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 con-sumption. 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 slur-ry pressure drops over a wide range of operating conditions and physical properties of different slur-ries. 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) meth-od is based on multi-phase flow modeling approach. This study attempts to modify this approach by con-sidering material-specific values of Durands equa-tion co-efficient and by defining flow regimes based on particle Reynolds number. When compared with experimental data, the modified Wasp method pro-posed 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 con-ditions.

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 design-ing 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.

Keywords: Particle size distribution, Slurry flow, volumetric concentration, Pressure drop, Wasp Method, Drag Coefficient, friction factor, critical velocity, Durands 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 formu-lation of the correlations for prediction of pressure drop and for delineation of flow regimes. The rheo-logical approach is best applicable to slurries of ultra-fine 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 describ-ing 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 pre-dicting the pressure drop. Slurry Flow in Pipe

For a pure liquid, the pressure drop in a pipe de-pends 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 mono-tonic (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 ve-locity 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.

Figure 1. Plot of transitional mixture velocity with pressure drop

To achieve the optimum performance i.e. mini-mum 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

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 re-duced 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 (salta-tion).

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

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

Research Work & Different Methods

Over the years, two principal research works have

been developedone around the DurandCondolios 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 two-layer 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.

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 im-portant parameter accounting for pressure drop calcu-lation. Wasp method is an improvement of Durand-Condolios 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 dis-tributions 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 viscosi-ty of the equivalent liquid vehicle. Wasp method also considers the effect of pipe diameter and pipe rough-ness on pressure drop in slurry pipes.

The iterative method proposed by Wasp et al. (1977) is summarized as follows:

1. By using Durands equation, the total size fraction is divided into a homogeneous and heterogeneous fraction.

2. The friction losses of the homogeneous frac-tion are calculated based on the rheology of the slurry, assuming Newtonian flow.

3. The friction losses of the heterogeneous frac-tion are calculated using Durands equation.

4. 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).

5. Based on the value of C/CA, the fraction size of solids in homogeneous and heterogeneous flows is determined.

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., Durands equation coef-ficient, is material-specific. For different ma-terials, 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 re-sults.

2. Different flow regimes have been defined based on the assumption that particle size having Reynolds number less than 2 will al-ways contribute toward homogeneous losses and particle size having Reynolds number greater than 525 will always contribute to-ward heterogeneous losses in horizontal slur-ry flow pipe for all flow velocities as pro-posed by Duckworth (Jacobs 2005).

These modifications have been described in the following section. We refer to these modifications as the Modified Wasp Model.

Algorithm for Modified Wasp et al. (1977) Method

The example below illustrates the modifications

proposed in this paper to the Wasp et al (1977) meth-od. 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 pipe-line with a roughness coefficient of 0.045 at a weighted concentration of 26.3%. The results of pres-sure drop versus velocity are presented in Table 1.

Table 1. Pressure drop versus Speed in a 159-mm ID Steel Pipe at a Weight Concentration of

26.3% Velocity

(m/s) Pressure drop

(Pa/m) 1.5 175


4

1.9 270 2.3 360 2.7 525 3.1 688 3.5 847 4.0 1046

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

Table 2. Particle Size versus Wt. % solids in the slurry

Particle Size (m)

Wt. %

-450 1.88 -200 2.2 -95 1.65 -61 1.17 -44 93.1

Step 1: Plot the PSD curve.

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

Table 3. PSD Data of the solids presents in the slurry

Particle Size (mm)

Weight %

Cumulative 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

Figure 3. Particle size distribution curve

Step 2: Determine cut size. dcut using Wasps 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-

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

[ ]

= 340416

6

25253 LLS.

vcut

/)(gD*C*).(VDd

(1)

Where

v = Operating velocity of slurry (m/s) D = Pipe inside diameter (m) Cv = Volumetric concentration of solid dcut = Cut size

s = Specific gravity of solid

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

Figure 4. Cut size for flow regime Case 1

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 contrib-ute 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 contin-uous 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 modi-fication, the new cut size dcut will now be dRe max. Case 2: Cut size - dcut < dRe min


Where, dRe min is the particle size having Reynolds

number 2.


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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 exist-ing wasp method. As a result of this modification, the new cut size dcut will now be dRe min. Case 3: Cut size - dcut lies between dRe max and dRe min.

