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.Int. J. Miner. Process. 63 2001 115145

www.elsevier.comrlocaterijminpro

Settling velocities of particulate systems: 12.Batch centrifugation of flocculated suspensions

R. Burger a, ), F. Concha ba

Institute of Mathematics A, Uniersity of Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germanyb

Department of Metallurgical Engineering, Uniersity of Concepcion, Casilla 53-C, Correo 3,

Concepcion, ChileReceived 10 May 2000; received in revised form 30 November 2000; accepted 30 November 2000

Abstract

In this contribution we show how the phenomenological theory of sedimentationconsolidation

processes can be extended to the presence of centrifugal field. The modelling starts from the basic

mass and linear momentum balances for the solid and liquid phase, which are referred to a rotating

frame of reference. These equations are specified for flocculated suspensions by constitutiveassumptions that are similar to that of the pure gravity case. The neglection of the influence of the

gravitational relative to the centrifugal field and of Coriolis terms leads to one scalar hyperbolic

parabolic partial differential equation for the solids concentration distribution as a function of

radius and time. Both cases of a rotating tube and of a rotating axisymmetric vessel are included.

A numerical algorithm to solve this equation is presented and employed to calculate numerical

examples of the dynamic behaviour of a flocculated suspension in a sedimenting centrifuge. The

phenomenological model is appropriately embedded into the existing theories of kinematic .centrifugation processes of ideal non-flocculated suspensions. q 2001 Elsevier Science B.V. All

rights reserved.

Keywords:phenomenological theory; sedimentationconsolidation process; flocculated suspension; centrifuga-

tion

1. Introduction

The enhanced body force obtained by the rotation of a solidfluid mixture in a

centrifuge or hydrocyclone permits the solidfluid separation of particles well below 1

mm in size, a task that gravitational forces alone are not able to meet. Slow large-diame-

)

Corresponding author. Tel.: q49-711-685-7647; fax: q49-711-685-5599. . .E-mail addresses:[email protected] R. Burger , [email protected] F. Concha .

0301-7516r01r$ - see front matter q2001 Elsevier Science B.V. All rights reserved. .P I I : S 0 3 0 1 - 7 5 1 6 0 1 0 0 0 3 8 - 2

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( )R. Burger, F. ConcharInt. J. Miner. Process. 63 2001 115145116

ter basket units use ratios of centrifugal to gravitational forces from 100 = g to 300 = g,

where g is the gravitational forces; most industrial centrifuges are in the range of

500 = g to 5000 = g; high-speed smaller decanter centrifuges can go upto 10 000= g,

while tabular laboratory units have values of 20 000= g with analytical ultracentrifuges

going up to 500 000 = g. Table 1 shows a comparison of the size of separation and the

ratio of centrifugal to gravitational field.The ability of centrifuges to adjust the external field to the particle size to be

separated has extended its use as analytical technique in the laboratory and to substitute

traditional industrial processes such as clarification, thickening and filtration.

Industrial centrifuges can be classified into two types: sedimenting and filtering

centrifuges. The principles underlying are the same of gravity sedimentation and

pressure filtration, respectively. The choice of using a sedimenting centrifuge instead of

filtering centrifuge depends on whether the suspension has a considerable amount of

material below 45 mm and whether the sediment is highly compressible, which makes

filtering centrifuges inapplicable. Sedimenting centrifuges are used extensively in min-eral processing operations such as: dewatering of materials with a significant fraction of

fines, such as thickener discharge of calcium carbonate or fine coal; for the classification

and degritting in the calcium carbonate and kaolin production; for the elimination of

small fractions of very fine solids in leaching, solvent extraction and ion exchange; for

the elimination of contaminants dissolved in mother liquor by separating and redilution

in several stages in solvent extraction.

Despite their extended use, the theoretical treatment of centrifuges lags behind that of

gravity thickening. Although theories of centrifugal separation have been presented by

several authors for ideal suspensions Baron and Wajc 1979; Anestis, 1981; Anestis and.Schneider, 1983; Greenspan, 1983; Schaflinger, 1990 , the most comprehensive publica-

.tion the field Leung, 1998 uses ad-hoc formulations and no general phenomenologicaltheory seems to have been presented for flocculated suspensions. We mention that

.overviews of the use centrifuges are also given by Day 1974 and in the recent . .handbooks by Wakeman and Tarleton 1999 and Rushton et al. 2000 , and that

centrifuges are of particular interest in biotechnical and medical applications, in which

small solidfluid density differences make centrifugal enhancement of hindered settling .mandatory Wiesmann and Binder, 1982; Lueptow and Hubler, 1991 .

In this paper, we present a phenomenological theory of centrifugal separation offlocculated suspensions in decanting, or sedimenting centrifuges, as an extension of the

phenomenological theory of gravity thickening. In a later paper, we will present a

similar theory for filtering centrifuges.

Table 1

Parameters of industrial solid liquid separation equipment

.Size mm Equipment F rFcentrifugal gravitational

100 Gravity thickener-clarifier 0

38 Large hydrocyclone 205 Small hydrocyclone, low- speed centrifuge 200

3 Industrial centrifuge 2000

1 Small high-speed centrifuge 20 000

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( )R. Burger, F. ConcharInt. J. Miner. Process. 63 2001 115145 117

This paper is organized as follows: in Section 2 the mathematical model is developed,

starting from the basic mass and linear momentum balances for the solid and the fluid

and using similar material specific constitutive assumptions as for the pure gravity case.

These equations are referred to a rotating frame of reference and are developed for the

presence of both centrifugal and gravity forces. We finally consider the case in which

both Coriolis and gravity effects are negligible and obtain one scalar hyperbolicpara-bolic partial differential equation describing the concentration distribution as a function

of radius and time. In Section 3 we briefly present a numerical algorithm for the solution

of this equation, which we apply in Section 4 to obtain a variety of centrifugation test

cases. Conclusions that can be drawn from this paper are summarized in Section 5.

2. Mathematical model

2.1. General balance equations

The basic assumptions are the same as those stated in our previous papers Burger.and Concha, 1998; Burger et al., 1999, 2000e and in the monograph by Bustos et al.

