Slurry-Flow Pressure Drop in Pipes With Modified Wasp Method (Ej) [MALI; KHUDABADI Et Al] [SME Annual Meeting; 2014-02] {13s}

SLURRY-FLOW PRESSURE DROP IN PIPES WITH MODIFIED WASP METHOD Tanaji Mali, Andritz Technologies Pvt. Ltd, Bangalore, India Vijay Khudabadi, Andritz Technologies Pvt. Ltd, Bangalore, India Rana A.S, Indian School of Mines, Dhanbad, India Arihant Vijay, Indian School of Mines, Dhanbad, India Adarsh M.R, University of Petroleum and Energy Studies, Dehradun, India

Presented at SME Annual Meeting/Exhibit, February 24-26, 2014, Salt Lake City, UT, USA

Abstract Over the last many decades, a significant amount of research has gone into the domain of slurry transport. However, design engineers still face many challenges with respect to prediction of pressure drop, critical velocity and other design parameters as a function of Solids % and Particle Size Distribution (PSD). The industry requirement is to transfer the slurry at the maximum concentration as possible (above 30% (volume %)) to make slurry transport more economically viable and to reduce water consumption. To facilitate the design and scale-up of slurry transport in pipelines and in process plants, there is a need for a correlation that can predict slurry pressure drops over a wide range of operating conditions and physical properties of different slurries. The objective of this study is to overcome the limited range of applicability and validity of existing correlations and to develop a generalized but more rigorous correlation applicable to a wider range of slurry systems. The existing Wasp et al. (1977) method is based on multi-phase flow modeling approach. This study attempts to modify this approach by considering material-specific values of Durand’s equation co-efficient and by defining flow regimes based on particle Reynolds number. When compared with experimental data, the modified Wasp method proposed in this study predicts the pressure drop for slurry flows more accurately than other available correlations. Also, the proposed method requires minimal test/experimental data for a particular slurry system and can be extended over different input conditions. An iterative computer algorithm is developed to calculate the critical settling velocity and pressure drop in a pipe as a function of Solids % and PSD. The solution method can easily be implemented in designing slurry pipes, design validation, and studying the different slurry transport scenarios. The modified method can also be extended to accurately predict pressure drops in dynamic pressure flow networks used in commercial process simulators.

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Keywords: Particle size distribution, Slurry flow, volumetric concentration, Pressure drop, Wasp Method, Drag Coefficient, friction factor, critical velocity, Durand’s equation

Introduction Slurry transport involves huge capital investment. Therefore, at present many organizations throughout the world are carrying out research and development to abate these costs. Literature survey reveals that studies on slurry transport have followed one of these three major approaches: (a) The empirical approach (b) The rheological based continuum approach (c) The multiphase flow modeling approach. Amongst the above mentioned approaches, the empirical approach is the simplest, and hence has been widely used and applied. This has led to formulation of the correlations for prediction of pressure drop and for delineation of flow regimes. The rheological approach is best applicable to slurries of ultrafine non-colloidal particles. The multiphase flow modeling approach, which considers liquid, particle and boundary interaction effects, requires significant computational effort and is best suitable for describing heterogeneous solid-liquid mixture flows. In this study, multiphase flow modeling approach has been followed. It considers various important design parameters such as particle size distribution, volumetric concentration and pipe roughness in predicting the pressure drop. Slurry Flow in Pipe For a pure liquid, the pressure drop in a pipe depends on the flow velocity. The change of pressure drop with respect to flow velocity is monotonic in nature. However, in case of slurries, it is not monotonic (Vanoni 1975; Govier and Aziz 1977), as shown in Figure 1. When the flow velocity is sufficiently high, all solid particles are suspended with the parti-

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cle distribution being homogeneous. As the velocity decreases below V4 (see Figure 1) (Govier and Aziz 1977), all of the solids are still suspended, but their distribution becomes heterogeneous. When the velocity further decreases to the critical velocity V3, some solids start to move along the pipe bottom as a “bed load.” At this point, the pressure drop is usually minimum. When the velocity decreases further very few solids are transported as the suspended load, and more amount of solid is transported as the bed load. At further reduced velocity, V2, the bed load starts to generate a stationary bed. The stationary bed further increases the apparent pipe fraction factor, resulting in increased pressure drop. Finally, at further reduced velocity, V1, all solids stop moving.

body forces as well as the viscous resistance of the particles. Heterogeneous flow When the slurry velocity decreases, intensity of turbulence and lift forces also decreases due to which, there is distortion of the concentration profile of the particles. In this flow regime more of the solids, particularly the larger particles are contained in the lower part of the pipe. Thus, there is a concentration gradient across the pipe cross section with a larger concentration of solids at the bottom. This flow is also called asymmetric flow Saltation flow In this regime the slurry velocity is low and the solid particles tend to accumulate on the bottom of the pipe, first in the form of separated “dunes” and then as a continuous moving bed. Stationary bed flow In this regime the slurry velocity is further reduced which leads to the lowermost particles of the bed being nearly stationary. Thus, the bed thickens and the bed motion is due to the movement of the uppermost particles tumbling over one another (saltation).

