Homogeneous Equilibrium Model Research Articles

Impact of modelling assumptions in cavitating flow of simplified injector

Here, three-dimensional Reynolds-averaged Navier–Stokes (RANS) simulations are employed for in-nozzle flow modelling. The aim is to evaluate the capabilities and sensitivities of simulating in-nozzle flow phenomena by using homogeneous phase change models. Two commonly used homogeneous models are employed – (i) Homogeneous Equilibrium Model (HEM) and (ii) Homogeneous Relaxation Model (HRM) A simplified nozzle with reference experimental data is modelled in the developing, super cavitation, and hydraulic flip regimes using OpenFOAM. The influence of the inlet boundary condition and turbulence models are investigated. With HEM, three barotropic compressibility models are tested. The research seeks to understand the strengths and limitations of each approach by comparing different sub-models and their interactions with the flow fields and phase change phenomena. Ultimately, this study contributes to a better understanding of how different modelling choices can influence the accuracy and reliability of cavitation simulations using homogeneous phase change models. While the simulations demonstrated reliable predictions for low-order velocity statistics, substantial discrepancies were identified in the prediction of the cavitating region by both homogeneous models. Additionally, the relaxation model predicted only minor cavitation. In contrast, acceptable flow predictions were achieved using different barotropic compressibility models and inlet boundary condition types, as well as most RANS turbulence models, where the RNG k-ɛ model deviated more from the other turbulence models. Therefore, the added value of the present study is summarised as: (a) Four different cavitation regimes are simulated with HEM and HRM in a single study. (b) The more comprehensive HRM approach does not produce better velocity or cavitation predictions for a low-speed flow using the default relaxation time. (c) The choice of turbulence model can radically affect the cavitation prediction, in addition to velocity fields. (d) The HEM flow simulation has little sensitivity to the barotropic compressibility model.

Two-phase simulation of transcritical CO$$_2$$ flow in nozzle and ejector using the homogeneous equilibrium model

Two-phase simulation of transcritical CO$$_2$$ flow in nozzle and ejector using the homogeneous equilibrium model

The Development of a Transient Analysis Platform of Near-Critical CO2 Thermodynamic Systems via an Enthalpy-Based Implicit Continuous Eulerian Approach

This work presents the development and validation of an enthalpy-based implicit continuous Eulerian (ICE) solver, termed the near-critical ICE solver (NICES), for the analysis of near-critical CO2 thermodynamic systems. Traditional approaches relying on pressure and temperature as main inputs for the analysis have limitations in handling CO2 near the critical point, which exhibits unique characteristics and frequent phase changes. To overcome these limitations, this study proposes using enthalpy as a more suitable mathematical modeling approach. The NICES methodology employs the homogeneous equilibrium model and the Span and Wagner equations of state for CO2. This solver demonstrates improved numerical stability and computational speed compared to explicit calculation methods, as validated by frictionless heated pipe scenarios involving phase transitions near the critical point. The enthalpy-based NICES platform can predict thermohydraulics, including multiphase flows, without requiring specialized two-phase flow models.

A homogeneous relaxation model algorithm in density-based formulation with novel tabulated method for the modeling of CO2 flashing nozzles

Accurate prediction of the flow physics within nozzles and ejectors working with two-phase carbon dioxide remains particularly challenging. Indeed, the flashing phenomenon in the device often comes along with the formation of the metastable phase, which can severely affect both the flow physics, and the flow rate. Due to the particular challenges associated with building and operating two-phase carbon dioxide experimental setups, computational fluid dynamics constitutes a compelling alternative to study and predict nozzle and ejector flows. In this work, the metastability effects are modeled through a Homogeneous Relaxation Model for the first time expressed in density-based formulation, within the SU2 solver. The thermodynamic relations for the equation of state are tabulated using a novel quad-tree algorithm and directly coupled to the solver, ensuring fast and accurate predictions. The simulation tool is then validated against experimental nozzle pressure profiles and used to compare the commonly used Homogeneous Equilibrium Model to the Homogeneous Relaxation Model in terms of local flow topology and mass flow rates. Most notably, it was found that the Homogeneous Relaxation Model was able to better fit the experimental pressure profiles within the first part of the expansion.

