Consolidation of Non-Colloidal Spherical Particles at Low Particle Reynolds Numbers †

When a system of identical spheres settles under conditions of negligible surface and inertial forces an idealised form of sediment consolidation unfolds amenable to a universal description. We have described this complex process using a simple constitutive model expressed as an elementary scaling law in time, t, applied at the local particle level. The free-volume surrounding a particle consists of two volume contributions occupied by fluid, one portion fixed and the other portion variable, the latter of which declines with t–2. A comprehensive system of analytical equations was derived using this one idea, and associated boundary conditions, to describe all aspects of the batch settling process. An experimental system exhibiting negligible surface and inertial forces was used to validate the model and hence assess the merits of the scaling law. Excellent agreement was achieved. The precise physics responsible for this scaling law, and the applicable boundary conditions, remain unclear at this stage. Hence this work is likely to motivate further work in this area, concerned with the dynamics of random consolidation of settling spheres.


Introduction
Concha F. and Bürger R. (2002) have provided a useful historical review of batch settling, evolving from a mere "art" into one of the most studied problems in solid-liquid separation.Batch settling has been used to provide the data needed to design industrial thickeners (Fitch B., 1979), and to examine the effectiveness of settling aids such as flocculants.Coe H. and Clevenger G. (1916) are credited with providing the first insights into the flux limits of continuous thickening, basing their approach on multiple batch settling tests conducted at different initial concentrations.
A batch settling test is generally used to develop an understanding of the sedimentation of a given feed, and its expected response to the thickening process.This test is illustrated in Fig. 1 and the batch settling test curve is presented in Fig. 2. Given most industrial feeds are subjected to flocculation or coagulation, the ultrafine particles form larger entities called flocs.These can be considered to be effectively of the same size and density, and to therefore act as distinct particles during the sedimentation.Thus, in general, batch settling is often examined in terms of the settling of particles of one size and density.
Consider an initially uniform suspension of mono-sized particles of one density (Fig. 1).The subsequent process of sedimentation can be divided into two stages.The first, termed "free-settling", involves minimal inter-particle contact, hence the net particle gravitational force in the fluid (taking into account gravity and buoyancy) is balanced entirely by the fluid drag force.The initial suspension height is H i and volume fraction of the particles ϕ i .The bulk of the particles will start to settle with a constant settling velocity, forming a supernatant or clear water layer as the suspension interface moves downwards.Meanwhile particles start to form sediment at the base of the container, producing a settled bed that rises upwards (Fig. 1).Ultimately the rising bed meets the falling suspension interface at time t o and height H o .The process of bed consolidation continues, causing the surface of the settled bed to decline gradually until the bed reaches its final bed height, H f .Kynch G. (1952) developed the first complete description of sedimentation linking the free-settling and consolidation zones to achieve a unified theory.His fundamental assumption was that the settling velocity of the particles is governed by the local volume fraction of solids only.
His model, which was based on the continuity equation, provided a basis for calculating the concentration of the particles at the upper interface of the sediment, allowing a thickener to be sized for the first time on the basis of a single batch settling test.
The analysis of Kynch neglected the compressive state of the sediment.A given particle within the sediment rests, at least in part, on the particles below, which rest at least in part on those even further below, which in turn rest on the bottom of the container.Thus, a particle within the sediment experiences a state of compression, and is not entirely free to settle.Rather, its movement is constrained by the presence of those particles below, and the subsidence of those particles.Therefore, the perceived settling velocity of the particles in the sediment at a given volume fraction of solids cannot be the same as the settling velocity observed when a uniform suspension is prepared at the same volume fraction of solids.
Since then many more complex models of the process of sedimentation and consolidation have been proposed.Buscall R. and White L. (1987) and later Howells I. et al. (1990) assumed that a complex, irreversible, network structure evolved within the consolidation zone, exhibit-ing a compressive yield stress condition.Bürger R. and Concha F. (1998) developed a one-dimensional transient model of an operating thickener incorporating the effects of compression while Usher S. et al. (2006) focused on the effects of compression over long time-scales.Dimitrova T. et al. (2000) applied the ideas of Buscall R. and White L. (1987) to describe the process of creaming, during which the particles settle in the upwards direction to form the cream, leaving clear serum below.Again, inter-particle forces result in a complex network structure that yields under the influence of a compressive load.They presented the following governing equation for describing the process of sedimentation and consolidation.That is, where K(ϕ) is the hindered settling factor, ζ the friction coefficient of an isolated particle in an infinite medium, Δρ the particle-liquid density difference, g the acceleration due to gravity and v the volume of one particle.The dimensionless particle pressure developed by the network structure is usually assumed to be a function of the volume fraction only, with p o ~ 0.002.We have solved their equation, setting p o = 0.0003, the smallest possible value for a stable solution, with the aim being to minimize the critical yield stress.That solution is compared later with the solution obtained using our new theoretical model.

