Pressure drop prediction for flow of solid-liquid mixtures in horizontal pipes

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FIFTH AUSTRALASIAN CONFERENCE

on

HYDRAULICS AND FLUID MECHANICS

at

University of Canterbury, Christchurch, New Zealand. 1974 December 9 to December 13

PRESSURE DROP PREDICTION FOR FLOW OF SOLID-LIQUID MIXTURES IN HORIZONTAI PIPES

by

A.D. Thomas* and L.R. Flint*+

SUMMARY

The flow of solid-liquid mixtures or slurries in horizontal pipes is discussed under two main headings, settling and non-settling slurries. The available methods of predicting the pressure drop for these two cases are reviewed and compared. In the flow of settling slurries there are a number of possible mechanisms of particle suspension. These are discussed in connection with the prediction methods.

* M.D. Research Co. Khartoum Rd, North Ryde. N.S.W. + Now with Bellambi Coal Co., Sydney. N.S.W.

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PRESSURE DROP PREDICTION FOR FLOW OF SOLID-LIQUID MIXTURES IN HORIZONTAL PIPES.

INTRODUCTION

Solid-liquid mixtures can be classified under two main headings; settling and non-settling. A settling mixture is one which solid particles will quickly settle out if the mixture is left standing. A non-settling mixture is one in which the solid particles or settle only very slowly. An often used rough criterion is that particles of diameter less than 50 microns can be classed as forming a non-settling mixture in water. A non-settling mixture will generally behave as a single liquid with viscosity and density different from that fluid, and will flow in a pipe in a homogeneous manner. The manner in which a settling mixture flows in a pipe will depend on the velocity. sufficiently high velocities all particles will be kept suspended although there will be a concentration gradient vertically across the pipe. This type of flow is termed heterogeneous flow and is a true two-phase flor in that the solid and liquid phases retain their identity. If the velocity is progressively reduced the concentration gradient will increase until a stage is reached when the motion of the liquid will no longer be sufficient to suspend all of the particies. At this velocity, deposit velocity, come particles will separate out and form a stationary or moving bed of solids at the bottom of the pipe. It should be noted that if there is size distribution the fine particles may be transported homogeneously whilst the coarser particles could be transported heterogeneously. In this paper various methods of predicting the pressure drop for the two types of mixtures will be reviewed and discussed in the light of the possible mechanisms by which the particles are suspended in a flowing liquid.

PRESSURE DROP PREDICTION FOR FLOW OF NON-SETTLING MIXTURES

Non-settling mixtures of solid granular particles in water often exhibit non-Newtonian behaviour such as a yield stress and a non-linear stress-shear rate relationship, especially at high solids concentration. The problem of pressure drop prediction of a homogeneous mixture is therefore one of pressure drop prediction for a non-Newtonian fluid. approaches to the prodiction of pressure drop in a commercial size pipeline. The first is to use viscometric results obtained under laminar flow conditions to establish a rheological model

Using this model the laminar pressure drop may then be calculated theoretically

and the turbulent pressure drop by semi-empirical methods. The second method is to use a scaleup procedure from small pipe data. In the laminar regime a chart of T usually correlate data for different pipe diameters and enable scale-uf. For turbulent pressure drop prediction the small scale laminar results are discarded and sma! scale turbulent data are used to scale-up in the turbulent regime. These two basic method will now be discussed

Pressure Drops from Viscometric Data From viscometric tests a rheological model of the mixture can be evaluated. Cheng (1) has found that a wide range of commercial slurries can be characterised b the generalised Binphan model in whích the shear stress is given by T= T ÷ r VI where T is the yield stress, K is the consistency index, Y is the rate of shear strain, is the behavioural index. For turbulen?' K and n are determined the laminar pressure drop can be predicted theoretical!!• flow it is necessary to resort to semi-empirical correlations. Most of these are

based on either the Blpsius type equation for the friction factor (2)

