What pressure gradient along the streamline dp ds

iv

2.3.2 Bourdon Gage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 2.3.3 Pressure Transducer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 2.3.4 Manometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

2.3.4.1 Piezometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 2.3.4.2 U-tube and inclined-tube manometers . . . . . . . . . . . . . . . . . 96

2.4 Forces on Plane Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 2.5 Forces on Curved Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 2.6 Buoyancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

2.6.1 Fully Submerged Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 2.6.2 Partially Submerged Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 2.6.3 Buoyancy effects within fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

2.7 Rigid-Body Motion of Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 2.7.1 Liquid with Constant Acceleration . . . . . . . . . . . . . . . . . . . . . . . . 131 2.7.2 Liquid in a Rotating Container . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

3 Kinematics and Streamline Dynamics 171 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 3.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

3.2.1 Tracking the Movement of Fluid Elements . . . . . . . . . . . . . . . . . . . . 175 3.2.2 The Material Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 3.2.3 Flow Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

3.3 Dynamics of Flow Along a Streamline . . . . . . . . . . . . . . . . . . . . . . . . . . 186 3.4 Applications of the Bernoulli Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 196

3.4.1 Flow through Orifices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 3.4.2 Flow Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

3.4.2.1 Pitot-static tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 3.4.2.2 Venturi meter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

3.4.3 Trajectory of a Liquid Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 3.4.4 Compressibility Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 3.4.5 Viscous Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 3.4.6 Branching Conduits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

3.5 Curved Flows and Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 3.5.1 Forced Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

3.5.1.1 Cylindrical forced vortex . . . . . . . . . . . . . . . . . . . . . . . . 218 3.5.1.2 Spiral forced vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

3.5.2 Free Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 3.5.2.1 Cylindrical free vortex . . . . . . . . . . . . . . . . . . . . . . . . . . 221 3.5.2.2 Spiral free vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

4 Finite Control Volume Analysis 253 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 4.2 Reynolds Transport Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 4.3 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

v

4.3.1 Closed Conduits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 4.3.2 Free Discharges from Reservoirs . . . . . . . . . . . . . . . . . . . . . . . . . . 261 4.3.3 Moving Control Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

4.4 Conservation of Linear Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 4.4.1 General Momentum Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 265 4.4.2 Forces on Pressure Conduits . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

4.4.2.1 Forces on reducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 4.4.2.2 Forces on bends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 4.4.2.3 Forces on junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

4.4.3 Forces on Deflectors and Blades . . . . . . . . . . . . . . . . . . . . . . . . . . 278 4.4.4 Forces on Moving Control Volumes . . . . . . . . . . . . . . . . . . . . . . . . 279 4.4.5 Wind Turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 4.4.6 Reaction of a Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 4.4.7 Jet Engines and Rockets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

4.5 Angular-Momentum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 4.6 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

4.6.1 The First Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . 304 4.6.2 Steady-State Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 305 4.6.3 Unsteady-State Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . 317

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

5 Differential Analysis 355 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 5.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

5.2.1 Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 5.2.2 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 5.2.3 Angular Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 5.2.4 Linear Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

5.3 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 5.3.1 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 5.3.2 The Stream Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

5.4 Conservation of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 5.4.1 General Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 5.4.2 Navier-Stokes Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 5.4.3 Nondimensionalized Navier-Stokes equation . . . . . . . . . . . . . . . . . . . 379

5.5 Solutions of the Navier-Stokes Equation . . . . . . . . . . . . . . . . . . . . . . . . . 383 5.5.1 Steady Laminar Flow between Stationary Parallel Plates . . . . . . . . . . . 383 5.5.2 Steady Laminar Flow between Moving Parallel Plates . . . . . . . . . . . . . 386 5.5.3 Steady Laminar Flow Adjacent to Moving Vertical Plate . . . . . . . . . . . . 388 5.5.4 Steady Laminar Flow through a Circular Tube . . . . . . . . . . . . . . . . . 392 5.5.5 Steady Laminar Flow through an Annulus . . . . . . . . . . . . . . . . . . . . 394 5.5.6 Steady Laminar Flow Between Rotating Cylinders . . . . . . . . . . . . . . . 397

5.6 Inviscid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 5.6.1 Bernoulli Equation for Steady Inviscid Flow . . . . . . . . . . . . . . . . . . . 402 5.6.2 Bernoulli Equation for Steady Irrotational Inviscid Flow . . . . . . . . . . . . 405

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5.6.3 Velocity Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 5.6.4 Two-Dimensional Potential Flows . . . . . . . . . . . . . . . . . . . . . . . . . 409

5.7 Fundamental and Composite Potential Flows . . . . . . . . . . . . . . . . . . . . . . 413 5.7.1 Principle of Superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 5.7.2 Uniform Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 5.7.3 Line Source/Sink Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 5.7.4 Line Vortex Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 5.7.5 Spiral Flow Towards a Sink . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 5.7.6 Doublet Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 5.7.7 Flow Around a Half-Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 5.7.8 Rankine Oval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 5.7.9 Flow Around a Circular Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . 435

5.8 Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 5.8.1 Occurrence of Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 5.8.2 Turbulent Shear Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 5.8.3 Mean Steady Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

5.9 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453

6 Dimensional Analysis and Similitude 479 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 6.2 Dimensions in Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 6.3 Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482

6.3.1 Conventional Method of Repeating Variables . . . . . . . . . . . . . . . . . . 485 6.3.2 Alternative Method of Repeating Variables . . . . . . . . . . . . . . . . . . . 487 6.3.3 Method of Inspection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488

6.4 Dimensionless Groups as Force Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . 490 6.5 Dimensionless Groups in Other Applications . . . . . . . . . . . . . . . . . . . . . . . 495 6.6 Modeling and Similitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

7 Flow in Closed Conduits 529 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 7.2 Steady Incompressible Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 7.3 Friction Effects in Laminar Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536 7.4 Friction Effects in Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540 7.5 Practical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549

7.5.1 Estimation of Pressure Changes . . . . . . . . . . . . . . . . . . . . . . . . . . 549 7.5.2 Estimation of Flow Rate for a Given Head Loss . . . . . . . . . . . . . . . . . 550 7.5.3 Estimation of Diameter for a Given Flow Rate and Head Loss . . . . . . . . . 551 7.5.4 Head Losses in Noncircular Conduits . . . . . . . . . . . . . . . . . . . . . . . 553 7.5.5 Empirical Friction-Loss Formulae . . . . . . . . . . . . . . . . . . . . . . . . . 553 7.5.6 Local Head Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 7.5.7 Pipelines with Pumps or Turbines . . . . . . . . . . . . . . . . . . . . . . . . 563

7.6 Water Hammer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565

vii

7.7 Pipe Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570 7.7.1 Nodal Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 7.7.2 Loop Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573

7.8 Building Water-Supply Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 7.8.1 Specification of Design Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 7.8.2 Specification of Minimum Pressures . . . . . . . . . . . . . . . . . . . . . . . 580 7.8.3 Determination of Pipe Diameters . . . . . . . . . . . . . . . . . . . . . . . . . 580

