Introduction to the Theory of Flow Machines

Introduction to the Theory of Flow Machines

1st Edition - January 1, 1966

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  • Author: Albert Betz
  • eBook ISBN: 9781483180908

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Introduction to the Theory of Flow Machines details the fundamental processes and the relations that have a significant influence in the operating mechanism of flow machines. The book first covers the general consideration in flow machines, such as pressure, stress, and cavitation. In the second chapter, the text deals with ducts; this chapter discusses the general remarks, types of flow, and mixing process. Next, the book tackles the types of cascades, along with its concerns. The closing chapter covers the flow machine and its components, such as turbine, wheels, engines, and propellers. The text will be of great use to mechanical engineers and technicians.

Table of Contents

  • Foreword

    Preface to the English Edition

    A. General Considerations

    1. Static and Dynamic Energy Transfer

    Transmission of Force by Pistons or by Movement of Aerofoils and Cascades

    2. Purposes and Classification of Flow Machines

    Machines That Supply Energy to the Fluid and Those That Extract Energy from It

    Energy Transmission

    Machines with Pressure Fall and Those with Pressure Rise

    Ducted and Non-Ducted Machines

    Axial, Radial, and Diagonal Forms of Construction

    3. Some Geometrical Concepts

    Steady and Unsteady Processes

    Streamlines and Particle Paths

    Plane ad Three-Dimensional Flows

    Two-Dimensional Flows

    4. Pressure

    Pressure in a Fluid at Rest

    Dependence of Specific Gravity and Density on Pressure and Temperature

    Compressible and Incompressible Fluids

    Gas Constant

    5. Adiabatic Changes of State

    Relations between Temperature, Pressure, Density, and Specific Gravity for Processes Involving No Heat Exchange

    6. Shear Stress

    Forces associated with Deformation


    Kinematic Viscosity

    7. Bernoulli's Equation

    Dependence of Pressure on Height and Speed

    Reduction to a Reference Height

    Total Pressure, Static Pressure, Dynamic Pressure

    Acceleration Term in Unsteady Flows

    Relations between Variables of State for Ideal Compressible Fluids

    Speed of Sound

    Critical Speed

    8. Cavitation

    Critical Pressure

    Cavitation Number

    Highest Permissible Speed

    Effect on Efficiency

    Destruction of the Material of the Wall

    9. Potential Flow, Rotation, Circulation

    B. Ducts

    Formation and Properties of Potential Flows


    Parallel Flow, Source, Vortex

    Circulation and Lift (Kutta-Joukowsky Theorem)

    Concept of Circulation for Aerofoils with Wakes

    Behavior of the Energy in a Vortex Field

    10. General Remarks

    Inlet Flow and Fully Established Pipe Flow

    Volume and Mass Flow Rates

    Mean Flow Velocity

    11. Types of Flow; Reynolds Number

    Laminar and Turbulent Flow

    Critical Reynolds Number

    Equivalent Diameter

    12. Laminar Flow

    Velocity Distribution in Pipe and Gap

    Pressure Fall and Resistance Coefficients for Circular and Rectangular Cross-Sections

    13. Turbulent Flow

    Velocity Distribution and Pressure Fall in Smooth Pipes

    Rough Walls

    Sand Roughness

    Pressure Fall and Velocity Distribution in Rough Pipes

    Behavior of Flow in Non-Circular Cross-Sections

    14. Conditions in the Inlet

    Displacement Thickness

    Momentum Thickness

    Transition from Laminar to Turbulent Boundary-Layer Flow

    Critical Boundary Layer Thickness, and Position of Transition Point

    Growth of Laminar and Turbulent Boundary Layers


    15. Changes in Cross-Section

    Change with Cross-Section of the Mean Velocity, of the Velocity Distribution across the Cross-Section, and of the Pressure

    Phenomena in Expanding Ducts (Diffusera)

    Boundary-Layer Separation

    Reduction in Pressure Rise (Diffuser Effect) with Non-Uniform Velocity Distribution

    Efficiency of Diffusera

    Favorable Effects of Bodies Producing Extra Resistance or of Rotors at End of Diffuser, of Boundary-Layer Suction, and of Swirl in the Flow Core of Dead Water

    16. Mixing Processes

    Pressure Rise and Energy Loss associated with Mixing

    Mixing of Two Streams of Different Velocity

    Sudden Expansion of a Duct

    Diffuser in Front of and behind a Mixing Process

    Combustion Processes

    17. Curved Ducts

    Stable and Unstable Velocity Distributions

    Forces on Channel Walls

    Behavior of the Boundary Layer

    Secondary Flow

    Energy Loss in Elbows for Turbulent and Lamina Reflow

    Reduction of Losses by Stators or by Cross-Sections in Which One Dimension is Much Larger than the Other

