A comparison of heterogenous and homogenous models of two-phase transonic compressible

Transcription

1 Home Search Collections Journals About Contact us My IOPscience A comparison of heterogenous and homogenous models of two-phase transonic compressible CO 2 flow through a heat pump ejector This content has been downloaded from IOPscience. Please scroll down to see the full text IOP Conf. Ser.: Mater. Sci. Eng ( View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: This content was downloaded on 06/10/2015 at 10:03 Please note that terms and conditions apply.

2 A comparison of heterogenous and homogenous models of two-phase transonic compressible CO 2 flow through a heat pump ejector Zbigniew Bulinski, Jacek Smolka, Adam Fic, Krzysztof Banasiak and Andrzej J. Nowak Institute of Thermal Technology, Silesian University of Technology, Konarskiego 22, Gliwice, Poland [email protected] Abstract. This paper presents mathematical model of a two-phase transonic flow occurring in a CO 2 ejector which replaces a throttling valve typically used in heat pump systems. It combines functions of the expander and compressor and it recovers the expansion energy lost by a throttling valve in the classical heat pump cycle. Two modelling approaches were applied for this problem, namely a heterogenous and homogenous. In the heterogenous model an additional differential transport equation for the mass fraction of the gas phase is solved. The evaporation and condensation process in this model is described with use of the Rayleigh-Plesset equation. In the homogenous model, phases are traced based on the thermodynamic parameters. Hence the heterogenous model is capable to predict non-equilibrium conditions. Results obtained with both models were compared with the experimental measurements. 1. Introduction An ejector was invented in 1858 by the Henry Giffard. Initially it was used to pump liquid water to the reservoir of the steam engine boilers. Since then ejectors have been extensively applied in different fields. Generally, the ejector can be defined as a non-volumetric type hydraulic pump which utilises the pressure drop during an expansion process of driving fluid to draw off and increase pressure of a driven fluid. Single component ejectors depending on the phase of the driving and driven fluid flows can be classified as: vapour jet, liquid jet, condensing or two-phase ejectors [8]. Giffard s device could be classified as a two-phase condensing ejector. In 1931 Gay proposed a CO 2 refrigeration cycle in which a throttling valve would be replaced with the the two-phase ejector [5]. This innovation leads to increase of the cycle performance by the reduction of the inherent throttling losses. Literature reports from 10 up to 20 per cent improvement of the coefficient of performance (COP) of the refrigeration cycle with ejector comparing to the cycle with the throttling valve [6, 11, 10]. In the last two decades this idea has experienced intensive renaissance. Huge technical progress has made possible to use transcritical ejectors making them very compact and usable in small heat pump systems or even in car airconditioning systems. Figure 1 presents simplified schema of the transcritical CO 2 refrigeration cycle and pressure-enthalpy diagram of thermodynamic processes constituting this cycle. Except already mentioned increase of the cycle COP, there are a few other advantages of using ejector in such a system: c 2010 Published under licence by Ltd 1

3 reduction of the vapour mass fraction at the inlet to the evaporator, which increase heat transfer efficiency and decrease pressure drop in the evaporator; increase of the compressor efficiency and reduction of the compression work, due to lowered pressure ratio. All aforementioned elements are making such ejectors more and more popular []. Figure 1. Simplified schema of the CO 2 heat pump cycle with transcritical ejector (a) and its p-h diagram (b). The transcritical CO 2 ejector consists of the convergent-divergent motive nozzle (MN) through which liquid/transcritical CO 2 flows, see Figure 2. The secondary fluid stream is sucked to the suction chamber (SC) preliminary expanded and mixed with the motive stream in the mixer chamber (MC). Further, mixed streams are slowed down and compressed in the diffuser (D) to the pressure level higher than the inlet pressure of the secondary fluid stream. From the Figure 2. Schematic view of the CO 2 transcritical ejector. design point of view a very important parameter for the ejector operation is the ratio of the mass flow rate ṁ s of the suction stream to the mass flow rate ṁ m of the motive stream, called 2

4 WCCM/APCOM 2010 entrainment ratio: ϕ= m s m m (1) Having constant thermodynamic parameters of the inflow and outflow ejector streams, the ejector efficiency depends linearly on the entrainment ratio. Hence, ejector designers tend to increase this parameter as high as it is possible. This paper reports the results of two mathematical models of the transcritical CO2 ejector, namely homogenous and heterogenous one. Those results are compared with the measurement date gathered on the experimental facility [4]. This is small part of the project which main aim is to develop robust mathematical model of the ejector appropriate for the optimization of its geometry. 2. Geometrical and numerical grid of the ejector Based on the information on the dimension of the test ejector a geometrical model was created and further discretised with use of the ANSYS Gambit software [1]. Since the geometry was not so straightforward a mixed mesh was applied it contained of tetrahedral and hexahedral cells. In Figure 3 the geometry and numerical mesh are shown. The geometrical model consisted from the ejector only. Moreover, the advantage of the one symmetry plane in the model was exploited. Therefore, only half of the model was considered. The mesh was made finer in the crucial parts of the model, especially near the walls and in the divergent part of the motive nozzle. Figure 3. Numerical grid for the transcritical ejector calculations. 3

