Graduate Theses and Dissertations

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1 Univesity of South Floida Schola Commons Gaduate Theses and Dissetations Gaduate School 009 Numeical heat tansfe duing patially-confined, confined, and fee liquid jet impingement with otation and chemical mechanical planaiation pocess modeling Joge C. Lallave-Cotes Univesity of South Floida Follow this and additional woks at: Pat of the Ameican Studies Commons Schola Commons Citation Lallave-Cotes, Joge C., "Numeical heat tansfe duing patially-confined, confined, and fee liquid jet impingement with otation and chemical mechanical planaiation pocess modeling" (009). Gaduate Theses and Dissetations. This Dissetation is bought to you fo fee and open access by the Gaduate School at Schola Commons. It has been accepted fo inclusion in Gaduate Theses and Dissetations by an authoied administato of Schola Commons. Fo moe infomation, please contact

2 Numeical Heat Tansfe Duing Patially confined, Confined, and Fee Liquid Jet Impingement with Rotation and Chemical Mechanical Planaiation Pocess Modeling by Joge C. Lallave Cotes A dissetation submitted in patial fulfillment of the equiements fo the degee of Docto of Philosophy Depatment of Mechanical Engineeing College of Engineeing Univesity of South Floida Co Majo Pofesso: Muhammad M. Rahman, Ph.D. Co Majo Pofesso: Ashok Kuma, Ph.D. Aydin K. Sunol, Ph.D. Caig P. Lusk, Ph.D. Gaett Matthews, Ph.D. Date of Appoval: Mach 17, 009 Keywods: steady state, tansient analysis, conjugate heat tansfe with spinning boundaies, sluy abasive paticles, and CMP tempeatue distibutions Copyight 009, Joge C. Lallave Cotes

3 Note to Reade The oiginal of this document contains colo that is necessay fo undestanding the data. The oiginal dissetation is on file with the USF libay in Tampa, Floida.

4 Dedication To my wife Milda, and my childen Lyann and Chistian fo thei unconditional love. To my paents Joge Lallave, S. and Camen Cotes and siste Lynnette Lallave fo encouaging me to pusue a pofessional caee with high standads and always un the exta mile. To my mothe in law Povidencia and fathe in law Luciano, fo thei suppot duing all these yeas. I m thankful and blessed fo having a family that believes in me.

5 Acknowledgements Fistly, I would like to thank my majo pofesso, D. Muhammad Mustafiu Rahman, fo giving me an oppotunity to pusue this wok unde his guidance and suppot thoughout my doctoal studies. I will like to thank D. Tapas Das, D. Ashok Kuma, D. Geoffey Okogbaa, and D. Giselle Centeno fo thei guidance and pofessional encouagement as pat of the STARS Pogam. I am gateful fo the financial assistance of the National Science Foundation though the GK 1 STARS Fellowship (Gant # ). I thank D. Ashok Kuma, D. Aydin K. Sunol, D. Amy L. Stuat and D. Caig Lusk, fo being membes of my dissetation committee and fo thei valuable suggestions to impove this wok. Many thanks also to my USF mentos, D. Abdul Lateef Gai and Cesa Henande, fo thei inspiation and achievements as pat of D. Muhammad M. Rahman s eseach goup. I would like to thank D. Son Hong Ho, whose guidance in the aeas of FEM and softwae modeling wee cucial fo completion of this dissetation. Special thanks go to all my fiends of the STARS pogam fo thei suppot and encouagement which helped me to fulfill this wok. I am extemely gateful to God fo all the oppotunities He povides thoughout my life.

6 Table of Contents List of Tables List of Figues List of Symbols Abstact iv v xvii xxi Chapte 1 Intoduction Configuation of Impinging Jets 4 1. Impinging Jet Chaacteistics and Zones Fee Suface Jets Confined Submeged Jets Patially confined Jets Chemical Mechanical Polishing Pocess Set up Expeimental Testing Set Up Poblem Unde Study Objectives 16 Chapte Liteatue Review 18.1 Fee Suface Jet Impingement 18. Jet Impingement with Spinning Boundaies 1.3 Tansient Jet Impingement 5.4 Confined Submeged Jet Impingement 7.5 Patially confined Jet Impingement 31.6 Chemical Mechanical Polishing Pocess 33 Chapte 3 Mathematical Models and Computation Fee Liquid Jet Impingement Model Govening Equations and Bounday Conditions: Steady State Cooling of Spinning Taget Govening Equations and Bounday Conditions: Tansient Cooling of Spinning Taget Confined Liquid Jet Impingement Model Govening Equations and Bounday Conditions: Steady State Cooling of Spinning Taget Govening Equations and Bounday Conditions: Steady State Cooling of Spinning Wall 48 i

7 3.3 Patially confined Liquid Jet Impingement Model Govening Equations and Bounday Conditions: Steady State Cooling of Spinning Taget Govening Equations and Bounday Conditions: Steady State Cooling of Co Rotating Taget and Confined Wall Govening Equations and Bounday Conditions: Tansient Cooling of Spinning Taget Thee Dimensional Chemical Mechanical Polishing Model Govening Equations and Bounday Conditions: Steady State Govening Equations and Bounday Conditions: Tansient Numeical Computation Fee Suface Liquid Jet Impingement Steady State and Tansient Pocess Confined Submeged Liquid Jet Impingement Steady State Pocess Stationay Confined Wall with Spinning Taget Spinning Confined Wall with Stationay Taget Patially confined Submeged Liquid Jet Impingement Steady State and Tansient Pocess Chemical Mechanical Polishing Steady State and Tansient Pocess Mesh Independence and Time Step Study Fee Liquid Jet Impingement Model Confined Liquid Jet Impingement Model Stationay Confined Wall with Spinning Taget Spinning Confined Wall with Stationay Taget Patially confined Liquid Jet Impingement Model Stationay Confined Wall with Spinning Taget Co Rotating Taget and Confined Wall Chemical Mechanical Polishing Model 86 Chapte 4 Fee Liquid Jet Impingement Model Results Steady State Cooling of Spinning Taget Tansient Cooling of Spinning Taget 106 Chapte 5 Confined Liquid Jet Impingement Model Results Steady State: Stationay Confined Wall with Spinning Taget 1 5. Steady State: Spinning Confined Wall with Stationay Taget 133 Chapte 6 Patially confined Liquid Jet Impingement Model Results Steady State Cooling of Spinning Taget Steady State Cooling of Spinning Confined Wall and Taget Tansient Cooling of Spinning Taget 179 Chapte 7 Chemical Mechanical Polishing Model Results Steady State Pocess Tansient Pocess 10 ii

8 Chapte 8 Conclusions and Recommendations Fee Liquid Jet Impingement Steady State Cooling of Spinning Taget Tansient Cooling of Spinning Taget Confined Liquid Jet Impingement Steady State Cooling of Stationay Confined Wall with Spinning Taget Steady State Cooling of Spinning Confined Wall with Stationay Taget Patially confined Liquid Jet Impingement Steady State Cooling of Spinning Taget Steady State Cooling of Spinning Confined Wall and Taget Tansient Cooling of Spinning Taget Chemical Mechanical Polishing Model Steady State and Tansient Pocess Futue Wok and Recommendations 47 Refeences 49 Appendices 6 Appendix A: CFD Fee Liquid Jet Impingement 63 Appendix B: CFD Confined Liquid Jet Impingement 71 Appendix C: CFD Patially confined Liquid Jet Impingement 77 Appendix D: CFD Chemical Mechanical Polishing Model 86 Appendix E: MATLAB Pogams fo 3 D Solution isualiation 9 Appendix F: Gid Convegence Index Analysis Sample 94 About the Autho End Page iii

9 List of Tables Table 1.1 CETR univesal tibomete specifications. 10 Table 1. Summay of poblems unde study. 15 Table 3.1 Constant themo physical popeties used fo computational analysis. 40 Table 3. Gid convegence study of figue Table 3.3 Gid convegence study of figue Table 3.4 Gid convegence study of figue Table 3.5 Gid convegence study of figue Table 3.6 Gid convegence study of figue Table 6.1 Local Nusselt numbe compaison between expeimental data of Oa et al. [44, 45] and pesent numeical esults (T j =93 K, q w =3 kw/m, b= m, Re=38, H n = m, p / d =0.5, b/d n =0.15, hin = p = m). 191 Table 7.1 Table 7. Table F.1 Aveage substate and pad heat tansfe convection coefficients and expeimental data of Boucki et al. [17] unde diffeent CMP paametes and vaiable input heat flux along the sufaces. 10 Aveage substate and pad heat tansfe convection coefficients unde diffeent CMP paametes and vaiable input heat flux at specific adial locations along the sufaces. 37 Tempeatue esults of figue 3.13 at a dimensionless distance of / n =8. 94 iv

10 List of Figues Figue 1.1 (a) Fee liquid jet impingement, (b) Confined liquid jet impingement, and (c) Patially confined liquid jet impingement. 4 Figue 1. Schematic side view epesentation of a CMP pocess. 9 Figue 1.3 Thee dimensional (3 D) CMP schematic. 14 Figue 3.1 Figue 3. Figue 3.3 Figue 3.4 Figue 3.5 Thee dimensional schematic of axis symmetic fee liquid jet impingement on a unifomly heated spinning disk. 39 Thee dimensional schematic of a confined axial jet impinging on a unifomly heated and spinning disk. 46 Thee dimensional schematic of axis symmetic confined spinning disk liquid jet impingement on a unifomly heated disk. 49 Thee dimensional schematic of axis symmetic patially confined liquid jet impingement on a unifomly heated spinning disk. 51 Thee dimensional schematic of axis symmetic patially confined liquid jet impingement on a unifomly heated disk with two spinning boundaies. 54 Figue 3.6 Thee dimensional CMP contol volume outline. 57 Figue 3.7 Wafe pad elative velocity pofile. 61 Figue 3.8 Axis symmetic fee suface liquid jet impingement mesh plot. 64 Figue 3.9 Axis symmetic confined liquid jet impingement mesh plot. 68 Figue 3.10 Axis symmetic patially confined jet impingement mesh plot. 70 Figue 3.11 Solid fluid inteface tempeatue fo diffeent numbe of elements in and diections (Re=1,500, b=0, d n =1. mm, Ek=.65x10 4, β=.67). 77 v

11 Figue 3.1 Figue 3.13 Figue 3.14 Figue 3.15 Figue 3.16 Figue 3.17 Figue 3.18 Figue 3.19 Figue 4.1 Figue 4. Figue 4.3 Time step independence study fo maximum dimensionless inteface tempeatue vaiation at diffeent time steps (Re=550, silicon disk, wate, b/d n =0.5, Ek=.65x10 4, q w =15 kw/m, β=.67). 79 Local dimensionless inteface tempeatue fo diffeent numbe of elements in and diections at constant flow ate (Re=1,500, Q=7.08x10 m 3 /s, b=0, Ek=.65x10 4, q w =50 kw/m, H n /d n = 5.33, Ω=15 RPM, H n =0.3 cm). 80 Dimensionless inteface tempeatue distibutions fo diffeent numbe of elements in and diections (Re=1,000, b=0.3 mm, d n =0.1 mm, Ek=1.06x10 3, β=.0). 81 Dimensionless inteface tempeatue distibutions fo diffeent numbe of elements in and diections (Re=750, p / d =0.667, b/d n =0.5, Ek=4.5x10-4, β=0.5). 83 Maximum dimensionless inteface tempeatue vaiation fo diffeent time steps with wate as the cooling fluid (Re=5, Ek=.13x10 4, β=0.5, silicon disk, b/d n =0.5, and p / d =0.667). 84 Dimensionless inteface tempeatue distibutions fo diffeent numbe of elements in and diections (Re=750, b/d n =0.5, Ek 1, =4.5x10 4, p / d =0.667, β=0.5). 85 Tempeatue distibution acoss the sluy egion beneath the substate suface fo vaious numbe of elements (Q sl =65 cc/min, Ω w =15 RPM, Ω p =150 RPM, COF=0.4, δ sl =50 µm, P=4.35 kpa, w =1.9 cm, q sl =7.4 to 10.1 kw/m ). 87 Gid topology of contol volume that includes the wafe, alumina sluy, and polishing pad. 87 elocity vecto distibution fo jet impingement on a silicon wafe with wate as the cooling fluid (Re=900, Ek=.65x10 4, β=.67, b/d n =0.5). 89 Fee suface height distibution fo diffeent Reynolds numbes with wate as the cooling fluid (Ek=.65x10 4, β=.67, b/d n =0.5). 90 Dimensionless inteface tempeatue and local Nusselt numbe distibutions fo a silicon wafe with wate as the cooling fluid fo diffeent Reynolds numbes (Ek=.65x10 4, β=.67, b/d n =0.5). 91 vi

12 Figue 4.4 Figue 4.5 Figue 4.6 Figue 4.7 Figue 4.8 Figue 4.9 Figue 4.10 Figue 4.11 Figue 4.1 Figue 4.13 Figue 4.14 Aveage Nusselt numbe and heat tansfe coefficient vaiations with Reynolds numbe fo a silicon wafe with wate as the cooling fluid (β=.67, b/d n =0.5). 9 Dimensionless inteface tempeatue and local Nusselt numbe distibutions fo a silicon wafe with wate as the cooling fluid at diffeent Ekman numbes (Re=1,500, β=.67, b/d n =0.5). 93 Aveage Nusselt numbe and heat tansfe coefficient vaiations with Ekman numbe fo a silicon wafe with wate as the cooling fluid (β=.67, b/d n =0.5). 94 Dimensionless inteface tempeatue and local Nusselt numbe distibutions fo diffeent wafe thicknesses with wate as the cooling fluid (Re=1,000, Ek=.65x10 4, β=.67). 95 Local Nusselt numbe distibution fo diffeent nole diametes fo a silicon wafe with wate as the cooling fluid (Re=1,000, Ek=6.6x10 5, β=.67, b/d n =0.5). 96 Dimensionless inteface tempeatue and local Nusselt numbe distibutions fo a silicon disk with wate as the cooling fluid fo diffeent nole to taget spacing (Re=750, Ek=.65x10 4, b/d n =0.5). 97 Local Nusselt numbe and dimensionless inteface tempeatue vaiations fo diffeent cooling fluids fo silicon as the disk mateial (Re=750, Ω=15 RPM, β=.67, b/d n =0.5). 98 Local Nusselt numbe and dimensionless inteface tempeatue vaiations fo diffeent solid mateials with wate as the cooling fluid (Re=1,500, Ek=.1x10 5, β=.67, b/d n =0.5). 99 Maximum to minimum tempeatue diffeence and maximum solid fluid inteface tempeatue (Re=1,500, d n =0.1 cm, β=.67, Ek=.1x10 5 ). 101 Aveage Nusselt numbe and heat tansfe coefficient vaiations with disk thickness (Re=1,500, Ek=.1x10 5 ). 10 Compaison of height of the fee suface with analytical pedictions of Watson [4] and expeimental data of Stevens and Webb [16] (Re=1,500, Ek=, b/d n =0.5). 103 vii

13 Figue 4.15 Local Nusselt numbe compaison with Liu and Lienhad [10] unde diffeent Ekman numbes (Re=1,500, β=.67, b/d n =0.5). 104 Figue 4.16 Figue 4.17 Figue 4.18 Figue 4.19 Figue 4.0 Figue 4.1 Figue 4. Figue 4.3 Figue 4.4 Figue 4.5 Figue 4.6 Compaison of pedicted aveage Nusselt numbes of equation 4.1 with pesent numeical data. 105 Dimensionless inteface tempeatue distibutions fo diffeent Fouie numbes (Re=500, Ek=.65x10 4, β=.67, silicon disk, wate, b/d n =0.5, q w =15 kw/m ). 107 Local Nusselt numbe distibutions fo diffeent Fouie numbes (Re=500, Ek=.65x10 4, β=.67, silicon disk, wate, b/d n =0.5, q w =15 kw/m ). 108 Dimensionless maximum tempeatue vaiations fo diffeent Reynolds numbes (Ek=.65x10 4, β=.67, silicon disk, wate, b/d n =0.5, q w =15 kw/m ). 109 Aveage Nusselt numbe vaiations fo diffeent Reynolds numbes (Ek=.65x10 4, β=.67, silicon disk, wate, b/d n =0.5, q w =15 kw/m ). 110 Dimensionless maximum tempeatue vaiations fo diffeent Ekman numbes (Re=750, β=.67, silicon disk, wate, b/d n =0.5, q w =15 kw/m ). 111 Aveage Nusselt numbe vaiations fo diffeent Ekman numbes (Re=750, β=.67, silicon disk, wate, b/d n =0.5, q w =15 kw/m ). 11 Dimensionless maximum tempeatue vaiations fo diffeent dimensionless disk thicknesses (Re=1,100, Ek=1.0x10 4, β=.67, silicon disk, wate, and q w =15 kw/m ). 113 Aveage Nusselt numbe vaiations fo diffeent dimensionless disk thicknesses (Re=1,100, Ek=1.0x10 4, β=.67, silicon disk, wate, and q w =15 kw/m ). 114 Dimensionless maximum tempeatue vaiations fo diffeent solid mateials (Re=650, Ek=.65x10 4, b/d n =0.5, β=.67, wate, and q w =15 kw/m ). 115 Aveage Nusselt numbe vaiations fo diffeent solid mateials (Re=650, Ek=.65x10 4, b/d n =0.5, β=.67, wate, and q w =15 kw/m ). 116 viii

14 Figue 4.7 Figue 4.8 Time equied to each steady state unde the effects of vaious mateial popeties and disk thickness (Re=1,100, Ek=1.0x10 4, β=.67, wate, and q w =15 kw/m ). 117 Time equied to each steady state unde the effects of diffeent Reynolds numbe (β=.67, wate, Ek=.65x10 4, silicon disk, b/d n =0.5, and q w =15 kw/m ). 118 Figue 4.9 Isothemal lines at diffeent instants (Re=1,100, Ek=1.0x10 4, β=.67, silicon disk, wate, q w =15 kw/m, and b/d n =0.5). 119 Figue 4.30 Isothemal lines at diffeent instants (Re=1,100, Ek=1.0x10 4, β=.67, silicon disk, wate, and q w =15 kw/m, and b/d n =1.67). 119 Figue 4.31 Figue 5.1 Figue 5. Compaison of pedicted aveage Nusselt numbe of equation 4. with pesent numeical data. 10 elocity vecto distibution fo a confined jet impingement on a silicon wafe with wate as the cooling fluid (Re=1,000, β=1.5, Ek=1.4x10 4, b/d n =0.5). 1 Local Nusselt numbe and dimensionless inteface tempeatue distibution fo a silicon wafe at diffeent Re, and wate as the cooling fluid (b=0.3 mm, H n =0.3 cm, Ek=.65x10 4, and q w =50 kw/m ). 14 Figue 5.3 Aveage Nusselt numbe and heat tansfe coefficient compaison fo diffeent Reynolds numbe at low, intemediate and high Ekman numbes (q w =50 kw/m, H n =0.3 cm). 14 Figue 5.4 Figue 5.5 Figue 5.6 Local Nusselt numbe and dimensionless inteface tempeatue plots fo a silicon wafe at diffeent Ekman numbes and wate as the cooling fluid (Re=750, Q=3.54x10 m 3 /s, b=0.3 mm, H n /d n =5.333, and q w =50 kw/m ). 16 Inteface tempeatue fo diffeent cooling fluids (Re=750, Q=3.54x10 m 3 /s, Ω=15 RPM, b=0.3 mm, H n /d n =.67, and q w =50 kw/m ). 17 Local Nusselt numbe fo diffeent cooling fluids (Re=750, Q=3.54x10 m 3 /s, Ω =15 RPM, b=0.3 mm, H n /d n =.67, and q w =50 kw/m ). 17 ix

15 Figue 5.7 Figue 5.8 Figue 5.9 Figue 5.10 Figue 5.11 Figue 5.1 Figue 5.13 Figue 5.14 Figue 5.15 Figue 5.16 Figue 5.17 Figue 5.18 Local Nusselt numbe and dimensionless inteface tempeatue distibutions fo diffeent solid mateials with wate as the cooling fluid (Re=1,500, Q=7.08x10 m 3 /s, q w =50 kw/m, Ω=15 RPM, H n =0.3 cm, q w =50 kw/m ). 18 Aveage Nusselt numbe coelation esults fo vaious studied paametes. 130 Local Nusselt numbe distibution fo a silve disk with FC 77 as the cooling fluid (H n /d n =4, q w =50 kw/m ). 131 Aveage Nusselt numbe coelation fo vaious Reynolds numbes and Ekman numbe and thee diffeent P values of liquid oil axis symmetic jet impingement. 13 Local Nusselt numbe and dimensionless inteface tempeatue distibutions fo a silicon disk with wate as the cooling fluid fo diffeent Reynolds numbes (Ek=4.5x10 4, β=.0, b/d n =0.5). 134 Aveage Nusselt numbe vaiations with Reynolds numbe at diffeent Ekman numbes fo a silicon disk with wate as the cooling fluid (β=.0, b/d n =0.5). 136 Local Nusselt numbe distibutions fo a silicon disk with wate as the cooling fluid at diffeent Ekman numbes (Re=750, β=3.0, b/d n =0.5). 137 Dimensionless inteface tempeatue distibutions fo a silicon disk with wate as the cooling fluid at diffeent Ekman numbes (Re=750, β=3.0, b/d n =0.5). 138 Dimensionless inteface tempeatue distibutions fo diffeent silicon disk thicknesses with wate as the cooling fluid (Re=1,500, Ek=1.5x10 4, β=.0). 139 Local Nusselt numbe distibutions fo diffeent silicon disk thicknesses with wate as the cooling fluid (Re=1,500, Ek=1.5x10 4, β=.0). 140 Dimensionless inteface tempeatue distibutions fo a silicon disk with wate as the cooling fluid fo diffeent nole to taget spacings (Re=750, Ek=4.5x10 4, b/d n =0.5). 141 Local Nusselt numbe distibutions fo a silicon disk with wate as the cooling fluid fo diffeent nole to taget spacings (Re=750, Ek=4.5x10 4, b/d n =0.5). 141 x

16 Figue 5.19 Figue 5.0 Figue 5.1 Figue 5. Dimensionless inteface tempeatue distibutions fo diffeent cooling fluids with silicon as the disk mateial (Re=1,000, β=.0, b/d n =0.5). 14 Local Nusselt numbe distibutions fo diffeent cooling fluids with silicon as the disk mateial (Re=1,000, β=.0, b/d n =0.5). 143 Local Nusselt numbe and dimensionless inteface tempeatue distibutions fo diffeent solid mateials with wate as the cooling fluid (Re=1,000, Ek=4.5x10 4, β=.0, b/d n =0.5). 144 Compaison of pedicted aveage Nusselt numbe of equation 5. with pesent numeical data. 146 Figue 5.3 Stagnation Nusselt numbe compaison of Scholt and Tass [6], Nakoyakov et al. [68], and Liu et al. [17] with actual numeical esults unde diffeent Reynolds and Ekman numbes (d n =1. mm, b=0.3 mm). 148 Figue 6.1 Figue 6. Figue 6.3 Figue 6.4 Figue 6.5 Figue 6.6 elocity vecto distibution fo a patially confined jet impingement on a silicon disk with wate as the cooling fluid (Re=475, Ek=4.5x10 4, p / d =0.5, β=0.5, b/d n =0.5). 149 Fee suface height distibution fo diffeent plate to disk confinement atio with wate as the cooling fluid (Re=450, Ek=4.5x10 4, β=0.5, b/d n =0.5). 151 Local Nusselt numbe and dimensionless inteface tempeatue distibutions fo a silicon disk with wate as the cooling fluid fo diffeent Reynolds numbes (Ek=4.5x10 4, β=0.5, b/d n =0.5, p / d =0.667). 15 Aveage Nusselt numbe vaiations with Reynolds numbe at diffeent Ekman numbes fo a silicon disk with wate as the cooling fluid (β=0.5, b/d n =0.5, p / d =0.667). 153 Local Nusselt numbe and dimensionless inteface tempeatue distibutions fo a silicon disk with wate as the cooling fluid at diffeent Ekman numbes (Re=540, β=0.5, b/d n =0.5, and p / d =0.667). 154 Local Nusselt numbe and dimensionless inteface tempeatue distibutions fo diffeent silicon disk thicknesses with wate as the cooling fluid (Re=450, Ek=4.5x10 4, β=0.5, p / d =0.667). 155 xi

17 Figue 6.7 Figue 6.8 Figue 6.9 Figue 6.10 Figue 6.11 Figue 6.1 Figue 6.13 Figue 6.14 Figue 6.15 Figue 6.16 Local Nusselt numbe and dimensionless inteface tempeatue distibutions fo a silicon disk with wate as the cooling fluid fo thee diffeent nole to taget spacing atio (Re=750, Ek=4.5x10 4, b/d n =0.5, p / d =0.667). 156 Local Nusselt numbe and dimensionless inteface tempeatue distibutions fo diffeent cooling fluids fo silicon as the disk mateial (Re=750, β=0.5, b/d n =0.5, p / d =0.667). 157 Local Nusselt numbe and dimensionless inteface tempeatue distibutions fo diffeent solid mateials with wate as the cooling fluid (Re=875, Ek=4.5x10 4, β=0.5, b/d n =0.5, and p / d =0.667). 158 Local Nusselt numbe and dimensionless inteface tempeatue distibutions fo diffeent plate to disk confinement atio (Re=450, Ek=4.5x10 4, β=0.5, b/d n =0.5). 159 Compaison of pedicted aveage Nusselt numbes of equation 6.1 with pesent numeical data. 160 Effects of Reynolds numbe on local Nusselt numbe and dimensionless solid fluid inteface tempeatue vaiation fo a silicon disk with wate as the cooling fluid (β=0.5, b/d n =0.5, p / d =0.667, Ek 1, =1.93x10 4 ). 16 Effects of Reynolds numbe on aveage Nusselt numbe at diffeent Ekman numbes fo a silicon disk with wate as the cooling fluid (β=0.5, b/d n =0.5, p / d =0.667, Ek =4.5x10 4 ). 164 Effects of Ek 1 vaiation on local Nusselt numbe and dimensionless inteface tempeatue distibutions fo a silicon disk with wate as the cooling fluid (Re=540, β=0.5, b/d n =0.5, p / d =0.667, Ek =4.5x10 4 ). 165 Effects of Ek vaiation on local Nusselt numbe and dimensionless inteface tempeatue distibutions fo a silicon disk with wate as the cooling fluid (Re=540, β=0.5, b/d n =0.5, p / d =0.667, Ek 1 =4.5x10 4 ). 167 Effects of thickness vaiation on local Nusselt numbe and dimensionless inteface tempeatue distibutions fo a silicon disk with wate as the cooling fluid (Re=450, β=0.5, Ek 1, =4.5x10 4, p / d =0.667). 169 xii

18 Figue 6.17 Figue 6.18 Figue 6.19 Figue 6.0 Figue 6.1 Figue 6. Figue 6.3 Figue 6.4 Figue 6.5 Figue 6.6 Effects of nole to taget spacing atio on local Nusselt numbe and dimensionless inteface tempeatue distibutions fo a silicon disk with wate as the cooling fluid (Re=900, b/d n =0.5, Ek 1, =4.5x10 4, p / d =0.667). 171 Effects of diffeent cooling fluids with silicon as the disk mateial on local Nusselt numbe and dimensionless inteface tempeatue (Re=750, β=0.5, b/d n =0.5, p / d =0.667). 173 Effects of diffeent solid mateials with wate as the cooling fluid on local Nusselt numbe and dimensionless inteface tempeatue (Re=875, Ek 1, =1.77x10 4, β=0.5, b/d n =0.5, and p / d =0.667). 174 Local Nusselt numbe and dimensionless inteface tempeatue distibutions fo diffeent plate to disk confinement atio (Re=450, Ek 1, =4.5x10 4, β=0.5, b/d n =0.5). 176 Compaison of pedicted aveage Nusselt numbes of equation 6. with pesent numeical data. 177 Compaison of numeical and expeimental local Nusselt numbe distibutions at diffeent spinning ates fo an aluminum disk with wate as the cooling fluid (T j =93 K, Re=38, H n = m, b= m, b/d n =0.15, p = m, and p / d =0.5). 178 Local Nusselt numbe and dimensionless inteface tempeatue distibutions fo a silicon disk with wate as the cooling fluid fo diffeent Fouie numbes (Re=75, Ek=4.5x10 4, β=0.5, b/d n =0.5, p / d =0.667). 180 Aveage Nusselt numbe and dimensionless tempeatue vaiations with time fo diffeent Reynolds numbes (Ek=4.5x10 4, β=0.5, silicon disk, b/d n =0.5, and p / d =0.667). 181 Aveage Nusselt numbe and dimensionless tempeatue vaiations with time fo diffeent Ekman numbes (Re=550, β=0.5, silicon disk, b/d n =0.5, and p / d =0.667). 183 Aveage Nusselt numbe and dimensionless tempeatue vaiations with time fo diffeent nole to plate spacing (Re=750, Ek=4.5x10 4, silicon disk, b/d n =0.5, and p / d =0.667). 184 xiii

19 Figue 6.7 Figue 6.8 Figue 6.9 Figue 6.30 Figue 7.1 Figue 7. Figue 7.3 Figue 7.4 Figue 7.5 Figue 7.6 Figue 7.7 Figue 7.8 Aveage Nusselt numbe and dimensionless tempeatue vaiations with time fo diffeent plate to disk confinement atios (Re=450, Ek=4.5x10 4, β=0.5, silicon disk, b/d n =0.5). 185 Aveage Nusselt numbe and dimensionless tempeatue vaiations with time fo diffeent solid mateials (Re=875, Ek=.13x10 4, b/d n =0.5, β=0.5, and p / d = 0.667). 186 Aveage Nusselt numbe and dimensionless tempeatue vaiations with time fo diffeent silicon disk thicknesses (Re=450, Ek=4.5x10 4, β=0.5, and p / d =0.667). 188 Compaison of pedicted aveage Nusselt numbe of equation 6.3 with pesent numeical data. 189 Steady state tempeatue contou plots fo alumina (sluy), the substate and pad sufaces at vaious sluy flow ates, (a) Q sl =15 cc/min, (b) Q sl =30 cc/min, and (c) Q sl =75 cc/min. 193 Coss sectional wafe and pad tempeatue distibutions and local heat tansfe convection coefficients along the cente of pad and substate sufaces fo two diffeent sluy flow ates. 194 Coss sectional wafe and pad tempeatue distibutions and local heat tansfe convection coefficients along the cente of pad and substate sufaces unde two chaacteistic CMP pessue loads. 197 Steady state wafe, and pad tempeatue contou distibutions fo two diffeent caie spinning ates equal to: (a) Ω c =15 RPM and (b) Ω c =75 RPM. 199 Coss sectional tempeatue distibutions and local heat tansfe convection coefficients along the cente of pad and substate sufaces unde two diffeent caie spinning ates. 00 Tempeatue distibutions along the cente of pad and substate sufaces fo thee chaacteistic sluy film thicknesses. 0 Local heat tansfe convection coefficient distibutions along the cente of pad and substate sufaces fo thee chaacteistic sluy film thicknesses. 03 Steady state wafe, and pad tempeatue contou distibutions fo two chaacteistic pad spinning ates equal to: (a) Ω p =100 RPM and (b) Ω p =00 RPM. 05 xiv

20 Figue 7.9 Figue 7.10 Figue 7.11 Figue 7.1 Figue 7.13 Figue 7.14 Figue 7.15 Figue 7.16 Figue 7.17 Figue 7.18 Coss sectional tempeatue distibutions and local heat tansfe convection coefficients along the cente of pad and substate sufaces unde two diffeent pad spinning ates. 06 Compaison of mean tempeatue ise of pad at diffeent sluy flow ates of pesent esults with expeimental esults of Boucki et al. [18]. 08 Sluy (alumina), wafe, and pad sufaces tempeatue contou distibutions fo a flow ate value of: (a) Q sl =30 cc/min and (b) Q sl =75 cc/min. 11 Tansient wafe tempeatue distibution and wafe pad tempeatue diffeences fo two diffeent flow ates at a adial location of / w =7/8 along the: (a) 3 o clock position and (b) 5 o clock position. 14 Wafe and pad tansient heat tansfe convection coefficients fo two diffeent flow ates at a adial location of / w =7/8 along the: (a) 3 o clock position and (b) 5 o clock position. 15 Sluy, wafe, and pad sufaces tempeatue contou distibutions unde a constant pessue value of: (a) 17.4 kpa and (b) kpa. 17 Tansient wafe tempeatue distibutions and wafe pad tempeatue diffeences at diffeent adial locations along the 1 o clock position unde a constant pessue value of: (a) P=17.4 kpa and (b) P=41.37 kpa. 0 Tansient wafe heat tansfe convection coefficient at diffeent adial locations along the 1 o clock position unde a constant pessue value of: (a) P=17.4 kpa and (b) P=41.37 kpa. 1 Sluy, wafe, and pad sufaces tempeatue contou plots fo a caie spinning ate equal to: (a) Ω c =15 RPM and (b) Ω c =75 RPM. 3 Tansient wafe tempeatue distibutions and wafe pad tempeatue diffeences fo two diffeent caie spinning ates at a: (a) Radial location of / w =1 along the 1 o clock position and (b) Radial location of / w =7/8 along the 3 o clock position. 4 xv

21 Figue 7.19 Figue 7.0 Figue 7.1 Figue 7. Figue 7.3 Figue 7.4 Figue 7.5 Figue 7.6 Figue 7.7 Wafe and pad tansient heat tansfe convection coefficients fo two diffeent caie spinning ates at a: (a) Radial location of / w =1 along the 1 o clock position and (b) Radial location of / w =7/8 along the 3 o clock position. 7 Sluy, wafe, and pad sufaces tempeatue contou distibutions unde a sluy film thickness equal to: (a) δ sl =40 µm and (b) δ sl =10 µm. 9 Tansient wafe tempeatue distibutions and wafe pad tempeatue diffeences fo 3 diffeent sluy film thicknesses at a adial location of / w =7/8 along the 1 o clock position. 30 Tansient wafe heat tansfe convection coefficients fo thee diffeent sluy film thicknesses at a adial location of / w =7/8 along the 3 o clock position. 31 Wafe, and pad tempeatue contou distibutions fo a pad spinning ate equal to: (a) Ω p =175 RPM and (b) Ω p =50 RPM. 3 Tansient wafe tempeatue distibutions and wafe pad tempeatue diffeences fo thee diffeent pad spinning ates at a adial location of / w =7/8 along the 5 o clock position. 33 Tansient wafe heat tansfe convection coefficients fo thee diffeent pad spinning ates at a adial location of / w =7/8 along the 5 o clock position. 34 Expeimental esults of pad suface tempeatue ise duing coppe polish at diffeent sluy flow ates fom Mudhivathi [146]. 36 Pesent numeical esults of pad tempeatue ise at the leading edge along the 5 o clock position fo thee diffeent sluy flow ates at (Ω p =150 RPM, Ω c =30 RPM, T sl =97 K, COF=0.4, P=4.35 kpa, δ sl =40 µm, w =1.9 cm, q sl =4.6 to 10.8 kw/m ). 36 xvi

22 List of Symbols b C p d n d w Disk thickness, [m] Specific heat [J/kg K] Diamete of the nole [m] Diamete of the wafe [m] g Acceleation due to gavity [m/s ] Ek Ekman numbe, ν f /(4 Ω d ) Ek 1 Ekman numbe of impingement disk, ν f /(4 Ω 1 d ) Ek Ekman numbe of confinement disk, ν f /(4 Ω d ) F o Fouie numbe, α f t /d n h Heat tansfe coefficient [W/m K], q int /(T int T j ) h av Aveage heat tansfe coefficient [W/m K], defined by equation havp,w Aveage pad and wafe heat tansfe convection coefficients [W/m K] H n Height of the nole fom the plate [cm] k Themal conductivity [W/m K] n n Nu Numbe of elements in adial diection Numbe of elements in axial diection Nusselt numbe, (h d n )/k f Nu av Aveage Nusselt numbe fo the entie suface, (h av d n )/k f p P Pessue [Pa] Pandtl numbe, (µ f Cp f )/ k f xvii

23 q Heat flux [W/m ] Q 0 1 1, d p olumetic flow ate (m 3 /s) o cc/min Radial coodinate [m] Pad to wafe cente distance, [m] CMP model adial coodinate system of efeence, [m] Disk adius [m] Plate adius [m] p / d Confinement plate to disk adius atio, (Confinement atio) Re t T Reynolds numbe, ( j d n )/ν f Time [s] Tempeatue [K] T 1 Solid disk sub laye tempeatue [K] T int Aveage inteface tempeatue [K], j Jet velocity [m/s] d T intd d 0,, θ elocity component in the, and θ diection [m/s] θ pc Co tangential elative velocity, [m/s], defined by equation , Axial coodinate [m] CMP model axial coodinate system of efeence, [m] Geek Symbols: α β δ δ sl Themal diffusivity [m /s] Dimensionless nole to plate spacing, H n /d n Fee suface height [m] Sluy film thickness [m] xviii

24 Γ ε Disk adius to nole diamete atio, d /d n Themal conductivity atio, k s /k f µ Dynamic viscosity [kg/m s] µ f Pad coefficient of fiction ν θ θ 1, Kinematic viscosity [m /s] Angula coodinate [ad] CMP model axial coodinate system of efeence, [m] Θ Dimensionless tempeatue, k f (T int T j )/ (q d n ) ρ Density [kg/m 3 ] σ Ω Ω 1 Ω Suface tension coefficient [N/m] Angula velocity [ad/s] Angula velocity of the impingement disk [ad/s] Angula velocity of the confinement disk [ad/s] Ω p,c Pad and caie angula velocity [ad/s] Subscipts: atm Ambient av c f i in int j Aveage Caie Fluid Initial condition Inlet Solid fluid inteface Jet o inlet xix

25 max Maximum n p pw s sl SS w Nole Plate o pad Polishing pad cente to wafe edge distance, [m] Solid disk Sluy Steady state Bottom suface of the impingement disk o wafe xx

26 Numeical Heat Tansfe Duing Patially confined, Confined, and Fee Liquid Jet Impingement with Rotation and Chemical Mechanical Planaiation Pocess Modeling Joge C. Lallave Cotes ABSTRACT This wok pesents the use of numeical modeling fo the analysis of tansient and steady state liquid jet impingement fo cooling application of electonics, and enegy dissipation duing a CMP pocess unde the influence of a seies of paametes that contols the tanspot phenomena mechanism. Seven thoough studies wee done to exploe how the flow stuctue and conjugated heat tansfe in both the solid and fluid egions was affected by adding a seconday otational flow duing the jet impingement pocess. Axis symmetical numeical models of ound jets with a spinning o static nole wee developed using the following configuations: confined, patially confined, and fee liquid jet impingement on a otating o stationay unifomly heated disk of finite thickness and adius. Calculations wee done fo vaious mateials, namely coppe, silve, Constantan, and silicon with a solid to fluid themal conductivity atio coveing a ange of 36.91, at diffeent lamina Reynolds numbes anging fom 0 to,000, unde a boad otational ate ange of 0 to 1,000 RPM (Ekman numbe= 3.31x10 5 ), nole to plate spacing (β= ), dimensionless disk thicknesses (b/d n = ), confinement atio ( p / d = ), and Pandtl numbe ( ) using NH 3, H O, xxi

27 FC 77 and MIL 7808 as woking fluids. An engineeing coelation elating the aveage Nusselt numbe with the above paametes was developed fo the pediction of system pefomance. The simulation esults compaed easonably well with pevious expeimental studies. The second majo contibution of this eseach was the development of a thee dimensional CMP model that shows the tempeatue distibutions pofile as an index of enegy dissipation at the wafe and pad sufaces, and sluy inteface. A finite element analysis was done with FIDAP package unde the influence of physical paametes, such as sluy flow ates ( cc/s), polishing pessues ( kpa), pad spinning ates ( RPM), caie spinning ates (15 75 RPM), and sluy film thicknesses (40 00 µm). Results in this study povide futhe insight of how the above paametes influence the themal aspects of pad and wafe tempeatue and heat tansfe coefficients distibutions acoss the contol volume unde study. Numeical esults suppot the intepetation of the expeimental data. xxii

28 Chapte 1 Intoduction The impinging jet can be defined as a high velocity mass flow ejected fom a nole o slot that impinges on the heat tansfe suface. The pincipal vitues of this method of cooling ae the lage ate of heat tansfe and the elative ease with which both the heat tansfe ate and distibution can be contolled. Impinging jets and spays have been demonstated to be an effective means of poviding high heat/mass tansfe ates in industial pocesses whee apid heating, cooling, o dying is necessay. Pocesses like annealing of metal and plastic sheets, tempeing glass, chemical vapo deposition, avionics cooling, cooling of tubine blades, and dying of textiles ae some examples whee we use this technique. In confined egions of aifoils such as the leading edge o tailing edge, span wise lines of impingement jets ae sometimes used to focus cooling on a pimay location of high extenal heat load, like the aifoil s stagnation egion. Nowadays, many of these pocesses have become moe complex and electonic poducts ae becoming smalle in sie, opening the doos to new techniques whee conventional methods ae inadequate o ineffective. The second pat of the investigation involved a thee dimensional model of the chemical mechanical polishing pocess using a finite element method (FEM) to examine futhe the outcome of a seies of expeimental set up chaacteistics as pat of the oveall pocess. Duing the last twenty yeas, CMP pocess has been geneally used in the micoelectonics industy due to its vesatility, simplicity, and cost effectiveness to 1

29 achieve global planaiation, patteing of metals, and dielectic layes in fewe steps than othe conventional methods. Nowadays, the modeling of the mateial emoval pocess duing CMP seems to have taken two distinctive types of appoaches based on two extemes in dealing with the inteactions between the pad and the wafe. The fist is puely a fluid mechanical appoach, in which the wafe and the pad ae assumed to be sepaated by a continuous fluid laye of sluy and the mateial emoval is viewed as a consequence of eosion, chemical emoval and paticles abasion of the sluy. These ae called wafe scale macoscopic analyses o hydodynamic contact modes, which may povide useful infomation about the influence of the shea and nomal stess on the emoval ate. The second appoach is based on solid to solid contact mechanics in CMP and the assumptions of plastic contact ove the wafe, abasive paticles, and pad inteface in which the mateial emoval is attibuted to abasive wea of the wafe suface in diect contact of the sluy paticles and the pad. The solid to solid contact mode is efeed to the mode whee the down pessue is elatively lage and the velocity is elatively small. This is consideed the most fequent mode in CMP pocess, in which the two body abasion dominates and the fluid flow effects ae immateial. Many eseaches extensively acknowledged that this appoach seems to be physically moe easonable in descibing expeimental esults. On the contay, thee has been expeimental evidence that vaious chemical effects obseved like oxidiing and change in concentation of the sluy ae esponsible fo the mateial emoval ate (MRR) and the quality of the suface finish. Such obsevations cannot be explained by changes in the suface chage because its effect is ielevant. Indeed, significant changes in the wafe suface exposed to the chemicals in

