Settling velocities of particulate systems: 12

Int. J. Miner. Process. 63 Ž2001. 115–145 www.elsevier.comrlocaterijminpro Settling velocities of particulate systems: 12. Batch centrifugation of flocculated suspensions a,) R. Burger , F. Concha b ¨ a Institute of Mathematics A, UniÕersity of Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany b Department of Metallurgical Engineering, UniÕersity of Concepcion, ´ Casilla 53-C, Correo 3, Concepcion, ´ Chile Received 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 sedimentation–consolidation 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 constitutive assumptions 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 Žnon-flocculated. suspensions. q 2001 Elsevier Science B.V. All rights reserved. Keywords: phenomenological theory; sedimentation–consolidation process; flocculated suspension; centrifugation 1. Introduction The enhanced body force obtained by the rotation of a solid–fluid mixture in a centrifuge or hydrocyclone permits the solid–fluid 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: buerger@mathematik.uni-stuttgart.de ŽR. Burger ¨ ., fconcha@udec.cl ŽF. Concha.. 0301-7516r01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 7 5 1 6 Ž 0 1 . 0 0 0 3 8 - 2 116 R. Burger, F. Conchar Int. J. Miner. Process. 63 (2001) 115–145 ¨ 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 mineral 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 ŽBaron and Wajc 1979; Anestis, 1981; Anestis and Schneider, 1983; Greenspan, 1983; Schaflinger, 1990., the most comprehensive publication the field ŽLeung, 1998. uses ad-hoc formulations and no general phenomenological theory seems to have been presented for flocculated suspensions. We mention that overviews of the use centrifuges are also given by Day Ž1974. and in the recent handbooks by Wakeman and Tarleton Ž1999. and Rushton et al. Ž2000., and that centrifuges are of particular interest in biotechnical and medical applications, in which small solid–fluid density differences make centrifugal enhancement of hindered settling mandatory ŽWiesmann and Binder, 1982; Lueptow and Hubler, 1991.. ¨ In this paper, we present a phenomenological theory of centrifugal separation of flocculated 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 Žmm. Equipment Fcentrifugal r Fgravitational 100 38 5 3 1 Gravity thickener-clarifier Large hydrocyclone Small hydrocyclone, low- speed centrifuge Industrial centrifuge Small high-speed centrifuge 0 20 200 2000 20 000 R. Burger, F. Conchar Int. J. Miner. Process. 63 (2001) 115–145 ¨ 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 hyperbolic–parabolic 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 ŽBurger ¨ and Concha, 1998; Burger et al., 1999, 2000e. and in the monograph by Bustos et al. ¨ Ž1999., 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 v s v k, where v is the scalar angular speed and k Žwith 5k 5 s 1. is the unit vector of its axis of rotation. Fig. 1 shows two possible cases for this system. The first ŽFig. 1a. consists of a tube rotating around an axis, such as for a small laboratory centrifuge, and the second ŽFig. 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 Ef q = P Ž f vs . s 0, Et = P q s 0, Ž 1. Ž 2. where f denotes the local solids volume fraction, t is time, vs is the solids phase velocity, and q is the volume average flow velocity of the mixture defined by q s f vs q Ž 1 y f . vf , where vf is the fluid phase velocity. Ž 3. R. Burger, F. Conchar Int. J. Miner. Process. 63 (2001) 115–145 ¨ 118 Fig. 1. Ža. Rotating tube with constant cross-section Žg s 0., Žb. rotating axisymmetric cylinder Žg s1.. The concentration zones are the clear liquid Ž f s 0., the hindered settling zone Ž0 - f F fc . and the compression zone f ) fc . 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 Ds Df ž ž Evs Et Evf Et / / q = vs P vs s = P Ts q Ds b s q m, Ž 4. q = vf P vf s = P Tf q D f b f y m, Ž 5. where Ds , D f , Ts and Tf are the constant mass densities and the stress tensors of the solid and the fluid, respectively, m is the solid–fluid interaction force, and b s and b f are the external body forces that, in a rotating frame of reference, take the respective forms b s s yg k y v = v = r y 2 v = vs , b f s yg k y v = v = r y 2 v = vf , Ž 6. 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 accelerations, both product of the moving frame of reference. The body forces b s and b f can be separated into two parts, a conservative force = ŽF q V ., where F and V are given by F s yg P r ' gz , 1 1 V s y Ž v = r. P Ž v = r. s y v 2 r 2 , 2 2 Ž 7. R. Burger, F. Conchar Int. J. Miner. Process. 63 (2001) 115–145 ¨ 119 and the non-conservative parts y2v = vs and y2v = vf , so that b s and b f can be written in the forms b s s y= Ž F q V . y 2 v = vs , b f s y= Ž F q V . y 2 v = vf . Ž 8. The ratio between the representative centrifugal and gravity components of b is expressed by the Froude number of the system ŽUngarish, 1993., F s v 2 Rrg, where R is a typical distance to the axis of rotation Žfor example, the outer radius of the container.. The two limiting cases are here F s 0, corresponding to gravity settling, and F s ` for a centrifugally dominated configuration. 