Structural and Thermal Analyses of Deepwater Pipes [1st ed.] 9783030535391, 9783030535407

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Chen An Menglan Duan Segen F. Estefen Jian Su

Structural and Thermal Analyses of Deepwater Pipes

Structural and Thermal Analyses of Deepwater Pipes

Chen An • Menglan Duan • Segen F. Estefen Jian Su

Structural and Thermal Analyses of Deepwater Pipes

Chen An College of Safety and Ocean Engineering China University of Petroleum-Beijing Beijing, China

Menglan Duan College of Safety and Ocean Engineering China University of Petroleum-Beijing Beijing, China

Segen F. Estefen COPPE - Ocean Engineering Department Federal University of Rio de Janeiro Rio de Janeiro, Brazil

Jian Su COPPE - Nuclear Engineering Department Federal University of Rio de Janeiro Rio de Janeiro, Brazil

ISBN 978-3-030-53539-1 ISBN 978-3-030-53540-7 (eBook) https://doi.org/10.1007/978-3-030-53540-7 © Springer Nature Switzerland AG 2021 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 imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To our families.

Preface

Producing oil and gas from offshore and deepwater by means of pipelines has gained a tremendous momentum in the energy industry in the past thirty years. At the time of this writing, the pipeline technology has been successfully used in areas with water depths of greater than 8000 feet. Pipelines, and more generally long tubular structures, are major oil and gas industry tools used in exploration, drilling, production, and transmission. Installing and operating tubular structures in deep waters places unique demands on them. The high pressures and elevated temperatures of the oil wells, the high ambient external pressures, the large forces involved during installation, and generally the hostility of the environment can result in a large number of limit states that must be addressed. This book was intended to cover the scope of limit strength, structural dynamics and thermal insulation of deepwater pipes, which presents its contents in the following four aspects: Chapter 1 presents overview of deepwater pipes including design criterion, manufacturing process, installation method, limit strength, fatigue and considerations in flow assurance; Part I provides limit strength of sandwich pipes, including finite element analysis using Python scripts, Collapse of sandwich pipes with cementitous/polymer composites, buckle propagation of sandwich pipes; Part II deals with the problem on dynamic behavior of subsea pipes, including theory of integral transform technique, flow-induced vibration of functionally graded pipes, two-phase flow-induced vibration of pipelines, vortex-induced vibration of freespanning pipelines; Part III presents thermal analysis of composites pipes, including theory of improved lumped models and enthalpy method, including thermal analysis of composite pipelines with active heating, thermal analysis of sandwich pipelines with phase change material layer. The book was aimed at the practicing professionals, but can also serve as a graduate level text for structural and thermal analysis of deepwater pipelines and risers. Each chapter deals with a specific mechanical or thermal problem that is analyzed independently, for the most part, of others. Entry-level graduate school background in structural mechanics and heat transfer, and working knowledge of issues in structural dynamic and numerical methods should make going through the analytical developments easier. vii

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Preface

Many of the results presented have originated from studies performed over the years in cooperation with colleagues and graduate students. Their contributions are reflected in the references cited in the work. We would like to thank especially Guangming Fu, Tong He, Jinlong Wang for their input in some of the chapters. Thanks are also due to Bingqi Liu, who reproduced several numerical results using modern finite element codes. Additional help received as well as photographs and drawings provided by individuals are acknowledged in the text. Special thanks go to Djane Cerqueira, Hui Wang, and Fangqiu Li for generating some figures and Tongtong Li for proofreading the book. Over the years we have had the pleasure and privilege of interacting with many university colleagues and many researchers and engineers from industry. Although too many to list by name here, we would like to acknowledge interactions with university colleagues Professors Theodoro Netto, Islon Pasqualino, Deli Gao and Jijun Gu. We also thank our colleagues in the Institute for Ocean Engineering and College of Mechanical and Transportation Engineering at the China University of Petroleum-Beijing, and in the Alberto Luiz Coimbra Institute of Graduate School and Research in Engineering (COPPE), Federal University of Rio de Janeiro (UFRJ) for providing a fertile atmosphere for intellectual growth, research and development. Finally, we acknowledge gratefully the financial support from Chinese and Brazilian funding agencies: National Key Research and Development Plan (Grant No. 2016YFC0303704), National Natural Science Foundation of China (grant No. 51879271, 51509258), Fundamental Research Funds from China University of Petrosleum-Beijing, the Chinese Scholarship Council, CNPq, FAPERJ, and CAPES. Beijing, China

Chen An

Beijing, China

Menglan Duan

Rio de Janeiro, Brazil

Segen Estefen

Rio de Janeiro, Brazil March 2019

Jian Su

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Limit Strength of Sandwich Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Dynamics of Fluid-Conveying Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Thermal Analysis of Multilayer Pipelines . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 2

Part I Limit Strength of Sandwich Pipes 2

Sandwich Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Core Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Thermal Insulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Passive Thermal Insulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Active Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Connection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Buckle Arrestor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 9 11 11 11 12 13 13

3

Sandwich Pipes Filled with Steel Fiber Reinforced Concrete . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Finite Element Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Material Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Element Type and Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Steel Tube-SFRC Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Load and Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Numerical Evaluation and Parametric Study. . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Numerical Analysis of Sandwich Pipes Under External Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Numerical Analysis of Sandwich Pipes Under Combined External Pressure and Bending . . . . . . . . . . . . . . . . 3.3.3 Parametric Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 17 18 24 25 26 27 27 27 30 34

ix

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Sandwich Pipes Filled with PVA Fiber Reinforced Cementitious Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Material Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Geometric Features of Tubes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Fabrication of SP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Collapse Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Experimental Results and Discussion . . . . . . . . . . . . . . . . . . . . . 4.3 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Finite Element Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Correlation of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Parametric Studies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 35 39 39 43 45 46 47 49 49 54 55 58

5

Buckle Propagation of Sandwich Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Finite Element Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Material Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Element Type and Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Interface and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Geometry of Initial Imperfections . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.5 Numerical Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.6 Verification of Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Parametric Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Influence Factors of Buckle Propagation Pressure . . . . . . . . 5.3.2 Adhesive Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Geometric Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 59 60 61 61 63 63 64 65 66 66 67 67 69 70

6

Sandwich Pipe: Reel-Lay Installation Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Large-Scale Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Small Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Comparisons Between Large-Scale Tests and Finite Element Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73 73 75 75 79 85 86

Part II Dynamics of Fluid-Conveying Pipes 7

Integral Transform Solutions of Solid and Structural Mechanics Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Integral Transform Solution Methodology . . . . . . . . . . . . . . . . . . . . . . . . . .

89 89 90

Contents

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7.3

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.3.1 Vibration of Axially Moving Euler Beams . . . . . . . . . . . . . . . . 91 7.3.2 Vibration of Axially Moving Timoshenko Beams . . . . . . . . 95 7.3.3 Vibration of Pipes Conveying Fluid . . . . . . . . . . . . . . . . . . . . . . . 100 7.3.4 Vibration of Axially Moving Orthotropic Plates . . . . . . . . . . 103 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.4 8

Pipes Conveying Gas–Liquid Two-Phase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Integral Transform Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Two-Phase Flow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Convergence Behavior of the Solution . . . . . . . . . . . . . . . . . . . . 8.4.3 Parametric Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.4 Volumetric-Flow-Rate Stability Envelope . . . . . . . . . . . . . . . . . 8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

109 109 111 113 116 116 117 118 121 124

9

Pipes Conveying Vertical Slug Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Integral Transform Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Two-Phase Flow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125 125 127 129 132 133 139

10

Pipes Conveying Horizontal Slug Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Integral Transform Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Two-Phase Flow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141 141 142 144 147 148 153

11

Axially Functionally Graded Pipes Conveying Fluid . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Integral Transform Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 Convergence Behavior of the Solution . . . . . . . . . . . . . . . . . . . . 11.4.2 Verification of the Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.3 Parametric Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155 155 157 159 161 161 163 165 171

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Part III Thermal Analysis of Multilayer Pipelines 12

Fundamentals of Thermal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Equation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Heat Conduction in Pipe Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Heat Conduction in a Single-Layer Pipe Wall . . . . . . . . . . . . 12.2.2 Heat Conduction in a Multilayer Pipe Wall . . . . . . . . . . . . . . . 12.3 Steady-State Temperature Distribution of Produced Fluid in an Unheated Pipeline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

175 175 177 177 179

13

Steady-State Thermal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Global Heat Balance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Analysis of Heat Medium Circulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Analysis of Direct Electrical Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

183 183 184 186 188 189 190

14

Steady-State Analysis of Heavy Oil Transportation . . . . . . . . . . . . . . . . . . . . 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

191 191 192 194 196

15

Analysis of Direct Electrical Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Global Heat Balance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Physical Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Electrical Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

197 197 198 199 199 200 202 204

16

Transient Analysis of Multilayer Composite Pipelines with Active Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 The Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Improved Lumped Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4 Numerical Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

205 205 207 209 212 215

181

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

Acronyms

CFDST CDP CHS ECC ESF FE FRC GITT HDPF HPFRCC HPHT HWC PC PEEK PIP PP PVA PVDF SDFV SFRC SHCC SHS SP SPP SS

Concrete-filled double skin steel tubular Concrete damaged plasticity Circular hollow sections Engineered cementitious composites Epoxy syntactic foam Finite element fiber reinforced cement Generalized integral transform technique High density polyimide foam High-performance fiber-reinforced cementitious composites High-pressure and high-temperature Heated water circulation Polycarbonate Polyetheretherketone Pipe-in-Pipe Polypropylene Polyvinyl alcohol polyvinylidene difluoride Solution-dependent field variable Steel fiber reinforced concrete Strain hardening cementitious composites Square hollow sections Sandwich Pipe Solid polypropylene Stainless steel

xiii

Chapter 1

Introduction

To meet the challenge due to high pressure, low temperature, complex internal and external flow, the limit strength, dynamical and thermal analysis are main concerns in designing the deepwater pipes. This chapter introduces the book, viz., signposts to the topics covered. For the sandwich pipes, the collapse behavior, the buckle propagation, and the reel-lay installation effects are presented. For the fluidconveying pipes, the effects of the internal two-phase flow, the slug flow, the external marine current, the axially functional graded material on the dynamic behavior are considered. For the thermal analysis of multilayer pipelines, the aspects on the steady state, the active heating, and the heavy oil transportation are discussed.

1.1 Limit Strength of Sandwich Pipes One of the challenges that the offshore oil industry faces as it moves to ultradeepwater is to design well-insulated pipelines and risers capable to withstand high internal and ambient external pressures. The concepts presented in Part I aim at combining structural strength and thermal insulation in an optimized sandwich pipe with three layers, which are able to work together to resist combined high external pressure and bending loads, typical of installation processes of pipelines in ultradeepwater. As the external pressure increases, the stiffness of pipes gradually decreases. The subsea pipeline may collapse if the limit state is reached due to the excessive external pressure. Furthermore, the local collapse can propagate along the pipeline for long distances in both directions till the external pressure magnitude becomes less than the propagation pressure. Therefore, it is necessary to improve the understanding of the collapse and buckle propagation behavior of sandwich pipes under external pressure.

© Springer Nature Switzerland AG 2021 C. An et al., Structural and Thermal Analyses of Deepwater Pipes, https://doi.org/10.1007/978-3-030-53540-7_1

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2

1 Introduction

Part I of the book is organized as follows: Chap. 2 gives a general idea of the concept of sandwich pipes; Chaps. 3 and 4 present the limit strength of sandwich pipes filled with steel fiber- reinforced concrete and PVA fiber- reinforced cementitious composites, respectively; Subsequently, buckle propagation of sandwich pipes under external pressure is investigated in Chap. 5; Finally, the reel-lay installation effects on the ultimate strength of sandwich pipes are presented in Chap. 6.

1.2 Dynamics of Fluid-Conveying Pipes Vibration induced by internal and external flow is a key issue for subsea production pipelines. Systematic and extensive investigations have been carried out in the past decades to understand the dynamic behavior of pipes conveying single-phase and two-phase flow, and under external cross flow. Many achievements have been made in understanding the dynamic characteristics of pipes conveying fluid. In Part II of the book, a hybrid analytical-numerical method, the Generalized Integral Transform Technique (GITT) is applied systematically to analyze the dynamical behavior of pipelines conveying single and two-phase flow. Chapter 7 presents the theoretical framework of GITT and its applications in solid and structural mechanics problems. In Chap. 8, the dynamic behavior of pipes conveying gas–liquid two-phase flow is analytically and numerically investigated. In Chap. 9, a fluid-structural model for analyzing the dynamic behavior of riser vibration subjected to simultaneous internal gas–liquid two-phase flow and external marine current is presented. In Chap. 10, dynamical behavior of horizontal subsea pipelines transporting twophase gas– liquid slug flow and subject to external marine current is analyzed. Finally, Chap. 11 analyzes the dynamical behavior of axially functional graded pipes conveying single-phase fluid.

1.3 Thermal Analysis of Multilayer Pipelines Keeping the thermodynamic state of produced fluid within adequate range is essential to meet the production requirements and minimize downtime due to possible pipeline blockage. For a given combination of hydrocarbons flowing into a multilayer pipeline, the range of pressure variation is largely determined by the wellhead and separator pressures, while the temperature distribution of the produced fluid is determined by the energy balance over the pipeline. For the thermal design of steady-state operation conditions, adequate thermal insulation system must be specified to meet the requirement of keeping the temperature above the wax appearance temperature. Under shut-in conditions, a reasonable long cooling-down time should be achieved to prevent the formation of gas hydrate. Part III of the book deals with the thermal analysis of multilayer composite pipelines. In Chap. 12, basic governing equations for thermal analysis of pipelines

1.3 Thermal Analysis of Multilayer Pipelines

3

are presented, together with the overall heat transfer coefficient for multilayered composite pipeline and the temperature distribution of the produced fluid along the pipeline under steady-state conditions. Chapter 13 presents a global thermal analysis of multilayer composite pipelines and establishes the requirement of active heating when the passive thermal insulation alone cannot meet the thermal design requirement. Active heating by hot medium circulation and direct electrical heating are discussed. The challenge of heavy oil production is addressed in Chap. 14. Chapter 15 analyzes direct electrical heating at steady-state operation in more details. Finally, Chap. 16 presents transient thermal analysis of multilayer composite pipelines with direct electrical heating, using finite difference method and lumped models.

Part I

Limit Strength of Sandwich Pipes

One of the challenges that the offshore oil industry faces as it moves to ultra deepwater is to design well-insulated pipelines and risers capable to withstand high internal and ambient external pressures. The concept presented in this Part aims at combining structural strength and thermal insulation in an optimized sandwich pipe with three layers, which are able to work together to resist combined high external pressure and bending loads, typical of installation processes of pipelines in ultra deepwater. As the external pressure increases, the stiffness of pipes gradually decreases. The subsea pipeline may experience collapse if the limit state is reached due to the excessive external pressure. Furthermore, the local collapse can propagate along the pipeline for long distances in both directions till the external pressure magnitude becomes less than the propagation pressure. Therefore, it is necessary to perform a thorough investigation to improve understanding of the collapse and buckle propagation behavior of sandwich pipes under external pressure. Part I of the book is organized as follows: Chap. 2 provides a general idea of the concept of sandwich pipes; Chaps. 3 and 4 present the limit strength of sandwich pipes filled with steel fiber reinforced concrete and PVA fiber reinforced cementitious composites, respectively; subsequently, buckle propagation of sandwich pipes under external pressure is investigated in Chap. 5; Finally, the reel-lay installation effects on the ultimate strength of sandwich pipes are presented in Chap. 6.

Chapter 2

Sandwich Pipes

2.1 Introduction Sandwich pipes (SP), considered to be an effective solution for the ultra-deepwater submarine pipeline combining high structural resistance with thermal insulation capability, have attracted considerable research interest over the last few years. Similar to other sandwich structures, SP possess the advantages of improved strength-to-weight and stiffness-to-weight ratios and tailorable characteristics to satisfy the specific requirements for subsea pipelines, flowlines, and risers. To design and develop deepwater pipes, three aspects should be considered: subsea environment, physical properties of crude oil, and installation loads, the details of which are described in Table 2.1. The current most-common industrial solution for deepwater pipes with the requirement of thermal insulation is pipe-in-pipe (PIP) systems, for which, typically, the space between the two pipes is either empty or contains insulation material which provides minimal mechanical support to the system (Kyriakides 2002; Kyriakides and Vogler 2002; Kyriakides and Netto 2004). The carrier pipe and the inner pipe are designed independently to resist the external environmental pressure and the internal hydrocarbon pressure, respectively. Dissimilar to PIP, the design of SP is constrained simultaneously by external collapse pressure, internal bursting pressure and sufficient thermal insulation. In other words, the core layer should withstand the mechanical loads in addition to satisfying the thermal insulation requirements. As shown in Fig. 2.1, several types of core material have been selected and applied to the laboratory prototypes of SP developed previously by Estefen et al. (2005a), Castello and Estefen (2007), Castello (2011b), and An et al. (2012c) successively, where load-bearing capacity analysis of SP subjected to external pressure was performed. Recently, huge subsalt reserves, at a water depth up to 2,400 m, were discovered off the coast of Brazil. Ultra-deepwater scenarios, at depths beyond 1,500 m, require very thick-walled steel pipes or PIP systems, which are expensive and difficult to © Springer Nature Switzerland AG 2021 C. An et al., Structural and Thermal Analyses of Deepwater Pipes, https://doi.org/10.1007/978-3-030-53540-7_2

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Table 2.1 Considerations for design and development of deepwater pipes Aspects Subsea environment Physical properties of crude oil Installation loads

Details High hydrostatic pressure, low temperature, seabed topography, soil condition, etc. High-pressure and high-temperature (HPHT), hydrate plug formation, wax deposition, etc. High bending moment, high clamping force, high tensile force, etc.

Fig. 2.1 SP family: (a) SP with polypropylene core (left) and with pure cement mortar (right) (Estefen et al. 2005a), (b) SP with polypropylene (Castello and Estefen 2007), (c) SP with polypropylene (Castello 2011b), and (d) SP with strain hardening cementitious composites core (An et al. 2012c)

install because of their excessive weight. In this context, SP is being engineered for overcoming the above-mentioned problem and matching all the requirements for deepwater pipes. Therefore, it is necessary to review the concepts and understanding about collapse behavior and buckle propagation of SP under external pressure. The main aim of this chapter is to present the studies performed on buckling, collapse,

2.2 Core Material

9

and buckle propagation analysis of SP under external pressure during the last 10 years, thereby providing a broad perspective of the current state of the art and trends in this field.

2.2 Core Material External pressure resistance and insulation capabilities, therefore, choosing the right core material is crucial for mechanical and thermal performance of sandwich pipes. With this criterion, solid polypropylene (SPP) (Estefen et al. 2005a; Castello and Estefen 2007, 2008), polyetheretherketone (PEEK) de Souza et al. (2007), polycarbonate (PC) (de Souza et al. 2007), epoxy syntactic foam (ESF) (Castello and Estefen 2008), high density polyimide foam (HDPF) (Castello and Estefen 2008) were preliminarily selected for evaluation, whose properties are listed in Table 2.2. With good thermal insulation properties and high compressive strength, SPP was given the most attention as the feasible core material of SP. The elastic behavior of SPP can be modeled by Ogden model (Ogden 1997) for incompressible isotropic hyperelastic materials, in which the material coefficients should be calibrated by the measured stress–strain data. The plastic behavior of SPP is usually neglected due to its yield strain value being much higher than the design strain level of pipes considered during installation process, such as the reel-lay method. Considering the relatively low thermal insulation capacity of SP with SPP core compared to pipe-in-pipe (PIP), other polymer-based materials with lower thermal conductivity can be employed, such as ESF and HDPF. ESF is composite material synthesized by filling an epoxy resin matrix with hollow glass microspheres, which has the almost same yield strength but only half of the thermal conductivity. Except for the high price, HDPF is an advanced thermoplastic elastomer with excellent mechanical and thermal properties required by SP systems, which is extensively used in the aerospace, automotive, and marine industries. Besides polymeric materials, also cement-based materials can be adopted to fill in the annulus of SP due to high compressive strength, relatively low thermal

Table 2.2 Properties of suitable polymeric core materials for SP (de Souza et al. 2007; Castello and Estefen 2008) Material SPP PEEK PC ESF HDPF

Density (Kg/m3) 900 646 679 720 500

a Maximum

Yield strength (MPa) 23 68 44 22 26

service temperature

Yield strain (%) 8.0 4.0 5.0 8.5 9.1

Elastic modulus Thermal conduc(MPa) tivity (W/mK) 1000 0.20 2331 0.18 1599 0.22 1580 0.12 521 0.066

Tmaxa (◦ C) 145 348 188 177 300

10

2 Sandwich Pipes

conductivity, availability, and low cost, such as pure cement mortar (Estefen et al. 2005a), steel fiber reinforced concrete (SFRC) (An et al. 2012a), strain hardening cementitious composites (SHCC) (An et al. 2014). The high tensile strain capacity of the latter two materials (SFRC and SHCC), known as specific versions of high-performance fiber reinforced cementitious composites (HPFRCC), make them particularly suitable for resisting bending loads during deepwater installation of SP. Pure cement mortar can be modeled by a simplified associative flow rule with isotropic hardening for the plastic regime. The yield surface, which is a function of hydrostatic stress and the von Mises equivalent stress, can be calibrated by the uniaxial compression tests and Brazilian indirect tensile tests (Estefen et al. 2005a). SFRC is proposed as the core material, since the cementitious composite possesses increased tensile strength and improved toughness under flexural loading in comparison to plain concrete, as a result of its superior resistance to cracking and crack propagation (Holschemacher et al. 2010). The Concrete Damaged Plasticity (CDP) model described in ABAQUS (2009a) is employed to simulate the inelastic behavior of damaged SFRC including stiffness degradation and crack opening. The material parameters are obtained via uniaxial tensile, uniaxial compression, and four-point bending tests. Different from the tension softening constitutive law and local cracking failure of SFRC under tension, the typical crack pattern of SHCC exhibits strain hardening behavior and multiple fine cracking characters, which allows for large energy absorption (Li and Xu 2010). The tensile strain of SHCC can reach to 3–7% with the crack width smaller than 0.1 mm under uniaxial tensile tests. The typical stress–strain response and crack pattern of SHCC specimens under monotonic tensile loading are shown in Fig. 2.2. Note that SHCC was formerly named as

Fig. 2.2 Typical stress–strain response and crack pattern of ECC specimens under monotonic tensile loading (Weimann and Li 2003)

2.3 Thermal Insulation

11

“Engineered Cementitious Composites (ECC)” by the original developers (Li 1994). SHCC are heterogeneous composites and have a natural multi-scale behavior. For the convenience of analysis and design, however, it is often considered as a homogeneous material at the macroscopic scale. Therefore, CDP model can be also used to simulate mechanical properties of SHCC (An et al. 2014).

2.3 Thermal Insulation Previous studies have been carried out to design sandwich pipes with the insulation of polypropylene and cement paste, which show the advantage of possessing higher mechanical strength than single wall pipes for an equivalent steel weight. It was also observed that the collapse pressure is very sensitive to the variation of geometric imperfections, and the lack of adhesion between the inner layer of the sandwich pipes and the core can reduce the capacity to withstand collapse by up to 64%. Eccentricity did not significantly affect the performance of collapse resistance. Polypropylene is adopted because it is reasonably inexpensive, with good mechanical properties and low thermal conductivity. In addition, the injection process to construct the sandwich pipes can be conducted without major difficulties.

2.3.1 Passive Thermal Insulation In the system, the insulation material fills the annular layer, being in contact with both internal and external surfaces of the steel pipes, Fig. 2.3. The interface bonding should be with good adhesion, enhancing the mechanical strength of the assembly. Such adhesion between metal and annular layer may be achieved by using vulcanization processes. The choice of material for the annular layer depends not only on the thermophysical properties, but also on the mechanical properties, such as tensile strength.

2.3.2 Active Heating The transient heat transfer analysis of sandwich pipes with polypropylene as the insulation material and with active heating was conducted, as shown in Fig. 2.4. The pipes contained 4 strips of electrical cables arranged axial symmetrically. A mathematical model based on finite difference scheme was used to calculate the heat conduction in the sandwich pipes and the energy carried by the fluid produced. It is found that the choice of the insulation thickness is important to ensure the efficiency of the system considering both the cost and the energy generation.

12

2 Sandwich Pipes

Fig. 2.3 Sandwich pipes with polypropylene insulation layer

Fig. 2.4 Sandwich pipes with polypropylene as the insulation material and active heating conductors

2.4 Connection The connections deserve special attention due to the fact that they need to meet the criteria of mechanical resistance, thermal insulation, and the ease of installation. In general, the connection is prone to stress concentration and thermal insulation loss. There are two basic types of connections: mechanical connection and welded connection. Mechanical connections can be employed to conduct the partial replacements of damaged pipes. The dimensional tolerances are indicated by the manufacturing standards. The main concerns for the connections are: ovalization, local imperfection, surface roughness, and straightness. For the welded connections, the pipes can be welded end-to-end directly or welded to one intermediate part. The efficiency and the cost of the process depend on the type of welding selected, while the strength and reliability depend almost entirely on the quality of the weld.

2.6 Fabrication

13

2.5 Buckle Arrestor The damage caused by collapse of sandwich pipes can be extended. The phenomenon is called “buckle propagation,” which is a progressive structural failure mode. Once a collapse initiates in the pipes, it will propagate in both directions under the condition that the operating pressure is greater than a minimum value defined as “propagation pressure.” The attractive way to restrict collapse is to mount the buckle arrestor along the length of the pipes. Studies on evaluating the dynamic performance of these elements show that the propagation occurs at the speed of hundreds of meters per second, depending on the geometry and material of the pipes, the external pressure, and the surrounding medium used. For PIP and sandwich pipes, the buckle arrestor should be designed to affect as little as possible on thermal insulation and flexural stiffness of the original pipes. The buckle arrestor can be also manufactured in epoxy, and can be injected locally in the annular space, before the PIP is transferred to the spool launch. There are several advantages for the epoxy buckle arrestor: Low influence on the overall heat transfer coefficient of PIP, Higher cross-over pressure compared to the “Clamp” type arrestor, Possibility of injecting epoxy onshore to avoid loss of time during installation. Other companies prefer to use the clamp-type arrestor, which is installed externally to the PIP after it is unrolled from the spool (Tough et al. 2001).

2.6 Fabrication During the manufacture process of sandwich pipes, certain geometric defects such as initial ovality are generated in the steel pipes, and due to the characteristics of cold forming, the bending residual stress is also caused. All of these factors can affect the structural performance of the sandwich pies. Similar to the single-layer pipes, each segment of sandwich pipes is assembled on land. Generally speaking, there are two main methods depending on the assembly of the middle layer: (1) Injection of the insulation material (foam or cement) into the annular space, (2) Wrapping the insulation material onto the inner tube in advance, then slipping into the outer tube. In detail, the main steps of the whole process include: (1) Blasting the surface of the steel pipe using the anti-corrosion coating FBE, and placing it on the fixture, (2) Cleaning the outer tube and fixing it, (3) The inner tube is covered with a layer of polypropylene, and the adhesive can be spread at the interface, (4) The inner tube is slipped into the outer tube.

Chapter 3

Sandwich Pipes Filled with Steel Fiber Reinforced Concrete

3.1 Introduction Collapse behavior for pipelines and risers has been investigated systematically due to their importance as common facilities for the offshore oil and gas transportation (Netto and Estefen 1994; Estefen et al. 1995; Estefen 1999; Netto et al. 2005). As an effective solution for the deep and ultra-deepwater submarine pipelines and risers, sandwich pipes (SP), which are composite structures consisting of two concentric steel tubes and a polymeric or cement-based core, have been developed with the capacity to combine high structural resistance and thermal insulation properties (Estefen et al. 2005a). Recently, Castello and Estefen (2007) analyzed numerically the ultimate strength of SP filled with solid polypropylene under external pressure and longitudinal bending, estimated the reeling effect on the ultimate strength and observed that the ultimate strength is strongly dependent on the inter-layer adhesion by performing the numerical simulation with a contact surface model. Extending previous work, Castello and Estefen (2008) conducted the collapse simulation of three SP employing different annular materials (solid polypropylene, epoxy foam, and polyimide foam) and showed that both steel weight and submerged weight are reasonably lighter than a pipe-in-pipe (PIP) system when designed for a hypothetic oil field with the specific requirements, such as inner diameter, maximum heat transfer coefficient, and water depth. In another study, Castello et al. (2009) investigated the effects of relative ovality direction and temperature-dependent polymer stiffness on the collapse mode. Moreover, Arjomandi and Taheri (2010) presented an analytical approach for estimating the elastic buckling capacity of sandwich pipes with different inter-layer bonding configurations under external hydrostatic pressure. Arjomandi and Taheri (2011a) studied the influence of certain structural parameters on the plastic buckling pressure capacity of sandwich pipelines based on the finite element approach and presented an optimization procedure on the material and geometry of the SP system to minimize a desired cost function. With © Springer Nature Switzerland AG 2021 C. An et al., Structural and Thermal Analyses of Deepwater Pipes, https://doi.org/10.1007/978-3-030-53540-7_3

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3 Sandwich Pipes Filled with Steel Fiber Reinforced Concrete

the finite element method, Arjomandi and Taheri (2011b) considered four design configurations for the SP with respect to the bonding properties between core layer and surrounding tubes, and proposed a simplified practical equation for calculating the pressure capacity of SP. Furthermore, Arjomandi and Taheri (2011c) developed a simplified equation suitable to evaluate the buckling capacity of SP with all kind of interface conditions, which was used to optimize SP configurations suitable for different operational water depths. Besides, Su et al. (2005) studied the transient heat transfer in sandwich pipelines with active electrical heating and showed that SP with active heating is a viable solution to meet severe flow assurance requirements of ultra-deepwater oil production even under unplanned and prolonged cool-down conditions. Based on the philosophy of the annular materials selection for SP, viz. low cost materials with high compression strength (Estefen et al. 2005a), cementbased material can be also adopted. This double skin sandwich structure was firstly introduced as a new form of construction for deepwater vessels to resist external pressure (Montague 1978; Goode et al. 1996), then used for submerged tube highway tunnel (Wright et al. 1991a,b), legs of offshore platforms (Wei et al. 1995a,b), and high-rise bridge piers (Yagishita et al. 2000). With the advantages of enhanced global and local stability, lighter weight, good damping characteristics, and good cyclic loading performance (Elchalakani et al. 2002), a similar concept, known as concrete-filled double skin steel tubular (CFDST) columns, has been widely investigated for their potential applications in building structures. Zhao et al. (2002) developed a plastic mechanism to predict the collapse behavior of concretefilled double skin stub columns, where confinement and strength degradation were considered for the concrete model. Based on eight compression tests, Elchalakani et al. (2002) presented the typical failure modes and an axial strength model for CFDST stub columns with circular hollow sections (CHS) as outer tubes and square hollow sections (SHS) as inner tubes. Han et al. (2004b) performed a series of compression and bending tests on CFDST stub columns, beams and beam-columns with SHS as outer tubes and CHS as inner tubes, and also developed mechanics models using the unified theory, where a confinement factor was introduced to describe the composite interaction between the steel tube and the sandwiched concrete. As a continuation of their research, CFDST stub columns and beamcolumns with CHS for both outer and inner tubes were experimentally studied by Tao et al. (2004). Furthermore, Han et al. (2009) suggested simplified models for the moment versus curvature response and the lateral load versus lateral deflection, respectively, based on the mechanics model predicting the behavior of CFDST beam-columns subjected to constant axial load and cyclically increasing flexural loading. In addition to experimental and analytical approach, the finite element (FE) method has been employed to predict the three-dimensional behavior of CFDST columns. With an equivalent stress–strain model presented by Han et al. (2007), Huang et al. (2010a) reported a finite element analysis of the compressive behavior of CFDST stub columns with SHS or CHS outer tubes and CHS inner tubes. To understand the non-uniformly confined concrete by fiber reinforced polymer, Yu

3.2 Finite Element Modeling

17

et al. (2010a) critically assessed the existing Drucker–Prager (D-P) type concrete plasticity models for confined concrete, and proposed a modified D-P type model implemented in FE program ABAQUS by modifying its Extended Drucker–Prager Model and making use of the facility of user-defined solution-dependent field variables (SDFV). The proposed model is unable to simulate the reduction of elastic stiffness during the loading process. To overcome the limitation, Yu et al. (2010b) developed an improved plastic-damage model within the theoretical framework of the Concrete Damaged Plasticity (CDP) model in ABAQUS, which is applied in FE models to investigate the behavior of confined concrete in various forms of columns. An accurate estimation on the collapse pressure of submarine pipeline subjected to combined external pressure and bending loads is of great importance for pipe design to assure the safe installation and operation (Zhang and Li 1997). In this, chapter the ultimate strength of sandwich pipes for combined external pressure and longitudinal bending is studied using FE model based on ABAQUS (2009a). Steel fiber reinforced concrete (SFRC) is proposed herein as the core material, since the cementitious composite possesses increased extensibility and tensile strength under flexural loading, as a result of its superior resistance to cracking and crack propagation (Balaguru and Shah 1992; Holschemacher et al. 2010). The material properties of SFRC used in the FE modeling are adopted from a recent evaluation by Velasco (2008), where a systematic study was performed on the mechanical characteristics of the self-consolidating concrete reinforced with high volumetric fractions of steel fibers. The CDP model is used to simulate the inelastic behavior of damaged SFRC including stiffness degradation and crack opening. The material parameters are obtained from the uniaxial compression test and four-point bending test. The pressure-curvature ultimate strength for SP with perfect adhesion and no adhesion interface condition is presented. Besides, a parametric study is also performed in order to investigate the effect of the thickness of each layer on the pressure-curvature collapse envelopes of SP.

3.2 Finite Element Modeling The same geometrical properties of the sandwich pipes analyzed by Castello and Estefen (2007) are employed here, as presented in Table 3.1, where Dn , t, Ri and Re present the nominal diameter, the pipe thickness, the inner and outer radius, respectively. An initial ovality (Δ0 = 0.2%) is introduced in the numerical model. Table 3.1 Geometrical properties of the sandwich pipes (Castello and Estefen 2007)

Dn (in) 6 58 Annular 8 58

Ri (mm) 77.75 84.15 103.15

Re (mm) 84.15 103.15 109.55

t (mm) 6.35 19 6.35

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3 Sandwich Pipes Filled with Steel Fiber Reinforced Concrete

3.2.1 Material Characteristics API X-60 steel is used for inner and outer tubes, with yield stress 414 MPa, Poisson coefficient 0.3, and Young modulus 205 GPa. It is modeled by Hooke’s law of elasticity theory and the J2 flow theory of plasticity associated with von Mises yielding criteria and isotropic hardening for the proposed model under combined external pressure and bending loads. The cementitious composite material, viz. SFRC with fiber content of 2%, is produced from self-consolidating matrices reinforced with high volumetric fractions of steel fibers (Velasco 2008). Mixture proportion for per cubic meter of SFRC is given in Table 3.2. The CDP model defined in ABAQUS (2009a) is used to simulate the mechanical properties of SFRC, which represents the inelastic behavior using the concepts of isotropic damaged elasticity in combination with isotropic tensile and compressive plasticity. The stress–strain relationship for the general threedimensional state is governed by the scalar damage elasticity equation: pl el pl σ = (1 − d)Del 0 : (ε − ε ) = D : (ε − ε ),

(3.1)

el where Del 0 is the initial elastic stiffness matrix of the material, D the degraded elastic stiffness matrix, and d the scalar stiffness degradation variable, varying from zero to one. In terms of effective stress, the yield function takes the form

F =

  1 pl q¯ − 3α p¯ + β(˜εpl )σˆ¯ max  − γ −σˆ¯ max  − σ¯ c (˜εc ) = 0, 1−α

(3.2)

with α and γ are dimensionless material constants, α = α(σb0 /σc0 ), σb0 /σc0 the ratio of initial equibiaxial compressive yield stress to initial uniaxial compressive pl pl pl yield stress, β = (1 − α)σ¯ c (˜εc )/σ¯ t (˜εt ) − (1 + α), σ¯ c (˜εc ) the effective pl tensile cohesion stress, σ¯ t (˜εt ) the effective compressive cohesion stress, σˆ¯ max the maximum principal effective stress, p¯ = − 13 trace(σ¯ ) the hydrostatic pressure stress, Table 3.2 Mixture proportions of SFRC (Velasco 2008)

Components Cement Silica fume Fly ash River sand Crushed stone Superplasticizer (%) Steel fiber (Dramix) Wollastonite Water

Materials 243.83 kg/m3 34.83 kg/m3 69.66 kg/m3 845.83 kg/m3 845.93 kg/m3 1.50 156 kg/m3 145 kg/m3 174.16 kg/m3

3.2 Finite Element Modeling

19

 3 ¯ ¯ ¯ q¯ = 2 (S : S) the Mises equivalent effective stress, and S the effective stress deviator. Plastic flow is governed by a flow potential function G(σ ) according to nonas) sociated flow rule dεpl = dλ ∂G(σ potential G used for the model is the ∂σ . The flow  Drucker–Prager hyperbolic function, G = ( σt0 tanψ)2 + q¯ 2 − ptanψ, ¯ where ψ is the dilation angle measured in the p − q plane at high confining pressure, σt0 the uniaxial tensile stress at failure, and a parameter, referred to as the eccentricity, that defines the rate at which the function approaches the asymptote (the flow potential tends to a straight line as the eccentricity tends to zero). The dilation angle ψ = 23 φ is adopted, where φ is the internal-friction angle as a critical parameter of the Mohr– Coulomb failure criterion model and can be measured from triaxial compression test. In ABAQUS/Standard 6.9-1, the stress–strain curves for uniaxial tension and compression are needed to define elastic, plastic, and damage behaviors, as shown in Fig. 3.1 (ABAQUS 2009a), where E0 is the initial (undamaged) elastic stiffness Fig. 3.1 Stress–strain curves of concrete under (a) uniaxial tension and (b) uniaxial compression (ABAQUS 2009a)

σt σto

E0

(1−dt)Ε0 ∼ pl εt

εt

εtel

(a)

σc σcu σc0

E0

∼ pl εc

(1-dc)E0

εc

εc0l (b)

20

3 Sandwich Pipes Filled with Steel Fiber Reinforced Concrete pl

pl

of the material, ε˜ t and ε˜ c the tensile and compressive equivalent plastic strains, respectively, dt and dc the uniaxial damage variables for tension and compression, respectively, σt0 the uniaxial tensile stress at failure, σc0 the initial compressive yield stress, and σcu the ultimate compressive stress. The stress–strain relations under uniaxial tension and compression are pl

(3.3)

pl

(3.4)

σt = (1 − dt )E0 (εt − ε˜ t ) σc = (1 − dc )E0 (εc − ε˜ c ).

