Introduction to the Theory of Flow Machines - 1st Edition - ISBN: 9780080114330, 9781483180908

Introduction to the Theory of Flow Machines

1st Edition

Authors: Albert Betz
eBook ISBN: 9781483180908
Imprint: Pergamon
Published Date: 1st January 1966
Page Count: 300
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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


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


No. of pages:
© Pergamon 1966
eBook ISBN:

About the Author

Albert Betz