Introduction to Convective Heat Transfer
John Wiley & Sons Inc (Verlag)
978-1-119-76676-6 (ISBN)
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.
Nevzat Onur is Emeritus Professor of Mechanical Engineering at Gazi University. He pursued his undergraduate studies in mechanical engineering at the University of California, Davis, USA where he received his B.S. degree in 1974. He then attended the Tennessee Technological University, Cookeville, USA completing his M.S. and Ph.D. degree in 1976 and 1980. He taught at different universities in Turkey and he retired from Gazi University in 2011. He has over thirty years’ experience in heat transfer research and development. His research interests have mainly been in viscous flow and convection heat transfer. He lives in Ankara, Turkey.
Preface xv
About the Author xvii
About the Companion Website xviii
1 Foundations of Convective Heat Transfer 1
1.1 Fundamental Concepts 1
1.2 Coordinate Systems 1
1.3 The Continuum and Thermodynamic Equilibrium Concepts 2
1.4 Velocity and Acceleration 3
1.5 Description of a Fluid Motion: Eulerian and Lagrangian Coordinates and Substantial Derivative 4
1.5.1 Lagrangian Approach 4
1.5.2 Eulerian Approach 5
1.6 Substantial Derivative 7
1.7 Conduction Heat Transfer 10
1.8 Fluid Flow and Heat Transfer 11
1.9 External Flow 11
1.9.1 Velocity Boundary Layer and Newton’s Viscosity Relation 11
1.9.2 Thermal Boundary Layer 12
1.10 Internal Flow 19
1.10.1 Mean Velocity 19
1.10.2 Mean Temperature 20
1.11 Thermal Radiation Heat Transfer 22
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 22
Problems 28
References 31
2 Fundamental Equations of Laminar Convective Heat Transfer 33
2.1 Introduction 33
2.2 Integral Formulation 33
2.2.1 Conservation of Mass in Integral Form 33
2.2.2 Conservation of Linear Momentum in Integral Form 34
2.2.3 Conservation of Energy in Integral Form 36
2.3 Differential Formulation of Conservation Equations 38
2.3.1 Conservation of Mass in Differential Form 38
2.3.1.1 Cylindrical Coordinates 41
2.3.1.2 Spherical Coordinates 41
2.3.2 Conservation of Linear Momentum in Differential Form 42
2.3.2.1 Equation of Motion for a Newtonian Fluid with Constant Dynamic Viscosity μ and Density ρ 45
2.3.2.2 Cartesian Coordinates (x, y, z) 45
2.3.2.3 Cylindrical Coordinates (r, θ,z) 46
2.3.2.4 Spherical Coordinates (r, θ, ϕ) 46
2.3.3 Conservation of Energy in Differential Form 47
2.3.3.1 Mechanical Energy Equation 53
2.3.3.2 Thermal Energy Equation 53
2.3.3.3 Thermal Energy Equation in Terms of Internal Energy 54
2.3.3.4 Thermal Energy Equation in Terms of Enthalpy 55
2.3.3.5 Temperature T and Constant Volume Specific Heat CV 55
2.3.3.6 Temperature and Constant Pressure Specific Heat cp 56
2.3.3.7 Special Cases of the Differential Energy Equation 58
2.3.3.8 Perfect Gas and the Thermal Energy Equation Involving T and cp 58
2.3.3.9 Perfect Gas and the Thermal Energy Equation Involving T and CV 58
2.3.3.10 An Incompressible Pure Substance 58
2.3.3.11 Rectangular Coordinates 59
2.3.3.12 Cylindrical Coordinates (r, θ, z) 59
2.3.3.13 Spherical Coordinates (r, θ, ϕ) 59
Problems 64
References 67
3 Equations of Incompressible External Laminar Boundary Layers 69
3.1 Introduction 69
