{"http:\/\/dx.doi.org\/10.14288\/1.0401094":{"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool":[{"value":"Applied Science, Faculty of","type":"literal","lang":"en"},{"value":"Engineering, School of (Okanagan)","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider":[{"value":"DSpace","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeCampus":[{"value":"UBCO","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/creator":[{"value":"Sisodiya, Deepanshi","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/issued":[{"value":"2021-07-29T15:37:33Z","type":"literal","lang":"en"},{"value":"2021","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#relatedDegree":[{"value":"Master of Applied Science - MASc","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeGrantor":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/description":[{"value":"Supersonic air ejectors are mechanical devices in which a supersonic primary air stream entrains\r\na secondary air stream by creating suction. Because of their ability to use low-grade energy\r\nfor their operation and the absence of moving parts, these ejectors are used in various sectors\r\nsuch as refrigeration and fluid transportation. The ejectors' performance is quantified using the\r\nentrainment ratio, which is the amount of secondary mass flow rate entrained per unit primary\r\nmass flow rate. Though many research studies evaluated the impact of operating conditions and\r\ngeometrical configuration on the ejector entrainment ratio, few studies characterized the loss\r\ngeneration in these ejectors. Although these studies attributed viscous dissipation as the major\r\nirreversibility contributing mechanism, there is limited information about the turbulence driven\r\nirreversibilities, the influence of eddy viscosity on mean flow entropy generation, and impact of\r\nthe shock strengths and shear layers on entropy generation. First, a one-dimensional analytical\r\nmodel is developed in the present study to estimate the entrainment ratio and the overall characteristics of the flow parameters within the ejector. This model is subject to some assumptions and cannot predict the flow intricacies in-depth. For detailed investigations, two-dimensional unsteady Reynolds-averaged Navier-Stokes simulations are performed to quantify entropy generation in a supersonic air ejector operating at different stagnation pressure ratios for a range of area ratios subject to two different mixing chamber geometries, namely, constant-pressure and constant-area mixing chambers. Local entropy generation is dominated by viscous dissipation in the flow at locations corresponding to flow separation zones, shear layer instabilities, recirculation zones, and shock structures. The results show that the influences of the turbulence and gas dynamics on the ejector irreversibilities are significant; however, the latter dominates. In\r\nthis thesis, several designs are proposed in which various arrangements of inclined plates are\r\ninserted into the diffuser aiming to reduce ejector entropy generation. These variants impact\r\nthe diffuser shock structure, shock location, and ejector choking conditions. The results show\r\nthat, at a stagnation pressure ratio of 3.5, an ejector modified with a single-inclined plate may\r\nreduce the entropy generation by 9% relative to its unmodified conditions.","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO":[{"value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/79140?expand=metadata","type":"literal","lang":"en"}],"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note":[{"value":"Entrainment Ratio and EntropyGeneration of Vacuum-ObjectiveSupersonic EjectorsbyDeepanshi SisodiyaB.Tech., National Institute of Technology - Surat, 2017A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE COLLEGE OF GRADUATE STUDIES(Mechanical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)July 2021\u00a9 Deepanshi Sisodiya, 2021The following individuals certify that they have read, and recommend to the College ofGraduate Studies for acceptance, a thesis\/dissertation entitled:Entrainment Ratio and Entropy Generation of Vacuum-Objective Supersonic Ejectorssubmitted by Deepanshi Sisodiya in partial fulfillment of the requirements of the degree ofMaster of Applied ScienceDr. Sina Kheirkhah, UBC, School of EngineeringSupervisorDr. Joshua Brinkerhoff, UBC, School of EngineeringCo-supervisorDr. Sunny Li, UBC, School of EngineeringSupervisory Committee MemberDr. Kevin Golovin, UBC, School of EngineeringSupervisory Committee MemberDr. Liwei Wang, UBC, School of EngineeringUniversity ExamineriiAbstractSupersonic air ejectors are mechanical devices in which a supersonic primary air stream en-trains a secondary air stream by creating suction. Because of their ability to use low-grade energyfor their operation and the absence of moving parts, these ejectors are used in various sectorssuch as refrigeration and fluid transportation. The ejectors\u2019 performance is quantified using theentrainment ratio, which is the amount of secondary mass flow rate entrained per unit primarymass flow rate. Though many research studies evaluated the impact of operating conditions andgeometrical configuration on the ejector entrainment ratio, few studies characterized the lossgeneration in these ejectors. Although these studies attributed viscous dissipation as the majorirreversibility contributing mechanism, there is limited information about the turbulence drivenirreversibilities, the influence of eddy viscosity on mean flow entropy generation, and impact ofthe shock strengths and shear layers on entropy generation. First, a one-dimensional analyticalmodel is developed in the present study to estimate the entrainment ratio and the overall char-acteristics of the flow parameters within the ejector. This model is subject to some assumptionsand cannot predict the flow intricacies in-depth. For detailed investigations, two-dimensionalunsteady Reynolds-averaged Navier-Stokes simulations are performed to quantify entropy gen-eration in a supersonic air ejector operating at different stagnation pressure ratios for a range ofarea ratios subject to two different mixing chamber geometries, namely, constant-pressure andconstant-area mixing chambers. Local entropy generation is dominated by viscous dissipationiiiAbstractin the flow at locations corresponding to flow separation zones, shear layer instabilities, recircu-lation zones, and shock structures. The results show that the influences of the turbulence andgas dynamics on the ejector irreversibilities are significant; however, the latter dominates. Inthis thesis, several designs are proposed in which various arrangements of inclined plates areinserted into the diffuser aiming to reduce ejector entropy generation. These variants impactthe diffuser shock structure, shock location, and ejector choking conditions. The results showthat, at a stagnation pressure ratio of 3.5, an ejector modified with a single-inclined plate mayreduce the entropy generation by 9% relative to its unmodified conditions.ivLay SummaryThe present study focuses on characterizing the losses because of fluid friction and tempera-ture difference between the mixing fluids and using the related insight to suggest improvementsto the design of vacuum-objective supersonic ejectors. Because of their ability to use low-gradeenergy and the absence of moving parts, these supersonic ejectors are used in various sectors ofengineering, such as refrigeration, high altitude testing, transportation, and desalination plants.Because of their versatility, design improvements will benefit various industries. This thesis con-ducts a series of analytical and numerical flow simulations to quantify the losses and identify theindividual contributions of the three principal parts of the ejector, namely, the primary nozzle,mixing chamber, and diffuser. For high pressure applications, the diffuser contributes more than50% of the losses. Hence, new diffuser designs are proposed that yield upto a 9% reduction inthe total ejector loss.vPrefaceThe research presented in this thesis was conducted by the author under the supervision ofProfessor Sina Kheirkhah and co-supervision of Professor Joshua Brinkerhoff in the Computa-tional Fluid Dynamics Laboratory at University of British Columbia. Parts of this thesis werepresented in the Annual Meeting of the APS Division of Fluid Dynamics in November 2020.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiNomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxixDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxxiiChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Ejector terminologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Research objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6viiTABLE OF CONTENTSChapter 2: Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Effect of operating conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.1 Stagnation pressure ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.2 Back pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.3 Primary stream temperature . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Effect of geometrical specifications . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.1 Mixing chamber geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 Nozzle exit position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.3 Suction chamber angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.4 Area ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Entropy generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.1 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.2 Ejector flow irreversiblities . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4 Computational modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4.1 1D analytical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4.2 2D and 3D computational models . . . . . . . . . . . . . . . . . . . . . . 182.4.3 Turbulence model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.5 Optimization techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.5.1 Supersonic diffuser designs . . . . . . . . . . . . . . . . . . . . . . . . . . 202.5.2 Flow over a wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.6 Literature Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Chapter 3: 2D Computational Methods . . . . . . . . . . . . . . . . . . . . . . . 253.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25viiiTABLE OF CONTENTS3.2 Flux schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.1 Discretization of convective terms . . . . . . . . . . . . . . . . . . . . . . 273.2.2 Discretization of gradient terms . . . . . . . . . . . . . . . . . . . . . . . . 303.2.3 Discretization of Laplacian terms . . . . . . . . . . . . . . . . . . . . . . . 313.3 Time integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.4 Solution approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.5 Solver verification in literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.6 Solver settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.7 Turbulence modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.8 Computational domain and boundary conditions . . . . . . . . . . . . . . . . . . 353.9 Spatial mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.10 Validation study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.11 Simulated cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Chapter 4: 1D Analytical Model of the Ejector . . . . . . . . . . . . . . . . . . . 444.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.2.1 Flow through the primary nozzle . . . . . . . . . . . . . . . . . . . . . . . 464.2.2 Secondary and primary flow through the mixing chamber . . . . . . . . . 484.2.3 Mixed flow upstream of the shock . . . . . . . . . . . . . . . . . . . . . . 494.2.4 Mixed flow downstream of the shock . . . . . . . . . . . . . . . . . . . . . 514.2.5 Solution method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Chapter 5: Characterization of the Ejector Performance . . . . . . . . . . . . . 59ixTABLE OF CONTENTS5.1 General performance parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2 Ejector internal flow features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.2.1 Constant-pressure mixing ejector . . . . . . . . . . . . . . . . . . . . . . . 635.2.2 Constant-area mixing ejector . . . . . . . . . . . . . . . . . . . . . . . . . 695.3 Entropy generation analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.3.1 Constant-pressure mixing ejector . . . . . . . . . . . . . . . . . . . . . . . 735.3.2 Constant-area mixing ejector . . . . . . . . . . . . . . . . . . . . . . . . . 79Chapter 6: Diffuser Design Modifications to Reduce Entropy Generation . . . 846.1 Design parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.2 Overall ejector performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.3 Spatial variation of the entropy generation and the flow features . . . . . . . . . 896.4 Influence of the design parameters on the ejector performance . . . . . . . . . . . 93Chapter 7: Conclusions and Future Steps . . . . . . . . . . . . . . . . . . . . . . 947.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 947.2 Future steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Appendix A: Domain and Mesh Coordinates . . . . . . . . . . . . . . . . . . . . . . . 111Appendix B: Total Pressure Boundary Condition . . . . . . . . . . . . . . . . . . . . . 114Appendix C: Sensitivity to Inlet Turbulence Intensity . . . . . . . . . . . . . . . . . . 116Appendix D: Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117xList of TablesTable 3.1 Thermophysical properties for the numerical setup. . . . . . . . . . . . . . 33Table 3.2 Constant values for k-\u000f turbulence model. . . . . . . . . . . . . . . . . . . 35Table 3.3 Wall functions for the k-\u000f turbulence model. . . . . . . . . . . . . . . . . . 38Table 3.4 Comparison of entrainment ratio for four different meshes. . . . . . . . . . 39Table 3.5 Comparison of the entrainment ratio reported in the literature and in thisvalidation study. The symbol * represents the results from Mazelli etal. For all the presented cases, 2D simulations are carried out using thestandard k-\u000f turbulence model. The relative error is calculated betweenthe Mazelli et al. numerical results and the validation study results. . . . 42Table 3.6 Simulated conditions for the constant-pressure mixing ejector. For all thepresented cases, p0s = 1 bar and pb = 1 bar. . . . . . . . . . . . . . . . . . 42Table 3.7 Simulated conditions for the constant-area mixing ejector. For all thepresented cases, p0s = 1 bar and pb = 1 bar. . . . . . . . . . . . . . . . . . 43Table 4.1 Ejector performance parameters. Comparison of the 2D numerical and1D analytical results for the constant-pressure mixing ejector. For all thepresented cases, D\/d = 2.5 and p0s = 1 bar. The symbol * represents theejector operation in the mixed flow regime. . . . . . . . . . . . . . . . . . 54xiLIST OF TABLESTable 5.1 Shock strength at different spatial locations of the constant-pressure mix-ing ejector with D\/d = 2.5. . . . . . . . . . . . . . . . . . . . . . . . . . . 78Table 5.2 Shock strength at different spatial locations for the constant-area mixingejector operating at p0p\/p0s = 2. . . . . . . . . . . . . . . . . . . . . . . . 82Table 6.1 Approximate freestream conditions used for finalizing the design parame-ters for the ejector operation at p0p\/p0s = 3.5 and D\/d = 3. . . . . . . . . 86Table 6.2 Analytical solution for the dowstream flowfield of flow past a supersonicwedge. pw1 is the static pressure upstream of the shock, and pw2 is thestatic pressure downstream of the shock. . . . . . . . . . . . . . . . . . . . 88Table 6.3 Comparison of the entrainment ratio and integrated entropy generationrate of the four different ejector diffuser designs with the base cases in aconstant-pressure mixing ejector for the ejector operation at p0p\/p0s = 2and p0p\/p0s = 3.5. For all the presented cases, D\/d = 3 and \u03b4 = 5\u25e6. . . . 88Table 6.4 Influence of the angle of inclination of the single-inclined plate on theejector performance for ejector operation at p0p\/p0s = 3.5 and D\/d = 3. . 93Table C.1 Comparison of the entrainment ratio and integrated entropy generationrate for three different turbulence intensities at p0p\/p0s = 5 and D\/d =2.5 of a constant-pressure mixing ejector. . . . . . . . . . . . . . . . . . . . 117xiiList of FiguresFigure 1.1 Schematic of the ejector components: (a) constant-pressure mixing cham-ber ejector and (b) constant-area mixing chamber ejector. . . . . . . . . . 3Figure 2.1 Influence of back pressure on the ejector entrainment ratio. . . . . . . . . 9Figure 2.2 Shocks structure in a supersonic ejector. . . . . . . . . . . . . . . . . . . 10Figure 2.3 Supersonic flow over a wedge and presentation of an oblique shock. . . . 22Figure 2.4 Supersonic flow over a wedge and presentation of a bow shock. . . . . . . 23Figure 3.1 Schematic of the computational domain. . . . . . . . . . . . . . . . . . . 36Figure 3.2 Mesh details at selected locations. . . . . . . . . . . . . . . . . . . . . . . 39Figure 3.3 Time-averaged centerline static pressure for four different meshes. . . . . 40Figure 3.4 Instantaneous Mach number profile inside the ejector for the validationstudy at p0p = 5 bars, p0s = 1 bar and pb = 1.2 bars. For the presentedcase, 2D simulation is carried out using the standard k-\u000f turbulence model. 41Figure 3.5 Centerline static pressure for the ejector operation at p0p = 5 bars, p0s =1 bar and pb = 1.2 bars. For the presented case, 2D simulation is carriedout using the standard k-\u000f turbulence model. . . . . . . . . . . . . . . . . 42xiiiLIST OF FIGURESFigure 4.1 Ejector schematic for the constant-pressure mixing model. Section 1\u20131represents the ejector inlet. Section 2\u20132 is at the primary nozzle throat.Section 3\u20133 represents the primary nozzle exit and the secondary streaminlet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Figure 4.2 Flow chart of the 1D ejector flow model solver. . . . . . . . . . . . . . . . 53Figure 4.3 Variation of the entrainment ratio versus the stagnation pressure ratio.D\/d is the area ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Figure 4.4 Variation of (a) pressure and (b) Mach number along the normalizedejector centerline axis. The solid and dashed curves are the predictionsof the 2D numerical and 1D analytical simulations, respectively. For thepresented case, p0p\/p0s = 5 and D\/d = 2.5. . . . . . . . . . . . . . . . . . 57Figure 4.5 Variation of (a) pressure and (b) Mach number along the normalizedejector centerline axis. The solid and dashed curves are the predictionsof the 2D numerical and 1D analytical simulations, respectively. For thepresented case, p0p\/p0s = 3.5 and D\/d = 2.5. . . . . . . . . . . . . . . . . 57Figure 5.1 Performance curves for the constant-pressure mixing ejector: (a) entrain-ment ratio versus stagnation pressure ratio, (b) integrated entropy gen-eration rate versus stagnation pressure ratio, and (c) entrainment ratioversus integrated entropy generation rate. . . . . . . . . . . . . . . . . . . 60Figure 5.2 Performance curves for the constant-area mixing ejector: (a) entrainmentratio versus stagnation pressure ratio, (b) integrated entropy generationrate versus stagnation pressure ratio, and (c) entrainment ratio versusintegrated entropy generation rate. . . . . . . . . . . . . . . . . . . . . . 61xivLIST OF FIGURESFigure 5.3 Time-averaged magnitude of the numerical schlieren contours for theconstant-pressure mixing ejector. (a), (b), and (c) pertain to p0p\/p0s= 2, 3.5, and 5, respectively. For all the presented cases, D\/d = 2.5. . . . 63Figure 5.4 Time-averaged Mach number profile for the constant-pressure mixing ejec-tor. The solid black line represents the dividing streamline and the solidwhite line represents the sonic line. (a), (b), and (c) pertain to p0p\/p0s =2, 3.5, and 5, respectively. For all the presented cases, D\/d = 2.5. . . . . 64Figure 5.5 Streamlines in the diffuser region of the constant-pressure mixing ejectorhighlighting the flow recirculation at p0p\/p0s = 5 and D\/d = 2.5. Thestreamlines are coloured by the local Mach number. . . . . . . . . . . . . 65Figure 5.6 Normalized and time-averaged temperature contours for the constant-pressure mixing ejector. (a), (b), and (c) pertain to p0p\/p0s = 2, 3.5, and5, respectively. For all the presented cases, D\/d = 2.5. . . . . . . . . . . 67Figure 5.7 Time-averaged Mach number profile for the constant-pressure mixing ejec-tor. (a), (b), and (c) pertain to D\/d = 2, 2.5 and 3, respectively. For allthe presented cases, p0p\/p0s = 2. . . . . . . . . . . . . . . . . . . . . . . . 68Figure 5.8 Time-averaged magnitude of the numerical schlieren for the constant-areamixing ejector. (a), (b), (c), (d), and (e) pertain to D\/d = 2.5, 3, 3.5, 4,5.5, respectively. For all the presented cases, p0p\/p0s = 2 . . . . . . . . . 70Figure 5.9 Time-averaged Mach number profile for the constant-area mixing ejector.The solid black line represents the dividing streamline and the solid whiteline represents the sonic line. (a), (b), (c), (d), and (e) pertain to D\/d =2.5, 3, 3.5, 4, 5.5, respectively. For all the presented cases, p0p\/p0s = 2. . 71xvLIST OF FIGURESFigure 5.10 Entropy generation number contour due to (a) heat transfer mechanismand (b) viscous dissipation mechanism for the constant-pressure mixingejector. For all the presented cases, p0p\/p0s = 5 and D\/d = 2.5 . . . . . 74Figure 5.11 Integral of the Bejan number along the vertical axis versus the normalizedejector axial distance at p0p\/p0s = 2, 3.5 and 5 for the constant-pressuremixing ejector. For all the presented cases, D\/d = 2.5. . . . . . . . . . . 75Figure 5.12 (a) Integral of overall entropy generation number (Ns) versus the normal-ized ejector axial distance and (b) entropy generation number of the meanflow (Ns,mf) and turbulent flow (Ns,tf) versus the normalized ejector axialdistance for the constant-pressure mixing ejector. For all the presentedcases, p0p\/p0s = 5 and D\/d = 2.5. . . . . . . . . . . . . . . . . . . . . . . 76Figure 5.13 (a) Integral of entropy generation number of the mean flow and (b)the turbulent flow versus the normalized ejector axial distance for theconstant-pressure mixing ejector. The results pertain to p0p\/p0s = 2, 3.5and 5. For all the presented cases, D\/d = 2.5. . . . . . . . . . . . . . . . 77Figure 5.14 (a), (b), and (c) are contributions of the mean flow to Ns at p0p\/p0s = 2,3.5, and 5, respectively. (d), (e), and (f) are contributions of the turbulentflow to Ns at p0p\/p0s = 2, 3.5, and 5, respectively. The results pertain tothe constant-pressure mixing ejector with D\/d = 2.5. . . . . . . . . . . . 79Figure 5.15 (a) Integral of entropy generation number of the mean flow and (b)the turbulent flow versus the normalized ejector axial distance for theconstant-pressure mixing ejector. The results pertain to D\/d = 2.5, 3,3.5, 4, and 5.5. For all the presented cases, p0p\/p0s = 2. . . . . . . . . . . 80xviLIST OF FIGURESFigure 5.16 (a), (b), (c), and (d) are contributions of the mean flow to Ns for D\/d =2.5, 3.5, 4 and 5.5, respectively. (e), (f), (g), and (h) are contributions ofthe turbulent flow to Ns for D\/d = 2.5, 3.5, 4 and 5.5, respectively. Theresults pertain to the constant-area mixing ejector operating at p0p\/p0s= 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Figure 6.1 Tested ejector diffuser designs of the constant-pressure mixing ejector:(a) single inclined plate placed inside the diffuser, (b) two inclined platesconnected at the leading edge placed inside the diffuser, (c) two divergingplates placed inside the diffuser and (d) two converging plates placedinside the diffuser. \u03b4 is the angle of inclination of the plates w.r.t. to thehorizontal line measured counter-clockwise for the top plate and clockwisefor the bottom plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Figure 6.2 Overall entropy generation number contours for the four different ejectordiffuser designs of the constant-pressure mixing ejector: (a) without anyinclined plates inside the diffuser (which is the base case), (b) single in-clined plate placed inside the diffuser, (c) two inclined plates connectedat the leading edge placed inside the diffuser, (d) two diverging platesplaced inside the diffuser, and (e) two converging plates placed inside thediffuser. For all the presented cases, p0p\/p0s = 2 and D\/d = 3. . . . . . . 