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OPERATION AND EXERGETIC ANALYSIS OF A SUPERSONIC R134A
EJECTOR BY LOW-REYNOLDS NUMBER TURBULENCE MODEL
Sergio CROQUER(*), Sébastien PONCET(*), Zine AIDOUN(**)
(*)
Université de Sherbrooke, Faculté de génie, Département de génie mécanique,
2500 Boulevard de l’Université, Sherbrooke (QC), J1K 2R1, Canada
[email protected]
(**)
CETC-Varennes, Natural Resources Canada, P.O. Box 4800,
1615 Boulevard Lionel Boulet, Varennes (QC), J3X 1S6, Canada
ABSTRACT
A single-phase supersonic ejector with R134a as working fluid was modeled using a Low-Reynolds Number
approach for the solution of the RANS equations. Results are validated against experimental data by García
del Valle et al. (2014) and compared with a High-Reynolds Number model previously analysed by Croquer et
al. (2015), showing losses dependency on shock-wave structure and operating conditions.
1.
INTRODUCTION
Ejectors are supersonic flow mixers, used to harvest the energy available in a high-pressure stream. A primary
or motive fluid (usually a gas) is accelerated in a converging-diverging nozzle to supersonic conditions, which
creates a vacuum and draws up a secondary stream. A momentum transfer takes place and the mechanical
energy of the secondary flow is increased. The mixing process typically occurs at supersonic conditions, and
is followed by a series of shock-waves. The resulting mixture is then decelerated in a subsonic diffuser to reach
the discharge conditions (Bartosiewicz et al., 2005).
Ejector performance is usually described by two parameters; the entrainment ratio w, which is the ratio of
secondary to primary mass flow rates and the compression ratio Pratio, which relates outlet to secondary inlet
static pressures. In normal operation, the motive flow acts as a wall forming an annular passage for the
secondary flow to be entrained and accelerated towards sonic conditions (Huang et al., 1999).
Supersonic ejectors have no moving parts, require low maintenance and can be driven by renewable or low
quality energy sources, such as solar energy or waste heat (Meyer et al., 2009). These features make their
application attractive from economical, operational and environmental perspectives, to replace or support
conventional compression-expansion devices (Sumeru et al., 2012). Currently, there is an extensive interest in
ejector design, performance and integration with refrigeration and energy recovery systems. García del Valle
et al. (2014) performed an experimental study of an ejector using R134a with three different mixing chamber
geometries over a wide range of operating conditions. Results show little performance dependency on inlet
flow superheating, whereas there is an optimum NXP (primary nozzle position with respect to constant area
section inlet, as shown in Fig.1) for which a maximum entrainment ratio is achieved. Density field
visualizations in the constant section area show shock-wave dependency on operating conditions (Zhu and
Jiang, 2014).
Given the small dimensions of ejectors (primary nozzle throat diameter under 3mm), Computational Fluid
Dynamics (CFD) has been necessary in order to relate the internal flow structure to the ejector global
performance. Pianthong et al. (2007) suggested the use of real gas property models and wall heat transfer to
increase accuracy of CFD ejector models. Bartosiewicz et al. (2005) found κ − ε RNG and SST models to
agree better with experimental data among six two-equation models. Pianthong et al. (2007) found that 3D and
2D axisymmetric models provide very similar results, hence the 3D behavior of the flow does not have a major
effect over the global ejector performance. Results by Mazzelli and Milazzo (2015) show that wall friction
effects on CFD accuracy are considerable at off-design conditions but negligible at on-design conditions.
The exergetic analysis performed by Bilir Sag et al. (2015) shows that an ejector-based expansion refrigeration
cycle has a greater overall COP by 7 to 12% over the ordinary TXV-based refrigeration cycle. Regarding
internal losses, flow irreversibilities are largely affected by the total mass flow (Banasiak et al., 2014), which
depends on operating conditions. Updated reviews on the state of art about ejectors such as those of
Aphornratana (1996), Sumeru et al. (2012) and Liu (2014) are available.
This paper describes the CFD analysis of a supersonic ejector in single-phase conditions using R134a as
working fluid, with a Low-Reynolds Number (LRN) approach for the RANS equations. The use of such
turbulence model avoids near-wall approximations, thereby improving the characterization of the whole flow.
A very good agreement is obtained with the experimental data of García del Valle et al. (2014). A comparison
of the resulting global ejector behaviour and shock-wave structure with a previous High-Reynolds Number
(HRN) analysis of the same ejector (Croquer et al., 2015) is carried out. Finally, inner flow losses and exergy
efficiency of the ejector for different operating conditions are assessed.
