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## Material Information- Title:
- Computational Investigation of Buoyancy Effect on Temperature Field over an Axisymmetric Microheater
- Creator:
- Singh, Shantanu
- Place of Publication:
- [Gainesville, Fla.]
Florida - Publisher:
- University of Florida
- Publication Date:
- 2017
- Language:
- english
- Physical Description:
- 1 online resource (68 p.)
## Thesis/Dissertation Information- Degree:
- Master's ( M.S.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Mechanical Engineering
Mechanical and Aerospace Engineering - Committee Chair:
- MEI,RENWEI
- Committee Co-Chair:
- HAHN,DAVID WORTHINGTON
- Committee Members:
- SCHEFFE,JONATHAN
## Subjects- Subjects / Keywords:
- axisymmetric -- microheater -- nucleation
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF - Genre:
- bibliography ( marcgt )
theses ( marcgt ) government publication (state, provincial, terriorial, dependent) ( marcgt ) born-digital ( sobekcm ) Electronic Thesis or Dissertation Mechanical Engineering thesis, M.S.
## Notes- Abstract:
- Steady state, laminar, external natural convection flow over a horizontal, circular, isothermal microheater is studied for Rayleigh number range up to 100 at Prandtl number of 1,5 and 10. Momentum and energy equations are solved using a finite-difference scheme for the fluid region over the heater. For the case of pure conduction; influence of domain size is studied and finite difference results are validated with the exact solution. Dependence of results on grid size is studied both for the case of pure conduction and convective flow. For conduction, grid dependency of results is checked against exact solution; whereas, for convective flow, Richardson extrapolation is used to check the results against extrapolated values. Finally, a correlation for dimensionless fluid temperature near the heater is developed as a function of dimensionless axial and radial coordinates, and Rayleigh number (Theta (Z, R, Ra)). Least square regression fit is implemented to get the correlation. The correlation is a 4th order unified polynomial valid for Rayleigh number from 0 to 100. Three correlations are developed for Prandtl number of 1,5 and 10. It is observed that the flow near the centerline of heater does not follow boundary layer type behavior; whereas, it follows boundary layer type behavior on moving away from the centerline. ( en )
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- In the series University of Florida Digital Collections.
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- Includes vita.
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- Includes bibliographical references.
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- Description based on online resource; title from PDF title page.
- Source of Description:
- This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
- Thesis:
- Thesis (M.S.)--University of Florida, 2017.
- Local:
- Adviser: MEI,RENWEI.
- Local:
- Co-adviser: HAHN,DAVID WORTHINGTON.
- Statement of Responsibility:
- by Shantanu Singh.
## Record Information- Source Institution:
- UFRGP
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- Applicable rights reserved.
- Classification:
- LD1780 2017 ( lcc )
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PAGE 1 COMPUTATIONAL INVESTIGATION OF BUOYANCY EFFECT ON TEMPERATURE FIELD OVER A N AXISYMMETRIC MICROHEATER By SHANTANU SINGH A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE R EQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2017 PAGE 2 2017 Shantanu Singh PAGE 3 To my parents PAGE 4 4 ACKNOWLEDGMENTS I would like to express my utmost gratitude to my advisor, Dr. Renwei Mei Professor of Mechanical and Aerospace Engineering at University of Florida for his constant support, inspiration and encouragement. This dissertation would have been impossible without his expert guidance and enthusiasm on the subject. I would also like to express my gratitude to Dr. David W. Hahn chair of Mechanical and Aerospace Engineering and Dr. Jonathan R. Scheffe Associate Professor of Mechanical and Aerospace Engineering for serving on my supervisory committee. I thank my fellow graduate student s Mr. Amish Gadigi, Mr. Kelvin Randhir and Mr. Chen Chen for their immense support and encouragement. I also acknowledge the support provided by the High Performance Computing Center at University of Florida. Finally, I am deeply grateful for the enduring and endless support and love of my family. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ ...... 4 LIST OF TABLES ................................ ................................ ................................ ................ 7 LIST OF FIGURES ................................ ................................ ................................ .............. 8 LIST OF ABBREVIATIONS ................................ ................................ ................................ 9 LIST OF SYMBOLS ................................ ................................ ................................ .......... 10 ABSTRACT ................................ ................................ ................................ ........................ 12 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ ........ 14 1.1 Background and Motivation ................................ ................................ .................. 14 1.2 Objective ................................ ................................ ................................ ............... 16 2 LITERATURE REVIEW ................................ ................................ .............................. 18 2.1 Natural Convection over Isothermal Surfaces ................................ ..................... 18 2.2 Natural Convection for Low Ra ................................ ................................ ............ 19 2.3 Domain Size for External Flows with Natural Convection ................................ ... 20 3 NUMERICAL MODELLING ................................ ................................ ........................ 22 3.1 Governing Equations ................................ ................................ ............................ 22 3.2 Boundary Conditions and Grid Stretching ................................ ........................... 24 3.3 Discretization Method ................................ ................................ ........................... 27 4 RESULTS AND DISCUSSIONS ................................ ................................ ................ 33 4.1 Influence of Domain Size ................................ ................................ ..................... 33 4.1.1 Valida tion Based on Exact Solution for Pure Conduction Case ............... 34 4.1.2 Validation Based on Comparison of Experimental and Numerical Isotherms and Streamlines ................................ ................................ ............... 35 4.2 Influence of Grid Size ................................ ................................ ........................... 36 4.2.1 Iterative Convergence ................................ ................................ ................. 36 4.2.2 Grid Convergence for the Case of Pur e Conduction ................................ 37 4.2.3 Grid convergence for Moderate Convective Flow ................................ ..... 38 4.3 Results ................................ ................................ ................................ .................. 40 4.3.1 Effects of Pr and Ra on Steady State Isotherm and Streamlines ............. 40 4.3.2 Developing Correlation for Thermal Field near the Heater ....................... 41 PAGE 6 6 4.3.3 Verification of Correlation ................................ ................................ ........... 43 5 CONCLUSIONS ................................ ................................ ................................ .......... 64 LIST OF REFERENCES ................................ ................................ ................................ ... 65 BIOGRAPHICAL SKETCH ................................ ................................ ................................ 68 PAGE 7 7 LIST OF TABLES Table page 3 1 Grid stretching parameters for different domain size ................................ ............ 32 4 1 Grid convergence and local order of accuracy for R=1, Z=0.026471 with Ra=100, Pr=1 ................................ ................................ ................................ ......... 56 4 2 Comparison of finite differenc e results for various grid size at R=0, with the extrapolated value for Ra=100, Pr=1 ................................ ................................ ..... 56 4 3 ................................ ......... 58 4 4 Polynomial coefficients in developing correlation for Pr=1 ................................ ... 58 4 5 Comparison between heat flux given by correlation and finite difference for the case of Ra=28, Pr=10 ................................ ................................ ...................... 63 PAGE 8 8 LIST OF FIGURES Figure page 3 1 Sectional view of cylindrical domain over an axisymmetric microheater with cylindrical coordinate system ................................ ................................ ................. 29 3 2 2 D axisymmetric axial plane with boundary conditions ................................ ....... 30 3 3 The point of discontinuity R=1 is halfway between the two grids for all grid sizes (161X401 is the present grid) ................................ ................................ ....... 31 4 1 Comparison between exact solution of temperature and finite difference results (domain size R =Z =10,50,100) for pure conduction case ...................... 46 4 2 Isotherms over heater for various domain size ................................ ..................... 46 4 3 Streamline pattern in exper imental study by Torrence and Orloff [29](Left) and present study by numerical modelling (Right) situation for Ra=560, Pr=0.7 ................................ ................................ ................................ ..................... 47 4 4 Comparison of steady state isotherms between Torrance et al [28] and present work for Pr=0.7 ................................ ................................ .......................... 48 4 5 Heat flux and vorticity residual plot for Pr=10 with Ra=28 with initial condition ................................ ................................ ................................ ............ 49 4 6 Comparison between exact solution of temperature and finite difference results for grid size s 81X201,161X401 and 321X801 for pure conduction case ................................ ................................ ................................ ........................ 49 4 7 Comparison between extrapolated solution of temperature and finite difference results for grid sizes 81X201,161X401 and 321X801 at R=0 for Ra=100, Pr=1 ................................ ................................ ................................ ......... 50 4 9 Steady state isotherms for Pr=1 ................................ ................................ ............ 51 4 10 Steady state isotherms for Pr=10 ................................ ................................ .......... 52 4 11 Steady state streamlines for Pr=10 ................................ ................................ ....... 53 4 12 Pr=10 ................................ ................................ ................................ ...................... 54 4 13 Variation of temperature in axial direction at various R locations for Ra=100 with Pr=1 and 10 ................................ ................................ ................................ .... 54 4 14 Comparison of temperature obtained by co rrelation and finite difference for Ra=55, Pr=10 at various radial locations ................................ .............................. 55 PAGE 9 9 LIST OF ABBREVIATIONS Gr Grashof number Nu Nusselt number Pr Prandtl number Ra Rayleigh number PAGE 10 10 LIST OF SYMBOLS Dimensionless temperature Stream function Non dimensional stream function Vorticity Non dimensional vorticity Thermal diffusivity Coefficient of thermal expansio n T sat Saturation temperature T wall Temperature of heater Kinematic Viscosity Axial coordinate in computational domain Radial coordinate in computational domain Parame ter in grid stretching Parameter in grid stretching Parameter in grid stretching Parameter in grid stretching Parameter in grid stretching Parameter in grid stretching Dimensionless heat flux Constant for separation of variables Heat flux on heater wall Infinite boundary in radial direction PAGE 11 11 Infinite boundary in axial direction r Radial coordinate z Axial coordinate R Non dimensional radial coordinate Z Non dimensional axial coordinate u r Radial velocity u z Axial velocity U Non Dimensional radial velocity V Non Dimensional axial velocity x C artesian x coordinate y Cartesian y coordinate Degree of superheat t Dimensional time t Non dimensional time p Order of convergence f 1 Flow field value for finest mesh f 2 Flow field value for fine mesh f 3 Flow field value for coarsest mesh GCI Grid convergence index PAGE 12 12 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Sc ience COMPUTATIONAL INVESTIGATION OF BUOYANCY EFFECT ON TEMPERATURE FIELD OVER A N AXISYMMETRIC MICROHEATER By Shantanu Singh August 2017 Chair: Renwei Mei Major: Mechanical Engineering Steady state laminar external natural convection flow over a hor izontal, circular, isothermal micro heater is studied for Rayleigh number range up to 100 at Prandtl number of 1,5 and 10. M omentum and energy equations are solved using a finite difference scheme for the fluid region over the heater For the case of pure c onduction; influence of domain size is studied and finite difference results are validated with the exact solution Dependence of results on g rid size is studied both for the case of pure conduction and convective flow. For conduction, grid dependency of r esults is checked against exact solution; whereas, for convective flow Richardson extrapolation is used to check the results against extrapolated values. Finally, a correlation for dimensionless fluid temperature near the heater is developed as a function of dimensionless axial and radial coordinate s, and Rayleigh number (Z, R, Ra)). Least square regression fit is implemented to get the correlation The correlation is a 4 th order unified polynomial valid for Rayleigh number from 0 to 100 Three correlations are developed for Prandtl number of 1,5 and 10 It is observed that the flow near the centerline of heater does not follow PAGE 13 13 boundary layer type behavior; whereas, it follows boundary layer type behavior on moving away from the centerline. PAGE 14 14 1 CHAPTER 1 1 INTRODUCTION 1.1 Background and M otivation There are many app lication of boiling heat transfer, such as in high heat flux electronic co oling refrigeration, nuclear reactor cooling and in heat exchangers [1] With a rapid rate of miniaturization of electronic components, there has been a n ever increasing demand to manage the high heat flux emanated from these compact devices. Presently, air cooling is the most commonly used technique for electronic cooling; however, poor thermal transport properties of air reduces its potential for high h eat flux thermal management. Boiling on the other hand is complemented by high heat transfer rate Liquid immersion cooling involving boiling heat transfer, with dielectric fluids such as Fluorinert (FC series by 3M), is a pragmatic solution for cooling of microelectronics and microsystem packaging [2] The phenomenon responsible for high heat transfer rate in boiling is bubble n ucleat ion It occurs when the surface is superheated and heat flux is below the critical heat flux. There have been many studies for b ubble incipience on superheated surface T h e conventional