90xviiLIST OF FIGURESFigure 6.3 Overall entropy generation number contours for the four different ejectordiffuser designs of the constant-pressure mixing ejector: (a) without anyinclined plates inside the diffuser (which is the base case), (b) single in-clined plate placed inside the diffuser, (c) two inclined plates connectedat the leading edge placed inside the diffuser, (d) two diverging platesplaced inside the diffuser, and (e) two converging plates placed inside thediffuser. For all the presented cases, p0p\/p0s = 3.5 and D\/d = 3. . . . . . 91Figure 6.4 Time-averaged Mach number profile for the two different ejector diffuserdesigns of the constant-pressure mixing ejector: (a) without any inclined-plates inside the diffuser (which is the base case), and (b) single-inclinedplate placed inside the diffuser. For all the presented cases, p0p\/p0s = 2and D\/d = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Figure 6.5 Time-averaged Mach number profile for the two different ejector diffuserdesigns of the constant-pressure mixing ejector: (a) without any inclined-plates inside the diffuser (which is the base case), and (b) single-inclinedplate placed inside the diffuser. For all the presented cases, p0p\/p0s = 3.5and D\/d = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Figure D.1 Variation of strength of the first diffuser shock with time for differenttime-averaging window sizes at p0p\/p0s = 5 and D\/d = 2.5 of a constant-pressure mixing ejector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119Figure D.2 Variation of pressure with time at the first diffuser shock location fordifferent time-averaging window sizes at p0p\/p0s = 5 and D\/d = 2.5 of aconstant-pressure mixing ejector. . . . . . . . . . . . . . . . . . . . . . . . 119xviiiLIST OF FIGURESFigure D.3 Variation of temperature with time at the first diffuser shock location fordifferent time-averaging window sizes at p0p\/p0s = 5 and D\/d = 2.5 of aconstant-pressure mixing ejector. . . . . . . . . . . . . . . . . . . . . . . . 120Figure D.4 Variation of velocity magnitude with time at the first diffuser shock loca-tion for different time-averaging window sizes at p0p\/p0s = 5 and D\/d =2.5 of a constant-pressure mixing ejector. . . . . . . . . . . . . . . . . . . 120Figure D.5 Variation of density with time at the first diffuser shock location for dif-ferent time-averaging window sizes at p0p\/p0s = 5 and D\/d = 2.5 of aconstant-pressure mixing ejector. . . . . . . . . . . . . . . . . . . . . . . . 121xixNomenclatureSymbolsAmix Cross-section area of constant-area duct of the mixing chamberAsk Cross-section area at the shock locationAp1 Cross-section area at the primary nozzle inletAp2 Cross-section area at the primary nozzle exitA\u2217p Cross-section area at the primary nozzle throatApx Primary nozzle area at x locationApy Primary stream area at the y-y plane (choking plane)Asy Secondary stream area at the y-y plane (choking plane)Be Bejan numberB Parameter used in Eq. (3.30)Co Courant numberCp Specific heat at constant pressureC\u00b5 Parameter used in Eqs. (3.23) and (3.25) and given in Table 3.2C\u03b51 Parameter used in Eq. (3.23) and given in Table 3.2C\u03b52 Parameter used in Eq. (3.23) and given in Table 3.2cf+ Speed of sound in the f+ directioncf\u2212 Speed of sound in the f\u2212 directionxxNOMENCLATURED Diameter of constant-area duct of the mixing chamberDne Primary nozzle exit diameterD\/d Area ratiod Primary nozzle throat diameter~dfN Connecting vector between the face center of the owner and neighbouringcells, and the cell centre of neighbouring cell~dpN Connecting vector between the cell centers of the owner and neighbouringcellsE Energye Specific internal energyFcv External forces acting on a control volumef Cell facef+ Outward flux direction at the cell facef\u2212 Inward flux direction at the cell facef2 Damping functiongf+ Parameter used in Eq. (3.15)h Specific enthalpyI Turbulence intensityk Turbulent kinetic energyL Length of mixing chamber\u2019s constant-area ductLsk Predicted shock locationLinlet Lateral dimension at the primary inlet` Integral length scalexxiNOMENCLATUREMa Mach numberM Molecular weightMap1 Primary flow inlet Mach numberMapx Mach number at the corresponding primary nozzle area (Apx)Mapy Primary stream Mach number at y-y plane (choking plane)Masy Secondary stream Mach number at y-y plane (choking plane)Mamix Mach number of the mixed streamMap Primary stream Mach numberMapx Local Mach number of the primary streamMa2 Mach number downstream of a wedge or shockMa1 Mach number upstream of a wedge or shockMne Primary nozzle exit Mach numberm\u02d9s\/m\u02d9p Entrainment ratiom\u02d9p Primary mass flow ratem\u02d9s Secondary mass flow ratem Mass entering or leaving a control volumeN Cell centre of the neighbouring cellNXP Nozzle exit positionNs Entropy generation numberNs,mf Entropy generation number of the mean flowNs,tf Entropy generation number of the turbulent flowP Cell centre of the owner cellPr Prandtl numberxxiiNOMENCLATUREP\u02d9s Entropy generation ratep Pressureppx Local static pressure of the primary streamppy Primary stream pressure at y-y plane (choking plane)psy Secondary stream pressure at y-y plane (choking plane)pbp Static pressure at the boundary patchp0 Stagnation pressurepb Back pressurep\u2217b Critical Back pressurep0p Primary stream stagnation pressurepp Primary stream static pressurep0s Secondary stream stagnation pressurep0mix Mixed stream stagnation pressurepmix Mixed stream static pressurep0mix1 Mixed stream stagnation pressure upstream of the shockp0mix2 Mixed stream stagnation pressure downstream of the shockp0p\/p0s Stagnation pressure ratiop1 Static pressure inlet to the shockp2 Static pressure downstream of the shockpw2\/pw1 Static pressure ratio for flow over a wedgepx Local static prresure in the diffuser region downstream of the shockQn Total number of data pointsqj Heat flux vectorxxiiiNOMENCLATURERs Shock strengthRet Turbulent Reynolds numberRene Primary nozzle exit Reynolds numberR Specific gas constantr Ratio of successive gradientsrx Mesh refinement ratioSf Face area vectorS\u02d9\u2032\u2032\u2032gen Entropy generation rate per unit volumeS\u02d9\u2032\u2032\u2032mf Entropy generation rate per unit volume in the mean flowS\u02d9\u2032\u2032\u2032tf Entropy generation rate per unit volume in the turbulent flowS\u02d9\u2032\u2032\u2032gen,heat Entropy generation rate per unit volume because of the heat transferS\u02d9\u2032\u2032\u2032gen,viscous Entropy generation rate per unit volume because of the viscous dissipation\u222bS\u02d9\u2032\u2032\u2032gendV Integrated entropy generation rateSij Strain ratet TimeT TemperatureT \u2217 Non-dimensional temperatureTpx Local static temperature of the primary streamTp Primary stream static temperatureT0p Primary stream stagnation temperatureT0s Secondary stream stagnation temperatureT0mix Mixed stream stagnation temperatureT0mix1 Mixed stream stagnation temperature upstream of the shockxxivNOMENCLATURET0mix2 Mixed stream stagnation temperature downstream of the shockTmix Mixed stream static temperatureTf+ Temperature in the outward direction at the cell face fTf\u2212 Temperature in the inward direction at the cell face fTpy Temperature of the primary stream at the y-y plane (choking plane)Tsy Temperature of the secondary stream at the y-y plane (choking plane)u x-velocity componentu+ Dimensionless velocityu Velocity vectoruf Velocity vector at face fui Velocity component in the ith directionu\u03c4 Friction velocityuavg Mean flow velocityumix Velocity of the mixed streamupx Local velocity of the primary streamupy Velocity of the primary stream at the y-y plane (choking plane)usy Velocity of the secondary stream at the y-y plane (choking plane)V Volumev y-velocity componentw z-velocity componenty+ y Pluszi Shock strength corresponding to the ith time-averaging window sizez\u00af Mean shock strengthxxvNOMENCLATUREGreek symbols\u03b1 Thermal diffusivity\u03b1t Turbulent thermal diffusivity\u03b2vl van Leer flux limiter\u03b2 Flux limiter\u0393 Diffusion coefficient\u03b3 Ratio of specific heats\u2206 Relative error\u2206t Time step\u03b4 Wedge angle\u03b4m\u02d9s Relative error in the secondary mass flow rate\u03b4\u03c9 Relative error in the entrainment ratio\u03b4ij Kronecker delta\u03b4max Critical wedge angle\u000f Dissipation rate of the turbulent kinetic energy\u000f\u00af Average rate of dissipation of the turbulent kinetic energy\u03b6 Time bound used for Reynolds averaging in Eq. (3.4)\u03b8 Angle of the deflected shock across a wedge\u03ba Thermal conductivity\u03bav von Karman constant\u00b5 Dynamic viscosity\u00b5t Eddy viscosity\u03bd Kinematic viscosityxxviNOMENCLATURE\u03c1 Density\u03c1p Density at the centre of the owner cell\u03c1px Local density of the primary stream\u03c1mix Mixed stream static density\u03c10p Primary stream stagnation density\u03c10s Secondary stream stagnation density\u03c10mix Mixed stream stagnation density\u03c1w Density near the wall\u03c1inlet Static density at the primary inlet\u2207\u03c1\u2217 Numerical Schlieren\u2207\u03c1 Density gradient\u039b Weighting coefficient\u03c3dev Standard deviation\u03c3ij Viscous stress tensor\u03c3k Parameter used in Eq. (3.23) and given in Table 3.2\u03c3\u03b5 Parameter used in Eq. (3.23) and given in Table 3.2\u03c4 Shear stress tensor\u03c4w Wall shear stress\u03a6(x, t) Dependent variable used in Eq. (3.4)\u03c6f Volumetric flux at the cell face f\u03c6p Volumetric flux at the centre of the owner cell\u03c6f+ Volumetric flux in the outward direction at the cell face f\u03c6f\u2212 Volumetric flux in the inward direction at the cell face fxxviiNOMENCLATURE\u03c6k Wall term used in Eq. (3.22)\u03c6\u03b5 Wall term used in Eq. (3.23)\u03c7 Parameter used in Eq. (4.24)\u03a8f Flow property value at the cell face\u03a8f+ Flow property value in the outward direction at the cell face f\u03a8f\u2212 Flow property value in the inward direction at the cell face f\u03a8P Flow property value at the cell centre of the owner cell\u03a8N Flow property value at the cell centre of the neighbouring cell\u03c8f+ Volumetric flux associated with the local speed of sound in the f+ direction\u03c8f\u2212 Volumetric flux associated with the local speed of sound in the f\u2212 direction\u03c8 Stream function\u03c8c Compressibility\u03c9f Weighting coefficientxxviiiAcknowledgementsI would like to extend my immense gratitude to my supervisors, Professor Sina Kheirkhahand Professor Joshua Brinkerhoff, for their motivation when I was anxious, guidance when Iwas clueless, frequent interactions, financial assistantship and consistent support at all times.Considering my limited capabilities and knowledge of computational modeling, it was challengingto have this Master\u2019s degree as a fruitful experience. Thanks a million to my Professors who werehappy to extend their assistance in developing my capabilities and equipped me with the relevantresources. Thanks a lot to Professor Joshua Brinkerhoff, whose course on Computational FluidDynamics helped me understand the ground concepts thoroughly required for this research.Over this journey of 2.5 years, I could see my skill-set reaching another level with a remarkableimprovement in my writing and presentation skills. Huge credits to my supervisors who aredetail-oriented and were never hesitant to teach or assist me. I would like to thank the financialsupport from the Natural Sciences and Engineering Research Council of Canada as well as theindustrial partner of this project (IVAC). I would also like to thank MITACS for the GraduateFellowship I received to supplement my financial capabilities. The motivation to pursue aMaster\u2019s degree came from internship supervisors, Professor Dhiman Chatterjee and ProfessorLuc Frechette, and my undergraduate Professor A.V. Doshi. Thank you so much Professors, formotivating me and recommending me to this program at the University of British Columbia.Over this journey as a Master\u2019s student, I came across a bunch of interesting people full ofxxixAcknowledgementsintellect and experts in various domains. Since I had two supervisors, I was a part of two differentlabs and was lucky enough to interact with so many captivating personalities. Thanks to mylabmates Sahar, Sajjad, Ramin, Sepher, Leslie, Morales, Shahab, Kasper, Najiba, Saeed, Aron,Dr. Wang, Dr. Sharma and Dr. Ismail, for being available and open to any help. WheneverI had issues figuring out MATLAB codes, Sajjad and Sahar were always available round theclock. I used to have long discussions with Morales and his inputs were always helpful. I owe mycontrolDict file to Shahab, who was always ready to help me with the postProcessing utilitiesin OpenFOAM. I am grateful to Dr. CS Wang for giving his inputs in the data interpretationand extending his help in generating presentable figures for my thesis. I am for sure going tomiss the tea breaks headed by Sajjad during pre-COVID times and my birthday celebrations.I would like to thank my friends Hitika, Archit, Nidhi, Aman, Varad, Gaurav, Adithiya,Baljinder, Rajvir, Vikas, Manjot, Sneha and Abhishek who are my party and gossip squad andalways ensure that I do not feel homesick and miss my family. Thanks to Pranav, my pseudo-psychologist and the English expert, for always hearing me out and helping me to deal withstress. He still awaits when our chats would no longer have the word \u201cthesis\u201d. I would also likeextend my deep gratitude to Alka Yagnik, Indian Playback Singer. Alka, you are just amazing!I used to listen to your songs while writing my thesis.Finally, I would like to extend my deep and sincere gratitude to my parents, Narayan SinghSisodiya and Meena Sisodiya, for always being on my side, constantly encouraging me and givingme all the opportunities and experiences. Love and regards to my brother, Kushank SinghSisodiya, the coding expert, for being an annoying friend and helping me with the programminglanguages. Mom, thanks for raising me strong, all my achievements are for you and because ofyou. You have always inspired me and taught me to get through difficult situations. WithoutxxxAcknowledgementsyour sacrifices and support, I wouldn\u2019t have been who I am.xxxiDedicationTo my lovely familyxxxiiChapter 1Introduction1.1 MotivationThe world\u2019s power consumption has drastically increased over the past decades. The recordssuggest this global power consumption is increasing at a rate faster than the world population,which ultimately increases the per capita power consumption rate [1]. The US Energy Infor-mation Administration\u2019s latest international outlook reported that, by 2050, the world energyconsumption is going to increase nearly by 50 percent [2]. These continued power demandsincrease consumption from the existing natural resources. Such surplus demands are raising thedepletion concerns and the hazards of global warming, which negatively affects sustainability.In 2019, approximately 84% of the world\u2019s energy was produced from the non-renewable fossilfuels [3]. This is predicted to remain about 80% in the following decades.Helping decrease the energy consumption, technologies are being developed relevant to manyengineering equipment, such as supersonic ejectors. These ejectors find their application in sev-eral engineering sectors such as refrigeration [4\u20136], high-altitude testing facilities [7], desalin-ization plants [8], fluid mixing [9], material transportation [10] and thrust augmentation [11].Depending on the application, these ejectors are designed either for vacuum creation [12] and\/orincreasing the pressure recovery [6]. Of relevance to the present study are those used in materialtransport [10] and refrigeration systems [4\u20136]. For these applications, the ejectors are designed11.1. Motivationwith the objective of increasing the amount of secondary flow induced in the ejector and thepressure recovery achieved at the ejector outlet. For material transport, ejector operation canbe facilitated using a high pressure and high temperature gas of other industrial processes.These ejectors are used in refrigeration systems to replace the compressor units [4]. Usually,the operation of these compressor units is power intensive. However, when the ejectors replacethese compressor units, the overall power consumption to run the refrigeration cycles is greatlyreduced. This is because ejectors have the ability to use low-grade waste heat from variousindustrial processes for their operation [6]. Furthermore, these ejectors lack moving parts; and,hence, their operational and maintenance costs are relatively low [6]. Because of these advan-tages, supersonic ejectors are a promising technology. Thus, there is a need to characterize andincrease their performance.A review of literature [13] shows that the supersonic ejectors are categorized as constant-pressure and constant-area mixing ejectors based on the mixing chamber geometries as shownin Figs. 1.1a and 1.1b, respectively. Both geometries are composed of a primary nozzle, amixing chamber, and a diffuser. A high-pressure stream directed from the primary nozzle speedsup to attain supersonic velocities. This high-speed stream generates suction which entrains asecondary stream into the ejector [13]. The two streams interact and undergo mixing whiletraversing through the mixing chamber. This is accompanied by the formation of a series ofcompression and expansion waves and shear layers that affect the mixing process. The diffuserat the end of the mixing chamber is used for the pressure recovery [13]. The mixing chamberof a constant-pressure mixing ejector has a variable area duct integrated with a constant-areaduct, but the mixing chamber of a constant-area mixing ejector comprise only a constant-areaduct. Both constant-pressure and constant-area mixing chamber geometries are used in various21.2. Ejector terminologiesapplications with a focus on the careful selection of the operating conditions to maximize theejector performance. Different parameters are used to quantify the ejector performance. In thepast, some authors assessed the ejector performance in terms of the entrainment ratio [14\u201318];however, some focused their attention on the ejector efficiency [16, 19]. The entrainment ratio(m\u02d9s\/m\u02d9p), which quantifies the capacity of the ejector to entrain the secondary mass flow rateper unit primary mass flow rate, is used to evaluate the ejector performance in the present study.(a)(b)Fig. 1.1. Schematic of the ejector components: (a) constant-pressure mixing chamber ejectorand (b) constant-area mixing chamber ejector.1.2 Ejector terminologiesThe terminologies used in the context of supersonic ejectors are presented below.\u0088 Choking: The flow is referred to be choked when it reaches sonic conditions, i.e. it attainsMach number equal to 1. The choking condition puts a limit on the secondary mass flow31.2. Ejector terminologiesrate that can be entrained into the mixing chamber.\u0088 Sonic line: It is the loci of points within the flow with Mach number equal to 1.\u0088 Critical flow regime: This defines the ejector operation under the conditions when boththe primary and secondary streams are choked. In this regime, the maximum possible sec-ondary flow rate is entrained in the mixing chamber for a given set of operating conditions.\u0088 Mixed flow regime: This is the regime for which the ejector operates under the conditionswhen the primary stream is choked; however, the secondary stream is not choked.\u0088 Non-mixed length: It is the length of the mixing chamber until which the primary andsecondary streams remain visually distinct. Beyond this length, the mixing is said to becomplete.\u0088 Back pressure: It is the pressure imposed at the ejector outlet.\u0088 Stagnation pressure ratio: It is the ratio of the stagnation pressure of the primary stream(p0p) to the stagnation pressure of the secondary stream (p0s).\u0088 Entrainment ratio: It is the ratio of the secondary mass flow rate (m\u02d9s) to the primarymass flow rate (m\u02d9p).\u0088 Area ratio: It is the ratio of the area of mixing chamber\u2019s constant-area duct (Amix) tothe primary nozzle throat area (A\u2217p). For the 2D computations, the area ratio can beapproximated by D\/d.\u0088 Over-expanded mode: In this mode, the primary nozzle exit pressure is smaller than theejector back pressure. As a result, there is a shock wave in the primary nozzle.41.3. Research objectives\u0088 Under-expanded mode: In this mode, the primary nozzle exit pressure is greater than theejector back pressure.\u0088 Fully-expanded mode: In this mode, the primary nozzle exit pressure equals the ejectorback pressure.1.3 Research objectivesThe present study focuses on characterizing of the losses and using that insight to suggestimprovements to the design of vacuum-objective supersonic ejectors. The study of ejector flowtopologies help to quantify the entropy generation rate, which is used to characterize the lossesin vacuum-objective supersonic ejectors. The objectives of the present work are listed below.1. Study the impact of stagnation pressure ratio (p0p\/p0p) and area ratio (D\/d) on the ejectorentrainment ratio (m\u02d9s\/m\u02d9p), entropy generation rate (S\u02d9\u2032\u2032\u2032gen) and flow topologies for twodifferent set of mixing chamber geometries, namely, constant-pressure and constant-areamixing ejectors;2. Develop knowledge related to the different entropy generation mechanisms and investigatethe effect of mean flow and turbulence on entropy generation rate;3. Study the location-dependent impact of the shocks, shear layers, nozzle expansion modesand diffuser inefficiency in determining the performance of the constant-pressure andconstant-area mixing ejectors operating in both the critical and mixed flow regimes;4. Suggest alternative designs that allow improving the ejector performance.51.4. Thesis outline1.4 Thesis outlineThis thesis is organized in 7 chapters. The general overview of the problem statementand research objectives are presented in Chapter 1. The background related to the presentstudy and the computational methodology is discussed in Chapters 2 and 3, respectively. Thedetails for setting up the 1D analytical model of ejector and the associated results\/explainationsare presented in Chapter 4. The 2D computational results are discussed in Chapters 5 and6. Chapter 5 pertains to the discussions related to the general performance parameters andflow topologies in the ejectors, addressing objective 1, listed in Subsection 1.3. The entropygeneration analysis is discussed in Chapter 5, addressing objectives 2\u20133, listed in Subsection 1.3.The alternative designs and modifications to improve the ejector performance and the associatedresults are presented in Chapter 6, addressing objective 4, listed in Subsection 1.3. Finally, theconcluding remarks and future scope of work are discussed in Chapter 7.6Chapter 2BackgroundThis chapter is organized in 6 subsections. In Subsections 2.1 and 2.2, the influences ofthe operating conditions and ejector geometry on the entrainment ratio and the internal flowstructure are reviewed. This is followed by the review of literature related to the mathematicalmodeling for entropy generation and the studies concerning entropy generation relevant to theejector flows in Subsection 2.3. Since this thesis follows a computational methodology, thebackground related to the computational modeling of ejector flow is also reviewed in Subsection2.4. The background related to the techniques used for altering the supersonic flow structures isdiscussed in Subsection 2.5. Finally, the areas that require further investigations are summarizedin Subsection 2.6.2.1 Effect of operating conditionsIn the literature, some studies used the secondary stream stagnation pressure (p0s) to normal-ize the primary stream stagnation pressure (p0p), while some studies reported the inlet pressureitself. The operating conditions, such as the stagnation pressure ratio (p0p\/p0s), the back pres-sure (pb), and the primary stream temperature (Tp), significantly influence the entrainment ratio(m\u02d9s\/m\u02d9p) and the ejector internal flow structure, which are discussed below.72.1. Effect of operating conditions2.1.1 Stagnation pressure ratioThe entrainment ratio and flow