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Sedimentation batch test: Application to deduce some parameters of aggregates in metal hydroxides suspensions
Powder Technology, 71 (1992) 217-227
217
Sedimentation batch test: Application to deduce some parameters aggregates in metal hydroxides suspensions
df
R. Font*, A. F. Marcilla and C. ZofFmann Divi.Sn
de Ingenierla Quimica, Facultad de Ciencias, Universidad de Alicante, Apartado 99, Alicante (Spain)
(Received October 11, 1990; in revised form February 14, 1992)
Abstract The size of aggregates and the amount of water in them were estimated from batch sedimentation tests of metal hydroxides suspensions. This method is suitable for slurries in which the initial solids concentration is less than the critical value (limit between the non-compression range and the compression range of solids concentration). Inertial effects were negligible for these suspensions. The aggregate diameter and the parameter N (ratio between the force of acceleration and the inertial effect) were strongly negative correlated.
Fundamentals
Introduction Sedimentation is a unit operation of liquid-solid separation which allows either the solids concentration in a suspension to be increased or a liquid clarification with nil or very small solids concentration to be obtained. In many industrial processes, wastewaters with a high metal ion content are obtained. Sometimes the recovery of the metals is required and on other occasions it is necessary to eliminate the metals contained in the suspension in order to obtain clear waters. These can either be recirculated in the industrial process (with the corresponding commercial profit) or be discharged into the public irrigation channels (in accordance with the existing regulations). Once the metal hydroxides have been precipitated, it is sometimes necessary to add flocculants to the formation of floes or aggregates with great density and/ or size. These aggregates can then be separated by sedimentation due to the force of gravity. In concentrated metal hydroxide suspensions, large aggregates are formed, which can sediment without addition of flocculants. The main aim of this paper is to characterize the aggregates formed from the data obtained in a sedimentation batch test. Two parameters of the aggregates are considered: a) the amount of water retained in an aggregate, and b) the diameter of the aggregate. The results obtained have been proven logical.
*Author to whom correspondence
should be addressed.
a) Basic relationships and the numerical method proposed for the determination of the aggregate diameter and the amount of water retained in an aggregate
Two zones can be distinguished in the sedimentation of stable aggregates (constant amount of solids in each aggregate): a) non-compression zone (or hindered settling zone or free settling zone) where the aggregates descend separately and the solids concentration controls the sedimentation rate, and b) compression zone, where the increase in solids concentration is due to the unbuoyed weight of the upper solids and taking into account the resistance force due to the liquid flow. The calculation of the thickener area per unit of solids flow sometimes depends only on the relationship between the settling rate and the solids concentration [l]. Some equations have been proposed to correlate the settling rate of aggregates and particles as a function of the solids concentration. The main relationships with physical significance are discussed as follows: a) Using dimensional analysis, Richardson and Zaki [2] developed a general correlation for the hindered settling and fluidization of nonflocculent particles of the form: V, = V&
(1)
where V, is the settling velocity, E is the suspension porosity, V0 is the intercept velocity at E= 1, and n is a function mainly of the particle Reynolds number, Re. For spherical particles, the intercept velocity was equal to the terminal, unhindered settling velocity of a single
0 1992 - Elsevier Sequoia. All rights reserved
218
particle, V,. For conditions of creeping flow (Re < 0.2) and a low ratio of particle diameter to settling column diameter, n was 4.65. b) Michaels and Bolger [3] observed that the settling units of the flocculated kaolinite suspensions examined in their investigation were individual aggregates, and the effective porosity of the suspension would, therefore, be interaggregate porosity. Equation (1) can be expressed as: V, = VU(1 - 4$.65
(2)
