Study on the influence of geometrical parameters to enhance heat transfer in a finned cylindrical segment, incorporating vortex generators

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(1)Instituto Tecnológico y de Estudios Superiores de Monterrey Campus Monterrey. School of Engineering and Sciences. Study on the influence of geometrical parameters to enhance heat transfer in a finned cylindrical segment, incorporating vortex generators. A thesis presented by. Anuar Samuel Chilaca Tarango Submitted to the School of Engineering and Sciences in partial fulfillment of the requirements for the degree of Master of Science in Energy Engineering. Monterrey, Nuevo León, May, 2018.

(2) Dedication. To god For having allowed me to achieve this goal and to give me health, intelligence and will, in addition to their infinite kindness and love. To my parents Samuel and Rosamaria, for having supported me at all times, for their advice, their values, for the constant motivation that has allowed me to be a good person, but more than anything, for their love. You are the fundamental pillar in everything I am, in all my education, both academic and life, To my family To my grandfather Juan (RIP), to whom I dedicate this work, you will always be in my mind and in my heart. To my grandmothers Enriqueta (RIP) and Rosa for the examples of perseverance and humility that characterize them, for the courage shown to get ahead and for their love. To my sister Fernanda, because despite being far away, you represent my greatest motivation. To my friends To all my classmates and friends, for the time we have spent together, now they are part of my fond memories and I hope that the years strengthen our friendship and that our future will be fulled of blessings.. iii.

(3) Acknowledgements. Firstly, I would like to express my sincere gratitude to Professor Dr. Alejandro García Cuéllar, for his continuous support and interest in this project, for his open doors, motivation and discussions. Also to my thesis committe members (Dr. Carlos Rivera Solorio and Dr. Jose Luis López Salinas), who took time from their busy schedules to give me guidance and I hope someday I could inspire a young engineer the way they have inspired to me. I would like to thank my friends and collages for the tuition and their infinitive patience, without their endless support and encouragement I would not be where I am today. To Alejandra for his continuous motivation and support throughout all ups and downs during the writing of this thesis. This will be a time we won’t forget. Special thank to Tecnologico de Monterrey for giving me the opportunity to continue my studies, without a doubt I am part of the best educational institution. And also to SENER-CONACyT for having given me the financial support during these two years.. iv.

(4) Study on the influence of geometrical parameters to enhance heat transfer in a finned cylindrical segment, incorporating vortex generators. by Anuar Samuel Chilaca Tarango Abstract The present work addresses the simulation of geometries considering forced convection of turbulent flow for the thermal optimization of a generator of a water-ammonia absorption refrigeration system, for which purpose, several simulations were carried out on ANSYS Fluent, varying the geometric parameters in order to define the optimal design for the generator. In the first part, a geometrical analysis of the previously geometry proposed for the construction of the generator is presented, evaluating those geometrical factors that enhance the heat transfer. The results obtained from the simulations are used to calculate the global heat transfer coefficient by convection, as well as the average Nusselt number. High heat transfer coefficients were found where geometries shows specific arrangements that modify the evolution of the flow, those changes in the flow contributes to the higher mixed and to the heat transfer. The second part of the thesis analyze the modification of arrangement and evaluate the introduction of different types of fin geometries. Realistic and manufacturable geometries were considered for maximization of thermal heat transfer coefficient and also the minimization of friction forces. In order to compare these various geometries, a set of standard conditions were required. Finally, the thesis contemplates the incorporation of Vortex Generators (VG) to enhance the heat transfer along the generator. Vortex generators is one of the passive methods to generate streamwise vortices that create high turbulence in fluid flow over heat transfer surfaces. VG have shown to be an effective way to increase the heat transfer coefficient, decreasing the thermal resistance of the sublayer adjacent to the wall immediately where the viscous effects of the sublayer are dominant. The increase of turbulence of the fluid flow in the main stream have shown positive effects on the heat transfer. The thesis evaluates the present research of VG and contemplate the simulation of the incorporation of an array of VG over the surface of a previously finned- cylindrical geometry of generator, contrasting the immersion of the VG’s to baseline geometry, the effects on the pressure drop are also studied. Subsequently, the incorporation of a modified annular winglet vortex generator over the generator surface was also evaluated. The results were compared to the no VG fin type geometry. The results show that the heat transfer increases considerably, but an increase on the pressure drop is also observed.. v.

(5) List of Figures 1.1 1.3 1.4. General design of the Generator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cylindrical geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diamond geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.1 2.2 2.3 2.4 2.5 2.6. Shear stress over the surface of a cube . . . . Boundary layer development along a flat plate. Velocity Contour of Flow through a Cylinder. Vapor Absorption Refrigeration System sketch Flow over concentric tubes . . . . . . . . . . Fluid Pathlines of a Delta Vortex Generator. .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. 9 15 17 25 27 36. 3.1 3.3 3.4 3.5 3.7 3.9 3.11 3.13 3.14 3.15 3.16 3.18 3.19 3.20 3.22 3.23 3.24 3.25 3.26 3.27 3.28 3.29 3.30 3.31 3.32 3.33. Geometry construction . . . . . . . . . . . . . . . . . . . . . . . . . Front view, parameters of the fins. . . . . . . . . . . . . . . . . . . . Panoramic view, parameters of the fins. . . . . . . . . . . . . . . . . . Geometries configuration for the δ variation . . . . . . . . . . . . . . Geometries configurations for the γ variation with w constant. . . . . Geometries configuration for the γ variation . . . . . . . . . . . . . . Geometries configuration for the λ variation . . . . . . . . . . . . . . Sketches of the different fin geometries types . . . . . . . . . . . . . Geometrical parameters of the DWP . . . . . . . . . . . . . . . . . . Front view, delta-type VG incorporated to the surface of the generator. Schematic flow representations of the Wang model. . . . . . . . . . . Geometrical parameters of the new geometry. . . . . . . . . . . . . . New Vortex Configuration over the surface of the heat exchanger. . . . Geometry and Fluid domain . . . . . . . . . . . . . . . . . . . . . . Inlet of the generator. . . . . . . . . . . . . . . . . . . . . . . . . . . External wall of finned tube. . . . . . . . . . . . . . . . . . . . . . . Internal wall of the generator jacket. . . . . . . . . . . . . . . . . . . Outlet of the generator. . . . . . . . . . . . . . . . . . . . . . . . . . External wall, zero heat transfer condition . . . . . . . . . . . . . . . Internal Wall, constant heat source. . . . . . . . . . . . . . . . . . . . Type of Geometry Analyzed, Segmented Tube . . . . . . . . . . . . . Type of Geometry Analyzed, Segmented Tube . . . . . . . . . . . . . Geometry mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representation of Temperatures. . . . . . . . . . . . . . . . . . . . . Geometry mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mesh of the VG model . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. 43 44 44 47 48 49 50 52 53 54 55 55 56 57 60 60 60 61 61 62 63 63 63 64 65 66. 4.1 4.2. Experimental assembly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometry the vortex generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 68 69. vi. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. 5 7 7.