2.2. ConstitutiÕe assumptions 2.2.1. Solid and fluid stress tensors The stress tensors of the components are assumed to take the forms Ts s ypf I q TsE , Tf s yps I q TfE , Ž 9. TsE TfE where ps and pf are the solid and fluid phase pressures and and are the viscous or extra stress tensors. A detailed discussion of the possible forms of TsE and TfE is provided by Burger et al. Ž2000e.. ¨ The theoretical variables ps and pf are replaced by two experimental quantities, the pore pressure p and the effective solid stress se . The experimental variables are those parts 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 Ž 10 . pt s ps q pf s p q se . Burger et al. Ž2000e. Žsee also Concha et al., 1996 and Bustos et al., 1999. show that the ¨ assumption that the local surface porosity of a cross section of the network equals the volume porosity f leads to the equations pf s Ž 1 y f . p, Ž 11 . ps s f p q se . 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 pe , which is the pore pressure less the static pressure ŽUngarish, 1993., ž pe :s p q D f Ž F q V . s p q D f gz y 1 2 / v2r2 . Ž 12 . As in our previous papers in this series ŽBurger et al., 1999; Garrido et al., 2000., se ¨ is given as a constitutive function se s se Ž f ., which depends on the local solids volume fraction only and satisfies se Ž f . ½ s0 )0 ½ d se Ž f . s 0 for f F fc , s X Ž f . :s for f ) fc , e )0 df for f F fc , for f ) fc . Ž 13 . R. Burger, F. Conchar Int. J. Miner. Process. 63 (2001) 115–145 ¨ 120 We do not require se to be continuous at f s fc . A typical constitutive equation is the three-parameter function frequently referred to as ‘power law’ Žsee e.g. Landman and White, 1994. se Ž f . s ½ for f F fc , 0 k s 0 Ž Ž frfc . y 1 . for f ) fc , s 0 ) 0, k ) 1. Ž 14 . 2.2.2. Solid–fluid interaction force We assume that the solid–fluid interaction force m is given by a constitutive equation linear in the concentration and the relative solid–fluid velocity: m s ya Ž f . vr q b=f , Ž 15 . where a Ž f . is the resistance coefficient, corresponding to the second constitutive function describing the material behaviour of the mixture. Inserting Eq. Ž8. and the constitutive assumptions Eqs. Ž9., Ž11. and Ž15. into the momentum balances Eqs. Ž4. and Ž5. yields Ds ž Evs Et / q = vs P vs s y2 Ds v Ž k = vs . q Ds f Ž yg k q v 2 r . y a Ž f . vr q b=f y f= p y p=f y = Ž se Ž f . . q = P TsE , Df ž Evf Et Ž 16 . / q = vf P vf s y2 D f v Ž k = vf . q D f Ž 1 y f . Ž yg k q v 2 r . q a Ž f . vr y b=f q p=f y Ž 1 y f . = p q = P TfE . Ž 17 . Considering Eq. Ž17. at equilibrium, that is setting vs s vf ' 0, vr ' 0 and since the pore pressure equals the hydrostatic, 1 = p s D f yg k q v 2 r , 2 ž / Ž 18 . and introducing Eq. Ž18. into Eq. Ž17. we conclude that b s p. Then, from Eqs. Ž16. and Ž17. the momentum balances are: Ds ž Evs Et / q = vs P vs s y2 Ds v Ž k = vs . q Ds f Ž yg k q v 2 r . y a Ž f . vr y f= p y = Ž se Ž f . . q = P TsE , Df ž Evf Et Ž 19 . / q = vf P vf s y2 D f v Ž k = vf . q D f Ž 1 y f . Ž yg k q v 2 r . q a Ž f . vr y Ž 1 y f . = p q = P TfE . Ž 20 . R. Burger, F. Conchar Int. J. Miner. Process. 63 (2001) 115–145 ¨ 121 2.3. Dimensional analysis Considerable simplification of the four field equations Eqs. Ž1., Ž2., Ž19. and Ž20. can be achieved by an order-of-magnitude study. Consider a typical length scale L 0 , a typical velocity U0 , a typical scalar angular velocity v 0 , a kinematic viscosity n 0 and assume that Ds is a typical density. Velocities are here referred to a rotating reference frame with angular velocity v 0 . A star will denote the dimensionless analogue of each variable. Moreover, t 0 s L0rU0 will be chosen as time scale for dimensionless time derivatives. Introducing the characteristic parameters in Eqs. Ž19. and Ž20., we obtain the following equations in terms of the dimensionless variables: Ds Df Fr ž Evs) Et ) sy q = ) vs) P vs) 2 Ds Fr D f Ro / v ) Ž k = vs) . q Ds Df y a ) Ž f . vr) y f= ) p ) y = Fr ž Evf) Et ) sy q = ) vf) P vf) 2 Fr Ro 2 f Ž yk q F Ž v ) . r ) . ) Ž se ) Ž f . . Fr Re ) = ) P Ž TsE . , Ž 21 . / 2 v ) Ž k = vf) . q Ž 1 y f . Ž yk q F Ž v ) . r ) . q a ) Ž f . vr) y Ž 1 y f . = ) p ) q Fr Re ) = ) P Ž TfE . . Ž 22 . In these equations, Fr [ U02rŽ gL. is the Froude number of the flow, Ro [ U0rŽ v 0 L0 . is the Rossby number and Re s L0 U0rn 0 is the Reynolds number of the flow. Typical numerical values for the constants and the characteristic parameters are g s 10 mrs 2 L0 s 0.1 m U0 s 10y4 mrs n 0 s 10y6 m2rs Žaccelaration of gravity., Žtypical size of sedimenting space in a centrifuge., Žsettling velocity of a particle., Žkinematic viscosity of water., from which we obtain Fr s 10y8 and FrrRe s 10y7 . The value of v 0 and thus those of F 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 the linear momentum balances, i.e. those terms which bear the coefficients Fr and FrrRe, respectively. Noting that we may express the phase velocities vs and vf in terms of vr s vs and q as vs s q q Ž1 y f .vr and vf s q y f vr , we obtain after rearranging and R. Burger, F. Conchar Int. J. Miner. Process. 