where +a is the aggregate volume fraction. Letting +a equal j& where j is the volume of the aggregate per volume of solids inside the aggregate and A is equal to the volume fraction of solids (m3 of solids mm3 of suspension), eqn. (2) becomes: V 1/4.65,V~/4.65_ Vol/4-65j~s s
(3)
The parameter ‘j’ is also known as the Aggregate Volume Index (AVI). The volume of water retained per unit of volume of solids in an aggregate equals ‘j- 1’. The amount of water per unit of any extensive property of the solids can easily be calculated from the value y- 1’ and the densities. Michaels and Bolger [3] extended the utility of eqn. (3) to the estimation of the size and density of the aggregates by expressing V, as Stokes’ Law for the terminal settling velocity of a spherical aggregate in creeping flow: V, =g(p, - pX/(l8~)
(4)
where g is the gravitational acceleration, d, is the aggregate diameter, p is the liquid viscosity, pa is the aggregate density and p, is the liquid density. The volume fraction of solids can be calculated by the equation: A = CJPS
(5)
where C, is the solids concentration, and p. is the solids density. For stable aggregates, the parameter j is equal to: j=m’
l/67@) = (d&,)3 aggregate mm3 solid = 1/67~7.Q( (6)
where d, is the diameter of a hypothetical particle which has the same amount of solids as the aggregate. The aggregate density pa equals Pa = (Ps + o’-
l>PJO
(7)
From eqns. (4), (6) and (7) we deduce that: K,=g(p,-pJdZpj”l(l8pj)
= Vwj-1’3
- ~JdzpIW4
v:/4.65 =(Vwj-l/3)1/4.65
_(V,j-l/3)l/4.65j,+s
(10)
If 7 (m3 of aggregate me3 of solids in an aggregate) is constant and the assumptions are acceptable, plotting V,1’4.65vs. 4, should give a linear variation. Frequently, this variation is not linear which could be due to the following reasons: a) Some of the experimental settling velocity data correspond to the compression zone where the sedimentation velocity depends on many factors, and consequently eqn. (2) cannot be used to correlate these data. b) The parameter ‘j’ is the ratio between the volume of an aggregate and thevolume of solids in this aggregate. As an aggregate is the result of the union of several particles, between these particles in each aggregate there is fluid which occupies the free space. When the aggregate descends, it has also a thin exterior layer of fluid 131, which can be large or small depending on the settling rate and on the proximity of the other aggregates. A variation of j, therefore, can be due to the variation of the thickness of the layer surrounding the aggregate. Some researchers have used plots of Vs1’4.65m. $s to deduce parameters of the aggregates formed during the sedimentation of wastewater sludges [4] and flocculated suspensions [5]. Some aspects must be considered when K1’4.65is plotted ZJS.& - Firstly, all the experimental data of settling rates must correspond to the hindered settling zone or noncompression zone. - Secondly, if the variation of V:‘4.65 vs. 4s is curved, it is not correct to determine the values of VW and ‘j’ from the slope and intercept of the tangent at a point in the curve observed. A mathematical discussion of this aspect can be found in [l]. c) Equations (4), (8) and (9) are valid in creeping flow (when Re < 0.2). For values of aggregate Reynolds number (Re = p,daV,/p) between 0.2 and 500, a correction factor must be taken into account (6). In this case, V,=g(p,-p,)~I(18p(1+0.15Re”~687))
(11)
On considering equations ‘(6) and (8), the Reynolds number equals: Re =p,daVolp = p,dpj”3V,j-1”lp
(8)
From equations obtain:
(9)
J’,, =g(ps - p,)$l((18N
where v,=g(ps
V, is the free settling velocity of a hypothetical solid particle which has the same amount of solids as an aggregate. From eqns. (3) and (8), therefore, we deduce that:
=p,d,,V,/p
(12)
(6), (7), (8), (11) and (12) we can + 0.15(p,d,V,lcL)“.687)))
(13)
219
d) Garside and Al-Dibouni [7] have corrected the exponent in eqn. (1) (which has the value, of 4.65 in eqn. (2)) and have proposed the following equation to calculate the value of 12: (5.09 - n)/(n - 2.73) = 0.104ReoS877
_
-j4$
+b4S+c4,2
INTERFACE
4 L=F(l) , SEDINENT SURFACE
(15)
If the thickness of the fluid layer which surrounds the aggregate decreases when the volume fraction of solids increases, a polynomial variation of j VS. 4. can be assumed according to the equation: j=a
n
H=F(T) , UPPER
“i
(14)
This equation has been obtained for porosities (1 - &J of suspensions between 0.8 and 0.5. Under this assumption, the following equation can be written: v,= V,j_‘“(l
H
(16)
