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Fluid Mechanics and Machinery

Fluid Mechanics and Machinery
C.P. Kothandaraman
R. Rudramoorthy
614 pages

Open: Fluid Mechanics and Machinery
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Introduction:

The flow of ideal non-viscous fluids was extensively studied and mathematical theories were developed during the last century. The field of study was called as ‘Hydrodynamics’. However the results of mathematical analysis could not be applied directly to the flow of real fluids. Experiments with water flow resulted in the formulation of empirical equations applicable to engineering designs. The field was called Hydraulics. Due to the development of industries there arose a need for the study of fluids other than water. Theories like boundary layer theory were developed which could be applied to all types of real fluids, under various conditions of flow. The combination of experiments, the mathematical analysis of hydrodynamics and the new theories is known as ‘Fluid Mechanics’. Fluid Mechanics encompasses the study of all types of fluids under static, kinematics and dynamic conditions.

The study of properties of fluids is basic for the understanding of flow or static condition of fluids. The important properties are density, viscosity, surface tension, bulk modulus and vapor pressure. Viscosity causes resistance to flow. Surface tension leads to capillary effects. Bulk modulus is involved in the propagation of disturbances like sound waves in fluids. Va pour pressure can cause flow disturbances due to evaporation at locations of low pressure. It plays an important role in cavitation studies in fluid machinery.

Various properties of fluids are discussed in detail, with stress on their effect on flow. Fairly elaborate treatment is attempted due to their importance in engineering applications. The basic laws used in the discussions are :

(i) Newton’s laws of motion,
(ii) Laws of conservation of mass and energy,
(iii) Laws of Thermodynamics, and
(iv) Newton’s law of viscosity.

A fluid is defined as a material which will continue to deform with the application of shear force however small the force may be.

TOC

1 Physical Properties of Fluids .. 1
1.0 Introduction .. 1
1.1 Three Phases of Matter.. 2
1.2 Compressible and Incompressible Fluids 2
1.3 Dimensions and Units. 3
1.4 Continuum . 4
1.5 Definition of Some Common Terminology . 4
1.6 Va pour and Gas .. 5
1.7 Characteristic Equation for Gases .. 6
1.8 Viscosity .. 7
1.8.1 Newtonian and Non Newtonian Fluids 10
1.8.2 Viscosity and Momentum Transfer 11
1.8.3 Effect of Temperature on Viscosity 11
1.8.4 Significance of Kinematic Viscosity 11
1.8.5 Measurement of Viscosity of Fluids .. 12
1.9 Application of Viscosity Concept .. 13
1.9.1 Viscous Torque and Power—Rotating Shafts .. 13
1.9.2 Viscous Torque—Disk Rotating Over a Parallel Plate .. 14
1.9.3 Viscous Torque—Cone in a Conical Support 16
1.10 Surface Tension 17
1.10.1 Surface Tension Effect on Solid-Liquid Interface . 17
1.10.2 Capillary Rise or Depression . 18
1.10.3 Pressure Difference Caused by Surface Tension on a Doubly
Curved Surface 19
1.10.4 Pressure Inside a Droplet and a Free Jet .. 20
1.11 Compressibility and Bulk Modulus . 21
1.11.1 Expressions for the Compressibility of Gases . 22
1.12 Va pour Pressure .. 23
1.12.1 Partial Pressure . 23
Solved Problems .. 24
Objective Questions 33
Review Questions . 38
Exercise Problems 39

2 Pressure Distribution in Fluids 42
2.0 Introduction 42
2.1 Pressure 42
2.2 Pressure Measurement 43
2.3 Pascal’s Law.. 45
2.4 Pressure Variation in Static Fluid (Hydrostatic Law) . 46
2.4.1 Pressure Variation in Fluid with Constant Density 47
2.4.2 Pressure Variation in Fluid with Varying Density .. 48
2.5 Manometers 49
2.5.1 Micromanometer 51
2.6 Distribution of Pressure in Static Fluids Subjected to Acceleration, as . 53
2.6.1 Free Surface of Accelerating Fluid . 54
2.6.2 Pressure Distribution in Accelerating Fluids along Horizontal Direction . 55
2.7 Forced Vortex 58
Solved Problems .. 60
Review Questions . 71
Objective Questions 71
Exercise Problems 74

3 Forces on Surfaces Immersed in Fluids 80
3.0 Introduction 80
3.1 Centroid and Moment of Inertia of Areas 81
3.2 Force on an Arbitrarily Shaped Plate Immersed in a Liquid .. 83
3.3 Centre of Pressure for an Immersed Inclined Plane . 84
3.3.1 Centre of Pressure for Immersed Vertical Planes 86
3.4 Component of Forces on Immersed Inclined Rectangles 87
3.5 Forces on Curved Surfaces 89
3.6 Hydrostatic Forces in Layered Fluids 92
Solved Problems .. 93
Review Questions .. 111
Objective Questions . 112
Exercise Problems . 115