63 (2001) 115–145 ¨ 122 inserting Eq. Ž22. into Eq. Ž21. explicit equations for the relative solid–fluid velocity and for the excess pore pressure, respectively: vr) s 1yf a)Žf. = ž fy 2 D D )f Ž yk q F Ž v ) . r ) . y = Ds Df / ž Ž 1 y f . Ž k = q) . y f 2 y ) Ž se ) Ž f . . Ds Df 2 q a)Žf. 1yf vr) y Fr 2 v ) Ro 1 y f 2v) Ro a ) Ž f . / Ž 1 y f . Ž k = vr) . q O Ž Fr Ž 1 q 1rRe . . , = ) pe) s Fr Ž 23 . Ž Ž k = q ) . y f Ž k = vf) . . q O Ž Fr Ž 1 q 1rRe . . . Ž 24 . The assumed numerical values which are independent of the applied angular velocity v provide a rationale for assuming that the O Ž Fr Ž1 q 1rRe .. terms are negligible. Before further reducing the momentum balances, we briefly discuss some properties of the system of equations formed by the continuity Eq. Ž1., which by using the definition of the slip velocity vr takes the form Ef Et q = P Ž f q q f Ž 1 y f . vr Ž f ,=f ,q . . s 0, Ž 25 . the condition = P q s 0, and the dimensional analogues of Eqs. Ž23. and Ž24. after deleting the O Ž Fr Ž1 q 1rRe .. terms. These equations provide a complete system of five scalar equations for the scalar quantities f and pe and the three components of the volume-average velocity q. This means 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. Ž24. Žwithout the Coriolis terms, and deleting these also in Eq. Ž23.. 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 ŽBurger et al., 2000e. or by modeling the coupling by boundary conditions ¨ ŽSchneider, 1982.. Implications of that coupling are discussed by Burger and Kunik ¨ Ž1999.. 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. F ) 1, such that it is reasonable to neglect the gravity terms in Eq. Ž23.; on the other hand, 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 F v F 1000 radrs, corresponding to a range between about 1000 and 10 000 rpm, R. Burger, F. Conchar Int. J. Miner. Process. 63 (2001) 115–145 ¨ 123 and R s L 0 s 0.1 m. We then have FrrRos 10y5 , F s 100 for v s 100 radrs and FrrRos 10y4 , F s 10 4 for v s 1000 radrs, respectively. This discussion is, of course, 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. Ž23. and Ž24.: 1yf vr) s a)Žf. = ) pe) s 2 D D )f Ž yk q F Ž v ) . r ) . y = a)Žf. 1yf ) Ž se ) Ž f . . , Ž 26 . Ž 27 . vr) . As in our previous work ŽBurger and Concha, 1998; Burger et al., 2000e., we replace ¨ ¨ the resistance coefficient a Ž f . by the corresponding Kynch batch flux density function f bk Ž f . :s y D D gf 2 Ž 1 y f . 2 , aŽf. Ž 28 . D D :s Ds y D f . We substitute this function into Eqs. Ž26. and Ž27. and return to a dimensional form. Then vr and pe are given by vr s f bk Ž f . D D gf 2 Ž 1 y f . yD Dfv 2 r q = Ž se Ž f . . , Ž 29 . = pe s D Dfv 2 r y = Ž se Ž f . . . Ž 30 . Due to the neglection of gravity, the equation for the solid–fluid relative velocity, Eq. Ž29., only uses the radius vector. Considering axisymmetric solutions leaves the radius r as unique space variable. The scalar versions of Eqs. Ž29. and Ž30. are Õr s E pe Er f bk Ž f . 2 D D gf Ž 1 y f . s D Dfv 2 r y yD Dfv 2 r q Ese Ž f . Er Ese Ž f . Er Ž 31 . , Ž 32 . . There are now two cases possible: Ža. Flow in a rotating tube with constant cross section ŽFig. 1a.: Eqs. Ž1. and Ž2. reduce to Ef Et Eq Er q E Er Ž f q q f Ž 1 y f . Õr . s 0, Ž 33 . Ž 34 . s 0. Since we consider only tubes, which are closed at their outward-pointing end during rotation, we obtain q ' 0. Inserting Eq. Ž29. into Eq. Ž33. yields the field equation Ef Et q E Er ž y f bk Ž f . v 2 r g / ž s E Er y f bk Ž f . seX Ž f . Ef D D gf Er / Ž 35 . R. Burger, F. Conchar Int. J. Miner. Process. 63 (2001) 115–145 ¨ 124 for f . Defining a Ž f . :s y f bk Ž f . seX Ž f . D D gf A Ž f . :s ; f H0 aŽ s . d s, Ž 36 . we may rewrite Eq. Ž35. as Ef Et q E Er ž y f bk Ž f . v 2 r g / s E2 Er 2 AŽ f . . Ž 37 . Žb. Flow in a batch cylindrical centrifuge ŽFig. 1b.: The governing equation is now Ef Et q 1 E Ž r Ž f q q f Ž 1 y f . Õr . . s 0, r Er Ž 38 . which we may rewrite in view of q ' 0 and using aŽ f . and AŽ f . as defined above as Ef Et q ž 1 E r Er y f bk Ž f . v 2 r 2 g / s 1 E ž r Er r E AŽ f . Er / Ž 39 . . Relating the centrifugal Kynch batch settling function f ck to the conventional gravity function f bk by fck Ž f .:s yf bk Ž f .rg and defining the parameter g s 0 for the rotating tube and g s 1 for the axisymmetric case, we obtain the partial differential equation Ef Et q 1 E r g 1 E Ž f Ž f . v 2 r 1q g . s r g Er ck Er ž rg E AŽ f . Er / Ž 40 . , which can be rewritten as Ef Et q E Er ž 2 f ck Ž f . v r y g AŽ f . r / s E2 Er 2 ž A Ž f . q g yfck Ž f . v 2 q AŽ f . r2 / . Ž 41 . Since aŽ f . ½ s0 )0 for 0 F f F fc and f s fmax , for fc - f - fmax , Ž 42 . it becomes evident that the governing equation for the centrifugation problem, Eq. Ž40. or Eq. Ž41., is a second order strongly degenerate partial differential equation Žwith source terms in the case g s 1.. It is of the first order hyperbolic type for 0 F f F fc , 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 ) fc . The first-order equation is equivalent Ž1991.. to that investigated by Anestis and Schneider Ž1983. and Lueptow and Hubler ¨ The location of the type-change interface, where f s fc is valid, is not known a priori. This 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 Ž2000. for details. ¨ R. Burger, F. Conchar Int. J. Miner. Process. 63 (2001) 115–145 ¨ 125 In both the rotating tube and cylindrical centrifuge cases, the equation E pe Er s D Dfv 2 r y Ese Ž f . Ž 43 . Er can be used to calculate the excess pore pressure pe a posteriori from the concentration distribution. 2.5. Initial and boundary conditions We finally have to specify initial and boundary conditions. We assume that the variable r varies between an inner radius R 0 and the outer radius R and assume that Õs s 0 at both r s R 0 and r s R. This implies the boundary conditions f ck Ž f . v 2 R 0 q f ck Ž f . v 2 R q E AŽ f . Er E AŽ f . Er Ž R 0 ,t . s 0, t ) 0, Ž 44 . Ž R ,t . s 0, t ) 0. Ž 45 . The initial condition is f Ž r ,0 . s f 0 Ž r . , Ž 46 . R 0 F r F R. For simplicity, we limit ourselves in this paper to the case that the initial concentration f 0 is constant. 