By numerical optimization, it is possible to calculate the best values of V,, a, b and c for the correlation of the experimental values of the settling rates as a function of the volume fraction of solids in the equation combined from eqns. (15) and (16). Value d, can be obtained from eqn. (13). In this way, an interval of values of j (a parameter implying the amount of water retained in the aggregate) can be calculated from eqn. (16) and the aggregate diameter d, is calculated from eqn. (6). b) Zones in a batch test of sedimentation
The settling velocity of aggregates for each solids concentration can be determined in two ways: a) Carrying out many batch tests at different solids concentration and measuring the velocity of the supematant-suspension interface in the period of time when this velocity is constant, and b) Performing a batch test in a long column and applying the Kynch [8] theory for calculating the relation between the settling rate and the solids concentration. With the first method mentioned previously, it is necessary to perform several batch tests with different solids concentration. All the suspensions must, however have been prepared and flocculated in the same way. With the second method, only from a batch test, it is possible to obtain a relation between the settling rate and the solids concentration in an interval of solids concentration (according to the hydrodynamics of the suspension and the value of the initial solids concentration). The batch test has been studied by several researchers [9,10,11,12] taking into consideration the compression zone which builds up on the bottom of the column. Consider the data plotted in Fig. 1, where the variation of the upper discontinuity interface height and the
t Fig. 1. Zones in a batch test.
sediment (compression zone), surface height are plotted TX time. An analysis of the different zones that can be distinguished can be found elsewhere [ll, 121. Kynch [8] developed a theory to calculate the relation between the settling rate of solids observed in the upper interface and the solids concentration. In this theory, it is assumed that the settling rate depends only on the solids concentration (in the settling zone, because there is no compression zone due to the fact that the suspension is considered non compressible). The same theorems are valid for zones ‘a’ and ‘b’ presented in Fig. 1, for compressible suspensions (10). According to these theorems [lo, 111, we can obtain the settling rates V, from the slopes dH/df of the curve H=f(t), and calculate the solids fractions or concentrations corresponding to each point by the equation: 4S= +H,IHi
or C, = CfioIH,
(17)
where Hi is the intercept of the tangent drawn with H-axis (Fig. 1). For zone ‘c’ (Fig. 1) the equation used to calculate A or C, is presented elsewhere [ 111. Nevertheless, when the variation of the slope dLJdt, corresponding to the variation of the sediment surface height zx. time, is not very great, the same eqn. (17) can be applied to calculate the solids concentration [ll]. The limit of solids concentration (critical concentration) between the hindered settling range (or noncompression range) and the compression range cannot be easily determined. This limit can be determined by Font’s method [ll] which considers the variations in the height of the supernatant-suspension discontinuity and the surface of the sediment in a batch test for determining the position of the point T as in Fig. 1 (point T is the intercept between the tangent to the sediment surface curve at the origin and the upper interface). In this way, it is possible to calculate the solids concentration C,, -solids mass/suspension volume - or 4, -volume fraction of solids - , corresponding
220
to the limit between the non-compression range and the compression range of solids concentration, by the equations:
where H,* is the height of the point T. Equation (18) is only a consequence of the material balance. Any relationship between the settling rate and the solids concentration has not been taken into account to obtain this equation. Shannon et al. [13] studied the batch thickening of suspensions of rigid spheres in water. They found that the final volume fractions of solids were between 0.63 and 0.65. Considering that with suspensions with different particle sizes the volume fraction of solids in a fixed bed can be somewhat larger (small particles can fill the holes between larger particles), the upper limit 0.65 has been taken into account as the volume fraction of aggregates at the critical concentration (limit between non-compression range and compression range). Nevertheless, similar conclusions to those stated in this paper could be obtained assuming a value of volume fraction of solids between 0.60-0.62 and 0.65-0.67.
