Introduction to Convective Heat Transfer E-Book

Nevzat Onur

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 1091

Veröffentlichungsjahr: 2023

Beschreibung

INTRODUCTION TO CONVECTIVE HEAT TRANSFER A highly practical intro to solving real-world convective heat transfer problems with MATLAB¯® and MAPLE In Introduction to Convective Heat Transfer, accomplished professor and mechanical engineer Nevzat Onur delivers an insightful exploration of the physical mechanisms of convective heat transfer and an accessible treatment of how to build mathematical models of these physical processes. Providing a new perspective on convective heat transfer, the book is comprised of twelve chapters, all of which contain numerous practical examples. The book emphasizes foundational concepts and is integrated with explanations of computational programs like MATLAB¯® and MAPLE to offer students a practical outlet for the concepts discussed within. The focus throughout is on practical, physical analysis rather than mathematical detail, which helps students learn to use the provided computational tools quickly and accurately. In addition to a solutions manual for instructors and the aforementioned MAPLE and MATLAB¯® files, Introduction to Convective Heat Transfer includes: * A thorough introduction to the foundations of convective heat transfer, including coordinate systems, and continuum and thermodynamic equilibrium concepts * Practical explorations of the fundamental equations of laminar convective heat transfer, including integral formulation and differential formulation * Comprehensive discussions of the equations of incompressible external laminar boundary layers, including laminar flow forced convection and the thermal boundary layer concept * In-depth examinations of dimensional analysis, including the dimensions of physical quantities, dimensional homogeneity, and dimensionless numbers Ideal for first-year graduates in mechanical, aerospace, and chemical engineering, Introduction to Convective Heat Transfer is also an indispensable resource for practicing engineers in academia and industry in the mechanical, aerospace, and chemical engineering fields.

Leseprobe

Table of Contents

Cover

Title Page

Copyright

Dedication

Preface

About the Author

About the Companion Website

1 Foundations of Convective Heat Transfer

1.1 Fundamental Concepts

1.2 Coordinate Systems

1.3 The Continuum and Thermodynamic Equilibrium Concepts

1.4 Velocity and Acceleration

1.5 Description of a Fluid Motion: Eulerian and Lagrangian Coordinates and Substantial Derivative

1.6 Substantial Derivative

1.7 Conduction Heat Transfer

1.8 Fluid Flow and Heat Transfer

1.9 External Flow

1.10 Internal Flow

1.11 Thermal Radiation Heat Transfer

1.12 The Reynolds Transport Theorem: Time Rate of Change of an Extensive Property of a System Expressed in Terms of a Fixed Finite Control Volume

Problems

References

2 Fundamental Equations of Laminar Convective Heat Transfer

2.1 Introduction

2.2 Integral Formulation

2.3 Differential Formulation of Conservation Equations

Problems

References

3 Equations of Incompressible External Laminar Boundary Layers

3.1 Introduction

3.2 Laminar Momentum Transfer

3.3 The Momentum Boundary Layer Concept

3.4 The Thermal Boundary Layer Concept

3.5 Summary of Boundary Layer Equations of Steady Laminar Flow

Problems

References

4 Integral Methods in Convective Heat Transfer

4.1 Introduction

4.2 Conservation of Mass

4.3 The Momentum Integral Equation

4.4 Alternative Form of the Momentum Integral Equation

4.5 Momentum Integral Equation for Two‐Dimensional Flow

4.6 Energy Integral Equation

4.7 Alternative Form of the Energy Integral Equation

4.8 Energy Integral Equation for Two‐Dimensional Flow

Problems

References

5 Dimensional Analysis

5.1 Introduction

5.2 Dimensional Analysis

5.3 Nondimensionalization of Basic Differential Equations

5.4 Discussion

5.5 Dimensionless Numbers

5.6 Correlations of Experimental Data

Problems

References

6 One‐Dimensional Solutions in Convective Heat Transfer

6.1 Introduction

6.2 Couette Flow

6.3 Poiseuille Flow

6.4 Rotating Flows

Problems

References

7 Laminar External Boundary Layers: Momentum and Heat Transfer

7.1 Introduction

7.2 Velocity Boundary Layer over a Semi‐Infinite Flat Plate: Similarity Solution

7.3 Momentum Transfer over a Wedge (Falkner–Skan Wedge Flow): Similarity Solution

7.4 Application of Integral Methods to Momentum Transfer Problems

7.5 Viscous Incompressible Constant Property Parallel Flow over a Semi‐Infinite Flat Plate: Similarity Solution for Uniform Wall Temperature Boundary Condition

7.6 Low‐Prandtl‐Number Viscous Incompressible Constant Property Parallel Flow over a Semi‐Infinite Flat Plate: Similarity Solutions for Uniform Wall Temperature Boundary Condition

7.7 High‐Prandtl‐Number Viscous Incompressible Constant Property Parallel Flow over a Semi‐Infinite Flat Plate: Similarity Solutions for Uniform Wall Temperature Boundary Condition

7.8 Viscous Incompressible Constant Property Parallel Flow over a Semi‐Infinite Flat Plate: Similarity Solution for Uniform Heat Flux Boundary Condition

7.9 Viscous Incompressible Constant Property Parallel Flow over a Semi‐Infinite Flat Plate: Similarity Solutions for Variable Wall Temperature Boundary Condition

7.10 Viscous Incompressible Constant Property Flow over a Wedge (Falkner–Skan Wedge Flow): Similarity Solution for Uniform Wall Temperature Boundary Condition

7.11 Effect of Property Variation

7.12 Application of Integral Methods to Heat Transfer Problems

7.13 Superposition Principle

7.14 Viscous Flow over a Flat Plate with Arbitrary Surface Temperature Distribution

7.15 Viscous Flow over a Flat Plate with Arbitrarily Specified Heat Flux

7.16 One‐Parameter Integral Method for Incompressible Two‐Dimensional Laminar Flow Heat Transfer: Variable U

∞

(x)

and Constant

T

w

 − T

∞

 = const

7.17 One‐Parameter Integral Method for Incompressible Laminar Flow Heat Transfer over a Constant Temperature of a Body of Revolution