Simulation study of key cross-sectional area dimensions of CO2 ejector on performance

In this paper, based on the homogeneous equilibrium model, the effects of nozzle inlet cross-sectional area, nozzle throat cross-sectional area, pre-mixing chamber cross-sectional area, mixing chamber cross-sectional area, and diffusion chamber outlet cross-sectional area on the performance of the ejector of a CO2 transcritical refrigeration system are analyzed by using Ansys Fluent. The simulation results show that the primary flow rate of the ejector is sensitive only to the nozzle throat cross-sectional area, while the entrainment ratio is strongly influenced by the nozzle throat and the mixing chamber cross-sectional area, and there is an optimum value. The nozzle inlet cross-sectional area has no significant effect on the performance of the ejector. The pre-mixing and diffusion chamber outlet cross-sectional areas do not have a significant effect on the performance within a limited range, but deviating too much from the optimum value can also lead to degradation of ejector performance.

Development of Two-Phase Frictional Pressure Gradient Correlation for Saturated Cryogenic Flow Boiling in Uniformly Heated Tubes

Experiments to determine two-phase frictional pressure drop for saturated cryogenic flow boiling in uniformly heated round tubes have been performed throughout the globe during the past sixty years for a variety of cryogens. However, the experimental data are scarce and either rarely published or published only for a few cryogens, with majority of the data remaining in the archives of authors, or in obscure technical reports of an organization or other inaccessible sources. In the present study, the Purdue University-Boiling and Two-Phase Flow Laboratory (PU-BTPFL) Cryogenic Total Pressure Drop Database is consolidated both for saturated adiabatic liquid-vapor and saturated flow boiling for cryogens from world literature dating back to 1964. With 474 total pressure drop data points for LHe, LH2, LNe, and LN2, it represents the largest total pressure drop database for cryogenic saturated two-phase flow assembled to date. Using this database, predictive accuracy of seminal frictional pressure gradient correlations employing both the Homogenous Equilibrium Model (HEM) and the Separated Flow Model (SFM) are assessed. The distinct fluid physics unique to cryogens is studied further using the database and new dimensionless criteria are developed to demarcate a dispersed flow dominant data subset from the separated flow dominant data subset, which are better represented using HEM and SFM, respectively. These criteria are subsequently used to develop a hybrid HEM-SEM frictional pressure gradient correlation for saturated cryogenic flow for the two distinct data subsets. The new hybrid correlation performs the best of all seminal correlations and provides excellent predictive agreement with the database, evidenced by a Mean Absolute Error (MAE) of 14.7 %.

Extension of the delayed equilibrium model to flashing flows of organic fluids in converging-diverging nozzles

Flashing flows of organic fluids find application in various energy systems, but the available modelling approaches rely on semi-empirical correlations calibrated for other fluids and operating conditions. This paper extends the state-of-the-art one-dimensional models developed for water to flashing of organic fluids, using R134a as a reference. First, the delayed equilibrium model combined with Grönnerud's friction correlation is identified as the most appropriate approach, reducing the error on mass flow rate with respect to experimental results from 48.3 %, obtained with the homogeneous equilibrium model, to 10.3 %. Then, the traditionally adopted incompressible flow assumption for the metastable liquid phase is replaced by a dedicated thermodynamic model, preventing the appearance of unphysical metastable phase temperatures. Moreover, the delayed equilibrium model is tailored to R134a by tuning the semi-empirical coefficients of the constitutive law. The proposed modifications to the state-of-the-art delayed equilibrium model result in an improvement in the prediction of the pressure profiles and the mass flow rate for flashing flows of R134a, with a decrease in the error on mass flow rate from 10.3 % to 6.1 % with respect to the original formulation.

