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Mead, Ryan M.
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Analysis of flow in a spray nozzle with emphasis on exiting jet free surface
h [electronic resource] /
by Ryan M Mead.
260
[Tampa, Fla.] :
University of South Florida,
2003.
502
Thesis (M.S.M.E.)University of South Florida, 2003.
504
Includes bibliographical references.
516
Text (Electronic thesis) in PDF format.
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System requirements: World Wide Web browser and PDF reader.
Mode of access: World Wide Web.
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Title from PDF of title page.
Document formatted into pages; contains 230 pages.
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ABSTRACT: A conical nozzle with two separate inlets within its top plate is analyzed. One of the inlets is in the center of the top plate, which is free to rotate, whereas the other inlet is positioned away from the center. The fluid entering through the outer inlet slot causes the top plate of the nozzle to spin. Several fluids including FC77, FC72, FC87, and Methanol running at different flow rates were investigated to observe the effect that their particular properties have on the geometry of the fluid's free surface exiting the nozzle. Another variation performed was the geometry of the nozzle. The outer inlet slot was positioned at various radial distances along the top plate. For this nozzle, the top plate remained stationary and swirling was introduced to the fluid at the inlets. It was observed that the faster flow rates caused an increase in the free surface height and cone angle. For the various radial locations of the outer inlet slot, it was noted that a position at approximately 75% of the nozzle radius produced the largest free surface height. The largest cone angle was produced when the outer inlet slot was positioned at the edge of the nozzle top plate. Another factor that increased the radial height and cone angle of the free surface was the working fluid used in the study. A larger Reynolds number produced a larger cone angle and larger free surface height (while a smaller Reynolds number produced a less significant cone angle and free surface height).
590
Adviser: Rahman, Muhammad
653
cone angle.
spray cooling.
atomizer.
liquidgas interface.
mixing length.
690
Dissertations, Academic
z USF
x Mechanical Engineering
Masters.
773
t USF Electronic Theses and Dissertations.
4 856
u http://digital.lib.usf.edu/?e14.138
PAGE 1
Analysis of Flow in a Spray Nozzle With Emphasis on Exiting Jet Free Surface by Ryan M. Mead A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Department of Mechanical Engineering College of Engineering University of South Florida Major Professor: Muhammad M. Rahman, Ph.D. Roger A. Crane, Ph.D. Autar K. Kaw, Ph.D. Date of Approval: November 4, 2003 Keywords: spray cooling, cone angle, mixi ng length, liquidgas interface, atomizer Copyright 2003, Ryan M. Mead
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i Table of Contents List of Tables iii List of Figures v Abstract xviii 1 Introduction 1 2 Nomenclature 6 3 Mathematical Model 9 4 Numerical Computation 18 5 Results and Discussion 21 5.1 Large Nozzle 22 5.1.1 Large Nozzle Â– FC77 24 5.1.2 Large Nozzle Â– FC72 30 5.2 Small Nozzle 36 5.2.1 Small Nozzle Â– FC77 38 5.2.2 Small Nozzle Â– FC72 48 5.2.3 Cavitation 54 5.2.4 Varied Nozzle Height 55 5.2.5 Extend Free Surface 57 5.2.6 Initial Mesh with Upward Slope 58 5.2.7 Initial Mesh with Downward Slope 59 5.2.8 Varying Outer Slot Location 60 5.2.8.1 R2 Equal to 2.50 x 104 m 61 5.2.8.2 R2 Equal to 4.00 x 104 m 81 5.2.8.3 R2 Equal to 5.50 x 104 m 100 5.2.8.4 R2 Equal to 7.20 x 104 m 119 5.2.8.5 Cavitation 149 5.2.8.6 Sectional Velocities for FC72 151 6 Conclusion 183 References 184
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ii Appendices 185 Appendix I: GAMBIT File for Large Nozzle 186 Appendix II: FIJOUR File for Large Nozzle 189 Appendix III: GAMBIT File for Small Nozzle 194 Appendix IV: FIJOUR File for Small Nozzle (1.262 x 107 m3/s and 2.524 x 107 m3/s) 197 Appendix V: FIJOUR File for Small Nozzle (4.416 x 107 m3/s and 5.678 x 107 m3/s) 201 Appendix VI: GAMBIT File for Varying Outer Slot Location 205 Appendix VII: FIJOUR File for Varying Outer Slot Location 208 Appendix VIII: Fluid Properties 211
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iii List of Tables Table 1: Angular velocity for give n flow rate in the smaller nozzle 12 Table 2: Angular velocity for give n flow rate in the larger nozzle 12 Table 3: Calculated spinning rate fo r given flow rate in larger nozzle 22 Table 4: Inlet velocities for the gi ven flow rates in the larger nozzle 23 Table 5: Reynolds number for each fluid and each flow rate 23 Table 6: Inlettooutlet pressure drop for each working fluid and each inlet flow rate 36 Table 7: Final free surface height for each working fluid and each flow rate 36 Table 8: Cone angle for each work ing fluid and each flow rate 36 Table 9: Calculated spinning rate fo r given flow rate in smaller nozzle 37 Table 10: Inlet velocities for the gi ven flow rates in the smaller nozzle 38 Table 11: Reynolds number for each fluid and each flow rate 38 Table 12: Cavitation number for FC77 and FC72 in the small nozzle 55 Table 13: Values of R1, R2, and R3 used in the investigation 61 Table 14: Inlet velocities for the given flow rates 61 Table 15: Free surface height for differe nt nozzle geometries, working fluids, and flow rates 142 Table 16: Cone angle for different nozzle ge ometries, working fluids, and flow rates 144 Table 17: Pressure drop for different nozzle geometries, working fluids, and flow rates 146 Table 18: Reynolds number for each fluid and each flow rate 149 Table 19: Cavitation number for all fluids at all flow rates (R2 = 2.50 x 104 m) 150
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iv Table 20: Cavitation number for all fluids at all flow rates (R2 = 4.00 x 104 m) 150 Table 21: Cavitation number for all fluids at all flow rates (R2 = 5.50 x 104 m) 151 Table 22: Cavitation number for all fluids at all flow rates (R2 = 7.20 x 104 m) 151 Table 23: Analyzed sections with corresponding axia l coordinate 152 Table 24: Average velocity and mome ntum components at each section (R2=2.50 x 104 m, Re=5904) 179 Table 25: Average velocity and mo mentum components at each section (R2=4.00 x 104 m, Re=5904) 180 Table 26: Average velocity and mome ntum components at each section (R2=5.50 x 104 m, Re=5904) 180 Table 27: Average velocity and mome ntum components at each section (R2=5.50 x 104 m, Re=7591) 181 Table 28: Average velocity and mome ntum components at each section (R2=7.20 x 104 m, Re=5904) 181 Table 29: Pressure drop comparison for various outer slot locations 182
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v List of Figures Figure 1: Velocity profile for various grid sizes 20 Figure 2: Dimensioned schematic for larger nozzle 21 Figure 3: Dimensioned schematic of smaller nozzle 22 Figure 4: Velocity vector plot for FC77 (Q = 1.262 x 106 m3/s) 24 Figure 5: Pressure contour plot for FC77 (Q = 1.262 x 106 m3/s) 25 Figure 6: Velocity vector plot for FC77 (Q = 2.524 x 106 m3/s) 26 Figure 7: Pressure contour plot for FC77 (Q = 2.524 x 106 m3/s) 27 Figure 8: Velocity vector plot for FC77 (Q = 3.785 x 106 m3/s) 28 Figure 9: Pressure contour plot for FC77 (Q = 3.785 x 106 m3/s) 28 Figure 10: Free surface profile for FC 77 at various inlet flow rates 29 Figure 11: Dimensionless free surface profile for FC77 (large nozzle) 29 Figure 12: Velocity vector plot for FC72 (Q = 1.262 x 106 m3/s) 31 Figure 13: Pressure contour plot for FC72 (Q = 1.262 x 106 m3/s) 31 Figure 14: Velocity vector plot for FC72 (Q = 2.524 x 106 m3/s) 32 Figure 15: Pressure contour plot for FC72 (Q = 2.524 x 106 m3/s) 32 Figure 16: Velocity vector plot for FC72 (Q = 3.785 x 106 m3/s) 33 Figure 17: Pressure contour plot for FC72 (Q = 3.785 x 106 m3/s) 34 Figure 18: Free surface profile fo r FC72 at various flow rates 34 Figure 19: Dimensionless free surface profile for FC72 (large nozzle) 35 Figure 20: Velocity vector plot for FC77 (Q = 1.262 x 107 m3/s) 39
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vi Figure 21: Pressure contour plot for FC77 (Q = 1.262 x 107 m3/s) 40 Figure 22: Streamline contour plot for FC77 (Q = 1.262 x 107 m3/s) 40 Figure 23: Velocity vector plot for FC77 (Q = 2.524 x 107 m3/s) 41 Figure 24: Pressure contour plot for FC77 (Q = 2.524 x 107 m3/s) 42 Figure 25: Streamline contour plot for FC77 (Q = 2.524 x 107 m3/s) 42 Figure 26: Velocity vector plot for FC77 (Q = 4.416 x 107 m3/s) 43 Figure 27: Pressure contour plot for FC77 (Q = 4.416 x 107 m3/s) 44 Figure 28: Streamline contour plot for FC77 (Q = 4.416 x 107 m3/s) 44 Figure 29: Velocity vector plot for FC77 (Q = 5.678 x 107 m3/s) 45 Figure 30: Pressure contour plot for FC77 (Q = 5.678 x 107 m3/s) 46 Figure 31: Streamline contour plot for FC77 (Q = 5.678 x 107 m3/s) 46 Figure 32: Free surface profile fo r FC77 at various flow rates 47 Figure 33: Magnified free surface profile for FC77 at various flow rates 47 Figure 34: Dimensionless free surfa ce profile for FC77 (small nozzle) 48 Figure 35: Velocity vector plot for FC72 (Q = 4.416 x 107 m3/s) 49 Figure 36: Pressure contour plot for FC72 (Q = 4.416 x 107 m3/s) 50 Figure 37: Streamline contour plot for FC72 (Q = 4.416 x 107 m3/s) 50 Figure 38: Velocity vector plot for FC72 (Q = 5.678 x 107 m3/s) 51 Figure 39: Pressure contour plot for FC72 (Q = 5.678 x 107 m3/s) 52 Figure 40: Streamline contour plot for FC72 (Q = 5.678 x 107 m3/s) 52 Figure 41: Free surface profile fo r FC72 at various flow rates 53 Figure 42: Magnified free surface profile for FC72 at various flow rates 53
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vii Figure 43: Dimensionless free surfa ce profile for FC72 (small nozzle) 54 Figure 44: Velocity vector plot for FC 77 (Q = 4.416 x 107 m3/s, L = 1.05 x 103 m) 56 Figure 45: Pressure contour plot for FC77 (Q = 4.416 x 107 m3/s, L = 1.05 x 103 m) 57 Figure 46: Streamline contour plot for FC77 (Q = 4.416 x 107 m3/s, L = 1.05 x 103 m) 57 Figure 47: Free surface profile fo r varied free surface length 58 Figure 48: Mesh plot of nozzle with in itial upward slope of the free surface 59 Figure 49: Mesh plot of nozzle with init ial downward slope of the free surface 59 Figure 50: Schematic of nozzle wi th varying outer slot radii 60 Figure 51: Velocity vector plot for FC77 (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m) 62 Figure 52: Pressure contour plot for FC77 (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m) 63 Figure 53: Streamline contour plot for FC77 (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m) 63 Figure 54: Vector velocity plot for FC77 (Q = 5.678 x 107 m3/s, R2 = 2.5 x 104 m) 64 Figure 55: Pressure contour plot for FC77 (Q = 5.678 x 107 m3/s, R2 = 2.5 x 104 m) 65 Figure 56: Streamline contour plot for FC77 (Q = 5.678 x 107 m3/s, R2 = 2.5 x 104 m) 65 Figure 57: Velocity vector plot for FC72 (Q = 4.416x 107 m3/s, R2 = 2.5 x 104 m) 66 Figure 58: Pressure contour plot for FC72 (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m) 67 Figure 59: Streamline contour plot for FC72 (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m) 67 Figure 60: Velocity vector plot for FC72 (Q = 5.678 x 107 m3/s, R2 = 2.5 x 104 m) 68 Figure 61: Pressure contour plot for FC72 (Q = 5.678 x 107 m3/s, R2 = 2.5 x 104 m) 69 Figure 62: Streamline contour plot for FC72 (Q = 5.678 x 107 m3/s, R2 = 2.5 x 104 m) 69
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viii Figure 63: Velocity vector plot for FC87 (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m) 70 Figure 64: Pressure contour plot for FC87 (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m) 71 Figure 65: Streamline contour plot for FC87 (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m) 71 Figure 66: Velocity vector plot for FC87 (Q = 5.678 x 107 m3/s, R2 = 2.5 x 104 m) 72 Figure 67: Pressure contour plot for FC87 (Q = 5.678 x 107 m3/s, R2 = 2.5 x 104 m) 73 Figure 68: Streamline contour plot for FC87 (Q = 5.678 x 107 m3/s, R2 = 2.5 x 104 m) 73 Figure 69: Velocity vector plot for Methanol (Q=4.416 x 107 m3/s, R2 = 2.5 x 104 m) 74 Figure 70: Pressure contour pl ot for Methanol (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m) 75 Figure 71: Streamline contour pl ot for Methanol (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m) 75 Figure 72: Vector velocity plot for Methanol (Q=5.678 x 107 m3/s, R2 = 2.5 x 104 m) 76 Figure 73: Pressure contour pl ot for Methanol (Q = 5.678 x 107 m3/s, R2 = 2.5 x 104 m) 77 Figure 74: Streamline contour pl ot for Methanol (Q = 5.678 x 107 m3/s, R2 = 2.5 x 104 m) 77 Figure 75: Free surface profile for all fluids (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m) 78 Figure 76: Magnified free surface profile for all fluids (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m) 78 Figure 77: Free surface profile for all fluids (Q = 5.678 x 107 m3/s, R2 = 2.5 x 104 m) 79 Figure 78: Magnified free surface profile for all fluids (Q = 5.678 x 107 m3/s, R2 = 2.5 x 104 m) 79 Figure 79: Dimensionless free surfa ce profile for all fluids (Q = 4.416x 107 m3/s, R2 = 2.5 x 104 m) 80
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ix Figure 80: Dimensionless free surface profiles for all fluids (Q = 5.678 x 107 m3/s, R2 = 2.5 x 104 m) 80 Figure 81: Velocity vector plot for FC77 (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m) 81 Figure 82: Pressure contour plot for FC77 (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m) 82 Figure 83: Streamline contour plot for FC77 (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m) 82 Figure 84: Velocity vector plot for FC77 (Q = 5.678 x 107 m3/s, R2 = 4.0 x 104 m) 83 Figure 85: Pressure contour plot for FC77 (Q = 5.678 x 107 m3/s, R2 = 4.0 x 104 m) 84 Figure 86: Streamline contour plot for FC77 (Q = 5.678 x 107 m3/s, R2 = 4.0 x 104 m) 84 Figure 87: Velocity vector plot for FC72 (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m) 85 Figure 88: Pressure contour plot for FC72 (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m) 86 Figure 89: Streamline contour plot for FC72 (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m) 86 Figure 90: Velocity vector plot for FC72 (Q = 5.678 x 107 m3/s, R2 = 4.0 x 104 m) 87 Figure 91: Pressure contour plot for FC72 (Q = 5.678 x 107 m3/s, R2 = 4.0 x 104 m) 88 Figure 92: Streamline contour plot for FC72 (Q = 5.678x 107 m3/s, R2 = 4.0 x 104 m) 88 Figure 93: Velocity vector plot for FC87 (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m) 89 Figure 94: Pressure contour plot for FC87 (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m) 90 Figure 95: Streamline contour plot for FC87 (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m) 90 Figure 96: Velocity vector plot for FC87 (Q = 5.678 x 107 m3/s, R2 = 4.0 x 104 m) 91 Figure 97: Pressure contour plot for FC87 (Q = 5.678 x 107 m3/s, R2 = 4.0 x 104 m) 92 Figure 98: Streamline contour plot for FC87 (Q = 5.678 x 107 m3/s, R2 = 4.0 x 104 m) 92
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x Figure 99: Velocity vector plot for Methanol (Q=4.416 x 107 m3/s, R2 = 4.0 x 104 m) 93 Figure 100: Pressure contour pl ot for Methanol (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m) 94 Figure 101: Streamline contour pl ot for Methanol (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m) 94 Figure 102: Velocity vector plot for Methanol (Q=5.678 x 107 m3/s, R2=4.0 x 104 m) 95 Figure 103: Pressure contour pl ot for Methanol (Q = 5.678 x 107 m3/s, R2 = 4.0 x 104 m) 96 Figure 104: Streamline contour pl ot for Methanol (Q = 5.678 x 107 m3/s, R2 = 4.0 x 104 m) 96 Figure 105: Free surface positi on for all fluids (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m) 97 Figure 106: Magnified free surface pr ofile for all fluids (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m) 97 Figure 107: Free surface profile for all fluids (Q = 5.678 x 107 m3/s, R2 = 4.0 x 104 m) 98 Figure 108: Magnified free surface pr ofile for all fluids (Q = 5.678 x 107 m3/s, R2 = 4.0 x 104 m) 98 Figure 109: Dimensionless free surface profile for all fluids (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m) 99 Figure 110: Dimensionless free surface profile for all fluids (Q = 5.678 x 107 m3/s, R2 = 4.0 x 104 m) 99 Figure 111: Velocity vector plot for FC77 (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m) 100 Figure 112: Pressure contour plot for FC77 (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m) 101 Figure 113: Streamline contour plot for FC77 (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m) 101 Figure 114: Velocity vector plot for FC77 (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m) 102
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xi Figure 115: Pressure contour plot for FC77 (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m) 103 Figure 116: Streamline contour plot for FC77 (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m) 103 Figure 117: Velocity vector plot for FC72 (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m) 104 Figure 118: Pressure contour plot for FC72 (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m) 105 Figure 119: Streamline contour plot for FC72 (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m) 105 Figure 120: Velocity vector plot for FC72 (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m) 106 Figure 121: Pressure contour plot for FC72 (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m) 107 Figure 122: Streamline contour plot for FC72 (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m) 107 Figure 123: Velocity vector plot for FC87 (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m) 108 Figure 124: Pressure contour plot for FC87 (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m) 109 Figure 125: Streamline contour plot for FC87 (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m) 109 Figure 126: Velocity vector plot for FC87 (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m) 110 Figure 127: Pressure contour plot for FC87 (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m) 111 Figure 128: Streamline contour plot for FC87 (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m) 111 Figure 129: Velocity vector plot for Methanol (Q=4.416 x 107 m3/s, R2=5.5 x 104 m) 112 Figure 130: Pressure contour pl ot for Methanol (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m) 113
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xii Figure 131: Streamline contour pl ot for Methanol (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m) 113 Figure 132: Velocity vector plot for Methanol (Q=5.678 x 107 m3/s, R2=5.5 x 104 m) 114 Figure 133: Pressure contour pl ot for Methanol (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m) 115 Figure 134: Streamline contour pl ot for Methanol (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m) 115 Figure 135: Free surface profile for all of the fluids (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m) 116 Figure 136: Magnified view of free su rface profile for all fluids (Q=4.416 x 107 m3/s, R2=5.5 x 104 m) 116 Figure 137: Free surface profile for all fluids (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m) 117 Figure 138: Magnified free surface pr ofile for all fluids (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m) 117 Figure 139: Dimensionless free surface profile for all fluids (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m) 118 Figure 140: Dimensionless free surface profile for all fluids (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m) 118 Figure 141: Velocity vector plot for FC77 (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m) 119 Figure 142: Pressure contour plot for FC77 (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m) 120 Figure 143: Streamline contour plot for FC77 (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m) 120 Figure 144: Velocity vector plot for FC77 (Q = 5.678 x 107 m3/s, R2 = 7.2 x 104 m) 121 Figure 145: Pressure contour plot for FC77 (Q = 5.678 x 107 m3/s, R2 = 7.2 x 104 m) 122
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xiii Figure 146: Streamline contour plot for FC77 (Q = 5.678 x 107 m3/s, R2 = 7.2 x 104 m) 122 Figure 147: Velocity vector plot for FC72 (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m) 123 Figure 148: Pressure contour plot for FC72 (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m) 124 Figure 149: Streamline contour plot for FC72 (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m) 124 Figure 150: Velocity vector plot for FC72 (Q = 5.678 x 107 m3/s, R2 = 7.2 x 104 m) 125 Figure 151: Pressure contour plot for FC72 (Q = 5.678x 107 m3/s, R2 = 7.2 x 104 m) 126 Figure 152: Streamline contour plot for FC72 (Q = 5.678 x 107 m3/s, R2 = 7.2 x 104 m) 126 Figure 153: Velocity vector plot for FC87 (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m) 127 Figure 154: Pressure contour plot for FC87 (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m) 128 Figure 155: Streamline contour plot for FC87 (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m) 128 Figure 156: Velocity vector plot for FC87 (Q = 5.678 x 107 m3/s, R2 = 7.2 x 104 m) 129 Figure 157: Pressure contour plot for FC87 (Q = 5.678 x 107 m3/s, R2 = 7.2 x 104 m) 130 Figure 158: Streamline contour plot for FC87 (Q = 5.678 x 107 m3/s, R2 = 7.2 x 104 m) 130 Figure 159: Velocity vector plot for Methanol (Q=4.416 x 107 m3/s, R2=7.2 x 104 m) 131 Figure 160: Pressure contour pl ot for Methanol (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m) 132 Figure 161: Streamline contour pl ot for Methanol (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m) 132
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xiv Figure 162: Velocity vector plot for Methanol (Q=5.678 x 107 m3/s, R2=7.2 x 104 m) 133 Figure 163: Pressure contour pl ot for Methanol (Q = 5.678 x 107 m3/s, R2 = 7.2 x 104 m) 134 Figure 164: Streamline contour pl ot for Methanol (Q = 5.678 x 107 m3/s, R2 = 7.2 x 104 m) 134 Figure 165: Free surface profile for all fluids (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m) 135 Figure 166: Magnified free surface pr ofile for all fluids (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m) 135 Figure 167: Free surface profile for all fluids (Q = 5.678 x 107 m3/s, R2 = 7.2 x 104 m) 136 Figure 168: Magnified free surface pr ofile for all fluids (Q = 5.678 x 107 m3/s, R2 = 7.2 x 104 m) 136 Figure 169: Dimensionless free surface plot for all fluids (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m) 137 Figure 170: Dimensionless free surface profile for all fluids (Q = 5.678 x 107 m3/s, R2 = 7.2 x 104 m) 137 Figure 171: Free surface profile for FC77 in all nozzle geometries (Q = 4.416 x 107 m3/s) 138 Figure 172: Magnified free surface profile for FC77 in all nozzle geometries (Q = 4.416 x 107 m3/s) 138 Figure 173: Free surface profile for FC72 in all nozzle geometries (Q = 4.416 x 107 m3/s) 139 Figure 174: Magnified free surface profile for FC72 in all nozzle geometries (Q = 4.416 x 107 m3/s) 139 Figure 175: Free surface profile for FC87 in all nozzle geometries (Q = 4.416 x 107 m3/s) 140 Figure 176: Magnified free surface profile for FC87 in all nozzle geometries (Q = 4.416 x 107 m3/s) 140
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xv Figure 177: Free surface profile for Methanol in all nozzle geometries (Q = 4.416 x 107 m3/s) 141 Figure 178: Magnified free surface profile for Methanol in all nozzle geometries (Q=4.416 x 107m3/s) 141 Figure 179: Free surface height for all fluids and all oute r slot locations (Q = 4.416 x 107 m3/s) 143 Figure 180: Free surface height for all fluids and all oute r slot locations (Q = 5.678 x 107 m3/s) 143 Figure 181: Cone angle for all flui ds at all outer slot locations (Q = 4.416 x 107 m3/s) 145 Figure 182: Cone angle for all flui ds at all outer slot locations (Q = 5.678 x 107 m3/s) 145 Figure 183: Pressure drop for all flui ds at all outer slot locations (Q = 4.416 x 107 m3/s) 147 Figure 184: Pressure drop for all flui ds at all outer slot locations (Q = 5.678 x 107 m3/s) 147 Figure 185: Coefficient of pressure for each nozzle with respect to the Reynolds number 148 Figure 186: Radial velocity component at various sections (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m) 153 Figure 187: Axial velocity component for various sections (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m) 154 Figure 188: Theta velocity component for various sections (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m) 155 Figure 189: Radial velocity component for various sections (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m) 156 Figure 190: Axial velocity component for various sections (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m) 157 Figure 191: Theta velocity component for various sections (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m) 158
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xvi Figure 192: Radial velocity component for various sections (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m) 159 Figure 193: Axial velocity component for various sections (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m) 160 Figure 194: Theta velocity component for various sections (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m) 161 Figure 195: Radial velocity component for various sections (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m) 162 Figure 196: Axial velocity component for various sections (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m) 163 Figure 197: Theta velocity component for various sections (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m) 164 Figure 198: Radial velocity component for various sections (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m) 165 Figure 199: Axial velocity component for various sections (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m) 166 Figure 200: Theta velocity component for various sections (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m) 167 Figure 201: Dimensionless radial velocity component at Section E (FC72, Q = 4.416 x 107 m3/s) 168 Figure 202: Dimensionless axial veloc ity component at Section E (FC72, Q = 4.416 x 107 m3/s) 168 Figure 203: Dimensionless theta veloc ity component at Section E (FC72, Q = 4.416 x 107 m3/s) 169 Figure 204: Dimensionless radial velocity component at Section F (FC72, Q = 4.416 x 107 m3/s) 169 Figure 205: Dimensionless axial veloc ity component at Section F (FC72, Q = 4.416 x 107 m3/s) 170 Figure 206: Dimensionless theta veloc ity component at Section F (FC72, Q = 4.416 x 107 m3/s) 170
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xvii Figure 207: Dimensionless radial velocity component at Section G (FC72, Q = 4.416 x 107 m3/s) 171 Figure 208: Dimensionless axial veloc ity component at Section G (FC72, Q = 4.416 x 107 m3/s) 171 Figure 209: Dimensionless theta veloc ity component at Section G (FC72, Q = 4.416 x 107 m3/s) 172 Figure 210: Dimensionless radial velocity component at various sections (R2 = 2.5 x 104 m) 173 Figure 211: Dimensionless axial veloc ity component at various sections (R2 = 2.5 x 104 m) 173 Figure 212: Dimensionless theta veloc ity component at various sections (R2 = 2.5 x 104 m) 174 Figure 213: Dimensionless radial velocity at various sections (R2 = 4.0 x 104 m) 174 Figure 214: Dimensionless axial veloc ity component at various sections (R2 = 4.0 x 104 m) 175 Figure 215: Dimensionless theta veloc ity component at various sections (R2 = 4.0 x 104 m) 175 Figure 216: Dimensionless radial velocity component at various sections (R2 = 5.5 x 104 m) 176 Figure 217: Dimensionless axial veloc ity component at various sections (R2 = 5.5 x 104 m) 176 Figure 218: Theta velocity com ponent at various sections (R2 = 5.5 x 104 m) 177 Figure 219: Dimensionless radial velocity component at various sections (R2 = 7.2 x 104 m) 177 Figure 220: Dimensionless axial veloc ity component at various sections (R2 = 7.2 x 104 m) 178 Figure 221: Dimensionless theta veloc ity component at various sections (R2 = 7.2 x 104 m) 178
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xviii Analysis of Flow in a Spray Nozzle With Emphasis on Exiting Jet Free Surface Ryan Mead Abstract A conical nozzle with two separate inlets within its top plate is analyzed. One of the inlets is in the center of the top plate, which is free to rotate, whereas the other inlet is positioned away from the center. The fluid entering through the outer inlet slot causes the top plate of the nozzle to spin. Seve ral fluids including FC77, FC72, FC87, and Methanol running at different flow rates were investigated to observe the effect that their particular properties have on the geometry of the fluidÂ’s free surface exiting the nozzle. Another variation performed was the geometry of the nozzle. The outer inlet slot was positioned at various radial distances along the top plate. For this nozzle, the top plate remained stationary and swirling was introduced to the fluid at the inlets. It was observed that the faster flow rates cause d an increase in the free surface height and cone angle. For the various radial locations of the outer inlet slot, it was noted that a position at approximately 75% of the nozzle radius pr oduced the largest free surface height. The largest cone angle was produced when the outer inlet slot was positioned at the edge of the nozzle top plate. Another factor that incr eased the radial height and cone angle of the free surface was the working fluid used in the study. A larger Reynolds number produced a larger cone angle and larger fr ee surface height (while a smaller Reynolds number produced a less significant cone angle and free surface height).
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1 1 Introduction Spraycooling is commonly used thr oughout the industry for heat transfer applications due to the high heat transfer co efficients that they provide. They are frequently used to anneal metals, cool effluents from pulp and paper mills, cool fission and fusion components, as well as cooling combustion walls and turbine blades. Another emerging application is the cool ing of electronics. Cooling electronic circuit integration is a vital part in mainta ining the efficiency and reliability of the circuitry. Undesirably high temperatures can severely strain the operational safety and effectiveness of the electronics. By spraying fluid on a plate that is shielding electronic sources, it is possible to carry heat away th rough the fluid and maintain an acceptable temperature. However, before determining the heat transfer propertie s of the system, it is important to determine the geometry of the sp ray exiting the swirlatomizer used to emit the fluid. Prediction of nozzle performance fo r design and analysis is critical in aiding designers to meet strict performance requir ements. For instance, the cone angle of a particular spray would be an important number to determine. A larger cone angle would mean that the spray would be covering a great er surface area, and thus cooling a larger portion of the electronics. A nother important factor is how wide the spray becomes after it exits the nozzle, or the radial height of th e free surface. Again, a greater radial height would signal that more surface are is being intr oduced to the spray. An increase in these two factors would provide better cooling for the electronics.
