Computational fluid-dynamics modeling of supersonic ejectors: Screening of turbulence modeling approaches

Accepted Manuscript Computational Fluid-Dynamics modeling of supersonic ejectors: screening of turbulence modeling approaches Giorgio Besagni, Fabio Inzoli PII: DOI: Reference:

S1359-4311(16)32829-0 http://dx.doi.org/10.1016/j.applthermaleng.2017.02.011 ATE 9891

To appear in:

Applied Thermal Engineering

Received Date: Revised Date: Accepted Date:

28 October 2016 22 January 2017 4 February 2017

Please cite this article as: G. Besagni, F. Inzoli, Computational Fluid-Dynamics modeling of supersonic ejectors: screening of turbulence modeling approaches, Applied Thermal Engineering (2017), doi: http://dx.doi.org/10.1016/ j.applthermaleng.2017.02.011

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Computational Fluid-Dynamics modeling of supersonic ejectors: screening of turbulence modeling approaches Giorgio Besagni*, Fabio Inzoli Politecnico di Milano, Department of Energy, Via Lambruschini 4, 20156 *Corresponding author: Giorgio Besagni, +39-02-2399-3826; [email protected]; Address: Politecnico di Milano, Department of Energy Via Lambruschini 4a, 21056 Milan, Italy.

ABSTRACT Ejector refrigeration has not been able to penetrate the market because of the low coefficient of performance and because of the high influence of ejector performance on the efficiency of the refrigeration system. Improving the performance of ejector refrigeration systems relies on the understanding of the fluid dynamic phenomena, which can be obtained through computational fluid-dynamics approaches. Unfortunately, no agreement has been reached yet on the closures for the turbulence modeling in the Reynolds Averaged Navier-Stokes (RANS) approach. This paper contributes to the existing discussion by presenting a numerical study of the turbulent compressible fluid in a supersonic ejector. Seven RANS turbulence closures have been compared and, in addition, the turbulence models were tested under different near-wall modelling options in order to investigate the wall treatment effect on the numerical results. The numerical results have been validated by literature data consisting in entrainment ratio and wall static pressure measurements, for different ejector geometries and operating conditions. As a result, the k– omega model shows better performance in terms of global and local flow phenomena predictions for the different ejector geometries and operating conditions: (a) on the global point of view, the entrainment ratio is well predicted with a maximum relative error equal to about 10%, (b) on the local point of view, the shock wave position, the pressure recovery and the wall static pressure values are well predicted. The results, taking into account previous numerical studies, suggest the use of the k–omega model to simulate ejector fluid dynamics.

Keywords: CFD; Ejector refrigeration; Computational Fluid Dynamics; Turbulence models; Modeling

1

NOMENCLATURE Acronyms CFD

Computational Fluid-dynamics

CFL

Courant–Friedrichs–Lewy

COP

Coefficient of performance

EWT

Enhanced-Wall-Treatment

FMG

Full multi-grid initialization

RANS

Reynolds Averaged Navier-Stokes

RMSE

Root mean square error, Eq. (2)

SWF

Standard-Wall-Function

Symbols ER

Relative error [-]

[-]

m

Mass flow rate in Figure 1a

[kg/s]

I

Turbulence intensity

[-]

p

Pressure

[Pa]

Q

Rate of heat in Figure 1b

[W]

T

Temperature

[°C]

Wp

Pump power in Figure 1b

[W]

X

Generic variable in Eq. (1) and Eq. (2)

Subscripts c

Condenser

CFD

Calculated value from the CFD model

e

Evaporator

exp

Experimental measurement

g

Generator

p

Primary flow

s

Secondary flow

w

Wall static measurement

2

1

INTRODUCTION

Ejector technology is receiving growing attention in thermal engineering and in refrigeration industries (e.g., building, industrial and commercial refrigeration—see, for example, refs. [1-3]) thanks to the many advantages: an ejector (Figure 1a) provides a combined effect of compression, mixing and entrainment, with no-moving parts and without limitations concerning working fluids. On the practical point of view, the simplicity of construction, the lack of any mechanically operated parts, the reliability, the little maintenance, the low cost and the long lifespan are some of the practical advantages in ejector technology. Ejector technology can be applied in building, industrial and commercial refrigeration: (a) in buildings refrigeration, heat-driven ejector refrigeration would allow a significant reduction of energy consumption (cooling energy and electricity demand); (b) in industrial refrigeration, heat-driven ejector refrigeration may contribute to the energy efficiency, by exploiting the low grade energy sources [4, 5]; (c) in commercial refrigeration, owing to the recent regulations concerning refrigerants (see ref. [6]), ejectors are receiving growing attention, to overcome the low efficiency of the trans-critical R744 systems in warm climates [7-9]. The simplest ejector refrigeration system is shown in Figure 1b and consists of a generator, a condenser, an evaporator, an ejector, a circulation pump and a throttle valve. The heat energy is delivered to the generator for the working fluid vaporization; subsequently, the high-pressure vapor (“primary flow”) flows out from the generator, enters inside the motive nozzle, and draws low-pressure vapor (“secondary flow”) from the evaporator. The primary and the secondary flows mix, the pressure of the mixed stream is raised in the diffuser, and the flow reaches the condenser where it changes phases from vapor to liquid rejecting heat. Once condensation takes place, the flow is splitted in two parts, one heading to the generator and the ejector and another to the evaporator. For further details, the reader may refer to the most recent reviews (see refs. [3, 10-12]) for a comprehensive literature survey concerning the different ejector refrigeration systems. Despite the simple component design, ejector refrigeration technologies were not been able to penetrate the market because of (a) the low coefficient of performance (COP) and (b) the high influence of the ejector performance on the efficiency of the whole system, especially in off-design operating conditions [12]. Indeed, the COP of the ejector-based system (the “refrigeration-system-scale”) relies, mainly, on the ejector performance (the “component-scale”, i.e., the primary and secondary mass flow rates, the pressure recovery,, the operation modes, …). Unfortunately, ejectors are characterized by extremely complex fluid dynamic interactions and, for this reasons, their correct design, operation, and scale-up (towards large-scale industrial applications) rely on the knowledge of the fluid dynamics at the “local-

3

scale” (i.e., boundary layers subject to adverse pressure gradients, shock waves, under-expanded jets, flow reparation, recirculation, turbulence mixing phenomena bounded by near-wall regions,…) and at the “component-scale” (i.e., the entrainment ratio—defined as the ratio between the secondary and the primary mass flow rates; it measures the recirculation ratio and it is a measure of the performance of the ejector). Indeed, the global behavior, at the “component-scale, is the result of the flow features inside the ejector at the “local-scale”. The many relationships between the different scales makes the estimation of ejector fluid dynamics a very challenging task; this task is even more complex owing to the many relationships between the ejector fluid dynamics and the various variables characterizing the system (i.e., geometrical parameters, operating temperature and pressures, refrigerant properties, …). In this respect, the prediction of ejector local flow phenomena is actually matter of intensive research and is usually approached by using Computational Fluid-Dynamics (CFD) method, within the Reynolds Averaged NavierStokes (RANS) approach. Unfortunately, RANS approaches are generally validated with global fluid dynamic properties (i.e. the entrainment ratio) [13, 14] in a limited range of operating conditions. However, local parameters are needed to verify the correspondence between the experiments and the numerical simulations: indeed validated CFD model may be able to correctly determine the global parameters, but it might not accurately estimate the local flow phenomena (i.e. mixing losses, friction losses, …) [15]. In particular, this issues was demonstrated by Hemidi et al. [16]: the k-ε Standard and k-ω SST predicted similar entrainment ratios (global flow features), but different local flow phenomena. Indeed, the ejector flow phenomena concerns shock waves, under-expanded jet and flow reparation requiring the validation performed with local data and the evaluation of different RANS models before studying a broader range of operating conditions [17-21]. In this respect, different studies were proposed in the last decades and a brief summary is proposed in the following. Bartosiewicz et al. [22] compared the performance of six RANS turbulence models (k-ε Standard, k-ε RNG, k-ε Realizable, k-ω Standard , k-ω SST, RSM). The k-ε RNG and k-ω Standard best predict the shock phase, strength, and the mean line of pressure recovery. The validated model has been used to reproduce different operation modes of a supersonic ejector [18]. Bouhanguel et al. [23] compared the performance of several RANS turbulence models in a supersonic air ejector. The results strongly depended on turbulence modeling and the authors observed that none of the RANS turbulence models tested was able to capture the shock reflection pattern in the nozzle exit region. ElBehery and Hammed [24] studied different turbulence models for the case of axi-symmetric diffuser. They have shown that the k–ω Standard, k–ω SST and v2-f models clearly performed better than other models when an adverse

4

pressure gradient was present. The RSM model shows an acceptable agreement with the velocity and turbulent kinetic energy profiles but it fails to predict the location of separation and attachment points. The k-ε Standard and the low-Re k-ε models give very poor results. Dvorak and Vit [25] compared different turbulence models with hot wire anemometry experimental measurements in terms of the static pressure at wall, velocity profile and turbulence intensity. They suggested the k-ε Realizable as the most promising turbulence model. Kolar and Dvorak [26] verified the k–ω SST turbulence model by comparison with experimental Schlieren picture: they found good agreement of the shock wave prediction and boundary layer separation, but the shear stress between streams was over-predicted. Bouhanguel et al. [27] used the laser sheet method to investigate shock structure, flow instabilities, and mixing process. The results of this visualization were used for the validation of the turbulence models by the same authors in ref. [23]. Hemidi et al. [28] compared the k-ε Standard and k-ω SST models for supersonic ejector working with air. The k-ε model provided better results for on-design conditions, with errors mostly less than 10%. The k-ω SST model, instead, yields errors often more than 20%, but with better prediction at off-design operation conditions. Thus, for the analysis of ejector performance in a wide range of operating conditions the k-ω SST or other models should be considered. In addition, Hemidi et al. [16], in the second part of their study, found that the k-ε Standard and k-ω SST predicted similar entrainment ratios (global flow features), but different local flow phenomena. C. Li and Y.Z. Li [29] investigated the behavior and performance of single-phase and two-phase ejectors with different working fluids. In their analysis, the k-ε Standard, the k-ω Standard and the k-ω SST models were considered. The k-ε model provides better results in terms of entrainment ratio prediction: the relative errors were less then 25% in almost all cases. Moreover, it appears that the k-ω SST model significantly over-predicts the entrainment ratio for small pressure difference between the entrained pressure and the discharge pressure. Ruangtrakoon, Thongtip et al. [30] investigated the effect of the primary nozzle geometries on the performance of an ejector. In this study, the k-ε Realizable and k-ω SST models were used and compared. It was found that the simulated results based on the k-ω SST model more closely corresponded to the experimental values than those based on the k-ε Realizable model. According to the authors, a possible reason is that the k-ε Realizable model is unable to accurately predict the performance of the ejector under a strong adverse pressure gradient. Gagan et al. [19] used the PIV technique for a supersonic ejector flow visualization and compared it with numerical results obtained for various turbulence models. They recommended the k-ε Standard model. The k-ε Standard, the k-ε Realizable and the RSM models predict accurately the entrainment ratio. The k-ε RNG, the RSM as well as the k–ω Standard do not predict vortex a region

