Slurry transport in inclined pipes

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Slurry transport in inclined pipes

Miedema, Sape

Publication date 2017

Document Version Final published version Published in

Proceedings Dredging Summit and Expo 2017

Citation (APA)

Miedema, S. (2017). Slurry transport in inclined pipes. In D. F. Hayes (Ed.), Proceedings Dredging Summit and Expo 2017 (pp. 218-232). Bonsall, CA, USA: Western Dredging Association.

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SLURRY TRANSPORT IN INCLINED PIPES

Sape A. Miedema1

ABSTRACT

Different approaches are present in literature for slurry transport in inclined pipes. Most models assume that the so-called solids effect, the hydraulic gradient of the mixture minus the hydraulic gradient of the carrier liquid, has to be multiplied by the cosine of the inclination angle to a certain power. These powers vary from 0.25 to 1.7. In addition, a potential energy term is added. The models are usually based on the heterogeneous flow regime and do not take different flow regimes into account. A homogeneous component is absent in the models, resulting in a zero solids effect in a vertical pipe.

In this paper an overview is given of the existing models, with the pro’s and con’s. In this paper, also a more fundamental model is derived for each flow regime separately. The advantage of this is also that shifts of the transition of the flow regimes become visible. The cosine and sine of the inclination angle still plays an important role, but more complicated than just a power. The new derived model is compared with data from literature.

Keywords: Slurry transport, inclined pipes, flow regimes.

INTRODUCTION

In dredging inclined pipes occur in ladders of cutter suction dredgers and suction pipes of trailing suction hopper dredgers. On land, inclined pipes occur going up and down slopes. So, inclined pipes may have positive and negative inclination angles up to 45º. The question is, what is the influence of the inclination angle on the hydraulic gradient, on the Limit of Stationary Deposit Velocity (LSDV) and on the Limit Deposit Velocity (LDV). The effect of inclined pipes is expressed based on the length of the pipe, not the horizontal distance. A number of cases, based on flow regimes, have to be distinguished. However, first some models from literature are discussed.

The Heterogeneous Flow Regime, Durand & Condolios and Gibert.

The basic equation for the solids effect of Durand & Condolios (1952) and Gibert (1960) is given by:

3/ 2 3/ 2 2 2 ls x ls x m l m l vt l vt p sd p sd v C v C i i 81 and i i 1 81 C i C g D R g D R  _  _  _   _   _ _ _ _ _ _    _ _    _ _ _  _  _   _ _ _   _ _   (1)

For inclined pipes, they modified the solids effect by adding the cosine of the inclination angle according to:

 





 





3/ 2 2 m, sd vt l ls x l vt p sd 3/ 2 2 ls x m, l vt sd vt p sd i sin 1 R C i v C 81 i C g D R cos v C i i 1 81 C sin 1 R C g D R cos              _  _    _ _  _     _  _ _ _   _ _   _  _ _  _          _ _    (2)

This can be written as:

1_{Associate Professor, Delft University of Technology, Mekelweg 2, 2628 CD, Delft, The Netherlands, T:} ++31-15-2788359, Email: s.a.miedema@tudelft.nl.

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 





_{ }

 





 

3/ 2 2 ls x m, l sd vt l vt p sd 3/ 2 2 3/ 2 ls x m, l sd vt l vt p sd v C i i sin 1 R C i 81 C g D R cos v C i i sin 1 R C i 81 C cos g D R      _                       _                      (3)

So, the solids effect has to be multiplied with the cosine of the inclination angle to the power of 3/2. This means the solids effect is decreasing with an increasing inclination angle, whether the inclination is upwards or downwards. It should be mentioned that the hydraulic gradient is based on the length of the pipe and not on the horizontal length component.

The Heterogeneous Flow Regime, Worster & Denny.

Worster & Denny (1955) have a slightly different approach. They state that the hydraulic gradient in an inclined pipe equals the sum of the hydraulic gradients of the horizontal component and the vertical component. This gives the following equation:

 

3/ 2 2 ls x m, l, l vt sd vt p sd v C i i i 81 C cos sin R C g D R     _       _ _            (4)

The difference with Durand & Condolios (1952) and Gibert (1960) is the power of the cosine. In both cases, the equations match the hydraulic gradient of a horizontal pipe if the inclination angle equals zero and a vertical pipe if the inclination angle equals 90 degrees, whether the inclination is upwards (positive inclination angle) or downwards (negative inclination angle). However, the Equivalent Liquid Model (ELM) component for a vertical pipe is missing.

