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Sedimentation: integral behaviour
ChemicalEngineeringScience,1976,Vol. 31, pp. 273475. PergamonPress. Printedin Great Britain
SEDIMENTATION: INTEGRAL BEHAVIOUR D. E. SMILES Division of Environmental Mechanics, CSIRO, Canberra, Australia (Received 23 April 1975;accepted in revisedfon
16 October 1975)
Abstract-Sedimentation is considered in terms of a non-linear Fokker-Planck equation in Lagrangian coordinates. For the boundary and initial conditions appropriate to the process there is no known solution in terms of anaiytical functions. Linearization, however, permits prediction of integral behaviour, and a method for calculating the appropriate mean d8usivity is tentatively suggested. Good experimental agreement with sediment& red mud is observed.
T is the depth of the slurry, ‘ycthe specific gravity of the solid, M the total volume of solid/unit area of column, and m the cumulative volume of solid/unit area from z = 0:
Sedimentation is probably the most important large-scale method used to separate liquid from solid in industrial effluent slurries. At the same time, the process is not well
understood and no completely satisfactory predictive theory is available. Conventionally, the problem is approached in terms of Stokes’ law and it is assumed that no sorting of particles occurs [ l-31. The weakness of the approach is that Stokes’ law fails when particles interact, and attempts to describe microscopically the consequences of this interaction do not appear to help in macroscopic prediction[4]. This note deals with application of the theory of water flow in swelling soils (Smiles and Rosenthal[S], Philip[6]) to the problem, identifies the non-linear Fokker-Planck equation as the appropriate description of transient flow, and demonstrates that in a linear form this equation can be used to predict early stages of accumulation of liquid above a sedimenting slurry. The theory of vertical flow of water in a saturated swelling system rests on: 1. Darcy’s law, 0 =
-K(4)a@/az,
m =
a6 -=-at
19)~’dz.
(4)
au am’
(5)
4. A knowledge of the characteristic properties q(S), K(4), and ‘yc,each of which may be simply determined experimentally. Combination of eqns (1) and (3)-(S) gives rise to the flow equation
(1)
The sedimentation process may be described by the solution to eqn (6) subject to the conditions a=&;
is the hydraulic the water, @, and K(6) conductivity/moisture ratio relation. 2. The observation that @ has three components[7], (a) a gravitational potential, z, because of position of the element of water in the gravitational field; (b) a component, VI,that arises because of the interaction of the water and the geometry and surfaces of the solid component of the system; (c) a component, a, that represents the work involved in vertical movement of solid and liquid, which accompanies water content change.
(2)
If we consider the vertical system shown in Fig. 1, and use as potential units the work per unit weight of water (m), it can be show@] that @=T++t((y,-l)(M-m).
( (1 t
3. A continuity equation in appropriate coordinates, viz.
in which u is the volume flux relative to the particles, a@/az is the space gradient of the total potential of
a=\utztn.
I
I9 = 40;
OSmsM; m=M;
t=O
(74
t>O
(7b)
g=U-y,)g;m =O;
t>O.
I +Z
2.0
(3) 273
Fii. 1. Diagramof sedimentation situation.