Problems

References

8 Laminar Momentum and Heat Transfer in Channels

8.1 Introduction

8.2 Momentum Transfer

8.3 Thermal Considerations in Ducts

8.4 Heat Transfer in the Entrance Region of Ducts

8.5 Fully Developed Heat Transfer

8.6 Heat Transfer in the Thermal Entrance Region

8.7 Circular Pipe with Variable Surface Temperature Distribution in the Axial Direction

8.8 Circular Pipe with Variable Surface Heat Flux Distribution in the Axial Direction

8.9 Short Tubes

8.10 Effect of Property Variation

8.11 Regular Sturm‐Liouville Systems

Problems

References

9 Foundations of Turbulent Flow

9.1 Introduction

9.2 The Reynolds Experiment

9.3 Nature of Turbulence

9.4 Time Averaging and Fluctuations

9.5 Isotropic Homogeneous Turbulence

9.6 Reynolds Averaging

9.7 Governing Equations of Incompressible Steady Mean Turbulent Flow

9.8 Turbulent Momentum Boundary Layer Equation

9.9 Turbulent Energy Equation

9.10 Turbulent Boundary Layer Energy Equation

9.11 Closure Problem of Turbulence

9.12 Eddy Diffusivity of Momentum

9.13 Eddy Diffusivity of Heat

9.14 Transport Equations in the Cylindrical Coordinate System

9.15 Experimental Work on the Turbulent Mean Flow

9.16 Transition to Turbulent Flow

Problems

References

10 Turbulent External Boundary Layers: Momentum and Heat Transfer

10.1 Introduction

10.2 Turbulent Momentum Boundary Layer

10.3 Turbulence Models

10.4 Turbulent Flow over a Flat Plate with Constant Free‐Stream Velocity: Couette Flow Approximation

10.5 The Universal Velocity Profile

10.6 Approximate Solution by the Integral Method for the Turbulent Momentum Boundary Layer over a Flat Plate

10.7 Laminar and Turbulent Boundary Layer

10.8 Other Eddy Diffusivity Momentum Models

10.9 Turbulent Heat Transfer

10.10 Analogy Between Momentum and Heat Transfer

10.11 Some Other Correlations for Turbulent Flow over a Flat Plate

10.12 Turbulent Flow Along a Semi‐infinite Plate with Unheated Starting Length: Constant Temperature Solution

10.13 Flat Plate with Arbitrarily Specified Surface Temperature

10.14 Constant Free‐Stream Velocity Flow Along a Flat Plate with Uniform Heat Flux

10.15 Turbulent Flow Along a Semi‐Infinite Plate with Arbitrary Heat Flux Distribution

10.16 Turbulent Transition and Overall Heat Transfer

10.17 Property Variation

Problems

References

11 Turbulent Internal Flow: Momentum and Heat Transfer

11.1 Introduction

11.2 Momentum Transfer

11.3 Fully Developed Turbulent Heat Transfer

11.4 HFD Thermally Developing Turbulent Heat Transfer

11.5 Analogies for Internal Flow

11.6 Combined Entrance Region

11.7 Empirical and Theoretical Correlations for Turbulent Flow in Channels

11.8 Heat Transfer in Transitional Flow

11.9 Effect of Property Variation

Problems

References

12 Free Convection Heat Transfer

12.1 Introduction

12.2 Fundamental Equations and Dimensionless Parameters of Free Convection

12.3 Scaling in Natural Convection

12.4 Similarity Solution for Laminar Boundary Layer over a Semi‐Infinite Vertical Flat Plate

12.5 Integral Method (von Karman–Pohlhausen Method): An Approximate Analysis of Laminar Free Convection on a Vertical Plate

12.6 Turbulent Free Convection Heat Transfer on a Vertical Plate

12.7 Empirical Correlations for Free Convection

12.8 Free Convection Within Parallel Plate Channels

12.9 Rectangular Enclosures

12.10 Horizontal Concentric Cylinders

12.11 Concentric Spheres

12.12 Spheres

Problems

References

Appendix: Thermophysical Properties of Matter

Index

End User License Agreement

List of Tables

Chapter 1

Table 1.1 Molecular diameter values for different gases.

Table 1.2 Typical ranges of Prandtl numbers.

Chapter 5

Table 5.1 Dimensions of common variables.

Table 5.14a Experimental data for Example 5.14.

Table 5.14b Reorganized experimental data for Example 5.14.

Table 5.E15a Experimental data.

Table 5.E15b Reorganized experimental data.

Table 5.E15c Reorganized experimental data.

Chapter 7

Table 7.1 Velocity distribution in laminar boundary layer.

Table 7.2 Falkner–Skan solutions with impermeable surface.

Table 7.E3 Solution of Eq. (7.61) for m = 1.

Table 7.3 Shear and shape functions correlated by Twaites [24].

Table 7.4 Solution of the energy equation for

Pr = 0.7

under ...

Table 7.5 The values of for various Prandtl numbers.

Table 7.6 The values of for low Prandtl numbers.

Table 7.7 Solution of the energy equation for uniform heat flux boundary co...

Table 7.8 Values of for uniform wall temperature boundary condition on a ...

Table 7.9 Values of

a

and

b

for different Prandtl numbers Pr.

Table 7.10 Values

C

1

,

C

2

, and

C

3

for different Prandtl numbers Pr.

Chapter 8

Table 8.1 Solutions for slug flow in circular duct under constant wall temp...

Table 8.2 The values for

Nu

D

, , and

θ

m

.

Table 8.3 Nusselt number for HFD and TFD laminar flow heat transfer in duct...

Table 8.4 The constants needed for the solution of the problem with uniform...

Table 8.5 Mean temperature and Nusselt numbers for thermal entry length in ...

Table 8.6 Eigenvalues and eigenfunctions.

Table 8.7 Nusselt numbers for uniform heat flux and parabolic velocity prof...

Table 8.8 The first 10 eigenvalues and the constants of the Cartesian Graet...

Table 8.9 Characteristic values and the constants.

Table 8.10 Local and average Nusselt numbers.

Table 8.11 Eigenvalues and constants for Eqs. (8.699) and (8.705).

Table 8.12 Constants for Eq. (8.707).

Table 8.13 Constants for Eq. (8.708).

Chapter 9

Table 9.1 Empirical exponents for the power law equation.

Chapter 10

Table 10.1 Selected universal velocity distribution models for the inner re...

Table 10.2 Values of A for different values of

Re

xc

.

Table 10.3 Values of A for different values of

Re

xc

.

Table 10.4 Selected eddy diffusivity models.

Chapter 11

Table 11.1 Selected eddy diffusivity models and velocity profiles.

Table E11.8 The ratio for fully developed turbulent flow in a pipe.

Table 11.2 Selected eigenvalues and constants for the turbulent Graetz prob...