3.2 Laminar Momentum Transfer 69
3.3 The Momentum Boundary Layer Concept 70
3.3.1 Scaling of Momentum Equation 71
3.4 The Thermal Boundary Layer Concept 76
3.4.1 Scaling of Energy Equation 77
3.5 Summary of Boundary Layer Equations of Steady Laminar Flow 82
Problems 82
References 83
4 Integral Methods in Convective Heat Transfer 85
4.1 Introduction 85
4.2 Conservation of Mass 85
4.3 The Momentum Integral Equation 87
4.3.1 The Displacement Thickness δ1 88
4.3.2 Momentum Thickness δ2 89
4.4 Alternative Form of the Momentum Integral Equation 90
4.5 Momentum Integral Equation for Two-Dimensional Flow 90
4.6 Energy Integral Equation 91
4.6.1 Enthalpy Thickness 93
4.6.2 Conduction Thickness 93
4.6.3 Convection Conductance or Heat Transfer Coefficient 93
4.7 Alternative Form of the Energy Integral Equation 94
4.8 Energy Integral Equation for Two-Dimensional Flow 94
Problems 94
References 96
5 Dimensional Analysis 97
5.1 Introduction 97
5.2 Dimensional Analysis 101
5.2.1 Dimensional Homogeneity 102
5.2.2 Buckingham π Theorem 102
5.2.3 Determination of π Terms 103
5.3 Nondimensionalization of Basic Differential Equations 116
5.4 Discussion 125
5.5 Dimensionless Numbers 125
5.5.1 Reynolds Number 125
5.5.2 Peclet Number 126
5.5.3 Prandtl Number 126
5.5.4 Nusselt Number 126
5.5.5 Stanton Number 126
5.5.6 Skin Friction Coefficient 126
5.5.7 Graetz Number 127
5.5.8 Eckert Number 127
5.5.9 Grashof Number 127
5.5.10 Rayleigh Number 127
5.5.11 Brinkman Number 127
5.6 Correlations of Experimental Data 128
Problems 136
References 147
6 One-Dimensional Solutions in Convective Heat Transfer 149
6.1 Introduction 149
6.2 Couette Flow 151
6.3 Poiseuille Flow 156
6.4 Rotating Flows 171
Problems 175
References 180
7 Laminar External Boundary Layers: Momentum and Heat Transfer 183
7.1 Introduction 183
7.2 Velocity Boundary Layer over a Semi-Infinite Flat Plate: Similarity Solution 183
7.2.0.1 x-Component of Velocity – u/ U∞ 190
7.2.0.2 Boundary Layer Thickness δ(x) 190
7.2.0.3 Wall Shear Stress τw 191
7.2.0.4 Local Skin Friction Coefficient cf (x) 191
7.2.0.5 drag Force d 192
7.2.0.6 Average Skin Friction Coefficient c̅f 192
7.2.0.7 Displacement Thickness δ1(x) 192
7.2.0.8 Momentum Thickness δ2(x) 192
7.3 Momentum Transfer over a Wedge (Falkner–Skan Wedge Flow): Similarity Solution 195
7.4 Application of Integral Methods to Momentum Transfer Problems 201
7.4.1 Laminar Forced Flow over a Flat Plate with Uniform Velocity 203
7.4.2 Two-Dimensional Laminar Flow over a Surface with Pressure Gradient (Variable Free Stream Velocity) 204
7.4.2.1 The Correlation Method of Thwaites 207
7.4.2.2 A Thwaites Type Correlation for Axisymmetric Body 212
7.5 Viscous Incompressible Constant Property Parallel Flow over a Semi-Infinite Flat Plate: Similarity Solution for Uniform Wall Temperature Boundary Condition 212
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 225
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 228
7.8 Viscous Incompressible Constant Property Parallel Flow over a Semi-Infinite Flat Plate: Similarity Solution for Uniform Heat Flux Boundary Condition 230
7.9 Viscous Incompressible Constant Property Parallel Flow over a Semi-Infinite Flat Plate: Similarity Solutions for Variable Wall Temperature Boundary Condition 237
