models and tools from USA, Europe, Russia, Japan, Iran, China are collected, studied and compared to each other in order to find advantages and drawbacks present in each work.the literature study is fundamental to understand in which direction this work should be directed. Chapter 3 summarizes the ESPSS library. It describes the models implemented in the European Space Propulsion System Simulation library, a collection of models for each engine s component used as starting point for this work. The chapter includes the way the library and the main components are modelled: fluid properties, pipes, volumes, valves and junctions components, pumps and turbines, combustion devices, cooling channels and nozzles. The assumptions behind each formulation as well as the way components interact with each other are analysed; advantages and limitations of each models are presented in this chapter. Chapter 4 illustrates the steady state modelling. It examines the most important engine components models developed for the creation of a steady state library. The main purpose of these components is the design and the analyses in steady state conditions liquid rocket engine systems cycles. The development of these models is used to enhance the system capabilities of the code and create a fundamental instrument to be used along the entire design period from pre-design phase to parametric studies for fine tuning of the engine parameters. The Steady State library presented within this chapter enables to perform iterative engine design loops and parametric studies in a reasonable computational time. 6

21 1. Introduction Chapter 5 concerns the transient modelling. It provides a description of all the new models developed to enhance and improve the simulation capabilities during transient phases of a propulsion system. Three new models are here implemented and discussed: a new injector plate model, a new formulation for the evaluation of the heat fluxes in combustion devices, finally a new model for the evaluation of thermal stratification on cooling channels is described. For each model a discussion of the derived equations is provided as well as test cases for validation purposes. Where available models simulations are compared with numerical test cases or experimental results. Chapter 6 describes the system level validation. It contains the main test case examined: the RL-10A-3-3A rocket engine which presents significant challenges due to its expander cycle configuration and its ignition settings. Modelling of the liquid rocket engine in the frame of its transient phases (start-up and shut-down) and the resulting simulation data are discussed. Chapter 7 concludes this thesis summarising the main achievements of the work performed and indicating those points which, in the frame of further work, could be improved and others which could be considered in a further extension of the models presented. 7

22 2. State of the Art 2.1. Engine cycles and their start-up and shut-down transients An engine cycle for turbopump-fed engines describes the specific propellant flow paths through the major engine components, the method of providing the hot gas to one or more turbines, and the method of handling the turbine exhaust gases. Depending of the propellant path considered we will have (among the main ones) a gas generator cycle, an expander cycle or a staged combustion cycle. Each system displays a specific sequence for start-up and shut-down phases. The sequences are tailored to fit general engine characteristics such as engine cycle, as well as more specific details such as ignition system type. A gas-generator engine will therefore have a different sequence to an expander cycle or a staged combustion engine Gas Generator Engine In the gas generator cycle the turbine inlet gas comes from a separate gas generator. Its propellants can be supplied from separate propellant tanks or can be bled off the main propellant feed system. This cycle is relatively simple; the pressures in the liquid pipes and pumps are relatively low (which reduces inert engine mass). It has less engine specific impulse than an expander cycle or a staged combustion cycle. The pressure ratio across the turbine is relatively high, but the turbine or gas generator flow is small (1 to 4% of total propellant flow) if compared to closed cycles. Alternatively, this turbine exhaust can be aspirated into the main flow through openings in the diverging nozzle section. This gas then protects the walls near the nozzle exit from high temperatures. Both methods can provide a small amount of additional thrust. The gas generator mixture ratio is usually fuel rich (in some 8

23 2. State of the Art engine it is oxidizer rich) so that the gas temperatures are low enough (typically 900 to 1350 K) to allow the use of uncooled turbine blades and uncooled nozzle exit segments [133, 94]. The European Vulcain and Vulcain 2 (see Figure 2.1) engines are two examples of gas generator cycle engines. The gas generator in these engines provides hot gases to power two turbines, one for each propellant pump. Figure 2.1.: Vulcain 2 schematic [46] Prior to commencing the start-up sequence the engines undergo a chill-down phase of all components with the exception of the main propellant valves and gas generator, to reduce thermal shock effects when the cold fuel is fed to the engine at start-up. This phase lasts ca. 2.5 hours. The engines are then started using a starter to move the turbine in order to provide a minimal amount of power to the pumps to feed the gas generator. A pyrotechnic igniter is then used to ignite 9

24 2. State of the Art the gas generator. Given the now full power available to the turbines to run the pumps, feeding of the main combustion chamber is possible. The fuel valves are opened first, the regenerative cooling channels fill with fuel which then irrupts into the combustion chamber which, with aid of another pyrotechnic igniter, is ignited under lean conditions. The combination of the fuel rich condition at ignition and the cooling performed by the regenerative cooling system increases the complexity of the ignition process of the main combustion chamber [91] Expander Engine In the expander cycle most of the engine coolant (usually the fuel) is fed to lowpressure-ratio turbines after having passed through the cooling jacket where it picked up energy. Part of the coolant, perhaps 5 to 15%, bypasses the turbine and rejoins the turbine exhaust flow before the entire coolant flow is injected into the engine combustion chamber where it mixes and burns with the oxidizer. The primary advantages of the expander cycle are good specific impulse, engine simplicity, and relatively low engine mass. In the expander cycle all the propellants are fully burned in the engine combustion chamber and expanded efficiently in the engine exhaust nozzle [133, 94]. The American RL-10 engine and the European Vinci engine (currently under development) represent two examples for this type of cycle (see Figure 2.2). The latter is one such expander engine which implements a regenerative cooling system based on hydrogen to cool the combustion chamber. Hot gases generated in an electric igniter are implemented as a pilot flame to start the main combustion chamber in which the temperature progressively increases as stable combustion is established. The liquid hydrogen from the tanks leaving the cooling system undergoes a phase change from liquid to supercritical during the ignition sequence and thus powers up the turbines as it gains velocity and pressure. For these very reasons, the Vinci engine unlike other European engines implements reaction rather than impulsive turbines [91, 38]. The use of LH 2 and LOX as propellants in the RL-10 engine, makes necessary a chill-down step to avoid propellant boiling and pump cavitation during operation. Cavitation must be avoided due to its propensity to cause erosion damage to the 10

25 2. State of the Art Figure 2.2.: Vinci schematic [45] blades. When the propellants are used for chill-down, the pre-start valves are opened to allow flow through the pumps and the tank head pressure provides the driving force. The LOX is sent through its entire circuit and fed into the main chamber. The LH 2, however, is flowed out through the cooldown valves prior to going through the regenerative cooling tubes. This will allow the cooling tubes to remain at a significantly higher temperature for start-up and avoid any possibility of inadvertent propellant mixing and igniting prior to start-up. When the engine start command is given, the cooldown valves are closed and the main fuel valve is opened. This allows for the LH 2 to flow through the regenerative cooling lines, which heat up and vaporise the fuel. Once the gaseous hydrogen begins to drive the turbine, the pumps take over providing the pressure gradient for flow. The main chamber is ignited using a pilot flame created from gaseous propellants tapped off from the main lines. One the combustion chamber is lit, additional heat is transferred into hydrogen flow form combustion process. This drives the turbine to its nominal power and eventually the engine reaches a steady state condition. When the shut-down command is issued, the main fuel shut-off valve is closed 11

26 2. State of the Art tu cut-off fuel flow into the main chamber. As a result, combustion in the combustion chamber and thrust are rapidly terminated. The prestart valves are closed to stop the propellant flow and the cooldown valves are opened to prevent over pressure of the engine due to trapped hydrogen Staged Combustion Engine In the staged combustion cycle, the coolant flow path through the cooling jacket is the same as that of the expander cycle. Here a high-pressure precombustor (gas generator) burns all the fuel with part of the oxidizer to provide high-energy gas to the turbines. The total turbine exhaust gas flow is injected into the main combustion chamber where it burns with the remaining oxidizer. This cycle lends itself to high-chamber-pressure operation, which allows a small thrust chamber size. The extra pressure drop in the precombustor and turbines causes the pump discharge pressures of both the fuel and the oxidizer to be higher than with open cycles, requiring heavier and more complex pumps, turbines, and piping. The turbine flow is relatively high and the turbine pressure drop is low, when compared to an open cycle. The staged combustion cycle gives the highest specific impulse, but it is more complex and heavy [133, 94]. A variation of the staged combustion cycle is used in the Space Shuttle Main Engine (SSME) (see schematic in Figure 2.3). The engine assembly consists of the engine powerhead and main combustion chamber/nozzle assembly. The powerhead uses two preburners, a main injector, and an oxidizer heat exchanger, all welded into the hot gas manifold. The preburners generate fuel-rich combustion gases to drive the LOX and LH2 turbopumps. Hydrogen fuel is also used to cool the main chamber and nozzle. Prior to starting the SSME engine, the start preparation phase takes place. This consists of purging and thermal conditioning followed again by purging. During the first purging phase, dry nitrogen and dry helium are used to remove moisture as well as air which would otherwise freeze along the oxidiser and fuel lines respectively. After thorough purging has been performed, thermal conditioning is undertaken by allowing propellants to flow into the engine down to the main fuel valve (MFV) on the LH 2 side and down to three oxidiser valves on the LOX side, i.e. the main oxidiser valve (MOV), the oxidiser preburner valve (OPOV) and fuel 12

27 2. State of the Art Figure 2.3.: Space Shuttle Main Engine schematic [149] preburner oxidiser valve (FPOV). LH 2 and LOX recirculation flows, through bleed valves, are maintained for about one hour to chill the four turbopumps down to cryogenic temperatures and to eliminate gas pockets in the propellant feed system. Once thermal conditioning is completed, a final dry helium purging of the fuel lines downstream the MFV is performed [78, 12]. At t = 0 s the start command is sent and the MFV is opened completely and the three spark igniters are provided with electrical power. The three main oxidiser valves, MOV, FPOV, and OPOV are then regulated in order to reach the target priming times for the new preburners and main combustion chamber (MCC). These are precisely selected to ensure a stable ignition process and are at 1.4 s for the fuel preburner (FPB), 1.5 s for the MCC, and 1.6 s for the oxidiser preburner (OPB). Pressure oscillations in the fuel system, arising form thermodynamic instabilities during the expander-cycle-like start-up, are closely monitored and the FPOV and OPOV positions are controlled to avoid high mixture ratios in the preburners which result in dangerously high temperatures for the turbines. The FPB is ignited during the second fuel system pressure dip and is followed by ignition of the PFB. At 1.25 s the rotational speed of the high pressure fuel 13

28 2. State of the Art Figure 2.4.: Space Shuttle Main Engine start-up sequence [12] turbopump is checked to ensure that hydrogen can be bumped through the system against the back-pressure created by the MCC oxidiser priming. As the drive power of both high pressure turbines increases, the chamber coolant valve (CCV) is throttled down to 70%. This conditon is maintained until 2.4 s at which point the control system measures the MCC pressure and regulates the OPOV, FPOV, and CCV in order to follow the pre-programmed chamber pressure ramp until the nominal operational point has been reached. Finally the FPOV is regulated to adjust the fuel mass flow rate until the nominaol mixture ratio is obtained. At 5 s stable operation has been achieved. During the shut-down phase (see Figure 2.5), the main goal of which is to ensure a safe and as quick as possible shut-down of the engine, the OPOV is the first valve to be closed with a closing rate not higher than 45%/s to avoid a too abrupt thrust decay which would endanger the orbiter s structural integrity. Closing of the FPOV follows. Positioning of both valves is monitored to maintain a low mixture ratio and maximum oxidiser pressure decay whilst avoiding 14

29 2. State of the Art Figure 2.5.: Space Shuttle Main Engine shut-down sequence [12] back-flow of hot gases into oxidiser lines. In order to compensate for the increased heat loads due to throttling, the chamber coolant valve is regulated to force more coolant into the main combustion chamber. Simulataneously the MOC is closed at a control rate to ensure a combusiton chamber pressure above the inlet turbine pressures. Finally after 1 s of additional MFV opening time to assure a very fuel-rich shut-down, the MFV and the CCV are closed [12, 91] Modelling: review of previous works The development of software tools for analyses of rocket engine systems is critical to the successful design and fine tuning of such systems. Typically, the ignition process of a liquid rocket engine involves non-linear interactions between multiple engine components with phenomena such as flow resistance, off-design turbopump operation, heat transfer, phase change, and combustion. Furthermore, the physical properties of liquid propellants and combustion products in such systems vary widely and in a rapid way. Developing tools for predicting the dynamical 15

30 2. State of the Art behaviour of an engine with such characteristics is a challenging but important task for engineers and researchers. Worldwide, various tools have been developed to simulate the transient behaviour of rocket engine systems. In 1990, Ruth et al. [124] developed the Liquid Rocket Transient Code (LRTC) the in-house code of The Aerospace Corporation. It represents one of the first attempts to simulate propulsion systems with a modular structure. LRTC models an engine through a modular scheme with the method of characteristics for a flow through line segments (pipes) connected by nodes (zero-dimensional components such as valves, orifices, pumps, branches etc.). Comparisons with Titan IV K-01 flight data of the Stage I start transient demonstrated general agreement. The Rocket Engine Transient Simulator (ROCETS) [13] was designed and developed during the 90s by Pratt & Whitney for NASA-MSFC; it allows for costeffective computer predictions of liquid rocket engine transient performance. The most popular application of ROCETS is the RL-10A-3-3A rocket engine [13, 14, 15], varying from start transient analysis to modelling of thrust increase with densified propellants [59, 58]. Another powerful tool created in the United States is the Generalized Fluid System Simulation Program (GFSSP) for modelling cryogenic fluids in a complex flow circuit [80]. Recently, other researchers and engineers have developed (or have started to) other codes, for the transient analysis of propulsion systems, but their work was mainly focused to only a part of an engine system [7, 22, 31]. In Japan, during the 90s, a quasi-steady simulation code for transient analysis of the original LE-7 engine was developed [70]. This was Japan s first attempt to develop a staged combustion cycle engine, and establishing a safe and reliable start-up and shut-down method was very important. Later in 2002, the Visual Integrated Simulator for Rocket Engine Cycle (VIS- REC) [6] was developed by Mitsubishi Heavy Industries. VISREC is a one dimensional flow and heat analysis program using the lumped parameter approach. It 16

31 2. State of the Art can analyse start-up and shut-down transient behaviour of many types of engine cycles such as expander, staged combustion, and gas generator. Together with the LE-7A rocket engine, the Rocket Engine Dynamic Simulator (REDS) [152] in 2004 was developed and applied to start-up and shut-down transient analyses. In China, in 2000, Kun et al. [77] have developed a tool to study rocket engine system transient based on the disassembly method whose principles are the following: the modules have independent physical function and mathematical model; there are uniform parameters exchange interfaces between each module, and finally, the engine system can be disassembled into modules by practical physical units. Also in India, during the 2000s, a first try to develop a dynamic simulator for liquid-propellant rocket engines has been accomplished; this tool is called CRESP-LP [134]. In Iran, only recently they have transient simulation tools performed by the University of Technology in Tehran. Prof. Karimi et al. have developed their own code and performed several analyses to study transient regime in rocket engines [71, 72, 73, 119] In Russia, extensive studies have been performed on engine transient behaviour. These studies have taken advantage of the extensive database generated during testing of the wide range of liquid rocket engines that Russia has developed. The problem of transient phase in liquid rocket engine systems has been studied in [68, 136]. The transient analysis of liquid rocket engine is illustrated in great detail in [10]. A simple ordinary differential equation (ODE) approach is presented and complemented with experimental-based empirical equations in those cases where ODEs are not sufficient to describe the occurring phenomena. Finally in Europe, CNES tested a dedicated library for the modelling and simulation of rocket engine system dynamics developed in the AMESim platform [121]. The platform carries multiple sub-systems of a rocket engine, such as tanks, pneu- 17

32 2. State of the Art matic lines, turbopump, regenerative circuit, combustion chamber, and starters. In cooperation with ONERA, CNES has also developed another tool called Carins [102, 84], an open platform featuring the symbolic manipulation method to simulate the transient behaviour of propulsion systems. The Astrium SMART code has been developed for the EPS start-up simulation [74] and recently, from 2008 a complete set of models able to simulate liquid propulsion system components called European Space Propulsion System Simulation (ESPSS) has been developed by a joint European team in the frame of a GSTP Programme for the European Space Agency [32]. 18

33 3. ESPSS: European Space Propulsion System Simulation This chapter documents the models of the propulsion system library implemented within the existing analysis software EcosimPro [40], used as a basis for this Ph.D. research. EcosimPro is an object-oriented visual simulation tool capable of modelling various kinds of dynamic systems represented by differential-algebraic equations (DAE) [18] or ordinary-differential equations (ODE) and discrete events. The modelling of physical components is based on the EcosimPro language (EL), an object-oriented programming language which is very similar to other conventional programming languages but is powerful enough to model continuous and discrete processes. It can be used to study both stationary states and transients. EcosimPro employs a set of libraries containing various types of components which can be interconnected to model complex dynamic systems: control math mechanical ports_lib thermal The European Space Propulsion System Simulation (ESPSS) consists of multiple libraries to represent a functional propulsion system, e.g. fluid properties, pipe networking including multi-phase fluid flow, two-phase two fluids tanks, non-adiabatic combustion chambers, chemistry, turbomachinery, etc: 19

34 3. ESPSS: European Space Propulsion System Simulation fluid_properties fluid_flow_1d comb_chambers tanks turbo_machinery The Libraries sections hereafter describe those libraries [43], focussing on their physical modelling, and on the main models used for this Ph.D thesis Fluid Properties Library fluid_properties is an EcosimPro library in charge of the calculation of fluid properties. The functions available on this library are mainly used by the fluid_flow_1d library for the simulation of fluid systems. The most important features are summarized as follows: Most of the fluids used for rockets applications are available Fluids are supported in different categories depending on the type used: - Perfect gases (transport and heat capacity properties obtained from CEA polynomials (temperature dependent)). Only used in the combustor/nozzle components - Perfect gases (transport and heat capacity properties interpolated from tables (temperature dependent)) - Simplified liquids interpolated from tables (temperature dependent) - Real fluids interpolated from tables considering either liquid, superheated, supercritical or two-phase flow (temperature and pressure dependent) - User-defined fluids are available. The properties must be defined in external data files and can be of any of the last three types previously mentioned 20

35 3. ESPSS: European Space Propulsion System Simulation Mixtures of a real fluid with a non-condensable gas are allowed. The homogeneous equilibrium model is used to calculate the properties (quality, void fraction, etc) in case of two phase flow. Mixtures of two real fluids are not allowed. Therefore, phenomena such as fractional distillation are not modelled. The fluid_properties library does not contain any component. It only provides a large collection of functions returning the value of a fluid property (or the complete thermodynamic state) by introducing relevant parameters (i.e. temperature, internal energy, pressure, density, heat transfer and friction correlations etc) Perfect Gas properties according to CEA The programming is based on the perfect gas state equation. The expressions used are summarized as follows: P = ρ R T MW Z = 1; β = 1/T ; κ = 1/P The expressions used for the energy calculation are based on the computation of the specific heat at constant pressure for ideal gases (Cp 0 ) as a function of temperature only (by means of polynomial expressions). The expression proposed was obtained from a very large database providing data for a very wide range of 21

36 3. ESPSS: European Space Propulsion System Simulation temperatures (between 200 and K) and is summarized as follows: Cp R = a 1T 2 + a 2 T 1 + a 3 + a 4 T + a 5 T 2 + a 6 T 3 + a 7 T 4 (3.1) T H = H(T 0 ) + Cp(T )dt = T ( 0 R T a 1 T 2 + a 2 T 1 T ln T + a 3 + a a T a T a T b ) 1 T (3.2) T Cp(T ) S = S(T 0, P 0 ) + dt R T 0 T MW log( P ) = R P 0 MW ( P log R ( a 1 T 2 a 2 T 1 T 2 + a 3 ln T + a 4 T + a 5 P a T a T b )) 2 T (3.3) The functions giving the viscosity and the thermal conductivity in case there are data available (mainly from NIST species database [88], in this case the NIST database is not as extensive as for the thermodynamic properties) are also based on polynomial expressions. The viscosity and thermal conductivity functions have respectively the following form: ln µ = A 1 ln T + B 1 T + C 1 T 2 + D 1; ln λ = A 2 ln T + B 2 T + C 2 T 2 + D 2 Otherwise the properties are estimated as follows: Viscosity: There are different estimation methods available. The approach used is subjected to the availability of property data: critical properties and the dipole moment. In this case the expression used [120] is: µ = [ T r exp( T r ) exp( T r ) ] F o P F o Q ζ 10 7 where T r is the reduced temperature computed as follows T r = T/T c, and FQ o are correction factors and ζ is the reduced inverse viscosity, calculated as follows: ( T c ζ = MW 3 Pc 4 ) 1/6 22

37 3. ESPSS: European Space Propulsion System Simulation F o P is a correction factor that mainly depends on the polarity of the molecule. It is computed as follows: F o P = 1 0 µ r F o P = (0.292 Z c) µ r F o P = (0.292 Z c) (T r 0.7) µ r F o Q is a correction factor used only in quantum gases. In the present case its value is 1. µ r is the relative dipole moment computed as follows: µ r = µ2 P c T 2 c where µ is the dipole moment in Debyes. In case critical properties and dipole moment are not available, other estimations must be done based on quantum formulation. The following expression will be used: µ k = η ns MW k T Ω k 10 7 where η ns is a constant ( in S.I.) and Ω k Thermal Conductivity: Ω k = ln ( 50 MW 4.6 ) k T 1.4 Similarly to viscosity, the thermal conductivities that are not available are estimated. The approach used is summarized as follows: λ k = µ k R ( (Cp k /R 2.5) ) MW k Perfect Gas interpolated properties For the state equation, the same expressions as in Chapter are used. The expressions used for the energy calculation are based on the table interpolation of 23

