# 7.2: Flow Regimes and Scenario’s


## 7.2.1 Introduction

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. We can think of the models of Durand, Condolios, Gibert, Worster, Zandi & Govatos, Jufin Lopatin, Fuhrboter, Newitt, Doron, Wilson, Matousek, Turian & Yuan and the SRC model. Some models are based on phenomenological relations and thus result in semi empirical relations, others 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 with and without sheet flow or suspension, sliding bed, heterogeneous transport, (pseudo) homogeneous transport and sliding flow. 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 authors 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. At high line speeds the volumetric spatial concentration (volume based) and the volumetric transport concentration (volume flux based) are almost equal, because all the particles are in suspension with a small slip related to the carrier liquid velocity. The difference of the head loss between the two concentrations will be within the margin of the scatter of the experiments. At low line speeds however, there may be a sliding or fixed bed, resulting in a big difference between the two concentrations and thus between laboratory and real life situations.

This chapter describes 8 flow regimes and 6 possible scenarios.

The flow regimes for constant spatial volumetric concentration Cvs are, from line speed zero with increasing line speed:

1: Fixed bed without suspension (fine particles) or sheet flow (coarse particles).

2: Fixed bed with suspension (fine particles) or sheet flow (coarse particles).

3: Fixed bed with suspension (fine particles) or sliding bed with sheet flow (coarse particles).

For fine to coarse particles d/Dp<0.015:

5: Heterogeneous transport CvsCvt.

5/6: Pseudo homogeneous transport, CvsCvt.

6: Homogeneous transport, CvsCvt.

For very coarse particles d/Dp>0.015:

7: Sliding flow.

The flow regimes for constant delivered volumetric concentration Cvt are, from line speed zero with increasing line speed:

8: Fixed bed with suspension (fine particles) or sheet flow (coarse particles).

4: Fixed bed with suspension (fine particles) or sliding bed with sheet flow (coarse particles).

For fine to coarse particles d/Dp<0.015:

5: Heterogeneous transport CvsCvt.

5/6: Pseudo homogeneous transport, CvsCvt.

6: Homogeneous transport, CvsCvt.

For very coarse particles d/Dp>0.015:

7: Sliding flow.

3 scenarios are based on a constant volumetric spatial concentration (usually in a laboratory) and 3 scenarios are based on a constant volumetric transport concentration (usually in real life). The flow regimes and scenarios are explained and examples of experiments are given. Based on the experimental evidence, one can conclude that the approach followed in this book gives a good resemblance with the reality.

## 7.2.2 Concentration Considerations

Based on an analysis of many experiments from literature, 8 flow regimes and 6 scenarios can be distinguished, which will be discussed in the next chapters. In order to understand these 8 flow regimes and 6 scenarios, the difference between the spatial volumetric concentration Cvs and the volumetric transport (delivered) concentration Cvt will first be discussed. In hydraulic transport, 2 definitions of the concentration are often used. Contractors are interested in the delivered volumetric concentration, also named the volumetric transport concentration Cvt. Cvt is defined as the ratio between the volume flow of solids and the volume flow of the mixture. In general one can say that the average solids velocity will be smaller than the average mixture velocity. The difference is called the slip velocity. The spatial volumetric concentration Cvs is defined as the volume of solids divided by the volume of the mixture containing these solids. So the spatial volumetric concentration is based on a volume ratio, while the delivered volumetric concentration is based on a volume flux ratio. Concentration (Cvs) is usually derived from density meter or U-loop readings as:

$\ \mathrm{C}_{\mathrm{v s}}=\frac{\rho_{\mathrm{m}}-\rho_{\mathrm{l}}}{\rho_{\mathrm{s}}-\rho_{\mathrm{l}}}(\text { density meter }) \quad\text{ or }\quad \mathrm{C}_{\mathrm{v t}}=\frac{\rho_{\mathrm{m}}-\rho_{\mathrm{l}}}{\rho_{\mathrm{s}}-\rho_{\mathrm{l}}}(\mathrm{U}-\mathrm{l o o p})$