Outline of the problem
In this paper we examine an idealised system consisting of smooth, incompressible, mono-sized, spherical particles settling within a Newtonian fluid.Both inertial forces and surface forces are assumed to be negligible, while viscous forces dominate.All particles exhibit some scale of surface roughness that places limits on the magnitude of the viscous lubrication force, hence even for "smooth" spheres some level of inter-particle friction will also play a role (Galvin K. et al., 2001).The system undergoes a complex, well defined, process of gradual particle packing, driven towards a state of minimum potential energy.The movement of a given particle is invariably constrained by the extent of the cumulative movement of the particles below, while occasionally promoted by the ad hoc presentation of local vacancies.
This paper seeks to reduce this problem of sediment consolidation to the most elementary description that is possible, recognising the existence of two volume contributions to the space surrounding a given spherical particle, one portion fixed and the other variable.The variable portion declines with time according to a simple, constitutive, scaling law due to the consolidation.This idea was first presented in a conference forum by Forghani M. et al. (2013), and has been built upon here into a more complete theory, sufficient for experimental validation.Their simple scaling law, inferred from the experimental findings of Galvin K. (1996), is used here to derive a comprehensive system of equations to describe the fall of the upper interface with time, the consolidation velocity of the particles versus time at any fixed elevation, and the changing concentration profile of the sediment.This description should be universal, within the constraints of the assumptions, and in the absence of surface asperities.An experimental system, reasonably consistent with the above assumptions, was used to validate the model.Galvin K. (1996) undertook an experimental study investigating the sediment consolidation of a model particle system.The particles, known as Sephadex, swelled in water to form spheres with a mean diameter of 220 μm and low density of 1018 kg m -3 .The relatively large particle size and hence gravitational force minimized the significance of surface forces, while the very low particle density minimized the particle Reynolds number.According to Evans I. and Lipps A. (1990), these particles settle to form a sediment with a volume fraction of 0.64.Within the limits of the experimental conditions, the particles can be considered incompressible hence the final volume fraction of the particles is not governed per se by the compressive load that arises from the gravitational force minus that of the buoyancy force of the sediment bed.
Galvin measured the velocity of individual particles at a fixed elevation during the process of consolidation, and repeated the approach at other fixed elevations.Within the field of view the particles appeared to move, largely en masse, in a downwards direction, while sometimes individual particles would momentarily free themselves from others and settle before re-contacting other particles.At each elevation the velocity of the particles decreased gradually, until there was virtually no movement at that elevation.He found that the particle velocities, V s , at each level scaled with time, t, as V s ∝ t -3 .This scaling law forms the basis of a simple constitutive relationship used in this study.Integration suggests that changes in height due to consolidation should scale with t -2 .
Our work is motivated by the curious proposition that a simple constitutive scaling law may underpin much of what is observed in this idealised form of batch settling.Thus our work is not motivated by the need to develop yet another method for sizing thickeners.Rather the findings should motivate new lines of enquiry, in particular the dynamics of random particle packing and consolidation using discrete element modelling, which should help explain the physical underpinnings of the assumed scaling law.Further, the transition to systems influenced by inertial and surface forces, and to systems with different particle sizes, should then follow.

Consolidation model
This study is concerned with the batch settling of initially uniform suspensions of particles of the same size and density.The settling commences via a process of free settling at the initial concentration, ϕ i , resulting in an upper interface H which falls at constant velocity V si (Fig. 1).The gravitational force on the particles during this free-settling stage is balanced entirely by the buoyancy and drag force.When the particles deposit onto the base of the container, evolving a bed of particles, much of the particle gravitational force minus that of the buoyancy force, become supported by the reaction force from off the base of the container.The system is then in a state of compression.
It is assumed that when material deposits onto the surface of the rising sediment there are three distinct components.These components may be described as the solids portion, ΔS, the fixed free-volume portion corresponding to the retained fluid, ΔR, and the variable free-volume portion corresponding to the expressible fluid, ΔW, all taken as a volume per unit bed cross-sectional area.This distinction between these three components is made in the knowledge that ultimately the quantum of solids, ΔS, will be associated in the end with a quantum of fluid, ΔR, corresponding to the maximum volume fraction of solids, ϕ m = ΔS/(ΔS + ΔR) at the completion of the consolidation.Thus the remaining portion, ΔW, decreases with time due to the consolidation.Clearly, during the consolidation of the bed, the fluid associated with the solids below is expressed through the void spaces surrounding the particles and hence in physical terms the fluid does not have an intimate association with the particles.Thus the portion of the so-called retained fluid, ΔR, reflects a portion of the void space around the particles that will always retain some fluid.The remainder of the void space, referred to as the expressible portion, ΔW, gradually decreases with time.Galvin K. (1996) examined in detail the consolidation of a model sediment at a fixed elevation, showing that the observed consolidation velocity, V s , scaled with the time, t, as V s ∝ t -3 .Thus, we hypothesize that the volume of expressible fluid, ΔW, decreases with time t according to the simple scaling law, Equation 3 is effectively a guess, which is hereby used to derive a system of equations to describe the consolidation.
It is shown later by Equation 38 that the resulting system of equations is in fact consistent with the scaling law identified by Galvin K. (1996).
A key objective of this paper is to show how this simple scaling law can be used to derive other properties of the consolidating system, such as the particle velocity and volume fraction as a function of time and height, and to test these predictions against experimental data.

