Development of Delayed Equilibrium Model for CO2 convergent-divergent nozzle transonic flashing flow
Introduction
Flashing is a phenomenon of intensive evaporation in flowing liquid due to a decrease in static pressure. The process of the vapour phase generation is usually accompanied by significant thermodynamic and mechanical non-equilibrium, which is mainly manifested by a difference in temperature and velocities of the phases. This article focuses on the flashing process that occurs in the flow through a convergent-divergent nozzle that is a part of a two-phase ejector. The schematic of the two-phase ejector with supercritical or subcooled liquid phase as a motive fluid and with a vapour phase as a secondary fluid is presented in Fig. 1a. The liquid phase due to depressurisation achieves saturation conditions and partly evaporates inside the motive nozzle so that two-phase flow emanates from the nozzle. Due to momentum transfer between the motive jet and the vapour, the vapour is entrained into the suction chamber and further into the mixing chamber. The progressing momentum transfer causes a mixing process that takes place together with formation of a two-phase shock wave. The appearance of the shock wave dramatically rises the static pressure of the mixture. Finally, the homogeneous two-phase mixture is additionally compressed in the diffuser achieving the discharge pressure pm, which is higher than the pressure of the entrained vapour p0.
A more detailed insight into the flashing process that occurs in the convergent-divergent nozzle is presented in Fig. 1b. The supercritical or subcooled liquid enters the motive nozzle and partly evaporates (mostly due to accelerational depressurisation). The rate of evaporation at the beginning of the flashing flow depends on two factors: the number of active nuclei and the superheat of liquid. However, the smaller nucleus the higher superheat is required for its activation. Therefore, despite that at a certain location, the equilibrium saturation pressure is achieved, the vaporization process does not start and a metastable liquid flow occurs up to a location where the metastable liquid is sufficiently superheated to activate the largest nuclei, Boure et al. (1976). From that location, the two-phase flashing flow is formed and the vapour phase growth rate is limited by the interphase heat transfer rate rather instead of mechanical expansion. However, under real operation conditions, both mechanical and thermal non-equilibrium exist simultaneously during flashing process inside the motive nozzle making strong difficulties to accurately predict critical mass flow rate through the motive nozzle. The appearance and growth of vapour bubbles drastically decreases the fluid density. Consequently, the velocity, Mach number and fluid compressibility dramatically increase. However, due to the presence of dissipative phenomena, the flow achieves the local sound speed (M=1) somewhere after the nozzle throat. The further decompression and Mach number increase take place in the divergent portion of the nozzle.
Besides nozzles, the flashing phenomenon is encountered in such common devices as valves, orifices and throttling devices. Therefore, the accurate prediction of the flashing process is of crucial importance for a design of apparatus that could be generally classified as devices of chemical and process engineering. Cases of such applications were described for example by Pangarkar (2017), Weber et al. (2018), Rahman et al. (2010), Gamisans et al. (2004). The flashing also occurs in accidentally/randomly appearing channels (or microchannels) as cracks or full bore ruptures in pressurized tanks and tubes. Thus, the process is also relevant for nuclear power plant and chemical plant safety where the accurate prediction of the leakage rate through the crack plays a crucial role. These problems were, in turn, investigated, for example, by Hanaoka et al. (1990), Bartosiewicz et al. (2010), Papini et al. (2011).
As a result of this huge range of process occurrence, the flashing flow became a topic of many types of research and many modelling approaches have been developed. Those approaches can be classified mainly in three groups: lumped-parameter models, one-dimensional models and CFD approaches, as noted by Lorenzo et al. (2017). Unfortunately, the vast majority of those approaches were developed for water. However, the recent studies on implementations of the ejector in CO2 systems such as heat pumps, refrigeration and air-conditioning devices, demonstrated significant improvement of those systems energy efficiency, as reported for example by Boccardi et al. (2017), Liu et al. (2016), Lucas et al. (2013). As a result, additional studies of carbon dioxide ejector refrigeration cycles have been conducted. Some of them included investigations on flashing flow through the convergent-divergent nozzle, e.g. Banasiak et al. (2013), Nakagawa et al. (2009). Consequently, some experimental data in this field are available providing an opportunity to study the predictions of CO2 two-phase transonic flows through the convergent-divergent nozzles.