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590

8 Turbomachines 613 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613 8.2 Mechanics of Turbomachines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 8.3 Hydraulic Pumps and Pumped Systems . . . . . . . . . . . . . . . . . . . . . . . . . 618

8.3.1 Flow Through Centrifugal Pumps . . . . . . . . . . . . . . . . . . . . . . . . 621 8.3.2 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626 8.3.3 Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627 8.3.4 Specific Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631 8.3.5 Performance Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635 8.3.6 System Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638 8.3.7 Limits on Pump Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640 8.3.8 Multiple-Pump Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 8.3.9 Variable-Speed Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647

8.4 Fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650 8.4.1 Performance Characteristics of Fans . . . . . . . . . . . . . . . . . . . . . . . 650 8.4.2 Affinity Laws of Fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 8.4.3 Specific Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651

8.5 Hydraulic Turbines and Hydropower . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 8.5.1 Impulse Turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 8.5.2 Reaction Turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 660 8.5.3 Practical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673

9 Flow in Open Channels 701 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701 9.2 Basic Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701

9.2.1 Steady-State Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . 702 9.2.2 Steady-State Momentum Equation . . . . . . . . . . . . . . . . . . . . . . . . 702

9.2.2.1 Darcy–Weisbach equation . . . . . . . . . . . . . . . . . . . . . . . . 706 9.2.2.2 Manning equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711 9.2.2.3 Velocity distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 716 9.2.2.4 Surface-Wave propagation . . . . . . . . . . . . . . . . . . . . . . . . 718

9.2.3 Steady-State Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 720 9.2.3.1 Energy grade line . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723 9.2.3.2 Specific energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723

9.3 Water-Surface Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734

viii

9.3.1 Profile Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734 9.3.2 Classification of Water-Surface Profiles . . . . . . . . . . . . . . . . . . . . . . 735 9.3.3 Hydraulic Jump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741 9.3.4 Computation of Water-Surface Profiles . . . . . . . . . . . . . . . . . . . . . . 747

9.3.4.1 Direct-integration method . . . . . . . . . . . . . . . . . . . . . . . . 750 9.3.4.2 Direct-step method . . . . . . . . . . . . . . . . . . . . . . . . . . . 751 9.3.4.3 Standard-step method . . . . . . . . . . . . . . . . . . . . . . . . . . 752 9.3.4.4 Practical considerations . . . . . . . . . . . . . . . . . . . . . . . . . 754

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 759

10 Drag and Lift 771 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771 10.2 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772

10.2.1 Friction and Pressure Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773 10.2.2 Drag and Lift Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774 10.2.3 Flow Over Flat Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777 10.2.4 Flow over Curved Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779

10.3 Estimation of Drag Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782 10.3.1 Drag on Flat Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782

10.3.1.1 Drag coefficient on hydrodynamically smooth surfaces . . . . . . . . 784 10.3.1.2 Drag coefficient on hydrodynamically rough surfaces . . . . . . . . . 784 10.3.1.3 Drag coefficient on intermediate surfaces . . . . . . . . . . . . . . . 785 10.3.1.4 Drag coefficient with flow normal to a flat plate . . . . . . . . . . . 786

10.3.2 Drag on Spheres and Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . 787 10.3.2.1 General characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 788 10.3.2.2 Analytic expressions for drag coefficients of spheres . . . . . . . . . 790 10.3.2.3 Terminal velocities of spheres and other bodies . . . . . . . . . . . . 791

10.3.3 Drag on Vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794 10.3.4 Drag on Ships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797 10.3.5 Drag on Two-Dimensional Bodies . . . . . . . . . . . . . . . . . . . . . . . . . 798 10.3.6 Drag on Three-Dimensional Bodies . . . . . . . . . . . . . . . . . . . . . . . . 799 10.3.7 Drag on Composite Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 801 10.3.8 Drag on Miscellaneous Objects . . . . . . . . . . . . . . . . . . . . . . . . . . 802 10.3.9 Added Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803

10.4 Estimation of Lift Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805 10.4.1 Lift on Airfoils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805 10.4.2 Lift on Airplanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809 10.4.3 Lift on Hydrofoils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814 10.4.4 Lift on a Spinning Sphere in Uniform Flow . . . . . . . . . . . . . . . . . . . 814

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819

11 Boundary-Layer Flow 843 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843 11.2 Laminar Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845

11.2.1 Blasius Solution for Plane Surfaces . . . . . . . . . . . . . . . . . . . . . . . . 845

ix

11.2.2 Blasius Equations for Curved surfaces . . . . . . . . . . . . . . . . . . . . . . 850 11.3 Turbulent Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852

11.3.1 Analytic Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852 11.3.2 Turbulent Boundary Layer on a Flat Surface . . . . . . . . . . . . . . . . . . 853

11.3.2.1 Flow in the viscous sublayer . . . . . . . . . . . . . . . . . . . . . . 854 11.3.2.2 Flow in the transition layer . . . . . . . . . . . . . . . . . . . . . . . 854 11.3.2.3 Flow in the turbulent layer . . . . . . . . . . . . . . . . . . . . . . . 855 11.3.2.4 One-seventh-power law velocity distribution . . . . . . . . . . . . . 858

11.3.3 Boundary Layer Thickness and Shear Stress . . . . . . . . . . . . . . . . . . . 860 11.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 862

11.4.1 Displacement Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 862 11.4.2 Momentum Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865 11.4.3 Momentum Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 866 11.4.4 General Formulations for Self-Similar Velocity Profiles . . . . . . . . . . . . . 870

11.5 Mixing-Length Theory of Turbulent Boundary Layers . . . . . . . . . . . . . . . . . 872 11.5.1 Smooth Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872 11.5.2 Rough Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873 11.5.3 Velocity-Defect Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874 11.5.4 One-Seventh-Power-Law Distribution . . . . . . . . . . . . . . . . . . . . . . . 875

11.6 Boundary Layers in Closed Conduits . . . . . . . . . . . . . . . . . . . . . . . . . . . 876 11.6.1 Smooth Flow in Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876 11.6.2 Rough Flow in Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 878 11.6.3 Notable Contributors to Understanding Flow in Pipes . . . . . . . . . . . . . 878

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 882

12 Compressible Flow 901 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 901 12.2 Principles of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 901 12.3 The Speed of Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 907 12.4 Thermodynamic Reference Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 915

12.4.1 Isentropic Stagnation Condition . . . . . . . . . . . . . . . . . . . . . . . . . 916 12.4.2 Isentropic Critical Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 921

12.5 Basic Equations of One-Dimensional Compressible Flow . . . . . . . . . . . . . . . . 922 12.6 Steady One-Dimensional Isentropic Flow . . . . . . . . . . . . . . . . . . . . . . . . . 925

12.6.1 Effect of Area Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925 12.6.2 Choked Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926 12.6.3 Flow in Nozzles and Diffusers . . . . . . . . . . . . . . . . . . . . . . . . . . . 927

