Int. J. Miner. Process. 55 Ž1999. 267–282
Settling velocities of particulate systems: 9. Phenomenological theory of sedimentation processes: numerical simulation of the transient behaviour of flocculated suspensions in an ideal batch or continuous thickener R. Burger ¨ b
c
a,)
, M.C. Bustos
b,1
, F. Concha
c,2
a Mathematisches Institut A, UniÕersity of Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany Department of Mathematical Engineering, UniÕersity of Concepcion, ´ Casilla 4009, Correo 3, Concepcion, ´ Chile Department of Metallurgical Engineering, UniÕersity of Concepcion, ´ Casilla 53-C, Correo 3, Concepcion, ´ Chile
Received 30 March 1998; revised 15 September 1998; accepted 4 November 1998
Abstract The transient behaviour of flocculated suspensions in an ideal thickener is simulated by numerical solution of the parabolic–hyperbolic equation of the phenomenological theory of sedimentation. The numerical results yield the expected sedimentation and consolidation behaviour for batch sedimentation and for the most important operations of continuous thickening: filling up, transition between steady states and emptying of a continuous thickener. q 1999 Elsevier Science B.V. All rights reserved. Keywords: ideal thickener; sedimentation; continuous thickener
1. Introduction In recent years, the authors contributed to the development of a general phenomenological theory of sedimentation of flocculated suspensions as two superimposed continu)
Corresponding author. Fax: q49-711-6855599; E-mail:
[email protected] Fax: q56-41-240280. 2 Fax: q56-41-230759; E-mail:
[email protected] 1
0301-7516r99r$ - see back matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 7 5 1 6 Ž 9 8 . 0 0 0 3 7 - 4
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ous media. For non-flocculated suspensions, this theory reduces to the classical purely kinematical sedimentation model by Kynch Ž1952. and its extensions to continuous thickening. The phenomenological model leads to a scalar conservation law, valid in the interior of the thickener, of hyperbolic type for hindered sedimentation Žcorresponding to Kynch’s theory. and of parabolic type for the compressible sediment layer. The location of the interface between both parts is in general unknown beforehand. This phenomenon constitutes the main mathematical difficulty of the model presented here. Initial and boundary conditions are formulated to describe the initial concentration and the feeding and discharge conditions. The object of this work is to present numerical transient solutions to the initial– boundary value problem of sedimentation processes, to compare the results with the work of others and to predict the behaviour of continuous thickeners when they are filled, emptied or their feeding and discharge conditions are changed. In Section 2, we briefly summarize the model describing the settling behaviour of a flocculated suspension in a continuous thickener, which results in the initial–boundary value problem mentioned above. In Section 3 we present numerical examples for transient batch and continuous sedimentation processes, and in Section 4, we discuss these results.
2. Mathematical model Consider a sedimentation process in a so-called Ideal Thickener ŽIT. ŽBustos et al., 1990a. as shown in Fig. 1. An IT is a cylindrical vessel showing no wall effects and in which all field variables are assumed to be constant on each cross section, so that they depend only on the variables height z and time t. The IT is an Ideal Continuous Thickener ŽICT. if at height z s L a surface source and at z s 0 a surface sink are provided for feeding and discharge in continuous
Fig. 1. Schematic representation of an Ideal Thickener.
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269
operation. An IT without feed and discharge is a settling column for batch sedimentation, which will be included as a special case. Furthermore, we assume that the flocs begin to touch each other at a critical volumetric solid concentration value fc , while they perform hindered settling for volumetric solid concentrations f F fc . The sedimentation of the flocculated suspension is described with the local mass and linear momentum balances for the solid and liquid component, respectively ŽConcha et al., 1996., for which additional kinematical and dynamical constitutive assumptions are introduced to consider specific choices of materials and flocculant, thickener design and manner of operation. Summing up, we require that the solid particles are small with respect to the sedimentation vessel and have the same density; that the solid and liquid components of the suspension are considered as incompressible elastic fluids and that there is no mass transfer between each other; the suspension is assumed to be entirely flocculated at the beginning of the sedimentation process; the solid component is allowed to perform a one-dimensional simple compression motion only and gravitation is the only body force. Since the settling velocity of a single solid floc in an unbounded medium y` and the height of the thickener L are of the orders of magnitude y` f 10y4 mrs and L f 1 m, respectively, the Froude number of the flow, defined by Fr [ y`2rŽ L P g ., is of the order of Fr f 10y9 < 1, which justifies neglecting the convective terms in the momentum balances. The result is the following field equation for the volumetric solid concentration f in the thickener as a function of height 0 F z F L and time 0 F t F T :
Ef Et
E q
Ez
E
Ž q Ž t . f q f bk Ž f . . s y
Ez
ž
f bk Ž f .
