WODCON XXI PROCEEDINGS
A COMPARISON OF DIFFERENT SLURRY TRANSPORT MODELS FOR SANDS & GRAVELS S.A. Miedema1 and R.C. Ramsdell2 ABSTRACT In dredging, sand and gravel are often transported through pipelines mixed with water. In order to determine the pumping power required, but also predict whether there will be a bed or not, the resistance (hydraulic gradient) and the Limit Deposit Velocity (LDV) have to be determined. The transport process is always an interaction between the pump and the pipeline resistance of the mixture transported. An important parameter is the LDV, the line speed above which there is no fixed or sliding bed. There are many models in literature to determine the resistance, expressed as the hydraulic gradient, and the LDV, but which models are best suited for the case considered. For the pipeline resistance models, the following categories can be distinguished: 2LM or 3LM sliding bed model, concentration distribution based models, semi-empirical heterogeneous flow regime models and homogeneous flow regime models. For the LDV about 25 models are compared, where most models are semi-empirical. The paper will give an overview of the most used models and show methods to compare these models and make a choice of the best suited model, the Delft Head Loss & Limit Deposit Velocity (DHLLDV) Framework. Keywords: Slurry transport, Limit deposit velocity, Two phase flow. INTRODUCTION The theoretical transition velocity between the heterogeneous regime and the homogeneous regime (the Equivalent Liquid Model or ELM is assumed for this) gives a good indication of the excess pressure losses due to the solids. For normal dredging applications with large diameter pipes and rather high line speeds, this transition velocity will be at a line speed higher than the Limit Deposit Velocity and will often be near the operating range of the dredging operations. The excess head losses are in some way proportional to this transition velocity. For a Dp=0.1 m diameter pipe most models match pretty well, due to the fact that most experiments are carried out in small diameter pipes and the models are fitted to the experiments. Although the different models may have a different approach, the resulting equations go through the same cloud of data points. Since there are numerous fit lines of numerous researchers it is impossible to cover them all, so a choice is made to compare Durand & Condolios (1952), Newitt et al. (1955), Fuhrboter (1961), Jufin & Lopatin (1966), Zandi & Govatos (1967), Turian & Yuan (1977), the SRC (Saskatchewan Research Counsel) model (1991) and Wilson et al. (1992) with the DHLLDV (Delft Head Loss & Limit Deposit Velocity) Framework as developed by Miedema & Ramsdell (2013). TRANSITION HETEROGENEOUS-HOMOGENEOUS MODELS CONSIDERED The theoretical transition velocity of the heterogeneous flow regime to the homogeneous flow regime (Equivalent Liquid Model or ELM) can be determined by making the relative excess hydraulic gradient of the heterogeneous flow regime and the homogeneous flow regime equal (see Figure 1 for the excess hydraulic gradient). Relative here means the excess hydraulic gradient is divided by the volumetric concentration Cv and the relative submerged density Rsd. This is possible if flow regime transition effects are omitted and the basic equations are applied. For the DHLLDV Framework, this equation implies that the transition velocity to the 4th power depends linear on the pipe diameter Dp and reversely on the viscous friction coefficient λl squared. Since the viscous friction coefficient λl depends reversely on the pipe diameter Dp with a power of about 0.2, the transition velocity will depend on the pipe diameter with a power of about (1.4/4) =0.35. The equation derived is implicit and has to be solved iteratively. This
1
Associate Professor, Dredging Engineering, Delft University of Technology, Mekelweg 2, 2628CD Delft, The Netherlands. Phone: 031-15-2788359, email:
[email protected].
2
Manager Production Department, Great Lakes Dredge & Dock, 2122 York Road, Oak Brook, IL 60523 USA.
WODCON XXI PROCEEDINGS
gives for the transition velocity between the heterogeneous flow regime and the homogeneous flow regime vls,hh for the latest version of the DHLLDV Framework: Liquid il curve
0.40
Fixed Bed Cvs=c.
0.35
Sliding Bed Cvs=c. Lower Limit Sliding Bed Cvs=c. Mean
im-il
0.30
Sliding Bed Cvs=c. Upper Limit Heterogeneous Flow Cvs=c.
0.25
im-il
Equivalent Liquid Model
im-il
0.15
Homogeneous Flow Cvs=Cvt=c.
im-il
0.20
im-il
Hydraulic gradient im, il (m water/m)
Hydraulic gradient im, il vs. Line speed vls
Resulting im curve Cvs=c. Resulting im curve Cvt=c.
0.10
Limit Deposit Velocity Cvs=c. Limit Deposit Velocity Cvt=c.
0.05
Limit Deposit Velocity 0.00 0
1
2
3
4
5
6
7
8
9
10
Line speed vls (m/sec) Dp=0.1524 m, d=1.500 mm, Rsd=1.585, Cv=0.300, μsf=0.420
© S.A.M.
Figure 1. The excess pressure gradient (im-il) due to the solids at a number of line speeds.