(6) 4.3 4.4 4.5 4.6. Constructed geometry . . . . . . . . . . . . . . . . Mesh of the model . . . . . . . . . . . . . . . . . . Velocity contours for different positions in the fluid. Results of the Variation of N um against Reynolds. .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 71 71 72 72. 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.17 5.19 5.20 5.21 5.23 5.24 5.25. N um /N u0 for δ variation . . . . . . . . . . . . . . . . . . . . . . . . f /f0 for δ variation . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature and Velocity distribution for δ = 1 . . . . . . . . . . . . N um /N u0 for γ variation with t constant . . . . . . . . . . . . . . . f /f0 for γ variation with t constant . . . . . . . . . . . . . . . . . . . Pathlines of velocity fluid along the tube. . . . . . . . . . . . . . . . . N um /N u0 for γ variation with N constant . . . . . . . . . . . . . . . f /f0 for γ variation with N constant . . . . . . . . . . . . . . . . . . f /f0 for γ variation with t constant . . . . . . . . . . . . . . . . . . . N um /N u0 for λ variation . . . . . . . . . . . . . . . . . . . . . . . . f /f0 for λ variation . . . . . . . . . . . . . . . . . . . . . . . . . . . Pathlines of the fluid, geometry λ = 0.38. . . . . . . . . . . . . . . . N um /N u0 for geometry-type variation . . . . . . . . . . . . . . . . . f /f0 geometry-type variation . . . . . . . . . . . . . . . . . . . . . . Evolution of the vorticity pathlines . . . . . . . . . . . . . . . . . . . Pathlines and temperature distribution of different geometries. . . . . N um /N u0 for vortex-distance collocation. . . . . . . . . . . . . . . . f r/f r0 for vortex-distance collocation . . . . . . . . . . . . . . . . . Vorticity pathlines in different planes over the surface of the generator Nu variation of the Annular VG. . . . . . . . . . . . . . . . . . . . . f variation of the Annular VG. . . . . . . . . . . . . . . . . . . . . . Vorticity pathlines in a segment of the geometry . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . .. 75 75 76 76 77 77 78 79 79 80 81 81 82 83 83 84 85 86 86 87 87 88. vii. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . ..

(7) List of Tables 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10. List of fin parameters . . . . . . . . . . . . . . Geometries variation for δ factor. . . . . . . . . Geometry variation for γ factor with t constant Geometry variation for γ factor with N constant Geometries variation for λ factor. . . . . . . . . Geometrical factors, second analysis . . . . . . Specific measurements for the vortex generator. Values of the geometrical design. . . . . . . . . Physical properties of the selected materials. . . Independence Mesh Analysis . . . . . . . . . .. . . . . . . . . . .. 44 46 48 49 50 51 54 56 58 64. 4.1 4.2. Fluid properties: Air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical results obtained from the CFD simulation. . . . . . . . . . . . . . . . . .. 70 73. viii. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . ..

(8) Contents. Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. v. List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. vii. List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii 1 Introduction . . . . . . . . . . . . . 1.1 Problem Statement and Context . 1.2 Thesis Outline . . . . . . . . . . 1.3 Perspective . . . . . . . . . . . 1.4 Research Objective . . . . . . . 1.5 Justification . . . . . . . . . . . 1.6 Solution Overview . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. 1 1 3 4 4 6 7. 2 Background . . . . . . . . . . . . . . . . . . . . . . 2.1 Definitions . . . . . . . . . . . . . . . . . . . . 2.2 Flux Properties . . . . . . . . . . . . . . . . . 2.3 Fluid Properties . . . . . . . . . . . . . . . . . 2.4 Governing Equations . . . . . . . . . . . . . . 2.5 Constitutive equations . . . . . . . . . . . . . . 2.6 Ideal Flow . . . . . . . . . . . . . . . . . . . . 2.6.1 Newtonian approximation. . . . . . . . 2.6.2 Non-compressible Governing Equations 2.6.3 Non-Newtonian Flow . . . . . . . . . . 2.7 Boundary Layer approximation . . . . . . . . . 2.7.1 Laminar Boundary Layer on a Flat Plate 2.7.2 Turbulent Boundary Layer . . . . . . . 2.7.3 Boundary Layer separation . . . . . . . 2.7.4 Vorticity and Circulation . . . . . . . . 2.7.5 Helmholtz Theorem . . . . . . . . . . . 2.7.6 Vorticity equation . . . . . . . . . . . . 2.7.7 Boundary conditions . . . . . . . . . . 2.8 Heat Transfer Mechanisms . . . . . . . . . . . 2.8.1 Conduction- Fourier’s Law . . . . . . . 2.8.2 Heat Diffusion Equation . . . . . . . . 2.8.3 Convection . . . . . . . . . . . . . . . 2.9 Absorption Refrigeration System . . . . . . . . 2.9.1 Heat exchanger Model . . . . . . . . . 2.9.2 Concentric Internal Flow . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. 8 8 8 9 11 12 13 13 14 14 15 15 16 16 17 18 18 19 21 21 21 22 25 26 26. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. ix. . . . . . . .. . . . . . . ..

(9) 2.10 Turbulence . . . . . . . . . . . . . . . . . 2.11 Turbulence models . . . . . . . . . . . . 2.11.1 Spalart-Allmaras Model . . . . . 2.11.2 Standard κ- Model . . . . . . . . 2.11.3 k − ω turbulence model . . . . . . 2.11.4 SST model . . . . . . . . . . . . 2.11.5 Reynolds Stress Turbulence model 2.12 Computational Fluid Dynamics . . . . . . 2.13 Influence of geometric parameters . . . . 2.14 Vortex Generators . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. 29 30 30 32 32 33 34 36 37 40. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. 42 42 42 44 45 45 46 57 57 57 58 58 59 64. 4 Validation of the Computational model . . . . . . . . . . . . . . . . . . . . . . . . . . 4.0.1 Summary of the Chunhua M. and collaborators experiment . . . . . . . . . . 4.0.2 Computational validation . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 67 67 70. 5 Computational Results . . . . . . . . . . . . . . . . . . . . 5.0.1 Analysis of δ variation. . . . . . . . . . . . . . . 5.0.2 Analysis of γ variation for constant thickness. . . 5.0.3 Analysis of γ variation fixed Number of fins. . . 5.0.4 Analysis of λ variation . . . . . . . . . . . . . . 5.0.5 Second analysis: New geometry selection . . . . 5.0.6 Third analysis: Introduction of vortex generators. 5.0.7 Third analysis: Modified annular vortex generator. . . . . . . . .. 74 74 76 78 80 82 85 87. 6 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 89 89 91. 3 Computational Fluid Dynamics Modeling . 3.1 Problem description . . . . . . . . . . . 3.1.1 First Geometry selection . . . . 3.1.2 Table of geometrical parameters 3.1.3 Characterization of the device . 3.1.4 Geometrical Analysis . . . . . . 3.1.5 Constructed geometries . . . . . 3.2 Physical Modeling . . . . . . . . . . . . 3.2.1 Considerations and assumptions 3.2.2 Domain of the problem: . . . . . 3.2.3 Governing equations . . . . . . 3.2.4 Turbulence model . . . . . . . . 3.2.5 Boundary conditions . . . . . . 3.2.6 Meshing . . . . . . . . . . . . .. x. . . . . . . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . ..

(10) 6.3. Lambda variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113. xi.

