Slurry Transport

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Slurry Transport

Fundamentals, a historical overview and the Delft Head Loss & Limit Deposit Velocity

Framework

Miedema, Sape; Ramsdell, RC

Publication date 2016

Document Version Final published version Citation (APA)

Miedema, S., & Ramsdell, RC. (Ed.) (2016). Slurry Transport: Fundamentals, a historical overview and the Delft Head Loss & Limit Deposit Velocity Framework. (1st ed.) SA Miedema / Delft University of Technology.

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Fundamentals, A Historical Overview &

The

Delft Head

Loss &

Limit Deposit Velocity Framework

By

Sape A. Miedema

Edited by

Robert C. Ramsdell

0.01 0.1 1 10 100 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

1.E-05 1.E-04 1.E-03 1.E-02 1.E-01

Particle diameter (mm) Line s pe e d (m/s ) Dur a nd Fr oud e num be r FL (-) Particle diameter d (m)

DHLLDV Flow Regime Diagram

© S.A.M. Dp=0.1524 m, Rsd=1.585, Cvs=0.175, μsf=0.416 Homogeneous (Ho) Heterogeneous (He) Sliding Bed (SB) Fixed/Stationary Bed (FB) Viscous Effects Sliding Flow (SF) LDV FB-SB (LSDV) Undefined He-Ho SB-He SB-SF FB-He

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Slurry Transport

Fundamentals, A Historical Overview &

The

Delft Head

Loss &

Limit Deposit Velocity Framework

1 st

Edition

By

Sape A. Miedema

Edited by

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All rights reserved. No part of this book may be reproduced, translated, stored in a database or retrieval system, or published in any form or in any way, electronically, mechanically, by print, photo print, microfilm or any other means without prior written permission of the author, dr.ir. S.A. Miedema.

Disclaimer of warranty and exclusion of liabilities: In spite of careful checking text, equations and figures, neither the Delft University of Technology nor the author:

 Make any warranty or representation whatever, express or implied, (A) with respect of the use of any information, apparatus, method, process or similar item disclosed in this book including merchantability and fitness for practical purpose, or (B) that such use does not infringe or interfere with privately owned rights, including intellectual property, or (C) that this book is suitable to any particular user’s circumstances; or  Assume responsibility for any damage or other liability whatever (including consequential damage)

resulting from the use of any information, apparatus, method, process or similar item disclosed in this book. Design & Production: Dr.ir. S.A. Miedema (SAM-Consult)

ISBN Book: 978-94-6186-293-8 ISBN EBook: 978-94-6186-294-5

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Preface

In dredging, trenching, (deep sea) mining, drilling, tunnel boring and many other applications, sand, clay or rock

has to be excavated. The productions (and thus the dimensions) of the excavating equipment range from mm3_/sec

- cm3_{/sec to m}3_{/sec. After the soil has been excavated it is usually transported hydraulically as a slurry over a short}

(TSHD’s) or a long distance (CSD’s). Estimating the pressure losses and determining whether or not a bed will occur in the pipeline is of great importance. Fundamental processes of sedimentation, initiation of motion and erosion of the soil particles determine the transport process and the flow regimes. In all cases we have to deal with soil and high density soil water mixtures and its fundamental behavior.

The book covers horizontal transport of settling slurries (Newtonian slurries). Pipelines under an angle with the horizontal and non-settling (non-Newtonian) slurries are not covered.

Although some basic knowledge about the subject is required and expected, dimensionless numbers, the terminal settling velocity (including hindered settling), the initiation of motion of particles, erosion and the flow of a liquid through pipelines (Darcy Weisbach and the Moody diagram) are summarized. In the theory derived, the Zanke (1977) equation for the settling velocity is used, the Richardson & Zaki (1954) approach for hindered settling is applied and the Swamee Jain (1976) equation for the Darcy-Weisbach friction factor is used, Moody (1944). The models developed are calibrated using these basic equations and experiments.

An overview is given of experiments and theories found in literature. The results of experiments are considered to be the physical reality. Semi empirical theories based on these experiments are considered to be an attempt to describe the physical reality in a mathematical way. These semi empirical theories in general match the experiments on which they are based, but are also limited to the range of the different parameters as used for these experiments. Some theories have a more fundamental character and may be more generic as long as the starting points on which they are based apply. Observing the results of many experiments gives the reader the possibility to form his/her own impression of the processes involved in slurry transport.

