Indian Journal of Engineering & Materials SciencesVol. 4, February 1997, pp. 1-9

Prediction of concentration and size distribution of solids in a slurry pipeline

V Seshadri", S N Singh" & Mukhtar Ahmed"

"Department of Applied Mechanics, Indian Institute of Technology, New Delhi 110 016, India

"Department of Mechanical Engineering, Jamia Millia Islamia, New Delhi 110 025, India

Received 24 May 1996; accepted 11 October 1991)

In any practical slurry pipeline application, the solids that are being transported are invariably multi-sized with sizes varying over three orders of magnitude. The prediction of solid distribution across thepipe cross-section is very complicated due to large number of parameters involved. It can be easily en-visaged that not only the solid concentration but also the size distribution of solids vary across the pipecross-section. This knowledge is important for optimum design of the pipeline. The existing experimen-tal data have been compared with the predictions based on the model proposed by Karabelas I. The rea-sons for discrepancy have been identified and modifications have been proposed. The dimensionlesseddy diffusivity is assumed to vary with solid concentration and an optimum relationship has beenderived using a semi-emperical approach. The modified model has been found to give more accuratepredictions.

Slurry transportation by pipelines is now a wellestablished mode for long distance haulage ofsolid materials as it offers many advantages overconventional modes of transport. This mode hasgained further importance in today's world due toincreased awareness of the harmful effects of envi-ronmental pollution. Design of slurry pipeline isvery complex and is dependent on many geometri-cal and hydraulic parameters. Besides these par-ameters the various phenomena that occur in theheterogeneous flow of suspension are too compli-cated. One of the most important parameterswhich is responsible for the complex phenomenais the concentration and the related particle sizedistribution. The distribution of solids across thecross-section of the pipe depends on many factorssuch as flow velocity, pipe diameter, particle size,its density and concentration. Some models areavailable in literature for the prediction of concen-tration profile but none is general enough for theprediction of the total concentration profile for alltypes of slurries. Ismail? taking a lead from theconcept of O'Brien3 and Rouse" for open channelflow, developed an expression for the predictionof concentration profile in a rectangular duct andcompared with his own experimental data toprove its validity. Wasps using his expression pre-dicted the concentration profile for coal slurries ina 300 mm NB pipe and found good agreementwith their own experimental results by makingsome simple assumptions. Encouraged by this

matching, Wasp et at." further analysed their ex-perimental data and using Ismail's data developeda modified equation for pipe flow.

... (1)

Shook and Daniel? identified some shortcomingsin the expression for concentration proposed byWasp et al'' In the late seventies, Karabelas I!deve-loped a semi-empirical model for the prediction ofconcentration profile as well as the distribution ofeach size fraction across the pipe cross-section fordilute suspensions. Seshadri et al.R made detailedmeasurements for zinc tailings slurry and com-pared them with the predictions based on Karabe-las model. They observed some major discrepan-cies. Based on their observations, they suggested anew modified model? which predicts the concen-tration profile reasonably well even at higher con-centrations. Shook et at.1O have recently suggesteda model for the prediction of concentration profileby solving a set of differential equations. However,the procedure is some what complex and requiressome assumptions on solid properties. Wilson andPugh" have also developed a method for the cal-culation of concentration profile as a continuousfunction of height from bottom of pipe. All themethods quoted are not universally applicable andare specific to the slurry for which they have been

2 INDIAN 1.ENG. MATER. set, FEBRUARY 1997

developed. Most of these studies are for equisizedparticles except that of Karabelas I and Shook etal.1O In the present study, Karabelas' model whichhas worked for dilute slurries has been adoptedand modified for the prediction of concentrationprofile and particle size distribution even at higherconcentration. The prediction of concentrationprofile across the pipe cross-section is a very im-portant input for calculating the total volume ofmaterial being transported through the pipe line. Itis also important for establishing the extent of un-even wear likely to take place along the peripheryof the pipe surface.

Brief Description of Karabelas' ModelKarabelas I presented closed form expressions

for predicting solids concentration distribution in avertical plane of multisized solid panicles in theturbulent core of pipe and channel.

