Solid-Liquid Two-Phase Flow in Centrifugal Pump 9819918219, 9789819918218

Table of contents :
Contents
1 Introduction
1.1 Two-Phase Flow Model and Calculation Method
1.2 Two-Phase Transport Performance and Prediction
1.3 Two-Phase Wear Properties
References
2 Solid–Liquid Two-Phase Calculation Model and Method
2.1 Solid–Liquid Two-Phase Model
2.1.1 Analysis of the Force of Moving Particles in Fluids
2.1.2 Calculation Model of Solid–Liquid Two-Phase Flow
2.1.3 Model Suitability Study
2.2 Collision Rebound Model
2.2.1 Wet and Dry Wall Collision Rebound
2.2.2 Rotating Wall Collision Rebound
References
3 Wear Characteristics of Static Walls
3.1 Calculation Method and Experiment
3.1.1 Computational Method and Geometric Model
3.1.2 Experimental System and Measurement Method
3.1.3 Results Comparison and Calculation Method Verification
3.2 Particle Motion and Wear Characteristics in the Bend Pipe
3.2.1 Analysis of Particle Motion in the Bend Pipe
3.2.2 Analysis of Wall Wear Characteristics
3.3 Effect of Wall Shape Change on Wear
3.3.1 Wear Analysis of Wall Surface with Bump in Bend Pipe
3.3.2 Wear Analysis of Wall Surface with Groove in Bend Pipe
References
4 Two-Phase Flow Characteristics and Transportation Performance in Centrifugal Pump
4.1 Calculation of Two-Phase Flow in Pump Based on Mixture Model
4.1.1 Calculation Scheme and Grid
4.1.2 Different Solid Particle Sizes Conditions
4.1.3 Different Solid Concentration Conditions
4.1.4 Different Flow Conditions
4.1.5 Different Solid Density Conditions
4.2 Two Phase Flow Calculation in Pump Based on DPM Model
4.2.1 Calculation Scheme
4.2.2 Different Solid Particle Sizes Conditions
4.2.3 Different Solid Concentration Conditions
4.2.4 Different Flow Conditions
4.3 Two Phase Flow Calculation in Pump Based on DEM Model
4.3.1 Calculation Method and Scheme
4.3.2 Analysis of Particle Motion Characteristics
4.3.3 Influence of Particle Mass Concentration on Flow Field in Centrifugal Pump
Reference
5 Wear Characteristics of the Wall Surface in Centrifugal Pump
5.1 Solid–Liquid Two-Phase Transport Experiment and Calculation Method Verification
5.1.1 Experimental Apparatus
5.1.2 Grid and Calculation Settings
5.1.3 Calculation Method Verification
5.2 Wear of the Centrifugal Pump Runner
5.2.1 Impeller Wear Analysis
5.2.2 Volute Wear Analysis
5.2.3 Wear Analysis of Front and Rear Wear Plates
5.3 Wear of the Pipeline Matched with Centrifugal Pump
5.3.1 Vertical Pipe Wear Analysis
5.3.2 Bend Pipe Wear Analysis
5.3.3 Horizontal Pipe Wear Analysis
5.3.4 Particle Motion Analysis Inside Pipes
5.4 Wear of the Flow Channel for Conveying Mixed Size Particle
5.4.1 Calculation Method and Scheme
5.4.2 Calculation Method Verification
5.4.3 Runner Wear Analysis
5.4.4 Relationship Analysis Between the Flow Characteristics and Wear Characteristics
Reference
6 Engineering Calculation of Solid–Liquid Two-Phase Pump
6.1 Single-Stage Centrifugal Pump
6.1.1 Geometric Model and Calculation Settings
6.1.2 Analysis of Solid–Liquid Two-Phase Flow Calculation Results
6.2 High Speed Centrifugal Pump with Compound Impeller
6.2.1 Geometric Model and Calculation Settings
6.2.2 Analysis of Solid–Liquid Two-Phase Flow Calculation Results
6.3 Two-Stage Centrifugal Pump
6.3.1 Geometric Model and Calculation Settings
6.3.2 Analysis of Solid–Liquid Two-Phase Flow Calculation Results

Citation preview

Fluid Mechanics and Its Applications

Zuchao Zhu Yi Li Zhe Lin

Solid-Liquid Two-Phase Flow in Centrifugal Pump

Fluid Mechanics and Its Applications Founding Editor René Moreau

Volume 136

Series Editor André Thess, German Aerospace Center, Institute of Engineering Thermodynamics, Stuttgart, Germany

The purpose of this series is to focus on subjects in which fluid mechanics plays a fundamental role. As well as the more traditional applications of aeronautics, hydraulics, heat and mass transfer etc., books will be published dealing with topics, which are currently in a state of rapid development, such as turbulence, suspensions and multiphase fluids, super and hypersonic flows and numerical modelling techniques. It is a widely held view that it is the interdisciplinary subjects that will receive intense scientific attention, bringing them to the forefront of technological advancement. Fluids have the ability to transport matter and its properties as well as transmit force, therefore fluid mechanics is a subject that is particulary open to cross fertilisation with other sciences and disciplines of engineering. The subject of fluid mechanics will be highly relevant in such domains as chemical, metallurgical, biological and ecological engineering. This series is particularly open to such new multidisciplinary domains. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of a field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity. Springer and Professor Thess welcome book ideas from authors. Potential authors who wish to submit a book proposal should contact Dr. Mayra Castro, Senior Editor, Springer Heidelberg, e-mail: [email protected] Indexed by SCOPUS, EBSCO Discovery Service, OCLC, ProQuest Summon, Google Scholar and SpringerLink

Zuchao Zhu · Yi Li · Zhe Lin

Solid-Liquid Two-Phase Flow in Centrifugal Pump

Zuchao Zhu Department of Energy and Power Engineering Zhejiang Sci-Tech University Hangzhou, Zhejiang, China

Yi Li Department of Energy and Power Engineering Zhejiang Sci-Tech University Hangzhou, Zhejiang, China

Zhe Lin Department of Energy and Power Engineering Zhejiang Sci-Tech University Hangzhou, Zhejiang, China

ISSN 0926-5112 ISSN 2215-0056 (electronic) Fluid Mechanics and Its Applications ISBN 978-981-99-1821-8 ISBN 978-981-99-1822-5 (eBook) https://doi.org/10.1007/978-981-99-1822-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed 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 Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Two-Phase Flow Model and Calculation Method . . . . . . . . . . . . . . . . 1.2 Two-Phase Transport Performance and Prediction . . . . . . . . . . . . . . . 1.3 Two-Phase Wear Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 8 12 16

2 Solid–Liquid Two-Phase Calculation Model and Method . . . . . . . . . . . 2.1 Solid–Liquid Two-Phase Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Analysis of the Force of Moving Particles in Fluids . . . . . . . 2.1.2 Calculation Model of Solid–Liquid Two-Phase Flow . . . . . . 2.1.3 Model Suitability Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Collision Rebound Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Wet and Dry Wall Collision Rebound . . . . . . . . . . . . . . . . . . . 2.2.2 Rotating Wall Collision Rebound . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23 23 24 31 33 35 36 45 60

3 Wear Characteristics of Static Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.1 Calculation Method and Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.1.1 Computational Method and Geometric Model . . . . . . . . . . . . 63 3.1.2 Experimental System and Measurement Method . . . . . . . . . . 68 3.1.3 Results Comparison and Calculation Method Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.2 Particle Motion and Wear Characteristics in the Bend Pipe . . . . . . . 76 3.2.1 Analysis of Particle Motion in the Bend Pipe . . . . . . . . . . . . . 77 3.2.2 Analysis of Wall Wear Characteristics . . . . . . . . . . . . . . . . . . . 85 3.3 Effect of Wall Shape Change on Wear . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.3.1 Wear Analysis of Wall Surface with Bump in Bend Pipe . . . 98 3.3.2 Wear Analysis of Wall Surface with Groove in Bend Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

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Contents

4 Two-Phase Flow Characteristics and Transportation Performance in Centrifugal Pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Calculation of Two-Phase Flow in Pump Based on Mixture Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Calculation Scheme and Grid . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Different Solid Particle Sizes Conditions . . . . . . . . . . . . . . . . 4.1.3 Different Solid Concentration Conditions . . . . . . . . . . . . . . . . 4.1.4 Different Flow Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.5 Different Solid Density Conditions . . . . . . . . . . . . . . . . . . . . . 4.2 Two Phase Flow Calculation in Pump Based on DPM Model . . . . . 4.2.1 Calculation Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Different Solid Particle Sizes Conditions . . . . . . . . . . . . . . . . 4.2.3 Different Solid Concentration Conditions . . . . . . . . . . . . . . . . 4.2.4 Different Flow Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Two Phase Flow Calculation in Pump Based on DEM Model . . . . . 4.3.1 Calculation Method and Scheme . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Analysis of Particle Motion Characteristics . . . . . . . . . . . . . . 4.3.3 Influence of Particle Mass Concentration on Flow Field in Centrifugal Pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Wear Characteristics of the Wall Surface in Centrifugal Pump . . . . . 5.1 Solid–Liquid Two-Phase Transport Experiment and Calculation Method Verification . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Grid and Calculation Settings . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Calculation Method Verification . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Wear of the Centrifugal Pump Runner . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Impeller Wear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Volute Wear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Wear Analysis of Front and Rear Wear Plates . . . . . . . . . . . . 5.3 Wear of the Pipeline Matched with Centrifugal Pump . . . . . . . . . . . . 5.3.1 Vertical Pipe Wear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Bend Pipe Wear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Horizontal Pipe Wear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Particle Motion Analysis Inside Pipes . . . . . . . . . . . . . . . . . . . 5.4 Wear of the Flow Channel for Conveying Mixed Size Particle . . . . . 5.4.1 Calculation Method and Scheme . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Calculation Method Verification . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Runner Wear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 Relationship Analysis Between the Flow Characteristics and Wear Characteristics . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121 121 121 123 128 129 135 137 138 138 143 147 150 150 154 156 166 167 167 167 168 170 171 171 174 177 181 181 185 189 192 194 194 195 196 206 208

Contents

6 Engineering Calculation of Solid–Liquid Two-Phase Pump . . . . . . . . . 6.1 Single-Stage Centrifugal Pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Geometric Model and Calculation Settings . . . . . . . . . . . . . . . 6.1.2 Analysis of Solid–Liquid Two-Phase Flow Calculation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 High Speed Centrifugal Pump with Compound Impeller . . . . . . . . . 6.2.1 Geometric Model and Calculation Settings . . . . . . . . . . . . . . . 6.2.2 Analysis of Solid–Liquid Two-Phase Flow Calculation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Two-Stage Centrifugal Pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Geometric Model and Calculation Settings . . . . . . . . . . . . . . . 6.3.2 Analysis of Solid–Liquid Two-Phase Flow Calculation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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209 209 209 211 216 217 218 223 224 225

Chapter 1

Introduction

Abstract There are many solid–liquid mixed transportation needs in energy, environmental protection, petrochemical, mining, and metallurgy, etc. In the process of solid–liquid two-phase mixed transportation, it is easy to cause wall wear, which leads to poor reliability. At the same time, the existence of solid particles will reduce transport efficiency and performance of the pump. However, due to the different parameters such as the size, density, and concentration of solids, the complex geometry of the flow channel in the solid–liquid multiphase pump, and the high-speed rotation of the impeller, it is very difficult to study the mechanism of solid–liquid two-phase flow. At present, researchers have carried out some computational and experimental studies on the solid–liquid multiphase pump. The research status of twophase model, two-phase hydraulic performance and wear performance of centrifugal pump are described in this chapter.

1.1 Two-Phase Flow Model and Calculation Method There are three kinds of modeling methods for solid particle–fluid two-phase flow at present: two-fluid model (TFM), combined continuum and discrete model (CCDM), and pseudo particle model (PPM). TFM, also known as the continuum model (CM), belongs to the Eulerian-Eulerian method. Fluid and solid particles are both described by Euler based on continuity assumption. The selected control volume size should be far larger than the singleparticle size, but also much smaller than the characteristic scale of the system. Therefore, it is necessary to provide the interphase interaction model and solid-phase constitutive model. The turbulence control equations based on TFM can be obtained by various averaging methods or kinetic methods of Euler equations. Liu and Shen [1] adopted the two-phase turbulent flow of dilute granular liquid K-ε-AP model to simulate the water–sediment movement in three dimensions. The results show that the method is more accurate than the single fluid turbulence model. Based on the Eulerian-Eulerian method, Feng et al. [2] developed a solid–liquid two-phase explicit algebraic stress model (EASM) with the ability to characterize turbulent anisotropy and found that the prediction effect is better than the k-ε-Ap model. Cheng et al. [3] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Z. Zhu et al., Solid-Liquid Two-Phase Flow in Centrifugal Pump, Fluid Mechanics and Its Applications 136, https://doi.org/10.1007/978-981-99-1822-5_1

1

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1 Introduction

distinguished the turbulent kinetic energy of solid phase and particle temperature, combined the kP model of solid-phase turbulence with the particle kinetic model, and proposed the model of fluid–solid particle two-phase turbulence k-ε-kP -Θ four equation model. Zheng et al. [4] also considered the dissipation rate of turbulent kinetic energy in the granular phase and developed a k-ε-kP -εp -Θ five equation model. Zeng et al. [5] adopted the unified second-order moment model (USM-Θ Model) and kε-kP -εp -Θ five equation model respectively to simulate the two-phase turbulent flow of power-law fluid and particle. The results show that the two-phase turbulence has strong anisotropy and the simulation effect of the USM-Θ model is better. Savage and Jeffrey [6] derived the stress model of smooth elastic sphere particles in uniform shear flow by using the Maxwell-type probability density function (PDF). Makkawi et al. [7] pointed out that for the dense particle phase, if the friction stress generated by the contact between particles is ignored, the porosity will be overestimated, which will lead to a large difference between the numerical simulation results and the experimental results. It is pointed out that it is necessary to establish the constitutive model of dense particles based on the friction theory [8]. Because the collision model based on Maxwell PDF cannot predict the influence of fluctuating velocity anisotropy and particle motion correlation, the improved collision model is obtained by using the Grad moment method [9]. Based on the above model research, Zaichik et al. [10] proposed a statistical collision model of double diffusion particles suitable for different sizes and densities, which can include the anisotropy of particle pulsation motion, the velocity correlation of adjacent particles, and the relative drift effect between different types of particles. It is mainly used in the above-mentioned two-phase flow cyclone and solid riser. CCDM belongs to the Eulerian–Lagrangian method, in which the fluid phase is regarded as a continuous medium to solve the N-S equation in the Eulerian coordinate system, while the solid particle is regarded as a discrete medium, and its motion equation is tracked and solved in the Lagrangian coordinate system. According to the different analytical degrees of discrete particles in the flow field, the corresponding methods can be divided into three categories according to complexity: point source particles, semi-analytical particles, and analytical particles. If the fluid description scale is larger than the single-particle size, the geometry and motion characteristics of a single particle cannot be completely distinguished, so it is necessary to use the interphase interaction model to seal the flow model. The particles in this method can be regarded as point source particles or semi-analytical particles, so a model should be provided to close the fluid-particle interaction: the effect of fluid on particles is directly applied to a single particle, and the local average of the particle to fluid effect is locally averaged into the fluid calculation grid. When semi-analytical particles are used in dense liquid–solid two-phase flow, the volume effect and particle collision must be considered, and the average volume of particles in the grid can be reflected in the fluid flow control equation as an explicit variable, and the collision rebound relationship of particles needs to be determined by the particle collision model.

1.1 Two-Phase Flow Model and Calculation Method

3

The earliest research on the collision rebound phenomenon is Hertz [11]’s experimental study on head-on collision of particles in the vertical direction. Through the analysis of experimental results, the relationship between instantaneous static contact stress and interaction force is pointed out. Johnson KL et al. [12] extended Hertz’s experiment to the oblique collision. They pointed out that the rebound trend of particles mainly depends on two dimensionless parameters: the dimensionless incident angle is related to the incident angle of collision, and the other parameter is related to the rotation radius of particle collision trajectory. Grant and Tabakoff [13] discussed the velocity, tangential velocity, normal velocity, and recovery coefficient of collision angle of particles with different diameters in the process of particle collision and rebound under the condition of different incident angles in gas–solid two-phase erosion wear experiment, and finally established the grant particle wall collision rebound mathematical model which is still widely used up to now. Later, the rebound model of particle wall collision was established under different experimental conditions. Joseph et al. [14] and others carried out the collision rebound experiment in a viscous fluid. The trajectory of particles was controlled by setting the initial inclination angle of the pendulum immersed in the fluid, and the resulting collision was monitored by a high-speed camera. The experimental results show that the recovery coefficient of particle collision velocity is highly related to the Stokes number in the process of motion and has little correlation with the properties of the impact wall material. At the same time, the surface roughness of particles and walls will affect the measured data of repeated experiments at low incident velocity. Fu et al. [15] studied the impact behavior of wet particles by measuring their coefficient of restitution and maximum contact area. In the experiment, they used a collision target plate with a size far larger than the diameter of the incident particle as a semi-infinite rigid target. The influence of these factors on the experimental results was studied by using the particles with different solid–liquid ratios, particle sizes, and materials. Two high-speed cameras are used to record the process of impact and rebound when the water-containing particles are released vertically and in contact with the target plate, and the collision recovery coefficient of particles is finally calculated by calculating the collision and rebound velocity. Mueller et al. [16] and others selected three kinds of spherical particles with different water content to carry out impact tests on hardened steel plates, and the granular materials represent plastic materials, elastic materials, and semi-elastic materials respectively. A falling body experimental device is constructed. The particles are accelerated by spring ejection. The whole process of incident, impact, and rebound is recorded by a high-speed camera, and the normal and tangential velocity recovery coefficients of particles are analyzed. The results show that the three materials exhibit elastic–plastic impact properties. Moisture content is an important parameter affecting the impact and rebound process of particles, and the recovery coefficient decreases significantly with the increase of water content. In the range of 0.25 and 2.25 m/s, the recovery coefficient increases with the increase of impact velocity, and decreases when the velocity is higher than 2.25 m/s. Sommerfeld and Huber [17] conducted experimental analysis by establishing a particle–wall collision mode. The particle sizes in the study were about 100 and 500 μm for the glass particle, and about 100 μm for the quartz particles. It is found

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1 Introduction

that the wall roughness significantly changes the rebound behavior of particles and leads to the dispersion of particles on average. Hastie [18] carried out the measurement experiment of the recovery coefficient of irregularly shaped particles impacting the horizontal surface and proposed an experimental method to capture the random behavior of irregular-shaped polyethylene particles in a three-dimensional environment by using high-speed video. The angular displacement and angular velocity of particles are determined by experiments, and the methods of determining the coefficient of restitution with various equations are given. Dong and Moys [19] used particles with initial rotation velocity to collide with plates with different tilt angles and used video analysis technology to simultaneously measure the center position and orientation of the ball. All kinds of image distortion are corrected. The research results show that when the incident particles collide with the wall at the initial rotational velocity, the contact mode between the particles and the wall is not like that in the prediction model of Maw and Walton, the large incident angle is rolling and the small incident angle is sliding, but there is a certain probability. Gibson et al. [20] studied the particle wall collision model under four boundary conditions when study the particle motion law in coal gasifiers. The particles used in the experiment are coke particles, polyethylene particles, and polystyrene particles. The collision walls are metal plates, low viscosity silica gel surfaces, high viscosity silica gel surfaces, and plates covered with some coke particles. The experiments demonstrate that the main factors affecting the coefficient of restitution are the material properties of the particles, such as the uniformity of the particle materials and the young’s modulus of the particles and the wall. At the same time, considering different types of wall conditions, it is found that the surface morphology of the wall, such as flatness, can affect the impact rebound angle of particles. Troiano et al. [21] and others further studied the change of recovery coefficient in the process of particle collision with the wall in a gasifier with high-temperature coal slag. They found that in the far wall region of the gasifier, the main existence mode of particles is a discrete phase, while in the near-wall region, particles change from solid to liquid due to high temperature, resulting in the reduction of recovery coefficient to a minimum. The impact of particles and wall materials, the impact of incident angle in normal and oblique collisions, the influence of fluid on the adhesion of particles, and the motion state and morphology of particles at the moment of collision are discussed directed against impact rebound experiment under dry and wet wall conditions. Recently, due to the continuous progress of experimental equipment and the updating of application scenarios, the motion of particle with liquid attached to the surface is affected by the surface force of the liquid, which has become a research hotspot. The main contents of the study are the collision between micro particles and the wall with liquid film and the particle wall collision rebound experiment in a humid environment. In the experiment of building a particle wall collision rebound model in a humid environment, Dong et al. [22] set up an experimental device composed of an air intake system, humidity control system, collision device, and high-speed camera system to study the rebound recovery coefficient of pulverized coal ash particles in the normal direction of the wall under different humidity conditions. In the experiment, the control variables are the particle incident velocity and particle diameter. It is found

1.1 Two-Phase Flow Model and Calculation Method

5

that the largest correlation factor of the normal recovery coefficient of small particles is the incident velocity. When the incident velocity is less than the impact velocity of the particle reaching the yield point, the normal recovery coefficient increases with the increase of the incident velocity; when the incident velocity decreases, the recovery coefficient of the impact velocity is greater than that of the normal velocity. Li et al. [23] extended Ming Dong’s research, using dust particles with a particle size of about 7 μm collected in Liaoning, Xinjiang, and Inner Mongolia to explore the normal collision recovery coefficient of particles under experimental conditions of different humidity. The normal impact velocity and rebound velocity of dust particles under different air humidity was measured by a high-speed camera, and a modified mathematical model for calculating the rebound coefficient under wetting conditions is proposed. Their study found that the normal coefficient of restitution decreases with the increase of relative humidity, particle density, and Young’s modulus. As for the research on the collision between particles and the wall covered with liquid film, Mueller et al. [24] and others studied the influence of particle incident velocity and liquid film thickness on the recovery coefficient through experiments. The experimental results show that the recovery coefficient decreases with the increase of liquid film thickness. In their work, they try to use Stokes number to characterize the viscous damping effect of liquid film on particle impact. The energy dissipated by the viscous force depends on the travel distance of particles in the liquid film and the viscosity of the liquid. Müller and Huang [25] studied the impact of particles on a flat plate with a thick liquid film on the surface, and also studied the energy loss during the collision process. The experimental results show that the normal recovery coefficient of the particles is linear with St-1 and proportional to the incident velocity. If the description scale of the fluid is smaller than the particle size, the fluid– solid coupling numerical simulation of the two-phase flow is carried out directly, so that the size, shape, and motion of the particles can be completely resolved. Due to the continuous movement of particles in the flow field, the spatial structure of the flow field changes at each time step. Therefore, special dynamic mesh technology is developed to adjust the mesh of each step to adapt to the moving phase interface through mesh overlap, mesh reconstruction, mesh deformation, and deformation reconstruction. In Ref. [26], the overlapping grid boundary definition technology is used to transfer data in an unstructured overlapped moving grid, and the extra boundary value of the subnet is obtained through the active grid in the overlapping area of the background grid. The mesh reconstruction method regenerates the mesh at each time step to adapt to the change in the calculation area. It is simple in concept and easy to handle the problems of large deformation and displacement. In Ref. [27], the problem of interfacial motion between immiscible liquids was studied by using the mesh reconstruction method. In Ref. [28], based on the N-S equation, the grid reconstruction combined with the mixed length model was used to calculate the forced vibration of turbine engine blades. It was found that there was a positive shock wave on the suction surface of the issued blade, which was very sensitive to small changes in flow conditions. Based on the finite element method [29], the dendritic growth process of binary alloy in fluid in different directions was simulated, and the moving grid technology was used to capture the movement of the complex interface

6

1 Introduction

in moving fluid. In Ref. [30], based on the N-S equation, the mesh reconstruction strategy was used to analyze the free interface flow, the moving spatial structure, and the deformed fluid–solid interface. However, the mesh reconstruction method will introduce additional interpolation dissipation error in each step of interpolation, and the calculation workload is too large. Deformable dynamic mesh technology mainly redistributes the mesh nodes while keeping the mesh topology unchanged. A point stretching method is proposed [31], which regards the connection between mesh nodes as equal strength springs, and the equilibrium state of springs is the optimized mesh distribution when the boundary constraint conditions are given. In the process of dynamic mesh generation, the spring balance state is destroyed due to the motion of the object, so the relevant mesh nodes are moved by relaxation iteration to keep the dynamic balance of the spring system. For the case of large deformation or displacement, the sheet stretch method is proposed and used to generate the dynamic mesh of wing flutter [32]. The method is also applied to the dynamic boundary treatment of object separation [33], free surface [34], aerodynamic shape optimization [35], and deal with aeroelastic problems [36–38]. An interpolation method can adapt to large-scale deformation without iterative calculation and obtain good dynamic mesh are proposed based on the Delaunay background grid [39, 40]. The method is used to simulate the crack growth [41]. The Delaunay background grid method was used to optimize the position of the aircraft cabin [42]. A mesh deformation method based on the reduced-order model of mesh optimization is proposed to improve its deformation efficiency [43]. Large deformation will lead to a sharp decline in the quality of mesh elements, even make the mesh intersect. It will affect the calculation accuracy, and even cause the calculation divergence failure. So, the mesh deformation method is difficult to deal with large deformation unsteady problems. Therefore, a new strategy of dynamic mesh generation is proposed. Firstly, the spring stretching method is used to move the mesh nodes for mesh generation, and then the mesh quality is detected. If the requirements are met, the above steps are repeated. If the mesh quality does not meet the requirements, local mesh reconstruction is performed [44, 45] which combines the advantages of mesh deformation and mesh reconstruction methods and can generate high-quality meshes with complex shapes. In Ref. [46], to improve the efficiency of dynamic mesh generation and enhance its applicability, the Delaunay background grid, spring stretching method, and local mesh reconstruction method were combined to study the external load separation problem. Despite the continuous efforts of many researchers, the moving grid technology inevitably needs to reconstruct the mesh at each time step, which brings a huge computational load. Moreover, when the old and new grids are alternated, it will bring about numerical instability. Therefore, the immersed boundary method (IBM) is proposed [47]. It is not only a mathematical modeling method but also a numerical discretization method. Its basic idea is to model the boundary of complex structure into a kind of physical force in Navier Stokes momentum equation and use the simple Cartesian grid, to effectively avoid the difficulty of body-fitted mesh generation and improve the computational efficiency. During the process of establishing the

1.1 Two-Phase Flow Model and Calculation Method

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connection between the solid boundary and the surrounding fluid grids, the immersed boundary method is divided into the continuous force method [48] and discrete force method [49, 50] according to the treatment of force source term. The force source term of the former [51] satisfies some general mechanical relation with analytic expression, which is mainly used to deal with elastic boundary problems δ function to establish the relationship between elastic boundary and fluid mesh; However, the force source terms of the latter [52, 53] are obtained from the discrete governing equations, and generally, there is no analytical expression, which is mainly used for the problem of rigid solid interface The solid shape of this kind of problem remains unchanged, and the flow field only exists outside (or inside) the solid. It is necessary to establish a correlation between the solid boundary and the fluid grid by interpolation. Based on the original immersed boundary method, an improved immersed boundary method was proposed [54]. A new projection operator was constructed to replace the Hamiltonian operator in the N-S equation, which effectively improved the volume conservation problem of the immersed boundary method. In Ref. [55], an adaptive immersed boundary method was proposed, in which adaptive mesh refinement was introduced. The IMB can be multi-grid adaptive; In Ref. [56], the calculation module near the immersed boundary is modified to a similar form dealing with boundary conditions to simulate the motion of a high Reynolds number. In recent years, the combination of immersed boundary method and various numerical methods has developed the combination of immersed boundary method and finite volume method [57, 58], the combination of immersed boundary method and finite element method [59, 60], and the combination of immersed boundary method and large eddy simulation method [61, 62]. Although the immersed boundary method has been developed for a long time, it is very complex to track the mark points of the immersion boundary in three-dimensional space after spatial discretization. So far, the immersed boundary method is mainly used in two-dimensional cases or axisymmetric motion, and only a few cases are used in three-dimensional space. Therefore, how to extend it from two-dimensional space to three-dimensional space needs further study. The third type of PPM treats the fluid as a granular phase. If the model of particle-to-particle coupling is relatively simple, then it is considered that the flow field between particles is simple. If it cannot be simplified to a dilute phase, the particle collision model is the key to the discrete particle model. Three models have been developed for numerical simulation of particle collision behavior: hard-sphere model (HSM) [63, 64], soft sphere model (SSM), discrete element method or distinct element method (DEM) [65–68] and direct simulation Monte Carlo (DSMC) model [69–72]. When the collision between particles tends to be rigid and the collision time between particles tends to be zero, it is suitable to calculate the collision time of particles. The motion of particles is described by Newton’s second law and the stress–strain law between particles. It can directly solve the time-varying impact force and allow multiple particle collisions. After the particle position and velocity are known, to accurately describe the motion of particles in the particle fluid system, the ideal method is to directly determine whether there is a collision between particles by the mutual position relationship between particles and track the trajectory of

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1 Introduction

each particle in the system. However, if there are many physical (actual) particles, the calculation workload is very large to accurately determine whether there is a collision between particles. Therefore, DSMC technology is proposed to overcome this difficulty. It does not directly determine the particle collision events but uses the probability sampling method to determine whether there is a collision between particles. In this method, the calculated particles used for simulation are regarded as the representatives of multiple physical particles, and the number of particles in the simulation system is far less than that in the physical (actual) system, so it can simulate more particles in the liquid–solid two-phase flow system. PPM belongs to the Lagrangian method, which not only treats particles as discrete phases but also discretizes the fluid into fluid micro clusters—fluid “particles”. Some classical phenomena and microscopic characteristics of two-phase flow are described and reproduced by simulating the interaction between fluid particles and solid particles. Because the model can describe the microstructure of the fluid–solid two-phase flow, in a sense, it points out the direction for the study of the fluid–solid two-phase flow. However, due to its huge demand for computing resources, the current simulation is still limited to some ideal cases, such as fluid around particles [73], particle to fluid drag force simulation [74], etc. In recent years, smoothed particle dynamics (SPH), as a meshless method, has also been introduced into the calculation of solid–liquid two-phase flow. In this case, the SPH method can effectively avoid the problem of interface accuracy, and its Lagrangian characteristics are conducive to the stability of calculation. Therefore, researchers have successfully simulated some examples [75–79] in recent years. To develop a high-precision SPH algorithm for multiphase flow, many scholars have done a lot of theoretical work and engineering attempts [80–82]. It can be found that the particle fluid two-fluid model has developed from sparse particle phase to dense particle phase, from single-phase turbulence analogy method to particle dynamics derivation method, from smooth particle elastic collision to rough particle inelastic collision, from isotropic turbulence to anisotropic turbulence, the applicability of the solid–liquid multiphase flow model has been verified by experiments. For the description of coarse particles, there is no interaction between discrete particles in the dilute phase at the beginning, and the interaction between particles and the two-way flow field is considered; the description of point source particles, it has been developed into various kinds of dynamic mesh direct fluid– structure coupling simulation considering the volume effect of particles and the immersion boundary method simulation based on virtual volume force.

1.2 Two-Phase Transport Performance and Prediction To study the influence of particles on pump performance, many scholars have done some research on numerical simulation, performance experiments, and flow observation experiments of solid–liquid two-phase flow pumps.

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At present, when calculating the internal flow of a solid–liquid two-phase flow pump at home and abroad, the solid particles are treated as fluid, and the two-fluid model is used for calculation. However, when the particle volume is smaller than the actual value, it cannot be ignored. In Ref. [83], the internal solid–liquid two-phase flow in a non-overload mud pump was numerically simulated by using the IPSA (inter-phase slip algorithm). It was found that there was backflow at the inlet and there was velocity slip between the two phases. At the intersection of pressure surface and suction surface, the turbulent kinetic energy and dissipation rate reached an extreme value. Reference [84] takes the impeller of a semi-open solid–liquid two-phase centrifugal pump with different clearance as the research object, compares the pump performance when conveying water, sand, and other media with different hardness, and summarizes the influence formula of particle size on pump performance. The deviation between the predicted value and measured value of the solid–liquid two-phase delivery head is between − 20% and + 15%. In Ref. [85], a centrifugal pump for conveying solid–liquid mixed medium including water, mortar and zinc tailings was experimentally investigated at different speeds. It is found that when the solid concentration is less than 20%, the relation between pump head and flow can be determined by the relation of clear water condition. When the solid concentration increases, it is necessary to consider the impact of solid particles on pump performance. In Ref. [86], PIV technology was used to analyze the velocity of slurry particles in the impeller of the centrifugal slurry pump with different volume fractions. It was found that the fluid flow separation phenomenon occurred on the suction surface of the blade tip, and the particle velocity increased with the increase of pump speed. Based on the mixture model, a low specific speed centrifugal pump was studied in Ref. [87] by using a numerical calculation method. It was found that the volume fraction, particle diameter, and density had a great influence on the hydraulic performance of the pump. With the increase of the particle diameter and volume fraction, the head and efficiency of the pump decreased, while the particle density had a relatively small impact on the pump performance, the blade suction surface wear is more serious than the pressure surface, and the jet wake structure appears near the volute tongue, and it becomes more obvious with the increase of volume fraction. In Ref. [88], the experimental study and internal flow simulation analysis of the ah-type mud pump was carried out. The numerical simulation results were compared with the experimental results, and the influence of the internal flow law of the impeller and solid volume fraction on pump performance was discussed. In Ref. [89], the dynamic grid method was used to simulate the startup process of a solid–liquid two-phase centrifugal pump. The user-defined function was used to write the time-varying relationship between the rotational speed and the flow rate obtained from the experiment into the calculation program. The results show that the transient performance of the pump for conveying clean water and solid–liquid mixture is quite different during the start-up process, the results show that the time of a steady flow is higher than that of clean water, and the axial force value, dynamic stress value and pressure fluctuation value of impeller are increased, while the pressure increase value is decreased. The influence of solid particles on the performance of the centrifugal pump was analyzed by numerical simulation

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1 Introduction

and experimental research [90]. Clear water and solid–liquid two-phase experiments were carried out for different types of mud pumps at rated speeds [91]. The solid mass fraction was 50–70%. The results showed that the head and efficiency of the centrifugal pump decreased with the increase of solid concentration, and the degree of influence was closely related to the slurry characteristics. The input power of the pump increases with the increase of solid concentration, while the performance parameters and performance of the screw pump are opposite to that of the centrifugal pump. At the rated speed, the head and efficiency increase with the increase of solid concentration. Liu et al. [92] used the Euler model to simulate the centrifugal pump under different conditions and particle sizes, and concluded that the distribution law of solid particles in the impeller channel is mainly affected by the particle size, which has a good reference value for the optimal design of the pump. Zhao et al. [93] carried out a numerical simulation on the non-clogging centrifugal pump with a doublechannel impeller and found that the particle distribution is very uneven under nonrated conditions. In the impeller flow channel, the particles are mainly concentrated on the pressure side, while in the volute basin, they are mainly concentrated in the area near the volute outlet. Wei et al. [94] studied WQS 120-60-45 sewage pump and found that reducing the impeller blade outlet angle is conducive to the flow of solid particles and can improve the efficiency of sewage pump. Salim et al. [95] studied the performance of centrifugal slurry pump under three slurry concentrations and three rotational speeds through experiments, and found that the performance of slurry pump was greatly affected by slurry type, concentration, and size. Wang et al. [96] carried out experiments on the double suction centrifugal pump, studied the change of hydraulic performance of double suction centrifugal pump by controlling the size and concentration of conveying particles, and proposed the corresponding functional relationship through the processing of experimental data, which is of great help to the pump manufacturing. Wenjie et al. [97] carried out a three-dimensional solid–liquid two-phase flow numerical simulation of the molten salt pump, and found that when the particle diameter ratio is small, there is a linear relationship between the particle volume fraction at the pump inlet and the pump performance parameters, and with the increase of particle diameter, the performance parameters of the pump decrease rapidly. Tressia et al. [98] studied the influence of abrasive particle size and PH value of water on the pump delivery performance and wear resistance. It was found that when the particle diameter exceeded the critical particle size, the linear ratio between mass loss and particle diameter decreased. Through the analysis of experimental data, the critical size was redefined, and it was also found that the decrease in pH value of aqueous solution would aggravate the mass loss. When Zhao et al. [99] studied the transient flow characteristics and pressure fluctuation of sediment-laden flow transported by a centrifugal pump, it was found that the sediment concentration had a great influence on the distribution of turbulent kinetic energy and the peak amplitude of fluctuation frequency, while the particle size had a relatively small impact. At the same time, it was also found that the dominant frequency of solid liquid was about 0.8 times that of pure water, and the transient characteristics of sediment-laden flow showed the characteristics of low frequency and high amplitude.

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Hitoshi et al. [100] carried out an experimental study on the performance of various types of small air system pumps for conveying Al2 O3 particles with a diameter of 3 mm. Liu et al. [101] studied the liquid–solid two-phase flow jet pump, calculated the nozzle and throat diameters of different shapes, and compared the experimental results to provide a reference for further optimization design. Angst and Kraume [102] and Wang et al. [103] studied the dispersed phase in multiphase flow. Tahsin and Mesut [104] The effects of particle size and particle size fraction on the solid volume fraction of the centrifugal pump were studied. Saleh et al. [105] simulated the gas–liquid two-phase flow discharged from the branched stratified area of 1/4 circular surface of a circular elbow. Caridad and Kenyery [106] studied the influence of slip parameters on the performance of turbomachinery and found that the slip parameters suitable for single-phase flow of most centrifugal pump impellers were obtained. Shi et al. [107] designed a new type of particle image velocimetry (PIV) device and used the device to photograph the motion of two-phase flow containing glass beads and two-phase flow containing oil droplets in a centrifugal pump, obtained the velocity distribution and external characteristic curve of the flow field in the pump, and analyzed the experimental results. Torabi and Nourbakhsh [108] studied a low specific speed centrifugal pump and solved the three-dimensional Navier–Stokes equation for the whole pump, including impeller, volute, pipe, front and rear side wall clearance, and balance hole. Comparing the simulation results with the experimental data, the results show that the fluid is strictly sensitive to the viscosity in the side wall gap. The reduction of internal leakage in the wear ring and balance hole leads to an increase in viscosity and an increase in volumetric efficiency, which leads to a significant decrease in the disc and overall efficiency. Tarodiya and Gandhi [109] used mixture model and Eulerian-Eulerian multiphase model to simulate centrifugal slurry pump. The experimental results show that compared with the mixture model, the Eulerian-Eulerian multiphase model is closer to the experimental results in predicting the influence of solid on pump performance. Peng et al. [110] studied and analyzed the slurry flow of heavy-duty slurry pump in grinding circuit of ore mill under various particle concentration and small volume flow rate by Eulerian Euler method. The results show that for a small flow rate, the flow rate is very unstable, and with the increase of particle concentration, the flow resistance increases, and the backflow increases, which leads to the local wall wear aggravation. Wu et al. [111] is based on RNG k-ε based on the turbulence model and dynamic mesh technology, the unsteady turbulent flow of solid–liquid two-phase three-dimensional full phase flow was simulated by Fluent software. The results show that the relative velocity fluctuation of solid flow at the impeller outlet is 7.6% less than that of liquid phase flow. The fluctuation of head and radial force is 8.1% and 85.7% respectively. The research results will provide a theoretical basis for improving efficiency and reducing hydraulic loss and wear. Huang et al. [112] considered the physical characteristics of solid particles and the interaction between particles and particles and structure, and combined DEM and CFD methods to calculate and analyze the transient solid–liquid two-phase flow in a single-stage centrifugal pump. The simulation results show the movement of solid particles in the pump, and the influence of particles on the flow

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1 Introduction

performance of the pump is mainly reflected in the change of head with time. In addition, the velocity field of the two-phase flow, the volume fraction distribution, and the particle trajectory of solid particles in the centrifugal pump is also given.

1.3 Two-Phase Wear Properties Particle wear refers to the phenomenon of surface material loss when solid particles contact and move relative to the solid wall. The wear of particles on the wall has an important impact on the service life and reliability of the equipment. Therefore, a large number of theoretical and experimental research have been carried out on wear and tear. The particle-to-wall collision wear model is the embodiment of the collision wear mechanism, and it is also a tool for wear prediction at present. Due to the diversity of the physical properties, motion parameters of the particles, and the properties of the worn wall materials, as well as the differences in research and analysis methods, there is no judgment on the wear mechanism of materials at present. Therefore, according to different assumptions and application ranges, there are many types of wear patterns of particles on the wall. At present, it is mainly divided into (1) semi-empirical and semi-theoretical wear models based on theoretical hypothesis, analysis, and experimental research, and (2) empirical wear models obtained from the experimental analysis. Based on the micro-cutting principle of plastic materials, the first wear calculation model of plastic materials was proposed [113, 114], and the volume of wall material loss was calculated. Then, combined with the experimental results, the model is improved. However, the model is not suitable for the case of particle impact angle of 90° and underestimates the wear of materials in large angle impact and overestimates the wear of materials in small-angle impact. Based on the micro-cutting mechanism of particle erosion and deformation wear theory, it is found that material wear is mainly caused by the joint action of cutting wear and deformation wear [115, 116]. The wear model proposed is suitable for all impact angles. When the impact angle is small, cutting wear dominates, and when the impact angle is large, deformation wear plays a major role. However, the complexity of the formula limits its use. The model proposed in Ref. [115] is simplified by carrying out two-phase flow experiments in Ref. [117], which improves the practicability. Based on the indentation damage theory and energy balance theory of single-particle collision, a wear model including deformation wear and cutting wear is proposed, but the hardness index in the model is controversial [118]. A two-stage collision wear theory was proposed [119]. In the first stage, particles collide with the material surface and the cutting wear was produced. In the second stage, the secondary impact wear was caused by the particle fragment produced in the first stage of collision. The total wear amount of the material is the sum of the two stages of wear. However, it is difficult to determine the degree of particle fracture under certain conditions, which limits the application scope of the model. Many

1.3 Two-Phase Wear Properties

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experiments of the impact wear of coal ash particles on metal walls were carried out [120]. An empirical wear equation based on the impact velocity and angle was proposed, which is suitable for wear prediction of aluminum, stainless steel, and titanium. Based on the mechanism of lamellar wear, the critical plastic strain is introduced to judge the wear degree of materials [121], and a wear model of normal collision of spherical particles is proposed. However, the volume fraction of plastic deformation indentation cannot be measured directly, and the variation of strain rate with particle size and velocity is not considered. Based on a large number of wear tests for carbon steel and aluminum [122], a multi-parameter wear equation including impact velocity, impact angle, Brinell hardness of materials, and particle shape was proposed. The wear equation is one of the most widely used wear models. Based on the above-mentioned wear model, the collision angle function and velocity index were modified by experimental analysis to be suitable for the wear prediction of Cr Ni Fe alloy 718 and other materials [123, 124]. Based on a series of experiments and a basic wear model, a new wear model was proposed for a variety of commonly used materials [125]. However, some important parameters, such as particle size, particle size, and particle size, are not provided in the literature [126, 127]. Chong et al. [128] studied the wear of gas–solid mixture on the annular cavity in the pipe. The particle diameter used in the experiment is 38 and 198 μm. Andrews et al. [129] studied the relationship between erosion and impact angle of stellite alloy 6 and 316 stainless steel slurry by impact wear experiment and observed the surface morphology by scanning electron microscope. The results show that the wear resistance of stellite alloy is stronger than that of 316 stainless plates of steel at the impact strength of 6. Kesana et al. [130] used a pitot tube with a diameter of 6.35 mm to sample sand at five different locations in a 0.0732 m diameter horizontal pipe with L/D = 150 to obtain the relationship between the distribution of sand flowing in the pipeline and particle size and liquid viscosity, and also studied the influence of sand concentration measurement on erosion. Arabnejad et al. [131] used a solid– liquid mixture containing different particles to carry out erosion wear experiments on stainless steel samples and studied the influence of particles with a different hardness on the wear of samples, and the particle size was 2–38 μm. Patel et al. [132] carried out spray erosion experiments on boiler tube steel with 50 μm alumina particles. It is found that the erosion rate of alumina at a 30° impact angle is higher than that at a 90° impact angle. Vieira et al. [133] measured the wear of the elbow by measuring the metal loss rate with a resistance probe and analyzed the influence of various parameters such as liquid flow rate, viscosity, particle size, and flow direction. The particle size used in the study was 20, 150, 300 μm. In the same year, Araoye et al. [134] studied the influence of flow parameters and particle size on the wear of carbon steel pipe in solid–liquid two-phase flow using experiment and numerical simulation. The particle size was 50–400 μm. Although the above models have some limitations, the accuracy of wear prediction can be guaranteed by choosing a reasonable wear model according to the actual working conditions.

