Damped Wave Transport and Relaxation (eBook)

Cover 1
Table of Contents 14
Preface 8
Acknowledgements 12
About the Author 13
The Damped Wave Conduction and Relaxation Equation 20
Origin 21
Wave Dominant and Fourier Regimes 25
Worked Example 1.1 - Temperature Variation of Thermal Conductivity 27
Worked Example 1.2- Wave and Fourier Regimes during Convection, Conduction and Relaxation 28
Boundary Layer in Time and Scaling of the Governing Equation 29
Poor Convergence of Fourier Series 30
Light Speed Barrier and Finite Speed of Heat Propagation 34
Hyperbolic, Parabolic and Elliptic PDE 35
Non-Fourier Equations for Heat Conduction 36
Third Order PDE to describe Non-Fourier Heat Conduction 48
Finite Speed Conduction, Diffusion and Momentum Transfer 51
Experimental Measurement of Relaxation Times 52
Case of Infinite Relaxation Time and D'Alambert's Solution 54
Summary 55
Exercises 57
Transient Heat Conduction and Relaxation 65
The Ballistic Term 67
Semi-Infinite Medium Subject to Constant Wall Temperature 68
Relativistic Transformation of Coordinates 68
Regimes of Heat Flux at a Interior Point inside the Medium 71
Approximate Solution 73
Extension to 3-Dimensions 76
Worked Example 2.1- Can c in w=c I 0 (sqrt(tau^2 - X^2))/ 2) be a function of time ? 77
Worked Example 2.2 - Verification of w = c I (sqrt(tau^2 - X^2))/ 2) as solution 78
Worked Example 2.3 - Verification of w = cSin(.X - .t) + dCos(.X - .tau) is a solution to Hyperbolic PDE without 79
Worked Example 2.4 - Transformation, . = sqrt(tau^2 - X^2), to convert hyperbolic PDE into Bessel equation 80
Worked Example 2.5 - Examine g in g(tau) I 0(tau/2) = exp(tau/2) 81
Worked Example 2.6 - Method of Laplace Transform Solution to Hyperbolic PDE Without the Damping Term 82
Relativistic Transformation of Cylindrical Coordinates in Infinite Medium 83
Worked Example 2.7-Governing Equation of Heat flux in Cylindrical Coordinates 86
Relativistic Transformation of Spherical Coordinates in a Infinite Medium 88
Worked Example 2.8 - Transient Temperature in a Semi-infinite Cone at Constant Apex Temperature 92
Worked Example 2.9 - Governing Equation of Heat flux in Spherical C 94
Finite Slab at Constant Wall Temperature 95
On the Use of Storage Coefficient 100
Hyperbolic PDE as Governing Equation 100
Parabolic PDE as Governing Equation 102
The Dual Phase Lag Model 103
Transient Temperature in a Sphere at CWT 107
Pulse Injection in an Infinite Medium 109
Temperature Overshoot 111
Convective Boundary Condition 112
Plane of Null Transfer 114
Temperature Dependent Heat Source 117
Critical Point of Zero Transfer 120
Worked Example 2.10 - Convective Heating in a Rod 124
Method of Laplace Transforms 126
Problem of Semi-infinite Medium in Cartesian Coordinates given a Step Change in Temperature at one of the Walls 128
Constant Wall Flux in a Semi-infinite Medium 130
Worked Example 2.11 - Transient Heat Conduction and Relaxation in a Ice Cube with Phase Change 131
Convective Heating in a Semi-infinite Medium 133
Average Temperature in a Finite Slab with Convective Boundary Condition 136
Worked Example 2.12 - Generalized Fourier Expression with up to the Second Derivative in Heat Flux and Average Temperature in a Finite Slab 140
Parabolic Solution 142
Worked Example 2.13 - Dual Phase Lag Model and Temperature in a Finite Slab 143
Worked Example 2.14 - Magnesium Oxychloride Sorrel Cement Coating and Subcritical Damped Oscillations 146
Summary 149
Exercises 152
Transient Mass Diffusion and Relaxation 165
Introduction 166
Worked Example 3.1 - Depletion of Molecules at the Surface and non-Fick Equation 171
Finite Speed Diffusion with Fast Chemical Reaction in a Semi-Infinite Catalyst 173
Worked Example 3.2 - Spatiotemporal Concentration when k* = 1 178
Worked Example 3.3 - Solution of Spatiotemporal Concentration by Method of Laplace Transforms 178
Worked Example 3.4 - Mass Diffusion and Relaxation and Simultaneous Zeroth Order Chemical Reaction 179
Finite Speed Diffusion with Fast Chemical Reaction in Infinite Catalyst in Cylindrical Coordinates 181
Finite Speed Diffusion with Fast Chemical Reaction in Infinite Catalyst in Spherical Coordinates 186
Finite Slab at Constant Wall Concentration 192
Finite Cylinder at Constant Wall Concentration 194
Finite Sphere with Constant Wall Concentration 196
Critical Radii of Nuclear Fuel Rod Neither Less than the Cycling Limit Nor Greater than the Shape Limit 198