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2 It has been recognized that a change in the geometry of the nozzle results in a change in geometry of the exiting spray. Jeng et al. (1998) perf ormed experiments on 15 different nozzle geometries with four differe nt flow rates. They used the ArbitraryLangrangianEulerian (ALE) method to calcul ate the position of th e free surface. The finite element predictions were in good ag reement with their experiments. They concluded that the geometry of the nozzle ha d a significant effect on the parameters of the exiting free surface that they were investigating. The idea that the nozzle geometry plays a major role in the nozzle performance was reiterated by Dumouchel et al. (1993). They applied numerical analysis to the velocity field throughout a swirl spray nozzle, and more specifical ly, at the nozzle orifice. They found that the conical liquid sheet pr oduced at the nozzleÂ’s orifice was mainly dependent on the shape of the nozzle. Also in agreement with this statement is Sakman et al. (2000). They studied the lengthtodiam eter ratio of the swirl chamber and orifice, stating that an increase in the lengthdodiameter ratio fo r both the swirl chamber and orifice resulted in a decrease in the cone angl e. However, an increase in the lengthtodiameter ratio for the swirl chamber produced an increase in film thickness; an increase in the lengthtodiameter ratio for the orifice re sulted in a decrease in the film thickness. Miller and Ellis (2000) investigated sp ray nozzles for agricultural uses, mainly focusing on spray characteristic s and droplet size. They c oncluded that the interaction between the physical properties of the spray liquid and the characteristics of the spray formed was a function of the nozzle design. While some of th e changes in spray formation could be related to the dynamic surface tension of the spray liquid, there was
PAGE 22
3 evidence to show that there were other phys ical parameters that influenced spray formation. Som and Biswas (1986) agree d, stating that the pertinent governing parameters regarding the spray dispersion in cluded the liquid velocity, liquid viscosity, liquid surface tension, the dens ity of the ambient atmosphere as well as the geometrical dimensions of the nozzle. Some other investigations were perf ormed that observed the effect some parameters had on the free surface position an d the cone angle of the fluid exiting the nozzle. Datta and Som (2000) studied ways to provide theoretical pred ictions of the cone angle produced by swirl spray pressure nozzl es using numerical co mputations of the flow. They found that an increase in the fl uid flow rate created a sharp increase in the cone angle of the fluid exiti ng the swirl nozzle. Rothe and Block (1977) examined the effect that the pressure of the ambient environment to which the fluid is being sprayed had on the shape of the liquid sheet. Their wo rk, which agrees with many other studies, found that an increase in ambien t pressure and nozzle pressure drop created an increase in contraction of the liquid sheet emanating from the nozzle. However, an increase in nozzle diameter aided in decreasing the amount of contraction. Gavaises and Arcoumanis (2001) state th at an accurate esti mation of the nozzle flow exit conditions are significant in the calcul ation of sprays ejected from the nozzle. Therefore, it is important to know the conditions at the location where the fluid exits the nozzle in order to truthfully predict the pos ition of the free surface, as well as other interesting variables. After the free surface of the fluid has been modeled correctly, the heat transfer potential can then be ev aluated. Ciofalo et al. (1999) performed
PAGE 23
4 experiments with full cone swirl atomizers onto a heated wall. They confirmed that the heat transfer coefficient and maximum heat flux was dependent of the mass flux of the spray, as well as the droplet velocity. For this problem, two differentsized c onical nozzles with tw o inlet slots and one outlet were analyzed. The top plate for both of the nozzles housed the two inlet slots Â– one inlet was in the center of the top plate, whereas the outer inlet slot was positioned at the edge of the top plate. Several different nozzle geometries were examined, as well as several different fluids such as FC77, FC72, FC87, and Methanol. Each of these fluids was run at various flow rates ranging from 1.262 x 107 m3/s to 5.678 x 107 m3/s in the smaller nozzle. For a larger nozzle that was studied, only FC77 and FC72 were analyzed at flow rates of 1.262 x 106 m3/s, 2.524 x 106 m3/s, and 3.785 x 106 m3/s. For the smaller nozzle that was studied, an investigation into the effect of the location of the outer inlet slot was performed. The outer in let slot was moved throughout the range of the top plate to observe any cha nges in the performance of the nozzle. What was analyzed during each trial was the hei ght of the free surface formed by the fluid exiting the nozzle. From this data, the cone angle of the spray could easily be calculated. As mentioned earlier, the radial height of the free surface along with the cone angle of the spray exiting the nozzle are important for later as pects of this project. Future studies will use the data collected in this investigation to analyze the heat transf er characteristics of these fluids as they are used to cool electr onics. Therefore, a larg er radial height and cone angle of the free surface is beneficial, because this would indicate that a greater
PAGE 24
5 fraction of the electronics would be coole d. This is significant for efficiency, and consequently, the cost of the design. However, the trouble with accurately pred icting the flow exiting the nozzle deals mainly with tracking the fluid/air interface. Commercially available software, FIDAP, which utilized the Galerkin finite element method, was used to solve for the position of the free surface. For the cases where the flow within the nozzle was deemed turbulent, the mixing length model was employed in FI DAP. This was done so that the NewtonRaphson solution method could be used to solve the flow problem.
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6 2 Nomenclature Arabic Symbols A Total area [m2] B Damping constant [nondim] Cp Coefficient of pressure [nondim] Ca Cavitation number [nondim] f Friction factor [nondim] g Gravitational constant [m/s2] hf Head loss [m] L Length of nozzle [m] Lf Length of free surface [m] l Mixing length [m] m Mass flow rate, Q [kg/s] inm Mass flow rate in [kg/s] outm Mass flow rate out [kg/s] r M Radial component of momentum [kg m/s2] z M Axial component of momentum [kg m/s2] M Theta component of momentum [kg m/s2] M Resultant vector of momentum [kg m/s2]
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7 n Unit normal vector [nondim] p Pressure [N/m2] P Inlet to outlet pr essure drop [kg/m s2] Q Volumetric flow rate [m3/s] r Radial coordinate [m] R Radial distance to nozzle wall [m] R1 Radial distance of central inlet [m] R2 Radial distance to outer inlet slot [m] R3 Radial distance to outside of outer inlet slot [m] Re Reynolds number, 2 Ur/ [nondim] Rout Radius of the nozzle outlet [m] Rtop Radius of the top plate [m] U Average velocity at the nozzle exit, [m/s] Vin Inlet velocity [m/s] Vout Outlet velocity [m/s] vr Radial velocity component [m/s] vz Axial velocity component [m/s] v Theta velocity component [m/s] v* Friction velocity [m/s] yn Normal distance from a node to the wall [m] yn + Nondimensional scale for yn [nondim] z Axial coordinate [m]
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8 Greek Symbols Radius of free surface height [m] Cone angle [degrees] Von Karman constant [nondim] Density [kg/m3] Absolute viscosity [kg/m s] Kinematic viscosity [m2/s] t Momentum eddy viscosity [m2/s] Surface tension or surface stress [kg/s2] Angular velocity of the top plate [rad/s] Subscripts atm Atmospheric avg Average i Unit vector direction j Unit vector direction min Minimum sat Saturation
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9 3 Mathematical Model Consider the axisymmetric conical nozzl e as shown in Figures 2 and 3 (pages 21 and 22). The top of the nozzle contains four inlets Â– one at the center and three outer inlets spaced equidistant from each other aroun d the top plate, which are drilled at a 45 degree angle. To model this flow in two dime nsions, the three outer inlets are modeled as a single inlet along the entire outer edge of the nozzle. The fluid entering the nozzle through the outer inlet at a 45 degree angle causes the top plate to rotate around the zaxis. This causes the fluid to have a swirlin g effect within the nozzle. To determine the rate at which the exiting fluid causes the disc to rotate, the angular momentum theorem is applied. These calculations were done for a threedimensional geometry, whereas the problem is simulated as twodimensional. According to White (1999), a control volume analysis can be useful to th e angular momentum relation: dt d H M (1) where M is the net moment about the center of mass of the system, H is the angular moment of the system about its center of mass, and t is time. Since this system involves nonrigid fluid particles that each has a differe nt velocity, the theory of mass moment of inertia has to be abandoned. For this system the instantaneous angular momentum has to be calculated by integrating H over the entire elemental masses, dm. To calculate the angular momentum about a point O, the following equation is used:
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10 dmsystem O ) (V r H (2) where r is the position vector from O to the elemental mass dm, and V is the velocity of that elemental mass. Applying the Reynolds transport theorem to Equation (2) reveals: CS CV system OdA V d dt d dt d n V V r V r Hr (3) where V d is the differential mass of the fluid and n Vr is the relative normal velocity component. From the angular momentum theorem, Equation (3) must be equal to the sum of all the moments about point O which is exerted on the control volume. Therefore, O O Odt dF r M H (4) where MO are the moments about point O, and F are the forces applied to the system. In this situation, the sum of the moments is to be taken at the center of the disc, point O. in in out out O Om m V r V r j T M (5) where TO is the retarding torque, and m is the mass flow rate of the system. For this study, the retarding torque, TO, is considered to be negligible, and therefore is set to zero. Applying the conditions of this system to the angular momentum theorem results in: Q m mout in (6) j A Q Vin (7) k j r A Q A Q Vout) 45 cos( ) 45 cos( (8)
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11 k j V r ) 45 cos( ) 45 cos(2A Q r A Q r rout (9) k V r A Q rin (10) where Q is the volumetric flow rate of the fluid, is the angular velocity of the disc, and A is the total inlet area. Since the disk is rotating solely about the jaxis, and it is fixed with respect to the other axes, then only the Â‘jtermsÂ’ of the cross products are of interest. Therefore, ) 45 cos( 02A Q r r QOM (11) ) 45 cos(2 A Q r r (12) r A Q ) 45 cos( (13) For the dimensions of the conical nozzle, we can calculate the total inlet area, A, and the position vector, r. 2 1 2 2 245 cos ) ( 2 R R R Atop (14) topR r (15) Hence, for the smaller nozzle: Q 3 9m 1 10 852 1 (16) where Q is the volumetric flow rate given in m3/sec. As for the larger nozzle, the spinning rate is obtained from the following equation.
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12 Q m 3 61 10 248 2 (17) where, again, Q is the volumetric flow rate given in m3/sec. The following is a table that lists several flow rates of interest, along with its co rresponding angular velocity, Table 1: Angular veloci ty for given flow rate in the smaller nozzle Inlet Flow Rate Q [m3/s] Angular Velocity [rad/sec] 1.262 x 107 233.67 2.524 x 107 467.34 4.416 x 107 817.84 5.678 x 107 1051.50 Table 2: Angular velo city for given flow rate in the larger nozzle Inlet Flow Rate Q [m3/s] Angular Velocity [rad/sec] 1.262 x 106 2.837 2.524 x 106 5.675 3.785 x 106 8.512 Next, the actual fluid mech anics of the problem must be solved. If the fluid is considered to be incompressible, the equa tions describing the conservation of mass and momentum in cylindrical coordinates can be expressed as (Burmeister, 1993):
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13 0 z v r v r vz r r (18) 2 2 2 21 1 r v z v r v r r r r p z v v r v r v vr r r t r z r r (19) 2 2 21 r v z v r v r r r z v v r v v r v vt z r r (20) 2 21 1 z v r v r r r z p g z v v r v vz z t z z z r (21) Now, the turbulence within the fluid must be considered. For the larger nozzle, all of the flow rates resulted in laminar flow However, for the smaller nozzle, only the flow rates of 1.262 x 106 m3/s and 2.524 x 106 m3/s resulted in laminar flow. For the remaining flow rates, the mixing length mode l was used for simulation of turbulence in this problem. The mixing length turbulence model is a zeroequation model which uses the following relationship to determine the turbulent viscosity. r v v lr r t 2 (22) B y y ln nexp 1 (23) where is the Von Karman constant ( = 0.4), yn is the normal distance from the node to the wall yn + is a scale used to nondimensionali ze the problem, and B is the damping constant. The Van Driest damping factor is located within the brackets [ ]. *v y yn n (24)
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14 where v* is the friction velocity. The previous equations are subject to the following boundary conditions: 0 , 0 : 10 00 1 0 0 At 3 v v v v r zin z r (25) 0 0 : 10 43 5 10 .00 1 0 At 3 3 z r v v r z (26) 0 45 cos 45 sin : 10 43 7 10 3 .4 5 0 At 3 3 v v v v v r zin z in r (27) 0 0 0 : R L z 0 At v v v rz r (28) 0 z 0 0 : 0 0 At z rv v v L z r (29) 0 : 0 At p r L L zf (30) 0 1 : L At 2 / 3 2 2 2 n v dz d dz d p p v v dz d L L z rt atm z r f (31) For the instance where the outer inlet slot on the smaller nozzle was varied, Equation 25 through Equation 27 of the boundary conditions became: 0 , 0 : 10 00 1 0 0 At 3 v v v v r zin z r (32) 0 0 0 : 10 .00 1 0 At 2 3 v v v R r zz r (33) in in z rv v v v v R r R z , 0 : 0 At 3 2 (34) 0 0 0 : 0 At 3 v v v R r R zz r (35)
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15 Also for the instance where the outer inle t slot on the smaller nozzle was varied, R1 remained constant at 1.00 x 103 m and four cases for R2 was chosen. The outer radial dimension for the outer inlet slot, R3, was then calculated. The outer inlet was simulated as a continuous slot around the nozzle. In ac tuality, there are three separate outer inlets, each having the same area as the central inlet. Therefore, R3 was determined so that the continuous slot had three times the area of the central inlet. Theref ore the inlet velocity would be the flow rate divided by the sum of the inlet areas. Since all four inlets have the same area, the total area can be taken as f our times the area of the central inlet. The following equations show how R3 and the inlet velocity were determined. 2 1 2 2 2 33R R R (36) 2 14R Q Vin (37) Often, due to limitations in computer resources it is necessary to simulate a given flow problem in a truncated computationa l domain. In such situations, a boundary condition must be supplied at the boundary typically an outlet boundary (z = L + Lf), of the truncated computational domain. In most situations, the stressfree boundary condition that arises naturally from the appl ication of the finite element method to the flow equations is quite adequate. However, in certain situations the natural outflow boundary condition is not suitable. This occurs when there is a free surface involved with the problem. The problem arises from the pr esence of the pressure term in the expression for the normal stress. At the outlet boundary, the surface stress vector on the boundary of a fluid element is defined in FIDAP by:
PAGE 35
16 j ij in (38) ji ij ij ijv v p (39) where ij is the stress tensor and nj is the unit normal vector at the boundary. The normal component of the stress vect or is given in FIDAP by: j e ji ij i i nn n v v p n (40) It is also desired to determine if cav itation is present at the location where the pressure is a minimum in the nozzle. Cavitation occurs when the pressure of the liquid falls below the saturation pressure for that part icular fluid. When this happens, the fluid begins to evaporate causing tiny bubbles to form on the boundary surface, which will eventually erode and destroy the syst em. The cavitation number is found by: 2 min5 0U p p Casat (41) Next, it is important to know the averag e value for each component of velocity within the system. With the average component of velocity it is easy to calculate the average component of momentum for the flui d. The equations to accomplish this are given for the radial component of velocity and momentum. R r avg rdA V A V0 ,1 (42) avg r avg rV m M, (43) where the axial and theta components of velo city and momentum ar e calculated in the same manner.
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17 The pressure coefficient can be calcula ted for each case. A larger pressure coefficient usually means that there is a la rge pressure drop. However, the pressure coefficient is also dependent on the veloci ty, as well as the density, of the fluid. 25 0U p Cp (44) To assure that the pressure drops from the inlet to the outlet of the nozzles are in accordance with what they should be, BernoulliÂ’s equation was used to calculate them. Theses calculated values were then compared to the values obtained from FIDAP. The pressure drop obtained using BernoulliÂ’s eq uation must be larger than the values produced by FIDAP. Bernoulli Â’s equation is given by: f in outh g z z g V V p1 2 2 22 (45) where g is the gravitational constant, and hf is the head loss due to friction. This value is given by: g V r z z f hout f2 22 1 2 (46) where f is the friction factor f ound in the Moody chart (White, 1999).
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18 4 Numerical Computation The governing equations, along with th e boundary and initial conditions were solved using the finiteelement method. Th e fluid region was divided into a number of quadrilateral elements. After the Galerkin formulation was used to separate the governing equations, the Ne wtonRaphson method was used to solve the ensuing algebraic equations. For the turbulent fl ows, the mixing length turbulence model was employed so that the NewtonRaphson method c ould be utilized. The equations for the conservation of mass and conser vation of momentum were then solved simultaneously. For this free surface problem, calculations we re performed for each time step until steady state conditions were attained. The results observed were the results that occurred at steady state. A variable time step was instituted to ensure efficiency due to large variations in the beginning and small vari ations toward the arrival of steady state conditions. In order to ensure that an accurate so lution was obtained, the number of elements that were used to mesh the geometry had to be deemed adequate. This was done by performing computations for several combinati ons of elements in the axial and radial directions covering the nozzle geometry. The velocity profiles at the nozzle outlet for these simulations are plotted in Figure 1. It was noted that the numerical solution became grid independent when the number of elements in the axial and radi al directions were greater than 172 and 32, respectively. Com putations with a 172x32 grid produced results
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19 that were identical up to f our significant figures to the values produced by a grid of 248x48. In order to conserve computer time while maintaining accuracy, a grid size of 172x32 was used for all of the final compuations. The quantitative difference in grid independence can be calculated using the following equation: e N D C V (47) where N is the number of elements along an ax is, and C, D, and e were constants to be evaluated. V is the ve locity at a given radial coordinate along the outlet of the nozzle. Equation (47) has three unknowns at three sets of velocities ta ken at three different grid sizes. The result is a set of nonlinear equations with three variables. An initial value of e is assumed, and after performing a number of iterations, a correct value for e is determined. By definition the value of e mu st be greater than one. At the radial coordinate of r = 4.8 x 105 m, e was found to be 8.056. After replacing e into Equation (47), the values of C and D were determined to be 851.7149598 and 7.1324174 x 1016, respectively. To obtain a percent error for the various computations, the following equation was used. 100 C C V (48) It was seen that at r = 4.8 x 105 m, the percent error fo r the grid involving 80 elements in the axial direction and 16 elements in the radial direction was approximately 3.906%. At the same radial location, the percent error for the 172x32 grid was calculated to be 8.206 x 103 %, whereas the percent error for th e 248x48 grid was determined to be 4.33 x 104 %. It is easily seen that these two gr ids produce nearly identi cal results. This
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20 calculation is also backed up by Figure 1 in which the velocity profile at the outlet of the nozzle is plotted for th e various grid sizes. 0 1 2 3 4 5 6 7 8 9 10 0.0E+002.0E054.0E056.0E058.0E051.0E041.2E041.4E04 Nz x Nr = 80 x 16 Nz x Nr = 172 x 32 Nz x Nr = 248 x 48 Figure 1: Velocity profile for various grid sizes Velocity [m/s] Radial Coordinate [m]
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21 5 Results and Discussion Computations were performed with two different general nozzle sizes. Figure 2 shows the dimensions of the larger nozzle, while Figure 3 shows the dimensions of the smaller nozzle. For the larger nozzle, two fl uids (FC77 and FC72) were tested at three different flow rates (1.262 x 106 m3/s, 2.524 x 106 m3/s, and 3.785 x 106 m3/s). For the smaller nozzle, FC77 and FC72 were used in addition with FC87 and methanol. These fluids were tested through a range of inle t flow rates that varied from 1.262 x 107 m3/s to about 5.678 x 107 m3/s. The inlet velocities were calcula ted so that the velocity of the fluid entering the center hole was equal to that of the fluid entering thr ough the outer slot. NOTE: ALL DIMENSIONS ARE IN METERS Figure 2: Dimensioned schematic for larger nozzle
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22 NOTE: ALL DIMENSIONS ARE IN METERS Figure 3: Dimensioned schematic of smaller nozzle 5.1 Large Nozzle The larger nozzle is to be analyzed firs t. The inlet flow en tering through the outer slot causes the top plate to rotate at a calcula ted rate. Table 3 depict s the spinning rate of the top plate as a result of the corresponding inlet flow rate in the larger nozzle. As would be expected, the increase in flow ra te results in an increased spinning rate. Table 3: Calculated spinning rate for given flow rate in larger nozzle Inlet Flow Rate Q [m3/s] Spinning Rate [rad/sec] 1.262 x 106 2.837 2.524 x 106 5.674 3.785 x 106 8.512
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23 As was mentioned previously, the fluid velocity through the center inlet and the outer slot are equal. Since th e fluid enters the nozzle throug h the outer slot at a 45 degree angle, the axial component of the velocity is equal to the total velocity multiplied by the cosine of 45 degrees, and the radial velocity is equal to the total velocity multiplied by the negative sine of 45 degrees. Table 4 shows the inlet flow rates with the equivalent inlet velocities. All of the flow rates were determined to be laminar in the nozzle. Table 4: Inlet velocities for the given flow rates in the larger nozzle Inlet Flow Rate Q [m3/s] Center Hole Vin [m/s] Axial Component for Outer Slot Vz [m/s] Radial Component for Outer Slot Vr [m/s] 1.262 x 106 0.0757 0.0535 0.0535 2.524 x 106 0.1513 0.1070 0.1070 3.785 x 106 0.2270 0.1605 0.1605 Table 5 shows the Reynolds number calculated at the nozzle exit for each fluid and each flow rate used with the large nozzle. The Reynolds number for FC72, which plays an important role in the shape of the fr ee surface, is almost twice the value used for FC77. This is because the viscosity of FC72 is approximately half of that for FC77. Table 5: Reynolds number for each fluid and each flow rate Inlet Flow Rate Q [m3/s] Working Fluid Reynolds Number at Nozzle Outlet [nondim] FC77 316 1.262 x 106 FC72 664 FC77 632 2.524 x 106 FC72 1328 FC77 949 3.785 x 106 FC72 1992
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245.1.1 Large Nozzle Â– FC77 The first fluid that was tested in the larger nozzle was FC77. This fluid has a density of 1780 kg/m3, a viscosity of 1.424x103 kg/ms, and a surface tension of 0.015 N/m. Figure 4 shows the velocity vector plot for FC77 with an inlet flow rate of 1.262 x 106 m3/s. For this case, the top plate is spinning at a rate of 2.837 rad/ sec. For each case the radial free surface heights, as well as th e cone angles, were analyzed to provide information for future projects concentrating on heat transfer characteristics involving these sprays. As can be observed from the ve locity vector plot, the free surface of the spray exiting the nozzle tends to rise as the axial distance increases. For this particular trial, the initial free surface height was 3.175 x 103 m and the final free surface height reached 3.427 x 103 m, which caused a cone angle of 11.75 degrees. The maximum velocity experienced wa s 0.0757 m/s, at each of the inlets. This is because a smaller area should have a grea ter velocity due to the conservation of momentum, which states that the inlet flow ra te must be equal to the outlet flow rate. Figure 4: Velocity vector plot for FC77 (Q = 1.262 x 106 m3/s). Units are cm/s.
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25 The pressure contour plot for this case is shown in Figure 5. The inlettooutlet pressure drop for this trial was 3.06 Pa, which is excellent. It is ideal to have a minimum pressure drop from the inlet of the nozzle to the outlet of the nozzle, so as to allow the fluid to flow freely through the nozzle. It was noted that the larger nozzle produced a much smaller pressure drop than did the smal ler nozzle. As will be seen later in the paper, the smaller nozzle forces the fluid to move very fast through the throat and essentially the outlet of the nozzle, which cau ses a decrease in the pressure at that location. Therefore, the slower moving fluid in the larger nozzle allows the pressure at the outlet to remain at a greater value causing a minute pressure drop. Figure 5: Pressure contour plot for FC77 (Q = 1.262 x 106 m3/s). Units are gm/cm s2 (x101 Pa). Next, Figure 6 shows the results of the velo city vector plot for FC77 with an inlet flow rate of 2.524 x 106 m3/s. This flow rate results in the top plate spinning at 5.674 rad/sec. The maximum velocity was discove red to be 0.1513 m/s, and similar to the previous trial, was located at the two nozzle in lets. Again, the initial free surface height
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26 was 3.175 x 103 m and the final free su rface height was 3.409 x 103 m. The phenomenon also resulted in a cone angle of 11.75 degrees. This result was counterintuitive in the sense that it the final free surface height for 2.524 x 106 m3/s was less than that of 1.262 x 106 m3/s. It was concluded that this was a numerical error within the computing program itself. The pressure contour plot for this cas e is shown in Figure 7. The maximum pressure was found to be 16.45 Pa, and the mi nimum pressure was found to be 8.99 Pa. The inlettooutlet pressure drop for this scen ario was 12.72 Pa, which is greater than that of the previous trial. All of the pressure contour plots for the larger nozzle reveal that the pressure is at a maximum in the area of the out er inlet slot, and then decreases as the axis of symmetry and the outlet of the nozzle are approached. Figure 6: Velocity vector plot for FC77 (Q = 2.524 x 106 m3/s). Units are cm/s.
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27 Figure 7: Pressure contour plot for FC77 (Q = 2.524 x 106 m3/s). Units are gm/cm s2 (x101 Pa). The velocity vector plot for an inlet flow rate of 3.785 x 106 m3/s is shown in Figure 8. For this case, the top plate was sp inning at a rate of 8. 512 rad/sec due to an inlet velocity of 0.227 m/s. Identical to the other two cases the maximum velocity in the nozzle is located at the inlets and equal to th e inlet velocity. Also identical to the other two cases, the initial free su rface height was 3.175 x 103 m; however, the final free surface height rose to 3.540 x 103 m with a cone angle of 15.00 degrees. This was to be expected, since an increase in flow rate re sults in a faster movi ng fluid, which should result in a larger cone angle. Figure 9 illustrates the pressure contour plot for this case. The maximum and minimum pressures calculated for this case we re 42.59 Pa and 18.77 Pa. The inlettooutlet pressure drop was determined to be a bout 33.74 Pa, which is quite higher than the previous two cases. Also, similar to the othe r trials, the pressure is a maximum at the outer inlet slot and gradually wanes towards th e axis of symmetry and the nozzle outlet.
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28 Figure 8: Velocity vector plot for FC77 (Q = 3.785 x 106 m3/s). Units are cm/s. Figure 9: Pressure contour plot for FC77 (Q = 3.785 x 106 m3/s). Units are gm/cm s2 (x101 Pa). Figure 10 shows the free surface profile for FC77 at the previous flow rates. It was noted that the flow rate of 3.785 x 106 m3/s produced the greatest radial height of the free surface, whereas 1.262 x 106 m3/s and 2.524 x 106 m3/s produced similar radial heights.
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29 Figure 10: Free surface profile for FC77 at various inlet flow rates A dimensionless free surface profile is shown in Figure 11 based on the Reynolds number calculated at the nozzle exit. It was observed that a higher Reynolds number provided a greater radial free surface height and a larger cone angle. 0.96 0.98 1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14 9.510.010.511.011.512.012.513.013.5 Re = 316 Re = 632 Re = 949 Figure 11: Dimensionless free surface profile for FC77 (large nozzle) 3.0E03 3.1E03 3.2E03 3.3E03 3.4E03 3.5E03 3.6E03 3.0E023.2E023.4E023.6E023.8E024.0E024.2E024.4E02 Q=1.262E06 m 3 /s Q=2.524E06 m 3 /s Q=3.785E06 m 3 /sAxial Coordinate [m] Radial Coordinate [m] Dimensionless Axial Coordinate, z/RoutDimensionless Radial Coordinate, r/Rout
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305.1.2 Large Nozzle Â– FC72 The above analysis was for the larger no zzle using FC77 as the working fluid. Now, the larger nozzle with FC72 introduced as the working fluid shall be discussed. FC72 has a density of 1680 kg/m3, a viscosity of 6.4 x 104 kg/ms, and a surface tension of 0.010 N/m. Similar to the investigation involving FC77, this investigation is to include trials involving 1.262 x 106 m3/s, 2.524 x 106 m3/s, and 3.785 x 106 m3/s as the various inlet flow rates. Th e inlet velocities for each case are the same as the previous cases, which in turn result in the same spinning rates for the top plate. Therefore, the maximum velocities, as well as the location of these velocities, are also the same as the previous cases where FC77 was used as the wo rking fluid. Again, the radial free surface heights, as well as the cone a ngles were analyzed. A larger free surface height and larger cone angle would be beneficial for this probl em, since it is desired to yield a spray that covers a maximum surface area. Figure 12 shows the velocity vector plot for 1.262 x 106 m3/s as the inlet flow rate. With the initial free surf ace height remaining at 3.175 x 103 m, the final free surface height reached 3.691 x 103 m. This is slightly greater than the final free surface height obtained for FC77. The cone angle fo r this case was calculated to be about 18.16 degrees, which is also greater than the valu e that resulted from FC77 as the working fluid. For this same case, Figure 13 shows the pressure contour plot. The maximum and minimum pressures obtained in the nozzle were 4.13 Pa and 2.93 Pa, respectively. The inlettooutlet pressure drop for this instan ce was 3.88 Pa. It was noted that for both working fluids, as the inlet flow rate increas ed, so did the inlettooutlet pressure drop.
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31 Figure 12: Velocity vector plot for FC72 (Q = 1.262 x 106 m3/s). Units are cm/s. Figure 13: Pressure contour plot for FC72 (Q = 1.262 x 106 m3/s). Units are gm/cm s2 (x101 Pa). The inlet flow rate of 2.524 x 106 m3/s was observed next. Figure 14 shows the velocity vector plot, which re vealed that the maximum veloci ty was again located at the nozzle inlets. The free surface of the spra y exiting the nozzle increased to 3.775 x 103 m from an original height of 3.175 x 103 m, which meant a cone angle of 20.61 degrees was
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32 present. Figure 15 shows the pressure contour plot for this case. The maximum pressure observed in the nozzle was 20.57 Pa and the mi nimum pressure was 9.06 Pa. The inlettooutlet pressure drop was determined to be 14.81 Pa, which is greater than the pressure drop for the same inlet flow rate using FC77 as the working fluid. Figure 14: Velocity vector plot for FC72 (Q = 2.524 x 106 m3/s). Units are cm/s. Figure 15: Pressure contour plot for FC72 (Q = 2.524 x 106 m3/s). Units are gm/cm s2 (x101 Pa).
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33 Figure 16 and Figure 17 show the velocity vector plot and pre ssure contour plot for 3.785 x 106 m3/s used as the inlet flow rate. The vector velocity plot continues to support the fact that greater fl ow rates result in a greater free surface height. For this case, the free surface height increased from 3.175 x 103 m to a height of 4.049 x 103 m. The cone angle for this situ ation was determined to be 26.75 degrees, which was the highest value obtained for the larger nozzle. The pressure contour plot shows that the maximum pressure within the nozzle was 52.90 Pa, and the minimum pressure was 23.93 Pa. The inlettooutlet pressure drop was calculated as 38.41 Pa. Comparing the two working fluids that were studied, it wa s observed that although both of the fluids resulted in similar pressure drops, FC72 was noted to resu lt in an increased pressure drop for each inlet flow rate. Figure 16: Velocity vector plot for FC72 (Q = 3.785 x 106 m3/s). Units are cm/s.