Depressurization of CO2 in a pipe: Effect of initial state on non-equilibrium two-phase flow

It is key in several industrial applications to accurately describe rapid depressurization of liquid and dense phase states. Examples include refrigeration systems, nuclear reactor cooling and CO2 capture and storage (CCS). It is expected that large-scale CO2 pipeline transportation must be deployed as a vital part of reaching net zero emissions by 2050. During rapid depressurization of liquid-like CO2, boiling will in many cases occur out of equilibrium, at a lower pressure than the local saturation pressure. Capturing the non-equilibrium effects is necessary to predict outflow rates and the resulting pressure and temperature inside the pipe. In the present work, we quantify the non-equilibrium effects by studying a series of CO2 pipe depressurization experiments from liquid-like states at initial temperatures from 10°C to 40°C. We compare the experimental results to predictions of the homogeneous equilibrium model (HEM) and a homogeneous relaxation-type non-equilibrium model (HRM*) where the mass-transfer rate from liquid to gas is tuned by a relaxation time. The relaxation time was found to decrease for increasing temperatures, and it was observed to be approximately 60 times longer for the coldest experiment than for the warmest one.

Sensitivity of TEHL Simulations to the Use of Different Models for the Constitutive Behaviour of Lubricants

This study compares the film thickness, lubricant temperature, and traction curves of two groups of commonly used constitutive models for lubricants in thermo-elastohydrodynamic lubrication (TEHL) modelling. The first group consists of the Tait equation of state, the Doolittle Newtonian viscosity model, and the Carreau shear thinning model. The second group includes the Dowson equation of state, the Roelands–Houpert Newtonian viscosity model, and the Eyring shear thinning model. The simulations were conducted using a Computational Fluid Dynamic and Fluid-Structure Interaction (CFD-FSI) approach, which employs a homogeneous equilibrium model for the flow simulation along with a linear elastic solver to describe the deformation of the solid materials. The simulations were conducted under a load range of 100 kN/m to 200 kN/m and a slide-to-roll-ratio (SRR) range between 0 and 2 using Squalane lubricant. The results show up to a 10% deviation in central film thickness, a 31% deviation in coefficient of friction (CoF), and a 38% deviation in maximum lubricant temperature when using the different constitutive models. This study highlights the sensitivity of TEHL simulation results to the choice of constitutive models for lubricants and the importance of carefully selecting the appropriate models for specific applications.

Simulation study on active cavitation suppression for a typical hydrofoil

An active cavitation suppression structure with jet holes on its surface is proposed for a typical hydrofoil (NACA66 MOD). Based on the homogeneous equilibrium model and the Zwart-Gerber-Belamri cavitation model, the cavitation characteristics around the hydrofoil before and after jet holes are provided on hydrofoil surface at a typical operating condition are investigated through simulation. Simulation results show that the jet holes on hydrofoil surface are capable of regulating and controlling the cavitation flow field, and to a certain extent, the jet holes on hydrofoil suction surface inhibit the development and shedding of cavitation. Four groups of different jet flow rates are set up, and it is found that with the increase of jet flow rate, the cavitation suppression effect gradually increases, and the lift-to-drag ratio gradually decreases. The research shows that the jet can achieve a cavitation suppression effect of 69.35% under the premise of ensuring that the range of hydrodynamic performance does not exceed 5%. By coupling the distribution of cavitation shape, re-entrant jet and pressure field of a hydrofoil with fluid flowing around and studying the periodic change of flow field, the mechanism of the jet to suppress cavitation is analyzed.

Simulation study on passive cavitation suppression for a typical hydrofoil

A passive cavitation suppression structure with flow barriers on its surface was proposed for a typical hydrofoil (NACA66 MOD). Based on the homogeneous equilibrium model and the Zwart-Gerber-Belamri cavitation model, the cavitation characteristics around the hydrofoil before and after flow barriers on hydrofoil surface at a typical operating condition are simulation investigated. Results show that the flow barriers provide on hydrofoil surface are capable of regulating and controlling the cavitation flow field, suppressing 47.1 percent of cavitation flow. By coupling the distribution of cavitation shape, re-entrant jet and pressure field of the hydrofoil with fluid flowing around, and analysing the periodic change of flow field, it is found that to a certain extent, the flow barrier provided on hydrofoil suction surface inhibited the development and shedding of cavitation.