The Heterogeneous Flow Regime, Wilson et al.

Wilson et al. (2006) derived the following equation for heterogeneous transport in horizontal pipes:

M sf 50 m l sd vt ls v i i R C 2 v            (5)

For inclined pipes, they modified the equation, matching the reasoning of Worster & Denny (1955), but with the use of the power M according to:

 

M M sf 50 m, l, sd vt sd vt ls v i i R C cos sin R C 2 v                    (6)

The power M has a value of 1.7 for uniform or narrow graded sands and decreases to 0.25 for very broad graded sands. For narrow graded sands the influence of the inclination angle is similar to the Durand & Condolios (1952) and Gibert (1960) approach with a power of 1.5 versus 1.7 for Wilson et al. (2006). For medium graded sands with a power around 1, the influence is similar to the Worster & Denny (1955) approach.

The Sliding Bed Regime, Doron et al.

Doron et al. (1997) investigated the influence of inclined pipes, based on their 2LM and 3LM models (LM=Layer Model). Basically, they multiplied the sliding friction with the cosine of the inclination angle and they added the potential energy term, which is proportional with the sine of the inclination angle. They carried out experiments with

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inclination angles from -7 to +7 degrees. The resulting data however is dominated by the potential energy term, because of the small inclination angles.

DISCUSSION OF LITERATURE

After adding the potential energy terms to the hydraulic gradient in a correct way, the pipe inclination effect can be considered, by multiplying the solids effect term with the cosine of the inclination angle to a power ranging from 1.0 to 1.7. Different researchers give different powers, most probably because the models are either empirical or have different physical backgrounds. This implies that the solids effect reduces to zero for a vertical pipe, which is doubtful, especially for very small particles giving homogeneous flow (ELM). One would expect an equation of the following form:

 



1



  2





 

m, l sd vs rhg sd vs sd vs

i _ i  1  R C sin   E R C cos    1R C sin  ₍₇₎

The first term on the right-hand side is the Darcy Weisbach friction, including the mobilized ELM (the homogeneous solids effect) corrected for the inclination angle. The second term is the heterogeneous solids effect corrected for the inclination angle. The third term is the potential energy term. So, where the heterogeneous solids effect decreases with the inclination angle, the homogeneous solids effect increases. In this form a vertical pipe shows mobilized/reduced ELM behavior, which is observed by Newitt et al. (1961). Other flow regimes were not considered.

The LSDV increases with increasing inclination angle to a maximum with 25%-30% increase for an inclination angle of 15 to 30 degrees. This probably depends on the spatial volumetric concentration, the particle size, the relative submerged density and the particle slip, but not enough data could be found to quantify this.

MODELING

In dredging and other industries often parts of a pump/pipeline system are inclined or vertical. So, it is interesting to see what the implications of an inclined pipe on the pressure losses are. Here an analytical solution to this problem is given.

Pure Carrier Liquid

First, the flow of pure carrier liquid. The equilibrium of forces on the liquid is:

 

l l dp A L O L A L g sin dx               (8)

The hydraulic gradient can now be determined with:

 

_{ }

l l l, l l l l A L g sin O L dp A L i i sin dx A L g A L g A L g                                (9)

So apparently, the hydraulic gradient increases with the sine of the inclination angle. Which also means that a downwards slope with a negative inclination angle gives a negative sine and thus a reduction of the hydraulic gradient. In this case the hydraulic gradient may even become negative.

Stationary Bed Regime

The equilibrium of forces on the layer of liquid above the bed is:

 

1 1 12 12 1 1 1 12 12 l 1 m l 1 O L O L dp A L O L O L A L g sin so: i dx A L g                               (10)

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Since the bed is not moving, the friction between the bed and the pipe wall compensates for the weight component of the bed. The hydraulic gradient can now be determined with:

 

1 1 1 12 12 m, m l 1 l 1 A L O L O L dp i sin i sin dx A L g A L g   _{ } _{ }    _{ }   _{ }       (11)

Which is the hydraulic gradient of a stationary bed in a horizontal pipe plus the sine of the inclination angle. The weight of the solids does not give a contribution to the hydraulic gradient, since the solids are not moving.