Experimental results and discussion a) Experimental results of a batch test where the sediment surjace is visible. Method for corrobating the procedure proposed for the characterization of the aggregates
A suspension of Zn(OH),, Ni(OH)2 and Cu(OH), was prepared from a solution of ZnSO,, NiSO, and CuS04, where Ca(OH), (70 wt.%) +Mg(OH), (3Owt.%) was added up to pH=11.7. The initial solids concentration was 8.4 kg mm3. With this suspension it was possible to observe the sedimentation surface. The test was carried out at 20 “C. This test is labeled with number 1 and more details can be observed in Tables 1 and 2. The variations of the supernatant-suspension discontinuity height and the sediment surface height are shown vs. time in Fig. 2. We can observe that the curvature of the sediment height vs. time variation is small. The tangent to the sediment height vs. time curve at the origin intercepts at the upper interface at point T. Between the points T and C* (critical point) the variation of the upper discontinuity is nearly linear. Applying the Kynch theory we can obtain the relationship between the solids volume fraction and the solids concentration from equation (17). In zone ‘c’, the slope dH/dz is nearly constant, so its corresponding concentration can also be calculated from eqn. (17), at the limit between zones ‘b’ and ‘c’.
In this case, the positions of the points C* and T have also been easily determined by considering the variation of the sediment surface height vs. time curve. At the critical point C* where the two curves intersect, we can observe a change in the slope of dH/dt. This criterion has been used to determine the position of the critical point in the other tests, presented later in this paper, where the sediment surface is not visible. When the data corresponding to the sediment curve are not available, the position of point T (intercept of the tangent to the sediment surface height vs. time curve at origin and the upper discontinuity) can only be estimated and considered to be close to the beginning of the linear variation of the upper discontinuity height vs. time (or when values -dH/dt are nearly constant at the section just before the critical point). This procedure for estimating the position of point T can be tested from the run shown in Fig. 2 and from the batch tests with calcium carbonate suspensions presented elsewhere [ll]. It is possible that with other flocculent suspensions with a behaviour distinct from that shown in Fig. 2, the estimated position of the point T is not close to that deduced by the method proposed in this paper. As the determinations are not exact, intervals for the position of the points C* and T are calculated for the following tests, where the sediment surface is not visible. If eqns. (14)-(16) are considered as representative of the relationship between the settling rate and the solids concentration in the non-compression range, and if we accept the modifications of the Kynch theory proposed in [ll], then the following relations must be fulfilled: 1) The values iA (= 4.) corresponding to the characteristics which intercept the upper interface in the batch test must be less than 0.65 (volume fraction of aggregates less than that when the aggregates are in contact). 2) The concentration C,, (limit between non-compression zone and compression zone) calculated from two independent procedures must take a similar value. These two procedures are the following: a) Estimating the position of the point T and applying eqn. (18). b) By the following expression: 0.65(volume fraction of aggregates when they are in contact)=+*, =jCS,/pS
(19)
At the critical concentration C,,, the aggregates are in contact and the volume fraction of aggregates, therefore, must be close to 0.65 [13]. The value 3’ considered corresponds to the value obtained at the higher solids concentration (and consequently close to that corresponding to E,,). From equation (19):
221 TABLE Test no.
1. Concentrations
of metals
.
Initial concentration of the mixture (ppm)
Precipitating agent
PH supematant
1
850 Ni+’ 850 Cu+’ 850 Zn+’
Ca(OH)a 70%wt. Mg(OH)2 30%wt.