(7c)
214
D. E
SMILES
Equation (7~) is the zero-flux statement appropriate to m = z = 0; eqn (7b) appears to be an appropriate description of the upper surface of a sedimenting slurry (but see [9]), & corresponds to 8 = 0. There is no solution to eqn (6) subject to conditions (7) in terms of analytical functions. In the interim, it is interesting to examine the solution to a linear form of the flow equation. This approach does not permit us to predict liquid content profiles, but it has been a useful approach in infiltration theory for predicting integral aspects of the process, and should confirm by implication the appropriateness of eqns (6) and (7). In soil physics it is conventional to linearise eqn (6) in terms of 9; in this situation, however, it appears more convenient to restate the flow equation in terms of @. Equations (1) and (3)-U) may be combined to give
If we make the reasonable assumption that Y,, = Y0 = 0, the solution of eqn (8) subject to eqn (9) is @ = (@- T)/(rc - l), = 5z0
(2n t 1jm2 sin ([2n t l][M - m]lr/2M)
X exp(-6[2n
t l]*&t/4M*),
(see, for example, Carslaw and Jaeger[lO]) and the cumulative volume of liquid, Q, appearing above the sedimenting solid is given by
or &‘Q
+.L(&i.jt),
= lqrc - l)Iw
2 (2n t l)-‘(-1) "SO
x (1 - exp [-D(2n + 1)*~*t/4MZ1). which takes the linear form Also, if Qm= lim Q, I-
(8) In this equation, the constant diffusivity, D, represents some appropriate mean value of [K/(1 t 4)][dY/d9] for the particular sedimentation process. The point is discussed further below. Also, eqns (7a, b, c) become, respectively, eqns (9a, b, c), viz. @=(Y”tT)t(rc-l)(M-m);
OcmSM;
t=o (9a)
@=(YotT); g=O;
m=M;
m =O;
t>o
(9b)
0.01
and m
r’Q/Qm = 32 zo(2” t l)-‘(-1) x (1- exp [-D(2n t 1)*1r2t/4MZ1). Figure 2 shows Q/Q.. plotted as a function of h/M’, together with some experimental data for red mud in which Q(t) was plotted on log/log graph paper. Two experiments were performed with de-airedt columns of mud: (a) 4, = 12.6; T = 0.866 m; M = 6.37 x lo-’ m. (b) 8” = 12.6; T = 0.176 m; M = 1.29 x 10e2m.
t >O.
tDe-airing the slurry is important. When a slurry is shaken violently before an experiment and not de-aired, the initial settling rate is low and appears to increase with time. This is consistent with solutiop of air bubbles attached to particles during shaking.
0.01
Q* = (rc - 1)I?M2/2D,
o-1
These conditions imply that T at m = 0 is 0.41 m (a) and 2.8 x lo-’ m (b), giving values of 6 of 3.75 and 4.4, respectively. As will be observed, the experimental data match the
Dt/M’
1.0
Fig. 2. Rate of accumulation, Q, of supematant in a sedimenting system, as a function of Dt/M* calculated as described in the text. Q.. is lim Q. The experimental points were graphed as Q vs t and matched to the calculated t-.X curve.
215
Sedimentation: integrai behaviour
linearised prediction quite well. Experiment (a) indicates a value of D = 2.5 x 10-9mZSC’, while for (b) D = 8.5 X lo-” m*s-‘. It is therefore concluded that eqns (8) and (9) satisfactorily described integral properties of sedimentation of red mud and if appropriate values of b can be selected, then they can be used predictively. This conclusion implies that eqns (6) and (7) might be used to describe all aspects of the process when methods of solution of eqn (6) are developed. As was noted above, D is related to t (or 4) through material characteristics K(9) and V(4). It is not simply related to @. For any process where M and T are constant, however, the limits of q (and 6) are defined and thus the limits on fi can be identified. We observe, for example, that the values of D obtained by the curve matching procedure in Fig. 2 are “consistent” with D,,,(6) obtained for this material by the method of Smiles and Harvey[l6] insofar as they lie within the range shown in Table 1. They are self-consistent insofar as the average water content for experiment (b) is less than that of (a), and the mean diffusivity, weighted towards values at the base of the column where water content changes are greatest, is correspondingly greater.