seX Ž f . Ef D D gf E z
/
.
Ž 1.
A similar equation was proposed by Bascur Ž1976. and Concha and Bascur Ž1977.. Here, q Ž t . F 0 is the volume-average velocity of the suspension, which can be prescribed by the discharge control, and D D ) 0 is the difference of solid and fluid mass densities. The choice of materials is reflected by the model functions f bk Ž f ., which denotes the Kynch batch flux density function and by seŽ f ., the effective solid stress, which have to be determined experimentally. We assume that f bk is a smooth function satisfying f bk Ž 0 . s f bk Ž 1 . s 0,
f bk Ž f . - 0 for 0 - f - 1.
Ž 2.
The effective solid stress is assumed to be constant for particles which do not touch each other, i.e., for f F fc , and to increase monotonically for fc - f . Hence, we assume that se satisfies
seX Ž f .
½
s0 )0
for for
f F fc , f ) fc .
Ž 3.
Combining the assumptions Ž2. and Ž3., we conclude that Eq. Ž1. is a second order quasilinear parabolic partial differential equation degenerating into first order hyperbolic type for the interval of solution values w0, fc x. The location of the change is solution-de-
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270
pendent and in general unknown beforehand. The hyperbolic equation corresponds to the sedimentation theory by Kynch Ž1952., whose extension to continuous sedimentation was studied thoroughly by Bustos and Concha Ž1988., Bustos et al. Ž1990a., Concha and Bustos Ž1991, 1992. and Bustos and Concha Ž1996.. Due to its nonlinearity, it is clear that solutions of Eq. Ž1. might develop discontinuities regardless of smooth initial and boundary data. The sedimentation process is determined by prescribing the initial concentration distribution,
f Ž z ,0 . s f 0 Ž z . ,
0 F z F L,
Ž 4.
a feeding flux density f F Ž t . s yQ F Ž t . f F Ž t .rS at z s L, where Q F Ž t . and f F Ž t . are the volume feed rate and the feed concentration, respectively, and S is the thickener cross-sectional area, or, equivalently, by prescribing the volume average velocity q Ž t . and the concentration f Ž L,t . defined by
f Ž L,t . s f 1 Ž t . ,
0-tFT ,
Ž 5.
both satisfying q Ž t . f 1 Ž t . q f bk Ž f 1 Ž t . . s f F Ž t . ,
0-tFT.
The boundary condition at z s 0 requires that the total solid volume flux reduces to its convective outflow part,
ž
q Ž t . f Ž 0,t . q f bk Ž f Ž 0,t . . 1 q
seX Ž f Ž 0,t . . Ef D D g f Ž 0,t . E z
/
Ž 0,t . s q Ž t . f Ž 0,t . ,
0-tFT , that is
ž
f bk Ž f . 1 q
seX Ž f . Ef D D gf E z
/
s 0,
0-tFT.
Ž 6.