theory of nucleate boiling states that vapor/gas is trapped in imperfections (cavities and scratches) on a surface, which then serve s as nuclei for bubble formatio n on su perheated surface during boiling Nucleation can be homogeneous or heterogeneous depending upon where bubble incipience occurs. If the bubble is formed in the bulk of liquid then nucleation is homogeneous, whereas if the bubble is formed at the flui d superheated wall interface then it is a case of heterogeneous nucleat ion. In absence of trapped vapor sources on the heating surface, h omogeneous nucleation occurs at high wall superheat s (homogeneous limit) ; while the PAGE 15 15 presence of trapped vapor sources o n the surface results in heterogeneous nucleation which occurs at relatively lower wall superheats (heterogeneous limit) [3] [4] Theoretical and experimental studies conducted during the 1970s and 1980s suggested that the only mode of nucleation possible on cavity free surfaces was homogeneous nucleation [5] [6] [7] Th e long held classic nucleation theory (CNT) of trapped vapo r giving rise to nucleation, was questioned in 2002 when Theofanous et al. [8] observed nucleation at low wall superheat over a cavity free surface ( nanoscopic level smooth heater free of micron scale cavities). Low wall superheat was supposed to be a characteristic of heterogeneous nucleation, a phenomenon which cannot occur on a cavity free surface according to the existing theory Subsequently, Qi and Klausner [9] were also able t o initiate boiling at low wall superheats on nanoscopic level smooth stainless steel and brass surfaces. Bon et al. [10] evaluated the nucleation wall superheat of two fluids (pentane and Hexane) over nine types of different su rfaces. They found these nucleation wall superheat s to be 30 to 50 % of the homogeneous limit for all cases. Even more similar studies [11] [12] [13] [14] demonstrated heterogeneous nucleation at low wall superheats ( ~10 K under atmospheric water saturated conditions) on a smooth hydrophilic surface. These experimental findings strengthened the opinion that heterogeneous nucleation could be f ound over smooth cavity free surfaces devoid of vapor traps. To explain the contradictions to the established nucleation theory, Jo et al. [15] in 2014 used a thermal boundary layer model They combined their boundary layer mo del with kinetics and dynamics of superheated liquid and thermodynamic stability of generated vapor. Using a 2 D boundary layer approximation they arrived at temperature PAGE 16 16 field of exponential form. However, their solution was formulated over a semi infinite smooth horizontal plate and they assumed the presence of thermal boundary layer Flows over microheater fall in low Ra range du e to the small chara cteristic length of microheater. However, most of the studies done in this regard focused on high Ra range, wherein they predicted the existence of a well developed boundary layer [16] [17] Petralanda [3] solved a 3 D conduction problem over a heater and came up with a two t erm simple correlation for temperature at centerline. However, influence of Ra was not clear. Suriano et al. [18] studied steady, laminar natural convection on a semi infinite plate for moderate Ra They predicted a semi bounda ry layer type of behavior over the heater for Ra 300 onwards. Again, there study was for a semi infinite plate. Therefore, a study of the temperature field over a finite geometry for low Ra range and different Pr would be helpful in describing the nature o f flow over microheater. 1.2 Objective P rimary objective of the present research is to study the temperature field over a horizontal isothermal microheater due to a steady laminar free convection flow over it. This study covers low Ra range up to 100. C or relation (Z, R, Ra) is to be obtained for Pr=1,5 and 10. During nucleation, a bubble forms on top of the heating surface. It is worth noting that a temperature gradient exists from the heating surface to the bulk fluid in its vicinity. This fluid near the heater is at saturation temperature. Since the vapor embryo forms in the bulk liquid on top of the heating surface, the temperature gradient exist s from the heater surface into the vapor bubble to all the way into the b ulk fluid. It is PAGE 17 17 important to investigate this temperature gradient to understand the thermal boundary layer and deduce how it effects bubble incipience. Moreover cooling of a microelectronic device is an external flow natural convection problem for low r ange of Ra Analysing the thermal field over the heater can reveal the dominant mode of heat transfer i.e. conduction or natural convection for these low Ra flows. PAGE 18 18 2 C HAPTER 2 2 LITERATURE REVIEW 2.1 Natural C onvection over I sothermal S urfaces Free convectiv e boundary layer over isothermal surfaces ha ve been extensively studied since 1958 [18] [16] [17] [19] [20] St udies conducted by Stewartson [19] showed that only one of the two possible flows : heated plate facing upwards (or a cooled plate facing downwards), or a cooled plate facing upwards (or a heated plate facing downwards), would g ive rise to a boundary layer type of solution. Later Gill et al. [20] and Rotem [21] showed that boundary layer solution could only be obtained for the first case i.e. heated plate facing upwards (o r a cooled one facing downwards) Petralanda [3] solved a 3 D conduction problem over a heater and came up with a two term simple correlation for temperature at centerline. However, influence of Ra was not clear. Jo et al. [14] formulated an exponential temperature profile over a semi infinite plate for natural convection case. They assumed 2 D boundary layer approximation. Corcione [22] summarized the published experimen tal and numerical data on free convection from upward facing horizontal plates It was shown that there existed large variation in the heat transfer correlations because they wer e highly dependent on the surface geometry and fluid. Temperature field is be ing investigated over a microheater. Now, Gr or Ra i s directly proportional to x 3 and characteristic length (x) of microheater is in microns. Thus, flow over a microheater happens to be in low Ra range. Moreover the previous studies assumed a semi infinit e geometry; whereas, finite size of the heater. is the focus of the present study. PAGE 19 19 2.2 Natural C onvection for L ow Ra As explained in S ection 2.1 small geometry results in low Ra flows, characterized by lower buoyancy forces and consequently lesser contribu tion by natural convec tion. Furthermore, past studies indicate the existence of threshold Ra in enclosed flows, below which natural convection does not occur. According to the existing correlations, heat transfer due to natural convection increases with in creasing Ra for external flows. However, Kostoglou et al. [23] showed that above is true only when Ra is above a threshold value. Below this value there is no natural convection. Kostoglou et al. [24 ] studied a miniature spheroid heater with water and FC 72 as cooling fluid to ascertain the existence of threshold Ra for external flows. They predicted a threshold Ra value of 40 for water, while they were unable to determine a specific value for FC 72 They reported that Nu for FC 72 was insensitive to the Ra range of 400 1000, yet there would exist a threshold value for FC 72. The existence o f low Gr and Ra sets this study in a unique region where the mode of heat transfer is not clearly defined. Suri ano et al. [18] carried out the study on horizontal plates for low Gr Although their study was for a semi infinite plate, nevertheless it can be insightful for analysis of finite geometry. They found that there wa s no effect o f viscous dissipation on the flow physics However, the i r conclusion was based on Pr=0.72 and 10, so the effect of dissipation on high Pr fluid is still unclear. Furthermore, they found three regions of interest based on the variation of Gr or Ra at cons tant Pr. The first region was for Ra up to unity, the second was for Ra up to 50 and the third for Ra up to 300. In first region conduction was the only heat transfer mechanism. As the Ra increased into the second region, convection began to be PAGE 20 20 impactful; however near the heating plate conduction was still more dominant. The Nu also started to increase beyond conduction. As the Ra was increased even further into the third region, boundary layer type flow start ed to develop Authors predicted that the flow behaviour would gradually tend to boundary layer type, as the Ra was increased beyond 300. The p resent study will investigate low Ra flow over a flat plate type microheater for different Pr The Ra range would be 0 to 100 and temperature field over microh eater would be studied. 