Liquid-solid flow

DOI: 10.1615/AtoZ.l.liquid-solid_flow

Liquid-solid flow represents the flow of a liquid continuum carrying dispersed solid panicles suspended and conveyed by the drag and pressure forces of the liquid acting on the particles. The aim of those slurry flows may be the transport of bulk-solids or physical or chemical processes between carrier liquid and solids. In reality such a flow comprises two very different flows: the total mixture flow characterized by the Pipe Reynolds Number Re and the relative flow between the solid particles and the carrier fluid characterized by the Particle Reynolds Number Res. The complete range of velocities is not possible with slurries as it is with pure liquids. The two superimposed phases influence each other and should be harmonized to flow without depositions or blockages.

Flow Behaviour, Flow Pattern and Flow Regimes

How the solid-particles behave in the mixture—whether they distribute evenly, and move suspended in the carrier-flow or segregate and deposit, depends as well on the solid properties (grain size, shape, density), on the properties of the carrier liquid (density, viscosity), on the operation parameters of pipe flow (velocity, pipe diameter, solid concentration) and on flow direction. Under some conditions, solid particles can change the rheologic behaviour of the slurry from Newtonian to non-Newtonian.

Rheologic Classification

With larger solid particles, the fluid and solid phases mostly retain their own identities because of the working inertial forces. Thus the increase of viscosity is relatively small, and the slurry flow behaviour then remains for any concentration Newtonian, like that of the Newtonian carrier fluid. Fine-grained slurries behave likewise while the solids concentration remains <25% because the distances between the suspended particles are still large enough to avoid intermolecular cohesive forces. By increasing the concentration, non-Newtonian flow behaviour will occur.

Flow Direction

A relatively low tendency towards segregation exists for vertical flow as a result of the symmetrical configuration of forces. Even for coarse material, a fairly uniform solid distribution can be expected in the pipe as long as the condition for conveyance is well-satisfied.

For horizontal conveyance gravity causes asymmetrical configuration of forces and segregation is always present, even when the conveyance condition for horizontal flow is well-satisfied. Horizontal slurry flows therefore show a solids concentration profile depending on the velocity and are called settling slurries.

Flow Regimes of Newtonian Slurries

Only very fine particles with Res <10−6, which can be conveyed by Brownian Molecular Movement, are kept in suspension without any Turbulence, the so-called colloidal dispersions.

Fine particles with 10−6 < Res < 0.1 can be easily held in suspension by hydraulic forces, and this tendency is supported by a low solid density and by a nonspherical particle shape. Only little turbulence is needed to keep those particles homogeneously suspended; so liquid velocity can be low in this homogeneous flow regime.

Particles with 0.1 < Res < 2 need some more turbulence and velocity to be held in suspension, but in the case of horizontal flow, completely uniform solid distribution cannot be reached. A certain degree of segregation is permitted. This type of suspension can exist at economically-feasible velocities and is called the pseudo-homogeneous flow regime.

To guarantee equal conditions for solids of any density relative to the concentration profile at these Res limits, the ratio of the settling velocity and the fluid velocity wso/uL must remain constant. Fluid velocities required for this condition can be obtained from Table 1.

Table 1. Limits for pseudohomogenous flow regime according to Weber (1978)

For coarser particles with Res > 2, the segregation is greater and the mixture flow is more heterogeneous. In this heterogeneous flow regime, lower velocities can lead to conveyance by saltation, and finally to the so-called critical deposition velocity, where the solid particles begin to settle out. At this point the pressure drop of the mixture is minimum.

The rheologic behaviour and the flow regimes are listed in Table 2.

Table 2. Classification of hydraulic flow regimes relying on Durand (1953)

A more detailed impression of the complex relationship between particle size, friction velocity, pipe diameter and the flow pattern of the solid particles in a Newtonian liquid-solid flow is given by the generalized phase diagram of Thomas (1962) in Figure 1. (ds is particle size and δ boundary layer thickness, Uo friction velocity and ν kinematic viscosity.)

Generalized phase diagram for horizontal Newtonian suspension transport.
Generalized phase diagram for horizontal Newtonian suspension transport.

Figure 1. Generalized phase diagram for horizontal Newtonian suspension transport.

Flow Regimes of Non-Newtonian Slurries

Highly concentrated non-Newtonian slurries, so called nonsettling slurries are homogeneous, and do not need turbulence to prevent settling out while conveyed through a relatively short pipe because the settling process is very slow. However, for long distance transport, turbulent flow should be recommended. Here the transition velocity from laminar to turbulent flow is referred to as the critical transition velocity.

Basic Relationships

Basically a heterogeneous liquid-solid flow can also be treated as a homogeneous mixture flow, like a pseudo-liquid that obeys the usual equations of single-phase flows. Then suitable weighted average pseudo properties can be formed, based on the properties and conditions of the two single-phases. Those typical virtual properties are, for instance, mean velocity, mean density, mean consistency of the mixture and so on. To proceed with pressure drop calculation in this way indeed depends on grain size and flow direction.

Vertical upward liquid-solid flow

Solid particles can be conveyed upward when the transport condition is well-satisfied; that means when fluid velocity exceeds the terminal settling velocity of the solids:


where ws is the hindered settling velocity at higher concentration and/or at nonspherical particle shape; wso is the single particle settling velocity; Res is the particle related Reynolds number; cν = VS/(VS = VL) is the local solid concentration by volume; ds50 is the grain size at 50% passing sieve; ρs is solid density; ρf is liquid density; and νL is kinematic liquid viscosity.

