Particle settling through a diffusive-type thermohaline staircase in the ocean

Deep-Sea

Pergamon PII: s0!%7-0637(!%)00119-7

I, Vol. 44, No. 3, pp. 399-412.1997 0 1997 Ekvier Scimce Ltd All rightsreserved. F’rintiin Great Britain 0967X137/97 $17.00+0.00

Research

Particle settling through a diffusive-type themohabe ocean ROSS C. KERR*

and GABRIELLE

staircase in the

S. KUIPER*

(Received 24 January 1996; in revisedform 29 July 1996; accepted 6 October 1996)

Abstract-Sedimentation through a “diffusive”-type thermohaline staircase in the ocean is investigated experimentally and theoretically. The experiments span a wide range in the density ratio that controls the stability of the interface between the convecting fluid layers (1.1 c R, < 2.5). In each experiment, it is found that the number of particles in a convecting layer decreases exponentially with time, with a decay rate equal to the settling velocity divided by the depth of the layer. This result is identical to that seen for sedimentation onto a rigid boundary, and is due to the small magnitude of the vertical component of the MIS convective velocity in the vicinity of the interface. A theoretical analysis is also used to quantify the rate of sedimentation through a staircase with several convecting layers. As the number of layers in the staircase is increased, the rate of sedimentation is found to steadily approach the rate of sedimentation in a non-convecting fluid. 0 1997 Elsevier Science Ltd

INTRODUCTION Particle settling occurs in a very diverse range of turbulently convecting systems. Important earth science applications include the deposition of sediments and algae in oceans and lakes (Smith, 1982; Reynolds, 1984; Hutchinson and Webster, 1994; Denman and Gargett, 1995), the deposition of crystals in crustal magma chambers (Martin and Nokes, 1988, 1989), and the chemical differentiation of the early Earth (Solomatov et al., 1993). Sedimentation onto a rigid boundary from a homogeneous, turbulently convecting fluid was the subject of a detailed theoretical and experimental study by Martin and Nokes (1988, 1989). They showed that the number of particles in suspension decreased exponentially with time, with a decay rate equal to the settling velocity divided by the depth of the layer. This result can be used either to predict algal accumulation rates in lakes (Smith, 1982; Reynolds, 1984), or to measure algal settling (or flotation) velocities in the laboratory (Hutchinson and Webster, 1994). While the former studies are adequate for understanding sedimentation in small water bodies that are often homogeneous, some uncertainty remains in predicting the behaviour of particles in large stratified water bodies such as the ocean. In parts of the ocean, this stratification consists of smooth gradients in temperature and salinity. However, in other regions of the ocean, these gradients are unstable, and double-diffusive staircases are formed (Turner, 1979; Schmitt, 1994). The staircases consist of well-mixed, turbulently convecting layers that are separated by sharp “diffusive” or “salt-finger” interfaces. The aim of the current paper is to explore sedimentation in such multi-layered convecting systems.

* Research School of Earth Sciences, The Australian National University, Canberra, ACT 0200, Australia. 399

400

R. C. Kerr and G. S. Kuiper

THEORETICAL

BACKGROUND

Thermal convection in a fluid layer is controlled by two dimensionless parameters, the Rayleigh number Ra

=-

guATh3 KV

(1)

’

and the Prandtl number

where g is the gravitational acceleration, h is the depth of the fluid layer, AT is the imposed temperature difference across the fluid layer, and ~1,K and v are respectively, the thermal expansion coefficient, thermal diffusivity and kinematic viscosity of the fluid (Turner, 1979). If the Rayleigh number is sufficiently large (> 105), the convective flow is fully turbulent. Sedimentation in a turbulently convecting fluid layer has been investigated theoretically and experimentally by Martin and Nokes (1988, 1989). They focused on the case where the rms convective velocity W is much larger than the settling velocity v, of the particles. In this limit, the particle distribution is uniform in the interior of the flow. However, near the rigid boundary at the base of fluid, convective velocities are negligible and the particles are able to settle with velocity v,. If the number of particles in suspension is N, the flux of particles onto the boundary is therefore given by Nv,/h, which leads to N decreasing exponentially with time t: N(t) = NO exp(-v,t/h).

(3)

This theoretical result, which had been predicted by Smith (1982), was confirmed by the laboratory experiments of Martin and Nokes, provided that three conditions are satisfied. First, the convective velocity must be sufficiently large in comparison to the settling velocity (S= v,/ W-cOS), so that the particles are uniformly distributed in the convective flow. Second, the particle concentration must be sufficiently small that the particles influence neither each other nor the flow (cf. Kerr, 1991; Kerr and Lister, 1992). Third, the particles must not be re-entrained into the fluid from the boundary (cf. Solomatov et al., 1993).