To be suitable as input of damaged plasticity model, the experimental stress– strain behaviors of tension and compression, obtained by Velasco (2008), are mathematically presented by the following ideal stress–strain curves. The tension behavior of SFRC is defined by two curves, including one stress–strain curve before crack nucleation and another postfailure stress-cracking displacement curve, illustrated in Fig. 3.2. The linear stress–strain relationship is expressed by σt (εt ) = E0 εt , εt ≤ εt0 and the tri-linear model for stress-cracking displacement is determined by the following points, (0, σto ), (w1 , σt1 ), (w2 , σt2 ), and (wu , 0), where σt1 = k1 σt0 , σt2 = k2 σt0 , w1 = wu /c1 , and w2 = wu /c2 . k1 and k2 are the empirical parameters that can better describe the postfailure softening behavior in uniaxial tension test, while c1 and c2 are the constants, c1 = 20 and c2 = 5, respectively (Velasco 2008). As proposed by Velasco (2008), the ideal stress–strain curve under compression is composed by three sections: (a) initial elastic branch, (b) damage-based plastic rising branch, and (c) damage-based plastic declining branch, illustrated in Fig. 3.3, and the relations are given by the following equations: ⎧ E0 εc , for εc ≤ εc0 , ⎪ ⎪ ⎪ ⎪

   ⎪ ⎪ εc η1 ⎨ σcu 1 − 1 − , for εc0 < εc ≤ εcu , σc (εc ) = εcu ⎪ ⎪

   ⎪ ⎪ εc − εcu η2 ⎪ ⎪ ⎩ σcu 1 − , for εcu < εc ≤ εcm , εcm − εcu

(3.5a,b,c)

where εcu is the strain corresponding to the ultimate stress, εcm = kc εcu the maximum strain in the ideal model, kc the empirical parameter obtained from the compression tests, η1 and η2 the exponentials describing the curvatures of rising and declining branches, respectively. η1 and η2 can be estimated from the experimental data using the Statistical “NonlinearFit” package of Mathematica 7.0 (Wolfram 2003). The damage parameters dt and dc employed in concrete compression hardening and tension stiffening curves can be calibrated through uniaxial compression and four-point bending tests. Based on the observation and verification on the simulation examples in ABAQUS (2009b), Wang and Chen (2006) concluded that there is a first-order decay exponential function relationship between normalized compressive

3.2 Finite Element Modeling

21

σt

σto

Eo

εto

εt

(a) σt

σt0 σt1

σt2

w1

w2

wU w (b)

Fig. 3.2 Ideal stress–strain curve for uniaxial tension of SFRC (Velasco 2008)

in damage variable Dcnorm and normalized compressive inelastic strain ε˜ cnorm , as follows:

Dcnorm = A0 e−˜εcnorm /t0 + B0 , in

(3.6)

22

3 Sandwich Pipes Filled with Steel Fiber Reinforced Concrete σc σcu σc0

E0

εco

εcu

εcm

εc

Fig. 3.3 Ideal stress–strain curve for uniaxial compression of SFRC (Velasco 2008). (a) Before crack initiation. (b) Postfailure

ε˜ in

in c where Dcnorm = dc , ε˜ cnorm = εcm , ε˜ cin = εc − Eσc0 , A0 = −1/t10 , B0 = − −1/t10 , e −1 e −1 and the only unknown is t0 . The stress–strain curves are obtained by simulating the uniaxial compression test with various values of t0 in ABAQUS, then the actual value of t0 can be determined by the agreement with the stress–strain curve from the tests. Similarly, the relationship between normalized tensile damage variable Dtnorm and normalized cracking displacement wnorm can be also fit with a first-order decay exponential function as follows:

Dtnorm = A1 e−wtnorm /t1 + B1 ,

(3.7)

where Dtnorm = dt , wtnorm = wwu , A1 = e−1/t11 −1 , B1 = − e−1/t11 −1 , and the only unknown is t1 . With the determined parameter t0 , the simulation of four-point bending test needs to be performed to calibrate the value of t1 by correlating the load-displacement curve. For the material definition of SFRC, the Poisson’s ratio ν is set to 0.2, based on the observation by Thomas and Ramaswamy (2007) that the effect of steel fibers on the Poisson’s ratio of concrete is not significant. The internal-friction angle φ can be adopted as 37◦ , since a trivial effect of the steel fiber reinforcement on this parameter was found by Lu and Hsu (2006). Besides, a small value for the viscosity parameter (μ = 0.0001) is defined to improve the convergence rate in the concrete softening and stiffness degradation regimes, following the suggestion from Barth and Wu (2006). Using the above-mentioned approach, the values of the variables t0 and t1 for compression and tension damages are estimated to be 10 and 0.5, respectively. The hardening and softening behaviors and the evolution of the scalar damage variables for compression and tension are presented in Table 3.3.

3.2 Finite Element Modeling

23

Table 3.3 Material parameters of CDP model for SFRC Young’s modulus E0 (Mpa) Dilation angle ψ (◦ ) σb0 /σc0 Viscosity parameter Stress 30.92 40.91 49.51 56.72 62.57 67.06 70.21 72.67 71.08 70.50 69.28 68.03 66.04 63.84 48.11 31.13 13.24 3.99 Stress 4.560 2.791 0.935 0.223

37673 24.7 1.16 0.0001 Crushing strain 0 0.000073062 0.000177135 0.000317789 0.000494683 0.000707398 0.000955386 0.001553581 0.002947045 0.003412622 0.004344204 0.005276020 0.006710993 0.008255655 0.019802627 0.029558802 0.039220713 0.044016885 Cracking strain 0 0.757463 3.317160 5.354480

Poisson ratio μ Eccentricity

K

0.2 0.1 0.6667

dc 0 0.001628997 0.003949000 0.007083635 0.011024611 0.015761663 0.021281523 0.034584562 0.065507880 0.075819450 0.096421534 0.116988087 0.148580648 0.182480784 0.405067257 0.621532123 0.831925126 0.934891454 dt 0 0.155465715 0.613804594 0.914807031

Crushing Strain 0 0.000073062 0.000177135 0.000317789 0.000494683 0.000707398 0.000955386 0.001553581 0.002947045 0.003412622 0.004344204 0.005276020 0.006710993 0.008255655 0.019802627 0.029558802 0.039220713 0.044016885 Cracking Strain 0 0.757463 3.317160 5.354480

Figure 3.4a shows the comparison between the numerical result predicted using ABAQUS/Standard 6.9-1 and the ideal model from experimental data measured by Velasco (2008) for the uniaxial compression test of SFRC, while Fig. 3.4b illustrates the correlation of the numerical result analyzed with the one obtained by Velasco (2008) using Diana (2000) for the four-point bending test. Notice that the SFRC specimens for compression test are cylindrical with diameter of 100 mm and height of 200 mm, and the dimensions of the ones for four-point bending tests are 100 × 100 × 400 mm with a free span of 300 mm. The good agreement between the present numerical outcomes and the previous results validates the accuracy of the proposed definition of material properties.

24

3 Sandwich Pipes Filled with Steel Fiber Reinforced Concrete

Fig. 3.4 Comparison of numerical results predicted by ABAQUS/Standard 6.9-1 with (a) stress–strain curve represented by ideal model and (b) load-displacement curve obtained by Velasco (2008)

3.2.2 Element Type and Mesh Generation The half ring model validated in Estefen et al. (2005a) and Castello and Estefen (2007) is adopted. ABAQUS C3D8R continuum-brick elements are used for modeling both the steel tubes and the SFRC core, which can be used for linear analysis and for complex nonlinear analyses involving contact, plasticity, and large deformations (Kim and Kuwamura 2007). A standard mesh-sensitivity analysis is carried out considering the effect of the element size on the collapse pressure (Pco ), observing that the results tend to converge for 60 elements in the circumferential

3.2 Finite Element Modeling

25

Fig. 3.5 A schematic view of the finite element mesh for SP

Y Z

X

direction, two elements for each steel layer, and four elements for the SFRC annular in the radial direction with lengths of 6.35 mm (z-direction), as shown in Fig. 3.5. Similar mesh-sensitivity analyses are carried out for the remaining studies presented in this chapter.

3.2.3 Steel Tube-SFRC Interface For SP, the adhesion between annulus and steel tubes, which exhibited strong influence on the ultimate strength (Estefen et al. 2005a; Castello and Estefen 2007), should be carefully examined. Therefore, two-layer interface conditions are simulated numerically, including perfect adhesion and no adhesion between steel tubes and SFRC. As proposed by Castello and Estefen (2007) and Huang et al. (2010a), the contact interaction model is applied to the steel tube-SFRC interface, which is defined by a contact pressure model in the normal direction and a Coulomb friction model in the tangential direction. For the unbonded condition, the hard contact relation with separation after contact allowed is selected as normal mechanical property, while no friction is considered as tangential behavior, as depicted in Fig. 3.6b. The fully bonded condition is simulated through hard contact relation with separation after contact not allowed for the normal behavior and the penalty enforcement method for the tangential behavior, respectively. For the latter case, no limit shear stress is adopted, with Coulomb elastic isotropic friction model, as presented in Castello and Estefen (2007).

26

3 Sandwich Pipes Filled with Steel Fiber Reinforced Concrete

Fig. 3.6 FE model of SP: (a) load and boundary condition and (b) interface and coupling condition

3.2.4 Load and Boundary Condition Ultimate strength analysis of SP subjected to external pressure and bending moment independently employs Riks method (the arc-length method) and automatic increment control (the load controlled Newton–Raphson method), respectively. Combined loading is initially implemented by fixed increments of external pressure, followed by incremental rotations until buckling failure is achieved. As shown in Fig. 3.6a, the external pressure is applied on the outer pipe through surface load. The bending moment is induced by defining the rotational displacement in the xdirection at the reference point located at the neutral axis. Automatically generated kinematic coupling equations are used to link the degrees of freedom of the nodes in the transverse plane to the reference point. The coupling of the translational freedom along the z-axis induces a plane strain state for the SP section in order to simulate a long pipe configuration (note that other five freedoms are kept free to move in order to allow the cross section of SP to ovalize but remain plane), as shown in Fig. 3.6b. The longitudinal and transversal symmetry conditions are assumed. Notice here that the longitudinal symmetry condition is applied on the side different from the one set by the coupling equation. Besides, the cross section is partitioned to remain one element at the Y-Z plane to be set by the symmetry condition, and the coupling options are applied at the other elements, through which over-constrained problem can be avoided.

3.3 Numerical Evaluation and Parametric Study

27

3.3 Numerical Evaluation and Parametric Study 3.3.1 Numerical Analysis of Sandwich Pipes Under External Pressure By simulating the behavior of sandwich pipes under hydrostatic pressure using ABAQUS/Standard 6.9-1, the Pco of the SP with perfect adhesion and no adhesion between steel tubes and SFRC are 97.3 and 65.1 MPa, respectively. The contour plots of the von Mises stress, compressive equivalent plastic strain, and compressive damage fields at the Pco for the two cases are represented in Figs. 3.7 and 3.8. Note that the annuluses for both cases are dominated by compressive stress, thus the results of tensile damage field are not shown herein. Through the comparisons between the results, it can be observed that the global stress level of outer tube is lower than the one of inner tube for both cases when collapse happens. Besides, for the SP with perfect adhesion, the stress level of both tubes is higher than the yield stress, but for the one with no adhesion, all of the stress value of both tubes is below the yield stress level. On the other side, considering the compressive damage of the annuluses for both cases, the damage level of the annulus for the SP with perfect adhesion is greatly higher than the one for the SP with no adhesion, where the results of elements set by the transversal symmetry condition are neglected. Comparing the compressive equivalent plastic strain with the data shown in Table 3.3, it can be clearly seen that at the Pco the stress level of the annulus for the SP with perfect adhesion is on the damage-based plastic declining branch, which means the annulus has experienced the ultimate compressive stress, however, the stress level of the annulus for the SP with no adhesion belongs to the damage-based plastic rising branch, not reaching the ultimate compressive stress. Remember that the flow potential G is the Drucker–Prager hyperbolic function, which can describe the lateral confinement effect on stress–strain relation of reinforced concrete, and as a result, the von Mises stress of annuluses for both cases are much greater than the stress obtained from uniaxial compression test of SFRC. The results show that lack of adhesion between steel tubes and annulus can significantly decrease the Pco of SP, or in other words, good adhesion can make each layer generate more contribution towards the overall stiffness of SP.

3.3.2 Numerical Analysis of Sandwich Pipes Under Combined External Pressure and Bending The ultimate strength analysis is performed for the SP with perfect adhesion and no adhesion between steel tubes and SFRC under combined external pressure and bending, and the pressure-curvature collapse envelopes are shown in Fig. 3.9. Each envelope is described by six points, corresponding to constant pressure of 0%, 20%, 40%, 60%, 80%, and 100% of the corresponding Pco . The results show that,

28

3 Sandwich Pipes Filled with Steel Fiber Reinforced Concrete

Fig. 3.7 The contour plots of the von Mises stress and compressive damage fields for SP with perfect adhesion at the Pco : (a) von Mises stress for tubes, (b) von Mises stress for annulus, (c) compressive equivalent plastic strain for annulus, and (d) compressive damage for annulus

the ultimate curvature for the SP with no adhesion between layers increases as the external pressure decreases, which equals to 0.33 in the pure bending mode. However, for the SP with perfect adhesion between layers, the ultimate curvature corresponding to 20%Pco (= 19.464 MPa) is greater than the one caused by the pure bending. Figure 3.10 shows the compressive and tensile damage fields for the SP with perfect adhesion between layers at the limit state corresponding to 0%, 20%, and 40%Pco . By carefully investigating the graphs, it can be observed that the tensile damage of SFRC for all of the cases examined is small compared to the compressive damage, and an interesting finding is the maximum compressive damage occurring at the upper part under the pure bending moves to the lower part when the SP subjected to combined external pressure (20%Pco ) and bending. The reason is that under the external pressure the upper part of SFRC is in three-axis compression, and the ultimate compressive stress is improved, which is different

3.3 Numerical Evaluation and Parametric Study

29

Fig. 3.8 The contour plots of the von Mises stress and compressive damage fields for SP with no adhesion at the Pco : (a) von Mises stress for tubes, (b) von Mises stress for annulus, (c) compressive equivalent plastic strain for annulus, and (d) compressive damage for annulus

from the stress state of the lower part of the annulus. The considerably greater compressive damage shift from the upper to lower region of the annulus can explain the non-monotonicity of the pressure-curvature collapse envelope of the SP with perfect adhesion between layers. The same thing does not happen to the SP with no adhesion between layers, due to the reason that the ultimate strength of this case under the pure bending condition is dominated by the steel tubes, while the one of the SP with perfect adhesion is dependent on the steel tubes together with SFRC annulus. The ultimate strength envelope of the SP cored by polypropylene with perfect adhesion from Castello and Estefen (2007), with the same geometrical properties analyzed herein, is also presented in Fig. 3.9. As can be seen, the Pco of the SP (SFRC) with perfect adhesion is approximately two times than the one of the SP (polypropylene), and even the Pco of the SP (SFRC) with no adhesion is higher than the latter. On the other hand, the ultimate curvature of SP (SFRC) with either perfect or no adhesion under the pure bending condition is far less than the one of SP (polypropylene) with perfect adhesion.

30 Fig. 3.9 Pressure-curvature ultimate strength for the SP with perfect adhesion and no adhesion between steel tubes and SFRC, and compared with the structural performance for the SP cored by polypropylene with perfect adhesion from Castello and Estefen (2007)

3 Sandwich Pipes Filled with Steel Fiber Reinforced Concrete 100 90 80 70 60 50 40 30 20 10 0 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

1.1 1.2

3.3.3 Parametric Studies To the different SP possessing same nominal internal diameter of inner pipe, thicknesses of inner pipe ti , annulus ta , and outer pipe te are the main parameters affecting the ultimate strength under combined pressure and bending. For the parametric study, thicknesses ti and te are assumed identical for simplicity and an initial out-of-roundness of 0.2% is considered. The geometric properties of the calculated SP in case 1 and 2 are presented in Table 3.4, where Dni and Dne are the inner and outer nominal diameters, ti and te the wall thicknesses. Figure 3.11 shows the pressure-curvature collapse envelopes for the SP with different thickness of the annulus ta . The Pco of SP2 with no adhesion and perfect adhesion between steel tubes and SFRC are 118.7 and 119.8 MPa, respectively, as shown in Table 3.5. To reflect the confinement characteristic of the SFRC, the ultimate curvature corresponding to 10%Pco is also calculated for SP2. It can be seen that the value difference between the Pco of SP1 with perfect adhesion condition and the one of SP1 with no adhesion condition is much greater than the gap between those values of SP2. Note that the thicknesses of steel tubes for both SP are the same, 6.35 mm. Compared with the thickness of the annulus for SP1, the one for SP2 is greater, about 2.4 times of the preceding value. When SP2 is subjected to external pressure, although the perfect adhesion condition between layers can increase the Pco , the Pco is actually dominated by the resistance of annulus, and that is why the Pco values of SP2 with no adhesion and perfect adhesion are so close to each other. On the other hand, for SP1 with no adhesion condition, the localized incompatibility between the deformation of steel tubes and that of SFRC causes the separation in the normal

3.3 Numerical Evaluation and Parametric Study

31

Fig. 3.10 The compressive and tensile damage fields for the annulus of the SP with perfect adhesion between layers at the limit state: (a) compressive damage at 0%Pco , (b) compressive damage at 20%Pco , (c) compressive damage at 40%Pco , (d) tensile damage at 0%Pco , (e) tensile damage at 20%Pco , and (f) tensile damage at 40%Pco

32

3 Sandwich Pipes Filled with Steel Fiber Reinforced Concrete

Table 3.4 Geometric properties of SP for parametric studies

Fig. 3.11 Pressure-curvature ultimate strength for SP filled with SFRC for Case 1 (Dni = 6 58 in, ti = te = 6.35 mm)

Dni (in) Case1 SP1 6 5/8 SP2 6 5/8 Case2 SP1 6 5/8 SP3 6 5/8 SP4 6 5/8

ti (mm)

Dne (in)

te (mm)

6.35 6.35

8 5/8 10 3/4

6.35 6.35

6.35 4.7752 8.7376

8 5/8 8 5/8 8 5/8

6.35 4.7752 8.7376

120 SP1, no adhesion SP2, no adhesion SP1, perfect adhesion SP2, perfect adhesion

110 100 90 80 70 60 50 40 30 20 10 0 0

0.1

0.2

0.3

0.4

0.5

direction and slippage in the tangential direction of the layers, thus the annulus and the inner tube do not contribute the strength to the overall SP strength as much as their contribution in the case of SP2 with perfect adhesion condition. It is worth to note that similar observation on the gaps can be found in the literature accomplished by Estefen et al. (2005a), where the SP annular was filled with pure cement mortar. It can be observed that the ultimate curvatures under pure bending for the both SP with perfect adhesion condition are greater than the ones with no adhesion condition, proving that the good adhesion between layers can improve the bending resistance of the structure. As shown in Fig. 3.11, except for SP1 with perfect adhesion, the relationship between the pressure (P) and ultimate curvature (K) for SP2 with perfect condition or no adhesion condition is also not monotonous, which is different from the results reported by Estefen et al. (2005a). For all of these three cases, the ultimate curvature firstly rises with the increasing of the pressure to a certain value even higher than that of pure bending, and then diminishes till the zero value is reached. Again, the phenomenon can be explained by the material modeling of SFRC, where the lateral confinement effect is described by the Drucker–Prager hyperbolic function. This behavior can be further understood as the confined SFRC

3.3 Numerical Evaluation and Parametric Study Table 3.5 Predicted collapse pressures of SP for parametric studies

Fig. 3.12 Pressure-curvature ultimate strength for SP filled with SFRC for Case 2 (Dni = 6 58 in, Dne = 8 58 in)

33

SP1 SP1 SP2 SP2 SP3 SP3 SP4 SP4

Steel tube-SFRC interface No adhesion Perfect adhesion No adhesion Perfect adhesion No adhesion Perfect adhesion No adhesion Perfect adhesion

Pco (MPa) 65.1 97.3 118.7 119.8 60.4 75.4 70.8 122.9

130 SP1, no adhesion SP3, no adhesion SP4, no adhesion SP1, perfect adhesion SP3, perfect adhesion SP4, perfect adhesion

120 110 100 90 80 70 60 50 40 30 20 10 0 0

0.2

0.4

0.6

dominating the overall structural behavior of the SP, since for SP2 the thickness of the annular is much greater than the ones of steel tubes. On the other hand, for SP1 with no adhesion condition the ultimate curvature decreases monotonously with the increasing of the external pressure. The reason is that the no adhesion condition weakens the contribution of the annulus to the overall structural strength. In other words, the overall structure represents the characteristics of the steel tubes more than the one of annulus. Figure 3.12 shows the pressure-curvature collapse envelopes for the SP filled with SFRC for Case 2, which have the same inner and outer nominal diameters but different wall thickness. From the results, it can be observed that both the Pco and the ultimate curvature in pure bending of SP with no adhesion condition rise with the value of ta increasing. Due to the perfect adhesion condition, the corresponding values of the Pco and the ultimate curvature increase for all of the SP in Case 2. Besides, the lateral confinement effect can also be clearly seen for the SP with perfect adhesion condition.

34

3 Sandwich Pipes Filled with Steel Fiber Reinforced Concrete

3.4 Conclusions The SP filled with SFRC are analyzed for ultimate strength under combined external pressure and bending, using a ring section model. To reduce the numerical instabilities due to the detection of cracks when using the smeared crack concrete model, the material behavior of SFRC is modeled by damaged plasticity model in Abaqus 6.9-1, where the experimental data of the uniaxial compression and tension tests are adopted. The damage parameters are defined by the first-order decay exponential functions. Good agreement between experimental measurement and numerical analysis shows the accuracy of the proposed model for SFRC. The SP with no adhesion and perfect adhesion interface condition are analyzed in relation to the collapse pressures and pressure-curvature collapse envelopes. The non-monotonicity of the pressure-curvature collapse envelope of the SP with perfect adhesion between layers can be explained by the lateral confinement effect provided by the inner and outer tubes. The parametric study shows that the thickness of each layer, the adhesion between layers and the lateral confinement effect on SFRC play a dominant role in the pressure-curvature collapse envelope of SP. Although the results from numerical simulations give some new understanding of this kind of SP filled with SFRC, further experimental studies may be performed to verify the numerical models and then confirm the results presented here.

Chapter 4

Sandwich Pipes Filled with PVA Fiber Reinforced Cementitious Composites

4.1 Introduction As the oil and gas industry shifts its attention to the deep and ultra-deepwater fields, pipes with new structural configurations, such as pipe-in-pipe (PIP), are demanded to satisfy the requirement of thermal insulation and mechanical integrity for untreated well fluids transportation, especially when considering long distance subsea tie-back flowlines (Singh et al. 2010). PIP, composed of two concentrically mounted steel tubes with the annular space filled with either circulating hot water or insulated materials, is designed to resist the ambient external pressure by the outer tube, to withstand the internal pressure by the inner tube, and to provide the thermal insulation by the core layer, independently. Recently, as alternative to the traditional PIP system (for which the core material is only used for thermal insulation), sandwich pipes (SP) were introduced to reduce the installation weight (Estefen et al. 2005a), the design of which is constrained simultaneously by external collapse pressure, internal bursting pressure, and sufficient thermal insulation. Similar to other sandwich structures, SP possess the advantages of improved strength-to-weight and stiffness-to-weight ratios and tailorable characteristics to satisfy the specific requirements (such as high structural resistance, good thermal insulation, etc.) for subsea pipelines, flowlines, and risers. Considerable studies have been performed on the structural behaviors of SP under external pressure, bending moment, and combination of these loads over the past several years. Kardomateas (2001) presented a closed-form elasticity solution for a cylindrical sandwich shell under external pressure, for which the two face sheets and the core were assumed to be orthotropic. In addition, an elasticity solution to the problem of buckling of sandwich long cylindrical shells subjected to external pressure was reported (Han et al. 2004a; Kardomateas and Simitses 2005), where the results were compared with the ones obtained using shell theory. The solution proposed can be employed as a benchmark for accurately assessing the limitations of shell theories in predicting stability loss in sandwich © Springer Nature Switzerland AG 2021 C. An et al., Structural and Thermal Analyses of Deepwater Pipes, https://doi.org/10.1007/978-3-030-53540-7_4

35

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4 Sandwich Pipes Filled with PVA Fiber Reinforced Cementitious Composites

shells. An et al. (2013) reviewed most of the research done in recent years (2002–2012) on the buckling, collapse, and buckle propagation of SP, which emphasized the development of theoretical, experimental, and numerical methods adopted to analyze such structural behavior of SP with different core material. Considering the problem as a two-dimensional plane strain problem, Arjomandi and Taheri (2010) recently presented the mathematical formulation of a long threelayer circular cylindrical shell under external pressure, and obtained analytical solutions by introducing a displacement potential function to simplify the governing equation. Estefen et al. (2005a) investigated the collapse behavior of small-scale SP filled with polypropylene (see Fig. 4.1a) under external pressure by the means of hyperbaric chamber tests and finite element (FE) modeling, and presented the pressure-curvature ultimate strength envelope of SP under combined loading of external pressure and longitudinal bending. Castello and Estefen (2007) analyzed numerically the ultimate strength of SP filled with solid polypropylene under external pressure and longitudinal bending, obtained the maximum shear stress of the metal-polymer interface by experimental tests (see Fig. 4.1b), estimated the reeling effect on the ultimate strength, and observed that the ultimate strength is strongly dependent on the inter-layer adhesion by performing the numerical simulation with a contact surface model. Arjomandi and Taheri (2011a) developed a simplified equation suitable to evaluate the buckling capacity of SP with all kinds of interface conditions, which was used to optimize SP configurations suitable for different operational water depths. In that investigation, a Python language script was written to generate and run the FE models with various geometric and material parameters automatically, which provided an effective technical way to conduct the parametric study for structural behavior of SP. Furthermore, Arjomandi and Taheri (2011b) studied the influence of certain structural parameters on the plastic buckling pressure capacity of sandwich pipelines based on the finite element approach and presented an optimization procedure on the material and geometry of the SP system to minimize a desired cost function. With the finite element method, Arjomandi and Taheri (2011c) considered four design configurations for the SP with respect to the bonding properties between core layer and surrounding tubes, and proposed a simplified practical equation for calculating the pressure capacity of SP. In another work, Arjomandi and Taheri (2012) investigated the influence of several significant structural parameters on the pre-buckling, buckling, and post-buckling response of SP subjected to pure bending. Besides, Su et al. (2005) studied the transient heat transfer in sandwich pipelines with active electrical heating and showed that SP with active heating is a viable solution to meet severe flow assurance requirements of ultra-deepwater oil production even under unplanned and prolonged cool-down conditions. The laboratory tests of SP under external pressure were conducted by Estefen et al. (2005a), Castello and Estefen (2007), and Castello (2011b), for which the specimens are shown in Fig. 4.1. In the past, most of the research that has been carried out focused on the structural behavior of SP with polypropylene core (Estefen et al. 2005a; Castello and Estefen 2007; Castello 2011b), as shown in Figs. 4.1a-left,b,c, respectively. However, the distribution uniformity of the interface

4.1 Introduction

37

Fig. 4.1 SP specimens for the laboratory tests: (a) SP with polypropylene core (left) and with pure cement paste (right) (Estefen et al. 2005a), (b) SP with polypropylene (Castello and Estefen 2007), (c) SP with polypropylene (Castello 2011b), and (d) SP with strain hardening cementitious composites core

adhesion between polypropylene and inner (or outer) tube along SP was difficult to control during the fabrication process, which were rather cumbersome to implement in the modeling procedures for FE analysis and caused inconsistency in collapse behaviors of full laboratory prototypes of SP with the same geometrical and material parameters (Castello 2011b). Cement-based materials can be filled into the core of SP as the substitute of polypropylene, assuring good distribution uniformity

38

4 Sandwich Pipes Filled with PVA Fiber Reinforced Cementitious Composites

of interface adhesion, which may significantly reduce the FE modeling efforts on collapse analysis due to the symmetrical characteristic of geometry (Estefen et al. 2005a; An et al. 2012c). The pure cement paste and conventional fiber reinforced concrete (FRC) can meet the compressive strength requirement for SP core material, however, they apt to crack during the installation process (e.g. reel-lay installation) due to the large bending moments induced. To overcome the cracking problem, PVA fiber reinforced cementitious composites (also named strain hardening cementitious composites—SHCC) were introduced as core material for SP due to their high tensile ductility (An et al. 2012a). Different from the tension-softening constitutive law and local cracking failure of FRC under tension, the typical crack pattern of SHCC exhibits strain hardening behavior and multiple fine cracking characteristics, which allow for large energy absorption (Li et al. 2001). The tensile strain of SHCC can reach to 3–7% with the average crack width smaller than 0.1 mm under uniaxial tensile tests. The typical stress–strain response and crack pattern of SHCC specimens under monotonic tensile loading are shown in Fig. 4.2. Note that this SHCC composite family was named as “Engineered Cementitious Composites (ECC)” by the original developers (Li and Leung 1992). Collapse behavior of offshore pipelines under external pressure is a primary concern for ultimate limit state design criteria of structural integrity (Kyriakides and Corona 2007a; Bai and Bai 2005). In this chapter, the ultimate strength behavior of sandwich pipes filled with SHCC subjected to external pressure is investigated using both experimental and numerical approaches, with an emphasis on the problem of collapse of the sandwich pipes filled with SHCC (the core material has been never considered before). The rest of the chapter is organized as follows. In the next section, the preparation procedure for SHCC and the fabrication process for SP are described, including determination of the material characteristics. Then, the fullscale laboratorial tests of SP performed using a hyperbaric chamber are presented. In Sect. 4.3, the FE modeling of SP with SHCC core is given, whose results are in good agreement with the ones obtained from the experiments. A parameter study is then performed to investigate the effects of the ovality, the inner/outer tube thickness, and the radius ratio on the collapse pressure of SP. Finally, the chapter ends in Sect. 4.5 with conclusions and perspectives. Fig. 4.2 Typical stress–strain response and crack pattern of SHCC specimens under monotonic tensile loading (Jun and Mechtcherine 2010)

4.2 Experiments

39

4.2 Experiments 4.2.1 Material Characteristics The materials, stainless steel (SS) and SHCC, used for SP are evaluated for mechanical characteristics through tension and compression tests.

4.2.1.1

Stainless Steel

The inner and outer steel tubes of SP are made of SS304 stainless steel with the nominal yield stress of 205 MPa. Uniaxial tensile testing in the longitudinal direction of the tubes are performed to obtain the actual mechanical properties used in the numerical model. The nominal thicknesses of the inner tube (with diameter of 6 inch) and the outer tube (with diameter of 8 inch) are 1.8 mm and 2.0 mm, respectively. Three specimens are tested for the inner/outer tube, which are taken from the tube at every 120◦ along the cross section. Tensile tests are conducted using an Instron 5566 tensile testing machine equipped with a non-contacting video extensometer. Each specimen is pulled in tension at a constant rate of around 0.1 mm/min. The typical true stress–strain curves for SS304 material of the inner and outer tubes are shown in Fig. 4.3. The difference in their stress–strain response is due to the variation of the components of raw material used for inner and outer pipes (although they comply with the standard specifications and labeled as “SS304” in the manufacturer’s product list). Fig. 4.3 Stress–strain relationship for SS304 material of the inner and outer tubes

600

Stress (MPa)

500

400

300

200

Inner tube Outer tube

100

0 0

2

4 Strain (%)

6

8

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Table 4.1 Mixture proportions of SHCC (unit content: kg/m3 ) C 488.1

4.2.1.2

S 516.13

FA 593.45

W 360.0

SP 30.0

VA 3.2

PVA 29.0

SHCC

The cementitious composite materials are made of commercially available materials in Brazil (except that PVA fibers “REC 15” with a volume content of 2% are produced by Kuraray, Japan). Mixture proportion for per cubic meter of SHCC is given in Table 4.1, where “C” is cement, “S” fine sand with maximum grain size equal to 0.212 mm, “FA” fly ash, “W” water, “SP” superplasticizer, and “VA” viscosity agent. Five tension specimens are prepared during the casting of each SP. The procedure of mixing SHCC is listed as follows: (a) Place all the solid ingredients of the matrix (cement, sand, fly ash, viscosity agent) into the mixer and stir for 1.5 min, (b) add all water within 1.5 min, (c) add 50% of superplasticizer within 1 min, (d) pause 1 min for cleaning the inner wall and bottom of the mixer, (e) add the remaining superplasticizer (50%) within 1 min, (f) mix for 1 min, (g) pause 2 min for cleaning the inner wall and bottom of the mixer, (h) add the PVA fibers within 2 min, (i) pause 2 min for cleaning the inner wall and bottom of the mixer, and (j) mix for 2 min. The tensile tests are performed with SHCC specimens to obtain the stress and strain at the first crack, at the post-cracking peak and at the maximum strain, respectively. This type of test has a high degree of difficulty, since any misalignment of the specimen with the axis of the load application can generate bending load or torque, causing the premature rupture of the specimen. The geometry of SHCC specimens and the setup of tensile test are illustrated in Fig. 4.4. The test is controlled by the cross-head displacement at a rate of 0.1 mm/min. The tested tensile stress–strain data and the fitted curve are given in Fig. 4.5a. The fitting scheme is described as follows: (a) pick up the yielding stress (first crack) “A,” the maximum stress “B,” and the ultimate stress “C,” (b) use the Young’s modulus obtained in the compression test to determine the related strain corresponding to “A” (since the Young’s modulus of compression and tension should be the same value in damaged plasticity model adopted in the numerical simulation), and (c) join the points by straight lines. Note that the descending branch of the curve does not accurately represent variation of the strain in the SHCC, since in this stage one single crack widens and dominates the deformation (the localization of the deformation at the crack tip). Five cylindrical specimens of 50 mm diameter and 100 mm height are casted for the compression tests, which are performed on a servo-controlled machine Shimadzu of 1000 kN with an axial deformation rate of 0.015 mm/min. The axial displacements are measured by two LVDT’s (linear variable differential transducer), positioned inside a frame which attaches the specimens. The data of loading and

4.2 Experiments

41

Fig. 4.4 Geometry of SHCC specimens (a) and the setup (b) of tensile test

axial displacements are recorded by a data acquisition system consisting of a conditioner ADS 2000 (16 bits) and a software AQDados version 7.02.08 from Lynx Company. As proposed by An et al. (2012c), the ideal stress–strain curve under compression is composed by three stages (initial elastic branch, plastic rising branch, and plastic declining branch), which are formulated by the following equations: ⎧ E ε , for εc ≤ εc0 , ⎪ ⎪ 0 c ⎪ ⎪

   ⎪ ⎪ εc η1 ⎨ , for εc0 < εc ≤ εcu , σcu 1 − 1 − σc (εc ) = εcu ⎪ ⎪

   ⎪ ⎪ εc − εcu η2 ⎪ ⎪ ⎩ σcu 1 − , for εcu < εc ≤ εcm , εcm − εcu

(4.1a,b,c)

where σc , εc are the compressive stress and strain, E0 the Young’s modulus, σcu , εcu the ultimate compressive stress and the corresponding strain, εcm = kc εcu the maximum strain in the ideal model, kc the empirical parameter obtained from the compression tests. The exponentials η1 and η2 describe the curvatures of rising and declining branches, respectively, which can be estimated with the above-mentioned five groups of data using the nonlinear least square method provided in the Statistical “NonlinearFit” package of Mathematica 7.0 (Wolfram 2003). After calculation, η1 = 2.13 and η2 = 0.52. The uniaxial compression stress–strain relationship and the fitted curve are given in Fig. 4.5b.

