Homogeneous Relaxation Research Articles

Modelling phase change in a novel turbo expander for application to heat pumps and refrigeration cycles

A novel turbo expander based on the Tesla turbine is proposed to be applied to a heat pump or refrigeration cycle to improve the overall cycle efficiency. Initial numerical modelling of this turbo expander at representative conditions was carried out using the homogeneous relaxation model (HRM) to assess the influence of phase change on performance. The presence of a dense cloud of liquid droplets within the rotor was predicted to produce a significant back pressure on the turbine nozzle postponing the phase change. This was expected to occur in the vicinity at the outlet of the nozzle, but high volume fractions of liquid was predicted to penetrate deeper inside the rotor, especially at higher RPM. The resulting lower velocities of the liquid flow at the inlet of the rotor was predicted to significantly degrades the performance of the turbine. It is thus important for a successful implementation of this concept to remove as much liquid droplets as possible before the flow enters the rotor in order to minimise the back pressure.

A thermodynamically consistent model of a liquid-vapor fluid with a gas

This work is devoted to the consistent modeling of a three-phase mixture of a gas, a liquid and its vapor. Since the gas and the vapor are miscible, the mixture is subjected to a non-symmetric constraint on the volume. Adopting the Gibbs formalism, the study of the extensive equilibrium entropy of the system allows to recover the Dalton’s law between the two gaseous phases. In addition, we distinguish whether phase transition occurs or not between the liquid and its vapor. The thermodynamical equilibria are described both in extensive and intensive variables. In the latter case, we focus on the geometrical properties of equilibrium entropy. The consistent characterization of the thermodynamics of the three-phase mixture is used to introduce two Homogeneous Equilibrium Models (HEM) depending on mass transfer is taking into account or not. Hyperbolicity is investigated while analyzing the entropy structure of the systems. Finally we propose two Homogeneous Relaxation Models (HRM) for the three-phase mixtures with and without phase transition. Supplementary equations on mass, volume and energy fractions are considered with appropriate source terms which model the relaxation towards the thermodynamical equilibrium, in agreement with entropy growth criterion.

Numerical investigation on flashing jet behaviors of single-hole GDI injector

In this paper, n-hexane flashing jets discharged from a single-hole gasoline direct injector (GDI) were studied numerically with the adoption of diffuse Eulerian framework and the homogeneous relaxation model (HRM). The fuel temperature ranged from 30 to 130 °C, and the ambient pressure varied from 20 to 101 kPa. The results showed that considerable vaporization started at the counter bore and a liquid core existed near the nozzle exit. Due to drastic vaporization, the pressure within the liquid core increased so the two-phase flow became under-expanded. Violent expansion then occurred and a low-pressure region was formed, which is believed as the origin of the spray collapse under flashing conditions for multi-hole GDI injectors. At high superheat levels, shock wave structures similar to those in highly under-expanded gaseous jets were identified. However, the transonic position located at some distances from the nozzle rather than at the throttle. Besides, vapor fraction played the dominant role in the onset of expansion, while the expansion was ended by the pressure difference between the two sides of the Mach disk.

Temperature-dependent modelling of a HNBR O-ring seal above and below the glass transition temperature

Sealing products can undergo many different atmospheric conditions and hence large variations in the in-service temperatures. These products, typically made of elastomer, are known to have heat sensitive mechanical characteristics. Since the reliability of sealing systems can be affected by cyclic temperature conditions, it is essential to predict seal performances depending on the temperature in order to prevent the occurrence of interfacial leaks. The aim of the present study is to simulate the thermo-mechanical behavior of a HNBR O-ring seal during a temperature cycle ranging from room temperature to low temperatures down to −34 °C. The main objective is to present the construction and the thermal dependency of a hyperelasto-visco-hysteresis (HVH) model so as to be able to predict the mechanical behavior of the elastomer in function of temperature. This phenomenological model is designed in the form of a combination of hyperelastic, viscoelastic and elasto-hysteretic stress contributions. The relevant parameters are identified using a material database based on the results of classical homogeneous isothermal relaxation tests and anisothermal relaxation tests. The predictions of the HVH model obtained shows good agreement with the experimental contact force during a 20% squeezed O-ring test under cyclic thermal loading below the glass transition temperature.