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1 Introduction

In Ref. [135], three kinds of jet pumps are calculated at the design flow point, two of which are traditional structures, and one is the latest design. The flow rate, pressure, and efficiency of the three kinds of pumps are analyzed. The results show that the new design of the jet pump has the best hydraulic performance. The particle trajectory tracking model and Finnie wear model were used to predict the wear, and the wear intensity and distribution in the jet pump were analyzed. In Ref. [136], the correctness of the wear model was confirmed by comparative analysis of the experiment and numerical simulation, and the erosion experiment of sand on cast iron and stainless steel. The results show that the wear coefficient increases with the increase of particle diameter and volume fraction, and decreases with the increase of impact velocity. The impact and rotation of particles can increase the wear data, but it is not obvious; The larger the particle diameter, the more serious the wear; With the increase of flow rate, the wear near the volute tongue will increase gradually. In Ref. [137], the influence of different design parameters on the inlet seal wear of the closed impeller was predicted by experiments, and the relevant model was established. In Ref. [138], PIV technology was used to study the velocity of the flow field in the tongue and adjacent flow channel of a solid–liquid two-phase pump with different rotational speeds and volume concentrations. It was found that with the increase of rotating speed, the wear of the vane pressure side, tongue, and pump cover increased, and the wear of the tongue was mainly caused by the collision caused by the kinetic energy fluctuation of solid particles. In the aspect of particle impact wear, it is concluded in Ref. [139] that the wear causes of flow passage parts are divided into the direct impact of particles, disordered impact, and sliding friction along the wall. The influence of the two-phase flow characteristics on the internal flow characteristics of the centrifugal pump is revealed by comparing with the experimental results in Ref. [140]. In Ref. [141], a wear prediction method based on rapid experiment and mathematical analysis was proposed to replace the traditional wear experiment. The comparison error between the prediction result and the experimental result is less than 3%. In Ref. [142], the effects of pump operating parameters, flow rate, rotating speed, particle diameter and geometry, tongue curvature, and volute width on particle erosion wear were analyzed by numerical calculation. It was found that the wear rate curve became gentler with the increase of flow rate. The wear rate decreases with the increase of the wear rate and with the decrease of rotating speed. Based on the theoretical analysis and numerical calculation of the solid–liquid two-phase flow of mud pump in Ref. [143], it is found that the flow parameters such as particle size, shape, and liquid velocity have a great influence on the erosion pits in the surface of flow passage parts, and the distortion degree and maximum stress of the pits increase with the increase of particle diameter. Zhang et al. [144] carried out a numerical simulation on the fluid transportation process, focusing on the prediction of the position of the maximum wear point of the elbow. The discrete element method (DEM) is used to describe the kinematic trajectory and particle–particle interaction of discrete particles. The effects of slurry velocity, bending direction, and elbow angle on the location of the maximum wear point are discussed. Duarte CAR et al. [145] proposed an erosion prediction model

1.3 Two-Phase Wear Properties

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based on CFD-DEM considering the interaction of liquid particle, particle–particle, and particle wall. The correctness of the prediction model was verified by the experimental data of Chen et al. [146] and Blanchard et al. [147]. At the same time, the wear of 45°, 60°, and 90° elbow was studied by using the model. It is found that the maximum wear position is near the exit of the three elbows, and the influence of the bending angle on the maximum erosion rate is very significant. Peng and Cao [148] used the Eulerian–Lagrangian two-way coupling method to solve the problem of solid–liquid two-phase flow in an elbow and compared the wear patterns of 450μm particles obtained in experiments. The results show the relationship between the Stokes number and the dynamic movement of the maximum erosion position, and three collision mechanisms are proposed to explain how the change of Stokes number affects the erosion location. Coker and Dan [149] considered the impact of particle collision on the wear of horizontal pipe, and the particle diameter was 50–300 μm. In the same year, Zhang et al. [150] examined the elbow erosion caused by the flow of a gas–liquid–solid mixture, which contained 20% water and had a particle size of 0.01–0.05 mm. In addition, the influence of Stokes number on particle trajectory and erosion scar is analyzed. Pei et al. [151] used computational fluid dynamics (CFD) to study the relationship between flow field, particle trajectory, maximum erosion area, and influencing factors (including particle size, pipe geometry, etc.), and verified the accuracy of the proposed method through experimental data available in the literature. The results show that the Stokes number of particle movement at the elbow has no decisive effect on the location of the maximum erosion area; Erosion location is directly related to particle size; Changing the radius of curvature will change the flow field in the pipeline, and then change the location of the maximum erosion area. Zhu and Li [152] studied a method of adding a rectangular baffle into a 90° elbow to reduce wear through the numerical simulation method of CFD-DPM. At the same time, the correctness of the method was verified by experiments. The results show that the rectangular baffle is used as a barrier element to protect the elbow from direct particle impact. When the rectangular baffle is located at 25°, its corrosion resistance is the best. Now many researchers have proposed new methods to predict two-phase wear more accurately and quickly. Uzi et al. [153] developed the ODEM model and conducted numerical simulation to predict the wear of bent pipes in gas–solid transportation. Compared with the three-dimensional CFD-DEM model, the ODEM model has a faster calculation speed and can ensure certain accuracy. Florio [154] studied the collision between particles and the wall by considering the rolling, deformation, sliding, and adhesion effects between particles, to predict the solid–liquid two-phase wear more efficiently. Wang et al. [155] established a new model to simulate the wall wear in the pipeline. Based on the relationship analysis between the secondary flow and the particle trajectory, it is found that the erosion mainly occurs in the area near the outlet of the bend. Beinert et al. [156] developed a new particle contact model, which considers the impact and torsion, shear, and rolling caused by rotation and translation velocity, which makes the particle–particle and particle wall collision more realistic and makes the subsequent wear distribution more reliable.

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Chapter 2

Solid–Liquid Two-Phase Calculation Model and Method

Abstract As described in Chap. 1, the computational fluid dynamics (CFD) methods have been used widely in the study of solid–liquid two-phase flow. According to the different modes of solid–liquid two-phase description, the current model methods can be divided into the Two-Fluid Model (TFM), Combined Continuum and Discrete Model (CCDM), and Pseudo Particle Model (PPM). TFM belongs to the Euler-Euler method, both fluid, and solid particles are described by Euler based on the assumption of continuity, and the selected control body scale should be much larger than the single-particle scale, and it needs to be much smaller than the characteristic scale of the system. CCDM belongs to the Euler–Lagrange method, the fluid phase is regarded as a continuous medium, the N-S equation under the Euler coordinate system is solved, the solid phase particles are treated as discrete media, and the equation of motion is tracked and solved under the Lagrange coordinate system. PPM belongs to the Lagrange method, which discretizes fluids into fluid micro-groups. By simulating the interaction between fluid “particles” and solid particles, the classical phenomena and microscopic characteristics in two-phase flow are studied and reproduced. Based on the actual application scenarios, this chapter studies the applicability of two solid– liquid two-phase models (TFM and CCDM) and a particle wall collision rebound model.

2.1 Solid–Liquid Two-Phase Model For solid–liquid two-phase fluid, due to the interaction between the mobile fluid phase and the particle phase, there is a transfer of momentum and energy, so the difficulty in studying the movement of the particle phase lies in how to determine the momentum transfer relationship at the interphase interface, and the interphase coupling must be based on the correct analysis of the interphase force. However, due to the complex diversity of the problem of two-phase flow, it is not yet possible to make a comprehensive and uniform definition of it, and there is no universally applicable two-phase model.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Z. Zhu et al., Solid-Liquid Two-Phase Flow in Centrifugal Pump, Fluid Mechanics and Its Applications 136, https://doi.org/10.1007/978-981-99-1822-5_2

23

24

2 Solid–Liquid Two-Phase Calculation Model and Method

2.1.1 Analysis of the Force of Moving Particles in Fluids 2.1.1.1

Classification of Forces

Taking a single spherical particle with a diameter d p and a density ρ p as an example, the force on a solid particle in a non-constant motion in the fluid is given. Take ρ f , μ and ν as the density, dynamics, and kinematic viscosity coefficients of the fluid, respectively. To make the expression of the force simple, take uf and up as tensor components of fluid and particle velocity respectively. The force is divided into the following three categories [1]: The first force Forces are independent of the relative motion between fluid and particles. Such as inertial force, gravity and differential pressure, etc. (1) Inertial force du p 1 Fin = − π d 3p ρ p 6 dt

(2.1)

(2) Gravity Fg =

1 3 πd ρpg 6 p

(2.2)

(3) Differential pressure force Due to the different pressures in the flow field, the unequal pressures acting on the surface of the particles. dp 1 F∇ p = − π d 3p 6 dx

(2.3)

Among them, ddpx is the pressure gradient. If this pressure gradient is caused by the gravitational action of the fluid, this force is buoyancy. The second force Forces are generated by the relative motion between the fluid and the particles. Its direction is in the direction of the relative motion—the longitudinal force. For example, resistance, additional mass forces, and Basset forces. (1) Resistance Viscous resistance is the most important force in the process of particle movement, in the actual two-phase flow, the size of the particle movement resistance is affected by many factors, it is not only related to the characteristic Reynolds number of particles but also related to the turbulent movement of the liquid flow, the incompressibility of

2.1 Solid–Liquid Two-Phase Model

25

the fluid, the shape of the particles, the concentration of the particles and the presence of the wall surface. ) ( FD = 3π μ u f − u p d p

(2.4)

If the particle shape is ellipsoidal (i.e., the vertical semi-axis lengths of the ellipsoid are a, b, and c), respectively, and satisfy a2 + b2 = c2 , the viscous resistance of the particle is [2]: ( ) FD = K · 3π μ u f − u p d p

(2.5)

{ √ [ ( ) ]}−1 Among them, K = 0.75 λ2 − 1 λ − λ2 − 1 cos λ , λ = bc = ( 2 )−0.5 a −1 . b2 When there is no theoretical solution to the viscous resistance for irregularly shaped particles, the particle shape coefficient can be determined experimentally to consider the influence of shape on resistance. Aiming at the resistance problem of n particles when moving at a low Reynolds number, the literature [1] considers the effect of particle concentration on the resistance as the fluid properties changes, and introduces the equivalent dynamic viscosity coefficient of the two-phase mixture to obtain: (

μeff

2.5α p = μ · exp 1 − Sα p

) (2.6)

Among them, α p is the particle phase concentration, S is a factor, and 1.35 < S < 1.95. When α p is very small, the above equation can be simplified to the Einstein formula: ) ( μeff = μ · 1 + 2.5α p

(2.7)

The above formula is the viscous resistance formula at low Reynolds numbers of particles (Rep < 1). Due to the flow within the pump, the relative motion of the particles does not necessarily satisfy the Stokes hypothesis, i.e., there may be a case where the particle Rep >> 1. The Reynolds number of particles in the common solid–liquid mixed transportation pump in the practical project is shown in Table 2.1. As can be seen from Table, the Reynolds number of particles in the pump varies widely, and in most cases, it is much greater than 1. To consider its implications, first, paraphrase the above resistance formula as follows: FD =

24 |u f −u p |d p ν

( )2 |( ) d p || 1 u f − u p| u f − u p · ρfπ 2 2

26

2 Solid–Liquid Two-Phase Calculation Model and Method

Table 2.1 Values of Reynolds number of particles in common solid–liquid mixed pumps Data sources

Solid-phase material

Specific gravity

Particle diameter (mm)

Impeller diameter (mm)

Pump rotate speed (r/min)

Particle Reynolds number

[3]

Sand 1

2.65

0.180

270

1000

3.05

[4]

Sand 2

2.65

0.230

270

1000

5.65

[4]

Sand 3

2.65

0.460

270

1000

27.52

[4]

Zinc

5.51

0.455

270

1000

54.44

[5]

Ilmenite

4.63

0.170

371

1300

5.16

[6]

Coal 1

1.48

0.900a

270

1450

47.81

[7]

Coal 2

1.716

15.300a

825

/

8073.48

[7]

Macadam

2.6

26.700a

825

/

26,318.57

[4]

Iron Ore 1

4.35

0.663

270

1000

95.01

[8]

Iron Ore 2

4.15

1.800

430

800

602.68

[8]

Granite

2.67

3.000

430

800

1019.46

a

weighted mean particle diameter; The remaining diameters are: mass median particle diameter

( )2 |( ) d p || 1 u f − u p| u f − u p = Cd ρ f π 2 2

(2.8)

24 , Rep is the Among them, the resistance coefficient Cd = |u −u24|d /ν = Rep f p p Reynolds number of particles in relative motion. The resistance coefficient obeys different laws according to the range of Rep . Because of the complex phenomena involving wake and flow separation, there are many correction formulas for Stokes viscous resistance, and several typical correction formulas for resistance coefficients are given in Table 2.2 [9–11].

(2) Additional mass force When particles accelerate in the fluid, they will surely drive part of the fluid around to accelerate. The force to promote the movement of particles not only increase the kinetic energy of the particles themselves, but also increase the kinetic energy of the fluid, so this force will be greater than the force required to accelerate the particles Table 2.2 Typical correction formulas for resistance coefficients Data sources [9] [10] [11]

Cd

) ( 3 Rep 1 + 16 ( ) 24 0.687 1 + 0.15Re Cd = Re p p ) ( 24 3 Cd = Rep Rep cos θ + 0.45 sin θ 1 + 16

Cd =

24 Rep

where θ is the flow separation angle of the particle

Re p 40 0.0524 α · Re p

(2.16)

/ ∂u Among them, α = d 2p · ∂ yf νRe p . For particle groups, additional mass forces, Basset forces, Magnus forces, and Saffman forces can be calculated by using the equivalent dynamic viscosity coefficient instead of the original viscosity coefficient by referring to the method of viscous resistance correction. In summary, the equation for the balance of forces acting on solid particles is: Fin + Fg + F∇ p + FD + Fvin + FB + Fl + FM + FS = 0

(2.17)

However, in the solution to the practical problem, not all of the above forces are of the same order of magnitude, so it is not necessary to count the effects of all forces. If you conditionally ignore the complex interphase forces of some expressions for a specific problem, you can greatly simplify the solution of the two-phase flow.

2.1.1.2

Force Magnitude Analysis

From the causes of the above forces, in the solid–liquid two-phase flow, the gravitational and inertial forces in the first type of force will have a great impact on the movement of particles and cannot be ignored. The viscous resistance in the second type of force is important and cannot be ignored. However, the differential pressure

2.1 Solid–Liquid Two-Phase Model

29

on the particles in the first type of force can be ignored in the case of a pressure gradient is not very large. At the same time, the lift in the third type of force is 0 for spherical particles, so the lift can be ignored. However, the importance of the additional mass and Basset forces in the second type of force and the Magnus force and the Saffman force in the third type of force is highly controversial, and the trade-offs in the different flow fields also need to be further discussed. Since the viscous resistance related to relative motion is important and cannot be ignored, it is meaningful to compare the above three forces with viscous resistance by order of magnitude to provide theoretical support for subsequent computational analysis. To simplify the analysis process, first, assume that the acceleration of motion of the particles relative to the fluid is constant and approximately expressed in the differential Eq. (2.18): ) ( d u f − up u f − up ≈ = const dt t − t0

(2.18)

Then the additional mass forces, Basset forces, Magnus forces, and Saffman forces are divided by the viscous resistance, and the magnitude is compared to obtain: 1 π d 3p ρ f 12

(

du f dt

−

du p dt

)

d 2p 1 ( ) ≈ · (2.19) 36ν t − t0 3π μ d p u f − u p ) ( t d u f − up dp dp 1 FB )∫ (2.20) = √ ( dτ ≈ √ √ FD dτ t −τ π ν(t − t0 ) 2 π ν u f − u p t0 ( ) 1 π d 3p ρ f ω p u f − u p d 2p ω p FM 8 ( ) = (2.21) = FD 24ν 3π μ d p u f − u p / )√ ( / √ 1.62d 2p ρ f μ u f − u p ξ dp Re p FS ( ) d p ξ = 0.17 dpξ = = 0.17 ν u f − up FD 3π μ d p u f − u p (2.22) Fvin = FD

| | | ∂u | Among them, the velocity gradient ξ = | ∂ yf |, and the Reynolds number of |u −u | particles Re p = f ν p d p . The discussion is as follows: (1) Additional mass force and Basset force. If the additional mass force and basset force are of the same magnitude as the d2

d

p 1 1 FB 1 viscous resistance, it is required FFvin ≈ 36νp · t−t ≥ 10 , FD ≈ √πν(t−t . ≥ 10 D 0 0) The kinematic viscosity of the common sparse solid–liquid two-phase flow is 1.792 × 10−6 m2 /s. When the particle diameter changes within a certain range, the feature time Δt = t − t0 is required to be less than the relevant value in Table 2.3.

30

2 Solid–Liquid Two-Phase Calculation Model and Method

Table 2.3 Maximum feature time (s) Particle diameter (mm)

0.05

0.1

0.2

0.5

Additional mass forces

0.0004

0.002

0.006

0.039

1

Basset force

0.04

0.18

0.7

4.4

17.8

0.5

1

0.155

2

5

0.62

3.875

71

444

Table 2.4 Maximum characteristic distance (m) Particle diameter (mm)

0.05

0.1

0.2

2

5

Additional mass forces

0.001

0.01

0.02

0.13

0.53

2.1

13.2

Basset force

0.1

0.6

2.4

15

61

241

1510

Table 2.5 Minimum particle spin angular velocity (rad/s) Particle diameter (mm)

0.05

0.1

0.2

0.5

1

2

5

Magnus force

1730

430

108

17

4

1

0.17

If the characteristic relative flow rate of the particles is u p = 3.4 m/s, the characteristic movement distance of the particles in the pump runner during the characteristic time is shown in Table 2.4. As can be seen from Tables 2.3 and 2.4, even when the particle diameter is 0.5 mm, the Basset force cannot be ignored, and the additional mass force should be taken into account after the particle diameter has increased to a certain extent. (2) Magnus force If the Magnus force and viscous resistance are of the same magnitude, that is d2 ωp

p 1 required FFMD = 24ν ≥ 10 . When the particle diameter changes within a certain range, the particle spin angular velocity ω p is required to be greater than the relevant value in Table 2.5. From the analysis data of Table 2.5, when the particle diameter is 0.05 mm, the angular rotation velocity of the particle is required to be greater than 1730 (rad/s), and the Magnus force can be the same magnitude as the resistance. When the pump is transported in the solid–liquid two-phase, due to the internal viscosity of the water, the particle angular velocity generally cannot meet the above conditions, that is, the Magnus force can be ignored. However, as the diameter of the particle increases, the required spin angular velocity is greatly reduced, and when the particle diameter increases to 5 mm, the spin angular velocity is required to be greater than 0.17 (rad/s), so the Magnus force can no longer be ignored at this time.

(3) Saffman force If the Saffman force √ and viscous resistance are of the same magnitude, that is d FS 1 required FD = 0.17 νp d p ξ ≥ 10 . When the particle diameter changes within a certain range, the speed gradient ξ is required to be greater than the relevant value in Table 2.6.

2.1 Solid–Liquid Two-Phase Model

31

Table 2.6 Particle velocity gradient minimum (1/s) Particle diameter (mm)

0.05

0.1

0.2

0.5

1

2

5

Saffman force

248

62

16

2.5

0.6

0.2

0.02

As can be seen from the data in Table 2.6, when the particle diameter is 0.05 mm, it is required that ξ is greater than 248 (1/s), and Saffman force talent and resistance are of the same magnitude. However, for the sparse two-phase flow inside the lowspeed rotating impeller mechanical flow field, there is no extreme shear zone, and the above conditions can generally only be satisfied in the boundary layer. However, as the diameter of the particle increases to 5 mm, ξ is required to be greater than 0.02 (1/s). Therefore, the effect of the Saffman force needs to be considered at this time. Therefore, in summary, in general, the solid phase particles in the two-phase flow in the pump can be regarded as only subject to resistance, gravity, inertial force, and Basset force, while the added mass force, Magnus force, and Saffman force should also be calculated accordingly after the particle diameter is increased to a certain diameter, and it should be counted accordingly and included in the force analysis.

2.1.2 Calculation Model of Solid–Liquid Two-Phase Flow 2.1.2.1

Model Based on Euler-Euler Method

The concept of the Euler-Euler method is to treat the particles as quasi-fluids, which are considered to occupy the same space as the fluid phase on a macroscopic level and penetrate each other. In the Euler coordinate system, the conservation equations of each phase are examined, and the system of time-controlled equations of solid– liquid two-phase flow can be obtained by averaging them according to the method of Reynolds of the single-phase flow instantaneous equation system, and the timeaveraging of the time-averaging equations for solid–liquid two-phase flow can be obtained. For convenience, remove the arrow above the vector (similar below), and the continuity equation and momentum equation of the mixed phase are, respectively: ∂ρ + ∇ · (ρv) = 0 ∂t

(2.23)

)] [ ( ∂ (ρv) + ∇ · (ρvv) = − ∇ p + ∇ · μ ∇v + ∇v T + ρg + F ∂t ⎛ ⎞ +∇ ·⎝



k= f, p

αk ρk vdr,k vdr,k ⎠

(2.24)

32

2 Solid–Liquid Two-Phase Calculation Model and Method

 where the μ is the mixed phase viscosity coefficient, μ = 2k=1 αk μk ; vdr,k is the drift speed,vdr,k = vk − v; F is the volume force; k is the number of phases; subscript f represents the fluid phase; subscript p represents the particle phase. The above system of equations is unclosed and requires the addition of model equations that simulate turbulence characteristics. The fluid phase is enclosed by an equation of k − ε, as follows: ( ) ( ) ∂ vjk ∂k 1 ∂ μe ∂k + Gk − ε + G p + = ∂t ∂x j ρ ∂ x j σk ∂ x j ( ) ( ) ) ] ε[ ( ∂ε ∂ v j ε 1 ∂ μe ∂ε + c1 G k + G p − c2 ε + = ∂t ∂x j ρ ∂ x j σε ∂ x j k

(2.25)

(2.26)

The particle phase is enclosed using the algebraic A p model: ) ( τr p −1 νp kp = 1+ = νt k τt

(2.27)

where μe is the effective viscosity coefficient, μe = μ + μt ; μt is the turbulence viscosity coefficient,( μt = cμ)ρk 2 /ε; G k is the turbulent kinetic energy genera∂v

tion term, G k = μt ∂∂vx ij + ∂ xij ∂∂vx ij ; G p is the term of fluid turbulence caused by the interaction of particle turbulence energy and fluid turbulence energy, G p =  ) √  ρp  ( ν p ∂n p k − i τr p 2 k − c p k · k p + vi σ p n p ∂ xi ; τr p is the granular dynamic response √ ρpd2 3/4 time, τr p = 18μp ; τt is the fluid turbulence pulsation time, τt = 23 cμ k/ε. The values of each common constant are as follows: cμ = 0.09, c1 = 1.44, c2 = 1.92, ckp = 0.75, σk = 1, σε = 1.33, σ p = 0.7.

2.1.2.2

Model Based on the Euler–Lagrange Method

When the diameter of the particle is large, the volume effect cannot be ignored, and the use of quasi-fluid concept error is large, at this time the particles need to be analyzed as a discrete phase, and the liquid phase is still treated as a continuous phase, that is, the Euler–Lagrange method is adopted. The governing equations of the liquid phase are like those of the mixed phases described above. ∂ρ + ∇ · (ρv) = 0 ∂t 

(2.28) 

( ) 2 ) ( ) ∂( ρ f v f + ∇ · ρ f v f v f = −∇ p + ∇ · μ f ∇v f + ∇v T − ∇ · v f I + ρ f g + F ∂t 3

(2.29)

2.1 Solid–Liquid Two-Phase Model

33

wherein, μ f is the liquid phase viscosity coefficient, I is the unit tensor, and F is the force between the particle and the liquid phase. Discrete phase particles are solved using the Lagrange method, which obtains its motion trajectory by integrating the momentum equation of the particle. The momentum equation for particles can be given directly by Newton’s second law: mp

 dv p = F dt

(2.30)

According to the magnitude analysis in Sect. 2.1.1, the force on the particle should contain gravity, resistance, and basset forces, so the particle momentum equation is simplified to ( ) t d u f − up ( ) 3 2 √ dv p 1 = m p g + 3π μ d p u f − u p + d p ρ f π ν ∫ ·√ mp · dτ dt 2 dτ t −τ t0 (2.31)

2.1.3 Model Suitability Study Based on the above two types of two-phase models, the two-phase flow of solid and liquid in a centrifugal pump is numerically calculated, and the calculated performance parameters are compared with the experimental results to confirm the applicability of the two-phase flow model. Centrifugal pumps have a flow rate of 150 m3 /h, a head of 80 m, and an impeller inlet diameter of 125 mm. In this study, the impeller and volute mesh were generated in the form of a hybrid mesh, and the mesh number of impeller and volute was determined to be 737,066 and 611,100, respectively, after the mesh independence check. The impeller inlet adopts the speed inlet, assuming that there is no velocity slippage between the granular phase and the liquid phase at the inlet, that is, the inlet velocity of the two phases is the same, and it is evenly distributed on the inlet crosssection, and the initial value of the given velocity is ν = 3.395 m/s. Export conditions are set to free outflow, assuming that the flow has been fully developed. The wall surface adopts the condition of a no-slip boundary, and the near wall area is treated with the standard wall function. The liquid phase is clean water at room temperature and the density ρ f is 1000 kg/m3 . The solid phase is sand grains and the material density ρ p is 2520 kg/m3 .

2.1.3.1

Performance Calculation Based on the Euler-Euler Method

In the design flow condition (Q = 150m 3 /h), the Mixture model based on the EulerEuler method was used to numerically simulate the solid–liquid two-phase flow of

34

2 Solid–Liquid Two-Phase Calculation Model and Method

Table 2.7 Error table of mixture model calculation and experimental results (absolute value)

Volume fraction

5 (%)

10 (%)

20 (%)

Head error (0.64 mm)

0.67

0.12%

1.19

Head error (1.27 mm)

6.24

7.62

5.02

Head error (2.2 mm)

11.81

12.52

11.90

Efficiency error (0.64 mm)

1.21

0.57

1.22

Efficiency error (1.27 mm)

7.06

6.19

11.57

Efficiency error (2.2 mm)

6.92

7.65

11.28

particles of different diameters and different volume fractions. The median particle size of the particles was 0.64 mm, 1.27 mm, and 2.2 mm, respectively. The solid phase volume fractions are 5%, 10%, 20%, and 30%, respectively. The comparison error between the calculation and experimental results is shown in Table 2.7. It can be seen from the result error Table 2.7 that when the particle size is small (0.64 mm), the numerical calculation results of the head ratio and the efficiency ratio are consistent with the experimental results, and the maximum error between the head calculation and the experimental results is 2.7%, the maximum error between the efficiency calculation and the experimental results is 1.22%, and the error values are very small, which shows that when the particle diameter is small, the volume effect is not obvious, so it is feasible to carry out a solid–liquid two-phase numerical simulation based on the Mixture model. However, when the particle size increases, the deviation between the numerical calculation results and the experimental results begins to increase, especially when the particle size increases to 2.2 mm, under different volume fraction working conditions, the numerical calculation results of the head ratio and the efficiency ratio and the experimental results have a very large error, basically, more than 10%, in which the maximum error between the head calculation and the experimental result is 12.52%, and the maximum error between the efficiency calculation and the experimental result is 11.28%, which shows that when the particle diameter increases, the volume effect occurs. At this time, it is no longer suitable for solid–liquid two-phase numerical simulation based on the Mixture model.

2.1.3.2

Performance Calculation Based on the Euler–Lagrange Method

From the previous calculation analysis, it is found that when the particle diameter size is small, the influence of the particle volume effect can be ignored, and the required accuracy can be achieved. When the particle diameter is increased to a certain order of magnitude or more, combined with the particle force analysis results (see Sect. 2.1.1), it is found that in the calculation process, ignoring the influence of basset force and other forces will seriously affect the calculation accuracy and lead to a larger error. In this study, the influence of Basset force is introduced in the calculation process through UDF (User-defined function), and the DPM model based on the Euler–Lagrange method is used to numerically simulate the solid–liquid

2.2 Collision Rebound Model Table 2.8 Error table of DPM model calculation and experimental results (absolute value)

35 Volume fraction

5 (%)

10 (%)

Head error (1.27 mm)

0.41

0.78

Head error (2.2 mm)

0.59

0.01

Efficiency error (1.27 mm)

1.20

1.08

Efficiency error (2.2 mm)

1.33

0.41

two-phase flow of two larger diameter particles at different volume fractions under the design flow condition. The median particle size of the particles is 1.27 mm and 2.2 mm, respectively. The solid phase volume fractions are 5%, 10%, and 20%, respectively. The comparison error between the calculation and experimental results is shown in Table 2.8. It can be seen from the error Table 2.8 that the numerical calculation results of the head ratio and efficiency ratio are consistent with the experimental results, and the calculation error of the head and efficiency is reduced by 5%–10% compared with that of using Mixture model. This shows that when the particles are large, the use of DPM models can significantly improve the calculation accuracy.

2.2 Collision Rebound Model In the process of solid–liquid mixed transport, the collision between particles and the wall surface will affect the subsequent movement trajectory of the particles, and at the same time cause impact and friction damage to the wall surface, so it is very important to obtain an appropriate particle–wall collision rebound model for accurating simulation of solid–liquid two-phase flow. The particle–wall collision rebound model is a mathematical model that describes the effect of a collision on particle motion and is an expression of the amount of change in certain motion parameters during the instantaneous process of particles touching the wall and bouncing up rapidly. In the current study, the collision recovery coefficient is considered a parametric indicator commonly used to model and evaluate collision motion. The usual research idea is to build a test bench under the corresponding working conditions and carry out many repeated single-particle collision experiments, through the acquisition and processing of experimental data combined with the calculation of various collision forces in the collision process, and finally, establish a parametric expression of the collision recovery coefficient. However, the previous research results mainly focused on fine particles, dry walls, static walls, etc., and there were fewer studies on wet wall collisions and moving wall collisions during solid–liquid mixed transport.

36

2 Solid–Liquid Two-Phase Calculation Model and Method

2.2.1 Wet and Dry Wall Collision Rebound The existing collision rebound models are mostly based on small particle gas–solid two-phase flow experiments and are corrected, which are different from the wet wall surface of solid–liquid mixed delivery. At the same time, when the particles are larger, their motion characteristics are also quite different from those of small particles, so this book conducts a visual experimental study on the collision rebound of large particles and wet walls to obtain a collision rebound model suitable for solid– liquid two-phase flow containing large particles. The ratio of the relative collision velocity of particles to the rebound velocity, that is, the recovery coefficient, is used to describe the collision rebound of particles.

2.2.1.1

Experimental Apparatus and Measurement Method

The particle–wall collision experimental device is shown in Fig. 2.1, which is mainly composed of a particle release device, a water tank, a test frame with adjustable angles, and a high-speed camera. The physical properties parameters of the particles and wall materials are shown in Table 2.9.

Fig. 2.1 Schematic diagram of the experimental apparatus

2.2 Collision Rebound Model

37

Table 2.9 Wall and particle material parameters Material

Density (kg/m3 )

Young’s modulus (GPa)

Poisson’s ratio

HBS

Particle diameter (mm)

316 stainless steel (wall)

8030

212

0.3

187

/

6061 aluminum alloy (wall)

2690

68.9

0.33

95

/

HT250 grey iron 7150 (wall)

135

0.3

209

/

No. 45 steel (particle)

210

0.269

197

3, 5, 7

7850

The particle collision process is taken by a CCD camera with a time interval of Δt = 0.5 ms for each of the two pictures. The three pictures before and after the collision are intercepted to obtain the particle incidence velocity and rebound velocity respectively, and the sphere is identified by the Matlab Imfindcircles function, the coordinates of the sphere center are obtained, and the particle velocity before and after the collision and the collision rebound angle are obtained after conversion (Fig. 2.2). The normal incidence velocity is: V p1 =

L pix sin β1 RRpix Δt

(2.32)

Among them, L pix is the pixel distance between two adjacent pictures; β 1 is the angle of incidence of the particle, which can be directly adjusted through the specimen holder; R is the particle radius; Rpix is the pixel radius of the particle; Δt is the time interval between two adjacent pictures. The rebound angle is: β2 = (90 − β1 ) − arctan((X 2 − X 1 )/(Y2 − Y1 ))

(2.33)

Among them, X 1 , X 2 , Y 1 , Y 2 is the pixel abscissa and ordinate coordinates of two adjacent frames when the particle bounces. In the same way, the tangential incidence velocity, normal rebound velocity, and tangential rebound velocity can be obtained (Fig. 2.2). As shown in Fig. 2.3, the elastic recovery coefficient of collision rebound between particles and wall surfaces is divided into two components: normal (en ) and tangential (et ). Represents the rate of change of momentum in the vertical wall direction and the wall tangent direction after the collision between the particle and the wall, respectively, ev is the total recovery coefficient, the angle recovery coefficient (eβ ) is the ratio of the rebound angle and the collision angle before and after the collision, and the calculation expression is as follows: en = v p1 /v p2

(2.34)

38

2 Solid–Liquid Two-Phase Calculation Model and Method

Fig. 2.2 Particle boundary identification diagram

2.2.1.2

et = u p1 /u p2

(2.35)

ev = V1 /V2

(2.36)

eβ = β2 /β1

(2.37)

Analysis of Experimental Results Under Wet and Dry Wall Conditions

Sommerfeld et al. [15] found that when the particle diameter is small (100–500 μm), the particle diameter and wall surface roughness have a significant impact on the recovery coefficient, the larger the particle size, the smaller the recovery coefficient, and as the surface roughness increases, the smaller the particle size has less influence on the recovery coefficient. The above conclusions are only for small particle sizes, but there is relatively little research on the collision rebound of large particles. This book studies the collision rebound of particles with large diameters (3–7 mm), and

2.2 Collision Rebound Model

39

Fig. 2.3 Schematic diagram of particle collision with the wall surface

the normal recovery coefficient curve of collision of 6061 aluminum alloy dry surface particles under different particle sizes is shown in Fig. 2.4. The study found that under different particle sizes, the normal and tangential recovery coefficient curves are almost the same, indicating that in the case of large particle sizes, the recovery coefficient of particle collision does not change with the change of particle size. To further analyze the influence of parameters such as material hardness and wall friction coefficient on the recovery coefficient, a 45 steel ball with a particle size of 7 mm was used to conduct collision experiments with dry and wet walls of different materials.

(a) Normal recovery coefficient

(b) Tangential recovery coefficient

Fig. 2.4 Curves of collision recovery coefficient at different particle sizes (6061 aluminum alloy, dry surface)

40

2 Solid–Liquid Two-Phase Calculation Model and Method

Figure 2.5 shows the particle collision angle and recovery coefficient curve under different collision conditions. Each collision angle is tested separately 30 times, taking the average of its recovery coefficients. With the change of collision angle, the normal recovery coefficient and angle recovery coefficient of 316 stainless steel is significantly greater than the normal and angle recovery coefficient of 6061 aluminum alloy, and slightly less than the normal and angle recovery coefficient of HT250 gray iron. This is related to the hardness of the material, the greater the hardness, the greater the normal recovery coefficient and the angle recovery coefficient. However, the tangential recovery coefficient has no obvious relationship with the hardness of the material. It is mainly related to the friction coefficient of the wall surface and the collision angle. There is a clear difference in the collision rebound characteristics of particles under immersion conditions and dry surface conditions, especially in the case of small collision angles or large collision angles. The three materials have the same change trend when the collision angle is less than 20°, the normal recovery coefficient, and angle recovery coefficient decrease with the increase of the collision angle, when the collision angle is greater than 20°, the normal recovery coefficient gradually tends to be stable, and the angle recovery coefficient gradually increases. The total recovery factor decreases gradually as the collision angle increases. When the collision angle is less than 70°, the tangential recovery coefficient gradually decreases with the increase of the collision angle. To apply the recovery coefficient measured by the experiment to the numerical simulation, the curves in Fig. 2.5 are polynomially fitted to obtain the polynomials shown below, and the constants in the polynomials are shown in Tables 2.10 and 2.11. The following functional relationship formulas are the collision rebound model of large particles and wall surfaces in water and air, which can be directly applied to the numerical calculation of solid–liquid two-phase flow, which is more in line with the large-particle solid–liquid two-phase flow conditions than the collision rebound model commonly used at present. en =



ai β1i−1

(2.38)

bi β1i−1

(2.39)

ci β1i−1

(2.40)

di β1i−1

(2.41)

i=1

et =

 i=1

eβ =

 i=1

ev =

 i=1

Among them, ai , bi , ci , d i are fitted coefficients. By solving the momentum equation and Coulomb’s law, the collision equation of the particles in the absence of slip and slip collisions is obtained, and the equation of the friction coefficient of the surface of the particle collision with the wall surface is obtained by the conversion to eliminate the influence of the particle speed and

2.2 Collision Rebound Model

41

(a) Normal recovery coefficient

(b) Tangential recovery coefficient

(c) Angle recovery coefficient

(d) Total recovery factor

Fig. 2.5 Curve of particle collision angle and recovery coefficient

particle diameter [16], as shown in Eq. (2.42). μ=

| | |u p1 − u p2 | (1 + en )vp1

(2.42)

The relationship between the different wall friction coefficients and particle collision angles is shown in Fig. 2.6. Figure 2.6a is a diagram of the friction coefficient and collision angle of different particle sizes colliding with HT250 gray iron under dry surface and immersion conditions. As can be seen from the figure, the change in the diameter of the particle does not cause a significant change in the friction coefficient and the collision angle curve.

42

2 Solid–Liquid Two-Phase Calculation Model and Method

Table 2.10 Collision fitting coefficients in water Wall material

6061 aluminum

a1

a2

a3

a4

a5

a6

a7

b1

b2

b3

b4

b5

b6

b7

c1

c2

c3

c4

c5

c6

c7

d1

d2

d3

d4

d5

d6

d7

0.9985 –

10.701

–

3.3165 1.0005 –

2.6915 –

0.9995 –

13.952

316 stainless steel 1.002

0.9977

1.0001 –

19.05

–

–

27.637

8.807

– 8.5475 15.359 – 0.3937 2.3302

4.0924 14.725

–

27.369 – 3.46

1.4714 7.9881

10.824

3.5462 0.166

–

18.589

0.4527

0.999

17.182

2.4947 13.445

10.6

5.8872

– 10.872

2.6972

– 4.9723 4.1083

– 1.2089 – – 1.1526

0.1687 1.0012 1.9118 1.0007 –

– 13.164

25.053

– 0.8018 0.5993

– 18.853

4.9806

– 0.1203 –

– –

0.0146 HT250 cast iron

1.003

–

– 0.9349 3.1692

2.9395 0.889

–

– 0.1693 0.6937

– 0.8428 0.2778

–

0.4914 0.9993 – 0.1745 1.0056 –

0.3606 0.2741

0.1923 –

–

0.2915 0.1531

–

0.5067 0.9995 – 0.183

– 0.139

0.1798

Sommerfeld et al. [15] found that when the particle diameter is small (100–500 μm), particle diameter has a significant impact on the friction coefficient, the larger the particle size, the smaller the friction coefficient. And in the small collision angle, due to the influence of wall roughness, the difference in friction coefficient is particularly obvious. As the collision angle increases, the coefficient of friction tends to a stable value. However, the study in this study found that when the particle diameter is large, this law does not apply, and the change in particle size does not cause a significant change in the friction coefficient and collision angle curve. This shows that in the case of large particle diameter, the particle diameter has no effect on the friction coefficient of the particle collision with the wall surface, so there is a large error in the calculation of the value of the solid–liquid two-phase flow of large-diameter particles. Figure 2.6b shows the effect of different wall materials on the coefficient of friction under dry surfaces and immersion conditions. As shown in the figure, the three materials have a similar tendency for the coefficient of friction to change with the collision angle. When the friction coefficient is 20–50 at the collision angle, the friction coefficient remains unchanged, but when the collision angle is greater than

2.2 Collision Rebound Model

43

Table 2.11 Collision fitting coefficients in air Wall material

6061 aluminum

a1

a2

a3

a4

a5

a6

a7

b1

b2

b3

b4

b5

b6

b7

c1

c2

c3

c4

c5

c6

c7

d1

d2

d3

d4

d5

d6

d7

0.9971 – 2.7248

6.427

1.0006 – 0.5034

3.2429 – 12.647

0.9998 – 3.4062 13.063

– 7.5429 – 25.317

4.1638 – 0.8608 21.643

– 16.393

25.425

– 12.312

0.9995 0.2001

–3.5525 8.8031

– 10.347

5.6068

316 stainless steel 1.0022 1.2414

–9.6573 17.105

– 12.019

2.9661

4.4902 2.2715 – 1.1192 –

0.9992 – 0.0818 –0.2784 0.0888

0.0321 –

–

1.0076 – 0.0872 –1.6094 2.5941

– 1.0119 –

–

1.0002 – 0.0063 –0.4778 – 0.842 HT250 cast iron

–

1.003

– 0.4914 –0.9349 3.1692

0.9993 – 0.1745 –0.1693 0.6937 1.0017 0.4268

–7.1048 21.493

0.9995 – 0.183

–0.139

0.1798

1.7525 – 0.7532

–

– 2.9359 0.889

–

– 0.8428 0.2778

–

– 27.699

16.598

– 0.2915 0.1531

– 3.7824 –

(a) Different media, different particle size particles collide with wall surface of HT250 gray iron (b) Different media, particle size 7 mm steel balls collide with different wall surfaces

Fig. 2.6 Relationship between impact angle and friction coefficient

50°, the friction coefficient increases with the increase of the angle. Because the dynamic friction coefficient does not change with the change of the collision angle, so when the collision angle is less than 50°, the particles are slip collisions, and when it is greater than 50°, the friction coefficient changes significantly with the change of the angle, indicating that the friction coefficient at this time is the static friction coefficient, so when the collision angle is greater than 50°, it is a non-slip collision.

44

2 Solid–Liquid Two-Phase Calculation Model and Method

The friction coefficient of 316 stainless steel is less than the friction coefficient of 6061 aluminum alloy and slightly greater than the friction coefficient of HT250. And the friction coefficient of the same wall material in the water is slightly less than the friction coefficient in the air.

2.2.1.3

Model Applicability Verification

Because the collision rebound between particles and walls has a great influence on the particle motion trajectory, the applicability of the proposed collision rebound model is verified by using the experimental results of the visualization of the particle motion trajectory in the bent tube. The visual experimental device is composed of a water tank, a centrifugal pump, an electromagnetic flowmeter, an electric control valve, a particle release device, and a plexiglass bend pipe, which pushes the particles into the pipe by squeezing the push rod in the particle release device downwards, and the particles enter the bend pipe after full development with the water flow and collide with the wall of the bend pipe (Fig. 2.7). The particles are 45 steel balls, with a particle size of 3 mm, the wall material is stainless steel, and the particle movement process is photographed by a high-speed camera (Fig. 2.7). Figure 2.8 shows the particle trajectory map obtained by visual experiments and calculations based on different collision rebound models, and the numerical calculation results using three different models are significantly different. The collision rebound model and normal and tangential recovery coefficients proposed by Grant et al. [17] are all constant models for numerical calculation, and the number and trajectory of collisions between particles and walls are significantly different from the visual experimental results. The particle trajectory diagram solved by using the model obtained by the collision rebound experiment between large particles and wet

Fig. 2.7 Experimental apparatus for visualization of particle motion trajectory

2.2 Collision Rebound Model

45

(a) Experimental particle trajectory (b) Collision rebound model obtained experimentally

(c) Grant collision rebound model

(d)en=0.7, et=0.7

Fig. 2.8 Calculation of the trajectory diagram of the experimental trajectory of the particles and the different collision rebound models

wall surfaces is closer to the actual working conditions. This is because, under the same working conditions, the rebound angle and rebound speed after the particle rebound are calculated by different models, so it will lead to different subsequent movement trajectories of the particles, so finding a collision rebound model suitable for the actual working conditions can accurately simulate the particle movement in the solid–liquid two-phase flow. At the same time, it also proves that the collision rebound model obtained by experimental means has good applicability to the numerical simulation of large particle solid–liquid two-phase flow.