4 Buoyancy Forces and Stability of Floating Bodies .. 119
4.0 Archimedes Principle . 119
4.1 Buoyancy Force . 119
4.2 Stability of Submerged and Floating Bodies .. 121
4.3 Conditions for the Stability of Floating Bodies .. 123
4.4 Metacentric Height . 124
4.4.1 Experimental Method for the Determination of Metacentric Height 125
Solved Problems 125
Review Questions .. 136
Objective Questions . 137
Exercise Problems . 139

5 Fluid Flow—Basic Concepts—Hydrodynamics . 142
5.0 Introduction . 142
5.1 Lagrangian and Eularian Methods of Study of Fluid Flow 143
5.2 Basic Scientific Laws Used in the Analysis of Fluid Flow .. 143
5.3 Flow of Ideal / Inviscid and Real Fluids . 143
5.4 Steady and Unsteady Flow . 144
5.5 Compressible and Incompressible Flow . 144
5.6 Laminar and Turbulent Flow 144
5.7 Concepts of Uniform Flow, Reversible Flow and Three Dimensional Flow. 145
5.8 Velocity and Acceleration Components .. 145
5.9 Continuity Equation for Flow—Cartesian Co-ordinates . 146
5.10 Irrotational Flow and Condition for Such Flows .. 148
5.11 Concepts of Circulation and Vorticity . 148
5.12 Stream Lines, Stream Tube, Path Lines, Streak Lines and Time Lines .. 149
5.13 Concept of Stream Line 150
5.14 Concept of Stream Function .. 151
5.15 Potential Function 153
5.16 Stream Function for Rectilinear Flow Field (Positive X Direction) .. 154
5.17 Two Dimensional Flows—Types of Flow 154
5.17.1 Source Flow 155
5.17.2 Sink Flow . 155
5.17.3 Irrotational Vortex of Strength K. 155
5.17.4 Doublet of Strength Λ .. 156
5.18 Principle of Superposing of Flows (or Combining of Flows) .. 157
5.18.1 Source and Uniform Flow (Flow Past a Half Body) .. 157
5.18.2 Source and Sink of Equal Strength with Separation of 2a Along x-Axis 157
5.18.3 Source and Sink Displaced at 2a and Uniform Flow (Flow Past a Rankine Body) . 158
5.18.4 Vortex (Clockwise) and Uniform Flow .. 158
5.18.5 Doublet and Uniform Flow (Flow Past a Cylinder) .. 158
5.18.6 Doublet, Vortex (Clockwise) and Uniform Flow .. 158
5.18.7 Source and Vortex (Spiral Vortex Counterclockwise).. 159
5.18.8 Sink and Vortex (Spiral Vortex Counterclockwise) .. 159
5.18.9 Vortex Pair (Equal Strength, Opposite Rotation, Separation by 2a) 159
5.19 Concept of Flow Net 159
Solved Problems 160
Objective Questions . 174
Exercise Problems . 178

6 Bernoulli Equation and Applications. 180
6.0 Introduction . 180
6.1 Forms of Energy Encountered in Fluid Flow.. 180
6.1.1 Kinetic Energy . 181
6.1.2 Potential Energy . 181
6.1.3 Pressure Energy (Also Equals Flow Energy) 182
6.1.4 Internal Energy 182
6.1.5 Electrical and Magnetic Energy 183
6.2 Variation in the Relative Values of Various Forms of Energy During Flow . 183
6.3 Euler’s Equation of Motion for Flow Along a Stream Line 183
6.4 Bernoulli Equation for Fluid Flow . 184
6.5 Energy Line and Hydraulic Gradient Line .. 187
6.6 Volume Flow Through a Venturimeter .. 188
6.7 Euler and Bernoulli Equation for Flow with Friction 191
6.8 Concept and Measurement of Dynamic, Static and Total Head 192
6.8.1 Pitot Tube 193
Solved Problems 194
Objective Questions . 213
Exercise Problems . 215

