Numerical and One-Dimensional Studies of Supersonic Ejectors for Refrigeration Application: the Significance of Wall Pressure Va

applied sciences

Article Numerical and One-Dimensional Studies of Supersonic Ejectors for Refrigeration Application: The Signiﬁcance of Wall Pressure Variation in the Converging Mixing Section

Eldwin Djajadiwinata 1 , Shereef Sadek 1, Shaker Alaqel 1 , Jamel Orﬁ 1,2 and Hany Al-Ansary 1,2,*

1 Mechanical Engineering Department, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia; [email protected] (E.D.); [email protected] (S.S.); [email protected] (S.A.); orﬁ[email protected] (J.O.) 2 K.A.CARE Energy Research and Innovation Center at Riyadh, Riyadh 11451, Saudi Arabia * Correspondence: [email protected]

Abstract: This paper studies the pressure variation that exists on the converging mixing section wall of a supersonic ejector for refrigeration application. The objective is to show that the ejector one- dimensional model can be improved by considering this wall’s pressure variation which is typically assumed constant. Computational Fluid Dynamics (CFD) simulations were used to obtain the pressure variation on the aforementioned wall. Four different ejectors were simulated. An ejector was obtained from a published experimental work and used to validate the CFD simulations. The other three ejectors were a modification of the first ejector and used for the parametric study. The secondary . mass flow rate, ms, was the main parameter to compare. The CFD validation results indicate that . the transition SST turbulence model is better than the k-omega SST model in predicting the ms. The Citation: Djajadiwinata, E.; Sadek, S.; results of the ejector one-dimensional model were compared before and after incorporating the wall Alaqel, S.; Orfi, J.; Al-Ansary, H. pressure variation. The comparison shows that the effect of the pressure variation is significant at Numerical and One-Dimensional certain operating conditions. Even around 2% change in the average pressure can give around 32% Studies of Supersonic Ejectors for . difference in the prediction of ms. For the least sensitive case, around 2% change in the average Refrigeration Application: The pressure can give around 7% difference in the prediction. Significance of Wall Pressure Variation in the Converging Mixing Section. Appl. Sci. 2021, 11, 3245. Keywords: ejector refrigeration; one-dimensional model; computational fluid dynamics (CFD); https://doi.org/10.3390/app11073245 transition SST turbulence model; converging mixing section; wall pressure variation

Academic Editor: Szabolcs Varga

Received: 16 March 2021 1. Introduction Accepted: 1 April 2021 Energy-saving topic gains more attention globally. This is a logical consequence of Published: 5 April 2021 the increase in people’s awareness all over the world regarding the limited fossil energy resources. It keeps depleting and eventually will become extinct. To solve this issue, many Publisher’s Note: MDPI stays neutral researchers have been searching for renewable energy resources as well as developing with regard to jurisdictional claims in efficient technologies to utilize that energy. These attempts have led researchers to various published maps and institutional affil- ideas such as an idea to use ejectors in refrigeration systems to compress the refrigerant. iations. Instead of using electricity, an ejector is able to utilize thermal energy to do the compression. An ejector is a device that entrains a fluid at low pressure (secondary flow) by ac- celerating another fluid at high pressure (primary flow) via a nozzle. Then, the primary and secondary flows mix inside the device and leave through an outlet at an intermediate Copyright: © 2021 by the authors. pressure. In an ejector refrigeration system, the secondary flow is the refrigerant leaving Licensee MDPI, Basel, Switzerland. the evaporator, and the ejector acts as a compressor through which this refrigerant is This article is an open access article compressed. A typical structure of an ejector can be seen in Figure1. In this work, the distributed under the terms and mixing section consists of a converging mixing section, also called constant pressure mixing conditions of the Creative Commons section, and a constant area mixing section, also called ejector throat (ET) (Figure1). Attribution (CC BY) license (https:// A supersonic ejector operation can be divided into three modes, i.e., critical (double- creativecommons.org/licenses/by/ choked) mode, sub-critical (single-choked) mode, and malfunction mode [1]. Figure2 4.0/).

Appl. Sci. 2021, 11, 3245. https://doi.org/10.3390/app11073245 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, x FOR PEER REVIEW 2 of 27

mixing section, and a constant area mixing section, also called ejector throat (ET) (Figure 1).

Appl. Sci. 2021, 11, 3245 2 of 26

Appl. Sci. 2021, 11, x FOR PEER REVIEW 2 of 27 shows various ejector operation modes at a ﬁxed primary and secondary inlet pressures. The ejector is in a critical mode operation when the primary and secondary ﬂows are both

choked. This means that both mass flow rates are constant with respect to the change of Figure 1. Schematic of a typical ejector. ejector outlet pressure, P4. The secondary flow is choked because it accelerates until it mixing section, andreaches a constant sonic speedarea throughmixing ase hypotheticalction, also converging called ejector annular throat duct formed(ET) (Figure by the outer 1). A supersonic ejector operation can be divided into three modes, i.e., critical (double-choked)boundary of the mode, primary sub-critical flow jet (singl ande-choked) the wall ofmode, the ejector and malfunction [2]. mode [1]. Figure 2 shows various ejector operation modes at a fixed primary and secondary inlet pressures. The ejector is in a critical mode operation when the primary and secondary flows are both choked. This means that both mass flow rates are constant with respect to the change of ejector outlet pressure, 𝑃 . The secondary flow is choked because it accelerates until it reaches sonic speed through a hypothetical converging annular duct formed by the outer boundary of the primary flow jet and the wall of the ejector [2]. As we increase the outlet pressure, the secondary flow will not be choked anymore (only the primary flow is choked). This pressure threshold is referred to as critical pressure, 𝑃. Beyond this pressure, the ejector will enter the sub-critical mode operation within which an increase in 𝑃 will decrease the secondary mass flow rate but has no effect on the primary mass flow rate. As the outlet pressure is increased further, it reaches a pressure, 𝑃, at which the ejector starts to malfunction. There will be no secondary flow or, even, reverse flow through the secondary inlet will occur. In practice, an ejector is usually required to Figure 1. Schematic of a typical ejector. Figure 1. Schematic operateof a typical in the ejector.critical mode/condition, and hence, it is also called the on-design condition.

A supersonic ejector operation can beSub-critical divided into three modes, i.e., critical mode (double-choked) mode, sub-criticalCritical mode (single-choked) mode, Malfunction and malfunction mode [1]. Figure 2 shows various ejector operation modes at a fixed primary and secondary inlet pressures. The ejector is in a critical mode operation when the primary and secondary flows are both choked. This means that both mass flow rates are constant with respect to the change of ejector outlet pressure, 𝑃 . The secondary flow is choked because it accelerates until it reaches sonic speed through a hypothetical converging annular duct formed by the outer boundary of the primary flow jet and the wall of the ejector [2]. As we increase the outlet pressure, the secondary flow will not be choked anymore

(only the primary flowEntrainmentER ratio, is choked). This pressure threshold is referred to as critical pressure, 𝑃. Beyond this pressure, the ejector will enter the sub-critical mode operation within which an increase in 𝑃 will decrease the secondary mass flow rate but has no Outlet pressure, effect on the primary mass flow rate. Figure 2. Various ejector operation modes at a fixed primary and secondary inlet pressures. Entrain- As the outlet Figurepressure 2. Various is increasejector operationed further, modes at ait fixe reachesd primary a and pressure, secondary inlet 𝑃 pressures., at which the ment ratio (the y-axis) is the ratio of secondary to primary mass flow rates. ejector starts to malfunction.Entrainment ratio (theThere y-axis) will is the beratio noof secondary secondary to primary flow mass or,flow even,rates. reverse flow through the secondaryAs inlet we increasewill occur. the outlet In pr pressure,actice, thean secondaryejector is flow usually will notrequired be choked to any- operate in the criticalmore mode/condition, (only the primary and flow henc is choked).e, it is Thisalso pressure called the threshold on-design is referred condition. to as critical pressure, Pcrit. Beyond this pressure, the ejector will enter the sub-critical mode operation within which an increase in P4 will decrease the secondary mass flow rate but has no effect on the primary massSub-critical flow rate. As the outlet pressuremode is increased further, it reaches a pressure, Pmal, at which the Criticalejector mode starts to malfunction. There will Malfunction be no secondary flow or, even, reverse flow through the secondary inlet will occur. In practice, an ejector is usually required to operate in the critical mode/condition, and hence, it is also called the on-design condition. Designing an ejector is an important task so that the system utilizing this device is not only able to operate functionally, but also efficiently. To know/predict the performance of an ejector, analytical and computational fluid dynamics (CFD) models are usually used before experiment. The one-dimensional analytical model has an advantage over the CFD for its simplicity and quick result. It does not require the time-consuming generation of the discretized EntrainmentER ratio,

Outlet pressure,

Figure 2. Various ejector operation modes at a fixed primary and secondary inlet pressures. Entrainment ratio (the y-axis) is the ratio of secondary to primary mass flow rates. Appl. Sci. 2021, 11, 3245 3 of 26

computational domain. Furthermore, the calculation time is also short, in the order of a few minutes or even seconds, compared to that of CFD which can take hours. Due to these advantages, the one-dimensional model is attractive to predict the global performance of . . an ejector such as predicting the primary and secondary mass flow rates (mp and ms) and the ejector outlet pressure (P4). To improve the model’s accuracy, the effects of local flow phenomena have to be included [1]. Such effects can only be analyzed if the detail of the flow is obtained either by experiment or CFD simulation or both. The works of Keenan and Neumann [3] as well as Keenan et al. [4] have become the basis of many researchers doing ejector one-dimensional modeling. Many attempts have been done to improve the model’s prediction of ejector behavior by considering the losses that occur inside it. Several loss coefficients are incorporated into the one-dimensional model and the values of these coefficients are obtained by matching the results of the model with experimental data. This approach was used by Huang et al. [2] and Chen et al. [5,6]. Kumar and Ooi [7] have also developed a semi-empirical one-dimensional model based on the model of Huang et al. [2] with a modification that considers the Fanno flow concept inside the mixing section to capture the effect of friction. An ejector mathematical model for critical mode operation has been developed by Zhu et al. [8] which was called the “shock circle model”. The model took into account the velocity profile of the secondary flow at the entrance of the constant area chamber. Al-Ansary [9,10] has pointed out the importance of the pressure variation along the converging mixing section and attempted to develop a mathematical model considering this pressure variation. However, the pressure variation function used was an assumption and has not been validated extensively. The previous works presented above have indicated that the studies of pressure variation along the wall of the converging mixing section are still limited. Moreover, many of the one-dimensional models do not consider this pressure variation. Instead, they usually assume that the pressure variation at this location is negligible and equal to the pressure of the secondary flow inlet. Therefore, more researches about its effect on the accuracy of the one-dimensional model need to be conducted. The current research is an attempt to comprehend that effect further through CFD analysis, especially for ejectors operating in critical conditions. Additionally, the k-omega SST model and the transition SST model were tested to find the best-suited turbulence model for the ejector flow in this research. Hence, it will also contribute to the turbulence modeling test data for flow within an ejector, especially the use of transition SST model. The steps used in this research are as follows. Computational fluid dynamics simulations were used to obtain the pressure variation on the wall of the converging mixing section. Four different ejectors were simulated. An ejector was obtained from a published experimental work and used to validate the CFD simulations. The other three ejectors were a modification of the first one and used for the parametric study. The predicted secondary . mass flow rates, ms, obtained using the one-dimensional analytical model were compared before and after incorporating the above-mentioned pressure variation into the model.

2. The Geometry of the Ejector Used The geometry of the ejector used in this research was mainly based on the work of Al- Ansary [9] and Al-Ansary and Jeter [11]. The procedure of the CFD modeling was validated by comparing its results with the results of Al-Ansary’s experiments. Afterward, for further simulations, it was decided to modify Al-Ansary’s ejector geometry to be simpler or more generic for the sake of ease of the parametric studies in the future.

geometry to be simpler or more generic for the sake of ease of the parametric studies in the future.