A radioactive density meter reads a density of the entire mass of slurry in the pipe, and thus is best suited to measuring Cvs. In addition the placement of the meter in horizontal or vertical pipe, can affect the readings. A U- tube device reads the delivered density and is thus best suited to measuring Cvt. In a closed loop system, we will know the volume of the closed loop and amount of material added, and thus can calculate Cvs directly, but not necessarily Cvt. The volumetric delivered (transport) concentration is:

$\ \mathrm{C_{v t}=\frac{\dot{V}_{s}}{\dot{V}_{m}}=\frac{Q_{s}}{Q_{m}}=\frac{v_{s} \cdot A_{s}}{v_{m} \cdot A_{p}}=\frac{v_{s} \cdot C_{v s} \cdot A_{p}}{v_{m} \cdot A_{p}}=C_{v s} \cdot \frac{v_{s}}{v_{m}}}$

With vs the average velocity of the solids and vm the average velocity of the mixture, also called the line speed vls.

The volumetric spatial concentration is based on the volume ratio solids/mixture according to:

$\ \mathrm{C}_{\mathrm{v s}}=\frac{\mathrm{V}_{\mathrm{s}}}{\mathrm{V}_{\mathrm{m}}}$

The slip velocity vsl is defined as the difference between the velocity of the mixture vm and the velocity of the solids vs:

$\ \mathrm{v}_{\mathrm{sl}}=\mathrm{v}_{\mathrm{m}}-\mathrm{v}_{\mathrm{s}}=\mathrm{v}_{\mathrm{m}} \cdot\left(\mathrm{1}-\frac{\mathrm{v}_{\mathrm{s}}}{\mathrm{v}_{\mathrm{m}}}\right)=\mathrm{v}_{\mathrm{m}} \cdot\left(\mathrm{1}-\frac{\mathrm{C}_{\mathrm{vt}}}{\mathrm{C}_{\mathrm{v} \mathrm{s}}}\right)$

Because of the fact that most experiments are carried out in a closed loop system, the concentration might be determined by the ratio of the volume of solids divided by the volume of the closed loop system.

$\ \mathrm{C}_{\mathrm{v s}}=\frac{\mathrm{V}_{\mathrm{s}}}{\mathrm{V}_{\mathrm{c l}}}$

This means that the concentration of solids in the liquid above the bed will be much smaller once a bed is formed. Now assume a bed with a porosity n of about 40% containing 50% of the solids, matching the v50 of Wilson (1997). This gives for the total bed volume in the closed loop system:

$\ \mathrm{V}_{\mathrm{b}}=\frac{\mathrm{V}_{\mathrm{s}}}{2} \cdot \frac{\mathrm{1}}{\mathrm{1 - n}}=\frac{\mathrm{C}_{\mathrm{v s}} \cdot \mathrm{V}_{\mathrm{c l}}}{\mathrm{2} \cdot(\mathrm{1}-\mathrm{n})}$

The volume of solids in suspension is the same, so the volume of the solids in the liquid is:

$\ \mathrm{V}_{\mathrm{s}, \mathrm{s}=} \frac{\mathrm{V}_{\mathrm{s}}}{\mathrm{2}}$

The volume of the mixture in suspension above the bed equals the closed loop volume minus the bed volume.

$\ \mathrm{V}_{\mathrm{m}, \mathrm{s}}=\mathrm{V}_{\mathrm{c l}}-\mathrm{V}_{\mathrm{b}}=\mathrm{V}_{\mathrm{c l}}-\frac{\mathrm{C}_{\mathrm{v s}} \cdot \mathrm{V}_{\mathrm{c l}}}{2 \cdot(1-\mathrm{n})}=\mathrm{V}_{\mathrm{c l}} \cdot\left(\frac{2 \cdot(1-\mathrm{n})-\mathrm{C}_{\mathrm{v} \mathrm{s}}}{\mathrm{2} \cdot(\mathrm{1}-\mathrm{n})}\right)$

The concentration of the solids in suspension is the volume of these solids, divided by the volume of the closed loop system minus the volume of the bed.