12.6.3.1 Converging nozzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 930 12.6.3.2 Converging-diverging nozzle . . . . . . . . . . . . . . . . . . . . . . 934

12.7 Normal Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 940 12.8 Steady One-Dimensional Non-Isentropic Flow . . . . . . . . . . . . . . . . . . . . . . 952

12.8.1 Adiabatic Flow with Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . 953 12.8.2 Isothermal Flow with Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . 966 12.8.3 Diabatic Frictionless Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 968 12.8.4 Application of Fanno and Rayleigh Relations to Normal Shocks . . . . . . . . 973

x

12.9 Oblique Shocks, Bow Shocks, and Expansion Waves . . . . . . . . . . . . . . . . . . 978 12.9.1 Oblique Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 978 12.9.2 Bow Shocks and Detached Shocks . . . . . . . . . . . . . . . . . . . . . . . . 985 12.9.3 Isentropic Expansion Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 998

A Units and Conversion Factors 1015 A.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015 A.2 Conversion Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016

B Fluid Properties 1019 B.1 Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1019 B.2 Air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1021 B.3 The Standard Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022 B.4 Common Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023 B.5 Common Gasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024 B.6 Nitrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025

C Properties of Areas and Volumes 1027 C.1 Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027 C.2 Properties of Circles and Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1028

C.2.1 Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1028 C.2.2 Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1029

C.3 Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1030

D Pipe Specifications 1031 D.1 PVC Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1031 D.2 Ductile-Iron Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1031 D.3 Concrete Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1032 D.4 Physical Properties of Common Pipe Materials . . . . . . . . . . . . . . . . . . . . . 1032

Chapter 4

Finite Control Volume Analysis

4.1 Introduction

The motion and properties of a fluid can be described using a variety of reference frames, with different reference frames giving different perspectives of the flow. The two reference frames that are most commonly used in fluid mechanics are: (1) a reference frame moving with a specified fluid element that contains a defined mass of fluid, and (2) a reference frame fixed in space. A reference frame moving with a fluid element containing a defined mass of fluid can be regarded as the fundamental reference frame, since the laws of motion and thermodynamics are directly applicable to a defined fluid mass as it moves within a fluid continuum. In contrast, a fixed reference frame describes the motion of (different) fluid elements as they pass by a fixed point in space, which is the usual perspective from which fluid motions are observed and measured. The equations governing the behavior of fluids are typically formulated by first applying the fundamental laws to a specified fluid element or a defined fluid mass, and then transforming the defined-fluid-mass equations into equivalent fixed-reference-frame equations to facilitate practical applications.

Lagrangian and Eulerian reference frames. A Lagrangian reference frame moves with a fluid element that contains a fixed mass of fluid, and fluid properties within the fluid element are only observed to change with time. For example, the velocity, v, of a fluid element in a Lagrangian reference frame is described in the form, v(t), where t is the time. An Eulerian reference frame is fixed in space, and changes in fluid properties are described at fixed locations. For example, the velocity field of a fluid in an Eulerian reference frame is described in the form v(x, t), where x is the location and t is the time. From a practical viewpoint, we are usually interested in the behavior of fluids at particular locations in space, in which case an Eulerian reference frame is preferable. The complicating factor in working with Eulerian reference frames is that the fundamental equations of fluid motion and thermodynamics are all stated for fluid elements in Lagrangian reference frames. For example, Newton’s second law states that the sum of the forces on any fluid element (as it moves within the fluid continuum) is equal to the mass of fluid within the fluid element multiplied by the acceleration of the fluid element. To transform the fundamental equations of fluid motion and thermodynamics into useful equations in Eulerian reference frames, it is necessary to understand the relationship between Lagrangian equations and Eulerian equations, and the transformation of Lagrangian equations into Eulerian equations is generally done using Reynolds transport theorem.

253

322

Q Pump ?

Intake Discharge

Figure 4.41: Flow through a pump

4.2. In the United States, flow rates through shower heads are regulated to be no greater than 9.5 L/min (2.5 gpm) under any water-pressure condition likely to be encountered in a home. Water pressures in homes are typically less than 550 kPa (80 psi). A practical shower head will deliver water at a velocity of at least 5 m/s. If nozzles in a shower head can be manufactured with diameters of 0.75 mm, what is the maximum number nozzles that would be required to make a practical shower head?

4.3. An air conditioner delivers air into a 200-m3 room at a rate of 3.5 m3/min. Air from the room is removed by a 0.6 m × 0.8 m duct. Assuming steady-state conditions and incompressible flow, what is the average velocity of air in the exhaust duct?

4.4. At a particular section in a 100-mm diameter pipe, the velocity is measured as 1200 m/s. Downstream of this section, the flow is expanded to a 300-mm pipe in which the velocity and density of the air flow is measured as 700 m/s and 1.1 kg/m3, respectively. If flow conditions are steady, estimate the density of the air at the upstream (100-mm diameter) section.

4.5. Air enters a heat exchanger at a rate of 200 kg/h and exits at a rate of 195 kg/h, as illustrated in Figure 4.42. The purpose of the heat exchanger is to dehumidify the air. Estimate the rate at which liquid water drains from the system.

200 kg/h air 195 kg/h airHeat Exchanger

Liquid water

Figure 4.42: Flow through a heat exchanger

4.6. A piston moving at 10 mm/s displaces air from a 100-mm diameter cylinder into a 25-mm discharge line as shown in Figure 4.43. Assuming that compressibility effects can be neglected, what is the volumetric flow rate and velocity in the discharge line?

10 mm/s100 mm

25 mm Piston Cylinder

Discharge line

Air

Figure 4.43: Air flow driven by a piston

323

4.7. Air enters a 1.2-m3 storage tank at a rate of 0.5 m3/s, and at a temperature of 20◦C and pressure of 101 kPa as shown in Figure 4.44. When air is released from the tank at a rate of 0.3 m3/s, the pressure of the released air is measured as 150 kPa, and a weight scale indicates that the weight of the air in the tank is increasing at a rate of 0.1 kg/s. Estimate the temperature and the density of the air that is being released from the tank.

0.5 m3/s 20oC 101 kPa

Air 0.3 m3/s

Weight scale

Air storage tank

150 kPa

Figure 4.44: Air storage in a tank

4.8. Pipelines containing gasoline, water, and methanol merge to create a mixture as shown in Figure 4.45. All fluids are at 20◦C, and the known diameters and fluid velocities in the merging pipes are also shown in Figure 4.45. (a) What is the volumetric flow rate of the mixture? (b) What is the density of the mixture?

Gasoline

Water

Methanol

Mixture

1

2

3

4

Diameter Velocity

Pipe (mm) (m/s) Fluid

1 20 0.30 Gasoline

2 30 0.25 Water

3 35 0.40 Methanol

4 50 ? Mixture

Figure 4.45: Mixing in pipes

4.9. Water flows from a garden hose into a sprinkler at a volume flow rate, Q, of 20 L/min. The flow exits the sprinkler via four rotating arms as shown Figure 4.46. The radius, R, of the sprinkler is 0.2 m, and the diameter of each discharge port is 10 mm. At what rate, ω, would the sprinkler have to rotate such that the absolute velocity of the discharged water is equal to zero. Can you visualize how this would look to a stationary observer?

