NUMERICAL COMPUTATION OF THE FLUID FLOW AND HEAT TRANSFER BETWEEN THE ANNULI OF CONCENTRIC AND ECCENTRIC HORIZONTAL CYLINDERS XU ZHIDAO B Eng Xi’an Jiaotong University A THESIS SUBMITTED FOR THE DEGREE OF MASTER ENGINEERING DEPARTMENT OF MECHICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2003 ACKNOWLEDGEMENT The author wishes to express my deepest gratitude to his supervisors, Associate Professor T S Lee and Associate Professor H T Low, for their invaluable guidance, supervision, encouragement and patience throughout the course of the investigation I would like to thank the National University of Singapore for the research scholarship, which supports this research work Support and encouragement from my wife will always be remembered and appreciated Additionally, I would like to acknowledge the moral support and encouragement from my parents and parents-in-law Thanks also go to the staff of the Fluid Mechanics Laboratory, who contributed their time, knowledge and effort towards the fulfillment of this work Finally, the author wished to express his gratitude to those who have directly or indirectly contributed to this investigation I SUMMARY Mixed convections of air with Prandtl number of 0.701 in concentric and eccentric horizontal annuli with isothermal wall conditions are numerically investigated The inner cylinder is stationary and at a higher temperature while the outer cylinder is rotating counter-clockwise The effects of various parameters such as the radius ratio of the annulus, the eccentricity of the annulus, the Rayleigh number and Reynolds number of the rotation of the inner cylinder are studied using a two-dimensional finite-difference model Overall and local heat transfer results are obtained The physics of the flow underlying the heat transfer behavior observed is revealed by the streamline and the isotherm plots of the numerical solutions For the case of concentric cylinders, the present numerical results are in good agreement with the similar results of other investigators where such results are available In particular, rotating outer cylinder in the concentric cylinders was investigated, the flow patterns were categorized into three types, and characteristics of flow patterns and heat transfer are elucidated For the flows in horizontal eccentric annulus, the flow and the heat transfer are strongly influenced by the orientation and eccentricity of inner cylinder Around the inner cylinder, there exists a stagnant area in the opposite direction of eccentricity The turbulence and instability situation is the aim which the next researchers need to search for and new turbulence model needs to be developed II CONTENTS Page Acknowledgement I Summary II Nomenclature V List of Figures VIII Chapter Introduction 1.1 Background 1.2 Literature Review 12 1.3 Objectives and Scope Chapter Problem Formulation 14 15 2.1 Governing Equations 2.1.1 Simplifying the Governing Equations 2.1.2 Stream-Function Vorticity Formulation 2.1.3 Non-Dimensionalization 19 2.2 Coordinate System 2.2.1 Concentric Geometry 2.2.2 Eccentric Geometry 21 2.3 Boundary Conditions 2.3.1 Velocity and Thermal Boundary Conditions 2.3.2 Vorticity Boundary Conditions 2.3.3 Stream-Function Boundary Conditions 24 2.4 Specification of Cases Studied Chapter Analysis and Numerical Solutions 26 3.1 Finite Difference Approaches 26 3.2 Solution Procedure 27 3.3 Numerical Methods 28 3.3.1 Parabolic Equations 3.3.2 Elliptic Equations 3.4 Upwind Differencing 31 3.5 Boundary Conditions 33 3.5.1 Vorticity Transport Equation 3.5.2 Stream-Function Vorticity Equaiton 3.5.3 Energy Equation 3.5.4 Progressive Built-up of Boundary Conditions III 3.6 Temperature Interpolation at the Wall 36 3.7 Convergence Criteria 37 3.7.1 Convergence of the Inner Iterations 3.7.2 Overall Convergence 3.8 Mesh Size and Time Step 40 3.9 Computation of the Overall Heat Transfer Coefficients 40 Chapter Results and Discussion