7 Flow in Closed Conduits (Pipes).. 219
7.0 Parameters Involved in the Study of Flow Through Closed Conduits 219
7.1 Boundary Layer Concept in the Study of Fluid Flow 220
7.2 Boundary Layer Development Over A Flat Plate 220
7.3 Development of Boundary Layer in Closed Conduits (Pipes) .. 221
7.4 Features of Laminar and Turbulent Flows .. 222
7.5 Hydraulically “Rough” and “Smooth” Pipes . 223
7.6 Concept of “Hydraulic Diameter”: (Dh) .. 223
7.7 Velocity Variation with Radius for Fully Developed Laminar Flow in Pipes .. 224
7.8 Darcy–Weisbach Equation for Calculating Pressure Drop 226
7.9 Hagen–Poiseuille Equation for Friction Drop 228
7.10 Significance of Reynolds Number in Pipe Flow . 229
7.11 Velocity Distribution and Friction Factor for Turbulent Flow in Pipes . 230
7.12 Minor Losses in Pipe Flow .. 231
7.13 Expression for the Loss of Head at Sudden Expansion in Pipe Flow 232
7.14 Losses in Elbows, Bends and Other Pipe Fittings 234
7.15 Energy Line and Hydraulic Grade Line in Conduit Flow .. 234
7.16 Concept of Equivalent Length.. 235
7.17 Concept of Equivalent Pipe or Equivalent Length .. 235
7.18 Fluid Power Transmission Through Pipes 238
7.18.1 Condition for Maximum Power Transmission .. 238

8 Dimensional Analysis.. 263
8.0 Introduction . 263
8.1 Methods of Determination of Dimensionless Groups. 264
8.2 The Principle of Dimensional Homogeneity 265
8.3 Buckingham Pi Theorem . 265
8.3.1 Determination of π Groups 265
8.4 Important Dimensionless Parameters 270
8.5 Correlation of Experimental Data . 270
8.5.1 Problems with One Pi Term . 271
8.5.2 Problems with Two Pi Terms .. 271
8.5.3 Problems with Three Dimensionless Parameters .. 273
Solved Problems 273
Objective Questions . 291
Exercise Problems . 293

9 Similitude and Model Testing . 296
9.0 Introduction . 296
9.1 Model and Prototype .. 296
9.2 Conditions for Similarity Between Models and Prototype . 297
9.2.1 Geometric Similarity 297
9.2.2 Dynamic Similarity 297
9.2.3 Kinematic Similarity 298
7.19 Network of Pipes .. 239
7.19.1 Pipes in Series—Electrical Analogy 240
7.19.2 Pipes in Parallel .. 241
7.19.3 Branching Pipes .. 243
7.19.4 Pipe Network. 245
Solved Problems 245
Objective Questions . 256
Exercise Problems . 259

9.3 Types of Model Studies . 298
9.3.1 Flow Through Closed Conduits . 298
9.3.2 Flow Around Immersed Bodies.. 299
9.3.3 Flow with Free Surface .. 300
9.3.4 Models for Turbomachinery . 301
9.4 Nondimensionalising Governing Differential Equations. 302
9.5 Conclusion . 303
Solved Problems 303
Objective Questions . 315
Exercise Problems . 317

10 Boundary Layer Theory and Flow Over Surfaces . 321
10.0 Introduction . 321
10.1 Boundary Layer Thickness . 321
10.1.1 Flow Over Flat Plate 322
10.1.2 Continuity Equation . 322
10.1.3 Momentum Equation 324
10.1.4 Solution for Velocity Profile . 325
10.1.5 Integral Method .. 327
10.1.6 Displacement Thickness . 330
10.1.7 Momentum Thickness . 331
10.2 Turbulent Flow .. 332
10.3 Flow Separation in Boundary Layers .. 334
10.3.1 Flow Around Immersed Bodies – Drag and Lift . 334
10.3.2 Drag Force and Coefficient of Drag . 335
10.3.3 Pressure Drag .. 336
10.3.4 Flow Over Spheres and Cylinders 337
10.3.5 Lift and Coefficient of Lift . 338
10.3.6 Rotating Sphere and Cylinder 339
Solved Problems 341
Objective Questions . 353
Exercise Problems . 356

11 Flow Measurements . 359
11.1 Introduction . 359
11.2 Velocity Measurements. 359
11.2.1 Pitot Tube 360
11.2.2 Vane Anemometer and Currentmeter .. 362
11.2.3 Hot Wire Anemometer. 362
11.2.4 Laser Doppler Anemometer . 363
11.3 Volume Flow Rate Measurement 364
11.3.1 Rotameter (Float Meter) 364
11.3.2 Turbine Type Flowmeter 364
11.3.3 Venturi, Nozzle and Orifice Meters . 365
11.3.4 Elbow Meter .. 367
11.4 Flow Measurement Using Orifices, Notches and Weirs .. 367
11.4.1 Discharge Measurement Using Orifices .. 367
11.4.2 Flow Measurements in Open Channels 368
Solved Problems 371
Review Questions .. 379
Objective Questions . 380
Exercise Problems . 381