Sliding Bed Regime

The equilibrium of forces on the layer of liquid above the bed is:

 

1 1 1 12 12 l 1 dp A L O L O L A L g sin dx                   (12)

The cross-section of the bed can be determined with:

vs 2 vb C A A C   ₍₁₃₎

The weight of the bed, including pore water is:

















b b 2 s vb l vb 2 s l vb l 2 l sd vs l 2 W A L g C 1 C A L g = C A L g = R C A L g A L g                                    (14)

The submerged weight of the bed can be determined with:





b,s s l vb 2 l sd vs

W     C A     L g R C    A L g (15)

This gives for the equilibrium of forces on the bed:

 

2 2 2 12 12 b sf b,s 2 2 2 12 12 b,s sf b,s l 2 dp A L O L O L W sin W cos dx dp

A L O L O L W sin W cos A L g sin

dx

                  

                         

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For the whole pipe cross section the two contributions can be added, giving:

 

1 1 2 2 l l sd vs sf l sd vs dp A L O L O L A L g sin dx + R C A L g sin R C A L g cos                                      (17)

This can also be written as:

 

1 1 2 2 m sf l sd vs dp A L O L O L A L g sin R C A L g cos dx                              (18)

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In terms of the hydraulic gradient this gives:





 

1 1 2 2 m, sd vs sf sd vs l l O L O L dp A L i 1 R C sin + R C cos dx g L A g L A   _{   }    _{   }               (19)

In Miedema (June 2016) it has been proven that the first term on the right-hand side almost equals the pure liquid hydraulic gradient il, without pipe inclination, so:





 







 



m, l sd vs sf sd vs m, l sd vs sf i i 1 R C sin + R C cos

i i sin R C cos sin

 

          

           (20)

The hydraulic gradient for pure liquid with pipe inclination can now be expressed as:

 

l, l

i _ i sin  ₍₂₁₎

Giving for the mixture hydraulic gradient with pipe inclination:

 





m, l, sd vs sf

i _i _R C   cos  sin  (22)

The relative excess hydraulic gradient Erhg,θ is now:

 

m, l, rhg, sf sd vs i i E cos sin R C             (23) Heterogeneous Regime

In the heterogeneous flow regime, the energy losses consist of the potential energy losses and the kinetic energy losses. The equation for the total head loss is:





rhg hr rs m l hr rs sd vs

E S S and i  i S S R C ₍₂₄₎

The potential energy contribution is:

vs t C hr ls C v 1 S v      _     (25)

The kinetic energy contribution is:

2 2 2/ 3 4/ 3 sl v t t rs t * v v v S c v d 11.6 u g d               _   _ _ _         (26)

In an inclined pipe the effective terminal settling velocity perpendicular to the pipe wall gives a potential energy term of:

 

vs t C hr, hr ls C v cos 1 S S cos v        _ _        (27)

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For the kinetic energy losses, the angle of attack has to be adjusted in an inclined pipe. The angle of attack is defined as the ratio between the terminal settling velocity and the velocity at the thickness of the viscous sub layer, giving:

 

2 4/ 3 2/ 3 t v t rs, * t v cos v S c d 11.6 u v sin g d              _ _ _ _ _ _  _ _ _       (28)

So, for very small particles with vt<<11.6·u*, the kinetic energy losses are proportional to the cosine of the inclination

angle to a power of 4/3. For larger particles, the term in the denominator becomes significant resulting in different behavior of a positive versus a negative inclination angle. Apart from this, also the lifting of the mixture has to be added, giving:

 



 



rhg, hr, rs, m, l, hr, rs, sd vs

E _ S _S _sin  and i _i _ S _S _sin  R C (29)

Literature shows a power of the cosine between 1 and 1.7. Here a more complicated formulation is found. Considering that the potential energy losses are much smaller than the kinetic energy losses, a power of about 4/3 is found for small particles, while larger particles will show a smaller power depending on the terminal settling velocity (see equation (28)). The higher the terminal settling velocity, the smaller the power. Theoretically this power may even become zero when nominator and denominator decrease in the same way with increasing inclination angle.