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Fig. 4.5 Tested stress–strain relationships and fitted curves: (a) Tension and (b) compression

4.2 Experiments

43

4.2.2 Geometric Features of Tubes The ovalities of the inner tube and outer tube of each SP are determined by the following method: Firstly, the cross sections of the tubes are marked along the length with increasing order of alphabet, as shown in Fig. 4.6a. Secondly, the measurement of maximum diameter (Dmax ), minimum diameter (Dmin ), and their corresponding directions at each section is performed with the aid of FARO’s portable measuring arms, which comprise optical encoders in the joints and a spherical contact probe for measuring the geometry, as shown in Fig. 4.6b. The equipment is connected to a computer and controlled by proprietary software (CAM2), through which data is managed and stored. The main advantage of using FARO arms over the conventional caliper measurement is to obtain large number of points along the circumferential direction at each section of the tubes and save them directly into the computer software. The obtained coordinates of the points are post-processed to calculate Dmax and Dmin of each section. Thirdly, the ovality of each section is computed i )/(D i i i by Δi = (Dmax − Dmin max + Dmin ), i = B, C, D, E, F, G, H. The ovality of i the tube is defined by Δ0 = max{Δ , i = B, C, D, E, F, G, H}. The geometric properties of the inner (denoted by “I ”) and outer (denoted by “O”) tubes of each SP are presented in Table 4.2, where Dn , r0 , Δ0 , and t are the nominal diameter, the measured outer radius of each tube, the initial ovality, and the tube thickness, respectively. Considering that the directions of Dmax of inner and outer tubes should be kept in the same direction during the fabrication process, the

Fig. 4.6 Measurement of tube ovality: (a) Mark the cross sections along the length, (b) measure the shape of cross section using FARO portable arm

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4 Sandwich Pipes Filled with PVA Fiber Reinforced Cementitious Composites

°

° °

°

°

°

Fig. 4.7 The relative positions between the weld lines of the inner and outer tubes for (a) SP1, (b) SP2, and (c) SP3

relative positions between the weld lines of the inner and outer tubes are determined and given in Fig. 4.7. The minimum length of SP for the test is evaluated numerically using ABAQUS/Standard 6.9-1 (ABAQUS 2009a), where the effect of additional rigidity to the SP ends caused by the steel plugs on the collapse resistance of SP is considered. As shown in Fig. 4.8, the sensitivity study shows that the length of SP should be six times greater than the outer tube diameter. The length of each tube is 1.75 m.

4.2 Experiments

45

Fig. 4.8 Sensitivity study of the length effect on the collapse pressure (Pco−min is the minimum value of the calculated Pco , L and OD are the length and the nominal outside diameter of SP, respectively)

4.2.3 Fabrication of SP Two plugs are fabricated to fix the tubes during the casting and curing processes, while the upper plug has holes in order to help to check if the annulus is fully filled with SHCC after casting. The machine oil is coated on the surfaces of the plugs to facilitate the pull-out after 48 h. The materials and the tools needed to fabricate the SP are shown in Fig. 4.9. Before starting mixing the SHCC components, inner and outer tubes should be mounted on the lower plug, as shown in Fig. 4.9a. To investigate the most critical case, the directions of Dmax of inner and outer tubes (e.g. position of Dmax of SP2-I (84◦ from its diameter-measurement starting point) and position of Dmax of SP2-O (92◦ from its diameter-measurement starting point), see Fig. 4.9b) should be aligned. Three fixers are employed to confine the relative displacement between inner and outer tubes during pouring the SHCC and vibration, as shown in Fig. 4.9b. The component materials of SHCC are shown in Fig. 4.9c and also listed in Table 4.1. The tools needed for casting SP are demonstrated in Fig. 4.9c, in which the sealing cap is to seal the inner tube during pouring SHCC to avoid the ingress of the material, and the funnel is attached to the outer tube and works as the guidance of SHCC slurry. The procedures of mixing the SHCC are described in Sect. 4.2.1.2. The SP are kept in a room with the temperature of 21◦ C. To keep the humidity, damp cloths are used to cover the upper and lower plugs. After 2 days, the plugs are removed. Then, the ends of SP should be continuously covered by damp cloths for curing until 28-day age (see Fig. 4.9d).

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4 Sandwich Pipes Filled with PVA Fiber Reinforced Cementitious Composites

Fig. 4.9 Materials and tools needed for casting SP

4.2.4 Collapse Experiment To assist the numerical analysis of the collapse pressure Pco of SP with SHCC core, collapse tests are performed on full-scale pipe samples. A total of three SP are tested under hydrostatic pressure. The nominal diameters of the inner and outer tubes are 6 and 8 inches, respectively. In the following, we present the detailed procedures for performing the test. The hyperbaric chamber at Subsea Technology Lab (LTS) of Federal University of Rio de Janeiro (UFRJ) is utilized to carry out the collapse tests of SP with SHCC core, as shown schematically in Fig. 4.10, which has been used in several full-scale collapse test programs. The hyperbaric chamber has the length of 5 meters, the inner diameter of 15 inch, and the operating pressure capacity of 50 MPa. The common method of conducting the collapse experiments is enclosing an endcapped specimen in a hyperbaric chamber and pressurizing the vessel. The outer tubes of SP specimens are marked with nine sections along the axial direction (see

4.2 Experiments

47

Fig. 4.10 Sketch of the setup used for the collapse test of SP with SHCC core

Fig. 4.6a) and with 36 uniformly distributed points on each section, taken as the reference of the deformed SP surface after testing. The test starts from placing the capped SP inside the hyperbaric chamber. Then, seal the vessel and fill the cavity completely with water. A pneumatic control system drives a positive displacement pump that pressurizes the chamber gradually. The pressure inside the chamber is monitored at the pump outlet using a pressure transducer, which is connected to a computer for data acquisition. The rate of pressurization employed in the tests is 0.4 MPa/min (60 psi/min). This scheme approximates “volume-controlled” pressurization. Collapse of the SP is recognized by a loudly audible “bong,” and a sudden decrease in pressure. After testing, the specimen is removed from the chamber, followed by another specimen being placed into the chamber.

4.2.5 Experimental Results and Discussion The maximum pressure recorded (Pco ) during the tests is 30.5 MPa for SP1, 30.6 MPa for SP2, and 29.7 MPa for SP3, respectively. The three experimental results have relatively low variability. The average value of Pco is 30.3 MPa and the standard deviation is 0.37 MPa. In general, the results show that the collapse pressure of each SP is consistent, which proves the reliability and repeatability of the test. The collapsed specimens are shown in Fig. 4.11. Structural behavior of each tested SP is carefully examined. On the surface of SP1, some pits can be clearly found, as shown in Fig. 4.12a. This is due to the appearance of air void during the casting process. The weld line of the outer tube is splitted after SP1 collapse, as shown in Fig. 4.12b. It should be noted that the maximum ovality of the outer tube for SP1 happens on section G (Δ0 = max{Δi , i = B, C, D, E, F, G, H} = ΔG ), and Dmin on section G appears in the direction of “7◦ -187◦ ,” which is almost the same direction of flattening (see Fig. 4.12c). Figure 4.12d and e show the collapse behavior of SP2. It can be seen that wrinkles are generated near section I (the right end of SP2) and E. As SP2 collapses, the damage of SHCC increases simultaneously, which induces the increase of compressive stress in the outer tube, leading to compressive buckling of

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4 Sandwich Pipes Filled with PVA Fiber Reinforced Cementitious Composites

Fig. 4.11 SP after collapse test: (a) SP1, (b) SP2, and (c) SP3

the outer tube in the axis direction. From Fig. 4.12f, it can be seen that the collapse propagates along the axial direction till the position of the sealing cap. The reason may be that the air in the cavity of SP3 is compressed suddenly when the collapse happens, which increases the cavity pressure to push the plug out of the specimen. For other specimens (SP1 and SP2) this situation is not observed, since the plugs are well sealed and installed onto their ends.

4.3 Numerical Simulation

49

Fig. 4.12 Structural response of each tested specimen: (a–c) SP1, (d–e) SP2, and (f) SP3

4.3 Numerical Simulation 4.3.1 Finite Element Modeling In this section, the ring model is adopted to simulate the structural behavior of SP subjected to external pressure. The measured maximum ovality of the nine sections is assumed to be uniformly distributed along each SP. The imperfection introduced in the model is w = r0 Δ0 cos2θ for both inner and outer tubes, where r0

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4 Sandwich Pipes Filled with PVA Fiber Reinforced Cementitious Composites

Table 4.2 Geometric properties of the inner and outer tubes of each SP

Label SP1-I SP1-O SP2-I SP2-O SP3-I SP3-O

Dn (inch) 6 8 6 8 6 8

r0 (mm) 76.2 101.4 76.3 101.5 76.3 101.5

Δ0 0.32% 0.41% 0.22% 0.47% 0.23% 0.39%

t (mm) 1.8 2.0 1.8 2.0 1.8 2.0

represents the measured outer radius of each tube, as given in Table 4.2. This initial imperfection is the more probable buckling mode shape in reality.

4.3.1.1

Material Model

Stainless steel 304 is used for inner and outer tubes, with yield stress 205 MPa, Poisson coefficient 0.3 and Young’s modulus 200 GPa. It is modeled by Hooke’s law of elasticity theory and the J2 flow theory of plasticity associated with von Mises yielding criteria and isotropic hardening law. The CDP model defined in ABAQUS/Standard 6.9-1 (ABAQUS 2009a) is used to simulate the mechanical properties of SHCC, which represents the inelastic behavior using the concepts of isotropic damaged elasticity in combination with isotropic tensile and compressive plasticity. The stress–strain relationship for the general three-dimensional state is governed by the scalar damage elasticity equation: pl el pl σ = (1 − d)Del 0 : (ε − ε ) = D : (ε − ε ),

(4.2)

el where Del 0 is the initial elastic stiffness matrix of the material, D the degraded elastic stiffness matrix and d the scalar stiffness degradation variable, varying from zero to one. In terms of effective stress, the yield function takes the form

F =

  1 pl q¯ − 3α p¯ + β(˜εpl )σˆ¯ max  − γ −σˆ¯ max  − σ¯ c (˜εc ) = 0, 1−α

(4.3)

where α and γ are dimensionless material constants, α = α(σb0 /σc0 ), σb0 /σc0 the ratio of initial equibiaxial compressive yield stress to initial uniaxial compressive pl pl pl yield stress, β = (1 − α)σ¯ c (˜εc )/σ¯ t (˜εt ) − (1 + α), σ¯ c (˜εc ) the effective pl tensile cohesion stress, σ¯ t (˜εt ) the effective compressive cohesion stress, σˆ¯ max the maximum principal effective stress, p¯ = − 13 trace(σ¯ ) the hydrostatic pressure stress,  ¯ the von Mises equivalent effective stress, and S¯ the effective stress q¯ = 32 (S¯ : S) deviator. Plastic flow is governed by a flow potential function G(σ ) according to nonas) sociated flow rule dεpl = dλ ∂G(σ potential G used for the model is the ∂σ . The flow  Drucker–Prager hyperbolic function, G = ( σt0 tanψ)2 + q¯ 2 − ptanψ, ¯ where ψ

4.3 Numerical Simulation

51

is the dilation angle measured in the p − q plane at high confining pressure, σt0 the uniaxial tensile stress at failure and a parameter, referred to as the eccentricity, that defines the rate at which the function approaches the asymptote (the flow potential tends to a straight line as the eccentricity tends to zero). The dilation angle ψ = 23 φ is adopted, where φ is the internal-friction angle as a critical parameter of the Mohr– Coulomb failure criterion model and can be measured from triaxial compression test. In ABAQUS/Standard 6.9-1, the stress–strain curves for uniaxial tension and compression are needed to define elastic, plastic, and damage behaviors. The uniaxial damage variables for tension and compression are neglected, which means that the damage plasticity concrete model is naturally a plasticity concrete model. The experimental stress–strain relationships of tension and compression are employed, as shown in Fig. 4.5. For the material definition of SHCC, the Poisson’s ratio ν is set to 0.2 according to Pereira et al. (2012), and the internal-friction angle φ is assumed to be 37◦ . In addition, a small value for the viscosity parameter (μ = 0.0001) is defined to improve the convergence rate in the concrete softening and stiffness degradation regimes, following the suggestion from Barth and Wu (2006).

4.3.1.2

Interface Model

For SP, the adhesion between annulus and steel tubes, which exhibited strong influence on the ultimate strength (Estefen et al. 2005a; Castello and Estefen 2007), should be carefully examined. Contact between SHCC layer and tubes is modeled by using surface-based contact, where the strict master-slave algorithm is adopted. In this scheme, the specified master surface is defined by the surface of tubes, and the slave surface is defined by the surface of SHCC. The contact direction is always normal to the master surface, and the slave nodes are constrained not to penetrate into the master surface. Small sliding option is used between the SHCC layer and the tubes. In this chapter, two types of interface conditions are simulated numerically, including perfect adhesion and no adhesion between steel tubes and SHCC. As proposed by Castello and Estefen (2007) and Huang et al. (2010a), the contact interaction model is applied to the steel tube-SHCC interface, which is defined by a contact pressure model in the normal direction and a Coulomb friction model in the tangential direction. For the unbonded condition, the “Hard Contact” relation with “Allow separation after contact” is selected as normal mechanical property, while the “frictionless” is taken as tangential behavior, as depicted in Fig. 4.13a. “Hard Contact” is described by the pressure-overclosure (p − h) model, where h is the overclosure between contact surfaces. No contact pressure means h < 0, while any positive contact pressure means h = 0. The fully bonded condition is simulated through the “Hard Contact” relation without “Allow separation after contact” for the normal behavior and the “Penalty” method for the tangential behavior, respectively.

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4 Sandwich Pipes Filled with PVA Fiber Reinforced Cementitious Composites

Fig. 4.13 FE model of SP: (a) interface and coupling condition, (b) mesh, and (c) load and boundary condition

For the latter case, friction coefficient = 1 and “No limit” for shear stress are set, as presented by Castello and Estefen (2007). It is worth mentioning that the “Tie constraint” was selected as our first option to simulate the fully bonded case. However, although it works well for the steel/polymeric material/steel sandwich pipes (Arjomandi and Taheri 2011c), convergence cannot be achieved for the steel/cementitious material/steel configurations (note that the core material is described by the CDP model). Therefore, surface-to-

4.3 Numerical Simulation

53

surface contact with the coefficient of friction of 1.0 is applied, following Castello and Estefen (2007) and An et al. (2012c).

4.3.1.3

Element Type and Mesh Generation

The ring model validated in Estefen et al. (2005a) and Castello and Estefen (2007) is adopted here, with a difference in adopting a quarter-ring model. ABAQUS C3D8R continuum-brick elements with reduced integration are used for modeling both the steel tubes and the SHCC core, which can be used for linear analysis and for complex nonlinear analyses involving contact, plasticity and large deformations (Kim and Kuwamura 2007). A standard mesh-sensitivity analysis is carried out considering the effect of the element size on the collapse pressure (Pco ), observing that the results tend to converge for 40 elements in the circumferential direction, 1 element for each steel layer and 8 elements for SHCC annular in the radial direction with length of 2 mm in the axial direction, as shown in Fig. 4.13b.

4.3.1.4

Load and Boundary Condition

The elastic-plastic stress analysis of SP under external pressure can be performed by the Newton–Raphson iterative algorithm. The standard rate-independent plasticity formulation for strain-softening materials can lead to a solution with the feature that the load-displacement diagram exhibits snapback. From the mathematical point of view, the cause is the so-called loss of ellipticity of the governing differential equation. For the sudden plastic collapse of the structure, the tangent stiffness matrix becomes singular due to loss of ellipticity after the strain-softening regime of the material is attained (de Borst and Mühlhaus 1992), causing ill-posed boundary value problem and severe convergence difficulties. Collapse analysis of SP subjected to external pressure employs Riks method (the arc-length method), which causes the Newton–Raphson equilibrium iterations to converge along an arc, thereby often preventing divergence, even when the slope of the load vs. deflection curve becomes zero or negative. As shown in Fig. 4.13c, the external pressure is applied on the outer pipe through surface load. Automatically generated kinematic coupling equations are used to link the degrees of freedom of the nodes in the transverse plane to the reference point. The coupling of the translational freedom along the z-axis induces a plane strain state for the SP section in order to simulate a long pipe configuration (note that other five freedoms are kept free to move in order to allow the cross section of SP to ovalize but remain plane), as shown in Fig. 4.13a. The longitudinal and transversal symmetry conditions are assumed. Besides, the distributed load highlighted by orange color represents the axial force subjected to the plugs when pressurizing the vessel (see Fig. 4.13c). Assume the magnitude of external pressure is P , then according to the force equilibrium relation in the axial direction, the pressure added in the axial direction can be

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4 Sandwich Pipes Filled with PVA Fiber Reinforced Cementitious Composites

calculated, Paxi = Aplug /ASP × P , where Aplug and ASP represent the area of plugs and the area of transverse section of SP, respectively. After calculation, it can be obtained that Paxi = 2.17P .

4.3.2 Correlation of Results By simulating the behavior of sandwich pipes under hydrostatic pressure using ABAQUS/Standard 6.9-1, the collapse pressures Pco of SP1 with unbonded and fully bonded condition between stainless steel tubes and SHCC are 31.6 and 35.6 MPa, respectively, as shown in Table 4.3. It can be seen that the gap between the Pco of SP1 with fully bonded condition and the one of SP1 with unbonded condition is small. Similar results can be found for SP2 and SP3. The reason is that Pco is dominated by the resistance of the annulus due to the small thicknesses of inner and outer tubes. It is worth to note that similar observation on the gap can be found in the literature accomplished by Estefen et al. (2005a), where the SP annular was filled with pure cement. Figures 4.14 and 4.15 show the contour plots of the von Mises stress, equivalent plastic strain, and displacement fields for SP1 with perfect adhesion and no adhesion at the Pco , respectively. For the case of SP1 with perfect adhesion, the higher stress appears at the maximum diameter region of the inner tube. Besides, the stress level of annulus attached to the inner tube at the maximum diameter region is much higher than the one attached to the outer tube. The stress and equivalent plastic strain level of tubes and annulus of SP1 with no adhesion are lower than those with perfect adhesion, indicating that the interface bonding can increase the collapse pressure of SP. Similar results can be found for SP2 and SP3. The Pco considering perfect bonding conditions predicted by proposed FE model is 35.6, 36.1, and 35.3, respectively, 16.7%, 18.0%, and 18.9% higher than the measured collapse pressure, while the Pco with unbonded interface conditions predicted by proposed FE model is 31.6, 32.0, and 32.1, respectively, 3.6%, 4.6%, and 8.1% higher than the measured collapse pressure. It can be observed that the predicted results by FE method with unbonded interface condition are closer to the experimental results, which means that the unbonded interface condition can reflect the actual interface condition of SP more reasonably (note that there is no adhesive on the interface and no special treatment of tube surface to increase the interfacial adhesion). The good agreement between predicted collapse pressure and measured results proves the accuracy of the quarter-ring model. The predictions are slightly higher than the measured collapse pressure, for which the error sources can be the following: (a) the tensile and compressive damage of SHCC is not considered in the CDP model, which may improve somewhat the predictions, (b) the assembly process of SP cannot guarantee SHCC to distribute uniformly along the SP length, which is proved by the appearance of void found after the tests (see Fig. 4.12a), and (c) the axial compressive load caused by the plugs is simulated by the uniformly distributed pressure in the FE modeling. Although

4.4 Parametric Studies

55

Fig. 4.14 The contour plots of the von Mises stress, equivalent plastic strain, and displacement fields for SP1 with perfect adhesion at the Pco : (a) von Mises stress for tubes, (b) von Mises stress for annulus (SHCC), (c) equivalent plastic strain for tubes, (d) equivalent plastic strain for annulus (SHCC), and (e) displacement fields for SP

some assumptions are introduced into the simulation, the FE model can provide reasonable results of collapse pressure of SP with SHCC core. To simulate the practical case of long distance pipeline, the predicted Pco of SP without compressive axial force is also given in Table 4.3.

4.4 Parametric Studies The main objective of this section is to discuss the factors affecting the collapse pressure of SP. Since the FE model with unbonded interface condition is more reasonable to predict the Pco , the parametric study is only performed for the cases with unboned interface condition. To simplify the analysis process, we assume that

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Fig. 4.15 The contour plots of the von Mises stress, equivalent plastic strain, and displacement fields for SP1 with no adhesion at the Pco : (a) von Mises stress for tubes, (b) von Mises stress for annulus (SHCC), (c) equivalent plastic strain for tubes, (d) equivalent plastic strain for annulus (SHCC), and (e) displacement fields for SP

the inner and outer tubes have the same thickness and same ovality. The geometry of SP with SHCC core is demonstrated in Fig. 4.16, where t is the thickness, and r1 , r2 are the radii of inner and outer tubes, respectively. The examined cases are listed in Table 4.4, where Case a-b-c-d describes different geometrical configurations: a = 6 means the diameter of inner tube is 6 inch; b = 8, 9, 0 means the diameters of outer tube are 8, 9, 10 inch, respectively; c = 1, 2, 3, 4 means the wall thickness/radiusof-inner-tube ratios are 0.02, 0.04, 0.06, and 0.08, respectively; d = 1, 2, 3, 4, 5 means the ovalities of inner and outer tubes are 0.2%, 0.4%, 0.6%, 0.8%, and 1.0%. Note that the material properties of SS and SHCC are the same as those described in Sect. 4.3.1.1. Firstly, we keep the thickness constant and investigate the relationship between Pco and ovality (Δ) for each SP with three different external diameters (8, 9, and 10 inch). From the results shown in Table 4.4, it can be clearly seen that the Pco

4.4 Parametric Studies

57

Table 4.3 Comparison between predicted and measured collapse pressure of each SP Label SP1 SP2 SP3 SP1 SP2 SP3

a (fully bonded, MPa) Pco 35.6 36.1 35.3 c (fully bonded, MPa) Pco 17.1 17.1 17.1

a (unbonded, MPa) Pco 31.6 32.0 32.1 c (unbonded, MPa) Pco 15.6 15.7 16.0

b (MPa) Pco 30.5 30.6 29.7

a Predicted

by FE method with compressive axial force in experiments c Predicted by FE method without compressive axial force b Measured

Fig. 4.16 Illustration of SP geometry in the parametric study

of SP with SHCC core decreases with the increasing ovality for all the three cases. The results show that the Pco of SP is sensitive to initial ovality. Besides, it can be clearly seen that the sensitivity to initial ovality enhances with increasing thickness. Note that the difference between the Pco of Case 6841 and the one of Case 6845 is much higher than the difference between Case 6811 and Case 6815. Then, we keep the ovality constant and investigate the relationship between Pco and wall thickness/radius-of-inner-tube ratio (t/r1 ) for each SP with different external diameters. From Table 4.4, it can be clearly seen that the Pco of SP increases in t/r1 for all the cases. Finally, we can also investigate the effect of radius ratio (r2 /r1 ) on Pco of SP from the data given in Table 4.4. It can be observed that the Pco of SP increases with the radius ratio (e.g. comparison among Case 6811 (r2 /r1 = 1.33), 6911 (r2 /r1 = 1.5) and 6011 (r2 /r1 = 1.67)). In other words, increasing the core thickness can increase the collapse pressure capacity of SP.

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Table 4.4 Predicted Pco of SP for parametric studies Case 6811 6812 6813 6814 6815 6911 6912 6913 6914 6915 6011 6012 6013 6014 6015

Pco (MPa) 14.2 13.9 13.8 13.5 13.1 15.8 15.5 15.1 14.7 14.3 17.0 16.9 16.3 16.0 15.7

Case 6821 6822 6823 6824 6825 6921 6922 6923 6924 6925 6021 6022 6023 6024 6025

Pco (MPa) 22.6 21.6 20.3 19.3 18.7 24.4 23.4 22.5 22.0 20.8 26.7 25.5 25.1 24.2 23.8

Case 6831 6832 6833 6834 6835 6931 6932 6933 6934 6935 6031 6032 6033 6034 6035

Pco (MPa) 29.3 26.7 25.7 23.7 21.2 31.5 30.7 29.8 29.1 28.3 32.3 31.9 29.9 28.9 28.5

Case 6841 6842 6843 6844 6845 6941 6942 6943 6944 6945 6041 6042 6043 6044 6045

Pco (MPa) 35.1 32.2 31.6 26.7 26.0 37.9 36.7 34.7 34.1 32.8 38.0 36.5 35.7 35.3 34.1

4.5 Conclusions In this chapter, the collapse behavior of SP with SHCC core under external pressure is investigated by both experiment and modeling. The SHCC preparation and the SP fabrication process are described in detail. Three SP are tested using a hyperbaric chamber to obtain the very similar collapse pressure, demonstrating the reliability of the experimental results. A quarter-ring model is adopted to perform the FE analysis of structural behavior of SP under external pressure, where SHCC is modeled by a concrete damaged plasticity model. The accuracy of the model is verified by the good agreement between the numerical and experimental results. Note that the dilation angle ψ adopted in the CDP model is chosen empirically. However, according to van Zijl (2004), the confined nature of SHCC in the double-pipe setup may require a zero dilation angle to avoid significant artificial strength build-up upon distortional inelasticity, which should be investigated further. Although the Poisson’s ratio of 0.2 for SHCC is employed based on the literature, experimental measurement should be conducted to determine the real value, and special attention should be paid for high values of Poisson’s ratio applicable for SHCC in the order of 0.3 < ν < 0.35, as mentioned by Boshoff and Van Zijl (2007). Besides, a parametric study is conducted to investigate the effects of ovality, wall thickness, and radius ratio on the Pco of SP with SHCC core, showing that the Pco increases with wall thickness and radius ratio, and decreases with ovality. Although the unbonded interface condition can give reasonable results of the Pco , experimental tests may be performed to further investigate the interface properties.

Chapter 5

Buckle Propagation of Sandwich Pipes

5.1 Introduction With the global marine resources development in full swing, the focus of offshore oil and gas exploration and development has shifted from shallow water to deepwater. Since the failure of deepwater pipeline may lead to enormous economic cost and even environmental disaster, research on pipeline behavior under typical loads, such as external hydrostatic pressure, is very important for oil/gas transportation. Buckle propagation phenomenon starts from a locally weakened section of the pipe, such as a dent induced by impact by a foreign object, a local buckle resulting from excessive bending during installation, or a wall thickness reduction caused by wear or corrosion (Kyriakides and Corona 2007a). During the buckle propagation along the pipelines, the pipe inner walls come into contact. As a result, the transportation capacity of the pipeline will degrade. Buckle and collapse due to external pressure play an important role in the design of such tubular structures. The minimum pressure required for buckling propagation is the propagation pressure (Pp ). One of the earliest attempts to determine the propagation pressure analytically was presented by Palmer and Martin (Païdoussis 1975). Kyriakides and Bobcock (Kyriakides and Babcock 1981) carried out extensive theoretical and experimental study on buckle propagation and arrest phenomena of externally pressurized pipes. After that, many experimental and numerical studies were conducted to investigate the buckle propagation pressures for different material characteristics and geometric parameters (Showkati and Shahandeh 2009; Albermani et al. 2011; Gong et al. 2012; Khalilpasha and Albermani 2013). The propagation pressure primarily depends on geometric characteristics and material properties of the pipes (Dyau and Kyriakides 1993) , which is usually 15–30% of the collapse pressure of intact pipes (Gong et al. 2013). Besides, buckle propagation of pipe-in-pipe systems were studied through extensive experimental studies and numerical simulations (Kyriakides 2002; Kyriakides and Vogler 2002; Kyriakides and Netto 2004).

© Springer Nature Switzerland AG 2021 C. An et al., Structural and Thermal Analyses of Deepwater Pipes, https://doi.org/10.1007/978-3-030-53540-7_5

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5 Buckle Propagation of Sandwich Pipes

To achieve flow assurance in extreme deepwater environment, sandwich pipes (SP), combining high structural resistance with thermal insulation capability, have attracted considerable research interest over the last few years. Sandwich pipe consists of two concentric metal pipes in which the annulus is filled with a proper material that combines structural strength and thermal insulation in an optimized design (Estefen et al. 2005a). An analytical approach using energy method was proposed by Arjomandi and Taheri (2010) for estimating the bucking capacity of sandwich pipes. The interfacial adhesion characteristics between annular core and pipes affect much the structural integrity of SP. Castello and Estefen (2007) studied the effect of interfacial adhesion degree between polypropylene core and API X-60 tubes on the collapse pressure of SP. Based on the results of 2400 finite element models developed using the parametric modeling procedure, He et al. (2015) proposed a simplified practical equations to calculate the pressure capacity of sandwich pipes with different inter-layer bonded strengths. Xu et al. (2016) conducted shear tests of SP section to determine the interface adhesion between the polypropylene core layer and surrounding steel pipes. Besides polymer, high performance cement-based material can be another choice for the annular core, based on the characteristics of high fracture toughness and good adhesion with metal. An et al. (2012c) investigate the collapse behavior of SP with steel fiber reinforced concrete under external pressure using finite element method. An et al. (2014) reported the experimental results of the hyperbaric chamber collapse tests for the full-scale SP filled with PVA fiber reinforced cementitious composites. Although some experimental and numerical results of buckle propagation of SP under external pressure were summarized by An and Su (2013), the influence of material characteristics, geometric parameters, and adhesive properties on the buckle propagation behavior needs to be further estimated. In this chapter, a threedimensional finite element model of buckle propagation for SP is developed and verified by comparing the numerical results with the one published previously. Using Python script language, 96 SP models with various parameters are generated, which are employed to analyze the influence of material characteristics, geometric parameters, and adhesive properties on the buckle propagation pressure of SP.

5.2 Finite Element Modeling The finite element method has been adopted to build a set of three-dimensional models. A Fortran program is developed to generate and manage the files from parametric studies. A Python code is established for extracting and processing the buckle propagation pressure from ABAQUS output files. The parametric modeling method can greatly improve the computational efficiency. For the quasi-static buckle propagation analysis of SP under external pressure, the Newton–Raphson iteration methods are not suitable for buckle problems and often fail in the neighborhood of critical points. To overcome this problem and trace the equilibrium paths through limit points into the post-critical range, the Arc-length

5.2 Finite Element Modeling

61

type method has been employed. Automatic increment control scheme with variable loading increments is adopted.

5.2.1 Material Characteristics For deepwater pipelines, carbon manganese steel is generally used as the pipe material, which has a significant yield point and plastic deformation capacity. Ramberg-Osgood (R-O) model described by Eq. 5.1 is used to define the constitutive relation of steel:

 σ 3

σ

n−1 ε= (5.1) 1+ E 7 σy where E is the elastic modulus, σ stress, ε strain,the yield stress and n the material hardening parameter. The R-O constitutive model can better fit the stress–strain relationship of steel when the strain is small, but has a gradually increasing deviation as the strain becomes large. To compensate the drawback of R-O model, a modified R-O model described by Eq. 5.2 is used to fit the stress–strain curve of steel, where the curve for strain larger than 0.015 is approximated using a straight line. ε=



⎧ ⎨ ⎩

σ E 1 E

1+



n−1 

3 σ 7 σy

ε < 0.015

(5.2)

(σ − σ1.5 + 0.015) ε ≥ 0.015,

where E  is defined by

E =

dσ | = dε ε=0.015

E

n−1 .

1 + 37 n σσy

(5.3)

Fig. 5.1 shows the modified stress–strain curve for X65 steel grade fitted by R-O and modified R-O models, where the yield stress σy is 448 MPa strain of 0.005 according to API 5L. The material characteristics of polypropylene for the annular core refers to Estefen et al. (2005a).

5.2.2 Element Type and Mesh Generation C3D27, a three-dimensional quadratic brick element with twenty-seven nodes, is chosen to model the inner and outer tubes. This type of element has an accurate solution of displacement and stress, and can overcome shear locking phenomenon

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5 Buckle Propagation of Sandwich Pipes

Fig. 5.1 Modified Ramberg-Osgood stress–strain curve for X65

Fig. 5.2 Finite element mesh of the SP

for the subsequent contact problems during the buckle propagation process. Besides, C3D27H, a three-dimensional, twenty-seven nodes, quadratic hybrid elements, is employed to model the annular layer. The mesh-sensitivity analysis is carried out to consider the effect of element sizes on the convergence of propagation pressure. Each steel pipe is discretized into 14 elements around the quarter circumference, 40 elements along the length, while the annular layer is discretized into 3 elements through the thickness. In addition, the element mesh is refined near =0◦ and =90◦ as shown in Fig. 5.2. The contact is simulated by using the surface-based contact model, which prevents the nodes on the inner surface of the SP from penetrating the planes of symmetry. A volume-controlled loading procedure is adopted using the hydrostatic fluid element F3D4, which can indicate the volume change inside a control region defined around the SP.

5.2 Finite Element Modeling

63

5.2.3 Interface and Boundary Conditions A small-sliding surface-based formulation is used for the contact between the inner surface of the internal pipe and the rigid plane. The contact relationships between pipes and annular layer are assumed to be unbonded and fully bonded which stands for no adhesion and perfect adhesion, respectively. For the unbonded model, nodes for pipes and annular layer at the interface are generated independently. For fully bonded case, pipes and annular layer share common nodes at the interface. The tangential behavior between master and slave surfaces is assumed to be frictionless or perfect adhesion. The collapsing layers do not allow separation after contact. The symmetrical boundary conditions are applied at planes X-Y and X-Z, and the Y-direction and Z-direction displacements of the nodes are fixed but free to expand in the X-direction at X = L.

5.2.4 Geometry of Initial Imperfections The geometric properties of SP are shown in Table 5.1 and Fig. 5.3. The values are extracted from the API 5L, where Dn is the nominal diameter, Ri the radius of internal surface, Re the radius of external surface, t the thickness, L the length. The cross section of SP is assumed to be circular along the length except that a local imperfection is added near one end of the SP at x ∈ ( 0, 0.5D), which is the way to initiating local collapse and subsequent buckle propagation. The local imperfection of the three layers is assumed to be identical and defined by Eq. 5.4.  W0 (θ ) = −Δ0

D 2



 x 2  cos 2θ, x ∈ ( 0, 0.5D), exp −β D

(5.4)

where W0 (θ ) is the radial displacement, D the tube diameter, θ the polar angular coordinate, x the axial coordinate, Δ0 the imperfection amplitude, β the extent of the imperfection. The applied imperfection pattern of SP is shown in Fig. 5.4.

Table 5.1 Geometric parameters of the SP Internal pipe Annular layer External pipe

Dd (in) 14 18

Ri (mm) 168.91 177.80 217.17

Re (mm) 177.80 217.17 228.6

t (mm) 8.89 39.37 11.43

L (mm) 5000 5000 5000

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5 Buckle Propagation of Sandwich Pipes

Fig. 5.3 Geometries of the SP

External pipe

Annular layer

Ri

Re

t1 tc

Internal pipe

(a)

β = 100

t2

(b)

Δ0 = 0.02

Fig. 5.4 Applied imperfection on the SP

5.2.5 Numerical Simulation Results Figure 5.5 shows the calculated pressure change in volume responses for SP, where V0 is the initial internal volume of the SP, and ΔV the absolute value of volume change evaluated for each deformed configuration. A sequence of deformed configurations of the SP is also demonstrated in the figure. The initial state of the SP is identified by the Roman number I, after that the pressure monotonically increases. The configuration II shows that the buckle collapse occurs to the SP, which is followed by a sharp decrease in pressure. The configuration III represents the SP with local collapse at the imperfection region. As the opposite walls of the internal pipe meet in the configuration IV, the pressure ceases to drop. The collapse is arrested locally, and the buckle starts to propagate along the downstream direction of the SP. The subsequent pressure plateau represents steady-state propagation of the buckle, and the configuration V shows the profile of buckle propagation of the SP with longer contact length. Eventually, as the buckle propagation terminated at the end of the pipe, which is flattened in the configuration VI.

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65

Fig. 5.5 Pressure change in volume responses for the SP Table 5.2 Geometric parameters of the SP Model SP.G1.P01a SP.G1.P02a SP.G2.P01a SP.G2.P02a

Di (mm) 45.83 46.27 46.43 47.06

ti (mm) 1.60 1.71 1.69 1.67

ta (mm) 11.50 11.15 4.82 4.41

tm (mm) 1.64 1.64 1.48 1.49

a Label

of SP used in Pasqualino et al. (2005a), where G1 and G2 represent two nominal geometric configurations

5.2.6 Verification of Model The proposed finite element model is verified by comparing its results with the ones obtained by Pasqualino et al. (2005a). The geometric parameters of the SP are shown in Table 5.2. The imperfection amplitude Δ0 is 0.02, and the extent β is 1. Based on the uniaxial tensile stress–strain curves determined by Netto et al. (2002), the hyperelasticity Marlow model in ABAQUS is used to describe the material behavior of polypropylene. As shown in Fig. 5.6, the pressure change in volume response curves of SP with G2 geometry calculated using the current model is compared with the ones presented by Pasqualino et al. (2005a). It can be seen that the two set of curves have the identical pressure changing trend and the similar pressure values. The calculated buckle propagation pressure is listed in Table 5.3. The maximum difference appears for the no adhesion case of SP.G2.P01, which is 2.68% The sources of errors are

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5 Buckle Propagation of Sandwich Pipes

Fig. 5.6 Pressure change in volume response curves of SP with G2 geometry

25

Current Model Pasqualino Model

20 Fully-bonded P (MPa)

15 10 5

Unbonded

0 0.0

0.1 ΔV/V0

0.2

Table 5.3 Comparison of the buckle propagation pressure of SP

Model SP.G1.P01 SP.G1.P02 SP.G2.P01 SP.G2.P02 a

Experimentsa (MPa) 8.39 7.89 2.48 2.28

No adhesiona (MPa) 8.30 8.03 2.61 2.62

No adhesion (MPa) 8.25 8.10 2.68 2.57

Perfect adhesiona (MPa) 25.48 24.87 13.81 14.33

Perfect adhesion (MPa) 25.28 25.13 14.12 13.92

Max error % 0.78 1.04 2.68 1.91

Results from Lourenço et al. (2008)

mainly attributed to the difference in the mesh resolution of the model along the radial, circumferential, and longitudinal directions.