Flow regimes are identified and theoretical models are developed for each main flow regime based on constant volumetric spatial concentration. The 5 main flow regimes are the fixed or stationary bed regime, the sliding bed regime, the heterogeneous regime or the sliding flow regime and the homogeneous regime. It is the opinion of the authors that the basic model should be derived for a situation where the amount of solids in the pipeline is known, the constant volumetric spatial concentration situation.

A new model for the Limit Deposit Velocity is derived, consisting of 5 particle size regions and a lower limit. Based on the Limit Deposit Velocity a (semi) fundamental relation is derived for the slip velocity. This slip velocity is required to determine constant volumetric transport concentration relations based on the constant volumetric spatial concentration relations. These relations also enable us to determine the bed height as a function of the line speed.

The concentration distribution in the pipe is based on the advection diffusion equation with a diffusivity related to the LDV.

Finally a method is given to determine relations for non-uniform sands based on the superposition principle. The last chapter is a manual on how to reproduce the Delft Head Loss & Limit Deposit Velocity model. The DHLLDV Framework is based on numerous experimental data from literature, considered to be the reality.

This book is supported by the website www.dhlldv.com containing many additional graphs and tables with

experimental data. The website also has spreadsheets and software implementing the model.

The name Delft in the title of the DHLLDV Framework is chosen because most of the modelling is carried out at the Delft University of Technology and in my home in Delft.

Another book by the author is: The Delft Sand, Clay & Rock Cutting Model.

Published by IOS Press, www.iospress.nl, in Open Access.

Modeling is an attempt to approach nature without

having the presumption to be nature.

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This book is dedicated to our wives

Thuy K.T. Miedema

and

Jennifer L. Ramsdell

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About the Author

Dr.ir. Sape A. Miedema (November 8th_{1955) obtained his}

M.Sc. degree in Mechanical Engineering with honors at the Delft University of Technology (DUT) in 1983. He obtained his Ph.D. degree on research into the basics of soil cutting in relation with ship motions, in 1987. From 1987 to 1992 he was Assistant Professor at the chair of Dredging Technology. In 1992 and 1993 he was a member of the management board of Mechanical Engineering & Marine Technology of the DUT. In 1992 he became Associate Professor at the DUT with the chair of Dredging Technology. From 1996 to 2001 he was appointed Head of Studies of Mechanical Engineering and Marine Technology at the DUT, but still remaining Associate Professor of Dredging Engineering. In 2005 he was appointed Head of Studies of the MSc program of Offshore & Dredging Engineering and he is also still Associate Professor of Dredging Engineering. In 2013 he was also appointed as Head of Studies of the MSc program Marine Technology of the DUT.

Dr.ir. S.A. Miedema teaches (or has taught) courses on soil mechanics and soil cutting, pumps and slurry transport, hopper sedimentation and erosion, mechatronics, applied thermodynamics related to energy, drive system design principles, mooring systems, hydromechanics and mathematics. He is (or has been) also teaching at Hohai University, Changzhou, China, at Cantho University, Cantho Vietnam, at Petrovietnam University, Baria, Vietnam and different dredging companies in the Netherlands and the USA.

His research focuses on the mathematical modeling of dredging systems like, cutter suction dredges, hopper dredges, clamshell dredges, backhoe dredges and trenchers. The fundamental part of the research focuses on the cutting processes of sand, clay and rock, sedimentation processes in Trailing Suction Hopper Dredges and the associated erosion processes. Lately the research focuses on hyperbaric rock cutting in relation with deep sea mining and on hydraulic transport of solids/liquid settling slurries.

About the Editor

Robert Ramsdell (23rd_{September 1964) obtained his BA}

degree in Mathematics at the University of California at Berkeley in 1985. Since 1989 he has worked for Great Lakes Dredge & Dock Company. Robert started as Field Engineer, eventually becoming a Project Engineer then Superintendent, working with Trailing Suction, Cutter Suction and Mechanical dredges on a variety of projects in the United States. From 1995 to 1996 Robert was the Project Engineer on the Øresund Link project in Denmark. In 1996 he joined the Great Lakes Dredge & Dock Production Department as a Production Engineer, becoming Production Engineering Manager in 2005. In the department Robert’s focus has been on developing methods and software for estimating dredge production, recruiting and training Engineers, and developing methods to analyze and improve dredge operations. A particular focus has been in modeling slurry transport for dredging estimating and production optimization.