Table 11.3 Values of

G, g

0

, and c.

Table 11.4 Selected eigenvalues and constants for the thermal entrance regi...

Table 11.5 Values of F in the Martinelli analogy for

Pr

t

 = 1

....

Table 11.6 Values of

(Tw − Tm)/(Tw − Tc)

...

Table 11.7 C and n constants for Eq. (11.375).

Table 11.8 Constants for Eqs. (11.395) and (11.396).

Table 11.9 Constants for Eq. (11.397).

Table 11.10 Constants for Eq. (11.398).

Table 11.11 Friction factor‐Reynolds number relationships to be used with E...

Chapter 12

Table 12.1 Computed parameters for free convection on an isothermal vertica...

Table 12.2

θ (0)

and

F″(0)

values for different Prandtl nu...

Table 12.3 Values of for different Prandtl numbers (UHF BC n=1/5).

Table 12.4 Critical angles.

Table 12.5 Constants C and n for Eq. (12.181).

Table 12.6 Average Nusselt numbers for flow in a vertical channel.

Table 12.7 Constants for use in Eq. (12.206).

Table 12.8 Constants for use in Eq. (12.204).

Table 12.9 Comparison of correlations for horizontal parallel plates.

Table 12.10 Constants a and b for Eq. (12.211).

Table 12.11 Critical angle for inclined enclosure.

Table 12.12 Constants C and m for Eq. (12.224).

List of Illustrations

Chapter 1

Figure 1.1 (a) Cartesian coordinate system. (b) Cylindrical coordinate syste...

Figure 1.2 Velocity components in the rectangular coordinate system.

Figure 1.E2 Illustration of Lagrangian approach.

Figure 1.E3 Illustration of Eulerian approach.

Figure 1.E7 Illustration of substantial derivative of pressure.

Figure 1.3 Illustration of sign convention in the Fourier law of heat conduc...

Figure 1.4 (a) Velocity boundary layer on a flat plate. (b) Typical velocity...

Figure 1.5 (a) Thermal boundary layer over a solid surface. (b) Temperature ...

Figure 1.6 The surface energy balance.

Figure 1.7 Definition of average heat transfer coefficient.

Figure 1.E10 (a) Problem description for Example 1.10. (b) Variation of loca...

Figure 1.8 Thickness ratio of momentum and thermal boundary layers for liqui...

Figure 1.9 Thickness ratio of momentum and thermal boundary layers for gases...

Figure 1.10 Thickness ratio of momentum and thermal boundary layers for wate...

Figure 1.11 Thickness ratio of momentum and thermal boundary layers for visc...

Figure 1.12 Coordinate system for axisymmetric flow in a circular tube.

Figure 1.E12 Problem description for Example 1.12.

Figure 1.13 (a) Laminar flow temperature distribution in a tube. (b) Turbule...

Figure 1.14 Radiation from a solid surface.

Figure 1.15 Differential element for extensive property B.

Figure 1.16 (a) System and control volume at time t, (b) System and control ...

Figure 1.17 Subregion 1 at time t.

Figure 1.18 Cylinder containing fluid.

Figure 1.19 Slanted cylinder containing fluid.

Figure 1.20 Subregion III at time

t + Δt

.

Figure 1.P1 A block sliding down the inclined plane.

Figur 1.P2 Liquid film flowing down the inclined plane.

Figur 1.P9 Flow over a triangular plate.

Figur 1.P15 Variation oil temperature as a function of distance y.

Chapter 2

Figure 2.1 System at time t for Newton's second law.

Figure 2.2 System for first law of thermodynamics.

Figure 2.3 Subregion at time t.

Figure 2.4 Three‐dimensional differential control volume element.

Figure 2.5 Cylindrical coordinates.

Figure 2.6 Spherical coordinates.

Figure 2.7 Convention for stress components on a differential control volume...

Figure 2.8 Heat conduction terms on a differential control volume element.

Figure 2.9 Differential control volume element for a viscous fluid.

Figure 2.10 Normal stress and x‐velocity component on x faces.

Figure 2.E2 Schematic of viscous fluid flow over an inclined plane.

Figure 2.E3 Pressure driven flow between parallel plates.

Figure 2.P6 Geometry and coordinate system for Couette flow.

Figure 2.P7 Geometry and coordinate system for laminar flow between infinite...

Figure 2.P8 Geometry and coordinate system for laminar flow in a circular pi...

Figure 2.P9 Geometry and coordinate system for laminar flow between infinite...

Figure 2.P10 Thermal boundary layer for low Prandtl number flow over a flat ...

Figure 2.P11 Geometry and coordinate system for liquid metal flow between in...

Figure 2.P12 Geometry and coordinate system for laminar viscous flow between...

Figure 2.P13 Geometry and coordinate system for laminar Couette flow.

Chapter 3

Figure 3.1 Two‐dimensional boundary layer.

Figure 3.2 Velocity boundary layer flow over a flat plate.

Figure 3.3 Thermal boundary layer for

Δ > δ

or

Pr 

...

Figure 3.4 Thermal boundary layer for

Δ < δ

or

Pr 

...

Figure 3.5 Similar triangles.

Figure 3.P1 Typical velocity boundary layer for laminar flow over a flat pla...

Chapter 4

Figure 4.1 Coordinate system and control volume in the boundary layer over a...

Figure 4.2 Control volume for the development of the integral equation conse...

Figure 4.3 Control volume for the development of the integral equation conse...

Figure 4.4 Control volume for the development of displacement and momentum t...

Figure 4.5 Coordinate system and control volume in the boundary layer over a...

Figure 4.6 Control volume for the development of the integral equation conse...

Figure 4.P4 Linear temperature profile for liquid metal flow over a flat pla...

Chapter 5

Figure 5.E1 A plot of H vs. x for fixed value of y.

Figure 5.E2 (a–d) A plot H vs. for fixed values of y and z.

Figure 5.1 A plot dimensionless

π

1

as a function of

π

2

.

Figure 5.E4 Forced flow over a sphere.

Figure 5.E7 Velocity and thermal boundary layers for flow over a flat plate....

Figure 5.E8 Forced flow across a cylinder.

Figure 5.E9 Problem description for Example 5.9.

Figure 5.E10 Free convection on a vertical plate.

Figure 5.E11 Problem description for Example 5.11.

Figure 5.E12 Coordinate system for laminar flow in a tube.

Figure 5.E13 Coordinate system for free convection on a vertical plate.