Variation of concentration with time
When the particles first join the bed surface, the initial sediment volume fraction of solids is, where ΔW o is the initial value of the variable fluid volume observed at the top of the sediment.In general, the volume fraction of solids at any other bed location k and time t is, where ΔW k is the value of the variable fluid volume at a time t = nΔt, directly associated with the particles that arrived at the surface of the rising sediment at time t k = kΔt.These time values are defined in Fig. 3. Introducing our fundamental constitutive scaling relationship, it follows that, The maximum volume fraction of the solids, ϕ m , is reached once all of the expressible fluid has left the portion of solids.Thus, We have used ϕ m ≈ 0.64, the accepted value for the non-interacting random packing of mono-sized spheres, as obtained by Shannon P. et al. (1964).It should be noted that for other systems the final volume fraction of the bed can be considerably lower due to specific attraction between the particles comprising the particle network (Franks G. et al., 1999).
A useful identity derived from ( 4) and ( 7) is, Substituting Eqn. 6 into Eqn.5 gives, As already noted, t k , is the time of arrival of the particles at the surface of the rising sediment, and t is the time since the commencement of the batch settling test.Again we refer the reader to Fig. 3. Rearranging Eqn. 9 while incorporating Eqn. 8 gives a relationship for the volume fraction of the particles as a function of the time spent in the sediment, defined by t k and t, Eqn. 10 is combined later with the integrated sum of the sediment height, relevant to the time values defined by t k and t, to produce a description of the concentration profile at time t.It is noted that the time, t k , is linked intimately with the particles that arrive at the sediment at that time.For instance, the time t k can be replaced with the time, t o , that the rising sediment meets the falling interface of the suspension.Eqn. 10 then provides a measure of the volume fraction of the particles, ϕ k , at the surface of the suspension as a function of time.

Variation of sediment height with time
Fig. 3 shows the deposition of particles onto the rising sediment.The sediment is discretised into the elements of material that are added to the sediment as a function of time, providing the framework for describing the accounting system.These elements arrive at a time t 1 = Δt, then t 2 = 2Δt, t 3 = 3Δt, and so on.Thus in general the kth layer of material joins the surface of the bed at a time, t k = kΔt.Each element carries the portions, ΔS solids, ΔR retained fluid, and ΔW o expressible fluid.Eqn.6 describes the variation in the expressible fluid with time.For example, at time interval n = 4 the expressible fluid in elements k = 1, 2, 3 and 4 are, Hence the height h k to the top of sediment layer k at a time t = nΔt (where n ≥ k) is, ( ) Incorporating the series summation, the height becomes, Noting that for large k the term k(k + 1)(2k + 1) → 2k 3 , Eqn. 13 becomes, ( ) Introducing the times t k = kΔt and t = nΔt, and incorporating Eqns.7 and 8 gives, Let H o be the total height of the sediment layer at the moment t o when the rising sediment meets the interface of the falling suspension.Invoking the boundary condition, t = t k = t o , gives, Thus the effective solids feed flux, G F , of the batch settling test is, At relatively high initial concentrations, ϕ i , there should be no discontinuity with the concentration just below the surface of the rising bed and hence ϕ o = ϕ i .In this case the feed flux is given by G F = (-V si + V po )ϕ i .Thus, in this case we can calculate V po and in turn locate the boundary of the rising sediment.This boundary condition is used later in the analysis of the batch settling tests.
Incorporating Eqn. 17 into Eqn.15 gives the height h k of particles that arrived at the sediment surface at time t k , at any given later time, t.That is, In batch settling, the last particles arrive at the sediment surface at the time, t k = t o .Hence the observed consolidation, defined in terms of the falling upper interface, proceeds according to, where H is the total height of the sediment bed at time  and that at time t = ∞ the final sediment height is, Substituting these conditions into Eqn.19 gives a simple dimensionless description for the height of the falling interface, applicable for t ≥ t o and H ≤ H o .That is, This equation is used formally to validate this model against a series of batch settling test results.

Description of the concentration profile
Sections 2.1 and 2.2 provided expressions for the concentration of the particles ϕ k and height of the particles h k as a function of time t, in terms of the time, t k , that these particles first reached the rising sediment.Combining Eqns. 10 and 18 gives, Simplifying Eqn.23 leads to, Incorporating Eqn. 10 again eliminates t k and gives the concentration profile of the sediment as a function of time.Here we drop the subscript k, noting that h = h(ϕ, t).

