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Although liquid propellant rocket engines are operational and have been studied for decades, cryogenic injection at supercritical pressures is still considered essentially not understood. This thesis intends to approach this problem in three steps: by developing a numerical model for real gas thermodynamics, by extending the present thermodynamic view of supercritical injection, and finally by applying these methods to the analysis of injection.
A new numerical real gas thermodynamics model is developed as an extension of the DLR TAU code. Its main differences to state-of-the-art methods are the use of a precomputed library for fluid properties and an innovative multi-fluid-mixing approach. This results in a number of advantages: There is effectively no runtime penalty of using a real gas model compared to perfect gas formulations, even for high fidelity equations of state (EOS) with associated high computational cost. A dedicated EOS may be used for each species. The model covers all fluid states of the real gas component, including liquid, gaseous, and supercritical states, as well as liquid-vapor mixtures. Numerical behavior is not affected by local fluid properties, such as diverging heat capacities at the critical point. The new method implicitly contains a vaporization and condensation model. In this thesis, oxygen is modeled using a modified Benedict-Webb-Rubin equation of state, all other involved species are treated as perfect gases.
A quantitative analysis of the supercritical pseudo-boiling phenomenon is given. The transition between supercritical liquid-like and gas-like states resembles subcritical vaporization and is thus called pseudo-boiling in the literature. In this work it is shown that pseudo-boiling differs from its subcritical counterpart in that heating occurs simultaneously to overcoming molecular attraction. In this process, the dividing line between liquid-like and gas-like, the so called Widom line, is crossed. This demarcation is characterized by the set of states with maximum specific heat capacity. An equation is introduced for this line which is more accurate than previous equations. By analyzing the Clausius-Clapeyron equation towards the critical limit, an expression is derived for its sole parameter. A new nondimensional parameter evaluates the ratio of overcoming molecular attraction to heating: It diverges towards the critical point but shows a significant pseudo-boiling effect for up to reduced pressures of 2.5 for various fluids.
It appears reasonable to interpret the Widom-line, which divides liquid-like from gas-like supercritical states, as a definition of the boundary of a dense supercritical fluid. This may be used to uniquely determine the radius of a droplet or the dense core length of a jet. Then, a quantitative thermodynamic analysis is possible. Furthermore, as the pseudo-boiling process may occur during moderate heat addition, this allows for a previously undescribed thermal jet disintegration mechanism which may take place within the injector.
This thermal jet break-up hypothesis is then applied to an analysis of Mayers and Branams nitrogen injection experiments. Instead of the constant density cores as predicted by theory, the majority of their cases show an immediate drop in density upon entering the chamber. Here, three different axial density modes are identified. The analysis showed that heat transfer did in fact take place in the injector. The two cases exhibiting a dense core are the cases which require the largest amount of power to reach the pseudo-boiling temperature. After this promising application of pseudo-boiling analysis, thermal break-up is tested numerically. By accounting for heat transfer inside the injector, a non dense-core injection can indeed be simulated for the first time with CFD.
Finally, the CFD model is applied to the A60 Mascotte test case, a reactive GH2/LOX single injector operating at supercritical pressure. The results are compared with experimental and other researchers numerical data. The flame shape lies well within the margins of other CFD results. Maximum OH* concentration is found in the shear layer close to the oxygen core and not in the shoulder, in agreement with experimental data. The axial temperature distribution is matched very well, particularly concerning position and value of the maximum temperature.

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... However, this assumption is physically justified in this work focused on supercritical oxygen-hydrogen combustion. Banuti et al. [60,65] explain that, for oxygen-hydrogen combustion at supercritical pressures, the LOx transitions from a liquid to a supercritical gas-like state under pure fluid conditions, where essentially all real fluid effects are confined to the LOx core. Thus, for the flames considered in the present work, the multi-fluid ideal-mixing model is appropriate. ...

... Computational domain[53] main features: each species has a dedicated equation of state and the properties of the mixture are determined by a multi-fluid mixing model, see[60][61][62] .Here the ideal gas equation of state is solved for H 2 , H, O, OH and H 2 O, and a real gas equation of state is solved for O 2 . The real gas properties of oxygen are computed from the high fidelity modified Benedict-Webb-Rubin (MBWR) equation of state (EOS) of Younglove ...