Figure 5.E14 Average Nusselt number versus Reynolds number for flow over a c...

Figure 5.E15a An experimental rig for investigating forced convection heat t...

Figure 5.E15b Average Nusselt number versus Reynolds number for flow in a tu...

Figure 5.P1 Laminar flow of liquid over an inclined plane.

Figure 5.P4 Rotating cylinder about its axis.

Figure 5.P5 Coordinate system for Problem 5.5.

Figure 5.P6 Coordinate system for Problem 5.6.

Figure 5.P7 Inclined enclosure.

Figure 5.P9 Free fall of a sphere.

Figure 5.P10 Description of Problem 5.10.

Figure 5.P12 Coordinate system of Problem 5.P12.

Figure 5.P14 An experimental rig to study forced convection heat transfer fr...

Figure 5.P16 Coordinate system and steady fully developed laminar flow in a ...

Figure 5.P18 Coordinate system and temperature field for steady slug flow in...

Figure 5.P19 Coordinate system and temperature field for steady slug flow be...

Figure 5.P25 Coordinate system for flow over a triangular plate.

Chapter 6

Figure 6.1 Streamlines between infinite parallel plates.

Figure 6.2 Concentric streamlines.

Figure 6.E1a Heat transfer in Couette flow.

Figure 6.E1b Dimensionless temperature distribution in Couette flow for diff...

Figure 6.E2a Coordinate system and problem description for Example 6.2.

Figure 6.E2b Dimensionless temperature distribution in flow between parallel...

Figure 6.E3 Coordinate system and problem description for Example 6.3.

Figure 6.E4 Coordinate system and problem description for Example 6.4.

Figure 6.E5 Coordinate system and problem description for Example 6.5.

Figure 6.E6 Circular Couette flow.

Figure 6.P2 Couette flow with convective boundary condition.

Figure 6.P3 Coordinate system and problem description.

Figure 6.P5 Coordinate system and problem description.

Figure 6.P6 Heat transfer for laminar flow with variable thermal conductivit...

Figure 6.P7 Coordinate system and problem description for Couette flow.

Figure 6.P8 Heat transfer in flow between rotating concentric pipes.

Figure 6.P9 Heat transfer in pipe flow with convective boundary condition.

Figure 6.P10 Heat transfer in liquid flow over an inclined plane with unifor...

Figure 6.P11 Heat transfer in liquid flow over an inclined plane with consta...

Figure 6.P13 Couette flow with constant pressure gradient.

Figure 6.P14 Heat transfer in Couette flow.

Figure 6.P15 Coordinate system and problem description.

Figure 6.P16 Heat transfer in Couette flow.

Chapter 7

Figure 7.1 Steady boundary layer flow over a flat plate.

Figure 7.2 Similar cubic velocity profiles.

Figure 7.3 Dimensionless velocity profile.

Figure 7.4 Comparison of longitudinal velocity profile with experimental dat...

Figure 7.E1 Variation of longitudinal velocity u as a function of vertical d...

Figure 7.5 (a–c) Wedge flow with various wedge angles.

Figure 7.E3 Flow across a cylinder.

Figure 7.6 Velocity profiles for different values of

Λ

.

Figure 7.7 Velocity and thermal boundary layer over a flat plate.

Figure 7.8 Dimensionless temperature as a function dimensionless distance fo...

Figure 7.9 Variation of as function of Prandtl number.

Figure 7.E5 Geometry and problem description for Example 7.5.

Figure 7.E6a Variation of local heat flux along the plate.

Figure 7.E6b Variation of longitudinal velocity u as a function of vertical ...

Figure 7.E6c Variation of temperature as a function of vertical distance y....

Figure 7.10 Velocity and Temperature profile for

Pr ≪ 1

.

Figure 7.11 Velocity and Temperature profile for

Pr ≫ 1

.

Figure 7.E8 Variation of local heat flux along the plate.

Figure 7.E9 Variation of surface temperature along the plate.

Figure 7.E10 Variation of local heat flux along the plate.

Figure 7.E11 Heat transfer at the stagnation point.

Figure 7.E12 Geometry and problem description for Example 7.12.

Figure 7.12 Laminar flow over a flat plate with an unheated length.

Figure 7.13 Laminar flow over a flat plate with constant surface temperature...

Figure 7.14 Laminar flow over a flat plate with an unheated length subjected...

Figure 7.15 Laminar flow over a flat plate subjected to uniform heat flux.

Figure 7.16 (a) A constant temperature flat plate in inviscid flow. (b) Step...

Figure 7.17 Wall temperature variation having two steps.

Figure 7.18 Wall temperature variation with three steps.

Figure 7.19 Wall surface temperature variation with a step at

x = ξ

...

Figure 7.20 Arbitrary temperature distribution on a surface.

Figure 7.21 Function with a discontinuity.

Figure 7.22 Function with several jumps.

Figure 7.23 (a) A constant heat flux over flat plate in inviscid flow. (b) S...

Figure 7.24 Heat flux variation with two steps.

Figure 7.25 Heat flux variation with triple steps.

Figure 7.26 Heat flux variation with single step along a flat plate.

Figure 7.27 Arbitrary heat flux distribution on a surface.

Figure 7.E16 Single step heat flux distribution along the plate.

Figure 7.E17 Linear heat flux distribution along the plate.

Figure 7.28 Flat plate with stepwise variation in the wall temperature.

Figure 7.E18 Stepwise temperature distribution along the plate.

Figure 7.29 Variation of heat flux.

Figure 7.E19 Double step heat flux distribution along the plate.

Figure 7.30 Semi‐infinite plate with variable surface temperature distributi...

Figure 7.E22 Arbitrary surface temperature distribution along the plate.

Figure 7.31 (a–b) Semi‐infinite plate with variable surface heat flux distri...

Figure 7.E23 Double step heat flux distribution along the plate.

Figure 7.E24 Single step heat flux distribution along the plate.

Figure 7.32 Flow over an axisymmetric body.

Figure 7.E20 Double step temperature distribution along the plate.

Figure 7.E21 A step‐ramp surface temperature distribution.

Figure 7.E25a Geometry and problem description for Example 7.25.

Figure 7.E25b Variation of local Nusselt number with angular position for ai...

Figure 7.E25c Comparison of experimental and theoretical study.

Figure 7.P11 Flow over a cylinder.

Figure 7.P12 Flow over a cylinder.

Figure 7.P17 Geometry and coordinate system for Problem 7.17.