(11) Chapter 1:. Introduction. 1.1. Problem Statement and Context Currently a large part of the total energy consumption is used in refrigeration and air conditioning equipments. Several authors have development available improvements in the energy efficiency and economics of industrial and residential refrigeration systems. The global contribution from buildings towards energy consumption, both residential and commercial, has steadily increased reaching figures between 20% and 40% in developed countries, and has exceeded the other major sectors: industrial and transportation [1]. Refrigeration and air conditioning play important roles in providing human comfort, food processing and storage and many other industrial processes. Generally those applications, utilize mechanical vapor compression systems, which typically are powered by electric energy [2]. Besides, these systems employ different types of refrigerants, such as Chlorofluorocarbons (CFCs), from which their high potential to harm to the environment has been proven [3, 4]. And also the electricity consumption is growing up as the commercial market of the refrigeration is expanding, which is corresponding to a increase in the fossil fuels burning and therefore as a consequence, pollution and the appearance of greenhouse gases in the atmosphere. On the other hand, solar energy is widely available and has no fuel cost, which makes the running of solar powered refrigeration a viable and economic proposition [2]. And in addition, a lot of industrial process uses thermal energy of the combustibles to produce heat. After the processes, part of the heat is rejected to the surrounding as waste. This waste heat can be converted to a useful refrigeration by using a heat operated refrigeration system [5]. Considering the previously detailed, one of the solutions for the energy demand due to refrigeration is the refrigeration based on absorption systems. Since 1970s solar refrigeration has received great interest and many projects for development or optimization of solar refrigeration technologies have been implemented. A variety of solar refrigeration technologies are now available in the market at much cheaper prices than ever [6]. The basic problem of absorption systems their low efficiency in contrast with the standard compression refrigeration and their high cost, therefore vapor compression systems still dominate all market sectors. In order to promote the use of absorption systems, further development is required to improve their performance and reduce its cost. In 1997, Dr. José Manrique patented the idea of an Ammonia-water Absorption Refrigeration System (AW-ARS), using the heat of hot water coming from solar collectors [7], currently it is operating in the Casa Solar at Tecnológico de Monterrey, Campus Monterrey. The thermal machine operates obtaining the cooling effect by the removal of heat, evaporating a low pressure refrigerant fluid which is a mixture of ammonia and water. Also Dr. Manrique developed the construction of the. 1.

(12) generator, which allowed to operate the AW-ARS up to three tons of refrigeration. The configuration of the cycle constructed by Dr. Manrique is pretend to be improve and for that purpose, one of the possible modification is to adapt the generator guaranteeing the highest heat transfer of the hot fluid to the Water-ammonia solution. There are a wide range of possible design modifications which can reduce the shaft power used in refrigeration cycles, and these modifications can be grouped into three categories: structure modifications, adjustment of operating conditions and modification of refrigerant composition [8]. Ideally, refrigeration cycles should be designed considering all three elements: the identification of the most suitable configuration, the selection of the most appropriate refrigerant (and refrigerant composition) and the determination of optimal operating conditions in an integrated and cost-effective manner. The present work pretends to define the geometrical and physical parameters for the best performance of the generator, which plays on an important role on the cycle. In order to do this, several simulations were carried out based on the Finite Volume method in the ANSYS Fluent, exploring the possibility of finding a better design with similar performance to the already constructed. The behavior of a vapor heat generator is basically corresponding to a heat exchanger, where the heat is transfer from the hot source (which in the ARC is hot water coming from solar collectors) to another fluid (water-amonnia solution) without fluid contact. In most heat exchangers, heat transfer between fluids takes place through a separating wall or into and out of a wall in a transient manner. The fluids are separated by a heat transfer surface, and ideally they do not mix or leak . By this application, heat exchangers are widely used in air conditioning, ventilation, refrigeration and thermal systems [9]. Computational fluid dynamics (CFD), is a science that can be helpful for studying fluid flow, heat transfer, chemical reactions, etc, by solving partial differential equations based on numerical analysis. It is equally helpful in designing heat exchanger systems by suggesting geometrical modifications, obtaining flow parameters as temperature distribution and velocity fields, in a shorter time at a lower cost than required from experimental work [10]. Improving the thermal performance of such systems is crucial to meet energy costs and environment impact. Many techniques for heat transfer augmentation have been widely adopted to obtain a compact and small size of the systems with lower operating costs. Those research are focused on the geometry adaptation like incorporating fins or diverse turbulence generators which enhance the heat transfer over the surface of the exchanger. In this research several new design geometries are proposed for improving the heat exchange obtaining the average Nusselt number as function of the geometry considering the same mass flow for each of the designs. Therefore, the problem is on the category of forced convection of turbulent internal flow.. 2.

(13) 1.2. Thesis Outline Chapter I, presents the introduction, the general objectives of the thesis and the justification of the selected problem; specifying its classification, nature and the necessary tools to solve it. Also the possible modifications according to literature review are enlisted. Chapter II shows the literature research accomplished based on the improvement in thermal systems for the increase in heat transfer (convection and conduction). The investigation details both computational methods (simulations based on finite element) and experimental techniques that central its sought in the optimization of the geometry of thermal equipments. An introduction to the fundamental theory underlying the problem, evaluating the classification of the problem, as well as the formulation of the governing equations and the physics related to problem. Criteria are established to select a turbulence model, and for the evaluation of the thermal performance in heat exchangers. Finally an introduction of the current literature of Vortex Generators and Fin-type geometries is presented. Chapter III explains the criteria for the construction of the geometries, detailing the geometrical factors that were selected for each geometry. Also the boundary conditions of the problem are stated, the digitalization aspects of the geometries, the configuration of the simulation, the meshing sequence, and post-processing of the results are enlisted. Chapter IV illustrates a computational validation of the selected turbulence model. using experimental data obtained for the analysis of a Rectangular Delta winglet Vortex Generator, simulations were carried out to compare two different turbulence models, therefore the numerical results shows a high acceptance over the experimental data. Validating the selection of the physical turbulence model of heat transfer and fluid dynamics for the simulation of the geometries. Chapter V details the results obtained of the simulations. The results explain the behavior of Nusselt number as function of the geometric parameters and the length of the generator, also correlations of the j factor are shown in therms of the related configuration. Finally Chapter VI list the most remarkable conclusions of the present work. A description of the geometric parameters that influences the increment on the heat transfer and the increment of drop pressure is detailed. Analyzing the effect of the introduction of Vortex Generators and generally the future work to extend this research.. 3.

(14) 1.3. Perspective Upon several previous works have attended the influence of the design of heat exchangers devices in order to optimize the heat transfer between fluids. Generally the incorporation of extended surfaces represents a sterling technique for the enhance of heat transfer. When talking about extended surface, it becomes reference to a solid that experiences energy transfer by conduction within its limits, as well as energy transfer by convection (and / or radiation) between its limits and the surroundings. The most frequent application in which extended surfaces are implemented is to increase the heat transfer coefficient of a contiguous fluid, the fins are used when heat transfer between a solid and a fluid needs to be improve. Basically this thesis pretends to construct different types geometries and evaluate its parameters that might affect positively the heat transfer over the surface of a generator and therefore optimize the global thermal performance of the ARC. For that purpose, CFD analysis was required in order to obtain the thermal and hydrodynamic variables in all constructed geometries, keeping the initial and boundary conditions similar. And then a relationship between the geometrical parameters and its effect specifically on heat transfer was established. Due to the results obtained was critical to define functions of the efficiency of the geometry, not only based on its thermal contribution, but based in its critical energy evaluate considering the friction effects. In the first part of the analysis, the geometry proposed by Manrique [7], was analyzed, and geometrical parameters where studied, looking for the best adaptation. Then in the second part, new geometries were proposed. And the evaluation of the incorporation of vortex generators is finally contemplated where 2 types of Vortex Generators are evaluated.. 1.4. Research Objective One of the newest technologies implemented for the improvement of energy savings is the absorption refrigeration cycle, which unlike the standard refrigeration cycles avoids the use of a compressor. The cycle of refrigeration by absorption allows the withdrawal of heat from a certain place, this cycle takes advantage of the substances absorb heat when changing state, from liquid to gaseous. In the case of absorption, the cycle is physically based on the ability of some substances, such as lithium bromide, to absorb another substance, such as water, in the vapor phase [11]. In particular, the design of the absorption refrigeration system built by Dr. Manrique [7] aims to use the heat from an arrangement of solar collectors to vaporize the mixture of water and ammonia and separating the refrigerant from the absorber. The design created allows 3 tons of refrigeration to be obtained. One way to improve the capacity of the cycle and with it the thermodynamic efficiency is to improve the heat transfer in the generator, where the heat of the hot fluid is transfer to the mixture,. 4.