R

Q ω

D

Figure 4.46: Rotating sprinkler

4.10. The flow in a pipeline is divided as shown in Figure 4.47. The diameter of the pipe at Sections

324

1, 2, and 3 are 100 mm, 75 mm, and 50 mm, respectively, and the flow rate at Section 1 is 10 L/s. Calculate the volumetric flow rate and velocity at Section 2.

10 L/s

V = 1 m/s

1 2

75 mm

50 mm 100 mm

3

Figure 4.47: Divided flow

4.11. Water at 20◦C is flowing in a 100-mm diameter pipe at an average velocity of 2 m/s. If the diameter of the pipe is suddenly expanded to 150 mm, what is the new velocity in the pipe? What are the volumetric and mass flow rates in the pipe?

4.12. A 200-mm diameter pipe divides into two smaller pipes, each of diameter 100 mm. If the flow divides equally between the two smaller pipes and the velocity in the 200-mm pipe is 1 m/s, calculate the velocity and flow rate in each of the smaller pipes.

4.13. Water flows steadily through the round pipe shown in Figure 4.48. The entrance velocity is constant, u = U0, and the exit velocity approximates turbulent flow with a velocity distribu- tion

u(r) = umax ( 1− r

R

) 1 7

(4.170)

(a) Use the velocity distribution given by Equation 4.170 to determine the volume flow rate in the pipe in terms of umax and R. (b) Determine the ratio U0/umax.

r R

x = 0 x = L

u(r)U0

Inflow Ou!low

Figure 4.48: Flow through a pipe

4.14. Consider the 1-m3 storage tank shown in Figure 4.49. Initially, the density of the liquid in the tank is 1000 kg/m3. At a particular instant, the inlet valve is opened and a fluid of density 900 kg/m3 flows into the tank at a rate of 100 L/min. At this same instant, the outlet valve is opened and adjusted such that the liquid level in the tank is maintained at a constant elevation, a mixer is also turned on to ensure that the liquid in the tank is well mixed. How long will it take after the valves are open and the mixer turned on for the density of the liquid

325

in the tank to decline from 1000 kg/m3 to 950 kg/m3? If the system is left operating long enough, what will be the ultimate density of the liquid in the tank?

100 L/min

900 kg/m3 1 m3

Storage tank

∞

Mixer

Figure 4.49: Flow through a tank

4.15. Water drains from a funnel that has a top diameter of 20 cm, a bottom (outlet) diameter of 0.5 cm, and a height of 25 cm. The exit velocity is related to the depth of water, h, in the funnel by the relation

V = √

2gh

Determine the relationship between the rate of change of depth (dh/dt) and the depth (h). What is the rate of change of depth when the funnel is half full?

4.16. Water enters the cylindrical reservoir shown in Figure 4.50 through a pipe at point A at a rate of 2.80 L/s, and exits through a 5-cm diameter orifice at B. The diameter of the cylindrical reservoir is 60 cm, and the velocity, v, of water leaving the orifice is given by

v = √

2gh

where h is the height of the water surface (in the reservoir) above the orifice. How long will it take for the water surface in the reservoir to drop from h = 2 m to h = 1 m? [Hint : You might need to use integral tables.]

60 cm

h

2.80 L/s A B

v = √2gh

Figure 4.50: Water exiting a cylindrical reservoir

4.17. Water leaks out of a 5-mm diameter hole in the side of a large cup as shown in Figure 4.51. Estimate how long it takes for the cup to go from being full to being half-full. [You can assume that the velocity at the outlet is

√ 2gh where h is the height of the water above the

outlet. Also, you might need to know that the volume of a cone is πr2h/3.]

326

50 cm

40 cm

20 cm

10 cm

Figure 4.51: Leaking cup

4.18. A 2-m high cone with a 1-m diameter base is turned upside down to form a reservoir and is filled with water. A drain hole in the side of the reservoir is opened to release water, and the release rate, Q, is given by

Q = 7.8 √ h− 0.02

where Q is in L/s and h is the height of the water surface above the bottom of the reservoir in meters. Estimate how long it will take for the reservoir to empty half of its volume. [The volume of a cone is πr2h/3]

4.19. A storage tank with a volume of 0.1 m3 contains compressed air. Air is released from the tank by opening a valve connected to a 10-mm diameter tube. At the instant that the valve is opened, air exits at a velocity of 200 m/s. Find the density and the rate of change of density of the air in the tank instant of valve opening under the following conditions: (a) the density of the air exiting the tank is 5 kg/m3, and (b) the air exiting the tank has a temperature of −15◦C and an absolute pressure of 350 kPa.

Section 4.4: Conservation of Linear Momentum

4.20. Calculate the momentum correction coefficient, β, for the following velocity distribution

v(r) = V0

[ 1−

( r R

)2]

where v(r) is the velocity at a distance r from the centerline of a pipe of radius R.

4.21. Water flows over a 0.2-m high step in a 5-m wide channel, as illustrated in Figure 4.52. If the flow rate in the channel is 15 m3/s and the upstream and downstream depths are 3.00 m and 2.79 m, respectively, calculate the force on the step.

327

15 m3/s 3 m

2.79 m

0.2 m

1 2

Figure 4.52: Flow over a step

4.22. Water at 20◦C flows at 30 m3/s in a rectangular channel that is 10 m wide. The flowing water encounters 3 piers as shown in Figure 4.53, and the flow depths upstream and downstream of the piers are 3 m and 2.5 m, respectively. Estimate the force on each pier.

30 m3/s

10 m

Pier

Side of channel

0.7 m

(a) Plan view

Water surface

3 m 2.5 m

Pier

Bo!om of

channel

(b) Eleva"on view

Figure 4.53: Piers in stream

4.23. The 3-m high and 7-m wide spillway shown in Figure 4.54 is designed to accommodate a flow of 35 m3/s.

10 m

6 m

1.5 m

35 m3/s Spillway

Piles

Soil

3 m

Figure 4.54: Flow over a spillway

328

Under design conditions the upstream and downstream flow depths are 6 m and 1.5 m re- spectively. The base length of the spillway is 10 m. (a) Estimate the force exerted by the water on the spillway and the shear force exerted by the spillway on the underlying soil. (2) If the shear force is resisted by piles, and each pile can resist a shear force of 100 kN, how may piles are necessary.

4.24. Air at standard atmospheric conditions flows past a structure as shown in Figure 4.55. The flow conditions do not change in the direction perpendicular to the plane shown in Figure 4.55. Upstream of the structure, the air velocity is uniform at 25 m/s, and downstream of the structure the air velocity is reduced linearly behind the structure as shown in Figure 4.55. A bounding streamline expands by an amount δ between locations upstream and downstream of the structure, and within the bounding streamline the volumetric flow rate remains constant. (a) Estimate the value of δ. (b) Estimate the force of the air on the structure per unit length perpendicular to the plane of the air flow.