Figure 7.P23 Geometry and coordinate system for Problem 7.23.

Figure 7.P24 Flow over a cylinder.

Figure 7.P25a Coordinate system for flow parallel to the axis of body of rev...

Figure 7.P25b Flow over a sphere.

Figure 7.P26 Flow over a sphere.

Figure 7.P29 Heat transfer from a flat plate with unheated section.

Figure 7.P31 Laminar flow over a flat plate with an unheated length.

Figure 7.P36 Stepwise surface temperature distribution.

Figure 7.P38 Temperature variation on the plate.

Figure 7.P40 Delayed ramp heat flux distribution along the plate.

Chapter 8

Figure 8.1 Growth of velocity boundary layer along a circular duct.

Figure 8.2 A differential momentum balance.

Figure 8.3 Development of the velocity profile for laminar flow between para...

Figure 8.4 Fully developed velocity profile and shear stress distribution in...

Figure 8.5 Control volume to study fully developed flow in a circular tube....

Figure 8.6 The ratio of the apparent Moody friction factor to fully develope...

Figure 8.7 Fully developed velocity profile and shear stress distribution fo...

Figure 8.8 Ratio of the apparent Fanning friction factor to the fully develo...

Figure 8.9 Thermal entrance region.

Figure 8.10 Fully developed thermal flow in a pipe.

Figure 8.11 Variation of heat transfer coefficient with axial position.

Figure 8.12 Control volume in a duct.

Figure 8.13 Axial variation of mean fluid temperature.

Figure 8.14 Control volume in a channel.

Figure 8.15 The log mean temperature difference (LMTD).

Figure 8.16 Qualitative behavior of the Nusselt number or the heat transfer ...

Figure 8.17 (a) Slug flow in circular tube. (b) Constant wall temperature.

Figure 8.18 Nusselt number as a function of dimensionless axial distance for...

Figure 8.19 Control volume.

Figure 8.20 Slug flow in entrance region of circular tube heated by uniform ...

Figure 8.21 Slug flow between parallel plates. (a) Thermal boundary layer de...

Figure 8.22 Nusselt number as a function of dimensionless axial position for...

Figure 8.23 Slug flow between parallel plates of constant heat flux. (a) The...

Figure 8.24 Nusselt number as a function of dimensionless.

Figure 8.25 Thermal boundary layer development for slug flow between paralle...

Figure 8.26 Thermal boundary layer development for slug flow between paralle...

Figure 8.27 Slug flow in a pipe.

Figure 8.28 Control volume.

Figure 8.29 Flow in a circular pipe.

Figure 8.30 Flow in a pipe.

Figure 8.31 Laminar flow between parallel plates.

Figure 8.32 (a) Parabolic velocity profile and developing temperature profil...

Figure 8.33 Location of eigenvalues.

Figure 8.34 Graetz functions as function of dimensionless distance

η

.

Figure 8.35 Control volume for energy balance.

Figure 8.36 Schematic drawing for the coordinate system.

Figure 8.37 (a) Poiseuille flow in circular tube being heated by a uniform h...

Figure 8.38 Location of characteristic values.

Figure 8.39 Graetz problem for parallel plates.

Figure 8.40 Location of eigenvalues.

Figure 8.41 (a) Graetz problem between parallel plates, (b) Step change in h...

Figure 8.42 Location of eigenvalues.

Figure 8.43 (a) Pipe surface temperature variation. (b) Step change in pipe ...

Figure 8.E9 Stepwise temperature distribution on tube surface in axial direc...

Figure 8.44 (a) Arbitrary variation of surface heat flux. (b) Heat flux step...

Figure 8.P1 Geometry and problem description for Problem 8.1.

Figure 8.P5 Thermal boundary layer development in the entrance region of a p...

Figure 8.P6 Thermal boundary layer development in the entrance region of a p...

Figure 8.P7 Geometry and problem description for Problem 8.7.

Figure 8.P8 Geometry and problem description for Problem 8.8.

Figure 8.P10 Geometry and problem description for Problem 8.10.

Figure 8.P11 Geometry and problem description Leveque problem.

Figure 8.P14 Geometry and problem description for Problem 8.14.

Figure 8.P15 Geometry and problem description for Problem 8.15.

Figure 8.P16 Geometry and problem description for Problem 8.16.

Figure 8.P17 Geometry and problem description for Problem 8.17.

Figure 8.P18 Geometry and problem description for Problem 8.18.

Figure 8.P19a Geometry and problem description for viscous flow between para...

Figure 8.P19b Solar collector.

Figure 8.P21 Geometry and problem description for Problem 8.21.

Figure 8.P22 Geometry and problem description for Problem 8.22.

Figure 8.P27 Geometry and problem description for Problem 8.27.

Figure 8.P28 Geometry and problem description for Problem 8.28.

Figure 8.P30 Geometry and problem description for Problem 8.30.

Figure 8.P31 Thermal boundary layer development in the entrance region of pa...

Figure 8.P32 Geometry and problem description for Problem 8.32.

Figure 8.P33 Flow in a circular pipe.

Figure 8.P34 Geometry and problem description for Problem 8.34.

Figure 8.P35 Geometry and problem description for Problem 8.35.

Figure 8.P37 Geometry and problem description for Problem 8.37.

Figure 8.P38 Geometry and problem description for Problem 8.38.

Chapter 9

Figure 9.1 Reynolds experiment.

Figure 9.2a Distribution of eddies.

Figure 9.2b Instantaneous velocity profile in channel flow.

Figure 9.2c Variation of

u

with time t.

Figure 9.2d Comparison of laminar and time‐averaged turbulent velocity profi...

Figure 9.3 Relative turbulence intensities in the flow along a smooth flat p...

Figure 9.4 Turbulent flow over flat plate.

Figure 9.5 Temperature profile in turbulent flow.

Figure 9.6 Coordinate system for flow in a circular pipe.

Figure 9.7 Apparent shear stress variation in a fully developed turbulent fl...

Figure 9.8 Dimensionless velocity profile for different Reynolds numbers....

Figure 9.9 Comparison of different velocity distribution equations a functio...

Figure 9.10 The universal velocity profile plotted in inner variable coordin...

Figure 9.11 Turbulent boundary layer velocity profile on a smooth and rough ...

Figure 9.12 Defect law plot of turbulent velocity profiles in the outer regi...

Figure 9.13 Universal velocity distribution for turbulent velocity profiles....

Figure 9.E16 Plot of Eq. (a).