.1999 , in which sedimentation under the influence of gravity was studied.

1. The solid particles are small with respect to the sedimentation vessel and have the

same density.

2. The constituents of the suspensions are incompressible.

3. The suspension is completely flocculated before the sedimentation begins.4. There is no mass transfer between the solid and fluid during sedimentation.

Further assumptions will be subsequently specified as constitutive equations.

Consider a sedimenting centrifuge as a rotating system with an angular velocity 5 5 .vs vk, where v is the scalar angular speed and k with k s1 is the unit vector of

.its axis of rotation. Fig. 1 shows two possible cases for this system. The first Fig. 1aconsists of a tube rotating around an axis, such as for a small laboratory centrifuge, and

.the second Fig. 1b is a bowl rotating around its axis, like an industrial decanting

centrifuge.The solid and the liquid are modeled as superimposed continuous media. We recall

that the local mass balances of both components or, equivalently, of the solid and of the

mixture can then be written as

Efq=P fv s0, 1 . .s

Et

=P q s0, 2 .

where f denotes the local solids volume fraction, t is time, v is the solids phases

velocity, and q is the volume average flow velocity of the mixture defined by

q sfv q 1y f v , 3 . .s f

where v is the fluid phase velocity.f

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. . . .Fig. 1. a Rotating tube with constant cross-section gs 0 , b rotating axisymmetric cylinder gs1 . The . .concentration zones are the clear liquid fs 0 , the hindered settling zone 0 - fFf and the compressionc

zone f) f .c

The linear momentum balances for the solid and the fluid phases can be written, for a

frame of reference rotating at velocity v, in the form

EvsD q =v P v s =P T q Db q m, 4 .s s s s s s /Et

EvfD q =v P v s=P T q D b y m, 5 .f f f f f f /Et

where D, D , T andT are the constant mass densities and the stress tensors of the solids f s f and the fluid, respectively, m is the solidfluid interaction force, and b and b are thes fexternal body forces that, in a rotating frame of reference, take the respective forms

b syg kyv=v= r y 2v= v , b syg k yv=v= r y2v= v , 6 .s s f f

where the first term denotes the gravitational force and the second and third terms

represent the inertial forces originating from the centripetal and the Coriolis accelera-

tions, both product of the moving frame of reference. The body forces b and b can bes f

.separated into two parts, a conservative force = FqV , where F and Vare given by1 1

2 2Fsyg P r ' gz , Vsy v= r P v= r sy v r , 7 . . .2 2

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and the non-conservative parts y2v= v and y2v= v , so that b and b can bes f s f written in the forms

b sy= Fq V y2v= v , b sy= FqV y 2v= v . 8 . . .s s f f

The ratio between the representative centrifugal and gravity components of b is

. 2expressed by the Froude number of the system Ungarish, 1993 , FFsv Rrg, where Ris a typical distance to the axis of rotation for example, the outer radius of the

.container . The two limiting cases are here FFs 0, corresponding to gravity settling, andFFs` for a centrifugally dominated configuration.

2.2. Constitutie assumptions

2.2.1. Solid and fluid stress tensors

The stress tensors of the components are assumed to take the forms

T syp I qT E , T syp I qT E , 9 .s f s f s f

where p and p are the solid and fluid phase pressures and T E and T E are the viscouss f s f or extra stress tensors. A detailed discussion of the possible forms of T E and T E iss f

.provided by Burger et al. 2000e .The theoretical variables p and p are replaced by two experimental quantities, thes f

pore pressure p and the effective solid stress s. The experimental variables are thoseeparts of the total pressure in the mixture which are supported by the solids network and

by the fluid filling the pores between the solid flocs, respectively. Consequently, we

have

p sp qp sp qs . 10 .t s f e

. .Burger et al. 2000e see also Concha et al., 1996 and Bustos et al., 1999 show that theassumption that the local surface porosity of a cross section of the network equals the

volume porosity f leads to the equations

p s 1 y f p , p sfp q s . 11 . .f s e

However, the flow in a porous medium, such as the flocculated solid network, does not

depend on the pore pressure itself, but rather on its difference to the hydrostatic

pressure. Therefore, the pore pressure p should be expressed in terms of the excess pore .pressure p , which is the pore pressure less the static pressure Ungarish, 1993 ,e

12 2p :sp q D FqV spq D gz y v r . 12 . .e f f /2

.As in our previous papers in this series Burger et al., 1999; Garrido et al., 2000 , s e .is given as a constitutive function s s s f , which depends on the local solidse e

volume fraction only and satisfiesd s f .s 0 for fFf , s 0 for fF f ,ec cX

s f s f :s 13 . . .e e ) 0 for f) f , ) 0 for f) f .dfc c

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We do not require s to be continuous at fs f. A typical constitutive equation is thee cthree-parameter function frequently referred to as power law see e.g. Landman and

.White, 1994

0 for fF f ,cs f s s ) 0, k) 1. 14 . .ke 0s frf y1 for f) f , . .0 c c

2.2.2. Solidfluid interaction force

We assume that the solidfluid interaction force m is given by a constitutive equation

linear in the concentration and the relative solidfluid velocity:

m sya f v qb=f, 15 . .r

.where a f is the resistance coefficient, corresponding to the second constitutivefunction describing the material behaviour of the mixture.