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34 Figure 17: Pressure contour plot for FC72 (Q = 3.785 x 106 m3/s). Units are gm/cm s2 (x101 Pa). Figure 18 shows the free surface profile obtained for each flow rate when FC72 was used as the working fluid. Agai n, the inlet flow rate of 3.785 x 106 m3/s produced the greatest free surface height. Figure 18: Free surface profile fo r FC72 at various flow rates 3.0E03 3.2E03 3.4E03 3.6E03 3.8E03 4.0E03 4.2E03 3.0E023.2E023.4E023.6E023.8E024.0E024.2E024.4E02 Q=1.262E06 m 3 /s Q=2.524E06 m 3 /s Q=3.785E06 m 3 /sAxial Coordinate [m] Radial Coordinate [m]
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35 Figure 19 shows the dimensionless plot for the free surface profile with FC72 used as the working fluid. Similar to the same plot involving FC 77, this figure shows that a higher Reynolds number produces a greater radial height for the free surface, as well as a larger cone angle, which will be important values for future investigations. 0.96 1.01 1.06 1.11 1.16 1.21 1.26 1.31 9.510.010.511.011.512.012.513.013.5 Re = 664 Re = 1328 Re = 1992 Figure 19: Dimensionless free surface profile for FC72 (large nozzle) In summary of the results for the larger nozzle, Tables 6 Â– 8 compare the pressure drop, final free surface height, and cone angle respectively, for the two working fluids. For this problem, it is advantageous to have a large final free surface height and a large cone angle. Since these sprays are going to be used to cool electronics, a larger value for these factors results in a cooli ng process that is more efficien t. It was also observed that an increase in the Reynolds number produced a larger free surface height and a larger cone angle. As the inlet velocity increased, so did the values for these variables. Also, as the viscosity decreased, these values increase d. This is because a fluid with lower viscosity has lower shear stress, which a llows the fluid to flow more freely. Dimensionless Axial Coordinate, z/Rout Dimensionless Radial Coordinate, r/ R out
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36 Table 6: Inlettooutlet p ressure drop for each working fl uid and each inlet flow rate Inlet Flow Rate Q [m3/s] FC77 FC72 1.262 x 106 3.06 Pa 3.88 Pa 2.524 x 106 12.72 Pa 14.81 Pa 3.785 x 106 33.74 Pa 38.41 Pa Table 7: Final free surfa ce height for each working fluid and each flow rate Inlet Flow Rate Q [m3/s] FC77 FC72 1.262 x 106 3.427 x 103 m 3.691 x 103 m 2.524 x 106 3.409 x 103 m 3.775 x 103 m 3.785 x 106 3.540 x 103 m 4.409 x 103 m Table 8: Cone angle for ea ch working fluid a nd each flow rate Inlet Flow Rate Q [m3/s] Working Fluid Cone Angle [deg] FC77 11.75 1.262 x 106 FC72 18.16 FC77 11.75 2.524 x 106 FC72 20.61 FC77 15.00 3.785 x 106 FC72 26.75 5.2 Small Nozzle The next variation analyzed involved usi ng a smaller nozzle. For this design, the central nozzle inlet remained the same size, whereas the other nozzle dimensions decreased. For this nozzle, FC77 was used as the working fluid for inlet flow rates of
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37 1.262 x 107 m3/s, 2.524 x 107 m3/s, 4.416 x 107 m3/s, and 5.678 x 107 m3/s. FC72 was also used, but with flow rates of only 4.416 x 107 m3/s, and 5.678 x 107 m3/s. For this nozzle geometry, no cone angles were produced, so therefore, none were reported. This smaller nozzle was also subject to some additional alterations, which will be discussed later in the paper. Similar to the large nozzl e, and for the same reasons, the free surface height and cone angle were analyzed for each variation as necessary. First, since the nozzle dimensions are mu ch smaller than the larger nozzle, the velocity of the fluid through the inlets need s to be large to produce the required flow rates. This large velocity in turn results in a very large spinning ra te for the top plate of the nozzle. Although the geometry of this no zzle is different than that of the larger nozzle tested, the velocities through the nozzle in lets are calculated in the same manner. The velocity through the central hole is equi valent to the velocity entering through the outer slot at a 45 degree angle. Table 9 and Table 10 depict the spinning rate and inlet velocities, respectively, for the various inlet flow rates. Table 9: Calculated spinning rate for given flow rate in smaller nozzle Inlet Flow Rate Q [m3/s] Spinning Rate [rad/sec] 1.262 x 107 233.67 2.524 x 107 467.34 4.416 x 107 817.84 5.678 x 107 1051.50
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38Table 10: Inlet velocities for the gi ven flow rates in the smaller nozzle Inlet Flow Rate Q [m3/s] Center Hole Vin [m/s] Axial Component for Outer Slot Vz [m/s] Radial Component for Outer Slot Vr [m/s] 1.262 x 107 0.2455 0.1736 0.1736 2.524 x 107 0.4911 0.3472 0.3472 4.416 x 107 0.8594 0.6077 0.6077 5.678 x 107 1.1049 0.7813 0.7813 Table 11 shows the Reynolds number calcula ted at the nozzle exit for each fluid and each flow rate used with the small nozzle. It can again be seen that FC72 produces a Reynolds number almost twice as much as FC 77. This is because their densities are very similar, but the viscosity of FC72 is approximately half of that for FC77. Table 11: Reynolds number for each fluid and each flow rate Inlet Flow Rate Q [m3/s] Working Fluid Reynolds Number at Nozzle Outlet [nondim] FC77 803 1.262 x 107 FC72 FC77 1607 2.524 x 107 FC72 FC77 2812 4.416 x 107 FC72 5904 FC77 3615 5.678 x 107 FC72 7591 5.2.1 Small Nozzle Â– FC77 Figure 20 shows the velocity vector plot for FC77 as the working fluid flowing at 1.262 x 107 m3/s. For this flow rate, the top plate was spinning at 233.67 rad/sec. Also
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39 for this flow rate, the flow throughout the nozzle was determined to be laminar. The maximum velocity was found to be around 3.25 m/s, located in the throat of the nozzle. Unlike the larger nozzle, the smaller dimensions of this nozzle force the fluid through the throat of the nozzle at a much faster rate than when it ente rs the through the inlets. The free surface began at an in itial height of 1.250 x 104 m, but decreased to a final height of 1.202 x 104 m. Figure 20: Velocity vector plot for FC77 (Q = 1.262 x 107 m3/s). Units are cm/s. Figure 21 and Figure 22 show the pre ssure and streamline contour plots, respectively. The maximum pressure w ithin the nozzle was found to be 9.48 x 103 Pa, while the minimum pressure was 1.47 x 103 Pa. The inlettooutlet pressure drop was calculated to be about 6.28 x 103 Pa. The streamline contour pl ot shows that most of the fluid entering the outer slot flows along the nozzle wall toward the outlet, while some of that fluid initially flowed to ward the center of the nozzle. The fluid entering through the
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40 central inlet flows slightly outward in the radi al direction as it moved toward the outlet. Still, some of the entering fluid was swirling next to the rotating top plate. Figure 21: Pressure contour plot for FC77 (Q = 1.262 x 107 m3/s). Units are gm/cm s2 (x101 Pa). Figure 22: Streamline contour plot for FC77 (Q = 1.262 x 107 m3/s) The next flow rate that was used with FC77 as the fluid was 2.524 x 107 m3/s. For this trial, the top plate was rotating at 467.34 ra d/sec. The velocity vector plot for
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41 this scenario is shown in Fi gure 23. The maximum velocity within the nozzle was found to be about 6.57 m/s. The fr ee surface of the liquid exiti ng the nozzle began at a height of 1.250 x 104 m, and steadily declined to a final height of 1.210 x 104 m. In Figure 23, it is observed that as the fluid enters the throat of the nozzle, the velo city of the fluid near the wall and the outer portion of the free surf ace have lower velocities than elsewhere in the flow. This phenomenon is due to the boundary condition of zero velocity at the nozzle walls. Figure 23: Velocity vector plot for FC77 (Q = 2.524 x 107 m3/s). Units are cm/s. Figure 24 and Figure 25 show the pressu re contour plot and the streamline contour plot for this case. The maxi mum pressure was found to be 3.48 x 104 Pa, whereas the minimum pressure was found to be 8.75 x 103 Pa. The pressure drop from the inlet to the outlet of the nozzle was calculated to be ap proximately 2.29 x 104 Pa. Similar to the other cases, the streamline cont our plot shows that most of the fluid that enters through the outer slot follows the nozzl e wall to the outlet, while some of the fluid
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42 move toward the center of the nozzle as it flows to the outlet. The fluid that enters through the central inlet has almo st a purely axial flow, while a portion of the fluid in the nozzle is caught swirling near the top plate. Figure 24: Pressure contour plot for FC77 (Q = 2.524 x 107 m3/s). Units are gm/cm s2 (x101 Pa). Figure 25: Streamline contour plot for FC77 (Q = 2.524 x 107 m3/s)
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43 Figure 26 reveals the velocity vector plot for the case where 4.416 x 107 m3/s is used as the inlet flow rate with FC77 as the working fluid. This resulted in the top plate of the nozzle spinning at 817.84 rad/sec. Th e maximum velocity was found to be around 11.20 m/s, and was also located in the throat of the nozzle. The free surface began at a height of 1.250 x 104 m, decreased to a height 1.217 x 104 m, then increased back to a height of 1.220 x 104 m. This flow was determined to be turbulent, and the mixing length turbulence model was employed for this case. Figure 26: Velocity vector plot for FC77 (Q = 4.416 x 107 m3/s). Units are cm/s. Figure 27 shows the pressu re contour plot for this case. The maximum and minimum pressures plotted were determined to be 1.29 x 105 Pa and 2.69 x 104 Pa, respectively. The inlettooutlet pressu re drop was calculated to be 1.19 x 105 Pa. The streamline contour plot is depicted in Figure 28. It shows that most of the fluid entering through the outer inlet slot follows the outer nozzle wall as it makes it way towards the outlet. The fluid entering th rough the central inlet moves sli ghtly outward in the radial
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44 direction before heading toward the outlet. However, for this flow rate, there is no portion of the fluid swirling at th e top plate. For this case, the top plate is rotating faster than the previous cases, which essentially pushes the fluid down the nozzle, thus eliminating the portion of fluid shown in the previous streamline plots. Figure 27: Pressure contour plot for FC77 (Q = 4.416 x 107 m3/s). Units are gm/cm s2 (x101 Pa). Figure 28: Streamline contour plot for FC77 (Q = 4.416 x 107 m3/s)
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45 The velocity vector plot fo r FC77 traveling at 5.678 x 107 m3/s, which increased the spinning rate of the top plate to 1051.5 rad/sec, is shown in Fi gure 29. This was also found to be turbulent, and again, the mixing length turbulence model was used. The maximum velocity, like the other trials, was f ound to be in the throat of the nozzle, and equivalent to about 14.30 m/s. Also similar to the previous trial, the free surface height started at 1.250 x 104 m, declined to 1.217 x 104 m, then rose to 1.221 x 104 m. It was observed from this data, that the inlet flow ra te had very little ef fect on the free surface height; however, out of all of the variations, the inlet flow rate affected the free surface height and the cone angle the most. Figure 29: Velocity vector plot for FC77 (Q = 5.678 x 107 m3/s). Units are cm/s. Figure 30 and Figure 31 show the pressure and streamline contour plots for this situation. The maximum pressure was determined to be 2.11 x 105 Pa, whereas the minimum pressure was determined to be 4.62 x 104 Pa. The two extremes were located at the inlet and outlet of the nozzle, respectiv ely. The pressure drop from inlet to outlet
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46 was calculated to be 1.95 x 105 Pa. The streamline plot show s that the fluid flows in a fairly straight path toward th e nozzle outlet. Most of the fl uid entering through the outer inlet slot flows along the nozzle wall, while the fluid entering through the central inlet flows along the line of symmetry. Figure 30: Pressure contour plot for FC77 (Q = 5.678 x 107 m3/s). Units are gm/cm s2 (x101 Pa). Figure 31: Streamline contour plot for FC77 (Q = 5.678 x 107 m3/s).
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47 Figure 32 and Figure 33 depict the prof ile of the free surface obtained when the working fluid was FC77. It was noted that th e inlet flow rate had little effect on the height of the free surface; however, 5.678 x 107 m3/s produced a sligh tly greater height. Figure 32: Free surface profile fo r FC77 at various flow rates Figure 33: Magnified free surface prof ile for FC77 at various flow rates 1.00E04 1.05E04 1.10E04 1.15E04 1.20E04 1.25E04 1.30E04 1.35E04 1.40E04 1.45E04 1.50E04 1.2E03 1.3E03 1.4E031.5E03 1.6E031.7E03 1.262E07 m 3 /s 2.524E07 m 3 /s 4.416E07 m 3 /s 5.678E07 m 3 /s 1.19E04 1.20E04 1.21E04 1.22E04 1.23E04 1.24E04 1.25E04 1.26E04 1.2E03 1.3E03 1.4E03 1.5E03 1.6E03 1.7E03 1.262E07 m 3 /s 2.524E07 m 3 /s 4.416E07 m 3 /s 5.678E07 m 3 /s Axial Coordinate [m] Radial Coor d inate [m] Axial Coordinate [m] Radial Coordinate [m]
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48 Figure 34 shows the dimensionless free surf ace profile for FC77 with respect to the Reynolds number. As was expected, it was shown that a higher Reynolds number produced a greater height for the free surface. Since FC77 was the only fluid analyzed for this plot, the greater Reynolds number is solely a product of a larger velocity. Therefore, it is reasonable to state that a larger velocity produces a larger radial free surface height. However, it was observed that for the cases when the Reynolds number was 2812 and 3615, the height did not vary a significant amount. 0.96 0.96 0.97 0.97 0.98 0.98 0.99 0.99 1.00 1.00 1.01 9.510.010.511.011.512.012.513.013.514.0 Re = 803 Re = 1607 Re = 2812 Re = 3615 Figure 34: Dimensionless free surfa ce profile for FC77 (small nozzle) 5.2.2 Small Nozzle Â– FC72 Next, FC72 was substituted as the workin g fluid, and trials were run for inlet flow rates of 4.416 x 107 m3/s and 5.678 x 107 m3/s. The top plate of the nozzle is still being simulated as a spinning disc. The case using FC72 at 4.416 x 107 m3/s through the nozzle inlets was analyzed first. Similar to the case involving FC77 as the working Dimensionless Axial Coordinate, z/Rout Dimensionless Radial Coordinate, r/ R out
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49 fluid, this flow rate caused the top plate to spin at 233.67 rad/sec. Figure 35 shows the velocity vector plot for this variation. The maximum velocity was found to be approximately 11.10 m/s. The free surface of the fluid exiting the nozzle began at a height of 1.250 x 104 m, decreased to 1.218 x 104 m, and then increased to a final height of 1.222 x 104 m. Figure 35: Velocity vector plot for FC72 (Q = 4.416 x 107 m3/s). Units are cm/s. Figure 36 and Figure 37 show the pressure and streamline contour plots for this case, respectively. The maximum pressure within the nozzle was found to be approximately 1.19 x 105 Pa, whereas the minimum pressure was about 2.77 x 104 Pa. The pressure drop from the inlet of the nozzl e to the outlet was calculated to be about 1.10 x 105 Pa. The streamline plot shows results sim ilar to the others obtained thus far. This particular nozzle geometry allows the flui d to flow in the same manner regardless of the working fluid or the inlet flow rate.
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50 Figure 36: Pressure contour plot for FC72 (Q = 4.416 x 107 m3/s). Units are gm/cm s2 (x101 Pa). Figure 37: Streamline contour plot for FC72 (Q = 4.416 x 107 m3/s). Next, the inlet flow rate of FC72 was increased to 5.678 x 107 m3/s. The velocity vector plot for this situation is shown in Figure 38. The maximum velocity was again located in the th roat of the nozzle, and had a valu e of about 14.25 m/s. The free
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51 surface for this case initiated at a height of 1.250 x 104 m, declined to 1.220 x 104 m, before increasing to a final radial height of 1.223 x 104 m. It was noted that throughout the various inlet flow rates, FC72 produced a greater radial height for the free surface over FC77. Figure 38: Velocity vector plot for FC72 (Q = 5.678 x 107 m3/s). Units are cm/s. Figure 39 and Figure 40 show the pressure and streamline contour plot for this situation. The maximum pressure with in the nozzle was found to be 1.96 x 105 Pa, while the minimum pressure was found to be about 4.66 x 104 Pa. The maximum pressure was present at the nozzle inlet, whereas the minimu m pressure was located at the throat of the nozzle. The inlettooutlet pressure dr op of the system was calculated to be approximately 1.81 x 105 Pa. The streamline contour plot shows that most of the fluid entering through the outer inlet slot follows the nozzle wall as it flows toward the inlet. However, some of that fluid does flow towa rd the line of symmetry of the nozzle as it
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52 flows toward the outlet. The fluid entering through the centra l inlet travels in an axial path through the nozzle. Figure 39: Pressure contour plot for FC72 (Q = 5.678 x 107 m3/s). Units are gm/cm s2 (x101 Pa). Figure 40: Streamline contour plot for FC72 (Q = 5.678 x 107 m3/s). The profile of the free surface obtained when FC72 was used as the working fluid is displayed in Figure 41 and Figure 42. It was noted that the inlet flow rate had a
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53 very minute effect on the radial height of the free surface. However, a flow rate of 5.678 x 107 m3/s did produce a slightly greater radial height. Figure 41: Free surface profile fo r FC72 at various flow rates Figure 42: Magnified free surface prof ile for FC72 at various flow rates 1.100E04 1.150E04 1.200E04 1.250E04 1.300E04 1.350E04 1.400E04 1.450E04 1.500E04 1.2E031.3E031.4E031.5E031.6E031.7E03 4.416E07 m 3 /s 5.678E07 m 3 /s 1.215E04 1.220E04 1.225E04 1.230E04 1.235E04 1.240E04 1.245E04 1.250E04 1.255E04 1.2E03 1.3E03 1.4E031.5E03 1.6E03 1.7E03 4.416E07 m 3 /s 5.678E07 m 3 /sAxial Coordinate [m] Radial Coordinate [m] Axial Coordinate [m] Radial Coordinate [m]
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54 The dimensionless free surface profil e based on the Reynolds number for FC72 is shown in Figure 43. It can be seen that the free surface heights for the two Reynolds numbers are not much different However, similar to all of the other cases, a greater Reynolds number produced a greater value for th e final radial free su rface height. It was noted that FC72 produced a larger free surf ace height than FC77. Since FC72 has a viscosity almost half of that for FC77, a la rger Reynolds number is present for FC72. Therefore, not only is the free surface height a function of velocity, but also of viscosity. A fluid with a lower viscosity has a lower shear stress. This permits the fluid to flow in a more free manner, which allows for an increas e in the final height of the free surface. 0.96 0.97 0.97 0.98 0.98 0.99 0.99 1.00 1.00 1.01 9.510.010.511.011.512.012.513.013.514.0 Re = 5904 Re = 7591 Figure 43: Dimensionless free surfa ce profile for FC72 (small nozzle) 5.2.3 Cavitation Cavitation is a phenomenon that occurs when the pressure of the liquid falls below the saturated pressure of that partic ular liquid corresponding to the temperature of Dimensionless Axial Coordinate, z/Rout Dimensionless Radial Coordinate, r/ R out
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55 that liquid. When this happens, the fluid begins to evaporate, which causes tiny bubbles to form at the location where the pressure is below the critical poi nt. These bubbles will eventually erode and destroy the boundary of the nozzle. Table 12 shows the value of the cavitation number for each fluid and each flow rate for this nozzle geometry. It was observed that only FC77 with an inlet flow rate of 1.262 x 107 m3/s (Re = 803) and FC72 with an inlet flow rate of 4.416 x 107 m3/s (Re = 5904) caused cavitation. There was no cavitation found to be present in any of the other cases that were investigated thus far. Table 12: Cavitation number for FC 77 and FC72 in the small nozzle Working Fluid Saturation Pressure Psat [Pa] Reynolds Number Re [nondim] Minimum Pressure Pmin [Pa] Pressure Difference Pmin Psat [Pa] Cavitation Number Ca [nondim] 803 1470 4150 0.353 1607 8750 3130 0.067 2812 26900 21280 0.148 FC77 5.62 x 103 3615 46200 40580 0.170 5904 27700 3200 0.024 FC72 30.9 x 103 7591 46600 15700 0.070 5.2.4 Varied Nozzle Height Another variation of the nozzle geomet ry was performed with FC77 as the working fluid and 4.416 x 107 m3/s as the inlet flow rate. The length of the nozzle, L, was changed from 1.22 x 103 m to 1.05 x 103 m. However, the sl ope of the outer wall remained the same, which caused the radius of the nozzle outlet to increase from 1.25 x 104 m to 2.23 x 104 m. The velocity vector plot for th is case is depicted in Figure 44. The top plate of the nozzle was still spinni ng at the same calculated rate, which was
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56 233.67 rad/sec. The maximum velocity was dete rmined to be 3.55 m/s within the throat of the nozzle. The free surface of the fluid exiting the nozzle began at a height of 2.230 x 104 m, and decreased to a height of 2.151 x 104 m. Figure 44: Velocity vector plot for FC 77 (Q = 4.416 x 107 m3/s, L = 1.05 x 103 m). Units are cm/s. Figure 45 and Figure 46 show the pressu re and streamline contour plots for the case involving the shortened nozzle. The ma ximum and minimum pr essure plotted was 9.90 x 103 Pa and 3.22 x 103 Pa, respectively. Again, the maximum value was located at the inlet of the nozzle, and the minimum valu e was located at the throat of the nozzle. The resulting inlettooutlet pressure drop was calculated to be 5.85 x 103 Pa, which is much less than the other trial involving th e longer nozzle with FC77 flowing at 4.416 x 107 m3/s. The streamline contour plot shows that the fluid entering the central inlet has almost a purely axial flow, wher eas most of the flow entering through the outer slot flows along the outer nozzle wall toward the outlet. Al so, a large portion of the fluid inside the nozzle is swirling next to the rotating top plate.
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57 Figure 45: Pressure contour plot for FC77 (Q = 4.416 x 107 m3/s, L = 1.05 x 103 m). Units are gm/cm s2 (x101 Pa). Figure 46: Streamline contour plot for FC77 (Q = 4.416 x 107 m3/s, L = 1.05 x 103 m) 5.2.5 Extended Free Surface To ensure that a variation in the length of the free surface did not affect the final position of the free surface, a trial was run with an extended free surface length. For this
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58 study, it was increased fro m a length of 4.8 x 104 m to 7.8 x 104 m. It was found that the variation in length did not alte r the position of the free surfac e. Figure 47 shows the free surface profile for this situati on. The profiles for the long er free surfaces are in very good comparison to the profiles involvi ng the shorter free surface lengths. Figure 47: Free surface profile for varied free surface length 5.2.6 Initial Mesh with Upward Slope To assure that the program was obtaini ng the correct free surface position of the fluid exiting the nozzle, the initial mesh of the free surface was given an upward slope. Figure 48 shows the mesh plot of this pa rticular system. After the program was complete, it was observed that this case pr oduced the same final position of the free surface as did the situation with an initially horizontal free surface. Therefore, it was noted that the initial free surface position did not affect the outcome that was produced. However, another mesh was crea ted to reiterate this point. 1.18E04 1.20E04 1.22E04 1.24E04 1.26E04 1.28E04 1.30E04 1.2E031.3E031.4E031.5E031.6E031.7E031.8E031.9E032.0E032.1E03 4.416E07 m 3 / s (short) 4.416E07 m 3 /s (long) 5.678E07 m 3 / s (short) 5.678E07 m 3 /s (long)Axial Coordinate [m] Radial Coordinate [m]
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59 Figure 48: Mesh plot of nozzle with in itial upward slope of the free surface. 5.2.7 Initial Mesh with Downward Slope For this trial, the mesh of the free surf ace was initially set with a downward slope. Figure 49 shows this mesh plot. Again, this produced no effect on the final free surface position that was created. The final free su rface had the same position as did the free surface obtained with an initial horizontal profile, along with an initial upward slope. Figure 49: Mesh plot of nozzle with init ial downward slope of the free surface.
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605.2.8 Varying Outer Slot Location Next, additional adjustments were made to the operating system, and the results that were produced were analyzed. The investigation of the spray exiting the nozzle involved altering the geometry of the nozzle. The most significan t variation in the geometry was the location of the outer inle t slot on the nozzle. Figure 50 shows a schematic of the nozzle and the placement of the outer inlet slot. The radius for the center inlet, R1, remained constant at 1.00 x 104 m, and then a value was chosen for R2. The location of R3 was then bounded by Equation 36. Figure 50: Schematic of nozzle with varying ou ter slot radii. All di mensions are in meters. Four different values for R2 were used in this study, w ith each variation subject to different fluids at differ ent flow rates. Table 13 lists the values of R2 that were used and the corresponding value of R3.
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61Table 13: Values of R1, R2, and R3 used in the investigation R1 R2 R3 1.00 x 104 m 2.50 x 104 m 3.04 x 104 m 1.00 x 104 m 4.00 x 104 m 4.36 x 104 m 1.00 x 104 m 5.50 x 104 m 5.77x 104 m 1.00 x 104 m 7.20 x 104 m 7.43 x 104 m The inlet velocity was then determined by Equation 37. The axial velocity is equal to Vin for both the center hole and the outer slot, whereas the outer slot has an additional theta velocity also equal to Vin. For this part of the investigation, the top plate of the nozzle remained stationary and swirl was introduced at the inlets. These problems were treated as turbulent, and the mixing length model was employed in FIDAP. This was done so that the NewtonRaphson met hod could be used. The following table illustrates the different inlet flow rates and the inlet velocities that they produce. Table 14: Inlet velocities for the given flow rates Inlet Flow Rate Q [m3/s] Inlet Velocity for Center Hole Vz [m/s] Axial Component of Inlet Velocity for Outer Slot Vz [m/s] Theta Component of Inlet Velocity for Outer Slot V [m/s] 4.416 x 107 3.514 3.514 3.514 5.678 x 107 4.519 4.519 4.519 5.2.8.1 R2 Equal to 2.50 x 104 m As was mentioned previously, this study involved the use of different fluids, as well as different flow rates. The different fluids used were FC77, FC72, FC87, and
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62 methanol. All of these fluids were chosen b ecause of their compatib ility with electronics, and the possibility of being used as a coo ling agent for these electronics. The first geometry that was observed was where R2 was 2.50 x 104 m. This was first observed with FC77 as the working fluid. Figure 51 shows the vector velocity plot for this geometry with an inlet flow rate of 4.416 x 107 m3/s. The maximum velocity was found to be approximately 8.75 m/s located within th e throat of the nozzle. An important value is the final height of the free surface. For th is case, the free surface began at a height of 1.250 x 104 m, gradually decreased to 1.215 x 104 m, but then rose again to end at a height of 1.239 x 104 m. This free surface produced a cone angle determined to be 1.51 degrees, which was the lowest value observe d for this particular nozzle geometry. Figure 51: Velocity vector plot for FC77 (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m). Units are cm/s. Figure 52 and Figure 53 show the pre ssure and streamline contour plots, respectively, for the same case involving FC77 with a flow rate of 4.416 x 107 m3/s. The pressure drop from the inlet to th e outlet was calculate d to be 7.25 x 104 Pa. As with all of the cases, the fluid increases speed in the throat of the nozzle, this in turn caused the
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63 pressure at that area to decrease. The st reamline contour plot s hows that as the fluid enters through the outer slot, some of the fl uid begins to swirl at the top of the nozzle before it follows the outer wall towards th e outlet. The fluid entering through the center hole briefly moves toward the outer wall, th en flows in the direction of the outlet. Figure 52: Pressure contour plot for FC77 (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m). Units are g/cm s2 (x101 Pa). Figure 53: Streamline contour plot for FC77 (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m)
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64 Next, the inlet flow rate of th e FC77 was increased to 5.678 x 107 m3/s and used in the nozzle having the same geometry as the above case. Figur e 54 shows the vector velocity plot for this trial. The maximum ve locity was again located in the throat of the nozzle; however, its value has increased to 11. 25 m/s. The height of the free surface began at 1.250 x 104 m, declined to 1.215 x 104 m, then increased to 1.243 x 104 m. This value is greater than that when the lesser flow rate was used. The cone angle for this case was calculated as about 1.73 degrees. It was expected that a faster moving fluid would provide a greater fr ee surface height, as well as a greater cone angle. Figure 54: Vector velocity plot for FC77 (Q = 5.678 x 107 m3/s, R2 = 2.5 x 104 m). Units in cm/s. Figure 55 shows the pressure contour plot for this case. The inlettooutlet pressure drop was determined to be 1.04 x 105 Pa. Again, there was high pressure towards the inlets of the nozzle, and then it decreased as the outlet was approached. Figure 56 details the streamline contour plot. It shows that the fluid is behaving the same way it did with 4.416 x 107 m3/s as the inlet flow rate. Af ter entering the inlets, the fluid
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65 tends to go towards the outer wall of the noz zle before making its way to the outlet. Again, some of the fluid begins to swirl in the top part of the nozzle. Figure 55: Pressure contour plot for FC77 (Q = 5.678 x 107 m3/s, R2 = 2.5 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 56: Streamline contour plot for FC77 (Q = 5.678 x 107 m3/s, R2 = 2.5 x 104 m)
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66 The next variation performed with this geometry was changing the working fluid to FC72. This fluid has a lower density, lower viscosity, and a lower surface tension than FC77. The first case to be examined involved was an inlet flow rate of 4.416 x 107 m3/s. Figure 57 shows the velocity vector plot for this scenario. The maximum velocity was about 8.72 m/s within the throat of the nozzle. The free surf ace height began at 1.250 x 104 m, gradually decreased to 1.216 x 104 m, and then increased to a final radial height of 1.248 x 104 m. The cone angle calculated fo r this scenario was determined to be 2.05 degrees. Figure 57: Velocity vector plot for FC72 (Q = 4.416x 107 m3/s, R2 = 2.5 x 104 m) Units are cm/s. The pressure contour plot is depicted in Figure 58. The inle ttooutlet pressure drop is extremely similar to the pressure drop found with FC 77 as the working fluid. The pressure drop for this case was found to be about 7.23 x 104 Pa. Figure 59 shows the streamline contour plot. This streamline plot is also similar to the trial with FC77 as the working fluid. It shows that as the fluid en ters the nozzle, a portion of it begins to swirl
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67 in the top region, whereas the rest of the fl uid moves towards the outer wall as it proceeds to the outlet. Figure 58: Pressure contour plot for FC72 (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 59: Streamline contour plot for FC72 (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m)
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68 The next variation in the study execut ed with the nozzle geometry having R2 equal to 2.50 x 104 m is using FC72 as the working flui d with an inlet fl ow rate of 5.678 x 107 m3/s. Figure 60 shows the velocity vector plot for this particular case. The maximum velocity was found to be about 11.21 m/s, which is very fast. The initial free surface height was 1.250 x 104 m. After exiting the nozzle, the fluid decreased to a height of 1.217 x 104 m, before gradually increasing to a final height of 1.251 x 104 m, which resulted in a cone angle of approximately 2.27 degrees. Figure 60: Velocity vector plot for FC72 (Q = 5.678 x 107 m3/s, R2 = 2.5 x 104 m). Units are cm/s. Figure 61 shows the pressure contour plot for this case. The maximum pressure found was 1.20 x 105 Pa, whereas the minimum pressure was found to be 2.04 x 104 Pa. The pressure drop from the inlet to the outlet of the nozzle was calculated to be approximately 9.85 x 104 Pa, which is very similar to the value obtained for FC77 with an inlet flow rate of 5.678 x 107 m3/s. A streamline contour plot for FC72 traveling at 5.678 x 107 m3/s is shown in Figure 62. The moveme nt of the fluid through the nozzle is
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69 observed to be very similar for all of the tria ls. After the fluid enters the nozzle through the outer slot, it moves toward the outer wall then down towards the outlet. The fluid entering through the center hole br iefly flows toward the outer wall before moving to the outlet of the nozzle. Figure 61: Pressure contour plot for FC72 (Q = 5.678 x 107 m3/s, R2 = 2.5 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 62: Streamline contour plot for FC72 (Q = 5.678 x 107 m3/s, R2 = 2.5 x 104 m)