Among all aspects of the flashing flow, the problem of accurate predictions of critical mass flow rate is currently the most widely studied. The most profound conclusion from these studies is that the aspects of mechanical non-equilibrium are in this matter a secondary meaning issue. This was first concluded through theoretical considerations, among others, by Boure et al. (1976) and Bilicki et al. (1990). However, there are also empirical observations stating that the flashing transonic flow has usually form of a fully dispersed flow, Brown et al. (2013). Consequently, application of a complex two-fluid models or even simpler slip ratio models in modelling of the flashing transonic flow is unjustified. The problem of accurate predictions of the critical mass flow rate can be reduced to a problem of obtaining a physically consistent prediction of the sound speed, as inferred by Bilicki et al. (1990), Attou et al. (1999), Lorenzo et al. (2017). The mentioned researchers investigated various types of non-equilibrium two-phase flow models. Despite they used different analytical approaches, they all have shown that the physically consistent predictions of the sound speed cannot come from equilibrium approaches and that the introduction of the thermal non-equilibrium effects into the modelling is in this matter crucial. Keeping this in mind, the authors focused their attention first on the Homogeneous Relaxation Model (HRM) which is a relatively simple approach accounting for thermal non-equilibrium effects, Angielczyk et al. (2010). Using the HRM, the authors calculated the decompression curves for the selected nozzle geometries and boundary conditions obtaining reasonable matching with the applied experimental data. However, subsequently Bartosiewicz et al. (2010) proved that the Delayed Equilibrium Model (DEM), which is also a relaxation model, is more accurate than HRM in terms of the critical mass flow rate prediction. The ability of the DEM to accurately predict the critical mass flow rate was also confirmed by Lorenzo et al. (2017) through confrontation with experimental data of a wide range (incorporating flows through long tubes, short tubes, and slits) that comes mainly from the famous Super Moby-Dick experiment (this experiment is still the most coherent experimental database concerning water transonic flashing flows; it contains information on critical mass flow rates, pressure distributions and void fraction distributions). The main conclusion of the above-mentioned investigations is that among tested models (HEM, HRM and lumped-parameter approaches of Moody and Hanry-Fauske), DEM is the most accurate in terms of both the critical mass flow rate predictions and pressure distribution predictions. Moreover, the DEM has been implemented in WAHA code by Bartosiewicz and Seynhaeve (2014) and in NEPTUNE_CFD code by Duponcheel et al. (2013). It was shown that the use of DEM in CFD codes allows a very good representation of the experimental data and the results obtained were very close to the ones obtained by application of the stationary version of the model. Therefore, the main aim of the subsequent investigation, Angielczyk et al. (2019), was to check if the Delayed Equilibrium Model with a closure law, originally developed for water, may be considered as suitable for modelling of CO2 transonic two-phase flows. Unfortunately, the mentioned investigations revealed that the original DEM closure law is not suitable for CO2 transonic flow calculations. The investigation revealed also that the applied Darcy friction factor approach has a significant impact on the simulation results. Hence, two friction factor approaches were tested, namely, the Lockhart–Martinelli and the Friedel method. It was shown that the Lockhart–Martinelli approach returns such high values of the frictional pressure gradient that the obtained results have no physical meaning. In the case of the Friedel approach, this situation happened rarely. Therefore, the Friedel approach seemed to be more proper for CO2 flows. Also, Aakenes et al. (2014) got a similar conclusion: among the investigated approaches (homogeneous friction-model, Friedel and Cheng approach), the Friedel approach turned out to be the most accurate for the case of CO2 tube flows. Therefore, the authors decided to use the Friedel approach as a basis for developing a CO2 dedicated approach. In the case of the saturation index evolution law, it was decided to develop a new formula on the basis of the original law.
The comparison of the proposed DEM setup with another DEM implementation is not possible since the developed approaches are the first proposition in the domain of DEM simulations of pure-CO2 flows. Thus, the Homogeneous Equilibrium Model is applied as a reference case. Moreover, since the HEM does not consider any of non-equilibrium effects, its application as a comparative case allows to assess how the introduction of the thermal non-equilibrium effects affects prediction abilities of the modelling.
Section snippets
Homogeneous Equilibrium Model (HEM)
Among the two-phase flow models, the HEM is distinguished by the smallest number of the closure equations. Namely, it requires only the equation for the transferred heat flux q and the equation for the wall shear stress τ. It is also related to an instantaneous mass transfer between fluid fractions. For those reasons, HEM is commonly treated as the reference case.
The HEM imposes the thermal equilibrium between vapour phase and liquid phase. Consequently, the temperatures and pressures of both
Solution procedure
In the case of the nozzle fed with fluid under supercritical or subcooled state, in order to obtain the solution describing the nozzle flow, a single-phase flow model has to be utilized first. The applied single-phase flow model consists of the system of equations (1-3) supplemented with the thermodynamic state equations that describe the supercritical or subcooled properties of the fluid (Subsection 2.3). This model operates until the saturation conditions are reached. Subsequently, a
The proposed law of the saturation index evolution
The flashing process occurs due to the static pressure drop below the saturation pressure in the flowing fluid. Therefore, the pressure drop rate determines the intensity of the phase change process that takes place in the expanding fluid. For the incompressible flows through channels with a constant cross-sectional area, the pressure drop occurs as a result of friction between the fluid layers (or elements) and between the fluid and the channel walls. However, in the case of the
The frictional pressure gradient
In the previous investigation, Angielczyk et al. (2019), and work of Aakenes et al. (2014) it was shown that among investigated approaches the Friedel frictional pressure gradient method turned out to be the most accurate for CO2 flows. Therefore, the approach proposed here is based on the Friedel method:
The coefficient a1 turned out to be strongly dependent on the reduced subcooling level:
The following
The Experiment
The unique experimental data concerning two-phase transonic CO2 flow through a de Laval nozzle were published by Nakagawa et al. (2009). The experiment was conducted as a blown-down test of CO2. The investigated nozzle was fed from CO2 tank and after passing through the nozzle and the back-pressure regulation valve, the fluid was exhausted to the atmosphere. Convergent-divergent stainless steel nozzles were applied in the experiment. All used nozzles had a rectangular cross-sectional area. The
Analysis of simulation results
After the preliminary calculations, three out of ten experimental cases were excluded from the investigation as they turned out to be cases of a single-phase flow. Those three cases are related to the supercritical inlet conditions and the following nozzle divergent part angles: 0.077 o, 0.153 o, 0.306 o.
Therefore, the proposed approaches were developed on the base of seven experimental static pressure distributions of CO2 two-phase transonic flows. The inlet conditions of the experimental
Conclusions
The presented statistical analyses proved that the proposed saturation index evolution law and the proposed frictional pressure gradient approach significantly improve the prediction ability of transonic flow pressure profiles of the Delayed Equilibrium Model: The mean maximum discrepancy between experimental and calculated static pressure values decreased almost 2 times, with the almost fourfold decrease in the standard deviation, in comparison to the referential Homogeneous Equilibrium Model.
Acknowledgements
The research was carried out as part of research work No. WZ/WM-IIM/1/2020 at the Bialystok University of Technology and financed from a subsidy provided by the Ministry of Science and Higher Education.
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