. . . .Inserting Eq. 8 and the constitutive assumptions Eqs. 9 , 11 and 15 into the . .momentum balances Eqs. 4 and 5 yields

Evs 2D q =v P v sy2Dv k = v q Df yg k qv r y a f v q b=f . . .s s s s s s r /Et

y f=p yp=fy = s f q =P T E , 16 . . .e s

Evf 2D q =v P v sy2D v k = v q D 1 y f yg k qv r q a f v . . . .f f f f f f r /Et

y b=fqp=fy 1 y f =p q =P T E . 17 . .f

.Considering Eq. 17 at equilibrium, that is settingv sv ' 0, v' 0 and since the pores f rpressure equals the hydrostatic,

12=p s D yg k q v r , 18 .f

/2 . . .and introducing Eq. 18 into Eq. 17 we conclude that bsp. Then, from Eqs. 16 and .17 the momentum balances are:

Evs 2D q =v P v sy2Dv k = v q Df yg k qv r y a f v y f=p . . .s s s s s s r /Et

y = s f q =P T E , 19 . . .e s

Evf 2D q =v P v sy2D v k = v q D 1 y f yg k qv r q a f v . . . .f f f f f f r /Ety 1y f =pq =P T E . 20 . .f

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2.3. Dimensional analysis

. . . .Considerable simplification of the four field equations Eqs. 1 , 2 , 19 and 20 canbe achieved by an order-of-magnitude study. Consider a typical length scale L , a0

typical velocity U , a typical scalar angular velocity v , a kinematic viscosity n and0 0 0assume that D is a typical density. Velocities are here referred to a rotating referencesframe with angular velocity v . A star will denote the dimensionless analogue of each0variable. Moreover, t sL rU will be chosen as time scale for dimensionless time0 0 0

. .derivatives. Introducing the characteristic parameters in Eqs. 19 and 20 , we obtainthe following equations in terms of the dimensionless variables:

D Ev )s s) ) )Fr q = v P vs s) /D Etf

2D Fr Ds s 2) ) ) )sy v k = v q f ykqFF v r . . .sD Ro Df f

Fr )) ) ) ) ) ) ) Eya f v yf= p y = s f = P T , 21 . . . . .r e s

Re

Ev )f) ) )Fr q= v P vf f) /Et

2 Fr2) ) ) )sy v k = v q 1 yf ykqFF v r . . . .f

Ro

Fr )) ) ) ) ) Eqa f v y 1 y f = p q = P T . 22 . . . .r f

Re

2 . .In these equations, Fr[ U r gL is the Froude number of the flow, Ro [ U r v L0 0 0 0is the Rossby number and Re sL U rn is the Reynolds number of the flow. Typical0 0 0numerical values for the constants and the characteristic parameters are

2 .g s 10 mrs accelaration of gravity , .L s0.1 m typical size of sedimenting space in a centrifuge ,0

y4 .Us10 mrs settling velocity of a particle ,0y6 2 .n s 10 m rs kinematic viscosity of water ,0

from which we obtain Frs10y8 and FrrRe s10y7. The value of v and thus those0of FF and Ro will be specified later.

On the basis of these estimates, and considering that the terms with stars are of the

order of one, we may neglect the viscous and the convective acceleration terms in thelinear momentum balances, i.e. those terms which bear the coefficients Fr and FrrRe,respectively. Noting that we may express the phase velocities v and v in terms ofs f

.v s v and q as v sq q 1 y f v and v s q yfv , we obtain after rearranging andr s s r f r

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. .inserting Eq. 22 into Eq. 21 explicit equations for the relative solidfluid velocityand for the excess pore pressure, respectively:

)1y f Fr 2 v2) ) ) ) ) )v s DD f ykqFF v r y= s f q . . . .r e) )a f Ro a f . .

=D Ds s 2) 2 )fy 1 yf k = q y f y 1 y f k = v . . . .r / /D Df f

qOO Fr 1 q 1rRe , 23 . . .

a ) f Fr 2 v) .) ) ) ) )= p s v y k = q yf k = v . . .e r f

1 yf Ro 1 yf

qOO Fr 1 q 1rRe . 24 . . .

The assumed numerical values which are independent of the applied angular velocity v ..provide a rationale for assuming that the OO Fr 1 q1rRe terms are negligible.

Before further reducing the momentum balances, we briefly discuss some properties .of the system of equations formed by the continuity Eq. 1 , which by using the

definition of the slip velocity v takes the formr

Efq=P fq qf 1 yf v f,=f,q s0, 25 . . . .r

Et

. .the condition =P q s 0, and the dimensional analogues of Eqs. 23 and 24 after ..deleting the OO Fr 1 q 1rRe terms.These equations provide a complete system of five scalar equations for the scalar

quantities f and p and the three components of the volume-average velocity q. Thisemeans that the Coriolis terms provide the necessary coupling between the concentration

field and the volume average flow field. This is a remarkable result, since in the pure . gravity case by taking the curl of Eq. 24 without the Coriolis terms, and deleting these

..also in Eq. 23 we obtain that f depends only on the vertical coordinate. In general,the resulting field equations will then not be sufficient to determine the quantities f, pe

and q, which can only be achieved by reconsidering viscous or advective accelaration .terms Burger et al., 2000e or by modeling the coupling by boundary conditions .Schneider, 1982 . Implications of that coupling are discussed by Burger and Kunik .1999 .

2.4. Final field equation

Our goal is now to obtain one scalar model equation for f. We assume that the

angular velocity v is chosen so large that centrifugal effects dominate gravity, i.e.

.FF) 1, such that it is reasonable to neglect the gravity terms in Eq. 23 ; on the otherhand, the Rossby number Ro should not be too small, in order to provide justification

for neglecting the Coriolis terms. We therefore consider the range of values 100

radrs FvF 1000 radrs, corresponding to a range between about 1000 and 10 000 rpm,

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and R sL s 0.1 m. We then have FrrRo s10y5, FFs 100 for vs 100 radrs and0FrrRos 10y4 , FFs 10 4 for vs 1000 radrs, respectively. This discussion is, ofcourse, not rigorous, but meant to illustrate that simultaneously neglecting both Coriolis

and gravity terms yields a reasonable approximation for the system of interest here. We . .then obtain from Eqs. 23 and 24 :

1y f2) ) ) ) ) )v s DD f ykqFF v r y= s f , 26 . . . . .r e)a f .

a ) f .) ) )= p s v . 27 .e r

1 yf

.As in our previous work Burger and Concha, 1998; Burger et al., 2000e , we replace .the resistance coefficient a f by the corresponding Kynch batch flux density function

22

DDgf 1 y f .f f :sy , DD:s D y D . 28 . .bk s f a f .