(15) Figure 1.1: General design of the Generator. Computational model of the design of the generator by Manrique. [7]. basically the generator has the behavior of a heat exchanger. In present literature details several alternatives to obtain an enhancement on heat transfer on heat exchangers, one of them is the introduction of extended surfaces or fins in order to increase the turbulence of the hot fluid and with them, improve the heat transfer over the surface of the exchanger. By this way, the present work pretends to analyze the effects of the incorporation of fins, considering the influence of forced convection over two concentric finned cylinders, establishing boundary and initial conditions. Likewise, the behavior of the dimensionless number (Nu, f). The general objective of the project is to understand the effects of the incorporation of fins in the design of the exchanger, and determine what of the evaluated geometries is a suitable configuration for the construction of the generator of the WA-ARS. According to it, simulations were carried out on Fluent using the forced convection mechanism on a segment of the tube, varying geometrical parameters. Firstly, the previously geometry constructed by Dr. Manrique [7], was studied, varying the size of the fins in proportion to the size of the fluid segment, also the proportion between the number of fins and the separation on each fin was studied, likewise the proportion between the incorporation of another set of fins consecutive to the followings, in accordance with the width of the fin. Those parameters were studied considering the same boundary and initial conditions. The second part of the thesis evaluates the introduction of new fin type geometries and the effect of incorporating Vortex Generators over the surface of the generator. Based on literature review, two types of VG were selected and added to the previous geometries, The purpose of the above was to establish if the presence of VG enhance significantly the heat transfer convection without obtaining great modifications on the pressure drop. In this manner, it was observed the dependence of geometric parameters on the heat transfer, specifically focusing on the Nusselt number and in the drop of pressure.. 5.

(16) 1.5. Justification Computational simulations represents an advantageous way to solve complex problems that involve solving partial differential equations. It must be taking in account that the equations of conservation of energy, quantity of moment and continuity are a system of coupled partial differential equations that generally turn out to be non-linear systems, based on the geometric characteristics of the system, the boundary and initial conditions of the problem it is possible to determine the solution for the phenomenon of study using a numerical approximation, the common numerical methods are finite difference and finite volume. Forced convection represents symbolically diverse technological applications, such as the design of heat exchangers, thermal condensers or cooling towers. The basis model for the absorption generator of the AW-ARS is a heat exchanger, where a constant heat source is required to raise the temperature of the fluid to a certain desired value. In this case the source of heat comes from the fluid previously heated by the solar collectors. It is desired to transfer as much energy as possible from the fluid to the binary mixture. One way to improve this, is to add fins to the surface of the generator, which porpoise is to induce turbulence [12]. The problem states that the heat transfer coefficient of the fluid is incremented by incorporating geometries, and as a consequence the heat transfer of the fluid to the geometry is also increased. Ansys Fluent is an advance software which allows solving phenomena as fluid dynamics and heat transfer coupled. The present work, is sought to define several geometries based on literature research that have been proved an improvement in the thermal efficiency on heat exchangers. Currently many authors have pointed out the incorporation of new geometry standards such as vortex generators to enhance the heat transfer in fluid, based on the fact that vortex allows to introduce mixing on the fluid. Fin geometries have been studied but not optimized to its best performance. Contrasting both approaches, the thesis presents a variation of the geometry proposed by Giner [13] and secondary several parameters are varied looking for its best thermal configuration, then an analysis of the incorporation of vortex generators is reached out, conjugating the hydrodynamic performance based on the pressure drop and Nusselt number. In today’s environment, designing an efficient thermal heat exchanger significant amount of analysis and trade study. Design is an iterative process and it is useful to know what level of numerical analysis is necessary in order to obtain a sufficient level of accuracy. When possible, it is advantageous to reduce time and cost by using dependable computational analysis as opposed to experimental data. The level of analysis can range from a steady 1D streamline code to a full 3D unsteady Navier-Stokes analysis. As the analysis becomes more complex, more resources are required. Therefore, determining the appropriate level of analysis required to obtain reasonable results is valuable, since under-analyzing the problem can lead to meaningless results, while an overly complex approach demands unnecessary effort.. 6.

(17) The hydrodynamic and thermal analysis for a heat exchanger is complex, The adverse pressure gradient, tip vortex structure and end walls can make an analysis very challenging. Historically, unsteady effects, turbulence and end wall secondary flows were handled with empirical correlations, but these can become unsuitable near stall [14]. Therefore simulations represent an advance technique to obtain thermal and hydrodynamic variables from a determine geometry and specific boundary and initial conditions.. 1.6. Solution Overview Heat transfer is a discipline of thermal engineering that deals with the generation, consumption, conversion and exchange of thermal energy between physical systems. Thermal energy is transferred from one system to another by various mechanisms, namely conduction, convection and thermal radiation. Each heat transfer mechanism has a unique phenomenon so is expressed by characteristic rate equations. Forced convection is a well known studied mechanism, and represents advanced applications in industry. In a crucial way, the comprehension and investigation of phenomena related with the heat transfer enhancement is relevant for energy savings and efficiency improvement. Thermal machines, such as heat exchangers, have a high dependence of the study of forced convection analysis, and the optimization has already affected the construction of this devices. The present investigation pretends to evaluate the global thermal characteristics, based on the geometrical modifications of fins in order to reach higher Nusselt numbers for a determined mass flow rate. Simulations will allow to understand the geometrical modifications in the design that affect considerably the heat transfer over the generator, and therefore decide if the contemplated modification represents benefice in energy savings and a possible construction of the generator could be achieved.. Figure 1.4: Diamond geometry. Figure 1.3: Cylindrical geometry. 7.

(18) Chapter 2:. Background. 2.1. Definitions Fluid Dynamics is a science present in many phenomena of daily life. The study of its mechanism is essentially driven by understanding the physics involved, as well as its control in diverse applications of engineering such as astrophysics, meteorology, oceanography, aerodynamics, hydrodynamics, lubrication, marine engineering, turbomachinery, reservoir engineering and combustion engineering [15]. Fluids belong to deformable continua science, physically this means that a material system forms a continuum or a continuum spectrum if it is filled with a continuous matter. As the matter is composed by molecules, the continuum hypothesis leads to the fact that a very small volume will contain a very large number of molecules [16].. 2.2. Flux Properties Some of the dynamical and thermodynamical properties of a fluid, are now enlisted: • Temperature T : Scalar magnitude that represents the internal activity, of a substance. The concept is linked to the transfer of energy by heat. Two regions that posses the same temperatures does not transmit heat. This condition is stated as the zero-law of thermodynamics. • Velocity V~ : Is a vector that represents the direction, angle and magnitude of the speed of the movement of a fluid. Hydrostatics studies the behavior of fluids where its velocity is zero. • Pressure: Fluid pressure is the measurement of the force per unit area on a object in the fluid or on the surface of a closed container. This pressure can be caused by gravity, acceleration, or by forces outside a closed container. Since a fluid has no definite shape, its pressure applies in all directions. Fluid pressure can also be amplified through hydraulic mechanisms and changes in the velocity of the fluid. • Stress τij : Considering two types of forces present in a fluid: body forces and surface forces. Surface Forces are due to the action of physical contact of an external material in the boundary of the control volume. Considering the net forces of a fluid portion dF , which acts on a infinitesimal area dA, and has a direction indicated by the unit vector n. F = dFnn + dFtt dF. 8. (2.1).