The important steps in the development ofKarabelas expression are briefly explained below:

The system of equations describing vertical dif-fusion of particles assuming solids diffusivity forall size fractions uniform and equal to the liquiddiffusivity is given by Hunt'? as

... (2)

where

n

Vy= I U1iCii-I

... (3)

For horizontal conduits or pipes, the system of Eq.(2) admits the following general solutions':

C(y')= Gjexp[UIj/(y)] .) 1 +Ie

iexp[ - U.J(y)]' j= 1,2,...,n ... (4)

where, f(y)= dy/Es(Y) and Gj is a set of character-istic coefficients of each size fraction but inde-pendent of space coordinates. In order to proceedin the development of the distribution functionC) (y) the following two assumptions are made:

(i) the dimensionless eddy diffusivity, ~ is a con-stant and independent of flow concentration andspace coordinates, that is

... (5)

(ii) the solids concentration is a function of ver-tical coordinate yonly. Using Eq. (5) for E; f(y)can be expressed as

... (6)

where, y' =y/R, +1 ~ y" ~ - 1. Substitution of thisvalue of fly) into Eq, (4) leads to

where

-J!JL .K, - ~ U*' 1,2,...,n ... (8)

A transformation is required in order to determinethe parameter Gj. For that, first the following localand mean relative concentration are defined as

C Ev.-- ) -~ (9))- - '\ - - ...1 L.,Ci 1 Cc

where, the bars denote quantities averaged overthe pipe cross-section in terms of the relative con-centration Vj (y), the solution (7) can be expressedas follows

~ (y')= Gj -expl - Kj y'); j= 1,2,...,n ... (10)

Now it is assumed that the flow is steady, there isno particle deposition at the bottom of pipe, hencethe mean concentration of each particle size Cj inthe pipe cross-section is constant and alreadyknown. The relative concentration Vj is constanttoo. Therefore, the integration of Eq. (10) over thepipe cross-sectional area A leads to:

or

VGj='E~K \; j= 1,2, ...,n

\ j/

... (11)

with the coefficients E( Kj ) defined as follows

E(Kj)= 11A f Aexp( - Kjy') dA; j= 1,2,...,n

... (12)

SESHADRI et al.: PREDICTION OF SOUD DISTRIBUTION IN A SLURRY PIPEUNE 3

Table I-Properties of solid material (Zinc tailings)

(i) Overall specific gravity of the solids =2.597

(ii) Specific gravity of the individual size fractions

B.S. mesh size Specific gravity

-25+36 252-36+52 2.71-52+72 2.74-72 + 100 2.74

- 100 + 120 2.78- 150 + 170 2.86- 170 + 200 2.88

(iii) Particle size distribution in the fresh sample (Wet sievingover B.S. 200 mesh + hydrometer analysis)

Particle diameter % finer (by wt.)

1180297150106755348383121161286

100.0091.6876.0361.1050.1440.5037.6235.3232.4226.7816.615.084.073.73

(iv) Static settling of zinc tailings slurry (fresh sample) initialconcentration = 28.63% (by wt.)

Time,min Settled concentration%bywt28.6729.1129.7230.3132.5838.6041.9444.1350.93

o5

10153060

150270

4320

The approximate form of E(Kj) can be obtainedby expanding exp( - Kj y' ).The result is

... (13)

The solution for the concentration distributionfor each size fraction as a function of y (vertical

distance measured from the bottom of the pipe) is

c, (y) =s[E(lj) exp( - s, y')]

X[1+I ~exP'(_Kjy')]-l ... (14);-2 E(Kj)

The theoretical closed form solution of concen-tration profile obtained by Karabelas 1 fitted themeasured concentration profile very well, with ~varying from 0.2 to 0.33. The average value of di-mensionless lateral particle diffusivity t deter-mined from his data is approximately 0.25. Itshould be noted that this value is much higherthan that for pure liquid flows. Further the theor-etical predictions were compared with experimen-tal data only for low solid concentrations due tolack of availability of data in the literature.

Experimental data used for comparisonExperimental data generated by Mukhtar" for

zinc tailings at different concentrations have beenused to verify the predictions for concentrationprofile and particle size distribution. The concen-tration profiles have been measured in 105 mmdiameter pipeline for which pilot plant loopfacility existed at lIT Delhi. Concentration profileshave been given for five efflux concentrationvarying between 9-48% by weight at three velocit-ies for each concentration. The PSD has also beenreported after dividing the sizes of particles in sizesix class intervals for two concentrations namely20.1 and 34.47% at three velocities each. Theproperties of the material used are given inTable 1.