2.3 Domain S ize for E xternal F low s with N atural Convection For flow over a microheater, the walls confining the flow are located at a scale much larger than the scale of the heater itself. Therefore, for flow over a microheater, the boundaries are essentially at infinity. Infinite boundaries have been a source of difficulty. For a semi infinite plate, Suriano et al. [18] acknowledged from an analytical perspective that analysis of infinite domain problem should be done by matching some near heater solution (analytical or numerical) to asymptotic solution at large distance from the heated surface. The only available asymptotic solution is the well known boundary layer far wake solution, where the velocity component opposite to the gravity field increases unboundedly with the distance from the heating surface. This is not feasible physically and more so in the case of low Ra flows. To define the flow boundaries for their numerical studies, Suriano et al. cho se a rectangular finite domain whose dimensional scale was large compared to the length scale of their heating surface. Since then, many studies used the same concept of sufficiently large domain size to simulate the effect of infinite boundaries [24] [16] [25] [26] PAGE 21 21 In the present study, a finite domain size was chosen and grid stretching technique was implemented to efficiently resolve the hi gh gradient reg ion. It is to be noted that the flow field and temperature field near the heater are independent of domain size after a certain value of domain size, as will be explained later. So, we mainly focus our study on the region near heater bounded by R=1, Z=1, i.e. 1X1 domain. PAGE 22 22 3 CHAPTER 3 3 NUMERICAL MODELLING 3.1 Governing Equations To model a steady, laminar free convection external flow, a vertical cylindrical domain with the microheater in the bottom center was assumed. Generally, the heatin g element in boiling heat transfer experiments is a planar array of several microheaters arranged in a certain pattern [12] [27] For studying a single bubble incipience, Chen et al [12] chose to keep only one of the 96 distinct microheaters active. Dimensions of a single microheater was taken as 250 m X 250 m in the study by Petralanda [3] With such a size of microheater it is safe to say that; the length scale of domain size was large enough compared to the length scale of heater to make the domain an infinite boundary. Figure 3 1 shows the schematic representation of the problem under study. The flow is axisymme tric and Boussinesq approximation is used in representing the buoyancy force. The stream function vorticity formulation is used in lieu of Navier Stokes equation. Due to axisymmetric nature of the flow, the modeling could be done in one half of a 2 D axial plane as shown in Figure 3 2 Other assumptions employed are: negligible viscous dissipation [1], no heat generation, and constant fluid properties viz. ) an [24] z), the velocity can be evaluated as ( 3 1 ) PAGE 23 23 where u r and u z are the r and z comp onent of the fluid velocity, respectively. This ensures that the continuity equation is automatically satisfied. From the definition of vorticity, in axisymmetric flow, the following relation is establ ished: ( 3 2 ) The momentum equations for u z and u r in the presence of buoyancy force is modeled using Boussinesq approximation, and it can be combined by cross differentiation to obtain the vorticity tr ansport equation: ( 3 3 ) The energy equation for temperature T in the axisymmetric flow is: ( 3 4 ) For a heater with radius L as characteristic leng th, the following dimensionless variables and parameters are introduced, The dimensionless relationship between velocity, stream function and vorticity becomes: ( 3 5 ) PAGE 24 24 ( 3 6 ) The dimensionless vorticity transport and energy equations becomes: ( 3 7 ) ( 3 8 ) Where Ra= and Pr= / 3.2 Boundary Conditions and Grid Stretching The boundary conditions were: ; ( 3 9 a ) ; ( 3 9 b ) ; ( 3 9 c ) R= ; ( 3 9 d ) Z= ; ( 3 9 e ) The first boundary condition of isothermal heater resulted in a temperature discontinuity. There was an option of giving the heater a constant heat flux condition or constant temperature con dition. However, it is preferable to maintain the heaters at a constant temperature for the following reasons [3] [12] [27] PAGE 25 25 1. Maintaining a constant surface temperature of heater reduces the complexity of the analysis because it eliminates the spatial changes in wall temperature. 2. Conduction between adjacent microheaters through substrate is minimized as they are at the same temperature 3. By eliminating conduction in the sub strate and providing a constant temperature boundary condition, numerical or analytical models of nucleate and transition boiling, and critical heat flux, can be simplified. 4. Data acquisition of critical heat flux and transition boiling regime can be done without the danger of heater dry out. 5. To identify the nature of boiling incipience a constant temperature condition is favoured because degree of superheat is a controlling parameter. To accommodate for a large domain ( large R and Z ) to reduce the impac t of the d omain size in the present study grid stretching is necessary in R and Z directions. Furthermore there is a discontinuity in boundary condition at the end of the heater; thus, smaller grid spacing is needed near R=1 to maintain a sufficient res olution. A recirculating flow is expected within the enclosure of large but finite domain. However, upon increasing Ra a boundary layer type flow could also be expected. In the case of a boundary layer flow, the thickness of the boundary layer decreases w ith an increase in Ra. To capture any boundary layer type of behavior, sufficient grid resolution should be maintained in the Z direction near the heater. To simulate far field boundaries, the length scale of domain size should be large enough than the len gth scale of heater. The characteristic length of a typical microheater is around 125 m; which is practically negligible when compared to the radius of the vessel in which boiling PAGE 26 26 heat transfer experiment i s carried out [3]. To define a domain, it was req uired to choose some value of R and Z so that an imaginary boundary wall at that position was sufficiently far enough not to affect the temperature field and flow pattern over microheater. Goldstein et al. [19] reported a mere increase of 2% in heat tra nsfer rate on doubling the size of the domain. In the present study, the effect of different domain size was observed on local flow field over the heater as will be explained later in section 4.1 Accordingly, a domain size of 10 was chosen so that the res ults in 1X1 region were relatively independent of domain size The physical domain represented by R and Z were transformed to a computational domain represented by a uniformly spaced grid with dimensions the grid: In R dimension : ( 3 10 ) w here ; is adjusted for proper grids; =0.25 ( 3 11 ) In Z dimension : (3 12 ) Parameter is adjusted for proper grids Table 3 1 s hows the v alue for grid stretching parameters used in the present study for different domain sizes PAGE 27 27 The above grid stretching ensures that the boundary condition discontinuity at R=1 always falls between the two adjacent grids for all grid sizes as shown in Figure 3 3 Having the discontinuity halfway between two grids does not results in a loss of resolution on increasing the grid size by a given factor (2 in the present study). 