30 the sluy will play a significant ole in the CMP pocess. As a esult of the pevious findings, an intemediate appoach fo the actual mateial emoval mechanism is intoduced as a semi diect contact appoach that is analogous to the tansition egion of the bounday laye appoach of Pandtl. This appoach pesents the emoval mechanism as a function of vaious factos that includes mechanical and chemical effects. In a non diect contact mode, the CMP pocess occus by the two body and thee body abasions occuing simultaneously at the inteface. As the oughness of the pad is in the ode of micons, and the sie of the abasive paticles is in the ode of nanometes, the two body abasions mainly occu acoss the wafe suface and the pad suface aspeities. Convesely, the pocess of CMP is moe complicated at times when the wafe, abasive paticles in sluy and pad sufaces come in contact constituting a thee body abasion at the inteface. In this model, it is assumed that the abasive paticles ae fully embedded acoss the wafe suface unde the applied pessue using the polishing pad, in which ploughing and cutting pocesses occu simultaneously, esulting in the mateial emoval fom the wafe suface. This model does not cove factos such as chemical effects, themal effects, pad wea, and so on. The model is based on the assumptions of the abasive wea theoy, in which the abasive indents and causes plastic defomation in the wafe. Due to the fiction between the paticles of the sluy, pad aspeity and wafe suface unde a given pessue yield to high tempeatues. The changes in fiction and wea, especially duing the un in peiod, ae stongly coelated to the blockage of enegy dissipation paths within the sliding mateials. As such, the pesevation of the tibological integity of a ubbing mateial depends mainly on the efficiency of 3

31 dissipation of the fiction induced themal enegy. Wafe, being moe themally conductive than the polymeic polishing pad, absobs a majo pat of the heat. Themal effects ae impotant in the CMP pocess, as the chemical eactions and pad popeties ae affected to a consideable extent. 1.1 Configuation of Impinging Jets Jets can be configued in diffeent ways. The two main qualitatively flow configuations ae fee suface impinging jet and confined submeged impinging jet. A thid physical configuation of patially confined impinging jet has been studied as pat of this investigation. The fluid dynamics of all cases ae diffeent. (a) (b) Ai Ai Liquid Liquid Liquid Liquid (c) Ai Liquid Liquid Ai Liquid Figue 1.1 (a) Fee liquid jet impingement, (b) Confined liquid jet impingement, and (c) Patially confined liquid jet impingement. 4

32 Figue 1.1 shows the two dimensional epesentation of all pevious cases. In tems of geomety, thee two main cases a plana case with the jet issuing fom a slot, and an axis symmetic case with a ound nole. Additional geometies such as jet issuing fom squae, ectangula o elliptical noles, o oblique jets ae also possible. 1. Impinging Jet Chaacteistics and Zones 1..1 Fee Suface Jets A fee suface jet is fomed when a liquid dischages fom a nole o oifice into the ambient ai o othe gaseous envionment. The fee suface develops instantly at the nole exit and emains thoughout the impingement pocess. When an axial fee suface jet impinges on a cicula disk, the fluid foms a bounday laye, which gows along the disk adius. The flow can be divided into two egions, the impingement o stagnation one and the wall jet egion. The jet flow is undeveloped up to six o seven times the nole diamete measued fom the lip. The stagnation one is chaacteied by pessue gadient, which stops the flow in the axial diection and tuns adially outwad. The bounday laye aound the stagnation point is lamina due to a favoable pessue gadient effect. The incease of the velocity along the wall keeps a thin bounday laye thickness. The wall jet one is fee of gadients of the mean pessue; the flow deceleates and speads adially. At the end, the stuctue of the jet fee suface depends on suface tension, and gavitational and pessue foces. The liquid jet sie, speed, and oientation detemine the magnitude of these foces. The inteaction of fee liquid jet impingement and taget otation esults in a complex and poweful flow capable of impoving the heat tansfe pocesses consideably. This method of cooling o heating can be used fo pocesses involving a apid heat dissipation ate o high heat flux. 5

33 1.. Confined Submeged Jets If the fluid is dischaged fom a nole o oifice attached to a confinement plate into a body of suounding fluid that is the same as the jet itself, then it is called confined submeged. Confined submeged liquid jets find use in both axis symmetic and plana configuation. Both configuations shae the common featue of a small stagnation one at the impingement suface whose sie is of the ode of the nole diamete o slot dimension, with the subsequent fomation of a wall jet egion. The model coves the entie fluid egion (impinging jet and flow speading out unde a stationay o spinning confined suface) and stationay o spinning solid disk as a conjugate poblem. The liquid jet consideed in this study is axis symmetic and submeged, with the jet issuing into a egion containing the same liquid. In most applications, the nole to plate distance is too small to enable the development of a jet flow condition. A thicke shea laye foms unde lamina conditions aound the nole, with a simila behavio as a plane shea laye. The shea laye thickness becomes compaable with the jet diamete downsteam, and the behavio of the laye changes consideably. The inteaction of otation and impingement ceates a complex and poweful flow capable of impoving heat tansfe pocesses consideably. This aangement is suitable fo micogavity applications whee centifugal foce due to disk otation can be used to foce the fluid ove the heated suface Patially confined Jets In a patially confined jet the nole o slot is attached to a confinement plate paallel to the impingement suface with a sepaation distance of H n. The diamete o length of the confinement egion is smalle than the impingement taget, and theefoe the 6

34 fluid comes out of confinement speading downsteam as a fee suface flow exposed to the ambient envionment. To achieve a eliable cooling system design with impinging jet one has to choose an appopiate jet configuation and suounding geomety. It is necessay to undestand that the heat tansfe ate fom an impinging jet onto a suface is a complex function of many paametes, such as flow ate, woking fluid popeties, nole stuctue and oientation, nole to taget spacing, confinement atio and displacement fom the stagnation point. The liquid jet consideed in this study is axis symmetic. Heat tansfe capabilities of jets impinging on a otating body ae of impotance in the themal analysis of vaious types of machineies and in a wide vaiety of applications in the aea of themal heating and cooling. Pocesses like micogavity flow delivey, annealing of metal, chemical vapo deposition, and electonics packaging can use this technique. Nowadays, many of these pocesses have become moe complex and electonic poducts ae becoming smalle in sie, opening the doos to new techniques whee conventional methods ae inadequate o ineffective. The pincipal vitues of this method of cooling ae the lage heat tansfe ate attainable elative to nonimpinging flows and the elative ease with which both the heat tansfe ate and distibution can be contolled. 1.3 Chemical Mechanical Polishing Pocess Set up A standad CMP pocess consists of thee main components. The fist component includes a polishing pad fastened to a cicula polishing platfom. The second component is a wafe caie (polishing head) that holds the substate with a etaining ing that is adjusted to geneate a unifom pessue pofile acoss the entie wafe to help offset excessive mateial emoval at the edges. Cuently the wafe pessue vesus etaining 7

35 ing pessue adjustment is done by tial and eo. This wafe is otated about its axis while being pessed down against a otating polymeic polishing pad commonly made of polyuethane, since the chemisty of this polyme allows the pad chaacteistics (such as hadness and poosity) to be tailoed to meet specific mateial popety needs in CMP Jaiath et al. [1]. Both pevious components of the pocess ae cicula and typically otate at simila speed and in the same diection but eccentically oiented, despite the fact that pessing the wafe against the pad suface by applying a load o foce which can be vaied. The thid component of the pocess is caied by the polishing pad and is the sluy, a liquid that contains a colloidal suspension of abasive paticles such as alumina (Al O 3 ) o silica SiO as well as specific chemicals chosen fo polishing. Finally, the suface laye of the polished mateial is emoved pogessively as a esult of the chemical and mechanical inteactions povided by the polishing sluy. The sluy chemisty, including chemical eagents and its concentation, modifies the popeties of the suface to be polished. The mechanical inteactions, on the othe hand, vay depending on solid loading, the sluy paticle sie, and distibution, in view of the fact that these factos ceate a dispaity in the load applied pe paticle. Othe empiical vaiables can be ecognied, such as the applied nomal foce (o down pessue), elative velocity of the wafe to the pad, and pad popeties (Young s Modulus, hadness and poosity, etc.). Howeve, due to the complexity of CMP by concuent polishing of multiple mateials and lumped paamete conditions, the fundamental polishing mechanism undelying the pocess ae not yet well undestood []. Figue 1. shows a schematic side view of a CMP pocess. 8

36 Ωc DOWN FORCE WAFER CARRIER SLURRY PAD WAFER RETAINING PRESSURE RING Ωp WAFER PLATEN Figue 1. Schematic side view epesentation of a CMP pocess Expeimental Testing Set Up The univesal tibological teste (UTT) technique is done with a bench top Cente fo Tibology Reseach (CETR) univesal tibomete to examine the natue of polish (acoustic emission vesus time) and the suface oughness of the sample. The eal time measuements of the data, along with othe measuements, ae used to calculate the wea and mateial emoval ate (MRR) of the sample. In addition, it helps to quantify the wea esistance of the mateial at diffeent pessue and otational velocity. Table 1.1 povides the basic specifications of an expeimental univesal tibomete. The UTT teste povides eal time measuements of the following tibological paametes of the polishing: caie and platen speed anging, minimum and maximum load, and contact pessue load esolution and coefficient of fiction as esults of acoustic fequency as function of time. 9

37 Table 1.1 CETR univesal tibomete specifications. Specifications Specimens dimensions Pad dimensions Speed anging Minimum load Maximum load (w/high load system) Contact Pessue Load esolution (w/high load system) Total sampling ate Measuement o Desciption 0.5 inch to 4 inches 1 inch to 9 inches 0.1 micon/s (0.001 RPM) up to 50 m/s (10,000 RPM) 0.1 mn (10 mg) 0.5 kn (50 Kg) 0.05 to 500 psi 50 nn (yes, the same as without the high load system! 0 kh 1.4 Poblem Unde Study The detailed infomation about conjugate heat tansfes fom a otating taget (Pob. #1) o spinning confinement disk (Pob. #) cooled by a confined liquid jet is cuently not available in open liteatue. Table 1. summaied the nine poblems unde study as pat of this wok. Past studies ae esticted to eithe cooling of a stationay disk by jet impingement o by pue otation. Additionally, most of these woks deal with aveage heat tansfe measuements athe than local distibutions. The intent of the following eseach is to study the conjugate heat tansfe effect with a steady flow ove a otating solid wafe unde confinement with constant fluid popeties (Pob. #1) and to study the conjugate heat tansfe effect of a spinning confinement disk ove a solid stationay disk unde tempeatue dependent popeties (Pob. #). Numeical esults wee done fo vaious flow ates o jet Reynolds numbes, spinning ates o Ekman numbes, diffeent disk thicknesses and nole to taget spacing atios. A boad ange of Pandtl numbes was coveed with the use of fou woking fluids, namely wate, 10

38 ammonia, FC 77, and MIL The themal conductivity effect was studied with the implementation of fou diffeent disk mateials: coppe, silicon, silve, and Constantan. The esults offe a bette undestanding of the fluid mechanics and heat tansfe behavio of liquid jet impingement unde an insulated spinning o stationay wall condition on a stationay o otating taget. In addition, the enhancements of heat tansfe duing liquid jet impingement ove a otating disk could be done by tiggeing the tubulence in the flow field by inceasing the flow ate o otational speed but it was not examine as pat of this study. The following studies wee limit to lamina flow conditions duing the pesent investigation. Even though no new numeical technique has been developed, esults obtained in the pesent investigation ae entiely new. The quantitative effects of diffeent paametes as well as the coelation fo aveage Nusselt numbes will be pactical guides fo enhancement of heat o mass tansfe unde micogavity. Thee have been only a few studies on tansient heat tansfe and most of them ae expeimental wok on fee jet impingement. None of these studies consideed tansient heat tansfe duing patially confined liquid jet impingement. Pob. #3 and #7 consideed only lamina flow conditions to addess the enhancement of heat tansfe emoval that is citical in space bone applications and accomplish the job with lowe fluid inventoy and hence lowe the mass of the cooling system by adding otation to the pocess. A highe ate of otation is expected to enhance heat tansfe at the impingement egion, but may esult in flow sepaation fom the heat tansfe suface futhe downsteam, which is not desiable. Theefoe, pesent studies ae significant in addessing the heat tansfe enhancement unde steady state (Pob. #3) and tansient conditions (Pob. #7) fo a patially confined liquid jet impingement ove a spinning 11

39 taget. The vaiation of disk tempeatue as well as local and aveage heat tansfe coefficients duing steady state (Pob. #3) and tansient heating pocess (Pob. #7) ae exploed fo diffeent combinations of flow ate, spin ate of the taget disk, nole to taget spacings, confinement atio, disk thickness and disk mateials. The numeical esults as well as the coelation fo aveage Nusselt numbe ae expected to be valuable towads the design of cooling o heating systems fo engineeing applications. A wealth of infomation exists on heat tansfe effects on the basic cases of individual and aay set up of fee and confined jet impingement. Howeve, newe and moe specific cases of cooling design equie additional infomation to account fo the heat tansfe effects of patially confined jet impingement. Pob. #4 consideed the simultaneous spinning of a confinement disk and taget suface unde lamina patially confined jet impingement. In addition, none of the studies have consideed the steady state otation of the nole cove plate and taget disk to futhe induce fluid motion at micogavity. Theefoe, the pesent study is significant in addessing heat tansfe enhancement unde cetain conditions. Calculations wee done unde five diffeent flow ates o jet Reynolds numbes, six spinning ates o Ekman numbes, five diffeent disk thicknesses and fou nole to taget spacings. A boad ange of Pandtl numbes was coveed with the use of fou woking fluids, namely wate (H O), ammonia (NH 3 ), flouoinet (FC 77) and MIL 7808 lubicating oil. The themal conductivity effect was studied with the implementation of five diffeent disk mateials: aluminum, Constantan, coppe, silicon, and silve. Even though jet impingement heat tansfe fom a stationay suface has been thooughly investigated, only a few attempted to poduce local heat tansfe distibution fo a otating disk in combination with a fee liquid jet impingement. 1

40 In addition, none of the studies have attempted to exploe conjugate heat tansfe effect in a otating taget duing axial fee jet impingement. The pesent study attempts to cay out a compehensive investigation of a steady state (Pob. #5) and (Pob. #6) tansient conjugate heat tansfe analysis fo a fee liquid jet impingement ove a spinning solid disk. Computations using wate (H O), ammonia (NH 3 ), flouoinet (FC 77), and oil (MIL 7808) as woking fluids wee caied out fo diffeent combinations of geometic and flow paametes and five diffeent disk mateials. The numeical model along with the esults fo steady state and tansient heat tansfe fo diffeent Reynolds numbes, Ekman numbes, disk thicknesses and solid mateial popeties is expected to be valuable towads the design of liquid jet impingement cooling o heating systems fo vaious engineeing applications. Most publications pimaily focus on othe aspects of the CMP pocess like film stess, patten dependencies, pad oughness, mateial emoval ate, abasive paticles sie, sluy film and pessue distibutions, and chemicals effects. Only a few examine the themal aspects duing the planaiation pocess ove the wafe suface. Howeve, in these eseach woks, the epoted tempeatue ise is eithe the aveage tempeatue on the pad suface, a pedicted aveage tempeatue on the wafe suface, o tempeatue ise at diffeent isolated locations on the wafe. These woks epot the oveall tempeatue ise but do not povide the infomation about the tempeatue distibution on the wafe suface. The tempeatue pofile on the wafe suface as a function of the adius unde the influence of the above paametes will povide valuable insight into the extent of tempeatue ise at diffeent locations on the wafe. Fo example, since the mateial emoval ate duing coppe CMP is sensitive to tempeatue, the tempeatue distibution 13

41 ove the entie wafe will significantly affect the unifomity of mateial emoval ove the entie wafe. Undestanding the tempeatue pofile of the substate will decease the with in wafe non unifomity and thus impoves yield by minimiing the numbe of faulty dies. The physical epesentation of a thee dimensional CMP schematic is shown in figue 1.3. Caie Down Foce Caie Rotation 1 Contol olume θ Load Senso Sluy 1 Wafe Pad θ 1 d w Spindle Rotation d p Figue 1.3 Thee dimensional (3 D) CMP schematic. Pob. #8 and #9 chaacteied the steady state and tansient tempeatue distibutions o pofile as the index of enegy dissipation at the wafe suface, sluy and pad inteface. By solving the numeical poblems, we pesent the tempeatue pofiles and heat tansfe convection coefficients on the pad and wafe sufaces unde the influence of diffeent physical paametes, such as sluy flow ate, sluy film thickness, wafe spinning ate, pad spinning ate, and polishing pessue. 14

42 The numeical modeling effots ae suppoted with a finite element analysis using the compute fluid dynamics of the FIDAP package. The coefficient of fiction values (µ f = ) at diffeent pessues and velocities equied to calculate the heat dissipation at the inteface ae obtained fom coppe polishing expeiments conducted on a CETR univesal tibomete bench top teste. To gain geate insight into this behavio, the themal dynamics associated with enegy ae futhe discussed in the esults by figues and paametes. Finally, tempeatue distibution and heat tansfe convection coefficients esults ae compaed with expeimental data unde the same pocess conditions, which wee found to be consistent. The main chaacteistics of the nine poblems epoted in this wok ae summaied in Table 1.. They ae pesented in an ode such that the difficulty fom a computational point of view inceases gadually. Table 1. Summay of poblems unde study. Poblem # Fluid wate, ammonia flouoinet and oil (MIL 7808) wate, ammonia flouoinet and oil (MIL 7808) wate, ammonia flouoinet and oil (MIL 7808) wate, ammonia flouoinet and oil (MIL 7808) wate, ammonia flouoinet and oil (MIL 7808) wate, ammonia flouoinet and oil (MIL 7808) 7 wate Model Axis symmetic, Confined submeged jet Axis symmetic, Confined submeged jet Axis symmetic, Patially confined submeged jet Axis symmetic, Patially confined submeged jet Axis symmetic, Fee suface jet Axis symmetic, Fee suface jet Axis symmetic, Patially confined submeged jet 8 Alumina 3 D 9 Alumina 3 D BC s and Fluid Popeties Taget otation and constant Confined disk otation and Tempeatue dependent Taget otation and Tempeatue dependent Taget and confined disk otation, Tempeatue dependent Taget otation and Tempeatue dependent Taget otation and Tempeatue dependent Taget otation and Tempeatue dependent Caie and pad spinning, and constant Caie and pad spinning, and constant Analysis Steady State Steady State Steady State Steady State Steady State Tansient Tansient Steady State Tansient Body foce Gavity Gavity Gavity Gavity Gavity Gavity Gavity Gavity Gavity 15

43 The level of difficulty includes model set up, time, and the computing esouces equied as pat of the simulation pocess. The computational difficulty fo each poblem appeas in diffeent aspects that can be simplified into the following ules: 1. 3 D models (pob. #8 and #9) ae moe difficult than axis symmetic (pob. #1 though #7).. Tansient analysis with fixed time steps (pobs. #6, #7, and #9) ae moe difficult than steady state analyses (pob. #1 though #5). 3. Fee suface jet (pob. #5 and #6) ae moe sensitive and difficult than confined and patially confined jet impingement (pob. #1 though #4). 1.5 Objectives The main objective of the fist pat of the pesent investigation is to undestand the elationship between fluid and solid as conjugated heat tansfe phenomena duing a pocess of fee, confined, and patially confined jet impingement unde steady state and tansient cooling conditions. Most of the above simulations conside tempeatue dependent popeties of the fluid egion in ode to pedict moe pecise esults cuently not available in the liteatue. In addition, this eseach examines the themal bounday laye behavios that contol the steady state and tansient convective heat tansfe unde the influence of a seconday otational flow. The pesent eseach focuses on addessing the effects of the following paametes on the steady state and tansient heat tansfe pocess. 1. Jet Reynolds numbe.. Ekman numbes of taget and confined wall at diffeent spinning ates. 3. Disk thicknesses. 16

44 4. Nole to taget spacings. 5. Confinement atio ( p / d ). 6. Pandtl numbe of diffeent fluids. 7. Themal conductivity of vaious mateials. 8. Fee, confined and patially confined jet impingement configuations. Additionally, a set of coelations fo aveage Nusselt numbes esults have been developed as a function of the above paametes to chaacteie the above heat tansfe pocesses. The quantitative effects of diffeent paametes ae attached to the exponent that coelates with actual numeical esults. These coelations ae expected to be valuable and pactical towads the design of cooling o heating systems unde jet impingement o micogavity engineeing applications. The second pat of this eseach includes the development of a thee dimensional heat tansfe model to estimate the steady state and tansient tempeatue distibutions at the wafe suface, sluy and pad inteface duing the CMP pocess. The model examines the index of enegy dissipation at the sluy inteface, wafe and pad sufaces as a diffeentiation technique of the CMP mechanism. These numeical studies captue the effects of the following paametes on the steady state and tansient chemical mechanical polishing pocess. 1. Sluy flow ates.. Diffeent polishing pessues. 3. aiable heat flux at the polishing suface. 4. Polishing pad and caie spinning ates. 5. Sluy film thickness. 17

45 Chapte Liteatue Review.1 Fee Suface Jet Impingement Heat tansfe fom a stationay suface by fee jet impingement has been well documented in the liteatue. One of the fist theoetical analyses of a cicula lamina impinging jet speading into a thin film was done by Glauet [3]. Solutions to the bounday laye equations wee sought fo a lamina flow using similaity tansfomation. Watson [4] consideed the flow due to jet speading out ove a plane suface, eithe adially o in two dimensions. Chaudhuy [5] pesented the heat tansfe aspect of Watson s poblem. Heat and mass tansfe chaacteistics of an impinging axis symmetic jet issuing fom a cicula nole has been studied by Scholt and Tass [6]. The theoetical and expeimental findings ae well coelated in the stagnation flow and in the wall jet egions. Metge et al. [7] expeimentally studied the effects of Pandtl numbes on heat tansfe by liquid jets on a unifom tempeatue bounday condition at the test suface. They pesented only suface aveage values of the Stanton numbe, detemined fom the measuement of the total heat flux, test suface tempeatue, and the adiabatic jet wall tempeatue. The coelations ae based on the data fo oil and wate, and thei coelations epesent 95% of the data fo disk adii up to 6.6 jet diametes to within ± 5%. Jiji and Dagan [8] caied out expeimental studies fo single jet and aays of jets using wate and FC 77 coolant fo vaious heat souce configuations. Theoetical flow solutions fo lamina axis symmetic liquid jet impingement ove a 18

46 stationay suface wee discussed by Adachi [9]. Liu and Lienhad [10] investigated cicula sub cooled liquid jet impinging on a suface maintained at unifom heat flux. They used an integal method to obtain analytical pedictions of tempeatue distibution in the liquid film and the local Nusselt numbe. They caied out expeiments to test the pedictions of the theoy. A eview of both analytical and expeimental studies on jet impinging on a flat suface has been pesented by Polat et al. [11]. Wang et al. [1, 13] pesented an analytical study of heat tansfe between an axis symmetic fee impinging jet and a solid flat suface with a non unifom wall tempeatue o wall heat flux. The esults obtained showed that the non unifomity of the wall tempeatue o heat flux has a consideable effect on the Nusselt numbe. Wolf et al. [14] pefomed expeiments on a plana, fee suface jet of wate to investigate the effects of non unifom velocity pofile on the local convective heat tansfe coefficient fo a unifom heat flux suface. The heat tansfe coefficient was measued fo diffeent heat fluxes and Reynolds numbes. ade et al. [15] measued tempeatue and heat flux distibutions on a flat, upwad facing, and constant heat flux suface cooled by a plana, impinging wate jet. The jet velocity, the fluid tempeatue, and heat flux wee vaied. They found that the stagnation convection coefficient exceeded those pedicted by lamina flow analysis and this was caused by the existence of fee steam tubulence. Stevens and Webb [16] consideed an axis symmetic fee liquid jet impinging on a flat unifomly heated suface. Thei expeimental study pesented the effects of Reynolds numbe, nole to plate spacing, and jet diamete. Liu et al. [17] pesented an analytical and expeimental investigation fo jet impingement cooling of unifomly 19

47 heated sufaces to detemine local Nusselt numbe fom the stagnation point to adii up to 40 diametes. Womac et al. [18] pesented coelating equations fo heat tansfe coefficient fo the cooling of discete heat souces by liquid jet impingement. Leland and Pais [19] pefomed an expeimental investigation to detemine the heat tansfe ate fo an impinging fee suface axis symmetic jet of lubicating oil fo a wide ange of Pandtl numbes, and fo conditions vaying inside the fluid film. They concluded that the heat tansfe suface configuation has an impotant effect on Nusselt numbe. Rahman et al. [0] pefomed a numeical simulation of a fee jet of high Pandtl numbe fluid impinging pependiculaly on a solid substate of finite thickness containing electonics on the opposite suface. Computed esults wee validated with available expeimental data. Chattopadhyay and Saha [1] pefomed a numeical study of tubulent flow and heat tansfe fom an aay of impinging hoiontal knife jets on a moving suface using lage eddy simulation with a dynamic sub gid stess model. Roy et al. [] epoted suface tempeatue measuements fo ectangula jet impingement heat tansfe on a vehicle windshield using liquid cystals. Chan et al. [3] epoted expeimental esults on heat tansfe chaacteistics of a heated slot jet impinging on a semi cicula convex suface. Aldabbagh and Seai [4] caied out a numeical investigation of the flow and heat tansfe chaacteistics of a lamina thee dimensional, squae twin jet impingement on a flat plate unde steady state condition. Thei esults showed that the flow stuctue is stongly affected by jet to plate distance. Chattejee et al. [5] studied lamina impinging flow heat tansfe fo a puely viscous inelastic fluid. Thei study demonstated that a small depatue fom Newtonian heology leads to qualitative changes in the Nusselt numbe distibution along the 0

48 impinging suface. Yilbas et al. [6] numeically examined the jet impingement onto a hole with a constant wall tempeatue using a contol volume appoach. Tong [7] numeically studied convective heat tansfe of a cicula liquid jet impinging onto a substate to undestand the hydodynamics and heat tansfe of the impingement pocess using the volume of fluid method to tack the fee suface of the jet. The effects of seveal key paametes on the hydodynamics and heat tansfe of an impinging liquid jet wee examined. Silveman and Nagle [8] epoted expeimental data on the application of jet impingement fo the cooling of acceleato tagets using wate as the coolant. Seai and Aldabbagh [9] investigated the stuctue of the flow field and its effect on the heat tansfe chaacteistics of a jet aay system in steady state fo Reynolds numbes between 100 and 400. Yang and Hwang [30] pesented the numeical simulations of flow chaacteistics of a tubulent slot jet impinging on a semi cylindical convex suface.. Jet Impingement with Spinning Boundaies The applications of liquid jet impingement ove a otating suface ae gowing in vaious pocesses encounteed in mechanical, manufactuing, electical and chemical engineeing. The high heat tansfe ate, along with the simplicity of hadwae equiements makes this cooling pocess an attactive option in a vaiety of applications. In addition, otation is used in metal etching, insing opeations to dissolve species, suface pepaation o coating, and micogavity fluid handling. The inteaction of liquid jet impingement and otation geneates a poweful flow capable of impoving themal diffusion and mass tansfe consideably in the absence of gavity. On all otating disks, whethe smooth o oughened, thee is an inheent pumping of fluid adially outwad along the disk suface. Ealy eseach wok on otational flow 1

49 confined between two infinite paallel disks, one at est and the othe otating was pefomed by Batchelo [31]. His analysis showed that thee flow egions develop at high otational ate, having the stuctue of two shea layes bounding an inviscid coe otating at constant angula velocity. An additional study on heat tansfe ate fom a otating disk was caied out by Keith et al. [3]. Thei eseach coveed a wide ange of otational Reynolds numbes (400 to 10,000) including lamina, tubulent and tansitional egimes. This type of flow is found in paallel disk viscometes, otay disk in a stationay housing of a oto, and the chemical mechanical polishing pocess whee the abasive polishing sluy inteacts with the pad and the wafe. The pesence of otation adds moe complexity to the flow field. Expeimental studies of a single ound jet impinging on a otating disk wee conducted by Metge and Gochowsky [33]. Tests wee conducted fo a ange of flow ates and disk otational speeds with vaious combinations of jet and disk sies. Flow visualiation using smoke addition to the jet flow evealed the pesence of a tansition egime. They concluded that highe otational speeds, lage impingement adii, and smalle jet flow ate favo a otationally dominated flow wheeas the opposite tends favo an impingement dominated flow. Heat tansfe ate was essentially independent of jet flow ate in the otationally dominated egime but inceased stongly with inceasing flow ate in the impingement dominated egime. Cape and Deffenbaugh [34] conducted expeiments to detemine the aveage convective heat tansfe coefficient fo the otating solid fluid inteface at unifom tempeatue, cooled by a single liquid jet of oil impinging nomal to the otating disk. Tests wee conducted fo a ange of Reynolds numbes fom 30 to 1,800 and fo vaious disk otational speeds. Cape et al. [35] conducted futhe expeiments to conside the Pandtl numbe effects on the aveage heat tansfe

50 coefficient at the otating disk. They documented the effects of otational Reynolds numbe on the aveage Nusselt numbe fo vaious liquid jet Reynolds numbes. Popiel and Boguslawski [36] epoted measuements of heat tansfe ate fo a ange of otational and jet Reynolds numbes. Metge et al. [37] employed liquid cystal fo mapping local heat tansfe distibutions on a otating disk with jet impingement. Bodesen et al. [38] expeimentally studied the flow field inteaction between an impinging liquid jet and a otating disk. Thei expeiments coveed sepaate measuements of the disk wall flow, the jet flow and inteaction between the two. Saniei et al. [39] investigated the heat tansfe coefficients fom a otating disk with jet impingement at its geometic cente. The ai jet was placed pependicula to the disk suface at fou diffeent distances fom the cente of the disk. Saniei and Yan [40] pesented local heat tansfe measuements fo a otating disk cooled with an impinging ai jet. Seveal impotant factos, such as otational Reynolds numbes, jet Reynolds numbes, jet to disk spacing, and the location of the jet cente elative to the disk cente, wee examined. Hung and Shieh [41] epoted expeimental measuements of heat tansfe chaacteistics of jet impingement onto a hoiontally otating ceamic based multichip disk. The chip tempeatue distibutions along with local and aveage Nusselt numbes wee pesented. Kang and Yoo [4] caied out an expeimental study using hotwie anemomety to investigate the tubulence chaacteistics of the thee dimensional bounday laye on a otating disk with liquid jet impingement at its cente. Shevchuk et al. [43] pesented an appoximate analytical solution using integal method fo jet impingement heat tansfe ove a otating disk. The chaacteiation of a thin film of 3

51 wate fom an axis symmetic contolled impinging jet ove stationay and otating disk sufaces wee expeimentally studied by Oa et al. [44, 45]. The authos measued the thickness of the liquid film on the disk suface by an optical method, including the chaacteiation of the hydaulic jump. They concluded that the effect of otation on heat tansfe was lage fo a lowe liquid flow ate and gadually deceases with the incement of liquid flow ate. Semi empiical coelations fo both local and aveage Nusselt numbes wee poposed based on thei expeimental esults. In a late study, Rice et al. [46] pesented an analysis of the liquid film and heat tansfe chaacteistics of a fee suface contolled liquid jet impingement onto a otating disk. Computations wee un fo a two dimensional axis symmetic Euleian mesh while the fee suface was calculated with the volume of fluid method. Iacovides et al. [47] epoted an expeimental study of impingement cooling in a otating passage of semi cylindical coss section. Cooling fluid was injected fom a ow of five jet holes along the centeline of the flat suface of the passage and impinged the concave suface. An integal analysis of hydodynamics and heat tansfe in a thin liquid film flowing ove a otating disk suface was pesented by Basu and Cetegen [48]. The model consideed constant tempeatue and constant heat flux bounday conditions ove a ange of Reynolds and Rossby numbes coveing both inetia and otation dominated egimes. Rahman and Lallave [49] numeically studied the convective heat tansfe pefomance of a fee liquid jet impinging on a otating and unifomly heated solid disk of finite thickness and adius. A genealied aveage Nusselt numbe coelation was developed fom numeical esults. 4

52 .3 Tansient Jet Impingement Tansient heat tansfe duing jet impingement has been the subject matte in only a few past studies. Moallemi and Naaghi [50] pefomed a seies of tansient expeiments to study the feeing of wate impinging vetically on a subeo disk though a cicula jet. Thei expeiments chaacteied the ice laye pofiles at diffeent times fo diffeent values of jet Reynolds numbe and Stefan numbe of the suface. an Teuen et al. [51] measued the local heat tansfe unde an aay of impinging jets employing a tansient method. They used a tempeatue sensitive coating consisting of thee encapsulated themo chomic liquid cystal mateials to detemine the local adiabatic wall tempeatue and the local heat tansfe coefficient ove the complete suface of a taget plate fo vaious Reynolds numbes. Steady state and tansient methods wee used by Owens and Libudy [5] in ode to study jet impingement cooling of sufaces. Themo chomic liquid cystals wee employed to measue the suface tempeatue which could be used to study the local heat tansfe coefficient distibution. Kumagai et al. [53] investigated tansient boiling heat tansfe ate of a two dimensional impinging wate jet on a ectangula suface fo jet sub cooling fom 14 K to 50 K. They discoveed that boiling occus at the moment of jet impingement and geneates vapo at that egion. Lachefski et al. [54] numeically analyed the velocity field and heat tansfe in ows of ectangula impinging jets in tansient state. Axial and adial jets coming out of ectangula noles wee consideed. Sahin et al. [55] investigated the themal chaacteistics of jet impingement dying of a moist poous solid using a one dimensional tansient model. Fujimoto et al. [56] pesented a numeical simulation of tansient cooling of a hot solid by an impinging cicula fee suface liquid jet. The flow and themal fields 5

53 in the liquid as well as the tempeatue distibution in the hot solid wee pedicted numeically by a finite diffeence method. Rahman et al. [57] pesented the tansient analysis of a fee jet of high Pandtl numbe fluid impinging on a stationay solid disk of finite thickness. Computed esults included the velocity, tempeatue, and pessue distibutions in the fluid and the local and aveage heat tansfe coefficients at the solid fluid inteface. Bula Silvea et al. [58] pesented infomation on tansient heat tansfe pocess of a fee slot jet of high Pandtl numbe fluid impinging pependiculaly on a solid flat substate of finite thickness containing discete electonics souces on the opposite suface. The geomety of the fee suface was detemined iteatively. The influences of diffeent opeating paametes, such as jet velocity, heat flux, plate thickness, plate mateial, and the location of the heat geneating electonics, wee investigated. Liu et al. [59] pesented a numeical simulation of tansient convective heat tansfe duing ai jet impingement cooling of a confined multichip module disk. They found that a lage ate of decease of chip tempeatue and aveage Nusselt numbe happens in the ealie pat of the tansient. Saghini and Ruocco [60] pesented a tansient numeical analysis of a plana jet impingement on a finite thickness substate at low volumetic flow ate, including the effects of buoyancy. They found that conduction plays a significant ole at the initial pat of the tansient. Fang et al. [61] epoted expeimental tansient mixed convection measuements of heat tansfe chaacteistics of jet impingement onto a hoiontally otating ceamic based multichip disk. Thei esults wee pesented in tems of tansient dimensionless tempeatue distibution on the chip, tansient heat flux distibution of input powe, and local and aveage Nusselt numbes. 6

54 .4 Confined Submeged Jet Impingement In the liteatue a easonable amount of heat tansfe infomation is available fo cicula disks. These data ae typically esticted to eithe cooling of a stationay disk by jet impingement o cooling by pue otation. The effect of the combination of otation and jet impingement has been consideed in only a small numbe of investigations. Additionally, most of these woks deal with aveage heat tansfe measuements athe than local distibutions. As in all convective heat tansfe situations, the flow field of an impinging liquid jet contols the heat tansfe chaacteistics. In suppot of this statement, much wok has been done on submeged confined liquid jets. The following povides a sample of some of the pevious eseach elated to this study. McMuay et al. [6] studied the convective heat tansfe of an impinging plane jet ove a unifom heat flux bounday condition at the wall. To fit thei data, they based heat tansfe coelations on the stagnation flow in the impingement one and on the flat plate bounday laye thickness in the unifom paallel flow one. Impinging slot jet techniques unde confinement with a plate paallel to the impingement suface wee studied by Koge and Kiek [63], Kumada and Mabuchi [64], Miyaaki and Silbeman [65], and Spaow and Wong [66], and many of them ae in pactice in vaious industial opeations. Heat tansfe fom a stationay suface by liquid jet impingement has been epoted by Saad et al. [67]. They investigated the effects of Reynolds numbe, distance between nole and impingement suface, diametes of impingement and confinement sufaces, and the shape of the velocity pofile at the nole exit. Nakoyakov et al. [68] studied, both theoetically and expeimentally, the hydodynamics and mass tansfe of a adial submeged liquid jet impinging onto a hoiontal plate. Thei studies measued the 7

55 wall shea stess, local and mean mass tansfe coefficients within the entie flow egion by an electo diffusion method in a wide ange of liquid flow ates. In addition, simple fomulas wee developed fo the calculation of fiction facto, liquid laye thickness, suface velocity, and convection heat tansfe coefficient at stagnation point as a function of dischage paametes. Ma et al. [69] epoted expeimental measuements fo local heat tansfe coefficient duing impingement of a cicula jet pependicula to a taget plate. Both confined and fee jet configuations wee used. Ethylene glycol and tansfome oil wee used as woking fluids. Polat et al. [70, 71] measued local and aveage heat tansfe coefficient fo a confined tubulent slot jet impinging on a pemeable suface and moving suface consideing though flow. Measuements wee caied out fo a wide ange of jet Reynolds and though flow velocity. Moeno et al. [7] investigated the mass tansfe behavio of a confined impinging jet applied to wet chemical pocesses such as wate insing and metal etching o platting, and the potential applicability to pinted wiing boad s fabication. Chang et al. [73] examined the local heat tansfe distibutions of submeged liquid jet unde confinement. Thei investigation confimed the local heat tansfe coefficients tend of a half bell shaped distibution with espect to adial distance fom the stagnation point. Hung and Lin [74] poposed an axis symmetic sub channel model fo evaluating local suface heat flux fo confined and unconfined cases. Thei models eveal that no significant deviation occus fo stagnation Nusselt numbes at nole to plate spacing (H n /d n ) while significant deviation exists when H n /d n <. Expeimental esults fo the distibution of local heat tansfe coefficient duing confined 8

56 submeged liquid jet impingement with FC 77 as the woking fluid wee pesented by Gaimella and Rice [75]. In addition, Webb and Ma [76] pesented a compehensive eview of studies on jet impingement heat tansfe. They concluded that heat tansfe in submeged jets is moe sensitive to nole to plate spacing than in fee jets, especially when the heat tansfe suface is beyond the potential coe of the jet. Gaimella and Nenaydykh [77], Fitgeald and Gaimella [78, 79], and Li et al. [80] all consideed a confining top plate such as the one used at the pesent study fo a submeged liquid jet using FC 77 as the woking fluid at diffeent volumetic flow ates. Howeve, no otation was used. Thei expeiments wee done to detemine the effects of the nole geomety on the local heat tansfe coefficients fom a small heat souce to a nomally impinging, axis symmetic, submeged and confined liquid jet at diffeent nole to plate spacing and Reynolds numbes. They concluded that the effect of the aspect atio becomes less ponounced as the nole to plate spacing is inceased. Ma et al. [81] investigated the adial distibution of the ecovey facto fo a confined impinging jet of high Pandtl numbe liquid by a numeical appoach, with emphasis on its physical mechanism. They found that the ecovey facto is stongly dependent on the Pandtl numbe, nole to plate spacing, and the velocity pofile at the nole exit, but basically independent of the Reynolds numbe. Abou Ziyan and Hassan [8] made an expeimental study on foced convection due to impingement of confined submeged and fully tubulent jets in elation to the cooling of engine cylinde heads by wate. They concluded that jet impingement can save between 50 and 9 pecent of the equied cooling wate compaed to simple foced 9

57 convection. Mois and Gaimella [83] computationally investigated the flow fields in the oifice and the confinement egions of a nomally impinging, axis symmetic, confined and submeged liquid jet. Teng et al. [84] numeically studied a seies of confined impinging tubulent slot jet models. Eight tubulence models, including one standad and seven low Reynolds numbe k ε models wee employed and tested to pedict the heat tansfe pefomance of multiple impinging jets. Chattejee and Devipasath [85] numeically investigated the heat tansfe to a lamina impinging jet at small nole to plate distances. Li and Gaimella [86] studied the effects of fluid themo physical popeties on heat tansfe fom a confined and submeged impinging jet. Local heat tansfe coefficients wee obtained expeimentally fom a discete heat souce. Genealied coelations fo heat tansfe wee epoted fo the Pandtl numbe ange of Rahman et al. [87] numeically evaluated the conjugate heat tansfe of a confined jet impingement ove a stationay disk using liquid ammonia as the coolant. Ichimiya and Yamada [88] pesented the heat tansfe and fluid flow chaacteistics of a single cicula lamina impinging jet, including buoyancy effect in a compaatively naow space with a confining wall. They identified the pesence of foced, mixed, and natual convection modes of heat tansfe as the flow moved downsteam in the adial diection. Tempeatue distibution and velocity vectos in the space wee obtained numeically. The flow and heat tansfe chaacteistics in the cooling of a heated suface by impinging slot jets wee investigated numeically by Sahoo and Shaif [89]. Computations wee done fo vetically downwad diected two dimensional slot jets impinging on a hot isothemal suface at the bottom and confined by a paallel adiabatic suface on top. The 30

58 local and aveage Nusselt numbes and skin fiction coefficients at the hot suface fo vaious conditions wee pesented. Qing Guang et al. [90] studied the flow chaacteistics associated with a thee dimensional lamina impinging jet issuing fom a squae pipe nole. The authos discussed the flow field chaacteistics fo diffeent nole to plate spacing and Reynolds numbes. El Gaby and Kaminski [91] pesented expeimental measuements of local heat tansfe distibution on smooth and oughened sufaces unde an aay of angled impinging jets. Liquid cystal video themogaphy was used to captue suface tempeatue data at five diffeent jet Reynolds numbes anging fom 15,000 to 35,000. Heat tansfe fom a ow of tubulent jets impinging on a stationay suface was investigated by Salamah and Kaminski [9]. The geometic paametes of the jet aay and the effects of Reynolds numbe wee examined as pat of this study. Rahman and Mukka [93] developed a numeical model fo the conjugate heat tansfe duing vetical impingement of a two dimensional (slot) submeged confined liquid jet using liquid ammonia as the woking fluid. Lin et al. [94] caied out a seies of expeimental investigations on tansient and steady state cooling pefomance of heat sinks with a confined slot jet impingement..5 Patially confined Jet Impingement Thomas et al. [95] measued the film thickness acoss a stationay and otating hoiontal disk using the capacitance technique, whee the liquid was deliveed to the disk by a contolled semi confined impinging jet. The aim was to povide an undestanding of the fundamental hydodynamics pocesses which occu in the flow. Rahman and Faghi [96] pesented the esults of a numeical simulation of the flow field and associated heat tansfe coefficient fo the fee suface flow of a thin liquid film 31