That is,
Eqn. 25 shows that the concentrations propagate from the origin as a fan (Lester D. et al., 2005) over the full range from ϕ o to ϕ m , reflecting the absence of a traditional network structure.Fig. 4a shows the concentration profile of the system as a function of time, obtained using the model of Dimitrova T. et al. (2000) with the parameter defining the strength of the particle pressure, p o = 0.0003, set at the lowest possible value for a stable numerical solution.The objective was to obtain the solution for a system with a very low critical yield stress.The profiles are reported for different dimensionless time values.It is evident that the bed concentration evolves towards the maximum possible volume fraction of particles, set equal to 0.64.Fig. 4b shows the solution given by Eqn. 25, and Eqn.22, obtained using the new theoretical model developed in this Section.Good agreement is evident.
Defining the propagation velocity of a wave of concentration ϕ as, it follows that the propagation velocity is a function of the solids concentration, ϕ, and is influenced strongly by the boundary conditions, V po , ϕ o , and ϕ m .

Consolidation velocity of bed particles
The consolidation velocity of the particles within the sediment, relative to the vessel, is governed by the reorganisation of the sediment, and hence the gradual release of expressible fluid from below.This particle velocity, defined as negative in the downwards direction, is obtained by differentiating the height, h k , in Eqn.18 with respect to time, t, with the value of t k held constant.Therefore, and Incorporating Eqn. 10 which describes the concentration as a function of time, and again dropping the subscript k and noting that V s = V s (ϕ), gives, While Eqn. 30 is a function of the local volume fraction of the particles, it is emphasized that this dependence is fundamentally different to that in the free settling zone, as it reflects the constrained nature of the consolidation.Indeed the consolidation velocity is always less than the free settling velocity.Eqns.27 and 30 conform to the requirements of continuity.That is, ( ) 2.5 Settling velocity at constant elevation Galvin K. (1996) examined the settling of the individual particles within the sediment at a fixed elevation.Particles passed downwards into and out of the field of view, and hence the data collected involved different particles.Firstly we set Then, combining Eqns.17, 29 and 32 gives, Returning to Eqn. 18, and incorporating Eqns.17, and 32 gives, Setting the height, h k , to be equal to the constant elevation, h c , provides the corresponding time, t.
A series of values of β k provide a common independent variable for calculating the consolidation velocity, V s , from Eqn. 33, and the value of the time, t, from Eqn. 35, revealing the scaling law, V s ∝ t -3 .Alternatively, Eqn.33 can be inserted into Eqn.35, giving Note that the consolidation velocity, V s , is numerically negative.During consolidation the value of (-V s ) 1/3 eventually dominates over the value of -V s , and hence the second term on the right hand-side of Eqn.36 can be neglected.It then follows that at fixed elevation, Eqn. 37 provides the asymptotic behaviour of the consolidation process within the sediment at fixed elevation, as observed by Galvin (1996).That is, Galvin K. (1996) obtained data on the settling velocity at various fixed elevations.The data sets collapsed to produce a single curve in accordance with Eqn.38.

