CFD Analysis and Geometrical Parameter Investigation For The Design of A High-Efficiency Supersonic Ejector

This paper presents a CFD analysis of a supersonic ejector model for refrigeration systems. The model is validated by comparing it with previous studies and used to design a high-efficiency ejector. The effects of geometrical parameters such as convergence angle, throat length, divergence angle, and diffuser length on the entrainment ratio are studied. The optimal values of these parameters are found and used to modify the base model. The performance of the optimized model is evaluated under different operating conditions. The paper aims to provide design guidelines for supersonic ejectors in refrigeration applications .


INTRODUCTION
Supersonic ejectors can be considered the simplest mechanical components in comparison with vacuum pumps and compressors, as they lack any moving mechanical parts and don't need lubricants to achieve the recompression and the mixing of the two overlapping fluid streams.Moreover, its ability to utilize low-temperature heat to produce cooling makes ejector refrigeration a promising solution for economical and environmentally safe technologies and makes it attractive for the industrial and automotive fields in wasteheat recovery appliances.
A long time ago, ejectors for compressible fluids were recognized and used in a variety of technical applications.In aeronautical engineering, they can be used to evaluate a propulsion system's altitude by lowering the pressure in a test chamber [1].In order to decrease the thermal signature, exhaust gases and fresh air are mixed using the pumping effect [2].
The utilization of supersonic ejectors for refrige-ration systems is the major focus of this paper.Undoub-tedly, it is one of the most significant applications for ejectors, according to numerous papers available on this subject.
The review papers by [3] and [4] provide a decent overview of the various applications in this topic, which piqued the interest of many experts in an effort to develop energy-efficient and eco-friendly techniques in reaction to present practices that are causing environmental damage such as ozone depletion and global warming.[5] Present and categorize a review of solardriven ejectors used in refrigeration systems, a development history, and improvements to ejector refrigeration systems.[6] and [7] Provided an overview of ejector refrigeration systems' historical and present impro-vements.
Work shown in [8] provides recent advancements in ejector refrigeration technology, reporting and categorizing a wide range of studies on topics such as geometric optimization, mathematical simulations, refrigerant selection, and operating condition optimization.
Many experimental and theoretical investigations have been conducted to understand better not just the fundamental mechanisms of heat transfer and fluid dynamics but also the operational behavior of the ejectors.
Early studies were based on one-dimensional models.[9] Set up a one-dimensional (1-D) model using air as a working fluid, and then [10] used refrigerants as a working fluid.These one-dimensional models provided a general understanding of ejectors, but they were unable to understand the flow physics that locally occurs along with the ejector, such as local interactions between boundary layers and shock waves and their influence on the recompression and mixing rate.A better understanding of this physics would allow a more accurate and reliable design.
The effect of different turbulence models on the simulation results using air as a working fluid was studied in [11], whereas [12] used refrigerant R142b and described its thermophysical properties using real properties rather than ideal gas properties.In [13], refrigerant R141b was used, as well as R236fa and R245fa as the working fluid and it was found that using standard k-epsilon as a turbulence model yielded good results during the simulation.Paper [14] investigated numerically Different shock generation and reflections in different types of supersonic nozzles.
Paper [15] studied the effect of shock wave structures on the ejector performance using nitrogen as the operating fluid and discovered that the pressure-based RNG k-ε model gave the best predictions of both the structures of shock waves and the mass flow rate.Paper [16] accounted for the real properties of R245fa refrigerant in the simulations using a density-based k-ω-SST turbulence model and recommended defining the ejector surface roughness accurately.The effect of geometric parameters on ejector performance has been investigated numerically and experimentally in many types of research.Paper [17] used the CFD to study the optimization of high-efficiency ejectors and provided Several earlier studies where a number of researchers attempted to understand how the ejector geometry affected the performance of the jet ejector and [18] Studied the Effects of suction chamber geometry and provided a decent review of previous work that concerned with the study of the Effects of various operating conditions on the ejector performance, such as nozzle velocity, pressure drop, and ejector geometry parameters.
After that [19] studied the effects of changing the converging angle of the mixing section and the position of the primary nozzle exit on the ejector performance and found that increasing the primary flow pressure causes the optimum converging angle and the optimum position of the primary nozzle exit to increase.
[20] Studied the effects of different geometries of the primary nozzle on the performance of a steam ejector used in the refrigeration cycle, using a densitybased k-omega-SST turbulence model to explain flow inside the ejector.Also, [21] and [22] studied the supersonic flow of air through a 2D nozzle Using a density-based k-ω SST turbulent model, which agreed well with experimental results.
[23] studied the influences of the mixing chamber and the diffuser section lengths on the performance of the CRMC ejector.[24] studied the effect of the Throat length on the Flow Patterns in supersonic nozzles Using the Spalart-Allmaras.[25] used the SST k-turbulence model with air as a working fluid to study the effect of jet nozzle geometry on supersonic ejector performance.
From the review, it could be concluded that many researchers studied the effect of the geometrical parameters for ejectors work with steam and R245.To the best of our knowledge, the work done on supersonic ejectors in refrigeration cycles using R142b had a little cover in research.The current work aims to investigate the influence of different geometrical parameters on the performance of this type of ejector.According to the literature review, an effective ejector performance optimization method requires geometric optimization under specific operating conditions.In this paper, different turbulence models using R141b as the working fluid were tried to investigate the performance of a supersonic ejector used in refrigeration applications.The CFD model validation is first presented, then the influence of different geometrical parameters on the ejector performance is discussed to determine the suitable mechanisms that contribute to improving ejector performance and to design a high-efficiency supersonic ejector.