    18. Behavior of Compressible Fluids; Laval Nozzle; Shock Waves

    Speed of Sound

    Critical Speed

    Laval Nozzles

    Propagation of Disturbances in Subsonic and Supersonic Flow

    Mach Lines

    Mach Number

    Normal and Oblique Shock Waves

    19. Behavior of a Gas Flow with Addition and Removal of Heat

    Consequences of Continuity Equation

    Temperature and Velocity Changes in Subsonic and Supersonic Flow

    Temperature Maximum

    20. Flow through Ducts in Rotating Rotors

    Unsteady Potential Flow or Steady Flow with Constant Rotation

    Straight and Curved Ducts without and with Expansion of Cross-Sections

    Point of Reversal of Velocity

    Separation Lines between Flow Passing through and Flow Coming from outside and Returning outside

    Coriolis Forces

    Increased Danger of Boundary-Layer Separation

    Secondary Flow

    21. Variable Volume Flow Rate; Hydraulic Ram

    Pressures When the Flow through the Duct is Accelerated or Retarded

    Speed of Propagation of Pressure Waves in Ducts with Elastic Walls

    Reflection of Pressure Waves at Points Where the Speed of Propagation Changes or Where the Cross-Section Changes

    Pressure Fluctuations at a Throttle Point

    Possible Damage to the Duct from the Pressure Fluctuations, and Means of Reducing The Danger

    Use of the Pressure Fluctuations in the Hydraulic Ram

    C. Cascades

    22. Straight and Circular Cascades

    Concept, Properties, and Purposes of a Cascade

    Impulse, Turbine, and Compressor Cascades

    23. Deflection without Losses through a Straight Cascade

    Behavior of Velocity Components in the Cascade Direction and Normal to This Direction, and Behavior of Pressure

    Peculiarities of Compressible Fluids

    Forces on the Blades

    Power and Energy Change for Incompressible and Compressible Fluids

    24. Deflection without Losses through a Circular Cascade

    Behavior of Velocity Components and Pressure for Incompressible and Compressible Fluids

    Power and Energy Change

    25. Investigation of Losses

    Efficiency of a Cascade

    Shaft Efficiency

    Relations for Compressible Fluids

    26. The Shape and Arrangement of the Blades

    Blades Far Apart from One Another and Those Close Together

    Processes at a Moderate Blade Spacing

    27. Cascades with Blades Very Close Together

    Velocity and the Distribution of Circulation as Functions of Slope of Blade

    Effect of Curvature of Channel

    Effect of Thicknesses of Blade and Boundary Layer

    28. Conditions at the Discharge End of the Blade

    Deviation of Flow Direction from Blade Slope at Discharge End

    Rotating Cascades with Radial Flow

    29. Conditions at the Inlet End of the Blade

    Ideal Inlet Conditions or Separation at Sharp Edge of an Inlet

    Consequences of Separation

    30. Cascades with Blades Very Far Apart

    Lift, Drag, and Glide Angle of an Isolated Aerofoil

    Disturbances from Neighboring Blades of Cascade

    31. The Forces on an Isolated Aerofoil

    Angle of Incidence, and Reference Direction

    Direction of Zero Lift, and Angle of Incidence for Zero Lift

    Lift Coefficient

    Thin, Flat Plates

    Circular Arc and S-Shaped Cambered Profiles

    Line of Action of Forces, Profiles with Fixed Center of Pressure

    Effect of Drag

    Representation of Aerofoil Properties by Polars

    Guiding Principles for Determining Effect of Various Shapes

    Transfer of Aerofoil Properties to Blades in Cascades

    32. Pressure Distribution on an Isolated Aerofoil

    Flat Plate

    Circular Arc and S-Shaped Cambered, Thin Profiles

    Effect of Wing Thickness

    33. Isolated Aerofoils in Compressible Fluids

    Prandtl-Glauert Rule and Krahn Rule for Subsonic Flow

    Relation between Slope and Velocity in Supersonic Flow

    Shock Waves at Points Where the Slope Increases Suddenly

    Lift and Drag Coefficients

    Detached Shock Wave

    34. Transition from a Very Large to a Moderate Blade Spacing

    Effect of Neighboring Blades

    Adapting Blade Shape to the Disturbed Flow

    Approximation to Disturbance Functions in Neighborhood of Origin

    35. Transition from a Very Small to a Moderate Blade Spacing

    of Significance Mainly for Circular Cascades

    Approximate Distribution of Circulation at Blade Ends for Inlet Conditions Ideal and Not Ideal, with Infinitesimally Thin Blades