59 adjacent to a hoiontal otating disk. The computation was pefomed fo diffeent flow ates and otational velocities using a thee dimensional bounday fitted coodinate system. Al Sanea [97] pesented a numeical model that studied thee cases: fee jet, semi confined jet and semi confined jet impingement though a cossflow fo lamina slot jet impinging on an isothemal flat suface. Rahman and Faghi [98] analyed the pocesses of heating and evapoation in a thin liquid film adjacent to a hoiontal disk otating about a vetical axis at a constant angula velocity. The fluid emanated axis symmetically fom a souce at the cente of the disk and was caied downsteam by inetial and centifugal foces. Faghi et al. [99] expeimentally, analytically, and numeically studied the heat tansfe effect fom a heated stationay o otating hoiontal disk to a liquid film fom a contolled impinging jet unde a patially confined condition fo diffeent volumetic flow ates and inlet tempeatues fo both supecitical and subcitical egions. Rahman [100] pesented a theoetical analysis of the gas absoption pocess of a thin liquid film fomed by the impingement of a patially confined liquid jet at the cente of the disk and the subsequent adial speading of the liquid along the suface of a hoiontal otating disk. Shi et al. [101] pesented a numeical study to examine the effects of themo physical popeties fo semi confined lamina slot jet. The fluid Pandtl numbe anged fom 0.7 to 71. Local, stagnation, and aveage values of the impingement Nusselt numbe wee epoted. Dano et al. [10] investigated the flow and heat tansfe chaacteistics of confined jet aay impingement with cossflow. Digital paticle image velocimety and flow visualiation wee used to detemine the flow chaacteistics. Lallave and Rahman 3

60 [103] numeically studied the conjugate heat tansfe fo a patially confined liquid jet impinging on a otating and unifomly heated solid disk of finite thickness and adius..6 Chemical Mechanical Polishing Pocess The CMP pocess was applied pimaily on silica (SiO ) and tungsten layes. The eve inceasing demand in the semiconducto industy fo high pefomance micoelectonics has esulted in the fabication of inceasingly complex, dense and miniatuied devices and cicuits [104]. This event has unlocked the doos to a lage vaiety of polishing mateials such as Al, Cu, Ti, TiN, Ta, W, and thei alloys, and insulatos such as Si 3 N 4, polysilicon and polymeic low κ mateials that ae cuently used as pat of the CMP pocess development. CMP has been adopted in the following thee aeas of integated cicuits (IC) fabications: The fist aeas includes the intelaye dielectic (ILD) and inte metal dielectic (IMD) planaiation to fom inteconnections between devices duing multilevel metalliation (MLM). The second aea coves the coppe damascene pocess and the thid aea involves the pocess of shallow tench isolation (STI). In fact, the CMP pocess in the cuent semiconducto device manufactuing industy needs to be optimied in all the aspects of polishing. Specifically, defects induced duing the polishing pocess such as non unifomity, dishing and eosion, need to be educed in ode to get good yields and thus lowe opeational costs. Impoving wafe scale unifomity would at the least educe many defects duing polishing. A citical step of the CMP pocess optimiation equies the pope undestanding of how diffeent paametes influence the complex function of planaiation. 33

61 The chaacteiation of chemical mechanical polishing (CMP) pocess in ecent yeas has taditionally focused on the use of Peston s equation to model the mechanics of the polishing pocess. Fu and Chanda [105] pesented an analytical expession fo the pessue distibutions at the wafe and pad inteface duing the CMP pocess. Thei pofiles wee used to detemine the MRR using Peston s Equations. Thei analytical model was compaed with the FEM simulations and expeimental data obsevations. The volume emoval ate popeties fo a floating polishing pocess unde diffeent lubicating conditions wee investigated by Su [106]. These lubicating conditions ae those that make the pad in non contact with the wok suface. Su s pape ties to confim the lubicating hypothesis and the two possible oles of the abasive paticles on the volume emoval ate (RR) of the film suface. Su [106] study esult suggests that the high emoval ate occus at the lubication nea the bounday between the iso viscous elastic (IE) and iso viscous igid (IR) egimes. Zhou et al. [107] expeimentally investigated the intefacial fluid pessue and fiction effects duing the polishing pocess. An analytical model was developed to pedict the magnitude and the distibution of this fluid pessue. The effects of pocess vaiables such as nomal load, elative velocity, pad suface oughness and modulus, fluid viscosity, and taget suface cuvatue, wee studied by compaing the 1D fluid pessue distibutions. The effects of the sub ambient fluid pessue on the mateial emoval ate and the pofile with themally gown SiO on single cystal silicon wafes wee shown as pat of thei esults. Luo and Donfeld [108] numeically investigated the abasion mechanism in solid to solid contact mode fo CMP pocess. Based on assumptions of plastic contact ove wafe abasive and pad abasive intefaces, the poposed model integates pocess 34

62 paametes like velocity and pessue. In addition, it integates input paametes, such as wafe and pad hadness, pad oughness, abasive sie and abasive geomety into the same fomulation to pedict the mateial emoval ate (MRR=ρ w Nol emoved ). The expeimental esults of the mateial emoval ate wee compaed with the suggested model, showing how accuately it pedicts the mateial emoval ate. Much wok has been done to incopoate the oughness effect into lubication. Pevious studies that quantify the suface oughness effect [109, 110] of lage systems with small topogaphies poved to be computationally exhaustive, even though getting the topogaphy inticate details of the system expeimentally could be difficult and impactical. Hence, some wok [ ] has been done to employ stochastic concepts to solve the poblem. Most of these models ae limited to one dimensional idges oiented eithe tansvesely o longitudinally. It is difficult to extend to thee dimensional o anisotopic oughness using the stochastic appoach. Thee is also a petubation method [116, 117] to model oughness in lubication. Fu et al. [118] pesented the behavio of the hydoxylated laye by a pefectly plastic mateial and mechanistic model fo the mateial emoval ate (MRR) duing a CMP pocess. The plasticity model was utilied to exploe the effects of vaious design paametes (e.g., abasive shape, sie and concentation, and pad igidity) on the MRR. Thei model took into account the dependence of pessue and elative velocity, plus delineated the effects of pad and sluy popeties. Thakuta et al. [119] pesented a thee dimensional chemical planaiation sluy model based upon the lubication theoy, using the genealied Reynolds equation that includes pad poosity and bending. Thei model calculated the sluy film thickness and sluy velocity distibution between the wafe and 35

63 pad, with the minimum sluy film thickness detemining the degee of contact between the wafe and pad. In addition, the minimum sluy film thickness was examined ove a ange of input vaiables, namely, applied pessue, caie and pad velocity, wafe adius and cuvatue, sluy viscosity, and pad poosity and compessibility. Yang [10] developed a model fo the CMP of coppe dual damascene based on the multi step, multi sluy pocess platfom. His model pedicted coppe dishing and ILD eosion fo thee steps coppe CMP. The fist step involved fast coppe emoval sluy, the second, a low pessue step fo coppe cleaing, and a final step fo diffusion baie emoval. Even though a numbe of publications have been consideed, most of them pimaily focus on othe aspects of the CMP pocess such as film stess, patten dependencies, pad oughness, mateial emoval ate, abasive paticles, sluy film taxonomy, chemicals effects, and pessue and velocity distibutions. Only a few examine the themal effects duing the planaiation pocess ove the wafe suface. The fist attempt to measue the tempeatue on the silicon coppe wafe was done by Sampuno et al. [11]. A diect tempeatue measuement set up was developed wheein a novel wafe caie was designed such that the tempeatue on the back side of the wafe was measuable using a themal imaging infa ed (IR) camea. Howeve, in all these eseach woks, the epoted tempeatue ise is eithe the aveage tempeatue on the pad suface, a pedicted aveage tempeatue on the wafe suface, o the tempeatue ise at diffeent isolated locations on the wafe. These woks epoted the oveall tempeatue ise but did not povide infomation about the tempeatue distibutions o contou plots along the substate and pad sufaces. Since the mateial emoval ate duing coppe CMP is so sensitive to tempeatue, tempeatue distibutions ove the entie wafe will significantly 36

64 affect the unifomity of mateial emoval at the substate suface. Undestanding the tempeatue pofile will decease the with in wafe non unifomity and theefoe impoving the yield by minimiing the numbe of faulty dies. The activation enegy of the coppe oxidation eaction in the sluy is vey low [1, 13]. Heat dissipation due to fiction can esult in a tempeatue ise at the inteface and a ise of about 10 K at the polishing inteface; it is high enough to double the emoval ate duing coppe polishing [14, 15]. Also, it has been noted that a change of 1 K can affect the pocess emoval ate duing polishing by 7% [14]. Factions of heat geneated at the inteface ae eithe conducted to the wafe and pad, o convected away by the sluy, which acts as a coolant at the inteface. The themal aspect of CMP even though it is a significant facto that affects the pocess output, has not been eseached as extensively as othe paametes like pessue, velocity, sluy flow ate, and chemical aspects. Reseach wok on polishing pads duing the intelaye dielectic (ILD) and metal polishing pocesses that includes the emoval ate dependence on tempeatue and the effect of sluy flow ate on wafe and pad tempeatue ise, etc., has been caied out in the ecent past to undestand the ole of tempeatue at the inteface on CMP pefomance [14, 16 19]. Boucki et al. [17, 18] developed a themal model fo ILD polishing and then modified it slightly to get a model fo coppe CMP, which was validated by compaing with tempeatue measuements on the pad duing metal CMP. They developed a theoetical undestanding of the themal aspects in thei eseach and pedicted the tempeatue on the pad fo the initial stages (fist 60 seconds) of polishing by evaluating the model based on tansient heat tansfe mechanism. White et al., [19] 37

65 have modeled the dynamic themal behavio, which explains the enegy exchange between the pad and sluy. Heat accumulation in the pad and the convection of heat to the sluy wee explained in thei eseach wok. In addition, a tansient themal model was poposed to explain the initial behavio obseved duing CMP. Soooshian et al. [130] investigated the effect of heat geneation and themal inputs on the fictional chaacteistics of intelaye dielectic (ILD) and coppe CMP pocesses. Thei coefficient of fiction esults indicated an inceasing tend fo ILD and coppe polishing tempeatue. The dynamic mechanical analysis of the polishing pads evealed links between the softening effects of the pad, with ising tempeatues, and the incement of shea foces esulting fom the contact of the pad and wafe duing polishing. Additional eseach woks on themal aspects that used the tempeatue change as an end point detection, and expeimental woks that involved the tempeatue ise on polishing pad [ ], can also be found in the liteatue. Howeve, infaed ed (IR) expeiments and the pesented numeical esults showed that themal behavio of the sluy aound the caie and acoss the pad wafe inteface is still a complex and dynamic pocess. The tempeatue pofiles on the pad and wafe sufaces as a function of adius unde the influence paametes like sluy flow ate, pad and caie spinning ates, sluy film thickness, and polishing pessue will povide valuable insight into the extent of tempeatue ise at diffeent locations on the wafe. 38

66 Chapte 3 Mathematical Models and Computation 3.1 Fee Liquid Jet Impingement Model The physical poblem coesponds to an axis symmetic liquid jet that impinges on a solid spinning disk, as shown in figue 3.1. The fee jet dischages fom the nole and impinges pependiculaly at the cente of the top suface of the disk while the bottom suface is subjected to a constant heat flux. The pesent study consideed an incompessible, Newtonian, and axis symmetic flow unde a steady state condition. Nole d n j H n θ =Ω b Ω θ q w Spinning disk Figue 3.1 Thee dimensional schematic of axis symmetic fee liquid jet impingement on a unifomly heated spinning disk. 39

67 Fluid popeties fo H O, NH 3, MIL 7808, and FC 77 wee obtained fom Bejan [134], Bady vendo, and 3M Specialty Fluids espectively. The fluid popeties such as density, viscosity, themal conductivity, and specific ae assumed to be constant fo the tempeatue ange encounteed in the system. The initial jet flow tempeatue condition of Ammonia was set to a lowe value due to the feasibility of the fluid to emains in the liquid state duing the pocess at nomal atmospheic pessue conditions. In tems of MIL 7808 the initial tempeatue value o jet flow conditions was set to a hotte tempeatue to educe the viscosity effect of the fluid and pevent any clogging issues on such small jets o nole to taget spacing atios. The themo physical popeties of the solid mateials used fo the numeical analysis ae assumed to emain constant ove the woking tempeatue ange, as shown in Table 3.1. Table 3.1 Constant themo physical popeties used fo computational analysis. Mateial Refeence Tempeatue T(K) Density ρ(kg/m 3 ) Dynamic viscosity µ (kg/m s) P Conductivity k(w/mk) Specific Heat C p (J/kgK) Constantan 303 8, Coppe 303 8, Silicon 303, Aluminum Silve , Wate x ,179 Ammonia x ,460 MIL ,159 FC , ,047 40

68 Govening Equations and Bounday Conditions: Steady State Cooling of Spinning Taget Due to otational symmety of the poblem the /θ tems could be omitted. The equations descibing the consevation of mass, momentum (,θ and diections espectively), and enegy can be witten as Schlichting [135]: 0 = (3.1.1) = f f θ 1 ν p ρ 1 (3.1.) = θ θ θ θ f θ θ θ 1 ν (3.1.3) = f f 1 ν p ρ 1 g (3.1.4) α = T f T f 1 f T f T f (3.1.5) The vaiation of themal conductivity of solids with tempeatue encounteed in the poblem was not significant. Theefoe, the consevation of enegy inside the solid can be chaacteied by the following equation: 0 T s T s 1 T s = (3.1.6) The following bounday conditions wee used to complete the physical poblem fomulation. 0 T s 0 : b 0, At = = (3.1.7)

69 T At = 0, 0 H : 0, 0, f n θ = = = = 0 (3.1.8) T At =, b 0 : s d = 0 (3.1.9) At = d, 0 δ : p = p atm (3.1.10) T At = b, 0 : k s d s = q w (3.1.11) At T s T = 0, 0 :, 0,T T,k k f d θ = Ω = = s = f s = f (3.1.1) At = H n, 0 d n : = j, = θ = 0,T f = T j (3.1.13) The bounday condition at the fee suface can be expessed as: At = δ, d n d dδ d d δ σ d s T =, p = p, 0, f atm = = 0 (3.1.14) 3 n n dδ 1 d whee s is the fluid velocity component along the fee suface and n is the coodinate nomal to the fee suface. The bounday conditions at the fee suface d n d include the kinematic condition and balance of nomal and shea stesses. The kinematic condition elates the velocity components to local slope of the fee suface. The nomal stess balance takes into account the effects of suface tension. In the absence of any significant esistance fom the ambient ai, the shea stess encounteed at the fee suface is essentially eo. Similaly, a negligible heat tansfe esults in eo tempeatue gadient at the fee suface. The local and aveage heat tansfe coefficients can be defined as: 4

70 k T T h = s int 1 (3.1.15) int 1 T T int j h av d = h( T T ) d ( ) int j (3.1.16) d Tint T j 0 whee T int is the aveage tempeatue at the solid liquid inteface. The local and aveage Nusselt numbes ae calculated accoding to the following expessions: h d Nu = n (3.1.17) k f Nu av h d = av n (3.1.18) k f 3.1. Govening Equations and Bounday Conditions: Tansient Cooling of Spinning Taget At t=0, the powe supply is tuned on and the heat is supplied to the bottom suface of the disk stating with an isothemal solid disk and fluid flow that has been established on the disk due to jet impingement. The pesent study consideed an incompessible, Newtonian, and axis symmetic fluid flow. The fluid popeties wee dependent on tempeatue only. The popeties of the above fluids in section 3.1 wee coelated accoding to the following equations. Fo wate between 300 K<T<411 K; Cp f =9.5x10 3. T T5098.1; k f = 7.0x10 6. T 5.8x10 3. T ; ρ f =.7x10 3. T T848.07; and ln (µ f ) = T. Fo ammonia between K<T<370 K; Cp f = T T9468; k f = x10 3. T; ρ f = T T ; and ln (µ f ) = T. Fo MIL 7808 between 303 K<T<470 K; Cp f = T; k f =0.18 1x10 4. T; ρ f = T; and ln (µ f ) = T. 43

71 44 Fo FC 77 between 73 K<T<380 K; Cp f = T; k f = x10 5. T; ρ f =, T; and ln (µ f ) = T. The initial jet flow tempeatue condition of Ammonia was set to a lowe value due to the feasibility of the fluid to emains in the liquid state duing the pocess at nomal atmospheic pessue conditions. In tems of MIL 7808 the inlet jet flow tempeatue value was set to a hotte tempeatue to educe the viscosity effect of the fluid and pevent any clogging issues on such small jets o nole to taget spacing atios. In these coelations, the absolute tempeatue T was used in K. Due to otational symmety of the poblem the /θ tems could be omitted. The equations descibing the consevation of mass, momentum (,θ and diections espectively), and enegy can be witten as Bumeiste [136]: ( ) ( ) 0 ρ f ρ f 1 t ρ f = (3.1.19) = p t ρ f θ µ f 3 µ f µ f 3 1 (3.1.0) = θ µ f θ µ f 1 θ θ θ t θ ρ f (3.1.1) = µ f 1 p ρ f g t f ρ µ f 3 (3.1.) ( ) ( ) = T f k f T f k f 1 Cp f T f Cp f T f t T f ρ f θ θ 1 µ f θ 1 (3.1.3)

72 T s t The consevation of enegy inside the solid can be defined as: T 1 T T = α s s s s (3.1.4) Equations ( ) wee subjected to the bounday conditions descibed by equations ( ). The solid disk was assumed to be at themal equilibium with jet fluid befoe the tansient heating of the plate was tuned on. The velocity field at this condition was detemined by solving only the continuity and momentum equations ( ) in the fluid egion. Thus, = (3.1.5) At t=0: T f =T s =T j, ( isothemal) i To complete the mathematical fomulation it is necessay to define diffeent elevant paametes, such as local and aveage heat tansfe coefficients, and local and aveage Nusselt numbes. The local and aveage heat tansfe coefficients and Nusselt numbes can be defined accoding to equations ( ). 3. Confined Liquid Jet Impingement Model A thee dimensional epesentation of the confined axial jet impinging pependiculaly on a unifomly heated spinning solid wafe coesponds to two paallel disks, as shown in figue 3.. The liquid jet is dischaged though an oifice at the cente of the top disk. The emainde of the top disk acts as an insulated stationay confinement plate. The bottom disk (wafe) is subjected to a unifom otational velocity. Heat souces ae located at the bottom of the wafe poducing a constant heat flux along the suface. Heat is conducted though the wafe and convected out to the fluid adjacent to the top suface of the wafe, as shown in figue 3.. The pesent study consideed an incompessible, Newtonian, and axis symmetic flow unde a steady state condition. 45

73 Nole d n Confined disk H n j θ =Ω b Ω θ d q w Spinning taget Figue 3. Thee dimensional schematic of a confined axial jet impinging on a unifomly heated and spinning disk. Fluid popeties fo H O, NH 3, MIL 7808, and FC 77 wee obtained fom Bejan (1995) [134], Bady vendo, and 3M Specialty Fluids espectively. The fluid popeties wee assumed to be constant fo the tempeatue ange encounteed in the system, as shown in Table Govening Equations and Bounday Conditions: Steady State Cooling of Spinning Taget Due to otational symmety of the poblem the /θ tems could be omitted. The equations descibing the consevation of mass, momentum (,θ and diections espectively), and enegy can be witten as [135]: = 0 (3..1) 46

74 47 = f f θ 1 ν p ρ 1 (3..) = θ θ θ θ f θ θ θ 1 ν (3..3) = f f 1 ν p ρ 1 g (3..4) = α f T f T 1 f f T f T (3..5) The vaiation of themal conductivity of solids with tempeatue encounteed in the poblem was not significant. Theefoe, the consevation of enegy inside the solid can be chaacteied by the following equation: 0 T s T s 1 T s = (3..6) To complete the set of equations to be solved, equations ( ) wee subjected to the following bounday conditions: 0 T s 0 : b 0, At = = (3..7) 0 T f 0, 0, θ : H n 0 0, At = = = = = (3..8) 0 T s 0 : b, At d = = (3..9) 0 p : H n 0, At d = = (3..10) q w T s k s : 0 b, At d = = (3..11)

75 At = H n, 0 d n : = j, = θ = 0,T f = T j (3..1) T At = H d : 0, f n, n d = = θ = = 0 (3..13) At T s T = 0, 0 :, 0,T T, k k f d θ = Ω = = s = f s = f (3..14) The local and aveage heat tansfe coefficients can be defined as: q h = w (3..15) T int T j h av d = h( T T ) d ( ) int j (3..16) d Tint T j 0 whee T int is the aveage tempeatue at the solid liquid inteface. The local and aveage Nusselt numbes ae calculated accoding to the following expessions: h d Nu = n (3..17) k f Nu av h d = av n (3..18) k f 3.. Govening Equations and Bounday Conditions: Steady State Cooling of Spinning Wall A schematic of the physical poblem is shown in figue 3.3. An axis symmetic liquid jet is dischaged though a nole and impinges at the cente of a stationay solid disk subjected to a unifom heat flux. The top plate acts as an insulated confinement suface spinning at constant angula velocity. Heat is conducted though the disk and convected out to the fluid adjacent to the top suface of the stationay disk, as shown in 48

76 49 figue 3.3. The pesent study consideed an incompessible, Newtonian, and axis symmetic flow unde a steady state condition. The vaiation of fluid popeties with local tempeatue was taken into account. Due to otational symmety of the poblem the /θ tems could be omitted. Figue 3.3 Thee dimensional schematic of axis symmetic confined spinning disk liquid jet impingement on a unifomly heated disk. The equations descibing the consevation of mass, momentum (,θ and diections espectively), and enegy can be witten as seen in [136]: ( ) ( ) 0 f ρ f ρ 1 = (3..19) = f µ 3 1 p f ρ θ f µ 3 f µ (3..0) Nole Stationay taget θ =Ω Spinning confined wall q w b d n Ω H n j θ d

77 50 = θ f µ θ f µ 1 θ θ θ f ρ (3..1) = f µ 1 p g f ρ f ρ f µ 3 (3..) ( ) ( ) = f T f k f T f k 1 f T f Cp f T f Cp f ρ θ θ 1 µ f θ 1 (3..3) The vaiation of themal conductivity of solids with tempeatue is not significant. Theefoe, the consevation of enegy inside the solid can be chaacteied by the following equation: 0 s T s T 1 s T s α = (3..4) To fulfill the physical fomulation of equations ( ) it is necessay to use the bounday conditions descibed by equations ( ) and update the bounday conditions of the taget and confined wall cuently defined by equations ( ). 0 T f 0,, θ : d n H n, At d = = = = Ω = (3..5) T f k f T s k s, T f T s 0, θ : 0 0, At d = = = = = = (3..6) The mathematical fomulation was completed with the definition of elevant paametes, such as local and aveage heat tansfe coefficients, and local and aveage Nusselt numbes accoding to equations ( ).

78 3.3 Patially confined Liquid Jet Impingement Model An axis symmetic liquid jet is dischaged though a nole and impinges at the cente of the top suface of a solid cicula disk o wafe spinning with a unifom angula velocity about the axis. w Figue 3.4 Thee dimensional schematic of axis symmetic patially confined liquid jet impingement on a unifomly heated spinning disk. The insulated confinement plate attached to the nole is smalle in diamete than the disk which allows the fomation of fee suface flow when the fluid exits the confined egion, as shown in figue 3.4. The pesent study consideed an incompessible, Newtonian, and axis symmetic lamina flow unde a steady state condition. The vaiation of fluid popeties with local tempeatue was taken into account. 51

79 Govening Equations and Bounday Conditions: Steady State Cooling of Spinning Taget Due to otational symmety of the poblem the /θ tems could be omitted. The equations descibing the consevation of mass, momentum (,θ and diections espectively), and enegy can be witten as [136]: ( ) ( ) 0 f ρ f ρ 1 = (3.3.1) = µ f 3 1 p θ ρ f f µ 3 f µ (3.3.) = θ f µ θ f µ 1 θ θ θ f ρ (3.3.3) = f µ 1 p g f ρ f ρ f µ 3 (3.3.4) ( ) ( ) = f T f k f T f k 1 f T f Cp f T f Cp f ρ θ θ 1 µ f θ 1 (3.3.5) The consevation of enegy inside the solid can be defined as: 0 s T s T 1 s T s α = (3.3.6) The following bounday conditions wee used. 0 T s 0 : b 0, At = = (3.3.7)

80 At = 0, T 0 H : 0, 0, f n θ = = = = 0 (3.3.8) At = d, T b 0: s = 0 (3.3.9) At = d, 0 δ : p = p atm (3.3.10) T At = b, 0 : k s d s = q w (3.3.11) At = H n, 0 d n : = j, = θ = 0,T f = T j (3.3.1) T At = H d : 0, f n, n p = = θ = = 0 (3.3.13) At T s T = 0, 0 :, 0,T T, k k f d θ = Ω = = s = f s = f (3.3.14) The bounday condition at the fee suface can be expessed as: At = δ, p d : dδ d d δ σ d s T =, p = p, 0, f atm = = 0 (3.3.15) 3 n n dδ 1 d whee S is the fluid velocity component along the fee suface and n is the coodinate nomal to the fee suface. The bounday conditions at the fee suface wee obtained by satisfying the kinematic condition elating the slope of the fee suface with velocity components as well as fom the balance of nomal and shea stesses at the fee suface. Fo steady flow of a Newtonian fluid nomal stess balance essentially educes to an 53

81 equation elating the pessue and suface tension as shown by White [137]. The shea stess encounteed fom the ambient gaseous medium is expected to be negligible. Similaly, the heat tansfe fom the fee suface to the ambient gas is also assumed to be negligible. Relevant paametes, such as local and aveage heat tansfe coefficients and local and aveage Nusselt numbes ae defined accoding to equations ( ) to complete the mathematical fomulation Govening Equations and Bounday Conditions: Steady State Cooling of Co Rotating Taget and Confined Wall An axis symmetic liquid jet is dischaged though a nole and impinges at the cente of a solid unifomly heated cicula disk that spins at constant angula velocity about the axis, as shown in figue 3.5. The insulated top plate acts as a confined spinning wall that ends allowing the exposue of the fluid to a fee suface bounday condition. w Figue 3.5 Thee dimensional schematic of axis symmetic patially confined liquid jet impingement on a unifomly heated disk with two spinning boundaies. 54

82 The pesent study consideed an incompessible, Newtonian, and axis symmetic flow unde a steady state condition. The vaiation of fluid popeties with local tempeatue was taken into account. Fluid popeties fo H O, NH 3, MIL 7808, and FC 77 wee obtained fom Bejan [134] and Bula [138]. The physical fomulation of the above poblem is defined in section The equations that descibed the consevation of mass, momentum ( ) and bounday conditions (3.3.7 though 3.3.1, and ) emain the same. The new bounday conditions at the taget and confined wall wee defined by the following equations: T At = H d : 0,, f n, n p = = θ = Ω = 0 (3.3.16) T s T At = 0, 0 : 0,,T T, k k f d = = θ = Ω1 s = f s = f (3.3.17) The mathematical fomulation was completed by the definition of elevant paametes, such as local and aveage heat tansfe coefficients, and local and aveage Nusselt numbes. In addition, the local and aveage heat tansfe coefficients and Nusselt numbes ae defined accoding to equations ( ) Govening Equations and Bounday Conditions: Tansient Cooling of Spinning Taget The tansient conjugate heat tansfe of both solid and fluid egions of a patially confined liquid jet impinging on a otating and unifomly heated solid disk of finite thickness and adius ae examined as pat of this study. A constant heat flux was imposed at the bottom suface of the solid disk at t=0 and heat tansfe was monitoed fo the entie duation of the tansient until the steady state condition was eached. 55

83 56 Afte an isothemal fluid flow has been established on the disk, at t=0, the powe souce is tuned on to delive a unifom heat flux at the bottom suface of the disk while the confinement plate is kept insulated. Due to symmety of the poblem about the axis of otation, all /θ tems can be dopped out. The equations fo the consevation of mass, momentum (,θ and diections espectively), and enegy fo incompessible flow of a Newtonian fluid with tempeatue dependent popeties can be witten as [136]: ( ) ( ) 0 ρ f ρ f 1 t ρ f = (3.3.18) = p t ρ f θ µ f 3 µ f µ f 3 1 (3.3.19) = θ µ f θ µ f 1 θ θ θ t θ ρ f (3.3.0) = µ f 1 p ρ f g t f ρ µ f 3 (3.3.1) ( ) ( ) = T f k f T f k f 1 Cp f T f Cp f T f t T f ρ f θ θ 1 µ f θ 1 (3.3.) The consevation of enegy inside the solid can be defined as: = T s T s 1 T s α s t T s (3.3.3) Equations ( ) ae subjected to the bounday conditions descibed by equations ( ). The solid disk was assumed to be at themal equilibium with

84 jet fluid befoe the tansient heating of the plate was tuned on. The velocity field at this condition was detemined by solving only the continuity and momentum equations ( ) in the fluid egion. Thus, = (3.3.4) At t=0: T f =T s = T j, ( isothemal) i To complete the mathematical fomulation it is necessay to define diffeent elevant paametes, such as local and aveage heat tansfe coefficients, and local and aveage Nusselt numbes. In addition, the local and aveage heat tansfe coefficients and Nusselt numbes ae defined accoding to equations ( ). 3.4 Thee Dimensional Chemical Mechanical Polishing Model The contolled volume unde study of the CMP pocess, sketched in figue 3.6, consists of the wafe suface, sluy inteface and polishing pad subjected to a vaiable heat flux bounday condition at its polished suface. The vaiable heat flux is diven by the pad coefficient of fiction, the down foce pessue, the adial distance measue fom the pad cente and the elative spinning ate of the pad and caie (q w =µ. f P. θ pc ). Contol olume Wafe Ω c Wafe cente 1 pw 1 Sluy outlet δ sl θ Sluy inlet θ 1 Polishing pad Ω p q w (aiable heat flux) d w Figue 3.6 Thee dimensional CMP contol volume outline. 57

85 Figue 3.6 shows that the inlet of the sluy (alumina) coves half of the wafe cicumfeence, and the othe half is consideed as being the flow outlet. The sluy flow is diven to the inlet by the spinning ate of the platen that holds the polishing pad. The centifugal motion assumption of the sluy is valid fo the closeness of the bounday laye thickness of the flow that is going to pass though a confined aea with a magnitude of the micomete scale sie. This obsevation is in ageement with wok done by Lallave and Rahman [103], and Bodesen et al. [38] that studied the chaacteistics of a pedominant otational diven flow vesus a jet impingement momentum flow. A contact aea of a flat pad suface was used as pat of the contol volume on this model. The gavity and suface tension effects and angula acceleation of the platen was taken into account as pat of the sluy film thickness desciption. The pesent model ignoes the non unifomity of the sluy paticles and thei height distibution, including the heat tansfe effect duing conditioning and all losses of heat along the wafe etaining ing. The offset thickness between the ing and wafe is not taken into account as pat of the CMP model set up. As pat of the numeical analysis and expeimental set up, this model stats with an isothemal sluy to substate bounday condition and a thin laye of sluy that has been established on the pad as pat of the polishing pocess. In addition, a vaiable heat flux is input into the numeical poblem as the poduct of the mechanical abasion of the pad, and chemical inteactions of the sluy at the substate suface Govening Equations and Bounday Conditions: Steady State The Navie Stokes equations wee used to simulate the fluid mechanics of an incompessible (constant density and viscosity) Newtonian flow that eaches the steady 58

86 59 state condition thoughout the CMP pocess. The fluid popeties used fo the numeical simulation such as density, viscosity, themal conductivity, and specific heat ae assumed to emain constant ove the woking tempeatue ange. Detailed explanations on the fomulation of the govening equations descibing the consevation of mass, momentum (,θ, and diections espectively), and enegy using cylindical coodinate system can be found in [136]: 0 1 = θ θ (3.4.1) θ θ = θ θ θ 1 1 ν p ρ 1 sl sl θ (3.4.) θ θ θ ρ = θ θ θ θ θ θ θ sl sl θ θ θ θ 1 1 ν p 1 (3.4.3) θ = θ θ sl sl 1 1 ν p ρ 1 g (3.4.4) The Navie Stokes equations ( ) ae useful fo the hydodynamic egime whee the combined oughness (s), of the two opposing sufaces is smalle than the film thickness, h sl, and thee is little o no contact between the aspeities of the sufaces. Fo the sluy film and oughness atio (h sl /s) >> 3, the oughness effects ae not impotant and the smooth film Navie Stokes equations ae sufficiently accuate. When the sluy film and oughness atio (h sl /s) ae equal to 3, the oughness effects become impotant. When h sl /s < 3, contacts between aspeities fom the opposing sufaces can occu and the system goes into the mixed lubication egime. In CMP, the sufaces involved ae a elatively flat and igid wafe beneath a ough and soft pad. Anothe sign

87 of intimate wafe and pad contact is when pad glaing is obseved. Thus, a mixed lubication appoach has to be taken [139]. The enegy equation fo incompessible sluy popeties and negligible viscous dissipation can be witten as: T sl θ Tsl Tsl 1 Tsl 1 Tsl Tsl = α sl θ (3.4.5) θ The enegy tansfeed in the contolled volume is due to mechanical abasion of the pad and sluy paticles on the wafe suface, and chemical enegy associated with sluy chemisty and enthalpy. Fo neutal sluies, the majo chemical enegy souce is the enthalpy. In a good numbe of themal systems, thee exist heat loss mechanisms that can be neglected as pat of the analysis. The themal losses fom the ubbe and plastic bladde between the wafe and steel polishing caie wee neglected due to thei lowe themal conductivity and insulato popeties. As we know, most of the pads used fo expeimentation, like IC 1,000 and FX 9, ae made of polyuethane, a mateial consideed to be a themal insulato as such. Theefoe, we neglect any loss fom conduction though the pad as pat of ou numeical analysis. The above analysis concentates on the heat loss mechanisms associated with download pessue, sluy flow ate, elative spinning velocity of pad and wafe, and sluy film thickness unde diffeent coefficients of fiction. To define the poblem completely, appopiate bounday conditions wee equied on all boundaies of the computational domain. The bounday conditions at the inlet, outlet, wafe suface, and pad suface espectively have the following fom: π π At (w pw) 1 pw, hsl 0, θ : = Ωp 1, = θ = 0,Tsl = Tj (3.4.6) 60

88 π π At w 0, hsl 0, θ : p = patm (3.4.7) At At T T 0,,0 :, 0,k sl = P k w w w θ π θc =Ωc = = sl =µ f θ w pc = (3.4.8) T p = hsl, (pw dw) 1 pw,0 θ π:θp =Ωp 1, = = 0, = 0 (3.4.9) Figue 3.7 Wafe pad elative velocity pofile. The elative co tangential velocity effect of both sufaces along the two axes of otation is shown in figue 3.7. The co tangential velocity effect (θ pc ) was used to detemine the magnitude of the vaiable heat flux input into the system. The caie pad elative velocity in cylindical coodinates system with the oigin at the pad cente can be deived as: θpc ( Ω Ω ) sin θ [( Ω Ω ) cosθ Ω ] = (3.4.10) p c p c p

89 The numeical equations ( ) wee subjected to the bounday conditions descibed by equations ( ) as pat of the mathematical fomulation of pesent model. Subsequently, the steady state tempeatue contous o pofile as the index of enegy dissipation along the cente of wafe suface, and pad inteface ae plotted as pat of the CMP model solution. In addition, the local heat tansfe coefficients fo wafe and pad ae calculated accoding to the following expessions: h w = µ f P θpc ( Tw Tsl ) (3.4.11) h p µ f P θpc = ( Tp Tsl ) 3.4. Govening Equations and Bounday Conditions: Tansient (3.4.1) The cuent thee dimensional CMP model of substate suface, fluid egion (sluy) and pad suface as a contol volume wee studied unde the exposue of a vaiable heat flux at t=0, due to the mechanical abasion of the pad and sluy paticles. The chemical inteactions of the sluy acts and supplies the heat onto the suface of the wafe, stating with an isothemal solid fluid bounday condition and a thin film of sluy that has been established on the wafe as pat of polishing pocess. The model desciption, including geomety, model set up, and assumptions ae descibed in section The Navie Stokes equations wee used to simulate the fluid mechanics duing the tansient stage of the CMP pocess. The sluy popeties (wate plus alumina) wee assumed to be constant fo the tempeatue ange encounteed in the system. Detailed explanations on the fomulation of the govening equations descibing the consevation 6

90 63 of mass, momentum (,θ, and diections espectively), and enegy using cylindical coodinate system can be found in [136]: The consevation of mass of the sluy can be witten in the most geneal fom as: 0 ) ( ) ( 1 ) ( 1 t sl θ sl sl sl = ρ θ ρ ρ ρ (3.4.13) The consevation of momentum (Navie Stokes equations) fo constant density and viscosity liquid o incompessible sluy popeties can be witten as: θ θ = θ θ θ 1 1 ν p ρ 1 t sl sl θ (3.4.14) θ θ θ ρ = θ θ θ θ θ θ θ sl sl θ θ θ θ θ 1 1 ν p 1 t (3.4.15) θ = θ θ sl sl 1 1 ν p ρ 1 g t (3.4.16) The enegy equation fo incompessible sluy popeties and negligible viscous dissipation can be witten as: θ = α θ θ sl sl sl sl sl sl sl sl T T 1 T 1 T T T t T (3.4.17) Equations ( ) wee subjected to the bounday conditions descibed by equations ( ). The wafe suface is assumed to be at themal equilibium with the alumina (Al O 3 ) sluy befoe the tansient heating of the polishing takes place. The velocity field at this condition is detemined by solving only the continuity and momentum equations ( ) in the fluid egion. Thus, At t=0: T sl =T p =T w, ( ) isothemal i = (3.4.18)

91 By solving the numeical poblem, the tansient tempeatue contous o pofile as the index of enegy dissipation of the wafe suface, the sluy and pad inteface, wee shown as pat of the esults. In addition, as pat of this study, tempeatue distibutions as function of time on isolated nodes wee examined fo the entie tansient pocess. This was done unde a diffeent set of physical paametes, such as sluy flow ates, polishing pessues, caie and pad spinning ates, and sluy film thicknesses. To complete the mathematical fomulation the local heat tansfe coefficients fo wafe and pad sufaces ae defined accoding to equations ( ). 3.5 Numeical Computation Fee Suface Liquid Jet Impingement Steady State and Tansient Pocess The govening equations ( ) of Pob. #5 and ( ) of Pob. #6 and the bounday conditions ( ) and (3.1.5) just fo the tansient conditions descibed in the peceding sections (3.1.1 and 3.1.), wee solved using the Galekin finite element method [140]. Fou node quadilateal elements wee used. A scaled dense gid distibution was used to adequately captue lage vaiations nea the solid fluid inteface of the meshed domain, as shown in figue 3.8. Figue 3.8 Axis symmetic fee suface liquid jet impingement mesh plot. 64

92 In each element, the velocity, pessue, and tempeatue fields wee appoximated which led to a set of equations that defined the continuum. The solution of the esulting non linea equations was caied out using the Newton Raphson method. The appoach used to solve the fee suface poblem descibed hee was to intoduce a new degee of feedom epesenting the position of the fee suface. This degee of feedom was intoduced as a new unknown into the global system of equations. Due to non linea natue of the govening tanspot equations, an iteative pocedue was used to aive at the solution fo the velocity and tempeatue fields. In ode to detemine the initial velocity field ( i ), the equations fo the consevation of mass and momentum wee solved. Since the solution of the momentum equation equied only two out of the thee bounday conditions at the fee suface, the thid condition that elates the slope of the fee suface to local velocity components at the fee suface was used to upgade the position of the fee suface at the end of each iteation step. The Newton Raphson solve used spines to tack the fee suface and e aanged gid distibution with the movement of the fee suface. These spines ae staight lines passing though the fee suface nodes and connecting the nodes undeneath the fee suface. The fee suface movement affected only nodes along the spine. Once the final fee suface height distibution and the flow field fo the isothemal equilibium condition wee eached, the powe of the heat souce was tuned on and heat began to flow. Then the computation domain included both solid and fluid egions. The continuity, momentum, and enegy wee solved simultaneously as a conjugate poblem taking into account the vaiation of fluid popeties with tempeatue. The computation coveed the entie tansient peiod all the way to the steady state 65

93 condition. Because of lage changes at the outset of the tansient and vey small changes when the solution appoached the steady state condition, a fixed time step was used to cove the ealie pat of the tansient up to 5 seconds, and a vaiable time step was used fo the est of the computation. At each time step, the solution was consideed conveged when elative change in field values fom a paticula iteation to the next, and the sums of the esiduals fo each vaiable became less than The chaacteistics of the flow ae contolled by thee majo physical paametes: the Reynolds numbe, Re j = j d n /ν f, the dimensionless nole to plate spacing atio, β=h n /d n, and the Ekman numbe, Ek=ν f /. 4. Ω d. The values of Reynolds numbe was limited to a maximum of 1,800 to stay within the lamina egion. The mateials popeties used fo the numeical simulation such as density, viscosity, themal conductivity, and specific heat ae assumed to emain constant ove the woking tempeatue ange. The popeties of the following solid mateials: Constantan, coppe, aluminum, silicon, and silve wee obtained fom Öisik [141]. The nole diamete opening was vaied ove the ange of 1.0 to 3.60 mm. The disk adius was kept at a constant value of 7.6 mm and the heat flux (q w ) was also kept constant at 50 kw/m fo steady state conditions (Pob. #5) and 15 kw/m tansient state conditions (Pob. #6). The incoming fluid jet tempeatue (T j ) was 310 K fo wate and FC 77, 303 K fo ammonia (at a pessue of 0 bas), and 375 K fo MIL The thickness of the disk was vaied ove the following values: 0.0, 0.40, 0.60, 0.90, 1., 1.5 and.0 mm. The jet impingement height o the distance between the nole and disk was set at the following values: 6.6x10 4, 9.0x10 4, 1.5x10 3,.4x10 3, 3.6x10 3, 4.8x10 3, and 6.0x10 3 m at (Pob. #5), convesely the jet impingement height was kept at a constant 66

94 value of 3. mm fo Pob. #6. Howeve, fo compaison with othe numeical and expeimental esults the impingement heights wee set to:.4, 1.5, 0.9, and 0.66 mm espectively. The spinning ate (Ω), and flow ate (Q) wee vaied fom to ad/s o 15 to 1,500 RPM and 3.360x10 7 to 1.133x10 6 m 3 /s; espectively at Pob. #5. The ange fo Reynolds numbe and Ekman numbe wee set at: Re=445 to 1,800 and Ek=.1x10 5 to.65x10 4. On Pob. #6 the spinning ate (Ω) was vaied fom 0 to 5.36 ad/s o 0 to 500 RPM, that coespond to the ange of Ekman numbe fom to 6.6x10 5. In addition, the flow ate was vaied fom 3.775x10 7 to 1.057x10 6 m 3 /s, fo a ange of Reynolds numbe fom 500 to 1,400. The possibility of getting into tubulent flow due to disk otation was checked using the lamina tubulent tansition citeion of Popiel and Boguslawski [36] and anyo [14]. All uns used in the study checked out to be lamina Confined Submeged Liquid Jet Impingement Steady State Pocess The govening equations along with the bounday conditions wee solved using the Galekin finite element method as demonstated by Fletche [140]. Fou node quadilateal elements wee used. Fo each element, the velocity, pessue, and tempeatue fields wee appoximated which led to a set of equations that defined the continuum. Due to non linea natue of the govening tanspot equations, an iteative pocedue was used to aive at the solution fo the velocity and tempeatue fields. The solution of the esulting non linea diffeential equations was caied out using the Newton Raphson method. The solution was consideed conveged when the field value did not change fom one iteation to the next and the sum of the esiduals fo all the dependent vaiables was less than a pedefined toleance value; in this case,