Homogeneous Regime

In the homogeneous flow regime, the relative excess hydraulic gradient is, including the lubrication effect:





rhg E l m l l E sd vs l E sd vs

E   i and i     i i R C     i 1 R C ₍₃₀₎

For an inclined pipe, only the lifting of the mixture has to be added, giving:

 









 





rhg, E l m, l, E l sd vs l E sd vs sd vs E i sin i i i sin R C i 1 R C sin 1 R C                            (31)

Sliding Flow Regime

The method for determining the Sliding Flow Regime is not affected by pipe inclination. Of course, the equations for a pipe with inclination for the sliding bed regime and the heterogeneous regime have to be applied.

The Limit Deposit Velocity

The Limit of Stationary Deposit Velocity is affected by the pipe inclination. In an ascending pipe, the cross sectional averaged line speed has to be higher compared to a horizontal pipe in order to make a bed start sliding. In a descending pipe this line speed is lower. It is even possible that in a descending pipe the bed will always slide because of gravity. The Limit Deposit Velocity is defined as the line speed above which there is no stationary of sliding bed, is determined by either the potential energy losses or a limiting sliding bed. In both cases this is affected by the cosine of the inclination angle, the component of gravity perpendicular to the pipe wall. Since in both cases the Limit Deposit Velocity depends on the cube root of this cosine, the Limit Deposit Velocity will decrease according to:

 

1/3 ls,ldv, ls,ldv

v _v cos  (32)

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DISCUSSION OF MODELING

For the stationary bed regime, only the potential energy term of the pure liquid has to be added to the hydraulic gradient of the mixture. For all other flow regimes, the potential energy term of the mixture has to be added, together with a correction of the so-called solids effect. The result of this is a higher line speed for the intersection point of the stationary bed curve and the sliding bed curve. So, in general an increase of the Limit of Stationary Deposit Velocity with increasing inclination angle. This may however also result in omission of the occurrence of a sliding bed for an inclined pipe, where a sliding bed would occur in a horizontal pipe. This makes sense, since a higher line speed is required to make a bed start sliding, there is the possibility that the bed is already fully suspended before it could start sliding.

In the sliding bed regime and the heterogeneous regime, the hydraulic gradient is lower for an inclined pipe compared with a horizontal pipe, if the potential energy term of the mixture (static head) is not considered, especially for small particles in the heterogeneous regime. For the heterogeneous regime, there is a difference between ascending and descending pipes, due to the term with the angle of attack in the kinetic energy losses. The decrease in an ascending pipe is smaller than in a descending pipe and could even be a small increase in an ascending pipe at low line speeds. The transition line speed of the heterogeneous flow regime to the homogeneous flow regime will also decrease with increasing inclination angle.

In case of a sliding bed one may expect more stratification in an ascending pipe compared to a descending pipe, due to the higher line speed in an ascending pipe to make the bed start sliding. In other words, a higher shear stress on the bed is required in an ascending pipe, resulting in a thicker sheet flow layer at the top of the bed.

The resulting curves in the following figures are determined by first determining the curves for each flow regime individually and then combining these curves to a resulting curve.

Figure 1 and Figure 2 show the hydraulic gradient in a horizontal pipe with Dp=0.1524 m and d=0.5 mm, an ascending

pipe with slope 30º and a descending pipe with slope -30º. In a horizontal pipe, for constant spatial concentration, a sliding bed will occur. The ascending pipe does not show a sliding bed, in fact the resulting curve just touches the sliding bed curve at a higher line speed than the start of a sliding bed in a horizontal pipeline. The descending pipe shows a sliding bed from zero line speed up to the line speed where the heterogeneous flow regime starts, which happens at a lower line speed than for the horizontal pipe.

Figure 3 and Figure 4 show the same phenomena in a pipe with Dp=0.762 m. Here a d=3 mm particle is required to

have a sliding bed. The figures also show a slight decrease of the Limit Deposit Velocity for the pipes with a slope. Figure 5 and Figure 6 show the same for a d=5 mm particle. Now the ascending pipeline also has a sliding bed, but it starts at a higher line speed and stops at a lower line speed, compared to the horizontal pipe. A sliding bed in an ascending pipe will encounter a higher bed shear stress and have a lower bed velocity, compared with a horizontal pipeline.