3.3 Discretization Method The stream function equation is elli ptic; while, the vorticity transport and energy equations are parabolic. To transform the physical grid to computational grids f ollowing terms were defined : Eq uation ( 3 6 ) to ( 3 8 ) are written in conservative form and they are discretized using three point central difference scheme Vorticity and temperature field on internal grids were advanced across a time step from t *n t o t* n+1 using semi implicit finite difference approxim ation to the above equations. The following discretization are implemented in the code in the given order: ( 3 13 ) PAGE 28 28 ( 3 14 ) ( 3 15 ) At the boundary, new vorticities are calculated from the stream function boundary condition as stated in Section 3.2. A first order vorticity approximation is obtained by ns calculated by Eq. ( 3 16 ) ( 3 16 ) Now we define the dimensionless heat flux at the heater su ( 3 17 ) The iterative solutions were converged when the L2 norm for heat flux and vorticity at the bottom wall, defined by Eq ( 3 18 ) and ( 3 19 ) became less than 10 10 ( 3 18 ) ( 3 19 ) PAGE 29 29 Figure 3 1 Sectional view of cylindrical domai n over an axisymmetric microheater with cylindrical coordinate system PAGE 30 30 Figure 3 2 2 D a xisymmetric axial pla ne with boundary conditions PAGE 31 31 Figure 3 3 The poin t of discontinuity R=1 is halfway between the two grids for all grid sizes (161X401 is the present grid) PAGE 32 32 Table 3 1 Grid stretching parameters for different domain size R=Z=10 R=Z=50 R=Z=100 a a 51.7 51.7 51 .7 0 0.25 0.25 0.25 0.75 0.7 0 0.94 188 188 188 PAGE 33 33 4 CHAPTER 4 4 RESULTS AND DISCUSSION S 4.1 Influence of Domain Size To simulate the effect of an infinite boundary a finite domain size should be chosen such that increasing the size any further does no t produce any meaningful change in the flow field near heater For this purpose, results were generated using domain size of 10,50 100 for the case of pure conduction. Figure 4 1 (a), (b) and (c) show that finite difference tempe rature solution for domain size R=10,50 and 100 follows the exact solution of temperature very closely, for a range of R values between 0 to 1. It is evident that increasing the domain size further than 10 had no considerable influence on the flow physics in the 1X1 region over heater. Based on above analyses it i s safe to conclude that i s a well enough domain size. to represent boundaries at infinity. Figure 4 2 shows isotherms over heater for various domain sizes. It was observed that the isotherms pattern remained unaffected by a change in domain size. With a proper choice of modelling parameters, the code was implemented to generate results. F ollowing methods were used for validation of numerical modelling resul ts : 1. Comparing exact temperature solution with finite difference results for pure conduction case 2. Comparing avai lable experimental results and modelling results. It is worth mentioning that patterns in entire computational domain were compared with reporte d literature only for validation It wa s known that the results far away from heater are domain size dependent, so they we re of no physical significance. PAGE 34 34 For the present study, it is only the results in vicinity of heater that matters. However, obtaining a similar streamline and isotherm pattern to that of reported literature w ould attest the validity of our modelling. 4.1.1 Validation Based on Exact Solution for Pure Conduction Case Energy equation for pure conduction in cylindrical coordinates reduce s to E q ( 4 20 ) ( 4 20 ) B oundary C ondition s : ( 4 21 a ) Z ( R Z ) = 0 ( 4 2 1 b ) R =0: ( 4 2 1 c ) R ( R Z ) = 0 ( 4 2 1 d ) Solution for the above case is of form: ( 4 22 ) Using E q. ( 4 21 a b, c, d ) the following solution is obtained: ( 4 23 ) At the centerline R=0, therefore E q. ( 4 23 ) becomes ( 4 24 ) Numerical integration i s used to get the (Z) The above equations we re used to generate exact so lution for different values of R. PAGE 35 35 To observe accuracy of finite difference solution and the effect of domain size; = =10, 50.100) we re compared with the generated exact solution for the respective R values. Figure 4 1 (a ), ( b), (c) shows the comparison between exact solution and finite difference results for domain sizes of 10,50,100. Finite differen ce results for different domain sizes followed the exact solution closely for a range of R values between 0 to 1. Moreover, the agreement between finite difference results and exact solution goes even up to Z =2. From the above observation, the following i s concluded: 1. The code is validated for pure conduction case. 2. Domain size s of 10, 50 and 100 produce similar temperature field near heater. Therefore, further discussions are based on results produced by domain size of 10. 4.1. 2 Validation B ased on Compari son of E xperimental and N umerical I sotherms and S treamlines Torr a nce and Rockett [28] used an identical mathematical model for study of natural convection in a cylindrical enclosure. They studied natural convection over a heat er for a range of Gr with air as the fluid (Pr=0.7). They used a normalized domain size of 1X1 with length of heater as 0.1. The domain used in present study was 10X10 with hea ter length of 1; therefore, result of the present study are scaled by a factor of 10. To validate the results of present study by comparing with the results of Torr a nce and Rockett, we scaled the Gr and Ra in proportion to domain size ratio of the two studies. Length ratio of the domain size in two studies was 10. Since the Gr is dir ectly PAGE 36 36 proportional to L 3 equivalent Gr in this study would be a thousandth of that used by Torr a nce et al Torr a nce et al. [29] also conducted experimental studies for Pr=0.7 Results were generated for Pr=0.7 in the present s tudy and they were compared to the existing literature. For a comparison with reported experimental results, modelling was done for Pr=0.7 with Gr=800 or Ra=560 (Gr=8X10 5 for reported literature) Figure 4 3 shows a good agreement between the experimental and numerical modelling The streamline patterns were in good agreement. This code was also run for the case of conduction and Gr= 40 or Ra=28 (Gr=4X10 4 for reported literature) to compare our numerical results to the numerical r esults in existing literature [28] Figure 4 4 shows a good agreement between the two max was also done. The results for max =7 07 1 a nd it occurred at R= 4.938, Z=5.422. When scaled to compare with the results of Torrance e t al. we get a value of max =0. 707 occurring at R= 0.4938, Z=0.5422. Torrance reported a value of max =0.719 and it occurred at almost R=Z=0.5 ; thus the obtained results are in close agreement with reported literature. 4.2 Influence of Grid Size 4.2.1 Iterative Convergence Equations ( 3 18 ) and ( 3 19 ) were used to check the iterative convergence as explained in section 3.3. Often false convergence results when there is a momentary plateau in convergence curve for a span of iterations. T o avoid this, convergence was not checked after every iteration for the whole iterative process. Instead, following criterion was implemented for convergence check : for iteration range 0 to 5,000 PAGE 37 37 convergence was checked after every iteration; for iterat ion range 5,000 to 50,000 convergence was checked after every 20 iterations; for iteration range 50,000 to 500,000 convergence was checked after every 100 iterations; for iteration range 500,000 to 3,000,000 convergence was checked after every 1000 i terations and for iterations above 3,000,000 convergence was checked after every 10,000 iterations. Figure 4 5 shows the residual plot for the iterative convergence for Pr=10, Ra=28. Once the convergence criteria for both type o f residuals goes under 10 10 steady state was achieved. Thus, the results can be assumed to be steady with respect to iterations of time. 4.2. 