(19) Figure 2.1: Shear stress over the surface of a cube Taken from [17] So, we can define stress, as the force that is exerted on a unitary area. We can define two types of stresses: τn =. dFn dA. τt =. dFt dA. (2.2). τn refers to the normal stress, and τt to shear stress. Considering a square body, of sides dx1 , dx2 , dx3 , the surface forces acts over all the six faces in both 3 directions x1 , x2 and x3 . These forces divided by its corresponding area return the stresses over every face, as shown in figure 2.1. To specify the state of stress at any particular point in the fluid, the values of the nine components of τij are needed which can also be represented in a matrix form: . . τ11 τ12 τ13   τij = τ21 τ22 τ23  τ31 τ32 τ33. (2.3). 2.3. Fluid Properties The following fluid properties, depends of other variables such as temperature, pressure, etc. • Density: The density of a fluid is defined as the mass per unit volume. Under flow conditions, particularly in gases, the density may vary greatly throughout the fluid. The density, ρ, at a particular point in the fluid is defined as: ρ = lim. ∆V →δV. ∆m ∆V. (2.4). Where ∆m is the mass contained in a volume ∆V , and δV is the smallest volume surrounding the point for which statistical averages are meaningful. The concept of the density at a mathematical point, i.e., at ∆V = 0 is seen to be fictitious; however, taking ρ = lim∆V →0 (∆m/∆V ) is extremely useful, as it allows us to describe fluid flow in terms of continuous functions. The 9.

(20) density, in general, may vary from point to point in a fluid and may also vary with respect to time as in a punctured automobile tire [18]. • Surface Tension: In the continuum limit where the transition layer becomes a mathematical surface separating one material from the other, the difference in molecular binding energy manifests itself as a macroscopic surface energy density. And where energy is found, forces are not far away.Since a larger area of the surface contains larger surface energy, external forces must perform positive work against internal surface forces to increase the total area of the surface. Mathematically, the internal surface forces are represented by surface tension, defined as the normal force per unit of length [19]. • Compressibility: Compressible flow appears in many natural and many technological processes. Define it as change in volume by the effect of the variation of pressure. This process can be isothermal, isentropic or any other [20]. Then the module of volumetric elasticity βc could be defined in particular ways. For the isentropic case: 1 ∂P ∂P = (2.5) βc = −vs ∂vs ρ ∂ρ Where vs is the specific volume. Another case occurs when the chance of the module of volumetric elasticity βD varies with the temperature at a constant pressure. βD =. 1 ∂ρ 1 ∂vs =− vs ∂T ρ ∂T. (2.6). • Viscosity: The viscosity of a fluid is a measure of its resistance to deformation rate, Tar and molasses are examples of highly viscous fluids; air and water, which are the subject of frequent engineering interest, are example of fluids with relatively low viscosities. An understanding of the existence of the viscosity requires an examination of the motion of fluid on a molecular basis. In solids, the resistance of deformation is the modulus of elasticity, the analogous description to relates the shear stress in a parallel, laminar flow to a property of the fluid is the Newton’s las of viscosity. Where viscosity is define as the ratio of the shear stress and the rate of shear strain. The property depends upon the temperature, composition, pressure of the fluid, but it is independent of the rate of shear strain [18]. Newton’s relation is: τ =µ. dU dy. (2.7). While the kinetic theory of gases is well developed, and the more sophisticated models of molecular interaction accurately predict viscosity in a gas, the molecular theory of liquids is 10.

(21) much less advanced. Hence the major source of knowledge concerning the viscosity of liquids is experiment. The difficulties in the analytical treatment of a liquid are largely inherent in the nature of the liquid itself. Whereas in gases the distance between molecules is so great that we consider gas molecules as interacting or colliding in pairs, the close spacing of molecules in a liquid results in the interaction of several molecules simultaneously. This situation is somewhat akin to an N-body gravitational problem. In spite of these difficulties, an approximate theory has been developed which illustrates the relation of the intermolecular forces to viscosity [18]. The viscosity of a liquid can be considered due to restraint caused by intermolecular forces. As a liquid heats up, the molecules become more mobile. This results in less restraint from molecular forces.. 2.4. Governing Equations The equations that govern the mechanics of fluids are obtained from the application of conservation principles to a certain fluid volume. The equations of fluid mechanics represent laws of conservation or balance of physical quantities.To obtain the equations is considered the continuous fluid and the conservation of a variable Φ, within a control volume can be represented as a balance between several processes that tend to increase or decrease the variable. The generalized equation relating the laws of mechanics to the fluid which is instantaneously contained within the control volume is given by the Leibniz–Reynolds transport theorem [21]: ∂ ZZZ dΦ ZZ = φρV · nds + φρdV (2.8) dt ∂t C.S. C.V. Where Φ is any extensive property such as mass, linear momentum, angular momentum, or stored energy, and φ is the extensive property per unit mass. Vi is the vector velocity field of the fluid and ρ the density of the fluid. The equations for continuity, linear momentum, angular momentum and energy, are obtained by substituting φ as N/M , V , r × V and E, respectively. Φ=. Z. φρdV. (2.9). V. The principle of the conservation of the mass, is one of the basic laws in the study of fluid mechanics, the differential form of this concept considers a volume of control of arbitrary form in the flow. the sum of the rapidity of the variation of the mass of the volume and the net mass output of the surface must be zero. So applying the Reynolds transport theorem to a body, considering the extensive property as mass and applying divergence theorem [22]. The continuity equation in differential expression states: ∂ρ ∂ (ρUi ) + =0 (2.10) ∂t ∂xi Analogous to Newton’s second Law, the equation of conservation of motion appears, where the sum of the forces on a particle is equal to the speed of variation of its lineal momentum. Considering a 11.

(22) field velocity Vi and the body forces and surfaces forces. The net balance per unit mass is established as: ∂τki ∂(ρUi ) ∂(ρUi Uj ) + = + ρfi (2.11) ∂t ∂xj ∂xk Where τki is known as the stress tensor that accounts for the distribution of internal stresses and stresses in the continuous medium. Equation 2.3 states the mathematical model of τij . The left side of the equation represent mass times acceleration, while the right side provides the vector sum of the applied surface and body forces per unit volume [22]. By including the continuity equation on the momentum balance, and defining the Total derivative: ρ. ∂Ui ∂Ui ∂τki DUi = + ρUj = + ρfi Dt ∂t ∂xj ∂xk. (2.12). The first law of thermodynamics establishes the conservation of energy. Considering a system, where the energy change is the sum of the energy input and outputs, and also the heat and of work introduced to the system. The energy of the system comprises the internal energy and the kinetic energy, if we consider the internal energy per unit mass as e in a volume differential element. The differential equation of energy conservation is established as follows. ρ. ∂e ∂e De ∂qi ∂Ui = ρ + ρUj = τji − Dt ∂t ∂xj ∂xj ∂xi. (2.13). where qi is the vector of heat flow. The left side of the equation represents the rate of change of the internal energy, the first term on the right side represents the action of the force on the deformation and the last term is the effect of the heat transfer [15].. 2.5. Constitutive equations The partial differential equations, (2.10, 2.12 and 3.12) are all based on the conservation law of fluid mechanics, they constitute a system of 5 non-linear partial differential equations, which unknown variables are the ρ, Vi (3 components), τij , (9 elements), e, qi (3 elements). The force of the body fi depends of external factors of the fluid. The previous equations are applicable in general, but it is necessary to specify the characteristics related to the nature of the medium. Usually τij and qi , are written as a function of other properties: τij = τij (ρ, Ui , e). qi = qi (ρ, Ui , e). 12. (2.14).