$\ \mathrm{C}_{\mathrm{vs}, \mathrm{s}}=\frac{\mathrm{V}_{\mathrm{s}}}{2 \cdot \mathrm{V}_{\mathrm{m}, \mathrm{s}}}=\frac{\mathrm{C}_{\mathrm{vs}}}{2 \cdot\left(\frac{2 \cdot(1-\mathrm{n})-\mathrm{C}_{\mathrm{vs}}}{2 \cdot(1-\mathrm{n})}\right)}=\frac{(1-\mathrm{n}) \cdot \mathrm{C}_{\mathrm{vs}}}{2 \cdot(1-\mathrm{n})-\mathrm{C}_{\mathrm{vs}}} \approx \frac{0.6 \cdot \mathrm{C}_{\mathrm{vs}}}{1.2-\mathrm{C}_{\mathrm{vs}}}=\frac{\mathrm{C}_{\mathrm{vs}}}{2-1.66 \cdot \mathrm{C}_{\mathrm{vs}}}$

Of course the closed loop will not consist of just horizontal parts where a bed may occur, but the above example is just meant to give an indication.

This implies that at low spatial volumetric concentrations Cvs, the concentration in the suspension phase, the heterogeneous transport phase, is 50% of the total volumetric concentration. At a high concentration of Cvs=0.3, the concentration of the heterogeneous phase is still reduced to 0.2. At a concentration of Cvs=0.6, the above equation results in a concentration of 0.6, which makes sense, since this is solid sand and there is no suspension anymore. When experiments are carried out it should be clear which concentration is used. Is it the concentration based on the volume of the closed loop system, giving some constant volumetric spatial concentration? Is it the concentration based on radio active density meters in the pipe section where also the hydraulic gradient is measured, resulting in a spatial volumetric concentration? Or is the concentration measured with a U-tube resulting in a volumetric transport concentration.

Now in a real life production situation there is not a closed loop system, but an open system. There is not a fixed amount of solids in the pipeline, which can be divided in a part in a bed and the rest in suspension. Instead, the supply at the suction mouth can vary from water to twice or more that the delivered concentration. In a stable situation, the production that enters the system is equal to the production that leaves the system. The concentration is determined at the suction mouth and although there may be a bed in part of the pipeline, this does not change the transport concentration, it just increases the line speed and concentration above the bed compared with a pipeline without a bed, due to the conservation of volume in the pipeline. The conclusion of the above considerations is that for a good interpretation of the results of experiments, the method of determining the concentration should be known. It is also important how the results are presented. Graf & Robinson (1970) for example, present their results based on a constant amount of solids in their closed loop system, while Doron & Barnea (1987) connect data points with constant volumetric transport concentration. The presentations of the results are different, while the physics are the same.

## 7.2.3 The 8 Flow Regimes Identified

In literature different flow patterns or flow regimes are distinguished. Durand & Condolios (1952) distinguish 4 regimes, based on the particle size. Abulnaga (2002) also distinguished 4 regimes based on the actual flow of particles and their size. Matousek (2004) in his lecture notes distinguishes 6 flow regimes. Here we will consider 8 flow regimes and 6 scenarios for laboratory and real life conditions. These are (Figure 7.2-1 gives some definitions of fully stratified flow in a pipe):

## 7.2.4 The 6 Scenario’s Identified

In pipes with small diameters the hydraulic gradient will be relatively high, resulting in relatively high hydraulic gradients when transporting a mixture. This results in hydraulic gradients approaching the hydraulic gradient required to create a sliding bed. In pipes with large diameters the hydraulic gradient will be relatively small, also resulting in relatively small hydraulic gradients when transporting a mixture. This results in hydraulic gradients too small compared with the hydraulic gradient required to create a sliding bed.

From the 8 flow regimes, 6 scenarios can be constructed, where a scenario does not have to contain all 8 flow regimes. A scenario describes the flow regime behavior when the line speed increases from zero to a certain maximum. This maximum is arbitrary but should be related to practical line speeds. So this maximum increases with increasing pipe diameter.

