Fans: Aerodynamic Design - Noise Reduction - Optimization 3658379588, 9783658379582

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Thomas Carolus

Fans

Aerodynamic Design - Noise Reduction Optimization

Fans

Thomas Carolus

Fans Aerodynamic Design - Noise Reduction Optimization

Thomas Carolus (emeritus) Applied Fluid Mechanics and Turbomachinery Institute for Fluid- and Thermodynamics University Siegen Siegen, Germany

ISBN 978-3-658-37958-2 ISBN 978-3-658-37959-9 https://doi.org/10.1007/978-3-658-37959-9

(eBook)

The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com). A subsequent human revision was done primarily in terms of content. # Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer Vieweg imprint is published by the registered company Springer Fachmedien Wiesbaden GmbH, part of Springer Nature. The registered company address is: Abraham-Lincoln-Str. 46, 65189 Wiesbaden, Germany

Preface

The number of air and gas transporting machines in use today is uncountable. These machines enable key areas of our daily lives by providing air for heating and cooling, delivering gases in chemical plants and food processing, assisting in generating electricity, propelling airplanes, and many more. This book deals with industrial fans, that is, with all type of fans with the exception of those for aircraft engines. In a recent article, S. Castegnaro (see the bibliography at the end of Chap. 4) pointed out how the aerodynamic design methods for fans developed historically: Before the beginning of the twentieth century, only empirical knowledge was available. Then, mathematicalphysical analytical theories became increasingly established. They are still in use today in ever more refined variants. The development of the modern computer from the 1950s onwards, however, made it possible to solve the fundamental equations of fluid mechanics with increasingly less simplifications. The key word is “computational fluid dynamics (CFD)”. Thus, numerical flow simulation has become the third pillar of fluid mechanics, together with the established experimental and analytical methods. Nowadays, CFD is often used to predict the aerodynamic operating behaviour of a new fan before a model or prototype is even built. A large number of design variants can be simulated. This paves the way for a real aerodynamic optimization, for instance of efficiency or fan power density. Optimization may yield more complex shapes than the classic design methods, but modern composite materials and additive manufacturing processes increasingly can cope with that. The aeroacoustics of turbomachinery in general is still a topic of strong current research interest. While aerodynamic design methods can be considered proven and mature, the “acoustic design” of fans is much more challenging. Accurate fan noise prediction is difficult but essential for designing low noise fans. In addition to the mere mitigation of flow-induced fan noise, the minimization of annoyance due to unavoidable fan noise and hence psychoacoustics metrics increasingly become the focus of attention. The book is arranged in three major sections. The first (Chaps. 1, 2, 3, and 4) provides the selection of a fan for a given duty in a plant as well as classic aerodynamic design methods for centrifugal and axial fans. The second (Chaps. 5, 6, 7, and 8) focuses on fan noise, especially on its generation and propagation, on its prediction, and on its mitigation

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Preface

by special design features. The last section (Chap. 9) provides an introduction into experimental and numerical methods including optimization. The German editions of this book have served as an educational text, but also as a professional reference for engineers who are concerned with the design and application of industrial fans. Many international readers and the publisher encouraged me to bring out this first English translation of the fourth German edition.

Acknowledgements

Many individuals and organizations have assisted me in working on this and the past editions of this book. I am especially grateful to all my former research assistants and Ph.D. students who were involved in turbomachinery research in my team: Dipl.-Ing. Bernd Homrighausen, Dr.-Ing. Michael Beiler, Dr.-Ing. Thomas Fuest, Dr.-Ing. Bernhard Schulze-Dieckhoff, Dr.-Ing. Robert. Basile. Dr.-Ing. Michael Stremel, Dr.-Ing. Marc Schneider, Dr.-Ing. Hauke Reese, Dr.-Ing. Julian Winkler, Dr.-Ing. Daniel Wolfram, Dr.Ing. Michael Kohlhaas, Dr.-Ing. Stephan Pitsch, Dr.-Ing. Tom Gerhard, Dr.-Ing. Michal Sturm, Dr.-Ing. Konrad Bamberger, Dr.-Ing. Sebastian Knirsch, Dr.-Ing. Ralf Starzmann, Dr.-Ing. Christoph Moisel, Dr.-Ing. Tao Zhu, Dr.-Ing. Nico Kaufmann, Dr.-Ing. Kevin Volkmer, Dr.-Ing. Farhan Manegar, Dr.-Ing. Carolin Feldmann, Dr.-Ing. Leonard Mackowski and M.Sc. Kathrin Stahl. Their contributions and discussions were indispensable. In addition, I appreciate very much the early technical discussions with Dr.-Ing. J. Franke from the former flow simulation group at the University Siegen and Dr.-Ing. Ş. Çağlar from the Karlsruhe Institute of Technology. Prof. Dr.-Ing. P. Pelz from the Technical University Darmstadt contributed to the section on efficiency upgrading. The section on the psychoacoustic evaluation of fan noise was written with the special collaboration of Dr. rer. nat. S. Töpken from the Carl von Ossietzky University Oldenburg and Dr.Ing. Carolin Feldmann. Dr.-Ing. Konrad Bamberger was the main author of the section on optimization. My basic knowledge for this textbook, however, is owed to my own former professors Dr.-Ing. H. Marcinowski, Dr.-Ing. Dr. techn. E.h. J. Zierep and Dr.-Ing. Dr. h.c. K.-O. Felsch during my time as a student at the Technische Hochschule Karlsruhe (now Karlsruhe Institute of Technology, KIT), Germany, as well as to Prof. Allan Pierce, Ph.D., whose student I had the privilege of being at the Georgia Institute of Technology in Atlanta, USA. I am indebted also to many individual companies in the fan industries that have supported my research for nearly 40 years. Many challenging projects were initiated by the German Forschungsvereinigung für Luft- und Trocknungstechnik (FLT) e.V. (German Research Association for Air and Drying Technology) which connects industrial to academic research in an excellent manner.

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Acknowledgements

The Fulbright Commission supported my two research sabbaticals in the Department of Aerospace Engineering at the Pennsylvania State University in State College, USA, which contributed to the progress of the book tremendously. I want to thank the Department of Mechanical Engineering at the University Siegen, Germany, for providing me and my team with a laboratory and computer facilities for nearly 30 years. Special thanks to all readers of the first three editions who gave constructive feedback and suggested additions and extensions. Finally, my great (literally and figuratively) family has supported me over many years when I disappeared behind my desk or in the laboratory. I am most grateful for their understanding and continuous encouragement.

Contents

1

2

3

Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Fan Performance Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Selection of the Fan: Demand of the Plant . . . . . . . . . . . . . . . . . . . 1.3 Aerodynamic Performance Characteristics . . . . . . . . . . . . . . . . . . . 1.4 Non-dimensional Parameters, Model Laws, Types of Fans . . . . . . . 1.4.1 Non-dimensional Parameters . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Model Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Scale-Up Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Systematics of Fan Types: The Cordier-Diagram . . . . . . . . 1.5 Practice Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Pressure Rise Requirement of a Plant and Fan Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Selection of the Type of Fan . . . . . . . . . . . . . . . . . . . . . . 1.5.3 From Model- to Full-Scale . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 1 3 4 5 5 6 8 9 11

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11 12 12 12

Blade Performance Parameters, Cascades of Blades, Kinematics, Losses and Efficiencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Blade Work, Blade Volume Flow Rate . . . . . . . . . . . . . . . . . . . . . 2.2 Flow Kinematics (Velocity Triangles) . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Cascade of Radial Blades . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Cascade of Axial Blades . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Losses and Efficiencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Losses in the Impeller and Efficiencies . . . . . . . . . . . . . . . 2.3.2 Losses in the Casing and Guide Vanes, Efficiencies . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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15 15 16 16 19 21 21 23 26

Design of Centrifugal Fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Blade Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Slip of Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 29 29

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3.1.2 3.1.3 3.1.4

The Slip Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selection of the Number of Blades . . . . . . . . . . . . . . . . . . Blockage of the Inlet and Outlet Due to the Finite Thickness of the Blade . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Summary: Blade Design for Centrifugal Impellers . . . . . . . 3.1.6 Further Empirical Geometry Parameters of the Centrifugal Impeller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Layout of a Volute Casing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 One-Dimensional Streamline Theory . . . . . . . . . . . . . . . . 3.2.2 Further Empirical Geometrical Parameters of the Simple Volute Casing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Practice Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Design of a Centrifugal Fan Impeller . . . . . . . . . . . . . . . . 3.3.2 Design of a Volute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Design of Axial Fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Flow Kinematics in the Axial Impeller: Radial Equilibrium . . . . . . . 4.1.1 Isoenergetic Loading Distribution . . . . . . . . . . . . . . . . . . . 4.1.2 Radius-Dependent Loading Distribution . . . . . . . . . . . . . . 4.1.3 Summary of Swirl Distributions . . . . . . . . . . . . . . . . . . . . 4.2 Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Blade Element Momentum (BEM) Method for Low-Pressure Axial Fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Derivation of the Key Equation . . . . . . . . . . . . . . . . . . . . 4.3.2 Summary: Blade Design for Low-Solidity Axial Impellers with the BEM Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Blade Skew . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 The Lieblein Method for High-Pressure Axial Fans . . . . . . . . . . . . 4.4.1 Blade Element Inlet Angle . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Blade Exit Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Camber and Mean Line . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Summary: Blade Design for High-Solidity Axial Impellers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Design Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 De Haller-Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Criterion of Strscheletzky . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Diffusion Coefficient According to Lieblein . . . . . . . . . . . 4.5.4 Further Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Practice Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Design of a Low-Pressure Axial Fan . . . . . . . . . . . . . . . . .

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51 52 55 58 60 60

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61 62

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65 68 70 71 72 73

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74 75 75 75 77 77 78 78

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4.6.2 Design of High-Pressure Axial Fan Stage . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79 80 80

5

Sound Generation and Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The Mechanisms of Sound Generation: An Overview . . . . . . . . . . . 5.2 Rotating Pressure Fields of Axial Fans . . . . . . . . . . . . . . . . . . . . . 5.2.1 The Rotating Pressure Field of an Isolated Impeller . . . . . . 5.2.2 Rotor-Stator Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Flow-Induced Sound from Lift-Generating Surfaces . . . . . . . . . . . . 5.4 Sound Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Radiation into the Free Field . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Sound Propagation in Ducts: Duct Modes . . . . . . . . . . . . . 5.4.3 Excitation of Duct Modes by a Fan . . . . . . . . . . . . . . . . . . 5.5 Significance of Sound Sources and Examples . . . . . . . . . . . . . . . . . 5.5.1 Rotor-Stator Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Turbulent Inflow, Stall . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Vortex Shedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4 Tip Clearance Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Practice Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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81 81 84 84 85 87 91 93 93 96 98 99 102 103 104 105 106

6

Sound Prediction Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Class I Sound Prediction Methods . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Formula of Madison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Regenscheit’s Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Estimation of the Octave Band Sound Power Level . . . . . . 6.3 Class II Sound Prediction Methods . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 The Sharland Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Spectral Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Duct Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Summary and Example . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Practice Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Acoustic Model Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Fan Acoustic Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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109 109 109 109 111 112 114 114 119 124 124 126 126 126 127 128

7

Psychoacoustic Assessment of Fan Noise . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Perception of Annoyance and Quality of Fan Sound . . . . . . . . . . . . .

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7.3 Two Psychoacoustic Metrics for Fan Noise . . . . . . . . . . . . . . . . . . . 7.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

133 136 138

Design Features of Noise Reduced Fans . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 General Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Reduction of the Circumferential Speed . . . . . . . . . . . . . . 8.1.2 Increasing the Spacing Between Stationary and Rotating Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3 Phase Shift of the Interaction Between Stationary and Rotating Components . . . . . . . . . . . . . . . . . . . . . . . . 8.1.4 Uneven Blade Spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.5 Wavy Leading Edge and Serrated Trailing Edge . . . . . . . . 8.1.6 Optimum Inlet Geometry and Turbulence Control Screen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Further Special Measures for Centrifugal Fans . . . . . . . . . . . . . . . . 8.2.1 Meridional Contour of Shroud . . . . . . . . . . . . . . . . . . . . . 8.2.2 Suppression of Resonance . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Further Special Measures for Axial Fans . . . . . . . . . . . . . . . . . . . . 8.3.1 Tuning the Number of Blades (Mode Propagation) . . . . . . 8.3.2 Skewed Blades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Influencing the Tip Gap Flow . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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141 141 141

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144 146 147

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149 151 151 152 153 153 153 155 156

Numerical and Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Numerical Flow Field Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Overview of CFD Methods . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Computational Domain and Numerical Grid . . . . . . . . . . . 9.1.3 Boundary and Initial Conditions . . . . . . . . . . . . . . . . . . . . 9.1.4 The Rotor-Stator Problem . . . . . . . . . . . . . . . . . . . . . . . . 9.1.5 Control Parameters, Convergence, Residuals and Termination of Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.6 Post Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.7 Validation and Verification . . . . . . . . . . . . . . . . . . . . . . . 9.1.8 Example: Axial Fan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Fan Test Rigs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Measurement of Flow Field Quantities: Measuring Probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Measurement of Acoustic Quantities . . . . . . . . . . . . . . . . . 9.3 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Optimization Procedures . . . . . . . . . . . . . . . . . . . . . . . . .

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159 160 160 164 167 169

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170 171 171 172 174 174

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9.3.2 9.3.3

Example: Optimization of an Axial Fan . . . . . . . . . . . . . . Example: Centrifugal Fan Impellers with Maximum Total-to-Static Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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191 192 194

10

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Effective Pressure Rise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Centrifugal Impeller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.2 Axial Impeller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.3 Axial Impeller with Outlet Guide Vanes and Diffuser . . . . 10.2 Airfoil Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Isolated Airfoil Section in Unbounded Flow . . . . . . . . . . . 10.2.2 Airfoil Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.3 Airfoil Sections in a Cascade . . . . . . . . . . . . . . . . . . . . . . 10.3 Some Basics of Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Lieblein Design Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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195 195 196 197 199 203 203 205 211 213 218 222 227

11

Answers to Practice Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Answer to Problem 1.5.1: Pressure Rise Requirement of a Plant and Fan Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Answer to Problem 1.5.2: Selection of the Type of Fan . . . . . . . . . 11.3 Answer to Problem 1.5.3: From Model- to Full-Scale . . . . . . . . . . . 11.4 Answer to Problem 3.3.1: Design of a Centrifugal Fan Impeller . . . 11.5 Answer to Problem 3.3.2: Design of a Volute . . . . . . . . . . . . . . . . 11.6 Answer to Problem 4.6.1: Design of Low-Pressure Axial Fan . . . . . 11.7 Answer to Problem 4.6.2: Design of High-Pressure Axial Fan Stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8 Answer to Problem 5.6: Axial Fan – Acoustic Modes . . . . . . . . . . . 11.9 Answer to Problem 6.4.1: Acoustic Model Law . . . . . . . . . . . . . . . 11.10 Answer to Problem 6.4.2: Fan Acoustic Power . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

229

. . . . . .

229 233 234 235 238 238

. . . . .

242 246 246 246 249

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

251

List of Symbols (Selection)

A (m2) b (m) B (m) BPF (Hz) c (m/s) c0 (m/s) CL (–) CD (–) D (m, N) DF (–) f (Hz, m) F (N) He (–) i (°) k (1/m) l (m) L (N) Lp (dB) Lr (dB) LW (dB) m_ (kg/s) n (rpm) Ma (–) p (Pa) p′ (Pa) P (W) Q (–) r (m) Re (–) s (m) Sr (–) TI (–)

area width of centrifugal impeller, airfoil section span width of volute casing blade passing frequency absolute flow velocity speed of sound lift coefficient drag coefficient impeller diameter, drag diffusion coefficient frequency, camber force Helmholtz number incidence angle wave number chord length lift sound pressure level rating level sound power level mass flow rate impeller shaft speed Mach number pressure fluctuating pressure, sound pressure power sound quality radius Reynolds number blade thickness, tip clearance Strouhal number turbulence intensity xv

xvi

List of Symbols (Selection)

t (m, s) u (m/s) V_ (m3/s) w (m/s) Y (W/(kg/s)) z (–) zSt (–)

blade spacing, time circumferential velocity volume flow rate relative flow velocity specific work or power number of blades (impeller) number of blades (stator) or guide vanes

α (°) β (°) βB (°) γ (°) δ (°, -) δr (m) Δp (Pa) ε (–) η (–) λ (–, m, –) Λ (m) μ (–) ν (–, m2/s) ρ (kg/m3) σ (–) φ (–, °) ψ (–) ω (rad/s) Ω (rad/s)

absolute flow angle, angle of attack relative flow angle blade angle (as distinct from flow angle) stagger angle deviation angle, sweep angle, specific diameter radial extension of blade element (BE) or elemental blade cascade (CA) pressure rise drag-to-lift ratio efficiency power coefficient, wavelength, aspect ratio turbulent length scale slip factor hub-to-tip ratio, kinematic viscosity fluid density specific speed, solidity (= l/t) volume flow rate coefficient, geometric camber angle pressure rise coefficient acoustic angular frequency angular shaft frequency

Subscripts

1 1 2 ac B c.o. Cas h hub Imp loss m oa opt S t tip ts u vol

vectorial mean, infinite number of blades, free field inlet (blade cascade) exit (blade cascade) acoustic blade cut off casing hydraulic hub impeller loss meridional overall (sound pressure, power) optimal shaft total (= total-to-total) (blade) tip effective (= total-to-static) circumferential volumetric

xvii

Subscripts

– .

spatial or temporal average or mean value time rate of change

xix

Abbreviations

BE BEM CA CAA CFD DES DNS LBM LES OF RANS URANS

Blade Element Blade Element Momentum method Cascade (of elemental blades) Computational Aeroacoustics Computational Fluid Dynamics Detached Eddy Simulation Direct Numerical Simulation Lattice-Boltzmann Method Large Eddy Simulation Objective Function Reynolds-Averaged Navier-Stokes Simulation Unsteady Reynolds-Averaged Navier-Stokes Simulation

xxi

1

Basics

This chapter shows how the fundamental performance parameters of a fan need to be specified – as a first step for selecting a fan that allows an economic and safe operation of a complete plant. Furthermore, with the aid of non-dimensional performance parameters – common in turbomachinery engineering – it is explained which fan type, e.g. axial or centrifugal, is appropriate for a given duty. Finally, the model laws are presented which allow scaling of an existing fan for other performance parameters without any new aerodynamic design.

1.1

Fan Performance Parameters

The most important aerodynamic performance parameters of a fan are the volume flow rate V_ of the gaseous fluid conveyed by the fan in units of m3/s and the total pressure rise Δpt in units of Pa. Throughout this book the gaseous fluid is considered incompressible.1 Given its density ρ, equivalent fan performance parameters are • the mass flow through the fan m_ = ρV_ in kg=s and

ð1:1Þ

1 According to VDI 2044 [1] (“VDI Code of Practice for Fans”) and ISO 5801 [2], this is assumed to hold true as long as the discharge velocity of the fluid does not exceed 20% of the speed of sound (Mach number Ma < 0.2). If compressibility is not negligible, please refer to the standards.

# Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 T. Carolus, Fans, https://doi.org/10.1007/978-3-658-37959-9_1

1

2

1 Basics

• the specific work done by the fan on the fluid2 Y t = Δpt =ρ in JðouleÞ=kg or WðattÞ=ðkg=sÞ:

ð1:2Þ

It is evident that Δpt, like Yt, is an energetic quantity. The total pressure rise is the difference in the total pressures between the fan inlet and exit. In general, it comprises a rise in both, static and dynamic pressure: Δpt 

pt,exit |ffl{zffl}



total pressure at the fan exit



- pt,in |{z} 

at the inlet

     = pst,exit þ pd,exit - pst,in þ pd,in = pst,exit - pst,in þ pd,exit - pd,in |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} static

ð1:3Þ

dynamic pressure rise

The dynamic pressure is always (ρ/2)c2 with the fluid velocity c = cin and cexit at the fan inlet and exit, respectively. The specification of the static pressure rise of a fan is meaningful only, if the location and the size of the respective through flow areas are specified as well. An important special case is when the fan is the last component in a plant and exhausts directly into the free atmosphere or a large room. The dynamic pressure of the discharge jet constitutes a loss. Accordingly, an effective (or total-to-static) pressure rise Δpts  Δpt - pd,exit = pst,exit - pt,in

ð1:4Þ

is defined. The usefulness of this quantity is demonstrated by the practice problem in Sect. 1.5.1.3 The shaft power a motor must supply to the fan impeller is calculated from the shaft torque MS and the impeller shaft speed n as

“Specific” means work done on unit mass (J/kg) of fluid; the equivalency J/kg = (J/s)/(kg/s) = W/ (kg/s) shows that Y can also be interpreted as power related to unit mass flow rate. 3 The terms “specific work”, “total pressure rise (or increase)”, “rise of static and dynamic pressure” and “effective pressure rise” are used in this way in VDI 2044 [1]. Occasionally and very sensibly the total pressure rise is more precisely called the total-to-total pressure rise, resulting in the symbol Δptt. Similarly, the effective pressure rise is a total-to-static pressure rise named Δpts. This index is used throughout this book for the effective pressure rise. In ISO 5801 [2], on the other hand, the total pressure rise is simply called the “fan pressure” with the symbol pf. The effective pressure rise is called “fan static pressure” pfs which must not be mistaken for the true static pressure rise as in Eq. (1.3). As a conclusion, it is imperative to ensure a clear understanding of the terms used when specifying or accepting fan performance data. 2

1.2

Selection of the Fan: Demand of the Plant

3

PS = M S 2πn:

ð1:5Þ

The total and effective efficiency of the fan is the ratio η=

1.2

_ t _ ts VΔp VΔp and ηts = , respectively: PS PS

ð1.6a, bÞ

Selection of the Fan: Demand of the Plant

A plant is a system of internal flow passages such as pipes and other equipment through which a specified volume flow rate of the gaseous fluid is to be conveyed by the fan. An example is a simple cooling unit in Fig. 1.1. In order to properly select the fan for such a system, the specific work or, equivalently, the total pressure rise required by the plant to provide the targeted volume flow rate V_ needs to be determined. The key relation comes from the somewhat simplified first law of thermodynamics: Δpt,plant,req = ðpII - pI Þ þ

 X ρ 2 cII - c2I þ Δploss 2 I → II

ð1:7Þ

Here, I and II represent the boundaries of a control surface around the plant. Fluid enters at station I and leaves at station II. The boundaries are chosen such that the values of the static pressures pI and pII and flow velocities cI and cII are known best possibly.

Fig. 1.1 Cooling unit as an example of a plant with a fan; I and II indicate the locations where fluid passes the boundaries of a chosen control surface (not shown here) around the plant

4

1 Basics

The pressure losses Δploss of the individual distinct plant components, such as pipe sections, elbows, cross-section jumps, heat exchangers, etc. are usually determined separately and then added to the overall pressure loss between I and II – leaving the fan out. The individual pressure losses are calculated via ρ Δploss = ζ c2 : 2

ð1:8Þ

c is the mean flow velocity at a reference cross-section of the component. The pressure loss coefficient ζ of each component can be obtained from references, such as IDELCHICK [3]. Although this approach neglects the interaction of the flow between the individual components, it is in most cases the only way to estimate the overall pressure loss. Eventually, the fan must be selected or designed in such a way that the demand of the plant is covered by the supply of the fan, i.e. Δpt = Δpt,plant,req

ð1:9Þ

_ at the targeted volume flow rate V:

1.3

Aerodynamic Performance Characteristics

The full operating behavior of a fan as a function of the flow rate is described in terms of performance characteristics. These are determined by operating the fan at various flow rates on a test rig as described in Sect. 9.2. In most cases, the shaft speed is kept constant for this purpose. Figure 1.2a shows schematically the total pressure rise and efficiency characteristics. A distinct point on the total pressure characteristic is the optimum point, i.e. where the fan operates at its peak efficiency. If the fan is operated to the left of its optimum, it is referred to as part-load and the right as overload operation. Operation at partload is sometimes problematic: Some fans exhibit a discontinuity in their characteristics that indicates stalled flow in the fan. Stalled flow with its massive flow separation is often unsteady and causes noise and vibrations, hence operating points close to, at and left of this stall point should be avoided. The total pressure rise required by the plant for any volume flow rate V_ can also be represented as a characteristic curve. All velocities in Eqs. (1.7) and (1.8) are proportional to the volume flow rate. Therefore, the plant characteristic is, to a good approximation, a quadratic parabola, Fig. 1.2b. The synthesis of the plant and the fan according to Eq. (1.9) yields the intersection of fan and plant characteristics as shown in Fig. 1.2c which is the operating point of the fan on its performance characteristic. Ideally, the operating point coincides with the fan’s optimal point. In practice, however, this is not always easy to achieve. In this representation, the operating volume flow rate V_ OP that actually is attained can be read off immediately.

1.4

Non-dimensional Parameters, Model Laws, Types of Fans

Fig. 1.2 Aerodynamic performance characteristics: (a) Fan characteristics, (b) Plant characteristic, (c) Synthesis of fan and plant with the resulting operating point (schematic)

5

a)

b)

c)

1.4

Non-dimensional Parameters, Model Laws, Types of Fans

1.4.1

Non-dimensional Parameters

The definitions of some important non-dimensional parameters in turbomachinery and thus also in fan engineering are compiled in Table 1.1. D is a representative dimension of the fan, usually the outer diameter of the impeller Dtip, n the impeller shaft speed in revolutions per second, ρ the density of the gaseous fluid.

6

1 Basics

Table 1.1 Important non-dimensional parameters; D is a representative dimension of the fan, usually Dtip, n the impeller shaft speed, ρ the density of the gaseous fluid Designation Volume flow rate coefficient

Definition _

φ  π2 V 3 4

(1.10)

D n

or ψ t  π2 Δp2t

ψ t  π2 Y2t

Pressure rise coefficient

2

λ  π4

Power coefficient

8

D n2

2

PS ρD5 n3

ρD n2

(1.11) (1.12)

Specific speed

σ

n 3=4 - 1=2 ð2π 2 Þ - 1=4 Y t V_

(1.13)

Specific diameter

δ

D - 1=4 _ 1=2 ð8=π 2 Þ1=4 Y t V

(1.14)

Total efficiency

_ t η  mY PS =

_ t VΔp PS

(1.6a)

φ, ψ and λ are mostly used for the non-dimensional presentation of performance data, whereas σ and δ rather serve to characterize the type of fan. Not all non-dimensional parameters are independent of each other, e.g. σ and δ can be calculated from φ and ψ as 1

3

1

1

σ = φ2 =ψ 4t or δ = ψ 4t =φ2 : There are also variants of these parameters (which are irrelevant here) and parameters such as the Reynolds number, which will be discussed later.

1.4.2

Model Laws

We consider two fans which are geometrically similar, i.e. the ratios of all geometric dimensions in both machines are the same. If, in addition, there is kinematic similarity (i.e. the velocity triangles in both machines are similar) and dynamic similarity (i.e. the force ratios in the flow, e.g. inertial to frictional forces, are the same in both machines), all non-dimensional parameters of both machines have the same values. As a consequence, from the definition of the non-dimensional coefficients one readily obtains the model laws V_  D3 n Y t  D2 n2 or Δpt  ρD2 n2 PS  ρD5 n3 :

ð1:15aÞ ð1.16a, bÞ ð1:17aÞ

The model laws allow converting each operating point when varying shaft speed, fluid density and machine size. If the machine size D and the density ρ of the fluid remain fixed, these laws reduce to the so called affinity laws

1.4

Non-dimensional Parameters, Model Laws, Types of Fans

V_  n Y t  n2 or Δpt  n2 PS  n3 :

7

ð1:15bÞ ð1.16c, dÞ ð1:17bÞ

These affinity laws are useful in practice for calculating the effect of change of shaft speed of a particular fan. The graph on the left in Fig. 1.3 shows, as an example, the dimensional performance characteristics of a centrifugal fan measured at different speeds. In the non-dimensional representation these performance data collapse to a single and universal curve. Conversely, from a non-dimensional characteristic curve, one can predict the dimensional performance for any speed.

Fig. 1.3 Total-to-static pressure rise of a centrifugal fan without casing; upper: Constant-speed characteristics obtained from experiments with air at different speeds of the fan with an impeller diameter D = 406 mm, the density of the air during the measurement is ρ = 1.2 kg/m3; lower: Non-dimensional representation of the data – all three dimensional characteristics coincide

8

1.4.3

1 Basics

Scale-Up Methods

When applying the model laws, e.g. for converting the performance data of a scaled- down prototype to a geometrically scaled-up full-size fan, the dynamic similarity is most probably violated. An indicator for this is the value of a characteristic Reynolds number. In fan engineering, the Reynolds number is often, but not always defined as ReD =

uD : ν

ð1:18Þ

u is the circumferential speed of the tip of the impeller (tip speed), D the outer diameter of the impeller and ν the kinematic viscosity of the fluid. Figure 1.4 shows schematically the effect of the Reynolds number on the φ /η performance characteristic of a fan. If the values of the Reynolds number for the model and the full-scale version are very different, the effect of Reynolds number on efficiency can be estimated by scale-up methods. A classic scale-up formula for the peak efficiency is due to Ackeret from 1948, see Spurk [4]:   α  ηfull scale - ηmodel  Δη = ð1 - ηmodel ÞV 1 - Re D,model = Re D,full scale

ð1:19Þ

ηmodel is the measured efficiency of the prototype model, V is the portion of the scalable losses (typically between 0.5 and 0.7, to be determined empirically), and the Reynolds number exponent α is between 0.5 and 0.7. If the Reynolds number of the model is very low (i.e. below 5105), Spurk recommends for α a value of approximately 0.5. A disadvantage of this scale-up formula is that a single chosen set of the empirical constants does not apply to the entire efficiency characteristic. Therefore, Hess and Pelz [6] investigated how the coefficient V varies with φ. Moreover, Pelz et al. [5] report a

Fig. 1.4 Efficiency curves for different Reynolds numbers; the optimum points lie in good approximation on a straight line (schematic, after Pelz et al. [5])

1.4

Non-dimensional Parameters, Model Laws, Types of Fans

9

generalized scale-up methodology. They attribute Δη at the optimum point to a change of the friction coefficient in the machine due to different surface roughness and/or Reynolds number. The experimentally observed shift of peak efficiency Δφ towards higher volume flow with increasing Reynolds number (Fig. 1.4) is explained by the boundary layer on the blades, which, depending on the Reynolds number, de facto modifies the effective blade contour and thus the optimum point. Hence, scaling up the complete efficiency characteristic results in a shift in η and φ. Contractors may stipulate that designs are accepted merely on the basis of model tests and a scale-up method (cp. “International Codes for Model Acceptance Tests”).

1.4.4

Systematics of Fan Types: The Cordier-Diagram

The possible values of the performance parameters, achievable with common single-flow and single-stage fan types, are limited. Expressing the optimal volume flow rate and total pressure rise of various fan types in terms of the specific diameter and speed δopt and σ opt and plotting δopt against σ opt, results in the so called Cordier-diagram, Fig. 1.5. All values of

Fig. 1.5 Systematics of fan types (impellers) in the Cordier-diagram, according to Cordier [7–10]

10

1 Basics

δopt and σ opt lie with a certain scatter in a relatively narrow band. Centrifugal fans with backward curved blades are low specific speed turbomachines, typically with σ opt-values ranging from 0.1 to 0.6, see Mode [10]. In contrast, axial fans with σ opt > 0.6 are high specific speed machines. The mixed-flow fans are in between. Only the centrifugal fan with forward curved blades lies outside this band. The range limits are only indicative and the ranges of the different types overlap. Nevertheless, the Cordier-diagram is very useful for selecting the most appropriate fan type, its size, shaft speed and, if necessary, the number of stages in series or parallel for a given duty. Figure 1.6 depicts the measured performance characteristics of three fans for optimum points φopt or σ opt covering a certain range in the Cordier-diagram. Centrifugal fans are suitable for a duty requiring a large total pressure rise and low volume flow rates, while axial fans are typically characterized by high volume flow rates with low pressure rise. σopt

φopt

Type of fan impeller centrifugal (backward curved blades)

0.41

0.04

D1/D2= 0.34 b/D2 = 0.15 centrifugal (backward curved blades)

0.60

0.15

D1/D2= 0.72 b/D2 = 0.34 axial

1.54

0.21

Dhub/Dtip= 0.45

Fig. 1.6 Measured performance characteristics of different fan types (centrifugal impellers with constant-width blade passage in a volute casing, axial impeller with guide vanes); D1/D2 is the ratio of the blade leading /trailing diameter (D2 is equivalent to the impeller diameter), b/D2 the ratio of the width to the diameter of the impeller, Dhub/Dtip the hub-to-tip ratio (diameter of hub/diameter of impeller)

1.5

Practice Problems

1.5

Practice Problems

1.5.1

Pressure Rise Requirement of a Plant and Fan Specification

11

Determine the pressure rise specification for the fan in each cooling system in Fig. 1.7, if a volume flow rate of air V_ = 14:4 m3 =s has to be delivered through the radiator. What would be the required shaft power of the respective fans? Further data: • • • •

Surface area of the radiator matrix: ARad = 12 m2. Cross-sectional area of the chimney-like exhaust duct in system A: ADuct = 1 m2. In system B, a diffuser with an exit area ADiff = 1.3 m2 is installed. The pressure loss coefficient of the radiator matrix and fan cowl is ζ = 200, related to the air velocity at the radiator inlet as the reference cross-section. • The pressure losses due to wall friction in the chimney-like duct in system A and the diffuser in system B are to be neglected for the sake of simplicity. • For simplicity, assume an average constant air density ρ = 1.00 kg/m3 throughout the system.

Fig. 1.7 Three cooling systems

12

1.5.2

1 Basics

Selection of the Type of Fan

Two different ventilation systems require operating values of the fan as in this table: V_ ½m3 =s 10 10

System D E

Yt [W/(kg/s)] 6000 600

For both, a typical single stage axial and centrifugal fan, estimate the impeller diameter and shaft speed required.

1.5.3

From Model- to Full-Scale

A full-scale fan (index “FS”) foreseen for conveying a gas with the density ρ = 0.80 kg/m3 is specified as: Impeller outer diameter Dtip,FS 1.828 m

Shaft speed nFS 627 1/min

Density ρFS 0.80 kg/m3

Volume flow rate V_ FS 31.5 m3/s

Total pressure rise Δpt,FS 800 Pa

The fan is to be built first as a scaled-down model (index “Mod”) and tested on a laboratory test bench with air of density ρ = 1.20 kg/m3. The main parameters of the model are: Impeller outer diameter Dtip,Mod 0.305 m

Shaft speed nMod 3000 1/min

a) Determine the values of volume flow rate, total pressure rise, shaft power and shaft torque to be expected in the model test. b) By how many percentage points is the maximum efficiency likely to differ between the model and the full-scale version?

References 1. VDI 2044: Abnahme- und Leistungstests an Ventilatoren – VDI-Ventilatorregeln (Acceptance and performance test of fans – VDI Code of Practice for Fans), Nov. 2018 2. ISO 5801:2017(E): Fans – Performance testing using standardized airways, 3rd edition 2017-09 3. Idelchick, I.E.: Handbook of Hydraulic Resistance, CRC Press 1994 4. Spurk, J.H.: Dimensionsanalyse. Springer-Verlag, Berlin-Heidelberg, 1992

References

13

5. Pelz, P.F., Stonjek, S, Matyschok, B.: The influence of Reynolds number and roughness on the efficiency of axial and centrifugal fans – a physically based scaling method. Fan2012, Senlis (Frankreich), 2012 6. Heß, M., Pelz, P.F.: On reliable performance prediction of axial turbomachines. ASME Paper GT2010-22290, 2010 7. Cordier, O.: Ähnlichkeitsbedingungen für Strömungsmaschinen, BWK, 5(10),1953, pp. 337–340. 8. Fister, W.: Fluidenergiemaschinen, Band 1: Physikalische Voraussetzungen, Kenngrößen, Elementarstufen der Strömungs- und Verdrängermaschinen. Springer-Verlag, Berlin-Heidelberg, 1984 9. Roth, H. W.: Optimierung von Trommelläuferventilatoren. Strömungsmechanik und Strömungsmaschinen – Mitteilungen des Instituts für Strömungslehre und Strömungsmaschinen, 29/81, Universität Karlsruhe (TH), 1981 10. Mode, F.: Ventilatoranlagen. 4. Auflage, Verlag Walter de Gryter, 1972

2

Blade Performance Parameters, Cascades of Blades, Kinematics, Losses and Efficiencies

The blade performance parameters are the pillars on which the design of any fan relays. The external moments on the fluid temporarily occupying the bladed passage of the impeller equals the rate at which the angular momentum of the fluid is changing. Here, the flow velocities at the inlet and exit of the bladed passage come into play. The blade performance parameters, however, are not readily measurable. The final fan performance is always lower due to various losses which are also dependent on characteristic flow velocities in the impeller and peripheral components such as the casing, guide vanes, etc. The purpose of this chapter is to make the reader familiar with the fundamental flow kinematics of radial and axial blade cascades and its link to the transfer of energy and frictional losses. The simple cascade of radial blades is treated first. Then the concept of segmentation of axial impellers into elemental blade cascades is explained, vital for the analysis and design of axial fans. For practical purposes losses are frequently expressed in terms of efficiencies. The chapter provides a number of efficiencies, each assessing the quality of a certain part of the fan. Ultimately, the product of all efficiencies will yield the total efficiency of the entire fan.

2.1

Blade Work, Blade Volume Flow Rate

Work is done to the fluid only by the rotating blades of a fan impeller. The blades perform a specific blade work on the fluid or – equivalently – a blade total pressure rise Y t,B or Δpt,B , respectively: Similarly, the mass or volume flow rate through all blade channels is

# Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 T. Carolus, Fans, https://doi.org/10.1007/978-3-658-37959-9_2

15

16

2 Blade Performance Parameters, Cascades of Blades, Kinematics, Losses and . . .

m_ B or V_ B , respectively: These are the design parameters for any targeted blade performance of axial and centrifugal blades.

2.2

Flow Kinematics (Velocity Triangles)

2.2.1

Cascade of Radial Blades

The centrifugal impeller in Fig. 2.1 comprises a row of blades which form a radial blade cascade with the cylindrical inlet and exit areas A1=2 = 2πr 1=2 b1=2 :

ð2:1Þ

Figure 2.1 on the left shows an example of the temporally averaged velocity field at A2 of a rotating centrifugal impeller as measured with a three-dimensional hot-film probe, WOLFRAM [1]. Obviously the velocity vectors of the fluid particles vary in magnitude and direction with location. Spatial averaging of flow velocities at A1 and A2 results in vectors of magnitude w1 and w2 which are independent of the location and hence representative, Fig. 2.1 right. Ultimately, this allows applying the one-dimensional streamline theory from inlet to exit. Spatio-temporal averages of flow quantities are used frequently throughout this text book unless aeroacoustic problems are treated. → Of interest are the velocity vector w as seen by an observer in the rotating frame of → → reference and c as seen by another observer in the stationary laboratory system. c is → → → termed the absolute, w the relative flow velocity. c is linked to w via the vector equation →

→

→

c = u þ w,

ð2:2Þ

Fig. 2.1 Left: Measured and temporally averaged flow velocity at the exit of a centrifugal impeller, right: Representative velocities at the inlet and exit by spatio-temporal averaging; from Wolfram [1]

2.2

Flow Kinematics (Velocity Triangles)

17

Fig. 2.2 Velocity triangle

Fig. 2.3 Left: Velocity triangles at the inlet and exit of a radial blade cascade, right: circular projection and meridional section of a centrifugal impeller →

where u is the circumferential velocity of the impeller at the radius under consideration. → → → c = u þ w can be visualized as a velocity triangle as in Fig. 2.2. Of particular importance → are the tangential (swirl) component cu and the meridional component cm of c as well as → → 1 the angle β between w and the circumferential velocity u : Figure 2.3 depicts a radial blade cascade with its inlet and exit velocity triangle. Due to the angular velocity of the impeller Ω = 2πn

ð2:3Þ

the circumferential velocities of the blade inlet and exit, referred to 1 and 2, respectively, become

Note that throughout this book β and α are measured from the circumferential direction. This is different from many Anglo-Saxon text books where frequently these angles are measured from the axial direction.

1

18

2 Blade Performance Parameters, Cascades of Blades, Kinematics, Losses and . . .

u1=2 = r 1=2 Ω = r 1=2 2πn:

ð2:4Þ

The absolute flow velocities are calculated as c1=2 =

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cm1=2 2 þ cu1=2 2

ð2:5Þ

with cm being the meridional and cu the circumferential (swirl) component. The relative flow velocities then become w1=2 =

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 cm1=2 2 þ u1=2 - cu1=2 :

ð2:6Þ

The flow angles according to Fig. 2.2 or Fig. 2.3 are  β1=2 = arctan

cm1=2 u1=2 - cu1=2

 ð2:7Þ

and   cm1=2 α1=2 = arctan : cu1=2

ð2:8Þ

An important special case is when the inflow is free of any swirl, i.e. cu1 = 0 and α1 = 90°.2 It is of utmost importance that cm and cu are related to blade volume flow rate and blade work via cm1=2 =

V_ B 2πr 1=2 b1=2

ð2:9Þ

and Y t,B = u2 cu2 - u1 cu1



 or Δpt,B = ρðu2 cu2 - u1 cu1 Þ :

ð2:10Þ

If it were true that the fluid follows the contour of the blade exactly, one would choose the flow angle β2 as the “metal” blade exit angle. Moreover, for the condition of “impact-free entry”, where the front stagnation point is located at the leading edge, one would choose β1 as the metal blade inlet angle. However, this thought model of a blade-congruent is merely a didactic simplification to understand the principle of turbomachinery. The design of blading, that yields the targeted blade performance parameters with minimal losses, require methods as for instance described in Chaps. 3 and 4. 2

2.2

Flow Kinematics (Velocity Triangles)

19

Equation (2.10) is referred to as the famous turbomachinery equation by Leonhard Euler and – in essence – describes the conservation of angular momentum. For a modern derivation see e.g. Zierep and Bühler [2].

2.2.2

Cascade of Axial Blades

In an axial impeller the representative velocities at the inlet and exit plane vary from hub to tip, since the circumferential velocity of the blades increases with radius r. Therefore, it is convenient to segment the impeller into coaxial strips, each with a small radial extension δr for each of which all velocities are assumed to be independent of r, Fig. 2.4. Each coaxial strip comprises a row of blade elements (index “BE”) which form an elemental blade cascade (index “CA”) with the small annular inlet and exit area δA = 2πrδr:

ð2:11Þ

In contrast to the radial blade cascade, the circumferential velocities and the meridional components of the absolute velocities at the inlet and exit of an elemental axial blade cascade are identical (Fig. 2.5): u1 = u2 = u = rΩ = r2πn

ð2:12Þ

cm1 = cm2 = cm :

ð2:13Þ

The flow angles α1/2 und β1/2 are calculated as in the previous section for the radial blade cascade. Of relevance is also the vectorial mean velocity of the relative flow velocities at inlet and exit w1 and w2

Fig. 2.4 Left: Axial impeller segmented into coaxial strips and hence elemental blade cascades, right: Representative (spatio-temporally averaged) velocities at the entrance and exit of a plane elemental blade cascade

20

2 Blade Performance Parameters, Cascades of Blades, Kinematics, Losses and . . .

Fig. 2.5 Velocity triangles at the inlet and exit of an elemental axial blade cascade; the lower vector diagram illustrates the definition of w1 as the vectorial mean of w1 and w2

  1 → → w1 þ w2 2

ð2:14Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 u þ w22 - c2m þ 4cm 2

ð2:15Þ

→

w1 =

with its magnitude 1 w1 = 2

and the angle measured from the circumferential direction, i.e. the impeller plane, ! 2cm pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : β1 = arctan u þ w22 - cm 2

ð2:16Þ

The mass and volume flow rate through the elemental blade cascade is δm_ CA and δV_ CA , respectively. Hence, cm in Eq. (2.16) is linked to δV_ CA via

2.3

Losses and Efficiencies

21

cm =

δV_ CA : 2πrδr

ð2:17Þ

Furthermore, each elemental blade cascade performs a specific work on the fluid according to EULER’s turbomachinery equation   Y t,CA = uðcu2 - cu1 Þ or Δpt,CA = ρuðcu2 - cu1 Þ :

ð2:18Þ

Finally, a summation over all elemental blade cascades from hub to tip yields the blade performance parameters Y t,B =

1 X Y t,CA δV_ CA _V B all CA

V_ B =

X

δV_ CA :

ð2:19Þ ð2:20Þ

all CA

2.3

Losses and Efficiencies

The performance parameters of the complete fan Yt and m_ or Δpt and V_ are obtained by subtracting various losses in the impeller and downstream casing or guide vanes from the blade performance parameters Yt,B and m_ B or Δpt,B and V_ B . For the design of a new fan, which is to have a certain performance, in principle two procedures are conceivable: • By estimating all losses a priori, one can extrapolate from Yt and m_ to Yt,B and m_ B and design the blades accordingly. • For an initial set of Yt,B and m_ B the blades are designed, then the losses are determined, and finally it is checked whether the targeted design values Yt and m_ of the complete fan have been achieved. The designer’s choice of several geometrical parameters determine the values of flow velocities in the machine which have a direct effect on the losses.