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Acknowledgements

The authors want to thank:

Ron Derammelaere, Edward Wasp and Ramesh Gandhi, Ausenco PSI, for reviewing the chapter about the Wasp model.

Baha E. Abulnaga, director of Splitvane Engineers Inc., for reviewing the chapter about the Wilson 2 layer model, the heterogeneous model and the Wilson & Sellgren 4 component model.

Randy Gillies, Pipe Flow Technology Centre SRC, for reviewing the chapter about the Saskatchewan Research Council (SRC) model.

Deo Raj Kaushal, Indian Institute of Technology in Delhi, for reviewing the chapter about the Kaushal & Tomita model.

Vaclav Matousek, Czech Technical University in Prague, for reviewing the chapter about the Matousek model. Their contributions have been very valuable for having a correct reproduction of their models.

The authors also want to thank all the reviewers of our conference and journal papers for their effort. This has improved the quality of our work.

A special thanks to the 147th_{board of “Gezelschap Leeghwater” (the student association of Mechanical}

Engineering of the Delft University of Technology) for letting us use their name for the Double Logarithmic Elephant Leeghwater.

Recommendations

In this book, the author’s intention is to introduce the slurry transport in pipelines to the readers by describing the relevant phenomena both physically and mathematically through underlying theories and governing equations. It is a focused work presented by the author to the upcoming generation of researchers and practitioners, in particular. The special feature of the book is that in a chapter, a specific phenomenon is presented starting with a physical description followed by a derivation and ending with discussion and conclusions. All throughout the book, coherence in presentation is maintained.

As a concluding remark, this book can effectively be used as a guide on slurry transport in pipelines. Professor Subhasish Dey, Indian Institute of Technology Kharagpur, India

Open Access

This book is intentionally published in Open Access with the purpose to distribute it world-wide. In this digital era publishing hard-copies has downsides. Usually it’s expensive and new editions take years. Using Open Access creates the possibility of a fast world-wide distribution and the possibility to update the document regularly. This book is published on ResearchGate, the CEDA website, the WEDA website, the IADC website, at

www.dhlldv.com and others.

If you have comments, additions or other remarks about this book, please send an email to:

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In dredging, the hydraulic transport of solids is one of the most important processes. Since the 50’s many researchers have tried to create a physical mathematical model in order to predict the head losses in slurry transport. One can think of the models of Durand & Condolios (1952) & Durand (1953), Worster & Denny (1955), Newitt et al. (1955), Gibert (1960), Fuhrboter (1961), Jufin & Lopatin (1966), Zandi & Govatos (1967) & Zandi (1971), Turian & Yuan (1977), Doron et al. (1987) & Doron & Barnea (1993), Wilson et al. (1992) and Matousek (1997). Some models are based on phenomenological relations and thus result in semi empirical relations, other tried to create models based on physics, like the two and three layer models. It is however the question whether slurry transport can be modeled this way at all. Observations in our laboratory show a process which is often non-stationary with respect to time and space. Different physics occur depending on the line speed, particle diameter, concentration and pipe diameter. These physics are often named flow regimes; fixed bed, shearing bed, sliding bed, heterogeneous transport and (pseudo) homogeneous transport. It is also possible that more regimes occur at the same time, like, a fixed bed in the bottom layer with heterogeneous transport in the top layer. It is the observation of the author that researchers often focus on a detail and sub-optimize their model, which results in a model that can only be applied for the parameters used for their experiments.

1.2 Flow Regimes Literature.

Based on the specific gravity of particles with a magnitude of 2.65, Durand (1953) proposed to divide the flows of non-settling slurries in horizontal pipes into four flow regimes based on average particle size as follows:

1. Homogeneous suspensions for particles smaller than 40 μm (mesh 325)

2. Suspensions maintained by turbulence for particle sizes from 40 μm (mesh 325) to 0.15 mm (mesh 100) 3. Suspension with saltation for particle sizes between 0.15 mm (mesh 100) and 1.5 mm (mesh 11) 4. Saltation for particles greater than 1.5 mm (mesh 11)

Due to the interrelation between particle sizes and terminal and deposition velocities, the original classification proposed by Durand has been modified to four flow regimes based on the actual flow of particles and their size (Abulnaga, 2002).