2.2.2 Rotating Wall Collision Rebound In the project, rotating machinery such as centrifugal pumps are commonly used to lift solid–liquid two-phase media, and its internal impeller flow channel always maintains a rotational movement. Under the rotating wall condition, the particle– stationary wall collision rebound model is not fully applicable, so it is necessary to carry out the collision rebound experiment under the rotating boundary condition to study the change of the particle motion state after the collision with the rotating wall.

46

2.2.2.1

2 Solid–Liquid Two-Phase Calculation Model and Method

Experimental Apparatus and Measurement Method

The experimental device for studying the collision and rebound of particles and wall surfaces under rotating wall conditions is shown in Fig. 2.9, which is mainly composed of the particle injection device, a rotating wall table, and a high-speed photography acquisition and processing system. The high-speed photography acquisition system is composed of two high-speed cameras placed vertically, shooting the moving pictures in the two directions of the front view and the side view respectively, and the shooting files of each camera are post-processed to obtain the motion parameters of the particles, and then fitted to obtain the motion parameters of the particles in the three-dimensional coordinate system. The incident particles are steel ball particles of 316 stainless steel with a diameter of 3 mm, 5 mm, and 7 mm respectively, and the collision wall surface is a circular wall surface made of 6061 aluminum alloy with a diameter of 100 mm. Among the various existing definitions of recovery coefficients, the Newton recovery coefficient and the Poisson recovery coefficient consider the linear velocity of the particles, but ignore the effects of particle rotation, and only apply to the ideal experimental conditions of particle impact on a smooth stationary plane. For the collision rebound model of particles and rotating walls, when the moving particles collide with the rotating wall, the spin motion is inevitably generated when they bounce up, so the rotational motion of the particles cannot be ignored. The particle rotation speed in this experiment is determined by counting the number of frames required n for the marker point on the particle to turn 180°, and the calculation

Fig. 2.9 Schematic diagram of the particle–wall collision rebound experimental apparatus under the condition of rotating boundary

2.2 Collision Rebound Model

47

formula is as follows: ωin/out =

π (n in/out − 1) ∗ Δti

(2.43)

where ω is the particle rotation speed and Δt i is the time interval between two adjacent frames of photographs. Since the direction of rotation of the wall is counterclockwise, the spin movement trend after a collision with the wall is always opposite to the direction of rotation of the wall. When the rotation speed of the wall surface changes, the angular velocity of the particle rotation changes, but the rotation direction does not change. The effect of the rotational motion of the wall surface on the collision rebound motion of the particles was studied by the following parameters: the energy recovery coefficient of the particle eenergy , the angle recovery coefficient of the particle eβ , and the deflection angle α generated by the rotating wall surface when the particle rebounded. The kinetic energy recovery coefficient of the particle eenergy characterizes the amount of kinetic energy change of the particle before and after the collision, defined as follows: eenergy =

E out E in

(2.44)

wherein, E in , E out are the total kinetic energy of the incident and rebound of the particles respectively, which are calculated as follows: 

E = E liner + E rotation

(2.45)

Among them, E liner and Rotation are the linear velocity component and rotational velocity component of the total kinetic energy of the particle. Since there is no rotation when the particle is ejected, and rotation occurs after contact with the rotating wall, the total amount of incident kinetic energy of the particle E in and the total amount of rebound kinetic energy of the particle E out can be defined as follows: 1 2 mv 2 in

(2.46)

1 2 1 2 mvout + I ωout 2 2

(2.47)

E in = E out =

Among them, m is the mass of the particle, vin and vout are the linear velocities of the particle incident and rebound, ωout is the rotational speed of the particle when the particle rebounds, and I is the moment of inertia of the particle. The angle recovery coefficient of the particle is defined as the ratio of the incident angle β in and the rebound angle β out of the particle. Before the particle reaches the

48

2 Solid–Liquid Two-Phase Calculation Model and Method

collision point, its motion is completely within the plane YOZ, which can be regarded as the motion within the two-dimensional plane. After contacting the rotated wall and bouncing up, its motion trajectory and velocity vector direction are in XYZ three-dimensional space. Define the angle between the rebound velocity vector vout of the particle and the plane XOY as its rebound collision angle β out , and the angle of incidence β in is still the angle between the incidence velocity vin and the Y-axis in the coordinate system. Thus, the particle collision angle β and the particle angle recovery coefficient eβ are defined as follows: βin/out = arctan eβ =

|Z in/out | vin/out

βout βin

(2.48) (2.49)

where the denominator is the incidence velocity or rebound velocity value of the particle correspondingly, and the numerator is the absolute value of component of the corresponding velocity vector on the Z-axis in Eq. (2.48). To characterize the instantaneous linear motion of particles in three-dimensional space after the collision, the deflection angle α is defined as the angle between the motion vector vout of the particle’s rebound speed and the plane YOZ. It is calculated as follows: α = arcsin

|X out | vout

(2.50)

where the denominator is the rebound velocity value of the particle, and the numerator is the absolute value of component of the rebound velocity vector on the X-axis.

2.2.2.2

Analysis of the Experimental Results Under Rotating Wall Conditions

Using 7 mm of 316 stainless steel granules and 6061 aluminum alloy disks, the rotation speed of the wall surface is selected as 0 (stationary), 10°/s, 30°/s, and 50°/s, and the speed is controlled by the stepper motor by the electrical signal, and the phase angle is precise. The angle of incidence is selected as 15°, 30°, 45°, 60°, 75°, and 90°. In this group of experiments, the relationship between the kinetic energy recovery coefficient eenergy of the particles, the collision angle recovery coefficient eβ of the particles, and the deflection angle of the particles α and the speed change of the wall surface were analyzed. Figure 2.10a depicts the kinetic energy recovery coefficients of 7 mm steel particles and 6061 aluminum alloy wall surfaces at wall speeds of 0, 10°/s, 30°/s, and 50°/s. It can be found that the kinetic energy recovery coefficient of the particles

2.2 Collision Rebound Model

49

under the different wall conditions decreases gradually with the increase of the incidence angle, which indicates that the kinetic energy loss of the particles at the large collision angle is more. In the figure, the kinetic energy recovery coefficient of the particle is the highest at the wall speed of 50°/s, followed by the wall speed of 30°/s and the wall speed of 10°/s, while the kinetic energy recovery coefficient curve of the particle is at the bottom under the static wall condition. The kinetic energy recovery coefficient of the particles increases with the rotational speed of the wall surface. This phenomenon indicates that the steel ball obtains an increase in kinetic energy during the collision of the rotating wall surface, and the increase value becomes larger with the speed of the rotating wall. Figure 2.10b, c illustrate the kinetic energy ratio of particles in the rebound process after the particles meet the rotating wall, that is, the ratio between linear velocity kinetic energy and total kinetic energy and the ratio between angular velocity kinetic energy and total kinetic energy respectively. For the solid lines that describe the kinetic energy ratio of the particle line speed, the curves of different wall speeds from top to bottom is inversely related to the trends of the wall speed, and for the dotted lines that describe the proportion of the kinetic energy of the particle line

(a) Kinetic energy recovery coefficient (b) Ratio of translational kinetic energy to total kinetic energy after particle collision (c) Ratio of rotational kinetic energy to total kinetic energy after particle collision Fig. 2.10 Effect of wall speed on particle kinetic energy

50

2 Solid–Liquid Two-Phase Calculation Model and Method

speed, the curves of different wall speeds from top to bottom is positively related to the trends of the wall speed. This shows that the greater the rotation speed of the rotating wall, the greater the proportion of the rotating part of the particles. A single curve shows a trend of increasing first and then decreasing, indicating that with the increase of the incident angle of particles, the speed of particles tends to increase first and then decrease. In the case of low wall speed, this trend is not obvious, and the curve is relatively gentle. Figure 2.11a depicts the collision angle recovery coefficient of the particles with the angle of incidence, and the four curves represent the results of the particles at speeds of 0, 10°/s, 30°/s, and 50°/s, respectively. It can be concluded from the figure that when the particles collide with the wall surface under different wall rotation speed conditions, the angle recovery coefficient tends to increase with the increase of the incidence angle. At the same time, the value of the angle recovery coefficient eβ is maintained at a high level in the range of 0.7 to 1.0, indicating that the angle of incidence and rebound angle before and after the collision change less, and the way the particles are in contact with the moving wall is relatively single. The coefficient curves of the three wall speeds are not significantly distinguished, indicating that the different speeds of the rotating wall surfaces have no significant effect on the collision angle recovery coefficient of the particles. Compared with the static wall condition, the collision angle recovery coefficient under the rotating wall condition is greater than the collision angle recovery coefficient under the static wall condition, and the main role of the angle recovery coefficient is to measure the collision posture of the particles when they are in contact with the wall, which indicates that the rotational motion of the wall affects the contact mode between the particles during the collision instantaneous and the wall. According to the tangential elastic compliance theory, the classification model of the particle motion state on the wall surface is described. The dimensionless collision

(a) Recovery coefficient of the collision angle

(b) Results analysis based on tangential compliance theory

Fig. 2.11 Effect of different wall rotation speeds on the collision angles of particles

2.2 Collision Rebound Model

51

angle is Ψ defined as follows: Ψin/out =

K1 tan βin/out μc

(2.51)

Among them, μc is the maximum static friction coefficient of the collision between the particle and the wall, that is the Coulomb coefficient; while K 1 represents the tangential elastic compliance, which represents the ratio of the tangential contact stiffness and the normal contact stiffness of the two materials of the particle and the wall. When the material of the particles and the material of the wall surface are different, K 1 is defined as follows: K1 =

1−υ1 G1 υ 1− 21 G1

+ +

1−υ2 G2 υ 1− 22 G2

(2.52)

Among them, υ i and Gi represent the Poisson’s ratio and Young’s modulus of particles and wall materials, respectively. The conclusion is that in the interval of Ψ in < 1, the contact mode between the particle and the wall surface is rolling; in the interval of 1 < Ψ in < 4χ-1, the contact mode transitions from the rolling of the particles to the sliding of the particles on the wall; in the interval of Ψ in > 4χ-1, the contact mode between the particles and the wall surface is mainly the state of a rigid body sliding. The value of χ is 7K 1 /4. According to the theory of tangential elastic compliance, the relationship between the dimensionless incident angle Ψ in and the dimensionless rebound angle Ψ out in Fig. 2.11b shows that under the static wall condition, when the value of Ψ in is between 0.5 and 1.0, there is a turning point in the fitted line, indicating that with the change of the incident angle, the contact mode between the particle and the wall surface tends to change from viscous to sliding. Under the rotating wall condition, the main contact method between the particles and the wall surface is sliding, that is, the particles will slip when the wall surface is rotated, the friction between the particles and the wall surface must be sliding friction. Figure 2.12 depicts the angle between the rebound velocity of the 7 mm particle and the ZOY plane after colliding with the wall surface at a wall speed of 10°/s, 30°/s, and 50°/s. The curve of the wall rotation speed of 10°/s is higher than the curves of 30°/s and 50°/s, and this distribution trend shows that the larger the wall rotation speed, the smaller the particle collision deflection angle, that is, the higher the wall speed, and the smaller the impact of the rotation speed on the deflection angle of the particle. At the same time, the collision deflection angle of the particles becomes larger with the angle of incidence, indicating that the incident angle of the particles also affects the particle deflection angle. According to the experimental results, it is found that the kinetic energy recovery coefficient of the particle is always less than 1, indicating that the loss of mechanical energy during the collision and rebound of the particle is inevitable. The analysis results of the kinetic energy components (see Fig. 2.10) show that there is a

52

2 Solid–Liquid Two-Phase Calculation Model and Method

Fig. 2.12 Effect of wall speed on deflection angle

phenomenon of the translational kinetic energy decreasing and the rotational kinetic energy increasing after the collision. Based on analysis of the collision rebound motion of particles in the Cartesian coordinate system the motion of particles is divided into two parts: the vertical incidence direction on the Z-axis and the tangent plane direction of the particle collision point on the XOY plane. As shown in Fig. 2.13, the energy loss in the normal direction is mainly the energy loss of the direct contact between the particles and the wall surface during the collision [18, 19]. The reason for this part of the loss is mainly that the elastic wave propagation loss inside the material between the particles and the wall during the collision process and the loss in the energy conversion between mechanical energy and internal energy, and some of the losses lead to plastic deformation on the wall. According to the theory of solid friction [19], the particles are in plastic contact with the surface of the wall surface, and the actual contact area accounts for only a small part of the apparent contact. The stress at the contact peak point under the action of the impact force reaches the yield limit of the pressure and produces plastic deformation. After the peak point is exceeded, the stress at the contact point no longer changes, and the continuous increase in load can only be borne by expanding the contact area. In the collision point tangent plane, the main cause of energy loss is the frictional force. According to the theory of viscous friction, the metal at the contact point is in a plastic flow state, so there is an adhesion effect between the two metal materials, and the bonding node has strong adhesion, and under the action of friction, the bonding node is sheared and slides. From the previous analysis, it can be found that when the particles collide with the rotating wall, the particles will always slide on the wall, so the sliding friction is the reason for the main kinetic energy loss of the particles in the XOY plane. Since

2.2 Collision Rebound Model

53

Fig. 2.13 Schematic diagram of the distribution of energy losses during collision rebound

the direction of sliding friction coincides with the plane XOY, and the contact point is a little at the edge of the particle, the sliding friction force also causes the rotational motion of the particle. Before analyzing the energy loss in both directions, it is necessary to perform a qualitative analysis of the experimental results of the normal recovery coefficient en and tangential recovery coefficient et of particle motion. en can be defined as the recovery coefficient in the direction of the Z-axis in three-dimensional space, that is, the ratio of the velocity component of the rebound velocity in the Z-axis direction to the velocity component of the incident velocity in the Z-axis direction. The tangential velocity of the particle after collision in three-dimensional space should be the velocity component of the rebound instantaneous tangent plane, and the value of the vector component in the Cartesian coordinate system is equal to the velocity component of the rebound velocity in the XOY plane, which is calculated as follows: | z | |v | | (2.53) en = || out vinz | et =

X OY vout vinX OY

(2.54)

wherein, the denominator represents the component of the incidence velocity in the XOY plane, since the incidence velocity vector is always within the plane YOZ, its t ; and the numerator represents the component of the rebound value is equal to vin velocity on the XOY plane. Figure 2.14 shows the particle normal recovery coefficient en and tangential speed recovery coefficient et with the angle of incidence under four working conditions of the wall rotation speed of 0, 10°/s, 30°/s, and 5°/s. From the figure, the normal recovery coefficient of the particle under four different operating conditions remains within a stable range at different angles of incidence, and the range of change is very small. The recovery coefficient of particle normal velocity under static wall conditions is smaller than that of rotating wall surface, but the change of normal recovery coefficient under different wall rotation speed conditions is not obvious.

54

2 Solid–Liquid Two-Phase Calculation Model and Method

It shows that the normal speed recovery coefficient is only related to the material characteristics between the particles and the wall. However, the tangential velocity recovery coefficient of the particles under different working conditions has the same trend of change, that is, as the incidence angle β in increases, et tends to decrease first and then increase, and this convergence indicates that the tangential recovery coefficient of the particles is related to the angle of incidence. The greater the rotation speed of the wall, the tangential velocity recovery coefficient is about close to the experimental result curve of the static wall, and the smaller the rotation speed of the wall, the smaller the et of the particles. This distribution indicates that the kinetic energy loss in the cutting plane of the particle is greater than that of the high wall speed at low wall speed, and the kinetic energy loss of the tangent plane under static wall conditions is relatively minimal. The Z-axis direction of vertical incidence is the normal direction of particle incidence and rebound velocity. The particle kinetic energy loss in this direction can be approximated as the contact energy loss before and after particle collision. The calculation method is as follows:

(a)Normal recovery factor (b) Tangential recovery factor Fig. 2.14 Velocity recovery coefficient of particles at different wall speeds

2.2 Collision Rebound Model

55

1 1 2 Z 2 m(vinZ ) − m(vout ) 2 2 1 2 1 2 = mvin sin2 βin − mvout sin2 βout 2 2 ( ) 1 2 = mvin sin2 βin 1 − e2n 2

E contact = E normal =

(2.55)

Similarly, the loss of kinetic energy before and after collision in the direction of the tangent plane can be expressed as: 1 1 2 X OY 2 ) m(vinX OY ) − m(vout 2 2 1 2 1 2 = mvin cos2 βin − mvout cos2 βout 2 2 ( ) 1 2 cos2 βin 1 − e2t = mvin 2

E tangent =

(2.56)

The amount of kinetic energy loss on the tangent plane includes both the energy loss of the particle rotation caused by the sliding friction after the collision and the energy loss of the sliding friction process. So, the sliding friction loss can be obtained by subtracting rotating kinetic energy from the total particle kinetic energy reduced before and after the collision on the tangent plane. The formula for calculating the kinetic energy of a spherical particle rotating around its sphere is as follows: E rotation =

1 2 Iω 2

(2.57)

where, I is the moment of inertia of the particle, and the value is I = 25 m R 2 when the spherical particle rotates around its center; R is the radius of the particle; m is the mass of the particle. Sliding friction loss kinetic energy is the total loss of kinetic energy on the tangent plane minus the energy that causes the particles to rotate, that is: E slip =E tangent − E rotation ( ) 1 1 2 cos2 βin 1 − et2 − I ω2 = mvin 2 2

(2.58)

Figure 2.15 depicts the trend of the spin angular velocity of the particle after colliding with the rotating wall surface with the angle of incidence. The three polylines in the figure represent the spin speed of the particles during the rebound process at different wall rotation speeds. It can be found that the spin speed of the 7 mm diameter particles during the rebound process is between 3 rad/s and 11 rad/s, and there is a clear positional difference between the three curves. Overall, two trends can be summarized in the chart. First, the rotational angular velocity of the particles increases with the increase of the angle of incidence, and there is a tendency

56

2 Solid–Liquid Two-Phase Calculation Model and Method

Fig. 2.15 Rotational velocity of particles after a collision at different wall speeds

to increase first and then decrease, which is consistent with the distribution of the proportion of kinetic energy in the particle rebound process in Fig. 2.10; second, the three curves of the rotation speed of different wall surfaces show a trend of decreasing from top to bottom, which shows that the rotation speed of the rotating wall surface is related to the rotational angular velocity of the particles after the collision, and the larger the rotation speed of the wall surface, the greater the spin speed during the particle rebound process. From the analysis of the theory of contact elastic compliance in Fig. 2.11b, sliding friction will occur when the moving wall surface meets the particle, so the influence of the particle rotation angular velocity is not the contact method is different, but only the size of the incident angle. According to the above formula, the distribution of the contact energy loss to the initial kinetic energy ratio and the sliding friction loss to the initial kinetic energy at the wall speed of 10°, 30°, and 50° per second are calculated respectively, as shown in Fig. 2.16. Figure 2.16a depicts the situation of contact energy loss, the abscissa of the plot is the angle of incidence β in , and the vertical coordinate is the ratio of the total kinetic energy in the normal direction to the total kinetic energy of the particle at incidence. In the figure, the black dotted line is the total kinetic energy ratio line, that is, y = 1, and the red dotted line is the trend line fitted to each point of the experimental results. Figure 2.16b depicts the ratio of sliding friction loss to initial kinetic energy during the collision, and the experimental results use the incident angle as the abscissa. In Fig. 2.16a, as the angle of incidence continues to increase, the contact energy loss caused by collisions between particles and wall continues to increase. And

2.2 Collision Rebound Model

57

(a) Proportion of contact energy loss

(b) Proportion of dynamic friction loss

Fig. 2.16 Ratio of particle kinetic energy loss to total kinetic energy at different wall speeds

the ratio range is between 0 and 0.7, indicating that contact loss is the main form of energy loss during the collision. However, there is no obvious distinction between the experimental result points of different wall speeds, and the trend is similar, indicating that the wall rotation speed has almost no impact on contact loss. The ratio of contact energy loss to total energy is positively correlated with the angle of incidence, and the data points in the plot are distributed in the same straight line. According to the experimental study of particle collision rebound under static wall conditions, the contact energy loss of particles in the normal direction is mainly related to the elastic coefficient of the interaction between particles and wall materials ε. The coefficient is defined as the ratio of the viscosity force to the stiffness of the solid, that is, the ratio of the force causing the deformation to the strength value of the solid against deformation. The coefficient ε characterizes the difficulty of deformation of the material, and is defined as follows: ε=

4K 2 μvin R 3/2

(2.59)

5/2

π x0

where x 0 is the initial distance between the particle and the colliding wall, vin is the incident velocity, R is the particle radius, and μ is the dynamic viscosity of the ambient fluid. K 2 is defined as follows: ( K2 =

1 − υ22 1 − υ12 + G1 G2

) (2.60)

58

2 Solid–Liquid Two-Phase Calculation Model and Method

where Gi represents Young’s modulus value of the particles and wall materials, and υi represents the Poisson ratio of the particles and wall materials. Therefore, when the wall material and the particle material remain unchanged, the contact energy loss of the particles under the rotating wall surface will not change with the wall speed. That is, contact energy loss is always only related to the material properties of the particles and walls. As can be seen from Fig. 2.16b, the relative amount of sliding friction loss is gradually decreasing as the angle of incidence increases. As can be seen in the figure, the sliding friction loss ranges from 0.3 to –0.3, and the negative value represents the positive work of the rotating wall facing the particle, that is, the friction between the particle and the wall surface increases the kinetic energy after the collision of the particle. In terms of the distribution of scattering results in the figure, there is a distinction between the scatters of particles at three wall speeds: the experimental result scatters of the wall speed of 10°/s are located at the highest point, the scatters of the wall speed experimental results of 50°/s are at the lowest, and the experimental results of the wall speed of 30°/s are located in between. This distribution indicates that the greater the wall speed, the less the relative amount of energy lost by sliding friction. Since this part of the energy loss occurs on the XOY surface of the tangent plane at the point of collision, the main force between the particles and the wall surface in this plane is the sliding friction force. Sliding friction occurs, then there must be a relative displacement between the particles and the wall surface in a very short period, and the greater the wall speed, the smaller the relative displacement at the same time, and the smaller the working distance of the sliding friction, the less kinetic energy loss (or increase) is generated. The greater the rotation speed of the wall surface, the less the slip distance of the collision instantaneously, which also leads to the phenomenon that the particle rebound deflection angle α decreases with the increase of rotation speed.

2.2.2.3

Model Applicability Verification

To verify the collision rebound model between particles and rotating wall surfaces, a rotating disc simulating a closed impeller was designed, and the speed adjustment was adjusted by connecting the frequency converter with the frequency conversion motor, which was 400 r/min, 500 r/min, 600 r/min, 700 r/min, and 800 r/min, respectively, and the experimental verification of solid–liquid two-phase flow wear inside the rotating disc was carried out (Fig. 2.17). To obtain the thickness loss of the wear wall surface, the wear depth of the wear surface is measured with a laser displacement sensor (Fig. 2.17). The internal solid–liquid two-phase numerical simulation of the rotating disc runner is based on the Euler–Lagrange method, the control equation is described in Sect. 2.1.2, and the collision rebound model selects the experimental results of the collision of the rotation boundary. The wear model proposed by Ahlert [20] has the following expressions:

2.2 Collision Rebound Model

59

Fig. 2.17 Rotating disc experimental bench

E R = A(B H )−0.59 Fs u npA f (θ )  f (θ ) =

aθ 2 + bθ θ ≤ θ0 x cos θ sin(wθ ) + y sin2 θ + zθ > θ0 2

(2.61)

(2.62)

wherein, A is the constant associated with the wall material; B H is the Brinell hardness of the wall material; Fs is the coefficient related to the shape of the particle (the spherical particle is taken to 0.2, the sharp particle is taken to 1); u p is the particle collision velocity; f (θ ) is the particle collision angle function. The pair of wear experiments and calculation results are shown in Fig. 2.18, and the coincidence between the experiment and the calculation results is very good. It can be found that the wear area develops with the increase of the speed to the blade entrance, and the thickness loss of the experimental aluminum sheet gradually increases from the inlet to the outlet under various working conditions, and the difference in the value of the thickness loss of the blade at each position increases after the speed increases. Therefore, the model obtained by the rotating wall collision rebound experiment can significantly improve the simulation accuracy of solid phase motion, thereby reducing the overall numerical calculation error of solid and liquid phases.

60

2 Solid–Liquid Two-Phase Calculation Model and Method

Fig. 2.18 Comparison of experimental and numerical wear on blades at different speeds

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11. Guoren D (1981) Turbulent flow mechanics[M]. People’s Education Press, Beijing (In Chinese) 12. Odar F (1966) Verification of the proposed equation for calculation of the forces on a sphere accelerating in a viscous fluid[J]. J Fluid Mech 25(03):591–592 13. Tsuji Y, Morikawa Y, Mizuno O (1985) Experimental measurement of the Magnus force on a rotating sphere at low Reynolds numbers[J]. J Fluids Eng 107(4):484–488 14. Mei R (1992) An approximate expression for the shear lift force on a spherical particle at finite Reynolds number. Int J Multiph Flow 18(1):145–147 15. Sommerfeld M, Huber N (1999) Experimental analysis and modelling of particle-wall collisions. Int J Multiph Flow 25(6):1457–1489 16. Davis RH, Serayssol JM, Hinch EJ (1986) The elastohydrodynamic collision of two spheres. J Fluid Mech 163(163):479–497 17. Grant G, Tabakoff W (1975) Erosion prediction in turbomachinery resulting from environmental solid particles. J Aircr 12(5):471–478 18. Clark HM, Wong KK (1995) Impact angle, particle energy and mass loss in erosion by dilute slurries. Wear, 186–187(part-P2): 454–464 19. Li X, Mi C (2019) Effects of surface tension and Steigmann–Ogden surface elasticity on Hertzian contact properties. Int J Eng Sci 145:103–165 20. Ahlert KR (1994) Effects of particle impingement angle and surface wetting on solid particle erosion of AISI 1018 steel. The University of Tulsa

Chapter 3

Wear Characteristics of Static Walls

Abstract When the solid–liquid two-phase mixture is transported, wear on flow parts occurs due to the particle impact and friction. This will reduce delivery reliability and component life. So it is particularly important to study the mechanism of solid–liquid two-phase flow and wear on flow parts. The effects of different flow rates, bend curvature and solid concentration on the solid–liquid two-phase flow characteristics in a bend pipe are analyzed in this chapter. And the realationship between the two-phase flow characteristics and the wear characteristics on the static wall surface of the bend pipe is studied. The influence of particle shape, different particle size combinations and different surface shape on the wear laws was discussed.

3.1 Calculation Method and Experiment 3.1.1 Computational Method and Geometric Model The solid–liquid two-phase flow is simulated using coupled the discrete element EDEM particle software and FLUENT software. The liquid phase water is treated as a continuous medium, the solid phase particles are regarded as discrete units. The Euler–Lagrange approach is used in numerical simulation. The flow field is solved using FLUENT and the particle motion is calculated using EDEM. As shown in Fig. 3.1, the flow field is calculated by Fluent and the converged computational result of the fluid is input into the EDEM, the particles generated by EDEM are filled into the model at a certain rate. The interphase force is taken into account when particle motion are calculated. After a certain number of calculation time steps, the calculation result of particle motion is returned to FLUENT from EDEM to calculate the flow field, and so forth, the particle motion in the pipe is calculated. The motion of the flow fluid can be obtained with the local mean variables according to the continuity and momentum conservation equations. The governing equations of the fluid are as follows:

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Z. Zhu et al., Solid-Liquid Two-Phase Flow in Centrifugal Pump, Fluid Mechanics and Its Applications 136, https://doi.org/10.1007/978-981-99-1822-5_3

63

64

3 Wear Characteristics of Static Walls

Fig. 3.1 EDEM–FLUENT coupling schematic

Continuity equation: ∂ ∂ (α f ρ f ) + (α f ρ f u j ) = 0 ∂t ∂x j

(3.1)

Momentum conservation equation:    ∂u j ∂u i ∂ ∂ ∂p ∂ α f μe f f (α f ρ f u i ) + (α f ρ f u i u j ) = − + + ∂t ∂x j ∂ xi ∂x j ∂x j ∂ xi + α f ρ f g + Fs

(3.2)

where ρ f represents the fluid density and is a constant because the fluid is assumed to be incompressible; u, the fluid velocity; p, the pressure of the fluid; μe f f , the effective viscosity; x, the coordinates; g, the acceleration due to gravity; Fs , the interaction term due to the force between the particles and the fluid; αf , the porosity near the particle and can be calculated as: αf = 1 −

n 

V p,i / Vcell

(3.3)

i=1

where V p,i , the volume of particle i in the selected CFD cell; n, the amount of particles inside the cell; V cell , the volume of the cell. The translational and rotational motions of the particles are calculated using Newton’s kinetic equations: m

 dV = mg + Fc + Fdrag + Fm + Fsl dt

(3.4)

3.1 Calculation Method and Experiment

I

dω  = Tc + T f dt

65

(3.5)

where Fc , Fdrag , Fm and Fsl respectively represent the contact force, fluid drag force, Magnus force and Saffman lift force; m and I are respectively the mass and moment of inertia of the particles; Tc and Tf respectively denote the contact torque and the torque generated by the fluid phase. The soft-sphere model of Zeng et al. [1] shown in Fig. 3.2a are used to illustrate the collisions between particles affected by fluid, the particle i contact with particle j at point C. The two particles are deformed in subsequent movements. Let δn and δt respectively denote the normal displacement and tangential displacement between particles, Fc,n and Fc,t represent the corresponding normal contact force and tangential contact force. Figure 3.2b shows the constitutive model of the interaction between the two particles. In the figure, the spring is used to simulate deformation damp, and a damper simulate the damping effect. When the diameter of one of the particles in model is infinite, the particle can be thought as a wall. So, the collision of the two particles can be simulated the collision between the particle and the wall. The contact force Fc are calculated as: Fc = Fc,n + Fc,t

(3.6)

Here Fc,n and Fc,t are solved using the linear spring–buffer model proposed by Cundall and Strack [2]: Fc,n = −kn δn − ηn Vn

(3.7)

Fig. 3.2 Schematic of the soft-sphere model. a Normal displacement δ n and tangential displacement δ t of particle collision. b Constitutive model of the interaction between two particles

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3 Wear Characteristics of Static Walls

Fc,t = −kt δt − ηt Vt

(3.8)

where Vn and Vt are the normal velocity and tangential relative velocity between the particles; k n and k t are respectively the normal and tangential stiffness of the springs; and ηn and ηt are respectively the normal and tangential damping coefficients. The contact torque Tc is equal to the torque Tt generated by the tangential contact force, which is defined as: Tt = r × Fc,t

(3.9)

where r is the radius vector from the circle center to the contact point. Forces from particles to fluid can be calculated as:

FS =

−

n  i i i=1 Fdrag + Fsl + Fm Vcell

(3.10)

where V cell is the volume of the cell and n represents the total number of particles in this cell. Equation (3.10) is based on the principle of Newton’s third law of motion such that the forces of the liquid acting on the particles will react on the liquid from the particles in each computational cell. The wear equation proposed by Archard [3] are used to calculate the wall wear, and the wear model is: W = Ks

P Pm

(3.11)

where W is the wear volume (mm3 ), s is the sliding distance, P is the applied load (N), and Pm is the hardness of the wall material (N/mm2 ). The ratio between P and Pm is considered the real contact area. K is the dimensionless wear constant related to the material itself. The wear constant (K = 3.685 × 10–4 ) is calculated from the data of Prasad et al. [4]. The variable curvature pipe model is shown in Fig. 3.3. To obtain fully developed flow, extend the length of inlet and outlet each 20 times that of the section side. Therefore, the calculation area includes the extended inlet section, the test section, and the extended outlet section. And the structured hexahedral mesh is adapted to ensure high stability and accuracy, the value of y+ is set to 30. The boundary employs the progressive grid, and the radial grid is 25-layer grid. The height of the first grid is 0.05 mm, and the growth factor is 1.2. According to the verification of grid independence, the final number of cells is determined to be 305,166. Using the standard K-ε model to analyze the internal flow, the standard wall function to deal with the flow near the walls. The no-slip boundary condition is adopted for the boundary of the wall surface. The inlet of the calculation domain is set to the “velocity-inlet”, and the outlet is set to the “outflow”. Sub-relaxation is selected and used for the iterative calculation of the algebraic equation. The convergence

3.1 Calculation Method and Experiment

67

Fig. 3.3 Schematic of the geometry and the CFD mesh of the bend

precision is 10−5 . The liquid phase enters the inlet of the vertical section of the bend, and a virtual space named particle factory is established at the inlet to generate particles. Specific parameters are given in Tables 3.1 and 3.2. Table 3.1 Parameters of solid phase in simulation

Parameters

Particles

Wall

Material

Soda-lime glass

Al alloy 6061

Poisson’s ratio

0.25

0.3

Shear modulus (GPa)

1.96

26.5

Density (kg/m3 )

2500

2700

Radius (mm)

1.5

/

68 Table 3.2 Parameters of liquid phase in simulation

3 Wear Characteristics of Static Walls Parameters

Value

Material

Water

Viscosity (kg/(m s))

0.001003

Density

(kg/m3 )

998.2

Initial average velocity (m/s)

11.77

Reynolds number

2.577 × 105

3.1.2 Experimental System and Measurement Method The experimental loop is shown in Fig. 3.4. To observe the wear situation conveniently, the experimental section is designed as a removable and detachable square pipe that can be fastened with screws. An aluminum sheet with surface roughness of Ra3.2 as a test sample is fixed on the wall by a groove in the bend. The cross-sectional dimensions of the bend are 22 mm × 22 mm. In the experiment, the direction of the bend is vertical. There are 600 mm square pipe sections before and after the test section. Fifteen groups of wear experiments are carried out under the condition of mass concentration varying from 1 to 15% in steps of 1%. Each group is carried out five times. The duration of a wear experiment is 25 min, and the mixture flow rate is about 11.77 m/s. Every test sheet is weighed before the experiment using electronic scale with accuracy of 0.01 g. After wear experiment, the test sheets are taken from the bend pipe for cleaning, drying and weighing. Finally, test pieces are ranked in ascending order of concentration for each working condition group, as shown in Fig. 3.5. A periodic corrugated pattern caused by the collision of particles exists on the surface of each test sheet.

Fig. 3.4 Test facility

3.1 Calculation Method and Experiment

69

Fig. 3.5 Experimental wear results

The degree of wear varies with working conditions. The severely worn areas mainly exist in the bend section within the interval of 4–12.5 cm. To obtain the maximum thickness loss rate of the test sheet, the thickness of the sheet is measured with a PX-7 DL Ultrasonic Thickness Gauge. The most-worn area in the region of 6–10 cm is measured. As shown in Fig. 3.6, divide the specimen into 6 and 12 equal parts respectively along the flow direction and the vertical flow direction, place the probe of the ultrasonic thickness gauge at the intersection of the horizontal and vertical lines, and then measure the thickness of the test sheet. The value of maximum thickness loss is obtained by comparing the minimum material thickness with the original value. Proceed to the next step, the maximum thickness loss rate can be obtained by dividing the maximum thickness loss by the wear time. The average thickness loss rate of the test sheet can be obtained by the following equation: v=

Δm ρst

(3.12)

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3 Wear Characteristics of Static Walls

Fig. 3.6 Specific measurement of thickness

where m is the mass loss of test sheet before and after the experiment, ρ is the density of the test sheet material, s is the surface area of the test sheet, and t is time during the wear experiment.

3.1.3 Results Comparison and Calculation Method Verification The computational accuracy of two-phase flow and wear can be verified by comparing experimental and simulated results. The specific analyses include the relationship between the experimental severe wear area and the simulated particle motion law, comparison of experimental and simulated wear rates, and surface topography at different particle mass concentrations.

3.1.3.1

Relationship Between Severe Wear and Particle Movement

The severely worn areas mainly locate in the bend section 4–12.5 cm, where is also the region that the most particle collide with the wall for the first time, as shown in Fig. 3.7. Particles move along the straight pipe section are shown in Fig. 3.7a. The first collision with the bend wall occurs on arc AB. However, due to the force of the fluid on particles, the collision area is offset along the flow direction by some distance, i.e., arc AC. Figure 3.7b shows particle velocity vectors for 15% mass concentration. The figure reveals that the trajectory shifts after particles enter the bend. Particle collisions therefore focus on the area of arc AC, which is the section 4–12.5 cm along the aluminum piece. Meanwhile, the arc AC is also the region, where the loss of particle velocity and energy is the maximum, also, wear is therefore most serious. From inlet to outlet of

3.1 Calculation Method and Experiment

71

Fig. 3.7 Particles motion. a Schematic diagram of particles moving in the bend. b Particle velocity vector results from CFD-DEM simulation (15% mass concentration)

the bend, select 130 particles randomly to calculate their average velocity to illustrate the movement of the particles in the bend after the flow is stable. As shown in Fig. 3.8, scattered blue dots denote discrete velocities of particles, and the red line denotes the mean velocity. Under the effect of gravity and the fluid drag force, particles move a certain distance in the pipe at higher speed. The particle velocity reduces suddenly when particles collide with the wall surface at this speed for the first time. The wall at this location is thus the most deformed section of wall and even starts to shed material. The particles then rebound from the wall surface at a lower speed and accelerate again under the effect of the drag force of the fluid until colliding a second time with the wall. However, because the acceleration process is short, the velocity is still low when particles collide with the wall for the second time, therefore, the wear caused by the first collision is the maximum.

3.1.3.2

Relationship Between Wear Rate and Particle Mass Concentration

Figure 3.9 shows the contrast of the average thickness loss rate with the maximum thickness loss rate of the test sheet for different mass concentrations. Results are presented on a logarithmic scale. Figure 3.10 shows the distribution of particles in the bend section when the particle mass concentration is 1, 6, 11 and 15%. At each concentration, the maximum wear rate obtained from the five groups of experiments was basically the same. But the average erosion amount fluctuated,

72

3 Wear Characteristics of Static Walls

Fig. 3.8 Particle velocity from CFD-DEM simulation

Fig. 3.9 Wear rates at different mass concentrations

especially under low concentration conditions. The numerical simulation shows that the maximum wear rate and the average wear rate are almost the same as the concentration changes. A comparison of experimental and numerical simulations of the thickness loss rate in Fig. 3.9 shows that the experimental and numerical simulation results are generally consistent. The maximum thickness loss rate under different

3.1 Calculation Method and Experiment

73

Fig. 3.10 Particle motions and positions at different mass concentrations results from CFD

working conditions is approximately 7–10 times the average thickness loss rate. As the mass concentration of the particles increases gradually from 1 to 9%, the wear rate of the test sheet increases. However, the growth rate gradually decreases with increasing mass concentration. When the mass concentration is greater than 9%, the magnitude of the wear rate tends to be stable, approximately 1 × 10–5 mm/s. The above phenomenon was called “buffer effect” proposed by Duarte et al. [5]. The mechanism of the buffer effect is as follows. Firstly, the particles carried by the fluid flow into the bend in a straight-line motion, in this process, the collision occurs mainly between particle and particle. Secondly, for the inertial of particles, their motion is decoupled from the fluid, and tend to colliding bend wall directly. After that, the particles are forced to change direction. In this step, the interaction between particles becomes important. Finally, a large number of particles accumulates on the bend wall, and the particles adjacent to the wall form a “virtual barrier”, which prevents the surrounding particles from hitting the bend wall directly. The buffering effect occurs at this stage.

74

3 Wear Characteristics of Static Walls

As the solid–liquid mixture flows in the pipeline, the factors affecting particle movement are mainly the drag force of the liquid phase, collisions between particles and the wall, and collisions among particles. Figure 3.10 shows that the particle velocity follows the same changing law for different mass concentrations, which means that the effect of the drag force of the liquid phase on particles is the same at the same water velocity. The particle velocity is thus almost the same value of about 10 m/s before particles enter the bend region. Therefore, the change of the wear law is mainly related to the number of collisions of particles and the energy loss (i.e., the deformation energy of the wall surface transmitted by particles). On account of the particle energy loss during the collision process, the particle velocity decreases, and the particles thus gather near the wall surface and travel forward slowly along the surface. Subsequent particles therefore first collide with previous particles, and many particles decelerate to form a barrier near the wall surface, which provides a buffering effect for particle–particle–wall collisions. The barrier becomes more and more obvious with the increase of the mass concentration. In Fig. 3.11, the kinetic energy lost due to the collision between the particles and the particles is almost the same for different concentrations, meanwhile, the average deformation energy of the wall surface obtained from the collision of a single particle with the wall surface is also almost the same for different concentrations. So, the wear quantity of the wall surface is directly related to the collision number. The collision number of the particles and wall increases with the increase of concentration when the concentration is lower. However, when the concentration increases to a certain level, the collision number of particles and particles increases sharply until the barrier layer are formed near the wall surface, the collision between the particles and the wall tends to be stable, the wear quantity thus tends to stabilize.

3.1.3.3

Relationship Between the Surface Morphology and Solid–liquid Mixture Motion

To explore the formation mechanism of the wavy streak on the worn wall surface, a Japanese Keith VHX-2000 super-depth microscope is applied to observe the microscopic appearance of the wear of the surface material. Figure 3.12 shows the surface topography at different distances from the entrance. The distance of 8 cm is almost at the beginning of the curve while the distance of 20 cm is near the outlet of the bend. The wear image shows a nearly periodic waveform along the flow direction. The change of amplitude of corrugation are not obviously in the same worn surface. Compare Fig. 3.12b, d, the surface topography is similar at different distances. However, the total width of the five waves of the two figures are 3.81 mm and 3.05 mm, respectively. The ripple frequency in Fig. 3.12d is higher than that in (b). Figure 3.13 shows that the wear patterns of the test sheet surface are same for all working conditions, but the degree of wear are different. The experimental and simulation results are nearly consistent. When the mass concentration is lower, the amplitude of the wear fringe is lower. The fringe amplitude increases as the mass

3.1 Calculation Method and Experiment

75

Fig. 3.11 Energy loss per particle, number of particles and erosion rate of particle–particle and particle–wall for different mass concentrations. E-lossp /E-lossp-min : Energy loss per particle/Energy loss per particlemin; No.-collision/No.-collisionmin : Number of particle collisions/Number of particle collisionsmin ; Er-ave/ Er-avemin : Average erosion of the wall/Average erosion of the wallmin

concentration increases. When the concentration reaches a certain level, the amplitude of the fringe keeps basically the unchanged, which is consistent with the results presented in the wear cloud diagram. The erosion ripple is caused by the disturbance of flow around particles according to Karimi and Schmid [6]. Due to the friction of wall-liquid interface, the velocity of fluid near the wall surface is much slower than that of the bulk fluid. Meanwhile, the particle concentration near the wall is larger than that in the main flow. With the coupling of these two effects, the velocity difference increases and generates shear forces, and thus the periodic disturbance of flow around particles come into being (see Fig. 3.14a). Under the effect of flow turbulence, particle will bounce forward (i.e., impact on the wall surface, rebound, and then impact again), and leave some dimples on it (see Fig. 3.14b). Over a period, there are large amount of bouncing particles impact on the surface, and thus makes the formation of the erosion ripple (see Fig. 3.14c). The waviness of the erosion ripple is dominated by the particle

76

3 Wear Characteristics of Static Walls

Fig. 3.12 Surface topography of the central position of the test sheet at different distances from the entrance

concentration. The higher the particle concentration is, the larger the surface waviness will be. Above all, the numerical simulation results are in good agreement with the experiments. So the coupled calculation method is accurate for solid–liquid two-phase flow simulation.