Figure 1. Plot of transitional mixture velocity with pressure drop To achieve the optimum performance i.e. minimum pump pressure requirement, the slurry should be transported at critical velocity (V3). When a higher proportion of particles start to move as bed load, a higher pump pressure is required to move them. If the pump pressure is not high enough, danger of plugging the pipeline arises. Thus, to overcome the risk of pipeline plugging, slurry transfer through the pipeline must be operated above the critical velocity, V3. Flow Regimes In slurry transport, different patterns of solid movement are observed depending upon the nature of the slurry and the prevailing flow condition. As shown in Figure 2, the patterns are dependent on the particle size, volumetric concentration of the solids and the flow velocity. In horizontal pipes, the patterns can be conveniently be classified into the following four regimes: Homogeneous flow This regime is also called symmetric flow. In this flow regime there is a uniform distribution of solids about the horizontal axis of the pipe, although it may not be exactly uniform. In this regime, turbulent and other lifting forces are capable of overcoming the net

Figure 2. Flow regimes of heterogeneous flows in terms of particle size vs. velocity (after Shen,1970) and pictorial representation of flows

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Where

4.

A represents saltation flow regime B represents heterogeneous flow regime C represents homogeneous flow regime 5.

Research Work & Different Methods Over the years, two principal research works have been developed—one around the Durand–Condolios approach and the other around the Newitt approach. The former evolved gradually and Wasp modified it for multilayer compound systems (Abulnaga, BE 2002). The latter gradually evolved to yield the twolayer model (Abulnaga, BE 2002). Wasp and Durand methods are useful tools for concentrations of coarse particles up to 20% (volume fraction) (Abulnaga, BE 2002). This covers, in fact, most dredged gravels and sands, coal in a certain range of sizes, as well as crushed rocks (Abulnaga, BE 2002). It is also worth noting that Zandi and Govatos (1967) worked on sand samples up to 22% (volume concentration). The two-layer models have made it possible to work with volumetric concentrations of 30% (Abulnaga, BE 2002). But these models have many limitations and still considerable amount of work has to be done to overcome these limitations.

Steps 2 to 5 are re-iterated until convergence of the friction loss. Although this method works well for water-coal mixture, it over predicts pressure drop for mineral and rock slurry systems. To overcome this limitation, the following two modifications have been proposed in the paper: 1. The value of k, i.e., Durand’s equation coefficient, is material-specific. For different materials, different values of k should be used and the value is calculated as described in algorithm given in next section (step 9). This modification is needed because the pressure drop predicted by the heterogeneous part does not match with the experimental results. 2. Different flow regimes have been defined based on the assumption that particle size having Reynolds number less than 2 will always contribute toward homogeneous losses and particle size having Reynolds number greater than 525 will always contribute toward heterogeneous losses in horizontal slurry flow pipe for all flow velocities as proposed by Duckworth (Jacobs 2005). These modifications have been described in the following section. We refer to these modifications as the Modified Wasp Model.

Wasp Method Wasp et al. (1977) method is the most widely used method for slurry transport applications around the world because it is applicable for all kinds of flow regime and accurately predicts the pressure drop, considering the PSD of the slurry which is an important parameter accounting for pressure drop calculation. Wasp method is an improvement of DurandCondolios approach which predicts pressure drop accurately for both slurry systems in which particles have narrow as well as wide size range. Wasp method accounts for large particle size distributions and pressure drop by dividing the slurry into homogenous (due to vehicle) and heterogeneous (due to bed formation) fractions. The solids in the homogenous fraction increase the density and viscosity of the equivalent liquid vehicle. Wasp method also considers the effect of pipe diameter and pipe roughness on pressure drop in slurry pipes. The iterative method proposed by Wasp et al. (1977) is summarized as follows: 1.

2.

3.

A ratio, C/CA, is defined for the size fraction of solids based on friction losses estimated in steps 2 and 3 (where C/CA is the ratio of volumetric concentration of solids at 0.08D from top to that at pipe axis). Based on the value of C/CA, the fraction size of solids in homogeneous and heterogeneous flows is determined.

Algorithm for Modified Wasp et al. (1977) Method The example below illustrates the modifications proposed in this paper to the Wasp et al (1977) method. This algorithm has been developed in MS Excel and all the cases described later in this work have been validated using this newly developed algorithm. Example 1 Nickel ore slurry was tested in a 159-mm pipeline with a roughness coefficient of 0.045 at a weighted concentration of 26.3%. The results of pressure drop versus velocity are presented in Table 1.

By using Durand’s equation, the total size fraction is divided into a homogeneous and heterogeneous fraction. The friction losses of the homogeneous fraction are calculated based on the rheology of the slurry, assuming Newtonian flow. The friction losses of the heterogeneous fraction are calculated using Durand’s equation.