1. Flow with a stationary bed

2. Flow with a moving bed and saltation (with or without suspension) 3. Heterogeneous

 Heterogeneous mixture with saltation and rolling  Heterogeneous mixture with all solids in suspension

4. Pseudo homogeneous and/or homogeneous mixtures with all solids in suspension

The four regimes of flow can be represented by a plot of the hydraulic gradient versus the average speed of the mixture as in Figure 1.2-1. The 4 transitional velocities are defined as:

• V1: velocity at or above which the bed in the lower half of the pipe is stationary. In the upper half of the pipe,

some solids may move by saltation or suspension. Below V1 there are no particles above the bed.

• V2: velocity at or above which the mixture flows as an asymmetric mixture with the coarser particles forming

a moving/saltating bed.

• V3: velocity at or above which all particles move as an asymmetric suspension and below which the solids

start to settle and form a moving bed.

• V4: velocity at or above which all solids move as an almost symmetric suspension.

Wilson (1992) developed a model, which will be discussed in detail later, for the incipient motion of granular

solids at V2, the transition between a stationary bed and a sliding bed. He assumed a hydrostatic pressure exerted

by the solids on the wall. Wilson also developed a model for heterogeneous transport with a V50, where 50% of

the solids are in a (moving/saltating) bed and 50% in suspension. This percentage is named the stratification ratio.

The transitional velocity V3 is extremely important because it is the speed at which the hydraulic gradient is at a

minimum. Although there is evidence that solids start to settle at lower line speeds in complex mixtures, operators and engineers often refer to this transitional velocity as the speed of deposition or critical velocity. Figure 1.2-3 shows the 4 regimes and the velocity and concentration profiles. At very high line speeds the pressure drop will reach an equivalent liquid curve asymptotically. Whether or not this occurs at practical line speeds depends on the particle diameter, pipe diameter and the concentration. For large particle diameters and concentrations it may seem like the pressure drop reaches the water curve asymptotically, but at higher line speeds the pressure drop will increase again up to an equivalent liquid model. Whether or not this equivalent liquid model contains the mixture density instead of the water density, or some value in between is still the question.

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Page 2 of 782 TOC Copyright © Dr.ir. S.A. Miedema Figure 1.2-1: The 4 regimes and transitional velocities (Abulnaga, 2002), Dp=0.15 m, d50=2 mm, Cvt=0.2. Figure 1.2-1 gives the impression that the 4 flow regimes will always occur sequentially. Starting from a line speed zero and increasing the line speed, first the fixed or stationary bed will occur without suspension, at a line speed

V1 part of the bed starts to erode and particles will be in suspension, at a line speed V2 the remaining bed will start

to slide while the erosion increases with the line speed, at a line speed V3 the whole bed is eroded and the heterogeneous regime starts and finally at a line speed V4 the heterogeneous regime transits to the (pseudo) homogeneous regime. In reality not all the regimes have to occur, depending on the particle size, the pipe diameter and other governing parameters.

Figure 1.2-2: Flow regimes according to Newitt et al. (1955).

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0 1 2 3 4 5 6 7 8 9 10 Hy dra ulic gra die nt im (m w a ter/ m pipe )

Line speed v_ls(m/sec)

Hydraulic gradient i

_m

versus Line speed v

_ls

Water Slurry

V1: Start Fixed Bed With Suspension V2: Start Moving Bed/Saltating Bed V3: Start Heterogeneous Transport V4: Start (Pseudo) Homogeneous Transport V1 V3 M ov ing B ed /S alt ati ng B ed V1 V2 V4 S tati on ary B ed Wi tho ut S us pe ns ion S tati on ary B ed W it h S us pe ns ion H etero g en eo u s T ran sp o rt P se ud o Ho moge ne ou s T rans po rt H o mo g en eo u s T ran sp o rt © S.A.M. 0.001 0.010 0.100 1.000 10.000 100.000 0 1 2 3 4 5 6 7 8 9 10 P a rtic le Dia me ter d (mm)

Line Speed vls(m/sec)

Flow Regimes according to Newitt et al. (1955) & Durand & Condolios (1952)