Figure 6. Cut size for flow regime Case - 3

In this case, particles having size less than dcut

will remain suspended and contribute toward homo-geneous losses, whereas particles with size greater than dcut will settle down and contribute toward heter-ogeneous losses. Case 4: Cut size - dcut is > d100 and dRe max > d100


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 cal-culated 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 ve-locity 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. 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):

40424 .)Re

()Re

(cD ++= (2)

The single particle settling velocity vt is calculat-ed using the Stokes equation:

)/(*)c

gd(*v LLS

D

Ret = 3

4 (3)

The Particle Reynolds number Re is then calcu-

lated :

L

Ret

v

dvRe = (4)

Where

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

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 re-gime will be considered. 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 Darcys formula:

Loss (Pa/m) = D 2

Vf m2

D (5)

Where

fD = Darcy friction factor. v = Mean velocity of slurry (m/s).

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 coeffi-cient of the homogeneous part of the mixture:

{ }29074573250

].mRe/../)i/Dlog[(

.Df

+= (6)

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 m as given below:


6

m

mmm

)DV(Re

= (7)

)6.16exp(00273.0

+05.10+5.2+1= 2l

m

vf

vfvf

C

CC

(8)

Where

Cvf = volumetric concentration of solids in ho-mogeneous part of mixture

l = liquid viscosity m = 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*( m / s ) (9)

Cv total = 26.3*(1244/ 4072) = 8.035%

Cvf = 0.95*Cv total = 7.633% Where

m = slurry density

s = solid density Cv total = total volumetric concentration of sol-ids Cw = total solids concentration by weight

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, m 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 Darcys 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. 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 distribu-tion 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

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.

Table 4. Redistributed particle size for heteroge-neous part of the mixture

Size ( mm)

Average Particle

size (mm)

Cv in Sol-

ids (%)

Cv bed in the slurry

(at overall

solids Cv of mixture at 8.035%)

(%) -0.85

+ 0.40 0.63 1.88 0.151

-0.40 + 0.20

0.30 2.20 0.177

-0.20 +0.152

0.18 0.92 0.074

Total 5 0.402

Step 6: Determine the particle Reynolds number and drag coefficient for each size range.

It is essential first to determine the drag coeffi-cient 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 m, viscosity of the slurry m and the speed of the carrier fluid V are used. The equation to calculate the particle Reynolds number is as fol-lowing:

m

mPP

VdRe

= (10)

Where

dp = the average particle size of each size frac-tion.

To calculate the drag coefficient, CD, of a sphere,

the Turton equation is used:

CD = ((PRe

24 )*(1+0.173*Rep0.657)) +

)(Re*11630+1413.0

09.1P

(11) Results are summarized in Table 5.


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Table 5. Drag Coefficient for single particle size

in Example 1 Particle size (mm)

Av-erage

Particle Size (mm)

Particle Reynolds number

(Rep)

Drag coefficient

(Cd)

-0.85 + 0.40

0.63 1160 0.455

-0.40 + 0.20

0.30 557 0.550

-0.20 + 0.152

0.18 327 0.662

Step 7: Calculate the friction losses of the heterogeneous fraction using Durands equation.

Wasp et al. (1977) recommend using Durands equation for each size fraction of solids to determine the increase in pressure losses due to moving bed:

51

2

.

D

LLsivbedLbed

CV/)(gD

CP82P

=

(12) Where

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

Ellis and Round (1963) indicated that Durands

equation coefficient of 82 is too high for nickel sus-pensions. Therefore, for this particular example, a value of 23 is used as the modified Durands 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.

Table 6. Calculated Losses for each fraction of

solids in moving bed Av-

erage Particle

Size (mm)

Parti-cle

Reyn-olds

number (Rep)

Drag coeffi-cient

(Cd)

Cv bed

(%)

Pres-sure Loss

(Pa/m)

0.63 1160 0.455 0.151

14.51

0.30 557 0.550 0.177

14.74

0.18 327 0.662 0.074

5.36

Total 34.61 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):

fx

t

A UKV.