. .We substitute this function into Eqs. 26 and 27 and return to a dimensional form.Then v and p are given byr e

f f .bk 2v s yDDfv r q = s f , 29 . . .r e2DDgf 1 y f .

=p s DDfv2 r y= s f . 30 . . .e e

Due to the neglection of gravity, the equation for the solidfluid relative velocity, Eq. .29 , only uses the radius vector. Considering axisymmetric solutions leaves the radius r

. .as unique space variable. The scalar versions of Eqs. 29 and 30 are

f f Es f . .bk e2 s yDDfv rq , 31 .r 2 ErDDgf 1 yf .

Ep Es f .e e2sDDfv ry . 32 .Er Er

.There are now two cases possible: a Flow in a rotating tube with constant cross . . .section Fig. 1a : Eqs. 1 and 2 reduce to

Ef Eq fq qf 1 yf s 0, 33 . . .r

Et Er

Eqs0. 34 .

Er

Since we consider only tubes, which are closed at their outward-pointing end during

. .rotation, we obtain q ' 0. Inserting Eq. 29 into Eq. 33 yields the field equationEf E f f v2 r E f f s

Xf Ef . . .bk bk e

q y s y 35 . / /Et Er g Er DDgf Er

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for f. Defining

f f sX

f . . fbk ea f :sy ; A f :s a s d s, 36 . . . .H

DDgf 0

.we may rewrite Eq. 35 asEf E f f v2 r E2 .bk

q y s A f . 37 . .2 /Et Er g Er . .b Flow in a batch cylindrical centrifuge Fig. 1b : The governing equation is now

Ef 1 Eq r fq qf 1 yf s 0, 38 . . . .r

Et r Er

. .which we may rewrite in view of q'

0 and using a f and A f as defined above as

Ef 1 E f f v2 r2 1 E EA f . .bkq y s r . 39 . / /Et r Er g r Er Er

Relating the centrifugal Kynch batch settling function f to the conventional gravityck . .function f by f f :syf frg and defining the parameter gs 0 for the rotatingbk ck bk

tube and gs1 for the axisymmetric case, we obtain the partial differential equation

Ef 1 E 1 E EA f .2 1qg g

q f f v r

s r , 40 . .

.ckg g /Et r Er r Er Er

which can be rewritten as

Ef E A f E2 A f . .2 2q f f v ryg s A f qg yf f v q . . . .ck ck 2 2 / /Et Er r Er r

41 .

Since

s 0 for 0F fF f and fs f ,c maxa f 42 . .) 0 for f - f- f ,c max .it becomes evident that the governing equation for the centrifugation problem, Eq. 40

. or Eq. 41 , is a second order strongly degenerate partial differential equation with.source terms in the case gs1 . It is of the first order hyperbolic type for 0 FfF f ,c

i.e. in the hindered settling zone, where the solid particles do not yet touch each other,

and of the second order parabolic type for f) f. The first-order equation is equivalentc . .to that investigated by Anestis and Schneider 1983 and Lueptow and Hubler 1991 .

The location of the type-change interface, where fsf is valid, is not known a priori.cThis unusual feature allows concentration discontinuities in the hindered settling zone

and requires a particular mathematical treatment within the framework of entropy .solutions. See Burger and Karlsen 2000 for details.

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In both the rotating tube and cylindrical centrifuge cases, the equation

Ep Es f .e e2sDDfv ry 43 .Er Er

can be used to calculate the excess pore pressure p a posteriori from the concentrationedistribution.

2.5. Initial and boundary conditions

We finally have to specify initial and boundary conditions. We assume that the

variable rvaries between an inner radius R and the outer radius R and assume that0 s 0 at both rsR and rsR. This implies the boundary conditionss 0

EA f .2f f v R q R ,t s 0, t) 0, 44 . . .ck 0 0

Er

EA f .2f f v R q R ,t s 0, t) 0. 45 . . .ck

Er

The initial condition is

f r,0 s f r , R F rFR . 46 . . .0 0

For simplicity, we limit ourselves in this paper to the case that the initial concentration

f is constant.0

3. Numerical algorithm

.To solve the initial-boundary value problem given by Eq. 41 together with the . .initial and boundary conditions Eqs. 44 46 numerically, we employ a modificationof the generalized upwind finite difference method. This method has been presented in

. detail by Burger and Karlsen 2001 for the case of a pure gravity field see also Burger .et al., 2000d , and shall be outlined only briefly here.

.Let J, NgN, D r[ R yR rJ, D t[ TrN, r[R qjD r, j s 1r2,1,3r2 , . . . ,J0 j 0n . 0 .y 1r2, J and f f f r, nD t . The computation starts by setting f [ f r forj j j 0 j

j s 0 , . . . ,J. Assume then that the solution values fn, j s 0 , . . . ,Jhave been calculatedjfor the time level t [ nD t. To compute the values fnq 1, we first compute then j

extrapolated values

D r D rL n n R n nf :sf y s , f :s f q s , j s 1 , . . . ,Jy 1, 47 .j j j j j j

2 2

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where the slopes s n can be calculated, for example, by the minmod limiter functionj .M P,P,P in the following way:

fn yfn fn yfn fn y fnj jy1 jq1 jy1 jq1 jns sMM u , ,u ,j /D r 2D r D rw xug 0,2 , j s 2 , . . . ,Jy2, 48 .

s n s s n s s n s s n s0, 49 .0 1 Jy1 J

4min a,b ,c ifa,b ,c ) 0,~MM a, b ,c :s . 50 . . 4max a, b ,c ifa,b ,c - 0,

0 otherwise

The extrapolated values fL and fR appear as arguments of the numerical centrifugalj jEO

.Kynch flux density function f P,P which, according to the EngquistOsher schemeck .Engquist and Osher, 1981 , is defined by

fEO u , :sfq u qfy , 51 . . . .ck ck ck

u X Xq yf u :sf 0 q max f s ,0 d s, f :s min f s ,0 d s. 52 4 4 . . . . . .H Hck ck ck ck ck

0 0

.The interior scheme, which approximates the field Eq. 40 and from which the interiorapproximate solution values fn, . . . ,fn are calculated, can then be formulated as1 Jy1

2v D tnq 1 n 1qg EO R L 1qg EO R Lf s f y r f f ,f yr f f ,f . .j j jq1r2 ck j jq1 jy1r2 ck jy1 jgr D rj

D tg n n g n nq r A f yA f y r A f yA f , . . . . . .jq1r2 jq1 j jy1r2 j jy1g 2r D rj

j s 1 , . . . ,Jy 1, 53 .