Limit of Stationary Bed Dp=1 inch

Limit of Stationary Bed Dp=6 inch

Limit of Moving Bed, All Dp Heterogeneous vs Homogeneous Dp=1 inch Heterogeneous vs Homogeneous Dp=6 inch © S.A.M. Flow w it h a S tati on ary B ed Dp = 1 inc h Dp = 6 inc h Dp=1 inch Dp=6 inch Flow with a Moving Bed

Flow as a Homogeneous Suspension Flow as a Heterogeneous

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Copyright © Dr.ir. S.A. Miedema TOC Page 3 of 782 Figure 1.2-2 shows the regimes according to Newitt et al. (1955). From this figure it is clear that not all regimes have to occur and that the transition velocities depend on the particle and the pipe diameter. The influence of the volumetric concentration is not present in this graph. Figure 1.2-3 shows the flow regimes as used by Matousek (2004) also showing velocity and concentration distributions.

Figure 1.2-3: Different mixture transport regimes.

1.3 The Parable of Blind Men and an Elephant.

Wilson et al. (1992), (1997) and (2006) refer to the old parable of 6 blind men, who always wanted to know what an elephant looks like. Each man could touch a different part of the elephant, but only one part. So one man touched the tusk, others the legs, the belly, the tail, the ear and the trunk. The blind man who feels a leg says the elephant is like a pillar; the one who feels the tail says the elephant is like a rope; the one who feels the trunk says the elephant is like a tree branch; the one who feels the ear says the elephant is like a hand fan; the one who feels the belly says the elephant is like a wall; and the one who feels the tusk says the elephant is like a solid pipe. They then compare notes and learn they are in complete disagreement about what the elephant looks like. When a sighted man walks by and sees the entire elephant all at once, they also learn they are blind. The sighted man explains to them: All of you are right. The reason every one of you is telling a different story is because each one of you touched a different part of the elephant. So actually the elephant has all the features you mentioned.

The story of the blind men and an elephant originated in the Indian subcontinent from where it has widely diffused. It has been used to illustrate a range of truths and fallacies; broadly, the parable implies that one's subjective experience can be true, but that such experience is inherently limited by its failure to account for other truths or a totality of truth. At various times the parable has provided insight into the relativism, opaqueness or inexpressible

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nature of truth, the behavior of experts in fields where there is a deficit or inaccessibility of information, the need for communication, and respect for different perspectives (source Wikipedia).

Figure 1.3-1: Flow regimes and the Double Logarithmic Elephant “Leeghwater”.

Figure 1.3-1 shows a comparison between the parable of the elephant and slurry flow. Slurry transport also has many truths, points of view. Experiments can be carried out with small versus large pipes, small versus large particles, low versus high concentrations, low versus high line speeds, low versus high particle diameter versus pipe diameter ratios, laminar versus turbulent flow, Newtonian versus non Newtonian liquids, low versus high solid densities, etc. Depending on the parameters used, experiments are carried out in different flow regimes, or maybe at the interface between flow regimes, resulting in different conclusions.

Wilson et al. (1992), (1997) and (2006) show with this parable that the research of slurry flow often focusses on different parts or aspects of the process, but not many times it will give an overview of the whole process. The starting point is that every researcher tells the truth, based on his/her observations. Combining these truths gives an impression of the aggregated truth, which is still not the whole truth. The 6 men for example cannot look inside the elephant, only touch the outside. The internal structure of slurry flow may however be very important to understand the slurry flow behavior. The 6 men cannot access the memory of the elephant, which is supposed to be very good. In long pipelines the overall behavior of the slurry flow does depend on the history, so the memory function is also important. The Double Logarithmic Elephant is named after the student association of Mechanical Engineering of the Delft University of Technology, Leeghwater, using the elephant as their symbol. Leeghwater stands for strength, precision and of course hydraulic transport through the proboscis.