The 8 flow regimes and 6 scenarios are shown in Figure 7.2-2, Figure 7.2-3, Figure 7.2-4, Figure 7.2-5, Figure 7.2-6 and Figure 7.2-7. Figure 7.2-2 and Figure 7.2-3 show the scenario’s L1 for laboratory conditions and R1 for real life conditions for fine sands. Figure 7.2-4 and Figure 7.2-5 show the scenario’s L2 for laboratory conditions and R2 for real life conditions for coarse sands. Figure 7.2-6 and Figure 7.2-7 show the scenario’s L3 for laboratory conditions and R3 for real life conditions for gravels.

The difference between laboratory conditions and real life conditions can be found at low line speeds where the volumetric spatial Cvs and transport Cvt concentrations differ substantially due to slip. At higher line speeds with heterogeneous and (pseudo) homogeneous transport it is assumed that the slip velocity vsl is small compared to the line speed vls.

Most of the graphs are made for sands and gravels with water as the carrier liquid, however the modelling is also valid for other solids and other liquids.

### 7.2.4.1 Scenarios L1 & R1

 Scenario L1 (the red solid line) Table 7.2-3: Indication of occurrence of L1. Starting at a line speed vls=0, there will be a stationary (fixed) bed (1). When the line speed is increased, there will not be erosion until the Shields parameter is high enough above the Shields curve. Increasing the line speed further will result in erosion and suspension or sheet flow of the particles (2). The upwards directed solid red curve becomes steeper. At a certain line speed, the macroscopic behavior will have a transition from fixed bed to heterogeneous (3). This also means that the excess pressure losses go from shear stress dominated to collision dominated. Increasing the line speed further results in heterogeneous transport (5). The solid red curve is downwards directed. The Limit Deposit Velocity is somewhere in the heterogeneous flow regime. Increasing the line speed further, results in a transition region between heterogeneous transport and (pseudo) homogeneous transport (5/6). At very high line speeds, the regime will be the homogeneous regime (6). Whether this regime will be reached with practical line speeds depends completely on the combination of the parameters involved. The solid red curve is upwards directed again. Scenario R1 (the green dashed line) Table 7.2-4: Indication of occurrence of R1. Starting at a line speed vls=0, there will be equilibrium between erosion and deposition, resulting in a certain bed height. Above the bed there will be heterogeneous transport or suspension. At very low line speeds, the hydraulic gradient is so high that a sliding bed may occur (8). At these low line speeds, it is also possible that only part of the bed is siding resulting in sliding friction that is not fully mobilized. The green dashed curve is slightly downwards directed. The height of the green dashed line depends on the Cvt. A smaller Cvt gives a higher green dashed line, because a smaller Cvt will have more slip. The red solid line (constant Cvs) is hardly influenced by the spatial concentration. At higher line speeds, the bed has a higher velocity and starts vaporizing, the hydraulic gradient drops, resulting in a transition towards heterogeneous behavior (4) and at higher line speeds full heterogeneous behavior (5). The dashed green line is downwards directed. The Limit Deposit Velocity is somewhere in the heterogeneous flow regime. Increasing the line speed further, results in a transition region between heterogeneous transport and (pseudo) homogeneous transport (5/6). At very high line speeds, the regime will be the homogeneous regime (6). Whether this regime will be reached with practical line speeds depends completely on the combination of the parameters involved. The green dashed line is upwards directed again. The solid red line and the dashed green line coincide.