38 3. ESPSS: European Space Propulsion System Simulation the specific heat at constant pressure (C 0 p) as a function of temperature only: S = S(T 0, P 0 ) + H = H(T 0 ) + C p R = C p R (T ) T T T 0 C p (T ) T 0 T C p (T )dt dt R MW log(p/p 0) The functions giving the viscosity and the thermal conductivity are also interpolated from the external user-defined property file as a function of temperature Simplified Liquid interpolated properties For simplified liquids, the formulation is based on the tables where, density, sound speed and specific heat are interpolated as functions of the temperature. Thus, the volumetric expansivity can be obtained as follows: β = 1 ρ dρ dt P =const This derivative is calculated numerically. Then, the following others thermodynamic derivatives can be calculated: C p C v = 1 + T β 2 νsound 2 /C p κ = 1 dρ C p = ρ dp T =cte ρ νsound 2 C v Once these properties have been interpolated, the equation of state can be applied assuming constant compressibility with pressure: ρ(p, T ) = ρ(t ) [ 1 + κ (P P ref ) ] The enthalpy is calculated integrating numerically the Cp: H = H ideal (T 0 ) + T T 0 Cp ideal (T )dt 24

39 3. ESPSS: European Space Propulsion System Simulation Viscosity and Thermal conductivity are also interpolated from the external data of the property file as a function of temperature Real Fluids interpolated properties For real fluids, FORTRAN functions will perform special searching techniques in 2-D property tables to interpolate the desired property in the nearest cell of the data tables. A single reading will be done the first time that a function s call of any property is made. The functions will identify if a certain fluid is liquid, vapour or two-phase flow by giving a pair of variables that can be ρ-t, ρ-u, s-h, P-H, P-T, etc. With the exception of two-phase flow or in the case of table extrapolation, no special hypothesis concerning the Equation of State and the properties has been done: all the properties are interpolated using the data tables of the properties file. Under two phase conditions the quality of the mixture is calculated from the saturation properties of the liquid and steam phases. Knowing the mixture (vapour/liquid) density and energy, the following two equations are used: quality = x = u u liq u vap u liq 1/ρ = 1/ρ liq + x (1/ρ vap 1/ρ liq ) An iterative process in pressure is needed. For each iteration, the saturation conditions will be calculated and the pressure fulfilling the two previous equations can be found. The void fraction is calculated as follows: void fraction = α = V vap /(V vap + V liq ) = (ρ liq ρ)/(ρ liq ρ vap ) The transport properties and the heat capacity in two-phase conditions are calculated in a simple way: µ = x µ vap + (1 x)µ liq (3.4) λ = x λ vap + (1 x)λ liq (3.5) Cp = x Cp vap + (1 x)cp liq (3.6) 25

40 3. ESPSS: European Space Propulsion System Simulation Nevertheless, the liquid and vapour saturation values of the transport properties, together with those of the heat capacities, densities and enthalpies (latent heat) will be returned for the calculation of other two-phase properties as the sound speed and the film coefficient. Under one-phase conditions the sound speed is directly given by the FORTRAN function interpolations. The returned value is equivalent to the following expression that uses other properties (Cp, β, κ) also returned by the properties function: C p v sound = ρκc p β 2 T (3.7) Under two-phase conditions the sound speed must be calculated. The equilibrium sound speed presents discontinuities at phase changes. In order to ensure system robustness the following approach is given by Wallis (1969) [147]: 1/v 2 sound = (αρ vap + (1 α)ρ liq )(α/ρ vap /v 2 sound,vap + (1 α)/ρ liq/v 2 sound,liq ) (3.8) Another frozen [28, 8] sound speed expression can be used instead, which is also continuous with the sound speed at phase changes: dp dt = v 2 sound = where: x Cp v + (1 x)cp l T (xβ v ν v + (1 x)β l ν l ) x ( ɛ Cp v T dp dt ν v ν : specific volume; T ( dp v ) 2 dt x: quality ( (1 + ɛ) βv κ v dp dt )) + (1 x) ( ɛ Cpl T dp dt ν l ( (1 + ɛ) βl κ l dp dt )) ɛ = 0 frozen sound speed ɛ = 1 equilibrium sound speed Sub indexes v and l indicate vapour and liquid saturated conditions. 26

41 3. ESPSS: European Space Propulsion System Simulation Perfect gas mixtures Perfect gas mixtures are calculated with linear mixing rules assuming the same temperature for all the constituents. x k = ρ k nchem k=1 ρ k ; nchem R T P k = ρ k ; P = MW k k=1 P k where, R is the gas constant = [J/kmol K], MW k is the molecular weight of the chemical constituent k, T is the mixture temperature [K], ρ k is the density of the chemical constituent k and x k is the mass fraction of the chemical constituent k. The molecular weight of the mixture MW mix and the molar fractions y k are calculated as follows: nchem 1 x k = ; MW mix MW k k=1 y k = molar fraction = x k MW mix MW k The energy properties are computed as follows: H = C p = nchem k=1 nchem k=1 x k Cp k (T ); C v = x k H k (T ); S = nchem k=1 The sound speed is calculated as follows: γ = nchem k=1 x k S k (T ) + x k C v,k (T ); nchem k=1 y k ln y k (T ) C p C p R/MW mix (3.9) v sound = γ R T/MW mix (3.10) Regarding the transport properties, the computation of the mixture viscosity is 27

42 3. ESPSS: European Space Propulsion System Simulation computed as follows [64]: µ mix = nchem i=1 y i + y i µ i nchem j=1 j i y j φ ij (3.11) where φ ij is the interaction parameter estimated with the following formulation: φ ij = 1 4 [ 1 + ( µi µ j ) 0.5 ( ) ] ( MWi 2MW j MW j MW i + MW j ) 0.5 Similarly to viscosity, the thermal conductivity for mixtures is computed as follows: λ mix = nchem i=1 y i + y i λ i nchem j=1 j i y j ψ ij (3.12) The interaction parameter for the thermal conductivity is based on the one computed for viscosity. The expression is as follows: ψ ij = φ ij [ (MW ] i MW j ) (MW i MW j ) (MW i + MW j ) Real Fluid - Perfect gas mixtures The field of mixture of flows has been and is still today the subject of intensive research. Various models exist in the literature that represent with variable accuracy the chemico-physical phenomena that occur in mixed flows. There are mainly two different mathematical formulations of mixed flows : The two-fluid models, where equations are written for mass, momentum and energy balances for each fluid separately. The mixture models, where equations for the conservation of physical properties are written for the two-phase mixture. Mixture models have a reduced number of balance equations compared to twofluid models, and may hence be considered as simplifications in terms of math- 28

43 3. ESPSS: European Space Propulsion System Simulation ematical and physical complexity. However, some mixture models, like the Homogeneous Equilibrium Model (HEM) [27] that is used within this work, are still of significant interest. The HEM formulation has indeed multiple advantages with respect to the other models : its convective part is unconditionally hyperbolic, which is not the case for other models; it is very similar to the Euler equations, thus it can benefit of all the numerical studies made for Euler equations; we do not need to derive nor to implement the mass, momentum and energy transfer between phases, as they cancel each other out in the HEM mixture formulation. This is why this very simple formulation is still worth being used, despite its inherent limitations. The most important restriction is that the flow should be in equilibrium, or at least close to it, in order for the HEM formulation to approximate correctly the physical behaviour of the flow. By equilibrium one means here that both phases have the same velocity, pressure and temperature. The homogeneous equilibrium model of a mixture of two fluids is then applied. One fluid must be real or a simplified liquid, and the other a perfect non-condensable gas (ncg). Assuming that: The Real Fluid in possibly subcooled liquid, saturated or superheated vapour state and the ncg form a homogeneous mixture with a uniform temperature. The Real Fluid, if present, occupies the entire volume. ncg, if present, occupies the same volume as the Real Fluid vapour according to the Gibbs Dalton Law. If ncg and Real Fluid Liquid are present, the Real Fluid vapour is saturated (relative humidity is equal to 1) If ncg is present, the Real Fluid liquid conditions are the subcooled conditions corresponding to P and T. (Liquid phase pressure = P vap + P ncg ), 29

44 3. ESPSS: European Space Propulsion System Simulation The ncg gas is insoluble in the Liquid Phase of the Real Fluid. There can be no ncg if the volume is filled with the liquid phase of the Real Fluid The following state equations are involved in presence of liquid: ρ liq, u liq = f state (fluid, P, T ) where P = P nc + P vap (subcooled conditions) ρ vap, u vap = f sat (fluid, P, T ) where P vap = f sat (fluid, T ) (saturated conditions) u is the internal energy. Subscript nc denotes the non-condensable fluid. f denote the corresponding pure fluid functions. In this system of equations, P nc and T are unknowns. Assuming that the volume density, ρ, the non-condensable mass fraction, x nc, and the mixture energy, u, are known, the following closing equations allow the calculation of the homogeneous temperature and the non-condensable pressure: ρ nc α = ρ x nc where ρ nc = f state (fluid nc, P nc, T ) u = (1 x nc )(x u vap + (1 x)u liq ) + x nc u nc Applying the definitions of the void fraction (α = V g / V tot ) and quality (x = M vap / (M vap + M liq )), it is possible to find an expression for the quality appearing in the equation above as a function of densities: α = (ρ liq ρ cond )/(ρ liq ρ vap ) x = α ρ vap /(ρ liq α(ρ liq ρ vap )) The variable ρ cond = (1 x nc )ρ refers to the condensable mass (liquid & vapour) divided by the total volume. The non-condensable density ρ nc and the vapour density are referring to the gas volume and not to the total volume, so ρ nc x nc ρ. For the sake of clarity, it can be seen that the last two equations are identities introducing the definition of: V g = V vap = V nc 30

45 3. ESPSS: European Space Propulsion System Simulation V tot = V g + V liq α = V g / V tot x = M vap / (M vap + M liq ) ρ = (M nc + M liq + M vap ) / V tot ρ cond = (M vap + M liq ) / V tot ρ nc = M nc / V g ρ liq = M liq / V liq ρ vap = M vap / V g The heat capacity, viscosity and thermal conductivity are calculated as a mixture of a liquid and a composed gas (the vapour and the non-condensable gas). The mixture properties are calculated in a simple way (weighing the pure fluid properties with the mass fractions): Cp = x mix Cp gas + (1 x mix )Cp liq (3.13) µ = x mix µ gas + (1 x mix )µ liq (3.14) λ = x mix λ gas + (1 x mix )λ liq (3.15) where the mixture quality (x mix ) is defined as the mass ratio of gas (vapour + non condensable) x mix = α ρ gas /(ρ liq α(ρ liq ρ gas )) The void fraction α has the same meaning than in a pure two-phase fluid, i.e. the gas volume divided by the total fluid volume. The gas mixture properties are calculated as follows: ρ gas = ρ vap + ρ nc Cp gas = (ρ vap Cp vap + ρ nc Cp nc )/ρ gas 31

46 3. ESPSS: European Space Propulsion System Simulation Similarly, the gas mixture transport properties and sound speed are calculated as follows: µ gas = (ρ vap µ vap + ρ nc µ nc )/ρ gas λ gas = (ρ vap λ vap + ρ nc λ nc )/ρ gas v 2 sound,gas = ρ gas/(ρ vap /v 2 sound,vap + ρ nc/v 2 sound,nc ) All individual properties have been computed with pure fluid functions. The sound speed is approximated as an equivalent two-phase mixture where the vapour phase is in fact a mixture of a non-condensable fluid with 100% of humidity: 1/v 2 sound = (αρ gas + (1 α)ρ liq )(α/ρ gas /v 2 sound,gas + (1 α)ρ liq/v 2 sound,liq ) 3.2. Fluid Flow 1D Library fluid_flow_1d is an EcosimPro library for 1-D transient simulations of two-fluid, two-phase systems. The most important features are the following: The conservation equations include gas, liquid and two-phase flow regimes for ideal or real fluids. The working fluid(s) can be easily selected from a large collection of fluids included in the fluid_properties library. The fluid phase will be automatically calculated. The homogeneous equilibrium model is used to calculate a real fluid under two phase conditions with or without a non-condensable gas mixture. Absorption/desorption is not yet considered. Flow inversion, inertia, gravity forces and high speed phenomena are considered in pipes, volumes and junctions, the pipes also incorporating an area-varying non-uniform mesh 1-D spatial discretisation into n (input data) volumes. Calculation of concentrated (valves) and distributed (pipes) load losses including two-phase wall friction correlations. 32

47 3. ESPSS: European Space Propulsion System Simulation Heat transfer between the walls and the fluid. Multiple thermo-hydraulic correlations and initialization options are included. Other special components such as check valves, pressure regulators, heatexchangers and Tees (for convergent and divergent flows) are available. 1-D Pipe flows can be simulated using some robust and accurate numerical techniques upwind (Roe) or centred schemes. Hydraulic or pneumatic systems where the heat transfer or system controls are coupled will be easily evaluated with the fluid_flow_1d library. Cavitation and priming phenomena under two-phase flow (with or without a non-condensable gas travelling in a liquid) will be calculated in pipes or other components. Besides, fluid_flow_1d will permit to analyse in great detail transient aspects due to inertia (water-hammer) and bubble collapse (priming) Components Classification The components of the fluid_flow_1d Library are listed in Figure 3.1 below, where the inheritance hierarchy is shown. In an ESPSS fluid network, every component is either a resistive component or a capacitive component. A resistive component receives the state variables (pressure, density, velocity, chemical composition and enthalpy) as input and gives back the flow variables (volumetric, mass and enthalpy flows) as output. A capacitive component receives the flow variables as input and gives back the state variables at output. To build a fluid network, the user must connect resistive components to capacitive ones, alternatively. So, from a computational point of view, components are divided into two classes: C (capacitive) elements, integrating the mass and the energy conservation equations. The thermodynamic functions will be used to calculate the complete thermodynamic state M (momentum) elements, calculating explicitly (inertia terms) the mass flows between C elements. Reverse flow is allowed This computational scheme prevents the appearance of algebraic loops and high index DAE (Differential Algebraic Equations) in the mathematical model of the pipe network. 33

48 3. ESPSS: European Space Propulsion System Simulation Figure 3.1.: Components in the fluid_flow_1d library The existing components of the fluid_flow_1d library have the following types: Volumes and Heat-exchangers components are C elements. Junctions, Valves, Filter and Jun_TMD components are M elements. Bound components (VolPT_TMD, VolPx_TMD, etc) are also C elements. (Here, TMD means time dependant). VolPsTsVs_TMD and VolPsTs_TMD are M elements because they calculate mass flow. In TMD elements, state variables are imposed in the experiment file (fixed or depending on the time) or by means of control library components connected to its control ports. Pipe, Tube, HeatExchanger and Nozzle components are 1-D models with dedicated numerical schemes comparable to a C element. Pipe_res and ColdThruster components are 1-D models with dedicated numerical schemes comparable to an M element. The ColdThruster incorporates an internal valve component. Others topological components such as the Tee component are M elements because they internally finish in junctions, even if they have some internal 34

49 3. ESPSS: European Space Propulsion System Simulation C components. The graphical symbols of the components provide information about the kind of computational element to which each port is connected. Ports belonging to a C element have a small dot in the middle of the arrow while ports belonging to an M element are just represented by the arrow (see Figure 3.1). It is noted that the Tube and Pipe components can simulate an area-varying non-uniform 1-D mesh, as it is the case for the ColdThruster and Nozzle components, even though, for simplicity, the symbol graphical representation does not indicate this capability on Pipes and Tubes components Junction/Valve This component represents a junction. It is a basic component where no mass accumulation is considered. The mass and enthalpy flows of the inlet and outlet ports are equal (no mass accumulation in junctions): ṁ 1 = ṁ 2 = ṁ ṁ 1,nc = ṁ 2,nc = ṁ nc ṁh 1 = ṁh 2 = ṁh where indexes 1 and 2 refer to the connected fluid ports. For each port, ṁ, ṁ nc, and ṁh are the total mass, non-condensable mass and the enthalpy flows respectively. These flows are calculated here taking into account the flow direction, so different temperatures and densities may exist at both sides of a junction. The following momentum balance equation dynamically calculates the mass flow per unit of area: ( (I 1 +I 2 ) A dg ) dt + GdA +lv dg dt dt = (P +0.5ρv2 ) 1 (P +0.5ρv 2 ) 2 0.5(ζ+ζ crit ) G G ρ up (3.16) where, P 1, P 2 are the static pressures at port 1 and 2, calculated by the connected volumes (0.5ρv 2 ) 1 2 represent the dynamic pressures at port 1 and 2, calculated by the connected volumes. ρ and v are the mean density and speed at the connected volumes; I 1, I 2 represent half inertia of the connected pipe ends 1 and 35

50 3. ESPSS: European Space Propulsion System Simulation 2, respectively. A is the instantaneous valve cross section; lv = sqrt(a ref ), this term (valve inertia) is very small but makes the equation no singular if A = 0. G is the mass flow per unit of area, ζ is pressure drop coefficient and ρ up is the upstream gas/liquid mixture density. The mass flow will be calculated as ṁ = GA. The use of G (mass flow per unit or area) instead of ṁ (mass flow) allows a complete closing (A = 0) of the valve without making the system of equations singular. The pressure drop contribution is quadratic with mass flow: 0.5(ζ +ζ crit )G G /ρ up. This term would make the momentum equation singular at zero flow because very small perturbations in pressure lead to non-negligible variations in mass flow ( G/ P ). Physically, what is happening is that pressures losses are linear with mass flow for laminar regimes. To account for that, the quadratic term G G is linearised for G < G lam in this way: k(g)g G < G lam G G = G 2 sign(g) G > G lam G lam = µre lam A; k(g)g 0 G lam k(g) is a smoothing factor to assure continuous transitions between laminar to turbulent regimes. µ is the upstream viscosity calculated by the connected volumes, Re lam is the minimum Reynolds number set at 2000 as default value. The sonic flow limitation is taken into account by adding a correction factor to the pressure loss coefficient, ζ crit, which limits the mass flow per unit area to be less than or equal ( ) to the critical flow per unit area. First, the flow under steady state conditions is obtained by cancelling the derivatives in the previous momentum equation: G st = 2ρ up [(P + 0.5ρv 2 ) 1 (P + 0.5ρv 2 ) 2 ] The added term is calculated in such a way that if the flow attempts to be greater than critical flow, the following extra term will limit the flow to the critical value: 36

51 3. ESPSS: European Space Propulsion System Simulation ζ crit = max ( (G st /G crit ) 2 ζ, 0 ) where G crit is the critical (sonic) flow per unit of area (ρc) crit calculated by the capacity components (see section 3.2.3) connected by the junction Capacity/Volume The Capacity component simulates a volume with several fluid ports named f[j]. It s the basic capacitive component containing the mass and energy conservation equations for this type of components. Here below the general equations for a non-adiabatic variable volume. It is assumed that the mixture (non-condensable plus main fluid in liquid, gas or two phase conditions) has an homogeneous temperature. Mass conservation V dρ dt + ρ dv dt = j P orts ṁ j (3.17) Non-condensable mass fraction x nc conservation ρ V dxnc dt Energy conservation ( + x nc V dρ dt + ρ dv ) = ṁ nc j (3.18) dt j P orts V dρ dt u + ρ dv du u + ρv dt dt = j P orts (ṁh) j + Q P dv (3.19) u = total specific energy = u st + v 2 /2 (3.20) where ρ, x nc and u are the fluid mixture (including two phase flow) density, the non-condensable mass fraction and the total energy respectively; ṁ j, ṁ nc j and ṁh j are the mass and enthalpy flows at port j calculated at the connected resistive type components (see section 3.2.2). V is the volume, which can change with time. Assuming that the volume V and its rate of change are known, previous conservation equations enable to calculate the derivatives of the mixture density, 37

52 3. ESPSS: European Space Propulsion System Simulation mixture energy and non-condensable mass fraction. These variables can be integrated, so they are known at any time. Assuming thermodynamic equilibrium, the conservation equations are always valid even if the fluid conditions are liquid, vapour or homogeneous two phase flow. Then, the complete thermodynamic state (partial pressures, temperature, quality...) can be calculated using the pure fluid thermodynamic routines or the homogeneous equilibrium model for mixtures of a non-condensable gas plus a real fluid: FL_state_vs_ru Volume average velocities are required for the total energy conservation equation, the evaluation of the wall frictional forces and of the wall heat transfer. For the calculation of the average velocity, volumes are considered to have two sides, side 1 and side 2. The total mass flow rate entering the volume at side 1 and 2 is: ṁ 1,in = ṁ j,in ; ṁ 2,in = ports in side1 ports in side2 ṁ j,in Port mass flows can be positive or negative. It is defined as positive when entering the volume. The average velocity in the volume is defined as: v = ṁin,1 ṁ in,2 2ρA where ρ is the average density in the volume, and A is the cross area of the volume. This average volume velocity is transmitted to the ports. The effective port velocity will be multiplied by the cosine of the port angle α because the lateral velocities do not compute in the total pressure: v(j) = v cos(α j ) The term Q appearing in the energy conservation equation of the capacity permits the exchange of heat through a thermal port. The walls (that can be represented by thermal components) are not included in this component: Q = h film A wall (tp.t T ) where tp is the name of the thermal port (with one node) connected to the 38

53 3. ESPSS: European Space Propulsion System Simulation Volume. tp.t behaves as the internal wall temperature, to be determined in the connected thermal component. The film coefficient is calculated using empirical correlations Tubes/Pipes These components simulate area-varying non-uniform mesh high resolved 1D fluid veins that exchanges heat with a 1D thermal port. They incorporate the 1D mass, energy and momentum equations in transient regime. The number of volumes in which the pipe is discretised will be a parameter. All kind of flows (compressible or nearly incompressible flows, single component or two-component flows, single phase or two-phase flows) can be simulated by using the following system of governing equations, here in area-scaled conservation form [138]: Mass conservation: A ρ t + ρva x = ρak P wall t (3.21) Non-condensable mass fraction x nc conservation: A ρxnc t + ρvxnc A x = ρx nc Ak wall P t (3.22) Momentum conservation: A ρv t + [(ρv2 + P )A] x = 1 dξ ρ v v A + ρga + P 2 dx ( ) da dx (3.23) Energy conservation: A ρe t + ρvha x = ( d Q w dx ) + ρgva (3.24) where ρ, x nc, P, u are the gas/liquid mixture density, the non-condensable mass fraction, the pressure and the total energy respectively. A is the variable flow area and v the velocity. This system of 4 equations represents the general case 39