or the Nikuradse type equation = A 10g (Re f', + c (3) In the evaluation of the Reynolds number, Re, various values for the viscosity have been used. One of the earliest correlations used the viscosity of the suspending medium. Others have used the plastic viscosity and some the limiting viscosity at infinite shear rate. and Metzner (2) replaced the conventional Reynolds number in equation (3) by a generalised Reynolds number Re' = D"V2-1p/K (4) after applying a dimensional analysis approach to a power law fluid. Cheng (1) has modified their method to accommodate a generalised Bingham fluid and has applied it to a large number of commercial pipelines. In a recent paper Kemblowski and Kolodziejski (?) have reviewed most previous correlations and have proposed a new one based on the Blasius equation. Their correlregion the pressure drops approach the Blasius value for a Newtonian fluid. Hanks and Dadil (4) employed Prandtl's mixing length approach and applied ation allows for the decrease of non-Newtonian behaviour at high levels of turbulence in which spreach and apied it it to Bingham fluid. Their theory

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has by Kenchington (5) to accommodate a generalised Bingham fluid. viscous interaction co-effic lent" to nudity theological parameters under turbulent conditions. Kenchington (5) has and kiln feed slurries for pipe diameters ranging from 16 um to 329 mm, above methods with experimental pressure drops for clay that all the above methods generally predicted turbulent pressure drops with comparable accuracy with no single method being significantly superior In the case of laminar flow he found that Cheng's (1) method gave reliable results except for the case of the clay slurry in the largest (329 mm) diameter pipe where it overpredicted by about four the measured pressure drop. He attributed this gross overprediction to the occurrence of "wall slip" in the large pipe which was not present can also occur in the viscometer and give erroneous effect is caused by the formation of a particle free layer near giving reduced values of the local shear stress. particle free layer has been observed in laminar tube flow (Segre and Silberberg (7)) in turbulent pipe flow (Roberts et al (8)), and in Couette cylindrical viscometers (Harris (9)). Harris presents a method of identifying slip in a co-axial cylinder viscometer. method (10 and 11) allows slip to be detected in laminar and turbulent pipe flows. Kenchington (5 and 12) found that slip did not significantly affect turbulent pipe flow prediction but could cause large errors in laminar pressure drop prediction. Pressure Drop Prediction by Scale-Up large pipes in both the laminar and turbulent Bowen (13) has presented a complete design procedure for scaling up small pipe results to In the laminar regime he uses the method of Metzner and Reed (14). is applicable to any purely viscous liquid ie., one which is time independent and inelastic. For such a liquid it can easily be shown that a log-log plot of wall shear stress T against 8 V/D will correlate the data for different pipes providing there is no slip effect. For a power law fluid this plot will be a straight line. wi!! need to be allowed for in any scale up procedure. If slip occurs plots of data from different pipes will be displaced and this effect This is discussed by Kenchington (12). For turbulent pressure drop prediction Bowen's method uses small scale turbulent data for scaling up. It has as it basis the assumption that for a non-Newtonian fluid the The Blasius friction factor f = 0.079 (D Vp /4 )-•

one obtains Ap/L = 2 A V2-b Combining this with the Farning the sleeti fon theodom riction fact 7 Westontan Fluid. be generalised. (6) (5)

where b =.25. For a particular Newtonian fluid (or Bingham plastic) p &u are constant Equation (7) becomes: AP/L = B V2-b D-1-b

Metzner (2), Re', into the Blasius equation ie., For a pseudo plastic ftuid Bowen substituted the generalised Reynolds number of Dodge and

- Once again combining with the Fanning equation and removing Re' = DR V2-n /K where K and n are power law constants P and K which are constant for a (9) particular pseudo plastic fluid.

- This equation is analagous to equation (8). In the case of a Newtonian fluid or a Bingham AP/L = B, V° Dd

plastic only two parameters, b and B, need to be determined from experimental data. case of a pseudo plastic fluid three parametersc, d and B, must be determined experimentally. Once these parameters have been determined by tests on a pilot plant scale the pressure loss written a full scale pipeline can be obtained. For pipes of diameters D, and D2 equation (10)can be

where J is the pressure gradient (11)

In the scale-up process any un ertainty in the small scale data will be magnified. Kenchington (15) has extended Bowen's method to allow calculation of confidence limits for the full scale results. He found that in any scale-up over a large diameter range (say 8 to 1) it is imperative to calculate the confidence limits as calculations based on the best fit value could give results seriously in error. Comparison of the Two Methods Criticism can be levelled at both methods. Harris (16) has argued that methods using laninar viscometric data to predict turbulent behaviour, in particular the Dodge-Metzner method, are theoretically unsound. pointed out by Kenchington (12), inherent in He advocates the Bowen scale-up procedure. the Bowen method is the assumption that the fluid Hovever, as obeys a particular law throughout the scale-up This may not be the case since the

"turbulent viscosity" of a power law fluid tends to become constant at very high slear rates.

