Pipe Flow of Solid-Liquid Suspensions With a Broad Size Range-M.G. Rasteiro-proceedings

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International Congress for Particle Technology

Alexandre R. G., M. G. Rasteiro Chemical Engineering Department Coimbra University Plo II, Pinhal de Marrocos 3030-290 Coimbra Portugal Proceeding Session: 23 Multiphase Flow/CFD Internal Number: 195 Pipe Flow of Solid-Liquid Suspensions with a Broad Size Range

Pipe Flow of Solid-Liquid Suspensions with a Broad Size Range Alexandre R.G., M.G. Rasteiro* Chemical Engineering Department, Coimbra University Coimbra- Portugal KEYWORDS: Solid-Liquid flow, Settling-Dispersion model, Polydispersed particles. 1- Introduct ion Designing two-phase flow systems requires a good understanding of the mechanisms of particle suspension. Several strategies can be adopted, going from the empirical approximation, to the approach that considers each particle individually and takes into account all the forces acting on the particles. The first approach is of limited application. In fact, though several empirical equations have been developed over the years, the correlation which is still most widely used to predict pressure drop in a hydraulic transport pipe is the Durand equation [1], because of the large range of experimental measurements that support it. On the other hand, the second approach is not yet capable of modelling real multiparticle systems, requiring large computing facilities. An alternative way to model these systems is to adapt the equations normally used for fluid flow, in order to take into consideration the presence of the particles. In this work we have followed this last strategy to model the flow of heterogeneous solid-liquid suspensions in pipes. The model adopted was the Settling-Dispersion model, which is a lumped parameter model, based on the balance that builds up in the cross-section of the pipe between the tendency of the particles to settle, due to gravity, and their tendency to disperse from the regions of higher concentration, due to random motions. Other authors have used momentum balance equations to particles and fluid to describe the flow of the suspension in the pipe [2]. Once again, they are faced with the need to define several parameters, which are difficult to estimate for multiparticle systems, to solve the resulting system of equations. The Settling-Dispersion model has proved adequate to predict both one and two dimensional solids distributions in the cross-section of the conveying pipe, that can be used to better calculate the pressure drop in the system. So far, the model has been used to describe the flow of solid-liquid suspensions with a narrow size solid phase, that can be represented by a single diameter (monodispersed model). However, when real systems are being modelled, there is always a range of particle diameters, the previous approach becoming inadequate. The results that will be reported were obtained by modifying the Settling-Dispersion model, in order to account for particulate phases with a wide particle size range (polydispersed model). By doing so, it was possible to calculate, for the cross-section of the conveying pipe, both the total solids distribution (one and two dimensional profiles) and the distribution of the various size fractions of the solid phase, with different particle diameters. The results obtained predict the existence, in the pipe, of segregation between the small and large particles. 2- The Sett l ing-Dispersion Model This model considers, as referred before, that for particles turbulently conveyed in a horizontal pipe, a dynamic equilibrium is reached between the tendency of the particles to settle due to gravity, and their tendency to

Nuremberg, Germany, 27-29 March 2001

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disperse from the regions of higher concentration, due to random motions. This behaviour is strongly controlled by the properties of the particles and by the geometry of the pipe. The settling-dispersion model was initially formulated by OBrien for the unidimensional situation [3] and, since then, was used and improved by other authors [4,5,6]. In previous works we have extended the model to the two-dimensional situation. Additionally, we have also shown elsewhere [7] that a better fit to the experimental data is obtained when the model parameters (settling velocity and dispersion coefficient) are considered dependent on the local conditions in the cross-section of the pipe, namely on local particle concentration. This model can be mathematically translated by the convection-dispersion equation which, in steady state, can be written as:

( ) ( ) 0=+ cvcD (1) where D is the dispersion tensor, c is the local particle concentration (v/v) and v is the velocity vector. This equation can be solved, for an horizontal pipe, in cylindrical coordinates (r,,z). The boundary conditions are described in more detail elsewhere [7]. In the present work we have reformulated the model to adapt it to the more realistic case where the solids exhibit a range of diameters. Rewriting equation (1) for the particles of class i, diameter dpi, for a system with N size classes, we have:

( ) ( ) 0=+ iiii cvcD (2) where D i is now the dispersion tensor for species i, v i is the velocity vector for the particles with a diameter dpi and ci is the local volumetric concentration of those particles. To obtain the distribution of the solids it is necessary to solve, simultaneously, the N equations for the different particle size classes of the solid phase. The simplifications introduced, including the assumption of isotropic dispersion, and the boundary conditions, were the same that were adopted for monodispersed systems [7]. Thus, the simplified convection-dispersion equation, in cylindrical coordinates, for the generic species i, is:

0

1coscos

2

2

22

2

2

=+

+

+

+

+

++

+

iii

i

iiiiiiisiisi

siisii

crD

rcD

rc

rDcD

rrc

rDc

rv

rc

sinvv

rc

rv

sinc(3)

where Di is the dispersion coefficient for species i and vsi is the settling velocity for the same species. The total volumetric concentration of the solids, for each nodal point of the cross-section of the pipe, is given by the sum of the concentration of all the species:

=

=N

iicc

1

(4)

Once again the model parameters were considered dependent on local conditions. Additionally, since the settling tendency of each particle will be influenced by the presence of the other species, the equation proposed by Masliyah [8] has been used to take into account those interactions:

(5) ( ) ( ) ( ) 11

1 1'1'1v

=

= ki noiNik

kk

noiisi cvccvc

In this equation n i is a function of the particle Reynolds number, v 'oi is the settling velocity of the particles with diameter dpi , for infinite dilution. The correlation selected for the calculation of the dispersion coefficient was the same that has been used for monodispersed systems [7], which was adopted by analogy with the liquid-liquid diffusion process:

( ) 20 nMi ccDDD == (6)


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In fact, as referred in previous publications, the empirical constant D0 does not depend on the type of particles and, though n 2 may be a function of the particle characteristics this relation is not yet explicit. In equation (6) cM is the packing concentration of the particles. The system of non-linear algebraic equations was solved by the Brent algorithm [9], which is a software package that uses a modification of the discrete Newton method. In previous works [10] we have developed a pressure drop correlation that takes into account the solids distribution in the pipe cross-section. That equation considers that the total head-loss is the result of three contributions, as happens in a fluidised bed:

44444 344444 2144 344 2144 344 21pE

eractionsparticleparticletoduelosshead

vE

lossheadenergyviscous

cE

lossheadenergykinetic

lossheadtotal

+

+

=

int/

This equation has been fitted to a set of 60 experimental measurements leading to the following correlation:

p2

vc3

susp

E105.539E382.447E105.825LP ++=

(7) Once the solids distribution in the pipe is known, the pipe cross-section will be horizontally discretized, and equation (7) will be applied to each slice of pipe, the total pressure drop being the sum of these contributions. In this paper we will also present, for comparison, pressure drop values calculated using the Durand equation. 3- Simulated Concentration Profi les In this section we will present a selection of the simulated results produced. We will start by showing two examples of solids distribution in the cross-section of a horizontal conveying pipe, that have been obtained assuming the particulate phase to be monodispersed, that is, associating a single diameter to the solid phase. For those examples we will show, in Fig. 1, both the 1D and 2D distributions, and also the calculated pressure drop values using either the Durand equation or equation (7). The 1D distributions and the pressure drop values will be compared with experimental data from the literature. The remaining figures show the simulated distributions obtained supposing the particles to be polydispersed (with a range of particle diameters), which have been calculated using equation (3). The tests presented show the influence of the particle size range on the solids distribution in the pipe, namely on the extent of the segregation that can develop in the cross-section of the pipe. The influence, on the calculated profiles, of the size of the class ranges which have been considered during the calculations, will also be discussed. The distributions obtained by assuming the particles to be polydispersed will be compared with the results from the monodispersed model. Furthermore, a comparison with experimental measurements will also be attempted, though the experimental data for polydispersed systems is rather difficult to obtain, and thus, quite scarce in the literature. 3.1- Monodispersed Systems Table 1 summarises the operating conditions of the experimental tests that we have used as input for the simulation studies [11]. The calculated profiles were obtained using the model for monodispersed systems.