3.2 Particle Motion and Wear Characteristics in the Bend Pipe The analysis of the flow characteristics of solid–liquid two-phase flow is very important to understand the movement of particles and flow field, and to understand the internal relationship between them, which also provides a more intuitive explanation for the understanding of solid–liquid two-phase wear. Based on the previously validated coupling calculation method, the solid–liquid two-phase flow in a 120-degree square bend pipe is calculated and analyzed (Table 3.3).

3.2 Particle Motion and Wear Characteristics in the Bend Pipe

77

Fig. 3.13 Surface topography (experiment), wear cloud (simulation) and particle position in the flow field (simulation) at 8 cm for different mass concentrations. a 1%. b 6%. c 11%. d 15%

3.2.1 Analysis of Particle Motion in the Bend Pipe 3.2.1.1

Analysis of Particle Motion Characteristics

Particle size is 2 mm, the particle concentration is 13%, the particle density is 4500 kg/m3 , the fluid velocity is 5 m/s, and the curvature radius of the bend pipe is 100 mm, the numerical simulation results are used to analyze the movement characteristics of the particle phase in the bend pipe. After the 0.1 s flow field calculation is stable, the particles begin to release. Figure 3.15a shows that the particles have just been generated from the particle factory at 0.1 s. The initial velocity of the particles is 0, while the velocity of the fluid in the flow field is 5 m/s. therefore, the particles in this region are accelerated. The difference in particle color in the figure indicates the size of particle velocity. Figure 3.15b shows

78

3 Wear Characteristics of Static Walls

Fig. 3.14 Schematic diagram of ripple formation

Table 3.3 Simulation related parameters Particle shape

Particle size (mm)

Particle density (kg/m3 )

Particle concentration (%)

Fluid velocity (m/s)

The curvature radius of the bend (mm)

Spherical

2

4500

7, 10, 13

5, 10, 15

50, 75, 100

that the particles accelerate to a certain speed under the combined action of fluid force and gravity. The fluid resistance brought by the velocity to the particles is equal to that of gravity and fluid force. The particles are kept in equilibrium, the acceleration process is finished, and the velocity of particles in this area keeps uniform motion. Figure 3.15c shows that many particles impact the wall at a certain speed and collide with the wall. At the same time, many particles collide with each other. The speed of particles decreases sharply, and the motion track changes. Figure 3.15d shows that the particles have been washed out of the bend, and the particles still move along the wall in this area due to the inertia of the particles. Figure 3.15e shows that under the action of gravity, the particle’s motion track disperses and moves slowly towards the inner wall as the velocity increases. Figure 3.15f shows the stable phase of particles, and all particles in the bend pipe keep this trajectory. The particles began to form in 0.1 s and began to exit from the bend outlet at 0.3 s. The retention time of particles in the whole bend was 0.2–0.25 s. After the particles are stabilized, the maximum velocity is 6.1 m/s, which is due to the acceleration effect of particles in the exit section because of gravity.

3.2 Particle Motion and Wear Characteristics in the Bend Pipe

79

Fig. 3.15 Position of particles in the bend pipe at different times

After 0.5 s and every 0.1 s, the velocity, angular velocity, collision frequency, and other related information of 100 particles were randomly extracted from the inlet (near the particle factory). A total of 10 groups of 1000 particles were extracted, that is, the cut-off time was 1.5 s. Calculate the average velocity of each group of particles, as shown in Fig. 3.16. It can be seen from the figure that the data of 10 groups of particles show that the changing trend of velocity is the same, that is, the particles first accelerate under the condition of the fluid force, and the duration of this acceleration phase is about 0.05 s. The velocity of group 3 decreases in the acceleration stage, which is caused by the collision between the particles of group 3 and the wall at this stage; With the increase of velocity, the acceleration becomes smaller and smaller, until the acceleration is 0, the particles keep almost constant velocity, and the duration of this uniform motion phase is about 0.04 s; After a period of uniform motion, the particles enter the bend area and collide with the particles gathered on the wall or near the wall. The velocity drops sharply, and the velocity direction changes. The particles will continue to roll and bounce near the wall and collide with the wall many times; Finally, the particles enter the exit section. In this stage, under the combined action of gravity and fluid force, the particles enter the acceleration stage again and remain unchanged after accelerating to a certain size until they move out. The changing trend of particle velocity is shown in Fig. 3.17. The blue discrete points represent the angular velocity values of 10 groups of particles, while the red line represents the fitting curve of the average angular velocity of these particles.

80

3 Wear Characteristics of Static Walls

Fig. 3.16 Velocity of particles in the bend pipe at different times

It can be seen from the figure that although the angular velocity of particles keeps increasing in the process of moving towards the bend pipe, the growth rate is not large. This is because, at this stage, most collisions only occur between particles, and the number of collisions is relatively small, so the angular velocity of particles in this process is small (the initial angular velocity of particles is set to 0 in the simulation). When the particles collide with the wall, the particles are subjected to the force given by the wall, which causes the particles to rotate greatly, and the average angular velocity of the particles reaches 1700 rad/s. After that, the probability of particle collision with the wall begins to decrease, so the angular velocity decreases slightly. Figure 3.18 shows the collision information of particles in 0.1–0.5 s. It can be seen from the figure that before the particles enter the bend pipe area and collide with the wall, a small number of collisions have taken place between particles and between particles and between particles and walls, which is caused by the different initial positions and motion states of particles. During this period, the number of collisions between particles and walls is greater than that between particles. After the particles enter the bend area, the particles have a buffering effect in this area, and the particles in front impact the wall to slow down, thus gathering near the wall. As a result, the number of collisions between particles is greater than that between particles and walls at this stage.

3.2.1.2

Effect of Different Fluid Velocities on Particle Motion

The effects of different fluid velocities on particle motion were studied. The velocity of clean water was 5 m/s, 10 m/s, and 15 m/s respectively. Simulation parameters are

3.2 Particle Motion and Wear Characteristics in the Bend Pipe

81

Fig. 3.17 Angular velocity of particles in the bend pipe at different times

Fig. 3.18 Collision times of particles in the bend pipe at different times

particle size of 2 mm, particle concentration of 13%, particle density of 4500 kg/m3 , and curvature radius of bend pipe of 100 mm. Select 100 particles near the particle factory under each working condition. Since the particles are just generated here, the particle velocity starts from 0. Record the velocity of these particles in the bend. It can be seen from Fig. 3.19 that the acceleration of particles at different fluid velocities is different. The higher the fluid velocity is, the greater the particle acceleration is, and the higher the final velocity is. With the increase in fluid velocity, the

82

3 Wear Characteristics of Static Walls

Fig. 3.19 Velocity variation of particles at different fluid velocities

average time of particles moving in the bend is 0.702 s, 0.126 s, and 0.091 s, respectively. This shows that the residence time of particles in the bend pipe decreases with the increase of fluid velocity, but the decreasing range is also decreasing. At each different speed, the movement law of particles is consistent: firstly, the particles are accelerated by the action of fluid force, and when the acceleration reaches a certain time, the particles will keep a uniform motion for a short time. However, since gravity slows down the particles in this region, the velocity of the uniform motion will be less than that of the fluid. After that, the particles collide with the wall of the bend pipe, and the velocity decreases. At the same speed, the particles start to move at a constant speed and then leave the pipe. At this time, the velocity of particles is higher than that of fluid because of the acceleration of gravity.

3.2.1.3

Influence of Different Curvature Radius on Particle Motion

In a circular bend, the ratio of curvature radius to pipe diameter is called the bend diameter ratio, which has a great influence on solid–liquid two-phase flow. In our study, because the research object is a square bend, the hydraulic diameter can be calculated according to the flow section of the pipe. The influence of curvature radius on the two-phase flow was studied by changing the curvature radius of the bend. The curvature radius of the bend was selected as 50 mm, 75 mm and 100 mm respectively. Other related parameters are particle size of 2 mm, particle concentration of 13%, particle density of 4500 kg/m3 , and flow rate of fluid of 5 m/s. Figure 3.20 shows the velocity distribution of particles with different radii of curvature. It is shown that the velocity of particles is the same before entering the

3.2 Particle Motion and Wear Characteristics in the Bend Pipe

83

Fig. 3.20 Velocity distribution of particles with different radii of curvature

bend. After the collision between the bend area and the bend wall, the movement law of particles in the bend with different radii of curvature is also different. The larger the radius of curvature is, the smaller the velocity change rate is after the collision between particles and the wall. The final velocity decreases with the increase of curvature radius. Figure 3.21 shows the trend of particle wall collision with time under the different radius of curvature. At first, the number of collisions between particles and the wall is very small before the particles enter the bend area. When the particles enter the bend, the collision frequency increases sharply. During this period, the growth rate of the number of collisions between particles and walls is the same under the different radius of curvature. After the whole particle field is stable, the number of collisions between particles and wall is 262, 363, and 461 respectively with the increase of curvature radius. It shows that the frequency of particle wall collision is proportional to the radius of curvature.

3.2.1.4

Effect of Particle Concentration on Particle Motion

The particle concentration is 7%, and the particle concentration is 10%. Simulation parameters are particle size of 2 mm, particle density of 4500 kg/m3 , fluid velocity of 5 m/s, and curvature radius of the bend of 100 mm. Figure 3.22 shows the distribution and trajectory of particles in the bend pipe when the particle mass concentration is 7, 10, and 13%. It can be seen from the figure that with the increase of particle mass concentration, the distribution density of particles in the bend pipe increases. Near the outlet of the bend, the position of

84

3 Wear Characteristics of Static Walls

Fig. 3.21 Collision between particle and wall under different curvature radius

particles is closer to the inner side with the increase of particle mass concentration. This is because when the concentration of particles increases, the number of particles moving near the outer wall increases, leading to the increase in a collision between particles, and then many particles are forced to move inward. Figure 3.23 shows the velocity variation of particles in the bend when the particle mass concentration is 7, 10, and 13%. It can be seen from the figure that although the

Fig. 3.22 Distribution of particles in the bend at different particle concentrations

3.2 Particle Motion and Wear Characteristics in the Bend Pipe

85

Fig. 3.23 Velocity distribution of particles at different mass concentrations

mass concentration is different, the particle velocity in each stage of the pipeline has little influence, and the particle velocity in the three conditions is consistent. This shows that although the particle concentration affects the volume ratio of solid–liquid two-phase, the collision frequency between particles changes, this effect is different from that of particle and wall collision, which will not directly change the trajectory of particles, so it has little effect on the change of particle velocity.

3.2.2 Analysis of Wall Wear Characteristics Many factors affect the solid–liquid two-phase wear in the bend pipe. Firstly, the orthogonal test method is used to analyze the influence of particle size, particle mass concentration, particle density, fluid velocity, and radius of curvature on the wall wear. Based on the control variable method, the influence of particle shape and particle size combination on the wear is analyzed.

3.2.2.1

Wear Analysis Based on Orthogonal Test

There are many kinds of wear-related factors, among which particle factors include particle concentration, particle density, particle shape, particle hardness, particle size, particle surface roughness, and different particle size combinations among particles; Fluid factors include fluid velocity, fluid density, and fluid viscosity; Geometric

86 Table 3.4 Values of each factor (Bend pipe)

3 Wear Characteristics of Static Walls Parameter

Value

Particle size (mm)

1, 2, 3, 4, 5

Particle mass concentration (%) 1, 4, 7, 10, 13 Particle density (kg/m3 )

1500, 2700, 4500, 6000, 7800

Fluid velocity (m/ s)

5, 10, 15, 20, 25

The radius of curvature (mm)

50, 62.5, 75, 87.5, 100

factors include pipe diameter, the radius of curvature, the ratio of bending to diameter, etc. In this paper, the main factors considered in the orthogonal experiment are particle size, particle mass concentration, particle density, fluid velocity, and radius of curvature. The values of various factors are shown in Table 3.4. According to the number of factors and levels to be studied, the orthogonal test table of 5 levels and 6 factors is selected for experimental design. The table is the smallest orthogonal table that can fully meet the requirements. 5 levels and 6 factors orthogonal test table is shown in Table 3.5. To avoid the systematic error caused by human factors, the levels of factors are not arranged according to the order of factor level values. The order of each factor is randomly disordered by drawing lots. Finally, the specific factor level table is as follows (Table 3.6). This study does not consider the interaction between the various factors, a factor occupies a column, and factor 6 is designed as a blank column. The blank column is also called the error column in variance analysis of orthogonal design, which does not affect the experimental scheme. Finally, according to the above steps, the design scheme is as follows (Table 3.7). The maximum wear rate at the outlet of the bend pipe is calculated, including the maximum wear rate at the outlet of the bend pipe. The average wear rate and maximum wear rate of the outer wall of the bend pipe are calculated. The specific results are shown in Table 3.8. Table 3.9 is the result of the processing analysis of Table 3.8. Ki in the table represents the sum of the corresponding experimental results when the horizontal number is i in any column (i = 1, 2, 3, 4 or 5 in this study). For example, in the third column where factor C is located, taking the average wear rate as the standard, C is taken as the level C1 in experiments 1, 10, 14, 18, and 22, so K1 is the sum of test results 1, 10, 14, 18 and 22, that is, K1 = 4.90 × 10–4 . ki in the table represents the arithmetic mean of the experimental results obtained when level I is taken as the factor in any column. The specific expression is ki = Ki / n, where n is the number of times each level appears in any column. In this study, n is equal to 5. R is the range. In any column, R = max{K 1 , K 2 . . . , K i }−min{K 1 , K 2 . . . , Ki }. The range of each column is generally not equal, which shows that the influence of the level change of various factors on the change of experimental results is not the same. The larger the range is, the greater the influence of these factors within the

3.2 Particle Motion and Wear Characteristics in the Bend Pipe

87

Table 3.5 Orthogonal test table of 5 levels and 6 factors Experiment number

Factor 1

Factor 2

Factor 3

Factor 4

Factor 5

Factor 6

1

1

1

1

1

1

1

2

1

2

2

2

2

2

3

1

3

3

3

3

3

4

1

4

4

4

4

4

5

1

5

5

5

5

5

6

2

1

2

3

4

5

7

2

2

3

4

5

1

8

2

3

4

5

1

2

9

2

4

5

1

2

3

10

2

5

1

2

3

4

11

3

1

3

5

2

4

12

3

2

4

1

3

5

13

3

3

5

2

4

1

14

3

4

1

3

5

2

15

3

5

2

4

1

3

16

4

1

4

2

5

3

17

4

2

5

3

1

4

18

4

3

1

4

2

5

19

4

4

2

5

3

1

20

4

5

3

1

4

2

21

5

1

5

4

3

2

22

5

2

1

5

4

3

23

5

3

2

1

5

4

24

5

4

3

2

1

5

25

5

5

4

3

2

1

Table 3.6 Factor level table Level

Particle size (A) (mm)

Particle mass concentration (B) (%)

Particle density (C) (kg/m3 )

Fluid velocity (D) (m/s)

Radius of curvature (E) (mm)

1

2

1

6000

20

87.5

2

3

13

4500

5

100

3

1

10

2700

10

75

4

5

4

1500

15

50

5

4

7

7800

25

62.5

88

3 Wear Characteristics of Static Walls

Table 3.7 Final design scheme Experiment number

A

B

C

D

E

Blank column

Experiment scheme

1

1

1

1

1

1

1

A1 B1 C1 D1 E1

2

1

2

2

2

2

2

A1 B2 C2 D2 E2

3

1

3

3

3

3

3

A1 B3 C3 D3 E3

4

1

4

4

4

4

4

A1 B4 C4 D4 E4

5

1

5

5

5

5

5

A1 B5 C5 D5 E5

6

2

1

2

3

4

5

A2 B1 C2 D3 E4

7

2

2

3

4

5

1

A2 B2 C3 D4 E5

8

2

3

4

5

1

2

A2 B3 C4 D5 E1

9

2

4

5

1

2

3

A2 B4 C5 D1 E2

10

2

5

1

2

3

4

A2 B5 C1 D2 E3

11

3

1

3

5

2

4

A3 B1 C3 D5 E2

12

3

2

4

1

3

5

A3 B2 C4 D1 E3

13

3

3

5

2

4

1

A3 B3 C5 D2 E4

14

3

4

1

3

5

2

A3 B4 C1 D3 E5

15

3

5

2

4

1

3

A3 B5 C2 D4 E1

16

4

1

4

2

5

3

A4 B1 C4 D2 E5

17

4

2

5

3

1

4

A4 B2 C5 D3 E1

18

4

3

1

4

2

5

A4 B3 C1 D4 E2

19

4

4

2

5

3

1

A4 B4 C2 D5 E3

20

4

5

3

1

4

2

A4 B5 C3 D1 E4

21

5

1

5

4

3

2

A5 B1 C5 D4 E3

22

5

2

1

5

4

3

A5 B2 C1 D5 E4

23

5

3

2

1

5

4

A5 B3 C2 D1 E5

24

5

4

3

2

1

5

A5 B4 C3 D2 E1

25

5

5

4

3

2

1

A5 B5 C4 D3 E2

experimental range, and the greater the influence of each factor on the results. The range of the blank column represents the error of the whole experiment. In this study, the range in the blank column is the smallest, which shows that the whole experiment is credible. As shown in Table 3.9, for different indicators, the influence degree of different factors is different. Based on the average wear rate, the range value R is RD > RB > RE > RA > RC , which shows that for the average wear rate, the influence of fluid velocity is the biggest, and the effect of particle density is the least. Based on the maximum wear rate, the range R is RD > RA > RC > RE > RB . This shows that for the maximum wear rate, the influence of fluid velocity is the biggest, the influence of particle density is the least, and the influence of particle concentration is the least. The best choice is based on wear rate. In the case of the two indicators, the number of

3.2 Particle Motion and Wear Characteristics in the Bend Pipe

89

Table 3.8 Statistics of simulation results Experiment number

A

B

C

D

E

Blank column

Average wear rate (mm/s)

Maximum wear rate (mm/s)

1

1

1

1

1

1

1

1.78 × 10–5

8.32 × 10–5

10–6

5.61 × 10–6

2

1

2

2

2

2

2

1.89 ×

3

1

3

3

3

3

3

1.17 × 10–5

2.70 × 10–5

4

1.73 ×

10–5

6.25 × 10–5

10–4

7.57 × 10–4

4

1

4

4

4

4

5

1

5

5

5

5

5

2.85 ×

6

2

1

2

3

4

5

3.04 × 10–6

2.34 × 10–5

1

5.61 ×

10–5

1.39 × 10–4

10–5

3.72 × 10–5

7

2

2

3

4

5

8

2

3

4

5

1

2

1.79 ×

9

2

4

5

1

2

3

6.13 × 10–5

3.17 × 10–4

4

8.35 ×

10–6

4.44 × 10–5

10–5

1.62 × 10–4

10

2

5

1

2

3

11

3

1

3

5

2

4

2.22 ×

12

3

2

4

1

3

5

6.66 × 10–5

1.22 × 10–4

1

1.33 ×

10–6

8.99 × 10–6

10–6

3.73 × 10–5

13

3

3

5

2

4

14

3

4

1

3

5

2

7.03 ×

15

3

5

2

4

1

3

3.49 × 10–5

9.08 × 10–5

3

3.28 ×

10–7

3.20 × 10–6

10–5

1.37 × 10–4

16

4

1

4

2

5

17

4

2

5

3

1

4

2.20 ×

18

4

3

1

4

2

5

6.00 × 10–5

3.08 × 10–4

1

1.50 ×

10–4

7.60 × 10–4

10–4

4.12 × 10–4

19

4

4

2

5

3

20

4

5

3

1

4

2

1.31 ×

21

5

1

5

4

3

2

7.53 × 10–6

9.49 × 10–5

3

3.97 ×

10–4

1.12 × 10–3

10–4

5.94 × 10–4

22

5

2

1

5

4

23

5

3

2

1

5

4

1.68 ×

24

5

4

3

2

1

5

1.04 × 10–6

5.31 × 10–6

1

1.05 ×

3.77 × 10–5

25

5

5

4

3

2

10–5

levels of different factors in the optimal scheme is consistent. When the particle size is 1 mm, the particle mass concentration is 1%, the particle density is 1500 kg/m3 , the fluid velocity is 5 m/s, and the curvature radius is 87.5 mm, the optimal scheme is that the average wear rate and maximum wear rate are the minima. In the orthogonal table, the experiment number based on average wear rate and maximum wear rate (minimum wear rate) is No.16 experiment A4 B1 C4 D2 E5 . Now the optimal scheme A3B1C4D2E1 obtained from the analysis is simulated and then compared with the results of experiment No. 16 A4 B1 C4 D2 E5 . The results show that the average wear rate and maximum wear rate of the optimized scheme are 1.38 × 10–7 m/s and 1.57 × 10–6 m/ s, which are smaller than the results of experiment 16 A4 B1 C4 d2 E5 . This shows that the optimal scheme obtained from the previous analysis is correct and reliable.

90

3 Wear Characteristics of Static Walls

Table 3.9 Analysis of simulation results Index Average wear rate (mm/s)

Maximum wear rate (mm/s)

A

B

C

D

E

Blank column

K1

3.33 × 10–4

5.09 × 10–5

4.90 × 10–4

4.44 × 10–04

9.35 × 10–5

2.36 × 10–4

K2

1.47 × 10–4

5.43 × 10–4

3.57 × 10–4

1.29 × 10–5

1.56 × 10–4

1.65 × 10–4

K3

1.32 × 10–4

2.58 × 10–4

2.22 × 10–4

5.43 × 10–5

2.44 × 10–4

5.05 × 10–4

K4

3.63 × 10–4

2.37 × 10–4

1.13 × 10–4

1.76 × 10–4

5.50 × 10–4

2.37 × 10–4

K5

5.84 × 10–4

4.69 × 10–4

3.77 × 10–4

8.71 × 10–4

5.16 × 10–4

4.15 × 10–4

k1

6.66 × 10–5

1.02 × 10–5

9.80 × 10–5

8.89 × 10–5

1.87 × 10–5

4.71 × 10–5

k2

2.93 × 10–5

1.09 × 10–4

7.15 × 10–5

2.59 × 10–6

3.12 × 10–5

3.31 × 10–5

k3

2.64 × 10–5

5.17 × 10–5

4.44 × 10–5

1.09 × 10–5

4.88 × 10–5

1.01 × 10–4

k4

7.27 × 10–5

4.73 × 10–5

2.25 × 10–5

3.52 × 10–5

1.10 × 10–4

4.75 × 10–5

k5

1.17 × 10–4

9.39 × 10–5

7.53 × 10–5

1.74 × 10–4

1.03 × 10–4

8.31 × 10–5

R

4.51 × 10–4

4.93 × 10–4

3.77 × 10–4

8.59 × 10–4

4.56 × 10–4

3.40 × 10–4

Main factor → Secondary factor

D→B→E→A→C

Excellent scheme

D2 B1 E1 A3 C4

K1

9.35 × 10–4

3.67 × 0–4

1.59 × 10–3

1.53 × 10–3

3.53 × 10–4

1.03 × 10–3

K2

5.62 × 10–4

1.52 × 10–3

1.47 × 10–3

6.75 × 10–5

8.31 × 10–4

5.86 × 10–4

K3

4.21 × 10–4

9.75 × 10–4

7.45 × 10–4

2.62 × 10–4

1.05 × 10–3

1.56 × 10–3

K4

1.62 × 10–3

1.18 × 10–3

2.62 × 10–4

6.96 × 10–4

1.62 × 10–3

1.00 × 10–3

K5

1.85 × 10–3

1.34 × 10–3

1.31 × 10–3

2.83 × 10–3

1.53 × 10–3

1.22 × 10–3

k1

1.87 × 10–4

7.34 × 10–5

3.18 × 10–4

3.06 × 10–4

7.06 × 10–5

2.06 × 10–4

k2

1.12 × 10–4

3.04 × 10–4

2.95 × 10–4

1.35 × 10–5

1.66 × 10–4

1.17 × 10–4

k3

8.42 × 10–5

1.95 × 10–4

1.49 × 10–4

5.25 × 10–5

2.10 × 10–4

3.11 × 10–4 (continued)

3.2 Particle Motion and Wear Characteristics in the Bend Pipe

91

Table 3.9 (continued) Index

A

B

C

D

E

Blank column

k4

3.24 × 10–4

2.37 × 10–4

5.24 × 10–5

1.39 × 10–4

3.25 × 10–4

2.00 × 10–4

k5

3.70 × 10–4

2.68 × 10–4

2.63 × 10–4

5.67 × 10–4

3.06 × 10–4

2.43 × 10–4

R

1.43 × 10–3

1.15 × 10–3

1.33 × 10–3

2.77 × 10–3

1.27 × 10–3

9.69 × 10–4

Main factor → Secondary factor

D→A→C→E→B

Excellent scheme

D2 A3 C4 E1 B1

To study the influence of each factor level, the following trend chart is made according to ki value in Table 3.9. According to the above trend chart, the influence of five factors on wear can be analyzed, and the influence trend of each factor on average wear rate and maximum wear rate is consistent. Figure 3.24a shows the relationship between particle size and average and maximum wall wear rates. Since the points in the graph are scattered, two fitting curves are made for the average wear rate and the maximum wear rate respectively. It can be seen from the curve that the average wear rate and maximum wear rate increase with the increase of particle size, and the growth rate of maximum wear rate is larger. This is because with the increase of particle size, the greater the kinetic energy carried by the particle itself, which results in the more intense collision with the wall, the more serious the wear. Figure 3.24b shows the relationship between particle concentration and average and maximum wall wear rates. It can be seen from the figure that the trend expressed by the two lines is relatively uniform: with the increase of particle concentration, the wear rate of particles also increases, but the growth rate of wear rate is smaller. This is due to the buffering effect caused by the aggregation of particles near the wall. This is consistent with the conclusion in Sect. 3.1.3. Figure 3.24c shows the relationship between particle density and average and maximum wall wear rates. As shown in the figure, with the increase of particle density, the average wear rate and maximum wear first increase and then decrease. This is because before the particle collides with the bend pipe, the direction of the fluid force and gravity on the particle is opposite. When the particle density increases, due to the same volume, the greater the gravity of the particles, while the fluid force is unchanged. The more kinetic energy the particles carry with the wall, the more kinetic energy the particles will not carry with the wall. However, when the density of particles increases to the same degree, the velocity of particles decreases to a very small degree due to the influence of gravity. It has been known that the velocity of fluid has the greatest influence on the wall wear in the previous optimization scheme.

92

3 Wear Characteristics of Static Walls

Fig. 3.24 Trend chart of 5 different factors

In addition, due to the same mass flow rate, the number of particles will also decrease. So, the wear will decrease with the increase of density. Figure 3.24d shows the relationship between the radius of curvature of the bend pipe and the average and maximum wear rate of the wall. It can be seen from the figure that with the increase of curvature radius, the average wear rate and maximum wear first decrease and then increase. Therefore, with the increase of curvature radius, the average wear rate and maximum wear rate are closer and closer, which indicates that the wall wear is more uniform with the increase of curvature radius. Figure 3.24e shows the relationship between fluid velocity and average and maximum wall wear rates. It can be found that with the increase of fluid velocity, the

3.2 Particle Motion and Wear Characteristics in the Bend Pipe

93

degree of wall wear increases, and the growth rate also increases. This is consistent with the previous optimal scheme in which the factor of fluid velocity has the greatest influence on the wall wear. This is because with the increase of fluid velocity, the fluid force obtained by particles is greater, and the velocity of particles also increases. This results in a very large amount of kinetic energy when the particles collide with the wall. In addition, the increase in particle velocity leads to the increase in particle impact frequency., When the fluid velocity is very high, the wear caused by particle impact on the wall is particularly serious.

3.2.2.2

Wear Analysis Based on Control Variable Method

In addition to the wear research based on orthogonal experimental design, the influence of particle shape and different particle size combinations on wear is also studied by the control variable method. Effect of Particle Shape on Wear To study the effect of particle shape on wear, four kinds of regular polyhedrons were used in this study, including tetrahedron, hexahedron, octahedron, and dodecahedron. The polyhedron is composed of various small particles overlapped, as shown in Fig. 3.25, while the specific properties of particles are shown in Table 3.10. In Table 3.10, the sphericity is used to describe the degree to which the particle shape approaches the sphere shape. The sphericity is defined as the ratio of the surface area of the sphere with the same volume as the object to the surface area of the object. The closer the value is to 1, the closer it is to the sphere, and the sphericity of the ball is equal to 1. It is shown that the average wear is the maximum when the particle shape and shape are different (Fig. 3.26). This is because the dodecahedron has the most edges

Fig. 3.25 Different regular polyhedrons and models

94

3 Wear Characteristics of Static Walls

Table 3.10 Particle parameters Shape

Edge length/diameter (mm)

Volume (mm3 )

Surface area (mm2 )

Sphericity

Tetrahedron

4.9319

14.1372

42.1389

0.6711

Regular hexahedron

2.4180

14.1372

35.0800

0.8060

Regular octahedron

3.1069

14.1372

33.4378

0.8456

Regular dodecahedron

1.2265

14.1372

31.0553

0.9105

Ball (3 mm)

3

14.1372

28.2743

1

and corners, and the frequency of collision with the wall is the largest. Therefore, the wear of the dodecahedron is the most serious; However, the spherical particles have no edges and corners, so the wear on the wall after the collision is the least (Fig. 3.26) The ratio of the wear rate produced by polyhedral particles to that produced by spherical particles is defined as the relative wear rate. In Fig. 3.27, the least square method is used to fit the average relative wear and the maximum relative wear, and the sum of squares of residual errors are 0.00534 and 0.0076 respectively, indicating that the regression equation can effectively reflect the influence of sphericity on the wear rate. When the sphericity is less than 0.74, the wear rate decreases with the increase of sphericity; When the sphericity is 0.74, the minimum wear occurs for the first time; When the sphericity is between 0.74 and 0.91, the wear rate increases with the increase of sphericity; When the sphericity is 0.91, the wear rate reaches the

Fig. 3.26 Wall wear in the bend pipes with different particle shapes

3.2 Particle Motion and Wear Characteristics in the Bend Pipe

95

Fig. 3.27 Regression equation of particle sphericity

maximum value; When the sphericity is greater than 0.91, the wear rate decreases with the increase of sphericity. When the sphericity is 1, the wear rate is the minimum. Zhou et al. [7] studied the effect of particle shape on wall wear in gas–solid two-phase flow, and the conclusion is consistent with this. Figure 3.28 shows the wear of the outer wall with different particle shapes at different angles. It can be seen from the figure that the wear rate increases with the increase of the angle at 0°–55°; At 60°–70°, the wear rate is the most serious in this area, and the changing trend fluctuates greatly. At 55° and 75°, the wear rate reaches a maximum; The wear rate decreases gradually at 70°–105°, and reaches a maximum at 55° and 75°; At 105°–120°, the wear rate decreases gradually, and at 55° and 75°, the wear rate caused by other particles except for spherical particles increases, while the spherical particles remain unchanged in this range. Effect of Different Particle Combinations on Wear To study the influence of particle size combination on wear, the particle size combination as shown in Table 3.11 is used in this paper. The five combinations shown in the table are all under the condition of mass flow of 0.08984 kg/s. The relative volume fraction of small particles (2 mm) is defined as the volume percentage of all particles (2 and 3 mm). Figure 3.29 shows the wear of the outer wall with different particle combinations. It can be seen from the figure that with the increase of the relative volume fraction of small particles, the average wall wear presents a linear downward trend, but the maximum wear presents a trend of almost constant at first and then increases. Figure 3.30 shows the relevant information on particle collision. We find that with the change of the relative volume fraction of small particles, the energy loss caused

96

3 Wear Characteristics of Static Walls

Fig. 3.28 Wall wear at different angles with different particle shapes Table 3.11 Different particle size combinations

Number of groups Particles (2 mm) (%) 3 mm particles (%) 1

100

0

2

75

25

3

50

50

4

25

75

5

0

100

Fig. 3.29 Wall wear with different particle combinations

3.3 Effect of Wall Shape Change on Wear

97

Fig. 3.30 Energy loss and number of collisions with different particle combinations in one time step

by single 2 and 3 mm particles colliding with the wall is the same, and the energy loss caused by 3 mm particles colliding with the wall is about 6 times of that of 2 mm particles. With the increase of the relative volume fraction of small particles by 25, the number of collisions of small particles increases by 22.11, while that of large particles decreases by 7.15. This shows that the total energy generated by particle impact on the wall decreases with the increase of the relative volume fraction of small particles, so the average wear of the wall decreases. Figure 3.31 shows the wear of the outer wall at different angles when the particle size combination is different. When the particle size combination is not allowed, the wear law at different angles is the same. However, the higher the content of small particles, the smaller the erosion of the bend wall at all angles, especially in the range of 40°–80°, this phenomenon is obvious.

3.3 Effect of Wall Shape Change on Wear The change of wall shape will directly affect the trajectory of particles after impacting the wall, and then change the distribution of the solid phase in the two-phase flow field, which has an important impact on wall wear. Therefore, it is necessary to study the influence of wall shape change on wear.

98

3 Wear Characteristics of Static Walls

Fig. 3.31 Wall wear with different particle combinations at different angles

3.3.1 Wear Analysis of Wall Surface with Bump in Bend Pipe The angle of the line between the midpoint of the bump and the center of curvature and the horizontal line to change the parameters, four different bump positions are set. The first bump is located at 30° counterclockwise of the entrance horizontal line, and the four positions are 30°, 35°, 40°, and 45° respectively. The height and width of the bump are set as 2 mm (Fig. 3.32). Considering that different particle mass flow rates may affect α. Therefore, 10 groups of different particle mass flow conditions were set up, from Qp = 0.0484 kg/s to Qp = 0.484 kg/s, corresponding to particle mass concentration of 1–10% (Fig. 3.32).

3.3.1.1

Effect of the Location of the Bump on Wear

Figure 3.33 shows the flow field velocity vector diagram of the prototype bend pipe and the bend pipe with different positions of the bump with the condition of the particle mass flow rate of 0.242 kg/s. The solid–liquid two-phase flow accelerated at the inner wall of the bend pipe and decelerated at the outer wall. The circulation area was formed behind the bump, where vector lines were rare and the flow was almost stationary, indicating that the bump provided a shelter for solid–liquid two-phase flow. The addition of the bump caused the acceleration area on the inside of the bend to spread to the bump, and the bump at different positions affected the flow field to form different acceleration areas. The wear on the outer bend wall was caused by the collision and friction between the particles and the wall, which were closely related to the velocity of the particles.

3.3 Effect of Wall Shape Change on Wear

99

Fig. 3.32 Schematic diagram of bump position

The left, middle, and right columns of Fig. 3.34 show the wear cloud diagram on the outer bend wall with different bump positions for the particle mass flow rate of 0.242 kg/s, the particle distribution and velocity diagram inside the flow passage, and the velocity variation diagram of the 12 particles selected to be tracked over time. In Fig. 3.34, the color of the cloud map in the left column represents the wear depth of the wall, the color of the particles in the middle column represents the velocity of the particles at the current position, the black spot in the right column represents the velocity scatter of the 12 particles at each moment, and the blue curve represents the average velocity of the 12 particles at each moment. The particle movement in the bend pipe could be divided into three stages. The first stage was from 0 s to about 0.05 s, which was a stable accelerated motion cycle. In the extension section of the entrance, the particle’s velocity accelerated from the initial zero velocity to 10 m/s. The second stage was from 0.05 to 0.09 s after particle formation. At that stage, the velocity of particles plummeted after the collision with the wall. As can be seen from the velocity variation in Fig. 3.34, the average velocity of the particles decreased more sharply after the collision with the wall in the curved pipe with a bump. At the same time, the bump disturbed the flow of water and the movement of particles in the curved pipe. Therefore, the velocity distribution of the particles after the rebound was more dispersed, and the velocity of some particles even decreased to zero. Thus, the particles temporarily accumulated near the bump, and the accumulated particles continued to block subsequent particles, generating a

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3 Wear Characteristics of Static Walls

Fig. 3.33 Flow field velocity vector in bend pipes

buffer layer effect in advance. An added bump on the outer bend wall could protect the wall from particle impact, thus reducing the wall wear. The third stage was from 0.09 to 0.12 s after the particle entered the channel. The particles followed the fluid in the outlet extension at a constant velocity. The position of the bump also had a certain influence on wear. When the bump was close to the inlet, that is, when αbump was small, the severe wear area on the outer bend wall mainly occurred after the bump. With the increase of αbump , that is, when the bump moved toward the outlet, the wear area behind the bump decreased, and the maximum wear point on the outer bend wall shifted to the front of the bump. The collision wear was the most severe on the side of the entry direction. This showed that when the particle size was large, the flow field change caused by the wall bump was not enough to affect the movement of the particles. The particle movement was generally affected by the inertia. After entering the curved pipe from the straight pipe section, the centrifugal force caused the particles to interact with the wall surface. The collision position was essentially unchanged; so, the most severely worn parts experienced little change. However, the presence of the bump made the buffer layer effect occur in advance, which was beneficial for reducing the maximum wear value. Figure 3.35 shows the relationship between the wear of the outer bend wall with the bump and the bump position angle α for the condition that the particle mass flow rate was 0.242 kg/s. Combined with Fig. 3.35, it could be found that the most severe

3.3 Effect of Wall Shape Change on Wear

101

Fig. 3.34 Relationship between wear rate and movement characteristics (left: wear cloud diagram on the outer bend wall; middle: particle distribution; right: particle velocity)

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3 Wear Characteristics of Static Walls

Fig. 3.35 Relationship between bump position and wear rate

wear was at the collision between the particles and the wall surface with different positions of the bump, and the position did not change, but the bump could reduce the wear amount at this position. With the increase of α bump , the maximum wear of the outer bend wall first decreased and then increased. The average wear of the outer bend wall always increased slightly, but it had less than the average wear of the outer bend wall of the prototype bend pipe. When the position of the bump was near the collision position between the particles and the wall surface, the maximum wear amount decreased the most significantly. Figure 3.36 shows the number of particles–outer wall surface collisions over time. The number of collisions in the prototype bend was significantly larger than that in the bend with the bump; the difference was about 150. In the bend with a bump, as the α bump became larger, the number of collisions decreased slightly. The existence of the bumps prevented a considerable part of the particles from directly impacting the outer wall surface, and the number of particles blocked by the different positions of the bumps was similar.

3.3.1.2

Effect of Particle Mass Flow

Particle mass flow Qp refers to the particle mass passing through the effective crosssection of a pipe per unit time, and it is directly related to the particle concentration. Increasing the particle mass flow will cause more particles to impact the outer wall. Figure 3.37 shows the wear cloud diagram of the outer wall surface of each bend pipe for different particle mass flow rates. With the increase of the particle mass flow, the wear area changed from scattered to connected. The severely worn area of the

3.3 Effect of Wall Shape Change on Wear

103

Fig. 3.36 Collision number between particles and wall surface

prototype bend pipe was concentrated in the front section of the bend pipe, where the particles first collided with the outer wall. As the α bump became larger, the bump divided the severely worn area and finally shifted the severely worn area toward the bend entrance. The wear area of the outer wall surface of the bend with α bump = 30° was concentrated behind the bump. The wear area of the outer wall surface of the bend with α bump = 45° was concentrated in front of the bump. The maximum wear rate of the outer wall surface of these two bend pipes was close to that of the prototype bend pipe. The wear area of the outer wall surface of the two kinds of bends with α bump = 35° and α bump = 40° appeared larger, but the maximum wear rate was smaller than that of the prototype bend pipe. Figure 3.38 shows the change of the maximum wear rate of the outer wall surface for five kinds of bends with the particle mass flow rates. When the particle mass flow rate was less than 0.2 kg/s, the maximum wear rate increased rapidly. When the particle mass flow rate was further increased from 0.2 to 0.5 kg/s, the increase in the maximum wear rate slowed down. When the particle mass flow was greater than 0.5 kg/s, the maximum wear rate showed a stable trend. When the particle concentration was higher than a certain level, the particles near the wall surface formed a shield effect, preventing the rear particles from directly impacting the wall surface; so, the number of particles impacting the wall surface did not increase as the particle concentration increased. In the two kinds of bends with α bump = 35° and α bump = 40°, the maximum wear rate of the outer wall surface was generally

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3 Wear Characteristics of Static Walls

Fig. 3.37 Wear distribution at different particle mass flow rates

smaller than that of the prototype bend pipe, which indicated that the bump at these two positions could reduce the wear of the outer wall surface of the bend. When the particle mass flow was less than 0.3 kg/s, the maximum wear rate of the outer wall surface of the bend with α bump = 40° was smaller than that with α bump = 35°. When the particle mass flow rate was greater than 0.3 kg/s, the maximum wear rate of the outer wall surface of the bend with α bump = 40° was larger than that with α bump = 35°. For different particle mass flow conditions, there were different bumps to set the optimal position. In the two kinds of bends with α bump = 30° and α bump = 45°, the maximum wear rate of the outer wall surface was almost the same as that of the prototype bend pipe. For the conditions of large particle mass flow, the maximum wear rate of the outer wall surface of the bend with α bump = 30° was smaller. The bump of α bump = 30° could block the rolling cutting abrasion caused by particles along the outer wall surface for the large particle concentration. The larger the particle mass flow was, the smaller the α bump had to be when setting the bump position. Figure 3.39 show the changes of the average wear on the inner wall of the curved tube, the total energy loss of the particles in the flow passage, the number of collisions, and the energy loss of the particles in a single collision with the particle mass flow at each time step (0.001 s), when the bump was located at α bump = 35° and α bump = 40°. The total energy loss of the particles in the two kinds of bend pipe tubes increased with the increase of the particle mass flow rate, but the increasing trend became more and more gentle, and the two curves almost coincided, which was also the reason why the average wear amount of the outer wall surface of the two angles in Fig. 3.39a was almost the same. The number of particle collisions with the wall surface increased with the linear increase of the particle mass flow. The number of collisions with the α bump = 35° bend pipe was always greater than that with the α bump = 40° bend pipe. The energy loss of the particles in a single collision decreased with the increase of the particle mass flow, and the downward trend became more and

3.3 Effect of Wall Shape Change on Wear

105

Fig. 3.38 Maximum wear rate at different particle mass flow rates

more gentle. In comparison with the prototype bend pipe, the bump at both positions had the effect of reducing the maximum wear rate. The α bump = 35° bend pipe had a smaller energy loss from single impact particles and a smaller maximum wear rate when the particle mass flow rate was large. Figure 3.40 shows the relationship between the wear rate at the bump and the particle mass flow rate. When the particle mass flow rate was less than 0.2 kg/s, the wear rate at the bump increased rapidly. When the particle mass flow rate continued to increase, the wear rate at the bump gradually decreased. The increase of the particle mass flow rate led to an increase in the number of particles impacting the bump. However, when the particle mass flow rate was greater than a certain critical value, the particles stayed in the vicinity of the bump. These stagnation particles protected the bump from the direct impact of subsequent particles; so, the wear rate at the bump dropped to a minimum value. This showed that under the condition of large particle mass flow, adding a bump to the bend could reduce the maximum wear value of the outer wall surface of the bend, and the wear of the bump itself was relatively small.