5.3 Parametric Study 5.3.1 Influence Factors of Buckle Propagation Pressure The pressure capacity of SP is dependent on the material properties, the geometric characteristics, and the relationship between layers, which can be represented as Eq. 5.5.   (5.5) Pp = F t1 , r1 , t2 , r2 , vp , vc , Ec , Ep , σy1 , σy2 , α, adh, imp . The Poisson’s ratios of the steel pipe and polymeric material are 0.3 and 0.5, respectively. adh stands for the adhesion properties between the layers, which is considered to be the perfect or no adhesion condition. Besides, the initial imperfection does not affect on the buckle propagation pressure. Therefore, the nondimensionalized form of Eq. 5.5 can be written as follows:

5.3 Parametric Study

67

Table 5.4 Comparison of the buckle propagation pressure of SP t1 /r1 0.03 0.05 0.07 0.09

t2 /r2 0.03 0.05 0.07 0.09

r1 /r2 0.489 0.580 0.636 0.727 0.818

Pp =f Ep

Ec /Ep 0.1 0.01 0.001



σy1 /Ep 0.001865 (X56) 0.002165 (X65) 0.002665 (X80) 0.003331 (X100) 0.003997 (X120)

σy2 /Ep 0.001865 (X56) 0.002165 (X65) 0.002665 (X80) 0.003331 (X100) 0.003997 (X120)

 t1 t2 r2 Ec σy1 σy2 , , , , , , α, adh . r1 r2 r1 Ep Ep Ep

α 25 50 100 1000 10000

(5.6)

Table 5.4 lists the value range for the selected parameters, and the internal surface radii of the inner and outer pipes are given in Table 5.1. The variation of the steel pipe’s t/r ratio is from 0.03 to 0.09, which covers most of the thickness values specified for API 5L pipes. The steel grades consisting of X56, X65, X80, X100, and X120 for inner and outer pipes are considered. Moreover, the strain hardening parameter α = Ep /Ep of the steel pipes is included, which can reflect the strain hardening effect of material in plastic deformation stage. Based on the Abaqus CAE package and the Python programming language, the series of models with different parameters shown in Table 5.4 are created and analyzed, respectively.

5.3.2 Adhesive Properties The interactions between core and steel pipes in the tangential and normal directions are depending on the smoothness of the contact surfaces for steel pipes and interlayer adhesive strength, respectively, which have an impact on the propagation pressure capacity of SP. Figure 5.7 shows the pressure-volume response of SP during the propagation process. It can be seen that the buckle propagation pressure of SP with fully bonded condition is about 9 times the value of the one with unbonded condition

5.3.3 Geometric Characteristics 5.3.3.1

Thickness-to-Radius Ratio of Steel Pipes

Based on API 5L Specification for Line Pipe, the thickness-to-radius ratio varies from 0.03 to 0.09, which is applied for investigating the influence of thickness-toradius ratio of steel pipes on the propagation pressure, viz. t1 /r1 and t2 /r2 varying from 0.03 to 0.09. Note that when varying the values of t1 /r1 , t2 /ris equal to 0.05,

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5 Buckle Propagation of Sandwich Pipes

Fig. 5.7 Pressure-volume response of SP during the propagation process

Table 5.5 Geometries of the internal pipe

D1 (mm) 10.75 12.75 14 16 18

t1 (in) 0.562 0.625 0.688 0.812 0.875

t1 /r1 0.052 0.049 0.049 0.051 0.049

r1 /r2 0.489 0.580 0.636 0.727 0.818

and vice versa. The propagation pressure of SP with different thickness-to-radius ratio is shown in Fig. 5.8. It can be seen that the buckle propagation pressure of SP increases with the thickness-to-radius ratio for both unbonded and fully bonded cases.

5.3.3.2

Core Thickness

In order to explore the core thickness on the propagation pressure of SP, different core thicknesses are selected by adjusting the diameter and thickness of internal pipe according to API 5L standard, as shown in Table 5.5 The diameter and thickness of the external pipe are 22 and 1.125 inch, respectively. Figure 5.9 shows the influence of the core thickness on the buckle propagation pressure of SP. The propagation pressure decreases with for both unbonded and fully bonded cases, which means that it increases with the core thickness.

5.3 Parametric Study

69

Fig. 5.8 Influence of the thickness-to-radius ratio on the buckle propagation pressure (a) SP with unbonded interface condition and (b) SP with fully bonded interface condition

5.3.4 Material Properties 5.3.4.1

Steel Grade

We consider the steel grades of inner and outer pipes from X56 to X120 to examine the effect of steel grade on the propagation pressure of SP. The simulated results of buckle propagation pressure with different steel grades are shown in Fig. 5.10, where σy1 and σy2 are yielding stresses of inner and outer pipes, respectively. It can be seen that the buckle propagation pressure of SP with the unbonded or fully

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5 Buckle Propagation of Sandwich Pipes

Fig. 5.9 Influence of the core thickness on the buckle propagation pressure (a) SP with unbonded interface condition, (b) SP with fully bonded interface condition

bonded interface condition increases with the steel grades of inner and outer pipes. However, from Figs. 5.10b and d, upgrading the steel grade of outer pipe from X100 to X120 does not improve the buckle propagation pressure of SP by a significant margin.

5.4 Conclusion This chapter presents the buckle propagating behavior of sandwich pipes under external pressure in the quasi-static steady-state conditions. A three-dimensional finite element model for sandwich pipes is developed to analyze the buckle

5.4 Conclusion

71

Fig. 5.10 Influence of the steel grade on the buckle propagation pressure (a) SP with unbonded interface condition, (b) SP with fully bonded interface condition, (c) SP with unbonded interface condition, (d) SP with fully bonded interface condition

propagation phenomenon. 96 sandwich pipes are parametrically built to study the influence of material properties, geometric characteristics, and adhesive properties on the propagation pressure. The following conclusions can be drawn: • The adhesive properties between the layers have significant effects on the buckle propagation pressure. We can obtain the maximum propagation pressure from fully bonded SP and the minimum propagation pressure from unbonded case. • Increasing the t/r ratio and steel grades of inner and outer pipes can improve the propagation pressure of the system. Besides, the greater steel grade and t/r ratio of internal pipe can increase the propagation pressure more obviously.

Chapter 6

Sandwich Pipe: Reel-Lay Installation Effects

6.1 Introduction Sandwich pipes (SPs) research has been published since 2002, where different core materials: cement and polypropylene (Pasqualino et al. 2002), fiber reinforced concrete (An 2012), and PVDF (Paz 2013) have been assessed Fig. 6.1. SPs are subsea pipelines with high structural performance specially appropriate for ultradeepwater. The structure comprises two concentric steel pipes and a core material in the annulus. These three layers work together to provide the structural strength and the core material is also responsible for thermal insulation. The SP concept is different than pipe-in-pipe (PIP) systems (Kyriakides 2002; Kyriakides and Vogler 2002) where each layer has its one functionality and is designed separately. In this chapter, the ultimate strength of the SPs under external pressure is assessed through experiments and numerical models. Special attention is paid to the reel-lay installation procedure in the collapse resistance of the SPs built with SHCC core material. Strain hardening cementitious composites (SHCC) with polyvinyl alcohol (PVA) fiber is a special FRC material categorized into engineered cementitious composites (ECC), where the ultimate tensile strain attained by ECC is 2%–6%, which is 200– 600 times greater than that of concrete (Zhang et al. 2006) with increasing tensile loading capacity accompanied by multiple crack formation (Yang and Fischer 2006), as shown in Fig. 6.2. An et al. (2012) analyzed a similar composition of SHCC to evaluate the capacity of SPs using this core material when subjected to external pressure. Nevertheless, the effect of the installation by reel-lay method of SP structures was only studied by Castello (2005) using polypropylene core. In his work, FE models were used without experimental tests.

© Springer Nature Switzerland AG 2021 C. An et al., Structural and Thermal Analyses of Deepwater Pipes, https://doi.org/10.1007/978-3-030-53540-7_6

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6 Sandwich Pipe: Reel-Lay Installation Effects

Fig. 6.1 Examples of SP structures using different core materials. (a) PP and cement (Estefen et al. 2005a), (b) PP (Castello and Estefen 2007), (c) PP (Castello 2011a), (d) SHCC (An 2012), (d) SHCC (ABAQUS 2009a), (e) PVDF (Paz 2013)

6.2 Experimental Setup

75

Fig. 6.2 A typical tensile stress–strain curve of ECC (Nawy 2008)

6.2 Experimental Setup 6.2.1 Large-Scale Tests Experimental tests were conducted with two pipe geometries (Table 6.1) and steel pipe materials. The nomenclature of the specimens is presented below. The nomenclature “R" was used to indicate the specimens where the reeling/straightening procedure was carried out prior to the collapse test. SP A B R

Sandwich Pipe Steel layers with stainless steel AISI-304 Steel layers with stainless steel AISI-316 Reeled specimen

Paz et al. (2014) describes in detail the manufacturing process of the SP with cementitious annular. The author also describes the different components of the SHCC material. The manufacturing process can be summarized as follows. The inner and outer pipes of the structure are assembled together in the vertical position. Centralizers are used to keep the steel pipes concentric. The SHCC material, mixed and homogenized, is introduced in the annular space through an opening in the top. Shaker is used to vibrate and remove the trapped air. After manufacturing and prior to testing, specimen rests for 28 days for curing the cement.

6.2.1.1

Reel-Lay Simulation Apparatus

The bending apparatus was designed to induce plastic deformations in pipes through bending and reverse bending over rigid surfaces similarly to what occurs during

76 Table 6.1 Nominal geometries

6 Sandwich Pipe: Reel-Lay Installation Effects Specimen SP-A SP-B

OD 203.2 219.1

ID 148.4 162.7

tan 23.4 28.2

tep 2.0 2.7

tip 2.0 2.7

OD—external diameter ID—internal diameter tan —annular thickness tep —external pipe thickness tip —inner pipe thickness

Fig. 6.3 Top view of bending apparatus

some installation procedures. It is composed of a main steel structure and dies with variable radii of curvature that are driven by two hydraulic actuators. A schematic view of the experimental setup is shown in Fig. 6.3. The pipe is placed between the bending and the straightening dies, each of which being part of two different die tools. They are mounted on two rails where they can slide on back and forth. Figure 6.4 depicts the bending and straightening processes as performed by the apparatus. When the actuators distend, the pipe is bent over the bending die. Similarly, the pipe is straightened when the actuators move back slightly beyond their original position. In all experiments, the radii of curvature of the reeling and straightening tools were 8 m and 40 m, respectively. The maximum nominal axial bending strain with these tools is 1.20% for SP-A geometry and 1.29% for SP-B geometry. The local strain history was monitored using electrical strain gauges mounted on the external surface of the specimens in the central section and close to both specimen ends. Figure 6.5 shows one specimen with strain gauges and connecting wires ready for test.

6.2 Experimental Setup

77

Fig. 6.4 Sequence of pipe deformed configurations during reeling/straightening test

Fig. 6.5 Strain gauge instrumentation

Fig. 6.6 Results of SP-A-R geometry

The experimental results are presented in strain versus time curves. Some of the assembled strain gauges were not presented in the plots. This is because some of the sensors may detach from the test specimen due to excessive plasticity, or the wire connecting the sensors to the DAQ may be cut during the displacement of the pipe inside the testing apparatus. The longitudinal strain acquired during the test is presented in Figs. 6.6 and 6.7, respectively.

6.2.1.2

Collapse Tests

The collapse behavior of intact and bended pipeline specimens was studied by testing under hydrostatic pressure in a 10000 psi hyperbaric chamber from Subsea Technology Laboratory (COPPE/UFRJ) (Fig. 6.8).

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6 Sandwich Pipe: Reel-Lay Installation Effects

Fig. 6.7 Results of SP-B-R geometry

Fig. 6.8 Hyperbaric chamber

Fig. 6.9 SP-A before collapse

Fig. 6.10 Collapsed SP-A specimen

Figures 6.9 and 6.10 show SP-A specimen: before and after collapse test. Figure 6.11 shows the results of four collapse tests carried out with intact and bended pipes.

6.2 Experimental Setup

79

Fig. 6.11 Collapse results

Results suggest that the detrimental effect of the reeling loads in the collapse pressure of the SPs developed with SHCC core material is not predominant for the tested geometries.

6.2.2 Small Scale Numerical simulations (Fu et al. 2014) using the commercial program ABAQUS (ABAQUS 2013) were developed to reproduce the mechanical behavior of sandwich pipes under bending loads and hydrostatic external pressure. FE models employ distinct material constitutive models to simulate the behavior of the stainless steel and SHCC layers. Thus, small-scale tests on these materials were done to calibrate its properties.

6.2.2.1

Stainless Steel Properties

The tensile properties of steels used to produce sandwich pipes were determined on uniaxial tensile test with an Instron (8802 series) hydraulic machine. Tests were carried out with a load rate of 0.3 mm/min. During the test, clip-gauges and strain gauges were used to acquire the stress x strain curves. Two distinct stainless steels AISI-304 and AISI-316 were used. Figures 6.12 and 6.13 show the true stress–strain curves obtained for both materials.

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6 Sandwich Pipe: Reel-Lay Installation Effects

Fig. 6.12 True stress versus plastic strain of stainless steel AISI-304

Fig. 6.13 True stress versus plastic strain of stainless steel AISI-316

6.2.2.2

SHCC Properties

To calibrate the SHCC was needed different tests: tensile (elastic model) and compression (plastic model). The tensile tests were performed to measure stress and strain at the first crack, the highest post-cracking stress, and the maximum allowable strain. During test preparation, the specimen was carefully assembled to the testing machine since any misalignment of the specimen with the loading axis would generate undesired bending or torque, causing premature rupture. A typical tensile failure process and fractured sample are shown in Fig. 6.14. The cracks appeared when the strain is near 1% (0.01) and the tensile stress reaches an average maximum value of 2.5 MPa. When the fracture occurs, the biggest strain is around 3.5% (0.035). Figure 6.15 shows the stress versus strain curves of tensile test.

6.2 Experimental Setup

81

Fig. 6.14 Failure processes of SHCC on tensile. (a) Crack formation during tensile failure. (b) Fractured sample

Fig. 6.15 Stress versus strain of SHCC

Although the addition of fiber increases the resistance on tensile deformation, many studies showed that the influence of the addition of fiber on compressive strength depends on the type of fiber. FRCs filled with steel and carbon fibers shows an increase resistance on compression, however, if filled with Kevlar and polypropylene fibers presents a decrease resistance on compression (Lima 2004). Thus, it is necessary calibrate the numerical model with these tests: tensile and compression. So unconfined and confined (triaxial) compression tests were performed on a specimen, made of cylindrical SHCC, with dimensions of 50 mm×100 mm. The unconfined test has axial load (σ1 ) controlled by a load piston and confined test consists of applying a hydrostatic state of stress and an axial load on a cylindrical test specimen. The confining pressure (σc = σ2 = σ3 ), which acts in all directions, is obtained through a test chamber filled with liquid. The average of stress x strain curves and the compression test setup shown in Fig. 6.16. The plasticity model adopted was Mohr–Coulomb (MC) failure criterion that is an appropriate description for materials like SHCC, where the behavior is usually characterized by the dependence of yield stress on the hydrostatic pressure. This criterion consists of an envelope line, which can be curved or straight, depending on the material, tangential to the Mohr circle, representing the critical condition of the combinations of the principal tensions. This criterion assumes that the failure of

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6 Sandwich Pipe: Reel-Lay Installation Effects

Fig. 6.16 Stress x strain curves of compression tests

a plan only occurs when the shear stress τ and the normal stress δn reach a critical combination. τ = c + σn tan φ,

(6.1)

where c is the cohesion and φ is the friction angle of the material. In this chapter, the criterion of Drucker–Prager (DP) which has been widely used to model the behavior of materials like concrete is used to model the SHCC material. The criterion for the DP said that plastic flow begins when the invariant J2 of the deviator stress state and the hydrostatic pressure p reach a critical combination, wherein the flow rule determines the direction of the plastic deformation; and the hardening rule defines as the yield surface evolves with the plastic deformation (Yu et al. 2010a). For the linear DP model, the yield surface is defined by Eq. 6.2: f = t − d − ptanβ, where t =

√

(6.2)

J2 , defined by Eq. 6.3:.    3  r 1 1 1 t = q 1+ + 1− , 2 k k q

(6.3)

where p is the first invariant of stress, and J2 is the second invariant of the stress deviator tensor; In Eq. 6.2, t and p are dependent functions of the principal stresses. K is a constant defined as the ratio of the yield stress in triaxial tension to the yield stress in triaxial compression, which must have values between 0.788 and 1.0 to ensure the convexity of the yield surface.

6.2 Experimental Setup

83

Fig. 6.17 Stress x strain curves of compression tests

dpl  t  hardening  d p

In Eq. 6.2, q is the von Mises equivalent stress, r is the third invariant of deviatoric stress (ABAQUS 2013), d cohesion of material, tanβ is the slope of the linear yield surface, referred to as the friction angle of the material. The yield surface of the DP model is presented as a cylindrical cone in the space of principal stress, smoothing the corners of the Mohr–Coulomb (MC) yield function to avoid associated singularities. For the linear DP model the plastic potential is defined by the dilatation angle ψ in the p − t plane, as shown in Eq. 6.4. g = t − ptanψ

(6.4)

When the plastic potential function g is equal to the plastic function f, the form of the flow rule is called an associated flow rule. When the flow is not associated, it implies that ψ ≡ β, consequently, the evolution of plastic strain is not normal to the yield surface, as illustrated in Fig. 6.17. If ψ = 0, the inelastic deformation is incompressible; if ψ ≥ θ , the material expands. Therefore,ψ is referred to as an expansion angle. From the values of the friction angle and cohesion of the Mohr–Coulomb model can be determined the Drucker–Prager parameters. The Mohr-Coulomb criterion assumes that failure is controlled by the maximum shear stress and that this failure shear stress depends on the normal stress. This can be represented by plotting Mohr’s circle for states of stress at failure in terms of the maximum and minimum principal stresses. The Mohr-Coulomb failure line is the best straight line that touches these Mohr’s circles. Using the results for the invariants MC model in triaxial compression and making these expressions identical to Drucker–Prager model for all values of σ1 and σ3 , this is possible by setting: K=

1 1+

1 3 tanβ

(6.5)

84

6 Sandwich Pipe: Reel-Lay Installation Effects

Fig. 6.18 Definition of the parameters in the Mohr–Coulomb failure model

By comparing the Mohr–Coulomb model with the linear Drucker–Prager model: tanβ =

6senφ 3 − senφ

(6.6)

σc0 = 2c

cos φ 1 − senφ

(6.7)

hence: K=

3 − senφ 3 + senφ

(6.8)

The K value of the linear Drucker–Prager model is restricted toK ≥ 0.778 so that the yield surface to remain convex, this implies that φ ≤ 22◦ . ABAQUS manual (ABAQUS 2013) recommend that when this material parameter is lower than this value the K = 0.778 should be used and the other parameters β and σc0 calculated from this. The parameters of the DP model for SHCC material were calculated by approximation with the Mohr–Coulomb parameters provided in ABAQUS manual (ABAQUS 2013). Figure 6.18 illustrates as the parameters φ and c are defined from the Mohr–Coulomb failure model through the triaxial compression tests. Thereafter, making use of Eqs. 6.6 and 6.8 is obtained the parameters β, K, for the linear model of DP. The internal friction angle is φ = 35◦ of the MC model, and using Eq. 6.6 is obtained a value of β = 54.81◦ for the model of DP. The dilatation angle ψ in the plane for the SHCC is chosen to be the same, so β = 54.81◦ . The constant K value is determined by Eq. 6.8, being equal to 0.677. But according to the recommendations in the manual ABAQUS (ABAQUS 2013), the recommended value K = 0.778 is taken.

6.3 Comparisons Between Large-Scale Tests and Finite Element Analysis

85

6.3 Comparisons Between Large-Scale Tests and Finite Element Analysis Numerical analyses (shown in Fu et al. (2014)) and experimental tests have been correlated for two specimens and the nominal geometries are considered. Figures 6.19 and 6.20 indicate the correlations of deformations during installation process between numerical model and experimental measurements and Fig. 6.21 the collapse pressures correlation. Sandwich pipes were tested to collapse for both intact pipe and bended pipe; after bending test the specimens revealed a high collapse pressure. These results show that the detrimental effect of the reeling processes by reel-lay method is not predominant in the collapse pressure for SPs with SHCC core material. The comparison between experimental (EXP) vs finite element analysis (FE) had an error equal to 3.5% on average and 7% on maximum. Fig. 6.19 Installation processes correlation (SP-A-R)

Fig. 6.20 Installation processes correlation (SP-B-R)

86

6 Sandwich Pipe: Reel-Lay Installation Effects

Fig. 6.21 Collapse pressure correlation

6.4 Conclusions This chapter describes the experimental program under development to assess the behavior of sandwich pipes built with a cementitious core material. The strain hardening cementitious composite (SHCC) used in the experiments has PVA fibers in the composition and is expected to offer economic advantages when compared to other core material previously studied. Moreover, the sandwich pipe manufacturing process using SHCC has proven to be easy and no significant flaws were observed in the tested specimens. The experimental tests involved samples of different geometries where the influence of the bending/straightening process related to the reel-lay installation method was studied. When the collapse pressure observed in the experiments conducted with bended pipes is compared with the intact pipe, no significant variation was observed, proving that the SHCC core material is suitable for application in sandwich pipes intended to be installed by the reel-lay method. Finite element numerical models calibrated through small-scale tests of the steel and cementitious materials previously presented to calculate the collapse pressure were compared with the experimental results. The numerical results presented good agreement with the test results.

Part II

Dynamics of Fluid-Conveying Pipes

Vibration induced by internal and external flow is a key issue for subsea production pipelines. The systematic and extensive investigations have been carried out in the past decades to understand the dynamic behavior of pipes conveying single-phase and two-phase flow, and under external cross-flow. Many achievements have been made in understanding the dynamic characteristics of pipes conveying fluid. In Part III of the book, a hybrid analytical–numerical method, the Generalized Integral Transform Technique (GITT) is applied systematically to analyze the dynamical behavior of pipelines conveying single- and two-phase flow. Chapter 7 presents the theoretical framework of GITT and its applications in solid and structural mechanics problems. In Chap. 8, the dynamic behavior of pipes conveying gas– liquid two-phase flow was analytically and numerically investigated. In Chap. 9, a fluid-structural model for analyzing the dynamic behavior of riser vibration subjected to simultaneous internal gas–liquid two-phase flow and external marine current is presented. In Chap. 10, dynamical behavior of horizontal subsea pipelines transporting two-phase gas–liquid slug flow and subject to external marine current is analyzed. Finally, Chap. 11 analyzes the dynamical behavior of axially functional graded pipes conveying single-phase fluid.

Chapter 7

Integral Transform Solutions of Solid and Structural Mechanics Problems

7.1 Introduction Recently, a hybrid numerical–analytical approach, known as Generalized Integral Transform Technique (GITT), has been successfully developed in heat and fluid flow applications (Cotta 1993; Cotta and Mikhailov 1997; Cotta 1998). The most interesting feature in this technique is the automatic and straightforward global error control procedure, which makes it particularly suitable for benchmarking purposes, and the only mild increase in overall computational effort with increasing number of independent variables. In order to perform the GITT approach, the steps to be followed are to define and solve auxiliary eigenvalue problem with homogeneous boundary conditions to obtain the eigenfunctions, eigenvalues, and norms to develop the direct and inverse transforms pairs, achieve the integral transform of the partial differential system in a coupled ordinary differential system, and obtain the original potentials through inversion formula. Since most of the structural behaviors such as bending, vibration, and buckling are described by the partial differential equations, it motivates us to utilize the GITT approach to analyze the structural problems such as the bending of orthotropic rectangular thin plates (An et al. 2015), the dynamic response of axially moving Euler beams (An and Su 2011), axially moving Timoshenko beams (An and Su 2014a), axially moving orthotropic plates (An and Su 2014b), axially functionally graded pipes conveying fluid (An and Su 2017), fluid-conveying pipes (Gu et al. 2013), pipes conveying gas–liquid two-phase flow (An and Su 2015), and vortexinduced vibration of long flexible cylinders (Gu et al. 2012). In this chapter, we review the GITT approach adopted to analyze some typical structural dynamic problems. In the next section, the general solution methodology to solve the vibration problems is presented. Special topics and applications including the vibration of axially moving Euler beams, axially moving Timoshenko beams, pipes conveying fluid, pipes conveying gas–liquid two-phase flow, and

© Springer Nature Switzerland AG 2021 C. An et al., Structural and Thermal Analyses of Deepwater Pipes, https://doi.org/10.1007/978-3-030-53540-7_7

89

90

7 Integral Transform Solutions of Solid and Structural Mechanics Problems

Fig. 7.1 Illustration of an axially moving beam with certain boundary conditions

axially moving orthotropic plates are presented in Sect. 7.3. Finally, the main conclusions are summarized in Sect. 7.4.

7.2 Integral Transform Solution Methodology The generalized vibration problem of an individual non-uniform member with an axially moving speed on an elastic foundation can be expressed as follows:  

 



(A(x)w ) + (B(x)w ) + C(x)w˙ + D(x)w¨ + E(x)w = F (x),

0 < x < L, (7.1)

subjected to the following boundary conditions: 







A(x)w − kRL w = 0 or

 

(A(x)w ) + kTL w = 0 at

x = 0,

(7.2)

x = L,

(7.3)

and A(x)w + kRR w = 0

or

 

(A(x)w ) − kTR w = 0

at

where w(x, t) is the transverse displacement, A(x) the flexural rigidity, L the length, kRL and kRR the rotational spring constants, kTL and kTR the translational spring constants, as shown in Fig. 7.1. According to Anderson (1969), if we introduce the following fourth-order Sturm–Liouville eigenvalue problem  

 

(A(x)Xi ) + (B(x)Xi ) + E(x)Xi = μ4i p(x)Xi ,

(7.4)

with the boundary conditions 



A(x)Xi − kRL Xi = 0 or and

 

(A(x)Xi ) + kTL Xi = 0 at

x = 0,

(7.5)

7.3 Applications 

91 

A(x)Xi + kRR Xi = 0

 

(A(x)Xi ) − kTR Xi = 0

or

at

x = L,

(7.6)

then the eigenfunction set {Xi (x)} is orthogonal with respect to the weight function p(x) over the internal (0, L), i.e. 

L

p(x)Xi (x)Xj (x)dx = δij Ni ,

(7.7)

0

with δij = 0 for i = j , and δij = 1 for i = j . L Applying an integral transformation to Eq. (7.1) with 0 p(x)Xi (x)dx, the following set of fourth-order differential terms  

 

L[w] = (A(x)w ) + (B(x)w ) + E(x)w,

(7.8)

can be removed from the partial differential equation Eq. (7.1), which is reduced to an ordinary different equation. The solution methodology proceeds towards the proposition of the integral transform pair for the potentials, the integral transformation itself, and the inversion formula. For the transverse displacement: 

L

w¯ i (t) =

p(x)Xi (x)w(x, t)dx,

transform,

(7.9a)

0

w(x, t) =

∞ 

Xi (x)w¯ i (t)/Ni , inverse.

(7.9b)

i=1

7.3 Applications 7.3.1 Vibration of Axially Moving Euler Beams 7.3.1.1

Mathematical Formulation

The dimensionless governing equation for linear free vibration of a tensioned Euler– Bernoulli beam traveling at constant speed v with two fixed supports can be written as (An and Su 2011): utt + 2vuxt − (1 − v 2 )uxx + ξ uxxxx = 0, together with the boundary conditions

0 < x < 1,

(7.10a)

92

7 Integral Transform Solutions of Solid and Structural Mechanics Problems

u(0, t) = 0,

∂u(0, t) = 0, ∂x

u(1, t) = 0,

∂u(1, t) = 0, ∂x

(7.10b–e)

where u is the dimensionless transverse displacement, ξ the dimensionless flexural rigidity of the beam. The initial conditions are defined as follows: u(x, 0) = 0,

7.3.1.2

u(x, ˙ 0) = v0 sin(π x).

(7.11a,b)

Integral Transform Solution

The auxiliary eigenvalue problem is adopted for the transverse displacement representation as follows: d4 Xi (x) = μ4i Xi (x), dx 4

0 < x < 1,

(7.12a)

with the following boundary conditions: Xi (0) = 0, Xi (1) = 0,

dXi (0) = 0, dx dXi (1) = 0, dx

(7.12b,c) (7.12d,e)

where Xi (x) and μi are, respectively, the eigenfunctions satisfying an orthogonality property and eigenvalues of problem (7.12). Problem (7.12) is readily solved analytically to yield ⎧ cos[μ (x − 1/2)] cosh[μ (x − 1/2)] i i ⎪ − , ⎪ ⎨ cos(μi /2) cosh(μi /2) Xi (x) = ⎪ ⎪ ⎩ sin[μi (x − 1/2)] − sinh[μi (x − 1/2)] , sin(μi /2) sinh(μi /2)

for i odd, (7.13a,b) for i even,

where the eigenvalues are obtained from the transcendental equations:  tanh(μi /2) =

− tan(μi /2), tan(μi /2),

for i odd, for i even,

(7.14a,b)

and the normalization integral is evaluated as Ni = 1,

i = 1, 2, 3, . . .

The integral transform pair for the transverse displacement is

(7.15)

7.3 Applications

93



1

u¯ i (t) =

X˜ i (x)u(x, t)dx,

transform,

(7.16a)

0

u(x, t) =

∞ 

X˜ i (x)u¯ i (t), inverse,

(7.16b)

i=1

where X˜ i (x) = Xi (x)/Ni is the normalized eigenfunction. The integral transfor1 mation process is now employed through operation of (7.10a) with 0 X˜ i (x)dx, to find the transformed transverse displacement system: 1/2

∞

∞

j =1

j =1

  du¯ j (t) d2 u¯ i (t) + (v 2 − 1) + 2v Aij Bij u¯ j (t) 2 dt dt + ξ μ4i u¯ i (t) = 0,

i = 1, 2, 3, . . . ,

(7.17)

where the coefficients are analytically determined from the following integrals:  Aij = 0

1

 X˜ i X˜ j dx,

 Bij = 0

1

 X˜ i X˜ j dx.

(7.18a,b)

In the similar manner, initial conditions are also integral transformed to eliminate the spatial coordinate, yielding u¯ i (0) = 0,

du¯ i (0) = v0 dt



1

X˜ i sin(π x)dx,

i = 1, 2, 3, . . .

(7.19a,b)

0

For computational purposes, the expansion is truncated at sufficiently large order N . Equations (7.17) and (7.19) in the truncated forms are then numerically handled by the NDSolve routine of the Mathematica system (Wolfram 2003). Once the transformed potential, u¯ i , has been numerically evaluated, the inversion formula Eq. (7.16b) is recalled to provide explicit analytical expressions for the original potential, the dimensionless transverse displacement u(x, t).

7.3.1.3

Results and Discussion

We now present numerical results for the transverse displacement w(x, t) of clamped axially moving beams by employing the GITT approach. For all the cases studied, v0 = 0.01 is employed in the initial conditions (7.11a,b). The solution of the system (7.17) is obtained with N ≤ 50 to analyze the convergence behavior. The profiles of the transverse displacement at different time for v = 0.15 and ξ = 0.1 are illustrated in Fig. 7.2 with different truncation orders. Through the parametric study, it indicates that the amplitudes of the system increase and the vibration frequencies decrease with the translating velocity, and increasing the flexural rigidity leads to a

94

7 Integral Transform Solutions of Solid and Structural Mechanics Problems

Fig. 7.2 GITT solutions with different truncation orders N for the dimensionless transverse displacement u(x, t) of a clamped axially moving beam at different time for v = 0.15 and ξ = 0.1. (a) t = 5. (b) t = 20. (c) t = 100

7.3 Applications

95

decrease in amplitudes and an increase in vibration frequencies of the system (see An and Su 2011).

7.3.2 Vibration of Axially Moving Timoshenko Beams 7.3.2.1

Mathematical Formulation

The dimensionless governing equations of transverse motion of an axially moving uniform Timoshenko beam can be derived as (An and Su 2014a): 

   2 ∂ 2w ∂ 2 w ∂θ ∂ 2w ∂ 2w 2∂ w +α 2 +ηξ cos(Ωτ ) = 0, + − v − +2v 2 2 2 ∂ξ ∂ξ ∂τ ∂τ ∂ξ ∂ξ ∂ξ

(7.20)

   2 ∂ 2θ ∂ 2θ ∂w ∂ 2θ 2∂ θ + γ −θ −β v + + 2v = 0. ∂ξ ∂ξ ∂τ ∂ξ 2 ∂τ 2 ∂ξ 2

(7.21)



with the dimensionless clamped–clamped boundary conditions, w = θ = 0,

at

ξ = 0 and

w = θ = 0,

ξ = 1,

at

(7.22a,b)

and the dimensionless simply-supported boundary conditions, w=

∂θ = 0, ∂ξ

at

ξ = 0 and

w=

∂θ = 0, ∂ξ

at

ξ = 1,

(7.23a,b)

respectively, where θ is the dimensionless angle of rotation, ξ the dimensionless space variable, τ the dimensionless time variable, α the dimensionless axial tension, η the dimensionless exciting load, Ω the dimensionless frequency of the distributed harmonic force, β the dimensionless moment of inertia, and γ the dimensionless flexural rigidity. The initial conditions are defined in the dimensionless form as follows: w = 0,

7.3.2.2

w˙ = 0,

θ = 0,

θ˙ = 0,

at

τ = 0.

(7.24)

Integral Transform Solution

For the clamped–clamped boundary conditions, the eigenfunctions can be obtained by solving analytically the additional eigenvalue problems for the transverse deflection and for the angle of rotation, respectively: Xi (ξ ) = sin(μi ξ ),

i = 1, 2, 3, . . . ,

(7.25)

96

7 Integral Transform Solutions of Solid and Structural Mechanics Problems

Yi (ξ ) = sin(φi ξ ),

i = 1, 2, 3, . . . ,

(7.26)

and the eigenvalues become μi = iπ,

i = 1, 2, 3, . . . ,

(7.27)

φi = iπ,

i = 1, 2, 3, . . .

(7.28)

For the simply-supported boundary conditions, the eigenfunctions for the transverse deflection are the same as the one above mentioned, while the eigenfunctions for the angle of rotation are Zi (ξ ) = cos(ψi ξ ),

i = 1, 2, 3, . . . ,

(7.29)

and the eigenvalues become ψi = iπ,

i = 1, 2, 3, . . .

(7.30)

The following integral transform pairs for the transverse deflection and the angle of rotation need to be established: 

1

w¯ i (τ ) =

X˜ i (ξ )w(ξ, τ )dξ,

transform,

(7.31a)

inverse,

(7.31b)

transform,

(7.31c)

inverse,

(7.31d)

0

w(ξ, τ ) = θ¯i (τ ) =



∞ 

X˜ i (ξ )w¯ i (τ ),

i=1 1

Γ˜i (ξ )θ (ξ, τ )dξ,

0

θ (ξ, τ ) =

∞ 

Γ˜i (ξ )θ¯i (τ ),

i=1

where X˜ i (ξ ) and Γ˜i (ξ ) (Γ˜i (ξ ) = Y˜i (ξ ) for clamped–clamped boundary conditions and Γ˜i (ξ ) = Z˜ i (ξ ) for simply-supported boundary conditions) are the normalized eigenfunctions. Now, to carry out the integral transform, the dimensionless equations (7.20) 1 1 and (7.21) are multiplied by the operator 0 X˜ i (ξ )dξ and 0 Γ˜j (ξ )dξ , respectively, where the inverse formulas (7.31b) and (7.31d) are applied. After some derivation, the following set of ODEs can be obtained:     ∞ ∞   dw¯ m (τ ) d2 w¯ i (τ ) 2 2 2 ¯ − μi w¯ i (τ ) + Ain θn (τ ) − v μi w¯ i (τ ) − 2v Bim dτ dτ 2 n=1

m=1

7.3 Applications

97

+ αμ2i w¯ i (τ ) − Fi η cos(Ωτ ) = 0, i = 1, 2, 3, . . . , (7.32a)     ∞ ∞  dθ¯n (τ ) d2 θ¯j (τ ) − Cj m w¯ m (τ ) − θ¯j (τ ) + β v 2 λ2j θ¯j (τ ) − 2v Dj n dτ dτ 2 m=1

n=1

− γ λ2j θ¯j (τ ) = 0,

j = 1, 2, 3, . . . ,

(7.32b)

where the coefficients are given from the following integrals:  Ain =

1

0

 Dj n =

0

1

 X˜ i Γ˜n dξ, Bim =

 Γ˜j Γ˜n dξ, Fi =



1 0



1

 X˜ i X˜ m dξ, Cj m =

X˜ i ξ dξ,

 0

1

 Γ˜j X˜ m dξ,

(7.32c–g)

0

and λj equals to φj and ψj for the clamped–clamped and simply-supported boundary conditions, respectively. Similarly, initial conditions Eq. (7.24) are also integral transformed through the operator, generating dw¯ i (0) = 0, i = 1, 2, 3, . . . , dτ dθ¯j (0) = 0, j = 1, 2, 3, . . . θ¯j (0) = 0, dτ

w¯ i (0) = 0,

(7.33a) (7.33b)

For computational purposes, the expansions for the transverse deflection and the angle of rotation are truncated to finite orders N W and N A, respectively. Equations (7.32) and (7.33) in the truncated series are subsequently calculated by the NDSolve routine of Mathematica (Wolfram 2003). Once the transformed potential, w¯ i and θ¯j , have been numerically evaluated, the inversion formulas Eqs. (7.31b) and (7.31d) are then applied to yield explicit analytical expressions for the dimensionless transverse deflection w(ξ, τ ) and the angle of rotation θ (ξ, τ ).