Supersonic ejector in a refrigeration cycle
The schematic drawing in Fig 1 illustrates a thermodynamic cycle of a supersonic ejector operating in one of the simplest forms of a refrigeration cycle.In this cycle, a supersonic ejector, vapor generator, and liquid feed pump replace the mechanical compressor.The primary fluid with the highest total energy from the generator is the motive one, while the other with the lowest total energy from the evaporator is the secondary or induced stream.
During regular operation of such systems, the primary fluid is accelerated to supersonic speed as it passes through a converging-diverging nozzle.This low-pressure primary flow exits the nozzle and entrains and accelerates the secondary fluid stream from the evaporator by an entertainment-induced effect.
The two streams interact and enter the mixing chamber, where they are re-compressed and mixed.Complicated interactions occur between the mixing layer and shocks.This indicates that there is a transfer of energy from the greatest energy level to the lowest energy level, with a mixing pressure value located between the driving pressure and induction pressure.
After the two streams are mixed, a series of shock waves occur, which reduce the flow velocity to subsonic speeds.The resulting subsonic flow and shock waves through the diffuser cause the flow pressure to increase to match that at the condenser at point C.After the condenser, there are two currents: one is returned to the generator via a feed pump, while the other is expanded to the evaporator through a throttling valve.

Ejector theory
An important variable to express the performance of a supersonic ejector is the entrainment ratio, which can be expressed as the ratio between the secondary mass flow rate and the primary mass flow rate as in (1).
Hence, the performance of the refrigeration system can be determined by means of the Coefficient of Performance (COP), which can be defined by dividing the cooling load by the total power and heat inputs.As shown in ( 4), the performance of the refri-geration system depends largely on the Entrainment Ratio.
During operation, it's found that by increasing the ejector back pressure (condenser pressure) for the same evaporator and generator pressures, the value of the entrainment ratio remains invariable until a critical back pressure (condenser pressure) (pc * ) is reached.Increasing condenser pressure above that critical pressure causes the entrainment ratio to drop rapidly until the ejector ceases to operate.
As illustrated in Fig 2, The ejector's performance curve is separated into three regimes: the critical (choked flow) regime, the subcritical (unchoked flow) regime, and the backflow (reversed flow) regime.As shown in the curve, a point called "critical condenser pressure" separates between the choked flow and the unchoked flow regions, and a point called "breakdown backpressure" separates between the reversed flow and the unchoked flow regions.
For the critical (choked flow) regime, the pressure of the condenser is below the critical value.The quantity of the secondary entrained fluid drawn from the evaporator is constant, which leads to a constant value of the entrainment ratio.In this mode, after the nozzle, the primary fluid expands, presenting a series of oblique shocks in the suction chamber and accelerating the secondary fluid to choke in the mixing chamber; then, in the diffuser, the mixed flow shocks again, which is known as a double choking phenomenon [10].The location of the shock in the mixing chamber depends on the condenser pressure.When the condenser pressure is increased, the oblique shock will shift upstream (toward the primary nozzle) without making any disturbance to the mixing process, as shown in Fig 3.For the subcritical (unchoked flow) regime, the ejector's condenser pressure is greater than the critical value.When the condenser pressure is increased above that critical pressure, the value of the entrainment ratio remarkably drops.
In the diffuser, the flow does not reach the choking condition (single-choking) [10].The choking of the flow no longer occurs in the mixing chamber.The oblique shock is thought to move upstream into the converging duct section and leads to disturbance in the mixing process between the primary fluid and the entrained fluid.
For the backflow (reversed flow) regime, the condenser pressure is higher than the breakdown value.In this region, the mixed flow direction is reversed back to the secondary flow inlet, and then the ejector finally malfunctions.
Changing the generator and evaporator conditions causes significant modifications in the entrainment ratio and the critical pressure.As shown in Fig 4, rising evaporator temperature (from T eva-1 = -10ºc to T eva-3 =10ºc) leads to greater values of the entrainment ratio (due to the greater amount of secondary mass flux) and critical pressures (as a result of the increase in the total pressure of the entrained flow).But when the generator temperature is (from T gen-1 =60ºc to T gen-2 =80ºc), the entrainment ratio is reduced as the increase in the primary mass flow is not followed by an increase in the entrainment of the secondary stream, whose mass flow rate stays almost constant [11].However, the larger the amount of energy content introduced by the primary flow, the higher the condenser (back) pressures and the critical pressure.