3.3.1.3

Effect of Particle Size

This section describes how the change in the particle size affected the wear morphology and wear rate at various locations at the outer wall for the same particle mass flow rate. For convenience, the particle size was defined as D. Figure 3.41 shows the wear cloud image of the outer wall of the prototype bend pipe, the bend with α bump = 35°, and the bend with α bump = 40° for different particle size conditions. In the prototype bend pipe, when the particle size increased, the wear

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3 Wear Characteristics of Static Walls

Fig. 3.39 Comparison of average wear rate and particle information

zone moved towards the entrance of the bend. In the bends with α bump = 35° and α bump = 40°, at D = 1 mm and D = 2 mm, there was almost no wear on the area behind the bump, and the location of the largest wear point was far behind the bump. At D = 3 mm, the outer wall wear around the bump became serious, and the maximum wear point was close to the bump or even appeared in front of the bump. This was because when the particle size was small, the particles followed better. There was a circulation area behind the bump with a lower speed. The particles directly followed the fluid and bypassed the circulation area directly to the outlet.

3.3 Effect of Wall Shape Change on Wear

107

Fig. 3.40 Maximum wear rate of two bump bend pipes

Fig. 3.41 Wear distribution on the outer wall surface for different particle sizes

Under the condition that the particle mass flow rate was determined, the smaller the particle size was, the more likely it was that corrugated wear would occur on the outer wall surface of the bend. As shown in Fig. 3.41, the wear cloud images of the outer wall surfaces of the bend with D = 1 mm and D = 2 mm all showed obvious corrugations, but for D = 3 mm, the corrugations were not obvious. Karimi and Schmid [6] pointed out that when a fluid moved near the wall surface, friction occurred with the wall surface, while the fluid velocity was smaller than the fluid velocity at the center of the bend. Due to the influence of inertia, the particle concentration near the outer wall surface was much larger than the particle concentration in the central area. The combined effect of the above two factors enhanced the periodic flow disturbance around the particles. Due to the flow disturbance around the particles, the particles near the wall surface repeated the rebound and impact movement on the wall surface. Each small impact created pits on the outer wall surface, and the pits generated by a large number of particles eventually formed corrugations.

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3 Wear Characteristics of Static Walls

The smaller the particle diameter, the smaller the Stokes number, and the better the following of the particles. At the same particle concentration, the number of particles was larger. Therefore, the corrugated wear was more obvious. Figure 3.42 shows the maximum wear rate of the outer wall surface of the three bends at different angles under different particle size conditions. The maximum wear point of the outer wall surface of the prototype bend pipe appeared at α = 45° (1 mm), α = 55° (2 mm), and α = 45° (3 mm). The maximum wear point of the outer wall surface of the bend with α bump = 35° appeared at α = 65° (1 mm), α = 60° (2 mm), and α = 55° (3 mm). The maximum wear point of the outer wall surface of the bend with α bump = 40° appeared at α = 65° (1 mm), α = 60° (2 mm), and α = 30° (3 mm). The maximum wear point of the outer wall surface of the prototype bend pipe changed within a range with the change of the particle size, and the angle of the maximum wear point of the outer wall surface of the bend with α bump = 35° and α bump = 40° changed as the particle size increases became smaller. By comparing the maximum wear of the three pipes with the same particle size, it could be found that when D = 1 mm and D = 2 mm, the maximum wear rate of the outer wall surface of the prototype bend pipe was less than the bend with α bump = 40°, at D = 3 mm, and the maximum wear rate of the outer wall surface of the prototype bend pipe was greater than the bend with α bump = 35° and α bump = 40°. This showed that the existence of a bump was less effective in reducing the maximum wear caused by small particles than that caused by large particles (Fig. 3.42). Figure 3.43 shows the change of the average wear rate of the outer wall surface of the three bends with the particle size. The bump reduced the average wear of the outer wall surface of the three types of bends, especially when the particle size was small. Under the condition of small particles, the number of particles was large, and the collision energy loss of the single particles was not very different. The previous study showed that the number of particles–outer wall collisions in bends with bumps was much smaller than that in the prototype bend pipe, which resulted in the average wear rate of the outer wall of bends with a bump being much smaller than that of the prototype bend pipe. In the prototype bend pipe, the average wear rate was only slightly affected by the particle size. The particle size increased by 1 mm, and the average wear rate increased by less than 5%, but the maximum wear rate increased by 50%. Therefore, the change of the particle size mainly affected the maximum wear rate of the outer wall surface.

3.3.1.4

Effect of Inlet Velocity on Wear Results

The size of the inlet velocity affects the velocity of the first particle impact on the outer wall of the bend pipe, and the inlet velocity increases. If the particle concentration is

3.3 Effect of Wall Shape Change on Wear

109

Fig. 3.42 Maximum wear rate on the outer wall surface for different particle sizes

kept the same, the particle mass flow rate will also increase accordingly, so the size of the inlet velocity will also affect the number of particle outer wall collisions. Outer Wall Wear at Different Inlet Velocities Figure 3.44 shows the change of outer wall wear of five different bend pipes with different inlet velocities. It can be seen from Fig. 3.44a that the maximum wear rate of the outer wall of the two kinds of bend pipes with 35° and 40° is less than that of the prototype bend pipe at different inlet velocities. But the difference between the maximum wear rate of the two kinds of bend pipes with 30° and 45° and that of the prototype bend pipe becomes more obvious when the inlet velocity is large. In general, the maximum wear rate of the outer wall of the prototype bend pipe increases with the increase of the velocity. From Fig. 3.44b, it is found that the average wear rate of the outer wall of the bend pipe is almost linearly related to the velocity, and the difference between the various bends is very small, which indicates that the average wear rate is not changed greatly by adding bumps. When the inlet velocity is 5 m/s, the average wear rate of the outside wall of the bend pipe is very small, almost close

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3 Wear Characteristics of Static Walls

Fig. 3.43 Average wear rate on the outer wall surface for different particle sizes

to 0. However, when the velocity increases by 5 m/s, the average wear rate increases by more than 100%, which indicates that the average wear rate of the outer wall is greatly affected by the inlet velocity, and when the inlet velocity is less than a critical value, it will not cause obvious wear on the wall. Influence of Inlet Velocity on Wear Factor It has been proved in the previous study that the wear of the outer wall of the bend pipe is affected by the number of particle outer wall collisions and the impact energy loss. Figure 3.45 shows the curves of the number of particle outer wall collisions and the energy loss of a single particle with the inlet velocity in five kinds of bend pipes. It can be seen from the figure that there is a linear correlation between the

Fig. 3.44 Wear rate on the outside wall surface at different inlet velocities. a Maximum wear rate. b Average wear rate

3.3 Effect of Wall Shape Change on Wear

111

number of collisions and the inlet velocity because when the particle concentration is determined, the number of particles is linearly related to the inlet velocity, but the slope of the number of collisions with the inlet velocity of the five bend pipes is different, When the inlet velocity is 5 m/s, the difference between the particle outer wall collision times in the prototype bend pipe and that in the bump bend pipe is less than 100. However, with the increase of the inlet velocity, the difference will become larger and larger. When the inlet velocity is 15 m/s, the difference has reached 200 times. The energy loss curve of a single particle is different from that of collision times. When the inlet velocity is 5 m/s, the energy loss of a single particle in each bend pipe is almost the same, which is at a very low value. When the inlet velocity increases to 15 m/s, the energy loss gap of a single particle in each bend pipe has widened. The smallest difference of single-particle energy loss in the bump bend pipe (α = 30°) is 50% larger than that in the prototype bend pipe. The maximum difference (α = 45°) is 150% larger than that in the prototype bend. So, the maximum wear of the outer wall of the bend (α = 45°) is greater than that of the prototype bend pipe. However, the average wear of the bump bend pipe has little change compared with that of the prototype bend pipe.

3.3.1.5

Relation Between Stokes Number and Maximum Wear

In the previous study, it is found that the maximum wear rate of the outer wall of the bend pipe is closely related to the flow velocity and particle size. Therefore, Stokes number is introduced to analyze the wear. Stokes number is defined as the ratio of particle relaxation time to fluid characteristic time, which can reflect the inertia force and resistance of particles. It’s a dimensionless number to characterize the curvilinear motion of solid particles. The calculation formula is as follows: St =

ρp d2p u 18 μD

(3.13)

Among them, ρ p is the density of the particle, d p is the diameter of the particle, and u is the particle velocity, μ is the dynamic viscosity of the liquid, D is the hydraulic diameter of the bend pipe section. Five different Stokes number simulation conditions are set up α = The inner flow of 40° bend pipe was simulated. Due to the fixed geometry of the bend pipe and the physical properties of the particles, the Stokes number is changed by changing the particle size and inlet velocity, as shown in Table 3.12. Figure 3.46 shows the change of maximum wear rate with Stokes number for different particle mass flow rates. To facilitate the comparison of different particle mass flow conditions, the ratio of the maximum wear rate to the maximum wear rate at the Stokes number of 795.45 is taken as the ordinate. It can be seen from the figure that the maximum wear in the bend pipe increases with the increase of Stokes number, and the changing relationship between them tends to be a linear correlation, and the change situation under different particle mass flow rates is almost the same, which

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3 Wear Characteristics of Static Walls

Fig. 3.45 Collision numbers and single particle energy loss Table 3.12 Working condition settings of different Stokes numbers

Inlet velocity (m/s)

Particle diameter (mm)

Stokes number

Group 1

10

1

63.13

Group 2

10

2

252.53

Group 3

10

3

568.18

Group 4

12

3

681.81

Group 5

14

3

795.45

3.3 Effect of Wall Shape Change on Wear

113

Fig. 3.46 Change of maximum wear rate with different Stokes numbers

indicates that the change of particle mass flow rate will not change the relationship between the maximum wear rate and the Stokes number.

3.3.2 Wear Analysis of Wall Surface with Groove in Bend Pipe To improve the research efficiency and find out the more influential factors, the orthogonal test method is used to study the effect of the groove on the outer wall of the bend pipe. The CFD-DEM coupling method is used to simulate the wear of the outer wall of the bend pipe after the change of various factors of the groove on the bend pipe.

3.3.2.1

Wall Wear Analysis Based on Orthogonal Test

Orthogonal Experiment In the study of multi-factor problems, orthogonal experimental design is a more convenient operation of the experimental design method, the core of which is that each level of each factor is matched with each level of another factor once or only once. Experiments are carried out according to the established orthogonal table. Compared with the comprehensive test, the orthogonal test can greatly reduce the workload. Before the numerical simulation, four main factors are proposed: groove depth,

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3 Wear Characteristics of Static Walls

groove width, groove position angle, and solid particle concentration. Figure 3.47 shows the profile of the outer wall of the bend pipe and the properties of the groove added to the outer wall. The length of the groove in the Z direction is fixed at 22 mm as the square section of the bend pipe. The values of various factors are shown in Table 3.13. Orthogonal table L9(34 ) is a classical orthogonal table that meets the research requirements. The details are shown in Table 3.14. To minimize the mutual influence among the factors, the values of each level of each factor are arranged by random sampling. The final factor level table is shown in Table 3.15. After filling in the specific values of each level in Table 3.14, the practical orthogonal design scheme is obtained (Table 3.16). Wear analysis of bend pipe wall based on orthogonal test. After the groove is added to the outer wall of the bend pipe, the maximum wear and average wear of the outer wall are still concerned in this paper. The final test table and simulation results are as follows: Table 3.18 is obtained by processing Table 3.17. The treatment methods are as follows: (1) the calculated results of all levels i (i = 1, 2, 3) under a column, and mark it as K i;

Fig. 3.47 Diagram of groove factor

Table 3.13 Values of each factor (Bend pipe with groove)

Parameter

Value

Groove depth (mm)

1, 2, 3

Groove width (mm)

2, 3, 4

Groove position angle (°)

30, 35, 40

Particle concentration (%)

1, 5, 10

3.3 Effect of Wall Shape Change on Wear

115

Table 3.14 Orthogonal table L9 (34 ) Try experiment number

Factor A

Factor B

Factor C

Factor D

Horizontal combination

1

1

1

1

1

A1 B1 C1 D1

2

1

2

2

2

A1 B2 C2 D2

3

1

3

3

3

A1 B3 C3 D3

4

2

1

2

3

A2 B1 C2 D3

5

2

2

3

1

A2 B2 C3 D1

6

2

3

1

2

A2 B3 C1 D2

7

3

1

3

2

A3 B1 C3 D2

8

3

2

1

3

A3 B2 C1 D3

9

3

3

2

1

A3 B3 C2 D1

Table 3.15 Horizontal factors Level

Groove depth (A) (mm)

Groove width (B) (mm)

Groove position angle (C) (°)

Particle concentration (D) (%)

1

1

2

40

5

2

2

4

35

10

3

3

3

30

1

Table 3.16 Orthogonal design scheme Experiment number

Groove depth (mm)

Groove width (mm)

Groove position angle (°)

Particle concentration (%)

1

1

2

40

5

2

1

4

35

10

3

1

3

30

1

4

2

2

35

1

5

2

4

30

5

6

2

3

40

10

7

3

2

30

10

8

3

4

40

1

9

3

3

35

5

(2) the arithmetic mean value of the calculation results when all levels under a column are i, and mark it as k i ; (3) Max {K i } − min {K i )} under a column to get the range of each column, which is recorded as R.

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3 Wear Characteristics of Static Walls

Table 3.17 Simulation results Experiment number

A

B

C

D

Average wear rate (m/s)

Maximum wear rate (m/s)

1

1

1

1

1

9.58 × 10–9

1.88 × 10–7

2

1

2

2

2

1.87 × 10–8

3.55 × 10–7

3

1.73 ×

10–9

6.97 × 10–8

10–9

4.48 × 10–8

3

1

3

3

4

2

1

2

3

1.70 ×

5

2

2

3

1

9.80 × 10–9

1.76 × 10–7

2

1.84 ×

10–8

3.27 × 10–7

10–8

2.70 × 10–7

6

2

3

1

7

3

1

3

2

1.85 ×

8

3

2

1

3

1.71 × 10–9

5.82 × 10–8

1

9.70 ×

1.57 × 10–7

9

3

3

2

10–9

For the average wear rate, RD > RC > RB > RA , which indicates that the D factor (particle concentration) has the greatest influence on the average wear rate, and the specific range values are far greater than RC , RB and RA , the other three factors have a negligible effect on the average wear rate. For the maximum wear rate, RD > RA > RB > RC , which indicates that the influence of the D factor (particle concentration) on the maximum wear rate is also the largest among the four factors, followed by factor A (groove depth), factor B (groove width), factor C (groove position angle). The optimum scheme of the maximum wear rate is D3 A3 B1 C3 , that is, when the groove depth is 3 mm, the width is 2 mm, and the position angle is 30°, it is the best scheme to set the groove.

3.3.2.2

Research on Wall Wear of Bend Pipe Based on Control Variable Method

It is known from the previous study that the groove depth has the greatest impact on the outer wall wear of the grooved bend. Therefore, this paper makes a more in-depth study on this factor and considers whether the groove depth has different effects on the outer wall wear under different particle mass flow rates. The groove depth is set from 0.5 to 3 mm in six different conditions, and the particle mass flow rate is 0.0484 kg/s, 0.242 kg/s, 0.484 kg/s and 0.726 kg/s, respectively. The groove width, groove position angle, inlet velocity, and particle size are 2 mm, 30°, and 10 m/s, respectively. The specific simulation scheme is shown in Table 3.19. The calculated results of the outer wall wear of the bend pipe are shown in Fig. 3.48. The results show that the change of average wear rate with groove depth is almost a horizontal line under various particle mass flow conditions, which indicates that the change of groove depth has little effect on the average wear of the outer wall. When the particle mass flow rate Qp = 0.0484 kg/s, the maximum wear rate of the outer wall does not change significantly with the groove depth, but increases at first and then remains unchanged; When the particle mass flow rate Qp = 0.242 kg/s,

3.3 Effect of Wall Shape Change on Wear

117

Table 3.18 Result processing table A

B

C

D

K1

3.00 × 10–8

2.98 × 10–8

2.97 × 10–8

2.91 × 10–8

K2

2.99 × 10–8

3.02 × 10–8

3.01 × 10–8

5.56 × 10–8

K3

2.99 × 10–8

2.98 × 10–8

3.00 × 10–8

5.14 × 10–9

k1

1.00 × 10–8

9.93 × 10–9

9.90 × 10–9

9.70 × 10–9

k2

9.97 × 10–9

1.01 × 10–8

1.00 × 10–8

1.85 × 10–8

k3

9.97 × 10–9

9.93 × 10–9

1.00 × 10–8

1.71 × 10–9

R

1.00 × 10–10

4.00 × 10–10

4.00 × 10–10

5.05 × 10–8

Main factor → secondary factor

D→C→B→A

Excellent scheme

D3 C1 B1 A2

K1

6.13 × 10–7

5.03 × 10–7

5.73 × 10–7

5.21 × 10–7

K2

5.48 × 10–7

5.89 × 10–7

5.57 × 10–7

9.52 × 10–7

K3

4.85 × 10–7

5.54 × 10–7

5.16 × 10–7

1.73 × 10–7

k1

2.04 × 10–7

1.68 × 10–7

1.91 × 10–7

1.74 × 10–7

k2

1.83 × 10–7

1.96 × 10–7

1.86 × 10–7

3.17 × 10–7

k3

1.62 × 10–7

1.85 × 10–7

1.72 × 10–7

5.77 × 10–8

R

1.28 × 10–7

8.60 × 10–8

5.70 × 10–8

7.79 × 10–7

Main factor → secondary factor

D→A→B→C

Excellent scheme

D3 A3 B1 C3

Factor Average wear rate (m/s)

Maximum wear rate (m/s)

Table 3.19 Simulation scheme of groove depth Particle mass flow rate (kg/s)

Groove depth (mm)

0.0484

0.5

1

1.5

2

2.5

3

0.242

0.5

1

1.5

2

2.5

3

0.484

0.5

1

1.5

2

2.5

3

0.726

0.5

1

1.5

2

2.5

3

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3 Wear Characteristics of Static Walls

the maximum wear rate of the outer wall first increases and then decreases with the increase of groove depth, and the change range is large; When the particle mass flow rate Qp = 0.484 kg/s, the maximum wear rate of the outer wall first decreases and then increases with the increase of groove depth, and the change range is very large; When the particle mass flow rate Qp = 0.726 kg/s, the maximum wear rate of the outer wall first decreases and then increases with the increase of groove depth, and the amplitude is generally stable with Qp = 0.0484 kg/s. With the increase of the mass flow rate Qp , the maximum wear rate of the outer wall changes more and more obviously with the increase of the groove depth. However, when the increase exceeds a certain value, the change becomes not obvious. When the particle mass flow rate is small, the minimum value of the maximum wear rate of the outer wall appears at the outer wall of the bend pipe with the corresponding groove depth of 0.5 mm, the minimum value of the maximum wear occurred on the outer wall of the bend pipe with a groove depth of 1.5 mm. Figure 3.49 is a curve of the number of collisions between particles and the outer wall in a time step with the depth of the groove. It is found that under the same particle mass flow rate, the change of groove depth has little effect on the number of collisions between particles and the outer wall. When the particle mass flow rate is less than 0.484 kg/s, the number of collisions is directly proportional to the particle mass flow rate. When the particle mass flow rate continues to increase, the number of collisions increases, but the increase amplitude decreases, which is due to the shielding effect. The particle mass flow rate of the bend pipe with groove is larger than that of the bend pipe with bump. Figure 3.50 shows the energy loss of a single particle impact on the outer wall with different groove depth under different particle mass flow conditions. When the particle mass flow rate Qp = 0.0484 kg/s, the particle energy loss decreases first and then increases with increase of groove depth. When the particle mass flow rate Qp = 0.242 kg/s, the energy loss of single particle impact on the outer wall is very small, and the numerical value is the smallest among the four particle mass flow conditions. When the mass flow rate of particles Qp = 0.484 kg/s, the energy loss of a single

Fig. 3.48 Wear rate on outer wall surface of bend pipes with different groove depth. a Maximum wear. b Average wear

3.3 Effect of Wall Shape Change on Wear

119

Fig. 3.49 Relationship between collision numbers groove depth in one time step

particle impact is also very small, and the minimum value appears at the groove depth of 1.5 mm. When the particle mass flow rate Qp = 0.726 kg/s, the energy loss of a single particle impact decreases first and then increases with the increase of groove depth. And it is the largest in the four particle mass flow conditions. It can be found that with the increase of particle mass flow rate, the energy loss of a single particle impact on the outer wall first decreases and then increases. The reason may be that with the increase of particles mass flow rate, the number of collisions between particles increases and the velocity of particles decreases. So, the energy loss of the single particle impact on the wall decreases. But when the particles increase to a certain extent, the number of collisions between the particles and the outer wall surface decreases due to the shielding effect. Only the particles with the fastest velocity can collide with the outer wall first, so the energy loss of the single particle impact on the wall will increase. When the groove depth is 1.5–2 mm, the energy loss of a single particle impact on the outer wall is the smallest.

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3 Wear Characteristics of Static Walls

Fig. 3.50 Energy loss of single particle impact on the outer wall

References 1. Zeng D, Zhang E, Ding Y et al (2018) Investigation of erosion behaviors of sulfur-particle-laden gas flow in an elbow via a CFD-DEM coupling method. Powder Technol 329:115–129 2. Cundall PA, Strack ODL (1983) Modeling of microscopic mechanisms in granular material. Stud Appl Mech 7:137–149 3. Archard JF (1953) Contact and rubbing of flat surfaces. J Appl Phys 24(8):981–988 4. Prasad BK, Prasad SV, Das AA (1992) Mechanisms of material removal and subsurface work hardening during low-stress abrasion of a squeeze-cast aluminium alloy-Al2 O3 , fibre composite. Mater Sci Eng A 156(2):205–209 5. Duarte CAR, Souza FJD, Salvo RDV et al (2017) The role of inter-particle collisions on elbow erosion. Int J Multiph Flow 89:1–22 6. Karimi A, Schmid RK (1992) Ripple formation in solid–liquid erosion. Wear 156(1):33–47 7. Zhou JW, Liu Y, Liu SY et al (2017) Effects of particle shape and swirling intensity on elbow erosion in dilute-phase pneumatic conveying. Wear 380–381:66–77

Chapter 4

Two-Phase Flow Characteristics and Transportation Performance in Centrifugal Pump

Abstract The existence of solid particles will change the flow field in the pump, which will be reflected in the external characteristics. Therefore, it is necessary to study the particle movement and obtain the two-phase flow law in the channel of the centrifugal pump through calculation, so as to obtain the influence law of various factors on the external characteristics of the pump. At present, there are several ways to deal with solid particles in the flow calculation of solid–liquid two-phase flow pump. When the diameter of solid particles is small, it can be treated as a continuous phase. So, the Two Fluid Model based on Euler-Euler method is used. When the solid particles increase to a certain extent, the numerical method based on TFM is no longer applicable. So, the particles need to be treated as discrete phases based on Lagrangian view point. In this chapter, according to different solid–liquid transportation conditions, three solid–liquid two-phase models are used to complete the internal flow simulation of the centrifugal pump, and the internal flow characteristics and mixed transportation performance changes of the centrifugal pump are analyzed under different conditions.

4.1 Calculation of Two-Phase Flow in Pump Based on Mixture Model According to the calculation results in Chapter 2, when the centrifugal pump is used for small particle solid–liquid two-phase transportation, the mixture model based on Euler-Euler method can meet the calculation accuracy requirements.

4.1.1 Calculation Scheme and Grid Based on FLUENT software, the numerical calculation of solid–liquid two-phase flow in centrifugal pump is completed for different particle sizes, concentrations, densities and flow rate conditions. The design flow rate for the centrifugal pump is

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Z. Zhu et al., Solid-Liquid Two-Phase Flow in Centrifugal Pump, Fluid Mechanics and Its Applications 136, https://doi.org/10.1007/978-981-99-1822-5_4

121

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Table 4.1 Numerical simulation scheme Particle diameter d (mm)

Concentration C V (%)

Flow rate Q (m3 /h)

Density ρ (kg/m3 )

0.05, 0.10, 0.15, 0.20, 0.25

1, 4, 7, 10, 13, 16

40, 100, 140

1100, 1300, 1500, 1700, 1900, 2100

100 m3 /h, the head is 19 m, the rotating speed is 2900 r/min, and the number of blades is 6. The calculation scheme is shown in Table 4.1. The rotating area of centrifugal impeller and the static area of volute are divided into tetrahedral unstructured grids, and the extended area of suction chamber and volute are structured grids. The total grid number of computing domain is 1.21 million (Fig. 4.1). Taking the velocity inlet as the inlet boundary condition, the inlet velocity can be calculated from the flow rate. Assuming that the flow has been fully developed when the fluid reaches the outlet boundary of the volute, the free outflow is set as the outlet boundary condition. Due to the influence of fluid viscosity, the non-slip wall boundary condition is used, and the standard wall function method is used in the near wall region.

(a) Impeller channel Fig. 4.1 Computational domain grid

(b) Volute channel

4.1 Calculation of Two-Phase Flow in Pump Based on Mixture Model

123

Fig. 4.2 Effect of particle size on pump external characteristics

4.1.2 Different Solid Particle Sizes Conditions 4.1.2.1

Effect of Particle Size on External Characteristics

The quantitative analyses of the influence of different particle diameters on the external characteristics of the pump are shown in Fig. 4.2. From Fig. 4.2, as the particle diameter increasing from 0.05 to 0.25 mm, the pump head decreases from 16.55 to 16.26 m, and the efficiency decreases from 70.89 to 69.72%. However, the shaft power is maintained at about 6.66 kw, and the change range is very small. The influence of particle size on shaft power is almost unchanged.

4.1.2.2

Effect of Particle Size on Particle Distribution

Figures 4.3, 4.4 and 4.5 respectively show the contour map of particle concentration distribution on the middle section of the impeller, the impeller surface and the blade surface at the design flow rate and different particle sizes. The particle size has a great influence on the distribution of particles in the impeller channel and the movement of particles. From the impeller inlet to the outlet, the solid volume fraction decreases. The particles are mainly concentrated at the impeller inlet, and the concentration reaches 10%, and then decreases. When reaching the impeller outlet, the particle concentration is 1%. The particles in the whole volute channel are relatively small, which are all 1%. Also, at the same particle concentration, the larger the particle size, the greater the particle concentration on the pressure surface of the blade, and the concentration value decreases from 10 to 1%. When the particle diameter is 0.05 mm, the concentration at the impeller inlet is 10% and then decreases to 1%, and the particle concentration in most areas of the volute channel is 1%. The particles

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4 Two-Phase Flow Characteristics and Transportation Performance …

(a) d= 0.05mm

(b) d= 0.1mm

(c) d= 0.25mm

Fig. 4.3 Particle distribution on the middle section of impeller

(a) d= 0.05mm

(b) d= 0.1mm

(c) d= 0.25mm

Fig. 4.4 Particle distribution on the surface of impeller calculation domain

are mainly concentrated at the inlet edge of the impeller, so the probability of the blade inlet being worn increases. When the particle diameter is 0.25 mm, the impeller inlet concentration is 10% and then decreases to 1%, but the high concentration area is smaller. The particles are also mainly concentrated in the impeller channel.

4.1.2.3

Effect of Particle Size on Static Pressure

Figure 4.6a and b show the influence of particle size on the static pressure distribution on different blade surfaces, where the abscissa represents the relative blade length, and the ordinate represents the static pressure value. Compared with Fig. 4.6a and b, the solid concentration at the blade inlet is greater than that at the blade outlet. The inlet and outlet pressure difference is large, and the static pressure difference reaches 184 kPa. No matter the suction surface or the pressure surface, the static pressure value at the same position of the blade delivering the particle diameter of 0.25 mm is

4.1 Calculation of Two-Phase Flow in Pump Based on Mixture Model

125

(a) d=0.05mm Pressure surface (c) d=0.1mm Pressure surface (e) d=0.25mm Pressure surface

(b) d=0.05mm Suction surface (d) d=0.1mm Suction surface (f) d=0.25mm Suction surface

Fig. 4.5 Particle distribution on blade surface

less than that delivering the particle diameter of 0.05 mm. Figure 4.6a shows that the maximum static pressure value is 155.19 kPa and the minimum static pressure value is – 7.83 kPa on the blade pressure surface. Figure 4.6b shows that the maximum static pressure is 136.43 kPa, while the minimum static pressure is – 48.56 kPa on the blade suction surface. Figure 4.6c shows that the solid concentration increases with the increase of particle size. The particle distribution at different positions of the impeller is different. The average particle concentration at the back cover plate is more than 4%, the particle concentration at the blade suction surface is more than 3% but less than 4%, and the particle concentration at the front cover plate and the pressure surface is less than 3%. The particles are mainly distributed at the back cover plate of the impeller, followed by the blade suction surface, the front cover plate and the blade pressure surface.

4.1.2.4

Effect of Particle Size on Velocity

Figure 4.7a shows the relative velocity distribution of blade suction surface and pressure surface when the solid particle diameter is 0.05 mm. From Fig. 4.7, it can be seen that the velocity increases gradually from the blade inlet to the outlet. The relative velocity of the pressure surface and the suction surface have the same overall change trend. The particle velocities on the pressure surface and suction surface at the impeller inlet are 10 and 9 m/s, respectively. So it is obvious that the velocity on the pressure surface is greater than that on the suction surface. They tend to be same at the impeller outlet and reach to 21 m/s. Figure 4.7b shows the mixure phase velocity distribution along the circumference of the interface between impeller and volute. The velocity peak appears at θ = 135°, and it is 12 m/s. The minimum velocity value appears between 180° and 225° and it is about 8 m/s. When the particle size is small, the influence of different particle sizes on the impeller outlet velocity can be almost ignored. It shows a certain regularity of periodicity and symmetry. Figure 4.7c show

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4 Two-Phase Flow Characteristics and Transportation Performance …

(a) Blade pressure surface

(b) Blade suction surface

(c) Particle distribution on the different positions surface of impeller Fig. 4.6 Effect of solid particle size on particle distribution curve

particle phase velocity distribution along the circumference of the interface between impeller and volute. When paritcle diameter is 0.05 mm, the particle phase velocity reaches 11 m/s near 270°. Though the particle size is different, the change trend is the same. When the particle diameter are larger and the position reaching the same velocity is more backward due to the greater mass with the greater inertia. Figure 4.8 shows the relative velocity distribution of liquid phase in the middle section under the design flow rate when the solid particle diameter is 0.05 mm, 0.10 mm and 0.25 mm respectively. From Fig. 4.8, the relative velocity of the liquid phase increases from the impeller inlet to the volute outlet. The relative velocity near the inlet is 5 m/s, and the relative velocity at the outlet reaches 55 m/s. The relative velocity gradient in the impeller is large, and the energy loss is relatively large. The flow field in the centrifugal pump is uniform and stable under the design condition.

4.1 Calculation of Two-Phase Flow in Pump Based on Mixture Model

127

(a) Relative velocity of suction surface and pressure surface( d= 0.05mm)

(b) Mixed phase velocity

(c) Particle phase velocity

Fig. 4.7 Effect of solid particle size on velocity distribution along the circumference interface between impeller and volute

Figure 4.9 shows the cloud diagram of solid phase concentration distribution in the middle section under the design flow rate when the solid particle diameter is 0.05 mm, 0.10 mm and 0.25 mm respectively. From Fig. 4.9, the concentration distribution of solid particles with different particle sizes is basically the same. The particle concentration near the impeller inlet is 5%, and the particle concentration in the impeller channel is 10%. The particle concentration inside the volute is relatively small, and the concentration value is 5%. Smaller particle size has little effect on particle concentration distribution.

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4 Two-Phase Flow Characteristics and Transportation Performance …

(a) d= 0.05mm

(b) d= 0.1mm

(c) d= 0.25mm

Fig. 4.8 Relative velocity of liquid phase with different particle sizes

(a) d= 0.05mm

(b) d= 0.1mm

(c) d= 0.25mm

Fig. 4.9 Solid particle distribution

4.1.3 Different Solid Concentration Conditions 4.1.3.1

Effect of Particle Concentration on External Characteristics

According to Fig. 4.10, when the particle concentration is 1%, the shaft power is the minimum value of 6.777 kW, and the efficiency and head are 74.4% and 18.47 m respectively. When the particle concentration increases to 16%, the shaft power increases to a maximum value of 7.49 kW, and the efficiency and head reduces to a minimum value of 60.13% and 15.34 m respectively. The influence of solid particle concentration on the head, efficiency and shaft power of centrifugal pump is different. As the concentration increases from 1 to 16%, the pump power increases, but the head and efficiency decrease. The decrease of head and the increase of power are due

4.1 Calculation of Two-Phase Flow in Pump Based on Mixture Model

129

Fig. 4.10 Effect of volume concentration on external characteristics of pump

to the increase of solid particle concentration, which increases the viscosity of the mixture and the friction between internal fluids, resulting in the increase of hydraulic loss.

4.1.3.2

Effect of Particle Concentration on Particle Distribution

Figure 4.11 shows the particle concentration distribution in the middle section of the impeller under the design flow rate with different initial solid concentration and particle size d = 0.10 mm. In Fig. 4.11a, the maximum concentration at the inlet side is 1.4%, and it is 1% at the suction side of impeller and outlet side when Cv is 4%. The maximum concentration at the inlet side is 11%, and that at the suction side of impeller and outlet side is 9% when Cv is 10%. In Fig. 4.11a, the maximum concentration at the inlet side is 1.4%, and the maximum value of the center of the impeller rear cover plate is 1.4% when Cv is 4%. The maximum concentration at the inlet side is 12%, and the maximum value of the center of the impeller rear cover plate is 12%. The particles are mainly concentrated on the blade suction surface inside the impeller channel.

4.1.4 Different Flow Conditions 4.1.4.1

Effect of Different Flow Rates on Particle Distribution

Figure 4.12 shows the relative velocity distribution of liquid phase in the middle section under different flow rates, when the flow rates are 0.4 Qopt , Qopt and 1.4 Qopt

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4 Two-Phase Flow Characteristics and Transportation Performance …

CV=4%

CV=10%

CV=16%

(a) The middle section of the impeller

CV=4%

CV=10%

CV=16%

(b) The impeller rear cover plate Fig. 4.11 Effect of different solid volume concentration on particle distribution

respectively. From Fig. 4.12a–c, the relative velocity of liquid phase from inlet to outlet gradually increases, the relative velocity near the inlet is 5 m/s, and the relative velocity at the outlet reaches 55 m/s. At the flow rate of 1.4 Qopt , the relative velocity gradient of liquid phase inside the impeller is large, and the energy loss is relatively large, especially at the tongue of volute. As shown in Fig. 4.12d, there is a significant velocity gradient. The relative velocity rapidly decreases from 50 to 10 m/s. Under small flow rate and design flow rate, the fluid flow in the centrifugal pump is uniform and stable, and there is no huge velocity gradient change at the tongue. Figure 4.13 shows the solid phase concentration distribution in the middle section under different flow rates, when the flow rates are 0.4 Qopt , Qopt and 1.4 Qopt respectively. According to the analysis of Fig. 4.13, the particle concentration distribution is basically the same under different flow rates when the particle diameter is 0.10 mm. Smaller particle size has little effect on particle concentration distribution.

4.1 Calculation of Two-Phase Flow in Pump Based on Mixture Model

(a) 0.4Qopt

(b) 1.0Qopt

131

(c) 1.4Qopt

(d) Partial enlargement near tongue at flow rate of 1.4Qopt

Fig. 4.12 Relative velocity of liquid phase in the middle section with different flow rates

(a) 0.4Qopt

(b) 1.0Qopt

Fig. 4.13 Particle concentration in the middle section at different flow rates

(c) 1.4Qopt

132

4.1.4.2

4 Two-Phase Flow Characteristics and Transportation Performance …

Effect of Flow Rate on Pressure Distribution

Figure 4.14 shows the static pressure distribution in the middle section of the impeller when the particle diameter of the solid phase is 0.10 mm, the solid phase concentration Cv is 10%, and the flow rates are 0.4 Qopt , Qopt and 1.4 Qopt respectively. At the flow rate of 0.4 Qopt , the maximum static pressure at the outlet is 300 kPa, the static pressure near the inlet side is 0 kPa. At design flow rate, the static pressure near the inlet side is 0 kPa, and there is a negative pressure near the suction surface, which is – 20 kPa. But at the flow rate of 1.4 Qopt , the high-pressure area is mainly concentrated in the lower half of the volute, and the negative pressure appears at the volute outlet and the tongue, and the static pressure is – 100 kPa. Comparing 4.14a–c, the static pressure from impeller inlet to outlet gradually increases, and the range of static pressure is from – 20 to 160 kPa. As shown in Fig. 4.14e, there is a large gradient of static pressure at the tongue at the flow rate of 1.4 Qopt , which changes from – 800 to – 100 kPa. The static pressure distribution in the impeller channel is uneven, and there is a large static pressure gradient in the blade channel near the volute tongue.

4.1.4.3

Effect of Flow Rate on Velocity Distribution

Figure 4.15 shows the relative velocity distribution of solid particles in the impeller under three different flow rates. From Fig. 4.15, the streamlines distribution inside the impeller is completely inconsistent under different working conditions. Under off design conditions, the streamline distribution is extremely uneven. At the small flow rate of 0.4 Qopt , as shown in Fig. 4.15a, a small vortex appears in the impeller channel near the volute tongue, and the flow trace at other positions is irregular. As shown in Fig. 4.15c, there is slight eddy at the impeller inlet. At the design flow rate, there is no vortex and secondary flow, and the velocity trace is evenly distributed. The results show that when the flow rate is less than the design flow rate, the vortex is easy to occur in the impeller channel near the volute tongue, and the vortex phenomenon becomes more serious with the decrease of flow rate. Figure 4.16 shows the vortex intensity along the circumferential interface between impeller and volute under different flow rates, which appears the same change law, the smaller the flow, the greater the vortex intensity. At the flow rate of 1.4 Qopt , the peak value of vorticity magnitude exceeds 16,000 near θ = 180°. at the design flow rate, the average turbulence intensity is 1698.45. Under off design conditions, the larger the flow Q deviates from the design point flow Qopt , the more the turbulence intensity distributed along the circumference deviates from the average one at the design flow rate. The maximum turbulence intensity average is 2129.2 when the flow rate is Qopt . The minimum turbulence intensity average is 3486.37 when the flow rate is 1.4 Qopt . So the internal flow in the centrifugal pump is more stable under the design condition.

4.1 Calculation of Two-Phase Flow in Pump Based on Mixture Model

(a) 0.4Qopt

(b) 1.0Qopt

(d) Blade inlet at flow rate of Qopt

133

(c) 1.4Qopt

(e) Tongue at flow rate of 1.4Qopt

Fig. 4.14 Static pressure distribution of middle section under different flow rates

(a) 0.4Qopt

(b) 1.0Qopt

(c) 1.4Qopt

Fig. 4.15 Relative velocity streamline of solid phase in middle section of impeller

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4 Two-Phase Flow Characteristics and Transportation Performance …

Fig. 4.16 Vorticity distribution along the circumferential interface between impeller and volute

4.1.4.4

Effect of Particle Size on Particle Distribution Under Low Flow Conditions

Figure 4.17 shows the relative velocity distribution of liquid phase in the middle section under low flow conditions. As can be seen from Fig. 4.17a–c, the relative velocity of the liquid phase increases from the inlet to the outlet. At the flow rate of 1.4 Qopt , the relative velocity gradient of liquid phase in the impeller is large, and the energy loss is relatively large, especially at the tongue of volute.

(a) d= 0.05 mm

(b) d= 0.1 mm

(c) d= 0.25 mm

Fig. 4.17 Relative velocity of liquid phase with different particle sizes at flow rate of 0.4 Qopt

4.1 Calculation of Two-Phase Flow in Pump Based on Mixture Model

135

Fig. 4.18 The external characteristics of the pump with the particle density

4.1.5 Different Solid Density Conditions 4.1.5.1

Effect of Particle Density on External Characteristics

Figure 4.18 shows the change curve of the external characteristics of the pump with the particle density. When the particle density is 1100 kg/m3 , the power is the minimum value of 6.59 kw, the efficiency is 74.8%, and the head is 16.82 m. When the particle density is 2100 kg/m3 , the power is the maximum value of 6.76 kw, the efficiency is 74.16%, and the head is 15.54 m. with the increase of particle density from 1100 to 2100 kg/m3 , the shaft power of centrifugal pump increases, while the head and efficiency decrease. In comparison, the solid particle density has less influence on the performance of the pump than the particle diameter and concentration.

4.1.5.2

Effect of Particle Density on Particle Distribution

Figure 4.19 shows the particle distribution in the middle section of the impeller under the conditions of different density particles. When the density is 1100 kg/m3 , the concentration at impeller inlet is 10%, and then decreases along the impeller channel, the particle concentration is reduced to 1% at impeller outlet. When the density is 1500 kg/m3 , the concentration at impeller inlet is 10%, and the range for the particle concentration of 1% is less than that under the condition of the density of 1100 kg/m3 . Also, it decreases along the impeller channel, and the particle concentration is reduced to 1% at impeller outlet. When the density is 1900 kg/m3 , and the range for the particle concentration of 1% is smaller than that under the condition of the density of 1500 kg/m3 , also the particle concentration is reduced

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4 Two-Phase Flow Characteristics and Transportation Performance …

to 1% at impeller outlet. With the increase of solid particle density, the particle distribution in the impeller is basically the same. From Fig. 4.20, the solid particle concentration increases with the increase of particle density. The average particle concentration on the rear cover plate of the impeller is 5%, the average value on the blade suction surface is 4.8%, and it is relatively small on the front cover plate and the blade pressure surface. The particle distribution at different positions of the impeller is different, mainly distributed at the rear cover plate of the impeller, followed by the blade suction surface, the front cover plate and the blade pressure surface.

(a) ρ= 1100 kg/m

(b) ρ= 1500 kg/m

(c) ρ= 1900 kg/m

Fig. 4.19 Influence of density on particle distribution in impeller cross section

Fig. 4.20 Particle distribution at different positions of impeller

4.2 Two Phase Flow Calculation in Pump Based on DPM Model

4.1.5.3

137

Effect of Particle Density on Static Pressure

Figure 4.21a and b show the influence of particle diameter on the static pressure distribution on different blade surfaces. The abscissa represents the relative blade length, and the ordinate represents the static pressure value. Under different solid particle density conditions, the change trend of static pressure of blade pressure surface and suction surface is basically the same. The static pressure on the blade pressure surface is greater than that on the blade suction surface. Whether the suction surface or the pressure surface, the static pressure under the condition of the density of 1900 kg/m3 is greater than that at the same position of the blade compared to the condition of the density of 1500 kg/m3 . From Fig. 4.21a, the maximum static pressure on the blade pressure surface is 154.38 kPa, while the minimum static pressure is – 8.2 kPa. From Fig. 4.21b, the maximum static pressure on the blade suction surface is 135.83 kPa and the minimum static pressure is – 49.66 kPa.

4.2 Two Phase Flow Calculation in Pump Based on DPM Model With the increase of particle diameter, the particle volume effect is obvious, and there is a large error in calculating and analyzing as a fluid continuous phase. The two-phase model based on Euler Lagrange method needs to be adopted. When the concentration is less than 10%, the DPM model ignoring the collision between particles can meet the calculation accuracy requirements. For the centrifugal pump with the same model as the previous section, the DPM two-phase model is used to calculate and analyze the internal flow characteristics of large particle solid–liquid two-phase transportation, and the internal flow characteristics of the solid–liquid two-phase flow pump are studied under different solid-phase parameters and different flow rates.