5

downstream nozzle. The k-ε Realizable and the k–ω SST model under predict the scale of the vortex area. Zhu and Jiang [31] investigated the entrainment performance and the shockwave structures in a 3D ejector. Four turbulence models are used: k-ε Standard, k-ε RNG, k-ε Realizable and k-ω SST. The results show that the k-ε RNG and k-ω SST models agree best with the experimental entrainment ratio and shockwave structures. The k-ε Standard model fails in predicting shock location after the second shock and, in the critical operating conditions, the k-ε Standard and k-ε Realizable models over-predict the first shock wavelength and fail in predicting the reflected shocks. Croquer et al. [32, 33] investigated a R134a ejector and compared four different turbulence models (k–ε Standard, k–ε Realizable, k– ω SST, and stress–ω RSM): they concluded that k–ω SST showed the best performance. The authors also tested different approaches to model equations of state of refrigerants. Mazzelli et al. [34] investigated a rectangular crosssection ejector and compared four different turbulence models (k–ε Standard, k–ε Realizable, k–ω SST, and stress–ω RSM): they found that the entrainment ratio was reasonably predicted at on-design operating conditions, whereas the numerical model poorly predicts the off-design operation (i.e., the entrainment ratio and the critical back pressure). Among the different turbulence models, the k–ω SST showed the best performance. At present, there are no clear recommendations on the selection of the numerical approaches to simulate the ejector fluid dynamics. In our previous paper we proposed a comparison of several turbulence closures (under different nearwall modelling options) for a subsonic ejector [35]. This paper further contributes to the present discussion and extends the previous numerical results to the case of a supersonic ejector by considering both global and local flow features. This work provides guidelines and a rational basis to simulate supersonic ejectors to be applied in refrigeration systems. In this respect, it is worth noting that future studies will consider benchmarks with other refrigerants. The paper is structured as follows. In Section 2 the numerical benchmark is presented. In Section 3 the CFD approach is outlined, and in Section 4 the numerical results are presented and discussed. First, the sensitivity analysis to set up the numerical model are presented (grid independency and near wall modeling). Secondly, the numerical approach is tested considering all the operating conditions of the benchmark used. Finally, the main conclusions are outlined.

2

THE NUMERICAL BENCHMARK

The validation of the numerical model has been performed by using the experimental dataset provided by Sriveerakul et al. [36]. In this study, both global (the entrainment ratio, representing the “component-scale”) and local (the wall

6

static pressure along the ejector, representing the “local-scale”) measurements are available for a complete validation of the approach (as discussed in the introduction). It is worth noting that wall static pressure measurements are considered local measurements as they highly depends on the local flow properties, i.e., boundary layers subject to adverse pressure gradients, shock waves, under-expanded jets, flow reparation, recirculation, turbulence mixing phenomena bounded by near-wall regions. For this reason, the proposed benchmark allows a complete validation of the CFD approach, owing to the multi-scale dataset employed. Unfortunately, the proposed benchmark does not consider local velocity profiles (as the data employed in our previous study, ref. [35]): future studies will be devoted to extend the screening of turbulence approaches proposed in the following to other numerical benchmarks. The experimental investigation of Sriveerakul et al. [36] considered a steam ejector that works under different working conditions (Tg = 120÷130°C, Te = 5÷15°C and Tc = 24÷40°C; where the subscripts stand for generator, evaporator and condensed respectively – See Figure 1b) and with different geometries. The main geometrical characteristics of the ejector and the operating conditions are listed in Table 1 and Table 2, respectively. The reader should refer to the original reference [36] for a complete description of the cases tested.

3

NUMERICAL MODEL

3.1 Mesh generation The mesh generations process has been handled with GAMBIT 2.4.6. A 2D axi-symmetric domain has been applied for modeling the ejector, based on its nature. Concerning the 2D axi-symmetric, the reader may also refer to the discussion proposed by Croquer et al. [32]. The three ejector geometries considered (Table 1) are modeled using three meshes and, after a grid independency study, the final meshes are composed of approximately 70000 quadrilateral elements and were refined on proximity of the wall. The independence of the results from the grid density has been analyzed and the results are presented in Section 4.1 and in Table 3 the mesh quality parameters are summarized. Please note that the dynamic solution-adaptive mesh refinement, performed during the simulations, ensures a good representation of the local flow phenomena (oblique shock-waves, boundary layer, mixing process). This point was also discussed by Besagni et al. [35].

7

3.2 Numerical setting 3.2.1 Solver The fluid flow in the ejector is compressible and turbulent and, in the present case, it is considered as steady state. The relationship between density, velocity and temperature is given by the conservation of continuity, momentum, energy equations, in the form of a set of partial differential equations. In this paper, the finite volume commercial code ANSYS Fluent Release 15.0 has been used to solve the steady state 2D axisymmetric Reynolds Averaged NavierStokes (RANS) equations for the turbulent compressible Newtonian fluid flow. The governing equations have been discretized as follows. Second order numerical schemes have been used for the spatial discretization, in order to limit the numerical diffusion. Second order schemes also for the turbulence quantities have been used. Gradients are evaluated by a least-squares approach. After the discretization process of the governing equations, a system of algebraic equations is obtained and it has been solved: due to the strong coupling between mass, momentum and energy equations, determined by the presence of a high speed compressible flow, a density-based coupled solver has been chosen and applied. IN ANSYS FLUENT, both density-based and pressure-based algorithms are available: a comparative discussion concerning density-based and pressure-based algorithms to solve ejector fluid dynamics was provided by Croquer et al. [32]. The Coupled algorithm applied solves the continuity, momentum, and energy (density is computed through the equation of state selected (See Section 3.2.3)) equations simultaneously. The formulation of the coupled algorithm requires setting up a Courant–Friedrichs–Lewy (CFL) condition; it was set CFL = 0.1 in order to run a stable simulation. Because of to the complex fluid dynamics and in order to achieve fast convergence, a full multi-grid (FMG) initialization scheme has been adopted.

3.2.2 Turbulence modeling and near wall treatment The turbulent behavior has been treated using the RANS approach. Seven RANS turbulence models have been tested and compared: Spalart-Allmaras, k-ε Standard, k-ε Realizable, k-ε RNG, k-ω Standard, k-ω Standard and RSM. These models are well known in the literature and a detailed description of their mathematical structure is beyond the scope of the present paper. For further information about their mathematical formulation the reader should refer to the discussions proposed by Wilcox [37]; for further information about their mathematical formulation and their application in ejector modeling the reader may refer to Mazzelli et al. [34] and Croquer et al. [32] proposed a brief and complete summary on the main turbulence models; For further information about their implementation in the

8

commercial code ANSYS-Fluent the reader should refer to ref. [38]. All the considered models have been used in their original implementation without further modifications. Beside turbulence models, the near wall formulation should be considered. The Spalart-Allmaras, k-ω Standard, k-ω Standard models do not need a near wall treatment because their mathematical structure already emphasizes on the flow close to the wall. However, some aspects, in their ANSYS FLUENT implementations, have to be considered. Concerning the Spalart-Allmaras model, its boundary conditions have been implemented so that the model may work on coarse mesh (such as would be appropriate for the wall function approach) and on a fine mesh in its low-Reynolds formulation. Concerning the k-ω Standard and the k-ω SST, a low-Reynolds implementation may be enabled in ANSYS FLUENT for fully resolver near wall grids. Conversely, the k-ε Standard, k-ε Realizable, k-ε RNG and RSM models require a near wall modeling. In the present study, a sensitivity analysis on the following wall treatments has been performed: •

Standard-Wall-Function (SWF). The viscous sub-layer is not resolved and the first grid cell need to be in the region 30 < y+ < 300. It becomes less reliable when the local equilibrium assumption is not valid (i.e. strong pressure gradient, large curvature, highly 3D flow).

•

Non-Equilibrium-Wall-Function. The Non-Equilibrium-Wall-Function has the same validity range of the SWF approach, but relaxes the local equilibrium assumption in the turbulent region of the wall-neighboring cells.

•

Enhanced-Wall-Treatment (EWT). The Enhanced-Wall-Treatment requires a very fine mesh near the wall, to +

resolve the viscous sub-layer (y < 1). This approach is indicated for low-Reynolds flows or flows with complex near-wall phenomena. The detailed investigation and the results of the sensitivity analysis on the near wall approaches can be found in section 4.1.2. As a result of the sensitivity analysis, the Standard Wall Functions approach is suggested and applied in the comparison of the turbulence models in Section 4.2. For the sake of clearness, here we can just state that the use of wall functions coupled with high-Reynolds turbulence models is based on the absence of detachment, reattachment of the flow, adverse pressure gradient and general non-equilibrium conditions. The absence of these phenomena allows using a y+ value higher than unity reducing the computational costs and high volume aspect ratio close to the walls. This observation agrees with the results discussed in ref. [35] .

9

3.2.3 Working fluid The working fluid employed in the benchmark is water vapour and, because of the low operating pressure, the ideal gas assumption has been considered. Furthermore, no phase transition/condensation effect has been considered. The thermophysical properties of water vapour, provided by the Fluent database, have been used and have been considered as constant (see Error! Reference source not found.Table 4); the density has been evaluated using the ideal gas law.