4 v ls ,h h
2 g Dp l
vt
C vs 1 0 .1 7 5 1
8 .5
v ls ,h h
2
l
g d vt
10 / 3
l g
2/3
(1)
The transition velocity between the heterogeneous regime and the homogeneous regime of Durand & Condolios (1952) and later Gibert (1960) is (The Wasp et al. (1977) related models also follow this equation):
3 v ls ,h h
K R sd
g D p R sd Cx
3/2
w ith :
K =85,
Cx
g d vt
(2)
The transition velocity between the heterogeneous regime and the homogeneous regime of Newitt et al. (1955) is: 3
v ls ,h h K
1
g Dp
vt
w ith :
K
1
1100
(3)
The transition velocity between the heterogeneous regime and the homogeneous regime of Fuhrboter (1961) is, using an approximation for the Sk value: 3
v ls ,h h
2 g Dp l R
sd
4 3 .5
1
Cx
R
sd
dm
l
g
1/3
2 g Dp Sk l R sd
(4)
The transition velocity between the heterogeneous regime and the homogeneous regime of Jufin & Lopatin (1966) is, with added terms to make the dimensions correct:
WODCON XXI PROCEEDINGS
3 v ls ,h h
2 82654
g d
3/4
vt
g Dp R sd
1/2
l g
2/3
C
vt
1/2
(5)
The transition velocity between the heterogeneous regime and the homogeneous regime of Zandi & Govatos (1967) is:
3 .8 6 v ls ,h h
g D p R sd 280 Cx
1 .9 3
1
(6)
R sd
The transition velocity between the heterogeneous regime and the homogeneous regime of Turian & Yuan (1977), with saltating transport can be described by:
v ls ,h h
g R sd D p
1 0 7 .1 C 1 .0 1 8 vt
l 4
0 .0 4 6 * 0 .4 2 1 3
CD
R sd C vt
1
1 / 1 .3 5 4
0 .5
(7)
The transition velocity between the heterogeneous regime and the homogeneous regime of Turian & Yuan (1977), with heterogeneous transport can be described by:
v ls ,h h
g R sd D p
3 0 .1 1 C 0 .8 6 8 vt
l 4
0 .2 0 0 * 0 .1 6 7 7
CD
R sd C vt
1
1 / 0 .6 9 3 8
0 .5
(8)
The simplified equation for heterogeneous transport of Wilson et al. (1992) for non-uniform PSD’s gives: 21
1
v ls ,h h 4 4 .1
sf g D p l
d 50
0 .3 5 1
(9)
The simplified equation for heterogeneous transport of Wilson et al. (1992) for uniform PSD’s gives: 2 1 .7
v ls ,h h
4 4 .1
1 .7
sf g D p l
d 50
0 .3 5 1 .7
(10)
It should be mentioned that this is based on the simplified model of Wilson et al. (1992) for heterogeneous transport. The full model as used in the graphs, may give slightly different results. The simplified equation for heterogeneous transport of Wilson et al. (2000) according to the Near Wall Lift model gives: 4
v ls ,h h
2 g Dp
2 l
d
1/2
(11)
The SRC model (Shook & Roco, 1991) consists of two terms regarding the excess hydraulic gradient. A term for the contact load and a term for the suspended load. Here only the term for the contact load is considered, assuming the contact load results in a small bed and there is no buoyancy from the suspended load. The SRC model results in an implicit relation if only the contact load is considered, this gives:
WODCON XXI PROCEEDINGS
0 .0 2 1 2 2
v ls ,h h e
v l s ,h h vt
sf
2 g Dp l
(12)
EXAMPLES TRANSITION HETEROGENEOUS TO HOMOGENEOUS The comparison is based on models for saltating or heterogeneous transport. As mentioned before, the transition line speed from heterogeneous to homogeneous transport is a good indicator for the excess pressure losses. A higher transition line speed indicates higher excess pressure losses. For a pipe diameter of 0.1016 m (4 inch) and medium sized particles (0.1 mm to 2 mm) all models are close for high concentrations (around 20%-30%), shown in Figure 3, which is used as a reference graph. This is caused by the fact that most experiments are carried out in small diameter pipes, resulting in a cloud of data points because of scatter. Many curves will fit through this cloud of data points. The Limit Deposit Velocity curve (according to DHLLDV), the particle size below which particles influence the viscosity, the transition to the sliding flow regime (d>0.015·Dp), see Wilson et al. (1992) and the intersection of a sliding bed with the ELM, are also shown giving the relevant area in the graph. The following graphs (Figure 2, Figure 3 and Figure 4) have a dimensionless vertical axis by dividing the transition line speed by the maximum transition line speed occurring in one of the 11 models. This maximum transition line speed (giving a relative line speed of 1) is shown in the lower right corner of each graph. For normal dredging operations, particle diameters from 0.1 mm up to 10 mm are of interest. The different models are compared with the DHLLDV Framework of Miedema & Ramsdell (2013). For very small pipe diameters (0.3 m) the Jufin & Lopatin (1966) model is representative, although this model tends to underestimate the pressure losses slightly for large pipe diameters and high concentrations. The Wilson et al. (1992) and the SRC models (Shook & Roco, 1991) have developed over the years and are based on many experimental data and can thus be considered to be representative in all cases for medium sized particles. For the 0.0254 m (1 inch) pipe, in the range of medium sized particles most models are very close. The Durand & Condolios model however seems to underestimate the transition velocity, while the Jufin & Lopatin and the Turian & Yuan 2 models overestimate the transition velocity. The SRC and the DHLLDV models are very close up to a particle size of about 2 mm. For the 0.1016 m (4 inch) pipe, most models are close for medium sized particles for higher concentrations. The Turian & Yuan 2 and the Jufin & Lopatin models depend on the concentration. The SRC and DHLLDV models match closely up to a particle diameter of about 1 mm. The Wilson et al. models matches both models for particle diameters close to 1 mm. For the 0.762 m (30 inch) pipe, the models deviate strongly. Durand & Condolios, Zandi & Govatos and Turian & Yuan 1 and 2 give high transition velocities. Fuhrboter and Wilson et al. -1.0 medium transition velocities, while DHLLDV, SRC, Wilson et al. -1.7 and Jufin & Lopatin (medium concentrations) give low transition velocities. Figure 5 shows the standard deviation of all 12 models. For pipes with a diameter close to 0.1016 m (4 inch) and particles with a diameter in the range of 0.3 mm to 1.5 mm the standard deviation is below 10% and the models give a comparable result in terms of the solids effect. For much smaller or larger pipes the standard deviation increases, which is also the case for very small or large particles. Figure 6 shows the standard deviation of the DHLLDV framework, the SRC model and the Wilson et al. models with a power of 1.7 and the near wall lift method (NWL). In the range of particle diameters from 0.2 mm to 1.5-4 mm the standard deviation is less than 10% and does not really depend on the pipe diameter. Very small particles show a high standard deviation, but the heterogeneous models are not applicable there. At normal operational line speeds, which are much higher than the transition line speeds found, the transport will be according to the homogeneous flow regime (ELM related). Very large particles show an increasing standard deviation, resulting from the fact that here the sliding bed or sliding flow regimes are applicable and not the heterogeneous flow regime. Although the physics behind the 4 models considered the DHLLDV framework, the SRC model and the 2 Wilson et al. models, are different, the 4 models give about the same result for medium sands in terms of the solids effect, irrespective of the pipe diameter.