2.
NUMERICAL MODELING
2.1. Geometry Modeling
The geometry and operating conditions modeled in this investigation are based on the experimental data
presented by García del Valle et al. (2014), who carried out an experimental analysis of three supersonic
ejectors (Models A, B and C) with refrigerant R134a as working fluid. Case A was selected for this study.
Figure 1 shows a schematic view of the chosen geometry, its main dimensions are summarized in Table 1.
Figure 1. Schematic geometry of the ejector.
The flow was modeled in steady state regime, and assumed to be 2D axisymmetric (Pianthong et al., 2007).
Table 1.Main dimensions of the ejector.
Parameter
Value
Primary Nozzle Throat Diameter, nd (mm)
2.00
Primary Nozzle Exit Diameter, d (mm)
3.00
Nozzle Exit Position, NXP (mm)
-5.38
Mixing Chamber Diameter, D (mm)
4.80
Mixing Chamber Length, l (mm)
41.39
Diffuser Exit Diameter, ed (mm)
20.00
2.2. Flow Solver
The flow was considered in steady state regime, 2D axisymmetric, compressible and supersonic. The
governing equations system was solved using the commercial software ANSYS Fluent v.15, via the finite
volume technique. The flow domain was divided into small elements wherein the governing equations in their
conservation form were solved. A second-order upwind scheme was used for the advective terms for each
equation, except for the pressure equation where the PRESTO! scheme was chosen. For this particular
application, this scheme proved to be more stable than other FLUENT built-in options. High-order term
relaxation was applied throughout the entire computation to ensure convergence smoothness.
The pressure-based algorithm with full pressure-velocity coupling was selected (Li and Li, 2011; Yazdani et
al., 2012; Zhu and Jiang, 2014). For this particular study, this algorithm was more stable than the densitybased solvers.
Two turbulence models have been considered. A standard k-ε model used in its high-Reynolds number version
(referred as HRN in the following) and a k-ω SST model based on low-Reynolds number approach (LRN).
Refrigerant properties were computed using the REFPROP 7.0 equation database. For R134a, this model is
based on the formulation of Tillner-Roth and Baehr (1994) which uses the Helmholtz free energy equation of
state. This equation depends on 21 parameters, obtained from statistical analysis and least-square fitting of the
most accurate R134a measurements available at the time. This model accurately represents real gas behavior
in the temperature range 170K-455K and up to 70 MPa. A thorough discussion on the choice of gas and HRN
turbulence model was previously done by Croquer et al. (2015).
Convergence was defined when stable values of static outlet temperature, total mass flow across the domain
and RMS residuals for all equations are under 10−4. It was typically reached shortly after 1000 iterations. All
the computations were performed using a workstation with 16 GB of RAM and a 4 core 3.40 GHz CPU. In
average, the computation of a single case took 60 min and 180 min for the HRN and LRN models respectively.
2.3. Mesh Sensitivity
A mesh sensitivity assessment was carried out to ensure that the results were independent of the spatial
discretization. Three grids were tested, ranging from 645000 to 890000 elements. Figure 2 shows the
proportion of CFD predicted entrainment ratio wcfd versus experimental entrainment ratio wexp for the three
meshes. Since the differences are negligible, the coarser grid was selected.
WCFD/WEXP [-]
1.1
Selected Mesh
1.0
0.9
600000
700000
800000
900000
Number of Elements [-]
Figure 2. Grid independence of the solution for the LRN approach.
The chosen grid consisted of 645000 elements. 18 element layers were concentrated near the wall region to
capture the linear and logarithmic sublayers. The average value for the wall coordinate was 0.79, with a
maximum of 3. Figure 3 shows details of the mesh for the LRN computations.
Figure 3. Grid refinements at three different locations within the ejector for the LRN approach.
2.4. Boundary and Initial Conditions
A pressure-inlet / pressure-outlet combination was used for the domain inlets and outlet respectively. Inlet and
outlet velocity values are negligible in comparison with the average values inside the ejector. Thus, static
pressure and temperature are considered equal to total temperature and total pressure (Bartosiewicz et al.,
2006; Pianthong et al., 2007). Turbulence intensity at the inlets was set at 5%. Walls were considered as
adiabatic and smooth surfaces.
Boundary values reflected the operating conditions used by García del Valle et al. (2014). At both inlets and
the outlet, the pressure was set as the saturation value corresponding to the Tsat temperature shown in Table 2.