Figure 9.P4 Geometry and problem description for Problem 9.4.

Chapter 10

Figure 10.1 Velocity profile on a flat plate.

Figure 10.2 Local friction coefficient for turbulent flow over a flat plate....

Figure 10.3 The temperature law of the wall, computed from Eq. (10.76a) with...

Figure 10.4 Comparison of Eq. (10.88) with experimental...

Figure 10.5 Coordinate system for turbulent flow over an external surface.

Figure 10.6 Boundary layer on a plate with an unheated starting length.

Figure 10.7 Control volume for the conservation of mass.

Figure 10.8 Control volume for the application of the momentum theorem.

Figure 10.9 Control volume for the force balance.

Figure 10.E7 Variation of local heat flux along the plate.

Figue 10.E8 Stepwise temperature distribution.

Figure 10.E9 Linear temperature distribution.

Figure 10.10 (a) Single step in heat flux at x = 0. (b) Single step in heat ...

Figure 10.E10 Stepwise heat flux distribution.

Figure 10.11 Step heat flux distribution on a surface.

Figure 10.E11 Linear heat flux distribution.

Figure 10.12 Nusselt number across the transition region for water flowing o...

Figure 10.13 Nusselt numbers measured across the transition region for air f...

Figure 10.14 Comparison of Eq. (10.88) to constant wall heat flux data of Bl...

Figure 10.P11 Stepwise temperature distribution along the plate.

Figure 10.P13 Step‐ramp temperature distribution.

Figure 10.P14 Delayed ramp wall temperature.

Figure 10.P15 Step heat input.

Figure 10.P16 Delayed ramp heat input.

Chapter 11

Figure 11.1 Mean velocity distribution.

Figure 11.2 Turbulent flows between infinite parallel plates.

Figure 11.3 Variation of total shear stress across a turbulent channel flow....

Figure 11.4 Turbulent flow in a circular pipe

Figure 11.5 Mean velocity as function of r/R.

Figure 11.6 Control volume for force balance in a pipe.

Figure 11.7 Typical turbulent flow velocity profiles.

Figure 11.8 Turbulent flow between parallel plates subjected to uniform heat...

Figure 11.9 Control volume for energy balance.

Figure 11.10 Dimensionless mean temperature distribution for the parallel pl...

Figure 11.11 Turbulent flow heat transfer a in a circular tube.

Figure 11.12 Control volume for energy balance.

Figure 11.13 Thermally developing flow.

Figure 11.E9 Geometry and problem description for Example 11.9.

Figure 11.E9a Variation of dimensionless mean temperature along the tube.

Figure 11.E9b Variation of local Nusselt number along the tube.

Figure 11.14 Coordinate system for turbulent flow in a pipe.

Figure 11.15 Coordinate system for turbulent flow in a pipe; Prandtl–Taylor ...

Figure 11.16 Coordinate system for turbulent flow in a pipe; von Karman anal...

Figure 11.E11 (c) Concentric annulus. (d) Semicircular duct. (e) Circular tu...

Figure 11.E12 Schematic diagram.

Figure 11.17 Schematic of inlet configurations.

Figure 11.P4 Geometry and problem description for Problem 11.4.

Figure 11.P8 Turbulent Couette flow.

Figure 11.P15 Turbulent flow between parallel plates.

Figure 11.P21 Geometry and problem description for Problem 11.21.

Figure 11.P22 Geometry and problem description for Problem 11.22.

Figure 11.P23 Geometry and problem description for Problem 11.23.

Figure 11.P44 Geometry and problem description for Problem 11.44.

Figure 11.P45 Geometry and problem description for Problem 11.45.

Chapter 12

Figure 12.1 The formation of velocity boundary layer in free convection.

Figure 12.2 Temperature boundary layers of free convection near a constant t...

Figure 12.3 Velocity boundary layers of free convection near a constant temp...

Figure 12.E1 (a) Variation of u velocity with y. (b) Variation of tempetratu...

Figure 12.4 Isothermal vertical plate.

Figure 12.5 Vertical plate under uniform heat flux.

Figure 12.E5 Variation of local surface temperature.

Figure 12.6 Free convection on the horizontal surface.

Figure 12.7 (a, b) Buoyancy driven flows over horizontal plates.

Figure 12.8 (a–b) Buoyancy driven flows over horizontal plates.

Figure 12.9 Concept of positive and inclination negative angles from vertica...

Figure 12.E8 Geometry and problem description for Example 12.8.

Figure 12.10 Vertical cylinder in free convection.

Figure 12.11 Horizontal cylinder in free convection.

Figure 12.12 Coordinate system for inclined cylinder in free convection.

Figure 12.13 Channel configuration for isothermal walls.

Figure 12.14 Prediction of Nusselt number in turbulent differentially heated...

Figure 12.E11 Geometry and problem description for Example 12.11.

Figure 12.15 Channel configuration under uniform heat flux.

Figure 12.16 Horizontal parallel plates.

Figure 12.17 Inclined channel configuration.

Figure 12.18 Nusselt number versus modified Rayleigh number.

Figure 12.E12 Geometry and problem description for Example 12.12.

Figure 12.19 Horizontal enclosure.

Figure 12.E13 Geometry and problem description for Example 12.13.

Figure 12.20 Fluid circulation in a vertical cavity.

Figure 12.E14 Geometry and problem description for Example 12.14.

Figure 12.21 Inclined rectangular enclosure.

Figure 12.22 (a) Regularly spaced convective cell. (b) Cellular flow.

Figure 12.E15 Geometry and problem description for Example 12.15.

Figure 12.P1 Free convection boundary layers between vertical parallel plate...

Figure 12.P6 Geometry and problem description for Problem 12.6.

Figure 12.P7 Heater in a tank.

Figure 12.P11 Geometry and problem description for Problem 12.11.

Figure 12.P13 Geometry and problem description for Problem 12.13.

Figure 12.P14 Geometry and problem description for Problem 12.14.

Figure 12.P16 Geometry and problem description for Problem 12.16.

Guide

Cover

Table of Contents

Title Page

Copyright

Dedication

Dedication

Preface

About the Companion Website

Begin Reading

Appendix: Thermophysical Properties of Matter

Index

End User License Agreement

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Introduction to Convective Heat Transfer

A Software-Based Approach Using Maple and MATLAB®

Nevzat Onur

Emeritus Professor, Department of Mechanical EngineeringGazi University, Turkey

This edition first published 2023

© 2023 John Wiley & Sons, Inc

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.