3. Numerical algorithm To solve the initial-boundary value problem given by Eq. Ž41. together with the initial and boundary conditions Eqs. Ž44. – Ž46. numerically, we employ a modification of the generalized upwind finite difference method. This method has been presented in detail by Burger and Karlsen Ž2001. for the case of a pure gravity field Žsee also Burger ¨ ¨ et al., 2000d., and shall be outlined only briefly here. Let J, N g N, D r [ Ž R y R 0 .rJ, D t [ TrN, r j [ R 0 q jD r, j s 1r2,1,3r2, . . . , J y 1r2, J and f jn f f Ž r j , nD t .. The computation starts by setting f j0 [ f 0 Ž r j . for j s 0, . . . , J. Assume then that the solution values f jn, j s 0, . . . , J have been calculated for the time level t n [ nD t. To compute the values f jnq 1 , we first compute the extrapolated values f jL :s f jn y Dr 2 s jn , f jR :s f jn q Dr 2 s jn , j s 1, . . . , J y 1, Ž 47 . R. Burger, F. Conchar Int. J. Miner. Process. 63 (2001) 115–145 ¨ 126 where the slopes s jn can be calculated, for example, by the minmod limiter function M ŽP,P,P . in the following way: ž s jn s MM u u g w 0,2 x , n n n n y f jy1 y f jn f jn y f jy1 f jq1 f jq1 , ,u , Dr 2D r Dr / Ž 48 . j s 2, . . . , J y 2, n s0n s s1n s s Jy1 s s Jn s 0, Ž 49 . °min a,b,c4 ¢0 if a,b,c ) 0, if a,b,c - 0, . otherwise MM Ž a,b,c . :s~max  a,b,c 4 Ž 50 . The extrapolated values f jL and f jR appear as arguments of the numerical centrifugal Kynch flux density function f ckEO ŽP,P . which, according to the Engquist–Osher scheme ŽEngquist and Osher, 1981., is defined by q f ckEO Ž u,Õ . :s f ck Ž u . q fcky Ž Õ . , q f ck Ž u . :s fck Ž 0 . q u H0 max  f X ck Ž 51 . Õ Ž s . ,0 4 d s, y f ck Ž Õ . :s H0 min  f X ck Ž s . ,0 4 d s. Ž 52 . The interior scheme, which approximates the field Eq. Ž40. and from which the interior n approximate solution values f 1n, . . . , f Jy1 are calculated, can then be formulated as f jnq 1 s f jn y q v 2D t r jgD r Dt r jgD r 2 L 1q g R 1q g r jq1r2 f ckEO Ž f jR , f jq1 f ckEO Ž f jy1 , f jL . . y r jy1r2 g n g n r jq1r2 . y A Ž f jn . . y r jy1r2 .. , Ž A Ž f jq1 Ž A Ž f jn . y A Ž f jy1 Ž 53 . j s 1, . . . , J y 1, where the function AŽP. was defined in Eq. Ž36.. The update formulas for the boundary values f 0n and f Jn follow from formula Eq. Ž53. for j s 0 and j s J by inserting the discrete analogues of the boundary conditions Eqs. Ž44. and Ž45., respectively. Moreover, 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 f 0n s f 0ny1 y f Jn s f Jny1 q y v 2D t r 1q g f EO R g0 D r 1r2 ck v 2D t R gD r Dt R gD r 2 Ž f 0n , f 1n . q Dt R g0 D r 2 g r 1r2 Ž A Ž f 1n . y A Ž f 0n . . , Ž 54 . n 1q g r Jy1r2 f ckEO Ž f Jy1 , f Jn . g n n r Jy 1r2 Ž A Ž f J . y A Ž f Jy1 . . . Ž 55 . R. Burger, F. Conchar Int. J. Miner. Process. 63 (2001) 115–145 ¨ 127 To ensure convergence of the numerical scheme to the entropy weak solution of the initial-boundary value problem Eqs. Ž41., Ž44. – Ž46., the CFL stability condition R v 2max< fckX Ž f . < f Dt Dr q 2 max a Ž f . f Dt Dr2 Ž 56 . F1 must be satisfied. In this work, this condition was ensured by selecting D r freely in every example and determining D t from Dts 0.9D r . 2 2 v u` q max a Ž f . Dr f R R0 Ž 57 . The parameter u s 1 was chosen in all examples, and the accuracy was J s 200, except for one case where a high resolution solution was computed with J s 1000. 4. Numerical examples 4.1. Kinematic model of centrifugation of ideal suspensions Anestis and Schneider Ž1983. studied the centrifugation of ideal suspensions in the framework of the theory of kinematic waves which, for suspension flows, is equivalent to Kynch’s theory ŽKynch, 1952.. Their model equation is obtained from Eq. Ž41. if we set se ' 0 and thus A ' 0: Ef Et q E Er Ž fck Ž f . v 2 r . s g Ž yfck Ž f . v 2 . . Ž 58 . To establish the relation to the phenomenological theory including compression effects, we briefly recall some properties of Eq. Ž58. Žsee Anestis, 1981; Anestis and Schneider, 1983; Lueptow and Hubler, 1991 for details.. To this end, we introduce new dimension¨ less variables t ) , r ) and f ck) by t ) [ trT0 , r ) [ rrR and f ck) Ž f . [ f ck Ž f .ru` , X Ž0., T0 [ 1rŽ v 2 u` ., and define R 0) [ R 0rR and T ) [ TrT0 . In these where u` [ f ck dimensionless variables, Eq. Ž58. reads Ef Et ) q E Er ) Ž fck) Ž f . r ) . s yg fck) Ž f . , R 0) F r ) F 1, 0 F t ) F T ) . Ž 59 . We immediately see that solutions of Eq. Ž59. do not depend on R or w. These solutions can be constructed using the well-known method of characteristics: the general solution is given in parametric form by 1 r ) s C1 Ž j . Ž fck) Ž f . . y 1q g , t) sy 1 f H 1qg f 0 du f ck) Ž u. q C2 Ž j . , Ž 60 . where the parameter j is constant along characteristics and the functions C1Ž j . and C2 Ž j . and the lower boundary of integration, f 0 , have to be determined from initial and boundary conditions. Note that, unlike the pure gravity case ŽBustos and Concha, 1988; R. Burger, F. Conchar Int. J. Miner. Process. 63 (2001) 115–145 ¨ 128 Ž1991., Concha and Bustos, 1991. and as is illustrated in Fig. 6 by Lueptow and Hubler ¨ the solution f is not constant along characteristics. It is well known that intersections of characteristics cause solutions of Eq. Ž59. to be discontinuous in general, and that the propagation speed s ) of a discontinuity S at a point Ž r 0) , t 0) . g S in the Ž r ) , t ) .