(a) Blade pressure surface

(b) Blade suction surface

Fig. 4.21 Effect of particles density on static pressure distribution on blade surface

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4 Two-Phase Flow Characteristics and Transportation Performance …

Table 4.2 Numerical calculation scheme

Solid particle diameter Concentration (%) Flow rate (m3 /h) (mm) 2.5, 3.75, 5, 8, 12

1, 5, 10

80, 100, 120

4.2.1 Calculation Scheme Because the rotational speed of the centrifugal pump used in the calculation is 2900 r/min, the blade number is 6, and the rotation period T is 0.0207 s/r. Each 1° rotation of the impeller is set as a time step, namely Δt is 5.747 × 10–5 s. The liquid phase is normal temperature clean water with a density of 1000 kg/m3 , and the solid phase is sand with a density of 1550 kg/m3 . The particles are assumed as spherical ones with the same diameter (Table 4.2). The inlet boundary condition is set as the velocity inlet and it is assumed that there is no velocity slip between the liquid phase velocity and the solid phase velocity at the inlet, so the inlet velocities of the two phases are the same. The inlet velocity is calculated according to the flow rate. The inlet turbulence is described by the turbulence intensity and the hydraulic diameter. The inlet turbulence intensity is set to 5%, and the hydraulic diameter is the inlet diameter of the pump impeller Dj (89 mm). The outlet boundary condition is set as free outflow. No slip boundary condition is adopted for the wall surface and standard wall function is adopted for the near wall area. The escape boundary condition is adopted for the particles at the inlet and outlet, while the boundary condition on the inner surface of each flow channel part is set as rebound and elastic collision. The type of particle jet source is surface jet source, and the initial position of incidence is on the inlet surface of the centrifugal pump. The particle incidence velocity vp is the same as the fluid velocity at the inlet and vp is 4.465 m/s.

4.2.2 Different Solid Particle Sizes Conditions Under the conditions of design flow rate Q of 100 m3 /h and solid concentration C v of 10%, the numerical simulation results under three particle diameters of 5 mm, 8 mm and 12 mm are analyzed respectively.

4.2.2.1

Effect of Particle Diameter on Liquid Phase Flow

Figure 4.22 is the liquid phase streamline in the flow channel with different particle diameters. The flow in the pump is smooth and there is no obvious vortex, indicating that the flow condition is good. Due to its obstruction at the tongue of the volute, part of the liquid-phase streamline will return to the volute and the impeller flow channel, and the particles will also return to the volute with it. The kinetic energy

4.2 Two Phase Flow Calculation in Pump Based on DPM Model

(a) d = 5mm

(b) d = 8mm

139

(c) d = 12mm

Fig. 4.22 Streamline of liquid phase with particle diameter in the flow channel of the pump

of the particles returning to the flow channel is very large, resulting in friction loss with the wall. The particles flowing back to the flow channel will increase the solid volume concentration in the flow channel, and the power required to be consumed under the same flow rate will also increase. The increase of particle diameter will lead to the increase of particle volume effect, the increase of inertia force, and the poor following of solid particles. Therefore, the larger the diameter, the more particles flow back to the volute with the liquid phase, and the particle concentration in the volute will increase. Under the same flow rate, the efficiency of centrifugal pump conveying two-phase medium will decrease.

4.2.2.2

Effect of Particle Diameter on Pressure and Velocity

Figure 4.23 shows the volute cross section with the tongue at 67.5° and the static pressure distribution curve of the interface between the impeller and the volute along the circumferential direction under different particle diameter conditions. From Fig. 4.23, the particle diameter has little influence on the pressure distribution on the whole interface, only during the range of 40°–80° and 110°–150°, the pressure distribution is greatly affected by the particle diameter. The two positions are near section VII and section II respectively. Since the shape of two flow channel sections changes greatly, the pressure distribution will also be affected. Along the circumference, the static pressure distribution changes periodically. The peak position of static pressure on the interface is at θ = 210.7°, and the maximum pressure is 154.74 kPa. The minimum pressure is at θ = 171.27°, and the minimum value is 8.29 kPa. Figure 4.24 shows the mixed phase velocity distribution curve along the circumferential direction at the interface of impeller and volute. The particle diameter has certain influence on the mixed phase velocity in circumferential direction θ = 60°–80° and 130°–150°, and these two positions are also near section VII and section II, which indicates that the position where the channel shape changes greatly has great influence on the mixed

140

4 Two-Phase Flow Characteristics and Transportation Performance …

phase velocity. The velocity distribution of other positions is basically the same. The maximum mixed phase velocity appears at θ = 297.39°, and the maximum value is 16.21 m/s. The minimum one appears in θ = 23.48°, and the minimum value is 6.10 m/s.

Fig. 4.23 Static pressure distribution curve at the interface between impeller and volute

Fig. 4.24 Mixed phase velocity distribution curve at the interface of impeller and volute

4.2 Two Phase Flow Calculation in Pump Based on DPM Model

(a) d = 5mm

(b) d = 8mm

141

(c) d= 12mm

Fig. 4.25 Variation of discrete phase velocity distribution with particle diameter in pump

4.2.2.3

Effect of Particle Diameter on Discrete Phase Velocity Distribution

Figure 4.25 is a scatter diagram of the discrete phase velocity distribution in the pump with different particle diameters and the color depth indicates the velocity magnitude. Under the working conditions of different particle diameters, the variation of the discrete phase velocity is basically similar. In the impeller channel, the particles are subjected to centrifugal force due to the rotating motion of the blades, and the discrete phase velocity will gradually increase from the inlet to the outlet of the impeller. The discrete phase velocity near the blades is the largest. When the particle diameter d is 5 mm, 8 mm and 12 mm, the maximum discrete phase velocity is 32.1 m/s, 34.27 m/s and 36.08 m/s respectively. From the impeller outlet to the volute outlet, the discrete phase velocity decreases. The discrete phase velocity near the outer edge wall of volute is smaller than that at other positions of volute channel. The blocking effect of the tongue position is the same as its effect on the liquid phase velocity. The particle velocity here will decrease, and the larger the diameter, the larger the particle concentration near the tongue, indicating that the tongue has a stronger blocking effect on the particles with smaller diameter.

4.2.2.4

Effect of Particle Diameter on Discrete Phase Distribution

Figure 4.26 shows the discrete phase distribution in the impeller and volute under different diameter conditions. The particle concentration in the volute is smaller than that in the impeller. With the increase of particle diameter, the discrete phase concentration in the impeller will increase, and the particle concentration in the volute will decrease. When the diameter is small, the particles in the volute are mainly distributed on the wall near the rear cover plate. When the diameter is large, the particles are distributed on the whole wall. The increase of solid particles in the impeller is mainly due to the larger

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4 Two-Phase Flow Characteristics and Transportation Performance …

(a) Impeller (d= 5mm)

(b) Volute (d= 5mm)

(c) Impeller (d= 8mm)

(d) Volute (d= 8mm)

(e) Impeller (d= 12mm)

(f) Volute (d= 12mm)

Fig. 4.26 The discrete phase distribution in the channel under different diameter conditions

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143

the diameter, the more obvious the volume effect, the greater the inertial force on the particles, the worse the particle following with the liquid movement, and the more solid particles retained in the impeller, the less the particles will flow into the volute. The particle concentration at the impeller inlet, the blade pressure surface close to the rear cover plate, the rear cover plate close to the blade pressure surface is large, and the particle concentration at the volute wall close to the rear cover plate is also large. The wear caused by the particles is more serious at these positions with large particle concentration.

4.2.3 Different Solid Concentration Conditions Under the conditions of design flow rate Q of 100 m3 /h and solid phase diameter d of 5 mm, the numerical simulation results were analyzed for three solid phase concentrations of 1, 5 and 10%

4.2.3.1

Effect of Solid Concentration on Liquid Streamline

Figure 4.27 shows the liquid phase streamline in the pump under different particle concentration conditions. The streamlines under the three conditions are smooth, indicating that the internal flow state is good and there is no obvious mutation in the streamlines. Currently, the loss in the pump is mainly hydraulic friction loss, and the fluid collision loss is small. Some streamlines do not flow out of the volute outlet diffuser analyzing from the streamline distribution near the tongue, but return to the volute or impeller from the tongue, which further increases the particle concentration in the flow channel. Meanwhile, the high-speed moving particles returning to the flow channel increase the friction loss, which leads to the decrease of the delivery efficiency of the pump.

4.2.3.2

Effect of Solid Concentration on the Discrete Phase Motion

Figure 4.28 is a scatter diagram of the discrete phase velocity distribution in the pump under different particle concentration conditions. The color depth indicates the velocity magnitude. The variation of the discrete phase velocity in the channel is basically similar under the three particle concentration conditions, and the particle concentration has little effect on the discrete phase velocity. In the impeller, the discrete phase velocity gradually increases from the impeller inlet to the outlet, and the maximum velocity is 32.05 m/s under the particle concentration of 10%. From the impeller outlet to volute outlet, the discrete phase velocity near the volute wall is small, and the particle velocity near the volute outlet decreases due to the blocking effect of the tongue.

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4 Two-Phase Flow Characteristics and Transportation Performance …

(a) Cv= 1%

(b) Cv= 5%

(c) Cv= 10%

Fig. 4.27 Variation of liquid streamline with particle concentration

(a) Cv= 1%

(b) Cv= 5%

(c) Cv= 10%

Fig. 4.28 Variation of discrete phase velocity distribution with particle diameter (m/s)

Figure 4.29 is the particle Reynolds number distribution in the pump under different particle concentration conditions. It can be inferred from the relationship between the particle Reynolds number and the two-phase slip speed that the particle Reynolds number is in proportion to the two-phase slip speed, so the particle Reynolds number in the figure can show the size of the two-phase slip speed. Under the three volume concentration conditions, the distribution of the two-phase slip velocity in the channel is basically the same, and the change with the concentration is not obvious. The two-phase slip velocity from the inlet to the outlet of the impeller gradually increases, and the maximum slip velocity is 26.03 m/s, 27.52 m/s and 28.16 m/s respectively. Due to the rotation movement of the blade, the solid phase and liquid phase are separated seriously, so the two-phase slip velocity near the impeller blade is large. In the volute, the two-phase slip velocity is large in the area affected by the blade rotation movement, and the two-phase slip velocity near the volute wall and the volute outlet will become small.

4.2 Two Phase Flow Calculation in Pump Based on DPM Model

(a) Cv= 1%

(b) Cv= 5%

145

(c) Cv= 10%

Fig. 4.29 Variation of particle Reynolds number distribution with particle concentration

Figure 4.30 shows the variation of the maximum particle velocity and the maximum two-phase slip velocity with the particle concentration. The two curves are approximately linear changes. Although the maximum value of particle velocity changes under different concentrations, the change is not obvious and can be ignored. The maximum value of two-phase slip velocity is more affected by the concentration than the maximum value of particle velocity.

Fig. 4.30 Discrete phase velocity and the maximum slip velocity vary

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4.2.3.3

4 Two-Phase Flow Characteristics and Transportation Performance …

Effect of Particle Concentration on Discrete Phase Concentration

Figures 4.31 and 4.32 are the particle concentration distribution in the impeller and the volute under three volume concentration conditions. It can be seen from the Figs. 4.31 and 4.32 that the particle concentration in the volute is smaller than that in the impeller. The concentration of the discrete phase in the volute and impeller channel increases with the increase of the particle concentration. In the impeller channel, the particles are mainly distributed at the impeller inlet, and the particle concentration at the outlet of the impeller pressure surface near the rear cover plate is larger. In the volute flow channel, the position where the particle concentration is large is on the volute wall near the rear cover plate. With the increase of the concentration, the particle concentration on this side

(a) Cv= 1%

(b) Cv= 5%

(c) Cv= 10%

Fig. 4.31 Discrete phase distribution in the impeller with different particle concentrations

(a) Cv= 1%

(b) Cv= 5%

(c) Cv= 10%

Fig. 4.32 Discrete phase distribution in the volute with different particle concentration

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147

is larger, and the particles are gradually distributed on the wall of the whole volute flow channel. The greater the impact wear caused by the particle movement at the positions with larger particle concentration, the more serious the wear of the flow channel with the increase of particle concentration.

4.2.4 Different Flow Conditions Under the conditions of particle diameter of 5 mm and particle concentration of 10%, the numerical simulation results are analyzed at flow rates Q = 80 m3 /h, Q = 100 m3 /h and Q = 120 m3 /h respectively.

4.2.4.1

Liquid Phase Flow on the Cross Section

Figure 4.33 shows the distribution of liquid-phase streamline in the cross section of the centrifugal pump flow channel under different flow rates. From Fig. 4.33, at the design flow rate (Q = 100 m3 /h), the streamline from the impeller to the volute is relatively smooth, which indicates that the internal flow state is good and there is no obvious mutation. At this time, the loss of liquid flow in the pump is mainly hydraulic loss, and the collision loss is very small. At the small flow rate (Q = 80 m3 /h), the flow in the impeller is stable, which is similar to the flow in the design flow rate. There is a sudden change for the streamline in the volute, and it collides with the volute wall for many times. At this time, the loss of liquid flow in the pump has collision loss in addition to the hydraulic loss, resulting in the reduction of the delivery efficiency of the centrifugal pump. At the large flow rate (Q = 120 m3 /h), the flow traces of the liquid phase at the impeller outlet and the

(a) Q= 80 m3/h

(b) Q= 100 m3/h

(c) Q= 120 m3/h

Fig. 4.33 Variation of liquid phase streamline with flow rate in the whole flow channel

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4 Two-Phase Flow Characteristics and Transportation Performance …

volute outlet are unevenly distributed and disordered, and the vortices appear near the tongue of the volute. The flow at the volute outlet is unstable and the backflow occurs, which will cause relatively large energy loss.

4.2.4.2

Relative Velocity of Mixed Phase in Horizontal Plane

Figure 4.34 shows the relative velocity distribution of the mixed phase in the horizontal section of the flow passage of the centrifugal pump X = 0 under different flow rates. From Fig. 4.34, the vortices appear inside the volute under the three flow rates. At the design flow (Q = 100 m3 /h), the mixed phase flow at the inlet is relatively smooth. When it reaches the impeller area, the velocity gradient from the impeller inlet to the impeller outlet is large, with the maximum value of 18 m/s and the minimum value of 2 m/s. The vortex appears at both ends of the volute flow channel near the wall surface, and the streamline at the volute outlet is evenly distributed, indicating that the mixed phase flow at the outlet is stable. At the large flow (Q = 120 m3 /h), there are vortices at both ends of the wall surface at the bottom of the volute. At this time, there is collision loss of the liquid flow in the pump. The velocity near the tongue has a minimum value and there are vortices at the tongue. The velocity gradient at the volute outlet is large, and there are vortices accompanied by backflow, indicating that the larger the flow, the more unstable the flow and the more serious the energy loss of the mixed phase. At the small flow (Q = 80 m3 /h), the change trend of relative velocity is basically similar to that at the design flow rate. However, in the volute channel, the vortex of mixed phase flow occurs in the middle of the volute channel, the flow at the volute outlet is smooth, and the relative velocity decreases.

(a) Q= 80 m3/h

(b) Q= 100 m3/h

(c) Q= 120 m3/h

Fig. 4.34 Variation of mixed phase velocity with flow rate in horizontal section (m/s)

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(a) Q= 80 m3/h

(b) Q= 100 m3/h

149

(c) Q= 120 m3/h

Fig. 4.35 Variation of static pressure distribution with flow rate in middle cross section (Pa)

4.2.4.3

Effect of Volume Flow Rate on Pressure Distribution

Figure 4.35 shows the static pressure distribution of the middle cross section in the flow passage of the centrifugal pump at different flow rates. From Fig. 4.35, the static pressure of the middle cross section decreases gradually with the increase of the flow rate. At the design flow rate (Q = 100 m3 /h), the static pressure from the impeller inlet to the outlet gradually increases, the minimum value is – 60 kPa, and the maximum value is 160 kPa, which appears at the impeller outlet. From the impeller outlet to the volute outlet, the static pressure increases, and the local static pressure decreases at the volute outlet. At the small flow (Q = 80 m3 /h), the static pressure distribution in the impeller is similar to that at the design flow rate. The pressure distribution from the impeller inlet to the outlet is consistent with that at the design flow rate. The static pressure from the impeller outlet to the volute outlet is larger than that at the design flow rate. At the large flow rate (Q = 120 m3 /h), the internal pressure distribution is unstable, and there is a local static pressure minimum area at the suction surface of the blade, where cavitation is easy to occur.

4.2.4.4

Particle Velocity Distribution

Figure 4.36 is the scatter diagram of discrete phase particle velocity distribution in the flow channel, and the color depth indicates the value of discrete phase velocity. From Fig. 4.36, the larger the flow rate, the larger the particle velocity. At the design flow rate (Q = 100 m3 /h), the particle velocity gradually increases from the inlet to the outlet of the impeller in the impeller channel, with the maximum value of 37.6 m/s and the minimum value of 0.31 m/s. From the impeller outlet to the volute outlet, the velocity gradually decreases. Due to the blocking effect of the volute tongue, the solid-phase velocity in the additional area will appear smaller. The variation of particle velocity at the large flow rate (Q = 120 m3 /h) and small flow rate (Q = 80 m3 /h) is basically similar to that at the design flow rate. The larger

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4 Two-Phase Flow Characteristics and Transportation Performance …

(a) Q = 80 m3/h

(b) Q= 100 m3/h

(c) Q= 120 m3/h

Fig. 4.36 Variation of discrete phase velocity with flow rate in flow channel (m/s)

the flow rate, the faster the liquid phase flow velocity, and the greater the relative velocity between the particles and the liquid phase movement under the centrifugal force near the impeller outlet. When the particles are thrown out of the impeller channel, there is still a tendency of centrifugal movement due to inertia, which is more obvious than that at the small flow rate, and the particle distribution here will be relatively sparse. At the volute tongue, the smaller the flow rate, the less the particle distribution, and the larger the flow rate, the more the particle distribution. At the volute outlet, the outlet flow is unstable in case of large flow rate, and the distribution of particles is sparse and irregular.

4.3 Two Phase Flow Calculation in Pump Based on DEM Model In order to improve the calculation accuracy, the DEM model considering the factors such as particle collision and particle rotation is used to conduct transient numerical simulation of the centrifugal pump to study the particles movement law and the influence of different particle mass concentration on the flow field of the centrifugal pump.

4.3.1 Calculation Method and Scheme 4.3.1.1

Particle Motion Equation

The equation of motion of particles is derived from Newton’s second law.

4.3 Two Phase Flow Calculation in Pump Based on DEM Model

∑ ⎫ m i u¨ i = F⎬ ∑ M⎭ Ii θ¨i =

151

(4.1)

The above formula is numerically integrated by the central difference method to obtain the updated velocity equation of particles. ⎧ ] [∑ F/m i N Δt (u˙ i ) N + 21 = (u˙ i ) N − 21 + ] ( ) [∑ ( ) M/Ii N Δt θ˙i N + 1 = θ˙i N − 1 + 2

(4.2)

2

Where u¨ i is acceleration of Particle i, m/s2 ; θ¨i is angular acceleration of particle ∑ i, 2 F is kg/m ; rad/s2 ; m i is mass of particle, kg; Ii is moment of inertia of particle, ∑ combined external force of particles at the mass center of mass, N; M is resultant torque at the mass center, N · m; t is time, s. During the movement of particles, according to the above formula, the velocity and displacement are constantly updated and repeated, so that the movement of particles at each time can be tracked.

4.3.1.2

Particle Contact Model and Contact Retrieval Process

During the movement of particles, there will be contact between particles and between particles and wall surface. Therefore, selecting a suitable contact model will directly determine the force and moment received by particles, which will not only affect the force transfer between particles, but also affect the flow field. Since this simulation does not consider the bonding between particles and the heat transfer between particles and between particles and the wall, the Hertz-Mindlin non- slip contact model is selected as the contact model between particles. The normal force component of the contact model is based on Hertz contact theory, and the tangential friction follows Coulomb friction law. For the collision between particles, it is first necessary to determine the position between particles, and then predict whether there will be collision between particles through calculation and analysis. In the calculation, it is divided into two steps to judge whether there will be contact between particles: the first step is to search the adjacent area for the particles to find the nearest particles or boundaries to them; The second step is to analyze and calculate the particles or boundaries closest to the particle and make a judgment on whether a collision occurs. The implementation of this method depends on the grid element method, that is, the whole simulation area is divided into several cubes, and the side length of the cube lbox and the maximum diameter of the particles dmax must meet the following relationship: dmax < lbox < 2dmax .

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4 Two-Phase Flow Characteristics and Transportation Performance …

Fig. 4.37 Computational domain grid

4.3.1.3

Computational Grid

The design parameters of the centrifugal pump calculated and analyzed are that flow rate Q = 70 m3 /h, head H = 14 m, rotating speed n = 1450 r/min. The centrifugal pump model grid is a hybrid grid, where the front wear-resistant plate domain, the rear wear-resistant plate domain, the front cavity domain, the rear cavity domain, the inlet and outlet extension domain are structural grids, and the volute domain and the impeller domain are unstructured grids. At the same time, in order to meet the calculation requirements, the length width ratio of the grid is controlled within 5, the minimum angle of the grid is 24°, the minimum quality of the grid is 0.3, and the grid scale increasing coefficient is 1.1. The number of grids is 2.43 million, as shown in Fig. 4.37.

4.3.1.4

FLUENT-EDEM Coupling Calculation Process

In the process of FLUENT-EDEM coupling simulation calculation, FLUENT is responsible for the calculation of clean water, and EDEM is responsible for the calculation of particles. The two softwares keep transmitting information until the end of the simulation process. Generally, the selection of time step can be determined by making the time step independent. However, due to the coupling of FLUENT-EDEM, the selection of time step needs to meet two requirements: the time for data transfer between two softwares and the time for particle retrieval of EDEM. At the same time, the time step should

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153

not be too small, otherwise the calculation time will be affected. The determination of time step in EDEM is related to Rayleigh wave. This is because 70% of the total energy consumption is consumed by Rayleigh wave when contact collision occurs between particles, and the total time consumed by the time step determined by Rayleigh wave during simulation is the least. The following formula is the time step formula determined by Rayleigh wave velocity. At the same time, for the stability of calculation, it is specified that the fixed time step is 20–40% of the Rayleigh time step, and the time step ratio between the two shall be controlled between 100:1 when the FLUENT-EDEM is coupled, that is, the time step of the FLUENT shall be greater than the time step of the EDEM. By comprehensively considering these requirements and considering that the simulation model is a rotating machine, the time step in FLUENT is taken as the time when the impeller rotates 2°, that is, t = 0.000,229,885 s. In order to ensure that the selection of the time step does not affect the simulation results, the independence of the time step is verified. In order to verify the independence of time steps, the time steps selected are 0.5°, 1°, 2°, 3° and 4° respectively. Three monitoring points T1, T2 and T3 are taken at the volute tongue and the pressure of these three points at different time steps are monitored. It can be seen from Fig. 4.38 that the results of the pressure at monitoring point T1 and monitoring point T2 of the volute tongue are basically unchanged under different time steps, while the pressure at monitoring point T3 decreases gradually with the decrease of time steps, but the downward trend is relatively slow. It can be approximated that the pressure at monitoring point T3 is stable within these time steps, Therefore, it is proved that the time step in FLUENT meets the simulation requirements. [

R Δt = π 0.163v + 0.877

Fig. 4.38 Verification of time step independence

/

ρ G

] (4.3) min

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where ρ is particle density, kg/m3 ; G is Shear modulus of particle materials, Pa; v is Poisson’s ratio of particle materials.

4.3.1.5

Calculation Scheme

Based on the coupling of FLUENT software and EDEM software, the numerical simulation of solid–liquid two-phase flow in centrifugal pump at the design flow rate (Q = 70 m3 /h) is completed. The liquid phase is clean water with a density of 1000 kg/m3 , and the solid phase concentration is 2%, 4% and 6% respectively. The specific parameters are shown in Table 4.3.

4.3.2 Analysis of Particle Motion Characteristics From the position of solid particles in the centrifugal pump at different times (Fig. 4.39), when the time is 0.2 s, the solid particles just enter the impeller inlet, and due to the influence of gravity, there are more particles near the bottom of the impeller inlet. With the rotation of the impeller, the solid particles are continuously sucked into the centrifugal pump. When t = 0.22 s, the solid particles first contact with the nuts on the impeller. Because the geometry of the nut is conical, the particles will be thrown away by the rotating nut, thus leading to the solid particles to collide with the impeller blade head, causing certain damage to the blade head and impact wear. Meanwhile, due to the inertia of the solid particles, the following property of the particles is not very good. Therefore, during the rotation of the impeller, the solid particles will collide with the middle area of the pressure surface of the impeller blade again. With the continuous rotation of the impeller, the wear area in the middle of the blade pressure surface increases. When t = 0.24 s, part of the particles adheres to the blade pressure surface and move together with the impeller. When t = 0.26 s, this part of solid particles will be separated from the impeller area by centrifugal force due to the rotation of the impeller, forming an impact on the volute and causing impact wear on the volute surface. Therefore, the wear area begins to appear on the volute surface Table 4.3 Solid phase parameters

Parameters

Values

Poisson’s ratio (particle)

0.25

Shear modulus (particle) (GPa)

1.96

Density (particle) (kg/m3 )

2700

Diameter (particle) (mm)

3

Poisson’s ratio (wall)

0.3

Shear modulus (wall) (GPa)

79.4

Density (wall)

(kg/m3 )

7800

4.3 Two Phase Flow Calculation in Pump Based on DEM Model

155

and increases with the rotation of the impeller. When t = 0.3 s, some particles have escaped from the centrifugal pump. So, it can be concluded that during the whole rotation process of the impeller, the blade pressure surface of the impeller is always the most severely worn area, and the blade suction surface is hardly affected by the particles. Meanwhile, when the particles move in the centrifugal pump, they collide with the impeller blades twice, and the volute surface wear area is partly caused by the impact of solid particles (Fig. 4.39). In order to study the influence of particle mass concentration on the movement process of particles, the velocities of 30 particles are randomly extracted from the particle factory in the extension section of the inlet of the centrifugal pump and their velocities are averaged when the particle mass concentration is 2%, 4% and 6%, respectively. Through comparison, it is found that, as shown in Fig. 4.40, when the particle mass concentration is 2, 4 and 6%, the particle velocity has two sudden changes. The time of the first sudden change in velocity is about 0.2 s, which is caused by the solid particles entering the impeller and colliding with the impeller blade head. The time of the second sudden change in velocity is about 0.25 s, and the degree of this sudden change in velocity is relatively small. This is because after the first collision, the kinetic energy obtained by the particles from the impeller decreases due to the resistance of the liquid and the inertia of the particles, so the velocity will be slightly reduced. At the same time, with the continuous rotation of the impeller,

Fig. 4.39 Particles position in centrifugal pump at different times

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Fig. 4.40 Particle velocity in centrifugal pump at different times

the middle and lower regions of the blade collide with the particles again, resulting in a small increase in the particle velocity. The two accelerations of the particle velocity coincide with the above analysis results of the positions of the particles at different times. After the second collision with the impeller, the particles begin to enter the volute. Since there is a certain angle between the liquid flow direction in the volute and the velocity direction of the particles entering the volute, the particle velocity is hindered to a certain extent, so the particle velocity begins to slow down. Meanwhile, the higher the mass concentration of particles, the earlier the collision time with the impeller. This may be due to the presence of particles, which reduces the passing area of the liquid and accelerates the liquid velocity. Therefore, the movement velocity of particles will be accelerated. So the particles collide with the impeller earlier as the concentration increases.

4.3.3 Influence of Particle Mass Concentration on Flow Field in Centrifugal Pump During the operation of the centrifugal pump, there is complex energy exchange, and these energies are converted in different forms in the flow field. For example, due to the high-speed operation of the centrifugal pump, the mechanical wear of the bearing and other components of the pump is caused, and a large amount of heat is generated, resulting in an increase in the temperature of the flow field in the centrifugal pump; Another example, due to the work done by the centrifugal pump, the conveyed medium obtains kinetic energy and potential energy. When the centrifugal pump transports the solid–liquid two-phase flow, due to the presence of solid particles, part of the energy will be transferred to the solid particles, resulting in the reduction of the energy obtained by the liquid. Meanwhile, because of various

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157

forces between the solid particles and the liquid, between the solid particles and the solid particles, and between the solid particles and the wall of the centrifugal pump, the internal flow field of the centrifugal pump will be affected. In order to analyze the influence of solid particles with different mass concentrations on the internal flow field of the centrifugal pump, the internal flow field of the centrifugal pump are compared and analyzed when the particle mass concentration is 2, 4 and 6% at the design flow rate.

4.3.3.1

Effect of Particle Mass Concentration on Flow Rate and Pressure of Impeller S3 Section

In order to analyze the influence of particle mass concentration on the internal flow state of the centrifugal pump, five S3 sections (A, B, C, D and E) of the impeller are selected, as shown in Fig. 4.41. The positions of sections A to E correspond to the impeller inlet, streamwise 0.2, streamwise 0.5, streamwise 0.8 and streamwise 0.99 respectively. Each S3 section is divided into 10 parts from the rear wear-resistant plate side to the front wear-resistant plate side on average to study the flow rate and pressure changes at the impeller of the centrifugal pump. Effect of particle mass concentration on the flow rate of S3 section of impeller By analyzing the flow rate distribution of several S3 sections of the centrifugal pump impeller at the design flow rate, as shown in Fig. 4.42, it can be found that the unit area flow rate near the front wear-resistant plate at the impeller inlet (section A) is the smallest under both clean water working conditions and solid–liquid two-phase flow working conditions with different mass concentrations. This is because the pressure at the blade suction surface of the impeller is lower than atmospheric pressure during the rotation of the impeller. Under the action of pressure, the liquid is sucked into the centrifugal pump and the liquid flow direction changes from axial flow to radial flow due to the characteristics of the impeller geometry of the centrifugal pump. Coupled Fig. 4.41 Schematic diagram of S3 section position

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4 Two-Phase Flow Characteristics and Transportation Performance …

with the inertia effect of the liquid, the unit area flow rate of the impeller inlet section near the rear wear-resistant plate is larger than that near the front wear-resistant plate, and the unit area flow rate of section A appears the maximum value in the area about 50% away from the rear wear-resistant plate. The area where the unit area flow rate distribution of section B and section C is relatively uniform is mainly concentrated in the area of 0.2–0.7. This is because the area is far from the solid boundary layer, and the disturbance is relatively small, so the flow is more stable. At the same time, it is found that since Section D and section E are close to the impeller outlet, and the centrifugal impeller is an open one, there is a certain gap between the impeller and the front and rear wear-resistant plates, which reduces the secondary flow in the impeller channel. Also, the unstable low-energy fluid in the boundary layer of the pressure surface enters the boundary layer of the blade suction surface in the same channel through the front and rear wear-resistant plates, and the low-energy fluid in the boundary layer is reduced. Therefore, the jet-wake structure at the impeller outlet is reduced, so the flow distribution at Section D and section E is relatively uniform. It is found that when the solid–liquid two-phase flow is transported, the unit area flow rate of section E is larger than that of Section D, which is just opposite to that when the clean water is transported. It is possible that the section E is close to the impeller outlet, and the flow velocity increases due to the presence of solid particles during the transportation of solid–liquid two-phase flow. The interaction between solid particles and liquid changes the flow characteristic of the liquid, making the internal flow field more uniform and reducing the generation of secondary flow at the section E. By comparing the mass concentration of solid particles on the unit area flow rate distribution of several sections of the centrifugal pump impeller, it is found that with the increase of the mass concentration of solid particles, the change of unit area flow rate at each section of the impeller is not large. This may be because the volume effect of particles and the collision between particles are not considered in DPM model. Effect of particle mass concentration on the pressure at the S3 section of the impeller Since section B is located between section A and section C, and its geometric position is not special, the pressure distribution on Section B can be obtained by analyzing section A and section C, so section B is not considered for the pressure analysis on section S3. From Fig. 4.43, the pressure near the front wear-resistant plate is lower than that near the rear wear-resistant plate, which is caused by the working mechanism of the centrifugal pump and the inertia of the medium. Also, it is found that the change trend of pressure distribution at Section A, Section C and Section D is consistent under the clean water condition and the solid–liquid two-phase flow condition. As for section E, because it is close to the impeller outlet, it is easy to be affected by the fluid in the volute. In addition, the interaction between solid particles and liquid causes the inconsistency between the pressure change at section E and the pressure change under the clean water condition. Meanwhile, it is found that the change of pressure value at Section A and section E decreases with the increase of the mass concentration of solid particles. This is because the solid particles cannot

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159

Fig. 4.42 Flow rate distribution of flow section

transfer the pressure, resulting in the loss of part of the pressure, thus causing the decrease of the overall pressure. However, there is no such trend at section C and section D. On the contrary, when the particle mass concentration is 2%, the pressure on the section is greater than that under clean water condition. This may be due to the existence of solid particles with low mass concentration, which suppresses the separation of boundary layer and the generation of secondary flow in the impeller channel and reduces the pressure loss. Therefore, the pressure is higher than that under the clean water condition. Moreover, the highest pressure at section C and section E appears at the position about 30% away from the rear wear-resistant plate. For section C, it may be caused by the back blade on the back of the impeller, while for section E, it may be affected by the jet wake structure. From the pressure value, the pressure value at Section A is negative pressure. This is because section A is close to the impeller inlet. In the process of centrifugal rotation of the impeller, negative pressure is generated, which sucks in the medium and does work to the conveying medium. It can be found that the pressure increases gradually from section A to section E. This is because the work done by the impeller converts kinetic energy into pressure energy, so that the pressure energy of the medium increases continuously in the impeller. Therefore, the closer to the impeller outlet, the larger the pressure on the section.

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Fig. 4.43 Pressure distribution of flow section

4.3.3.2

Effect of Particle Mass Concentration on Internal Flow Field

Four sections are selected in the inlet extension section of the centrifugal pump. By comparing the pressure changes on the sections, the influence of the mass concentration of solid particles on the internal flow field of the inlet extension section is analyzed. The vortex identification method proposed by Liu [1] is used to analyze the internal flow field of the centrifugal pump. And Ω = 0.52 during analysis. Vortex analysis in internal flow field From Fig. 4.44, under the clean water condition, a small part of vortex appears between section 3 and section 4 at the extension section of the pump inlet. This may be because this region is close to the impeller inlet, and the axial vorticity is formed at the impeller inlet due to the rotation of the impeller, which has an impact on the flow field near the impeller inlet, resulting in a small part of vortex between the section 3 and section 4 of the extension section of the centrifugal pump inlet. At the same time, it is found that with the increase of the mass concentration of solid particles, the vortex between section 3 and section 4 gradually disappears, while the vortex between section 2 and section 3 gradually increases, which indicates that the presence of solid particles reduces the vorticity of the flow field near the impeller inlet, but also increases the risk of vortex in the inlet extension section of the centrifugal pump. This is because the presence of solid particles makes the flow field near the impeller inlet more stable and reduces the reverse pressure gradient in the impeller inlet region. It

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161

can also be seen from the figure that the vortex at region A increases with the increase of the mass concentration of solid particles. This is because when solid particles are added, the particles enter the volute under the centrifugal force of the impeller and flow along the volute wall together with the water flow. The structure of region A is an inwardly convex curved surface. When solid particles arrive at region A, they will move along the tangential direction of the convex surface due to the inertia of solid particles, resulting in the increase of the reverse pressure gradient in the region and the forward movement of the separation point, thus accelerating the separation of the boundary layer, forming many separation vortices accompanied by the generation of backflow, thus causing the increase of vortices in region A. As for region C, the vortices in this region have increased when the particle mass concentration is 2%. This may be because a large number of solid particles impact region C due to the collision with the impeller blades when the solid particles move in the impeller channel, resulting in the turbulence of the flow field in region C and the increase of the vortices. However, as the mass concentration of solid particles continues to increase, the vortex in region C begins to decrease. This may be because the kinetic energy obtained by a single particle decreases with the continuous increase of the mass concentration of solid particles under the action of the gravity field, so the impact on region C of the volute decreases, resulting in the reduction of vortex. In the outlet extension section, when the mass concentration of solid particles is 4 and 6%, a small part of vortices appear in the flow field. The generation of the vortices is caused by the separation of solid particles and liquid in region C, causing the turbulence of the flow field in the nearby region, causing the separation point to move forward, and the area of backflow and separation vortices to expand continuously, resulting in the generation of vortices in the outlet extension section during the transportation of these turbulent fluids. Figure 4.45 is the vortex cloud diagram in the impeller under different working conditions, and the pressure is dimensionless and expressed by CP . The pressure on the blade pressure surface of the impeller is generally higher than that on the blade suction surface, and there is a minimum pressure at the blade suction surface

Fig. 4.44 Steady vortex distribution in full flow channel (Ω = 0.52)

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at the inlet of the impeller blade, which is caused by the centrifugal operation of the impeller. Under the condition of clean water, the vortex is mainly generated on the blade suction surface, which is due to the rotation of the impeller. A certain degree of pre swirl is generated at the impeller inlet, forming a certain degree of positive attack angle, so that the liquid separates at the blade suction surface and generates a separation vortex, and this separation will develop toward the trailing edge of the blade suction surface, resulting in turbulence of the flow field. However, under the solid–liquid two-phase flow condition, the vortices are generated not only at the blade suction surface, but also at the blade pressure surface, especially at the blade pressure surface of the No. 3 blade. With the increase of the mass concentration of solid particles, the vortex structure in the impeller near the rear wear-resistant plate increases continuously. This may be because when conveying solid–liquid two-phase flow, the mass of solid particles is relatively large, St > 1, resulting in poor follow-up of particles. Therefore, during the rotation of the impeller, the particles will collide and slip with the blade pressure surface, thus destroying the stability of the boundary layer on the blade pressure surface and causing vortex generation. The vortex on the suction surface side of No. 2 blade is concentrated near the front wear-resistant plate side. This may be because the low-energy micro clusters in the boundary layer of the pressure surface of No. 1 blade enter the boundary layer of the suction surface of No. 2 blade through the front wear-resistant plate, causing the boundary layer on the suction surface of the No. 2 blade to become thicker and generate a large reverse pressure gradient. This is also consistent with the flow mechanism of the internal flow field of the impeller under clean water conditions. However, for the suction surface of No. 1 blade and the suction surface of No. 3 blade, the vortex appears near the rear wear-resistant plate, and there is almost no vortex generated near the front wear-resistant plate. It can be judged that this is caused by the action of solid particles on the fluid. And through further analysis, it can be judged that during the operation of the centrifugal pump, the particles should be mainly concentrated in the impeller A channel and the impeller C channel, and the solid particles in the impeller B channel have a relatively weak impact on the fluid. Section pressure analysis of inlet pipeline In order to analyze the influence of particle mass concentration on the internal pressure distribution in the inlet extension, the pressure is dimensionless for the convenience of comparison. From Fig. 4.46, the pressure in the inlet extension section is all negative pressure under both the clean water condition and the solid–liquid twophase flow condition, and the pressure distribution on the four sections is relatively uniform under the clean water condition. The pressure distribution on the crosssection becomes more disordered with the increase of solid particle mass concentration under the solid–liquid two-phase flow condition. For section 1, since it is close to the inlet, the pressure distribution is very uniform under clean water condition. Through comparison, it is found that the absolute value of pressure gradually increases from the wall surface to the center of the cross-section, indicating that the pressure at the center of the circle is the lowest. This is because the existence of the boundary layer on the wall of the inlet extension section causes the velocity

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163

Fig. 4.45 Steady vortex distribution in impeller channel (Ω = 0.52)

of viscous fluid near the wall to be low, thus causing the pressure at the wall to be higher than that at the center of the cross-section. As the distance to the impeller inlet decreases, the pressure distribution at section 2, section 3 and section 4 changes, and the absolute value of the pressure at the center of the section is lower than the that at the wall, which indicates that the pressure at the wall surface is low. This may be due to the influence of impeller rotation, which causes pre swirl in the inlet extension section, and verifies the generation of vortex in the above. When conveying solid–liquid two-phase flow, the lowest point of pressure at section 1 and section 2 continuously moves up with the increase of particle mass concentration, which may be because the gravity factor of solid particles reduces the liquid flow rate in the downstream region of the inlet extension section. The pressure change at section 3 is greatly affected by the rotation of the impeller because it is close to the impeller inlet. And the pressure at section 4 changes little with the particle mass concentration, which may be because the No. 4 section passes through the impeller nut, and the pressure distribution here is mainly affected by the rotation movement of the impeller. Streamline analysis on volute flow section In order to study the effect of solid particle mass concentration on the internal flow field of the volute, a total of 10 flow sections of the volute are taken from the volute throat and volute outlet for streamline analysis. From Fig. 4.47, the liquid flow velocity of No. 1 flow section is relatively large under the clean water condition

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Fig. 4.46 Pressure distribution on the inlet extension section

and the solid–liquid two-phase flow condition, which is because the area of No. 1 flow section is relatively small. With the increase of the flow section area, the flow velocity of the liquid gradually decreases, but local high-speed fluid appears in the bottom area of the No. 5 flow section and the No. 8 flow section. Through analysis, it is found that since the multi coordinate system is adopted for the steady-state numerical simulation, the relative physical position of the impeller during the simulation process does not change. Meanwhile, the positions of the No. 5 flow section and the No. 8 flow section are located at the blade C and the blade A respectively, which causes high-speed flow in the local area. And the fluid velocity of the ten flow sections near the volute wall is relatively low, which is caused by the existence of the boundary layer. Under the condition of clean water, it is found that except for No. 2 and No. 9 flow sections, the vortices are generated at other flow sections. The reason for this is

4.3 Two Phase Flow Calculation in Pump Based on DEM Model

165

Fig. 4.47 Velocity streamline distribution on volute flow section

not only related to the geometric structure of the centrifugal pump, but also related to the difference between the liquid flow angle at the blade outlet and the placement angle of the tongue. The vortices on No. 3, No. 4 and No. 5 flow sections all appear on the side close to the front cavity, the vortices of No. 8 flow section appear on the side close to the rear cavity, and a pair of Dean vortices appear on No. 6 and No. 7 flow sections respectively. This is because the flow channel of the volute is a threedimensional curved surface, and the centrifugal force produces symmetric vortices that move according to a certain law and have opposite directions. Therefore, it can be considered that the spatial motion of the fluid in the volute is a rotating forward flow, like a twisted “fried dough twist” and two relatively large “fried dough twist” structures exist in the flow field. Under the solid–liquid two-phase flow condition, when the particle mass concentration is 2%, the vortex on the No. 1 flow section near the front cavity disappears, the vortex intensity on the No. 3, No. 4, No. 5 and No. 8 flow sections weakens, and the vortex on the No. 6 flow section becomes one near the rear cavity. These changes are due to the interaction between solid particles, volute wall and the fluid inside the volute, which suppresses the shedding of the boundary

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layer and reduces the adverse pressure gradient near the wall, thus reducing the occurrence of secondary flow. Compared with No. 8 flow section under clean water condition, the high-speed fluid in the bottom area disappears, because the particles slip at the tail area of blade A and directly impact the fluid in the volute at No. 8 flow section, resulting in a decrease in the velocity of the fluid in this area. For the No. 5 flow section, due to the influence of the gravity of solid particles, the particles will directly enter the volute after colliding with the blade C and will not slip at its tail. Therefore, the impact on the bottom area of No. 5 flow section is relatively small, and the fluid velocity in this area is still relatively high. Except that the streamlines on the flow sections at the volute throat and volute outlet change significantly, and the changes on other flow sections are relatively small with the increase of the mass concentration of solid particles. It can be seen from the figure that with the increase of the particle mass concentration, the vortices on the flow section of the volute throat increase continuously, and the positions where the vortices are generated are close to the rear cavity. This is because the solid particles enter the centrifugal pump together with the water flow and shift to the rear cavity due to the inertia. Meanwhile, since the No. 9 flow section is in the volute throat, the geometry of the volute from this area to the diffusion section changes, and there is a G2 continuous curved surface. When the particles move to this area, they will move along the tangential direction somewhere in this area, resulting in the separation of solid particles and liquid, and promoting the separation vortex and backflow in this area. With the increase of particle mass concentration, the influence of particles on the liquid is expanding, causing the vortex generated on the flow section at the centrifugal pump outlet to become larger. Through the analysis of the simulation results of solid–liquid two-phase flow, it is found that: (1) The presence of solid particles will reduce the occurrence of backflow in the impeller channel to a certain extent. However, with the increase of solid particle mass concentration, the pressure in the impeller channel is also decreasing. (2) The influence of particle mass concentration on the internal flow field of centrifugal pump is analyzed by vortex identification method. With the increase of particle mass concentration, the vortex in the inlet extension section disappears at the impeller inlet and tends to move forward. The existence of particles promotes the generation of vortices at the diffusion section of the volute, and it is found that the particles distribution in the impeller is uneven through the analysis of the vortices in the three impeller channels. (3) By analyzing the velocity streamline on the flow section of the volute, it is found that the presence of particles will affect the boundary layer near the wall in the volute, and to some extent prevent the formation of vortices, which will have an important impact on the wear of the channel.