7.3.2.3

Results and Discussion

We now present the convergence behavior of numerical results for the dimensionless transverse deflection w(ξ, τ ) and the angle of rotation θ (ξ, τ ) of an axially moving Timoshenko beam calculated using the GITT approach. For the case examined, the following dimensionless parameters are taken in Eqs. (7.20) and (7.21): α = 1.0, η = 1.0, β = 1.0, γ = 1.0, and v = 0.8. The solution of the system, Eqs. (7.32a) and (7.32b), is obtained with NW ≤ 160 and NA ≤ 60 to analyze the convergence behavior. Figures 7.3 and 7.4 present the GITT results with different truncation orders N W for time history of the transverse deflection at τ ∈ [15, 20] and τ ∈ [35, 40] of axially moving Timoshenko beams with clamped–clamped and simplysupported boundary conditions, respectively. The good convergence behavior of

98

7 Integral Transform Solutions of Solid and Structural Mechanics Problems

Fig. 7.3 GITT solutions with different truncation orders N W for time history of the dimensionless transverse deflection at (a) τ ∈ [15, 20] and (b) τ ∈ [35, 40] of an axially moving Timoshenko beam with clamped–clamped boundary conditions (α = 1.0, η = 1.0, β = 1.0, γ = 1.0, and v = 0.8)

7.3 Applications

99

Fig. 7.4 GITT solutions with different truncation orders N W for time history of the dimensionless transverse deflection at (a) τ ∈ [15, 20] and (b) τ ∈ [35, 40] of an axially moving Timoshenko beam with simply-supported boundary conditions (α = 1.0, η = 1.0, β = 1.0, γ = 1.0, and v = 0.8)

100

7 Integral Transform Solutions of Solid and Structural Mechanics Problems

our method is clearly exhibited. Through the parametric study, it indicates that the maximum absolute amplitude values of the system do not vary monotonously with the axial speed, and the beating phenomenon can occur at a certain value of the axial speed (e.g., v = 0.7). Besides, the maximum absolute amplitude value of the system decreases with the axial force and increases proportionally with the amplitude of external distributed force (see An and Su 2014a).

7.3.3 Vibration of Pipes Conveying Fluid 7.3.3.1

Mathematical Formulation

The dimensionless governing equation for the dynamic response of an elastic pipe conveying fluid of a constant velocity U is given by Gu et al. (2013): 2  ∂ 2y ∂ 2y ∂ 4y 2∂ y + u + 2 βu + = 0, ∂z∂t ∂t 2 ∂z2 ∂z4

z ∈ (0, 1),

(7.34)

∂y(1, t) = 0, ∂z

(7.35)

with the boundary conditions: y(0, t) = 0,

y(1, t) = 0,

∂y(0, t) = 0, ∂z

where y is dimensionless transverse displacement, z the dimensionless space variable, t the dimensionless time, u the dimensionless conveying speed, and β the dimensionless mass density of the fluid. The initial conditions are defined as follows: y(z, 0) = 0,

∂y(z, 0) = O(10−3 ), ∂t

(7.36)

where O(10−3 ) represents a random noise.

7.3.3.2

Integral Transform Solution

Following the principle of GITT, the eigenfunctions and the eigenvalues are selected, which is the same as the ones given in Sect. 7.3.1.2. The integral transform pair for the transverse displacement is  y¯i (t) = 0

1

φ˜ i (z)y(z, t)dz, transform,

(7.37a)

7.3 Applications

101

y(z, t) =

∞ 

φ˜ i (z)y¯i (t), inverse,

(7.37b)

i=1

where φ˜ i (z) = φi (z)/Ni is the normalized eigenfunction. The next step is to accomplish the integral transformation of the partial differential system. For this 1 purpose, Eqs. (7.34)–(7.36) are multiplied by 0 φ˜ i (z)dz and the inverse formula given by Eq. (7.37b) is employed: 1/2

∞ ∞    d2 y¯i (t) dy¯i (t) 2 + u + 2 βu A Bij y¯i (t) ij dt dt 2 j =1

+ λ4i y¯i (t) = 0,

j =1

i = 1, 2, 3, . . . ,

(7.38)

where the coefficients of the ordinary differential system are given by the following expressions: 

1

Aij = 0

φ˜ i (z)

dφ˜ j (z) dz, dz



1

Bij = 0

φ˜ i (z)

d2 φ˜ j (z) dz, dz2

(7.39)

and λi is the eigenvalues. The integral transformation of the initial conditions is y¯i (0) = 0,

dy¯i (0) = dt



1

φ˜ i (z)O(10−3 )dz,

(7.40)

0

System (7.38) is now in an appropriate format for numerical solution through dedicated routines for initial value problems. The subroutines DIVPAG from IMSL Library (IMSL 2003) are well tested and capable of handling such situations, offering an automatic accuracy control scheme with the selected error 10−6 . For this computational purpose, the expansions are then truncated to N orders, so as to reach the user requested accuracy target in final solution. The related coefficients given by Eq. (7.38) above are also handled through IMSL Library. Once y¯i (t) have been numerically evaluated, the analytical inversion formula (7.37b) recovers the dimensionless function y(z, ˜ t).

7.3.3.3

Results and Discussion

The convergence behavior of integral transform solution for dynamic response of pipe conveying fluid is examined for increasing truncation orders N = 4, 8, 16, 32, and 64. Figure 7.5 illustrates the convergence of dimensionless time trace in an interval t ∈ [10, 13] with a combination of u = 1.5, 4.5 and β = 0.5, 1.0. It is noticed that the convergence is quite favorable. Through the modal separation analysis, it indicates that the dynamic response is dominated by the first mode. The

102

7 Integral Transform Solutions of Solid and Structural Mechanics Problems

Fig. 7.5 GITT solutions with different truncation orders N for the dimensionless time history y(z, t). (a) u = 4.5, β = 1.0. (b) u = 4.5, β = 0.5. (c) u = 1.5, β = 0.5

7.3 Applications

103

deflection ratio increases as the flow velocity increases, and decreases as the mass ratio increases. Meanwhile, natural angular frequency decreases monotonically at first mode but increases monotonically at mode 2, 3, 4, and 5 when mass ratio increases (see Gu et al. 2013).

7.3.4 Vibration of Axially Moving Orthotropic Plates 7.3.4.1

Mathematical Formulation

The dimensionless governing equation of transverse motion of an axially moving orthotropic plate can be derived as (An and Su 2014b): ∂ 4 w 2α ∂ 4 w β ∂ 4w ∂ 2w ∂ 2w ∂ 2w 2 + + + + 2γ − ζ ) = q sin(Ωt), + (γ ∂x∂t ∂x 4 ξ 2 ∂x 2 ∂y 2 ξ 4 ∂y 4 ∂t 2 ∂x 2 (7.41) with the dimensionless CCCC boundary conditions, ∂w = 0, ∂x

w=

at

x = ±1 and

w=

∂w = 0, ∂y

at

y = ±1,

(7.42a,b)

and the dimensionless CCSS boundary conditions, w=

∂ 2w = 0, ∂x 2

at

x = ±1

and

w=

∂ 2w = 0, ∂y 2

at

y = ±1, (7.43a,b)

respectively, where α is the dimensionless effective torsional rigidity, ζ the dimensionless in-place force per unit length, β the dimensionless flexural rigidities about the x axis, γ the dimensionless axial speed, q and Ω the dimensionless amplitude and the dimensionless frequency of the uniformly distributed harmonic force. The initial conditions are defined in the dimensionless form as follows: w = w˙ = 0,

7.3.4.2

at

t = 0.

(7.44)

Integral Transform Solution

For the CCCC boundary conditions, the spatial coordinates “x” and “y” are eliminated through integral transformation, and the related eigenfunctions and eigenvalues are the same as the ones given in Sect. 7.3.1.2. For the CCSS boundary conditions, the related eigenvalue problem for the coordinate “x” is the same as the one above mentioned, while the eigenfunctions and eigenvalues for the coordinate “y” are

104

7 Integral Transform Solutions of Solid and Structural Mechanics Problems

 Y2i (y) =

cos(φ2i y),

for i odd,

sin(φ2i y),

for i even,

(7.45a,b)

and ⎧π ⎪ ⎨ (2i − 1), for i odd, 2 φ2i = π ⎪ ⎩ i, for i even, 2

(7.46a,b)

respectively. The integral transform pair for the transverse displacement is  w¯ ij (t) =

1



1

−1 −1

X˜ i (x)Y˜j (y)w(x, y, t)dxdy,

w(x, y, t) =

∞ ∞  

transform,

(7.47a)

inverse,

(7.47b)

X˜ i (x)Y˜j (y)w¯ ij (t),

i=1 j =1

where X˜ i (x) and Y˜j (y) are the normalized eigenfunctions. Now, to perform the integral transform process, the dimensionless equation (7.41) is multiplied by 1 1 the operator −1 −1 X˜ i (x)Y˜j (y)dxdy and the inverse formula (7.47b) is applied. After some mathematical manipulations, the following set of ordinary differential equations (ODEs) can be obtained: ∞  ∞   m=1 n=1

 d2 w¯ mn (t) dw¯ mn (t) Eij mn + (γ 2 − ζ )Hij mn + 2γ Gij mn dt dt 2   2α β +μ4m Eij mn + 2 Fij mn + 4 φn4 Eij mn w¯ mn (t) ξ ξ = Dij q sin(Ωt),

i = 1, 2, 3, . . . , j = 1, 2, 3, . . . ,

(7.48a)

where the coefficients are analytically determined from the following integrals:  Eij mn =  Fij mn =

−1 1

−1

 Gij mn =

1

1 −1

X˜ i X˜ m dx  X˜ i X˜ m dx

 X˜ i X˜ m dx

 

1 −1 1

−1



1 −1

Y˜j Y˜n dy,

(7.48b)

 Y˜j Y˜n dy,

(7.48c)

Y˜j Y˜n dy,

(7.48d)

7.3 Applications

105

 Hij mn =

1 −1

 X˜ i X˜ m dx



Dij =

1 −1



1 −1

X˜ i dx

Y˜j Y˜n dy,



1

−1

Y˜j dy,

(7.48e) (7.48f)

and φn equals to φ1n and φ2n for the CCCC and CCSS boundary conditions, respectively. In the similar manner, initial conditions Eq. (7.44) are also integral transformed to eliminate the spatial coordinate, yielding Eij mn w¯ mn (0) = 0,

Eij mn

dw¯ mn (0) = 0, dt

i = 1, 2, 3, . . . , j = 1, 2, 3, . . . (7.49a,b)

For computational purposes, the expansions are truncated to a sufficiently large finite order N W . Equations (7.48) and (7.49) in truncated form are then numerically handled by the NDSolve routine of the Mathematica system (Wolfram 2003). Once the transformed potential, w¯ mn , has been numerically evaluated, the inversion formula Eq. (7.47b) is recalled to provide explicit analytical expressions for the original potentials, the dimensionless transverse displacement w(x, y, t).

7.3.4.3

Results and Discussion

We now present the convergence behavior of numerical results for the transverse displacement w(x, y, t) of an axially moving orthotropic plate calculated using the GITT approach. For the case examined, the following dimensionless parameters are taken in Eq. (7.41): γ = 1.2, ψ = 0.5, q = 3.0, Ω = 5.0, ξ = 0.8, α = 2.0, and β = 1.0 (note that the values of the parameters presented here can be chosen arbitrarily). The solution of the system, Eqs. (7.48) and (7.49), is obtained with N W ≤ 20 to analyze the convergence behavior. The convergence behavior of the integral transform solution is examined for increasing truncation terms NW = 4, 8, 12, 16, and 20 at t = 5, 20, and 50, respectively. The investigation shows that, the solutions for CCCC boundary condition case converge essentially at a reasonably low truncation order (NW ≤ 12), while the ones for CCSS boundary condition case converge at relatively low truncation orders (N W ≤ 8). The profiles of transverse displacement at t = 20 along the symmetric lines (x = 0 or y = 0) are illustrated in Figs. 7.6 and 7.7 with different truncation orders. Through the parametric study, it indicates that the amplitudes of the system increase with the decreasing flexural rigidity ratio, the increasing moving velocity, and the increasing aspect ratio for both CCCC and CCSS boundary condition cases (see An and Su 2014b).

106

7 Integral Transform Solutions of Solid and Structural Mechanics Problems

Fig. 7.6 GITT solutions with different truncation orders N W for the dimensionless transverse displacement profiles (a) w(x, y, t)|y=0,t=20 and (b) w(x, y, t)|x=0,t=20 of an axially moving orthotropic plate with CCCC boundary conditions (γ = 1.2, ψ = 0.5, q = 3.0, Ω = 5.0, ξ = 0.8, α = 2.0, and β = 1.0)

7.3 Applications

107

Fig. 7.7 GITT solutions with different truncation orders N W for the dimensionless transverse displacement profiles (a) w(x, y, t)|y=0,t=20 and (b) w(x, y, t)|x=0,t=20 of an axially moving orthotropic plate with CCSS boundary conditions (γ = 1.2, ψ = 0.5, q = 3.0, Ω = 5.0, ξ = 0.8, α = 2.0, and β = 1.0)

108

7 Integral Transform Solutions of Solid and Structural Mechanics Problems

7.4 Conclusions In this chapter, the GITT application on the vibration problems such as the dynamics of axially moving Euler beams, axially moving Timoshenko beams, pipes conveying fluid, pipes conveying gas–liquid two-phase flow, and axially orthotropic plates are reviewed. Excellent convergence, good long-time numerical stability, and high accuracy can be obtained for all relevant cases. The proposed approach can be further employed to perform the bending, vibration, and buckling analysis of other typical structures, yielding sets of reference results with controlled accuracy.

Chapter 8

Pipes Conveying Gas–Liquid Two-Phase Flow

8.1 Introduction Internal flow-induced structural vibration has been experienced in numerous fields, including heat exchanger tubes, chemical plant piping systems, nuclear reactor components, and subsea production pipelines. The systematic and extensive investigations have been carried out in the past decades to understand the dynamic behavior of pipes conveying single-phase flow (Païdoussis and Li 1993; Païdoussis 1998; Kuiper and Metrikine 2005; Païdoussis 2008). Recently, many achievements have been made in understanding the dynamic characteristics of pipes conveying fluid. Kang (2000) investigated the effects of rotary inertia of concentrated masses on the free vibrations and system instability of the clamped-supported pipe conveying fluid, and concluded that introduction of rotary inertia can influence the higher natural frequencies and mode shapes of fluid-conveying pipes. Sinha et al. (2005) utilized a nonlinear optimization method to predict the flow-induced excitation forces and the structural responses of the piping system, whose validation was presented through a simulated experiment on a long straight pipe conveying fluid. Based on the principle of eliminated element-Galerkin method, Huang et al. (2010b) reduced the binary partial differential equation to an ordinary differential equation using the separation of variables, then utilized Galerkin method to obtain the natural frequency equations of pipeline conveying fluid with different boundary conditions, and found that the effect of Coriolis force on natural frequency is inappreciable. Li et al. (2011) presented a Timoshenko beam theory-based model for the dynamics of pipes conveying fluid, and proposed the dynamic stiffness method to analyze the free vibration of multi-span pipe. Zhai et al. (2011) determined the natural frequencies of fluid-conveying Timoshenko pipes via the complex modal analysis, and calculated the dynamic response of pipeline under random excitation by the pseudo excitation algorithm. Zhang and Chen (2012) analyzed the internal resonance of pipes conveying fluid in the supercritical regime, in which the straight pipe equilibrium configuration becomes unstable and bifurcates into two possible © Springer Nature Switzerland AG 2021 C. An et al., Structural and Thermal Analyses of Deepwater Pipes, https://doi.org/10.1007/978-3-030-53540-7_8

109

110

8 Pipes Conveying Gas–Liquid Two-Phase Flow

curved equilibrium configurations. As the extension of the previous work, Zhang and Chen (2013) and Chen et al. (2014) studied the nonlinear forced vibration of a viscoelastic pipe conveying fluid around the curved equilibrium configuration resulting from the supercritical flow speed, where the frequency and amplitude relationships of stead-state responses in external and internal resonances were derived. However, the literature on the dynamic behavior of the pipes subjected to twophase internal flow, which commonly occurs in piping elements, is still sparse. A few studies have been conducted to investigate the vibration behavior of pipes conveying gas–liquid two-phase flow. Hara (1977) performed the experimental and theoretical analyses of the excitation mechanism of air–water two-phase flowinduced vibrations of a straight horizontal pipe, and found that vibrations were dominantly excited by parametric excitation and by natural vibration resonance to the external force through the instability analysis of a system of Mathieu equations. Fluidelastic instability occurs at a critical velocity for which the energy from the fluid is no longer dissipated entirely by the damping of the piping system (Riverin and Pettigrew 2007), which is the most important of several flow-induced vibration excitation mechanisms that can cause large oscillations and early pipe failure. Monette and Pettigrew (2004) reported the results of a series of experiments to study the fluidelastic instability behavior of cantilevered flexible tubes subjected to two-phase internal flow, where a modified two-phase model was proposed to formulate the characteristics of two-phase flows. Pettigrew and Taylor (1994) outlined cylinders experienced fluidelastic instabilities in liquid flow in the form of buckling or oscillations, and fluidelastic instabilities can be affected by several parameters such as flow velocity, void fraction, flexural rigidity, end conditions, and annular confinement. Pettigrew et al. (1998) analyzed the relative importance of flow-induced vibration excitation mechanisms (fluidelastic instability, periodic wake shedding, turbulence-induced excitation, and acoustic resonance) for liquid, gas, and two-phase axial flow. Riverin et al. (2006) measured the time-dependent forces resulting from a two-phase air–water mixture flowing in an elbow and a tee, and explored a relation between the fluctuating forces caused by the two-phase flow and the characteristics of the flow. Riverin and Pettigrew (2007) conducted an experimental study to investigate the governing vibration excitation mechanism governing in-plane vibrations observed on U-shaped piping elements. Zhang and Xu (2010) carried out an experimental study of pipe vibrations subjected to internal bubbly flow. Zhang et al. (2010) examined the characteristics of channel wall vibration induced by internal bubbly flow, and developed a mathematical model that can well predict the spectral frequencies of the wall vibrations and pressure fluctuations, the corresponding attenuation coefficients and propagation phase speeds. Recently, flow-induced vibration of pipelines conveying high-pressure oil and gas has received attention since it causes damage to supporting structures or pipeline ruptures leading to costly shutdown and severe environmental problems (Guo et al. 2014). Patel and Seyed (1989) examined the contribution of time varying internal slug flow to the dynamic excitation forces applied to a flexible riser. Seyed and Patel (1992) deduced an analytical model for calculating the pressure and

8.2 Mathematical Formulation

111

internal slug flow-induced forces on flexible risers. Ortega et al. (2012) reported a computational method to analyze the interaction between an internal slug flow and the dynamic response of flexible risers. Extending previous work, Ortega et al. (2013) studied the interaction between an internal slug flow plus an external regular wave and the dynamic response of a flexible riser. In this chapter, the dynamic behavior of pipes conveying gas–liquid two-phase flow is analytically and numerically investigated on the basis of the generalized integral transform technique (GITT), which has been successfully developed in heat and fluid flow applications (Cotta 1993; Cotta and Mikhailov 1997; Cotta 1998) and further applied in solving the dynamic response of axially moving beams (An and Su 2011), axially moving Timoshenko beams (An and Su 2014b), axially moving orthotropic plates (An and Su 2014a), damaged Euler–Bernoulli beams (Matt 2013a), cantilever beams with an eccentric tip mass (Matt 2013b), fluid-conveying pipes (Gu et al. 2013), the wind-induced vibration of overhead conductors (Matt 2009), and the vortex-induced vibration of long flexible cylinders (Gu et al. 2012). The most interesting feature in this technique is the automatic and straightforward global error control procedure, which makes it particularly suitable for benchmarking purposes, and the only mild increase in overall computational effort with increasing number of independent variables. The main contribution is the application of the effective tool, the generalized integral transform technique, to developing an analytical–numerical approach to determine the dynamic characteristics of pipes conveying gas–liquid two-phase flow, aiming at providing more reference data in this topic for future comparison. The chapter is organized as follows: In Sect. 8.2, the mathematical formulation of the transverse vibration problem of pipes conveying gas–liquid two-phase flow is presented. In Sect. 8.3, the hybrid numerical–analytical solution is obtained by carrying out integral transform. Numerical results of proposed method including transverse displacements and their corresponding convergence behavior are presented in Sect. 8.4. A parametric study is then performed to investigate the effects of the volumetric qualities on natural frequencies and vibration amplitudes of pipes conveying air–water two-phase flow, respectively. Besides, the volumetric-flow-rate stability envelope for the system is also presented. Finally, the chapter ends in Sect. 8.5 with conclusions and perspectives.

8.2 Mathematical Formulation We consider a vertical pipe with clamped–clamped boundary conditions subjected to gas–liquid two-phase internal flow, as illustrated in Fig. 8.1, where the internal flow of the pipe is constituted by the discrete gaseous bubbles in a continuous liquid. If internal damping, external imposed tension and pressurization effects are either absent or neglected, the equation of motion of the pipe can be derived following the Newtonian derivation by means of decomposing an infinitesimal pipe-fluid element into the pipe element and the fluid element under the assumptions of the Euler–

112

8 Pipes Conveying Gas–Liquid Two-Phase Flow

Fig. 8.1 Illustration of a clamped–clamped pipe conveying gas–liquid two-phase flow, where U1 and U2 express the velocities of the gas and the liquid phases

z

x

L

U

U

g

1

2

Bernoulli beam theory, according to the procedures given by Païdoussis (1998) and Monette and Pettigrew (2004):   2 2  ∂ 2w ∂ w ∂ 4w  2∂ w + M k Uk + 2 M k Uk Mk + m EI 4 + ∂x∂t ∂x ∂x 2 ∂t 2 k

k

k

   ∂ 2 w ∂w  = 0, + Mk + m g (x − L) 2 + ∂x ∂x

(8.1a)

k

subjected to the clamped–clamped boundary conditions w(0, t) = 0,

∂w(0, t) = 0, ∂x

w(L, t) = 0,

∂w(L, t) = 0, ∂x (8.1b–e)

8.3 Integral Transform Solution

113

where w(x, t) is the transverse displacement, EI is the flexural rigidity of the pipe which depends upon both Young’s modulus E and the inertial moment of cross section area I , Mk and Uk are, respectively, the masses per unit length and the steady flow velocities of the gas and the liquid phases (k takes the values of 1 for gas (G) and 2 for liquid (L)), m is the mass of the pipe per unit length, L is the pipe length, and g is the gravitational acceleration. Note that the terms of Eq. (8.1a) represent respectively: flexural force, centrifugal force, Coriolis force, inertia force, and gravity, where centrifugal and Coriolis forces are generated by both gas and liquid phases. The following dimensionless variables are introduced x x = , L ∗

 Γ k = Uk L

Mk , EI

w w = , L ∗

t τ= 2 L



EI , k Mk + m  k Mk + m 3 γ = gL . EI

Mk , k Mk + m

βk = 



(8.2a–c)

(8.2d–f)

Substituting Eq. (8.2) into Eq. (8.1) gives the dimensionless equation (dropping the superposed asterisks for simplicity) 2  ∂ 4w  2 ∂ 2w ∂ 2w 1/2 ∂ w + Γk +2 Γk βk + 4 2 ∂x∂τ ∂x ∂x ∂τ 2 k

k

 ∂ 2 w ∂w  = 0, +γ (x − 1) 2 + ∂x ∂x

(8.3a)

together with the boundary conditions w(0, τ ) = 0,

∂w(0, τ ) = 0, ∂x

w(1, τ ) = 0,

∂w(1, τ ) = 0, ∂x

(8.3b–e)

The initial conditions are defined as follows: w(x, 0) = 0,

w(x, ˙ 0) = v0 sin(π x),

(8.4a,b)

where the sign “. ” denotes the time derivative of the dimensionless transverse displacement.

8.3 Integral Transform Solution According to the principle of the generalized integral transform technique, the auxiliary eigenvalue problem needs to be chosen for the dimensionless governing equation (8.3a) with the homogenous boundary conditions (8.3b–e). The spatial

114

8 Pipes Conveying Gas–Liquid Two-Phase Flow

coordinate “x” is eliminated through integral transformation, and the related eigenvalue problem is adopted for the transverse displacement representation as follows: d4 Xi (x) = μ4i Xi (x), dx 4

0 < x < 1,

(8.5a)

with the following boundary conditions: Xi (0) = 0, Xi (1) = 0,

dXi (0) = 0, dx dXi (1) = 0, dx

(8.5b,c) (8.5d,e)

where Xi (x) and μi are, respectively, the eigenfunctions and eigenvalues of problem (8.5). The eigenfunctions satisfy the following orthogonality property 

1

Xi (x)Xj (x)dx = δij Ni ,

(8.6)

0

with δij = 0 for i = j , and δij = 1 for i = j . The norm, or normalization integral, is written as 

1

Ni = 0

Xi2 (x)dx.

(8.7)

Problem (8.5) is readily solved analytically to yield ⎧ cos[μ (x − 1/2)] cosh[μ (x − 1/2)] i i ⎪ − , for i odd, ⎪ ⎨ cos(μi /2) cosh(μi /2) Xi (x) = ⎪ sin[μi (x − 1/2)] sinh[μi (x − 1/2)] ⎪ ⎩ − , for i even, sin(μi /2) sinh(μi /2)

(8.8a,b)

where the eigenvalues are obtained from the transcendental equations:  tanh(μi /2) =

− tan(μi /2), tan(μi /2),

for i odd, for i even,

(8.9a,b)

and the normalization integral is evaluated as Ni = 1,

i = 1, 2, 3, . . .

(8.10)

8.3 Integral Transform Solution

115

Therefore, the normalized eigenfunction coincides, in this case, with the original eigenfunction itself, i.e. Xi (x) X˜ i (x) = 1/2 . Ni

(8.11)

The solution methodology proceeds towards the proposition of the integral transform pair for the potentials, the integral transformation itself, and the inversion formula. For the transverse displacement: 

1

w¯ i (τ ) =

X˜ i (x)w(x, τ )dx,

transform,

(8.12a)

0

w(x, τ ) =

∞ 

X˜ i (x)w¯ i (τ ), inverse.

(8.12b)

i=1

The integral transformation process is now employed through operation of (8.3a) 1 with 0 X˜ i (x)dx, to find the transformed transverse displacement system: ∞

  dw¯ j (τ ) d2 w¯ i (τ ) 1/2 + μ4i w¯ i (τ ) +2 Γk βk Aij 2 dτ dτ j =1

k

+



Γk2

Bij w¯ j (τ ) + γ

j =1

k

+γ

∞ 

∞ 

∞ 

Cij w¯ j (τ )

j =1

Aij w¯ j (τ ) = 0,

i = 1, 2, 3, . . . ,

(8.13a)

j =1

where the coefficients are analytically determined from the following integrals: 

1

 X˜ i (x)X˜ j (x) dx,

(8.13b)

 X˜ i (x)X˜ j (x) dx,

(8.13c)

 X˜ i (x)(x − 1)X˜ j (x) dx.

(8.13d)

Aij = 

0 1

Bij =  Cij =

0 1

0

In the similar manner, initial conditions are also integral transformed to eliminate the spatial coordinate, yielding

116

w¯ i (0) = 0,

8 Pipes Conveying Gas–Liquid Two-Phase Flow

dw¯ i (0) = v0 dt



1

X˜ i (x) sin(π x)dx,

i = 1, 2, 3, . . .

(8.14a,b)

0

For computational purposes, the expansion for the transverse deflection is truncated to finite orders NW . Equations (8.13) and (8.14) in the truncated series are subsequently calculated by the NDSolve routine of Mathematica (Wolfram 2003). Once the transformed potential, w¯ i (τ ) has been numerically evaluated, the inversion formulas Eq. (8.12b) are then applied to yield explicit analytical expression for the dimensionless transverse deflection w(x, τ ).

8.4 Results and Discussion 8.4.1 Two-Phase Flow Model Three fundamental parameters of gas–liquid two-phase flow, the volumetric gas fraction εG , the void fraction α, and the slip factor K, are defined as follows: εG =

QG , QG + QL

(8.15)

α=

AG , AG + AL

(8.16)

UG , UL

(8.17)

and K=

where QG and QL are the volumetric flow rates of the gas and liquid phases, respectively, AG and AL are the area occupied by the gas and the liquid in the inner cross section of the pipe, respectively. The flow velocities of the gas and the liquid are given by UG =

QG AG

and

UL =

QL . AL

(8.18)

The void fraction α is related to the volumetric gas fraction G through the slip factor K:   1 − εG 1−α 1 = . (8.19) εG α K

8.4 Results and Discussion

117

In this chapter, the slip factor K is calculated from the model proposed by Monette and Pettigrew (2004):  K=

εG 1 − εG

1/2 .

(8.20)

As pointed out by Monette and Pettigrew (2004), the slip factor K calculated from Eq. (8.20) is consistent with observed flow patterns and is not very different from Chisholm’s slip ratio (Chisholm 1983).

8.4.2 Convergence Behavior of the Solution We now present the convergence behavior of numerical results for the transverse displacement w(x, τ ) of a pipe conveying air–water two-phase flow calculated using the GITT approach. To make the analysis computationally tractable, the parameters of the pipe and the two-phase flow within the parameter range given by Monette and Pettigrew (2004) are taken in Eq. (8.1). The inner and outer diameters of pipe cross section are d = 12.7 mm and D = 15.9 mm, respectively. The pipe is considered to be a flexible tube (Tygon® R-3606) with the density of 1180 kg/m3 , the Young’s modulus of 3.6 MPa, and the length of 1 m. For the flow condition, the volumetric air and water flow rates are known: QG = 0.0005 m3 /s and QL = 0.0002 m3 /s and with the densities of the air and the water (ρG = 1.2 kg/m3 and ρL = 1000 kg/m3 ), the masses of each phase per unit length can be obtained by MG = AG ρG and ML = AL ρL . The dimensionless variables can be obtained through Eq. (8.2). In addition, v0 = 1.0 is employed in the initial conditions (8.4). The solution of the system, Eqs. (8.13) and (8.14), is obtained with N W ≤ 20 to analyze the convergence behavior. The dimensionless transverse displacement w(x, τ ) at different positions, x = 0.1, 0.3, 0.5, 0.7, and 0.9, of the pipe conveying air–water two-phase flow is presented in Table 8.1. The convergence behavior of the integral transform solution is examined for increasing truncation terms NW = 4, 8, 12, 16, 20, and 24 at τ = 5, 20, and 50, respectively. Note that the option MaxSteps for NDSolve, which specifies the maximum number of steps that NDSolve will ever take in attempting to find a solution, is set to be 106 . It can be observed that almost all the solutions converge to the values with two significant digits, while some of them converge to the values with three significant digits with a truncation order of N W = 20. The results at τ = 50 indicate that the convergence behavior of the integral transform solution does not change with time, verifying the good long-time numerical stability of the scheme. For the same cases, the profiles of the transverse displacement at τ = 30 and τ = 50 are illustrated in Fig. 8.2 with different truncation orders. In addition, the time histories for τ ∈ [25, 30] and τ ∈ [45, 50] of the transverse displacement at the central point of the pipe are shown in Fig. 8.3, which also demonstrates

118

8 Pipes Conveying Gas–Liquid Two-Phase Flow

Table 8.1 Convergence behavior of the dimensionless transverse displacement w(x, τ ) of a pipe conveying air–water two-phase flow x τ =5 0.1 0.3 0.5 0.7 0.9 τ = 20 0.1 0.3 0.5 0.7 0.9 τ = 50 0.1 0.3 0.5 0.7 0.9

NW = 4

NW = 8

N W = 12

N W = 16

N W = 20

N W = 24

0.00127 0.0121 0.0233 0.0226 0.00560

−0.00676 −0.0479 −0.102 −0.112 −0.0258

−0.00498 −0.0360 −0.0770 −0.0873 −0.0213

−0.00468 −0.0340 −0.0729 −0.0832 −0.0206

−0.00462 −0.0336 −0.0719 −0.0822 −0.0204

−0.00460 −0.0334 −0.0716 −0.0819 −0.0204

0.00516 0.0298 0.0703 0.0892 0.0228

0.00756 0.0504 0.0855 0.0552 0.00682

0.0105 0.0768 0.157 0.136 0.0216

0.0103 0.0773 0.161 0.142 0.0227

0.0103 0.0772 0.161 0.143 0.0230

0.0102 0.0772 0.162 0.143 0.0231

0.00673 0.0569 0.134 0.121 0.0215

−0.000719 0.00513 0.0396 0.0685 0.0170

−0.00826 −0.0705 −0.157 −0.147 −0.0257

−0.00673 −0.0613 −0.146 −0.143 −0.0259

−0.00623 −0.0583 −0.141 −0.139 −0.0256

−0.00604 −0.0573 −0.140 −0.138 −0.0254

that the convergence is achieved at a truncation order of N W = 20. In order to identify the frequency content of the structural response, the Fast Fourier Transform (FFT) amplitude spectrum of the time history for τ ∈ [0, 50] of the transverse displacement at x = 0.5 is shown in Fig. 8.3c, from which it can be clearly seen that the three peaks in the amplitude spectrum appear at the frequencies of 0.520 (the fundamental frequency), 8.34, and 18.2, respectively.

8.4.3 Parametric Study In this section, transverse displacement of pipes conveying air–water two-phase flow with clamped–clamped boundary conditions are analyzed to illustrate the applicability of the proposed approach. The variations of the fundamental frequency ω1 and the vibration amplitude with the volumetric gas fraction εG and the liquid flow rate QL are studied. In the following analysis, we use a relative high truncation order, N W = 20, for a sufficient accuracy.

8.4 Results and Discussion Fig. 8.2 GITT solutions with different truncation orders N W for the transverse displacement profiles (a) w(x, τ )|τ =30 and (b) w(x, τ )|τ =50 of a pipe conveying air–water two-phase flow

119

120 Fig. 8.3 Time histories (a) w(0.5, τ )|25≤τ ≤30 and (b) w(0.5, τ )|45≤τ ≤50 of the transverse displacement at the central point of a pipe conveying air–water two-phase flow; (c) amplitude spectrum of the structural response w(0.5, τ )|0≤τ ≤50

8 Pipes Conveying Gas–Liquid Two-Phase Flow

8.4 Results and Discussion

8.4.3.1

121

Variation of the Fundamental Frequency

To obtain the fundamental circular frequency for the transverse vibration of the system, the coupled ODEs, Eq. (8.13a), can be represented in the matrix form as follows: ¨ ˙ Mw(t) + Cw(t) + Kw(t) = F(t).

(8.21)

Consider the generalized eigenvalue problem of Eq. (8.21), and the fundamental circular frequency can be obtained by using standard eigenvalue routine for a complex general matrix. The dimensionless fundamental natural frequencies of the pipe conveying air–water two-phase flow for different volumetric gas fractions 0 ≤ εG ≤ 1 and liquid flow rates QL = 0.0001, 0.0002, 0.0003 m3 /s are calculated, as shown in Fig. 8.4, where the other geometrical and physical parameters are same as in Sect. 8.4.2. It can be seen that the fundamental frequency decreases with the volumetric gas fraction and the liquid flow rate. When εG = 0, ω1 equal to 43.9, 36.6, and 22.7 for QL = 0.0001, 0.0002, and 0.0003 m3 /s, respectively, which represent the fundamental frequencies of the pipe conveying liquid. When the volumetric gas fraction reaches a critical value (εG = 0.974, 0.721 and 0.125), the fundamental frequencies equal to zero for QL = 0.0001, 0.0002, and 0.0003 m3 /s, respectively, which means the dynamic system loses its stability by divergence.