2.3.1

Losses in the Impeller and Efficiencies

Figure 2.6 illustrates the mass and energy fluxes in a centrifugal impeller by means of an equivalent circuit diagram and a schematic Sankey diagram.

22

2 Blade Performance Parameters, Cascades of Blades, Kinematics, Losses and . . .

Fig. 2.6 Exemplary equivalent circuit diagram for mass and energy fluxes in a centrifugal impeller (left) and schematic Sankey diagram (right); powers which is independent of the fluid mass flow rate _ is designated with P, otherwise with mY Fig. 2.7 Leakages through gaps of fan assemblies

The shaft power PS is fed into the impeller via the shaft. Due to mechanical losses Ploss,mech in bearings and seals, however, only the impeller power PImp is available to drive the impeller. The torque due to fluid skin friction on other surfaces than the blading - for instance the disk where the blades are mounted - decreases the useful power further by Ploss,Sf. This remaining power3 is available as blade power m_ B Y t,B , which subsequently is reduced by the hydraulic (frictional) losses in the blade channels m_ B Y loss,B : The power in the fluid at the immediate exit of the impeller is m_ B Y t,Imp . In addition, the shaft bearing and sealing gaps of an impeller/casing assembly - in this book always assigned to the impeller - cause volumetric losses resulting in a net mass flow rate through the complete impeller m_ = m_ B - m_ Leak < m_ B , Fig. 2.7. Hence, including the unavoidable leakage the fluid power at the exit of the impeller eventually becomes _ t,Imp . Each partial loss corresponds to an efficiency according to Table 2.1. mY 3

This step of power conversion is given by Euler’s turbomachinery equation, Eq. (2.10).

2.3

Losses and Efficiencies

23

Table 2.1 Impeller efficiencies Efficiency Mechanical (“m”) Skin friction (“Sf”) Hydraulic (“h”) Inner (“i”)

Volumetric (“vol”) Total of the entire impeller (“Imp”)

2.3.2

assesses the... ... mechanical quality of the bearing ... torque due to skin friction on other surfaces than the blading ... hydraulic quality of the blading ... hydraulic quality of the entire impeller ... leakage through gaps ... quality of the entire impeller including bearings and sealing gaps

Definition ηm =

PS - Ploss,mech PS

=

PImp PS

ηSf =

PImp - Ploss,Sf PImp

=

_ B Y t,B m PImp

ηh =

m_ B Y t,Imp m_ B Y t,B

ηi =

m_ B Y t,Imp PImp

= 

ηvol =

mY _ t,Imp m_ B Y t,Imp

ηImp =

_ t,Imp mY PS



=

m_ B ðY t,B - Y loss,B Þ _ B Y t,B m

=

V_ B Δpt,Imp PImp

=

m_ m_ B



(2.21)

=



=

_ t,Imp VΔp PS

V_ B Δpt,B PImp

=

Y t,Imp Y t,B





= ηSf ηh = ηi V_ V_ B

(2.22)

=



Δpt,Imp Δpt,B



(2.23) (2.24)



= ηm ηSf ηh ηvol = ηm ηi ηvol

(2.25) 

(2.26)

Losses in the Casing and Guide Vanes, Efficiencies

Further losses occur in an optional casing and in guide vanes due to friction and secondary flows. They are summarized in terms of the casing efficiency in Eq. (2.27) of Table 2.2. Eventually, the efficiency of the complete fan is calculated according to Eq. (1.6a). In the following paragraphs these losses are discussed in more detail. Guide vanes and diffuser losses in an axial fan Fig. 2.8 depicts a typical axial fan with outlet guide vanes and tail cone diffuser. Here, guide vanes and diffuser are counted as part of the fan.4 In general, the determination of any pressure loss ρ Δploss = ζ c2 2 requires a pressure loss coefficient ζ and a reference speed c. For the pressure loss in outlet guide vanes the relevant reference velocity is

4

For inlet guide vanes and a standard diffuser the approach is similar.

24

2 Blade Performance Parameters, Cascades of Blades, Kinematics, Losses and . . .

Table 2.2 Efficiency of casing and complete fan Efficiency Casing (“Cas”) Total of the complete fan (no index!)

assesses the... ... overall hydraulic quality of the casing ... quality of the complete fan

Definition ηCas = η=

_ t mY PS

_ t,Imp - mY _ loss,Cas mY _ t,Imp mY



=

_ t VΔp PS

=

Yt Y t,Imp

= ηImp ηCas





=

Δpt Δpt,Imp



(2.27) (1.6a)

Fig. 2.8 Axial fan consisting of the impeller, outlet guide vanes and tail cone diffuser; if the cone is omitted, the sudden increase of through-flow area immediately downstream of the guide vanes forms a Carnot diffuser

c2 =

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2u2 þ c2m2 :

Since cu2 and cm2 are generally functions of the radius, the dynamic pressure needs to be averaged.5 The idea behind an energetic mean as applied here is that the power from a mean quantity results in the actual power: ! 1 2 c m_ B = 2

Z  A

 1 2 c ρcm dA 2

If, as is always the case here, ρ is constant, one immediately obtains c2 =

1 V_ B

Z c2 cm dA: A

Thus, the pressure loss in the outlet guide vanes becomes

5

For the problem of averaging inhomogeneous flow fields, see, e.g. [3–5].

ð2:28Þ

2.3

Losses and Efficiencies

25

  ρ ρ Δploss,Guide vanes = ζ Guide vanes c22 = ζ Guide vanes c2m2 þ c2u2 2 2 ! c2u2 ρ 2 = ζ Guide vanes 1 þ cm2 : c2 2

ð2:29Þ

m2

The pressure loss in the diffuser is ρ ρ Δploss,Diffuser = ζ Diffuser c23 = ζ Diffuser c2m2 : 2 2

ð2:30Þ

Which individual parameters determine the pressure loss coefficient ζ Diffuser, is explained in Sect. 10.1.3. Thus, the casing efficiency for the axial fan according to Fig. 2.8 can be expressed as ηCas = 1 -

 

c2 ζ Guide vanes 1 þ 2u2 þ ζ Diffuser ρ2 c2m2 cm2

Δpt,Imp

:

ð2:31Þ

Here, it becomes evident that the individual casing losses depend on flow velocities and their radial distribution. These parameters can only be determined within the impeller design process. Therefore, the casing losses are estimated upon the aerodynamic design of the impeller. Volute casing losses in a centrifugal fan In the volute casing, Fig. 2.9, at least two loss mechanisms can be identified: Firstly, the abrupt change of area from the impeller exit with its inside width b2 to the volute casing with B > b2 results in a loss due to the sudden expansion, a typical discharge loss. The second loss mechanism is the internal and skin friction the fluid experiences in the volute itself. The discharge loss is determined by a fraction of the dynamic pressure associated with the meridional velocity cm2: ρ Δploss,Discharge = ζ Discharge c2m2 2

ð2:32Þ

The frictional losses in the volute casing, on the other hand, are based on the volute through-flow velocity

26

2 Blade Performance Parameters, Cascades of Blades, Kinematics, Losses and . . .

Fig. 2.9 Centrifugal impeller in volute casing with width B > b2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2ffi b 2 , c3 = c2u2 þ c B m2 and hence become Δploss,Volute = ζ Volute

c2u2 c2m2

 2 ! b ρ 2 þ 2 c : 2 m2 B

ð2:33Þ

Eventually, the centrifugal volute casing efficiency for the radial fan according to Fig. 2.9 is obtained as ηCas = 1 -

 ζ Discharge þ ζ Volute

c2u2

c2m2

Δpt,Imp

þ

b 2 2

B



ρ 2 2 cm2

:

ð2:34Þ

Again, these casing losses depend on flow velocities that are a consequence of the impeller and casing design. Note that for convenience the bars denoting a spatio-temporal mean are often omitted.

References 1. Wolfram, D.: Experimentelle Untersuchung des stationären und instationären Stromfeldes am Austritt eines freilaufenden Radialventilatorrades. Bericht Nr. F104/1B des Instituts für Fluidund Thermodynamik der Universität Siegen; also final report of project L 204 of the Forschungsgemeinschaft Luft- und Trocknungstechnik e.V. (FLT), Frankfurt, 2005

References

27

2. Zierep, J., Bühler, K.: Grundzüge der Strömungslehre. Verlag Springer-Vieweg, 2018 3. Traupel, W.: Thermische Turbomaschinen. Erster Band, Springer-Verlag, Berlin-Heidelberg, 3. Auflage, 1977 4. VDI 4675, Blatt/Part 1: Bilanzgerechte Mittelung inhomogener Strömungsfelder – Einführung (Balance-based averaging of inhomogeneous flow fields – Introduction). Sept. 2012 5. VDI 4675, Blatt/Part 2: Bilanzgerechte Mittelung inhomogener Strömungsfelder – Anwendungen (Balance-based averaging of inhomogeneous flow fields – Applications). April 2019

3

Design of Centrifugal Fans

The aim of any blading design method is to determine a geometry which provides the change of the fluid velocity as quantified by the velocity triangles. Moreover, this change of velocity in the blading passage ought to be achieved with minimum losses. The velocity triangles are known from a previous calculation step. As described in Chap. 2, they are a function of the targeted blade performance parameters, the shaft speed and a few fundamental geometrical parameters like impeller diameter, width, etc. The nomenclature of the centrifugal impeller is illustrated in Fig. 3.1. This chapter summarizes Pfleiderer’s classic theory that provides the blade inlet and exit angles βB1 and βB2. One has to keep in mind that several other geometrical parameters, for instance the number of blades and hence the blade spacing, are not provided by this theory. They have to be set with the help of empirical correlations. The method as presented here is limited to single-curved, non-airfoil shaped blades (circular arc or logarithmic spiral blades). Those simple shapes are frequently used in centrifugal fan design. The design of more complex 3D shaped blades can be found, for example, in Pfleiderer [1]. Likewise, only the algorithm for designing a simple spiral casing with parallel side walls is presented.

3.1

Blade Design

3.1.1

Slip of Flow

Initially, one assumes a flow field according to Fig. 3.2 with the features: • The relative flow in the blade channel is perfectly guided by the blades, i.e. the flow angles coincides with the geometric blade angles, e.g. β2 = βB2; such an ideal blade# Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 T. Carolus, Fans, https://doi.org/10.1007/978-3-658-37959-9_3

29

30

3

Design of Centrifugal Fans

D1/2 = blade leading/trailing edge diameter b1/2 = blade width t1/2 = blade spacing βB1/2 = blade inlet/exit angle s = thickness of blade

Fig. 3.1 Nomenclature of the centrifugal impeller

Fig. 3.2 Blade-congruent ideal relative velocity in the blade channel of a radial blade cascade

congruent flow field can be imagined as being produced by an infinite number of infinitesimally thin blades (index “1 “). • The ideal velocity field is assumed to be one-dimensional as depicted in Fig. 2.1 right. • The ideal pressure field is rotationally symmetric as the velocity field, thus there is no difference of pressure between front- and back-side of the blades. In a second step, we take into account that a real impeller is made of a limited number of blades. Due to the space in between the blades, the real relative flow receives less guidance from the blades, it “slips”. Moreover, the thickness of the blades partly blocks the throughflow area. As a consequence, the relative flow in the real blade channel shows considerable deviations from the ideal. In particular, one finds: • The velocity is non-uniform across the blade channel. The vectors transverse to the main flow direction vary significantly in terms of magnitude and direction (Fig. 3.3). A velocity profile is established in the circumferential direction (recognizable in Fig. 3.4 from the different line spacing of the spark curtains). • The flow angle is smaller than the geometric blade angle, e.g. at the channel outlet δ  βB2 – β2 > 0 (Fig. 3.4). The deflection of the real flow is less as compared to the ideal.

3.1

Blade Design

31

Fig. 3.3 Relative flow in the blade channel of a centrifugal impeller (result of a CFD (CFD = computational fluid dynamics; here a numerical flow field simulation with a threedimensional Navier-Stokes method) simulation)

Fig. 3.4 Relative flow in the blade channel of an centrifugal compressor impeller; visualization utilizing spark curtains; courtesy of [2]

Fig. 3.5 Real relative flow field in a radial blade cascade thought as superposition of the plain through-flow and the relative channel vortex

Pfleiderer [1] explains the reduced deflection with the superposition of two elementary flow fields, the simple through-flow and the relative channel vortex (Fig. 3.5). The relative channel vortex occurs because the rotation of the impeller is not transferred to the fluid, i.e. for an observer in the relative system the channel vortex rotates against the direction of the impeller rotation. This holds true even under frictionless conditions. If the flow in the blade channel is assumed to be isoenergetic (i.e. the total pressure transverse to the direction of flow is constant), a blade pressure and suction side develops: On the blade side with the higher relative velocity, the static pressure is lower than on the opposite side with lower velocity.

32

3

Design of Centrifugal Fans

Fig. 3.6 Exit velocity triangle for ideal blade-congruent (index “1 “) and real flow in a centrifugal impeller

Figure 3.6 shows the real (no index) and the ideal (index “1 “) velocity triangles at the impeller exit. The blades are assumed to be thin. The real circumferential component of c2 is smaller than the ideal, i.e. cu2 < cu21. Since the specific blade work Yt,B depends on cu2, the specific blade work is reduced by ΔY  Y t,B1 - Y t,B :

ð3:1Þ

The reduction of blade work is not a loss in the energetic sense. Slip is basically an inviscid phenomenon and reduces the shaft power as well. That means that the efficiency of the fan is mainly unaffected by the slip. When designing a fan, it is essential to compensate for the slip by exaggerating the blade exit angle. Otherwise the targeted pressure rise will not be met. In the case of thin blades, blockage of the through-flow passage does not exist. Then the blade exit angle is to be set to βB2 = β21 :

ð3:2Þ

The determination of β21 from the design performance parameters is described in the next sections.

3.1.2

The Slip Factor

The difference between the ideal and the real blade work is expressed in the form of the slip factor μ

Y t,B : Y t,B1

ð3:3Þ

μ is always less than 1. Occasionally, an alternative definition of the slip factor is used:

3.1

Blade Design

33

Table 3.1 The empirical constant ψ’ in Pfleiderer’s Eqs. (3.5 and 3.6) as a function of the downstream component of the fan assembly [1]; βB2 in degrees; valid for blades of constant thickness and standard number of blades (see Sect. 3.1.4) ψ’

Downstream component Outlet guide vanes (bladed plate diffuser)


 βB2 0:6 1 þ 60 °


 βB2 ð0:65 to 0:85Þ 1 þ 60 °
 βB2 ð0:85 to 1:0Þ 1 þ 60 °

Volute casing Plate diffuser without vanes

σ

u2 ‐csl c = 1‐ sl ð < 1Þ u2 u2

(3.7) (3.8) (3.9)

ð3:4Þ

with the slip velocity csl as in Fig. 3.6. σ can be converted into μ. Numerous efforts have been made to obtain accurate predictions of the slip factor. Pfleiderer [1] was among the first to predict the slip factor as a function of only a few parameters.1 For the centrifugal fan with single-curved blades and D1/D2 > 0.5 0 2ψ 0

1-1

A μ = @1 þ
z 1 - ðD1 =D2 Þ2

:

ð3:5Þ

z is the number of blades and D1/D2 the diameter ratio, both fixed in a prior step. The empirical constant ψ ′ strongly depends on the actual blade angle βB2 and to some extent on the type of downstream component such as casing or diffuser, Table 3.1 and Fig. 3.7. ψ’ must not be confused with the non-dimensional pressure rise coefficient ψ. Eq. (3.5) in combination with (3.7) to (3.9) reflects exactly the expectations: For large numbers of blades, the slip factor approaches 1 because of the nearly perfect guidance of the fluid, whereas large blade angles βB2, i.e. large targeted deflection, cause small values of μ. If the blades are extended towards the center in such a way that D1/D2 ≤ 0.5, Pfleiderer recommends  -1 8 ψ0 μ= 1 þ , 3 z

ð3:6Þ

independent of the value of D1/D2.

1

The method of blade design employing slip factors is called Pfleiderer’s method. Deviating from here, Pfleiderer in [1] originally formulated the slip factor as (2ψ ′)/(z(1 - (D1/D2)2)).

34

3

Design of Centrifugal Fans

Fig. 3.7 Top: Volute casing, bottom: Downstream bladed plate diffusers, parallel-walled or diverging

Empirical investigations of centrifugal fan of centrifugal fan impellers with backward curved blades in a volute casing seemed to confirm the validity of Eqs. (3.5) and (3.6) (Bommes [3]). Eck [4] also developed semi-empirical correlations, which are compiled in Table 3.2. Other similar approaches such as by Stodola (1924), Busemann (1924) and many others are compiled by Wiesner in [5] and Bommes et al. in [6].

3.1.3

Selection of the Number of Blades

A simple theory-based procedure for identifying the most appropriate number of blades for centrifugal impellers does not exist. In general, more blades reduce the slip but at the same Table 3.2 Slip factors according to Eck [4]; βB2 in degrees Applicable, if cu1 = 0 (no inlet swirl) walls of shroud and backplate parallel (b1 = b2) βB2 ≤ 45° cu1 = 0 (no inlet swirl) walls of shroud and backplate converging (b2 < b1) for cm(r) = const. 45° < βB2 ≤ 90°

μ  1þz

β

1:4þ2:790B2° ð1 - ðD1 =D2 Þ2 Þ

 1 þ 2z

-1

π sin βB2 ð1 - ðD1 =D2 Þ2 Þ

-1

(3.10)

(3.11)

3.1

Blade Design

35

time increase the wetted surfaces which might results in an increase of friction losses. The optimum of these competing factors must be sought empirically utilizing experimental or CFD-methods. Nevertheless, three common first guesses are: • According to Eck [4] for βB2 ≤ 90° (i.e. backward-curved blades) z = 10

sin βB2 1 1- D D2

ð3:12aÞ

• According to Bommes [3], following Pfleiderer, for βB2 ≤ 90° (i.e. backward- curved blades) " z = 5 to 8

1 1þD D2

1-

D1 D2

# sin ½0:5ðβB1 þ βB2 Þ ,

ð3:12bÞ

where the factor 5 to 8 is recommended for types of fans with specific speed in the range of σ opt = 0.63 to 0.19, respectively. In other words, the recommended number of blades decreases as σ opt increases. • According to Roth [7] for βB2 > 90° (i.e. forward-curved blades) z = 40

3.1.4

ð3:12cÞ

Blockage of the Inlet and Outlet Due to the Finite Thickness of the Blade

So far, details of the blade design have not been included. One refinement of the design method is to include the blockage effect due to the finite thickness of the blades. Thick blades reduce the cross-sectional are of the internal flow passages. Figure 3.8 shows a section through an impeller with blades of constant thickness s. At the inlet of the blade channel the cross-sectional area, obstructed by a single blade, is su1 =

s : sin βB1

ð3:13Þ

In this equation the impeller width is omitted for convenience. The spacing between two blades is

36

3

Design of Centrifugal Fans

Fig. 3.8 Centrifugal impeller with blades of constant thickness s

t1 =

πD1 : z

ð3:14Þ

The meridional velocity immediately upstream of the blade channel entrance is denoted by cm1 and just inside the blade channel by c0m1 . Then, with the assumption ρ = const., the continuity equation yields cm1 t 1 = c0m1 ðt 1 - su1 Þ, i.e. c0m1 = cm1

t1 : t 1 - su1

ð3:15Þ

The circumferential velocity immediately upstream of the blade channel inlet and just inside the blade channel is identical: u01 = u1; further assuming swirl-free inflow c0u1 = cu1 = 0), one readily can draw the velocity triangles relevant for both locations, right upstream of and right inside the blade channel, Fig. 3.9 left. It becomes clear that the obstruction due to the finite blade thickness • provides an acceleration of the meridional flow velocity from cm1 to c0m1 in the blading leading edge region; • as a consequence, the inlet angle of the inflow immediately in the bladed inlet region is increased by

3.1

Blade Design

37

Fig. 3.9 Inlet and outlet velocity triangles in the vicinity of the blade leading and trailing edges of thin and thick blades

Δ1 = β01 - β1 = arctan



 c0m1 ‐β1 : u1

ð3:16Þ

From this, one can derive the rule: For tangential inflow to the blade leading edge, the inlet angle of the thick blade (marked by the index “with”) should be chosen as βB1,with = βB1,w=o þ Δ1

ð3:17Þ

(the index “w/o” meaning without blockage and indicates the theoretical limit of infinitely thin blades). By analogy, at the exit of the blade channel c0m2 = cm2

t2 t 2 - su2

ð3:18Þ

holds true. c0m2 is the meridional velocity in the blading trailing edge region and cm2 immediately downstream of the blading. With u02 = u2 and c0u2 = cu2 the exit velocity triangles in Fig. 3.9 right show that the obstruction by the finite blade thickness provides for • a deceleration of the meridional velocity from c0m2 to cm2 and thus for a • reduction of the downstream flow angle by Δ2 = β02 - β2 = arctan



 c0m2 ‐β2 : u2 - cu2

ð3:19Þ

Hence, to achieve the desired flow deflection, the blade exit angle for the thick blades ought to be exaggerated by Δ2:

38

3

Design of Centrifugal Fans

βB,with = βB2,w=o þ Δ2

ð3:20Þ

Some designers sharpen the blade towards the trailing edge and achieve a more gradual reduction from c0m2 to cm2.

3.1.5

Summary: Blade Design for Centrifugal Impellers

Figure 3.10 summarizes the blading design with the help of a flow chart. It becomes obvious that some of the quantities sought require some simple iterations.

3.1.6

Further Empirical Geometry Parameters of the Centrifugal Impeller

Geometrical parameters not considered so far have to be determined empirically. Some hints from the literature are summarized in this section. However, as with the slip factor and the number of blades, it is always advisable to include one’s own data base if available. Blade angle, diameter and width ratio, gap geometry For the centrifugal impeller with backward curved blades, Bommes recommends in [8] (see also [6]) setting the diameter ratio of the impeller as a function of the type and hence non-dimensional coefficients as D1 0:5 = 1:27 ψ t,opt σ opt0:83 : D2

ð3:21Þ

In order to avoid flow separation when the flow is deflected from the axial direction in the suction mouth into the radial direction of the blade channels, the deceleration must be limited; for this purpose, the area ratio A1/AS (Fig. 3.11 left) can be dimensioned according to Bommes [3] and [8] as A1 πD1 b1 1=6 = π 2 = 2:16 σ opt : AS D 4 S

ð3:22aÞ

Often it is advantageous to choose DS = D1, so that b1 1=6 = 0:54 σ opt : D1

ð3:22bÞ

This assumes a contour of the fixed inlet nozzle, the gap geometry and the contour of the shroud as shown in Fig. 3.11 right and Table 3.4.

3.1

Blade Design

39

Fig. 3.10 Flow chart: Design of single-curved blades for the centrifugal impeller

The size of the outlet area A2 controls the deceleration of the relative flow from the blade inlet to exit w2/w1. Having already fixed D2, the exit width b2 of the impeller can be readily derived from the recommendations by Bommes [3]: cm2/cm1 ≥ 0.6 (for σ opt < 0.25), cm2/cm1 ≥ 0.8 (for σ opt > 0.50) In Table 3.3, some further geometrical parameters for the centrifugal impeller with backward-curved blades are compiled. They origin from various sources in the open literature.

40

3

Design of Centrifugal Fans

Fig. 3.11 Further geometry parameters of the centrifugal impeller with backward curved blades and the inlet nozzle, according to Bommes [3], see also Table 3.4

Table 3.3 Empirical geometrical parameters of the centrifugal impeller with backward-curved blades βB1 ≤ 35°

30° to 40°b ≥ 45°c

βB2

D1/D2 pffiffiffiffiffiffiffiffi ≥ 1:194 3 φopt

b1/D1

0:5 σ 0:83 1:27 ψ t,opt t,opt opt

0:54 σ opt

= βB1 for βB1 > 25° = βB2 + 10° else = βB1 = βB1

cm2/cm1a 1=6

≥ 0.6 for σ opt < 0.25 ≥ 0.8 for σ opt < 0.50

Reference Eck [4] Bommes [8] Bommes [3]

Pohl [10] Bommes [3]

For determining b2 Also for conveying gases with abrasive solid particles c When conveying gases with a high dust content to avoid dust deposits on the blade surface a

b

For the impeller with forward-curved blades empiric parameters according to Tables 3.4 and 3.5 have proved successful. The large blade inlet angle of 80° is surprising, which by no means corresponds to an impact-free tangential inflow to the blades. Attempts to design this blade angle for tangential inflow or to round the blade leading edges in the manner of airfoils frequently failed to improve the aerodynamic characteristics, see e.g. Haber [9]. Apparently, the strong deflection in the short blade channels of this type of blading requires a highly turbulent flow triggered by local flow separation immediately in the leading edge region. Blade shapes Once the blade inlet and exit angles have been determined, the shape of the blade must be specified. Two different strategies are common: • The choice of a geometrically simple blade contour.

3.1

Blade Design

41

Table 3.4 Empirical geometrical parameters of the impeller inlet, compare Fig. 3.11 Blade shape Backwardscurved

Suction diameter DS 1.0 D1

Nozzle radius rD 0.14 D1 0.2 to 0.3 D1 for D1/ D2 > 0.7

Forwardcurved

Axial gap overlap ls 0 to 0.08 D1

Reference Bommes [3]

Axial gap permissible up to 0.03 D2

Grundmann in [6]

Table 3.5 Empirical geometrical parameters of the centrifugal impeller with forward-curved blades βB2 170°

βB1 80°

D1 0.8 D2

b1 (= b2) 0.4 D2

Reference Roth [7]

• The design of the blade channel for e.g. a gradually constant deceleration of the flow In industrial fans frequently simple blade contours are used, for instance circular arc blades, sections of a logarithmic spiral or even plane blades, Fig. 3.12. Especially the first two are aerodynamically relatively satisfying. Occasionally, backward-curved blades are also airfoil-shaped. Table 3.6 gives an overview of the characteristics of different blade shapes, Table 3.7 of their typical application (Pohl [10]). It is noteworthy that in logarithmically shaped blades the blade angle is the same at each radius, in particular βB1 = βB2 = βB. Construction of the circular arc blade The construction steps are illustrated in Fig. 3.13. The radius of curvature of the circular arc blade is calculated as

ρ=


 1 - ðD1 =D2 Þ2

D2 4 cos βB2 -

D1 D2

cos βB1

,

ð3:23Þ

the blade length (arc length) for βB1 < βB2 "

! !# D1 sin βB2 D2 sin βB1 lS = ρ arctan - arctan : 1 2 Dρ2 - cos βB2 2 Dρ2 - D D2 cos β B1

ð3:24Þ

Splitter blades Occasionally, when the diameter D1 is small, so called splitter blades (SB) are used. Frequently, they are a shortened version of the main blades (MB) as depicted in Fig. 3.14. The primary aim is to avoid excessive blockage at the entrance but still ensure

42

3

Design of Centrifugal Fans

Fig. 3.12 Alternative blade shapes for centrifugal impellers, after Pohl [10]

Table 3.6 Design features of common blades for centrifugal fans, according to Pohl [10] Shape 1

2 3 4 5 6

Blade contour Circular arc, backward curved Circular arc, radially ending Circular arc, forward curved Logarithmic spiral Flat plate, radially upright Flat plate, inclined backward

Blade length lS see Eq. (3.22b)

βB2 Arbitrary

90°

see Eq. (3.22b) see Eq. (3.22b)

Arbitrary = βB1 (= βB)

D2 - D1 2 sin βB

= 90° (= βB1)

D2 - D1 2


arccos

D1 D2

Comment Inlet and exit angle freely selectable

cos βB1



βB1 always not tangential to inflow, inflow not impact-free

D2 sin ðβB2 - βB1 Þ 2 cos βB1

maximum flow deflection in the bladed passage. Moreover, the use of splitter blades can also save material and thus costs. Based on simulations and experiments, Basile [11] determined an optimum ratio of blade lengths l*min = lSB/lMB. His study was confined to impellers with shroud and backplate being parallel (b1 = b2) and logarithmic spiral blades (i.e. βB1 = βB2 = βB). lMB and lSB denote the real arc length and not the chord length of the blades as customary otherwise. The optimum splitter blade length is defined as the arc length of the splitter blade relative to the main blade, above which an extension does not significantly improve the

3.1

Blade Design

43

Table 3.7 Remarks on the use of different blade shapes for centrifugal fans according to Fig. 3.12, after Pohl [10] Shape 1 2 3 4 5

6

Comments Results in machines with high efficiencies; blades easy to manufacture Lower efficiencies than with 1 or 4, higher pressure rise coefficients, total pressure rise characteristic relatively steep Low efficiencies, highest pressure rise coefficients Results in machines with high efficiencies, pressure rise characteristic has a maximum; also utilized for conveying gases containing abrasive solid particles βB1 always wrong, lower values of pressure rise coefficient, lowest efficiencies, pressure rise characteristic very steep, low risk of caking, even self-cleaning effect; suitable for both directions of rotation when used without volute casing (reverse operation) In comparison to 4: Due to a larger blade exit angle always higher pressure rise, steep pressure rise characteristic; low risk of caking of dust

1. Straight line from center at EB1 + EB2 2. Find intersection with circle D1 Ÿ C 3. Chord A-B through C and B 4. Perpendicular bisector on chord A-B 5. Straight line from B at

EB2 6. Find intersection point P = center of the circular arc blade

Fig. 3.13 Steps for the construction of the circular arc blade

aerodynamic optimum parameters of the impeller. For a fixed diameter ratio D1/D2 = 0.33 and blade thickness s/D2 = 0.01, the optimum splitter blade length is shown in Fig. 3.15 as a function of the main blade number zMB, the blade angle βB, and the aspect ratio b1/D1. The example in the plot shows that for a main blade angle βB = 40° and b1/D1 = 0.40 the optimum splitter blade length is about 64% of the main blade length for six main blades (l*min = 0.64), 40% for eight and only about 28% for ten. According to [11], the increase in total pressure rise at the optimum point due to the installation of optimum splitter blades amounts to 7 to 12% for zMB = 6 and 2% for zMB = 10, compared to a design without any splitter blades. In this case, the optimal volume flow rate stays constant or even shifts to slightly higher values. Basile [11] recommends to install the splitter blade approximately in the middle between two main blades.

44

3

Design of Centrifugal Fans

Fig. 3.14 Centrifugal impeller with splitter blades (SB)

Fig. 3.15 Design diagram for splitter blades for a centrifugal impeller with D1/D2 = 0.33, s/D2 = 0.01; the rectangles are results of a numerical simulation utilizing RANS-CFD; from Basile [11]

3.2

Layout of a Volute Casing

The volute casing of centrifugal fans has the task of collecting the fluid downstream of the impeller with minimum losses and recovering static pressure from its kinetic energy. Utilization of a good volute casing usually provides fan assemblies of high efficiency. In the case of centrifugal impellers with forward-curved blades, a volute casing is even indispensable for building up the static pressure. The reason is that forward-curved blades typically have a very low degree of reaction, i.e. the static pressure rise is very small while the exit flow velocity is large.

3.2

Layout of a Volute Casing

3.2.1

45

One-Dimensional Streamline Theory

In this section the theory behind the simple volute casing with parallel side walls is described. It is based on first principles like conservation of mass and angular momentum. Secondary flows, similar to those occurring in a pipe elbow, and friction are neglected. Design methods for arbitrary cross-sectional contours of volute casings, and taking friction into account, can be found elsewhere (e.g. Pfleiderer [1], Eck [4]). In a first step, a casing is considered with the same width B as the impeller exit, B = b2, Fig. 3.16 left. Figure 3.17 illustrates that the angle of the stream line leaving the impeller - measured against the circumferential direction - is determined by the velocity components: tan αS =

cm cu

ð3:25aÞ

It is assumed that the angular momentum of the fluid particles downstream of the impeller is preserved: rcu = const: = r 2 cu2 :

ð3:26Þ

The continuity equation in radial direction (conservation of mass) gives   2πrBcm = V_ r = const: = 2πr 2 b2 cm2 :

ð3:27Þ

Combining with Eq. (3.25a) shows that αS is independent of r, i.e. the angle of the streamlines – measured against the circumferential direction – remains constant at all distances away from the impeller: tan αS =

r2 r cm2 r2 r cu2

=

cm2 = tan α2 = const: cu2

ð3:25bÞ

The tangent of the angle can also be expressed in terms of the co-ordinates φ and r as

Fig. 3.16 Centrifugal impeller in volute casing, left: B = b2, right: B > b2

46

3

Design of Centrifugal Fans

Fig. 3.17 Streamline (blue dashed) and velocity components downstream of the impeller (schematically)

tan αS =

dr : rdφ

ð3:25cÞ

Separation of the variables (tanαSdφ = dr/r) and integration results in ð tan αS Þφ = ln ðr=r 2 Þ and eventually _ _  _ r φ = r 2 e φ tan αS φ in radians :

ð3:28Þ

Hence, the fluid particles follow exactly a logarithmic spiral. If we now require that such a streamline becomes the casing wall, the contour of the volute casing is given by Eq. (3.28) with (3.25b). It is decisive that the spiral angle of the streamline α2 and hence the casing αS does depend on the impeller exit velocity components: αS = α2 = arctan

cm2 cu2

ð3:25dÞ

Because cm2  V_ and cu2  Yt, B, it becomes clear that for other than the design operating point the contour of the casing no longer matches the streamlines. This leads to increased losses at the off-design operating points, which effect the final performance characteristics of the complete fan assembly. Large values of α2 result in radially large housings. More compactness can be achieved by selecting B > b2, Fig. 3.16 right. Two strategies are conceivable:

3.2

Layout of a Volute Casing

47

• B ≠ f(φ) which leads to _ _  r φ = r 2  e φ tan αS with tan αS =

b2 cm2 : B cu2

ð3:29Þ ð3:30Þ

• Increasing of B with φ; these casings have the shape similar to a snail shell.

3.2.2

Further Empirical Geometrical Parameters of the Simple Volute Casing

The logarithmic spiral is bounded by the tongue and the upper wall of the discharge port, Fig. 3.18. Similar to the impeller, remaining geometric parameters of the casing are empirical. For example, the tongue distance sz, the tongue radius rz and the tongue angle φz can be selected according to Tables 3.8 and 3.9, whereby attention is already paid to mitigation of the flow induced noise, see Sect. 8.1.2. Frequently it is desirable to make the cross-sectional area of the discharge and intake port the same size. This can be achieved by controlling the width of the casing.

Fig. 3.18 Further geometrical parameters of the simple spiral casing

48

3

Design of Centrifugal Fans

Table 3.8 Empirical geometry parameters of the volute casing for centrifugal impellers with backward-curved blades B 2 to 4 b2 2 to 3 b2

sz 0.125 to 0.167 D2 0.125 to 0.167 D2

rz approx. 0.025 to 0.05 D2

φz 45° 55° to 65°

Reference Bommes [3] Grundmann in [6]

Table 3.9 Empirical geometry parameters of the volute casing for centrifugal fans with forwardcurved blades B 1.2 b2 1.2 to 1.5 b2

αS 5° to 6° 5° to 7°

sz 0.08 D2 0.08 D2

rz 0.05 D2 0.05 D2

3.3

Practice Problems

3.3.1

Design of a Centrifugal Fan Impeller

φz 65° 65°

Reference Roth [7] Grundmann in [6]

A centrifugal impeller with backward curved circular arc blades, backplate and shroud according to Fig. 3.1 is to be designed. The targeted design performance parameters for the impeller including a stationary inlet nozzle are specified as V_ = 0:48 m3 =s Y t,Imp = 1000 W=ðkg=sÞ: Further data given: • • • • • • • •

Impeller diameter D2 = 300 mm Intake diameter D1 = 180 mm Blade exit width b2 = 70 mm Blade inlet width b1 = 84 mm Blade thickness s = 3 mm Shaft speed n = 3000 rpm Swirl-free inflow to the impeller Only volumetric and hydraulic impeller losses are taken into account, here ηh = 83% and ηvol = 94%. • The impeller is to be designed for operation in a volute casing; losses in the volute casing, however, are not to be considered at this stage of the design.

References

3.3.2

49

Design of a Volute

For the impeller from problem 3.3.1, determine the spiral contour for a suitable simple volute casing. Assume: a) Width of the volute casing B equals the impeller width b2. b) Width of the volute casing B is three times the impeller width b2.

References 1. Pfleiderer, C.: Die Kreiselpumpen für Flüssigkeiten und Gase. Springer-Verlag, BerlinHeidelberg, 5. Auflage 1961 2. Fister, W.: Fluidenergiemaschinen. Band 2, Springer-Verlag, Berlin-Heidelberg, 1984 3. Bommes, L., Kramer, C. and 11 co-authors: Ventilatoren – mit ausgewählten Problemlösungen für den Geräte- und Anlagenbau. Bd. 292 Kontakt&Studium Maschinenbau. expert-Verlag Ehningen, 1990 4. Eck, B.: Ventilatoren. Springer-Verlag, Berlin-Heidelberg, 1972 5. Wiesner, F. J.: A review of slip factors for centrifugal impellers. Trans ASME, J. Eng. for Power (1967), pp. 558–572 6. Bommes, L., Fricke, J., Grundmann, R. (Ed.): Ventilatoren. Vulkan-Verlag, Essen, 2003 7. Roth, H. W.: Optimierung von Trommelläuferventilatoren. Strömungsmechanik und Strömungsmaschinen – Mitteilungen des Instituts für Strömungslehre und Strömungsmaschinen, 29/81, Universität Karlsruhe (TH), 1981 8. Bommes, L.: Problemlösungen bei der Gestaltung von Radialventilatoren. HLH Bd. 25 Nr. 12, 1974 9. Haber, J.: Untersuchung von Trommelläuferventilatoren mit unterschiedlichen Breitenverhältnissen und Schaufelwinkeln. Diplomarbeit (Final thesis) No. 89/13, FH Karlsruhe, 1989 10. Pohl, C.: Schaufelformen bei Industrie-Radialventilatoren. Training documents VDI-Ventilatortagung Braunschweig, 2001 11. Basile, R.: Aerodynamische Untersuchungen von Zwischenschaufeln in Laufrädern spezifisch langsamläufiger Radialventilatoren (Aerodynamic investigation of splitter blades in impellers for low-specific speed centrifugal impellers). Fortschr.-Ber. VDI Reihe 7 Nr. 424, VDI-Verlag Düsseldorf, 2002, ISBN 3-18-343407-X (also Ph.D. thesis University Siegen)

4

Design of Axial Fans

Typically, “high-pressure” axial fans, characterized by large values of their non-dimensional pressure rise coefficient ψ t,opt, require impellers with many highly cambered blades, placed closely together. It is said that the solidity of such a blade cascade is large. The opposite is true for “low-pressure” fans. The effect of solidity on the degree of interference between neighboring blades is illustrated by two exemplary maps of relative flow velocities in Fig. 4.1. In the case of a low-pressure blade cascade the velocity in the vicinity of the blade surface is affected by the neighboring blade only marginally, here relatively close to the trailing edge on the blade suction sides. By contrast, each blade in the high-pressure cascade senses the neighbor already from about midchord.1 The conclusion is that the interference of the blades needs to be taken into account for high- but may be neglected for low-pressure axial blade cascades. This is the reason for presenting in this chapter two different blade design methods. As in Chap. 3 for the centrifugal blading, the outcome of both methods is the geometry of that blading which provides the flow velocities according to the velocity triangles necessary for the targeted fan performance. Despite different approaches, both blade design methods yield a qualitatively identical result: The geometric camber of the blade must always be greater than the desired deflection of the flow. In this respect, the result is similar to Pfleiderer’s method for centrifugal fans. However, axial blading design is based more consistently on mathematical-physical models and sound empirical data. One indicator is that the number of blades required is a direct outcome, no empiricism is required. The chapter is structured as follows: First, the problem of the radius-dependent velocity distributions is addressed. As a result, the inlet and exit velocity triangles in the bladed

1 A similar effect is known from historical aircraft construction. As a consequence, due to interference, the total lift force of both wings of a biplane does not correspond to the sum of the forces of the isolated wings - it is usually lower.

# Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 T. Carolus, Fans, https://doi.org/10.1007/978-3-658-37959-9_4

51

52

4 Design of Axial Fans

Fig. 4.1 Field of relative flow velocities in a midspan section of a low- (top) and a high-pressure axial impeller (bottom); result of a numerical simulation (RANS-3D)

annulus are obtained as a function of the radial position. It is worth noting that the variation of velocity triangles from hub to tip ultimately yields the twist typical for any axial blade. Then, the blade element momentum method (BEM) for low-pressure and the Lieblein method for high-pressure blade cascades are described. In case the reader is unfamiliar with families of classical airfoil sections and the basics of airfoil aerodynamics, he is referred to Sect. 10.2. Finally, some design criteria are presented, serving to check the aerodynamic feasibility of a blade design in an early stage.

4.1

Flow Kinematics in the Axial Impeller: Radial Equilibrium

Ideally, the meridional flow through an axial fan stage operated at its design point is purely axial. All streamlines should lie on coaxial cylindrical surfaces, i.e. without inclination or curvature in- or outwards. No radial velocity component should exist. To ensure this “twodimensionality” of the flow the radial forces acting on all fluid elements passing the impeller must be in equilibrium. An analytic formulation of the radial equilibrium inside the blade channels is very difficult. However, knowledge of the flow kinematics in a plane immediately up- or downstream of the impeller or guide vanes is often sufficient for the blade design. Here, the swirling flow only in a plane downstream of the impeller and in

4.1

Flow Kinematics in the Axial Impeller: Radial Equilibrium

53

Fig. 4.2 Left: Fan stage with an exemplary element of fluid in the gap between impeller and outlet guide vanes (plane 2), right: radial forces on the element of fluid (inviscid flow)

front of the guide vanes (station 2 in Fig. 4.2 left), both placed in a duct-type housing, is analyzed. It is assumed that • the flow field is rotationally symmetric, i.e. circumferentially homogeneous in plane 2, • the flow is inviscid in the gap between impeller and guide vanes. The forces acting on a mass element of fluid dm in radial direction are depicted in Fig. 4.2 right. Due to its angular velocity Ω0 = cu2 =r it experiences a centrifugal body force. Furthermore, the static pressure exerts radial forces on the element. In order to ensure no tilting or bending of the streamline these forces must be in balance: p2 dA - ðp2 þ dp2 ÞdA þ rΩ02 dm = 0

ð4:1Þ

Replacing dm by ρdAdr yields dp2 c 2 = ρ u2 : dr r It is important to note that both p2 and cu2 are functions of radius r. The total pressure in plane 2 is

ð4:2Þ

54

4 Design of Axial Fans

 ρ ρ
pt2 = p2 þ c2 2 = p2 þ cu2 2 þ cm2 2 : 2 2 Differentiating with respect to r,   dpt2 dp2 dc dc = þ ρ cu2 u2 þ cm2 m2 , dr dr dr dr ( pt2 and cm2 are also functions of r) and replacing the gradient of the static pressure by the right hand side of Eq. (4.1) yields dpt2 c 2 dc dc = ρ u2 þ ρcu2 u2 þ ρcm2 m2 dr r dr dr |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} =ρ

cu2 dðcu2 rÞ dr r

or, after some rearrangement, 1 dpt2 cu2 dðrcu2 Þ dc þ cm2 m2 : = dr ρ dr r dr

ð4:3Þ

pt2 in this equation can be expressed in terms of the specific work done by all blades in an elemental blade cascade CA at a radial location via Y t,CA ðr Þ =

1 1 ðp - pt1 Þ ηh ρ t2

and eventually in terms of cu2 via Euler’s Eq. (2.18) for cu1 = 0: Y t,CA ðr Þ = ucu2 ðr Þ = 2πnrcu2 ðr Þ

ð4:4Þ

Assuming2 ηh ≠ f(r) and pt1 ≠ f(r) gives dðrcu2 ðr ÞÞ dpt2 : = ρηh 2πn dr dr Finally, inserting this expression into Eq. (4.3) yields the important working equation

2

Increased losses in the hub and blade tip region could be taken into account by allowing ηB = f(r).