1.4 The Delft Head Loss & Limit Deposit Velocity Framework.

In the following chapters the different models from literature will be analyzed, leading to a new integrated model based on a new classification of the flow regimes. This new model is named the Delft Head Loss & Limit Deposit Velocity Framework (DHLLDV Framework). The Framework is integrated in a way that all flow regimes are described in a consistent way showing the transition velocities. The model is validated by many experiments from literature and experiments carried out in the Delft University Dredging Engineering Laboratory for particles ranging from 0.05 to 45 mm, pipe diameters ranging from 0.0254 to 0.9 m and relative submerged densities ranging

from 0.24 to 4 ton/m3_{. The model does not just give hydraulic gradient relations, but also Limit Deposit Velocity}

relations, slip ratio relations (the relation between the volumetric spatial concentration and the volumetric delivered concentration), bed height relations and a concentration distribution model. The Framework also gives a tool to determine the influence of the grading of the sand or gravel. The starting point of the model is a uniform sand or gravel and a constant volumetric spatial concentration. Based on the hydraulic gradient and slip ratio relations, the volumetric delivered concentration hydraulic gradient relations are derived. The latter is very important for practical applications.

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1.5 Approach of this book.

The book covers horizontal transport of settling slurries (Newtonian slurries). Pipelines under an angle with the horizontal and non-settling (non-Newtonian) slurries are not covered.

The book has the following approach:

1. Chapter 1 explains the context of slurry flow, based on flow regimes as identified in literature.

2. Chapter 2 gives definitions of the dimensionless numbers and other important parameters as used in the book. Definitions are the language of engineers and scientists and are thus essential for the understanding.

3. Chapter 3 deals with homogeneous Newtonian liquid flow through horizontal circular pipes. Equations and graphs are given to determine the Darcy Weisbach friction factor. The Swamee Jain (1976) equation for the Darcy Weisbach (Moody (1944)) friction factor is used in this book. Also the influence of the concentration of very fine particles on the liquid properties is discussed.

4. Chapter 4 explains the terminal settling velocity of particles, including hindered settling. In the theory derived, the Zanke (1977) equation for the settling velocity is used and the Richardson & Zaki (1954) approach for hindered settling is applied.

5. Chapter 5 shows the basics of the initiation of motion of particles and shells, which is important to understand the behavior of the interface between a bed and the liquid flow above the bed, especially for the stationary and sliding bed regimes. Initiation of motion is the start of sediment motion, but at higher flow velocities also erosion and/or sediment transport will occur. The basics of sediment transport as bed load and suspended load are discussed for open channel flow and pipe flow.

6. Chapter 6 gives an overview of the historical developments of models to predict head losses in slurry flow. The overview starts with the early history, followed by empirical and semi empirical models. The models are given, analyzed and discussed and issues of the models are addressed. The models for the Limit Deposit Velocity (LDV) are discussed, analyzed and compared. Conclusions are drawn regarding the behavior of the LDV related to the solids, liquid and flow parameters. A number of 2 layer models (2LM) and 3 layer models (3LM) based on physics are given and analyzed, as well as other physical models.

7. Chapter 7 describes the new Delft Head Loss & Limit Deposit Velocity (DHLLDV) Framework. The DHLLDV Framework is based on uniform sands or gravels and constant spatial volumetric concentration. This chapter starts with an overview of 8 flow regimes and 6 scenarios. The new models for the main flow regimes, the stationary bed regime without sheet flow and with sheet flow, the sliding bed regime, the heterogeneous regime, the homogeneous regime and the sliding flow regime, are derived and discussed. A new model for the Limit Deposit Velocity is derived, consisting of 5 particle size regions and a lower limit. Based on the LDV a method is shown to construct slip velocity or slip ratio curves from zero line speed to the LDV and above. Based on the slip ratio, the constant delivered volumetric concentration curves can be constructed. Knowing the slip ratio, the bed height for line speeds below the LDV can be determined. New equations are derived for this. The transition from the heterogeneous regime to the homogeneous regime requires special attention. First of all, this transition line speed gives a good indication of the operational line speed and allows to compare the DHLLDV Framework with many models from literature. Secondly the transition is not sharp, but depends on 3 velocities. The line speed where a particle still fits in the viscous sub layer, the transition line speed heterogeneous-homogeneous and the line speed where the lift force on a particle equals the submerged weight of the particle. Finally the grading of the Particle Size Distribution (PSD) is discussed. A method is given to construct resulting head loss, slip velocity and bed height curves for graded sands and gravels.

8. Chapter 8 summarizes the DHLLDV Framework. The essential equations are given, with reference to the original equations, to reproduce the DHLLDV Framework, accompanied with flow charts.