### 7.2.4.2 Scenarios L2 & R2

 Scenario L2 (the red solid line) Table 7.2-6: Indication of occurrence of L2. Starting at a line speed vls=0, there will be a stationary (fixed) bed (1). When the line speed is increased, there will not be erosion until the Shields parameter is high enough above the Shields curve. Increasing the line speed further will result in erosion and suspension or sheet flow of the particles (2). The upwards directed solid red curve becomes steeper. At a certain line speed, the hydraulic gradient is high enough to make the bed to start sliding (3). Increasing the line speed further will result in an increase of the velocity of the bed and an increase of the erosion. The relative excess hydraulic gradient remains constant, because the weight of the suspension and the bed is a constant, resulting in an almost constant friction force. Increasing the line speed further will give a transition to heterogeneous transport (5). The solid red line is first horizontal and then downwards directed. The Limit Deposit Velocity is somewhere at the right of the sliding bed regime. Increasing the line speed further, results in a transition region between heterogeneous transport and (pseudo) homogeneous transport (5/6). At very high line speeds, the regime will be the homogeneous regime (6). Whether this regime will be reached with practical line speeds depends completely on the combination of the parameters involved. The solid red curve is upwards directed again. Scenario R2 (the green dashed line) Table 7.2-7: Indication of occurrence of R2. Starting at a line speed vls=0, there will be equilibrium between erosion and deposition, resulting in a certain bed height. Above the bed there will be heterogeneous transport or suspension. At very low line speeds, the hydraulic gradient is so high that a sliding bed may occur (8). At these low line speeds, it is also possible that only part of the bed is siding resulting in sliding friction that is not fully mobilized. The green dashed curve is slightly downwards directed. The height of the green dashed line depends on the Cvt. A smaller Cvt gives a higher green dashed line, because a smaller Cvt will have more slip. The red solid line (constant Cvs) is hardly influenced by the spatial concentration. At higher line speeds, the bed has a higher velocity and starts vaporizing, the hydraulic gradient drops, resulting in a transition towards heterogeneous behavior (4) and at higher line speeds full heterogeneous behavior (5). The dashed green line is downwards directed. The Limit Deposit Velocity is somewhere at the right of the sliding bed regime. Increasing the line speed further, results in a transition region between heterogeneous transport and (pseudo) homogeneous transport (5/6). At very high line speeds, the regime will be the homogeneous regime (6). Whether this regime will be reached with practical line speeds depends completely on the combination of the parameters involved. The solid red curve is upwards directed again.

### 7.2.4.3 Scenarios L3 & R3

 Scenario L3 (the red solid line) Table 7.2-9: Indication of occurrence of L3. Starting at a line speed vls=0, there will be a stationary (fixed) bed (1). When the line speed is increased, there will not be erosion until the Shields parameter is high enough above the Shields curve. Increasing the line speed further will result in erosion and suspension or sheet flow of the particles (2). The upwards directed solid red curve becomes steeper. At a certain line speed, the hydraulic gradient is high enough to make the bed to start sliding (3). Increasing the line speed further will result in an increase of the velocity of the bed and an increase of the erosion. The relative excess hydraulic gradient remains constant, because the weight of the suspension and the bed is a constant, resulting in an almost constant friction force. Increasing the line speed further will give a transition to sliding flow (7). The solid red line is first horizontal and then slightly downwards directed. The Limit Deposit Velocity is somewhere at the right of the sliding bed regime or in the sliding flow regime. Increasing the line speed further, results in a transition region between sliding flow and (pseudo) homogeneous transport (5/6). At very high line speeds, the regime will be the homogeneous regime (6). Whether this regime will be reached with practical line speeds depends completely on the combination of the parameters involved. The solid red curve is upwards directed again. Scenario R3 (the green dashed line) Table 7.2-10: Indication of occurrence of R3. Starting at a line speed vls=0, there will be equilibrium between erosion and deposition, resulting in a certain bed height. Above the bed there will be heterogeneous transport or suspension. At very low line speeds, the hydraulic gradient is so high that a sliding bed may occur (8). At these low line speeds, it is also possible that only part of the bed is siding resulting in sliding friction that is not fully mobilized. The green dashed curve is slightly downwards directed. The height of the green dashed line depends on the Cvt. A smaller Cvt gives a higher green dashed line, because a smaller Cvt will have more slip. The red solid line (constant Cvs) is hardly influenced by the spatial concentration. At higher line speeds, the bed has a higher velocity and starts vaporizing, the hydraulic gradient drops, resulting in a transition towards sliding flow behavior (7) and at higher line speeds full heterogeneous behavior (5). The dashed green line is slightly downwards directed. The Limit Deposit Velocity is somewhere at the right of the sliding bed regime or in the sliding flow regime. Increasing the line speed further, results in a transition region between heterogeneous transport and (pseudo) homogeneous transport (5/6). At very high line speeds, the regime will be the homogeneous regime (6). Whether this regime will be reached with practical line speeds depends completely on the combination of the parameters involved. The solid red curve is upwards directed again.