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Thus experiments carried out in small pipes where shear rates are high may give erroneous results for a large pipe where lower shear rates prevail. Kenchington (12) has compared the two methods and came to the conclusion that they both predicted pressure drops in the turbulent regime with accuracies of the order of 30% Of course one big advantage of using viscometric tests is that they are a lot less costly than pipe loop tests. However Kenchington points out that the presence of slip, especially under laminar flow conditions, can cause large errors which could be overlooked when using a viscometric method. A pilot test would enable this slip to be detected. Kenchington recommends the use of both methods so that any discrepancy between the two can be investigated.

PRESSURE DROPS FOR FLOW OF SETTLING SUSPENSIONS

in the introduction the character of the flow of settling suspensions depends

on the velocity. Large heavy particles may never be suspended and may be transported by rolling or sliding along the bottom of the pipe. Smaller particles will be maintained in suspension provided the mean pipe velocity is high enough. Only the turbulent regime is of interest since no suspension of particles will occur in the laminar regime. At present no purely theoretical methods are available for pressure drop prediction with settling suspensions. Numerous empirical equations have been proposed which correlate data in specific instances but no equation has yet been proposed which will allow accurate prediction of pressure drop from the properties of the liquid and solids phases. Because of this it is necessary to resort to pilot scale tests. However if the pipe diameter of the pilot plant is significantly smaller than the full scale pipeline there is not even a reliable scaling up proced re which can be used vith confidence. Some of the correlations which have been proposed will now be briefly discussed. 1. Empirical Correlations of the form $= K *™ (12) where $ = I-Jw (13) and (14)

g D (S-1)

K is a constant, J and Jw are the head loss per unit length of pip for the mixture and for water alone respectively, & is the delivered volumetric concentrat on, V is the average mixture velocity in the pipe, Cd is the steady state drag co-effic ent of a particle, D is the pipe diameter, and S is the ratio of the density of the solids to tha: of water. Durand (17) was first to propose a correlation of this form. From experiments with a number of materials of grain size from 2.5 mn to 80 mm in pipes of diameters from 40 mm to 700 mm he gave the value of the index m as-l.5. Zandi and Govatos (13) gave two values of m = -.35 and m = -1.93. The choice of which one to use depended on the mixture velocity and concentration. Hayden and Stelson (19) gave a value of m = -1.3. Inherent in all correlations of this form is the assumption that the increase in the head loss over that for water is proportional to the delivered concentration. Babcock (20) questioner this assumption and performed accurate experiments which showed that whilst it was true in some cases it was invalid in others. He also found that in some cases a better correlation could be obtained bi not including the drag co-efficient. urthermore he found that the Froude Number vi adequately correlate data from pipes of widely differing diameters. type of correlation remains one of the most used. shortcomings there is no doubt that a rough correlation does exist between $ and V and this 2. Correlations Derived from a Dimensional Analysis Approac! a friction factor due to the presence of sölids. an entirely experimental approach employing only the techniques of Jinensional analysis. assumed Recognising the limitations of the above correlations Rose and Duckworth (21) attempted that the mixture friction factor f was given by the sum of the fluid friction f• and They They then included all varlables Which they thought could possibly affect fs and obtained the following non-dimensional expression. ie., f (15)