11.67
0 70 Ni+’ 0:75 cu+2 0.82 Zn+2
8.4
2
250 Ni+’ 250 cu+2 250 Zn+2
NaOH
1M
8.08
1 10 Ni+2 0:73 Cu+2 0.82 Zn+’
1.50
3
250 Ni+’ 250 cu+2 250 Zn+2
NaOH
1M
8.18
0.66 Ni+’ <0.5 Cu+2 0.44 Zn+’
1.42
4
250 Ni+’ 250 Cu+’ 250 Zn+’
NaOH
1M
8.79
<0.2 Ni+’ <0.5 Cu+2 <0.2 Zn+’
1.43
5
250 NiC2 250 Cu+2 250 Zn+’
Ca(OH)2 70%wt. Mg(OH)2 30%wt.
8.62
0.27 Ni+’ <0.5 cu+r <0.2 Zn+2
1.99
6
87.5 87.5 87.5 87.5 87.5 87.5
Ni+2 Cu+2 Zn+’ Pb+’ Fe+’ Cr+’
NaOH
1M
6.78
17 0 Nit2 0:50 cu+2 1.32 Znf2 <50 Pb+2 0.24 Fe+’ 1 Cr+’
0.756
7
175 175 175 175 175 175
NiC2 cu+2 Zn+2 Pb+’ Fe+3 Cr+3
NaOH
1M
<0.5 Ni+2 <0.5 cu+2 <0.2 Zn+2 <50 Pb+’
1.40
8
875 87:5 87.5 87.5 87.5 87.5
Ni+2 CU+~ Zn+’ Pb+2 Fe+3 Cr+’
Ca(OH)2 70%wt. Mg(OH)2 30%wt.
8.50
0.5 Ni+’ 0.5 cu+2 0.2 Zn+’ <50 Pb+’ 1 Fe+’ 1 Crc3
1.61
9
87 5 87:5 87.5 87.5 87.5 87.5
Ni+2 Cu+’ Zn+2 Pb+2 Fe+’ Cr+’
CaW-02
6.68
35.5 Ni+’ 0.5 Cu+2 11.3 Zn+’ <50 Pb+’ 5 Fe+3 1 Cr+’
0.822
10
87.5 87.5 87.5 87.5 87.5 87.5
Ni+’ Cu+’ Zn+2 Pb+’ Fe+3 Cr+3
WOW2
7.90
11
334 Al+’ 100 cl640 so;
HCI because initial pH was 11
7.0
1 Al+’ 580 Cl573 SO;
2.24
12
334 Al+3 100 cl640 SO;
HISO, because initial pH was 11
7.0
1 Al+3 100 cl1064 SO;
2.24
10.3
Concentration of metals in supematant (ppm)
Solids concentration of the initial mixture (kg m-‘)
0.910
222 TABLE Test no.
2. Parameters of correlation dp
b
a
C
Cm)
Range
Range
i
d.
O.F.