### 7.2.4.4 Conclusions & Discussion

From Figure 7.2-2, Figure 7.2-3, Figure 7.2-4, Figure 7.2-5, Figure 7.2-6 and Figure 7.2-7. it is clear that the characterization of flow regimes of Durand (1952), Abulnaga (2002) or Matousek (2004) is not adequate enough to identify all possible scenarios. Flow regime graphs like the ones published by Newitt (1955) or King (2002) (based on Turian & Yuan (1977)) already give a better understanding. These graphs however do not show the difference between laboratory (Cvs) and real life (Cvt) conditions and do not take the sliding flow effect into account, probably because the volumetric concentrations were not high enough.

## 7.2.5 Verification & Validation

The Relative Excess Hydraulic Gradient Erhg is the contribution of the solids to the relative hydraulic gradient. The word relative is used here because the hydraulic gradient is divided by the volumetric concentration Cv and the relative submerged density Rsd in order to determine the Erhg. The Erhg can be applied for all flow regimes. The relative submerged density Rsd is defined as:

$\ \mathrm{R}_{\mathrm{s d}}=\frac{\rho_{\mathrm{s}}-\rho_{\mathrm{l}}}{\rho_{\mathrm{l}}}$

The Slip Relative Squared Srs is the Slip Velocity of a particle vsl divided by the Terminal Settling Velocity of a particle vt squared and this Srs value is a good indication of the excess pressure losses due to the kinetic energy losses of the solids. The Settling Velocity Hindered Relative Shr is the ratio between the hindered settling velocity vt·(1-Cv/κ)β and the line speed vls, divided by the relative submerged density Rsd and the volumetric concentration Cv. The Shr value gives a good approximation of the potential energy losses of the solids. The Shr and Srs are derived and can be applied for the heterogeneous regime.

$\ \mathrm{E}_{\mathrm{rhg}}=\frac{\mathrm{i}_{\mathrm{m}}-\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{sd}} \cdot \mathrm{C}_{\mathrm{v}}}=\mathrm{R}_{\mathrm{ss}}=\mathrm{S}_{\mathrm{hr}}+\mathrm{S}_{\mathrm{rs}}=\left(\frac{\mathrm{v}_{\mathrm{t}} \cdot\left(\mathrm{1}-\frac{\mathrm{C}_{\mathrm{v}}}{\mathrm{\kappa}_{\mathrm{C}}}\right)^{\beta}}{\mathrm{v}_{\mathrm{l s}}}\right)+\left(\frac{\mathrm{v}_{\mathrm{s} \mathrm{l}}}{\mathrm{v}_{\mathrm{t}}}\right)^{2}$

The Stratification Ratio of the Solids R is a measure for the level of stratification of slurry as introduced by Wilson et al. (1997). A high stratification ratio means that the slurry is (almost) fully stratified; the liquid phase and the sediment (bed) phase are almost separated.

Under laboratory conditions with constant volumetric spatial concentration Cvs, the Erhg is limited by the value of the sliding friction coefficient μsf. The lower limit of the Erhg is when the heterogeneous regime transits to the (pseudo) homogeneous regime. Also here the Srs is derived and can be applied for the heterogeneous regime only. Resuming, the Erhg is valid for all flow regimes, the Srs is valid for the heterogeneous regime and the friction factor μsf is valid for the sliding bed regime. In the following examples the Erhg(il) graph will be used. The advantage of the Erhg(il) graph is that this type of graph is almost independent of the values of concentration Cvs and relative submerged density Rsd, but also almost independent of the pipe wall roughness and the temperature (kinematic viscosity). A disadvantage may be that it will take more effort to transform this graph back to real life data, the hydraulic gradient or pressure versus the line speed.