Table 1. Characteristics of the red mud used in this study -9 b 30)
6
0,
cm* a-3
5.97x
10
10 -8
1.15 x 10-8
1 -2 9.8 x 10
2.22 x 10-9
1 x 10-2
9.4
T Expeinental~ta
* lo-l0 1
*1x10 <1x10
-3 -3
6
8.82 x 10-11
7
1.82 x 10-l'
II
8
3.80 x 10-12
II
9
8.55 x 10-13'
I,
10
2.07 x lo-l3
II
11
5.83 * lo-l4
I,
12
2.70 * lo-l4
Finally, it is noteworthy that the theory demonstrates not only that the rate of accumulation of liquid above the settling solid is dependent on the liquid content at the top of the solid, but that it is dependent also on M, the total amount of solid in the column. This dependence does not appear previously to have been recognised. NOTATION
Dtll liquid ditfusivity in the Lagrangian coordinate system m*s-’
b mean klue of D m*s-’ K
hydraulic conduc%ity, m SC’
II mean value of K, m SC’ m cumulative volume of solid per unit area of crosssection from z = 0 to z, m total volume of solid per unit area of cross-section, m Q cumulative volume of liquid appearing above sedimenting solid, m t time, s T total depth of slurry, m volumetric flux density of water relative to the D solid, m SC’ space coordinate, m 2
M
Greek symbols yc solid specific gravity 6 volumetric liquid content per unit volume of solid Cp total potential of liquid, m \It liquid potential component of @, m fl “overburden” potential component of @, m REFERENCES [l] Kynch G. J., Trans. Faraday Sot. 1952 48 166. [2] Fitch E. B., Tram. AZME 1962 223 129. [3] Dell C. C. and Kelleghan W. T. H., Powder Technol. 19737 189. [4] Batchelor G. K., Ann. Reo. Fluid Mech. 1974 6 227. [S] Smiles D. E. and Rosenthal M. J., Amt. J. Soil Res. 1%8 6 231. [6] Philip J. R., Wafer Resourc. Rex 19706 1248. [7] Philip J. R., Amt. J. Soil Res. 1%9 7 99. [8] Smiles D. E., Soil Sci. 1974117 288. [9] Smiles D. E., Separ. Sci. 197611 1. [lo] Carslaw H. S. and Jaeger J. C., Conduction of Heat in Solids, 2nd Edn. Oxford University Press, Oxford 1959. 1111Smiles D. E. and Harvey A. G., Soil Sci. 1973116 391.
SEDIMENTATION: INTEGRAL BEHAVIOUR D. E. SMILES Division of Environmental Mechanics, CSIRO, Canberra, Australia (Received 23 April 1975;accepted in revisedfon
16 October 1975)
Abstract-Sedimentation is considered in terms of a non-linear Fokker-Planck equation in Lagrangian coordinates. For the boundary and initial conditions appropriate to the process there is no known solution in terms of anaiytical functions. Linearization, however, permits prediction of integral behaviour, and a method for calculating the appropriate mean d8usivity is tentatively suggested. Good experimental agreement with sediment& red mud is observed.
T is the depth of the slurry, ‘ycthe specific gravity of the solid, M the total volume of solid/unit area of column, and m the cumulative volume of solid/unit area from z = 0:
Sedimentation is probably the most important large-scale method used to separate liquid from solid in industrial effluent slurries. At the same time, the process is not well
understood and no completely satisfactory predictive theory is available. Conventionally, the problem is approached in terms of Stokes’ law and it is assumed that no sorting of particles occurs [ l-31. The weakness of the approach is that Stokes’ law fails when particles interact, and attempts to describe microscopically the consequences of this interaction do not appear to help in macroscopic prediction[4]. This note deals with application of the theory of water flow in swelling soils (Smiles and Rosenthal[S], Philip[6]) to the problem, identifies the non-linear Fokker-Planck equation as the appropriate description of transient flow, and demonstrates that in a linear form this equation can be used to predict early stages of accumulation of liquid above a sedimenting slurry. The theory of vertical flow of water in a saturated swelling system rests on: 1. Darcy’s law, 0 =
-K(4)a@/az,
m =
a6 -=-at
19)~’dz.