Table 1. Pressure drop versus Speed in a 159mm ID Steel Pipe at a Weight Concentration of 26.3% Velocity Pressure drop (m/s) (Pa/m) 1.5 175 3

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1.9 2.3 2.7 3.1 3.5 4.0

homogeneous) at a given operating flow velocity of the slurry. Any particle with size greater than dcut will settle down (heterogeneous):

270 360 525 688 847 1046

  V6 d cut = D   3 6 1.404 * [2 gD( ρ S − ρ L ) / ρ L ]   ( 3.525 ) * C v (1)

The particle size distribution of the ore is presented in Table 2.

Where Table 2. Particle Size versus Wt. % solids in the slurry Particle Size (μm)

Wt. %

-450 -200 -95 -61 -44

1.88 2.2 1.65 1.17 93.1

v D Cv dcut

ρs

= Operating velocity of slurry (m/s) = Pipe inside diameter (m) = Volumetric concentration of solid = Cut size = Specific gravity of solid

ρ L = Specific gravity of the liquid Based on the cut size, i.e., dcut, four cases have been considered. They are as following: Case 1: Cut size - dcut > dRe max and dRe max < d100

Step 1: Plot the PSD curve. Based on the Particle size distribution (PSD) of solids present in the slurry, plot the PSD curve. Particle size vs. cum. wt. % passing is presented in Table 3 and Figure 3 shows the PSD curve.

Figure 4. Cut size for flow regime – Case 1

Table 3. PSD Data of the solids presents in the slurry Particle Size Weight Cumulative (mm) % Weight % Passing -0.85 + 0.40 1.88 100 -0.40 + 0.20 2.20 98.12 -0.20 + 0.105 1.65 95.92 -0.105 + .044 1.17 94.27 -0.044 93.1 93.1 Total 100.00

Where, dRe max is the particle size having Reynolds number 525 and d100 is the maximum particle size present in the slurry. In this case, particles having size greater than dRe max will remain in the heterogeneous part and contribute moving bed losses, whereas particles having size less than dRe max will remain suspended and contribute toward homogeneous losses. This is based on the assumption proposed by Duckworth that the minimum Reynolds number at which particles will settle by saltation without continuous suspension is approximately 525 (Jacobs 2005). Hence particles having Reynolds number greater than or equal to 525 will always be in the heterogeneous part of the mixture. This is the proposed modification in the existing wasp method. As a result of this modification, the new cut size dcut will now be dRe max. Case 2: Cut size - dcut < dRe min

Figure 3. Particle size distribution curve Step 2: Determine cut size. dcut using Wasp’s modified Durand equation. This is the most important step as it determines the flow regime of the slurry. The following is the correlation given by Wasp et al. (1977) to calculate the cut size, dcut, i.e., the maximum size of the particle that will remain suspended in the slurry (pseudo-

Figure 5. Cut size for flow regime – Case 2 Where, dRe min is the particle size having Reynolds number 2.

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The single particle settling velocity vt is calculated using the Stoke’s equation:

In this case, particles having size less than dRe min will remain suspended in the slurry and contribute toward the losses, whereas particles having size greater than dRe min will settle down and contribute toward the heterogeneous losses. This is another proposed modification in the existing wasp method. As a result of this modification, the new cut size dcut will now be dRe min.

vt =

The Particle Reynolds number Re is then calculated : vt d Re (4) Re = vL Where

Case 3: Cut size - dcut lies between dRe max and dRe min.

=Particle size having Reynolds number Re dRe (Re min = 2 and Re max = 525) = kinematic liquid viscosity νL

Figure 6. Cut size for flow regime – Case - 3

To calculate the value of particle size having Reynolds number 2 and 525, substitute the values in equations (4), (3) and (2) and calculate dRe. For this particular example, dRe max = 2.2 mm and dRe min = 0.150 mm; therefore, Case 3 of the flow regime will be considered.

In this case, particles having size less than dcut will remain suspended and contribute toward homogeneous losses, whereas particles with size greater than dcut will settle down and contribute toward heterogeneous losses. Case 4: Cut size - dcut is > d100 and dRe max > d100

Step 4: Calculate the friction losses of the homogeneous fraction based on the rheology of the slurry, assuming Newtonian flow. Friction losses of the homogeneous fraction are calculated using Darcy’s formula: 2 Loss (Pa/m) = f DV ρ m

Figure 7. Cut size for flow regime – Case 4

2D

In this case, no particles will settle, and the total loss will be due to the homogeneous part only and losses due to the heterogeneous part will be zero. No iteration is needed in this case; therefore, loss is calculated directly considering pseudo-homogeneous flow. This usually happens when velocities are very high and particle size present in the slurry is very small. For this particular example, the mean slurry velocity under consideration is 1.9 m/s, solid specific gravity = 4.074, liquid specific gravity = 1, and pipe diameter is 0.159 m. Thus, calculated cut size, dcut = 152 micron.