for Concentric Cylinders 41 4.1 Flow Mechanics and General Patterns 42 4.2 Effects of Reynolds Number 44 4.2.1 Streamline and Isothermal plots 4.2.2 Local equivalent thermal conductivity Keql 46 4.3 Effects of Rayleigh Number 4.3.1 Low Rayleigh number 4.3.2 Increasing Rayleigh number 51 4.4 Cases with higher Radius Ratio 4.4.1 Flow and temperature distribution 4.4.2 The local equivalent thermal conductivity 54 Chapter Results and Discussion for Eccentric Cylinders 5.1 Effects of Orientation of the Inner Cylinder 55 5.2 Effects of Eccentricity 57 5.3 Increasing Rayleigh Number to 10 58 60 Chapter Conclusions and Recommendations 6.1 Concentric Cases 60 6.2 Eccentric Cases 61 6.3 Recommendations 62 References 63 Figures 71 IV Nomenclature A Area C Constant in the transformation equations from Bipolar coordinate systems to Cartesian Coordinate System D Diameter e Magnitude of the eccentricity vector, e (= | e | ) e Eccentricity Vector, e =( e h , ev ) er Magnitude of the eccentricity ratio vector e , ( = | e | ) e Eccentricity ratio vector, ( = e /L ) g Gravitational acceleration, ( = | g | ) − −r − − − −r −r − − g Gravitational acceleration vector − hξ , hη Metric or scale factors in the Bipolar coordinate system ig Unit vector in the direction of the gravity vector, i g = g /g k Thermal conductivity k eqe Overall equivalent thermal conductivity k eql Local equivalent thermal conductivity L ‘Mean’ clearance between the two cylinders, ( = ro − ri ) Nu D Nusselt number based on the diameter of the heated cylinder, p, P Pressure Pr Prandtl number, (= υ / α ) Ra D Rayleigh number based on the diameter of the heated cylinder, (= Ral , Ra − Reyleigh number based on the mean clearance L, ( = gβ∆T ' L3 υα − ( = hD/k ) gβ∆T ' D υα ) V ) Re D Re L , Re Reynolds number based on the diameter of the heated cylinder, ( = Reynolds number based on the mean clearance L, (= ri Ω i L υ t Time T Temperature U Velocity vector x, y Coordinate variables in the Cartesian coordinate system r Radius Ri , ri Radius of inner cylinder Ro , ro Radius if outer cylinder − ri Ω i D υ ) ) Greek α Thermal diffusivity β Coefficient of thermal expansion γ Angle measured clockwise from the upward vertical through the center of the heat cylinder ε Total emissivity of a surface ζ Vorticity η,ξ Coordinate variables in the Bipolar coordinate system θ Angular coordinate in the Bipolar coordinate system υ Kinematic viscosity ρ Density σ Stefan-Boltzmann’s constant φ Angular position of the gravity vector relative to the negative y-axis measured in the clockwise direction ψ Stream function Ω, ϖ Angular speed VI Subscripts i for inner cylinder; also used as an indexing integer variable for the mesh points o for outer cylinder r for reference quantity; also used as an indexing integer variable for the mesh points w for wall Superscripts k number of iteration n number of the time step or global iteration ‘ the ‘prime’ symbol emphasizes the dimensional form of a variable as distinct from its non-dimensional usage VII List of Figures Fig 4.1 Streamline and Isotherm plots with Ra= × 10 and Radius Ratio = 2.6 for different Reynolds numbers 73 Fig 4.2 The effects of rotation on the local heat transfer coefficient, Radius Ratio=2.6, Ra= × 10 74 Fig 4.3 Streamline and Isotherm plots with Ra= 10 and Radius Ratio = 2.6 for different Reynolds numbers 75 Fig 4.4 Streamline and Isotherm plots with Ra= 10 and Radius Ratio = 2.6 for different Reynolds numbers 77 Fig 4.5 Streamline and Isotherm plots with Ra= 10 and Radius Ratio = 2.6 for different Reynolds numbers 79 Fig 4.6 The effects of rotation on the local heat transfer coefficient, Radius Ratio=2.6, Ra= 10 82 Fig 4.7 The effects of rotation on the local heat transfer coefficient, Radius Ratio=2.6, Ra= 10 83 Fig 4.8 Transitional Reynolds Numbers at different Rayleigh Numbers 84 Fig 4.9 The effects of rotation on the overall heat transfer coefficient at various Ralyeigh