2 Grid Convergence for the Case of Pure Conduction The code used in present study implemented grid interpolation to advance the result o n a specific grid system as an initial condition for the next finer grid size. Grid refinement was done by a factor of two. Thus, mesh size was refined from 41X101 to 81X201 to 161X401 to 321X801. Finite difference temperature soluti on obtained by different grid size w ere various locations of Z. Figure 4 6 s hows the comparison. It is observed that even till Z=2 and R=2, r esults from grids 81X201, 161X401 and 321X801 are similar and follow the exact solution very closely. Thus, the results can be considered converged with respect to grid size at grid size of 161X401 for pure conduction case and we can save the ad ditional computational cost required in case of 321X801 grids PAGE 38 38 4.2. 3 Grid convergence for Moderate Convective F low For Ra=100 and Pr=1; g was carried out and flow filed values at various locations of the domain were compared with the respective extrapolated values. Eq ( 4 25 ) and ( 4 26 ) are used for extrapolation ( 4 25 ) ( 4 26 ) Where, p is order pf accuracy f1, f2, f3 are the flow field values for mesh size 1,2,3 with 1 being the finest and 3 most coarse r is the grid refinement ratio which is 2 for present case For different grid size s, values of finite difference vorticity, stream function, theta, radial and axial velocities were compared with extrapolated values for the case of Pr=1 and Ra = 100 Table 4 1 shows the grid convergence and order of accuracy analysis for a specific location R=1 and Z= 0.026471 Results show that the deviation between extrapolated values and finite difference values given by 161 X 401 grids is v ery less. Moreover, the local order of accuracy is close to 2. Another similar analysis was conducted at radial location R=0 i.e. centerline and R=0.7 for Z going from 0 to 1. At R=0, local order of accuracy of temperature was calculated f or the axial gr id locations common in 81X201, 161X401 and 321X801 grids. Error and Grid convergence index (GCI) between the grids w ere also calculated using E q ( 4 27 ) and ( 4 28 ) As defin ed by Roache [30] GCI provides an estimate of the discretization error in finest gri d solution relative to the converged numerical solution. PAGE 39 39 The converged numerical solution lies in the interval [f 1 (1 GCI),f 1 (1+GCI)] with a 95 % confidence level for a typical factor of safety Fs=1.25 ( 4 27 ) ( 4 28 ) Where, e 21 is error between grids 161X401 and 321X801 F s is factor of safe ty (1.25 in present study) r 21 is grid refinement ratio between grids 161X401 and 321X801 (2 in present study) p is order of convergence GCI 21 is the grid convergence index between grid s 161X401 and 321X801 Table 4 2 shows the local order of accuracy and GCI for temperature at R=0 with respect to Z locations (Z<1). It is observed that the local order of accuracy is almost 2 which is consistent with the discretization scheme used to model the study. Moreover, GCI2 1 and GCI32 are significantly small ; suggesting that : even results obtained by grids 161X401 are almost as accurate as extrapolated solution. Figure 4 7 and Figure 4 8 show s that all the grid sizes considered in the present study follow the extrapolated solution closely at location R=0 and R=0.7 Based on above analyses for conduction and convection case it is observed that results from 161X401 are sufficiently accurate and they can be considered converged with respect to grid size. Therefore, further results are based on a grid size of 161X401. PAGE 40 40 4 .3 Results Present study cover s a range of Ra from 0 to 100 for the fluid Pr of 1 ,5 and 10 The domain size, heater position a nd grid layout was as explained in the preceding text. Objective was to develop a correlation for the temperature field over microheater. 4.3.1 Effects of Pr and Ra on Steady S tate I sotherm and S treamlines As explained in Section 4.1.2, it wa s only to show the similarity with the previous reported literature s that the isotherms and streamlines pattern in the whole domain we re presented Even though the numerical values outside 1X1 domain cannot be relied upon, yet observing the whole domain gives us an idea about th e nature of flow. Figure 4 9 Figure 4 10 and Figure 4 11 shows s teady state isotherms and streamlines plotted for Pr=1 and Pr=10 with Ra=1X10 8 ,28,50,7 5 and 100 Stream lines were plotted as ratio max It was observed that the flow pattern formed a rolling max at the center. Table 4 3 shows the variation in value and max with respect to Ra Circulation of fluid became stronger as max increased with Ra It was observed that with an increase in Ra the vortex center gradually moves up with the radial position remaining almost same. Although there is only subtle difference between streamline pattern for conducti o n and R a =100, but a numerical observation showed that the streamlines near the centerline, top of domain and right hand side of domain moved closer to each other. Reason of this observation was higher velocities in these re gions as Ra was increased. When the fluid gets heated it formed a circulation pattern over heater. A rising column of hot fluid forms over the heater (near centerline). When this rising column encounters the boundary at infinity, the warm layer of fluid t urns along the top boundary in form of a wall jet to flow radially. This moving fluid wa s once again turned by the wall PAGE 41 41 at R=R Z = Z The momentum of descending fluid maintains it s motion despite of the retarding effect of buoyancy As the fluid slows dow n it is entrained towards the heater once again and the height to which it rises is decided by it residual buoyancy. For Pr=10, t (Z, R=0) was observed for different Ra. Figure 4 12 s hows that for a given Pr ; the temperature profile becomes less steep. with an increase in Ra. Upon increasing Ra convection increases and the fluid is moving upwards away from the heater wall making the temperature profile less steep. This phenomenon at centerline canno t be described by any conventional boundary layer theory, suggesting the absence of boundary layer behavior near centerline. To see the influence of Pr when Ra is fixed; temperature profile at different radial locations were plotted for Pr=1 and 10 with Ra =100 From Figure 4 13 it is observed that temperature gradient in Z direction is steeper for the case of lower Pr i.e. 1 for R<0.3 whereas a steeper temperature gradient occurs for higher Pr i.e. 10 for R>0.3. For region 0.3 PAGE 42 42 this result in E q. ( 3 8 ) gives Therefore, coefficient of Z 2 is zero in Eq ( 4 29 ) or ( 4 30 ) From the finite difference results a ( Z, R, Ra) of the form of Eq ( 4 29 ) or ( 4 30 ) was developed Following procedure was implemented to develop the correlation. For each Pr=1,5 and 10; data was generated for Ra=0,14,28,36,50,63,75,90,100 Fo r a given Pr: 1. For any particular Ra ; ifference was fitted by least square regression using MATLAB 2016a, to get an equation of form: ( ) =1+ a 1 + a 3 3 + a 4 4 This process was done for various radial locations within the heater. 