(16)

diametels of the particles are the mass flow rates of solids and fluid respectively d and Dare the and the pipe respectively, k is a roughness factor, 2 is a shape factor, and B is a parameter defining the spread of the sizes of the particles. were then performed on a wide range of materials in air and water in different pipe sizes up Experiments of 2 and B was not determined. to 75 mm diameter but only for spherical The influence of pipe roughness was also not determined but particles of closely graded size so that the influence deduced that the effect of k on f because they obtained good correlation from pipes of widel! varying roughness they

s was relatively small. Their final results are given in

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graphical form. The large number of variables affecting the pressure drop of slurry flow and the wide range of values that these variables can have makes any purely experimental approach immensely difficult. necause of this many of the graphs presented are drawn through only a few points and no gral accuracy is possible using their method. Its main value lies in the link which it provid:s between the fields of hydraulic and pneumatic solids transport. Turian et al (2) also approached the problem using dimensional analysis but reduced the number of non-dimensional groúps by assuming that some of the groups could be combined. By considering only spherical particles of closely graded size flowing in smooth pipes their initial analysis indicated the following groups:

f = F DewY (17)

gD

By postulating that the free settling velocity of particles in still water was an important variable and after consideration of the forces involved they reduced the number of groups thus

(18)

They then assumed the functional relationship I-Ew = Kea tw° Ca [BIs - 1) D/V]° (19)

and obtained the values of K, a,b,c and e by non-linear least squares analysis of 1511 data points. The final form of their correlation was

for $51 (20)

whera 1 = 20.19 f f. Wo 0.3281-36. d -011 for $ <1 (8D(S - 1)/N270.90 (22) (21)

The data used in obtaining this correlation was obtained for a reasonable range of particle microns co 4.38 mm) and particle densities (from 2.3 to 11.3 g/cc). only a limited range of pipe sizes was covered (12 mm, 25 um and 50 mm diameter) and its use outside this range is of course highly suspect. The correlations of the type= KY

or previous section can be compared with this by writing them in friction factor form 1= w m/2 [gD (S - 1) / VI'"I (23)

3. Semi-Theoretical Correlations A number of correlations have been developed by applying some theoretical reasoning about the flow situation. Some of these will now be discussed. Newitt et al (23) reasoned that the work done in maintaining particles of settling velocity V in suspension would be proportional to their effective vei nt and their settling velocity. Equating the work done on and by the particles they obtained the equation

From their data they gave K, * = Ky (S - 1) (V /V) (8D/V, to be a constant equal to 1100. Wasp et al (24 and 25) have (24)

developed a systematic method for pressure drop prediction in the heterogenecus flow regime which they have successfully applied to commercial size coal-water pipelines. Their method splits the pressure drop up into two fractions - that due to the solids which are transported homogeneously and that due to solids transported heterogeneously. They then use single phase Newtonian methods for estimating the pressure drop of the homogeneous portion and employ the Durand correlation (equation 12) for estimating the heterogeneous pressure drop. The homogeneous portion increases the effective density and viscosity of the "carrier fluid" and the consequent reduction in the settling velocity of the coarse particles reduces the heterogeneous pressure drop compared to what it would have been if the carrier fluid were pure water. This is an attempt to allow for the observed fact that as the amount of fine material in a turbulent stream is increased the carrying capacity for coarse material is increased. fraction of solids in the homogeneous and heterogeneous regimes they employ the theory of Ismail (26) to predict the in-situ concentration q at a height y = 0.8D above the bottom of the pipe. This theory employs the equation

€ dg + V 9=0 (25)

which equates the, rate of upward transfer of particles resulting from turbulent exchange with the rate of settling. settling velocity. Equation (25) is integrated by assuming that € is the mass transfer co-effictert and 5 equal to the momentum transfer co-efficient. The concentration at 0.8D is then assumed tổ be the portion of the solids which is homogeneously distributed. the solids have a wide size distribution they are broken up into 5 to 10 size fractions and the above calculations are performed for each