Experimental method
Sephadex G-200 grade granules, made by cross-linking dextran with epichlorohydrin under alkaline conditions, were sourced from Pharmacia.The dry particles, with density of 1538 kg m -3 , were sieved to obtain the -75 +63 μm size fraction.These particles were added to a beaker of distilled water and allowed to swell.Evans I. and Lips A. (1990) studied a number of different grades of Sephadex particles and found that the settled bed packing fraction was always close to 0.64, as expected for random tight packing of spheres (Shannon P. et al., 1964).Galvin K. (1996), who used the same system of particles, reported the particles swelled over a 24 h period to a diameter of 220 μm, forming a settled bed of 30 cm 3 per 1 g of dry Sephadex.Assuming the settled bed volume fraction was 0.64 and that the volumes of Sephadex and water were additive, this implies that the swollen particles had a density of 1018 kg m -3 .
In the present work, the same batch of Sephadex particles was re-used multiple times over a prolonged period.Fig. 5 shows that whilst much of the swelling was complete within the first 24 h, there was clear evidence of additional swelling over the following months as the final settled bed height was observed to increase from around 2.0 cm/g (volume of 39 cm 3 /g) to over 2.4 cm/g (47 cm 3 /g).Again, assuming a constant final settled bed volume fraction of 0.64, this suggests the density of the swollen particles decreased from 1014 kg m -3 down to 1012 kg m -3 .Thus, for internal consistency, it was necessary to limit comparisons between experiments to insure that the degrees of swelling were similar.
Eventually, the observed diameter of the spheres was around 250 μm, with a density of about 1013 kg m -3 .Hence these particles settled in water according to Stokes' law (Stokes, 1901) with a very low particle Reynolds number of ~ 0.1.Although the final particles were in principle deformable, within the limits of the experimental conditions, the particles did not compress or deform.Evans I. and Lips A. (1990) formed the same conclusions while also noting that surface forces were also negligible.Thus this experimental system is a very close approximation to the ideal system of non-deformable, mono-sized spherical particles settling at low Reynolds numbers with negligible surface forces.
A set of batch settling tests was conducted, focused initially on the fall of the upper interface versus time.A given series of experiments involved the same mass of the Sephadex granules.In the first series the mass was 12 g.
Then this batch of material was halved to perform a series of tests using 6 g of Sephadex, and then halved again to test 3 g of Sephadex.Then all this material was recombined and an additional 12 g of Sephadex added in order to perform a series of tests using 24 g.Within each series, different initial suspension concentrations were obtained by adding or removing different quantities of distilled water.
At the start of an experiment the Sephadex spheres were suspended within a 1.25 m high Perspex cylinder of internal diameter 50 mm using a perforated plunger, and the initial height, H i , noted.A quiescent state was formed within seconds.A ruler located down the wall of the cylinder provided elevations accurate to within 1 mm.During the experiment the height of the upper interface, H, was recorded versus the time, t.The final height of the sediment, H f , was recorded after the material had been left to consolidate for at least 24 h.Based on the findings of Evans and Lips (1990), it was assumed the final volume fraction of the spheres was ϕ m = 0.64.Hence the initial volume fraction was calculated to be ϕ i = (H f /H i )ϕ m .Experiments were conducted using initial volume fractions in the range 0.28 to 0.55.
Following the initial set of experiments, approximately 12 months later additional experiments were conducted using the same batch of Sephadex in order to study the consolidation velocity of spherical particles at fixed elevations.A CCD video camera was positioned at a specific elevation during the experiment.The movement of different particles within a height range of about 1 mm was recorded over an extended time period.Experiments were repeated in order to observe the internal consolidation at different elevations.In addition, the height of the upper interface versus time was recorded during each of these experiments in order to properly characterise the settling process.
Finally, about 24 months after the first set of experiments, additional experiments were conducted in order to examine the evolution of the concentration profile versus time.Suspension concentration was measured using the light extinction method (Hinds W., 1982).The experimental system shown in Fig. 6 was located in a dark room, free of spurious light sources.A 5 mW 532 nm laser beam was directed horizontally into the suspension at a given elevation and the light exiting the opposite side recorded using a CCD camera.The laser and camera were then moved in unison to a new elevation in order to record the transmitted light at that point.A set of these measurements at different elevations was recorded within a relatively brief time period, allowing a single time value to be assigned to a given concentration profile data set.This process was repeated at multiple nominal time values during the batch settling.
Each image was later analysed to measure the average Extent of swelling of the Sephadex particles over 24 months, accounting for different bed volumes in experiments conducted at different times.Legend shows the mass of dry Sephadex in the system.Line is drawn to guide the eye.
intensity of the transmitted light.The signal obtained at the start of the batch settling test corresponded to the initial volume fraction, ϕ i , while the signal obtained after one week near the base of the sediment, corresponded to ϕ m .Tests showed that the transmission for intermediate concentrations could be obtained by linear interpolation between these two extremes.Thus it was possible to obtain the volume fraction value corresponding to a given signal.

Boundary condition between free-settling and consolidating bed
The initial suspension exists in a state of uniform concentration hence the particle gravitational force, less that of the buoyancy force is fully supported by the hydrodynamic drag force.Thus, the initial settling velocity of the particles is a well-defined function of the volume fraction of the particles, referred to as the hindered settling function.Fig. 7 shows a typical batch settling test curve ranging in height from the initial value, H i through to the final height, H f .The suspension settles at the initial velocity, V si , while the sediment rises at the velocity V po , producing an initial sediment height, H o , in a time period t o .
The analysis of the batch settling test commences with the identification of the boundary condition (t o , H o ), something that can be difficult to achieve objectively.The approach adopted in this study was to locate this point using the experimental data in conjunction with the consolidation model.The feed flux, G F , given by Eqn.17, can be equated with the magnitude of the initial settling flux, relative to the velocity of the rising sediment.That is, ( ) where the initial settling velocity, V si , has a negative value indicating its downwards direction.Rearranging and setting With reference to Fig. 7, it follows that It is tacitly assumed that the concentration at the surface of the rising sediment is equal to the initial concentration, ϕ o = ϕ i .This assumption appears to be valid when the initial volume fraction is sufficiently high, and inertial forces are negligible.Thus the condition appears to apply in this study.Eqn.40 then becomes, It is noted that this equation potentially approaches a singularity when ϕ i → ϕ m .This singularity can only be avoided if V si approaches 0 as ϕ i → ϕ m .Hence the Richardson-Zaki equation (given later by Eqn.45) cannot be used near ϕ i = 0.64 because it incorrectly predicts that V si still has finite values up to the limit ϕ i = 1.0.However, as the highest ϕ i value used in this work was 0.55, this limitation was not an issue here.
Substituting Eqn.41 into Eqn.42 to eliminate V si gives, and hence the value of t o is, The boundary condition defining the point at which the rising sediment meets the falling interface of the suspension can be obtained through trial and error.The first few data points can be used to estimate the initial settling velocity, V si , which in turn can be used to obtain t o via Eqn.44.Then the initial settling velocity can be based on a linear best-fit to all of the data covering the range of time values up to t o .The resultant value, V si , can then be used to update the value of t o , leading to confirmation of the boundary condition, (t o , H o ), and the value of V po .This objective approach insures the boundary condition is consistent with the consolidation model.