Towards an engineering tool for the prediction of running ductile fractures in CO2 pipelines

A prerequisite for the deployment of CO2 capture and storage (CCS) is to establish a large network of high-pressure transport pipelines. It is then vital to assess new and existing pipeline designs for running ductile fracture (RDF). RDF is a phenomenon in which a defect develops into a crack propagating along the pipeline, sustained by the pressure forces from the escaping fluid. The most common engineering method for RDF, the Battelle two-curve method (BTCM), was originally developed for natural gas (NG) and has proved non-conservative for CO2. In this work we examine the BTCM in the light of available RDF experiments with CO2-rich mixtures. We present an improved material curve, in which the change in fluid properties when replacing NG with CO2 results in a new effective toughness correlation. Furthermore, we present an improved method for calculating the crack-tip pressure. This delayed homogeneous equilibrium model (D-HEM) accounts for the non-equilibrium thermodynamics due to the rapid depressurization, resulting in boiling pressures below the saturation pressure. Together, the adaptation of the material and fluid treatment yields improved results, and is a step towards a viable engineering tool for the prediction of RDF in CO2 pipelines.

Evaluation of Large-Eddy Simulation Coupled with an Homogeneous Equilibrium Model for the Prediction of Coaxial Cryogenic Flames under Subcritical Conditions

Large Eddy Simulations of liquid O2/gaseous H2 coaxial flames at subcritical pressure conditions are reported in this paper. These simulations reproduce the experimental Mascotte cases A1, A10 and A30, operating at 1, 10 and 30 bar, respectively, and for which temperature measurements and experimental visualisations are available. The main objective of this work is to assess the accuracy of the multi-fluid Homogeneous Equilibrium Model (HEM) described in Pelletier et al. (Computers & Fluids, 2020) for rocket engine applications. Of particular interest is the comparison with the experimental temperature measurements from Grisch et al. (Aerospace science and technology, 2003). To that purpose, numerical simulations are conducted with care, in order to ensure a proper statistical convergence and estimate the influence of the grid resolution for each case. Despite the crude assumptions—no surface tension and no atomisation model, for instance—that are made with the HEM used in this work, results are found to be in reasonable agreements with the measurements for case A10, even with the coarser grid. For case A30, a fine mesh resolution is required to capture the low intensity recirculation zone downstream of the inner jet necessary to reproduce the shape of the experimental profile. Finally, case A1 simulations, with the lowest Weber number, show large departures with the experimental measurements. This is expected to be due to a deficiency of the model to properly reproduce the two-phase dispersed flow.

Modelling of two-phase expansion in a reciprocating expander

Two-phase expansion devices can be applied in a multitude of cycles which allow for an increase of the overall system efficiency in currently used heat and power cycles. The expanders that are mainly investigated for the purpose of two-phase expansion are the Lysholm and the reciprocating expanders. From these two, the reciprocating expander is challenging to model as there is a need to describe the non-equilibrium state of the phases during the expansion process. This study aims to fill the knowledge gap that exists for two-phase reciprocating expanders, by constructing a predictive model to describe the expansion process and the non-equilibrium phenomena occurring. The simulation framework makes use of a modified homogeneous relaxation model to describe the thermal non-equilibrium between the phases within the isolated working chamber. The model results are verified and compared to experimental data in literature with water as working fluid. Subsequently, Cyclopentane is selected as a more suitable candidate and a thorough analysis is presented. The simulation results predict a 15% lower work and a 10% higher mass intake per stroke of the expander compared to equilibrium working conditions. • A model is constructed to predict the evaporation rate of a two-phase mixture in a reciprocating expander. • The homogeneous relaxation model is applied and compared to the homogeneous equilibrium model. • The evaporation rate in a reciprocating expander is lower than expected, resulting in metastable conditions during the process. • The updated disequilibrium model predicts around 25% more mass intake and 8% lower work compared to the equilibrium model.