7.9.1 Superposition Principle 245
7.10 Viscous Incompressible Constant Property Flow over a Wedge (Falkner–Skan Wedge Flow): Similarity Solution for Uniform Wall Temperature Boundary Condition 249
7.11 Effect of Property Variation 252
7.12 Application of Integral Methods to Heat Transfer Problems 253
7.12.1 Viscous Flow with Constant Free Stream Velocity Along a Semi-Infinite Plate Under Uniform Wall Temperature: With Unheated Starting Length or Adiabatic Segment 256
7.12.1.1 The Plate Without Unheated Starting Length 262
7.12.2 Viscous Flow with Constant Free Stream Velocity Along a Semi-Infinite Plate with Uniform Wall Heat Flux: With Unheated Starting Length (Adiabatic Segment) 262
7.12.2.1 The Plate with No Unheated Starting Length 265
7.13 Superposition Principle 265
7.13.1 Superposition Principle Applied to Slug Flow over a Flat Plate: Arbitrary Variation in Wall Temperature 266
7.13.1.1 Boundary Condition: Single Step at X = 0 266
7.13.1.2 Boundary Condition: Two Steps at X = 0 and X =ξ1 268
7.13.1.3 Boundary Condition: Three Steps at X = 0, X =ξ1 , and X =ξ2 268
7.13.2 Superposition Principle Applied to Slug Flow over a Flat Plate: Arbitrary Variation in Wall Heat Flux 272
7.13.2.1 Boundary Condition: Single Step at X = 0 273
7.13.2.2 Boundary Condition: Two Steps at X = 0 and X =ξ1 274
7.13.2.3 Boundary Condition: Triple Steps at X = 0, X =ξ1 , and X =ξ2 275
7.13.3 Superposition Principle Applied to Viscous Flow over a Flat Plate: Stepwise Variation in Wall Temperature 278
7.13.3.1 First Problem 278
7.13.3.2 Second Problem 279
7.13.3.3 Heat Flux for 0 < X < ξ 279
7.13.3.4 The Heat Flux for X > ξ 1 280
7.13.4 Superposition Principle Applied to Viscus Flow over a Flat Plate: Stepwise Variation in Surface Heat Flux 282
7.13.4.1 First Problem 282
7.13.4.2 Second Problem 283
7.14 Viscous Flow over a Flat Plate with Arbitrary Surface Temperature Distribution 284
7.15 Viscous Flow over a Flat Plate with Arbitrarily Specified Heat Flux 289
7.16 One-Parameter Integral Method for Incompressible Two-Dimensional Laminar Flow Heat Transfer: Variable U ∞ (x) and Constant Tw − T∞ = const 293
7.17 One-Parameter Integral Method for Incompressible Laminar Flow Heat Transfer over a Constant Temperature of a Body of Revolution 295
Problems 299
References 310
8 Laminar Momentum and Heat Transfer in Channels 313
8.1 Introduction 313
8.2 Momentum Transfer 313
8.2.1 Hydrodynamic Considerations in Ducts 313
8.2.2 Fully Developed Laminar Flow in Circular Tube 318
8.2.3 Fully Developed Flow Between Two Infinite Parallel Plates 323
8.3 Thermal Considerations in Ducts 326
8.4 Heat Transfer in the Entrance Region of Ducts 335
8.4.1 Circular Pipe: Slug Flow Heat Transfer in the Entrance Region 337
8.4.1.1 Heat Transfer for Low-Prandtl-Number Fluid Flow (Slug Flow) in the Entrance Region of Circular Tube Subjected to Constant Wall Temperature 337
8.4.1.2 Heat Transfer to Low-Prandtl-Number Fluid Flow (Slug Flow) in the Entrance Region of the Circular Tube Subjected to Constant Heat Flux 345
8.4.1.3 Empirical and Theoretical Correlations for Viscous Flow Heat Transfer in the Entrance Region of the Circular Tube 350
8.4.2 Parallel Plates: Slug Flow Heat Transfer in the Entrance Region 355