2.1. Ejector for the CFD Validation (Ejector G1) Al-Ansary [9] used a coordinate measuring machine (CMM) to measure the internal contour of the ejector accurately. The measurement results were presented in terms of piece-wise equations and were used in the current research to build the CFD computational domain. The equations for the geometry were divided into three sets, i.e., the primary nozzle section set, the mixing section set, and the diffuser section as shown in Equations (1) to (3). The radius, 𝑅, was presented as a function of the axial distance, 𝑥, measured from the primary flow inlet-plane. Both variables are in millimeter. This geometry, also called ejector G1, was used for the CFD validation. The contour can be seen in Figure 3 below. For the primary nozzle section: • nozzle converging section: 𝑅 = −0.1782864𝑥 + 10.6096; 0𝑥6.2 𝑅 = 9.504; 6.2 𝑥 16.6 𝑅 = −1.37414𝑥 + 32.315; 16.6 𝑥 18.0 𝑅 = −0.16645𝑥 + 10.576; 18.0 𝑥 39.7 (1) • nozzle throat: 𝑅 = 3.968; 39.7 𝑥 43.2 • nozzle diverging section: 𝑅 = 0.093565𝑥 − 0.074; 43.2 𝑥 72.9 For the mixing section: • converging mixing section: 𝑅 = −0.423313𝑥 + 49.0172; 80.7 𝑥 97.0 Appl. Sci. 2021, 11, 3245 4 of 26 𝑅 = 6.98 + 0.97𝑒(..); 97.0 𝑥 110.0 (2) • constant area mixing section (ejector throat): 𝑅 = 6.999; 110.0 𝑥 174.0 to (3).For The the radius, diffuserR, section: was presented as a function of the axial distance, x, measured from the primary ﬂow inlet-plane. Both variables are in millimeter. This geometry, also called 𝑅 = 179.3466 − 2.583078𝑥 + 0.01261313𝑥 − 0.0000198874𝑥; 174.0 𝑥 229.0 (3) ejector G1, was used for the CFD validation. The contour can be seen in Figure3 below.

Figure 3. The internal contour of Al-Ansary’s ejector [9] (ejector G1) used for the computational ﬂuid dynamics (CFD) Figure 3. The internal contour of Al-Ansary’s ejector [9] (ejector G1) used for the computational fluid dynamics (CFD) validation in the current study. validation in the current study. For the primary nozzle section: 2.2. Ejectors for the CFD Parametric Study • nozzle converging section : R = −0.1782864x + 10.6096; 0 ≤ x ≤ 6.2 R = 9.504; 6.2 ≤ x ≤ 16.6 R = −1.37414x + 32.315; 16.6 ≤ x ≤ 18.0 R = −0.16645x + 10.576; 18.0 ≤ x ≤ 39.7 (1) • nozzle throat : R = 3.968; 39.7 ≤ x ≤ 43.2 • nozzle diverging section : R = 0.093565x − 0.074; 43.2 ≤ x ≤ 72.9

For the mixing section:

• converging mixing section : R = −0.423313x + 49.0172; 80.7 ≤ x ≤ 97.0 R = 6.98 + 0.97e(29.311−0.302x); 97.0 ≤ x ≤ 110.0 (2) • constant area mixing section (ejector throat) : R = 6.999; 110.0 ≤ x ≤ 174.0

For the diffuser section:

R = 179.3466 − 2.583078x + 0.01261313x2 − 0.0000198874x3; 174.0 ≤ x ≤ 229.0 (3)

2.2. Ejectors for the CFD Parametric Study There were three ejectors for the parametric study. The ﬁrst ejector was the base ejector while the other two ejectors were the variation of the base ejector. These three ejectors were referred to as ejector G2, G2.1, and G2.2, respectively.

2.2.1. The Base Ejector (Ejector G2) The base ejector for the parametric study was still based on Al-Ansary’s ejector [9,11] but made simpler. The modiﬁcation was done by changing the shape of the end of the converging mixing section from a curved conical shape into a straight conical shape. Another difference was that the inlet of the converging mixing section was extended Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 27

There were three ejectors for the parametric study. The first ejector was the base ejector while the other two ejectors were the variation of the base ejector. These three ejectors were referred to as ejector G2, G2.1, and G2.2, respectively.

2.2.1. The Base Ejector (Ejector G2) The base ejector for the parametric study was still based on Al-Ansary’s ejector [9,11] Appl. Sci. 2021, 11, 3245 but made simpler. The modification was done by changing the shape of the end of5 ofthe 26 converging mixing section from a curved conical shape into a straight conical shape. Another difference was that the inlet of the converging mixing section was extended upstreamupstream to to the the primary primary nozzle nozzle outlet outlet plane. plane. Hence, Hence, for for ejector ejector G2, G2, the the primary primary nozzle nozzle outletoutlet plane plane and and the the inlet inlet of of the the converging converging mixing mixing section section became became coplanar. coplanar. The The other other partsparts of of the the ejector ejector are are not not changed. changed. To To be be cl clearer,earer, the comparison is shown in Figure4 4..

Figure 4. The zoomed-in ejector geometry around the converging mixing section. The base geometry Figure 4. The zoomed-in ejector geometry around the converging mixing section. The base geometryfor the parametric for the parametric study, ejector study, G2, ejector (solid G2, grey (sol lines)id grey is superimposed lines) is superimposed on the geometry on the forgeometry the CFD forvalidation, the CFD ejectorvalidation, G1, (theejector solid G1, green (the linessolid connectinggreen lines theconnecting green dots). the green dots). 2.2.2. The Variation of Ejector G2 2.2.2. The Variation of Ejector G2 For the parametric study, two additional ejectors were created based on ejector G2. All theFor partsthe parametric of the additional study, two ejectors additional have the ejectors same were shape created as that based of ejector on ejector G2 except G2. All the parts of the additional ejectors have the same shape as that of ejector G2 except the the radius of the converging mixing section inlet, RCV,in. It was decreased by 3.3 mm and radius6.7 mm of fromthe converging the initial radiusmixing of section 18.1577 inlet, mm. 𝑅, The. ﬁrstIt was and decreased the second by 3.3 variation mm and of 6.7 the Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 of 27 mmejectors from are the called initial ejector radius G2.1 of and18.1577 G2.2, mm. respectively. The first The and comparison the second of variation all three ejectorsof the ejectorsis depicted are called in Figure ejector5. G2.1 and G2.2, respectively. The comparison of all three ejectors is depicted in Figure 5.

Figure 5. The variation in ejector geometry for the parametric study. Figure 5. The variation in ejector geometry for the parametric study. 3. The CFD Settings 3.3.1. The Numerical CFD Settings Set-Up The CFD simulations were conducted using ANSYS Fluent 19.0. The mesh was 3.1. Numerical Set-Up generated using ICEM CFD 19.0, part of ANSYS software package. The solver chosen wasThe density-based CFD simulations solver withwere theconducted ﬂow assumed using ANSYS to be 2-D Fluent axisymmetric 19.0. The andmesh steady. was generatedThe 2-D axisymmetric using ICEM assumptionCFD 19.0, part has of been AN shownSYS software by some package. researchers The to solver give accuratechosen was density-based solver with the flow assumed to be 2-D axisymmetric and steady. The 2-D axisymmetric assumption has been shown by some researchers to give accurate results, especially when the ejector operates in the critical conditions (choked secondary flow) [12,13]. The spatial discretization schemes were all second order upwind and the gradient scheme used was Green-Gauss cell-based. The implicit solution scheme and Roe-FDS flux scheme were chosen to solve the set of discretized equations.

3.2. Turbulence Model One of the challenges in CFD modeling of flow inside an ejector is determining the turbulence model. Although many researchers have tested various turbulence models to predict an ejector flow, the results have not been conclusive as to which turbulence model gives the best prediction. Many of them have found that the k-omega SST turbulence model is the best one [12,14–17]. However, some others have found that RNG k-epsilon [18,19] or even the standard k-epsilon models are better [20–22]. Some other researchers [23,24] have also found that the realizable k-epsilon model can predict an ejector flow well, and even better if used in conjunction with the enhanced wall treatment [25]. Zhu and Jiang [19] have tested four turbulence models: standard k-epsilon, realizable k-epsilon, RNG k-epsilon, and k-omega SST models. They have found that the RNG k-epsilon model results matched the experimental data (the mass flow rate and the shock structure) better than the others while the k-omega SST model was the second best. Mohamed et al. [26] have used RNG k-epsilon turbulence model in their work and found that the CFD results predicted the experimental entrainment ratios with an error of no greater than 20%, and the predicted values tended to be higher than those of the experiment. Sriveerakul et al. [23] applied the realizable k-epsilon model in their CFD work and found a good agreement between the CFD results and the experimental data at critical flow conditions (choked secondary flow) but not as good in the sub-critical conditions (unchoked secondary flow). In the current research, we tested two turbulence models during the validation, i.e., the k-omega SST and the transition SST turbulence models with the default values of model constants given in ANSYS Fluent. The best between those two was then used for Appl. Sci. 2021, 11, 3245 6 of 26

results, especially when the ejector operates in the critical conditions (choked secondary ﬂow) [12,13]. The spatial discretization schemes were all second order upwind and the gradient scheme used was Green-Gauss cell-based. The implicit solution scheme and Roe-FDS ﬂux scheme were chosen to solve the set of discretized equations.

3.2. Turbulence Model One of the challenges in CFD modeling of flow inside an ejector is determining the turbulence model. Although many researchers have tested various turbulence models to predict an ejector flow, the results have not been conclusive as to which turbulence model gives the best prediction. Many of them have found that the k-omega SST turbulence model is the best one [12,14–17]. However, some others have found that RNG k-epsilon [18,19] or even the standard k-epsilon models are better [20–22]. Some other researchers [23,24] have also found that the realizable k-epsilon model can predict an ejector flow well, and even better if used in conjunction with the enhanced wall treatment [25]. Zhu and Jiang [19] have tested four turbulence models: standard k-epsilon, realizable k-epsilon, RNG k-epsilon, and k-omega SST models. They have found that the RNG k-epsilon model results matched the experimental data (the mass flow rate and the shock structure) better than the others while the k-omega SST model was the second best. Mo- hamed et al. [26] have used RNG k-epsilon turbulence model in their work and found that the CFD results predicted the experimental entrainment ratios with an error of no greater than 20%, and the predicted values tended to be higher than those of the experiment. Sriveerakul et al. [23] applied the realizable k-epsilon model in their CFD work and found a good agreement between the CFD results and the experimental data at critical flow conditions (choked secondary flow) but not as good in the sub-critical conditions (unchoked secondary flow). In the current research, we tested two turbulence models during the validation, i.e., the k-omega SST and the transition SST turbulence models with the default values of model constants given in ANSYS Fluent. The best between those two was then used for the parametric study. Transition SST turbulence model was chosen as one of the candidates because flow in an ejector may have boundary layer separation and circulation at one or more locations such as at the wall of the diverging section of the primary nozzle (before the outlet) [10–12] and at the converging mixing section wall [11,25]. This may cause re-laminarization of the flow at a certain region which might be better predicted by using the transition SST model. It is worth pointing out that only a few researchers have used the transition SST model for an ejector flow as done by Varga et al. [27].

3.3. Working Fluid Properties The air properties used are shown in Table1. The thermal conductivity, κ, and viscosity, µ, were obtained from the air properties database in Engineering Equation Solver (EES) [28]. The temperature range of these two properties is 100–350 K.

Table 1. The properties of air used in the CFD simulation.

Properties Values/Equations Density, ρ Ideal gas Speciﬁc heat, cp 1006.43 J/kg.K κ = 1.039718 × 10−3 + 8.985199 × 10−5 T− Thermal conductivity, κ 2.609688 × 10−8 T2 [W/m.K]; where T is in Kelvin µ = −1.874901 × 10−6 + 1.024389 × 10−7 T− Viscosity, µ 1.568282 × 10−10 T2 + 1.415525 × 10−13 T3 [Pa.s]; where T in Kelvin

3.4. Boundary Conditions for the CFD Simulations The boundary conditions at the primary ﬂow inlet and the secondary ﬂow inlet were both pressure inlet, while the outlet was set to pressure outlet. All walls were assumed Appl. Sci. 2021, 11, 3245 7 of 26

smooth with no-slip condition and adiabatic. Axisymmetric boundary condition was applied at the axis of the ejector. The boundary condition values were based on the experimental results of Al-Ansary and Jeter [9,11], either from the direct measurements that they did or from calculated values based on these measurements. The primary flow inlet pressure, P0, was varied from 308 kPa to 618 kPa. The secondary flow inlet pressure, P2, was ambient pressure (100.8 kPa). The ejector outlet pressure, P4, was also near ambient pressure. The temperature of the primary and secondary flow inlet (T0 and T2) was 294 K while and ejector outlet, T4, was around 292 K in average. The values of the turbulence intensity, TI, at the primary inlet, secondary inlet, and ejector outlet (denoted as TI0, TI2, and TI4, respectively) were calculated from the experimental data. Since the mass flow rates were all measured and the dimensions of the ejector were available, the turbulence intensity values can be estimated by using Equation (4) [29].

− 1 TI = 0.16Re 8 × 100% (4) Dh

where ReDh was the Reynolds number based on the hydraulic diameter. The turbulence intensities obtained through Equation (4) were ranging from 3.44% to 4.66%.