12 Flow in Open Channels .. 383
12.0 Introduction . 383
12.1.1 Characteristics of Open Channels 383
12.1.2 Classification of Open Channel Flow . 384
12.2 Uniform Flow: (Also Called Flow at Normal Depth) .. 384
12.3 Chezy’s Equation for Discharge .. 385
12.4 Determination of Chezy’s Constant .. 386
12.4.1 Bazin’s Equation for Chezy’s Constant . 386
12.4.2 Kutter’s Equation for Chezy’s Constant C.. 387
12.4.3 Manning’s Equation for C . 388
12.5 Economical Cross-Section for Open Channels 390
12.6 Flow with Varying Slopes and Areas 395
12.6.1 Velocity of Wave Propagation in Open Surface Flow .. 395
12.6.2 Froude Number 397
12.6.3 Energy Equation for Steady Flow and Specific Energy.. 397
12.6.4 Non Dimensional Representation of Specific Energy Curve 400
12.7 Effect of Area Change 404
12.7.1 Flow Over a Bump . 404
12.7.2 Flow Through Sluice Gate, from Stagnant Condition . 406
12.7.3 Flow Under a Sluice Gate in a Channel 407
12.8 Flow with Gradually Varying Depth 409
12.8.1 Classification of Surface Variations 410
12.9 The Hydraulic Jump (Rapidly Varied Flow) 411
12.10 Flow Over Broad Crested Weir 414
12.11 Effect of Lateral Contraction. 415
Solved Problems 416
Review Questions .. 430
Objective Questions . 430
Exercise Problems . 432

13 Dynamics of Fluid Flow.. 435
13.0 Introduction . 435
13.1 Impulse Momentum Principle .. 435
13.1.1 Forces Exerted on Pressure Conduits 436
13.1.2 Force Exerted on a Stationary Vane or Blade .. 438
13.2 Absolute and Relative Velocity Relations . 439
13.3 Force on a Moving Vane or Blade .. 439
13.4 Torque on Rotating Wheel .. 443
Solved Problems 445
Exercise Questions 450

14 Hydraulic Turbines. 452
14.0 Introduction . 452
14.1 Hydraulic Power Plant.. 452
14.2 Classification of Turbines 453
14.3 Similitude and Model Testing .. 453
14.3.1 Model and Prototype. 457
14.3.2 Unit Quantities 459
14.4 Turbine Efficiencies 460
14.5 Euler Turbine Equation 461
14.5.1 Components of Power Produced 462
14.6 Pelton Turbine 464
14.6.1 Power Development .. 466
14.6.2 Torque and Power and Efficiency Variation with Speed Ratio.. 470
14.7 Reaction Turbines 472
14.7.1 Francis Turbines . 473
14.8 Axial Flow Turbines 480
14.9 Cavitation in Hydraulic Machines . 482
14.9 Governing of Hydraulic Turbines .. 484
Worked Examples . 486
Review Questions .. 513
Objective Questions . 514
Exercise Problems . 515

15 Rotodynamic Pumps 519
15.0 Introduction . 519
15.1 Centrifugal Pumps .. 519
15.1.1 Impeller 521
15.1.2 Classification . 521
15.2 Pressure Developed by the Impeller . 522
15.2.1 Manometric Head .. 523
15.3 Energy Transfer by Impeller . 523
15.3.1 Slip and Slip Factor .. 525
15.3.3 Losses in Centrifugal Pumps .. 525
15.3.4 Effect of Outlet Blade Angle 526
15.4 Pump Characteristics. 527
15.5 Operation of Pumps in Series and Parallel .. 529
15.6 Specific Speed and Significance .. 531
15.7 Cavitation . 532
15.8 Axial Flow Pump .. 533
15.9 Power Transmitting Systems 535
15.9.1 Fluid Coupling.. 535
15.9.2 Torque Converter 536
Solved Examples 538
Revierw Questions 556
Objective Questions . 556
Exercise Problems . 557

16 Reciprocating Pumps .. 560
16.0 Introduction . 560
16.1 Comparison .. 560
16.2 Description and Working . 560
16.3 Flow Rate and Power 562
16.3.1 Slip .. 563
16.4 Indicator Diagram 564
16.4.1 Acceleration Head .. 565
16.4.2 Minimum Speed of Rotation of Crank 569
16.4.3 Friction Head 570
16.5 Air Vessels 572
16.5.1 Flow into and out of Air Vessel .. 575
16.6 Rotary Positive Displacement Pumps . 576
16.6.1 Gear Pump.. 577
16.6.2 Lobe Pump .. 577
16.6.3 Vane Pump . 577
Solved Problems 578
Review Questions .. 587
Objective Questions . 587
Exercise Problems . 587
Appendix . 590
Index .. 595