THEORY

FOR A DIFFUSIVE

INTERFACE

Consider now sedimentation in the simple two-layer system shown schematically in Fig. 1. The system consists of a warm salty fluid that is overlain by a fluid that is colder, less salty, and particle-laden. A sharp “diffusive interface” separates the two turbulently convecting fluid layers (Turner, 1979). Convection in both layers is driven by a temperature difference AT = (Tl - T&2.

(4)

The behaviour of the interface is controlled by the ratio of the stabilizing compositional density difference to the destabilizing thermal density difference R

_

P-

S(G - Gl) WI - Tu) ’

(5)

Particle settling through a diffusive-type thermohaline staircase

401

I I I I hu

doublediiusive

??

v

intefface

C"

cn

h,

Fig. 1. Schematic diagram illustrating the simple case investigated of sedimentation in a multilayered convecting system. Two thermally convecting fluid layers are separated by a sharp doublediffusive interface. The lower layer is a fluid of temperature T,, salinity Cl, density pt, viscosity p, and depth hl. The upper layer is a fluid of temperature r, (less than q), salinity C,, (less than Cl), density pU(less than pt), a viscosity uUand depth h,. The upper layer initially contains a dilute suspension of dense particles.

where p is the compositional “coefficient of expansion” of the fluid. In particular, the heat flux q through the diffusive interface is accurately estimated by “‘4PZ

3.8 P

(see Turner, 1979, p. 274), where qp is the heat flux for solid plane boundaries

and cp is the specific heat of the fluid. The rrns convective velocities in the layers are then related to q by W = 0.7 gz (

Ii3 >

(8)

(Martin and Nokes, 1989). As eqn (6) indicates, a significant feature of the diffusive interface is that the heat flux through it increases as R, is decreased towards 1. Turner suggested that “this increase can be associated with two effects: the interface behaves more like a free surface, so that there is a weaker constraint on the horizontal motion, and it also supports waves which can break, so increasing the effective surface area.” The motivation of the current experiments is to determine whether these effects also influence the rate of particle settling in this multilayered convecting system.

402

R. C. Kerr and G. S. Kuiper

EXPERIMENTAL

EQUIPMENT

AND TECHNIQUES

The experimental particles were spherical polystyrene beads, which were sieved so that their diameters lay mostly in the range 0.25-0.3 1 mm. The beads were then introduced into a linear density gradient, from which were extracted those particles with a density of 1.043 +O.OOl g/cm3. Measurement under a microscope of the diameter of 72 randomly chosen beads gave a mean of 0.285 mm, a standard deviation of 0.028 mm, and a standard deviation of the mean of 0.003 mm. The settling velocity of a bead with the mean diameter was then found using the empirical equations of Morsi and Alexander (1972). The fluids were aqueous NaCl solutions, whose physical properties were obtained from tabulations in Washburn (1926) and Weast (1989). The upper layer fluid was about 2.5 wt.% NaCl and initially at about 20°C while the lower layer fluid was about 3.5 wt.% NaCl and initially at about 40°C. To assist wetting of the beads by the solutions, a few drops of the photographic surfactant Photo-F10 600 were used. Photo-F10 600 (Kodak Pty. Ltd.) is primarily composed of 70% water, 25% ethylene glycol and 5% ptertiary-octylphenoxy polyethyl alcohol. The apparatus used for the experiments was an acrylic tank of length 30 cm, width 15 cm and height 65 cm. The tank was designed with a movable metal barrier at a height of 30 cm, which could be slid horizontally to divide the tank into two regions. The desired two-layer configuration was created as follows. First, the tank was filled to the height of the barrier with the lower fluid. The barrier was then inserted and the particle-laden upper fluid was added. The barrier was then carefully removed, which ensured that there was minimal mixing of the two fluids. Throughout the experiments, thermistors placed in each layer were used to monitor their evolving temperatures. Fluid samples taken during the experiments showed that the change in salinity of the two layers was not significant. The number of particles in the upper layer as a function of time was determined by a photographic technique. The upper half of the tank was illuminated by two electronic flash guns shining through l-cm vertical slits in the 15-cm sides of the tank, and short exposure photographs were taken through one of the 30-cm walls. The back of the tank was covered with black paper (a “Wet or Dry” coated abrasive) and the room was darkened to produce sharp photographs. The photographic negatives were then converted using a Nikon slide scanner into digital images on a computer. The number of beads was then counted using the computer package “NIH Image”.