3.5. The Convergence Criteria The criteria used to judge the convergence of the simulation were: 1. The convergence of physical quantities monitored at 10 different locations. 2. The mass and energy imbalance. 3. Attempt to get scaled residual values of equal or less than 1 × 10−6. The first two criteria were the main criteria and the last was a secondary criterion. For the first criterion, the physical quantities were considered to have been converged if the difference between the minimum and the maximum values, within 1000 iterations, was no greater than 0.05%. The monitored quantities in this first criterion were as follows: 1. The average Mach number (facet average) along the axis located between the primary nozzle outlet plane and the ejector throat inlet plane. 2. The average Mach number (facet average) along the axis of the diffuser. 3. The secondary mass flow rate at the inlet. 4. The mass flow rate at the outlet. 5. The average wall shear stress (area-weighted average) along the primary nozzle. 6. The average wall shear stress (area-weighted average) along the converging mixing section. 7. The average wall shear stress (area-weighted average) along the constant-area mixing section. 8. The average wall shear stress (area-weighted average) along the diffuser. 9. The average static pressure (area-weighted average) along the wall of the converging mixing section. 10. The static pressure at a point located at the end of converging mixing section. For criterion no. 2, the mass and energy imbalance within the ejector should be no greater than 0.1% compared to the lowest mass and energy flow rates through the inlets and outlet. The third criterion, i.e., scaled residual of equal or less than 1 × 10−6, was regarded as a secondary criterion due to the difficulties reaching such values. However, many of the simulations were able to reach this residual value and only a few of them had residual values stalled in the order of 1 × 10−4 or 1 × 10−5. The scaled residual used here was the “globally scaled” residual defined by ANSYS fluent whose details can be seen in the user’s guide [29]. Appl. Sci. 2021, 11, 3245 8 of 26

4. Mesh Independence and Validation of the CFD Simulation 4.1. Mesh Independence Study The mesh created was quadrilateral structured mesh. For accurate results, k-omega SST and transition SST turbulence models require near-wall meshes with y+ ∼ 1 and y+ < 1, respectively [29]. Therefore, the final meshes used for the simulations were generated with y+ average value (i.e., area-weighted average) of less than 1. The y+ maximum value was greater than but close to 1 such as 1.27 to mention a value from one of the geometries. To obtain the mesh mentioned above, four meshes were created with different numbers of cells for the grid independence study. They were referred to as coarse (112,959 cells), medium (151,834 cells), semi-fine (452,760 cells), and fine (902,069 cells) meshes. The parameters used to judge if the CFD results were mesh independent were: 1. The secondary . . mass flow rate, ms; 2. The primary mass flow rate, mp; 3. The axis static pressure along the ejector; 4. The wall static pressure along the ejector; 5. The pressure profiles across the primary nozzle outlet; 6. The pressure profiles across a location at the end of the converging mixing section (i.e., before the ejector throat inlet). The primary inlet pressure, P0, used for the grid independence study was 618 kPa. The comparison of the secondary and primary mass flow rates (parameters no. 1 and 2) for these four meshes can be seen in Table2. The mass flow rates of all mesh sizes are nearly identical with a maximum change of around 0.61% for the secondary mass flow rate and nearly no change in the primary mass flow rate.

Table 2. The comparison of mass ﬂow rates change with mesh size increase.

Mesh Number k-Omega SST Transition SST k-Omega SST Transition SST Name of Cells . . . . . 1 . 2 . 1 . 2 mp[kg/s] ms[kg/s] mp[kg/s] ms[kg/s] ∆mp [%] ∆ms [%] ∆mp [%] ∆ms [%] Coarse 112,959 0.07104 0.01212 0.07107 0.01156 0.00 0.00 0.00 0.00 Medium 151,834 0.07104 0.01211 0.07108 0.01157 0.00 −0.08 0.01 0.09 Semi-fine 452,760 0.07102 0.01212 0.07106 0.01161 −0.03 0.00 −0.01 0.43 Fine 902,069 0.07102 0.01211 0.07106 0.01163 −0.03 −0.08 −0.01 0.61 1 . . . . . . ∆mp = mp − mp,coarse × 100/mp,coarse [%]; mp and mp,coarse are the primary mass flow rates of a certain mesh and the coarse mesh, 2 . . . . . . respectively. ∆ms = ms − ms,coarse × 100/ms,coarse [%]; ms and ms,coarse are the secondary mass flow rates of a certain mesh and the coarse mesh, respectively.

For parameters no. 3, 4, 5, and 6, it was found that they did not change significantly with respect to the mesh size increase. As an example, Figure6 shows the static pressure profile along the axis of the ejector for the four meshes. Based on these comparisons and a reasonable computational time needed, we decided to choose the medium mesh (151,834 cells) for further CFD study. The sample of the chosen mesh (the medium mesh) around the outlet of the primary nozzle as well as the inlet of the secondary flow is presented in Figure7. Appl. Sci. 2021, 11, x FOR PEER REVIEW 9 of 27

Appl. Sci. 2021, 11, x FOR PEER REVIEW 9 of 27

cells) for further CFD study. The sample of the chosen mesh (the medium mesh) around the outlet of thecells) primary for further nozzle CFD as well study. as theThe inletsample of theof the secondary chosen mesh flow (theis presented medium mesh)in around Figure 7. the outlet of the primary nozzle as well as the inlet of the secondary flow is presented in Figure 7. Table 2. The comparison of mass flow rates change with mesh size increase. Table 2. The comparison of mass flow rates change with mesh size increase. Number of k-Omega SST Transition SST k-Omega SST Transition SST Mesh Name 𝒎 Number of𝒎 k-Omega𝒎 SST 𝒎 Transition∆𝒎 SST1 ∆𝒎 2k-Omega∆𝒎 SST1 ∆𝒎 Transition2 SST MeshCells Name 𝒑 [kg/s] 𝒔 [kg/s] 𝒑 [kg/s] 𝒔 [kg/s] 𝒑 [%] 𝒔 [%] 𝒑 [%] 𝒔 [%] 1 2 1 2 Coarse 112,959 0.07104Cells 0.01212𝒎 𝒑 [kg/s] 0.07107𝒎 𝒔 [kg/s] 0.01156𝒎 𝒑 [kg/s] 𝒎0.00 𝒔 [kg/s] ∆𝒎 0.00 𝒑 [%] ∆𝒎 0.00 𝒔 [%] ∆𝒎0.00 𝒑 [%] ∆𝒎 𝒔 [%] Medium 151,834Coarse 0.07104112,959 0.012110.07104 0.071080.01212 0.011570.07107 0.000.01156 −0.080.00 0.01 0.00 0.09 0.00 0.00 Semi-fine 452,760Medium 0.07102151,834 0.012120.07104 0.071060.01211 0.011610.07108 −0.011570.03 0.000.00 −0.01−0.08 0.430.01 0.09 Semi-fine 452,760 0.07102 0.01212 0.07106 0.01161 −0.03 0.00 −0.01 0.43 Fine 902,069 0.07102 0.01211 0.07106 0.01163 −0.03 −0.08 −0.01 0.61 1 Fine 902,069 0.07102 0.01211 0.07106 0.01163 −0.03 −0.08 −0.01 0.61 Appl. Sci. ∆𝑚2021 =𝑚, 11, 3245 −𝑚 , × 100 𝑚 , [%]; 𝑚 and 𝑚 , are the primary mass flow rates of a certain mesh and9 of 26 1 ∆𝑚 =𝑚 −𝑚 × 100 𝑚 [%]; 𝑚 and 𝑚 are the primary mass flow rates of a certain mesh and the coarse mesh, respectively. 2 ∆𝑚, =𝑚 −𝑚 ,×100 𝑚 [%,]; 𝑚 and 𝑚 are the secondary mass the coarse mesh, respectively. 2, ∆𝑚 =𝑚 −𝑚 ,×100 𝑚 [%,]; 𝑚 and 𝑚 are the secondary mass flow rates of a certain mesh and the coarse mesh, respectively. , , , flow rates of a certain mesh and the coarse mesh, respectively.

650 650 600 600 Coarse 550 550 Coarse Medium 500 500 Medium 450 450 Semi-fine Semi-fine 400 400 Fine Fine 350 350 300 300 250 250 200 200 150 150 Axis static pressure, abs [kPa] Axis static pressure, abs [kPa] 100 100 50 50 0 0 0 20 40 60 80 100 120 140 160 180 200 220 0 20 40 60 80 100 120 140 160 180 200 220 x [mm] x [mm] 15 15 10

10 [mm] 5 [mm] 5 R 0

R 0 0 20 40 60 80 100 120 140 160 180 200 220 0 20 40 60 80 100 120 140 160 180 200 220 x [mm] x [mm]

Figure 6. AbsoluteFigure static 6. Absolute pressure static along pressure the axis along (top). the Theaxis ejector(top). The internal ejector contourinternal (bottom)contour (bottom) is is also Figure 6. Absolutepresented static pressure to see the along corresponding the axis (top). location The ejectorof the static internal pressure. contour The (bottom) turbulence is also model is the also presented to see the corresponding location of the static pressure. The turbulence model is the presented to see thetransition corresponding SST model. location of the static pressure. The turbulence model is the transitiontransition SST SST model. model.

Appl. Sci. 2021, 11, x FOR PEER REVIEW 10 of 27

(a) (a)

(b)

Figure 7. TheFigure sample 7. The of the sample chosen of mesh the chosen (medium mesh mesh) (medium around mesh) the exit around of the theprimary exit ofnozzle the primary as well as nozzle the inlet as of the secondary wellflow asis shown the inlet in of(a) the and secondary the “Detail ﬂow A” is shown in in (b (a).) and the “Detail A” is shown in (b).

4.2. Validation To ensure the validity of the CFD simulations, a comparison with experimental data was conducted. The experimental results used for this comparison were those of Al-Ansary [9]. The data selected were only those when the ejector was operating in the critical mode. This was done because the current work focuses on an ejector operating at this mode. Comparing with the experimental data, it was found that the 𝑚 from CFD results using the transition SST turbulence model agree better than those using the k-omega SST model. This finding confirms the results of Varga et al. [27]. The comparison can be seen in Table 3. It should be noted that Al-Ansary [9] has found experimentally that 𝑃 = 308 kPa is the minimum pressure at which the secondary flow started to choke. Therefore, at this pressure or above, the ejector operates in critical mode. Al-Ansary [9] mentioned that the uncertainty of the measured mass flow rates was no greater than 4%. Hence, almost all of the CFD results using the transition SST model are within the experimental uncertainty while it is the opposite for those using the k-omega SST (Table 3). Regarding the 𝑚 , the results of CFD using the transition SST model are comparable to those using the k-omega SST. The difference between the CFD results and the experimental data is less than 6% for all operating conditions and the average is only around 4.2% (Table 4).

Table 3. Comparison of the secondary mass flow rates obtained from CFD and experiments.

Secondary Mass Flow Rate, 𝒎 [kg/s] ∆𝒎 1 [%] Sim. 𝑷 𝒔 𝒔,𝑪𝑭𝑫 𝟎 Transition k-Omega no. [kPa] Exp. Transition SST k-Omega SST SST SST 0 618 0.01189 0.01157 0.01211 -2.69 1.85 1 584 0.01206 0.01195 0.01315 -0.91 9.04 2 549 0.01240 0.01177 0.01284 -5.08 3.55 3 515 0.01286 0.01204 0.01290 -6.38 0.31 4 480 0.01308 0.01281 0.01349 -2.06 3.13 5 446 0.01320 0.01340 0.01433 1.52 8.56 6 412 0.01342 0.01350 0.01439 0.60 7.23 7 377 0.01365 0.01362 0.01472 -0.22 7.84 8 343 0.01382 0.01383 0.01491 0.07 7.89 Appl. Sci. 2021, 11, 3245 10 of 26

Table 3. Comparison of the secondary mass ﬂow rates obtained from CFD and experiments.

. . 1 Secondary Mass Flow Rate, ms [kg/s] ∆ms,CFD [%] Sim. no. P0 [kPa] Transition k-Omega Transition k-Omega Exp. SST SST SST SST 0 618 0.01189 0.01157 0.01211 -2.69 1.85 1 584 0.01206 0.01195 0.01315 -0.91 9.04 2 549 0.01240 0.01177 0.01284 -5.08 3.55 3 515 0.01286 0.01204 0.01290 -6.38 0.31 4 480 0.01308 0.01281 0.01349 -2.06 3.13 5 446 0.01320 0.01340 0.01433 1.52 8.56 6 412 0.01342 0.01350 0.01439 0.60 7.23 7 377 0.01365 0.01362 0.01472 -0.22 7.84 8 343 0.01382 0.01383 0.01491 0.07 7.89 9 308 0.01399 0.01425 0.01527 1.86 9.15 1 . . . . . . ∆ms,CFD = ms,CFD − ms,exp × 100/ms,exp [%]; ms,CFD and ms,exp are the secondary mass ﬂow rates obtained from the CFD and experiment, respectively.

Al-Ansary [9] mentioned that the uncertainty of the measured mass ﬂow rates was no greater than 4%. Hence, almost all of the CFD results using the transition SST model are within the experimental uncertainty while it is the opposite for those using the k-omega SST (Table3). . Regarding the mp, the results of CFD using the transition SST model are comparable to those using the k-omega SST. The difference between the CFD results and the experimental data is less than 6% for all operating conditions and the average is only around 4.2% (Table4) .