4.1

Flow Kinematics in the Axial Impeller: Radial Equilibrium

ηh 2πn

dðrcu2 Þ cu2 dðrcu2 Þ dc = þ cm2 m2 : dr dr r dr

55

ð4:5Þ

That means, for the streamlines to lie on coaxial surfaces, Eq. (4.5) must be satisfied. A common practice is choosing a function for rcu2(r) and from this calculating the unknown function cm2(r). Eventually, this determines the velocity triangles at each radius and ultimately, in a broader sense, the twist of the blade. rcu2(r) is the radial distribution of the swirl. From rearranging Eq. (4.4) it becomes clear that rcu2(r) also corresponds to the distribution of specific work done by the elemental blade cascades

Yt ,CA (r )

rcu 2 (r )2S n .

Yt,CA (r) is called the loading distribution. Two special cases, a radius-constant (i.e. isoenergetic) and a radius-dependent loading distribution are now discussed in more detail.

4.1.1

Isoenergetic Loading Distribution

If one decides for an isoenergetic loading distribution, i.e. Yt,CA = const. = Yt,B, then Eq. (4.4) yields the required swirl distribution rcu2 =

Y t,B = const:ð bÞ: 2πn

ð4:6Þ

The constant is abbreviated b in view of a later generalization and is fixed by the choice of Yt,B at a given rotational speed n. With Eq. (4.5) follows immediately that cm2dcm2/dr = 0 and hence cm2 = const:

ð4:7aÞ

It is worth to note: • The distribution of the meridional velocity in the impeller exit plane 2 is radially uniform. Because of V_ B V_ B =   cm2 =  2 2 π 2 2 π r tip - r hub D D tip hub 4

ð4:7bÞ

56

4 Design of Axial Fans

Fig. 4.3 Velocity triangles in the blade tip and hub region (left) and blade twist (right, top view from tip to hub, schematically)

its value is fixed by the choice of V_ B and the through-flow area of the bladed annulus. • The isoenergetic loading distribution requires considerably larger values of cu2 in the hub region, where r is small, than in the tip region, Fig. 4.3 left. • A first estimate of the alignment of the blade sections with respect to the impeller plane is given by the direction of the vectorial mean velocity w1 according to Eq. (2.14). Figure 4.3 right illustrates the resulting blade twist. Of occasional interest are the radial distributions of the static pressure p2 and the reaction R

Δpst,CA static pressure rise = : total pressure rise CA Δpt,CA

ð4:8Þ

If the swirl distribution Eq. (4.6) is inserted into Eq. (4.2), the static pressure between hub and tip is obtained by integration: p2 ð r Þ = k 1 -

k2 , k , k = const: r2 1 2

ð4:9Þ

This means that the static pressure increases from hub to tip as depicted in Fig. 4.4. Utilizing Euler’s equation, R can be approximated in terms of velocity components as

4.1

Flow Kinematics in the Axial Impeller: Radial Equilibrium

57

Fig. 4.4 Isoenergetic loading distribution: Distribution of cu2, cm2 and p2 in plane 2 from hub to tip

R≈1-

1 cu1 þ cu2 : 2 u

ð4:10aÞ

With the additional assumption of swirl-free inflow, i.e. cu1 = 0, the radial distribution of the reaction becomes Rðr Þ = 1 -

1 cu2 ðr Þ k = 1 - 23 , k 3 = const: 2 uð r Þ r

ð4:10bÞ

The plot in Fig. 4.5 illustrates that the elemental blade cascade at the hub produces the lowest rise of static pressure, at the blade tip the highest.3 The designer’s choice of a swirl distribution rcu2 = const. is convenient since Yt,CA and cm2 are simply independent of radius. However, this swirl distribution also has some disadvantages: • It results in a strong blade twist, especially for blades with large blade height, i.e. small hub-to-tip ratios. • The flow deflection required in the hub region is comparably large; large values of cu2, with u being small at the same time, bear the risk of flow separation and hence increased losses.

The tip gap leakage flow is driven by the static pressure differential across the clearance. The choice of a blade loading distribution with reduced loading of the tip cascades may be a strategy for mitigating tip leakage losses - see the next section for arbitrary loading distributions. 3

58

4 Design of Axial Fans

Fig. 4.5 Distribution of the reaction R from hub to tip for the example of an isoenergetic loading distribution and the set boundary condition R(rtip) = 0.9

• The exit flow angle α2 which is equivalent to the entrance angle to outlet guide vanes, is also a function of r, meaning that guide vanes must be twisted as well; this might cause additional cost of manufacturing. • From an aerodynamic point of view, rcu2 = const. corresponds to a free vortex downstream of the impeller. A free vortex occurs very often in natural flows in combination with a forced vortex with solid-body rotation, forming the core; Fig. 4.6 shows a tornado and the schematic distributions of velocity and pressure. This combined vortex is called a Rankine vortex. See also the discussion of the wake downstream of an impeller hub in Sect. 4.5.2.

4.1.2

Radius-Dependent Loading Distribution

In contrast to the isoenergetic distribution, Yt,CA is now chosen to vary along the blade height. This allows the share in energy transfer of individual elemental blade cascades to be weighted and potentially the blade twist reduced. The consequence is a nonuniform

4.1

Flow Kinematics in the Axial Impeller: Radial Equilibrium

59

Fig. 4.6 Tornado as an example of a Rankine vortex (Historic photograph of a tornado on its way to the town of Vulcan, Canada, 8 July 1927; Historic National Weather Service Collection, USA)

through-flow velocity cm. As an example, a generalization of the swirl distribution according to Eq. (4.6) is rcu2 ðr Þ = ar þ b, a, b = const:,

ð4:11Þ

i.e. the loading distribution Yt,CA (r) = 2πn(ar + b). Substituting this swirl distribution into Eq. (4.5) and integrating, one obtains the meridional velocity distribution sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi ab cm2 ðr Þ = 2 ηh 2πnar - a2 log r þ þ k , a,b,k = const: r

ð4:12Þ

As a check, setting a = 0 results in the special case of radius-independent meridional velocity as before. Within the design process, the constants a, b, k must be specified such that the targeted blade performance parameters V_ B and Yt,B according to Eqs. (2.19 and 2.20) are achieved.

60

4.1.3

4 Design of Axial Fans

Summary of Swirl Distributions

Other swirl distributions are given and discussed by Horlock [1], e.g. the generalization rcu2 ðr Þ ¼ ar n þ b, a,b,n ¼ const:

ð4:13Þ

Table 4.1 gives an overview. It must be pointed out, that, in general, a designer is free to choose any other arbitrary swirl distribution, not necessarily in an analytical form as listed in this table.

4.2

Segmentation

As already pointed out in Sect. 2.2.2 the bladed annulus of the axial impeller between hub (index “hub”) and blade tip (or casing wall, index “tip”) needs to be segmented into elemental blade cascades. Each elemental blade cascade is represented by a coaxial reference section, allocated in the center of the cascade. Different strategies of segmentation are possible. A number of radially equidistant elemental blade cascades from hub to blade tip would be a simple solution. Frequently, however, it is preferred to segment in such a way that the through-flow area δA of all elemental blade cascades with the exception of the tip and hub cascades is identical. In this case, the radii at which the sections are located are easily determined by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ðj - 1Þ r 2tip - r 2hub r j = r 2hub þ , j = 1, . . . ,n n-1

ð4:14Þ

with n equal to the number of coaxial sections chosen. The hub and tip section always coincides with the hub and blade tip radius rhub and rtip, respectively. The through-flow area of each elemental blade cascade becomes then 8   2 2 > π r r > tip hub > > < , n- 1  δAj = > 2 2 > π r r > tip hub > : 0:5 , n-1 and their radial extension

j = 2, . . . , n - 1 j = 1, n

ð4:15Þ

4.3

The Blade Element Momentum (BEM) Method for Low-Pressure Axial Fans

61

Table 4.1 Overview of various analytic swirl distributions; a, b, n are constants rcu2 (r) b ar + b arn + b

Comment Frequently used; results in cm2(r) = const.; reaction R(r) increases with r, results in strongly twisted blades See ahead Generalization according to Eq. (4.11)

(4.6) (4.11) (4.13)

Fig. 4.7 Example: Segmentation of an axial impeller with n = 4 blade sections; the dash-dotted red lines indicate the reference sections allocated to each elemental blade cascade

δr j =

δAj : 2πr j

ð4:16Þ

It is advised to check whether the complete impeller has been correctly segmented. The complete annular through-flow area between hub and blade tip must be recovered by the sum of the elemental through-flow areas:  Z  X n A = dA ≈ δAj A

ð4:17Þ

j=1

The example in Fig. 4.7 illustrates the case of n = 4 blade sections: Two full elemental blade cascades with reference sections at r2 and r3 and two halves at r1 = rhub and r4 = rtip.

4.3

The Blade Element Momentum (BEM) Method for Low-Pressure Axial Fans

The method presented here was probably first applied for fans by Keller [2] in 1934. Pfleiderer [3], however, refers to an early work by Bauersfeld in 1922, who obviously proceeded similarly for his design of water turbines and propellers. Today, in the AngloSaxon literature the method is often referred to as the blade element momentum (BEM) method and also used for the design of modern axial wind turbines.

62

4 Design of Axial Fans γ= stagger angle of the blade element t = blade spacing l = chord length of the blade element l/t =σ= solidity of elemental blade cascade

Fig. 4.8 Nomenclature for the elemental blade cascade

Essentially, the result of the BEM method is the ratio of blade chord length to blade spacing l/t and the stagger angle γ of an elemental blade cascade, Fig. 4.8. l/t is termed the “solidity” σ of the blade cascade. The reciprocal ratio t/l is called “space-chord ratio”. If the solidity is known and e.g. the number of blades is chosen, the required chord length l of each blade element is fixed, or vice versa.

4.3.1

Derivation of the Key Equation

The derivation of the key equation of the BEM method starts with the analysis of the flowinduced force δFBE on one single blade element in an elemental blade cascade. This force can be expressed in two different ways: (i) Applying the principle of conservation of momentum on the elemental blade cascade, Fig. 4.9, and (ii) in terms of lift and drag of an airfoil section equivalent to the blade element, Fig. 4.10. i) Conservation of momentum Conservation of momentum states that the sum of the inertia forces and the net external forces acting on a fluid mass element is zero. External forces acting on the mass in the control volume in Fig. 4.9 are • the blade force δFBE and • shear stress forces on the control volume surfaces; they are neglected here. In this context it is sufficient to focus on the circumferential components only. Inertia _ Conservation of forces due to the flow through the blade channel are cu2 δm_ and cu1 δm. momentum immediately gives

4.3

The Blade Element Momentum (BEM) Method for Low-Pressure Axial Fans

63

Fig. 4.9 BEM: Force and its circumferential component on a blade element due to the flow through the elemental blade cascade; the height of the elemental blade cascade is δr

Fig. 4.10 BEM: Force and its components on an equivalent airfoil section; the height of the section is δr

_ jδF BE,u j = ðcu2 - cu1 Þδm:

ð4:18Þ

With δm_ = ρcm tδr and Euler’s Eq. (2.18), cu2 - cu1 = Yt, CA/u, one obtains the circumferential component of the force on the blade element jδF BE,u j =

ρY t,CA cm tδr: u

ð4:19Þ

ii) Lift and drag acting on an equivalent airfoil section The flow-induced net force on an airfoil section in Fig. 4.10 is δFBE. Its component perpendicular to the inflow velocity is the lift δL, and parallel the drag δD. For any good airfoil section δD is much smaller than δL. Hence the angle ε is small. That means that tan ε = δD/ δL ≈ ε and ε is identical to the drag-to-lift ratio of the airfoil section. Moreover, the approximation |δFBE| ≈ |δL| holds true. Figure 4.10 reveals that the circumferential component of δFBE is δF BE,u = sin ðβ1 þ εÞδF BE : Replacing δFBE by δL = CL ρ2 w21 lδr yields

ð4:20Þ

64

4 Design of Axial Fans

ρ δF BE,u = sin ðβ1 þ εÞC L w21 lδr: 2

ð4:21Þ

Differently from an isolated airfoil section in a wind tunnel, the vectorial mean of the inlet and exit velocity according to Eqs. (2.14 and 2.15) is taken as the relevant flow velocity w1.4 The inflow angle is β1 according to Eq. (2.16), and α is the angle of attack. The next step in the derivation is equating the right hand sides of Eqs. (4.19) and (4.21). Implementing the relation from the velocity triangles cm = wm = w1 sin β1, the trigonometric identity sin(β1 + ε) = cos ε sin β1 + sin ε cos β1 as well as the approximation sin ε ≈ ε and cos ε ≈ 1 eventually yields Specific work done by all blade elements in the elemental blade cascade zffl}|ffl{ Y t,CA l   = : ð σ Þ t|fflfflffl{zfflfflffl} 1 ε w1 u C L 1 þ 2|fflffl{zfflffl} tan β1 Geometric |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} Velocity Aerodynamic parameters triangles airfoil parameters of elemental

ð4:22Þ

blade cascade This is the key equation of the BEM method. It links the solidity σ of the elementary blade cascade, i.e. the essential geometric parameter, to the design work Yt,CA, the velocity triangles and aerodynamic airfoil parameters. As a prerequisite, the solidity of the elemental blade cascade must be low in order to avoid aerodynamic interference of adjacent blades. From experience, a valid range is σ ≤ 0.7; some authors, such as Eck [4] and Sabersky et al. [5], recommend σ ≤ 0.35. Then CL and ε can be taken from the many wellknown catalogues of isolated airfoil data. CL and ε not only depend on the airfoil shape chosen but also on the angle of attack α. Three different strategies for choosing the design angle of attack α are conceivable: • The airfoil section operates at its smallest possible lowest drag-to-lift ratio ε; this promises the best hydraulic efficiency of the blading. • The airfoil section operates close to its maximum lift coefficient CL; this attains the smallest solidity of the blade cascade, i.e. either the shortest chord length or alternatively the largest blade spacing, with the consequence of the smallest number of blades; the

4

This is particularly justified for blade cascades designed for relatively small deflection of the relative flow, which are typical for low-pressure fans.

4.3

The Blade Element Momentum (BEM) Method for Low-Pressure Axial Fans

65

Fig. 4.11 Simplified schematic relationship between the profile operating point (top left) and the operating point of the fan in its volume flow-total pressure characteristic curve (bottom)

results are low manufacturing cost, but the design point is not from the stall point, hence part load operation of the fan becomes critical. • A compromise between lowest drag-to-lift ratio and maximum lift coefficient. Somewhat simplified, Fig. 4.11 establishes the link between the polar curve of the airfoil section or equivalently blade element and the fan aerodynamic characteristic: Part-load operation of the fan is equivalent to a steeper direction of w1 approaching the blade, resulting in a larger angle of attack. The opposite is true for overload operation. The closer the design airfoil lift coefficient is set to stall point, the greater is the danger of driving the fan into stalled operation.5

4.3.2

Summary: Blade Design for Low-Solidity Axial Impellers with the BEM Method

In summary, the procedure for designing a blading employing the BEM method is as follows: 5

In fact, the causes of stall in the entire fan stage are much more complex. Among others, the separation of the casing wall boundary layer (see De Haller [6]) and the tip gap flow (see e.g. Saathof et al. [7]) play a role.

66

4 Design of Axial Fans

Step 1: Blade design parameters Start is the specification of the blade design parameters Δpt,B and V_ B for a given fluid density ρ. Estimating at least the expected volumetric and hydraulic efficiencies of the fan the Cordier-diagram allows a first guess of the impeller outer diameter Dtip and the rotational speed n. The hub diameter Dhub needs to be estimated as well, an early-stage feasibility check of its value will be addressed in Sect. 4.5. Step 2: Segmentation Segmentation into a number of elemental blade cascades, represented by coaxial sections, e.g. according to Eq. (4.14). Step 3: Loading distribution, determination of velocity triangles Selecting a loading distribution according to Sect. 4.1.3 yields Yt,CA for each elemental blade cascade and eventually allows determining the velocity triangles as described in Sect. 2.2.2. Step 4: Calculation of the solidity The solidity for each elemental blade cascade is obtained from Eq. (4.22) with the input of • the allocated work Yt,CA of the elemental blade cascade, • the corresponding velocity triangles, • the chosen airfoil section parameters CL and ε. Step 5: Number of blades and chord length Choosing a number of blades z, the blade spacing in each elemental blade cascade is obtained t=

2πr z

ð4:23Þ

and eventually the required chord length l = σt of each blade element. In general, the airfoil polar is also affected by the chord-based Reynolds number Rel =

w1 l : v

ð4:24Þ

Therefore, it is now to be checked whether the Reynolds number assumed for the airfoil polar from which CL(α) was taken, agrees with its value based on the calculated chord length. If not, it needs to be modified by choosing a different number of blades, or the solidity has to be recalculated based on a polar with a better guess of Rel. Step 6: Stagger angle and airfoil coordinates The stagger angle of each blade element, Fig. 4.12, is the sum of the vectorial mean flow angle β1 and the chosen angle of attack α: γ = β1 þ α:

ð4:25aÞ

4.3

The Blade Element Momentum (BEM) Method for Low-Pressure Axial Fans

67

Fig. 4.12 BEM: Nomenclature for blade element

The coordinates of the airfoil(s) used are documented in catalogues, typically for a standard chord length of unity. The coordinates of the blade sections are obtained by scaling up the chord to their individual length l and rotating by their stagger angle γ. A potential refinement is the projection of the plane blade section onto the respective coaxial cylinder surface. Step 7: Complete blade The complete blade is obtained by “stacking” all blade elements onto a stacking line. The simplest stacking line is a radial beam. Many state-of-the art designs, however, introduce a skew, see Sect. 4.3.3. The point of intersection of the stacking line with each blade element can be chosen at the center of gravity or in the vicinity of l/4 from the leading edge which roughly is the point of action of lift and drag; or pragmatically so that the impeller fits into a given installation space, unless structural reasons speak against this. Step 8: Further refinements of the basic BEM method Occasionally, the basic BEM method described so far is refined: • Unlike the isolated airfoil section in an unbound region, e.g. of a wind tunnel, the blade element in a blade cascade operates in a deflected flow. In order to take this into account the “camber of the flow”   β2 - β1 f 1 , = tan l 1 2 4

ð4:26aÞ

68

4 Design of Axial Fans

Horlock [1], can be added to the geometric camber of the original airfoil f/l: f f f = þ : l modified l l 1

ð4:26bÞ

• Similar to the radial impeller design, the blockage effect of the blades with their maximum thickness d can be taken into account. Marcinowski [8] developed a semiempirical model for this purpose and proposed the thickness correction

    d αd = 0:5 γ - arctan tan γ : t cos γ

ð4:27Þ

The stagger angle of the blade element eventually becomes γ = β 1 þ α þ αd :

ð4:25bÞ

This correction αd needs to be determined iteratively, since it is a function of γ.

4.3.3

Blade Skew

Many state-of-the art blades are skewed. An important goal of skewing is the reduction of flow induced noise, see Sect. 8.3.2. Moreover, it turned out that proper blade skew shifts the stall point on the fan performance characteristic to smaller volume flow rates, hence increases the useful range of operation of the fan as e.g. pointed out in [9]. Skewing a blade means that the elemental blade elements are not stacked onto a straight radial line but on a curved from hub to tip, Fig. 4.13. Fig. 4.13 Example of an impeller with swept blades; backward sweep in the hub area, forward sweep in the blade tip (= casing) region

4.3

The Blade Element Momentum (BEM) Method for Low-Pressure Axial Fans

69

Fig. 4.14 Effect of blade skew on the normalizedmeridional velocity distribution cm2(r); AG = unskewed, AR = backward skew, AV = forward skew; results of a numerical RANS simulation; from Beiler [10]

For simplicity, many blades in axial fans are skewed circumferentially by shifting blade elements in the circumferential direction. The skew angle δ(r), i.e. the angle between the actual threading line and a radial beam, lies in the impeller plane. This, however, results in a combination of sweep and dihydral. A swept wing of an aircraft is a wing with a swept leading and trailing edge. On the other hand, dihydral is the upward angle of a wing as compared to a horizontal line. In analogy, δ(r) of a purely swept blade lies in the twisted w1 – plane, otherwise the blade exhibits also dihydral. Sweep targets at reducing the velocity component perpendicular to the blade leading and trailing edges which is beneficial for acoustic reasons, whereas dihedral causes additional forces acting on the fluid in radial direction. According to Beiler [10] the main effects of blade skew on the flow field in the impeller are: • The inclined blade surfaces cause radial forces on the fluid so that the velocity distributions (Fig. 4.14) and thus also the distribution of the blade load along the radius is modified as compared to unskewed blades; • The local total blade pressure rise is reduced as a function of the sweep angle as Δpt,CA = Δpt,CA δ = 0 ð cos δÞ0:62 ;

ð4:28Þ

• the lift coefficient of swept blade elements changes close to the wall; if – e.g. seen from the casing – there is a backward sweep (i.e. the blade is swept forward), the lift coefficient decreases, and vice versa.

70

4 Design of Axial Fans

When designing skewed blades, at least the reduction of the total pressure rise according to Eq. (4.28) should be compensated for. This can be accomplished, for instance, by increasing the angle of attack α, but of course without exceeding the stall limit of the airfoil.

4.4

The Lieblein Method for High-Pressure Axial Fans

High-pressure axial fans and and even more compressors, i.e. machines with small specific speed, require a blading with large solidity. Lift coefficient and drag-to-lift ratio of the isolated airfoil are no longer valid as the mutual interference of the blades becomes relevant. The basis of the method described in this section are systematic measurements of the deflection properties of cascades of two-dimensional plane airfoil section by Lieblein [11, 12]. These cascades were made of NACA 65(A10)-series airfoil sections with a relative thickness6 d/l = 0.10, mounted in a special wind tunnel and operated at Rel ≈ 2 × 105. The family of NACA 65(A10) airfoils is described in Sect. 10.2.2. From the measurements, Lieblein deduced design charts which give the blade geometry necessary to obtain a targeted flow deflection with minimum losses. They are compiled in Sect. 10.5. Lieblein’s design charts are applicable for the design of elementary blade cascades with a solidity in the range of approximately 0.4 ≤ σ ≤ 2.0. In contrast to the BEM method, it is expedient to start with the solidity as an input parameter and then obtain the following important quantities (Fig. 4.15): • The incidence angle i; this is the angle between the direction of the relative inlet velocity vector and the tangent to the blade mean camber line at the leading edge: i  βB1 - β1

ð4:29Þ

• The deviation angle δ, which is the angle between exit flow vector and the tangent to the mean camber line at the trailing edge: δ  βB2 - β2 ,

ð4:30Þ

• The blade element camber angle φ, i.e. the difference between angles of tangents to mean camber line at leading and trailing edges: φ  βB2 - βB1

ð4:31Þ

For thicker or thinner airfoils, correction factors can be found in [11]. In general, i0 becomes larger for thicker airfoils and smaller for thinner. 6

4.4

The Lieblein Method for High-Pressure Axial Fans

71

Fig. 4.15 Lieblein’s method: Nomenclature for the elemental blade cascade (as drawn, δ and i are positive)

The design charts not only can be used for the design of the rotating blades of an impeller but also for stationary guide vanes. In the case of outlet guide vanes β1 is replaced by α2 and β2 by α3.

4.4.1

Blade Element Inlet Angle

Lieblein’s experiments revealed that the losses in a cascade are minimal if the so called design incidence angle iA(≠0) is maintained. iA is expressed as a function of the camber angle φ and the design incidence angle for an uncambered blade element i0 via iA = i0 þ nφ:

ð4:32Þ

i0 = f(β1, σ) is plotted in Fig. 10.22, and the proportionality factor n = f(β1, σ) in Fig. 10.23, both in Chap. 10. The original cascade experiments were carried out with a stationary blade cascade mounted in a special wind tunnel. To account for the partially three-dimensional flow conditions in a rotating impeller, a radius-dependent correction term (ic – i2D) is proposed in [11]:

72

4 Design of Axial Fans

Table 4.2 Correction term (ic – i2D) and (δc – δ2D) as a function of the relative blade height according to Lieblein Relative blade height

r - r hub rtip - r hub

(ic – i2D) (δc – δ2D)

0.1 (near the hub) +1.6° +1.0°

iA

0.3 +0.2° -0.1°

0.5 -1.0° -0.5°

0.7 -1.8° -0.5°

i0  nM  (ic  i2 D )

0.9 (near the casing) -2.6° -0.5°

ð4:33Þ

This correction term was obtained from a comparison of cascade and rotating impeller measurements and is tabulated in Table 4.2. Eventually, the blade element inlet angle (related to the equivalent circular arc mean line) becomes

E B1 E1  i0  nM  (ic  i2 D ).

ð4:34Þ

The correction term (ic – i2D) is omitted for guide vanes.7

4.4.2

Blade Exit Angle

By analogy the design deviation angle is δA = δ0 þ mφ

ð4:35Þ

with δ0 = f(β1, σ), for an uncambered blade element and the proportionality factor m = f (β1, σ), again both plotted in Figs. 10.24 and 10.25 in Chap. 10. Again, Lieblein recommends additional corrections for the blade elements in a rotating impeller

G A G 0  mM  (ic  i2 D )

wG wi

E1

 (G c  G 2 D )

ð4:36Þ

with ∂δ = f ðβ 1 , σ Þ ∂i β1

7

Instead of these Lieblein corrections, Stark [13] rather recommends corrections directly from the velocity deficit in the boundary layers at the hub and housing walls.

4.4

The Lieblein Method for High-Pressure Axial Fans

73

Fig. 4.16 Lieblein’s method: Nomenclature for blade element

from Fig. 10.26 and (δc – δ2D) according to Table 4.2. However, wake measurements by Schiller [14] showed that for highly loaded blading these deviation angle corrections (δc – δ2D) should be increased by an additional 2.5° near the hub, by 2.0° between 15% and 85% of the relative blade height and by 0.5° to 1.0° near the casing in order to actually achieve the desired flow exit angle. Eventually, for the desired flow angle β2 the blade element exit angle becomes

EB 2

4.4.3

E 2  G 0  mM  (ic  i2 D )

wG wi

E1

 (G c  G 2 D ).

ð4:37Þ

Camber and Mean Line

All preceding relations involve the geometric camber angle φ  βB2 – βB1, Fig. 4.16, which is unknown so far. However, it can be determined a priori from the velocity triangles more specifically from the desired flow deflection (β2 – β1): Subtracting Eq. (4.34) from Eq. (4.37) "

# ∂δ βB2 - βB1 = β2 þ δ0 þ mφ þ ðic - i2D Þ þ ðδc - δ2D Þ ∂i β1 - ½β1 þ i0 þ n  φ þ ðic - i2D Þ, identifying βB2 – βB1 as φ, and solving for φ immediately yields

74

4 Design of Axial Fans

φ=

ðβ2 - β1 Þ þ ðδ0 - i0 Þ þ ðic - i2D Þ

  ∂δ 1 þ ðδc - δ2D Þ ∂i β 1

1-m þ n

,

ð4:38Þ

Finally, the stagger angle results from the blade angles as γ=

βB1 þ βB2 , 2

ð4:39Þ

and the radius of curvature of the equivalent circular arc mean line is ρ=

4.4.4

l : 2 sin φ2

ð4:40Þ

Summary: Blade Design for High-Solidity Axial Impellers

In summary, the procedure for designing a blading employing Lieblein’s method is as follows. Steps 1 to 3 and 7 are identical with those described for the BEM method, the reader is referred to Sect. 4.3.2. Step 4: Calculation of the solidity The number of blades z and for each elemental blade cascade a chord length l are chosen. It has to be checked whether the chord-based Reynolds number is ≥2 × 105. If not, the chord length l and possibly z need to be modified. The resulting solidity is σ = l/(2πr/z). σ must be in the range 0.4 ≤ σ ≤ 2.0. Step 5: Calculation of inlet and exit blade angles For each elemental blade cascade βB1 and βB2 are determined with the help of Lieblein’s design charts. Step 6: Stagger angle and blade section coordinates The blade element camber angle according to Eq. (4.38) allows the determination of the theoretical lift coefficient CfL, which - for this purpose - is merely a parameter to compute the NACA (A10) mean line according to Eq. (10.25). Superposition of a thickness distribution from Table 10.6, scaling up to chord length l and rotating by the stagger angle γ yields the blade section coordinates. The stagger angle of each blade element is obtained via Eq. (4.39). A potential refinement is the projection of the plane blade section onto the respective coaxial cylinder surface.

4.5

Design Criteria

75

Step 8: Further refinements of the Lieblein method There are no further refinements. In particular, the blockage effect in the cascade due to the finite thickness of the airfoils and the camber of the flow are inherently included and must not be corrected.

4.5

Design Criteria

Both design methods for axial bladings are valid only within certain limits. On the one hand, fundamental physics, such as the maximum lift of an airfoil, the phenomena of boundary layer separation and stall, or the effect of the Reynolds number, impose limits. On the other hand, the range of parameters for establishing a data base, as for instance the solidity in the Lieblein cascade measurements, limits its applicability. In the next sections some design criteria are presented which allow a check of feasibility already in an early stage of the design processes.

4.5.1

De Haller-Criterion

One of the difficulties encountered in axial fans is the formation of a separated boundary layer at the casing walls due to the positive pressure gradient. In order to avoid the lateral contraction of the through-flow passage it is sufficient to limit the diffusion in the blading – in the case of the impeller according to the work by De Haller [6] at the representative hub region, where typically the diffusion in the blading is maximal: w2 ! ≥ gDH w1 hub

ð4:41Þ

In the original work by De Haller, gDH was found to be 0.75. Later on, Marcinowski in [14] recommended for single-stage fans gDH = 0.55 to 0.60. The latter range was confirmed more recently by Schiller [15].

4.5.2

Criterion of Strscheletzky

In a swirling pipe flow, a dead-water core is formed in which the fluid rotates similarly to a rigid body. In the core back flow may occur, Horlock [1]. The diameter DCore of the deadwater core is a function of the ratio of the meridional to the circumferential swirl component cm/cu. The dead-water core becomes larger as the swirl increases, disappearing only in the case of a pure through-flow (cu = 0). Figure 4.17 shows the development of the core for _ ð2πrcu r a Þ , were Dtip is the pipe a swirling flow with rcu = const. as a function of V= _ diameter and V / cm : If the length of the swirling flow is limited by downstream guide

76

4 Design of Axial Fans

Fig. 4.17 Development of the dead-water core in a swirling flow

vanes (axially limited swirl), the dead-water core in the immediate vicinity of the guide device is bigger than in the case of an axially unlimited swirl flow. In an axial flow machine, the wake downstream the impeller is called the hub deadwater. The harmful effect of the hub dead-water is not only a partial blockage of the through-flow area and hence an increase of the meridional velocity but also the dissipation of energy due the interaction of the main flow with the wake. In addition, the shear layer between dead-water and main flow is quite unstable and subject to substantial fluctuations. According to Fig. 4.17, the diameter of the hub dead-water increases with reducing the volume flow rate, which eventually can affect the shape and stability of the fan performance characteristic in part-load operation. According to Strscheletzky and Marcinowski (see Horlock [1]), the core of the hub dead-water should be smaller than the impeller hub, Dhub ≤ DCore. Assuming a blade design with rcu = const. the relation cm2 ! ≥ gSt cu2 hub

ð4:42Þ

serves as a criterion. For a single-stage fan (corresponding to an axially unlimited swirl) gSt = 0.8, for a multi-stage fan, e.g. impeller and downstream guide vanes (corresponding to an axially limited swirl) gSt = 1.0. The choice of the hub diameter Dhub plays an important role in satisfying this Strscheletzky-criterion. Frequently, an increase of Dhub helps. As an alternative the formation of hub dead-water can be influenced by selecting another swirl distribution than rcu = const.

4.5

Design Criteria

4.5.3

77

Diffusion Coefficient According to Lieblein

The so-called diffusion coefficient was derived from the analysis of the boundary layer on airfoils (see e.g. Scholz [15]):     w2 1 Δwu d DF = 1 þ þ 2σ w1 l w1 |{z} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflffl{zfflfflfflfflffl} Relative blade Diffusion Measure for the blade load ðcontains C L Þ

ð4:43aÞ

thickness

Because this criterion is often applied to airfoils with 10% relative thickness, a special diffusion coefficient is used:   w2 1 Δwu DF 0:1 = 1 = DF - 0:1 þ 2σ w1 w1

ð4:43bÞ

NACA measurements showed the range of the diffusion coefficient DF0.1 should not exceed 0.6. Otherwise, an increase of the airfoil losses is to be expected. Again Schiller [16] found that especially for highly loaded axial fans this limit can be replaced by 0.7, depending on the hub-to-tip ratio and the radial position of the elemental blade cascade.

4.5.4

Further Limitations

Reynolds number The Reynolds number at which the individual elemental blade element operates should not be too low, as otherwise the airfoil and thus blade losses will increase sharply. A guide line is 

 w1 l ! Rel  ≥ 1:5 × 105 : v For small, slow-running fans, however, this cannot always be achieved. Mach number Although the Mach number is irrelevant for fans for which the fluid is assumed to be incompressible, it should be mentioned that losses increase significantly with increasing Mach number.

78

4 Design of Axial Fans

4.6

Practice Problems

4.6.1

Design of a Low-Pressure Axial Fan

A low-pressure axial fan without outlet guide vanes and diffuser (impeller-only fan) is to be designed for the performance parameters – Air volume flow rate V_ = 31 m3 =s. – Effective pressure rise Δpts = 250 Pa. – Air with a density ρ = 1.2 kg/m3 and a kinematic viscosity ν = 15.1 × 10-6 m2/s. Further parameters are given: • • • • • •

Impeller diameter Dtip = 1.8 m Hub diameter Dhub = 0.78 m Rotational speed n = 675 1/min Number of blades z = 9 Swirl-free inflow to the impeller The gap between blade tips and casing is small.

Hints: • The fan is an impeller-only fan, hence the casing efficiency ηCas is set to 100%. Thus, the fan effective pressure rise Δpts is equivalent to the impeller effective pressure rise Δpts,Imp. • The blade total pressure rise Δpt,B which is required for the blade design can be estimated from the effective pressure rise utilizing Eqs. (10.5, 10.6 and 10.7) in Chap. 10. • Choose a swirl distribution rcu2 = const. • Since the gap between blade tips and casing is small, the volumetric efficiency is estimated as ηvol = 95%.

4.6

Practice Problems

79

• The hydraulic efficiency shall be assumed to be ηh = 90% along the complete blade height.

4.6.2

Design of High-Pressure Axial Fan Stage

A high-pressure axial-flow fan consisting of an impeller and outlet guide vanes is to be designed for the performance parameters – Volume flow rate V_ = 2:65 m3 =s. – Total pressure rise Δpt = 3024 Pa. – Air with a density ρ = 1.2 kg/m3 and a kinematic viscosity ν = 15.1 × 10-6 m2/s.

Further parameters are given: • • • • • • • •

Impeller diameter Dtip = 0.40 m Hub diameter Dhub = 0.28 m Rotational speed n = 4500 1/min Number of blades of the impeller: z = 20 Number of blades of the outlet guide vanes (= stator): zSt = 26 Radius-independent chord length of impeller blades and guide vanes: l = lSt = 0.060 m Swirl-free inflow to the impeller The gap between blade tips and casing is very small.

Hints: • Since the gap between blade tips and casing is very small, the volumetric efficiency is estimated as ηvol = 98%. • The casing efficiency is mainly determined by the losses in the outlet guide vanes. Thus, ηCas is set to 90%. • The hydraulic efficiency of the impeller blading is assumed to be ηh = 80% along the complete blade height.

80

4 Design of Axial Fans

References 1. Horlock, J. H.: Axialkompressoren. Verlag G. Braun, Karlsruhe, 1967 2. Keller, C.: Axialgebläse vom Standpunkt der Tragflügeltheorie. Dr.-Ing. Diss. ETH Zürich, 1934 3. Pfleiderer, C., Petermann, H.: Strömungsmaschinen. Springer-Verlag, Berlin-Heidelberg, 1991 4. Eck, B.: Ventilatoren. Springer-Verlag, Berlin-Heidelberg, 1972 5. Sabersky, R. H., Acosta, A. J., Hauptmann, E. G.: Fluid flow. Mcmaillian Publishing Company, New York, 1989 6. de Haller, P.: Das Verhalten von Tragflügelgittern in Axialverdichtern und im Windkanal. Brennstoff-Wärme-Kraft, Bd. 5, Heft 10, 1953, pp. 333–337 7. Saathoff, H., Deppe, A., Stark, U., Rohdenburg, M., Rohkamm, H., Wulff, D., Kosyna, G.: Steady and unsteady casing wall flow phenomena in a single stage low speed compressor at part load. Proc. 9th Symposium on Transport Phenomena and Dynamics of Rotating Machinery, Honolulu, Hawaii, 2002 8. Marcinowski, H.: Strömungsmaschinen II. Lecture notes, Universität (TH) Karlsruhe, 1975 9. Carolus, Th., Stremel, M.: Sichelschaufeln bei Axialventilatoren, HLH Bd. 51 (2000) Nr. 8, pp. 33–39 10. Beiler, M.: Untersuchung der dreidimensionalen Strömung durch Axialventilatoren mit gekrümmten Schaufeln. Fortschr.-Ber. VDI Reihe 7 Nr. 298, VDI-Verlag Düsseldorf, 1996 (also Ph.D. thesis University Siegen) 11. Members of the Staff of Lewis Research Center: Aerodynamic design of axial-flow compressors. NASA SP-36, Washington, D.C., 1965 12. Lieblein, S.: Incidence and Deviation-Angle Correlations for Compressor Cascades. Trans. of the ASME. J. of Basic Engineering, Sept. 1960, pp. 575–587 13. Stark, U., TU Braunschweig. Private Communication from July 10, 2003 14. Marcinowski, H.: Optimalprobleme bei Axialventilatoren. Ph.D. thesis TH Karlsruhe, 1956 15. Scholz, N.: Aerodynamik der Schaufelgitter. Band 1. Braun-Verlag Karlsruhe, 1965 16. Schiller, F.: Theoretische und experimentelle Untersuchungen zur Bestimmung der Belastungsgrenze bei hochbelasteten Axialventilatoren. Ph.D. thesis Universität Braunschweig, 1983

Further Reading Bommes, L., Fricke, J., Grundmann, R. (Editors): Ventilatoren. Vulkan-Verlag, Essen, 2003 Castegnaro, S.: Aerodynamic design of low-speed axial-flow fans: A historical overview. Designs 2018, 3, 20; doi:https://doi.org/10.3390/designs2030020

5

Sound Generation and Propagation

The chapter begins by introducing the fundamental mechanisms of flow induced sound generation in fans. Then the propagation of sound into a duct system or the free field is discussed. The terms sound and noise are frequently used interchangeably in acoustics, but have different connotations. Sound refers to the physics of the generation and propagation of acoustic waves whereas noise is defined as unwanted sound. In this and the following chapters, with a few exceptions, the term sound is used. By contrast, Chap. 7 will mainly deal with the effect of fan sound on humans, hence noise.

5.1

The Mechanisms of Sound Generation: An Overview

Sound from fans can either be flow induced or mechanical sound from bearings, vibrations due to unbalanced impellers, etc. In this text book, exclusively the aerodynamically generated sound in fans is of interest. In general, sound in flows can be generated by three mechanisms: • By fluid displacement (example: Expanding and imploding cavitation bubbles in a flow of water), • By forces on solid surfaces immersed in a flow (example: The Kármán vortex street behind a cylinder produces force fluctuations on the cylinder surfaces and finally the so called aeolion tone), • By turbulence in the free flow (example: Sound from the jet of a jet engine).

# Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 T. Carolus, Fans, https://doi.org/10.1007/978-3-658-37959-9_5

81

82

5

Sound Generation and Propagation

Fig. 5.1 Forces F on a rotating elemental blade cascade (schematic); the flow velocity relevant for the rotating blades is the relative velocity w1; p is the surface pressure; the wavy arrows symbolize unsteadiness of the quantities due to turbulence

Theoretical considerations1 lead to the corresponding terms “monopole”, “dipole” and “quadrupole” sources of sound. In principle, all three mechanisms are active in a fan (see Fig. 5.1): • Displacement of fluid by the finitely thick blades as they move in the fluid, “thickness noise” • Forces on the wetted surfaces, in particular on the blades, due to various flow phenomena • Turbulence in the free flow, e.g. in the exhaust Industrial fans, however, are characterized by clearly subsonic flow velocities - or equivalently low characteristic Mach numbers. Theoretical and experimental investigations show that for this case the fluid displacement and the turbulence in the free flow contribute comparatively little to the total sound generation. Most relevant are the flow induced forces acting on the wetted surfaces.

1

Sound generation by turbulence and surfaces in arbitrary motion has been described, for example, by Ffowcs Williams-Hawkings [1].

5.1

The Mechanisms of Sound Generation: An Overview

83

Figure 5.1 visualizes the flow velocities and the various resulting forces on an elemental blade cascade. It is useful to distinguish between stationary and unsteady forces (F , F′, respectively) as well as blade-bound and blade-unbound forces. The blades of the impeller experience blade-bound lift and drag forces F associated with a pressure distribution on the wetted surfaces p: They are unavoidable, since the fundamental task of a fan, namely to increase the pressure, requires fluid deflection due to a force. These forces are stationary in the rotating frame of reference. By contrast, unsteady and bladeunbound forces F′ arise, when the inflow to the fan impeller is spatially and temporally non-uniform. The blades encounter different flow regimes while spinning, e.g. the wake of downstream struts or inlet guide vanes. Or they interact with the asymmetric flow in a volute casing. An occasionally used generic term is “rotor-stator interaction”. Other unsteady forces are blade-bound such as the footprint of the turbulent boundary layer on the blade surfaces or the forces induced by the turbulent wake itself. All unsteady blade forces are always the result of a distribution of surface pressure fluctuations p’. Table 5.1 gives a summary of these mechanisms and the frequently used terms like “loading noise” etc. Table 5.1 Flow induced forces as sources of fan noise Flow induced force Lift and drag forces on blades F blade-bound F′(t) Spatially non-uniform inflow to rotor due to upstream guide vanes, struts, asymmetrical inlet, interaction of a volute casing with the impeller → blade-unbound Unsteady, stochastic inflow due to turbulence → blade-bound Stochastic velocity fluctuations in the turbulent boundary layers above wetted surfaces → blade-bound Turbulent wake and vortex shedding at trailing edges, etc. (blade-bound) → blade-bound Boundary layer separation → blade-bound Due to secondary flows Flow separation at the hub and casing walls Tip gap flow (axial impeller) Gap flow (centrifugal fan) Rotating stall Flow over cavities (bores, blowholes)b

Type of noise tonal tonal

broadband broadband

Denomination of sound Steady loading noise (Gutina sound) Unsteady loading noise, rotor-stator interaction

Turbulence ingestion or turbulent inflow noise Self-noise

tonal and broadband

Self-noise

broadband

Stall noise

tonal and broadband

Secondary flow noise Tip clearance noise

tonal

Cavity noise

L. Gutin in 1936 was probably the first to develop a prediction method for the sound of a propeller with the model of a rotating force. His contribution was translated from Russian into English [2]; see also [3] b For cavity noise in turbomachinery see the monograph [4] a

84

5

Sound Generation and Propagation

5.2

Rotating Pressure Fields of Axial Fans

5.2.1

The Rotating Pressure Field of an Isolated Impeller

An axial impeller has z blades, which are arranged equidistantly in the circumferential direction on the hub. The shaft speed n leads to the impeller’s angular velocity Ω = 2πn:

ð2:1Þ

Linked to the blade-bound blade forces, which are stationary in the rotating frame of reference, is a pressure field that • also rotates with the angular velocity Ω and • has the period in circumferential direction 2π/z, Fig. 5.2. This rotating pressure field is called a spinning mode. An observer in the stationary laboratory system would measure a pressure oscillation (or fluctuation) p’. In reference planes up- and downstream of the impeller, the pressure fluctuation is - harmonically idealised - dependent on the angular position ϑ and time t2 p0 ðϑ, t Þ = a cos ½zϑ - zΩt  = a cos ½zðϑ - Ωt Þ:

ð5:1Þ

The frequency of the pressure fluctuation, the so called blade-passing or blade passage frequency (BPF) in Hertz, is BPF = zn =

zΩ : 2π

ð5:2Þ

In reality, the pressure fluctuations are not sinusoidal, the details of the flow field ultimately determine the shape of the 2π/z-periodic pattern. However, via a Fourier analysis they can be represented as a simple discrete spectrum of harmonic components,

In general, the function cos[aϑ - bΩt] = cos [a(ϑ - b/aΩt)] has the period 2π/a in ϑ-direction and the angular frequency bΩ. It describes a pattern which rotates with the angular velocity Ωb/a and thus propagates like a wave in +ϑ-direction (compare the typical argument (x-c0t) of a plane sound wave traveling in +x- direction with the speed of sound c0). 2

5.2

Rotating Pressure Fields of Axial Fans

85

Fig. 5.2 Rotating pressure field - spinning mode -, associated with a rotating axial impeller with z = 5 blades (schematic)

p0 ðϑ, t Þ =

1 X j=1

p0j =

1 X

  aj cos jzðϑ - Ωt Þ þ Φj , j = 1,2, . . . ,

ð5:3Þ

j=1

i.e. they consist of a series of rotating pressure fields, each of which rotates at the impeller angular velocity Ω and is periodic in the circumferential direction with 2π/( jz).3 The index j denotes the order of the harmonics. j = 1 is the fundamental tone with the blade passing frequency BPF, j > 1 are the higher harmonics. The fundamental tone and its higher harmonics form the sound related to the rotation of the impeller. It will be shown in Sect. 5.4.2, that only those rotating pressure fields which rotate at supersonic circumferential velocity, propagate as sound in ducts. By definition, the impeller of a fan rotates at low subsonic circumferential speeds. Hence, the fundamental bladebound forces are not the cause for the dominant tonal sound. This is often misunderstood. Nevertheless, the concept of rotating pressure fields or spinning modes is important, since rotor-stator interaction phenomena can be constructed from a series of modes which partly rotate at supersonic circumferential velocity.