3.2.4 Boundary conditions The boundary conditions at the inlets and at the outlet (See the notations in Figure 1a and Figure 1b) have been assigned as the saturation temperatures and saturation pressures, accordingly to the benchmark operating conditions (as displayed in Table 2). In particular, pressure-based boundary conditions (in the ANSYS FLUENT implementation) have been applied at the primary flow inlet, at the secondary flow inlet, as well as at the outlet section (see the notation in Figure 1b). In order to ensure a stable solution, the value of the pressure-outlet boundary condition was discretely increased until convergence during the initialization process. The boundary conditions at the walls have been set as adiabatic and no-slip. The turbulence intensity and hydraulic diameter have been chosen as turbulence boundary conditions; unfortunately, no turbulence measurements have been performed and the same approach as the one described in our previous paper [35] has been applied: (i) the hydraulic diameter and (ii) a turbulence intensity equal to I = 5% for the primary flow and equal to I = 2% for the secondary flow. Future studies will be devoted to study the effects of uncertainties on turbulence boundary conditions on the prediction of the numerical model (See, for example, ref. [39]).

3.3 Convergence criteria The numerical solution is considered as converged when all the following converging criteria were satisfied simultaneously: (a) a decrease in numerical residuals by six orders of magnitude; (b) the normalized difference of mass -7

flow rates at the inlet and at the outlet passing through the modelled ejector had to be less than 10 ; (c) the areaweighed-averaged value for inlet velocity of primary and secondary flow is constant. Please refer also to the discussion concerning physical and numerical convergence in ref. [35].

4

NUMERICAL RESULTS

Herein the numerical results are presented and discussed considering both the global (i.e., the entrainment ratio – the “component-scale”) and the local (i.e., wall static pressure along the ejector - the “local-scale”) quantities. First, the

10

sensitivity analysis to set up the numerical model are presented. Secondly, the numerical approach is tested considering all the operating conditions of the benchmark (Table 2). Finally, some considerations on the convergence capability and the computational effort are proposed, for the sake of clarity. The effectiveness of the numerical results (compared with the corresponding experimental data) has been evaluated in terms of the relative error ER defines as follows:

ER ( X ) =

Xexp − XCFD Xexp

⋅100

(1)

Where X is the considered variable, Xexp and XCFD are the experimental measurement and the CFD model estimates, respectively. Moreover, to better asses the validity of the turbulence models, the mean squared deviation has been computed as follows:

RMSE( X) = ∑( XCFD − Xexp )

2

(2)

n

4.1 Sensitivity anlaysis In this section, the sensitivity analyses are presented and discussed: first, (a) the mesh independency analysis and, second, (b) the near-wall treatment sensitivity analysis. These sensitivity analyses are needed to set up the numerical model that has been used for a broader range of operating conditions in Section 4.2. The results are discussed considering the global (i.e., entrainment ratio - Table 5 and Table 6) and the local (i.e., wall static pressure profiles Figure 2 and Figure 3) measurements.

4.1.1 Grid independency analysis In this study, mesh grids having approximately 70000 quadrilateral elements have been used for the detailed analyses performed in Section 4.2. The applied mesh density is the result of a grid convergence analysis, carried out to obtain a numerical solution not affected by discretization errors. To this end, simulations have been performed on three different meshes (see the details in Table 3): (a) “coarse” mesh (approximately 40000 quadrilateral elements); (b) “medium” mesh (approximately 70000 quadrilateral elements); (c) “fine” mesh (approximately 280000 quadrilateral elements). Please notice that the above-mentioned mesh densities do not consider the mesh refinement procedure considered during the simulations. In order to assess the grid sensitivity of the results, the simulations corresponding to the case “RUN 1” have been performed with the three different meshes (the “coarse”, the “medium” and the “fine”

11

mesh) and considering all seven turbulence models. The results of the grid convergence analysis are summarized in Table 5 (global quantities: the entrainment ratio – the “component-scale”) and in Figure 2 (local quantities: wall static pressure along the ejector - the “local-scale”). Considering the entrainment ratios (Table 5), the results enlighten that an asymptote has been reached with ”medium” and the ”fine” meshes for all turbulence models, expect for the Spalart-Allmaras model. However, in the case of the Spalart-Allmaras model, the variation of the predicted entrainment ratio between the “medium” mesh and the “fine” mesh is almost negligible and within the expected range of the experimental uncertainty. Similarly, an asymptotic convergence is obtained by considering the wall static pressure profiles for all turbulence models, except for the Spalart-Allmaras model. In particular, agreement between the “medium” mesh and the “fine” mesh is very good (the results for the “medium” and the “fine” meshes are almost overlapping) in the case of: (a) k-ε Standard (b) k-ε Realizable, (c) k-ω Standard, (d) k-ω Standard, (e) k-ω SST and (f) RSM. Conversely, a fair agreement between the “medium” mesh and the “fine” mesh is observed for the k-ε RNG model and slightly worst agreement is observed in the case of Spalart-Allmaras model. To explain these performances of the Spalart-Allmaras model and, in general, when discussing about non-asymptotic convergence, it is worth noting that the high degree of coupling between the equations and closure relations together with the inherent non-linearity of the equations are such that the results hardly display a monotonic convergence of the resolved variables, regardless of the mesh resolutions used. In conclusion, the grid independency has been verified for the 70000 “medium” grid for all the turbulence models: therefore, the ”medium” mesh represents the right trade-off between accuracy and number of cells.

4.1.2 Wall treatment analysis To investigate the effects of the near-wall treatments on the numerical results, a near-wall treatment sensitivity analysis has been performed. In particular, the near-wall treatment sensitivity analysis has been performed on two meshes: (a) the “medium” mesh (described above and in Table 3) with the grid point spacing at approximately y+ ≈ 40, and the fine mesh (y+ < 1 ≈ 0.25 to 0.75), derived from the “medium” one by refining the region near the wall boundaries. The first mesh has been used to study the wall-function approaches: (a) the Standard-Wall-Function (SWF), and (ii) the Non-Equilibrium-Wall-Function. The second mesh has been used to study the influence of Enhance Wall Treatment (EWT) approaches. It is worth noting that the k–ε Standard, k–ε RNG, k–ε Realizable, and the RSM models require a wall-function approaches (SWF), Non-Equilibrium-Wall-Function, (EWF). Conversely, the k–ω Standard, k–ω SST, and Spalart–Allmaras models do not consider a wall function approach because of their

12

mathematical formulation: (a) the k–ω Standard and the k–ω SST have been implemented with the low-Reynolds correction; (b) the Spalart-Allmaras model is a low Reynolds model and do not need any further specification for the wall treatment. When considering these results, it should be considered that the refinement of the grid near the wall involves an increase of the computational cost due to the growth of the cell number. The results of the near-wall treatment sensitivity analysis are summarized in Table 6 (global quantities: the entrainment ratio – the “component-scale”) and in Figure 3 (local quantities: wall static pressure along the ejector the “local-scale”). As expected, a change in the wall treatment method resulted in a modification of the numerical predictions, especially for the wall static pressure profiles along the ejector (which is expected, as these data are strictly related to the near wall modeling). Comparing the experimental and the numerical entrainment ratios (Table 6), it is found that the use of Non-Equilibrium-Wall-Function instead of SWF had little effects on the numerical results: (a) the prediction of the entrainment ratio by applying the k-ε Standard model with Non-Equilibrium-Wall-Function improved compared with the use of SWF (the relative error decreased from +13.59% to 3.56%); (b) the prediction of the entrainment ratio by applying the k-ε Realizable model with Non-Equilibrium-Wall-Function is similar to the results obtained by using SWF (the relative error for both near wall treatments is +2.27%); (c) the use of NonEquilibrium-Wall-Function in the k-ε RNG and RSM models results in a worsens of the convergence capability -3

(residuals for the continuity equation do not fall below 10 ). These issues in the convergence behavior are in agreement with the results reported by Besagni et al. [35]. As consequence, Non-Equilibrium-Wall-Function were not considered in the followings. Conversely, EWT approaches allow achieving generally better results (in particular, when considering the RSM and the k-ε models that involve wall functions). However, the differences between the SWF approach and the EWT approach were negligible in most of the cases. The Spalart-Allmaras model provides very similar results between the different meshes, due to its low-Reynolds formulation: it does not need wall treatment because already implemented in its mathematical structure and, therefore, it has a slight benefit from the near-wall grid refinement. Similarly, the Low-Reynolds correction of the k-ω SST model does not improve the results between the different meshes. Moreover, the independence of the k-ω SST model from the wall treatment, in this range of y+ (y+ < 40), is a great advantage compared with the other turbulence models and, for these reasons (also considering the outcomes of Section 4.1.1), is applied in the following. Similar conclusions were drawn by Besagni et al. [35] when considering a convergent-nozzle ejector.

13

4.2 Analysis of the different operating conditions: comparison of the turbulence models 4.2.1 Global quantities: the entrainment ratio Table 7 compares the experimental entrainment ratios with the entrainment ratios computed from the CFD model, for the different RUNs. Table 7 also summarizes the relative errors and the RMSE (Eq. (2)) committed in the prediction of the entrainment ratios. In addition, Figure 3 displays the frequency distribution of the relative errors for the seven turbulence models employed. According to the results, all the turbulence models are able to predict the global quantities (the entrainment ratio) with acceptable errors and in line with the literature [28, 29]. The maximum relative error has been obtained with the k-ε Standard model (equal to 24.5%) and, in general, the k-ε models have lower performance than the other turbulence models. The k-ω Standard model has achieved a maximum error equal to 16.3%, but the half of the relative errors is less than 5%. The k-ω SST and RSM models, instead, attain errors less than 12%. However, the k-ω SST model is the best turbulence model in terms of prediction ability of the entrainment ratio of the ejector. This result is supported by the comparison between the RMSE: the minimum RMSE value is achieved by the k-ω Standard (0.064) and the k-ω SST (0.092) models. The RSM model showed low RMSE value but, in the RUN 5 it could con convergence. The other turbulence models showed higher value of the RMSE, ranging from 0.147 (k-ε Realizable) to 0.201 (k-ε Standard).