Table 4. Comparison of the primary mass ﬂow rates obtained from CFD and experiments.

. . 1 Primary Mass Flow Rate, mp [kg/s] ∆mp,CFD [%] Sim. No. P0 [kPa] Transition k-Omega Transition k-Omega Exp. SST SST SST SST 0 618 0.06713 0.07108 0.07104 5.88 5.82 1 584 0.06464 0.06717 0.06713 3.91 3.85 2 549 0.06087 0.06314 0.06310 3.73 3.66 3 515 0.05716 0.05923 0.05919 3.62 3.55 4 480 0.05325 0.05520 0.05516 3.66 3.59 5 446 0.04931 0.05129 0.05125 4.02 3.93 6 412 0.04543 0.04738 0.04733 4.29 4.18 7 377 0.04150 0.04335 0.04331 4.46 4.36 8 343 0.03768 0.03943 0.03940 4.64 4.56 9 308 0.03397 0.03540 0.03537 4.21 4.12 1 . . . . . . ∆mp,CFD = mp,CFD − mp,exp × 100/mp,exp [%]; mp,CFD and mp,exp are the primary mass ﬂow rates obtained from the CFD and experiment, respectively. Appl. Sci. 2021, 11, 3245 11 of 26

Based on the above comparison, the transition SST turbulence model was chosen over the k-omega SST for the CFD study in this work. However, it should be noted that although . the ms obtained from CFD using k-omega SST are less accurate, they also show a similar trend compared to those from the experiments. Moreover, their discrepancies are not too bad, i.e., only around 9% or less, concerning the complexity of the flow. This means that the k-omega SST turbulence model should not be excluded as an option in predicting an ejector flow. Ideally, to increase the soundness of the CFD simulations, comparison between the CFD results and experimental data should also be done for other variables such as wall pressure or axis Mach number distribution. However, such data are not available in the work of Al-Ansary [9]. Despite this limitation, an attempt was done to compare qualitatively our CFD results with other researchers’ results obtained either experimentally or numerically. Han et al. [25] have done experimental and numerical work using ANSYS Fluent on an ejector flow, focusing their work on the boundary layer separation inside the mixing section. They have measured the wall pressure along the ejector’s mixing section up to the diffuser to validate their CFD results. Several turbulence models have been tested (k-epsilon standard, k-epsilon realizable, k-epsilon RNG, and k-omega SST) combined with two wall functions, i.e., the standard wall function (SWT) and enhanced wall function (EWT). To do the qualitative comparison mentioned above, the current CFD results are compared to the CFD and experimental results of Han et al. [25] which is presented in Figure8. It should be noted that Han et al. have named the mixing section differently than what we have defined. They have given the term mixing section only for the converging mixing section while the constant area duct is simply termed the ejector throat. In the Appl. Sci. 2021, 11, x FOR PEER REVIEW 12 of 27 current work, however, the term mixing section encompasses both the converging section and the ejector throat.

1.20

1.05

0.90 [-]

2 0.75 The beginning

/P of shock (in the w diffuser) P 0.60 Near the inlet of the 0.45 throat

0.30

0.15 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 axial position, x [m] (a) (b)

FigureFigure 8.8.Qualitative Qualitative comparison comparison of theof the non-dimensional non-dimensional wall wall pressure pressure (Pw/ P(2𝑃)along/𝑃) along the mixing the mixing section section up to the up diffuser to the diffuser between Han et al.’s work (a) and the current work (turbulence model: transition SST; 𝑃 = 618 kPa) (b). The left between Han et al.’s work (a) and the current work (turbulence model: transition SST; P0 = 618 kPa) (b). The left ﬁgure (a) 𝑦 isfigure reprinted (a) is from reprinted the work from of the Han work et al. of [25 Han] with et theal. wall[25] pressurewith the (wally-axis) pressure is non-dimensionalized ( -axis) is non-dime by thensionalized secondary by inlet the secondary inlet pressure of their work (𝑃 = 3170 Pa). pressure of their work (P2 = 3170 Pa).

AsAs cancan bebe seenseen inin FigureFigure8 ,8, the the trend trend of of both both results results is is nearly nearly the the same. same. There There is is a a relativelyrelatively sharpsharp pressure pressure drop drop near near the ejectorthe ejector throat throat inlet which inlet iswhich caused is by caused the secondary by the ﬂowsecondary acceleration flow acceleration through a constricted through a constricted hypothetical hypothetical annular duct. annular This duct duct. is This formed duct by is theformed primary by the jet primary ﬂow and jet the flow wall and of thethe ejector.wall of Inthe our ejector. CFD results,In our CFD the shock results, train the enters shock thetrain ejector enters throat the ejector further, throat while further, in Han while et al.’s in work, Han theet al.’s shock work, train the strength shock hastrain decreased strength has decreased before it enters the throat. The shock train inside the throat causes the hypothetical annular duct to constrict periodically along the x-direction. This is the reason why the wall pressure inside the ejector throat fluctuates in the present work while it does not in the work of Han et al. Despite this difference, the general trend of the wall pressure inside the throat is the same, i.e., the pressure gradually increases until the flow enters the subsonic diffuser at a speed greater than or equal to sonic speed. Therefore, the flow expands sharply since, for a supersonic flow, a subsonic diffuser acts as a nozzle. At some point inside the diffuser, a shock appears (circled in black) which is indicated by a sharp increase in pressure. The Mach number contour plot for M ≥ 1 is presented in Figure 9 to clarify the discussion while for the Mach number contour plot of Han et al., one can refer directly to their paper.

Figure 9. The Mach number contour plot for M ≥ 1 showing the periodically constricted hypothetical annular duct (turbulence model: transition SST; 𝑃 = 618 kPa).

5. The One-Dimensional Analytical Model Appl. Sci. 2021, 11, x FOR PEER REVIEW 12 of 27

1.20

1.05

0.90 [-]

2 0.75 The beginning

/P of shock (in the w diffuser) P 0.60 Near the inlet of the 0.45 throat

0.30

0.15 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 axial position, x [m] (a) (b)

Figure 8. Qualitative comparison of the non-dimensional wall pressure (𝑃/𝑃) along the mixing section up to the diffuser between Han et al.’s work (a) and the current work (turbulence model: transition SST; 𝑃 = 618 kPa) (b). The left figure (a) is reprinted from the work of Han et al. [25] with the wall pressure (𝑦-axis) is non-dimensionalized by the secondary inlet pressure of their work (𝑃 = 3170 Pa).

As can be seen in Figure 8, the trend of both results is nearly the same. There is a relatively sharp pressure drop near the ejector throat inlet which is caused by the Appl. Sci. 2021, 11, 3245 secondary flow acceleration through a constricted hypothetical annular duct. This duct12 is of 26 formed by the primary jet flow and the wall of the ejector. In our CFD results, the shock train enters the ejector throat further, while in Han et al.’s work, the shock train strength has decreased before it enters the throat. The shock train inside the throat causes the beforehypothetical it enters annular the throat. duct The to shockconstrict train periodically inside the throatalong causesthe x-direction. the hypothetical This is annularthe ductreason to constrictwhy the periodicallywall pressure along inside the thex-direction. ejector throat This isfluctuates the reason in whythe present the wall work pressure insidewhile it the does ejector not in throat the work fluctuates of Han in et the al. presentDespite workthis difference, while it does the general not in the trend work of the of Han etwall al. pressure Despite thisinside difference, the throat the is the general same, trend i.e., the of thepressure wall pressuregradually inside increases the until throat the is the same,flow enters i.e., the the pressure subsonic gradually diffuser increases at a speed until greater the flow than enters or equal the subsonic to sonic diffuser speed. at a speedTherefore, greater the thanflow orexpands equal tosharply sonic speed.since, for Therefore, a supersonic the flowflow, expands a subsonic sharply diffuser since, acts for a supersonicas a nozzle. flow,At some a subsonic point inside diffuser the diffuser, acts as a a nozzle. shock appears At some (circled point in inside black) the which diffuser, is a shockindicated appears by a sharp (circled increase in black) in pressure. which is indicatedThe Mach bynumber a sharp contour increase plot in for pressure. M ≥ 1 is The Machpresented number in Figure contour 9 to clarify plot for the M discussion≥ 1 is presented while for in the Figure Mach9 number to clarify contour the discussion plot of whileHan et for al., the one Mach can refer number directly contour to their plot paper. of Han et al., one can refer directly to their paper.

FigureFigure 9. The Mach number contour contour plot plot for for M M ≥≥ 1 1showing showing the the periodically periodically constricted constricted hypothetical hypothetical annular duct (turbulence model: transition SST; 𝑃 = 618 kPa). annular duct (turbulence model: transition SST; P0 = 618 kPa).

5.5. The One-Dimensional Analytical Analytical Model Model In this section, the one-dimensional analytical model is presented. It is based on Al- Ansary’s work [9] with some re-arrangement. This will be in the ﬁrst subsection. The ejector schematic drawing in Figure1 is used for the derivation of the model. This analytical model will be referred to as the standard model. The equations are solved using Engineering Equation Solver (EES) [28]. Afterward, the modiﬁed one-dimensional analytical model is presented in the second subsection.

5.1. The Standard One-Dimensional Model To simplify the one-dimensional ejector model, several common assumptions are made as shown below: 1. The flow is one-dimensional and steady all over the ejector. 2. The fluid is an ideal gas with constant specific heats (air is used in this work as the fluid). 3. Friction at the walls is negligible. 4. The kinetic energy at the inlet of primary flow, inlet of secondary flow and outlet of the ejector is negligible. 5. The flow at the outlet of the primary nozzle is supersonic, i.e., the flow through the primary nozzle is choked. This assumption is merely based on the practical point of view, i.e., an ejector is usually required to operate at the critical conditions at which both primary and secondary flow are choked. 6. The ejector is adiabatic, rigid (no boundary work involved), and impermeable (no flow entering or leaving the ejector besides flow through the inlets and outlet of the ejector). 7. The isentropic efficiency of the primary nozzle and the subsonic diffuser (ηN and ηD) is known beforehand. In this work, the values are assumed to be constant with values of 0.95 and 0.8 for the nozzle and diffuser, respectively. These values are based on the typical efficiency of the devices. However, it should be kept in mind that actually, these fixed values of efficiency cannot be applied in all operating conditions of an ejector, and they will be one of the sources of error in the model. Therefore, a study to estimate the efficiencies of the nozzle and especially, the diffuser should be conducted Appl. Sci. 2021, 11, 3245 13 of 26

in the future. By doing so, their efﬁciencies can be estimated more accurately by just knowing the geometry and operating conditions of the ejector. 8. For an ejector that operates in the critical mode, a normal shock is assumed to happen at the exit of the mixing section (station 3). Therefore, the Mach number at that location, M3, will be an input and have a value of one. For an ejector that operates in the sub-critical mode, the outlet static pressure, P4, will be an input while the M3 will be an output. However, in this work, the focus will be on the critical mode operation and, therefore, M3 = 1 will act as an input and P4 as an output. It should be noted that this assumption (M3 = 1) has been found to be inaccurate at certain operating conditions. Yet, it is simple to apply and is thought to be a good initial approximation. In fact, the value of M3 can either be greater or less than one, depending on where the shock starts. This is a challenge that remains to be solved. . . The main targets of this model are to obtain the ms, mp, and P4. The given/known parameters are as follows: 1. The primary and secondary stagnation conditions (i.e., properties at station 0 and 2 in Figure1). 2. Ejector geometry: Appl. Sci. 2021, 11, x FOR PEER REVIEW 14 of 27 Appl. Sci. 2021, 11, x FOR PEER REVIEW • Primary nozzle: Throat diameter, Dth, and outlet diameter, D1. 14 of 27

• Converging mixing section: Inlet diameter, DCV,in. • Constant area mixing section (ejector throat): Diameter, D3. 3. Either the 𝑀 =1 (for critical mode operation) or the 𝑃 (for sub-critical mode 3.3. EitherEither thethe M𝑀 ==11 (for(for critical critical mode mode operation) operation) or theor Pthe(for 𝑃 sub-critical (for sub-critical mode operation).mode operation). However,3 the current work will focus on the4 critical mode operation of operation).However, theHowever, current the work current will focuswork onwill the focus critical on modethe critical operation mode ofoperation an ejector of and, an ejector and, therefore, 𝑀 =1 will act as an input and 𝑃 will be an output an ejector Mand,= therefore, 𝑀 =1 will act Pas an input and 𝑃 will be an output (assumptiontherefore, no.3 8).1 will act as an input and 4 will be an output (assumption no. 8). 4. (assumptionThe values of no.η N8).and ηD are 0.95 and 0.8, respectively (assumption no. 7). 4. The values of 𝜂 and 𝜂 are 0.95 and 0.8, respectively (assumption no. 7). 4. The values of 𝜂 and 𝜂 are 0.95 and 0.8, respectively (assumption no. 7). The governing governing equations equations (continuity, (continuity, mome momentum,ntum, and and energy energy balance balance equations), equations), as as The governing equations (continuity, momentum, and energy balance equations), as wellwell as equation of of state state of of an an ideal ideal gas, gas, will will be beapplied applied whenever whenever needed needed to construct to construct the the well as equation of state of an ideal gas, will be applied whenever needed to construct the model. There There are are five ﬁve control control volumes volumes required required to todevelop develop the the model model and and these these control control model. There are five control volumes required to develop the model and these control volumes will be be shown shown in in Figures Figures 10–12. 10–12 . volumes will be shown in Figures 10–12.