-plane, which separates two concentration values fy and fq is given by the Rankine–Hugoniot condition s )s dr) dt) S r 0) Ž f ck) Ž fq . y f ck) Ž fy . . s fqy fy Ž 61 . . By the change of variables r ) s r ) Ž j , f . and t ) s t ) Ž j , f ., it is not difficult to derive from Eqs. Ž60. and Ž61. the ordinary differential equation Ž r Ž j ,f . q ) dj df f ck) q y Ž f . y r Ž j ,f . ) fck) Žf .. y sy Ž r ) Ž j , fq . fck) Ž fq . y r ) Ž j , fy . fck) Ž fy . . q f s f or f s f y Et ) Ef Et ) Ej q y q Ž f yf . y Ž fqy fy . Er ) Ef , Er ) Ej Ž 62 . describing the shock front in the Ž j , f .-plane. This front can be transformed into the Ž r ) , t ) .-plane by Eq. Ž60.. ½ & m u f 1yf . , f bk Ž f . s ` Ž 0 for 0 - f - fmax :s 0.66, otherwise. Ž 63 . We now construct the exact solution of a simple case of Anestis and Schneider Ž1983., to compare this result later with numerical solutions of the phenomenological model. To be specific, we take the Kynch batch flux density function ŽRichardson and & & Zaki, 1954. where u` s y u` rg with u` s y0.0001 mrs and m s 5. The constant initial datum, f 0 s 0.07, has been chosen in such a way that the chords joining the point Ž f 0 , f bk Ž f 0 .. with the points Ž0, 0. and Ž fmax , 0. respectively, both lie above the graph of f bk in an f bk Ž f . versus f plot, according to a the case Ia of Anestis and Schneider Ž1983. for centrifugation and that of a mode of sedimentation MS-1 by Bustos et al. Ž1999. and Burger and Tory Ž2000.. ¨ In this case, two kinematic shocks will form: a shock S 1 separating the suspension from the clear liquid zone Ž f s 0. which is forming on the inner wall and a shock S 2 separating the suspension from the sediment of concentration f s fmax . These shocks will meet at the critical time t c) and merge into a third stationary shock S 3 . Unlike the gravity case, the concentration of the bulk suspension between S 1 and S 2 is not constant for 0 F t ) - t c) . In fact, by using the initial condition f Ž r ) , 0. s f 0 , we obtain from Eq. Ž60. 1 r ) s j Ž f ck) Ž f 0 . rfck) Ž f . . t) sy 1 f H 1qg f 0 du f ck) Ž u. . 1q g , Ž 64 . Ž 65 . R. Burger, F. Conchar Int. J. Miner. Process. 63 (2001) 115–145 ¨ 129 From the equation for t ) which, for flux density functions given by Eq. Ž63. with u` - 0 and integers n, takes the explicit form t ) s t Ž f . y t Ž f0 . , t Ž u . :s 1 ln 1qg ž 1yu u / ny1 yÝ js1 u ny1 1 j j 1yu ž / ž / j , Ž 66 . 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 S 1 and S 2 , we integrate Eq. Ž62. to obtain r ) s R 0) Ž f 0rf . 1 1 1q g 1q g for S1 , r ) s Ž Ž fmax y f 0 . r Ž fmax y f . . for S2 , Ž 67 . where f is a parameter and t ) is given by Eq. Ž65.. The parameter f runs from f 0 to the value f̂ s fmax 1 q Ž Ž fmaxrf 0 . y 1 . Ž R 0) . Ž 68 . y Ž1q g . obtained by interesting the shock curves S 1 and S 2 . In this example, we obtain f̂ s 0.015298 for g s 0 and s 0.0031174 for g s 1. The corresponding exact solutions are 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. Ž41. containing AŽ f . in comparison to those present in Eq. Ž58.. To this end, we define the dimensionless integrated diffusion coefficient A) Ž f .:s AŽ f .rAŽ fmax .. Note that the monotonicity of A implies that 0 F A) Ž f . F 1. Using the same dimesionless variables as before, we may rewrite Eq. Ž41. in dimensionless form as Ef Et ) q E Er ) s yg Ž fck) Ž f . r ) . f ck) Žf. q 1 Pe g ž A) Ž f . Ž r). 2 y E Er ) ž A) Ž f . r) // q E 2A ) Ž f . EŽ r ) . 2 , Ž 69 . 130 R. Burger, F. Conchar Int. J. Miner. Process. 63 (2001) 115–145 ¨ Fig. 2. Exact solution of a kinematic centrifugation problem. The lines S1 , S 2 and S 3 denote kinematic shocks; the vertical lines are iso-concentration lines corresponding to the annotated values. where the Peclet number Pe is defined by Pe s R 2v 2 u` rAŽ fmax ., in analogy to the ´ Peclet number introduced by Auzerais et al. Ž1988. 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 se of the material considered. Eq. Ž69. is again considered for R 0) F r ) F 1 and 0 F t ) F T ) . However, solutions of Eq. Ž69. 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. Ž59. depicted in Fig. 2 possible. In particular, for given functions AŽ f . and f ck Ž f . and a vessel of fixed outer radius R, we observe that 1rPe ™ 0 for v ™ `. This means that the solutions of Eq. Ž59., considered on the appropriately scaled time interval w0, T ) x, are the limit case obtained R. Burger, F. Conchar Int. J. Miner. Process. 63 (2001) 115–145 ¨ 131 from the phenomenological sedimentation–consolidation model when the angular velocity v approaches infinity. To illustrate this, we now include the compression terms in Eqs. Ž41. and Ž69. by choosing the effective solid stress function se Ž f . s ½ for f F fc :s 0.1, 0 9 5.7 Ž Ž frfc . y 1 . Pa for f ) fc , Ž 70 . corresponding to a calcium carbonate slurry ŽDamasceno et al., 1992; Burger et al., ¨ 2000b.. The gravity Kynch batch flux density function f bk Ž f . defined by Eq. Ž63. and the effective solid stress function se given by Eq. Ž70. are plotted in Fig. 3. We choose the outer radius R s 0.3 m, the corresponding inner radius R 0 s 0.06 m, and again the initial concentration f 0 s 0.07. Three different values of the angular velocity v are chosen in such a way that the centrifugal force R v 2 at the bottom of the vessel equals 100 = g, 1000 = g and 10 000 = g, respectively. The corresponding times T0 are 30, 3 and 0.3 s. Fig. 4 shows the numerical result for both the rotating tube Žg s 0. and the cylindrical vessel Žg s 1. as settling plots Žiso-concentration lines in an r vs. t plot., while Fig. 5 displays selected concentration profiles at different times for the rotating tube case, together with the concentration profiles at corresponding times of the exact solution for the case se ' 0 depicted in Fig. 2a.. Both Figs. 4 and 5 illustrate that the sedimentation–consolidation 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 supernate–suspension and the sediment–suspension interfaces have met. As v is increased, the compression zone becomes thinner, and the final Žmaximum. concentration increases, which is well visible in Fig. 5b–e. 