In these graphs always the lines for fixed/stationary bed (constant Cvs, thick solid red upwards on the left), ELM (thin dashed dark blue, upwards), homogeneous transport (thick dashed dark brown, upwards), heterogeneous transport (Cvs thick solid red downwards, Cvt thick dashed green downwards), sliding bed (Cvs thick solid red horizontal and constant Cvt thick dashed green slightly downwards) are drawn, in order to form a reference system.

The graphs show 3 additional lines. The thin dotted black line shows the ratio of the potential energy to the kinetic energy for heterogeneous transport. The thin blue dash-dot-dot line shows the heterogeneous curve without transitions to other flow regimes. The thin brown dash-dot-dot line shows the mobilization of homogeneous transport. The latter two lines added give the transition from the heterogeneous to the homogeneous flow regime.

All data are compared with the DHLLDV Framework.

### 7.2.5.1 L1: Fixed Bed & Heterogeneous, Constant Cvs

These experiments clearly show the transition of a fixed bed (flow regimes 1 and 2) to heterogeneous transport (flow regime 5) at a constant volumetric spatial concentration. The solid lines are drawn for a Cvs=0.036 & 0.17 and may differ slightly for other concentrations. The transition is smoother than the DHLLDV Framework predicts.

## 7.2.5.2 R1: Heterogeneous, Constant Cvt

These experiments show that at line speeds below the Limit Deposit Velocity and constant Cvt, the heterogeneous line is still followed. The grading of the sand makes the heterogeneous curve less steep, but this depends on the grading, the particle size and the pipe diameter.

### 7.2.5.3 L2: Fixed & Sliding Bed – Heterogeneous & Sliding Flow, Constant Cvs

These experiments at constant Cvs, show a fixed bed to sliding bed to heterogeneous behavior. In both figures it is clear that the curve for graded sands match the data points better. The effect of the grading is different for different particle sizes.

### 7.2.5.4 R2, R3: Sliding Bed & Sliding Flow, Constant Cvt

These experiments show the constant Cvt behavior at small line speeds, with a sliding/fixed bed and sliding flow behavior. The DHLLDV Framework gives a good prediction for both heavy (MnO2) and light (Acetal) solids.

## 7.2.5.5 L1, R1, L2, R2:, Homogeneous

These experiments show homogeneous behavior, which occurs with small particles at relatively high line speeds. The Thomas (1976) graph also includes Thomas (1965) viscosity for very small particles.

## 7.2.5.6 L3, R3: Sliding Bed & Sliding Flow, Constant Cvs

These experiments show the sliding bed and sliding flow regimes. Beyond the intersection point between a sliding bed and heterogeneous flow, the sliding flow curve is followed.

## 7.2.6 Discussion & Conclusions

The experimental graphs are given without a lot of explanation, because they should speak for themselves. These graphs show the different flow regimes and sometimes more than one flow regime. In general there is a lot of scatter. This is caused by the way experiments were carried out and specifically the accuracy of the concentration measurements. Sometimes concentrations within a certain bandwidth (for example 10-15%) are given with an average mentioned on the graph (for example 12.5%). But in spite of the scatter, the graphs clearly show the different regimes.

From these graphs and the regime and scenario definitions, it should be clear that experiments carried out in very small pipelines, like 1 inch diameter pipelines, can never be compared with experiments in very large pipelines, like 1 m diameter pipelines. In a 1 m diameter pipeline it is difficult to get a sliding bed regime, while in a 1 inch diameter pipeline it is very difficult not to get a sliding bed regime, due to the high hydraulic gradients. It is like comparing laminar and turbulent flow.