Free-settling velocity-The hindered settling function
In order to properly describe the velocity of propagation of the rising sediment, given by Eqn.42, we need to firstly describe the initial free-settling velocity, V si .The empirical hindered settling function of Richardson J. and Zaki W. (1954), ( ) provides an accurate description.The exponent, n, varies with the particle Reynolds number while the terminal velocity, V t , of an isolated spherical particle settling according to Stokes' law is: Here, d is the particle diameter, ρ s the particle density, ρ the fluid density, g the acceleration due to gravity, and μ the fluid viscosity.Fig. 8 shows batch settling test results for experiments conducted using 3 g of Sephadex particles, using different initial concentrations.Additional results are shown in Forghani M. (2015).Fig. 9 shows the initial settling velocities of the particles used in the batch settling tests versus the initial volume fractions of the particles.These data can be described very well using Eqn.45 with V t = -399 μm/s and n = 5.23.As mentioned above, Eqn.45 incorrectly predicts finite values of V si for ϕ > 0.64.However, within the range of the experimental conditions up to ϕ i = 0.55, the prediction is sufficiently accurate.
Fig. 10 shows the data for velocity of propagation, V po , versus the initial volume fraction of the particles, sourced directly from the batch settling tests.The solid curve passing through the data was formed by substituting the expression for V si (Eqn.45) into Eqn.42.The singularity as ϕ i approaches 0.64 is clearly evident.However, within the range of experimental conditions the data are reasonably consistent with the curve, given the inherent scatter.By comparison, Kynch theory predicts a propagation velocity: which combined with Eqn. 45 gives, ( ) The result obtained using Eqn.48 is also shown in Fig.  10.It is evident the actual velocities were higher than those predicted using Kynch theory, reflecting the constrained nature of the consolidation process compared to free-settling.

Consolidation
According to the theoretical model, the consolidation process within the bed is characterised by the normalised function given by Eqn.22.That is, where H f is the final sediment height after infinite time.It is noted that the pair of starting values required in Eqn.
22, H o and t o , were set equal to the values at the end of the free-settling period as described in Section 4.1.Fig. 11, which is based on the experimental data shown in Fig. 8, shows the full collapse of the data sets along the diagonal from the upper coordinate (1, 1) to the lower coordinate (0, 0).These data provide strong evidence to support the proposed scaling law in this study.Additional data sets are provided in Forghani M (2015) for different initial quantities of Sephadex.