The hydraulic gradients of the inclined pipes are determined per meter of inclined pipe and not per meter of horizontal pipe. The potential energy term is included in all figures, resulting in an offset at zero line speed of the inclined pipe hydraulic gradient. In all graphs, the thick lines are for a horizontal pipe, while the thin lines represent the inclined pipe.

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Figure 1: The hydraulic gradient in an ascending pipeline, d=0.5 mm, Dp=0.1524 m.

Figure 2: The hydraulic gradient in a descending pipeline, d=0.5 mm, Dp=0.1524 m.

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 0 1 2 3 4 5 6 7 8 9 10 Hy dra ulic gra die nt im , il (m w a ter/ m)

Line speed vls(m/sec)

Hydraulic gradient i

_m

, i

_l

vs. Line speed v

_ls

Liquid il curve Equivalent Liquid Model Homogeneous Flow Cvs=Cvt=c. Sliding Bed Resulting im curve Cvs=c. Resulting im curve Cvt=c.

Limit Deposit Velocity Liquid il curve with slope

Equivalent Liquid Model with slope Homogeneous Flow Cvs=Cvt=c. with slope Sliding Bed with slope Resulting im curve Cvs=c. with slope Resulting im curve Cvt=c. with slope Limit Deposit Velocity with slope © S.A.M. Dp=0.1524 m, d=0.500 mm, Rsd=1.585, Cv=0.175, μsf=0.416, Slope=30º -0.70 -0.60 -0.50 -0.40 -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.50 0 1 2 3 4 5 6 7 8 9 10 Hy dra ulic gra die nt im , il (m w a ter/ m)

Hydraulic gradient i

_m

, i

_l

vs. Line speed v

_ls

Equivalent Liquid Model with slope Homogeneous Flow Cvs=Cvt=c. with slope Sliding Bed with slope Resulting im curve Cvs=c. with slope Resulting im curve Cvt=c. with slope Limit Deposit Velocity with slope

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Figure 3: The hydraulic gradient in an ascending pipeline, d=3 mm, Dp=0.762 m.

Figure 4: The hydraulic gradient in a descending pipeline, d=3 mm, Dp=0.762 m.

Hydraulic gradient i

_m

, i

_l

vs. Line speed v

_ls

Hydraulic gradient i

_m

, i

_l

vs. Line speed v

_ls

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Figure 5: The hydraulic gradient in an ascending pipeline, d=5 mm, Dp=0.762 m.

Figure 6: The hydraulic gradient in a descending pipeline, d=5 mm, Dp=0.762 m.

Hydraulic gradient i

_m

, i

_l

vs. Line speed v

_ls

Hydraulic gradient i

_m

, i

_l

vs. Line speed v

_ls

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VALIDATION

Doron et al. (1997) investigated the influence of inclined pipes, based on their 2LM and 3LM models. Basically, they multiplied the sliding friction with the cosine of the inclination angle and they added the potential energy term, which is proportional with the sine of the inclination angle. Their experiments were carried out with inclination angles from -7 to +7 degrees. The resulting data however is dominated by the potential energy term.

They also investigated the Limit of Stationary Deposit Velocity (LSDV), the start of a sliding bed. Ascending pipes show an increasing LSDV with a maximum for an inclination angle of about 15 degrees, while descending pipes show a sharp decrease of the LSDV, because gravity becomes the driving force. At a certain negative inclination angle the bed will start sliding downwards because of gravity. It should be mentioned that Doron et al. (1997) use the delivered volumetric concentration, while LSDV models are usually based on spatial volumetric concentration. Especially in the stationary and sliding bed regimes the difference is significant.

Figure 7, Figure 8, Figure 9 and Figure 10 show the data of Doron et al. (1997) versus the DHLLDV Framework for a horizontal pipe and a 4º and 7º ascending pipe and a 4º and 7º descending pipe. The solid red lines show the hydraulic gradient for the spatial volumetric concentration. The solid and dashed green lines show the hydraulic gradient for the delivered volumetric concentration. In general, the theoretical curves and the experimental data match well, although at very low line speeds the experimental points are lower than the theoretical curves. The theoretical curves are based on a sliding bed and it is possible that at very low line speeds there is a stationary bed, resulting in smaller hydraulic gradients, since the sliding friction is not fully mobilized. The difference between a horizontal pipe and the inclined pipes is dominated by the potential energy term (the sine), since the cosine is larger than 0.99 for the inclination angles considered, while the sine has a value of 0.12 for a 7º inclination angle. Much larger inclination angles are required to see the influence of the cosine on the sliding friction.

