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Optimization for Maximum Heat Transfer across an Enclosure Filled with a Fluid

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Title:
Optimization for Maximum Heat Transfer across an Enclosure Filled with a Fluid
Creator:
Huang, Miao
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
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english
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1 online resource (36 p.)

Thesis/Dissertation Information

Degree:
Master's ( M.S.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mechanical Engineering
Mechanical and Aerospace Engineering
Committee Chair:
SHERIF,SHERIF AHMED
Committee Co-Chair:
CHUNG,JACOB NAN-CHU
Committee Members:
MIREHEI,SEYED MOUSSA

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Subjects / Keywords:
cooling -- fin -- genetic -- optimization
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
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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:
The problem of cooling of electronics has been extensively studied in the last few decades. Recently, studies considering heterogeneous enclosures filled with a fluid and solid bodies heated from the side, in which an isothermal hot wall at a higher temperature, an isothermal cold wall at a lower temperature; and, adiabatic horizontal surfaces were considered, were reported. Although significant progress has been made in understanding the processes involved in heating and cooling of such containers, the effect of extended surfaces (fins) and their profile on heat transfer from the containers has not been considered. In this paper, the effect of an extended hot wall surface on the cooling process is studied; and, the fin profile with maximum cooling effect is determined using the Genetic Algorithm. Left and right walls of the enclosure are considered isothermal and at T_h andT_c (T_h>T_c), respectively, while the top and bottom surfaces are considered adiabatic. The porosity (fluid volume fraction) of the enclosure is set at 0.84; and, Rayleigh and Prandtl numbers are set at 107 and 1, respectively. Optimization is performed for fins with polynomial profiles, for polynomial degrees zero and one. The heat transfer process is monitored in terms of total heat flux at the cold surface. Results are reported in terms of fin profiles; with, pertaining total heat flux; and, instantaneous isotherm and streamline distributions. Findings elucidate the effect of changes in the fin profile on heating and cooling of containers by natural convection. ( en )
General Note:
In the series University of Florida Digital Collections.
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Includes vita.
Bibliography:
Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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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: SHERIF,SHERIF AHMED.
Local:
Co-adviser: CHUNG,JACOB NAN-CHU.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2018-06-30
Statement of Responsibility:
by Miao Huang.

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Embargo Date:
6/30/2018
Classification:
LD1780 2017 ( lcc )

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FIN PROFILE OPTIMIZATION FOR MAXIMUM HEAT TRANSFER ACROSS AN ENCLOSURE FILLED WITH A FLUID By MIAO HUANG A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2017

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2017 Miao Huang

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To my parents

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4 ACKNOWLEDGMENTS Dr. S. Moussa Mirehei advised this research. I would like to thank him, for that and serving on the thesis committee. I would also like to thank Dr. S.A. Sherif for serving as the committee chair, and Dr. Chung for serving on the thesis co mmittee. I thank Yiqing Chen for helping me.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF FIGURES ................................ ................................ ................................ .......... 6 NOMENCLATURE ................................ ................................ ................................ .......... 7 ABSTRACT ................................ ................................ ................................ ................... 10 C HAPTER 1 INTRODUCTION AND LITERATURE REVIEW ................................ ..................... 12 Methods of Analysis for Enclosure with a Solid and a Fluid ................................ .... 12 Effects from Geometries and Properties of Fins on Heat Transfer Performance .... 13 Use of Genetic Algorithms in Heat Transfer ................................ ............................ 14 Effect of Local Convective Heat Transfer Coefficient on Accuracy ......................... 15 2 MATHMATICAL MODELING AND NUMERICAL SIMULATION ............................. 17 Problem Formulation ................................ ................................ ............................... 17 Mathematical Model ................................ ................................ ................................ 17 Numerical Simulation ................................ ................................ .............................. 21 3 RESULT S AND DISCUSS I O NS ................................ ................................ ............. 25 Sensitivity Analysis of and with Total Heat Flux ................................ ........... 25 Effect of on non dimensional heat flux at right wall ................................ ..... 25 Effect of on non dimensional heat flux at right wall ................................ ..... 26 Optimal Profile and Position of Rectan gular Fin and Optimized Heat Flux at the Right Wall ................................ ................................ ................................ ............ 26 Analysis of Isotherm Contour and Streamline Contour ................................ ........... 27 Mesh test ................................ ................................ ................................ ................ 28 4 CONCLUSIONS AND RECOMMENDATIONS ................................ ....................... 31 Conclusions ................................ ................................ ................................ ............ 31 Recommendations ................................ ................................ ................................ .. 31 LIST OF REFERENCES ................................ ................................ ............................... 33 BIOG RAPHICAL SKETCH ................................ ................................ ............................ 36