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size fraction and the pressure drops added. The method of Waspst al, employing as it does the Durand corr lation, is subject to the 0.8D as representing the homogeneous portion of the slurry is rather arbitrary. shortcomings as the Durand correlation. In addition the choice of the concentration at Wasp et al (25) themselves admit its success stems in part from In spite of its partly theoretical background the method is largely empirical and "fitting" of arbitrary constants to experimental data. None the less it is of importance because of its successful use in the design of commercial pipelines. where turbulence is the dominant form of suspension by assuming Shook and Daniel (2) have developed a theory for the flow of that the suspension behaves suspensions of fine particles. essentially as a variable density single phase fluid. the method of Julian and Dukler (28) in which it is hypothesized that the solids make their As the basis of their analysis they used influence felt primarily by modifying the local turbulence in the fluid. solids contribute to the increased pressure drop by increasing the eddy the pipe in the vertical direction rather than use the average value. fluid. Unlike Julian and Dukler however they allowed the solids con entration to vary across slurry where f is the friction factor for a Newtonian f and f the friction factor f is "flowing in the pipe at the same pressure gradient as the slurry. is then a measure of the degree to which the slurry departs (VE = 11/VD - (1-n'fluid of the same der sity and viscosity as the The difference between by its Bulk properties because of the variation of concentration acress the pipe. mentioned previouster needs to be determined independently and Shook and Daniel suggest from the behaviour predicted using the method of Ismail The value of allowed for it is not likely to be any better significantly superior to Using experimentally determined values of p the Durand equation. than the Durand method. However when the coror in predicting o is the method was shown to be interactions became important. Shook and Daniel (29) also investigated flows at high concentrations where particle-particle suggested that in such a situation particles may be concentrations above 15% by volume and found that when such a suspension supported by Bagnold stresses. Bagnold (30) performed experiments on coarse particles at hich was sheared there was a normal or dispersive stresspset up, the magnitude of which was proportional to the shearing Under conditions of coarse particles at high concentrations Bagnold stresses may be responsible for particle support. concentration q across the pipe-cross-section and T In such a situation P will be equal to the integral of the in-situ will be calculable from equation (27). alone will be impossible to calculate. To overcome this Shook and Daniel confined their in general the she stest essese to the fluid stresse due to Bagnold stresses investigation to flow in a channel with a stationary bed present and a steep they found it possible to estimate the above the bed and essentially clear liquid towards the top gradient above the bed such that there was a moving layer of high con entration immediately of the chanel. In such a situation J= EV12gh + (5 - 1)ấ k where h is the channel height, fluid shear stress To at the lop of the channel and where a is the average concentration above the stationar: led. A series of experiments was performed for sand, lead and nickel particles of various sies. It vas found that in all cases where turbulence was unlikely to be responsible for particle support (ie., for coarse, heavy particles) the values of K were within the range 0.4 to 0.75 which compare well with Bagnold's similar to equation (28) for flow with a sliding bed which has been found to apply equally well Newitt et al (23) using an entirely different line of reasoning proposed an equation to situations where there is a stationary bed present and to where there is no bed at all. Shook and Daniel suggest that this could indicate that their derivation based on a sliding bed is questionable and that this lends support to the Bagnold dispersive stress hypothesis.

w+ks & (s- 1) V (29)

where subscripts m and w refer to the mixture and the liquid respectively. from a dimensional analysis and reflects the increased friction due to the increase in the The first term arises alone. density and viscosity. a to differ from Any change in the turbulence structure due to the presence of the solids will cause is related to the co-efficient in the f-Re relation for limid flowing maintain the particles in suspension and is similar to that obtained by Nevitt et al (equation The second term results from a consideration of the work done to Vocadlo and Charles present a means" 24). Suspensions of coarse particles often exhibit Newtonian type belaviour in laminar shear flow and in such a case the term "m F'alculating this ratin or alternatively + could be is dependent only on the particle concentration.

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meusured in a viscometer. To compare the method with experimental results and with other prediction methods they put a enly predicted the pressure Badiew, t=1.80 and k, = 10 and found that the method consistwith more accuracy than the other methods. However all comparisons were for a 25 mm diameter pipe only. From equation (29) for two pipes of diameters

Pml

Dy and D2 12-K52Č, (S = 1) •(· (3) 2-t P m2 t-1 (30)

This equucion offers possibilities as a scale-up procedure in a manner analagous to that of If in addition it is assumed that m Bowen 's method described earlier. For a non-settling slurry the terms k depend only on the bulk delivered concentration.