Comparison of Measured and Predicted Concen-tration Profiles based on Karabelas' Model

Computations of concentration profiles havebeen done for all five concentrations of zinc tail-ings slurry at three different velocities for eachconcentration with a constant value of dimension-less diffusivity ~= 0.25. The number of size frac-tions chosen for the analysis are the same as chos-en for experimental analysis of concentration dis-tribution in the pipe. The number of size fractionsin experimental analysis were six as the range ofparticle size in the zinc tailings sample was quitewide. Unhindered settling velocity for each sizefraction and friction velocity V *, have been ob-tained from Mukhtar's experimental investigation 13

on zinc tailing slurry.

4

1·0

0-8

o O'-s:0'

0·2

(),O

0

1·0

0·&0

()'60-s: 0"1.

()'2

0·00

INDIAN 1. ENG. MATER. SCL, FEBRUARY 1997

( a)

I\0..,

"5 6

(b)

,'\,-, o

2 3 4Cyl Cyl

1·0

().S

O~g-s: ~

(}2

O<l0

(e)

o

3 42Cy/Cyf

"Fig. I-Comparison between measured and predicted concen-tration profile at Cwf = 9.97% by wt. (Cvf = 4.09"10). [(a)Vm=1.57 mis, (b) Vm=1.96 mls and (c) Vm=2.89 m/s] [(0)

Experimental, (--) Karabelas' and (-) modified)

Fig. 1 shows the comparison of predicted con-centration distribution with experimental concen-tration profiles for 9.97% concentration at threevelocities namely 1.57, 1.96 and 2.89 m/s. It isobserved that concentration profile obtained fromKarabelas' model are more symmetric than thoseobtained experimentally. This is true for all threevelocities at this concentration. There is underpre-diction of approximately 46% at the bottom of thepipe (hID= 0.05) and overprediction by about 76%at the top of the pipe (hID= 0.95) for two velocit-ies 1.57 and 1.96 m/s. But as the velocity goes up

1-0 (c )

0'8

0 (}6-s:0-4

02

3·0

1·0 (b)

0'&

e ()'6

s:

0'4

0·2

0

0'6 1'0 1·5 2·0r-y levl

1-0 (e)o ,\

0-8 \0\

\\

0·6 \e \L \

0'4 \\ 0\

0'2 ~0

,06 1'0 1·5 2·0

c..,jCvl

Fig. 2-Comparison between measured and predicted concen-tration profile at Cwf=20.l% by wt, (Cvf=8.83%) [(a)Vm = 1.48 mis, (b) Vm = 2.05 rn/s and (c) Vm = 3.05 m/s] [(0)

Experimental, (--) Karabelas' and (-) modified)

to 2.89 m/s the symmetry in the experimentalconcentration profile increases so there is syste-matic decrease in deviation between predicted andexperimental values. Similar observations can bemade from Figs 2 to 4 at higher concentrations. Itcan also be seen that matching between predictedand experimental values is well within expected in-accuracies of measurement for 23 and 35% con-centration. At 34.47% (Fig. 4) the predicted valuesof concentration match very well with the experi-mental values especially at 1.6 rn/s flow velocity.But at higher velocities at this concentration, thereis reversal in the trend of the deviations between

SESHADRI et al.: PREDICTION OF SOUD DISTRIBUTION IN A SLURRY PIPELINE

°O~----~~----~2~------~3c, ICvf

Fig. 3-Comparison between measured and predicted concen-tration profile at Cwf=23.39% by wt. (Cyf= 10.52%) [(a)Vm= 1.76 rn/s, (b) Vm=2.25 m/s and (c) Vm=3.05 m/s] [(0)

Experimental, (--) Karabelas' and (-) modified)

',00

0·8

o~-.J::.

04

0·2

00

'·0

0·8

0 O·-s:0'4

0·2

00

\·00

0·8

00.6-s:0'4

0-2

(OJ

o3

CvlCvl

(b)

-c 0-c,-, o

2 3Cy I Cyl

(c)

predicted and measured profiles. This could bedue to the failure of the basic assumption that par-ticle diffusivity, ~ is constant across the pipe cross-section along the vertical diameter. At lower con-centration, the skewness in the concentration pro-file is more and therefore prediction with constant~ results in over prediction in the top of the pipeand under prediction at the bottom. As concentra-tion is increased, the skewness in concentrationprofile reduces due to increased collision betweenparticles and therefore the prediction show a bet-ter agreement in terms of over prediction in thetop of the pipe which has reduced from 36% tonearly zero. Similarly, in the bottom of the pipeunder prediction of 46% reduced also to zero

5

1·0r----------------:"1(0 )

00·6-0·4

0-2

0·8 ',0Cy/Cyl

I·Or---q--------~(:-b~)\\\

0\\\0\,

"o

0'&

00'6.

