Book contents
- Frontmatter
- Contents
- Preface
- Frequently Used Notation
- 1 Thermophysical and Transport Fundamentals
- 2 Boundary Layers
- 3 External Laminar Flow: Similarity Solutions for Forced Laminar Boundary Layers
- 4 Internal Laminar Flow
- 5 Integral Methods
- 6 Fundamentals of Turbulence and External Turbulent Flow
- 7 Internal Turbulent Flow
- 8 Effect of Transpiration on Friction, Heat, and Mass Transfer
- 9 Analogy Among Momentum, Heat, and Mass Transfer
- 10 Natural Convection
- 11 Mixed Convection
- 12 Turbulence Models
- 13 Flow and Heat Transfer in Miniature Flow Passages
- APPENDIX A Constitutive Relations in Polar Cylindrical and Spherical Coordinates
- APPENDIX B Mass Continuity and Newtonian Incompressible Fluid Equations of Motion in Polar Cylindrical and Spherical Coordinates
- APPENDIX C Energy Conservation Equations in Polar Cylindrical and Spherical Coordinates for Incompressible Fluids With Constant Thermal Conductivity
- APPENDIX D Mass-Species Conservation Equations in Polar Cylindrical and Spherical Coordinates for Incompressible Fluids
- APPENDIX E Thermodynamic Properties of Saturated Water and Steam
- APPENDIX F Transport Properties of Saturated Water and Steam
- APPENDIX G Properties of Selected Ideal Gases at 1 Atmosphere
- APPENDIX H Binary Diffusion Coefficients of Selected Gases in Air at 1 Atmosphere
- APPENDIX I Henry's Constant, in bars, of Dilute Aqueous Solutions of Selected Substances at Moderate Pressures
- APPENDIX J Diffusion Coefficients of Selected Substances in Water at Infinite Dilution at 25°C
- APPENDIX K Lennard–Jones Potential Model Constants for Selected Molecules
- APPENDIX L Collision Integrals for the Lennard–Jones Potential Model
- APPENDIX M Some RANS-Type Turbulence Models
- APPENDIX N Physical Constants
- APPENDIX O Unit Conversions
- APPENDIX P Summary of Important Dimensionless Numbers
- APPENDIX Q Summary of Some Useful Heat Transfer and Friction-Factor Correlations
- References
- Index
5 - Integral Methods
- Frontmatter
- Contents
- Preface
- Frequently Used Notation
- 1 Thermophysical and Transport Fundamentals
- 2 Boundary Layers
- 3 External Laminar Flow: Similarity Solutions for Forced Laminar Boundary Layers
- 4 Internal Laminar Flow
- 5 Integral Methods
- 6 Fundamentals of Turbulence and External Turbulent Flow
- 7 Internal Turbulent Flow
- 8 Effect of Transpiration on Friction, Heat, and Mass Transfer
- 9 Analogy Among Momentum, Heat, and Mass Transfer
- 10 Natural Convection
- 11 Mixed Convection
- 12 Turbulence Models
- 13 Flow and Heat Transfer in Miniature Flow Passages
- APPENDIX A Constitutive Relations in Polar Cylindrical and Spherical Coordinates
- APPENDIX B Mass Continuity and Newtonian Incompressible Fluid Equations of Motion in Polar Cylindrical and Spherical Coordinates
- APPENDIX C Energy Conservation Equations in Polar Cylindrical and Spherical Coordinates for Incompressible Fluids With Constant Thermal Conductivity
- APPENDIX D Mass-Species Conservation Equations in Polar Cylindrical and Spherical Coordinates for Incompressible Fluids
- APPENDIX E Thermodynamic Properties of Saturated Water and Steam
- APPENDIX F Transport Properties of Saturated Water and Steam
- APPENDIX G Properties of Selected Ideal Gases at 1 Atmosphere
- APPENDIX H Binary Diffusion Coefficients of Selected Gases in Air at 1 Atmosphere
- APPENDIX I Henry's Constant, in bars, of Dilute Aqueous Solutions of Selected Substances at Moderate Pressures
- APPENDIX J Diffusion Coefficients of Selected Substances in Water at Infinite Dilution at 25°C
- APPENDIX K Lennard–Jones Potential Model Constants for Selected Molecules
- APPENDIX L Collision Integrals for the Lennard–Jones Potential Model
- APPENDIX M Some RANS-Type Turbulence Models
- APPENDIX N Physical Constants
- APPENDIX O Unit Conversions
- APPENDIX P Summary of Important Dimensionless Numbers
- APPENDIX Q Summary of Some Useful Heat Transfer and Friction-Factor Correlations
- References
- Index
Summary
An integral method is a powerful and flexible technique for the approximate solution of boundary-layer problems. It is based on the integration of the boundary-layer conservation equations over the boundary-layer thickness and the assumption of approximate and well-defined velocity, temperature, and mass-fraction profiles in the boundary layer. In this way, the partial differential conservation equations are replaced with ODEs in which the dependent variable is the boundary-layer thickness. The solution of the ODE derived in this way then provides the thickness of the boundary layer. Knowing the boundary-layer thickness, along with the aforementioned approximate velocity and temperature profiles, we can then easily find the transport rates through the boundary layer. The integral technique is quite flexible and, unlike the similarity solution method, can be applied to relatively complicated flow configurations.
Integral Momentum Equations
Let us first consider the velocity boundary layer on a flat plate that is subject to the steady and uniform parallel flow of a fluid, as shown in Fig. 5.1. We define a control volume composed of a slice of the flow field that has a thickness dx and height Y. We choose Y to be large enough so that it will be larger than the boundary-layer thickness throughout the range of interest. The inflow and outflow parameters relevant to momentum and energy are also depicted in Fig. 5.1.
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- Convective Heat and Mass Transfer , pp. 151 - 176Publisher: Cambridge University PressPrint publication year: 2011