54 3. ESPSS: European Space Propulsion System Simulation of a mixture of two fluid components, for which the first one can be either one phase or two-phase, and the second one is always a non-condensable gas. This set of equations is closed by a thermodynamic equation of state (EoS), which is described in the fluid properties library, and hereafter written under general form: p = p(ρ, u) (3.25) The choice of density ρ and internal energy u as independent thermodynamic variables is the most efficient one regarding CPU-time when the EoS is left under arbitrary form. The different source terms are the following: In the first equation governing the mixture mass conservation, a source term responsible for the wall compressibility effect of the mixture, determinant in water hammer simulations, is included; k wall is the wall compressibility. Assuming linear elasticity for the pipe wall material, we have three configurations: pipe anchored with expansion joints throughout: k wall = D in /t/m E pipe anchored at its upstream end only: k wall = D ( ) in 5 t M E 4 η pipe fully anchored: k wall = D in t M E (1 η 2 ) M E is the Young s modulus of elasticity, η is the Poisson s ratio and t is the wall thickness. The wall compressibility shall be multiplied by P/ t to account for the volume change. For this purpose is calculated from the current state variables (density and energy) and the thermodynamic derivatives: ( P ρ t = t ρ ( u h P t + P ρ ρ 2 t )) ( ρ / P 1 h ρ ρ ) h P In the second governing equation, the non-condensable mass conservation (if any), a similar source term is included; In the third governing equation, the mixture momentum conservation, a source term represents the friction (dξ, proportional to dx, is the pressure 40

55 3. ESPSS: European Space Propulsion System Simulation drop coefficient derived by empirical correlations), another one takes into account the gravity, and the last one is responsible for the area variation. The equivalent distributed friction, ξ i is calculated as follows: ξ i = k add + bend,j hdc bend (α bend,j, R bend,j, D i, ɛ) + x i D i hdc fric (D i, ɛ, Re i ) where k add is an input data representing concentrated load losses to be distributed along the pipe; Function hdc bend calculates the bend pressure drop coefficient; Function hdc fric calculates the friction factor including laminar and turbulent regimes. g represents the gravitational acceleration, if any. It is computed as the scalar product of the gravity vector (g x, g y, g z ) with the direction of the pipe in the global axis system ( x, y, z), which are the difference of position of the tube tips. In the last governing equation, the mixture energy conservation, a source term Q w takes into account the heat transfer with the wall when it is included: Q w = h film dx[p inner (tp_in.t T ) + P outer (tp_out.t T )] where P inner and P outer are the wet perimeters; tp_in, tp_out are the names of the thermal ports connected to the tube with the same number of nodes as the fluid vein. The port temperatures tp_... T behave as the wall internal temperatures, to be determined in the connected thermal components. The film coefficient for each node is calculated using empirical two-phase correlations: h film = htc(x, D h, λ, Re, P r...) Re = G D h µ P r = µ Cp λ 41

56 3. ESPSS: European Space Propulsion System Simulation where G = ρv is the mass flow rate per unit area, D h is the hydraulic diameter of the pipe D h = 4A/P w. In case of circular cross section D h is equal to the geometric diameter of the cross section; otherwise it represents a reasonable characteristic length of the cross section. Another source term, ρgva, takes into account the gravity work. The tube and pipe components are discretised by either a centred or an upwind numerical scheme. Figure 3.2 describes the pipe discretisation. The inner fluxes are computed using one or the other of these schemes, and the first and last junctions (1 and n + 1) ones are given by the fluid ports, as they are calculated at resistive type components using momentum equation with sonic flow limitation. Note that the first and last half-nodal inertia are included in the junction component equations. Figure 3.2.: Pipe discretisation Using the centred scheme, a staggered mesh approach is applied, for which the state variables (pressure, density, velocity, chemical composition and enthalpy) are associated with the n nodes, and the flow variables (volumetric, mass and enthalpy flows) are calculated at the internal junctions (each junction has associated two half volume inertias). With this scheme, the various fluxes to be computed at the inner junctions are simply the flow variables, except for the mixture momentum 42

57 3. ESPSS: European Space Propulsion System Simulation flux that is associated to the n nodes: f mass,j = ṁ j f nc,j = ṁ nc j (3.26) f mom,j = (P i + ρ i v 2 i + qn i )A i f ene,j = ṁh j The momentum flux term includes an artificial dissipation term qn i calculated as follows: qn i = Damp ṁi+1 ṁ i v sound,i A As an alternative to the centred scheme, an adequate upwind scheme has been developed to improve the discontinuities resolution of these transient two-phase flows in quasi-1d pipe networks. Using that scheme, a collocated mesh approach is applied, for which all variables are discretised on the n nodes, even the mass flow. All the fluxes f(u) are discretised at the junctions and include a central part and an upwind part, following the initial Roe scheme [123]. Please refer to Appendix A for a detailed description of the scheme Turbomachinery Library turbo_machinery is an EcosimPro library for the simulation of pumps, turbines and compressors. The most important features are the following: Pump model provided with user-defined dimensionless turbo-pump characteristic curves adapted to positive and negative speeds and flow zones Turbine and Compressor components provided with user-defined dimensionless performance maps as a function of the reduced axial speed and pressure ratio Special turbomachinery components allowing simple calculation of generalized performance maps as a function of the nominal performances and other significant design data 43

58 3. ESPSS: European Space Propulsion System Simulation Programming of the turbomachinery components is non-dependant on the working fluid type: the properties of the selected working fluid (calculated inside the corresponding thermodynamic function) will depend on the fluid, but the non-dimensional parameters of the performance maps are equally defined for all kind of fluids. The generalized performances maps of the turbo_machinery library allows robustly analysing the transients during the start up and shutting down processes, where the reduced axial speed and flow are far away from the nominal values. The turbo_machinery components can be connected to fluid_flow_1d components with the aim of simulate a complete rocket engine cycle. The components of the turbo_machinery library are listed below, and represented in Figure 3.3: - Compressor - Pump - Turbine - Compressor generic - Pump generic - Turbine generic - Pump vacuum Two different types of pumps/turbines are available: one generic model if the off-design characteristics are unknown and one specific model which can only be used with tables, for well defined turbo machinery. All these components behave externally as resistive elements (see section 3.2.1). Components ports are resistive because they calculate the mass flow. Nevertheless, every model of a turbo-machinery component includes an internal capacitive element receiving/giving the mechanical work. Since in turbopump-fed engine cycles only pumps and turbines are present, only these two class of components will be discussed in this section. 44

59 3. ESPSS: European Space Propulsion System Simulation Figure 3.3.: Components in the turbo_machinery library Pump & Generic Pump This component simulates a pump for liquids. It is provided with fixed or userdefined dimensionless turbo-pump characteristic curves adapted to positive and negative speeds and flow zones, and valid for several types of pumps. Usually it is rather difficult to find pump curves including the non normal zones, so this is the reason to include in this component a set of curves [25] covering all zones of the pump operation for 3 different specific speeds: Ns = 25 corresponding to a pure centrifugal pump, Ns=147 corresponding to mixed pump, and Ns=261 corresponding to an axial pump. The specific speed is defined as: Ns = rpm Q/n s (TDH/n st ) 0.75 where, n s is number of suctions, n st is the number of stages, Q [m 3 /s] is volumetric flow and TDH [m] is the actual total dynamic head of the pump. The pump model makes use of performance maps for head and resistive torque. The pump curves are introduced by means of fixed 1-D data tables defined as functions of a dimensionless variable θ that preserves homologous relationships 45

60 3. ESPSS: European Space Propulsion System Simulation in all zones of operation. θ parameter is defined as follows: θ = π + arctan(ν/n) (3.27) where ν and n are the reduced flow and speed parameters respectively: ν = Q Q R = ṁin/ρ in Q R n = ω ω R (3.28) The dimensionless characteristics (head and torque) are defined as follows: h = TDH / TDH R n 2 + ν 2 β = τ / τ R n 2 + ν 2 (3.29) τ and TDH are the torque and the total dynamic head respectively. Sub index R means rated (nominal) conditions. The nominal torque is calculated from the other nominal parameters: τ R = g ρ up TDH R Q R η R ω R (3.30) This method eliminates most concerns of zero quantities producing singularities. To simplify the comparison with generic map curves, these relations are normalized using the head, torque, speed and volumetric flow at the point of maximum pump efficiency. In case there are no user defined curves, the ones already implemented in the component at different specific speeds will be used and interpolated as function of the actual Ns and θ: h = h_vs_theta_ns(ns, θ) β = β_vs_theta_ns(ns, θ) dimensionless pump TDH dimensionless pump torque using the definition of h and β, the actual torque τ and the pressure rise TDH (expressed as the total dynamic head in meters) will be calculated. The mechanical balance allows the calculation of the axial speed dynamically: I mech dω dt = τ shaft τ (3.31) 46

61 3. ESPSS: European Space Propulsion System Simulation where ω is mechanical speed, I mech is the mechanical inertia and τ is the torque calculated using the non-dimensional performances pump. The enthalpy flow rise is a function of the absorbed power while the evaluation of the mass flow rate is performed through an ODE. (ṁ h) out = τ ω (ṁ h) in I dṁ dt = ( P ρv2) out ( (3.32) P ρv2) in gρ in TDH In case performance maps of a particular pump are known, a different approach is used; the pump curves (head and torque) are introduced by means of input data tables: Independent variables: - Mass flow coefficient: φ + = ṁ/(ρ in ω) Dependent variables: - Head rise coefficient: ψ + = P/(ρ in ω 2 ) - Reduced torque: C + = τ/(ρ in ω 2 ) The head rise and needed torque coefficients are computed with 1-D tables, as a function of mass flow coefficient only. Rotational speed is not taken into account. The pump model described here has the disadvantage of being less general than the Pump_gen component: the coefficients used (φ +, ψ + and C + ) are not dimensionless. Hence, the characteristics differ for each pump, even for geometrically similar pumps (with impellers having the same angles and proportions) Turbine & Generic Turbine This component simulates a turbine for gases. It is provided with calculated (no input) but adjustable dimensionless characteristic curves adapted to positive and negative speeds and flow zones, and valid for several types of turbines. Adjustable dimensionless pressure ratio curves are function of beta design parameter, reduced 47

62 3. ESPSS: European Space Propulsion System Simulation speed and inlet Mach number. Efficiency curves are function of reduced speed and pressure ratio parameters. The value of the power W is obtained using GasTurbo_pow function. This function basically estimates the efficiency as a function of the reduced speed and pressure ratio parameters. It also needs the inlet mass flow to calculate the power. The rest of the arguments are input data (η nom, N nom,...) or calculated values in the connected pipes components (bound pressures and enthalpies, etc.). As for the pump, the mechanical balance allows the calculation of the axial speed dynamically: I mech dω dt = τ shaft τ The mechanical work (τ ω) extracted from the turbine is simulated as an enthalpy flow: (ṁ h) out = (ṁ h) in τ ω The inlet mass flow equation is expressed dynamically in accordance with the turbine pressure drop as follows: I dṁ (P dt = + 12 ) ρv2 (P + 12 ) ρv2 (Π nom 1) P in dp_rel in out dp_rel is obtained using GasTurbo_dpTurb function. This function basically estimates the relative pressure drop of the turbine as a function of the beta design parameter, the reduced speed and the inlet Mach number. The inlet Mach number M in is related to the inlet mass flow: M in = ρ in A in C o C o is the sound speed calculated at the outlet volume or at inlet according to the vsound_outlet option. When specific turbine performance maps are introduced, the Turbine_Gen component cannot be used anymore and the Turbine component is then necessary adopted. Turbine map curves (dimensionless torque and mass flow coefficients) are input data tables depending on the dimensionless speed and pressure ratio coefficients. The performance maps (mass flow coefficient and specific torque) are introduced by means of 2D input data tables: Independent variables: ṁ in 48

63 3. ESPSS: European Space Propulsion System Simulation - Speed coefficient: N = r ω C o - Total pressure ratio: Π = P 01 / P 02 Dependent variables: - Mass flow coefficient: Q + = ṁmap C o r 2 P 01 - Specific torque: ST = τ r ṁ map C o Independent variables are easily calculated because they are obtained from the dynamic rotational speed and from the boundary pressures. Then, the mass flow and torque will be computed interpolating into the 2D input data tables representing the turbine maps. Compared to the generic component model, only the mass flow equation is different. The inlet mass flow equation is expressed dynamically in accordance with a time delay (inertia terms): τ dṁ dt = (ṁ map ṁ) ; τ = I r 2 /C o 3.4. Combustion Chambers Library comb_chambers is an EcosimPro library for the simulation of rocket engines. The most important features are the following: The properties of the combustion gases (transport and heat capacity) are obtained from the CEA [55] coefficients (see Section 3.1.1) for an arbitrary mixture or chemical s reactants. The equilibrium molar fractions at of a mixture of reactants are derived from the Minimum Gibbs energy method Non adiabatic 1D Combustor component: the equilibrium combustion gases are calculated using previous capabilities. The chamber conditions will be derived from the general transient conservation equations along a 1D spatial discretisation Inclusion of another more advanced non-equilibrium, non adiabatic 1D combustor component where a model for the liquid droplets evaporation and for the global reaction time are also considered. The non-equilibrium 49

64 3. ESPSS: European Space Propulsion System Simulation model is only a first approach; it does not include finite rate chemistry but only time delay parameters Pre-burners and main combustion chambers are topologically built by means of a combustor, two injectors and two cavity components. They can work either under liquid, two-phase or gas injection conditions. The bubble collapse calculation is included in the model of cavities The combustion gases generated in a chamber can be conducted (using standard fluid_flow_1d components) to the turbines or even to other chambers where any of the previously combusted gases will be considered a new reactant. Cooling jacket component: Several models are available: one with a complete 3D wall temperature distribution and another also including the injection tores Modelling of solid propellant starters, igniters and thermal coating protection are available using combustor components Ideal or non adiabatic exhaust nozzles provided with a 1D spatial discretisation. A special nozzle component allows simple simulation of the film cooling injection The comb_chambers components can be connected with fluid_flow_1d, tanks or turbo_machinery components for the simulation of a complete rocket engine cycle. Models where one or more chambers are present (staged engines) can be evaluated. This library permits to analyse in great detail the transients during the start-up and shut-down processes, where the valve sequences are decisive. The numeric method used for the resolution of the subsonic sections of a combustor is based on the transient conservation equations. ESPSS combustor models calculate the chamber pressure and the mixture ratio as a function of the combustor geometry (design parameters) and the physical boundaries whereas in CEA code those variables were imposed. Advantages of the ESPSS methods are: 50

65 3. ESPSS: European Space Propulsion System Simulation Transient phenomena (including pressure/temperature peaks at the start up and shut down processes) are taken in to account The number of implicit equations is reduced, the state variables being dynamic The wall heat exchange (non adiabatic terms) and the pressure drops are taken into account Transient formulation allows to include vaporization / non-equilibrium phenomena, as it done in the Combustor_rate component The drawbacks of this method are: The characteristic time (integration time step) can be very small. Nevertheless, numerical instabilities are normally smoothed, the integration being faster than using an implicit method The total pressure is not strictly conserved along the 1D combustor volumes. Typical errors are between 0.5 and 1%, mainly produced near the throat where the Mach number is close to 1 Concerning the supersonic sections of nozzles, a resolution method based on the transient conservation equations would have numeric problems (passage from subsonic to supersonic regime, shock waves if non-adapted conditions, etc), so a 1D quasi-steady implicit method has been implemented for these components, including non-isentropic effects under frozen or equilibrium conditions. The components of the comb_chambers library are listed in Figure 3.4, where the inheritance hierarchy is shown. The existing components have the following types (see section 3.2.1): Cavities, Regenerative Circuit and Nozzle components are C (capacitive) elements Injectors and Chemical inflator components are M (resistive) elements Propellant fluid ports of Preburners and CombustChamberNozzle components are capacitive, and can be connected to any fluid_flow_1d library 51

66 3. ESPSS: European Space Propulsion System Simulation resistive component. Thermal ports should be connected to a Cooling Jacket component or other thermal component Figure 3.4.: Components in the comb_chambers library ABS_Combustor, Injector and Inj_Cavity components should only be used for building Combustion Chambers and Preburners because they must be interconnected to each other. Preburner components include a combustor, two injectors and cavities components, and an outlet resistive fluid port where the mass flow is calculated. This port must be connected to a fluid_flow_1d capacitive component so the combustion gases can be conducted to another combustor. CombustChamberNozzle components include a combustor, two injectors and cavities components, and a non-adiabatic 1D Nozzle. They have a special exit port so a new nozzle extension component (with or without film cooling injection) can be connected. 52

67 3. ESPSS: European Space Propulsion System Simulation Th_mux, Th_demux are multiplexer/demultiplexer components allowing splitting or jointing a vectorized thermal port into several ones, so that different sized Cooling Jacket components can be connected to a chamber Injector Cavity Inherited from a Capacity (fluid_flow_1d library, see Section 3.2.1) this component represents the combustion cavities upstream the injectors with thermal ports allowing heat exchange. The formulation of the conservation equations are the same as in Section they are valid even if the fluid conditions are liquid, vapour or homogeneous two phase flow. This component is also charged of the calculation of the propellant molar fractions from the injected fluids: N chem = 1; N k = y k ; MW mix = Nchem k=1 y k MW k ; (case of pure fluid) (Case of previously burned gases) In case of a pure fluid, chem is the chemical corresponding to the main fluid. In case of previously combusted gases, y k are the molar fractions at the inlet of the cavity calculated by an upstream combustor. MW k is the molecular weight of the chemical constituent k Combustor Equilibrium In the combustion chamber the merged, mixed and atomized propellants are vaporized and burned. In doing so, the chemical bound energy from the propellants is transformed into thermal energy. Hence, it follows an increase of the combustion chamber temperature T c, which also involves a pressure increase P c in the chamber. This component represents a non adiabatic 1D combustion process inside a chamber for liquid or gas propellants. The transient conditions (pressures, temperatures, mass flows and heat exchanged with the walls) will be derived from general 1D transient conservation equations. 53

68 3. ESPSS: European Space Propulsion System Simulation The equilibrium combustion gases are calculated using the minimum Gibbs energy method as a function of the propellant s mixture molar fractions and enthalpies, and the chamber pressure. A mixture equation between the injected propellants and the combustion gases is applied. From the definition of the mixture ratio (MR) and derivation, the following dynamic equation gives the MR evolution: MR = Mass ox /Mass fu Mass fu = (ρv ) chamber 1 + MR ṁ ox = MR ṁ fu + dmr (ρv ) chamber dt 1 + MR (3.33) The implementation of the 1D set of equations is done according to a staggered grid in which the P /ρ/x variables are defined in the centre of the volumes and the mass flows at the junction between the volumes. Gas mixture mass conservation equation: A ρ t + ρva x = 0 (3.34) Gas mixture momentum conservation equations: A ρv t + [(ρv2 + P )A] x Gas mixture energy conservation equation: A ρe t = 1 dξ 2 dx ρ v v A + P + ρvha x = ( ) d qw dx ( ) da dx (3.35) (3.36) At injector level (i=0), the junction mass flows will be calculated by the Injector components. The effective liquid mass flow entering into the chamber will be computed as follows: Burning = FALSE, only the injected vapours and non-condensable gases contribute to the chamber pressurisation. Injected liquids are supposed to be lost 54

69 3. ESPSS: European Space Propulsion System Simulation Burning = TRUE, it is supposed that all the injected liquid will be vaporised within a delay time, τ v (injected gas is not modified): ( ( = (1 x fu )ṁ fu tanh 10 T ) ) n T tr,fu ṁ fu /τ v dt T tr,fu ( ( dṁ ox,liq = (1 x ox )ṁ ox tanh 10 T ) ) n T tr,ox ṁ ox /τ v dt T tr,ox dṁ fu,liq where ṁ ox,liq, ṁ fu,liq are the effective injected mass flow; ṁ ox, ṁ fu are the total injectors mass flows; x fu and x ox are the gas (vapours and noncondensable gases) mass fractions in the cavities. The hyperbolic tangent term is added to produce a relaxation of the injected liquid mass flow at very low temperatures (ignition process) Total enthalpies h jun are calculated using the upstream cell conditions: h jun,i = h i 1 = (u + P/ρ) i 1. Again, at injector level the enthalpy flows will be computed using the effective liquid and gas mass flows multiplied by the corresponding cavity enthalpy (liquid and vapour). In this way injection conditions (liquid, gas or two-phase flow) will be taken into account. The starter terms (starter_m and starter_mh) will be added to the mass and energy conservation equations of the first chamber volume. The composition of the solid propellant gases is an input data within a predefined set of chemicals starter_mh = f(starter_t, powder_composition); starter_m, starter_t and powder_composition being input data. In the momentum fluxes an artificial dissipation qn is added, and is calculated as follows: qn i = Dampṁjun(i + 1) ṁ jun (i) v sound (i) A Damp is a global input data of the fluid_flow_1d library. Momentum equations are applied to the exit of any volume in which the combustor is discretised with the exception of the last volume that should end with the throat to avoid numerical problems (transition from subsonic to supersonic flow). The outlet mass flow will be calculated at the throat. Node velocities v are calculated using the adjacent junction mass flows and the 55

70 3. ESPSS: European Space Propulsion System Simulation mean junction densities: v i = 1 A i ( ṁjun,i 1 ρ i + ρ i 1 + ṁjun,i ρ i + ρ i+1 ) i = 2, n; v 1 = ṁjun,1 A 1 ρ 1 (3.37) For the first volume, it is supposed that only the outlet junction will account for the node velocity. For the last volume, the throat density is used. The vapours and the non-condensable conservation equations take into account the mixture process: ( ) xfu ρ t ( ) xox ρ t ( x nc ) ρ t ( xpw ρ t A + x fuṁ jun x A + x oxṁ jun x A + xnc ṁ jun x ) A + x pwṁ jun x = 0 = 0 = 0 = 0 where, x fu is the reducer vapour mass fraction, and x ox is the oxidizer vapour mass fraction ; x nc is the non-condensable mass fraction and x pw represents the solid propellant gases mass fraction. The mass flow of the injected vapours (m fu, m ox ) and injected non-condensable gases will be added as source terms to the respective conservation equation of the first chamber volume. In the same way, the mass flow of the solid propellant gases, starter_m, will be added as source a term to the solid propellant gases conservation equation of the first chamber volume. Under burning conditions, only the mass fractions of the first chamber volume will be used to compute the molar fraction of the reactants. Subsequent volumes will use as reactant the product of the upwind volume. For each chamber volume the combustion gases properties (product s molar fraction, heat and transport properties) are calculated using the Minimum Gibbs energy method as a function of the propellant molar fractions, pressure and specific enthalpy: 56