. .where the function A P was defined in Eq. 36 . The update formulas for the boundary

n n .values f and f follow from formula Eq. 53 for j s0 and j sJ by inserting the0 J . .discrete analogues of the boundary conditions Eqs. 44 and 45 , respectively. More-

over, we do not use extrapolated values for the boundary schemes in order to avoid

referring to auxiliary solution values. We end up with the boundary formulas

v2D t D tn ny1 1qg EO n n g n nf sf y r f f ,f q r A f yA f , 54 . . . . .0 0 1r2 ck 0 1 1r2 1 0g g 2R D r R D r0 0

v2D tn ny1 1qg EO n nf s

f q

r f f ,f .J J Jy1r2 ck Jy1 JgR D r

D tg n ny r A f yA f . 55 . . . .Jy 1r2 J Jy1g 2R D r

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To ensure convergence of the numerical scheme to the entropy weak solution of the . . .initial-boundary value problem Eqs. 41 , 44 46 , the CFL stability condition

D t D tX2 <

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. .Concha and Bustos, 1991 and as is illustrated in Fig. 6 by Lueptow and Hubler 1991 ,the solution f is not constant along characteristics.

.It is well known that intersections of characteristics cause solutions of Eq. 59 to bediscontinuous in general, and that the propagation speed s ) of a discontinuity SS at a

) ) . ) ) . ypoint r , t gSS in the r , t -plane, which separates two concentration valuesf0 0and fq is given by the RankineHugoniot condition

) ) ) q ) yd r r f f yf f . . .0 ck ck )s s s . 61 .

) q yd t f yfSS) ) . ) ) .By the change of variables r s r j, f and t s t j, f , it is not difficult to

. .derive from Eqs. 60 and 61 the ordinary differential equation

Et) Er)) q ) q ) y ) y q yr j,f f f y r j,f f f q f y f . . . . . .ck ck d j Ef Ef

sy ,) )

Et Erdf ) q ) q ) y ) y q yr j,f f f y r j,f f f y f y f . . . . . .ck ck Ej Ej

fs fq or fs fy 62 .

.describing the shock front in the j, f -plane. This front can be transformed into the ) ) . .r , t -plane by Eq. 60 .

& mu f 1 y f , for 0 - f- f :s 0.66, .` maxf f s 63 . .bk 0 otherwise.

We now construct the exact solution of a simple case of Anestis and Schneider .1983 , to compare this result later with numerical solutions of the phenomenological

model. To be specific, we take the Kynch batch flux density function Richardson and& &

.Zaki, 1954 where u sy u rg with u sy0.0001 mrs and m s5. The constant` ` `initial datum, f s0.07, has been chosen in such a way that the chords joining the point0 .. . .f , f f with the points 0, 0 and f , 0 respectively, both lie above the graph0 bk 0 max

.of f in an f f versus fplot, according to a the case Ia of Anestis and Schneiderbk bk .1983 for centrifugation and that of a mode of sedimentation MS-1 by Bustos et al. . .1999 and Burger and Tory 2000 .

In this case, two kinematic shocks will form: a shock SS separating the suspension1 .from the clear liquid zone fs 0 which is forming on the inner wall and a shock SS2

separating the suspension from the sediment of concentration fsf . These shocksmaxwill meet at the critical time t) and merge into a third stationary shock SS .c 3

Unlike the gravity case, the concentration of the bulk suspension between SS and1) ) )SS is not constant for 0 F t - t . In fact, by using the initial condition f r ,2 c

. .0 sf , we obtain from Eq. 6001

1qg) ) )

r sj f f rf f , 64 . . . .ck 0 ck 1 d uf

)t sy . 65 .H )1 qg f u .f ck0

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) .From the equation for t which, for flux density functions given by Eq. 63 withu - 0 and integers n, takes the explicit form`

ny1 j1 1 yu 1 un y1)t s t f y t f , t u :s ln y , . . . 0 /

/ /j1

qg u j 1

yu

js166 .

we see that the concentration in the suspension varies with time, but not with the

radius r. .To obtain explicit expression for the shock curves SS and SS , we integrate Eq. 621 2

to obtain

11

1qg1qg) ) )

r sR frf for SS , r s f y f r f y f for SS , . . . .0 0 1 max 0 max 267 .

) .where f is a parameter and t is given by Eq. 65 . The parameterf runs from f to0the value

fmaxfs 68 . .y 1qg)1 q f rf y 1 R . . .max 0 0

obtained by interesting the shock curves SS and SS . In this example, we obtain1 2fs 0.015298 for gs 0 and s0.0031174 for gs1. The corresponding exact solutionsare drawn as iso-concentration lines in Fig. 2.

4.2. Comparison with the kinematic model

Having recalled the known results from Anestis and Schneider, we come back to the

phenomenological model of sedimentation with compression. Our emphasis is now on

. .the order of magnitude of the terms of Eq. 41 containing A f in comparison to those .present in Eq. 58 . To this end, we define the dimensionless integrated diffusion

) . . .coefficient A f :sA frA f . Note that the monotonicity of A implies thatmax) .0 FA f F1.

.Using the same dimesionless variables as before, we may rewrite Eq. 41 indimensionless form as

Ef E) )q f f r . .ck) )Et Er

) ) 2 )1 A f E A f E A f . . .)sygf f q g y q , 69 . .ck ) )2 2) ) / /Pe Er rr E r . .

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Fig. 2. Exact solution of a kinematic centrifugation problem. The lines SS , SS and SS denote kinematic1 2 3

shocks; the vertical lines are iso-concentration lines corresponding to the annotated values.