Research on condensation characteristics of supercritical carbon dioxide in Laval nozzle considering working condition of power cycles

The supercritical carbon dioxide (sCO2) thermal power cycle is a solution for generation IV nuclear energy systems. The potential condensation in the key equipment of the power system poses a serious threat to the safe operation of the system. Aiming at the condensation that may occur inside the sCO2 compressor, this paper numerically simulates the condensation that occurs in a convergent-divergent nozzle. The wet steam model and the homogeneous equilibrium model (HEM) are present and compared with the experimental data. The HEM is used to calculate the influence of the inlet conditions of the fluid into the nozzle and the different expansion rates corresponding to different nozzle shapes on the condensation phenomenon. The results show that the HEM can represent the two-phase flow of carbon dioxide in the nozzle. The nozzle inlet boundary conditions and nozzle expansion rate have an important influence on the condensation characteristics, the former influences the condensation position, while the expansion rate has an influence on the outlet humidity and pressure drop.

Experiments and modelling of choked flow of CO2 in orifices and nozzles

In order to accelerate the deployment of CO2 capture and storage (CCS), engineers need experimentally validated models, among other things, to predict the mass flow rate in process equipment and flow restrictions like valves, nozzles and orifices. There are few available, relevant data for choked CO2 flow in such geometries. To amend the situation, in this work, we report on six pipe-depressurization experiments from a pressure of 12 MPa and a temperature of 25 °C through three sizes (4.5, 9.0 and 12.7 mm) of orifices and nozzles. The results indicate that for the present cases, the choke point is at a non-equilibrium state. In order to predict quasi-steady choked flow in restrictions, the homogeneous equilibrium model (HEM) and the Henry–Fauske (HF) model are commonly used. The HEM often underpredicts the mass flow rate because it does not account for delayed phase transition. Here we develop a delayed HEM (D-HEM) where evaporation starts at the superheat limit described using classical nucleation theory. We then employ the HEM, D-HEM, and HF model in 1D CFD pipe simulations to describe the outflow of depressurization experiments and we also compare with experimental data for converging–diverging nozzles. In the CFD simulations, HF gave the best results, while HEM consistently underpredicted the mass flux. For the nozzle calculations, we found D-HEM to be the best model with a relative absolute error of 2.5 % for the predicted mass flux.

One-dimensional mathematical modeling of two-phase ejectors: Extension to mixtures and mapping of the local exergy destruction

The ejector is a process equipment frequently used in refrigeration processes. There is currently a knowledge gap on the efficiency of ejectors operating with mixtures. To address this knowledge gap, we present a one-dimensional ejector model for mixtures, defined by spatially distributed mass-, energy- and momentum-balances for different zones, which together constitute the full ejector geometry. The recently developed delayed homogeneous relaxation model is used to describe the two-phase transition in the motive and suction nozzles. For evaporation of pure CO 2 in the throat of the motive nozzle, the model yields an average error of 2.6% in the critical mass flow rate, which is significantly lower than the homogeneous equilibrium model that has an average error of 8.4%. A comparison to new experimental data shows that both models give excellent predictions of critical mass flow rates where condensation occurs in the throat, with an average error below 0.9%. New experimental data with CO 2 are presented, which are used to regress two parameters in correlations that describe the momentum transfer between the primary and secondary stream in the mixer and diffuser sections. This leads to accurate reproduction of the pressure lift in five different ejector geometries, with a mean error of 2.3%. By using nonequilibrium thermodynamics, we derive formulae for the local exergy destruction in the ejector. The largest exergy destruction is located in the mixer, and originates in transfer of momentum between the primary and secondary streams. The local exergy destruction profiles through the mixer and diffuser are highly non-uniform, and deviate from the established guidelines for energy-efficient design characterized by equipartition of exergy destruction. This reveals a potential to increase the performance of ejectors by suitable adjustments to their geometric design. The mathematical model validated for pure CO 2 is assumed to also be valid for mixtures rich in CO 2 . We show that minute concentrations of a second component can have a significant influence on the ejector performance. When fixing the inlet conditions and ejector geometry, we demonstrate that adding 2% H 2 to a mixture of CO 2 decreases the critical mass flow rate by 20%, while adding 2% SO 2 increases the pressure lift by 1 bar. • A state-of-the-art, one-dimensional ejector model for mixtures is presented. • New experimental data for ejector operation with CO2 improves the model. • Critical mass flow rates and pressure lifts from experiments reproduced within 3%. • Local mapping of exergy destruction with nonequilibrium thermodynamics. • Minute concentrations of a second component alter the ejector performance.