EXPERIMENTAL

RESULTS

The main experimental parameters and results are summarized in Table 1. The depth of the fluid layers and temperature difference between them were sufficient to ensure a large Rayleigh number and fully-developed turbulent convection. In addition, the ratio of the particle settling velocity to the rms convective velocity was small enough (ScO.5) for the particles to be uniformly distributed in the convecting layers. In the 15-min duration of each experiment, the heat flux through the interface reduced the temperature difference between the layers from about 20°C to about 10°C. As a consequence, the density ratio R, doubled during this time (Fig. 2), which increased greatly the stability of the interface. In Fig. 3a-c, the number of suspended particles in the upper layer Ni is shown as a function of time. Also shown in the figures are error bars that correspond to a 95.45%

Particle settling through a diffusive-type thermohaline staircase Table 1.

403

Parameters for the three experiments, which characterize the

vigour of the upper layer convection, the nature of the double-dijjiive interface, and the rate ofparticle settling

Parameter

hu T-T, VS W Ra Pr s R, h

Expt 1

Expt 2

31.8 29.5 17.1-10.1 16.2-8.9 0.143 0.152 0.71Xk40 0.81-0.44 (4.8-3.1) x 10’ (6.2-4.0) x lo9 6.1-5.7 5.5-5.3 0.20-0.36 0.19-0.35 1J-2.2 1.1-2.0 0.88 0.92

1.0 * 0

200

400

600

Expt 3

Units

30.8 18.6-10.0 0.159 0.7SO.39 (6.8-4.0) x 109 5.6-5.4 0.20-0.40 1.3-2.5 0.90

cm “C cm/s cm/s

800

1000

t (a) Fig. 2.

The density ratio R, as a function of time t during experiment 3.

confidence interval (i.e. an error in Ni of two standard deviations: Nt f 2&Q. In each figure, Ni is found to decrease exponentially with time, despite the large change in the stability of the interface during each experiment. The exponential decay times in the experiments can also be used to obtain a dimensionless decay rate

where td is the measured exponential decay time. The value of h was about 0.90 (Table l), which indicates that the particles in the experiments settled at a rate about 10% slower than eqn (3) suggests. We note however that the experiments of Martin and Nokes (1989) yielded a similar mean value for h of 0.89. In both cases, these results are probably due to the range of particle sizes present in the experiments, which causes settling over several dimensionless time units to be biased towards the slower-settling particles, which are preferentially remaining in suspension (see Martin and Nokes, 1989, figure 9).

404

R. C. Kerr and G. S. Kuiper

(bl

__

0

200

400

600

800

1000

t (4 Fig. 3. The fraction of particles suspended in the upper layer, N1(t)/No, as a function of time t. (a) In experiment 1, the fraction decayed exponentially with a decay time rd=233 s. (h) In experiment 2, the fraction decayed exponentially with a decay time td=226 s. (c) In experiment 3, the fraction decayed exponentially with a decay time cd= 216 s.

THEORETICAL

ANALYSIS OF SEDIMENTATION CONVECTING LAYERS

FROM MANY

The experiments described in the last section have demonstrated that the number of suspended particles in a convecting double-diffusive layer decreases exponentially with time. This result is now used to quantify the rate at which sedimentation occurs from a series of n convecting layers each of thickness h. For simplicity, the density difference between the particles and the fluid is assumed to be much greater than differences within the fluid, which implies that the particle settling velocity v, is independent of depth. In addition, all the (No) particles are assumed to be initially in the uppermost layer, which is designated as layer 1. (A second case of an initially uniform particle distribution is treated in the Appendix.) The number of particles in this layer as a function of time I is hence given by N,(r) = NO exp(-v,t/h).

(10)

In the underlying layer (layer 2), the particle fluxes into and out of this layer imply that

405

Particle settling through a diffusive-type themohaline staircase

$=(N, -N*)$

647)

Since Nz(0) = 0, eqn (11) has the solution N*(t) = NO;

exp(-v,t/h).

(12)

-N,$,

(13)

In layer 3, the differential equation is

z=

(N*

and its solution for Ns(0) = 0 is N3(t) =

NoQ$f exp(-v,t/h).

(14)

Finally, in layer n, the number of particles N,, must satisfy %=(N+,

-N.);

(15)

and N,(O) = 0, and is given by Nn(t) = No’:‘“‘;,r’

n

.

exp(-vst/h).