h4=2(no~?J100
Cm) 2.07E-4 2.06E-4
1.390E-2
5.27
580 455
1.3E-3 1.2E-3
l.O85E-3
1.47
3.944E6
551 475
9.4E-4 8.9E-4
l.l27E-3
1.50
-2.174E5
2.473E6
625 513
1.3E-3 1,2E-3
2.128E-4
0.65
377.0
- 5.535E4
1.831E6
325 239
9.0E-4 8.1E-4
6.676E-3
3.65
1.229E-4
927.2
-3.865E5
2.091E5
831 570
ME-3 l.OE-3
5.161E-3
3.33
7
8.310E-5
702.5
- 1.983E5
8.759E5
592 452
7.0E-4 6.4E-4
4.461E-3
2.99
8
1.273E-4
822.2
-3.05oE5
1.231E6
692 538
l.lE-3 l.OE-3
l.O22E-3
1.43
9
l.O56E-4
726.0
- 2.327E3
- 6.433E5
603 520
8.9E-4 8.6E-4
3.753E-3
2.74
10
2.721E-4
1171.0
1.56OE5
2.141E5
1361 1129
3.OE-3 2.8E-3
6.607E-5
0.36
11
9.340E-5
574.9
- 1.445E5
3.970E6
468 406
7.3E-4 7.0E-4
1.088E-4
0.47
12
1.048E-4
636.1
- 2.OOOE5
4.356E6
487 400
8.2E-4 7.7E-4
2.263E-4
0.67
220.5
69.7 70.7
1
5.013E-5
69.04
2
1.61OE-4
731.6
-2.068E5
1.897E6
3
1.150E-4
651.2
-1.516E5
4
1.490E-4
749.1
5
1.306E-4
6
Cs,=0.65pslj
33.99
(20)
The two procedures for estimating c, are independent. It the values calculated are similar, it can be concluded that the procedure proposed for correlating the experimental data is correct. In addition, this permits the estimation of the mean aggregate diameter and the average amount of water in the aggregate to be obtained. In addition, at the critical point of a batch test the mean solids concentration C, can be calculated by the equation: c, = H&,/K
(21)
where H, is the sediment height at the critical point. The value C, must be greater than C,, due to the compression of the sediment. Logically the value C, calculated by eqn. (21) must always be greater than the value C,, calculated by eqn. (18) but must also be greater than the value C,, obtained by eqn. (20). b) Other experimental results and dik.kon
In order to test the relationships presented previously, we carried out several batch runs with metal hydroxide suspensions precipitated with different bases. Table 1
shows the composition of the initial mixture (content in cations), the precipitating agent, the pH in the supematant, and the solids concentration of the suspension when the metal ions have been precipitated. The first test shown in Table 1 corresponds to that discussed in the previous section. Tests were carried out at 20 “C. In Figs. 3 and 4, the experimental results corresponding to two tests can be observed (the variations of settling rate US. time can be compared to the Tory plots presented by Fitch [14], where the settling rate data are presented VS. a reduced height. Although the magnitudes of the X-axes are different in both plots, some similarities can easily be deduced). The procedure for estimating the position of point T (intercept between the tangent to the sediment curve at origin and the upper discontinuity curve) and the critical point C* has been the following: a) values of - dH/dt have been calculated by fitting the considered value Hk, the previous one Hk_-l and the following one Hk+ 1 to second order polynomial variation with respect to the time t, and calculating the derivate at the intermediate point, b) calculated values of - dH/dt are plotted VS. time as indicated in Figs. 3 and 4. Point T is located where
dH/dt
0.8
lo4
CM/ s)
-2
H (14
0.8
o.j, , , ‘-1 6000
4000
2000
8000
T IflE t (s) Fig. 4. Batch test number 12.
H=f(t)=a,+a,t+a,tZ+a,t3+ 0 0
12
3
4
5
6
7
f(s)xlo-3 Fig. 2. Batch test number 1.
H CM)
0.2
3
2 0,l 1
0 1000
2000
3000
T I ME t W Fig. 3. Batch test number 4.
there is a marked decrease of -dHldr and point C* is located at the end of the zone where the values of -dH/dt are nearly constant or with small variation (this procedure has only been used for estimating the position of the points T and C*). The computer calculation proceeds as follows: a) Correlate the upper interface height with time by a polynomial expression such as:
8
...
(22)
In this way the oscillations of the settling rate values obtained as explained in the previous section are avoided. b) Calculate the values of the settling rates dH/dt by differentiating H=f(t) with respect to time t, and the corresponding values of solids concentration by eqn. (17). The interval considered is between the end of the initial straight line (of the upper discontinuity) and the critical point. c) Optimize by a Flexible Simplex Program, the values of a, b, c and V,,\ according to eqns. (15) and (16). Calculate, then, the value of d, corresponding to V, using eqn. (13). The Objective Function is: OF = 5 ((Vex, - L,)I(K, n-l
+ L))”
(23)
where each I/e._ is an experimental value of the settling rate, and each I’_, is a calculated value. A magnitude M, which indicates an upper value of the mean relative error is: M= 200((OF/N)1~)
(24)
The results of the correlations are shown in Table 2, where the values of d,, a, b and c, intervals of j and d, are presented. The values of it4, which are also presented in Table 2, are low as a consequence of the good agreement between the experimental and calculated values of the settling rate. We can observe that the aggregate diameter varies slightly around 67%. This’small variation is a consequence of the assumption that 7 (AVI) decreases when the solids concentration increases due to a small variation of the water thickness which surrounds the descending aggregate. This
224 TABLE
3. Comparison
No. ref.