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6 L IST OF FIGURES Figure page 2 1 Schematic of the enclosure ................................ ................................ ................ 24 3 1 Effect of on the non dimensional heat flux at right wall at , , ................................ ................................ 28 3 2 Effect of on the non dimensional heat flux at right wall at , , ................................ ................................ 29 3 3 Shots of isotherm distributions of enclosure for optimized rectangular fin, steady state, , ................................ ...................... 29 3 4 Shots of streamlines of enclosure for optimized rectangular fin, steady state, , ................................ ................................ ........... 30

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7 NOMENCLATURE fluid specific heat FLUENT fluid specific heat solid specific heat FLUENT solid specific heat gravitational acceleration ( FLUENT gravitational acceleration ( heat transfer coefficient e nclosure height fluid thermal conductivity FLUENT fluid thermal conductivity s olid thermal conductivity FLUENT solid thermal conductivity e nclosure width normal direction non dimensional coordinate normal direction coordinate pressure n on dimensional pressure dimensional distance from top of the fin to top of the enclosure ( non dimensional distance from top of the fin to top of the enclosure dimensional distance from top of the fin to bottom of the fin ( non dimensional distance from top of the fin to bottom of the fin Prandtl number

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8 heat flow of fluid heat flow of solid Rayleigh number enclosure aspect ratio temperature of fluid temperature of solid left wall temperature right wall temperature horizontal velocity component n on dimensional horizontal velocity component vertical velocity component n on dimensional vertical velocity component horizontal coordinate n on dimensional horizontal coordinate vertical coordinate n on dimensional vertical coordinate fluid thermal diffusivity solid thermal diffusivity isobaric coefficient of volumetric thermal expansion FLUENT isobaric coefficient of volumetric thermal expansion N on dimensional fluid temperature N on dimensional solid temperature solid to fluid thermal conductivity ratio

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9 FLUENT dynamic viscosity dynamic viscosity kinematic viscosity fluid density FLUENT fluid density solid density FLUENT solid density solid to fluid volumetric heat capacity ratio

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10 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Master of Science FIN PROFILE OPTIMIZATION FOR MAXIMUM HEAT TRANSFER ACROSS AN ENCLOSURE FILLED WITH A FLUID By Miao Huang December 2017 Chair: S.A. Sherif Major: Mechanical Engineering The problem of cooling of electronics has been extensively studied in the last few decades. Recently, studies considering heterogeneous enclosures filled with a fluid and solid bodies heated from the side, in which an isothermal hot wall at a higher temperature, an isothermal cold wall at a lower temperature and adiabatic horizontal surfaces were considered, were reported. Although significant progress has been made in understanding the proc esses involved in heating and cooling of such containers, the effect of extended surfaces (fins) and their profile on heat transfer from the containers has not been considered. In this thesis the effect of an extended hot wall surface on the cooling proce ss is studied and the fin profile with maximum cooling effect is determined using g enetic a lgorithm s The l eft and right walls of the enclosure are consi dered isothermal and at and ( ), respectively, while the top and bottom surfaces are considered adiabatic. The porosity (fluid volume fraction) of the enclosure is set at 0.84 and the Rayleigh and Prandtl numbers are set at and 1, respectively. Optimization is performed for fins with rectangular profile The heat transfer process is m onitored in terms of total heat flux at the cold surface. Results are reported in terms of

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1 1 fin profiles with the associated total heat flux and instantaneous isotherm s and streamline s Findings elucidate the effect of changes in the fin profile on heating and cooling of containers by natural convection.