(16)

For a series of n layers, the total number of particles Nl(t) in suspension is therefore NT(t) = N1 + Nz + N3 + . . . . . . .Nnr

(17)

and is given by + ... + (vst’h)“-l (n - l)!

exp(-v st/h) .

Using the above equations, a comparison can be made between sedimentation two-layered system, where

NT(~)= No( 1 +

F?exp(-v,t/h),

(18) from a

(19)

and sedimentation from a single layer of thickness 2h, where N(t) = NO exp(-v,t/2h)

(20)

(Fig. 4). It is seen that sedimentation is initially slower from the two-layer system, where the initial absence of particles in the lower layer ensures that dN,/dt = 0 at t = 0. However, after r = i, sedimentation is faster from the two-layer system, and it is this system that is the first to deposit 90% of its particles. In Fig. 5, the dimensionless time (r = tv,/nh) to sediment 90% and 99% of the particles from n convecting layers is shown. As the number of layers increases, it is found that the dimensionless deposition times decrease steadily from the onelayer values towards the value (i.e. z = 1) for complete sedimentation from a non-convecting fluid.

406

R. C. Kerr and G. S. Kuiper

Fig. 4. The fraction of particles suspended (NT/No) as a function of dimensionless time (r = rvJ2h), for a two-layer convecting system (solid line) and for the equivalent single convecting layer of thickness 2h (dashed line). Initially, all the Nc particles are in the upper layer.

Fig. 5. The dimensionless time (T= tvdnh) for particles to settle through n convecting layers, for the case where all the A’,,particles are initially in the upper layer. The figure shows both the time required for 90% of the particles to settle (m), and the time for 99% of the particles to settle (a).

DISCUSSION

AND OCEANOGRAPHIC

APPLICATION

In this paper a simple experimental study has been made that quantifies particle settling in a “diffusive’‘-type thermohaline system. In the experiments, the values of both the Rayleigh number and the density ratio are comparable to the values (Ra=(0.1-5.4) x IO’, and R, = 1.3-7.7) that have been observed in the ocean (Kelley, 1984). The experiments show that the number of particles in suspension in a convecting layer decreases exponentially with time, with a decay rate equal to the settling velocity divided by

Particle settling through a diffusive-type themohaline

staircase

407

Particlesettlingthrougha diffusive-typetherrnohalinestaircase.

409

the depth of the layer. This exponential law holds despite the presence of horizontal motion and waves at a “diffusive” interface, in contrast to the previously studied case of a rigid boundary where velocities must go to zero (cf. Martin and Nokes, 1988, 1989). To help explain this somewhat surprising result of our experiments, we first observe that horizontal motion at the interface does not affect the vertical sedimentation of particles. Second, although waves increase the effective interfacial surface area available for the vertical and horizontal diffusion of heat, they do not change the effective surface area for vertical sedimentation. Third, and most importantly, the strong vertical density gradient through the interface can reduce the vertical component of the rms convective velocity in the vicinity of the interface to a value that is much smaller than the particle settling velocity (Fig. 6). This conclusion, which is consistent with detailed horizontal velocity measurements of zero-mean-flow turbulence through a density interface (McDougall, 1979), implies that the turbulent motion at the interface can result in a net particle flux that is negligible compared to downward particle flux due to particle settling. This result should only break down when the value of R, (or alternatively of the interfacial Richardson number) is sufficiently low that the rms fluid velocity at the interface becomes comparable to the particle settling velocity. The results of this study should be useful in understanding sedimentation in the Arctic and Antarctic oceans. In both locations, “diffusive’‘-type thermohaline staircases have been observed over very large areas (Padman and Dillon, 1988; Muench et al., 1990; Schmitt, 1994). Staircases at depths of 200400 m appear to be a ‘ubiquitous feature under most of the Arctic ice field away from boundaries” (Schmitt, 1994). In the Antarctic, staircases are commonly found at depths of 100-500 m over much of the Weddell Sea, and have also been observed near 10”E within about 100 km of the Antarctic coast (Muench et al., 1990). These locations also feature substantial deposition of particles (Jacobs, 1989) derived from episodic biogenic production (Nelson et al., 1989; Dunbar et al., 1989) and from the seasonal melting of glacial ice containing terrigenous sediment (Anderson et al., 1984). The particles range widely in size from gravels and sands, down to silts, clays and diatomaceous material. The behaviour of these particles will depend on how their settling velocity compares with the convective velocity in the centre of the double-diffusive layers (which is estimated to be about 1-3 mm/s for the layers described by Muench et al., 1990). For gravels and sands, the settling velocity is much larger than the convective velocity, so the particles will fall uniformly and not be affected by the double-diffusive convection. In contrast, for clays, silts, phytoplankton and some fecal pellets (Honjo and Roman, 1978), the settling velocity is smaller than the convective velocity, so the convection will uniformly distribute the particles in each layer. Exponential decay laws will then control the transport of these particles between each layer, and the total time to sediment 90% or 99% of the particles will be increased. For phytoplankton and other small organisms, the exponential decay laws are of particular importance since an infinite amount of time is required for their complete sedimentation from the convecting water. As a consequence, a very small number of them can remain alive in suspension for very long times and provide the seed for a future phytoplankton bloom (Smith, 1982). If the particles are released suddenly, then the analysis presented in the previous section will describe the particle distribution as a function of time (see also Figs 4 and 5). In contrast, if particles are released constantly at the top of the water column, then a uniform particle distribution will eventually be created in the water, with an equal loss of particles to the floor (regardless of the presence or absence of double-diffusive staircases in between). If the