zs 0
The initial–boundary value problem given by Eq. Ž1. and conditions Ž4. – Ž6. was analyzed mathematically by Burger and Wendland Ž1998a., where existence and unique¨ ness of generalized solutions under certain smoothness and compatibility assumptions on the data and coefficients of the problem is shown, and in ŽBurger and Wendland, ¨ 1998b., where discontinuity and entropy boundary conditions are derived. Of course, treating the type degeneracy of Eq. Ž1. adequately formed the main difficulty for the mathematical analysis. The characteristics along which the solution values of the hyperbolic conservation law are propagated may intersect the boundary of the computational domain from the interior. Therefore, the prescribed value f 1Ž t . at z s L is not always assumed exactly by the generalized solution and the boundary condition might be violated. This problem is treated by means of the entropy boundary condition which stipulates the admissible boundary values. More details on the mathematical analysis of the sedimentation process can be obtained in the works of Burger and Wendland Ž1998a,b.. ¨
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3. Numerical results Denote by aŽ f . the diffusion coefficient on the right hand side of Eq. Ž1.: a Ž f . s yf bk Ž f . seX Ž f . r Ž D Df g . , then the equation to be studied is the convection–diffusion equation Ef E E Ef q aŽ f . . Ž q Ž t . f q f bk Ž f . . s y Et Ez Ez Ez
ž
/
The approximate solution to the initial and boundary value problem is obtained by an operator splitting procedure. Eq. Ž1. is split into a second order diffusion equation, a linear convective equation and a nonlinear first order hyperbolic equation, which are solved numerically for each time step by an implicit finite difference method, a second order upwind method and a second order total-variation diminishing method, respectively. For details on these methods, see Burger and Concha Ž1997, 1998.. ¨ 3.1. Experimental data and parameters Consider a Kynch batch flux density function of the type proposed by Richardson and Zaki Ž1954., f bk Ž f . s y` P f P Ž 1 y f .
Cq 1
,
and choose for the effective solid stress se a function of the type
se Ž f . s
½
0
for
f F fc ,
a exp Ž bf .
for
f ) fc
Ž 7.
with parameters obtained from experimental measurements on Chilean copper ore tailings ŽBecker, 1982.:
y` s y6.05 = 10y4 mrs, C s 11.59, fc s 0.23, a s 5.35 Nrm2 , b s 17.9.
Ž 8.
The remaining constants are L s 6 m, D D s 1500 kgrm3 and g s 9.81 mrs 2 . The model for se leads to a discontinuous diffusion coefficient aŽ f .. In Burger and Concha ¨ Ž1997, 1998., calculations of sedimentation processes are presented in which this discontinuity is smoothed out by a cubic interpolation polynomial in order to satisfy the regularity assumptions of the mathematical analysis ŽBurger and Wendland, 1998a,b. of ¨ the initial–boundary value problem. However, the numerical results have turned out satisfactory for the discontinuous case as well, hence the jump at f s fc implied by Eq. Ž7. will not be smoothed out here. Fig. 2 shows the Kynch batch flux-density function f bk Ž f . together with the diffusion coefficent aŽ f .. 3.2. Example 1: batch settling of a uniform suspension Batch settling is interesting because it provides a simple method to test models by comparing experiments with simulations. Consider a suspension of initial concentration f 0 s 0.123. Fig. 3 shows the settling plot and Fig. 4 some selected concentration profiles for a simulated sedimentation time of T s 250 000 s f 69 h. The 400 discrete
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Fig. 2. Kynch batch flux density function f bk Ž f . and diffusion coefficient aŽ f . constructed with the parameters given in Eq. Ž8..
solution points obtained for the concentration profiles in Fig. 4 have been interpolated linearly. In Figs. 3, 5, 7 and 9, the small dots correspond to characteristic lines in the settling region and to lines of concentration in the sediment layer. The fat dots represent the line of critical concentration. 3.3. Example 2: comparison with experimental measurements To demonstrate the predictive power of the sedimentation model and the numerical method, the experimental data of Been and Sills Ž1981. will be compared with simulation of the sedimentation of a suspension of soft soil in a settling column of height L s 1.742 m with an initial concentration of f 0 s 0.05264. The experimental
Fig. 3. Settling plot for batch settling showing sedimentation with compression. The dotted lines correspond to f 1 s 0.123, f 2 s 0.14, f 3 s 0.17, f4 s 0.20, f5 s fc s 0.23 Žfat dots., f6 s 0.26, f 7 s 0.29, f 8 s 0.32, f 9 s 0.35, f 10 s 0.38 and f 11 s 0.41.