Figure 7: The data of Doron et al. (1997) versus

the DHLLDV Framework for a horizontal and a 7º ascending pipe. -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 H y d ra u li c g ra d ie n t im , il (m w a te r/ m )

Hydraulic gradient i

_m

, i

_l

vs. Line speed v

_ls

Equivalent Liquid Model with slope Homogeneous Flow Cvs=Cvt=c. with slope Sliding Bed with slope Resulting im curve Cvs=c. with slope Resulting im curve Cvt=c. with slope Limit Deposit Velocity with slope Beta=0 degrees Beta=7 degrees Beta=-7 degrees Beta=4 degrees Beta=-4 degrees © S.A.M. Dp=0.0500 m, d=3.000 mm, Rsd=0.210, Cv=0.130, μsf=0.480, Slope=7.0º

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the DHLLDV Framework for a horizontal and a 4º ascending pipe.

the DHLLDV Framework for a horizontal and a 4º descending pipe. -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 H y d ra u li c g ra d ie n t im , il (m w a te r/ m )

Hydraulic gradient i

m

, i

l

vs. Line speed v

ls

Equivalent Liquid Model with slope Homogeneous Flow Cvs=Cvt=c. with slope Sliding Bed with slope Resulting im curve Cvs=c. with slope Resulting im curve Cvt=c. with slope Limit Deposit Velocity with slope Beta=0 degrees Beta=7 degrees Beta=-7 degrees Beta=4 degrees Beta=-4 degrees © S.A.M. Dp=0.0500 m, d=3.000 mm, Rsd=0.210, Cv=0.130, μsf=0.480, Slope=4.0º -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 H y d ra u li c g ra d ie n t im , il (m w a te r/ m )

Hydraulic gradient i

_m

, i

_l

vs. Line speed v

_ls

Equivalent Liquid Model with slope Homogeneous Flow Cvs=Cvt=c. with slope Sliding Bed with slope Resulting im curve Cvs=c. with slope Resulting im curve Cvt=c. with slope Limit Deposit Velocity with slope Beta=0 degrees Beta=7 degrees Beta=-7 degrees Beta=4 degrees Beta=-4 degrees © S.A.M. Dp=0.0500 m, d=3.000 mm, Rsd=0.210, Cv=0.130, μsf=0.480, Slope=-4.0º

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the DHLLDV Framework for a horizontal and a 7º descending pipe. CONCLUSIONS

Models from literature multiply the so-called solids effect with the cosine of the inclination angle to a power between 1 and 1.7, based on the heterogeneous flow regime. These models do not take the (reduced) Equivalent Liquid Model (ELM) into consideration for very high inclination angles. The consequence is, that there is no solids effect in vertical pipes, which is doubtful.

A proper model for inclined pipes should consider the different flow regimes individually and then combine the flow regime hydraulic gradients, based on which flow regime will occur at which line speed. The effect of the inclination angle may be different for the different flow regimes.

Since the occurrence of the different flow regimes depends strongly on the pipe diameter and the line speed, this also has a dominant effect on the occurrence of the flow regimes in inclined pipes.

The use of the cosine of the inclination angle on the solids effect for the sliding bed regime seems appropriate. However, for the heterogeneous flow regime this is more complicated, resulting in a difference for an ascending compared to a descending pipe.

In an ascending pipe a bed will start sliding at a higher line speed and transit to heterogeneous flow at a lower line speed, with the possibility that there is no sliding bed at all, while there would be in a horizontal pipe. In a descending pipe the opposite will occur, with the possibility that there is a sliding be from line speed zero up to the transition to heterogeneous transport.

The validation with the Doron et al. (1997) experiments show a good correlation. However, it should be stated that the potential energy terms are dominating and it is very difficult to identify the exact behavior of the solids effect.