5.2.2

Rotor-Stator Interaction

When an impeller (“rotor”) rotates upstream or downstream of a stationary obstacle (guide vanes, struts, etc., in short a “stator”), the rotating flow field interacts with that of the stator. Several mechanisms of this rotor-stator interaction are possible:

3

A radius-dependent pressure distribution can be modelled by aj → aj(r) and Φj → Φj(r).

86

5

Sound Generation and Propagation

• The wake4 of the rotor blades encounter struts, outlet guide vanes or the asymmetrical inner wall of the volute casing including the tongue. • The wake of inlet guide vanes or struts is “chopped” by the blades of the rotor. • In a broader sense, the interaction of the rotor with any type of a non-uniformity in its vicinity can be regarded as rotor-stator interaction. When a rotor blade encounters a stator blade then both “coincide”. The consequence is the generation of a pressure pulse. For the following analysis it is assumed that the z rotor blades and the zSt stator blades are arranged equidistantly along the circumference. Tyler and Sofrin [5] showed that the pressure pulses create a pressure field p0 ðϑ, t Þ =

1 X j=1

p0j =

 h  i jz zSt amj cos m ϑ - Ωt þ Φmj , m m= -1

1 i X X j=1

ð5:4Þ

where m can only take the following values: m = jz þ kzSt , k = . . . , - 1,0,1, . . .

ð5:5Þ

Eq. (5.4) states that the pressure field - as in the case of the isolated rotor - is composed of a series of harmonics p0j . Each harmonic consists of another series of elementary rotating pressure fields of circumferential order m, each of which • rotates with different (because m-dependent) angular velocity jzΩ/m, and • is 2π/m-periodic in circumferential direction Only values satisfying the condition Eq. (5.5) can be mode orders m. The amplitudes amj and the phase angles Φmj are determined by the details of the flow in the bladed region of the rotor and stator. Table 5.2 shows some orders m of the pressure fields resulting from the interaction of a 4-bladed rotor with a 1- to 5-bladed stator. Only the tone with j = 1 (= BPF) is considered. Figure 5.3 illustrates the pressure pulses which occur due to coincidences during 1/4 turn of the rotor. The following is noteworthy and generalizable: • Rotating pressure fields occur which rotate with a multiple or infinite multiple of the rotor speed against (negative m) or in (positive m) the direction of rotor rotation. This is

The wake behind a body in a flow is the region with a velocity deficit, see Fig. 4.1. The wake behind a cascade of blades causes the flow to be rotationally asymmetric. It is an effect of both inviscid (potential) flow and friction (turbulent wake behind an airfoil). The further downstream the distance from the cascade, the more the flow becomes spatially homogeneous, i.e. the wake dies away. 4

5.3

Flow-Induced Sound from Lift-Generating Surfaces

87

Table 5.2 Rotor-stator interaction: Order and angular velocity of some rotating pressure fields of the tone with BPF ( j = 1) for z = 4 rotor blades and zSt = 1 to 5 downstream stator blades Number of stator blades zSt 1

2

3

4

5

k -4 -3 -2 -2 -1 0 -2 -1 0 -1 0 +1 -2 -1 0

m = z + k zSt 0 +1 +2 0 +2 +4 -2 +1 +4 0 +4 +8 -6 -1 +4

Angular velocity of the pressure field zΩ/m 1 +4 Ω ( See Fig. 5.3 +2 Ω 1 +2 Ω ( See Fig. 5.3 +1 Ω -2 Ω +4 Ω ( See Fig. 5.3 +1 Ω 1 +1 Ω ( See Fig. 5.3 +0.50 Ω -0.67 Ω -4 Ω ( See Fig. 5.3 +1 Ω

relevant since the circumferential speed of the pressure field determines whether it propagates as sound into a duct or not (see Sect. 5.4.2). • If the number of stator and rotor blades differs by an integer multiple, the number of coincidences is particularly high; e.g. four coincidences per 1/4 turn in the case of zSt = 3, but eight in the case of zSt = 4, and only six in the case of zSt = 5. Ratios zSt/z that lead to an unnecessarily high number of coincidences must be avoided, as they can be a cause of high tonal sound pressures.

5.3

Flow-Induced Sound from Lift-Generating Surfaces

Fundamental elements of a fan like the blades of an impeller or guide vanes are in principle lift-generating surfaces. Decisive for the generation of sound from lifting surfaces is the unsteady fluid velocity relative to the surface. It is irrelevant whether the surface is stationary or moving. Since the character of the flow field strongly depends on location, the sound source may also depend on location. For instance, the flow over an airfoil from leading to trailing edge varies from accelerated to decelerated. Depending on chordwise position, a laminar/turbulent transition zone, a laminar separation bubble, flow detachment etc. may exist each producing its own sound. In addition, any edge of a surface (e.g. the

88

5

Sound Generation and Propagation

Fig. 5.3 Pressure pulses due to coincidences of rotor and stator blades during 1/4 turn of the rotor (example for z = 4 rotor and zSt = 1 to 5 downstream stator blades)

trailing edge of an airfoil or blade) is relevant for sound generation. The most important sound generation mechanisms are explained in the following paragraphs using the example of the airfoil section in a flow depicted in Fig. 5.4.

5.3

Flow-Induced Sound from Lift-Generating Surfaces

89

Fig. 5.4 Airfoil section in a flow (schematic)

Fig. 5.5 Inflow turbulence; left: Aerodynamically compactness (eddy dimensions are larger than chord length), right: Non-compactness (eddy dimensions significantly smaller than chord length)

Turbulent Inflow • Case 1: The chord length l is smaller than the length scale of the turbulent structure Λ (aerodynamic compactness,5 Fig. 5.5 left) of the inflow: l< Λ Then predominantly the pressure fluctuations in the vicinity of the leading edge emit sound. The distinction of whether the condition for aerodynamic and/or acoustic compactness is satisfied or not, is important, because the characteristic effect of e.g. the flow velocity on the sound generation and radiation is different. Turbulent Boundary Layer Pressure fluctuations on a surface beneath the turbulent boundary layer radiate broadband sound. But their contribution to the overall sound is usually small.7 However, turbulence structures convected over a trailing edge generate sound very effectively due to the interaction of the turbulent boundary layer with the edge.

6

Acoustic compactness: The condition of acoustic compactness can also be expressed in terms of the wave number k = ω=c0 = 2πf =c0

ð5:9Þ

kl < < 2π:

ð5:8bÞ

as

If one assumes a quarter wavelength as the limit in Eq. (5.8a), then acoustic compactness is ensured up to an upper frequency of f upper =

7

1 c0 : 4 l

ð5:8cÞ

In case of a predominantly laminar unstable boundary layer on either, suction or pressure side or both, a tonal sound can be generated. A small separation bubble and a feedback loop amplify the tones to high levels, see Yakhina et al. [6]. Such conditions, however, are rather rare in fans.

5.4

Sound Propagation

91

Fig. 5.6 Schematic spectra of the flow-induced sound radiated from an stationary airfoil section (schematically after Blake [7])

At large angles of attack, the boundary layer detaches and stall occurs, with massive unsteadiness of the flow. As a result, low frequency broad band sound is radiated at high levels. Vortex Shedding in the Wake Similar to the Kármán vortex street behind a cylinder in a flow, vortex shedding is observed from trailing edges. Parameters are the chordwise Reynolds number, the boundary layer and the geometric thickness (= bluntness) of the trailing edge. Acoustic Spectra Acoustic spectra of some of the sounds are presented schematically in Fig. 5.6. Turbulent inflow causes rather low-frequency broadband sound, whereas the trailing edge sound is more high-frequency. The acoustic signature of vortex shedding from a blunt trailing edge is tonal. Blades of an axial impeller can be thought a number of airfoil sections piled up from hub to tip. Naturally, the face different flow velocities, since they move at different radiusdependent circumferential velocities. This broadens the bandwidth of the overall resulting tonal sound.

5.4

Sound Propagation

The fundamental sources are independent of whether the fan emits sound into the free field or into an attached duct. However, the sound perceived by a listener depends strongly on the installation of the fan as well on frequency, i.e. the wavelength λ. As an example the installation of an axial fan stage is considered. Two ranges can be distinguished, Fig. 5.7: Installation in a short duct If an axial fan stage of diameter D is installed in a very short duct-type casing of axial length L, i.e. L < D, and/or L < λ, the sound field is not affected significantly by the duct. Here a listener in the stationary frame of reference can feel a

92

5

Sound Generation and Propagation

Fig. 5.7 Effect of a duct on the sound field; a short duct allows the sound sources, i.e. the blade forces, forming the free sound field by superposition (left), while a long duct rearranges and guides the sound field (right); after Roger [8]

temporal pressure fluctuation and hence hear sound. Since, in general, the sound sources, i.e. the “blades forces”, are moving at different distances from the listener at different speed, differences in propagation time and the Doppler effect play a role in the formation of sound in the free field. Installation in a long duct If, on the other hand, the fan stage radiates sound into a very long duct (L > D and L > > λ), the duct acts as a wave guide due successively reflections at its walls. The features of those duct modes is discussed below. Sound at very high frequency - less typical in fan acoustics - is the realm of ray acoustics not treated here. A special feature applies to broadband sound sources. Here, their rotation can be disregarded if the so called circumferential Mach number (the ratio of the source circumferential speed and the speed of sound) Mau = u=c0 = rΩ=c0

ð5:10Þ

Ma2u > < - 19:001 ak = > - 5:548 > > : - 0:060

ð6:32Þ

qffiffiffiffiffiffiffiffiffiffiffiffi c0 2 =c2 into Eq. (6.31) the working formula for

the power spectral density of the turbulent velocity fluctuations becomes Sc ðf Þ = cTI 2 Λ10F=10 :

ð6:33Þ

Although Sc is based on experimental data such as from a turbulent pipe flow, Költzsch postulated its validity for the turbulence ingested by the impeller of a fan: Sw ðf Þ = Sc ðf Þ:

ð6:34Þ

Thus - with knowledge of the turbulence intensity TI and the length scale of the incoming turbulent eddies Λ - the sought power spectral density of the inflow turbulence to the blades is known. TI and Λ must be determined experimentally. Turbulent boundary layer Mugridge’s [11] formulation of the power spectral density of the lift force fluctuations due to the turbulent boundary layer above the plate surfaces was guided by experimental results:

122

6

8 1 1 > lw1 Sp ðf Þlb > > 5π f > > > < 2 2 1 SA,2 ðf Þ = 5π 2 w1 2 Sp ðf Þlb for f > > > > > 61 1 > : 4 w1 3 3 Sp ðf Þlb π l f

Sound Prediction Methods

π Sr l ≤ 2 2 ≤ π Sr l ≤

15 π

ð6:35Þ

15 ≤ π Sr l π

Here, Sp 

dp0 2 df

ð6:36Þ

is the power spectral density of the wall pressure fluctuations. Equation (6.35) reflects the fact that the size of the correlation area AC depends on frequency and hence on Strouhal number, now defined with the chord length Sr l 

fl : w1

ð6:37Þ

AC decreases with f -1 at low and with f -3 at high frequencies, in between with f -2. The power spectral density of the wall pressure fluctuations is still missing. Wall pressure fluctuations under the turbulent boundary layer above a flat, stationary plate or a stationary airfoil section had been determined in wind tunnel experiments by many authors. In order to minimize the effect of inflow turbulence, attention was always paid to a very low turbulence intensity of the wind tunnel flow. Keith et al. compiled extensive data in [12]. They collapse fairly well in a universal band, if they are non-dimensionlized with the so called outer variables – these are the free-field velocity outside the boundary layer w1 and the displacement thickness of the boundary layer δ*:   Sp fδ = f w1 ρ2 w21 δ

ð6:38Þ

The independent variable is the Strouhal number Sr δ 

f δ : w1

Figure 6.7 presents the non-dimensionalized data in terms of levels

ð6:39Þ

6.3

Class II Sound Prediction Methods

123

Fig. 6.7 Non-dimensional wall pressure fluctuations due to the turbulent boundary layer

LSp ðSr δ Þ = 10 log

Sp : ρ2 w31 δ

ð6:40Þ

Költzsch [10] reconstructed the fluctuating pressures on the rotating blades of axial fans from sound pressure measurements came to the conclusion that they must be considerably higher than those at the stationary flat plate or airfoil. Assuming the same spectral shape as the universal “band” by Keith, Költzsch proposes LSp ðSr δ Þ = 10 log

Sp = 10 log ðGðSr δ ÞÞ ρ2 w31 δ

ð6:41Þ

with the curve fit GðSr δ Þ =

0:01 , 1 þ 4:1985Sr δ þ 0:454Sr δ 6

ð6:42Þ

which is also shown in Fig. 6.7. With this, the power spectral density of the wall pressure fluctuations can now be predicted: Sp ðf Þ = ρ2 w21 δ 10G=10

ð6:43Þ

As a first approximation the boundary layer displacement thickness δ* can be taken from the well-known relationship for the turbulent boundary layer on the flat plate: δ = 0:05x Re l- 0:2 : l

ð6:44Þ

124

6

Sound Prediction Methods

It is worth noting that the boundary layer displacement thickness can be considerably larger in a flow over an airfoil with its adverse pressure gradients (typically by a factor of about 2 to 4). Nevertheless, all approximations inserted in Eq. (6.26) finally result in the desired relation for the calculation of the sound power spectral density SPac,2 ( f ) due to the “turbulent boundary layer” on the blade surface. Vortex shedding in the turbulent wake The prediction of spectral sound power due to vortex shedding are difficult because several mechanisms may be responsible. For example, blunt trailing edges may produce a pronounced whistling sound - the geometric details of the trailing edge play an important role. Here we refer to the experimental work of Brooks, Pope and Marcolini [13] and based on this Lowson’s approximation in [1], who investigated a NACA 0012 airfoil section with a sharp trailing edge in a wind tunnel. After all, the acoustic power caused by vortex shedding in the turbulent wake is comparatively low if other mechanisms are active.

6.3.3

Duct Model

So far, reflecting surfaces from casings etc., such as a duct casing, have not been taken into account. Based on the work by Morfey [14], Költzsch [10] proposed a replacement for Eq. (6.10) SPac ðf Þ = z

π f A S ðf Þ, 4 ρc2 r tip ð1 - ν2 Þ2 ± L 0

ð6:45Þ

that applies to an axial fan impeller as a sound source in a hard-walled duct with anechoic terminations. Radial modes in the duct are excluded. z is the number of blades, rtip the outer radius of the impeller and ν the hub-to-tip ratio rhub/rtip. A± (also called radiation function) contains the effect of a flow in the duct, which is superimposed on the sound field; however, it can be set approximately 1 for low flow velocities (Ma < 0.5). In Eq. (6.45), the previously given expression for SL(f) can be used.

6.3.4

Summary and Example

As a summary, in Table 6.3 the essential equations described above are listed. Not included are the combinations with for the acoustic duct model Eq. (6.45). Figure 6.8 shows an example of the predicted broadband sound spectrum from a low-pressure impeller-only axial fan (Schneider and Carolus [15]). The input parameters are compiled in Table 6.4. The fan operates in a duct casing, hence the duct model Eq. (6.45) is chosen. The statistics of the inflow turbulence was taken from hot-wire

6.3

Class II Sound Prediction Methods

125

Table 6.3 Summary of the equations for the Class II broadband sound prediction methods Mechanism Turbulent inflow

Sharland method Eq. (6.15)

Turbulent boundary layer

Eq. (6.19)

Vortex shedding

Eq. (6.20)

Spectral method Eq. (6.29) with Eqs. (6.32) to (6.34)

Eq. (6.26) with Eqs. (6.35), (6.42), (6.43)

Fig. 6.8 Example: Broadband sound of an axial low-pressure fan impeller; prediction with a class II method and comparison with experimental data, Schneider and Carolus [15]

measurements; quantities varying over the inflow cross-sectional plane were replaced by their spatial mean values. The results in Fig. 6.8 show that - as expected - the turbulent inflow contributes most at lower frequencies, while the turbulent boundary layer sound on the blade surfaces dominates at higher frequencies. The agreement with the measured acoustic power spectrum, which is also shown, is more or less satisfactory. Strictly speaking, due to the inherent assumption of acoustic compactness, the upper-frequency limit is fupper = 1440 Hz. Naturally, the method cannot determine the tonal components, which result from an interaction of the spatially inhomogeneous inflow with the impeller (see Sect. 5.5.1).

126

6

Sound Prediction Methods

Table 6.4 Parameters for the example Geometry Impeller diameter Dtip = 300 mm Blade length x blade height lxb = 0.061 × 0.08 m2 Number of blades z = 6 Operating point and kinematics Volume flow V_ = 0:67 m3 =s Mean relative velocity at an blade section at 70% of the relative blade height w1 = 41m/s Fluid data (air) Density ρ = 1.2 kg/m3 Speed of sound c0 = 346 m/s Kinematic viscosity ν = 15.1 × 10-6 m2/s Turbulence parameters of the inflow Turbulence intensity Tu = 4% Turbulent length scale Λ = 35 mm

6.4

Practice Problems

6.4.1

Acoustic Model Law

Assuming the validity of Madison’s approach, by how many dB does the sound power level of a fan change, a) if increasing the rotor speed by 20% b) choosing a similar fan which is 1.5 times larger?

6.4.2

Fan Acoustic Power

Given is a low-pressure impeller-only axial fan with the following parameters: • • • • • • • • •

Volume flow rate V_ = 0:6 m3 =s Total pressure rise Δpt = 300 Pa Total efficiency η = 80% Impeller diameter Dtip = 300 mm Rotor speed n = 3000 rpm Number of blades z = 6 Blade length x blade height lxb = 0.06x0.08 m2 Mean relative velocity at 70% of relative blade height w1 = 40.8 m/s Slope of the α/CL-curve of the blade profile Φ = 2.8.

References

127

The fan delivers air of density ρ = 1.2 kg/m3, kinematic viscosity ν = 15.1x10-6 m2/s and with speed of sound c0 = 346 m/s. a) Estimate the discharge duct sound power level according to VDI 3731. b) Estimate the sound power level radiated into intake and discharge and the unweighted octave band spectrum according to VDI 2081 Part 1. c) Using the SHARLAND method, calculate the sound power and sound power level for the inflow turbulence intensities TI = 2%, 4%, 8%.

References 1. Lowson, M. V.: Assessment and prediction of wind turbine noise. Flow Solutions Report 92/19, ETSU W/13/00284/REP, pp. 1–59, Dec. 1992 2. Madison, R. D.: Fan Engineering (Handbook), 5th Edition. Buffalo Forge Company, Buffalo N. Y., 1949 3. VDI 2081, part 1: Raumlufttechnik – Geräuschmessung und Lärmminderung (Air-conditioning – Noise generation and noise reduction), March 2019 4. Eck, B.: Ventilatoren. Springer-Verlag, Berlin-Heidelberg, 1991 5. VDI-Richtlinie 3731, part 2: Emissionskennwerte technischer Schallquellen/Ventilatoren (Characteristic noise emission values of technical sound sources; fans), Nov. 1990 6. Grundmann, R., Reinartz, D.: Vergleich von Geräuschmessung und -prognose nach VDI 3731. Heiz. Lüft.-Klima Haustech. Vol. 44, No. 8, 1993 7. Sharland, I. J.: Sources of noise in axial flow fans. J. of Sound and Vibration, Vol. 1, No. 3, pp. 302–322, 1964 8. Doak, P. E.: Acoustic radiation from a turbulent fluid containing foreign bodies. Proc. Royal Society of London, Series A, pp. 129–145, 1960 9. Curle, S. N.: The influence of solid boundaries upon aerodynamic sound. Proc. Roy. Soc. (London), Series A, Vol. 231, pp. 505–514, 1955 10. Költzsch, P.: Ein Beitrag zur Berechnung des Wirbellärms von Axialventilatoren. Jahrestagung der Deutschen Akustischen Gesellschaft 1993; also published in: Bommes, L., Fricke, J., Klaes, K. (Hrsg.): Ventilatoren. Vulkan-Verlag, Essen, 1994 11. Mugridge, B. D.: Broadband noise generation by aerofoils and axial flow fans. AIAA Paper 73-1018, 1973 12. Keith, W. L., Hurdis, D. A., Abraham, B. M.: A comparison of turbulent boundary layer wallpressure spectra. Transactions of the ASME, J. of Fluids Engineering, Vol. 114, pp. 338–347, Sept. 1992 13. Brooks F. T., Pope, D. S., Marcolini, M. A.: Airfoil self-noise and prediction. NASA RP-1218, 1989 14. Morfey, C. L.: The acoustics of axial flow machines. J. of Sound and Vibration 22 (4), 1972, pp. 445–466 15. Schneider, M., Carolus, Th.: Ventilatorbreitbandgeräusch – Berechnung des breitbandigen aeroakustischen Geräuschspektrums von Axialventilatorlaufrädern aus Stromfeldgrößen. Final report for Forschungsgemeinschaft für Luft- und Trocknungstechnik (FLT e.V.), Heft L193, 2002

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Further Reading Sound in Turbomachinery in General Költzsch, P.: Ein Beitrag zur Lärmbekämpfung an Maschinen und Anlagen. Freiberger Forschungshefte A 697, VEB Deutscher Verlag für Grundstoffindustrie, Leipzig, 1984 Költzsch, P.: Geräusch von Strömungsmaschinen. Freiberger Forschungshefte A 697, VEB Deutscher Verlag für Grundstoffindustrie, Leipzig, 1984 Grosveld, F. W.: Prediction of broadband noise from horizontal axis wind turbines. J. of Propulsion and Power. Vol. 1, No. 4, pp. 292–299, 1985 Költzsch, P., Gruhl, S., Biehn, K. u. a.: Berechnung der Schalleistung von axialen Strömungsmaschinen. Freiberger Forschungshefte A 721, VEB Deutscher Verlag für Grundstoffindustrie, Leipzig, 1986 Blake, W. K.: Mechanics of flow induced sound and vibration. Vol. I & II, Academic Press, 1986 Lowson, M. V., Fiddes, S. P.: Design prediction model for wind turbine noise: 1. Basic aerodynamic and acoustic models. Flow Solution Report 93/06, W/13/00317/00/00, pp. 1–46, Nov. 1993 Wagner, S., Bareiss, R., Guidati, G.: Wind turbine noise. Springer-Verlag, Berlin-Heidelberg, 1996 Roger, M.: Noise in turbomachines. In: Breugelmans, F. A. E., Anthoine, J. (Editor): Noise in turbomachines. Lecture Series 2000–02. Von Karman Institute for Fluid Dynamics, 2000

Acoustic Model Laws for Fans Weidemann, J.: Beitrag zur Analyse der Beziehungen zwischen den akustischen und strömungstechnischen Parametern am Beispiel geometrisch ähnlicher Radialventilator-Laufräder. DLR Research report 71-12, 1971

Inflow Turbulence Ffowcs Williams, J. E., Hawkings, D. L.: Sound generation by turbulence and surfaces in arbitrary motion. Philosophical Transactions of the Royal Society of London, Vol. 264, No. A 1151, 1969 Sevik, M.: Sound radiation from a subsonic rotor subjected to turbulence. Int. Symp. Fluid Mech. Des. Turbomachinery, Pennsylvania State University, University Park, NASA SP-304, 1974, pp. 493–511 Amiet, R. K.: Acoustic radiation from an airfoil in a turbulent stream. J. of Sound and Vibration, Vol. 41, No. 4, pp. 407–420, 1975 Carolus, Th., Stremel, M.: Blade surface pressure fluctuations and acoustic radiation from an axial fan rotor due to turbulent inflow. Acta Acustica, Vol. 88, pp. 472–482, 2002

Trailing Edge Sound Ffowcs Williams, J. E., Hall, L. H.: Aerodynamic sound generation by turbulent flow in the vicinity of a scattering half plane. J. of Fluid Mechanics, Vol. 40, No. 4, pp. 657–670, 1970 Paterson, R. W., Amiet, R. K.: Acoustic radiation and surface pressure characteristics of an airfoil due to incident turbulence. AIAA Paper No. 76-571, 1976 Amiet, R. K.: Noise due to turbulent flow past a trailing edge. J. of Sound and Vibration, Vol. 47, No. 3, pp. 387–393, 1976 Longhouse, R. E.: Vortex shedding noise of low tip speed, axial flow fans. J. of Sound and Vibration, Vol. 53, No. 1, pp. 25–46, 1977 Amiet, R. K.: Effect of the incident surface pressure field on noise due to turbulent flow past a trailing edge. J. of Sound and Vibration, Vol. 57, No. 2, pp. 305–306, 1978

References

129

Howe, M. S.: A review of the theory of trailing edge noise. J. of Sound and Vibration, Vol. 61, No. 3, pp. 437–465, 1978 Schlinker, R. H., Amiet, R. K.: Helicopter rotor trailing edge noise. NASA Contractor Report 3470, pp. 1–145, 1981 Ganz, U., Glegg, S. A. L., Joppa, P.: Measurement and prediction of broadband fan noise. AIAA98-2316, pp. 675–687, 1998

7

Psychoacoustic Assessment of Fan Noise

7.1

Introduction

In the previous two chapters the acoustics of fans was described in terms of physical quantities such as sound pressure and sound power. Figure 7.1 depicts schematically the overall sound power level emitted by a fan as the volume flow rate is varied. Each value of LW,oa is associated with a spectrum of sound power or sound pressure. Figure 7.2 shows a typical sound pressure spectrum obtained from microphone measurements at a certain distance from the fan and averaged over a given time interval.1 In many cases, the tone at blade-passing frequency BPF = nz and multiples as higher harmonics are clearly audible. The overall level and the spectral composition are strongly dependent on the operating point of the fan. It is well-known that the overall level and even the spectrum do not fully reflect the perception of fan sound by humans. Usually the sound of fans is annoying. Hence it is named noise. But noise can have different dimensions. This chapters gives a short introduction into the psychoacoustic assessment of fan noise.

7.2

Perception of Annoyance and Quality of Fan Sound

In general, according to Guski and Felscher-Suhr [1], the perception of annoyance has three characteristics: The compulsion to do something one does not want to do, the cognitive and emotional evaluation of a noise situation, and the feeling of being partially helpless in the face of a noise situation. 1/3 octave band spectra and overall levels are often

1

The representation can be either narrow band or in proportional frequency bands such as standardized 1/3-octave or octave bands, see Sect. 10.3. # Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 T. Carolus, Fans, https://doi.org/10.1007/978-3-658-37959-9_7

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Fig. 7.1 Aerodynamic and acoustic characteristics of a fan (schematic)

Fig. 7.2 Typical noise spectrum of a fan at an operating point in narrow and 1/3 octave band representation. OA denotes the total level, i.e. the characteristic points in Fig. 7.1

frequency weighted (for instance A-weighted) to get an approximation for the perception of loudness. However, several studies indicate differences in judgments on sounds having the same A-weighted level Lp,A, especially if they differ in their spectral content or time structure, Töpken and van de Par [2]. The German Technical Instructions on Noise (TA Lärm [3]) recommends penalties for tonality, impulsiveness, etc. on top of the temporal mean value of the sound pressure level to obtain a rating level

7.3

Two Psychoacoustic Metrics for Fan Noise

Lr = Lp,A þ Penalties dB:

133

ð7:1Þ

It may serve as a first measure for the perception of annoyance of a particular noise. An inherent disadvantage, however, is that most penalties according to the TA noise need to be estimated by subjective judgment, only the tonality can be determined from measurements according to DIN 45681 [4]. An alternative view on the effect of sounds on humans is based on the so called product sound quality, defined as the adequacy of a sound in relation to the product, Blauert and Jekosch [5]. How well does the sound of a product match the user’s expectations? Here, an active interaction with a product is assumed, i.e., a person is not “being at the mercy” of a situation. Sound quality is often predicted by empirical models. In order to create such a model, representative jury test participants are asked to describe a class of sounds (or sound sources, e.g. vacuum cleaners) via adjective scales. In parallel, psychoacoustic metrics such as loudness, sharpness, tonality, roughness, etc. are derived from the objectively measured acoustic signatures of the sounds.2 Eventually, the sound quality of a given sound is a linear combination of these psychoacoustic metrics. Their individual weights are the results of the jury listening test. In general, the more the class of sounds and hence products and evaluation contexts is confined (e.g. only vacuum cleaners in the context of “cleaning a living room”), the higher is the prediction quality – at the cost of lacking transferability to other products and contexts.

7.3

Two Psychoacoustic Metrics for Fan Noise

An advanced fan-specific rating level has been developed by Töpken and van de Par [2, 10]. For this purpose, 37 different fan sounds were evaluated with a semantic differential (Table 7.1, left) by test persons in jury listening tests. From these, preferenceequivalent levels were determined. The final model Lr = Lp:A þ ð15:7 - 8:4N ratio Þ dB

ð7:2Þ

depends on Nratio which is the ratio of the loudness in certain frequency ranges. Nratio can easily be calculated according to a rule given in [10]. The basis is the loudness of the sound according to ISO 532-1 ([6]), which can be derived from a measured sound pressure spectrum. For the class “fans and air handling units with comparatively low sound power level” such as heat pumps, air cleaners, kitchen exhaust hoods, electrical-control cabinet ventilation, heat exchangers and the “small” fans typical in these applications, Feldmann [11]

2

Algorithms for determining the psychoacoustic metrics are either standardized, e.g. in [4, 6–9], or subject of ongoing research.

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Table 7.1 Two possible collections of adjective scales for characterizing the noise of fans and air handling units No. 1 2

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

According to Töpken and van der Par for fans in general [10] soft – loud (in German: leise – laut) low – high (tief – hoch) unpleasant – pleasant (unangenehm – angenehm) not noise like – noise like (nicht rauschend – rauschend) jarring – dull (schrill – dumpf) not disturbing – disturbing (nicht störend – störend) not droning – droning (nicht dröhnend – dröhnend) not humming – humming (nicht brummend – brummend) not annoying – annoying (nicht nervig – nervig) non-vibrating – vibrating (nicht vibrierend – vibrierend) bassless – bassy (basslos – bassig) agitated – calm (unruhig – ruhig) not buzzing – buzzing(nicht surrend – surrend) unbearable – bearable (unerträglich – erträglich) non-reverberating – reverberating (nicht hallend – hallend) single sound – mixture of sounds (Einzelgeräusch – Geräuschgemisch) obtrusive – negligible (penetrant – ausblendbar) not roaring – roaring (nicht röhrend – röhrend) inconspicuous – conspicuous (unauffällig – auffällig) not squeaking – squeaking(nicht fiepend – fiepend)

According to Feldmann for air handling units with low sound power level [11] completely disturbing – not disturbing at all (in German: gar nicht störend – völlig störend) cannot be blanked out at all – can be blanked out completely (gar nicht ausblendbar – völlig ausblendbar) totally annoying – not annoying at all (völlig lästig – gar nicht lästig) unpleasant – pleasant (unangenehm – angenehm) obtrusive – unobtrusive (aufdringlich – unaufdringlich) completely humming – not humming at all (völlig brummend – gar nicht brummend) dark – light (dunkel – hell) completely roaring – not roaring at all (völlig röhrend – gar nicht röhrend) low – high (tief – hoch) completely booming – not booming at all (völlig dröhnend- gar nicht dröhnend) heavy – light (schwer – leicht) completely fluctuating – not fluctuating at all (völlig fluktuierend – gar nicht fluktuierend) instationär – stationär (unsteady – steady) completely varying – not varying at all (völlig schwankend – gar nicht schwankend) moving – static (bewegt – statisch) uneven – even (ungleichmäßig – gleichmäßig) weak – strong (schwach – stark) low performance – high performance (leistungsschwach – leistungsstark) powerless – powerfull (kraftlos – kräftig) completely hissing – not hissing at all (völlig zischend – gar nicht zischend) (continued)

7.3

Two Psychoacoustic Metrics for Fan Noise

135

Table 7.1 (continued) No. 21 22 23 24 25 26 27 28 29

According to Töpken and van der Par for fans in general [10] undamped – damped (ungedämpft – gedämpft) not noisy – noisy (nicht lärmend – lärmend) not-propeller like – propeller-like (nicht propellerartig – propellerartig) not booming – booming (nicht wummernd – wummernd) monotonous – varied (monoton – abwechslungsreich) irregular – regular (ungleichmäßig – gleichmässig) slow – fast (langsam – schnell) powerless – powerful (kraftlos – kräftig) hollow – full (hohl – voll)

According to Feldmann for air handling units with low sound power level [11] completely rustling – not rustling at all (völlig rauschhaft – gar nicht rauschhaft) completely whistling – not whistling at all (völlig pfeifend – gar nicht pfeifend) completely grinding – not grinding at all (völlig schleifend – gar nicht schleifend)

developed a sound quality prediction model. She started with 37 adjective scales in a listening test, the reduction to 23 adjective scales in Table 7.1 right was the result of a principal component analysis upon the listening test. The quality prediction model became Q = 10:84 - 0:396N ½sone - 2:40S ½acum - 1:24T ½tuHMS - 0:137H ½bit :

ð7:3Þ

Influencing parameters are the loudness N, the sharpness S, and the tonality T, each with a specific weight. The metric “roughness” is here replaced by the the Shannon entropy H.3 Q is an interval-scaled quantity. By definition, pink noise with a loudness of 11 sone corresponds to a sound quality of 1. More pleasantly perceived sounds have a higher, more unpleasant sounds a lower value. In principle, the scale is open at both ends. It is interesting noting that in the listening test with the comparatively small number of 69 test participants, three different groups emerged. The judgments of the largest group led to the prediction model Eq. (7.3). By means of parallel interviews also different groups could be identified:

3

Oetjen et al. [12] showed on the basis of automotive noise that the psychoacoustic metric roughness does frequently not correlate with the perception on an adjective scale from “rough” to “smooth”. The modulation of technical sounds is often random – the more random, the lower is the roughness perceived by human hearing. The Shannon entropy is commonly used in information theory to measure the randomness of a distribution. Here, the Shannon entropy allows a unified roughness prediction over different sound groups, Feldmann [11] and Feldmann et al. [13].

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About 40% of all test individuals stated that they preferred a “hum” to a “rustle”, while about 19% preferred the “rustle”. Another 41% preferred a “not too dark, not dominant hum”. From this, it can be seen that the criteria which quality judgments are based on can vary from individual to individual. Other annoyance or quality prediction models for vacuum cleaners are known, e.g. from Altinsoy [14] and Hülsmeier et al. [15], and for air conditioning systems from Susini et al. [16] and Jeon et al. [17].

7.4

Examples

Example 1 A domestic ventilation unit is considered. At a certain distance, the sound pressure spectrum shown in black in Fig. 7.3 is measured and A-weighted. The overall A-weighted sound pressure level is 62.4 dB(A). Due to a design modification, the sound of the device is lowered above 2 kHz, the red spectrum in Fig. 7.3. The A-weighted total sound pressure level, however, remains unchanged. The fan-specific rating level according to Eq. (7.1) is calculated as shown in Table 7.2 and is thus improved by about 4 dB by this design modification.

Fig. 7.3 Domestic ventilation unit: Sound pressure spectrum of the original (black) and the acoustically improved unit (red)

Table 7.2 Numerical values for Example 1 Noise spectrum of a domestic ventilation unit Lp,A from measurement Nratio according to Töpken and van de Par [10] Lr according to Eq. (7.1) (Töpken and van de Par [10])

Original 62. 4 dB(A) 0.96 70.0 dB

Modified 62. 4 dB(A) 1.41 66.3 dB

7.4

Examples

137

Fig. 7.4 Sound quality Q of 486 fans and air handling units as a function of the unweighted (left) and the A-weighted (right) overall sound pressure level

Example 2 The noise quality of a total of 486 fans and air handling units is determined from measured spectra with model Eq. (7.3) (Feldmann [11]). Figure 7.4 left depicts Q as a function of the unweighted, Fig. 7.4 right of the A-weighted overall sound pressure level. Firstly it is evident and expected, that the sound quality decreases with increasing overall sound pressure level. But clearly, the unweighted level Lp,oa is a worse measure for Q than the A-weighted Lp,oa,A. Fans and air handling units with the same measured Lp,oa can have a significantly different sound quality. By contrast, the correlation of Q with the A-weighted sound pressure level is considerably better. This can be explained by the fact that of course Lp,oa,A is more related to the psychoacoustic metric loudness N than Lp,oa. Thus, Lp,oa,A can be considered a first rough approximation for sound quality. Nevertheless, the sound quality scatters for a given value of Lp,oa,A by approximately five quality units. This again illustrates the fact that the A-weighted overall sound power level does not fully represent the annoyance. The margin at constant Lp,oa,A leaves plenty of room for improving the sound quality. Example 3 The sound quality of the domestic ventilation unit in Example 1 is determined from the spectra using the model Eq. (7.3). The device in its original state exhibits the sound quality Q = 0.5 (point A in Fig. 7.5). By reducing the noise above 2 kHz, ultimately the value of the psychoacoustic metric “sharpness” S is reduced. This increases the noise quality by ΔQ = 1.3 to Q = 1.8, point B in Fig. 7.5. A second option (of others) is to lower all spectral components by the same amount while keeping the shape of the spectrum unchanged, i.e. to “simply” shift the black spectrum to lower levels. To arrive at the same improved sound quality Q = 1.8, one would have to decrease the A-weighted overall sound pressure level by 3.4 dB (A). This then results in point C in Fig. 7.5.

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Fig. 7.5 Two possible options for improving the sound quality of a domestic ventilation unit: from A to B by spectral modification according to Fig. 7.3, from A to C by decreasing the level of all spectral components by the same amount; the grey dots are taken from Fig. 7.4 for information only

Taking into account inherent uncertainties, both, the models of Töpken and van de Par [2, 10] and Feldmann [11], obviously predict the effect of this spectral modification on psychoacoustic annoyance and quality in good agreement.

References 1. Guski, R, Felscher-Suhr, U: The concept of noise annoyance: How international experts see it. Journal of Sound and Vibration (1999) 223(4), 513–527 2. Töpken, S., van de Par, S.: Determination of preference-equivalent levels for fan noise and their prediction by indices based on specific loudness patterns, J. Acoust. Soc. Am. 145 (6), June 2019, pp. 3399–3409, doi:https://doi.org/10.1121/1.5110474 3. Deutsches Bundesministerium für Umwelt, Naturschutz und Reaktorsicherheit: Sechste Allgemeine Verwaltungsvorschrift zum Bundes-Immissionsschutzgesetz: Technische Anleitung zum Schutz gegen Lärm – TA Lärm. Berlin 1998 4. DIN 45681. Akustik – Bestimmung der Tonhaltigkeit von Geräuschen und Ermittlung eines Tonzuschlages für die Beurteilung von Geräuschimmissionen (Acoustics – Determination of tonal components of noise and determination of a tone adjustment for the assessment of noise immissions). Berlin: Beuth-Verlag, 2005 5. Blauert, J., Jekosch, U.: Sound-quality evaluation – a multi layered problem. Acustica united with Acta Acustica Nr. 5, Jg. 83 (1997), pp. 747–753 6. ISO 532-1. Acoustics – Methods for calculating loudness – Part 1: Zwicker method. Berlin: Beuth-Verlag, 2017 7. DIN 45692. Messtechnische Simulation der Hörempfindung Schärfe (Measurement technique for the simulation of the auditory sensation of sharpness). Berlin: Beuth-Verlag, 2009 8. ECMA International European association for standardizing information and communication systems: ECMA 74 (15th edition). Measurement of airborne noise emitted by information technology and telecommunications equipment. Genf: ECMA International, 2018

References

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9. ANSI American national standards institute: ANSI S1.13-2005/A7. Measurement of sound pressure level in air. Washington D.C., 2005 10. Töpken, S., van der Par, S.: Perceptual dimensions of fan noise and their relationship to indexes based on the specific loudness. Acta Acustica United with Acustica. Vol. 105 (2019). pp. 195–209 11. Feldmann, Carolin: Ein psychoakustisches Prognosemodell für die Geräuschqualität lufttechnischer Geräte mit niedrigem Schallleistungspegel, Shaker Verlag, Düren, 2019, ISBN 978-3-8440-6678-4 (also Ph.D. thesis University Siegen) 12. Oetjen, A., Letens, U., van de Par, S., Verhey, J. L., Weber, R.: Roughness calculation for randomly modulated sounds. Proc. of 40th Annual Conference on Acoustics ant the 39th German Annual Conference on Acoustics (DAGA) Berlin: DEGA, 2013, pp. 1122–1124 13. Feldmann, C., Lhonneur, G., Carolus, T., Schneider, M.: Time structure analysis of fan sounds. Proc. of the International Conference FAN2018, Darmstadt, 2018 14. Hülsmeier, D., Schell-Majoor, L., Rennies, J., van de Par, S.: Perception of sound quality of product sounds: A subjective study using a semantic differential. Proc. Internoise. Melbourne, 2014 15. Altinsoy, M. E.: Towards an European sound label for household appliances: Psychoacoustical aspects and challenges. Proc. of 4th International Workshop on Perceptual Quality of Systems. Wien, 2013, pp. 85–90 16. Susini, P., McAdams, S., Winsberg, S., Perry, I., Vieillard, S., Rodet, X.: Characterizing the sound quality of air conditioning noise. Appl. Acoust. 65 (2004), pp. 763–790 17. Jeon, J. Y., You, J., Jeong, C. I., Kim, S. Y., Jho M. J.: Varying the spectral envelope of air-conditioning sounds to enhance indoor acoustic comfort. Build. Env. 46 (2011), pp. 739–746

8

Design Features of Noise Reduced Fans

This chapter presents a selection of design features aiming at low-noise fans. In contrast to attenuators in the flow path up- or downstream of the fan these design features are intended to mitigate the noise generation itself. Hence they are primary noise reduction measures. Actually, the dominant sound source of a fan must be known before implementing a certain measure. However, a careful source analysis usually requires considerable effort and is not always possible in everyday industrial practice. Moreover, design measures aiming at weakening a particular sound source often have an effect on other mechanisms as well. This illustrates the problematic nature of many explanations concerning the effect of a particular noise reduction measure. Ultimately, the reduction of aerodynamically generated sound in fans is always a complex and difficult task that often requires extensive test series in addition to theoretical analyses. The examples compiled in this chapter are to be understood as a collection of ideas. Unfortunately, the occasionally given numerical values of a level reduction can rarely be transfered to other applications. It is worth noting that prior to any design modification it should always be checked whether the fan operates at or close to its acoustic optimum operating point. The dimensioning and selection of the fan is of great importance not only from an efficiency perspective, but also from an acoustic point of view.