    More Exact Relations

    Effect of Finite Blade Thickness

    36. Cascades of Flat Plates

    Deflection Produced by a Cascade of Flat Plates

    Conformal Mapping of Strip of Cascade on to Exterior of a Circle

    Important Geometrical Relations for That Purpose

    Calculation of Velocities

    37. Cascades with Arbitrary Blade Shapes

    Equivalent Cascade of Flat Plates

    Approximate Procedure for Conformal Mapping of Strip of Width Equal to Pitch of Cascade on to a Corresponding Strip of Flow about an Isolated Aerofoil

    Calculation of Velocity Distribution

    Processes in Compressible Fluids

    38. Imperfect Cascades

    Concepts and Examples

    39. Cascades with Finite Blade Span

    Idealization of Cascade by Surface with Pressure Jump

    Pressure Equalization, and Conversion of Pressure Differences into Velocity Differences

    Mean Flow Velocity through Cascade

    Energy Loss

    Loading Factor

    40. Conditions at the Tips of the Blades

    Most Favorable Distribution of Lift at Blade Tips

    Equivalent Surface with Pressure Jump

    41. Conditions at the Gap between Blade and Ducting

    Gap Loss and Gap Resistance

    Most Favorable Behavior of Circulation

    Reduction of Gap Loss

    Effect of Friction at Wall of the Ducting

    42. Cascades with Non-Parallel Bounding Walls

    Replacement of the Conical Walls by Plane Walls with a Source Distribution

    A Simple Approximate Solution

    D. The Flow Machines

    43. Survey

    Axial-Flow, Radial-Flow, and Diagonal-Flow Rotors

    Connection with Corresponding Cascades and Deviations from them

    Importance of Simple Methods for Obtaining Estimates for Significant Quantities

    44. Pumps, Fans, Compressors

    Increase of Total Pressure and of Static Pressure

    Pump Characteristic

    Pressure Rise and Flow Coefficients

    Equivalent Nozzle Crosssection

    Loading Factor and Throttle Coefficient

    Rotational Instability in Flow When Severe Throttling is Present

    Pressure Rise and Flow Coefficients for Compressible Fluids

    Unstable Operating Relationships (Surging)

    Velocity Ratio

    45. Pumps with A Mainly Axial Flow Direction

    Approximately Plane Cascade Flow in a Developed Cylindrical Section

    Deviations from Plane Flow

    Maximum Pressure Rise

    Effect of Hub Diameter

    Utilization of Swirl Energy in Guide-Vanes

    Significance of Swirl before Inlet

    aspects of Design of Rotor Blades

    Qualification for Large Volume Flow Rates and Small Pressures

    Multi-Stage Pumps


    Effective Radius

    46. Design of an Axial Fan

    Losses in Fan and in following Diffuser

    Most Favorable Value of Rotor Diameter

    Calculations of Dimensions of Rotor Stator Blades

    47. Centrifugal Pumps with a Radial Flow Direction

    Pressure Increase without Loss from Centrifugal Force

    Maximum Theoretical Pressure Rise

    Loss associated with Conversion of Velocity into Pressure

    Ratio of Pressure Rise without Loss to Total Pressure Rise for Various Forms of Construction and Operating Conditions

    Requisite Number of Blades

    Losses through Deflection of an Incoming Axial Flow into Radial Direction

    Improvement by Use of Rotating Entry Vane

    Gap Losses or Friction Losses associated with a Cover

    Spiral Casing

    48. Rotors with Conical Flow (Diagonal-Flow Rotors)

    Intermediate Form between Axial and Radial Rotors

    Cone Angles Different inside and outside

    Meridianally Accelerated Rotors

    Mapping of Conical Flow on to a Plane

    Flow Directed Obliquely to Blades

    Diffuser behind Rotor

    49. Hydraulic Power Plants

    Practical Diameter of Rotor

    Maximum Volume Flow Rate

    Maximum Power


    Absorption Capacity

    Regulation by Stators

    Characteristic Quantities for Volume Flow Rate, Head, Loading, Velocity Ratio, and Size

    50. The Kaplan Turbine

    The Special Velocity Relations

    Cavitation Danger

    51. The Francis Turbine

    Radial Incoming Flow and Deflection into Axial Direction

    Involved Nature of Flow

    Mapping into Plane Flow

    52. The Pelton Wheel

    Processes at Nozzle and Bucket

    Forces, Power Transmission, and Efficiency

    53. The Föttinger Transmission; the Vulkan Coupling

    Change of the Rev/Min by Connection of Pump, Stator, and Turbine, One behind the Other