topologies are influenced by the stagnation pressure ratio[20\u201324]. The studies of [20\u201322] reported an inverse relation between the stagnation pressureratio and the entrainment ratio. Arun et al. [23] and Jia et al. [24] concluded that increasingthe stagnation pressure ratio, the entrainment ratio increases, attains its maximum value, andthen decreases. These observed trends can be explained by the primary nozzle expansion modeswhich can either operate in the over-expanded mode or the under-expanded mode. In the over-expanded mode, the primary nozzle exit pressure is smaller than the ejector back pressure; andhence, there is a shock in the primary nozzle. In the under-expanded mode, the primary nozzleexit pressure is greater than the ejector back pressure. The stagnation pressure ratio leading toan over-expanded nozzle mode produces a higher entrainment ratio than the under-expandedmode [20\u201322]. Further, the non-mixed length decreases in the over-expanded mode and viceversa in the under-expanded mode relative to the fully-expanded mode [20\u201322].2.1.2 Back pressureA limiting value of the back pressure (pb), referred to as the critical back pressure (p\u2217b), marksthe switching of the ejector operation between the on-design and off-design regimes [25, 26]. Thevariation of the entrainment ratio as a function of the back pressure is shown in Fig. 2.1, asconcluded by the investigations of Aidoun et al. [25] and Wang et al. [26]. When the backpressure is less than the critical back pressure, the ejector operates in the on-design regime inwhich the entrainment ratio is insensitive to the back pressure variations. However, when theback pressure is larger than the critical back pressure, the ejector operates in the off-designregime. For the off-design operation, the entrainment ratio decreases with increasing the back82.1. Effect of operating conditionspressure.Fig. 2.1. Influence of back pressure on the ejector entrainment ratio.Chen et al. [27] investigated the reason for the trend shown in Fig. 2.1. Their explainationis linked to the internal flow structure of the ejector, specifically, the effective area, which isthe cross-section area of the region between the primary stream and the ejector wall at thelocation where the secondary stream chokes in the mixing chamber, as shown in Fig. 2.2 [27].When the backpressure is significantly smaller than the critical back pressure, two independentshock trains are formed downstream of the primary nozzle: one in the mixing chamber and thesecond in the diffuser, as shown in Fig. 2.2. At this operating condition, the static pressureupstream of the effective area section remains unaffected. Since the pressure upstream of wherethe secondary stream chokes remains unchanged, the entrainment ratio is constant. As thebackpressure increases, the second shock train moves upstream and finally merges with the firstshock train at pb = p\u2217b, without affecting the upstream static pressure of the effective areasection. Further increase of the back pressure increases the static pressure upstream of theeffective area section, and the mass flow rate of the induced secondary stream decreases. As a92.1. Effect of operating conditions(#1)(#2)Fig. 2.2. Shocks structure in a supersonic ejector.result, the entrainment ratio decreases.2.1.3 Primary stream temperatureYapici et al. [16] and Han et al. [28] investigated the effect of the primary stream temper-ature on the ejector entrainment ratio. They showed that, as the primary stream temperatureincreases, the entrainment ratio increases until it attains a maximum value [16, 28]. Furtherincreasing the primary stream temperature decreases the entrainment ratio [16, 28]. Han etal. [28] elaborated this using the ejector internal flow structure and showed that the primarystream temperature influences the length of the choking zone. This zone is defined as a regionof the mixing chamber where the static pressure is constant and the velocity does not featuresignificant changes, as shown in Fig. 2.2. It is desirable to maximize the length of the chokingzone to facilitate the most optimized energy exchange process between the primary and thesecondary streams. The most optimized energy exhange process will result in the most efficientmixing, thereby optimizing the entrainment ratio [28]. Prior to reaching the maximum entrain-ment ratio, the length of the choking zone and the primary stream temperature are positivelyrelated. Beyond this, increasing the primary stream temperature decreases the mixing efficiencywhich in turn decreases the choking length [28]. As a result, the entrainment ratio decreases.102.2. Effect of geometrical specifications2.2 Effect of geometrical specificationsThe geometrical parameters, specifically the mixing chamber geometry, nozzle exit position(NXP ), suction chamber wall half-angle, and the area ratio (Amix\/A\u2217p) influence the entrainmentratio and the ejector internal flow structure, which are discussed below. The area ratio is definedas the ratio of the area of the constant-area duct (Amix) of the mixing chamber to that of thearea of the primary nozzle throat (A\u2217p).2.2.1 Mixing chamber geometryThe studies of Chen et al. [29] and Liu et al. [30] respectively showed that the lengthand diameter of the mixing chamber significantly influence the ejector entrainment ratio. Chenat al. [29] observed that, as the length of the mixing chamber increases, the entrainment ratioincreases and then plateaus. Liu et al. [30] observed that, as the diameter of the mixing chamberincreases, the entrainment ratio increases. Both studies [29, 30] provided an explaination forsuch trends using the mixing efficiency and effective flow area (see Fig. 2.2) of the secondaryflow. Chen et al. [29] stated that, as the length of the mixing chamber increases, the contacttime between the primary and secondary streams increases which increases the mixing efficiency,increasing the effective area of the secondary flow in the mixing chamber thereby increasing theentrainment ratio [29]. However, beyond a certain length, the effective area does not change;and, as a result, the entrainment ratio remains constant [29]. Liu et al. [30] stated that, as thediameter of the mixing chamber increases, the mixing efficiency increases which increases theeffective area of the secondary flow thereby increasing the entrainment ratio.112.2. Effect of geometrical specifications2.2.2 Nozzle exit positionChen et al. [29], Ramesh et al. [31], Arun et al. [23] and Chen et al. [14] investigatedthe relationship between NXP and the ejector entrainment ratio. They [14, 23, 29, 31] foundthat, for a fixed set of operating conditions, the entrainment ratio increases first, attains itsmaximum value and then decreases, as the nozzle moves away from the mixing chamber. Thereason behind such observed behaviour is explained using the concept of effective flow area (seeFig. 2.2) for the secondary stream in the mixing chamber. Arun et al. [23] show that, forvery large values of NXP (i.e. the nozzle recessed from the mixing chamber), the expansionof the primary nozzle flow decreases the effective flow area of the secondary flow decreasing theentrainment ratio. Similarly, for conditions that the nozzle is positioned excessively into themixing chamber, the nozzle obstructs the secondary flow, decreasing the effective area of thesecondary flow; and, as a result, decreasing the entrainment ratio.2.2.3 Suction chamber angleRamesh et al. [31] investigated the effect of the suction chamber wall half-angle on theejector entrainment ratio. They [31] concluded that, as this angle increases, the entrainmentratio first increases, attains its maximum value, and then decreases. Study of [31] shows thatchanging the half-angle of the chamber wall influences the shape of a virtual nozzle formed inbetween the wall and the primary jet shear layers. For relatively small values of the chamber wallhalf-angle, the virtual nozzle cross-section area either increases or remains unchanged movingalong the secondary flow direction. Thus, the flow is not accelerated and the entrainment ratiois rather small. For the large values of the chamber wall half-angle, although the virtual nozzlecross section area decreases along the secondary flow direction, the length of the nozzle is rather122.2. Effect of geometrical specificationsshort; and, as a result, the flow cannot accelerate and the entrainment ratio remains small.Ramesh et al. [31] showed that there exists an optimal angle at which the entrainment ratio ismaximized.2.2.4 Area ratioYapici et al. [16] and Liu et al. [30] investigated the relationship between the area ratioand the ejector entrainment ratio. They [16, 30] concluded that there exists an optimum arearatio for each set of operating conditions that maximizes the entrainment ratio. Liu et al. [30]provided an explaination for such trends in terms of the mixing efficiency. They [30] found thatthe mixing efficiency (which is used to quantify the energy loss in the mixing process) improvesby 200% by changing the area ratio from 0.025 to 0.8. The mixing efficiency is affected by howmuch time the secondary flow takes to be accelerated to the sonic velocities; and, a longer timedeclines the mixing efficiency, thereby decreasing the entrainment ratio [30].The area ratio can be increased either by increasing the diameter of the constant-area ductof the mixing chamber or by decreasing the diameter of the primary nozzle throat. Liu et al. [30]suggested that the preferrable approach of increasing the area ratio is to increase the diameterof the constant-area duct of the mixing chamber. Decreasing the area of the primary nozzlethroat decreases the effective flow area of the primary stream which increases the effective areaof the secondary stream. Despite this increase (which seems desirable), the small area of theprimary nozzle suggests the kinetic energy of the primary stream is small; and, as a result, theentrainment ratio is not maximized.132.3. Entropy generation2.3 Entropy generationIn addition to the entrainment ratio, the ejector performance can be quantified and lateroptimized by estimating the entropy generation [32\u201334]. The background on the mathematicalmodels for entropy generation, and the entropy generation in supersonic ejectors is discussedbelow.2.3.1 Mathematical modelDesign optimization by entropy generation minimization is used to identify the local flowirreversibilities [33\u201338]. Studies of [33\u201338] broadly classify entropy generating mechanisms intotwo primary categories. It includes the local entropy generation because of heat transfer acrossa finite temperature difference [39, 40] and the local entropy generation because of viscous dissi-pation [39, 40]. Depending on the application, either or both can have significant contributions.The entropy generation rate (P\u02d9s) is given byP\u02d9s =\u03ba\u2207T \u00b7 \u2207TT 2+1T(\u03c4 : \u2207u) , (2.1)where \u03ba is the fluid themal conductivity, T is the fluid temperature, u is the fluid velocity vectorand \u03c4 is the shear stress tensor. \u2207 is the vector derivative expressed as \u2207 = i\u22071 + j\u22072 + k\u22073 =i ddx + jddy + kddz . Equation (2.1) is derived using the conservation of mass, momentum andenergy, the Clausius-Duhem inequality [41], Gibbs Free Energy equation [42] and Fourier\u2019s Law[43]. The terms in Eq. (2.1) can be expanded and re-arranged using the mathematical relationsfor the eddy viscosity model [44]. This leads to Eq. (2.2) which is used to calculate the entropygeneration rate per unit volume (S\u02d9\u2032\u2032\u2032gen) for turbulent shear flows [39, 40, 44, 45]. This parameteris given by142.3. Entropy generationS\u02d9\u2032\u2032\u2032gen =\u03baT 2[(\u2202T\u2202x)2+(\u2202T\u2202y)2+(\u2202T\u2202z)2]\ufe38 \ufe37\ufe37 \ufe38Term 1+\u00b5+ \u00b5tT{(\u2202u\u2202y+\u2202v\u2202x)2+(\u2202u\u2202z+\u2202w\u2202x)2+(\u2202v\u2202z+\u2202w\u2202y)2+ 2[(\u2202u\u2202x)2+(\u2202v\u2202y)2+(\u2202w\u2202z)2]}\ufe38 \ufe37\ufe37 \ufe38Term 2+\u03b1\u03b1t\u03baT 2[(\u2202T\u2202x)2+(\u2202T\u2202y)2+(\u2202T\u2202z)2]\ufe38 \ufe37\ufe37 \ufe38Term 3+\u03c1\u000fT\ufe38\ufe37\ufe37\ufe38Term 4,(2.2)where \u00b5 is the fluid dynamic viscosity, \u00b5t is the eddy viscosity, \u03b1 is the fluid thermal diffusivity,\u03b1t is the turbulent thermal diffusivity, \u03c1 is the fluid density, and \u000f is the average rate of thedissipation of the turbulent kinetic energy. u, v and w are the velocity components in the x\u2013,y\u2013 and z\u2013directions, respectively. Terms 1 and 2 in Eq. (2.2) are used to model the entropygeneration due to heat transfer and viscous dissipation in the mean flow, respectively. Terms 3and 4 in Eq. (2.2) are used to model the effects of turbulence on entropy generation because ofheat transfer and viscous dissipation, respectively. Equation (2.2) is applicable for modeling theentropy generation rate in compressible flows [45]. However, it is only valid for the interaction ofsingle-phase Newtonian fluids and applies to solving Reynolds-averaged Navier-Stokes equationsusing the eddy viscosity models [44, 45].2.3.2 Ejector flow irreversiblitiesSierra-Pallares et al. [33], Lamberts et al. [34] and Omidvar et al. [35] performed theentorpy generation analysis to quantify the ejector losses. These studies [33\u201335] followed differentapproaches for quantifying the entropy generation in supersonic ejectors. Sierra-Pallares et al.152.3. Entropy generation[33] studied the impact of ejector mixing chamber designs on the entrainment ratio and entropygeneration rate and concluded that the viscous dissipation in the turbulent flow is the importantmechanism responsible for the ejector flow irreversibilities. They [33] showed that the entropygeneration is mainly due to the shear layer formed at the exit of the primary nozzle and theshock trains and further added that the contribution because of the shear layers and shocktrains is location-independent. Lamberts et al. [34] used an exergy-based analysis to quantifythe ejector flow irreversibilities and study the impact of back pressure on the exergy lossesand identified the viscous dissipation mechanism responsible for the ejector flow irreversibilities.They [34] concluded when the ejector operates close to the critical back pressure, the secondarystream\u2019s net exergy gain is maximum; and, hence, at this operating back pressure, the ejectorflow irreversibilities are smaller than the rest of the tested back pressures. Omidvar et al. [35]studied the impact of the condensor temperature on the entropy generation for a solar-drivenvariable geometry ejector. The effect of the condenser temperature on the ejector operationis similar to the back pressure. They [35] concluded when the condenser temperature is lessthan the critical temperature, the contribution from the shocks dominates the ejector flowirreversibilities; and, when the condenser temperature is more than the critical temperature,the contribution from the mixing phenomenon dominates the ejector flow irreversibilities. They[35] obsevered a strong correlation between the entrainment ratio and the entropy generationand concluded that the operating condition with the minimum entropy generation leads to themaximized entrainment ratio.162.4. Computational modeling2.4 Computational modelingThe background related to the one-dimensional (1D) ejector flow model, the relative accu-racy of two-dimensional (2D) and three-dimensional (3D) computational methodology, and thesuitability of different turbulence models for the ejector flow prediction is discussed below.2.4.1 1D analytical modelHuang et al. [46] and Chen et al. [47] used a 1D analytical model to predict the ejectorentrainment ratio using the constant-pressure mixing model. They [46, 47] used their experi-mentally measured values of the entrainment ratio and predictions based on the isentropic flowto estimate the isentropic efficiencies. They utilized these efficiency calculations to improve theaccuracy of their 1D analytical model. Similarly, Petrovic et al. [48] used the ejector componentefficiencies in the 1D analytical model to improve its accuracy. Selvaraju et al. [49] developeda 1D analytical model to account for the friction losses and the variation in the specific heats.Chen et al. [14] developed a 1D model capable of predicting the qualitative relationship be-tween the NXP and the entrainment ratio. They [14] concluded that when the primary nozzlemoves close to the converging-section of the mixing chamber, the secondary flow rate attainsits limiting value, and further nozzle movement downstream does not increase the entrainmentratio.Eames et al. [50] assumed a constant rate of change of momentum in the diffuser region whichaims to remove the thermodynamic shock in the diffuser region thereby increasing the totalpressure recovery and minimizing the diffuser losses. They [50] concluded that the entrainmentratio and the pressure recovery at the diffuser exit improves with their new desing when comparedto the conventional design. The study of Eames et al. [50] was extended by Kumar et al. [51]172.4. Computational modelingwith some modifications. Specifically, Kumar et al. [51] used the constant rate of change ofmomentum aprroach for the diffuser region and considered the localized frictional effects in theadiabatic flow and showed an improved agreement between the predictions of their 1D modeland those of their computational simulations.2.4.2 2D and 3D computational modelsMazelli et al. [52], Hemidi et al. [53] and Pianthong et al. [54] investigated the accuracy ofthe 2D and 3D computational techniques to predict the ejector entrainment ratio by comparingthe computational results against the experimental data. They [52\u201354], concluded that forthe ejector operation in the on-design conditions, both of the computational techniques showa good agreement with the experimental data; however, for the off-design conditions, the 3Dcomputational approach is required to estimate the flow charactersitics accurately. They [52\u201354] suggested that the agreement between the 2D and 3D results for the on-design conditions,compared to the off-design conditions, is because the value of the velocity component in thethird direction is minimal; and, as a result, the 3D effects are negligible.Gagan et al. [55] investigated the accuracy of the 2D and 3D computational techniques forpredicting the ejector flow by comparing the computational results against the experimentaldata. They [55] concluded that for the 2D and 3D computational models, there is a negligibledifference between the axial distribution of the static pressure profile between the simulationresults and those of the experiments. Their results [55] show that the axial velocity at locationsfar away from the supersonic jet predicted from the simulations is different from that of theexperiments. Nevertheless, the disagreement related to the flow charateristics at large distancesfrom the supersonic flow is not expected to influence the prediction of the ejector performance.In the present study, for all of the tested conditions, the ejector operates in the on-design mode;182.4. Computational modelingand hence, a 2D computational model is used.2.4.3 Turbulence modelIn the past, researchers have used different turbulence models to account for the flow tur-bulence in supersonic ejectors. There does not exist a consensus on one turbulence model thatis best suited for simulating the ejector flow behaviour. Wang et al. [56], Hakkaki-fard et al.[57], Gagan et al. [55] and Riffat et al. [58] recommended that the standard k-\u000f model maypredict the entrainment ratio and flow turbulence accurately, and concluded that the standardk-\u000f model predicts the overall flow behaviour with reasonable accuracy. Contrary to this, Chenet al. [29] recommended that the renormalization-group (RNG) k-\u000f model and Bartosiewicz etal. [59] recommended the shear-stress transport (SST) k-\u03c9 model for accounting for accuratelysimulating turbulence in the supersonic ejectors.Zhu et al. [18] compared the relative accuracy of the standard k-\u000f, realizable k-\u000f, RNGk-\u000f and SST k-\u03c9 turbulence models to predict the entrainment ratio and resolve shocks, andrecommended the use of the realizable k-\u000f model. Mazelli et al. [52] verified the suitability of thek-\u000f, realizable k-\u000f, SST k-\u03c9 and stress-\u03c9 Reynolds stress models for the ejector flow prediction,and concluded that the value of the entrainment ratio shows a minor variation with changesin the turbulence model, with the SST k-\u03c9 model producing the most accurate results. They[52] also concluded that at low primary flow pressure, \u000f-based models predict the ejector flowand performance more accurately when compared to the \u03c9-based models. Riffat et al. [60] usedthe RNG k-\u000f, standard k-\u000f and Reynolds stress models to predict the ejector entrainment ratioand recommened that the RNG k-\u000f yields improved results. Garcia del Valle et al. [61] usedthe standard k-\u000f and k-\u03c9 models to predict the ejector flow and entrainment ratio in the mixedand critical regime operations. They [61] showed that, for the critical regime operation, both192.5. Optimization techniquesk-\u000f and k-\u03c9 models produce accurate results when compared against the experimental data;however, SST k-\u03c9 yields more accurate results. For the mixed regime operation, both k-\u000f andk-\u03c9 models could not yield reliable results for the entrainment ratio; however, the prediction forthe pressure profiles agree.2.5 Optimization techniquesThe background related to the techniques used for controlling the supersonic flow structuresis discussed below.2.5.1 Supersonic diffuser designsThe supersonic diffusers translate the kinetic energy into potential energy minimizing thetotal pressure loss. They, for example, are used in supersonic wind tunnels [62] and supersonicengine intakes [62], where the flow should slow down with a focus on the pressure recovery. Thispressure recovery is achieved either through a single compression wave or a series of compressionand expansion waves. These waves generate adverse pressure gradients which make the bound-ary layer susceptible to the flow separation. Such flow separations contribute to the entropygeneration. The experimental studies verify that a supersonic flow\u2019s continuous compression tosubsonic flow occurs only in the presence of shocks. Though these shocks cannot be avoided,their strength can be decreased by incorporating the diffuser design modifications with an aimto maximize the total pressure recovery.Martin et al. [62] and Wen et al. [63] investigated the diffuser designs and their impact onthe diffuser flow behaviour. Both of these studies incorporated simple geometrical changes in thediffuser designs to modify the supersonic flow structure. Martin et al. [62] visualized the shocks202.5. Optimization techniquesand the flow separation zones in a supersonic diffuser of a supersonic ejector. They [62] increasedthe total pressure at the primary inlet to generate a series of oblique shocks and facilitate theshock-boundary layer interactions. They [62] concluded that an optimum value of the diffusercross-section area maximizes the total pressure recovery. However, the diffuser design used intheir study [62] is a constant-area duct with the main focus on the total pressure recovery.Wen et al. [63] investigated the influence of the back pressure on the diffuser performance andthe total pressure recovery. They [63] used a conical and a curved-wall as well as a second-throat diffuser design. They [63] concluded that the second-throat diffusers result in the mostfavourable pressure recovery by limiting the expansions of the shock trains, and the diffusers areless sensitive to the back pressure changes.2.5.2 Flow over a wedgeAn approach for optimizing the diffuser design is to insert a bluff-body in the diffuser [64].This bluff body is expected to alter the supersonic shock structure in the diffuser. There areanalytical and experimental studies that show the impact of the wedge-type shapes on thesupersonic flow structure, see for example [65\u201367]. Ommi et al. [68] used a design algorithmto optimize the shock arrangement in a supersonic diffuser of a supersonic engine intake. Suchintake comprises a supersonic diffuser, a throat section and a diffuser section. They [68] modifiedthe shape of the supersonic diffuser by implementing a wedge-shape which alters the shockarrangement and maximizes the total pressure recovery. Their [68] results compare well againsttheir analytical model.The upstream Mach number (Ma1) and the wedge angle (\u03b4) are the two main parameterswhich affect the shock train and the downstream Mach number of the flow [66, 67]. The wedgeangle determines whether the generated shock is an oblique or a bow shock. The analytical212.5. Optimization techniquesrelationship between the upstream Mach number, wedge angle and angle of deflected obliqueshock (\u03b8) is given by [66, 67]tan \u03b4 =[2tan \u03b8(Ma21 sin2 \u03b8 \u2212 1)][Ma21(\u03b3 + cos 2\u03b8) + 2] . (2.3)Equation (2.3) yields two solutions for \u03b8. If the oblique shock is weak, the flow downstreamof the shock is supersonic, as shown in Fig. 2.3(a). If the oblique shock is powerful, the flowdownstream to the shock is subsonic, as shown in Fig. 2.3(a). The oblique shock is expected tooccur at the leading edge of the wedge. If the wedge angle is greater than a critical wedge angle(\u03b4max) there exists a bow shock instead of an oblique shock, as shown in Fig. 2.4. A bow shockdoes not attach to the leading edge of a wedge [66, 67].Fig. 2.3. Supersonic flow over a wedge and presentation of an oblique shock.222.6. Literature GapFig. 