A computational approach to predict external spray characteristics for flashing and cavitating nozzles

A computational approach to predict external spray characteristics for flashing and cavitating nozzles was presented and validated. It is developed as a fully Eulerian and compressible two-phase flow solver that simulates vaporization and condensation of the fuel using a Homogeneous Relaxation Model (HRM). The flow solver together with the interface area density model was applied to predict spray spreading angle. A method to identify spray plume boundary from the predicted flow field that offers a meaningful comparison to experimental definition was discussed in detail. Using the experimental data available in literature, a comparison between axi-symmetric and asymmetric nozzles was made to assess the nature of the influence of the nozzle geometry and the operating conditions on the ensuing spray. On the basis of this comparison, it was inferred that the spray plume angle of asymmetric nozzles is largely geometry dependent for a wide range of pressure ratios.

Numerical simulation of subcooled and superheated jets under thermodynamic non-equilibrium

Modelling and simulating the rapid pressure drop inside nozzles is a significant challenge because of the complexity of the multiple associated phenomena. In the present study, FlashFOAM a compressible solver for calculating the phase change within various nozzle geometries undergoing rapid pressure drops has been developed in the frame of the open source Computational Fluid Dynamics (CFD) code OpenFOAM. FlashFOAM accounts for the inter-phase heat transfer with the Homogeneous Relaxation Model (HRM). The work describes the development of a pressure equation within a different formulation than in other studies. The surface forces due to liquid-gas interfacial instabilities are modelled here in a novel coupling of HRM with the volume of fluid method giving rise to a conservative method for modelling primary atomisation. This new pressure equation is validated with published experimental measurements. A validation series dedicated to long nozzles is included for the first time. Novel additional tests for the flow characteristics and vapour generation in cryogenic liquid cases are included showing that the solver can be employed to gain some new insights into the physics of the flow regimes of sudden depressurising cryogenic liquids. The dependency of the geometry of the nozzles, pressure and subcooled degree on the vapour generation has been analysed including the effect of turbulence on the nozzle flow avoiding the laminar flow scenarios of previous validation studies. The validation study has demonstrated that FlashFOAM can be used to simulate flash boiling scenarios accurately and predict the properties of flash atomisation.

Optical nutation in acetylene-filled hollow-core photonic crystal fiber

We report detailed experimental results on the optical nutation effect in acetylene-filled hollow-core photonic crystal fibers (HC-PCFs). This coherent quantum optical effect is manifested by the appearance of attenuated oscillations in the profiles of transmitted step-like coherent optical pulses. The experiments used 15 ns optical pulses with peak powers up to ∼4 W and centered at 1530.37 nm, which corresponds to the acetylene (12C2H2) vibration-rotation resonance absorption line P9. The all-fiber cell made from a HC-PCF with internal diameter ∼10.3 μm contained a gas pressure of about 0.12 Torr. The Maxwell-Bloch equations are used to analyze the influence of the homogeneous relaxation, the inhomogeneous Doppler broadening of the absorption line, and the random orientation of the acetylene molecules. By comparing the experimental results with numerical simulations, we obtain the longitudinal and the transverse relaxation times to be close to 10 ns and find that the transition dipole moment of the acetylene P9 absorption line is ∼1.36×10−32 Cm.

A deterministic-stochastic method for computing the Boltzmann collision integral in $\mathcal{O}(MN)$ operations

We developed and implemented a numerical algorithm for evaluating the Boltzmann collision integral with \begin{document}$O(MN)$\end{document} operations, where \begin{document}$N$\end{document} is the number of the discrete velocity points and \begin{document}$M . At the base of the algorithm are nodal-discontinuous Galerkin discretizations of the collision operator on uniform grids and a bilinear convolution form of the Galerkin projection of the collision operator. Efficiency of the algorithm is achieved by applying singular value decomposition compression of the discrete collision kernel and by approximating the kinetic solution by a sum of Maxwellian streams using a stochastic likelihood maximization algorithm. Accuracy of the method is established on solutions to the problem of spatially homogeneous relaxation.