WODCON XXI PROCEEDINGS
Transition Heterogeneous - Homogeneous Particle diameter d (mm) 0.01 1.00
0.1
1
10
100 Newitt et al.
0.95
Fuhrboter
Relative line speed vls,hh (-)
0.90 0.85
Durand & Condolios
0.80
Jufin & Lopatin
0.75
Turian & Yuan 1
0.70
Turian & Yuan 2
0.65
Zandi & Govatos
0.60
Wilson et al. - 1.0
0.55
Wilson et al. - 1.7
0.50
Wilson et al. NWL
0.45
SRC
0.40 0.35
DHLLDV
0.30
DHLLDV LDV
0.25
Transition Stokes=0.03
0.20
Transition d=0.015·Dp
0.15
Upper Limit SB=ELM
0.10
Standard Deviation 12
0.05 0.00 1.E-05
Standard Deviation 3 1.E-04
1.E-03
1.E-02
1.E-01
Particle diameter d (m) Dp=0.0254 m, Rsd=1.585, Cvs=0.300, μsf=0.416
© S.A.M.
vls,hh,max=5.6 m/sec
Figure 2. The transition line speed He-Ho for Dp=0.0254 m and Cv=0.30.
Transition Heterogeneous - Homogeneous Particle diameter d (mm) 0.01 1.00
0.1
1
10
100 Newitt et al.
0.95
Fuhrboter
Relative line speed vls,hh (-)
0.90 0.85
Durand & Condolios
0.80
Jufin & Lopatin
0.75
Turian & Yuan 1
0.70
Turian & Yuan 2
0.65
Zandi & Govatos
0.60
Wilson et al. - 1.0
0.55
Wilson et al. - 1.7
0.50
Wilson et al. NWL
0.45
SRC
0.40 0.35
DHLLDV
0.30
DHLLDV LDV
0.25
Transition Stokes=0.03
0.20
Transition d=0.015·Dp
0.15
Upper Limit SB=ELM
0.10
Standard Deviation 12
0.05 0.00 1.E-05
Standard Deviation 3 1.E-04
1.E-03
1.E-02
1.E-01
Particle diameter d (m) © S.A.M.
Dp=0.1016 m, Rsd=1.585, Cvs=0.300, μsf=0.416
vls,hh,max=9.5 m/sec
Figure 3. The transition line speed He-Ho for Dp=0.1016 m and Cv=0.30.
WODCON XXI PROCEEDINGS
Transition Heterogeneous - Homogeneous Particle diameter d (mm) 0.01 1.00
0.1
1
10
100 Newitt et al.
0.95
Fuhrboter
Relative line speed vls,hh (-)
0.90 0.85
Durand & Condolios
0.80
Jufin & Lopatin
0.75
Turian & Yuan 1
0.70
Turian & Yuan 2
0.65
Zandi & Govatos
0.60
Wilson et al. - 1.0
0.55
Wilson et al. - 1.7
0.50
Wilson et al. NWL
0.45
SRC
0.40 0.35
DHLLDV
0.30
DHLLDV LDV
0.25
Transition Stokes=0.03
0.20
Transition d=0.015·Dp
0.15
Upper Limit SB=ELM
0.10
Standard Deviation 12
0.05
Standard Deviation 3
0.00 1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
Particle diameter d (m) Dp=0.7620 m, Rsd=1.585, Cvs=0.300, μsf=0.416
© S.A.M.
vls,hh,max=19.6 m/sec
Figure 4. The transition line speed He-Ho for Dp=0.762 m and Cv=0.30.
Standard Deviation at Transition He-Ho, 12 Models
Dp=0.0254 m
1.00
Dp=0.0508 m
0.95 0.90 0.85
Homogeneous
Dp=0.1016 m
Sliding Bed
0.80
Dp=0.1524 m
Standard Deviation (-)
0.75 0.70
Dp=0.2032 m
Heterogeneous
0.65 0.60
Dp=0.2540 m
0.55 0.50
Dp=0.3000 m
0.45
Dp=0.4000 m
0.40 0.35
Dp=0.5000 m
0.30 0.25
Dp=0.6000 m
0.20 0.15
Dp=0.8000 m
0.10
0.05 0.00 0.00001
Dp=1.0000 m 0.00010
0.00100
0.01000
Particle Diameter (m)
Figure 5. The standard deviation of 12 models.
0.10000
Dp=1.2000 m
WODCON XXI PROCEEDINGS
Standard Deviation at Transition He-Ho, 4 Models
Dp=0.0254 m
1.00
Dp=0.0508 m
0.95 0.90 0.85
Homogeneous
Dp=0.1016 m
Sliding Bed
0.80
Dp=0.1524 m
Standard Deviation (-)
0.75 0.70
Dp=0.2032 m
Heterogeneous
0.65 0.60
Dp=0.2540 m
0.55 0.50
Dp=0.3000 m
0.45
Dp=0.4000 m
0.40 0.35
Dp=0.5000 m
0.30 0.25
Dp=0.6000 m
0.20 0.15
Dp=0.8000 m
0.10
0.05 0.00 0.00001
Dp=1.0000 m 0.00010
0.00100
0.01000
Particle Diameter (m)
0.10000
Dp=1.2000 m
Figure 6. The standard deviation of 4 models. THE LIMIT DEPOSIT VELOCITY The Limit Deposit Velocity is defined here as the line speed above which is no stationary bed or sliding bed, Thomas (1962). Below the LDV there may be either a stationary or fixed bed or a sliding bed. For the critical velocity often the Minimum Hydraulic Gradient Velocity (MHGV) is used, Wilson (1942). For higher concentrations this MHGV may be close to the LDV, but for lower concentrations this is certainly not the case. Yagi et al. (1972) reported using the MHGV, making the data points for the lower concentrations too low, which is clear from Figure 7. Another weak point of the MHGV is, that it depends strongly on the model used for the heterogeneous flow regime. Durand and Condolios (1952), Fuhrboter (1961), Jufin and Lopatin (1966) and others will each give a different MHGV. In dredging the process is instationary, meaning a constantly changing PSD and concentration in long pipelines, making it almost impossible to determine the MHGV. Wilson (1979) derived a method for determining the transition velocity between the stationary bed and the sliding bed, which is named here the Limit of Stationary Deposit Velocity (LSDV). Since the transition stationary bed versus sliding bed, the LSDV, will always give a smaller velocity value than the moment of full suspension or saltation, the LDV, one should use the LDV, to be sure there is no deposit at all. For small particles it is also possible that the bed is already completely suspended before the bed could ever start sliding (theoretically). In that case an LSDV does not even exist. This is the reason for choosing the LDV as the critical velocity, independent of the head loss model and always existing Miedema & Ramsdell (2015A). The Froude number FL is often used for the LDV, because it allows comparison of the LDV for different pipe diameters Dp and relative submerged densities Rsd without having to change the scale of the graph, this is defined as:
WODCON XXI PROCEEDINGS
FL
v ls , ld v 2g R
sd
(13)
Dp
It should be noted that sometimes the 2 and sometimes the relative submerged density Rsd are omitted. Because there are numerous data and equations for the critical velocity (LSDV, LDV or MHGV), some equations based on physics, but most based on curve fitting, a selection is made of the equations and methods from literature. The literature analyzed are from Wilson (1942), Durand and Condolios (1952), Newitt et al. (1955), Jufin and Lopatin (1966), Zandi and Govatos (1967), Charles (1970), Graf et al. (1970), Wilson and Judge (1976), Wasp et al. (1977), Wilson and Judge (1977), Thomas (1979), Oroskar and Turian (1980), Parzonka et al. (1981), Turian et al. (1987), Davies (1987), Schiller and Herbich (1991), Gogus and Kokpinar (1993), Gillies (1993), Berg (1998), Kokpinar and Gogus (2001), Shook et al. (2002), Wasp and Slatter (2004), Sanders et al. (2004), Lahiri (2009), Poloski et al. (2010) and Souza Pinto et al. (2014). The research consisted of two parts, analyzing the experimental data and analyzing the resulting models based on these data. First the experimental data will be discussed.