(5)

Where fD = Darcy friction factor. v = Mean velocity of slurry (m/s). ρ m = Density of carrier fluid (kg/m3) (including the particles less than dcut). D = Inside diameter of pipe (m). The Swain-Jain equation may be used in the range of 5000 < Re < 107 to determine the friction coefficient of the homogeneous part of the mixture: fD =

Step 3: Calculate dRe max and dRe min to determine the flow regime of the slurry. In order to calculate the particle size having Reynolds number 2 and 525, the following equation are used: The drag co-efficient cD is calculated using the Reynolds number by the following equation (Manfred Weber): cD = (

d g 4 * ( Re ) * ( ρ S − ρ L / ρ L ) (3) 3 cD

{

0.25

}

(6)

2 log[( ∈ /Di ) / 3.7 + 5.74 / Rem0.9 ]

Where

∈ = Roughness coefficient (m). Di = Inside diameter of the pipe (m). Rem is the Reynolds number for the slurry which is calculated using Thomas (1965) correlation for slurry viscosity µm as given below:

24 4 )+( ) + 0.4 (2) Re Re

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Re m =

( ρ mV m D )

µm

size fraction loss gives the combined loss due to the moving bed or heterogeneous part of the mixture. In this particular example, dcut is 152 micron, therefore, particle size between 850 micron and 152 micron contributed toward heterogeneous losses. Redistributed particle size is presented in Table 4.

(7)

μm = 1 + 2.5Cvf + 10.05Cvf2 + μl (8) 0.00273 exp(16.6Cvf )

Table 4. Redistributed particle size for heterogeneous part of the mixture Size Average Cv in Cv bed in ( mm) Particle Solthe slurry size (mm) ids (at (%) overall solids Cv of mixture at 8.035%) (%) -0.85 0.63 1.88 0.151 + 0.40 -0.40 0.30 2.20 0.177 + 0.20 -0.20 0.18 0.92 0.074 +0.152 Total 5 0.402

Where Cvf = volumetric concentration of solids in homogeneous part of mixture µl = liquid viscosity µm = slurry viscosity For this particular example, dcut, as determined in step 2 is 152 micron. Therefore, from the PSD curve, 95% by weight solids are less than 152 micron; hence, they contribute to the homogeneous losses. Next, calculate the volumetric concentration Cvf of solids in the homogeneous part of the mixture as (Abulnaga, BE 2002): Cv total = Cw*( ρ m / ρ s ) (9)

Step 6: Determine the particle Reynolds number and drag coefficient for each size range. It is essential first to determine the drag coefficient and the particle Reynolds number for each size fraction to calculate the loss due to each size fraction in the heterogeneous part of the mixture. To calculate the particle Reynolds number, the density of the slurry ρm, viscosity of the slurry µm and the speed of the carrier fluid V are used. The equation to calculate the particle Reynolds number is as following:

Cv total = 26.3*(1244/ 4072) = 8.035% Cvf = 0.95*Cv total = 7.633% Where

ρm ρs

= slurry density = solid density = total volumetric concentration of sol-

Cv total ids Cw = total solids concentration by weight

ReP =

Therefore, out of total 8.035% solids by volume in slurry, 7.633% solids by volume are in suspension and contribute toward the homogeneous losses. Next, µm can be calculated using the Cvf and it is calculated as 1.2598 and corresponding Rem is 2956192. Substituting the value of Rem in Swain-Jain equation to determine the fD, which is equal to 0.017. Therefore, loss (Pa/m) due to homogeneous part of the mixture from Darcy’s formula is calculated as 238.46 Pa/m. The lab test measured 270 Pa/m; the losses due to the moving bed are therefore 31.54 Pa/m.

Vd P ρ m

µm

(10)

Where dp = the average particle size of each size fraction. To calculate the drag coefficient, CD, of a sphere, the Turton equation is used: CD = (( 24 )*(1+0.173*Rep0.657)) + Re P

0.413 1 + 11630 * (Re P

Step 5: Calculate the redistributed particle size considering the cut size dcut. To calculate the losses due to the moving bed or heterogeneous part of the mixture, the size distribution of the heterogeneous part (size greater than dcut) is divided into various size fraction and then loss due to each size fraction is calculated and sum of all the

1.09

)

(11)

Results are summarized in Table 5.

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Table 5. Drag Coefficient for single particle size in Example 1 Particle AvParticle Drag size erage Reynolds coefficient (mm) Particle number (Cd) Size (Rep) (mm) -0.85 + 0.63 1160 0.455 0.40 -0.40 + 0.30 557 0.550 0.20 -0.20 + 0.18 327 0.662 0.152

Table 6. Calculated Losses for each fraction of solids in moving bed AvPartiDrag Cv Preserage cle coeffisure bed Particle Reyncient ( Loss Size olds (Cd) %) (Pa/ (mm number m) ) (Rep) 0.63 1160 0.455 0. 14.51 151 0.30 557 0.550 0. 14.74 177 0.18 327 0.662 0. 5.36 074 Total 34.61

Step 7: Calculate the friction losses of the heterogeneous fraction using Durand’s equation. Wasp et al. (1977) recommend using Durand’s equation for each size fraction of solids to determine the increase in pressure losses due to moving bed:

∆ Pbed

 gD ( ρ − ρ ) / ρ  L L  = 82 ∆PL C vbed  i s2   V CD (12)

Step 8: Based on the value of C/CA, determine the fraction size of solids in homogeneous and heterogeneous flows. By comparing with the measured 270 Pa/m, the calculations for the bed are higher and can be refined by the method of concentration1 using equation (Abulnaga, BE 2002):

1.5

Where

log10 [

= Pressure drop due to flow of carrier ∆PL liquid. = Volumetric concentration of bed porCv bed tion of a particular size fraction. Di = Inside diameter of the pipe v = Mean slurry velocity. CD = Drag coefficient of a particular size fraction.