9. In chapter 9 the DHLLDV Framework is compared with other models from literature.

10. Chapter 10 shows how to apply the DHLLDV Framework on the hydraulic transport of a cutter suction dredge. 11. Chapter 11 gives the journal and conference publications of the authors on which this book is based.

The DHLLDV Framework models have been verified and validated with numerous experimental data.

The results of experiments and calculations are shown in standard graphs showing Hydraulic Gradient versus

Line Speed i(vls), the Relative Excess Hydraulic Gradient versus the Line Speed Erhg(vls) and the Relative

Excess Hydraulic Gradient versus the Liquid Hydraulic Gradient (the clean water resistance) Erhg(il). The advantage of the Erhg(il) graph is that this type of graph is almost independent of the values of the spatial

concentration Cvs and relative submerged density Rsd. The advantage of the im(vls) graph is that is clearly shows

head losses versus flow and thus gives an indication of the required power and specific energy, combined with pump graphs. Most experimental data is shown in the Relative Excess Hydraulic Gradient versus the Liquid

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1.6 Nomenclature.

Cv Volumetric concentration -

Cvs Volumetric spatial concentration -

Cvt Volumetric transport/delivered concentration -

d Particle/grain diameter m

d50 50% passing particle diameter m

Dp Pipe diameter m

Erhg Relative Excess Hydraulic Gradient -

i, il, iw Hydraulic gradient liquid m.w.c./m

im Hydraulic gradient mixture m.w.c./m

LSDV Limit of Stationary Deposit Velocity m/s

LDV Limit Deposit Velocity m/s

m.w.c. Meters water column, pressure expressed in m.w.c.(10 m.w.c.=100 kPa=1 bar) m

Rsd Relative submerged density -

v Line speed m/s

vls Line speed m/s

V1 Transition fixed bed without suspension – fixed bed with suspension m/s

V2 Transition fixed bed with suspension – sliding bed with suspension m/s

V3 Transition sliding bed with suspension – heterogeneous transport m/s

V4 Transition heterogeneous transport – (pseudo) homogeneous transport m/s

V50 Velocity with 50% stratification according to Wilson m/s

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Chapter 2: Dimensionless Numbers & Other Parameters.

A number of dimensionless numbers and other important parameters are be used in this book. This short chapter gives an overview of these dimensionless numbers and parameters.

2.1 Definitions.

2.1.1 The Friction Velocity or Shear Velocity u

*

.

The term friction velocity comes from the fact that √(τ12/ρl) has the same unit as velocity and it has something to

do with the friction force. The bottom shear stress τ12 is often represented by friction velocity u*, defined by:

1 2 l * l s l u v 8       (2.1-1)

2.1.2 The Thickness of the Viscous Sub Layer δ

v

.

Very close to the pipe wall the flow is laminar in the so called viscous sub layer. The thickness of the viscous sub layer is: l v * v * v l 1 1 .6 u u 1 1 .6            (2.1-2)

2.2 Dimensionless Numbers.

2.2.1 The Reynolds Number Re.

In fluid mechanics, the Reynolds number (Re) is a dimensionless number that gives a measure of the ratio of inertial (resistant to change or motion) forces to viscous (heavy and gluey) forces and consequently quantifies the relative importance of these two types of forces for given flow conditions. (The term inertial forces, which characterize how much a particular liquid resists any change in motion, are not to be confused with inertial forces defined in the classical way.)

The concept was introduced by George Gabriel Stokes in 1851 but the Reynolds number is named after Osborne Reynolds (1842–1912), who popularized its use in 1883.

Reynolds numbers frequently arise when performing dimensional analysis of liquid dynamics problems, and as such can be used to determine dynamic similitude between different experimental cases.

They are also used to characterize different flow regimes, such as laminar or turbulent flow: laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant liquid motion; turbulent flow occurs at high Reynolds numbers and is dominated by inertial forces, which tend to produce chaotic eddies, vortices and other flow instabilities.

The gradient of the velocity dv/dx is proportional to the velocity v divided by a characteristic length scale L.

Similarly, the second derivative of the velocity d2_v/dx2_{is proportional to the velocity v divided by the square of}

the characteristic length scale L.