]CC[log

81

10 = (13)

Where

C/CA = the ratio of volumetric concentration of solids at 0.08D from top to that at pipe axis,

= the dimensionless particle diffusivity and is taken as 1.0

Kx = 0.4 and is defined as von Karman coeffi-cient

Uf = the friction velocity calculated from the pressure drop in first iteration Vt = settling velocity of a particular size parti-cle calculated using standard drag relationships.

Where Vt is defined as the single particle size terminal velocity using the Stokes equation:

DL

gLSt C

gd)(V

34

= (14)

The equivalent fanning factor (Abulnaga, BE 2002) is calculated as:

Total loss = if /DVf 22 (15)

2itotal

N 2VDP

f

= (16)

To calculate Uf, the following Equation (Abulnaga, BE 2002) is used:

Uf = V (fN /2) (17)


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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 heterogene-ous 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. Step 9: Determine the value of Durands equation coefficient, k.

The value of the Durands 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, pres-sure drop using k=82 is calculated. If the total calcu-lated pressure drop does not match with the experi-mental pressure drop then a modification in the value of k is required. To determine the modified value of k, a ratio R is defined.

R = total experimental loss / total calculated loss (for k=82).

An Iterated value of k is determined by multiply-ing k by ratio R, which is re-iterated until the value of R converges to 1. After iteration, a value of k is ob-tained for which R = 1, which is then defined as the modified Durands equation coefficient.

Validation of Modified Wasp et al (1977) Method

Case study 1: Verification of Modified Wasps 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.

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

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 Wasps 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 grav-ity was 2.47. Results of this case study are presented in Table 15

Vertical Slurry Flow in Pipes

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

SwV >> Where

)c(vw VtS = 1 (18)

Sw = Hindered settling velocity of particle = constant. vt = Single particle terminal velocity (from

equation 14)

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).

Figure 8. Influence of the concentration on settling velocity according to Maude and Whitmore (1958) replotted from Weber (1974).

Reynolds number is calculated using equations (2),(3) and (4) corresponding to particle diameter d100 (=dRe). Using this Reynolds number, the value of is calculated from the above curve. Hindered settling velocity of the particle is calculated corresponding to the value of 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


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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 calculat-ed by Wasp Modified Durands equation (19).

L

Ls/

.vcri DgD

dc.u

= 25253

611002340

(19)

Homogeneous or pseudo-homogenous flow re-

gime occurs when mean slurry velocity (V) is greater than the critical deposition velocity (ucri). At this con-dition 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).

vertmvertm

D LgDL

Vfp +

= 22 (20)

Where

p =total pressure drop for vertical slurry flow us = velocity of solids

vertL = length of vertical section under con-sideration


Table 7. 1st iteration results Average Particle Size

(mm)

Drag Coeffi-cient

(Cd)

Terminal Velocity

(m/s)

-1.8 vt/KxUf

C/CA Iterated Cv bed (%)

Iterated Pressure loss (Pa/m)

0.63 0.455 0.201 -0.0844 0.823 0.125 11.94 0.30 0.550 0.127 -0.0532 0.884 0.157 13.03 0.18 0.662 0.089 -0.0371 0.918 0.068 4.92 Total 29.89

Table 8. 2nd iteration results Average Particle Size (mm)

Drag Coef-ficient

(Cd)

Terminal Velocity

(m/s)

-1.8 vt/KxUf



0.63 0.455 0.201 -0.0837 0.824 0.1248 11.96 0.30 0.550 0.127 -0.0528 0.886 0.1569 13.05 0.18 0.662 0.089 -0.0368 0.919 0.068 4.92 Total 29.89

Table 9. 3rd iteration results

Average Particle Size (mm)

Drag Coef-ficient

(Cd)

Terminal Velocity

(m/s)

-1.8 vt/KxUf



0.63 0.455 0.201 -0.0837 0.824 0.1248 11.96 0.30 0.550 0.127 -0.0528 0.886 0.1569 13.05 0.18 0.662 0.089 -0.0368 0.919 0.068 4.92 Total 29.89

Table 10. Pressure drop vs. speed in a 159-mm ID Steel pipe at a weight concentration of 26.3% at 20o C

Cv (%)

Veloci-ty

(m/s)

Di (m) S

(kg/m3)

L (kg/m3)

L (mPa.s)

Experi-mental Loss

(Pa/m)