4.2.2 Local quantities: static wall pressure profiles The validation of CFD models with global parameters (Section 4.2.1) provides preliminary assessment of the RANS turbulence models, but it is not enough: the CFD model may be able to correctly determine the entrainment ratios (global parameters), but it might not accurately estimate the local flow phenomena (i.e. mixing losses and friction losses), as demonstrated by Hemidi et al. [16]. Indeed, every RANS model was developed for particular test cases, flow phenomena and the prediction of the local flow features should be always verified [37]. In this respect, we have performed a comparison between the local measurements (wall static pressure along the ejector) and the local numerical results. For each simulation, we have summarized the results in a table, compared the pressure profiles by means of a graph and reported the flow field, in terms of the Mach number, for all the turbulence models. The results are displayed in Tables 8-15 and Figures 5-20. Tables 8-15 also summarize the relative errors and the RMSE (Eq. (2)) committed in the prediction of the entrainment ratios.

14

It is observed that the overall trend of the wall static pressure distribution is quite well predicted by all the turbulence models and the predictions are similar to those presented by Sriveerakul et al. (2007) [36] and other literature studies (see ref. [40, 41]). This observation suggests that the overall CFD modeling approach is suitable to predict the ejector fluid dynamics. Considering the influence of the operating conditions, the prediction of the CFD approach increases as the outlet pressure decreases; conversely, for higher outlet pressures (i.e., RUN 5, Table 12, Figure 13 and Figure 15) the results are slightly worse for all the turbulence models and, in particular, the RSM has convergence issues in this case. In this case, having high outlet pressure, it is possible to observe that the ‘shock starts’, corresponding to the rapid increase in pressure, is predicted in a downstream position respect with the experimental measurements. In the previous paragraph, some general comments on the CFD model predictions are provided. Now, we better compare the predictions of the different turbulence approaches. The Spalart-Allmaras model, characterized by low computational costs, allows achieving satisfactory performance, especially in cases, RUN 2 (Table 9, Figure 7 and Figure 8), RUN 3 (Table 10, Figure 9 and Figure 10) and RUN 8 (Table 15, Figure 19 and Figure 20) with a low outlet pressure (pc = 3000 Pa). In the case of RUN 1 (Table 8, Figure 5 and Figure 6), it over predicts the pressure before the shock wave, but it is able to correctly predict the pressure recovery. In the other cases, it largely underestimates the pressure recovery. The k-ε Standard model predicts a normal shock wave at the end of the mixing chamber wave and an increase in the static pressure is observed at the inlet of the diffuser for the RUN 1 (Table 8, Figure 5 and Figure 6), RUN 2 (Table 9, Figure 7 and Figure 8), RUN 3 (Table 10, Figure 9 and Figure 10), and RUN 6 (Table 13, Figure 16 and Figure 17). A similar observation was made by Croquer et al. [32] . These cases correspond to the lower outlet pressures on geometry G1 (Table 3). Conversely, the predictions of the k-ε Standard model are similar to the k-ω Standard model and the k- ω SST model for geometry G1 (RUN 4— Table 11, Figure 11 and Figure 12— and RUN 5— Table 12, Figure 13 and Figure 14) and to the k- ε RNG model, k- ε Realizable model, and the RSM model for the geometries G2 and G3 (RUN 7— Table 14, Figure 17 and Figure 18— and RUN 8— Table 15, Figure 19 and Figure 20). The k- ε RNG model, k- ε Realizable model, and the RSM model showed similar performances: expect for the case RUN 5 (discussed above), they generally predicts the shock wave in a upstream position compared with the experimental measurements and they overpredicts the pressure recovery. Conversely, the k-ω Standard model and the k- ω SST model better fit the experimental results than the other turbulence models (in terms of shock wave position, pressure recovery and fitting of the experimental data before the shock wave). These results are supported by the comparison between the RMSE (see the values in Tables 8-15): the minimum RMSE value is achieved by the k-ω Standard and the

15

k-ω SST models. The other turbulence models showed higher value of the RMSE. The overall better performance of the k-ω SST model also emerges from the literature [30, 32-34, 42, 43] and from our previous study concerning a subsonic ejector [35]. A more detailed understanding of the CFD model results is provided considering the Mach contours. Generally, the flow patterns show a supersonic jet exited from the nozzle outlet and extended into the mixing chamber of the ejector. Thus, shockwaves occur in a determined position in the mixing chamber. According to the value of the discharge pressure, the shock is more or less close to the diffuser. The flow downstream of the shock wave is subsonic. The under-expanded wave at the nozzle exit is well described by the turbulence models for all the simulated geometries and operating conditions. The comparison among the different cases shows that the expansion angle depends also on the secondary flow conditions. For example, Figure 21 shows the Mach contour lines for the cases RUN 1 and RUN 2 computed with the k-ω SST model. It can be observed that an increase of the secondary flow pressure determines a smaller shocks region (jet-flow core) and thus a greater secondary mass flow rate can be entrained in the mixing chamber. The major difference among the turbulence models consists in the prediction of the flow behavior in correspondence of the shocking position. It can be observed that the k-ε models predict the shock in advance position respect to the other turbulence models for all the simulated cases. The k-ω SST model seems to have a greater relevance to the actual physical behavior and to several literature analyses [44-46]. According to ref. [47], concerning an investigation about the effect of the downstream pressure, shows that the shock will not affect the mixing behavior of the two streams because the discharge pressure does not exceed the critical backpressure (outlet pressure). Indeed, the flow structures in front of the shocking position are unchanged and the size of the primary jet core remained constant and independent from downstream conditions. However, the shocking position changes with the value of the backpressure: an increase of the discharge pressure moves the shock position to the upstream of the ejector (Figure 22). In conclusion, according to the results, the best agreement with both global and local parameters is achieved with the k-ω SST model. To the author’s opinion, the better performance of the k–ω SST model can be explained by its inherent structure, designed to be accurate for both near-wall and free-stream regions, and calibrated to yield better results for transonic to moderate supersonic regimes (as also remarked by Mazzelli et al. [34] and Croquer et al. [32]) .The interested reader may refer to the discussion proposed by Croquer et al. [32] concerning the turbulence quantitates and the local flow phenomena. In future studies, the predictions of the RANS models will be improved by modification

16

of the model constants (i.e., the production and dissipation terms), a correction in the model diffusion coefficients and/or the inclusion of cross-diffusion contributions, as also suggested in our previous paper [35]. It is worth noting that the proposed CFD approach has been also applied to study the local and the global fluid dynamics in R134a ejector [48].

4.3 Convergence capability and computational effort According to the convergence criteria reported in Section 3.3, we have performed a convergence analysis in order to compare the models from this aspect. Considering fixed the discretization scheme, the grid density, and the numerical methods the computing time mainly depends upon the turbulence model used [35]. For this purpose, we have selected for all the turbulence models the fine mesh (280000 elements), used in the grid sensitivity analysis. For a rigorous analysis, the main factors that influence the convergence capability and the computing time (i.e. grid resolution, discretization scheme, numerical methods, CFL and under-relaxation factor) must be the same for all the simulations. Thus, we have changed these parameters (i.e. increase of the CFL number, first/second order method switch) in the same way for every turbulence model. In all the simulated cases, the reduction of the mass and energy residuals has been the most difficult to achieve. In particular, these residuals have not fallen below 10-4 with the RSM. The convergence problems of this model are mainly due to the high degree of non-linearity. Conversely, the SpalartAllmaras, k-ω Standard and k-ω SST models easily reach convergence. From a computational point of view, the Spalart-Allmaras is the fastest model to converge, while the most onerous model is the RSM. This is due to the number of equations that must be resolved. The results of the analysis are reported in the Table 16. The computational effort is calculated based on the number of iterations needed to achieve convergence (which is related to the interaction between the turbulence model and the flow field, see Appendix A in ref. [35]), in relative terms, taking as reference the Spalart-Allmaras model. These results can be also compared with the convergence study proposed in our previous paper [35] and the one presented by El-Behery and Hamed [24].

5

CONCLUSIONS

In this paper, we have presented numerical results concerning a supersonic ejector. The numerical approach has been validated against global and local measurements. In particular, we have compared different turbulence models (Spalart-Allmaras, k-ε Standard RNG k- ε, Realizable k- ε, k-ω Standard , SST k- ω, RSM) and the different turbulence models have been tested under different near-wall modelling options in order to investigate the wall treatment effect on the numerical results. The present paper is intended as a complete evaluation of turbulence approaches in order to

17

provide guidelines and a rational basis to simulate supersonic ejectors to be applied in refrigeration systems. As a result, the k- ω SST has showed the best agreement with the experimental measurements concerning both global and local flow quantities. The results show that the k-ω SST model performs better than the other employed models both in global and local parameters prediction. Indeed, the entrainment ratio is well predicted under different geometries and operating conditions with a maximum relative error equal to about 10%. Moreover, the predicted wall pressure distribution is quite well fitted with the experimental data. Considering the results in our previous paper (a complete evaluation of turbulence approaches for a subsonic ejector), as well as the other studies from the literature, we suggest the k- ω SST model for future numerical investigations of supersonic ejectors (i.e., ejectors applied in refrigeration systems). Future studies should be devoted to extend the screening of turbulence approaches presented in this paper and in ref. [35], to take into account the (a) effects of other refrigerants (i.e., the behavior of real gases, see refs. [32, 48]) and (b) phase change inside ejector (in steam ejectors [49, 50] and in ejectors using other refrigerants [51]).

ACKNOWLEDGMENTS The authors thank Giuseppe Di Leo for performing and post-processing the simulations, during its master of science thesis. The authors would like to thank the anonymous reviewers for their valuable comments.