Table 1: Summary of experimental variables and model parameters PARTICLE

CHARACTERISTICS FLUID

CHARACTERISTICS OPERATING CONDITIONS

TEST s (kg/m3)

Diameter (m)

vo (m/s)

Rep L (kg/m3)

105(Ns/m2)

Pipe Diam. (m)

vf(m/s)

Av. Conc. (% vol.)

1 2650 297 0.039 10.24 999.13 114 0.0532 3.05 15 2 2650 525 0.078 36.10 999.13 114 0.0532 3.05 15

MODEL PARAMETERS TEST Do 104

(m2/s) n2 Do/vo 102

(m) Calculated Average Concentration (% vol)

1 6.25 1.2 1.59 14.99 2 11.5 1.2 1.47 15.25


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The obtain the best agreement between the simulated and the expeconvIn Fprofexpeprofgoodsmalthe pthanMakthe ithe mcrosdistrmod

Conc

entr

atio

n (%

Vol.

)Co

ncen

trat

ion

(%Vo

l.)

Nuremvalue of Do was adjusted for each test in order to

rimental profiles. However, it was obvious a coherent variation of that parameter as a function of the eying velocity, larger conveying velocities implying larger values for Do.

ig. 1 the simulated profiles for the operating conditions of the tests in Table 1 are compared with measured iles. The comparison is based on the vertical variation of the chord average concentration, since most of the rimental data available is presented in that manner. The deviation between the simulated and experimental iles is also shown in each graph (Dev.). Looking at those graphs it is obvious that, generally speaking, a agreement between simulated and experimental profiles is obtained. This adjustment is better for the ler particles and, other studies have shown that it is also better for the lower concentrations. For these tests, ressure drop calculated using the simulated profiles is also very near the experimental value, more close

the values obtained with the Durand equation. ing a more detailed analysis of the results in Fig. 1, we can conclude that the model is capable of translating nfluence of the operating conditions on the solids distribution in the cross-section of the pipe. For instance, odel is adequate to describe the effect of the diameter of the particles on the solid phase distribution in the

s-section of the horizontal pipe: larger settling velocities, i.e., larger particles, correspond to less uniform ibutions for the same conveying velocity. Fig. 1 shows also the 2D profiles that can be obtained with the el, giving a full picture of the solids distribution in the cross-section of the pipe.

60

90

120

TEST 14 5 TEST 1

0,0 0,2 0,4 0,6 0,8 1,00

10

20

30

40

50

60

y/D

Pressure Drop (Pa/m ): M esured= 2074.62 S im ulated= 2058.53 D urand E q.= 1849.71Dev= 17.9%

TEST 2

S im ul. Exper.

0

30

60

90

120

150

180

210

240

270

300

330

3.00%

5.00%

10.00%

15.00%

20.00%25.00%

35.00%40.00%45.00%

TEST 2

(d) 2D Simulated concentration profiles (% vol.)

(b) Variation of the chord average concentration in the vertical direction

0. 0 0 .2 0 .4 0. 6 0. 8 1 .00

5

1 0

1 5

2 0

2 5

3 0

3 5

4 0Pressure Drop (P a/m ): M easured= 1822.45 S im ulated= 1984.205 D urand Eq.= 1662.93Dev= 10.4%

y/D

S im ul. Exper.

0

30150

180

210

240

270

300

330

3.00%

5.00%

10.00%

15.00%

20.00%25.00%

35.00%40.00%

(a) Variation of the chord average concentration in the vertical direction (c) 2D Simulated concentration

profiles (% vol.)