Zone of Zero Transfer in a Ziegler Natta Polymerization Catalyst 202
Worked Example 3.5 - Zone of Zero Transfer in a Zeigler Natta Catalyst when the Bulk Reaction Rate in First Order 205
Worked Example 3.6 - Mass Diffusion and Relaxation from a Dissolving Drop 208
Worked Example 3.7 - Method of Laplace Transform Solution in a Finite Slab During Simultaneous Diffusion, Relaxation and Reaction 210
Worked Example 3.8 - Spatiotemporal Concentration from a Perfume Container at the Shopping Mall in Spherical Coordinates 212
Worked Example 3.9 - Spatiotemporal Concentration from a Perfume Container at the Shopping Mall in Spherical Coordinates in the Creeping Transfer Limit 214
Worked Example 3.10 - Diffusion and Relaxation from a Dissolving Pill in Spherical Coordinates in 1 Dimension 216
Worked Example 3.11 - Chemisorption of Organic Sulfides in Automobile Gasoline onto Palladium 219
Worked Example 3.12 - Diffusion and Relaxation from a Dissolving Pill in Spherical Coordinates in 3 Dimensions 220
Diffusion, Relaxation and Convection 223
Worked Example 3.13 225
Worked Example 3.14 230
Summary 233
Exercises 236
Transient Momentum Transfer and Relaxation 241
34 Different Flow Types 243
Damped Wave Momentum Transfer and Relaxation 246
Worked Example 4.1 - Emptying of a Filled Pipe using the Extended Euler Equation 251
Worked Example 4.2 - Emptying of a Filled Pipe with Multiple Ports 253
Flow Near a Horizontal Wall Suddenly Set in Motion 255
Worked Example 4.3 - General Substitution =Z+ (1 - )^(1/2) to Transform Hyperbolic PDE to Parabolic PDE 261
Worked Example 4.4 - Emptying of a Filled Pipe using the Extended Euler Equation with Porous Packing 263
Worked Example 4.5 - Emptying of a Filled Pipe using the Extended Euler Equation with Porous Packing with Positive Permitivity 264
Shear Flow Between Two Plates Moving in Opposite Direction at Constant Velocity with Separation Distance 2a 266
Worked Example 4.6 - Shear Flow Between Two Vertical Plates Moving In Opposite Direction at Constant Velocity with Separation Distance 2a with Porous Packing 271
Transient Laminar Flow in a Circular Conduit 274
Worked Example 4.7 - Oscillations in a U Tube Manometer from the Extended Euler Equation including the Ballistic Term 278
Flow of Blood in Capillaries 279
Tangential Flow Induced by a Rotating Cylinder 284
Worked Example 4.8 - Tangential Flow Induced by a Rotating Cylinder at Short Distances 287
Worked Example 4.9 - Tangential Flow Induced by a Rotating Cylinder from Shell Balance 289
Transient Flow Past a Sphere 291
Worked Example 4.10 - Radial Flow between Two Concentric Spheres 293
Worked Example 4.11 - Room Cooled by a Ceiling Fan 296
Worked Example 4.12 - Outward Radial Squeeze Flow between Two Circular Disks 300
Worked Example 4.13 - Conversion of Hyperbolic PDE into a PDE with Similarity Transformation Variable and Time Variable 303
Worked Example 4.14 - Response to a Oscillating Pressure Gradient in a Circular Tube 305
Friction Factors 309
Summary 312
Exercises 314
Applications 323
Convective and Conductive Contributions in Circulating Fluidized Bed Heat Transfer 326
CPU Overheating and the Storage Coefficient 332
Cosinuous Concentration Profile during Scale-up of Grafting in Helical Ribbon Agitated CSTR 334
Thermal Barrier Coating 337
Asymptotic Analysis of Governing Equation 339
Average Temperature (UL 1709 Test) 341
Exact Solution 342
Infinite Order PDE and Solution by Method of Laplace Transforms 344
Maxwellian Basis for Secondary Pressure Fluctuations in Gas-Solid Bubbling Fine Particle Fluidized Beds 348
Delivery of Drugs and Finite Speed Diffusion 353
Response to a Pulse in Mass Flux 354
Cylindrical Coordinates 356
Spherical Coordinates 359
Solute Binding and Elimination 364
Worked Example 5.7.1 - Effective Diffusivity and Binding Constant 367
Maternal Effect Genes 371
Drug Penetration in the Tissue 372
Neurotransmission across the Synaptic Cleft 373
Enzymatic Reactions 374
Transdermal Delivery Systems 378
Diffusion Through Cylindrical Membranes 383
Matrix Delivery System 386
Krogh Tissue Cylinder 389
Worked Example 5.7.2 - Anaroxic Regions in Cartesian Coordinates 391
Summary 393
Exercises 395
Sources, References and Further Reading 405
Appendix A 432
Appendix B 439
Appendix C 443
Answers to Excercises 450
Subject Index 454