(4)
au am’
(5)
4. A knowledge of the characteristic properties q(S), K(4), and ‘yc,each of which may be simply determined experimentally. Combination of eqns (1) and (3)-(S) gives rise to the flow equation
(1)
The sedimentation process may be described by the solution to eqn (6) subject to the conditions a=&;
is the hydraulic the water, @, and K(6) conductivity/moisture ratio relation. 2. The observation that @ has three components[7], (a) a gravitational potential, z, because of position of the element of water in the gravitational field; (b) a component, VI,that arises because of the interaction of the water and the geometry and surfaces of the solid component of the system; (c) a component, a, that represents the work involved in vertical movement of solid and liquid, which accompanies water content change.
(2)
If we consider the vertical system shown in Fig. 1, and use as potential units the work per unit weight of water (m), it can be show@] that @=T++t((y,-l)(M-m).
( (1 t
3. A continuity equation in appropriate coordinates, viz.
in which u is the volume flux relative to the particles, a@/az is the space gradient of the total potential of
a=\utztn.
I
I9 = 40;
OSmsM; m=M;
t=O
(74
t>O
(7b)
g=U-y,)g;m =O;
t>O.
I +Z
2.0
(3) 273
Fii. 1. Diagramof sedimentation situation.
(7c)
214
D. E
SMILES
Equation (7~) is the zero-flux statement appropriate to m = z = 0; eqn (7b) appears to be an appropriate description of the upper surface of a sedimenting slurry (but see [9]), & corresponds to 8 = 0. There is no solution to eqn (6) subject to conditions (7) in terms of analytical functions. In the interim, it is interesting to examine the solution to a linear form of the flow equation. This approach does not permit us to predict liquid content profiles, but it has been a useful approach in infiltration theory for predicting integral aspects of the process, and should confirm by implication the appropriateness of eqns (6) and (7). In soil physics it is conventional to linearise eqn (6) in terms of 9; in this situation, however, it appears more convenient to restate the flow equation in terms of @. Equations (1) and (3)-U) may be combined to give
If we make the reasonable assumption that Y,, = Y0 = 0, the solution of eqn (8) subject to eqn (9) is @ = (@- T)/(rc - l), = 5z0
(2n t 1jm2 sin ([2n t l][M - m]lr/2M)
X exp(-6[2n
t l]*&t/4M*),
(see, for example, Carslaw and Jaeger[lO]) and the cumulative volume of liquid, Q, appearing above the sedimenting solid is given by
or &‘Q
+.L(&i.jt),
= lqrc - l)Iw
2 (2n t l)-‘(-1) "SO
x (1 - exp [-D(2n + 1)*~*t/4MZ1). which takes the linear form Also, if Qm= lim Q, I-
(8) In this equation, the constant diffusivity, D, represents some appropriate mean value of [K/(1 t 4)][dY/d9] for the particular sedimentation process. The point is discussed further below. Also, eqns (7a, b, c) become, respectively, eqns (9a, b, c), viz. @=(Y”tT)t(rc-l)(M-m);
OcmSM;
t=o (9a)
@=(YotT); g=O;
m=M;
m =O;
t>o
(9b)
0.01
and m
r’Q/Qm = 32 zo(2” t l)-‘(-1) x (1- exp [-D(2n t 1)*1r2t/4MZ1). Figure 2 shows Q/Q.. plotted as a function of h/M’, together with some experimental data for red mud in which Q(t) was plotted on log/log graph paper. Two experiments were performed with de-airedt columns of mud: (a) 4, = 12.6; T = 0.866 m; M = 6.37 x lo-’ m. (b) 8” = 12.6; T = 0.176 m; M = 1.29 x 10e2m.
t >O.
tDe-airing the slurry is important. When a slurry is shaken violently before an experiment and not de-aired, the initial settling rate is low and appears to increase with time. This is consistent with solutiop of air bubbles attached to particles during shaking.
