Fluid Power, Mathematical Design of Several Components

MECHANICAL ENGINEERING THEORY AND APPLICATIONS FLUID POWER, MATHEMATICAL DESIGN OF SEVERAL COMPONENTS No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services MECHANICAL ENGINEERING THEORY AND APPLICATIONS Additional books in this series can be found on Nova‟s website under the Series tab Additional e-books in this series can be found on Nova‟s website under the e-book tab MECHANICAL ENGINEERING THEORY AND APPLICATIONS FLUID POWER, MATHEMATICAL DESIGN OF SEVERAL COMPONENTS JOSEP M BERGADA AND SUSHIL KUMAR New York Copyright © 2014 by Nova Science Publishers, Inc All rights reserved No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers‟ use of, or reliance upon, this material Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works Independent verification should be sought for any data, advice or recommendations contained in this book In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services If legal or any other expert assistance is required, the services of a competent person should be sought FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS Additional color graphics may be available in the e-book version of this book Library of Congress Cataloging-in-Publication Data ISBN:  (eBook) Published by Nova Science Publishers, Inc † New York CONTENTS Preface About the Authors Chapter Introduction Josep M Bergada 1.1 Fluid, A Molecular Point of View 1.2 Fluid, A Thermodynamic Point of View 1.3 Fluid, A Mechanical Point of View 1.4 Continuum Theory 1.5 Local Thermodynamic Equilibrium 1.6 Fluid Properties 1.6.1 Bulk Modulus of a Fluid 1.6.2 Thermal Expansion Coefficient 1.6.3 Relation between Fluid Volume, Bulk Modulus and Thermal Expansion Coefficient 1.6.4 Effective Bulk Modulus 1.6.5 Surface Tension 1.6.6 Definition of Viscosity 1.7 Fluid Kinematics 1.7.1 Concept of Material or Total Derivative 1.7.2 Concept of Convective Flow 1.7.3 Circulation 1.7.4 Streamlines, Path Lines and Streaklines 1.7.4.1 Pathlines 1.7.4.2 Streaklines 1.7.4.3 Streamlines 1.7.5 Concept of Vorticity and Non Rotational Flow 1.7.6 Kinematic Study of a Fluid Particle 1.8 Nomenclature 1.9 References xiii xv 1 4 5 11 12 15 15 16 17 17 18 19 19 21 23 26 28 vi Contents Chapter Main Fluid Mechanics Equations Josep M Bergada 2.1 Introduction Reynolds Transport Equation 2.2 Continuity Equation, Integral Form 2.2.1 Continuity Equation, Differential Form 2.3 Momentum Equation, Integral Form 2.3.1 Momentum Equation, Differential Form 2.4 Momentum Equations for a Non Inertial Coordinate System, Integral Form 2.4.1 Momentum Equations for a Non Inertial Coordinate System, Differential Form 2.5 Equation of Angular Momentum for an Inertial Coordinate System Integral Form 2.5.1 Application of the Angular Momentum Equation to Turbomachinery 2.5.2 Equation of Angular Momentum for Non Inertial Coordinate Systems 2.6 Energy Equation Integral Form 2.6.1 Composition of the Mechanical Work 2.6.2 Energy Equation Applied to Turbomachinery, Case Thermal and Hydraulic Machines 2.6.3 Energy Equation Differential Form 2.7 Application of Differential Equations: Flow under Dominant Viscosity 2.7.1 Flow between Two Parallel Plates 2.7.1.1 Plane Couette - Poiseulle Flow 2.7.1.1.1 Couette Flow 2.7.1.1.2 Hagen-Poiseulle or Plane Poiseulle Flow 2.7.2 Time Dependent Flow, Rayleich Flow 2.7.3 Stationary Flow inside Circular Ducts 2.7.3.1 Poiseulle Flow 2.7.4 Flow between Annular Tubes 2.7.4.1 Example Flow between Two Concentric Pipes Boundary conditions 1a 2.7.4.2 Example Flow between Two Concentric Pipes Boundary conditions 2.7.4.3 Example Flow between Two Concentric Pipes Boundary conditions 2b 2.7.5 Flow between Concentric Rotating Tubes 2.7.5.1 Example Case Boundary Conditions 2.7.5.2 Example Case Boundary Conditions 2.7.5.3 Example Case Boundary Conditions 2.7.5.4 Example Case Boundary Conditions (Modified) 2.8 Introduction to Flow with Negligible Acceleration 2.8.1 Introduction 2.8.2 Reynolds Lubrication Theory Hydrodynamic Plane Journal Bearings 29 29 34 34 36 37 42 45 46 48 50 51 