2.4. Supersonic flow over a wedge and presentation of a bow shock.2.6 Literature GapThe review of literature suggests that the understanding related to the spatial variationof the entropy generation rate in supersonic ejectors and how this changes by changing theejector geometry and operational conditions is limited. Few studies [33\u201335] considered thedesign improvements by entropy generation minimization in supersonic ejectors. Although thesestudies attributed viscous dissipation as the major irreversibility contributing mechanism, thereis limited information about the turbulence driven irreversibilties, the influence of eddy viscosityon mean flow entropy generation, and location-dependent impact of the shock strengths andshear layers. The mathematical model used for the entropy generation analysis by Sierra-Pallares et al. [33] is similar to the one used in the present study. However, Sierra-Pallareset al. [33] mathematical model did not account for the eddy viscosity effects on the meanflow. The mathematical model used in the present study takes into account the effects ofturbulence in addition to the mean flow irreversibilities which can be monitored separately.Further, despite several investigations have been performed for understanding the gas dynamics232.6. Literature Gapof supersonic flows over wedges, it is unclear how these can be used to control the flow topologyin the supersonic ejectors in order to minimize the entropy generation and possibly enhance theejectors performance. All these literature gaps are addressed in this thesis.24Chapter 32D Computational MethodsIn this chapter, the governing equations, the fluxes schemes used for the discretization ofconvective terms, gradient terms and Laplacian terms, the time integration scheme, verifica-tion of the solver, solver settings, turbulence modeling, computational domain and boundaryconditions, spatial mesh, validation study, and the simulated conditions are presented.3.1 Governing equationsReynolds-averaging is applied to the Navier-Stokes equations for simplifying turbulence inthe fluid flow problem. The unsteady Navier-Stokes equations for compressible flows consistingof the conservation of mass, momentum and energy [69] are given by\u2202\u03c1\u2202t+\u2202\u2202xi(\u03c1ui) = 0 , (3.1)\u2202\u2202t(\u03c1ui) +\u2202\u2202xj(\u03c1ujui) = \u2212 \u2202p\u2202xi+\u2202\u03c3ji\u2202xj, (3.2)\u2202\u2202t[\u03c1(e+12uiui)]+\u2202\u2202xj[\u03c1uj(h+12uiui)]=\u2202\u2202xj(ui\u03c3ij)\u2212 \u2202qj\u2202xj, (3.3)where \u03c1, p, ui, e and h are the density, pressure, velocity component in the ith direction of253.1. Governing equationsCatersian coordinates, specific internal energy and specific enthalpy, respectively. The specificenthalpy is expressed as h = e+ p\/\u03c1. The viscous stress tensor (\u03c3ij) is expressed as \u03c3ij = \u03c3ji =2\u00b5(Sij\u2212 13 \u2202uk\u2202xk \u03b4ij), where Sij is the strain rate and is given by Sij = 12(\u2202ui\u2202xj+ \u2202ui\u2202xi). The subscriptsi and j denote the axis normal to the surface and the direction of the stress, respectively. \u03b4ij isthe Kronecker delta which is equal to 1 for i = j and is equal to 0 for i 6= j. qj is the heat fluxvector expressed as qj = \u2212\u03ba \u2202T\u2202xj , where \u03ba is the thermal conductivity. Using the ideal gas law,the pressure is calculated from p = \u03c1RT = (\u03b3 \u2212 1)\u03c1e.The Reynolds averaging for a dependent variable, \u03a6(x, t), is given by [70]\u03a6\u00af(x, t) =1\u03b6\u222b\u03b6\u03a6(x, t)dt . (3.4)\u03a6(x, t) can be split into a mean and a fluctuating term using \u03a6 = \u03a6\u00af + \u03a6\u2032. The density-weightedtime average, referred to as Favre average, is defined as \u03a6\u02dc = \u03c1\u03a6\u03c1\u00af and Favre decomposition isexpressed as \u03a6 = \u03a6\u02dc + \u03a6\u2032\u2032. Reynolds decomposition and Favre decomposition can be used todecompose the flow variables as follows: \u03c1 = \u03c1\u00af + \u03c1\u2032, p = p\u00af + p\u2032, q = q\u00af + q\u2032, ui = u\u02dci + u\u2032\u2032i ,e = e\u02dc+ e\u2032\u2032, and h = h\u02dc+ h\u2032\u2032. Using these decompositions along with the governing Navier-StokesEqs. (3.1\u20133.3) and further time-averaging lead to the unsteady compressible Reynolds-averagedNavier-Stokes equations as follows [71, 72]\u2202\u03c1\u00af\u2202t+\u2202\u2202xi(\u03c1\u00afu\u02dci) = 0 , (3.5)\u2202\u2202t(\u03c1\u00afu\u02dci) +\u2202\u2202xj(\u03c1\u00afu\u02dciu\u02dcj) = \u2212 \u2202p\u00af\u2202xi+\u2202\u03c3ij\u2202xj\u2212 \u2202\u2202xj(\u03c1u\u2032\u2032i u\u2032\u2032j), (3.6)263.2. Flux schemes\u2202\u2202t[\u03c1\u00af(e\u02dc+12u\u02dciu\u02dci)+12\u03c1u\u2032\u2032i u\u2032\u2032i]+\u2202\u2202xj[\u03c1\u00afu\u02dcj(h\u02dc+12u\u02dciui)+ u\u02dcj\u03c1u\u2032\u2032i u\u2032\u2032i2]=\u2202\u2202xj[u\u02dci(\u03c3\u00afij \u2212 \u03c1u\u2032\u2032i u\u2032\u2032j)\u2212 q\u00afj \u2212 \u03c1u\u2032\u2032jh\u2032\u2032 + \u03c3jiu\u2032\u2032i \u221212\u03c1u\u2032\u2032ju\u2032\u2032i u\u2032\u2032i].(3.7)The mean pressure is given by p\u00af = (\u03b3\u2212 1)\u03c1\u00afe\u02dc. The viscous stress tensor (\u03c3\u00afij) is approximated as\u03c3\u00afij \u2248 2\u00b5\u02dc(S\u02dcij \u2212 13 \u2202u\u02dck\u2202x\u02dck \u03b4ij), wtih \u00b5 being the local dynamic viscosity [71, 72]. In the momentumequation, Eq. (3.6), \u03c1u\u2032\u2032i u\u2032\u2032j represents the Reynolds stresses. In the present study, Reynoldsstresses are modeled using the eddy viscosity model which is described in Subsection 3.7. Theseunsteady compressible Reynolds-averaged Navier-Stokes equations are solved in the Eulerianframe of reference using the rhoCentralFoam solver of OpenFOAM v5.0 [73].3.2 Flux schemesGreenshields et al. [73] describe the development of the rhoCentralFoam solver and imple-mentation of its governing equations. This solver uses a finite volume method based on the fluxschemes of Kurganov and Tadmor [74]. The discretization process of convective terms, gradientterms, and Laplacian terms based on Kurganov and Tadmor\u2019s flux schemes is discussed in thissection.3.2.1 Discretization of convective termsThe general representation of the convective term is \u2207 \u00b7 [u\u03a8], where \u03a8 can be \u03c1, (\u03c1u), (\u03c1E)or p. For the numerical calculations, the entire computational domain is discretized into smallercomputational cells. The convective term is integrated over each control volume defined by eachcomputational cell as follows [73, 75]273.2. Flux schemes\u222bV\u2207 \u00b7 [u\u03a8]dV =\u222bSdS \u00b7 [u\u03a8] \u2248\u2211f[Sf \u00b7 uf ] \u03a8f =\u2211f\u03c6f\u03a8f , (3.8)where Sf is the face area vector and the volumetric flux (\u03c6f) is calculated using \u03c6f = Sf \u00b7uf . Thevelocity at the cell face (uf) is obtained by using a central-differencing scheme to interpolatevalues between the cell centroids of neighbouring cells. The volume integral in Eq. (3.8) isconverted to the surface integral using Gauss divergence theorem [76]. This surface integral isfurther approximated as a linear summation (\u2211f) over the faces of control volume. In finitevolume methods, the quantities are obtained at the cell centroids by interploating the values atthe cell faces. rhoCentralFoam uses a central-differencing scheme to interpolate values from cellfaces to cell centroids, which is second-order accurate [73]. Upwind-differencing scheme is usedto calculate \u03a8f . The linear interpolation of \u03a8f is obtained using\u03a8f = \u03c9f\u03a8P + (1\u2212 \u03c9f) \u03a8N , (3.9)where \u03c9f is the weighting coefficient calculated using \u03c9f = | ~Sf \u00b7 ~dfN|\/| ~Sf \u00b7 ~dPN|. The subscriptP denotes the cell centre of the owner cell, and subscript N denotes the cell centre of theneighbouring cell. ~dfN is the connecting vector between the face center of owner and neighbouringcells, and the cell centre of the neighbouring cell N. ~dPN is the connecting vector between the cellcenter of the owner cell P and the neighbouring cell N. \u03a8p and \u03a8N denote the value of the flowproperty at points P and N, respectively. The following equation is used for the discretizationof the convective terms\u2211f\u03c6f\u03a8f =\u2211f\u039b\u03c6f+\u03a8f+ +\u2211f(1\u2212 \u039b)\u03c6f\u2212\u03a8f\u2212 +\u2211f\u03c9f (\u03a8f\u2212 \u2212\u03a8f+) . (3.10)283.2. Flux schemesIn Eq. (3.9), f+ and f\u2212 denote the outward and inward flux directions at the cell face f,respectively. The first two terms on the right-hand side of the Eq. (3.10) are the fluxes in thedirections of f+ and f\u2212, and the third term accounts for the discontinuity existing at the face f.In Eq. (3.10), \u03c9f = \u039b max(\u03c8f+ , \u03c8f\u2212). Kurganov and Tadmor flux scheme is a central-differencingand \u039b is equal to 0.5 [74]. The volumetric fluxes \u03c8f+ and \u03c8f\u2212 corresponding to the local speedsof transmission of discontinuities in the f+ and f\u2212 directions are, respectively, given by\u03c8f+ = max (cf+ |Sf |+ \u03c6f+ , cf\u2212 |Sf |+ \u03c6f\u2212) . (3.11)\u03c8f\u2212 = max (cf+ |Sf | \u2212 \u03c6f+ , cf\u2212 |Sf | \u2212 \u03c6f\u2212) . (3.12)The local speed of sound at cell face f is given bycf\u00b1 =\u221a\u03b3RTf\u00b1 , (3.13)where R is the specific gas constant, \u03b3 is the ratio of specific heats and T is the local gastemperature.Since shock waves tend to produce spurious oscillations near the locations of pressure discon-tinuities, to improve numerical stability, the interpolation scheme uses the van Leer flux limiter,\u03b2vl(r), for all the flow variables. The van Leer flux limiter is given by [77]\u03b2vl(r) =r + |r|1 + |r| , (3.14)where r is the ratio of sucessive gradients. For a polyhedral mesh, the value of r in f+ direction iscalculated as r = 2d\u00b7(\u2207\u03a8)p(\u2207d\u03a8)f \u2212 1 [73]. This flux limiting scheme is total variation diminishing and293.2. Flux schemessymmetric [78]. A symmetric flux limiter follows \u03b2(r)r = \u03b2(1r). The flux limiter switches to thefirst-order accuracy at pressure discontinuities for solutions that have higher-order accuracy [79].For instance, when using the flux limiter, the solver uses the following equation for interpolationof flow variables in f+ direction\u03a8f+ = (1\u2212 gf+) \u03a8P + gf+\u03a8N . (3.15)In Eq. (3.15), gf+ is given by gf+ = \u03b2vl(r) (1\u2212 \u03c9f).3.2.2 Discretization of gradient termsThe discretization of gradient terms is second-order accurate. The gradient terms are dis-cretized using [73, 75]\u222bV\u2207\u03a8dV =\u222bSdS\u03a8 \u2248\u2211fSf\u03a8f . (3.16)To obtain the right-hand-side of Eq. (3.16), the cell values are interpolated in the f+ and f\u2212directions using\u2211fSf\u03a8f =\u2211f[\u039bSf\u03a8f+ + (1\u2212 \u039b)Sf\u03a8f\u2212 ] . (3.17)When a discontinuity is encountered, the gradient terms are limited in the same manner asconvective terms using the van Leer flux limiter [73, 75].303.3. Time integration3.2.3 Discretization of Laplacian termsThe discretization of Laplacian terms is second-order accurate. For obtaining the diffusioncoefficient (\u0393f) correponding to each face f, the values at the cell centroids are linearly interpo-lated using [73, 75]\u222bV\u2207 \u00b7 [\u0393\u2207\u03a8]dV =\u222bSdS[\u0393\u2207\u03a8] \u2248\u2211f\u0393fSf \u00b7 [\u2207\u03a8] . (3.18)3.3 Time integrationA second-order accurate backward time integration scheme is used in the present simulations.In this scheme, the value of the flow variable for the current time step is calculated using thevalues from the previous two time steps as follows [80]\u2202\u2202t\u222b\u222b\u222bV\u03c1\u03c6dV =3 (\u03c1P\u03c6PV )i \u2212 4 (\u03c1P\u03c6PV )i\u22121 + (\u03c1P\u03c6PV )i\u221222\u2206t, (3.19)where superscript i refers to the value from the current time step, i \u2212 2 and i \u2212 1 refer to thevalues from previous time steps, \u2206t is the time step, and \u03c1p is the density at the centre of theowner cell.3.4 Solution approachThe rhoCentralFoam solver sequentially solves the governing equations and hence is calleda segregated solver [73]. The predictor-corrector method is used to solve the equations forthe conservation of momentum and energy at each time step [73]. In the predictor step, thecontinuity equation, Eq. (3.5), is first explicitly solved followed by the momentum equation, Eq.313.5. Solver verification in literature(3.6), with viscous terms equated to zero. This inviscid solution is used in the corrector stepfor solving the momentum equation with the viscous terms implicitly. The energy equation, Eq.(3.7) is solved similarly to estimate the enthalpy. Then, the density, temperature and pressureare obtained from the continuity equation, enthalpy, and equation of state, respectively. Finally,the flow calculation proceeds to the next time step.3.5 Solver verification in literatureSeveral studies in the past verified the reliability of the rhoCentralFoam solver for compress-ible flow applications [73, 81, 82]. Zang et al. [81] have verified the capability of rhoCentralFoamto capture shock waves, shear layers and mixing charactersitics with high-accuracy. Greenshieldset al. [73] and Azadboni et al. [82] analyzed the shock capturing ability of this solver. Besidesthis, rhoCentralFoam is a transient solver and is expected to produce more accurate resultswhen compared with steady-state solvers. The capabilities of the unsteady Reynolds-averagedNavier-Stokes equations for predicting the qualitative unsteady flow characterisitcs and quan-titative time-averaged flow properties have been assessed in the literature [83, 84]. The flowinside supersonic ejectors is compressible and encompasses the formation of shock waves andshear layers. Moreover, OpenFOAM is an open-source CFD software and offers reduced cost forsolving engineering problems. Hence, the rhoCentralFoam solver is well-suited for the presentstudy.3.6 Solver settingsThe discretization of divergence, gradient and Laplacian schemes is second-order accurate,as discussed in Subsection 3.2. The rhoCentralFoam solver has an explicit behaviour, and hence323.7. Turbulence modelingto ensure numerical stability, the Courant-Friedrichs-Lewy (Co) number is set to 0.5 for all thesimulated conditions. This is an acceptable Co number for simulation of supersonic flows [85].Co number is a function of the fluid velocity (u), mesh size (\u2206x) and time step (\u2206t) and ismathematically expressed as Co = u\u2206t\u2206x [86]. In the present study, \u2206t is adjusted maintaining aconstant value for the Co number. The value of \u2206t varies 2\u00d710\u22128\u20135\u00d710\u22128 s. The convergenceis reached when the mass flow imbalance indicated by the globally integrated continuity erroris less than or equal to 10\u22126 kg\/s. The working fluid in this study is air. The ratio of specificheats (\u03b3) is 1.4. The thermophysical properties for air areTable 3.1. Thermophysical properties for the numerical setup.M(g\/mol) Cp(J\/(kg \u00b7K)) \u00b5(Pa \u00b7 s) Pr29 1005 1.831\u00d7 10\u22126 0.705In Table 3.1, M and Pr are the molecular weight of air, and the Prandtl number, respectively.The Prandtl number is given byPr =Cp\u00b5\u03ba. (3.20)The value of Cp and \u00b5 for air is a function of the temperature. Since the temperature variationthroughout the computational domain is not significant in the present study, the values of Cpand \u00b5 at T = 298 K are used.3.7 Turbulence modelingThe entrainment process in the supersonic ejectors is governed by the turbulent mixingbetween primary and secondary streams [87], further validated by the primary stream Reynoldsnumber at the nozzle exit, which is of the order of 105, for various simulated conditions in333.7. Turbulence modelingthe present study. To account for the flow\u2019s turbulent characteristics, it becomes imperative todeploy a turbulence model. The present study uses the standard k-\u000f model [69, 72]. This model iseasy to implement and computationally inexpensive when compared to other turbulence models.The k-\u000f model is used extensively and validated for various applications ranging from modelingenvironmental to industrial flows [88]. This model is well-suited for studies with confined flowswherein Reynolds shear stresses play a significant role [69], which is similar to the present study.Though it can accurately predict the flow properties in a fully developed turbulent region, it haspoor performance in the near-wall region [72]. This limitation of inaccuracy in predicting theboundary layer is solved to an extent by using wall functions. The choice and implementationof wall functions are described later in Subsection 3.8. A review of literature shows that the k-\u000fmodel is extensively validated for predicting the flow in supersonic ejectors and gives a goodapproximation of the mean flow quantities and the ejector entrainment ratio [55\u201358].The standard k-\u000f model is a two-equation eddy viscosity model based on the Boussinesqhypothesis [89]. This hypothesis is used to model the eddy viscosity (\u00b5t) as follows\u03c4ij = \u2212\u03c1u\u2032\u2032i u\u2032\u2032j = 2\u00b5t(Sij \u2212 13\u2202u\u02dck\u2202xk\u03b4ij)\u2212 23\u03c1\u00afk\u03b4ij , (3.21)Originally developed by Launder et al. [90], this model solves two additional transport equationsto predict the turbulent kinetic energy (k) and turbulent dissipation rate of kinetic energy (\u000f),respectively, given by\u2202(\u03c1k)\u2202t+\u2202\u2202xj[\u03c1uj\u2202k\u2202xj\u2212(\u00b5+\u00b5t\u03c3k)\u2202k\u2202xj]= \u03c4ijSij \u2212 \u03c1\u000f+ \u03c6k , (3.22)and343.8. Computational domain and boundary conditions\u2202(\u03c1\u000f)\u2202t+\u2202\u2202xj[\u03c1uj\u000f(\u00b5+\u00b5t\u03c3\u03b5\u2202\u000f\u2202xj)]= C\u03b51\u000fk\u03c4ijSij \u2212 C\u000f2f2\u03c1\u000f2k+ \u03c6\u03b5 , (3.23)where \u03c4ij represents the Reynolds stresses and can be calculated using Eq. (3.21). The eddyviscosity is calculated using \u00b5t = \u03c1C\u00b5k2\u000f . The damping function (f2) is given by f2 = 1 \u2212 0.3 \u00b7exp(\u2212Re2t), where Ret is the turbulent Reynolds given by Ret \u2261 \u03c1k2\u03b5\u00b5 . The wall terms (\u03c6k and\u03c6\u03b5) are given by \u03c6k = 2\u00b5(\u2202\u221ak\u2202y)2and \u03c6\u03b5 = 2\u00b5\u00b5t\u03c1(\u22022us\u2202y2)2, respectively, where us is the meanvelocity. The values of the remaining constants are listed in Table 3.2. The results obtainedusing this turbulence model are compared against the experimental data from the literature [52],which is discussed later in Subsection 3.10.Table 3.2. Constant values for k-\u000f turbulence model.C\u00b5 \u03c3k \u03c3\u000f C\u000f1 C\u000f20.09 1.0 1.3 1.45 1.923.8 Computational domain and boundary conditionsFigure 3.1 presents the computational domain\u2019s schematic and the set boundary conditions.The details regarding the domain and mesh coordinates are provided in Appendix A. The totalpressure and total temperature conditions are imposed at both primary (p0p, T0p) and secondary(p0s, T0s) inlets, and fixed static pressure is imposed at the outlet (pb). The details regarding theappropriate syntax to be used for the total pressure boundary condition suitable for the presentstudy is provided in Appendix B. The walls are adiabatic with no-slip boundary conditions. Thecomputational domain is 2D. Half of the computational domain is simulated with a symmetryboundary condition to reduce the computational cost. The turbulence intensity (I) is specifiedas 5% for both the inlets. The details regarding the sensitivity of the numerical results to I are353.8. Computational domain and boundary conditionsprovided in Appendix C. The boundary values of the turbulent kinetic energy and the turbulentdissipation rate are calculated, respectively, usingkx=0 =32Iu2avg , (3.24)and\u000fx=0 = C3\/4\u00b5k3\/2`, (3.25)where uavg and ` are the mean flow velocity, and integral length scale, respectively. The integrallength scale is taken equal to the lateral dimension of the primary inlet.Fig. 3.1. Schematic of the computational domain.The boundary layer development is approximated using wall functions which are utilized toprovide boundary conditions for the turbulence model for the near-wall region. When usingwall functions, the first step is to calculate the friction velocity (u\u03c4 ) and wall shear stress (\u03c4w).The calculated friction velocity is used to obtain boundary conditions for the turbulent kineticenergy and the rate of dissipation of turbulent kinetic energy at the location of the first gridpoint next to the wall. The calculated wall shear stress is used to estimate the diffusion termsin Eqs. (3.1\u20133.3) at the location of the first grid point next to the wall. The friction velocity isgiven by363.8. Computational domain and boundary conditionsu\u03c4 =\u221a\u03c4w\u03c1w, (3.26)where \u03c1w is the fluid density evaluated at the center of the first grid point next to the wall.In the near-wall region, the viscous sub-layer is closest to the wall, which is followed bythe buffer region and the log-law region [91]. The wall functions bridge the gap between theviscous sub-layer and the fully developed turbulent region away from the wall. For an accurateapproximation of the boundary layer, the centre of the first boundary cell should lie in the log-law region. This means that the value of y+ should vary between 30\u2013300 [92], which is followedin the present study. y+ is a non-dimensional quantity given by [93]y+ =yu\u03c4\u03bd, (3.27)where y is the distance between the wall and the center of a given cell. The dimensionlessvelocity (u+) is used to identify the various regions of a boundary layer and is given byu+ =uu\u03c4. (3.28)In the viscous sub-layer,u+ = y+ . (3.29)The equation for the velocity profile in the log-law region is given byu+ =1\u03bavln(y+)+B , (3.30)373.9. Spatial meshwhere \u03bav is the von Karman constant which varies from 0.40 to 0.41 [91]. B is the additionalconstant ranging from 4.9 to 5.5 [91]. In the absence of the wall functions, the centre of thefirst boundary cell should lie in the viscous sub-layer. This increases the mesh size significantly,adding to the computational cost. Hence, the application of wall function helps with decreasingthe mesh size and the computational cost significantly.The wall functions used in the present study are listed in Table 3.3 [93]. The epsilonWall-Function and kqRWallFunction pertain to high Reynolds number flows and are accurate forsimulating the flow in supersonic ejectors. The nutkWallFunction and alphatWallFunction areinsensitive to the Reynolds number.Table 3.3. Wall functions for the k-\u000f turbulence model.Wall function DescriptionepsilonWallFunction High Reynolds number flowkqRWallFunction High Reynolds number flow, Pure zero gradient boundary conditionnutkWallFunction Turbulent kinematic viscosity boundary condition based on turbulent kinetic energyalphatWallFunction Turbulent thermal diffusivity boundary condition3.9 Spatial meshThe computational domain is discretized using a structured mesh, as shown in Fig. 3.2. Themesh is refined at the boundaries and at interior regions where steep gradients are expected toprevail because of the primary and secondary streams\u2019 interactions.To ensure mesh independency, the solution is computed for four different meshes, for theejector operation at p0p\/p0s = 5 and pb\/p0s = 1. Such operating conditions result in complexflow features such as shock trains, shear layer instabilities, and recirculation zones; and, as aresult, these conditions are well-suited for the mesh independence study. Initially, the mesh iscontinuously refined in the y-direction to fulfill the y+ criterion. This procedure results in Mesh383.9. Spatial meshFig. 