Each regime has its own physical and mathematical model. The fixed bed regime can be modeled with flow through a restricted cross section using the Televantos (1979) method for determining the friction factor. The sliding bed regime and partly the sliding flow regime can be modeled using the Newitt et al. (1955) method, with the appropriate friction factor (0.35-0.7). The heterogeneous regime can be modeled with one of the existing equations or with the Miedema et al. (2013) model. The homogeneous regime can be modeled using the equivalent liquid model, using 100% of the solids or for example using 60% of the solids, like some authors do. For the fixed bed/sliding bed regimes below the Limit Deposit Velocity, a 2 layer or 3 layer model can be used, but the Durand & Condolios (1952) approach, considering a flow Froude number which is equal to the flow Froude number at the Limit Deposit Velocity, also gives good results.

Just carrying out some curve fits and drawing conclusions is very dangerous, because the experiments may cover 2 or more regimes. For example, if 50% of the experiments are in the heterogeneous regime and 50% of the experiments are in the homogeneous regime, a curve fit would give a horizontal line in the Erhg(il) graph. If we look at experiments where 50% is in the fixed bed regime (constant Cvs) and 50% is in the heterogeneous regime (for example Figure 7.2-8 & Figure 7.2-12), the result of a curve fit is also a horizontal line. The cases however are completely different.

Recognizing the different regimes and especially the transitions between the different regimes is crucial in understanding what is physically happening.

The DHLLDV Framework gives a good match with most of the experiments shown here. In this book many more experiments are shown, usually compared to the DHLLDV Framework. The DHLLDV Framework gives good results for fine sands up to fine gravel in small to large pipes, compared to the experiments. For very coarse particles however still more research is required.

## 7.2.7 Nomenclature Flow Regimes & Scenario’s

 A2 Cross section of the bed in the pipe m2 A1 Cross section of the mixture in suspension above the bed m2 A, Ap Cross section of the pipe m2 A2 Cross section of the solids in the pipe m2 Cv Volumetric concentration - Cvs Volumetric spatial concentration - Cvs,s Volumetric spatial concentration of the mixture in suspension above a fixed or sliding bed - Cvt Volumetric transport (delivered) concentration - d Particle diameter mm d50 Median particle diameter - 50% by weight is smaller mm Dp Pipe diameter m Erhg Relative Excess Hydraulic Gradient - im Mixture head loss m/m il Liquid hydraulic gradient m/m iw Water hydraulic gradient m/m n Porosity - O1 Circumference liquid-pipe wall m O2 Circumference bed-pipe wall m O12 Width liquid-bed interface m $$\ \mathrm{Q}_{\mathrm{m}} \text{ or } \dot{\mathrm{V}}_{\mathrm{m}}$$ Flow rate of mixture m3/s $$\ \mathrm{Q}_{\mathrm{s}}\text{ or }\dot{\mathrm{V}}_{\mathrm{s}}$$ Flow rate of solids m3/s Rsd Relative submerged density - R Stratification ratio Wilson - Srs Slip Relative Squared or Stratification Ratio Solids - Shr Settling velocity Hindered Relative - vls Velocity of the slurry, line speed m/s vls,e Effective line speed. Line speed above the fixed or moving bed. m/s vm Velocity of the slurry, line speed (same as vls) m/s vs Average velocity of the solids m/s vsl Slip velocity of the solids relative to the mixture m/s Vb Volume of the bed m3 Vcl Volume of the closed loop m3 Vm Volume of the mixture in a pipe m3 Vm,s Volume of the mixture in suspension above the bed m3 Vs Volume of the solids in a pipe m3 Vs,s Volume of the solids in suspension above the bed m3 vt Terminal settling velocity of the particles m/s β Richardson & Zaki hindered settling power - λ1 Darcy-Weisbach friction factor between liquid and pipe wall - λ2 Darcy-Weisbach friction factor between liquid and pipe wall in bed - λ22 Darcy-Weisbach friction factor between liquid and bed - κC Concentration eccentricity factor. - μsf Friction coefficient for a sliding bed - ρl, ρw Density of the liquid ton/ m3 ρm Density of the mixture ton/ m3 ρs Density of the solids ton/ m3

7.2: Flow Regimes and Scenario’s is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Sape A. Miedema via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.