    Regulation by Adjustment of Stator

    Avoidance of Diffuser Components with their Losses

    Vulkan Coupling with Slip Regulation

    Föttinger Transmission in Land Vehicles

    54. Heat Engines

    Cycles for Conversion of Heat into Mechanical Work

    Adiabatic, Irreversible Processes


    Increase of Entropy in Irreversible Processes

    Limitation of Efficiency by Insufficient Control over High Combustion Temperatures and by Mechanical Losses

    Advantages of Processes in Steam-Engine

    Difficulties associated with Gas Turbine

    55. Steam Turbines

    Control of High Speeds in Steam

    Laval Turbine

    Parsons Turbine

    Velocity Stages

    Pressure Stages

    Degree of Reaction

    Curtis Stages

    56. Gas Turbines

    Utilizable Temperatures

    Method of Operation of Gas Turbines

    Example to Demonstrate Composition of Losses

    Attainable Efficiency

    Favorable Effect of Heat Exchangers

    Advantages of Gas Turbine, and Technical Difficulties

    57. Means of Propulsion

    Momentum and Energy Considerations

    Maximum Theoretical Efficiency

    Mean Flow Velocity through Propeller

    Disc Loading P. 206 ; Screw Propellers

    Tunnel Screws

    Voith-Schneider Propeller

    Limitation of Applicability of Air-Screws Because of Approach to Speed of Sound

    Modern Methods of Propulsion

    58. The Screw Propeller

    Rate of Advance, Disc Loading, Thrust and Torque Coefficients

    Additional Losses

    Contraction of Slipstream

    Helicopter Screw

    Swirl Losses

    Calculation of Blade Properties

    Determination of Main Dimensions

    Adaptation to Different Operating Conditions

    The Air-Screw at Great Heights

    Avoidance of Speed of Sound and of Cavitation

    59. Interference between Propeller and Vehicle; the Ducted Propeller

    Propeller in the Wake

    Improvement of Maximum Theoretical Efficiency

    Utilization of Energy in Wake

    Propeller in Disturbed Potential Flow

    The Ducted Propeller

    Thrust on Ducting

    Advantages and Disadvantages of the Ducting

    60. The Paddle-Wheel

    Method of Operation

    Comparison of Requisite Dimensions for Paddlewheels and Screw Propellers

    61. Rockets

    Thrust, Decrease in Mass, and Velocity

    Energy Source, Utilizable Power, and Losses

    Increase and Subsequent Decrease of Kinetic Energy of Rocket


    Difference between Rocket and Jet-Engine

    62. The Pulsating Jet-Engine

    Method of Operation

    Maximum Speed


    Recent Developments

    63. The Jet-Engine

    Historical Remarks

    Construction and Method of Operation

    Relation to and Differences from Gas Turbine and Rocket

    Efficiency and Favorable Region of Velocity

    Poor Efficiency, but Low Weight of Propulsive Mechanism

    64. Modifications to the Jet-Engine

    Two-Stream Jet-Engine


    65. The Ram-Jet

    Apparatus for Subsonic Flow and for Supersonic Flow


    Ram-Jet Combined with Pulsating Jet-Engine

    66. Wind-Driven Rotors

    Peculiarity of Economic aspect

    Maximum Theoretical Power

    Axial Force

    Blade Area Required

    Velocity Ratio

    Effect of Wind Fluctuations

    Importance of Energy Storage

    Effect of Wind Speed and Rotor Size on Economics

    Special Forms of Construction

    E. Appendix

    67. Tables

    Material Values (Density, Specific Gravity, Viscosity, Gas Constant) for Liquids and Gases

    Properties of Rocket Propellants

    68. Figures

    Pressure Fall in Ducts

    Variables of State for Air and Superheated Steam

    Effect of Shock Waves

    Effect of Cascade for Compressible Fluids

    Lift-Drag Polare

    Velocity Distribution for Flow pastBodies in Compressible Fluids

    Disturbing Influence of Neighboring Blades in a Series of Blades

    Circulation round and Deflection Produced by Blades in Cascades

    Conformal Mapping of Cascades of Flat Plates

    Behavior of Imperfect Cascades

    69. List of Commonly Used Symbols

    70. List of References

    Author Index

    Subject Index

Product details

  • No. of pages: 300
  • Language: English
  • Copyright: © Pergamon 1966
  • Published: January 1, 1966
  • Imprint: Pergamon
  • eBook ISBN: 9781483180908

About the Author

Albert Betz

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