δ

δ

4 m

25 m/s

25 m/s

y

Structure

Bounding streamline

Figure 4.55: Air flow past a structure

4.25. A jet pump such as that illustrated in Figure 4.56 is used in a variety of practical applications, such as in extracting water from ground-water wells. Consider a particular case in which a jet of diameter 150 mm with a velocity of 40 m/s is used to drive the flow of water in a 350-mm diameter pipe where the velocity of flow in the pipe at the location of the jet is 4 m/s. At the location of the jet, Section 1, the pressure is approximately the same across the entire cross section, and at the downstream location, Section 2, the jet momentum is dissipated and the velocity is the same across the entire cross section. Estimate the increase in the water pressure between Section 1 and Section 2. Assume water at 20◦C.

Jet Flow

1 2

150 mm

350 mm40 m/s

4 m/s

Figure 4.56: Jet pump

329

4.26. A reducer is to be used to attach a 400-mm diameter pipe to a 300-mm diameter pipe. For any flow rate, the pressures in the pipes upstream and downstream of the reducer are expected to be related by

p1 γ

+ V 21 2g

= p2 γ

+ V 22 2g

where p1 and p2 are the upstream and downstream pressures and V1 and V2 are the upstream and downstream velocities. Estimate the force on the reducer when water flows through the reducer at 200 L/s and the upstream pressure is 400 kPa.

4.27. The vertical nozzle shown in Figure 4.57 is attached to a pipe via a thread connection. The diameter of the source pipe is 25 mm, the diameter of the nozzle exit is 10 mm, the length of the nozzle is 100 mm, and the mass of the nozzle is 0.2 kg. The nozzle is oriented vertically downward and discharges water at 20◦C. What force will be exerted on the thread connection when the flow through the nozzle is 20 L/min?

10 mm

25 mm

100 mm Nozzle

Thread

connec!on

Flow

Figure 4.57: Flow through a vertical nozzle

4.28. Air flows in a 150-mm diameter duct, and at a particular section the velocity, pressure, and temperature are measured as 70 m/s, 600 kPa, and −23◦C, respectively. At a downstream section, the pressure and temperature are measured as 150 kPa and −73◦C. Estimate: (a) the average velocity at the downstream section, and (b) the wall friction between the upstream and downstream sections.

4.29. Water under a pressure of 350 kPa flows with a velocity of 3 m/s through a 90◦ bend in the horizontal plane. If the bend has a uniform diameter of 300 mm, and assuming no drop in pressure, calculate the force required to keep the bend in place

4.30. Water flows at 100 L/s through a 200-mm diameter vertical bend, as shown in Figure 4.58. If the pressure at Section 1 is 500 kPa and the pressure at Section 2 is 450 kPa, then determine the horizontal and vertical thrust on the support structure. The volume of the bend is 0.16 m3.

330

100 L/s

1

2

500 kPa

450 kPa

200 mm

Support

g

Ver!cal bend

Figure 4.58: Flow through a vertical bend

4.31. The bend shown in Figure 4.59 discharges water into the atmosphere. Determine the force components at the flange required to hold the bend in place. The bend lies in the horizontal plane, the interior volume of the bend is 0.25 m3, and the mass of the bend material is 250 kg.

60 cm

30 cm

10 m/s

60oFlow

Atmosphere

Bend Flange

Figure 4.59: Bend

4.32. The fire-hose nozzle shown in Figure 4.60 is to be held by two firemen who (working together) can support a force of 1 kN. (a) Determine the maximum flow rate in the fire hose that can be supported by the firemen. Give your answer in liters per minute. If the firemen were to let go of the nozzle, in what direction would it move? (b) If the nozzle is pointed vertically upward, how high will the water jet rise?

5 cm

Flow

Nozzle

Jet

25 cm

Fire hose

Figure 4.60: Nozzle on fire hose

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4.33. Water at 20◦C flows through a 300-mm diameter pipe that discharges into the atmosphere as shown in Figure 4.61. When a partially open valve extends 200 mm from the top of the pipe, the flow rate in the pipe is 0.2 m3/s and the (gage) pressure just upstream of the valve is 600 kPa. Estimate the force exerted by the water on the partially open valve.

0.2 m3/s

600 kPa

100 mm

300 mm Atmosphere

Valve

Figure 4.61: Flow past a partially open valve

4.34. A 30◦ bend connects a 250-mm diameter pipe (inflow) to a 400-mm pipe (outflow). The volume of the bend is 0.2 m3, the weight of the bend is 400 N, and the pressures on the inflow outflow sections are related by

p1 γ

+ V 21 2g

= p2 γ

+ V 22 2g

The pressure at the inflow section is 500 kPa, the bend is in the vertical plane, and the bend- support structure has a maximum allowable load of 18 kN in the horizontal direction and 40 kN in the vertical direction. Determine the maximum allowable flowrate in the bend.

4.35. Determine the force required to restrain the pipe junction illustrated in Figure 4.62. Assume that the junction is in the horizontal plane and explain how your answer would differ if the junction were in the vertical plane.

250 mm p = 420 kPa

150 mm p = 350 kPa

200 mm p = 400 kPa

30o

40o

40 L/s

60 L/s

Flow

Pipe junc!on

Figure 4.62: Pipe junction

4.36. The pipe connection shown in Figure 4.63 is to be used to split the flow from one pipe into two pipes.

332

1

2

3

2 m

2 m

40o

30o

g

200 mm p = 400 kPa

0.04 m3/s

160 mm

120 mm

Figure 4.63: Pipe connection

The pipe connection weighs 200 lb, the pressure at the incoming section (Section 1) is 400 kPa, the flow at Section 1 is 0.04 m3/s, the flow divides such that 60% goes to Section 2 and 40% goes to Section 3, and the temperature of the water is 20◦C. Determine the force required to support the connection.

4.37. Find the x- and y-force components on the horizontal T-section shown in Figure 4.64. Neglect viscous effects.

60 mm 60 mm

40 mm

12 m/s

8 m/s

500 kPa

x

y

Figure 4.64: T-section of pipe

4.38. Water is flowing into and discharging from a pipe U-section as shown in Figure 4.65. At Flange 1, the total absolute pressure is 200 kPa, and 30 kg/s flows into the pipe. At Flange 2, the total pressure is 150 kPa. At Location 3, 8 kg/s of water discharges to the atmosphere, which is at 100 kPa. The center of the pipe at Flange 2 is located 4.0 m above the center of the pipe at Flange 1. The weight of the water in the bend is 280 N, and the weight of the bend is 200 N. Determine the total x- and z- forces (including their directions) to be supported by the two flanges connecting the pipe. Take the momentum-flux correction factor to be 1.03.