18

REFERENCES [1] J. Godefroy, R. Boukhanouf, S. Riffat, Design, testing and mathematical modelling of a small-scale CHP and cooling system (small CHP-ejector trigeneration), Applied Thermal Engineering, 27 (2007) 68-77. [2] R. Ben Mansour, M. Ouzzane, Z. Aidoun, Numerical evaluation of ejector-assisted mechanical compression systems for refrigeration applications, International Journal of Refrigeration, (2014). [3] G. Besagni, R. Mereu, F. Inzoli, Ejector refrigeration: a comprehensive review, Renewable and Sustainable Energy Reviews, 53 (2016) 373-407. [4] A.B. Little, S. Garimella, Comparative assessment of alternative cycles for waste heat recovery and upgrade, Energy, 36 (2011) 4492-4504. [5] G. Besagni, R. Mereu, G. Di Leo, F. Inzoli, A study of working fluids for heat driven ejector refrigeration using lumped parameter models, International Journal of Refrigeration, 58 (2015) 154-171. [6] EU 517/2014, REGULATION (EU) No 517/2014 OF THE EUROPEAN PARLIAMENT AND OF THE COUNCIL of 16 April 2014 on fluorinated greenhouse gases and repealing Regulation (EC) No 842/2006 (Text with EEA relevance), 2014., in, 2014. [7] F. Liu, E.A. Groll, J. Ren, Comprehensive experimental performance analyses of an ejector expansion transcritical CO2 system, Applied Thermal Engineering, 98 (2016) 1061-1069. [8] M. Palacz, J. Smolka, W. Kus, A. Fic, Z. Bulinski, A.J. Nowak, K. Banasiak, A. Hafner, CFD-based shape optimisation of a CO2 two-phase ejector mixing section, Applied Thermal Engineering, 95 (2016) 62-69. [9] P. Gullo, A. Hafner, G. Cortella, Multi-ejector R744 booster refrigerating plant and air conditioning system integration – A theoretical evaluation of energy benefits for supermarket applications, International Journal of Refrigeration, (2017). [10] J. Chen, S. Jarall, H. Havtun, B. Palm, A review on versatile ejector applications in refrigeration systems, Renewable and Sustainable Energy Reviews, 49 (2015) 67-90. [11] S. Elbel, N. Lawrence, Review of recent developments in advanced ejector technology, International Journal of Refrigeration, 62 (2016) 1-18. [12] A.B. Little, S. Garimella, A critical review linking ejector flow phenomena with component- and system-level performance, International Journal of Refrigeration, 70 (2016) 243-268. [13] S. He, Y. Li, R.Z. Wang, Progress of mathematical modeling on ejectors, Renewable and Sustainable Energy Reviews, 13 (2009) 1760–1780. [14] D.A. Arias, T.A. Shedd, CFD analysis of compressible flow across a complex geometry venturi, Journal of Fluids Engineering, 129 (2007) 1193-1202. [15] G. Besagni, R. Mereu, E. Colombo, CFD study of ejector efficiencies, in: ASME 2014 12th Biennal Conference on Engineering Systems Design and Analysis, Vol. Dynamics, Vibration and Control; Energy; Fluid Engineering; MIcro and Nano Manufacturing, ESDA2014-20053, Copenhagen, Denmark, July 25-27, 2014, 2014, pp. V02T11A004. [16] A. Hemidi, F. Henry, S. Leclaire, J.-M. Seynhaeve, Y. Bartosiewicz, CFD analysis of a supersonic air ejector. Part II: Relation between global operation and local flow features, Applied Thermal Engineering, 29 (2009) 2990-2998. [17] S. Ghahremanian, B. Moshfegh, Evaluation of RANS models in predicting low reynolds, free, turbulent round jet, Journal of Fluids Engineering, 136 (2013) 011201-011201.

19

[18] Y. Bartosiewicz, Z. Aidoun, Y. Mercadier, Numerical assessment of ejector operation for refrigeration applications based on CFD, Applied Thermal Engineering, 26 (2006) 604-612. [19] J. Gagan, K. Smierciew, D. Butrymowicz, J. Karwacki, Comparative study of turbulence models in application to gas ejectors, International Journal of Thermal Sciences, 78 (2014) 9-15. [20] M. Yazdani, A.A. Alahyari, T.D. Radcliff, Numerical modeling and validation of supersonic two-phase flow of CO2 in converging-diverging nozzles, Journal of Fluids Engineering, 136 (2013) 014503-014503. [21] J. García del Valle, J. Sierra-Pallares, P. Garcia Carrascal, F. Castro Ruiz, An experimental and computational study of the flow pattern in a refrigerant ejector. Validation of turbulence models and real-gas effects, Applied Thermal Engineering, 89 (2015) 795-811. [22] Y. Bartosiewicz, Z. Aidoun, P. Desevaux, Y. Mercadier, CFD-experiments integration in the evaluation of six turbulence models for supersonic ejectors modeling, in: Proceedings of Integrating CFD and Experiments, Vol. 26, Glasgow, 2003, pp. 71-78. [23] A. Bouhanguel, P. Desevaux, E. Gavignet, 3D CFD simulation of supersonic ejector, in: International Seminar on Ejector/jet-pump Technology and Application, Vol. CD, Paper No. 15, Louvain-La-Neuve, Belgium 2009. [24] S.M. El-Behery, M.H. Hamed, A comparative study of turbulence models performance for turbulent flow in a planar asymmetric diffuser, World Academy of Science, Engineering and Technology, 53 (2009) 769-780. [25] V. Dvorak, T. Vit, Experimental and numerical study of constant area mixing, in: 16th International Symposium on Transport Phenomena, Prague, 2006. [26] J. Kolář, V. Dvořák, Verification of K-ω SST turbulence model for supersonic internal flows, World Academy of Science, Engineering and Technology, 81 (2011) 262-266. [27] A. Bouhanguel, P. Desevaux, E. Gavignet, Flow visualisation in supersonic ejectors using laser tomography techniques, in: International Seminar on Ejector/jet-pump Technology and Application, Vol. CD, Paper No. 16, LouvainLa-Neuve, Belgium 2009. [28] A. Hemidi, F. Henry, S. Leclaire, J.-M. Seynhaeve, Y. Bartosiewicz, CFD analysis of a supersonic air ejector. Part I: experimental validation of single-phase and two-phase operation, Applied Thermal Engineering, 29 (2009) 1523-1531. [29] C. Li, Y.Z. Li, Investigation of entrainment behavior and characteristics of gas–liquid ejectors based on CFD simulation, Chemical Engineering Science, 66 (2011) 405-416. [30] N. Ruangtrakoon, T. Thongtip, S. Aphornratana, T. Sriveerakul, CFD simulation on the effect of primary nozzle geometries for a steam ejector in refrigeration cycle, International Journal of Thermal Sciences, 63 (2013) 133-145. [31] Y. Zhu, P. Jiang, Experimental and numerical investigation of the effect of shock wave characteristics on the ejector performance, International Journal of Refrigeration, 40 (2014) 31-42. [32] S. Croquer, S. Poncet, Z. Aidoun, Turbulence modeling of a single-phase R134a supersonic ejector. Part 1: Numerical benchmark, International Journal of Refrigeration, 61 (2016) 140-152. [33] S. Croquer, S. Poncet, Z. Aidoun, Turbulence modeling of a single-phase R134a supersonic ejector. Part 2: Local flow structure and exergy analysis, International Journal of Refrigeration, 61 (2016) 153-165. [34] F. Mazzelli, A.B. Little, S. Garimella, Y. Bartosiewicz, Computational and experimental analysis of supersonic air ejector: Turbulence modeling and assessment of 3D effects, International Journal of Heat and Fluid Flow, 56 (2015) 305-316. [35] G. Besagni, R. Mereu, P. Chiesa, F. Inzoli, An Integrated Lumped Parameter-CFD approach for off-design ejector performance evaluation, Energy Conversion and Management, 105 (2015) 697-715.

20

[36] T. Sriveerakul, S. Aphornratana, K. Chunnanond, Performance prediction of steam ejector using computational fluid dynamics: Part 1. Validation of the CFD results, International Journal of Thermal Sciences, 46 (2007) 812-822. [37] D.C. Wilcox, Turbulence modeling for CFD, DCW industries La Cañada, CA, 2006. [38] Ansys FLUENT 13 - Theory guide, Ansys FLUENT, 2010. [39] X. Han, P. Sagaut, D. Lucor, On sensitivity of RANS simulations to uncertain turbulent inflow conditions, Computers & Fluids, 61 (2012) 2-5. [40] K. Chunnanond, S. Aphornratana, An experimental investigation of a steam ejector refrigerator: the analysis of the pressure profile along the ejector, Applied Thermal Engineering, 24 (2004) 311-322. [41] I.W. Eames, S. Wu, M. Worall, S. Aphornratana, An experimental investigation of steam ejectors for applications in jet-pump refrigerators powered by low-grade heat, Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy, 213 (1999) 351-361. [42] Y. Bartosiewicz, Z. Aidoun, P. Desevaux, Y. Mercadier, Cfd-experiments integration in the evaluation of six turbulence models for supersonic ejector modeling, in, 2003. [43] Y. Bartosiewicz, Z. Aidoun, P. Desevaux, Y. Mercadier, Numerical and experimental investigations on supersonic ejectors, International Journal of Heat and Fluid Flow, 26 (2005) 56-70. [44] K. Pianthong, W. Seehanam, M. Behnia, T. Sriveerakul, S. Aphornratana, Investigation and improvement of ejector refrigeration system using computational fluid dynamics technique, Energy Conversion and Management, 48 (2007) 2556-2564. [45] N. Sharifi, M. Sharifi, Reducing energy consumption of a steam ejector through experimental optimization of the nozzle geometry, Energy, 66 (2014) 860-867. [46] S. Alimohammadi, T. Persoons, D.B. Murray, M.S. Tehrani, B. Farhanieh, J. Koehler, A validated numericalexperimental design methodology for a movable supersonic ejector compressor for waste-heat recovery, Journal of Thermal Science and Engineering Applications, 6 (2014) 021001. [47] T. Sriveerakul, S. Aphornratana, K. Chunnanond, Performance prediction of steam ejector using computational fluid dynamics: Part 2. Flow structure of a steam ejector influenced by operating pressures and geometries, International Journal of Thermal Sciences, 46 (2007) 823-833. [48] G. Besagni, R. Mereu, F. Inzoli, Numerical investigation of R-134a ejector, in: R. B. (ed.) st International Conference IIR of Cryogenics and Refrigeration Technology, ICCRT 2016, Vol. 22-25-June-2016, Bucharest; Romania, 2016, pp. 124-133. [49] Q.L. Dang, R. Mereu, G. Besagni, V. Dossena, F. Inzoli, Simulation of R718 flash boiling flow inside motive nozzle of ejector, in: R. B. (ed.) 1st International Conference IIR of Cryogenics and Refrigeration Technology, ICCRT 2016;, Vol. 22-25-June-2016, Bucharest; Romania, 2016, pp. 116-123. [50] F. Giacomelli, G. Biferi, F. Mazzelli, A. Milazzo, CFD Modeling of the Supersonic Condensation Inside a Steam Ejector, Energy Procedia, 101 (2016) 1224-1231. [51] G. Biferi, F. Giacomelli, F. Mazzelli, A. Milazzo, CFD Modelling of the Condensation Inside a Supersonic Ejector Working with R134a, Energy Procedia, 101 (2016) 1232-1239.