8.1

General Measures

8.1.1

Reduction of the Circumferential Speed

The acoustic models of Chap. 6 show that the circumferential speed of the impeller determines the sound power substantially. For this reason, it is advisable to design fan impellers with the lowest possible circumferential speed. However, since the aerodynamic # Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 T. Carolus, Fans, https://doi.org/10.1007/978-3-658-37959-9_8

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specification in terms of V_ and Δpt must be reached, this means fans with the largest possible values of the non-dimensional parameters φ and ψ t. The consequence is a high aerodynamic loading of the blades. As a drawback, the self-noise as listed in Table 5.1 can then become the dominant noise source, Roger [1]. The consequences of a high blade loading can be well explained using the example of a centrifugal elemental blade cascade with forward-curved blades, Fig. 8.1: In the blade channel, the flow detaches on the suction side of the blades, large zones of dead-water exist; ultimately, the downstream flow becomes very non-uniform with a jet/wake structure. The centrifugal fan with such an aerodynamically highly loaded blade cascade has a large aerodynamic power for a given size of the fan (see Fig. 1.5), but its overall specific acoustic power, e.g. according to Madison’s LWspec,M, is higher than that of a centrifugal fan with backward curved blades (see Table 6.1). There is no lack of attempts to deflect the flow in blade cascades with less separation aiming at reduced pressure fluctuations. Petrov et al. [3], for example, covered the inlet of a centrifugal impeller with co-rotating screens. They were intended to increase the turbulence and thus to energize the flow in the blade channels in order to reduce the risk of separation. However, in addition to an improvement in the sound spectrum, efficiency losses had to be accepted. Turbulence generators on the blade suction sides of an impeller with forward curved blades were not successful, in contrast to an isolated airfoil section in a wind tunnel, Lavrich [2]. Fig. 8.1 Jet/wake structure of the flow downstream of an aerodynamically highly loaded elemental blade cascade (schematic according to measurements, from Lavrich [2])

8.1

General Measures

143

Fig. 8.2 Tandem blades in a centrifugal impeller (schematic)

Fig. 8.3 Slots in the blade of a centrifugal impeller with forward curved blades (some blades not drawn for greater clarity); schematic after Saeki et al. [9]

Another attempt to allow an increase of aerodynamic load without boundary layer detachment are so called tandem blades, Fig. 8.2 [4–7]. They had been investigated for both, axial and centrifugal fans. The gap somewhere in the blade allows a jet from the blade pressure to the suction side supplying additional kinetic energy to the suction side boundary layer. However, as shown by Embleton [8], the effect of this measure on the overall noise is rather small. A small acoustic and aerodynamic benefit was achieved by Saeki et al. [9] with slot-type openings near the leading edge of a centrifugal impeller with forward curved blades, Fig. 8.3. The slot width was 1 mm for an impeller with diameter of 170 mm. The slots – according to the result of their CFD study – eliminated a recirculation on the blade suction side. Carolus [10] investigated tandem blades in a low pressure axial fans with comparatively little effect. In order to homogenize at least the non-uniform flow immediately downstream of centrifugal impellers (compare the jet/wake structure in Fig. 8.1), Petrov et al. [11] mounted co-rotating screens over the outlet. A noise reduction was observed, but mainly in cases where the impeller flow was inherently poor.

144

8.1.2

8

Design Features of Noise Reduced Fans

Increasing the Spacing Between Stationary and Rotating Components

If the spacing between stationary and rotating components is increased, the interaction of the potential fields of the components is weakened. In addition, the further downstream, the more frictional wake effects are leveled out. As a consequence, downstream components face a more homogenized flow and a reduction of both, tonal and broadband noise can be expected. For acoustic reasons the axial distance from impeller to stator should always be as large as possible (Fig. 8.4). The same applies to struts and motor mountings. However, if outlet guide vanes are placed too far downstream of the impeller, the pressure recovery is less than expected, since part of the swirl is already dissipated. In centrifugal fans, the spacing from the impeller to the casing tongue (Fig. 8.4 right) is particularly critical. Strong pressure fluctuations are induced at the casing tongue, as the impeller exit flow passes by. This causes an effective sound radiation at blade passing frequency BPF and higher harmonics. However, too great a distance of the tongue from the impeller degrades the volumetric efficiency.

8.1.3

Phase Shift of the Interaction Between Stationary and Rotating Components

By inclining the guide vanes or struts of an axial fan in the circumferential direction (Fig. 8.5 left), the overlap is not linear but becomes point-like. This avoids that the blade wakes interact in phase over the entire blade height with the guide vanes. Alternatively, the impeller blades can be inclined by skewing, see Sect. 8.3.2. In the case of the centrifugal impeller, an oblique tongue (Fig. 8.5 right) or inclined trailing edges of the blades relative to the impeller axis target at the same phenomenon [8]. In all cases the transient forces are reduced and in the acoustic near field destructive interference and a reduction of BPFrelated noise is achieved.

Fig. 8.4 Acoustically critical spacings

8.1

General Measures

145

Fig. 8.5 Point-by-point overlap of stationary and rotating components; left: Inclined struts, right: Inclined tongue in a centrifugal fan

Fig. 8.6 Stationary sinusoidally contoured inflow disturbance element for generating a secondary BPF-tone that destructively interferes with the primary BPF-tone; after Gérard et al. [12, 13]

Gérard et al. [12, 13] made use of the interference even more specifically to minimize BPF-related tones. They considered an axial fan for automotive engine cooling, which, due to its installation with struts close to the impeller and partial blocking of the downstream flow, generates tones – the sound generation mechanism “unsteady loading noise” (Table 5.1) is active. They mounted an additional stationary plate with sinusoidal outer contour in front of the fan (Fig. 8.6), which generates a secondary interaction tone. By choosing the circumferential and axial position of the plate in relation to the struts and the blocking, the modal structure of the secondary sound field can be phase-shifted in such a way that destructive interference and thus reduction of tones occurres. Potential critical side effects are the excitation of higher harmonics and a degradation of aerodynamic efficiency. An experimental method for optimizing the position of the inflow element was described by Goth et al. [14]. There and in Gérard et al. [15] this measure was also applied to the minimization of tones in centrifugal impellers.

146

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Fig. 8.7 Non-uniform blade spacing for a five-bladed axial impeller; top: minimal deviation from uniform blade spacing, unbalanced; bottom: naturally balanced; after Mellin and Sovran [16]

8.1.4

Uneven Blade Spacing

The tonal content of the sound can be modified by uneven circumferential blade spacing. For instance, the sound can be tuned towards “white noise” (all spectral components have the same energy) which is perceived as less annoying.1 In many cases, this does not change the overall sound power level. A classical proposal was made by Mellin and Sovran [16] in 1970. They considered steady loading noise from an isolated axial impeller and determined blade spacing distributions with which the sound power of the BPF-related tone is reduced to that of the next strongest harmonic. A distribution with the smallest possible deviation from a uniform spacing results in an impeller that is statically unbalanced. Here the centrifugal forces of the blades G do not form a closed vector train, so additional balancing weights are required. In addition, Mellin and Sovran also came up with “naturally balanced” distributions. Numerical values are compiled in Tables 10.7 and 10.8 of Chap. 10. Figure 8.7 shows both versions of blade spacing for a five-bladed impeller. Although the underlying blade-bound steady blade forces are rarely the dominant noise source (see Table 5.5 and Chap. 5), the spacing distribution of Mellin and Sovran provide a good first design, if the BPF-related tone has to be mitigated. These blade spacing distributions have also been applied to centrifugal impellers with similar benefit, Lauchle et al. [17].

1

Such a psycho-acoustically driven “sound design” is also addressed in Chap. 7.

8.1

General Measures

147

Fig. 8.8 Theoretical redistribution of the sound power level with sinusoidally modulated pitch of the impeller blades in a fan stage; after [18]

Tones due to rotor-stator interaction can also be reduced by non-uniform spacing. Due to the wakes of the impeller blades, the downstream stator blades experience alternating forces whose harmonic content can be affected by disturbing the circumferential periodicity of the impeller wake. The aim of the work by Duncon and Dawson [18] was to make the tonal noise components subjectively more pleasant by a sinusoidally modulated spacing of the impeller blades and even to reduce the sound emission from the stator by utilizing interference effects in the near field. They investigated a small axial fan with 32 impeller blades and 22 guide vanes. The impeller spacing was sinusoidally modulated as θi =

h i 360∘ 2πi , i = 1,2, . . . ,z: i þ a sin z z

ð8:1Þ

θι is the angular position of the i-th blade, measured from the origin of an impeller-fixed coordinate system, a is a constant, in their study set to 0.5. Figure 8.8 shows the theoretical redistribution of the sound power due to this measure, which could be validated experimentally with a few limitations. In [19], the authors also showed a way to reduce the tone via a non-uniform spacing of only the guide vanes.

8.1.5

Wavy Leading Edge and Serrated Trailing Edge

Howe [20] was one of the first to point out the potential of a serrated trailing edge, Fig. 8.9, targeting at the mitigation of trailing edge noise (see Table 5.1). Gruber et al. [21] performed systematic variations of the geometry of the serrations. A study by Carolus et al. [22] on an airfoil segment, operated at Rel = 3.5 × 105 in a wind tunnel, showed up to 3 dB noise reduction in a certain frequency range, if the trailing edge serrations had the geometry parameters h1/δ = 3.0 and h2/h1 = 1.00. δ is the boundary layer thickness at the trailing edge of the airfoil segment on the suction side. The best angle inclination (flap angle) φ was found to be +5°. Stahl et al. [23] investigated a different airfoil at a much higher Reynolds number Rel = 1.2 × 106. Trailing edge serrations with h1/ δ = 2.0 and h2/h1 = 0.35 brought up to 5 dB noise reduction in a certain frequency range. The optimum angle φ was also positive, but a function of the angle of attack α.

148

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Design Features of Noise Reduced Fans

Fig. 8.9 Trailing edge serrations on an airfoil segment

Fig. 8.10 Vision of an axial fan impeller with blades based on Polacsek [24]; the sinusoidally corrugated leading edges of the blades aim at reducing the blade leading edge noise

By contrast, given a poor inflow quality to the impeller, turbulence ingestion noise (cp. Table 5.1) may be mitigated by means of a wavy leading edge of the impeller blades, Fig. 8.10. Polacsek et al. [24] investigated an airfoil segment at rest in the wind tunnel with a turbulence-generating grid mounted upstream. For maximum decorrelation of sound sources along the leading edge, h2 should be approximately twice the correlation length of the incoming turbulence across the leading edge of the blade. (h1 and h2 are the geometric parameters in analogy to the serrated trailing edge.) h1 should be chosen as large as possible from an acoustic point of view, but for aerodynamic reasons Polacsek et al. recommend a maximum of 20% of the chord length. A comprehensive and differentiated overview of the effect of wavy leading edges on aerodynamics and acoustics for both the airfoil segment and a low-pressure axial fan was provided by Biedermann [25].

8.1

General Measures

149

Gruber et al. [26] investigated a tandem airfoil arrangement with serrated leading and trailing edges. They succeeded in reducing both the trailing edge sound of both airfoils and the leading edge sound from the downstream airfoil.

8.1.6

Optimum Inlet Geometry and Turbulence Control Screen

The spatio-temporal uniformity of the inflow to the impeller is crucial for low noise fan installations. Spatial inhomogeneities cause unsteady loading noise, whereas temporal are setting off turbulence ingestion noise. In the case of an intake from the free undisturbed environment, a well-designed bellmouth-type inlet nozzle is of utmost importance. Figure 8.11 shows the effect of increasing the bellmouth’s size on the aerodynamic and acoustic characteristics. (It is worth noting that in this example the BPF-related tones of the assembly with the small bellmouth may origin from an unfavorable rotor-stator interaction due to the proximity of the impeller to the struts). Too small a bellmouth or – even worse – an inlet with sharp edges creates flow separation which degrades the inflow quality substantially. Occasionally, low pressure axial fan assemblies with a very short bellmouth show their best noise performance if the impeller is mounted such that the blades protrude slightly Fig. 8.11 Influence of the bellmouth on the aerodynamic and acoustic characteristics of a low pressure axial fan

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Design Features of Noise Reduced Fans

from the nozzle, Stütz [27]. This is surprising and the underlying mechanism seems still to be not fully understood. Ultimately, disturbed inflow can be homogenized by external upstream devices. It was recognized early in aircraft engine technology that the acoustic signature of an engine is different when tested on ground or attached to a flying aircraft. In the former case, large and small vortical structures, which always occur inevitably near the ground, are sucked in. Therefore, the acoustic emission of an engine on the ground is always higher than in flight in the undisturbed free atmosphere. To eliminate this effect, today’s acoustic ground tests are usually performed with a hemispherical turbulence control screen in front of the engine inlet [28], Fig. 8.12 left. In [29–31] dimensions and design suggestions for efficient turbulence screens can be found. Schneider [32] developed and tested a turbulence control screen for a small axial fan according to Fig. 8.12 right. It is made of a combination of flexible honeycomb sections and a wire mesh. The honeycombs have a cell length of 30 mm and an average cell diameter of about 3 mm. The diameter of the mesh wire is 0.36 mm with a mesh size of 1 mm. According to [31], the ratio of the outer screen diameter to the inlet diameter immediately in front of the impeller (here 300 mm) should be between 2 and 5, here it was chosen as 3. This screen very efficiently homogenizes the inflow with the consequence of an almost complete elimination of “stator-rotor interaction” tones, otherwise observed with the fan on a laboratory test rig, see also Carolus et al. [33], Sturm and Carolus [34] and Sturm [35]. Like the bellmouth inlet, a turbulence control screen is not an immediate component of the fan. Nevertheless, some manufacturers already integrate simplified turbulence control screens into their (smaller) fan-assemblies [36]. Moreover, even if a laboratory fan test rig has been designed according to the aerodynamic and acoustic standards – for instance an

Fig. 8.12 Left: Turbulence control screen at the inlet of an aircraft engine for ground tests, from a European patent specification [28]; right: Inlet nozzle with turbulence control screen for an axial fan with 300 mm impeller diameter (not drawn), from Schneider [32]

8.2

Further Special Measures for Centrifugal Fans

151

anechoic suction chamber -, the inflow provided is rarely fully homogeneous and hence affects the sound signature of a fan. Therefore, it would be expedient for future standardized acoustic laboratory measurements to generally employ a well-designed turbulence control screen.

8.2

Further Special Measures for Centrifugal Fans

8.2.1

Meridional Contour of Shroud

Fehse and Neise [37] showed that low-frequency broadband sound form centrifugal fans with backward-curved blades is caused by flow separation in the impeller. Correlation of unsteady flow and acoustic quantities revealed that the sound sources are located at the shroud and the blade suction sides. A contour of the shroud as in Fig. 8.13 with the geometry parameters according to Table 8.1 proved to be acoustically advantageous. Whether the gap geometry between the stationary inlet nozzle and the impeller – which in itself is not entirely uncritical – plays an acoustic role was not investigated in more detail. Fig. 8.13 Centrifugal impeller with a shroud for minimum low-frequency noise, according to Fehse and Neise [37]

Table 8.1 Geometric parameters of a centrifugal impeller, according to Fehse and Neise [37] βB1 21°

βB2 48° – 42° (at back plate and shroud, respectively)

D1/D2 0.75

b2/D2 0.25

rD/D2 0.17

152

8.2.2

8

Design Features of Noise Reduced Fans

Suppression of Resonance

The noise of centrifugal fans with a bladed diffuser can increase significantly at certain impeller rotational speeds. Sugimura et al. [38] attributed this to a resonance problem. The interaction of the impeller blades with outlet guide vanes can produce standing pressure waves in both, the blade channels of the impeller and in the bladed diffuser. As in an organ pipe, one side of the air column experiences an excitation by an alternating pressure, the other side can be considered acoustically open. In the outlet guide vane channel in Fig. 8.14 the column of air is excited with the angular frequency ω = jzΩ, j = 1, 2, . . ., i.e. BPF and its higher harmonics. The effective length of the oscillating air column l’ is somewhat longer than the geometric length l. This is accounted for by an end-correction l0 =

  b 1 2 2a þ ln : 2 π π b

ð8:2Þ

a is the longer side of the rectangular cross-section formed by two guide vanes and shroud and backplate. Assuming plane wave propagation in the channel resonance occurs if   ω 1 ðl þ l0 Þ = i þ π, i = 0,1,2, . . . : c0 2

ð8:3Þ

i = 0 corresponds to a quarter, i = 1 a three-quarter wavelength in the blade channel, etc. The critical angular velocities of the impeller where resonance is to be expected become

Fig. 8.14 Standing pressure wave in the outlet guide vane channel of a bladed diffuser and compensation hole for damping, after Sugimura et al. [38]

8.3

Further Special Measures for Axial Fans

Ωcrit,i =

  c0 1 i þ π, i = 0,1,2, . . . , 0 2 ðl þ l Þjz

153

ð8:4Þ

j = 1 corresponds to BPF, j = 2, 3, . . . to its higher harmonics. In the original paper, the flow velocity in the duct was also taken into account. The authors propose compensating holes in the guide vanes for suppressing resonance, Fig. 8.14.

8.3

Further Special Measures for Axial Fans

8.3.1

Tuning the Number of Blades (Mode Propagation)

By wise choice of the number of rotor and stator blades (or struts), modes in a duct-type casing can be excited, which are incapable to propagate. The theory was discussed in Sects. 5.4.2 and 5.4.3. This is a frequently validated concept of preventing certain BPF-related tones from being emitted from the openings into the free environment.

8.3.2

Skewed Blades

Historically, blade skew was frequently applied to propellers. Most probably, the fundamental patent “Low noise fan” by Gray [39] in 1980 introduced blade skew into the axial fan industry. Since then it has become an important design feature for small to large low noise axial fans. The number of scientific publications investigating the effect of skew, or more specifically sweep and dihydral, are countless. Skewed blades have already been addressed in Sect. 4.2.2 from an aerodynamic point of view. Skewing of the blades, however, influences several of the fundamental tonal and broadband generation mechanisms described in Sect. 5.1. This is the content of the following paragraphs. Effect of skew on tonal noise Frequently, a destructive interference effect is attributed to skewing. The idea is that phase-shifted blade-bound stationary forces at different radii are superposed. However, this is true only for fast rotating propellers with high circumferential Mach number (see [40, 41]), but hardly for axial fans with their relatively slow rotational speed. An investigation by Stütz [42] seem to support this: He tried to predict the power of the tonal sound of an isolated axial fan impeller. His computed thickness and steady loading noise was 15 dB below experimental data. In addition, the prediction turned out to be independent of blades skew. Instead, blade skew in axial fans seems to address the unsteady loading noise, i.e. the mechanism “blade-unbound unsteady forces due to spatially non-uniform inflow”. This already became clear in the work of Cumming et al. [43], who investigated skewed ship propellers. The inflow to the propeller is spatially strongly non-uniform due to the ship’s stern. If a disturbance hits the blade synchronously over the full span, the resulting blade

154

8

Design Features of Noise Reduced Fans

force becomes maximal. Hence, the skillful phase mismatch of disturbances colliding with the blade leading edges is mandatory for minimizing the peak blade force occurring during one impeller revolution. Depending on the spatial distribution of the inflow, blade skew can be an option. Hayden [44] confirmed the success of this measure for in case of an aircraft propeller, Stütz [42] for a fan impeller downstream of several purely radial struts. Effect of skew on broadband noise According to an argument by Wright et al. [46] the fluid in the near wall region on the blades is forced to move outwards due to the centrifugal forces. The distance covered by fluid particles form leading to trailing edge is larger if the blade is backward and, vice versa, shorter if forward skewed. As a consequence, a thicker or thinner turbulent boundary layer develops on the blade surfaces with higher or lower levels of the wall pressure fluctuations and hence sound emission, respectively. Ffowcs Williams and Hall [47] published in 1970 a theory for the sound generation of a turbulent flow over a stationary plate with an edge. They found the sound power to be proportional to sin2κ where κ is the angle between the edge and the flow direction (note that sinκ < 1 for |κ| < 90°). The authors concluded: “. . . this does suggest that noise from a sharp-edged surface can be considerably reduced by giving it a swept wing characteristic”. Hayden [44] and Brown [45] refer to the interaction of the turbulent boundary layer with the blade trailing edge and the vortex formation collectively as “vortex shedding” phenomena. They hypothesized that the flow velocity component wn normal to the edge is decisive for the sound generation by vortex shedding. The sound power is then proportional to cos2δ, where δ is the angle of inclination of the blade trailing edge relative to the radial direction. Experiments with a propeller operated with two directions of rotation and thus different trailing edge inclination seemed to confirm their hypothesis, Fig. 8.15. For an airfoil Kerschen and Envia [48] modelled the sound generation by turbulent inflow and the influence of leading edge inclination. According to their theory, at low Mach numbers of the inflow even a small leading edge inclination provides a destructive interference of the sound pressure generated at different spanwise locations. In the case of a rotating axial fan blade the leading edge inclination must be increased by a factor of 2–4 compared with the results from their simple theory [45]. Effect of skew on the stall point Well-designed skewed blades can also extend a fan’s “healthy” range of operation without running into stall (Carolus and Stremel [49]). The two graphs in Figs. 8.16 show the measured characteristics of the effective pressure rise and the Fig. 8.15 On the influence of the blade trailing edge angle on the edge-normal flow velocity component, after Brown [45]

8.3

Further Special Measures for Axial Fans

155

Fig. 8.16 Skewed and unskewed blades: Comparison of pressure rise and sound power performance characteristics; upper diagram: High pressure axial flow fan with outlet guide vanes (Dtip = 305 mm, n = 1800 rpm, ρ = 1.16 kg/m3); lower diagram: Low pressure axial rotor-only axial fan (Dtip = 305 mm, n = 3000 rpm, ρ = 1.16 kg/m3); from [49]

intake duct sound power level of both a high- and low-pressure fan. For both fans two versions of impeller blades were tested, unskewed and essentially forward-skewed. The inflow was more or less homogeneous, the turbulence intensity low. As expected, the experimental data show that almost over the entire operating range the skewed blading is barely quieter than the conventional unskewed. However, the stall point, which is indicated by the more or less pronounced discontinuity of the characteristic curves, is moved significantly to lower volume flow rates when skewing the impeller blades. As a conclusion, if the unskewed fan is operated left from the stall point, the skewed fan still operates in the healthy range of its performance characteristic curves with a considerable benefit in noise emission.

8.3.3

Influencing the Tip Gap Flow

In order to reduce the noise caused by the secondary flow in the tip gap region between the rotor blade and the casing, a number of design features have been proposed and investigated, Fig. 8.17. Longhouse [50] and Yapp et al. [51] aimed at reducing the secondary flow at the blade tips by means of a rotating shroud. Moreover, to rectify the unavoidable recirculation flow in the tip region, Longhouse added a rotating bellmouth to the shroud, and Yapp et al. placed small guide vanes into the gap. Kameier and Neise [52] identified a broad hump in the spectrum as the acoustic footprint of the tip gap flow which was significantly reduced by a brush-like turbulence generator attached to the casing wall next to the blade tip. The same effect showed a solid or ablatable protrusion with an

156

8

Design Features of Noise Reduced Fans

Fig. 8.17 Design measures to reduce tip clearance noise; (a) Co-rotating shroud and inlet nozzle according to Longhouse [50], (b) Controlled recirculation of the leakage volume flow according to Yapp et al. [51], (c) Turbulence generator in the gap area according to Kameier and Neise [52], (d) Protrusion and blade groove according to Zhu and Carolus [53], (e) Winglets according to [54]

optional complementary groove in the blade, Zhu and Carolus [53]. The utility model [54] describes the idea of winglets, i.e. plate-shaped elements on the tip of each blade.

References 1. Roger, M.: On the noise of open rotors. In: Breugelmans, F. A. E., Anthoine, J. (Hrsg.): Noise in turbomachines. Lecture Series 2000–02, von Karman Institute for Fluid Dynamics, 2000 2. Lavrich, P. L.: Three component velocity measurements behind a low speed, forward leaning centrifugal fan rotor. United Technology Research Center Report No. 90-27, 1990 3. Petrov, Yu. I., Khoroshev, G. A., Novoshilov, S. Ya.: Reduction of the noise level in centrigugal fans by means of transition meshes. Aus dem Russischen in das Englische übersetzt in NAVSTIC-Transl. 3322, 1972 4. Klassen, H. A., Wood, J. R., Schumann, L. F.: Experimental performance of a 13.65- centimetertip-diameter tandem-bladed sweptback centrifugal compressor designed for a pressure ratio of 6. NASA Technical Paper 1091, pp. 1–24, 1977 5. Bammert, K., Staude, R.: New features in the design of axial-flow compressors with tandem blades. ASME-Paper Nr. 81-GT-113, pp. 1–13, 1981 6. Matsumiya, H., Shirakura, M.: The performance of slotted blades in cascade. JSME, No. 182, pp. 1320–1334, 1980 7. Fryml, T., Heisler, V., Pavulach, L.: The influence of auxiliary blades on the characteristics and efficiency of centrifugal pump impellers. Proc. 7th Conf. on Fluid Machinery, Vol. 1, pp. 247–255, Budapest, 1983 8. Embleton, T. F. W.: Experimental study of noise reduction in centrifugal blowers. J. Acoust. Soc. Am. 35, pp. 700 – 705, 1963 9. Saeki, N., Kamiyama, K., Uomoto, M., Ishihara, Y.: Development of low noise blower fan. Paper No. 971842, Proc. VTMS 1997 Conference, Indianapolis, pp. 573–578, 1997

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10. Carolus, Th.: Experimentelle Untersuchung von Niederdruckventilatoren mit Spaltschaufeln. VDI-Forschung im Ingenieurwesen, Bd. 60, Nr. 7/8, pp. 173–179, 1994 11. Petrov, Yu. I., Khoroshev, G. A.: Improving the noise-level of centrifugal fans. Russian Engineering Journal 51, pp. 42–44, 1971 12. Gérard, A., Berry, A., Masson, P., Gervais, Y.: Experimental validation of tonal noise control from subsonic axial fans using flow control obstructions. J. of Sound and Vibration 321 (2009) 8–25 13. Gérard, A., Berry, A., Masson, P., Gervais, Y.: Modelling of tonal noise control from subsonic axial fans using flow control obstructions. J. of Sound and Vibration 321 (2009) 26–44 14. Gérard, A., Besombes, M., Berry, A., Masson, P., Moreau, S.: Tonal noise control from centrifugal fans using flow control obstructions. Fan2012, Senlis, Frankreich, 2012 15. Goth, Y., Besombes, M., Chassaignon, C., Gérard, A.: Fan tonal noise reduction using calibrated obstructions in the flow: An experimental approach. Fan2012, Senlis, Frankreich, 2012 16. Mellin, R. C., Sovran, G.: Controlling the tonal characteristics of the aerodynamic noise generated by fan rotors. Trans. of the ASME, J. of Basic Engineering, pp. 143–154, March 1970 17. Lauchle, G., Brungart, T.: Modifications of a vacuum cleaner for noise control. Internoise 2000, Nizza, France, 2000 18. Duncan, P. E., Dawson, B.: Reduction of interaction tones from axial flow fans by suitable design of rotor configuration. J. of Sound and Vibration, Vol. 33, No. 2, pp. 134– 154, 1974 19. Duncan, P. E., Dawson, B.: Reduction of interaction tones from axial flow fans by nonuniform distribution of the stator vanes. J. of Sound and Vibration, Vol. 38, No. 3, pp. 357–371, 1975 20. Howe, M. S.: Noise produced by a sawtooth trailing edge. J. Acoust. Soc. Am. 90 (1), pp. 482–487, July 1991 21. Gruber, M., Joseph, P.F., Chong, T.P.: On the mechanisms of serrated airfoil trailing edge noise reduction, 17th AIAA/CEAS Aeroacoustics Conference, Portland, Oregon, 2011 22. Carolus, T., Manegar, F., Thouant, E., Volkmer, K.: An experimental parametric study of airfoil trailing edge serrations. 7th International Meeting on Wind Turbine Noise, Rotterdam, 2–5 May 2017 23. Stahl, K., Manegar, F., Carolus, T., Binois, R.: Experimental investigation self-aligning trailing edge serrations for airfoil noise reduction. Proc. 8th International Conference on Wind Turbine Noise, Lisbon, Portugal, 2019 24. Polacsek, C., Reboul, G., Clair, V., Le Garrec, T., Deniau, H.: Turbulence-airfoil interaction noise reduction using wavy leading edge: An experimental and numerical study. InterNoise 2011, Osaka, Japan, 2011 25. Biedermann, T. M.: Aeroacoustic transfer of leading edge serrations from single aerofoils to low-pressure fan applications. Ph.D. thesis Technical University Berlin, 2019 26. Gruber, M., Joseph, P., Chong, T. P.: Noise reduction using combined trailing edge and leading edge serrations in a tandem airfoil experiment. 18th AIAA/CEAS Aeroacoustics Conference (33rd AIAA Aeroacoustics Conference), Colorado Springs, 2012 27. Stütz, W.: Untersuchungen zu der Wechselwirkung zwischen Einlaufdüse und Axialventilator. VDI-Berichte Nr. 1249, pp. 259–274, 1996 28. Inflow Control Device for Engine Testing. EP0249253, 1987 29. Scoles, J., Ollerhead, J. B.: An experimental study of the effects of an inlet flow conditioner on the noise of a low speed axial flow fan. National Gas Turbine Establishment, Report No. AT/2170/ 049/XR, University of Technology, Loughborough, 1981 30. Scheimann, J., Brooks, J.D., Comparison of experimental and theoretical turbulence reduction from screens, honeycomb and honeycomb-screen-combinations, J. of Aircraft 18 (8), 1981 31. Society of Automotive Engineers (SAE): Methods of Controlling Distortion of Inlet Airflow during static Acoustical Tests of Turbofan Engines and Fan Rigs. Aerospace Information Report AIR 1935, Warrendale, USA, February 1985

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32. Schneider, M.: Der Einfluss der Zuströmbedingungen auf das breitbandige Geräusch eines Axialventilators. Fortschritt-Berichte VDI Reihe 7 Nr. 478, VDI Verlag GmbH, Düsseldorf, 2006 (also Ph.D. thesis University Siegen) 33. Carolus, T., Schneider, M., Reese, H.: Axial flow fan broad-band noise and prediction, J. Sound and Vibration 300 (2007) pp. 50–70 34. Sturm, M., Carolus, T.: Tonal fan noise of an isolated axial fan rotor due to inhomogeneous coherent structures at the intake. Noise Control Engineering Journal, Vol. 60, No. 6, pp. 669–706, 2012 35. Sturm, M.: Tonal noise of axial fans induced by large-scale inflow distortions. Shaker- Verlag ISBN 978-3-8440-4279-5, 2016 (also Ph.D. thesis University Siegen) 36. Fan Flow Grid. US-Patent US D782645S, März 2017 37. Fehse, K.-R., Neise, W.: Entstehungsursachen tieffrequenter Druckschwankungen bei Radialventilatoren. VDI-Bericht 1249, pp. 139–153, 1996 38. Sugimura, K., Watanabe, M.: A study on surpressing acoustic resonance of interaction tones from a centrifugal motor fan. Proc. 7th Int. Congr. on Sound and Vibration. Garmisch Partenkirchen, pp. 1259–1266, 2000 39. Gray, L. M.: Low noise fan. U.S. Patent Nr. 4 358 245, 1982 40. Hanson, D. B.: Influence of propeller design parameters on far-field harmonic noise in forward flight. AIAA Journal Vol. 18, No. 11, Nov. 1989, pp.1313–1319 41. Mikkelson, D. C., Mitchell, G. A., Bober, L. J.: Summary of recent NASA propeller research. NASA Technical Memorandum 83733, 1984 42. Stütz, W.: Einfluß der Sichelung auf das aerodynamische und akustische Verhalten von Axialventilatoren. Strömungsmechanik und Strömungsmaschinen – Mitteilungen des Instituts für Strömungslehre und Strömungsmaschinen, 44/92, Universität Karlsruhe (TH), 1992 43. Cumming, R. A., Morgan, W. B., Boswell, R. J.: Highly skewed propellers. Trans. SNME, Vol. 80, 1972, pp. 98–135 44. Hayden, R. E.: Some advances in design techniques for low noise operation of propellers and fans. Noise-Con77, NASA Langley Research Center, 1977 45. Brown, N. A.: The use of skewed blades for ship propellers and truck fans. Noise and Fluids Eng., presented at Winter Annual Meeting of ASME, Atlanta, 1977 46. Wright, T., Simmons, W. E: Blade sweep for low-speed axial fans. ASME J. of Turbomachinery, Vol. 112, pp. 151–158, 1990 47. Ffowcs Williams, J. E., Hall, L. H.: Aerodynamic sound generation by turbulent flow in the vicinity of a scattering half plane. J. Fluid Mech. Vol. 40, Part 4, pp. 657–670, 1970 48. Kerschen, E. J., Envia, A.: Noise generation by a finite span swept airfoil. AIAA-Paper No. 83-0768, AIAA 8th Aeroacoustics Conference, 1983 49. Carolus, Th., Stremel, M.: Sichelschaufeln bei Axialventilatoren. HLH Bd. 51, August, pp. 33 – 39, 2000 50. Longhouse, R. E.: Control of tip clearance noise of axial flow fans by rotating shrouds. J. of Sound and Vibration, Vol. 58, No. 1, pp. 201–214, 1978 51. Yapp, M., Van Houten, R., Hickey, R.: Housing with recirculation control for use with banded axial flow fans. International patent WO 95/06822, 1994 52. Kameier, F., Neise, W.: Verfahren zur Reduzierung der Schallemission sowie zur Verbesserung der Luftleistung und des Wirkungsgrades bei einer axialen Strömungsmaschine. German Patent P 43 10 104.6, 1993 53. Zhu, T., Carolus, T.: Axial fan tip clearance noise: Experiments, Lattice–Boltzmann simulation, and mitigation measures. International Journal of Aeroacoustics 2018, Vol. 17(1–2) 159–183 54. DE 20 2004 005 548 U1: Fan (with winglet), German utility model, 2004

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Numerical and Experimental Methods

The next steps upon model based design of a fan involve manufacturing, either a full-size prototype or a scaled-down model, and experimental testing. Acceptance tests are preferably performed on standardized fan performance test rigs or, if impossible, in situ. A frequent intermediate step is the simulation of the fan performance with the help of numerical flow field simulation methods (Computational Fluid Dynamics, CFD). A test rig directly provides the integral parameters volume flow rate, pressure rise, torque, etc. In contrast, a CFD simulation always yields the detailed flow field, from which the integral parameters must be derived by integration and/or averaging. The fact that the flow field is always the starting point, is of advantage. Flow field variables are costly to obtain by measurement, but are of great value for the evaluation of flow details and ultimately for the intuitive improvement of the fan geometry. The same applies to the sound field and the sound power using computational aeroacoustics (CAA) methods. Instead of an intuitive improvement of a design one can also perform a systematic variation of geometric parameters and search for the best variant. This leads to – preferably CFD-based – optimization methods. CFD per se is not a fan design method. It only predicts the performance of a given fan geometry similarly to an experiment on a test rig. But on the other hand, CFD in the loop of an optimization scheme is in principle capable of extending or even replacing the classic fan design methods. The aim of this chapter is to provide a brief introduction into relevant numerical and experimental methods. An exhaustive treatment is out of the scope of this text book. The reader is referred to the literature including the documentation of commercial software, and also to the latest standards concerning test rigs and acceptance measurements.

# Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 T. Carolus, Fans, https://doi.org/10.1007/978-3-658-37959-9_9

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9.1

Numerical Flow Field Simulation

9.1.1

Overview of CFD Methods

Numerical and Experimental Methods

At each point in a flow a unique value of the six field variables • • • •

flow velocity (three components in three spatial directions) static pressure density temperature

exists. In case of unsteady flows these variables also depend on time. Predominantly two types of numerical methods are currently used for flow field simulation in fans: • The fluid is assumed to be a continuum and the system of conservation equations of mass and momentum (and energy) is solved; generally, these methods are called Navier-Stokes methods. • The fluid is modeled as a set of discrete fluid particles interacting with each other by convection and collision. However, the particles are not considered individually, but in terms of distribution functions of the particle density. Finally, the macroscopic flow field variables can be calculated from the distribution functions, among others. This numerical method is called the Lattice-Boltzmann-Method (LBM), lattice for “grid”. LBM is an inherently unsteady method and the compressibility of the fluid is taken into account. This makes LBM particularly suitable for computational aeroacoustics (CAA), i.e. for determining the flow-induced sound field. Depending on the simplification of the basic equations, the Navier-Stokes methods are classified. In the following, a short overview is given. In the flow through a fan with its typically low pressure rise and flow velocities (as compared to e.g. a turbo-compressor), the density and temperature variations are small – the fluid can be assumed to be incompressible. If one aims only at the aerodynamic performance characteristics, only the field variables “velocity” and “static pressure” are of interest. Hence, the solution of the fundamental continuity and Navier-Stokes-equations ∂ci =0 ∂xi

ð9:1Þ

9.1

Numerical Flow Field Simulation


 2 ∂ðρci Þ ∂ ρci cj ∂p ∂ c þ =þ μ 2i þ ρf i ∂t ∂xj ∂xi ∂xj

161

ð9:2Þ

is sufficient,1 and the energy equation is irrelevant. ci are the three components of the flow velocity, p the static pressure. fi stands for external forces and μ is the dynamic viscosity. When transforming the Navier-Stokes-equations into a rotating reference system – e.g. a fan rotor – fi represents, among others, the Coriolis forces, which have a significant effect on the flow field. The flow in fans is predominantly turbulent. This requires special strategies for solving the basic equations, which will be briefly explained. Direct numerical simulation – DNS If the initial equations are solved numerically without further simplifications, one speaks of a direct numerical simulation (DNS). Here, as schematically indicated in the lower row of Fig. 9.1, the complete kinetic energy spectrum of the fluctuation quantities E(k) is resolved in the computational domain. k is the wave number. The computational cost of a DNS is immense and increases with the third power of the Reynolds number. Therefore, a DNS of the flow in a fan is rather unrealistic. The DNS-method is mentioned here mainly for orientation. Large Eddy simulation – LES In the class of LES, the coarse structures in the flow field (large eddies) are resolved in space and time, whereas the fine structures, which have a universal, i.e. case-independent character, are modeled, Fig. 9.1. For example, a wellknown model is the subgrid-scale model by Smagorinsky [2]. LES is a compromise between DNS and the methods with a higher degree of approximation discussed below. Nevertheless, the numerical cost for a LES is huge, making parallelization of the computational code essential. A LES is particularly suitable for phenomena, where the unsteadiness of the flow is of relevance, like for stall phenomena or acoustics. Reynolds-Averaged Navier-Stokes simulation – RANS Resolving the details of a turbulent flow field is not needed in many applications, for instance for the prediction of the aerodynamic performance characteristics of a fan away from the stall point. Following the idea of Reynolds, it is then sensible simulating the flow field of time-averaged variables only. The Reynolds-averaged fundamental equations become

1

Partly for historical reasons, partly still today with some justification, occasionally further simplifications are introduced. E.g. the assumption of inviscid flow, restriction to two dimensions, and others. However, since the three-dimensional Navier-Stokes methods have become widely accepted – also in the form of commercial software – only these are treated in more detail here.

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Fig. 9.1 Resolved and modelled turbulent energy spectrum in CFD-methods with various degrees of approximation; for instance, LES is a low, RANS a high degree approximation method; schematically after Sagaut [1]

9.1

Numerical Flow Field Simulation

163

Fig. 9.2 Illustration of time averages. Example: Flow velocity at the exit of a free-running centrifugal impeller, measured with a hot-wire probe in the fixed frame of reference (“original”). The time interval of 0.4 s, corresponding to one revolution of the impeller, is chosen as an ensemble; ensembleaveraging over 1000 ensembles clearly reveals the wakes of the six blade channels. Time-averaging over one ensemble or, as here, over 1000 ensembles eventually yields the temporal “average” or mean value of the velocity

∂ci =0 ∂xi

ð9:3Þ

  2 ∂ðρci Þ ∂ ∂p ∂ ci þ ρci cj þ ρc0i c0j = þμ þ ρf i : ∂t ∂xj ∂xj ∂xi

ð9:4Þ

If the flow is steady except for the turbulent fluctuations, the time-averaged quantities are also steady and the time derivative term in Eq. (9.4) disappears. Eqs. (9.3) and (9.4) are then the initial equations for a Reynolds-averaged Navier-Stokes- (= RANS-) simulation. If the flow is deterministically unsteady at comparatively low frequency (e.g. the flow impinging on a stator, which has an exact time pattern due to the jet/wake structure of the upstream rotor), the pure time averaging leading to a mean value is to be replaced by ensemble averaging as illustrated in Fig. 9.2. Then in Eq. (9.4) the time derivative term is also relevant and the numerical procedure is called Unsteady Reynolds-averaged NavierStokes- (= URANS-) simulation, Sagaut [1].2 Each averaging produces a term ρc0i c0j , which is called the Reynolds stress tensor. It describes the influence of turbulence on the averaged flow field and contains the unknown turbulent velocity fluctuations, so that the system of Eqs. (9.3) and (9.4) is not closed.

2

Occasionally, URANS-simulations are also carried out when the solution has no clear temporal periodicity due to an external influence. According to Sagaut [1] for principal reasons those results are questionable.

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Numerical and Experimental Methods

Similar to a LES, the closure is performed by a turbulence model. Compared to the LES, however, the requirements for the turbulence model are considerably more demanding, since the entire energy spectrum (i.e. not only the universal fine structures, but also the anisotropic coarse structures) must be modeled, Fig. 9.1, upper row. The development of a single turbulence model that is applicable to a wide variation of flow problems, has so far proved very difficult. Therefore, a variety of models exist, including the k‐ε‐and the k‐ω‐turbulence model. The k‐ε‐model is particularly suitable for the approximation of turbulence in a free flow, the k‐ω‐model has advantages in the wall region. Combinations such as the shear-stress-transport model (SST, Menter et al. [3]) attempt to combine the advantages of both models. The RANS-method is currently the most widely used for standard tasks. Hybrid methods: Detached eddy simulation (DES): Turbulence models in URANS methods partially fail in detached flows. However, an accurate prediction of regions with separated flow including detachment and re-attachment point is desirable. Spalart et al. [4] proposed a hybrid method that combines a classical RANS with the elements of a LES. This approach, also called Detached Eddy Simulation (DES), uses a RANS-type turbulence model in the boundary layer region and switches to a LES in the detached flow. Figure 9.3 shows a comparison of the visualized results of a flow simulation with three different methods. An axial fan with an upstream generic safety grille that generates a highly turbulent inflow to the impeller serves as a test case. Without going into quantitative results, the plots reveal that the lower the degree of approximation of the simulation method, the more details of the flow field are resolved. Unfortunately, the computational effort and the degree of approximation are opposed to each other.

9.1.2

Computational Domain and Numerical Grid

The numerical solution of the flow field equations is performed in a predefined computational domain. The choice of its boundaries is important, since it may have an influence on the result of the simulation. As a rule, it is insufficient to confine the computational domain for instance to the blade passages. Adjacent flow regions or components, through which the flow passes, must also be included. As an example, Fig. 9.4 shows the computational domain for the simulation of a free-running centrifugal fan impeller. The computational domain at the exhaust and inflow region exceeds by far the size of the impeller. For a deeper discussion the reader is referred to the section on boundary conditions. The discrete locations in the computational domain at which the field variables are to be determined are defined by the numerical grid, essentially a discrete representation of the complete computational domain. For this purpose, the geometry model of the complete fan is passed over to a “grid generator”, i.e. a code for generating the numerical grid in the computational domain. A distinction is made between unstructured and structured grids. Three-dimensional unstructured grids are often constructed from tetrahedra (= polyhedra

9.1

Numerical Flow Field Simulation

165

Fig. 9.3 Test case “Axial fan with upstream ‘safety’ grille”: Comparison of unsteady numerical simulation methods; snapshots of the absolute velocity on a coaxial cylindrical surface at midspan and the static pressure on all solid walls; (a) Test configuration, (b) URANS, (c) DES, (d) LES; from Reese [5]

with four planar faces), whereas structured grids consist of hexahedra (= polyhedra with six planar bounding faces), Fig. 9.5. With structured grids continuous grid lines are associated. Tetrahedral grids are very flexible in meshing complex geometries, hexahedral less, but lead to a simpler data structure. In comparison, a larger number of tetrahedral than hexahedral cells are often needed to sufficiently resolve the computational domain. This mainly explains the longer computation time on a tetrahedral grid. In contrast, hexahedral

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Numerical and Experimental Methods

Fig. 9.4 Computational domain for the CFD-simulation of the flow in a blade channel of a freerunning centrifugal impeller

Fig. 9.5 Tetrahedron (left) and hexahedron (right) as elements of numerical grids

Fig. 9.6 Block-structured grid: Selected grid plane in the blade channel of a centrifugal impeller; the inflow and exhaust regions are also meshed; for illustration, the red line is one of the interfaces between two blocks

cells can be stretched in the wall region, saving grid elements. Since they can be aligned in the flow direction, also the so called numerical diffusion can be lower than with tetrahedral grids. Block structured grids combine the advantages of both types of grids. Here, different blocks of the computational domain are meshed with different grids. Examples of blockstructured grids are given in Figs. 9.6 and 9.7. The coupling of the different grids requires interfaces, where in some cases an interpolation of the flow field data is necessary.