THE SUPERSONIC EJECTOR CFD MODEL SETUP
The CFD technique depends on two main functions: creating and preparing the physical model (domain of calculation) and solving the resulting set of mathematical equations.In this study, a commercially available CFD package fluent 18.2 has been used to form and solve a two-dimensional, axis-symmetric model.The ejector with its general schematic drawing is created, as shown in Fig 5.
The geometry was designed in 2D, where it is assumed that the supersonic ejectors are axis-symmetric, and only the top half of the ejector was modeled and created, as shown in Fig 6.In Figure 7, after the geometry, the model was meshed, and named selections were made for one primary stream inlet, one secondary stream inlet, and the ejector outlet.The first layer size at the direction normal to the wall was at first assumed, then from CFD results using post-processor, the y + value was extracted.In the first trial, the y + value at the wall was high (between 5 and 22), and according to ( [26], [27] ), it is recommended to get a y + value less than 5. Therefore, the cell near the wall is refined using the adaptation technique until we have the desired y + values.The variation of y + values during refinement on the ejector wall, the inside wall of the nozzle, and the outside wall of the nozzle is shown in Fig 8,  The grid independence was analyzed by comparing the results of different meshes using the Realizable k-ε model.The grid independence check was performed on the model of geometry AB (refer to Table 2) and the inlet boundary conditions as model AB1(refer to Table 3) and an outlet pressure of 0.06 MPa.The results showed (refer to Table 1) that there are only slight changes in the entrainment ratio beyond 33,000 cells.So, for further study, a cell number of around 33,000 has been used.

Model Governing equations
The fluid flow within the ejector is governed by a steady-state two-dimensional compressible form of the fluid flow conservation equations where derivatives with respect to time can be neglected.As a result, the governing equations can be expressed as follows in their compact Cartesian form: where E is the total energy ( ρ 2 ) and S h is the source term.The second-order upwind scheme was used to discretize the convective terms.For the pressure field, the coupled algorithm was applied.The scalable wall function was employed for the near-wall treatment.The model's working fluid was R141b, with the use of the EES (Engineering Equation Solver pro 9) database to calculate the thermophysical properties of R141b.The ideal gas relationship was used to determine the fluid density.The viscosity, thermal conductivity, and specific heat were assumed to remain constant over the whole solution domain.

Model Geometry and Operating Boundary Conditions
The CFD model's geometry was first set to simulate the AG and AB geometries of [10].The primary and secondary inlet boundary conditions were considered as pressure inlets and their pressure values were taken as the saturation pressure of the corresponding temperature taken directly from [10] (refer to Table 3); the outlet boundary condition was considered as a pressure outlet of 0.06 MP and a corresponding saturation temperature of 17.79 º C. Table 2 provides an overview of the geometry using (Fig 5) as a guide.For all solid surfaces in the model, the no-slip boundary condition was used.The z-axis was a symmetry axis.Table 3 summarizes the operating conditions, reference results, and CFD results for different turbulence models.