Reference 1. Liu CQ, Wang YQ, Yang Y et al (2016) New omega vortex identification method. Sci China Phys Mech Astron 59(8):684711

Chapter 5

Wear Characteristics of the Wall Surface in Centrifugal Pump

Abstract When the centrifugal pump is transported in the solid–liquid two-phase flow, the particle movement wears the overcurrent wall surface, thereby reducing the service life and reliability of the centrifugal pump. This chapter uses experimental and numerical simulation methods to study the flow and wear of centrifugal pumps and pump tube systems during solid–liquid two-phase transportation. The wear of centrifugal pump impeller, volute, front wear plate, rear wear plate, vertical pipes, bend pipes and horizontal pipes were analyzed under different working conditions. At the same time, because collisions occur between solid particles and between particles and the wall surface of centrifugal pump during two-phase flow transportation, particles will be broken to varying degrees, which will affect the wear and transportation performance of the centrifugal pump. Therefore, the characteristics of mixed size particles flow in the centrifugal pump are also studied. Its relationship with wear laws is discussed simultaneously.

5.1 Solid–Liquid Two-Phase Transport Experiment and Calculation Method Verification 5.1.1 Experimental Apparatus The experimental apparatus can be divided into five parts, solid–liquid two-phase medium stirring device, system flow regulation, and test device, speed, and torque test device, centrifugal pump head test device, centrifugal pump power input device, and pipeline conveying device (Fig. 5.1) This experimental study is mainly divided into centrifugal pump piping system performance experiment and wear experiment. The first performance experiment is the centrifugal pump performance experiment, which includes the performance experiment of clean water at different flow rates, different mass fractions (C m = particle mass/two-phase flow mass), and two-phase performance experiments at different flow rates. The variables of the performance experiment were flow rate and particle mass fraction, flow rate from dead point flow (0 m3 /h) to large flow rate 100 m3 /h, and particle mass fraction from C m = 0% to C m = 10%. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Z. Zhu et al., Solid-Liquid Two-Phase Flow in Centrifugal Pump, Fluid Mechanics and Its Applications 136, https://doi.org/10.1007/978-981-99-1822-5_5

167

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5 Wear Characteristics of the Wall Surface in Centrifugal Pump

Fig. 5.1 Centrifugal pump two-phase conveying test bench

After the performance experiment of the centrifugal pump is completed, the wear experiment of the centrifugal pump piping system is carried out, and the wear experiment is carried out under the rated flow rate, and the experimental variable is the particle mass fraction. Wear experiments start at low concentrations, starting with C m = 1%, then gradually increasing the concentration, divided into 7 times to increase the concentration to C m = 7%, and wear experiments at each concentration for 48 h, after each experiment, observe the wear position, and use ultrasonic thickness gauge devices to measure the thickness loss at the punctuation position.

5.1.2 Grid and Calculation Settings The mixed meshing of the computational domains, where the impeller and volute regions use an unstructured tetrahedral mesh, and the rest of the hydraulic structure uses a structured hexahedral mesh. The calculation domain mesh and its mesh independence verification results are shown in Figs. 5.2 and 5.3, when the number of meshes increases to 8,368,198 (i.e., the 6th group of meshes), the external characteristic changes tend to be stable, and after mesh quality check, the mesh quality adopted and its minimum angle meet the calculation requirements. The liquid phase calculation is carried out in the FLUENT software, and the noslip function is used at the wall surface, the standard wall function is selected for the low Reynolds number area near the wall surface, and the slip mesh is used at the dynamic and static interface to exchange data. Set the speed inlet condition, the speed direction of the vertical inlet boundary, the speed size according to the centrifugal pump rated flow rate and the diameter of the inlet pipe calculation, the initial flow rate of the fluid at 2.47 m/s, and the outlet boundary is the free flow outlet (outflow),

5.1 Solid–Liquid Two-Phase Transport Experiment and Calculation …

169

Fig. 5.2 The geometry and its mesh

Fig. 5.3 Mesh independence verification

the solver uses the SIMPLEC algorithm to achieve the coupling relationship between speed and pressure. The solid phase calculation is carried out in the EDEM software, and the particles are started at the inlet pipe at 0 s, the initial velocity of the particles and the fluid velocity are consistent, which is 2.47 m/s, and the particles follow the fluid movement into the centrifugal pump and flow out of the outlet pipe. The rotation

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Table 5.1 Material parameter settings

Parameter

Wall

Particle

Material

Steel

Glass

Poisson’s ratio

0.25

0.3

Modulus of shear (GPa)

1.96

79.4

Density (kg/m3 )

2700

7800

Mass flow (kg/s)

/

0.58227, 0.97045, 1.35864

speed of the centrifugal pump is 1450 r/min, that is, the rotation period is 0.041,379 s, and the calculation time in this study is 75 r/min and 3.103 s. To accurately capture the particle–particle, particle–wall contact behavior, the time step of the solid phase particle calculation is much smaller than the time step of the liquid phase, ensuring that the particles are very small in a separate solid phase calculation time step. The calculation time step is set in FLUENT is 2.298,85 × 10–4 , the time step is set in EDEM is 2.298,85 × 10–6 , the ratio of the two is 100:1, and the Rayleigh time step is 38% (Table 5.1).

5.1.3 Calculation Method Verification The presence of solid phase particles affects the energy exchange inside the centrifugal pump and changes the external characteristics of the centrifugal pump. Figure 5.4 is a calculation and experimental comparison diagram of the external characteristics of the centrifugal pump at different mass concentrations under the design flow point, C m = 0% for the clean water condition. From the comparison chart, the results of the two are in good agreement, which verifies the accuracy of the calculation method. The influence of particle mass concentration on the performance of centrifugal pumps was analyzed, and it was found that as the particle concentration increased, the head and efficiency decreased with the increase in concentration, which was due to the increase in the number of particles, the more energy required for particle movement, and the particle movement energy was provided by a fluid drag force, and the density and viscosity of the solid–liquid two-phase flow medium were higher than that of the clear water medium, and the energy loss increased, so the centrifugal pump efficiency and head decreased with the increase of particle mass concentration. The centrifugal pump shaft power is calculated from the rotor torque and speed, which shows a slight upward and slow downward trend, and the shaft power reaches its maximum value at C m = 7%. Overall, the centrifugal pump head and efficiency are greatly affected by the particle concentration, which decreases with the increase of the particle concentration, and the shaft power change is not obvious.

5.2 Wear of the Centrifugal Pump Runner

171

Fig. 5.4 Centrifugal pump performance compared between calculation and experiment

5.2 Wear of the Centrifugal Pump Runner The relationship between the numerical calculation results and the performance experiment of the centrifugal pump and the wear data of the overcurrent wall surface is compared to the wear law of the internal flow channel of the centrifugal pump and the particle movement process.

5.2.1 Impeller Wear Analysis The impeller is a high-speed rotating part of the centrifugal pump, and the particles collide with the impeller at a very high speed, resulting in severe blade wear. Figure 5.5 is the experimental result of impeller wear under three kinds of mass concentrations, and it can be found that there are the most serious on the hub near the blade leading edge and the blade pressure surface. The paint on these area falls off, and even the material of the blade falls off. The wear degree of the blade suction surface is light. Blade wear begins at the tail of the blade and then gradually moves towards the blade leading edge. At the leading edge of the blade, particles with large axial velocities cause impact wear on the bottom area of the blade and the center area of the hub. At C m = 3%, all of paints on the hub disappears, and obvious pits appear,

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especially in those areas with material defects. The wear on the other parts is fishscale distribution. The wear degree gradually deepens along the rotating direction of the impeller. With the accumulation of time and the increase of concentration to C m = 5%, the wear morphology changes. The wear on the hub deepens, and the pit at the material defect becomes shallower. The previous fish- scale wear gradually smooth, and the new fish- scale wear appears near the suction surface of the blade at the hub edge. When the concentration increases to C m = 7%, the fish- scale morphology on the entire hub is basically disappeared, which is consistent with the simulation results. The wear of the blade pressure surface is gradually aggravated from the leading edge to the tail, and the second area of severe wear is the hub near the impeller inlet. The particle velocity direction in the impeller is shown in Fig. 5.6d. The particle and the wall impact angle are large, the particle velocity direction is almost perpendicular to the blade. The wear is mainly impact wear. The closer to the tail of the blade, the greater the particle velocity, and the more serious the corresponding wear. The impeller does a periodic high-speed rotation movement in the centrifugal pump, and the number of collisions between the particles and the impeller and the instantaneous wear characteristics all show periodic changes. The pellets begin to be stocked at 0 s and the first particles arrive at the impeller inlet at about 0.55 s (centrifugal pump rotation cycle 13–14). Figures 5.7 and 5.8 show the number of collisions between the blade pressure surface and the particles in the 20th cycle (0.827,6 s) to the 25th cycle (1.034,5 s) of the centrifugal pump and the instantaneous thickness loss rate, the data show that the blade pressure surface and the particle

Fig. 5.5 Impeller wear experiment results

5.2 Wear of the Centrifugal Pump Runner

173

Fig. 5.6 Calculation of impeller wear (t = 3.103 s)

collision number and instantaneous thickness loss rate appear periodic peak-like, and there is a fixed phase difference between the number of collisions and the peak position of the wear of the blade, and the phase difference is related to the number of blades. The maximum number of collisions and the instantaneous thickness loss rate is consistent, and the maximum number of collisions is the greatest amount of wear. According to the time of wear peak, when the blade is found to be above the 5–6 section of the volute, the position is shown in Fig. 5.9, wear on the blade pressure surface is the most. When the impeller rotates away from this position, the instantaneous thickness loss rate and the number of particle collisions are gradually reduced. Because of the obvious gravity effect of the large particles, the particles sink directly when they enter the centrifugal pump from the inlet pipe. The number of particles is the largest above the 5–6 sections. So, the above phenomenon occurs.

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Fig. 5.7 Number of collisions between the blade pressure surface and the particles

Fig. 5.8 Instantaneous thickness loss rate of the blade pressure surface

5.2.2 Volute Wear Analysis The particle movement in the volute is different from that in the impeller. So, the wear mechanism on the inner wall of the volute is also different. The contribution of impact wear and cutting wear changes. As shown in Fig. 5.10, at the three mass concentrations of 3%, 5% and 7%, the wear trend on the volute is consistent. The wear is most severe in the tongue area, outlet area, and area between the 5th and 8th sections. The degree of wear increases with the increase of mass concentration, and the average thickness loss between the 6–7 sections reaches the maximum. Particles enter the volute after acceleration from the impeller channel, and most of the particles move along the wall after colliding with the wall surface of the volute. Their velocity direction is parallel to the circumferential tangent of the wall surface. Figure 5.11b is the velocity vector distribution of particles between the 6–7 cross-section (C m = 7%, t = 3.103 s). It can be found that the particle velocity gradient here is very large, and there is a layer of particles near the wall surface. The particle sliding along the wall causes cutting wear of the wall surface. Due to the gravity of the particles, the particles deposit more at the bottom of the volute. And the particles rotate along

5.2 Wear of the Centrifugal Pump Runner

175

Fig. 5.9 Blade position when the instantaneous thickness loss rate of the blade is greatest

the rotation direction of the impeller, reach the maximum speed at the outer edge of the blade. And then particles slow down after impact with the wall of the volute. The area between the 6–7 section is in the key impact part of the transition from high-speed particles to low-speed particles, so the wear here is the most serious. The minimum thickness loss rate between the 4–5 sections is due to the least of moving particles here. Most of the particles begin to sink here and are driven to the outlet by the impeller. Only a small number of particles enter the volute again through the tongue. As can be seen from Fig. 5.11a, the maximum and average wear values of the volute tend the same basically. The wear of each component increases as the mass concentration increase. The average wear rate reaches a maximum at C m = 7% and a minimum at C m = 1% at each location. When the concentration increases, the difference between the maximum and the minimum gradually decreases. which is due to the movement of particles along the wall surface of the volute. As shown in Fig. 5.11b, the particles accumulate near the wall to form a buffer layer. In the buffer layer, particles move slowly and steadily. The collision near the wall side occurs between the particles and the wall surface, causing cutting wear to the wall surface. And the collision near the impeller side occurs mainly between particles so that the particles slow down and redirect into the buffer layer. Therefore, the particle

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5 Wear Characteristics of the Wall Surface in Centrifugal Pump

Fig. 5.10 Volute wear diagram (t = 3.103 s)

Fig. 5.11 Average wear and maximum wear distribution in different areas of the volute (t = 3.103 s)

mass concentration increases linearly, but the amplitude of the increasing wall wear decreases. The wear experiment (C m = 7%) results of the volute are shown in Fig. 5.12. It can be seen that the numerical simulation results are consistent with the experimental trend. There is less wear between the 1–4 sections. The wear gradually intensifies from the 4th section. In this area, the wear is mainly concentrated in the middle area of the volute. At this area, not only does the paint come off, but the material is also lost. On both sides of the volute there is still a small amount of residual paint. The wear morphology on the wall surface of the volute is relatively flat and smooth, which is caused by the small collision angle and small sliding speed of the particles in the volute. At the tongue region, wear intensified by the violent collision of particles. The impact crater and plastic deformation of the material appear here. Such pits will affect the fluid flow, bring disturbance to the stable flow in the centrifugal pump, and affect the stable operation of the centrifugal pump. In the diffusion section, when the

5.2 Wear of the Centrifugal Pump Runner

177

Fig. 5.12 Volute wear experimental results (C m = 7%)

fluid carries particles through this point, the particles move along the tangent line of the wall. Due to the curvature transition of the flow channel, the particle deflection is serious. There are almost no particles in contact with the wall surface in the area above the tangent line of the curvature transition. There is less wear, as shown in the area B. Figure 5.13 shows the comparison of calculation results and experimental results in the 3–7 sections. As a whole, the number of particles gradually increases along the flow direction, the contact time between the particles and the wall surface becomes longer, and the wear gradually increases. The internal wear of the volute is uneven, and the wear near the outlet side of the impeller is serious. There is still residual paint near the inlet side, as shown in the marked part in the Fig. 5.13a. Particles flow into the volute channel from the rotating impeller channel. The wear characteristics on the wall surface of volute shows a cyclical trend. But the diffusion section and the outlet pipe area are not connected to the rotating parts, there is no periodicity trend of the wear at this area. As shown in Fig. 5.14, the number of cycles in which the wear value changes between the 6–7 section is basically equal to the product of the blade numbers and the number of rotation cycles after the particles reach the volute. When the impeller rotates a certain number of turns, the wear tends to be stable. The instantaneous thickness loss rate fluctuates up and down at a fixed value, and the minimum value gradually increases and stabilizes.

5.2.3 Wear Analysis of Front and Rear Wear Plates The front and rear wear plates are stationary fixed components, and there is a certain gap between the wear plate and impeller. When the impeller rotates, the particles are easy to enter the gap between the blade and the wear plate, driven by the impeller with high-speed circular movement. Due to the friction of particles in the gap, wear plates occur plastic deformation. The wear on the underside of the wear plate is intense for severe deposition of particle. And the wear of the rear plate is more

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5 Wear Characteristics of the Wall Surface in Centrifugal Pump

Fig. 5.13 Volute section wear analysis (C m = 7%)

Fig. 5.14 Instantaneous thickness loss characteristics of Sections 6–7 and diffusion sections

serious because the particles have axial speed and collides with the rear wear plate, as shown in Figs. 5.15 and 5.17. It can be found that the wear morphology is related to the rotation direction of the impeller, which is consistent with the experimental results. Figures 5.16 and 5.18 are the thickness loss rate of the front wear plate and the rear wear plate at the different mass concentrations. The thickness loss rate of the wear plate under the three mass concentrations is inconsistent. At C m = 3%, the wear rate is relatively stable, the inner and outer rings of the front wear plate are relatively concentrated, and the wear rate is larger. The wear rate has a wide range of pulsations at C m = 5% and C m = 7%, and the curves of wear rate cross

5.2 Wear of the Centrifugal Pump Runner

179

Fig. 5.15 Simulation results of the front-wear plate (t = 3.103 s)

each other, which is related to the experimental time and the material performance of the wear plate. At a small concentration of wear for a short time, the surface of the material begins to undergo plastic deformation, and then the material falls off. As the concentration increases, a striped wear morphology appears on the wall surface. This results in cluttered wear rate curves from measurements.

Fig. 5.16 Experimental data of the front-wear plate

Fig. 5.17 Simulation results of rear-wear plate wear (t = 3.103 s)

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5 Wear Characteristics of the Wall Surface in Centrifugal Pump

Fig. 5.18 Experimental measurement data of rear-wear plate wear

Figures 5.19 and 5.20 are the thickness loss and instantaneous thickness loss rate of the wear plate under different particle mass concentrations. For the limited volume of the gap between the wear plate and the impeller, the particle mass concentration has no obvious impact on the wear of the wear plate. The thickness loss rate change amplitude is concentrated near the size of 1 × 10–10 m/s. But it can still be seen that the increase of the concentration will promote the wear degree of the plate, and the intercept of the thickness loss curve is also larger.

Fig. 5.19 Instantaneous thickness loss and thickness loss rate of the front-wear plate

5.3 Wear of the Pipeline Matched with Centrifugal Pump

181

Fig. 5.20 Instantaneous thickness loss and thickness loss rate of the rear-wear plate

5.3 Wear of the Pipeline Matched with Centrifugal Pump In the inlet pipeline of centrifugal pump, the particle speed is low because the particles are not accelerated by the impeller. The wear of the inlet pipeline is lighter. Therefore, the study of the pipeline wear focuses on the outlet pipeline. To facilitate the analysis of pipeline wear, the outlet pipe of centrifugal pump is divided into four parts: a vertical pipe, a bend pipe 1, a horizontal pipe and a bend pipe 2. The segmentation of outlet pipe is as shown in Fig. 5.21.

5.3.1 Vertical Pipe Wear Analysis This vertical pipe is the first section of the centrifugal pump outlet pipe, which is directly connected to the pump volute outlet. Figure 5.22 is the pipeline location schematic diagram and wear cloud diagram. As can be seen from Fig. 5.22a, the wear of the pipe wall surface is irregular, and a maximum value appears at the outlet of the volute, which is caused by the large particle velocity here. Figure 5.22b is the velocity distribution of particles in the pipe at a concentration of C m = 7%. When the particles come out of the volute, they do not evenly fill the entire pipe, and the particles are biased to the left side of the pipe, which is related to the rotation direction of the impeller. When the particles flow for a distance, particles begin to fill the entire

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5 Wear Characteristics of the Wall Surface in Centrifugal Pump

Fig. 5.21 Schematic diagram of the segmentation of the outlet pipe

flow channel. The movement of particles are not parallel to the axial direction of the pipeline, but spirals upward. But because large particles have poor fluid followability, the spiral motion of particles is not synchronized with the spiraling fluid at this time. In the process of rising, due to the downward direction of gravity is the opposite of the direction of motion, the particles acceleration reduces. The speed reaches the maximum value at 5–10 d2 (d2 is the diameter of the outlet pipe) from the pump outlet. And then particles continue to decelerate and rise. The speed reaches the minimum value at 0–10 d2 from the inlet of the bend pipe 1. The closer to the wall, the more obvious is particle deceleration. Figure 5.23 is a graph of the maximum wear value of each pipe cross-section along the pipe axial length (0 m at the inlet of the vertical pipe, and the vertical pipe length is 2.14 m.). It can be seen from the figure that the wear law is consistent under the three concentrations. But the degree of wear is related to the concentration of the particles, and the wear is more serious when the concentration is larger. At the inlet of the pipeline, particles just flow out from the pump outlet, the particle speed is larger, where the maximum wear appears. In the 0–0.5 m segment, the particles begin to spread to the middle of the pipeline from the near-wall area, and the number of particles in contact with the wall decreased. So, the wear in this section also tends to decline. In the 0.5–1.5 m segment, the particles fill the entire pipeline and spiral upward. The walls of the pipe are worn all around, and the wear increases. After that, the particle velocity decreases, and the maximum wear value gradually decreases. Figure 5.24 is the result of the wear experiment under three mass concentrations of the particle, the wear value is converted over time (experimental wear time is 48 h). The vertical coordinate represents the maximum wear value under different lengths, the maximum value is positive. The horizontal coordinate represents the length of the pipeline. The vertical pipe of the pump outlet is composed of three

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183

Fig. 5.22 Vertical pipe wear (t = 3.103 s)

segments (paragraph 3, paragraph 4, and paragraph 5), and the pipes are connected by flanges. Compared with Fig. 5.23, the magnitude order and trend of the wear experiment results are consistent with that of the calculation results, which also verifies the accuracy of the calculation method. As can be seen from the figure, the wear of vertical pipe fluctuates greatly in the 3rd section (near the pump outlet). The closer to the pump outlet, the greater the wear. Then the wear slightly decreased. In this section the particle velocity is larger, and the disorder degree of the two-phase medium increases due to the impact of the centrifugal pump. The collision frequency between the particles and the pipeline wall increases, so the wear is serious.

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5 Wear Characteristics of the Wall Surface in Centrifugal Pump

Fig. 5.23 Vertical pipe thickness loss (t = 3.103 s)

Fig. 5.24 Experimental wear data of vertical pipe (t = 3.103 s)

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185

5.3.2 Bend Pipe Wear Analysis To accurately analyze the wear in the bend pipe, the bend part is decomposed as shown in Fig. 5.25. The circular angle of the bend pipe is set to Φ, that is, 0° ≤ Φ ≤ 90°. The angle of the bend pipe cross section is set to θ, θ is the positive angle on the outer side of the bend pipe, and the angle on the inner side is negative. Figure 5.26 is a wear cloud diagram of bend pipe 1 and bend pipe 2 at three concentrations (at t = 3.103 s). Overall, the wear on the wall surface of the bend pipe is not symmetrically distributed, and the wear gradually intensifies along the flow direction. The wear area is mainly concentrated in the outer arch of the bend pipe, and the inner wall of the bend pipe and the cheeks are less worn. With the increase of concentration, the wear area on the bend pipe gradually expands and intensifies. At C m = 3%, there is only slight wear on the cheeks at the entrance of bend pipe 1. At C m = 7%, the wear here deepens. The interactions of particle–particle and particle–wall are enhanced with the particle concentration increasing. The disorder degree of fluid increases simultaneously, which eventually leads to increased wear. The wear of bend pipe 2 is significantly less than that of bend pipe 1, but the two change trends are consistent. Due to the change in the flow direction of fluid at the bend pipe, the solid phase particles collide directly with the bend pipe under the action of the fluid wrapping. A most wear area occurs on the outside of the bend pipe. Figure 5.27 is the particle distribution in bend 1. The particles enter the bend through the vertical pipe and collide directly with the outside of the bend pipe. The wear on the wall surface of the bend pipe is also aggravated from here. In the 0° < Φ < 30° segment, the particle movement direction changes less. Only a small part of the particles is in contact with the bend wall, and the degree of wear is lighter. In the 30° < Φ < 90° segment, Fig. 5.25 Schematic diagram of bend pipe

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5 Wear Characteristics of the Wall Surface in Centrifugal Pump

Fig. 5.26 Calculated wear of bend pipe (t = 3.103 s)

lots of particles collide directly with the wall surface, and there are also repeated collisions. A non-particle zone appears near the arch inside the bend pipe. The solid phase particles in this section mainly begin to do a spiral motion with the secondary flow, which will increase the number of collisions between the particles and the pipe wall surface and prolong the wear area. Figure 5.28 shows the thickness loss and thickness loss rate of bend pipe 1 over time. The particles are accelerated by the centrifugal pump from the inlet pipe to

Fig. 5.27 Particle distribution of bend pipe 1 (t = 3.103 s)

5.3 Wear of the Pipeline Matched with Centrifugal Pump

187

reach the inlet of bend 1 for about 1.25 s and begin to produce wear on the wall surface of the bend pipe. In the 1.25–1.6 s segment, due to the gradual increase of particle number entering the bend pipe, the collision number of particles gradually increases. The thickness loss rate continues to increase, but the growth trend becomes slow. After 1.6 s, the two-phase flow in the bend pipe is basically stable, and the number of particles is unchanged. So, the flow trend will not be abruptly changed, and the thickness loss rate of the wall surface of the bend pipe fluctuates near a fixed value. At C m = 3%, the thickness loss rate is maintained at 1.0 × 10–10 m/s, and the thickness loss has been increasing almost linearly with time. Comparing the thickness loss rate and thickness loss under the three concentrations, it can be found that the change trends of two variables are consistent under the different concentrations. The thickness loss continues to increase as the particle concentration increases. At t = 3.103 s, the thickness loss rate of C m = 3% concentration is 3.95 × 10–11 m/s, C m = 5% concentration is 1.88 × 10–10 m/s, C m = 7% concentration is 7.89 × 10–11 m/s. It can be found that when the particle concentration increased, the thickness loss rate of the bend pipe increased to a certain extent and began to remain unchanged, which is because the number of particles in the bend pipe increases to form a buffer layer. As can be seen from Fig. 5.27, near the outer arch wall of the bend pipe, there is a layer of particles with low speed, the thickness of this particle layer does not thicken with the increase of particle concentration. This particles in this buffer layer move along the bend pipe wall, causing cutting wear of the wall. These particles also prevents more particles to impact on the bend wall. So, the thickness loss rate will not continue to increase with the concentration increasing but maintain a relatively stable value. Figure 5.29 is the calculation data of the wear on bend 1 at different positions (at t = 3.103 s), and Fig. 5.30 is the data of thickness loss measured after the wear experiment of bend 1 (wear time is 48 h). The curves θ = 0° and θ = 180° are the curves at the two cheeks of the pipe. Due to the unevenness of the particle distribution, the thickness loss of the corresponding part of θ = 0° is 0 m, so it is expressed using a scatter plot; θ = 90° and θ = –90° are the curves of the outer arch and the inner side of the bend pipe, respectively. Since the curve data of θ = –90° are all 0 in the calculation result, there are only two curves and one set of scatter plots in the calculated data. Judging from the comparison results of the two figures, the changing trend and wear magnitude are the same. On the θ = 90° curve, the wear gradually increases with Φ. In the experimental result plot, its rising curve fluctuates greatly and even has a negative value. The main reason for the negative value is that the value of thickness loss is the thickness difference measured before and after the wear experiment. Because there is plastic deformation of the pipe wall, the ripple morphology appears on the wall surface of pipeline. The thickness loss at the corrugated bulge is negative. Comparing the two curves of θ = 0° and θ = 180°, it can be found that the two curves are not consistent. In the experimental data plot, the two curves have been fluctuating around zero, and there is no obvious upward or downward trend, which also shows that the particles in the bend pipe are not uniformly distributed and flows completely along the pipe wall. The particles are spiral flowing forward in the bend

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5 Wear Characteristics of the Wall Surface in Centrifugal Pump

Fig. 5.28 Bend pipe 1 wear over time

pipe. Part of this area is the high-frequency region where the particles collide with the wall. Some areas are non-particle regions, and the wear data is 0 in these areas. Curve of θ = 180° is a slight rise and then a fall and reaches a peak between 45° < Φ < 50°. From the previous particle distribution, it can be known that the particles from the vertical pipe directly collide with the wall of bend pipe 1, and the pipe wall is seriously worn. Change trends of curves at C m = 3% and C m = 5% are more similar. The curve of C m = 7% is not the same, θ = 90° curve does not show an upward trend but fluctuates up and down around a fixed value. It is indicated that the wear on outer arch of the bend pipe do not increase with the increase of particle concentration. Because at this time there is a highly clustered state of particles near the pipe wall, a stable low-speed particle buffer layer is formed on the wall surface of bend pipe 1, which can be seen from the previous analysis of particle distribution. This layer of particles exacerbates the interaction between particles and the particle energy is reduced. Due to energy loss of particles, the interaction between the particles and the wall is weaken. More particles are prevented from touching the wall, which eventually leads to the wear of wall surface unchanged or even decreased. However, as the particle mass concentration increases, the particles diffuse to the cheeks, and the curves of θ = 0° and θ = 180° both increase with the particle mass fraction.

5.3 Wear of the Pipeline Matched with Centrifugal Pump

189

Fig. 5.29 Bend pipe 1 wear (t = 3.103 s)

5.3.3 Horizontal Pipe Wear Analysis The particle movement in the pipe is different from that in the flow channel of centrifugal pump. Because there is no centrifugal force provided by the impeller rotation, the particle movement is mainly driven by the fluid drag force and is affected by its gravity. Figure 5.31 is the wall wear and particle distribution in horizontal pipes (C m = 7%). As can be seen from the figure, the wear on the wall surface of the pipe is directly related to the location of the particle aggregation. The wear exists on both cheeks at the entrance of the horizontal pipe. With the concentration of the particles increasing, the degree of wear increases and the wear area is also wider. In the latter

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5 Wear Characteristics of the Wall Surface in Centrifugal Pump

Fig. 5.30 Experimental wear data of bend pipe 1

half of the horizontal pipeline, the wear is mainly concentrated on the lower side of the pipeline. From Fig. 5.31b, it can be seen that the particle at the inlet of the horizontal pipe is more uniformly distributed and are accumulated in the bottom of the pipe at the middle and rear parts, which is mainly caused by: (1) When the particles pass through the bend pipe, the rotation speed of the particles increases

5.3 Wear of the Pipeline Matched with Centrifugal Pump

191

Fig. 5.31 Horizontal pipe wear (t = 3.103 s)

because of the effect of collision with the wall and the change of flow direction. The force produced by the particle rotation increases correspondingly. In the bend pipe, the particles with higher speed move to the outlet of the bend pipe under the action of inertia and drag force. The particles with low Reynolds number do a movement around the inner wall of pipe under the influence of rotational lift. So, the particle distribution at the horizontal pipe inlet is basically uniform. (2) Due to the density difference between the solid phase and liquid phase, in the middle and rear part of the pipeline, the gravity results in a high concentration of particles at the bottom of the pipeline and a low concentration at the top. The particles are distributed in a stepped shape in the pipeline. Figure 5.32 is a curve of the maximum wear value of the pipe wall surface on the unit length coordinate of the horizontal pipe (On the horizontal coordinate, 0 m is located at the entrance of the horizontal pipe. The length of the horizontal pipe is 4.46 m). Because the closer to the pipe inlet, the greater the particle velocity, the pipeline wear shows a trend of first growth and then decline. In the pipeline inlet section, there are two wear maximums. The first wear maximum appears at the two cheeks of the horizontal pipe, which is caused by the spiral flowing out of the bend 1. The second maximal value appears at the bottom of the horizontal pipe, which is caused by the accumulation of particles at the bottom of the pipe. The two wear maximums have similar values, the first maximum is mostly contributed by cutting wear, and the second maximum has a higher proportion of impact wear. In the section from the pipe inlet, the region of 0–15 d2 (d2 is the diameter of the centrifugal pump outlet pipe) is a severe wear area, which is a key protection area. Figure 5.33 is the time-converted wear experimental data of the horizontal pipeline. The wear value shows a slight downward trend with the length of the pipeline, basically consistent with the calculation results. Due to the limited data points, the experimental data curve fluctuates greatly.

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5 Wear Characteristics of the Wall Surface in Centrifugal Pump

Fig. 5.32 Calculation results of horizontal pipe thickness loss (t = 3.103 s)

5.3.4 Particle Motion Analysis Inside Pipes The wear on overcurrent wall is closely related to the particle motion in the solid– liquid two-phase flow. The particle movement in the flow channel is analyzed. All particles in the first-time step of calculation are selected, and the average velocity of these particles in the computational domain is calculated and shown in Fig. 5.34. The particle velocity of different concentrations is consistent. In the initial stage, the particle velocity is consistent with the fluid. Due to the poor flow followability of large particles, the particle velocity gradually decreases. At the impeller inlet of the centrifugal pump, the particle velocity drops to the lowest value, about 1.65 m/s. The particles collide with the impeller inlet at this speed. The particle velocity continues to decline slightly for the energy loss of collision. Then the particles enter the impeller and are driven by the high-speed rotated fluid in the impeller. The particle speed increases sharply, and the particle velocity at the outlet of the impeller is up to 2 times the velocity at the impeller inlet. When the particles move to the blade tail and collide with it, the centrifugal acceleration effect of the particles disappears, and the acceleration trend of the particles slows down. At the outlet of the centrifugal pump, the particle speed reaches the maximum value, greater than 5 m/s. And then

5.3 Wear of the Pipeline Matched with Centrifugal Pump

193

Fig. 5.33 Horizontal pipe wear experimental data (t = 3.103 s)

the particles decelerate in the vertical pipe due to the gravity. A minimum value of particle speed appears at about 5 to 10 d2 (d2 is the diameter of the centrifugal pump outlet pipe) from the inlet of the bend pipe 1. After the particles enter bend pipe 1, the number of vortices in the fluid increases. The particles begin to accelerate and reach an extreme value at the region of 0 to 15 d2 from the outlet of bend pipe 1, but this speed maximum is lower than the previous extreme value at the outlet of the pump. After about 1.65 s, the particles gradually decelerate in the horizontal pipe, the speed change is not obvious. Subsequently, the particles enter bend pipe 2 and gradually accelerate. Because the gravity direction is consistent with that of the particle motion, the particle acceleration here is more obvious than in the bend pipe 1. Comparing the changes of particle velocity before entering the impeller inlet, it can be found that the particle velocity changes insignificantly at three concentrations. The three curves almost overlap. The particles in this section are mainly driven by fluid drag force. When particles move out of the centrifugal pump, the particles speed of the small concentration reach the maximum first. And the value is higher than the maximum speed of the large concentration. This indicates that the fluid driving effect of particles is more obvious under small concentration, and the stagnation time in the centrifugal pump is shorter.

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5 Wear Characteristics of the Wall Surface in Centrifugal Pump

Fig. 5.34 Average velocity of particles

5.4 Wear of the Flow Channel for Conveying Mixed Size Particle Collision between particles and between particles and the wall surface will cause particles to be broken in different degrees. The change of particle size make the flow in centrifugal pump more complicated and affect the external performance of the pump. Therefore, it is particularly important to study the flow characteristics and wear characteristics of centrifugal pump under the conditions of the mixed size particles.

5.4.1 Calculation Method and Scheme The numerical simulation of the two-phase flow and wear in a centrifugal pump is carried out based on the DPM model, which treats the solid phase particle into a discrete phase and adds the collision rebound model of the particles to solve the

5.4 Wear of the Flow Channel for Conveying Mixed Size Particle

195

Table 5.2 The parameters in the wear model A

nA

θ

1.599 × 10–7

1.73

π/12

0

a

b

x

y

z

w

− 38.4

22.7

3.147

0.3609

2.532

1

interaction between the particles and the wall. According to the material characteristics of the overcurrent wall, the wear model proposed by Ahlert [1] is selected (see Eqs. 2.61 and 2.62). Depending on the wall material, the parameters used in the equations are shown in Table 5.2. The centrifugal pump parameters are the same as the experimental pump (see Sect. 5.1). In the simulation, the inlet boundary condition was velocity inlet, which was set to 2.47 m/s, and this velocity was calculated by rated flow and inlet diameter. To observe the fully developed internal flow, the corresponding turbulence parameters on the boundary conditions were set, which are the turbulence intensity (I = 0.033,907,305,907,0281) and the hydraulic diameter (100 mm). The file form was selected for particle injection on the inlet surface, and 64 particle injection points were evenly distributed on the inlet surface. The particle inject speed is the same as the fluid at 2.47 m/s. This velocity was considered to have the same velocity as the flow. The impeller zone condition used mesh motion and the rotating speed was set to 1450 r/min, which was the same rotating speed as the centrifugal pump impeller in the experiment. The outlet boundary condition was outflow. The collision model and the wear model in the wall of the impeller and volute were the models mentioned above. The setting of all boundary conditions in the simulation was determined based on experimental conditions. Under this condition, the time step was set to 0.000,114,9425,5 s, which was the time of the impeller to rotate through 1°, and the convergence criterion of the residual value in each time step was set to 10−4 . The simulation was considered complete when the number of particles in the centrifugal pump gradually stabilized.

5.4.2 Calculation Method Verification To verify the applicability of the numerical simulation method, the performance and wear experiments by transporting solid–liquid two-phase flow with mixed size particle are carried out. Both the numerical calculation and the two-phase experiment are carried out based on the experimental pump (see Sect. 5.1). The numerical simulation results are compared with the experimental results, and Fig. 5.35 shows the numerical results and the experimental data with a ratio of 3:7 (between large particles and small particles). In the Fig. 5.35, the changing trends of the experimental value and the calculated value are the same. The efficiency experimental value is generally less than the calculated value, which is because the power loss caused by volume leakage and mechanical friction is not considered in the numerical calculation. And due to the

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5 Wear Characteristics of the Wall Surface in Centrifugal Pump

Fig. 5.35 Performance comparison of the experiment and the simulation

influence of experimental uncertainty, the experimental curve is slightly jumpy, while the calculation curve is relatively smooth. The maximum error of the head at the rated flow rate is 3.01%, and the maximum error of the efficiency is 7.23%. This result is within a reasonable margin of error, so numerical simulation results can be considered reliable.

5.4.3 Runner Wear Analysis After 24 h wear experiments, the wear area and wear degree of the centrifugal pump runner are different under different mixing ratios and different mass fraction conditions.

5.4.3.1

Impeller Wear

For the solid–liquid two-phase flow with a 3:7 mixing ratio, the simulated wear results under different mass fractions and the experimental wear results are compared, as shown in Fig. 5.36. From the wear experiment results under the 3:7 mixing ratio, with the increase of the particle mass fraction, the wear of the impeller wall surface becomes more serious. The disappearance area of indicated paint on each wall is increasing. The leading edge of the blade is the most severely worn, with almost all metal surfaces exposed at a mass fraction of 1%. As the mass fraction increases, the wear part

5.4 Wear of the Flow Channel for Conveying Mixed Size Particle

197

Fig. 5.36 Comparison of wear results of the impeller at a ratio 3:7

expands from the leading edge of the blade to the tail edge, and the wear range of the blade tail also increases rapidly. At a mass fraction of 9%, the leading edge of the blade becomes rounded due to the collision of a large number of particles. Comparing the simulation results, it can be found that the wall thickness loss obtained by the numerical simulation is consistent with the experimental wear results. With the increase of the mass fraction, the thickness loss and wear range of the flow channel increase rapidly. Under different mass fraction conditions, the vanishing sequence of paint coating on the suction surface of impeller is analyzed to find that the wear initially occurs at the leading edge of the blade. As the mass fraction increases, the tail of the suction surface also is worn, and the wear area and the wear degree are also increasing. However, when the mass fraction is increased to a certain amount (7% in this study), the wear area on the suction surface does not continue to increase but maintains a relatively stable range. This is also reflected in the simulation results. The continuous increase in mass fraction of solid phase has not caused continuous expand of the wear area on the suction surface of the blade, which may be related to the buffer layer generated by the mass particle aggregation. From the wear experiment results on the wall of the impeller hub, it can be found that the wear initially occurs in the position

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5 Wear Characteristics of the Wall Surface in Centrifugal Pump

of the outer edge of the hub near the suction surface of the blade. Under the mass fraction of 1%, the first layer of green paint in this position has been worn off, and the second layer of white paint coating has appeared, while the other positions are relatively intact. With the gradual rise of the mass fraction, the wear range is getting larger and larger, from the outer edge of the hub to the impeller inlet, from the suction surface to the pressure surface, and gradually extending to the entire hub wall. And it can be seen that staring at 5% mass fractions, the indicative paint coating on the hub wall has completely disappeared. The topside of the blade begins to have dents from 3% mass fraction. As the particle mass fraction continues to rise, the dents become more and more obvious, and even two grooves appear on the top of the blade. This may be because there is a gap between the wear plates and the impeller. And during the rotation of the impeller, the fine particles of 0.3 mm particle enter the leaf top gap. With the rotation of the impeller, the leaf top occurs serious wear. As the particle mass fraction continues to rise, the number of fine particles in the leaf top gap continues to increase. This increases the wear degree on the topside of the leaf. Enlarge the wear pattern of the hub wall, as shown in Fig. 5.37. It can be seen that when the mass fraction is low, the wear on the hub wall is mainly the thickness loss caused by the particle collision. The pit produced by the particle collision with hub wall surface can be seen. As the particle mass fraction rises to the mass fraction of 3%, there are gradually slight scratches in addition to the pit produced by the particle collision. This phenomenon is more obvious as the mass fraction continues to increase to 5 and 7%. The scratches are mainly extended along the two red lines of a and b. When the mass fraction rises to 9%, there is a fish scale-like wear on the wheel wall. Analyzing the experimental results under the condition of a 7:3 mixing ratio, it can be found that the wear trend of each wall surface of the impeller is similar to the

Fig. 5.37 Hub wall wear at a ratio 3:7

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199

Fig. 5.38 Comparison of wear on impellers at a ratio 7:3

result of the mixing ratio of 3:7, but the wear result on the wall surface of the hub is slightly different at low concentration. This is shown in Fig. 5.38. As can be seen from Fig. 5.38, the impeller suction surface is very worn at a mixing ratio of 7:3. Starting from the 1% mass fraction, the two layers of paint on the impeller suction surface and the hub wall surface are polished. The metal surfaces are exposed. This is because that the proportion of large particles is relatively high under the mixing ratio of 7:3. After two phases flow into the impeller basin, the larger particles directly impact the wheel hub wall due to inertia, and constantly hit the paint coating of hub wall until it falls off, resulting in a large area of paint coating shedding. The simulation results of the hub wall surface at a 7:3 ratio are similar to the experimental results. The wear of the hub wall surface occurs from the area near the suction surface to the equilibrium hole position. The indicator paint on the suction surface of the blade is almost completely disappeared at the 5% mass fraction at the 7:3 ratio. But at the 3:7 ratio, the indicative paint on the suction surface of the blade is almost completely disappeared until the 9% mass fraction. This is because that when the two-phase flow enters the impeller basin, the large particles are less affected by the flow and directly impact the surface of the suction surface of the blade, while the small particles are greatly affected by the flow, and a part of small

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5 Wear Characteristics of the Wall Surface in Centrifugal Pump

particles directly follow the fluid movement and will no longer collide with the hub wall. Therefore, the wear of the suction surface of the blade in the 7:3 ratio is more serious than in the 3:7 ratio. Under the mixing ratio of 7:3, the wear of the leaf top is similar to that of the 3:7 ratio, but the wear degree is slightly lower at the same particle mass fraction. In the 3:7 ratio, the initial wear of the leaf roof has begun to appear at a mass concentration of 3%, while in the 7:3 ratio, this phenomenon occurs at a mass concentration of 5%. It can be considered that when the mixing ratio is 3:7, the proportion of small particles is large. The probability of small particles entering the leaf top gap is greater. So, wear occurs earlier. The wear on blade pressure surface at a mixing ratio of 3:7 is shown in Fig. 5.39. As can be seen from the figure, pressure surface wear at a 1% mass fraction occurs first at the tail of the blade close to the root position. As the mass fraction rises, the residual area of indicator paint on the pressure surface continues to shrink. At the 7% mass fraction, the first layer of green paint has been completely polished, leaving only the white paint on the bottom layer. When the mass fraction increases to 9%, indicate paint coatings are all gone. The simulation results are consistent with the experimental results. Wear starts from the tail of the pressure surface near the root area of the leaf, and gradually spreads to the entire pressure surface. Observing the remaining paint coating and the exposed metal surface of the pressure surface, it can be seen that there are pits due to particle impact. And as the concentration of particles rises, fish- scale wear occurs. The pressure surface wear at a mixing ratio of 7:3 is shown in Fig. 5.40. As can be seen from Fig. 5.40, there are many pits on the blade pressure surface caused by particle collisions. These pits almost cover the tail of the blade pressure surface. Comparing the experimental results of the two mixing ratios, when the mass fraction is low (less than 5% in this experiment), the primary position and degree of wear on the pressure surface are basically the same. However, as the mass fraction

Fig. 5.39 Comparison of wear on the pressure surface at a ratio of 3:7

5.4 Wear of the Flow Channel for Conveying Mixed Size Particle

201

Fig. 5.40 Comparison of wear on pressure surfaces at a ratio 7:3

rises to 7%, the wear at 7:3 ratio is more severe than the wear at 3:7 ratio. The indicative paint coating on the pressure surface of the blade falls off completely. From the wear experiment results, it can be found that the thickness loss of the pressure surface with a higher proportion of large particles is greater at high concentration. It is opposite at low concentration. And it can be observed from the experimental photos that at a ratio of 7:3, the pressure surface of the blade at a mass concentration of 1% is covered with pits produced by the particle collision. At the 3:7 ratio, the blade pressure surface has appeared fish- scale wear early before the 7:3 ratio. With the increase of the mass fraction, the difference between the two ratios decreases. When the mass fraction rises to a certain extent, the wear of the 7:3 ratio begins to be more serious than the wear of the 3:7 ratio. The number of small particles at the 3:7 ratio is more under the same mass fraction. So, the buffer layer phenomenon results in a relatively light wear on the pressure surface of the blade.