8.4.2.1 Heat Transfer to a Low-Prandtl-Number Fluid (Slug Flow) in the Entrance Region of Parallel Plates: Both Plates Are Subjected to Constant Wall Temperatures 355
8.4.2.2 Heat Transfer for Low-Prandtl-Number Fluid Flow (Slug Flow) in the Entrance Region of Parallel Plates: Both Plates Are Subjected to UHF 358
8.4.2.3 Heat Transfer for Low-Prandtl-Number Fluid Flow (Slug Flow) in the Entrance Region of Parallel Plates: Upper Plate Is Insulated While the Lower Plate Is Subjected to Constant Wall Temperature 363
8.4.2.4 Heat Transfer for Low-Prandtl-Number Fluid Flow (Slug Flow) in the Entrance Region of Parallel Plates: Upper Plate Is Insulated While the Lower Plate Is Subjected to Constant Heat Flux 367
8.4.2.5 Empirical and Theoretical Correlations for Viscous Flow Heat Transfer in the Entrance Region of Parallel Plates 370
8.5 Fully Developed Heat Transfer 372
8.5.1 Circular Tube 372
8.5.1.1 HFD and TFD Laminar Forced Convection Heat Transfer for Slug Flow in a Circular Pipe Subjected to Constant Wall Heat Flux 372
8.5.1.2 HFD and TFD Laminar Forced Convection Heat Transfer for Viscous Flow in a Circular Tube Subjected to Constant Wall Heat Flux 375
8.5.1.3 HFD and TFD Laminar Forced Convection Heat Transfer for Viscous Flow in a Circular Tube Subjected to Constant Wall Temperature 378
8.5.2 Infinite Parallel Plates 382
8.5.2.1 HFD and TFD Laminar Forced Convection Heat Transfer for Viscous Flow Between a Parallel Plate Channel. Both Plates Are Subjected to Constant Wall Heat Flux Boundary Condition 383
8.6 Heat Transfer in the Thermal Entrance Region 387
8.6.1 Circular Tube 388
8.6.1.1 Graetz Problem: HFD and Thermally Developing Flow in a Circular Tube under Constant Wall Temperature Boundary Condition 388
8.6.1.2 The Leveque Solution: UWT Boundary Condition 401
8.6.1.3 Graetz Problem: HFD and Thermally Developing Flow for Viscous Flow in Circular Tube Under Uniform Wall Heat Flux Boundary Condition 406
8.6.1.4 Empirical and Theoretical Correlations for Viscous Flow in the Thermal Entrance Region of the Pipe 415
8.6.2 Two Infinite Parallel Plates 419
8.6.2.1 Graetz Problem: HFD and Thermally Developing Flow Between Parallel Plates Subjected to Constant Wall Temperature 419
8.6.2.2 Graetz Problem: HFD and Thermally Developing Flow Between Parallel Plates Subjected to Constant Wall Heat Flux 428
8.6.2.3 Empirical and Theoretical Correlations for Viscous Flow in Thermal Entrance Region of Parallel Plates 436
8.7 Circular Pipe with Variable Surface Temperature Distribution in the Axial Direction 438
8.8 Circular Pipe with Variable Surface Heat Flux Distribution in the Axial Direction 443
8.9 Short Tubes 446
8.10 Effect of Property Variation 448
8.11 Regular Sturm-Liouville Systems 449
Problems 450
References 463
9 Foundations of Turbulent Flow 465
9.1 Introduction 465
9.2 The Reynolds Experiment 465
9.3 Nature of Turbulence 466
9.4 Time Averaging and Fluctuations 467
9.5 Isotropic Homogeneous Turbulence 470
9.6 Reynolds Averaging 470
9.7 Governing Equations of Incompressible Steady Mean Turbulent Flow 474
9.8 Turbulent Momentum Boundary Layer Equation 477
9.9 Turbulent Energy Equation 478
9.10 Turbulent Boundary Layer Energy Equation 479