numbers, Radius Ration=2.6 84 Fig 4.10 Streamline and Isotherm plots with Ra= 5× 10 and Radius Ratio = 5.0 for different Reynolds numbers 85 Fig 4.11 The effects of rotation on the local heat transfer coefficient, Radius Ratio=5.0, Ra= 5× 10 87 Fig 5.1.1 Streamline and Isotherm plots with er = / , Ra = 10 , Radius Ratio = 2.6, and Φ = for different Reynolds numbers 88 Fig 5.1.2 Streamline and Isotherm plots with er = 1/ , Ra = 10 , Radius Ratio = 2.6, and Φ = π / for different Reynolds numbers 90 Fig 5.1.3 Streamline and Isotherm plots with er = / , Ra = 10 , Radius Ratio = 2.6, and Φ = π for different Reynolds numbers 92 Fig 5.1.4 Streamline and Isotherm plots with er = 1/ , Ra = 10 , Radius Ratio = 2.6, and Φ = 3π / for different Reynolds numbers 94 Fig 5.1.5 The effects of rotation on the local heat transfer coefficient, 96 VIII with er = 1/ , Ra = 10 , Radius Ratio = 2.6, and Φ = for different Reynolds numbers Fig 5.1.6 The effects of rotation on the local heat transfer coefficient, with er = / , Ra = 10 , Radius Ratio = 2.6, and Φ = π / for different Reynolds numbers 97 Fig 5.1.7 The effects of rotation on the local heat transfer coefficient, with er = / , Ra = 10 , Radius Ratio = 2.6, and Φ = π for different Reynolds numbers 98 Fig 5.1.8 The effects of rotation on the local heat transfer coefficient, with er = 1/ , Ra = 10 , Radius Ratio = 2.6, and Φ = 3π / for different Reynolds numbers 99 Fig 5.2.1 Streamline and Isotherm plots with er = 1/ , Ra= 10 , Radius Ratio = 2.6, and Φ = for different Reynolds numbers 100 Fig 5.2.2 Streamline and Isotherm plots with er = 1/ , Ra= 10 , Radius Ratio = 2.6, and Φ = π / for different Reynolds numbers 102 Fig 5.2.3 Streamline and Isotherm plots with er = 1/ , Ra= 10 , Radius Ratio = 2.6, and Φ = π for different Reynolds numbers 104 Fig 5.2.4 Streamline and Isotherm plots with er = 1/ , Ra= 10 , Radius Ratio = 2.6, and Φ = 3π / for different Reynolds numbers 106 Fig 5.2.5 The effects of rotation on the local heat transfer coefficient, with er = 1/ , Ra = 10 , Radius Ratio = 2.6, and Φ = for different Reynolds numbers 108 Fig 5.2.6 The effects of rotation on the local heat transfer coefficient, with er = 1/ , Ra = 10 , Radius Ratio = 2.6, and Φ = π / for different Reynolds numbers 109 Fig 5.2.7 The effects of rotation on the local heat transfer coefficient, with er = 1/ , Ra = 10 , Radius Ratio = 2.6, and Φ = π for different Reynolds numbers 110 Fig 5.2.8 The effects of rotation on the local heat transfer coefficient, with er = 1/ , Ra = 10 , Radius Ratio = 2.6, and Φ = 3π / for different Reynolds numbers 111 Fig 5.3.1 Streamline and Isotherm plots with er = / , Ra= 10 , Radius Ratio = 2.6, and Φ = for different Reynolds numbers 112 IX Keql γ ω Re=0 Re=5 Re=50 Re=100 Re=200 -180 -90 90 Angular Position 180 a) Outer Cylinder Keql γ Re=0 Re=5 Re=50 Re=100 ω Re=200 -180 -90 90 Angular Position 180 a) Outer Cylinder Fig 5.5.7 The effects of rotation on the local heat transfer coefficient, with er = / , Ra = 10 , Radius Ratio = 2.6, and Φ = π for different Reynolds numbers 146 Keql γ ω Re=0 Re=5 Re=50 Re=100 Re=200 -180 -90 90 Angular Position 180 a) Inner Cylinder Keql γ Re=0 ω Re=5 Re=50 Re=100 Re=200 -180 -90 90 Angular Position 180 b) Outer Cylinder Fig 5.5.8 The effects of rotation on the local heat transfer coefficient, with er = / , Ra = 10 , Radius Ratio = 2.6, and Φ = 3π / for different Reynolds numbers 147 Streamline Isotherms a) Re=0 b) Re=5 Isotherms Streamline Streamline Isotherms Re=50 Fig 5.6.1 Streamline and Isotherm plots with er = / , Ra= 10 , Radius Ratio = 2.6, and Φ = for : a) Re=0; b) Re=5; c) Re=50 148 Streamline Isotherms d) Re=100 Streamline Isotherms e) Re=200 Fig 5.6.1(Continued) Streamline and Isotherm plots with er = / , Ra= 10 , Radius Ratio = 2.6, and Φ = for : d) Re=100; e) Re=200 149 Streamline Isotherms a) Re=0 Streamline Isotherms b) Re=5 Streamline Isotherms c) Re=50 Fig 5.6.2 Streamline