5 3. Mathematical model In this paper two modelling approaches were considered. In the first one two phases were considered, namely liquid and gaseous. For the mixture of these two phases a set of compressible fluid flow equations were solved. Additionally, an extra transport equation for the vapour quality and kinetic model of the evaporation/condensation process were introduced. Although this model assumed thermodynamic and mechanical equilibrium between phases it is capable to predict existence of a subcooled vapour or superheated liquid. That is why it is called heterogenous model. In the second model, called homogeneous one, only a set of flow equations for the compressible fluid was considered and each phase was identified base on the value of thermodynamic parameters (pressure and temperature) in a given point. In both approaches steady state averaged equations were considered. Most important properties for the compressible fluid flow i.e.: density and speed of sound were calculated with use of highly accurate libraries REFPROP 8.0 (see [9]). Calculations were carried out with the commercial CFD software ANSYS Fluent Heterogenous model This mathematical model consisted of the following governing equations: continuity equation for the two-phase mixture (ϱ w) = 0 (2) where ϱ and w stands appropriately for the density and velocity vector of two-phase CO 2 mixture. The mixture density is calculated as a weighted average according to the volume fractions of both phases: 1 ϱ = 1 f ϱ l + f (3) ϱ g where f is the vapour mass fraction (vapour quality). the transport equation of the vapour mass fraction: (fϱ w) = ṁ v,e ṁ v,c (4) where ṁ v,e and ṁ v,c are volumetric mass sources appropriately due to evaporation and condensation. They were calculated with use of the simplified solution of the Rayleigh- Plesset equation [3]: 3α nuc (1 α) 2 (p sat p) ṁ v,e = C e (5) R b 3ϱ l 3α 2 (p p sat ) ṁ v,e = C c (6) R b 3ϱ l where C e and C c are evaporation and condensation constants, they were assumed equal to 10 3 and Notation α refers to the volume fraction of the gas phase, α nuc is the volume fraction of the nucleation sites, it was assumed equal to The p sat is the saturation pressure for given temperature of the mixture and R b is the radius of the nucleation bubbles, the value 10 7 m was used. Between volume and mass fraction of the gas phase the following relationship holds: α = f ϱ ϱ g (7) 4

6 momentum equation for the two-phase mixture (ϱww) = p + τ (8) where p is the pressure shared by all the phases, τ is the stress tensor. turbulence model equations authors in [2] reports that the RNG k ε model perform well in such applications, hence this model was employed here: (ϱkw) = [α k µ eff k] + G k + G b ϱε Y M (9) ε (ϱεw) = [α ε µ eff ε] + C 1ε k (G k + C 3ε G b ) C 2ε ϱ ε2 k R ε (10) where k is the turbulence kinetic energy and ε turbulence dissipation rate, G k represents the generation of the turbulence kinetic energy due to the mean velocity gradients, G b stands for the generation of the turbulence kinetic energy due to buoyancy and Y M is the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate. Model parameters were equal C µ = , C ε1 = 1.42, C ε2 = 1.68, α k = α ε = More details concerning this turbulence model can be found in [1, 12]. energy equation [ w ( ϱ ( h + w 2 2 ) + p )] = (τw + k eff T ) + (ṁ v,c ṁ v,e ) h vap (T ) (11) where h is the specific enthalpy, T refers to the temperature in given point and h vap stands for the specific enthalpy of vaporisation Homogenous model As it was mentioned above in this model only flow governing equations were solved i.e.: Eq. (2) and Eq. (8-11). The phases were identified at the postprocessing stage based on the values of pressure and temperature. Although this approach is very simplified it may give some insight in the processes that take place in the ejector. 4. Results Results obtained with use of both described models were compared with the measurement data gathered on the special test facility, cf. [4]. In Figure 4 contours of the pressure at the ejector symmetry plane are presented. It can be seen that for the heterogenous model the lowest pressure occurs just behind the throat (i.e. the smallest cross-section diameter of the motive nozzle) and it is more than one order lower than the minimal pressure obtained in the homogenous model, cf. Figure 5. This the place where the most intense evaporation of the liquid CO 2 takes place, see Figure 6b. In the case of homogenous model the lowest pressure appears just behind the outlet from the motive nozzle. The plots of pressure along the ejector axis shows that in the mixing section and in the diffuser the pressure variation are close in both models and they agree quite well with the measurements data (Figure 5). Unfortunately, these data do not allow us to verify which of the presented model preforms better, as no experimental pressure distribution is available inside the motive nozzle (such a measurement would be extremely difficult because the motive nozzle throat had only around 1 mm in diameter). The pressure field obtained with the heterogenous model gradually increase along mixing section and diffuser, what seems to be more realistic than quite fast increase at the beginning of the mixing section observed in the homogenous model, cf. Figure 5. 5