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70 Next, FC87 was used as the working fluid with 4.416 x 107 m3/s as the inlet flow rate. Figure 63 shows the veloci ty vector plot for this situation. The maximum velocity was again located in the throat of the no zzle and had a value of about 8.75 m/s. Beginning at a height of 1.250 x 104 m, the free surface dipped to a minimum height of 1.217 x 104 m before rising again to a final height of 1.251 x 104 m, which caused a cone angle of 2.27 degrees to be present. Th is proved to be the largest value of the free surface height and cone angle where the parameters included R2 equal to 2.50 x 104 m and the inlet flow rate equal to 4.416 x 107 m3/s. Figure 63: Velocity vector plot for FC87 (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m). Units are cm/s. The pressure and streamline contour plot s are shown in Figure 64 and Figure 65. The maximum pressure within the nozzle was determined to be 7.06 x 104 Pa, whereas the minimum pressure was found to be 1.21 x 104 Pa. The pressure drop from the inlet to the outlet of the nozzle was calculated as 7.18 x 104 Pa. The streamline contour plot shows that the fluid entering through the out er inlet slot flows toward the line of
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71 symmetry, and then flows toward the outer wall as it moves through the nozzle. A pocket of swirling fluid formed just to the outside of the outer inlet slot. A portion of the fluid entering through that slot got caught in that pocket. The fluid entering through the central inlet flowed almost purely in the axial direction to the outlet of the nozzle. Figure 64: Pressure contour plot for FC87 (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 65: Streamline contour plot for FC87 (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m)
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72 Still using FC87 as the working fluid, the flow rate was in creased to 5.678 x 107 m3/s. The velocity vector plot for this s cenario is shown in Figure 66. The maximum velocity within the nozzle was found to be approximately 11.20 m/s. The free surface began at a radial height of 1.250 x 104 m, decreased to 1.217 x 104 m, and then increased to a final height of 1.253 x 104 m. The cone angle for thes e parameters was calculated as 2.38 degrees. Again, this fluid produced the gr eatest radial height of the free surface, as well as the greatest cone angle for th e studies including similar parameters. Figure 66: Velocity vector plot for FC87 (Q = 5.678 x 107 m3/s, R2 = 2.5 x 104 m). Units are cm/s. Figure 67 and Figure 68 show the pressu re contour plot and the streamline contour plot, respectively. The maximum pr essure within the noz zle was found to be 1.17 x 105 Pa, while the minimum pressure was found to be 2.07 x 104 Pa. The inlettooutlet pressure drop was then ca lculated to be about 1.19 x 105 Pa. This value is slightly larger than the values obtained from the ot her fluids. The streamline contour plot shows results that are similar to all of the streamline plots for this nozzle geometry. Some of the
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73 fluid entering through the outer inlet slot bega n to swirl near the t op plate of the nozzle, while the rest of the fluid flowed toward th e outer wall before heading toward the outlet. Figure 67: Pressure contour plot for FC87 (Q = 5.678 x 107 m3/s, R2 = 2.5 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 68: Streamline contour plot for FC87 (Q = 5.678 x 107 m3/s, R2 = 2.5 x 104 m)
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74 The final variations performed on th is particular nozzle geometry involved Methanol as the working fluid. This fluid is the least dense fluid used in the investigation, but it has the gr eatest surface tension out of a ll of the fluids that were studied. Figure 69 shows the velocity vector pl ot for Methanol with an inlet flow rate of 4.416 x 107 m3/s. Again, the maximum velocity within the nozzle occurred at the location of the throat and had a magnitude of about 8.75 m/s. The radial height of the free surface was initially at 1.250 x 104 m before it decreased to about 1.216 x 104 m, and then rose again to a final height of 1.241 x 104 m, which produced a cone angle of only 1.62 degrees. Figure 69: Velocity vector plot for Methanol (Q=4.416 x 107 m3/s, R2 = 2.5 x 104 m). Units are cm/s. Figure 70 and Figure 71 show the pressu re contour plot and the streamline contour plot for this situation. The maximu m pressure within the nozzle was found to be 3.44 x 104 Pa, while the minimum pressure was 5.34 x 103 Pa. These values are much smaller than any of the other trials. The pressure drop was then calculated to be about
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75 3.44 x 104 Pa, which is about half of that for the other trials using the same flow rate and nozzle geometry. The streamline contour plot shows results very similar to the other trials. The fluid entering th rough the central inlet flows axially toward the outlet, whereas the fluid entering thr ough the outer inlet sl ot first flows toward the nozzle wall. Figure 70: Pressure contour plot for Methanol (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 71: Streamline contour plot for Methanol (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m)
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76 Finally, the last variation for this nozzle geometry included Methanol as the working fluid with an inlet flow rate of 5.678 x 107 m3/s. Figure 72 shows the velocity vector plot for this situation. Similar to the other cases, the maximum velocity had a value of about 11.25 m/s and was located in the throat of the nozzle. The free surface of the fluid exiting the nozzle be gan at a height of 1.250 x 104 m, decreased to 1.216 x 104 m, and then increased to a height of 1.245 x 104 m. A cone angle of 1.95 degrees was observed for this particular study. Figure 72: Vector velocity plot for Methanol (Q=5.678 x 107 m3/s, R2 = 2.5 x 104 m). Units are cm/s. The pressure contour plot and streamline contour plot for this case are shown in Figures 73 and 74. The maximum pre ssure was found to be about 5.66 x 104 Pa, while the minimum pressure was found to be 9.08 x 103 Pa. The pressure drop from the inlet of the nozzle to the outlet of th e nozzle was calculated as 5.11 x 104 Pa. Similar to the case involving 4.416 x 107 m3/s as the inlet flow rate, this pressure drop is about half of that for the other cases involving the same nozzle geometry and inlet flow rate. The
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77 streamline plot shows that the fluid enteri ng through the central inlet has almost pure axial motion as it flows toward the outlet. Th e fluid entering through the outer inlet slot flows around a swirling pocket as it moved to the nozzle wall, a nd then toward the outlet. Figure 73: Pressure contour plot for Methanol (Q = 5.678 x 107 m3/s, R2 = 2.5 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 74: Streamline contour plot for Methanol (Q = 5.678 x 107 m3/s, R2 = 2.5 x 104 m)
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78 Figure 75 and Figure 76 show the free surf ace profile for all of the fluids entering at 4.416 x 107 m3/s for this nozzle geometry. It was noted that FC87 produced the highest free surface position with a height of 1.251 x 104 m. Figure 75: Free surface profile for all fluids (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m) Figure 76: Magnified free surface prof ile for all fluids (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m) 1.00E04 1.05E04 1.10E04 1.15E04 1.20E04 1.25E04 1.30E04 1.35E04 1.40E04 1.45E04 1.50E04 1.2E03 1.3E03 1.4E03 1.5E03 1.6E03 1.7E03 FC77 FC72 FC87 Methanol 1.21E04 1.22E04 1.23E04 1.24E04 1.25E04 1.26E04 1.27E04 1.28E04 1.2E03 1.3E03 1.4E03 1.5E03 1.6E03 1.7E03 FC77 FC72 FC87 Methanol Axial Coordinate [m] Radial Coordinate [m] Axial Coordinate [m] Radial Coordinate [m]
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79 Figure 77 and Figure 78 show the free surf ace profile for all of the fluids entering at 5.678 x 107 m3/s for this nozzle geometry. Ag ain, it was observed that FC87 produced the highest free surface position with a height of 1.253 x 104 m. Figure 77: Free surface profile for all fluids (Q = 5.678 x 107 m3/s, R2 = 2.5 x 104 m) Figure 78: Magnified free surface prof ile for all fluids (Q = 5.678 x 107 m3/s, R2 = 2.5 x 104 m) 1.00E04 1.05E04 1.10E04 1.15E04 1.20E04 1.25E04 1.30E04 1.35E04 1.40E04 1.45E04 1.50E04 1.2E031.3E03 1.4E03 1.5E03 1.6E03 1.7E03 FC77 FC72 FC87 Methanol 1.21E04 1.22E04 1.23E04 1.24E04 1.25E04 1.26E04 1.27E04 1.28E04 1.2E031.3E03 1.4E03 1.5E03 1.6E03 1.7E03 FC77 FC72 FC87 MethanolAxial Coordinate [m] Radial Coordinate [m] Axial Coordinate [m] Radial Coordinate [m]
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80 Figure 79 and Figure 80 show the dime nsionless free surface profile for this nozzle geometry based on the Reynolds number produced for each fluid at each flow rate. It is noted that th e larger Reynolds number produced the greatest free surface height. 0.95 0.97 0.99 1.01 1.03 1.05 9.510.010.511.011.512.012.513.013.514. 0 Re = 2812 Re = 5904 Re = 8093 Re = 3212 Figure 79: Dimensionless free surface profile for all fluids (Q = 4.416x 107 m3/s, R2 = 2.5 x 104 m) 0.95 0.97 0.99 1.01 1.03 1.05 9.510.010.511.011.512.012.513.013.514.0 Re = 3615 Re = 7591 Re = 10406 Re = 4130 Figure 80: Dimensionless free surface pr ofiles for all fluids (Q = 5.678 x 107 m3/s, R2 = 2.5 x 104 m) Dimensionless Axial Coordinate, z/Rout Dimensionless Radial Coordinate, r/ R out Dimensionless Axial Coordinate, z/Rout Dimensionless Radial Coordinate, r/ R out
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815.2.8.2 R2 Equal to 4.00 x 104 m The next geometry that was used was when R2 was equal to 4.00 x 104 m and R3 was equal to 4.36 x 104 m. The same fluids and flow rates were used as the previous geometry. First, the case of FC77 with an inlet flow rate of 5.678 x 107 m3/s was analyzed. Figure 81 shows the vector velocity plot for this particular scenario. The maximum velocity was found to be 8.83 m/s locat ed within the throat of the nozzle. The free surface began at a height of 1.250 x 104 m, decreased to 1.216 x 104 m, but then rose to a final height 1.245 x 104 m. The cone angle was f ound to be 1.84 degrees. This value is slightly more than the height when R2 was equal to 2.50 x 104 m. Figure 81: Velocity vector plot for FC77 (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m). Units are cm/s. Figure 82 and Figure 83 show the pressure contour plot and streamline contour plot, respectively. Again, th e pressure decreases from the inlet to the outlet as the velocity of the fluid increases. The pre ssure drop from the inlet to the outlet was determined to be 7.93 x 104 Pa. The streamline contour plot illustrates the path of the
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82 fluid as it flows through the nozzle. After entering the nozzle, a portion of the fluid begins to swirl near the top, while the rest of the fluid flows toward the outer wall, and eventually, the outlet. However, since the location of the outer slot has moved more towards the out edge of the nozzle, the portion of the fluid that is swirling has decreased. Figure 82: Pressure contour plot for FC77 (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 83: Streamline contour plot for FC77 (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m)
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83 The inlet flow rate of the FC77 was then increased to 5.678 x 107 m3/s. The velocity vector plot for this case is show n in Figure 84. The maximum velocity was found to be in the throat of the nozzle with a value of a bout 11.33 m/s. The free surface is initially at a height of 1.250 x 104 m, and then gradually decreases to 1.216 x 104 m, before rising to a height of 1.249 x 104 m. It was noted that for both flow rates, this geometry with the outer slot located further away from the center, has produced a more pronounced free surface profile than the previous geometry. The cone angle for this situation increased to a value of 2.05 degrees. Figure 84: Velocity vector plot for FC77 (Q = 5.678 x 107 m3/s, R2 = 4.0 x 104 m). Units are cm/s. The pressure contour plot is shown in Fi gure 85. The inlettooutlet pressure drop was calculated as 1.20 x 105 Pa, which is slightly gr eater than when 4.416 x 107 m3/s was used as the flow rate. The maximum pressure was 1.34 x 105 Pa, and the minimum pressure was 2.11 x 104 Pa. Figure 86 shows the streamli ne contour plot for this case. The fluid behaves the same with this flow rate as it did when 4.416 x 107 m3/s was used
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84 as the flow rate. Most of th e fluid entering the nozzle throu gh the outer slot flow toward the outer wall before moving to the outlet; but some of the fluid moves toward the center of the nozzle before heading to the outlet. Th ere is also still a portion of the fluid which begins to swirl in the top region of the nozzle. Figure 85: Pressure contour plot for FC77 (Q = 5.678 x 107 m3/s, R2 = 4.0 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 86: Streamline contour plot for FC77 (Q = 5.678 x 107 m3/s, R2 = 4.0 x 104 m)
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85 The next study involves the use of the sa me geometry and flow rates, but with FC72 as the working fluid. Figure 87 shows the vector velocity plot for FC72 as the working fluid trav eling at 4.416 x 107 m3/s. The maximum velocity within the nozzle was found to be about 8.79 m/s. The fluid exit ing the nozzle began at a height of 1.250 x 104 m. This free surface then began to decline to a height of 1.217 x 104 m before rising to a final height of 1.256 x 104 m, which formed a cone angl e of 2.48 degrees. This is greater than the values obtained from either flow rate using FC77 as the working fluid. Figure 87: Velocity vector plot for FC72 (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m). Units are cm/s. Figure 88 shows the pressure contour plot for this case. The maximum pressure was found to be 7.66 x 104 Pa, whereas the minimum pressure was found to be approximately 1.39 x 104 Pa. The pressure drop from the inlet to the outlet was also calculated and determined to be 5.43 x 104 Pa, which is less than the value obtained with FC77 as the working fluid. The streamline co ntour plot for this s ituation is shown in Figure 89. This graphic shows that the flui d is moving along the same paths with this
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86 geometry, independent of the flow rate. Mo st of the fluid entering through the inlets moves toward the outer wall as it moves thr ough the nozzle, while a fraction of the fluid begins to swirl. Figure 88: Pressure contour plot for FC72 (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 89: Streamline contour plot for FC72 (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m)
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87 The next variation for this geometry was using FC72 as th e working fluid and increasing the inlet flow rate to 5.678 x 107 m3/s. This flow rate provided an axial inlet velocity of 4.519 m/s to both the center inle t and outer slot. Th e outer slot also maintained a theta velocity component, whic h was also equal to 4. 519 m/s. Figure 90 shows the velocity vector plot for this case. The maximum velocity in the nozzle was located in the throat and found to have a value of 11.30 m/s. The free surface height began at 1.250 x 104 m, decreased to 1.218 x 104 m, then increased to 1.259 x 104 m. The resulting cone angle was calculated as 2.70 degrees. Figure 90: Velocity vector plot for FC72 (Q = 5.678 x 107 m3/s, R2 = 4.0 x 104 m). Units are cm/s. The pressure contour plot is shown in Figure 91. The maximum pressure was determined to be 1.27 x 105 Pa, whereas the minimum pressure was determined to be 2.43 x 104 Pa. The inlettooutlet pressure drop was calculated as 1.02 x 105 Pa, which is just slightly larger than for FC77 unde r the same parameters. Figure 92 shows the streamline contour plot. It shows the same im age as the rest of the trial involving the
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88 nozzle where R2 is equal to 4.00 x 104 m. Most of the flui d that enters the nozzle through the outer slot flows toward the outer wall before going to the outlet. The fluid entering through the center inle t moves to in the radial direction briefly before going towards the outlet, with some of the fluid swirling in the top portion of the nozzle. Figure 91: Pressure contour plot for FC72 (Q = 5.678 x 107 m3/s, R2 = 4.0 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 92: Streamline contour pl ot for FC72 (Q = 5.678x 107 m3/s, R2 = 4.0 x 104 m).
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89 Next, the working fluid was changed to FC87 and run at the same inlet flow rates. Figure 93 shows the velo city vector plot for the case where the inlet flow rate was 4.416 x 107 m3/s. The maximum velocity within th e nozzle was found to be 8.79 m/s. The free surface formed by the fluid exiting th e nozzle began at a height of 1.250 x 104 m. It then decreased to 1.218 x 104 m before it grew to a final height of 1.259 x 104 m, which produced a cone angle of 2.70 degrees. FC87 produced the greatest radial height of the free surface for this nozzle geometry as well as for all other geometries. Figure 93: Velocity vector plot for FC87 (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m). Units are cm/s. Figure 94 and Figure 95 show the pressu re contour plot and the streamline contour plot, respectively. The maximum pr essure within the noz zle was found to be 7.44 x 104 Pa, whereas the minimum pressure was found to be about 1.44 x 104 Pa. The inlettooutlet pressure drop was then calculated as approximately 7.53 x 104 Pa. The streamline plot shows that th e fluid entering through the oute r inlet slot must maneuver around a pocket of swirling flui d before it flows to the nozzle wall, while the fluid
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90 entering through the central inlet has almost pur e axial motion to the nozzle outlet. Since this geometry has the outer inlet slot further outward in a radial sens e, the location of the swirling pocket of fluid is also located further outward in the radial direction. Figure 94: Pressure contour plot for FC87 (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 95: Streamline contour plot for FC87 (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m)
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91 With FC87 still being used as the worki ng fluid within the nozzle, the flow rate was increased to 5.678 x 107 m3/s. The velocity vector plot for this situation is shown in Figure 96. The maximum velocity was again found to be located in the throat of the nozzle and had a magnitude of approximately 11.30 m/s. The free surface began at a height of 1.250 x 104 m, decreased to 1.218 x 104 m, and then increased to a final height of 1.261 x 104 m. The cone angle that resulted from this flow was determined to be about 2.81 degrees. Again, this was the larg est radial height for the free surface observed from the studies involving the same nozzle geometry and inlet flow rate. Figure 96: Velocity vector plot for FC87 (Q = 5.678 x 107 m3/s, R2 = 4.0 x 104 m). Units are cm/s. Figure 97 shows the pressure contour pl ot for this situation. The maximum pressure within the nozzle was found to be 1.23 x 105 Pa, while the minimum pressure was found to be 2.46 x 104 Pa. The pressure drop from the inlet of the nozzle to the outlet was calculated as approximately 1.25 x 105 Pa. The streamline contour plot is shown in Figure 98. This plot shows results that are similar to the other streamline plots
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92 for this nozzle geometry. The fluid entering through the central inlet flowed straight to the nozzle outlet, while the fluid entering through the outer inlet slot flowed toward the nozzle wall before heading toward the outlet. Again, there was a pocket of swirling fluid present near the top of the nozzle outside of where the outer inlet slot was positioned. Figure 97: Pressure contour plot for FC87 (Q = 5.678 x 107 m3/s, R2 = 4.0 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 98: Streamline contour plot for FC87 (Q = 5.678 x 107 m3/s, R2 = 4.0 x 104 m)
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93 The final variations performed on this nozzle geometry involv ed using Methanol as the working fluid. Figure 99 shows the veloc ity vector plot for Methanol with an inlet flow rate of 4.416 x 107 m3/s. The maximum velocity for this situation was found to be about 8.82 m/s. The free surface wa s initially at a he ight of 1.250 x 104 m before it decreased to 1.216 x 104 m. It then increased to a final height 1.247 x 104 m, which resulted in a cone angle of 1.95 degrees. Th ese values were second only to the values obtained from FC77 as the lowest free surf ace radial height and lowest cone angle. Figure 99: Velocity vector plot for Methanol (Q=4.416 x 107 m3/s, R2 = 4.0 x 104 m). Units are cm/s. Figure 100 and Figure 101 show the pressure contour pl ot and streamline contour plot for this scenario, resp ectively. The maximum pressu re within the nozzle was found to be about 3.59 x 104 Pa, while the minimum pressure was determined to be approximately 5.39 x 103 Pa. The inlettooutlet pressure drop was then calculated as 3.57 x 104 Pa which is about half of the pressu re drop obtained from the other working fluids. The streamline contour plot shows th e fluid entering through the central inlet has
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94 almost a pure axial motion as it flows toward the outlet. It also shows that the fluid entering through the outer inlet slot had to fl ow around a swirling pocket of fluid to reach the nozzle wall. However, there is a portion of the fluid entering through the outer inlet slot that flows toward the line of symmetry before heading toward the outlet. Figure 100: Pressure contour plot for Methanol (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 101: Streamline contour plot for Methanol (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m)
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95 Finally, Methanol was used on this nozzle geometry with an inlet flow rate of 5.678 x 107 m3/s. The velocity vector plot for thes e parameters is shown in Figure 102. The maximum velocity was found to be abou t 11.32 m/s, and was located within the throat of the nozzle. The free surface was originally at a height of 1.250 x 104 m, and then it decreased to 1.217 x 104 m before it increased again to a final height of 1.251 x 104 m. The cone angle was determined to be a mere 2.16 degrees. Again, only FC77 produced a lower free surface radi al height and cone angle. Figure 102: Velocity vector plot for Methanol (Q=5.678 x 107 m3/s, R2=4.0 x 104 m). Units are cm/s. The pressure contour plot and streamline contour plot for this situation are shown in Figure 103 and Figure 104. The maximum pressure within the nozzle was found to be about 5.92 x 104 Pa, whereas the minimum pressure was found to be 9.78 x 103 Pa. The pressure drop from the inlet of the nozzle to the outlet of the no zzle was calculated as 5.90 x 104 Pa. The streamline plot shows that as the fluid enters through the outer inlet slot, it must maneuver around a pocket of swir ling fluid located at the top of the nozzle
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96 just outside of the where the outer inlet slot is located. The fluid then flows to the outer nozzle wall as it heads toward the outlet. Again, the fluid entering through the central inlet has almost pure axial movement as it fl ows straight to the outlet of the nozzle. Figure 103: Pressure contour plot for Methanol (Q = 5.678 x 107 m3/s, R2 = 4.0 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 104: Streamline contour plot for Methanol (Q = 5.678 x 107 m3/s, R2 = 4.0 x 104 m)
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97 Figure 105 and Figure 106 show the free surface profile for this geometry with all of working fluids with an inlet flow rate of 4.416 x 107 m3/s. FC87 resulted in the highest free surface position, with a value of 1.259 x 104 m. Figure 105: Free surface position for all fluids (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m) Figure 106: Magnified free surface profile for all fluids (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m) 1.00E04 1.05E04 1.10E04 1.15E04 1.20E04 1.25E04 1.30E04 1.35E04 1.40E04 1.45E04 1.50E04 1.2E03 1.3E03 1.4E03 1.5E03 1.6E03 1.7E03 FC77 FC72 FC87 Methanol 1.21E04 1.22E04 1.23E04 1.24E04 1.25E04 1.26E04 1.27E04 1.28E04 1.2E03 1.3E03 1.4E03 1.5E03 1.6E03 1.7E03 FC77 FC72 FC87 MethanolAxial Coordinate [m] Radial Coordinate [m] Axial Coordinate [m] Radial Coordinate [m]
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98 Figure 107 and Figure 108 show the free surface profile for this geometry with all of working fluids with an inlet flow rate of 5.678 x 107 m3/s. Again, it was noted that FC87 resulted in the highest free surf ace position, with a value of 1.261 x 104 m. Figure 107: Free surface profile for all fluids (Q = 5.678 x 107 m3/s, R2 = 4.0 x 104 m) Figure 108: Magnified free surface profile for all fluids (Q = 5.678 x 107 m3/s, R2 = 4.0 x 104 m) 1.00E04 1.05E04 1.10E04 1.15E04 1.20E04 1.25E04 1.30E04 1.35E04 1.40E04 1.45E04 1.50E04 1.2E031.3E03 1.4E03 1.5E03 1.6E03 1.7E03 FC77 FC72 FC87 Methanol 1.21E04 1.22E04 1.23E04 1.24E04 1.25E04 1.26E04 1.27E04 1.28E04 1.2E031.3E03 1.4E03 1.5E03 1.6E03 1.7E03 FC77 FC72 FC87 Methanol Axial Coordinate [m] Radial Coordinate [m] Axial Coordinate [m] Radial Coordinate [m]
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99 Figure 109 and 110 show the dimensionle ss free surface profile for all of the fluids within this nozzle geometry. Once ag ain, a larger Reynolds number produced a larger radial height and cone angle for th e free surface of the fluid exiting the nozzle. 0.95 0.97 0.99 1.01 1.03 1.05 9.510.010.511.011.512.012.513.013.514.0 Re = 2812 Re = 5904 Re = 8093 Re = 3212 Figure 109: Dimensionless free surface pr ofile for all fluids (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m) 0.95 0.97 0.99 1.01 1.03 1.05 9.510.010.511.011.512.012.513.013.514. 0 Re = 3615 Re = 7591 Re = 10406 Re = 4130 Figure 110: Dimensionless free surface pr ofile for all fluids (Q = 5.678 x 107 m3/s, R2 = 4.0 x 104 m) Dimensionless Axial Coordinate, z/RoutDimensionless Radial Coordinate, r/ R out Dimensionless Axial Coordinate, z/Rout Dimensionless Radial Coordinate, r/ R out
PAGE 119
1005.2.8.3 R2 Equal to 5.50 x 104 m The next geometry that was used was for the nozzle having R2 equal to 5.50 x 104 m and R3 equal to 5.77 x 104 m. The same fluids were us ed to investigate their effects on this nozzle geometry. The first situation to be discussed is when FC77 was used as the working fluid with an inlet flow rate of 4.416 x 107 m3/s. Figure 111 shows the velocity vector plot for this instance. Th e maximum velocity was determined to be 19.77 m/s, which was almost twice as high as this value for the other geometries. The free surface began at 1.250 x 104 m, declined to 1.219 x 104 m, then increased again to 1.261 x 104 m and had a cone angle of about 2.29 de grees. This location of the outer slot provided the most increase in the fr ee surface height of the fluid. Figure 111: Velocity vector plot for FC77 (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m). Units are cm/s. Figure 112 and Figure 113 show the pressure and streamline contour plots for this scenario. The maximum pressure plotted was found to be 3.99 x 105 Pa, whereas the minimum pressure was found to be 7.94 x 104 Pa. The inlettooutlet pressure drop was
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101 calculated as 3.32 x 105 Pa. The streamline contour plot sh ows that most of the fluid that enters through the outer slot flows toward the center of the nozzle as it makes it way towards the outlet, with a portion of the ente ring fluid swirling in th e upper region. The fluid entering through the central in let has almost purely axial motion. Figure 112: Pressure contour plot for FC77 (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 113: Streamline contour plot for FC77 (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m)
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102 Next, FC77 was still being used as th e working fluid, but the flow rate was increased to 5.678 x 107 m3/s. Figure 114 shows the velocity vector plot for this case. The maximum velocity was found to be about 25 .40 m/s within the thro at of the nozzle. Again, this value was close to double the va lue of the maximum velocity for the other nozzle geometries. Similar to the other trials, the free surface decrease d in height at first before rising to its final position. For this case, the free surface began at 1.250 x 104 decreased to 1.220 x 104 m, then increased to a height of 1.263 x 104 m. The cone angle was calculated to be 2.43 degrees, which al ong with Methanol, was the lowest value. Figure 114: Velocity vector plot for FC77 (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m). Units are cm/s. The pressure contour plot for this case is shown in Figure 115. The maximum pressure within the nozzle was found to be 6.59 x 105 Pa, and the minimum pressure was found to be 1.38 x 105 Pa. The inlettooutlet pressure drop was determined to be approximately 5.58 x 105 Pa. Figure 116 shows the str eamline contour plot for this scenario. This plot is very similar to the streamline plot for the last trial. Most of the
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103 entering fluid flows toward the center of th e nozzle as it moves th roughout the nozzle. Still, a fraction of the fluid gets caught swir ling in an upper region of the nozzle. This region is smaller than the other nozzle geometri es because the location of the outer slot is located more toward the outer edge of the nozzle. Figure 115: Pressure contour plot for FC77 (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 116: Streamline contour plot for FC77 (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m)
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104 Next, the investigation was continued by changing the working fluid from FC77 to FC72. Figure 117 shows the velocity ve ctor plot for the nozzle geometry having R2 equal to 5.50 x 104 m, FC72 as the working fluid, and 4.416 x 107 m3/s as the inlet flow rate. The maximum velocity for this scenario was determined to be 19.77 m/s, which is the same maximum velocity that was found when FC77 was operating under the same circumstances. The free surface height for this case began at 1.250 x 104 m, gradually dropped to 1.220 x 104 m, and then rose to a final height of 1.268 x 104 m and had a cone angle of 2.72 degrees. Ov erall, this location of the ou ter inlet slot resulted in the most significant increase in the free surface heig ht for all of the fluids that were analyzed. Figure 117: Velocity vector plot for FC72 (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m). Units are cm/s. Figure 118 shows the pressure contour pl ot for this situation. The maximum pressure was found to be 3.75 x 105 Pa, while the minimum pressure within the nozzle was 8.46 x 104 Pa. The calculated value for the pressure drop from the inlet of the nozzle to the outlet was determined to be 3.39 x 105 Pa. The streamline contour plot for
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105 this case is shown in Figure 119. Again, the fluid enteri ng the nozzle thr ough the central inlet has almost pure axial motion throughout. However, most of the fluid entering through the outer slot flows toward the center of the nozzle as it moves toward the outlet. Figure 118: Pressure contour plot for FC72 (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 119: Streamline contour plot for FC72 (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m)
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106 Next, the inlet flow rate for FC72 was increased to 5.678 x 107 m3/s. Figure 120 shows the velocity vector plot for this inst ance. The maximum velocity, similar to the other trials, was found to be located in the th roat of the nozzle and have a value of about 25.40 m/s. This value, again, is much great er than any of the maximum velocity values obtained from the other geometries. The free surface began at height of 1.250 x 104 m, decreased to 1.220 x 104 m, and then increased to a final height of 1.269 x 104 m. The resulting cone angle was calculated to be 2. 72 degrees, which is the same value obtained for this fluid traveling at 4.416 x 107 m3/s within this same nozzle geometry. Figure 120: Velocity vector plot for FC72 (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m). Units are cm/s. The pressure contour plot and the str eamline contour plot are shown in Figure 121 and Figure 122, respectively. The maximum pr essure within the nozzle was determined to be about 6.20 x 105 Pa, while the minimum pressure was found to be 1.43 x 105 Pa. The inlettooutlet pressure drop was cal culated to be appr oximately 6.51 x 105 Pa. Again, the streamline plot shows that the fluid entering through the central inlet has
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107 almost a pure axial motion as it flows to the nozzle outlet. However, most of the fluid entering through the outer inlet slot flows inward toward the line of symmetry before it heads toward the outlet. Some of the fluid entering through the outer inlet slot must flow around a pocket of swirling fluid as it reaches the nozzle wall. Figure 121: Pressure contour plot for FC72 (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 122: Streamline contour plot for FC72 (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m)
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108 The next variation with this nozzle geom etry involved changing the working fluid to FC87. Figure 123 shows the velocity vector plot of the first case to be analyzed, which was FC87 entering at 5.678 x 107 m3/s. The maximum velocity within the nozzle was found to be 19.77 m/s. The height of the free surface began at 1.250 x 104 m, decreased to 1.220 x 104 m, then increased to a height of 1.270 x 104 m, which resulted in a cone angle of 2.72 degrees. This valu e for the final free surface height was the largest value produced from any of the fluids that were used. Figure 123: Velocity vector plot for FC87 (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m). Units are cm/s. Figure 124 and Figure 125 show the pressu re contour plot and the streamline contour plot, respectively. The maximum pr essure within the noz zle was found to be about 3.64 x 105 Pa, whereas the minimum pressure was found to be 8.45 x 104 Pa. The pressure drop from the inlet of the nozzl e to the outlet was ca lculated as 3.82 x 105 Pa. The streamline plot shows that some of th e fluid entering through the outer slot moves toward the outer wall, while some of the fl uid moves toward the cen ter of the nozzle.