71 3. ESPSS: European Space Propulsion System Simulation Reducer contribution: y k,fu N k,i = x fu,i ; MW mix,fu = MW mix,fu Oxidiser contribution: y k,ox Nchem N k,i = N k,i + x ox,i ; MW mix,ox = MW mix,ox Solid propellant gases contribution: y k,pw N k,1 = N k,1 + x pw,i ; MW mix,pw = MW mix,pw Non-condensable gases contribution: N nc,i = N nc,i + xnc i MW nc k=1 Nchem k=1 y k,fu MW k,fu Nchem k=1 y k,ox MW k,ox y k,pw MW k,pw N chem is extended to any chemical treated by the fluid_properties library; MW k is the molecular weight of the chemical constituent k; x nc 1 is the mass fraction of non-condensable gas at volume 1. x fu,i, x ox,i, x pw,i are mass fractions of vapours and solid propellant gases; y k,fu ; y k,ox ; y k,pw are the molar fraction of chemical k of the reducer (oxidizer) mixture and N k,1 is number of moles of the chemical constituent k of the reactant mixture at volume 1. The propellants (reducer and oxidizer mixtures) molar fractions have been calculated by the cavity components. Propellants can be formed by any allowed fluid mixture using the fluid_flow_1d library, including those of a previous combustion. Then, the number of moles, N k is normalized. With this entry calculated and using the dynamic enthalpy value obtained from the conservation equations of the first combustor volume it is possible to call the Minimum Gibbs energy method to obtain the equilibrium temperature and the molar fraction of the products: (y k_eq, T eq ) = f mingibbs (N k,1, h 1 v 2 1/2, P 1 ) 57

72 3. ESPSS: European Space Propulsion System Simulation Two possibilities are foreseen calling previous function: - Equilibrium - Frozen flow In the last case (no ignition), molar fractions remain constant: y k_eq,1 = N k,1. Under frozen conditions, the molar fractions of the subsequent volumes will be calculated as for the first volume. The temperature calculation from the enthalpy value will require an iteration procedure, in this case lesser complicated than in equilibrium conditions. Under equilibrium conditions, subsequent volumes will consider that the molar fraction of the products of the previous volume will act as the inlet propellant mixture, so the Minimum Gibbs energy method can be applied to any combustor volume. (y k_eq, T eq ) i = f mingibbs (N k,i 1, h i vi 2 /2, P i ) The effective combustion gas constants (R i, Cp i, λ i, µ i ) will be derived using the mixture properties equations as a function of y k_eq,1, see Section The pressure is obtained from the perfect gas state equation: P i = ρ i R i T i η (3.38) where η is the combustor efficiency. Note that the pressure equation and the Minimum Gibbs function become an algebraic loop. Finally, the molar fraction of the products of the last volume will be transmitted to the outlet port to be used by the Nozzle component or in another possible chamber. The term q w appearing in the energy conservation equation permits the exchange of convective and radiative heat through a thermal port. The walls (that can be represented by thermal components or by the Cooling Jacket component) are not included in this component: q w = h c A wet (T aw tp.t ) + σa wet (T 4 tp.t 4 ) (3.39) tp is the name of the thermal port (with n nodes in axial direction) connected to 58

73 3. ESPSS: European Space Propulsion System Simulation the Combustor. σ is the Stefan-Boltzmann constant = [W m 2 K 4 ]. The heat exchanged with the fluid is transmitted through this port: tp. q(i) = q w,i ; tp.t(i) behaves as the internal wall temperature, to be determined in the connected wall component. T aw is the adiabatic wall temperature defined as: T aw = T ( 1 + Pr 0.33 ref ) γ 1 M 2 ; Pr ref = 2 ( ) Cpλ The reference conditions (ref) are calculated at a temperature halfway between the wall and the free stream static temperature. It is supposed that the mixture composition do not change between these two temperatures. The film coefficient for each volume is calculated using empirical correlations according to Bartz [9]: µ ref ( ) 0.6 ( ) 0.1 h c = µ 0.2 λref ref Cp 0.4 ref µ (ṁ th) 0.8 /A 0.9 πdth (3.40) ref 4R curv where, µ ref : viscosity of combusted gases at volume no. i and T ref temperature λ ref : conductivity of combusted gases at volume i and T ref temperature R curv : Curvature radius of the throat D th : Throat diameter ṁ th : Throat mass flow A: Cross section at volume i Combustor rate This component represents non-equilibrium, non adiabatic quasi 1-D combustor component for liquid or gas propellants. The transient chamber conditions (pressures, temperatures, mass flows and heat exchanged with the walls) will be derived from general quasi 1-D transient conservation equations. The nonequilibrium model is only a first approach; it does not include finite rate chemistry but only time-delay parameters. The mass, energy and momentum equations include those of the equilibrium Combustor component plus the ones concerning the vaporization model. Liquid phase conservation equations are also included. 59

74 3. ESPSS: European Space Propulsion System Simulation Gas mixture mass conservation equation: A ρ t + ρva x = ṁvap_fu A fu V fu + ṁvap_ox A ox V ox (3.41) Gas mixture momentum conservation equations: A ρv t + [(ρv2 + P )A] x = 1 dξ 2 dx ρ v v A + P ( ) da dx (3.42) Gas mixture energy conservation equation: A ρe t + ρvha x = d q w dx + q vap_fu A fu + q vap_ox A ox (3.43) V fu V ox where, ρ and E are the gas mixture density and total energy; v is the mean velocity; ṁ vap_ox and m vap_fu represent the vaporized liquid mass flow of oxidizer and of the reducer. At injector level, the gas mass and enthalpy flows (m jun,0 ) will be calculated by the Injector components taking into account the quality calculated in the Cavities components. The liquid contributions will be considered in the droplets conservation equations. ṁ fu = ṁ fu,inj x fu ; ṁ ox = ṁ ox,inj x ox where ṁ ox,inj, ṁ fu,inj are the injectors mass flow (liquid or gas) and x fu and x ox are the vapour mass fractions in the cavities. Then ṁ jun,0 = ṁ fu + ṁ ox. An hyperbolic tangent term is added to produce a relaxation of the injected vapour mass flow at very low temperatures. Since a centred scheme with a staggered mesh is adopted, total enthalpies h jun are calculated using the upstream cell conditions: h jun,i = h i 1 = (u + P/ρ) i 1. Starter terms are in included in the same way as in the combustor_eq component. Also, the node velocities v are calculated as in the combustor_eq component. The burned gases production rate does not contribute explicitly to the gas mixture equation because the mass is conserved in the chemical reactions (no condensation). 60

75 3. ESPSS: European Space Propulsion System Simulation Vaporization flows, m vap, and enthalpies flows, q vap (source term for the gas mixture conservation equations), are calculated by the droplet vaporization models. Two models are available: User defined model : The vapour mass flows in each volume are calculated assuming a characteristic vaporization time modulated with user defined vaporization factors: ṁ vap_fu, = f vap M liq_fu /τ vap ; ṁ vap_ox = f vap M liq_ox /τ vap q vap_fu = ṁ vap_fu h(t liq_fu ); q vap_ox = ṁ vap_ox h(t liq_ox ) where, M liq is the liquid mass at the volume i; T liq are the liquid droplets temperatures, τ vap is the characteristic vaporization time and f vap represents the vaporization factor (time dependant input data) at the volume i. Droplets vaporization model : Assuming a very thin saturated layer between the droplets and the surrounding gases, the conservation equations establish that the sum of convective heat plus enthalpy mass flow are the same at both sides of the layer. Then, the following set of equations is applied at each combustor volume i, allowing the calculation of the mass and energy exchanges through this layer: ṁ vap_fu = A liq_fu ( ) h c (T T sat_fu ) + h c,liq_fu (T liq_fu T sat_fu ) (h vap_fu h liq_fu ) q vap_fu = ṁ vap_fu h vap_fu A liq_fu gas h c (T T sat_fu ) h c,liq_fu = 2 λ liq_fu /D droplet_fu A liq_fu = f vap 6 M liq_fu /ρ liq_fu /D droplet_fu 61

76 3. ESPSS: European Space Propulsion System Simulation ṁ vap_ox = A liq_ox ( ) h c (T T sat_ox ) + h c,liq_ox (T liq_ox T sat_ox ) (h vap_ox h liq_ox ) q vap_ox = ṁ vap_ox h vap_ox A liq_ox gas h c (T T sat_ox ) h c,liq_ox = 2 λ liq_ox /D droplet_ox A liq_ox = f vap 6 M liq_ox /ρ liq_ox /D droplet_ox where, T are the gas temperatures, h c is the heat exchange coefficient at gas side. Same value as for the wall is used. T sat is the saturation temperature calculated at the partial vapour pressure of volume i; h vap and h liq represent the saturation enthalpies calculated at the partial vapour pressure of volume i; D droplet is the mean droplets diameter and A liq is the equivalent exchange area between the droplets and the gas. We point out that the droplet diameter has been modulated by the vaporization factor. This factor is an input data depending on the time and on the volume number. In theory, assuming a known droplet size at the injection plate, the droplet diameter evolution could be determined by simple equations relating the evaporated mass flow with the liquid mass conservation equations. Nevertheless, due to the high penetration and break up of the liquid jets, it seems more realistic to assume a known droplet size at each chamber volume, the number of droplets being determined by the current liquid mass. In both methods (used defined and droplet model), the vapour mass flow is also weighted with a factor to prevent vaporisation at very low temperatures (frozen liquid): ( ṁ vap = ṁ vap tanh 10 T T ) sat T sat In normal situations, T > T sat, making the value of the hyperbolic tangent be equal to one. Burning Rate : The burned gases mass flow (second source term for the vapour conservation 62

77 3. ESPSS: European Space Propulsion System Simulation equations, see Eq. 3.44a and Eq. 3.44b) is calculated assuming a global characteristic burning time [9]. It is also supposed that any species (vapour or burned gas) present in gas mixture contributes to the global reaction rate, so the burning rate will be proportional to the total gas mixture density: ṁ bu = f bu ρ V τ bu where τ bu is the characteristic burning time and f bu is the burning factor at the volume i. The burning factors are automatically set to one if the burning conditions are true: mixture ratio within the allowed limits and ignition flag activated. Otherwise the burning factors are set to zero. vapours / non-condensable / solid propellant gases mass equations : The vapours and non-condensable mass conservation equations take into account the burned gases production and the vaporization terms previously calculated: (x fu ρ) t (x ox ρ) t (x nc ρ) t (x pw ρ) t + (x fuρva) x + (x oxρva) x + (xnc ρva) x + (x pwρva) x = ṁvap_fu A fu V fu = ṁvap_ox A ox V ox ṁ bu x fu A fu V fu ṁ bu x ox A ox V ox (3.44a) (3.44b) = 0 (3.44c) = 0 (3.44d) The mass flow of the injected vapours (ṁ fu, ṁ ox ) and injected non-condensable gases will be added as source terms to the respective conservation equation of the first chamber volume. In the same way, the mass flow of the solid propellant gases, starter_m, will be added as source a term to the solid propellant gases conservation equation of the first chamber volume. 63

78 3. ESPSS: European Space Propulsion System Simulation Liquids conservation equations : The liquid mass and enthalpy flows are calculated assuming that v gas = v liq and neglecting Cp liq derivatives: M liq_fu t + (M liq_fu v) x = ṁ vap_fu t (M T ) liq_fu + x [(M T ) liq_fuv] = q vap_fu /Cp liq_fu M liq_ox t + (M liq_ox v) x = ṁ vap_ox t (M T ) liq_ox + x [(M T ) liq_oxv] = q vap_ox /Cp liq_ox Combustion gases properties calculation : It is supposed that any molar fraction follows a global reaction rate accordingly with the previously mentioned burning time: dy k,bu dt = (y k,eq y k,bu ) τ bu where y k,eq is the equilibrium molar fraction of the chemical constituent k at each volume i. y k_bu is the actual burned molar fraction of the chemical constituent k at each volume i, and τ bu is characteristic burning time. For each chamber volume i, the combustion gases equilibrium composition (needed for the calculation of the actual burned gases composition) is calculated using the Minimum Gibbs energy method as a function of the gas mixture molar fractions, pressure and the enthalpy. The gas mixture molar fractions in each volume are calculated as follows: Reducer vapours contribution: y k,fu N k = x fu ; MW mix,fu = MW mix,fu Nchem k=1 y k,fu MW k,fu 64

79 3. ESPSS: European Space Propulsion System Simulation Oxidiser vapours contribution: y k,ox N k = N k + x ox ; MW mix,ox = MW mix,ox Solid propellant gases contribution: y k,pw N k = N k + x pw ; MW mix,pw = MW mix,pw Non-condensable gases contribution: Burned gases contribution: y k,bu N nc = N nc + x nc MW nc N k = N k + x bu ; MW mix,bu = MW mix,bu Nchem k=1 Nchem k=1 Nchem k=1 y k,ox MW k,ox y k,pw MW k,pw y k,bu MW k,bu The vapour (reducer and oxidizer mixtures) molar fractions have been calculated by the cavity components and can include any allowed fluid mixture using the fluid_flow_1d library, including that of a previous combustion. The burned molar fractions, y k_bu, are calculated dynamically. Once the number of moles of the reducer/oxidizer/burned gases mixture has been evaluated, and using the dynamic enthalpy value obtained from the conservation equations, it is possible to call to the Minimum Gibbs energy method to obtain the equilibrium combustion gases composition: (y k_eq, T eq ) i = f mingibbs (N k,i, h i v 2 i /2, P i ) Two possibilities are foreseen calling the previous function: Equilibrium and frozen flow. In the last case (no ignition) the molar fractions remain constant: y k_eq,i = (N k,i ). The effective combustion gas constants (R i, Cp i, λ i, µ i ) will be derived using the mixture properties equations as a function of y k_bu,i see Section The pressure is obtained from the perfect gas equation: P i = ρ i R i T i. With 65

80 3. ESPSS: European Space Propulsion System Simulation respect to the Combustor_eq component, the pressure equation and the Minimum Gibbs function do not become an algebraic loop because the actual molar fractions (y k_bu,i ) of the mixture are dynamic variables. The molar fraction of the products of the last volume will be transmitted to the outlet port to be used by the Nozzle component or in another possible chamber Nozzle This component represents 1D supersonic nozzle in quasi-steady conditions. Two possibilities are foreseen in the main body of this component: an ideal nozzle using a variable gamma approximation a non-adiabatic, non-isentropic nozzle The mathematical model explained below can be applied either to a complete nozzle or to a nozzle extension. The important point is to know the inlet total conditions calculated from the upstream enthalpy/entropy conditions and from the mass flow calculated in the complete nozzle component: - If the nozzle is connected to a chamber (case of a complete nozzle), then the total conditions will be those of the exit of the chamber transmitted by the nozzle port. - If the nozzle is connected to another nozzle (case of a nozzle extension), then the total conditions will be those of the exit of the upwind nozzle. Inlet/outlet mass flows are the same. The throat calculation (choked mass flow, see below) will be only implemented in the complete nozzle. The choked throat conditions (P th, T th and v th ) can be calculated with the following three equations assuming that the total enthalpy and the static entropy are known (those of the last combustor volume): h ch,tot = h(n k,th, T th ) + v 2 th /2 s ch = s(n k,th, P th, T th ) v th = γ th R th T th ; ṁ th = A th v th P th /(R th T th ) 66

81 3. ESPSS: European Space Propulsion System Simulation where, γ th is the isentropic coefficient. This is a function of P th, T th, N k,th ; s is the entropy, a function of P th, T th and N k,th (see section 3.1.1); h is the static enthalpy. This one is a function of T th and N k,th (see section 3.1.1). N k,th is the number of moles of the chemical constituent k of the burned gases at throat. It is calculated from the Minimum Gibbs energy method: (N k, T ) th = f mingibbs (N k,ch, s ch, P th ) where sub index ch denotes the conditions at the exit of the combustor. Two possibilities are foreseen calling the previous function: Equilibrium and frozen flow. In the last case, molar fractions remain constant. The equations above are solved iterating in pressure and in temperature. To cover subsonic conditions, the effective throat mass flow is calculated using a dynamic momentum equation (see Section 3.2.2) where the left pressure corresponds to the combustor exit, and the right one to the external pressure. Of course, under normal steady conditions, the dynamic mass flow will be limited to the critical mass flow, ṁ th, previously calculated. Ideal supersonic nozzle : The nozzle is divided in sections. Assuming an isentropic frozen expansion between the node i and i+1 the temperature and pressure for each section can be calculated as follows: θ i = T tot T i = 1 + ( γi 1 2 ) M 2 i ; δ i = P tot P i = θ γi γ i 1 (3.45) where, γ i is the burned gases isentropic coefficient at section no.i. This is a function of T M i is burned gases Mach number at section no.i. T tot is the burned gases total temperature (input). P tot is the burned gases total pressure (input). The closing condition to calculate the Mach number knowing the area ratio is: ( ) A γ i +1 i 2θi 2(γ i 1) M i = ; γi = f(t i ) (3.46) A th γ i

82 3. ESPSS: European Space Propulsion System Simulation The equations above are solved iterating in Mach and in temperature. Non-adiabatic, non-isentropic supersonic nozzle : It is assumed that the non-adiabatic process takes place into two separated steps: Heat losses : From section i to i+1, the following losses in enthalpy and entropy will take place: h tot,i+1 = h tot,i q w,i /ṁ th (3.47) s i+1 = s i q w,i /ṁ th /T i (3.48) where q w,i is calculated with Equation 3.39 (using Bartz correlations). For the first nozzle station (i = 1) the total enthalpy and the static entropy are known (those of the exit of upstream component). Expansion : Assuming now that the isentropic relations are valid we have: h tot,i+1 = h(n k,i+1, T i+1 ) + v 2 i+1/2 (3.49) s i+1 = s(n k,i+1, P i+1, T i+1 ) (3.50) ṁ th = ρ i+1 v i+1 A i+1 (3.51) N k,i is the number of moles of the chemical constituent k at the nozzle station i. It is calculated from the Minimum Gibbs energy method: (N k, T ) i+1 = f mingibbs (N k,i, s i, P i+1 ). The three equations above with three unknowns (pressure, temperature and speed) are solved iterating in pressure and temperature. Two possibilities are foreseen calling the previous function: Equilibrium and frozen flow. In the last case, molar fractions remain constant. The following numeric derivatives are done to calculate the isentropic coefficient 68

83 3. ESPSS: European Space Propulsion System Simulation at equilibrium conditions: ln ν ln T = 1 + ln N tot,dt / ln(1 + ɛ) ln ν ln P = 1 + ln N tot,dp / ln(1 + ɛ) where N tot,dt, N tot,dp are the total number of moles (increments with respect to one) due to separated perturbations in P/T ( P = ɛp ; T = ɛt, ɛ = 1e 4): MinGibbsEnergy_PT ( mix, N k, T (1 + ɛ), P, N k,dt, N tot,dt ) MinGibbsEnergy_PT ( mix, N k, T, P (1 + ɛ), N k,dp, N tot,dp ) N k is the number of moles of the chemical constituent k at equilibrium conditions. N k,dt, N k,dp are the number of moles after perturbation in T, P. The calculation of the specific heat at constant pressure, Cp, has two terms: Cp fr and Cp re. The frozen one is calculated as in Section and the reactive term is calculated as follows: where, Cp = Cp fr + Cp re Cp re = ( ) ln Nk ln T k N h h k T MW mix ( ) ( ln Nk ln(nk,dt /MW mix,dt ) ln(n k /MW mix ) ) = ln T ln(1 + ɛ) Then calculation of the specific heat at constant volume and the isentropic coefficient is: R Cv = Cp + MW mix γ = Cp Cv ( ln ν/ ln P ) ( ln ν/ ln T )2 ln ν/ ln P Under frozen conditions, ( ln ν/ ln P ) is equal to one. Regarding the heat exchanged with the walls, it is used the same formulation present in the Combustor model (see Equation 3.39). In the case of an ideal nozzle it is supposed that the heat exchanged with the walls is a small quantity with 69

84 3. ESPSS: European Space Propulsion System Simulation respect to the enthalpy flow, so the isentropic equations are supposed still valid. In order to calculate the thrust of the engine, here below the expressions needed to calculate the thrust F and the Isp at section i: F i = ṁ th M i γi R gas T i + (P i P out ) A i ISP i = F i /ṁ th where P out is the external boundary condition in pressure; R gas represents the burned gases constant at nozzle exit ṁ is the mass flow at throat Cooling Jacket components These components represent a Regenerative Circuit of a Chamber. Two models are here described: the CoolingJacket component and the CoolingJacket_simple component. For the first one a 3-D geometry (built by means of several 3-D walls around the channels) is taken into account. For the second one, a simplified wall geometry is considered. Cooling Jacket component It is constructed by aggregation of one Tube (fluid_flow_1d library) representing the channels and five 3-D walls around them. The cooling jacket is divided into a variable number of sections in axial direction. Every section is made of: one fluid node of the Tube component (fluid_flow_1d library), which is simulating the cooling channels five slides of the wall_3d components, which are simulating the metallic walls. They are arranged according to Figure 3.5 Channels: they are simulated by only one Tube component making its num data equal to the number of channels. The mesh size of each node is function of the axial position through the non-dimensional geometry tables. 70

85 3. ESPSS: European Space Propulsion System Simulation Figure 3.5.: Cooling jacket wall mesh [43] The rectangular channel geometry (widths, heights, wet areas) is similarly calculated but using the interpolated widths and heights values as follows: A wet,i = 2 (a i + b i ) l ch,i a i = w ch interp(x i /L, wc_vs_l) b i = t ch interp(x i /L, tc_vs_l) Heat conduction: the wall_3d components used to simulate the walls will calculate the heat conduction in every direction including the axial direction. This thermal component features thermal ports in radial and in azimuth directions allowing an exact calculation of heat conduction through the channel corners. The walls are divided in 5 different 3-D components as shown in Figure 3.5. Each component has a 3-dimensional discretisation in tangential, radial and longitudinal direction (dx, dy, dz), respectively. The formulation for this component is the typical one for 3-dimensional conduction elements; the thermal capacitance for each volume is defined as: C i,j,k = ρ Cp (i,j,k) dx dy dz (3.52) 71