95 The numbe of elements equied fo accuate esults was detemined fom a gid independence study. Figue 3.9 shows an unstuctued gid of the confined egion in which the sie of the elements nea the solid fluid inteface was made smalle to adequately captue the lage vaiations in velocity and tempeatue nea wall. Figue 3.9 Axis symmetic confined liquid jet impingement mesh plot Stationay Confined Wall with Spinning Taget The top disk emains stationay while the bottom disk otates at a unifom angula velocity (Ω) of 5.36 to adians/sec o 50 to 1,000 RPM to cove diffeent scenaios. The values of Reynolds numbe was limited ove 750 to avoid any fluid boiling condition up to a maximum of,000 to stay within the lamina egion. The oifice nole and the solid wafe disk have adii of 0.3 and 7.6 mm espectively; additionally the solid wafe thickness was kept at a value of 0.3 mm. The jet impingement height was vaied fom: 7x10 4 to 3.x10 3 m. The heat flux (q w ) and jet tempeatue wee kept constant at 50 kw/m and 310 K espectively. The fluid and solid mateial popeties ae assumed to be constant fo the tempeatue ange encounteed in the system, as shown in Table

96 3.5.. Spinning Confined Wall with Stationay Taget The bottom disk emains stationay while the top disk otates at a unifom angula velocity. The values of Reynolds numbe was limited to a maximum of 1,500 to stay within the lamina egion. The nole opening and the solid wafe disk have adii of 0.6 and 6.0 mm espectively. The heat flux (q w ) was kept constant at a value of 50 kw/m. The incoming fluid jet tempeatue (T j ) was 310 K fo wate and FC 77, 303 K fo ammonia (at a pessue of 0 bas), and 375 K fo MIL The thickness of the disk was vaied ove the following values: 0.3, 0.6, 0.9, 1., 1.5 and.0 mm. The jet impingement height o the distance between the nole and disk was set at the following values: 3x10 4, 6x10 4, 9.0x10 4, 1.x10 3,.4x10 3, 3.6x10 3, 4.8x10 3, and 6x10 3 m. The spinning ate (Ω) was vaied fom 0 to ad/s o 0 to 750 RPM. The flow ate was vaied fom 3.78x10 7 to 1.13x10 6 m 3 /s. The ange fo Reynolds numbe and Ekman numbe anged fom: Re=500 to 1,500 and Ek=7.08x10 5 to espectively. Using the lamina tubulent tansition citeion used by Popiel and Boguslawski [36] and anyo [14], all uns used in the pape checked out to be lamina. The simulation was caied out fo a numbe of disk mateials, namely Constantan, coppe, silicon, and silve. The popeties of solid mateials wee obtained fom Öisik [141]. Fluid popeties fo H O, NH 3, MIL 7808, and FC 77 wee obtained fom Bejan [134], the Bady vendo, and 3M Specialty Fluids espectively. The popeties of the above fluids wee coelated accoding to the equations shown in section In these coelations, the absolute tempeatue T was used in K. 69

97 3.5.3 Patially confined Submeged Liquid Jet Impingement Steady State and Tansient Pocess The pupose of a finite element method is to beak down the continuum poblem, of essentially an infinite numbe of degees of feedom, to a finite numbe of degees by discete siing the continuum into a numbe of simple shaped elements. The govening equations along with the bounday conditions of section (3.3.1 to 3.3.3) wee solved using the Galekin finite element method [140]. Fou node quadilateal elements wee used. Fo each element, the velocity, pessue, and tempeatue fields wee appoximated which led to a set of discetied equations that defined the continuum. In ode to detemine the initial velocity field ( i ), the equations fo the consevation of mass and momentum wee solved. The numbe of elements equied fo accuate esults was detemined fom a gid independence study. The sie of the elements nea the solid fluid inteface was made smalle to adequately captue lage vaiations in velocity and tempeatue in that egion, as shown in figue Figue 3.10 Axis symmetic patially confined jet impingement mesh plot. 70

98 Due to non linea natue of the govening diffeential equations the Newton Raphson method was used to aive at the solution fo the velocity and tempeatue fields. The solve used spines to tack the fee suface and e aanged gid distibution with the movement along the fee suface. The movement of the fee suface affected only the nodes along the spine. The appoach used to solve the fee suface poblem descibed hee was to intoduce a new unknown δ epesenting the position of the fee suface in the global system of equations. In ode to stat the computation, initial values of δ wee assigned to all nodes at the fee suface. A linea distibution with δ=h n at = p to δ H n / at = d was used as the initial guess. Since the solution of the momentum equation equied only two out of the thee bounday conditions at the fee suface, the thid condition in equation (3.3.15) was used to upgade the position of the fee suface at the end of each iteation step. Then the velocity components at the fee suface wee used to check the fulfillment of the kinematic condition (the fist condition in equation ). The value of the fee suface height (δ) was upgaded by applying a coection obtained fom the equied slope of the fee suface at each fee suface node. In ode to peseve the numeical stability duing this iteative solution fo δ a elaxation facto of 0.1 was used. Once a new location fo the fee suface node has been detemined, the location of all fluid nodes undeneath the fee suface extending to the solid fluid inteface wee adjusted keeping the same gid atio. It may be noted that the adjustment was done only in the vetical diection (along the axis) and only in the egion of p < < d and 0 < <δ. The iteative solution fo the detemination of the fee suface height distibution was continued by solving the consevation of mass and momentum equations and upgading the gid stuctue undeneath the fee suface. 71

99 Once the final fee suface height distibution was obtained no futhe change in δ was needed and the flow field fo the isothemal equilibium condition was eached, the powe souce was tuned on and the heat began to flow. Then the enegy equation (3.3.5) was solved simultaneously, along with the consevation of mass and momentum equations ( ) as a conjugate poblem taking into account the vaiation of fluid popeties with tempeatue to detemine the final distibution of velocity, pessue, and tempeatue. The computation coveed the entie tansient peiod all the way to the steady state condition. Because of lage changes at the outset of the tansient and vey small changes when the solution appoached the steady state condition, a fixed time step was used to cove the ealie pat of the tansient up to 5 seconds, and a vaiable time step was used fo the est of the computation. The solution was consideed conveged when elative change in field values fom a paticula iteation to the next, and the sums of the esiduals fo each vaiable became less than The consevation of mass was independently checked by calculating the flow ate at the outlet (= d ) fom computed velocity field and compaing that with fluid intake at the nole (=H n ). The diffeence was essentially eo. The chaacteistics of the flow ae contolled by thee majo physical paametes: the Reynolds numbe, Re j = j d n /ν f, the dimensionless nole to plate spacing atio, β=h n /d n, and the Ekman numbe, Ek 1, =ν f /. 4. Ω. 1, d. The values of Reynolds numbe was limited to a maximum of 900 to stay within the lamina egion. The nole opening and the heated taget disk have adii of 0.6 and 6.0 mm espectively. The heat flux (q w ) was kept constant at 15 kw/m. The incoming fluid jet tempeatue (T j ) was 310 K fo wate and FC 77, 303K fo ammonia (at a pessue of 0 bas), and 375 K fo MIL The 7

100 thickness of the disk (b) was vaied ove the values of: 0.30, 0.60, 1.0, 1.5 and.0 mm. The jet impingement height o the distance between the nole and disk was set at the following values: 3x10 4, 6x10 4, 9.0x10 4, and 1.x10 3 m. The spinning ate (Ω) was vaied fom 0 to ad/s o 0 to 750 RPM. The flow ate was vaied fom 6.65x10 7 to.7x10 6 m 3 /s. These values coves the ange of Ekman numbes of Ek 1, =7.08x10 5 to and Reynolds numbes of 0 to 900 espectively. The lamina tubulent tansition citeion of Popiel and Boguslawski [36] and anyo [14] confims that all uns in this study wee lamina. The solid and fluid popeties wee obtained fom Öisik [141], Bejan [134], and Bula [138].The fluid popeties ae coelated accoding to the following equations. Fo wate between 300 K<T<411 K; Cp f =9.5x10 3 T 5.999T5098.1; k f = 7.0x10 6 T 5.8x10 3 T ; ρ f =.7x10 3 T T848.07; and ln(µ f ) = T. Fo ammonia between K<T<370 K; Cp f =0.083T T9468; k f = x10 3 T; ρ f = T T ; and ln (µ f ) = T. Fo MIL 7808 between 303 K<T<470 K; Cp f = T; k f =0.18 1x10 4 T; ρ f = T; and ln (µ f ) = T. Fo FC 77 between 73 K<T<380 K; Cp f = T; k f = x10 5 T; ρ f =, T; and ln (µ f ) = T. In these coelations, the absolute tempeatue T is in K Chemical Mechanical Polishing Steady State and Tansient Pocess Fo a poblem unde study, the govening equations and the bounday conditions wee solved using the finite element method (FEM). The FI GEN module of FIDAP (Fluent, 005) and the softwae GAMBIT (Fluent, 006) wee used fo geometic modeling and mesh geneation. In FEM, the computational domain is discetied into 73

101 elements. Fou node quadilateal elements wee used. In each element, velocity components, pessue, and tempeatue fields, if any, wee appoximated by using the Galekin FEM pocedue [140] that leads to a set of algebaic equations that defines the discetied continuum. Fo 3 D models, the numbe of elements and nodal points ae usually so lage that the use of a fully coupled algoithm may equie computing esouces that exceed those available. To avoid that type of poblem, the solution of the esulting non linea equations was caied out using the segegated method. The segegated solution algoithm avoids the diect fomation of a global system matix. Instead, in each iteation, only one unknown is solved fo, while the othes keep thei pevious values. The next iteation is used to solve fo the next unknown. Due to its sequential and uncoupled natue, the segegated appoach equies less disk stoage but moe iteations than the fully coupled appoach. The fomulation of the segegated algoithm is quite involved and can be found in FIDAP Documentation (Fluent, 005). The pesent CMP model solution was consideed conveged when the elative change in field values fom a paticula iteation to the next, and the sums of the esiduals fo each vaiable became less than The technical computing pogam Matlab (The MathWoks, 007) was used to compute and geneate 3 D visualiations contou plots fo the numeical solutions fom FIDAP impoted into MATLAB though the neutal files (*.FPNEUT). The polishing pad and heated wafe of pesent investigation had a adius of 7.65 cm and 1.9 cm espectively. The souce of heat flux (q sl ) in the model was fom the non unifom shea fiction and it was vaied ove a ange of 3.75 to 3.1 (kw/m ). The 74

102 incoming sluy tempeatue (T sl ) was set to 97 K fo alumina (Al O 3 ). The sluy film thickness was vaied fom 40 to 00µm. The pad and caie spinning ate (Ω p,c ) was vaied fom 8.38 to 5.13 ad/s and 1.57 to 7.85 ad/s espectively. The flow ate was vaied fom cc/s. The possibility of getting into a tubulent flow due to disk otation was checked using the lamina tubulent tansition citeion of Popiel and Boguslawski [36] and anyo [14]. All uns used in the study checked out to be lamina. 3.6 Mesh Independence and Time Step Study Fee Liquid Jet Impingement Model The examination of the spatial convegence of a simulation is a staight fowad method that detemines the odeed of the discetiation eo in a CFD simulation. The method involves pefoming the simulation on two o moe successively fine gids. As pat of this study, a quantitative diffeence of gid independence was calculated by the accuacy of code using the asymptotic ange of convegence of Roache s methodology [143]. The Gid Convegence Index (GCI) was used to measue the numeical esults pecentage of accuacy in tems of the asymptotic numeical value of the exact solution. The GCI indicates an eo band and how fa the solution is fom the asymptotic value. It indicates how much the solution would change with a futhe efinement of the gid. A small value of GCI indicates that the computation is within the asymptotic ange. The GCI can be computed using two levels of gid; howeve, thee levels ae ecommended to detemine the ode of convegence and to check if the solutions ae within the asymptotic ange of convegence. The GCI on the gid is defined as: T T1 Fs T1 GCI = 100 (3.6.1) ( p 1) 75

103 whee F S is a facto of safety. The efinement may be spatial o in time. The safety facto of F S =3.0 is ecommended fo two gids. On the othe hand, a safety facto of F S =1.5 is ecommend fo thee o moe gids. It is impotant that each gid level yield to a solution that is in the asymptotic ange of convegence of the mesh. This can be checked by obseving two of the GCI values as computed ove thee gids, GCI 3 = p GCI 1 (3.6.) As the gid spacing educes, the tempeatue values at the inteface appoach to the asymptotic eo gid spacing value. We can detemine the local ode of convegence fom these esults, that diect evaluation of p can be obtained fom a thee gid solution using the gid efinement atio, equal to the numbe of elements of the fine gid (M n1 ) divided by the numbe of elements of the coase gid (M n ). T3 T ln T T1 p = (3.6.3) N3 ln N1 The local ode of accuacy is the ode of the stencil epesenting the discetiation of the equation at one location (/ d ) in the gid. The global ode of accuacy consides the popagation and accumulation of eos outside the stencil. This popagation causes the global ode of accuacy to be, in geneal, one degee less than the local ode of accuacy. To fulfill the analysis of the Gid Convegence Index (GCI), it was necessay to use Richadson s extapolation method fo highe ode. The Richadson s extapolation method was used to estimate the continuum value at eo gid spacing fom a seies of lowe ode discete values. The continuum value at eo gid spacing and the pecentage eo can be genealied fo a p th ode methods and value of gid atio (which does not have to be an intege) defined by the following expessions: 76

104 T1 T Th = 0 = T1 (3.6.4) p 1 T int T h = 0 Th =0 x100 (3.6.5) Additionally, to detemine the numbe of elements fo accuate numeical solution, computation was pefomed fo seveal gids o combinations of numbe of elements in the hoiontal and vetical diections coveing the solid and fluid egions, as shown in figue Dimensionless Inteface Tempeatue,Θint Dimensionless Radial Distance, / d n x n =14x38 n x n =0x51 n x n =x76 n x n =31x60 n x n =3x81 n x n =6x85 n x n =6x93 n x n =31x93 n x n =3x15 Figue 3.11 Solid fluid inteface tempeatue fo diffeent numbe of elements in and diections (Re=1,500, b=0, d n =1. mm, Ek=.65x10 4, β=.67). The numeical solution becomes gid independent when the numbe of divisions equal to 6x85 in the axial () and adial () diections espectively is used. Numeical esults fo a 6x85 gid gave almost identical esults compaed to those using x76 and 77

105 3x15 gids. The aveage diffeence was 0.69%. Theefoe, futhe computations wee caied out using a 6x85 gid. Subsequently, the GCI method was used to confim the accuacy of the chosen mesh. The GCI s values, the pecent eo, and T h=0 at / d =0.75 ae shown in Table 3.. Thus, the above set of equations (3.6.1 to 3.6.4) was solved to detemine the local accuacy of convegence o GCI s values fo the following thee gids. The last column of Table 3. shows that a gid of 6x85 is the most accuate with a pecent eo of 3.36x10 3 fom its asymptotic numeical exact solution. The pupose of the GCI method is to point out an eo band and how fa the local solution of the mesh is fom its asymptotic value. Table 3. Gid convegence study of figue Run # MSH GCI (%) T int at / d = 0.75 (K) T h=0 at / d = 0.75 (K) Eq (%eo) 1 x76 GCI 1 = x85 GCI 3 = x x15 GCI 31 = Computations wee also pefomed to calculate a suitable fixed time step to detemine its sensitivity on the tansient solution. Figue 3.1 plots the vaiation of maximum dimensionless solid fluid inteface tempeatue fo diffeent time incements as a function of Fouie numbe (F o ) as a dimensionless numbe to epesent the time. It may be noted that the solution is not susceptible to the sie of the time step o incement when an incement of seconds o less is chosen. A time incement of 0.05 seconds was selected to ensue a smooth vaiation. Notice how the maximum dimensionless tempeatue inceases apidly all the way to the steady state condition. 78

106 Maximum Dimensionless IntefaceTempeatue, Θmax Time Incement = s Time Incement = 0.05 s Time Incement = s Time Incement = s Time Incement = s Time Incement = 0.50 s Fouie Numbe, Fo Figue 3.1 Time step independence study fo maximum dimensionless inteface tempeatue vaiation at diffeent time steps (Re=550, silicon disk, wate, b/d n =0.5, Ek=.65x10 4, q w =15 kw/m, β=.67) Confined Liquid Jet Impingement Model Stationay Confined Wall with Spinning Taget The solid fluid dimensionless inteface tempeatues fo diffeent numbe of gids ae plotted in figue Seveal gids wee used to detemine the numbe of elements needed fo accuate numeical solution. It was obseved that the numeical solution becomes gid independent when the gids each a numbe of divisions equal to 35x79 in the axial () and adial () diections espectively. Numeical esults fo a 35x79 gid gave almost identical esults compaed to a 64x76 gid fo an impingement height (H n ) equal to 0.3 cm. Theefoe, the chosen gid was 35x79 that caied an aveage eo magin of 0.43% compaed to 64x76 gids. 79

107 Solid-Fluid Dimensionless Inteface Tempeatue, Θ NZxNR =x48 NZxNR =x79 NZxNR =35x79 NZxNR =64x Dimensionless Radial Location, / n Figue 3.13 Local dimensionless inteface tempeatue fo diffeent numbe of elements in and diections at constant flow ate (Re=1,500, Q=7.08x10 m 3 /s, b=0, Ek=.65x10 4, q w =50 kw/m, H n /d n = 5.33, Ω=15 RPM, H n =0.3 cm). The GCI s values, the pecent eo, and T h=0 at / n =8 ae shown in Table 3.3. Thus, the above set of equations ( ) was solved to detemine the local accuacy of convegence o GCI fo each of the following mesh domains. Table 3.3 Gid convegence study of figue Run # MSH GCI (%) T int at / n = 8 (K) T h=0 at / n = 8 (K) Eq (%eo) 1 x79 GCI 1 = 3.x x x79 GCI 3 = 8.1x x x76 GCI 31 = The last column of Table 3.3 shows that a gid of 35x79 is the most accuate with a pecent eo of.55x10 3 fom its asymptotic numeical exact solution. The pupose of 80

108 the GCI method is to point out an eo band and how fa the local solution of each paticula gid is fom its asymptotic value Spinning Confined Wall with Stationay Taget Fo the spinning confined wall with stationay taget, seveal gids o combinations of a numbe of elements wee used to detemine the accuacy of the numeical solution. Dimensionless solid fluid o inteface tempeatues at the heated plate fo seveal gids ae plotted in figue Dimensionless IntefaceTempeatue, Θ int n x n = 6 x 68 n x n = 38 x 68 n x n = 40 x 70 n x n = 38 x 8 n x n = 46 x Dimensionless Radial Distance, / d Figue 3.14 Dimensionless inteface tempeatue distibutions fo diffeent numbe of elements in and diections (Re=1,000, b=0.3 mm, d n =0.1 mm, Ek=1.06x10 3, β=.0). The numeical solution becomes gid independent when the numbe of divisions equal to 40x70 in the axial () and adial () diections espectively is used. Numeical esults fo a 40x70 gid gave almost identical esults compaed to 38x8 and 46x8 gids fo an impingement height (H n ) equal to 0.4 cm. The aveage diffeence was equal to 81

109 0.%. Theefoe, futhe computations wee caied out using 40x70 elements. The sie of the elements vaies with dense distibution at the solid fluid inteface and at the nole axis. Scaling atios of 1.5 and 1.6 wee used in adial and axial diections espectively. Table 3.4 Gid convegence study of figue Run # MSH GCI (%) T int at / d = 0.4 (K) T h=0 at / d = 0.4 (K) Eq (%eo) 1 38x8 GCI 1 =.4x x x8 GCI 3 =.5x x70 GCI 31 = 9.4x x10 5 A quantitative diffeence in local gid convegence was calculated using equation (3.6.1 to 3.6.4) fo the tempeatue at the solid fluid inteface T int at a given / d location of the taget disk fo each gid. The GCI s values and the exact solution fo a gid sie of eo spacing at / d =0.4, ae shown in Table 3.4. In addition, the last column of Table 3.4 shows the calculated pecent eo obtained by equation fo a gid of 40x70. These esults show that the chosen gid of 40x70 is the most accuate. The pupose of the GCI method is to point out an eo band and how fa the local solution of the mesh is fom its asymptotic value. It can be found that the numeical solution becomes gid independent when the numbe of divisions equal to 8x63 in the axial () and adial () diections, as shown in figue Compaing the numeical esults fo the 3x7 and 45x100 gids with a 8x63 gid shows an aveage diffeence of 0.7%. Theefoe, futhe computations wee caied out using a 8x63 gid. 8

110 3.6.3 Patially confined Liquid Jet Impingement Model Stationay Confined Wall with Spinning Taget The numbe of elements equied fo accuate numeical solution was detemined fom a systematic gid independence study. 0.4 Dimensionless Inteface Tempeatue, Θ int Dimensionless Radial Location, / d n x n = 14 x 39 n x n = 7 x 55 n x n = 4 x 59 n x n = 8 x 59 n x n = 36 x 59 n x n = 8 x 63 n x n = 30 x 63 n x n = 3 x 7 n x n = 45 x 100 Figue 3.15 Dimensionless inteface tempeatue distibutions fo diffeent numbe of elements in and diections (Re=750, p / d =0.667, b/d n =0.5, Ek=4.5x10-4, β=0.5). Table 3.5 Gid convegence study of figue Run # MSH GCI (%) T int at / d = 0.5 (K) T h=0 at / d = 0.5 (K) Eq (%eo) 1 8x63 GCI 1 = 6.7x x10 5 3x7 GCI 3 = 3.6x x x100 GCI 31 = 3.7x The GCI s values, the pecent eo, and T h=0 at / d =0.5 ae shown in Table 3.5. Thus, the above set of equations ( ) was solved to detemine the local accuacy 83

111 of the following gids 3x7 and 45x100 in compaison with a 8x63 gid. In addition, the last column of Table 3.5 shows the calculated pecent eo obtained by equation fo a gid of 8x63 is the most accuate in compaison with its asymptotic numeical exact solution. The pupose of the GCI method is to point out an eo band and how fa the local solution of the mesh is fom its asymptotic value Maximum Dimensionless IntefaceTempeatue, Θmax Time Incement = s Time Incement = 0.05 s Time Incement = s Time Incement = s Time Incement = s Fouie Numbe, Fo Figue 3.16 Maximum dimensionless inteface tempeatue vaiation fo diffeent time steps with wate as the cooling fluid (Re=5, Ek=.13x10 4, β=0.5, silicon disk, b/d n =0.5, and p / d =0.667). To detemine the sensitivity of the tansient solution futhe computations wee pefomed to calculate a suitable fixed time step, as shown in figue These tansient computations showed that the vaiation of the tempeatue is not sensitive to time step sie when an incement of seconds o less is chosen. Fo this study, the time incement of 0.05 seconds was selected to ensue a smooth vaiation. 84

112 Co Rotating Taget and Confined Wall Fo the co otating taget and confined wall, seveal gids o combinations of numbe of elements wee used to detemine the accuacy of the numeical solution, as shown in figue The numeical solution becomes gid independent when the numbes of divisions used wee equal to 34x63 in the axial () and adial () diections espectively. Compaing the numeical esults fo the 34x59 and 36x64 gids with a 34x63 gid showed an aveage diffeence of 0.159%. Dimensionless Inteface Tempeatue, Θint n x n = x 59 n x n = 8 x 63 n x n = 3 x 7 n x n = 34 x 59 n x n = 34 x 63 n x n = 36 x 59 n x n = 36 x Dimensionless Radial Location, / d Figue 3.17 Dimensionless inteface tempeatue distibutions fo diffeent numbe of elements in and diections (Re=750, b/d n =0.5, Ek 1, =4.5x10 4, p / d =0.667, β=0.5). The GCI s values, the pecent eo, and T h=0 at / d =0.4 ae shown in Table 3.6. The mesh convegence fo diffeent gids was calculated using the following equations (3.6.1 to 3.6.5). In addition, the last column of Table 3.6 shows that a gid of 34x63 is the most accuate with a pecent eo of 5.70x10 3 fom its asymptotic numeical exact 85

113 solution. The pupose of the GCI method is to point out an eo band and how fa the local solution of each paticula gid is fom its asymptotic value. Table 3.6 Gid convegence study of figue Run # MSH GCI (%) T int at / d = 0.4 (K) T h=0 at / d = 0.4 (K) Eq (%eo) 1 34x59 GCI 1 = 8.8x x x63 GCI 3 = 8.8x x x64 GCI 31 = 7.9x x Chemical Mechanical Polishing Model The distibution of an element sie in a computational domain is detemined fom a mesh independence study by systematically changing the element density in all space diections to obtain a mesh of acceptable accuacy. Seveal gids o combinations of numbe of elements wee used to detemine the flow field and wafe inteface tempeatue distibution, as shown in figue The numeical solution becomes gid independent fo the numbe of elements equal to 1,344. Numeical esults fo 1,344 elements gave almost identical esults compaed to those using 780 and 1,600 elements. The aveage magin of eo was 0.044%. A set of tempeatue distibutions acoss the sluy egion just below the wafe suface along the film thickness was used to chaacteie the accuacy of the mesh model. Theefoe, all futhe computations wee caied out using a gid of 1,344 elements. 86

114 Coss sectional Wafe Tempeatue,K 30 elements 780 elements 1344 elements 1600 elements 304 elements Leading Edge Tailing Edge Dimensionless adial wafe location, / w Figue 3.18 Tempeatue distibution acoss the sluy egion beneath the substate suface fo vaious numbe of elements (Q sl =65 cc/min, Ω w =15 RPM, Ω p =150 RPM, COF=0.4, δ sl =50 µm, P=4.35 kpa, w =1.9 cm, q sl =7.4 to 10.1 kw/m ). θ Figue 3.19 Gid topology of contol volume that includes the wafe, alumina sluy, and polishing pad. 87

115 The mesh gid topology of a sluy film thickness of 50 µm is plotted at figue A unifom and dense distibution of elements was used at the cente of the contol volume egion to captue the themal effect of the constict alumina sluy, as shown in figue

116 Chapte 4 Fee Liquid Jet Impingement Model Results 4.1 Steady State Cooling of Spinning Taget A typical velocity vecto distibution is shown in figue 4.1. It can be seen that the velocity emains almost unifom at the potential coe egion of the jet. The velocity deceases and the fluid jet diamete inceases as the fluid gets close to the plate duing the impingement pocess. Figue 4.1 elocity vecto distibution fo jet impingement on a silicon wafe with wate as the cooling fluid (Re=900, Ek=.65x10 4, β=.67, b/d n =0.5). The diection of motion of the fluid paticles shifts by as much as 90 o. Afte this, the fluid acceleates ceating a egion of minimum sheet thickness. This is the stat of the 89

117 bounday laye one. It can be noticed that as the bounday laye thickness inceases with adius, the fictional esistance fom the wall is eventually tansmitted to the entie film thickness. This is called fully viscous one. The thee diffeent egions obseved in the pesent investigation ae in ageement with the expeiments of Liu et al. [17]. Axial Distance (cm) Re=445 Re=600 Re=900 Re=100 Re=1500 Re= Radial Distance (cm) Figue 4. Fee suface height distibution fo diffeent Reynolds numbes with wate as the cooling f1uid (Ek=.65x10 4, β=.67, b/d n =0.5). Figue 4. pesents the fee suface height distibution fo diffeent Reynolds numbes when the jet stikes the cente of the disk while it is spinning at a ate of 15 RPM. It can be seen that the fluid speads out adially as a wavy thin film. As the Reynolds numbe inceases the film diminishes in thickness unde the same constant spinning ate due to a lage impingement velocity that tanslates to a highe fluid velocity in the film. Fo the conditions consideed in the pesent investigation, the flow was supecitical and a hydaulic jump did not occu within the computation domain. These obsevations ae in ageement with the expeimental wok of Metge et al. [37]. 90

118 Local Nusselt Numbe,Nu Re=445 Re=600 Re=900 Re=100 Re=1500 Re=1800 Temp, Re=445 Temp, Re=600 Temp, Re=900 Temp, Re=100 Temp, Re=1500 Temp, Re= Dimensionless Inteface Tempeatue,Θ int Dimensionless Radial Distance, /d n Figue 4.3 Dimensionless inteface tempeatue and local Nusselt numbe distibutions fo a silicon wafe with wate as the cooling fluid fo diffeent Reynolds numbes (Ek=.65x10 4, β=.67, b/d n =0.5). Figue 4.3 shows the dimensionless inteface tempeatue and local Nusselt numbe distibutions as a function of dimensionless adial distance (/d n ) along the solid fluid inteface at diffeent Reynolds numbes fo a otational ate of 15 RPM. The cuves in figue 4.3 eveal that the dimensionless inteface tempeatue deceases with jet velocity (o Reynolds numbe). The dimensionless inteface tempeatue has the lowest value at the stagnation point (undeneath the cente of the axial opening) and inceases adially eaching the highest value at the end of the disk. The local Nusselt numbe distibutions, as shown in figue 4.3 inceases apidly ove a small distance (coe egion) measued fom the stagnation point, eaching a maximum aound /d n =0.40, and then deceases along the adial distance as the bounday laye develops futhe downsteam. The location of the maximum Nusselt numbe can be associated with the tansition of the 91

119 flow fom the vetical impingement to hoiontal displacement whee the bounday laye stats to develop. Figue 4.3 confims to us how an inceasing Reynolds numbe contibutes to a moe effective cooling by the enhancement of the convective heat tansfe coefficient Aveage Nusselt Numbe, Nu av Nu at at Ek=.65x Nu at at Ek= Ek=7.79x x Nu at at Ek=.1x h h at at Ek=.65x h h at at Ek=7.79x h h at at Ek=.1x Reynolds Numbe, Re Aveage Heat Tansfe Coefficient, h av (W/m K) Figue 4.4 Aveage Nusselt numbe and heat tansfe coefficient vaiations with Reynolds numbe fo a silicon wafe with wate as the cooling fluid (β=.67, b/d n =0.5). Figue 4.4 plots the aveage Nusselt numbe and aveage heat tansfe coefficient as a function of Reynolds numbes fo low, intemediate, and high Ekman numbes o otational ates. It may be noted that aveage Nusselt numbe inceases with Reynolds numbe. As the flow ate (o Reynolds numbe) inceases, the magnitude of fluid velocity nea the solid fluid inteface that contols the convective heat tansfe ate inceases. Futhemoe, at a paticula Reynolds numbe the gaphical values ae shifted gadually upwad due to an incement of the spinning ate. This behavio confims the positive 9

120 influence of the otational ate on the aveage Nusselt numbe and aveage heat tansfe coefficient. Local Nusselt Numbe,Nu Ek=infinti = Ek=7.79x10-5, = 7.8x10-5 Ek=3.3x10-5, = 3.x10-5 Temp, at Ek=34.68 = Temp, at Ek=7.79x10-5 = 7.8x10-5 Temp, at Ek=3.3x10-5 = 3.x10-5 Ek=.65x10-4, = -4 Ek=4.57x10-5, = 4.6x10-5 Ek=.1x10-5 =.x10-5 Temp, at Ek=.65x10-4 = -4 Temp, at Ek=4.57x10-5 = 4.6x10-5 Temp, at Ek=.1x10-5 =.x Dimensionless IntefaceTempeatue,Θint Dimensionless Radial Distance, /d n Figue 4.5 Dimensionless inteface tempeatue and local Nusselt numbe distibutions fo a silicon wafe with wate as the cooling fluid at diffeent Ekman numbes (Re=1,500, β=.67, b/d n =0.5). The otational ate effects of the solid wafe on the dimensionless inteface tempeatue and local Nusselt numbe ae illustated in figue 4.5. It can be noted that Nusselt numbe distibution does not change dastically with the vaiation of otational ate o Ekman numbe in figue 4.5. Diffeences ae seen only at lage adial location of the disk whee the magnitude of the centifugal foce encounteed by the liquid film is highe. This clealy indicates that at Re=1,500 the flow field is dominated by the momentum of the impinging jet. Howeve, the dimensionless inteface tempeatue changes along the entie disk adius with the vaiation of Ekman numbe. It can be noted 93

121 that dimensionless inteface tempeatue deceases with the incement of the otational ate due to the enhancement of local fluid velocity adjacent to the wafe. The aveage Nusselt numbe and heat tansfe coefficient vaiations as a function of Ekman numbe at high, intemediate, and low Reynolds numbes ae shown in figue 4.6. As the Ekman numbe deceases fom.65x10 4 to.1x10 5 the aveage Nusselt numbe and heat tansfe coefficient inceases by an aveage 7.15% unde high Reynolds numbe (Re=1,500) and 13.19% unde low Reynolds numbe (Re=750) with an oveall incement of 0.17% in geneal Aveage Nusselt Numbe,Nu av Nu at Re=1500 Nu at Re=750 Nu at Re=1000 h at Re=1500 h at Re=750 h at Re= E00 5.E-05 1.E-04.E-04.E-04 3.E-04 3.E-04 Ekman Numbe,Ek Aveage Heat Tansfe Coefficient, h av (W/m K) Figue 4.6 Aveage Nusselt numbe and heat tansfe coefficient vaiations with Ekman numbe fo a silicon wafe with wate as the cooling fluid (β=.67, b/d n =0.5). The effects of disk thickness vaiation on the solid fluid dimensionless inteface tempeatue and local Nusselt numbe ae shown in figue

122 Local Nusselt Numbe,Nu b=0.mm b/d n = 0.17 b=0.4mm b/d n = 0.33 b=0.6mm b/d n = 0.50 b=0.9mm b/d n = 0.75 b=1.mm b/d n = 1.00 b=1.5mm b/d n = 1.5 b=mm b/d n = 1.67 Temp temp, at b/d b=0.mm n = 0.17 Temp temp, at b/d b=0.4mm n = 0.33 Temp temp, at b/d b=0.6mm n = 0.50 Temp temp, at b/d b=0.9mm n = 0.75 Temp temp, at b/d b=1.mm n = 1.00 Temp temp, at b/d b=1.5mm n = 1.5 Temp temp, at b/d b=mm n = Dimensionless Radial Distance, /d n Dimensionless Inteface Tempeatue,Θint Figue 4.7 Dimensionless inteface tempeatue and local Nusselt numbe distibutions fo diffeent wafe thicknesses with wate as the cooling fluid (Re=1,000, Ek=.65x10 4, β=.67). In these plots, silicon has been used as the disk mateial and wate as the cooling fluid. The dimensionless solid fluid inteface tempeatue distibution in figue 4.7 inceases fom the impingement egion all the way to the end of the disk. It may be also noted that the cuves intesect with each othe at a dimensionless adial distance of /d n =3.75. Thicke disks geneate moe unifom dimensionless inteface tempeatue due to a lage adial conduction within the disk. In addition, the dimensionless solid fluid inteface tempeatue and local Nusselt numbe distibutions did not change much beyond a disk thickness of 0.60 mm, o dimensionless thickness b/d n =0.50 indicating that the oveall heat tanspot eached a convection conduction equilibium condition at the solid fluid inteface. Local Nusselt numbe distibutions do not change significantly with the 95

123 vaiation of disk thickness. Highe local Nusselt numbe values ae obseved at a dimensionless adial distance (/d n ) of less than 0.5 fo all cuves. These steep Nusselt numbes values wee geneated by a highe ate of heat emoval at the impingement one. Local Nusselt Numbe,Nu Γ=6.333 Γ=5.066 Γ=3.60 Γ=3.167 Γ=.533 Γ= Dimensionless Radial Distance, /d n Figue 4.8 Local Nusselt numbe distibution fo diffeent nole diametes fo a silicon wafe with wate as the cooling fluid (Re=1,000, Ek=6.6x10 5, β=.67, b/d n =0.5). Figues 4.8 shows the local Nusselt numbe distibutions as a function of dimensionless adial distance (/d n ) along the solid fluid inteface fo diffeent disk adius to nole diamete atio (Γ) fom.11 to 6.33 unde a Reynolds numbe of 1,000 and otational ate of 500 RPM. Nusselt numbe inceases apidly ove a small distance at the coe egion measued fom the stagnation point, eaches a maximum aound /d n =0.40, and then deceases along the adial distance as the bounday laye develops futhe downsteam. The location of the maximum Nusselt numbe can be associated with the tansition of the flow fom the vetical impingement to hoiontal displacement 96

124 whee the bounday laye stats to develop. It can be noticed that local Nusselt numbe is geate fo highe Γ in the impingement one wheeas it is somewhat lowe at highe Γ in the bounday laye one. Fo a constant Reynolds numbe and constant disk adius, a lage value of Γ is ealied at highe jet velocity. Theefoe it povides a highe ate of convective heat tansfe in the stagnation egion whee the jet diectly stikes the disk. Since highe Γ is also associated with lowe jet diamete. The fluid has to tavel ove a longe path in the bounday laye egion whee it losses its momentum esulting in lowe convective heat tansfe ate by the time it exists the disk suface Local Nusselt Numbe,Nu ß=5.0 ß=4.0 ß=3.0 ß=.0 ß=1.5 ß=0.75 ß=0.55 Temp, ß=5.0 Temp, ß=4.0 Temp, ß=3.0 Temp, ß=.0 Temp, ß=1.5 Temp, ß=0.75 Temp, ß= Dimensionless Inteface Tempeatue, Θint Dimensionless Radial Distance, /d n 0 Figue 4.9 Dimensionless inteface tempeatue and local Nusselt numbe distibutions fo a silicon disk with wate as the cooling fluid fo diffeent nole to taget spacing (Re=750, Ek=.65x10 4, b/d n =0.5). The solid fluid dimensionless inteface tempeatue and local Nusselt numbe distibutions fo seven diffeent nole to taget spacing fo wate as the coolant at a spinning ate of 15 RPM and Reynolds numbe of 750 ae shown in figue 4.9. It may 97

125 be noticed that the impingement height quite significantly affects the dimensionless inteface tempeatue as well as the Nusselt numbe only at the smalle adii that contain the stagnation egion and the ealy pat of the bounday laye egion. At lage adii the values ae identical fo all impingement heights. It is quite expected since the impingement height essentially contols the change in velocity the fluid paticles encounte duing the fee fall fom nole exit to taget disk suface and theefoe affects aeas contolled by diect impingement. This obsevation is somewhat simila to a pevious study by Owosina [143] fo fee jet impingement ove a stationay disk. Local Nusselt Numbe,Nu Nu, NH 3 NH3(P1.9),(P=1.9) 1 Nu, H O,(P=5.49) H0(P=5.49) 0.9 Nu, FC-77,(P=3.66) FC-77(3.66) Nu, MIL-7808,(P=14.44) MIL-7808(P=14.44) 0.8 NH3 3, Temp H0 H O, Temp 0.7 FC-77, Temp 0.6 Mil-7808 MIL-7808, Temp Temp Dimensionless Radial Distance, /d n Dimensionless Inteface Tempeatue, Θint Figue 4.10 Local Nusselt numbe and dimensionless inteface tempeatue vaiations fo diffeent cooling fluids fo silicon as the disk mateial (Re=750, Ω=15 RPM, β=.67, b/d n =0.5). Figues 4.10 compaes the solid fluid inteface tempeatue and local Nusselt numbe distibution esults of ou pimay woking fluid (wate) with thee othe coolants that have been consideed in pevious themal management studies, namely ammonia 98

126 (NH 3 ), flouoinet (FC 77) and oil (MIL 7808). It may be noticed that wate pesents the lowest inteface tempeatue and second lowest Nusselt numbe distibution in compaison with FC 77, NH 3 and MIL The highest Nusselt numbe is obtained when FC 77 is used as the woking fluid. This is pimaily because of its lowe themal conductivity compaed to the othe fluids. Thee esults ae fo a constant Reynolds numbe of 750 while the disk is spinning at a ate of 15 RPM. Local Nusselt Numbe,Nu Silve Coppe Aluminum Silicon Constantan Temp, Silve Temp, Coppe Temp,Aluminum Temp, Silicon Temp, Constantan Dimensionless Radial Distance, /d n Dimensionless Inteface Tempeatue, Θ int Figue 4.11 Local Nusselt numbe and dimensionless inteface tempeatue vaiations fo diffeent solid mateials with wate as the cooling fluid (Re=1,500, Ek=.1x10 5, β=.67, b/d n =0.5). Figue 4.11 shows the dimensionless inteface tempeatue and local Nusselt numbe distibution plots as a function of a dimensionless adial distance (/d n ) fo diffeent solid mateials with wate as the woking fluid. The studied mateials wee silicon, silve, aluminum, coppe, and Constantan, having diffeent themo physical popeties. Constantan shows the lowest dimensionless tempeatue at the impingement 99

127 one and the highest at the outlet in compaison with othe solid mateials. Coppe and silve show a moe unifom distibution and highe tempeatue values at the impingement one due to thei highe themal conductivity. The dimensionless tempeatue and local Nusselt numbe distibutions of these two mateials ae almost identical due to thei simila themal conductivity values. The coss ove of cuves fo all five mateials occued due to a constant fluid flow and heat flux ate that eaches a themal enegy balance. Solid mateials with lowe themal conductivity show highe maximum local Nusselt numbe. Figue 4.1 pesents the maximum tempeatue and maximum to minimum tempeatue diffeence at the inteface fo all five disk mateials studied unde diffeent disk thicknesses with wate as the woking fluid. The tempeatue contol is cucial in the design of electonic packages. The maximum tempeatue at the inteface as well as the maximum solid disk tempeatue deceases as the disk thickness inceases. It maybe noticed that effects ae faily lage at smalle thicknesses indicating that it is a cucial paamete in maintaining the tempeatue unifomity. On the othe hand inceasing the disk thickness beyond cetain limit, fo each solid mateial, may not be useful. The choice of disk mateial is also cucial in detemining the magnitudes of these tempeatues. A mateial with lage themal conductivity will facilitate a faste ate of heat tansfe, and theefoe will esult in a lowe maximum tempeatue at the solid fluid inteface and inside the solid. The tempeatue diffeence at the inteface is an indication of the level of tempeatue non unifomity at the impingement suface, while the maximum tempeatue inside the solid indicates the themal esistance geneated by the disk mateial. When the disk thickness is negligible, the inteface tempeatue is contolled by the heat flux 100

128 condition at the heate. Adequate thickness povided a moe unifom inteface tempeatue due to adial heat speading within the solid. The maximum to minimum tempeatue diffeence is stongly affected by themal conductivity of the disk mateial deceasing it as the disk themal conductivity inceases. These findings ae in ageement with Rahman et al. [0] who studied fee liquid jet impingement ove a stationay disk. Maximum to Minimum Inteface Tempeatue Diffeence, (K) Silicon, Tmax-Tmin Aluminum, Tmax-Tmin Coppe, Tmax-Tmin Constantan,Tmax-Tmin Silve, Tmax-Tmin Silicon, Tmax Aluminum, Tmax Coppe, Tmax Constantan, Tmax Silve, Tmax Maximum IntefaceTempeatue,T max (K) Disk Thickness, b(cm) Figue 4.1 Maximum to minimum tempeatue diffeence and maximum solid fluid inteface tempeatue (Re=1,500, d n =0.1 cm, β=.67, Ek=.1x10 5 )