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109 Again, the flow entering through the central inlet has almost pure axial movement throughout the nozzle. Figure 124: Pressure contour plot for FC87 (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 125: Streamline contour plot for FC87 (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m)
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110 Next, the flow rate of the entering FC87 for this nozzle geometry was increased to 5.678 x 107 m3/s. Figure 126 shows the velocity vector plot for these parameters. The maximum velocity was found at the throat of the nozzle and had a va lue of approximately 25.40 m/s, which is extremely fast. The free surface began at a height of 1.250 x 104 m, declined to 1.220 x 104 m, and then rose to final height of 1.271 x 104 m. This radial height proved to be the greatest value for any of the trials performed in this study. The cone angle corresponding to the free surface pos ition was determined to be 2.72 degrees. Figure 126: Velocity vector plot for FC87 (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m). Units are cm/s. Figure 127 shows the pressure contour pl ot for this situation. The maximum pressure was found to be about 6.02 x 105 Pa, whereas the minimum pressure was found to be about 1.42 x 105 Pa. The pressure drop from the inlet of the nozzle to the outlet was calculated as 6.33 x 105 Pa. Figure 128 shows the stre amline contour plot. It shows results that are similar to the other stream line plots obtained for this nozzle geometry. The fluid entering through the outer inlet slot must maneuver around a pocket of swirling
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111 fluid located at the outer edge of the nozzle near the top plate. Since the location of the outer inlet slot has moved closer to the outer edge of the top plate, the pocket of swirling fluid trapped between it and the noz zle wall has decreased in size. Figure 127: Pressure contour plot for FC87 (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 128: Streamline contour plot for FC87 (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m)
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112 Next, methanol was used as the working fluid entering the nozzl e with a flow rate of 4.416 x 107 m3/s. Figure 129 shows the velocity vect or plot for this situation. The maximum velocity is found to be in the throat of the nozzle with a value of 19.77 m/s. The height of the free surface began at 1.250 x 104 m, declined gradually to 1.220 x 104 m, then increased to a height of 1.262 x 104 m. The cone angle resulting from this trial was calculated as 2.29 degrees. These values were only greater than the values obtained with FC77 used as the working fluid. Figure 129: Velocity vector plot for Methanol (Q=4.416 x 107 m3/s, R2=5.5 x 104 m). Units are cm/s. The pressure contour plot for this s ituation is shown in Figure 130. The maximum pressure within the nozzle was found to be 1.76 x 105 Pa, and the minimum pressure was 3.61 x 104 Pa. The inlettooutlet pressure drop was calculated to be about 1.83 x 105 Pa, which is one of the lower values found in this investigation. The streamline contour plot is shown in Figure 131. Similar to the other streamline plots, this plot shows that some of the fluid entering through the outer slot moves in the radial
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113 direction toward the outer wall and some move s in the radial direction toward the center of the nozzle as the flui d moves through the nozzle. Figure 130: Pressure contour plot for Methanol (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 131: Streamline contour plot for Methanol (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m)
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114 The final variation performed with this geometry had Methanol as the working fluid with an inlet flow rate of 5.678 x 107 m3/s. Figure 132 shows the results of the velocity vector plot for this scenario. Identical to the other trials performed with this nozzle geometry, the maximum velocity was found to be about 25.40 m/s. The free surface originated at a height of 1.250 x 104 m, decreased to 1.220 x 104 m, and then increased to a final height of 1.265 x 104 m. This free surface position produced a cone angle of approximately 2.43 degrees. Figure 132: Velocity vector plot for Methanol (Q=5.678 x 107 m3/s, R2=5.5 x 104 m). Units are cm/s. The pressure contour plot for this case is shown in Figure 133. The maximum pressure within the nozzle was found to be about 2.90 x 105 Pa, whereas the minimum pressure was found to be 6.24 x 104 Pa. The inlettooutlet pressure drop was calculated to be approximately 3.03 x 105 Pa. It was noted that Meth anol provided the lowest inlettooutlet pressure drops of a ll of the fluids that were studied. Figure 134 shows the streamline contour plot for this case. Most of the fluid entering the outer inlet slot flows
PAGE 134
115 toward the line of symmetry in the radial direction before it heads toward the nozzle outlet. However, some of the fluid entering from the outer inlet slot flows toward the nozzle wall as it makes it way toward the outlet. Figure 133: Pressure contour plot for Methanol (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 134: Streamline contour plot for Methanol (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m)
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116 Figure 135 and Figure 136 show the free surface profile for all of the fluids that were tested in the nozzle with this geomet ry. This figure depicts the results obtained solely for the inlet flow rate of 4.416 x 107 m3/s. It was observed that FC87 resulted in the highest free surface, where FC77 resulted in the lowest. Figure 135: Free surface profile for all of the fluids (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m) Figure 136: Magnified view of free surf ace profile for all fluids (Q=4.416 x 107 m3/s, R2=5.5 x 104 m) 1.00E04 1.05E04 1.10E04 1.15E04 1.20E04 1.25E04 1.30E04 1.35E04 1.40E04 1.45E04 1.50E04 1.2E031.3E03 1.4E03 1.5E03 1.6E03 1.7E03 FC77 FC72 FC87 Methanol 1.22E04 1.23E04 1.24E04 1.25E04 1.26E04 1.27E04 1.28E04 1.2E031.3E03 1.4E03 1.5E03 1.6E03 1.7E03 FC77 FC72 FC87 MethanolAxial Coordinate [m] Radial Coordinate [m] Axial Coordinate [m] Radial Coordinate [m]
PAGE 136
117 Figure 137 and Figure 138 show the free surface profile for all of the fluids that were tested in the nozzle with this geometry at a flow rate of 5.678 x 107 m3/s. It was observed that FC87 resulted in the highest free surface, where FC77 had the lowest. Figure 137: Free surface profile for all fluids (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m) Figure 138: Magnified free surface profile for all fluids (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m) 1.00E04 1.05E04 1.10E04 1.15E04 1.20E04 1.25E04 1.30E04 1.35E04 1.40E04 1.45E04 1.50E04 1.2E03 1.3E03 1.4E03 1.5E03 1.6E03 1.7E03 FC77 FC72 FC87 Methanol 1.22E04 1.23E04 1.24E04 1.25E04 1.26E04 1.27E04 1.28E04 1.2E03 1.3E03 1.4E03 1.5E03 1.6E03 1.7E03 FC77 FC72 FC87 MethanolAxial Coordinate [m] Radial Coordinate [m] Axial Coordinate [m] Radial Coordinate [m]
PAGE 137
118 Figure 139 and Figure 140 each show the dimensionless free surface profile for each fluid flowing in this nozzle geometry. The larger Reynolds number at the nozzle exit produced the larger free su rface height; however, the hei ght does not vary too much for the different Reynolds numbers that were evaluated. 0.95 0.97 0.99 1.01 1.03 1.05 9.510.010.511.011.512.012.513.013.514. 0 Re = 2812 Re = 5904 Re = 8093 Re = 3212 Figure 139: Dimensionless free surface pr ofile for all fluids (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m) 0.95 0.97 0.99 1.01 1.03 1.05 9.510.010.511.011.512.012.513.013.514. 0 Re = 3615 Re = 7591 Re = 10406 Re = 4130 Figure 140: Dimensionless free surface pr ofile for all fluids (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m) Dimensionless Axial Coordinate, z/RoutDimensionless Radial Coordinate, r/ R out Dimensionless Axial Coordinate, z/Rout Dimensionless Radial Coordinate, r/ R out
PAGE 138
1195.2.8.4 R2 Equal to 7.20 x 104 m The final variation in nozzle geometry was having R2 equal to 7.20 x 104 m, which put R3 at 7.43 x 104 m. This location put the outer in let slot at the very edge of the nozzle. Figure 141 shows the velocity vect or plot for FC77 traveling at 4.416 x 107 m3/s. The maximum velocity was found to be about 9.44 m/s within the throat of the nozzle. This is the main reason that the fina l height of the free surface decreased for this geometry. The free surface height began at 1.250 x 104 m, decreased to 1.219 x 104 m, then increased to a final height of 1.244 x 104 m. This value is almost identical to the height obtained when R2 was equal to 4.00 x 104 m. The cone angle for this trial was calculated to be a mere 1.62 degrees. Figure 141: Velocity vector plot for FC77 (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m). Units are cm/s. Figure 142 and Figure 143 show the pressure and streamline contour plots. The pressure data revealed that the inlettooutlet pressure drop was about 6.87 x 104 Pa. The streamline contour plot shows that some of th e fluid entering through the outer slot of the
PAGE 139
120 nozzle flows toward the center of the nozzle, then toward the outlet, while some of the fluid flows along the outer wall towards the out let. The flow ente ring through the central inlet almost has a pure axial motion. Cont rary to the other nozzle geometries, this configuration does not produce a portion of swirling fluid. Figure 142: Pressure contour plot for FC77 (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 143: Streamline contour plot for FC77 (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m)
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121 FC77 was kept as the working fluid as the inlet flow rate increased from 4.416 x 107 m3/s to 5.678 x 107 m3/s. Figure 144 shows the velocity v ector plot for this scenario. The maximum velocity within the nozzle was located in the throat and had a value of 12.16 m/s. The free surface bega n at a height of 1.250 x 104 m, gradually declined to 1.220 x 104 m, before climbing to a final height of 1.250 x 104 m and had a corresponding cone angle of 1.95 degrees. Th ese values were less than those obtained when R2 was equal to 5.50 x 104 m and FC77 was tr aveling at 5.678 x 107 m3/s. Figure 144: Velocity vector plot for FC77 (Q = 5.678 x 107 m3/s, R2 = 7.2 x 104 m). Units are cm/s. The pressure contour plot for this case can be seen in Fi gure 145. The maximum pressure that was plotted was 1.61 x 105 Pa, whereas the minimum pressure was found to be 2.69 x 104 Pa. The pressure drop from the in let to the outlet of the nozzle was calculated as 1.24 x 105 Pa. Figure 146 shows the streamlin e contour plot. Again, this plot shows that some of the fluid entering the nozzle through the outer inlet slow travels along the top plate of the nozzle toward the center. There, it makes it way toward the
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122 outlet of the nozzle. The other portion of th e fluid entering through th e outer slot travels along the outer wall toward the outlet. And si milar to the other nozzle configurations, the flow entering through the central in let has almost pure axial motion. Figure 145: Pressure contour plot for FC77 (Q = 5.678 x 107 m3/s, R2 = 7.2 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 146: Streamline contour plot for FC77 (Q = 5.678 x 107 m3/s, R2 = 7.2 x 104 m)
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123 The next variation performed with th is nozzle geometry involved changing the working fluid to FC72 and running the fluid at 4.416 x 107 m3 /s. Figure 147 shows the velocity vector plot for this situation. The maximum velocity was found to be about 9.42 m/s. The free surface started out at a height of 1.250 x 104 m, decreased to a height of 1.221 x 104 m, and then rose to a final height of 1.260 x 104 m. The cone angle that resulted from this particular free surf ace was determined to be 2.59 degrees. Figure 147: Velocity vector plot for FC72 (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m). Units are cm/s. Figure 148 and Figure 149 show the pressu re contour plot and the streamline contour plot for this case, respectively. The maximum pressure was found to be 9.22 x 104 Pa, while the minimum pressure was found to be 1.62 x 104 Pa. The inlettooutlet pressure drop was calculated as approximately 6.99 x 104 Pa. The streamline plot reveals that the fluid behaves the same way with this nozzle geometry independent of the working fluid. The fluid entering the central inlet has almost pure axial motion, while the fluid entering through the outer slot of the nozzle fl ows in the radial di rection as it moves
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124 toward the outlet. Again, this nozzle geom etry does not produce any pockets of swirling fluid. The other nozzle geometries produced swirling on the outer side of the outer inlet slot location. Since this geom etry has the outer slot location at the edge of the top plate, there is no place for the fluid to begin to swirl. Figure 148: Pressure contour plot for FC72 (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 149: Streamline contour plot for FC72 (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m)
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125 Next, FC72 was used at an inlet flow rate of 5.678 x 107 m3/s. The velocity vector plot for this situation is shown in Figure 150. The maximum velocity was again located within the throat of the nozzle and had a value of about 12.10 m/s, which is the same maximum velocity found for other fluids within this geom etry and traveling at this flow rate. The free surface for this case began at a height of 1.250 x 104 m, decreased to 1.221 x 104 m, and then increased to a final height of 1.265 x 104 m. This was the second greatest radial height behind FC87. The value fo r the cone angle corresponding to this free surface position was found to be 2.92 degrees. Figure 150: Velocity vector plot for FC72 (Q = 5.678 x 107 m3/s, R2 = 7.2 x 104 m). Units are cm/s. The pressure contour plot for this case is shown in Figure 151. The maximum pressure was determined to be about 1.53 x 105 Pa, whereas the minimum pressure was found to be about 2.92 x 104 Pa. The inlettooutlet pressure drop was calculated to be approximately 1.47 x 105 Pa. The streamline contour plot is shown in Figure 152. Some of the fluid entering through the outer inlet slot flows toward the line of symmetry before
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126 flowing toward the nozzle outlet. Still, some of the fluid flows along the nozzle wall toward the outlet. Identical to the other resu lts obtained from this geometry, there are no pockets of swirling fluid present within the nozzle. Figure 151: Pressure contour plot for FC72 (Q = 5.678x 107 m3/s, R2 = 7.2 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 152: Streamline contour plot for FC72 (Q = 5.678 x 107 m3/s, R2 = 7.2 x 104 m)
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127 The next working fluid that was used on this nozzle geometry was FC87. Figure 153 shows the velocity vector plot for the situation where FC87 is entering the nozzle with a flow rate of 4.416 x 107 m3/s. Similar to the other fluids discussed previously, the maximum velocity was found to be about 9.40 m/ s. The free surface began at a height of 1.250 x 104 m, declined to a value of 1.221 x 104 m, and then inclined to a final height of 1.266 x 104 m, which formed a cone angle of 2.92 degrees. Again, for this nozzle geometry, FC87 produced the greatest radial fr ee surface height out of all of the working fluids. Figure 153: Velocity vector plot for FC87 (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m). Units are cm/s. The pressure contour plot and streamline contour plot for this situation are shown in Figure 154 and Figure 155, respectively. The maximum pressure was found to be about 8.97 x 104 Pa, while the minimum pressure was found to be about 1.74 x 104 Pa. The pressure drop from the inlet to the outle t of the nozzle was calculated to be about 8.66 x 104 Pa. The streamline plot shows results that are similar to the other streamline
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128 plots obtained for this nozzle geometry. Th e fluid entering through the central inlet has almost pure axial motion as it flows through th e nozzle, while the fluid entering from the outer inlet slot flows in the radial dir ection, as well as the axial direction. Figure 154: Pressure contour plot for FC87 (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 155: Streamline contour plot for FC87 (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m)
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129 Next, FC87 was forced through the inlets of the nozzle at a flow rate of 5.678 x 107 m3/s. Figure 156 shows the velocity vector plot for this situation. The maximum velocity within the nozzle was found to be about 12.12 m/s and again was located in the throat. The free surface for this case began at a radial height of 1.250 x 104 m, declined to a value of 1.221 x 104 m, before increasing to a final height of 1.270 x 104 m. The resulting cone angle was determined to be 3.13 degrees. FC87 once again produced the largest radial height of the free surface a nd largest cone angle for the given geometry. Figure 156: Velocity vector plot for FC87 (Q = 5.678 x 107 m3/s, R2 = 7.2 x 104 m). Units are cm/s. The pressure contour plot for this scen ario is depicted in Figure 157. The maximum pressure within the nozzle was determined to be about 1.49 x 105 Pa, whereas the minimum pressure was found to be about 3.07 x 104 Pa. The pressure drop from the inlet to the outlet of the nozzle was calculated as approximately 1.45 x 105 Pa. Figure 158 shows the streamline contour plot for this situation. Some of the fluid entering through the outer inlet slot tended to flow al ong the top plate of the nozzle toward the
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130 center line before heading toward the outlet. The rest of that fluid followed the nozzle wall to the outlet. It was also observed that the flui d entering through the central inlet flowed through the nozzle in a re latively straight, axial motion. Figure 157: Pressure contour plot for FC87 (Q = 5.678 x 107 m3/s, R2 = 7.2 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 158: Streamline contour plot for FC87 (Q = 5.678 x 107 m3/s, R2 = 7.2 x 104 m)
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131 The final working fluid studied on this geometry was Methanol. The velocity vector plot for Methan ol flowing at 4.416 x 107 m3/s is shown in Figure 159. The maximum velocity was found to be about 9.45 m/s. The free surface height started at 1.250 x 104 m, decreased to a value of 1.220 x 104 m, before rising again to a final height of 1.247 x 104 m. The radial height produced by Methanol was, again, the second lowest behind FC77. Also second lowest behind FC77 was the cone angle, which was found to be approximately 1.84 degrees. Figure 159: Velocity vector plot for Methanol (Q=4.416 x 107 m3/s, R2=7.2 x 104 m). Units are cm/s. Figure 160 shows the pressure contour plot for this scenario. The maximum pressure was found to be about 4.30 x 104 Pa, while the minimum pressure was found to be about 7.09 x 103 Pa. The inlettooutlet pressure was calculated to be approximately 4.00 x 104 Pa. This pressure drop was less than half of that value obtained for the other fluids operating in this nozzle geometry. The streamline contour plot is shown in Figure 161. It shows the same results as the previ ous trials performed on this geometry. Some
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132 of the fluid entering through the outer inlet slot flows in the radial direction toward the line of symmetry before heading axially toward the nozzle outlet, while some of the fluid flows along the wall before heading toward the outlet. Figure 160: Pressure contour plot for Methanol (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 161: Streamline contour plot for Methanol (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m)
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133 The final variation performed on this nozzle geometry had Methanol entering with an inlet flow rate of 5.678 x 107 m3/s. Figure 162 shows the velocity vector results for these parameters. Similar to the values obtained from the other fluids operating within this geometry, the maximum velocity was found to be approximately 12.10 m/s. The free surface began at a height of 1.250 x 104 m, receded to a value of 1.220 x 104 m, and then increased to a final height of 1.253 x 104 m. The corresponding cone angle was found to be 2.16 degrees. FC77 was the onl y working fluid to pr oduce greater values. Figure 162: Velocity vector plot for Methanol (Q=5.678 x 107 m3/s, R2=7.2 x 104 m). Units are cm/s. Figure 163 and Figure 164 show the pr essure and streamline contour plots, respectively. The maximum pressure within the nozzle was found to be approximately 7.11 x 104 Pa, whereas the minimum pressure was found to be 1.20 x 104 Pa. The inlettooutlet pressure drop for this case was determined to be 6.70 x 104 Pa. The streamline contour plot shows that the fl uid entering through the central inlet has almost pure axial motion as the fluid flows thr ough the nozzle. As previously discussed, this geometry
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134 provided no pockets of swirling fluid. Agai n, the reason for the absence of the swirling fluid is due to the fact that the location of the outer inlet slot provides no place for a pocket of swirling fluid to form. Figure 163: Pressure contour plot for Methanol (Q = 5.678 x 107 m3/s, R2 = 7.2 x 104 m). Units are gm/cm s2 (x101 Pa). Figure 164: Streamline contour plot for Methanol (Q = 5.678 x 107 m3/s, R2 = 7.2 x 104 m)
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135 Figure 165 and Figure 166 s how the free surface profile for the different fluids with an inlet flow rate of 4.416 x 107 m3/s in this nozzle geometry. It was noted that FC87 produced the greatest free surface hei ght, while FC77 produced the lowest. Figure 165: Free surface profile for all fluids (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m). Figure 166: Magnified free surface profile for all fluids (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m). 1.00E04 1.05E04 1.10E04 1.15E04 1.20E04 1.25E04 1.30E04 1.35E04 1.40E04 1.45E04 1.50E04 1.2E03 1.3E03 1.4E03 1.5E03 1.6E03 1.7E03 FC77 FC72 FC87 Methanol 1.22E04 1.23E04 1.24E04 1.25E04 1.26E04 1.27E04 1.28E04 1.2E03 1.3E03 1.4E03 1.5E03 1.6E03 1.7E03 FC77 FC72 FC87 Methanol Axial Coordinate [m] Radial Coordinate [m] Axial Coordinate [m] Radial Coordinate [m]
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136 Figure 167 and Figure 168 s how the free surface profile for the different fluids with an inlet flow rate of 5.678 x 107 m3/s performed with this nozzle geometry. It was again noted that FC87 produced the great est radial height for the free surface Figure 167: Free surface profile for all fluids (Q = 5.678 x 107 m3/s, R2 = 7.2 x 104 m). Figure 168: Magnified free surface profile for all fluids (Q = 5.678 x 107 m3/s, R2 = 7.2 x 104 m) 1.00E04 1.05E04 1.10E04 1.15E04 1.20E04 1.25E04 1.30E04 1.35E04 1.40E04 1.45E04 1.50E04 1.2E03 1.3E03 1.4E03 1.5E03 1.6E03 1.7E03 FC77 FC72 FC87 Methanol Axial Coordinate [m] Radial Coordinate [m] Radial Coordinate [m] 1.22E04 1.23E04 1.24E04 1.25E04 1.26E04 1.27E04 1.28E04 1.2E031.3E03 1.4E03 1.5E03 1.6E03 1.7E03 FC77 FC72 FC87 MethanolRadial Coordinate [m]
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137 Figure 169 and Figure 170 show the dimens ionless free surface profile for this nozzle geometry based on the Reynolds number at the nozzle exit. Again, the larger Reynolds number produced the greatest free surfa ce height and cone angle for the fluid. 0.95 0.97 0.99 1.01 1.03 1.05 9.510.010.511.011.512.012.513.013.514. 0 Re = 2812 Re = 5904 Re = 8093 Re = 3212 Figure 169: Dimensionless free surface plot for all fluids (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m) 0.95 0.97 0.99 1.01 1.03 1.05 9.510.010.511.011.512.012.513.013.514. 0 Re = 3615 Re = 7591 Re = 10406 Re = 4130 Figure 170: Dimensionless free surface pr ofile for all fluids (Q = 5.678 x 107 m3/s, R2 = 7.2 x 104 m) Dimensionless Axial Coordinate, z/RoutDimensionless Axial Coordinate, z/Rout Dimensionless Radial Coordinate, r/ R out Dimensionless Radial Coordinate, r/ R out
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138 Figure 171 and Figure 172 bot h show the free surface profile for all of the geometries that used FC77 as the working fluid. These figures show the free surface profile solely for the flow rate of 4.416 x 107 m3/s. The results for 5.678 x 107 m3/s are identical except for a slight change in the values. Figure 171: Free surface profile for FC77 in all nozzle geometries (Q = 4.416 x 107 m3/s) Figure 172: Magnified free surface profile for FC77 in all nozzle geometries (Q = 4.416 x 107 m3/s) 1.00E04 1.05E04 1.10E04 1.15E04 1.20E04 1.25E04 1.30E04 1.35E04 1.40E04 1.45E04 1.50E04 1.2E031.3E03 1.4E03 1.5E03 1.6E03 1.7E03 R2 = 2.5E04 m R2 = 4.0E04 m R2 = 5.5E04 m R2 = 7.2E04 m 1.21E04 1.22E04 1.23E04 1.24E04 1.25E04 1.26E04 1.27E04 1.2E031.3E03 1.4E03 1.5E03 1.6E03 1.7E03 R2 = 2.5E04 m R2 = 4.0E04 m R2 = 5.5E04 m R2 = 7.2E04 mAxial Coordinate [m] Radial Coordinate [m] Axial Coordinate [m] Radial Coordinate [m]
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139 Figure 173 and Figure 174 bot h show the free surface profile for all of the geometries that used FC72 as the working fluid. These figures show the free surface profile for the flow rate of 4.416 x 107 m3/s. Again, the results for 5.678 x 107 m3/s are identical except for a slight change in the values. Figure 173: Free surface profile for FC72 in all nozzle geometries (Q = 4.416 x 107 m3/s) Figure 174: Magnified free surface profile for FC72 in all nozzle geometries (Q = 4.416 x 107 m3/s) 1.00E04 1.05E04 1.10E04 1.15E04 1.20E04 1.25E04 1.30E04 1.35E04 1.40E04 1.45E04 1.50E04 1.2E031.3E03 1.4E03 1.5E03 1.6E03 1.7E03 R2 = 2.5E04 m R2 = 4.0E04 m R2 = 5.5E04 m R2 = 7.2E04 m 1.21E04 1.22E04 1.23E04 1.24E04 1.25E04 1.26E04 1.27E04 1.28E04 1.2E031.3E03 1.4E04 1.5E04 1.6E04 1.7E04 R2 = 2.5E04 m R2 = 4.0E04 m R2 = 5.5E04 m R2 = 7.2E04 m Axial Coordinate [m] Radial Coordinate [m] Axial Coordinate [m] Radial Coordinate [m]
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140 Figure 175 and Figure 176 bot h show the free surface profile for all of the geometries that used FC87 as the working fluid. These figures show the free surface profile for the flow rate of 4.416 x 107 m3/s. Again, the results for 5.678 x 107 m3/s are identical except for a slight change in the values. Figure 175: Free surface profile for FC87 in all nozzle geometries (Q = 4.416 x 107 m3/s) Figure 176: Magnified free surface profile for FC87 in all nozzle geometries (Q = 4.416 x 107 m3/s) 1.00E04 1.05E04 1.10E04 1.15E04 1.20E04 1.25E04 1.30E04 1.35E04 1.40E04 1.45E04 1.50E04 1.2E03 1.3E03 1.4E03 1.5E03 1.6E03 1.7E03 R2 = 2.5E04 m R2 = 4.0E04 m R2 = 5.5E04 m R2 = 7.2E04 m 1.21E04 1.22E04 1.23E04 1.24E04 1.25E04 1.2E04 1.27E04 1.2E031.3E03 1.4E03 1.5E03 1.6E03 1.7E03 R2 = 2.5E04 m R2 = 4.0E04 m R2 = 5.5E04 m R2 = 7.2E04 mRadial Coordinate [m] Axial Coordinate [m] Axial Coordinate [m] Radial Coordinate [m]
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141 Figure 177 and Figure 178 bot h show the free surface profile for all of the geometries that used Methanol as the worki ng fluid. These figures show the free surface profile only for the fl ow rate of 4.416 x 107 m3/s. Again, the results for 5.678 x 107 m3/s are identical except for a slight change in the values. Figure 177: Free surface profile for Methanol in all nozzle geometries (Q = 4.416 x 107 m3/s) Figure 178: Magnified free surface profile for Methanol in all nozzle geometries (Q=4.416 x 107m3/s) 1.00E04 1.05E04 1.10E04 1.15E04 1.20E04 1.25E04 1.30E04 1.35E04 1.04E04 1.45E04 1.50E04 1.2E031.3E03 1.4E03 1.5E03 1.6E03 1.7E03 R2 = 2.5E04 m R2 = 4.0E04 m R2 = 5.5E04 m R2 = 7.2E04 m 1.21E04 1.22E04 1.23E04 1.24E04 1.25E04 1.26E04 1.27E04 1.2E031.3E03 1.4E03 1.5E03 1.6E03 1.7E03 R2 = 2.5E04 m R2 = 4.0E04 m R2 = 5.5E04 m R2 = 7.2E04 mAxial Coordinate [m] Radial Coordinate [m] Axial Coordinate [m] Radial Coordinate [m]
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142 Table 15 summarizes the final height of the free surface with respect to the nozzle geometry, working fluid, and flow rate. It was expected that for the hi gher flow rates, an increase in free surface height is evident. Also, as the outer inlet slot was moved outward in the radial direction, it was noted that the free surface height increased until the outer slot was located at the very edge of the no zzle. The maximum velocity within the nozzle for the geometry having R2 equal to 5.50 x 104 m was about twice that for any of the other nozzle geometries. Furthe r research is required to determine why this happened. Table 15: Free surface height for different no zzle geometries, working fluids, and flow rates Free Surface Height Outer Slot Location R2 [m] Working Fluid 4.416 x 107 m3/s 5.678 x 107 m3/s FC77 1.239 x 104 m 1.243 x 104 m FC72 1.248x 104 m 1.251 x 104 m FC87 1.251 x 104 m 1.253 x 104 m 2.50 x 104 Methanol 1.241 x 104 m 1.245x 104 m FC77 1.245x 104 m 1.249 x 104 m FC72 1.256 x 104 m 1.259 x 104 m FC87 1.259 x 104 m 1.261 x 104 m 4.00 x 104 Methanol 1.247 x 104 m 1.251 x 104 m FC77 1.261x 104 m 1.263 x 104 m FC72 1.268x 104 m 1.269 x 104 m FC87 1.270 x 104 m 1.271 x 104 m 5.50 x 104 Methanol 1.262 x 104 m 1.265 x 104 m FC77 1.244 x 104 m 1.250 x 104 m FC72 1.260 x 104 m 1.265 x 104 m FC87 1.266 x 104 m 1.270 x 104 m 7.20 x 104 Methanol 1.247 x 104 m 1.253 x 104 m
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143 Figure 179 and Figure 180 show the height of the free surface for each flow rate with respect to fluid and outer inlet slot loca tion. It was observed th at a greater Reynolds number produced a greater free surface height. This is easily seen in these figures. Figure 179: Free surface height for all fluids and all outer slot locations (Q = 4.416 x 107 m3/s) Figure 180: Free surface height for all fluids and all outer slot locations (Q = 5.678 x 107 m3/s) 1.23E04 1.24E04 1.25E04 1.26E04 1.27E04 1.28E04 2.0E04 3.0E04 4.0E04 5.0E04 6.0E04 7.0E04 8.0E04 FC77 FC72 FC87 Methanol 1.24E04 1.25E04 1.26E04 1.27E04 1.28E04 2.0E043.0E04 4.0E04 5.0E04 6.0E04 7.0E04 8.0E04 FC77 FC72 FC87 MethanolRadial Coordinate [m] Outer Slot Location, R2 [m] Radial Coordinate [m] Outer Slot Location, R2 [m]
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144 The cone angle for each of the cases included in the investigation involving the varying outer slot radii is listed in Table 16. It was observed that although the last nozzle geometry (R2 equal to 7.20 x 104 m) produced the greatest cone angles for the free surface, the nozzle geometry having R2 equal to 5.50 x 104 m produced the greatest radial height of the free surface. Table 16: Cone angle for different nozzle ge ometries, working fluids, and flow rates Cone Angle Outer Slot Location R2 [m] Working Fluid 4.416 x 107 m3/s 5.678 x 107 m3/s FC77 1.51deg 1.73 deg FC72 2.05 deg 2.27 deg FC87 2.27 deg 2.38 deg 2.50 x 104 Methanol 1.62 deg 1.95 deg FC77 1.84 deg 2.05 deg FC72 2.48 deg 2.70 deg FC87 2.70 deg 2.81 deg 4.00 x 104 Methanol 1.95 deg 2.16 deg FC77 2.29 deg 2.43 deg FC72 2.72 deg 2.72 deg FC87 2.72 deg 2.72 deg 5.50 x 104 Methanol 2.29 deg 2.43 deg FC77 1.62 deg 1.95 deg FC72 2.59 deg 2.92 deg FC87 2.92 deg 3.13 deg 7.20 x 104 Methanol 1.84 deg 2.16 deg Figure 181 and 182 show the cone angles produced by the various fluids traveling at both flow rates through all four of the various nozzle geom etries. Again, it was seen
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145 that a fluid with a greater Reynolds number produced a greater cone angle. FC87 produced the largest cone angle of approximately 3.13 degrees when R2 was equal to 7.20 x 104 m. Figure 181: Cone angle for all fluids at all outer slot locations (Q = 4.416 x 107 m3/s) Figure 182: Cone angle for all fluids at all outer slot locations (Q = 5.678 x 107 m3/s) 1.0 1.5 2.0 2.5 3.0 3.5 2.0E043.0E04 4.0E04 5.0E04 6.0E04 7.0E04 8.0E04 FC77 FC72 FC87 Methanol 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 3.1 3.3 2.0E043.0E04 4.0E04 5.0E04 6.0E04 7.0E04 8.0E04 FC77 FC72 FC87 Methanol Cone Angle [deg] Outer Slot Location, R2 [m] Cone Angle [deg] Outer Slot Location, R2 [m]
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146 Another important factor is the inlettooutlet pressure drop of the nozzle. Table 17 depicts the pressure drop for each of the variations, including nozzle geometry, working fluid, and flow rate. As was expected an increased flow rate caused an increase in the pressure drop. Also, the pressure dr op resulting when FC72 was the working fluid was less than the pressure drop obtained when FC77 was used as the working fluid. Table 17: Pressure drop for different nozzle geometries, working fluids, and flow rates Pressure Drop Outer Slot Location R2 [m] Working Fluid 4.416 x 107 m3/s 5.678 x 107 m3/s FC77 7.25 x 104 Pa 1.04 x 105 Pa FC72 7.23 x 104 Pa 9.85 x 104 Pa FC87 7.18 x 104 Pa 1.19 x 105 Pa 2.50 x 104 Methanol 3.44 x 104 Pa 5.11 x 104 Pa FC77 7.93 x 104 Pa 1.20 x 105 Pa FC72 7.67 x 104 Pa 1.02 x 105 Pa FC87 7.53 x 104 Pa 1.25 x 105 Pa 4.00 x 104 Methanol 3.57 x 104 Pa 5.90 x 104 Pa FC77 4.07 x 105 Pa 5.58 x 105 Pa FC72 3.39 x 105 Pa 6.51 x 105 Pa FC87 3.82 x 105 Pa 6.33 x 105 Pa 5.50 x 104 Methanol 1.83 x 105 Pa 3.03 x 105 Pa FC77 9.65 x 104 Pa 1.24 x 105 Pa FC72 9.21 x 104 Pa 1.47 x 105 Pa FC87 8.66 x 104 Pa 1.45 x 105 Pa 7.20 x 104 Methanol 4.00 x 104 Pa 6.70 x 104 Pa Figure 183 and Figure 184 show the pressure drop from the inlet of the nozzle to the outlet for both 4.416 x 107 m3/s and 5.678 x 107 m3/s, respectively. It was observed
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147 that the pressure drop peaked when R2 was set at 5.50 x 104. It was also observed that Methanol provided about half of the pressure dr op as the other fluids The pressure drop for the three other fluids were almost iden tical for the three other nozzle geometries. Figure 183: Pressure drop for all fluids at all outer slot locations (Q = 4.416 x 107 m3/s) Figure 184: Pressure drop for all fluids at all outer slot locations (Q = 5.678 x 107 m3/s) 0.0E+00 5.0E+04 1.0E+05 1.5E+05 2.0E+05 2.5E+05 3.0E+05 3.5E+05 4.0E+05 4.5E+05 2.0E04 3.0E04 4.0E04 5.0E04 6.0E04 7.0E04 8.0E04 FC77 FC72 FC87 Methanol 0.0E+00 1.0E+05 2.0E+05 3.0E+05 4.0E+05 5.0E+05 6.0E+05 7.0E+05 2.0E04 3.0E04 4.0E04 5.0E04 6.0E04 7.0E04 8.0E04 FC77 FC72 FC87 MethanolPressure Drop [Pa] Outer Slot Location, R2 [m] Pressure Drop [Pa] Outer Slot Location, R2 [m]
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148 Figure 185 shows the coeffici ent of pressure with resp ect to the Reynolds number for all of the nozzle geometries, fluids, and flow rates involved in this investigation. The plots appear to be somewhat of a sinusoidal wave that dampens out to a relatively constant value at higher Reynolds numbers. The larger nozzle has a higher coefficient of pressure than the smaller nozzl e and most of the nozzle geom etries where the outer inlet slot location was varied. Howe ver, the nozzle geometry having R2 equal to 5.50 x 104 m produced the largest coefficient of pressure. Th is is because the fluid at the outlet of this nozzle was moving almost twice as fast as the other nozzles, which caused the pressure at that location to be low and helped increase the pressure drop. Figure 185: Coefficient of p ressure for each nozzle with respect to the Reynolds number 0 1 2 3 4 5 6 7 0200040006000800010000 12000 Large Nozzle Small Nozzle R2 = 2.5E04m R2 = 4.0E04m R2 = 5.5E04m R2 = 7.2E04mReynolds Number, ReCoefficient of Pressure, Cp
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149 Table 18 shows the inlet flow rate of each fluid, and the corresponding Reynolds number that was calculated at the nozzle exit. It was noted that FC87 produced a much greater Reynolds number than any of the other fluids. This is because its density is relatively high and its viscosity is fairly low. Table 18: Reynolds number for each fluid and each flow rate Inlet Flow Rate Q [m3/s] Working Fluid Reynolds Number at Nozzle Outlet [nondim] FC77 2812 FC72 5904 FC87 8093 4.416 x 107 Methanol 3212 FC77 3615 FC72 7591 FC87 10406 5.678 x 107 Methanol 4130 5.2.8.5 Cavitation Cavitation occurs when the pressure of the liquid falls below the saturated pressure of that liquid corresponding to that temperature. When this happens, the fluid begins to evaporate, which causes tiny bubbles to form at the location where the pressure is below the critical point. These bubbles will eventually erode and destroy the boundary of the nozzle. Table 19 through Table 22 show the values of the cavitation number for each fluid, each flow rate, and each nozzle geometry. It was observed that the nozzle geometry having R2 equal to 5.50 x 104 m had no cavitation present with any of the working fluids. The cavitation number for the trials having R2 equal to 2.50 x 104 m and