O~

0·2

o~ ~ ~_~0·5 1·0

c, I Cvl1·5 1·7

\·or----------------=(-c):-1

-.J::.

0·8

00·6

0·4

Fig. 4-Comparison between measured and predicted concen-traction proile at Cwf= 34.47% lby wt. (Cyf= 16.84%) [(a)Vm= 1.6 mis, (b) Vm = 2.12 m/s and (c) Vm= 3.08 m/s [(0) Ex-

perimental, (~:) Karabelas! and (-) modified)

(Fig. 4a). Further, with increase in velocity, theconcentration becomes more uniform and the pre-dictions start showing over prediction in the bot-tom of the pipe and under prediction at the top ofthe pipe (Fig. 4c). Fig. 5 shows the comparison ofpredicted and experimental concentration profilefor 47.55% concentration. From these three fi-gures it is observed that there is overprediction atthe bottom and underprediction at the top of thepipeline. This means that now the predicted pro-file has become more asymmetric than the mea-sured one which is opposite of what was observedat lower efflux concentrations. These deviations

6 INDIAN J. ENG. MATER. SCI., FEBRUARY 1997

1·0 loO

OR 0.•

o (}6 0.6s: ~

O'L, ~()o4

~0·2

-,

~

-,()oJ<,

<,<,

005........... ......

015 0 4 a

( b)

1·2 1-3 1·4 6 8 10 12 14Cvi/Cvit

1.0(C)

(el

0·8

~0-6s:

OL,

02

10 r----------:---.

08

\\\\\\ 0\

006s:

04" -, -,

<,<,

<,

(}O~~~-~-~-~~-~0·8 0·9 12 13

0·2

Fig. 5-Comparison between measured and predicted concen-tration profile at Cwf=47.45% by wt. (Cvi=25.8%) [(a)Vm= 1.47 mis, (b) Vm= 1.76 mls and (c) Vm=2.9 m/s] [(0)

Experimental, (--) Karabelas' and (-) modified]

could be due to some of the simplifying assump-tions made by Karabelas' while deriving the rela-tionships. Similar observations were made bySeshadri et al.9 from the analysis of data availableto them at that time. The general observation fromthe present comparison is that the agreement be-tween measured and predicted values of concen-tration is good at medium range of concentration(20-35% by weight). However, at lower and highervalues of concentrations, the deviations are largeIt is also seen that at high concentrations the pre-dicted concentration profiles were more asymmet-ric as compared to measured ones (Fig. 5).

(0)

12 16 20Cyi/eyif

24 28

(b)

tv;/tY;1ltl Vmr).() 5 mts

of diffirent sin fractions at <:Wf:.20·1"1. by wt

Fig. 6-Concentration profile of different size fractions atCwf=20.1% by wt (Cyf=8.83%) [(a) Vm= 1.48 rn/s, (b)Vm=2.05 m/s and (c) Vm=3.05 m/s] [B.S. Mesh (0)- 14 + 50 (~) - 50 +100, (0) -100 + 150, (V) - 150 + 200,

(e) -200+300 and (x) -300J

Description of Modified Karabelas' ModelThe two main reasons for the failure of Karabe-

las' model, as discussed by Seshadri et al.8, are theuse of unhindered settling velocity in the calcul-ations and the assumption of dimensionless diffu-sivity ; to be constant at all concentrations andfor all size fractions. The boundary layer forma-tion at the edges of the pipe section and the ensu-ing eddies and interaction effects might also possi-bly be the other causes of the deviations. On thebasis of extensive analysis of the experimental dataand deviations in the predictions, Seshadri et aL9

proposed an empirical correlation for the calcula-

SESHADRI et al.: PREDICTION OF SOUD DISTRIBUTION IN A SLURRY PIPEUNE 7

'·0..-----------,(0)

o

0·1

(b)

c"i ICvitlbl'lm_2·12 m/5

(C)

0.1I:>...•.~

Fig. 7-Concentration profile of different size fractions atCwf- 34.47% wt. (Cvf'"' 16.84%) [(a) Vm -1.6 mis, (b)Vm - 2.12 m/s and (c) Vm - 3.05 m/s] [B.S. Mesh (0)-14 + 50 (a) - 50 + 100, (0) - 100 + ISO, (V) - 150 + 200,

(e) -200+300 and (x) -300]

tion of dimensionless diffusivity ~. The new empir-ical equation for determination of ~ as proposedby them is of the form,

... (16)

where K, and K2 are constants which depend onphysical properties of the material and C Y55 is thestatic settled concentration of slurry.