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 f s 0.42 is present in Fig. 4e only. In the hindered Fig. 3. Gravity Kynch batch flux density function Žleft. and effective solid stress function Žright. used for comparison with the kinematic model. 132 R. Burger, F. Conchar Int. J. Miner. Process. 63 (2001) 115–145 ¨ Fig. 4. Numerical simulation of a sedimentation–consolidation process Ža, c, e. in a rotating tube and Žb, d, f. in a cylindrical vessel at angular velocities Ža, b. v s 57.184 radrs, Žc, d. 180.83 radrs and Že, f. 571.84 radrs. The concentration lines correspond to the annotated values. Fig. 4 Ž continued .. R. Burger, F. Conchar Int. J. Miner. Process. 63 (2001) 115–145 ¨ 133 R. Burger, F. Conchar Int. J. Miner. Process. 63 (2001) 115–145 ¨ Fig. 4 Ž continued .. 134 R. Burger, F. Conchar Int. J. Miner. Process. 63 (2001) 115–145 ¨ 135 Fig. 5. Concentration profiles in a rotating tube at t s t i s iPT0 r3, is 0, . . . ,6 and t 7 s 4T0 : Ža. exact solution of Eq. Ž59., Žb. to Žd.: numerical solutions of Eq. Ž41. with R 0 s 0.06 m, Rs 0.3 m and different values of v . settling zone, the numerical scheme accurately reproduces Žto within numerical errors. the exact solution of Fig. 2. 4.3. Comparison with results by Sambuichi et al. (1991) Sambuichi et al. Ž1991. published experimental results of the centrifugation of three different 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 bk Ž f . or f ck Ž f .. Moreover, compression data obtained by both the compression–permeability cell method and the settling method ŽShirato et al., 1970. determined a unique effective solid stress function seŽ f . for each material. R. Burger, F. Conchar Int. J. Miner. Process. 63 (2001) 115–145 ¨ 136 In this paper, we choose the published experimental data referring to the clay suspension. Sambuichi et al. Ž1991. approximated the measured gravity settling velocities u g for the clay experiments by three different connecting straight segments in a log u g versus log f plot, depending on whether f falls into an assumed dilute Ž0.02 F f F 0.056., intermediate Ž0.056 F f F 0.107. or a concentrated region Ž f G 0.107.. Consequently, we obtain for each of these segments u g Ž f . s b 1 f b 2 with suitable real constants b 1 and b 2 . Converting the function u g into f bk via f bk Ž f . s f u g Ž f . Žwhereby we take into account that the measured settling velocities are propagation velocities of the clear liquid-suspension interface., and smoothly connecting the first segment with the origin by a second-order parabola, we obtain the continuous, piecewise differentiable function Žthe precise representation has been cut here to five significant digits. °0 Ž 23.229f f bk for f F 0 or f G fmax :s 0.5, 2 y 2.1673f . = 10 y8 Ž f . s~y5.8558 = 10 y9 f mrs y10 y0.57139 f y1.3869 = 10 ¢y6.68 = 10 0.72715 y1 0 0.132 f mrs mrs for 0 - f F 0.02, for 0.02 - f F 0.056, for 0.056 - f F 0.107, for 0.107 - f - fmax . Ž 71 . The last expression in Eq. Ž71., corresponding to the concentrated segment, has been proposed by Sambuichi et al. Ž1991.. Since that expression does not assume the value zero, the flux function had to be cut at a maximum concentration fmax . This value has been chosen here as 0.5. The actual choice of this value in a reasonable range Žsay greater than 0.3. does not influence the result for sedimentation with compression, since the maximum concentration possible with compression essentially depends on the Fig. 6. Gravity Kynch batch flux density function of Žleft. and effective solid stress function Žright. used for comparison with experimental data by Sambuichi et al. Ž1991.. R. Burger, F. Conchar Int. J. Miner. Process. 63 (2001) 115–145 ¨ 137 Fig. 7. Comparison of clear liquid–suspension Žcircles. and suspension–sediment Žblack dots. interfaces, measured by Sambuichi et al. Ž1991. during centrifugation of a clay suspension using three different angular velocities, with numerical solution of the phenomenological model. R. Burger, F. Conchar Int. J. Miner. Process. 63 (2001) 115–145 ¨ 138 behavior of se on intervals of f that are close to the critical concentration fc , and only to a small degree on the value of fmax . However, this value needs to be well-defined for our purpose of a comparison with the purely kinematic model. The published constitutive equation ŽFig. 3 of Sambuichi et al., 1991., e s 0.86 y 0.74 = 10y3 p˜s , Ž 72 . where e s 1 y f is the porosity and p˜s is Sambuichi et al.’s compressive solid pressure measured in pascals, can be converted into the effective solid stress function se Ž f . s ½ for f F fc , 0 a 1 Ž f y fc . k for f ) fc , Ž 73 . where the constants have the values fc s 0.14, a 1 s 1.142 = 10 7 Pa and k s 3.509. The density difference for the material was r s 1600 kgrm3. The functions f bk Ž f . and se Ž f . defined by Eqs. Ž71. and Ž73., respectively, are plotted in Fig. 6. Fig. 7 shows numerical solutions of the phenomenological model calculated with these parameters and functions Eqs. Ž71. and Ž73.. The initial concentration, f 0 s 0.089, and the three different angular velocities v s 146.4 radrs, 167.76 radrs and 230.59 radrs have also been chosen according to Sambuichi et al. Ž1991.. The maximum time for each diagram was chosen as 25 000 P Ž vrŽ146.4 radrs.. 