129 Aveage Nusselt Numbe, Nu av Nu, Silicon Nu, Aluminun Nu, Coppe Nu, Constantan Nu, Silve h, Silicon h, Aluminum h, Coppe h, Constantan h, Silve Aveage Heat Tansfe Coefficient, h av (W/m K) Thickness to nole diamete atio, b/d n 1500 Figue 4.13 Aveage Nusselt numbe and heat tansfe coefficient vaiations with disk thickness (Re=1,500, Ek=.1x10 5 ). The effects of disk thickness on the aveage heat tansfe coefficient and Nusselt numbe fo all five mateials can be obseved in figue It shows that the aveage heat tansfe coefficient and aveage Nusselt numbe attain constant values at b/d n geate than 0.50 fo all the mateials. Constantan has the highest aveage heat tansfe coefficient value among these mateials. This behavio is in ageement with the local Nusselt numbe distibution shown in figue The adial conduction becomes stonge as the disk thickness inceases geneating a bette heat distibution at the inteface. Howeve, the incement of solid thickness beyond cetain limit ceates moe themal esistance, which ends up cippling the heat tansfe pocess. Two of the papes used fo the validation of this numeical study wee the expeimental wok caied out by Stevens and Webb [16] and analytical studies by 10

130 Watson [4]. Computations wee caied out fo a wate jet that impinges pependiculaly at the cente of a stationay solid disk at vaious nole to taget spacing atios. Figue 4.14 compaes the calculated local fee suface height distibutions with the pofiles epoted by the expeimental studies of Stevens and Webb [3] and the analytical esults of Watson [4] at diffeent nole diametes (d n =.1 and 4.6 mm. The numeical values compae easonably well with the measued fee suface heights and Watson s analytical pedictions. Axial Impingement Distance (cm) ß = ß = 1.5 ß = 0.75 ß = 0.55 Watson[4], dn=.1 mm Watson[4], dn=4.6 mm Stevens and Webb [16], at Re=15800, dn=.1mm Dimensionless Radial Distance, /d n Figue 4.14 Compaison of height of the fee suface with analytical pedictions of Watson [4] and expeimental data of Stevens and Webb [16] (Re=1,500, Ek=, b/d n =0.5). In addition, the pesent numeical simulation esults at steady state wee compaed with the steady state test data acquied by Leland and Pais [19] fo a disk with no otation. The aveage heat tansfe coefficient fom the pesent numeical simulation using MIL 7808 as the woking fluid fo diffeent combinations of Reynolds numbe and 103

131 input heat flux wee compaed with the expeimental measuements of Leland and Pais [19]. The pecent diffeence of pesent aveage heat tansfe coefficient esults was defined in the following fom: % diff = ((h num h exp )/h exp ) 100. The pecent diffeence was in the ange of 0.41% 5.53%. Consideing the uncetainty of expeimental measuements and ound off and discetiation eos in numeical computation, the oveall compaison between test data and numeical esults can be consideed quite satisfactoy. The thid pape used fo compaison of this numeical study was the analytical wok caied out by Liu and Lienhad [10]. They obtained an integal solution fo the heat tansfe coefficient in the bounday laye and similaity egions fo Pandtl numbe geate than the unity fo a stationay disk. Local Nusselt Numbe,Nu Liu and and Lienhad [][10] Ek=34.68, = Ω=1x10-4 pm Ek=.65x10-4, = -4 Ω=15 pm Ek=7.79x10-5, = 7.80x10-5 Ω=45pm Radial Displacement (cm) Figue 4.15 Local Nusselt numbe compaison with Liu and Lienhad [10] unde diffeent Ekman numbes (Re=1,500, β=.67, b/d n =0.5). 104

132 A gaphical epesentation of the Nusselt numbe coelation fom Liu and Lienhad [10] and pesent numeical esults at diffeent spinning ates with wate as the woking fluid ae shown in figue The pecent diffeence of pesent local Nusselt numbe was defined in the fom: % diff = ((Nu num Nu analy )/Nu analy ) 100. The esults shown fo a stationay disk compae within an aveage diffeence of 3.5% with Liu and Lienhad coelation. The local Nusselt numbe unde spinning ates at 15 and 45 RPM coelates with an aveage magin of 3.49% and 1.83% espectively. In geneal, the oveall aveage diffeence of local Nusselt numbes was equal to 9.5%. A bette compaison fo stationay disk and highe deviation with highe spinning ate is expected, since the coelation in [10] was developed fo a stationay disk. 35 Numeical Aveage Nusselt Numbes, Nu av Pedicted Aveage Nusselt Numbes Figue 4.16 Compaison of pedicted aveage Nusselt numbes of equation 4.1 with pesent numeical data. 105

133 One of the goals of this dissetation was to develop a pedictive tend of the aveage heat tansfe coefficient. A gaphical compaison of the coelating equation and the numeical aveage Nusselt numbes obtained fom this computational analysis is shown in figue A coelation fo the aveage Nusselt numbe was developed as a function of themal conductivity atio, nole to plate spacing, Pandtl numbe, Ekman numbe, and Reynolds numbe to accommodate most of the tanspot chaacteistics of a fee liquid jet impingement cooling pocess. A coelation that best fitted the numeical data can be placed in the following fom: Nu av =Re Ek P 0.4 β ε 0.5 (4.1) The anges of the dimensionless vaiables used ae the following: 445 Re 1,800,.65x10 4 Ek.1x10 5, P=5.49, 0.55 β 5.0, 7.6 ε The Pandtl numbe exponent was taken fom Matin s equation [144] fo single ound nole impinging jet. The Aveage Nusselt numbe data wee then coelated in ode to detemine the othe exponents of equation 4.1 using the least squaes cuve fitting method. The pecent diffeence of the pedicted aveage Nusselt numbe was defined as: % diff = ((Nuav ped Nuav num )/Nuav num ) 100. The diffeences between numeical and pedicted aveage Nusselt numbe values ae in the ange of 19.63% to 17.83%. It should be noted fom figue 4.16 that a lage numbe of data points ae well coelated with equation 4.1 and only a few ae nea the limits. 4. Tansient Cooling of Spinning Taget The following section pesents the tansient conjugate heat tansfe of a fee liquid jet impinging on a otating solid disk of finite thickness and adius. Figue 4.17 illustates the dimensionless inteface tempeatue fo diffeent time instants. It can be obseved that 106

134 at the ealy pat of the tansient heat tansfe pocess, the solid fluid inteface maintains a moe unifom tempeatue. The diffeence of dimensionless maximum and minimum tempeatue at the solid fluid inteface inceases fom 0.01 at Fo=0.005 to 0.18 when the steady state condition eached at Fo= This patten is due to the themal stoage in the fluid that is necessay to develop the themal bounday laye since an isothemal condition was pesent at the beginning of the tansient heat tansfe pocess. As time goes on, the thickness of the themal bounday laye inceases and theefoe the tempeatue ises. The inteface tempeatue esponds to the bounday laye thickness that inceases downsteam. Theefoe, the tempeatue becomes minimum at the impinging point and maximum at the oute edge of the disk. Dimensionless Inteface Tempeatue,Θint Re=15 Fo = 0.339, Θ max - Θ min (int) = 0.18 Re=450 Fo = 0.154, Θ max - Θ min (int) = Re=675 Fo = 0.09, Θ max - Θ min (int) = Re=900 Fo = 0.051, Θ max - Θ min (int) = Re=5 Fo = 0.015, Θ max - Θ min (int) = 0.06 Re=100 Fo = 0.005, Θ max - Θ min (int) = Dimensionless Radial Location, / d Figue 4.17 Dimensionless inteface tempeatue distibutions fo diffeent Fouie numbes (Re=500, Ek=.65x10 4, β=.67, silicon disk, wate, b/d n =0.5, q w =15 kw/m ). 107

135 Figue 4.18 shows the vaiation of local Nusselt numbe along the solid fluid inteface at diffeent time instants. The local Nusselt numbe deceases with time until it eaches the steady state equilibium distibution. The local Nusselt numbe is contolled by local tempeatue and local heat flux at the solid fluid inteface. Both of these quantities incease with time. The local Nusselt numbe shows a highe value at ealy stages of the tansient pocess due to smalle tempeatue diffeence between the liquid jet and disk solid fluid inteface. This essentially means that all heat eaching the solid fluid inteface via conduction though the solid is moe efficiently convected out as the local fluid tempeatue is low eveywhee at the inteface. The local Nusselt numbe, as shown in figue 4.18 inceases apidly ove a small distance (coe egion) measued fom the stagnation point, eaching a maximum aound /d n =0.04, and then deceases along the adial distance as the bounday laye develops futhe downsteam. 160 Local Nusselt Numbe,Nu Seies1 Fo = 0.339, Nu avg = 10.8 Re=5 Fo = 0.154, Nu avg = Re=450 Fo = 0.09, Νu avg = Re=675 Fo = 0.051, Νu avg = Re=900 Fo = 0.015, Νu avg = 16.5 Re=100 Fo = 0.005, Nu avg = Dimensionless Radial Location, / d Figue 4.18 Local Nusselt numbe distibutions fo diffeent Fouie numbes (Re=500, Ek=.65x10 4, β=.67, silicon disk, wate, b/d n =0.5, q w =15 kw/m ). 108

136 The location of the maximum Nusselt numbe can be associated with the tansition of the flow fom the vetical impingement to hoiontal displacement whee the bounday laye stats to develop. The vaiation of dimensionless maximum tempeatue at the inteface, maximum tempeatue inside the solid, and maximum to minimum tempeatue diffeence at the inteface fo diffeent Fouie numbes with wate as the cooling fluid at diffeent Reynolds numbes ae shown in figue The contol of maximum tempeatue is impotant in many citical themal management applications including electonic packaging. As expected, the tempeatue inceases eveywhee with time stating fom the initial isothemal condition. A apid incement is seen at the ealie pat of the tansient, and it levels off as the themal stoage capacity of the solid diminishes and become eo at the steady state condition. Dimensionless Tempeatue, Θ Tmax,solid Θ max (solid), Re = 500 Tmax,sol, Θ (solid), Re = 800 Re=800 Tmax,sol, Θ (solid), Re = 1100 Re=1100 Re=15 Θ max (int), Re = 500 Re=450 Θ max (int), Re = 800 Re=675 Θ max (int), Re = 1100 Re=900 Θ max - Θ min (int), Re = 500 Re=5 Θ max - Θ min (int), Re = 800 Re=100 Θ max - Θ min (int), Re = Fouie Numbe, Fo Figue 4.19 Dimensionless maximum tempeatue vaiations fo diffeent Reynolds numbes (Ek=.65x10 4, β=.67, silicon disk, wate, b/d n =0.5, q w =15 kw/m ). 109

137 It maybe noted that the time equied to each the steady state condition is lowe at a highe Reynolds numbe because the highe velocity of the fluid helps to enhance the convective heat tansfe pocess. The maximum to minimum tempeatue diffeence at the inteface inceases with time as moe heat flows thoughout the solid disk and tansmitted to the fluid. 38 Aveage Nusselt Numbe, Nu av Re = 500 Ek= Re = 800 Ek=.65x10-4 Re = 1100 Ek=9.46x Fouie Numbe, Fo Figue 4.0 Aveage Nusselt numbe vaiations fo diffeent Reynolds numbes (Ek=.65x10 4, β=.67, silicon disk, wate, b/d n =0.5, q w =15 kw/m ). Figue 4.0 povides the integated aveage Nusselt numbe vaiations fo vaious Reynolds numbe with wate as the cooling fluid at diffeent time instants. As expected, the aveage Nusselt numbe is lage at the ealy pat of the tansient and monotonically deceases with time ultimately eaching the value fo the steady state condition. A highe Reynolds numbe inceases the magnitude of fluid velocity nea the solid fluid inteface that contols the convective heat tansfe and theefoe inceases the aveage Nusselt 110

138 numbe. These obsevations ae in line with the pevious studies by Rahman and Faghi [96, 98] and Saniei et al. [39]. Dimensionless Tempeatue, Θ Re=15 Θ max (int), Ek = Re=450 Θ max (int), Ek = 1.3x10-4 Re=675 Θ max (int), Ek = 6.6x10-5 Tmax,solid Θ (solid), Ek = Tmax,sol, Θ (solid), Ek = 1.3x10-4 Re=800 Tmax,sol, Θ max (solid), Ek = 6.6x10-5 Re=1100 Re=900 Θ max - Θ min (int),ek = Re=5 Θ max - Θ min (int),ek=1.3x10-4 Re=100 Θ max - Θ min (int),ek=6.6x Fouie Numbe, Fo Figue 4.1 Dimensionless maximum tempeatue vaiations fo diffeent Ekman numbes (Re=750, β=.67, silicon disk, wate, b/d n =0.5, q w =15 kw/m ). Figue 4.1 shows the esults fo the dimensionless maximum tempeatue vaiation at the inteface, maximum tempeatue inside the solid and maximum to minimum tempeatue diffeence at the inteface fo diffeent time instants with wate as the cooling fluid fo vaious Ekman numbes. The maximum tempeatue within the solid was encounteed at the outlet adjacent to the heated suface (= b, = d ). The tempeatues ise with time as the solid disk and the fluid stoe heat showing a apid esponse at the ealie pat of the heating pocess until the themal stoage capacity eaches its limit at steady state. The maximum to minimum tempeatue diffeence at the inteface inceases with time as moe heat flows though the solid disk and tansmitted to 111

139 the fluid. It may be noted that the magnitude of the dimensionless tempeatue as well as the time equied to each the steady state condition becomes smalle as the Ekman numbe deceases. This is because the magnitude of fluid velocity neas the solid fluid inteface that contols the convective heat tansfe ate inceases with the incement of the otational ate of the disk o the eduction of Ekman numbe. 5 Aveage Nusselt Num be, Nuav Ek = Ek= Ek =.65x10-4 Ek=.65x10-4 Ek = 1.3x10-4 Ek=1.3x10-4 Ek = 8.83x10-5 Ek=8.83x Ek = 6.6x10-5 Ek=6.6x Fouie Numbe, Fo Figue 4. Aveage Nusselt numbe vaiations fo diffeent Ekman numbes (Re=750, β=.67, silicon disk, wate, b/d n =0.5, q w =15 kw/m ). The aveage Nusselt numbe vaiations with time fo vaious Ekman numbes ae shown in figue 4.. As expected, the aveage Nusselt numbe is lage at the ealy pat of the tansient and monotonically deceases with time ultimately eaching the value fo the steady state condition. Thoughout the tansient heating pocess, the aveage Nusselt numbe is geate at lage spinning ate o smalle Ekman numbe. As the Ekman numbe deceases fom to 6.6x10 5 the aveage Nusselt numbe inceases by an 11

140 aveage of 0.81% when the Reynolds numbe is kept constant at 750. This obsevation is in ageement with Rice et al. [46]. Anothe impotant facto that contols the tansient heat tansfe pocess is the thickness of the disk. Its effect on the dimensionless maximum tempeatue vaiation at the inteface, maximum tempeatue inside the solid and maximum to minimum tempeatue diffeence at the inteface fo diffeent time instants with wate as the cooling fluid is pesented in figue 4.3. The plate thickness significantly affects the tempeatue distibution. It may be noted that as the thickness of the disk inceases, the time needed to achieve the steady state condition inceases. This is due to moe stoage capacity of heat within the solid Dimensionless Tempeatue, Θ Tmax,solid Θ max (solid), b/d n = 0.50 Tmax,sol, Θ max (solid), b/d n = 1.67 Re=1100 Re=15 Θ max (int), b/d n = 0.50 Re=675 Θ max (int), b/d n = 1.67 Re=900 Θ max - Θ min (int),b/d n = 0.50 Re=100 Θ max - Θ min (int),b/d n = Fouie Numbe, Fo Figue 4.3 Dimensionless maximum tempeatue vaiations fo diffeent dimensionless disk thicknesses (Re=1,100, Ek=1.0x10 4, β=.67, silicon disk, wate, and q w =15 kw/m ). 113

141 Also, the tempeatue at the solid fluid inteface emains lowe and moe unifom due to highe themal esistance of the solid to the path of heat flow and highe oppotunity fo adial conduction within the disk. 8 Aveage Nusselt Numbe, Nuav b/d n = 0.17 Ek= b/d n = 0.50 Ek=.65x10-4 b/d n = 1.00 Ek=1.3x10-4 b/d n = 1.5 Ek=8.83x10-5 b/d n = 1.67 Ek=6.6x Fouie Numbe, Fo Figue 4.4 Aveage Nusselt numbe vaiations fo diffeent dimensionless disk thicknesses (Re=1,100, Ek=1.0x10 4, β=.67, silicon disk, wate, and q w =15 kw/m ). Figue 4.4 shows the aveage Nusselt numbe vaiation as a function of time fo five distinct plate thicknesses using silicon as the solid mateial. The aveage Nusselt numbe is highe fo a thinne disk. A thinne disk offes lowe themal esistance to the path of the heat flow. In addition, the local convective heat tansfe coefficient at the impingement egion tuns out to be highe because of less smoothing out of intefacial tanspot due to lowe oppotunity fo adial conduction within the disk. Thee esult in highe aveage Nusselt numbe fo a thinne disk. 114

142 The effect of solid mateial popeties on tansient heat tansfe is pesented in figue 4.5. The studied mateials wee aluminum, Constantan, coppe, silicon, and silve having diffeent themo physical popeties. Fo all mateials, the tempeatue changes occu faste at the ealie pat of the heating pocess and the slope gadually decays when the steady state appoaches. The change of slope shows the themal enegy balance esponse of the tansient conduction convection heat tansfe at the solid fluid inteface. It can be obseved that a mateial having a lowe themal conductivity such as Constantan maintains a highe tempeatue at the solid disk inteface and within the solid as the themal conductivity contols how effectively the heat flows and distibutes though the mateial. Dimensionless Tempeatue, Θ Tmax,solid Θ max (solid), Constantan Tmax,sol, Θ max (solid), Aluminum Re=800 Tmax,sol, Θ max (solid), Silve Re=1100 Re=15 Θ max (int), Constantan Re=450 Θ max (int), Aluminum Re=675 Θ max (int), Silve Re=900 Θ max - Θ min (int),constantan Re=5 Θ max - Θ min (int),aluminum Re=100 Θ max - Θ min (int),silve Fouie Numbe, Fo Figue 4.5 Dimensionless maximum tempeatue vaiations fo diffeent solid mateials (Re=650, Ek=.65x10 4, b/d n =0.5, β=.67, wate, and q w =15 kw/m ). 115

143 Fo the same eason, the maximum tempeatue within the solid and that at the inteface ae significantly diffeent fo Constantan, wheeaeas about the same fo both silve and aluminum. The themal diffusivity of the mateial also contibutes to the tansient heat tansfe pocess. As noticed, silve and aluminum each the steady state faste than Constantan due to thei highe themal diffusivity. The values of themal diffusivity fo the mateials consideed hee at 303K ae α silve =1.74 x10 4 m /s, α aluminum =8.33 x10 5 m /s, and α Constantan = 6.0 x10 6 m /s.the magnitude of the tempeatue non unifomity at the inteface at steady state is contolled by themal conductivity of the mateial. It may be noted that the themal conductivity of Constantan (k Constantan =.7 W/m. K) has an aveage maximum to minimum tempeatue diffeence of 3.39 K, wheeaeas the themal conductivity of silve (k silve = 49 W/m. K) has only an aveage 6. K tempeatue diffeence at the inteface. Aveage Nusselt Numbe, Nuav Fouie Numbe, Fo AL CU CNT Silicon Silve Aluminum Coppe Constantan Silicon Silve Figue 4.6 Aveage Nusselt numbe vaiations fo diffeent solid mateials (Re=650, Ek=.65x10 4, b/d n =0.5, β=.67, wate, and q w =15 kw/m ). 116

144 Figue 4.6 shows the distibution of aveage Nusselt numbe with time fo the five mateials used in this study. Constantan shows a highe aveage heat tansfe coefficient compaed to the othe mateials ove the entie tansient pocess due to its lowe themal conductivity. The aveage Nusselt numbe distibutions of coppe and silve ae almost identical due to thei simila themal conductivity values. It will be also impotant to know how the mateials esponded in eaching themal equilibium based on thei thickness. Figue 4.7 pesents the steady state Fouie numbe (Fo ss ) fo these mateials at diffeent plate thicknesses. The steady state Fouie numbe (Fo ss ) was defined as the time needed to appoach 99.99% of the steady state local Nusselt numbe ove the entie solid fluid inteface. As the thickness inceases in value, the time to each steady state also inceases. The adial conduction becomes stonge as the disk thickness inceases geneating a moe unifom heat distibution at the inteface. Steady State Fouie Numbe, Foss AL CU CNT Aluminum Coppe Constantan Silicon Silicon Silve Silve Dimensionless Disk Thicknesses, b/d n Figue 4.7 Time equied to each steady state unde the effects of vaious mateial popeties and disk thickness (Re=1,100, Ek=1.0x10 4, β=.67, wate, and q w =15 kw/m ). 117

145 Howeve, the incement of solid thickness ceates moe themal esistance to the heat tansfe pocess. The themal diffusivity of the solid plays a significant ole in detemining the duation of the tansient heat tansfe pocess. Constantan takes longe in eaching steady state due to its lowe themal diffusivity compaed to the othe mateials. Figue 4.8 pesents the time equied to each the steady state condition as a function of Reynolds numbe. The duation of the tansient heat tansfe deceases as the Reynolds numbe inceases. This is due to quicke dissipation of heat with highe flow ate and lowe themal bounday laye thickness. 0.5 Steady State Fouie Numbe, Fo ss Reynolds Numbe, Re Figue 4.8 Time equied to each steady state unde the effects of diffeent Reynolds numbe (β=.67, wate, Ek=.65x10 4, silicon disk, b/d n =0.5, and q w =15 kw/m ). Figues 4.9 and 4.30 show the development of isothemal lines within the solid at diffeent time instants. It is impotant to notice that at ealy stages of the tansient heat tansfe pocess, the isothemal lines gow paallel to the bottom heated suface of the solid disk. 118

146 Figue 4.9 Isothemal lines at diffeent instants (Re=1,100, Ek=1.0x10 4, β=.67, silicon disk, wate, q w =15 kw/m, and b/d n =0.5). Figue 4.30 Isothemal lines at diffeent instants (Re=1,100, Ek=1.0x10 4, β=.67, silicon disk, wate, and q w =15 kw/m, and b/d n =1.67). 119

147 As time goes on, the isothemal lines stat moving upwad towad lowe tempeatue egions until they each the solid fluid inteface. Afte that, they stat to fom concentic lines nea the stagnation point and expand futhe down into the solid until a steady state condition is achieved. The tempeatues inside the solid fo figue 4.9 ae lowe compaed to figue Also, isothemal lines have lage slope in figue 4.9. The incement of solid thickness ceates moe themal esistance and povides a moe unifom inteface tempeatue due to adial heat speading within the solid. Based on ou numeical data, a coelation fo the aveage Nusselt numbe was developed as a function of themal conductivity atio, Ekman numbe, Reynolds numbe, and Fouie numbe to accommodate most of the tanspot chaacteistics of the tansient heat tansfe duing axial fee liquid jet impingement on a thick solid disk spinning at a constant angula velocity. 5 Numeical Aveage Nusselt Numbes Pedicted Aveage Nusselt Numbes Figue 4.31 Compaison of pedicted aveage Nusselt numbe of equation 4. with pesent numeical data. 10

148 Figue 4.31 gives a gaphical compaison between the numeical aveage Nusselt numbes to the aveage Nusselt numbes pedicted by equation 4.. The coelation that best fitted the numeical data can be placed in the following fom: Nu av =1.965 Re Ek ε 0.5 Fo 0.01 (4.) In developing this coelation, all aveage Nusselt numbe data coesponding to the vaiation of diffeent paametes wee used. The least squaes cuve fitting method was used. The pecent diffeence of the pedicted aveage Nusselt numbe was defined as: % diff = ((Nuav ped Nuav num )/Nuav num ) 100. The diffeences between numeical and pedicted aveage Nusselt numbe values ae in the ange of 10.58% to 13.83%. In geneal, the oveall aveage diffeence of aveage Nusselt numbes was equal to 6.59%. The values of the dimensionless vaiables used in this study ae: 500 Re 1,100, 6.6x10 5 Ek.65x10 4, β=.67, P=5.49, 7.6 ε and Fo It should be noted fom figue 4.31 that a lage numbe of data points ae well coelated with equation 4.. This coelation povides a convenient tool fo the pediction of aveage heat tansfe coefficient duing the tansient heat tansfe pocess. 11

149 Chapte 5 Confined Liquid Jet Impingement Model Results 5.1 Steady State: Stationay Confined Wall with Spinning Taget The numeical esults of a confined liquid jet impingement on top of a spinning taget o wafe ae pesented in tems of dimensionless solid fluid inteface tempeatue distibution and local as well as aveage Nusselt numbe vaiation. A chaacteistic velocity vecto distibution is shown in figue 5.1. Figue 5.1 elocity vecto distibution fo a confined jet impingement on a silicon wafe with wate as the cooling fluid (Re=1,000, β=1.5, Ek=1.4x10 4, b/d n =0.5). It can be seen that the velocity emains almost unifom at the potential coe egion of the jet. The velocity deceases and the jet diamete inceases as the fluid gets close to the plate duing the impingement pocess. Theeafte, the fluid stikes the solid suface at 1

150 which point thee is a apid deceleation while the flow changes diection paallel to the solid disk. Afte this, thee is a bief acceleation stating the development of bounday laye. It can be noticed that the bounday laye thickness inceases along the adius. The fluid between the bounday laye one and confined top plate emains quasi stagnant with a flow velocity ten times less than the inlet velocity. The poximity of the spinning confined plate geneates a eciculation patten in this egion. Figue 5.1 shows the vaiation of local Nusselt numbe distibutions and solid fluid dimensionless inteface tempeatue plots fo diffeent Reynolds numbe unde a low otational ate of 15 RPM. All local Nusselt numbe distibutions ae half bell shaped with a peak at the stagnation point. It may be noted, howeve, that due to spinning steamlines ae not aligned along the disk adius, athe the fluid moves at an angle based on the ate of otation. The plots in figue 5. eveal that dimensionless inteface tempeatue deceases with jet velocity (o Reynolds numbe). At any Reynolds numbe, the inteface tempeatue has the lowest value at the stagnation point (undeneath the cente of the axial opening) and inceases adially along the adius eaching the highest value at the end of the disk. This is due to the development of themal bounday laye as the fluid moves downsteam fom the cente of the disk. The thickness of the themal bounday laye inceases with adius and causes the inteface tempeatue to incease. Figues 5. confim to us how an inceasing Reynolds numbe contibutes with a moe effective cooling. 13

151 Local Nusselt Numbe,Nu Re=500 Re=750 Re=1000 Re=1500 Re=000 Temp, Re=500 Temp, Re=750 Temp, Re=1000 Temp, Re=1500 Temp, Re= Dimensionless Inteface Tempeatue,Θ int Dimensionless Radial Location, / n 0 Figue 5. Local Nusselt numbe and dimensionless inteface tempeatue distibution fo a silicon wafe at diffeent Re, and wate as the cooling f1uid (b=0.3 mm, H n =0.3 cm, Ek=.65x10 4, and q w =50 kw/m ). Aveage Nusselt Numbe, Nu av Ek=1.06x10-3 = 1.06x10-3 Nu Ek at = Ek= x10-4 Nu Ek at = Ek= x10-4 h h, at Ek= = 1.06x10-3 h h, at Ek= = 3.78x10-4 h h, at Ek= =.65x Reynolds Numbe, Re Aveage Convective Heat Tansfe Coefficient, h av (W/m K) Figue 5.3 Aveage Nusselt numbe and heat tansfe coefficient compaison fo diffeent Reynolds numbe at low, intemediate and high Ekman numbes (q w =50 kw/m, H n =0.3 cm). 14

152 Figue 5.3 plots the aveage Nusselt numbe and aveage heat tansfe coefficient as a function of Reynolds numbe fo low, intemediate, and high Ekman numbes o otational ates. It may be noted that aveage Nusselt numbe inceases with Reynolds numbe. As the flow ate (o Reynolds numbe) inceases, the magnitude of fluid velocity nea the solid fluid inteface that contols the convective heat tansfe ate inceases. It may be also noted that at lowe Reynolds numbe (1,500 1,700) the aveage heat tansfe coefficient (o aveage Nusselt numbe) deceases with Ekman numbe (o inceases with disk spinning ate). Theefoe, spinning povides a positive influence on convective heat tansfe at this Reynolds numbe ange. The gaphs intesect aound 1,750 and at highe Reynolds numbe (1,800,000) a highe Nusselt numbe is obseved at a lowe spinning ate. The intesection of all gaphs indicates the pesence of the liquid jet momentum dominated egion at Reynolds numbes geate than 1,750. The otational ate effects on the local Nusselt numbe and solid fluid dimensionless inteface tempeatue ae illustated in figue 5.4. All cuves on figue 5.4 potay a half bell shaped pofile with cest at the stagnation egion. This tend matches with pevious studies by Webb and Ma [76] and Chang et al. [73]. It may be noted that otational effect inceases local Nusselt numbe and geneates lowe tempeatue ove the entie solid fluid inteface with somewhat less intensity in compaison with the Reynolds numbe effect. An exception is the case with Ek=3.31x10 5 whee the local Nusselt numbe distibution shows significantly highe values up to / n =6 and aftewad it becomes lowe in compaison with othe plots in figue 5.4. In this paticula case the otation geneates a positive effect at smalle adial locations, wheeas at highe adial locations the bounday laye sepaates fom the wall and causes an ineffective cooling. 15

153 This type of behavio is consistent with the esults of Popiel and Boguslawski [36] whee in otation dominated egime the impinging jet stated being undescoed by the fluid ejection of the otating disk. Local Nusselt Numbe,Nu Nu,@Ek=6.6x10-4 = -4 Nu,@Ek=.65x10-4 = -4 Nu,@Ek=9.46x10-5 = -5 Nu,@Ek=6.6x10-5 = -5 Nu,@Ek=3.31x Temp,@Ek=6.6x10-4 Temp,@Ek=.65x Temp,@Ek=9.46x Temp,@Ek=6.6x Temp,@Ek=3.31x Dimensionless Inteface Tempeatue,Θint Dimensionless Radial Location, / n Figue 5.4 Local Nusselt numbe and dimensionless inteface tempeatue plots fo a silicon wafe at diffeent Ekman numbes and wate as the cooling f1uid (Re=750, Q=3.54x10 m 3 /s, b=0.3 mm, H n /d n =5.333, and q w =50 kw/m ). Figues 5.5 compaes the solid fluid inteface tempeatue esults of the pesent woking fluid (wate) with thee othe coolants that have been consideed in pevious themal management studies, namely ammonia (NH 3 ), flouoinet (FC 77) and oil (MIL 7808). Figue 5.6 shows the coesponding Nusselt numbe distibutions. It may be noticed that wate pesents lowe inteface tempeatue and Nusselt numbe distibution in compaison with FC 77, NH 3 and MIL 7808 at most locations. Ammonia on the othe hand has the oveall highest inteface tempeatue. 16

154 Solid-Fluid Inteface Tempeatue,K AMMONIA (P=1.9) WATER (P=5.49) FC-77 (P=3.66) MIL-7808 (P=14.44) Dimensionless Radial Distance, / n Figue 5.5 Inteface tempeatue fo diffeent cooling fluids (Re=750, Q=3.54x10 m 3 /s, Ω=15 RPM, b=0.3 mm, H n /d n =.67, and q w =50 kw/m ). Local Nusselt Numbe,Nu AMMONIA (P=1.9) WATER (P=5.49) FC-77 (P=3.66) MIL-7808 (P=14.44) Dimensionless Radial Location, / n Figue 5.6 Local Nusselt numbe fo diffeent cooling fluids (Re=750, Q=3.54x10 m 3 /s, Ω =15 RPM, b=0.3 mm, H n /d n =.67, and q w =50 kw/m ). 17

155 Local Nusselt Numbe,Nu Constantan Coppe Plain Suface Silicon Silve Temp, Constantan Temp, Coppe Temp, Plain Suface Temp, Silicon Temp, Silve Solid-Fluid Dimensionless Inteface Tempeatue,Θint Dimensionless Radial Location, / n Figue 5.7 Local Nusselt numbe and dimensionless inteface tempeatue distibutions fo diffeent solid mateials with wate as the cooling fluid (Re=1,500, Q=7.08x10 m 3 /s, q w =50 kw/m, Ω=15 RPM, H n =0.3 cm, q w =50 kw/m ). Figue 5.7 shows the dimensionless solid fluid inteface tempeatue and local Nusselt numbe distibution plots espectively as a function of the dimensionless adial distance measued fom the axis symmetic impingement axis fo diffeent solid mateials with wate as the woking fluid. The numeical simulation was caied fo a set of mateials, namely coppe, silve, Constantan and silicon, having diffeent themo physical popeties. Results fo plain suface (eo thickness of the disk) ae also plotted to identify the extent of conjugate effects. The tempeatue distibution plots eveal how the themal conductivity of the solids affects the heat flux distibution that contols the local inteface tempeatue. It may be noted that Constantan has the lowest tempeatue at the impingement axis and the highest at the oute edge of the disk. This lage inteface 18

156 tempeatue vaiation is due to its lowe themal conductivity. As the themal conductivity inceases, the themal esistance within the solid becomes lowe and the inteface tempeatue becomes moe unifom as seen in the plots coesponding to coppe and silicon. The coss ove of the cuves of the fou mateials and plain suface occued due to a constant fluid flow and heat flux ate that povides a constant themal enegy tansfe fo all cicumstances. Naow and elevated bell shape patten is seen in figue 5.4 fo all solid mateials with low themal conductivity. Convesely high themal conductivity mateials like coppe and silve potay a moe unifom Nusselt numbe distibution in geneal. Consideing the tends of heat tansfe enhancement as functions of themal conductivity atio, nole to plate spacing, Pandtl numbe, Ekman numbe, and Reynolds numbe and by accommodating most of the tanspot chaacteistics of a confined liquid jet impingement cooling, a coelation was developed in the following fom: Nu av =Re 1.6 Ek P.58 β 0.5 ε 0.65 (5.1) Figue 5.8 gives the compaison between the numeical aveage Nusselt numbes to aveage Nusselt numbes pedicted by equation 5.1. The pecent diffeence of the pedicted aveage Nusselt numbe was defined as: % diff = ((Nuav ped Nuav num )/Nuav num ) 100. The diffeences between numeical and pedicted aveage Nusselt numbe values ae in the ange 0.36% to 14.47%. The mean value of the eo was 7.70%. The anges of the dimensionless vaiables in this study ae: 750 Re,000, 6.6x10 5 Ek.65x10 4, P=5.49, 7.6 ε The Pandtl numbe exponent was deived fom 19

157 Matin s equation [144] fo single ound nole impinging jet. It should be noted fom figue 5.8 that a lage numbe of data points ae well coelated with equation 5.1. Numeical Aveage Nusselt Numbes, Nu av ß 5.33,(Re=1500,Ω=500pm,ε=7.6) 1.17 ß 5.33,(Re=1500,Ω=500pm,ε=7.6) 1.17 ß 5.33,(Re=1500,Ω=15pm,ε=7.6) 15 Ω 500pm,(Re=750,ß=5.33,ε=7.6) 7.6 ε 697.5,(Re=750,Ω=15pm,ß=5.33).67 ß 5.33,(Re=750,Ω=350pm,ε=7.6) 1.17 ß 5.33,(Re=1000,Ω=350pm,ε=7.6) 750 Re 000,(Ω=350pm,ß=4.0,ε=7.6) Pedicted Aveage Nusselt Numbes Figue 5.8 Aveage Nusselt numbe coelation esults fo vaious studied paametes. The deviation is pimaily in the coe egion whee the heat tansfe values ae lage unde lage Reynolds numbe and diffeent spinning ates. This coelation povides a convenient tool fo the pediction of aveage heat tansfe coefficient unde confined liquid jet impingement on top of a spinning disk. The majo diffeence between past studies and the pesent investigation is the accounting fo conduction within the solid wafe and fluid fo vaious mateials, plus the nole to plate spacing atio as a pat of the coelation. One of the papes used fo the validation of this numeical study was the expeimental wok caied out by Gaimella and Rice [75] using flouoinet (FC 77) as the coolant. This liquid was tested fo heat emoval unde confined liquid jet 130

158 impingement on a stationay disk (Ek= ). The simulation attempted to duplicate the exact conditions of that expeiment. Figue 5.9 compaes the vaiations of local Nusselt numbe distibution along the solid fluid inteface obtained fom the simulation with the coelation developed fom the expeimental data. The pecent diffeence of the pedicted aveage Nusselt numbe was defined as: % diff = ((Nuav ped Nuav num )/Nuav num ) 100. Consideing the eos inheent in any expeimental measuements (the epoted uncetainty ange fom.46% to 3.3%) as well as discetiation and ound off eos in the simulation, the compaison is quite satisfactoy. A simila pofile has also been documented by Ma et al. [69]. 400 Re=4000 Local Nusselt Numbe,Nu Gaimella and Rice, [75]coelation at Re= /d n Figue 5.9 Local Nusselt numbe distibution fo a silve disk with FC 77 as the cooling fluid (H n /d n =4, q w =50 kw/m ). The expeimental wok caied out by Cape et al. [35] to detemine the aveage heat tansfe coefficient of a otating disk, with an appoximately unifom suface tempeatue, cooled by a single oil liquid jet impinging nomal to the suface, was also 131

159 used fo the validation of the numeical esults. The authos pesented coelations that elated the aveage Nusselt numbe to otational Reynolds numbe, jet Reynolds numbe, and Pandtl numbe. Aveage Nusselt numbe, Nu av Cape and Deffenbaugh [34],P=70 Cape et al.[35],p=87 Cape et al.[35],p=70 Cape et al.[35],p=400 P=87, CFD Model P=70, CFD Model P=400, CFD Model Reynolds Numbe, Re Figue 5.10 Aveage Nusselt numbe coelation fo vaious Reynolds numbes and Ekman numbe and thee diffeent P values of liquid oil axis symmetic jet impingement. The simulation has attempted to duplicate numeically the exact conditions of that expeiment. The computation was conducted fo thee nominal values of T j of 375, 331 and 30 K esulting in values of P of 87, 70 and 400 espectively. The otational Reynolds numbe was kept constant at a value equal to 6,000. As a esult of these behavio thee distinct angula velocities values (Ω) had to be used: 140, 480 and 730 RPM coesponding to the Pandtl numbes of 87, 70 and 400 espectively. The disk had a diamete of 10 cm and thickness of.54 cm and was made of 7075 T6 Aluminum, a 13

160 mateial with a elatively high themal conductivity of 11.4 W/mK. As seen in figue 5.10, the ageement of the esults fom the aveage Nusselt numbe coelation of Cape et al. [35] with the pesent data is quite good. Thee diffeent plots based on this coelation have been included in ode to make a qualitative and quantitative compaison. The pecent diffeence of the expeimental aveage Nusselt numbe was defined in the fom: % diff = ((Nuav num Nuav exp )/Nuav exp ) 100. The aveage Nusselt numbe uncetainties of Cape et al. [35] ange fom 6% to 3.96% fo all Pandtl numbes. An additional aveage Nusselt numbe plot was included fom Cape and Deffenbaugh [34] fo Pandtl numbe of 70. The aveage Nusselt numbe uncetainties fo Cape and Deffenbaugh [34] coelation ange fom 9.1% to 18.34%. This validation with available expeimental data may povide good level of confidence on the numbes obtained duing pesent numeical simulation. 5. Steady State: Spinning Confined Wall with Stationay Taget This section descibes the heat tansfe chaacteistics of a confined liquid jet impingement unde a spinning confinement disk. Figue 5.11 shows the vaiation of local Nusselt numbe and solid fluid dimensionless inteface tempeatue distibutions fo diffeent Reynolds numbe unde a low otational ate (Ek=4.5x10 4 ). All local Nusselt numbe distibutions ae half bell shaped with a peak at the stagnation point. It may be noted, howeve, that due to spinning steamlines ae not aligned along the disk adius, athe the fluid moves at an angle based on the ate of otation. The positive influence of the spinning of the confinement disk can be obseved paticulaly at Re=750, at which point the Nusselt numbe at / d > 0.6 becomes highe in compaison with that of the 133

161 Reynolds numbe of 1,000. The plots in figue 5.11 eveal that dimensionless inteface tempeatue deceases with jet velocity (o Reynolds numbe). Local Nusselt Numbe,Nu Re=500 Re=750 Re=1000 Re=150 Re=1500 Temp, Re=500 Temp, Re=750 Temp, Re=1000 Temp, Re=150 Temp, Re= Dimensionless Inteface Tempeatue,Θint Dimensionless Radial Distance, / d Figue 5.11 Local Nusselt numbe and dimensionless inteface tempeatue distibutions fo a silicon disk with wate as the cooling fluid fo diffeent Reynolds numbes (Ek=4.5x10 4, β=.0, b/d n =0.5). At any Reynolds numbe, the dimensionless inteface tempeatue has the lowest value at the stagnation point (undeneath the cente of the axial opening) and inceases adially downsteam eaching the highest value at the end of the disk. This is due to the development of themal bounday laye as the fluid moves downsteam fom the cente of the disk. The thickness of the themal bounday laye inceases with adius and causes the inteface tempeatue to incease. The incement of the dimensionless inteface tempeatue coincides with the thickening of themal bounday laye. A lowe inteface tempeatue distibution at Re=750 is attained in compaison to Re=1,000 fo the dimensionless adial distance, / d > 0.6. This is due to the fact that the tangential velocity 134

162 fom the top plate penetates into the themal bounday laye thickness adjacent to the heated stationay disk. This effect emains stonge when the momentum of the jet fluid is lowe. At highe Reynolds numbes (i.e., Re 1,000), the jet fluid momentum ovecomes the tangential velocity effects and inceases the dimensionless inteface tempeatue. Figues 5.11 confim to us how an inceasing Reynolds numbe contibutes to a moe effective cooling. Simila pofiles have been documented by Gaimella and Nenaydykh [77] and Ma et al. [69, 81]. Figue 5.1 plots the aveage Nusselt numbe as a function of Reynolds numbe fo low, intemediate, and high Ekman numbes. It may be noted that aveage Nusselt numbe inceases with Reynolds numbe. As the flow ate (o Reynolds numbe) inceases, the magnitude of fluid velocity nea the solid fluid inteface that contols the convective heat tansfe ate inceases. Futhemoe, at a paticula Reynolds numbe, the Nusselt numbe decease with Ekman numbe (o gadually inceases with the incement of disk spinning ate). This behavio confims the positive influence of the otational ate on the aveage Nusselt numbe down to Ek=1.5x10 4 that coesponds to a spinning ate of 350 RPM. Howeve at Ek=1.06x10 4 (spinning ate of 500 RPM) the aveage Nusselt numbe is lowe than the stationay disk (Ek= ). At high spinning ate the themal bounday laye thickness inceases due to suction ceated by the spinning motion of the confinement plate. Theefoe the heat tansfe coefficient deceases compaed to the stationay disk; the aveage Nusselt numbes deceases by 39% at Re=750 and by % at Re= It may be also noticed that the aveage Nusselt numbe plots gets close to 135