According to the above correlation, as the con-centration of slurry approaches zero, the value of~ tends to 0.07, which is the value of dimension-less diffusivity for pipeflow of a pure liquid. Asthe value of C, increases, the ratio C; ICySS ap-

1.0

~ o.s~s.iii::;)

0·6......Q

IIIIII 0.4'"~z0Viz 0·2'"~Q

00 1.0

Fig. 8- Variation of dimensionless diffusivity, ~ with (C, I Cyss)

proaches a value of unity. Therefore, dimension-less diffusivity will also be increasing with increasein concentration (C.). An increase in the value of~ would result in the increase in E s and hence areduction in the concentration gradients in thepipeline as can be seen from Eq. (2). Thus at high-er concentrations, it would tend to get uniformsolid distribution across the pipe cross-sectionwhich is borne out from the experimental observ-ations. Hence, the correlation for ~ given in Eq.(16) takes into account the inhibition of settling ofsolids due to interference effects at higher concen-tration. The second reason identified by Seshadriet 0/.8 was the use of unhindered settling velocityin the calculation of concentration profile by Kar-abelas. Unhindered settling velocity is calculatedon the basis of specific gravity and median particlesize of each size fraction. However, he did notconsider the effect of concentration, particle sizedistribution and duct walls. In the present investi-gation, hindered settling velocity has been usedwhich besides being function of specific gravityand particle size is also a function of solid concen-tration and particle size distribution". For this pur-pose correlations available in the standard litera-ture have been used 13. With the above modifica-tion incorporated in the Karabelas' model, betterpredictions can be expected even at higher con-centrations.

C:omparison between Measured and PredictedConcentration Profiles based on ModifiedKarabelas' Model

In order to predict the concentration profile itis first necessary to evaluate the values of the con-stants K) and K2 in Eq. (16). The analysis of datashowed that the best values for K 1 and K 2 are as

8 INDIAN J. ENG. MATER. SCI" FEBRUARY 1997

follows

K, = 0.576 and K2 = 3.290

The values are of the same order of magnitude asthose given by Seshadri et at." The optimum varia-tion of ~ with (C, ICyss) is given in Fig. 8. Usingthe above values of K I and K2 for the predictionof ~ at any efflux concentration, concentrationprofile of each size fraction as well as overall con-centration profiles have been computed.

Figs 1-5 also show the comparison between themeasured concentration profile and the concentra-tion profile obtained with the help of modifiedmodel using variable diffusivity. It is observedfrom the Figs 1-5 that there is general improve-ment in the predicted values for all concentrationsat all velocities as compared to Karabelas' model.It is observed from Fig. l(a) that for 9.97% con-centration, at 1.57 ml s flow velocity, Karabelas'model underpredicts the concentration at bottomof the pipe by 88% whereas the modified modeloverpredicts it by 22% only. At 1.96 mls (Fig. Ib)and 2.89 mls (Fig. Ic] flow velocities Karabelas'model underpredicts by about 90% whereas modi-fied model overpredicts these concentrations byonly 25% at the bottom of the pipe. Similar im-provements in the predicted concentrations areobserved at the top of the pipeline (hID=0.95)when modified model is used. This trend of im-provement in predicted concentrations profile isobserved at all concentrations. The predictions at47.4 5% efflux concentration by modified modelshow an impressive improvement. The agreementin the experimental and predicted values are quitegood even at lower velocity, i.e., at 1.47 ml s flowvelocity (Fig. Sa). As the velocity goes up thesevalues at the top and bottom of the pipe deviateslightly from the experimental values. The devi-ations in the predicted values and experimentalvalues have come down from about 30% to just4% (Figs 5-b&c) of overprediction with the modi-fied model. Similar improvements in predicted va-lues at the top of the pipe (hID= 0.95) can beseen from the Fig. 5a,b,c. This shows that the useof diffusivity values dependent on solids concen-tration is justified. It is needless to state that fur-ther work on different materials is still required inorder to formulate a reliable model for the predic-tion of concentration profiles in straight pipelines.