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273
Fig. 4. Concentration profiles for batch settling, sedimentation with compression. The concentration profiles shown correspond to t 0 s 0 s, t1 s 5000 s, t 2 s10 000 s, t 3 s15 000 s, t 4 s 20 000 s, t5 s 25 000 s, t6 s 50 000 s, t 7 s 75 000 s, t 8 s100 000 s and t 9 sT s 250 000 s.
data, which were presented graphically by Been and Sills, lead to the following constitutive equations: f bk Ž f . s
½
y1.39 = 10y4f Ž 1 y f .
28 .59
mrs
for
f F fc s 1r12,
y8.0 = 10 f exp Ž 0.7675 Ž 1 y f . rf . mrs
for
f ) fc
y9 2
and
se Ž f . s
½
0
for
f F fc ,
2
for
f ) fc .
4.0 exp Ž 21.265f . Nrm
The result is shown in Fig. 5 where Been and Sills’ settling plot is compared with our predictions. 3.4. Steady states in a continuous thickener As was shown elsewhere ŽConcha and Barrientos, 1993; Concha et al., 1996., the analysis of the steady state in a continuous thickener provides a method for thickener design. Furthermore, it provides the initial and end conditions of a transient state. Assume that to a continuous thickener a suspension of concentration f F Ž t . is fed at a rate f F Ž t . s yQ F Ž t . f F Ž t . rS where Q F Ž t . is the volume feed rate of pulp and S is the thickener cross-sectional area. The concentration f 1 at z s L Žsee Fig. 1. is obtained solving the algebraic equation ŽConcha et al., 1996. f F Ž t . s q Ž t . f 1 Ž t . q f bk Ž f 1 Ž t . . .
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Fig. 5. Comparison of the simulation Ža. with the experimental settling plot of Been and Sills Ž1981. Žb..
At the discharge the solid flux density is f D Ž t . s q Ž t . f Ž 0,t . . At steady state, the feed and discharge flux densities must be equal: fD sfF . The conditions for the existence of a steady state can be obtained by considering time-independent solutions of Eq. Ž1., i.e., of the ordinary differential equation d d f bk Ž f . seX Ž f . d f y Ž qf q f bk Ž f . . s dz dz D D gf dz
ž
/
with general solution qf q f bk Ž f . q
f bk Ž f . seX Ž f . d f
s C. D D gf dz Assuming that a desired discharge concentration is f D , from boundary condition Ž6. at z s 0 we conclude that the integration constant must be C s qf D . Thus, the concentration profile in the compression zone is the solution of the ordinary boundary value problem d D D gf s y X Ž qf q f bk Ž f . y qf D . , for z ) 0 df se Ž f . f bk Ž f . Ž 9. f Ž 0. s f D . Eq. Ž9. is integrated until the critical concentration fc is reached at height z s z c . For sedimentation to take place, d frd z F 0 and the condition qf q f bk Ž f . F qf D for fe w fc , f D x
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275
Fig. 6. The set of admissible discharge concentrations and the construction of the boundary datum f 1 for f D s 0.36 for f Ž f . [ qf q f bk Ž f . with q sy1.5=10y5 mrs and fc s 0.23.
must be satisfied, so that the right-hand side of Eq. Ž9. is nonpositive: if the flux function possesses a local maximum at f s f M , then the condition fc - f D - f D ma x , where f D ma x is obtained from qf M q f bk Ž f M . s qf D ma x , must be satisfied. In the case of a monotonically decreasing flux function, f D ma x is given by qfc q f bk Ž fc . s qf D ma x . The boundary value at z s L is obtained by solving for f 1 the equation qf 1 q f bk Ž f 1 . s qf D ma x ,
f1 - f M .
It is clear that f 1 is the constant value of the steady state concentration profile in the hindered settling zone z c - z F L. Fig. 6 shows an example for the construction of f D ma x and of f 1 for a given value of f D . Examples of steady states are given in Table 1.
Table 1 Parameters of the steady states for several values of q
Example 3 Example 4
i
q i 10y5 Žmrs.
f Di
f 1i
f Fi s f Di 10y5 Žmrs.
Approximate sediment height li Žm.