Primitive Path Analysis and Stress Distribution in Highly Strained Macromolecules.

Polymer material properties are strongly affected by entanglement effects. For long polymer chains and composite materials, they are expected to be at the origin of many technically important phenomena, such as shear thinning or the Mullins effect, which microscopically can be related to topological constraints between chains. Starting from fully equilibrated highly entangled polymer melts, we investigate the effect of isochoric elongation on the entanglement structure and force distribution of such systems. Theoretically, the related viscoelastic response usually is discussed in terms of the tube model. We relate stress relaxation in the linear and nonlinear viscoelastic regimes to a primitive path analysis (PPA) and show that tension forces both along the original paths and along primitive paths, that is, the backbone of the tube, in the stretching direction correspond to each other. Unlike homogeneous relaxation along the chain contour, the PPA reveals a so far not observed long-lived clustering of topological constraints along the chains in the deformed state.

Investigations of effect of phase change mass transfer rate on cavitation process with homogeneous relaxation model

Abstracts The super high fuel injection pressure and micro size of nozzle orifice has been an important development trend for the fuel injection system. Accordingly, cavitation transient process, fuel compressibility, amount of non-condensable gas in the fuel and cavitation erosion have attracted more attention. Based on the fact of cavitation in itself is a kind of thermodynamic phase change process, this paper takes the perspective of the cavitation phase change mass transfer process to analyze above mentioned phenomenon. The two-phase cavitating turbulent flow simulations with VOF approach coupled with HRM cavitation model and U-RANS of standard k-e turbulence model were performed for investigations of cavitation phase change mass transfer process. It is concluded the mass transfer time scale coefficient in the Homogenous Relaxation Model (HRM) representing mass transfer rate should tend to be as small as possible in a condition that ensured the solver stable. At very fast mass transfer rate, the phase change occurs at very thin interface between liquid and vapor phase and condensation occurs more focused and then will contribute predictably to a more serious cavitation erosion. Both the initial non-condensable gas in fuel and the fuel compressibility can accelerate the cavitation mass transfer process.

Modified homogeneous relaxation model for the R744 trans-critical flow in a two-phase ejector

The proposed modified homogeneous relaxation model (HRM) applied to the numerical model of a CO2 two-phase ejector was numerically investigated and developed based on the optimisation of the relaxation time (RT) correlation. The optimisation procedure was performed using a genetic algorithm. The study of the RT definition on model accuracy was carried out using literature correlations and constant relaxation time, and by comparing the developed modified HRM with experimental results. The modified HRM showed the higher accuracy of the motive nozzle mass flow rate (MFR) than that of the other available numerical models for the subcritical operating regimes and similar high accuracy as the homogeneous equilibrium model (HEM) for the trans-critical operating regimes. The application range of the modified HRM was defined for the motive nozzle pressure above 59 bar to predict the motive nozzle MFR with the relative error below 15%. For the motive nozzle pressure level below 59 bar, the modified HRM improved the accuracy of the motive MFR prediction by 5%–10% compared to the HEM formulation.

A low noise discrete velocity method for the Boltzmann equation with quantized rotational and vibrational energy

A discrete velocity method is developed for gas mixtures of diatomic molecules with both rotational and vibrational energy states. A full quantized model is described, and rotation–translation and vibration–translation energy exchanges are simulated using a Larsen–Borgnakke exchange model. Elastic and inelastic molecular interactions are modeled during every simulated collision to help produce smooth internal energy distributions. The method is verified by comparing simulations of homogeneous relaxation by our discrete velocity method to numerical solutions of the Jeans and Landau–Teller equations, and to direct simulation Monte Carlo. We compute the structure of a 1D shock using this method, and determine how the rotational energy distribution varies with spatial location in the shock and with position in velocity space.