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150 4.00 x 104 m were almost identical, while the case where R2 was equal to 7.20 x 104 m produced similar values for the cavitation nu mber. It was noted that FC77 never produced cavitation within any of the nozzle geometries. FC 87 was the only other fluid beside FC77 not to have produced cavitation when R2 was equal to 7.20 x 104 m. Table 19: Cavitation number for all fluids at all flow rates (R2 = 2.50 x 104 m) Working Fluid Saturation Pressure Psat [Pa] Reynolds Number Re [nondim] Minimum Pressure Pmin [Pa] Pressure Difference Pmin Psat [Pa] Cavitation Number Ca [nondim] 2812 12100 6480 0.040 FC77 5.62 x 103 3615 20300 14680 0.062 5904 12100 18800 0.138 FC72 30.9 x 103 7591 20400 10500 0.047 8093 12100 69000 0.479 FC87 81.1 x 103 10406 20700 60400 0.254 3212 5340 4660 0.034 Methanol 10.0 x 103 4130 9080 920 0.004 Table 20: Cavitation number for all fluids at all flow rates (R2 = 4.00 x 104 m) Working Fluid Saturation Pressure Psat [Pa] Reynolds Number Re [nondim] Minimum Pressure Pmin [Pa] Pressure Difference Pmin Psat [Pa] Cavitation Number Ca [nondim] 2812 13900 8280 0.037 FC77 5.62 x 103 3615 21100 15480 0.065 5904 13900 17000 0.125 FC72 30.9 x 103 7591 24300 6600 0.029 8093 14400 66700 0.463 FC87 81.1 x 103 10406 24600 56500 0.237 3212 5390 4610 0.034 Methanol 10.0 x 103 4130 9780 220 0.001
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151Table 21: Cavitation number for all fluids at all flow rates (R2 = 5.50 x 104 m) Working Fluid Saturation Pressure Psat [Pa] Reynolds Number Re [nondim] Minimum Pressure Pmin [Pa] Pressure Difference Pmin Psat [Pa] Cavitation Number Ca [nondim] 2812 84600 78980 0.512 FC77 5.62 x 103 3615 138000 132380 0.556 5904 84600 53700 0.395 FC72 30.9 x 103 7591 143000 112100 0.499 8093 84500 3400 0.024 FC87 81.1 x 103 10406 142000 60900 0.573 3212 36100 26100 0.192 Methanol 10.0 x 103 4130 62400 52400 0.140 Table 22: Cavitation number for all fluids at all flow rates (R2 = 7.20 x 104 m) Working Fluid Saturation Pressure Psat [Pa] Reynolds Number Re [nondim] Minimum Pressure Pmin [Pa] Pressure Difference Pmin Psat [Pa] Cavitation Number Ca [nondim] 2812 16200 10580 0.071 FC77 5.62 x 103 3615 26900 21280 0.089 5904 16200 14700 0.108 FC72 30.9 x 103 7591 29200 1700 0.008 8093 17400 63700 0.082 FC87 81.1 x 103 10406 30700 50400 0.105 3212 7090 2910 0.175 Methanol 10.0 x 103 4130 12000 2000 0.084 5.2.8.6 Sectional Velocities for FC72 Next, the radial velocity component, Vr, the axial velocity component, Vz, and the theta velocity component, V, were all analyzed for the smaller nozzle where the radial
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152 location of the outer inlet slot varied. The three components of velocity were examined for the situation where FC72 was used as the working fluid and 4.416 x 107 m3/s was used as the inlet flow rate. In addition, when the outer inlet slot was located at the position that produced the largest cone angle (R2 = 5.50 x 104 m and R3 = 5.77 x 104 m), the results for the flow rate of 5.678 x 107 m3/s were observed. This was done to observe how the fluid was behaving within the no zzle and compare it to how the fluid was behaving once it exited the nozzle. The three velocity components were an alyzed at seven different locations throughout the nozzle. The first section was lo cated at about 10% of the nozzle length. The second section was located at about 50% of the nozzle length, whereas the third section was at the point where the nozzle converg ed at the throat. The fourth section was located half way down the throat, while the fift h section was located at the nozzle outlet. The sixth section was located at 1/3 the length of the free surface, and finally, the seventh section was located at 2/3 th e length of the free surface. The velocity components were not studied at the end of the free surface, because many numerical assumptions are made at that location, whic h could jeopardize the accuracy of the numbers. Table 23 shows the sections that were investigated, as well as their axial coordinate. Table 23: Analyzed sections wi th corresponding axial coordinate Section Planes A B C D E F G Axial Coordinate [m] 1.00 x 104 5.00 x 104 1.07 x 103 1.15 x 103 1.22 x 103 1.35 x 103 1.54 x 103
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153 Figure 186 displays the result for the radi al velocity component with FC72 as the working fluid and the outer inlet slot positioned at R2 equal to 2.50 x 104 m, and R3 equal to 3.04 x 104 m. The boundary conditions within the problem state that the radial velocity at the line of symmetry, as well as the wall, is equal to zero. This is why all of the radial velocities within the nozzle begin and end at a value of zero. At section A, the radial velocity took on a sinusoidal shape. Afte r Section A, all of th e sections within the nozzle produce negative radial velocities. The maximum radial velocity observed was located at section C, which is where the co nical shape of the noz zle converges at the throat. Here, the radial velocity was determined to be approximately 2.12 m/s. However, after the fluid exits the nozzle, the radial velocity becomes a positive value. As the fluid moved away from the nozzle out let, its radial velocity decreased. Figure 186: Radial velocity component at various sections (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m) 2.5 2.0 1.5 1.0 0.5 0.0 0.5 0.0E04 1.0E04 2.0E04 3.0E04 4.0E04 5.0E04 6.0E04 7.0E04 8.0E04 Section A Section B Section C Section D Section E Section F Section G Radial Coordinate [m] Radial Velocity [m/s]
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154 Figure 187 shows the axial velocity component for the same trial. The values at Sections A and B are the lowest out of the sect ions that were analyzed. This is because the other sections had a much lower radial he ight, which forced the fluid to move at a higher speed to provide the same flow rate th at had entered the nozzle. At some points towards the outer reaches of Section A where some swirling occurred the axial velocity became negative. The maximum axial veloci ty was found to be about 8.60 m/s located just as the fluid exited the nozzle. Where the nozzle converges at the throat, the axial velocity was observed to be fairly even th roughout the entire ra nge, before the boundary condition sent the value to zero. Again, the fluid within th e nozzle is constrained by the requirement that the velocity along the nozzle wall is equal to zero, which is why the halfparabola shape is depicted in the figure for Sections D and E. However, after the fluid exited the nozzle, the ax ial velocity decreased graduall y, but did not go to zero. Figure 187: Axial velocity component for various sections (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m) 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 2.0E04 3.0E04 Section A Section B Section C Section D Section E Section F Section G 1.0E04 0.0E04 4.0E04 5.0E04 6.0E04 8.0E04 7.0E04 Axial Velocity [m/s] Radial Coordinate [m]
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155 Finally, the values obtained for the th eta component of velocity is shown in Figure 188. Again, due to the boundary conditio n constraining the velo city to zero at the nozzle walls, all of the plots for the sections within the nozzle have a parabolic shape, which begins and ends at zero. After the fluid exited the nozzle, the theta velocity component became more of a linear pattern. Si nce Section A was close to the inlet of the nozzle, its values peaked when the radial value was 2.50 x 104 m, which is where the outer inlet slot was located. The value of the theta component had a maximum value of about 3.40 m/s at the location where the nozzl e converges, and then it declined minutely as the fluid moved through the throat of the nozzle. Figure 188: Theta velocity component for various sections (Q = 4.416 x 107 m3/s, R2 = 2.5 x 104 m) The next variation that was investigated continued to utilize FC72 as the working fluid with a flow rate of 4.416 x 107 m3/s. However, the location of the outer inlet slot 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0E04 1.0E04 2.0E04 3.0E04 4.0E04 5.0E04 6.0E04 7.0E04 8.0E04 Section A Section B Section C Section D Section E Section F Section GRadial Coordinate [m] Theta Velocity [m/s]
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156 was positioned so that R2 was equal to 4.00 x 104 m and R3 was equal to 4.36 x 104 m. Figure 189 shows the radial velocity values for this situation. Similar to the previous trial, the results for Section A provided somewh at of a sinusoidal wave Again, all of the values for the radial velocity within the nozzl e were negative. At Section C, where the nozzle converges to form the throat, the radial velocity reached its maximum value in the study. At that section, the value dipped dow n to about 2.10 m/s, which is quite large compared to the other observed locations. Once again, as the fluid exited the nozzle, the radial velocity became positive and slowly d ecreased as the distance from the nozzle increased. Figure 189: Radial velocity component for various sections (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m) Figure 190 shows the axial ve locity component values obtained for the various sections. Similar to the radial velocity component, the axial velocity at Section A 2.5 2.0 1.5 1.0 0.5 0.0 0.5 Section A Section B Section C Section D Section E Section F Section G 4.0E04 5.0E04 6.0E04 7.0E04 8.0E04 0.0E04 1.0E04 2.0E04 3.0E04Radial Velocity [m/s] Radial Coordinate [m]
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157 resembled a sinusoidal wave. Again, th e maximum axial velocity obtained was determined to be approximately 8.70 m/s located just as the fluid exited the nozzle. The axial velocity of the fluid in Section C is about the same throughout its extent before it drops off to zero, whereas the axial velocity of the fluid in the throat of the nozzle is a maximum at the line of symmetry and then declin es gradually to zero at the nozzle wall. After the fluid exited the nozzle, it was observe d that the axial veloc ity decreased as the axial distance from the nozzle increased. It wa s also noted that as the radial distance from the line of symmetry increased the axial velocity decreased. Figure 190: Axial velocity component for various sections (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m) Figure 191 reveals the values for the theta velocity component for this scenario. Again, the values for all of the sections within the nozzle form parabolas due to the boundary conditions. However, after the fluid exited the nozzle, it was observed that the 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 0.0E04 1.0E04 2.0E04 4.0E04 5.0E04 6.0E04 7.0E04 8.0E04 Section A Section B Section C Section D Section E Section F Section G 3 0 E0 4Axial Velocity [m/s] Radial Coordinate [m]
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158 theta velocity values formed a linear model. The maximum value was again located at Section C, where the nozzle converges to form the throat. This value was determined to be about 8.75 m/s. As the fluid moved thr ough the throat of the nozzle, this maximum value decreased slightly. For this case, the th eta velocity values at Section A and Section B both peaked at the approximate location of the outer inlet slot. Figure 191: Theta velocity component for various sections (Q = 4.416 x 107 m3/s, R2 = 4.0 x 104 m) Next, the nozzle geometry with R2 equal to 5.50 x 104 m and R3 equal to 5.77 x 104 m was used to analyze the three componen ts of velocity. Since this particular geometry produced the greatest cone angle, the velocity values were investigated for both of the flow rates used Â– 4.416 x 107 m3/s and 5.678 x 107 m3/s. The first flow rate to be studied was 4.416 x 107 m3/s. Figure 192 shows the radial velocity component for this scenario with FC72 as the working fluid. Ag ain, Section A took on a sinusoidal pattern. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0E04 1.0E04 2.0E04 3.0E04 4.0E04 5.0E04 6.0E04 7.0E04 8.0E04 Section A Section B Section C Section D Section E Section F Section GRadial Coordinate [m] Theta Velocity [m/s]
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159 It was noted that the radial velocity for this geometry was greater than that for the other geometries. The radial velocity at Secti on C was more than double that of the other geometries. It was at this location where the radial velocity co mponent was a maximum with a value of nearly 4.66 m/s. The other nozzle geometries produced a maximum radial velocity of roughly 2.10 m/s. As with the other nozzles, all of the radial velocity values within the nozzle were negative, while they became positive once the fluid exited the nozzle. Figure 192: Radial velocity component for various sections (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m) Figure 193 shows the axial velocity com ponent for the different sections with 4.416 x 107 m3/s as the inlet flow rate. Because the nozzle is larger at Sections A and B, their respective axial velocities are smaller than the other sections. However, it was observed that all three component s of velocity were much la rger than the same results 5.0 4.0 3.0 2.0 1.0 0.0 1.0 0.0E04 1.0E04 2.0E04 3.0E04 4.0E04 5.0E04 6.0E04 7.0E04 8.0E04 Section A Section B Section C Section D Section E Section F Section G Radial Velocity [m/s] Radial Coordinate [ m ]
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160 found in the other nozzles. The maximum axial velocity for this geometry was found at the nozzle outlet and had a value of about 19.60 m/s. This valu e is a little over 10.00 m/s faster than the maximum axial velocity for the other geometries. This is the reason for this nozzle geometry producing the largest radial free surface height. The patterns of the axial velocity throughout the different sections for this geometry are almost identical to the patterns observed for the other geometries. In this case, however, the values are much greater. Figure 193: Axial velocity component for various sections (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m) The theta velocity component is shown in Figure 194. Comparable to the other velocity components for this geometry, the theta velocity component was much larger than those for the other nozzle geometries. The theta velocity for all of the sections within the nozzle produced a pa rabolic shape due to the no slip condition at the nozzle 2.0 3.0 8.0 13.0 18.0 23.0 0.0E04 1.0E04 2.0E04 3.0E04 4.0E04 5.0E04 6.0E04 7.0E04 8.0E04 Section A Section B Section C Section D Section E Section F Section G Axial Velocity [m/s] Radial Coordinate [m]
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161 wall, and the boundary conditions at the line of symmetry. However, after the fluid exited the nozzle, it was noted that the thet a velocity component formed a linear pattern with its value increasing as the radial dist ance increased. The maximum value was about 9.30 m/s located at Section C. The theta ve locity value decreased slightly from the maximum value as the fluid moved through the throat of the nozzle. Figure 194: Theta velocity component for various sections (Q = 4.416 x 107 m3/s, R2 = 5.5 x 104 m) This same nozzle geometry (R2 = 5.50 x 104 m and R3 = 5.77 x 104 m) was analyzed with higher fl ow rate of 5.678 x 107 m3/s. Figure 195 shows the radial velocity component for this situation. As expected, the values for all three velocity components were greater than the other cases that were st udied. This is primarily due to the increased flow rate, as well as the nozzle geometry. Fo r the radial velocity component, the patterns for each section appeared the same as the ot her trials, only this trial produced larger 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 0.0E04 1.0E04 2.0E04 3.0E04 4.0E04 5.0E04 6.0E04 7.0E04 8.0E04 Section A Section B Section C Section D Section E Section F Section GRadial Coordinate [m] Theta Velocity [m/s]
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162 values. For this case, the maximum radial ve locity was again located at Section C where the nozzle converges to form the throat. Th is value was determined to be about 6.00 m/s. Once more, the radial velocities with in the nozzle initiate and terminate at zero because of the no slip condition on the no zzle walls, and the boundary condition of zero radial velocity along the line of symmetry. However, as th e fluid exits the nozzle, it is no longer in contact with the nozzle wall. This is why the radial velocity outside of the nozzle does not terminate at zero. It was also noted that th e region outside of the nozzle is the only region to produce all positive values for the radial velocity component. Figure 195: Radial velocity component for various sections (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m) The axial velocity component for this cas e is shown in Figure 196. The values for Section A and Section B were the lowest. From the streamline c ontour plot, a portion near the outer nozzle wall along Section A tends to swirl, which causes the value for the 7.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0 1.0 0.0E04 1.0E04 2.0E04 3.0E04 4.0E04 5.0E04 6.0E04 7.0E04 8.0E04 Section A Section B Section C Section D Section E Section F Section G Radial Velocity [m/s] Radial Coordinate [m]
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163 axial velocity to temporarily become negativ e. The maximum axial velocity was found to be where the fluid exited the nozzle and maintained a va lue of about 25.25 m/s. After the fluid exited the nozzle, th e axial velocity decr eased slightly, but remained at about 20.00 m/s. However, the axial velocity of the fluid within the throat of the nozzle decreased to a value of zero al ong the nozzle wall to form the sh ape of half of a parabola. The fluid converging from the conical nozzle sh ape to the throat of the nozzle remained at a fairly constant axial velocity throughout the range. Figure 196: Axial velocity component for various sections (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m) Finally for this geometry, the theta velo city component is shown in Figure 197. Again, Sections A and B produce the lowest values for the velocity component. The value for section A peaked at a radial distance of about 2.00 x 104 m, whereas the value for Section B peaked at a radial distance of about 4.00 x 104 m. The maximum theta 5.0 0.0 5.0 10.0 15.0 20.0 25.0 30.0 0.0E04 1.0E04 2.0E04 3.0E04 4.0E04 5.0E04 6.0E04 7.0E04 8.0E04 Section A Section B Section C Section D Section E Section F Section G Axial Velocity [m/s] Radial Coordinate [m]
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164 velocity value was found at Section C with a magnitude of about 12.10 m/s. Similar to the other trials, this maximum value decrease d slightly as the fluid moved through the throat of the nozzle Â– through Sections D and E. Once the fluid exited the nozzle, it was no longer bound by the nozzle wall, and th erefore, no longer bound by the noslip condition. This is why the patte rn of the theta velocity component for the fluid outside of the nozzle is in the shape of a line rather than a parabola. Figure 197: Theta velocity component for various sections (Q = 5.678 x 107 m3/s, R2 = 5.5 x 104 m) The final nozzle geometry that was investigated had the outer inlet slot positioned so that R2 was equal to 7.20 x 104 m and R3 was equal to 7.43 x 104 m. For this case, FC72 was still used as the working fluid, and the only inlet flow rate was 4.416 x 107 m3/s. Figure 198 shows the radial velocity resu lts obtained for this case. It was again noted that most of the values for the radial velocity component within the nozzle were negative, whereas the values were all positive after the fluid exited the nozzle. From the 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 0.0E04 1.0E04 2.0E04 3.0E04 4.0E04 5.0E04 6.0E04 7.0E04 8.0E04 Section A Section B Section C Section D Section E Section F Section GRadial Coordinate [ m ] Theta Velocity [m/s]
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165 positive and negative values observed along S ection A, it was noted that the fluid entering from the central inlet flowed outward in the radial direction toward the nozzle wall. However, the fluid entering from the outer inlet slot began to flow toward the symmetry line as it moved toward the nozzle outlet. The radial velocity reached a maximum value of approximately 2.50 m/s at Section C, before it decreased as it went through the throat of the nozzle. Figure 198: Radial velocity component for various sections (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m) Figure 199 shows the axial ve locity component for this situation. It was noted that for this geometry, all of the velocity components have decreased from the values provided by the geometry having R2 equal to 5.50 x 104 m to the same order of magnitude of the values for the first two ge ometries. Unlike the other geometries, the 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.5 0.0E04 1.0E04 2.0E04 3.0E04 4.0E04 5.0E04 6.0E04 7.0E04 8.0E04 Section A Section B Section C Section D Section E Section F Section G Radial Velocity [m/s] Radial Coordinate [m]
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166 axial velocity for a very small portion of th e fluid along Section A becomes negative. This is because the outer inlet slot is positi on at the edge of the t op plate, which provides no means for the fluid to swirl. This can be confirmed from viewing the streamline contour plot for these parameters. The maxi mum axial velocity had a value of about 3.85 m/s at the location where the fluid exited the no zzle. The fluid within the nozzle declined gradually to a value of zero due to the boundary c onditions placed on the system. The axial velocity of the fluid outside of the no zzle also decreased gra dually, but never went below 7.50 m/s. Figure 199: Axial velocity component for various sections (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m) The last case that was studied involved the theta velocity component for the various sections, which is shown in Figure 200. Again, the theta velocity value for all of the sections within the nozzle form a para bola because of the bounda ry conditions placed on the system. Section A and Section B pr ovided the lowest values with Section A 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 0.0E04 1.0E04 2.0E04 3.0E04 4.0E04 5.0E04 6.0E04 7.0E04 8.0E04 Section A Section B Section C Section D Section E Section F Section GRadial Coordinate [m] Axial Velocity [m/s]
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167 peaking at a radial di stance of about 2.00 x 104 m, and Section B peaking at a radial distance of about 3.00 x 104 m. The maximum theta velocity value was approximately 4.10 m/s located at Section C. Similar to the other trials, this value decreased insignificantly as the fluid moved through the th roat of the nozzle. Once the fluid exited the nozzle, it was free of conditional requ irements along the nozzle wall, which allowed the theta velocity to take on a linear form For these sections, the theta velocity component increased as the radial distance increased. Figure 200: Theta velocity component for various sections (Q = 4.416 x 107 m3/s, R2 = 7.2 x 104 m) Compounding on the previous plots, it is de sired to obtain a magnified view of the velocity component plots for the outlet of the nozzle and the exiting liquid sheet. Therefore, each velocity co mponent at Sections E, F, and G were plotted in dimensionless form in Figure 201 through Figu re 209. These plots simply provide a clearer view of what is occurr ing at these particular sections of the nozzle, and it is for 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0.0E04 1.0E04 2.0E04 3.0E04 4.0E04 5.0E04 6.0E04 7.0E04 8.0E04 Section A Section B Section C Section D Section E Section F Section GRadial Coordinate [m] Theta Velocity [m/s]
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168 this reason that a description of the occurrenc es has previously been discussed. These descriptions can be referenced from the prev ious pages. However, it can be easily seen that the all of the velocities al all of the sections were at a maximum when the ratio of the outer slot location to the top plate radius was equal to 0.74 (R2 equal to 5.50 x 104 m). 0.03 0.025 0.02 0.015 0.01 0.005 0 0.005 00.20.40.60.811.2 R2/Rtop = 0.34 R2/Rtop = 0.54 R2/Rtop = 0.74 R2/Rtop = 0.97 Figure 201: Dimensionless radial velocity component at Section E (FC72, Q = 4.416 x 107 m3/s) 0 0.5 1 1.5 2 2.5 00.20.40.60.811.2 R2/Rtop = 0.34 R2/Rtop = 0.54 R2/Rtop = 0.74 R2/Rtop = 0.97 Figure 202: Dimensionless axial velocity component at Section E (FC72, Q = 4.416 x 107 m3/s) Dimensionless Radial Coordinate, r/Rout Dimensionless Radial Velocity Component, Vr/U Dimensionless Radial Coordinate, r/Rout Dimensionless Axial Velocity Component, Vz/U
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169 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00.20.40.60.811.2 R2/Rtop = 0.34 R2/Rtop = 0.54 R2/Rtop = 0.74 R2/Rtop = 0.97 Figure 203: Dimensionless theta velocity component at Section E (FC72, Q = 4.416 x 107 m3/s) 0 0.005 0.01 0.015 0.02 0.025 0.03 00.20.40.60.811.2 R2/Rtop = 0.34 R2/Rtop = 0.54 R2/Rtop = 0.74 R2/Rtop = 0.97 Figure 204: Dimensionless radial velocity component at Section F (FC72, Q = 4.416 x 107 m3/s) Dimensionless Radial Coordinate, r/Rout Dimensionless Theta Velocity Component, V/U Dimensionless Radial Coordinate, r/Rout Dimensionless Radial Velocity Component, Vr/U
PAGE 189
170 0 0.5 1 1.5 2 2.5 00.20.40.60.811.2 R2/Rtop = 0.34 R2/Rtop = 0.54 R2/Rtop = 0.74 R2/Rtop = 0.97 Figure 205: Dimensionless axial velocity component at Section F (FC72, Q = 4.416 x 107 m3/s) 0 0.2 0.4 0.6 0.8 1 1.2 00.20.40.60.811.2 R2/Rtop = 0.34 R2/Rtop = 0.54 R2/Rtop = 0.74 R2/Rtop = 0.97 Figure 206: Dimensionless theta velocity component at Section F (FC72, Q = 4.416 x 107 m3/s) Dimensionless Radial Coordinate, r/Rout Dimensionless Axial Velocity Component, Vz/U Dimensionless Radial Coordinate, r/Rout Dimensionless Theta Velocity Component, V/U
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171 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 00.20.40.60.811.2 R2/Rtop = 0.34 R2/Rtop = 0.54 R2/Rtop = 0.74 R2/Rtop = 0.97 Figure 207: Dimensionless radial velocity component at Section G (FC72, Q = 4.416 x 107 m3/s) 0 0.5 1 1.5 2 2.5 00.20.40.60.811.2 R2/Rtop = 0.34 R2/Rtop = 0.54 R2/Rtop = 0.74 R2/Rtop = 0.97 Figure 208: Dimensionless axial velocity component at Section G (FC72, Q = 4.416 x 107 m3/s) Dimensionless Radial Coordinate, r/Rout Dimensionless Radial Velocity Component, Vr/U Dimensionless Radial Coordinate, r/Rout Dimensionless Axial Velocity Component, Vz/U
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172 0 0.2 0.4 0.6 0.8 1 1.2 00.20.40.60.811. 2 R2/Rtop = 0.34 R2/Rtop = 0.54 R2/Rtop = 0.74 R2/Rtop = 0.97 Figure 209: Dimensionless theta velocity component at Section G (FC72, Q = 4.416 x 107 m3/s) Figure 210 through Figure 221 are dime nsionless plots that compare each component of velocity at each section for FC72 in all of the various nozzle geometries. Again, these plots were obtai ned to acquire a more understa ndable view of how the fluid is behaving at certain sections for the va rious nozzle geometries. As was previously mentioned, the zerovelocity boundary conditions along the nozzle wall do not apply to the fluid after it has exited the nozzle. This is why the velocities at Sections F and G do not have a value of zero at its outermost radial node. It was also not ed that as the fluid moved further away from the no zzle outlet, its associated ve locities became more linear in nature. The following twelve figures depi ct each velocity component for each of the different nozzle geometries. Dimensionless Radial Coordinate, r/Rout Dimensionless Theta Velocity Component, V/U
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173 0.01 0.005 0 0.005 0.01 0.015 00.20.40.60.811.2 Section E Section F Section G Figure 210: Dimensionless radial veloci ty component at various sections (R2 = 2.5 x 104 m) 0 0.2 0.4 0.6 0.8 1 1.2 00.20.40.60.811.2 Section E Section F Section G Figure 211: Dimensionless axial veloci ty component at various sections (R2 = 2.5 x 104 m) Dimensionless Radial Coordinate, r/Rout Dimensionless Radial Velocity Component, Vr/U Dimensionless Radial Coordinate, r/Rout Dimensionless Axial Velocity Component, Vz/U
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174 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 00.20.40.60.811.2 Section E Section F Section G Figure 212: Dimensionless theta veloci ty component at various sections (R2 = 2.5 x 104 m) 0.01 0.005 0 0.005 0.01 0.015 00.20.40.60.811.2 Section E Section F Section G Figure 213: Dimensionless radial velocity at various sections (R2 = 4.0 x 104 m) Dimensionless Radial Coordinate, r/Rout Dimensionless Theta Velocity Component, V/U Dimensionless Radial Coordinate, r/Rout Dimensionless Radial Velocity Component, Vr/U
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175 0 0.2 0.4 0.6 0.8 1 1.2 00.20.40.60.811.2 Section E Section F Section G Figure 214: Dimensionless axial veloci ty component at various sections (R2 = 4.0 x 104 m) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 00.20.40.60.811.2 Section E Section F Section G Figure 215: Dimensionless theta veloci ty component at various sections (R2 = 4.0 x 104 m) Dimensionless Radial Coordinate, r/Rout Dimensionless Axial Velocity Component, Vz/U Dimensionless Radial Coordinate, r/Rout Dimensionless Theta Velocity Component, V/U
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176 0.03 0.02 0.01 0 0.01 0.02 0.03 0.04 00.20.40.60.811.2 Section E Section F Section G Figure 216: Dimensionless radial veloci ty component at various sections (R2 = 5.5 x 104 m) 0 0.5 1 1.5 2 2.5 00.20.40.60.811.2 Section E Section F Section G Figure 217: Dimensionless axial veloci ty component at various sections (R2 = 5.5 x 104 m) Dimensionless Radial Coordinate, r/Rout Dimensionless Radial Velocity Component, Vr/U Dimensionless Radial Coordinate, r/Rout Dimensionless Axial Velocity Component, Vz/U
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177 0 0.2 0.4 0.6 0.8 1 1.2 00.20.40.60.811.2 Section E Section F Section G Figure 218: Theta velocity co mponent at various sections (R2 = 5.5 x 104 m) 0.015 0.01 0.005 0 0.005 0.01 0.015 00.20.40.60.811.2 Section E Section F Section G Figure 219: Dimensionless radial veloci ty component at various sections (R2 = 7.2 x 104 m) Dimensionless Radial Coordinate, r/Rout Dimensionless Theta Velocity Component, V/U Dimensionless Radial Coordinate, r/Rout Dimensionless Radial Velocity Component, Vr/U
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178 0 0.2 0.4 0.6 0.8 1 1.2 00.20.40.60.811.2 Section E Section F Section G Figure 220: Dimensionless axial veloci ty component at various sections (R2 = 7.2 x 104 m) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 00.20.40.60.811.2 Section E Section F Section G Figure 221: Dimensionless theta veloci ty component at various sections (R2 = 7.2 x 104 m) Dimensionless Radial Coordinate, r/Rout Dimensionless Axial Velocity Component, Vz/U Dimensionless Radial Coordinate, r/Rout Dimensionless Theta Velocity Component V/U
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179 It is also important to know the aver age for each velocity component at each section throughout the nozzle. Tables 2428 de pict the average velocity components at each of the nozzle sections that were investigated. From these averages, the average component of momentum can be found at each section. It was observed that both the average velocity component and the average momentum component peaked at Section F, which is just after the fluid exited the nozzle. These tables confirm the previous graphs in the sense that the values calculated for the nozzle having R2 equal to 5.50 x 104 m are almost twice as much as the values obtained for the other outer slot location geometries. This could explain why that particular nozzle geometry produced the greatest radial free surface height during this investigation. Table 24: Average velocity and momen tum components at each section (R2=2.50 x 104 m, Re=5904) Section Vr, avg [m/s] Vz, avg [m/s] V avg [m/s] avg r,M [kg m/s2] avg z,M [kg m/s2] avg ,M [kg m/s2] M [kg m/s2] Inlet 0.000 0.433 0.433 0.000 3.21 x 104 3.21 x 104 4.54 x 104 A 0.125 0.270 1.430 9.24 x 105 2.00 x 104 1.06 x 103 1.08 x 103 B 0.216 0.524 0.662 1.60 x 104 3.88 x 104 4.91 x 104 6.46 x 104 C 1.024 6.677 2.361 7.60 x 1044.95 x 103 1.75x 103 5.31 x 103 D 0.509 3.248 2.441 3.78 x 1052.41 x 103 1.81 x 103 3.01 x 103 E 0.459 6.971 2.298 3.41 x 1055.17 x 103 1.70 x 103 5.45 x 103 F 0.267 7.645 2.263 1.98 x 105 5.67 x 103 1.68 x 103 5.92 x 103 G 0.398 7.559 2.193 2.95 x 105 5.61 x 103 1.63 x 103 5.84 x 103
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180Table 25: Average velocity and momen tum components at each section (R2=4.00 x 104 m, Re=5904) Section Vr, avg [m/s] Vz, avg [m/s] V, avg [m/s] avg r,M [kg m/s2] avg z,M [kg m/s2] avg ,M [kg m/s2] M [kg m/s2] Inlet 0.000 0.680 0.680 0.00 5.04 x 104 5.04 x 104 7.13 x 104 A 0.122 0.314 1.822 9.04 x 1052.33 x 104 1.35 x 103 1.37 x 103 B 0.163 0.518 1.761 1.21 x 1043.84 x 104 1.31 x 103 1.37 x 103 C 1.014 6.753 2.677 7.52 x 1045.01 x 103 1.99 x 103 5.44 x 103 D 0.994 7.093 2.723 7.37 x 1055.26 x 103 2.02 x 103 5.64 x 103 E 0.259 7.136 2.650 1.92 x 1055.29 x 103 1.97 x 103 5.64 x 103 F 0.808 7.615 2.483 5.99 x 105 5.65 x 103 1.84 x 103 5.94 x 103 G 0.450 7.568 2.404 3.34 x 105 5.62 x 103 1.78 x 103 5.89 x 103 Table 26: Average velocity and momen tum components at each section (R2=5.50 x 104 m, Re=5904) Section Vr, avg [m/s] Vz, avg [m/s] V, avg [m/s] avg r,M [kg m/s2] avg z,M [kg m/s2] avg ,M [kg m/s2] M [kg m/s2] Inlet 0.000 0.846 0.846 0.00 6.28 x 104 6.28 x 104 8.88 x 104 A 0.526 0.634 1.290 3.90 x 104 4.70 x 104 9.57 x 104 1.14 x 103 B 0.280 1.145 4.234 2.07 x 104 8.50x 104 3.14 x 103 3.26x 103 C 2.314 15.88 6.799 1.72 x 103 1.78 x 102 5.04 x 103 1.86 x 102 D 0.222 14.88 6.925 1.65 x 104 1.10 x 102 5.14 x 103 1.22 x 102 E 0.664 16.89 6.670 4.93 x 105 1.25 x 102 4.95 x 103 1.35 x 102 F 0.182 17.14 6.178 1.35 x 104 1.27 x 102 4.58 x 103 1.35 x 102 G 0.115 17.02 5.990 8.50 x 105 1.26 x 102 4.44 x 103 1.34 x 102
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181Table 27: Average velocity and momen tum components at each section (R2=5.50 x 104 m, Re=7591) Section Vr, avg [m/s] Vz, avg [m/s] V, avg [m/s] avg r,M [kg m/s2] avg z,M [kg m/s2] avg ,M [kg m/s2] M [kg m/s2] Inlet 0.000 1.088 1.088 0.00 8.07 x 104 8.07 x 104 1.14 x 103 A 0.688 0.814 4.082 5.10 x 104 6.04 x 104 3.03 x 103 3.13 x 103 B 0.351 1.524 5.104 2.61 x 104 1.13 x 103 3.79 x 103 3.96 x 103 C 2.965 20.41 8.847 2.20 x 103 1.51 x 102 6.56 x 103 1.67 x 102 D 0.282 16.38 9.023 2.09 x 104 1.22 x 102 6.69 x 103 1.39 x 102 E 0.885 16.62 8.694 6.57 x 105 1.23 x 102 6.45 x 103 1.39 x 102 F 0.233 22.02 8.051 1.73 x 104 1.63 x 102 5.97 x 103 1.74x 102 G 0.150 21.86 7.805 1.11 x 104 1.62 x 102 5.79 x 103 1.72 x 102 Table 28: Average velocity and momen tum components at each section (R2=7.20 x 104 m, Re=5904) Section Vr, avg [m/s] Vz, avg [m/s] V, avg [m/s] avg r,M [kg m/s2] avg z,M [kg m/s2] avg ,M [kg m/s2] M [kg m/s2] Inlet 0.000 0.568 0.568 0.00 4.21 x 104 4.21 x 104 5.96 x 104 A 0.217 0.250 1.953 1.61 x 1041.86 x 104 1.45 x 103 1.47 x 103 B 0.170 0.563 1.892 1.26 x 1044.17 x 104 1.40 x 103 1.47 x 103 C 1.185 7.748 2.939 8.79 x 1045.75 x 103 2.18 x 103 6321 x 103 D 0.981 7.577 2.881 7.28 x 1055.62 x 103 2.14 x 103 6.01 x 103 E 0.474 7.667 2.810 3.52 x 1055.69 x 103 2.08 x 103 6.06 x 103 F 0.811 8.208 2.719 6.02 x 105 6.09 x 103 2.02 x 103 6.42 x 103 G 0.488 8.111 2.638 3.62 x 105 6.02 x 103 1.96 x 103 6.33 x 103
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182 Next, the pressure drop obtained from FI DAP for this scenario was compared to the calculated pressure drop according to Bernou lliÂ’s equation. It was known that the pressure drop obtained from FIDAP must be larger than the value calculated using BernoulliÂ’s equation. Table 29 shows the pr essure drop obtained from the program, as well as the calculated pressure drop. It was not ed that, in fact, all of the trials obeyed the requirement by having the pressure drop from the program greater than the calculated pressure drop. Table 29: Pressure drop comparison for various outer slot locations Outer Slot Location R2 [m] Inlet Flow Rate Q [m3/s] FIDAP Pressure Drop [Pa] Calculated Pressure Drop [Pa] 2.50 x 104 4.416 x 107 7.25 x 104 5.45 x 104 4.00 x 104 4.416 x 107 7.67 x 104 5.77 x 104 4.416 x 107 3.39 x 105 3.29 x 105 5.50 x 104 5.678 x 107 6.51 x 105 3.38 x 105 7.20 x 104 4.416 x 107 9.21 x 104 6.64 x 104
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183 6 Conclusion The conclusions gathered from the results of this investigation can be summarized as follows: 1. The larger nozzle produced a greater radial height of the free surface and a greater cone angle. 2. The radial height of the free surface increased as the Reynolds number increased. 3. The cone angle produced by the fluid exiting the nozzle became larger as the Reynolds number became larger. 4. The cone angle increased as the radial va lue of the outer inlet slot also increased. 5. Increasing the radial distance of the outer inlet slot resulted in an increase in free surface height for all of the fluids that were investigated. However, the last scenario where the outer inlet slot was pos ition on the edge of the nozzle resulted in a lower free surface height than the previous position. 6. The pressure drop from the inlet of th e nozzle to the outlet increased with an increase in the flow rate. 7. The pressure drop also increased as the radial distance of the outer inlet slot increased. However, the pressure drop decreased when the location of the outer inlet slot reached the edge of the nozzle. 8. Overall, all of the variations produced small changes in the free surface position.