Durand Froude number FL (-) vs. Particle diameter d (m) 2.0
DHLLDV Jufin-Lopatin
1.8
Transition He-SF
Durand Froude number FL (-)
1.6
Thomas Kokpinar
1.4
Graf & Robinson Fuhrboter
1.2
Newitt 1.0
Sinclair
Blatch 0.8
Sassoli Silin
0.6
Gibert 0.4
Gillies Poloski
0.2 0.0 1.E-06
Smith Dp=0.075 m Smith Dp=0.050 m 1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
Particle diameter d (m) © S.A.M.
Dp=0.1524 m, Rsd=1.585, Cvs=0.175, μsf=0.416
Yagi Dp=0.0807 m Yagi Dp=0.1003 m Yagi Dp=0.1500 m
Figure 7. The Durand & Condolios Froude number FL, experimental data. Figure 7 shows many data points of various authors for sand and gravel in water. Each column of data points shows the results of experiments with different volumetric concentrations, where the highest points were at volumetric concentrations of about 15%–20%, higher concentrations gave lower points. The experimental data also shows that smaller pipe diameters, in general, give higher Durand and Condolios (1952) Froude FL numbers. The two curves in the graph are for the Jufin and Lopatin (1966) equation, which is only valid for sand and gravel, and the DHLLDV Framework which is described by Miedema & Ramsdell (2015A). Both models give a sort of upper limit to the LDV. The data points of the very small particle diameters, Thomas (1979) and Poloski (2010), were carried out in very small to medium diameter pipes, while the two curves are constructed for a 0.1524 (6 inch) pipe, resulting in slightly lower curves. Data points above the DHLLDV curve are in general for
WODCON XXI PROCEEDINGS
pipe diameters smaller than 0.1524 m. Some special attention is given to the relation between the Durand Froude FL number and the pipe diameter Dp. Both Thomas (1979) and Wasp et al. (1977) carried out research with a d = 0.18 mm particle in 6 pipe diameters. These experiments show a slight decrease of the FL value with increasing pipe diameter with a power close to –0.1, meaning the LDV increases with the pipe diameter to a power close to 0.4 and not to 0.5 as assumed by Durand and Condolios (1952).
Durand Froude number FL (-) vs. Particle diameter d (m) 2.5
Transition Stokes=0.03 Transition d=0.015·Dp Half Pipe Diameter
DHLLDV LDV DHLLDV LSDV
DHLLDV SB-He
Durand Froude number FL (-)
2.0
Jufin-Lopatin Wasp 1 Wasp 2 1.5
Wasp-Slatter Souza Pinto et al. Hepy Gogus-Kokpinar
1.0
Kokpinar-Gogus
van den Berg Turian et al.
0.5
Gillies Shook Poloski
0.0 1.E-05
Wilson LSDV 1.E-04
1.E-03
1.E-02
1.E-01
Particle diameter d (m)
D & C Dp=0.1524 m Yagi Dp=0.1524 m
© S.A.M.
Dp=0.0254 m, Rsd=1.585, Cvs=0.175, μsf=0.416
Poloski Converted
Figure 8. LDV FL curves for a 0.0254 m (1 inch) pipe. Figure 8, Figure 9 and Figure 10 show the Limit Deposit Velocities of DHLLDV, Durand and Condolios (1952), Jufin and Lopatin (1966), Wasp et al. (1970), Wasp and Slatter (2004), Souza Pinto et al. (2014), Hepy et al. (2008), Gogus and Kokpinar (1993), Kokpinar and Gogus (2001), Berg (1998), Turian et al. (1987) and Gillies (1993) for 3 pipe diameters. The figures also show the LSDV and the transition sliding bed to heterogeneous transport. The intersection between these two curves gives the smallest particle diameter where a sliding bed can exist. The curves of Hepy et al. (2008), Gogus and Kokpinar (1993) and Kokpinar and Gogus (2001) show a maximum FL value for particles with a diameter near d = 0.5 mm. However these models show an increasing FL value with the pipe diameter, which contradicts the numerous experimental data, showing a slight decrease. The models of Turian et al. (1987) , Wasp et al. (1970), Wasp and Slatter (2004) and Souza Pinto et al. (2014) show an increasing FL value with increasing particle diameter and a slight decrease with the pipe diameter. Jufin and Lopatin (1966) show an increase with the particle diameter and a slight decrease with the pipe diameter (power –1/6). The model of van den Berg (1998) shows an increasing FL with the particle diameter, but no dependency on the pipe diameter. Durand and Condolios (1952) did not give an equation but a graph. The data points as derived from the original publication in (1952) and from Durand (1953) are shown in the graphs. The data points show a maximum for d = 0.5 mm. They did not report any dependency on the pipe diameter. The model of Gillies (1993) tries to quantify the Durand and Condolios (1952) data points, but does not show any dependency on the pipe diameter for the FL Froude number. The increase of the FL value with the pipe diameter of the Hepy et al. (2008), Gogus and Kokpinar (1993) and Kokpinar and Gogus (2001) models is probably caused by the forced d/Dp relation. With a strong relation with the particle diameter and a weak relation for the pipe diameter, the pipe diameter will follow the particle diameter. Another reason may be the fact that they used pipe diameters up to 0.1524 m (6 inch) and the smaller the
WODCON XXI PROCEEDINGS
pipe diameter the more probable the occurrence of a sliding bed and other limiting conditions, due to the larger hydraulic gradient helping the bed to start sliding. The figures show that for medium pipe diameters all models are close. The reason is probably that most experiments are carried out with medium pipe diameters. Only Jufin and Lopatin (1966) covered a range from 0.02 m to 0.9 m pipe diameters. Recently Thomas (2014) gave an overview and analysis of the LDV (or sometimes the LSDV). He repeated the findings that the LDV depends on the pipe diameter with a power smaller than 0.5 but larger than 0.1. The value of 0.1 is for very small particles, while for normal sand and gravels a power is expected between 1/3 according to Jufin and Lopatin (1966) and 1/2 according to Durand and Condolios (1952). Most equations are one term equations, making it impossible to cover all aspects of the LDV behavior. Only Gillies (1993) managed to construct an equation that gets close to the original Durand and Condolios (1952) graph.