2 , 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 Ž f 0 , f bk Ž f 0 . can no longer be connected with the point Ž fmax , 0. by a straight line lying above the graph of f bk Žsee Fig. 6. implies that the bulk suspension is no longer separated from the sediment by a single kinematic shock. Rather, the sediment with f s fmax s 0.5 is separated from the initial concentra- Fig. 8. High accuracy reference solution without compression Ž A' 0. for the centrifugation experiment by Sambuichi et al. Ž1991. and the settling plots of Fig. 7. R. Burger, F. Conchar Int. J. Miner. Process. 63 (2001) 115–145 ¨ 139 tion f 0 s 0.089 by a contact discontinuity at f s 0.107, followed upwards by a centred rarefraction wave. The transition between f 0 s 0.089 and the time-dependent concentration 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 bk , which is marked by a circle in Fig. 9. The sedimentation process of that figure is one of Type II according to the classification of Anestis and Schneider Ž1983.. Due to the obvious difficulties related to measuring concentration profiles in a rotating system, Sambuichi et al. Ž1991. could only measure the propagation of the supernate–suspension and of the suspension–sediment interfaces. These experimental data are plotted in Fig. 7. The numerical solution, displayed with additional iso-concentration 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 Ž A' 0. for the centrifugation experiment by Sambuichi et al. Ž1991. and of numerical the simulation of centrifugation with compression with v s 230.59 radrs Žsee Fig. 7Ža... 140 R. Burger, F. Conchar Int. J. Miner. Process. 63 (2001) 115–145 ¨ 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 ŽSchaflinger 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 sedimentation–consolidation 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 lower and upper limit of the angular velocities possible with a given centrifuge. However, these assumptions are also inherent in the simpler kinematic treatments for centrifugation ideal suspensions ŽAnestis, 1981; Anestis and Schneider, 1983; Lueptow and Hubler, 1991., to which the phenomenological theory has been compared explicitly. As ¨ in the gravity case, this theory leads to one scalar hyperbolic–parabolic degenerate convection–diffusion 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 ŽBurger et al., 2000c; Burger and Karlsen, 2000.. In ¨ ¨ particular, it must be ensured that the numerical scheme applied to such an equation approximates the right discontinuous solution. However, this is the case with the presented modification Žrelated to the rotating frame of reference. of the generalized upwind scheme, which is computationally simple and correctly approximates the supernate–suspension and suspension–sediment interfaces without the necessity to track these explicitly. This fact sharply contrasts with the numerical solution procedure advanced by Sambuichi et al. Ž1991.: their algorithm is essentially based on alternately solving Eq. Ž58. in the hindered settling zone and the equation Žin our notation. EseŽ f .rEf s D f r v 2 , which is obtained from Eq. Ž31. by assuming that Õr is negligible in the compression zone. The appropriate supernatersuspension and suspensionrsediment interfaces are determined by a trial-and-error repetition of these solution procedures, 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 Õr in the compression zone, since our dimensional analysis does not provide justification to do so. As has become apparent in the example of comparison with the experimental data of Sambuichi et al. Ž1991., the current phenomenological formulation is limited to those cases where Coriolis terms are indeed negligible. However, retaining these terms in Eq. Ž23. and assuming that the flow variables depend on the radius only will again produce an explicit Žthough more complicated. equation for Õr , and a corresponding hyperbolic–parabolic partial differential equation for the volumetric solids concentration. For ideal suspensions, obtained by letting se ' 0 in our theory, such treatments R. Burger, F. Conchar Int. J. Miner. Process. 63 (2001) 115–145 ¨ 141 have already been performed by Schaflinger et al. Ž1986. and Schaflinger Ž1987.. An analogous 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. Ž1996., are in preparation. Finally, we mention that Burger et al. Ž2000a. derive a mathematical model of ¨ pressure filtration from the phenomenological theory of sedimentation–consolidation processes. 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 ŽSambuichi et al., 1987., to which we will come back in one of the next articles in this series. 