0.01
Q* = (rc - 1)I?M2/2D,
o-1
These conditions imply that T at m = 0 is 0.41 m (a) and 2.8 x lo-’ m (b), giving values of 6 of 3.75 and 4.4, respectively. As will be observed, the experimental data match the
Dt/M’
1.0
Fig. 2. Rate of accumulation, Q, of supematant in a sedimenting system, as a function of Dt/M* calculated as described in the text. Q.. is lim Q. The experimental points were graphed as Q vs t and matched to the calculated t-.X curve.
215
Sedimentation: integrai behaviour
linearised prediction quite well. Experiment (a) indicates a value of D = 2.5 x 10-9mZSC’, while for (b) D = 8.5 X lo-” m*s-‘. It is therefore concluded that eqns (8) and (9) satisfactorily described integral properties of sedimentation of red mud and if appropriate values of b can be selected, then they can be used predictively. This conclusion implies that eqns (6) and (7) might be used to describe all aspects of the process when methods of solution of eqn (6) are developed. As was noted above, D is related to t (or 4) through material characteristics K(9) and V(4). It is not simply related to @. For any process where M and T are constant, however, the limits of q (and 6) are defined and thus the limits on fi can be identified. We observe, for example, that the values of D obtained by the curve matching procedure in Fig. 2 are “consistent” with D,,,(6) obtained for this material by the method of Smiles and Harvey[l6] insofar as they lie within the range shown in Table 1. They are self-consistent insofar as the average water content for experiment (b) is less than that of (a), and the mean diffusivity, weighted towards values at the base of the column where water content changes are greatest, is correspondingly greater.
Table 1. Characteristics of the red mud used in this study -9 b 30)
6
0,
cm* a-3
5.97x
10
10 -8
1.15 x 10-8
1 -2 9.8 x 10
2.22 x 10-9
1 x 10-2
9.4
T Expeinental~ta
* lo-l0 1
*1x10 <1x10
-3 -3
6
8.82 x 10-11
7
1.82 x 10-l'
II
8
3.80 x 10-12
II
9
8.55 x 10-13'
I,
10
2.07 x lo-l3
II
11
5.83 * lo-l4
I,
12
2.70 * lo-l4
Finally, it is noteworthy that the theory demonstrates not only that the rate of accumulation of liquid above the settling solid is dependent on the liquid content at the top of the solid, but that it is dependent also on M, the total amount of solid in the column. This dependence does not appear previously to have been recognised. NOTATION
Dtll liquid ditfusivity in the Lagrangian coordinate system m*s-’
b mean klue of D m*s-’ K
hydraulic conduc%ity, m SC’
II mean value of K, m SC’ m cumulative volume of solid per unit area of crosssection from z = 0 to z, m total volume of solid per unit area of cross-section, m Q cumulative volume of liquid appearing above sedimenting solid, m t time, s T total depth of slurry, m volumetric flux density of water relative to the D solid, m SC’ space coordinate, m 2
M
Greek symbols yc solid specific gravity 6 volumetric liquid content per unit volume of solid Cp total potential of liquid, m \It liquid potential component of @, m fl “overburden” potential component of @, m REFERENCES [l] Kynch G. J., Trans. Faraday Sot. 1952 48 166. [2] Fitch E. B., Tram. AZME 1962 223 129. [3] Dell C. C. and Kelleghan W. T. H., Powder Technol. 19737 189. [4] Batchelor G. K., Ann. Reo. Fluid Mech. 1974 6 227. [S] Smiles D. E. and Rosenthal M. J., Amt. J. Soil Res. 1%8 6 231. [6] Philip J. R., Wafer Resourc. Rex 19706 1248. [7] Philip J. R., Amt. J. Soil Res. 1%9 7 99. [8] Smiles D. E., Soil Sci. 1974117 288. [9] Smiles D. E., Separ. Sci. 197611 1. [lo] Carslaw H. S. and Jaeger J. C., Conduction of Heat in Solids, 2nd Edn. Oxford University Press, Oxford 1959. 1111Smiles D. E. and Harvey A. G., Soil Sci. 1973116 391.