Durand Froude number FL (-) vs. Particle diameter d (m) 2.5
Transition Stokes=0.03 Transition d=0.015·Dp Half Pipe Diameter
DHLLDV LDV DHLLDV LSDV
DHLLDV SB-He
Durand Froude number FL (-)
2.0
Jufin-Lopatin Wasp 1 Wasp 2 1.5
Wasp-Slatter Souza Pinto et al. Hepy Gogus-Kokpinar
1.0
Kokpinar-Gogus
van den Berg Turian et al.
0.5
Gillies Shook Poloski
0.0 1.E-05
Wilson LSDV 1.E-04
1.E-03
1.E-02
1.E-01
Particle diameter d (m)
D & C Dp=0.1524 m Yagi Dp=0.1524 m
© S.A.M.
Dp=0.1524 m, Rsd=1.585, Cvs=0.175, μsf=0.416
Poloski Converted
Figure 9. LDV FL curves for a 0.1524 m (6 inch) pipe. The models analyzed result in a number of dominating parameters. These are the particle diameter d, the pipe diameter Dp, the liquid density ρl and kinematic viscosity νl, the solids density ρs, the sliding friction coefficient μsf, the bed concentration Cvb and the spatial volumetric concentration Cvs. Derived parameters are the relative submerged density Rsd, the Darcy Weisbach friction factor for pure liquid flow λl and the thickness of the viscous sub-layer δv. Dimensionless numbers are not considered at first, since they may lead to wrong interpretations. Lately Lahiri (2009) performed an analysis using artificial neural network and support vector regression. Azamathulla and Ahmad (2013) performed an analysis using adaptive neuro-fuzzy interference system and gene-expression programming. Although these methodologies may give good correlations, they do not explain the physics. Lahiri (2009) however did give statistical relations for the dependency on the volumetric concentration, the particle diameter, the pipe diameter and the relative submerged density. Resuming, the following conclusions can be drawn for sand and gravel: The pipe diameter Dp: The LDV is proportional to the pipe diameter Dp to a power between 1/3 and 1/2 (about 0.4) for small to large particles (Thomas (1979), Wasp et al. (1977), Lahiri (2009) and Jufin and Lopatin (1966)) and a power of about 0.1 for very small particles (Thomas (1979), Wilson and Judge (1976), Sanders et al. (2004) and Poloski et al. (2010)).
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The particle diameter d: The LDV has a lower limit for very small particles, after which it increases to a maximum at a particle diameter of about d = 0.5 mm (Thomas (1979), Thomas (2014), Durand and Condolios (1952), Gillies (1993) and Poloski et al. (2010)). For medium sized particles with a particle size d > 0.5 mm, the FL value decreases to a minimum for a particle size of about d = 2 mm (Durand and Condolios, 1952; Gillies, 1993; Poloski et al., 2010). Above 2 mm, the FL value will remain constant according to Durand and Condolios (1952) and Gillies (1993). For particles with d/Dp > 0.015, the Wilson et al. (1992) criterion for real suspension/saltation, the FL value increases again. This criterion is based on the ratio particle diameter to pipe diameter and will start at a large particle diameter with increasing pipe diameter. Yagi et al. (1972) reported many data points in this region showing an increasing FL value. The relative submerged density Rsd: The relation between the LDV and the relative submerged density is not very clear, however the data shown by Kokpinar and Gogus (2001) and the conclusions of Lahiri (2009) show that the FL value decreases with increasing solids density and thus relative submerged density Rsd to a power of –0.1 to –0.2. The spatial volumetric concentration Cvs: The volumetric concentration leading to the maximum LDV is somewhere between 15% and 20% according to Durand and Condolios (1952). Lahiri (2009) reported a maximum at about 17.5%, while Poloski et al. (2010) derived 15%. This maximum LDV results from on one hand a linear increase of the sedimentation with the concentration and on the other hand a reduced sedimentation due to the hindered setting. These two counteracting phenomena result in a maximum, which is also present in the equation of the potential energy of the DHLLDV Framework. For small concentrations a minimum LDV is observed by Durand and Condolios (1952). This minimum LDV increases with the particle diameter and reaches the LDV of 20% at a particle diameter of 2 mm with a pipe diameter of 0.1524 m (6 inch). For the dredging industry the Jufin and Lopatin (1966) equation gives a good approximation for sand and gravel, although a bit conservative. The model of van den Berg (1998) is suitable for large diameter pipes as used in dredging for sand and/or gravel, but underestimates the LDV for pipe diameters below 0.8 m. Both models tend to underestimate the LDV for particle diameters below 1 mm.
Durand Froude number FL (-) vs. Particle diameter d (m) 2.5
Transition Stokes=0.03 Transition d=0.015·Dp Half Pipe Diameter
DHLLDV LDV DHLLDV LSDV
DHLLDV SB-He
Durand Froude number FL (-)
2.0
Jufin-Lopatin Wasp 1 Wasp 2 1.5
Wasp-Slatter Souza Pinto et al. Hepy Gogus-Kokpinar
1.0
Kokpinar-Gogus
van den Berg Turian et al.
0.5
Gillies Shook Poloski
0.0 1.E-05
Wilson LSDV 1.E-04
1.E-03
1.E-02
1.E-01
Particle diameter d (m)
D & C Dp=0.1524 m Yagi Dp=0.1524 m
© S.A.M.