163 each othe as the Reynolds numbe inceases indicating that cuves will intesect at highe Reynolds numbes Aveage Nusselt Numbe, Nu av Ek= = 13.5 Ek=.65x10-4 = 4.5x Ek=9.46x10-5 = 1.5x10-4 Ek=6.6x10-5 = 1.06x Reynolds Numbe,Re Figue 5.1 Aveage Nusselt numbe vaiations with Reynolds numbe at diffeent Ekman numbes fo a silicon disk with wate as the cooling fluid (β=.0, b/d n =0.5). These intesections indicate the pesence of a liquid jet momentum dominated egion at highe Reynolds numbes. Fom the numeical esults it was obseved that the heat tansfe is dominated by impingement when Re. Ek> 0.11 and dominated by disk otation when Re. Ek< In between thee limits, both of these effects play impotant oles in detemining the vaiations of aveage Nusselt numbe. This type of behavio is consistent with the expeimental esults of Bodesen et al. [38] whee the atio of jet and otational Reynolds numbes was used to chaacteie the flow egime. 136

164 The otational ate effects on the local Nusselt numbe and solid fluid dimensionless inteface tempeatue ae illustated in figues 5.13 and 5.14 fo a Reynolds numbe of 750 and dimensionless nole to taget spacing (β) equal to Local Nusselt Numbe,Nu Ek=oo = Ek=.65x10-4 = 4.5x10-4 Ek=1.04x10-4 =.13x10-4 Ek=9.46x10-5 = 1.4x10-4 Ek=7.36x10-5 = 1.06x10-4 Ek=5.5x10-5 = 8.50x10-5 Ek=4.4x10-5 = 7.08x Dimensionless Radial Distance, / d Figue 5.13 Local Nusselt numbe distibutions fo a silicon disk with wate as the cooling f1uid at diffeent Ekman numbes (Re=750, β=3.0, b/d n =0.5). It may be noted that otational effect inceases local Nusselt numbe and geneates lowe tempeatue ove the entie solid fluid inteface with somewhat less intensity in compaison with the Reynolds numbe effect. Figue 5.14 shows that dimensionless inteface tempeatue deceases with the incement of the otational ate; as the Ekman numbe deceases fom to 1.4x10 4 the local Nusselt numbe inceases by an aveage 5.56% in figue 5.13 and the dimensionless inteface tempeatue deceases by an aveage.3% in figue 5.14 unde a Reynolds numbe of

165 Dimensionless Inteface Tempeatue, Θint Ek=6.6x10-4 = Ek=.65x10-4 = 4.5x10-4 Ek=1.04x10-4 =.13x10-4 Ek=9.46x10-5 = 1.4x10-4 Ek=7.36x10-5 = 1.06x10-4 Ek=6.6x10-5 = 8.50x10-5 Ek=3.31x10-5 = 7.08x Dimensionless Radial Distance, / d Figue 5.14 Dimensionless inteface tempeatue distibutions fo a silicon disk with wate as the cooling fluid at diffeent Ekman numbes (Re=750, β=3.0, b/d n =0.5). The enhancement of Nusselt numbe due to otation is pimaily caused by enhancement of local fluid velocity adjacent to the heated disk suface. The tangential velocity due to otation combined with axial and adial velocities due to jet momentum esults in an inceased magnitude of velocity vecto stating fom the cente of the disk. The effects of disk thickness vaiation on the solid fluid dimensionless inteface tempeatue and local Nusselt numbe ae shown in figues 5.15 and 5.16 espectively. In these plots, silicon has been used as the disk mateial and wate as the cooling fluid fo Reynolds numbe of 1,500 and otational ate of 350 RPM (Ek=1.5x10 4 ). The dimensionless solid fluid inteface tempeatue inceases fom the impingement egion all the way to the end of the disk. 138

166 Dimensionless Inteface Tempeatue,Θint b=0.1mm b/d n = 0.5 b=0.3mm b/d n = 0.50 b=0.5mm b/d n = 0.75 b=1mm b/d n = 1.00 b=1.5mm b/d n = 1.5 b=mm b/d n = Dimensionless Radial Distance, / d Figue 5.15 Dimensionless inteface tempeatue distibutions fo diffeent silicon disk thicknesses with wate as the cooling fluid (Re=1,500, Ek=1.5x10 4, β=.0). When tempeatue is lowe in the stagnation egion a highe outflow tempeatue is obtained. This is quite expected since the total heat tansfeed to the disk as well as the fluid flow ates ae the same fo all the cases. It may be noted that the disk thickness vaiation esults intesect with each othe aound dimensionless adial distance of / d =0.65. Thicke disks geneate moe unifom dimensionless inteface tempeatue due to lage adial conduction within the disk. The local Nusselt numbe plots in figue 5.16 change significantly with the vaiation of disk thickness. In all cases, it is evident that the Nusselt numbe is sensitive to the solid thickness especially at smalle adii whee highe Nusselt numbe ae obtained due to apid development of themal bounday laye. 139

167 Local Nusselt Numbe,Nu b=0.1mm b/d n = 0.5 b=0.3mm b/d n = 0.50 b=0.5mm b/d n = 0.75 b=1mm b/d n = 1.00 b=1.5mm b/d n = 1.5 b=mm b/d n = Dimensionless Radial Distance, / d Figue 5.16 Local Nusselt numbe distibutions fo diffeent silicon disk thicknesses with wate as the cooling fluid (Re=1,500, Ek=1.5x10 4, β=.0). Eight diffeent nole to plate spacing atio (β) fom 0.5 to 5 wee modeled fo wate as the coolant and silicon as the disk mateial. The effects of nole to taget spacing on the dimensionless inteface tempeatue and local Nusselt numbe distibutions at a spinning ate of 15 RPM o (Ek=4.5x10 4 ) and Reynolds numbe of 750 ae shown in figues 5.17 and It may be noticed that the impingement height quite significantly affects the Nusselt numbe distibution paticulaly at the stagnation egion. It may be noticed that a highe local Nusselt numbe at the stagnation egion is obtained when the nole is bought close to the heated disk (β=0.5). The spinning motion of the confinement disk eally penetates though the themal bounday laye adjacent to the heated stationay disk and povides a lage fluid velocity and theefoe a lage ate of convective heat tansfe. 140

168 Dimensionless Inteface Tempeatue,Θint ß=0.5 ß=0.50 ß=0.75 ß=1.00 ß=.00 ß=3.00 ß=4.00 ß= Dimensionless Radial Distance, / d Figue 5.17 Dimensionless inteface tempeatue distibutions fo a silicon disk with wate as the cooling fluid fo diffeent nole to taget spacings (Re=750, Ek=4.5x10 4, b/d n =0.5). Local Nusselt Numbe,Nu ß=0.5 ß=0.50 ß=0.75 ß=1.00 ß=.00 ß=3.00 ß=4.00 ß= Dimensionless Radial Distance, / d Figue 5.18 Local Nusselt numbe distibutions fo a silicon disk with wate as the cooling fluid fo diffeent nole to taget spacings (Re=750, Ek=4.5x10 4, b/d n =0.5). 141

169 As the nole is moved away fom the disk (β=0.5 1), the local Nusselt numbe deceases. This is due to smalle effect of the otational velocity of the confinement disk. Also, as the spacing is inceased, the jet fluid needs to tavel a lage distance though the existing fluid column between taget and confinement disks and theeby loses its momentum. The minimum stagnation Nusselt numbe is seen fo β=1 and also the shape of the cuve somewhat changes. The nole to taget atio of β= geneates an optimal mix of the impinging jet flow with the otationally induced flow esulting in highe heat tansfe ate. Thee is only small change in Nusselt numbe values at spacings geate than β=. This obsevation is in line with the pevious study by Hung and Lin [74] fo a confined jet impingement with a stationay disk. Dimensionless Inteface Tempeatue, Θint AMMONIA (P=1.9, Ek=3.3x10-5 ) WATER (P=5.49, Ek=1.5x10-4 ) FC-77 (P=3.66, Ek=1.5x10-4 ) MIL-7808 (P=14.4, Ek=1.7x10-3 ) Dimensionless Radial Distance, / d Figue 5.19 Dimensionless inteface tempeatue distibutions fo diffeent cooling fluids with silicon as the disk mateial (Re=1,000, β=.0, b/d n =0.5). 14

170 Figues 5.19 compaes the dimensionless solid fluid inteface tempeatue esults of the pesent woking fluid (wate) with thee othe coolants that have been consideed in pevious themal management studies, namely ammonia (NH 3 ), flouoinet (FC 77), and oil (MIL 7808) unde a Reynolds numbe of 1,000. Even though the otational ate (Ω) fo the top confining wall was set at 350 RPM the vaiation of Ekman numbe occued since the density (ρ) and dynamic viscosity (µ) ae diffeent fo each fluid. The inteface tempeatue distibution of figue 5.19 shows simila esults fo FC 77 and MIL It may be noticed that both ammonia and wate pesent highe dimensionless inteface tempeatue distibution in compaison with MIL 7808 and FC 77. Wate shows a lage vaiation of dimensionless inteface tempeatue along the adius of the disk. The wate and ammonia cuves intesect at a dimensionless adial distance of / d =0.65. Local Nusselt Numbe,Nu AMMONIA (P=1.9, Ek=3.3x10-5 ) WATER (P=5.49, Ek=1.5x10-4 ) FC-77 (P=3.66, Ek=1.5x10-4 ) MIL-7808 (P=14.4, Ek=1.7x10-3 ) Dimensionless Radial Distance, / d Figue 5.0 Local Nusselt numbe distibutions fo diffeent cooling fluids with silicon as the disk mateial (Re=1,000, β=.0, b/d n =0.5). 143

171 Figue 5.0 shows the coesponding local Nusselt numbe distibutions. It may be noticed that MIL 7808 pesents the highest local Nusselt numbe values in compaison with wate, NH 3 and FC 77 fo a dimensionless adial distance, / d < Only FC 77 exhibits a highe heat emoval ate beyond this point. MIL 7808 shows the lagest vaiation of local Nusselt numbe pimaily because of its lage vaiation of viscosity with tempeatue. Ammonia povides the lowest Nusselt numbe because of its small Pandtl numbe. Highe Pandtl numbe fluids lead to a thinne themal bounday laye, and moe effective heat emoval ate at the inteface. Pesent woking fluid esults ae in ageement with Li et al. [80] findings whee lage Pandtl numbe coesponded to a highe ecovey facto. Thus, diffeent Pandtl numbes epesent diffeent themal bounday laye thicknesses and diffeent heat geneations by viscous dissipation of the fluids. Local Nusselt Numbe,Nu Plain Suface Constantan Silicon Coppe Silve Temp, Plain Suface Temp, Constantan Temp, Silicon Temp, Coppe Temp, Silve Dimensionless Inteface Tempeatue,Θint Dimensionless Radial Distance, / d Figue 5.1 Local Nusselt numbe and dimensionless inteface tempeatue distibutions fo diffeent solid mateials with wate as the cooling fluid (Re=1,000, Ek=4.5x10 4, β=.0, b/d n =0.5). 144

172 The dimensionless solid fluid inteface tempeatue and local Nusselt numbe distibution plots as a function of a dimensionless adial distance (/ d ) measued fom the axis symmetic impingement axis fo diffeent solid mateials with wate as the woking fluid ae plotted in figue 5.1. The numeical simulation was caied out fo a set of mateials, namely coppe, silve, Constantan, and silicon, having diffeent themo physical popeties at Reynolds numbe of 1,000 and Ekman numbe of 4.5x10 4. Results fo plain suface (eo thickness of the disk) ae also plotted to identify the extent of conjugate effects. The tempeatue distibution plots eveal how the themal conductivity of the solid affects the heat flux distibution that contols the local inteface tempeatue. It may be noted that plain suface has the lowest tempeatue at the impingement axis and the highest at the oute edge of the disk. The inteface tempeatue vaiation fo Constantan is also quite lage due to its lowe themal conductivity. As the themal conductivity inceases, the themal esistance within the solid becomes lowe and the inteface tempeatue becomes moe unifom as seen in the plots coesponding to coppe and silicon. The coss ove of the cuves of the fou mateials and plain suface occued due to a constant fluid flow and heat input ate that povides a constant themal enegy tansfe fo all cicumstances. Naow and elevated bell shape patten is seen in figue 5.1 fo solid mateials with lowe themal conductivity. As the themal conductivity is inceased, a moe unifom Nusselt numbe distibution is obtained. One of the goals of this wok was to develop a pedictive tend of the aveage heat tansfe coefficient. A coelation fo the aveage Nusselt numbe was developed as a function of themal conductivity atio, nole to taget spacing, disk thickness, Ekman 145

173 numbe, and Reynolds numbe to accommodate most of the tanspot chaacteistics of a confined liquid jet impingement cooling pocess. A coelation that best fitted the numeical data can be placed in the following fom: Nu av =1.976 β 0.01 Re 0.75 Ek (b/d n ) 0.05 ε 0.69 (5.) In developing this coelation, all aveage Nusselt numbe data coesponding to the vaiation of diffeent paametes wee used. Only data points coesponding to wate as the fluid wee used because the numbe of aveage heat tansfe data points fo othe fluids was small. Figue 5. gives the compaison between the numeical aveage Nusselt numbes to aveage Nusselt numbes pedicted by equation 5.. Numeical Aveage Nusselt Numbe, Nu av Re 1500,β=, b/d n =0.5,Ek = 4.5x10-4, ε=7.6 Re, 500 Re 1500,β=, b/d n =0.5,Ek = 1.4x10-4, ε=7.6 ßRe=750,0.5 β 5, b/d n =0.5,Ek = 4.5x10-4, ε=7.6 Re=750,β=3, 15 Ω 500pm,(Re=750,ß=5.33,ε=7.6) b/d n =0.5,1.06x10-4 Ek 4.5x10-4, ε= ε 697.5,(Re=750,Ω=15pm,ß=5.33) Re=750,β=0.5, b/d n =0.5,1.06x10-4 Ek 4.5x10-4, ε=7.6 Re=1500,β=,.67 ß 5.33,(Re=750,Ω=350pm,ε=7.6) 0.5 b/d n 1.67,Ek=1.06x10-4, ε=7.6 Re=1000,β=, 1.17 ß 5.33,(Re=1000,Ω=350pm,ε=7.6) b/d n =1.67,Ek=4.5x10-4, 7.6 ε 67.6 Seies14 Re=750,β=, 0.5 b/d n 1.67,Ek=4.5x10-4, ε= Pedicted Aveage Nusselt numbe Figue 5. Compaison of pedicted aveage Nusselt numbe of equation 5. with pesent numeical data. 146

174 The pecent diffeence of the pedicted aveage Nusselt numbe was defined as: %diff = ((Nuav ped Nuav num )/Nuav num ) 100. The aveage Nusselt numbe deviates in the ange of 13.8% to 15.3% fom the aveage Nusselt numbes pedicted by equation 5.. The mean deviation of the coelation was equal to 6.8%. The anges of the dimensionless vaiables used ae: 500 Re 1,500, 4.5x10 5 Ek 1.06x10 4, 0.5 β 5, P=5.49, 0.5 b/d n 1.67, and 7.6 ε It should be noted fom figue 5. that a lage numbe of data points ae well coelated by equation 5.. This coelation povides a convenient tool fo the pediction of aveage heat tansfe coefficient unde liquid jet impingement with a spinning confinement disk. The majo diffeence between past studies and the pesent investigation is the accounting fo conduction within the solid wafe and fluid fo vaious mateials, the spinning ate of the confinement disk, and the nole to taget spacing atio as a pat of the coelation. Thee othe papes used fo the validation of this numeical study wee the analytical woks caied out by Scholt and Tass [6], Nakoyakov et al. [68], and Liu et al. [17] using fluids with Pandtl numbe geate than unity as coolants. The fluids wee tested fo heat emoval unde fee liquid jet impingement on a heated flat suface maintained at unifom heat flux. The gaphical epesentation of actual numeical Nusselt numbe esults at the stagnation point at diffeent Reynolds numbe ae shown in figue 5.3. The esults shown in figue 5.3 wee on aveage within 8.17% of Scholt and Tass [6], within 6.67% of Nakoyakov et al. [68], and within 6.75% of Liu et al. [17]. 147

175 Stagnation Nusselt numbe, Nuo Ek = 1.5x10-4 Ek= = 1.06x10-4 Scholt and Tass, [6] (Ek = ) Nakoyakov et al., [68] (Ek = ) Liu et al., [17] (Ek = ) Reynolds Numbe, Re Figue 5.3 Stagnation Nusselt numbe compaison of Scholt and Tass [6], Nakoyakov et al. [68], and Liu et al. [17] with actual numeical esults unde diffeent Reynolds and Ekman numbes (d n =1. mm, b=0.3 mm). The pecent diffeence of the pedicted of local Nusselt numbe at the stagnation was defined as: %diff = ((Nuo num Nuo exp )/ Nuo exp ) 100. The local Nusselt numbe unde Reynolds numbes of 750, 1,000, 1,50 and 1,500 coelates with an aveage diffeence magin of 11.83%, 6.31%,.6%, and 8.40% espectively. Consideing the inheent discetiation and ound off eos, this compaison of Nusselt numbe at the stagnation point is also quite satisfactoy. 148

176 Chapte 6 Patially confined Liquid Jet Impingement Model Results 6.1 Steady State Cooling of Spinning Taget The numeical esults of conjugate heat tansfe of a steady lamina flow by a patially confined liquid jet impingement on a unifomly heated and spinning disk of finite thickness and adius ae pesented in tems of its dimensionless inteface tempeatue distibutions and local as well as aveage Nusselt numbe vaiation. The examine paametes ae: seveal flow ates o jet Reynolds numbes, six spinning ates o Ekman numbe, five diffeent disk thicknesses and fou nole to taget spacings. Figue 6.1 elocity vecto distibution fo a patially confined jet impingement on a silicon disk with wate as the cooling fluid (Re=475, Ek=4.5x10 4, p / d =0.5, β=0.5, b/d n =0.5). A typical velocity vecto distibution is shown in figue 6.1. It can be seen that the velocity emains almost unifom at the potential coe egion of the jet. The velocity 149

177 deceases as the fluid jet expands in the adial diection as it appoaches the taget plate duing the impingement pocess. The diection of motion of the fluid paticles shifts by as much as 90 o. Afte this, the fluid acceleates ceating a egion of high velocity wall jet within the confined fluid medium. It can be noticed that as the bounday laye thickness inceases downsteam and the fictional esistance fom the walls ae eventually tansmitted to the entie film thickness. This effect is obseved once the fluid leaves the confined egion and moves downsteam with a fee suface on the top. The vectos in the viscous one show a paabolic pofile going fom a minimum value at the solid fluid inteface to a maximum at the fee suface. The bounday laye develops apidly and the velocity of the fluid deceases as it speads adially along the disk. It may be noted, howeve, that due to spinning steamlines ae not aligned along the disk adius, athe the fluid moves at an angle based on the ate of otation. The thee diffeent egions obseved in the pesent investigation ae in ageement with the expeiments of Liu et al. [17]. Figue 6. pesents the fee suface height distibution fo diffeent plate to disk confinement atios when the jet stikes the cente of the disk while it is spinning at a ate of 15 RPM (Ek=4.5x10 4 ). It can be seen that the fluid speads out adially as a thin film. The film thickness deceases as the plate to disk confinement atio deceases unde the same spinning ate and flow ate. This behavio occus due to dominance of suface tension and gavitational foces that fom the fee suface as the fluid leaves the confinement one and moves downsteam. When p is inceased, the fictional esistance fom both walls slows down the momentum and esults in geate film thickness. Fo the conditions consideed in the pesent investigation, a sudden dop in fluid height occus fo p / d < because the equilibium film height fo fee suface motion is 150

178 significantly lowe than confinement height. In this situation, liquid may not cove all the way to the end of the confinement disk and fee suface may stat to fom within the confinement egion to povide a smooth steamline fo the fee suface. At p / d 0.5, the confinement egion is fully coveed with fluid and a smooth tansition is seen in film height distibution afte exit. Axial Distance, (cm) /5 p / d = 0.0 1/4 p / d = 0.5 1/3 p / d = / p / d = 0.50 /3 p / d = /4 p / d = Dimensionless Radial Location, / d Figue 6. Fee suface height distibution fo diffeent plate to disk confinement atio with wate as the cooling f1uid (Re=450, Ek=4.5x10 4, β=0.5, b/d n =0.5). Figue 6.3 shows the local Nusselt numbe and the dimensionless inteface tempeatue vaiation fo diffeent Reynolds numbe unde a otational ate of 15 RPM (Ek=4.5x10 4 ). The plots eveal that dimensionless inteface tempeatue deceases with jet velocity (o Reynolds numbe). At any Reynolds numbe, the dimensionless inteface tempeatue has the lowest value at the stagnation point (undeneath the cente of the axial opening) and inceases adially downsteam eaching the highest value at the end of 151

179 the disk. This is due to the development of themal bounday laye as the fluid moves downsteam fom the cente of the disk. The thickness of the themal bounday laye inceases with adius and causes the inteface tempeatue to incease. All local Nusselt numbe distibutions ae half bell shaped with a peak at the stagnation point. Figue 6.3 confim to us how an inceasing Reynolds numbe contibutes to a moe effective cooling. Simila pofiles have been documented by Gaimella and Nenaydykh [77] and Ma et al. [69, 81] Local Nusselt Numbe,Nu Re=0 Re=360 Re=540 Re=70 Re=900 Temp, Re=0 Temp,Re=360 Temp, Re=540 Temp, Re=70 Temp, Re= Dimensionless Inteface Tempeatue,Θint Dimensionless Radial Location, / d Figue 6.3 Local Nusselt numbe and dimensionless inteface tempeatue distibutions fo a silicon disk with wate as the cooling fluid fo diffeent Reynolds numbes (Ek=4.5x10 4, β=0.5, b/d n =0.5, p / d =0.667). Figue 6.4 plots the aveage Nusselt numbe as a function of Reynolds numbe fo low, intemediate, and high Ekman numbes. It may be noted that aveage Nusselt numbe inceases with Reynolds numbe. As the flow ate (o Reynolds numbe) inceases, the magnitude of fluid velocity nea the solid fluid inteface that contols the 15

180 convective heat tansfe ate inceases. Futhemoe, at a paticula Reynolds numbe, the Nusselt numbe gadually inceases with the incement of disk spinning ate. 16 Aveage Nusselt Numbe, Nu av Ek = Ek= Ek = 4.5x10-4 Ek=.65x10-4 Ek = 1.93x10-4 Ek=9.46x10-5 Ek = 1.5x10-4 Ek=6.6x Reynolds Numbe,Re Figue 6.4 Aveage Nusselt numbe vaiations with Reynolds numbe at diffeent Ekman numbes fo a silicon disk with wate as the cooling fluid (β=0.5, b/d n =0.5, p / d =0.667). This behavio confims the positive influence of the otational ate on the aveage Nusselt numbe down to Ek=1.5x10 4 that coesponds to a spinning ate of 45 RPM. It may be also noticed that the aveage Nusselt numbe plots get close to each othe as the Reynolds numbe inceases indicating that cuves will intesect at highe Reynolds numbes. These intesections indicate the pesence of a liquid jet momentum dominated egion at highe Reynolds numbes. Fom the numeical esults it was obseved that the heat tansfe is dominated by impingement when Re. Ek> 0.14 and dominated by disk otation when Re. Ek< In between thee limits, both of these effects play an impotant ole in detemining the vaiations of aveage Nusselt numbe. This type of 153

181 behavio is consistent with the expeimental esults of Bodesen et al. [38] whee the atio of jet and otational Reynolds numbes was used to chaacteie the flow egime. Local Nusselt Numbe,Nu Re=0 Ek = Re=540 Ek =.13x10-4 Re=900 Ek = 1.06x10-4 Temp, Re=0 Ek = Temp, Re=540 Ek =.13x10-4 Temp, Re=900 Ek = 1.06x10-4 Re=360 Ek = 4.5x10-4 Re=70 Ek = 1.4x10-4 Re=100 Ek= 7.08x10-5 Temp,,Re=360 Ek = 4.5x10-4 Temp, Ek Re=70 = 1.4x10-4 Temp, Ek Re=100= 7.08x Dimensionless Inteface Tempeatue,Θint Dimensionless Radial Location, / d 0.00 Figue 6.5 Local Nusselt numbe and dimensionless inteface tempeatue distibutions fo a silicon disk with wate as the cooling f1uid at diffeent Ekman numbes (Re=540, β=0.5, b/d n =0.5, and p / d =0.667). The otational ate effects on the local Nusselt numbe and dimensionless inteface tempeatue ae illustated in figue 6.5 fo a Reynolds numbe of 540 and dimensionless nole to plate spacing (β) equal to 0.5. It may be noted that otational effect inceases local Nusselt numbe and geneates lowe tempeatue ove the entie solid fluid inteface with somewhat less intensity in compaison with the Reynolds numbe effect. In addition, figue 6.5 shows that as the Ekman numbe deceases fom to 7.08x10 5 the local Nusselt numbe inceases by an aveage of 4.0% and the dimensionless inteface tempeatue deceases by an aveage of 8.34%. The enhancement of Nusselt numbe due to otation is pimaily caused by enhancement of local fluid 154

182 velocity adjacent to the otating disk suface. The tangential velocity due to otation combined with axial and adial velocities due to jet momentum inceases the magnitude of the velocity vecto stating fom the cente of the disk Local Nusselt Numbe,Nu Seies1 b/d n = 0.5 Seies b/d n = 0.50 Seies3 b/d n = 0.83 Seies4 b/d n = 1.5 Seies5 b/d n = 1.67 Temp, Re=0 b/d n = 0.5 Temp,,Re=360 b/d n = 0.50 Temp, Re=540 b/d n = 0.83 Temp, Re=70 b/d n = 1.5 Temp, Re=900 b/d n = Dimensionless Inteface Tempeatue,Θint Dimensionless Radial Location, / d Figue 6.6 Local Nusselt numbe and dimensionless inteface tempeatue distibutions fo diffeent silicon disk thicknesses with wate as the cooling fluid (Re=450, Ek=4.5x10 4, β=0.5, p / d =0.667). The effects of disk thickness vaiation on the dimensionless inteface tempeatue and local Nusselt numbe ae shown in figue 6.6. The dimensionless inteface tempeatue inceases fom the impingement egion all the way to the end of the disk. It may be noted that the cuves intesect with each othe at a dimensionless adial distance of / d =0.55. The thicke disks geneate moe unifom dimensionless inteface tempeatue due to lage adial conduction within the disk. The local Nusselt numbe plots change slightly with the vaiation of disk thickness. In all cases, it is evident that the Nusselt numbe is moe sensitive to the solid thickness at the coe egion whee highe 155

183 values ae obtained. Fo a lowe stagnation tempeatue, the outlet tempeatue tends to be elatively highe unde constant flow ate and heat flux conditions. This is quite expected because of the oveall enegy balance of the system. This phenomenon has been documented by Lachefski et al. [54] Local Nusselt Numbe,Nu ß = 0.5 ß = 0.75 ß = 1.0 Temp, ß = 0.5 Temp, ß = 0.75 Temp, ß = Dimensionless Inteface Tempeatue,Θint Dimensionless Radial Location, / d Figue 6.7 Local Nusselt numbe and dimensionless inteface tempeatue distibutions fo a silicon disk with wate as the cooling fluid fo thee diffeent nole to taget spacing atio (Re=750, Ek=4.5x10 4, b/d n =0.5, p / d =0.667). Thee diffeent nole to taget spacing atios (β) fom 0.5 to 1 wee modeled and the esults ae shown in figue 6.7. It may be noticed that the impingement height quite significantly affects the Nusselt numbe distibution. A highe local Nusselt numbe is obtained when the nole is bought close to the heated disk (β=0.5). The smalle gap between the nole and the taget disk avoids loss of momentum as the jet tavels though the confined fluid medium and esults in a lage fluid velocity and theefoe a lage ate of convective heat tansfe. As the nole is moved away fom the disk, the local Nusselt 156

184 numbe deceases. This obsevation is in line with the pevious study by Hung and Lin [74] fo a confined jet impingement on a stationay disk. Local Nusselt Numbe,Nu AMMONIA (P=1.9, Ek=3.30x10-5 ) WATER (P=5.49, Ek=1.5x10-4 ) FC-77 (P=3.66, Ek=1.50x10-4 ) MIL-7808 (P=14.4, Ek=1.70x10-3 ) Temp, AMMONIA Temp, WATER Temp, FC-77 Temp, MIL Dimensionless Inteface Tempeatue,Θint Dimensionless Radial Location, / d Figue 6.8 Local Nusselt numbe and dimensionless inteface tempeatue distibutions fo diffeent cooling fluids fo silicon as the disk mateial (Re=750, β=0.5, b/d n =0.5, p / d =0.667). Figue 6.8 compaes the dimensionless inteface tempeatue and local Nusselt numbe esults of the pesent woking fluid (wate) with thee othe coolants, namely ammonia (NH 3 ), flouoinet (FC 77) and oil (MIL 7808) unde a Reynolds numbe of 750. Even though the otational ate (Ω) fo the impingement disk was set at 350 RPM the vaiation of Ekman numbe occued since the density (ρ) and dynamic viscosity (µ) ae diffeent fo each fluid. It may be noticed that MIL 7808 pesents the highest dimensionless inteface tempeatue and wate has the lowest value. Ammonia shows the most unifom distibution of tempeatue along the adius of the disk. MIL 7808 pesents the highest local Nusselt numbe values ove the entie adial distance. Ammonia on the 157

185 othe hand povides the lowest Nusselt numbe. Highe Pandtl numbe fluids lead to a thinne themal bounday laye and theefoe moe effective heat emoval ate at the inteface. Pesent woking fluid esults ae in ageement with Li et al. [80] findings whee a lage Pandtl numbe coesponded to a highe ecovey facto Local Nusselt Numbe,Nu Aluminum Constantan Coppe Silve Silicon Temp, Aluminum Temp,Constantan Temp, Coppe Temp, Silve Temp, Silicon Dimensionless Inteface Tempeatue,Θint Dimensionless Radial Location, / d 0 Figue 6.9 Local Nusselt numbe and dimensionless inteface tempeatue distibutions fo diffeent solid mateials with wate as the cooling fluid (Re=875, Ek=4.5x10 4, β=0.5, b/d n =0.5, p / d =0.667). Figue 6.9 shows the dimensionless inteface tempeatue and local Nusselt numbe distibution plots as a function of dimensionless adial distance (/ d ) measued fom the axis symmetic impingement axis fo diffeent solid mateials with wate as the woking fluid. The dimensionless tempeatue distibution plots eveal how the themal conductivity affects the heat flux distibution. Constantan shows the lowest tempeatue at the impingement one o stagnation point and the highest dimensionless tempeatue at the outlet in compaison with othe solid mateials. Coppe and silve show a moe 158

186 unifom distibution and highe tempeatue values at the impingement one due to thei highe themal conductivity. The dimensionless tempeatue and local Nusselt numbe distibutions of these two mateials ae almost identical due to thei simila themal conductivity values. The coss ove of cuves fo all five mateials occued due to a constant fluid flow and heat flux ate that eaches a themal enegy balance. A solid mateial with lowe themal conductivity shows highe maximum local Nusselt numbe. Local Nusselt Numbe,Nu Seies1 p / d = 0.0 Seies3 p / d = Seies5 p / d = Temp, p / d = 0.0 Temp,0 p / d = Temp,oeess9 p / d = Seies p / d = 0.5 Seies4 p / d = 0.50 Seies11 p / d = 0.75 Temp, p / d = 0.5 Temp, p / d = 0.50 Seies1 Temp, p / d = Dimensionless Inteface Tempeatue,Θint Dimensionless Radial Location, / d 0 Figue 6.10 Local Nusselt numbe and dimensionless inteface tempeatue distibutions fo diffeent plate to disk confinement atio (Re=450, Ek=4.5x10 4, β=0.5, b/d n =0.5). Six diffeent plate to disk confinement atios ( p / d ) fom 0. to 0.75 wee modeled fo wate as the coolant and silicon as the disk mateial. The effects of plate to disk confinement atio on the dimensionless inteface tempeatue and local Nusselt numbe ae shown in figue The dimensionless inteface tempeatue inceases with the incement of the plate to disk confinement atio ( p / d ). This incement coincides 159

187 with the incement of liquid film thickness in the fee jet egion as seen in figue 6.. Unde the same spinning and flow ates, when p is inceased the highe fictional esistance fom the confinement disk slows down the fluid momentum. In addition, a thinne film thickness fo the same flow ate esults in highe fluid velocity nea the solid fluid inteface esulting in a highe ate of convective heat tansfe. This is seen in the distibution of local Nusselt numbe which inceases with the decease of plate to disk confinement atio. 4 Numeical Aveage Nusselt Numbe, Nu av Pedicted Aveage Nusselt numbe Figue 6.11 Compaison of pedicted aveage Nusselt numbes of equation 6.1 with pesent numeical data. Figue 6.11 gives the compaison between the numeical aveage Nusselt numbes to aveage Nusselt numbes pedicted by equation 6.1. A coelation fo the aveage Nusselt numbe was developed as a function of confinement atio, themal conductivity atio, and dimensionless nole to taget spacing atio, Ekman numbe, and Reynolds 160

188 numbe to accommodate most of the tanspot chaacteistics of a semi confined liquid jet impingement cooling pocess. The coelation that best fitted the numeical data can be placed in the following fom: Nu av =1.948 β 0.1 Re 0.75 Ek 0.1 ε 0.7 ( p / d ) 0.05 (6.1) In developing this coelation, all aveage Nusselt numbe data coesponding to the vaiation of diffeent paametes wee used. Only data points coesponding to wate as the fluid wee used because the numbe of aveage heat tansfe data fo othe fluids wee small. The least squae cuve fitting technique was used in developing this equation. The sign of the exponents was detemined fom the tend of vaiation of aveage Nusselt numbe with each paamete. In addition, the pecent diffeence of the pedicted aveage Nusselt numbe was defined as: % diff = ((Nuav ped Nuav num )/Nuav num ) 100. The aveage Nusselt numbe deviates in a ange of 15.13% to 15.61% fom the aveage numeical esults pedicted by equation 6.1. The mean deviation of the above coelation was equal to 6.94%. The anges of the dimensionless vaiables in this study ae: 360 Re 900, 1.06x10 4 Ek 4.5x10 4, 0.5 β 1, 0. p / d 0.75, P=5.49, 7.6 ε A lage numbe of data points ae well coelated with equation 6.1, as shown in figue This coelation povides a convenient tool fo the pediction of aveage heat tansfe coefficient fo a patially confined liquid jet impingement on top of a spinning disk. 161

189 6. Steady State Cooling of Spinning Confined Wall and Taget Figue 6.1 shows the vaiation of the dimensionless inteface tempeatue and the local Nusselt numbe distibutions fo diffeent Reynolds numbes unde a otational ate of 75 RPM (Ek 1, =1.93x10 4 ) Local Nusselt Numbe,Nu Re=0 Re=360 Re=540 Re=70 Re=900 Temp, Re=0 Temp,Re=360 Temp, Re=540 Temp, Re=70 Temp, Re= Dimensionless Inteface Tempeatue,Θint Dimensionless Radial Location, / d Figue 6.1 Effects of Reynolds numbe on local Nusselt numbe and dimensionless solid fluid inteface tempeatue vaiation fo a silicon disk with wate as the cooling fluid (β=0.5, b/d n =0.5, p / d =0.667, Ek 1, =1.93x10 4 ). The plots in figue 6.1 eveal that dimensionless inteface tempeatue deceases with jet velocity (o Reynolds numbe). At any Reynolds numbe, the dimensionless inteface tempeatue has the lowest value at the stagnation point (undeneath the cente of the axial opening) and inceases adially downsteam eaching the highest value at the end of the disk. At a Reynolds numbe of 0, the tempeatue becomes pactically unifom afte / d > The thickness of the themal bounday laye inceases with adius and causes the inteface tempeatue to incease. The incement of the dimensionless inteface 16

190 tempeatue up to the end of its confinement coincides with the thickening of the themal bounday laye. Aftewad it becomes moe unifom beneath the fee suface. As noted in figue 6., thee is a significant e adjustment of fluid laye thickness as the flow comes out of the confinement and moves downsteam with a fee suface at the top. Figue 6.1 shows how the local Nusselt numbe distibutions inceases ove a small distance (coe egion) measued fom the stagnation point, eaching a maximum aound / d =0.05, and then deceases along the adial distance as the bounday laye develops futhe downsteam up to the end of the confined spinning plate o p / d Afte this location, the Nusselt numbe inceases downsteam and eaches a unifom value at lage adial locations of the disk. The location of the maximum Nusselt numbe can be associated with the tansition of the flow fom the vetical impingement to hoiontal displacement whee the bounday laye stats to develop. The incease of Nusselt numbe afte the exit fom the confinement is a esult of significant decease of film thickness that also deceases the thickness of the themal bounday laye until it eaches a new equilibium. It may be noticed that at low values of Reynolds numbe (Re=0 in paticula), local Nusselt numbe emains almost constant ove a good potion of the disk including a potion within the confinement egion. This is because at low Reynolds numbe, the jet momentum dies down and the flow is diven by otational motion of the disks. Figue 6.1 confims to us how an inceasing Reynolds numbe contibutes to a moe effective cooling. The obsevations ae in line with the pevious studies by Gaimella and Nenaydykh [77] and Saniei et al. [39]. Figue 6.13 plots the aveage Nusselt numbe as a function of Reynolds numbe fo low, intemediate, and high Ekman numbes of the solid disk. The spinning of the 163

191 confined plate was done at a constant ate of 15 RPM o Ek =4.5x10 4. It may be noted that aveage Nusselt numbe inceases with Reynolds numbe. As the flow ate (o Reynolds numbe) inceases, the magnitude of fluid velocity nea the solid fluid inteface that contols the convective heat tansfe ate inceases. Futhemoe, at a paticula Reynolds numbe, the Nusselt numbe gadually inceases with the incement of disk spinning ate. This behavio confims the positive influence of the otational ate of the solid disk on the aveage Nusselt numbe down to Ek 1 =1.5x10 4 that coesponds to a spinning ate of 45 RPM. 16 Aveage Nusselt Numbe, Nuav Ek 1 = Ek= Ek=.65x10-4 Ek 1 = 4.5x10-4 Ek=9.46x10-5 Ek 1 = 1.93x10-4 Ek=6.6x10-5 Ek 1 = 1.5x Reynolds Numbe,Re Figue 6.13 Effects of Reynolds numbe on aveage Nusselt numbe at diffeent Ekman numbes fo a silicon disk with wate as the cooling fluid (β=0.5, b/d n =0.5, p / d =0.667, Ek =4.5x10 4 ). It may be also noticed that the aveage Nusselt numbe plots gets close to each othe as the Reynolds numbe inceases indicating that cuves will intesect at highe Reynolds numbes. These intesections indicate the pesence of a liquid jet momentum 164

192 dominated egion at highe Reynolds numbes. Fom the numeical esults it was obseved that the heat tansfe is dominated by impingement when Re. Ek 1 > and dominated by disk otation when Re. Ek 1 < In between thee limits, both of these effects play an impotant ole in detemining the vaiation of aveage Nusselt numbe. This type of behavio is consistent with the expeimental esults of Bodesen et al. [39] whee the atio of jet and otational Reynolds numbes was used to chaacteie the flow egime. Local Nusselt Numbe,Nu Seies1 Ek 1, = Seies3 Ek 1 =.13x10-4 Seies5 Ek 1 = 1.06x10-4 Temp, Ek Re=0 1, = Temp, Ek Re=540 1 =.13x10-4 Temp, Ek Re=900 1 = 1.06x10-4 Seies Ek 1 = 4.5x10-4 Seies4 Ek 1 = 1.4x10-4 Seies11 Ek 1 = 7.08x10-5 Temp,,Re=360 Ek 1 = 4.5x10-4 Temp, Ek Re=70 1 = 1.4x10-4 Seies1 Temp, Ek 1 = 7.08x Dimensionless Inteface Tempeatue,Θint Dimensionless Radial Location, / d 0 Figue 6.14 Effects of Ek 1 vaiation on local Nusselt numbe and dimensionless inteface tempeatue distibutions fo a silicon disk with wate as the cooling f1uid (Re=540, β=0.5, b/d n =0.5, p / d =0.667, Ek =4.5x10 4 ). The otational ate effects of the solid disk unde the influence of a constant spinning ate of the confinement plate on the local Nusselt numbe and dimensionless inteface tempeatue ae illustated in figue 6.14 fo a Reynolds numbe of 540 and 165

193 dimensionless nole to plate spacing (β) equal to 0.5. It may be noted that the local Nusselt numbe emains the same ove the distance 0 </ d < 0.35 and inceases with otational ate (deceases with Ekman numbe) futhe downsteam. This is because the flow is highly dominated by jet inlet momentum at / d < 0.35, and the centifugal foces geneated by otation of the disks can influence the tanspot only at / d > It may also be noted that a highe otational ate povides a lesse amount of undeshoot in Nusselt numbe and a highe equilibium value at lage disk adii. Figue 6.14 shows that dimensionless inteface tempeatue deceases with the incement of the otational ate in compaison with the stationay case due to the enhancement of local fluid velocity adjacent to the disk. The local Nusselt numbe inceases by an aveage of 33.78% in figue 6.14; as the Ekman numbe of solid spinning disk deceases fom to 7.08x10 5 unde the influence of a constant spinning ate of 15 RPM (Ek =4.5x10 4 ) of the top confinement disk. The dimensionless inteface tempeatue deceases by an aveage of 10.85% in figue 6.14 unde a Reynolds numbe of 540. The enhancement of Nusselt numbe due to otation is pimaily caused by enhancement of local fluid velocity adjacent to the otating disk suface. The tangential velocity due to otation combined with axial and adial velocities due to jet momentum inceases the magnitude of the velocity vecto. Figue 6.15 shows the otational ate effects of the top confinement disk in conjunction with a constant spinning ate of the solid impingement disk on local Nusselt numbe and dimensionless inteface tempeatue distibutions fo a Reynolds numbe of 540 and dimensionless nole to plate spacing (β) equal to 0.5. It may be noted that otational effect up to a spinning ate of 375 RPM o (Ek =1.4x10 4 ) inceases the local 166