21

FIGURE CAPTION Figure 1

Ejector refrigeration system

Figure 2

Wall static pressure distribution along the ejector – Grid sensitivity analysis.

Figure 3

Wall static pressure distribution along the ejector – Enhanced wall treatment performance.

Figure 4

Absolute frequency distribution of the errors.

Figure 5

Wall static pressure distribution along the ejector (RUN 1).

Figure 6

Mach contours of the ejector flow field (RUN 1).

Figure 7

Wall static pressure distribution along the ejector (RUN 2).

Figure 8

Mach contours of the ejector flow field (RUN 2).

Figure 9

Wall static pressure distribution along the ejector (RUN 3).

Figure 10

Mach contours of the ejector flow field (RUN 3).

Figure 11

Wall static pressure distribution along the ejector (RUN 4).

Figure 12

Mach contours of the ejector flow field (RUN 4).

Figure 13

Wall static pressure distribution along the ejector (RUN 5).

Figure 14

Mach contours of the ejector flow field (RUN 5).

Figure 15

Wall static pressure distribution along the ejector (RUN 6).

Figure 16

Mach contours of the ejector flow field (RUN 6).

Figure 17

Wall static pressure distribution along the ejector (RUN 7).

Figure 18

Mach contours of the ejector flow field (RUN 7).

Figure 19

Wall static pressure distribution along the ejector (RUN 8).

Figure 20

Mach contours of the ejector flow field (RUN 8).

Figure 21

Mach contour lines comparison.

Figure 22

Mach contours of the ejector flow field – Effect of the discharge pressure (SST k-ω ω).

22

(a) Ejector layout

Ejector [Tg,pg]

Qc

[pc] Condenser [Te,pe]

Boundary conditions in the CFD approach

Primary flow

Throttle valve Evaporator Pump

Secondary flow

Generator Wp

Qg (b) Ejector refrigeration system

23

Spalart-Allmaras model

3000 Exp Spalart-Allmaras (coarse mesh)

2500

Spalart-Allmaras (medium mesh) Spalart-Allmaras (fine mesh)

2000

1500

3000 Exp

Standard k-e (medium mesh)

1500

500

500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0

0.4

Standard k-e (fine mesh)

2000

1000

0

Standard k-e (coarse mesh)

2500

1000

0

Standard k-e model

3500

Static pressure [Pa]

Static pressure [Pa]

3500

0

0.05

0.1

0.15

Distance along ejector [m]

RNG k-e model

3000 Exp RNG k-e (coarse mesh) 2500

RNG k-e (medium mesh) RNG k-e (fine mesh)

2000

1500

0.15

0.2

0.25

0.3

0.35

0

0.05

0.1

0.15

Static pressure [Pa]

Standard k-w (coarse mesh) Standard k-w (medium mesh) Standard k-w (fine mesh)

1500

0.15

0.2

0.25

0.3

0.35

1500

0

0.4

SST k-w (fine mesh)

0

0.05

0.1

0.15

Distance along ejector [m]

0.2

RSM

3000 Exp RSM (coarse mesh) RSM (medium mesh) RSM (fine mesh)

2000

1500

1000

500

0

0

0.05

0.25

0.3

Distance along ejector [m]

3500

2500

0.4

SST k-w (medium mesh)

500

0.1

SST k-w (coarse mesh)

2000

1000

0.05

0.35

Exp 2500

500

0

0.3

3000

1000

Static pressure [Pa]

Static pressure [Pa]

Exp

0

0.25

SST k-w model

3500

3000

2000

0.2

Distance along ejector [m]

Standard k-w model

2500

0.4

1500

Distance along ejector [m]

3500

0.35

Realizable k-e (fine mesh)

0

0.4

Realizable k-e (medium mesh)

2000

500

0.1

0.4

Realizable k-e (coarse mesh)

500

0.05

0.35

Exp 2500

1000

0

0.3

3000

1000

0

0.25

Realizable k-e model

3500

Static pressure [Pa]

Static pressure [Pa]

3500

0.2

Distance along ejector [m]

0.1

0.15

0.2

0.25

0.3

Distance along ejector [m]

24

0.35

0.4

k-e model

3000

Exp k-e (SWF)

2500 k-e (EWT) 2000

1500

3000 Exp

RNG k-e (EWT) 2000

1500

1000

500

500

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0

0.4

RNG k-e (SWF)

2500

1000

0

RNG k-e model

3500

Static pressure [Pa]

Static pressure [Pa]

3500

0

0.05

0.1

Distance along ejector [m]

Static pressure [Pa]

Static pressure [Pa]

Exp Realizable k-e (SWF) Realizable k-e (EWT) 2000

1500

0.15

0.2

0.25

0.3

0.35

0

0.05

0.1

Distance along ejector [m]

Static pressure [Pa]

Static pressure [Pa]

Exp Spalart-Allmaras Spalart-Allmaras (EWT) 2000

1500

0.15

0.2

0.25

0.3

0.35

0.4

Distance along ejector [m]

0.35

0.4

SST k-w SST k-w (EWT)

1500

500

0.1

0.4

2000

1000

0.05

0.3

Exp 2500

500

0

0.25

3000

1000

0

0.2

SST k-w model

3500

3000

2500

0.15

Distance along ejector [m]

Spalart-Allmaras model

3500

0.35

1500

0

0.4

RSM (SWF) RSM (EWT)

500

0.1

0.4

2000

500

0.05

0.35

Exp 2500

1000

0

0.3

3000

1000

0

0.25

RSM

3500

3000

2500

0.2

Distance along ejector [m]

Realizable k-e model

3500

0.15

0

0

0.05

0.1

0.15

0.2

0.25

0.3

Distance along ejector [m]

25

RUN 1

Static pressure [Pa]

3500

3000

Exp Spalart-Allmaras k-e RNG k-e Realizable k-e k-w SST k-w RSM

2500

2000

1500

1000

500

0

0

0.05

0.1

0.15

0.2

0.25

0.3

Distance along ejector [m]

26

0.35

0.4

Spalart-Allmaras

Standard k-ε

RNG k-ε

Realizable k-ε

Standard k-ω

SST k-ω

RSM

Mach Number

27

RUN 2

Static pressure [Pa]

3500

Exp Spalart-Allmaras k-e RNG k-e Realizable k-e k-w SST k-w RSM

3000

2500

2000

1500

1000

500

0

0.05

0.1

0.15

0.2

0.25

0.3

Distance along ejector [m]

28

0.35

0.4

Spalart-Allmaras

Standard k-ε

RNG k-ε

Realizable k-ε

Standard k-ω

SST k-ω

RSM

Mach Number

29

RUN 3

Static pressure [Pa]

4000

3500

Exp Spalart-Allmaras k-e RNG k-e Realizable k-e k-w SST k-w RSM

3000

2500

2000

1500

1000

500

0

0.05

0.1

0.15

0.2

0.25

0.3

Distance along ejector [m]

30

0.35

0.4

Spalart-Allmaras

Standard k-ε

RNG k-ε

Realizable k-ε

Standard k-ω

SST k-ω

RSM

Mach Number

31

RUN 4

4500

Static pressure [Pa]

4000 Exp Spalart-Allmaras k-e RNG k-e Realizable k-e k-w SST k-w RSM

3500 3000 2500 2000 1500 1000 500

0

0.05

0.1

0.15

0.2

0.25

0.3

Distance along ejector [m]

32

0.35

0.4

Spalart-Allmaras

Standard k-ε

RNG k-ε

Realizable k-ε

Standard k-ω

SST k-ω

RSM

Mach Number

33

RUN 5

5000

Static pressure [Pa]

4500 Exp Spalart-Allmaras k-e RNG k-e Realizable k-e k-w SST k-w

4000 3500 3000 2500 2000 1500 1000 500

0

0.05

0.1

0.15

0.2

0.25

0.3

Distance along ejector [m]

34

0.35

0.4

Spalart-Allmaras

Standard k-ε

RNG k-ε

Realizable k-ε

Standard k-ω

SST k-ω

Mach Number

35

RUN 6

Static pressure [Pa]

3500

3000

Exp Spalart-Allmaras k-e RNG k-e Realizable k-e k-w SST k-w RSM

2500

2000

1500

1000

500

0

0

0.05

0.1

0.15

0.2

0.25

0.3

Distance along ejector [m]

36

0.35

0.4

Spalart-Allmaras

Standard k-ε

RNG k-ε

Realizable k-ε

Standard k-ω

SST k-ω

RSM

Mach Number

37

Static pressure [Pa]

3500

3000

2500

2000

Exp Spalart-Allmaras k-e RNG k-e Realizable k-e k-w SST k-w RSM

1500

1000

500

0 0.05

0.1

0.15

0.2

0.25

0.3

Distance along ejector [m]

38

0.35

0.4

Spalart-Allmaras

Standard k-ε

RNG k-ε

Realizable k-ε

Standard k-ω

SST k-ω

RSM

Mach Number

39

RUN 8

Static pressure [Pa]

3500

3000 Exp Spalart-Allmaras k-e RNG k-e Realizable k-e k-w SST k-w RSM

2500

2000

1500

1000

500

0

0

0.05

0.1

0.15

0.2

0.25

0.3

Distance along ejector [m]

40

0.35

0.4

Spalart-Allmaras

Standard k-ε

RNG k-ε

Realizable k-ε

Standard k-ω

SST k-ω

RSM

Mach Number

41

Run 1: pg = 270280 Pa, pe = 872.5 Pa, pc = 3000 Pa

Run 2: pg = 270280 Pa, pe = 1228.1 Pa, pc = 3000 Pa

42

Run 2: pg = 270280 Pa, pe = 1228.1 Pa, pc = 3000 Pa

Run 3: pg = 270280 Pa, pe = 1228.1 Pa, pc = 3500 Pa

Run 4: pg = 270280 Pa, pe = 1228.1 Pa, pc = 4000 Pa

Run 5: pg = 270280 Pa, pe = 1228.1 Pa, pc = 4500 Pa

Mach Number

43

Geometry code name Nozzle throat diameter [mm] Nozzle exit diameter [mm] Mixing chamber inlet diameter [mm] Throat length [mm]