9.1

Numerical Flow Field Simulation

167

Fig. 9.7 Some grid planes for the simulation of the axial fan from Fig. 9.3; only every third grid line is shown for clarity

9.1.3

Boundary and Initial Conditions

As with any initial/boundary value problem, so called boundary conditions must be defined for all variables in the basic equations at the boundaries of the computational domain. At walls The velocity of a fluid in contact with any stationary impermeable wall is zero. The blade channels of most axial impellers is bound by the stationary casing wall. This leads to a peculiarity in turbomachinery CFD: In the rotating frame of reference this casing wall moves with the circumferential speed in opposite direction of the impeller rotation which requires setting the wall boundary condition accordingly. The velocity profile of a turbulent flow above a solid wall can be divided into a viscous sublayer and a logarithmic wall region. In the viscous sublayer, the velocity increases linearly with the wall distance. However, this region is usually very small, so that this layer can only be resolved with a very fine computational grid. In order to avoid this high numerical effort involved, the flow in the wall region is often approximated by a so called logarithmic wall function. At an inlet At the inlet, where the fluid enters the computational domain, the velocity distribution or the mass flow and the inflow direction need to be specified. Depending on the turbulence model, turbulence variables such as the turbulence intensity or the turbulent length scales are also requrired. Note that the outlet boundary conditions selected can influence the choice of inlet boundary conditions. Periodicity If the geometry features periodicity, e.g. due to the circumferential sequence of blade channels in an impeller, then the computational domain can be confined to a single blade channel, in particular for RANS-simulations, Figs. 9.4, 9.6, 9.9, and 9.10. The pressure and velocity distributions are simply equated at opposite planes (“periodic boundary conditions”). However, this prevents the prediction of an axially asymmetrical flow, which might exist despite perfect rotational symmetry of the geometry. Figure 9.8 shows an

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Numerical and Experimental Methods

Fig. 9.8 Centrifugal impeller: Snapshot of the streamlines in a plane near the impeller back-plate, coloured with the relative velocity; SAS (Menter and Egorov [8] suggested another hybrid method: A URANS that exhibits LES-like behavior. In this so called Scale-Adaptive Simulation (SAS), the turbulence model automatically adapts dynamically to the computed flow field, without information about the nature of the numerical grid.); from Wolfram [7]

Fig. 9.9 Boundary conditions for a RANS-simulation of the flow in the blade channel of a free-running radial impeller; 1 inlet, 2 outlet, 3 wall in the rotating system, 4 wall in the stationary system, 5 periodic, 6 symmetric, 7 frozen-rotor; from Basile [9]

example of a snapshot of an unsteady vortex in the suction port of a centrifugal impeller, the center of which is displaced upwards with respect to the geometric center at this particular time instance. Such a phenomenon can only be resolved if the computational domain comprises the complete impeller. Symmetry If a geometric plane of symmetry is present in a computational domain, only one half of the domain through which the flow passes might be simulated, thus reducing the

9.1

Numerical Flow Field Simulation

169

computational effort. For this purpose, the symmetry plane is replaced by a frictionless wall. However, this again prevents the formation of an asymmetric flow. At an outlet As a result of the simulation, pressure and velocity distributions should adjust themselves at the outlet. For this purpose, e.g. the pressure at a single grid point is specified as reference. The calculated pressure field is then always related to this reference pressure. Note that for incompressible flow, the result is independent of the absolute pressure level. Other boundary conditions are also possible. For LES-methods, non-reflecting boundary conditions may be required in order to prevent non-physical wave reflection at the boundaries of the computational domain. Initial conditions At the beginning of the simulation, initial values are required at all grid points. Sometimes they are set to zero. When calculating the performance characteristics of a fan, it is often time-saving to use the flow field data of a neighboring operating point as initial values. In transient simulation, the starting conditions must represent a solution to the system of equations. Sometimes a RANS-simulation is used to produce starting conditions for a LES.

9.1.4

The Rotor-Stator Problem

Guide vanes are frequently called a “stator”, but other stationary components such as an ensemble of struts, a volute casing, a diffuser, etc. can be understood as stator-type as well. In a turbomachine, there are always two frames of reference, the absolute (laboratory) of a stator and the rotating of the impeller (or rotor). In principle, one could always perform a CFD simulation in either one. However, simulation of the rotor flow in the absolute frame of reference would require a constantly changing computational grid, a so called dynamic mesh, since the blades move through the computational domain at rest. Therefore, it is more convenient to simulate the impeller flow in the rotating frame of reference, although further external forces must then be added in the basic equations (see 9.1.1). The following possible procedures for simulating an interaction of rotor and downstream stator are proposed by Scheurer [6]: • Separate analysis of rotor and stator: Steps: (i) Calculation of the flow in the rotating rotor, (ii) Spatial circumferential averaging of the exit field variables, (iii) Use as input boundary conditions for the stationary stator, (iv) Calculation of the flow in the stationary stator. Advantages of this approach are that the calculation is easy to set up and the computation time is comparatively short. This procedure is useful when rotor and stator are far apart. Disadvantageous is the neglect of the stator on the upstream rotor flow. • Stationary rotor-stator interaction: In contrast to the previous procedure no circumferential averaging is performed, rather the instantaneous position of rotor and stator is

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Numerical and Experimental Methods

taken into account (“frozen rotor”). This is the minimum modelling effort to simulate the flow in rotationally asymmetric configurations like an impeller/volute assembly. The transient term in the fundamental equations is neglected. Advantages are the numerical robustness of the method and short computation times. However, it is still not possible to resolve the flow very accurately; the result depends on the position of the rotor in relation to the stator. It should be noted that a series of stationary calculations cannot provide the same result as a single transient simulation. • Transient rotor-stator interaction: Here the transient term in the basic equations is retained. The numerical grids of rotor and stator are moved relative to each other. Thus, the details of the entire flow field are well captured. Disadvantages are the higher effort in setting up the calculation, long computation times, and the higher effort in postprocessing the transient data. Depending on the choice of turbulence modeling, a simple URANS up to a LES can be used. The coupling of the grid blocks in different frames of reference (rotating/stationary) requires special interfaces or a dynamic grid overlap, depending on the method and the computer code.

9.1.5

Control Parameters, Convergence, Residuals and Termination of Iteration

The fundamental equations are nonlinear partial differential equations and cannot be solved in closed form. The essence of every numerical method is the approximation of these equations by discretization. Discretization schemes such as finite differences, finite volumes or finite elements provide a system of algebraic equations whose size depends, among other things, on the number of grid points or elements. The solution of the algebraic system of equations is done by iterative numerical methods. The user of CFD codes can choose the numerical method with its own discretization error. For example, methods with first order discretization error have good iterative convergence behavior, but are generally too inaccurate. In addition, the influence of the grid size is greater than for second-order methods. However, if convergence problems occur, one can, for example, start the calculation with a first-order method and use the converged solution as initial values for a second-order method. Each code has further numerical parameters with which accuracy and convergence behavior can be optimized, such as damping or relaxation factors and time step size. However, these parameters can influence the result. Convergence is measured by the course of so called residuals for each iteration step. Residuals are a measure for the variation of the field variables between two successive iterations. One distinguishes, among others, between the maximum residual in the entire computational domain and the L2-norm of the residual (a quadratic mean value of the residuals at all grid points, Oertel and Laurien [10]). The maximum residual gives information about local convergence problems, the L2-residual about global ones. The

9.1

Numerical Flow Field Simulation

171

theoretical goal of the iteration is to bring the residual to zero. Practically, to terminate the iteration, the residual must have decreased to a pre-selected value to ensure an acceptable iteration error.

9.1.6

Post Processing

A CFD-method always provides the flow field variables at each grid point or element – in the case of unsteady methods for each time step. From these data, all desired quantities must be determined. For example, from RANS-results following quantities which are of interest for the designer of a fan can be determined: • Circumferentially averaged flow velocities and flow angles at each section of the bladed cascade, particularly at the impeller inlet and exit planes; these quantities can be compared with the kinematics from the underlying design method (Chaps. 3 and 4). • Integral3 quantities such as volume flow rate and pressure rise; it is also interesting to calculate the shaft torque by integrating the static pressure and the shear stress on all wetted surfaces, which simply results in the shaft power and ultimately the inner efficiency; if results are available not only for the design point but for several operating points, the performance characteristics of the fan can be determined. • Visualized velocity fields in special planes (meridional, coaxial cylindrical), “wall” streamlines, pressure distributions. The data volume for transient simulations (URANS, LES) can become very large. In addition to spatial averaging, temporal averaging (yielding e.g. the quadratic mean of the fluctuation variables) may also be required. The frequency content at a monitoring point in the field can be determined with a straight-forward spectral analysis via Fast Fourier Transformation (FFT).4 In principle, all methods of digital signal processing can be applied. Transient flow field variables can be visualized as snapshots or movies.

9.1.7

Validation and Verification

Validation Every mathematical model of a physical problem requires validation, regardless of whether the mathematics involved is analytical or numerical. Statements must be

For the problem of averaging inhomogeneous flow fields see Sect. 2.3.2. It should be noted that in the Fast Fourier Transform (FFT), the frequency resolution and frequency limits are determined by the time step size and the length of the time interval in which data is available. 3 4

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made about the model error, i.e. the difference between reality and the answer of the mathematical model. When applying CFD methods, typical questions arise: • Have the underlying equations been appropriately selected (incompressible/compressible, turbulence model, steady-state/unsteady-state, time resolution, etc.)? • Are the computational domain and the boundary and initial conditions adequate? See also the previously discussed rotor/stator problem. Validation is ultimately only possible on the basis of experiments. Verification Whether the equations on which the model is based were solved correctly, must be verified. One can distinguish between two errors: • Discretization error, i.e. the difference between the exact solution of the fundamental equations and the exact solution of the system of algebraic equations produced by the discretization. • Convergence error, i.e. the difference between the terminated and the exact solution of the algebraic system of equations. An important criterion for assessing the discretization error is an analysis of the extent to which the solution is grid-independent. However, sometimes the computer capacities are already exhausted when using a comparatively coarse grid, so that a systematic refinement is no longer possible. In such cases it has to be critically questioned whether at least all rules (e.g. for a high quality grid) are fulfilled. For the assessment of the convergence error the question has to be answered to what extent the convergence criteria are fulfilled. Wellfounded guidelines for quality assurance in the application of CFD methods can be found in [11, 12].

9.1.8

Example: Axial Fan

For illustration purposes, a few results of a RANS of the flow in an axial impeller with swept blades are shown. The geometric data, which originate from one of the previously described analytical design procedures, are transferred to a “grid generator”, here with the aid of a CAD (Computer Aided Design) software. There the computational domain is defined and the numerical grid is created, Fig. 9.10. Then the boundary conditions are set. With the inlet boundary condition, the operating point of the fan, i.e. the volume flow rate for which the simulation is carried out, is fixed. Finally, the remaining control parameters are set and the simulation is started. If the userspecified convergence criterion is met, the iteration is aborted. Now the flow field data is evaluated. Figure 9.11 shows an example of the near-wall streamlines on the pressure and suction side of a blade and in the tip gap as well as in the hub region. In Fig. 9.12 the

9.1

Numerical Flow Field Simulation

173

Fig. 9.10 CAD-model of the impeller (left) and computational domain (right) with selected grid lines on bounding surfaces; Dtip = 0.300 m, Dhub = 0.168 m

Fig. 9.11 RANS-calculated near-wall streamlines; left: Blade suction side, right: blade pressure side; the casing is not shown; note also the streamlines on the blade tip (i.e. in the gap region); operating point = design point, n = 3000 rpm, ρ = 1.2 kg/m3

Fig. 9.12 Comparison of data from a RANS simulation and a design method as in Chap. 4; circumferentially averaged radial (= spanwise) distribution of the meridional and circumferential velocity in the exit plane of the axial impeller from Fig. 9.10; operating point = design point

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Fig. 9.13 Validation: Measured and RANS-predicted effective pressure rise and efficiency performance characteristics of the axial impeller from Fig. 9.10

RANS-calculated circumferentially averaged meridional and circumferential velocities downstream of the impeller are plotted from hub to tip. In the comparison, differences between the underlying analytical design and the numerical simulation can be observed: Secondary flows (vortices) in the hub and tip regions are revealed in Fig. 9.11. Naturally, they could not be taken into account in the preceding analytic design. Finally, Fig. 9.13 shows the pressure and efficiency performance characteristics of the fan. The comparison with test rig measurements ultimately confirms that the simulation results are valid.

9.2

Experimental Methods

9.2.1

Fan Test Rigs

The performance of fans is affected by the inflow conditions. Fan test rigs (occasionally also called airways) should provide a swirl-free and uniform inflow to the fan in order to eventually ensure reproducible performance data. Commonly used are duct and chamber test rigs. In case of a duct test rig the fan inlet (suction side) and/or the fan exit (pressure side) are connected to ducts with similar cross-sectional areas as the fan inlet and outlet ports. In contrast, a suction-side chamber test rig provides the inflow to the fan via a large chamber. More rarely, in a pressure-side chamber test rig the fan exhausts into a large chamber. Three examples of test rigs are presented in Fig. 9.14. Typical test rigs comprise: • An undisturbed intake region • A volume flow rate meter (inlet nozzle, orifice plate or Venturi meter) with an undisturbed free intake region

9.2

Experimental Methods

175

Fig. 9.14 Fan test rigs for measuring the aerodynamic performance characteristics of fans (schematic); top: Suction-side chamber test rig, middle: Suction-side duct test rig, bottom: Pressure-side duct test rig; 1 undisturbed inflow region (no obstruction, ideal inflow), 2 volume flow rate meter (inlet nozzle, orifice or Venturi meter), 3 flow straightener, 4 controllable auxiliary fan (optional), 5 adjustable throttle valve, 6 chamber with internal screen or duct, 7 test fan, 8 free exhaust region; dimensions and further details for a design as a standard test rig in the relevant standards

• A flow control unit (adjustable throttle valve with optional controllable auxiliary fan to compensate for the test rig’s pressure losses) • A settling chamber with internal screens and pressure tapping or a duct with pressure tapping and flow straightener • The actual fan to be tested. For the dimensions and details of the components the reader is referred to the latest standards.

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When determining a pressure rise of the fan from measured static pressure differentials ΔpChamber or ΔpDuct it must be carefully distinguished whether the total (total-to-total) or effective (total-to-static) pressure rise is aimed at. Pressure rise when testing on a suction-side chamber test rig Since, per definition, the diameter of the chamber is large, the flow velocity in the chamber is negligibly small. Then the effective pressure rise corresponds directly to the measured difference of the static pressures between chamber and laboratory: ρ Δpts = ΔpChamber - c2Chamber ≈ ΔpChamber 2|fflfflfflfflffl{zfflfflfflfflffl}

ð9:5Þ

very small

From this the total pressure rise is calculated as ρ Δpt = Δpts þ c2 2 2

ð9:6Þ

with the mean discharge velocity c2 =

V_ , A2

ð9:7Þ

where A2 is the fan outlet area of the fan5 . Pressure rise when testing on a suction-side duct test rig The effective pressure rise is

5

This approach is pragmatic but contradicts the analysis in Sections 2.3.2 and 10.1 in two respects: • The dynamic pressure is correctly calculated in Sections 2.3.2 and 10.1 as the energetic average c22 

1 V_

Z c22 cn2 dA A

(cn2 is the measured velocity component normal to the exit plane). Only if c2 is independent of the location on A2 and normal to this plane, then the equality c22 = c2 2 holds exactly true. • An axial fan without outlet guide vanes or a free-running centrifugal impeller cause a swirling flow in A2. With this procedure, however, the cu2 component is not taken into account. Hence, only a pseudo-total pressure rise, which is smaller than the physically correct one, results from Eqs. (9.6) and (9.7).

9.2

Experimental Methods

177

Δpts = ΔpDuct -

ρ 2 c : 2 Duct

ð9:8Þ

In contrast to a test on a chamber test rig, the dynamic pressure of the flow velocity in the duct is by no means negligible. More exact results would even require the consideration of the pressure loss of a possibly existing flow straightener between test fan and pressure tap. The total pressure rise is calculated as: ρ Δpt = Δpts þ c2 2 2 The remark in the footnote also applies here. Pressure rise when testing on a pressure-side duct test rig The effective pressure rise is Δpts = ΔpDuct ,

ð9:9Þ

whereas the total pressure rise becomes ρ Δpt = Δpts þ c2 2 : 2

Duct test rigs on the exit side of a fan are only feasible if the outflow from the fan is almost swirl-free. Only then is the distribution of the static pressure uniform in the plane of the pressure tapping. In the case of axial fans, for example, this requires outlet guide vanes. Moreover, the condition of swirl-free exhaust is confined to the vicinity of the design operating point. This means that in the vast majority of cases a performance measurement with this test rig is limited.

9.2.2

Measurement of Flow Field Quantities: Measuring Probes

When developing a fan, it is occasionally necessary to measure flow field variables such as pressure and flow velocity. In principle, a distinction can be made between point and planar measurement methods. A further distinguishing feature is the temporal resolution. Measurements in the flow can be intrusive or non-intrusive. Non-intrusive velocity measurements are mainly based on laser-optical methods (Laser Doppler Anemometry, LDA, Particle Image Velocimetry, PIV, etc.). Then usually particles have to be added to the flow. Laser-optical methods are not discussed here; the reader is referred to the special literature. In this section only simple and common intrusive methods are presented.

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Fig. 9.15 Hydraulic probes: (a) Total pressure probe (Pitot-tube), (b) Static pressure probe, (c) Prandtl-tube for determining the flow velocity

Fig. 9.16 Total pressure probe (PITOT-tube): Head shapes and permissible flow angle δ for a maximum error of 1% of the total pressure at low Mach number; according to Wuest [13] and Beiler [14]

Hydraulic probes Classical hydraulic probes measure a flow quantity at one point in the flow field. They are intrusive and must not be used in the immediate vicinity of walls. Figure 9.15 depicts probes for the total pressure (Pitot-tube), the static pressure and the flow velocity (Prandtl-tube) in a free flow field. All quantities are traced back to the measurement of one or two static pressures. The head bore hole of a total pressure probe points against the direction of the flow (Fig. 9.15a). In the hole, the velocity of the incoming fluid is decelerated to zero. Therefore, the measured static pressure equals the stagnation or total pressure in the flow. A probe, which is mounted in the stationary system of the laboratory system, faces an unsteady flow velocity when placed immediately downstream of a rotating fan. Magnitude and direction of the fluid velocity vary considerably with the instantaneous position of the passing blade channel with respect of the probe. Figure 9.16 shows various head shapes and their effect on the sensitivity to oblique flows. If an oblique flow to the probe is expected, a probe which is as insensitive to direction as possible, such as the Kiel-probe, is advisable. Figure 9.17 shows two variants of the Prandtl-tube. The stagnation pressure is measured as with the total pressure probe. The static pressure in the flow is taken at lateral bore holes.

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Experimental Methods

179

a b d r nst

Variant 1 (AVA)

Variant 2 (ASME)

20D 5D 0.3D 5D 4

25D 8D 0.5D 3D 8

Fig. 9.17 Prandtl-tube, see Wuest [13] Fig. 9.18 Five-hole probes: Left: Hemispherical, right: spherical probe, according to Wuest [13]

The difference between the two pressures corresponds to the dynamic pressure, from which the flow velocity can be calculated if the air density is known. Multi-hole probes yield also information about the direction of flow, Fig. 9.18. The probe is rotated in the flow field until the static pressures at opposite holes are equal. Alternatively, the flow direction can also be found without rotation of the probe by means of prior calibration. Probes with two pairs of bores are required for three-dimensional direction detection. If required, a total pressure bore can be added in the centre of the probe, resulting in the five-hole probe shown. Often the actual pressure transducer is connected to hydraulic probes via elastic plastic hoses. The entire system is characterized by a transfer function. Typically, only stationary or low-frequency signals can be record without distortion, Carolus [15]. To increase the resolvable frequency range, miniature pressure sensors can be integrated directly into the probe (lower design in Fig. 9.16). Hot-wire anemometry Figure 9.19 shows a Constant Temperature Anemometer (CTA). The decisive element is a fine wire6 which is heated. The fluid with its velocity c cools the heated hot-wire; the bridge voltage UB is readjusted so that the temperature of the wire remains constant. UB is a direct measure of the flow velocity. With the single-wire probe, only the velocity component perpendicular to the wire can be measured correctly. If two or three wires are arranged spatially in different positions on the probe head, two- or even three-dimensional velocity fields can be analyzed as well, Fig. 9.20. However, the evaluation of the probe signals of multi-wire probes is complex, Lekakis et al. [17]. Depending on the wire dimensions and other parameters, hot-wire anemometry can easily be used to Typical: Tungsten wire with a thickness of 2.5 μm and a length of 1 mm; also a quartz substrate coated with platinum (hot film).

6

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Fig. 9.19 Hot-wire probe and measuring bridge (CTA = Constant Temperature Anemometer), after Nitsche and Brunn [16]

Fig. 9.20 Head of a triple hot-wire probe

record comparatively high-frequency velocity fluctuations. This makes it interesting with regard to all transient (e.g. turbulent) and even acoustic flow phenomena. Hot-wire probes must always be calibrated, especially after a repair when a broken wire has been replaced. Pressure sensors The static pressure in the flow can be measured via the lateral bore holes of a static pressure probe or Prandtl-tube (Fig. 9.15b, c), the static pressure distribution on stationary walls (e.g. along the wall of a volute casing) by means of bore holes, flush with the surface. The measurement of the pressure distribution on rotating blade surfaces is more problematic. For this purpose, miniature pressure transducers on a piezo resistive basis can be applied. Miniature microphones are also increasingly being used as fluctuating pressure sensors [7, 18].

9.2.3

Measurement of Acoustic Quantities

According to the former DIN 45635 T38 [19], sound power levels for fans are to be differentiated according to Table 9.1 and Fig. 9.21. For the precise determination of these sound power levels, frequently an anechoic, semianechoic or reverberation room or a duct with anechoic termination is utilized. In-situ methods with a lower degree of accuracy (engineering or survey methods) are also common. This section summarizes most important features of a few selected measurement methods. More detailed information can be found in the relevant standards and text books on technical acoustics. The reader is also referred to Sect. 10.3, where some basic acoustic terms are explained.

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Table 9.1 Definitions of various sound power levels for fans (DIN 45635 T38 [19]) Designation Fan overall sound power level LW1 Casing sound power level LW2 Intake and exhaust duct sound power level LW3 and LW4, respectively Free intake and free discharge sound power level LW5, LW6, respectively Casing as well as free intake and casing as well as free discharge sound power level LW7, LW8, respectively

Measure of the sound power that is radiated via . . . . . . the intake and exhaust port and the casing into the free environment . . . the casing into the environment . . . the intake or exhaust ports into attached ducts . . . the intake or exhaust ports into the free environment . . . the casing as well as the intake or exhaust ports into the free environment

Fig. 9.21 Definitions of some sound power levels for fans

Reverberation room The sound source, i.e. the fan, is placed in a reverberation room. The walls of a reverberation room are reflecting and sometimes not parallel in order to keep the sound field diffuse. The adequacy of the test room must be established according to the standards. A decisive factor is the equivalent sound absorbing area of the room. It is determined a priori once via the reverberation time in the room. The sound pressure associated with the sound source is measured with microphones at several points in the

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Fig. 9.22 Determination of the intake sound power level LW3 of a fan in a semi-anechoic room; the dimensions r and R are needed for the discussion of the effect of the reflecting flow on the sound pressure at microphone X

room and the spatial mean value is taken. From this, the sound power of the source is calculated. Alternatively, the sound power of the sound source can be determined by comparison with a calibrated reference sound source. This method is also suitable for in-situmeasurements in a less defined reverberating environment. With the reverberation room method, all fan-relevant sound power levels can be determined except for LW3 and LW4. It is relatively simple, and the number of measuring points is comparatively small. The measurement of tonal sound, however, can be problematic. As a matter of principle the directivity of the sound field cannot be determined. Anechoic and semi-anechoic rooms In an anechoic room the sound over the frequency range of interest is absorbed by all walls. Free-field conditions are obtained. In a semianechoic room, the floor is hard and reflecting, whereas all other surfaces are absorbing. The sound source is placed in the anechoic measurement room, or – in case of fans frequently more practical – on the floor of a semi-anechoic room. Depending on the geometry of the sound source, a hypothetical sound source surface is defined – in

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Experimental Methods

183

Fig. 9.22, for example, a parallelepiped comprising the fan inlet and the reflecting floor. Then the actual measurement surface is defined – in the example of Fig. 9.22 in such a way that it surrounds the hypothetical sound source at a distance where the acoustic far field begins. The surface sound pressure Lp is obtained by measuring the sound pressure on the measurement surface at several points and spatial averaging. Eventually, the sound power becomes   S LW = Lp þ 10 log dB S0

ð9:10Þ

with S being the area of the measurement surface and S0 = 1 m2 the references area. Further information on how to define the geometry of the hypothetical sound source and the measurement surface, on the positions and number of microphones required as well as on corrections for possible extraneous noise and an imperfect measuring environment is given in the standards.7 With this method, all fan-relevant sound power levels can be determined except for LW3 and LW4. In general, the number of microphone measuring points required for this method is comparatively high. On the other hand, there are no restrictions on the measurement of tonal sound, and the directivity pattern of the sound source can always be determined. The use of the sound pressure level at some monitoring point away from the source in lieu of the sound power level is not recommended, but sometimes convenient for the sake of quickness, especially during a development process. If the sound pressure level is determined in a semi-anechoic room, the reflection from the floor can affect its spectrum considerably. As a theoretical example, consider the pressure signal at microphone X in Fig. 9.22. In order to take the reflected sound into account, one can simply superimpose the sound field of the original with that of its mirror sound source. If one assumes that the fan inlet radiates into the room like a spherically radiating point source,8 then the difference in sound pressure becomes  2  r ΔLp  Lp,semi - anechoic - Lp,anechoic = 10 log 1 þ e - ikðR - rÞ dB, R

ð9:11Þ

Carolus [20]. The frequency f is hidden in the wave number k = 2πf/c0. Within this example Eq. (9.11) is evaluated for the numerical values h = 1.35 m, r = 1.3 m,

7

If the measurement environment does not feature laboratory conditions, e.g. in the case of in-situ measurements, then the sound power can be determined by scanning an enveloping measurement surface around the fan with a sound intensity probe. A sound intensity probe records the acoustic part of the fluid velocity and the sound pressure simultaneously, so that the vectors of the acoustic intensity are known. In this method, (stationary) background noise and reflections are cancelled out. 8 See Sect. 10.3 for the basic acoustic terms.

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Fig. 9.23 Theoretical difference of the sound pressure level measured of the same source in a full anechoic and in a semi-anechoic room; according to Eq. (9.11) for the dimensions h = 1.35 m, r = 1.3 m in Fig. 9.22

Fig. 9.24 Duct test rig (schematic) for simultaneous measurement of the intake and exhaust duct sound power levels and the aerodynamic performance; 1 undisturbed inflow region (no obstruction, ideal inflow), 2 volume flow rate meter (nozzle), 3 anechoic termination, 4 transverse measuring plane for sound pressure, 5 pressure tapping, 6 test fan, 7 variable cone throttle, 8 flow straightener

i.e. R = 3 m. It is obvious from the graph in Fig. 9.23 that the sound pressure level at microphone X is amplified as a function of frequency by up to 3 dB or reduced by 5 dB, compared to a measurement in a full anechoic room. In contrast, when determining the spectral sound power as described above, the effect of the floor is of course irrelevant, since microphone signals from many different positions on the measurement surface are averaged. Duct test rig Figure 9.24 shows a double-sided duct test rig for simultaneous measurement of intake and exhaust duct sound power levels LW3 and LW4, respectively. Due to the anechoic terminations, an almost frequency-independent acoustic impedance is achieved at both ends, which corresponds to an infinitely long duct. In a defined transverse measuring plane of the duct, the sound pressure is measured at several points. The spatially averaged sound pressure levels, together with the area of the measurement plane SDuct, yield the sound level

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185

  SDuct LW = Lp þ 10 log dB: S0

ð9:12Þ

A particular challenge is the measurement of the sound pressure in a flow. To shield the microphone from turbulence-induced pseudo-sound, frequently a slotted probe (Friedrichtube [21, 22]) is often placed in front of the actual measuring microphone capsule. The in-duct method is relatively simple; the number of points on the measurement surface is small. However, extensive corrections of the sound power level obtained via Eq. (9.12) are required for standard-compliant results. The achievable accuracy is not as high as with the other methods. In general, fans should be tested in a duct with an anechoic termination on both ends. In many cases, however, this is impossible because upstream components of a fan assembly do not permit a duct on the suction side. Finally, a remark on the emission of sound from an opening. Assuming the sound power in a duct is known, which portion is radiated into the free environment? Especially at low frequencies, a large part of the sound energy is reflected at the opening and thus not emitted into the free field. According to the guideline VDI 3731 [23]9 the difference in sound power levels ΔLW = LW,Free field - LW,Duct is a function of the so called Helmholtz number He = k

DSource πfDSource = 2 c0

ð9:13Þ

with DSource as the diameter of the opening. VDI 3731 recommends the empirical correlation  ΔLW = 10 log

2:3He2 1 þ 2:3He2

 dB,

ð9:14Þ

which is plotted in Fig. 9.25.

9

VDI 3731 considers circular or square cross-sections; for arbitrary rectangular cross-sections see VDI 2081 [24].

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Fig. 9.25 Difference of sound power levels in a duct and emitted into the free environment through an opening, Eq. (9.14)

9.3

Optimization

Numerical simulation as well as experimental methods can be used to search for an optimal fan design by systematic variation of individual parameters. Optimal in the strict sense that there is no better design within the permissible range of parameters. In a more abstract sense, “optimal” is understood as the extreme value of a specified objective function. In general, the first step in an optimization procedure is the definition of one or more objective functions. The objective function uniquely assigns a numerical value to each combination of input variables (also commonly called parameters, frequently the geometric parameters) and thus quantifies the goodness of each parameter combination. The parameters are then systematically varied and the minimum or maximum value of the objective function(s) is sought. Example: The maximum possible efficiency of a centrifugal impeller is, among other parameters, a function of the number of blades. Note that in PFLEIDERER’s empirical design method (Chap. 3) the optimum number of blades cannot be determined analytically. In an optimization the objective function “efficiency” as a function of the parameter “number of blades” would be maximized by a systematic variation of the number of blades. The objective function must be evaluated for numerous combinations of the input variables, i.e. many real or “numerical” experiments are required. In fluid mechanics, the objective function is usually a nonlinear function of the input parameters chosen for minimizing or maximizing. Therefore, so called nonlinear optimization methods are necessary. Nonlinear optimization problems typically have the following properties: • Many local optima exist. • No analytical solution exists, so iterative solution procedures must be used. Since the aerodynamic performance of fans can be predicted quite well with a CFD simulation, optimization methods that are based on the automated prediction of their

9.3

Optimization

187

performance by a CFD simulation are particularly suitable. Experimental optimization is also possible, but often very costly and time-consuming.10 In this section, exemplary optimization methods are briefly presented. A more detailed description of these and other optimization methods is provided, for example, by Nelles [26].

9.3.1

Optimization Procedures

A typical formulation of the objective function OF is     X   2 → → → OF p = OF  p ± αi Z i,target - Z i,actual p :

ð9:15Þ

i

OF* is the primary objective function, e.g. ηts of a fan. It depends on all components of → the parameter vector p that are allowed to be varied, for instance the geometrical parameters of an impeller. OF* can be modified into OF by a series of penalty terms. The penalties contain the weighted deviation from constraints Zi, e.g. in the case of efficiency optimization the deviation from the targeted pressure rise of the fan. The weights αi are necessary to initially bring the numerical values of OF and Zi into a similar order of magnitude. Furthermore, fine-tuning of αi enables controlling the relevance of each constraint. The more a constraint is violated, the worse the value of the objective function becomes. On this basis, the correct sign of the second term in Eq. (9.15) can be determined: For maximization problems (e.g. increasing efficiency), the penalty terms enter the objective function negatively, for minimization problems (e.g. reducing sound radiation) positively. Gradient-based optimization algorithms These algorithms require initial values for all parameters. The nearest local optimum is typically found by calculation of the gradient at the starting point. The gradient is used to determine the direction in which the parameters are to be varied. In the simplest case, this is the direction of the steepest descent. However, more efficient algorithms also use the information from the gradient of the previous iteration, e.g. the “conjugate gradient” or “quasi-Newton” method. Figure 9.26 shows an example of an objective function with only one parameter p to be varied, a chosen initial value and the gradient (derivative) at this initial value. Obviously, depending on the starting value, the global optimum cannot necessarily be found. Therefore, one can try to find the global optimum by so called multi-start methods, in which optimizations are executed for

10

An example has been described by Kohlhaas et al. [25]. Components of the tonal sound due to rotor-stator-interaction of an axial fan stage were eliminated completely filling the wake immediately downstream of the impeller blades. The optimal spanwise distribution of the air blown out through slots close to the blade trailing edges was found by a true automated experimental optimization.

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Fig. 9.26 Gradient-based optimization

several initial values. Especially for problems with a large number of parameters the choice of suitable initial values is difficult. Evolutionary Algorithms These algorithms were developed following the biological evolution theory: “Bad” individuals in a generation die out, while the “better” individuals mate and reproduce. By mixing the genes of two parent individuals, the offspring is created. If the offspring has positive characteristics, it has a good chance of reproducing, otherwise it is very likely to die out. In addition to the mixing of genes, mutations also play a decisive role. These are random changes in the genes that can have both positive and negative effects on the goodness of an individual and thus increase or decrease its chances of reproduction. In the technical implementation, the genes correspond to the parameters. A set of parameters describes an individual. A generation consists of a selectable number of individuals. The genes of the first generation are chosen to produce individuals that are as diverse as possible but sensible. The selection of parents for the reproduction of the next generation can be done by a methodology described by Thévenin and Janiga [27]: First, individuals are ranked, with rank one corresponding to the best value of the objective function and rank N to the worst. Once the rank ri of each individual i is known, it can be converted into a fitness value Fi =

N - ri þ 1 : N
 P N - rj þ 1

ð9:16Þ

j=1

The better (lower) the rank of an individual, the greater its fitness value. Often the best ranked individual passes directly into the next generation and is also exempt from the mutations described below. This is known as the “survival of fittest”. Each of the other individuals in the next generation require two parent individuals. The probability of an individual being chosen as a parent for reproduction is proportional to its fitness value.

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Optimization

189

Individuals of the next generation are created by randomly mixing the parent genes. Then the genes of the new individual are subjected to mutation. This is important so that new genes can emerge that were either not present in the initialization or had already died out. Usually the global optimum can be found with this algorithm, but there is no guarantee. Optimization for multiple objective functions If the target, for instance, is to maximize pressure rise and efficiency of a fan with a given diameter and rotational speed, one is confronted with two individual objective functions. Two ways are conceivable: One can form one single objective function as a weighted sum of individual objective functions OF =

X

αi OF i :

ð9:17Þ

i

The result of the optimization is then exactly one optimal solution. A second way is to count how often an individual is dominated by other individuals. Dominant means that all objective function values are better than those of another individual. The more often an individual is dominated, the worse its rank. Thus, the optimization procedure finds many optimal individuals that form the so called Pareto front, Fig. 9.27. For Pareto-optimal individuals it is impossible to improve one objective function value without worsening the other(s).

Fig. 9.27 Individuals (black) and Pareto-optimal individuals (red) for two objective functions to be maximized

190

9.3.2

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Numerical and Experimental Methods

Example: Optimization of an Axial Fan

In this example, an axial impeller-only fan in a duct casing is optimized with respect to its total-to-static efficiency ηts. The targeted non-dimension design point is φ = 0.2, ψ ts = 0.2. Parameters which are allowed to be varied within the optimization are: Number of blades z, hub diameter Dhub, distribution of blade sweep angle from hub to blade tip (see Sect. 4.3.3), distributions of chord length and stagger angle as well as parameters of the chosen fourdigit NACA airfoil sections (maximum camber, location of maximum camber, maximum thickness, see Sect. 10.2.2). A starting geometry of the impeller had been obtained via the classic blade element momentum method (Sect. 4.3). In this example, instead of continued CFD simulations, a metamodel is used. The underlying database was created beforehand by RANS-predicted performance characteristics of 14,000 individuals in the class of axial fan impellers. The combination of geometrical parameter of these individuals were determined systematically with a method of Design of Experiment (DoE, see e.g. Montgomery [30]). The metamodel is based on an artificial neural network. Training with preliminary data enables the metamodel to predict the efficiency and other aerodynamic characteristics of an impeller with any reasonable combination of geometric parameters with high accuracy. This metamodel is then coupled with an optimization algorithm, in this example an evolutionary algorithm. The advantage of this approach is that the costly simulation effort is made only once initially. The actual optimization is performed with the metamodel and hence very time efficient. Figure 9.28 presents a comparison of the optimized and the fan designed with classic method. The pressure performance characteristics of both designs meet the targeted non-dimensionless design point φ = ψ ts = 0.2. With the help of optimization, the

Fig. 9.28 (a) Classic design (blade element momentum method) and (b) optimized design; solid lines: CFD-predicted performance, dotted: experimental results; from Bamberger [28] and Bamberger and Carolus [29]

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Optimization

191

maximum total-to-static efficiency has increased from about 58% to 60%. In addition, the operating range with high efficiency is significantly larger and the stall point is shifted to substantially smaller volume flow rates (φ-values). This progress made is mainly attributed to the fact that in the classic blade element momentum method all secondary flow phenomena in the hub and tip gap region were disregarded, whereas with the RANSbased optimization, these are captured and included for optimization of the geometry. For more details, see Bamberger [28] and Bamberger and Carolus [29].

9.3.3

Example: Centrifugal Fan Impellers with Maximum Total-to-Static Efficiency

Thanks to the use of optimization methods, it can be investigated, which maximum efficiencies are achievable for a class of fan types. In a work by Bamberger et al. [31] most of the geometrical parameters of centrifugal impellers shown in Fig. 9.29 were allowed to be varied. The major restriction was that the blades were always non-airfoil shaped circular arc blades. Only the impeller itself without casing was considered. For any design points σ opt/δopt, feasible for the class of centrifugal fans,11 the variant with maximum total-to-static efficiency ηts was identified via optimization. Similarly to the previous example, a metamodel was used, here based on approx. 4000 individuals, whose performance was determined via automated RANS-simulations. The result is the carpet-plot in Fig. 9.30. The colors mark the maximum achievable total-to-static efficiencies of any pair σ opt/δopt, each representing a particular design of a centrifugal impeller in the entire class following the design principle in Fig. 9.29. Without addressing the difference of total-to-total and

Fig. 9.29 Geometrical parameters varied in the search for optimum centrifugal fan impellers; λ denotes the angle between back plate and blade (in a standard design λ = 90°); from Bamberger et al. [31]

The feasible range is guided by the Cordier-band in Fig. 1.5; the index “opt” refers to the optimum or – here equivalently – design point, cp. Fig. 1.2.

11

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Fig. 9.30 Maximum achievable total-to-static efficiencies of the entire class of centrifugal fan impellers according to Fig. 9.29; RANS prediction; the index “opt” refers to the optimum or here equivalently design point, see Fig. 1.2; from Bamberger et al. [31]

total-to-static efficiency it is extremely satisfying to see that optimization yields optimal designs for pairs of σ opt/δopt that Cordier had already identified purely empirically, see Fig. 1.5. For each design, one can now read out the underlying geometrical parameters (not shown here). It should be noted that refined designs, e.g. airfoil-shaped or 3D-curved blades, can further increase the peak efficiency of centrifugal impellers.

References 1. Sagaut, P.: Large eddy simulation for incompressible flows. Springer-Verlag, BerlinHeidelberg, 2001 2. Smagorinsky, J.: General circulation experiments with the primitive equations. Monthly Weather Review, Vol. 91, pp. 99–164, 1963 3. Menter, F. R.: Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications. AIAA Journal, Vol. 32(8), pp. 1598–1605, 1994 4. Spalart, P. R., Jou, W. H., Strelets, M., Allmaras, S. R.: Comments on the feasibility of LES for wings and on a hybrid RANS/LES approach. 1st AFOSR Int. Conf. On DNS/LES Ruston LA, 1997. In: Advances in DNS/LES, C. Lui und Z. Liu Eds., Greyden Press, Columbus, OH 5. Reese, H.: Anwendung von instationären numerischen Simulationsmethoden zur Berechnung aeroakustischer Schallquellen bei Ventilatoren. Fortschr.-Ber. VDI Reihe 7, Nr. 489, VDI Verlag, Düsseldorf, 2007 6. Scheurer, G.: Numerische Lösung der Navier-Stokes-Gleichungen für Turbomaschinen. Kurzlehrgang Strömungsmaschinen, Friedrich-Alexander-Universität Erlangen, 19–21. Feb. 2007 7. Wolfram, D.: Analyse des Entstehungsmechanismus von Drehtönen bei gehäuselosen Radialventilatoren. Fortschritt-Bericht VDI Reihe 7 Nr. 496, VDI-Verlag, 2009 (also Ph.D. thesis University Siegen)

References

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8. Menter, F.R., Egorov, Y.: A scale-adaptive simulation model using two-equation models. AIAA Paper, AIAA2005-1095, 2005 9. Basile, R.: Aerodynamische Untersuchungen von Zwischenschaufeln in Laufrädern spezifisch langsamläufiger Radialventilatoren. Fortschr.-Ber. VDI Reihe 7 Nr. 424, VDI-Verlag Düsseldorf, 2002 (zugl. Dr.-Ing. Diss. Univ. Siegen) 10. Oertel, H. jr., Laurien, E.: Numerische Strömungsmechanik. Springer-Verlag 1995; siehe auch: Laurien, E., Oertel, H. jr.: Numerische Strömungsmechanik. Vieweg + Teubner, Wiesbaden, 2009 11. Celik, I., Ghia, U., Roache, P. J., Freitas, C. J., Coleman, H., Raad, P. E.: Procedure for estimation and reporting of uncertainty due to discretization in CFD applications. Transactions of the ASME, Vol. 130, July 2008 12. Casey, M., Wintergerste, T. (Hrsg.): ERCOFTAC SIG „Quality and Trust in Industrial CFD“: Best Practice Guidelines. ERCOFTAC, 2000 13. Wuest, W.: Strömungsmesstechnik. Friedrich Vieweg + Sohn Verlag, Braunschweig, 1969 14. Beiler, M.: Untersuchung der dreidimensionalen Strömung durch Axialventilatoren mit gekrümmten Schaufeln. Fortschr.-Ber. VDI Reihe 7 Nr. 298, VDI-Verlag Düsseldorf, 1996 (also Ph.D. thesis University Siegen) 15. Carolus, Th.: Kunststoffleitungen zwischen Druckmeßstelle und Druckaufnehmer als Fehlerquelle bei der Messung instationärer Drucksignale. Forsch. Ingenieurw. Bd. 52 (1986) Nr. 6, pp. 191 – 197 16. Nitsche, W., Brunn, A.: Strömungsmesstechnik. Springer-Verlag 2006 17. Lekakis, I. C., Adrain, R. J., Jones, B. G.: Measurement of velocity vectors with orthogonal and non-orthogonal triple-sensor probes. Experiments in Fluids, No. 7, 1989, pp. 228–240 18. Carolus, T., Stremel, M.: Blade surface pressure fluctuations and acoustic radiation from an axial fan rotor due to turbulent inflow. Acta Acustica united with ACUSTICA, Vol. 88(2002), pp. 472 – 482 19. DIN 45635 T38: Measurement of airborn noise emitted by machines; enveloping surface method, reverberation room method and in-duct method; fans (Geräuschmessung an Maschinen; Luftschallemission; Hüllflächen-, Hallraum- und Kanal-Verfahren; Ventilatoren). April 1986 20. Carolus, Th.: The influence of a reflecting floor in a semi-anechoic room on sound pressure. Internal Report No. B27 100 001 B of the Institute for Fluid- and Thermodynamics, University Siegen, 2007 21. Friedrich, J.: Ein quasischallunempfindliches Mikrophon für Geräuschmessungen in turbulenten Luftströmungen. Technische Mitteilung RFT, 11. Jahrgang, Heft 1/1967, pp. 30–34 22. Wang, J. S., Crocker, M.: Tubular windscreen design for microphones for in-duct fan sound power measurement. J. Acoust. Soc. Am., Vol. 55, No. 3, March 1974, pp. 568–575 23. VDI 3731, Blatt/Part. 2: Emissionskennwerte technischer Schallquellen; Ventilatoren (Characteristic noise emission values of technical sound sources; fans). Nov. 1990 24. VDI 2081, Blatt/Part 1: Raumlufttechnik – Geräuscherzeugung und Lärmminderung (Air-conditioning – Noise generation and noise reduction). März 2019 25. Kohlhaas, M. Bamberger, K. Carolus, T.: Acoustic Optimization of Rotor-Stator Interaction Noise by Trailing-Edge Blowing. AIAA 2013–2294. Published online: 24 May 2013, https:// doi.org/10.2514/6.2013-2294 26. Nelles, O.: Nonlinear system identification. Springer-Verlag GmbH, Heidelberg, 2001 27. Thévenin, D., Janiga, G., Optimization and Computational Fluid Dynamics. Springer-Verlag, Heidelberg, 2008 28. Bamberger, K.: Aerodynamic optimization of low-pressure axial fans. Shaker Verlag 2015, ISBN 978-3-8440-4031-9 (also Ph.D. thesis University Siegen)

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29. Bamberger, K., Carolus, T.: Development, application, and validation of a quick optimization method for the class of axial fans. ASME J. Turbomachinery, Nov. 2017, Vol. 139 / 111001-1 – 10, https://doi.org/10.1115/1.4036764 30. Montgomery, D. C.: Design and analysis of experimen. Wiley, Hoboken, NY, 2005 31. Bamberger, K., Carolus, T., Belz, J., Nelles, O.: Development, validation and application of an optimization scheme for impellers of centrifugal fans using CFD-trained metamodels. ASME J. Turbomachinery, Nov. 2020, Vol. 142 / 111005-1 – 7, https://doi.org/10.1115/1.4048022

Further Reading Ferziger, J. H., Perić, M., Street, R.: Computational Methods for Fluid Dynamics. Springer-Verlag, Berlin-Heidelberg, 2019

10

Appendix

10.1

Effective Pressure Rise

When the fan is the last component in a plant and discharges directly to the free atmosphere or a large space, it is common and practical to deal with the effective (or total-to-static) pressure rise of the fan1 Δpts = Δpt - pd,exit :

ð1:4Þ

Within this context, two issues arise frequently: • If the effective pressure rise of an impeller Δpts,Imp is specified as the design target, how large is the total pressure rise required by the blading Δpt,B? This latter quantity is the relevant design parameter for the blades. • If the total pressure rise of an impeller Δpt,Imp is known from the design process, what effective pressure rise of the complete fan Δpts is to be expected when peripheral components such as outlet guide vanes, a diffuser, a volute casing, etc. are added?