3.2. Mesh details at selected locations.1 with 35,806 cells, as seen in Table 3.4. The subsequent two mesh arrangements are obtainedby refining Mesh 1 in the x-direction with a refinement ratio (rx) of 1.5. Mesh 2 and Mesh 3comprise 53,076 cells and 79,156 cells, respectively. The refinement ratio in the x-direction isreduced to 1.2 to obtain Mesh 4 with 94,836 cells. The refinement ratio is decreased to savethe computational cost. The entrainment ratio for the four meshes is tabulated in Table 3.4. Itis observed that the entrainment ratio is constant for all four meshes. Also, the time-averagedstatic pressure along the ejector centerline for the four mesh systems is presented in Fig. 3.3.It is observed that Mesh 3 and Mesh 4 result in approximately similar quantitative values forthe time-averaged centerline static pressure. All four meshes result in approximately the samevalues except for the diffuser region, where there is evidence of shock-induced flow separation.The meshes differ in predicting the shock location, but the differences are minimal between Mesh3 and Mesh 4. Hence, Mesh 3 with 79,156 cells is used to simulate all the cases.Table 3.4. Comparison of entrainment ratio for four different meshes.Mesh # Mesh 1 Mesh 2 Mesh 3 Mesh 4no. of cells 35,806 53,076 79,156 94,836m\u02d9s\/m\u02d9p 0.248 0.248 0.248 0.248393.10. Validation study0 10 20 30 40 50 60012345 105212'1'12'21'Fig. 3.3. Time-averaged centerline static pressure for four different meshes.3.10 Validation studyThe computational model is validated against the experimental and numerical data usingthe results of Mazelli et al. [52]. The numerical results of Mazelli et al. [52] are based on the2D simulations with the standard k-\u000f turbulence model. Figure 3.4 presents the instantaneousMach number profile obtained from the validation study. The geometrical configuration used inthe present study, as shown in Fig. 3.1, is very similar to the validation study, as shown in Fig.3.4. Thus, it can provide a suitable test case for validating the employed computational method.The solver settings in this validation study are as discussed earlier in Subsections 3.1\u20133.9, andthe spatial mesh has an equivalent resolution as Mesh 3 as shown in Table 3.4.403.10. Validation studyPrimary inlet Secondary Inlet OutletSecondary InletMa0.0e+00 2.3e+000.5 1 1.5 2Fig. 3.4. Instantaneous Mach number profile inside the ejector for the validation study at p0p= 5 bars, p0s = 1 bar and pb = 1.2 bars. For the presented case, 2D simulation is carried outusing the standard k-\u000f turbulence model.The centerline static pressure variation for the ejector operation at a primary flow pressureof 5 bars, secondary flow drawn from ambient and back pressure set to 1.2 bars is presentedin Fig. 3.5. The selected operation condition results in a shock-induced flow separation in thediffuser. The static pressure profile predicted from the validation study differs in predicting thediffuser-shock location in the amplitude of static pressure in the diffuser-shock vicinity. It isspeculated that this inaccuracy is due to the shock waves producing spurious oscillations andintroducing numerical errors, and as a result contaminating the solution. Similar observationis made by Lusher et al. [94] while analysing the shock-wave\/boundary-layer interactions intransitional rectangular duct flows.The entrainment ratio is compared for three sets of operating conditions, as presented inTable 3.5. Two of the tested conditions correspond to an on-design ejector operation, and onecorresponds to an off-design ejector operation. The relative error in predicting the entrainmentratio for on-design ejector operation is about 3.2%. This relative error increases to about 14%when the ejector operates in an off-design mode. Literature suggests that the discrepanciesbetween the experimental and numerical results for an off-design ejector operation are signifi-cantly higher than an on-design ejector operation [52]. Therefore, the results predicted by ourvalidation study are within acceptable limits.413.11. Simulated cases0 0.2 0.4 0.6 0.8 1 1.2 1.4012345105Fig. 3.5. Centerline static pressure for the ejector operation at p0p = 5 bars, p0s = 1 bar andpb = 1.2 bars. For the presented case, 2D simulation is carried out using the standard k-\u000fturbulence model.Table 3.5. Comparison of the entrainment ratio reported in the literature and in this validationstudy. The symbol * represents the results from Mazelli et al. For all the presented cases,2D simulations are carried out using the standard k-\u000f turbulence model. The relative error iscalculated between the Mazelli et al. numerical results and the validation study results.Operating Conditions m\u02d9s\/m\u02d9p Relative errorMode p0p (bar) p0s (bar) pb (bar) Experimental\u2217 Numerical\u2217 Validation Study (%)On-design 5 1 1.2 0.497 0.515 0.513 0.38On-design 3.5 1 1.2 0.794 0.786 0.811 3.18Off-design 2 1 1.2 0.591 0.727 0.827 13.753.11 Simulated casesTable 3.6. Simulated conditions for the constant-pressure mixing ejector. For all the presentedcases, p0s = 1 bar and pb = 1 bar.p0p\/p0sD\/d = 2 D\/d = 2.5 D\/d = 3Mne Rene \u00d7105 Mne Rene \u00d7105 Mne Rene \u00d71052 0.85 5.51 1.14 5.49 1.70 4.392.5 1.53 5.86 1.67 5.58 1.71 5.493 1.68 6.67 1.70 6.55 1.71 6.583.5 1.71 7.62 1.71 7.66 1.71 7.684 1.71 8.68 1.71 8.77 1.71 8.734.5 1.71 9.85 1.71 9.88 1.71 9.875 1.71 10.94 1.71 10.93 1.71 10.99In this thesis, a total of 45 cases are simulated to understand the impact of the operating423.11. Simulated casesconditions and geometrical parameters on the ejector\u2019s performance. Tables 3.6 and 3.7 tabulatepressure ratios and diameter ratios for the constant-pressure and constant-area mixing ejectors,respectively. In both the tables, Mne denotes the primary nozzle exit centerline Mach number,and Rene is the primary nozzle exit centerline Reynolds number, which is calculated based on thediameter of the primary nozzle and the velocity, dynamic viscosity, and density at the centerlineof the primary nozzle exit. For all the simulated cases, the secondary stream has an imposedinlet total pressure of 1 bar and the ejector outlet static pressure is set to 1 bar. Please notethat the test conditions presented in Tables 3.6 and 3.7 pertain to those investigated in bothChapters 5 and 6. For the test conditions discussed in Chapter 5, no inclined plate is used. InChapter 6, either one or two inclined plates are inserted in the diffuser.Table 3.7. Simulated conditions for the constant-area mixing ejector. For all the presented cases,p0s = 1 bar and pb = 1 bar.D\/dp0p\/p0s = 2 p0p\/p0s = 3.5 p0p\/p0s = 5Mne Rene \u00d7105 Mne Rene \u00d7105 Mne Rene \u00d71052 1.71 4.40 1.71 7.61 1.80 10.822.5 1.75 4.39 1.71 7.83 1.80 10.963 1.71 4.40 1.71 7.69 1.80 10.683.5 1.71 4.40 1.71 7.69 1.80 10.524 1.68 5.24 1.71 7.54 1.80 10.464.5 1.37 5.68 1.71 7.69 1.80 10.685 1.01 5.44 1.71 7.55 1.80 10.985.5 0.90 5.42 1.71 7.73 1.80 10.5643Chapter 41D Analytical Model of the EjectorIn this chapter, the development and implementation of a one-dimensional (1D) analyticalmodel of the ejector are discussed. The model objectives and assumptions are discussed inSubsection 4.1. The governing equations, implementation, and solution methods are discussedin Subsection 4.2. Finally, the results are presented in Subsection 4.3.4.1 Model descriptionBefore conducting the 2D computational analysis, a 1D analytical model of the ejector wasdeveloped based on certain assumptions. As discussed in Subsection 2.4.1, a 1D analyticalmodel is simpler and computationally less expensive than a full solution to the compressibleNavier-Stokes equations described in Subsection 3.1; and, yet, the 1D model yields reasonablepredictions for critical flow quantities, including pressure, density, temperature, velocity, andMach number. The literature has extensive information on the validity of the 1D analyticalmodel for predicting ejector entrainment ratio, as highlighted in Subsection 2.4.1 [14, 46\u201351].Also, the 1D analytical model can predict the qualitative trends of the ejector entrainment ratioas a function of the operating configurations, which is of interest for industrial applications.This model is useful as it allows to understand the overall ejector performance capturing thevariation of flow quantities when resolving the flow intricacies is not a critical requirement. The444.1. Model descriptionresults from the 1D model will be used for comparisons with those of the 2D simulations.The assumptions used to develop the 1D analytical model are provided below.1. The flow is steady-state. That is, flow properties remain constant irrespective of the timevariations.2. The flow is 1D. That is, flow characteristics in the direction perperndicular to the fluidflow are neglected.3. The process is adiabatic, and isentropic relations apply, i.e. the isentropic efficiency forthe compressible flow inside the ejector nozzle, mixing chamber, and diffuser is 100%.4. The working fluid is calorically perfect, i.e., the ratio of specific heats (\u03b3) and the specificheat capacity at constant pressure (Cp) is constant throughout the computational domain.The working fluid is air with \u03b3 = 1.4 and R = 287 J\/(kg \u00b7K).5. The ejector walls are adiabatic.6. A normal shock occurs after the mixing section, matching the ejector back pressure value(pb) of 1 bar.7. The ejector operates in the critical regime, i.e., the primary and secondary streams arechoked during the ejector operation.8. The primary jet initially flows without mixing with the secondary flow and forms a hy-pothetical throat, as shown in Fig. 4.1. The secondary flow chokes at the minimumcross-section area to attain Mach number of 1. The secondary flow is followed by the mix-ing between the two flows at the choking plane. The mixing process occurs at a uniformpressure to facilitate the constant-pressure mixing.454.2. Governing equationsFig. 4.1. Ejector schematic for the constant-pressure mixing model. Section 1\u20131 represents theejector inlet. Section 2\u20132 is at the primary nozzle throat. Section 3\u20133 represents the primarynozzle exit and the secondary stream inlet.The 1D thermodynamic model is developed in MATLAB. It is governed by the principles ofgas dynamics and conservation laws and is used to estimate the variations of the flow propertieswithin the ejector at a given set of operating conditions.4.2 Governing equationsThe 1D model\u2019s governing equations and their solution are discussed below.4.2.1 Flow through the primary nozzleFollowing to assumption 6 above, the ejector operates in the critical regime, which meansboth primary and secondary flows are choked. The primary inlet flow is initially subsonic,attains Ma=1 at the nozzle throat, and accelerates along the nozzle diverging section andbecomes supersonic. The subscripts p and s in the following denote the quantities associatedwith the primary and secondary flows, respectively. The subscript 0 denotes the stagnationcondition. The Mach number at the inlet and outlet of the primay nozzle is estimated using thearea-Mach number relation as follows [95]464.2. Governing equationsApxA\u2217p=(\u03b3 + 12)\u2212 \u03b3+12(\u03b3\u22121)(1 + \u03b3\u221212 Ma2px) \u03b3+12(\u03b3\u22121)Mapx, (4.1)where A\u2217p is the nozzle throat area, and Mapx is the Mach number at the corresponding primarynozzle area (Apx). The primary flow rate (m\u02d9p) for the nozzle choked condition is calculatedusing [95]m\u02d9p =Ap1p0p\u221aT0p\u221a\u03b3RMap1(1 +\u03b3 \u2212 12Ma2p1)\u2212 \u03b3+12(\u03b3\u22121), (4.2)where Ap1, p0p, T0p, Map1 are the nozzle inlet area, stagnation pressure of the primary flow,stagnation temperature of the primary flow, and the primary flow inlet Mach number, respec-tively. The stagnation pressure and stagnation temperature conditions at the primary inlet ofthe ejector are known. The stagnation temperature is set to 298 K for seven different operatingstagnation pressures of 2, 2.5, 3, 3.5, 4, 4.5 and 5 bar. The stagnation density of the primaryflow (\u03c10p) is calculated using the ideal gas equation of statep0p = \u03c10pRT0p . (4.3)Using the known primary nozzle inlet conditions, Eqs. (4.4\u20134.6) are used to obtain local staticpressure (ppx), temperature (Tpx) and density (\u03c1px) which are, respectively, given byppxp0p=(1 +\u03b3 \u2212 12Ma2px)\u2212 \u03b3\u03b3\u22121, (4.4)TpxT0p=(1 +\u03b3 \u2212 12Ma2px)\u22121, (4.5)474.2. Governing equations\u03c1px\u03c10p=(1 +\u03b3 \u2212 12Ma2px) \u22121\u03b3\u22121. (4.6)The local Mach number (Mapx) and local temperature (Tpx) are used to calculate the localvelocity (upx), usingupx = Mapx\u221a\u03b3RTpx . (4.7)4.2.2 Secondary and primary flow through the mixing chamberAfter the primary flow exits the nozzle, it forms a hypothetical duct and continues to expanduntil it reaches the constant-area section of the mixing chamber. The inlet stagnation pressure(p0s) and stagnation temperature (T0s) of the secondary flow are known, because the secondaryflow is assumed to be drawn from the ambient surroundings, where p0s = 1 bar and T0s = 298 K.Following to the assumption 6 above, the secondary flow is choked; and, thus, it attains a Machnumber of Masy = 1 at the entrance plane of the constant-area section of the mixing chamber.The secondary flow pressure (psy) at the y-y plane is obtained usingpsyp0s=(1 +\u03b3 \u2212 12Ma2sy)\u2212 \u03b3\u03b3\u22121. (4.8)Since the primary and secondary flows mix at y-y plane, the primary flow pressure (ppy) isassumed to equal the secondary flow pressure. The primary Mach number (Mapy) at the y-yplane is obtained using Eq. 4.4 by substituting ppx = ppy and Mapx = Mapy. Equations (4.5\u20134.7) are used to calculate the flow parameters at the y-y plane. The primary flow area (Apy) atthis plane is calculated using484.2. Governing equationsApy =m\u02d9p\u221aT0pp0p\u221aR\u03b31Mapy(1 +\u03b3 \u2212 12Ma2py) \u03b3+12(\u03b3\u22121). (4.9)The cross-section area (Ay) at the y-y plane is known, and the secondary flow area (Asy) at they-y plane is given byAsy = Ay \u2212Apy . (4.10)The secondary mass flow rate (m\u02d9s) is estimated fromm\u02d9s =Asyp0s\u221aT0s\u221a\u03b3RMasy(1 +\u03b3 \u2212 12Ma2sy)\u2212 \u03b3+12(\u03b3\u22121). (4.11)Finally, the entrainment ratio is given byEntrainment ratio = m\u02d9s\/m\u02d9p . (4.12)4.2.3 Mixed flow upstream of the shockThe subscript \u201cmix\u201d is used to denote mixed flow. Conservation of momentum is used tocalculate the resultant velocity (umix) of the mixed flow using Eq. (4.13) [47]. Following theconservation of linear momentum, the rate of change of momentum of a control volume is equal tothe summation of net external forces acting on the control volume (Fcv), i.e.,ddt(mu)cv =\u2211Fcv[96]. For the present 1D analytical model, the y-y plane is a fixed plane at the entrance of theconstant-area section of the mixing chamber (see Fig. 4.1). Considering the constant-pressuremixing assumption, the static pressures of the primary, secondary and fully-mixed flow are equalat the y-y plane. Hence, assuming an infinitesimally small control volume around the y-y plane,494.2. Governing equationsthe net external forces (pressure forces) exerted to the control volume is zero.umix =m\u02d9pupy + m\u02d9susym\u02d9p + m\u02d9s. (4.13)The conservation of energy is used to calculate the resultant temperature (Tmix) of the mixedflow using [47](m\u02d9p + m\u02d9s)CpTmix +(m\u02d9p + m\u02d9s)2u2mix = m\u02d9pCpTpy + m\u02d9sCpTsy +m\u02d9pu2py2+m\u02d9su2sy2. (4.14)Since the mixing between primary and secondary stream at y-y plane is occuring at constant-pressure, the pressure energy is neglected in Eq. 4.14. The Mach number of the mixed flow(Mamix) is calculated fromMamix =umix\u221a\u03b3RTmix. (4.15)The stagnation parameters of the mixed flow are obtained fromp0mix = pmix(1 +\u03b3 \u2212 12Ma2mix) \u03b3\u03b3\u22121, (4.16)T0mix = Tmix(1 +\u03b3 \u2212 12Ma2mix)1, (4.17)\u03c10mix = \u03c1mix(1 +\u03b3 \u2212 12Ma2mix) 1\u03b3\u22121, (4.18)504.2. Governing equationswhere p0mix, T0mix and \u03c10mix are the stagnation pressure, stagnation temperature, and stagnationdensity of the mixed flow, respectively.4.2.4 Mixed flow downstream of the shockAn iterative process is used to predict the shock location (Lsk, see Fig. 4.1), to accommodatethe back pressure effect. This is performed through matching the ejector back pressure to 1 bar.Initially, the normal shock is assumed to be located at the diffuser entrance, i.e. Lsk = 0. Forthe formulations presented below, subscript 1 refers to upstream of the shock, and subscript 2refers to downstream of the shock. The Mach number evaluated upstream of the shock (Ma1)is estimated fromMa1(1 +\u03b3 \u2212 12Ma21)\u2212 \u03b3+12(\u03b3\u22121)=(m\u02d9s + m\u02d9p)\u221aT0mix1Askp0mix1R\u03b3, (4.19)where Ask is the cross-section area at the shock location. Equation (4.19) is a quadratic equationand has two solutions. The mixed flow entering the diffuser for the tested operating pressuresis supersonic; and, hence, the solution that yields Ma1 > 1 is used for the calculations, and theother solution is discarded. The total temperature remains constant across a shock; however,the total pressure changes [95]. The stagnation pressure downstream of the shock (p0mix2) isestimated fromp0mix2 = p0mix1[(\u03b3 + 1)Ma21(\u03b3 \u2212 1)Ma21 + 2] \u03b3\u03b3\u22121[(\u03b3 + 1)2\u03b3Ma21 \u2212 (\u03b3 \u2212 1)] 1\u03b3\u22121, (4.20)where p0mix1 and Ma1 are the stagnation pressure, and Mach number upstream of the shock,respectively. The Mach number downstream of the shock (Ma2) is obtained from514.2. Governing equationsMa22 =(\u03b3 \u2212 1)Ma21 + 22\u03b3Ma21 \u2212 (\u03b3 \u2212 1). (4.21)The normal shock produces subsonic flow downstream [95]. Thus, the solution that yieldsMa1 < 1 is used for the further calculations and the other solution is discarded. The staticpressure downstream of the shock (pmix2) is given bypmix2 = p0mix2(1 +\u03b3 \u2212 12Ma22)\u2212 \u03b3\u03b3\u22121. (4.22)The variation of the static pressure in the diffuser region downstream of the shock (px) is givenbypx = p0mix2(1 +\u03b3 \u2212 12Ma2x)\u2212 \u03b3\u03b3\u22121, (4.23)where p0mix2 and Max are the stagnation pressure of the mixed flow downstream of the shock,and Mach number at the corresponding location, respectively. Provided the calculated staticpressure at the ejector outlet does not match the back pressure, the shock location is increasedby 10 \u00b5m steps, i.e. Lsknew = Lskold + 0.00001, and the above procedure in this section isrepeated with the new shock location.4.2.5 Solution methodFigure 4.2 shows the overview of the sequential path adopted in solving the governing equa-tions of the 1D analytcial model. The studies of Huang et al. [46] and Chen et al. [47] serve asbasis for the design algorithm used to setup the 1D analytical model in the present study.524.3. Results and DiscussionsFig. 4.2. Flow chart of the 1D ejector flow model solver.4.3 Results and DiscussionsThe quality of the present study\u2019s 1D analytical model\u2019s predictions is compared to thepredictive accuracy of the other 1D models in the literature. The 1D analytical results of Huanget al. [46], Chen et al. [47] and Selvaraju et al. [49] are used from the literature. For comparisons,the relative error (\u2206) is defined as\u2206 =\u03c7analytical \u2212 \u03c7exp.\/num.\u03c7exp.\/num., (4.24)where \u03c7 denotes the quantity of interest. Huang et al. [46] used a 1D analytical model andexperiments to estimate the entrainment ratio for their ejector operation in the critical flowregime. In [46], the isentropic efficiencies obtained from the experiments were used in the 1D534.3. Results and Discussionsanalytical model to improve its accuracy. The results of [46] show that the maximum relativeerror is less than 23%. Chen et al. [47] predicted the coefficient of performance of a refrigerationcycle with an ejector operating in the critical flow regime and mixed regime. They [47] used thereal gas properties in the 1D model to improve its accuracy. The maximum relative error of [47]is less than 25%. Selvaraju et al. [49] predicted the ejector entrainment ratio in the critical flowregime. The variations in the specific heat as well as the frictional losses were accounted for intheir [49] model. The maximum relative error of [49] is less than 15%.Table 4.1 shows the quantitative comparisons of the primary mass flow rate, the secondaryflow rate and the entrainment ratio obtained from the 1D analytical model and 2D numericalresults of the present study. The 2D numerical results are discussed in detail in Chapter 5.The 1D analytical results are compared to the 2D numerical results for the ejector operating atp0p\/p0s = 2, 2.5, 3, 3.5, 4, 4.5 and 5 and with D\/d = 2.5. For p0p\/p0s = 2 and 2.5, the ejectoroperates in the mixed flow regime, and for the rest of the tested stagnation pressure ratios, theejector operates in the critical flow regime. The secondary flow is drawn at a stagnation pressureof 1 bar.Table 4.1. Ejector performance parameters. Comparison of the 2D numerical and 1D analyticalresults for the constant-pressure mixing ejector. For all the presented cases, D\/d = 2.5 and p0s= 1 bar. The symbol * represents the ejector operation in the mixed flow regime.Operating Condition 1D Analytical 2D Numerical Error %p0p (bar) m\u02d9s (kg\/s) m\u02d9p (kg\/s) m\u02d9s\/m\u02d9p m\u02d9s (kg\/s) m\u02d9p (kg\/s) m\u02d9s\/m\u02d9p |\u2206\u03c9| |\u2206m\u02d9s |2\u2217 0.0057 0.0088 0.648 0.005 0.0076 0.658 1.52 142.5\u2217 0.0052 0.0109 0.477 0.005 0.0096 0.521 8.45 43 0.0047 0.0131 0.359 0.0052 0.0114 0.456 21.27 93.5 0.0042 0.0153 0.275 0.005 0.0132 0.379 27.44 164 0.0037 0.0175 0.211 0.005 0.0152 0.329 35.86 264.5 0.0032 0.0197 0.162 0.005 0.017 0.294 44.89 365 0.0028 0.0219 0.128 0.0046 0.0214 0.215 40.46 39.13The results in Table 4.1 show that the maximum relative error for predicting the entrainmentratio is less than 45%. The individual values of the primary mass flow rate and secondary mass544.3. Results and Discussionsflow rate are also compared. The maximum relative error for predicting the secondary massflow rate is less than 40%. In contrary to what the literature suggests, the 1D analytical modelin the present study is able to predict fairly accurately the secondary mass flow rate and theentrainment ratio for the ejector operation in the mixed flow regime. That is the relative errorin predicting the secondary mass flow rate and the entrainment ratio is less than 14% and9%, respectively. The flow losses in the mixed flow regime are expected to be significantlysmaller when compared to the critical flow regime. For the ejector operation in the critical flowregime, increasing the primary stagnation pressures, increases the relative error in predictingthe secondary mass flow rate and the entrainment ratio. It is expected that, at large primarystagnation pressures, the losses are significant when compared to those of the smaller primarystagnation pressures. Since the 1D analytical model does not account for the flow losses andisentropic efficiencies, it overpredicts the value of the secondary flow rate. Hence, for quantitativepredictions, the 1D analytical model in the present study performs inferior to the 1D analyticalmodels presented in the literature. The values predicted by the 1D analytical for the primarymass flow rate also deviate from the 2D numerical predictions. As the primary stream flowsthrough the primary inlet section, it encounters a sudden decrease in the cross-section area at theprimary nozzle throat. This sudden decrease leads to the flow separation at the sharp corners,forming a vena-contracta near the primary nozzle throat. As a result, the primary flow, infact, chokes further downstream of the primary nozzle throat, which reduces the actual chokingcross-section area. The 1D ejector flow model does not account for this type of flow behaviour,and as a result, the primary mass flow rate obtained from the 2D numerical analysis is smallerthan the 1D analytical predictions. This deviation in the prediction for the primary mass flowrate also incurs an added error in the prediction of the entrainment ratio. The incapability of554.3. Results and Discussionsthe 1D analytical model to account for the vena-contract formation also induces an error inpredicting the primary nozzle exit Mach number. The maximum relative error of Mane is 8.5%.2.5 3 3.5 4 4.5 5 5.500.10.20.30.40.50.60.7Fig. 4.3. Variation of the entrainment ratio versus the stagnation pressure ratio. D\/d is thearea ratio.Although the 1D analytical model does not facilitate accurate quantitative prediction of theejector performance, it can be used to study the performance qualitatively. Figure 4.3 showsthe variation of the entraiment ratio versus the stagnation pressure ratio for several area atiosand for the constant-pressure mixing ejector. The 2D numerical and 1D analytical results areshown by the solid and dashed curves, respectively. As can be seen, the 1D analytical modelaccurately captures the inverse relations between the entrainment and stagnation pressure ratios.Figures 4.4 and 4.5 show the comparisons of the 2D numerical and analytical predictions for thevariations of the pressure and Mach number along the normalized ejector centerline axis (x\/d).In the 1D analytical model, a single normal shock is assumed to be formed after the flows are564.3. Results and Discussions(a)0 10 20 30 40 50 600246 1050 5005105(b)0 10 20 30 40 50 6001230 5002Fig. 4.4. Variation of (a) pressure and (b) Mach number along the normalized ejector centerlineaxis. The solid and dashed curves are the predictions of the 2D numerical and 1D analyticalsimulations, respectively. For the presented case, p0p\/p0s = 5 and D\/d = 2.5.