The quantitative comparison of experimental data and results from CFD simulations is still an ongoing challenge in the investigation of high pressure combustion in rocket combustion chambers. This is due to the extreme environment which develops in liquid propellant rocket engines, which represent a challenge for experimental data collection. OH* radiation emitted from the flame has often been designated as an indicator of the combustion zone, because of its relative ease of detection with appropriate cameras. A method was developed to compare OH* radiation originating from cryogenic oxygen-hydrogen flames in an experimental combustor with the CFD simulation results. Pseudo-OH* images were obtained from CFD results of two combustors. The method consists in obtaining the path of a ray of light by a reverse ray tracing algorithm and sampling the thermodynamic properties along the path of the ray, simulating the emission and absorption spectra in the wavelength range of interest, in this case of OH* emission during combustion. The spectral radiance is then determined by solving the differential radiative transfer equation. Finally, the total radiance is calculated integrating the spectral radiance. The results obtained applying this method are then compared with former results of two test cases, a laminar and a turbulent flame, and with the related experimental data. An improvement of the comparison with the experimental data was achieved in terms of the prediction of self-absorption, which was underestimated in previous works by a factor of 15, and in terms of radiance near the injection plane, where difference is estimated to be about 40% when including refraction. The method allows for more direct comparison between 3D CFD results and 2D experimental images collected by the optical setup and probes.

... To investigate the response of the BKH study elements to an imposed acoustic disturbance, a model of a single injector under acoustic forcing has been developed. The model consists of steady and unsteady computations using a specialized version of the DLR TAU code for modeling real gas properties [9,10]. A steady-state Reynolds-averaged NavierStokes (RANS) computation is used to model the single injector under steady-state operation without acoustic forcing. ...

... To account for these real gas e¨ects, a new real gas mixture model has been developed and implemented in the DLR TAU code. The model and its validation are described in more depth in [9,10]. ...

An experimental combustor, designated BKH, is operated at DLR Lampoldshausen to investigate high-frequency combustion instability phenomena. The combustor operates with liquid oxygen (LOx) and gaseous or liquid hydrogen propellants at supercritical conditions analogous to real rocket engines. An externally imposed acoustic disturbance interacts with a series of 5 coaxial injection elements in the center of the chamber. A combination of experimental analysis and numerical modeling is used to provide further insight and understanding of the BKH experiments. Optical data from the BKH experiments are analyzed to extract the response of the §ame at the excitation frequency. A new method for reconstructing the acoustic ¦eld inside the chamber from dynamic pressure sensor data is used to describe the evolution of the acoustic mode and the local disturbance in the §ame zone. An Unsteady Reynolds-Averaged Navier�Stokes (URANS) model of a single BKH injection element subjected to representative transverse acoustic velocity excitation has been computed using a specialized release of the DLR TAU code. The single-element model reproduces the retraction of the dense LOx core during transverse velocity excitation as observed experimentally. The model also provides further insight into the §attening and §apping of the §ame. The §apping is identi¦ed as the oxygen core being transported by the transverse acoustic velocity.

... Summarized [17], "The acceptance of this change in understanding is perhaps best reflected in the shift of boundary conditions posed by the Rocket Combustion Modeling (RCM) workshops 2001 [18] and 2006 [19]. For the same configuration of a single injector combustion chamber (but at a different oxidizer/fuel ratio) at supercritical pressures, the 2001 workshop specified a spectrum of oxygen droplets to be prescribed in the CFD calculation. ...

Flows in liquid propellant rocket engines (LRE) are characterized by high pressures and extreme temperature ranges, resulting in complex fluid behavior that requires elaborate thermo-physical models. In particular, cubic equations of state and dedicated models for transport properties are firmly established for LRE simulations as a way to account for the non-idealities of the high-pressure fluids. In this paper, we review some shortcomings of the current modeling paradigm. We build on the common study of property errors, as a direct measure of the density or heat capacity accuracy, to evaluate the quality of cubic equations of state with respect to pseudo boiling of rocket-relevant fluids. More importantly, we introduce the sampling error as a new category, measuring how likely a numerical scheme is to capture real fluid properties during a simulation, and show how even reference quality property models may lead to errors in simulations because of the failure of our numerical schemes to capture them. Ultimately, a further evolution of our non-ideal fluid models is needed, based on the gained insight over the last two decades.

... In aeronautics and astronautics, in the regenerative cooling of rocket engines and hypersonic vehicles, the fuel first flows through the combustion chamber walls to absorb heat and then enters the combustion chamber [40,136], during which the fuel undergoes a complex heat transfer process at supercritical pressure (see Fig. 1.4 a). Meanwhile, the fuel injection process is also a complex hydrodynamic process from supercritical pressure to supercritical or subcritical pressure [21,118] (see Fig. 1.4 b). ...