52 54 57 59 59 62 63 63 64 69 69 72 74 75 78 80 83 84 86 88 91 91 94 Contents 2.8.3 Reynolds Lubrication Equation in Cartesian Coordinates, Case Two Dimensional Flow 2.8.4 Reynolds Lubrication Equation in Cartesian Coordinates and for Two Directional Three Dimensional Time Independent Flow 2.8.5 Reynolds Lubrication Equation in Cartesian Coordinates and for Two Directional Three Dimensional Time Dependent Flow 2.8.6 Flow with Negligible Acceleration, Case Cylindrical Journal Bearings Statically Loaded 2.8.7 Reynolds Equation of Lubrication in Cylindrical Coordinates 2.9 Nomenclature 2.10 References Chapter Introduction to Computer Fluid Dynamics (CFD) Sushil Kumar 3.1 Step by Step Numerical Formulation 3.1.1 Selecting an Appropriate Grid and Integration Formulation 3.1.2 Selection of an Appropriate Reference of Frame for the Problem 3.1.3 Selecting Appropriate Boundary Conditions for the Problem 3.2 Basic Fluid Dynamic Equations and Their Physical Interpretation 3.2.1 Understanding Momentum Equation as Flux Equation 3.3 Discretization of Momentum Equation 3.3.1 Temporal Discretization of Generalized Momentum Equation 3.3.2 Spatial Discretization of Generalized Momentum Equation Using Finite Volume Method 3.3.2.1 Source Term Linearization 3.3.3 Spatial Discretization of Generalized Momentum Equation Using Finite Difference Method 3.3.4 Pressure and Velocity Coupling for Finite Volume and Finite Difference Method 3.3.5 Spatial Discretization of Generalized Momentum Equation Using Finite Element Method 3.3.5.1 Weak form of NVS 3.3.5.2 Galerkin Finite Element Approximation 3.4 Solving a Finite Element Problem via Finite Volume Through Coordinate Transformation 3.4.1 Source Term Linearization for Transformed NVS equations 3.4.2 Spatial Discretization of Generalized Transformed Momentum Equation vii 100 101 102 104 111 117 119 121 121 121 122 123 124 125 126 126 127 130 131 132 134 135 137 138 139 140 viii Contents 3.4.3 Different Transformed Momentum Equations in Discrete Form 3.4.4 Pressure and Velocity Coupling for Transformed Equation 3.5 Convergence Criteria 3.5.1 Grid Independency Test 3.6 Closing Remarks 3.6.1 Solving a Steady and Transient Flow Problem 3.6.2 Mesh Topology 3.6.3 Mesh less Method 3.7 Nomenclature 3.8 References Chapter Valves Josep M Bergada 4.1 Introduction 4.2 Conical Seat Relief Valves 4.2.1 Previous Research on Conical Seat Relief Valves 4.2.2 Mathematical Development Based on Laminar Flow across a Conical Valve Seat 4.2.2.1 Theoretical Background 4.2.2.2 Force on a Conical Poppet Assuming Laminar Flow 4.2.3 CFD modelling 4.2.4 Experimental Results 4.2.5 Conclusion 4.2.6 References 4.3 Some Measured Steady State Characteristics on Proportional Directional Control Valves 4.4 Servovalve Performance 4.4.1 Introduction to the Four Nozzle Two Flapper Single Stage Servovalve 4.4.2 Directional Control Four Nozzle Two Flapper First Stage Servovalve 4.4.2.1 Forces Acting onto the Flappers 4.4.2.2 Servovalve Discharge Coefficients 4.4.2.3 Flow Instability 4.4.2.4 Servovalve Erratic Performance 4.4.2.5 Conclusion 4.4.2.6 Servovalve Static Performance Curves 4.4.2.7 References 4.5 Nomenclature Chapter Pumps and Motors Josep M Bergada, Sushil Kumar and John Watton 5.1 Introduction 5.1.1 General Classification of Pumps and Motors 5.1.2 Axial Piston Pump under Research 142 143 146 146 147 147 147 148 148 149 151 151 154 154 158 158 162 167 169 172 173 174 177 177 179 179 188 190 192 204 205 206 209 211 211 212 213 ...MECHANICAL ENGINEERING THEORY AND APPLICATIONS FLUID POWER, MATHEMATICAL DESIGN OF SEVERAL COMPONENTS No part of this digital document may be reproduced, stored in a retrieval... APPLICATIONS FLUID POWER, MATHEMATICAL DESIGN OF SEVERAL COMPONENTS JOSEP M BERGADA AND SUSHIL KUMAR New York Copyright © 2014 by Nova Science Publishers, Inc All rights reserved No part of this book... Bergada 1.1 Fluid, A Molecular Point of View 1.2 Fluid, A Thermodynamic Point of View 1.3 Fluid, A Mechanical Point of View 1.4 Continuum Theory 1.5 Local Thermodynamic Equilibrium 1.6 Fluid Properties