86 3. ESPSS: European Space Propulsion System Simulation the internal heat flows are evaluated by: q x(i,j,k) = k i,j,k dy dz (T i 1,j,k T i,j,k )/dx q y(i,j,k) = k i,j,k dx dz (T i,j 1,k T i,j,k )/dy q z(i,j,k) = k i,j,k dx dy (T i,j,k 1 T i,j,k )/dz (3.53) while the energy equation is: C i,j,k dt i,j,k dt = q x(i,j,k) q x(i+1,j,k) + q y(i,j,k) q y(i,j+1,k) + q z(i,j,k) q z(i,j,k+1) (3.54) As shown in Figure 3.5 only half channel has been considered because of symmetry reasons, with left and right sides adiabatic: q out,right_r = 0 q out,int_l = 0 q out,int_right_r = 0 q out,ext_l = 0 q out,ext_right_r = 0 The heat flux to the external side is calculated as the sum of all wall nodes connected to the ambient: N N q amb (i) = 2 n ch q out,ext (j) + q out,ext_right (j) j=1 j=1 The temperatures of the external walls (for each section i) in contact with the exterior are supposed to be the same: { Tw,amb (i) = T out,ext (j) T w,amb (i) = T out,ext_right (j) The same procedure is applied for the heat fluxes related to the internal side, the combustion chamber: N q in (i) = 2 n ch q in,int (j) + j=1 T w,in (i) = T in,int (j) T w,in (i) = T in,in_right (j) N q in,int_right (j) j=1 72

87 3. ESPSS: European Space Propulsion System Simulation The wet surfaces of the walls (wall_int, wall_right and wall_ext) in contact with the channel coolant, are supposed to be at three different temperatures (one for each side): T ch,in = T wall,int T ch,lat = T wall,right T ch,out = T wall,ext The corresponding heat flux (for each section i) between the channel and the walls around it are calculated as follows: N q ch,in = 2 q wall,int (j) q ch,lat = 2 q ch,out = 2 j=1 N q wall,right (j) j=1 N q wall,ext (j) j=1 The channel heat fluxes are also calculated by the Tube component using as input the wall/fluid temperatures and two phase hydraulic correlations for the film coefficient evaluation. Then, a set of implicit non linear equations is formed which is solved by EcosimPro. This also applies for the combustion chamber side, where the heat fluxes are calculated by the Combustor component connected to the CoolingJacket component. Concerning the external side (ambient), the heat flow shall be defined according to the thermal component connected to this port. Cooling Jacket simple component This component is constructed by aggregation of one Tube (fluid_flow_1d library) representing the channels and three 1D bars around them. The philosophy is the same as in the CoolingJacket component with the difference that here the five 3D-walls are replaced by three 1D-bars with only one thermal node per section. So, the cooling jacket is divided into a variable number of sections in axial direction. Every section is made of: one fluid node of the Tube component (fluid_flow_1d library), which is 73

88 3. ESPSS: European Space Propulsion System Simulation Figure 3.6.: Simplified Cooling Jacket wall disposition [43] simulating the cooling channels three slides of the bar_1d components, which are simulating the metallic walls. They are arranged according to Figure 3.6 Figure 3.6 shows the different temperatures and heat fluxes within the cooling channel. There T c is the temperature of the hot combustion gas, T W is the temperature of either the combustion chamber - or nozzle wall, T R is the temperature of the cooling rib, T CH is the temperature of the cooling fluid, T S is the temperature of the surface and T amb is the ambient temperature. For the model it is assumed that the temperatures are constant over the separate volumes. Heat always flows from regions with a higher potential to regions with a lower potential. Starting with the heat flux Q CC,W from the hot combustion gas to the chamber - or nozzle wall, the heat flux then splits up in a smaller part which goes into the channel ribs Q W,R and the main part which goes into the cooling fluid Q W,CH. From the ribs the main part flows into the cooling channel Q R,CH, but also a small part flows to the surface Q R,S. The heat from the channel is carried downstream with the fluid, but a small fraction might flow to the surface Q CH,S. The surface itself transfers heat to 74

89 3. ESPSS: European Space Propulsion System Simulation Figure 3.7.: Channel with relevant areas and surfaces for heat flux calculation the surrounding Q rad. There Q CC,W and Q rad transfer heat by radiation and convection. For the calculation of Q W,CH, Q R,CH and Q CH,S one solely has to consider the convective heat transfer. The heat fluxes Q W,R and Q R,S within the solid material are based on the conductive heat transfer. Prior to the listing of the heat fluxes the geometry of the cooling channels are explained. Hence, the cross-sectional area of the cooling channel is given once more in Figure 3.7. Additionally, the relevant areas for the heat flux calculation are entered. A int, A rib, A ext and A ch describe the cross-sectional areas of the wall, the cooling rib, the surface and the cooling channel. They can be calculated as follows: A int = ( π D ) n + w ti ch 4 2t c + t i + t e A rib = w r ( 2 A ext = π D ) n + w te ch 4 A ch = w c t c 75

90 3. ESPSS: European Space Propulsion System Simulation S w, S w,r, R, S ch,int, S ch,rib, S ch,ext, S r,s, and S ext describe the intermediate surfaces between the internal space of the combustion chamber or nozzle, the wall, the cooling rib, the cooling fluid, the surface and the surrounding. They can be calculated as follows: S w = S ext = l (w ch + 2w r ) S w,r = l wr 2 + t 2 i S r,s = l wr 2 + t 2 e S ch,int = S ch,ext = l w ch S ch,rib = l t ch By means of these values, the equations for the heat fluxes and the energy conservation can be set up. In the following the heat fluxes and subsequent the equations for the energy conservation are listed for a segment i: q wall = h c A wet (T aw tp.t ) + σa wet (Tc 4 tp.t 4 ) q cha = lπd λ int 2n t i /2 (tp.t T w,int) q coo,int = lw c λ int 2 t i /2 (Channel.tp in.t T w,int ) q coo,ext = lw c λ ext 2 t e /2 (Channel.tp out.t T w,ext ) λ int ( ) q coo,rib = lw c Channel.tplat.T T w,rib w r /2 λ int ( ) q rib,int = S w,r Tw,rib T w,int (t i + t c + w c )/2 λ ext ( ) q rib,ext = S r,s Tw,rib T w,ext (t e + t c + w c )/2 (3.55) where λ int and λ ext are the heat conductivities of the internal walls/ribs and external walls, respectively. Please note that the heat flux q wall equals the heat flux from the combustion chamber (Eq. 3.39). Additionally to the radial heat fluxes (Equations 3.55), the longitudinal heat fluxes need to be regarded. Since the cooling channel has four solid cross-sectional areas (AS, 2 x AR, AW), four longitudinal heat fluxes are calculated at every junction. Figure 3.8 illustrates these heat fluxes for a segment i. 76

91 3. ESPSS: European Space Propulsion System Simulation Figure 3.8.: Longitudinal heat fluxes for a segment i The longitudinal heat fluxes are based solely on the heat transfer mechanism conduction. In the following the associated equations are given: 2λ int q z,int (i) = A int (T w,int (i 1) T w,int (i)) l i 1 + l i 2λ ext q z,ext (i) = A ext (T w,ext (i 1) T w,ext (i)) l i 1 + l i 2λ int ( q z,rib (i) = A rib Tw,rib (i 1) T w,rib (i) ) l i 1 + l i (3.56) In order to calculate the energy conservation of the cooling jacket, one has to regard the different volumes separately. Subsequent the equations are given for the wall, the cooling rib and the surface: dt w,int (i) (ρ int V int ) i Cp(i) = q cha (i) q coo,int (i) q rib,int (i) + q z,int (i) q z,int (i + 1) dt dt w,ext (i) (ρ ext V ext ) i Cp(i) = q ext (i) q coo,ext (i) q rib,ext (i) + q z,ext (i) q z,ext (i + 1) dt dt w,rib (i) (ρ int V rib ) i Cp(i) = q rib,int (i) + q rib,ext (i) q coo,rib (i) + q z,rib (i) q z,rib (i + 1) dt (3.57) 77

92 4. Steady State Library The already available propulsion library ESPSS can be used to study both stationary states and transients of a propulsion system. Unfortunately its use for steady state applications is not trivial because of the complexity of the transient models there implemented. Therefore, simplified pre-design and parametric studies are difficult and time-consuming. The library shown hereafter is designed specifically for steady state purposes, providing a helpful and fast tool for the pre-design phase (feasibility analysis) allowing for parametric studies. To this aim, the available fluid properties and combustion modelling functions of ESPSS have been implemented in an adequate form into new libraries. Additionally, fluid dynamic, combustion and heat transfer models have been developed to simulate the physical steady state behaviour of the main components of a propulsion system, as pipes, valves, turbines, pumps, orifices, combustion chamber and nozzle. These components are suited for both launcher and spacecraft applications Components Overview The set of models developed for the State state library are able to represent most of the components of a liquid propulsion system for spacecraft or rockets. Figure 4.1 shows an overview of the components model developed for the steady state library Ports Ports are used to connect components to each other, in order to guarantee the propagation of the variables. Two new ports have been created: the Steady State fluid port [34], that represents the basis and the rationale on which the 78

93 4. Steady State Library Figure 4.1.: Components in the Steady State library 79

94 4. Steady State Library whole library is coded, and the nozzle port, to better manage nozzle connections, similarly to the transient nozzle port. Each port can connect two or more components at once. SUM variables, such as mass flow rate, will be summed at the ports to ensure mass flow conservation; EQUAL variables, such as stagnation pressure, will be propagated to all components connected to the same port. A standard flow component should have two ports, one IN and one OUT, thus defining the mass flow direction. IN ports ensure calculation of the enthalpy flow ṁh, while enthalpy h is computed in the OUT ports. Similarly to the original ESPSS fluid port, the Steady State fluid port can propagate the molar fraction of chemical species to allow the correct evaluation of fluid flow of combustion products in systems where this is required (e.g. staged combustion cycles) The type switch Most components have a type switch, that enable switching the model between Design and Off-Design mode. Design mode. Geometrical construction data for junctions and valves are an output of the components. They are calculated from a given P. The combustion chamber requests chamber pressure, mixture ratio MR and throat diameter as main design inputs. The propellant mass flows and heat fluxes are its main outputs. Turbomachinery components in design mode evaluate performance parameters as torque, power and rotational speed using the mass flow and pressures coming from the ports. Off-Design mode. This mode can be used for the analysis of a given cycle with fixed geometry and main characteristics. Here, junctions and valves have a given geometry. Mass flow or P are calculated, depending on the relative placement of the components. Combustion chambers have a given nozzle throat diameter (as in design 80

95 4. Steady State Library mode), but mass flows are given from the inlet ports. Chamber pressure and MR are calculated accordingly. From this general description some additional options are given in Design mode, in order to fine-tune the models depending on the cycle studied. Therefore, mass flow ratios through splits can be user-fixed (inputs), or calculated as outputs. Likewise, turbine pressure ratios can be fixed or calculated from the given cycle D pipes Tubes components are able to evaluate a one-dimensional flow in steady state conditions. The tube takes into account the enthalpy variation due to external heat fluxes and the pressure drop due to the friction along the pipe. As in the transient version of the component, the steady state tube is divided in volumes and junctions. Pressure drops and enthalpy variations are calculated at the end of each volume, on the junction. The governing equations based on the one-dimensional steady state model are the following: Mass conservation d (ρva) = 0 (4.1) dx Momentum conservation dp dx dv + ρv dx + 1 fr ρv2 = 0 (4.2) 2 D i where the friction factor fr is a function of Re and relative roughness, as defined in Ref. [43]. Energy conservation ρva dh 0 dx = h c (T w T ) P w (4.3) 81

96 4. Steady State Library Variable Description Unit Inputs ṁ Mass flow rate [kg/s] h 1 Inlet enthalpy [kj/kg] P 2 Outlet Pressure [Pa] Parameters P o Initial pressure [Pa] T o Initial temperature [K] rho o Initial density [kg/m 3 ] m o Initial mass flow [kg/s] rug Roughness [m] k f Multiplier of the friction factor [-] alpha b end Bend angle [deg] R b end Ratio of curvature bend [m] fld a dd Additional losses in fl/d [-] ht_option Heat transfer option [-] L Tube length [m] D Nominal tube inner diameter [m] Table 4.1.: 1-D pipe element The heat flux is evaluated using the thermal port and connecting the component with all components inside the EcosimPro thermal library. It has been decided to use the original transient EcosimPro thermal library, but still allowing to be interfaced with the steady state library. This choice enables a relaxation in the overall steady state model stiffness thanks to the first order differential equations present in the thermal components. Of course, due to the specifications of the steady state library, the time variable will have no physical meaning anymore, and it must rather be regarded as an integration constant. The pipe component is inherited from the tube. Additionally it features a 1-D wall model for the evaluation of the heat fluxes and heat capacities. As in the corresponding transient component, it includes a material pipe thermally connected to the tube, and permits simple convection with the ambient by using a constant convective coefficient h c. 82

97 4. Steady State Library D components: junctions & valves 0-D components represent concentrated pressure loss in a propulsion system, such as orifices and valves. These components are based on a similar model; the valve component differs from the orifice only because it enables the variation of the throat area while the orifice presents a constant one. Both the components feature a Design and an Off-Design mode. Variable Description Unit Inputs ṁ Mass flow rate [kg/s] h 1 Inlet enthalpy [kj/kg] P 2 Outlet Pressure [Pa] Outputs h 2 Outlet enthalpy [kj/kg] P 1 Inlet Pressure [Pa] A Valve Area [m 2 ] Parameters P Pressure loss (Design mode) [Pa] ζ Pressure drop coefficient [-] m o Initial mass flow [kg/s] Table 4.2.: 0-D Junction element In the Design mode the pressure drop is not related to the mass flow but is an input coming from the ports or from the user and the geometric parameters of the junction/valve are calculated: K jun = 2 P ρ ṁ 2 K valve = 2 P ρ ṁ 2 pos 2 ζ A = K In the second case, the Off-Design mode, the model calculates the concentrated pressure drop as: ṁ 2 P = ζ 2ρA 2 (4.4) 83

98 4. Steady State Library and the mass flow is calculated implicitly from Equation Combustion Chambers The thrust chamber is composed by two different components following the same idea developed in the transient library. A combustor component and a nozzle component are linked together to create the thrust chamber. Combustor This component represents a non adiabatic 1-D combustion process inside a convergent chamber (up to the throat section). It is a steady state, one dimensional, isoenthalpic combustion chamber. The equilibrium combustion products are calculated using the minimum Gibbs energy method [55] as a function of the propellant s mixture molar fractions and enthalpies and of the chamber pressure. The chamber geometry allows for precise chamber contour definitions and non homogeneous node discretisation. Thermodynamic properties along the chamber sections are evaluated using isoenthalpic correlations in frozen conditions. Heat fluxes are calculated with the Bartz correlation in closed form [9]. The chamber works only in ignited mode, as it is not required for a steady state model to show transitions between non burning and burning state. The compositions of the combustion products are evaluated from the injected fluids. It is possible to use either pure fluids or combustion products from a previous combustor (preburner). Along with the Design/Off-Design type switch, another switch is responsible for choosing the combustor type, which can be either a Main Combustion Chamber (MCC) or a Preburner (PB_GG). The main difference between the two combustor types is the following: - MCC. For a Main Combustion Chamber in Design mode, chamber pressure is user given (P c = P design ) - PB_GG. For a Preburner in Design mode, chamber pressure is taken from the outlet port (P c = f_out.p ) 84

99 4. Steady State Library Variable Description Unit Inputs h ox Oxidiser Inlet enthalpy [kj/kg] h fu Fuel Inlet enthalpy [kj/kg] Outputs ṁ Mass flow rate [kg/s] ṁ ox Oxidiser mass flow rate [kg/s] ṁ fu Fuel mass flow rate [kg/s] P c Chamber pressure [Pa] T c Chamber temperature [K] T w Chamber wall temperature [K] Parameters Pc o Chamber pressure [initial if Off-D; assigned if D] [Pa] Tc o Initial combustion temperature [K] Tc o x Initial Oxidiser combustion temperature [K] Tc f u Initial Fuel combustion temperature [K] MR o Mixture Ratio [initial if Off-D; assigned if D] [-] η c Combustion efficiency [kg/s] Lc Chamber length of subsonic part [m] Dt Nozzle throat diameter [m] Table 4.3.: Combustor element 85

100 4. Steady State Library In Design mode, the combustor component calculates inlet mass flows from given chamber pressure, MR and nozzle throat diameter. The mass flow conservation is written as: ṁ = ρ thv th A th η c (4.5) where the subscript th refers to throat conditions, and η c is the combustion efficiency. In Off-Design mode, the component evaluates the equilibrium composition in the first section to obtain thermodynamic and transport properties, and in the throat to evaluate the chamber pressure and the mass flow rate by an iterative loop. This equation is actually used in the overall loop to determine the chamber pressure P c implicitly. The ideal gas equation is written twice, for stagnation chamber and for throat conditions. Isentropic throat conditions are calculated iteratively assuming shifting equilibrium and variable isentropic coefficient γ. For each node i, the relevant characteristics (Mach number M i, P i, T i, ρ i, sound speed v sound,i, v i ) are calculated assuming isentropic flow conditions. This simplification is acceptable since these variables are only needed for assessing the heat transfer coefficient within the Bartz equation. Nozzle The component represents a 1-D supersonic nozzle in steady state conditions. The choked throat conditions (P th, T th and v th ) are evaluated in the combustor component and communicated through the nozzle port. Stagnation conditions are calculated from the throat conditions. Static conditions are evaluated in each section using isentropic correlations and assuming frozen chemistry. The heat flux in each section is evaluated using the semi-empirical correlation of Bartz in a closed form. 86

101 4. Steady State Library Variable Description Unit Inputs h Enthalpy at throat [J/kg] s Entropy at throat [J/kg K] P Pressure at throat [Pa] T Temperature at throat [K] ρ Density at throat [kg/m 3 ] Cp Specific heat at throat [J/ kg k] ṁ Mass flow rate [kg/s] Outputs P i Nozzle pressure profile [Pa] T i Nozzle temperature profile [K] T w Nozzle wall temperature [K] Isp Specific impulse [s] Thrust Thrust [N] Parameters Pc o Initial nozzle pressure [Pa] Tc o Initial nozzle temperature [K] η Cf Nozzle efficiency [-] P ext External pressure [Pa] Ld Nozzle length of supersonic part [m] Dt Nozzle throat diameter [m] Table 4.4.: Nozzle element 87

102 4.7. Cooling Channels 4. Steady State Library Cooling channels component features two different models depending on the complexity of the propulsion system: cooling jacket regenerative circuit The cooling jacket component is inherited from the tube component. It permits the modelling of a combustion chamber cooling jacket. Mass flow, pressure drop and temperature rise of the coolant are evaluated using the same governing equations as for the pipes and the tubes (see Equations 4.1, 4.2, 4.3). Since the direction of the flow in the Steady State library must be given at the schematic design stage, two components are foreseen: a co-flow and a counterflow cooling jacket. They are constructed by aggregation of one tube representing the channels and a simplified 3D geometry (built by means of several bars around the channels) around them (see Figure 4.2). The regenerative circuit component features a pre-defined pressure drop and a pre-assigned hot gas side wall temperature T w,hot profile along the combustion chamber. Mass flow value is coming from the ports; the component sends the T w,hot variable to the combustion chamber through the thermal port. In this way it is possible to evaluate the chamber heat flux q w (see Eq. 4.6). Using this variable, the enthalpy rise along the channel is evaluated and so the outlet temperature. Choosing a material for the chamber wall, the chamber wall thickness is an output of the design (see Eq. 4.7). The channel height is a user given input for the model, while the channel width is evaluated from the fixed number of channels. q w = h c A wet (T aw T w,hg ) (4.6) q w = λ t A wet (T w,hg T w,cf ) (4.7) 88

103 4. Steady State Library Variable Description Unit Inputs ṁ Mass flow rate [kg/s] h 1 Inlet enthalpy [kj/kg] P 2 Outlet pressure [Pa] Outputs P 1 Inlet pressure [Pa] h 2 Outlet enthalpy [kj/kg] T 2 Outlet temperature [K] T cool Coolant temperature [K] T w Channel wall temperature [K] q w Chamber heat flux [W] Parameters n ch Number of channels [-] P o Initial pressure [Pa] T o Initial temperature [K] rho o Initial density [kg/m 3 ] m o Initial mass flow [kg/s] rug Roughness [m] k f Multiplier of the friction factor [-] alpha b end Bend angle [deg] R b end Ratio of curvature bend [m] fld a dd Additional losses in fl/d [-] ht_option Heat transfer option [-] w ch Channel widths [m] t ch Channel heights [m] th i Jacket inner wall thickness [m] th e Jacket outer wall thickness [m] mat i Jacket internal material [-] mat e Jacket external material [-] L Channel length [m] D Nominal channel inner diameter [m] Table 4.5.: Cooling jacket element 89

104 4. Steady State Library Variable Description Unit Inputs ṁ Mass flow rate [kg/s] h 1 Inlet enthalpy [kj/kg] P 2 Outlet pressure [Pa] Outputs P 1 Inlet pressure [Pa] h 2 Outlet enthalpy [kj/kg] T 2 Outlet temperature [K] T cool Coolant temperature [K] q w Chamber heat flux [W] t w Internal wall thickness [m] Parameters n ch Number of channels [-] P o Initial pressure [Pa] T o Initial temperature [K] rho o Initial density [kg/m 3 ] m o Initial mass flow [kg/s] dp design Design pressure drop [Pa] T w Channel wall temperature [K] w ch Channel widths [m] t ch Channel heights [m] th e External wall thickness [m] mat i Jacket internal material [-] mat e Jacket external material [-] L Channel length [m] D Nominal channel inner diameter [m] Table 4.6.: Regenerative circuit element 90