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184 References Burmeister, L.C., Convective Heat Transfer, Wiley, New York, 1993. Ciofalo, M., DiPiazza, I., Brucato, V., Â“Invest igation of Cooling of Hot Walls by Liquid Water Sprays,Â” International Journal of Heat and Mass Transfer, Vol. 42, No. 7, 1999, pp. 11571175. Datta, A., and Som, S.K., Â“Numerical Predic tion of Air Core Diameter Coefficient of Discharge and Spray Cone Angle of a Swirl Spray Pressure Nozzle,Â” International Journal of Heat and Fluid Flow, Vol. 21, No. 4, 2000, pp. 412419. Dumouchel, C., Blook, M. I. G., Dimbro wski, N., Ingham, D. B., and Ledoux, M., Â“Viscous Flow in a Swirl Atomizer,Â” Chemical Engineering Science, Vol. 48, No. 1, 1993, pp. 8187. Gavaises, M., Arcoumane, C., Â“Modeling of Sprays from HighPressure Swirl Atomizers,Â” International Journal of Engine Research, Vol. 2, No. 2, 2001, pp. 95117. Jeng, S. M., Jog, M. A., and Benjamin, M. A., Â“Computation and Experimental Study of Liquid Sheet Emanating from Simplex Fuel Nozzle,Â” AIAA Journal, Vol. 36, No. 2, 1998, pp. 201207. Miller, P.C.H., and Ellis, M.C. Butler, Â“Effects of Formulation of Spray Nozzle Performance for Applications from GroundBased Boom Sprayers,Â” Crop Protection, Vol. 19, No. 810, 2000, pp. 609615. Rothe, P.H., and Block, J.A., Â“Aer odynamic Behavior of Liquid Sprays,Â” International Journal of Multiphase Flow, Vol. 3, No. 3, 1977, pp. 263272. Sakman, A. T., Jog, M. A., Jeng, S. M., and Benjamin, M. A., Â“Parametric Study of Simplex Fuel Nozzle Internal Flow and Performance,Â” AIAA Journal, Vol. 38, No. 7, 2000, pp. 12141218. Som, S.K., and Biswas, G., Â“Dispers ion of Spray from Swirl Nozzles,Â” Chemical Engineering and Processing, Vol. 20, No. 4, 1986, pp. 191200. White, Frank M. Fluid Mechanics: Fourth Edition, Boston, McGrawHill, 1999.
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185 Appendices
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186 Appendix I: GAMBIT File for Large Nozzle solver select "FIDAP" undo begingroup vertex create coordinates 0 0 0 undo endgroup undo begingroup vertex create coordinates 4.2 0 0 undo endgroup undo begingroup vertex create coordinates 4.2 0.3175 0 undo endgroup undo begingroup vertex create coordinates 3.0988 0.3175 0 undo endgroup undo begingroup vertex create coordinates 2.7178 0.3175 0 undo endgroup undo begingroup vertex create coordinates 0 1.88722 0 undo endgroup undo begingroup vertex create coordinates 0 1.68656 0 undo endgroup undo begingroup vertex create coordinates 0 0.10033 0 undo endgroup undo begingroup vertex create coordinates 1 2.4648 0 undo endgroup undo begingroup vertex create coordinates 1 0 0 undo endgroup edge create straight "vertex.1" "vertex.2" edge create straight "vertex.2" "vertex.3" edge create straight "vertex.3" "vertex.4" edge create straight "vertex.4" "vertex.5" edge create straight "vertex.5" "vertex.6" edge create straight "vertex.6" "vertex.7" edge create straight "vertex.7" "vertex.8"
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187Appendix I: (Continued) edge create straight "vertex.8" "vertex.1" edge create straight "v ertex.6" "vertex.9" edge create straight "vertex.9" "vertex.10" edge create straight "vertex.10" "vertex.1" face create wireframe "edge.1" "edge.2" "e dge.3" "edge.4" "edge.5" "edge.6" "edge.7" "edge.8" real face create wireframe "edge.9" "edge.10" "edge.11" "edge.8" "edge.7" "edge.6" real undo begingroup edge picklink "edge.1" edge mesh "edge.1" successi ve ratio1 1 intervals 110 undo endgroup undo begingroup edge picklink "edge.2" edge mesh "edge.2" successi ve ratio1 1 intervals 28 undo endgroup undo begingroup edge picklink "edge.3" edge mesh "edge.3" successi ve ratio1 1 intervals 30 undo endgroup undo begingroup edge picklink "edge.4" edge mesh "edge.4" successi ve ratio1 1 intervals 10 undo endgroup undo begingroup edge picklink "edge.5" edge mesh "edge.5" successi ve ratio1 1 intervals 70 undo endgroup undo begingroup edge picklink "edge.6" edge mesh "edge.6" successi ve ratio1 1 intervals 6 undo endgroup undo begingroup edge picklink "edge.7" edge mesh "edge.7" successi ve ratio1 1 intervals 16 undo endgroup undo begingroup edge picklink "edge.8" edge mesh "edge.8" successi ve ratio1 1 intervals 6 undo endgroup
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188Appendix I: (Continued) undo begingroup edge picklink "edge.9" edge mesh "edge.9" successi ve ratio1 1 intervals 20 undo endgroup undo begingroup edge picklink "edge.10" edge mesh "edge.10" successive ratio1 1 intervals 28 undo endgroup undo begingroup edge picklink "edge.11" edge mesh "edge.11" successive ratio1 1 intervals 20 undo endgroup face mesh "face.1" map size 1 face mesh "face.2" map size 1 physics create "symmetry" btype "PLOT" edge "edge.1" physics create "outlet" btype "PLOT" edge "edge.2" physics create "free" btype "SURFACE" edge "edge.3" physics create "wall1" btype "PLOT" edge "edge.4" physics create "wall2" btype "PLOT" edge "edge.5" physics create "inlet1" btype "PLOT" edge "edge.6" physics create "top" btyp e "PLOT" edge "edge.7" physics create "inlet2" btype "PLOT" edge "edge.8" physics create "wall3" btype "PLOT" edge "edge.9" physics create "inlet" btyp e "PLOT" edge "edge.10" physics create "symm" btype "PLOT" edge "edge.11" physics create "fluid" ctype "FLUID" face "face.1" "face.2" export fidap Â“large.FDNEUTÂ”
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189 Appendix II: FIJOUR File for Large Nozzle FICONV( NEUT, INPU, RESU ) INPUTFILE( FILE = large.FDNEUT" ) OUTPUTFILE( FILE, DELE ) END( ) FIPREP( ) PROBLEM( ADD, CYLI, INCO, TRANS, LAMI, NONL, NEWT, MOME, ISOT, FREE, SING ) EXECUTION( ADD, NEWJ ) ENTITY( ADD, NAME = "fluid ", FLUI, PROP = "fc77" ) ENTITY( ADD, NAME = "wall1", PLOT ) ENTITY( ADD, NAME = "wall2", PLOT ) ENTITY( ADD, NAME = "symmetry", PLOT ) ENTITY( ADD, NAME = "inlet1", PLOT ) ENTITY( ADD, NAME = "inlet2", PLOT ) ENTITY( ADD, NAME = "outlet", PLOT ) ENTITY( ADD, NAME = "top", PLOT ) ENTITY( ADD, NAME = "free", SURFAC E, DEPTH = 0, SPINES, STRAIGHT ) SOLUTION( ADD, N.R. = 70, KINE = 25, VELC = 0.00001, RESC = 0.001, SURF = 0.0001 ) BODYFORCE( ADD, CONS, FZC = 981, FRC = 0, FTHETA = 0 ) DATAPRINT( ADD, CONT ) OPTIONS( ADD, UPWINDING ) UPWINDING( ADD, STREAMLINE ) RELAXATION( ) 0.65, 0.65, 0.65, 0, 0, 0.4 PRESSURE( ADD, MIXED = 1E11, DISC ) DENSITY( ADD, SET = "fc77", CONS = 1.78 ) /Density for FC72 /DENSITY(ADD, SET = Â“fc72Â”, CONS = 1.68) VISCOSITY( ADD, SET = "fc77", CONS = 0.01424 ) /Viscosity for FC72 /VISCOSITY( ADD, SET = Â“fc72Â”, CONS = 0.0064) SURFACETENSION( ADD, SET = "fc77", CONS = 15 ) /Surface Tension for FC72 /SURFACETENSION( ADD, SET = Â“fc72Â”, CONS = 10 )
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190Appendix II: (Continued) TIMEINTEGRATION( ADD, BACK, NSTEPS = 2000, TSTART = 0, DT = 0.000001, VARI, NOFIXED = 10, WIND = 0.90 ) /For an inlet flow rate of 1.262 x 106 m3/s ICNODE( ADD, URC, ENTI = "inlet1", CONS = 5.35 ) ICNODE( ADD, UZC, ENTI = "inlet1", CONS = 5.35 ) ICNODE( ADD, URC, ENTI = "inlet2", CONS = 0 ) ICNODE( ADD, UZC, ENTI = "inlet2", CONS = 7.566 ) BCNODE( ADD, URC, ENTI = "i nlet1", CONS = 5.35 ) BCNODE( ADD, UZC, ENTI = "inlet1", CONS = 5.35 ) BCNODE( ADD, URC, ENTI = "inlet2", CONS = 0 ) BCNODE( ADD, UZC, ENTI = "inlet2", CONS = 7.566 ) /For an inlet flow rate of 2.524 x 106 m3/s /ICNODE( ADD, URC, ENTI = "inlet1", CONS = 10.7 ) /ICNODE( ADD, UZC, ENTI = "inlet1", CONS = 10.7 ) /ICNODE( ADD, URC, ENTI = "inlet2", CONS = 0 ) /ICNODE( ADD, UZC, ENTI = "inlet2", CONS = 15.132 ) /BCNODE( ADD, URC, ENTI = "inlet1", CONS = 10.7) /BCNODE( ADD, UZC, ENTI = "inlet1", CONS = 10.7 ) /BCNODE( ADD, URC, ENTI = "inlet2", CONS = 0 ) /BCNODE( ADD, UZC, ENTI = "inlet2", CONS = 15.132 ) /For an inlet flow rate of 3.785 x 106 m3/s /ICNODE( ADD, URC, ENTI = "inlet1", CONS = 16.05 ) /ICNODE( ADD, UZC, ENTI = "inlet1", CONS = 16.05 ) /ICNODE( ADD, URC, ENTI = "inlet2", CONS = 0 ) /ICNODE( ADD, UZC, ENTI = "inlet2", CONS = 22.698 ) /BCNODE( ADD, URC, ENTI = "inlet1", CONS = 16.05 ) /BCNODE( ADD, UZC, ENTI = "inlet1", CONS = 16.05 ) /BCNODE( ADD, URC, ENTI = "inlet2", CONS = 0 ) /BCNODE( ADD, UZC, ENTI = "inlet2", CONS = 22.698 ) /Boundary Conditions for all flow rates BCNODE( ADD, VELO, ENTI = "wall1 ", CONS = 0, X, Y, Z ) BCNODE( ADD, VELO, ENTI = "wall2 ", CONS = 0, X, Y, Z ) BCNODE( ADD, URC, ENTI = "symmetry", CONS = 0 ) BCNODE( ADD, URC, ENTI = "top", CONS = 0 ) BCNODE( ADD, UZC, ENTI = "top", CONS = 0 ) BCNODE( SURFACE, ZERO, NODE = 146 ) BCNODE( SURFACE, ZERO, NODE = 213 ) BCNODE( COORDINATE, NODE = 213 )
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191Appendix II: (Continued) BCNODE( COORDINATE, NODE = 170 ) /Initial Conditions for the top plate fo r an inlet flow rate of 1.262 x 106 m3/s ICNODE( ADD, UTHETA, NODE = 10, CONS = 5.297 ) ICNODE( ADD, UTHETA, NODE = 11, CONS = 4.968 ) ICNODE( ADD, UTHETA, NODE = 12, CONS = 4.639 ) ICNODE( ADD, UTHETA, NODE = 13, CONS = 4.309 ) ICNODE( ADD, UTHETA, NODE = 14, CONS = 3.980 ) ICNODE( ADD, UTHETA, NODE = 15, CONS = 3.651 ) ICNODE( ADD, UTHETA, NODE = 16, CONS = 3.321 ) ICNODE( ADD, UTHETA, NODE = 17, CONS = 2.992 ) ICNODE( ADD, UTHETA, NODE = 18, CONS = 2.663 ) ICNODE( ADD, UTHETA, NODE = 19, CONS = 2.333 ) ICNODE( ADD, UTHETA, NODE = 20, CONS = 2.004 ) ICNODE( ADD, UTHETA, NODE = 21, CONS = 1.675 ) ICNODE( ADD, UTHETA, NODE = 22, CONS = 1.346 ) ICNODE( ADD, UTHETA, NODE = 23, CONS = 1.016 ) ICNODE( ADD, UTHETA, NODE = 24, CONS = 0.387 ) ICNODE( ADD, UTHETA, NODE = 25, CONS = 0.358 ) /Initial conditions for the top plate fo r an inlet flow rate of 2.524 x 106 m3/s /ICNODE( ADD, UTHE, NODE = 10, CONS = 10.594 ) /ICNODE( ADD, UTHE, NODE = 11, CONS = 9.936 ) /ICNODE( ADD, UTHE, NODE = 12, CONS = 9.277 ) /ICNODE( ADD, UTHE, NODE = 13, CONS = 8.618 ) /ICNODE( ADD, UTHE, NODE = 14, CONS = 7.96 ) /ICNODE( ADD, UTHE, NODE = 15, CONS = 7.301 ) /ICNODE( ADD, UTHE, NODE = 16, CONS = 6.643 ) /ICNODE( ADD, UTHE, NODE = 17, CONS = 5.984 ) /ICNODE( ADD, UTHE, NODE = 18, CONS = 5.326 ) /ICNODE( ADD, UTHE, NODE = 19, CONS = 4.667 ) /ICNODE( ADD, UTHE, NODE = 20, CONS = 4.008 ) /ICNODE( ADD, UTHE, NODE = 21, CONS = 3.35 ) /ICNODE( ADD, UTHE, NODE = 22, CONS = 2.691 ) /ICNODE( ADD, UTHE, NODE = 23, CONS = 2.033 ) /ICNODE( ADD, UTHE, NODE = 24, CONS = 1.374 ) /ICNODE( ADD, UTHE, NODE = 25, CONS = 0.715 ) /Initial conditions for the top plate fo r an inlet flow rate of 3.785 x 106 m3/s /ICNODE( ADD, UTHE, NODE = 10, CONS = 15.893 ) /ICNODE( ADD, UTHE, NODE = 11, CONS = 14.905 ) /ICNODE( ADD, UTHE, NODE = 12, CONS = 13.917 )
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192Appendix II: (Continued) /ICNODE( ADD, UTHE, NODE = 13, CONS = 12.929 ) /ICNODE( ADD, UTHE, NODE = 14, CONS = 11.941 ) /ICNODE( ADD, UTHE, NODE = 15, CONS = 10.953 ) /ICNODE( ADD, UTHE, NODE = 16, CONS = 9.965 ) /ICNODE( ADD, UTHE, NODE = 17, CONS = 8.977 ) /ICNODE( ADD, UTHE, NODE = 18, CONS = 7.989 ) /ICNODE( ADD, UTHE, NODE = 19, CONS = 7.001 ) /ICNODE( ADD, UTHE, NODE = 20, CONS = 6.013 ) /ICNODE( ADD, UTHE, NODE = 21, CONS = 5.025 ) /ICNODE( ADD, UTHE, NODE = 22, CONS = 4.037 ) /ICNODE( ADD, UTHE, NODE = 23, CONS = 3.049 ) /ICNODE( ADD, UTHE, NODE = 24, CONS = 2.061 ) /ICNODE( ADD, UTHE, NODE = 25, CONS = 1.073 ) /Boundary conditions for the top plate for an inlet flow rate 1.262 x 106 m3/s BCNODE( ADD, UTHETA, NODE = 10, CONS = 5.297 ) BCNODE( ADD, UTHETA, NODE = 11, CONS = 4.968 ) BCNODE( ADD, UTHETA, NODE = 12, CONS = 4.639 ) BCNODE( ADD, UTHETA, NODE = 13, CONS = 4.309 ) BCNODE( ADD, UTHETA, NODE = 14, CONS = 3.980 ) BCNODE( ADD, UTHETA, NODE = 15, CONS = 3.651 ) BCNODE( ADD, UTHETA, NODE = 16, CONS = 3.321 ) BCNODE( ADD, UTHETA, NODE = 17, CONS = 2.992 ) BCNODE( ADD, UTHETA, NODE = 18, CONS = 2.663 ) BCNODE( ADD, UTHETA, NODE = 19, CONS = 2.333 ) BCNODE( ADD, UTHETA, NODE = 20, CONS = 2.004 ) BCNODE( ADD, UTHETA, NODE = 21, CONS = 1.675 ) BCNODE( ADD, UTHETA, NODE = 22, CONS = 1.346 ) BCNODE( ADD, UTHETA, NODE = 23, CONS = 1.016 ) BCNODE( ADD, UTHETA, NODE = 24, CONS = 0.387 ) BCNODE( ADD, UTHETA, NODE = 25, CONS = 0.358 ) /Boundary conditions for the top plate fo r an inlet flow rate of 2.524 x 106 m3/s /BCNODE( ADD, UTHE, NODE = 10, CONS = 10.594 ) /BCNODE( ADD, UTHE, NODE = 11, CONS = 9.936 ) /BCNODE( ADD, UTHE, NODE = 12, CONS = 9.277 ) /BCNODE( ADD, UTHE, NODE = 13, CONS = 8.618 ) /BCNODE( ADD, UTHE, NODE = 14, CONS = 7.96 ) /BCNODE( ADD, UTHE, NODE = 15, CONS = 7.301 ) /BCNODE( ADD, UTHE, NODE = 16, CONS = 6.643 ) /BCNODE( ADD, UTHE, NODE = 17, CONS = 5.984 ) /BCNODE( ADD, UTHE, NODE = 18, CONS = 5.326 )
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193Appendix II: (Continued) /BCNODE( ADD, UTHE, NODE = 19, CONS = 4.667 ) /BCNODE( ADD, UTHE, NODE = 20, CONS = 4.008 ) /BCNODE( ADD, UTHE, NODE = 21, CONS = 3.35 ) /BCNODE( ADD, UTHE, NODE = 22, CONS = 2.691 ) /BCNODE( ADD, UTHE, NODE = 23, CONS = 2.033 ) /BCNODE( ADD, UTHE, NODE = 24, CONS = 1.374 ) /BCNODE( ADD, UTHE, NODE = 25, CONS = 0.715 ) /Boundary conditions for the top plate fo r an inlet flow rate of 3.785 x 106 m3/s /BCNODE( ADD, UTHE, NODE = 10, CONS = 15.893 ) /BCNODE( ADD, UTHE, NODE = 11, CONS = 14.905 ) /BCNODE( ADD, UTHE, NODE = 12, CONS = 13.917 ) /BCNODE( ADD, UTHE, NODE = 13, CONS = 12.929 ) /BCNODE( ADD, UTHE, NODE = 14, CONS = 11.941 ) /BCNODE( ADD, UTHE, NODE = 15, CONS = 10.953 ) /BCNODE( ADD, UTHE, NODE = 16, CONS = 9.965 ) /BCNODE( ADD, UTHE, NODE = 17, CONS = 8.977 ) /BCNODE( ADD, UTHE, NODE = 18, CONS = 7.989 ) /BCNODE( ADD, UTHE, NODE = 19, CONS = 7.001 ) /BCNODE( ADD, UTHE, NODE = 20, CONS = 6.013 ) /BCNODE( ADD, UTHE, NODE = 21, CONS = 5.025 ) /BCNODE( ADD, UTHE, NODE = 22, CONS = 4.037 ) /BCNODE( ADD, UTHE, NODE = 23, CONS = 3.049 ) /BCNODE( ADD, UTHE, NODE = 24, CONS = 2.061 ) /BCNODE( ADD, UTHE, NODE = 25, CONS = 1.073 ) END( )
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194 Appendix III: GAMBIT File for Small Nozzle solver select "FIDAP" undo begingroup vertex create coordinates 0 0 0 undo endgroup undo begingroup vertex create coordinates 0.17 0 0 undo endgroup undo begingroup vertex create coordinates 0.17 0.0125 0 undo endgroup undo begingroup vertex create coordinates 0.122 0.0125 0 undo endgroup undo begingroup vertex create coordinates 0.107 0.0125 0 undo endgroup undo begingroup vertex create coordinates 0 0.0743 0 undo endgroup undo begingroup vertex create coordinates 0 0.0543 0 undo endgroup undo begingroup vertex create coordinates 0 0.01 0 undo endgroup edge create straight "vertex.1" "vertex.2" edge create straight "vertex.2" "vertex.3" edge create straight "vertex.3" "vertex.4" edge create straight "vertex.4" "vertex.5" edge create straight "vertex.5" "vertex.6" edge create straight "vertex.6" "vertex.7" edge create straight "vertex.7" "vertex.8" edge create straight "vertex.8" "vertex.1" face create wireframe "edge.1" "edge.2" "e dge.3" "edge.4" "edge.5" "edge.6" "edge.7" "edge.8" real
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195Appendix III: (Continued) undo begingroup edge picklink "edge.1" edge mesh "edge.1" successi ve ratio1 1 intervals 172 undo endgroup undo begingroup edge picklink "edge.2" edge mesh "edge.2" successive ra tio1 1 ratio2 1.1 intervals 32 undo endgroup undo begingroup edge picklink "edge.3" edge mesh "edge.3" successi ve ratio1 1 intervals 36 undo endgroup undo begingroup edge picklink "edge.4" edge mesh "edge.4" successi ve ratio1 1 intervals 24 undo endgroup undo begingroup edge picklink "edge.5" edge mesh "edge.5" successi ve ratio1 1 intervals 112 undo endgroup undo begingroup edge picklink "edge.6" edge mesh "edge.6" successi ve ratio1 1 intervals 8 undo endgroup undo begingroup edge picklink "edge.7" edge mesh "edge.7" successi ve ratio1 1 intervals 16 undo endgroup undo begingroup edge picklink "edge.8" edge mesh "edge.8" successi ve ratio1 1 intervals 8 undo endgroup /Mesh the face face mesh "face.1" map size 1 physics create "symmetry" btype "PLOT" edge "edge.1" physics create "outlet" btype "PLOT" edge "edge.2" physics create "free" btype "SURFACE" edge "edge.3" physics create "wall1" btype "PLOT" edge "edge.4" physics create "wall2" btype "PLOT" edge "edge.5" physics create "inlet1" btype "PLOT" edge "edge.6" physics create "top" btyp e "PLOT" edge "edge.7"
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196 physics create "inlet2" btype "PLOT" edge "edge.8" physics create "fluid" ct ype "FLUID" face "face.1" export fidap Â“small.FDNEUTÂ”
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197 Appendix IV: FIJOUR File for Small Nozzle (1.262 x 107 m3/s and 2.524 x 107 m3/s) FICONV( NEUT, INPU, RESU ) INPUTFILE( FILE = small.FDNEUT" ) OUTPUTFILE( FILE, DELE ) END( ) FIPREP( ) PROBLEM( ADD, CYLI, INCO, TRANS, LAMI, NONL, NEWT, MOME, ISOT, FREE, SING ) EXECUTION( ADD, NEWJ ) ENTITY( ADD, NAME = "fluid ", FLUI, PROP = "fc77" ) ENTITY( ADD, NAME = "wall1", PLOT ) ENTITY( ADD, NAME = "wall2", PLOT ) ENTITY( ADD, NAME = "symmetry", PLOT ) ENTITY( ADD, NAME = "inlet1", PLOT ) ENTITY( ADD, NAME = "inlet2", PLOT ) ENTITY( ADD, NAME = "outlet", PLOT ) ENTITY( ADD, NAME = "top", PLOT ) ENTITY( ADD, NAME = "free", SURFAC E, DEPTH = 0, SPINES, STRAIGHT ) SOLUTION( ADD, N.R. = 70, KINE = 25, VELC = 0.0001, RESC = 0.01, SURF = 0.001 ) BODYFORCE( ADD, CONS, FZC = 981, FRC = 0, FTHETA = 0 ) DATAPRINT( ADD, CONT ) OPTIONS( ADD, UPWINDING ) UPWINDING( ADD, STREAMLINE ) RELAXATION( ) 0.6, 0.6, 0.6, 0, 0, 0.1 PRESSURE( ADD, MIXED = 1E18, DISC ) DENSITY( ADD, SET = "fc77", CONS = 1.78 ) VISCOSITY( ADD, SET = "fc77", CONS = 0.01424 ) SURFACETENSION( ADD, SET = "fc77", CONS = 15 ) POSTPROCESS( ADD, NBLOCK = 2 ) 1 401 400 402 800 1
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198Appendix IV: (Continued) PRINTOUT( ADD, NBLOCK = 2 ) 1 401 400 402 800 1 TIMEINTEGRATION( ADD, BACK, NSTEPS = 800, TSTART = 0, DT = 0.0000001, VARI, NOFIXED = 10, WIND = 0.90 ) /Initial conditions for an in let flow rate of 1.262 x 107 m3/s ICNODE( ADD, URC, ENTI = "inlet1", CONS = 17.362 ) ICNODE( ADD, UZC, ENTI = "inlet1", CONS = 17.362 ) ICNODE( ADD, URC, ENTI = "inlet2", CONS = 0 ) ICNODE( ADD, UZC, ENTI = "inlet2", CONS = 24.553 ) ICNODE( ADD, UTHETA, NODE = 10, CONS = 12.688 ) ICNODE( ADD, UTHETA, NODE = 11, CONS = 12.041 ) ICNODE( ADD, UTHETA, NODE = 12, CONS = 11.394 ) ICNODE( ADD, UTHETA, NODE = 13, CONS = 10.747 ) ICNODE( ADD, UTHETA, NODE = 14, CONS = 10.100 ) ICNODE( ADD, UTHETA, NODE = 15, CONS = 9.453 ) ICNODE( ADD, UTHETA, NODE = 16, CONS = 8.806 ) ICNODE( ADD, UTHETA, NODE = 17, CONS = 8.159 ) ICNODE( ADD, UTHETA, NODE = 18, CONS = 7.512 ) ICNODE( ADD, UTHETA, NODE = 19, CONS = 6.866 ) ICNODE( ADD, UTHETA, NODE = 20, CONS = 6.219 ) ICNODE( ADD, UTHETA, NODE = 21, CONS = 5.572 ) ICNODE( ADD, UTHETA, NODE = 22, CONS = 4.915 ) ICNODE( ADD, UTHETA, NODE = 23, CONS = 4.278 ) ICNODE( ADD, UTHETA, NODE = 24, CONS = 3.631 ) ICNODE( ADD, UTHETA, NODE = 25, CONS = 2.984 ) /Initial conditions for an in let flow rate of 2.524 x 107 m3/s /ICNODE( ADD, URC, ENTI = "inlet1", CONS = 34.723 ) /ICNODE( ADD, UZC, ENTI = "inlet1", CONS = 34.723 ) /ICNODE( ADD, URC, ENTI = "inlet2", CONS = 0 ) /ICNODE( ADD, UZC, ENTI = "inlet2", CONS = 49.106 ) /ICNODE( ADD, UTHE, NODE = 10, CONS = 25.377 ) /ICNODE( ADD, UTHE, NODE = 11, CONS = 24.083 ) /ICNODE( ADD, UTHE, NODE = 12, CONS = 22.789 ) /ICNODE( ADD, UTHE, NODE = 13, CONS = 21.495 ) /ICNODE( ADD, UTHE, NODE = 14, CONS = 20.201 ) /ICNODE( ADD, UTHE, NODE = 15, CONS = 18.907 ) /ICNODE( ADD, UTHE, NODE = 16, CONS = 17.613 ) /ICNODE( ADD, UTHE, NODE = 17, CONS = 16.319 )