Figs 6 and 7 show some representative resultsof particle size distribution as predicted by modifi-ed model for 20.1% and 34.4% efflux concentr-ations. A comparison of these figures with themeasured concentration profile of individual size

fractions (Mukhtar'") shows a striking similaritybetween the two sets. The profiles have very simi-lar shapes thereby enhancing the confidence in themodel. However, there are still some deviations inthe actual values in the two sets. It is seen fromthese graphs (Figs 6 and 7) when compared withthe experimental results that (Cyj ICyjf) values forfiner particles, i.e., below 150 B.S. mesh size, thereis good agreement between predicted values andexperimental values. The deviation is seen to in-crease with increase in particle size. But this devi-ation narrows down even for coarser particleswhen the velocity is increased. This shows thatmodified model even predicts the particle size dis-tribution quite accurately at higher velocities. Theaccuracy of prediction decreases with decrease invelocity of flow and increase in particle size. Thisconfirms the doubt expressed by Seshadri et aL9

about the use of same constant value for dimen-sionless diffusivity (~) in Karakelas' model for allsize fractions of solid particles in a multisized par-ticulate slurry. Therefore more materials are re-quired to be tested for a wide data base whichmay help in having a better understanding of thecomplex nature of dimensionless diffusivity forslurries of multisized particles.

ConclusionIt is noted that the modifications proposed in

the model proposed by Karabelas have made itmore versatile and the accuracy of prediction hasvastly improved over a wide range of solid con-centration. However, it should be recognised thatthe model is still at its initial stage of development.At present, it is assumed that eddy diffusivity is afunction of efflux concentration only and is con-stant across the pipe cross-section. However, thesolid concentration and partide size distributionvary across the pipe cross-section. Thus, there ex-ists a need for a more rigrous analysis supportedby a wider data base on different materials. Atpresent, efforts are being pursued in this direction.

NomenclatureC =concentration by volume at 0.08 d from top of the

pipe= concentration by volume at pipe axis= local concentration of jth particle size fraction=concentration by volume=concentration by volume of "h size fraction in emux= static settled concentration by volume•• particle diameter- pipe diameter=mass transfer coefficient=height from bottom of pipe in vertical plane= radius of pipe

CACj, Cvi

Cv

Cvir

C,••dDE,hR

SESHADRI et al.: PREDICTION OF SOUD DISTRIBUTION IN A SLURRY PIPEUNE 9

U, = settling velocityV • = friction velocityVy, Vy = flow velocity

{J = ratio of mass transfer coefficient to momentumtransfer coeffieient

~ =dimensionless diffusivity~I =viscosity of liquidk = Von Karman constant

References1 Karabelas AJ, A IChE J, 23 (1977) 426.2 Ismail H M, Proc ASCE, 117 (1952) 409.3 0 Brien M P, Trans Arn Geophys Union; 14 (1933).4 Rouse H, Trans ASCE, 102 (1937)463.5 Wasp E J, Pipeline News, 35 (1963) 20.6 Wasp E J, Aude T C, Kenny J P, Seiter R H & Jacques R

B, Deposition velocities, transition velocities and spatial dis-

tribution of solids in slurry pipelines (Proc Hydrotransport1, BHRA Fluid Engineering, Cranfield, Bedford, Eng-land),1970.

7 Shook C A & Daniel S M, Can J Chern Eng, 47 (1969).8 Seshadri V, Malhotra R C & Sunder K S, Concentration

and size distribution of solids in a slurry pipeline (Proc11th National Conference on Fluid Mechanics and FluidPower, BHEL, Hyderabad, India), 1982.

9 Seshadri V, Malhotra R C & Adisheshu C, Prediction ofdistribution of solids across the cross section of a slurrypipeline (Proc 13th National Conference on Fluid Me-chanics and Fluid Power, Trichurapalli, India), 1984.

10 Shook C A, Gillies R, Hars D B, Husband W H W &Small M,J Pipeline, 3 (1982) 13.

11 Wilson K C & Pugh F J, Can J Chern Eng,66 (1988).12 Hunt J N, J Mech Appl Math, 22 (1979) 235.13 Mukhtar A, Investigations of the flow of multisized hetero-

geneous slurries in straight pipe and pipe bends, PhD The-sis, Indian Institute of Technology, Delhi, 1991.