Calculat-ed

Loss(Pa/m)

% Error Flow Regime

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 4.0 .159 4074 1000 1 1046 1008 -3.6 Case 4

Table 11. Calculated pressure drop for case study 2

Cv (%)

Velocity (m/s)

Di (m) S

(kg/m3)

L (kg/m3)

L (mPa.s)

Experi-mental Loss

(Pa/m)

Calculated Loss

(Pa/m)

% Error Flow Regime

4 1.2 .105 2820 1000 1 232 227 -1.81 Case 3 4 2.8 .105 2820 1000 1 1006 897 -10.9 Case 4 4 3.2 .105 2820 1000 1 1316 1165 -11.4 Case 4 4 3.6 .105 2820 1000 1 1652 1469 -11.0 Case 4 4 4 .105 2820 1000 1 2013 1808 -10.1 Case 4

8.1 1.6 .105 2820 1000 1 438 366 -16.5 Case 3 8.1 2 .105 2820 1000 1 619 500 -19.5 Case 4 8.1 3.6 .105 2820 1000 1 1781 1579 -11.3 Case 4 8.1 4 .105 2820 1000 1 2168 1942 -10.4 Case 4

12.8 1.6 .105 2820 1000 1 468 448 -4.3 Case 3


11

12.8 3.6 .105 2820 1000 1 1953 1700 -12.9 Case 4 12.8 4 .105 2820 1000 1 2317 2092 -9.7 Case 4 19.1 2 .105 2820 1000 1 722 638 -11.6 Case 3 19.1 3.6 .105 2820 1000 1 1962 1868 -4.74 Case 4 19.1 4 .105 2820 1000 1 2375 2298 -3.22 Case 4 26 2 .105 2820 1000 1 774 741 -4.23 Case 3 26 3.2 .105 2820 1000 1 1703 1636 -3.98 Case 4 26 3.6 .105 2820 1000 1 2104 2059 -2.14 Case 4 26 4 .105 2820 1000 1 2529 2531 0.04 Case 4

Table 12. Literature sources for pressure drop

No Author No Author 1 Wilson (1942) 11 Gillies et al. (1983) 2 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

Pipe Dia (m)

Particle Dia (micron)

Liquid den-sity

(kg/m3)

Solids den-sity

(kg/m3)

Liquid viscosity (mPa.s)

Velocity U (m/s)

Solids Conc. (fraction) f

0.019-0.495 38.3-13000 1000-1250 1370-2844 0.12-4 0.86-4.81 0.014-0.333 [1] Slurry system: coal/water, copper ore/water, sand/water, gypsum/water, glass/water, gravel/water.

Table 14. Results of data points collected from open literature having flow regime of case 4 Cv Veloci-

ty (m/s)

Di (m)

S (kg/m3)

L (kg/m3)

L (mPa.s)

Experi-mental

Loss (Pa/m)

Calcu-lated Loss

(Pa/m)

% Error

Flow Regime

0.3 3 0.0549 2470 1000 0.85 1990 2210 -11.1 Case 4 0.3 4 0.0549 2470 1000 0.85 3430 3726 -8.6 Case 4 0.3 5 0.0549 2470 1000 0.85 5350 5599 -4.7 Case 4 0.4 3 0.0549 2470 1000 0.85 2230 2706 -21.3 Case 4 0.4 4 0.0549 2470 1000 0.85 3790 4536 -19.7 Case 4 0.4 5 0.0549 2470 1000 0.85 6390 6786 -6.2 Case 4 0.5 3 0.0549 2470 1000 0.85 3410 3645 -6.9 Case 4 0.1 1.1 0.019 2840 1000 0.85 1250 1114 10.9 Case 4 0.1 1.11 0.0526 2330 1000 1 294 321 -9.2 Case 4 0.3 1.3 0.0526 2330 1000 1 543 609 -12.2 Case 4 0.3 2.59 0.2085 1370 1000 1 267 318 -19.1 Case 4 0.3 2.34 0.2085 1370 1000 1 226 263 -16.4 Case 4 0.3 2.01 0.2085 1370 1000 1 177 196 -10.7 Case 4 0.3 1.78 0.2085 1370 1000 1 147 158 -7.5 Case 4 0.3 1.59 0.2085 1370 1000 1 123 129 -4.9 Case 4