1 2 3 4
y1.5 y0.5 y1.5 y3.0
0.39 0.42 0.36 0.35
0.01077641 0.00360123 0.00983189 0.02139377
y0.585 y0.210 y0.540 y1.050
2.544 2.498 1.008 1.568
276
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3.5. Example 3: transient behaÕiour of a continuous thickener: filling and emptying of a continuous thickener Consider a continuous thickener full of water with a volume average velocity q Ž t . s q 1 [ y1.5 = 10y5 mrs. The thickener will first be filled, will attain a steady state with a discharge concentration of f D s 0.39 and will finally be emptied by manipulating the feed flux density f F , maintaining the volume average velocity q constant during the whole simulation. To fill up the thickener we choose f F s f F0 [ y1.756 = 10y5 mrs at t s 0 resulting in a concentration at z s L of f 1Ž t . s f 10 [ 0.06. In Fig. 7 we observe the formation of the sediment layer and the lines of constant concentration in both the suspension and the sediment. Fig. 8Ža. shows the concentration profiles in the thickener for several times during the filling interval. At t s 70 000 s f 19 h, the height of the sediment surpasses the value l1 s 2.544 m necessary to obtain a discharge concentration of f D s 0.39, therefore the feed flux density is adjusted to the corresponding steady state value f F1 s y0.585 = 10y5 mrs Žsee Table 1.. The sediment level decreases reaching the steady state at about t s 375 000 s f 104 h, with a concentration at z s L of f 11 s 0.01077641. After operating at steady state for about 35 h, the feeding flux is set to zero at t s 500 000 s, and the thickener
Fig. 7. Settling plot for filling and emptying an ICT. The dotted lines correspond to f 1 s 0.11, f 2 s 0.14, f 3 s 0.17, f4 s 0.20, f5 s fc s 0.23 Žfat dots., f6 s 0.26, f 7 s 0.29, f 8 s 0.32, f 9 s 0.35 and f 10 s 0.38. In the hindered settling zone, the rarefaction wave between f 0 s 0 and f 10 located between the characteristic lines s 0 and s 1 , along which these concentration values propagate, is shown. The lines s 2 and s 3 are the discontinuities Žshock lines. separating zones of concentrations f 10 and f 11 and f 11 and zero, respectively. Additionally, the changes of f F Ž t . and f 1Ž t . are shown.
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277
Fig. 8. Concentration profiles for continuous thickening: Ža. Filling up of an ICT from empty state. The concentration profiles shown correspond to t i s iP10 000 s, is 0,1,2, . . . ,10. Žb. Emptying of an ICT after steady state feeding and discharge conditions. The concentration profiles shown correspond to t11 s 500 000 s, t12 s 550 000 s, t13 s600 000 s, t14 s650 000 s, t 15 s 700 000 s and t 16 sT s 750 000 s.
starts to empty. Concentration profiles are shown in Fig. 8Žb. between t s 500 000 s and t s 750 000 s for the emptying period of the thickener. Note in Fig. 7 that at t s 0, a rarefaction wave between the characteristics belonging to the values zero and f 1Ž t . s 0.06 which travel at speeds X s 0 [ q 1 q f bk Ž 0 . s y6.2 = 10y4 mrs
and X s 1 [ q 1 q f bk Ž 0.06 . s y0.6952 = 10y4 mrs, respectively, starts to propagate from z s L into the thickener, and that the subsequent changes of f 1Ž t . propagate as discontinuities into the thickener with velocities given by the Rankine–Hugoniot condition
s2 [ q1 q
f bk Ž f 10 . y f bk Ž f 11 .
f 10 y f 11
s y2.3782 = 10y4 mrs
for the change between f 10 and f 11 and
s3 [ q1 q
f bk Ž f 11 .