General backgrounds and basic concepts are introduced in this chapter, including critical phenomenon, critical anomalies, and the applications of supercritical pressure fluids. The coupled heat and mass transfer is explained briefly. A literature review is also provided, followed by the motivation and outline of this book.

... This differs from the subcritical boiling, under which a constant temperature is kept and all the energy is used for phase change. The determination of T and T + can be found in Ref. [40,42]. Both i and k=i/i pc increases with increase of pressures, ...

The accurate prediction of heat transfer deterioration (HTD) is important to ensure the safe operation of scCO2 cycles driven by various heat sources. Here, the scCO2 heat transfer experiment is performed in a 10 mm diameter vertical tube, covering the ranges of pressures 7.51-21.1 MPa, mass fluxes 488-1500 kg/m²s and heat fluxes 43.7-488 kW/m². Both uniform heating and non-uniform heating cases are dealt with, but more attention is paid on non-uniform heating. We show that non-uniform heating displays strong circumference angles dependent heat transfer characteristic. Normal heat transfer (NHT) displays gentle rise of wall temperatures along flow length, but for HTD, wall temperature peak is detected ahead of pseudo-critical point. Pseudo-boiling is introduced to deal with scCO2 heat transfer. Heat added to scCO2 is decoupled into a temperature rise part and a phase change part. Flow structure includes a vapor-like fluid near tube wall and a liquid-like fluid in tube core. The analogy between subcritical boiling and supercritical heat transfer results in a supercritical-boiling-number SBO to govern the vapor layer thickness. Sudden change from NHT to HTD is found when crossing a critical SBOcr, which is 5.126×10⁻⁴ for uniform heating based on our experimental data and other data in the literature, but becomes 8.908×10⁻⁴ for non-uniform heating using our experimental data. Compared to uniform heating, non-uniform heating is found to delay the occurrence of HTD. The criterion presented here is useful to avoid the occurrence of HTD in the design and operation of scCO2 cycles.

... Based on Refs. [30,31] , specific heat for liquid-like fluid, c p , l , is defined at 3 4 T pc (see point ¯ A in Fig. 2 c). Specific heat for vapor-like fluid is ...

... Based on Refs. [30,31] , specific heat for liquid-like fluid, c p , l , is defined at 3 4 T pc (see point ¯ A in Fig. 2 c). Specific heat for vapor-like fluid is ...

... The formation of a spray is generally controlled by the nozzle geometry, the characteristics of the atomized fluid and of the receiving medium, and the thermodynamic conditions of the system and of the components involved. 17,18 The atomization process can be divided into two main steps: first is the primary atomization, in which the disruptive forces are almost comparable to the cohesive forces; second is the secondary atomization, in which the disruptive forces have completely overcome the cohesive forces, resulting in the complete atomization. More deeply, four atomization regimes are classically described in the literature for the study of a liquid injected in a gas: Rayleigh break-up, first-wind-induced or sinuous wave break-up, second-wind-induced or wave-like break-up, and full atomization. ...

Water jets in systems at low pressure have been frequently studied and some models for the prediction of spray cone angle (θ/2), intact-surface break-up length (L1) and intact-core break-up length (L2) have been developed. Complete studies of jets in supercritical receiving medium have not been carried out; for this reason, in this work, an experimental study on the observation of water jet break-up in supercritical carbon dioxide has been proposed. Different operating conditions were tested and jet characteristics in terms of θ/2, L1, and L2 were studied. The obtained data were first fitted using the models proposed in literature; then, a model was developed in this work, which takes into account the influence of pressure in the evaluation of the spray cone angle. The proposal of a pressure-dependent correlation revealed to be effective in the description of the sprays generated in supercritical conditions.

... This real gas implementation has two main features: each species has a dedicated equation of state and the properties of the mixture are determined by a multi-fluid mixing model. 6,7,10 Here the ideal gas equation of state is solved for H 2 , H, O, OH and H 2 O, and a real gas equation of state is solved for O 2 . The real gas transport properties of oxygen are computed form the high fidelity EOS for O 2 as proposed by Lemmon et al. 29 In the current release of TAU code, an adaptive tabulation of high-accuracy equation of state (EOS) is used, first introduced by Dumbser et al. 30 for cavitating flows. ...