105 4. Steady State Library Figure 4.2.: Cooling jacket channels wall mesh [43] 4.8. Turbomachinery Pump The pump component features a simple model using isentropic relations and constant, user-given efficiency η p to calculate pump conditions. The isentropic enthalpy rise is calculated assuming an isentropic transformation between inlet and outlet pressure. The Design type parameter decides whether the pump pressure rise is fixed from the ports (in Design mode) or calculated (in Off-Design mode). In both modes, given a assigned specific speed Ns, the shaft rotation speed ω is calculated: Ns = ω Q/n s (TDH/n st ) 0.75 (4.8) where n s is the number of suctions, n st is the number of stages of the pump, Q is the volumetric flow and TDH represents the actual total dynamic head of the pump. In Design mode, the inlet entropy s is calculated from inlet pressure P in and enthalpy h in. Then, the isentropic enthalpy rise dh is is calculated with an isentropic transformation, with known inlet and outlet pressures P in and P out, 91

106 4. Steady State Library Variable Description Unit Inputs h 1 Inlet enthalpy [J/kg] P 1 Inlet pressure [Pa] T 1 Inlet temperature [K] ṁ Mass flow rate [kg/s] Outputs P 2 Outlet pressure [Pa] ω Rotational speed [rad/s] τ Torque [N m] W Power [W] Parameters P o Initial pressure [Pa] T o Initial temperature [K] m o Initial mass flow [kg/s] rpm o Initial rotational speed [rad/s] η p Pump efficiency [-] Table 4.7.: Pump element and inlet entropy s in. Finally, the real enthalpy rise dh is calculated as: dh = dh is η p (4.9) and subsequently, the shaft rotational speed ω and torque τ are then linked with the actual enthalpy rise by the power balance equation: ω τ = ṁ h (4.10) In Off-Design mode, the inlet entropy s in is calculated (as for the Design mode) from inlet pressure P in and enthalpy h in. The real enthalpy rise dh is given by the power balance equation, and the isentropic enthalpy rise dh is is given by: dh is = η dh (4.11) Therefore, the outlet pressure P out is calculated with an isentropic transformation, with known isentropic enthalpy rise dh is and entropy s. As for the Design case, 92

107 4. Steady State Library it is possible to compute torque and power from it (thank to an assigned specific speed Ns). In the near future the off-design mode (with assigned specific speed) will be upgraded with the capability of using performance maps for efficiency and pressure head Turbine The turbine component is a model using isentropic relations and constant, usergiven efficiency η t to calculate turbine conditions. The isentropic enthalpy fall is calculated assuming an isentropic transformation between inlet and outlet pressure. The construction parameter turbine_type decides, in Design mode, whether the mass flow ṁ is assigned from the ports (known_mflow) thus calculating the upstream pressure, or the pressures are assigned from the ports (known_pressures, known_pi_tt), thus calculating the mass flows. This switch must be carefully set depending on the cycle studied: turbine_type = known_mflow. For closed cycles, in most cases, the mass flow is determined by the preburner (staged combustion) or by bypass valves (expander), therefore it is given from the ports, and the pressure ratio should be calculated in the turbine component. turbine_type = known_pressures. For open cycles (gas generator) or for one of the turbines in closed cycles with parallel turbines, the pressures should be fixed (known_pressures), and the model will find the mass flow that equilibrates the pump power. turbine_type = known_pi_tt. For open cycles, especially in the pre-design phase to have a first attempt result and to facilitate parametric studies changing the pressure ratio Π tt ; the model will find the mass flow that equilibrates the pump power. In all working modes (Design or Off-Design, known_mflow or known_pressures, known_pi_tt), torque, shaft rotation speed ω and power are given from the mechanical port. 93

108 4. Steady State Library Variable Description Unit Inputs h 1 Inlet enthalpy [J/kg] P 2 Outlet pressure [Pa] T 1 Inlet temperature [K] W Power [W] ṁ Mass flow rate (if know_mflow) [kg/s] P 1 Inlet pressure (if known_pressures) [Pa] PI tt Pressure ratio (if known_pi_tt) [-] Outputs ṁ Mass flow rate (if know_pressures or PI_tt) [kg/s] P 1 Inlet pressure (if know_mflow) [Pa] ω Rotational speed [rad/s] τ Torque [N m] Parameters P o Initial pressure [Pa] T o Initial temperature [K] m o Initial mass flow [kg/s] PI_tt o Initial pressure ratio [-] rpm o Initial rotational speed [rad/s] η t Turbine efficiency [-] Table 4.8.: Turbine element 94

109 4. Steady State Library Design mode - known_mflow. The mass flow is used to calculate the real enthalpy fall (Eq. 4.12). dh = P ower/ṁ (4.12) From this value, the isentropic enthalpy fall is calculated with the fixed efficiency η (Eq. 4.13). dh is = dh/η (4.13) The inlet entropy s in is calculated from the outlet pressure P out and the ideal outlet enthalpy h out = h in dh is. Then, the inlet pressure is calculated from entropy s in and inlet enthalpy h in. - known_pressures. The inlet entropy s in is calculated from the inlet pressure P in and the inlet enthalpy h in. Then, the isentropic enthalpy fall dh is is calculated knowing the inlet enthalpy h in, the outlet pressure P out and the inlet entropy s in (Eq. 4.14). dh is = h in f(s in, P out ) (4.14) Thereafter, the real enthalpy fall is given by using the turbine efficiency η (Eq. 4.15). dh = η dh is (4.15) The mass flow is finally estimated from Eq known_pi_tt. When this turbine type is chosen, the calculation procedure differs from the one used in known_pressures only by the inlet pressure P in given as an output from P out times Π tt. Off-Design mode. The procedure is roughly the same as the Design mode case with turbine_type = known_pressures Validation Several test cases have been performed in order to evaluate the reliability of the Steady State library. Following a step by step approach, first each component 95

110 4. Steady State Library singularly and then more complex systems were validated Component validations The schematics shown hereafter are only the graphical interfaces of mathematical models where all variables of each component are considered. The tool is able of calculating the steady state of the mathematical model by solving the non-linear algebraic equation system that results from the built schematic by means of the Newton-Raphson or the Minpack method [40]. In order to have a flexible and robust tool, each component model is carefully coded to provide the most suitable variables that can be used to break the nonlinear algebraic loops. The choice of the correct variables that enable the equations system to converge represents one of the most important achievements of this work. Pipeline test case The purpose of this test case is to validate the Steady State pipe component and demonstrate its proper function compared to a transient component. The schematic shown in Figure 4.3 has been also built to check the correct behaviour of Steady State components in long pipelines. A long pipeline is modelled twice, with standard ESPSS transient components and with Steady State components. Pressure drop distribution along the pipeline and mass flow rates are compared between the two models. Please note the absence of the volume between two junctions. Purely capacitive components are not needed in the Steady State library, and it is possible to chain multiple resistive components in series. The schematic describes a series of pipes linked together by junctions. A pressure difference has been imposed between inlet and outlet. The input data shown in Table 4.9 represent the inputs implemented in each component (steady or transient). The initial conditions have been taken equal for each pipe, with atmospheric pressure and low initial mass flow. These conditions are quite distant from the solution, and the convergence of the steady state code in this case is an indicator of its robustness. 96

111 4. Steady State Library Figure 4.3.: Schematic of the Pipeline test case. Purple: steady state components. Cyan: transient components Name Description Value Units P in Total Pressure at inlet 50 [bar] T in Total Temperature at inlet 300 [K] P out Total Pressure at outlet 30 [bar] P o Initial Total pressure in the pipe 1 [bar] T o Initial Total temperature in the pipe 300 [K] m o Initial mass flow in the pipe (guess value) 0.2 [kg/s] rug Roughness 5e-05 [m] L Pipe length 1 [m] D Pipe internal diameter 0.01 [m] nodes Pipe nodes discretisation 5 [-] Ao Junction area 7e-05 [m 2 ] ζ Loss coefficient 1 [-] fluid Working fluid Real H2O [-] Table 4.9.: Pipeline input data 97

112 4. Steady State Library The simulation results are summarized in Table 4.10, showing a very accurate mass flow calculation and pressure drop distribution. Name Value Transient Value Steady State Error m [kg/s] % P 1 [bar] % P 2 [bar] % P 3 [bar] % P 4 [bar] % Table 4.10.: Pipeline output data Combustion Chamber The purpose of this test case is to calculate the main characteristics of combustion chamber and nozzle components. Its schematic is shown in Figure 4.4 for both steady state and transient models. The only difference between the two models (besides the different modelling approach) is the absence of the injector capacity inside the Steady State combustion chamber injection plate. Relevant input data are listed in Table The initial conditions are the same for the steady state and the transient components. The test compares the propellant mass flows and the chamber pressure and temperatures between the two models. Other important characteristics as heat fluxes, wall temperatures and adiabatic wall temperatures have been evaluated as well, but are not reported here for simplicity. The output data from the two models are compared in Table It is evident that the steady state results are very similar to the respective transient results. 98

113 4. Steady State Library Figure 4.4.: Schematic of Combustion Chamber test case. Purple: steady state components. Cyan: transient components Name Description Value Units P in,ox Ox. Total Pressure at inlet 70.8 [bar] T in,ox Ox. Total Temperature at inlet 94.7 [K] P in,fu Fu. Total Pressure at inlet 71 [bar] T in,fu Fu. Total Temperature at inlet [K] N sub Number of subsonic nodes 5 [-] N sup Number of supersonic nodes 5 [-] L cc Chamber length of subsonic part 0.5 [m] D th Nozzle throat diameter 0.10 [m] P cc Initial Chamber pressure 1 [bar] T cc Initial Chamber temperature 300 [K] Table 4.11.: CC input data 99

114 4. Steady State Library Name Transient Value Steady State Value Error m ox [kg/s] % m fu [kg/s] % m tot [kg/s] % MR [-] % P cc [bar] % T cc [K] % Mach [-] % Table 4.12.: CC output data Subsystem validations HM7B Turbopump subsystem This test case was used during the ESPSS Industrial Evaluation from Astrium Bremen to validate the ESPSS library for liquid rocket engine cycles [41]. The schematic shown in Figure 4.5 represents the turbomachinery power pack of the upper stage engine of the Ariane 5 launcher, the HM7B engine, including the gas generator and both turbopumps. Figure 4.6 shows the equivalent schematic implemented with Steady State components. They are very similar to each other. Only volume components and non-condensable fluid lines are absent. The first ones are not needed for the same reasons stated in Section 4.9.1; the latter have been eliminated since there is no need to model the Helium purging phases in a steady state simulation. The steady state model has been used in Off-Design mode. The chosen input data are collected in Table 4.13; Table 4.14 summarizes the main system variables results performed by the transient and the steady state models. The steady state model matches very well the transient one for fluid flow, turbomachinery and gas generator main parameters. 100

115 4. Steady State Library Name Description Value Units P in,ox Total pressure in LOX tank 2.0 [bar] P in,fu Total pressure in LH 2 tank 3.0 [bar] P out,ox Total pressure at Pump outlet/gas Generator inlet 50.0 [bar] P out,fu Total pressure at Pump outlet/gas Generator inlet 55.0 [bar] P cc Initial chamber pressure 20.0 [bar] T cc Initial chamber temperature 900 [K] ω p,ox Initial LOX pump speed 1000 [rpm] ω p,fu Initial LH 2 pump speed 6000 [rpm] Table 4.13.: HM7B Turbopump input [44] and initial data Name Nominal Value [44] Error Transient/Steady State ṁ gg,ox [kg/s] 0.3% ṁ gg,fu [kg/s] 0.07% MR [-] 0.5% P gg [bar] 1.3% T gg [K] 0.08% ṁ t [kg/s] 0.3% ω t [rpm] % τ t [N m] % ṁ p,ox [kg/s] % P ox [bar] % ω p,ox [rpm] % τ p,ox [N m] 10.4% ṁ p,fu [kg/s] % P fu [bar] % ω p,fu [rpm] % τ p,fu [N m] 6.0% Table 4.14.: HM7B Turbopump output data 101

116 4. Steady State Library Figure 4.5.: Turbopump test case: HM7B power pack transient schematic Figure 4.6.: Turbopump test case: HM7B power pack steady state schematic 102

117 4. Steady State Library HM7B Chamber subsystem This test case represents the combustion chamber subsystem of the HM7B engine. The aim of this test case is to validate the behaviour of the combustion chamber and cooling jacket components when they are coupled together in a simulation, by comparing results with the transient model simulation. Figures 4.7 and 4.8 show the schematics of the combustion chamber subsystem using the ESPSS transient library and the steady state model, respectively. As in the previous test case, described in Section 4.9.2, the similarity of the two schematics shown hereafter is evident. The only difference for the steady state model is the absence of non-condensable fluid lines and capacitive components such as volumes. Also here the steady state model has been used in Off-Design mode. Figure 4.7.: Chamber test case: HM7B Combustion Chamber transient schematic In Table 4.15 the main input data for both systems are collected; in Table 4.16 the main system variables results are summarized, performed by the transient and the steady state models. As reported in the table the steady state model matches the transient results, showing very good agreement between the values of the combustion chamber, and of the cooling channel model. 103

118 4. Steady State Library Figure 4.8.: Chamber test case: schematic HM7B Combustion Chamber steady state Name Description Value Units P in,lox Total pressure at pump outlet/chamber inlet 50.0 [bar] P in,h2 Total pressure at pump outlet/chamber inlet 55.0 [bar] P cc Nominal chamber pressure 36.6 [bar] n ch Numbers of channels 128 [-] P i,cc Initial chamber pressure 30 [bar] T i,cc Initial chamber temperature 1000 [K] P o Initial total pressure in the channels 49 [bar] T o Initial total temperature in the channels 30 [K] m o Initial mass flow in the channels 2 [kg/s] Table 4.15.: HM7B CC input [44] and initial data Name Nominal Value [44] Error Transient/Steady State m ox [kg/s] % m fu [kg/s] % m tot [kg/s] % MR [-] % P cc [bar] % T cc [K] 0.88% m ch [kg/s] % P ch [bar] 9.9% T out,ch [K] 6.7% Table 4.16.: HM7B CC output data 104

119 4. Steady State Library Engine cycle designs At the beginning of a design analysis, a set of performance parameters must be chosen as assumption to define the engine class and the initial condition of the engine: - Propellants - Tank pressure and temperatures - Chamber Pressure P c - Chamber Mixture Ratio MR - Combustion efficiency η c - Throat Diameter Dt - Pump efficiencies η p - Pump specific speeds Ns - Turbine efficiencies η t Subsequently, it is possible to evaluate other important characteristics such as the contour of the chamber (by use of L* and simple geometrical correlations) and injector pressure drops. HM7B rocket engine system The HM7B is a gas generator cycle with single turbine and geared pumps. The two subsystems modelled in the previous work have been updated to the current library implementation and linked together to build the HM7B engine system model. Figure 4.9 shows the schematic of the HM7B engine using the Steady State library. All model components are in Design mode. In Table 4.17 the main input data for the system are collected; in Table 4.18 the main system variables results are summarized and compared with nominal data. Where nominal values are shown, they are taken from the open literature. 105

120 4. Steady State Library Figure 4.9.: HM7B engine system schematic 106

121 4. Steady State Library Name Description Value Units P in,lox Total pressure in LOX tank 2.0 [bar] T in,lox Total temperature in LOX tank 91.2 [K] P in,lh2 Total pressure in LH 2 tank 3.0 [bar] T in,lh2 Total temperature in LH 2 tank 21.0 [K] P cc Nominal chamber pressure 36.6 [bar] MR Nominal chamber mixture ratio 5. [-] n ch Numbers of channels 128 [-] T i,cc Initial chamber temperature 1000 [K] P o Initial total pressure in the channels 50 [bar] T o Initial total temperature in the channels 30 [K] m o Initial mass flow in the channels 2 [kg/s] P c,gg Initial gas generator pressure 20 [bar] T ch Initial gas generator temperature 900 [K] rpm ox Initial LOX pump speed 1000 [rpm] N s,ox LOX pump specific speed [-] rpm fu Initial H 2 pump speed 6000 [rpm] N s,fu H 2 pump specific speed 9.29 [-] m o,tu Initial Turbine mass flow 0.2 [kg/s] Table 4.17.: HM7B input [44] and initial data Pressure drop has been fixed in each valve and junction as well as turbomachinery efficiency and gas generator mixture ratio. For the turbine, the known_pi_tt type is chosen, while the pump specific speed Ns has been calculated from design data. The main chamber pressure and mixture ratio are fixed. Propellant mass flows to the main chamber are calculated, and fed back to the upstream components. The gas generator mass flow is not fixed. Only its mixture ratio is fixed, in accordance with the maximum allowable temperature in the turbine. The mass flow is then calculated by an algebraic equation system resulting automatically from the connection of gas generator and turbopumps. The needed shaft power drives the total gas generator mass flow rate, since the turbine pressure ratio is fixed by design. Initial values such as mass flow rates and shaft speed are chosen by rough engineering assessments. The robustness of the Steady State library is demonstrated 107

122 4. Steady State Library Name Nominal Value [44] Error ṁ ox,gg [kg/s] 0.59% ṁ fu,gg [kg/s] 0.23% ṁ t,gg [kg/s] 0.3% P gg [bar] 0% T gg [K] 1.03% m cc,ox [kg/s] 0.88% m cc,fu [kg/s] 0.83% m cc,t [kg/s] % m ch [kg/s] 2.36% P ch [bar] 0.5% T out,ch [K] 6.7% ω t [rpm] % τ t [N m] 4.48% W t [W] 4.32% T in,t [K] 11.3% ṁ p,lox [kg/s] 0.96% P LOX [bar] % ω p,ox [rpm] % τ p,ox [N m] 1.39% ṁ p,lh2 [kg/s] 7.56% P LH2 [bar] % ω p,fu [rpm] % τ p,fu [N m] 6.0% Table 4.18.: HM7B engine system output data 108

123 4. Steady State Library by the stability of the simulation in a wide range of initial conditions. From Table 4.18 a very good agreement with nominal data is recognisable. Few parameters have an higher percentage error: the fuel mass flow rate in the pump does not take into account the dump and the tap-off mass flow rate vented from the engine. The cooling jacket exit temperature takes into account the temperature increase in the injector dome. If compared to the LH2 injector dome temperature, the error decreases to 2.47%. Turbomachinery parameters show quite good results; the differences are mainly due to the turbine inlet temperature that is lower then the nominal one. 109

124 4. Steady State Library RL-10A-3-3A rocket engine system The RL-10 is an expander cycle with single turbine and geared pumps. Being a closed cycle, the cycle design is more difficult than for an open cycle, because all parameters are strongly dependent from each other. In this test case a model of the RL-10A-3-3A rocket engine system has been created and simulated in Design mode. Model results from the steady state calculations have been compared with the typical engine performance parameters at nominal operating conditions [15, 4]. Pressure drops of the main valves and junctions are taken from engine typical values [15] as well as for the tank pressure and temperature conditions. Pumps and turbine efficiency are fixed to a constant value and taken from open literature [15, 4]. No calibration has been adopted for the control valves: the Oxidiser Control Valve (OCV) aperture ratio has not been trimmed because the mixture ratio is assigned in the combustion chamber. Mass flows are given by combustion chamber conditions. Turbomachinery power is regulated by the Thrust Control Valve (TCV) that is open at its nominal open area ratio of 9% [58]. In Table 4.19 the main input data for the system is collected; in Table 4.20 the main system variables Name Description Value Units P in,lox Total pressure in LOX tank 2.43 [bar] T in,lox Total temperature in LOX tank [K] P in,lh2 Total pressure in LH 2 tank 1.86 [bar] T in,lh2 Total temperature in LH 2 tank [K] P cc Nominal chamber pressure [bar] MR Nominal chamber mixture ratio [-] n ch Numbers of cooling channels 180 [-] T i,cc Initial chamber temperature 1000 [K] m o Initial mass flow in the channels 2 [kg/s] Table 4.19.: RL-10A-3-3A input and initial data results are summarized, performed by the steady state model and compared with 110

125 4. Steady State Library Figure 4.10.: Schematic of the RL-10 engine 111

126 4. Steady State Library performance parameters at nominal condition. Name Nominal Value Error m cc,ox [kg/s] % m cc,fu [kg/s] % m cc,t [kg/s] % m ch [kg/s] % T out,ch [K] % ω t [rpm] % ṁ t [kg/s] % W t [kw] % ṁ p,lox [kg/s] % ω p,ox [rpm] % W p,ox [kw] % ṁ p,lh2 [kg/s] % ω p,lh2 [rpm] % W p,lh2 [kw] % T hrust [kn] % Isp [s] % Table 4.20.: RL-10A-3-3A engine system output data The chamber pressure is assigned together with the mixture ratio. The pressure cascade for both propellant lines is calculated by the model according to design parameters such as valve pressure drops. The pressure rise for the LOX pump results directly from the calculated LOX pressure cascade, whereas an algebraic loop is solved for calculating the needed H2 pump pressure rise and the turbine pressure ratio. In parallel, several other algebraic loops are solved, determining key variables such as cooling channel outlet temperature and turbine mass flow. The nominal fuel bypass flow ratio [58] has been fixed using the split component. The turbine evaluates the required pressure ratio and the mass flow is then evaluated via the Thrust Control Valve (TCV). The cooling channel component evaluates iteratively with the combustion chamber component the heat fluxes and the wall temperatures. Finally, pumps 112

127 4. Steady State Library power is evaluated from the required pressure rise and mass flow, and rotation speed from the given specific speed Ns. The RL10 design model here described has proven to be very robust with respect to initial conditions. The comparison of initial conditions in Table 4.19 and results in Table 4.20 demonstrates this assertion. For example, the initial mass flow in the cooling jacket m o is 2 kg/s, whereas the simulation result yields 2.76 kg/s. From Table 4.20 a very good agreement with nominal values is recognisable. 113