f 11
s y5.4285 = 10y4 mrs
for the change between f 11 and zero, respectively. 3.6. Example 4: transient behaÕiour of a continuous thickener: transition between steady states In the fourth example, we consider the transition between the three steady states characterized by Ž q 2 , f F2 , f 12 , f D2 , l2 ., Ž q3 , f F3 , f 13, f D3 , l3 . and Ž q 4 , f F4 , f 14 , f D4 , l 4 ., respectively. Let F 2 , F 3 and F4 denote the corresponding steady state concentration
278
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Fig. 9. Settling plot for continuous thickening: transition between approximate steady states. The dotted lines correspond to f 1 s 0.11, f 2 s 0.14, f 3 s 0.17, f4 s 0.20, f 5 s fc s 0.23 Žfat dots., f6 s 0.26, f 7 s 0.29, f 8 s 0.32, f 9 s 0.35, f 10 s 0.38 and f 11 s 0.41. In the hindered settling zone, the discontinuities Žshock lines. s4 separating the concentration values f 12 and f 123 and s 9 separating f 134 and f 14 are plotted, as well as the rarefaction waves between the values f 123 and f 13 Žlocated between the characteristic lines s5 and s6 , along which f 123 and f 13 propagate. and between f 13 and f 134 Žanalogously located between s 7 and s 8 .. Note that s5 and s6 almost coincide. Additionally, the changes of f F Ž t ., q Ž t . and f 1Ž t . are shown.
profiles. The simulation starts from the second steady state given in Table 1. Thus f 0 Ž z . s f 2 Ž z . for 0 F z F L and f 1Ž t . s f 12 for 0 F t - 50 000 s. At t s 50 000 s we increase the discharge opening changing q Ž t . from q 2 to q 3 without changing f F2 . The concentration at z s L changes from f 12 to f 123 s 0.003537748, which is obtained solving q 3f 123 q f bk Ž f 123 . s f F2 .
Ž 10 .
Since the feed flux density yf F2 is less than the value of yf F3 needed to obtain a steady state Žsee Table 1., the sediment level starts to fall at t s 50 000 s. At about t s 190 000 s the feed flux density is changed from f F2 to f F3. The sediment level reaches the level l3 , that is the desired second steady state, at t f 200 000 s. At that time the concentration should assume the value f 13 above the sediment level to ensure a constant l3 . Taking into account that the change of the feeding flux from f F2 to f F3 causes a rarefaction wave between the values f 123 and f 13 propagating down the thickener, the speed at which the value f 13 propagates is X s4 [ q 2 q f bk Ž f 12 . s y4.9616 = 10y4 mrs,
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279
Fig. 10. Concentration profiles for continuous thickening: Ža. Transition between approximate steady states f 1 and f 2 . The concentration profiles correspond to t i s iP50 000 s, is 0,1, . . . ,6. Žb. Transition between approximate steady states f 2 and f 3. The concentration profiles correspond to t6 s 300 000 s, t 7 s 450 000 s, t 8 s 500 000 s and t 9 s600 000 s.
hence the feeding flux should have been changed to f F3 at t s 200 000 s q Ž L y l3 . rs3 f 189 940 s to maintain the level l3 of the sediment. We observe that the sediment level remains approximately at constant height and that the discharge concentration is approximately f D3 s 0.36, the desired one. At t s 400 000 s we proceed to the next steady state by changing the feeding flux density from f F3 to f F4 without touching q 3. Similarly to Eq. Ž10., the next boundary datum f 134 is calculated from q 3f 134 q f bk Ž f 134 . s f F4
Ž 11 .
f 134 s 0.022326177.