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12 CHAPTER 1 INTRODUCTION AND LITERATURE REVIEW Methods of Analysis for Enclosure with a Solid and a Fluid Cooling of solid parts distributed in an enclosure filled with fluid is relevant in a lot of engineering applications, especially cooling of electronics. Commonly the enclosure consists of a solid and a fluid. One way to calculate the heat transfer perform ance of such an enclosure is to consider it as a porous medium. The problem can be solved by 2]. In this approach, there are only conservation equations of a porous medium instead of separate conservation equations of th e fluid and the solid. These equations can be derived by averaging the conservation equations of mass of fluid and solid, applying the volume theorem [3], and applying the conservation of energy between the fluid and the solid. A r epresentative elementary volume need s to be determined in this case. However a representative elementary volume may become invalid when the solid portion is relatively large which would eventually make the porous continuum approach invalid. For th is reason another approach is chosen which uses separate conservation equations of the solid and fluid. At the interface of the fluid and solid an appropriate compatibility condition is applied. Although significant progress has been made in understanding the proc esses involved in heating and cooling of such containers and numerical simulations of heat transfer in enclosure s employing this approach can be accurate [ 4 13 ], the effect of extended surfaces (fins) and their profile on heat transfer from the containers has not been considered. Periodic horizontal heating of enclosed disconnected solid bodies saturated with a fluid has been investigated by Mirehei et al [14] and simulations and analyses of

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13 natural convection inside heterogeneous containers heated by sola r radiation has been reported b y Mirehei [ 15] Effects from G eometries and P roperties of F in s on Heat Transfer Performance Huang et al. [16] studied the dynamic characteristics of rectangular fin arrays. Gravity is taken into consideration and downward flow is prove n to have negative effects on fins because of the thicker boundary layer. Heat transfer performance wi ll increase with the fin length. Harahap et al. [17] pointed out that the orientation of the length of the fins parallel to the shorter side of the base plate w ould result in higher heat transfer efficiency than what would have been obtained using the longer side. Heat transfer efficiency drop s drastically if the fi n base area i s smaller than the optimal area Al Doori et al. [18] showed that perforations positively affect the heat transfer efficiency. Also, as the diameter and number of perforations increase the heat transfer performance w ould increase Dannelley [1 9] showed for most fractal geometries that they can enhance the heat transfer per unit mass without too much increase in the mass compared to traditional fin shapes. And for extended surface s certain fractal geometries can increase the heat transfer per un it mass and thus reduce the mass of the fin, e specially under natural convection conditions Azarkish et al. [20] studied the improvement of heat transfer employing a number of fin arrays and selecting the right fin geometry. R esults indicated that changin g the fin geometry does not result in any change in the number of optimal fin array s For one dimensional heat transfer, an optimized fin geometry is better than a non optimized or conventional fin geometry. d. If two dimensional heat transfer is considere d, the fin geometry starts to play a role Bobaru et al. [21] studied the relation between optimal fin shape s and the thermal conductivity of the material. High conductivity result s in a sharp pointed optimal

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14 geometry. Low conductivity material s will resul t in a blunt and wide base geometry Kang et al. [22] focus ed on trapezoidal fins. As the ratio of the convection characteristic number increases or the fin base radius decrease s the opti mal heat transfer rate will increase. Hamadneh et al. [23] studied different conventional fin shape s including square, rectangle, circle and elliptical fins. The ir results showed that a square is the worst shape while a rectangle is the best one from the standpoint of the rate of entropy generation. Use of Genetic Algori thm s i n Heat Transfer Genetic algorithm is a method to solve optimization problem s It can be applied in a multi objective or a single objective optimization situation This algorithm is similar to the natural selection process in nature It will generate a generation with random individuals for the model which need s to be optimized. A fitness function will be applied on each individual to evaluate how good the individual is. Then individuals in this generation may crossover or mutate by chance. These two behavior s will either produce new individuals or change the existing individuals. The next generation will be selected by choosing the best individuals from the most recent generation according to the value of their fitness function The crossover, mutation and select ion process will repeat for iterations until stopping conditions are met. As the mutation probability increase s, the optimization will be more random so low mutation probability is recommende d. Low crossover probability will make the gen eration more stable and will get the results faster but less accurate. High crossover probability (>0.9) is also recommended [30]. Common stopping conditions are the limits on the number of iteration s and /or stall