Figure 10. 10. TheThe dashed dashed (blue) (blue) line line and anddotted dotted (red) (red)lines are lines showing are showing control controlvolumes volumes A and B, A and B, Figure 10. The dashed (blue) line and dotted (red) lines are showing control volumes A and B, respectively. respectively.respectively.

Figure 11. The dotted line is showing control volume C. FigureFigure 11. The dotted dotted line line is is showing showing control control volume volume C. C.

Figure 12. The dotted (red) and dashed (blue) lines are showing control volumes D and E, Figure 12. The dotted (red) and dashed (blue) lines are showing control volumes D and E, respectively. respectively. For control volume A shown in Figure 10, the mass and energy balance will be For control volume A shown in Figure 10, the mass and energy balance will be applied. The mass balance will result in Equation (5). applied. The mass balance will result in Equation (5). 𝑚 =𝑚 =𝑚 +𝑚 = (1+𝐸𝑅)𝑚 (5) 𝑚 =𝑚 =𝑚 +𝑚 = (1+𝐸𝑅)𝑚 (5) where 𝐸𝑅 stands for entrainment ratio and defined as where 𝐸𝑅 stands for entrainment ratio and defined as Appl. Sci. 2021, 11, x FOR PEER REVIEW 14 of 27

3. Either the 𝑀 =1 (for critical mode operation) or the 𝑃 (for sub-critical mode operation). However, the current work will focus on the critical mode operation of an ejector and, therefore, 𝑀 =1 will act as an input and 𝑃 will be an output (assumption no. 8). 4. The values of 𝜂 and 𝜂 are 0.95 and 0.8, respectively (assumption no. 7). The governing equations (continuity, momentum, and energy balance equations), as well as equation of state of an ideal gas, will be applied whenever needed to construct the model. There are five control volumes required to develop the model and these control volumes will be shown in Figures 10–12.

Figure 10. The dashed (blue) line and dotted (red) lines are showing control volumes A and B, respectively.

Appl. Sci. 2021, 11, 3245 14 of 26

Figure 11. The dotted line is showing control volume C.

Figure 12. TheThe dotted dotted (red) (red) and and dashed dashed (blue) (blue) lines lines are are showing showing control control volumes volumes D D and and E, E, respectively. respectively. For control volume A shown in Figure 10, the mass and energy balance will be applied. The massFor control balance volume will result A shown in Equation in Figure (5). 10, the mass and energy balance will be applied. The mass balance will result in Equation (5)...... m4 = m3 = mp + ms = (1 + ER)mp (5) 𝑚 =𝑚 =𝑚 +𝑚 = (1+𝐸𝑅)𝑚 (5)

where ER𝐸𝑅 stands stands for entrainment ratio ratio and and defined defined as as . . ER = ms/mp (6) . In order to obtain mp, the compressible flow equation for a choked flow (assumption no. 5) is used as presented in Equation (7) below.

γ+1 . r γ 2 2(γ−1) mp = Ath P0 (7) RT0 γ + 1

where γ and R are the specific heat ratio and gas constant of air and have values of 1.4 and 287 J/kg-K, respectively. The Ath denotes the primary nozzle throat area. The energy balance applied to the control volume A in conjunction with assumptions no. 2 and 4 gives Equation (8). (T0 + ERT2) = (1 + ER)T4 (8) . . . Up to now, there are five unknowns, i.e., ER, mp, ms, m4, and T4 and four equations which are Equations (5) to (8). Thus, another equation is needed to solve for these five unknowns. This equation will be obtained later via control volumes C and D. Next, control volume B, also shown in Figure 10, will be considered. Through this control volume, the velocity, pressure, and Mach number at the outlet of the primary nozzle, denoted by v1, P1, and M1, respectively, will be found. Applying energy balance to control volume B along with assumptions no. 2, 4, and 7, we will obtain Equation (9). v u " γ−1 # u P1 γ v1 = t2 ηN cp T0 1 − (9) P0

As stated in assumption no. 7, the isentropic efﬁciency of the primary nozzle, ηN, is given which is assumed to be 0.95. However, there are two unknowns (v1 and P1) and only one equation, i.e., Equation (9). Thus, we need another equation. To obtain another equation, the area ratio relation of isentropic compressible ﬂow will be used as follows. Mach number at the outlet of the primary nozzle (station 1), M1, can be calculated from (assuming there is no boundary layer separation between the throat and primary nozzle outlet):

γ+1 2(γ−1) A1 1 2 γ − 1 2 = 1 + M1 (10) Ath M1 γ + 1 2

where A1 is the area at station 1 (the cross sectional area of the primary nozzle outlet). Hence, the pressure at the nozzle outlet, P1, can be obtained from Appl. Sci. 2021, 11, 3245 15 of 26

γ γ−1 P0 γ − 1 2 = 1 + M1 (11) P1 2

Now, there are three unknowns and three equations which are v1, P1, and M1 and Equations (9) to (11). Therefore, these parameters can be found. For control volume C (see Figure 11 above and Figure 13 below), momentum equation will be applied to obtain a useful equation. It should be noted that the area at station 1, A1, and station 2, A2, are not overlapping. Area at station 1 is the outlet area of the primary nozzle and station 2 is the annulus area of the inlet of the converging mixing section. Thus, Appl. Sci. 2021, 11, x FOR PEER REVIEW 16 of 27 the total area at the inlet plane of control volume C is A1 + A2 which is also referred to as the inlet area of the converging mixing section, ACV,in.

FigureFigure 13. Forces and and momentum momentum flow ﬂow on on control control volume volume C. C.

The Newton second second law law states states that that the the sum sum of offorces forces acting acting on onan anobject object should should be be equalequal to the time rate rate of of change change of of momentum momentum of ofthat that object. object. This This statement statement transforms transforms to to thethe followingfollowing equation: equation:

. . . 𝑃𝐴 +𝑃𝐴 −𝑃𝐴 −𝑅 =𝑚 𝑣 −𝑚 𝑣 +𝑚 𝑣 (12) P1 A1 + P2 A2 − P3 A3 − Rx = m3v3 − mpv1 + msv2 (12) where where ( ) R𝑅x ==𝑃PCV,,,w,avg(𝐴A1++𝐴A 2−−𝐴A3 ) (13) (13)

InIn thisthis equation,equation,R 𝑅xand andP CV𝑃,,,w,avg are are the thex-direction x-direction of theof the reaction reaction force force and averageand staticaverage pressure static onpressure control on volume control C, respectively,volume C, respectively, done by the walldone of by the the converging wall of mixingthe section.converging It should mixing be section. noted that It thisshould pressure be noted is averaged that this (area-weighted) pressure is basedaveraged on the x(area-weighted)-direction of the based reaction on forcethe x and-directionx-direction of the of reaction the converging force and wall x-direction area. Also, of note the that . . . . mconvergingp and ms (Equation wall area. (12))Also, are note equal that to 𝑚m 1andand 𝑚m 2 (Equation(Figure 13 ),(12)) respectively. are equal to 𝑚 and 𝑚 (FigureIf it is assumed13), respectively. that the average static pressure on the wall of the converging mixing sectionIf it is is equal assumed to the that static the average pressure static at station pressure 2, then on the the wall reaction of the force converging in the xmixing-direction, Rsectionx, can is be equal written to the as staticRx = pressureP2(A1 + atA 2station− A3) 2,. Thisthen isthe illustrated reaction force in Figure in the 13 x-direction,. It should be noted𝑅, can that be writtenP2 has alsoas 𝑅 been =𝑃 assumed(𝐴 +𝐴 −𝐴 previously). This is to illustrated be equal toin theFigure stagnation 13. It should pressure be of thenoted secondary that 𝑃 hasﬂow also inlet. been Recall assumed assumption previously no. 4to where be equalv2 isto assumedthe stagna totion be pressure negligible of and, thus,the secondaryP2 is the stagnationflow inlet. Recall pressure assumption at station no. 2. 4 where 𝑣 is assumed to be negligible and, Manipulatingthus, 𝑃 is the stagnation Equation (12)pressure in conjunction at station 2. with Equations (5), (6), (13), and the Manipulating Equation (12) in conjunction with Equations (5), (6), (13), and the assumption of PCV,w,avg equals P2, the following equation, Equation (14), will be found. assumption of 𝑃,, equals 𝑃, the following equation, Equation (14), will be found. . . mpv + P A + P (A − A ) = mp(1 + ER)v + P A (14) 𝑚 1𝑣 +𝑃1 𝐴1 +𝑃2(𝐴3 − 𝐴1)=𝑚 (1 + 𝐸𝑅)𝑣 +𝑃3 𝐴3 3 (14)

After that,that, EquationEquation (15)(15) is is obtained obtained by by applying applying the the mass mass conservation conservation across across station 3station along 3 with along the with ideal the gas ideal equation gas equation of state. of state. 𝑚. (1+𝐸𝑅)𝑅𝑇 m (1 + ER)RT 𝑃𝐴 = p 3 (15) P3 A3 = 𝑣 (15) v3 If the previous equations, i.e., Equations (5) to (8), are considered in conjunction with EquationsIf the (14) previous and (15), equations, the number i.e., Equationsof unknowns (5) will to (8), be areeight considered while the innumber conjunction of with Equations (14) and (15), the number of unknowns will be eight while the number equations will be six. The unknowns are 𝐸𝑅, 𝑚 , 𝑚. , 𝑚. (or. 𝑚 ), .𝑇, 𝑣, 𝑇, and 𝑃. of equations will be six. The unknowns are ER, m , m , m (or m ), T , v , T , and P . Note that 𝑣 and 𝑃 are not included because they havep beens 4 obtained3 before4 3 through3 3 NoteEquations that v(9)1 andto (11).P1 areTherefore, not included to solve because these eight they unknowns, have been two obtained more equations before through are Equationsneeded and (9) one to of (11). them Therefore, can be obtained to solve from these control eight unknowns,volume D (Figure two more 12). equationsThe other are neededequation and is found one of based them canon the be obtaineddefinition from of Mach control number volume and D (Figurethe assumption 12). The of other knowing the value of 𝑀 (assumption no. 8). Applying energy balance on control volume D together with assumptions no. 2 and 4, Equation (16) is obtained. 𝑣 𝑐 𝑇 + 𝐸𝑅 𝑐 𝑇 = (1+𝐸𝑅) 𝑐 𝑇 + (16) 2 Using the definition of Mach number at station 3, we will get Equation (17).

𝑣 =𝑀𝛾𝑅𝑇 (17) Appl. Sci. 2021, 11, 3245 16 of 26

equation is found based on the deﬁnition of Mach number and the assumption of knowing the value of M3 (assumption no. 8). Applying energy balance on control volume D together with assumptions no. 2 and 4, Equation (16) is obtained. ! v2 c T + ER c T = (1 + ER) c T + 3 (16) p o p 2 p 3 2

Using the deﬁnition of Mach number at station 3, we will get Equation (17).