Model validation
The proposed was validated by comparing its results with those of [10] and [28], which included experimental studies and one-dimensional and two-dimensional axis-symmetric CFD models of supersonic ejectors.For all geometries and ejector models used in this study, a comparison between the numerically calculated and experimentally measured entrainment ratios is shown in Table 3.In comparison with the experimental results, the pressure-based Realizable k-ε model was the best suited to predict the entrainment ratio, which agrees with [29], which states the ability of the realizable k-ε model to accurately predict the spreading rate of both curved and planar jets and its ability to provide excellent performance for flows that involve boundary layers subjected to strong adverse pressure gradients and separation.Also, Table 3 and Fig 11 show the 1D model results of [10] and the CFD model results of [28].The solid line shows a complete correlation between the modeling and experimental data.The results of the CFD models are shown as points.The points closest to the solid line represent a better comparison.
The differences in entrainment ratios between the experimental and predicted results of the pressure-based Realizable k-ε model are between -4.8% and 8.4%; for the 1-D simulation of [10], the differences range from -22.9% to 12.8%; and for the CFD results of [28], the differences range from -4.9% to 10.6%.The suggested model is, therefore, comparable to that of [28].

RESULTS AND DISCUSSION
CFD techniques were used to determine the optimal geometry and operating conditions for the design of a high-efficiency jet ejector.The geometrical parameters that affect performance, such as angle of convergence, throat section length, and diffuser section, have been studied.The variation of the entrainment ratio of the ejector was computed independently based on the new geometry design conditions of the ejector.Only one dimension of the concerned geometrical parameter was changed at each time to study its effect on the ejector performance.Paper [17] Provided some optimum dimension values for different ejector components suggested by different investigators, such as: For the convergence section, according to [30], an angle greater than 20 degrees is recommended for a conical or tapered entrance, as the nozzle jet that has a typical angle of around 20 degrees will not generate objectionable shock and eddy losses at the convergence entrance.Moreover, [31] discovered that the best convergence angle is about 25 degrees.
For the throat section, according to [32], its length must be accurately designed.It should be long enough to generate a uniform velocity profile before the divergence section entrance, as the uniform velocity reduces the energy losses in the divergence section; thus, a better pressure recovery is obtained.The opti-mum length for the throat section, according to two literature studies cited in [32] and [9], is around 7 times the throat diameter.On the other hand, [33] proposed a different optimum length of 7.5 times the throat diameter.
For the divergence section, according to [32], it is not recommended to have an excessively rapid divergence after the throat section, and it indicated that an angle θ for the divergent section is recommended to have a range of 4 to 10 degrees.For better pressure recovery, its length is recommended to be from 4 to 8 times the throat diameter.
While [18] provided some of the optimum dimension values of components of conical ejectors suggested by previous investigators compiled in Table 4.
For the throat section, according to [46], the opti-mum ratio between throat section length and diameter is between 7 and 10.And according to other publications given in references ( [47][48][49][50][51]), the optimum ratio between throat section length and diameter is between 7 and 11.
For the divergence section, according to ( [52]; [53]), the optimal diffuser angle of divergence for vapor and steam ejectors is 5 to 7 degrees.
Taking the validated model AB1 from Table 3 as a case study and changing its dimensions according to the above recommendation by taking the convergence angle to be 25 degrees, the throat length to be 7.5 times the throat diameter (l 3 /2r 5 = 7.5), the divergence angle θ to be 10 degrees, and the divergence section length to be 6 times the throat diameter (l 4 /2r 5 = 6).The optimized dimensions of this geometof Fig 5) are showry (refer to the schematic illustration n in Table 5, and it was found that the 'entrainment ratio' increased from 0.3935 to 0.8548.Then, a parametric study was performed on this model (Opti-1) for more performance enhancement.

Effect of the angle of the convergent section
The effect of changing the convergence angle α on the entrainment ratio was numerically predicted using the validated model at the on-design condition by changing the angle α (as shown in Table 5) from ( 18  .The variation in convergence angle below or above that value decreases the ejector's entrainment ratio.

Effect of throat section length
The mixing section of the ejector system is a critical geometrical component that is subjected to substantial shock trains as a result of the intensive interaction between the supersonic primary flow and the subsonic entrained flow.According to several literature sources, the ejector mixing length should be 7 to 9 times the throat diameter for the best performance of the ejector.
According to [33], any mixing length between 4 and 14 throat diameters will provide around 4% of optimum performance.From Table 5, to study the effect of the mixing length on the entrainment ratio, the value of the ratio (l 3 /2r 5 ) was changed from (4:10); this effect was shown in Fig 9 for different convergence angles, where the results showed that the maximum entertainment ratio was achieved at (l 3 /2r 5 = 4).