Study on supercritical CO2 critical flow through orifices under power cycle operating conditions

For prediction of critical flow occurring under supercritical carbon dioxide (S-CO2) condition, one-dimensional (1-D) analytical critical flow models are evaluated in this study. Specifically, homogeneous equilibrium model (HEM) and a newly proposed non-equilibrium model are evaluated. The new non-equilibrium model adopts Moody’s slip ratio for mechanical non-equilibrium and a newly proposed correlation of supercooling for thermal non-equilibrium. For evaluation of the models over various thermodynamic states, S-CO2 critical flow experiment through an orifice is conducted to supplement and expand a range of previously published S-CO2 critical flow experiment data. From the evaluation results with the expanded S-CO2 critical flow experiment database, the new non-equilibrium model shows better prediction performance than HEM in terms of the prediction of critical mass flux and critical pressure. For practical applications, discharge coefficients for each 1-D analytical critical flow model are suggested for the orifice geometry.

Transcritical carbon dioxide heat pump cycle: Validation and comparison of one-dimensional models of ejector with independent data sets

Two typical 1-D homogeneous equilibrium models from the literature (constant and variable pressure mixing models) were used to study the ejector expansion transcritical CO 2 heat pump cycle. A sensitivity analysis was performed to study the impact of the isentropic efficiency of the compressor and of each component of the ejector on the overall system performance. Two groups of independent experimental data from the literature were used to validate the two models. It was found that the cycle performance was more sensitive to the isentropic efficiency of the compressor than that of the individual components of the ejector. With appropriate input value of the compressor efficiency, both 1-D models accurately predicted the ejector outlet pressure and vapor quality. However, for the ejector entrainment ratio and cycle COP, the deviations were more than 50 %. These deviations can be significantly minimized to 10 % by eliminating the imbalance occurring in the system operation using a data post-processing method to correct the pseudo steady state in the real operation system. By comparing the simulation results of the two models, it has been shown that there is minor difference on the prediction accuracy. The research can serve as a practical reference for the operation of two-phase-flow ejector expansion transcritical heat pump cycle, and can also provide an insight and orientation for the subsequent optimization of 1-D ejector numerical models. • Two 1-D homogeneous equilibrium models of the ejector are experimental validated. • Neglecting the pressure variation during the mixing has no significant impact. • The liquid imbalance is a predominant source of the modeling deviation. • An innovative data post-processing method is proposed. • The system performance is more sensitive to the compressor isentropic efficiency.

Computational fluid-dynamic investigation of a centrifugal compressor with inlet guide vanes for supercritical carbon dioxide power systems

A centrifugal compressor for large-scale supercritical carbon dioxide applications (∼50 MW) is analyzed by means of RANS simulations. The purposely designed compressor includes adjustable inlet guide vanes, an open impeller, and a wedge-shaped vaned diffuser. The upstream total state is located at 78.70 bar and 305.15 K, close to the thermodynamic critical point, whereby non-ideal effects and two-phase flows are significant. To account for these effects, the computational model implements a homogeneous equilibrium model (using pressure and specific enthalpy as state variables) complemented with a state-of-the-art equation of state explicit in the Helmholtz free energy and specific correlations for transport properties. It is found that the compressor can provide the entire pressure ratio required by the cycle (∼3.25) with high efficiency (87.9%, excluding parasitic losses). Three inlet vane rotations are considered: −10deg, 20deg, and 40deg. The preswirl has a limited effect on efficiency, which is reduced by 0.3% points away from choked conditions with vane rotation of 40deg. Nonetheless, the preswirl affects the extent of the two-phase region at the impeller intake, which has a detrimental impact on the compressor flexibility by setting an early choking.