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199Appendix IV: (Continued) /ICNODE( ADD, UTHE, NODE = 18, CONS = 15.025 ) /ICNODE( ADD, UTHE, NODE = 19, CONS = 13.731 ) /ICNODE( ADD, UTHE, NODE = 20, CONS = 12.437 ) /ICNODE( ADD, UTHE, NODE = 21, CONS = 11.143 ) /ICNODE( ADD, UTHE, NODE = 22, CONS = 9.849 ) /ICNODE( ADD, UTHE, NODE = 23, CONS = 8.555 ) /ICNODE( ADD, UTHE, NODE = 24, CONS = 7.261 ) /ICNODE( ADD, UTHE, NODE = 25, CONS = 5.967 ) /Boundary conditions for an inlet flow rate of 1.262 x 107 m3/s BCNODE( ADD, URC, ENTI = "i nlet1", CONS = 17.362 ) BCNODE( ADD, UZC, ENTI = "inlet1", CONS = 17.362 ) BCNODE( ADD, URC, ENTI = "inlet2", CONS = 0 ) BCNODE( ADD, UZC, ENTI = "inlet2", CONS = 24.553 ) BCNODE( ADD, VELO, ENTI = "wall1 ", CONS = 0, X, Y, Z ) BCNODE( ADD, VELO, ENTI = "wall2 ", CONS = 0, X, Y, Z ) BCNODE( ADD, URC, ENTI = "symmetry", CONS = 0 ) BCNODE( ADD, URC, ENTI = "top", CONS = 0 ) BCNODE( ADD, UZC, ENTI = "top", CONS = 0 ) BCNODE( SURFACE, ZERO, NODE = 146 ) BCNODE( SURFACE, ZERO, NODE = 205 ) BCNODE( COORDINATE, NODE = 205 ) BCNODE( COORDINATE, NODE = 170 ) BCNODE( ADD, UTHETA, NODE = 10, CONS = 12.688 ) BCNODE( ADD, UTHETA, NODE = 11, CONS = 12.041 ) BCNODE( ADD, UTHETA, NODE = 12, CONS = 11.394 ) BCNODE( ADD, UTHETA, NODE = 13, CONS = 10.747 ) BCNODE( ADD, UTHETA, NODE = 14, CONS = 10.100 ) BCNODE( ADD, UTHETA, NODE = 15, CONS = 9.453 ) BCNODE( ADD, UTHETA, NODE = 16, CONS = 8.806 ) BCNODE( ADD, UTHETA, NODE = 17, CONS = 8.159 ) BCNODE( ADD, UTHETA, NODE = 18, CONS = 7.512 ) BCNODE( ADD, UTHETA, NODE = 19, CONS = 6.866 ) BCNODE( ADD, UTHETA, NODE = 20, CONS = 6.219 ) BCNODE( ADD, UTHETA, NODE = 21, CONS = 5.572 ) BCNODE( ADD, UTHETA, NODE = 22, CONS = 4.915 ) BCNODE( ADD, UTHETA, NODE = 23, CONS = 4.278 ) BCNODE( ADD, UTHETA, NODE = 24, CONS = 3.631 ) BCNODE( ADD, UTHETA, NODE = 25, CONS = 2.984 ) /Boundary conditions for an inlet flow rate of 2.524 x 107 m3/s /BCNODE( ADD, URC, ENTI = "inlet1", CONS = 34.723 )
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200Appendix IV: (Continued) /BCNODE( ADD, UZC, ENTI = "inlet1", CONS = 34.723 ) /BCNODE( ADD, URC, ENTI = "inlet2", CONS = 0 ) /BCNODE( ADD, UZC, ENTI = "inlet2", CONS = 49.106 ) /BCNODE( ADD, VELO, ENTI = "wa ll1", CONS = 0, X, Y, Z ) /BCNODE( ADD, VELO, ENTI = "wa ll2", CONS = 0, X, Y, Z ) /BCNODE( ADD, URC, ENTI = "symmetry", CONS = 0 ) /BCNODE( ADD, URC, ENTI = "top", CONS = 0 ) /BCNODE( ADD, UZC, ENTI = "top", CONS = 0 ) /BCNODE( SURF, ZERO, NODE = 146 ) /BCNODE( SURF, ZERO, NODE = 205 ) /BCNODE( COOR, NODE = 205 ) /BCNODE( COOR, NODE = 170 ) /BCNODE( ADD, UTHE, NODE = 10, CONS = 25.377 ) /BCNODE( ADD, UTHE, NODE = 11, CONS = 24.083 ) /BCNODE( ADD, UTHE, NODE = 12, CONS = 22.789 ) /BCNODE( ADD, UTHE, NODE = 13, CONS = 21.495 ) /BCNODE( ADD, UTHE, NODE = 40, CONS = 20.201 ) /BCNODE( ADD, UTHE, NODE = 15, CONS = 18.907 ) /BCNODE( ADD, UTHE, NODE = 16, CONS = 17.613 ) /BCNODE( ADD, UTHE, NODE = 17, CONS = 16.319 ) /BCNODE( ADD, UTHE, NODE = 18, CONS = 15.025 ) /BCNODE( ADD, UTHE, NODE = 19, CONS = 13.731 ) /BCNODE( ADD, UTHE, NODE = 20, CONS = 12.437 ) /BCNODE( ADD, UTHE, NODE = 21, CONS = 11.143 ) /BCNODE( ADD, UTHE, NODE = 22, CONS = 9.849 ) /BCNODE( ADD, UTHE, NODE = 23, CONS = 8.555 ) /BCNODE( ADD, UTHE, NODE = 24, CONS = 7.261 ) /BCNODE( ADD, UTHE, NODE = 25, CONS = 5.967 ) END( )
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201 Appendix V: FIJOUR File for Small Nozzle (4.416 x 107 m3/s and 5.678 x 107 m3/s) FICONV( NEUT, INPU, RESU ) INPUTFILE( FILE = small.FDNEUT" ) OUTPUTFILE( FILE, DELE ) END( ) FIPREP( ) PROBLEM( ADD, CYLI, INCO, TRAN, TURB, NONL, NEWT, MOME, ISOT, FREE, SING ) EXECUTION( ADD, NEWJ ) ENTITY( ADD, NAME = "fluid ", FLUI, PROP = "fc77" ) ENTITY( ADD, NAME = "wall1", WALL ) ENTITY( ADD, NAME = "wall2", WALL ) ENTITY( ADD, NAME = "symmetry", PLOT ) ENTITY( ADD, NAME = "inlet1", PLOT ) ENTITY( ADD, NAME = "inlet2", PLOT ) ENTITY( ADD, NAME = "outlet", PLOT ) ENTITY( ADD, NAME = "top", PLOT ) ENTITY( ADD, NAME = "free", SURF, DEPT = 0, SPIN, STRA ) SOLUTION( ADD, N.R. = 70, KINE = 25, VELC = 0.0001, RESC = 0.01, SURF = 0.001 ) BODYFORCE( ADD, CONS, FZC = 981, FRC = 0, FTHE = 0 ) DATAPRINT( ADD, CONT ) OPTIONS( ADD, UPWI ) UPWINDING( ADD, STRE ) RELAXATION( ) 0.6, 0.6, 0.6, 0, 0, 0.1 PRESSURE( ADD, MIXE = 1e16, DISC ) DENSITY( ADD, SET = "fc77", CONS = 1.78 ) /Density for FC72 /DENSITY( ADD, SET = Â“f c72Â”, CONS = 1.68 ) VISCOSITY( ADD, SET = "fc 77", MIXL, CONS = 0.01424 ) /Viscosity for FC72 /VISCOSITY( ADD, SET = Â“fc72Â”, CONS = 0.0064 ) SURFACETENSION( ADD, SET = "fc77", CONS = 15 ) /Surface Tension for FC72 /SURFACETENSION( ADD, SET = Â“fc72Â”, CONS = 10 )
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202Appendix V: (Continued) POSTPROCESS( ADD, NBLO = 2 ) 1, 301, 300 302, 600, 1 PRINTOUT( NBLO = 2 ) 1, 301, 300 302, 600, 1 TIMEINTEGRATION( ADD, BACK, NSTE = 600, TSTA = 0, DT = 1e07, VARI, NOFI = 10, WIND = 0.9 ) RENUMBER( ADD, PROF ) EDDYVISCOSITY( ADD, SPEZ ) /Initial conditions for 4.416 x 107 m3/s ICNODE( ADD, URC, ENTI = "inlet1", CONS = 60.765 ) ICNODE( ADD, UZC, ENTI = "inlet1", CONS = 60.765 ) ICNODE( ADD, URC, ENTI = "inlet2", CONS = 0 ) ICNODE( ADD, UZC, ENTI = "inlet2", CONS = 85.935 ) ICNODE( ADD, UTHE, NODE = 10, CONS = 44.409 ) ICNODE( ADD, UTHE, NODE = 11, CONS = 42.145 ) ICNODE( ADD, UTHE, NODE = 12, CONS = 39.88 ) ICNODE( ADD, UTHE, NODE = 13, CONS = 37.616 ) ICNODE( ADD, UTHE, NODE = 14, CONS = 35.351 ) ICNODE( ADD, UTHE, NODE = 15, CONS = 33.087 ) ICNODE( ADD, UTHE, NODE = 16, CONS = 30.822 ) ICNODE( ADD, UTHE, NODE = 17, CONS = 28.558 ) ICNODE( ADD, UTHE, NODE = 18, CONS = 26.296 ) ICNODE( ADD, UTHE, NODE = 19, CONS = 24.029 ) ICNODE( ADD, UTHE, NODE = 20, CONS = 21.765 ) ICNODE( ADD, UTHE, NODE = 21, CONS = 19.5 ) ICNODE( ADD, UTHE, NODE = 22, CONS = 17.236 ) ICNODE( ADD, UTHE, NODE = 23, CONS = 14.972 ) ICNODE( ADD, UTHE, NODE = 24, CONS = 12.707 ) ICNODE( ADD, UTHE, NODE = 25, CONS = 10.443 ) /Initial conditions for an in let flow rate of 5.678 x 107 m3/s /ICNODE( ADD, URC, ENTI = "inlet1", CONS = 78.127 ) /ICNODE( ADD, UZC, ENTI = "inlet1", CONS = 78.127 ) /ICNODE( ADD, URC, ENTI = "inlet2", CONS = 0 ) /ICNODE( ADD, UZC, ENTI = "inlet2", CONS = 110.488 ) /ICNODE( ADD, UTHE, NODE = 10, CONS = 57.096 ) /ICNODE( ADD, UTHE, NODE = 11, CONS = 54.185 ) /ICNODE( ADD, UTHE, NODE = 12, CONS = 51.274 )
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203Appendix V: (Continued) /ICNODE( ADD, UTHE, NODE = 13, CONS = 48.362 ) /ICNODE( ADD, UTHE, NODE = 14, CONS = 45.451 ) /ICNODE( ADD, UTHE, NODE = 15, CONS = 42.54 ) /ICNODE( ADD, UTHE, NODE = 16, CONS = 39.628 ) /ICNODE( ADD, UTHE, NODE = 17, CONS = 36.717 ) /ICNODE( ADD, UTHE, NODE = 18, CONS = 33.806 ) /ICNODE( ADD, UTHE, NODE = 19, CONS = 30.894 ) /ICNODE( ADD, UTHE, NODE = 20, CONS = 27.983 ) /ICNODE( ADD, UTHE, NODE = 21, CONS = 25.072 ) /ICNODE( ADD, UTHE, NODE = 22, CONS = 22.16 ) /ICNODE( ADD, UTHE, NODE = 23, CONS = 19.249 ) /ICNODE( ADD, UTHE, NODE = 24, CONS = 16.338 ) /ICNODE( ADD, UTHE, NODE = 25, CONS = 13.426 ) /Boundary conditions for an inlet flow rate of 4.416 x 107 m3/s BCNODE( ADD, URC, ENTI = "i nlet1", CONS = 60.765 ) BCNODE( ADD, UZC, ENTI = "inlet1", CONS = 60.765 ) BCNODE( ADD, URC, ENTI = "inlet2", CONS = 0 ) BCNODE( ADD, UZC, ENTI = "inlet2", CONS = 85.935 ) BCNODE( ADD, VELO, ENTI = "wall1 ", CONS = 0, X, Y, Z ) BCNODE( ADD, VELO, ENTI = "wall2 ", CONS = 0, X, Y, Z ) BCNODE( ADD, URC, ENTI = "symmetry", CONS = 0 ) BCNODE( ADD, URC, ENTI = "top", CONS = 0 ) BCNODE( ADD, UZC, ENTI = "top", CONS = 0 ) BCNODE( SURF, CONS = 0, NODE = 146 ) BCNODE( SURF, CONS = 0, NODE = 205 ) BCNODE( COOR, NODE = 146 ) BCNODE( COOR, NODE = 170 ) BCNODE( ADD, UTHE, NODE = 10, CONS = 44.409 ) BCNODE( ADD, UTHE, NODE = 11, CONS = 42.145 ) BCNODE( ADD, UTHE, NODE = 12, CONS = 39.88 ) BCNODE( ADD, UTHE, NODE = 13, CONS = 37.616 ) BCNODE( ADD, UTHE, NODE = 14, CONS = 35.351 ) BCNODE( ADD, UTHE, NODE = 15, CONS = 33.087 ) BCNODE( ADD, UTHE, NODE = 16, CONS = 30.822 ) BCNODE( ADD, UTHE, NODE = 17, CONS = 28.558 ) BCNODE( ADD, UTHE, NODE = 18, CONS = 26.296 ) BCNODE( ADD, UTHE, NODE = 19, CONS = 24.029 ) BCNODE( ADD, UTHE, NODE = 20, CONS = 21.765 ) BCNODE( ADD, UTHE, NODE = 21, CONS = 19.5 ) BCNODE( ADD, UTHE, NODE = 22, CONS = 17.236 ) BCNODE( ADD, UTHE, NODE = 23, CONS = 14.972 )
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204Appendix V: (Continued) BCNODE( ADD, UTHE, NODE = 24, CONS = 12.707 ) BCNODE( ADD, UTHE, NODE = 25, CONS = 10.443 ) /Boundary conditions for an inlet flow rate of 5.678 x 107 m3/s /BCNODE( ADD, URC, ENTI = "inlet1", CONS = 78.127 ) /BCNODE( ADD, UZC, ENTI = "inlet1", CONS = 78.127 ) /BCNODE( ADD, URC, ENTI = "inlet2", CONS = 0 ) /BCNODE( ADD, UZC, ENTI = "inlet2", CONS = 110.488 ) /BCNODE( ADD, VELO, ENTI = "wa ll1", CONS = 0, X, Y, Z ) /BCNODE( ADD, VELO, ENTI = "wa ll2", CONS = 0, X, Y, Z ) /BCNODE( ADD, URC, ENTI = "symmetry", CONS = 0 ) /BCNODE( ADD, URC, ENTI = "top", CONS = 0 ) /BCNODE( ADD, UZC, ENTI = "top", CONS = 0 ) /BCNODE( SURF, CONS = 0, NODE = 146 ) /BCNODE( SURF, CONS = 0, NODE = 205 ) /BCNODE( COOR, NODE = 146 ) /BCNODE( COOR, NODE = 170 ) /BCNODE( ADD, UTHE, NODE = 10, CONS = 57.096 ) /BCNODE( ADD, UTHE, NODE = 11, CONS = 54.185 ) /BCNODE( ADD, UTHE, NODE = 12, CONS = 51.274 ) /BCNODE( ADD, UTHE, NODE = 13, CONS = 48.362 ) /BCNODE( ADD, UTHE, NODE = 14, CONS = 45.451 ) /BCNODE( ADD, UTHE, NODE = 15, CONS = 42.54 ) /BCNODE( ADD, UTHE, NODE = 16, CONS = 39.628 ) /BCNODE( ADD, UTHE, NODE = 17, CONS = 36.717 ) /BCNODE( ADD, UTHE, NODE = 18, CONS = 33.806 ) /BCNODE( ADD, UTHE, NODE = 19, CONS = 30.894 ) /BCNODE( ADD, UTHE, NODE = 20, CONS = 27.983 ) /BCNODE( ADD, UTHE, NODE = 21, CONS = 25.072 ) /BCNODE( ADD, UTHE, NODE = 22, CONS = 22.16 ) /BCNODE( ADD, UTHE, NODE = 23, CONS = 19.249 ) /BCNODE( ADD, UTHE, NODE = 24, CONS = 16.338 ) /BCNODE( ADD, UTHE, NODE = 25, CONS = 13.426 ) END( )
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205 Appendix VI: GAMBIT File for Varying Outer Slot Location solver select "FIDAP" undo begingroup vertex create coordinates 0 0 0 undo endgroup undo begingroup vertex create coordinates 0.17 0 0 undo endgroup undo begingroup vertex create coordinates 0.17 0.0125 0 undo endgroup undo begingroup vertex create coordinates 0.122 0.0125 0 undo endgroup undo begingroup vertex create coordinates 0.107 0.0125 0 undo endgroup undo begingroup vertex create coordinates 0 0.0743 0 undo endgroup undo begingroup vertex create coordinates 0 0.0304 0 /For R3 equal to 4.36 x 104 m /vertex create coordinates 0 0.0436 0 /For R3 equal to 5.77 x 104 m /vertex create coordinates 0 0.0577 0 /Omit this command for R3 equal to 7.43 x 104 m undo endgroup undo begingroup vertex create coordinates 0 0.025 0 /For R2 equal to 4.00 x 104 m /vertex create coordinates 0 0.04 0 /For R2 equal to 5.50 x 104 m /vertex create coordinates 0 0.055 0 /For R2 equal to 7.20 x 104 m /vertex create coordinates 0 0.072 0 undo endgroup undo begingroup vertex create coordinates 0 0.01 0 undo endgroup
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206Appendix VI: (Continued) edge create straight "vertex.1" "vertex.2" edge create straight "vertex.2" "vertex.3" edge create straight "vertex.3" "vertex.4" edge create straight "vertex.4" "vertex.5" edge create straight "vertex.5" "vertex.6" edge create straight "vertex.6" "vertex.7" edge create straight "vertex.7" "vertex.8" edge create straight "vertex.8" "vertex.9" edge create straight Â“vertex.9Â” Â“vertex.10Â” /Omit previous command for R2 equal to 7.20 x 104 m face create wireframe "edge.1" "edge.2" "e dge.3" "edge.4" "edge.5" "edge.6" "edge.7" "edge.8" Â“edge.9Â” real undo begingroup edge picklink "edge.1" edge mesh "edge.1" successi ve ratio1 1 intervals 172 undo endgroup undo begingroup edge picklink "edge.2" edge mesh "edge.2" successive ra tio1 1 ratio2 1.1 intervals 32 undo endgroup undo begingroup edge picklink "edge.3" edge mesh "edge.3" successi ve ratio1 1 intervals 36 undo endgroup undo begingroup edge picklink "edge.4" edge mesh "edge.4" successi ve ratio1 1 intervals 24 undo endgroup undo begingroup edge picklink "edge.5" edge mesh "edge.5" successi ve ratio1 1 intervals 112 undo endgroup undo begingroup edge picklink "edge.6" edge mesh "edge.6" successi ve ratio1 1 intervals 20 /For R2 equal to 4.00 x 104 m /edge mesh Â“edge.6Â” succe ssive ratio1 1 intervals 16 /For R2 equal to 5.50 x 104 m /edge mesh Â“edge.6Â” succe ssive ratio1 1 intervals 12
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207Appendix VI: (Continued) /For R2 equal to 7.20 x 104 m /edge mesh Â“edge.6Â” succe ssive ratio1 1 intervals 4 undo endgroup undo begingroup edge picklink "edge.7" edge mesh "edge.7" successi ve ratio1 1 intervals 4 /For R2 equal to 7.20 x 104 m /edge mesh Â“edge.8Â” succe ssive ratio1 1 intervals 24 undo endgroup undo begingroup edge picklink "edge.8" edge mesh "edge.8" successi ve ratio1 1 intervals 4 /For R2 equal to 4.00 x 104 m /edge mesh Â“edge.8Â” succe ssive ratio1 1 intervals 8 /For R2 equal to 5.50 x 104 m /edge mesh Â“edge.8Â” succe ssive ratio1 1 intervals 12 /For R2 equal to 7.20 x 104 m /edge mesh Â“edge.8Â” succe ssive ratio1 1 intervals 4 undo endgroup undo begingroup edge mesh Â“edge.9Â” successi ve ratio1 1 intervals 4 /Omit for R2 equal to 7.20 x 104 m undo endgroup face mesh "face.1" map size 1 physics create "symmetry" btype "PLOT" edge "edge.1" physics create "outlet" btype "PLOT" edge "edge.2" physics create "free" btype "SURFACE" edge "edge.3" physics create "wall1" btype "PLOT" edge "edge.4" physics create "wall2" btype "PLOT" edge "edge.5" physics create "inlet1" btype "PLOT" edge "edge.6" physics create "top" btyp e "PLOT" edge "edge.7" physics create "inlet2" btype "PLOT" edge "edge.8" physics create "fluid" ct ype "FLUID" face "face.1" export fidap Â“025.FDNEUTÂ” /For R2 equal to 4.00x 104 m /export fidap Â“04.FDNEUTÂ” /For R2 equal to 5.50 x 104 m /export fidap Â“055.FDNEUTÂ” /For R2 equal to 7.20 x 104 m /export fidap Â“072.FDNEUTÂ”
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208 Appendix VII: FIJOUR File for Varying Outer Slot Location FICONV( NEUT, INPU, RESU ) INPUTFILE( FILE = small.FDNEUT" ) OUTPUTFILE( FILE, DELE ) END( ) FIPREP( ) PROBLEM( ADD, CYLI, INCO, TRAN, TURB, NONL, NEWT, MOME, ISOT, FREE, SING ) EXECUTION( ADD, NEWJ ) ENTITY( ADD, NAME = "fluid ", FLUI, PROP = "fc77" ) ENTITY( ADD, NAME = "wall1", WALL ) ENTITY( ADD, NAME = "wall2", WALL ) ENTITY( ADD, NAME = "symmetry", PLOT ) ENTITY( ADD, NAME = "inlet1", PLOT ) ENTITY( ADD, NAME = "inlet2", PLOT ) ENTITY( ADD, NAME = "outlet", PLOT ) ENTITY( ADD, NAME = "top1", WALL ) ENTITY( ADD, NAME = "top2", WALL ) ENTITY( ADD, NAME = "free", SURF, DEPT = 0, SPIN, STRA ) SOLUTION( ADD, N.R. = 70, KINE = 25, VELC = 0.0001, RESC = 0.01, SURF = 0.001 ) BODYFORCE( ADD, CONS, FZC = 981, FRC = 0, FTHE = 0 ) DATAPRINT( ADD, CONT ) OPTIONS( ADD, UPWI ) UPWINDING( ADD, STRE ) RELAXATION( ) 0.6, 0.6, 0.6, 0, 0, 0.1 PRESSURE( ADD, MIXE = 1e16, DISC ) DENSITY( ADD, SET = "fc77", CONS = 1.78 ) /Density for FC72 /DENSITY( ADD, SET = Â“f c72Â”, CONS = 1.68 ) /Density for FC87 /DENSITY( ADD, SET = Â“f c87Â”, CONS = 1.63 ) /Density for Methanol /DENSITY( ADD, SET = Â“met hanolÂ”, CONS = 0.7855 ) VISCOSITY( ADD, SET = "fc 77", MIXL, CONS = 0.01424 )
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209Appendix VII: (Continued) /Viscosity for FC72 /VISCOSITY( ADD, SET = Â“fc72Â”, MIXL, CONS = 0.0064 ) /Viscosity for FC87 /VISCOSITY( ADD, SET = Â“fc87Â”, MIXL, CONS = 0.00453 ) /Viscosity for Methanol /VISCOSITY( ADD, SET = Â“methanolÂ”, MIXL, CONS = 0.0055 ) SURFACETENSION( ADD, SET = "fc77", CONS = 15 ) /Surface Tension for FC72 /SURFACETENSION( ADD, SET = Â“fc72Â”, CONS = 10 ) /Surface Tension for FC87 /SURFACETENSION( ADD, SET = Â“fc87Â”, CONS = 9.5 ) /Surface Tension for Methanol /SURFACETENSION( ADD, SET = Â“methanolÂ”, CONS = 22.2 ) POSTPROCESS( ADD, NBLO = 2 ) 1, 351, 350 352, 4000, 1 PRINTOUT( NBLO = 2 ) 1, 351, 350 352, 4000, 1 TIMEINTEGRATION( ADD, BACK, NSTE = 4000, TSTA = 0, DT = 1e07, VARI, NOFI = 10,WIND = 0.9 ) RENUMBER( ADD, PROF ) EDDYVISCOSITY( ADD, SPEZ ) /Initial conditions for an in let flow rate of 1.262 x 107 m3/s ICNODE( ADD, UZC, ENTI = "inlet1", CONS = 351.439 ) ICNODE( ADD, UTHE, ENTI = "inlet1", CONS = 351.439 ) ICNODE( ADD, UZC, ENTI = "inlet2", CONS = 351.439 ) /Inlet conditions for an in let flow rate of 5.678 x 107 m3/s /ICNODE( ADD, UZC, ENTI = "inlet1", CONS = 451.85 ) /ICNODE( ADD, UTHE, ENTI = "inlet1", CONS = 451.85 ) /ICNODE( ADD, UZC, ENTI = "inlet2", CONS = 451.85 ) /Boundary conditions for an inlet flow rate of 1.262 x 107 m3/s BCNODE( ADD, UZC, ENTI = "inlet1", CONS = 351.439 ) BCNODE( ADD, UTHE, ENTI = "inlet1", CONS = 351.439 ) BCNODE( ADD, UZC, ENTI = "inlet2", CONS = 351.439 ) BCNODE( ADD, VELO, ENTI = "wall1 ", CONS = 0, X, Y, Z ) BCNODE( ADD, VELO, ENTI = "wall2 ", CONS = 0, X, Y, Z ) BCNODE( ADD, VELO, ENTI = "top1" CONS = 0, X, Y, Z )
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210Appendix VII: (Continued) BCNODE( ADD, VELO, ENTI = "top2" CONS = 0, X, Y, Z ) BCNODE( ADD, URC, ENTI = "symmetry", CONS = 0 ) BCNODE( SURF, CONS = 0, NODE = 205 ) BCNODE( SURF, CONS = 0, NODE = 146 ) BCNODE( COOR, NODE = 146 ) BCNODE( COOR, NODE = 170 ) /Boundary conditions for an inlet flow rate of 5.678 x 107 m3/s /BCNODE( ADD, UZC, ENTI = "inlet1", CONS = 451.85 ) /BCNODE( ADD, UTHE, ENTI = "inlet1", CONS = 451.85) /BCNODE( ADD, UZC, ENTI = "inlet2", CONS = 451.85 ) /BCNODE( ADD, VELO, ENTI = "wa ll1", CONS = 0, X, Y, Z ) /BCNODE( ADD, VELO, ENTI = "wa ll2", CONS = 0, X, Y, Z ) /BCNODE( ADD, VELO, ENTI = "t op1", CONS = 0, X, Y, Z ) /BCNODE( ADD, VELO, ENTI = "t op2", CONS = 0, X, Y, Z ) /BCNODE( ADD, URC, ENTI = "symmetry", CONS = 0 ) /BCNODE( SURF, CONS = 0, NODE = 205 ) /BCNODE( SURF, CONS = 0, NODE = 146 ) /BCNODE( COOR, NODE = 146 ) /BCNODE( COOR, NODE = 170 ) END( )
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211 Appendix VIII Â– Fluid Properties FC77 Density: 1780 kg/m3 Viscosity: 0.001424 kg/m s Surface Tension: 0.015 N/m FC72 Density: 1680 kg/m3 Viscosity: 0.00064 kg/m s Surface Tension: 0.010 N/m FC87 Density: 1630 kg/m3 Viscosity: 0.000453 kg/m s Surface Tension: 0.0095 N/m Methanol Density: 785.5 kg/m3 Viscosity: 0.00055 kg/m s Surface Tension: 0.0222 N/m