194 Nusselt numbe and geneates lowe tempeatue ove the entie solid fluid inteface with less intensity in compaison with the Reynolds numbe effect shown in figue 6.1 and the solid disk otational ate effect unde a constant spinning of the confinement plate shown in figue Local Nusselt Numbe,Nu Seies1 Ek 1, = Seies3 Ek =.13x10-4 Seies5 Ek = 1.06x10-4 Temp, Re=0 Ek 1, = Temp, Re=540 Ek =.13x10-4 Temp, Re=900 Ek = 1.06x10-4 Seies Ek = 4.5x10-4 Seies4 Ek = 1.4x10-4 Seies11 Ek = 7.08x10-5 Temp,,Re=360 Ek = 4.5x10-4 Temp, Re=70 Ek = 1.4x10-4 Seies1 Temp, Ek = 7.08x Dimensionless Inteface Tempeatue,Θint Dimensionless Radial Location, / d Figue 6.15 Effects of Ek vaiation on local Nusselt numbe and dimensionless inteface tempeatue distibutions fo a silicon disk with wate as the cooling f1uid (Re=540, β=0.5, b/d n =0.5, p / d =0.667, Ek 1 =4.5x10 4 ). Figue 6.15 shows that dimensionless inteface tempeatue deceases with the incement of the otational ate up to a spinning ate of 375 RPM (Ek =1.4x10 4 ) in compaison with the stationay case due to the enhancement of local fluid velocity adjacent to the disk. The local Nusselt numbe inceases by an aveage of 5.9% and the dimensionless inteface tempeatue deceases by an aveage of 0.40% in figue 6.15; as the Ekman numbe of the top confined plate deceases fom to 1.4x10 4 unde the influence of a constant spinning ate of 15 RPM (Ek 1 =4.5x10 4 ) of the solid 167

195 impingement disk. Howeve, exceptions occu fo spinning ates of 500 and 750 RPM (Ek =1.06x10 4 and 7.08x10 5 ) whee highe values fo dimensionless inteface tempeatue and lowe values fo Nusselt numbe ae found fo the most pat of the solid fluid inteface. In these paticula cases, the otation geneates a negative effect within the confined egion. At these high otational ates of the top disk (4 and 6 times compaed to the bottom disk) the themal bounday laye stuctue at the heated bottom disk tends to get swept away by the stong otational motion of the top disk. Theefoe a lowe Nusselt numbe is achieved compaed to othe cases in the confined egion. Howeve, when the flow gets out of the confinement at (/ d =0.667), the added momentum exeted by the top disk esults in ise of heat tansfe coefficient fom this point all the way to the end of the disk. Theefoe, the pope selection of two spinning ates is cucial in a design pocess. This type of behavio is consistent with the obsevations of Popiel and Boguslawski [36]. The effects of disk thickness vaiation on the dimensionless inteface tempeatue and local Nusselt numbe ae shown in figue In these plots, silicon has been used as the disk mateial and wate as the cooling fluid fo Reynolds numbe of 450 and spinning ate of 15 RPM (Ek 1, =4.5x10 4 ). The dimensionless inteface tempeatue inceases fom the impingement egion all the way to the end of the disk. It may be noted that the disk thickness vaiation cuves fom the 0.5 to 1.67 intesect with each othe at a dimensionless adial distance of / d =0.55. The thicke disks geneate moe unifom dimensionless inteface tempeatue due to a lage adial conduction within the disk. 168

196 Local Nusselt Numbe,Nu Seies1 b/d n = 0.5 Seies b/d n = 0.50 Seies3 b/d n = 0.83 Seies4 b/d n = 1.5 Seies5 b/d n = 1.67 Temp, Re=0 b/d n = 0.5 Temp,,Re=360 b/d n = 0.50 Temp, Re=540 b/d n = 0.83 Temp, Re=70 b/d n = 1.5 Temp, Re=900 b/d n = Dimensionless Radial Location, / d Dimensionless Inteface Tempeatue,Θint Figue 6.16 Effects of thickness vaiation on local Nusselt numbe and dimensionless inteface tempeatue distibutions fo a silicon disk with wate as the cooling f1uid (Re=450, β=0.5, Ek 1, =4.5x10 4, p / d =0.667). Since the flow ate and heat input at the bottom of the disk ae kept constant, the global enegy balance dictates that aveage inteface tempeatue changes only slightly as the themal esistance offeed by the disk changes with the vaiation of disk thickness. It may be obseved fom figue 6.16 that aveage inteface tempeatue slightly inceases with the incement of disk thickness. The local distibution of inteface tempeatue is pimaily contolled by the e distibution of input heat within the solid. A thinne plate offes a smalle oppotunity fo heat flux e distibution and theefoe a lage vaiation contolled by convection and local fluid tempeatue is seen. Fo a thicke plate, moe oppotunity fo adial conduction esults in highe inteface heat flux in the impingement egion whee the fluid is coole and gadually smalle inteface heat flux as the fluid moves downsteam. This esults in moe unifom inteface tempeatue as shown in 169

197 figue The combined effects only slight change in aveage inteface tempeatue wheeas lage change in local distibution with the vaiation of thickness esults in plots intesecting each othe in figue Local Nusselt numbe plots in figue 6.16 change slightly with the vaiation of disk thickness. In all cases, it is evident that the Nusselt numbe is sensitive to the solid thickness especially at the coe egion whee highe Nusselt numbe values ae obtained. It may be noted that local Nusselt numbe was calculated by using local tempeatue and local heat flux at the inteface, both of which became lage in the impingement egion with incease of disk thickness. Theefoe the net effect was almost same Nusselt numbe distibution fo all the thicknesses. This phenomenon has also been documented by Lachefski et al. [54] fo jet impingement on a stationay disk. Fou diffeent nole to taget spacing atio (β) fom 0.5 to 1 wee modeled using wate as the coolant and silicon as the disk mateial. The effects of nole to taget spacing on local Nusselt numbe and dimensionless inteface tempeatue at a spinning ate of 15 RPM (Ek 1, =4.5x10 4 ) and Reynolds numbe of 900 ae shown in figue It may be noticed that the impingement height quite significantly affects the Nusselt numbe distibution paticulaly at the stagnation egion. A highe local Nusselt numbe at the coe egion is obtained when the nole is bought close to the heated disk (β=0.5). 170

198 Local Nusselt Numbe,Nu ß = 0.5 ß = 0.5 ß = 0.75 ß = 1.0 Temp, ß = 0.5 Temp, ß = 0.50 Temp, ß = 0.75 Temp, ß = Dimensionless Inteface Tempeatue,Θint Dimensionless Radial Location, / d Figue 6.17 Effects of nole to taget spacing atio on local Nusselt numbe and dimensionless inteface tempeatue distibutions fo a silicon disk with wate as the cooling fluid (Re=900, b/d n =0.5, Ek 1, =4.5x10 4, p / d =0.667). A lowe distance between nole and impingement plate povides lowe loss of momentum as the jet tavels fo a shote distance though the suounding liquid medium. In addition, a smalle gap povides quicke popagation of centifugal foce fom the spinning disks into the fluid medium inceasing the net tanspot ate. It may be also noticed that in figue 6.17, cuves fo β= ae close togethe wheeas, at β=1, a highe tempeatue is obtained all along the disk. In figue 6b, it can be noticed that minimum in Nusselt numbe moves downsteam with incease in gap and no minimum is obseved at β=1. Theefoe, otational effects cannot popagate well when the gap between impingement and confinement plates is lage. The local maximum is associated with the tansition of flow stuctue fom vetical stagnation flow to hoiontal 171

199 bounday laye flow adjacent to the heated disk. The Nusselt numbe is maximum at the stat of the themal bounday laye. The minimum is associated with the tansition fom jet momentum dominated flow to otation dominated flow. As the fluid moves downsteam, bounday laye gows in thickness and jet momentum diminishes. On the othe hand, the centifugal foce geneated by disk otation inceases as the fluid moves to a lage adial location. The balance of these simultaneous effects esults in the minimum in local Nusselt numbe. As both disks ae otating, a smalle vetical gap between disks causes a stonge popagation of otational effects to the fluid and theefoe ealie tansition fom momentum dominated to otation dominated flow. Figue 6.18 compaes the dimensionless inteface tempeatue esults of the pesent woking fluid (wate) with thee othe coolants that have been consideed in pevious heat tansfe studies, namely ammonia (NH 3 ), flouoinet (FC 77) and oil (MIL 7808) unde a Reynolds numbe of 750. Even though the otational ate (Ω 1, ) fo the impinging solid disk and confinement plate was set at 350 RPM the vaiation of Ekman numbe occued since the density (ρ) and dynamic viscosity (µ) ae diffeent fo each fluid. It may be noticed that MIL 7808 pesents the highest dimensionless inteface tempeatue and ammonia has the lowest value. Ammonia shows the most unifom distibution of tempeatue along the adius of the disk. 17

200 Local Nusselt Numbe,Nu AMMONIA (P=1.9, Ek 1, =3.30x10-5 ) WATER (P=5.49, Ek 1, =1.5x10-4 ) FC-77 (P=3.66, Ek 1, =1.50x10-4 ) MIL-7808 (P=14.4, Ek 1, =1.70x10-3 ) Temp, AMMONIA Temp, WATER Temp, FC-77 Temp, MIL Dimensionless Inteface Tempeatue,Θint Dimensionless Radial Location, / d Figue 6.18 Effects of diffeent cooling fluids with silicon as the disk mateial on local Nusselt numbe and dimensionless inteface tempeatue (Re=750, β=0.5, b/d n =0.5, p / d =0.667). Figue 6.18 shows the coesponding local Nusselt numbe distibutions. It may be noticed that MIL 7808 pesents the highest local Nusselt numbe values ove the entie dimensionless adial distance. Ammonia on the othe hand povides the lowest Nusselt numbe. The Nusselt numbe tend is well coelated with the vaiation of Pandtl numbe. A highe Pandtl numbe fluid leads to a thinne themal bounday laye and theefoe moe effective heat emoval ate at the inteface. The pesent woking fluid esults ae in ageement with Li et al. [80] and Ma et al. [81] findings whee a lage Pandtl numbe coesponded to a highe ecovey facto. 173

201 Local Nusselt Numbe,Nu Aluminum Constantan Coppe Silve Silicon Temp, Aluminum Temp,Constantan Temp, Coppe Temp, Silve Temp, Silicon Dimensionless Inteface Tempeatue,Θint Dimensionless Radial Location, / d 0 Figue 6.19 Effects of diffeent solid mateials with wate as the cooling fluid on local Nusselt numbe and dimensionless inteface tempeatue (Re=875, Ek 1, =1.77x10 4, β=0.5, b/d n =0.5, and p / d =0.667). Figue 6.19 shows the dimensionless inteface tempeatue and local Nusselt numbe distibution plots as a function of dimensionless adial distance (/ d ) measued fom the axis symmetic impingement axis fo diffeent solid mateials with wate as the woking fluid. The studied mateials wee aluminum, Constantan, coppe, silicon, and silve having diffeent themo physical popeties. The dimensionless tempeatue distibution plots eveal how the themal conductivity affects the heat flux distibution. Constantan shows the lowest tempeatue at the impingement one o stagnation point and the highest dimensionless tempeatue at the outlet in compaison with othe solid mateials. Coppe and silve show a moe unifom distibution and highe tempeatue values at the impingement one due to thei highe themal conductivity. The dimensionless tempeatue and local Nusselt numbe distibutions of these two mateials 174

202 ae almost identical due to thei simila themal conductivity values. The coss ove of cuves fo all five mateials occued aound / d This coss ove is expected because of themal enegy balance fo constant fluid flow and heat input ates. A solid mateial with a lowe themal conductivity (Constantan) shows a highe maximum local Nusselt numbe. Fo all solid mateials, the local Nusselt numbe distibution inceases apidly ove a small distance (coe egion) measued fom the stagnation point, eaches a maximum aound / d =0.50, and then deceases along the adial distance up to p / d Futhe downsteam when the film encountes a fee suface at the top along with the otation of the solid disk at the bottom, the local Nusselt values fo all mateials gadually incease due to the incement of the tangential velocity and thinne themal bounday laye that enhances the heat tansfe on the solid disk suface. Six diffeent plate to disk confinement atios ( p / d ) fom 0. to 0.75 wee modeled using wate as the coolant and silicon as the disk mateial. The effects of plate to disk confinement atio on the dimensionless inteface tempeatue and local Nusselt numbe at a spinning ate of 15 RPM o Ek 1, =4.5x10 4 and Reynolds numbe of 450 ae shown in figue 6.0. The plots in figue 6.0 eveal that the dimensionless inteface tempeatue inceases with the incement of the plate to disk confinement atio ( p / d ). This incement coincides with the incement of liquid film thickness in the fee jet egion seen in figue 6.. A thinne film thickness fo the same flow ate esults in highe fluid velocity nea the solid fluid inteface esulting in a highe ate of convective heat tansfe. This is seen in the distibution of local Nusselt numbe plotted in figue 6.0. The local Nusselt numbe inceases with the decease of plate to disk confinement atio. 175

203 Local Nusselt Numbe,Nu Seies1 p / d = 0.0 Seies3 p / d = Seies5 p / d = Temp, p / d = 0.0 Temp,0 p / d = Temp,oeess9 p / d = Seies p / d = 0.5 Seies4 p / d = 0.50 Seies11 p / d = 0.75 Temp, p / d = 0.5 Temp, p / d = 0.50 Seies1 Temp, p / d = Dimensionless Inteface Tempeatue,Θint Dimensionless Radial Location, / d Figue 6.0 Local Nusselt numbe and dimensionless inteface tempeatue distibutions fo diffeent plate to disk confinement atio (Re=450, Ek 1, =4.5x10 4, β=0.5, b/d n =0.5). A coelation fo the aveage Nusselt numbe was developed as a function of confinement atio, themal conductivity atio, dimensionless nole to taget spacing, Ekman numbe, Reynolds numbe, and confinement plate to disk adius atio to accommodate most of the tanspot chaacteistics of a semi confined liquid jet impingement cooling pocess. The coelation that best fitted the numeical data can be placed in the following fom: Nu av =1.948 β 0.01 Re 0.75 Ek Ek ε 0.69 ( p / d ) 0.05 (6.) In developing this coelation, all aveage Nusselt numbe data coesponding to the vaiation of diffeent paametes wee used. Only data points coesponding to wate as the fluid wee used because the numbe of aveage heat tansfe data fo othe fluids 176

204 wee small. Also data points coesponding to both disks otating at the same ate wee used. 4 Numeical Aveage Nusselt Numbe, Nu av Pedicted Aveage Nusselt numbe Figue 6.1 Compaison of pedicted aveage Nusselt numbes of equation 6. with pesent numeical data. Figue 6.1 gives the compaison between the numeical aveage Nusselt numbes to aveage Nusselt numbes pedicted by equation 6.. The pecent diffeence of the pedicted aveage Nusselt numbe was defined as: % diff = ((Nuav ped Nuav num )/Nuav num ) 100. The pedicted aveage Nusselt numbe values fom equation 6. deviates in a ange of 14.76% to 13.08% fom the actual numeical esults obtained in pesent dissetation study. The mean deviation of the pedicted aveage Nusselt esults was equal to 6.37%. The anges of the dimensionless vaiables in this study ae: 360 Re 900, 4.5x10 4 Ek x10 5, 4.5x10 4 Ek 7.08x10 5, 0.5 β 1, 0. p / d 0.75, P=5.49, 7.6 ε It should be noted fom figue 6.1 that a lage numbe of data points 177

205 ae well coelated with equation 6.. This coelation can be a convenient tool fo the pediction of aveage heat tansfe coefficient Local Nusselt Numbe, Nu pm 100 pm 00 pm Oa et al.[44,45], 50pm Oa et al.[44,45], 100pm Oa et al.[44,45], 00pm Rice et al.[46], 50pm Rice et al.[46], 100pm Dimensionless Radial Location, / d Figue 6. Compaison of numeical and expeimental local Nusselt numbe distibutions at diffeent spinning ates fo an aluminum disk with wate as the cooling fluid (T j =93 K, Re=38, H n = m, b= m, b/d n =0.15, p = m, and p / d =0.5). Figue 6. shows a compaison of local Nusselt numbes obtained in pesent numeical simulation with the expeimental data obtained by Oa et al. [44, 45] and numeical esults of Rice et al. [46] at vaious otational speeds. A otating disk with a heat flux of 3kW/m, cooled by a ound single wate jet impingement at a flow ate of 3 lite/min (Re=38) and spinning at speeds of 50, 100, 00 RPM wee compaed. The computation was conducted fo jet tempeatue (T j ) of 93 K; the nole to taget spacing was set to m, with a nole diamete of m and fo colla (o confinement) that extended ove a adial distance of m. The spinning disk had a diamete of 178

206 m and thickness of m. The disk was made of aluminum, a mateial with a themal conductivity of 0.4 W/mK. As seen in figue 6., the ageement of the local Nusselt numbe esults of Oa et al. [44, 45] and Rice et al. [46] with the pesent numeical simulation is satisfactoy. In those studies highe Nusselt numbes wee found at the inne potions of the disk, close to the colla, and deceased towads the oute edge. This was due to the adial spead of the flow, and lowe convective heat tansfe emoval of the liquid due to a moe ponounced backflow effect on the uppe confinement plate at a lage atio of confinement, including the sluggish development of the themal bounday laye thickness. The same behavio was obseved as pat of ou numeical study just with a slight cutback effect on local Nusselt numbe distibutions at lage atios of confinement. The pecent diffeence of pesent local Nusselt numbe esults was defined as: % diff = ((Nu num Nu exp )/Nu exp ) 100. The diffeence in local Nusselt numbe between Oa et al. [44, 45] and the pesent simulation is in the ange of 18.55% to.07% with an aveage diffeence of 1.33%. The diffeence in local Nusselt numbe between Rice et al. [46] and the pesent simulation falls in the ange of.08% to 5.01% with an aveage diffeence of 14.9%. The Nusselt numbe at the stagnation egion was compaed with the stagnation Nusselt numbe coelation developed by Liu et al. [17] fo liquid jet impingement ove a stationay disk. Fo the Reynolds numbe and otational ates consideed in ou study, the aveage diffeence was 13.14%. The otation always enhances the stagnation Nusselt numbe compaed to the stationay disk. 6.3 Tansient Cooling of Spinning Taget Figue 6.3 shows the local Nusselt numbe and the dimensionless inteface tempeatue vaiation fo diffeent time instants. It can be obseved that at the ealie pat 179

207 of the tansient heat tansfe pocess, the solid fluid inteface maintains a moe unifom tempeatue. The diffeence of dimensionless maximum and minimum tempeatue at the solid fluid inteface inceases fom at Fo=0.051 to 0.05 when the steady state condition eached at Fo= Afte the powe is tuned on, the heat is fist absobed by the solid as it is tansmitted though the solid and dissipated to the fluid. At the solid fluid inteface, the fluid absobs heat and caies it as it moves downsteam. At the stat of the tansient, the thickness of the themal bounday laye is eo. As time goes on, the thickness of the themal bounday laye inceases and theefoe the tempeatue ises. The inteface tempeatue esponds to the bounday laye thickness that inceases downsteam. Theefoe, the tempeatue becomes minimum at the impinging point and maximum at the oute edge of the spinning disk. Local Nusselt Numbe,Nu Nu(Fo Seies1= 0.369) Nu(Fo Re=450 = 0.09) Θ(Fo T,Fo=0.369 = 0.369) Θ(Fo T,Fo=0.09 = 0.09) Nu(Fo Re=5 = 0.174) Nu(Fo Re=675 = 0.051) Θ(Fo T,Fo=0.174 = 0.174) Θ(Fo T, Fo=0.051 = 0.051) Dimensionless Inteface Tempeatue,Θint Dimensionless Radial Location, / d 0 Figue 6.3 Local Nusselt numbe and dimensionless inteface tempeatue distibutions fo a silicon disk with wate as the cooling fluid fo diffeent Fouie numbes (Re=75, Ek=4.5x10 4, β=0.5, b/d n =0.5, p / d =0.667). 180

208 The local Nusselt numbe is contolled by local tempeatue and heat flux at the solid fluid inteface. It shows a highe value at ealy stages of the tansient pocess due to smalle tempeatue diffeence between the liquid jet and disk solid fluid inteface. This essentially means that all heat eaching the solid fluid inteface via conduction though the solid is moe efficiently convected out as the local fluid tempeatue is low eveywhee at the inteface. The local Nusselt numbe is maximum at the cente of the disk, and deceases along the adial distance as the bounday laye thickness inceases downsteam. The local Nusselt numbe deceases with time until it eaches the steady state equilibium distibution. Aveage Nusselt Numbe, Nu av Nu, Re = 75 Ek= Nu, Re = 550 Ek=.65x10-4 Nu, Re = 900 Ek=9.46x10-5 Seies4 Θ max (solid), Re = 75 Seies5 Θ max (solid), Re = 550 Seies6 Θ max (solid), Re = 900 Seies7 Θ max (int), Re = 75 Seies8 Θ max (int), Re = 550 Seies9 Θ max (int), Re = 900 Seies10 Θ max - Θ min (int), Re = 75 Seies11 Θ max - Θ min (int), Re = 550 Seies1 Θ max - Θ min (int), Re = Fouie Numbe, Fo Dimensionless Tempeatue, Θ Figue 6.4 Aveage Nusselt numbe and dimensionless tempeatue vaiations with time fo diffeent Reynolds numbes (Ek=4.5x10 4, β=0.5, silicon disk, b/d n =0.5, and p / d =0.667). The integated aveage Nusselt numbe and the vaiation of dimensionless maximum tempeatue at the inteface, maximum tempeatue inside the solid and 181

209 maximum to minimum tempeatue diffeence at the inteface fo diffeent Fouie numbes at diffeent values of Reynolds numbe ae shown in figue 6.4. The aveage Nusselt numbe is lage at the ealy pat of the tansient and monotonically deceases with time ultimately eaching the value fo the steady state condition. A highe Reynolds numbe inceases the magnitude of fluid velocity nea the solid fluid inteface that contols the convective heat tansfe and theefoe inceases the aveage Nusselt numbe. The contol of maximum tempeatue is impotant in many citical themal management applications including electonic packaging. As expected, the tempeatue inceases eveywhee with time stating fom the initial isothemal condition. A apid incement is seen at the ealie pat of the tansient, and it levels off as the themal stoage capacity of the solid diminishes and become eo at the steady state condition. It maybe noted that the time equied to each the steady state condition is lowe at a highe Reynolds numbe because the highe velocity of the fluid helps to enhance the convective heat tansfe pocess. This is due to quicke dissipation of heat with highe flow ate. The steady state Fouie numbe (Fo ss ) was defined as the time needed to appoach 99.99% of the steady state local Nusselt numbe ove the entie solid fluid inteface. It was found that Fo ss deceases fom at Re=75 to at Re=900. Figue 6.5 povides the vaiations of aveage Nusselt numbe and the dimensionless maximum tempeatue at the inteface, maximum tempeatue inside the solid, and maximum to minimum tempeatue diffeence at the inteface with the pogession of time at diffeent Ekman numbes. The aveage Nusselt numbe is lage at the ealy pat of the tansient and monotonically deceases with time ultimately eaching 18

210 the value fo the steady state condition. Thoughout the tansient heating pocess, the aveage Nusselt numbe is moe at lage spinning ate o smalle Ekman numbe. Aveage Nusselt Numbe, Nu av Ek= Nu, Ek = Ek=.65x10-4 Nu, =.13x10 4 Ek=1.3x10-4 Nu, = 1.06x10 4 Ek=8.83x10-5 Nu, = 7.08x10 5 Seies1 Θ max (int), Ek = Seies6 Θ max (int), Ek = 1.4x10 4 Seies7 Θ max (int), Ek = 7.08x10 5 Seies8 Θ max (solid), Ek = Seies9 Θ max (solid), Ek = 1.4x10 4 Seies10 Θ max (solid), Ek = 7.08x10 5 Seies11 Θ max Θ min (int), Ek = Seies1 Θ max Θ min (int), Ek= 1.4x10 4 Seies13 Θ max Θ min (int), Ek= 7.08x Fouie Numbe, Fo Dimensionless Tempeatue, Θ Figue 6.5 Aveage Nusselt numbe and dimensionless tempeatue vaiations with time fo diffeent Ekman numbes (Re=550, β=0.5, silicon disk, b/d n =0.5, and p / d =0.667). As the Ekman numbe deceases fom to 7.08x10 5 the aveage Nusselt numbe inceases by an aveage of 7.47% when the Reynolds numbe is kept constant at 550. The maximum tempeatue within the solid was encounteed at the outlet adjacent to the heated suface (= b, = d ). The tempeatues ise with time as the solid disk and the fluid stoe heat showing a apid esponse at the ealie pat of the heating pocess until the themal stoage capacity eaches its limit at the steady state condition. It may be noted that the magnitude of the dimensionless tempeatue as well as the time equied to each the steady state condition become smalle as the Ekman numbe deceases. This is because the magnitude of fluid velocity nea the solid fluid inteface that contols the 183

211 convective heat tansfe ate inceases with the incement of the otational ate of the disk o the eduction of Ekman numbe. These obsevations ae in ageement with the numeical solutions of Rice et al. [46]. The effects of nole to taget spacing fo wate as the coolant and silicon as the disk mateial at a spinning ate of 15 RPM o Ek=4.5x10 4 and Reynolds numbe of 750 is demonstated in figue 6.6. It may be noticed that a highe aveage Nusselt numbe and a smalle maximum tempeatue ae obtained ove the entie tansient pocess when the nole is bought close to the heated disk. Aveage Nusselt Numbe, Nu av Ek= Nu, β = 0.5 Nu, Ek=.65x10-4 β = 0.50 Nu, Ek=1.3x10-4 β = 0.75 Ek=8.83x10-5 Nu, β = 1.0 Seies1 Θ max (int), β = 1.0 Seies6 Θ max (int), β = 0.63 Seies7 Θ max (int), β = 0.5 Seies8 Θ max (solid), β = 1.0 Seies9 Θ max (solid), β = 0.63 Seies10 Θ max (solid), β = 0.5 Seies11 Θ max Θ min (int), β = 1.0 Seies1 Θ max Θ min (int), β = 0.63 Seies13 Θ max Θ min (int), β = Fouie Numbe, Fo Dimensionless Tempeatue, Θ Figue 6.6 Aveage Nusselt numbe and dimensionless tempeatue vaiations with time fo diffeent nole to plate spacing (Re=750, Ek=4.5x10 4, silicon disk, b/d n =0.5, and p / d =0.667). The smalle gap between the nole and the taget disk avoids the loss of momentum as the fluids tavels though the confined medium. This esults in a lage ate of convective heat tansfe with highe fluid velocity. As the nole to taget spacing deceases fom 1 184

212 to 0.5 the aveage Nusselt numbe inceases by an aveage of 1.71% when the Reynolds numbe is kept at 750. Diffeent plate to disk confinement atios ( p / d ) fom 0. to 0.75 wee investigated fo wate as the coolant and silicon as the disk mateial. The effects of plate to disk confinement atio on the vaiation of dimensionless maximum tempeatue at the inteface, maximum tempeatue inside the solid, and maximum to minimum dimensionless tempeatue diffeence at the inteface and aveage Nusselt numbe ae shown in figue 6.7. Aveage Nusselt Numbe, Nu av Ek= Nu, p / d = Nu, Ek=.65x10-4 p / d = 0.33 Nu, Ek=1.3x10-4 p / d = Nu, Ek=8.83x10-5 p / d = 0.75 ΘSeies1 max (int), p / d = ΘSeies6 max (int), p / d = ΘSeies7 max (int), p / d = 0.75 ΘSeies8 max (solid), p / d = ΘSeies9 max (solid), p / d = ΘSeies10 max (solid), p / d = 0.75 ΘSeies11 max Θ min (int), p / d = ΘSeies1 max Θ min (int), p / d = 0.50 ΘSeies13 max Θ min (int), p / d = Fouie Numbe, Fo Dimensionless Tempeatue, Θ Figue 6.7 Aveage Nusselt numbe and dimensionless tempeatue vaiations with time fo diffeent plate to disk confinement atios (Re=450, Ek=4.5x10 4, β=0.5, silicon disk, b/d n =0.5). The aveage Nusselt numbe inceases with the eduction of the plate to disk confinement atio. When p is inceased, the fictional esistance fom both walls slows down the momentum and esults in highe film thickness at the fee suface egion fo 185

213 any given spin ate and flow ate. A lowe fluid velocity obviously esults in smalle convective heat tansfe ate. As the plate to disk confinement atio deceases fom 0.75 to 0., the aveage Nusselt numbe inceases by an aveage of 18.07% when the Reynolds and Ekman numbes ae kept constant at 450 and 4.5x10 4 espectively. When the atio of confinement was educed fom 0.75 to 0.5, unde the same numeical conditions like flow and spinning ates, it was found that the maximum tempeatue inside the solid deceases by 8.96%. The effects of solid mateial popeties on tansient heat tansfe ae pesented in figue 6.8. The studied mateials wee aluminum, Constantan, coppe, silicon, and silve having diffeent themo physical popeties. Aveage Nusselt Numbe, Nu av Ek= Nu, Aluminum Nu, Ek=.65x10-4 Coppe 0.0 Nu, Ek=1.3x10-4 Constantan Nu, Ek=8.83x10-5 Silicon Seies14 Nu, Silve 0.16 ΘSeies1 max (int), Constantan ΘSeies6 max (int), Aluminum 0.1 ΘSeies7 max (int), Silve ΘSeies8 max (solid), Constantan ΘSeies9 max (solid), Aluminum 0.08 ΘSeies10 max (solid), Silve ΘSeies11 max Θ min (int),constantan 0.04 ΘSeies1 max Θ min (int),aluminum ΘSeies13 max Θ min (int),silve Fouie Numbe, Fo Dimensionless Tempeatue, Θ Figue 6.8 Aveage Nusselt numbe and dimensionless tempeatue vaiations with time fo diffeent solid mateials (Re=875, Ek=.13x10 4, b/d n =0.5, β=0.5, and p / d = 0.667). 186

214 Fo all mateials the tempeatue changes occu faste at the ealie pat of the heating pocess and the slope gadually decays when the steady state conditions appoaches. It can be obseved that a mateial having a lowe themal conductivity such as Constantan maintains a highe tempeatue at the solid disk inteface and within the solid as the themal conductivity contols how effectively the heat flows and distibutes though the mateial. Fo the same eason, the maximum tempeatue within the solid and that at the inteface ae significantly diffeent fo Constantan, wheeaeas about the same fo both silve and aluminum. The themal diffusivity of the mateial also contibutes to the tansient heat tansfe pocess. Silve and aluminum each the steady state faste than Constantan due to thei highe themal diffusivity. The values of themal diffusivity fo these mateials at 303 K ae α silve =1.74 x10 4 m /s, α aluminum =8.33 x10 5 m /s, and α Constantan =6.0x10 6 m /s.the magnitude of the tempeatue non unifomity at the inteface at steady state is contolled by themal conductivity of the mateial. It may be noted that Constantan (k Constantan =.7 W/m. K) has an aveage maximum to minimum tempeatue diffeence of 17.4 K, wheeaeas silve (k silve =49 W/m. K) has only an aveage 3.34 K tempeatue diffeence at the inteface. Figue 6.8 also shows the distibution of aveage Nusselt numbe with time fo the five mateials used in this study. Constantan shows a highe aveage heat tansfe coefficient compaed to the othe mateials ove the entie tansient pocess due to its lowe themal conductivity. Anothe impotant facto that contols the tansient heat tansfe pocess is the thickness of the disk. Its effects on the vaiation of the dimensionless maximum tempeatue at the inteface, maximum tempeatue inside the solid, and maximum to minimum dimensionless tempeatue diffeence at the inteface and aveage Nusselt 187

215 numbe ae shown in figue 6.9. In these plots, silicon has been used as the disk mateial and wate as the cooling fluid. The disk thickness significantly affects the tempeatue distibution. It may be noted that as the thickness of solid disk inceases, the time needed to achieve the steady state condition inceases. This is due to moe stoage capacity of heat within the solid. The adial conduction becomes stonge as the disk thickness inceases geneating a moe unifom heat distibution at the inteface. Howeve, the incement of solid thickness ceates moe themal esistance to the heat tansfe pocess. The aveage Nusselt numbe is highe fo a thinne disk Aveage Nusselt Numbe, Nu av b/d n = 0.5 Ek= b/d n = 0.50 Ek=.65x10-4 b/d n =0.83 Ek=9.46x10-5 b/d n =1.67 Seies4 Θ max (int), b/d n = 0.5 Seies5 Seies6 Θ max (int), b/d n = 1.67 Seies7 Θ max (solid), b/d n = 0.5 Seies10 Θ max (solid), b/d n = 1.67 Θ max - Θ min (int), b/d n = 0.5 Seies8 Θ max - Θ min (int),b/d n = 1.67 Seies Dimensionless Tempeatue, Θ Fouie Numbe, Fo Figue 6.9 Aveage Nusselt numbe and dimensionless tempeatue vaiations with time fo diffeent silicon disk thicknesses (Re=450, Ek=4.5x10 4, β=0.5, and p / d =0.667). Based on ou numeical data, a coelation fo the aveage Nusselt numbe was developed as a function of confinement atio, themal conductivity atio, dimensionless disk thickness, nole to taget spacing, Ekman numbe, Reynolds numbe, and Fouie 188

216 numbe to accommodate most of the tanspot chaacteistics of a tansient patially confined liquid jet impingement cooling pocess on a thick solid disk spinning at a constant angula velocity. The coelation that best fitted the data can be placed in the following fom: Nu av =1.948 β 0.01 Re 0.74 Ek 0.1 ε 0.7 (b/d n ) 0.05 ( p / d ) 0.05 Fo 0.01 (6.3) In developing this coelation, all aveage Nusselt numbe data coesponding to the vaiation of diffeent paametes wee used. The least squaes cuve fitting method was used. Figue 6.30 gives a gaphical compaison between the numeical aveage Nusselt numbes to the aveage Nusselt numbes pedicted by equation Numeical Aveage Nusselt Numbe Pedicted Aveage Nusselt Numbe Figue 6.30 Compaison of pedicted aveage Nusselt numbe of equation 6.3 with pesent numeical data. The pecent diffeence of the pedicted aveage Nusselt numbe was defined as: %diff = ((Nuav ped Nuav num )/Nuav num ) 100. The pedicted aveage Nusselt numbe 189

217 diffeences between numeical and pedicted values ae in the ange of 9.1% to 13.61%. The aveage Nusselt coelation mean diffeence was equal to 4.98%. The values of the dimensionless vaiables used fo this coelation ae: 5 Re 900, 7.08x10 5 Ek 4.5x10 4, 0.5 β 1.0, P=5.49, 7.6 ε 376.7, 0.5 b/d n 1.67, 0. p / d 0.75, and Fo 0.7. It should be noted fom figue 6.30 that a lage numbe of data points ae well coelated with equation 6.3. This coelation povides a convenient tool fo the pediction of aveage heat tansfe coefficient duing the tansient heat tansfe pocess. A compaison of the pesent numeical esults with the expeimental data of Oa et al. [44, 45] fo vaious spinning ates of the taget disk is pesented in Table 6.1. To match with the expeimental conditions, the combination of the paametes used was: q w =3 kw/m, Re=38, T j =93 K, H n = m, d n = m, p =0.051 m, d =0.03 m and b= m. Wate was used as the woking fluid. The disk was made of aluminum, a mateial with a themal conductivity of 0.4 W/mK. The local pecent diffeence of pesent numeical Nusselt numbe esults of Table 6.1 was defined in tem of: % diff = ((Nu num Nu exp )/Nu exp ) 100. As seen in Table 6.1, the diffeences in the value of local Nusselt numbe esults wee in the ange of 14.14% to 4.34% with an aveage diffeence of 6.91%. In Oa s eseach [44, 45] highe Nusselt numbes wee found at the inne potions of the disk, close to the colla, and deceased towads the oute edge. This was due to the adial spead of the flow, and lowe convective heat tansfe emoval of the liquid due to a moe ponounced backflow effect on the uppe confinement plate at a lage atio of confinement, including the sluggish development of the themal bounday 190

218 laye thickness. The same behavio was obseved as pat of ou numeical study just with a slight cutback effect on local Nusselt numbe distibutions at lage atios of confinement. Consideing the uncetainty of expeimental measuements and ound off and discetiation eos in numeical computation, the oveall compaison between test data and numeical esults can be consideed to be quite satisfactoy. Table 6.1 Local Nusselt numbe compaison between expeimental data of Oa et al. [44, 45] and pesent numeical esults (T j =93 K, q w =3 kw/m, b= m, Re=38, H n = m, p / d =0.5, b/d n =0.15, hin = p = m). Confinement Spinning ate, 50 RPM Pecent atio, / hin Numeical Expeimental diffeence Spinning ate, 100 RPM Numeical Expeimental Spinning ate, 00 RPM Numeical Expeimental Aveage

219 Chapte 7 Chemical Mechanical Polishing Model Results 7.1 Steady State Pocess A thee dimensional steady state FEM model was used to acquie the tempeatue pofile of the substate and pad sufaces duing a CMP pocess. Figue 7.1a shows the steady state maximum and minimum tempeatue contou distibutions fo the contol volume unde a sluy flow ate of Q sl =15 cc/min. It can be seen fom figue 7.1a that a consideable egion of the tailing edge along the, 3, and 4 o clock positions eaches up to a tempeatue diffeence of 6 degees at the wafe and pad sufaces, including the sluy egion. A second numeical un with the same polishing conditions except fo the amount of sluy flow ate of (Q sl =30 cc/min) ae shown in figue 7.1b. Figue 7.1b shows a tempeatue diffeence of 4.5 degees that extends along the tailing edge fom the to 4 o clock positions at the wafe and pad sufaces along the sluy inteface. Figue 7.1c illustates the same patten as figues 7.1a and 7.1b with a tempeatue diffeence slightly smalle, aound 4 degees, at few aeas of the tailing edge, along the, 3, and 4 o clock positions at the wafe and pad sufaces as well as the sluy egion. The steady state tempeatue contou plots of figue 7.1 wee done fo an abasive film thickness of 40 µm, unde a constant pessue load of P=4.35 kpa, and coefficient of fiction (µ f =0.4), with a vaiable heat flux that anges fom 3.6 to 8.3 (kw/m ), unde a pad and caie spinning ate of 10 and 30 RPM espectively. 19

220 (a) (b) (c) Figue 7.1 Steady state tempeatue contou plots fo alumina (sluy), the substate and pad sufaces at vaious sluy flow ates, (a) Q sl =15 cc/min, (b) Q sl =30 cc/min, and (c) Q sl =75 cc/min. 193

221 Heat tansfe convection coeff., h (w/m K) w15 hw, Qsl=30 cc/min w75 hw, Qsl=75 cc/min p15 hp, Qsl=30 cc/min p75 hp, Qsl=75 cc/min Tw15 Tw, Qsl=30 cc/min Tw75 Tw, Qsl=75 cc/min Tp15 Tp, Qsl=30 cc/min Tp75 Tp, Qsl=75 cc/min Tailing Edge Leading Edge Suface Tempeatue, T (K) Dimensionless adial location, / w Figue 7. Coss sectional wafe and pad tempeatue distibutions and local heat tansfe convection coefficients along the cente of pad and substate sufaces fo two diffeent sluy flow ates. Figue 7. shows the coss sectional wafe and pad sufaces tempeatue distibutions and the local heat tansfe convection coefficients along the dimensional adial distance fom the leading to the end of the tailing edge of the contol volume unde study, fo two chaacteistic sluy flow ates unde the same polishing conditions descibed in figues 7.1. The substate and pad tempeatue distibutions fo a highe sluy flow ate ae slightly smalle compaed to the lowe sluy flow ate esults in figue 7.. The tempeatue diffeence can be seen at a wafe dimensionless adial distance aound / w =0.7 towads the end of its tailing edge, whee the tempeatue of the pad and wafe dops and then inceases significantly, due to the backflow effect of the sluy obseved by Muldowney [147]. A backflow effect is linked to the otational motion 194

222 of the sluy and the shea effect of fictional foces due to the suface tension of the sluy paticles along such a small gap. The heat tansfe convection coefficients fo wafe and pad sufaces follow a highe pofile patten that stas at the leading edge and deceases along the adial distance up to the tailing edge of both sufaces. The wafe heat tansfe convection coefficient values ange fom 130 to 4 (W/m K). The aveage values of the heat tansfe convection coefficient fo the wafe along the suface wee appoximately equal to and (W/m K) unde lowe and highe flow ate conditions. The pad heat tansfe convection coefficient values ange fom 170 to 30 (W/m K). The aveage values of the heat tansfe convection coefficient fo the pad along the suface wee appoximately equal to 51.5 and 51.95(W/m K) unde lowe and highe flow ate conditions. The tempeatue contou plots in figue 7.1 and adial suface tempeatue distibutions of figue 7. eveal that the wafe and pad tempeatue pofile deceased by a slightly magin with the incement of the sluy velocity. In addition, the figues eveal that the heat tansfe convection coefficients ae highe at the pad suface than the substate suface, which is due to its lowe themal conductivity that esults in a lowe tempeatue gadient between the incoming sluy and pad suface. This effect esults in highe convective coefficients fo the pad by an aveage magin of 17.1% unde lowe and highe sluy ates. Pesent numeical esults ae in ageement with Sampuno et al. [11]. Figue 7.3 shows the coss sectional wafe and pad tempeatue ise and the local heat tansfe convection coefficient distibutions along the dimensional adial distance fom the leading to end of the tailing edge of the substate and pad sufaces fo two chaacteistic pessue loads of 17.4 kpa (.5psi) and kpa (6psi) espectively. The 195

223 steady state tempeatue esults wee done fo an abasive film of alumina though a film thickness of 40 µm, unde a constant sluy flow ate of Q sl =85 cc/min, with a pad coefficient of fiction of µ f =0.4, unde a pad and caie spinning ate of 00 and 30 RPM espectively. The change in pessue will diectly affect the amount of heat dispesed beneath the wafe as a esult of the geate sluy, pad, and substate shea stess inteaction duing polishing. Fo a load of 17.4 kpa the heat flux input into the system coves a ange of (q sl = kw/m ) along the leading edge towads the end of the wafe tailing edge. The incement of the load up to kpa as pat of the modeling set up, will incease the limits of a vaiable heat flux fom 10 to 3.4 (kw/m ) along the dimensionless adial distance of the contol volume unde study. The tempeatue ise of the wafe unde a load of and 17.4 kpa wee appoximately equal to 8. and 3.85 degees espectively at the end of the wafe tailing edge. The substate and pad tempeatue distibutions inceased unde a highe pessue load due to the incement of the heat flux geneated pe unit aea. Taking into consideation this tempeatue gadient, we can expect that the MRR will incease by a facto of 7% and 55% at the tailing edge egion along the to 4 o clock positions accoding to expeimental measuements of Li et al. [14], whee the incement of 1 C o K inceased the MRR by 7%. In addition, figue 7.3 shows that the backflow effect on the tempeatue ise was quite gone unde the CMP paametes input fo this numeical un. One easonable explanation fo the absence of the backflow effect obseved by Muldowney [147] is that the incement of pad spinning ate ovecame the suface tensional foces beneath the substate and pad, as shown in figue 7.3. The hotte sluy was diven out towads the end of the platen at a faste ate. 196