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100 mm

50 mm

30 kg/s

22 kg/s

8 kg/s

30 mm

1

2 3

x

z g

4.0 m

Atmosphere

p = 200 kPa

p = 150 kPa

(p = 100 kPa)

Flange

Figure 4.65: Pipe U-section

4.39. Water flowing at 20 L/s in a 150-mm diameter pipeline is delivered to two separate pipelines via the connection shown in Figure 4.66. At the connection, one outflow pipeline has a diameter of 100 mm, makes an angle of 45◦ with the inflow pipeline, and carries 8 L/s, while the other outflow pipeline has a diameter of 120 mm, makes an angle of 60◦ with the inflow pipeline, and carries 12 L/s. The pressure in the inflow pipe is 500 Pa and the connection is in the horizontal plane. (a) Estimate the pressures in the outflow pipelines, and (b) determine the force on the connection.

150 mm

100 mm

120 mm

45o

60o 20 L/s

8 L/s

12 L/s

500 Pa

Flange

Figure 4.66: Pipe junction

4.40. The thin-plate orifice shown in Figure 4.67 causes a large pressure drop. For 20◦C water flow at 500 gal/min, with a pipe D = 100 mm and orifice d = 60 mm, p1 − p2 ≈ 145 kPa. If the wall friction is negligible, estimate the force of the water on the orifice plate.

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1 2

Orifice plate

100 mm60 mm500 gpm

Figure 4.67: Thin-plate orifice

4.41. A nozzle with a quarter-circle shape discharges water at 4 m/s as shown in Figure 4.68. The radius of the quarter-circle is 300 mm and the height of the nozzle is 25 mm. Estimate the force required to hold the nozzle in a fixed position.

(a) Plan view

Inflow

Ou!low

(b) Eleva"on view

300 mm

25 mm 4 m/sNozzle

Figure 4.68: Quarter-circular nozzle

4.42. An jet of water at 20◦C exits a nozzle into air and and strikes a stagnation tube as shown in Figure 4.69. If head losses are neglected, estimate the mass flow in kg/s and the height h of the fluid in the stagnation tube.

h 4 cm

Jet

Stagna!on tube

Flow

12 cm

110 kPa

Figure 4.69: Water striking stagnation tube

4.43. A jet of water at 10 m3/s is impinging on a stationary deflector that changes the flow direction of the jet by 45◦. The velocity of the impinging jet is 10 m/s and the velocity of the deflected jet is 9 m/s. What force is required to keep the deflector in place?

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4.44. The concrete structure shown in Figure 4.70 is to be overturned by a jet of water striking at the center, P, of the vertical panel. The structure is expected to overturn along the side QR. The diameter of the jet is 200 mm, and it can be assumed that the density of concrete is 2300 kg/m3. Estimate the velocity of the jet that will be required to overturn the structure. The panel is made of concrete.

Water jet

1.4 m 1.4 m

0.9 m

1.2 m

0.3 m

P

Concrete structure

Q

R

0.2 m

Figure 4.70: Jet impinging on concrete structure

4.45. Water at 20◦C is contained in a pressurized tank that is supported by a cable as shown in Figure 4.71. The (gage) air pressure above the water in the tank is 600 kPa, the depth of water in the tank is 1 m, the diameter of the tank is 1 m, the weight of the tank is 1 kN, and there is a 50-mm diameter orifice in the bottom of the tank. The discharge coefficient of the orifice is 0.8. Estimate the tension is the cable supporting the tank.

600 kPa

1 m 1 m

50 mm

Tank

Orifice

Air

Cable

Figure 4.71: Tank supported by cable

4.46. A 100-mm diameter water jet strikes a flat plate as shown in Figure 4.72(a). Prior to striking the plate, the jet has a velocity of 20 m/s and is oriented normal to the plate. What support force must be provided to keep the plate from moving? Assume water at 20◦C.

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100 mm

20 m/s

Water jet

Flat plate

(a) Without orifice

100 mm

20 m/s

Water jet

(b) With orifice

Orifice

50 mm

Flat plate

Figure 4.72: Water jet striking flat plate

4.47. A 100-mm diameter water jet strikes a flat plate with a 50-mm diameter orifice whose center is aligned with the water jet as shown in Figure 4.72(b). Prior to striking the plate, the jet has a velocity of 20 m/s and is oriented normal to the plate. What support force must be provided to keep the plate from moving? Assume water at 20◦C.

4.48. Water-jet cutters are used to cut a wide variety of materials using a very high-pressure water jet. A particular water-jet cutter uses water at 20◦C at a flow rate of 4 L/min and a jet diameter of 0.25 mm. If this jet is used to cut a flat plate where the plate is oriented normal to the jet, what is the force per unit area exerted by the jet on the plate?

4.49. Air at 20◦C and 1 atm flows in a 25-cm diameter duct at 15 m/s as shown in Figure 4.73; the exit is choked by a 90◦ cone. Estimate the force of the air flow on the cone.

90o 40 cm Air flow

25 cm

1 cm

Cone

Figure 4.73: Air flow onto a cone

4.50. An air tank is hovers above a solid surface supported only by the air being released through a 15-mm diameter orifice that is located immediately below the center of gravity of the tank, as shown in Figure 4.74. It is known that the temperature of the air in the tank is 23◦C, and the weight of the tank plus air in the tank is 100 N. The flow of air through the orifice can be assumed to be incompressible and frictionless, and atmospheric pressure can be taken as 101 kPa. Estimate the air pressure in the tank.

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15 mm

Orifice

Tank T = 23oC p = ?

Solid surface

Air

Figure 4.74: Tank supported by air jet

4.51. A blade attached to a turbine rotor is driven by a stream of water that has a velocity of 18 m/s. The blade moves with a velocity of 8 m/s and deflects the stream of water through an angle of 85◦; the entrance and exit flow areas are each equal to 1.5 m2. Estimate the force on the moving blade and the power transferred to the turbine rotor.

4.52. A circular jet of water at 20◦C impinges on the vane shown in Figure 4.75. The incident jet has a velocity of 30 m/s and a diameter of 150 mm. The vane divides the incident jet equally, and the jet exits the vane as two jets that each make an angle of 15◦ with the incoming jet. (a) Determine the force on the vane when the vane is moving at a speed of 10 m/s in the same direction as that of the incident jet. (b) Determine the force on the vane when the vane is moving at a speed of 10 m/s in the opposite direction to that of the incident jet. Assume water at 20◦C.

30 m/s

Moving vane

150 mm

15o

V v

Incident jet

Figure 4.75: Force on moving vane

4.53. Water flows out the 6-mm slots as shown in Figure 4.76. Calculate the angular velocity (ω) if 20 kg/s is delivered by the two arms. [Note: The velocities of the exit jets might not be uniform as shown in the figure.]