Analysis of the internal consolidation
The analysis in the previous section was focused on the falling height of the upper interface versus time.While this work provided support for the consolidation model, the strength of the validation is limited given the relatively small changes in the measured height versus time, especially towards the end of the consolidation.Thus, in this section a much more thorough validation of the model is presented.Two approaches are taken.The first is based on the original approach of Galvin K. (1996).
In this work the vertical motion of the actual Sephadex particles during the consolidation was examined versus time at a fixed elevation.Initially the particles consolidated at a relatively high velocity.But over time, the consolidation velocity decreased by a significant factor.Eqns.33 and 35 describe the consolidation velocity versus time, at fixed elevation h c .This second phase of the study occurred some 12 months after the previous work using the same particles of Sephadex.It is noted that the particles underwent additional swelling during this period resulting in larger particles of marginally lower density and hence a final sediment height greater than obtained previously.The increase in the final sediment height was consistent with the increase in the observed size of the particles.Thus we are satisfied the final sediment volume fraction of the particles remained at the standard random packing value of 0.64.
Fig. 12 shows the batch settling test curve obtained using 24 g of Sephadex, with an initial volume fraction of 0.40.The smooth curve through the data was based on the theoretical consolidation model, with V si = 20.7 μm/s and V po = 59.5 μm/s.Fig. 13 shows log-log plots of the observed particle velocity within the system at fixed elevation.The velocities were obtained by observing the movement of specific particles present at that moment in the field of view.The quality of the images was not high, but was sufficient for identifying the boundaries of the particles, and hence sufficient for measuring the consolidation velocity.In general, movement occurred en masse and therefore the velocity of a given particle was representative of the rest.The data sets correspond to observations at elevations of 15, 30, and 45 cm above the base of the vessel, well below the final sediment height of approximately 58 cm.
The results in Fig. 13 are very significant.Effectively they describe the consolidation velocity as a function of both time and position.Moreover, it is evident that initially the velocity is constant, approximately 20 μm/s, reflecting the corresponding free-settling of the particles.The velocity then decreases to lower and lower values as the consolidation proceeds.The exact predictions of the theoretical consolidation model are given by the three solid curves.These curves show the formal scaling according to V s ∝ t -3 , consistent with the original finding of Galvin (1996).While the agreement between the model and the experimental data is weakest at 15 cm, the agreement appears to strengthen at the higher elevation of 30 cm, and become very strong at 45 cm.The improvement in the scaling with distance above the base is a reasonable expectation, given that the scaling should fail at the fixed base of the system.These results are remarkable, predicting the consolidation velocity to within 1 μm/s after a period of 20000 s at a significant elevation in the bed of 45 cm.
The knowledge of the consolidation velocity as a function of time and position provides arguably the most definitive description that is possible.Correct prediction of the consolidation velocities should to the correct prediction of the liquid expression and thus the volume fractions of the particles as a function of time and position.
Fig. 14 shows the concentration profiles of the consolidation zone versus time, measured using the experimental method outlined in Section 3. The theoretical predictions based on Eqn. 25 show excellent overall agreement, though there is some discrepancy again near the base.These findings provide comprehensive validation of the The experimental data is for 24 g of Sephadex at an initial concentration ϕ i = 0.40.Lines show theoretical predictions of Eqn. 25 which uses the V po value obtained from Fig. 12 and the assumption that ϕ o = ϕ i .model describing sediment consolidation of an ideal system consisting of mono-sized, non-deformable spheres at low Reynolds number and with negligible inter-particle forces.
In the above analysis we have assumed that there is full continuity between the free settling and consolidation zones.Thus the volume fraction of solids just below the surface of the rising sediment, ϕ o , is equated with the initial volume fraction, ϕ i , used in the batch settling.In turn the model requires that the particles at the surface of the sediment consolidate at the free settling velocity, even though the particles at the surface of the rising sediment have achieved formal contact with the structure below.It should noted, however, that the consolidation velocity of the particles within the sediment varies with the volume fraction according to the consolidation model, and not according to the free settling function.Thus, the agreement between the consolidation and free settling velocities only exists at the interface.
This paper requires further discussion concerning the boundary condition at the surface of the rising sediment.In most studies, the minimum sediment concentration is defined by the so-called gel concentration, an input into the model.The theory presented in this paper does not offer a general solution for determining the relevant volume fraction just below the surface of the sediment.In the context of the present problem there is scope for developing a Discrete Element Model (DEM) to improve our understanding of the boundary condition.We have recently established, using a Monte Carlo approach, that a network formed through a simple random deposition model, with fixed attachment between the particles, produces a bed with a volume fraction of approximately 0.194.Clearly a higher concentration is expected in the absence of particleparticle attachment.
Although Kynch theory has its limitations, the approach offers some insight here into the effects of an increase in the particle Reynolds number.According to drift flux theory (Wallis G., 1962) there is an inflection in the flux curve at the volume fraction, ϕ I = 2/(n + 1), where n is the exponent defined by the Richardson and Zaki equation.When the initial volume fraction of the particles, ϕ i , is higher than the inflection value, ϕ I , there is, in principle, no discontinuity in the concentration between the free settling particles and the surface of the rising sediment.This study has focused on such a concentration regime, ϕ i > ϕ I .The nature of the problem becomes a strong function of the particle Reynolds number, Re t , with n ≈ 2.4 applicable for large Re t and n = 5.2 applicable for the very low Re t used in this study.The corresponding concentrations at the flux curve inflexions are ϕ I = 2/ (1 + 2.4) = 0.59 for large Re t with n = 2.4, and ϕ I = 2/ (1 + 5.2) = 0.32 for small Re t with n = 5.2.Thus large spherical glass marbles settling in water at a high Re t should form densely packed sediment, with the volume fraction of the particles ranging from at least ϕ I = 0.59 just below the surface of the sediment.Thus, for large glass spheres, the sediment forms with a bed concentration ranging from at least 0.59 through to 0.64.Here there is little prospect for particle re-ordering and hence consolidation, as the high inertial forces drive the particles rapidly towards the lowest potential energy state, and a high volume fraction.For highly viscous systems, however, the volume fraction of the particles just below the surface of the sediment should range from at least 0.32.Thus viscous systems present the most interesting problem of sediment consolidation, with the potential for significant particle re-ordering towards the lowest potential energy state, slowed by significant viscous forces.
Further work is needed to account for (i) the boundary condition at the interface between the free settling and consolidating sediment bed and (ii) the scaling law that underpins the model developed in this study.Why does the volume of expressible fluid associated with the particles decay proportional to t -2 ?A discrete element model could be used to investigate both of these issues.