(23) 2.6. Ideal Flow A way to simplify the governing equations is considering ideal flow, where the behave of the fluid despise the viscous effects and also friction forces of the fluid. A more adequate way to define the equations of the ideal flow is based on the definition of the stress tensor, in this way, said tensor will be isotropic. This means that the normal stress is independent of the orientation of the surface ds as given by a normal vector n. This reduces τij to hydrostatic case [22]. Therefore: τij = −pδij (2.15) Where p is the pressure of the fluid, and δij is the identity tensor. The conservation equations reduces to the related equations, called Euler equations. ∂ρ ∂ (ρUi ) + =0 ∂t ∂xi ∂Ui ∂P ∂Ui + ρUk =− + ρfi ρ ∂t ∂xk ∂xi ∂e ∂e ∂Ui ∂qi ρ + ρUk = −P − ∂t ∂xk ∂xi ∂xi. 2.6.1. (2.16). Newtonian approximation.. Nature of fluids always opposes to resistance to shear deformation, and this can not be explained by perfect fluid models. The simplest useful constitutive model to represent viscous effects is based by the Newton’s viscous law (eq. 2.7). In the case of a fluid that posses viscosity, in addition to normal stresses, the shear stresses are now important. The stress tensor model for a Newtonian fluid is given by the following expression: ∂Ui ∂Uj ∂Uk +µ + τij = −pδij + λδij ∂xk ∂xj ∂xi. !. (2.17). This expression is established in this way because it must meet the conditions: When the fluid is at rest, the effort is due to the pressure exerted by its weight. When it does not depend on the fluid rotation the stress tensor is therefore isotropic. In this way the constitutive equation will depend on the values of λ and µ, which are coefficients of viscosity and are properties of the fluid. The case of µ, Newton viscosity law can be applied to calculate it. While for the case of lambda, the following expression is used: τi=j = −3P + (3λ + 2µ) 13. ∂Ui ∂xi. (2.18).

(24) Considering the average of the normal efforts does not depend on the viscosity 3λ + 2µ = 0. (2.19). The set of fluids that satisfies this relationship is known as Stokes fluids. (Including monoatomic gases), in the case of an incompressible fluid, the stress tensor is reduced to: ∂Ui ∂Uj τij = −pδij + µ + ∂xj ∂xi. 2.6.2. !. (2.20). Non-compressible Governing Equations. It is deeply known that most fluids in nature are non-Newtonian, but the behavior of fluids such as water and air can be modeled congruently with the viscous effects of Newtonian fluids. Considering the relationships previously established for the stress tensor in fluids and assuming the fluid is not compressible, we can establish a set of equations of fluid motion called the Navier-Stokes equations. Continuity Equation: ∂ρ ∂ (ρUi ) + =0 ∂t ∂xi Momemtum equation (Navier Stokes Equations). (2.21). ". ∂Ui ∂Ui ∂p ∂ ∂Ui ∂Uk ρ + ρUj =− + µ + ∂t ∂xj ∂xi ∂xk ∂xk ∂xk. !#. + ρfi. (2.22). Energy Conservation ". #. ∂e ∂e ∂Uj ∂qi ∂Ui ∂Uk ∂Uk ρ + ρUj = −p +µ + + ∂t ∂xj ∂xj ∂xk ∂xj ∂xi ∂xi. 2.6.3. (2.23). Non-Newtonian Flow. A non-Newtonian fluid is a fluid that does not follow Newton’s Law of Viscosity. Most commonly, the viscosity of non-Newtonian fluids is dependent on shear rate or shear rate history. However, still exhibit normal stress-differences or other non-Newtonian behavior. Specifically two types of non-Newtonian fluids are commonly studied: The Reopessici fluids: are substances which tend to increase their viscosity with the passing of time when subjected to shear forces and such variation of viscosity depends on the speed with which that force is applied, and Thixotropic-fluids: these are substances which, in contrast to the Reopessici fluids, they tend to decrease their viscosity with the passing of time when subjected to shear forces.. 14.

(25) 2.7. Boundary Layer approximation The boundary layer theory is the asymptotic theory of the Navier Stokes equations for high Reynolds numbers. This theory was developed by Ludwig Prandtl in 1904, since the inviscid flow solution does not satisfy the no-slip condition at the wall, boundary layer theory is called a singular perturbation method. The concept of the boundary layer, therefore, implies that flows at high Reynolds numbers can be divided up into two unequally large regions. In the bulk of the flow region, the viscosity can be neglected, and the flow corresponds to the inviscid limiting solution. This is called the inviscid outer flow. The second region is the very thin boundary layer at the wall where the viscosity must be taken into account [23].. 2.7.1. Laminar Boundary Layer on a Flat Plate. The decelerated fluid particles in the boundary layer do not in all cases, remain in the thin layer which adheres to the body along the whole whetted length of the wall. Considering a flat border in a flow. For viscous effects there is a boundary layer. In Figure 2.2, the distance between the x axes and the boundary represents the thickness of the boundary layer δ(x). The liquid layer begins at x = 0, where δ = 0. In other words, it is the place where the flow meets the boundary and the viscous and affective effects are presented. Downstream the boundary layer develops and the thickness grows with x. Frequently the boundary is arbitrarily given as being at the point where the velocity reaches a certain percentage of the outer velocity, e.g. 99%. For clarity, an index is often used, e.g δ99 [23]. For laminar plate boundary layers the boundary–layer thickness can easily be estimated, in the boundary layer the inertial forces and the friction forces are in equilibrium. Using scale analysis, it can be shown that the equations of motion can be reduced in order to find the boundary layer . For a plate of length x, ∂u is thickness. Inertial forces per unit volume are expressed as: ρu ∂u ∂x ∂x U∞ ∂τ proportional to x , where U∞ is the velocity of the outer flow. The friction force per unit is equal ∂y 2. and in the laminar flow considering the Navier Stokes Equation. µ ∂∂yu2 , so the friction force per unit is. Figure 2.2: Boundary layer development along a flat plate. Taken from [24]. 15.

(26) proportional to. µU∞ δ2. Therefore: µ. 2 U∞ ρU∞ ∼ δ2 x. (2.24). The boundary layer thickness is: s. δ∼. 2.7.2. vx U∞. (2.25). Turbulent Boundary Layer. The boundary layer on a plate does not always remain laminar. After a certain distance the boundary layer becomes turbulent. The boundary layer on a plate is laminar close to the leading edge and becomes turbulent further downstream. The treatment of turbulent boundary layers is far more difficult due to the time-dependent variation of the flow properties. One of the most widely used techniques in which turbulent flows are tackled is to apply Reynolds decomposition.. 2.7.3. Boundary Layer separation. The phenomenon of boundary layer separation is intimately connected with the pressure distribution in the boundary layer. The classical example of the phenomena consider a circular cylinder, as shown in figure 2.3, In the frictionless flow , the fluid particles are accelerated on the upstream half from the point D to E, and decelerated on the downstream half from E to F. Hence the pressure decreases from D to E and increases from E to F. when the flow is started up to motion in the first instant is very nearly frictionless, and remains so as long as the boundary layer remains this. Outside of the boundary layer a transformation occurs, of the pressure into kinetic energy along D to E. The reverse taking place along E to F. so that a particle arrives at F with the same velocity as it land at D. A fluid particle which moves in the immediate vicinity of the wall in the boundary layer remains under the influence of the same pressure field as that existing outside, because the external pressure is impressed on the boundary layer. Owing to the large friction forces in the thin boundary layer such a particle consumes so much of its kinetic energy on its path from D to E that the remainder is too small to surmount the pressure hill from E to F. Such particle cannot move far into the region of increasing pressure between E and F and its motion is eventually arrested. The external pressure causes it the to move in opposite direction. The pressure increases along the body contour from left to right. The vortex becomes separated shortly afterwards and moves downstream in the fluid. This circumstances changes completely the flied of the flow in the wake. and the pressure distribution suffers a radical change as compared with frictionless flow. In the eddying region behind the cylinder there is a considerable suction. This suctions causes a large pressure drag on the body. At a large distance from the body it is possible to discern a regular pattern of vortices, which are known as Kármán Vortex [23]. Its well known that a circulation near-wall layer in the separated flow behind a backward-facing step significantly affects heat transfer process. The flow past a downward step is known to be a. 16.