This effective pressure rise of the fan can be directly measured and thus reliably determined on various common test rigs, see Sect. 9.2.1. In contrast, the total pressure rise is frequently determined from the effective pressure rise purely by calculation. If a fan consists solely of an impeller without guide vanes, volute casing, etc., the discharge flow velocity has a swirl component, with the consequence that the true total pressure rise can only be determined with a great deal of effort, e.g. by measuring the complete exit flow velocity field. 1

# Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 T. Carolus, Fans, https://doi.org/10.1007/978-3-658-37959-9_10

195

196

10

Appendix

10.1.1 Centrifugal Impeller The effective pressure rise is Δpts,Imp = ηh Δpt,B -


 ρ 2 ρ 2 c2 = Δpt,Imp cm2 þ c2u2 : 2 2

ð10:1Þ

The energetic mean was already introduced with Eq. (2.28) and results in c2m2 

1 V_ B

Z c2m2 cm2 dA and c2u2  A2

1 V_ B

Z c2cu2 cm2 dA, respectively: A2

For the derivation of an analytical relationship between Δpts,Imp and Δpt,B we assume c2 and its components to be independent of the location on the exit plane A2 in terms of magnitude and direction, Fig. 10.1. The reader is also referred to Sect. 2.2.1 where representative velocities were introduced. Then it follows • with cm2 =

V_ B A2

• and with cu2 =

=

V_ B πD2 b2

≠ f ðA2 Þ: c2m2 =




2 V_ B πD2 b2

Δpt,B ρπD2 n

≠ f ðA2 Þ from Euler’s Eq. (2.10) in conjunction with the assumption
2 Δp of swirl-free inflow (cu1 = 0): c2u2 = ρπDt,B2 n :

Inserting these two relations into Eq. (10.1) yields Δpts,Imp = ηh Δpt,B -

 2  2 V_ B ρ ρ Δpt,B : 2 πD2 b2 2 ρπD2 n

By solving this quadratic equation the desired Δpt,B as a function of Δpts,Imp is obtained:

Fig. 10.1 Centrifugal impeller: Representative exit velocity and its components

10.1

Effective Pressure Rise

197

k Δpt,B = 1 2

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 21 - k2 , 4

ð10:2Þ

where k1 = - ηh 2ρðπD2 nÞ2 , (  ) 2 ρ V_ B þ Δpts,Imp 2ρðπD2 nÞ2 k2 = 2 πD2 b2

ð10:3Þ ð10:4Þ

10.1.2 Axial Impeller This section deals with an isolated axial impeller without guide vane and diffuser, Fig. 10.2. The dynamic pressure associated with the discharge depends on the radial distribution of the exit velocity. For an impeller design with a swirl distribution rcu2(r) = const. and cm2(r) = const. (see Sect. 4.4.1) it follows: • Because of cm2(r) ≠ f(r): cm2 =

V_ B V_ B  ≠ f ðA2 Þ, =
A2 π r 2tip - r 2hub

so that the energetic mean becomes

Fig. 10.2 Representative exit velocity and its components

198

10

0

Appendix

12

V_ B

A : c2m2 = @
π r 2tip - r 2hub Equation (4.6) immediately yields cu2 ðr Þ =

Δpt,B , 2πrnρ

hence 1 c2u2 = V_ B

Zrtip c2u2 cm2 2πrdr = rhub

r tip rhub r 2tip - r 2hub

2 ln



Δpt,B ρ2πn

2 :

Inserting these two relations into Eq. (10.1) results in 0

Δpts,Imp = ηh Δpts,Imp -

1

2 r tip  ρ@ ρ 2 ln rhub Δpt,B
A : 2 π r2 - r2 2 r 2tip - r 2hub ρ2πn tip hub V_ B

By solving this quadratic equation the desired Δpt,B as a function of Δpts,Imp is obtained: k Δpt,B = 3 2

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 23 - k4 4

ð10:5Þ

where
 ηh ρ r 2tip - r 2hub ð2πnÞ2   k3 = ln r tip =r hub

ð10:6Þ

and (  ρ 2

k4 =

V_ B π ðr2tip - r2hub Þ

2

)


 þ Δpts,Imp ρ r 2tip - r 2hub ð2πnÞ2   ln r tip =r hub

:

ð10:7Þ

10.1

Effective Pressure Rise

199

10.1.3 Axial Impeller with Outlet Guide Vanes and Diffuser In this section the expected effective pressure rise of various axial fan assemblies with optional guide vanes and diffuser as depicted in Fig. 10.3 is determined. Given is the total pressure rise of the impeller. Diffuser pressure loss coefficients Referring to Sect. 2.3.2, the pressure loss coefficients of the tail cone and Carnot diffuser (i.e. with and without cone, assemblies A and B in

Fig. 10.3 Axial fan assemblies: D denotes an impeller-only assembly, the assemblies A, B and C feature outlet guide vanes, assembly A a tail cone and B a Carnot diffuser

200

10

Appendix

Fig. 10.3) are determined first. ζ Diffuser is to be expressed by the so called diffuser efficiency ηDiffuser, which is frequently documented in the literature. The diffuser efficiency is the ratio of the pressure recovery in the actual diffuser with viscid flow and in the ideal, where the flow is assumed inviscid: ηDiffuser 

ðp4 - p3 Þjactual ðp4 - p3 Þjideal

ð10:8Þ

The pressure recovery in the ideal diffuser results immediately from Bernoulli’s equation ρ ρ p3 þ c23 = p4 þ c24 2 2 as ðp4 - p3 Þjideal =

 ρ 2 2 c 2ν - ν4 : 2 3

v = DhubDtip is equivalent to the hub-to-tip ratio of the impeller. Thus the pressure recovery in the actual diffuser becomes  ρ  ðp4 - p3 Þjactual = ηD c23 2ν2 - ν4 : 2 As seen in Fig. 10.3, c3 equals cm2. Setting the pressure loss in the real diffuser   ρ Δploss,Diffuser = ðp4 - p3 Þjideal - ðp4 - p3 Þjactual = ζ Diffuser c2m2 , 2 inserting the previous expressions and comparing yields    ζ Diffuser = 1 - ηDiffuser 2ν2 - ν4 :

ð10:9Þ

As an example, ηDiffuser can be taken from Fig. 10.4. Effective pressure rise The effective pressure rise expected from each assembly in Fig. 10.3 then becomes: • A (Complete stage comprising the impeller, outlet guide vanes and tail cone diffuser): " Δpts,A = Δpt,Imp - ζ Guide Vanes 1 þ

c2u2 c2m2

!

# þ ζ Tail cone diffuser

ρ 2 ρ c - c2 2 m2 2 4

ð10:10Þ

10.1

Effective Pressure Rise

201

• B (Impeller with outlet guide vanes and extended duct section): " Δpts,B ¼ Δpt,Imp - ζ Guide Vanes 1 þ

c2u2 c2m2

!

# þ ζ CARNOT Diffuser

ρ 2 ρ c - c2 2 m2 2 4

ð10:11Þ

• C (Impeller with outlet guide vanes, but no extended duct):

= Δpt,Imp - ζ Guide Vanes

!

ρ 2 ρ c - c2 2 m2 2 m2 ! # c2u2 ρ 1þ þ 1 c2m2 2 2 c

Δpts,C = Δpt,Imp - ζ Guide Vanes 1 þ "

c2u2

c2m2

ð10:12Þ

m2

• D (Impeller-only; identical to the fan in Sect. 10.1.2):
 ρ ρ 2 Δpts,D = Δpt,Imp - c22 = Δpt,Imp cm2 þ c2u2 2 2

ð10:13Þ

Numerical example An axial impeller with diameter Dtip = 300 m and Dhub = 135 m is designed for Δpt,Imp = 300 Pa at V_ = 0:65 m3 =s: The swirl distribution is rcu2 = const. The shaft speed is n = 3000 1/min, the density of the gas ρ = 1.2 kg/m3. The blade tip gap is so small, that the leakage flow is negligible. The blade efficiency is estimated to be ηh = 92%. Hence, the blades are designed for Δpt,B = Δpt,Imp/ηh = 326 Pa and V_ B = V_ = 0:65 m3 =s: The expected effective pressure rise of all four assemblies in Fig. 10.3 is obtained as follows: From the impeller design, the exit flow velocities are known and hence the dynamic pressures: •

ρ 2 2 cm2

= 80 Pa and ρ2 c2u2 = 40 Pa

The dynamic pressure at the outlet of the extended duct with the diameter 300 mm is •

ρ 2 2 c4

= 51 Pa:

The diffuser efficiencies are taken from Fig. 10.4 for v = Dhub/Dtip = 0.45 as ηTail cone diffuser = 0.900 and ηCARNOT diffuser = 0.752. The pressure loss coefficients become then ζ Tail cone diffuser = 0.036 and ζ CARNOT diffuser = 0.100 (Eq. 10.9). The pressure loss coefficient of the outlet guide vanes is assumed to be ζ Guide vanes = 0.10. Inserting these figures into

202

10

Appendix

Fig. 10.4 Efficiency of the tail cone and Carnot diffuser according to Fig. 10.3; after Wallis [1]

Fig. 10.5 Effective (= total-to-static) pressure rise of four different fan assemblies; the two left bars of total blade and impeller pressure rise are for information only

Eqs. (10.11, 10.12, and 10.13) yields the effective pressure rise of each assembly. The graphical representation in Fig. 10.5 clearly shows that the effective pressure rise of the impeller-only assembly (D) is only about 55% of the total pressure rise for which the blading was designed. This value can be expected when measuring the performance on a suction-side chamber test rig (Carolus et al. [2]). On the other hand, in this example outlet guide vanes and a tail cone diffuser increase the effective pressure rise Δpts of the assembly to 70% of Δpt,B. Obviously, the pressure recovery is greater than the frictional losses in these components. It is noteworthy that the shaft power required to run the impeller is determined by Δpt,B and V_ B as well as by potential mechanical and disc friction losses, and not by any of the effective pressure rises.

10.2

10.2

Airfoil Sections

203

Airfoil Sections

Similar to an airplane wing, blades of an axial impeller can be thought as made of a number of airfoil sections. This is surprising at first glance, since the motion of a wing is generally translational, while in a fan impeller rotational. Historically, however, airfoil section data were not determined in a wind tunnel, but with a whirling arm device, Fig. 10.6. It was Otto Lilienthal in 1871 who experimentally determined lift and drag of differently shaped airfoil sections utilizing this early experimental test rig.

10.2.1 Isolated Airfoil Section in Unbounded Flow An isolated airfoil section in an unbound steady flow with the freestream velocity w1 under an angle of attack α (Fig. 10.7) experiences a force F. Important components of F are lift

Fig. 10.6 Whirling arm device (Rundlaufapparat) by Otto Lilienthal from 1871, described in “Der Vogelflug als Grundlage der Fliegekunst (The Flight of Birds as the Basis of the Art of Flying), R. Gärtners Verlagsbuchhandlung, Berlin 1889”, see also [3, 4]; by courtesy of Deutsches Museum Munich

Fig. 10.7 Flow induced force and its components lift and drag on an airfoil section in a steady flow field; the airfoil section has a span b; the length of the drag force vector D is exaggerated substantially as compared to L

204

10

Appendix

L and drag D. By definition, L is the perpendicular, D the parallel component with respect to the direction of w1. A non-dimensional lift and drag coefficient is defined such that ρ L = C L w21 lb, 2

ð10:14Þ

ρ D = CD w21 lb, 2

ð10:15Þ

respectively. The product of chord length l and span b is the section area. The lift and drag coefficients CL and CD are function of the airfoil’s geometry, angle of attack α, the Reynolds number Rel = w1l/ν, surface quality and the turbulence intensity of the unbounded flow. In general, catalogue data of CL and CD refer to a low-turbulence freestream flow field. A measure of the aerodynamic quality of an airfoil section is the drag-to-lift ratio2 ε

D CD : = L CL

ð10:16Þ

The smaller ε, the better the section or the more favorable the operating point. The operating point of a given airfoil section is a function of the angle of attack α. For all common airfoils the drag is much smaller than the lift. Therefore, the angle δ between F and L is also very small, which leads to tan δ ≈ δ = ε:

ð10:17Þ

CL and CD are determined in a wind tunnel or via numerical flow simulation and are represented in the form of so called polars, CL = f(CD) (Lilienthal polars, Fig. 10.8) or CL = f(α) (resolved polars, Fig. 10.9). From the CD – CL polar curve one can also read off the optimum lift coefficient CL,opt of an airfoil, i.e. where it operates at its smallest drag-tolift-ratio. The critical value of the angle of attack αcrit is another important parameter. Starting from αcrit, an increase of the angle of attack no longer leads to an increase of the lift coefficient. On the contrary, depending on the airfoil, it decreases more or less abruptly due to flow separation. When polars are determined experimentally in wind tunnel tests, a sample section with an aspect ratio λ = b/l = 5/1 to 6/1 is usually used. Depending on the test setup, the result may also contain the induced drag due to the vortices at the tips. In the case of ducted fans, hub and casing prevent these vortices. Hence, the polars for λ = 1 are required. A conversion to an infinite aspect ratio can be done according to Prandtl (see e.g. Fister [5, 6]) via The drag-to-lift ratio represents the slope of the flight path of a gliding airfoil section; in aircraft design, the reciprocal of ε is used instead and called the glide ratio. 2

10.2

Airfoil Sections

205

Fig. 10.8 Lilienthal polars of an airfoil segment (schematic)

Fig. 10.9 Resolved polars (schematic)

CL 2 πλ

ð10:18Þ

57:3CL , πλ

ð10:19Þ

C D ðλ = 1Þ = CD ðλÞ and αðλ = 1Þ = αðλÞ where α is expressed in degrees.

10.2.2 Airfoil Families In this section a few common families of airfoils are described. A family is a series of airfoils developed according to the same design philosophy. Of only historical importance are the Joukowsky airfoils, defined and analyzed using conformal mapping. Later airfoils are based on empirical or theoretical design methodologies, for instance

206

10

Appendix

Fig. 10.10 Upper: Mean camber line, middle: Thickness distribution, lower: Combination of thickness distribution and mean camber line. f = maximum camber; xf = location of maximum camber; d = maximum thickness; xd = location of maximum thickness

• • • •

the Göttingen series NACA3 – airfoils C – airfoils (British) modern airfoils e.g. from Eppler, Wortmann, Althaus etc.

The Göttingen series Göttingen airfoils were designed empirically; in the context of turbomachinery, data of some selected airfoils can be found e.g. in Pfleiderer/Petermann [7]; today, this family is of less significance. NACA airfoils NACA airfoils have been systematically developed, studied, and cataloged; the four- and five-digit airfoils series are classified according to the parameters defining their geometry. Most airfoils can be defined as a combination of a thickness distribution and a mean camber line. The chord is the distance between the endpoints of the camber line (leading and trailing edge), Fig. 10.10. Four-digit NACA airfoils These airfoils (Jacobs et al. [8], Abbott and von Doenhoff [9]) are weakly cambered. The mean camber line and thickness distribution can be varied separately, which is very advantageous for fan blade design. Their development was significantly influenced by popular airfoils known at the time, such as the Göttingen 398 or the Clark Y. Their thickness distributions are very similar if the mean camber line is converted into a straight line and normalized to the same maximum thickness; hence, the thickness distribution of the four-digit NACA airfoils was chosen as

NACA = National Advisory Committee for Aeronautics, predecessor of the US aerospace agency NASA. 3

10.2

Airfoil Sections

207

d 0:20 rffiffiffi

2
3
4

x x x x x - 0:3516  0:2969 - 0:1260 þ 0:2843 - 0:1015 : ð10:20Þ l l l l l

yd =

This type of thickness distribution for different maximum thicknesses d can be visualized by plotting uncambered airfoils, Fig. 10.11. The mean camber line consists of two parabolic arcs with the two parameters f (maximum camber) and xf (location of maximum camber):

xf x
x2 ys f 1 x xf = x 2 2 for 0 ≤ ≤ l f l l l l l l

ð10:21Þ

l



xf xf x
x2 xf ys f 1 x 1-2 þ 2 =  ≤ ≤1 for l 1 - xf  2 l l l l l l l l

ð10:22Þ

The designation of four-digit NACA airfoils is based solely on these few geometry parameters, Table 10.1. For example, NACA 4509 means an airfoil with a maximum camber of 4% of the chord l, the location of maximum camber at 0.5 l, and a maximum thickness of 9% of the chord l. The same airfoil without camber would have the designation 0009. The effect of variation of the geometric parameters f and xf on the airfoil contour are visualized in Fig. 10.12.

Fig. 10.11 Four-digit NACA airfoils, uncambered, after Jacobs et al. [8]

Table 10.1 Designation of the four-digit NACA airfoils NACA WXYZ W X YZ

Maximum camber f in % of l Location of maximum camber xf in 1/10 of l Maximum thickness d in % of l

208

10

Appendix

Fig. 10.12 Four-digit NACA airfoils: Variation of maximum camber, location of maximum camber, maximum thickness, after Jacobs et al. [8]

The four-digit NACA airfoils have been modified in many ways. Thickness distributions with different locations of maximum thickness and different trailing edge thicknesses are tabulated in Table 10.5 in Sect. 10.4 (Abbott and von Doenhoff [9], Loftin and Cohen [10], Marcinowski [11]). The designation is then, for example, NACA 4509–64, which corresponds to the airfoil already mentioned, but with the modified thickness No. -64 in the leading edge region, tabulated in Table 10.5. The standard thickness of the trailing edge is 2yh/d = 0.02. The initially measured polars are documented in Jacobs et al. [8] for a Reynolds number of approximately   w1 l Rel  = 3 × 106 : ν Data for other Reynolds numbers are given in more recent profile catalogues, e.g. in Althaus [12], Althaus and Wortmann [13].

10.2

Airfoil Sections

209

For any slender (d/l ≤ 0.1) and slightly cambered airfoil ( f/l ≤ 0.1) airfoil Marcinowski [11] and Fister [6] recommend as first approximations " f C L ≈ 0:1α þ 10 and C L ≈ 0:092 α þ l

! #
 100 xf 2 f , 82 þ l 1 þ 5 dl l

ð10:23Þ

respectively, where α is inserted in degrees. Similarly, Pfleiderer [14] found for the drag-tolift ratio in the vicinity of the optimum lift coefficient d f ε = 0:012 þ 0:02 þ 0:08 : l l

ð10:24Þ

This simple relationship clearly illustrates the observation that increasing the thickness of an airfoil results in a less favorable drag-to-lift ratio, as does an increase of camber. Complete polars of almost any airfoil can be calculated accurately with – for instance – the airfoil design and analysis program XFOIL4 from Drela [15]. Here, the Reynolds number and even the Mach number are also taken into account here. NACA 65-series This family is very relevant for turbomachinery, since systematic studies of cascades composed of these airfoil sections were carried out. In contrast to the four-digit NACA profiles, their aerodynamic data are also documented for a larger camber (in a cascade assembly), Abbott et al. [16], Riegels [17], Herring et al. [18]. The most common mean camber line NACA (A10) has the form 

i CfL h
ys x x x x ln 1 þ ln : =1l l l l l 4π

ð10:25Þ

CfL is the theoretical lift coefficient of an airfoil in inviscid flow, φ!
 2π 1 - cos 2 2π φ φ C fL = tan , = ln 2 ln 2 4 sin 2

ð10:26Þ

which is solely a function of the geometric camber angle

4

XFOIL is free software under GNU General Public License, see also http://web.mit.edu/drela/ Public/web/xfoil/ (as of Oct. 2021).

210

10

φ = βB2 - βB1

Appendix

ð10:27Þ

(Fig. 10.13). The A10-mean camber line is based on theoretical considerations. Mean- and chord line are orthogonal at the leading and trailing edge. For quantification of the blade inlet and exit angles βB1 and βB2 it is therefore common to use an equivalent circular arc of the same maximum camber and chord length, Fig. 10.13. Some thickness distributions of the NACA 65-series are tabulated in Table 10.6 in Sect. 10.4. The designation of NACA 65-series airfoils contains both the (theoretical) aerodynamic parameter CfL and the geometric parameter d, Table 10.2 and Fig. 10.14. More recent airfoils Many airfoils have been developed in recent times, among others for use in sailplanes, model aircrafts and wind turbines. Examples are the Eppler (E-), Wortmann (FX-), Althaus (AH-) airfoils, named after their designer. They are often optimized for a certain range of Reynolds number. With these airfoils, separate variation of camber and thickness distribution is no longer possible. Their coordinates and polars are compiled in various catalogues ([12, 13, 19, 21, 22]). Figure 10.15 shows polars of the E392, obtained via XFOIL, and Fig. 10.16 measured polars of the FX60–126. The large value of the drag coefficient and the poor drag-to-lift ratio at very low Reynolds numbers in Fig. 10.15 are due to the fact that a laminar boundary layer tends

Fig. 10.13 NACA 65 series: Original and equivalent circular arc camber line and camber angle φ (schematically)

Table 10.2 Designation of the basic NACA 65-series airfoils NACA 65-XYZ or 65-(WX)YZ X or WX YZ

10CfL Maximum thickness d in % of l

10.2

Airfoil Sections

211

Fig. 10.14 NACA 65-series airfoils: Variation of CfL (and thus the camber)

Fig. 10.15 E392 airfoil: Polars at different Reynolds numbers Rel (predicted with XFOIL from Drela [15])

to separate earlier than a turbulent one. A separated boundary layer is a significant source of pressure drag. Albring [20] showed that for small Reynolds numbers a cambered thin plate may be aerodynamically more favorable than an airfoil-shaped, Fig. 10.17.

10.2.3 Airfoil Sections in a Cascade Essentially, the blades of a turbomachine form a cascade of airfoil sections. The flow field in a cascade differs in some respect from that around an isolated airfoil: • A blade cascade causes a deflection of the fluid that is maintained in the wake far downstream. • Lift and drag are potentially affected by the neighboring airfoils, as illustrated in Fig. 4.1. This becomes particularly clear when analyzing the pressure distribution

212

10

Appendix

Fig. 10.16 FX60–126 airfoil: Measured α/CL polars. (From Althaus [12], by courtesy of Neckar-Verlag, VS-Villingen)

Fig. 10.17 Effect of Reynolds number on the polars of a cambered flat plate and an airfoil; after Albring [20]

around the airfoil. In the blade cascade of an axial fan the static pressure p1 typically increases to p2 > p1 due to the deceleration of the fluid, Fig. 10.18.5 This pressure gradient is superimposed on the pressure distribution of the isolated airfoil. As a result, the adverse pressure gradient from the suction peak to the trailing edge of the airfoils in 5

An exception is the blade cascade in the impeller of a Schicht-blower, where a conical design of the hub contour ensures w2 = w1 and thus p2 = p1.

10.3

Some Basics of Acoustics

213

Fig. 10.18 Pressure distribution on the isolated airfoil ( p2 = p1) and on an airfoil in a cascade where the flow is decelerated ( p2 > p1)

the cascade is larger than for the isolated airfoil in an unbounded flow. This explains the fundamental difficulty in the design of any fan: The boundary layer is much more prone to separation than in a turbine, where the fluid is accelerated.

10.3

Some Basics of Acoustics

Sound pressure, sound pressure level The sound pressure p’ is the fluctuating acoustic pressure superimposed on the atmospheric pressure during the propagation of sound waves. The sound pressure level is defined as p0 2 Lp = 101og p0 2

! dB

ð10:28Þ

with the mean squared value of the sound pressure 1 p = T

t 0 þT Z

02

p0 ðt Þdt 2

t0

(the time interval T being an integer multiple of half periods or, more practical, reasonably large) and the standardized reference pressure p0 = 2 × 10-5 Pa for airborne sound. The sound pressure and its level depend, among others, on the distance to the sound source. The sound pressure or sound pressure level is decisive for assessing the effect of sound on a listener.

214

10

Appendix

Sound power, sound power level The sound power Pac of a sound source is the sound energy per unit of time that passes through an enveloping surface S enclosing the source. This surface is perpendicular to the direction of sound propagation. In the acoustic far field, the sound power is linked to the sound pressure via Z Pac = S

p0 2 dS: ρc0

ð10:29Þ

The sound power level is defined as   P LW = 101og ac dB P0

ð10:30Þ

with the standardized reference power P0 = 10-12 W for airborne sound. Sound power and its level are independent of the measurement location and characterize the strength of sound source. Although both, sound pressure and power level, are given in dB, in general their numerical values differ. If the sound source radiates with a spherical directivity, i.e., if the sound pressure is independent of the location on S, the sound pressure level on S can be calculated immediately from the sound power level: Lp = LW - 10 log

    ðρc0 Þk S - 10 log dB S0 ρc0

with the references S0 = 1 m2, ðρc0 Þk = 400 in the fluid.

kg m2 s,

ð10:31Þ

and ρ the density and c0 the speed of sound

Frequency spectra Noise is usually not mono-frequency; a spectral analysis provides the amplitude and phase of each frequency component. In acoustics, the phase is usually of secondary importance; the power spectrum is more important. A distinction is made in its representation: • Narrowband spectra; the entire frequency range is divided into narrow frequency bands of same bandwidth Δf in which the sound power (or sound power level) is specified. • Octave and one-third octave band spectra; here the frequency axis is divided into non-overlapping frequency bands whose upper frequency is twice or 21/3 times the lower; the frequency axis is therefore non-linear (logarithmic), see Fig. 10.19. The frequency bands are standardized and compiled in Table 10.3. • Spectral sound power density, i.e. the power per (very small) frequency band is

10.3

Some Basics of Acoustics

215

Fig. 10.19 Different spectral representations of a sound signal; logarithmic frequency axis; the band width Δf = 3.125 Hz is just an example for an arbitrary narrow band spectrum

SPac =

dPac averaged acoustic power Pac in frequency band Δf ≈ Δf df

h

i W : Hz

ð10:32Þ

A narrowband spectrum with the bandwidth Δf = 1 Hz corresponds to the sound power density. Narrowband spectra of arbitrary bandwidth or one-third octave and octave spectra can be calculated very easily from the spectral sound power density. Frequency weighting A classic first measure of the effect of annoyance is obtained by frequency weighting of the physically correctly measured sound pressure. For this purpose, the levels are weighted by a standardized frequency-dependent function, for instance in case of the A-weighting with ΔLA according to Table 10.3: Lp,A = Lp þ ΔLA dB

ð10:33Þ

Among different others, the A-weighting is the most commonly used. Combining of levels The overall sound pressure level of n incoherent (i.e. without interference effects) radiating sound sources is calculated as Lp,OA = 10 log

n X

! 10Lp,i =10

dB:

ð10:34Þ

i=1

This formula also applies to the calculation of the overall sound pressure level of a single source from band levels. As a good approximation, the following rule of thumb applies for combining levels of two different sound sources with L2 > L1: L1 þ L2 = L2 þ C ðΔLÞ: with the C-values according to Table 10.4.

ð10:35Þ

216

10

Appendix

Table 10.3 Center, lower and upper frequencies ( f0, f1 and f2) for standard one-third octave and octave bands and level corrections for A-weighting, according to DIN EN 61260 [23] Standard frequencies One-third octave bands f0 [Hz] f1 [Hz] 10 8.9 12.5 11.2 16 14.1 20 17.8 25 22.4 31.5 28.2 40 35.5 50 44.7 63 56.2 80 70.7 100 89.1 125 112 160 141 200 178 250 224 315 282 400 355 500 447 630 562 800 708 1000 891 1250 1122 1600 1413 2000 1778 2500 2239 3150 2818 4000 3548 5000 4467 6300 5623 8000 7079 10,000 8913 12,500 11,220 16,000 14,130 20,000 17,780

f2 [Hz] 11.2 14.1 17.8 22.4 28.2 35.5 44.7 56.2 70.7 89.1 112 141 178 224 282 355 447 562 708 891 1122 1413 1778 2239 2818 3548 4467 5623 7079 8913 11,220 14,130 17,780 22,390

Octave bands f0 [Hz] f1 [Hz]

f2 [Hz]

16

11

22

31.5

22

44

63

44

88

125

88

177

250

177

355

500

355

710

1000

710

1420

2000

1420

2840

4000

2840

5680

8000

5680

11,360

16,000

11,360

22,720

ΔLA [dB] -70.4 -63.4 -56.7 -50.5 -44.7 -39.4 -34.6 -30.2 -26.2 -22.5 -19.1 -16.1 -13.4 -10.9 -8.6 -6.6 -4.8 -3.2 -1.9 -0.8 +0.0 +0.6 +1.0 +1.2 +1.3 +1.2 +1.0 +0.5 -0.1 -1.1 -2.5 -4.3 -6.6 -9.3

10.3

Some Basics of Acoustics

Table 10.4 Combining of levels

217 ΔL = L2 – L1 [dB] 0 and 1 2 to 4 4 to 9 ≥ 10

C(ΔL ) [dB] 3 2 1 0

Propagation of sound from point sources Frequently, a spherically radiating point source is a good initial idealization of a more or less freely radiating sound source. The source may emit a sound power of level LW. If the radiation pattern – unhindered by surrounding reflecting walls – is symmetrical in all directions, then the sound pressure level at distance r from the source in the acoustic far field6 is obtained by  2 4πr Lp = LW - 10 log dB S0

ð10:36Þ

with the reference area S0 = 1 m2. Obviously, a doubling of the distance r from the source is equivalent to a decrease of sound pressure level by 10log22 = 6 dB. If the sound source is placed on a hard-walled floor, the sound power can only be radiated over half the spherical surface 2πr2. The sound pressure level at the same distance r from the source is 3 dB higher than without the reflecting floor. Figure 10.20 illustrates how this idea can be applied to the sound radiation of a fan. It was shown in Chap. 4 that in reality the radiation pattern of a fan is not completely symmetric. Silencers In ventilation systems, absorption silencers are generally used as a secondary noise reduction measure. An important measure for any sound attenuators is the insertion loss D. It is the difference in sound power levels transmitted through an air duct or opening with and without a silencer. Figure 10.21 shows a fan with splitter silencer in the discharge duct. The discharge duct sound power level LW4 of the fan is reduced by the insertion loss of the silencer to LW4 = LW4 - D dB:

ð10:37Þ

Manufacturers usually specify the attenuation of their silencer in octave bands. When dimensioning a silencer, care must be taken to ensure that the attenuation is not nullified by the flow-induced noise in the silencer itself.

Theoretically, far field condition is ensured for a distance r away from the source which is much larger than the characteristic wave length λ of the sound. According to [24], however, for most practical sources the far field can be assumed to exist for distances larger than 2× the largest dimension of the source. 6

218

10

Appendix

Fig. 10.20 Idealization of the discharge opening as symmetrically radiating point source Fig. 10.21 Centrifugal fan with splitter silencer in the discharge duct

10.4

Tables

The choice of the thickness distribution often only affects details of the shape of the polars, see for example the polars of NACA 0012 and NACA 0012–64 in Abbott and von Doenhoff [9] (Tables 10.5, 10.6, 10.7 and 10.8).

yd/l in % NACA0010-63b NACA0010a 0 0 1.578 1.60 2.179 2.21 2.962 3.01 3.500 3.57 3.902 3.97 4.454 4.51 4.781 4.81 5.001 5.00 4.836 4.87 4.87d 4.412 5.51 4.51 3.803 3.93 3.93 3.053 3.18 3.18 2.186 2.27 2.27 1.206 1.23 1.26 0.672 0.68 0.75 0.105 0.10 0.25 Relative radius of leading edge r/l = 1.10% 4.87d 4.51 3.93 3.18 2.29 1.40 0.95 0.50

NACA0010-64c 0 1.511 2.044 2.722 3.178 3.533 4.056 4411 4.856 5.000 4.856 4.86d 4.433 4.43 3.733 3.73 2.767 2.77 1.556 1.56 0.856 0.90 0.100 0.25 4.86d 4.43 3.73 2.77 1.64 1.07 0.50

NACA0010–65c 0 1.467 1.967 2.589 2.989 3.300 3.756 4.089 4.578 4.889 5.000 4.867 4.87d 4.389 4.39 3.500 3.50 2.100 2.10 1.178 1.10 0.100 0.25

4.87d 4.39 3.50 2.10 1.30 0.50

b

Standard thickness distribution of four-digit NACA profiles according to Eq. (10.20), see also Jacobs et al. [8] and Abbott and von Doenhoff [9] The thickness distribution 0010–63 with its location of maximum thickness at 30% of the chord length corresponds almost (but not exactly) to the standard airfoil 0010 (see p. 117 in Abbott and von Doenhoff [9]) c Values of these thickness distributions with their location of maximum thickness at 40% and 50% of the chord length from Appendix I in Abbott and von Doenhoff [9] d The thickened tail sections are from Marcinowski [11]

a

x/l in % 0 1.25 2.5 5.0 7.5 10 15 20 30 40 50 60 70 80 90 95 100

Table 10.5 Selected thickness distributions of the four-digit NACA airfoils

10.4 Tables 219

220

10

Appendix

Table 10.6 Thickness distributions of some NACA 65-series airfoils

x/l in % 0 0.50 0.75 1.25 2.5 5.0 7.5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 r/l of LE in % a

Normal NACA 65-010a yd/l in %

NACA 65(216)010b yd/l in %

NACA 65(216)010 with 1% TE radiusc yd/l in %

0 0.772 0.932 1.169 1.574 2.177 2.647 3.040 3.666 4.143 4.503 4.760 4.924 4.996 4.963 4.812 4.530 4.146 3.682 3.156 2.584 1.987 1.385 0.810 0.306 0 0.687

0 0.752 0.890 1.124 1.571 2.222 2.709 3.111 3.746 4.218 4.570 4.824 4.982 5.057 5.029 4.870 4.570 4.151 3.627 3.038 2.451 1.847 1.251 0.749 0.354 0.150 0.666

0 0.752 0.890 1.124 1.571 2.222 2.709 3.111 3.746 4.218 4.570 4.824 4.982 5.057 5.029 4.870 4.570 4.175 3.768 3.362 2.955 2.549 2.142 1.735 1.329 0 0.666

NACA 65-006d yd/l in % 0 0.476 0.574 0.717 0.956 1.310 1.589 1.824 2.197 2.482 2.697 2.852 2.952 2.998 2.983 2.900 2.741 2.518 2.246 1.935 1.594 1.233 0.865 0.510 0.195 0 0.240

NACA 65–008d yd/l in %

NACA 65–012d yd/l in %

NACA 65–015d yd/l in %

0 0.627 0.756 0.945 1.267 1.745 2.118 2.432 2.931 3.312 3.599 3.805 3.938 3.998 3.974 3.857 3.638 3.337 2.791 2.553 2.096 1.617 1.131 0.664 0.252 0 0.434

0 0.923 1.109 1.387 1.875 2.606 3.172 3.647 4.402 4.975 5.406 5.716 5.912 5.997 5.949 5.757 5.412 4.943 4.381 3.743 3.059 2.345 1.630 0.947 0.356 0 1.000

0 1.124 1.356 1.702 2.324 3.245 3.959 4.555 5.504 6.223 6.764 7.152 7.396 7.498 7.427 7.168 6.720 6.118 5.403 4.600 3.744 2.858 1.977 1.144 0.428 0 1.505

Source: Herrig et al. [18] Source: Herrig et al. [18]; also referred as “65–010 blower blade section” due to the finite trailing edge thickness that is technically feasible c Source: Pfleiderer and Petermann [7]; according to this source, the (aerodynamic) deterioration due to the thicker trailing edge is small d Source: E.g. Pfleiderer and Petermann [7] b

10.4

Tables

221

Table 10.7 Blade spacing that reduces the strength of the tone at BPF; rotor not naturally balanced; according to Mellin and Sovran [25] Number of blades z 4 5 6 7 Spacing angle of blades [deg] 71 58.6 49.9 43.5 109 75.5 58.7 48.7 109 91.8 71.4 57.0 71 75.5 71.4 61.6 58.6 58.7 57.0 49.9 48.7 43.5

Σ 360

Σ 360

Σ 360

Σ 360

8

9

10

11

12

13

14

15

38.7 41.8 47.3 52.2 52.2 47.3 41.8 38.7

34.9 36.9 40.6 44.5 46.2 44.5 40.6 36.9 34.9

31.7 33.1 35.7 38.7 40.8 40.8 38.7 35.7 33.1 31.7

29.1 30.1 31.9 34.2 36.2 37.0 36.2 34.2 31.9 30.1 29.1

26.9 27.6 29.0 30.7 32.4 33.4 33.4 32.4 30.7 29.0 27.6 26.9

25.00 25.55 26.55 27.90 29.25 30.35 30.80 30.35 29.25 27.90 26.55 25.55 25.00

23.39 23.79 24.56 25.60 26.71 27.69 28.26 28.26 27.69 26.71 25.60 24.56 23.79 23.39

Σ 360

Σ 360

Σ 360

Σ 360

Σ 360

Σ 360

Σ 360

21.96 22.27 22.87 23.68 24.59 25.44 26.05 26.28 26.05 25.44 24.59 23.68 22.87 22.27 21.96 Σ 360

Table 10.8 Blade spacing that reduces the strength of the tone at BPF; rotor naturally balanced; according to Mellin and Sovran [25] Number of blades z 4 5 6 7 Spacing angle of blades [deg] 68 46 50.7 40.,7 112 102 78.6 68.3 68 46 50.7 52.5 112 83 50.7 37.0 83 78.6 54.0 50.7 67.4 40.1

Σ 360

Σ 360

Σ 360

Σ 360

8

9

10

11

12

13

14

15

35.6 54.4 54.4 35.6 35.6 54.4 54.4 35.6

32.3 44.4 51.1 37.0 30.6 37.0 50.9 44.7 32.1

29.3 37.8 45.8 37.8 29.3 29.3 37.8 45.8 37.8 29.3

26.9 33.0 40.5 37.3 29.2 26.2 29.2 37.3 40.6 32.9 26.9

24.9 29.4 35.7 35.7 29.4 24.9 24.9 29.4 35.7 35.7 29.4 24.9

23.22 26.56 31.74 33.41 29.17 24.50 22.79 24.50 29.16 33.42 31.75 26.55 23.23

21.76 24.32 28.49 30.85 28.49 24.32 21.77 21.76 24.32 28.49 30.85 28.50 24.31 21.77

Σ 360

Σ 360

Σ 360

Σ 360

Σ 360

Σ 360

Σ 360

20.47 22.47 25.85 28.37 27.42 24.08 21.23 20.21 21.24 24.08 27.42 28.37 25.85 22.46 20.48 Σ 360

222

10

Appendix

Lieblein Design Diagrams7

10.5

Approximation function7 i0 = Aβ14 + B β13 + C β12 + D β1 + E with A = a6σ −6 + a5σ −5 + a4σ −4 + ... + a1σ −1 + a0 B = b6σ −6 + ...

a b c d e

6 -0.000000 11800450 0.000028 53716261 -0.002307 37991224 0.049462 05693000 1.221058 37277223

5 0.000001 27534563 -0.000308 25969476 0.0252339 0057628 -0.587451 17513630 -10.787006 84058033

4 -0.000005 60379879 0.001350 22210320 -0.111414 07630877 2.758158 36890554 38.327936 05744849

3 0.000012 82918392 -0.003074 43963361 0.254781 49313722 -6.595844 82427565 -71.225727 24273792

etc.

2 -0.000016 22871723 0.003863 88200689 -0.320867 08139299 8.561681 92061435 76.364394 24576079

1 0.000010 89422811 -0.002581 50775619 0.215044 35808000 -5.821037 10444651 -50.369081 58317770

0 -0.000003 15099628 0.000748 90393705 -0.063197 92932981 1.676996 32094571 21.588591 10857168

Fig. 10.22 Design incidence angle for an uncambered NACA 65-series airfoil i0 = f(β1,σ)

7

All approximation functions and diagrams after Fabre [26]. For reasons of accuracy all decimals should be used when encoding the approximation functions (Figs. 10.22, 10.23, 10.24, 10.25, and 10.26).

10.5

Lieblein Design Diagrams

223

Approximation function n = Aβ14 + B β13 + C β12 + D β1 + E with A = a6σ −6 + a5σ −5 + a4σ −4 + ... + a1σ −1 + a0 ,

a b c d e

6 -0.000000 04036158 0.000011 61755597 -0.001103 67458947 0.039443 58647366 -0.440737 79462212

5 0.000000 33176160 -0.000096 77584894 0.009269 00505236 -0.332458 28635265 3.698307 29620734

4 -0.000001 06270929 0.000316 16926091 -0.030639 04763052 1.104712 24912578 -12.218666 34452193

3 0.000001 66420315 -0.000510 92263494 0.050404 87663224 -1.831325 55294223 20.056385 14378197

Fig. 10.23 Proportionality factor n = f(β1,σ)

B = b6σ 6 + ... etc.

2 -0.000001 28744415 0.000418 52620986 -0.042555 08039721 1.563525 54658467 -16.692844 87830261

1 0.000000 41954548 -0.000155 52972437 0.016801 24746197 -0.627150 55678273 6.050872 93541413

0 -0.000000 04005738 0.000020 97442840 -0.002612 38458175 0.107660 94718801 -1.133293 01802607

224

10

Appendix

Approximation function δ 0 = Aβ14 + B β13 + C β12 + Dβ1 + E with A = a6σ −6 + a5σ −5 + a4σ −4 + ... + a1σ −1 + a0 ,

a b c d e

6 0.000000 526880 -0.000145 822902 0.014150 862349 -0.546528 959709 6.377298 418967

5 -0.000004 649611 0.001286 548446 -0.124814 670375 4.825851 345903 -56.764897 922376

4 0.000016 503679 -0.004563 823765 0.442564 269957 -17.137568 315117 203.713382 407748

B = b6σ 6 + ... etc.

3 -0.0000 30163647 0.008334 533282 -0.808079 934229 31.385815 516950 -378.987042 674248

2 0.0000 29942340 -0.008272 897373 0.803463 643823 -31.440726 767558 390.261099 506032

1 -0.000015 345926 0.004253 894585 -0.416387 089699 16.623105 421290 -218.395601 505197

0 0.000003 281827 -0.000920 082410 0.092298 719314 -3.902757 016196 59.160496 283446

Fig. 10.24 Design deviation angle for an uncambered NACA 65-series airfoil δ0 = f(β1,σ)

10.5

Lieblein Design Diagrams

225

Approximation function m = Aβ14 + B β13 + C β12 + D β1 + E with A = a6σ −6 + a5σ −5 + a4σ −4 + ... + a1σ −1 + a0 ,

a b c d e

6 -0.000000 06072358 0.000015 00101768 -0.001292 19672948 0.044913 03258376 -0.571043 43191183

5 0.000000 53243234 -0.000131 06041845 0.011258 06078928 -0.390718 61628836 4.937038 72284831

4 -0.000001 84665634 0.000452 98917311 -0.038809 40999166 1.345309 49151767 -16.900986 07844754

3 0.000003 21460414 -0.000786 15890614 0.067211 46815111 -2.328830 97902829 29.127689 52950583

Fig. 10.25 Proportionality factor m = f(β1,σ)

B = b6σ 6 + ... etc.