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MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This work's use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

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Hardback ISBN: 9781119766766 [LCCN 2022058932]

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To my wife: Ayfer

To our children: Hakan and Funda

Erdem and Gökcen

To our grandchildren: Ada, Aslan, and Mila

Science is the most genuine guide for spiritual and intellectual enlightenment in life

Mustafa Kemal Atatürk

Preface

This textbook covers sufficient material to support a one‐semester graduate‐level course in convective heat transfer. Topics such as boiling and condensation are not included, since these are beyond the scope of this book. This textbook represents the teaching methodology I developed over 30 years of my experience in teaching engineering students. The textbook focuses on teaching the fundamental concepts of convective heat transfer.

In this textbook, students will see a careful balance of theory and engineering applications. Special effort has been made to provide a physical interpretation of topics covered in the textbook.

Careful attention has been given to the derivation of dimensionless parameters and theoretical as well as semiempirical equations to establish the students' confidence. This process will help students to develop their judgment concerning fundamental concepts. The main goal of this textbook is to provide students with the capability, tools, and confidence to solve engineering problems in convective heat transfer, starting from basic principles.

I believe that an understanding of the physics and development of mathematical models is essential to the analysis and design of engineering processes. In classroom, emphasis should be placed on teaching the students how to develop mathematical models of physical phenomena. Students should learn how to formulate the convection problems and gain physical insight.

It is evident that the mechanisms of heat transfer and their operation in a given engineering system are very complex. A general representation for convective heat transfer is revealed through the development of basic equations in differential forms. Many students have difficulties in understanding and developing solutions to these differential equations. Solution methods, such as Bessel functions, Laplace transforms, separation of variables, Duhamel's theorem, and eigenfunction expansion, are required to solve these differential equations. Most students face difficulties in using such methods in convective heat transfer, and solutions can be extremely difficult. Because of this, there is a tendency toward a descriptive approach in most textbooks on convective heat transfer. Some textbooks present a survey of literature, skipping the details of formulations and solutions. In the author's opinion, this is not desirable from a teaching standpoint, and this approach does not provide engineering students with adequate fundamental knowledge to understand and interpret the solutions.

The book is devoted to a comprehensible exposition of the principles of convective heat transfer as well as the mathematical formulation and solution of processes encountered in convection heat transfer. The treatment of subjects is based on a background in basic heat transfer, thermodynamics, fluid mechanics, and mathematics. The problems provided at the end of each chapter require the students to think critically, but are not difficult to comprehend.

The textbook differs from existing convective heat transfer textbooks in many ways. First, to improve readability, the derivations of continuity, momentum, and energy equations are presented completely without skipping any steps. Sentences such as “it can be done,” “it can be shown,” etc., are eliminated as much as possible. To reduce students' frustration, solutions of governing equations of the problems considered in the textbook are presented completely without skipping any steps, and this will improve readability.

An important advantage of the textbook as compared to existing convective heat transfer textbooks is the integration of modern computational tools such as Maple and MATLAB. The specific commands associated with these software packages are used in the solution of examples. It is easy for students to use the extensive symbolic and numerical capabilities of Maple and MATLAB. Students using the computational software tools will find it easier to overcome mathematical difficulties and obtain analytical and/or numerical solutions. I believe that, using Maple and MATLAB, students will be able to explore convective heat transfer without getting bogged down in the complicated numerical and analytical methods required to solve convective heat transfer problems.

Almost none of the problems encountered in convective heat transfer can be solved with a hand calculator. For this reason, engineering students should be proficient with computational tools, and should make full use of their capabilities. The equations developed in the analysis can be solved easily by use of computational tools and, in fact, this process is a motivator to many students. The computational software tools used in this book have existed for more than three decades and are commonly used in academy and industry. It is not likely that they will disappear. The use of these tools should not present an economic difficulty for any engineering college or student, since academic versions of these software packages are available. I believe that, in time, engineering education will evolve in this way. Both simple as well complex problems can be solved by using Maple and MATLAB. This approach will provide students with ample time to concentrate on understanding all the steps involved in the physics of convection heat transfer.

This textbook is organized into 12 chapters. Chapter 1 reviews fundamental concepts of heat transfer and presents a cursory development of the Reynolds transport theorem. Chapters 2 and 3 cover fundamental equations of laminar convective heat transfer. Chapter 4 is devoted to the development of integral equations for boundary layer flows. Dimensional analysis, nondimensionalization of differential equations, and experimental heat transfer are covered in Chapter 5. Fundamental concepts are presented in terms of one‐dimensional solutions in Chapter 6. The conclusions drawn from one‐dimensional solutions are broad and valuable. Laminar external boundary layers of momentum and heat transfer are discussed in Chapter 7. The superposition principle is explained in terms of slug flow. Variable surface temperature and heat flux are introduced. Similarity solutions are also introduced in this chapter. Solutions are obtained by numerical and integral methods. Chapter 8 includes laminar momentum and heat transfer in ducts. An introduction to turbulent flows is presented in Chapter 9. Some available experimental data for internal and external turbulent flows are introduced. Turbulent external boundary layers for momentum and heat transfer are discussed in Chapter 10. Turbulent internal flows are discussed in Chapter 11, where analogies between momentum and heat transfer are also presented. Finally, Chapter 12 is devoted to free convection heat transfer, and concepts of internal and external natural convection are discussed in this chapter.

I gratefully acknowledge the help of my teachers, students, and colleagues. Their contributions and criticisms are reflected in the pages of this book. The inconsistencies, obscurities, and errors that remain in the textbook are all mine. I have taught convective heat transfer from several textbooks over the years and have incorporated material from these books in my course notes. Over the past many years, the origin of the material has been forgotten. I hope that readers will bring to my attention the material in the textbook to warrant acknowledgment to the original sources.

About the Author

Nevzat Onur is Emeritus Professor in the Department of Mechanical Engineering at Gazi University. He pursued his undergraduate studies in mechanical engineering at the University of California, Davis (CA, USA), where he received his BS degree in 1974. He then attended the Tennessee Technological University, Cookeville (TN, USA), where he completed his MS and PhD degrees in 1976 and 1980, respectively. His 40 years of academic experience includes appointments at different universities in Turkey. All of his university appointments have been in teaching and research in the thermal sciences (thermodynamics, fluid mechanics, and heat transfer), along with administrative duties. These administrative duties include positions such as the Chairman of the Department of Mechanical Engineering and the Dean of Engineering Faculty. He has authored or coauthored several refereed research papers, and is well‐qualified to write on the topic of convective heat transfer. He lives in Ankara, Turkey.