Figure 1: Comparison of simulated and experimental profiles- Monodispersed Model

berg, Germany, 27-29 March 2001

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3.2-Polydispersed Systems As referred previously, the Settling-Dispersion model has been adapted in order to incorporate the handling of a solid phase with a wide particle size distribution. In this case, the solid phase size distribution is divided in a certain number of fractions, an average diameter being attributed to each fraction (species diameter). Fig. 2 presents the profiles (1D and 2D) obtained considering the particle size distribution to be described by three size classes (ternary system). Tables 2 and 3, summarise the operating conditions and model parameters used in the simulation of the multispecies systems (tests 3 and 4). The diameter of the pipe was 410-2 m. The graphs in Fig. 2 refer both to the total solids distribution and to the distribution of the individual species. Tests 3 and 4 correspond to different particle size ranges, and it is apparent from the analysis of Fig. 2 that the larger the size range (test 4) the more evident is the segregation that pushes the smaller particles towards the top of the pipe (compare Figs. 2 (c) and (f) ). Altough the ratio Do/vo was kept the same in both tests, which means that the conveying velocity has been increased for the wider distribution, with a larger average settling velocity, segregation is more pronounced in the second case and the total solids distribution is less uniform, the smaller particles showing a maximum in the central zone of the cross-section. The model calculates also the total solids load in the pipe and the average concentration of each species, as shown in table 3.

Table 2: Suspension characteristics/Ternary systems: Tests 3 and 4 TEST 3 TEST 4

SPECIES 1 2 3 1 2 3 Diameter-dpi (m) 500 300 250 750 500 250

Terminal Settling Velocity - voi (m/s) 0.0402 0.0190 0.0142 0.0678 0.0402 0.0142 Particle Reynolds Number- Repi 7.13 2.03 1.26 18.05 7.13 1.26

Average Terminal Settling Vel. - ov (m/s) 0.0245 0.0407

Average Particles Diameter (m) 342 500

Table 3: Operating conditions and model parameters/Ternary systems: Tests 3 and 4 SPECIES CONC. (%VOL.) VOL. FRACTION OF EACH SPECIES

(%) Total av.

conc.(%vol.) Do104 (m2/s)

Do/ ov 102 (m)

Test 1 2 3 1 2 3 3 5.94 9.31 5.86 28.14 44.10 27.76 21.11 5.00 2.04 4 6.12 8.45 5.53 30.40 42.13 24.47 20.13 8.31 2.04

The solids distributions obtained assuming a polydispersed particulate phase have also been compared with the profiles calculated supposing a monodispersed solid, with a particle diameter equal to the average diameter of the multispecies system. Table 4 summarises the conditions used in those studies. The size range of the particles in test 5 is the same as in test 3 (see table 3), for test 6 the size range is equal to the one used in test 4. Moreover, table 4 presents data for binary and ternary particulate phases, since the results that will be presented in Fig. 3 were obtained applying the polydispersed model both to a two and three species system. Fig. 3 compares the one and two dimensional distributions obtained for those two situations with the results for the monodispersed system. Looking at this figure, the differences between the results obtained with the polydispersed and monodispersed models are obvious, even though the maximum size range in the polydispersed system is only 3.0 (dpmax/dpmin) in test 6. These differences increase when the size range increases (compare Figs. 3 (a) and (c)), being more pronounced in the regions of higher concentration. For the particle size ranges tested, going from a binary to a ternary distribution does not alter the overall solids distribution significantly.

Table 4: Operating conditions and suspension characteristics: Tests 5 and 6 TEST 5 TEST 6 Monod Bin. Tern. Monod Bin. Tern.

Average Concentration (%vol.) 21.09 21.16 21.11 20.00 20.09 20.13 Particle Reynolds Number 2.89 7.30

Average Terminal Settling Vel. - ov (m/s) 0.0226 0.0272 0.0245 0.0407 0.0410 0.0407

Average Particles Diameter (m) 345 350 340 505 502 507 Do (m2/s) 104 4.81 5.55 5.00 8.30 8.36 8.31 Do / ov (m) 102 2.04 2.04 2.04 2.04 2.04 2.04


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(d) Two-dimensional distribution of the three species (%vol.)

60

90

12010.00%

180

0

30

60

90

120

150

210

240

270

300

330

4.50%

4.70%4.80%4.90%

5.00%

4.00%

7.00%

13.00%

8.00%

10.00%

7.00%

4.00%

3.00%

2.00%

16.00%18.00%

TEST 4

Species1 Species2 Species3

Conc

entr

atio

n (%

Vol.