Run Geometry Tg [°C] pg [Pa] Te [°C] pe [Pa] Tc [°C] pc [Pa]

1 G1 130 270280 5 872.5 24.08 3000

G1 2 8 24 95

2 G1 130 270280 10 1228.1 24.08 3000

3 G1 130 270280 10 1228.1 26.67 3500

Geometry Number of elements Aspect ratio worst value >8 Size-change worst value > 1.1 Cell surface worst value > 0.2 Equi-angle skew worst value > 0.65

G2 2 8 24 57

4 G1 130 270280 10 1228.1 28.96 4000

5 G1 130 270280 10 1228.1 31.01 4500

G1 71289 10.1 0.06 % 1.27 0.37 % 0.22 1.91 % 0.86 0.13 %

G3 2 8 19 95

6 G1 120 198670 10 1228.1 24.08 3000

7 G2 130 270280 5 872.5 24.08 3000

G2 68129 10.1 0.06 % 1.27 0.39 % 0.22 2.00 % 0.86 0.14 %

8 G3 130 270280 5 872.5 24.08 3000

G3 71289 10.1 0.06 % 1.19 0.27 % 0.22 1.91 % 0.86 0.13 %

Density

Molecular mass

Cp

Termal conductivity

Vicosity

[kg/m3]

[kg/kmol]

[J/(kg⋅⋅K)] 2014.0

[W/(m⋅⋅K)] 0.0261

[kg/(m⋅⋅s)]

Ideal gas law

Coarse mesh Medium mesh Fine mesh Experimental data

18.01534

Spalart-Allmaras 0.298 0.303 0.302 0.309

k-ε Standard 0.351 0.349 0.304 0.309

k-ε RNG 0.313 0.312 0.309 0.309

k-ε Realizable 0.302 0.302 0.306 0.309

1.34 ⋅ 10-5

k-ω Standard 0.262 0.273 0.292 0.309

k-ω SST 0.298 0.303 0.303 0.309

RSM 0.273 0.273 0.284 0.309

Experimental data = 0.309 [-] Mesh

Medium mesh y+ ≈ 40

Wall treatment Standard Wall function (SWF)

Non Equilibrium Wall Function

Spalart-Allmaras

0.303 (ER = 1.94%) (ii)

k-ε Standard k-ε RNG k-ε Realizable k-ω Standard k-ω SST 0.349 (12.94%)

0.320 (3.56%)

44

0.312 (0.97%) (i)

0.302 (2.27%)

0.302 (2.27%)

0.273(ii) (11.65%)

RSM

0.273 0.303(ii) (11.65%) (1.94%) (i)

Refined mesh Enhanced wall treatment (EWT) +

y <1 (i) (ii) (iii) (iv)

0.301 (ER = 2.59%) (iv)

0.322 (4.21%)

0.314 (1.62%)

0.301(iii) (2.59%)

0.319 (3.24%)

0.302(iii) (2.27%)

0.318 (2.91%)

-3

Difficulties in convergence: residuals for the continuity equation do not fall below 10 Near wall treatment not needed because already implemented in the mathematical structure of the model Low-Re correction for the k-ω models activated in the viscous panel [31] The Spalart-Allmaras model, in its mathematical formulation, is a low Reynolds model and do not need any further specification for the wall treatment

Experimental data Spalart-Allmaras

Entrainment ratio [] Entrainment ratio [] Relative error ER [%] Entrainment ratio [k-ε Standard ] Relative error ER [%] Entrainment ratio [k-ε RNG ] Relative error ER [%] Entrainment ratio [k-ε Realizable ] Relative error ER [%] Entrainment ratio [k-ω Standard ] Relative error ER [%] Entrainment ratio [k-ω SST ] Relative error ER [%] RSM Entrainment ratio [] Relative error ER [%] *Convergence was not reached

Run 1 0.309

Run 2 0.397

Run 3 0.400

Run 4 0.400

Run 5 0.403

Run 6 0.527

Run 7 0.301

Run 8 0.172

0.303

0.477

0.477

0.477

0.477

0.656

0.304

0.169

1.94 0.349

20.15 0.338

19.25 0.302

19.25 0.390

18.36 0.387

24.48 0.622

1.00 0.307

1.74 0.177

12.94 0.312

14.86 0.485

24.50 0.487

2.50 0.487

3.97 0.487

18.03 0.577

1.99 0.312

2.91 0.176

0.97 0.302

22.17 0.470

21.75 0.469

21.75 0.469

20.84 0.469

9.49 0.574

3.65 0.304

2.33 0.172

2.27 0.273

18.39 0.394

17.25 0.394

17.25 0.394

16.38 0.394

8.92 0.531

1.00 0.252

0.00 0.158

11.65 0.303

0.76 0.438

1.50 0.438

1.50 0.438

2.23 0.438

0.76 0.476

16.28 0.302

8.14 0.169

1.94 0.273

10.33 0.413

9.50 0.413

9.50 0.414

8.68 n.c.*

9.68 0.550

0.33 0.274

1.74 0.158

11.65

4.03

3.25

3.50

n.c.*

4.36

8.97

8.14

RMSE, Eq. (2) -

0.201

0.155

0.181

0.147

0.064

0.092

0.058

** Value obtained excluding Run 5

45

x [mm]

Axial position

Experimental data

Wall static pressure [Pa] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%]

SpalartAllmaras

k-ε Standard

k-ε RNG

k-ε Realizable

k-ω Standard

k-ω SST

RSM

30

70

90

110

150

177.5

205

245

280

315

350

RMSE, Eq. (2)

661

488

564

752

755

854

653

1130

2076

2532

2892

-

1011

901

941

902

979

1019

1068

798

1979

2535

2846

52.95

84.63

66.84

19.95

29.67

19.32

-63.55

29.38

4.67

-0.12

1.59

718

537

507

756

870

937

1001

673

2530

2828

2942

-8.62

10.04

10.11

-0.53

15.23

-9.72

-53.29

40.44

21.87

11.69

-1.73

738

552

550

857

1055

1209

1437

1899

2654

2889

2963

11.65

13.11

2.48

13.96

39.74

41.57

120.06

68.05

27.84

14.10

-2.46

753

911

809

1382

533

626

811

965

1062

1211

1806

2655

2910

2965

13.92

-9.22

10.99

-7.85

27.81

24.36

-85.45

59.82

27.89

14.93

-2.52

791

724

731

723

734

618

716

1334

1946

2446

2807

19.67

48.36

29.61

3.86

2.78

27.63

-9.65

18.05

6.26

3.40

2.94

751

531

580

729

813

761

805

1381

1998

2468

2836

3.76

2.53

1.94

1165

484

350

13.62

-8.81

-2.84

3.06

-7.68

10.89

-23.28

22.21

768

656

834

901

971

1061

1192

1889

2568

2858

2960

16.19

34.43

47.87

19.81

28.61

24.24

-82.54

67.17

23.70

12.88

-2.35

46

1202

x [mm]

Axial position

Experimental data

Wall static pressure [Pa] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%]

SpalartAllmaras

k-ε Standard

k-ε RNG

k-ε Realizable

k-ω Standard

k-ω SST

RSM

x [mm]

Axial position

Experimental data

Wall static pressure [Pa] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%]

SpalartAllmaras

k-ε Standard

k-ε RNG

k-ε Realizable

k-ω Standard

k-ω SST

RSM

30

70

90

110

150

177.5

205

245

280

315

350

884

877

864

792

849

993

840

1332

1867

2466

2840

1020

911

946

899

977

1020

1059

796

1989

2548

2852

15.08

-2.72

-26.07

40.24

-6.53

-3.33

-0.42

1321

831

2414

2786

2920

12.98

-2.82

642

15.38

-3.88

-9.49

13.51

957

692

917

956

1177

1232

38.63

24.07

-57.26

37.61

29.30

1060

-8.26

21.09

-6.13

20.71

738

552

550

857

1055

1209

1439

1865

2644

2889

2963

21.75

-71.31

40.02

41.62

17.15

-4.33

16.52

37.06

36.34

-8.21

24.26

1015

1326

842

963

982

1228

1455

1704

1424

2395

2837

2932

14.82

3.99

11.46

23.99

44.64

46.53

102.86

-6.91

28.28

15.04

-3.24

1089

1034

973

861

882

889

867

1244

1851

2377

2767

23.19

17.90

12.62

-8.71

-3.89

10.47

-3.21

6.61

0.86

3.61

2.57

1032

953

913

867

968

1024

1077

1306

1953

2431

2793

-3.12

-28.21

1.95

-4.61

1.42

1.65

1265

342

344

16.74

-8.67

-5.67

-9.47

14.02

1064

993

998

1170

1170

1328

1470

1714

2452

2796

2943

20.36

13.23

15.51

47.73

37.81

33.74

-75.00

28.68

31.33

13.38

-3.63

30

70

90

110

150

177.5

205

245

280

315

350

820

869

811

781

846

959

720

1877

2545

3072

3375

1011

902

941

901

979

1019

1068

1720

2723

3199

3412

23.29 780

-3.80

15.36 918

15.72 1030

-6.26

-48.33

8.36

-6.99

-4.13

-1.10

749

16.03 841

1059

1087

1307

3225

3413

3471

4.88

13.81

-3.70

2397

2619

26.72 3186

11.10 3377

-2.84

955

10.43 1604

30.37

697

21.75 1371

-50.97

996

17.54 1072

3455

21.46 1030

19.79

17.76 975

37.26 986

62.06 1282

67.26 1399

232.92 1582

39.53 2564

25.19 3220

-9.93

-2.37

3397

3460

25.61 1089

-3.57

26.25 861

51.54 909

45.88 890

119.72 879

36.60 1764

26.52 2494

10.58 3347

-2.52

1034

20.22 972

3375

32.80 1032

18.99 953

19.85 914

10.24 868

-7.45

7.19

-22.08

6.02

2.00

-8.95

0.00

968

1023

1077

1879

2599

3097

3397

25.85 1064

-9.67

11.14 1184

14.42 1175

-6.67

-49.58

-0.11

-2.12

-0.81

-0.65

993

12.70 997

1328

1470

2540

3149

3378

3461

29.76

14.27

22.93

51.60

38.89

38.48

104.17

35.32

23.73

-9.96

-2.55

900

RMSE, Eq. (2) -

47

1194

RMSE, Eq. (2) -

534

1062

2175

1510

507

470

1409

x [mm]