128 5. Transient Modelling In this chapter we would like to introduce the new models developed for a better assessment of the phenomena occurring in the subsystems of liquid rocket engines during start-up. Three new, more complex and accurate models will be presented in this chapter: the first one for the injector plate, a second one for the evaluation of the heat transfer coefficient on the hot gas side of the thrust chamber, and the third one for the evaluation of the thermal stratification inside high aspect ratio cooling channels (HARCC) Injector Plate model The injector head s main task is to merge, mix and atomize the oxidizer coming from the main valve with the fuel coming from the cooling channels. Figure 5.1 shows a schematic illustration of an arbitrary injector head, in order to clarify a common structure. At first both propellants enter separate volumes in which they are uniformly distributed among the injector elements. Afterwards the injector elements dose the amount of propellant mass flow, by a defined pressure drop and atomize the propellants. For the computer model and for the mathematical model formulations, respectively, the configuration of both propellant lines in the injector head are simplified in that way, that one volume and one orifice are assumed in each line. The volume represents a collector where the propellant is distributed among the injector elements. Following, the orifices represent the in- and outlet of all the injector elements. In order to calculate the right flow velocity and Reynolds number in the injector element, one has to take the associated mass flow rate into account. Figure 5.2 (a) shows the component wise connection of the above mentioned volumes and orifices. Additionally, the convective and radiative heat transfer from the combustion chamber into the injector head has to be regarded. In this process 114

129 5. Transient Modelling Figure 5.1.: Schematic illustration of an arbitrary injector head a heat flux is transferred from the combustion chamber into the fuel cavity first and subsequently into the oxidizer cavity. The original injector plate model present in the ESPSS library features a very simplified thermal model. Indeed the injector plate topology takes into account the effects of the radiative heat transfer, but the conductive and convective heat fluxes are evaluated using only a virtual conductance. As in Figure 5.2 (a) the original injector plate is built by a radiative and a conductive component linked upstream in parallel directly to the combustor hot gases. These two components are linked to a capacitive component, used to simulate the total thermal inertia of the injector cavity walls. This heat capacity is then connected to the two fluid cavities. The original model is here described: q cond = λ c,hg (T core T cap ) q rad = σ (Tcore 4 Tcap) 4 q cap = q cond + q rad + q cav,ox + q cav,fu 115

130 5. Transient Modelling (a) topology schematic of the original injector plate (b) topology schematic of the new injector plate Figure 5.2.: Schematics of the injector plates 116

131 5. Transient Modelling where heat flux of the capacitive component is and for the cavities components q cap = CpM dt cap dt q cav,ox = h c (T w,cav T cav ) q cav,fu = h c (T w,cav T cav ) In this way it is not possible to evaluate the presence of convective heat transfer on injector face plate and to take into account the correct effect of the conductive and capacitive behaviour of the injector plate material. Moreover, the core temperature in the first volume of the chamber is considered as the wall temperature of the injector plate, and this is unrealistic. For this reason, an upgraded version of the injector plate topology inside the ESPSS library has been implemented [35]. The aim of this new model is to take into account the convective and radiative heat transfer between the fluid in the first volume of the chamber and the face plate, and evaluate the conductive and capacitive effect of the injector walls in more accurate way, representative of a generic injector head. The new structure of the injector plate (see Figure 5.2 (b)) wants to maintain the level of simplicity of the original model in order to keep the computational cost low, and to be applicable to several different injector geometries (impinging, coaxial, etc... ), but in the same time wants also to improve the heat transfer characteristics from the chamber to the injector cavities. In the first volume of the chamber the convective and radiative heat fluxes to the injector face are evaluated: q conv = h c,hg (T aw T w,hg ) q rad = σ (T 4 core T 4 w,hg ) using the mass and the material properties of the injector plate (heat capacity, thermal conductivity) the model evaluates the conductive heat transfer and capacity effect of the walls: 117

132 5. Transient Modelling ( λ q cond ox,fu = and for the capacitive components t ox,fu ) (T w,hg T w,cav ) q cap,hg = q cond,ox + q cond,fu + q conv + q rad q cap,ox = q cond,ox + q cav,ox q cap,fu = q cond,fu + q cav,fu q cap,k = CpM k dt dt with k = hg, ox, fu; The thermal conductivity and heat capacity values are function of the chosen material of the injector plate and of its temperature. For simplicity reasons the injector plate is assumed to be made of only one material. The thickness t used for the evaluation of the conductive heat flux has to be considered as a characteristic injector head thickness or width, and presents two different values, one for the ox side and the other for the fuel side. The capacitive components for each propellant side are divided in two parts in order to obtain three different temperatures: T w,hg, T ox,cav, T fu,cav, respectively the temperature of the injector plate on the hot side, the oxidizer and the fuel cavity wall temperature on the cold fluid side Qualitative behaviour In order to validate the behaviour of the new injector plate model, a numerical approach has been used because no experimental results were found in open literature. A pressure fed propulsion system has been modelled and tested with both injector plate models. The test case represents a typical spacecraft propulsion system supplied by nitrogen tetroxide (NTO) as oxidiser, and monomethylhydrazine (MMH) as fuel. The system is designed in order to reach a chamber pressure of 10 bar with a mixture ratio of 1.65 at steady state conditions. Figure 5.3 compares the two models by assessing the thermal behaviour inside the injector cavities and the injector plate walls. Table 5.1 summarizes the major features for each side of the injector plate evaluated by the new model at steady state conditions. It is evident that the temperature at injector plate wall in the original version of 118

133 5. Transient Modelling Figure 5.3.: Temperature profiles from original and new model Table 5.1.: Injector plate variables comparison Variable Fuel Oxidiser Chamber Input Propellant MMH N 2 O 4 Injector material Titanium Titanium Injector head mass [kg] 1.5 Injector head thickness [m] Injector head area [m 2 ] Chamber pressure [bar] 9.85 Chamber temperature [K] Mixture ratio [ ] 1.65 h c coefficient [W/m 2 K] 450 Inlet temperature [K] Output Inj. heat flux [W] Cavities T [K] Wall c,hg temperature [K]

134 5. Transient Modelling the model would have been unrealistic (T hg = T c = K). Only the presence of the virtual conductance allows the injector cavities not to increase the fluid temperature to unrealistic values. Using the newly developed model, the software is able to deliver reasonable outputs with physically valid geometries. Moreover it is possible to obtain different cavities wall temperatures for each propellant side, while before it could not occur Hot Gas side heat transfer coefficient models Models implemented In the ESPSS library the heat transfer coefficient inside the combustion chamber is evaluated using the well-known Bartz correlation [9]. The original formulation of this equation does not take into account several aspects, such as the combustion zone due to atomization, vaporization and combustion delays in the proximity of the injector plate, the boundary layer growth through the cylindrical part of the chamber, the correct evaluation of the flow acceleration in the convergentdivergent part of the nozzle, etc... The heat flux in ESPSS takes into account the convective and radiative phenomena: q w = h c A wet (T aw T w ) + σa wet (Tcore 4 Tw) 4 (5.1) Since many correction factors used in literature are based on Stanton type correlations, it was decided to use this kind of dimensionless number to evaluate the heat transfer coefficient. St = h c ρ v C p,ref = q ρ v C p,ref (T aw T w ) (5.2) the Stanton number represents the ratio between heat transferred to a fluid and the thermal capacity of this fluid. In the combustion chamber model three different correlations have been implemented [35]: Original Bartz Equation Modified Bartz Equation 120

135 5. Transient Modelling Pavli Equation In order to use the Bartz equation with Stanton type correction factors, the Bartz equation has been rewritten as a Stanton type equation: St Bartz = ( µ 0.2 ref ) (λref Cp,ref 0.6 µ ref ) 0.6 ( ) 0.1 (ṁ) 0.2 A 0.1 Dth π/4 (5.3) R curv where the thermodynamic and transport properties are calculated at the so-called film temperature calculated as: T ref = 0.5 (T st + T w ). To improve the behaviour of the original Bartz equation, a temperature correction factor K T was added, taking into account that the new reference temperature is calculated halfway between the wall and the free stream static temperature. Moreover, since the geometric reference parameter in the original Bartz equation was the throat diameter, a further correction factor K x was added for the consideration of the boundary layer growth in the cylindrical part and in the nozzle [5]: K T = ( Taw T ref ) a ( ) b x K x = (5.4) x th hence, the Stanton type modified Bartz equation becomes: St Bartz,mod = St Bartz K T K x (5.5) Because of its simplicity, the Pavli equation has been implemented as well. The Pavli equation including the two correction factors discussed before is [103], [5]: ( ) e ( ) f St P avli = Re 0.2 P r 0.6 Taw x (5.6) T ref x th The Reynolds number Re is calculated with respect to the local chamber diameter and the property reference temperature is an averaged boundary layer temperature. In this equation the temperature correction factor and the streamwise correction factor are also included. In order to improve the heat flux model in the combustion chamber another correction factor was added taking into account the vaporization phenomenon near the injector plate. In the so-called combustion zone the heat flux decreases when getting closer to the injector plate. This behaviour is due to the incomplete 121

136 5. Transient Modelling mixing and reaction of the flow for the given injector and combustion chamber. The mixing region has a finite length where the combustion is less effective, therefore the heat fluxes are lower. This correction factor is applied by using a Stanton type correlation derived from Bartz or Pavli equations. The combustion length can be given as an input or computed by a geometrical correlation. In the latter case, to generate a specific correction factor two steps are required: first, the length of this combustion zone x max has to be calculated based on the injector plate geometry; then a functional dependency of the heat flux in the range x 0 x x max has to be found. Once the combustion zone length is evaluated it is possible to calculate a correction factor by means of a tangential Stanton number dependency [5]: St (x) St(x max ) = 1 [ ( )] x 4 arctan (5.7) x max The last correction factor added to achieve a better agreement between the numerical and the experimental results is a correction factor K acc related to the flow acceleration. In fact, the measured heat fluxes are lower than the calculated ones upstream and downstream the nozzle throat. The behaviour is caused probably by the nozzle contour and therewith due to the flow acceleration (bigger boundary layer thickness). Instead of using the local velocity gradient to develop the correction factor, a more practicable way is to use the absolute value of the first derivative of the chamber radius with respect to the streamwise coordinate, dr/dx. The correction factor K acc requires two boundary conditions. In the cylindrical part K acc should be equal to 1. The other boundary condition is described by K acc = 0 and represents the disappearance of convective heat transfer due to flow separation. The following correction is used to take into account both conditions [5]: St = St K acc = St 1 dr dx (5.8) 122

137 5. Transient Modelling Validation In the years , in the frame of an ESA GSTP-2 contract, Astrium performed a series of experiments with a water cooled calorimetric combustion chamber [5]. The different correlations aforementioned based on Bartz ([5],[9]) and Pavli ([5],[103]) have been implemented to simulate the calorimetric combustion chamber tests and their results have been compared with the experimental results from this calorimetric chamber test campaign [57, 56]. The calorimetric chamber is a sub-scale, water cooled thrust chamber with twenty segments [116]. Each segment features an independent water feed system with volume flow measurement. For each segment the heat flux is measured individually. The described calorimetric system has been modelled using the following components [116]: 1 combustion chamber with 21 nodes (component to validate) 20 regenerative circuits with 5 nodes each 20 mass flow regulated water feed lines (with the necessary junctions and boundary conditions) mass flow regulated propellant feed lines The simulation was performed using the couple liquid oxygen/gaseous hydrogen as propellants, at an O/F ratio varying from 5 to 7 and at a total pressure from 35 to 70 bar in the combustor; for each test point the propellant mass flows are chosen in order to get the desired pressure and mixture ratio. In order to get the right pressure drop through the cooling circuit, the roughness of the cooling channels had to be adapted. Values between 3.2 and 25 µm were chosen. This tuning was necessary because of the partially unknown layout of the cooling circuit and its feed lines (pipes, fittings,... ). The heat fluxes calculated with every correlation described in the previous chapter are plotted in Figure 5.4 (a) for the nominal case (p = 60 bar, MR = 6); in Figure 5.4 (b) simulation results are compared to experimental data obtained varying the chamber pressure (p = 35, 60, 70 bar, MR = 6), Figure 5.4 (c) shows the simulation and experimental comparison for tests with constant chamber 123

138 5. Transient Modelling pressure but different mixture ratio (p = 60 bar, MR = 5, 7), while in Figure 5.4 (d) the hot gas wall temperature trend is shown at different chamber pressures. The correlations are all plotted with lines and experimental data with symbols. (a) Heat fluxes at MR = 6, p c = 60 bar (b) Heat fluxes at MR = 6, p c = 35, 60, 70 bar (c) Heat fluxes at MR = 5, 7, p c = 60 bar (d) Wall temperatures at MR = 6, pc = 35, 60, 70 bar Figure 5.4.: Heat fluxes and wall temperatures results The Combustion zone correction factor is able to represent the lower heat fluxes in the first part of the chamber. Unfortunately, the introduced correction cannot be considered predictive (that is, a correction that would give good results 124

139 5. Transient Modelling in a different combustion chamber and injector face): it would require experimental data with different calorimetric chambers and different injector configurations. This is out of the scope of a 0-D/1-D investigation. The heat fluxes in the divergent part are always overpredicted. This is a characteristic of the Bartz model and needs to be kept in mind when interpreting the results. However, introducing the flow acceleration correction factor is possible to achieve a better agreement with the experimental results. Moreover, unlike the combustion zone correction factor, its behaviour is not peculiar of the experiment considered so it can be used for different chamber configurations and performance conditions. No tuning has been performed on the Bartz and Pavli parameters, the constants have been taken as C = 0.026, and C = respectively as recommended by Bartz and Pavli. For each correlation, some remarks follow: Simple Bartz correlation. Here, the heat fluxes are underpredicted (around 30% in the cylindrical part) and the decreasing heat flux in the cylindrical part is not shown, but the shape of the curve in the convergent divergent nozzle region is similar to the experimental one. Modified Bartz correlation. Here, the heat fluxes are slightly overpredicted, but using the temperature and the streamwise correction it has the advantage of following very accurately the experimental data in the cylindrical part. Therefore, the model without a combustion zone correction factor can be applied only to part of the combustion chamber, after the mixing has taken place. Pavli correlation. This correlation is able to follow the experimental trend but in a different way of the Modified Bartz correlation. In fact, the Pavli correlation underestimates the heat fluxes while the Bartz correlation overestimates them. Pressure dependency. Using the modified Bartz correlation, test cases at different pressures have been modelled in EcosimPro. The results shown in Figure 5.4 (b) indicate a very good agreement with experimental heat flux 125

140 5. Transient Modelling values. Therefore, this correlation can be considered reliable for LOX/H 2 combustion at MR = 6. Mixture ratio dependency. The same approach has been taken for the mixture ratio dependency. Test cases at MR varying from 5 to 7 have been modelled in the code. As can be seen in Figure 5.4 (c), the results present a diverging behaviour. In particular, an increase in MR yields a general increase in experimental heat fluxes, while the modified Bartz correlation shows the opposite trend. It is difficult to indicate a clear explanation for these results. The main drivers for the convective heat fluxes are the mixture heat capacity at the reference temperature C p,ref and the temperature gradient (T aw T w ). When MR increases, the heat capacity decreases (because of less hydrogen in the mixture), whereas the temperature gradient increases. In the modified Bartz model, it seems that of these two counteracting properties, the variation in C p,ref is predominant. In the experiment, local MR variations at the wall might be responsible for the opposed trend Q-2D stratification model for HARCC For new engines the use of High Aspect Ratio Cooling Channels (HARCC) is necessary. Indeed, the use of these kinds of channels permits a lower wall temperature and a longer life. Beside these advantages, the HARCC have as usual also drawbacks: the pressure drop is higher and thermal stratification occurs within them. In order to optimize the design of this kind of channels it is fundamental to evaluate the thermal stratification effect and so the heat absorption of the coolant. It is therefore necessary to refine the models developed in the system modelling tools in order to obtain more capabilities, using specific models for each cooling system adopted Model description As compared to two different papers from the Department of Mechanics and Aerospace Engineering (DIMA) of Sapienza University of Rome [106] and the German Aerospace Center (DLR) [150] that found their own way to analyse the 126

141 5. Transient Modelling HARCC, a new approach [35] is here proposed to evaluate thermal stratification in system tools such as EcosimPro. Starting from the one-dimensional governing equations present in the ESPSS library: where u t + f(u) x = S(u) (5.9) ρ ρv ρx nc ρvx nc u = A ; f(u) = A ρv ρv 2 ; (5.10) + P ρe ρvh ρak wall ( P/ t) ρx nc Ak wall ( P/ t) S(u) = (5.11) 0.5(dξ/dx)ρ v v A + ρga + P (da/dx) q w (da wet /dx) + ρgva The new code presents an unsteady Q-2D model and can be considered as an evolution of the two inspiring works presented by DIMA and DLR. The control volumes are divided in slices, one on top of the other linked together longitudinally by the momentum and energy viscous fluxes. The mass conservation equation is written in a one-dimensional form but it is calculated for each slice, while the momentum and energy conservation equations are written in a quasi-2d form taking into account friction, longitudinal viscous transport, wall heat flux and longitudinal fluid heat flux respectively. Equations (5.9) and (5.10) have been modified in the following way, to obtain inside each channel several longitudinal fluid veins one on top of the other and linked by the momentum and energy viscous fluxes: where u t + f(u) x + g(u) y = S(u) (5.12) 127

142 5. Transient Modelling 0 0 v g(u) = A wet ; τ xy = µ t τ xy y ; q c q c = λ t T y (5.13) The turbulent conductivity coefficient λ t is evaluated using the empirical correlation of Kacynski [66]. By the use of a constant turbulent Prandtl number we obtain the turbulent viscosity. λ t λ = Re0.9 P r t = 0.9 µ t = P r t λ t c p (5.14) Hence each slice has his own velocity, and no empirical correlations are used to evaluate the velocity profile being automatically related to the viscous fluxes and the longitudinal heat flux. To accurately describe the wall heat flux also the wall temperature is assumed to vary along the y direction. All thermodynamic properties such as temperature, density and enthalpy depend on x, y and time. (a) 1D Fluid Element (b) Q-2D Fluid Element integrated with walls Figure 5.5.: left: 1-D fluid element and energy balance used for conventional 1-D method; right: control volumes of the Q-2D approach integrated in 3D wall elements 128

143 5. Transient Modelling Initial and boundary conditions of the cooling channel are the typical ones for capacitive components: a capacitive component receives the flow variables (mass and enthalpy flows) as input in inlet and outlet and gives back the state variables (pressure and enthalpy) as output. The cooling channel model is built from 3 components: one quasi-2d tube and two volumes, one at the inlet and the other one at the outlet, representing the manifold volumes of a typical cooling jacket. Each slice is connected directly to the volumes. The quasi-2d tube is a resistive component: it receives the state variables as input in inlet and outlet and gives back the flow variables as output. Please note that no velocity profile in the y direction has been assigned, but each fluid vein is affected by viscous fluxes and wall friction. Moreover a real time-dependent integration has been performed, in order to evaluate the thermal stratification through the time for unsteady analysis. The evaluation of the friction factor of each cell has been done by using a peculiar hydraulic diameter defined as function of the wet channel surface and the perimeter of each volume: D h,i = 4A i P wet,i To our knowledge it is the first time that a quasi-2d approach is implemented for pipe flows in a system tool for transient analysis; with this model we are able to evaluate not only the stratification effect but also the time that the coolant needs to show this stratification during the transient phase of the engine ignition. 3-D cooling channels walls The 3D wall components used to simulate the walls are part of the original ESPSS library [42]. They will calculate the heat conduction in every direction including the axial direction. This thermal component features thermal ports in radial and in azimuth directions allowing an exact calculation of heat conduction through the channel corners. The model has been modified in order to allow the connection between its thermal ports and the quasi-2d channel ports. The walls are divided in 5 different 3-D components as shown in Figure 5.6. Each component has a 3-dimensional discretisation in tangential, radial and longitudinal direction 129

144 5. Transient Modelling Figure 5.6.: Cooling jacket wall mesh (dx, dy, dz), respectively. The formulation for this component is the typical one for conduction elements; the thermal capacitance for each volume is defined as: the internal heat flows are evaluated by: C i,j,k = ρ C p(i,j,k) dx dy dz (5.15) q x(i,j,k) = k i,j,k dy dz (T i 1,j,k T i,j,k )/dx (5.16) q y(i,j,k) = k i,j,k dx dz (T i,j 1,k T i,j,k )/dy (5.17) q z(i,j,k) = k i,j,k dx dy (T i,j,k 1 T i,j,k )/dz (5.18) while the energy equation is: C i,j,k dt i,j,k dt = q x(i,j,k) q x(i+1,j,k) + q y(i,j,k) q y(i,j+1,k) + q z(i,j,k) q z(i,j,k+1) (5.19) As shown in Figure 5.6 only half channel has been considered because of symmetry reasons, with left and right sides adiabatic: 130

145 5. Transient Modelling q out,right_r = 0 (5.20) q out,int_l = 0 q out,int_right_r = 0 (5.21) q out,ext_l = 0 q out,ext_right_r = 0 (5.22) Numerical validation The Q-2D model for cooling channels has been validated by comparison with a numerical test case performed by DIMA [107, 108] of a turbulent flow of methane in a straight channel with asymmetric heating. These calculations have been compared with ESPSS 1D calculations and with the new Q-2D model object of this validation. The channel is smaller than the ones used in actual rocket channels. Indeed, the geometric and the boundary conditions have been chosen by DIMA to obtain small values of the Reynolds number, because the computational grid size of the 3D CFD code is function of this parameter [109]. In order to validate the correct behaviour of the new transient model, two different aspect ratios of the channel have been investigated, a first channel with aspect ratio 1 and a second one with aspect ratio 8. The length and the cross section area of the channel have been kept the same among the two different channels. Both channels are 27 mm long and have a cross section of 0.08 mm 2. The boundary conditions are the same for both channels and for all models: a stagnation inlet temperature of 220 K, a stagnation inlet pressure of 90 bar, a constant temperature of 600 and 220 K at the bottom and at the top of the walls, respectively. Along the lateral side of the channel a linear temperature distribution is applied from 600 to 220 K. At the inlet of the channel a pressure source provided the inlet pressure, while at the outlet a mass flow controlled component forced the mass flow rate. The outlet pressure is an output of the model. Three different temperature sources provided the correct temperature distribution for the bottom side, the lateral side and top side of the channel respectively. To ensure a correct trend of thermal sources during the transient phase, a conductive and a capacitive component have been linked between each temperature source and the thermal ports of the channel. The same configuration has been applied for the aspect ratio 1 and the aspect ratio 131