(a)0 10 20 30 40 50 6001234 1050 50024 105(b)0 10 20 30 40 50 6000.511.520 50012Fig. 4.5. Variation of (a) pressure and (b) Mach number along the normalized ejector centerlineaxis. The solid and dashed curves are the predictions of the 2D numerical and 1D analyticalsimulations, respectively. For the presented case, p0p\/p0s = 3.5 and D\/d = 2.5.574.3. Results and Discussionsmixed, based on the fixed static pressure of 1 bar at the ejector outlet. This is compared withthe 2D numerical results, for which a series of oblique shock waves in the diffuser are formed tomaintain the outlet back pressure at 1 bar. The 1D analytical model cannot resolve the mixingchamber shock trains and diffuser flow separation phenomena. Though the 1D analytical resultsprovide a reasonable qualitative estimation of the overall ejector performance and flow behaviour,it is essential to perform 2D numerical simulations to investigate the detailed flow characteristics.This is discussed in the following chapters.58Chapter 5Characterization of the EjectorPerformanceIn this chapter, first, influences of the operational conditions and geometrical parameterson the ejector performance (entrainment ratio) and entropy generation rate are presented inSubsection 5.1. The results are discussed for two configurations: constant-pressure and constant-area mixing chamber ejectors. In Subsection 5.2, the flow topology related to the investigatedconditions and configurations is presented. Finally, in Subsection 5.3, it is elaborated how thegenerated flow topologies influence the entropy generation and the entrainment ratio.5.1 General performance parametersThis section presents the results of the ejector CFD simulations following the methodologydescribed in Chapter 3. A total of 45 simulations were performed, as described in Subsection3.11. The performance parameters that are of interest are the entrainment ratio and the overallentropy generation rate. The variations of the entrainment ratio (m\u02d9s\/m\u02d9p) and the entropygeneration rate integrated over the ejector volume (\u222bS\u02d9\u2032\u2032\u2032gendV ) versus the stagnation pressureratio (P0p\/P0s) are presented in Figs. 5.1(a) and 5.2(a), respectively, for the constant-pressuremixing ejector, and for several values of area ratio (D\/d). For the constant-pressure mixing595.1. General performance parametersejector, the variation of m\u02d9s\/m\u02d9p versus\u222bS\u02d9\u2032\u2032\u2032gendV is presented in Fig. 5.1 (c) for all the studiedconditions. The results pertaining to the constant-area mixing ejector are presented in Fig. 5.2.For the results presented in Figs. 5.1 and 5.2,\u222bS\u02d9\u2032\u2032\u2032gendV was estimated using Eq. (2.2). In thisequation, the values of the air thermal conductivity (\u03ba), the specific heat at constant pressure(Cp), and the dynamic viscosity (\u00b5) are set to 0.02624 W\/(m \u00b7K), 1005 J\/K and 0.00001831Pa \u00b7 s, respectively. The simulation results show that the temperature varies from a minimum of140 K to a maximum of 298 K. This temperature variation does not lead to significant variationsof \u03ba, Cp and \u00b5. Thus, these parameters were assumed to remain constant and the correspondingvalues were estimated at 298 K. The rhoCentralFoam solver allows to calculate the values of thetemperature (T ), velocity components (u, v, and w), turbulent thermal diffusivity (\u03b1t), density(\u03c1), and mean dissipation rate of the turbulent kinetic energy (\u000f).1 2 3 4 5 60.10.30.50.70.91.11 2 3 4 5 60246810120 5 10 150.10.30.50.70.91.1Fig. 5.1. Performance curves for the constant-pressure mixing ejector: (a) entrainment ratio ver-sus stagnation pressure ratio, (b) integrated entropy generation rate versus stagnation pressureratio, and (c) entrainment ratio versus integrated entropy generation rate.605.1. General performance parameters0 2 4 600.511.50 2 4 602468100 2 4 6 8 1000.511.5Fig. 5.2. Performance curves for the constant-area mixing ejector: (a) entrainment ratio versusstagnation pressure ratio, (b) integrated entropy generation rate versus stagnation pressure ratio,and (c) entrainment ratio versus integrated entropy generation rate.The results in Fig. 5.1(a and b) show that, for a constant-pressure mixing ejector, increas-ing the stagnation pressure ratio decreases the entrainment ratio, but increases the integratedentropy generation rate for all the studied values of D\/d. It can be seen that, at a fixed valueof the stagnation pressure ratio, increasing D\/d increases the entrainment ratio and decreasesthe integrated entropy generation rate. Similar results are also observed for the constant-areamixing ejector, as shown in Fig. 5.2(a and b). The results in Fig. 5.2(a) also agree with thosein [20\u201323]. It is anticipated that increasing D\/d should increase the effective flow area for thesecondary stream, which should ultimately increase the entrainment ratio. However, the resultsof Liu et al. [30] and Jia et al. [24] suggest that increasing D\/d first increases the entrain-ment ratio, and then decreases this parameter. They [30] attributed the increase of entrainmentratio with increasing D\/d to the increased effective flow area of the secondary stream. How-ever, further increasing D\/d causes the ejector to perform in the off-design operation mode;and, hence, the entrainment ratio decreases [30]. The results presented in Figs. 5.1(a and b)and Figs. 5.2(a and b) show that, for both the tested configurations, increasing D\/d generallyincreases the entrainment ratio, except at p0p\/p0s = 2 and for D\/d \u2265 4 for the constant-area615.2. Ejector internal flow featuresmixing ejector. It is speculated that, for these conditions, the ejector operates in the off-designmode. This is similar to the results presented in [16, 97]. The results in Figs. 5.1(c) and 5.2(c)show that for all the simulated conditions and configurations, the entrainment ratio and theintegrated entropy generation rate are negatively related. Similar observations are reported in[33, 35]. This negative correlation is somewhat expected because the physical processes leadingto entropy generation rate decrease the available work of the system and degrade the mixingefficiency. These physical processes arise as a result of the flow separation phenomenon, shearlayer instablities and formation of shock structures. Despite the behaviour shown in Figs. 5.1and 5.2 is expected, how such behaviour arise as a result of local flow irreversibilties is not clearand requires investigations. Also, the results in Fig. 5.2(c) show that, for the constant-areamixing ejector and at p0p\/p0s = 5 and p0p\/p0s = 3.5, the entrainment ratio features a largesensitivity to\u222bS\u02d9\u2032\u2032\u2032gendV . The reason for this behaviour is unknown to the author, and this alsorequires further investigations.5.2 Ejector internal flow featuresTo facilitate elaborating the performance trends presented in Figs. 5.1 and 5.2, the time-averaged flow characterisitics for several conditions and for both the constant-pressure andconstant-area mixing ejectors are presented in Subsections 5.2.1 and 5.2.2, respectively. Thedetails regarding the statistical analysis of the time-averaged flow variables are provided inAppendix D.625.2. Ejector internal flow features5.2.1 Constant-pressure mixing ejectorFigure 5.3 shows the contours of the magnitude of the time-averaged numerical schlieren forp0p\/p0s = 2, 3.5 and 5. For the results shown in the figure, D\/d = 2.5. The numerical schlieren(\u2207\u03c1\u2217) is given by\u2207\u03c1\u2217 = | \u2207\u03c1 | Linlet\u03c1inlet, (5.1)where \u2207\u03c1 is the gradient of the density, Linlet is the vertical length of the primary inlet, and\u03c1inlet is the static density at the primary inlet. The magnitude of the numerical schlieren allowsto visualize both the shock structures and the shear layers. For p0p\/p0s = 2, the ejector operatesin the mixed flow regime, and for p0p\/p0s = 3.5 and 5, the ejector operates in the critical flowregime. The results in Fig. 5.3 show that the stagnation pressure ratio substantially influences(a) p0p\/p0s = 2(b) p0p\/p0s = 3.5(c) p0p\/p0s = 5\u2207\u03c1\u22170.0e+00 4.0e+001 2 3Fig. 5.3. Time-averaged magnitude of the numerical schlieren contours for the constant-pressuremixing ejector. (a), (b), and (c) pertain to p0p\/p0s = 2, 3.5, and 5, respectively. For all thepresented cases, D\/d = 2.5.the shocks and their locations within the ejector. For p0p\/p0s = 2, there exists a normal shock at635.2. Ejector internal flow featuresthe exit of the primary nozzle because of the over-expansion mode and one normal shock withinthe diffuser. As the stagnation pressure ratio increases, the primary nozzle mode of operationchanges from the over-expanded to under-expanded. This gives rise to the first shock trainin the mixing chamber downstream of the primary nozzle, and the second shock train in thediffuser, see Fig. 5.3(b). Further increasing the stagnation pressure ratio from 3.5 to 5 leadsto the diffuser shock train move downstream wherein the shocks are interconnected by a Machreflection. However, in the mixing chamber, increasing p0p\/p0s from 3.5 to 5 causes the shocktrain elements (which are connected via Mach discs) to connect via regular reflections.Figure 5.4 shows the time-averaged Mach number contours superimposed by the dividingstreamline and sonic line for the ejector operation at p0p\/p0s = 2, 3.5 and 5. The Mach number(a) p0p\/p0s = 2Shock # 1 Shock # 2(b) p0p\/p0s = 3.5S Shock Train # 1 Shock Train # 2    Recirculation Zones(c) p0p\/p0s = 5Shock Train # 1 Shock Train # 2 Recirculation Zones  Ma0.0e+00 1.6e+000.5 1Fig. 5.4. Time-averaged Mach number profile for the constant-pressure mixing ejector. Thesolid black line represents the dividing streamline and the solid white line represents the sonicline. (a), (b), and (c) pertain to p0p\/p0s = 2, 3.5, and 5, respectively. For all the presentedcases, D\/d = 2.5.645.2. Ejector internal flow featurescontours allow understanding the development of supersonic and subsonic regions within theejector. The dividing streamline, which originates from the primary nozzle lip and extendsdownstream, is defined as the locus of points where the stream function (\u03c8) equals zero [98].The streamline serves as a virtual boundary that separates the primary and secondary streams,and represents the mixing layer\u2019s edge. The sonic line is the locus of points where the Machnumber equals to 1. This line allows to locate the choking position of the secondary stream. Atthe choking location, the secondary stream reaches the minimum flow area.Separation Point Diffuser Wall Recirculation Zone # 1 Recirculation Zone # 2Ejector Centerline  Ma0.0e+00 1.6e+000.5 1Fig. 5.5. Streamlines in the diffuser region of the constant-pressure mixing ejector highlightingthe flow recirculation at p0p\/p0s = 5 and D\/d = 2.5. The streamlines are coloured by the localMach number.For all three simulated stagnation pressure ratios presented in Figs. 5.3 and 5.4, the diffuserflow is separated because of an oblique shock. For the ejector operation at p0p\/p0s = 2, therecirculation zones are absent in the flow-separated region, as can be seen in Fig. 5.4(a). In thisregion, the Mach number is close to zero. Since the ejector operation is in the mixed regime forthis operating condition, the secondary stream does not choke. For higher stagnation pressureratios, when the ejector is operating in the critical regime, the shock-induced flow separation in655.2. Ejector internal flow featuresthe diffuser leads to generation of recirulation zones in the flow-separated region. The streamlinesassociated with this region are color coded based on the flow Mach number and are shown inFig. 5.5. As can be seen, two recirculation zones exist in this region. For the ejector operationat p0p\/p0s = 3.5, the dividing streamline and the sonic line diverge in the mixing chamber throatand the secondary stream attains minimum area at the diffuser entry plane, see Fig. 5.5. For theejector operation at p0p\/p0s = 5, the dividing streamline and the sonic line initially diverge insidethe mixing chamber throat until the choking location is reached. Then, moving downsteam, thedividing streamline and sonic line converge. As can be seen in Fig. 5.4(b and c), increasingp0p\/p0s from 3.5 to 5 moves the location of the secondary flow choking upstream.The normalized and time-averaged temperature (T \u2217) contours for the ejector operation atp0p\/p0s = 2, 3.5 and 5 are shown in Figs. 5.6(a), (b), and (c), respectively. T\u2217 is given byT \u2217 =TpT0p(5.2)where Tp is the static temperature at given point in the computational domain, and T0p is theprimary inlet stagnation temperature. The normalized and time-averaged temperature contoursallow to understand the quality of mixing between the primary and secondary streams. Theresults in Fig. 5.6 show that the uniformity of the temperature distribution within the mixingchamber decreases with increasing the stagnation pressure ratio. For p0p\/p0s = 2, the results inFig. 5.6(a) show that both the primary and secondary streams entering the mixing chamber aresubsonic. These subsonic streams mix relatively well as the flow moves downstream and towardsthe diffuser. This mixing reduces the initial primary stream temperature of the air, reducing thespeed of sound, increasing the Mach number, and as result, transitioning the flow to a supersonicstate. However, since the flow must decelerate at the end of the diffuser (accomodating for the665.2. Ejector internal flow featuresback pressure condition), a normal shock occurs in the diffuser. As shown in Figs. 5.6(b andc), the mixing between the primary and secondary streams is poor for p0p\/p0s = 3.5 and 5compared to that for p0p\/p0s = 2. Thus, the relatively uniform and low temperature condition(and as a result small speed of sound) is achieved at farther locations from the secondary flowentrance. Thus, the shocks locations are pushed farther downstream, as shown in Figs. 5.6(band c). The improved mixing related to p0p\/p0s = 2 is desired and facilitates achieving largem\u02d9s\/m\u02d9p as shown in Fig. 5.1(a). Aiming to optimize the ejector operation, the flow features atthis stagnation pressure ratio and for several values of D\/d were investigated.(a) p0p\/p0s = 2(b) p0p\/p0s = 3.5(c) p0p\/p0s = 5T \u22175.0e-01 1.0e+000.6 0.7 0.8 0.9Fig. 5.6. Normalized and time-averaged temperature contours for the constant-pressure mixingejector. (a), (b), and (c) pertain to p0p\/p0s = 2, 3.5, and 5, respectively. For all the presentedcases, D\/d = 2.5.Figure 5.7 presents the time-averaged Mach number contours superimposed with the dividingstreamline and the sonic line for the ejector operation at p0p\/p0s = 2 and for D\/d = 2, 2.5 and675.2. Ejector internal flow features3. For all of the simulated area ratios at this stagnation pressure ratio, the ejector operates inthe over-expanded mode. The secondary stream does not choke as can be seen by the dividingstreamlines and the sonic lines. Hence, the ejector is operating in the mixed regime. As thearea ratio increases, the effective flow area available for the secondary stream increases, andthis increases the entrainment ratio. The increase in the area ratio decreases the Mach numberdifference between the primary and secondary streams and the flow attains more uniform Machnumber distribution in the mixing chamber.(a) D\/d = 2(b) D\/d = 2.5(c) D\/d = 3Ma0.0e+00 1.6e+000.5 1Fig. 5.7. Time-averaged Mach number profile for the constant-pressure mixing ejector. (a), (b),and (c) pertain to D\/d = 2, 2.5 and 3, respectively. For all the presented cases, p0p\/p0s = 2.In essence, the results suggest that p0p\/p0s = 2 and D\/d = 3 leads to an improved mixingbetween the primary and secondary flows, absence of shock trains and recirculation zones in themixing chamber and the diffuser, and as a result, the exergy loss is expected to be minimized.Thus, such operating condition is desirable, and is recommended for operation of the constant-pressure mixing ejector. This is of relevance and importance to the industrial partner of this685.2. Ejector internal flow featuresproject.5.2.2 Constant-area mixing ejectorFigure 5.8 presents the contours of the time-averaged magnitude of the numerical schlieren forD\/d = 2.5\u20135.5. For the results shown in the figure, p0p\/p0s = 2, and the configuration pertainsto the constant-area mixing ejector. The primary nozzle operates in the under-expanded modefor D\/d = 2.5, 3, and 3.5. For D\/d \u2265 4, the ejector operates in the over-expanded mode. Ascan be seen, two independent shock trains are formed in the ejector: one in the mixing chamberand one in the diffuser for the under-expanded mode. It can also be seen that for this mode,the compression and expansion waves in the mixing chamber extend into the secondary stream.As the area ratio increases, the length of the first shock train decreases, and the diffuser shocktrain moves upstream, see Figs. 5.8(a\u2013c).Figure 5.9 presents the time-averaged Mach number contours superimposed with the dividingstreamline and sonic line for the conditions shown in Fig. 5.8. In agreement with the resultsshown in Fig. 5.8, for D\/d = 2.5, 3 and 3.5, the primary nozzle operates in the under-expandedmode. However, the primary nozzle operates in the over-expanded mode for D\/d = 4 and 5.5.Further increasing D\/d beyond 5.5 causes the primary nozzle shock location move upstream(not presented here). The results presented in Fig. 5.4 suggested that operation in the over-expanded mode condition is desirable in terms of the entrainment ratio. The results in Fig. 5.9show that such operation mode can be achieved for D\/d > 3.5. On one hand large values ofD\/d are desirable (for leading to ejector operation at the over-expanded mode); however, onthe other hand, too large values of D\/d may not facilitate enough momentum for the primarystream to induce sufficient secondary stream. Also, too large values of D\/d are of industrialdesign concerns. For the stagnation pressure ratio of 2, several values of D\/d were simulated695.2. Ejector internal flow featuresand it was obtained that, at D\/d \u2248 4, the entrainment ratio is maximized. This follows theresults presented in Fig. 5.2(a).(a) D\/d = 2.5(b) D\/d = 3(c) D\/d = 3.5(d) D\/d = 4(e) D\/d = 5.5\u2207\u03c1\u22170.0e+00 3.5e+015 10 15 20 25Fig. 5.8. Time-averaged magnitude of the numerical schlieren for the constant-area mixingejector. (a), (b), (c), (d), and (e) pertain to D\/d = 2.5, 3, 3.5, 4, 5.5, respectively. For all thepresented cases, p0p\/p0s = 2The reason for the optimized operation of the constant-area ejector at D\/d = 4 is furtherelaborated below. As shown in Fig. 5.9, the dividing streamline and the sonic line help to identifythe choking location of the secondary stream. For the under-expanded mode (conditions withD\/d . 4), the secondary stream remains supersonic throughout the mixing chamber. For the705.2. Ejector internal flow featuresover-expanded mode (conditions with D\/d & 4), the Mach number contours become ratheruniform, which indicate improved mixing. This was confirmed using the temperature contours(not shown here) similar to the results presented in Fig. 5.6. As can be seen in Fig. 5.9, the mostuniform Mach contours are achieved for D\/d = 4 condition. In essence, the results presented inthis section suggest that the best performance (maximized entrainment ratio) is achieved whenthe shock trains do not exist in the mixing chamber, the ejector operated in the over-expandedmode, and a subsonic primary stream entrains a subsonic secondary stream.(a) D\/d = 2.5(b) D\/d = 3(c) D\/d = 3.5(d) D\/d = 4(e) D\/d = 5.5Ma0.0e+00 1.8e+000.5 1 1.5Fig. 5.9. Time-averaged Mach number profile for the constant-area mixing ejector. The solidblack line represents the dividing streamline and the solid white line represents the sonic line.(a), (b), (c), (d), and (e) pertain to D\/d = 2.5, 3, 3.5, 4, 5.5, respectively. For all the presentedcases, p0p\/p0s = 2.715.3. Entropy generation analysis5.3 Entropy generation analysisAlthough in the previous subsection the flow topology that leads to improved performance ofthe ejector was studied, the exergy loss at this condition and how this influences the entrainmentratio is unclear and needs to be investigated. In this subsection, the knowledge of the time-averaged flow fields in the ejector for different operating configurations is used to quantify theejector losses in the constant-pressure mixing ejector and constant-area mixing ejector. Entropygeneration analysis is used to quantify the ejector losses. The entropy generation rate is reportedusing a non-dimensional parameter called the entropy generation number (Ns) which is given byNs =S\u02d9\u2032\u2032\u2032genm\u02d9pCpAp1, (5.3)where S\u02d9\u2032\u2032\u2032gen is the entropy generation rate per unit volume, m\u02d9p is the primary mass flow rate forthe ejector operation at p0p\/p0s = 2, Cp is the specific heat of air at constant pressure and, Ap1is the primary nozzle inlet area. The discussions presented in [39, 40, 44, 45] suggest that theentropy generation rate per unit volume has two contributing sources, which are related to themean (S\u02d9\u2032\u2032\u2032gen,mf) and turbulent (S\u02d9\u2032\u2032\u2032gen,tf) flows and are, respectively, given byS\u02d9\u2032\u2032\u2032gen,mf =\u03baT 2[(\u2202T\u2202x)2+(\u2202T\u2202y)2+(\u2202T\u2202z)2]\ufe38 \ufe37\ufe37 \ufe38Term 1+\u00b5+ \u00b5tT{(\u2202u\u2202y+\u2202v\u2202x)2+(\u2202u\u2202z+\u2202w\u2202x)2+(\u2202v\u2202z+\u2202w\u2202y)2+ 2[(\u2202u\u2202x)2+(\u2202v\u2202y)2+(\u2202w\u2202z)2]}\ufe38 \ufe37\ufe37 \ufe38Term 2.(5.4)725.3. Entropy generation analysisS\u02d9\u2032\u2032\u2032gen,tf =\u03b1\u03b1t\u03baT 2[(\u2202T\u2202x)2+(\u2202T\u2202y)2+(\u2202T\u2202z)2]\ufe38 \ufe37\ufe37 \ufe38Term 3+\u03c1\u000fT\ufe38\ufe37\ufe37\ufe38Term 4.