Experimental inlet streams included a 10 K overheat to prevent condensation (e.g.: primary fluid inlet
temperature for OP 1 was 362.52K).
Table 2. Operating conditions for the cases modeled in this study.
Operating Point
Tsat Primary Tsat Secondary Outlet, Tsat Experimental
OP
Inlet (K)
Inlet (K)
(K)
w (-)
1
352.52
283.15
302.56
0.494
2
357.54
283.15
305.63
0.398
3
362.30
283.15
308.56
0.339
A preliminary result with the Perfect Gas model was set as the initial flow field for the definitive simulations,
which used the REFPROP database equation.
2.5. Exergy Analysis
The exergetic analysis allows a comparison of the energy performance of a device or thermodynamic systems
in terms of the energy quality rather than the energy amount. Through the ejector process, heat and work
transfers with the surroundings are negligible. Hence, the exergy analysis becomes important to understand
the occurring losses (Bilir Sag et al., 2015; Khennich et al., 2014). Exergy represents the maximum amount of
work theoretically available between any specific state and reference dead state (environmental conditions).
The exergy of a stream and exergy efficiency of a process can be computed from equations (1) and (2)
respectively (Bilir Sag et al., 2015):
=
ℎ−ℎ
−
−
(1)
=
(2)
where χ is the total stream exergy (W), m the mass flow rate (kg/s), h the specific enthalpy (kJ/kg), s the
specific entropy (kJ/kg/K) and ηχ the exergy efficiency. The subscript o refers to dead state conditions.
According to the theorem of Guy Stodola, the exergy destruction rate can be related the entropy generation
across a given section (Aghazadeh Dokandari et al., 2014):
=
=
−
(3)
where Ḋ is the exergy destruction rate (kW) and Ṡ the entropy rate (kW/K).
Ḋ can be used to pinpoint the losses in energy quality across a specific process. This is readily done with help
of the exergy destruction index, ξ, as defined by Equation (4) (Banasiak et al. 2014):
=
!"!#
(4)
$
where the subscripts i and ejector represent the exergy destruction rate for an i-section and the total ejector
respectively. In this analysis, the following dead stated was assumed: To = 300K and Po = 1atm.
3.
RESULTS AND DISCUSSION
3.1. Ejector Global Performance and Experimental Validation
Figure 4 compares the numerically predicted operating curve with both the HRN and LRN models and the
experimental data reported by García del Valle et al. (2014). There is a good overall agreement between both
models and the experimental results. The difference is less than 4% along the on-design conditions (Tsat < 306
K). According to both numerical models, the critical operation point is at Tsat = 306K, which marginally differs
from the experimental value (Tsat = 305.5 K). Thus, both the HRN and LRN approaches are capable of well
reproducing the ejector global performance, and the operating curve.
3.2. Flow Field: Shock-Wave Structure
Figure 5 depicts the profiles of pressure, temperature and Mach number along the ejector centerline. Results
are very similar for both models, up to the second shock region (x = 20 mm). The motive stream jet exits the
NXP through a small shock, and develops into a jet that entrains the secondary flow. After mixing, a shock
train occurs (x = [20mm – 40mm]). In this region, the HRN model predicts a simple normal shock-wave
occurring towards the end of the mixing section. On the other hand, the LRN model predicts the existence of
a shock train in the second half of the mixing chamber.
Afterwards, the subsonic mixture continues along the diffuser. Figure 5, shows that most of the pressure rise
in the subsonic diffuser occurs within its first half. Beyond a certain point, the diffuser effect is negligible and
could incur a negative effect in terms of friction losses. Further investigations on the geometry of this section
(e.g.: length and divergence angle) should be considered.
Figure 4. Ejector operating curve. CFD models vs. experimental data. Primary flow Tsat = 352.52K.
Figure 5. Comparisons between the HRN and LRN models in terms of pressure, temperature and Mach
number values at the centerline of the ejector. Primary inlet Tsat = 352.52 K, secondary inlet Tsat = 283.15 K,
outlet Tsat = 302.56 K.
Figure 6 compares the shock-wave structure obtained by both LRN and HRN models for the ejector operating
under critical conditions (outlet Tsat = 306 K). The colormap is clipped to the Mach number Ma greater than 1
to show the choked flow regions. The shock pattern occurring at the primary nozzle exit is very similar for
both models. In the mixing section, the effective area for the entrainment of the secondary flow is also very
similar in both cases. The entrainment ratio greatly depends on the available secondary flow passage,
explaining the similar global behaviour of both models (Ruangtrakoon et al., 2013). The oblique shock train
obtained with the LRN model better agrees with flow experimental visualizations (Zhu and Jiang, 2014).