2 2 .where the Peclet number Pe is defined by Pe sRv u rA f , in analogy to the ` max .Peclet number introduced by Auzerais et al. 1988 in the pure gravity case. Here, Pe

characterizes the order of magnitude of the convective centrifugal hindered settling

terms to those of centrifugal compression. The actual value of Pe depends, of course, on

the effective solid stress function s of the material considered.e . ) ) ) )Eq. 69 is again considered for R F r F 1 and 0 F t F T . However, solutions0 .of Eq. 69 now do depend on the values of R and w, and we have to specify these

.quantities to make comparison with the solutions of Eq. 59 depicted in Fig. 2 possible. . .In particular, for given functions A f and f f and a vessel of fixed outer radius R,ck

.we observe that 1rPe0 for v`. This means that the solutions of Eq. 59 ,w ) xconsidered on the appropriately scaled time interval 0, T , are the limit case obtained

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from the phenomenological sedimentationconsolidation model when the angular veloc-

ity v approaches infinity. . .To illustrate this, we now include the compression terms in Eqs. 41 and 69 by

choosing the effective solid stress function

0 for fF f :s0.1,cs f s 70 . .9e 5.7 frf y1 Pa for f) f , . .c ccorresponding to a calcium carbonate slurry Damasceno et al., 1992; Burger et al.,

. . .2000b . The gravity Kynch batch flux density function f f defined by Eq. 63 andbk .the effective solid stress function s given by Eq. 70 are plotted in Fig. 3.e

We choose the outer radius R s0.3 m, the corresponding inner radius R s 0.06 m,0and again the initial concentration f s 0.07. Three different values of the angular0velocityvare chosen in such a way that the centrifugal force R v2 at the bottom of the

vessel equals 100 = g, 1000 = g and 10 000 = g, respectively. The corresponding timesT are 30, 3 and 0.3 s. Fig. 4 shows the numerical result for both the rotating tube0 . . gs 0 and the cylindrical vessel gs 1 as settling plots iso-concentration lines in an

.r vs. tplot , while Fig. 5 displays selected concentration profiles at different times forthe rotating tube case, together with the concentration profiles at corresponding times of

.the exact solution for the case s' 0 depicted in Fig. 2a .eBoth Figs. 4 and 5 illustrate that the sedimentationconsolidation process terminates

in very short time. From Fig. 4 we observe that the iso-concentration lines of the

compression zone become horizontal very soon after the supernatesuspension and the

sedimentsuspension interfaces have met. As v is increased, the compression zone .becomes thinner, and the final maximum concentration increases, which is well visiblein Fig. 5be. However, it is worth noting that the rotating tube produces somewhat

higher bottom concentrations than the cylindrical vessel, which becomes apparent by the

fact that the iso-concentration line fs 0.42 is present in Fig. 4e only. In the hindered

. .Fig. 3. Gravity Kynch batch flux density function left and effective solid stress function right used for

comparison with the kinematic model.

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. .Fig. 4. Numerical simulation of a sedimentationconsolidation process a, c, e in a rotating tube and b, d, f i . .vs57.184 radrs, c, d 180.83 radrs and e, f 571.84 radrs. The concentration lines correspond to the annotate

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.

Fig.

4

continued

.

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.

Fig.

4

continued

.

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.Fig. 5. Concentration profiles in a rotating tube at ts t s iPT r3, is 0,...,6 and t s 4T : a exact solutioni 0 7 0 . . . .of Eq. 59 , b to d : numerical solutions of Eq. 41 with R s0.06 m, Rs0.3 m and different values of v.0

.settling zone, the numerical scheme accurately reproduces to within numerical errorsthe exact solution of Fig. 2.

( )4.3. Comparison with results by Sambuichi et al. 1991

.Sambuichi et al. 1991 published experimental results of the centrifugation of threedifferent aqueous suspensions, namely of limestone, yeast, and clay, in a cylindrical

sedimenting centrifuge. For each material, the measurement of gravitational settling

. .velocities led to a function that can be transformed into our functions f f or f f .bk ck Moreover, compression data obtained by both the compressionpermeability cell method

.and the settling method Shirato et al., 1970 determined a unique effective solid stress .function s f for each material.e

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In this paper, we choose the published experimental data referring to the clay .suspension. Sambuichi et al. 1991 approximated the measured gravity settling veloci-

ties u for the clay experiments by three different connecting straight segments in aglog u versus log f plot, depending on whether f falls into an assumed diluteg . . 0.02 FfF 0.056 , intermediate 0.056 FfF 0.107 or a concentrated region fG

. . b20.107 . Consequently, we obtain for each of these segments u f s b f withg 1suitable real constants b and b . Converting the function u into f via f1 2 g bk bk . . f sfu f whereby we take into account that the measured settling velocities areg

.propagation velocities of the clear liquid-suspension interface , and smoothly connectingthe first segment with the origin by a second-order parabola, we obtain the continuous,

piecewise differentiable function the precise representation has been cut here to five.significant digits

0 for fF 0 or fGf :s0.5, max2 y823.229f y2.1673f = 10 mrs for 0- fF 0.02, .

y9 0.72715~y5.8558 = 10 f mrs for 0.02 - fF0.056,f f s .bky10 y0.57139y1.3869 = 10 f mrs for 0.056 - fF 0.107,

y1 0 0 .132y6.68 = 10 f for0.107 - f- f .max71 .

.The last expression in Eq. 71 , corresponding to the concentrated segment, has been .proposed by Sambuichi et al. 1991 . Since that expression does not assume the value

zero, the flux function had to be cut at a maximum concentration f . This value hasmaxbeen chosen here as 0.5. The actual choice of this value in a reasonable range say

.greater than 0.3 does not influence the result for sedimentation with compression, sincethe maximum concentration possible with compression essentially depends on the

. .Fig. 6. Gravity Kynch batch flux density function of left and effective solid stress function right used for .comparison with experimental data by Sambuichi et al. 1991 .