146 5. Transient Modelling 8 channel. Results Figures 5.7 (c,d) show the bulk evolution of the pressure and temperature along the channel for the aspect ratio 1 case, while Figures 5.7 (e,f) show the pressure and temperature evolution for the aspect ratio 8 channel. When the stratification effect is not so evident, as in the aspect ratio 1 case, the 1D model and Q-2D model have a similar trend; but when stratification occurs, as in the aspect ratio 8 channel, the differences among 1D and Q-2D model are evident, and the Q-2D results are closer to the 3D-CFD ones. Figure 5.7 (a,b) compares the cross-section temperature contours at the channel outlet, for each studied model and for both aspect ratios discussed here. The AR = 1 case features some temperature stratification in the 3D simulation. This has not been observed with the Q-2D model described in Section 5.3, which shows virtually no stratification. On the other hand, for AR = 8, where a consistent stratification is expected, a very good agreement can be observed between the 3D simulations and the new Q-2D model Experimental validation The DLR Lampoldshausen test bench features a cylindrical combustion chamber segment with four different cooling channel geometries used to investigate thermal stratification [151, 132]. Its test results have been used to validate our Q-2D model for high aspect ratio cooling channels. The combustion chamber was designed at DLR institute of Space Propulsion particularly for studies with interchangeable segments. The combustor has a combustion chamber internal diameter of 80 mm and a nozzle throat diameter of 50 mm. Liquid hydrogen is supplied to the combustor at temperatures as low as K while supply pressures are in the range of bar. The HARCC segment is a single cylindrical segment with a diameter of 80 mm and 209 mm length. The test segment has on its circumference four different cooling channel geometries, in each 90 sector the cooling ducts have a different aspect ratio. 132

147 5. Transient Modelling (a) Temperature stratification, AR = 1 (b) Temperature stratification, AR = 8 (c) Bulk temperature, AR = 1 (d) Bulk pressure, AR = 1 (e) Bulk temperature, AR = 8 (f) Bulk pressure, AR = 8 Figure 5.7.: Methane bulk variables evolution along channel axis 133

148 5. Transient Modelling Figure 5.8.: Design of the 4 sector HARCC segment section height [mm] width [mm] channels number AR Table 5.2.: Cooling channels geometries Figure 5.8 shows the construction of the HARCC-segment with different cooling channel geometries. The experiment was performed using the couple liquid oxygen/gaseous hydrogen as propellants, and liquid hydrogen as coolant. Two pressure configurations have been simulated: the first with a chamber pressure of about 88 bar, and the second with a chamber pressure of 58 bar. Two sectors have been investigated: Quadrant 4, with channel aspect ratio 9.2 and Quadrant 3 with channel aspect ratio 30. The geometry of the investigated cooling channels as well as the number of channels referred to circumference of the chamber are given in Table 5.2. For each Quadrant, four sets of thermocouples have been positioned along the channels. In each group, 5 thermocouples have been arranged with different 134

149 5. Transient Modelling Position Distance from leading edge of the segment [mm] Thermocouple TE 1 TE 2 TE 3 TE 4 TE 5 Distance from the hot gas wall [mm] Table 5.3.: Positioning of themocouples distances from the hot gas side wall. Location and distance from the wall of the thermocouples are summarised in Table 5.3. Such temperature measurements at different locations provide important information regarding the development of stratification along the channels. Modelling The test bench has been modelled using EcosimPro [36]. As shown in Figure 5.9, two mass flow sources provide the correct mass flow rate of oxygen and hydrogen to the combustion chamber component. Because the HARCC test segment represents only a portion of the cylindrical part of the combustor, the first segment has been modelled as adiabatic. Thermal demux components connect the combustion chamber to the HARCC segment. At the inlet of the channel a pressure source provided the inlet pressure, while at the outlet a mass flow source forced the mass flow rate. The channel walls features three different materials: the inner side and the fins are in copper alloy; the external wall is built with another copper alloy and a jacket in Nickel alloy. Results Figures 5.10 and 5.11 refer to Quadrant 4 with channel aspect ratio of 9.2 and show the simulation results for the p c = 88 bar test and the p c = 58 bar test, respectively. Figures 5.12 and 5.13 refer to Quadrant 3 with channel aspect ratio of 30 instead and show the simulation results for the p c = 88 bar test and the p c = 58 bar test, respectively. Figures 5.10 (a,b) compare the wall temperatures in the cooling channels and 135

150 5. Transient Modelling Figure 5.9.: Schematic of the experimental test case 136

151 5. Transient Modelling the fluid temperatures, respectively, obtained by 1D and Q-2D simulations for the high pressure test case, while Figures 5.10 (c,d,e,f) show the temperature values at thermocouples positions, comparing experimental values with Q-2D and 1D simulation results. Figure 5.11 shows the same variables for the low pressure test case. Hydrogen enters the channels in supercritical conditions. The hot gas side heat transfer correlation described by Eq. (6.14) was slightly adapted to the calculated experimental hot gas side heat fluxes. In Equation 6.14, an adapted value of was taken. Hence representative hot gas conditions have been modelled in terms of heat transfer coefficient and combustion chamber temperatures. From Figures 5.10 (a,b) it is evident that the behaviour of the 1D model is completely different from the Q-2D model. The Q-2D model is able to obtain a more representative temperature trend in the radial and in the longitudinal direction. From the contour plot it is clear that the 1D model provides a very homogeneous temperature profile also in the walls because it is not able to take into account the occurring of thermal stratification. The validity and the usefulness of the Q-2D model is enhanced by the comparisons shown in Figures 5.10 (c,d,e,f): when high aspect ratio is used, 1D models are not adequate any more. Figures 5.12 and 5.13 show the same variables for channel aspect ratio 30. In these figures the difference between the Q-2D and 1D behaviour compared to experimental data is once more evident. Looking at Figures 5.12 (c,d,e,f) the maximum percentage error obtained by the Q-2D model, when compared to the experimental data of the first thermocouple, does not exceed 10%, while the percentage error for the 1D model is around 40%. Better results we obtain if we compare the percentage error of the same variable in the 58 bar test case, where Q-2D model error does not exceed 5% and 1D model error is around 39%. 137

152 5. Transient Modelling (a) Wall temperatures, AR = 9.2, p c = 88 bar (b) Fluid temperatures, AR = 9.2, p c = 88 bar (c) Thermocouples temperatures, x = 52 mm, p c = 88 bar (d) Thermocouples temperatures, x = 85 mm, p c = 88 bar (e) Thermocouples temperatures, x = 119 mm, p c = 88 bar (f) Thermocouples temperatures, x = 152 mm, p c = 88 bar Figure 5.10.: Wall and fluid thermal stratification,ar = 9.2, p c = 88 bar 138

153 5. Transient Modelling (a) Wall temperatures, AR = 9.2, p c = 58 bar (b) Fluid temperatures, AR = 9.2, p c = 58 bar (c) Thermocouples x = 52 mm,p c = 58 bar temperatures, (d) Thermocouples temperatures, x = 85 mm, p c = 58 bar (e) Thermocouples x = 119 mm,p c = 58 bar temperatures, (f) Thermocouples temperatures, x = 152 mm, p c = 58 bar Figure 5.11.: Wall and fluid thermal stratification, AR = 9.2, p c = 58 bar 139

154 5. Transient Modelling (a) Wall temperatures, AR = 30, p c = 88 bar (b) Fluid temperatures, AR = 30, p c = 88 bar (c) Thermocouples x = 52 mm,p c = 88 bar temperatures, (d) Thermocouples temperatures, x = 85 mm, p c = 88 bar (e) Thermocouples x = 119 mm,p c = 88 bar temperatures, (f) Thermocouples temperatures, x = 152 mm, p c = 88 bar Figure 5.12.: Wall and fluid thermal stratification, AR = 30, p c = 88 bar 140

155 5. Transient Modelling (a) Wall temperatures, AR = 30, p c = 58 bar (b) Fluid temperatures, AR = 30, p c = 58 bar (c) Thermocouples x = 52.5 mm,p c = 58 bar temperatures, (d) Thermocouples temperatures, x = 85.8 mm, p c = 58 bar (e) Thermocouples x = mm,p c = 58 bar temperatures, (f) Thermocouples x = mm, p c = 58 bar temperatures, Figure 5.13.: Wall and fluid thermal stratification, AR = 30, p c = 58 bar 141

156 6. Integrated Validation: RL-10 design and analysis The RL-10 engine is based on an expander cycle, in which the fuel (H 2 ) is used to cool the main combustion chamber and the thermal energy added to the fuel drives the turbopumps. The RL-10 rocket engine is an important component of the American space infrastructure. Two RL-10 engines form the main propulsion system for the Centaur upper stage vehicle, which boosts commercial, scientific and military payloads from a high altitude into Earth orbit. The RL-10A-3-3A developed by Pratt & Whitney under contract to NASA, incorporates component improvements with respect to the initial RL-10A-1 engine. A cryogenic expander cycle engine involves a strong coupling between the different subsystems. This coupling is even stronger during the start and shut-down transients, when non-linear interactions between subsystems play a major role. In addition complex phenomena such as combustion, heat transfer, turbopump operation, phase change, valve maneuverings are concerned, as well as important changes in the thermodynamic properties of the fluids involved. A transient model helps to reduce the number of engine tests by allowing to perform a certain amount of parametric studies in advance of the test campaign, and thus plays an important role in the cost and risk reduction. The RL-10 engine has been used extensively as object of simulations in the past years [15, 14, 13, 59, 58]. In this chapter we want to show the improvements made in terms of modelling with respect to the other tools; indeed, the model presented here features a 1-D discretisation not only in the cooling jacket model, but also for most of the other components, such as the combustion chamber, the Venturi duct and the other pipes. In previous works [15], the combustion chamber has been modelled as a built-in set of hydrogen/oxygen combustion tables. Here, a fully 1-D discretised chamber 142

157 6. Integrated Validation: RL-10 design and analysis and nozzle features a chemical equilibrium model based on Gibbs energy minimization for each section along the chamber. The present model also contains the injector plate model described in chapter 5.1 representative not only of the capacitive effect of the injector dome mass but also of the convective and radiative heat fluxes from the chamber to the injector and of the conductive heat flux between the fuel and oxidiser injector domes. The thermal model used for the cooling jacket component is modelled as a real one and a half counterflow cooling jacket. Finally it is important to mention that, to the best of the author s knowledge, chill down and pre-start procedures were never simulated before with transient system tools for the whole engine. In the present work, the cool-down (prestart) procedure has been simulated in order to obtain a accurate and complete engine state at start signal (t=0). The pre-start simulation results are in very good agreement with the few experimental data available Overview of the RL-10A-3-3A rocket engine The RL10A-3-3A includes seven engine valves as shown in Figure 6.2. The propellant flows to the engine can be shut off using the Fuel Inlet Valve (FINV) and the Oxidizer Inlet Valve (OINV). The fuel flow into the combustion chamber can be stopped by the Fuel Shut-off Valve (FSOV) located just upstream of the injector plenum. The FSOV is a helium operated, two position, normally closed, bullet-type annular gate valve. The valve serves to prevent fuel flow into the combustion chamber during the cool-down period and provide a rapid cut-off of fuel flow during engine shut-down [110]. The fuel interstage and discharge cool-down valves (FCV-1 and FCV-2) are pressure-operated, normally open sleeve valves. The purposes of these valves are the following [110]: allow overboard venting of the coolant for fuel pump cool-down during engine pre-chill and pre-start provide first stage fuel pump bleed control during the engine start transient (for the FCV-1) provide fuel system pressure relief during engine shut-down 143

158 6. Integrated Validation: RL-10 design and analysis The Thrust Control Valve (TCV) is used to control thrust overshoot at start and maintain constant chamber pressure during steady-state operation. TCV is a normally closed, servo-operated, closed-loop, variable position bypass valve used to control engine thrust by regulation of turbine power. As combustion chamber pressure deviates from the desired value, action of the control allows the turbine bypass valve to vary the fuel flow through the turbine [110]. The Oxidizer Control Valve (OCV) has two orifices: one regulates the main oxidizer flow (OCV-1) and the other controls the bleed flow required during engine start (OCV-2). The main-flow orifice in the OCV is actuated by the differential pressure across the LOX pump. The OCV valve is a normally closed, variable position valve. The valve controls oxidiser pump cool-down flow during the engine pre-start cycle and during engine start transient [110]. The Venturi upstream of the turbine is designed to help stabilize the thrust control. Ducts and manifolds in the RL10 are generally made out of stainless steel and are not insulated. The combustion chamber and nozzle walls are composed of cooling tubes. A silver throat is cast in place for the RL10A-3-3A and increases the expansion ratio for higher specific impulse. The inject has 216 coaxial elements; the oxidiser is located in the center of each element and hydrogen through the annulus. Onehundred-sixty-two of the LOX injector elements have ribbon flow-swirlers that provide enhanced combustion stability. The regenerative cooling jacket serves several functions in the RL10 engine. The basic configuration is a pass-and-a-half stainless-steel tubular design. Fuel enters the jacket via a manifold located just below the nozzle throat. A set of 180 short tubes carry coolant to the end of the nozzle. At the nozzle exit plane, a turn-around manifolds directs the flow back through a set of 180 long tubes. The long tubes are interspersed with the short tube in the nozzle section and comprise the chamber cooling jacket above the inlet manifold. Coolant flow exits through a manifold at the top of the chamber. The cooling tubes are brazed together and act as the inner wall of the combustion chamber and nozzle. The fuel pump consists of two stages, separated by an interstage duct, which is vented via the interstage cool-down valve (FCV-1) during start. Both fuel pump 144

159 6. Integrated Validation: RL-10 design and analysis stages have centrifugal impellers, vaneless diffusers and conical exit volutes; the first stage also has an inducer. The LOX pump consists of an inducer and a single centrifugal impeller, followed by a vaneless diffuser and conical exit volute. The LOX pump is driven by the fuel turbine through the gear train. The turbopump speed sensor is located on the LOX pump shaft [111]. The RL-10 turbine is a two stage axial-flow, partial admission, impulse turbine. Downstream of the turbine blade rows, exit guide vanes reduce swirling of the discharged fluid. The turbine is driven by hydrogen and powers both fuel and oxidiser pumps. There are a number of shaft seals which permit leakage from the pump discharge in order to cool the bearings. The fuel pump and the turbine are on a common shaft; power is transferred to the LOX pump through a series of gears. The seals, bearings, gear train all contribute to rotordynamic drag on the turbopump. Figure 6.1.: RL-10A-3-3A engine schematic [115] 145

160 6. Integrated Validation: RL-10 design and analysis Name Value Units Fuel Turbopump 1st stage impeller diameter [mm] 1st stage exit blade height 5.8 [mm] 2nd stage impeller diameter [mm] 2nd stage exit blade height [mm] Oxidiser Turbopump Impeller diameter [mm] Exit blade height [mm] Turbine Mean line diameter [mm] Mass moment of inertia [kg m 2 ] Ducts & Valves FINV flow area [m 2 ] FCV-1 flow area [m 2 ] FCV-2 flow area [m 2 ] Pump discharge duct [m 2 ] Venturi (inlet - throat) [m 2 ] TCV flow area a 1.01E 5 b [m 2 ] Turbine discharge housing (inlet - exit) [m 2 ] Turbine discharge duct [m 2 ] FSOV flow area [m 2 ] OINV flow area [m 2 ] OCV flow area a 3.96E 4 b [m 2 ] Cooling jacket Number of short tubes 180 [-] Number of long tubes 180 [-] Channel width at throat [mm] Channel height at throat [mm] Total coolant volume [m 3 ] Typical hot wall thickness [mm] HGS effective surface area [m 2 ] Thrust chamber Chamber diameter [m] Throat diameter [m] Nozzle area ratio 61 [-] Chamber/nozzle length [m] Number of injectors 216 [-] Injector assembly weight 6.72 [kg] Table 6.1.: RL-10A-3-3A construction data [15] a values at nominal full-thrust condition b this flow area includes the discharge coefficient for the orifice, which is unknown 146

161 6. Integrated Validation: RL-10 design and analysis Figure 6.2.: RL-10A-3-3A engine diagram 6.2. Design procedure The development of the RL-10 engine transient model has been conducted with EcosimPro and the ESPSS library, in the upgraded version including all the relevant models developed and described in Chapter Turbomachinery modelling Pumps The pump model makes use of performance maps for head and resistive torque. The pump curves are introduced by means of fixed 1-D data tables defined as functions of a dimensionless variable θ that preserves homologous relationships in all zones of operation. θ parameter is defined as follows: θ = π + arctan(ν/n) (6.1) 147

162 6. Integrated Validation: RL-10 design and analysis where ν and n are the reduced flow and the reduced speed respectively: ν = Q Q R = ṁin/ρ in Q R n = 30 ω /π rpm R (6.2) The dimensionless characteristics (head and torque) are defined as follows: h = TDH / TDH R n 2 + ν 2 β = τ / τ R n 2 + ν 2 (6.3) this method eliminates most concerns of zero quantities producing singularities. To simplify the comparison with generic map curves, these relations are normalized using the head, torque, speed and volumetric flow at the point of maximum pump efficiency. These maps have been created as a combination of available test data provided by Pratt&Whitney [15] and generic pump performance curves [25] (see Figures 6.3, test data range in grey). Additional maps were established (not shown here), giving a corrective factor on the pump torque, function of the rotational speed ratio (also provided by P&W). The enthalpy flow rise is a function of the absorbed power while the evaluation of the mass flow rate is performed through an ODE. (ṁ h) out = τ ω (ṁ h) in I dṁ (P dt = + 12 ) ρv2 (P + 12 ) ρv2 out in gρ in TDH Because of the presence of the FCV-2 valve between the first and the second stage, the fuel pump has been modelled with two separated pump components, one for each stage. Since the oxidiser pump has only one stage, it has been modelled with one component instead. For each pump model the main nominal parameters have been calculated by a numerical code specifically developed to find the nominal value of the outlet pressure, the pump torque τ p, the total dynamic head TDH, the pump efficiency η p and the specific speed Ns by use of the Pump head and Pump efficiency curves provided by Pratt & Whitney [15]. The development of a dedicated tool for the evaluation of the pump nominal parameters has been necessary since there was a discrepancy between the definition of total dynamic 148

163 6. Integrated Validation: RL-10 design and analysis (a) Extended Head map for LOX and Fuel pumps (b) Extended Torque map for LOX and Fuel pumps Figure 6.3.: Pumps performance maps head used in the pump model and the one used in P&W maps: TDH tool = h out h in ; TDH ESP SS = P out P in ; (6.4) g ρ in g ( Pout TDH P &W = P ) / in g (6.5) ρ out ρ in The first one represents the head rise given by the enthalpy difference between the inlet and the outlet conditions; this definition has been used to match the requested power of the pump. The second one is the head given by the pressure difference between inlet and outlet and the inlet density; this is the definition used in the ESPSS pump model (see Eq. 3.32). The third one is defined using the difference between the pressure on density ratio at outlet and inlet and used to define the numerical value from the P&W maps. These three definitions of the dynamic head can be considered the same only in the ideal case of a pure incompressible fluid (ρ in = ρ out ). As real fluids in the pump component are used, even if the fluid is in liquid conditions, the density difference between inlet and outlet generates a discrepancy between the aforementioned definitions. Moreover, using in the tool the Euler equation of turbomachinery to calculate the power, and comparing it with Eq

164 6. Integrated Validation: RL-10 design and analysis we obtain: W = ṁ h = ṁ h is = τ ω Q R ρ in g (h out h in ) is η p η = τ R ω (6.6) The term (h out h in ) is = TDH R TDH is not the same of the one η η present in inlet mass flow equation (see Eq. 3.32); in fact, comparing the definition of TDH in Eq with the definition of TDH in Eq we obtain: TDH R Eq = P ρg (h out h in ) = TDH R Eq (6.7) The mismatch present in the use of two different versions of the total dynamic head could affect the results of the simulations and the performances of pump itself. For this reason the code developed is able to calculate a modified pump efficiency in order to match either the pressure rise either the pump torque in the ESPSS pump model. Since no official values of the propellants leak to the gear box were found, an iterative procedure was adopted to find the correct value of the mass flow rate and the outlet pressure in each stage. h = W/ṁ h out NO Input W, ṁ 0, P in, T in P out? h out,is = TDH calc = TDH P &W f(h out,is, s in) YES P out, ṁ h out, η TDH φ = f(ṁ) TDH P &W η P &W NO Figure 6.4.: Iterative procedure for determining pump parameters Turbine The turbine performance maps provided by Pratt&Whitney depict the combined performance of the two stages (see Figures 6.5 (a,b)). The first one describes the effective area (area times discharge coefficient) as a function of velocity 150

165 6. Integrated Validation: RL-10 design and analysis ratio (U/C o ) for several different pressure ratios. The second one describes the combined two-stage turbine efficiency as a function of velocity ratio (U/C o ) as well. (a) Efficiency map for the Turbine (b) Effective Area map for the Turbine Figure 6.5.: Turbine performance maps from P&W [15] In the present study, Pratt & Whitney performance maps are transformed to obtain the turbine performance maps used in the ESPSS turbine model. These maps (mass flow coefficient and specific torque) are introduced by means of 2-D input data tables as a function of velocity ratio and pressure ratio (see Figures 6.6 (a,b)): N = r ω C o Π = P 01 / P 02 (6.8) and the mass flow coefficient and specific torque are defined as: Q + = ṁmap C o r 2 P 01 ST = τ r ṁ map C o (6.9) According to Eq. 6.9 and to the power balance equation τ ω = ṁ η h is we obtain the non-dimensional parameters as function of velocity ratio and pressure 151

166 6. Integrated Validation: RL-10 design and analysis ratio using data from the P&W maps: τ ω = ṁ η(π) h is ST r ṁ C o ω = ṁ η(π) h is ST C 2 o N = η(π) h is ST (Π, N) = η(π) h is C 2 o N and for the Q + parameter we just need to calculate the turbine mass flow as function of N and Π: A eff = C D A = f(π, N) 2γ ] ṁ = C D A P 01 ρ 01 [Π 2 γ Π γ+1 γ γ 1 This formulation is based on the assumption that no chocking conditions occur during the transient and at steady conditions of the turbine component. (a) Specific Torque map for the Turbine (b) Mass Flow coefficient map for the turbine Figure 6.6.: Turbine performance maps Thrust chamber and cooling jacket modelling The thrust chamber component, inherited from the original ESPSS library [43], represents a non adiabatic 1-D combustion process inside a chamber for liquid 152