1.8Vt C ]=− (13) CA β K xU f

Where = the ratio of volumetric concentration C/CA of solids at 0.08D from top to that at pipe axis, ẞ = the dimensionless particle diffusivity and is taken as 1.0 Kx = 0.4 and is defined as von Karman coefficient Uf = the friction velocity calculated from the pressure drop in first iteration Vt = settling velocity of a particular size particle calculated using standard drag relationships.

Ellis and Round (1963) indicated that Durand’s equation coefficient of 82 is too high for nickel suspensions. Therefore, for this particular example, a value of 23 is used as the modified Durand’s equation coefficient. The value of the constant is determined by iteration based on the test results. Step 9 explains how to determine the value of this constant. Results of the calculation for this particular case are presented in Table 6: The total friction loss is therefore 239.46 Pa/m + 34.61 Pa/m = 274.07 Pa/m. This value is compared with the experimental pressure drop, i.e., 270 Pa/m, it is observed that these values do not match. Therefore, an iterative technique is used to refine the value of pressure drop. This is explained in the following steps.

Where Vt is defined as the single particle size terminal velocity using the Stoke’s equation: Vt =

4( ρ S − ρ L )gd g 3ρ L C D

(14)

The equivalent fanning factor (Abulnaga, BE 2002) is calculated as: Total loss = 2 f f V 2 ρ/Di (15)

fN =

∆Ptotal Di

(16) 2V 2 ρ To calculate Uf, the following Equation (Abulnaga, BE 2002) is used: Uf = V (fN /2) ½ (17)

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in slurry flow was collected and Modified Wasp pressure drop was calculated and compared with the experimental results. To validate this method for different volumetric concentration of solids in slurry, data points from the open literature was collected and pressure drop using the Modified Wasp’s method was calculated. The Cv value ranged from 9.4% to 51.7% and the speed ranged from 1 m/s to 5 m/s. The Pipe inside diameter for this case was 54.9 mm and the solid specific gravity was 2.47. Results of this case study are presented in Table 15

By determining the value of C/CA ,new Cv bed for each size fraction is calculated directly by multiplying original Cv bed by C/CA. Iterated pressure drop for each size fraction is calculated using this new Cv bed. This is repeated till the value converges. For this particular example, the refined pressure loss after 1st, 2nd and 3rd iteration is represented in table 7, 8 and 9 respectively. After iteration, pressure loss due to heterogeneous part converges to 29.93 Pa/m, and hence total loss due to the slurry flow is 238.46 + 29.93 = 268.4 Pa/m which is very close to experimental value, i.e., 270 Pa/m.

Vertical Slurry Flow in Pipes

Step 9: Determine the value of Durand’s equation coefficient, k. The value of the Durand’s equation coefficient is material-specific. To determine its value, just one test result data is required and the modified value of k is calculated using iteration. For the 1st iteration, pressure drop using k=82 is calculated. If the total calculated pressure drop does not match with the experimental pressure drop then a modification in the value of k is required. To determine the modified value of k, a ratio R is defined.

Solid particles can be moved upward when the fluid velocity (V) exceeds the hindered settling velocity of the solids (wS).

V >> w S Where

w S = v t ( 1 − cV )γ (18) = Hindered settling velocity of particle = constant. vt = Single particle terminal velocity (from equation 14)

wS

γ

R = total experimental loss / total calculated loss (for k=82). An Iterated value of k is determined by multiplying k by ratio R, which is re-iterated until the value of R converges to 1. After iteration, a value of k is obtained for which R = 1, which is then defined as the modified Durand’s equation coefficient.

Hindering effect of solids concentrations must be taken into account while calculating the pressure drop. Fig.4 gives the influence of the concentration according to Maude and Whitmore (1958).

Validation of Modified Wasp et al (1977) Method Case study 1: Verification of Modified Wasp’s method for different flow regime. The modified Wasp method was verified for all the cases of flow regime, i.e., case 1, case 2, case 3, and case 4 and results are presented in Table 10. The calculated pressure drop was very accurate and within 10% error in prediction. Figure 8. Influence of the concentration on settling velocity according to Maude and Whitmore (1958) replotted from Weber (1974).

Case study 3: Verification of data sets available from open literature. In this case study, various experimental points have been collected from several sources spanning the years 1942-2002 [refer to Table 12]. This wide range of databases includes experimental information from different physical systems. Table 13 suggests the wide range of the collected databank for pressure drop. Results of the case study are presented in Table 14.