410

R. C. Kerr and G. S. Kuiper

1.0 0.8 0.6 0.4 0.2 0

Fig. 7. The fraction of particles suspended (Nr/ZNc) as a function of dimensionless time (r = tvJZh), for a two-layer convecting system (solid line) and for the equivalent single convecting layer of thickness 2h (dashed line). Initially there are iVs particles in both layers.

Fig. 8. The dimensionless time (r = tv,/nh) for particles to settle through n convecting layers, for the case where there are No particles initially in each layer. The figure shows both the time required for 90% of the particles to settle (m), and the time for 99% of the particles to settle (a). As the number of layers increases, these dimensionless deposition times decrease steadily from the one-layer values towards the corresponding values (i.e. T= 0.9 and r = 0.99, respectively) for sedimentation from a non-convecting fluid.

source of particles is then abruptly stopped (e.g. by the end of seasonal melting or biological growth), then the analysis given in the Appendix will describe the particle distribution as a function of time (see also Figs 7 and 8). As an example of how these results can be applied, consider the large quantities of particles that are produced in the ice-edge zone of the northwestern Weddell Sea (Nelson et al., 1989). In this zone, the staircase consists of about 20 layers between 100 and 180 m, and of 6-8 layers between 180 and 500 m (Muench et al., 1990). In the event of a rapid phytoplankton bloom or ice melting event, the results shown in Fig. 5 indicate that the time

Particle settling through a diffusive-type thennohaline staircase

411

required for 99% of the finer particles to escape the staircase region will be about twice the time it would take in the absence of a convecting staircase. We note however that the time to deposit 99% of the particles on the ocean floor will be increased by only about lo%, since the staircase makes up only about 10% of the total water column. Finally, we remark that further research on sedimentation in multi-layered convecting systems is required to understand deposition in other aquatic situations. For example, a study could be made of sedimentation in the “finger’‘-type thermohaline staircases that occur in parts of the Atlantic and the Mediterranean (Schmitt, 1994). Another interesting case is sedimentation in the three-region system that consists of an upper mixed layer, an intermediate stagnant thermocline, and a lower benthic mixed layer (Condie and Bormans, in press). Acknowledgements-We

thank ‘Scott Condie, Ross Griffiths, Paul Hutchinson, Andrew Kiss, Stewart Turner and George Veronis for helpful discussions. The technical assistance with the experiments of Tony Beasley, Derek Corrigan and Ross Wylde-Browne is also gratefully acknowledged. We also acknowledge financial support from an Australian Research Council Fellowship (R.C.K.) and an ANU Vacation Scholarship (G.S.K.).

REFERENCES Anderson, J. B., Brake, C. F. and Myers, N. C. (1984) Sedimentation on the Ross Sea Continental Shelf, Antarctica. Marine Geology, 57, 295-333. Condie S. A. and Bormans M. (in press) The influence of stratification on particle settling, dispersion and population growth. Journal of Theoretical Biology. Denman, K. L. and Gargett, A. E. (1995) Biological-physical interactions in the upper ocean: the role of vertical and small scale transport processes. Annual Review of Fluid Mechanics, 27, 225-255. Dunbar, R. B., Leventer, A. R. and Stockton, W. I. (1989) Biogenic sedimentation in McMurdo Sound, Antarctica. Marine Geology, 85, 155179. Honjo, S. and Roman, M. R. (1978) Marine copepod fecal pellets: production, preservation and sedimentation. Journal of Marine Research, 36, 45-57.