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. .Fig. 7. Comparison of clear liquidsuspension circles and suspensionsediment black dots interfaces, .measured by Sambuichi et al. 1991 during centrifugation of a clay suspension using three different angular

velocities, with numerical solution of the phenomenological model.

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behavior of s on intervals offthat are close to the critical concentration f , and onlye cto a small degree on the value off . However, this value needs to be well-defined formaxour purpose of a comparison with the purely kinematic model.

.The published constitutive equation Fig. 3 of Sambuichi et al., 1991 ,

es 0.86 y 0.74 = 10y3

p , 72 .s

where es1 y f is the porosity and p is Sambuichi et al.s compressive solid pressuresmeasured in pascals, can be converted into the effective solid stress function

0 for fFf ,cs f s 73 . .e ka fy f for f) f , .1 c c

where the constants have the values fs 0.14, a s 1.142 = 107 Pa and ks 3.509.c 13 .The density difference for the material was rs 1600 kgrm . The functions f f andbk

. . .s f defined by Eqs. 71 and 73 , respectively, are plotted in Fig. 6.eFig. 7 shows numerical solutions of the phenomenological model calculated with

. .these parameters and functions Eqs. 71 and 73 . The initial concentration, f s0.089,0and the three different angular velocities vs146.4 radrs, 167.76 radrs and 230.59

.radrs have also been chosen according to Sambuichi et al. 1991 . The maximum time ..2for each diagram was chosen as 25 000P vr 146.4 radrs , such that, in a similar way

as in the previous example, all three plots can compared with the high-accuracy solution

for the case A ' 0 depicted in Fig. 8.

The solution shown in Fig. 8 displays some additional features as compared to those

.given in Fig. 2. The fact that the point f , f f can no longer be connected with the0 bk 0 . .point f , 0 by a straight line lying above the graph of f see Fig. 6 implies thatmax bk

the bulk suspension is no longer separated from the sediment by a single kinematic

shock. Rather, the sediment with fs f s0.5 is separated from the initial concentra-max

.Fig. 8. High accuracy reference solution without compression A' 0 for the centrifugation experiment by .Sambuichi et al. 1991 and the settling plots of Fig. 7.

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tion f s0.089 by a contact discontinuity at fs0.107, followed upwards by a centred0rarefraction wave. The transition between f s 0.089 and the time-dependent concentra-0tion value of the bulk suspension also takes place continuously, as is well visible in the

concentration profiles plotted in Fig. 9. Moreover, we observe that the minimum

nonzero concentration of the system is 0.056, corresponding to the local minimum of

f , which is marked by a circle in Fig. 9. The sedimentation process of that figure isbk .one of Type II according to the classification of Anestis and Schneider 1983 .

Due to the obvious difficulties related to measuring concentration profiles in a .rotating system, Sambuichi et al. 1991 could only measure the propagation of the

supernatesuspension and of the suspensionsediment interfaces. These experimental

data are plotted in Fig. 7. The numerical solution, displayed with additional iso-con-

centration lines, approximates well both interfaces for small times and correctly predicts

the final heights of the sediments. However, the simulated settling process takes place

somewhat faster than the observed, and this discrepancy consistently increases with v.

This phenomenon has, however, a simple explanation: the model equation solved was

.Fig. 9. Simulated concentration profiles of the high accuracy reference solution without compression A' 0 .for the centrifugation experiment by Sambuichi et al. 1991 and of numerical the simulation of centrifugation

..with compression with vs230.59 radrs see Fig. 7 a .

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formulated under the assumption that Coriolis terms are negligible. Since it is well

known that Coriolis effects produce a retrograde rotation of the solid phase and thereby .enhance the separation time Schaflinger et al., 1986; Schaflinger, 1987 , it is not

surprising that our model underpredicts the separation time, as visible in Fig. 7.

5. Conclusions

The present work shows how the phenomenological theory of sedimentationcon-

solidation processes, which had been formulated so far a gravity field only, can be

extended to a rotating system in order to provide a rational model for the centrifugation

of flocculated suspensions. In this paper, we consider the simple case where the

gravitational field and the Coriolis force are negligible compared to the centrifugal force,

but that also Coriolis forces are negligible. Clearly, these restrictions imply both a lowerand upper limit of the angular velocities possible with a given centrifuge. However,

these assumptions are also inherent in the simpler kinematic treatments for centrifuga-tion ideal suspensions Anestis, 1981; Anestis and Schneider, 1983; Lueptow and

.Hubler, 1991 , to which the phenomenological theory has been compared explicitly. Asin the gravity case, this theory leads to one scalar hyperbolicparabolic degenerate

convectiondiffusion equation with a type-change interface marking the sediment level.

Solutions of such equations are discontinuous in general and require a treatment within a .suitable entropy solution framework Burger et al., 2000c; Burger and Karlsen, 2000 . In

particular, it must be ensured that the numerical scheme applied to such an equationapproximates the right discontinuous solution. However, this is the case with the

.presented modification related to the rotating frame of reference of the generalizedupwind scheme, which is computationally simple and correctly approximates the

supernatesuspension and suspensionsediment interfaces without the necessity to track

these explicitly. This fact sharply contrasts with the numerical solution procedure .advanced by Sambuichi et al. 1991 : their algorithm is essentially based on alternately

. .solving Eq. 58 in the hindered settling zone and the equation in our notation . 2 .Es frEfs Dfrv , which is obtained from Eq. 31 by assuming that is negligiblee r

in the compression zone. The appropriate supernatersuspension and suspensionrsedi-ment interfaces are determined by a trial-and-error repetition of these solution proce-

dures, combined with global mass balance considerations, several times during each

time step. We doubt not only the efficiency of this algorithm, but also the validity of

neglecting in the compression zone, since our dimensional analysis does not providerjustification to do so.

As has become apparent in the example of comparison with the experimental data of .Sambuichi et al. 1991 , the current phenomenological formulation is limited to those

cases where Coriolis terms are indeed negligible. However, retaining these terms in Eq.