9.11 Closure Problem of Turbulence 480
9.12 Eddy Diffusivity of Momentum 481
9.13 Eddy Diffusivity of Heat 482
9.14 Transport Equations in the Cylindrical Coordinate System 483
9.15 Experimental Work on the Turbulent Mean Flow 484
9.15.1 Turbulent Flow in Pipe: Velocity Profiles 485
9.15.2 Turbulent Flow over a Flat Plate: Velocity Profiles 491
9.16 Transition to Turbulent Flow 496
Problems 498
References 504
10 Turbulent External Boundary Layers: Momentum and Heat Transfer 507
10.1 Introduction 507
10.2 Turbulent Momentum Boundary Layer 507
10.3 Turbulence Models 508
10.3.1 Zero-Equation Models 508
10.3.1.1 Boussinesq Model 508
10.3.1.2 Prandtl’s Mixing-Length Model 508
10.3.1.3 Van Driest Model 509
10.4 Turbulent Flow over a Flat Plate with Constant Free-Stream Velocity: Couette Flow Approximation 510
10.4.1 Inner Region 510
10.5 The Universal Velocity Profile 511
10.5.1 Three-Layer (von Karman) Model for the Velocity Profile 511
10.5.2 Other Velocity Models 514
10.6 Approximate Solution by the Integral Method for the Turbulent Momentum Boundary Layer over a Flat Plate 514
10.7 Laminar and Turbulent Boundary Layer 519
10.8 Other Eddy Diffusivity Momentum Models 521
10.9 Turbulent Heat Transfer 522
10.10 Analogy Between Momentum and Heat Transfer 529
10.10.1 Reynold’s Analogy 529
10.10.2 Chilton–Colburn Analogy 531
10.10.3 Prandtl–Taylor Analogy 532
10.10.4 Von Karman Analogy 535
10.11 Some Other Correlations for Turbulent Flow over a Flat Plate 539
10.12 Turbulent Flow Along a Semi-infinite Plate with Unheated Starting Length: Constant Temperature Solution 542
10.13 Flat Plate with Arbitrarily Specified Surface Temperature 550
10.14 Constant Free-Stream Velocity Flow Along a Flat Plate with Uniform Heat Flux 553
10.15 Turbulent Flow Along a Semi-Infinite Plate with Arbitrary Heat Flux Distribution 554
10.16 Turbulent Transition and Overall Heat Transfer 558
10.17 Property Variation 564
Problems 564
References 569
11 Turbulent Internal Flow: Momentum and Heat Transfer 573
11.1 Introduction 573
11.2 Momentum Transfer 573
11.2.1 Momentum Transfer in Infinite Two Parallel Plates 573
11.2.1.1 The Entrance Region 574
11.2.1.2 The HFD Region 575
11.2.1.3 Prandtl’s Mixing-Length Model 578
11.2.1.4 Buffer Region 579
11.2.1.5 The Mean Velocity 582
11.2.1.6 Skin Friction Coefficient or Fanning Friction Factor cf 582
11.2.2 Momentum Transfer in Circular Pipe Flow 585
11.2.2.1 Entrance Region 585
11.2.2.2 The HFD Region 586
11.2.2.3 Average Velocity V 589
11.2.2.4 Skin Friction Factor cf 589
11.2.2.5 Moody Friction Factor f 589
11.2.2.6 Prandtl Mixing-Length Model 590
11.2.2.7 Laminar Sublayer 591
11.2.2.8 Buffer Region 591
11.2.2.9 Turbulent Region 591
11.2.2.10 Moody Friction Factor 592
11.2.2.11 Fanning Friction Factor 593
11.2.2.12 The Power Law Velocity Distribution 596
11.3 Fully Developed Turbulent Heat Transfer 597
11.3.1 TFD and HFD Turbulent Flow Between Parallel Plates Subjected to UHF 598
11.3.1.1 Mean Stream Temperature 602
11.3.2 TFD and HFD Turbulent Flow in a Pipe Subjected to UHF 605
11.3.2.1 Laminar Viscous Sublayer: 0 < y+ < 5 609
11.3.2.2 Buffer Layer: 5 < y+ < 30 610
11.3.2.3 Turbulent Region: y+ > 30 610
11.4 HFD Thermally Developing Turbulent Heat Transfer 618