2 l 2 2 2 l l l ₂ d v v I n e r t i a l f o r c e s _{d x} d v v d v v v L R e w i t h : V i s c o u s f o r c e s _{d v} d x L _{d x} _L d x               _(2.2-1)

(35)

The Reynolds number is a dimensionless number. High values of the parameter (on the order of 10 million) indicate that viscous forces are small and the flow is essentially inviscid. The Euler equations can then be used to model the flow. Low values of the parameter (on the order of 1 hundred) indicate that viscous forces must be considered.

2.2.2 The Froude Number Fr.

The Froude number (Fr) is a dimensionless number defined as the ratio of a characteristic velocity to a gravitational wave velocity. It may equivalently be defined as the ratio of a body's inertia to gravitational forces. In fluid mechanics, the Froude number is used to determine the resistance of a partially submerged object moving through water, and permits the comparison of objects of different sizes. Named after William Froude (1810-1879), the Froude number is based on the speed–length ratio as defined by him.

C h a r a c t e r is t ic v e lo c it y v

F r

G r a v it a t io n a l w a v e v e lo c it y g L

 

 (2.2-2)

Or the ratio between the inertial force and the gravitational force squared according to:

2 l l d v v I n e r t i a l f o r c e _{d x} v F r G r a v i t a t i o n a l f o r c e g g L          (2.2-3)

The gradient of the velocity dv/dx is proportional to the velocity v divided by a length scale L.

Or the ratio between the centripetal force on an object and the gravitational force, giving the square of the right hand term of equation (2.2-2):

2 2 C e n t r i p e t a l f o r c e m v / L v F r G r a v i t a t i o n a l f o r c e m g g L       (2.2-4)

2.2.3 The Richardson Number Ri.

The Richardson number Ri is named after Lewis Fry Richardson (1881-1953). It is the dimensionless number that expresses the ratio of the buoyancy term to the flow gradient term.

s d 2 g L R b u o y a n c y t e r m R i f l o w g r a d i e n t t e r m v      _{ } _   (2.2-5)

The Richardson number, or one of several variants, is of practical importance in weather forecasting and in investigating density and turbidity currents in oceans, lakes and reservoirs.

2.2.4 The Archimedes Number Ar.

The Archimedes number (Ar) (not to be confused with Archimedes constant, π), named after the ancient Greek scientist Archimedes is used to determine the motion of liquids due to density differences. It is a dimensionless number defined as the ratio of gravitational forces to viscous forces. When analyzing potentially mixed convection of a liquid, the Archimedes number parameterizes the relative strength of free and forced convection. When Ar >> 1 natural convection dominates, i.e. less dense bodies rise and denser bodies sink, and when Ar << 1 forced convection dominates. 3 s d 2 l g L R G r a v i t a t i o n a l f o r c e s A r V i s c o u s f o r c e s      (2.2-6)

Slurry Transport

Slurry Transport

Fundamentals, a historical overview and the Delft Head Loss & Limit Deposit Velocity

Framework

Fundamentals, A Historical Overview &

The

Delft Head

Loss &

Limit Deposit Velocity Framework

By

Sape A. Miedema

Edited by

Robert C. Ramsdell

DHLLDV Flow Regime Diagram

Slurry Transport

Fundamentals, A Historical Overview &

The

Delft Head

Loss &

Limit Deposit Velocity Framework

1

st

Edition

By

Sape A. Miedema

Edited by

Preface

Modeling is an attempt to approach nature without

having the presumption to be nature.

This book is dedicated to our wives

Thuy K.T. Miedema

and

Jennifer L. Ramsdell

About the Author

About the Editor

Acknowledgements

Recommendations

Open Access

Table of Contents

Chapter 1: Introduction.

1.1 Introduction.

1.2 Flow Regimes Literature.

Hydraulic gradient i

versus Line speed v

1.3 The Parable of Blind Men and an Elephant.

1.4 The Delft Head Loss & Limit Deposit Velocity Framework.

1.5 Approach of this book.

1.6 Nomenclature.

Chapter 2: Dimensionless Numbers & Other Parameters.

2.1 Definitions.

2.1.1 The Friction Velocity or Shear Velocity u

.

2.1.2 The Thickness of the Viscous Sub Layer δ

.

2.2 Dimensionless Numbers.

2.2.1 The Reynolds Number Re.

2.2.2 The Froude Number Fr.

2.2.3 The Richardson Number Ri.

2.2.4 The Archimedes Number Ar.