-0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 H y d ra u li c g ra d ie n t im , il (m w a te r/ m )

Hydraulic gradient i

_m

, i

_l

vs. Line speed v

_ls

Equivalent Liquid Model with slope Homogeneous Flow Cvs=Cvt=c. with slope Sliding Bed with slope Resulting im curve Cvs=c. with slope Resulting im curve Cvt=c. with slope Limit Deposit Velocity with slope Beta=0 degrees Beta=7 degrees Beta=-7 degrees Beta=4 degrees Beta=-4 degrees © S.A.M. Dp=0.0500 m, d=3.000 mm, Rsd=0.210, Cv=0.130, μsf=0.480, Slope=-7.0º

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NOMENCLATURE

A,Ap Cross section pipe m2

A1 Cross section restricted area above the bed m2

A2 Cross section bed m2

c Proportionality constant -

Cvb Bed volumetric concentration -

Cvs Spatial volumetric concentration -

d Particle diameter m

Erhg Relative excess hydraulic gradient without pipe inclination -

Erhg,θ Relative excess hydraulic gradient with pipe inclination -

g Gravitational constant (9.81) m/s2

il Hydraulic gradient liquid without pipe inclination -

il,θ Hydraulic gradient liquid with pipe inclination -

im Hydraulic gradient mixture without pipe inclination -

im,θ Hydraulic gradient mixture with pipe inclination -

L Length of pipe m

O1 Circumference restricted area above the bed in contact with pipe wall m

O2 Circumference of bed with pipe wall m

O12 Width of the top of the bed m

p Pressure in pipe kPa

Rsd Relative submerged density of solids -

Shr Settling velocity Hindered Relative without pipe inclination -

Shr,θ Settling velocity Hindered Relative with pipe inclination -

Srs Slip Ratio Squared without pipe inclination -

Srs,θ Slip Ratio Squared with pipe inclination -

u* Friction velocity m/s

vls Line speed m/s

vt Terminal settling velocity m/s

vsl Slip velocity solids m/s

vls,ldv Limit Deposit Velocity without pipe inclination m/s

vls,ldv,θ Limit Deposit Velocity with pipe inclination m/s

Wb Weight of the bed ton

Wb,s Submerged weight of the bed ton

x Distance in pipe length direction m

αE Homogeneous lubrication factor -

β Richardson & Zaki hindered settling power -

δv Thickness viscous sub-layer m

ρb Density of the bed including pore water ton/m3

ρs Density of the solids ton/m3

ρl Density of the liquid ton/m3

ρm Mixture density ton/m3

τ1 Shear stress between liquid and pipe wall kPa

τ12 Shear stress on top of the bed kPa

θ Inclination angle (positive upwards, negative downwards) º

μsf Sliding friction coefficient -

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REFERENCES

Doron, P., Simkhis, M., & Barnea, D. (1997). Flow of solid liquid mixtures in inclined pipes. International Journal

of Multiphase Flow, Vol. 23, No. 2., 313-323.

Durand, R., & Condolios, E. (1952). Etude experimentale du refoulement des materieaux en conduites en particulier des produits de dragage et des schlamms. Deuxiemes Journees de l'Hydraulique., 27-55.

Gibert, R. (1960). Transport hydraulique et refoulement des mixtures en conduites. Annales des Ponts et Chausees., 130(3), 307-74, 130(4), 437-94.

Miedema, S. A. (June 2016). Slurry Transport: Fundamentals, A Historical Overview & The Delft Head Loss & Limit

Deposit Velocity Framework. (1st Edition ed.). (R. C. Ramsdell, Ed.) Miami, Florida, USA: Delft University

of Technology.

Newitt, D. M., Richardson, J. F., & Gliddon, B. J. (1961). Hydraulic conveying of solids in vertical pipes. Transactions

Institute of Chemical Engineers, Vol. 39., 93-100.

Wilson, K. C., Addie, G. R., Sellgren, A., & Clift, R. (2006). Slurry transport using centrifugal pumps. New York: Springer Science+Business Media Inc.

Worster, R. C., & Denny, D. F. (1955). Hydraulic transport of solid materials in pipelines. Institution of Mechanical

Engineers (London), 563-586.

CITATION

Miedema, S.A., “Slurry transport in inclined pipes”. WEDA Dredging Summit & Expo ’17, Vancouver, Canada, June 2017.