(27) Figure 2.3: Velocity Contour of Flow through a Cylinder. simplest example of flow with separation. The flow past a rib is more complicated since it displays an additional separation in the upstream region of the obstacle. Therefore the main idea is the possibility of heat transfer intensification or suppression by modifying the shape of obstacles or by introducing artificial perturbations into the approaching flow [25].. 2.7.4. Vorticity and Circulation. The presence of vorticity in a fluid always implies the rotation of the fluid particles, accompanied or not by some transverse deformation. In a real fluid its existence is intimately linked to the tangential tensions. The equation for studying the kinetics of this field (called the vorticity transport equation) is obtained by taking the rotational on both sides of the momentum equation of the Navier-Stokes equations and expressing the local derivative in terms of the substantial derivative. Firstly the circulation contained in a closed curve within the flow is defined by the integral around the curve, where the component of the velocity tangent to the curve is described by a contour element: Γ=. I. U · ds. (2.26). Using the Stokes theorem, we can convert this integral to a surface integral. where n is the normal vector to the element dA. Z Γ = (∇ × U ) · dA (2.27) By this way we can define the vorticity vector as the curl of velocity vector. ~ ω ~ =∇×U. (2.28). Using the Einstein notation the vector is defined by: ∂Uj (2.29) ∂xk One of the conclusions of quantifying the circulation in a fluid, is to determine if the fluid is irrotational, for it: Z Γ = ω · ndA = 0 ωi = jki. A. 17.

(28) 2.7.5. Helmholtz Theorem. Considering the previous definition of vorticity in equation 2.28, if we define that the vector ω is a selenoidal field, (also called incompressible or null divergence field) in a given domain. Using Gauss Theorem in an arbitrary volume V. Z. ∇ · ωdA = V. Z. ω · ndA = 0. (2.30). A. This implies that if the product ω · n is zero at the borders, it can be concluded that: Γ1 = Γ2. (2.31). ω 1 A 1 = ω 2 A2. (2.32). This theorem implies that: This theorem is valid for any type of fluid and warns that the generation vortices is a closed system and analogous to the equation of continuity, the vortices end in some border.. 2.7.6. Vorticity equation. For the case of an incompressible fluid in a conservative force field the navier Stokes equations take the form of the equation 2.22. Using a vector identity, the Navier Stokes equations can be written as follows: ~ |2 ~ |U ∂U ~ × U ~ ×ω ~ + ρ∇ − ρU ~ = −∇p + µ∇2 U (2.33) ∂t 2 Applying the curl to the previous expression we can arrive to an equation in terms of the vorticity.. ρ. ρ. ∂~ω − ρ∇ × (~u × ω ~ ) = µ∇2 ω ~ ∂t. (2.34). Regrouping the equation and using a vector identity, we can applying the condition of incompressibility similar to the expression of Helmholtz, and therefor we arrive at an expression that indicates the rate of change of the vorticity of a fluid as a function of the gradient of velocity and the diffusion of viscous effects. D~ω ~ + µ∇2 ω ρ = ρ (~ω · ∇) U ~ (2.35) Dt The vorticity originates fundamentally in the solid contours because the fluids are not able to slide on them, and then it propagates to the interior of the fluid following the law of variation described by Equation 2.35. The first term corresponds to the variation of vorticity by deformation of the vortic lines. This phenomenon occurs in both viscous and non-viscous fluids, however it is a remarkable fact that when the fluid is non-viscous (ideal) this is the only way in which vorticity can vary. The second term of Equation 2.35, which unlike the first is only evaluated in viscous fluids, corresponds to the variation of vorticity by viscous diffusion and has an analogy (similar differential 18.

(29) equation) with the phenomenon of heat conduction in solids. Due to this phenomenon, particles that do not have vorticity acquire it from neighboring particles that do, producing a vorticity diffusion into the fluid. Truthfully the last term encompasses that viscosity is the ability of particles to infect their vorticity and that depending on it, the fluid will be more or less dominated by vorticity.. 2.7.7. Boundary conditions. A concrete problem of solids or fluids that involves to solve partial differential equations, it is necessary to the boundary conditions imposed on the body. In Fluid Mechanics this last aspect is delicate and it is worth devoting a section to describe the main possible types. • Dirichlet condition: In parts of the boundary of the material the velocity profile is known, v(x, t), An example of it, is the velocity profiles at the inlet in a pipe., in this case is well known that the velocity profile is parabolic. The mathematical expression for this kind of boundary conditions for a determined area Γv : x ∈ Γv ∀t. v(x, t) = v(x, t). (2.36). • Impenetrability Contour Condition: It expresses that the flow can not penetrate the walls of the material that contains it. In this case, and if the wall of the container has a normal vector n, the velocity of the fluid must be tangential to this surface, therefore: (v(x, t) − v(x, t)) · n(x) = 0. x ∈ Γv ∀t. (2.37). • No slip Condition : The viscous fluids not only can not penetrate the walls of the containers but their particles in contact with the walls must have a relative velocity null with respect to the position of these last ones. In this case, then, the contour condition throughout the wall Γv : v(x, t) = 0. x ∈ Γv ∀t. (2.38). • Pressure or Tension Contour condition: Besides the Velocity profile, the values of the Tensions in parts of the contour can also be known. If the stress vectors have a known value t on the contour Γt then the boundary conditions are of the form: σ(x, t)n(x) = t(x, t). x ∈ Γv ∀t. (2.39). In order to simplify this expression, a known value of pressure at the border is usually assumed. p(x, t) = p(x, t). x ∈ Γv ∀t. (2.40). In order to simplify this expression, a known value of pressure at the border is usually assumed. p(x, t) = p(x, t) 19. x ∈ Γv ∀t. (2.41).

(30) Coupled boundary condition: It is also possible to impose mixed boundary conditions that include the velocity and pressure. For example, pressure can be imposed on one side of the contour and tangential velocity components on the other. Alternatively, the tangential component of the tension and the normal component of the speed could be imposed. • Free surface Condition: It is used to determine the evolution of the free surface of a fluid in motion. Unlike the previous boundary conditions, this is used to determine the free contour Γl , which is not fixed. There are two ways to establish this condition. The first is to impose that the free surface is formed by those points of the fluids whose pressure is equal to the fluid surrounds it. In the case of a fluid located outdoors, the condition is expressed as: Γl = x : p(x, t) = patm Where patm is the atmospheric pressure.. 20. (2.42).