Two hindering effects must be taken into account: the solids concentration and the particle shape. Figure 2 gives the influence of the concentration according to Maude and Whitmore (1958) and Figure 3, the shape which is described by the sphericity ψ, being the surface of a volume equivalent sphere related to the real surface of the particle.

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

Figure 2. Influence of the concentration on settling velocity according to Maude and Whitmore (1958) replotted from Weber (1974).

Influence of the sphericity Ψ on settling velocity replotted from Weber (1974).

Figure 3. Influence of the sphericity Ψ on settling velocity replotted from Weber (1974).

Horizontal liquid-solid flow

Solid particles can be conveyed horizontally without deposition when the transport condition for horizontal flow is satisfied, which is true when fluid velocity exceeds the critical deposition velocity:


The critical deposition velocity can be taken from experimental results according to Wasp (1977), correlating to the following equation:


where D is pipe diameter.

Pressure Drop Calculation

The mentioned pseudo-liquid method can easily be applied to vertical and horizontal homogeneous and pseudo-homogeneous mixture flows, because there is only little slip between uL and us, so that um ≈ uL ≈ us and cT ≈ cv. The pressure drop should be calculated in this case by the following equation:


where λL = to one-quarter of the Fanning Friction Factor of the pure liquid; ρm = cνρs + (1 − cνL is the mean mixture density; is the mean mixture velocity; and is the delivered or transport concentration by volume.

In the heterogeneous vertical flow regime, the solids are also homogeneously distributed but a considerable slip can exist. Therefore, the local concentration cv has to be calculated for each velocity with respect to the delivered concentration and to the given solid mass flow rate before Equation 8 can be applied.


The pressure drop of horizontal liquid-solid flow can be calculated for any flow regime according to Weber (1986), by applying the generalized Durand equation:


where f is the fine solids (Res < 2); ρLs = ρs fcT + (1 − fcTL is the density of the carrier fluid enriched by “fines”; s = ρsLs is the specific solid density; m = 2 − (ds90 ds10)−.04 is a correction factor for grain size distribution; ds90 is grain size at 90% passing sieve; and ds10 is grain size at 10% passing sieve.

It should be stressed that when applying the fine part of Equation 10 the quantities ρLs, and s are directly influenced. The quantities ds90, ds50, ds10 are influenced by separating the fine solids part and adding it to the carrier fluid, as shown in Figure 4. In consequence, the correction factor m is changing because of ρLs and ds50 and also because of CD.

Effect of fine panicles added to the carrier fluid on the remainder effective particle size.

Figure 4. Effect of fine panicles added to the carrier fluid on the remainder effective particle size.

For horizontal heterogeneous liquid-solid flow, the local solid concentration by volume related to the delivered concentration can be calculated by the following equation:


In Figure 5 calculated results for an heterogeneous Newtonian slurry flow are compared with experimental results.

Heterogeneous Newtonian iron ore slurry flow with “fines” calculated by Equation 10. Source: Experimental data according to Report III (1973).

Figure 5. Heterogeneous Newtonian iron ore slurry flow with “fines” calculated by Equation 10. Source: Experimental data according to Report III (1973).

Relationships for non-Newtonian slurries

Since a multitude of non-Newtonian or nonsettling slurries are of the Bingham type, the rheological law of Binghamian slurries is relevant here:


where τ is the shear stress of the slurry; τo is the yield stress of the slurry; μB is the Bingham viscosity; and dv/dy is the velocity gradient.

Laminar Binghamian slurry flow

For short pipe lengths, very economical low velocities within the subcritical laminar flow regime can be chosen. The critical transition velocity for binghamian slurries can be found using the following equation according to Durand and Condolios (1952):


where is the Hedström number; Recrit = umcritmB is the critical transition Reynolds number; and umcrit = RecritμB/(Dρm) is the critical transition velocity.

To ensure the stability of the laminar slurry flow, no particles should be greater than


Dedegil (1986) refers to this diameter, which is critical between settling and suspending. Otherwise, overcritical turbulent flow should be realized to avoid settling out of solids.

The pressure gradient of laminar Binghamian slurry may be calculated by the Buckingham equation, but as this can not be done explicitely, an approximate solution of it which neglects the four-power terms may be used instead, according to Wasp, (1977):


Turbulent Binghamian slurries

Since in the case of turbulent Binghamian slurries the solid particles are pushed by turbulence eddies, in this turbulent flow regime the yield stress loses its effect. Therefore, the relationships of Newtonian slurries can be used, but the Reynolds numbers have to incorporate Binghamian viscosity. The friction factor for instance is λL = φ(ReB) and the drag coefficient is cD = f(ResB). This means the usual drag curve and the usual friction curve can be used. Only the Reynolds numbers, changed by the higher non-Newtonian consistencies, have to be applied.


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Maude, A. D. and Whitmore, R. L. (1958) A generalized theory of sedimentation. British Journal of Appl. Physics. 9. Dec.

Thomas, D. G. (1962) Transport characteristics of suspensions: Part VI. Minimum transport velocity for large particle size suspensions in round horizontal pipes. AlChE Journal. 8. 3.

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