)

0

30

60

90

120

150

180

210

240

270

300

330

4.00%

4.50%

5.00%

5.50%

5.50%6.00%

7.00%

8.00%

9.00%

2.00%

4.00%

6.00%

8.00%

10.00%

13.00%15.00%

TEST 3

Species1 Species2 Species3

60

90

120

(a) Two-dimensional distribution of the three species (%vol.)

In FdistrTablFig. the ehas b

Nurem

30150

12.00%

15.00%00

5

10

15

20

25

30

35

40

Conc

entr

atio

n (%

Vol.

)

0,0 0,2 0,4 0,6 0,8 1,00

5

10

15

20

25

30

35 Species1 Species2 Species3 Total

y/D

0180

210

240

270

300

330

19.00%

26.00%

30.00%

32.00%

(c) One dimensional distributions

(b) Total two-dimensional distribution of the solid phase (%vol.)

ig. 4 the 1D experimental distribution corresponding to teibutions obtained both with the monodispersed model ae 5 summarises the model parameters for the polydispers4 that, altough the size range is not too large (dpmax/dpmixperimental points. Fig. 4 shows also the 2D distributioeen divided in two fractions) segregation of the small pa

Figure 2: Simulated concentration

berg, Germany, 27-29 March 2001

30150(f) One dimensional distributions

(e) Total two-dimensional distribution of the solid phase (%vol.)

180

0

210

240

270

300

330

30.00%

25.00%

20.00%

15.00%

35.00%

,0 0,2 0,4 0,6 0,8 1,0

Species1 Species2 Species3 Total

y/D

st 1 (table 1) is compared, now, with the simulated nd with the polydispersed model (binary system). ed model (test 7). It is obviuos, from the analysis of n=2.5) the polydispersed model gives a better fit to n of the two species (the particle size distribution

rticles (species 1) being again apparent.

profiles- Polydispersed Model

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(d) Total two dimensional concentration profiles (% vol.)

0

30

60

90

120

150

180

210

240

270

300

330

10.00 %

15.00 %

20.00 %

25.00 %

30.00 %

35.00 %

10.00 %

15.00 %

18.00 %

25.00 %

30.00 %

35.00 %

10.00 %

15.00 %

20.00 %

25.00 %

30.00 %

35.00 %

TEST 6

Tern. Bin. Monod.

0,0 0,2 0,4 0,6 0,8 1,05

10

15

20

25

30

45

50 TEST 6

Tern. Bin.

onod.

Conc

entr

atio

n (%

Vol.

)

y/D

(c) One dimensional distributions of the total solid phase

0

30

60

90

120

150

180

210

240

270

300

330

26.00%

8.00%

10.00%

12.00%

15.00%

25.00%

30.00%35.00%

40.00%30.00%

12.00%

15.00%

20.00%

25.00%

32.00%

12.00%

15.00%

19.00%

30.00%

32.00%

Tern. Bin. Monod.

0,0 0,2 0,4 0,6 0,8 1,05

10

15

20

25

30

35

40

45 Tern. Bin. Monod.

Conc

entr

atio

n (%

Vol.

)

y/D (b) Total two dimensional concentration profiles (% vol.) (a) One dimensional distributions of the total

solid phase

Figure 3: Comparison of monodispersed and polydispersed profiles

0

10

20

30

40

50

Conc

entr

atio

n (%

Vol.

)

60

90

120 Species1

NuremTEST 7

35

40 M0,

bTEST 5

0 0,2 0,4 0,6 0,8 1,0

Pressure Drop (Pa/m) Measured= 1822.45 Simulated= 1996.34 Durand Eq.= 1662.93Dev. Bin.= 8.40% Dev. Mond.= 10.40%

y/D(a) One dimensional distributions of the total solid phase

Exper. TotalBin. Monod.