Dp=0.7620 m, Rsd=1.585, Cvs=0.175, μsf=0.416
Figure 10. LDV FL curves for a 0.762 m (30 inch) pipe.
Poloski Converted
WODCON XXI PROCEEDINGS
FLOW REGIME DIAGRAMS Now knowing the transition velocity between the heterogeneous and the homogeneous flow regimes and also knowing the LDV and the definition of the LSDV, what does this mean for the occurrence of the flow regimes. At low line speeds there will be a stationary bed. When the line speed increases, for small particles there will be a transition directly to the heterogeneous regime, while for larger particles a sliding bed will occur. The sliding bed will show a transition to the heterogeneous regime for large particles and to the sliding flow regime for very large particles with further increasing line speed. When the line speed increases more, the heterogeneous regime will show a transition to the homogeneous regime. In the homogeneous regime, very small particles may influence the viscosity and density of the carrier liquid. There are no absolute values for the transition particle diameters of very small to small to large to very large particles. These transition values depend on liquid properties, the pipe diameter, the relative submerged density and the volumetric concentration.
DHLLDV Flow Regime Diagram Particle diameter (mm) 0.01 3.0
0.1
1
10
100
Undefined 6.0
2.5
He-Ho Sliding Flow (SF)
5.0
Homogeneous (Ho)
SB-SF
Heterogeneous (He) 1.5
4.0
LDV
Viscous Effects
3.0
SB-He Sliding Bed (SB) 1.0 2.0
FB-SB (LSDV) 0.5
FB-He 1.0
Fixed/Stationary Bed (FB)
0.0 1.E-05
1.E-04
1.E-03
1.E-02
Particle diameter d (m) © S.A.M.
Dp=0.1524 m, Rsd=1.585, Cvs=0.175, μsf=0.416
Figure 11. The flow regime diagram for a 0.1524 m (6 inch) pipe, with LDV data of Durand & Condolios, Yagi et al. and Poloski in a 0.1524 m pipe.
0.0 1.E-01
Line speed (m/s)
Durand Froude number FL (-)
2.0
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Figure 11 and Figure 12 show the flow regime diagrams for a medium 0.1524 (6 inch) pipe and a large 0.762 m (30 inch) pipe, at a spatial volumetric concentration of 0.175 for sands and gravels, based on the DHLLDV Framework. The figures show that for the medium pipe, a sliding bed will occur for particles larger than 0.35 mm. For the large pipe this starts for particles larger than 2 mm. In fact, for the large pipe the sliding bed region is much smaller compared to the medium pipe. At operational line speeds just above the LDV (7 m/s) for the large pipe, medium to large particles (0.25-10 mm) will show heterogeneous transport and only small particles, smaller than 0.25 mm, will show homogeneous transport. Very large particles (>10 mm) will show sliding flow transport. For the medium pipe, particles smaller than 0.2 mm will show homogeneous transport at an operational line speed of about 3.3 m/s. Particles from 0.2 mm to 2 mm will show heterogeneous transport. Larger particles (>2 mm) will show sliding flow transport. This shows how the transition particle diameters shift as a function of the pipe diameter. Increasing the spatial volumetric concentration will increase the sliding bed region, but decrease the LDV curve.
DHLLDV Flow Regime Diagram Particle diameter (mm) 0.01 3.0
0.1
1
10
100
14.0
Undefined
13.0 2.5
12.0
11.0
Homogeneous
9.0
8.0 1.5
Viscous Effects
Heterogeneous 7.0
6.0
LDV 5.0
1.0
4.0
Sliding Bed LSDV
3.0
0.5 2.0
Stationary Bed 1.0
0.0 1.E-05
1.E-04
1.E-03
1.E-02
Particle diameter d (m) © S.A.M.
Dp=0.7620 m, Rsd=1.585, Cvs=0.175, μsf=0.416
Figure 12. The flow regime diagram for a 0.762 m (30 inch) pipe, with LDV data of Durand & Condolios, Yagi et al. and Poloski in a 0.1524 m pipe.
0.0 1.E-01
Line speed (m/s)
Durand Froude number FL (-)
10.0
Sliding Flow
2.0
WODCON XXI PROCEEDINGS
CONCLUSIONS The transition velocity of the heterogeneous regime equations with the ELM equation seems to be a good indicator for comparing the different models. A requirement is of course that the heterogeneous component can be isolated. The hydraulic gradient or head losses can be easily determined at the transition velocity, since the ELM is valid in this intersection point. The solids effect of the hydraulic gradient or head losses is proportional to the transition velocity squared. Based on the amount of experimental data, the SRC model and the Wilson et al. models seem to be the most reliable models from literature. The DHLLDV Framework is very close to these models for medium sized particles, although DHLLDV is based on kinetic energy losses, while both SRC and Wilson et al. are based on a diminishing bed. Models are based on the terminal settling velocity or models are based on the particle Froude number. The difference between the two groups is, that the terminal settling velocity continuously increases with the particle diameter, while the particle Froude number and the particle drag coefficient have a constant asymptotic value for large particle diameters. So models from the first group tend to overestimate the transition line speed for large particles. This is however not surprising, since both Newitt et al. (1955), SRC and Wilson et al. (1992) use a 2LM or sliding bed model for this case, so the large particle part of the curves is not relevant. The small particle part of the graphs is also not relevant, since the transport regime will be homogeneous at normal operational line speeds, which are much higher than the transition line speeds found for small particles. In general one can say that the transition line speed is proportional to the pipe diameter with a power of 0.3 to 0.4, based on the Wilson et al. (1992) models, the SRC model and the DHLLDV Framework. It depends weakly on the spatial volumetric concentration. There is no or hardly no influence of the sliding friction coefficient. There is a weak or no direct dependency on the relative submerged density. Some models give a weak indirect dependency on the particle Froude number. In general one can say that the more negative the power of the line speed in the heterogeneous hydraulic gradient equation, the smaller the transition velocity between the regimes. One has to take this into account interpreting the graphs. The Limit Deposit Velocity (LDV) and the Limit of Stationary Deposit Velocity (LSDV) are distinguished. Wilson et al. (1992) use the LSDV, while Durand & Condolios (1952) and the DHLLDV Framework use the LDV. The LDV will always give higher FL values than the LSDV. Not because of scatter or because of a safety factor, but because of a different definition. The LDV (the line speed above which there is no stationary of sliding bed) always exists, the LSDV only at the lower boundary of a sliding bed region. It should be noted that for small particles, the LDV and the transition of the heterogeneous regime to the ELM are very close. For very small particles however viscous effects may play a role, which is not taken into consideration here. The flow regime diagrams are an application of knowing all the different transition line speeds, the LDV and the LSDV. The examples for a medium and a large pipe show that both flow regime diagrams contain all regions, but of different size and at different boundaries. So knowing the flow regime diagrams is very important for identifying the flow regime that may be expected for different particle sizes and line speeds. Other spatial volumetric concentrations will give different flow regime diagrams. The DHLLDV Framework is a very good alternative to the well-known models from Wilson et al. (1992), SRC and others. The underlying theory is described in the book “Slurry Transport: Fundamentals, a Historical Overview and the Delft Head Loss & Limit Deposit Velocity Framework” from the authors of this paper. More graphs and other information can be found on www.dhlldv.com.