11.4.1 Circular Duct with UWT 618
11.4.2 Circular Duct with Uniform Wall Heat Flux 625
11.4.2.1 Solution for Fully Developed Temperature Distribution θ1 626
11.4.2.2 Solution for the Entry Region Temperature Distribution θ2 627
11.5 Analogies for Internal Flow 629
11.5.1 Reynolds Analogy 629
11.5.2 Colburn Analogy 631
11.5.3 Prandtl–Taylor Analogy 631
11.5.3.1 Laminar Sublayer 632
11.5.3.2 Turbulent Core 632
11.5.4 von Karman Analogy 633
11.5.4.1 Laminar Sublayer: 0 ≤ y+ ≪ 5 634
11.5.4.2 Buffer Layer: 5 ≤ y+ ≪ 30 635
11.5.4.3 Turbulent Core: y+ ≥ 30 635
11.5.5 The Analogy of Kadar and Yaglom 636
11.5.6 The Analogy of Yu et al. 637
11.5.7 Martinelli Analogy 639
11.6 Combined Entrance Region 641
11.7 Empirical and Theoretical Correlations for Turbulent Flow in Channels 642
11.7.1.1 Colburn Correlation 645
11.7.1.2 Dittus and Boelter Correlation 646
11.7.1.3 Sieder–Tate Correlation 646
11.7.1.4 Hausen Correlations 647
11.7.1.5 Petukhov Correlation 647
11.7.1.6 Gnielinski Correlation 649
11.7.1.7 Gnielinski Correlation with Modification 650
11.7.1.8 Sleicher and Rouse Correlation 650
11.7.1.9 Nusselt Correlation 651
11.8 Heat Transfer in Transitional Flow 652
11.8.1 Friction Factor in the Transitional Flow 653
11.8.2 Heat Transfer in the Transition Region 654
11.8.2.1 Tam and Ghajar Approach 654
11.8.2.2 Churchill Approach 655
11.8.2.3 Gnielinski Approach 656
11.8.2.4 Abraham et al. Approach 657
11.9 Effect of Property Variation 660
Problems 660
References 670
12 Free Convection Heat Transfer 675
12.1 Introduction 675
12.2 Fundamental Equations and Dimensionless Parameters of Free Convection 675
12.3 Scaling in Natural Convection 679
12.4 Similarity Solution for Laminar Boundary Layer over a Semi-Infinite Vertical Flat Plate 681
12.4.1 Constant Wall Temperature 681
12.4.2 Uniform Heat Flux 688
12.5 Integral Method (von Karman–Pohlhausen Method): An Approximate Analysis of Laminar Free Convection on a Vertical Plate 695
12.5.1 Constant Wall Temperature 697
12.5.2 Uniform Heat Flux 700
12.6 Turbulent Free Convection Heat Transfer on a Vertical Plate 702
12.7 Empirical Correlations for Free Convection 704
12.7.1 Vertical Plate 704
12.7.2 Horizontal Plate 712
12.7.3 Inclined Plates 715
12.7.4 Vertical Cylinders 719
12.7.5 Horizontal Cylinder 722
12.7.6 Inclined Cylinder 723
12.7.7 Free Convection from Vertical Cylinders of Small Diameter 724
12.8 Free Convection Within Parallel Plate Channels 725
12.8.1 Vertical Parallel Plate Channel 725
12.8.2 Horizontal Parallel Plate Channel 731
12.8.3 Inclined Parallel Plate Channel 732
12.9 Rectangular Enclosures 735
12.9.1 Horizontal Rectangular Enclosure (θ=0) 735
12.9.2 Vertical Rectangular Enclosure 737
12.9.3 Inclined Rectangular Enclosure 740
12.10 Horizontal Concentric Cylinders 743
12.11 Concentric Spheres 744
12.12 Spheres 744
Problems 745
References 752
Index 755
| Erscheinungsdatum | 01.06.2023 |
|---|---|
| Verlagsort | New York |
| Sprache | englisch |
| Gewicht | 2041 g |
| Themenwelt | Technik ► Maschinenbau |
| ISBN-10 | 1-119-76676-1 / 1119766761 |
| ISBN-13 | 978-1-119-76676-6 / 9781119766766 |
| Zustand | Neuware |
| Haben Sie eine Frage zum Produkt? |
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