Appl. Sci. 2021, 11, x FOR PEER REVIEW p 17 of 27 v3 = M3 γRT3 (17)

Thus, now, the system of equations is closed with eight equations (Equations (5) to (8) Thus, now, the system of equations is closed with eight equations. . (Equations. . (5) to and Equations (14) to (17)) and eight unknowns (ER, m , m , m (or m ), T , v , T , and P ). (8) and Equations (14) to (17)) and eight unknowns (𝐸𝑅, 𝑚 , 𝑚 ,p 𝑚 s(or 𝑚4 ), 𝑇 , 3𝑣 , 𝑇4 , 3 3 3 Control volume E, i.e., the diffuser (Figure 12), will be used to get the last equation that and 𝑃). willControl solve forvolume pressure E, i.e., atthe the diffuser diffuser (Figure outlet 12), will (i.e., be the used ejector to get outlet),the last equationP4. Applying energy thatbalance will solve on thisfor pressure control at volume the diff alonguser outlet with (i.e., assumptions the ejector outlet), no. 2, 𝑃 4,. Applying and 7, we will obtain energyEquation balance (18). on this control volume along with assumptions no. 2, 4, and 7, we will γ−1 obtain Equation (18). 2 P3 γ v ηD 3 T4 = T4 + (18) 𝑃 P4 𝑣 𝜂 2cp 𝑇 =𝑇 + (18) 𝑃 2𝑐 Hence, by using Equation (18), we can get the value of P4 since all other parameters haveHence, been by obtained using Equation previously. (18), we Recalling can get the assumption value of 𝑃 no. since 7, all the other diffuser parameters isentropic efficiency, have been obtained previously. Recalling assumption no. 7, the diffuser isentropic ηD, is known and assumed to have a value of 0.8. efficiency, 𝜂, is known and assumed to have a value of 0.8. 5.2. The Modified One-Dimensional Model 5.2. The Modified One-Dimensional Model TheThe modified modified one-dimensional one-dimensional analytical analytical model has model the same has theset of same equations set of as equations that as that of ofthe the standard standard model model exceptexcept onon thethe convergingconverging mixing mixing sectio sectionn wall wall pressure. pressure. The The converging convergingwall pressure wall pressure (PCV,w,avg (𝑃,,) used) toused calculate to calculate the the reaction reaction force, force,R 𝑅x,, inin EquationsEquations (12) and (13) (12)is not and anymore (13) is not assumed anymore to assumed equal P to2. Instead,equal 𝑃. theInstead, “actual” the “actual” average average wall pressure wall values will pressurebe utilized. values These will be average utilized. pressure These aver valuesage pressure are obtained values fromare obtained the CFD from simulations. the To find CFDout simulations. if the converging To find wallout if pressurethe convergi hasng a wall significant pressure effecthas a significant on the one-dimensional effect on model the one-dimensional model results, these average wall pressure values from the CFD results, these average wall pressure values from the CFD results are directly substituted results are directly substituted into the equation. The flow chart of the model is shown in Figureinto the14. equation. The flow chart of the model is shown in Figure 14.

, , Equation (7)

Equation (10) and (11)

Equation (9)

, , , , , Equation (5) ,(6), (12), (13), and (15) - (17)

Note: , , (or ), , , 1. is negligible (assumption no. 4). 2. for the standard one- Equation (8) dimensional model. 3. (assumption no. 8)

Equation (18)

End

FigureFigure 14. 14. FlowchartFlowchart of the of one-dimensional the one-dimensional model. model.

6. Results and Discussion 6.1. Pressure Variation on the Wall of the Converging Mixing Section Appl. Sci. 2021, 11, 3245 17 of 26

Appl. Sci. 2021, 11, x FOR PEER REVIEW 18 of 27

6. Results and Discussion 6.1. PressureFigure 15 Variation shows onthe the pressure Wall of thevariation Converging along Mixing the wall Section of the converging mixing sectionFigure (80.7 15 ≤ shows𝑥 ≤ 110 the mm) pressure of ejector variation G1. along The thewall wall of ofthe the converging converging mixing mixing section section consists(80.7 ≤ xof≤ two110 contour mm) of ejectorpatterns. G1. OneThe is wall the of linear/straight the converging contour mixing sectionwith a consistssteeper ofslope two (80.7contour mm patterns. ≤ 𝑥 ≤ 97One mm) is and the linear/straightthe other is thecontour curved withcontour a steeper with a slope decreasing (80.7 mm slope≤ x(97≤ mm97 mm) ≤ 𝑥 and≤ 110 the mm). other is the curved contour with a decreasing slope (97 mm ≤ x ≤ 110 mm).

120 115 110 105 100 95

90 G1-sim 00 G1-sim 01 85 G1-sim 02 G1-sim 03 Static presure,abs [kPa] 80 G1-sim 04 G1-sim 05 G1-sim 06 G1-sim 07 75 G1-sim 08 G1-sim 09 70 80 85 90 95 100 105 110 x [mm] 15

[mm] 10 R

5 80 85 90 95 100 105 110 x [mm] FigureFigure 15. 15. PressurePressure variation variation along along the thewall wall of the of converging the converging mixing mixing section section (80.7 mm (80.7 ≤ x mm≤ 110≤ mm)x ≤ of110 ejector mm) G1 of ejector(top) and G1 (top)the geometrical and the geometrical contour of contour the converging of the converging mixing section mixing wall section (bottom). wall (bottom). The operatingThe operating conditions conditions of sim of sim00 to 00 09 to can 09 canbe seen be seen in Table in Table 3 3(corresponding (corresponding to to simulation simulation no. no. 0 0 to 9).

InIn general, general, all all these these pressure pressure curves curves have have the the same same pattern. pattern. They They start start to to increase increase to to reachreach a a peak peak and and then then decrease, decrease, except except for for some some curves, curves, most most notably notably the the curves curves of sim of sim 01 and01 and 02. 02.Note Note that that the the operating operating conditions conditions for for sim sim 00 00 to to 09 09 in in Figure Figure 15 15 correspondcorrespond to to simulationsimulation no. no. 0 0 to to 9 9 in in Table Table 33,, respectively.respectively. AfterAfter thethe peak,peak, thesethese twotwo curvescurves decreasedecrease relativelyrelatively a a little little and and then then increase increase again again to to finally finally decrease. decrease. Hence, Hence, there there are are two two peaks. peaks. SuchSuch a a pattern pattern also also exists, exists, but but is is less less obvious, obvious, in in sim sim 03 03 and and 05. 05. This This different different pattern pattern (the (the twotwo peaks peaks pattern) pattern) is is mainly mainly because because of of the the di differentfferent position position and and size size of of the the shock shock train train cells.cells. Figure Figure 1616 compares simulationsimulation resultsresults havinghaving one one peak peak pressure pressure variation variation (sim (sim 00) 00) to tothat that having having two two peaks peaks (sim (sim 01). 01). The The first first pressure pressure peak peak that that occurs occurs on the on convergingthe converging wall wallis due is due to the to impingementthe impingement of the of secondarythe secondary flow flow on the on wall.the wall. After After the impingement,the impingement, part partof the of fluidthe fluid flows flows into theinto ET the and ET the and remaining the remaining diverted diverted into the into steep the converging steep converging wall to wallform to a form circulation. a circulation. Regarding Regarding the second the second peak that peak exists that onexists the on other the curves other curves (ex: sim (ex: 01 simin Figure 01 in Figure15), it is 15), due it to is thedue different to the different position position and size and of the size cells of ofthe the cells shock of the train. shock The train.outer surfaceThe outer of thesurface shock of train the and shock the ejectortrain and wall the create ejector a hypothetical wall create annular a hypothetical duct that annularis continuously duct that expanding is continuously and contracting expanding along and the xcontracting-direction. Whenalong the the cross-sectional 𝑥-direction. Whenarea of the this cross-sectional annular duct area increases of this (the annular duct duct diverges), increases the flow(the duct inside diverges), it will decelerate the flow insideand, hence, it will thedecelerate wall pressure and, hence, increases the wall (see thepressure second increases pressure (see peak the in second Figure pressure 16). The peakopposite in Figure will happen 16). The if the opposite cross-sectional will happen area of if the the duct cross-sectional decreases (the area duct of converges). the duct decreases (the duct converges). A similar phenomenon is also seen in the flow of ejectors G2, G2.1, and G2.2, which will be presented next. Appl. Sci. 2021, 11, 3245 18 of 26

Appl. Sci. 2021, 11, x FOR PEER REVIEW 19 of 27 A similar phenomenon is also seen in the ﬂow of ejectors G2, G2.1, and G2.2, which will be presented next.

Figure 16. Streamlines plot (colored by pressure) started from the secondary flow inlet for ejector FigureFigure 16. 16. StreamlinesStreamlines plot plot (colored (colored by by pressure) pressure) star startedted from from the thesecondary secondary flow flow inlet inlet for ejector for ejector G1 G1 with operating conditions of sim 00 (𝑃 = 618 kPa; top), and 01 (𝑃= 584 kPa; bottom). Mach with operating conditions of sim 00 (P0= 618 kPa; top), and 01 (P0= 584 kPa; bottom). Mach number number contour lines of the primary flow for 𝑀1 (black lines) are superimposed onto it to see contour lines of the primary flow for M ≥ 1 (black lines) are superimposed onto it to see the shock the shock cells position and size. cells position and size. In the parametric study, the standard and the modified one-dimensional models were appliedIn the parametric to the three study, ejectors the standard(G2, G2.1, and and the G2.2), modified and the one-dimensional results were compared. models were wereapplied applied to the to three the three ejectors ejectors (G2, G2.1,(G2, G2.1 and, G2.2),and G2.2), and theand results the results were were compared. compared. Allthese All these ejectors were simulated at the critical conditions (𝑃 308 kPa). The results of All these ejectors were simulated at the critical conditions (𝑃 308 kPa). The results of ejectors were simulated at the critical conditions (P0 ≥ 308 kPa). The results of the modified the modified model are encouraging. From Figures 17–19, there are two important model are encouraging. From Figures 17–19, there are two important observations to point observations to point out. Firstly, looking at the curves in general, one can see that the 𝑚 observations to point out. Firstly, looking at the curves in general, one. can see that the 𝑚 out. Firstly, looking at the curves in general, one can see that the ms from CFD simulation from CFD simulation decreases as the 𝑃 increases. This means that it has a negative decreases as the P increases. This means that it has a negative slope. On the other hand, the slope. On the other0 hand, the standard model shows the opposite, i.e., a positive slope. standard model shows the opposite, i.e., a positive slope. However, the curve slope of the However, the curve slope of the modified model gets closer to that of the CFD results. modified model gets closer to that of the CFD results. This means that the one-dimensional This means that the one-dimensional model is improved. The second observation to pointmodel out is improved.is that, at Thehigh second 𝑃 , the observation modified tomodel point improves out is that, the at prediction high P0, the of modified 𝑚 point out is that, at high 𝑃 , the modified. model improves the prediction of 𝑚 significantly.model improves Interestingly, the prediction such improvement of ms significantly. is obtained Interestingly, by only a slight such change improvement in the is averageobtained pressure by only of a the slight converging change in mixing the average section pressure wall as can of the be convergingseen in Table mixing 5. As an section example,wall as can for besimulation seen in Table no. 0’5. of As ejector an example, G2, a pressure for simulation difference no. 0’of ofonly ejector 1.91% G2, leads a pressure to 31.59%difference improvement of only 1.91% on the leads one-dimensional to 31.59% improvement model prediction on the of one-dimensionalthe secondary mass model flowprediction rate. of the secondary mass flow rate.

0.016 0.015 0.014 0.013 0.012

[kg/s] 0.012 [kg/s] s

s 0.011

ṁ 0.011 ṁ 0.010 0.009 CFD 0.008 1-D model, standard 1-D model, modified 0.007 1-D model, modified 0.006 250 300 350 400 450 500 550 600 650 P0 [kPa] P0 [kPa] FigureFigure 17. 17. TheThe secondary secondary mass mass flow ﬂow rates rates of ejector of ejector G2 obtained G2 obtained from from various various methods methods for different for different primaryprimary pressures. pressures. Appl.Appl. Sci. Sci. 20212021,, 1111,, x 3245 FOR PEER REVIEW 2019 of of 27 26 Appl. Sci. 2021, 11, x FOR PEER REVIEW 20 of 27

0.016 0.015 0.014 0.013 0.012 [kg/s] [kg/s] s 0.011 s 0.011 ṁ ṁ 0.010 0.009 0.009 CFD 0.008 0.008 1-D model, standard 0.007 1-D model, modified 0.006 250 300 350 400 450 500 550 600 650 P0 [kPa] P0 [kPa] FigureFigure 18. 18. TheThe secondary secondary mass mass flow ﬂow rates of ejector G2.1G2.1 obtainedobtained fromfrom various various methods methods for for different differentprimary primary pressures. pressures.

0.016 0.015 0.014 0.013 0.012 [kg/s] [kg/s] s 0.011 s 0.011 ṁ ṁ 0.010 0.009 0.009 CFD 0.008 1-D model, standard 0.007 1-D model, modified 0.006 250 300 350 400 450 500 550 600 650 P0 [kPa] P0 [kPa] Figure 19. The secondary mass flow rates of ejector G2.2 obtained from various methods for FigureFigure 19. 19. TheThe secondary secondary mass mass flow ﬂow rates of ejectorejector G2.2G2.2 obtainedobtained fromfrom various various methods methods for for different different primary pressures. differentprimary primary pressures. pressures.