Reynolds number is calculated using equations (2),(3) and (4) corresponding to particle diameter d100 (=dRe). Using this Reynolds number, the value of γ is calculated from the above curve. Hindered settling velocity of the particle is calculated corresponding to the value of γ obtained, using equation (18). If the fluid velocity is lesser than hindered settling velocity of particle (ws), chocking condition occurs and there is no movement of solids along the vertical section. If

Case study 4: Kaushal and Tomita {2003} data for different volumetric concentration of glass beads

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the fluid velocity is greater than the hindered settling velocity of particle (ws), then the pressure drop is calculated as described in the next section.

Pressure Drop Calculation The flow regime of the vertical section depends on critical deposition velocity (ucri), which is calculated by Wasp Modified Durand’s equation (19). 1/ 6

d  u cri = 3.525cv0.234  100   D 

2 Dg

ρs − ρL ρL

(19)

Homogeneous or pseudo-homogenous flow regime occurs when mean slurry velocity (V) is greater than the critical deposition velocity (ucri). At this condition there is almost no slip between particles and fluid, therefore particle velocity is approximately equal to mean slurry velocity. Pressure drop for the vertical homogenous regime is calculated using the following equation (20).

∆p = f D

ρm 2

V2

∆Lvert + ρ m g∆Lvert D (20)

Where

∆p =total pressure drop for vertical slurry flow us = velocity of solids ∆Lvert = length of vertical section under consideration

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Average Particle Size (mm) 0.63 0.30 0.18 Total

Drag Coefficient (Cd) 0.455 0.550 0.662 29.89

Average Particle Size (mm) 0.63 0.30 0.18 Total

Average Particle Size (mm)

Table 7. 1st iteration results Terminal -1.8 C/CA Velocity vt/βKxUf (m/s) 0.201 -0.0844 0.823 0.127 -0.0532 0.884 0.089 -0.0371 0.918

Drag Coefficient (Cd) 0.455 0.550 0.662 29.89

Drag Coefficient (Cd)

0.63 0.30 0.18 Total

0.455 0.550 0.662 29.89

Iterated Cv bed (%) 0.125 0.157 0.068

Iterated Pressure loss (Pa/m) 11.94 13.03 4.92

Table 8. 2nd iteration results Terminal -1.8 C/CA Velocity vt/βKxUf (m/s) 0.201 -0.0837 0.824 0.127 -0.0528 0.886 0.089 -0.0368 0.919

Iterated Cv bed (%) 0.1248 0.1569 0.068

Iterated Pressure loss (Pa/m) 11.96 13.05 4.92

Table 9. 3rd iteration results Terminal -1.8 C/CA Velocity vt/βKxUf (m/s)

Iterated Cv bed (%)

Iterated Pressure loss (Pa/m)

0.201 0.127 0.089

-0.0837 -0.0528 -0.0368

0.824 0.886 0.919

0.1248 0.1569 0.068

11.96 13.05 4.92

Table 10. Pressure drop vs. speed in a 159-mm ID Steel pipe at a weight concentration of 26.3% at 20o C Cv VelociDi ExperiCalculat- % Error Flow ρL µL ρS (%) ty (m) mental Loss ed Regime 3 (kg/m3) (kg/m ) (mP (m/s) (Pa/m) Loss(Pa/ a.s) m) 8 1.5 .159 4074 1000 1 175 195 11 Case 2 8 1.9 .159 4074 1000 1 270 268 -0.7 Case 3 8 2.3 .159 4074 1000 1 360 354 -1.6 Case 3 8 2.7 .159 4074 1000 1 525 472 -10 Case 4 8 3.1 .159 4074 1000 1 688 616 -10 Case 4 8 3.5 .159 4074 1000 1 847 779 -8 Case 4 8

Cv (%)

4 4 4 4 4 8.1 8.1 8.1 8.1 12.8

4.0

Velocity (m/s)

1.2 2.8 3.2 3.6 4 1.6 2 3.6 4 1.6

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.159

Di (m)

.105 .105 .105 .105 .105 .105 .105 .105 .105 .105

4074

1000

1

1046

1008

Table 11. Calculated pressure drop for case study 2 ExperiCalculated ρL µL ρS mental Loss 3 (kg/m3) (kg/m ) (mPa.s) Loss (Pa/m) (Pa/m) 2820 1000 1 232 227 2820 1000 1 1006 897 2820 1000 1 1316 1165 2820 1000 1 1652 1469 2820 1000 1 2013 1808 2820 1000 1 438 366 2820 1000 1 619 500 2820 1000 1 1781 1579 2820 1000 1 2168 1942 2820 1000 1 468 448

-3.6

Case 4

% Error

Flow Regime

-1.81 -10.9 -11.4 -11.0 -10.1 -16.5 -19.5 -11.3 -10.4 -4.3

Case 3 Case 4 Case 4 Case 4 Case 4 Case 3 Case 4 Case 4 Case 4 Case 3

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12.8 12.8 19.1 19.1 19.1 26 26 26 26

3.6 4 2 3.6 4 2 3.2 3.6 4

.105 .105 .105 .105 .105 .105 .105 .105 .105

No 1 2

Pipe Dia (m) 0.019-0.495 [1]