Hutchinson, P. A. and Webster, I. T. (1994) On the distribution of blue-green algae in lakes: Wind-tunnel tank experiments. Limnology and Oceanography, 39, 374-382. Jacobs, S. S. (1989) Marine controls on modern sedimentation on the Antarctic Continental Shelf. Marine Geology, 85, 121-153.

Kelley, D. (1984) Effective diffusivities within thermohaline staircases. Journal of Geophysical Research, 89, 1048410488.

Kerr, R. C. (1991) Erosion of a stable density gradient by sedimentation-driven convection. Nature, 353,4233425. Kerr, R. C. and Lister, J. R. (1992) Further results for convection driven by the differential sedimentation of particles. Journal of Fluid Mechanics, 243, 227-245. Martin, D. and Nokes, R. (1988) Crystal settling in a vigorously convecting magma chamber. Nature, 332, 534 536.

Martin, D. and Nokes, R. (1989) A fluid-dynamical study of crystal settling in convecting magmas. Journal of Petrology, 30, 1471-1500.

McDougall, T. J. (1979) Measurements

of turbulence in a zero-mean-shear

mixed layer. Journal of Fluid

Mechanics, 94, 409-431.

Morsi, S. A. and Alexander, A. J. (1972) An investigation of particle trajectories in two-phase Bow systems. Journal

of Fluid Mechanics, 55, 193-208.

Muench, R. D., Fernando, H. S. J. and Stegen, G. R. (1990) Temperature and salinity staircases in the northwestern Weddell Sea. Journal of Physical Oceanography, 20, 295-306. Nelson, D. M., Smith, W. O., Muench, R. D., Gordon, L. I., Sullivan, C. W. and Husby, D. M. (1989) Particulate matter and nutrient distributions in the ice-edge zone of the Weddell Sea: relationship to hydrography during late summer. Deep-Sea Research, 36, 191-209. Padman, L. and Dillon, T. J. (1988) On the horizontal extent of the Canada Basin diffusive staircase. Journal of Physical Oceanography, 20, 295-306.

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Reynolds C. S. (1984) The ecofogy offreshwaterphytoplkton. Cambridge University Press, Cambridge, 384 pp. Schmitt, R. W. (1994) Double diffusion in oceanography. Annual Review of Fluid Mechanics, 26, 255-285. Smith, I. R. (1982) A simple model of algal deposition. Freshwater Biology, 12, 445449. Solomatov, V. S., Olsen, P. and Stevenson, D. J. (1993) Entrainment from a bed of particles by thermal convection. Earth and Planetary Science Letters, 120,387-393. Turner J. S. (1979) Buoyancy effects influidr. Cambridge University Press, Cambridge, 368 pp. Washburn E. W., editor (1926) International critical tables of numerical data: physics, chemistry and technology. McGraw-Hill, New York. Weast R. C., editor (1989) CRC handbook of chemistry andphysics. CRC Press, Boca Raton, FL.

APPENDIX MULTI-LAYERED

SEDIMENTATION WITH UNIFORM DISTRIBUTION

INITIAL

PARTICLE

Ifthe initial particle distribution is uniform, with Ns particles in each layer, the number of particles in layer 1 as a function of time t is still given by Nt (t) = NOexp(-v,t/h).

(Al)

In the underlying layer (layer 2), the solution to eqn (11) that satisfies N,(O) = No is

Nz(t) = Ne(1

+$

exp(-v,t/h).

(42)

In layer 3, the solution to eqn. (13) that satisfies N,(O) = No is vst (vst’h)2 exp(-v st/h) N3(t)=No 1 +~+--j-Finally, in layer n, the solution to eqn (15) that satisfies N,(O) = No is (A4)

For a series of n layers, the total number of particles Ndt) in suspension is then given by

Using the above equations, a comparison can again be made between sedimentation from a two-layered system, where NT(t) = No(2+%?

exp(-v,t/h),

(A6)

and sedimentation from a single layer of thickness 2h with 2Ns particles, where N(t) = 2Ne exp(-v.t/2h).

(47)

It is found that the two-layer system sediments faster than the one-layer system (Fig. 7). This enhanced settling arises because particles can only travel downwards through the interlace, which therefore acts to constrain them to the shallower lower layer where sedimentation is more rapid. This enhanced settling effect increases as the number of layers is increased (Fig. 8).