.23 and assuming that the flow variables depend on the radius only will again produce .an explicit though more complicated equation for , and a correspondingr

hyperbolicparabolic partial differential equation for the volumetric solids concentra-

tion. For ideal suspensions, obtained by letting s' 0 in our theory, such treatmentse

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. .have already been performed by Schaflinger et al. 1986 and Schaflinger 1987 . Ananalogous extension of the present work to the presence of Coriolis terms, as well as

.comparisons with additional experiments such as those of Eckert et al. 1996 , are inpreparation.

.Finally, we mention that Burger et al. 2000a derive a mathematical model ofpressure filtration from the phenomenological theory of sedimentationconsolidationprocesses. This model leads again to a scalar partial differential equation of the

mentioned mixed type, but with a free boundary modeling the movement of the mixture

top boundary. In view of these advances, it is feasible now to unify the models of

centrifugation and filtration in a phenomenological theory of filtering centrifuges .Sambuichi et al., 1987 , to which we will come back in one of the next articles in thisseries.

6. List of symbols

.Variables that occur in both dimensional and dimensionless starred forms are listedhere only in their dimensional version.

Latin symbols .a f diffusion coefficient .A f integrated diffusion coefficient

b body force per unit massb , bs f external body forces

B .negative potential ofb . .C j ,C j1 2 integration constants .f fbk Kynch batch flux density function .f fck centrifugal Kynch batch flux density function

EO .f u, ck numerical Kynch batch flux functionFF Froude number of the system

Fr Froude number of the flow

g acceleration of gravityg gravity force

I identity tensor

j space index

J integer defining spatial discretization

k .exponent in a constitutive equation s ss fe ek upwards pointing unit vector

L0 typical length

m solidfluid interaction force

.MM a, b, c Minmod limiterm Exponent of the RichardsonZaki flux density function

n time index

N integer defining time step

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p pore pressure

pe excess pore pressure

p , ps f solid and fluid phase pressures

ps .compressive solid pressure Sambuichi et al., 1991pt total pressure

q volume average flow velocityr radius

rj value of r in numerical method

r radius vector of the system

R outer radius of the container

R0 .inner radius suspension surfaceRe Reynolds number of the flow

Ro Rossby number

snj slopes in numerical method

SS, SS , SS , SS1 2 3 symbols denoting kinematic shockst time

t0 time scale

tc critical time

T, Ts f solid and fluid stress tensors

T E, T Es f solid fluid extra stress tensors

u` .coefficient of f fck&

u` .coefficient of f fbkug gravity settling rate

U0 typical velocityv, v, , s f s f solid and fluid phase velocities

v, r r solidfluid relative velocity

z height

z axis of rotation

Greek symbols

a1 .coefficient in equation for s fe .a f resistance coefficient

b coefficient in the approach for m

b , b1 2 .parameters in the equation for u fgg parameter indicating rotating tube or axisymmetric case

D r space step of numerical method

D t time step of numerical method

DD solidfluid mass density difference

f volumetric solids concentration

f value of fat intersection of kinematic shocksfy, fq approximate limits of fat a discontinuity

f0 initial concentration

fc critical concentration

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fnj value of f in numerical method

fL , fRj j extrapolated numerical values of f

fmax maximum solids concentration

F potential of gravity force

m , ms f phase viscosities

n0 kinematic viscosity of waterD, Ds f solid and fluid mass densities

s propagation velocity of a discontinuity

s0 .coefficient in a constitutive equation s ss fe e .s fe effective solid stress function

.t u .auxiliary function defined in 66u parameter in numerical method

v scalar angular velocity

v0 typical value of v

v angular velocityV potential of centrifugal force

Acknowledgements

We acknowledge support by the Collaborative Research Programme Sonder-

.forschungsbereich 404 at the University of Stuttgart and by the Fondef ProjectD97-I2042 at the University of Concepcion.

References

Anestis, G., 1981. Eine eindimensionale Theorie der Sedimentation in Absetzbehaltern veranderlichen Querschnitts und in Zentrifugen. Doctoral Thesis, Technical University of Vienna, Austria.

Anestis, G., Schneider, W., 1983. Application of the theory of kinematic waves to the centrifugation of

suspensions. Ing.-Arch. 53, 399407.

Auzerais, F.M., Jackson, R., Russel, W.B., 1988. The resolution of shocks and the effects of compressible

sediments in transient settling. J. Fluid Mech. 195, 437462.

Baron, G., Wajc, S., 1979. Behinderte Sedimentation in Zentrifugen. Chem. -Ing. -Techn. 51, 333.

Burger, R., Concha, F., 1998. Mathematical model and numerical simulation of the settling flocculatedsuspensions. Int. J. Multiphase Flow 24, 10051023.

Burger, R., Karlsen, K.H., 2000. A strongly degenerate convectiondiffusion problem modeling centrifugationof flocculated suspensions. Submitted for publication in the proceedings volume of the Eighth International

Conference on Hyperbolic Problems, Magdeburg, Germany, February 28March 3, 2000.

Burger, R., Karlsen, K.H., 2001. On some upwind difference schemes for the phenomenological sedimenta-tionconsolidation model. J. Eng. Math., to appear.

Burger, R., Kunik, M., 1999. On boudary conditions for multidimensional sedimentationconsolidation processes in closed vessels. Preprint No. 544, Weierstra Institute for Applied Analysis and Stochastics,

Berlin.

Burger, R., Tory, E.M., 2000. On upper rarefaction waves in batch settling. Powder Technol. 108, 7487.Burger, R., Bustos, M.C., Concha, F., 1999. Settling velocities of particulate systems: 9. Phenomenological

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Shirato, M., Kato, H., Kobayashi, K., Sakazaki, H., 1970. Analysis of settling of thick slurries due to

consolidation. J. Chem. Eng. Japan 3, 98104.

Ungarish, M., 1993. Hydrodynamics of Suspensions. Springer Verlag, Berlin.

Wakeman, R.J., Tarleton, E.S., 1999. Filtration: Equipment Selection, Modelling and Process Simulation.

Elsevier Science, Oxford, UK.

Wiesmann, U., Binder, H., 1982. Biomass separation from liquids by sedimentation and centrifugation. Adv.

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