8.4.3.2

Variation of the Vibration Amplitude

The dimensionless vibration amplitudes at the central point of the pipe conveying air–water two-phase flow for different volumetric gas fractions 0 ≤ εG ≤ 1 and liquid flow rates QL = 0.0001, 0.0002, 0.0003 m3 /s are calculated, as shown in Fig. 8.5, where the other geometrical and physical parameters are same as in Sect. 8.4.2. The amplitude is the maximum absolute value found from the calculated time-history response of pipe conveying gas–liquid two-phase flow for τ ∈ [0, 50]. It can be seen that the vibration amplitudes increase with the volumetric gas fraction and the water flow rate. When εG = 0, wmax |x=0.5 equal to 0.0237, 0.0258, and 0.0327 for QL = 0.0001, 0.0002, and 0.0003 m3 /s, respectively, which represent the vibration amplitudes at the central point of the pipe conveying liquid. For each case, the vibration amplitudes tends to infinity (viz., instability of the dynamic system occurs) when the liquid flow rate reaches a critical value, which is the same value given in Fig. 8.4, proving the coherence of the analysis.

8.4.4 Volumetric-Flow-Rate Stability Envelope When the water flow rate is specified, the critical value of the volumetric gas fraction for the onset of instability of the dynamic system can be calculated as mentioned

122 Fig. 8.4 Variation of the dimensionless frequency (ω1 ) with the volumetric gas fraction (εG )

Fig. 8.5 Variation of the dimensionless vibration amplitude (wmax |x=0.5 ) with the volumetric gas fraction (εG )

8 Pipes Conveying Gas–Liquid Two-Phase Flow

8.4 Results and Discussion

123

Fig. 8.6 Normalized volumetric-flow-rate stability envelope for the pipe conveying air–water two-phase flow, and its comparison with that calculated using the method proposed by An et al. (2012b)

in Sect. 8.4.3.1. Following the same process, the normalized volumetric-flow-rate stability envelope for the pipe conveying air–water two-phase flow can be obtained, as demonstrated in Fig. 8.6. QG,max = 0.00823 m3 /s and QL,max = 0.000372 m3 /s represent the critical volumetric gas and liquid flow rates, respectively, when considering the single-phase internal flow. It should be noted that all points located outside the envelope represent a two-phase flow with the specified volumetric gas and liquid flow rates that will cause the dynamic system to lose its stability. To verify the results given in Fig. 8.6, the method to calculate the dimensionless critical velocity of each phase and the dimensionless frequency proposed by Monette and Pettigrew (2004) (Section 2. Fluidelastic instability theory in Monette and Pettigrew (2004)) is employed here. The classical mode summation response formulation can be expressed as w(x, τ ) =

∞ 

ai X˜ i (x)eiωτ .

(8.22)

i=1

Then the system of equations, Eq. (8.13a), is expressed as a summation of modes ∞    1/2 [(μ4j − ω2 )δij + 2 Γk βk Aij ωi + Γk2 Bij + γ Cij + γ Aij ]aj = 0. j =1

k

k

(8.23) To find the critical frequency of the tube at instability, the imaginary part of the frequency term is set to zero. The solution of the system of equations is obtained by allowing the determinant of the coefficients of aj to be equal to zero. In other words,

124

8 Pipes Conveying Gas–Liquid Two-Phase Flow

the real and the imaginary part of the determinant are equal to zero, respectively, which yields two equations. By combining the model for slip factor K (Eq. (8.20)), the three unknowns including the dimensionless velocity of each phase and the dimensionless frequency can be solved. The results calculated using the method proposed by Monette and Pettigrew (2004) are also shown in Fig. 8.6. The excellent agreement between the results obtained and the ones given by GITT indicates the validity of the proposed method.

8.5 Conclusions The generalized integral transform technique (GITT) has shown in this chapter to be a good approach for the analysis of dynamic behavior of pipes conveying gas–liquid two-phase flow, providing an accurate numerical–analytical solution for the natural frequencies and transverse displacements. The solutions confirm good convergence and long-time numerical stability of the scheme. The parametric studies indicate that the fundamental frequency decreases with the volumetric gas fraction and the water flow rate. When the water flow rate reaches a critical value, the dynamic system loses its stability by divergence. The vibration amplitudes increase with the volumetric gas fraction and the water flow rate. In addition, the normalized volumetric-flow-rate stability envelope for the pipe conveying air–water two-phase flow is obtained. The proposed approach can be employed to predict the dynamic behavior of pipes conveying gas–liquid two-phase flow in associated with other more two-phase models for future investigation.

Chapter 9

Pipes Conveying Vertical Slug Flow

9.1 Introduction Internal flow-induced structural vibration has been experienced in numerous fields, including heat exchanger tubes, chemical plant piping systems, nuclear reactor components, and subsea production pipelines. Systematic and extensive studies have been carried out in the past decades to understand the dynamic behavior of pipes conveying single-phase flow (Païdoussis 1998, 2001). An interesting subject in this context is the vibration of marine risers conveying fluid exposed to severe ocean current environments, which can produce a high level of fatigue damage in a relatively short time (Païdoussis 2005). Recently, many achievements have been made in understanding the dynamic characteristics of marine riser conveying internal fluid. By use of the small deflection theory, Guo et al. (2000) derived the lateral vibration equation for marine riser conveying fluid, and studied the effect of internal flow velocity and top tension on the natural frequency of the riser with the finite element method. In another works, Guo et al. (2004, 2008) derived governing equation for the vortex-induced vibration (VIV) of the top tensioned riser considering the effect of the internal flowing fluid and the external marine environmental condition, where the near-wake dynamics is described by the wake oscillator model. Considering the flow inside the pipe is inviscid, irrotational, and incompressible with constant velocity, Chatjigeorgiou (2010a,b) investigated the complete 3D nonlinear dynamic problem of extensible, catenary risers conveying fluid, and solved the system using a finite-difference numerical scheme. Dai et al. (2013) examined the VIV of a hinged–hinged pipe conveying fluid, considering the internal fluid velocities ranging from the subcritical to the supercritical regions. Furthermore, Dai et al. (2014b) investigated the VIV of a long flexible pipe conveying fluctuating flows, where the internal fluid velocity is assumed to have a harmonically varying component superposed on a steady mean velocity. Dai et al. (2014a) studied the nonlinear dynamical responses of a vertical riser conveying fluid subjected to combined VIV and base excitations. © Springer Nature Switzerland AG 2021 C. An et al., Structural and Thermal Analyses of Deepwater Pipes, https://doi.org/10.1007/978-3-030-53540-7_9

125

126

9 Pipes Conveying Vertical Slug Flow

Internal gas–liquid two-phase flows can generate dynamic fluid forces which may induce structural vibration. Excessive vibration may cause risers failures due to fatigue and fretting-wear. However, the literature on the dynamic behavior of marine risers subjected to two-phase internal flow, especially gas–liquid slug flow, is still sparse. Blanco and Casanova (2010) developed a fluid-structural model for analyzing the dynamical behavior of a riser subjected to simultaneous internal slug flow and an external marine current uniform flow, where a simplified slug flow model considering the slug unit composed by the liquid slug zone and the film zone is implemented. Nair et al. (2011) presented a methodology for analyzing slugging induced fatigue of multi-planar rigid jumper systems, where computational fluid dynamics is used to simulate the flow within the jumper and provide pressure fluctuations on the internal pipe wall for the vibration analysis. Ortega et al. (2012) analyzed the influence from slug flow on the structural dynamic response of a lazy wave flexible riser, where a computational tool for analysis of interaction between unsteady internal two-phase flow and structural dynamics was built. Furthermore, Ortega et al. (2013) modeled the interaction between an internal slug flow plus an external regular wave and the dynamic response of a flexible riser by resolving the fluid conservation equations for the internal two-phase flow, the Airy theory for the external wave, and the dynamic equilibrium equation for the structure response. In this chapter, a fluid-structural model for analyzing the dynamic behavior of riser vibration subjected to simultaneous internal gas–liquid two-phase flow and external marine current is proposed. Slug flow regime is considered as it causes most violent vibrations. An analytical model is adopted for the prediction of important flow characteristics of the gas–liquid slug flow. A wake oscillator is employed to model the vortex shedding behind the riser. The dynamic behavior of risers is analytically and numerically investigated by using the generalized integral transform technique (GITT), which has been successfully developed in heat and fluid flow applications (Cotta 1993; Cotta and Mikhailov 1997; Cotta 1998) and further applied in solving the dynamic response of axially moving beams (An and Su 2011), axially moving Timoshenko beams (An and Su 2014b), axially moving orthotropic plates (An and Su 2014a), damaged Euler–Bernoulli beams (Matt 2013a), cantilever beams with an eccentric tip mass (Matt 2013b), fluid-conveying pipes (Gu et al. 2013), the wind-induced vibration of overhead conductors (Matt 2009), and the vortex-induced vibration of long flexible cylinders (Gu et al. 2012). The most interesting feature in this technique is the automatic and straightforward global error control procedure, which makes it particularly suitable for benchmarking purposes, and the only mild increase in overall computational effort with increasing number of independent variables. The transverse vibration equation governing the riser dynamics is transformed into a coupled system of second-order differential equations in the temporal variable. Parametric studies are performed to analyze the effects of the superficial velocities of liquid and gas on the dynamic behavior of risers.

9.2 Mathematical Formulation

127

9.2 Mathematical Formulation We consider a riser with pinned–pinned boundary conditions subjected to internal gas–liquid slug flow and uniform external cross-flow with velocity V , as illustrated in Fig. 9.1. The mass of the gas is neglected. If internal damping and structural damping are either absent or neglected, the equation of motion for the riser conveying slug flow subjected to the vortex-induced force takes the following form, according to the procedures given by Hara (1977), Païdoussis (1998), and Guo et al. (2008): EI

2 ∂ 4w ∂ 2w 2∂ w + m (x, t)U (x, t) − T l l ∂x 4 ∂x 2 ∂x 2

+ P (x, t)A

∂w ∂ 2w ∂ 2w + Cf + 2ml (x, t)Ul (x, t) 2 ∂x∂t ∂t ∂x

+ (mr + ml (x, t) + mf )

∂ 2w 1 = ρf V 2 DO CL0 q, 4 ∂t 2

∂q α ∂ 2w ∂ 2q 2 2 + ω + εω (q − 1) q = f f ∂t Do ∂t 2 ∂t 2

Fig. 9.1 Illustration of a pinned–pinned riser conveying slug flow, with U representing the velocity of the slug flow and V the current speed

(9.1)

(9.2)

128

9 Pipes Conveying Vertical Slug Flow

subjected to the pinned–pinned boundary conditions w(0, t) = 0,

∂ 2 w(0, t) = 0, ∂x 2

w(L, t) = 0,

∂ 2 w(L, t) = 0, ∂x 2

(9.3)

q(0, t) = 0,

∂ 2 q(0, t) = 0, ∂x 2

q(L, t) = 0,

∂ 2 q(L, t) = 0, ∂x 2

(9.4)

and

where w(x, t) is the transverse displacement, EI is the flexural rigidity, CM is the added mass coefficient, ρf is the external fluid density, Do is the outer diameter of the riser, ml (x, t) is the variable mass per unit length of liquid, Ul (x, t) is the flow velocity, T is the top tension, p(x, t) is the pressure fluctuation of internal fluid due to slug flow, A is the internal cross-sectional area of the riser, Cf = ζ ωf ρf Do2 is the external fluid damping, ζ = CD /(4π St ) is a coefficient related to the mean sectional drag coefficient of the riser CD , St is the Strouhal number which is dependent on the Reynolds number, mr is the mass per unit length of the riser, mf = CM ρf Do2 π/4 is the external fluid added mass per unit length, CL0 is the reference lift coefficient which can be obtained from observation of a fixed structure subjected to vortex shedding, V is the current speed, q(x, t) = 2CL (x, t)/CL0 is the wake variable which may be interpreted as a reduced lift coefficient, CL (x, t) is the lift coefficient, ωf = 2π St V /Do is the vortex-shedding frequency, ε and α are the parameters that can be derived from experimental results (Facchinetti et al. 2004). The following dimensionless variables are introduced  EI x w t ∗ ∗ ∗ x = , w = , t = 2 , L Do mr L (9.5)   m m r r Ul∗ (x, t) = Ul (x, t)L , ωf∗ = ωf L2 . EI EI Substituting Eq. (9.5) into Eqs. (9.1) and (9.2) gives the dimensionless equations (dropping the superposed asterisks for simplicity) ∂ 4 w ml (x, t)Ul (x, t)2 ∂ 2 w T L2 ∂ 2 w + − mr EI ∂x 2 ∂x 4 ∂x 2 +

P (x, t)AL2 ∂ 2 w 2ml (x, t)Ul (x, t) ∂ 2 w + EI mr ∂x∂t ∂x 2

Cf L2 ∂w mr + ml (x, t) + mf ∂ 2 w + +√ mr ∂t 2 EI mr ∂t =

ρf V 2 CL0 L4 q, 4EI

(9.6)

9.3 Integral Transform Solution

129

∂ 2q ∂q ∂ 2w 2 2 + ω + εω (q − 1) q = α f f ∂t ∂t 2 ∂t 2

(9.7)

together with the boundary conditions w(0, t) = 0,

∂ 2 w(0, t) = 0, ∂x 2

w(1, t) = 0,

∂ 2 w(1, t) = 0, ∂x 2

(9.8)

q(0, t) = 0,

∂ 2 q(0, t) = 0, ∂x 2

q(1, t) = 0,

∂ 2 q(1, t) = 0. ∂x 2

(9.9)

and

The initial conditions are defined as follows: w(x, 0) = 0,

w(x, ˙ 0) = 0,

(9.10)

q(x, 0) = 1,

q(x, ˙ 0) = 0.

(9.11)

and

9.3 Integral Transform Solution According to the principle of the generalized integral transform technique, the auxiliary eigenvalue problem needs to be chosen for the dimensionless governing equations (9.6) and (9.7) with the homogeneous boundary conditions (9.8) and (9.9). The spatial coordinate “x” is eliminated through integral transformation, and the related eigenvalue problem is adopted for the transverse displacement representation as follows: d4 Xi (x) = μ4i Xi (x), dx 4

0 < x < 1,

(9.12a)

with the following boundary conditions

Xi (0) = 0,

d2 Xi (0) = 0, dx 2

(9.12b,c)

Xi (1) = 0,

d2 Xi (1) = 0, dx 2

(9.12d,e)

where Xi (x) and μi are, respectively, the eigenfunctions and eigenvalues of problem (9.12). The eigenfunctions satisfy the following orthogonality property:

130

9 Pipes Conveying Vertical Slug Flow



1

Xi (x)Xj (x)dx = δij Ni ,

(9.13)

0

with δij = 0 for i = j , and δij = 1 for i = j . The norm, or normalization integral, is written as  Ni = 0

1

Xi2 (x)dx.

(9.14)

Problem (9.12) is readily solved analytically to yield Xi (x) = sin(μi x),

(9.15)

where the eigenvalues are given by μi = iπ,

i = 1, 2, 3, . . .

(9.16)

and the normalization integral is evaluated as Ni = 1/2,

i = 1, 2, 3, . . .

(9.17)

Therefore, the normalized eigenfunction coincides, in this case, with the original eigenfunction itself, i.e. Xi (x) X˜ i (x) = 1/2 . Ni

(9.18)

The solution methodology proceeds towards the proposition of the integral transform pair for the potentials, the integral transformation itself, and the inversion formula. For the transverse displacement:  w¯ i (t) =

1

X˜ i (x)w(x, t)dx,

transform,

(9.19a)

0

w(x, t) =

∞ 

X˜ i (x)w¯ i (t), inverse.

(9.19b)

i=1

For the wake variable:  q¯j (t) =

1

X˜ j (x)q(x, t)dx,

transform,

(9.20a)

0

q(x, t) =

∞  j =1

X˜ j (x)q¯j (t), inverse.

(9.20b)

9.3 Integral Transform Solution

131

The integral transformation process is now employed through operation of (9.6) 1 and (9.7) with 0 X˜ k (x)dx to eliminate the spatial coordinate x, resulting in the transformed governing equations: ∞ 

∞

Aik

i=1

d2 w¯ i (t)  dw¯ i (t) dw¯ k (t) + C0 + Bik 2 dt dt dt i=1

+

∞ 

Dik w¯ i (t) +

i=1

∞ 

Eik w¯ i (t) +

i=1

+ μ4k w¯ k (t) = G0 q¯k (t),

∞ 

Fik w¯ i (t)

i=1

k = 1, 2, 3, . . . ,

(9.21a)

d2 q¯k (t) dq¯k (t) − εωf 2 dt dt ∞  1 ∞   dq¯j (t) ˜ ˜ + εωf Xk (x)Xj (x)( X˜ j (x)q¯j (t))2 dx dt 0 j =1

+ ωf2 q¯k (t) = α

j =1

d w¯ k (t) , dt 2 2

k = 1, 2, 3, . . . ,

(9.21b)

where the coefficients are analytically determined from the following expressions:  Aik =

1

X˜ i (x)X˜ k (x)

0

 Bik =

0

1

mr + ml (x, t) + mf dx, mr

2ml (x, t)Ul (x, t)  dx, X˜ i (x) X˜ k (x) mr

Cf L2 C0 = √ , EI mr  1 ml (x, t)Ul (x, t)2  Dik = dx, X˜ i (x) X˜ k (x) mr 0  T L2 1 ˜  Eik = − Xi (x) X˜ k (x)dx, EI 0  1 P (x, t)AL2  dx, Fik = X˜ i (x) X˜ k (x) EI 0 G0 =

ρf V 2 CL0 L4 . 4EI

(9.21c) (9.21d) (9.21e) (9.21f) (9.21g) (9.21h) (9.21i)

In the similar manner, initial conditions are also integral transformed to eliminate the spatial coordinate, yielding

132

9 Pipes Conveying Vertical Slug Flow

w¯ k (0) = 0, 

1

q¯k (0) = 0

X˜ k (x)dx,

dw¯ k (0) = 0, dt

k = 1, 2, 3, . . .

(9.22a,b)

dq¯k (0) = 0, dt

k = 1, 2, 3, . . .

(9.22c,d)

For computational purposes, the expansions for the transverse deflection and the wake variable are truncated to finite orders NW and N Q. Equations (9.21) and (9.22) in the truncated series are subsequently calculated by the NDSolve routine of Mathematica (Wolfram 2003). Once the transformed potentials, w¯ i (t) and q¯j (t) have been numerically evaluated, the inversion formulas Eqs. (9.19b) and (9.20b) are then applied to yield explicit analytical expressions for the dimensionless transverse deflection w(x, t) and the wake variable q(x, t).

9.4 Two-Phase Flow Model In order to solve the Eqs. (9.21a) and (9.21b), the pressure fluctuation of internal fluid due to slug flow P (x, t) and the variable mass per unit length of liquid ml (x, t) should be calculated firstly using the vertical slug flow model. According to Taitel (1986), the translational velocity Ut of a Taylor bubble is assumed to be given by Nicklin et al. (1962)  Ut = 1.2Us + 0.35 gDi ,

(9.23)

where Us is the superficial mixture velocity given by Us = Uls + Ugs , Uls and Ugs are the superficial velocities of the liquid and the gas, g is the gravitational acceleration, and Di is the internal diameter of the riser. The liquid velocity in the liquid slug is obtained by Ull = Us − U0 (1 − Rs ),

(9.24)

where U0 = 1.53(σ g(ρl − ρg )/ρl2 )0.25 is the relative bubble rise velocity, σ is the surface tension of the liquid, ρl and ρg are the densities of the liquid and the gas, Rs = 1 − αls is the liquid holdup in the liquid slug, αls is the void fraction in the liquid slug. The film velocity around the Taylor bubble is expressed as a function of the liquid film thickness:  Uf i =

(δ/Di )1−m 1/3  2  3 k μl / Di g(ρl − ρg )ρl (4ρl Di /μl )m

1

m

,

(9.25)

where k = 0.0682 and m = 2/3 are the empirical coefficients (Fernandes et al. 1983), μl is the dynamic viscosity of the liquid, δ is the liquid film thickness. δ can

9.5 Results and Discussion

133

be solved by a liquid mass balance relative to a coordinate system that moves with the translational velocity Rf (Ut + Uf i ) = Rs (Ut − Ull ),

(9.26)

where Rf = 4δ/Di − 4(δ/Di )2 is the liquid holdup in the cross-sectional area of the Taylor bubble and the liquid film. The ratio of the liquid slug length over the slug unit length βs is given by βs =

Uls + Uf i Rf ls = , lu (Us − U0 (1 − Rs ))Rs + Uf i Rf

(9.27)

where ls is the liquid slug length and lu is the slug unit length. Obviously, the ratio of Taylor bubble length over the slug unit length is βt = 1 − βs . The length of slug unit lu is given as an empirical parameter, lu = 20Di . Therefore, the length of the Taylor bubble lt and the length of the liquid slug ls are lt = βt lu and ls = βs lu , respectively. According to Barnea (1990), the total pressure loss across one slug unit ΔPt is ΔPt = ΔPa + ΔPf + ΔPg ,

(9.28)

where ΔPa = (Uls − Uf i )(Ut − Uls )ρl Rs is the pressure drop associated with the acceleration of the slow moving liquid in the film to the liquid velocity within the liquid slug. ΔPf = 2fs ρs Us2 ls /Di is the pressure loss due to frictional effects in the liquid slug, where ρs = ρl Rs + ρg (1 − Rs ), fs = 0.184Re−0.2 is the friction factor based on the mixture Reynolds number within the liquid slug, Re = ρs Us Di /μs is the Reynolds number, μs = (Uls μl + Ugs μg )/Us , and μg is the dynamic viscosity of the gas. ΔPg = ρs gls is the pressure drop due the hydrostatic head. There is no pressure drop across the Taylor bubble, and the pressure drop across each liquid slug is thus ΔPt . The pressure distribution as a function of spatial coordinate x and time t, P (x, t), is thus determined by the ΔPt , the slug length ls , the bubble length lb , and the slug frequency fs = Ut / lu . Similarly, the instantaneous liquid mass per unit length ml (x, t) and the instantaneous liquid velocity Ul (x, t) are given by the hydrodynamic parameters of the steady slug flow model and the slug frequency.

9.5 Results and Discussion We now present the numerical results for the transverse displacement w(x, t) of a riser with pinned–pinned boundary conditions subjected to internal gas–liquid slug flow and uniform external cross-flow using the GITT approach. To make the analysis computationally tractable, the parameters of the marine riser and the external flow given by Kaewunruen et al. (2005) and the properties of the internal slug flow are

134

9 Pipes Conveying Vertical Slug Flow

Table 9.1 General geometric and physical properties of marine riser, internal slug flow, and external current Parameter list Marine riser Top tension Riser length Outer diameter Thickness of riser Specific weight of riser in air Modulus of elasticity External current Specific weight of sea water Added mass coefficient Drag coefficient Reference lift coefficient Strouhal number Nonlinear part parameter Hydrodynamic parameter Internal slug flow Dynamic viscosity of liquid Dynamic viscosity of gas Surface tension of liquid Specific weight of liquid Specific weight of gas Empirical coefficient Empirical coefficient Void fraction in the liquid

Symbol

Value

Unit

T L Do t ρr E

476.198 300 0.26 0.03 7850 207,000

kN m m m kg/m3 MPa

ρf CM CD CL0 Sr ε α

1025 1 1.2 0.3 0.2 0.3 12

kg/m3

μl μg σ ρl ρg k m αls

0.001 1.81 × 10−5 0.0727 998 1.21 0.0682 2/3 0.3

kg/(s m) kg/(s m) N/m kg/m3 kg/m3

presented in Table 9.1. The current velocity is assumed to be 1.5 m/s. The dynamic response of the system, Eq. (9.21), is obtained with N W = 5 and N Q = 5 with different superficial velocities of liquid (Uls ) and gas (Ugs ). Figure 9.2 demonstrates the dimensionless dynamic deflection of the riser at t = 1.025 for Uls = 1.6 m/s and different superficial velocities of gas Ugs = 2.4, 3.2 and 4.0 m/s. Figure 9.3 shows the dimensionless midpoint dynamic deflection of the riser at x = 0.5 for different superficial velocities of gas, from which it can be seen that the amplitude of the system increases with the superficial velocity of gas. Figure 9.4 shows the dimensionless dynamic deflection of the riser for different superficial velocities of gas. Figure 9.5 demonstrates the dimensionless dynamic deflection of the riser at t = 1.025 for Ugs = 3.2 m/s and different superficial velocities of liquid Uls = 0.8, 1.6 and 2.4 m/s. Figure 9.6 shows the dimensionless midpoint dynamic deflection of the riser at x = 0.5 for different superficial velocities of liquid, from which it can be seen that the amplitude of the system increases with the superficial velocity of liquid. Figure 9.7 shows the dimensionless dynamic deflection of the riser for different superficial velocities of liquid.

9.5 Results and Discussion

135

Fig. 9.2 GITT solutions for the dimensionless transverse displacement w(x, t) of a riser subjected to internal gas–liquid slug flow and uniform external cross-flow at t = 1.025 for Uls = 1.6 m/s and different superficial velocities of gas Ugs Fig. 9.3 GITT solutions for the dimensionless transverse displacement w(x, t) of a riser subjected to internal gas–liquid slug flow and uniform external cross-flow at x = 0.5 for Uls = 1.6 m/s and different superficial velocities of gas Ugs

136 Fig. 9.4 GITT solutions for the dimensionless transverse displacement w(x, t) of a riser subjected to internal gas–liquid slug flow and uniform external cross-flow for Uls = 1.6 m/s and different superficial velocities of gas Ugs . (a) Ugs = 2.4 m/s. (b) Ugs = 3.2 m/s. (c) Ugs = 4.0 m/s

9 Pipes Conveying Vertical Slug Flow

9.5 Results and Discussion Fig. 9.5 GITT solutions for the dimensionless transverse displacement w(x, t) of a riser subjected to internal gas–liquid slug flow and uniform external cross-flow at t = 1.025 for Ugs = 3.2 m/s and different superficial velocities of liquid Uls

Fig. 9.6 GITT solutions for the dimensionless transverse displacement w(x, t) of a riser subjected to internal gas–liquid slug flow and uniform external cross-flow at x = 0.5 for Ugs = 3.2 m/s and different superficial velocities of liquid Uls

137

138 Fig. 9.7 GITT solutions for the dimensionless transverse displacement w(x, t) of a riser subjected to internal gas–liquid slug flow and uniform external cross-flow for Ugs = 3.2 m/s and different superficial velocities of liquid Uls . (a) Uls = 0.8 m/s. (b) Uls = 1.6 m/s. (c) Uls = 2.4 m/s

9 Pipes Conveying Vertical Slug Flow

9.6 Conclusions

139

9.6 Conclusions A fluid-structure model for analyzing the dynamic behavior of riser vibration subjected to simultaneous internal gas–liquid two-phase flow and external marine current is proposed. The generalized integral transform technique (GITT) is utilized to the analytical solution of vibration of marine risers conveying gas–liquid twophase flow. An analytical model is adopted for the prediction of important flow characteristics of the gas–liquid slug flow. The parametric study indicates that the amplitude of the system increases with the superficial velocities of liquid and gas. The proposed approach can be employed to analyze the dynamic behavior of risers conveying gas–liquid two-phase flow in associated with two-phase models for future investigation.

Chapter 10

Pipes Conveying Horizontal Slug Flow

10.1 Introduction Internal flow-induced structural vibration has been experienced in numerous fields, including heat exchanger tubes, chemical plant piping systems, nuclear reactor components, and subsea production pipelines. Systematic and extensive studies have been carried out in the past decades to understand the dynamic behavior of pipes conveying single-phase flow (Païdoussis 1998, 2001). An interesting subject in this context is the vibration of pipelines conveying fluid exposed to severe ocean current environments, which can produce a high level of fatigue damage in a relatively short time (Païdoussis 2005). Recently, many achievements have been made in understanding the dynamic characteristics of pipelines conveying internal fluid. Xu et al. (1999) developed the fatigue damage models for multi-span pipelines detailed both in time and frequency domain approaches. Pantazopoulos et al. (1993) put forward a Fourier transformation based methodology to study the VIV of free span submarine pipelines. Bryndum and Smed (1998) carried experiments in the VIV of submarine free spans under different boundary conditions. Furnes and Berntsen (2003) formulated time domain model of a free span pipeline subjected to ocean currents in which the in-line and cross-flow deflections are coupled. Internal gas–liquid two-phase flows can generate dynamic fluid forces which may induce structural vibration. Excessive vibration may cause pipelines failures due to fatigue and fretting-wear. However, the literature on the dynamic behavior of marine pipelines subjected to two-phase internal flow, especially gas–liquid slug flow, is still sparse. By introducing nonlinear characteristics of seabed supports, Casanova and Blanco (2010) presented numerical model combining fluid equations for predicting slug characteristics and a structural finite element model for the pipelines transporting slugs. Nair et al. (2011) presented a methodology for analyzing slugging induced fatigue of multi-planar rigid jumper systems, where computational fluid dynamics is used to simulate the flow within the jumper and © Springer Nature Switzerland AG 2021 C. An et al., Structural and Thermal Analyses of Deepwater Pipes, https://doi.org/10.1007/978-3-030-53540-7_10

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provide pressure fluctuations on the internal pipe wall for the vibration analysis. Deka et al. (2013) provide a methodology, combining transient dynamic, harmonic, and modal finite element analysis with the VIV tool SHEAR7, for assessing subsea jumpers for vibration induced fatigue. Bossio et al. (2014) studied the interaction between slug flow-induced vibration in horizontal pipelines and cross-flow response due to vortex shedding. In this chapter, a fluid-structural model for analyzing the dynamic behavior of pipeline vibration subjected to simultaneous internal gas–liquid two-phase flow and external marine current is proposed. Slug flow regime is considered as it causes most violent vibrations. An analytical model is adopted for the prediction of important flow characteristics of the gas–liquid slug flow. A wake oscillator is employed to model the vortex shedding behind the pipeline. The dynamic behavior of pipelines is analytically and numerically investigated by using the generalized integral transform technique (GITT), which has been successfully developed in heat and fluid flow applications (Cotta 1993; Cotta and Mikhailov 1997; Cotta 1998) and further applied in solving the dynamic response of axially moving beams (An and Su 2011), axially moving Timoshenko beams (An and Su 2014b), axially moving orthotropic plates (An and Su 2014a), damaged Euler–Bernoulli beams (Matt 2013a), cantilever beams with an eccentric tip mass (Matt 2013b), fluid-conveying pipes (Gu et al. 2013), the wind-induced vibration of overhead conductors (Matt 2009), and the vortex-induced vibration of long flexible cylinders (Gu et al. 2012). The most interesting feature in this technique is the automatic and straightforward global error control procedure, which makes it particularly suitable for benchmarking purposes, and the only mild increase in overall computational effort with increasing number of independent variables. The transverse vibration equation governing the pipeline dynamics is transformed into a coupled system of second-order differential equations in the temporal variable. Parametric studies are performed to analyze the effects of the superficial velocities of liquid and gas on the dynamic behavior of pipelines.

10.2 Mathematical Formulation We consider a pipeline with pinned–pinned boundary conditions subjected to internal gas–liquid slug flow and uniform external cross-flow with velocity V . The mass of the gas is neglected. If internal damping and structural damping are either absent or neglected, the equation of motion for the pipeline conveying slug flow subjected to the vortex-induced force takes the following form, according to the procedures given by Hara (1977), Païdoussis (1998), and Guo et al. (2008):

10.2 Mathematical Formulation

EI

143

2 ∂ 4w 2∂ w + m (x, t)U (x, t) l l ∂x 4 ∂x 2

+ P (x, t)A

∂w ∂ 2w ∂ 2w + 2m (x, t)U (x, t) + Cf l l ∂x∂t ∂t ∂x 2

+ (mr + ml (x, t) + mf )

∂ 2w 1 = ρf V 2 DO CL0 q, 2 4 ∂t

∂ 2q ∂q α ∂ 2w 2 2 + ω + εω (q − 1) q = f f ∂t Do ∂t 2 ∂t 2

(10.1)

(10.2)

subjected to the pinned–pinned boundary conditions w(0, t) = 0,

∂ 2 w(0, t) = 0, ∂x 2

w(L, t) = 0,

∂ 2 w(L, t) = 0, ∂x 2

(10.3)

q(0, t) = 0,

∂ 2 q(0, t) = 0, ∂x 2

q(L, t) = 0,

∂ 2 q(L, t) = 0, ∂x 2

(10.4)

and

where w(x, t) is the transverse displacement, EI is the flexural rigidity, CM is the added mass coefficient, ρf is the external fluid density, Do is the outer diameter of the pipeline, ml (x, t) is the variable mass per unit length of liquid, Ul (x, t) is the flow velocity, p(x, t) is the pressure fluctuation of internal fluid due to slug flow, A is the internal cross-sectional area of the pipeline, Cf = ζ ωf ρf Do2 is the external fluid damping, ζ = CD /(4π St ) is a coefficient related to the mean sectional drag coefficient of the pipeline CD , St is the Strouhal number which is dependent on the Reynolds number, mr is the mass per unit length of the pipeline, mf = CM ρf Do2 π/4 is the external fluid added mass per unit length, CL0 is the reference lift coefficient which can be obtained from observation of a fixed structure subjected to vortex shedding, V is the current speed, q(x, t) = 2CL (x, t)/CL0 is the wake variable which may be interpreted as a reduced lift coefficient, CL (x, t) is the lift coefficient, ωf = 2π St V /Do is the vortex-shedding frequency, ε and α are the parameters that can be derived from experimental results (Facchinetti et al. 2004). The following dimensionless variables are introduced  EI w t w∗ = , t∗ = 2 , Do mr L   mr mr , ωf∗ = ωf L2 . Ul∗ (x, t) = Ul (x, t)L EI EI x x∗ = , L

(10.5)

Substituting Eq. (10.5) into Eqs. (10.1) and (10.2) gives the dimensionless equations (dropping the superposed asterisks for simplicity)

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10 Pipes Conveying Horizontal Slug Flow

∂ 4 w ml (x, t)Ul (x, t)2 ∂ 2 w + mr ∂x 4 ∂x 2 +

P (x, t)AL2 ∂ 2 w 2ml (x, t)Ul (x, t) ∂ 2 w + EI mr ∂x∂t ∂x 2

Cf L2 ∂w mr + ml (x, t) + mf ∂ 2 w + +√ mr ∂t 2 EI mr ∂t =

ρf V 2 CL0 L4 q, 4EI

(10.6)

∂ 2q ∂q ∂ 2w 2 2 + ω + εω (q − 1) q = α f f ∂t ∂t 2 ∂t 2

(10.7)

together with the boundary conditions w(0, t) = 0,

∂ 2 w(0, t) = 0, ∂x 2

w(1, t) = 0,

∂ 2 w(1, t) = 0, ∂x 2

(10.8)

q(0, t) = 0,

∂ 2 q(0, t) = 0, ∂x 2

q(1, t) = 0,

∂ 2 q(1, t) = 0. ∂x 2

(10.9)

and

The initial conditions are defined as follows: w(x, 0) = 0,

w(x, ˙ 0) = 0,

(10.10)

q(x, 0) = 1,

q(x, ˙ 0) = 0.

(10.11)

and

10.3 Integral Transform Solution According to the principle of the generalized integral transform technique, the auxiliary eigenvalue problem needs to be chosen for the dimensionless governing equations (10.6) and (10.7) with the homogeneous boundary conditions (10.8) and (10.9). The spatial coordinate “x” is eliminated through integral transformation, and the related eigenvalue problem is adopted for the transverse displacement representation as follows: d4 Xi (x) = μ4i Xi (x), dx 4

0 < x < 1,

(10.12a)

10.3 Integral Transform Solution

145

with the following boundary conditions

Xi (0) = 0,

d2 Xi (0) = 0, dx 2

(10.12b,c)

Xi (1) = 0,

d2 Xi (1) = 0, dx 2

(10.12d,e)

where Xi (x) and μi are, respectively, the eigenfunctions and eigenvalues of problem (10.12). The eigenfunctions satisfy the following orthogonality property 

1

Xi (x)Xj (x)dx = δij Ni ,

(10.13)

0

with δij = 0 for i = j , and δij = 1 for i = j . The norm, or normalization integral, is written as  Ni = 0

1

Xi2 (x)dx.

(10.14)

Problem (10.12) is readily solved analytically to yield Xi (x) = sin(μi x),

(10.15)

where the eigenvalues are given by μi = iπ,

i = 1, 2, 3, . . .

(10.16)

and the normalization integral is evaluated as Ni = 1/2,

i = 1, 2, 3, . . .

(10.17)

Therefore, the normalized eigenfunction coincides, in this case, with the original eigenfunction itself, i.e. Xi (x) X˜ i (x) = 1/2 . Ni

(10.18)

The solution methodology proceeds towards the proposition of the integral transform pair for the potentials, the integral transformation itself, and the inversion formula. For the transverse displacement:  w¯ i (t) = 0

1

X˜ i (x)w(x, t)dx,

transform,

(10.19a)

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10 Pipes Conveying Horizontal Slug Flow ∞ 

w(x, t) =

X˜ i (x)w¯ i (t), inverse.