(5.5)In Eqs. (5.4) and (5.5), Terms 1 and 2 are used to model the entropy generation rate due tothe heat transfer and viscous dissipation in the mean flow, respectively, and Terms 3 and 4 areused to model the effects of turbulence on entropy generation rate because of the heat transferand viscous dissipation, respectively.5.3.1 Constant-pressure mixing ejectorThe results presented in Fig. 5.1 showed that, the constant-pressure mixing ejector oper-ating at p0p\/p0s = 5 generates the smallest entrainment ratio. This condition is investigatedto understand the entropy generation mechanisms here. As discussed in Chapter 2, entropygeneration features contributions from the heat transfer and viscous dissipation mechanisms.Both mechanisms can feature contributions from the mean flow and the turbulent flow. Figure5.10 presents the contours of the time-averaged entropy generation number due to heat trans-fer and viscous dissipation mechanisms. These contours highlight the flow topologies centralto the ejector irreversibilties. The legend scales are different for both the entropy generationrate mechanisms; and, as can be seen, the heat transfer contribution is rather negligible. Thelocations that significantly contribute to the entropy generation rate correspond to the primarynozzle throat, converging-section of the mixing chamber and the diffuser. Specially, formationof the vena-contracta at the primary nozzle throat due to the upstream flow separation, shocksin the diffuser as well as the shear layer downstream of the primary nozzle and in the diffuserplay significant roles in the entropy generation rate, as shown in Fig. 5.10.735.3. Entropy generation analysis(a)0.0e+00 3.5e-010.1 0.15 0.2 0.25(b)Ns0.0e+00 6.5e+032000 3000 4000 5000Fig. 5.10. Entropy generation number contour due to (a) heat transfer mechanism and (b)viscous dissipation mechanism for the constant-pressure mixing ejector. For all the presentedcases, p0p\/p0s = 5 and D\/d = 2.5Bejan number (Be) [99] is used to quantify the relative dominance of the heat transfer andviscous dissipation mechanisms on the entropy generation rate for the ejector operation. Themathematical expression for the Bejan number is given byBe =S\u02d9\u2032\u2032\u2032gen,heatS\u02d9\u2032\u2032\u2032gen,total, (5.6)where S\u02d9\u2032\u2032\u2032gen,heat is the entropy generation rate due to the heat transfer mechanism, and S\u02d9\u2032\u2032\u2032gen,totalis the total entropy generation rate. Figure 5.11 presents the integral of the Bejan number alongthe vertical axis versus the normalized ejector axial distance for p0p\/p0s = 2, 3.5 and 5. Ascan be seen, the integral of the Bejan number is negligible, which suggests that the entropygeneration rate due to heat transfer can be neglected. This observation is independent of thestagnation pressure ratio value. Hence, the viscous dissipation mechanism dominates the entropygeneration rate within the ejector, and the heat transfer mechanism can be ignored.745.3. Entropy generation analysis0 10 20 30 40 50 6010-810-610-410-2Fig. 5.11. Integral of the Bejan number along the vertical axis versus the normalized ejectoraxial distance at p0p\/p0s = 2, 3.5 and 5 for the constant-pressure mixing ejector. For all thepresented cases, D\/d = 2.5.Figure 5.12(a) shows the integral of the overall entropy generation number versus the no-ramlized ejector axial distance at p0p\/p0s = 5. The entropy generation number at the primarynozzle throat is comparable to the entropy generation number in the diffuser, which is attirbutedto both regions undergoing flow separation. The flow separation phenomenon at the primarynozzle throat is because of the formation of vena-contracta due to the sharp corners encounteredby the subsonic primary flow. This flow separation pertains to a small area in the vicinity of theprimary nozzle throat. This is contrasting to the shock-induced flow separation in the diffuserwhich generates shear layer instabilities and pertains to a large portion of the diffuser.The contributions of the mean flow and turbulent flow to Ns at p0p\/p0s = 5 are estimatedusing Eqs. (5.3\u20135.5), and the results are presented in Fig. 5.12(b). The values of the entropygeneration pertaining to both contributions were integrated along the axial distance. The resultssuggest that the mean flow contributes approximately 67% and the flow turbulence contributesapproximately 33%. The diffuser section contributes more than 50% to the overall entropy gen-eration rate, which is due to the shock-induced flow separation phenomenon. The constant-areainlet, primary nozzle, mixing chamber convergent-section and mixing chamber throat contribute1%, 11%, 14% and 14% to the overall entropy generation rate, respectively.755.3. Entropy generation analysis(a)0 10 20 30 40 50 6000.10.20.30.40.50.60.7(b)0 10 20 30 40 50 600.10.20.30.40.50.60.7Fig. 5.12. (a) Integral of overall entropy generation number (Ns) versus the normalized ejectoraxial distance and (b) entropy generation number of the mean flow (Ns,mf) and turbulent flow(Ns,tf) versus the normalized ejector axial distance for the constant-pressure mixing ejector. Forall the presented cases, p0p\/p0s = 5 and D\/d = 2.5.Figures 5.13(a and b) present the variation of the mean flow entropy generation numberand the turbulent flow entropy generation number versus the normalized ejector axial distance,respectively. Three stagnation pressure ratios of 2, 3.5, and 5 are considered. The results arepresented in logarithmic scale for clarity. As can be seen, the stagnation pressure ratio doesnot significantly impact the mean flow entropy generation number at the primary nozzle throat.However, the turbulent flow entropy generation number increases with increasing the stagnationpressure ratio in the convergent section and in the mixing chamber throat. The contribution ofthe mean flow to Ns is rather unchanged by changing the stagnation pressure ratio and in thenozzle section. However, the contribution of the turbulent flow to Ns increases with increasingp0p\/p0s in this section. The effect of p0p\/p0s on the contributions of mean and turbulent flow on765.3. Entropy generation analysisNs depends on x\/d. The integrated values of Ns (for both mean and turbulent flow contributions)over the entire ejector computational domain suggest decreasing p0p\/p0s from 5 to 2 decreasesNs by 80%.(a)0 10 20 30 40 50 6010-2100(b)0 10 20 30 40 50 6010-2100Fig. 5.13. (a) Integral of entropy generation number of the mean flow and (b) the turbulentflow versus the normalized ejector axial distance for the constant-pressure mixing ejector. Theresults pertain to p0p\/p0s = 2, 3.5 and 5. For all the presented cases, D\/d = 2.5.The results presented in Figs. 5.12 and 5.13 show that the shocks can significantly increasethe entropy generation. In order to investigate this, the shock strength (Rs) was estimated forthe results in Fig. 5.13. Rs is defined as the ratio of the static pressure rise across the shock tothe static pressure inlet to the shock, and is given byRs =p2 \u2212 p1p1, (5.7)where p1 and p2 are the pressure at inlet to the shock and pressure downstream of the shock,respectively. The values of Rs for the shocks occuring in the ejector were calculated and the775.3. Entropy generation analysisresults are presented in Table 5.1. For the over-expanded mode, i.e. p0p\/p0s = 2, there existsa primary nozzle shock. No primary nozzle shock was detected for the under-expanded mode,i.e. p0p\/p0s = 3.5 and 5. The values of Rs in Table 5.1 suggest, for the under-expanded mode,the mixing chamber shocks are relatively weak in comparison to the first shock in the diffuser.However, the strength of the subsequent diffuser shock is comparable to the mixing chambershocks. Also, the results show that as the stagnation pressure ratio increases, the strength ofthe first diffuser shock increases.Table 5.1. Shock strength at different spatial locations of the constant-pressure mixing ejectorwith D\/d = 2.5.Nozzle Mixing Chamber DiffuserShock#1 Shock#1 Shock#2 Shock#3 Shock#4 Shock#1 Shock#2p0p\/p0s = 2 1.706 - - - - 1.315 -p0p\/p0s = 3.5 - 1.453 0.906 0.544 0.298 3.220 0.557p0p\/p0s = 5 - 0.433 0.312 - - 4.855 0.403Figures 5.14(a\u2013c) and (d\u2013f) present contributions of turbulent and mean flow to Ns. Theresults in Figs. 5.14(a and d), (b and e) as well as (c and f) correspond to p0p\/p0s = 2,3.5, and 5, respectively. The results in Fig. 5.14 generally confirm that the contribution ofboth mean flow and turbulent flow to entropy generation are significant, their cummulativecontributions are related to the formation of shock trains and shear layers (which are due to theseparated flow either upstream of the primary nozzle throat or the diffuser shock). Analysis ofthe individual contributions suggests that the entropy generation rate due to the formation ofshocks is attributed to the mean flow effect and turbulence has negligible effect on this. Usually,the turbulence dominant regions correspond to the shear layers at the flow separated zones andin the mixing chamber. It can be generally observed that increasing the stagnation pressureratio increases the entropy generation due to the shock formation in the diffuser which may beattributed to the increased value of Rs, as presented in Table 5.1.785.3. Entropy generation analysis(a) p0p\/p0s = 2(b) p0p\/p0s = 3.5(c) p0p\/p0s = 5(d) p0p\/p0s = 2(e) p0p\/p0s = 3.5(f) p0p\/p0s = 5Ns0.0e+00 1.6e+03500 1000Fig. 5.14. (a), (b), and (c) are contributions of the mean flow to Ns at p0p\/p0s = 2, 3.5, and 5,respectively. (d), (e), and (f) are contributions of the turbulent flow to Ns at p0p\/p0s = 2, 3.5,and 5, respectively. The results pertain to the constant-pressure mixing ejector with D\/d = 2.5.5.3.2 Constant-area mixing ejectorThe mean and turbulent flow contributions to Ns are presented in Figs. 5.15(a and b),respectively. The results pertain to a constant-area mixing ejector and are presented for D\/dranging from 2.5 to 5.5. As can be seen, the entropy generation (either due to the mean flowor the turbulent flow) is rather insensitive to the stagnation pressure ratio. For the mean flow795.3. Entropy generation analysiscontribution, increasing D\/d generally decreases the entropy generation number, as shown inFig. 5.15(a). Similar observation is made for the contribution of the turbulent flow and in thediffuser as shown in Fig. 5.15(b). The contribution of the turbulent flow to Ns and in the mixingchamber depends on x\/d.(a)0 10 20 30 40 50 6010-2100(b)0 10 20 30 40 50 6010-5100Fig. 5.15. (a) Integral of entropy generation number of the mean flow and (b) the turbulentflow versus the normalized ejector axial distance for the constant-pressure mixing ejector. Theresults pertain to D\/d = 2.5, 3, 3.5, 4, and 5.5. For all the presented cases, p0p\/p0s = 2.The results presented in Table 5.1 and Fig. 5.14 suggested that there exists a relationbetween the shock strength as well as the entropy generation number for constant-pressuremixing ejectors. This is speculated to extend to constant-area mixing ejectors as well. Thus,variation of the shock strength were estimated and presented in Table 5.2. The contours ofentropy generation number pertaining to mean and turbulent flow of constant-area ejectors arepresented in Figs. 5.16(a\u2013e) and (f\u2013j), respectively. As can be seen in the figure and Table 5.2,the values of Ns pertaining to the shock train in the mixing chamber do not alter significantly805.3. Entropy generation analysis(a) D\/d = 2.5(b) D\/d = 3.5(c) D\/d = 4(d) D\/d = 5.5(e) D\/d = 2.5(f) D\/d = 3.5(g) D\/d = 4(h) D\/d = 5.5Ns0.0e+00 1.1e+03200 400 600 800Fig. 5.16. (a), (b), (c), and (d) are contributions of the mean flow to Ns for D\/d = 2.5, 3.5,4 and 5.5, respectively. (e), (f), (g), and (h) are contributions of the turbulent flow to Ns forD\/d = 2.5, 3.5, 4 and 5.5, respectively. The results pertain to the constant-area mixing ejectoroperating at p0p\/p0s = 2.815.3. Entropy generation analysisTable 5.2. Shock strength at different spatial locations for the constant-area mixing ejectoroperating at p0p\/p0s = 2.Nozzle Mixing Chamber DiffuserShock#1 Shock#1 Shock#2 Shock#3 Shock#4 Shock#1 Shock#2 Shock#3D\/d = 2.5 - 0.286 0.198 0.144 0.104 2.114 0.863 0.167D\/d = 3 - 0.277 0.188 0.141 - 1.093 0.653 0.152D\/d = 3.5 - 0.270 0.199 0.146 - 1.400 0.465 0.108D\/d = 4 1.939 - - - - - - -D\/d = 5.5 1.514 - - - - - - -by changing D\/d for the under-expanded modes. This trend is similar to the values of Rspresented in Table 5.2. However, increasing D\/d decreases the value of Ns as shown in Fig. 5.16for the under-expanded mode. A similar observation is made for the values of Rs presented inTable 5.2 and the diffuser section pertaining to the under-expanded mode. Thus, it is speculatedthat the existence of the shock train in the diffuser plays a major role in the entropy generation ofthese ejectors. For the simulated conditions pertaining to the over-expanded mode, the diffusershock does not exist and the entropy generation number is decreased significantly. Analysis ofthe individual contributions suggest that the entropy generation rate due to the formation ofshocks is attributed to the mean flow effect and turbulence has negligible effect on this, whichis similar to the observation made for the constant-pressure mixing ejectors. The turbulencedominant regions correspond to the shear layers at the flow separated zones.The operation of supersonic ejectors is accompanied by the formation of flow topologiesincluding shear layers, shock trains, flow separated regions and recirculation zones which sig-nificantly influence the ejector performance quantified in terms of the entrainment ratio andvolumetric entropy generation rate. In addition to these flow features, the primary nozzleexpansion modes also significantly influence the the ejector performance which are mainly cat-egorized as the under-expanded and over-expanded modes of operation. As inferred from thepresent analysis, the mixing chamber shock trains and shear layers contribute relatively insignif-825.3. Entropy generation analysisicantly towards the entropy generation rate when compared to those of the shock trains andshear layers formed in the diffuser for an under-expanded mode of operation in both constant-area and constant-pressure mixing ejectors. Thus, it is desirable to operate the ejectors in theover-expanded mode to avoid the formation of flow separated regions in the diffuser and shocktrains in the mixing chamber. In addition to this, the analysis in this chapter allows to identifya strong correlation between the entrainment ratio and the volumetric entropy generation rate.Considering the applications wherein a given set of operating conditions or ejector geometrycannot be avoided and the ejector operation at lower stagnation pressure ratios is not possible,the analysis presented in this chapter helps to find out critical areas requiring localized designmodifications to improve the ejector performance. For ejector operation at higher stagnationpressure ratios, the major area requiring design changes is the diffuser. This is investigated inChapter 6.83Chapter 6Diffuser Design Modifications toReduce Entropy GenerationThe results presented in Chapter 5 showed that the entrainment ratio features a negativecorrelation with the integrated entropy generation rate which is greatly influenced by the flowtopologies prevailing in the supersonic ejector. This integrated entropy generation rate is de-pendent on the phenomena such as shocks and shear layers, which in turn, are related to theoperation conditions and the ejector geometrical design. In this chapter, geometrical modifi-cations are made to the ejector in order to decrease the entropy generation rate without anynegative impact on the entrainment ratio. Since the results discussed in Chapter 5 showedthat the diffuser plays the primary role in the entropy generation rate, the proposed designchanges\/modifications are related to the diffuser. These changes to the diffuser are discussedin Subsection 6.1. The impacts of these design modifications on the overall ejector operationand how this relates to the entropy generation rate are discussed in Subsections 6.2 and 6.3,respectively. Finally, the optimized design parameters are discussed and recommendations aresummarized in Subsection 6.4. Since the constant-pressure mixing ejectors are most relevant tothe industrial partner of this thesis, this configuration is studied here.846.1. Design parameters6.1 Design parametersThe results discussed in Chapter 5 showed that depending on the test conditions, supersonicflow may enter the diffuser, and a shock occurs later downstream in the diffuser. It was shownthat the diffuser may account for approximately 50% of the total entropy generation rate. This(a)(b)(c)(d)Fig. 6.1. Tested ejector diffuser designs of the constant-pressure mixing ejector: (a) single in-clined plate placed inside the diffuser, (b) two inclined plates connected at the leading edgeplaced inside the diffuser, (c) two diverging plates placed inside the diffuser and (d) two con-verging plates placed inside the diffuser. \u03b4 is the angle of inclination of the plates w.r.t. to thehorizontal line measured counter-clockwise for the top plate and clockwise for the bottom plate.significant entropy generation rate is associated with the shock structures inside the diffuser.856.1. Design parametersThe diffuser entropy generation rate is undesirable for applications demanding a higher pressurerecovery, such as refrigeration applications where the ejectors are used to replace the compressors.It may be possible to reduce the entropy generation rate by modifying the strength and structureof the compression waves that occur within the diffuser. Such changes can be triggered inthe flow by applying specific design modifications to the diffuser. A variety of diffuser designmodifications are presented in Fig. 6.1. The modifications introduce one or two oblique platesnear the entrance of the diffuser to break down the strong shock into a series of weaker shocks.Four designs are suggested and summarized as follows: Design 1: a single inclined plate insertedat the entrance of the diffuser; Design 2: two inclined plates connected at the leading edgeinserted at the entrance of the diffuser; Design 3: two inclined and diverging plates (in the formof a supersonic nozzle) inserted at the entrance of the diffuser; and (d) Design 4: two inclinedand converging plates inserted at the entrance of the diffuser.Table 6.1. Approximate freestream conditions used for finalizing the design parameters for theejector operation at p0p\/p0s = 3.5 and D\/d = 3.Ma p (Pa) T (K) \u03c1 (kg\/m3)1.53 61303.60 216.38 0.99To select\/design the plates dimensions, assumptions and approximations are made which areprovided below. First, the angle of inclination of the plates is approximated using theoreticalcalculations. The condition with no bluff body is used to approximate the Mach number,pressure, temperature, and density at the diffuser entrance as shown in Table 6.1. As discussedin Subsection 2.5.2, a wedge could either result in an oblique shock or a bow shock depending onthe wedge angle and the upstream Mach number [66]. A detached\/bow shock is stronger than anoblique shock [66]. To ensure an oblique shock occurs in the given upstream flow conditions, thetheoretical angle of inclination of the wedge (\u03b4) can be 5\u25e6, as deduced from oblique shock tables866.2. Overall ejector performance[66]. A relevant section from the oblique shock tables is shown in Table 6.2 for clarity. Second,the inclined plate\u2019s length is calculated by taking the perpendicular dimension of the plate\u2019slength equal to one-fourth of the primary nozzle exit diameter (Dne). Third, the entrainmentratio is compared for the four tested ejector diffuser designs at different area ratios (D\/d) of aconstant-pressure mixing ejector. For the ejector operation at p0p\/p0s = 3.5, the entrainmentratio is compared for D\/d = 2, 2.5 and 3 where the secondary stream is drawn at the ambientconditions and the ejector outlet pressure is set at 1 bar. It is observed that when the inclinedplate is inserted in the diffuser for D\/d = 2 and 2.5, the ejector operates in the backflow regime,i.e. the secondary inlet behaves as an outlet. To operate the ejector in the on-design regime, thearea ratio is varied by increasing the diameter of the constant-area section of the mixing chamber.It can be concluded that there is a limiting area ratio for the proposed ejector diffuser designsbelow which the ejector operates in the backflow regime. At D\/d = 3, the proposed ejectordiffuser designs operate in the on-design regime; and hence, this area ratio is used for furtheranalysis. For all the discussed cases, p0p\/p0s = 2 and 3.5. Finally, to reduce the complexity ofcalculation, the inclined plates thickness is assumed to be smaller than the mesh size, i.e. theplate thickness is neligible.6.2 Overall ejector performanceFor all the cases discussed here, the boundary conditions are identical to those discussed inChapter 3; and, additionally, the no-slip adiabatic wall boundary condition is imposed on theinclined plates. Table 6.3 tabulates the mass flow rate values of the primary and secondarystreams, entrainment ratio and integrated entropy generation rate.As shown in Table 6.3, the value of the entrainment ratio for the new ejector diffuser designs876.2. Overall ejector performanceTable 6.2. Analytical solution for the dowstream flowfield of flow past a supersonic wedge. pw1is the static pressure upstream of the shock, and pw2 is the static pressure downstream of theshock.Ma1 \u03b4 (deg.) Ma2 \u03b8 (deg.) pw2\/pw11.0 5 Detached wave1.5 5 1.33 47.89 1.281.0 10 Detached wave1.5 10 1.11 56.68 1.661.0 15 Detached wave1.5 15 Detached wave1.0 20 Detached wave1.5 20 Detached wave1.0 25 Detached wave1.5 25 Detached wave2.0 25 Detached waveTable 6.3. Comparison of the entrainment ratio and integrated entropy generation rate of thefour different ejector diffuser designs with the base cases in a constant-pressure mixing ejectorfor the ejector operation at p0p\/p0s = 2 and p0p\/p0s = 3.5. For all the presented cases, D\/d =3 and \u03b4 = 5\u25e6.Base case Design 1 Design 2 Design 3 Design 4p0p\/p0s = 2 m\u02d9p (kg\/s) 0.0075 0.0075 0.0075 0.0075 0.0075m\u02d9s (kg\/s) 0.0074 0.006 0.0058 0.0052 0.0032m\u02d9s\/m\u02d9p 0.97 0.79 0.77 0.69 0.42\u222bS\u02d9\u2032\u2032\u2032gendV J\/(K \u00b7 s) 3.026 2.385 2.587 2.329 2.326p0p\/p0s = 3.5 m\u02d9p (kg\/s) 0.0132 0.0132 0.0132 0.0132 0.0132m\u02d9s (kg\/s) 0.007 0.007 0.0064 0.0068 0.0028m\u02d9s\/m\u02d9p 0.53 0.53 0.48 0.51 0.21\u222bS\u02d9\u2032\u2032\u2032gendV J\/(K \u00b7 s) 8.041 7.310 8.146 7.610 7.935is either smaller than or equal to the entrainment ratio of the corresponding base cases. Of thefour proposed diffuser designs, Design 1 leads to the maximum entrainment ratio and producesthe least integrated entropy generation rate. For Design 1, the integrated entropy generationrate is reduced by 21.18% and 9%, when compared to the corresponding base cases, for theejector operation at p0p\/p0s = 2 and p0p\/p0s = 3.5, respectively. The primary choking flow rateis the same for the base case and Designs 1\u20134. The secondary flow rate decreases by 18.91%when compared to the base case for ejector operation at p0p\/p0s = 2, and remains the same for886.3. Spatial variation of the entropy generation and the flow featuresthe base case and Design 1 for the ejector operation at p0p\/p0s = 3.5. From these observations,it is concluded that a single-inclined plate at the diffuser entrance allows to reduce the ejectorentropy generation rate while the entrainment ratio remains constant when compared to thebase case for ejector operation at p0p\/p0s = 3.5. Thus, Design 1 is used for further optimization,as discussed in detail in Subsection 6.4.6.3 Spatial variation of the entropy generation and the flowfeaturesFigures 6.2 and 6.3 present the spatial variation of the entropy generation number contoursfor the base case and Designs 1\u20134, for the ejector operation at p0p\/p0s = 2 and p0p\/p0s =3.5, respectively. The results in Fig. 6.2 show that, for ejector operation at p0p\/p0s = 2,Design 1 allows to reduce the overall entropy generation number in the mixing chamber andthe area occupied by the diffuser shock is approximately half of the diffuser shock area of thecorresponding base case. For ejector operation at p0p\/p0s = 3.5, Design 1 allows to reduce theoverall entropy generation number in the mixing chamber shear layer and alters the diffuseroblique shock profile, which is similar to the observations made for p0p\/p0s = 2. For both thestagnation pressure ratios, the base case and Designs 2\u20134 result in larger mixing chamber anddiffuser entropy generation numbers when compared to Design 1.The effect of the inclined-plates on the diffuser shock structure can be seen in the Machnumber contours for the ejector operation at p0p\/p0s = 2 and p0p\/p0s = 3.5, which are shownin Figs. 6.4 and 6.5, respectively. These Mach number contours suggest that a flow asymmetryarises because of a single-inclined plate inserted in the diffuser.896.3. Spatial variation of the entropy generation and the flow features(a) Base Case(b) Design 1(c) Design 2(d) Design 3(e) Design 4Ns0.0e+00 1.6e+03500 1000Fig. 6.2. Overall entropy generation number contours for the four different ejector diffuserdesigns of the constant-pressure mixing ejector: (a) without any inclined plates inside the diffuser(which is the base case), (b) single inclined plate placed inside the diffuser, (c) two inclined platesconnected at the leading edge placed inside the diffuser, (d) two diverging plates placed insidethe diffuser, and (e) two converging plates placed inside the diffuser. For all the presented cases,p0p\/p0s = 2 and D\/d = 3.906.3. Spatial variation of the entropy generation and the flow features(a) Base Case(b) Design 1(c) Design 2(d) Design 3(e) Design 4Ns0.0e+00 6.5e+032000 3000 4000 5000Fig. 6.3. Overall entropy generation number contours for the four different ejector diffuserdesigns of the constant-pressure mixing ejector: (a) without any inclined plates inside the diffuser(which is the base case), (b) single inclined plate placed inside the diffuser, (c) two inclined platesconnected at the leading edge placed inside the diffuser, (d) two diverging plates placed insidethe diffuser, and (e) two converging plates placed inside the diffuser. For all the presented cases,p0p\/p0s = 3.5 and D\/d = 3.916.3. Spatial variation of the entropy generation and the flow features(a) Base case(b) Design 1Ma0.0e+00 1.7e+000.5 1Fig. 6.4. Time-averaged Mach number profile for the two different ejector diffuser designs ofthe constant-pressure mixing ejector: (a) without any inclined-plates inside the diffuser (whichis the base case), and (b) single-inclined plate placed inside the diffuser. For all the presentedcases, p0p\/p0s = 2 and D\/d = 3.