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REFERENCES Azamathulla, H. M., & Ahmad, Z. (2013). Estimation of critical velocity for slurry transport through pipeline using adaptive neuro-fuzzy interference system and gene-expression programming. Journal of Pipeline Systems Engineering and Practice., 131-137. Berg, C. H. (1998). Pipelines as Transportation Systems. Kinderdijk, the Netherlands: European Mining Course Proceedings, IHC-MTI. Charles, M. E. (1970). Transport of solids by pipeline. Hydrotransport 1. Cranfield: BHRA. Davies, J. T. (1987). Calculation of critical velocities to maintain solids in suspension in horizontal pipes. Chemical Engineering Science, Vol. 42(7)., 1667-1670. Durand, R. (1953). Basic Relationships of the Transportation of Solids in Pipes - Experimental Research. Proceedings of the International Association of Hydraulic Research. Minneapolis. 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. Fuhrboter, A. (1961). Über die Förderung von Sand-Wasser-Gemischen in Rohrleitungen. Mitteilungen des FranziusInstituts, H. 19. Gibert, R. (1960). Transport hydraulique et refoulement des mixtures en conduites. Annales des Ponts et Chausees., 130(3), 307-74, 130(4), 437-94. Gillies, R. G. (1993). Pipeline flow of coarse particles, PhD Thesis. Saskatoon: University of Saskatchewan. Gogus, M., & Kokpinar, M. A. (1993). Determination of critical flow velocity in slurry transporting pipeline systems. Proceeding of the 12th International Conference on Slurry Handling and Pipeline Transport. (pp. 743-757). Bedfordshire, UK.: British Hydraulic Research Group. Graf, W. H., Robinson, M., & Yucel, O. (1970). The critical deposit velocity for solid-liquid mixtures. Hydrotransport 1 (pp. H5-77-H5-88). Cranfield, UK: BHRA. Hepy, F. M., Ahmad, Z., & Kansal, M. L. (2008). Critical velocity for slurry transport through pipeline. Dam Engineering, Vol. XIX(3)., 169-184. Jufin, A. P., & Lopatin, N. A. (1966). O projekte TUiN na gidrotransport zernistych materialov po stalnym truboprovodam. Gidrotechniceskoe Strojitelstvo, 9., 49-52. Kokpinar, M. A., & Gogus, M. (2001). Critical velocity in slurry transport in horizontal pipelines. Journal of Hydraulic Engineering, Vol. 127(9)., 763-771. Lahiri, S. K. (2009). Study on slurry flow modelling in pipeline. Durgapur, India: National Institute of Technology, Durgapur, India. Miedema, S. A. (2013). An overview of theories describing head losses in slurry transport. A tribute to some of the early researchers. OMAE 2013, 32nd International Conference on Ocean, Offshore and Arctic Engineering. (p. 18). Nantes, France: ASME. Miedema, S. A. (2014A). An overview of theories describing head losses in slurry transport - A tribute to some of the early researchers. Journal of Dredging Engineering, Vol. 14(1), 1-25. Miedema, S. A. (2014B). An analysis of slurry transport at low line speeds. ASME 2014 33rd International Conference on Ocean, Offshore and Arctic Engineering, OMAE. (p. 11). San Francisco, USA.: ASME. Miedema, S. A. (2014C). An analytical approach to explain the Fuhrboter equation. Maritime Engineering, Vol. 167(2)., 1-14.
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Miedema, S. A. (2015A). A head loss model for homogeneous slurry transport. Journal of Hydrology & Hydrodynamics, Vol. 1., 14 pages. Miedema, S. A. (2015B). Head Loss Model for Slurry Transport in the Heterogeneous Regime. Journal of Ocean Engineering, Vol. 106, 360-370. Miedema, S. A. (2015C). The slip ratio or holdup function in slurry transport. Dredging Summit and Expo 2015. (p. 12). Houston, Texas, USA.: WEDA. Miedema, S. A. (2015D). A method to compare slurry transport models. 17th International Conference on Transport & Sedimentation of Solid Particles. (p. 8). Delft, The Netherlands, September 2015.: T&S17. Miedema, S. A., & Matousek, V. (2014). An explicit formulation of bed friction factor for sheet flow. International Freight Pipeline Society Symposium, 15th. (p. 17 pages). Prague, Czech Republic: IFPS. Miedema, S. A., & Ramsdell, R. C. (2013). A head loss model for slurry transport based on energy considerations. World Dredging Conference XX (p. 14). Brussels, Belgium: WODA. Miedema, S. A., & Ramsdell, R. C. (2014A). The Delft Head Loss & Limit Deposit Velocity Model. Hydrotransport (p. 15). Denver, USA.: BHR Group. Miedema, S. A., & Ramsdell, R. C. (2014B). An Analysis of the Hydrostatic Approach of Wilson for the Friction of a Sliding Bed. WEDA/TAMU (p. 21). Toronto, Canada: WEDA. Miedema, S. A., & Ramsdell, R. C. (2015A). The Limit Deposit Velocity Model, a New Approach. Journal of Hydrology & Hydromechanics, Vol 63(4), 273-286. Miedema, S. A., & Ramsdell, R. C. (2015B). The Delft Head Loss & Limit Deposit Velocity Framework. Journal of Dredging Engineering, Vol. 15(2)., pp. 28. Newitt, D. M., Richardson, M. C., Abbott, M., & Turtle, R. B. (1955). Hydraulic conveying of solids in horizontal pipes. Transactions of the Institution of Chemical Engineers Vo.l 33., 93-110. Oroskar, A. R., & Turian, R. M. (1980). The hold up in pipeline flow of slurries. AIChE, Vol. 26., 550-558. Parzonka, W., Kenchington, J. M., & Charles, M. E. (1981). Hydrotransport of solids in horizontal pipes: Effects of solids concentration and particle size on the deposit velocity. Canadian Journal of Chemical Engineering, Vol. 59., 291-296. Poloski, A. P., Etchells, A. W., Chun, J., Adkins, H. E., Casella, A. M., Minette, M. J., & Yokuda, S. (2010). A pipeline transport correlation for slurries with small but dense particles. Canadian Journal of Chemical Engineering, Vol. 88., 182-189. Ramsdell, R. C., & Miedema, S. A. (2013). An overview of flow regimes describing slurry transport. WODCON XX (p. 15). Brussels, Belgium.: WODA. Sanders, R. S., Sun, R., Gillies, R. G., McKibben, M., Litzenberger, C., & Shook, C. A. (2004). Deposition Velocities for Particles of Intermediate Size in Turbulent Flows. Hydrotransport 16 (pp. 429-442). Santiago, Chile.: BHR Group. Schiller, R. E., & Herbich, J. B. (1991). Sediment transport in pipes. Handbook of dredging. New York: McGrawHill. Shook, C. A., Gillies, R. G., & Sanders, R. S. (2002). Pipeline Hydrotransport with Application in the Oil Sand Industry. Saskatoon, Canada: Saskatchewan Research Council, SRC Publication 11508-1E02. Shook, C., & Roco, M. (1991). Slurry Flow, Principles & Practice. Boston: Butterworth Heineman.