C and that a concentration! = "m2 equation (30)"becomes" Identical to equation® (11) for slurries of equal

Discussion to one or more of the following interactions. (a) Fluid-wall interaction, (b) Fluid-particle The pressure drop experienced by a turbulent suspension flowing through a pipe will be due interaction, (c) Particle-particle interaction, (d) Particle-wall interaction. The relative importance of each of these depends on firstly the concentration of the particles and secondly on the size of the particle with respect to the scale of turbulence in the fluid. 1. Flow at low concentration. At low concentrations type (a) and (b) interactions will predon. late. If the particles are large compared with the scale of turbulence the particles will onl follow the larger eddies and the main effect of the turbulence on each particle will be to alter its flow resistance. Uhlher and Sinclair (32) and Clift and Gauvin (33) have shown that in most cases the drag coefficient of a sphere is increased in a turbulent field although in SOMe cases a decrease can occur. Many of the correlations presented for flow of settling suspensions involve a particle settling velocity and invariably no account is taken of the eifect of turbulence on this settling velocity. Furthermore with non-spherical particles the settling velocity in still water will indicate the drag for one particular orientation only whereas the orientation between a particle and the fluid in a turbulent field will be changing continuously. the particles are small compared with the smallest scale of turbulence they will tend to follow all turbulent fluctuations. At very low concentrations (less than 5%) pressure drops lower than that of water alone have been observed (See Zandi (34)) however in general the pressure drop is greater than that of water. The variable density model of Shook and Daniel described previously is applicable for fine particles at low concentrations. This model assumes that the particle-fluid interactions increase the turbulent fluctuations and thus the friction drop. As pointed out by Wiles et al (35) this method does not take into account the phenomenon of particle migration away from the wall and this effect may be significant. Elow at high concentrations. At high concentrations the effects due to particle-particle aid particle-wall interactions become increasingly important. With fine particles the suspension may exhibit non-Newtonian effects. With large particles collision between particles and the rolling of particles along the bottom of the pipe may help to form large intensive eddies which in turn help to support the particles. Kazanskij (36 and 37) has proposed this phenomenon following experiments in which he compared the intensity of the large scale turbulent fluctuations in water with and without coarse sand present. For concentr.cions of sand above 15% he found a large increase in intensity over that for pure water. No such increase was observed with fine sand. At very high concentrations particle-particle and particle-wall effects would be expected to become dominant, and the Bagnold dispersive stress discussed earlier may become increasingly important. Shook and Daniel (29) employ this reasoning in the correlation for flow with a stationary deposit which was discussed previously. In a later paper Shook et al (38) again investigated Bagnold stresses. This time they performed experiments on fully suspended flow in a channel with fine and coarse sand and nickel. They measured the concentration profiles and compared them with concentration profiles obtained using the theory of Ismail (26) assuming turbulent diffusion was responsible for particle support. For the fine particles they obtained good agreement but with the coarser particles at higher concentrations the agreement became progressively worse. This discrepancy could be qualitatively explained by the presence of Bagnold stresses.

CONCLUSIONS

For a non-settling slurry there are two basic approaches available both of which give acceptable results. One of these methods requires only bench scale tests although to confidently predict pressure drops in large scale pipes it would seem necessary to perform pilot scale tests.

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The can be used situation with settling slurries is still very unsatisfactory and there is no method which to confidently predict pressure drops. Until the fundamental processes involved can be identified and understood the situation is not likely to improve. In the interim may be possible to modify the method of VocadIo and Charles (31) to allow scale-up from pilot scale