(10.19b)

i=1

For the wake variable: 

1

q¯j (t) =

X˜ j (x)q(x, t)dx,

transform,

(10.20a)

0

q(x, t) =

∞ 

X˜ j (x)q¯j (t), inverse.

(10.20b)

j =1

The integral transformation process is now employed through operation of (10.6) 1 and (10.7) with 0 X˜ k (x)dx to eliminate the spatial coordinate x, resulting in the transformed governing equations: ∞ 

∞

Aik

i=1

d2 w¯ i (t)  dw¯ i (t) dw¯ k (t) + C0 + Bik dt dt dt 2 i=1

+

∞ 

Dik w¯ i (t) +

i=1

∞ 

Fik w¯ i (t)

i=1

+ μ4k w¯ k (t) = G0 q¯k (t),

k = 1, 2, 3, . . . ,

(10.21a)

d2 q¯k (t) dq¯k (t) − εωf 2 dt dt ∞  1 ∞   dq¯j (t) ˜ ˜ + εωf Xk (x)Xj (x)( X˜ j (x)q¯j (t))2 dx dt 0 j =1

+ ωf2 q¯k (t) = α

j =1

d w¯ k (t) , dt 2 2

k = 1, 2, 3, . . . ,

(10.21b)

where the coefficients are analytically determined from the following expressions:  Aik =

1

X˜ i (x)X˜ k (x)

0

 Bik =

0

1

mr + ml (x, t) + mf dx, mr

2ml (x, t)Ul (x, t)  dx, X˜ i (x) X˜ k (x) mr

Cf L2 C0 = √ , EI mr

(10.21c) (10.21d) (10.21e)

10.4 Two-Phase Flow Model

 Dik =

1

0

 Fik =

0

G0 =

1

147 2

ml (x, t)Ul (x, t)  dx, X˜ i (x) X˜ k (x) mr

(10.21f)

2

P (x, t)AL  dx, X˜ i (x) X˜ k (x) EI

ρf V 2 CL0 L4 . 4EI

(10.21g) (10.21h)

In the similar manner, initial conditions are also integral transformed to eliminate the spatial coordinate, yielding w¯ k (0) = 0, 

1

q¯k (0) = 0

X˜ k (x)dx,

dw¯ k (0) = 0, dt

k = 1, 2, 3, . . .

(10.22a,b)

dq¯k (0) = 0, dt

k = 1, 2, 3, . . .

(10.22c,d)

For computational purposes, the expansions for the transverse deflection and the wake variable are truncated to finite orders N W and N Q. Equations (10.21) and (10.22) in the truncated series are subsequently calculated by the NDSolve routine of Mathematica (Wolfram 2003). Once the transformed potentials, w¯ i (t) and q¯j (t) have been numerically evaluated, the inversion formulas Eqs. (10.19b) and (10.20b) are then applied to yield explicit analytical expressions for the dimensionless transverse deflection w(x, t) and the wake variable q(x, t).

10.4 Two-Phase Flow Model In order to solve the Eqs. (10.21a) and (10.21b), the pressure fluctuation of internal fluid due to slug flow P (x, t) and the variable mass per unit length of liquid ml (x, t) should be calculated firstly using the horizontal slug flow model. According to Orell (2005), the translational velocity Ut of a Taylor bubble is assumed to be given by Nicklin et al. (1962)  Ut = 1.2Us + 0.35 gDi ,

(10.23)

where Us is the superficial mixture velocity given by Us = Uls + Ugs , Uls and Ugs are the superficial velocities of the liquid and the gas, g is the gravitational acceleration, and Di is the internal diameter of the pipeline. A liquid mass balance relative to a coordinate system that travels at the translational velocity of the slug unit yields (Ut − Uf )Hf = (Ut − Us )Hs .

(10.24)

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10 Pipes Conveying Horizontal Slug Flow

The pressure at a given pipe cross section within the film zone is uniform. A momentum balance on the liquid and gas can be derived as τf Sf − τi Si τG SG − τi Si = . Af AG

(10.25)

The shear stresses τf , τG , and τi are defined by: 1 ff ρL Uf2 , 2 1 2 τG = fG ρG UG , 2 τf =

τi =

1 fi ρG (UG − Uf )2 , 2

(10.26a) (10.26b) (10.26c)

where ff , fG , and fi are the Fanning friction factors of the liquid film, gas bubble, and the gas–liquid interface, respectively. The governing equations and the auxiliary equations can result in a set of six simultaneous equations that contain six unknowns: θ , Uf , UG , Ut , ff , fG , and ls / l. The equations are solved analytically and provide the following slug flow characteristic variables:Us , Uf , UG , Ut , Hs , Hf , ls / l, and lf /f . The solution procedure requires the following input variables: D, Usl , UsG , ρl , ρG , μl , uG , and σ . The average pressure gradient in a slug unit is obtained by performing a momentum balance over a global control volume of the slug unit that yields ΔPt = 2

τf Sf − τi Si lf ls fs ρs Us2 + , D l Af l

(10.27)

where fs is the slug friction factor, ρs = ρl Hs + ρG Es and Es is the void fraction in the aerated slug. There is no pressure drop across the Taylor bubble, and the pressure drop across each liquid slug is thus ΔPt . The pressure distribution as a function of spatial coordinate x and time t, P (x, t), is thus determined by the ΔPt , the slug length ls , the bubble length lb , and the slug frequency fs = Ut / lu . Similarly, the instantaneous liquid mass per unit length ml (x, t) and the instantaneous liquid velocity Ul (x, t) are given by the hydrodynamic parameters of the steady slug flow model and the slug frequency.

10.5 Results and Discussion We now present the numerical results for the transverse displacement w(x, t) of a pipeline with pinned–pinned boundary conditions subjected to internal gas–liquid slug flow and uniform external cross-flow using the GITT approach. To make the

10.5 Results and Discussion

149

Table 10.1 General geometric and physical properties of subsea pipeline, internal slug flow, and external current Parameter list Subsea pipeline Top tension Pipeline length Outer diameter Thickness of pipeline Specific weight of pipeline in air Modulus of elasticity External current Specific weight of sea water Added mass coefficient Drag coefficient Reference lift coefficient Strouhal number Nonlinear part parameter Hydrodynamic parameter Internal slug flow Dynamic viscosity of liquid Dynamic viscosity of gas Surface tension of liquid Specific weight of liquid Specific weight of gas Empirical coefficient Empirical coefficient Void fraction in the liquid

Symbol

Value

Unit

T L Do t ρr E

476.198 300 0.26 0.03 7850 207,000

kN m m m kg/m3 MPa

ρf CM CD CL0 Sr ε α

1025 1 1.2 0.3 0.2 0.3 12

kg/m3

μl μg σ ρl ρg k m αls

0.001 1.81×10−5 0.0727 998 1.21 0.0682 2/3 0.3

kg/(s m) kg/(s m) N/m kg/m3 kg/m3

analysis computationally tractable, the parameters of the marine pipeline and the external flow given by Kaewunruen et al. (2005) and the properties of the internal slug flow are presented in Table 10.1. The current velocity is assumed to be 1.5 m/s. The dynamic response of the system, Eq. (10.21), is obtained with N W = 5 and N Q = 5 with different superficial velocities of liquid (Uls ) and gas (Ugs ). Figure 10.1 demonstrates the dimensionless dynamic deflection of the pipeline at t = 0.5 for Uls = 1.6 m/s and different superficial velocities of gas Ugs = 2.4, 3.2, and 4.0 m/s. Figure 10.2 shows the dimensionless midpoint dynamic deflection of the pipeline at x = 0.5 for different superficial velocities of gas, from which it can be seen that the amplitude of the system increases with the superficial velocity of gas. Figure 10.3 shows the dimensionless dynamic deflection of the pipeline for different superficial velocities of gas. Figure 10.4 demonstrates the dimensionless dynamic deflection of the pipeline at t = 0.5 for Ugs = 3.2 m/s and different superficial velocities of liquid Uls = 0.8, 1.6, and 2.4 m/s. Figure 10.5 shows the dimensionless midpoint dynamic deflection of the pipeline at x = 0.5 for different

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10 Pipes Conveying Horizontal Slug Flow

Fig. 10.1 GITT solutions for the dimensionless transverse displacement w(x, t) of a pipeline subjected to internal gas–liquid slug flow and uniform external cross-flow at t = 0.5 for Uls = 1.6 m/s and different superficial velocities of gas Ugs

Fig. 10.2 GITT solutions for the dimensionless transverse displacement w(x, t) of a pipeline subjected to internal gas–liquid slug flow and uniform external cross-flow at x = 0.5 for Uls = 1.6 m/s and different superficial velocities of gas Ugs

superficial velocities of liquid, from which it can be seen that the amplitude of the system decreases with the superficial velocity of liquid. Figure 10.6 shows the dimensionless dynamic deflection of the pipeline for different superficial velocities of liquid.

10.5 Results and Discussion

151

Fig. 10.3 GITT solutions for the dimensionless transverse displacement w(x, t) of a pipeline subjected to internal gas–liquid slug flow and uniform external cross-flow for Uls = 1.6 m/s and different superficial velocities of gas Ugs . (a) Ugs = 2.4 m/s. (b) Ugs = 3.2 m/s. (c) Ugs = 4.0 m/s Fig. 10.4 GITT solutions for the dimensionless transverse displacement w(x, t) of a pipeline subjected to internal gas–liquid slug flow and uniform external cross-flow at t = 1.025 for Ugs = 3.2 m/s and different superficial velocities of liquid Uls

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10 Pipes Conveying Horizontal Slug Flow

Fig. 10.5 GITT solutions for the dimensionless transverse displacement w(x, t) of a pipeline subjected to internal gas–liquid slug flow and uniform external cross-flow at x = 0.5 for Ugs = 3.2 m/s and different superficial velocities of liquid Uls

Fig. 10.6 GITT solutions for the dimensionless transverse displacement w(x, t) of a pipeline subjected to internal gas–liquid slug flow and uniform external cross-flow for Ugs = 3.2 m/s and different superficial velocities of liquid Uls . (a) Uls = 0.8 m/s. (b) Uls = 1.6 m/s. (c) Uls = 2.4 m/s

10.6 Conclusions

153

10.6 Conclusions A fluid-structure model for analyzing the dynamic behavior of pipeline vibration subjected to simultaneous internal gas–liquid two-phase flow and external marine current is proposed. The generalized integral transform technique (GITT) is utilized to the analytical solution of vibration of marine pipelines conveying gas–liquid two-phase flow. An analytical model is adopted for the prediction of important flow characteristics of the gas–liquid slug flow. The parametric study indicates that the amplitude of the system increases with the superficial velocities of liquid and gas. The proposed approach can be employed to analyze the dynamic behavior of pipelines conveying gas–liquid two-phase flow in associated with two-phase models for future investigation.

Chapter 11

Axially Functionally Graded Pipes Conveying Fluid

11.1 Introduction Pipelines conveying fluid exist widely in many application fields, particularly in nuclear power plants, chemical plants, aeronautic, oil transportation, water supply, heat exchanger devices, human circulation, etc. The high velocity internal flow may cause severe flow-induced vibration of piping systems, which may further result in leakages, fatigue failures, high noise, fire, and explosions of the pipes (Sadeghi and Karimi-Dona 2011). Extensive investigations have been carried out in the past decades to understand the dynamical behavior of pipes conveying fluid, as described by Païdoussis and Li (1993) and Païdoussis (1998, 2008). Similar to other structural dynamic problems, the earliest concern of fluidconveying pipes was the free vibration response (Long 1955). Research reveals that the boundary conditions can affect significantly the natural frequencies of the dynamic systems (Païdoussis and Issid 1974; Païdoussis 1975). Some numerical methods such as homotopy perturbation method (Xu et al. 2010) and precise integration method (Liu and Xuan 2010) were developed to analyze the effect of fluid flow velocity on the natural frequencies. Although there are many studies considering the flow velocity as constant, the flow velocity varies with time for the actual industrial problems. The unsteady flow is usually modeled by the superposition of the steady flow and a time-dependent harmonic component, which may induce the dynamic instability due to parametric resonances (Ginsberg 1973). For instance, Jin and Song (2005) investigated the effect of some physical parameters of the system, such as damping, mean flow velocity, mass ratio, tension, and gravity, on the three regions of parametric resonances of pipes with supported ends conveying pulsating fluid. Panda and Kar (2008) analyzed the nonlinear planar vibration of a hinged–hinged pipe conveying fluid with harmonic flow velocity pulsation in the presence of internal resonance. With the development of material technology and application, new materials exhibiting viscoelastic behaviors such as polymer matrix composites are now widely used for pipes. Zhao et al. (2001) investigated the © Springer Nature Switzerland AG 2021 C. An et al., Structural and Thermal Analyses of Deepwater Pipes, https://doi.org/10.1007/978-3-030-53540-7_11

155

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11 Axially Functionally Graded Pipes Conveying Fluid

dynamic behavior and stability of Maxwell viscoelastic pipes conveying fluid with simply-supported ends. Zhang et al. (2001) presented a viscoelastic finite element approach to the vibration analysis of viscoelastic Timoshenko pipes conveying fluid. Wang et al. (2005) studied the vibration and stability of viscoelastic curved pipes conveying fluid using normalized power series method. Yang et al. (2007) investigated the dynamic stability for the transverse vibrations of pipes conveying fluid using the method of multiple scales. To avoid the failure caused by resonance due to the excitation of external forces, many researchers considered the forced vibrations of pipes conveying fluid. Gulyayev and Tolbatov (2002) carried out the numerical modeling of self-excited vibrations of tubes containing inner flows of non-homogeneous boiling fluid. Seo et al. (2005) presented the finite element method to predict the forced vibration response of a pipe conveying harmonically pulsating fluid. Liang and Wen (2011) studied the forced responses with both an internal resonance and an external periodic excitation of the constant-fluidconveying pipe by the multidimensional Lindstedt–Poincaré method. In practice, most of the load applied on the industrial pipes is random, therefore, the dynamic response of pipes conveying fluid subjected to random excitation was studied by Zhai et al. (2011) and Zhai et al. (2013). Since the high-temperature environment should be confronted in some industries such as nuclear reactors, space planes and chemical plants, the vibration behaviors of pipes conveying fluid under thermal loads have been studied in recent years (Kadoli and Ganesan 2004; Ganesan and Kadoli 2004; Sheng and Wang 2008; Qian et al. 2009; Hosseini and Fazelzadeh 2011). Due to the advantages of being able to withstand severe high-temperature gradient while maintaining structural integrity, functionally graded materials (FGMs) have attracted great interest in a broad range of applications including biomechanical, automotive, aerospace, mechanical, civil, nuclear, and naval engineering (Shen 2009; Zhong et al. 2012). As it is known that FGMs are a novel class of composite materials whose composition and/or function is designed to change continuously within the solid. The composites are usually made from a mixture of metals and ceramics to ensure the elastic and toughness properties gradually vary in space, which can prevent delamination and stress concentration in traditional multilayer, laminated composites. For pipes conveying fluid, Sheng and Wang (2008) reported the result of an investigation into the coupled vibration characteristics of fluid-filled functionally graded cylindrical shells, while Hosseini and Fazelzadeh (2011) investigated the thermomechanical stability of functionally graded thin-walled cantilever pipes conveying flow and loading by compressive axial force. Both of the above-mentioned investigations assumed that the material properties vary along the thickness direction of pipes, however, dynamic behaviors of axially functionally graded systems (structures with material graduation through the longitudinal directions) should be also concerned, as reported by Huang and Li (2010), Huang et al. (2013), Shahba et al. (2011), Shahba and Rajasekaran (2012), Simsek et al. (2012), and Alshorbagy et al. (2011). Nevertheless, the literature dealing with the dynamic behavior of fluid-conveying pipes made of axially FGMs is very limited, which motivates the continuing research on this topic.

11.2 Mathematical Formulation

157

In this chapter, the dynamic behavior of axially functionally graded (FG) pipes conveying fluid is analytically and numerically investigated on the basis of the generalized integral transform technique (GITT), which has been successfully applied in solving the dynamic response of axially moving beams (An and Su 2011), axially moving orthotropic plates (An and Su 2014a), fluid-conveying pipes (Gu et al. 2013) and pipes conveying gas–liquid two-phase flow (An and Su 2015), the wind-induced vibration on overhead conductors (Matt 2009), the vortexinduced vibration of long flexible cylinders (Gu et al. 2012), and the transverse vibrations of a cantilever beam with an eccentric tip mass in the axial direction (Matt 2013b). From an engineering viewpoint, this chapter generates the reliable reference data on the dynamic behavior of axially FG conveying fluid, while from a mathematical viewpoint, the chapter provides a feasible numerical solution of the variable coefficient partial differential equations governing the phenomenon. The chapter is organized as follows: In the next section, the mathematical formulation of the transverse vibration problem of axially FG pipes conveying fluid is presented. In Sect. 11.3, the hybrid numerical–analytical solution is obtained by carrying out integral transform. Numerical results of proposed method including transverse displacements and their corresponding convergence behavior and verification are presented in Sect. 11.4. A parameter study is then performed to investigate the effects of material distributions and mass ratios on natural frequencies and vibration amplitude of pipes conveying fluid, respectively. Besides, the variation of the dimensionless frequencies with Young’s modulus ratio, power exponent and flow velocity are also presented. Finally, the chapter ends in Sect. 11.5 with conclusions and perspectives.

11.2 Mathematical Formulation We consider a fluid-conveying pipe made of axially functionally graded (FG) material based on Euler–Bernoulli beam theory, as illustrated in Fig. 11.1. If gravity, internal damping, external imposed tension, and pressurization effects are either absent or neglected, the equation of motion of the FG pipe can be derived following

Fig. 11.1 Illustration of an axially functionally graded pipe conveying fluid, the transverse displacement of which is described by w(x, t)

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11 Axially Functionally Graded Pipes Conveying Fluid

the Newtonian derivation by means of decomposing an infinitesimal pipe-fluid element into the pipe element and the fluid element, according to the procedure given by Païdoussis (1998):   ∂ 2w 2 ∂2  ∂ 2w  ∂ 2w 2∂ w + m E(x)I + m v + 2m v + ρ(x)A = 0, f f f ∂x∂t ∂x 2 ∂x 2 ∂x 2 ∂t 2 (11.1a) subjected to the clamped–clamped boundary conditions w(0, t) = 0,

∂w(0, t) = 0, ∂x

w(L, t) = 0,

∂w(L, t) = 0, ∂x (11.1b–e)

where w(x, t) is the transverse displacement, E(x)I is the flexural rigidity of the pipe which depends upon both Young’s modulus E(x) and the inertial moment of cross-sectional area I , mf is the mass of fluid per unit length, v is the steady flow velocity, ρ(x)A is the mass of the pipe per unit length which depends upon mass density ρ(x) and cross-sectional area A, and L is the pipe length. Note that for the axially FG pipe, E(x) and ρ(x) are functions of the axial coordinate x. In addition, we assume that the coefficient of thermal expansion and the thermal conductivity of the FG material are constant along the pipe, therefore, the influence of the thermal environment can be ignored. The following dimensionless variables are introduced x x = , L ∗

α(x ∗ ) =

  mf E0 I t t = 2 , v ∗ = vL , E0 I L mf + ρ0 A  mf + ρ(x ∗ )A mf , γ (x ∗ ) = , β= mf + ρ0 A mf + ρ0 A

w w = , L ∗

E(x ∗ ) , E0

∗

(11.2a–d) (11.2e–g)

where E0 and ρ0 are the corresponding Young’s modulus and mass density at the end x ∗ = 0. Substituting Eq. (11.2) into Eq. (11.1) gives the dimensionless equation (dropping the superposed asterisks for simplicity) 2 ∂2  ∂ 2w ∂ 2w  ∂ 2w 2∂ w + γ (x) α(x) + v + 2βv = 0, ∂x∂t ∂x 2 ∂x 2 ∂x 2 ∂t 2

(11.3a)

together with the boundary conditions w(0, t) = 0,

∂w(0, t) = 0, ∂x

w(1, t) = 0,

∂w(1, t) = 0, ∂x

(11.3b-e)

The initial conditions are defined as follows: w(x, 0) = 0,

w(x, ˙ 0) = v0 sin(π x).

(11.4a,b)

11.3 Integral Transform Solution

159

11.3 Integral Transform Solution To conduct the integral transform technique, the additional should be introduced for the governing equation (11.3a) conditions (11.3b-e). The coordinate “x” is eliminated transformation, and the eigenvalue problem is employed displacement as following: d4 Xi (x) = μ4i Xi (x), dx 4

eigenvalue problem with the boundary by using integral for the transverse

0 < x < 1,

(11.5a)

with the boundary conditions Xi (0) = 0, Xi (1) = 0,

dXi (0) = 0, dx dXi (1) = 0, dx

(11.5b,c) (11.5d,e)

where Xi (x) and μi are the eigenfunctions and eigenvalues of problem (11.5), respectively. The orthogonality property should be satisfied by the eigenfunctions: 

1

Xi (x)Xj (x)dx = δij Ni ,

(11.6)

0

with δij = 0 when i = j , and δij = 1 when i = j . The norm is defined as 

1

Ni = 0

Xi2 (x)dx.

(11.7)

Eigenvalue problem (11.5) can be solved analytically to generate ⎧ cos[μ (x − 1/2)] cosh[μ (x − 1/2)] i i ⎪ − , ⎪ ⎨ cos(μi /2) cosh(μi /2) Xi (x) = ⎪ ⎪ ⎩ sin[μi (x − 1/2)] − sinh[μi (x − 1/2)] , sin(μi /2) sinh(μi /2)

for i odd, (11.8a,b) for i even,

where the eigenvalues are calculated through the transcendental equations:  tanh(μi /2) =

− tan(μi /2), tan(μi /2),

for i odd, for i even,

(11.9a,b)

and the normalization of integral is Ni = 1,

i = 1, 2, 3, . . .

(11.10)

160

11 Axially Functionally Graded Pipes Conveying Fluid

Then, the normalized eigenfunction yields Xi (x) X˜ i (x) = 1/2 . Ni

(11.11)

For the transverse displacement, the integral transformation and the inversion equations are as follows:  w¯ i (t) =

1

X˜ i (x)w(x, t)dx,

transform,

(11.12a)

0

w(x, t) =

∞ 

X˜ i (x)w¯ i (t), inverse.

(11.12b)

i=1

Now, the integral transformation procedure is conducted by operation of (11.3a) 1 with 0 X˜ i (x)dx, to obtain the transformed transverse displacement equation system: ∞    d2 w¯ j (t) dw¯ j (t)  2 4 Aij + 2βvB C + D + 2E + μ F ) w ¯ (t) + v ij ij ij ij ij j j dt dt 2 j =1

= 0,

i = 1, 2, 3, . . . , (11.13a)

where the coefficients are given by the integrals below 

1

Aij =

γ (x)X˜ i (x)X˜ j (x)dx,

0

(11.13b) 

1

Bij =

 X˜ i (x)X˜ j (x) dx,

0

(11.13c) 

1

Cij =

 X˜ i (x)X˜ j (x) dx,

0

(11.13d)  Dij =

1

  α(x) X˜ i (x)X˜ j (x) dx,

0

(11.13e)

11.4 Results and Discussion

161



1

Eij =

  α(x) X˜ i (x)X˜ j (x) dx,

0

(11.13f) 

1

Fij =

α(x)X˜ i (x)X˜ j (x)dx.

0

(11.13g) In the similar way, the initial conditions can be also transformed to eliminate the “x” coordinate: w¯ i (0) = 0,

dw¯ i (0) = v0 dt



1

X˜ i (x) sin(π x)dx,

i = 1, 2, 3, . . .

(11.14a,b)

0

In the computational process, the expansion for the transverse displacement is truncated to finite orders NW . Equations (11.13) and (11.14) in the truncated series are calculated using the NDSolve in Mathematica (Wolfram 2003). Once the values of w¯ i are determined, the inversion formulas Eq. (11.12b) are subsequently employed to yield explicit expression for the transverse displacement w(x, t).

11.4 Results and Discussion 11.4.1 Convergence Behavior of the Solution We now present the convergence behavior of numerical results for the transverse displacement w(x, t) of a functionally graded pipe conveying fluid calculated using the GITT approach. For the case examined, the geometrical parameters adopted by Zhai et al. (2011) are taken in Eq. (11.1): L = 1010 mm, D = 22.85 mm, and d = 19.65 mm, where D and d are the outer and inner diameters of pipe cross section, respectively. It is assumed that the material properties of the pipe, such as Young’s modulus E and mass density ρ, vary continuously as a power law through the pipe axis (Alshorbagy et al. 2011): P (x) = (PL − PR )(1 − x/L)k + PR , where PR and PL are the corresponding material of the right and the left side of the pipe, and k is the non-negative power-law exponent which dictates the material variation profile through the pipe axis. In the following calculations, aluminum and zirconia are chosen for the corresponding material of the left and the right sides of the pipe, respectively, the material properties of which are (Huang and Li 2010): Ea = 70 GPa, ρa = 2702 kg/m3 for aluminum and Ez = 200 GPa, ρz = 5700 kg/m3 for zirconia. The fluid density conveying in the pipe is ρf = 1000 kg/m3 . The dimensionless variables can be obtained through Eq. (11.2). The solution of the system, Eqs. (11.13) and (11.14), is obtained with N W ≤ 16 to analyze the convergence behavior.

162

11 Axially Functionally Graded Pipes Conveying Fluid

Table 11.1 Convergence behavior of the dimensionless transverse displacement w(x, t) of FG pipes conveying fluid for v = 1.0 and k = 1.0 x t = 10 0.1 0.3 0.5 0.7 0.9 t = 20 0.1 0.3 0.5 0.7 0.9 t = 30 0.1 0.3 0.5 0.7 0.9

NW = 4

NW = 8

N W = 12

N W = 16

0.0044244 0.0191635 0.0207420 0.0127285 0.0021788

0.0037878 0.0206171 0.0244299 0.0124189 0.0012293

0.0039159 0.0206690 0.0245583 0.0123877 0.0012868

0.0039563 0.0207072 0.0246031 0.0124042 0.0013476

−0.0060519 −0.0291388 −0.0358924 −0.0235454 −0.0040988

−0.0062440 −0.0296250 −0.0382805 −0.0256259 −0.0044978

−0.0060664 −0.0298559 −0.0382784 −0.0257693 −0.0042350

−0.0060803 −0.0298584 −0.0382978 −0.0257332 −0.0042428

0.0064297 0.0325025 0.0405411 0.0251618 0.0040959

0.0066167 0.0313819 0.0390336 0.0234384 0.0038549

0.0064907 0.0315394 0.0384158 0.0238947 0.0037553

0.0064724 0.0315556 0.0383750 0.0239246 0.0037439

The dimensionless transverse displacement w(x, t) at different positions, x = 0.1, 0.3, 0.5, 0.7, and 0.9, of axially FG pipes conveying fluid is presented in Tables 11.1 and 11.2, respectively. The convergence behavior of the integral transform solution is examined for increasing truncation terms N W = 4, 8, 12 and 16 at t = 10, 20, and 30, respectively. For the dimensionless transverse displacement with v = 1.0 and k = 1.0, it can be observed that convergence is achieved essentially with a reasonably low truncation order (N ≤ 8). For a full convergence to three significant digits, more terms (e.g., N ≤ 12) are required. The results at t = 30 indicate that the excellent convergence behavior of the integral transform solution does not change with time, verifying the good long-time numerical stability of the scheme. For the dimensionless transverse displacement with v = 3.0 and k = 1.0, convergence to three significant digits is achieved with truncation order N ≤ 16, which demonstrates that the increasing of v can make the solution with a relatively slow convergence. For the same cases, the profiles of the transverse displacement at t = 20 are illustrated in Fig. 11.2 with different truncation orders, where it can be clearly seen that the convergence behavior of the integral transform solution for the case of v = 1.0 is better than the case of v = 3.0.

11.4 Results and Discussion

163

Table 11.2 Convergence behavior of the dimensionless transverse displacement w(x, t) of FG pipes conveying fluid for v = 3.0 and k = 1.0 x t = 10 0.1 0.3 0.5 0.7 0.9 t = 20 0.1 0.3 0.5 0.7 0.9 t = 30 0.1 0.3 0.5 0.7 0.9

NW = 4

NW = 8

N W = 12

N W = 16

0.0030463 0.0083268 0.0025960 −0.0013099 −0.0006775

0.0024006 0.0100507 0.0057039 −0.0021990 −0.0011992

0.0026071 0.0099890 0.0059138 −0.0022931 −0.0010060

0.0026145 0.0099936 0.0059497 −0.0023027 −0.0010314

−0.0019432 −0.0071319 −0.0055658 −0.0028723 −0.0005661

−0.0013335 −0.0088004 −0.0118234 −0.0073600 −0.0009220

−0.0015234 −0.0088014 −0.0123907 −0.0070684 −0.0011155

−0.0015166 −0.0088167 −0.0124569 −0.0071240 −0.0011267

0.0033760 0.0113849 0.0093804 0.0054698 0.0008500

0.0043476 0.0171511 0.0150589 0.0066738 0.0005556

0.0041930 0.0168645 0.0157554 0.0066753 0.0008388

0.0041604 0.0169404 0.0157545 0.0066903 0.0009427

11.4.2 Verification of the Solution The influence of fluid velocity on the first five dimensionless natural frequencies of axially FG pipes conveying fluid with the power-law exponent k = 1.0 is presented in Table 11.3. To obtain the natural circular frequencies for the transverse vibration of the system, the coupled ODEs, Eq. (11.13), can be represented in the matrix form as follows: ¨ ˙ Mw(t) + Cw(t) + Kw(t) = F(t).

(11.15)

The fluid velocities of 0, 10, 20, 30, 40, and 50 m/s are considered, and with the increasing of the velocity, all of the five natural frequencies of the system decrease. To demonstrate the validity and accuracy of the proposed GITT approach, GITT solution for dimensionless free vibration frequencies for clamped–clamped pipes with mf = 0 and k = 0 is calculated to compare with the results presented in the literature (Leissa and Qatu 2011), where the excellent agreement between them can be found, as shown in Table 11.3.

164 Fig. 11.2 GITT solutions with different truncation orders N W for the dimensionless transverse displacement profiles (a) w(x, t)|t=20 for v = 1.0 and (b) w(x, t)|t=20 for v = 3.0 of FG pipes conveying fluid

11 Axially Functionally Graded Pipes Conveying Fluid

11.4 Results and Discussion

165

Table 11.3 Influence of fluid velocity on the first five dimensionless natural frequencies of FG pipes conveying fluid (k = 1.0) fluid velocity (m/s) 0 0 0 10 20 30 40 50

ω1 22.373a 22.373b 26.866 26.850 26.801 26.720 26.605 26.457

ω2 61.673a 61.673b 74.323 74.306 74.255 74.171 74.052 73.899

ω3 120.903a 120.903b 145.936 145.918 145.865 145.777 145.653 145.493

ω4 199.859a 199.859b 241.576 241.558 241.506 241.419 241.296 241.139

ω5 298.556a 298.556b 361.572 361.556 361.508 361.428 361.315 361.171

a Dimensionless

free vibration frequencies for clamped–clamped beams (Leissa and Qatu 2011) solution for dimensionless free vibration frequencies for clamped–clamped pipes (mf = 0 and k = 0)

b GITT

11.4.3 Parametric Study In this section, transverse displacement of axially FG pipes conveying fluid with clamped–clamped boundary conditions is analyzed to illustrate the applicability of the proposed approach. Different values of the mass ratio β, the Young’s modulus ratio Eratio , the material distribution k, the dimensionless flow velocity v are chosen to assess their effects on the dynamic behavior of the system. In the following analysis, we use a relative high truncation order, NW = 16, for a sufficient accuracy.

11.4.3.1

The Effect of Young’s Modulus Variation

The first three dimensionless natural frequencies of the axially FG pipe conveying fluid for different Young’s modulus ratios, power-law exponent, and mass ratios are tabulated in Tables 11.4, 11.5, and 11.6, where the following parameters are adopted: El = 70 GPa, Eratio = EL /ER , ρf = 1000 kg/m3 , ρratio = ρL /ρR = 1.0 and ρL /ρf = 4.0. The mass ratio is calculated by Eq. (11.2) with the specified value of pipe thickness t and d = 19.65 mm. The flow velocity of the fluid is 20 m/s. For the specified modulus ratio and power exponent, all of the first three dimensionless natural frequencies increase with the decrease of the mass ratio. The effect of the modulus ratios on the fundamental frequency of fluid-conveying pipe with the mass ratio of 0.796 is presented in Fig. 11.3. It is observed that, the fundamental frequency decreases significantly with increasing of modulus ratio especially for large power exponent. On the other hand, no significant changes can be seen in the fundamental frequency for different modulus ratios for lower value of power exponent. Figure 11.4 illustrates the variation of the fundamental frequency with the power exponent for fluid-conveying pipe with the mass ratio of 0.796, which shows that the increase in power exponent causes the increase in frequency for Eratio < 1, the decrease in frequency for Eratio > 1 and no changes occur for Eratio = 1.