Table 5. The discrepancy (in percent) between the. 𝑚 obtained from the one-dimensional analytical models (std and TableTable 5. The discrepancydiscrepancy (in(in percent) percent) between between the them 𝑚obtained obtained from from the the one-dimensional one-dimensional analytical analytical models models (std and(std mod)and mod) and CFD for various ejectors. s mod)and CFD and forCFD various for various ejectors. ejectors. Ejector G2 Ejector G2.1 Ejector G2.2 EjectorEjector G2 G2 Ejector Ejector G2.1 G2.1 Ejector Ejector G2.2 G2.2 Sim no. 𝑷 [kPa] ∆𝒎 2 [%] ∆𝒎 2 [%] ∆𝒎 2 [%] Sim no. 𝑷𝟎 [kPa] 1 ∆𝒎 𝒔,𝟏𝑫. 2 [%] 1 ∆𝒎 𝒔,𝟏𝑫. 2 [%] 1 ∆𝒎. 𝒔,𝟏𝑫 2 [%] 𝟎 ∆𝑷𝑪𝑽 1 [%]1 𝒔,𝟏𝑫 2 ∆𝑷𝑪𝑽 1 [%]1 𝒔,𝟏𝑫 2 ∆𝑷𝑪𝑽 1 [%]1 𝒔,𝟏𝑫2 Sim no. P [kPa]∆𝑷 ∆ P [%] ms,1D [%] ∆𝑷 ∆ P [%] ms,1D [%] ∆𝑷∆P [%] ms,1D [%] 0 𝑪𝑽 CV 1-D, std 1-D, mod 𝑪𝑽 CV 1-D, std 1-D, mod 𝑪𝑽 CV 1-D, std 1-D, mod [%] 1-D, std 1-D, mod[%] 1-D, std 1-D, mod[%] 1-D, std 1-D, mod 0’ 618 1.91 43.43 11.84 2.85 41.03 12.62 3.87 30.47 13.41 1’ 0’584 618 1.61 1.91 27.86 43.43 2.91 11.84 2.32 2.85 24.82 41.03 3.30 12.62 2.96 3.87 30.47 17.20 13.414.90 2’ 1’549 584 1.19 1.61 9.39 27.86 −7.19 2.91 1.62 2.32 6.42 24.82 −7.00 3.30 1.79 2.96 17.20 1.89 4.90−4.89 2’ 2’549 549 1.19 1.19 9.39 9.39 −7.19−7.19 1.62 1.62 6.42 6.42 −7.00−7.00 1.79 1.79 1.89 1.89 −−4.894.89 3’ 515 1.11 2.77 −12.57 1.49 −0.08 −12.39 1.11 −8.51 −12.54 3’ 3’515 515 1.11 1.11 2.77 2.77 −12.57−12.57 1.49 1.49 −0.08−0.08 −12.39−12.39 1.11 1.11 −−8.518.51 −−12.5412.54 4’ 4’480 480 1.01 1.01 −2.45−2.45 −16.53−16.53 1.34 1.34 −4.93−4.93 −16.11−16.11 1.33 1.33 −−10.7910.79 −−15.8115.81 − − − − − − 5’ 5’446 446 0.64 0.64 −14.5914.59 −22.9022.90 0.72 0.72 −16.4916.49 −22.0922.09 0.39 0.39 −19.5719.57 −20.9720.97 6’ 6’412 412 0.42 0.42 −22.28−22.28 −27.60−27.60 0.33 0.33 −24.69−24.69 −27.19−27.19 −−0.840.84 −−29.1829.18 −−26.4026.40 6’ 7’412 377 0.42 0.13 −22.28−30.00 −27.60−31.54 0.33−0.08 −24.69−31.61 −27.19−31.03 −−0.841.22 −−34.3029.18 −−30.2326.40 7’ 8’377 343 0.13−0.06 −30.00−36.01 −31.54−35.29 −0.08−0.36 −31.61−37.28 −31.03−34.73 −−1.221.72 −−39.5534.30 −−33.8830.23 8’ 9’343 308 −0.06−0.28 −36.01−42.85 −35.29−39.55 −0.36−0.70 −37.28−43.91 −34.73−39.02 −−1.722.25 −−45.5639.55 −−38.2833.88 9’ 1 308 −0.28 −42.85 −39.55 −0.70 −43.91 −39.02 −2.25 −45.56 −38.28 9’ ∆PCV =308P CV,w,avg −−0.28P2 × 100/P−242.85[%]; where −P39.552 = 100.8 kPa− and0.70P CV,w,avg−43.91is the average −39.02 wall pressure − of2.25 the converging−45.56 mixing section−38.28 1 2 . . . . . . 1obtained ∆𝑃 =(𝑃 from,, CFD. −𝑃∆ms,1)D ×= 100ms⁄,1 𝑃D− [%]ms CFD; where× 100 /𝑃m s =CFD 100.8[%]; mkPas,1D andandm s𝑃CFD,,are the is secondary the average mass ﬂow wall rates pressure obtained fromof the the ∆𝑃 =(𝑃,, −𝑃) × 100⁄ 𝑃 [%], ; where 𝑃 ,= 100.8 kPa and ,𝑃,, is the average wall pressure of the 2 convergingone-dimensional mixing model sectio andn CFD, obtained respectively. from CFD. 2 ∆𝑚 , =(𝑚 , −𝑚 ,) × 100⁄ 𝑚 , [%]; 𝑚 , and 𝑚 , are the converging mixing section obtained from CFD. ∆𝑚 , =(𝑚 , −𝑚 ,) × 100⁄ 𝑚 , [%]; 𝑚 , and 𝑚 , are the secondary mass flow rates obtained from the one-dimensional model and CFD, respectively. Appl. Sci. 2021, 11, x FOR PEER REVIEW 21 of 27

The fact that there is only a slight improvement at lower 𝑃 leads to the following conclusions. Firstly, the pressure variation on the converging wall is too small which means that the average (area-weighted) pressure is close to 𝑃. This can be seen in Table 5 above. Secondly, it means that other phenomena, which are beyond the scope of this research, play a more significant role in this pressure region. Despite the slight improvement in the lower 𝑃 region, it is important to note that the one-dimensional prediction of 𝑚 is very sensitive to the change of the converging wall pressure, regardless what the 𝑃 values are. Figure 20 helps to explain the statement. In Figure 20, ∆𝑃 has the same definition as that in Table 5. The 𝑚 ,, is defined as 𝑚 ,, −𝑚 ,, × 100 𝑚 , in percent where 𝑚 , is the Appl. Sci. 2021, 11, 3245 secondary mass flow rate obtained from CFD, and 𝑚 ,, and 𝑚 ,, are the 20 of 26 secondary mass flow rates obtained from the standard and modified one-dimensional models, respectively. In this figure, it can be concluded that the least sensitive ejector to wall pressure change is ejector G2.2. Yet, it is still something that cannot be ignored even at lower pressures. As an example, for the lowest 𝑃 (308 kPa), when the converging wall The fact that there is only a slight improvement at lower P0 leads to the following pressureconclusions. decreases Firstly, by the2.25%, pressure the predicted variation mass on flow the convergingrate increases wall by is7.28%. too small The most which means sensitive to the wall pressure change is ejector G2. At the lowest 𝑃 (308 kPa), an only that the average (area-weighted) pressure is close to P2. This can be seen in Table5 above. 0.28% decrease in the wall pressure results in 3.30% increase in the predicted mass flow Secondly, it means that other phenomena, which are beyond the scope of this research, play rate. a more signiﬁcant role in this pressure region. Despite the slight improvement in the lower These results suggest that, instead of simply assuming 𝑃,, to be equal. to 𝑃, a goodP0 region, estimate it is of important the pressure to noteshould that be the found one-dimensional and utilized. This prediction finding of alsoms isproves very sensitive whatto the has change been suggested of the converging by Little walland Garime pressure,lla regardless[1], i.e., more what local the flowP0 valuesphenomena are. Figure 20 shouldhelps tobe explainincluded the in the statement. one-dimensional model to improve it further.

10 Lower 5

0 -3-2-1012345 -5 [%] -10

G2 -15 s,1D,mod-std

ṁ G2.1 -20 Δ G2.2 -25 At the -30 highest (618 kPa) -35 ΔPCV [%] FigureFigure 20. 20. TheThe effect effect of the of theconverging converging wall pressure wall pressure change changeon the change on the of change the secondary of the secondarymass mass flowflow rate rate obtained obtained from from the theone-dimensional one-dimensional model. model. Figures 21 and 22 are presented to show how the pressure varies along. the In Figure 20, ∆PCV has the same definition as that in Table5. The ms,1D,mod−std is converging mixing . section wall. for ejector G2 and. G2.1, respectively. Ejector. G2.2 results defined as ms D mod − ms D std × 100/ms,CFD in percent where ms,CFD is the secondary are similar; hence,,1 they, are not,1 presented, here.. Like what. has been observed in ejector G1,mass the flow pressure rate obtainedstarts to vary from when CFD, the and prmimarys,1D,std flowand jetm sgets,1D,mod closerare to the the secondary wall. The mass flow pressurerates obtained initially fromincreases the standardto reach a and peak modified and then one-dimensional decreases as the flow models, is about respectively. to In enterthis figure,the throat. it can This be is concludedbecause the secondar that they least flow sensitiveimpinges ejectorthe converging to wall wall pressure as can change is beejector seen in G2.2. Figure Yet, 23, it hence is still the something peak in the that pressure cannot variation be ignored curve. evenAround at lowerthis region, pressures. As thean pressure example, is forat maximum. the lowest TheP0 (308impinging kPa), whenflow is thethen converging divided into wall two pressure streams; decreasesone by 2.25%, the predicted mass flow rate increases by 7.28%. The most sensitive to the wall pressure change is ejector G2. At the lowest P0 (308 kPa), an only 0.28% decrease in the wall pressure results in 3.30% increase in the predicted mass flow rate. These results suggest that, instead of simply assuming PCV,w,avg to be equal to P2, a good estimate of the pressure should be found and utilized. This finding also proves what has been suggested by Little and Garimella [1], i.e., more local flow phenomena should be included in the one-dimensional model to improve it further. Figures 21 and 22 are presented to show how the pressure varies along the converging mixing section wall for ejector G2 and G2.1, respectively. Ejector G2.2 results are similar; hence, they are not presented here. Like what has been observed in ejector G1, the pressure starts to vary when the primary flow jet gets closer to the wall. The pressure initially increases to reach a peak and then decreases as the flow is about to enter the throat. This is because the secondary flow impinges the converging wall as can be seen in Figure 23, hence the peak in the pressure variation curve. Around this region, the pressure is at maximum. The impinging flow is then divided into two streams; one enters the ET and the other stream diverts to the converging wall and forms circulation. Such flow circulation has been observed by Han et al. [25] in their CFD simulations. Appl. Sci. 2021, 11, x FOR PEER REVIEW 22 of 27

enters the ET and the other stream diverts to the converging wall and forms circulation. Such flow circulation has been observed by Han et al. [25] in their CFD simulations. Paying attention again to Figures 21 and 22, one can see that the curve peak (“mountain” curve size) decreases gradually as the 𝑃 decreases. It is also observed that the curve pattern for simulation no. 4’ (sim 04’) is rather different, i.e., the decrease in wall pressure after the peak is not as steep as the others. The explanation for these two phenomena (the gradual decrease of the “mountain” curve size and “anomaly” of sim 04’ curve) is as follows in conjunction with Figure 24. At higher 𝑃, the secondary flow has higher momentum as it enters the ejector. This is because the shear effect between the two flows (primary and secondary) gets stronger. However, due to the larger diameter of the shock cell, the secondary flow will not be able to directly enter the ejector throat. Rather, it will impinge the converging mixing section wall before the flow divided into two streams (Figure 23) as explained before. Thus, the dynamic pressure (kinetic energy) will be converted to pressure as the flow hits the wall. The higher the momentum of the secondary flow is, the higher the pressure at the converging wall will be, and vice versa. Appl. Sci. 2021, 11, 3245 21 of 26 This explains the gradual decrease in the “mountain” size in the two figures as the 𝑃 decreases.

125 120 115 110 105 100 95 G2-sim 00' G2-sim 01' 90 G2-sim 02' G2-sim 03' 85 G2-sim 04' G2-sim 05' 80 G2-sim 06' G2-sim 07' Static presure,abs [kPa] 75 G2-sim 08' G2-sim 09' 70 70 75 80 85 90 95 100 x [mm] 20

[mm] 10 R

5 70 75 80 85 90 95 100 x [mm] FigureFigure 21. 21. PressurePressure variation variation along along the the wall wall of of the the converging converging mixing mixing section section (72.9 (72.9 ≤ ≤x ≤x 99.259≤ 99.259 mm) mm) of Appl. Sci. 2021, 11, x FOR PEER REVIEW 23 of 27 ejectorof ejector G2 G2(top) (top) and and the thegeometrical geometrical contour contour of the of converging the converging mixing mixing section section wall wall(bottom). (bottom). The The operatingoperating conditions conditions of of sim sim 00’ 00’ to to 09’ 09’ can can be be seen seen in in Table Table 55 (correspond (correspond to to sim sim no. no. 0’ 0’to to 9’). 9’).