Effect of angle and length of the divergent section
The diffuser section of the ejector converts the momentum and kinetic energy of the incoming mixture from the mixing section into pressure energy.It allows a higher pressure discharge from the ejector system than the suction pressure.The entrainment ratio at different divergence angles (θ = 4 º :20 º ) was numerically predicted at the on-design condition to investigate the divergence angle effect on the ejector performance.The effect of the divergence angle on the entrainment ratio is shown in Fig 13 .The results showed that the entrainment ratio is constant from (θ = 7 º :16 º ).The variation in diffuser angle below or above decreases the entrainment ratio.For the diffuser length effect, as mentioned above, the diffuser length is recommended to be between 4 and 8 times the throat diameter.The ejector performance at (L 4 /2r 5 = 2:30) was numerically predicted at the on-design condition.The effect of diffuser lengths on the entrainment ratio is shown in Fig 14, where the results show that the entrainment ratio decreases below (L 4 / 2r 5 = 4).

Optimum geometry and operating conditions
The above study assisted in establishing important ejector geometry aspects that affect overall performance to produce an optimized geometry of a prechosen case study (model AB1 from Table 3); the optimized dimensions are shown in Table 6.For the same inlet conditions and at the on-design outlet condition (condenser back pressure under critical value), it was found that the 'entrainment ratio' increased from 0.3935 to 1.2001.The new optimized model was tested in a variety of operating conditions, as shown in Table 7 and Figure 15.

CONCLUSIONS
The first section of the paper focused on the validation of a numerical CFD model created using the ansys fluent 18.2 workbench package for simulation, analysis, and design of highly efficient supersonic ejectors for refrigeration applications.The validation was primarily focused on the choice of the more suitable turbulence model in terms of having an acceptable accuracy in the quantitative predictions for primary and secondary mass flow rates (Entrainment ratio) and convergence time.
The different turbulence models were then applied to different geometries and operating conditions for experimental investigations.It has been shown that the pressure-based Realizable k-ε model results agreed well with the corresponding experimental results.The largest difference between the pressure-based Realizable k-ε model results and the experimental results was 9 %.The geometrical parameters, such as angle of convergence, throat section length, and diffuser section (divergence angle and diffuser length), have been optimized, and their effect on entrainment ratio has been investigated.The study's most important conclusions are presented below.
1.The optimal convergence angle was about 23 º ; increasing or decreasing the angle from that value decreases the ejector performance (entrainment ratio).
2. The optimal length of the throat section is about four times the throat diameter.
3. The divergence angle of the diffuser section is recommended to be from 8 º to 15 º .4. The diffuser length is recommended to be larger than 4 times the throat diameter.
The above study assisted in producing an optimized geometry of a prechosen case study where the entrainment ratio' increased from 0.3935 to 1.2001 for the same operating conditions.
The performance of the optimized geometry design was predicted over a range of operating conditions.

Figure 1 :
Figure 1: Schematic view of the thermodynamic cycle and the refrigeration system using ejector

Figure 2 :
Figure 2: Effect of changing the condenser pressures on the ejector performance acquired through CFD simulation for a generator temperature of 60ºc and an evaporator temperature of 5ºC

Figure 3 :
Figure 3: Mach number contours illustrate the oblique shock motion as the condenser pressure increases

Figure 4 :
Figure 4: An example of the ejector operation curve at different evaporator and generator inlet conditions

Figure 7 :Figure 8 :Figure 9 :
Figure 7: Example of the base grid for domain discretization

Figure 10 :
Figure 10: The behavior of y+ in the outside wall of the nozzle

2. 2
Numerical modelingAnsys fluent 18.2 workbench package was used for solving, pre-processing, and post-processing this compressible fluid.The geometry was created in Design Modeler and divided into grid elements in Ansys meshing.The non-linear governing equations were solved with the trial of using the pressure-based and implicit density-based solvers using the model of the ideal gas and three turbulence models: Realizable k-ε model, Standard k-ε model, Renormalization-group (RNG) k-ε model (for both pressure and density-based solver) were tested in order to solve the high-speed supersonic flow within the ejector.
º : 28 º ) and as shown in Fig 12 the maximum value of entrainment ratio is achieved at a convergence angle of α =23 º

Figure 12 :
Figure 12: Effect of convergence angle on Entrainment ratio

Figure 131 :
Figure 131: Effect of divergence angle on Entrainment ratio

Figure 14 :Figure 15 :
Figure 14: Effect of changing the diffuser section length on the ejector performance