Axial position

Experimental data

Wall static pressure [Pa] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%]

SpalartAllmaras

k-ε Standard

k-ε RNG

k-ε Realizable

k-ω Standard

k-ω SST

RSM

[mm]

Axial position

Experimental data

Wall static pressure [Pa] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%]

Spalart-Allmaras

k-ε Standard

k-ε RNG

k-ε Realizable

k-ω Standard

k-ω SST

RSM

30

70

90

110

150

177.5

205

245

280

315

350

RMSE, Eq. (2)

870

871

866

942

1221

1664

2127

3192

3582

3828

4096

-

1011

901

941

901

980

1020

1068

2521

3410

3803

3940

16.21

-3.44

-8.66

4.35

19.74

38.70

49.79

21.02

4.80

0.65

3.81

1090

1074

1033

897

921

932

1265

2476

3297

3767

3942

25.29

23.31

19.28

4.78

24.57

43.99

40.53

22.43

7.96

1.59

3.76

1224

1083

1203

1257

1403

1514

1914

3578

3849

3940

3977

40.69

24.34

38.91

33.44

14.91

9.01

10.01

12.09

-7.45

-2.93

2.91

1029

1458

1452

855

902

983

1003

1229

1348

2198

3504

3808

3914

3966

18.28

-3.56

13.51

-6.48

-0.66

18.99

-3.34

-9.77

-6.31

-2.25

3.17

1089

1034

972

861

883

889

1283

2417

3235

3749

3938

25.17

18.71

12.24

8.60

27.68

46.57

39.68

24.28

9.69

2.06

3.86

1032

953

913

867

968

1024

1156

2588

3322

3789

3950

18.62

-9.41

-5.43

7.96

20.72

38.46

45.65

18.92

7.26

1.02

3.56

1064

993

997

1181

1174

1377

2461

3419

3782

3913

3968

22.30

14.01

15.13

25.37

3.85

17.25

15.70

-7.11

-5.58

-2.22

3.13

30

70

90

110

150

177.5

205

245

280

315

350

RMSE, Eq. (2)

879

1044

953

696

1689

2506

3231

4217

4655

4727

4860

-

1010

901

941

902

980

1020

1166

3397

4116

4369

4454

41.98

59.30

63.91

19.45

11.58

7.57

8.35

567

1507

1383

662

2883

14.90

13.70

1.26

29.60

1092

1046

973

874

892

1166

1964

3242

4051

4373

4462

47.19

53.47

39.21

23.12

12.98

7.49

8.19

24.23

-0.19

-2.10

25.57

1224

2390

1083

1204

1257

1403

1514

1913

4078

4349

4439

4477

39.25

-3.74

26.34

80.60

16.93

39.58

40.79

3.30

6.57

6.09

7.88

1311

1244

1193

1130

1215

1389

1782

2917

3735

4249

4438

49.15

19.16

25.18

62.36

28.06

44.57

44.85

30.83

19.76

10.11

8.68

1090

1034

973

860

882

1169

1951

3228

4045

4371

4461

47.78

53.35

39.62

23.45

13.10

7.53

8.21

968

1026

2054

3393

4118

4394

4464

42.69

59.06

36.43

19.54

11.54

7.04

8.15

n.c*

n.c*

n.c*

n.c*

n.c*

n.c*

n.c*

n.c*

n.c*

n.c*

n.c*

n.c*

n.c*

n.c*

24.00

0.96

-2.10

23.56

1032

953

914

867

17.41 n.c*

n.c*

n.c*

24.57 n.c*

n.c*

n.c*

n.c*

n.c*

8.72

4.09

*Convergence was not reached

48

1909

2643

2405

2322

n.c*

x [mm]

Axial position

Experimental data

Wall static pressure [Pa] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%]

SpalartAllmaras

k-ε Standard

k-ε RNG

k-ε Realizable

k-ω Standard

k-ω SST

RSM

30

70

90

110

150

177.5

205

245

280

315

350

RMSE, Eq. (2)

981

921

942

948

931

925

879

1666

2302

2688

2893

-

1011

901

941

902

980

1020

1068

798

1978

2535

2846

21.50

52.10

14.07

5.69

1.62

967

-3.06

2.17

0.11

4.85

-5.26

10.27

719

536

508

756

871

939

1003

670

2529

2832

2947

59.78

-9.86

-5.36

-1.87

1236

26.71

41.80

46.07

20.25

6.44

-1.51

14.11

738

552

550

857

1055

1209

1437

1900

2654

2889

2963

30.70

63.48

14.05

15.29

-7.48

-2.42

994

24.77

40.07

41.61

9.60

13.32

750

534

631

809

967

1061

1215

1803

2659

2921

2967

38.23

-8.22

15.51

-8.67

-2.56

811

23.55

42.02

33.01

14.66

-3.87

14.70

1090

1044

998

965

829

801

933

1758

2360

2761

2937

11.11

13.36

-5.94

-1.79

10.96

13.41

-6.14

-5.52

-2.52

-2.72

-1.52

751

533

580

728

813

761

808

1381

1997

2468

2836

23.45

42.13

38.43

23.21

12.67

17.73

8.08

17.11

13.25

8.18

1.97

764

659

832

909

969

1066

1195

1893

2572

2852

2963

-4.08

15.24

35.95

13.63

11.73

-6.10

-2.42

280

809

22.12

28.45

11.68

4.11

49

638

x [mm]

Axial position

Experimental data

Wall static pressure [Pa] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%]

SpalartAllmaras

k-ε Standard

k-ε RNG

k-ε Realizable

k-ω Standard

k-ω SST

RSM

x [mm]

Axial position

Experimental data

Wall static pressure [Pa] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%] Wall static pressure [Pa] Relative error ER [%]

SpalartAllmaras

k-ε Standard

k-ε RNG

k-ε Realizable

k-ω Standard

k-ω SST

RSM

70

90

110

150

177.5

205

245

280

315

350

385

RMSE, Eq. (2)

452

488

639

683

824

739

1679

2269

2602

2714

2893

-

744

647

509

696

682

712

1078

1962

2544

2879

2984

64.60

32.58

20.34

-1.90

17.23

3.65

35.80

13.53

2.23

-6.08

-3.15

742

632

536

859

1053

1156

1662

2622

2891

2964

2991

64.16

29.51

16.12

25.77

27.79

56.43

1.01

15.56

11.11

-9.21

-3.39

733

624

542

859

1065

1158

1157

2587

2893

2965

2993

62.17

27.87

15.18

25.77

29.25

56.70

31.09

14.01

11.18

-9.25

-3.46

741

643

516

788

931

1035

1413

2627

2884

2941

2987

63.94

31.76

19.25

15.37

12.99

40.05

15.84

15.78

10.84

-8.36

-3.25

789

741

725

713

603

676

1347

1929

2405

2777

2967

74.56

51.84

13.46

-4.39

26.82

8.53

19.77

14.98

7.57

-2.32

-2.56

746

653

518

731

707

748

1315

1919

2417

2829

2983

65.04

33.81

18.94

-7.03

14.20

-1.22

21.68

15.43

7.11

-4.24

-3.11

765

694

663

891

1040

1064

1722

2457

2839

2958

2997

69.25

42.21

-3.76

30.45

26.21

43.98

-2.56

-8.29

-9.11

-8.99

-3.59

802

808

950

756

716

674

707

30

70

90

110

150

177.5

205

245

280

315

350

RMSE, Eq. (2)

461

251

169

340

392

519

534

1222

1983

2273

2518

-

746

489

436

517

540

1237

1987

2575

2904 15.33

462

352

61.82

-84.06

108.28

-43.82

-11.22

0.39

-1.12

-1.23

-0.20

13.29

729

547

463

681

784

905

1027

2015

2731

2917

2967

58.13

117.93

173.96

100.29

100.00

74.37

92.32

64.89

37.72

28.33

17.83

729

546

472

675

782

910

1022

1909

2700

2922

2971

58.13

117.53

179.29

-98.53

-99.49

75.34

91.39

56.22

36.16

28.55

17.99

735

525

454

653

755

797

879

1796

2724

2936

2972

59.44

109.16

168.64

-92.06

-92.60

53.56

64.61

46.97

37.37

29.17

18.03

763

481

428

497

627

1347

1947

2441

2814

10.23

1.82

-7.39

11.76

465

428

65.51

-85.26

153.25

-41.47

-9.18

4.24

17.42

745

512

479

539

620

1394

1952

2466

2879

14.08

1.56

-8.49

14.34

468

363

61.61

-86.45

114.79

-50.59

-22.19

-3.85

16.10

752

498

449

619

711

729

815

1819

2525

2868

2966

-98.41

165.68

-81.38

40.46

52.62

48.85

27.33

26.18

17.79

63.12

-82.06

50

651

1647

1587

1477

606

639

1316

Computational effort

Spalart-Allmaras

k-ε Standard

k-ε RNG

k-ε Realizable

k-ω Standard

k-ω SST

RSM

1

1.33

1.53

1.45

1.42

1.84

3.73

TABLE CAPTION Table 1

Main geometrical parameters of the ejectors.

Table 2

Operating conditions of numerical simulations.

Table 3

Mesh quality parameters.

Table 4

Water vapour properties.

Table 5

Grid sensitivity analysis – Entrainment ratio prediction.

Table 6

Run 1 – Turbulence models and wall treatments.

Table 7

CFD models prediction of the entrainment ratio.

Table 8

Run 1 - Experimental and numerical results of the wall static pressure.

Table 9

Run 2 – Experimental and numerical results of the wall static pressure.

Table 10

Run 3 – Experimental and numerical results of the wall static pressure.

Table 11

Run 4 – Experimental and numerical results of the wall static pressure.

Table 12

Run 5 – Experimental and numerical results of the wall static pressure.

Table 13

Run 6 – Experimental and numerical results of the wall static pressure.

Table 14

Run 7 – Experimental and numerical results of the wall static pressure.

Table 15

Run 8 – Experimental and numerical results of the wall static pressure.

Table 16

Computational effort of CFD simulations with different turbulence models.

51

Highlights • The fluid dynamics in a supersonic ejector for refrigeration systems was studied • Different turbulence modeling approaches were compared • Different mesh sizes and near wall modeling approaches were compared • A computational fluid dynamics model was validated

52