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Souza Pinto, T. C., Moraes Junior, D., Slatter, P. T., & Leal Filho, L. S. (2014). Modelling the critical velocity for heterogeneous flow of mineral slurries. International Journal of Multiphase Flow., 31-37. Thomas, A. D. (1979). Predicting the deposit velocity for horizontal turbulent pipe flow of slurries. International Journal of Multiphase Flow, Vol. 5., 113-129. Thomas, A. D. (2014). Slurries of most interest to the mining industry flow homogeneously and the deposit velocity is the key parameter. HydroTransport 19. (pp. 239-252). Denver, Colorado, USA.: BHR Group. Thomas, D. G. (1962). Transport Characteristics of Suspensions: Part VI. Minimum velocity for large particle size suspensions in round horizontal pipes. A.I.Ch.E. Journal, Vol.8(3)., 373-378. Turian, R. M., & Yuan, T. F. (1977). Flow of slurries in pipelines. AIChE Journal, Vol. 23., 232-243. Turian, R. M., Hsu, F. L., & Ma, T. W. (1987). Estimation of the critical velocity in pipeline flow of slurries. Powder Technology, Vol. 51., 35-47. Wasp, E. J., & Slatter, P. T. (2004). Deposition velocities for small particles in large pipes. 12th International Conference on Transport & Sedimentation of Solid Particles, (pp. 20-24). Prague, Czech Republic. Wasp, E. J., Kenny, J. P., & Gandhi, R. L. (1977). Solid liquid flow slurry pipeline transportation. Transactions Technical Publications. Wasp, E. J., Kenny, J. P., Aude, T. C., Seiter, R. H., & Jacques, R. B. (1970). Deposition velocities transition velocities and spatial distribution of solids in slurry pipelines. Hydro Transport 1, paper H42. (pp. 53-76). Coventry: BHRA Fluid Engineering. Wilson, K. C. (1979). Deposition limit nomograms for particles of various densities in pipeline flow. Hydrotransport 6 (p. 12). Canterbury, UK: BHRA. Wilson, K. C., & Judge, D. G. (1976). New Techniques for the Scale-Up of Pilot Plant Results to Coal Slurry Pipelines. Proceedings International Symposium on Freight Pipelines. (pp. 1-29). Washington DC, USA: University of Pensylvania. Wilson, K. C., & Judge, D. G. (1977). Application of Analytical Model to Stationary Deposit Limit in Sand Water Slurries. Dredging Technology (pp. J1 1-12). College Station, Texas, USA.: BHRA Fluid Engineering. Wilson, K. C., Addie, G. R., & Clift, R. (1992). Slurry Transport using Centrifugal Pumps. New York: Elsevier Applied Sciences. Wilson, K. C., Sellgren, A., & Addie, G. R. (2000). Near-wall fluid lift of particles in slurry pipelines. 10th Conference on Transport and Sedimentation of Solid Particles. Wroclav, Poland: T&S10. Wilson, W. E. (1942). Mechanics of flow with non colloidal inert solids. Transactions ASCE Vol. 107., 1576-1594. Yagi, T., Okude, T., Miyazaki, S., & Koreishi, A. (1972). An Analysis of the Hydraulic Transport of Solids in Horizontal Pipes. Nagase, Yokosuka, Japan.: Report of the Port & Harbour Research Institute, Vol. 11, No. 3. Zandi, I., & Govatos, G. (1967). Heterogeneous flow of solids in pipelines. Proc. ACSE, J. Hydraul. Div.,, 93(HY3)., 145-159.
CITATION Miedema, S.A. & Ramsdell, R.C. “A comparison of different slurry transport models for sands and gravels”. Proceedings of the Twenty-First World Dredging Congress, WODCON XXI, Miami, Florida, USA, June 1317, 2016.
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NOMENCLATURE CD Cvt Cvs Cvb Cx d Dp FL g il im K K1 Rsd Sk vls vls,hh vls,ldv vt β λl μsf νl ζ
Particle drag coefficient Delivered (transport) volumetric concentration Spatial volumetric concentration Spatial volumetric concentration bed (1-n) Durand & Condolios reversed particle Froude number Particle diameter Pipe diameter Durand Froude number Gravitational constant 9.81 m/s2 Pure liquid hydraulic gradient Mixture hydraulic gradient Durand & Condolios constant Newitt constant Relative submerged density Fuhrboter particle coefficient Line speed Intersection velocity heterogeneous-homogeneous regimes Limit Deposit Velocity Terminal settling velocity particle Richardson & Zaki hindered settling power Darcy Weisbach friction factor liquid Sliding friction factor Kinematic viscosity liquid Fit function Fuhrboter
m m N m/s2 m/m m/m
-
m/s m/s m/s m/s m2/s -