2. The same proce ss is repeated for Ra=0,14,28,36,50,63,75,90,100. Thus we get coefficient a 1 a 2 a 3 for a combination of different R and Ra as shown in Table 4 4 for a Pr =1 3. For the data of form of Table 4 4 a 4 t h order multivariable regression was done in MATLAB u sing least square regression to develop correlation of form ( 4 29 ) or ( 4 30 ) The same process was repeated to get correlation for thermal field near t he heater for Pr=1,5 and 10. The correlation is valid for R<0.7 as beyond this region there is significant effect of strong singularity present at R=1. The adjusted R square value was almost 0.998 or greater for all the fits thereby suggesting a good fit. ( 4 29 ) ( 4 30 ) PAGE 43 43 Equations ( 4 31 ) ( 4 32 ) and ( 4 33 ) are the correlation for thermal field near heater ( ) for Pr of 1,5 and 10 valid for Ra range of 0 to 100. ( 4 31 ) ( 4 32 ) ( 4 33 ) 4.3.3 Verification of Correlation technique was used: For Pr=10, any random Ra=55 (say) was taken and given by finite difference was compared to Z) obtained by correlation at R= 0.4, 0.5,0.6 and 0.7 Fi gure 4 14 shows the comparison. From Eq ( 4 29 ) and ( 4 30 ) it is observe d that wall heat flux is the coefficient b 1 itself. Thus in Table 4 5 we compare the coefficient b 1 with the wall heat PAGE 44 44 flux obtained by finite difference, for various values of R for the case of Pr=10, Ra=28. Both Fi gure 4 14 and Table 4 5 s hows the that the temperature obtained by correlation follows the finite difference temperature results closely. PAGE 45 45 (a) (b) PAGE 46 46 ( c ) Figure 4 1 Comparison between exact solution of temperature and finite difference results (domain size R =Z =10,50 100) for pure conduction case (a) R=0 and 0.99 (b) R=0.6 ( c ) R=0.31 and 0.80 Figure 4 2 .Isotherms over heater for various domain size (a) (b) (c) PAGE 47 47 Figure 4 3 Streamline pattern in experimental study by Torrence and Orloff [29] (Left) and present study by numerical modelling (Right) situation for Ra=560 Pr=0.7 PAGE 48 48 Figure 4 4 Comparison of steady state isotherms between Torrance et al. [28] and present work for Pr=0.7 (a) & (b) for c a se of conduction (Ra~0) (c) & (d) Ra=28 PAGE 49 49 Figure 4 5 Heat flux and vorticity residual plot for Pr=10 with Ra=28 with initial condition Figure 4 6 Comparison between exact solution of temperature and finite difference results for grid sizes 81X201, 161X401 and 321X801 for pure conduction case PAGE 50 50 Figure 4 7 Comparison between extrapolated solution of temperature and finite difference results for grid sizes 81X201,161X401 and 321X801 at R=0 for Ra=100, Pr =1 Figure 4 8 Comparison between extrapolated solution of temperature and finite difference results for grid sizes 81X201,161X401 and 321X801 at R=0.7 for Ra=100, Pr=1 PAGE 51 51 Figure 4 9 Steady state isotherms for Pr=1 with (a)Ra= 1X10 8 (conduction) (b)Ra=28 (c) Ra=50 (d) Ra=75 (e) Ra=100 PAGE 52 52 Figure 4 10 Steady state isotherms for Pr=10 with (a)Ra= 1X10 8 (conduction) ( b)Ra=28 (c) Ra=50 (d) Ra=75 (e) Ra=100 PAGE 53 53 Figure 4 11 Steady state streamlines for Pr=1 0 with (a)Ra= 1X10 8 (conduction) (b)Ra=28 (c) Ra=50 (d) Ra=75 (e) Ra=100 PAGE 54 54 Figure 4 12 .Temperature profile (Z, R=0) at centerline for different values of Ra at Pr =10 Figure 4 13 Variation of temperature in axial direction at various R location s for Ra=100 with Pr=1 and 10 PAGE 55 55 Fi gure 4 14 Comparison of temperature obtained by correlation and finite difference for Ra=55, Pr=10 at various radial locations PAGE 56 56 Table 4 1 Grid convergence and local order of accuracy for R=1, Z= 0.026471 with Ra= 100, Pr=1 Grids V orticity S tream function T heta U V 321X801 8.841803 0.003635 0.466303 0.264470 0.002424 161X401 8.840194 0.003636 0.466320 0.264428 0.002424 81X201 8.833707 0.003637 0.466393 0.2 64241 0.002425 Local order of accuracy 2.011734 1.539540 2.064130 2.157078 2.588845 Extrapolated Value 8.842333 0.003635 0.466297 0.264482 0.002424 % deviation of 16 1X 4 01 grid value from extrapolated value 0.024192 0.022781 0.004915 0.020392 0.007348 Table 4 2 Comparison of finite diffe rence results for various grid size at R=0, with the extrapolated value for Ra=100, Pr=1 Z (f 3 )81X201 grids (f 2 )161X401 grids (f 1 )321X401 grids p e21 e32 GCI21 GCI32 0.00 000 1.00000 1.00000 1.00000 2.0000 0.00000 0.00000 0 0 0.00866 0.99234 0.99239 0.99240 2.30093 0.00001 0.00005 0.00000 0.00002 0.01748 0.98454 0.98464 0.98466 2.30058 0.00002 0.00010 0.00001 0.00003 0.02647 0.97659 0.97674 0.97678 2.30012 0.00003 0.000 16 0.00001 0.00005 0.03564 0.96849 0.96870 0.96875 2.29956 0.00004 0.00022 0.00001 0.00007 0.04499 0.96025 0.96052 0.96057 2.29890 0.00006 0.00028 0.00002 0.00009 0.05453 0.95187 0.95219 0.95226 2.29816 0.00007 0.00034 0.00002 0.00011 0.06426 0.94333 0 .94371 0.94379 2.29735 0.00008 0.00041 0.00003 0.00013 0.07419 0.93465 0.93509 0.93518 2.29647 0.00010 0.00047 0.00003 0.00015 0.08432 0.92583 0.92633 0.92643 2.29554 0.00011 0.00054 0.00004 0.00017 0.09466 0.91685 0.91742 0.91753 2.29458 0.00013 0.0006 2 0.00004 0.00020 0.10522 0.90773 0.90836 0.90849 2.29360 0.00014 0.00069 0.00005 0.00022 0.11600 0.89847 0.89916 0.89931 2.29263 0.00016 0.00077 0.00005 0.00025 0.12702 0.88906 0.88982 0.88998 2.29168 0.00018 0.00086 0.00006 0.00028 0.13827 0.87950 0. 88034 0.88051 2.29076 0.00019 0.00095 0.00006 0.00030 0.14976 0.86981 0.87071 0.87089 2.28991 0.00021 0.00104 0.00007 0.00033 0.16151 0.85997 0.86094 0.86114 2.28914 0.00023 0.00113 0.00007 0.00036 0.17352 0.84999 0.85104 0.85125 2.28848 0.00025 0.00123 0.00008 0.00040 0.18580 0.83987 0.84099 0.84122 2.28792 0.00027 0.00133 0.00009 0.00043 0.19835 0.82962 0.83082 0.83106 2.28751 0.00030 0.00144 0.00010 0.00046 0.21120 0.81923 0.82050 0.82077 2.28725 0.00032 0.00155 0.00010 0.00050 0.22434 0.80871 0.8 1006 0.81034 2.28716 0.00034 0.00167 0.00011 0.00054 0.23779 0.79806 0.79949 0.79978 2.28725 0.00037 0.00179 0.00012 0.00058 PAGE 57 57 0.25156 0.78729 0.78879 0.78910 2.28754 0.00039 0.00191 0.00013 0.00062 0.26566 0.77639 0.77798 0.77830 2.28805 0.00042 0.00204 0.00013 0.00066 0.28010 0.76537 0.76704 0.76738 2.28877 0.00044 0.00217 0.00014 0.00070 0.29489 0.75424 0.75598 0.75634 2.28973 0.00047 0.00231 0.00015 0.00074 0.31005 0.74299 0.74482 0.74519 2.29093 0.00050 0.00245 0.00016 0.00079 0.32558 0.73164 0.73 354 0.73393 2.29237 0.00053 0.00260 0.00017 0.00083 0.34151 0.72018 0.72217 0.72257 2.29407 0.00056 0.00274 0.00018 0.00088 0.35785 0.70863 0.71069 0.71111 2.29603 0.00059 0.00290 0.00019 0.00093 0.37461 0.69699 0.69912 0.69956 2.29825 0.00062 0.00305 0 .00020 0.00097 0.39180 0.68526 0.68746 0.68791 2.30073 0.00065 0.00321 0.00021 0.00102 0.40946 0.67344 0.67572 0.67619 2.30348 0.00068 0.00337 0.00022 0.00107 0.42758 0.66156 0.66391 0.66438 2.30649 0.00072 0.00354 0.00023 0.00112 0.44619 0.64960 0.652 02 0.65251 2.30977 0.00075 0.00370 0.00024 0.00117 0.46532 0.63759 0.64007 0.64057 2.31331 0.00078 0.00387 0.00025 0.00122 0.48497 0.62552 0.62806 0.62857 2.31710 0.00081 0.00404 0.00025 0.00127 0.50518 0.61340 0.61600 0.61652 2.32114 0.00084 0.00421 0. 00026 0.00132 0.52596 0.60125 0.60390 0.60443 2.32543 0.00087 0.00438 0.00027 0.00137 0.54733 0.58906 0.59176 0.59230 2.32995 0.00091 0.00456 0.00028 0.00141 0.56933 0.57686 0.57960 0.58014 2.33469 0.00094 0.00473 0.00029 0.00146 0.59198 0.56463 0.5674 1 0.56796 2.33964 0.00097 0.00490 0.00030 0.00151 0.61530 0.55240 0.55522 0.55577 2.34480 0.00100 0.00507 0.00031 0.00155 0.63932 0.54018 0.54302 0.54358 2.35013 0.00103 0.00523 0.00031 0.00160 0.66408 0.52796 0.53083 0.53139 2.35563 0.00105 0.00540 0.0 0032 0.00164 0.68961 0.51577 0.51865 0.51921 2.36127 0.00108 0.00556 0.00033 0.00168 0.71593 0.50360 0.50650 0.50706 2.36704 0.00111 0.00572 0.00033 0.00172 0.74309 0.49147 0.49438 0.49494 2.37290 0.00113 0.00588 0.00034 0.00176 0.77112 0.47939 0.48230 0.48286 2.37884 0.00116 0.00603 0.00034 0.00179 0.80006 0.46737 0.47027 0.47083 2.38483 0.00118 0.00617 0.00035 0.00183 0.82996 0.45541 0.45831 0.45886 2.39083 0.00120 0.00631 0.00035 0.00186 0.86085 0.44353 0.44641 0.44696 2.39681 0.00122 0.00645 0.00 036 0.00189 0.89279 0.43173 0.43459 0.43513 2.40274 0.00124 0.00658 0.00036 0.00192 0.92582 0.42002 0.42286 0.42340 2.40858 0.00126 0.00671 0.00037 0.00195 0.96000 0.40842 0.41123 0.41175 2.41430 0.00128 0.00683 0.00037 0.00197 0.99537 0.39693 0.39970 0.40022 2.41986 0.00130 0.00694 0.00037 0.00199 1.03200 0.38555 0.38829 0.38879 2.42521 0.00131 0.00705 0.00038 0.00202 PAGE 58 58 Table 4 3 for Pr=10 Ra Z R 0 4.2 04 4.938 1.497E 05 28 4.894 4.938 7.796 50 5.169 4.938 13.637 75 