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15 generation. The limit on the number of i teration s refers to the stopping the algorithm after a certain number of iterations have been executed The s tall generation limit refers to the case when the fitness function value of the best individual in the current generation does not change by a sign ificant extent after a certain number of generations has been created This limit is called the convergence stability percentage in ANSYS and is chosen as the convergence criterion for this work Hajabdollahi et al. [24] used genetic algorithm s to optimiz e a one dimensional pin fin. They use d Bezier [24] curves to describe the fin profile. Total heat transfer and fin efficiency are the two objects. The ir results identified the optimum fin shapes and their heat transfer performance. Higher dimensional cases are not in volved. In this thesis two dimensional cases will be analyzed. Wang and Wang [25] optimize d the conical fin by divid ing it into finite elements and mak ing every element transfer the maximum heat flux. Fin effectiveness was not considered and the heat transfer rate was the only object. Results can be generated faster and better by a step calculation method than b y traditional genetic algorithm s Najafi et al. [26] also used genetic algorithm s to maximize the heat transfer performance while simu ltaneously minimiz ing the cost of material s A set of solutions is achieved by combining these two conflict ed objectives using multi objective genetic algorithm s Genetic algorithm s ha ve been prove n to be an effective and accurate way for such optimiz ation problems In the study upon which this thesis reports, a multi objective genetic algorithm is used. Effect o f Local C onvective Heat Transfer Coefficient o n Accuracy Fabbri [27] showed that fin profile optimization can be effective when the convective heat transfer coefficient is not too high. The l ocal convective heat transfer

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16 coefficient was not used instead a constant coefficient was used. Naturally, using th e local convection coefficient will make the result s more accurate Dialameh et al. [28] studied the relation between the convection heat transfer coefficient and several parameters in the cooling of fin arrays. They found that t he coefficient increase s wh en the temperature difference or fin spacing increase. They also found that i t decrease s as the fin length increase s a nd that the average convection heat transfer coefficient is not sensitive to fin height and thickness Th e work upon this thesis reports w ill take the local convective heat transfer coefficient into consideration.

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17 CHAPTER 2 MATHMATICAL MODELING AND NUMERICAL SIMULATION In this chapter, the problem is described then a mathematical model will be derived. The problem will be solved numerically using ANSYS FLUENT. Problem Formulation This is a steady state heat transfer optimization problem. Consider a two dimensional rectangular enclosure filled with a Newtonian fluid and having a fin on the left surface (Fig. 2 1). The l ength of the bottom wa ll is L and the length of left wall is H. To simplify the analysis, H and L will be assumed equal. Both the l eft and right wall s are assumed isothermal. The t emperature at the left wall is while that at the right wall is kept ( ) The t op an d bottom wall s of the enclosure are assumed adiabatic. The fin base which is also the left side of the fin is on the left wall of the enclosure. The t op and bottom side s of the fin are perpendicular to the fin base. We will only consider a rectangular fin in this work The goal is to determine the profile and position of the fin which can provide the maximum heat transfer through the enclosure. Gravity is taken into consideration in this natural convection process and the classical Boussinesq model is applied where the density of fluid in the buoyancy term of the mom entum equations changes [29] For a rectangular fin, the first dimensional parameter is the distance from the top of the fin to the top of the enclosure. The second dimensional parameter is the distance from top of the fin to bottom of the fin. Mathematical Model The d imensionless governing equations and boundary conditions will be derived in this section based on Mirehei s [15] work. This model is a steady state heat transfer

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18 proble m so all terms in the equations are time independent. The model will not be solved analytically but instead solved numerically for the convenience of the optimization part that follows In this steady state problem, all time dependent derivative terms are all 0. Conservation equation of mass of fluid : ( 2 1) Conservation of momentum of fluid in the x direction: (2 2 ) Conservation of momentum of fluid in the y direction Use for density in the buoyancy term based on the Boussinesq approximation [29] After rearranging with ( 2 3) Conservation of energy of the fluid: ( 2 4) Conservation of energy of the solid: ( 2 5) Boundary conditions: Non slip condition at solid surface: ( 2 6) Left and right wall:

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19 at ( 2 7) at ( 2 8) Top and bottom wall: at ( 2 9) Compa tibility condition at the solid fluid interface: ( 2 10) ( 2 11) ( 2 12) Definitions of non dimensional variables are listed below: ( 2 13) Definitions of non dimensional parameters:

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20 ( 2 14) The non dimensional conservation equations are: Conservation equation of mass: ( 2 15) Conservation o f momentum in the x direction: ( 2 16) Conservation of momentum in the y direction : ( 2 17) Conservation of energy of the fluid: ( 2 18 ) Conservation of energy of th e solid : ( 2 19) Non dimensional boundary conditions are: ( 2 20) ( 2 21) ( 2 22) ( 2 23) Non dimensional compatibility conditions: ( 2 24) ( 2 25)

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21 ( 2 26) Numerical Simulation From ANSYS FLUENT Theory Guide we can find FLUENT use conservation equations below in simulation [ 14 ]: ( 2 27 ) ( 2 28 ) ( 2 29 ) ( 2 30 ) ( 2 31 ) Notice because of steady state. The coefficients and constants needed for using ANSYS FLUENT to numerically solve non dimensional conservation equations (E quations 2 15 to 2 19 and Equations 2 27 to 2 31) were obtained by comparing the equivalent terms in the dimensional and non dimensional conservation equations. The values and relations, that were used, are as follows: is set to 1, is set to is set to is set to is set to is set to is set to is set to Units of all coefficients are not changed. All values assigned are c hosen arbitrarily in this optimization. When applying this optimization to certain cases which

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22 include certain materials and certain gravity, the only change that need s to be made in these assigned values are and In the comparison betwee n the y momentum conservation Equation ( 2 17 ) and Equation ( 2 19 ) plays the same role of as the total pressure. The value 0 is chosen for to make the comparison easier since will not affect the problem which is in steady state [15, 31] The value of gravity is positive when the direction of gravity is upward which is the same direction of the y axis. If the direction of gravity is downward, the value of gravity is negative. In the numerical si mulation the followin g parameters are set as b elow: , The d imensionless temperature is set as follows: To avoid ANSYS error when is set to 0 at the boundary because the unit of temp erature is Ke lvin, we keep the difference of between and Solid area fraction of the enclosure is 16%. Mesh elements are uniform and mesh size is set to 0.004. The algorithm is chosen as the pressure velocity coupling scheme [15] The a verage heat flu x at the right wall is defined as: ( 2 32) The c orresponding non dimensional average heat flux at the right wall is defined as: ( 2 33)

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23 The value of non dimensional heat flux at the right wall is equal to the non dimensional average heat flux at the right wall in this case because is assumed. ( 2 34) In the optimization section, multiple objectives genetic algorithm (MOGA) is chosen as the optimization method. The n umber of initial samples is 100. The n umber of samples p er iteration is set to 100. The number of samples in each iteration will affect the optimization process and sometimes determine s if th e optimization will converge. If the number per iteration is too small there will not be enough chromosome s for achieving the final results. Sometimes it may still get the right results but most likely the optimization w ould converge to a local maximum due to the lack of good samples. However, the process will be slower as the number per iteration increases. It will take longer time for one generation to produce the next generation. The p roper number of samples per iteration should be pre selected A value of 100 is recommended by FLUENT The c onvergence S tability P ercentage is set to 2%. The Convergence Stabil ity Percentage criterion looks for population stability, based on the mean and standard deviation of the output parameters. When a population is stable with regards to the previous one, the optimization is converged. If the Convergence S tability P ercentage is set too high, the samples will be considered converged too fast when they still have some potential to evolve. If the Convergence S tability P ercentage is set lower, the optimization result s will be more accurate and at the same time will take more time If it is set to 0, all samples will be the same in the last iteration. A value of 0 is the best value for the Convergence S tability P ercentage but 2% is recommended for saving time and several candidate s for samples will be selected. The Mutation probabi lity is set

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24 to 1%. The Crossover probability is set 98%. Mutation is the reason the optimization process can keep moving. Some good genes that do not exist now can only be created by mutation. As the mutation probability increase s the optimization will be more random Therefore, low mutation probability is recommended but it cannot be set to 0 Low crossover probability will make the generation more stable and will get the results faster but less accurat e. High crossover probability (>0.9) is also recommend ed [30]. For a rectangular fin, the first parameter is the non dimensional distance from top of the fin to top of the enclosure. The second parameter is the non dimensional distance from top of the fin to bottom of the fin. To avoid incompatible errors in ANSYS FLUENT, the upper limit of both parameters is set to 0.995 and lower limit is set to 0.005 to make sure the parameters are greater than the minimum mesh size. Figure 2 1. Schematic of the enclosure