5.4.3.2

Volute Wear

After a continuous acceleration in the impeller basin, the solid–liquid two-phase flow enters the volute basin and causes wear. The wear simulation results of the wall surface of the volute are shown in Fig. 5.41. From Fig. 5.41, it can be seen that the wear on the volute occurs at the 1% mass fraction. As the mass fraction gradually rises, the volute wears gradually become more severe. The wear area tends to be closer to the posterior cavity of volute. From the left view of the volute under various mass fractions, the wear on wall surface can be seen from the bottom of the volute towards the direction of the tongue. There is a tendency to extend gradually from the middle of the volute wall to the rear cavity. A similar phenomenon can be seen in the right view. This phenomenon occurs because when the two-phase fluid enters the impeller basin, the particle and the liquid are simultaneously close to the side of the rear cavity due to its inertia and initial velocity. During the rotation of the impeller, the two-phase

202

5 Wear Characteristics of the Wall Surface in Centrifugal Pump

Fig. 5.41 Simulation results of wear on the volute at a ratio of 3:7

flow in the runner is constantly pushed to the side of the back cavity, resulting in more severe wear near the back cavity. This is because that the volute section is pear-shaped. The cross-sectional area continues to increase with the counterclockwise direction from the position of the tongue and finally reaches the maximum position near the diffusion section. So, the tendency of the particles approaching the rear cavity will gradually weaken with the gradual increase of the overcurrent area (Fig. 5.42). In the wear experiment at the high concentration, the indicative paint coating on each part of the volute disappears completely after a long-time wear experiment. So,

Fig. 5.42 Experimental results of volute wear at a ratio 3:7 (1% mass fraction)

5.4 Wear of the Flow Channel for Conveying Mixed Size Particle

203

the experimental results of the 1% mass fraction are selected to show the wear degree of different parts on the wall surface of the volute. From the wear experiment results of the volute, the wear near the back cavity side is all very serious, while the wear away from the back cavity is relatively light and a part of the indicator paint is still retained. As the observation position approaches the diffusion section, that is, the position of the 8-section, it can be seen that the wear range of the volute in each area is getting larger and larger. At the region between the 8–0 section, the indicator paint on the entire volute wall has almost completely disappeared. This phenomenon is the same as in the simulation results.

5.4.3.3

Wear Analysis of Front and Rear Wear Plate

The simulation results of the front-wear plate at a mixing ratio of 3:7 are shown in Fig. 5.43. From the figure, the front- wear plate gradually begins to wear from the bottom, and the wear area gradually expands along the counterclockwise direction and approaches the middle part of the plate as the increase of the mass fraction. The fixed position of the front-wear plate is located at the top of the impeller. The particles are subjected to gravity when entering the impeller basin and are deposited at the bottom of the impeller flow channel. So, the wear on the front- wear plate appears on the bottom of the wear plate first. At the same time, the particles are made circular motions under the influence of the rotating flow field and rub against the wear plate. The wear form is shown in the Fig. 5.43. Figure 5.44 is a graph of the measurement results of the thickness loss rate of the front- wear plate. It can be seen that the thickness loss rate of the front-wear

Fig. 5.43 Simulation results of wear on front-wear plate at a ratio 3:7

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5 Wear Characteristics of the Wall Surface in Centrifugal Pump

plate at various locations is relatively stable under the condition of a 1% mass fraction. However, as the mass fraction increased, the thickness loss begins to fluctuate violently. When the mass fraction is increased to 9%, the thickness loss rate at various locations on the front-wear plate becomes relatively stable again. This is related to the material properties of the wear plate. In the case of low concentration, the particles collide with the wall surface and lead to the shedding of a small amount of wall material. After the mass concentration increases, a certain degree of deformation occurs on the wall surface due to the increase of the number of particle collisions. And a hardening protective film is formed on the surface, which results in uneven changes in thickness. As the mass concentration is further increased, the contact area between the particles and the wall surface continues to increase and the number of collisions continues to increase. The deformation of the wall surface at various locations tends to be consistent. As a result, the thickness loss rate becomes relatively stable. From Fig. 5.45, the rear- wear plate also has serious wear. The initial position of wear is the lower-side of the rear- wear plate. As the increase of mass fraction, the wear area on the rear- wear plate gradually extends in a counterclockwise direction. The performance is most obvious at 9% mass fraction. Meanwhile the wear area also continuously extends along the circumference. This phenomenon is related to the position of the rear-wear plate and the structure of the impeller. The wear area of the rear-wear plate is close to the bottom of the impeller. When the solid–liquid two-phase flow enters the impeller basin from the

Fig. 5.44 Measurement of the thickness loss of the front-wear plate at a ratio 3:7

5.4 Wear of the Flow Channel for Conveying Mixed Size Particle

205

Fig. 5.45 Simulation results of wear on rear-wear plate at a ratio 3:7

inlet, parts of the fluid will directly impact the rear-wear plate after passing through the hub wall. The particles trajectory is offset for the gravity. So, wear first appears at the bottom of the rear-wear plate. The reason for the wear area extending in a counterclockwise direction is that the impeller is also rotated counterclockwise, the rest of the particles accelerated by the impeller is constantly close to the side of the rear cavity. The wear effect on the rear- wear plate is shown in the figure. Figure 5.46 is a distribution map of the thickness loss rate of the rear-wear plate. It can be seen that the actual wear position plate is close to the inner ring of the rear-wear plate at 1% mass fraction. Due to the gravitational action, the particles accumulate at the bottom of the rear- wear plate, the wear here is very serious. With the increase of the mass fraction, the thickness loss rate of the various positions on the rear-wear plate fluctuates in disorder. However, the wear tends to stabilize as the mass fraction increases to 9%. The wear of the innermost ring of the rear-wear plate is more serious than other measurement positions at this mass concentration. The wear characteristics of the rear-wear plate are generally similar to that of the front-wear plate. But because the rear-wear plate is located in the rear-cavity of the impeller, the two phases of flow will bring the particles directly to impact the wall of the hub. The wear of the rear-wear plate is often more serious than the front-wear plate. This is both reflected in the experimental results and simulation results.

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5 Wear Characteristics of the Wall Surface in Centrifugal Pump

Fig. 5.46 Measurement results of thickness loss of rear-wear plate at a ratio 3:7

5.4.4 Relationship Analysis Between the Flow Characteristics and Wear Characteristics The wear on the wall surface of the centrifugal pump runner is closely related to the particle movement, and the movement of the particles is directly affected by the flow field. So, the analysis of the two-phase flow field in the pump is necessary to study the wear performance of the centrifugal pump. The analysis of the two-phase flow field inside the centrifugal pump is carried out at a mixing ratio 3:7 and a 1% mass fraction. To display the flow field more intuitively, the impeller basin is expanded to a blade row. In the blade row diagram, the Span value is increased from 0.1 to 0.9. The larger the value is, the closer the cross-section position is to the hub, and conversely, the smaller the value is, the closer the cross-sectional position is to the leaf top. As can be seen from Fig. 5.47, the streamlines near the leaf top and the middle position of the impeller (Span = 0.1, 0.3, 0.5) are very smooth. But in the position close to the wall of the hub (Span = 0.9), there is a vortex in the two-phase condition, and its main distribution position is in the middle of the pressure surface and near the leading edge. The velocity gradient under the clear water condition is smaller than that of the two-phase condition, and the overall velocity distribution is gentler. Under the twophase working condition, there is always a high-speed area on the suction side and a

5.4 Wear of the Flow Channel for Conveying Mixed Size Particle

207

Fig. 5.47 Velocity streamline of clear water and two-phase flow

low-speed area on the pressure side, which makes the exchange of flow more intense in the overall flow channel. The speed of each flow channel gradually decreases from the leaf top to the bottom of the leaf. Whether it is single-phase or two-phase, the streamlined trend is similar in the three flow channels A, B, and C. As can be seen from Fig. 5.48, at a 3:7 ratio that small particles account for a large number of particles, there are particles in all three channels. As the cross-sectional position continues to decline, the number of particles is increasing. There are a large number of particles in the three channels close to the hub wall position (Span = 0.9). The particles are constantly gathered here, and the number of particles at the inlet of the flow channel is the largest. This is because when the two-phase fluid enters the impeller basin, the large particles are gradually deposited due to gravity, while the small particles have a good follow-up with the fluid movement. So, the small particles are distributed in the various channels of the impeller, resulting in more uniform wear of the hub wall surface in the three channels at a 3:7 ratio. When the particles with a certain initial velocity enter the flow channel of impeller, particles will directly impact the hub wall surface, the rear- wear plate, and the other

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5 Wear Characteristics of the Wall Surface in Centrifugal Pump

Fig. 5.48 Particle distribution at a ratio 3:7 and a 1% mass fraction

wall surfaces close to the rear cavity. The hub wall surface is seriously worn for a large number of particles gathering. Because the circumferential velocity is greater on the hub wall far from the axis, the wall surface is worn out from the outside first. There are always many particles in the leading edge of the blade at each crosssectional position, especially in the position close to the hub wall (Span = 0.9). Such a particle distribution will make the blade directly impacted by the particles at the leading edge, resulting in more serious wear. As the concentration increases and the wear time increases, the leading edge of the blade becomes further rounded and deformed severely. In addition to particle aggregation on the hub wall and on the leading edge of the blade, there is also particle aggregation at the pressure surface of the blade. In the Span = 0.5, 0.7, 0.9, there is always a certain number of particles in front of the blade pressure side. As the cross-sectional position decreases, the number of particles in this area is further rising, which results in the wear of the pressure surface occurring from the tail edge to the leaf bottom area. It is consistent with the wear experiment results.

Reference 1. Ahlert KR (1994) Effects of particle impingement angle and surface wetting on solid particle erosion of AISI 1018 steel. The University of Tulsa

Chapter 6

Engineering Calculation of Solid–Liquid Two-Phase Pump

Abstract The external characteristics and wear of a centrifugal pump are significantly affected by its internal solid–liquid two-phase flow. Revealing the flow mechanism will improve the transportation efficiency and reliability of a solid–liquid mixed transportation centrifugal pump. In this chapter, the analyses are focused on the internal flow and wear characteristics of centrifugal pumps with different structural forms under the condition of solid–liquid mixed transportation. Three engineering calculation examples are respectively a single-stage pump for conveying zinc sulfide, zinc sulfate, and sulfuric acid pulp (without water) mediums, a high speed centrifugal pump with compound impeller for conveying a zirconia catalyst solution, a two-stage centrifugal pump for conveying residual oil at the bottom of the tower.

6.1 Single-Stage Centrifugal Pump The single-stage pump is a solid–liquid mixed transportation pump for conveying zinc sulfide, zinc sulfate, and sulfuric acid pulp (without water) mediums. The flow rate of this pump is 40 m3 /h, the head is 80 m, the speed is 1480 r/min, and the impeller material is stainless steel.

6.1.1 Geometric Model and Calculation Settings The geometric model of the centrifugal pump is shown in Fig. 6.1. This centrifugal pump consists of the shroud, volute, impeller and hub. To ensure that the fluid entering the centrifugal pump has been fully developed during the calculation, the inlet pipe of the centrifugal pump is extended to six times the length of the pipe diameter, and the outlet pipe is similar. The centrifugal pump fluid calculation domain and mesh division are shown in Figs. 6.2 and 6.3. The entire computational domain inside the centrifugal pump is divided by a hexahedral mesh, in which the impeller edge and the area near the tongue of the volute are locally encrypted. The overall mesh quality index maximum-skewness is 0.46, which meets the calculation requirements. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Z. Zhu et al., Solid-Liquid Two-Phase Flow in Centrifugal Pump, Fluid Mechanics and Its Applications 136, https://doi.org/10.1007/978-981-99-1822-5_6

209

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6 Engineering Calculation of Solid–Liquid Two-Phase Pump

Fig. 6.1 Geometric model of centrifugal pump

Fig. 6.2 Centrifugal pump 50JZJ-46 hydraulic model

The velocity inlet, free outflow outlet, and no-slip wall boundary conditions are used in the calculations, and the values of k and ε for the walls are taken according to the standard wall function method. The RNG k-ε turbulence model is used for the turbulence model. The SIMPLE algorithm is used to solve the pressure–velocity coupling problem, and the first-order format is used in the numerical method of the turbulence calculation. The effective values of the computational convergence

6.1 Single-Stage Centrifugal Pump

211

Fig. 6.3 Mesh diagram of the fluid calculation domain

accuracy of each item in the numerical calculation are set to 10– 5 . The temperature variation is not considered in the calculation. In this calculation, the mixed model multiphase flow model (the Mixture model) is applied for the solid–liquid two-phase calculation, and the multiple reference system model is used to couple the moving and static components.

6.1.2 Analysis of Solid–Liquid Two-Phase Flow Calculation Results The internal flow field of the centrifugal pump is simulated and the results are analyzed under the mixed solid–liquid two-phase transport conditions. The flow rate is 40 m3 /h for the design condition, the liquid phase is zinc sulfide, zinc sulfate, and sulfuric acid pulp (without water) with a density of 1280 kg/m3 , the solid phase is zinc concentrate with a density of 4300 kg/m3 , the particle mass concentration is 15%, and the particle diameter is 70 mesh.

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6 Engineering Calculation of Solid–Liquid Two-Phase Pump

Fig. 6.4 Absolute velocity distribution in the middle section

6.1.2.1

Absolute Velocity Analysis

Figure 6.4 shows the absolute velocity distribution in the internal flow channel area of the centrifugal pump. As can be seen from the figure, in the impeller flow channel, the absolute speed from the impeller inlet to the impeller outlet gradually increases, the absolute speed at the outlet of the impeller is not uniformly distributed, and the absolute speeds at the middle and rear of the impeller pressure surface are larger. Specifically, the absolute speed at the impeller outlet near the part of the tongue is very large, which can easily produce more serious wear.

6.1.2.2

Solid Phase Volume Distribution Analysis

Figure 6.5 shows the distribution of the solid phase concentration in the flow channel area inside the centrifugal pump. It can be seen from the figure that the solid phase concentration is higher at the impeller inlet, where it is mainly subject to impact wear to a more serious extent. In the volute flow channel, with the action of centrifugal force, most of the particles gather on the volute wall. In particular, the particle concentration is very high near the tongue, which is also the reason for the serious wear of this pump volute and tongue. In addition, the solid phase concentration at the bottom of the volute is larger due to the influence of gravity settlement.

6.1 Single-Stage Centrifugal Pump

213

Fig. 6.5 Solid phase concentration distribution in the middle section

6.1.2.3

Energy Analysis

Figure 6.6 shows the turbulent energy distribution in the flow channel area inside the centrifugal pump. It can be seen from the figure that the turbulent energy inside the impeller channel changes very little. The turbulent energy near the impeller inlet is larger compared with the turbulent energy inside the impeller channel. The turbulent energy at the tongue and the channel outlet exhibits a large change, and the highest turbulent energy value appears at the impeller outlet near the tongue. This indicates that the fluid flow is more stable and the energy dissipation is low inside the impeller channel. In addition, because of the highest turbulent energy value is found at the impeller outlet near the tongue, which indicates that the fluid flow in the impeller flow path is more stable and the energy dissipation is low. The turbulent energy dissipation is mainly concentrated at the top of the blade and the tongue, where the hydraulic loss is serious. Figure 6.7 shows the kinetic energy distribution of the internal flow channel of the centrifugal pump. From the figure, it can be seen that the kinetic energy in the impeller inlet region is the smallest, and the kinetic energy changes gradually become larger along the blade outlet direction. The top of the blade has a great kinetic energy, which is consistent with the velocity distribution results shown in Fig. 6.4. The kinetic energy on the pressure side of the blade is larger compared with the suction side, and the kinetic energy in the area near the tongue of the volute is relatively small.

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Fig. 6.6 Turbulent kinetic energy distribution in the middle section

Fig. 6.7 Kinetic energy distribution in the middle section

6.1.2.4

Vorticity Analysis

Figure 6.8 shows the vorticity distribution of the internal flow channel of the centrifugal pump. It can be seen that the vorticity value near the impeller inlet is larger for the flow disturbance due to flow from axial to radial. In the impeller flow channel near the tongue, the flow instability phenomenon increases due to the flow obstruction by the tongue. At the middle of the blade pressure surface, the vorticity has a larger value, while away from the tongue flow channel, the flow is relatively stable; and the vorticity value is smaller.

6.1 Single-Stage Centrifugal Pump

215

Fig. 6.8 Vorticity distribution in the middle section

6.1.2.5

Wall Shear Stress Analysis

Figure 6.9a and b show the shear stress distribution diagram of the impeller wall surface of the centrifugal pump. As can be seen from Fig. 6.9a and b, the maximum shear stress distribution is in the impeller inlet hub wall surface and blade outlet area, with a maximum value of about 4500 Pa. Combined with the solid phase volume fraction distribution and the influence of gravity settlement, particles in the flow channel below increase, so the solid–liquid two-phase mixture concentration is high and the shear stress on the blade increases. The blade outlet position is the most affected. Figure 6.9c shows the centrifugal pump volute wall shear stress distribution. As can be seen from the Fig. 6.9c, because the two-phase mixture from the impeller channel into the volute with a large flow velocity along the volute wall sliding, the volute wall shear stress is very large, and the actual engineering volute wall has correspondingly large wear. In the diffusion section, the fluid velocity is reduced, so the shear stress is also reduced significantly. In summary, through the analysis of the flow characteristics of the solid–liquid two-phase flow inside the mixed transfer pump, it is found that the solid-phase concentration is not uniformly distributed in the impeller flow channel. When the particles from the axial inlet pipe move into the radial impeller flow channel, with the action of inertia and the hub wall back cover plate impact, in the vicinity of the impeller inlet, the solid phase concentration of the blade leading edge and suction surface is high, and the collision between the particles and the blade leading edge produce serious wear. At the suction surface, although there is a large number of particles and wall sliding friction, there is low flow velocity, so the wear is not serious. In the volute flow channel, the overall shear stress is higher, and the high

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6 Engineering Calculation of Solid–Liquid Two-Phase Pump

Fig. 6.9 Wall shear stress

concentration of two-phase flow generated by the settlement below the volute wall leads to particularly severe wear on the volute wall surface, especially the wall surface near the tongue.

6.2 High Speed Centrifugal Pump with Compound Impeller This example is used to analyze the internal flow and wear performance of a high speed centrifugal pump transporting a zirconia catalyst solution. The design flow rate of the centrifugal pump is 5 m3 /h, the head is 400 m, the speed is 13500 r/min, and the wall material is zirconium alloy Zr705.

6.2 High Speed Centrifugal Pump with Compound Impeller

217

6.2.1 Geometric Model and Calculation Settings In order to ensure the stability and full development of the inlet flow during calculation, the length of the calculated inlet pipe section is extended by 5–10 D (D is the diameter of the inlet pipe), and the outlet pipe is extended accordingly. The calculation domain and the grid division of the centrifugal pump are shown in Fig. 6.10. In this calculation, FLUENT software and EDEM software are coupled. The specific calculation process of coupling can be divided into two parts. Firstly, the flow field distribution in the centrifugal pump is calculated in FLUENT software. At each time step, when the iterative calculation of the flow field is finished, the EDEM software is invoked to start the calculation, and the particle force, instantaneous velocity and position are obtained. Then the updated particle information is transmitted to FLUENT for the next iteration calculation. For the calculation, the RNG K-ε model is used for the turbulence model, the DEM model is used for the two-phase flow model, the standard wall function is used for the low Reynolds number region near the wall, and the sliding grid is used for data exchange at the dynamic and static interface. The velocity inlet condition is set, and the velocity direction is perpendicular to the inlet boundary. The velocity is calculated according to the relationship between the rated flow rate of the centrifugal pump and the inlet pipe diameter, that is, the initial flow velocity is 0.71 m/s, and the outlet boundary is Outflow. The particles are placed at the inlet of the pipeline at t = 0 s. The initial velocity of the particles is consistent with the velocity of the fluid, which is 0.71 m/s. In order to accurately capture the contact behavior between one particle and another particle, and between a particle and the wall, the time step calculated in EDEM is required to be longer than that in FLUENT to ensure that the particles do not move over a large distance in one EDEM time step. The speed of the centrifugal

Outlet

Volute Impeller

Inlet

(a) Flow domain

(b) Grid diagram

Fig. 6.10 Flow domain and grid diagram of high-speed centrifugal pump

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6 Engineering Calculation of Solid–Liquid Two-Phase Pump

pump is 13500 r/min; that is, the rotation period is about 0.0044 s. The calculated time step in FLUENT is the time required for the impeller to rotate 2°, that is, 2.469 × 10–5 s. Considering the particle size and speed, in order to make EDEM more accurately simulate the movement of particles, the time step is set in EDEM to 1.2345 × 10–8 s, and the ratio of the two is 2000:1, which can meet the requirements.

6.2.2 Analysis of Solid–Liquid Two-Phase Flow Calculation Results The relevant parameters during calculation are as follows: The liquid density is 1350 kg/m3 , the particle size range is 0.2–0.3 mm, the particle density is 5850 kg/m3 , the particle hardness is 8.5 HM, and the particle volume concentration is 7%.

6.2.2.1

Analysis of Flow Field Velocity and Particle Motion

Figure 6.11a shows the flow diagram of the section in the flow channel of the centrifugal pump, and the flow line is stained by the fluid velocity. It can be seen that there are vortices on the pressure surface of each blade in the impeller channel. Specifically, the flow near the volute tongue is very disorderly, while the flow in the volute channel is relatively uniform. In general, the flow of the centrifugal pump is good, and only a small number of streamlines do not flow out. The velocity of the section in the flow channel is shown in Fig. 6.11b. It can be seen that the velocity distribution of each flow channel is relatively uniform, and there is a velocity instability region near the tongue, which is basically consistent with the flow disorder phenomenon shown in Fig. 6.11a at the tongue. The particle velocity distribution is shown in Fig. 6.12. It can be seen that the flow in the inlet pipe is pre-rotated with the influence of the impeller rotation, and the particles are enveloped by the pre-swirling fluid and begin to rotate along the inner surface of the pipe. The closer the particles are to the inlet of the impeller, the greater the influence will is. Specifically, the particle velocity increases, and the velocity of the particles rotating on the inner wall reaches the maximum near the inlet of the impeller. As shown in Fig. 6.12b, in the impeller channel, the particles move close to the pressure surface of the long blade; and accumulate slightly near the tongue due to the sudden change of the flow channel section. In the volute channel, with the action of the centrifugal force, most of the particles move along the volute wall, which can easily to lead to wall sliding and wear. In the diffusion section, the particles move along the spiral trajectory from the volute wall to the left pipe wall.

6.2 High Speed Centrifugal Pump with Compound Impeller

219

Velocity Streamline m/s 72.30

Velocity m/s 70

54.22

55

36.15

40

18.07

25

0.00

10

(a) Velocity streamline distribution

(b) Velocity distribution

Fig. 6.11 Streamline and velocity field distribution of the middle section in the flow channel Particle Velocity 70

m/s

55 40

25

10

(a) Overall flow channel

(b) Frontal flow channel

Fig. 6.12 Particle velocity distribution

6.2.2.2

Flow Field Vorticity and Turbulent Kinetic Energy Analysis

Figure 6.13a shows the vorticity of the middle section in the flow channel of the centrifugal pump. It can be seen that the larger vorticity inside the impeller mainly exists in the impeller inlet and the short blade inlet. It is because that the impeller inlet has a flow from axial to radial and the short vane inlet has a sudden narrowing of the flow channel. So, the flow field changes greatly. It results in an unstable flow field,

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6 Engineering Calculation of Solid–Liquid Two-Phase Pump

Turbulence Kinetic Energy m2s-2 20

Vorticity s-1 6000

4750

15

3500

10

2250

5

1000

0 (a) Vorticity diagram

(b) Turbulent kinetic energy diagram

Fig. 6.13 Vorticity and turbulent kinetic energy in the middle section

and the vorticity becomes large. In addition, there are also more vortices generated in the diffusion section inlet, which is due to the influence of the tongue, coupled with the sudden change in the shape of the flow channel, and the velocity change in the flow field is therefore also greater. Figure 6.13b shows the turbulent kinetic energy of the middle section. It can be seen that the turbulent kinetic energy distribution in each impeller channel is basically the same, the turbulent kinetic energy in the impeller channel near the tongue has a great change, and the highest turbulent kinetic energy value appears at the tongue. This indicates that the flow in most areas of the impeller channel is more stable and the energy dissipation is low. Additionally, the flow affected by the tongue is unstable and the energy dissipation is higher there. In the middle area of the diffusion section of the volute, there is a high point of the turbulent kinetic energy value, which may be due to the sudden expansion of the flow channel from the tongue to the diffusion section, resulting in the increase of the instability of the flow.

6.2.2.3

Wall Shear Stress Analysis

Figure 6.14a and b show the shear stress distribution on the impeller wall surface. It can be seen that the larger shear stress mainly appears in the impeller inlet hub surface, the long blade leading edge to the middle, the long blade trailing edge, and the short blade trailing edge, the maximum value is close to 600 Pa. The shear stress distribution on each blade wall surface is basically the same, and the difference is not

6.2 High Speed Centrifugal Pump with Compound Impeller

221

Wall Shear Pa 600

450

300 (a)Impeller front view 150

0

(b)Impeller end view

(c)Volute front view

Fig. 6.14 Wall shear stress

large. However, due to the effected of the abrupt change of the flow channel near the tongue, the two-phase flow is more turbulent, and the particles collide and rub against the flow channel, so the shear stress in the tongue area is very large. Figure 6.14c shows the shear stress distribution on the volute wall. It can be seen that the shear stress values in the volute flow channel are large, which is caused by the friction between the solid-phase particles entering the volute at a large speed. In the front of the diffusion section, there is high shear stress due to the large velocity of the solid phase particles. But the shear stress gradually becomes smaller as the overall flow velocity in the diffusion section decreases.

6.2.2.4

Wear Analysis

Figure 6.15 displays the scatter plot of the average wear amount of different parts of the centrifugal pump (the simulation calculation time is 0.6 s). It can be seen from the figure that compared with other parts of the centrifugal pump, the average thickness loss of the blade part is the most obvious. The magnitude of the thickness loss is approximately 10–7 mm, while the average thickness loss of the tongue and the volute middle wall surface is 10–10 mm, and the thickness loss order of the inlet pipe section and the volute diffusion section is smaller in comparison, around 10–12 mm. Figure 6.16 shows the distribution cloud of the impeller thickness loss. It can be seen that the long blade leading edge and the long blade pressure surface are compared with the other parts with the most serious wear. The lock nut wall surface

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6 Engineering Calculation of Solid–Liquid Two-Phase Pump

(a) Inlet pipe wall surface and volute wall

(b) Impeller

Fig. 6.15 Average thickness loss of different parts of the centrifugal pump

and hub wall surface near the impeller inlet experience the direct impact of particles, and the wear is more serious. The use of a flexible base and hard surface combination form to reduce the impact wear can be considered. In addition, the thickness loss of the short blade whether for the leading edge or pressure surface, is much smaller than the thickness loss of the long blade. It is clear that the position of the blades has a direct effect on the wear thickness loss. This is because the long blade leading edge is in the impeller inlet, where the direction of particle movement in the axial to radial turning moment, so the particle impacts on the long blade cause serious wear, and the short blade leading edge is in the middle of the flow channel, where the movement of the particles is relatively smooth. Figure 6.17 shows the volute thickness loss. It can be seen that the thickness loss on the volute wall surface is more serious except for the diffusion section, and the thickness loss on the volute wall surface is serrated, which may be related to

Thickness loss(mm) 3.470E-6 2.603E-6 1.735E-6 8.675E-7 0 (a) Impeller (front view) Fig. 6.16 Impeller thickness loss

(b) Impeller (oblique view)

6.3 Two-Stage Centrifugal Pump

223

Thickness loss(mm) 0

3.20E-10

6.40E-10

9.60E-10

1.28E-9

View 1

(a)

(b)

(a) View 1

View 2

(c)

(b) Oblique view

(c) View 2

Fig. 6.17 Volute wall thickness loss

the inverted “convex” outlet of the flow channel in the blade part (as shown in the red box in Fig. 6.16b. In addition, combined with Fig. 6.15, it can be seen that the thickness loss of the diffusion section is significantly smaller than that of the other volute walls, which may be due to the fact that small particles are more likely to follow fluid motion, so the thickness loss in the diffusion section area is less obvious. Figure 6.18 shows the inlet pipe thickness loss. Combined with the velocity of the particles in Fig. 6.12a and the scatter diagram of the thickness loss in different parts of centrifugal pump in Fig. 6.15, it can be seen that in the latter part of the inlet pipe section, the movement of the particles is influenced by the pre-spin movement of the fluid, and there are some particles against the inner wall surface of the inlet pipe section for rotational movement, which causes spiral thickness loss on the inner surface of the inlet pipe section. Through the above analysis, compared with the other flow channel wall, it can be seen that the average thickness loss of the blade is the largest, especially the blade pressure surface. The thickness loss of the long blade is more obvious than that of the short blade. As a result, a reasonable blade profile for reducing the average thickness loss amount is very important.

6.3 Two-Stage Centrifugal Pump This example is uses to study the internal flow and wear performance of a two-stage centrifugal pump for conveying residual oil at the bottom of the tower. The flow

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6 Engineering Calculation of Solid–Liquid Two-Phase Pump

Thickness loss(mm) 3.47E-12

Flow direction

2.60E-12 (a) Inlet pipe front side

1.74E-12 8.68E-13 0

(b) Inlet pipe back side

Fig. 6.18 Inlet pipe thickness loss

rate of the two-stage centrifugal pump is 375 m3 /h, the head is 240 m, the speed is 1485 r/min, and the impeller material is stainless steel.

6.3.1 Geometric Model and Calculation Settings Figure 6.19 shows a three-dimensional model of a two-stage centrifugal pump. Figure 6.20 shows the middle section of the first-stage and the second-stage flow channel, where 1 and 2 are the inner and outer flow channels of the first-stage volute, respectively, 3 and 4 are the first-stage volute tongue and internal and external channel separation, 5 and 6 are the inner and outer flow channels of the second-stage volute, respectively, 7 and 8 are the separation of the tongue and the internal and external flow channels of second-stage volute, respectively.

First stage impeller

Second stage impeller

(a) Front view

(b) End view

Fig. 6.19 Three-dimensional model

(c) Sectional view

6.3 Two-Stage Centrifugal Pump

225

6

4

5

7

2 8

1 3 (a) First-stage channel

(b) Second-stage channel

Fig. 6.20 Shape of section in runner

In this calculation, the mixed model multiphase flow model (the Mixture model) is used to calculate the solid–liquid two-phase flow, and the multiple reference frame model is used to couple the dynamic and static components to carry out the steady numerical simulation of the three-dimensional full flow field of the internal flow of the centrifugal pump. The inlet boundary condition is set to velocity-inlet, and it is assumed that there is no tangential velocity or radial velocity on the inlet boundary, the axial velocity is evenly distributed, and the inlet velocity can be calculated according to the flow rate. With the assumption that the flow is fully developed when the fluid reaches the volute outlet boundary, a boundary piece of free outflow is set up. The solid wall boundary conditions are defined as no-slip boundary conditions, and k and ε of the wall surfaces can be evaluated according to the standard wall function method. Figure 6.21 shows the grid map of the whole basin of the two-stage centrifugal pump. The total number of grids is 11613091.

6.3.2 Analysis of Solid–Liquid Two-Phase Flow Calculation Results During the calculation, the liquid phase consists of the bottom oil of the reduced pressure tower with a density of 1084 kg/m3 , and the solid phase consists of coke asphalt particles, with an average particle size of 0.03 mm, a particle density of 1824 kg/m3 , and a solid phase concentration of 25%.

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6 Engineering Calculation of Solid–Liquid Two-Phase Pump

Fig. 6.21 Total basin grid of two-stage centrifugal pump

6.3.2.1

Comparison of External Characteristics and Performance

Nine different flow conditions are selected for the simulation calculation. The specific flow values are shown in Table 6.1. According to the numerical simulation results, the performance data under various working conditions are obtained and compared with the experimental data. The external characteristic curve of the centrifugal pump is drawn as shown in Fig. 6.22. The data in the figure include the flow-head curve, flow-efficiency curve, and flowpower curve. As shown in Fig. 6.22, the overall trends of the calculated curve and the experimental curve are basically the same under different flow conditions. The power calculation curve is in good agreement with the experimental curve under different flow conditions. In the low-flow condition and the rated condition, the calculated efficiency curve is close to the experimental curve data, and the error is small. Beyond the rated flow rate, the calculated efficiency curve starts to deviate from the experimental curve. Similarly, the calculated head curve and the experimental curve are in good agreement before the rated flow rate, and the calculated head and the experimental head show a large deviation after the rated flow rate is exceeded. The performance data under nine working conditions are shown in Table 6.2. Table 6.1 Values of different flow conditions No

1

2

3

4

5

6

7

8

9

Flow rate (m3 /h)

3.75

75

150

225

300

375

400

450

525

6.3 Two-Stage Centrifugal Pump

227

Fig. 6.22 External characteristic curve of centrifugal pump under two-phase working conditions

Table 6.2 Performance data comparison table

6.3.2.2

Flow (m3 /h)

Head (m)

3.75

260.12

0.91

317.82

75

251.34

17.61

315.92

150

254.85

32.39

348.32

225

254.83

42.65

396.67

300

248.61

49.91

440.96

375

242.75

55.75

481.82

400

244.73

58.33

495.23

450

247.00

61.67

531.83

525

244.22

65.55

577.17

Efficiency (%)

Shaft power (kW)

Speed Analysis

Figure 6.23 shows the flow diagrams of the middle section of the first-stage and second-stage channels under the rated working conditions. The flow lines are stained by the fluid velocity. It can be seen that in the first-stage and second-stage flow channels, the streamline distribution of the inner and outer flow channels of the volute is relatively uniform, and the streamline velocity of the inner flow channel is greater

228

6 Engineering Calculation of Solid–Liquid Two-Phase Pump

(a) First-stage channel

(b) Second-stage channel

Fig. 6.23 Flow diagram of the middle section of the channel

than that of the outer flow channel. The flow is disorderly near the tongue of the first-stage volute, but there is no such phenomenon in the tongue of the second-stage volute. In the impeller flow channel, whether for the first-stage or the second-stage flow channels, there are large whirlpools in the impeller near the separation point of the inner and outer flow channels. In addition, the second-stage impeller also has a vortex near the septum tongue, and the flow loss in the second-stage channel is large. In general, the pass ability of the first-stage and second-stage flow channels is good, and only a small number of flow lines do not flow out. Figure 6.24 shows the cross-sectional velocity distribution of the first-stage and second-stage flow channel under the rated conditions. It can be seen that the velocity at the pressure surface of the blade is generally smaller than that at the suction surface of the blade, and the maximum velocity is near the pressure surface at the outlet of the impeller. In the first-stage impeller, there is a velocity instability area near the tongue and there is a peak velocity in this area, which is consistent with the flow disorder phenomenon in the cross section streamline of the first-stage channel shown in Fig. 6.23a. In the second-stage impeller, the flow is relatively uniform near the tongue and the velocity change is stable, which is consistent with the uniform flow line of the second-stage channel flow line diagram in Fig. 6.23b. In the secondstage impeller channel, the local high velocity region appears near the impeller outlet, especially the boundary point of the inner and outer channels and the impeller channel near the tongue.

6.3 Two-Stage Centrifugal Pump

229

Velocity 57.859 43.394 28.930 14.465 0.000

(a) first-stage channel

(b) second-stage channel

Fig. 6.24 Velocity cloud diagram of section in channel

6.3.2.3

Analysis of Solid Phase Volume Distribution

Figure 6.25 shows the solid volume distribution on the wall of the first-stage impeller and the second-stage impeller under the rated conditions. It can be seen that on the surface of the first-stage impeller, the solid phase is mainly concentrated on the suction surface, which is due to the small particle size and the tendency of the particles to attach to the suction surface. Specifically, the solid phase volume at the middle and outlet of the blade suction surface rotating to the separation of the internal and external channels is relatively higher. On the suction surface of the same blade, the distribution of the solid volume concentration is also different. Due to the good follow ability of small particles, only part of the solid medium is distributed in the middle and near the outlet of the blade. However, near the separation of internal and external flow channels, the solid concentration on the suction surface of the blade increases, and there is a high concentration phenomenon in the middle and rear sections. This is because the flow channel at the separation becomes narrower, which hinders the solid phase movement and leads to the aggregation of particles. However, on the wall of the second-stage impeller, the solid phase is mainly concentrated on the pressure surface of the blade. With the acceleration of the first impeller, the speed of the two-phase mixed medium is further accelerated, and the solid phase flows out rapidly along the pressure surface. At the separation of internal and external flow channels of the second-stage impeller, the solid volume of the blade pressure surface is particularly high. By observing the hub wall of the two-stage flow channel, it is found that the solid volume concentration of the pump is less than that of the blade, indicating that the solid particles of the pump have a good pass ability, and the solid phase can quickly flow out after the impeller is accelerated, rather than settlement gathering on the wall. Figure 6.26 shows the solid phase volume distribution of the middle section of the first-stage and second-stage flow channels under the rated working conditions. It can

230

6 Engineering Calculation of Solid–Liquid Two-Phase Pump

(a) First-stage channel

(b) Second-stage channel

Fig. 6.25 Solid phase volume distribution on impeller wall

be seen that in the first-stage channel, the solid phase volume has a high concentration on the outside wall of the inner channel in the volute, and the solid phase volume distribution in the outer channel is uniform. The solid phase volume fraction is larger at the rear side of the outer channel and the outlet of the volute. In the second-stage channel, the volume fraction of the solid phase in the outer channel of the volute is significantly higher than that in the inner channel, and the distribution of the solid phase in the outer channel is more uniform. In the inner channel of the second-stage volute, the solid phase distribution is similar to that in the inner channel of the firststage volute, which is higher on the outside wall of the inner channel of both volutes. In summary, the solid particles are mainly distributed at the outside wall of the volute channel under the influence of centrifugal force, and the solid particles mainly flow out through the outer channel of the second-stage volute.

6.3.2.4

Turbulent Kinetic Energy Analysis

Figure 6.27 shows the turbulent energy distribution of the middle section in the firststage and second-stage channels at the rated conditions. As shown in Fig. 6.27, the turbulent kinetic energy is basically unchanged in the first-stage impeller channel, which indicates that the fluid flow is relatively stable and the energy dissipation is low in the first-stage impeller channel. The turbulent kinetic energy changes greatly at the tongue, the separation of the inner and outer flow channels and the outlet of the flow channel, and the maximum value of the turbulent kinetic energy appears at the separation of the inner and outer flow channels. This is because there are obstacles at the top of the blade, the tongue and the separation of the inner and outer flow

6.3 Two-Stage Centrifugal Pump

231

Volume Fraction 0.280 0.253

0.212

0.177 0.150

(a) first-stage channel

(b) second-stage channel

Fig. 6.26 Solid phase volume distribution of the middle section in the flow channel

channels, and there is a mutation of the flow channel section and other factors, that lead to instability of the fluid flow and high energy dissipation. The distribution of the turbulent kinetic energy in the second-stage impeller flow channel is significantly different from that in the first-stage flow channel. The turbulent kinetic energy in the impeller flow channel varies greatly near the separation point of the inner and outer flow channels of the volute, and the region of the turbulent Turbulence Kinetic Energy 15.000

11.053

7.105

3.158 0.000

(a) First-stage channel Fig. 6.27 Velocity cloud diagram of section in channel

(b) Second-stage channel

232

6 Engineering Calculation of Solid–Liquid Two-Phase Pump

kinetic energy maximum appears, indicating that the flow instability in the secondstage impeller flow channel increases. The area of the local high value of turbulent kinetic energy is the same as the area of large whirlpools in Fig. 6.23b. Similarly, similar to the flow channel of the first-stage impeller, the turbulent kinetic energy increases at the abrupt flow channel positions of the blade tip of the second-stage flow channel, the tongue, the separation of the inner and outer flow channels, and the outlets of the impeller and volute.

6.3.2.5

Wall Shear Force Analysis

Figure 6.28 shows the wall shear force distribution of the first-stage and second-stage volutes under the rated conditions. As shown in Fig. 6.28, both the first-stage volute and the second-stage volute have large shear stress on the inner flow channel wall, especially the inner flow channel wall near the tongue, while the shear stress on the outer flow channel wall is small. This is because the high-speed two-phase fluid flows out of the first-stage impeller channel, first flowing directly into the inner flow channel. Additionally, there is the high-speed two-phase medium sliding friction on the inner flow channel wall, resulting in large wall shear stress, and the solid concentration of the outside of the wall of the inner flow channel is very high, so this part of the area has serious wear. Then the velocity of the fluid decreases and then flows into the outer flow channel, so the shear stress of the outer flow channel wall is significantly smaller than that of the inner flow channel. Figure 6.29 shows the wall shear stress force distribution of the first-stage and second-stage impellers under the rated conditions. As shown in Fig. 6.29, both the first-stage impeller and the second-stage impeller have large shear stress on the suction surface side of the blade close to the flow channel near the tongue. This is mainly because of the barrier effect of the tongue on the two-phase fluid which leads to the aggregation of the solid medium in these positions and increases the wall shear stress. At the wall of the first-stage impeller, the shear stress in a small area at the

Fig. 6.28 Volute wall shear stress force

6.3 Two-Stage Centrifugal Pump

233

Fig. 6.29 Impeller wall shear force

outlet of the blade pressure surface is also large, but the wall shear force at the hub is relatively small. In the second-stage impeller wall, the area with a large wall shear force is mainly concentrated at the outlet of the blade pressure surface, and there is a large shear force and a wide range of areas. At the separation point of the inner and outer flow channels, the shear force at the hub is significantly greater than that of the other parts because of the aggregation of solid media. In addition, it can be seen that the shear force on the wall of the second-stage impeller is much larger than that on the wall of the first-stage impeller due to the secondary acceleration of the two-phase fluid. In conclusion, the wear area of the first-stage impeller is mainly concentrated on the suction surface of the blade, especially at the outlet and middle parts of the suction surface. The wear area of the second-stage impeller is mainly concentrated in the pressure surface of the blade, and the wear is also greater at the outlet and the middle of the pressure surface of the blade. In addition, the shear stress of the second-stage impeller wall is much greater than that of the first-stage impeller wall, so the wear resistance design of the second-stage impeller is more important. The wear of the volute mainly exists in the inner flow channel, and the wear degree of the outer flow channel is very light. In addition, according to the solid phase distribution of the flow channel, the solid phase medium can flow out quickly after being accelerated by the impeller, rather than settling and gathering on the wall, and the overall pass ability of this pump is good.