REFERENCES

1. Cheng, D.C.H., Proc. Ist Int. Conf. on the Hydraulic Transport of Solids in Pipes, JS Organised by Brit. Hydromech. Res. Assoc. University of Warwick, England. (Sept Ist - 4th 1970). Dodge, D.W. and Metzner, A.B. A.I.Ch.E. Jnl, 5, p.189 (June, 1959). 4. Kemblowski, 2. and Kolodziejski, J., Int.. Chem. Engng. 13, p.265, (1973). Hanks, R.W. and Dadia, B.H. A.I.Ch. Eng. Jnl, I, p.554 (May 1971). 5. Kenchington, J. Proc. 3rd Int. the Hydraulic Transpört of Solids in Pipes, Paper Colo, U.S.A. (15th-17th May, Fl., Organised by Brit. Hydromech. 1974). Assoc. held at Colorado School of Mines, Golden, 7. 0. Krishna Murthy, V.R. and Zandi, I., Jnl of the Engng. Mech. Div. A.S.C.E. EMI p.271 (Feb '69 Segre, G. and Silberberg, A. Nature 189, p. 209 (1961). 8. 10. 9. Roberts Bli Ken en po (171): Ol royd, J.G. J. Bull. Brit. J. Colloid. Sci., 4, P.333 (1949). M. I.T. Hydrodynamics Report No. 103 (1967). 11. Olroyd, J.G. 12. Bowen, R.L. (Jnr), 1961, Series of Articles in Chem. Engng. June 12, P.243, June 26, p. 127, E., University of Strathclyde, Glasgow, April 1974. 13. 15. 14. July 10, p.147, July 24, p.143, August 7, P.129, August 21, p. 119. Metzner, A.B. and Reed, Kenchington, J. Proc. J.C. A.I.Ch.E. Jnl, 1, p.434 (1955). 10. Harris, J. Rheologica organised by Brit. Hydromech. Res. 2nd Int. Conf. Hydraulic Transport of Solids in Pipes. Paper C4, Acta, 7, p. 228 (1968). Assoc. Univ. of Warwick, England (Sept 20-22 1972). 17. 18. Zandi, I. and Govatos, G., A.S.C.E. Proc, Hyd. Div. 93, (HY3), p. 145 (1967). Durand, R. Proc. Minnesota Int. Hydraulics Convention, p.89. Int. Assoc. for Hyd. Res. (1953 19. Hayden, J.W. and Stelson T.E., Int. Symp. on solid-liquid flow in pipes, Uni. of Penn, Phil, U.S.A. (March 1968). 20. 1908). Babcock, H., Int. Symp. on solid-liquid flow in pipes, Univ. of Penn, Phil., U.S.A. (March 21. Rose, I.E. and Duckworth R.A. 1969, The Engineer, 227, (5903), p. 392; 227, (5904), p. 430; 221, (5905), p.478. 23. 22. Newitt, D.M. Richardson, J.F., Abbott, M. and Turtle, R.B. Trans. Turian, R.M., Yuan T.F., and lauri G., A.I.Ch.E. Jnl, 12, 4, p.809, (1971). p.93, (1955). Inst. Chem. Engrs, 33, 24. p. 20 (1963). Wasp, E.J. Regan, T.J., Withers, J., Cook, P.A.C. ard Clancey, J.T., Pipeline News, 35, 25.

Phil, U.S.A. (March

25. 27. Shook, C.A. and Daniel, Isnail, H.M., Trans. A.S.C.E., | 's.M. Can. 112, p.409 (1952). 28. 29. Julian, F.M. and Dukler, A.E. A.I.Ch.E. Jnl, 11, 5, p.853 (1965). Shook, C.A. and Daniel, S.M. Can. Jnl. Chem, Eng. 46, p.56 (1965). 30. Bagnold, R.A. Proc. Roy. Soc., London, 225, p.49 (1954). 31. Vocadlo, J.J. and Charles, M.E., Int. Conf. on the Hyd. Trans. of solids in pipes, Paper Cl, organised by Brit. Hydromech. Res. Assoc. Univ. of Warwick, England (Sept 20-22 1972). 33, Engrs, Clift, R., and Gauvin,.N.H. Uhlherr, Melbourne and Sydney, P.H.T. and Sinclair, C.G. Chemeca '70 Conf., Aust. Acad. of Sci., and Inst. Chem. Chemeca '70 Conf. Aust. Acad. of Sci., and Inst. Chem, Engrs., Session 1, (1970). Melbourne and Sydney, 34. 35. Wiles, Zandi, I., Int. Symp. R.J., Nicklin, on solid-liquid flow in pipes, Univ. of Penn., Phil, U.S.A. (March D.J. and Leung; L.S. Proc. Aust. Inst. Min. Met No.239, p.31 (Sept 36. Kazanskij, H., 2nd Int. Conf. on the Hyd. Trans. of solids in pipes, Paper A2, 37. Kazanskij, I, Bruhl, H., organised by Brit. Hydromech, Res. Assoc. Univ. Warwick, England (Sept 20-22 1972). & Hinsch, J., 3rd Int. Conf. on the Hyd. Trans. of solids in pipes, Goiden, Col., U.S.A. (15th-17 May 1974). Paper D2, organised by Brit. Hydromech. Res. Assoc, and held at Colorado School of Mines, (1968). Shook, C.A. Daniel, S.M. Scott, J.A. & Holgate, J.P. Can. Jnl. Chem. Engng, 46, P. 238