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25 CHAPTER 3 RESULT S AND DISCUSS I O NS The optimal parameters are and The n on dimensional maximum heat flux at the right wall is 14.822. In this chapter, sensitivity analysis will be done on and to observe the ir effect on the total heat flux. The e ffects from these two different parameters will be compared. The o ptimized fin profile and position will be shown as well as the temperature and streamline distribution of the optimal case. A m esh test is performed to ensure the accuracy of the resul ts. Sensitivity Analysis of and with Total Heat F lux Effect of on non dimensional heat flux at right wall The effect of on the non dimensional heat flux at values of , , is shown in Figure 3 1. When is fixed at a certain value, the e ffect of on non dimensional heat flux at the right wall can be observed. As the value of increase, the non dimensional heat flux at the right wall will decrease at , When as increase s the non dimensional heat flux at the right wall will also increase and will reach its maximum near Then the non dimensional heat flux at the right wall will drop This trend is observed for all other curves Comparing results from the different value of the non dimensional heat flux at the right wall is found to be higher at smaller value s of

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26 Effect of on non dimensional heat flux at right wall The effect of on the non dimensional heat fl ux at , , is shown in Figure 3 2. As increase the non dimensional heat flux at the right wall will also decrease. and have the same trend when a ffecting the non dimensional heat flux at the right wall However, seems to have greater effect than on the non dimensional heat flux at the right wall and are independent parameters but notice that cannot be any value between its upper and lower limit for a specific The upper and lower limit s are overall limit s for parameters. There is a certain limit for at each value. Optim al Profile and Position of Rectangular Fin and Optimized Heat Flux at the Right Wall Considering the parameters and objectives in the numerical simulation part, multi objective genetic algorithm (MOGA) is selected as the optimization method. Instead of consider ing the non dimensional area of the fin being 0.16 as an object, we will consider it as a constraint to make the optimization process fast er. To achieve this, the fin length and height are set as two dependent parameters to make sure the area of the fin is 0.16 and one of them is suppressed to avoid unnecessary work for ANSYS. The model is simulated using the following parameters: , After optimization and The m aximum heat flux at the right wall is 14.8.

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27 Analysis of Isotherm Contour and Streamline Contour Figure 3 3 shows the optimized profile and po sition for the rectangular fin along with the distribution of the isotherm s inside the enclosure. Figure 3 4 shows the distribution of the streamlines for this case. The gradient of temperature is high near the hot and cold wall. T he isotherm s are perpendicular to the top and bottom wa ll because the top and bottom walls of the enclosure are adiabatic and the temperature change of the fluid in the middle part of the enclosure is smoother than that near the hot and cold wall s For the fluid near the hot and cold wall s, the temperature gra dient is higher than that of the solid fin because of convection. However, co nduction in the s o lid makes the temperature gradient higher than that in the fluid which is not near the hot and cold wall s where convection is low Th e fin divides the fluid geometrically into a higher part and a lower part. For the fluid above th e fin, the cold fluid is first heated at the bottom left corner of this part by both the hot wall and the fin. Then it goes up to the top wall because buoyancy and gravity are both ta ken into consideration and the density of fluid decrease s as the temperature increase. The top wall is adiabatic so no heat from the fluid will dissipate through the wall. The stream will only be cooled by the cold fluid below it which will result in small temperature gradient s and horizontal isotherm s in the middle of this part of the enclosure After moving right to the cold wall the stream will dissipate heat quickly and the fluid become denser Then most of the fluid cooled by the cold wall will go dow nward to the fin which has a higher temperature because of conduction. The fluid stream is heated by the fin again while migrating left until it returns to the bottom left corner of this part of the enclosure A s imilar process will occur in the part of th e enclosure below the fin.