ω

5 cm 15 cm

2 cm

Sprinkler rotorExit jet

Figure 4.76: Sprinkler

4.54. The rocket shown in Figure 4.77 is launched vertically from high in the atmosphere where aerodynamic drag on the rocket is negligible. The rocket has an initial mass of 600 kg,

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burns fuel at a rate of 7 kg/s, and the velocity of the exhaust gasses relative to the rocket is 2800 m/s. (a) Determine the velocity and acceleration of the rocket 10 seconds after launch. (b) Compare the acceleration of the rocket at 10 seconds with the acceleration at launch.

gRocket

v e

Exhaust gas

Figure 4.77: Rocket with vertical trajectory

4.55. A 60-mm diameter jet of water impacts a vane at a velocity of 30 m/s as shown in Figure 4.78. The vane has a mass of 100 kg and deflects the jet through a 50◦ angle. The vane is constrained to move in the x direction, and there is no force opposing the motion of the vane. Determine the acceleration of the vane when it is first released. Assume water at 20◦C.

50o

30 m/s

60 mm

Moving vane

Figure 4.78: Motion of a vane caused by a water jet

4.56. A wind turbine is being proposed for commercial development in Miami, Florida. The pro- posed turbine will have a hub height of 50 m and rotor diameter of 70 m. Miami is approxi- mately at sea level. Estimate the wind power density and assess the feasibility of using wind as a source of energy in Miami.

4.57. Estimate the maximum wind power density available at any location within the land area of the 48 contiguous United States. For such a location, provide a rough layout of a 1 mi2 wind farm, where each wind turbine would have a 90-m diameter rotor. What is the peak power that could be produced from this wind farm? Assume standard temperature and pressure.

4.58. A 3-cm diameter orifice is 1 m below the surface of a 1.5 m diameter barrel containing water at 20◦C. Estimate the force on the barrel when water is flowing freely out of the orifice.

4.59. A vertical jet discharges water onto a reservoir, and water discharged from the same reservoir via an opening that is inclined at 45◦ to the horizontal as shown in Figure 4.79. The inflow

339

jet has a diameter of 25 mm, the outflow jet has a diameter of 35 mm, and conditions are at steady state. (a) What is the net force on the reservoir? (b) How high will the discharge jet rise above its discharge elevation?

35mm

25 mm

Reservoir

1.2 m

45o

Inflow jet

Ou!low jet

Figure 4.79: Inflow and outflow jets in a reservoir

4.60. A jet engine is mounted on an aircraft that is operating at an elevation of 8 km in a standard atmosphere, and the speed of the aircraft is 250 m/s. The intake area of the engine is 1.5 m2, the fuel consumption rate is 25 kg/s, and the exhaust gas exits at a speed of 200 m/s relative to the moving aircraft. The pressure of the exhaust gas is approximately equal to the ambient atmospheric pressure. Under these conditions, what thrust is expected from the engine?

4.61. The specifications of a jet engine indicate that it has an intake diameter of 2.57 m, consumes fuel at a rate of 1.5 L/s, and produces an exhaust jet with a velocity of 800 m/s. If the jet engine is mounted on an airplane that is cruising at 845 km/h (525 mph) where the air density is 0.400 kg/m3, estimate the thrust produced by the engine. Assume that the jet fuel has a density of 820 kg/m3.

4.62. Jet engines are usually tested on a static thrust stand as shown in Figure 4.80. In such tests, the pressures of the inflow and outflow gasses are usually expressed as gage pressures. In a particular test of an engine with an inflow area of 1.2 m2, the inflow air has a velocity of 250 m/s, a (gage) pressure of 50 kPa, and a temperature of −50◦C. The exhaust gas has a velocity of 550 m/s and is at atmospheric pressure, which is 101 kPa (absolute). (a) Estimate the thrust on the test stand assuming that the mass flow rate of the exhaust gasses is equal to the mass flow rate of the inflow air. (a) Estimate the thrust on the test stand taking into account that fuel is supplied to the engine at a mass flow rate equal to 2% of the air mass flow rate. Which estimated thrust is likely to be more accurate?

340

Jet engine

Open atmosphere

50 kPa

-50oC

250 m/s 550 m/s 0 kPa

Sta!c thrust stand

Thrust

Figure 4.80: Jet engine on test stand

4.63. The performance of a rocket with a 200-mm diameter nozzle exit is tested on a stand as shown in Figure 4.81. Under the conditions of a particular test, atmospheric pressure is 101 kPa, the exhaust gas has an exit velocity of 1500 m/s, a pressure of 150 kPa absolute, and a mass flow rate of 10 kg/s. (a) Estimate the thrust generated by the rocket. (b) How is the mass flow rate of the exhaust gas related to the fuel consumption rate of the rocket?

Test stand

Rocket

200 mm 1500 m/s 150 kPa 10 kg/s

Exhaust gas

Figure 4.81: Rocket on a test stand

4.64. A rocket weighs 6000-kg, burns fuel at a rate of 40 kg/s, and has an exhaust velocity of 3000 m/s. Estimate the initial acceleration of the rocket and the velocity after 10 seconds. Neglect the drag force of the surrounding air and assume that the pressure of the exhaust gasses is equal to the pressure of the surrounding atmosphere.

Section 4.5: Angular-Momentum Principle

4.65. Water flows through a 12-cm diameter pipe that consists 3-m long riser and a 2-m long horizontal section with a 90◦ elbow to force the water to be discharged downward. Water discharges to the atmosphere at a velocity of 4 m/s, and the mass of the pipe section when filled with water is 15 kg per meter length. (a) Determine the moment acting at the intersection of the vertical (riser) and horizontal pipe sections (i.e. point A in Figure 4.82). (b) What would be the moment if the flow were discharged upward instead of downward? (c) Which discharge orientation creates the greatest stress on the elbow at A?

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2 m

3 m

12 cm

4 m/s

A

Riser (pipe)

Elbow

Figure 4.82: Pipe system

4.66. Water flows through the pipe bend and nozzle arrangement shown in Figure 4.83 which lies with its axis in the horizontal plane. The water issues from the nozzle into the atmosphere as a parallel jet and friction may be neglected. Find the moment of the resultant force due to the water on this arrangement about a vertical axis through the point X.

1 cm

8 cm

10 cm

20 cm

Flow

Nozzle

Flexible

joint

128 kPa

16 m/s

Pipe

X

Figure 4.83: Force on nozzle

4.67. Water at 20◦C flows through the pipe bend in Figure 4.84 at 4000 gallons per minute, where the bend is in the horizontal plane. (a) Compute the torque required at point B to hold the bend stationary, and (b) determine the location of the line of action of the resultant force.

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27 cm

13 cm

194 kPa B

50 cm

50 cm

C

Air

Water jet

Figure 4.84: Pipe bend

4.68. The portable 2-inch diameter pipe bend shown in Figure 4.85 is to be held in place by a 150-lb person. The entrance to the bend is at A and the water discharges freely (underwater) at B. If the person holding the bend can resist a 100-lb force being exerted on the bend, what is the maximum flow rate that should be used? Determine the support location, C, where the person would experience zero torque. Assume that the head loss in the bend is 2V 2/2g, where V is the velocity in the bend.

Pipe bendA

B

C

1 !