5.338 4.938 18.510 100 5.443 4.938 22.316 Table 4 4 Polynomial coefficients in d eveloping correlation for Pr=1 a 1 a 3 a 4 R Ra 0.99960 0.5 4471 0.25315 0.00000 0 1.00231 0.55103 0.25730 0.06278 0 1.01042 0.57001 0.26975 0.12269 0 1.02334 0.60072 0.29002 0.17983 0 1.04072 0.64302 0.31820 0.23425 0 1.06234 0.69717 0.35467 0.28604 0 1.08804 0.76370 0.40000 0.33525 0 1.11773 0.84333 0.45495 0.38198 0 1.15134 0.93689 0.52036 0.42627 0 1.18881 1.04527 0.59710 0.46822 0 1.23010 1.16932 0.68606 0.50790 0 1.27515 1.30985 0.78807 0.54536 0 1.32385 1.46751 0.90384 0.58070 0 1.37610 1.64276 1.03396 0.61397 0 1.431 75 1.83583 1.17879 0.64525 0 1.49058 2.04666 1.33847 0.67462 0 1.55237 2.27486 1.51285 0.70214 0 0.96154 0.54481 0.26156 0.00000 14 0.96455 0.55142 0.26584 0.06278 14 0.97359 0.57128 0.27866 0.12269 14 0.98795 0.60340 0.29956 0.17983 14 1.00721 0.64763 0.32862 0.23425 14 1.03108 0.70421 0.36624 0.28604 14 1.05933 0.77368 0.41305 0.33525 14 1.09183 0.85675 0.46980 0.38198 14 1.12844 0.95425 0.53734 0.42627 14 1.16905 1.06702 0.61657 0.46822 14 1.21358 1.19591 0.70836 0 .50790 14 1.26191 1.34167 0.81352 0.54536 14 1.31392 1.50489 0.93276 0.58070 14 PAGE 59 59 1.36944 1.68598 1.06659 0.61397 14 1.42830 1.88509 1.21535 0.64525 14 1.49026 2.10206 1.37911 0.67462 14 1.55505 2.33644 1.55766 0.70214 14 0.93584 0.54792 0.27001 0.00000 28 0.93916 0.55481 0.27440 0.06278 28 0.94914 0.57550 0.28757 0.12269 28 0.96495 0.60896 0.30904 0.17983 28 0.98611 0.65503 0.33893 0.23425 28 1.01223 0.71395 0.37765 0.28604 28 1.04305 0.78625 0.42586 0.33525 28 1.0783 3 0.87264 0.48434 0.38198 28 1.11791 0.97393 0.55395 0.42627 28 1.16162 1.09096 0.63559 0.46822 28 1.20932 1.22453 0.73014 0.50790 28 1.26085 1.37535 0.83838 0.54536 28 1.31604 1.54396 0.96099 0.58070 28 1.37471 1.73068 1.09845 0.61397 2 8 1.43663 1.93559 1.25103 0.64525 28 1.50155 2.15846 1.41875 0.67462 28 1.56916 2.39873 1.60134 0.70214 28 0.92438 0.54968 0.27424 0.00000 36 0.92787 0.55672 0.27869 0.06278 36 0.93835 0.57785 0.29205 0.12269 36 0.95495 0.61203 0.31383 0.17983 36 0.97713 0.65909 0.34416 0.23425 36 1.00446 0.71926 0.38348 0.28604 36 1.03665 0.79308 0.43244 0.33525 36 1.07342 0.88126 0.49185 0.38198 36 1.11458 0.98459 0.56259 0.42627 36 1.15993 1.10392 0.64554 0.46822 36 1.20929 1.2400 1 0.74159 0.50790 36 1.26250 1.39354 0.85151 0.54536 36 1.31936 1.56504 0.97595 0.58070 36 1.37966 1.75478 1.11537 0.61397 36 1.44317 1.96279 1.27002 0.64525 36 1.50961 2.18879 1.43986 0.67462 36 1.57867 2.43220 1.62461 0.70214 36 0.90 765 0.55277 0.28093 0.00000 50 PAGE 60 60 0.91142 0.56006 0.28548 0.06278 50 0.92274 0.58193 0.29913 0.12269 50 0.94065 0.61730 0.32142 0.17983 50 0.96452 0.66599 0.35248 0.23425 50 0.99387 0.72825 0.39279 0.28604 50 1.02832 0.80460 0.44303 0.33525 50 1.06755 0.89575 0.50401 0.38198 50 1.11129 1.00249 0.57663 0.42627 50 1.15932 1.12563 0.66180 0.46822 50 1.21141 1.26591 0.76036 0.50790 50 1.26735 1.42397 0.87309 0.54536 50 1.32691 1.60025 1.00059 0.58070 50 1.38985 1.79499 1.143 29 0.61397 50 1.45590 2.00812 1.30138 0.64525 50 1.52478 2.23930 1.47477 0.67462 50 1.59615 2.48785 1.66311 0.70214 50 0.89486 0.55565 0.28649 0.00000 63 0.89888 0.56315 0.29112 0.06278 63 0.91095 0.58566 0.30504 0.12269 63 0.93000 0.62 208 0.32777 0.17983 63 0.95535 0.67220 0.35948 0.23425 63 0.98646 0.73629 0.40066 0.28604 63 1.02288 0.81486 0.45202 0.33525 63 1.06425 0.90863 0.51440 0.38198 63 1.11024 1.01836 0.58870 0.42627 63 1.16058 1.14486 0.67582 0.46822 63 1. 21501 1.28882 0.77662 0.50790 63 1.27328 1.45085 0.89184 0.54536 63 1.33514 1.63133 1.02205 0.58070 63 1.40031 1.83042 1.16765 0.61397 63 1.46851 2.04801 1.32876 0.64525 63 1.53943 2.28369 1.50527 0.67462 63 1.61273 2.53670 1.69674 0.702 14 63 0.88480 0.55827 0.29117 0.00000 75 0.88904 0.56596 0.29589 0.06278 75 0.90175 0.58903 0.31003 0.12269 75 0.92181 0.62636 0.33316 0.17983 75 0.94847 0.67775 0.36544 0.23425 75 0.98112 0.74345 0.40739 0.28604 75 PAGE 61 61 1.01926 0.82398 0.4 5975 0.33525 75 1.06249 0.92006 0.52337 0.38198 75 1.11043 1.03244 0.59916 0.42627 75 1.16278 1.16190 0.68803 0.46822 75 1.21924 1.30911 0.79081 0.50790 75 1.27952 1.47463 0.90824 0.54536 75 1.34334 1.65879 1.04086 0.58070 75 1.41042 1. 86171 1.18901 0.61397 75 1.48045 2.08321 1.35280 0.64525 75 1.55310 2.32281 1.53204 0.67462 75 1.62803 2.57969 1.72627 0.70214 75 0.87400 0.56148 0.29654 0.00000 90 0.87849 0.56939 0.30135 0.06278 90 0.89198 0.59313 0.31577 0.12269 90 0.91325 0.63155 0.33936 0.17983 90 0.94146 0.68445 0.37232 0.23425 90 0.97594 0.75207 0.41520 0.28604 90 1.01613 0.83495 0.46876 0.33525 90 1.06155 0.93379 0.53388 0.38198 90 1.11180 1.04934 0.61147 0.42627 90 1.16650 1.18234 0.70245 0.4 6822 90 1.22532 1.33344 0.80763 0.50790 90 1.28794 1.50312 0.92772 0.54536 90 1.35405 1.69167 1.06322 0.58070 90 1.42334 1.89914 1.21444 0.61397 90 1.49548 2.12527 1.38142 0.64525 90 1.57012 2.36950 1.56392 0.67462 90 1.64690 2.63095 1 .76143 0.70214 90 0.86766 0.56356 0.29986 0.00000 100 0.87233 0.57162 0.30472 0.06278 100 0.88632 0.59578 0.31932 0.12269 100 0.90835 0.63490 0.34321 0.17983 100 0.93755 0.68876 0.37661 0.23425 100 0.97320 0.75762 0.42010 0.28604 100 1. 01469 0.84201 0.47444 0.33525 100 1.06151 0.94263 0.54052 0.38198 100 1.11321 1.06022 0.61928 0.42627 100 1.16940 1.19549 0.71162 0.46822 100 1.22971 1.34908 0.81835 0.50790 100 PAGE 62 62 1.29380 1.52143 0.94015 0.54536 100 1.36135 1.71279 1.07752 0.58070 100 1.43201 1.92316 1.23071 0.61397 100 1.50546 2.15223 1.39974 0.64525 100 1.58133 2.39941 1.58433 0.67462 100 1.65925 2.66375 1.78392 0.70214 100 PAGE 63 63 Table 4 5 Comparison between heat fl ux given by correlation and finite difference for the case of Ra=28, Pr=10 R b 1 Q % Difference 0.00000 0.93560 0.93966 0.43404 0.04969 0.93792 0.93705 0.09286 0.11180 0.94745 0.94600 0.15259 0.17391 0.96466 0.96506 0.04180 0.24845 0.99641 0.99859 0.21 906 0.33540 1.05122 1.05268 0.13877 0.42236 1.13021 1.12910 0.09852 0.53416 1.28302 1.28204 0.07675 PAGE 64 64 5 CHAPTER 5 5 CONCLUSIONS Steady state, laminar, external natural convection flow over an isothermal, circular microheater is studied for Rayleigh numbe r range up to 100 at Prandtl number of 1,5 and 10 Finite difference scheme is used to model the fluid flow over heater. Influence of domain size is observed on the flow field near the heater for pure conduction case. Grid dependence studies for the case o f pure conduction and moderate convective flows are conducted by extrapolating the finite difference results using Richardson extrapolation. Results obtained by finite difference are fitted using least square regression fit to obtain a correlati on for temp erature field near heater. Three 4 th order unified polynomial s are developed to describe dimensionless temperature field near heater ; each for the case of Pr=1,5 and 10. This temperature (Z, R, Ra) for any given Pr; where Ra 00 an d The correlation obtained i s checked for Ra=55 Pr=10 with the finite difference solutions of temperature and it was also checked against wall heat flux obtained by finite difference solution for Ra=28, Pr=10. Results obtained by correlatio n shows good agreement with finite difference results. 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Roache, A Method for Uniform Reporting of Grid Refinement Studies," J. Fluids Eng, vol. 116, no. 3, pp. 405 413, 1994. PAGE 68 68 BIOGRAPHICAL SKETCH Shantanu Singh was born in Lucknow, India in 1990. He obtained his Bachelor of Technology in M echanical E ngineering from Indian Institute of Technology, (BHU) Varanasi (IIT Varanasi(BHU) ), India in May 2013. Thereafter, he worked in Bharat Petroleum Corporation Limited (BPCL ), India ; as Research & Development engineer till August 2015. He joined the graduate program at the University of Florida in August 2015. He has been working under Dr. Renwei Mei in the Department of Mechanical and Aerospace Engineering to obtain his Master of Science degree His current research interest inclu des computational fluid dynam ics and scientific computing. |