Conclusions
This work commenced with a simple idea, which in turn led to the development of a comprehensive theoretical model of the consolidation that occurs during batch settling.This model provided a description of the fall in the height of the upper interface with time, a description of the internal consolidation velocity as a function of time and position, and a description of the evolving concentration profile.An experimental study then followed, providing validation of the new model.
In this model a given portion of solids is associated with a void space that is occupied by a fixed portion of retained fluid and a further portion of expressible fluid.The space occupied by the expressible fluid decays according to a scaling law proportional to t -2 .As noted, further work is recommended based on a discrete element modelling approach to further understand the physical processes that result in this scaling law.

Fig. 1
Fig.1Diagram of the zones formed during batch settling illustrating some of the key variables.B = free settling zone of initial uniform suspension of concentration ϕ i and settling velocity V si .A = clear supernatant.S = sediment bed which propagates upwards at a velocity V po with a constant volume fraction of solids ϕ o at its upper surface until time t o , the critical settling point, when the rising bed meets the falling interface between the free-settling and supernatant.The maximum volume fraction of the solids is ϕ m .

Fig. 3
Fig. 3 Schematic of the proposed discretised model for the bed formation and consolidation process up to time t = 4Δt when four layers identified k = 1 to 4 have settled to form a consolidating bed.Each layer has a varying volume of expressible fluid ΔW (given by Eqn. 6) which results in each layer having a variable thickness Δh.

Fig. 4
Fig. 4 Comparison of predicted concentration profiles as a function of time from (a) the model of Dimitrova T. et al. (2000) with p o = 0.0003, and (b) the model developed in this work.The profile curves are given by dimensionless time values, based on common initial values of the suspension height, concentration, settling velocity, and maximum concentration.

Fig. 6
Fig. 6 Photograph of the experimental set up which consisted of a vertical 1.25 m high Perspex column with internal diameter of 50 mm.A 5 mW 532 nm laser was mounted on one side and a camera mounted on the opposite side on a frame with adjustable height.

Fig. 7
Fig. 7 Representation of the batch settling test showing the initial constant settling rate followed by the consolidation phase.The rise velocity of the sediment is V po = H o /t o .

Fig. 8
Fig. 8 Batch settling test results obtained using 3 g of Sephadex, with different initial concentrations.The curves through the data are based on the model developed in this paper.

Fig. 9
Fig. 9 Initial free-settling velocity of the particles versus the initial volume fraction of the particles.The curve denotes the best fit of Eqn. 45 to the experimental data with V t = -399 μm/s and n = 5.23.

Fig. 10
Fig. 10 Propagation velocity of the rising sediment versus the suspension initial volume fraction.The solid curve is based on the theoretical consolidation model (Eqn.42) and the dashed curve is based on the theory of Kynch (Eqn.48).

Fig. 11
Fig. 11 Normalized plots of the batch settling data shown in Fig. 8 based on 3 g of Sephadex and different initial volume fractions.The linearity between (0,0) and (1,1) provides strong support for the consolidation model, and hence the underlying scaling law.

Fig. 14
Fig. 14 Concentration profiles at different times, showing the height versus the volume fraction of the particles.The experimental data is for 24 g of Sephadex at an initial concentration ϕ i = 0.40.Lines show theoretical predictions of Eqn. 25 which uses the V po value obtained from Fig.12and the assumption that ϕ o = ϕ i .

Fig. 12
Fig. 12 Batch settling test with 24 g of Sephadex used to validate the model for describing the internal consolidation within the sediment.

Fig. 13
Fig. 13 Consolidation velocity versus time based on observations at elevations of h c = 15 cm, 30 cm, and 45 cm for experiments with 24 g of Sephadex and ϕ i = 0.40.Line shows prediction of Eqns.33 and 35.
gravity (m s -2 ) H height of suspension upper interface (m) H i initial suspension height (m) H f final height of sediment (m) H o height where upper interface of settling suspension meets rising sediment bed (m) h c a constant elevation within the bed (m) h k height of layer k within the sediment bed (m) propagation velocity of initial sediment surface concentration (m s -1 ) V p ϕ upwards propagation velocity of layer of concentration ϕ (m s -1 ) V s consolidation velocity of particles (m s -1 ) V si initial settling velocity of particles (m s -1 ) V t terminal free settling velocity (m s -1 ) ΔW volume of expressible fluid (m 3 m -2 ) ΔW o initial volume of expressible fluid (m 3 m -2 ) β k defined by Eqn.32 as (t k /t) 3 solids at upper surface of rising sediment bed for t < t o (-) μ fluid viscosity (Pa s) ρ fluid density (kg m -3 ) ρ s solid density (kg m -3 )