(31) 2.8. Heat Transfer Mechanisms 2.8.1. Conduction- Fourier’s Law. Joseph Batist Fourier published his remarkable book Théorie Analytique de la Chaleur in 1822, where he explained a complete exposition of the heat conduction theory. One of its outstanding proposal is that the heat flux resulting from thermal conduction is proportional to the magnitude of the temperature gradient and opposite to it in sign [26]. dT (2.43) dx The constant k is called thermal conductivity, the heat flux q is a vector quantity, if temperature decreases with x, q will be positive flowing in x-direction; if T increases with x, q will be negative, flowing in opposite direction. In either case, q flows from higher temperatures to lower temperatures. The three dimensional Fourier equation states: q = −k. qi = −k. 2.8.2. ∂T ∂xi. (2.44). Heat Diffusion Equation. A main objective behind heat conduction analysis, is to determine the Temperature Distribution T (x, y, z, t) in a medium knowing temperatures or heat transfer rates on the surfaces of the material. Consider the first law of thermodynamics, as the conservation of energy statement. assuming there is no mass transfer, or radiation sources, but a power source Q. ∆U = Q − W. (2.45). The conservation of energy states that the energy flowing into a control volume must be compensated either by energy flowing out of the volume, by a change in the total energy, or a combination of both . Neglecting the effects of kinetic energy. The quantity of heat flowing out of the control volume of dx dy dz dimensions in the direction of positive x during a time dt must be determined using a Taylor expansion [27]: (qx )out = qx (x + dx) = qx +. ∂qx ∂ 2 qx dx2 dx + + ... ∂x ∂x2 2!. (2.46). Due to energy conservation, since the volume dx dy dz remains constant over the interval dt, the only way to decrease or increase the energy is to affect the temperature (T), specific heat (c), density (ρ), or some combination of all the three. Assuming this the energy equation can be expressed as: ∂(ρcT ) ∂qi + =0 ∂t ∂xi 21. (2.47).

(32) It is normally assumed that the density and specific heat do not change with time in which case, it simplifies to the heat diffusivity equation where the phenomenological equation that completes the derivation is the Fouriers Law, equation 2.44, inserted in equation 2.47. ∂ 2T ∂T +α 2 =Q ∂t ∂xi. (2.48). where α = k/ρc is the thermal diffusivity,. 2.8.3. Convection. Convection is one of the three forms of heat transfer. It is characterized because it is produced by means of a fluid (liquid, gas or plasma) that transports heat between zones with different temperatures. Convection occurs only through materials, the evaporation of water or fluids. Convection itself is the transport of heat through the movement of fluid. Heat transfer involves the transport of heat in a volume and the mixing of macroscopic elements of hot and cold portions of a gas or a liquid. Convective heat transfer is expressed by Newton’s Law of Cooling, The heat flux for convection over a surface, through a fluid crossing is defined by: q˙00 = h(Ts − T∞ ). (2.49). Where h (W/m2 K) is the heat convection coefficient of the fluid. Ts (K) is the surface temperature, and T∞ (K) is the free current temperature of the fluid [28]. The h coefficient varies upon different factors, such as the geometrical properties of the surface and the fluid physical parameters. If the motion of the fluid is sustained by a force in the form of pressure difference created by an external device, pump or fan, the term of "forced convection is used [28]. If the motion of the fluid is sustained by the presence of a thermally induced density gradient, then the term of "natural convection" is used. So the main problem of the convection phenomenon lies in calculating the value of h. For this, the equation below relates the value of the heat transfer coefficient in a energy balance of a contact region of fluid and a solid. 00 00 q̇conduction = q̇convection ∂T = hx (T |y=0 − T∞ ) kf ∂x −kf ∂T ∂x h= T |y=0 − T∞. (2.50). And for the global coefficient, the integral is defined as: h=. Z. hx dx. 22. (2.51).

(33) To determine the distribution of h, therefore is necessary to compute the temperature distribution of analysis region, and that means that it is necessary to solve the governing equations to obtain temperature distribution, In particular it is necessary to modify the energy equation to generalize for temperature equation. Analogous to the Heat diffusion equation 2.48, the energy equation 2.23 for Newtonian fluids, can also be expressed in therms of the Fourier’s Law 2.44, and using the thermodynamic relation of energy and enthalpy, the thermal energy equations is defined as: ∂T ∂T + ρUj Cp = βT ρCp ∂t ∂xj. ∂p ∂p + ∂t ∂xj. !. ". #. ∂Ui ∂Uk ∂Uk ∂ ∂T +µ + + k ∂xk ∂xj ∂xi ∂xi ∂xi. !. (2.52). Where β is the compressibility factor of the fluid, in incompressible fluid β = 0. It is well known that the movement of the fluid increases the heat transfer with respect to conduction mechanism. One of the most important parameter when solving heat transfer problems is the Nusselt number (Nu). This term is the traditional dimensionless form of h (convective heat transfer coefficient). In other words, it is equal to the dimensionless temperature gradient at the surface and it provides a measure of the convective heat transfer occurring at the surface. It can also be defined as the ratio of convection heat transfer to fluid conduction heat transfer under the same conditions. N uL =. L qw00 (Convection) =h 00 qw (Conduction) k. (2.53). A Nusselt number of order unity would indicate a sluggish motion little more effective than pure fluid conduction: for example, laminar flow in a long pipe. A large Nusselt number means very efficient convection: For example, turbulent pipe flow yields Nu of order 100 to 1000. Reynolds number is also an important parameter on this study. It is the ratio of inertia forces to viscous forces in the fluid. At large Re numbers, the inertia forces, which are proportional to the density and the velocity of the fluid, are large relative to the viscous forces; thus the viscous forces cannot prevent the random and rapid fluctuations of the fluid (turbulent regime). Re =. ρV δ Inertiaf orces = V iscousf orces µ. (2.54). For the problem of heat convection along a definite surface, as a pipe or a as a geometric variation (fins), it is also known that Nusselt number varies as a function the geometry, the material properties, the velocity of the fluid, the viscosity etc. Described in therms of dimensionless parameters Nusselt could depends on: N u = f (Re, P r, ρ, µ, G) (2.55) Where Pr is known as the Prandtl number, defined as the quotient between moment diffusivity (viscosity) and thermal diffusivity. G are geometrical factors that have an impact on the fluid behavior. Several correlations for Nu number have been proposed for different problems.. 23.

(34) Free Convection Free convection, or natural convection, is a spontaneous flow arising from non homogeneous fields of volumetric (mass) forces (gravitational, centrifugal, Coriolis, electromagnetic, etc.). Free-convective flows may be laminar and turbulent. A flow past a solid surface, the temperature of which is higher (lower) than that of the surrounding flowing medium, is the most widespread type of free convection. In the theoretical analysis of FC flows and heat transfer the laws of momentum, mass and energy conservation at certain boundary conditions are used. The Boussinesq approximation of "weak" thermal convection is widely applied, i.e., density deviations from a mean value are considered to be negligible in all the equations, except for the equation of motion where they are taken into account in the buoyancy force term. Forced Convection Forced convection is a special type of heat transfer in which fluids are forced to move, in order to increase the heat transfer. For instance, the forced convection action can be achieved with a ceiling fan, a pump, suction device, or other. Convection is a complex heat transfer method, but can be expressed by Newton’s Law of Heating and Cooling, equation(2.49). To determine the distribution temperatures, fluid velocity, pressure gradient of a fluid forced convection problem is necessary to solve the conservation partial differential equations establishing the boundary conditions of the problem. A heat exchanger is coupled convection and conduction thermal analysis, The present thesis center to solve the convection problem for a definite geometry and boundary conditions in effort to determine the contribution of the heat convection coefficient. Newton’s Law of Heating and Cooling is a primarily approach to understand the nature of convection, the heat convection coefficient changes depending on whether or not the convection is forced. Several correlations have been reported for both analysis for specific fluid behavior (laminar or turbulent) and also for different type of geometries. Natural and Forced convection are eventually a difficult problems of interest due to the fact that the governing equations must be solved, also the problem could be even more complex if the interest fluid is not laminar, or Newtonian. In some cases the properties of the fluid varies with the temperature and pressure, and therefore a couple solution must be applied. That is why the use of computational tools allows to obtain a solution of the partial differential equations by the finite element method. Fluent represents a considerable advantage for the solution of complex thermal problems.. 24.