The current work aims to study the influence of the numerical setup on a URANS model of a single co-axial injection element under acoustic forcing. The single injector model is representative of an experimental rocket combustor with multiple injection elements, named BKH and operated at DLR Institute of Space Propulsion. The configurations presented in this work are fully 3D for the 1L and 1T mode excitation and compared with previous configurations.14, 34The flame zone is studied with an unstructured and hybrid mesh. An influence of the numerical setup is visible.

The thermophysical properties of argon, ethylene, parahydrogen, nitrogen, nitrogen trifluoride and oxygen are presented. Properties are given in tables and a standard set of equations is described. The tables list pressure, density, temperature, internal energy, enthalpy, entropy, heat capacity at constant volume, heat capacity at constant pressure, and sound velocity. Also included are viscosity, thermal conductivity, and dielectric constant, for some of the fluids. The equation and related properties of this report represent a compilation from the cooperative efforts of two research groups: the NBS Thermophysical Properties Division and the Center for Applied Thermodynamics Studies of the University of Idaho.

This project looks at injection processes of a dense jet simulating oxygen core flow with nitrogen of a coaxial injector used in cryogenic rocket engines. The rocket engine performance is highly dependent on the injection processes such as mixing and jet dissipation of propellants in the supercritical regime. Experimental data at various temperatures and injection velocities taken by Raman imaging and Shadowgraphy were compared to computational models allowing comparisons of density, length scales and jet spreading angles providing insight into mass mixing and jet dissipation.

Contenido: Vectores y tensores; Propiedades de equilibrio; Mecánica estadística; La ecuación del estado de los gases a densidad baja y moderada; La ecuación del estado de gases densos y líquidos; Equilibrio del líquido-vapor y fenómenos críticos; Teoría cuántica y la ecuación de estado; La teoría cinética de los gases; Fenómenos de transporte y gases; Propiedades de transporte de gases densos y líquidos; Teoría cuántica y fenómenos de transporte; Aplicaciones hidrodinámicas de las ecuaciones de cambio; Bases electromagnéticas de las fuerzas intermoleculares; La teoría de las fuerzas intermoleculares; Cálculos de mecánica cuántica de fuerzas intermoleculares.

The mechanism of atomization, which is still unknown, was investigated by the simultaneous use of two photographic techniques. The initial transient was observed with a 10**6 frames/sec camera and the steady state by a technique similar to spark photography. It was found, for example, that: jet divergence begins progressively closer to the nozzle exit as the gas density increases until it reaches the exit with no evidence of abrupt change; the divergence angle (spray angle) increases with increasing gas density, and sharpness of nozzle inlet and with decreasing liquid viscosity and nozzle length; divergence angle and jet intact length are quasi-steady with respect to upstream pressure changes which occur on time scales greater than 10 to 30 mu s; aerodynamic effects, liquid turbulence, jet velocity profile rearrangements, and liquid pressure oscillations, each could not alone be the mechanism of atomization.

Detailed coverage of advanced combustion topics from the author of Principles of Combustion, Second Edition Turbulence, turbulent combustion, and multiphase reacting flows have become major research topics in recent decades due to their application across diverse fields, including energy, environment, propulsion, transportation, industrial safety, and nanotechnology. Most of the knowledge accumulated from this research has never been published in book form-until now. Fundamentals of Turbulent and Multiphase Combustion presents up-to-date, integrated coverage of the fundamentals of turbulence, combustion, and multiphase phenomena along with useful experimental techniques, including non-intrusive, laser-based measurement techniques, providing a firm background in both contemporary and classical approaches. Beginning with two full chapters on laminar premixed and non-premixed flames, this book takes a multiphase approach, beginning with more common topics and moving on to higher-level applications. In addition, Fundamentals of Turbulent and Multiphase Combustion: Addresses seven basic topical areas in combustion and multiphase flows, including laminar premixed and non-premixed flames, theory of turbulence, turbulent premixed and non-premixed flames, and multiphase flows Covers spray atomization and combustion, solid-propellant combustion, homogeneous propellants, nitramines, reacting boundary-layer flows, single energetic particle combustion, and granular bed combustion Provides experimental setups and results whenever appropriate Supported with a large number of examples and problems as well as a solutions manual, Fundamentals of Turbulent and Multiphase Combustion is an important resource for professional engineers and researchers as well as graduate students in mechanical, chemical, and aerospace engineering.