:

Figure 4: Comparison of concentration profile

erg, Germany, 27-29 March 2001 TEST 5

0

30150

180

210

240

270

300

330

5.00%5.50%

6.00%6.50%7.00%

1.00%

2.00%

10.00%

15.00%

20.00%25.00%30.00%35.00%

Species2

(b) Two dimensional distributions of the two species (% vol.)

s (experimental / polydispersed model)

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Table 5: Model parameters / Binary system: Test 7 TEST Do 104

(m2/s) vo (m/s) Do/vo 102

(m) Calculated Average Concentration (% vol)

7 9.8 0.0546 1.79 15.87 4- Conclusions It has been shown here that in a hydraulic conveying system , where a heterogeneous suspension is flowing, the concentration profiles which build up in the cross-section of the pipe can be adequately described by the Settling-Dispersion model, based on the balance between the tendency of the particles to settle and their tendency to disperse from the regions of higher concentration. Additionally, a new pressure drop correlation, which takes into account the distribution of the solids in the pipe, was applied to the simulated profiles, having produced calculated pressure drop values very close to the experimental ones. However, a more realistic approach to the flow of a true suspension has to consider the particle size range of the solid phase. The Settling-Dispersion model has been modified in order to cover that situation. The results presented seem quite promising, though the experimental validation is still scarce. In fact, the existence of good quality measurements supplying a reliable picture of the local conditions in the conveying pipe (solids and velocity distributions) is considered of crucial importance. To summarise, it can be said that by being able to predict, in an accurate way, the solids distribution in the conveying pipe, it is possible to improve the design of such systems. The model presented here provides a relatively simple and flexible tool that can be used to interpret and predict experimental data on hydraulic transport systems. Acknowledgements This project was funded by PRAXIS XXI and by the Portuguese Foundation for Science and Technology (FCT). One of the authors would also like to acknowledge the receipt of a scholarship from FCT.

REFERENCES 1- Durand, R.; Basic Relationship of the Transport of Solids in Pipes- Experimental Research, in Proc. Int.

Ass. For Hydraulic Research, Minneapolis, USA (1953). 2- Roco, M.C., Shook, C.A.; New Approach to Predict Concentration Distribution in Fine Particle Slurry

Flow; Physico Chem. Hydrodynamics, 8, 1, p. 43 (1987). 3- OBrien, M.P.; Review of the Theory of Turbulent Flow and its Relationship to Sediment Transportation,

Trans. Am. Geophysical Union, Section of Hydrology, p. 487 (1933). 4- Shook, C.A., Daniel, S.M., Scott, J.A., Holgate, J.P.; Flow of Suspensions in Pipelines, part. 2: Two

Mechanisms of Particle Suspension, The Can. J. Chem. Eng., 46, 4, p. 238 (1968). 5- Karabelas, A.J.; Vertical Distribution of Dilute Suspensions in Turbulent Pipe Flow, AIChE J., 23, 4, p.

426 (1977). 6- Wilson, K.C., Pugh, F.J.; Dispersive-Force Modelling of Turbulent Suspension in Heterogeneous Slurry

Flow, The Can. J. Chem. Eng., 66, p. 721 (1988). 7- Rasteiro, M.G., Figueiredo, M.M., Maia, M.C.,Scarlett, B.; Modelling of Solid/Liquid Flow in Pipes,

Powder Handling & Processing, 5, 3, p. 253 (1993). 8- Masliyah, J.H.; Hindered Settling in a Multi-Species Particle System, Chemical Eng. Sc., 34, p. 1166

(1979). 9- Mor, J.J., Cosnard, M.Y.; BRENTM, a Fortran Subroutine for the Numerical Solution of Systems of Non-

Linear Equations, ACM Trans. On Math. Software, 6, 2, p.241 (1980). 10- Rasteiro, M.G., Figueiredo, M.M., Franco, H.; Pressure Drop for Solid/Liquid Flow in Pipes, Particulate

Science and Technology, 11, p. 147 (1993 11- Gillies, R.G., Shook, C.A; Concentration Distribution of Sand slurries in Horizontal Pipe Flow,

Particulate Science and Technology, 12, p. 45 (1994).


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Pipe Flow of Solid-Liquid Suspensions With a Broad Size Range-M.G. Rasteiro-proceedings