2 -0.000002 93179864 0.000715 50995694 -0.061116 68612276 2.120672 64505931 -26.513375 13995404

1 0.000001 30699069 -0.000318 55840235 0.027218 23576047 -0.948342 05359658 12.126156 41651276

0 -0.000000 22197800 0.000053 65296314 -0.004522 34834295 0.153297 61246803 -1.808845 81128943

226

10

Approximation function ∂δ = Aσ +5 + Bσ +4 + Cσ +3 + Dσ +2 + Eσ + F ∂i β 1 with A = a5 β15 + a4 β14 + a3 β13 + a2 β12 + a1β1 + a0 B = b5 β15 + ... etc.

a b c d e f

5 0.000000 00706137 -0.000000 03625705 0.000000 06503613 -0.000000 04417064 0.000000 00812804 -0.000000 00120024

4 -0.000002 14249972 0.000010 90055430 -0.000019 50882308 0.000013 58025849 -0.000002 77341781 0.000000 26963238

3 0.000245 90343872 -0.001238 44390677 0.002201 79197316 -0.001546 54142184 0.000330 95326833 -0.000023 29880970

  ¼ f ðβ 1 , σ Þ Fig. 10.26 Correction ∂δ ∂i β 1

2 -0.013332 88553173 0.066264 56460090 -0.116235 39134481 0.080969 06004907 -0.017279 27972125 0.000949 82547125

1 0.330833 81514517 -1.609239 92913710 2.740538 17809155 -1.824341 39319269 0.347906 44726185 -0.017537 09579136

0 -2.931439 15434537 13.807540 21102258 -22.548376 93858622 14.469500 48135243 -3.499065 80767630 1.125663 98859899

Appendix

References

227

References 1. Wallis, R. A.: Axial flow fans and ducts. John Wiley&Sons, 1983 2. Carolus, Th., Zhu, T., Sturm, M.: A low pressure axial fan for benchmarking prediction methods for aerodynamic performance and sound. Noise Control Engr. J. 63 (6), November–December 2015, pp. 1–9 3. Heinzerling, W., Trischler, H.: Otto Lilienthal. Deutsches Museum, München, 1991 4. Anderson, John D., Jr.: A history of aerodynamics. Cambridge University Press, 2000 5. Fister, W.: Fluidenergiemaschinen. Band 1, Springer-Verlag, Berlin-Heidelberg, 1984 6. Fister, W.: Fluidenergiemaschinen. Band 2, Springer-Verlag, Berlin-Heidelberg, 1984 7. Pfleiderer, C., Petermann, H.: Strömungsmaschinen. Springer-Verlag, 6. Auflage, 1991 8. Jacobs, E., Ward, K.E., Pinkerton, R.M.: The characteristiscs of 78 related airfoil sections from tests in the variable-density wind tunnel. NACA Report No. 460, 1933 9. Abbott, I. H., von Doenhoff, A. E.: Theory of wing sections, including a summary of airfoil data. Second edition, Dover Publications, 1959 10. Loftin, L. K., Cohen, K. S.: NACA Technical Note No. 1591, 1948 11. Marcinowski, H.: Strömungsmaschinen II. Skript zur Vorlesung an der Universität (TH) Karlsruhe, 1975 12. Althaus, D.: Profilpolaren für den Modellflug. Neckar-Verlag, VS-Villingen, 1980 13. Althaus, D., Wortmann, F.X.: Stuttgarter Profilkatalog I, Friedr. Vieweg & Sohn, Wiesbaden, 1981 14. Pfleiderer, C.: Die Kreiselpumpen für Flüssigkeiten und Gase. 5. Edition, Springer-Verlag, Berlin-Heidelberg, 1961 15. Drela, M.: XFOIL: An analysis and design system for low Reynolds number airfoils. Low Reynolds number aerodynamics, edited by T.J. Mueller, Vol. 54 of Lecture Notes on Engineering, Springer-Verlag, New York, June 1989, pp. 1-12 16. Abbott, H., von Doenhoff, A. E., Stivers, L. S.: Summary of airfoil data. NACA-Report Nr. 824, 1945 17. Riegels, F. W.: Aerodynamische Profile. R. Oldenbourg, München, 1958 18. Herrig, L. J., Emery, J. C., Erwin, J. R.: Systematic Two-dimensional cascade tests of NACA 65-series compressor blades at low speeds. 16], 1957 19. Thies, W.: Eppler-Profile. 8. Auflage, Verlag für Technik und Handwerk, Baden-Baden, 1981 20. Albring, W.: Angewandte Strömungslehre. 4. Auflage, Verlag Theodor Steinkopff, Dresden, 1970 21. Althaus, D.: Niedriggeschwindigkeitsprofile. Friedr. Vieweg & Sohn, Wiesbaden, 1996 22. Eppler, R.: Airfoil design and data. Springer-Verlag, Berlin-Heidelberg, 1990 23. DIN EN 61260: Elektroakustik – Bandfilter für Oktaven und Bruchteile von Oktaven. March 2003 24. ISO 3745-1977: Acoustics – Determination of sound power levels of noise sources – Precision methods for anechoic and semi-anechoic rooms. 1977 25. Mellin, R. C., Sovran, G.: Controlling the tonal characteristics of the aerodynamic noise generated by fan rotors. Trans. of the ASME, J. of Basic Engineering, pp. 143–154, March 1970 26. Fabre, A.: Parametrisierte Näherungsfunktionen für die experimentell ermittelten aerodynamischen Eigenschaften von NACA-Tragflügelgittern nach LIEBLEIN. Individual project No. A00 020 004, Universität Siegen, 2001

Answers to Practice Problems

11.1

11

Answer to Problem 1.5.1: Pressure Rise Requirement of a Plant and Fan Specification

General strategy The key is the evaluation of Eq. (1.7) Δpt,plant,req = ðpII - pI Þ þ

 X ρ
2 c - c2I þ Δploss : 2 II I → II

The fan must then deliver Δpt = Δpt,plant,req as stated with Eq. (1.9). Solution for system A The stations I and II are chosen here in such a way that a thought streamline starts in the free atmosphere with the air at rest and ends, when the discharge jet has mixed with the ambient atmosphere and the jet velocity has completely decayed, Fig. 11.1.1 Thus, cI = cII = 0. The pressure at I and II is the imposed atmospheric pressure p0, so that pII – pI = 0. The mean flow velocity through the radiator with its (reference) cross-section ARad = 12 m2 is

Alternative positions of the stations I and II are possible, resulting in different values for pI, pII, cI and cII and Δploss. In any case, however, the total pressure rise required by the plant is independent of the choice of the stations I and II. 1

# Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 T. Carolus, Fans, https://doi.org/10.1007/978-3-658-37959-9_11

229

230

11

Answers to Practice Problems

Fig. 11.1 Control surfaces/ stations for system A

cRad =

14:4 V_ = 1:2 m=s: = 12 ARad

Therefore, the pressure loss in the radiator matrix and fan cowl becomes ρ 1:00 2 Δploss,Rad = ζ c2T = 200 1:2 = 144 Pa: 2 2 The second loss relevant in this example is the loss due to the jet dissipating in the ambient atmosphere, the so called discharge loss. The discharge loss always corresponds to a pressure loss coefficient ζ Dis = 1. Hence, one gets ρ 1:00 14:4 Δploss,A = ζ Dis c2A = 1 14:42 = 104 Pa with cA = = 14:4 m=s: 2 2 1 Eventually, the total pressure rise requirement by system A becomes

11.1

Answer to Problem 1.5.1: Pressure Rise Requirement of a Plant and Fan . . .

Δpt,plant,req =

X I → II

231

Δploss = Δploss,Rad þ Δploss,A = 144 Pa þ 104 Pa = 248 Pa:

This means that the fan needs to be specified for a total pressure rise Δpt = Δpt,plant,req = 248 Pa at a volume flow rate V_ = 14:4m3 =s: Solution for system B The only difference to system A is that the exit duct has been replaced by a diffuser. This reduces the discharge loss to ρ V_ = 11 m=s Δploss,A = c2A = 60 Pa because of cA now being 2 AD and the total pressure requirement of the plant to Δpt,plant,req =

X

Δploss = 144 Pa þ 60 Pa = 204 Pa:

I → II

This means that the fan needs to be specified for a total pressure rise Δpt = Δpt,plant,req = 204 Pa at a volume flow rate V_ = 14:4m3 =s: Solution for system C The combination of Eqs. (1.7) and (1.9) gives ρ Δpt = ΔpVerl,Rad þ c2A : 2 Because the fan is an axial fan without a guide vane, the discharge jet is a swirling jet, Fig. 11.2. This makes the discharge loss very difficult to determine by the manufacturer.2 However, in this configuration, where the fan is the last component downstream in the system, the discharge loss of the system ðρ=2Þc2A is identical to the discharge loss of the fan ðρ=2Þc22 :

Typically, manufacturer calculate the total pressure rise from the effective pressure rise measured on many standardized test rigs. 2

232

11

Answers to Practice Problems

Fig. 11.2 Control surfaces/ stations for system C

ρ 2 2 c2



Δpts þ

ρ 2 c 2 A

 

Δp |{z}t

¼ Δploss,Rad þ

Hence, it is sensible to specify the effective rather the total pressure rise of the fan Δpts = Δploss:Rad = 144 Pa at a volume flow rate V_ = 14:4m3 =s: This procedure saves the detour via the total pressure rise avoiding the two possible errors; (a) due to an inaccurate total pressure rise of the fan, and (b) an erroneous estimate of the discharge loss in the system. Note on the required fan shaft power for all three systems In addition to the total and effective pressure rise, manufacturers frequently specify the total (-to-total) and the total-tostatic efficiency for their fans, here for example η = 85% and ηts = 40%, respectively. The shaft power is always PS =

_ t _ ts VΔp VΔp , = η ηts

resulting in 4.2 kW for the fan in system A, 3.5 kW in system B, and 5.2 kW in system C. In terms of energy consumption plant B is clearly superior to A because the diffuser reduces the discharge loss. The shaft power of the fan in system C is the highest because the fan has no guide vanes which recovers the energy in the swirl of the discharge jet.

11.2

Answer to Problem 1.5.2: Selection of the Type of Fan

11.2

233

Answer to Problem 1.5.2: Selection of the Type of Fan

With the help of the Cordier-diagram in Fig. 11.3, typical fans are chosen: • the axial with σ opt = 2.0 and δopt = 1.3 • the centrifugal with σ opt = 0.2 and δopt = 5.4 By rearranging Eqs. (1.13 and 1.14) one obtains n = σ opt

h


 - 1=4 3=4 - 1=2 2π 2 Y t V_

i

 1=4  8 - 1=4 _ 1=2 Yt and Dtip = δopt V : π2

Inserting the values on the right hand sides of these equations yields: Fan For plant D E

Axial Dtip [m] 0.44 0.80

n [1/min] 12,000 2200

u [m/s] 276 92

Centrifugal Dtip [m] 1.8 3.3

n [1/min] 1200 220

u [m/s] 113 38

The axial fan for system D is small but must rotate at high speed. Therefore, the centrifugal fan seems to be more suitable. Conversely, the size of the centrifugal fan for

Fig. 11.3 The two types of fans in the Cordier-diagram

234

11

Answers to Practice Problems

plant E is extremely large. The axial fan with its medium size and realistic speed might be the better choice. The table also shows the circumferential speeds u = πDtipn. In case of the axial fan for system D, it is so high that incompressibility of the fluid in the fan cannot assumed any longer.

11.3

Answer to Problem 1.5.3: From Model- to Full-Scale

a) The model laws in terms of proportionalities (1.15) to (1.17) readily yield the relationships between model and full scale

V_ Mod

  D3tip n 0:3053 x3000 m3 Mod = 31:5x = 0:70 = V_ FS 3  s 1:828 x627 D3tip n FS

Δpt, Mod

  D2tip n2 ρ 0:3052 x30002 x1:2 Mod = 800x = Δpt, FS = 765 Pa:  1:8282 x6272 x0:8 D2tip n2 ρ FS

These are the fan performance parameters expected in the model test. Assuming 85% efficiency of the model, the shaft power required to drive the model becomes, according to Eq. (1.5), PS,Mod =

3 0:70 ms x765 Pa V_ Mod Δpt,Mod = 630 W: = ηMod 0:85

The shaft torque is given by M S,Mod =

PS,Mod 630 W = 2 Nm: = 2πnMod 2πx3000=60 1=s

b) The ratio of the Reynolds numbers in the model and the full-scale version is 

Re Mod = Re Gr

  nD2tip  2 z}|{ = Mod = 3000x0:3052 = 0:133:  627x1:828 nD2tip 

uDtip  ν Mod u = πDtip n



uDtip  ν FS

FS

Utilizing ACKERET’s scale-up formula (1.19) with chosen constants V = 0.6 and α = 0.25, the difference of model- to full-scale efficiency is

11.4

Answer to Problem 3.3.1: Design of a Centrifugal Fan Impeller

235



α   Re Mod ηFS - ηMod = ð1 - ηMod Þ  V 1 = ð1 - 0:85Þx0:6 1 - 0:1330:25 Re FS = 0:036 = 3:6%: Hence, if the efficiency of the model is determined experimentally as 85%, the full-scale fan is expected to perform at 88.6%.

11.4

Answer to Problem 3.3.1: Design of a Centrifugal Fan Impeller

Step 1: Design performance parameters (blades) via Eqs. (2.23) and (2.25) V_ 0:48 m3 V_ B = = 0:511 and = ηvol 94% s Y t,Imp 1000 W Y t,B = = = 1205 ηh 83% kg=s

Step 2: Calculation of the velocity triangles according to Sect. 2.2.1 • At blade inlet u1 = 28.3 m/s, cm1 = 10.8 m/s, (cu1 = 0 m/s due to swirl-free inflow) and the flow angle β1 = 20.8°. • At blade exit u2 = 47.1 m/s, cm2 = 7.7 m/s, with Euler’s equation cu2 = Yt, B/u2 = 25.6 m/s, flow angle β2 = 19.8°. Step 3: Estimation of the number of blades A first guess of the number of blades is z = 10. Step 4: Iterative calculation of the blade exit angle without blade blockage Because of D1/D2 > 0.5 the slip factor Eq. (3.5) is used. ψ’ is determined via Eq. (3.8) since the impeller is to be designed for operation in a volute casing. With the initial value β21 = β2 = 19.8°, the iteration proceeds according to Table 11.1. Eventually, the blade exit angle without obstruction becomes βB2, w/o = β21 = 30.5 °. Step 5: Checking the estimated number of blades. The flow angle at the blade inlet is chosen as the blade inlet angle without obstruction, βB1, w/o = β1 = 20.8 °. Then

19.8 ° 28.7 ° ... 30.5 °

β21

0.930 1.035 ...

0

  βS2 ψ = 0:7 1 þ 60 ∘ 0.775 0.756 ...

μ= 1 þ

2ψ 0 zð1 - ðD1 =D2 Þ2 Þ

-1

Table 11.1 Iteration for calculating the blade exit angle without obstruction Y Sch μ

115 W(kg/s) 1596 W(kg/s) ...

Y Sch1 =

Y Sch1 u2

33.0 m/s 33.8 m/s ...

cu21 =

cm2 u2 - cu21 28.7 ° ↵ 30.2 ° ↵ ... 30.5 °

arctan

β21 =

236 11 Answers to Practice Problems

11.4

Answer to Problem 3.3.1: Design of a Centrifugal Fan Impeller

Table 11.2 Iteration for calculating the blade inlet angle with blockage  0  cm1 s t1 0 Δ = arctan s = 1 c = c u1 m1 t 1 - su1 u1 - cu1 - β1 βB2,with sin βB1,with m1 20.8° 0.0084 m 12.6 m/s 3.3 ° ... ... ... ... 23.7°

237

βB1, with = βB1, w/o + Δ1 24.1° ↵ ... 23.7°

z = 10 sin βDB21 = 12:7 and 1 D1 D2  1þD 2 sin ½ 0:5 ð β þ β Þ  = 8:7 to 13:7: z = 5 to 8 D1 B1 B2

• Equation (3.12a) yields • Equation (3.12b)

1-D

2

Thus, the initial guess z = 10 is within the range of the two correlations; fine-tuning of this figure needs either experiments or a numerical flow field and performance simulation. Step 6: Inlet blade angle with blockage taken into account The blade spacing at the inlet is t1 = πD1/z = 0.057 m. With the initial value βB1,with = βB1,w/o = 20.8° the iteration for calculating the inlet angle with blockage taken into account proceeds according to Table 11.2. Eventually, the converged solution becomes βB1,with = 23.7°. Step 7: Exit blade angle with blockage taken into account An analogous iteration yields the blade exit angle with blockage taken into account to βB2,with = 31.7°. Step 8: Construction of the circular arc blade With the geometry sizes now known, the circular arc blades can be drawn as outlined in Fig. 3.14. Step 9: Summary Figure 11.4 left depicts this simple centrifugal impeller. The figures in Table 11.3 illustrate the exaggeration of the blade shape relative to the flow. Naturally, the compensation Fig. 11.4 Left: Centrifugal impeller; the shroud is transparent for didactic reasons; right: Contour of volutes for two different spiral angles

238

11

Answers to Practice Problems

Table 11.3 Summary of the results Flow angles β2 β1 20.8° 19.8°

Blade angles without blockage βB1,w/o βB2,w/o 20.8° 30.5°

Blade angles with blockage βB1,with βB2,with 23.7° 31.7°

required for the finite thickness of the blade is somewhat greater at the inlet with 2.9° than at the outlet with only 1.2°. Anyway, the most substantial difference between flow and blade angle at the blade channel exit with 30.5° – 19.8° = 10.7° is due to the slip.

11.5

Answer to Problem 3.3.2: Design of a Volute


_ _ a) The volute contour is given by Eq. (3.28): r φ = r 2 e φ tan αS : The spiral angle αS is 16.7°, since tan αS =

cm2 cu2

=

7:7 m=s 25:6 m=s

= 0.301 (Eq. 3.25b). r2 is 0.150 m (= D2/2).

b) In the case of a casing width B different from the impeller width b2, αS is to be replaced by αS, Eq. (3.30): tan αS =

b2 cm2 B cu2

=

1 7:7m=s 3 25:6m=s

= 0:100. The spiral angle αS is now much

flatter at 5.7°. This means that the radial dimensions of the casing become much more feasible, Fig. 11.3 right.

11.6

Answer to Problem 4.6.1: Design of Low-Pressure Axial Fan

Step 1: Design performance parameters Dtip and n are given. The reader may confirm that corresponding values of σ opt and δopt are well within the Cordier-band. In this practice problem the value of Dhub is given as well, but will be checked with the help of the design criteria in step 3. • Blade volume flow rate Eq. (2.25) V_ 31 m3 = 32:6 : = V_ B = ηvol 95% s

• Since the effective pressure rise Δpts,Imp is given, the blade total pressure rise Δpt,B must be obtained via Eqs. (10.5, 10.6, and 10.7). The prerequisite of a swirl distribution rcu2 = const. is part of the specification and hence satisfied. With ηh = 90% and ηvol = 95% one obtains k1 = 4245.3 Pa and k2 = 1.885 × 106 Pa2 and eventually

11.6

Answer to Problem 4.6.1: Design of Low-Pressure Axial Fan

239

Δpt,B = 504 Pa or Y t,B = 420 W=ðkg=sÞ: Note the substantial difference between the targeted effective pressure rise of the fan Δpts = 250 Pa and the value of the total pressure rise the blades have to be designed for! Step 2: Segmentation Equation (4.14) is used to determine the blade sections. For didactic reasons only n = 4 elemental blade cascades and hence blade sections are chosen. Step 3: Blade loading, velocity triangles
 t,CA The stipulated swirl distribution is r j cu2j r j = const: = Y2πn with Yt,CA = Yt,B. This inherently results in the radius-independent meridional flow velocity cm2j(rj) = const. = _

VB : ðDtip 2 - Dhub 2 Þ The meridional flow velocity at the inlet and exit of each elemental blade cascade is cm1j = cm2j = cmj. Due to the swirl-free inflow, c1j = cmj applies. Table 11.4 shows the values of the remaining quantities of the velocity triangles according to the formulas from Sect. 2.2.2. Some of the design criteria can now be checked: The De Haller-criterion Eq. (4.41) can   be regarded as satisfied with ww21  = 0:63 (the hub corresponds to j = 1), the Strscheletzky hub cm2  criterion Eq. (4.42) with cu2  = 1:0 as well. In conclusion, the given size of the hub

π 4

hub

diameter Dhub is a reasonable choice.

Table 11.4 Velocity triangles

Radius Impeller circumferential velocity impeller Meridional flow velocity, absolute flow velocity inlet Circumferential component of c2 Absolute flow velocity exit Relative flow velocity inlet Relative flow velocity exit Vectorial mean Flow angle inlet Flow angle exit Vectorial mean

rj uj cmj, c1j cu2j c2j w1j w2j w1j β1j β2j β1j

m m/s m/s

j=1 (hub) 0.39 27.6 15.8

2 0.61 43.1 15.8

3 0.77 54.3 15.8

4 (tip) 0.9 63.6 15.8

m/s m/s m/s m/s m/s deg deg deg

15.2 21.9 31.8 20.0 25.4 29.8 52.0 38.4

9.7 18.6 45.9 36.9 41.3 20.1 25.3 22.5

7.7 17.6 56.6 49.2 52.9 16.2 18.7 17.4

6.6 17.1 65.5 59.2 62.4 13.9 15.5 14.7

240

11

Answers to Practice Problems

Table 11.5 Blade design with E 392 airfoil sections αj εj CLj σj tj lj Relj × 10-5 γj

Angle of attack (chosen) Drag-to-lift ratio Lift coefficient Solidity Blade spacing for z = 9 blades Chord length Rel ≥ 5 × 105? Stagger angle

deg – – – m m – deg

6.5 0.01 1.14 1.037 0.272 0.282 4.8 (✓) 44.8

5.5 0.009 1.05 0.439 0.425 0.187 5.1 ✓ 28.0

5.0 0.008 1.00 0.285 0.537 0.153 5.4 ✓ 22.4

5.0 0.008 1.00 0.205 0.628 0.129 5.3 ✓ 19.7

Step 4: Calculation of the solidity The airfoil section E 392 at a Reynolds number of about 500,000 is chosen along the blade. The corresponding CL-α polar curves can be found in Fig. 10.15. The airfoil should operate mainly at its most favorable, i.e. smallest drag-to-lift ratio ε. In order not to let the chord length at the hub become too large, a somewhat larger angle of attack is selected at the hub. The angles of attack α, listed in Table 11.5, are a reasonable choice. The corresponding values of the lift coefficient CLj follow immediately. The solidity according to Eq. (4.22) σj =

Y t,CA  1 2 w1j uj C Lj 1 þ

εj tan β1j



is calculated for each elemental blade cascade as in Table 11.5. Step 5: Number of blades and chord length The specified number of blades z = 9 yields the blade spacing via Eq. (4.23) tj =

2πr j z

and eventually the chord lengths lj = σ j t j : Now the chord Reynolds numbers can be calculated via Eq. (4.24). The values in Table 11.5 indicate that the Reynolds number is always in the range of 500,000. Hence, the airfoil polar curve for Rel = 500,000 is applicable. Moreover, the number of blades chosen is appropriate. Step 6: Stagger angle and airfoil coordinates The stagger angle is given by Eq. (4.25a) as γ j = β1j + aj.

11.6

Answer to Problem 4.6.1: Design of Low-Pressure Axial Fan

241

Fig. 11.5 Low pressure axial fan; left: Result of the practice problem, here with a segmentation into 15 instead of four elemental blade cascades; right: Modified version: Impeller with truly swept (not skewed) blades; blade load shifted towards the blade tip, sweep angle of stacking line at hub: δ = -30° (backward sweep), at tip: 55° (forward sweep); for structural reasons seven instead of nine blades

The coordinates of the basic E 392 can now be taken from Reference [22] in Chap. 10, so that the blade profile can be drawn for each section. Step 7: Complete blade The complete blade and the complete impeller is depicted in Fig. 11.5. Final remarks: • For low-pressure fans with a small hub-to-tip ratio, the blade regions close to the hub are often problematic, as is the case here: Strictly speaking the relatively large solidity at the hub σ = 1.037 requires a method which takes into account the interference of adjacent blades. The limit for the BEM method was σ = 0.7. • For cost or structural health reasons, one could also choose a lower number of blades. Since σ is fixed, this would lead to larger chord lengths, but would also increase the axial extension of the blades and the impeller. From an aerodynamic perspective the choice of a lower number of blades is to be preferred, since the Reynolds number increases and thus the airfoil aerodynamic polars tend to improve. • The strong blade twist and the large differences in blade chord lengths at the hub and at the casing can be reduced by shifting blade load towards the tip (see Sect. 4.1.2). Ultimately, this measure can also allow reducing the hub diameter. This can bei beneficial for minimizing the impeller exit flow velocity and the exit kinetic energy, which affects the effective pressure rise. • The E 392 airfoil has a taper trailing edge that is difficult to manufacture. There are two options to avoid this: Shorten the airfoil from the trailing edge or thicken it. The influence of thickening the airfoil has been addressed in [1].

242

11.7

11

Answers to Practice Problems

Answer to Problem 4.6.2: Design of High-Pressure Axial Fan Stage

a) Design of the impeller Step 1: Design performance parameters Dtip and n are given. The reader may confirm that corresponding values of σ opt and δopt are well within the Cordier-band. In this practice problem the value of Dhub is given as well, but will be checked with the help of the design criteria in step 3. • Blade volume flow rate Eq. (2.25) V_ 2:65 m3 V_ B = = 2:70 = : ηvol 98% s

• Blade total pressure rise via Eqs. (2.27) and (2.23) Δpt 3024 = = 3360 Pa and hence ηCas 90% Δpt,Imp 3024 Δpt,B = = = 4200 Pa or ηh 80% Δpt,Imp =

Y t,B = 3500 W=ðkg=sÞ: Note the difference between the targeted total fan pressure rise Δpt = 3024 Pa and this value of the total pressure rise the blades have to be designed for! Step 2: Segmentation Equation (4.14) is used to determine the blade sections. For didactic reasons only the hub and tip sections are considered in the next steps. Step 3: Blade loading, velocity triangles The stipulated swirl distribution is rjcu2j(rj) = const. = Yt, B/(2πn) which automatically


 results in the radius-independent meridional flow velocity cm2j r j = V_ B = π4 Da 2 - Di 2 : The meridional flow velocity at the inlet and exit of each elemental blade cascade is cm1j = cm2j = cmj. Due to the swirl-free inflow, c1j = cmj applies. Table 11.6 shows the remaining quantities of the velocity triangles according to the formulas from Sect. 2.2.2. Some of the design criteria can now be checked: The De Haller  criterion Eq. (4.41) can be regarded as just satisfied with ww21  = 0:56 (the hub corresponds Hub

11.7

Answer to Problem 4.6.2: Design of High-Pressure Axial Fan Stage

243

Table 11.6 Velocity triangles impeller Radius Circumferential velocity Meridional flow velocity, absolute flow velocity inlet Circumferential component of c2 Absolute flow velocity exit Relative flow velocity inlet Relative flow velocity exit Flow angle inlet Flow angle exit Flow angle absolute velocity exit

rj uj cmj, c1j cu2j c2j w1j w2j β1j β2j α2j

  to j = 1), the Strscheletzky-criterion Eq. (4.42) with ccm2  u2

Hub

m m/s m/s m/s m/s m/s m/s deg deg deg

0.14 (hub) 66.0 42.2 53.1 67.8 78.3 44.1 32.6 73.0 38.5

0.20 (tip) 94.2 42.2 37.1 56.2 103.3 71.0 24.1 36.5 48.6

= 0:8 as well. In conclusion, the

proposed size of the hub diameter Dhub is a reasonable choice. Step 4: Calculation of the solidity The actual solidity is obtained from the specified chord length and number of blades: σj =


l  : 2πr j =z

Table 11.6 shows that the solidity does not exceed the range 0.4 ≤ σ ≤ 2.0. Moreover, the values of the chord based Reynolds numbers, Eq. (4.24), are always ≥ 2 × 105. Hence, Lieblein’s design charts are applicable. Step 5: Calculation of inlet and exit blade angles βB1 and βB2 are determined with the help of Lieblein’s design charts as described in Sects. 4.4.1, 4.4.2, and 4.4.3. All intermediate variables and the final results are compiled in Table 11.7. Step 6: Stagger angle and blade section coordinates The theoretical lift coefficient CfL as a parameter of the NACA (A10) mean line according to Eq. (10.25) is determined. Input is the camber angle φ according to Eq. (4.38). Superposition of a thickness distribution, e.g. NACA 65(216)-010 from Table 10.6, scaling up the chord length l and rotating by the stagger angle γ yields the blade section coordinates. The stagger angle is obtained via Eq. (4.39).

244

11

Answers to Practice Problems

Table 11.7 Impeller blade design (z = 20 blades, l = 0.060 m) Relative blade height

 ðr - rtip Þ  ðrhub - rtip Þj

–

0.0

1.0

Solidity From Fig. 10.22 From Fig. 10.23

σj i0j nj

– deg –

From Fig. 10.24 From Fig. 10.25 From Fig. 10.26

δ0j mj

deg – –

1.364 ≈6 ≈0.17 ≈ 2 .1 ≈ 0.22 ≈ 0.05

0.955 ≈ 4.8 ≈0.29 ≈ 2.1 ≈ 0.30 ≈ 0.14

3D-correction from Table 4.2 3D-correction from Table 4.2 Profile camber angle Eq. (4.38) Blade inlet angle Eq. (4.34) Blade exit angle Eq. (4.37) Stagger angle Eq. (4.39) Theoretical lift coefficient NACA 65 airfoil Eq. (10.26) Diffusion coefficient Eq. (4.43b) Rel ≥ 2 × 105? Radius equivalent circular arc mean line Eq. (4.40)

(ic - i2D)|j (δc - δ2D)|j φj βB1j βB2j γj CfLj

deg deg deg deg deg deg –

≈ 1.6 ≈ 1.0 58.8 30.2 89.0 59.6 2.38

≈ -2.6 ≈ -0.5 27.6 18.3 45.9 32.1 1.10

DF0.1j Reljx10–5 ρj

– – m

0.68 ✓ 2.3 ✓ 0.061

0.50 ✓ 3.4 ✓ 0.126

  ∂δ=∂ijβ1 

j

b) Design of the outlet guide vanes The outlet guide vanes are designed analogously. w1, w2 and β1 are replaced by c2, c3 = cm and α2. The flow exit angle α3 is chosen to be 90°, i.e., the outlet guide vanes are designed for swirl-free exhaust flow, Fig. 11.6. The numerical results are compiled in Table 11.8. In

Fig. 11.6 Section of impeller and outlet guide vanes and nomenclature for flow velocities

11.7

Answer to Problem 4.6.2: Design of High-Pressure Axial Fan Stage

245

Table 11.8 Design of outlet guide vanes (= stator, index “St”), zSt = 26, lSt = 0.060 m 0.14 (hub) 38.5

0.20 (tip) 48.6

deg – deg

90 1.78 ≈7 ≈0.10 ≈ 2.1 ≈ 0.15 62.1

90 1.25 ≈4 ≈0.11 ≈ 1.2 ≈ 0.17 53.5

deg deg

39.3 101.4

46.8 100.3

deg

70.4

73.6

–

2.52

2.16

–

0.60

0.51

rj α2j

m deg

α3 j σj i0 j nj

deg – deg –

From Fig. 10.24 From Fig. 10.25 Camber angle

δ0 j mj

Guide vane inlet angle Guide vane exit angle

αB2j = α2j + i0j + njφj αB3j = α3j þ δ0j þ mj φj
 γ Stj = αB2j þ αB3j =2 CfL j DF0.1 j

Radius Flow angle inlet (= flow angle impeller exit!) Flow angle exit for swirl-free exhaust flow Solidity From Fig. 10.22 From Fig. 10.23

Stagger angle Theoretical lift coefficient NACA 65 airfoil Eq. (10.26) Diffusion coefficient Eq. (4.43b)

φj =

ðα3j - α2j Þþðδ0j - i0j Þ 1 - mj þnj

Fig. 11.7 High-pressure axial fan stage; here for a segmentation into n = 15 elemental blade cascades; in this example the outlet guide vanes are made of thin sheet metal with constant thickness

contrast to the impeller, some of the 3D-corrections are unnecessary (see Sects. 4.4.1 and 4. 4.2). The complete fan stage assembly is depicted in Fig. 11.7.

246

11

Answers to Practice Problems

Final remarks: • The diffusion number at the hub of the impeller exceeds the conservative limit of 0.6, but should still be tolerable according to the discussion in Sect. 4.5.3. • While the meridional velocity cm was assumed to be radius-independent, a more refined design would take into account the velocity deficits at the hub and casing wall, in the tip gap region, etc.; see the contribution by Kosyna et al. in [2]. • As expected the flow angle a3 = 90° immediately downstream of the guide vanes requires a metal exit angle αB3 (= 100.3°) > 90°. If the guide vane assembly is manufactured from one piece, e.g. injection-molded plastic or die-cast, an undercut may require expensive tools. In these case the target of a completely swirl-free exhaust is occasionally dispensed and αB3 is accordingly. • Instead of airfoil shaped guide vanes, thin sheet metal vanes are often used without any excessive loss of efficiency, see the article by Kosyna et al. in [2] and Fig. 11.6.

11.8

Answer to Problem 5.6: Axial Fan – Acoustic Modes

The impeller speed n = 995 rpm corresponds to an angular velocity Ω = 104.2 rad/s. With z = 6 impeller blades the blade passing frequency becomes BPF = 99.5 Hz, Eq. (5.2). All further results are summarized in Table 11.9. In summary, except at BPF, there are propagating modes for all higher harmonics of BPF; this is in agreement with the spectrum in Fig. 5.12, where 35 m downstream of the fan, tones of frequency 2 × BPF, 3 × BPF, 4 × BPF, etc. are clearly visible as distinct peaks, but not the tone at BPF. The negative sign for some circumferential modes indicates that they rotate against the direction of the impeller. But this has no influence on their ability to propagate or not.

11.9

Answer to Problem 6.4.1: Acoustic Model Law

Assuming a value of exponent α = 5, Eq. (6.3b) yields: a) For doubling the rotor speed: ΔLW,oa = 10 × 5 × log1.2 = 4 dB, b) When the fan is increased by a factor of 1.5: ΔLW,oa = 10 × (5 + 2) × log1.5 = 12.3 dB.

11.10 Answer to Problem 6.4.2: Fan Acoustic Power a) In Table 6.2 one finds for the axial fan without guide vanes LWspec,R = 96.6 dB and m = 3.16. The circumferential Mach number of the impeller is Mau,tip = 0.14. Thus, with Eq. (6.5c),

j=4 ω = 2500 rad/s f (= 4xBPF) = 398 Hz

j=3 ω = 1875 rad/s f (= 3xBPF) = 298.5 Hz

j=2 ω = 1250 rad/s f (= 2xBPF) = 199 Hz

j=1 ω = 635 rad/s (Eq.(5.16)) f (= BPF) = 99.5 Hz

Order of the harmonic j

k -3 -2 -1 0 +1 -3 -2 -1 0 +1 -3 -2 -1 0 +1 -3 -2 -1 0 +1

Rotating pressure field excited by the fan

Table 11.9 Results m Eq. (5.5) -33 -20 -7 +6 +19 -27 -14 -1 +12 +25 -21 -8 +5 +18 +31 -15 -2 11 24 37

Mau,pf Eq. (5.18b) -0.0794 -0.1310 -0.3743 +0.4367 +0.1379 -0.1941 -0.3743 -5.2399 +0.4367 +0.2096 -0.3743 -0.9825 +1.5720 +0.4367 +0.2535 -0.6987 -5.2399 +0.9527 +0.4367 +0.2832

Õ propagates, since Mau > Mau,c.o.

Õ propagates, since Mau > Mau,c.o.

Õ propagates, since Mau > Mau,c.o.

None of the modes propagate, since always Mau < Mau,c.o. (Mau,c.o. for hub/tip ratio ν = 0 from Table 5.4)

Corresponding acoustic duct modes

11.10 Answer to Problem 6.4.2: Fan Acoustic Power 247

248

11

Answers to Practice Problems



LW,oa

 


 V_ Δpt 1 = LWspec,R þ 10 log - 1 þ 10m log Mau,tip _V 0 Δp0 η = 96:6 þ 15:6 - 27:2 dB = 85 dB

b) In Table 6.1 one finds LWspec,M = 40 dB for the axial fan without guide vanes. Thus, with Eq. (6.2b), LW,oa = LWspec,M þ 10 log

Δpt V_ þ 20 log = 40 - 2:2 þ 49:5 dB = 87:3 dB _V 0 Δp0

Similarly, Table 6.2 yields C = + 0.10. With the relations in Sect. 6.2.3, the octave band spectrum is obtained as depicted in Fig. 11.8. c) Inserting Eq. (6.15) into Eq. (6.16) gives Pac,oa,1 ≈ z

1 ρ lbΦ2 w21 TI 2 : 48π c0 3

The Reynolds number is Rel = 1.62 × 105 which is required for the sound power due to vortex shedding in the turbulent wake in Eq. (6.20). Table 11.10 shows all results. In this example the prediction suggests that for TI > 4% the dominant mechanism for the broadband sound is the turbulent inflow. In summary, the total sound power levels calculated according to VDI 3731 and 2081 differ by only 2.3 dB, which is quite satisfactory. Sharland’s method reveals the effect of increasing turbulent intensity of the inflow. Given that the influence of the inflow Fig. 11.8 Estimated octave band spectrum according to VDI 2081

References

249

Table 11.10 Predicted sound power levels

TI % 2 4 8

Turbulent inflow Turbulent boundary layer Pac,oa,1 LW,oa,1 [W] [dB] Pac,oa,2 [W] LW,oa,2 [dB] 79 0.38 × 10-6 56 80 × 10-6 -6 85 320 × 10 1300 × 10-6 91

Vortex shedding in the turbulent wake Pac,oa,3 LWao,3 [W] [dB] 84 × 10-6 79

All three mechanisms Pac,oa [W] 164 × 10-6 404 × 10-6 1384 × 10-6

LW,oa [dB] 82 86 91

turbulence is not explicitly considered in VDI 3731 and 2081, their levels are in the range predicted with Sharland’s method.

References 1. Carolus, T., Starzmann, R.: An aerodynamic design methodology for low pressure axial fans with integrated airfoil polar prediction. Proceedings of the ASME Turbo Expo 2011, GT2011-45243 2. Bommes, L., Fricke, J., Grundmann, R. (eds.): Ventilatoren. Vulkan-Verlag, Essen, 2003

Index

A

C

Ackeret, see Scale-up Anechoic, 100, 124, 180 ff termination, 184 ff Annoyance, 131 ff, 215 Angle of attack, 64 ff, 89, 116 ff, 203 ff blade, 18, 29 ff, 72 ff camber, 70, 209 deviation, 70 flow, 18 ff incidence, 70 metal (see Blade) skew, 68 ff, 144, 153 ff stagger, 62 ff, 74, 190 A-weighting, 132, 136 ff, 215 ff

Camber, 51, 67 ff, 190, 206 angle, 70, 209 fan test rig, 174 ff line, 206 ff Chord line, 210 CFD, 159 ff Compactness acoustic, 90 aerodynamic, 89 Carnot, see Diffuser Cordier, 9 ff, 66, 191 ff Cut-off, 95 ff Cut-on, 94 ff

D B Blade cascade, 15 ff circular arc, 41 element, 19 ff elemental cascade, 19 ff element momentum method, 61 passing frequency, 84 shape, 41 ff spacing, 30, 62 ff, 146 ff splitter, 41 ff tandem, 143, 149 twist, 55 ff work, 15 Blockage, 35 ff, 41, 68, 75 BPF, see Blade, passing frequency

De Haller, 75 Design parameter, 16 Dihydral, see Skew Diffuser, 11, 23 ff, 33 ff, 152, 195 ff Diffusion coefficient, 77 Drag-to-lift ratio, 63 ff, 204 ff Drag, 62 ff, 83 coefficient, 204 ff Duct fan test rig, 174 ff, 184 ff

E Effective, 2 ff Efficiency casing, 24 ff effective, 3 hydraulic, 23, 24

# Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2022 T. Carolus, Fans, https://doi.org/10.1007/978-3-658-37959-9

251

252 Efficiency (cont.) impeller, 23 total, 3 volumetric, 23 Energetic mean, 24 Eppler, 210

F Frequency band narrow, 214 octave, 214 ff one-third octave, 214 ff Friedrich-tube, 185 FX-airfoil, 210

G Guide vanes, 23, 53, 58, 71 ff, 76, 83, 85 ff, 99 ff, 111, 144, 147, 152, 169, 195 ff

Index Model laws acoustic, 110 ff aerodynamic, 6 ff

N NACA, 206 ff Noise cavity, 83 loading, 83 ff, 145, 149, 153 thickness, 82 self-, 83, 114, 142 tip clearance, 83, 104 ff, 156 turbulent inflow (see Noise, turbulence ingestion) turbulence ingestion, 83, 148, 149 Non-dimensional parameters, 5 ff Numerical grid, 164 ff

O

Hot wire anemometry, 179 ff

Optimization, 186 ff Optimum point, 4 ff, 9 ff Overload, 5

I

P

Insertion loss, 217

Pareto-optimal, 189 Part-load, 5 Performance characteristics, 4 ff, 8, 10, 68, 155, 161, 171, 175 ff, 190 parameters, 1, 9, 15, 21 Pitot tube, 178 ff Prandtl tube, 178 ff Pressure loss, 4 Pressure rise effective, 2, 195 ff total, 2 total-to static (see Total) total-to-total (see Effective) coefficient (see Non-dimensional parameters) Plant, 1, 3 Power blade, 22 coefficient (see Non-dimensional parameters) shaft, 2 ff Psychoacoustics, 131 ff, 146

H

L LBM, 160 Leakage, 22 ff, 57, 156 LES, 161 ff Lieblein design diagrams, 222 ff diffusion coefficient, 77 methods, 70 ff Lift, 62 ff, 83, 87 ff, 203 ff coefficient, 63 ff, 69, 204 ff Loading distribution, 55 isoenergetic, 55 ff radius-dependent, 58 ff Losses, 21 ff

M Mach number, 1, 77 circumferential, 92, 97 ff, 111 ff Madison, 109 ff Mass flow rate, 1 Mean camber line, 206 Mode, 84 ff, 94 ff, 153

R Radial equilibrium, 52 ff Rankine vortex, 58 ff

Index RANS, 161 ff Rating level, 132 ff, 136 Reaction, 44, 56 ff Reynolds number, 8 ff, 66, 77, 91, 204, 208, 211 ff Rotating stall, 83, 101 Rotor-stator interaction, 83, 85 ff, 96 ff, 99, 147 problem, 169 ff

253 Tail cone, see Diffuser Tip gap, 65, 83, 104 ff, 155 ff, 172, 191 Tip clearance, see Tip gap Thickness distribution, 206 ff Total, 2 ff Total-to-static, see Effective Total-to-total, see Total Turbulence control screen, 150 Turbulent inflow, 83, 89 ff, 99, 102 ff, 115 ff, 119, 125

S Segmentation, 60 ff Serration, 147 ff Scale-up, 8 ff Silencer, 217 ff Slip, 29 ff factor, 32 ff Skew, 68 ff, 144, 153 ff Specific speed, see Non-dimensional parameters Specific diameter, see Non-dimensional parameters Splitter, see Blade Solidity, 51, 62 ff, 70 Sound power level, 181 ff, 214 pressure level, 214 prediction, 109 ff quality, 133 ff Stacking line, 67 ff Stall, 4, 65, 68 noise, 83, 91, 99 Strscheletzky, 75 ff Sweep, see Skew Swirl distribution, see Loading distribution

U URANS, see RANS

V Validation, 171 ff Verification, 171 ff Velocity triangles, 6, 16 ff, 32, 36 ff, 56, 64 Volume flow rate, 1 Volume flow rate coefficient, see Non-dimensional parameters Volute, 25 ff, 33 ff, 44 ff

W Work blade, 15, 32 specific, 2, 21, 54 ff, 64 Wortmann, see FX-airfoil

X T Tandem, see Blade

XFOIL, 209, 211