Figure 7 shows the evolution of the shock-wave structure inside the ejector with varying outlet conditions and
the LRN approach. The critical point is for an outlet saturation temperature equal to Tsat = 306 K. A shock train
exists up to the critical conditions. Inside the mixing section, the flow is supersonic and therefore both the
primary and secondary streams are choked. As the outlet pressure rises, the shock train shrinks towards the
NXP. In off-design conditions, the energy difference is not enough to reach sonic conditions in the mixing
chamber. The motive flow turns from supersonic to subsonic through a shock-wave just after the NXP. The
drawing capability of the primary flow reduces and the entrainment ratio w drops drastically.
Figure 6. Comparison between HRN and LRN models in terms of the isocontours of the Mach number in the
mixing region at critical conditions. Primary inlet Tsat = 352.52 K, secondary inlet Tsat = 283.15 K, outlet Tsat
= 306 K.
Figure 7. Isocontours of the Mach number Ma for different outlet conditions. Primary inlet Tsat = 352.52 K,
secondary inlet Tsat = 283.15 K.
3.3. Ejector Exergetic Performance
Figure 8 represents the variation of ξ and Ma along the ejector for the three operating conditions defined in
Table 2, with the LRN model. Most of the exergy losses inside the ejector take place in two regions: mixing
area and the second shock train area. Irreversibilities in the mixing section are due to two processes: firstly the
contact and mixing of both fluids, and secondly, the motive stream’s mild shocks (shown in Figure 6) and its
continuing expansion into the mixing chamber. At section 2, the sudden drop in Ma number reveals the
presence of the second shock train. This abrupt change in the fluid regime leads to a sudden increase in losses.
Table 3 summarizes the contribution of these two sections to the overall ejector performance, for the operating
points defined in Table 2. For increasing motive flow pressure (OP 1 through OP 3), losses in the mixing
section maintain at roughly 40 %. However, the importance of the shock-train losses ranges from 28 % to 34 %
of the total exergy losses. This trend is associated with the Ma drop across the shock train, which is 0.59, 0.64,
and 0.71 for operating conditions 1, 2 and 3 respectively. Hence, an increase in the motive flow energy leads
to a more intensive shock, meaning greater losses. However, this effect, is negligible for the ejector overall
performance. The exergy efficiency varies under 1% with the considered operating conditions.
Figure 8. Average values of the exergy destruction index and the Mach number within the ejector.
Table 3. Ejector exergetic efficiency for the three operating conditions.
OP 1
OP 2
OP 3
Mixing (%)
39.9
40.0
41.2
Shock Train (%)
28.0
32.1
34.3
Ejector Exergetic Efficiency (%)
79.7
79.5
79.4
4.
CONCLUSIONS
The flow structure and performance characteristics of a supersonic ejector with R134a were studied using a
low-Reynolds number turbulence model. Results deviate by up to 4 % from experimental values at on-design
conditions. Comparisons with a high-Reynolds number model and an exergy analysis were also carried out.
•
•
•
•
•
Compared to the low-Reynolds number approach, the high-Reynolds number turbulence model fails to
predict the appearance of the oblique shock-wave and its correct position within the constant section area.
At on-design conditions, the second shock train moves toward the primary nozzle exit with increasing
outlet pressure. Constant w reveals that both fluid streams remain choked and the flow structure just
beyond NXP does not vary.
Most of the compression in the subsonic diffuser is achieved within the first half, suggesting an oversize
length.
Exergetic balance shows that the primary flow jet into the constant area section, mixing and shock trains
account for more than 60% of the overall losses generated inside the ejector. These phenomena should be
further studied to increase ejector work recovery capabilities. Losses along the inlets and the subsonic
diffuser represent less than 20% of the overall performance.
Increasing motive stream inlet energy, leads to higher Mach numbers inside the constant area section and
a more intense shock train. Hence, shock train associated losses become more important.
5.
AKNOWLEDGEMENTS
This project is part of the research program of the Industrial NSERC Chair in Energy Efficiency, established
at the Université de Sherbrooke in 2014, with the support of Hydro-Québec, Canmet-Energie, Rio Tinto Alcan
and the Natural Sciences and Engineering Research Council of Canada.
6.
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