125 120 115 110 105 100 95 G2.1-sim 00' G2.1-sim 01' G2.1-sim 02' G2.1-sim 03' 90 G2.1-sim 04' G2.1-sim 05' 85

Static presure, abs [kPa] abs presure, Static G2.1-sim 06' G2.1-sim 07' 80 G2.1-sim 08' G2.1-sim 09' 75 70 75 80 85 90 95 100 x [mm] 15

10 [mm] R 5 70 75 80 85 90 95 100 x [mm] FigureFigure 22. 22. PressurePressure variation variation along along the the wall wall of of the the converging converging mixing mixing section section (72.9 (72.9 ≤ ≤x ≤x 99.259≤ 99.259 mm) mm) of ejectorof ejector G2.1 G2.1 (top) (top) and and the thegeometrical geometrical contour contour of the of theconverging converging mixing mixing section section wall wall (bottom). (bottom). The The operatingoperating conditions conditions of of sim sim 00’ 00’ to to 09’ 09’ can can be be seen seen in in Table Table 55 (correspond(correspond toto simsim no.no. 0’0’ toto 9’).9’).

Figure 23. Streamlines plot (colored by pressure) started from the secondary flow inlet for ejector G2 with operating condition of simulation no. 0’ (Table 5), i.e., 𝑃 = 618 kPa.

Regarding the “anomaly” of simulation no. 4’ (sim 04’), it is related to the shock cells position and size inside the ejector, particularly in this case, inside the converging mixing section. Looking at sim 04’ in Figure 24, one can see that the end of one of the shock cells is located just before the ET inlet. Thus, the hypothetical annular duct formed is diverging. This will hinder the flow to accelerate. The decrease in the flow acceleration shows up in the pressure curve of sim 04’ as a less steep pressure drop after the peak. Observing the Mach contour plot of the other simulations in this figure, it is clearly seen that the hypothetical annular duct near the ET inlet is converging and, therefore, the pressure drop in the curve is steeper because the flow has higher acceleration. Appl. Sci. 2021, 11, x FOR PEER REVIEW 23 of 27

125 120 115 110 105 100 95 G2.1-sim 00' G2.1-sim 01' G2.1-sim 02' G2.1-sim 03' 90 G2.1-sim 04' G2.1-sim 05' 85

Static presure, abs [kPa] abs presure, Static G2.1-sim 06' G2.1-sim 07' 80 G2.1-sim 08' G2.1-sim 09' 75 70 75 80 85 90 95 100 x [mm] 15

10 [mm] R 5 70 75 80 85 90 95 100 x [mm]

Appl. Sci. 2021, 11, 3245 Figure 22. Pressure variation along the wall of the converging mixing section (72.9 ≤ x ≤ 99.259 22mm) of 26 of ejector G2.1 (top) and the geometrical contour of the converging mixing section wall (bottom). The operating conditions of sim 00’ to 09’ can be seen in Table 5 (correspond to sim no. 0’ to 9’).

FigureFigure 23.23. StreamlinesStreamlines plotplot (colored(colored by by pressure) pressure) started started from from the the secondary secondary ﬂow flow inlet inlet for for ejector ejector G2 = withG2 with operating operating condition condition of simulation of simulation no. 0’no. (Table 0’ (Table5), i.e., 5), Pi.e.,0 𝑃618 = kPa.618 kPa.

PayingRegarding attention the “anomaly” again to Figures of simulation 21 and no.22, 4’ one (sim can 04’), see thatit is related the curve to the peak shock (“moun- cells tain”position curve and size) size decreases inside the gradually ejector, particular as the P0 lydecreases. in this case, It is inside also observed the converging that the mixing curve patternsection. for Looking simulation at sim no. 04’ 4’ in (sim Figure 04’) 24, is rather one can different, see that i.e., the the end decrease of one of in the wall shock pressure cells afteris located the peak just is before not as steepthe ET as inlet. the others. Thus, Thethe explanationhypothetical for annular these twoduct phenomena formed is (the gradual decrease of the “mountain” curve size and “anomaly” of sim 04’ curve) is as diverging. This will hinder the flow to accelerate. The decrease in the flow acceleration follows in conjunction with Figure 24. At higher P , the secondary flow has higher momen- shows up in the pressure curve of sim 04’ as a 0less steep pressure drop after the peak. tum as it enters the ejector. This is because the shear effect between the two flows (primary Observing the Mach contour plot of the other simulations in this figure, it is clearly seen and secondary) gets stronger. However, due to the larger diameter of the shock cell, the that the hypothetical annular duct near the ET inlet is converging and, therefore, the secondary flow will not be able to directly enter the ejector throat. Rather, it will impinge pressure drop in the curve is steeper because the flow has higher acceleration. the converging mixing section wall before the flow divided into two streams (Figure 23) as explained before. Thus, the dynamic pressure (kinetic energy) will be converted to pressure as the flow hits the wall. The higher the momentum of the secondary flow is, the higher the Appl. Sci. 2021, 11, x FOR PEER REVIEW 24 of 27 pressure at the converging wall will be, and vice versa. This explains the gradual decrease in the “mountain” size in the two figures as the P0 decreases.

FigureFigure 24. 24. MachMach number number contour contour plot plot (M (M ≥ 1)1) of of CFD CFD results results of ejector G2.1.G2.1. TheTheoperating operating conditions conditionsof sim 00’ of to sim 09’ can00’ to be 09’ seen can in be Table seen5 (correspondin Table 5 (correspond to sim no. 0’to tosim 9’). no. 0’ to 9’).

6.2. Secondary Mass flow Rate Comparison

The effect of reducing the radius of the converging mixing section inlet, 𝑅,, is analyzed. Reducing this radius means that the angle of the converging mixing section is reduced. The angles for ejector G2, G2.1, and G2.2 are 22.94°, 16.60°, and 9.60°, respectively. The CFD results show that the secondary mass flow rate increases as the radius decreases or, in other words, as the angle decreases. This trend can be seen in Table 6. The finding agrees with that of Jeong et al. [32], however, they have found an optimum angle which will result in a maximum 𝑚 . Thus, the 𝑚 keeps increasing as the angle decreases until it reaches a certain angle (the optimum angle). A further decrease in the angle will decrease the 𝑚 . In the current work, such an optimum angle of converging mixing section has not been found because, most likely, the range of the angle variation is not wide enough. Jeong et al. [32] concluded that the optimum angle is 1° while the ejector considered in the current research has a minimum angle of 9.60°. Moreover, the dimension of the other parts of their ejector is different which may cause the difference in the optimum angle. Zhu et al. [18] have also found an optimum converging mixing section angle. Their results show that the optimum angle cannot be generalized to meet all operating conditions. Therefore, they suggested that research to develop an adjustable ejector should be conducted, especially an ejector with an adjustable primary nozzle exit position and adjustable converging mixing section angle.

Appl. Sci. 2021, 11, 3245 23 of 26

Regarding the “anomaly” of simulation no. 4’ (sim 04’), it is related to the shock cells position and size inside the ejector, particularly in this case, inside the converging mixing section. Looking at sim 04’ in Figure 24, one can see that the end of one of the shock cells is located just before the ET inlet. Thus, the hypothetical annular duct formed is diverging. This will hinder the flow to accelerate. The decrease in the flow acceleration shows up in the pressure curve of sim 04’ as a less steep pressure drop after the peak. Observing the Mach contour plot of the other simulations in this figure, it is clearly seen that the hypothetical annular duct near the ET inlet is converging and, therefore, the pressure drop in the curve is steeper because the flow has higher acceleration.

6.2. Secondary Mass Flow Rate Comparison

The effect of reducing the radius of the converging mixing section inlet, RCV,in, is analyzed. Reducing this radius means that the angle of the converging mixing section is reduced. The angles for ejector G2, G2.1, and G2.2 are 22.94◦, 16.60◦, and 9.60◦, respectively. The CFD results show that the secondary mass ﬂow rate increases as the radius decreases or, in other words, as the angle decreases. This trend can be seen in Table6. The ﬁnding agrees with that of Jeong et al. [30], however, they have found an optimum angle . . which will result in a maximum ms. Thus, the ms keeps increasing as the angle decreases until it reaches a certain angle (the optimum angle). A further decrease in the angle will . decrease the ms. In the current work, such an optimum angle of converging mixing section has not been found because, most likely, the range of the angle variation is not wide enough. Jeong et al. [30] concluded that the optimum angle is 1◦ while the ejector considered in the current research has a minimum angle of 9.60◦. Moreover, the dimension of the other parts of their ejector is different which may cause the difference in the optimum angle. Zhu et al. [18] have also found an optimum converging mixing section angle. Their results show that the optimum angle cannot be generalized to meet all operating conditions. Therefore, they suggested that research to develop an adjustable ejector should be conducted, especially an ejector with an adjustable primary nozzle exit position and adjustable converging mixing section angle.

Table 6. Comparison of secondary mass ﬂow rates from CFD simulation for ejectors having different inlet radius/converging angle of the converging mixing section.

. . Secondary Mass Flow Rate, ms [kg/s] Increase of ms Relative to that of Ejector G2 [%] Sim. no. P0 [kPa] Ejector G2 Ejector G2.1 Ejector G2.2 Ejector G2.1 Ejector G2.2 0’ 618 0.00997 0.01014 0.01096 1.71 9.93 1’ 584 0.01066 0.01092 0.01163 2.44 9.10 2’ 549 0.01182 0.01215 0.01269 2.79 7.36 3’ 515 0.01193 0.01227 0.01340 2.85 12.32 4’ 480 0.01186 0.01217 0.01297 2.61 9.36 5’ 446 0.01275 0.01304 0.01354 2.27 6.20 6’ 412 0.01315 0.01357 0.01443 3.19 9.73 7’ 377 0.01360 0.01392 0.01449 2.35 6.54 8’ 343 0.01383 0.01411 0.01464 2.02 5.86 9’ 308 0.01426 0.01453 0.01497 1.89 4.98

7. Conclusions A research on supersonic ejectors for refrigeration application using computational fluid dynamics (CFD) and one-dimensional analytical models has been conducted. Several ejectors with varying radius of converging mixing section inlet were studied. The objective was to show that the ejector one-dimensional analytical model can be improved by considering wall pressure variation in the converging mixing section. Typically, the pressure on this wall is assumed to be constant and equal to the secondary flow inlet pressure. Two turbulence models, i.e., k-omega SST and transition SST, have been tested to find which Appl. Sci. 2021, 11, 3245 24 of 26

one gives the closest results to the experimental data. The predicted secondary mass flow . rates, ms, obtained from the one-dimensional model have been compared before and after incorporating the above-mentioned pressure variation into the model. The conclusions of this work are summarized as follows: 1. In the CFD analysis, the transition SST turbulence model is better than the k-omega . SST model in terms of how close the predicted ms is to the experimental data. Most of the transition SST model results are within the experimental uncertainty of Al- Ansary’s work [9], i.e., no greater than 4%. On the other hand, many of the errors of k-omega SST results are ranging from 7% to 9%. These results are obtained when the ejector is operating in the critical region. 2. The authors have found that the effect of wall pressure variation in the converging mixing section is actually prominent in the prediction of the one-dimensional model. Even a small amount of change in the converging wall pressure can result in a . significant improvement of the model prediction of ms, especially at high P0. However, . regardless of the operating conditions, the sensitivity of ms is high with respect to the change of the converging wall pressure, i.e., a small change in the pressure can result . in a high change in ms. It has also been found that, in general, the inclusion of the . wall pressure variation will improve the trend (slope) of the predicted ms. Based on these findings, this wall pressure variation should be considered in future modeling of ejectors. 3. The pressure increase on the wall of the converging mixing section is due to the impingement of the secondary flow on that wall. This happens because the shock train of the primary jet partially blocks the secondary flow as it tries to enter the ET. On the other hand, the pressure decrease at the end of that converging wall is due to the acceleration of the secondary flow as it enters the ET. The effects of these two phenomena compete each other that result in the increase or decrease of the average wall pressure. Moreover, we have also found that the behaviors of these two phenomena are linked to the primary jet speed, and the cell size and position of the shock train. 4. The CFD results show that the secondary mass flow rate increases as the angle of the converging mixing section decreases. However, due to the limited range of angles simulated, the current results have not verified the existence of an optimum angle value that has been observed by other researchers. The future work will be an effort to compare other different ejector geometries and to obtain a mathematical relation that links the operating conditions (P0, P2, and P4) and ejector geometry to the pressure of the converging mixing section wall. Moreover, other local flow phenomena should be analyzed to find out which ones that have significant effects on the one-dimensional model.

Author Contributions: Conceptualization, H.A.-A.; methodology, S.S., H.A.-A. and E.D.; software, E.D. and S.S.; validation, E.D., S.S. and S.A.; investigation, E.D., S.S. and S.A.; writing—original draft preparation, E.D.; writing—review and editing, S.S., H.A.-A. and E.D.; supervision, H.A.-A., S.S. and J.O. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by “The Deanship of Scientiﬁc Research at King Saud University, through research group no. RG-1440-103”, which is truly appreciated. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: The authors thank the Deanship of Scientiﬁc Research and RSSU at King Saud University for their technical support. Appl. Sci. 2021, 11, 3245 25 of 26

Conﬂicts of Interest: The authors declare no conﬂict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

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