obtaining The sediment level starts to rise after the corresponding rarefaction wave has arrived. The next steady state sediment level, l4 , is reached approximately at t s 447 500 s. Then q Ž t . is changed from q 3 to q 4 which causes f 1Ž t . to change from f 134 to f 14 . The concentration lines become horizontal and the third steady state is approximated well by the numerical solution at t s T s 600 000 s, when the simulation ends. Fig. 9 shows the settling plot for the whole simulation and indicates the choices of f F Ž t ., f 1Ž t . and q Ž t .. Fig. 10 shows some concentration profiles for the transition between F 1 and F 2 and between F 2 and F 3 . 4. Discussion This paper continues our series on the Settling Õelocities of particulate systems in which the settling behaviour of ideal and flocculated suspensions has been studied. A phenomenological theory of sedimentation has evolved slowly from restricted ad-hoc
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theories proposed by numerous research workers starting with Kynch in 1952. During the last 15 years, the work of the group formed by researchers from the University of Stuttgart, in Germany, and the University of Concepcion, ´ in Chile, has put sedimentation processes under a solid and unified mathematical background. The special case of an ideal suspension was studied thoroughly and was published in this series and in specialized journals ŽBustos and Concha, 1988; Bustos et al., 1990a,b; Concha and Bustos, 1991, 1992; Bustos et al., 1996.. The more recent results for flocculated suspensions are summarized in several publications. The phenomenological theory is presented in Concha et al. Ž1996., the details of mathematical analysis are contained in Burger and Wendland Ž1998a,b., and additional numerical examples are presented in ¨ Burger and Concha Ž1997, 1998.. ¨ The first two examples presented here Žbatch settling. show that the sedimentation behaviour predicted by numerical solution of the phenomenological model agrees well with experimental measurements by Been and Sills Ž1981. and Auzerais et al. Ž1990. and with numerical studies by Auzerais et al. Ž1988. and Eckert et al. Ž1996.. These results encouraged us to extend the work to the simulation of the transient behaviour of continuous thickeners. Examples 3 and 4 show that the essential operations of a continuous thickener, namely filling up, operating at steady state and emptying, can be modelled by merely choosing appropriate values of the feeding flux f F Ž t . and the average velocity q Ž t .. In each case, the transient and steady state values of the concentration in the thickener can be observed. In Example 4 we observe a stabilization effect. If the feeding and discharge conditions belong to steady states, the sediment assumes the corresponding steady state profile even from a disturbed initial concentration. However, our observation of the sediment level growth rate is still based on numerical experimentation. Based on numerous examples of changes between steady states, we have observed ŽBurger and Concha, 1997, 1998. that the growth of the ¨ sediment level is almost linear. We have used this information to calculate the time when changes of feed or discharge rate are necessary to approximate the next steady state. The mathematical proof of this fact is still lacking. The sediment growth rate cannot be calculated from the adjacent concentration values as easily as in the control model for the hyperbolic case presented by Bustos et al. Ž1990a,b., since the Rankine– Hugoniot condition, determining the jump speed for the parabolic–hyperbolic model presented here, also involves the derivative of the concentration with respect to z ŽBurger and Wendland, 1998b., which is unknown beforehand. ¨ Nevertheless, all these examples show that the phenomenological model of flocculated suspensions and the numerical solution predict reasonably well the behaviour of flocculated suspensions for most applications. 5. List of symbols aŽ f . C f bk Ž f . f DŽ t . fFŽt.
diffusion coefficent exponent in the flux density function Kynch batch flux density function discharge solid flux density feeding flux density
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Fr g L qŽ t . q 1, q 2 , q 3, q 4 QF Ž t . S t T y` z zc a, b DD l1 , l2 , l3 , l4 f f 0Ž z . f 1Ž t . f 11, f 12 , f 123, f 13, f 134 , f 14 F 2 Ž z ., F 3 Ž z ., F4Ž z . fc fD f D2 , f D3 , f D4 f D ma x fFŽt. fM s 0 , s 1 , s5 , s6 , s 7 , s 8
s 2 , s 3 , s4 , s 9 seŽ f .
281
Froude number acceleration of gravity settling column heightrheight of the surface source in an ICT volume-average flow velocity of the suspension values of q Ž t . in Examples 3 and 4 volume feed rate thickener cross-sectional area time end of the time interval settling velocity of a single floc in an unbounded medium height height of critical concentration parameters in the effective solid stress function difference of solid and fluid mass densities sediment heights in Examples 3 and 4 volumetric solid concentration initial concentration distribution prescribed concentration at z s L values of f 1Ž t . in Examples 3 and 4 steady state concentration profiles in Example 4 critical concentration value discharge concentration values of f D in Example 4 maximum discharge concentration feed concentration local maximum of the flux function concentration value propagation velocities in Examples 3 and 4 jump propagation velocities in Examples 3 and 4 effective solid stress function
Acknowledgements This work was supported by the German Research Foundation ŽDFG. through the Collaborative Research Programme ŽSonderforschungsbereich. 404 Žproject A2. at the University of Stuttgart, by the Fundacion ´ AndesrAlexander-von-Humboldt-Stiftung through project C-13131 and by Fondecyt project 89r352 at the University of Concepcion. ´ References 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.
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