3 !

Air

Water

2 in.

Flow

Figure 4.85: Flow in bend

4.69. The pipe double-bend shown in Figure 4.86 is commonly used in pipeline systems to avoid obstructions. In a particular application, the pipe has a diameter of 200 mm and carries water at a flow rate of 28 L/s and at a temperature of 15◦C. The pressure at the entrance to the bend is 25 kPa and the head loss within the bend can be neglected. If the bend is supported at P, estimate the magnitude of the force and the moment on the support. What is the location of the resultant force exerted by the flowing water on the bend?

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0.30 m

P 1.00 m

0.30 m

0.50 m

200 mm

28 L/s

g

Flow

Pipe double bendEntrance

Exit

Figure 4.86: Pipe double-bend

4.70. Water at 20◦C flows through the 100-mm diameter pipe-bend section that is supported at S as shown in Figure 4.87. The velocity in the pipe is 5 m/s, the inflow pressure is 550 kPa, and the outflow pressure is 450 kPa. The centerline of the inflow pipe is 80 mm from the support, and the centerline of the outflow pipe is 300 mm from the support. The pipe bend is in the horizontal plane. Determine the horizontal-plane moment that is exerted on the support by the flowing water.

100 mm

5 m/s

550 kPa

450 kPa

Support

80 mm

300 mm Pipe bend

S

Figure 4.87: Pipe bend in horizontal plane

4.71. Water at 20◦C is discharged through a slot in a 225-mm diameter pipe as shown in Figure 4.88. The slot has a length of 0.8 m and a width of 30 mm, and the velocity through the slot increases linearly from 6 m/s to 18 m/s over the length of the slot. The (gage) pressure in the pipe just upstream of the slot is 80 kPa and the slot discharges to the open atmosphere. Estimate the component forces and moment that are exerted on the support section.

225 mm

0.8 m

18 m/s6 m/s

Slot discharge Flow

Support sec!on

80 kPa

Figure 4.88: Discharge through a slot

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4.72. A fan blows air using the rotor shown in Figure 4.89. The rotor has an inner diameter of 0.2 m, an outer diameter of 0.4 m, a height of 30 mm, and a blade angle of 25◦ with the outflow surface. When the rotor turns at 1800 rpm, the inflow velocity in normal to the inflow surface and the flow rate through the fan is 10 m3/min. Estimate (a) the required blade angle at the inflow surface so that the inflow velocity is normal to the inflow surface, and (b) the power that the fan delivers to the air. Assume air at standard conditions.

25o

0.2 m 0.4 mω Rotor

Ou!low surface

Inflow surface

Typical blade Blade discharge angle

Figure 4.89: Air flow through a fan

4.73. A water-pump impeller has an inner diameter of 0.1 m, an outer diameter of 0.4 m, and a height of 0.08 m. When the impeller is rotating at 1800 rpm, the flow rate through the impeller is 1000 L/s, the inflow velocity is normal to the inflow surface, and the outflow velocity is 18 m/s as illustrated in Figure 4.90. Estimate the head added by the pump.

0.1 m 0.4 mω Impeller

Ou!low surface

Inflow surface

18 m/s

Figure 4.90: Flow through a pump

4.74. The water-turbine runner shown in Figure 4.91 is driven by an inflow velocity that makes an angle of 50◦ with the inflow surface. The outflow velocity is normal to the outflow surface and has a magnitude of 15 m/s. The outer diameter of the runner is 4 m, the inner diameter is 1.5 m, and the runner rotates at 60 rpm in the anticlockwise direction. Estimate the head extracted by the turbine. Would your answer be any different if the fluid were air instead of water?

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50o

1.5 m 4 m

Runner

Inflow velocity

15 m/s

Figure 4.91: Flow through a turbine

4.75. Water enters a turbine runner that has an inner diameter of 0.8 m and an outer diameter of 2 m as shown in Figure 4.92. Water crosses the inflow surface at an angle of 25◦ and with an absolute flow velocity of 15 m/s. The outflow velocity is normal to the outflow surface. The height of the runner is 0.4 m and the runner is rotating at a rate of 180 rpm. What power is being generated by the runner?

25o 15 m/s

0.8 m 2 mω

Runner

Inflow surface

Ou!low surface

Figure 4.92: Flow through a turbine runner

4.76. A three-arm sprinkler is used to water a garden by rotating in a horizontal plane by the impulse caused by water flow. Water enters the sprinkler along the axis of rotation at a rate of 40 L/s and leaves the 1.2-cm-diameter nozzles in the tangential direction. The bearing applies a retarding torque of 50 N·m due to friction at the anticipated operating speeds. For a normal distance of 40 cm between the axis of rotation and the center of the nozzles, determine the rate of rotation (in rpm) of the sprinkler shaft.

4.77. The two-arm sprinkler shown in Figure 4.93 is connected to a hose that delivers 10 L/min to the sprinkler. The radius of the sprinkler is 175 mm, and the nozzles of the sprinkler have a diameter of 5 mm and are directed inward at an angle of 15◦ to the tangent of the circle of rotation. What is the torque exerted on the sprinkler support?

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175 mm5 mm

Sprinkler 15o

40 rpm

Hose 10 L/min

Support

Figure 4.93: Two-arm sprinkler

4.78. The four-arm water sprinkler shown in Figure 4.94 has a radius-of-rotation of 0.7 m. The nozzles at the four outlets of the sprinkler each have an area of 30 mm2, and are oriented at an angle of of 40◦ to the circle-of-rotation. Consider the case where water is supplied to the sprinkler at a rate of 8 L/s. (a) What torque would be necessary to hold the sprinkler rotors stationary so that they do not rotate? (b) If the restraining torque is equal to zero, at what speed would the sprinkler rotate?

40o

Nozzle

0.7 m

Figure 4.94: Four-arm sprinkler

4.79. Consider the system shown in Figure 4.95 where a water sprinkler is used to generate electric power by rotating a shaft that is connected to an electric generator. Each arm of the sprinkler has a length of 10 cm, the nozzle at the end of each arm has a diameter of 1 cm, the total flow through the sprinkler is 8 L/s, and the rotational speed of the sprinkler rotor can be set by the sprinkler operator, using a device called a governor. (a) At what rotational speed, in rpm, would the maximum power be generated? (b) What is the maximum power that can be generated by the sprinkler-generator?

Nozzle

10 cm

Ou!low

Figure 4.95: Sprinkler for electricity generation

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Section 4.6: Conservation of Energy

4.80. (a) Air flows at 0.05 kg/s through an insulated circular duct such that the (gage) pressure immediately downstream of a blower is 10 kPa. If the pressure at a downstream location within the duct is measured as 8 kPa, estimate the change in the temperature of the air that would be expected at the downstream location. You may neglect the impact of elevation change in your analysis. (b) If a blower were to be used to raise the pressure in the duct described in Part (a) by 2 kPa at a single location, estimate the power required to drive the blower if it has an efficiency of 90%.