Unified gas-kinetic scheme for diatomic molecular flow with translational, rotational, and vibrational modes

The unified gas-kinetic scheme (UGKS) is a direct modeling method for both continuum and rarefied flow computations. In the previous study, the UGKS was developed for diatomic molecular simulations with translation and rotational motions. In this paper, a UGKS with non-equilibrium translational, rotational, and vibrational degrees of freedom, will be developed. The new scheme is based on the phenomenological gas dynamics model, where the translational, rotational, and vibrational modes get to the equilibrium with different time scales with the introduction of rotational and vibrational collision numbers. This new scheme is tested in a few cases, such as the homogeneous flow relaxation, shock structure, shock tube problem, and flow passing through a circular and semi-circular cylinders. The analytical and DSMC solutions are used for the validation of the UGKS, and reasonable agreements have been achieved.

Homogeneous two-phase flow models and accurate steam-water table look-up method for fast transient simulations

The accurate simulation of fast steam-water transients requires precise algorithms for calculating fluid properties. The system of the governing flow equations must be closed with an Equation of State (EoS) to calculate the pressure as a function of the system conservative variables. For water, accurate analytical EoS for this purpose are not available yet. The aim of this paper is to show an efficient and very accurate algorithm to calculate water properties when the independent variables of the EoS are the density and the specific internal energy. Our algorithm uses a new table look-up method with bicubic interpolation based on the IAPWS-IF97 EoS formulation, and it is able to account for metastable states. The liquid metastability domain is extended until the spinodal curve, here determined and compared with other formulations.The EoS algorithm is coupled to two classical homogeneous two-phase flow models, namely the Homogeneous Equilibrium Model (HEM) and the Homogeneous Relaxation Model (HRM). HEM and HRM are used to simulate fast depressurization, waterhammer and steam explosion problems. Comparison of the numerical results with available experimental data shows the good performance of the proposed algorithms.

Influence of fuel properties on internal nozzle flow development in a multi-hole diesel injector

Fuel physical properties are known to influence in-nozzle flow behavior, in turn affecting spray formation in internal combustion engines. A series of 3D simulations was performed to model the internal nozzle flow in a five-hole mini-sac diesel injector. The goal of the study was to evaluate the behavior of two gasoline-like fuels (full-range naphtha and light naphtha) and compare them against n-Dodecane, selected from a palette used as a diesel surrogate. Simulations were carried out using a multi-phase flow representation based on the mixture model assumption with the Volume of Fluid (VOF) method, and including cavitation effects by means of the Homogeneous Relaxation Model (HRM). Validated methodologies from our previous studies were employed to account for full needle motion. Detailed simulations revealed the influence of the fuel properties on injector performance, injected fuel energy and propensity to cavitation. The three fuels were compared with respect to global parameters such as mass flow rate and area contraction coefficients, and local parameters such as pressure and velocity distribution inside the sac and orifices. Parametric investigations were also performed to understand the fuel response to changes in the fuel injection temperature, injection pressure, and geometry details. Cavitation magnitude was observed to be strongly associated with the values of saturation pressure. Owing to their higher volatility, the two gasoline-like fuels were observed to cavitate more than n-Dodecane across all the investigated conditions. While at full needle lift cavitation was reduced for all fuels, during the injection transients the gasoline-like fuels showed more propensity to cavitate inside the orifice and seat regions. This is expected to have a profound influence on nozzle erosion. Although full-range and light naphtha have lower densities compared to n-Dodecane, owing to their lower viscosity, the mass flow rate differences between the naphtha fuels and n-Dodecane were small. The analysis of fuel energy content showed that the higher lower heating value (LHV) of light naphtha helped compensate for the slightly lower total delivered mass.

Numerical analysis of deposit effect on nozzle flow and spray characteristics of GDI injectors

Injector deposit is a common phenomenon for gasoline direct ignition (GDI) engines that greatly affects the spray behavior and consequently the combustion performance and emissions. In this study, the effect of deposit on both the in-nozzle flow dynamics and downstream spray behaviors was numerically investigated. High-resolution X-ray microtomographic scans were performed first to obtain nozzle and deposit morphologies and topology. In-nozzle simulation was then carried out in the Euler-Euler framework with cavitation taken into account by a homogeneous relaxation model (HRM). Finally, the effect of deposit on spray behaviors was evaluated in the Euler-Lagrangian framework, coupled with the in-nozzle simulation results. Results of the nozzle flow simulations highlight that the rough surface of the deposits leads to additional cavitation inception and restricts the flow area, causing mass flow rate losses. Deposits inside the counterbore act as an extension to the inner orifice and constrain the air recirculation. Turbulent levels at the exit of the counterbore are lower for the coked injector due to the reduced air/fuel interaction. Spray simulations have shown that deposits would lead to longer spray penetration, a smaller spray cone angle and larger droplets diameters. Simulation results agree reasonably well with the available experimental data.