12

0.3 1.37 0.2085 1370 1000 1 99 98 1.8 Case 4 0.1 1.66 0.0515 2650 1000 1 666 671 -0.8 Case 4 0.2 1.66 0.0515 2650 1000 1 900 791 12.1 Case 4 0.3 1.66 0.0515 2650 1000 1 1136 930 18.1 Case 4 0.1 2.9 0.263 2650 1000 1 261 272 -4.0 Case 4 0.1 3.5 0.263 2650 1000 1 334 392 -17.4 Case 4 0.2 2.9 0.263 2650 1000 1 305 313 -2.4 Case 4 0.2 3.5 0.263 2650 1000 1 382 443 -15.9 Case 4 0.3 2.9 0.263 2650 1000 1 355 353 0.5 Case 4 0.3 3.5 0.263 2650 1000 1 453 505 -11.4 Case 4 0.3 2.9 0.263 2650 1000 1 414 392 5.2 Case 4 0.3 3.5 0.263 2650 1000 1 526 560 -6.4 Case 4 0.1 3.16 0.495 2650 1000 1 143 151 -5.7 Case 4 0.1 3.76 0.495 2650 1000 1 186 211 -13.6 Case 4 0.2 3.07 0.495 2650 1000 1 157 161 -2.5 Case 4 0.3 3.76 0.495 2650 1000 1 254 271 -6.4 Case 4 0.1 2.5 0.1585 2650 1000 1.3 475 421 11.4 Case 4 0.3 2.5 0.1585 2650 1000 1.3 630 532 15.6 Case 4 0.1 3 0.1585 2650 1000 0.12 648 526 18.9 Case 4 0.1 1.9 0.0507 2650 1000 1 1175 1140 3.0 Case 4 0.2 2.8 0.04 2270 1250 4 3926 4022 -2.4 Case 4 0.1 2.7 0.04 2270 1250 4 3580 3487 2.6 Case 4 0.1 2.01 0.04 2270 1250 4 2217 1939 12.5 Case 4

Table 14. Results of the case study

Cv Velocity (m/s)

Di(m) S (kg/m3)

L (kg/m3)

L (mPa.s)

Experimental Loss (Pa/m)

WASP modified

loss (Pa/m)

% Error

Flow Regime

9.4 1 0.0549 2470 1000 1 261 222 14.9 Case 4 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


13

49.24 3 0.0549 2470 1000 1 3082 3689 -19.7 Case 4 Heterogeneous vertical flow regime occurs when mean slurry velocity (V) is less than critical deposi-tion 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 concentra-tion and given solid mass flow rate, before calculating the pressure drop.

+

+=

Vv

c4Vv

11Vv

V2Vc tvtotal

2tt

tV

(21) The new value of volumetric concentration is used to

calculate the slurry viscosity by using Thomas corre-lation (eq. 8). This results in a modified value of slur-ry 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).

Results and Discussion

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 pres-sure 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.

Figure 9. Experimental vs. Predicted pressure drop

k is corrected (Durands equation coefficient) for

calculating pressure drop using this method. The k value for any specific slurry system can be deter-mined 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

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. The Modified Wasp Model:

i. Is easy to implement, has direct calculations involving iterations.

ii. 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.

iii. Effectively predicts pressure drop as a func-tion of Particle Size Distribution (PSD) and solid percentage.

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

1. Abulnaga, Baha E (2002), Slurry Systems Handbook, McGraw-Hill.

2. Shou George, Solid-liquid flow system simulation and validation, Pipeline Systems Incorporated, USA

3. Manfred Weber, Liquid-Solid Flow 4. Lahiri SK and Ghanta KC,Prediction of

Pressure drop of slurry flow in Pipeline by Hybrid Support vector Regression and Ge-netic Algorithm Model, Department of Chemical Engineering, NIT, Durgapur, In-dia.

5. 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.

6. Kaushal and Tomita, Effect of particle size distribution on pressure drop and concentra-tion profile in pipeline flow of highly con-centrated slurry, Department of Mechanical Engineering, Kyushu Institute of Technolo-gy, Japan.

7. Jacobs BEA (2005), Design of Transport Systems, Elsevier Applied Science, England.


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

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