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28 Mesh test The optimization is performed at a mesh size of 0.004. To ensure the result is independent from the mesh size, a finer mesh is applied to the same optimization process. The new mesh size is 0.002 which is half of the fo rmer one. The optimal case is at: and The m aximum heat flux at the right wall is 14.3. Deviation of all parameters and maximum heat flux at the right wall from the former result s is less than 5%. We believe that the optimization proc ess as well as the results generated are fairly independent when the mesh size is 0.004. Figure 3 1 Effect of on the non dimensional heat flux at right wall at , ,

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29 Figure 3 2 Effect of on the non dimensional heat flux at right wall at , , Figure 3 3 Shots of isotherm distributions of enclosure for optimized rectangular fin, steady state, ,

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30 Figure 3 4 Shots of streamlines of enclosure for optimized rectangular fin, steady state, ,

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31 CHAPTER 4 CONCLUSION S AND RECOMMENDATIONS Conclusion s In this thesis, a rectangular fin based on a hot plate inside an enclosure is optimized to its optimal geometry and position to get the maximum heat transfer. Gravity and two dimensional heat transfer and the local convective heat transfer coefficient are all accounted for Since the heat transfer performance improve s as the fin area increase s, which result s in more cost for the fin material, the area of the fin is kept at 0.16. An optimal case is obtained through the use of genetic algorithm s All variables are non dimensionalized for future design when it comes to different material s and diffe rent enclosure geometr ies The values of and will be set at this optimization. The optimal parameters are and The n on dimensional maximum heat flux at right wall is 14.822. Recommendations Th e current study pertai ns to optimization for only one fin. Fin arrays are prove n to be more effective than single fin s and we can still use genetic algorithm s to get the optimal number of fins and optimal geometries for fin arrays. Also, t he area of the fin was kept constant in th e current study However this area may not be the optimal area which can produce the best ratio of cost and heat transfer performance. If the area as along with heat transfer were allowed to change in a multi objective genetic algorithm, a wider set of optimal cases c ould be achieved. Also, three dimension analysis will be more applicable in electron ic cooling and other industries.

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32 In the current study, g ravity is incorporated in the analysis through the Rayleigh number This can be expanded in a future investigation This will make it easier to perform such optimization when solving problems related to space where gravity is different from that on the surface of the earth.

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33 LIST OF REFERENCES [1] Merrikh, A. A., and J. L. Lage. continuum to porous continuum: the visual resolution impact on modeling natural convection in heterogeneous media ." D.B. Ingham, I. Pop (Eds.), Transport Phenomena in Porous Media, Chapter 3, Elsevier Science, England (2005) [2] Merrikh, A. A., and J. L. Lage. "Natural convection in nonhomogeneous heat generating media: Comparison of continuum and porous continuum models." Jou rnal of Porous Media 8.2 (2005): 149 163 [3] Whitaker, S "Theory and applications of transport in porous media: The method of volu me averaging." Kluwer Academic Publishers, The Netherlands (1999) [4] Merrikh, A. A. "Blockage effects in natural convection in differentially heated enclosures." Journal of En hanced Heat Transfer 8.1 (2001): 55 72 [5] Merrikh, A. A., and J. L. Lage. "Natural convection in an enclosure with disconnected and conducting solid blocks." International Journal of Heat and Mass Transfer 48.7 (2005): 1361 1372. [6] Massarotti, N., P. Nithiarasu, and A. Carotenuto. "Microscopic and macroscopic approach for natu ral convection in enclosures filled with fluid saturated porous medium." International Journal of Numerical Methods for Heat & Fluid Flow 13.7 (2003): 862 886. [7] Braga, Edimilson J., and Marcelo JS De Lemos. "Heat transfer in enclosures having a fixed am ount of solid material simulated with heterogeneous and homogeneous models." International Journal of Heat and Mass Transfer 48.23 (2005): 4748 4765. [8] Braga, Edimilson J., and Marcelo JS de Lemos. "Laminar natural convection in cavities filled with circ ular and square rods." Internatio nal Communications in Heat and Mass T ransfer 32.10 (2005): 1289 1297. [9] Pourshaghaghy, A. Hakkaki Fard, A. and Mahdavi Management 48.5 (2007): 1579 1589 [10] Jamalud Din, S. D., Rees, D. A. S., Reddy, B. V. K., and Narasimhan, A.. Prediction of natural convection flow using network model and numerical simulations inside enclosure with distributed solid blocks. Heat and Mass Transfer 46 .3 (2010) : 333 343. [11] Narasimhan, Arunn, and B. V. K. Reddy. "Natural convection inside a bidisperse porous medium enclosure." Journal of Heat Transfer 132.1 (2010): # 012502.