7 Figure 4. Contours of the static pressure (Pa) at the symmetry plane: (a) heterogenous model; (b) homogenous model. Figure 6 shows contours of vapour mass fraction (vapour quality) at the symmetry plane of the ejector. It can be noticed that the liquid stream which flows through the motive nozzle in the mixing section and diffuser also occupies the central part of the ejector and only minor mixing with the vapour takes place. The average vapour quality at the outlet equals and mainly it depends on the ratio of the mixing streams (i.e. entrainment ratio, see Eq. 1) because only around 3.1 per cent of the inflowing liquid CO 2 evaporates in the ejector. In Figure 6 plots of the vapour quality along ejector axis and outlet diameter are drawn. Although the most intense evaporation takes place near the motive nozzle throat, it can be seen that it is also very high downstream in the mixing section and diffuser. It is very well reflected in the vapour quality plot along the ejector axis. Obviously at the ejector inlet it is zero, the rapid increase is at the motive nozzle throat, further in the motive nozzle evaporation slows down and again fast increase is observed in the mixing section. The minimal value of the vapour quality at the outlet is just in the middle and equals In both models predicted flow through the motive nozzle is subsonic (Figure 8). Unfortunately, this can significantly depends on the method of calculation of the speed of sound for the two-phase mixture. In this work it was calculated simply as a weighted average with respect to the mass fraction of each phase. The bubbles which appears in the liquid CO 2 as it flows through the motive nozzle may cause significant decrease of the speed of sound resulting 6

8 WCCM/APCOM 2010 Figure 5. Profiles of the the static pressure along the ejector axis, results of calculations are compared with measurement data. in the supersonic flow of the whole mixture. Closer examination of the velocity field shows that in the case of the homogenous model the CO2 outflows through the secondary inlet instead of sucking in vapour CO2 from the evaporator which disagrees with the measurement data. The heterogenous model predicts correctly flow through ejector, however it gives too low entrainment ratio comparing to the measurements. In the experiment observed entrainment ratio was equal to 0.54, while value 0.20 was obtained from the model. 5. Conclusions This article presents preliminary results of the mathematical modelling of the transcritical CO2 nozzle, which is a part of the compact heat pump system. Two models were proposed two-phase Eulerian like heterogenous one and simplified homogenous one in which just one phase was considered. Numerical calculations results were compared with the measurements data. Considerably better agreement was observed for the heterogenous model. Validation of the numerical results was carried out mainly based on the pressure measurement along the ejector mixing section and diffuser. Unfortunately, it appears that such an information is not appropriate for the validation of the presented model because although homogenous model predicts completely wrong value of the entrainment ratio, the predicted pressure profile along the ejector agreed reasonably well with measurement data. On the other hand it came out that the entrainment ratio was very sensitive on any changes of the mathematical model set up. Hence, this parameter suits better to validation purposes. Now there is a new modelling 7

9 Figure 6. Contours of the vapour mass fraction (a) and the evaporation mass source flow rate, kg/(m 3 s) (b). Figure 7. Plots of the vapour mass fraction along the ejector axis (a) and outlet diameter (b). approach under development, in which main independent variable would be total enthalpy and 8

10 Figure 8. Contours of the velocity (m/s) at the symmetry plane: (a) heterogenous model; (b) homogenous model.. pressure. It is expected that it should preform better for this application. 6. Acknowledgments This work is financed by the Polish-Norwegian Research Fund, acknowledged herein. this support is greatly References [1] ANSYS Inc 2009 ANSYS Fluent Documentation [2] Bartosiewicz Y, Aidoun Z and Mercadier Y 2006 App. Thermal Eng [3] Bilus I and Predin A 2009 Int. J. Numer. Meth. Heat Fluid Flow [4] Drescher M, Hafner A and Banasiak K th IIR Gustav Lorentzen Conference on Natural Working Fluids [5] Gay N Y 1931 Refrigerating system U.S. Patent No. 1, [6] Elbel S and Hrnjak P 2004 International Refrigeration and Air Conditioning Conference [7] Elbel S and Hrnjak P 2008 Int. J. Refrig [8] Elbel S 2009 Eurotherm Seminar No. 85, International seminar on ejector/jet-pump technology and applications [9] Lemmon E W, Huber M L and McLinden M O 2007 NIST Reference Fluid Thermodynamic and Transport Properties - REFPROP ver User Guid [10] Nehdi E, Kairouani L and Bouzaina M 2007 Int. J. Energy Research [11] Ozaki Y et al th IIR Gustav Lorentzen Conference on Natural Working Fluids [12] Versteeg H K and Malalasekera W An Introduction to Computational Fluid Dynamics. The Finite Volume Method (Harlow UK: Pearson Education Limited) 9