2820 2820 2820 2820 2820 2820 2820 2820 2820

1000 1000 1000 1000 1000 1000 1000 1000 1000

1 1 1 1 1 1 1 1 1

1953 2317 722 1962 2375 774 1703 2104 2529

1700 2092 638 1868 2298 741 1636 2059 2531

-12.9 -9.7 -11.6 -4.74 -3.22 -4.23 -3.98 -2.14 0.04

Case 4 Case 4 Case 3 Case 4 Case 4 Case 3 Case 4 Case 4 Case 4

Table 12. Literature sources for pressure drop Author No Author Wilson (1942) 11 Gillies et al. (1983) Durand & Condolios(1952) 12 Roco & Shook(1984)

3

Newitt et al. (1955)

13

Roco & Shook(1985)

4

Zandi & Govatos (1967)

14

Ma (1987)

5

Shook et al.(1968)

15

Hsu (1987)

6

Schriek et al. (1973)

16

Doron et al. (1987)

7

Scarlett & Grimley (1974)

17

Ghanta (1996)

8

Turian & Yuan (1977)

18

Gillies et al. (1999)

9

Wasp et al. (1977)

19

Schaan et al.(2000)

10

Govier & Aziz (1982)

20

Kaushal and Tomita(2002)

Table 13. Slurry system[1] and parameter range from the literature data Particle Dia Liquid denSolids denLiquid Velocity U (micron) sity sity viscosity (m/s) (kg/m3) (kg/m3) (mPa.s)

Solids Conc. (fraction) f

38.3-13000

0.014-0.333

1000-1250

1370-2844

0.12-4

0.86-4.81

Slurry system: coal/water, copper ore/water, sand/water, gypsum/water, glass/water, gravel/water.

Cv

0.3 0.3 0.3 0.4 0.4 0.4 0.5 0.1 0.1 0.3 0.3 0.3 0.3 0.3 0.3

Table 14. Results of data points collected from open literature having flow regime of case 4 VelociDi ExperiCalcu% Flow ρL µL ρS ty (m) mental lated Error Regime 3 3 (kg/m ) (kg/m ) (mPa.s) Loss (Pa/m) (m/s) Loss (Pa/m) 3 0.0549 2470 1000 0.85 1990 2210 -11.1 Case 4 4 5 3 4 5 3 1.1 1.11 1.3 2.59 2.34 2.01 1.78 1.59

0.0549 0.0549 0.0549 0.0549 0.0549 0.0549 0.019 0.0526 0.0526 0.2085 0.2085 0.2085 0.2085 0.2085

2470 2470 2470 2470 2470 2470 2840 2330 2330 1370 1370 1370 1370 1370

1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

0.85 0.85 0.85 0.85 0.85 0.85 0.85 1 1 1 1 1 1 1

3430 5350 2230 3790 6390 3410 1250 294 543 267 226 177 147 123

3726 5599 2706 4536 6786 3645 1114 321 609 318 263 196 158 129

-8.6 -4.7 -21.3 -19.7 -6.2 -6.9 10.9 -9.2 -12.2 -19.1 -16.4 -10.7 -7.5 -4.9

Case 4 Case 4 Case 4 Case 4 Case 4 Case 4 Case 4 Case 4 Case 4 Case 4 Case 4 Case 4 Case 4 Case 4

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0.3 0.1 0.2 0.3 0.1 0.1 0.2 0.2 0.3 0.3 0.3 0.3 0.1 0.1 0.2 0.3 0.1 0.3 0.1 0.1 0.2 0.1 0.1

1.37 1.66 1.66 1.66 2.9 3.5 2.9 3.5 2.9 3.5 2.9 3.5 3.16 3.76 3.07 3.76 2.5 2.5 3 1.9 2.8 2.7 2.01

Cv

Velocity (m/s)

0.2085 0.0515 0.0515 0.0515 0.263 0.263 0.263 0.263 0.263 0.263 0.263 0.263 0.495 0.495 0.495 0.495 0.1585 0.1585 0.1585 0.0507 0.04 0.04 0.04

Di(m)

1370 2650 2650 2650 2650 2650 2650 2650 2650 2650 2650 2650 2650 2650 2650 2650 2650 2650 2650 2650 2270 2270 2270

ρS (kg/m3)

1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1250 1250 1250

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1.3 1.3 0.12 1 4 4 4

99 666 900 1136 261 334 305 382 355 453 414 526 143 186 157 254 475 630 648 1175 3926 3580 2217

Table 14. Results of the case study Experimental ρL µL Loss (Pa/m) 3 (kg/m ) (mPa.s)

98 671 791 930 272 392 313 443 353 505 392 560 151 211 161 271 421 532 526 1140 4022 3487 1939

WASP modified loss (Pa/m) 222

1.8 -0.8 12.1 18.1 -4.0 -17.4 -2.4 -15.9 0.5 -11.4 5.2 -6.4 -5.7 -13.6 -2.5 -6.4 11.4 15.6 18.9 3.0 -2.4 2.6 12.5

Case 4 Case 4 Case 4 Case 4 Case 4 Case 4 Case 4 Case 4 Case 4 Case 4 Case 4 Case 4 Case 4 Case 4 Case 4 Case 4 Case 4 Case 4 Case 4 Case 4 Case 4 Case 4 Case 4