and Isotherm plots with er = / , Ra= 10 , Radius Ratio = 2.6, and Φ = π / for : a) Re=0; b) Re=5; c) Re=50 150 Streamline Isotherms d) Re=100 Streamline Isotherms e) Re=200 Fig 5.6.2(Continued) Streamline and Isotherm plots with er = / , Ra= 10 , Radius Ratio = 2.6, and Φ = π / for : d) Re = 100; e) Re=200; 151 Isotherms Streamline a) Re=0 Isotherms Streamline b) Re=5 Streamline Isotherms c) Re=50 Fig 5.6.3 Streamline and Isotherm plots with er = / , Ra= 10 , Radius Ratio = 2.6, and Φ = π for : a) Re = 0; b) Re = 5; c) Re=50 152 Streamline Isotherms d) Re=100 Streamline Isotherms e e) Re=200 f) Fig 5.6.3(Continued) Streamline and Isotherm plots with er = / , Ra= 10 , Radius Ratio = 2.6, and Φ = π for : d) Re = 100; e) Re = 200 153 Isotherms Streamline a) Re=0 b) Re=5 Streamline Isotherms Isotherms Streamline h) Re=50 Fig 5.6.4 Streamline and Isotherm plots with er = / , Ra= 10 , Radius Ratio = 2.6, and Φ = 3π / for : a) Re = 0; b) Re = 5; c) Re=50 154 Isotherms Streamline d) Re=100 Streamline Isotherms e) Re=200 Fig 5.6.4(Continued) Streamline and Isotherm plots with er = / , Ra= 10 , Radius Ratio = 2.6, and Φ = 3π / for : d) Re = 100; e) Re = 200 155 Keql γ ω Re=0 Re=5 Re=50 Re=100 Re=200 -180 -90 90 Angular Position 180 a) Inner Cylinder Keql Re=0 Re=5 γ Re=50 Re=100 ω Re=200 -180 -90 90 Angular Position 180 b) Outer Cylinder Fig 5.6.5 The effects of rotation on the local heat transfer coefficient, with er = / , Ra = 10 , Radius Ratio = 2.6, and Φ = for different Reynolds numbers 156 Keql γ ω Re=0 Re=5 Re=50 Re=100 Re=200 -180 -90 Angular Position 180 90 Keql a) Inner Cylinder 4.5 γ Re=0 Re=5 Re=50 3.5 Re=100 ω Re=200 2.5 1.5 0.5 -180 -90 90 Angular Position 180 b) Outer Cylinder Fig 5.6.6 The effects of rotation on the local heat transfer coefficient, with er = / , Ra = 10 , Radius Ratio = 2.6, and Φ = π / for different Reynolds numbers 157 γ Keql ω Re=0 Re=5 Re=50 Re=100 Re=200 -180 -90 90 Angular Position 180 a) Inner Cylinder Keql γ Re=0 Re=5 Re=50 ω Re=100 Re=200 -180 -90 90 Angular Position 180 b) Outer Cylinder Fig 5.6.7 The effects of rotation on the local heat transfer coefficient, with er = / , Ra = 10 , Radius Ratio = 2.6, and Φ = π for different Reynolds numbers 158 Keql γ Re=0 Re=5 ω Re=50 Re=100 Re=200 -180 -90 90 Angular Position 180 4.5 Keql a) Inner Cylinder γ Re=0 Re=5 Re=50 Re=100 Re=200 ω 3.5 2.5 1.5 0.5 -180 -90 90 Angular Position 180 b) Outer Cylinder Fig 5.6.8 The effects of rotation on the local heat transfer coefficient, with er = / , Ra = 10 , Radius Ratio = 2.6, and Φ = 3π / for different Reynolds numbers 159 Overall Keq 2.8 2.6 2.4 2.2 1.8 1.6 er=(0.333,0) er=(-0.333,0) er=(0.5,0) er=(-0.5,0) er=(0.667,0) er=(-0.667,0) 1.4 1.2 er=(0,0.333) er=(0,-0.333) er=(0,0.5) er=(0,-0.5) er=(0,0.667) er=(0,-0.667) Reynolds Number 50 100 Fig 5.7 The effects of rotation outer cylinder on the overall heat transfer coefficient at Ra = 10 with various eccentricity, Radius Ration=2.6 3.9 Overall Keq 3.7 er=(0.333,0) er=(-0.333,0) er=(0.5,0) er=(-0.5,0) er=(0.667,0) er=(-0.667,0) 3.5 3.3 er=(0,0.333) er=(0,-0.333) er=(0,0.5) er=(0,-0.5) er=(0,0.667) er=(0,-0.667) 3.1 2.9 2.7 Reynolds Number 2.5 50 100 Fig 5.8 The effects of rotation outer cylinder on the overall heat transfer coefficient at Ra = 10 with various eccentricity, Radius Ration=2.6 160 ... Introduction 2) The numerical model is then extended for the study of the effects of rotation of the outer cylinder on the flow and the heat transfer in concentric and eccentric annuli Furthermore,... effects of various system parameters such as the geometry of the annulus, the properties of the fluid and the rotation rate of the outer cylinder on the flow and the heat transfer in the annular... three types, and characteristics of flow patterns and heat transfer are elucidated For the flows in horizontal eccentric annulus, the flow and the heat transfer are strongly influenced by the orientation