224 Heat tansfe convection coeff., h (w/m K) 40 w15 hw, P1=17.4 kpa 0 w75 hw, P=41.37 kpa 00 p15 hp, P1=17.4 kpa Tailing Edge p75 hp, P=41.37 kpa 180 Tw15 Tw, P1=17.4 kpa Tw75 Tw, P=41.37 kpa 160 Tp15 Tp, P1=17.4 kpa 140 Tp75 Tp, P=41.37 kpa Leading Edge Suface Tempeatue, T (K) Dimensionless adial location, / w Figue 7.3 Coss sectional wafe and pad tempeatue distibutions and local heat tansfe convection coefficients along the cente of pad and substate sufaces unde two chaacteistic CMP pessue loads. The heat tansfe convection coefficients fo wafe and pad sufaces follow the same patten as shown in figue 7.. The wafe heat tansfe convection coefficient values ange fom 159 to 31(W/m K) of both loads of applied pessue. Additionally, the figues eveal that the wafe heat tansfe convection coefficients decease by an aveage magin of 80% fo both loads of 17.4 and kpa espectively, once it eaches the wafe oute edge at the tailing egion. The aveage values of the wafe heat tansfe convection coefficient along the suface wee appoximately equal to 43.3 and 54.3 (W/m K) unde a load of 17.4 and kpa espectively. The pad heat tansfe convection coefficient values ange fom 15 to 30 (W/m K). In addition, the figues eveal that the pad heat tansfe convection coefficients decease by an aveage magin of 85% fo both 197

225 loads of 17.4 and kpa espectively, once it eaches the wafe oute edge at the tailing egion. The aveage pad values of the heat tansfe convection coefficient along the suface wee appoximately equal to and 65.37(W/m K) fo the smalle and highe load espectively. Low heat tansfe convection coefficients at a paticula egion indicate the pesence of a lage tempeatue gadient between the incoming sluy at the pad o substate sufaces. The aveage heat tansfe convection coefficients obtained ae in ageement with the ealie expeimental woks of Boucki et al. [17, 18]. The tempeatue contou distibutions of the wafe and pad sufaces unde two chaacteistic caie spinning ates of 15 and 75 RPM ae shown in figues 7.4a and 7.4b. The steady state tempeatue contou plots wee done fo an abasive film thickness of 40 µm of alumina, unde a constant sluy flow ate of (Q sl =60 cc/min), with a pad coefficient of fiction (µ f =0.4), fo a constant load of kpa that geneates a vaiable heat flux (q sl = kw/m ), and pad spinning ate of 150 RPM. The steady state tempeatue contou distibutions in figue 7.4a each up to a tempeatue diffeence of 5.5 degees fo a small faction of the uppe egion at the 1, 3 and 4 o clock positions of the tailing edge of the wafe and pad sufaces including the sluy. Figue 7.4b illustates a tempeatue gadient of 5 degees fo a caie spinning ate of 75 RPM. The tempeatue gadient extends at few aeas aound the 1, and 3 to 4 o clock positions of the tailing edge of the wafe and pad sufaces among the sluy. 198

226 TE W 305 LE P 97 Steady State Time= (a) TE W 305 LE P 97 Steady State Time= (b) Figue 7.4 Steady state wafe, and pad tempeatue contou distibutions fo two diffeent caie spinning ates equal to: (a) Ω c =15 RPM and (b) Ω c =75 RPM. 199

227 Heat ansfe convection coeff., h (w/m K) 00 w15 hw, Ωc=15 pm 180 w75 hw, Ωc=75 pm p15 hp, Ωc =15 pm 160 p75 hp, Ωc=75 pm 140 Tw75 Tw, Ωc=15 pm Tw15 Tw, Ωc=75 pm Tp75 Tp, Ωc=15 pm 10 Tp15 Tp, Ωc=75 pm Tailing Edge Leading Edge Suface Tempeatue, T(K) Dimensionless adial location, / w Figue 7.5 Coss sectional tempeatue distibutions and local heat tansfe convection coefficients along the cente of pad and substate sufaces unde two diffeent caie spinning ates. The coss sectional wafe and pad suface tempeatues ise, and the local heat tansfe convection coefficient distibutions along the dimensional adial distance that extend fom the leading to the end of the tailing edge of both sufaces fo two chaacteistic caie spinning ates of 15 and 75 RPM, ae shown in figue 7.5 unde the same polishing conditions of figues 7.4a and 7.4b. The substate and pad tempeatue distibutions fo a caie spinning ate (Ω c ) of 75 RPM ae slightly smalle compaed to a caie spinning ate (Ω c ) of 15 RPM, as shown in figue 7.5. The tempeatue diffeence could be seen at a wafe dimensionless adial distance aound / w =0.8 towads the end of tailing edge, whee the tempeatue of the pad and wafe dops fo a caie spinning ate (Ω c ) of 75 RPM, and ise fo a caie spinning ate (Ω c ) of 15 RPM. The caie that 00

228 spins at 75 RPM dives out the hot sluy fom the backflow egion ovecoming the suface tensional foces caused by the shea stess. Convesely, the caie that spins at a spinning ate of 15 RPM allows a geate hot sluy eciculation at the backflow egion obseved by Muldowney [147], causing an incement of 0.85 degees at the 3 o clock position of the substate tailing edge. A backflow effect is linked to the otational motion of the sluy and the shea effect of fictional foces due to the suface tension and viscosity of the sluy paticles along such a small gap. The aveage tempeatue ise of the wafe and pad unde a caie spinning ate of 75 and 15 RPM wee appoximately equal to 3.4, 3.5, and 3 degees espectively along the wafe and pad tailing edges. The heat tansfe convection coefficients fo wafe and pad sufaces follow a highe heat tansfe ate patten that stats at the sufaces leading edge and deceases along the adial distance up to the sufaces tailing edge. The wafe heat tansfe convection coefficient values ange fom 16 to 4 (W/m K). The aveage values of the heat tansfe convection coefficient fo the wafe along the suface wee appoximately equal to 4.61 and (W/m K) unde lowe and highe caie spinning ate conditions. The pad heat tansfe convection coefficient values ange fom 190 to 5 (W/m K). The aveage values of the heat tansfe convection coefficient fo the pad along the suface wee appoximately equal to and 5.43(W/m K) unde caie spinning ates of 15 and 75 RPM. This effect esults in highe convective coefficients fo the pad by an aveage magin of 14.93% unde a lowe and highe caie spinning ate espectively. The coss sectional wafe and pad sufaces tempeatue ise and the local heat tansfe convection coefficient distibutions along the dimensional adial distance of the 01

229 contol volume unde study fo thee chaacteistic sluy film thicknesses ae shown in figues 7.6 and 7.7. The tempeatue distibution and the local heat tansfe convection coefficients wee set fo a constant alumina sluy flow ate of 65 cc/min, with a pad coefficient of fiction (µ f =0.4), unde a constant load of P= 8 kpa, fo a vaiable heat flux ate of 5 to kw/m, with pad and caie spinning ates of 150 and 40 RPM. Figue 7.6 shows how the tempeatue distibution along the pad and wafe suface deceases with the incement of the sluy film thickness. Suface Tempeatue, T(K) Tw100 Tw, δsl=40 µm Tp100 Tp, δsl=40 µm Tw175 Tw, δsl =10 µm Tp175 Tp, δsl=10 µm Tw50 Tw, δsl=00 µm Tp50 Tp, δsl=00 µm Tailing Edge 97 Leading Edge Dimensionless adial location, / w Figue 7.6 Tempeatue distibutions along the cente of pad and substate sufaces fo thee chaacteistic sluy film thicknesses. The aveage wafe tempeatue esults of figue 7.6 along the adial distance wee appoximately equal to 99.1, 99.86, and degees K fo the abasive film thicknesses (δ sl ) of 00, 10, and 40 µm espectively. As seen in figue 7.6, the wafe 0

230 tempeatue distibutions of thicke film ae lowe and moe unifom due to an incement of the volumetic flow ate of the sluy that moves beneath the substate and pad sufaces. The incement of the volumetic flow ate inceases the heat advection pe unit aea, theefoe inceasing the heat tansfe effect on the pad and substate sufaces. This effect educes the oveall tempeatue of the wafe and pad at the contol volume unde study. Heat tansfe convection coeff., h (w/m K) Leading Edge w100 hw, δsl=40 µm p100 hp, δsl=40 µm w175 hw, δsl =10 µm p175 hp, δsl=10 µm w50 hw, δsl=00 µm p50 hp, δsl=00 µm Tailing Edge Dimensionless adial location, / w Figue 7.7 Local heat tansfe convection coefficient distibutions along the cente of pad and substate sufaces fo thee chaacteistic sluy film thicknesses. Figue 7.7 illustates the incement of the heat tansfe convection coefficients of pad and wafe unde thicke sluy films. The aveage heat tansfe convection coefficients of the pad sufaces in figue 7.7 wee equal to 55.91, 50.6, and W/m K and the aveage heat tansfe convection coefficients of the substate sufaces wee equal to 46.49, 44.13, and 39.9 W/m K fo the abasive film thicknesses (δ sl ) of 03

231 00, 10, and 40 µm espectively. The aveage pad heat tansfe convection coefficients obtained wee highe than the aveage wafe heat tansfe convection coefficients by 0.6%, 14.71% and 4.69% fo the sluy film thicknesses (δ sl ) of 00, 10, and 40 µm espectively. Pesent esults ae in ageement with Mudhivathi [146] and Boucki [17] findings whee the incement of the film thickness educed the mechanical contact, and inceased the amount of sluy inteaction esulting in lowe tempeatue pofiles, inceasing the heat tansfe convection along the wafe egion exposed to the abasive pad inteface. Figues 7.8a and 7.8b illustate the tempeatue contou distibutions of the wafe and pad sufaces unde two chaacteistic pad spinning ates of 100 and 00 RPM. The steady state tempeatue contou plots wee set fo an abasive film thickness of 40 µm of alumina, at constant sluy flow ate of 50 cc/min, unde a constant load of 4.35 kpa, fo a vaiable heat flux ange of 6 to 8.5 kw/m, and a caie spinning ate of 30 RPM. Figue 7.8a shows the steady state tempeatue diffeence of 6 degees at a small egion of the sluy, wafe, and pad sufaces at the and 3 o clock positions fo a pad spinning ate of 100 RPM. Figue 7.8b shows a tempeatue gadient of 4.5 degees fo a pad spinning ate of 00 RPM. The tempeatue gadient extends at a small faction of the 3 and 4 o clock positions of the tailing edge of the substate suface and a small potion of the sluy egion beneath the same wafe oientation. 04

232 LE W P TE Steady State Time= (a) LE W P TE Steady State Time= (b) Figue 7.8 Steady state wafe, and pad tempeatue contou distibutions fo two chaacteistic pad spinning ates equal to: (a) Ω p =100 RPM and (b) Ω p =00 RPM. The tempeatue ise and the local heat tansfe convection coefficient distibutions of the wafe and pad sufaces unde two chaacteistic pad spinning ates of 05

233 100 and 00 RPM, ae shown in figue 7.9 unde the same polishing conditions descibed in figues 7.8a and 7.8b. Heat tansfe convection coeff., h (w/m K) 180 p50 hp, Ωp=00 pm 165 w50 hw, Ωp=00 pm p hp, Tailing Edge Ωp =100 pm w100 hw, Ωp=100 pm 135 Tw50 Tw, Ωp=00 pm 10 Tp50 Tp, Ωp=00 pm Tw Tw, Ωp=100 pm Tp100 Tp, Ωp=100 pm Leading Edge Suface Tempeatue, T (K) Dimensionless adial location, / w Figue 7.9 Coss sectional tempeatue distibutions and local heat tansfe convection coefficients along the cente of pad and substate sufaces unde two diffeent pad spinning ates. The substate and pad tempeatue distibutions fo a pad spinning ate (Ω p ) of 00 RPM deceased in compaison with the esults of a pad spinning ate (Ω p ) of 100 RPM, as shown in figue 7.9. The pad that spins at 00 RPM dive out the hot sluy fom the backflow egion ovecoming the suface tension foces caused by the shea stess. Convesely, the pad that spins at a spinning ate of 100 RPM allows a majo hot sluy eciculation at the backflow egion, causing an incement of 1.5 degees at the 3 o clock position of the substate tailing edge. A backflow effect is linked to the otational motion of the sluy and the shea effect of fictional foces due to the suface 06

234 tension and viscosity of the sluy paticles along such a small gap. The aveage tempeatue ise of the wafe and pad unde a pad spinning ate of 00 and 100 RPM wee appoximately equal to 3.4, 3.5, and 3 degees espectively along the wafe and pad tailing edges. The heat tansfe convection coefficients fo wafe and pad sufaces follow the same patten as figues 7., 7.3, 7.5 and 7.6. The wafe heat tansfe convection coefficient values ange fom 130 to 14 (W/m K). The aveage values of the heat tansfe convection coefficient fo the wafe along the suface wee appoximately equal to and (W/m K) unde lowe and highe pad spinning ate conditions espectively. The pad heat tansfe convection coefficient values ange fom 180 to 15 (W/m K). The aveage values of the heat tansfe convection coefficient fo the pad along the suface wee appoximately equal to and 64.03(W/m K) unde a pad spinning ate of 100 and 00 RPM espectively. This effect esults in highe convective coefficients fo the pad by an aveage magin of 15.06% and 0.67% unde lowe and highe pad spinning ates. A lowe pad spinning ate of Ω p =100 RPM inceased the backflow eciculation fo a dimensionless adial distance 0.75 < / w 1. Howeve, the incement of the pad spinning ate educes the backflow effect obseved by Muldowney [147], as shown in the pad and wafe tempeatue distibutions in figue 7.9 fo a pad spinning ate of 00 RPM. This themal effect is consistent with the findings of Hong et al. [13] that point out that polishing tempeatue vaies in paallel with thei speed integal. The incement of the angula velocity of the platen geneates moe heat dissipation duing the chemical mechanical polishing pocess due to the incement of the tangential velocity at the 07

235 themal bounday laye thickness of the sluy that is foce out though the polishing, which is eplaced simultaneously with fesh, cool sluy that entes beneath the polished wafe aound its peimete. The minimum heat tansfe sluy wafe inteaction occued close to the wafe tailing edge aea. Hot spots can be obseved along the tailing edge and some inne egions whee pat of the sluy got tapped due to the emeging otational flow pattens. Simila tempeatue pofile pattens have been documented by Boucki et al. [17, 18] Mean Tempeatue Rise, K Pad suface Expeimental esults Boucki et al. [18] Seies Seies Sluy Flow Rate, Qsl (cc/min) Figue 7.10 Compaison of mean tempeatue ise of pad at diffeent sluy flow ates of pesent esults with expeimental esults of Boucki et al. [18]. Figue 7.10 shows the numeical esults of mean pad tempeatue ise at diffeent flow ates along the substate edge compaed to the esults fom the expeimental wok by Boucki et al. [18]. As seen in figue 7.10, the ageement of the expeimental mean tempeatue ise obtained by Boucki et al. [18] with the pesent numeical data is quite 08

236 good. The tempeatue ise unde sluy flow ates of 60, and 80 cc/min coelates with an aveage magin of 6.9%, and 4.73% espectively. Note that numeical pedictions ae within an aveage pecentage off eo of 5.83%. One of the papes used fo the validation of this numeical study was the expeimental wok by Boucki et al. [17] using a JSR Cop. flat pad with a commecial silica sluy unde a flow ate equal to Q sl =60 cc/min. The nominal wafe pessues used wee.5 and 6 (psi) o (17.4 and kpa) espectively, and the co otation ates fo the caie and pad ange fom 10 to 140 RPM. The aveage heat tansfe convection coefficients fom the pesent numeical simulation fo diffeent combinations of CMP paametes and input heat flux ae listed in Table 7.1. The heat tansfe convection coefficients fo the JSR Cop. flat pad wee coelated with the pesent numeical esults of uns #1 and #13 vesus expeimental esults pesented by Boucki et al. [17] on uns #14 and #15 unde the CMP paametes descibed in Table 7.1. Pesent numeical heat tansfe esults of uns #1 and #13 coelate with an aveage magin of 8.69%, and 5.57% espectively. Note that numeical pedictions ae within an aveage pecentage off eo of 7.13%. The numeical esults eveal a bette coelation at lage flow ates. The facto of woking at the micomete scale unde the influence of two spinning sufaces about diffeent axis of otation, the complexity of flow unde such type of bounday conditions, and the ange of flow paametes may contibute to the discepancy between expeimental and numeical data. In addition, computational eos can be intoduced because of ound off and discetiation of the mesh. Consideing these factos, the oveall compaison with test and numeical esults of pevious studies is satisfactoy. 09

237 Table 7.1 Aveage substate and pad heat tansfe convection coefficients and expeimental data of Boucki et al. [17] unde diffeent CMP paametes and vaiable input heat flux along the sufaces. 7. Tansient Pocess Tansient tempeatue pofiles and the heat tansfe convection coefficients of substate and pad sufaces duing a CMP pocess wee acquied using a thee dimensional FEM model. Figue 7.11a shows the maximum and minimum tempeatue contou distibutions fo the contol volume unde study fo a sluy flow ate of Q sl =30 cc/min. Duing the ealy pat of the tansient pocess the sluy, the wafe and pad sufaces eached a tempeatue diffeence of 6 degees at a consideable egion of the wafe tailing edge at the, 3, and 4 o clock positions. Afte a shot peiod of 100 sec the tansient tempeatue diffeence of 6 degees emains the same at smalle aeas of the wafe tailing edge along the, 3, and 4 o clock positions. A second numeical un with 10

238 the same polishing conditions except fo the amount of sluy flow ate (Q sl =75 cc/min) ae shown in figue 7.11b. Figue 7.11b shows a tempeatue diffeence of 10 degees at a small faction of the uppe egion of the tailing edge duing the entie polishing pocess. (a) (b) Figue 7.11 Sluy (alumina), wafe, and pad sufaces tempeatue contou distibutions fo a flow ate value of: (a) Q sl =30 cc/min and (b) Q sl =75 cc/min. In geneal, the tempeatue distibutions of the wafe pad inteface ae smalle fo a sluy ate of Q sl =75 cc/min just with the exception of two egions close to the tailing edge of the wafe exposed to the backflow effect of the sluy obseved by Muldowney [147]. A backflow effect is linked to the otational motion of the sluy and the shea 11

239 effect of fictional foces due to the suface tension of the sluy paticles along such a small gap. The tansient substate tempeatue vaiations and wafe pad tempeatue diffeence fo two diffeent flow ates at the 3 and 5 o clock positions ae shown in figues 7.1a and 7.1b espectively. The tansient tempeatue esults wee done fo an abasive film thickness of 40 µm, at a dimensionless adial distance of / w =7/8 unde a constant pessue load of P=4.35 kpa, fo a vaiable heat flux (q sl = kw/m, unde a pad and caie spinning ate of 150 and 30 RPM espectively. The vaiable heat flux ate (q sl ) used fo this analysis is a function of the pessue load, pad coefficient of fiction, the adial distance measued fom the cente of the platen, and the elative pad wafe spinning ate. The wafe tempeatue esults of figue 7.1a fo a sluy flow ate of (Q sl =75 cc/min) ae slightly lowe in compaison with an alumina flow ate of 30 cc/min. That slight change in tempeatue can be confimed with the compaison of the aveage tansient wafe tempeatue diffeences of both flow ates duing the entie pocess. An aveage tansient wafe tempeatue diffeence of 4.35 degees was attained fo a lowe sluy flow ate vesus the 3.98 degees acquied unde a highe sluy flow ate. The wafe pad tempeatue diffeences showed in figue 7.1a eveals that the pad tempeatue values ae lowe compaed to the substate suface esults. The wafe pad tempeatue diffeences examined ange up to 1. degees K o C fo the lowe sluy ate and up to 0.6 degees fo the highe sluy flow ate at the adial location unde study (/ w =7/8). The tempeatue diffeence of a sluy flow ate of 75 cc/min was lowe and moe stable compaed with a sluy flow ate of 30 cc/min. 1

240 In contast, figue 7.1b shows lowe tempeatue esults fo a sluy ate of 30 cc/min at the 5 o clock position fo a adial distance / w =7/8 of the wafe. That slight change in tempeatue can be cooboated with the compaison of aveage tempeatue diffeences of both sluy flow ates duing the entie pocess. An aveage wafe tempeatue diffeence of 4.5 degees was attained fo a lowe sluy flow ate vesus a 4.38 degees acquied unde a highe flow ate of alumina. The wafe tempeatue at the 3 o clock position is about 1 degee highe than the 5 o clock position unde both sluy flow ates. This adial tempeatue vaiation is elated to the sluy flow and the heat tansfe beneath the wafe suface. Fesh, cool sluy entes beneath the polished wafe aound its peimete. As seen in figue 7.1b, that the 5 o clock position is in themal advantage because it is facing the leading edge of the sluy and it is close to the cente of the platen that holds that pad. In contast, the 3 o clock position is facing the tailing edge fa away fom the cente of the platen; consequently it had moe heat to tansfe due to fictional inteaction of pad sluy paticles beneath the wafe. The wafe pad tempeatue diffeences examined ange up to 0.9 degees K o C fo the lowe sluy flow ate and up to 0.37 degees fo the highe sluy flow ate at a adial location of / w =7/8 along the 5 o clock position, as shown in figue 7.1b. The tempeatue diffeence of a 75 cc/min sluy flow ate was lowe and moe stable compaed with a lowe sluy flow ate of 30 cc/min. The tempeatue contou plots in figues 7.11a and 7.11b, and tempeatue vaiation of figues 7.1a and 7.1b eveal that the wafe tempeatue pofile deceases with the incement of the sluy velocity. Pesent numeical esults ae in ageement with Sampuno et al. [11]. 13

241 Wafe Tempeatue,T (K ) T, Qsl = 30 cc/min 50 T, Qsl = 75 cc/min dt100 T, Qsl = 30 cc/min dt50 T, Qsl = 75 cc/min LE Time (sec) TE 3 o clock Wafe-Pad Tempeatue diffeence, T (K) (a) Wafe Tempeatue,T (K ) T, Qsl = 30 cc/min 50 T, Qsl = 75 cc/min dt100 T, Qsl = 30 cc/min dt50 T, Qsl = 75 cc/min LE 5 o clock Time (sec) TE Wafe-Pad Tempeatue diffeence, T (K) (b) Figue 7.1 Tansient wafe tempeatue distibution and wafe pad tempeatue diffeences fo two diffeent flow ates at a adial location of / w =7/8 along the: (a) 3 o clock position and (b) 5 o clock position. 14

242 Wafe heat tansfe coef., h(w/m K) w100 Wafe,Qsl =30 cc/min w50 Wafe,Qsl =75 cc/min p, Pad,Qsl w100 =30 cc/min p, Pad,Qsl w50 =75 cc/min LE TE 3 o clock Pad heat tansfe coef., h(w/m K) Time (sec) 0 (a) Wafe heat tansfe coef., h(w/m K) w100 Wafe,Qsl = 30 cc/min w50 Wafe,Qsl = 75 cc/min p, Pad,Qsl w100 = 30 cc/min p, Pad,Qsl w50 = 75 cc/min LE 5 o clock TE Pad heat tansfe coef., h(w/m K) Time (sec) 0 (b) Figue 7.13 Wafe and pad tansient heat tansfe convection coefficients fo two diffeent flow ates at a adial location of / w =7/8 along the: (a) 3 o clock position and (b) 5 o clock position. 15

243 Figues 7.13a and 7.13b show the wafe and pad heat tansfe convection coefficients at the 3 and 5 o clock positions espectively fo the same conditions of figues 7.1a and 7.1b. The heat tansfe convection coefficients in figue 7.13a ae slightly highe at the pad than the substate suface that is due to a lowe tempeatue diffeence between sluy and pad. This validates the esults obtained in figue 7.1a and 7.1b whee the tempeatue diffeence between the pad and wafe substate wee aound 1 C o K smalle. That effect esults in highe convective coefficients fo the pad by 5.6% and 8.61% unde lowe and highe sluy flow ates. The aveage values of the heat tansfe convection coefficient fo the pad attained at this location wee appoximately equal to 6.8 and 9.19 W/m K unde lowe and highe sluy flow ate conditions. The heat tansfe convection coefficients in figue 7.13b ae slightly highe at the pad than the substate suface due to a lowe tempeatue gadient between the incoming sluy and pad suface. This effect esults in highe convective coefficients fo the pad by 6.86% and 9.83% unde lowe and highe sluy flow ates. The aveage values of the heat tansfe convection coefficients fo the pad attained at this location wee about.8 and 3.97 W/m K unde lowe and highe sluy flow ates. Figues 7.14a and 7.14b show the maximum and minimum tempeatue contou distibutions fo the contol volume unde study at a constant load of 17.4 kpa (.5psi) and kpa (6psi) espectively. The tansient tempeatue contou plots wee done fo an abasive film thickness of 40 µm, unde a constant sluy flow ate of Q sl =85 cc/min, with a pad coefficient of fiction of µ f =0.4, unde a pad and caie spinning ate of 00 and 30 RPM espectively. This change in pessue diectly affects the amount of heat dispesed beneath the wafe as esult of the geate sluy, pad, and substate shea stess 16

244 inteaction duing the polishing. Fo a load of 17.4 kpa the heat flux input into the system coves a ange of (q sl = kw/m ) along the leading to the tailing edge of the wafe, as shown in figue 7.14a. Duing the ealy pat of the tansient pocess the sluy, the wafe and pad eached up to a tempeatue diffeence of 8 degees at a small faction of the uppe egion of the tailing edge. Late on, afte a shot peiod of 100 seconds the tansient tempeatue diffeence deceases slightly to 7 degees and extends along the tailing edge in small aeas fom the 1 to 4 o clock positions. (a) (b) Figue 7.14 Sluy, wafe, and pad sufaces tempeatue contou distibutions unde a constant pessue value of: (a) 17.4 kpa and (b) kpa. 17

245 A second tial unde study is shown in figue 7.14b fo the same polishing conditions except fo an incement of the applied load to kpa as pat of the modeling set up. This new applied load set the limits of a vaiable heat flux that ange fom to 3.1 (kw/m ) along the leading to the tailing edge of the wafe. Figue 7.14b illustates a tempeatue diffeence of 14 degees duing the ealy pat of the pocess at a consideable egion of the wafe located at the 1 to 5 o clock position of the tailing edge. Afte a elative shot peiod of 100 seconds the tempeatue diffeence deceases to 1 degees at vaious consticted aeas moe likely at the edge of the wafe, due to the amount of heat tansfe with a sluy flow ate unde a highe tempeatue afte being exposed to the shea stess and fictional foces as pat of the tansient CMP pocess. The effect of adding moe pessue to the CMP pocess poduced lage tempeatue gadients at the wafe pad inteface as a esult of moe contact to contact abasion mode of the pad with sluy paticles and substate, as shown in figues 7.14a and 7.14b. Pesent esults ae in ageement with Sikde et al. [148]. Thei expeimental esults using an acoustic senso evealed that the coefficient of fiction deceased unde a lowe applied pessue. Figues 7.15a and 7.15b show the vaiable heat tansfe effect of two diffeent applied loads of pessue fom the univesal bench top tibomete duing the polishing pocess on the local tansient tempeatue distibutions at diffeent specific adial distances measued fom the cente of the substate. The tansient wafe tempeatue esults of figue 7.15a unde a load of 17.4 kpa ae lowe in compaison with the esults obtained at figue 7.15b unde the same dynamic polishing conditions of the model unde study in figues 7.14a and 7.14b. The aveage change in tempeatue of figue 7.15a 18

246 compaed with figue 7.15b at the 1 o clock location fo each of the study adial distances wee appoximately equal to.5 degees at / w =/3 of the wafe, 4.7 degees at / w =7/8 and 7.1 degees along the wafe edge. Aveage wafe pad tempeatue diffeences of 0.74,.4, and 0.69 degees wee attained at each of the following adial locations of /3, 7/8, and 1 espectively unde a pessue load of 17.4 kpa as shown in figue 7.15a. The wafe pad tempeatue diffeences shown in figues 7.15a and 7.15b ange up to 4.5 to 6 degees espectively. Pad tempeatue values ae lowe than wafe substate esults at the polishing suface. The wafe pad tempeatue diffeences at a adial distance aound the edge (/ w =1) wee lowe and moe stable in compaison with the othe two adial locations in figues 7.15a and 7.15b. The tempeatue diffeential at the adial location of / w =7/8 was less stable. This instability is pat of the fluid dynamics of the sluy that is in continuous e ciculating motion entapped beneath the wafe and polishing pad sufaces. As pat of the mechanics of the CMP pocess, fesh and cool sluy it is tanspoted continuously fom the cente of the pad to the suface beneath the substate, causing a majo fluctuation of the tempeatue gadient. This effect is moe ponounced when the heat sluy is getting close to the stating backflow egion undeneath the sluy. Convesely, this effect is less ponounced once the outgoing sluy eaches the substate edge at the outlet and mixes up with fesh and cool sluy. Aveage wafe pad tempeatue diffeences of 1.76, 3.38, and 1.6 degees wee attained at each of the following adial locations /3, 7/8, and 1 espectively unde a pessue load of kpa, as shown in figue 7.15b. 19

247 Wafe Tempeatue,T (K ) Τ, /w=/3 10 T, /w=7/8 40 T, /w=1 dt40 T, /w=/3 dt10 T, /w=7/8 dt00 T, /w=1 LE 1 o'clock Time (sec) TE Wafe-Pad Tempeatue diffeence, T (K) (a) Wafe Tempeatue,T (K ) T, /w=/3 10 T, /w=7/8 00 T, /w=1 dt40 T, /w=/3 dt10 T, /w=7/8 dt00 T, /w=1 LE Time (sec) (b) 1 o'clock TE Wafe-Pad Tempeatue diffeence, T (K) Figue 7.15 Tansient wafe tempeatue distibutions and wafe pad tempeatue diffeences at diffeent adial locations along the 1 o clock position unde a constant pessue value of: (a) P=17.4 kpa and (b) P=41.37 kpa. 0

248 70 Wafe heat tansfe coef., h(w/m K) / w = /3 /w=/3 / w = 7 /8 / w =1 /w=7/8 /w=1 LE 1 o'clock TE Time (sec) (a) 00 Wafe heat tansfe coef., h(w/m K) /w=/3 /w=7/8 /w=1 / w = /3 / w = 7 /8 / w =1 LE 1 o'clock TE Time (sec) (b) Figue 7.16 Tansient wafe heat tansfe convection coefficient at diffeent adial locations along the 1 o clock position unde a constant pessue value of: (a) P=17.4 kpa and (b) P=41.37 kpa. 1

249 The wafe heat tansfe convection coefficients at thee specific adial locations fo two distinctive pessue loads ae shown in figues 7.16a and 7.16b. Figue 7.16a shows that the pessue effect on the vaiable heat flux is moe intense at the tailing edge of the wafe that is futhe away fom the cente of the pad, causing and an uneven heating effect on the substate suface. The convective heat tansfe coefficients effect was moe ponounced at a adial distance of / w =/3 with an aveage value of 3.14 W/m K. The aveage values of the pad heat tansfe convection coefficient attained wee aound 1.3 and 17.6 W/m K fo the adial locations of / w =7/8 and 1 espectively. Figue 7.16b shows the same patten of figue 7.16a, whee the heat tansfe convection coefficient effect was moe ponounced at a adial distance of / w =/3 with an aveage value equal to 4.75 W/m K. The heat tansfe convection coefficient effect deceased by an aveage magin of 45.05% once it eaches the wafe oute edge at the tailing egion. A lowe heat tansfe convection coefficient at a paticula location indicates the pesence of a hot spot o a lage tempeatue gadient between the incoming sluy at the pad o substate sufaces. The aveage heat tansfe convection coefficients obtained ae in ageement with the ealie expeimental woks of Boucki et al. [17,18]. The tempeatue contou distibutions of the wafe and pad sufaces unde two chaacteistic caie spinning ates of 15 and 75 RPM ae shown in figues 7.17a and 7.17b. The tansient tempeatue contou plots wee done fo an abasive film thickness of 40 µm of alumina, unde a constant sluy flow ate of (Q sl =60 cc/min), with a pad coefficient of fiction (µ f =0.4), unde a constant load of kpa, fo a vaiable heat flux (q sl = kw/m ), and pad spinning ate of 145 RPM. Duing the ealy pat of the tansient pocess in figue 7.17a the sluy, the wafe and pad eached up to a

250 tempeatue diffeence of 9 degees at a small faction of the uppe egion of the tailing edge. Late on, the tansient tempeatue diffeence deceases slightly to 8 degees and extends along the tailing edge to seies of small aeas along the 1 to 3 o clock positions afte a peiod of 100 seconds, as shown in figue 7.17a. Figue 7.17b illustates a tempeatue gadient of 5 degees duing the ealy pat of the pocess at small aeas aound the to 4 o clock positions of the tailing edge. Afte a shot peiod of 100 seconds the tempeatue gadient of 5 degees emains the same aound the 3 o clock position, as shown in figue 7.17b. (a) (b) Figue 7.17 Sluy, wafe, and pad sufaces tempeatue contou plots fo a caie spinning ate equal to: (a) Ω c =15 RPM and (b) Ω c =75 RPM. 3

251 Wafe Tempeatue, T(K ) T, Ωc=15 pm 50 T, Ωc=75 pm dt100 T, Ωc=15 pm dt50 T, Ωc=75 pm LE 1o clock Time (sec) TE Wafe-Pad Tempeatue diffeence, T (K) (a) Wafe Tempeatue,T(K ) T, Ωc=15 pm 50 T, Ωc=75 pm dt100 T, Ωc=15 pm dt50 T, Ωc=75 pm LE Time (sec) TE 3 o clock Wafe-Pad Tempeatue diffeence, T (K) (b) Figue 7.18 Tansient wafe tempeatue distibutions and wafe pad tempeatue diffeences fo two diffeent caie spinning ates at a: (a) Radial location of / w =1 along the 1 o clock position and (b) Radial location of / w =7/8 along the 3 o clock position. 4

252 Figues 7.18a and 7.18b show the tansient substate tempeatue vaiations and wafe pad tempeatue diffeences fo two distinctive caie spinning ates at the 1 and 3 o clock positions with the same polishing conditions descibed in figues 7.17a and 7.17b. The aveage wafe tempeatue esults fo a caie spinning ate of (Ω c =75 RPM) ae.59 degees lowe compaed to a caie spinning ate of 15 RPM, as shown in figue 10a. The wafe pad tempeatue diffeences obtained in figue 7.18a ange up to 0.86 degees K o C fo the lowe caie spinning ate and up to degees fo the caie at highe spinning ate at the adial location unde study of / w =1. An oveall aveage tansient wafe tempeatue diffeence of 8.56 degees was obtained unde a caie spinning ate of 15 RPM vesus the 5.97 degees diffeential acquied fo a caie spinning ate of 75 RPM. The wafe pad tempeatue diffeences examined ange up to 0.86 degees K o C fo the lowe caie spinning ate and up to degees fo the highe caie spinning ate at the adial location unde study of / w =1. Figue 7.18b shows that the aveage tansient tempeatue esults fo a caie spinning ate of (Ω c =75 RPM) wee appoximately 1.55 degees lowe compaed to a caie spinning ate of 15 RPM along the 3 o clock position at a adial distance of 7/8 of the wafe adius. The oveall aveage tansient wafe tempeatue diffeential of 6.63 degees was attained fo a lowe caie spinning ate vesus the 5.08 degees diffeential acquied fo a caie at highe spinning ate. Figue 7.18b kept the same wafe pad tempeatue diffeence patten obseved in figue 7.18a. The wafe pad tempeatue diffeences examined ange up to degees K o C fo the caie at a lowe spinning ate and up to 0.31 degees fo the caie at a highe spinning ate at the adial location unde study. The wafe tempeatue at the 1 o clock position in figue 7.18a is about 1.5 5

253 degees highe than the tempeatue obseved at the 3 o clock position fo both caie spinning ates. In geneal, the adial tempeatue vaiations ae elated to the sluy flow ate and the heat tansfe beneath the wafe. Figue 7.18a illustates that the 1 o clock position at / w =1 is in themal disadvantage because it is at the back potion of the leading edge of the sluy. In contast, the 3 o clock position is between the leading and tailing edge at a adial distance of / w =7/8, theefoe it had less heat to tansfe fom the inteaction of pad sluy paticles beneath the wafe. Additionally, the wafe pad tempeatue diffeences fo a caie spinning ate of Ω c =75 RPM wee lowe and moe stable than a caie unde a spinning ate of 15 RPM, as shown in figues 7.18a and 7.18b. Figues 7.19a and 7.19b show the wafe and pad heat tansfe convection coefficients at the 1 and 3 o clock positions unde the same numeical paametes of figues 7.18a and 7.18b. In addition, figues 7.19a and 7.19b show a simila heat tansfe convection coefficient tend pofile, as shown in figues 7.13a and 7.13b. Theefoe, highe heat tansfe convection coefficients ae obseved in figue 7.19a fo the pad by a magin of 14.15% and 6.98% fo a caie at highe and lowe spinning ates espectively. The aveage heat tansfe convection coefficients fo the pad along the 1 o clock position at a adial location of / w =1 wee about 7.77 and W/m K unde highe and lowe caie spinning ates, as shown in figue 7.19a. 6

254 Wafe heat tansfe coef., h(w/m K) w100 Wafe,Ωc=15 pm w50 Wafe,Ωc=75 pm p, Pad,Ωc=15 w100 pm p, Pad,Ωc=75 w50 pm LE 1o clock TE Pad heat tansfe coef., h(w/m K) Time (sec) (a) Wafe heat tansfe coef., h(w/m K) w100 Wafe,Ωc=15 pm w50 Wafe,Ωc=75 pm p, Pad,Ωc=15 w100 pm p, Pad,Ωc=75 w50 pm LE Time (sec) (b) TE 3 o clock Pad heat tansfe coef., h(w/m K) Figue 7.19 Wafe and pad tansient heat tansfe convection coefficients fo two diffeent caie spinning ates at a: (a) Radial location of / w =1 along the 1 o clock position and (b) Radial location of / w =7/8 along the 3 o clock position. 7

255 Figue 7.19b eveals that the heat tansfe convection coefficients fo the pad inceased by 7.% and 3.1% fo a caie at highe and lowe spinning ates espectively. The aveage values of the heat tansfe convection coefficient fo the pad along the 3 o clock position at a adial location of / w =7/8 wee appoximately equal to 40.4 and W/m K unde highe and lowe caie spinning ate conditions, as shown in figue 7.19b. The tempeatue contou plots of the wafe and pad sufaces unde two diffeent sluy film thicknesses of (δ sl =40 and 10 µm) ae shown in figues 7.0a and 7.0b espectively. The tansient tempeatue contou wee pepaed fo a constant alumina sluy flow ate of 65 cc/min, with a pad coefficient of fiction (µ f =0.4), unde a constant load of P= 8 kpa, fo a vaiable heat flux ate of 5.6 to 1.30 kw/m, with a pad and caie spinning ate of 150 and 40 RPM espectively. Duing the ealy pat of the tansient pocess, as shown in figue 7.0a the wafe and pad eached up to a tempeatue diffeence of 9 degees at a consideable egion of the wafe along the 1 to 5 o clock positions of the tailing edge. This themal effect is consistent with the findings of Hong et al. [13], which pointed out that polishing tempeatue vaies in paallel with thei speed integal. Thei finding eveals that the location of the highest pedicted tempeatue by the speed integal match the highest measued tempeatue on the substate suface. Afte a peiod of 100 sec the tempeatue gadient of 9 degees emained the same at a few small aeas aound the 3 to 4 o clock positions of the tailing edge of the wafe, as shown in figue 7.0a. Figue 7.0b shows a tempeatue diffeence of 6 degees duing the ealy pat of the pocess at a small faction of the uppe egion (1 o clock) of the tailing edge and a significant egion along the 3 to 4 o clock positions close to the 8

256 cente of the substate. Late on, the tansient tempeatue diffeence educed slightly to 5 degees to thee consticted aeas nea the tailing edge of the wafe at the 1, 3 and 4 o clock positions. In addition, the tempeatue gadient fo a thicke sluy film at the pad suface showed an oveall tempeatue diffeence of 3 degees appoximately. (a) (b) Figue 7.0 Sluy, wafe, and pad sufaces tempeatue contou distibutions unde a sluy film thickness equal to: (a) δ sl =40 µm and (b) δ sl =10 µm. Figue 7.1 shows the tansient substate tempeatue vaiations and wafe pad tempeatue diffeences fo thee diffeent sluy film thicknesses at the 1 o clock position, fo the same conditions descibed in the peceding tempeatue contou plots. 9

257 The aveage tansient wafe tempeatue esults of figue 7.1 at a adial distance of / w =7/8 along the 1 o clock position wee appoximately equal to , , and degees K fo the abasive film thicknesses (δ sl ) of 00, 10, and 40 µm. The wafe pad tempeatue diffeences examined ange up to 3.03, 1.95, and 0.67 degees K o C fo the following abasive film thickness (δ sl ) of 00, 10, and 40 µm. Convesely, the wafe tempeatue distibutions of thicke film ae lowe due to an incement of the volumetic flow ate of the sluy that moves beneath the substate and pad sufaces. Wafe Tempeatue,T (K ) Τ, δsl = 40 µm 10 T, δsl =10 µm 00 T, δsl =00 µm dt00 T, δsl = 00 µm dt10 T, δsl = 10 µm dt40 T, δsl = 40 µm Time (sec) LE 1o'clock TE Wafe-Pad Tempeatue diffeence, T (K) Figue 7.1 Tansient wafe tempeatue distibutions and wafe pad tempeatue diffeences fo 3 diffeent sluy film thicknesses at a adial location of / w =7/8 along the 1 o clock position. Figue 7. shows the wafe heat tansfe convection coefficients at a adial distance of / w =7/8 along the 1 o clock position fo thee diffeent sluy film thicknesses. The aveage heat tansfe convection coefficients of the substate sufaces in 30

258 figue 7. attained wee equal to 8.51, 4.19, and 0.0 W/m K fo the abasive film thicknesses (δ sl ) of 00, 10, and 40 µm espectively. The incement of volumetic flow ate inceased the heat advection pe unit aea; theefoe it emoves moe heat in less time. This effect educes the oveall tempeatue of the wafe and pad at the contol volume unde study. Pesent esults ae in ageement with the findings of Mudhivathi [146] and Sikde et al. [148], whee the incement of the film thickness educed the mechanical contact, and inceased the amount of sluy inteaction, esulting in lowe tempeatue pofiles aound the tailing aea of the wafe egion exposed to the abasive pad inteface. 140 Wafe heat tansfe coef., h(w/m K) g04δsl =40 µm g1δsl =10 µm g0 δsl =00 µm Time (sec) Figue 7. Tansient wafe heat tansfe convection coefficients fo thee diffeent sluy film thicknesses at a adial location of / w =7/8 along the 3 o clock position. 31

259 The tempeatue contou distibutions of the wafe and pad sufaces unde two chaacteistic pad spinning ates of 175 and 50 RPM ae shown in figues 7.3a and 7.3b. (a) Figue 7.3 Wafe, and pad tempeatue contou distibutions fo a pad spinning ate equal to: (a) Ω p =175 RPM and (b) Ω p =50 RPM. (b) The tansient tempeatue contou plots wee set fo an alumina abasive film thickness of 40 µm, at a constant sluy flow ate of 50 cc/min, unde a constant load of 4.35 kpa, fo a vaiable heat flux ange of 3.9 to 9.14 kw/m, and a caie spinning ate of 30 RPM. Duing the ealy pat of the tansient pocess in figue 7.3a the sluy, wafe 3