6. List of symbols Variables that occur in both dimensional and dimensionless Žstarred. forms are listed here only in their dimensional version. Latin symbols diffusion coefficient aŽ f . AŽ f . integrated diffusion coefficient b body force per unit mass b s , b f external body forces Žnegative. potential of b B C1Ž j .,C2 Ž j . integration constants f bk Ž f . Kynch batch flux density function f ck Ž f . centrifugal Kynch batch flux density function f ckEO Ž u, Õ . numerical Kynch batch flux function F Froude number of the system Fr Froude number of the flow g acceleration of gravity g gravity force I identity tensor j space index J integer defining spatial discretization k exponent in a constitutive equation se s se Ž f . k upwards pointing unit vector typical length L0 m solid–fluid interaction force MM Ž a, b, c . Minmod limiter m Exponent of the Richardson–Zaki flux density function n time index N integer defining time step 142 R. Burger, F. Conchar Int. J. Miner. Process. 63 (2001) 115–145 ¨ p pore pressure excess pore pressure pe ps , pf solid and fluid phase pressures p̃s compressive solid pressure ŽSambuichi et al., 1991. total pressure pt q volume average flow velocity r radius value of r in numerical method rj r radius vector of the system R outer radius of the container R0 inner radius Žsuspension surface. Re Reynolds number of the flow Ro Rossby number slopes in numerical method s jn S , S 1 , S 2 , S 3 symbols denoting kinematic shocks t time time scale t0 critical time tc Ts , Tf solid and fluid stress tensors TsE , TfE solid fluid extra stress tensors u` coefficient of f ck Ž f . & u` coefficient of f bk Ž f . gravity settling rate ug typical velocity U0 vs , vf , Õs , Õf solid and fluid phase velocities solid–fluid relative velocity vr , Õr z height z axis of rotation Greek symbols a1 coefficient in equation for seŽ f . a Ž f . resistance coefficient b coefficient in the approach for m b 1 , b 2 parameters in the equation for u g Ž f . g parameter indicating rotating tube or axisymmetric case Dr space step of numerical method Dt time step of numerical method DD solid–fluid mass density difference f volumetric solids concentration f̂ value of f at intersection of kinematic shocks fy, fq approximate limits of f at a discontinuity initial concentration f0 critical concentration fc R. Burger, F. Conchar Int. J. Miner. Process. 63 (2001) 115–145 ¨ f jn f jL , f jR fmax F ms , m f n0 Ds , D f s s0 seŽ f . t Ž u. u v v0 v V 143 value of f in numerical method extrapolated numerical values of f maximum solids concentration potential of gravity force phase viscosities kinematic viscosity of water solid and fluid mass densities propagation velocity of a discontinuity coefficient in a constitutive equation se s seŽ f . effective solid stress function auxiliary function defined in Ž66. parameter in numerical method scalar angular velocity typical value of v angular velocity potential of centrifugal force Acknowledgements We acknowledge support by the Collaborative Research Programme ŽSonderforschungsbereich. 404 at the University of Stuttgart and by the Fondef Project D97-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, 399–407. 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, 437–462. 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 flocculated ¨ suspensions. Int. J. Multiphase Flow 24, 1005–1023. Burger, R., Karlsen, K.H., 2000. A strongly degenerate convection–diffusion problem modeling centrifugation ¨ of flocculated suspensions. Submitted for publication in the proceedings volume of the Eighth International Conference on Hyperbolic Problems, Magdeburg, Germany, February 28–March 3, 2000. Burger, R., Karlsen, K.H., 2001. On some upwind difference schemes for the phenomenological sedimenta¨ tion–consolidation model. J. Eng. Math., to appear. Burger, R., Kunik, M., 1999. On boudary conditions for multidimensional sedimentation–consolidation ¨ processes in closed vessels. Preprint No. 544, Weierstraß Institute for Applied Analysis and Stochastics, Berlin. Burger, R., Tory, E.M., 2000. On upper rarefaction waves in batch settling. Powder Technol. 108, 74–87. ¨ Burger, R., Bustos, M.C., Concha, F., 1999. Settling velocities of particulate systems: 9. Phenomenological ¨ 144 R. Burger, F. Conchar Int. J. Miner. Process. 63 (2001) 115–145 ¨ theory of sedimentation processes: numerical simulation of the transient behaviour of flocculated suspensions in an ideal batch or continuous thickener. Int. J. Mineral Process. 55, 267–282. Burger, R., Concha, F., Karlsen, K.H., 2000a. Phenomenological model of filtration processes: 1. Cake ¨ formation and expression. Preprint 2000r24, Sonderforschungsbereich 404, University of Stuttgart. Burger, R., Concha, F., Tiller, F.M., 2000b. Applications of the phenomenological theory to several published ¨ experimental cases of sedimentation processes. Chem. Eng. J. 80, 105–117. Burger, R., Evje, S., Karlsen, K.H., 2000c. On strongly degenerate convection–diffusion problems modeling ¨ sedimentation–consolidation processes. J. Math. Anal. Appl. 247, 517–556. Burger, R., Evje, S., Karlsen, K.H., Lie, K.-A., 2000d. Numerical methods for the simulation of the settling of ¨ flocculated suspensions. Chem. Eng. J. 80, 91–104. Burger, R., Wendland, W.L., Concha, F., 2000e. Model equations for gravitational sedimentation–consolida¨ tion processes. Z. Angrew. Math. Mech. 80, 79–92. Bustos, M.C., Concha, F., 1988. On the construction of global weak solutions in the Kynch theory of sedimentation. Math. Methods Appl. Sci. 10, 245–264. Bustos, M.C., Concha, F., Burger, R., Tory, E.M., 1999. Sedimentation and Thickening. 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