% Error

Flow Regime

14.9

Case 4

9.4

1

0.0549

2470

1000

1

261

10.06

2

0.0549

2470

1000

1

847

781

7.7

Case 4

10.41

3

0.0549

2470

1000

1

1754

1642

6.3

Case 4

10.44

4

0.0549

2470

1000

1

2868

2783

3.0

Case 4

10.93

5

0.0549

2470

1000

1

4153

4230

-1.9

Case 4

19.22

1

0.0549

2470

1000

1

341

263

22.6

Case 4

20.48

2

0.0549

2470

1000

1

1051

932

11.3

Case 4

20.4

3

0.0549

2470

1000

1

1981

1937

2.2

Case 4

19.52

4

0.0549

2470

1000

1

3263

3225

1.1

Case 4

20.45

5

0.0549

2470

1000

1

4666

4927

-5.6

Case 4

30.3

1

0.0549

2470

1000

1

373

323

13.4

Case 4

30.02

2

0.0549

2470

1000

1

1037

1101

-6.2

Case 4

31.19

3

0.0549

2470

1000

1

2037

2330

-14.4

Case 4

30.75

4

0.0549

2470

1000

1

3291

3889

-18.2

Case 4

30.24

5

0.0549

2470

1000

1

4851

5783

-19.2

Case 4

38.95

3

0.0549

2470

1000

1

2420

2731

-12.9

Case 4

40.64

4

0.0549

2470

1000

1

3865

4760

-23.2

Case 4

39.56

5

0.0549

2470

1000

1

5761

6933

-20.0

Case 4

51.7

2

0.0549

2470

1000

1

2099

2002

4.6

Case 4

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49.24

3

0.0549

2470

1000

Heterogeneous vertical flow regime occurs when mean slurry velocity (V) is less than critical deposition velocity (ucri). Although the distribution of solids is homogeneous, considerable slip exists. Therefore, the local concentration cV has to be calculated for each velocity with respect to the delivered concentration and given solid mass flow rate, before calculating the pressure drop.

1

3082

v   1 − t V 

2

  + 4 c vtotal 

Results and Discussion

Case 4

The Modified Wasp Model:

 vt  V   (21)

The new value of volumetric concentration is used to calculate the slurry viscosity by using Thomas correlation (eq. 8). This results in a modified value of slurry Reynolds number (eq. 7) and Darcy friction factor (eq. 6). This value of Darcy friction factor (fD) is used to calculate pressure drop using equation (20).

-19.7

drop for different pipe sizes, solid % and PSD. The applicability of this method is thus not constrained due to unavailability of extensive experimental data. It can be easily applied to new slurry systems with minimal laboratory effort.

i.  V  vt cV = −1+ 2Vt  V 

3689

Is easy to implement, has direct calculations involving iterations. Accurately predicts the pressure drop over a wide range of input parameters for all types of slurry system like coal/water, nickel ore/water, sand/water, copper ore/water, etc. Effectively predicts pressure drop as a function of Particle Size Distribution (PSD) and solid percentage.

ii.

iii.

Thus, with the modifications proposed in this study, this method delivers a comparatively smaller error percentage as compared to other existing models for prediction of pressure drop in slurry flows.

References

The parity plot for experimental and predicted pressure drop for all the data points considered in this study is shown in Figure 9. It is observed that pressure drop calculation by the Modified Wasp method gives better prediction and is in good agreement with experimental data. The best fit is the straight line having slope = 1, which means the predicted values are equal to the experimental values. It is observed that most of the predicted values lie very close to straight line and maximum error observed is less than 15%, which confirms that prediction is better using this method.

1. 2.

3. 4.

5.

6.

Figure 9. Experimental vs. Predicted pressure drop 7. k is corrected (Durand’s equation coefficient) for calculating pressure drop using this method. The k value for any specific slurry system can be determined with method described by Wasp. This method requires one experimental value of pressure drop for a unit length at a given solid % and PSD. Once this slurry specific ‘k’ is available the same value can be used for predicting the pressure

Abulnaga, Baha E (2002), “Slurry Systems Handbook”, McGraw-Hill. Shou George, “Solid-liquid flow system simulation and validation”, Pipeline Systems Incorporated, USA Manfred Weber, “Liquid-Solid Flow” Lahiri SK and Ghanta KC,”Prediction of Pressure drop of slurry flow in Pipeline by Hybrid Support vector Regression and Genetic Algorithm Model”, Department of Chemical Engineering, NIT, Durgapur, India. Kaushal and Tomita, “Solids concentration profiles and pressure drop in pipeline flow of multisized particles slurries”, Department of Mechanical Engineering, Kyushu Institute of Technology, Japan. Kaushal and Tomita, “Effect of particle size distribution on pressure drop and concentration profile in pipeline flow of highly concentrated slurry”, Department of Mechanical Engineering, Kyushu Institute of Technology, Japan. Jacobs BEA (2005), Design of Transport Systems, Elsevier Applied Science, England.

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