Spatially homogeneous relaxation of CO molecules with resonant VE transitions

In this paper, we study vibrational relaxation of CO molecules with excited electronic states. We consider three electronic terms and account for VV exchanges of vibrational energy within each electronic term, VT transitions of vibrational energy into a translational one, and VE exchange of vibrational energy between electronic terms. The initial vibrational state of the gas is strongly nonequilibrium. The effect of VE exchange on the vibrational relaxation of CO molecules is estimated for different kinds of initial vibrational distributions, in particular, the Treanor and Gordiets ones generalized for gases with electronically excited states. The set of equations of state-to-state vibrational kinetics, together with the gas dynamic equations, is solved numerically in the zero-order approximation of the Chapman–Enskog method for the case of spatially homogeneous relaxation. The following results are obtained: neglecting VE exchanges leads to an incorrect assessment of the number density for each electronic level; however, the error is small for the ground electronic state. It is shown that VE exchanges qualitatively affect the time dependence of the vibrational temperature.

Adaptive covariance relaxation methods for ensemble data assimilation: experiments in the real atmosphere

Covariance inflation plays an important role in the ensemble Kalman filter because the ensemble‐based error variance is usually underestimated due to various factors such as the limited ensemble size and model imperfections. Manual tuning of the inflation parameters by trial and error is computationally expensive; therefore, several studies have proposed approaches to adaptive estimation of the inflation parameters. Among others, this study focuses on the covariance relaxation method which realizes spatially dependent inflation with a spatially homogeneous relaxation parameter. This study performs a series of experiments with the non‐hydrostatic icosahedral atmospheric model (NICAM) and the local ensemble transform Kalman filter (LETKF) assimilating the real‐world conventional observations and satellite radiances. Two adaptive covariance relaxation methods are implemented: relaxation to prior spread based on Ying and Zhang (adaptive‐RTPS), and relaxation to prior perturbation (adaptive‐RTPP). Both adaptive‐RTPS and adaptive‐RTPP generally improve the analysis compared to a baseline control experiment with an adaptive multiplicative inflation method. However, the adaptive‐RTPS and adaptive‐RTPP methods lead to an over‐dispersive (under‐dispersive) ensemble in the sparsely (densely) observed regions compared with the adaptive multiplicative inflation method. We find that the adaptive‐RTPS and adaptive‐RTPP methods are robust to a sudden change in the observing networks and observation error settings.

Spatially homogeneous relaxation of CO molecules with resonant V E transitions

Spatially homogeneous relaxation of CO molecules with resonant V E transitions

HEM and HRM accuracy comparison for the simulation of CO2 expansion in two-phase ejectors for supermarket refrigeration systems

In this study, the accuracies of the homogeneous equilibrium (HEM) and homogeneous relaxation models (HRM) were compared. Both models were implemented in the ejectorPL computational tool. The HEM and HRM were used to simulate the carbon dioxide flow in ejectors that were designed for supermarket refrigeration systems. The model accuracy was evaluated by comparing the computational results with the experimental data. The discrepancy between the measured and computed motive nozzle mass flow rates was analysed. In addition, the difference between the experimental and computational mass entrainment ratios was calculated. The operating regimes in this study ranged from 47bar to 94bar and from 6°C to 36°C for the pressure and temperature, respectively. The model accuracy strongly depends on the distance between the operating regime and the critical point of the refrigerant. The discrepancy for the selected operating regimes ranged from 0.3% to 43.3% and 0.7% to 42.0% for HEM and HRM, respectively. For lower pressures and temperatures, the HRM has higher accuracy than the HEM. The errors of the HRM results were approx. 5% lower than those of the HEM results. The accuracy improvement of the HRM was considered unsatisfactory. The low accuracy improvements were possibly caused by the relaxation time formulation in the homogeneous relaxation model.