(a) Base case(b) Design 1Ma0.0e+00 2.0e+000.5 1 1.5Fig. 6.5. Time-averaged Mach number profile for the two different ejector diffuser designs ofthe constant-pressure mixing ejector: (a) without any inclined-plates inside the diffuser (whichis the base case), and (b) single-inclined plate placed inside the diffuser. For all the presentedcases, p0p\/p0s = 3.5 and D\/d = 3.926.4. Influence of the design parameters on the ejector performance6.4 Influence of the design parameters on the ejectorperformanceThe results discussed in Subsection 6.2 suggested that the integrated entropy generation rateis smaller for design compared to corresponding base cases. The sensitivity of the entrainmentratio and the integrated entropy generation rate to the plate inclination angle is assessed here,and the results are tabulated in Table 6.4. Five different values of \u03b4 = 5\u25e6, 10\u25e6, 15\u25e6, 25\u25e6 and35\u25e6 are considered. As can be seen, increasing \u03b4 decreases the secondary mass flow rate, and aresult decreases the entrainment ratio. Thus, \u03b4 = 5\u25e6 is desired.Table 6.4. Influence of the angle of inclination of the single-inclined plate on the ejector perfor-mance for ejector operation at p0p\/p0s = 3.5 and D\/d = 3.\u03b4 5\u25e6 10\u25e6 15\u25e6 25\u25e6 35\u25e6m\u02d9p (kg\/s) 0.0132 0.0132 0.0132 0.0132 0.0132m\u02d9s (kg\/s) 0.007 0.0065 0.0061 0.0055 0.0052m\u02d9s\/m\u02d9p 0.54 0.5 0.46 0.42 0.39S\u02d9\u2032\u2032\u2032gendV J\/(K \u00b7 s) 7.310 7.358 7.800 8.129 8.593Analysis of the results suggest that the supersonic ejectors, when used in the applicationswhere the entrained secondary mass flow rate and pressure recovery are central to the ejector\u2019sperformance, Design 1 features some benefits. It allows decreasing the entropy generation ratesignificantly without having an impact on the entrained secondary flow rate for high-pressureapplications. For example, this design can improve the system\u2019s performance coefficient withoutcompromising the secondary mass flow rate when used in refrigeration applications at highoperating pressures. This design is a preliminary approach to optimize the performance ofsupersonic ejectors and reduce the entropy generation rate during their operation. For practicalimplementation, this design needs to be further modified to ease its utility for the industrialapplications.93Chapter 7Conclusions and Future Steps7.1 ConclusionsBoth 1D analytical and 2D numerical simulations were performed to investigate and improvethe performance of supersonic ejectors. The 1D analytical ejector flow model (governed by theprinciples of gas dynamics) was developed in MATLAB for predicting the entrainment ratio ofa constant-pressure mixing ejector and for several area ratios and operating stagnation pressureratios. Though the 1D analytical model could not capture the detailed flow intricacies andprovide accurate quantitative predictions of the ejector entrainment ratio, it could capture thequalitative trends relatively accurately. The results show that the 1D analytcial model couldcapture the inverse relation between the stagnation pressure ratio and the entrainment ratio.The maximum relative error in predicting the entrainment ratio and the secondary mass flowrate is less than 45% and 40%, respectively. Further, in contrary to what the literature suggests,the 1D analytical model in the present study predicts fairly accurately the secondary mass flowrate and the entrainment ratio for the ejector operation in the mixed flow regime. This is thefirst contribution of this thesis.For precise quantitative predictions, the impact of stagnation pressure ratio and area ratioon ejector performance for the constant-pressure and constant-area mixing ejector geometries isanalyzed using the 2D Reynolds-averaged Navier-Stokes simulations. The ejector performance947.1. Conclusionsquantified in terms of entrainment ratio and overall entropy generation rate is associated withthe complex flow features forming in the ejector. The entropy generation analysis gives an insightinto the ejector losses and linked them to the flow topologies within the ejector. Such relationsbetween the flow topologies, entropy generation, and the entrainment ratio in supersonic ejectorsare the second contributions of this thesis. Followings summarize the key observations from the2D numerical simluations, which explain the localized ejector losses in terms of the entropygeneration rate and the prevailing flow topologies.1. As the stagnation pressure ratio increases, the entrainment ratio decreases; however, theintegrated entropy generation rate increases for both constant-pressure and constant-areamixing ejectors. For a constant-pressure mixing ejector, the integrated entropy generationrate reduced by 80% when the p0p\/p0p was reduced from 5 to 2.2. Increasing the area ratio generally increases the entrainment ratio for both constant-pressure and constant-area mixing ejectors, except at p0p\/p0p = 2 and for D\/d > 4 for theconstant-area mixing ejector. It is speculated that, for these conditions, the constant-areaejector operates in the off-design mode. The entropy generation generation rate decreaseswith increasing the area ratio of both constant-pressure and constant-area mixing ejectors.3. The entropy generation rate due to the heat transfer is negligible, and this rate is mainlydriven by the viscous dissipation mechanism. In addition to this, the entropy genera-tion rate due to the mean flow effect is larger in comparison to the turbulent flow. Forthe constant-pressure mixing ejector operating at p0p\/p0p = 5, the mean flow contribu-tion towards the integrated entropy generation rate was approximately 67% and the flowturbulence contribution was approximately 33%.957.1. Conclusions4. The local flow behaviour associated with the shear layer instabilities and shocks resultingfrom the flow separation phenomenon is the major contributor towards the ejector en-tropy generation rate and leads to significant diffuser inefficiencies. For some operatingconditions, diffuser losses can be as high as over 50% of the overall entropy generationrate. These losses are very large in comparison to the mixing chamber losses. The flowseparation phenomenon at the primary nozzle throat is because of the formation of vena-contracta due to the sharp corners encountered by the subsonic primary flow. This flowseparation pertains to a small area in the vicinity of the primary nozzle throat. This iscontrasting to the shock-induced flow separation in the diffuser which generates shear layerinstabilities and pertains to a large portion of the diffuser. From these observations, itcan be concluded that the entropy generation rate is large at the flow separation zonesirrespective of the cause and the extent of the flow separation.5. Occurence of the primary nozzle modes, which are the under-expanded mode and the over-expanded mode, depend on the stagnation pressure and area ratios. The over-expandednozzle mode produces a higher entrainment ratio when compared to the under-expandedmode. For the under-expanded mode of operation, the entropy generation rate in themixing chamber is smaller than the entropy generation rate in the diffuser.6. The entropy generation rate because of the shocks and shear layers is location-dependent.The contribution of shocks towards entropy generation rate is investigated using the shockstrength. For the under-expanded mode, the mixing chamber shocks are relatively weakerin comparison to the first diffuser shock. However, the strength of the subsequent diffusershock is comparable to the mixing chamber shocks. As the area ratio increases or thestagnation pressure ratio decreases, the strength of the first diffuser shock decreases. A967.2. Future stepsstronger diffuser shock generates larger entropy generation rate in the shear layer of theflow separated region.7. The entropy generation rate due to the formation of shocks is attributed to the mean flowand turbulence has negligible effect on viscous dissipation due to the formation of shocks.The turbulence dominant regions correspond to the shear layers at the flow separatedzones.Based on the observations pertinent to the flow topologies and localized entropy generationrate in the supersonic ejectors, it is concluded that the mixing phenomenon is not central tothe ejector entropy generation for the ejector operation in the under-expanded mode and, it isthe diffuser that generates majority of the ejector irreversibilities. Hence, four different ejectordiffuser modifications are proposed, and their influences on the entrainment ratio and integratedentropy generation rate are studied. The ejector diffuser design with a single-inclined plateinserted at the diffuser entrance reduces the entropy generation rate without inducing majorchanges in the entrainment ratio. For the ejector operation at a stagnation pressure ratio of 3.5,the integrated entropy generation rate decreases by 9% for the final proposed ejector diffuserdesign with a single-inclined plate relative to its unmodified conditions and the entrainmentratio remains the same for both of the designs.7.2 Future stepsThe present work can be carried forward with the vision of deeper analysis to address thequestions listed below.1. It can be concluded from the analysis carried out in the present study that the entropy977.2. Future stepsgeneration rate in supersonic ejectors due to the mean flow dominates but is comparableto that of the turbulent flow. Hence, it is essential to assess the capability of differentturbulence models to predict the entropy generation rate in supersonic ejectors.2. In the present study, the viscous effects are numerically modeled using a non-zero valuefor the dynamic viscosity. It is unclear how inviscid effects influence the ejector entropygeneration rate. Hence, Euler equations should be solved to highlight the contribution ofinviscid effects on the entropy generation rate in supersonic ejectors.3. The computational domain used in the present study is 2D. The 3D effects could potentiallyinfluence the mixing phenomenon thereby influencing the entropy generation rate. Hence,analyzing the 3D ejector geometry will help to understand the importance of 3D effects inquantifying the entropy generation rate.4. The data obtained in the present study can be further post-processed to relate the entropygeneration rate with the mixing layer growth rate, highlighting the qualitative influence ofthe ejector operational conditions and geometry on the mixing phenomenon in supersonicejectors.5. The ejector diffuser design proposed in the present study is a preliminary version andrequires to be further modified to ensure an easy implementation for the industrial appli-cations.6. 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Pergamon Press.[99] Paoletti, S., Rispoli, F., and Sciubba, E., 1989. \u201cCalculation of exergetic losses in compactheat exchanger passages\u201d. In ASME AES, Vol. 10, pp. 21\u201329.110AppendicesAppendix A: Domain and Mesh CoordinatesGmsh 4.4.1 is used to generate ejector geometry and subsequently mesh the computationaldomain. A structured mesh is generated using this tool. The ejector geometry is divided intoseveral blocks to ease the process of mesh generation. The points defining the ejector domainand additional blocks are saved in the .geo format. This file is loaded in Gmsh and the pointsare connected by lines. This is followed by defining plane surfaces and transfinite surfaces,which are divided into small segments to generate a structured mesh. The boundary patchesare defined for the allocation of the boundary conditions. A sample portion of the .geo file forconstant pressure mixing ejector, corresponding to D\/d = 2.5, is shown below. This segment ofthe .geo file (which is loaded in Gmsh) contains the information of the points and any additionalzones created to ensure the local refinement at the walls and the other regions of interest. Allthe dimensions are expressed in terms of nozzle throat diameter (d). The followings are thenotations used in the .geo fileL1: Jet inlet lengthL2: Primary nozzle converging section lengthL3: Primary nozzle diverging section lengthL4: Mixing chamber convergent-section lengthL5: Mixing chamber constant-area section lengthL6: Diffuser lengthD1: Jet inlet diameterD2: Primary nozzle throat diameterD3: Primary nozzle exit diameterD4: Secondary nozzle diameter111AppendicesD5: Constant-area mixing chamber diameterD6: Diffuser outlet diameterL1 = 12d;L2 = 0;L3 = 3.5d;L4 = 10d;L5 = 13d;L6 = 25d;D1 = 2.75d;D2 = d;D3 = 1.5d;D4 = 3.25d;D5 = 2.5d;D6 = 5.5d;\/\/Points\/\/Top levelPoint(1) = {0,D1\/2,0};Point(2) = {L1,D1\/2,0};Point(3) = {L1+L2,D2\/2,0};Point(4) = {L1+L2+L3,D3\/2,0};Point(5) = {L1+L2+L3,D4\/2,0};Point(6) = {L1+L2+L3+L4,D5\/2,0};Point(7) = {L1+L2+L3+L4+L5,D5\/2,0};112AppendicesPoint(8) = {L1+L2+L3+L4+L5+L6,D6\/2,0};\/\/Bottom levelPoint(9) = {L1+L2+L3+L4+L5+L6,0,0};Point(10) = {L1+L2+L3+L4+L5,0,0};Point(11) = {L1+L2+L3+L4,0,0};Point(12) = {L1+L2+L3,0,0};Point(13) = {L1+L2,0,0};Point(14) = {0,0,0};\/\/Extra zonesPoint(15) = {0,D2\/2,0};Point(16) = {L1+L2+L3+L4,D3\/2,0};Point(17) = {L1+L2+L3+L4+L5,D3\/2,0};Point(18) = {L1+L2+L3+L4+L5+L6,D3\/2,0};Point(19) = {0,D1\/3,0};Point(20) = {L1,D1\/3,0};Point(21) = {L1+L2+L3,D4\/3,0};Point(22) = {L1+L2+L3+L4,D4\/3,0};Point(23) = {L1+L2+L3+L4+L5,D4\/3,0};Point(24) = {L1+L2+L3+L4+L5+L6,D4\/3,0};Point(25) = {3*L1\/4,0,0};Point(26) = {3*L1\/4,D2\/2,0};Point(27) = {3*L1\/4,D1\/3,0};Point(28) = {3*L1\/4,D1\/2,0};113AppendicesAppendix B: Total Pressure Boundary ConditionThe total pressure boundary condition is imposed at the primary and secondary inlets. Air isused as the working fluid at T = 298 K and hence the ratio of specific heats (\u03b3) is taken equal to1.4. To ensure that the solver uses this assigned value of \u03b3, appropriate syntax for total pressureboundary condition should be used. The solver calculates the static pressure at the boundarypatch (pbp) from the imposed stagnation pressure value (p0). The method of calculation for thestatic pressure is dependent on the flow characteristics. Hence, appropriate syntax should beused depending upon the flow category. This syntax determines the dimensions of convectiveflux. The followings are the four possible syntaxes that could be used. In the present study, thelast syntax is used. The details related to all the syntaxes are provided below.(a) Subsonic incompressibleThe static pressure value at the boundary patch is calculated by subtracting the dynamicpressure from the total pressure. The dimensions of the convective flux in this case is m3\/s.pbp = p0 \u2212 12|u|2 , (1)where u is the fluid velocity vector.Syntax:<patchName>{type totalPressure;rho none;p0 uniform <value>;value uniform <value>;114Appendices}(b) Subsonic compressibleThe static pressure value at the boundary patch is calculated by subtracting the dynamicpressure from the total pressure. The dimensions of the convective flux in this case is kg\/s.pbp = p0 \u2212 12|\u03c1u|2 , (2)where \u03c1 is the fluid density.Syntax:<patchName>{type totalPressure;p0 uniform <value>;value uniform <value>;}(c) Transonic compressibleThe value of \u03b3 in this case is automatically considered equal to 1. The following equation isused to calculate the static pressure at the corresponding patchpbp =p01 + 12\u03c8c|u|2, (3)where \u03c8c is the compressibility.Syntax:<patchName>{115Appendicestype totalPressure;psi psi;p0 uniform <value>;value uniform <value>;}(d) Supersonic compressibleThis is used in the present study. The value of \u03b3 is greater than 1 in this case. The followingequation is used to calculate static pressure at corresponding patchpbp =p0(1 + \u03b3\u221212\u03b3 \u03c8c|u|2) \u03b3\u03b3\u22121. (4)Syntax:<patchName>{type totalPressure;psi psi;gamma <value>;p0 uniform <value>;value uniform <value>;}Appendix C: Sensitivity to Inlet Turbulence IntensityThe sensitivity of the entrainment ratio and integrated entropy generation rate (over theentire ejector computational domain) to the inlet turbulence intensity (I) is assessed for the116Appendicesejector operation at a stagnation pressure ratio (p0p\/p0s) of 5. Here, the turbulence intensity isdefined as the RMS of the velocity to mean velocity evaluated at the inlet to the primary nozzle.The selected operating condition is chosen since this condition leads to formation of the mostcomplex flow topologies. Effects of three different values of I (2%, 5% and 8%) on the aboveparameters are evaluated. In the following, the values of the entrainment ratio and integratedentropy generation rate evaluated for these turbulence intensities are provided in Table C.1.Table C.1. Comparison of the entrainment ratio and integrated entropy generation rate for threedifferent turbulence intensities at p0p\/p0s = 5 and D\/d = 2.5 of a constant-pressure mixingejector.I m\u02d9s\/m\u02d9p\u222bS\u02d9\u2032\u2032\u2032gendV J\/(K \u00b7 s)2% 0.2554 10.47435% 0.2553 10.52508% 0.2540 10.4863The results in Table C.1 suggest that the entrainment ratio and integrated entropy generationrate are not affected by the turbulence intensities tested here.Appendix D: Statistical AnalysisThe numerical data is time-averaged for different time-averaging windows to ensure thatthe numerical solution has reached a statistically steady state, i.e., the solution is independentof the time variations and size of the time-averaging window. For the statistical analysis, anoperating stagnation pressure ratio (p0p\/p0s) of 5 is chosen, since this operating condition leadsto formation of the most complex flow topologies. The ejector outlet pressure is set to 1 bar.First, the shock strength (Rs) is used to evaluate the effect of time-averaging window size onthe numerical solution. Rs is calculated using117AppendicesRs =p2 \u2212 p1p1, (5)where p1 and p2 are the pressure at inlet to the shock and pressure downstream of the shock,respectively. The time-averaging is started once the mass flow imbalance reaches 10\u22126 kg\/s(at t = 0.01 s for the present case). For example, the data point collected at t = 0.015 s istime-averaged over the time interval window of t = 0.01 s to 0.015 s. Beyond t = 0.01 s, a totalof 30 time-averaging window sizes are selected and the shock strength of the first diffuser shockis evaluated for each of the time-averaging windows. Figure D.1 shows the time-averaged shockstrength of the first diffuser shock plotted against time (the mean shock strength value is alsomarked). The standard deviation is calculated using\u03c3dev =\u221a\u2211(zi \u2212 z\u00af)2Qn, (6)where zi is the ith value of the shock strength, z\u00af is the mean shock strength calculated fromthe mean pressure values, and Qn is the total number of data points collected. The standarddeviation for the shock strength is 0.0581. Similar statistical analysis is extended to the otherflow parameters of interest (pressure, temperature, velocity magnitude and density). FiguresD.2, D.3, D.4 and D.5 show the time-averaged pressure, temperature, velocity magnitude anddensity (at the location upstream of the first diffuser shock) plotted against time, respectively(respective mean values are also marked). The standard deviations for time-averaged pressure,temperature, velocity magnitude and density are 0.0000104541, 0.002702, 0.13798 and 230.1843,respectively. From the values of standard deviations of the time-averaged quantities, it can beconcluded that the numerical solution is independent of the size of the time-averaging window.In this thesis, for the ejector operation at p0p\/p0p = 5 and D\/d = 2.5, the results are reported118Appendicesfor t = 0.04 s.0.01 0.015 0.02 0.025 0.03 0.035 0.044.74.754.84.854.94.9555.05Fig. D.1. Variation of strength of the first diffuser shock with time for different time-averagingwindow sizes at p0p\/p0s = 5 and D\/d = 2.5 of a constant-pressure mixing ejector.0.01 0.015 0.02 0.025 0.03 0.035 0.042.62.652.72.75 104Fig. D.2. Variation of pressure with time at the first diffuser shock location for different time-averaging window sizes at p0p\/p0s = 5 and D\/d = 2.5 of a constant-pressure mixing ejector.119Appendices0.01 0.015 0.02 0.025 0.03 0.035 0.04170.588170.59170.592170.594170.596170.598Fig. D.3. Variation of temperature with time at the first diffuser shock location for differenttime-averaging window sizes at p0p\/p0s = 5 and D\/d = 2.5 of a constant-pressure mixing ejector.0.01 0.015 0.02 0.025 0.03 0.035 0.04564.8565565.2565.4565.6565.8Fig. D.4. Variation of velocity magnitude with time at the first diffuser shock location fordifferent time-averaging window sizes at p0p\/p0s = 5 and D\/d = 2.5 of a constant-pressuremixing ejector.120Appendices0.01 0.015 0.02 0.025 0.03 0.035 0.040.543060.543070.543080.543090.54310.543110.54312Fig. D.5. Variation of density with time at the first diffuser shock location for different time-averaging window sizes at p0p\/p0s = 5 and D\/d = 2.5 of a constant-pressure mixing ejector.121","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/hasType":[{"value":"Thesis\/Dissertation","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#dateIssued":[{"value":"2021-09","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt":[{"value":"10.14288\/1.0401094","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/language":[{"value":"eng","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeDiscipline":[{"value":"Mechanical Engineering","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/provider":[{"value":"Vancouver : University of British Columbia Library","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/publisher":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/rights":[{"value":"Attribution-NonCommercial-NoDerivatives 4.0 International","type":"literal","lang":"*"}],"https:\/\/open.library.ubc.ca\/terms#rightsURI":[{"value":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/","type":"literal","lang":"*"}],"https:\/\/open.library.ubc.ca\/terms#scholarLevel":[{"value":"Graduate","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/contributor":[{"value":"Kheirkhah, Sina","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/title":[{"value":"Entrainment ratio and entropy generation of vacuum-objective supersonic ejectors","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/type":[{"value":"Text","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#identifierURI":[{"value":"http:\/\/hdl.handle.net\/2429\/79140","type":"literal","lang":"en"}]}}