β 0.796

0.672

0.587

t (mm) 1.0

2.0

3.0

Eratio 0.25 0.50 1.00 2.00 4.00 0.25 0.50 1.00 2.00 4.00 0.25 0.50 1.00 2.00 4.00

k = 0.0 22.1909 22.1909 22.1909 22.1909 22.1909 22.3008 22.3008 22.3008 22.3008 22.3008 22.3333 22.3333 22.3333 22.3333 22.3333

k = 0.1 25.2702 23.3497 22.1909 21.5289 21.1690 25.3695 23.4554 22.3008 21.6413 21.2828 25.3989 23.4867 22.3333 21.6746 21.3165

k = 0.2 27.1859 24.1498 22.1909 20.9829 20.2790 27.2788 24.2525 22.3008 21.0977 20.3968 27.3063 24.2829 22.3333 21.1317 20.4318

k = 0.5 30.5936 25.6193 22.1909 19.8618 18.3220 30.6755 25.7159 22.3008 19.9830 18.4519 30.6998 25.7445 22.3333 20.0189 18.4904

k = 1.0 33.4550 26.8588 22.1909 18.8849 16.5444 33.5282 26.9498 22.3008 19.0138 16.6913 33.5499 26.9767 22.3333 19.0520 16.7348

k = 2.0 36.1690 28.0211 22.1909 17.9785 14.9059 36.2347 28.1068 22.3008 18.1167 15.0763 36.2543 28.1322 22.3333 18.1575 15.1266

k = 5.0 39.0072 29.1967 22.1909 17.0936 13.3283 39.0667 29.2778 22.3008 17.2418 13.5273 39.0844 29.3018 22.3333 17.2856 13.5858

k = 10.0 40.4687 29.7777 22.1909 16.6791 12.5955 40.5260 29.8571 22.3008 16.8313 12.8067 40.5430 29.8807 22.3333 16.8762 12.8688

Table 11.4 The variation of the first natural angular frequencies for different material distributions k and mass ratios β when flow velocity is 20 m/s, Eratio = El /Er , ρratio = ρl /ρr = 1.0 and ρl /ρf = 4.0

166 11 Axially Functionally Graded Pipes Conveying Fluid

β 0.796

0.672

0.587

t (mm) 1.0

2.0

3.0

Eratio 0.25 0.50 1.00 2.00 4.00 0.25 0.50 1.00 2.00 4.00 0.25 0.50 1.00 2.00 4.00

k = 0.0 61.5004 61.5004 61.5004 61.5004 61.5004 61.5959 61.5959 61.5959 61.5959 61.5959 61.6276 61.6276 61.6276 61.6276 61.6276

k = 0.1 69.4713 64.4750 61.5004 59.8213 58.9166 69.5577 64.5668 61.5959 59.9190 59.0156 69.5864 64.5973 61.6276 59.9515 59.0485

k = 0.2 74.6411 66.6003 61.5004 58.4126 56.6441 74.7221 66.6895 61.5959 58.5124 56.7467 74.7490 66.7191 61.6276 58.5456 56.7809

k = 0.5 84.2515 70.6875 61.5004 55.3901 51.4604 84.3233 70.7715 61.5959 55.4955 51.5737 84.3471 70.7995 61.6276 55.5305 51.6114

k = 1.0 92.8974 74.4028 61.5004 52.5094 46.2746 92.9617 74.4821 61.5959 52.6216 46.4032 92.9830 74.5085 61.6276 52.6589 46.4459

k = 2.0 101.469 78.0533 61.5004 49.6838 41.1824 101.527 78.1281 61.5959 49.8040 41.3315 101.546 78.1529 61.6276 49.8439 41.3809

k = 5.0 110.495 81.7840 61.5004 46.9156 36.2901 110.547 81.8547 61.5959 47.0441 36.4620 110.564 81.8782 61.6276 47.0868 36.5192

k = 10.0 114.810 83.5138 61.5004 45.7453 34.2703 114.860 83.5830 61.5959 45.8770 34.4515 114.876 83.6060 61.6276 45.9207 34.5117

Table 11.5 The variation of the second natural angular frequencies for different material distributions k and mass ratios β when flow velocity is 20 m/s, Eratio = El /Er , ρratio = ρl /ρr = 1.0 and ρl /ρf = 4.0

11.4 Results and Discussion 167

β 0.796

0.672

0.587

t (mm) 1.0

2.0

3.0

Eratio 0.25 0.50 1.00 2.00 4.00 0.25 0.50 1.00 2.00 4.00 0.25 0.50 1.00 2.00 4.00

k = 0.0 120.726 120.726 120.726 120.726 120.726 120.822 120.822 120.822 120.822 120.822 120.855 120.855 120.855 120.855 120.855

k = 0.1 135.934 126.374 120.726 117.564 115.872 136.020 126.466 120.822 117.663 115.972 136.049 126.498 120.855 117.697 116.006

k = 0.2 145.948 130.466 120.726 114.895 111.598 146.028 130.555 120.822 114.996 111.702 146.056 130.586 120.855 115.031 111.738

k = 0.5 164.997 138.522 120.726 109.017 101.632 165.068 138.606 120.822 109.124 101.747 165.092 138.635 120.855 109.160 101.786

k = 1.0 182.693 146.073 120.726 103.185 91.1682 182.757 146.153 120.822 103.298 91.2969 182.779 146.180 120.855 103.336 91.3409

k = 2.0 200.618 153.635 120.726 97.3618 80.6355 200.677 153.711 120.822 97.4820 80.7826 200.697 153.737 120.855 97.5231 80.8330

k = 5.0 219.333 161.360 120.726 91.6997 70.6481 219.386 161.432 120.822 91.8275 70.8185 219.405 161.457 120.855 91.8713 70.8770

k = 10.0 227.933 164.876 120.726 89.3743 66.6398 227.984 164.946 120.822 89.5047 66.8177 228.002 164.970 120.855 89.5494 66.8789

Table 11.6 The variation of the third natural angular frequencies for different material distributions k and mass ratios β when flow velocity is 20 m/s, Eratio = El /Er , ρratio = ρl /ρr = 1.0 and ρl /ρf = 4.0

168 11 Axially Functionally Graded Pipes Conveying Fluid

11.4 Results and Discussion

169

Fig. 11.3 Variation of the dimensionless frequency (ω1 ) with Young’s modulus ratio

Fig. 11.4 Variation of the dimensionless frequency (ω1 ) with power exponent (k)

11.4.3.2

The Effect of Material Distribution

To examine the effect of material distribution on the frequencies of the axially FG pipe conveying fluid, the integral transform solutions are obtained based on the material properties given in Sect. 11.4.1 and the fluid velocity of 20 m/s. The first five natural angular frequencies for different mass ratios and power exponents are reported in Table 11.7. All of the natural frequencies (ω1 , ω2 ,. . . , ω5 ) increase with the power exponent for the specified mass ratio. Note that the natural frequencies increase with the decrease of the mass ratio for the case of k = 0.0, which means the pipe is made of single-component, aluminum. However, when considering the

β 0.796

0.672

0.587

t (mm) 1.0

2.0

3.0

ωi i=1 i=2 i=3 i=4 i=5 i=1 i=2 i=3 i=4 i=5 i=1 i=2 i=3 i=4 i=5

k = 0.0 22.1835 61.5036 120.732 199.693 298.392 22.2976 61.5972 120.825 199.781 298.477 22.3317 61.6283 120.856 199.812 298.508

k = 0.1 23.8868 65.7479 128.668 212.486 317.258 23.8315 65.3592 127.785 210.932 314.839 23.7692 65.1117 127.252 210.013 313.428

k = 0.2 24.9026 68.3906 133.700 220.721 329.420 24.6962 67.5855 131.993 217.798 324.951 24.5496 67.1047 131.000 216.115 322.400

k = 0.5 26.4715 72.7928 142.428 235.305 351.361 25.9020 71.0306 138.872 229.340 342.322 25.5605 70.0359 136.890 226.033 337.340

k = 1.0 27.4513 76.1318 149.573 247.679 370.905 26.4864 73.3695 144.125 238.617 357.127 25.9443 71.8650 141.179 233.735 349.740

k = 2.0 28.1186 79.0050 156.161 259.404 389.776 26.7497 75.2171 148.770 247.159 371.143 26.0152 73.2123 144.870 240.712 361.370

k = 5.0 28.8294 81.8223 162.514 270.748 407.194 27.1061 77.0180 153.121 255.195 383.736 26.2112 74.5377 148.272 247.171 371.651

k = 10.0 29.4111 83.2525 165.285 275.362 413.686 27.5738 78.0331 155.007 258.328 388.144 26.6270 75.3643 149.752 249.620 375.089

Table 11.7 The variation of natural angular frequencies for different material distributions k and mass ratios β when flow velocity is 20 m/s

170 11 Axially Functionally Graded Pipes Conveying Fluid

11.5 Conclusions

171

Table 11.8 The variation of vibration amplitudes for different material distributions k and mass ratios β when flow velocity is 20 m/s t (mm) 1.0 2.0 3.0

β 0.796 0.672 0.587

k = 0.0 0.05064 0.05071 0.05076

k = 0.1 0.04742 0.04779 0.04794

k = 0.2 0.04562 0.04607 0.04657

k = 0.5 0.04265 0.04365 0.04431

k = 1.0 0.04067 0.04226 0.04315

k = 2.0 0.03966 0.04176 0.04276

k = 5.0 0.03864 0.04093 0.04237

k = 10.0 0.03787 0.04042 0.04192

Fig. 11.5 Variation of the dimensionless frequency (ω1 ) with dimensionless flow velocity (v)

FG material with the power exponents k = 0.1, 0.2,. . . , 10, the natural frequencies decrease with the decrease of the mass ratio. The variation of vibration amplitudes for different material distributions and mass ratios is listed in Table 11.8. It can be seen that the vibration amplitude decreases with the increase of the power exponent and increases with the decrease of the mass ratio. In addition, the effect of material distribution on the critical velocity of fluidconveying pipe with the material properties given in Sect. 11.4.1 is analyzed, as shown in Fig. 11.5, which exhibits the variation of the fundamental frequency with the flow velocity of fluid for different power exponents. It can be clear seen that the fundamental frequency decreases with the flow velocity for the specified power exponent, and the critical velocity (ω1 = 0) increases with the power exponent.

11.5 Conclusions The generalized integral transform technique (GITT) has proved in this chapter to be a good approach for the analysis of dynamic behavior of an axially FG pipe conveying fluid, providing an accurate numerical–analytical solution for the natural

172

11 Axially Functionally Graded Pipes Conveying Fluid

frequencies and transverse displacements. The investigation shows that the solutions converge to the values with three significant figures at a reasonable low truncation order N ≤ 12 for v = 1.0, and the increasing of v can make the solution with a relatively slow convergence. The numerical results obtained are in good agreement with the ones presented in the literature. The parametric studies indicate that the fundamental frequency decreases significantly with increasing of modulus ratio especially for large power exponent, while no significant changes can be seen in the fundamental frequency for different modulus ratios for lower value of power exponent. The increase in power exponent causes the increase in frequency for Eratio < 1, the decrease in frequency for Eratio > 1 and no changes occur for Eratio = 1. The natural frequency increases with the power exponent, and the natural frequency decreases with the decrease of the mass ratio when considering the pipe is made of the FG material with the power exponents k = 0.1, 0.2,. . . , 10. The vibration amplitude decreases with the increase of the power exponent and increases with the decrease of the mass ratio. The critical velocity of fluid-conveying pipe increases with the power exponent. For future investigation, the proposed approach can be employed to predict the dynamic behavior of a transversally FG pipe conveying fluid, and for more general boundary conditions.

Part III

Thermal Analysis of Multilayer Pipelines

Keeping the thermodynamic state of produced fluid within adequate range is essential to meet the production requirements and minimize downtime due to possible pipeline blockage. For a given combination of hydrocarbons flowing into a multilayer pipeline, the range of pressure variation is largely determined by the wellhead and separator pressures, while the temperature distribution of the produced fluid is determined by the energy balance over the pipeline. For the thermal design of steady-state operation conditions, adequate thermal insulation system must be specified to meet the requirement of keeping the temperature above the wax appearance temperature. Under shut-in conditions, a reasonable long cooling-down time should be achieved to prevent the formation of gas hydrate. Part III of the book deals with the thermal analysis of multilayer composite pipelines. In Chap. 12, basic governing equations for thermal analysis of pipelines are presented, together with the overall heat transfer coefficient for multilayered composite pipeline and the temperature distribution of the produced fluid along the pipeline under steady-state conditions. Chapter 13 presents a global thermal analysis of multilayer composite pipelines and establishes the requirement of active heating when the passive thermal insulation cannot meet the thermal design requirement. Active heating by hot medium circulation and direct electrical heating are discussed. The challenge of heavy oil production is addressed in Chap. 14. Chapter 15 analyzes direct electrical heating at steady-state operation in more details. Finally, Chap. 16 presents transient thermal analysis of multilayer composite pipelines with direct electrical heating, using finite difference method and lumped models, respectively.

Chapter 12

Fundamentals of Thermal Analysis

The objective of thermal design of deepwater pipelines is to keep the produced hydrocarbons in a designed thermodynamic state to avoid the occurrence of undesired thermodynamic or transport processes, such as gas hydrate formation and paraffin deposition in the pipeline.

12.1 Equation of Energy The foundation of thermal analysis of pipelines is the first law of thermodynamics, that is, the law of conservation of energy, which can be expressed in the following rate form: dE ˙ − W˙ , =Q dt

(12.1)

˙ is the rate of heat transfer from the where E is the total energy of a system, Q environment into the system through the boundary surface of the system, and W˙ is the rate of the work done by the system to the environment. We first separate the total energy E of system into internal energy U and energy of other form Eg , d(U + Eg ) ˙ − W˙ , =Q dt

(12.2)

dEg dU ˙ − W˙ − =Q . dt dt

(12.3)

or

© Springer Nature Switzerland AG 2021 C. An et al., Structural and Thermal Analyses of Deepwater Pipes, https://doi.org/10.1007/978-3-030-53540-7_12

175

176

12 Fundamentals of Thermal Analysis

The rate of change of the other forms of energy is expressed as a heat generation term: ˙g = − Q

dEg . dt

(12.4)

Thus, the first law of thermodynamics for a system can be written as dU ˙ +Q ˙ g − W˙ . =Q dt

(12.5)

For flowing system with one inlet and one outlet such as a pipeline, we have the following equation for mass conservation: dM =m ˙ in − m ˙ out , dt

(12.6)

where M is the total mass in the open system or control volume, m ˙ in is the mass flow rate into the control volume, and m ˙ out is the mass flow rate exiting from the control volume. For steady-state flow system, we have dM = 0. dt

(12.7)

m ˙ in = m ˙ out = m. ˙

(12.8)

That is

For a flow system, the first law of thermodynamics is written as dU ˙ +Q ˙g + m =Q ˙ in uin − m ˙ out uout + pin Ain Vin − pout Aout Vout , dt

(12.9)

where u is the specific internal energy of the fluid, p is the pressure, A is crosssectional area, and V is the velocity of the fluid. The other forms of work are neglected. Using m ˙ = ρ A V,

(12.10)

p , ρ

(12.11)

and h=u+

where h is the specific enthalpy of the fluid, we rewrite the first law of the flowing system as: dU ˙ +Q ˙g + m ˙ in hin − m ˙ out hout . =Q dt

(12.12)

12.2 Heat Conduction in Pipe Wall

177

12.2 Heat Conduction in Pipe Wall 12.2.1 Heat Conduction in a Single-Layer Pipe Wall In this subsection, we consider one-dimensional, steady-state heat conduction in a single-layer pipe with convective boundary conditions at both inner and outer pipe walls, as illustrated in Fig. 12.1. The thermal conductivity of the pipe material k is considered constant. There is no volumetrical heat generation in the pipe. The governing heat conduction equation is written in cylindrical coordinate system as: k d r dr



dT r dr

 = 0,

r1 < r < ro ,

(12.13a)

with boundary conditions: dT = h1 (T − Tf ), at r = r1 , dr dT −k = ho (T − Ta ), at r = ro , dr k

(12.13b) (12.13c)

where h1 is the convective heat transfer coefficient at inner pipe wall r1 , ho is the convective heat transfer coefficient at outer pipe wall ro , and Tf and Ta are, respectively, the inner and outer fluid temperatures. Equation (12.13a) is integrated in r to get the heat flux: qr = −k

Fig. 12.1 Cross section of a single-layer pipe

A1 dT = . dr r

(12.14)

178

12 Fundamentals of Thermal Analysis

Equation (12.14) is integrated again in r to get the temperature distribution: T (r) = −

A1 ln r + A2 + Ta . k

(12.15)

It should be noted that for future convenience we separate the outside fluid temperature Ta from the second integration constant A2 . Equations (12.14) and (12.15) are used in the boundary conditions, Eqs. (12.13b) and (12.13c): −

A1 A1 ln r1 + A2 + Ta − Tf ), = h1 (− r1 k

(12.16a)

A1 A1 ln ro + A2 ). = ho (− ro k

(12.16b)

Equations (12.16a) and (12.16b) are solved for A1 and A2 : T f − Ta

A1 =  A2 =

1 r1 h1

1 ln ro + k ro ho



+

1 k

ln rro1 +

1 ro ho

(12.17a)

,

T f − Ta 1 r1 h1

+

ln rro1 +

1 k

1 ro ho

(12.17b)

.

The temperature distribution in the single-layer pipe is thus obtained as T (r) = −

=

Tf − Ta 1 r1 h1

+

1 k

ln rro1 +

1 ro ho



Tf − T a 1 r1 h1

+

1 k

ln

ro r1

+

1 ro ho

ln r + k



T f − Ta 1 r1 h1

+

1 k

1 ro 1 ln + k r ro ho

ln rro1 +

1 ro ho

ln ro 1 + k ro ho

 + Ta (12.18)

 + Ta .

The heat flux at the inner pipe wall r1 is obtained by using Eqs. (12.17a) and (12.17b) in Eq. (12.19): qr1 =

T f − Ta 1 r1 h1

+

1 k

ln

ro r1

+

1 ro ho

1 = r1

T f − Ta 1 h1

+

r1 k

ln rro1 +

r1 ro ho

= U (Tf − T a), (12.19)

where the global heat transfer coefficient U is given by U=

1 1 h1

+

r1 k

ln rro1 +

r1 ro ho

.

(12.20)

12.2 Heat Conduction in Pipe Wall

179

Fig. 12.2 Cross section of a M layer composite pipeline

12.2.2 Heat Conduction in a Multilayer Pipe Wall In this subsection, we extend the one-dimensional, steady-state heat conduction in a single-layer pipe to a multilayer composite pipe, consisted of M layers, as illustrated in Fig. 12.2. The inner and outer radii of i-th layer are ri and ri+1 , respectively. The thermal conductivity of the i-th layer material ki is considered constant. There is no heat generation in any of the layers. The pipe is subject to convective heat transfer at the inner pipe wall r1 and the outer pipe wall ro = rM+1 , with h1 being the convective heat transfer coefficient at inner pipe wall, ho the convective heat transfer coefficient at outer wall, and Tf and Ta , respectively, the inner and outer fluid temperatures. The governing heat conduction equation is written as ki d r dr



d Ti r dr

 = 0,

for

ri < r < ri+1 ,

i = 1, . . ., M

(12.21a)

r = r1 ,

(12.21b)

with inner and outer boundary conditions:

k1 −kM

d T1 = h1 (T1 − Tf ), dr

d TM = ho (TM − Ta ), dr

at

at

r = ro ,

(12.21c)

i = 1, . . ., M − 1,

(12.21d)

and interface boundary conditions:

ki

d Ti d Ti+1 = ki+1 , dr dr Ti = Ti+1 ,

at

at

r = ri+1 ,

r = ri+1 ,

for

for

i = 1, . . ., M − 1,

where Ti is the temperature in i-th layer of the pipe.

(12.21e)

180

12 Fundamentals of Thermal Analysis

Equation (12.21a) is integrated in r to get the heat flux: qi = −ki

Bi d Ti = , dr r

i = 1, . . ., M.

for

(12.22)

It can be seen from the continuity condition of the heat flux at the interfaces, Eq. (12.21d), that all the constants Bi take the same value: B 1 = B 2 = . . . = Bi = . . . = BM .

(12.23)

Equation (12.22) is integrated again in r to get the temperature distribution: Ti (r) = −

B1 ln r + Ci + Ta , ki

i = 1, . . ., M.

for

(12.24)

Equations (12.22), (12.23), and (12.24) are used in boundary conditions, Eqs. (12.21b) and (12.21c), and the temperature continuity conditions at the interfaces, Eqs. (12.21e) −

−

B1 B1 = h1 (− ln r1 + C1 + Ta − Tf ), r1 k1

(12.25)

B1 B1 = ho (− ln ro + Cm ). ro kM

(12.26)

B1 B1 ln ri+1 + Ci + Ta = − ln ri+1 + Ci+1 + Ta , ki ki+1 at r = ri+1 ,

for i = 1, . . ., M − 1.

(12.27)

The system of M + 1 linear equations formed by Eqs. (12.25), (12.26), and (12.27) is solved for the M + 1 unknown constants, B1 and Ci , i = 1, . . ., M: Tf − Ta , M 1 ri+1 + i=1 ki ln ri + ro1ho   1 ln ro B1 + CM = k ro ho   1 1 , i = M − 1, . . ., 1. − Ci = Ci+1 + B1 ln ri+1 ki ki+1 B1 =

1 r1 h1

(12.28a)

(12.28b) (12.28c)

The heat flux at the inner pipe wall r1 is obtained by using Eqs. (12.17a) and (12.17b) in Eq. (12.22): qr1 =

B1 = r1

1 r1 h1

Tf − Ta M 1 ri+1 + i=1 ki ln ri +

1 ro ho

1 = U (Tf − T a), r1

(12.29)

12.3 Steady-State Temperature Distribution of Produced Fluid in an Unheated. . .

181

where the global heat transfer coefficient U for a M-layer composite pipe is given by U=

1 h1

+

M

1

r1 i=1 ki

ln ri+1 ri +

r1 ro ho

.

(12.30)

12.3 Steady-State Temperature Distribution of Produced Fluid in an Unheated Pipeline Let us consider the temperature distribution of produced fluid in an unheated pipeline. At steady state, the one-dimensional energy transport in the produced fluid is uncoupled from the heat conduction in the pipeline wall, and governed by the following equation: m ˙ cpf

d Tf = −2π r1 U (Tf − T a), ds

for

0 < s < L,

(12.31)

with the inlet condition, Tf (0) = Tf,in ,

at

s = 0,

(12.32)

where m ˙ is the mass flow rate of the produced fluid, cpf the specific heat, and Tf,in the temperature of the produced fluid at the inlet of the pipeline. The global heat transfer coefficient U for a M-layer composite pipe is given by Eq. (12.30). Equation (12.31) is readily solved with Eq. (12.32) to obtain the temperature distribution of the produced fluid along the pipeline:   2π r1 U s . Tf (s) = Ta + (Tf,in − Ta ) exp − m ˙ cpf

(12.33)

Chapter 13

Steady-State Thermal Analysis

13.1 Introduction Flow assurance in deepwater conditions is an essential part of pipeline technology for offshore oil and gas production (Harrison and Herring 2000). Flow assurance requires a carefully considered plan to insure that (1) subsea well production can be as nearly continuous as feasible during normal operation, with minimum hydrate formation and wax deposition and (2) production can be rapidly suspended when necessary and later resumed with minimum difficulty and downtime. With increasing water depth and tie-back distance, pipeline insulation has turned to be mandatory in all deepwater developments. Active heating is required when passive thermal insulation alone is not sufficient to prevent wax deposition and hydrate formation. Two methods of active heating of deepwater pipelines have been studied in recent years: electrical heating and heating by a circulating heat medium. Norwegian oil companies, cable manufacturers, and pipeline installation companies have conducted studies on electrical heating of multiphase subsea pipelines and risers to prevent hydrate formation and wax plugs (Lervik et al. 1997, 1998; Halvorsen et al. 2000). The evaluation of technical feasibility and cost estimates have been done for a 50 Hz direct resistive heating system and for a system based on electromagnetic induction. The electrical rating of the systems depends on the heat requirement, pipe material, and the pipeline. The feasibility of the concepts has been verified through full-scale subsea tests. Results from the measurements were used to determine the characteristic parameters of the two systems on fields in the North sea. British Petroleum (BP) planned to develop the King project in Mississippi Canyon Block 85 (MC85) via a dual well, subsea production scheme (Harrison and Herring 2000). The active heating system employs hot water circulating in the closed loop PIP (Pipe-in-Pipe) annulus. External insulation around the jacket pipe and burial/backfill provide thermal insulation of the flowlines. Heat is extracted from the exhaust waste heat of three, 3.9 MW, electrical generator turbine drivers. © Springer Nature Switzerland AG 2021 C. An et al., Structural and Thermal Analyses of Deepwater Pipes, https://doi.org/10.1007/978-3-030-53540-7_13

183

184

13 Steady-State Thermal Analysis

It is mandatory that passive thermal insulation be used in all deepwater flowlines for multiphase production of oil and gas as it is the most economical method to prevent hydrate formation and wax deposition. The decision as which additional flow assurance measures is suitable for a particular field development depends on engineering, economical, and environmental considerations. The electrical heating method requires additional electrical power generating capacity in the platform, for 10–30 MW depending on system configuration. No significant difficulties are expected for the mechanical design and the installation of the pipeline with electrical heating. On the other hand, the heated water circulation (HWC) system is superior from a thermodynamic point of view as discharged waste heat from the turbine drivers can be recovered without additional energetic requirement. However, the mechanical design and the installation are more expensive and involved. In this chapter, we first present a global heat balance analysis of typical deepwater pipelines. We show that active heating is required for long pipelines due to technical limitations of passive thermal insulation systems. We then examine two methods of combined active heating and passive insulation of pipelines. For the heated water circulation in pipe-in-pipe method (HWC+PIP), we provide a closed-form analytical solution for the temperature distributions of the produced fluid and heat medium under certain simplifying hypothesis. Finally, we consider the active heating by electrical resistance on the inner steel pipe. For this case, we propose an optimum heating method to minimize electrical power requirement for a given pipeline configuration and desired minimum temperature of the produced fluid. Numerical results are shown for the last case. Significant reduction in power requirement is achieved.

13.2 Global Heat Balance Analysis Let us consider a typical insulated pipeline for deepwater production of oil and gas. Normally, the desired thermal profile of the produced fluid and the overall heat transfer coefficient are used to specify an adequate pipeline configuration. The overall heat transfer coefficient for a multilayered composite pipeline is given by U=

1 h1

+

N

r1 i=1 ki

1 ln ri+1 ri +

r1 rN+1 ha

,

(13.1)

where ri and ri+1 are the inner and outer radii of the i-th layer, respectively, ki the thermal conductivity, h1 is the heat transfer coefficient between the innermost layer and the produced fluid flowing inside it, and ha is the heat transfer coefficient between the outermost layer and the ambient fluid. By the first law of thermodynamics, the global energy balance of the produced fluid can be written as

13.2 Global Heat Balance Analysis

M˙ f cp,f (Tf,in − Tf,out ) = U 2π r1 LΔTm ,

185

(13.2)

where M˙ f is the mass flow rate of the produced fluid, cp,f is the specific heat capacity, Tf,in and Tf,out are the inlet (wellhead) and the outlet (TLP) temperatures of the produced fluid, respectively, r1 is the inner diameter of the innermost pipe, L is the total length of the pipeline. The average temperature difference between the produced fluid and the environmental fluid is defined as ΔTm =

(Tf,in − Tf,out ) . ln [(Tf,in − Ta )/(Tf,out − Ta )]

(13.3)

For a given inner radius of the pipeline r1 , a given temperature of the environmental fluid, and a desired thermal profile (Tf,in , Tf,out and thus ΔTm ), the maximum pipeline length Lmax that can meet the thermal profile requirement is a function of the overall heat transfer coefficient U and the heat duty of the produced fluid M˙ f cpi . From Eq. (13.2), we solve Lmax =

M˙ f cp,f (Tf,in − Tf,out ) M˙ f cp,f = ln [(Tf,in − Ta )/(Tf,out − Ta )]. U 2π r1 ΔTm U 2π r1 (13.4)

As the heat duty of the produced fluid, M˙ f cp,f , is usually given by the field develop plan, the maximum length of a pipeline that can meet the thermal profile requirement is inversely proportional to the overall heat transfer coefficient of the pipeline, U . The smaller the overall heat transfer coefficient, the longer the pipeline length up to which the outlet temperature of the produced fluid can meet the flow assurance requirement. However, there are technical limitations on reducing further the overall heat transfer coefficient. For a pipeline with a length L larger than Lmax given by Eq. (13.4), passive thermal insulation alone is not sufficient to keep the temperature of the produced fluid above a required minimum. Active heating is then required in combination with the passive thermal insulation. If active heating is used, for a given pipeline length L, we can determine the power requirement of active heating that is necessary to meet the desired thermal profile. The global energy balance is written as, for this case ˙ = U 2π r1 LΔTm , M˙ f cp,f (Tf,in − Tf,out ) + Q

(13.5)

˙ stands for the total power requirement of active heating. We can solve Q ˙ where Q from Eq. (13.5) ˙ = U 2π r1 LΔTm − M˙ f cp,f (Tf,in − Tf,out ) Q

(13.6)

Obviously, if all other parameters are the same, the longer the pipeline, the larger the power requirement.

186

13 Steady-State Thermal Analysis

The average linear heat input to the pipeline can be calculated as q˙av =

˙ M˙ f cp,f (Tf,in − Tf,out ) Q = U 2π r1 ΔTm − . L L

(13.7)

13.3 Analysis of Heat Medium Circulation We consider here an active heating, dual production flowline system similar to that considered by Harrison and Herring (2000). The active heating system employs heated water circulating in the closed loop pipe-in-pipe (PIP) annulus. External insulation around the jacket pipe provides passive thermal insulation of the pipeline. For the downflow leg in which the heated water flows downward and the produced fluid flows upward, a mathematical model for one-dimensional heat transport in the produced fluid and the heated water is written as M˙ f cp,f

dTf (x) = −Uf h 2π r1 L(Tf − Th ) dx

(13.8)

dTh (x) M˙ h cp,h = −Uf h 2π r1 L(Tf − Th ) dx + Uha 2π r1 L(Th − Ta )

(13.9)

Similarly, for the upflow leg in which both fluids flow upward, the onedimensional heat transport equations are M˙ f cp,f

dTf (x) = −Uf h 2π r1 L(Tf − Th ) dx

(13.10)

dTh (x) M˙ h cp,h = −Uf h 2π r1 L(Tf − Th ) dx + Uha 2π r1 L(Th − Ta ),

(13.11)

where Th (x) is the temperature of the heated water, M˙ h its mass flow rate and cp,h is the specific heat capacity. The ordinary differential equations are to be solved with two boundary conditions Tf (0) = Tf,wh

(13.12)

Th (L) = Th,T LP

(13.13)

13.3 Analysis of Heat Medium Circulation

187

Numerical methods for boundary value problems of ordinary differential equations such as shooting method and Runge–Kutta method may be applied to solve the equations. We present here a closed-form analytical solution of the system of ordinary differential equations, which is inspired by the works of Prof. Peter Heggs of UMIST on heat exchangers. The downflow problem is to be solve first which consists of Eqs. (13.8) and (13.9). We introducing the temperature difference to the environmental fluid as new dependent variables θ1 = T f − T a

(13.14)

θ2 = Th − Ta

(13.15)

and rewrite the governing equations and boundary conditions in a more convenient form dθ1 = a11 θ1 + a12 θ2 dx

(13.16)

dθ2 = a21 θ1 + a22 θ2 dx

(13.17)

θ1 (0) = Tf,wh − Tm

(13.18)

θ2 (L) = Th,T LP − Tm

(13.19)

where the coefficients are defined by a11 = −

Uf h 2π r1 L M˙ f cp,f

a12 = −a11 a21 =

−Uf h 2π r1 L M˙ h cp,h

a22 = −a21 +

Uha 2π r1 L M˙ h cp,h

(13.20) (13.21) (13.22)

(13.23)

The general solution of this system of homogeneous ordinary differential equations with constant coefficients is readily obtained as θ1 (x) = c1 ξ11 eλ1 x + c2 ξ21 eλ2 x

(13.24)

188

13 Steady-State Thermal Analysis

θ2 (x) = c1 ξ12 eλ1 x + c2 ξ22 eλ2 x ,

(13.25)

where λ is the eigenvalue and ξ is the eigenvector. The constants c1 and c2 are determined by using the boundary conditions (13.18) and (13.19). Once the downflow problem is solved, the upflow flow is straightforward as it is a standard initial value problem with inlet temperatures of both produced fluid and heated water. An important application of the closed-form analytical solution is in integrated thermal–structural analysis of the pipe-in-pipe system (Solano 2001). Based on the analytical solution of the produced fluid and the heated water, the temperature fields in both pipes are readily obtained, also in analytical form.

13.4 Analysis of Direct Electrical Heating In this section we consider the direct electrical heating of a multilayered composite pipeline. Assuming that the thermophysical properties of the produced fluid and the pipeline materials are constant, we can write the one-dimensional energy transport equation of the produced fluid as M˙ f cp,f

dTf (x) = −U 2π r1 L(Tf − Tm ) + q(x), ˙ dx

(13.26)

where q(x) ˙ is the linear heat input rate of direct electrical heating to the pipeline. For a given specification of the linear heat input, the temperature distribution along the pipeline can be readily obtained by solving Eq. (13.26). We are interested in determining an optimized linear heat rate distribution q(x) ˙ that minimizes the total energy consume but yet maintains the temperature of the produced fluid above a specified minimum. Instead of formulating a constrained, continuous optimization problem which is complex to be solved, we appeal to physical insights to obtain an engineering solution of the problem. We propose only begin to heat the pipe when the temperature of the produced fluid reaches a pre-specified minimum temperature, Tmin . The heating is to keep the produced fluid temperature at a constant temperature, this is, Tmin . The required linear heating rate is obtained by equating the right-hand side of Eq. (13.26) to zero q˙ = U 2π r1 L(Tmin − Tm )

(13.27)

13.5 Results and Discussion

189

13.5 Results and Discussion In lack of more realistic pipeline and production well data, we simulated an hypothetical pipeline with a length of 27 km and an inner diameter of 6 in.(0.1524 m). The overall heat transfer coefficient of the pipeline is 5.35 W/m2 K based on the internal pipeline diameter. The wellhead temperature of produced fluid is taken as 76.0 ◦ C and the sea water is 4.0 ◦ C. A mass flow rate of 14.72 kg/s is assumed with a constant density of 800.0 kg/m3 and a specific heat of 2700.0 J/kg ◦ C. The required minimum temperature of the produced fluid is 30.0 ◦ C. In Fig. 13.1 we show a comparison of the temperature distribution obtained by uniform heating of all pipeline length and the temperature distribution obtained by the optimum heating method which is our proposal. By the first method, a linear heat rate of 41.52 W/m is required to keep the temperature of produced fluid above the pre-specified minimum, Tmin = 30.0 ◦ C. The total power requirement is 1.12 MW. By the proposed heating method, no heat input is given for the first 15,804.7 meters of the pipeline. From that point onward, a linear heat rate of 66.60 W/m is required to keep the temperature of produced fluid above the pre-specified minimum, Tmin = 30.0 ◦ C. The total power requirement now is 745.6 kW, which represents a 33.4% reduction in the heating power requirement. It can be seen clearly from Fig. 13.1 that by the first method the produced fluid is unnecessarily overheated as the area between the two curves represents the lost energy to the environment. From a thermodynamic point of view, the energy requirement is minimized if the temperature difference between the produced fluid and the sea water be minimized, which is achieved by no heating the first part of the pipeline in our method. Fig. 13.1 Comparison of two heating methods

80.0

60.0

Tf (C)

40.0

20.0

Heating all pipeline length Heating when Tf = Tmin

0.0 0

3000 6000 9000 12000 15000 18000 21000 24000 27000 x (m)

190

13 Steady-State Thermal Analysis

13.6 Conclusions In this chapter, we first present a global heat balance analysis of typical deepwater pipelines for oil and gas production and show that active heating is necessary for long pipelines due to technical limitations of passive thermal insulation systems. Two methods of combined active heating and passive insulation are then considered. By the first method, active heating is provided by circulating heat medium in annulus. Under certain simplifying hypothesis, we provide a closed-form analytical solution for the temperature profiles of the produced fluid and heat medium. By the second method, active heating is provided by electrical resistance on the inner steel pipe. For this case, we propose a heating method that minimizes the power requirement for a given minimum temperature of produced fluid. Numerical results are shown for the two thermal insulation systems. For the example shown in this chapter, the proposed heating method reduces by 33.4% from the amount if uniform, all through heating is adopted. We conclude that our heating method will significantly reduce the power requirement of flow assurance by direct electrical heating.

Chapter 14

Steady-State Analysis of Heavy Oil Transportation

14.1 Introduction The tremendous increase of oil demand and the depletion of low-viscosity oils have led to the rapid development of very large world resources of heavy oils. Heavy crudes account for a large fraction of the potentially recoverable oil reserves in the world. Production of heavy crudes is expected to increase significantly in the near future as low-viscosity crudes are being depleted. Production and transport of these heavy oils have been challenging because of their very high viscosities. The viscosities of these crudes at 25 ◦ C vary from 1000 cP to more than 100,000 cP. Conventional pipelining is not suitable for transporting these heavy crudes from the reservoir to the refinery because of the high viscosities involved. Generally, crude viscosities less than 200 cP are desired for pipelining. Several alternative transportation methods were proposed, including dilution with lighter crudes, injection of a water sheath around the viscous crude, or preheating the crude with subsequent heating of the pipeline. Each of these methods may have logistic, technical, or economic drawbacks for a given application. Another promising pipeline technique is the transport of viscous crudes as concentrated oil-in-water emulsions. The technical viability of this method was demonstrated in an Indonesian pipeline in 1963 and in a 13 mile long, 8 inch diameter pipeline in California. This chapter presents the use of electrically heated pipelines for heavy oil transportation. In the past years, active electrical heating has been studied as a feasible alternative for oil and gas production in deep and ultra-deepwater, especially when associated with multilayered composite pipelines (Su 2003). Laoutr and Denntel (2001) presented a heated pipe-in-pipe system in which heating is generated by resistive cables using low voltage, which complements the insulating performance of pipe-in-pipe. Su and Cerqueira (2001) developed a mathematical model for analysis of transient heat transfer in multilayered composite pipeline during warm-up and cool-down. Su et al. (2002) presented a thermal analysis of combined active heating and passive insulation of deepwater pipelines and proposed © Springer Nature Switzerland AG 2021 C. An et al., Structural and Thermal Analyses of Deepwater Pipes, https://doi.org/10.1007/978-3-030-53540-7_14

191

192

14 Steady-State Analysis of Heavy Oil Transportation

a heating method that minimizes the power requirement for a given minimum temperature of produced fluid. Su et al. (2003) proposed a new concept to combine thermal insulation and active heating by inserting strips of electrical resistance into the sandwich pipes. Initially a global heat balance analysis to determine the energy input requirement was carried out and then a steady-state thermal analysis to determine the temperature distribution in a cross of the sandwich pipes under typical production conditions in ultra-deepwater. A mathematical model was developed for the analysis of steady-state heat transfer in the sandwich pipes. The analysis of transient heat transfer in the sandwich pipelines with active electrical heating was presented by Cerqueira et al. (2004). The mathematical model governing the heat conduction in the composite pipeline and the energy transport in the produced fluid was solved by using finite difference methods. As unplanned cool-down of the pipelines is most critical to safe and economical operation of pipelines in deep and ultra-deepwater conditions, numerical results of computational simulation of cool-down for three sandwich pipeline configurations under typical production conditions were also presented. In this chapter, we present a simplified thermal-hydraulic analysis of heated sandwich pipelines for heavy oil transportation. The temperature distribution of the produced fluid is determined by solving the energy transport equation for different heating rates. The pressure drop across the pipeline length is then determined by integrating the local pressure gradient as a function of the local temperature of the produced fluid. We show through a numerical example that active heating can greatly enhance the production rate of a pipeline for a given pressure drop.

14.2 Analysis We consider the transport of a heavy oil in a multilayer composite pipeline with direct electrical heating. The thermodynamic and transport properties of the produced fluid are specified through the density ρf , the thermal conductivity k(T ), the specific heat cp,f (T ), and the viscosity μ(T ). The geometrical configuration and material properties of the multilayered composite pipeline are specified. The composite medium consists of n concentrically cylindrical layers, as shown in Fig. 12.2. Each layer is assumed to be homogeneous, isotropic, and with constant thermal properties. The adjacent layers are assumed to be in perfect thermal contact. The overall heat transfer coefficient for a multilayered composite pipeline is defined as U=

1 h1

+

N

r1 i=1 ki

1 ln ri+1 ri +

r1 rN+1 ha

,

(14.1)

where ri and ri+1 are the inner and outer radii of the i-th layer, respectively, ki is the thermal conductivity, h1 is the heat transfer coefficient between the innermost

14.2 Analysis

193

layer and the produced fluid flowing inside it, and ha is the heat transfer coefficient between the outermost layer and the environmental fluid. The one-dimensional energy transport equation of the produced fluid is written as M˙ f cp,f (T )

dTf (z) ˙ = −U 2π r1 (Tf − Tm ) + q(z), dz

0