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34 [12] Junqueira, S. L., De Lai, F. C., Franco, A. T., and Lage, J. L. "Numerical investigation of natural convection in heterogeneous rectangular enclosures." Heat Transfer Engineering 34.5 6 (2013): 460 469. [13] Qiu, H., Lage, J. L., Junqueira, S. L., and Franco, A. T. Predicting the nusselt number of heterogeneou s (porous) enclosures using a generic form of the berkovsky polevikov correlations. J ournal of Heat Transfer 135.8 (2013): # 082601. [14] Mirehei, S. M., Franco, A. T., Ju nqueira, S. L. M., and Lage, J. L Periodic Horizontal Heating of Enclosed Disconnec ted Solid Bodies Saturated With a Fluid. In ASME 2014 International Mechanical Engineering Congress and Exposition (2014): V08AT10A034 [15] Mirehei, S. M. Simulations and analyses of natural convection inside heterogeneous conta iners heated by solar radi Diss. Southern Methodist University (2015). [16] Huang, Guei Jang, and Shwin Chung Wong. "Dynamic characteristics of natural convection from horizontal rectangular fin arrays." Applied Thermal Engineering 42 (2012): 81 89. [17] Harahap, Filino, Herr y Lesmana, and IKT Arya Sume Dirgayasa. "Measurements of heat dissipation from miniaturized vertical rectangular fin arrays under dominant natural convection cond itions." Heat and Mass T ransfer 42.11 (2006): 1025 1036. [18] Al Doori, W. H. A. R. "Enhancement of natural convection heat transfer from the rectangular fins by circular perforations." International Journal of Automotive and Mechanical Engineering 4 (2011): 428 436. [19] Dannelley, Daniel. Enhancement of extended surface heat transfe r u sing fractal Diss. University of Alabama ( 2013 ) [20] Azarkish, H., S. M. H. Sarvari, and A. Behzadmehr. "Optimum design of a longitudinal fin array with convection and radiation heat transfer using a genetic algorithm." International Jou rnal of Thermal Sciences 49.11 (2010): 2222 2229. [21] Bobaru, Florin, and Srinivas Rachakonda. "Optimal shape profiles for cooling fins of high and low conductivity." International Journal of Heat and Mass Transfer 47.23 (2004): 4953 4966. [22] Kang, Hyung Suk, and Dwight C. Look Jr. "Optimization of a trapezoi dal profile annular fin." Heat Transfer E ngineering 30.5 (2009): 359 367. [23] Hamadneh, N., Khan, W. A., Sa thasivam, S., and Ong, H. C. "Design optimization of pin fin geometry using particle sw arm optimization algorithm." PloS one 8.5 (2013): e66080.

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35 [24] Hajabdollahi, F., Rafsanjani, H. H. Hajabdollahi, Z., and Hamidi, Y Multi objective optimization of pin fin to determine the optimal fin geometry using genetic algorithm. Applied Mathematic al Modelling, 36 .1 (2012): 244 254. [25] Wang, Jiansheng, and Xiao Wang. "The heat transfer optimization of conical fin by shape modification." Chinese Journal of Chemical Engineering 24.8 (2016): 972 978. [26] Najafi, Hamidreza, Behzad Najafi, and Pooya Hoseinpoori. "Energy and cost optimization of a plate and fin heat exchanger using genetic algorithm." Applied Thermal Engineering 31.10 (2011): 1839 1847. [27] Fabbri, Giampietro. "A genetic algorithm for fin profile optimization." International Journal o f Heat and Mass Transfer 40.9 (1997): 2165 2172. [28] Dialameh, L., M. Yaghoubi, and O. Abouali. "Natural convection from an array of horizontal rectangular thick fins with short length." Applied Thermal Engineering 28.17 (2008): 2371 2379. [29] Tritton, D J. Von Nostrand Reinhold, New York (1979) [30] Fluent, Ansys. "12.0 Theory Guide." Ansys Inc 5 (2009) [31] Mirehei, S. M., and J. L. Lage. "Periodic natural convection inside a fluid saturated porous medium made of disconnecte d solid obstacles: A continuum approach." ASME 2016 Heat Transfer Summer Conference. 7. ( 2016 )

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36 BIOGRAPHICAL SKETCH M iao Huang 2017. University in the spring of 2015.