About the Companion Website

This book is accompanied by a companion website:

www.wiley.com/go/introtoconvectiveheattransfer

The instructor website includes

The solutions manual

Maple and MATLAB

®

files

The student website includes

Maple and MATLAB

®

files

1Foundations of Convective Heat Transfer

1.1 Fundamental Concepts

Heat transfer is an energy transfer process because of a temperature gradient or difference. This temperature difference is called a driving force that causes heat to flow from a high‐temperature region to a low‐temperature region. There are basic mechanisms or modes for heat transfer: conduction, convection, and radiation. In this chapter, we will present their fundamental equations.

1.2 Coordinate Systems

In the solution of heat transfer problems, we need to be able to provide the geometric identification of various points in the system under study. This is done by the use of a coordinate system. If the coordinate system is not accelerating and rotating, it is called an inertial system. The most common right‐hand coordinate systems are illustrated in Figure 1.1. The transformation equations for Cartesian (rectangular) and cylindrical coordinate systems are given as

(1.1a)

(1.1b)

(1.1c)

The spherical coordinate system may be transformed into the rectangular coordinate system. Transformation equations for Cartesian and spherical coordinate systems are given as

(1.2a)

(1.2b)

(1.2c)

Figure 1.1 (a) Cartesian coordinate system. (b) Cylindrical coordinate system. (c) Spherical coordinate system.

1.3 The Continuum and Thermodynamic Equilibrium Concepts

The convection heat transfer depends on the properties of material, such as density, thermal conductivity, pressure, and specific heat. These properties are assumed to be well defined at infinitely small points, and these properties are assumed to vary continuously from one point to another. In convective heat transfer, we will use the continuum model. In the continuum model, we deal with the macroscopic or average effect of molecules. The continuum assumption is very useful since it erases the molecular discontinuities by averaging the microscopic quantities on a small sampling volume. Macroscopic quantities such as density ρ, pressure p, temperature T, etc. are assumed to vary continuously and smoothly from point to point within the flow. The state of continuum may be described by continuous functions such as density ρ = ρ(x, y, z, t), temperature T = T(x, y, z, t), and velocity . The continuum and thermodynamic equilibrium concepts are discussed in [1–3]. Continuum assumption breaks down under certain conditions.

The criterion for the validity of continuum assumption depends on the Knudsen number Kn, and the Knudsen number is defined as

(1.3)

where λ is the molecular mean free path length, which is the average distance traveled by molecules before they collide. The characteristic length Le can be the diameter of a pipe or equivalent diameter of a channel. In other words, the length scale, Le, represents the overall dimension of the flow.

In general, the physics of liquid flow in micro devices is not well known. Suppose we restrict our discussion to gases. The kinetic theory of gases provides an expression for this mean free path as follows:

(1.4)

where n is the number density (the number of molecules per unit volume) and σ is the molecular diameter. This number represents an effective diameter of collusions for gas molecules. The values of σ for different gases are presented in [4]. Table 1.1 gives the values of σ for some gases.

The number density n is given as

(1.5)

where p is the absolute pressure, T is the absolute temperature, and k = 1.38065 × 10−23 J/K is the Boltzmann constant. When we combine these two equations, we get the equation for mean free path length λ

(1.6a)

A different form of Eq. (1.6a) for mean free path λ is

(1.6b)

where R is the specific gas constant and μ is the dynamic viscosity of the gas. In this model, as discussed in [3], ideal gas is modeled as rigid spheres. The continuum model is valid as long as λ ≪ Le. If this condition is violated, flow is not in equilibrium; the relation between stress and rate of strain and the no‐slip boundary condition are not valid anymore. In other words, the continuum Navier–Stokes model is valid when λ is much smaller than a characteristic dimension Le of the flow. In a similar way, the linear relation between heat flux and temperature gradient and the no‐jump temperature boundary condition at the solid–fluid interface are also not valid. Note that the continuum concept forms the basis of mass, momentum, and energy equations. This means that fluid and flow properties are distributed in space, and point properties can be defined; derivatives for these properties may be determined. We will now present the classification of flow regimes.

Table 1.1 Molecular diameter values for different gases.

Gas

σ × 10

10

 (m)

Air

3.66

N

2

3.70

CO

2

4.53

O

2

3.55

The continuum regime

: In this regime, the Knudsen number is

Kn < 10

−3

. Flows in the continuum regime can be modeled by the conservation of mass, momentum, and energy with no‐slip and no‐jump boundary conditions.

The slip flow regime

: In this regime, the Knudsen number is in

10

−3

 < Kn < 10

−1

range; the conservation of mass, momentum, and energy is valid with slip and temperature jump boundary conditions. The no‐slip and no‐jump assumptions are not valid anymore.

The transition regime:

The Knudsen number is in

10

−3

 < Kn < 10

−1

range. In this regime, the conservation of mass, momentum, and energy is not valid. Flow must be solved using molecular‐based models such as the Boltzmann equation.

Free molecule regime

: In this regime,

Kn > 10

. The collusions between molecules are neglected, and collisionless Boltzmann transport equations are required.

The continuum approach also requires that the sampling volume be in thermodynamic equilibrium. Under thermodynamic equilibrium, we have

(1.7)

(1.8)

If the fluid is gas, Eqs. (1.7) and (1.8) will not be valid if the mean free path length λ is not much less than the flow length scale Le. If Kn ≪ 1, slip is negligible. If Kn = O(10−1), there is slip. If Kn = O(1) or greater, the slip concept is not valid anymore. These points are discussed in [3, 5].

Example 1.1

This problem involves the mean free path and the Knudsen number. Compute the Knudsen number for a small probe in air at a temperature of 288.15 K for a characteristic length of 0.001 m. The air pressure is 101 325 Pa.

Solution

Using Eq. (1.6a), we obtain:

The analysis in this book is based on continuum and thermodynamic equilibrium. The conservation of mass, momentum, and energy equations is valid as long as continuum assumption is valid. The no‐velocity slip and no‐temperature jump at the solid boundary are valid if thermodynamic equilibrium is justified.

1.4 Velocity and Acceleration

It is important to describe the time rate of change of position of a particular element of fluid. This time rate of change of position, , is called velocity, and it is denoted by the symbol , i.e.

(1.9)

In a rectangular coordinate system, the velocity field is given as

(1.10)