of values
Range i
0.42 0.62 0.55 0.57 0.66 0.53 0.58 0.50 0.42 0.47 0.49 0.49
69.7-70.7 580-455 551475 625-513 325-239 831-570 592-452 692-538 603-520 1361-1129 468-406 487-400
1 2 3 4 5 6 7 8 9 10 11 12
G, by eqn. (20)
G, by eqn. (18)
C, of the sediment at the critical point
(kg m-‘)
(kg m-‘)
(kg m-‘)
22.7 3.920 3.683 3.422 5.670 3.340 3.700 4.528 3.780 1.930 4.950 5.020
21.0 3.5-3.7 3.2-3.3 3.4-3.6 5.2-5.9 2.3-2.5 3.4-3.6 4.14.2 4.W.2 2.2-2.4 5.0-5.8 6.4-7.1
25.2 3.94.1 3.6-3.7 3.9-4.3 6.1-6.4 2.7-2.9 3.9-4.1 4.7-4.9 4.9-5.8 2.5-2.7 6.1-6.4 7.4-8.0
decrease is cu. 0.05-0.08 mm, which is an admissible small value. Table 3 supplies further information: - The values of @Jmax corresponding to the last point considered in each test in the non-compression zone, are always between 0.42-0.66. This agrees with our earlier comment that the maximum value of (jA),,, at the critical point must be less than 0.65. Note that the only mathematical restriction introduced in the calculation of the data is that j4S must be less than 1. - The values of C,, and C, have been calculated as indicated previously. As the determination of the point T is not exact, an interval for the values of C,, is presented. We observe a good agreement between the two values of C,, obtained in different ways. This confirm thevalidity of the assumption and hypothesis considered. The third value C, corresponds to the mean solids concentration of the sediment at the critical point. This latter value must be greater than the values of C,, obtained by the two procedures explained before as occurs in all the tests. Consequently, the procedure proposed for the characterization of the aggregates (diameter and amount of water retained in each aggregate) does not present any contradictory aspect and the similarity of the values of C,, confirms the physical significance of the model’s parameters obtained. Other interesting conclusions can be deduced from the data corresponding to the metal hydroxide aggregates formed with different hydroxides, NaOH, Ca(OH), and Ca(OH), + Mg(OH),. - The aggregates formed with Ca(OH), are similar to those formed with NaOH, when the Ca(OH), is not introduced in excess. This can be deduced by comparing test numbers 6 and 9. If the Ca(OH), is added in excess, the aggregates formed are greater and contain a larger amount of water (see number 10). This agrees
Ca(OH), used (without metal ions) requires the addition of flocculants or a large initial solids concentration. - When Ca(OH), (70% wt.)+Mg(OH), (30% wt.) is used, the formation of larger crystals is possible because of the slower rate of precipitation due to the Mg(OH), [15]. This effect is more pronounced in test numbers 1 and 5 (hydroxides of Ni+‘, Cu+’ and Zn+‘), where the amount of water retained is less than in other cases. Figure 5 shows the variation of the mean aggregate diameter d, VS. the mean value j. Considering the parameters corresponding to test numbers 1 and 9, we can observe a general tendency of variation. This variation is in accordance with others proposed in the
with
Fig. 5. Aggregate
the
exnerimental
fact
that
the
BreciDitate
I
I
of
looh
1
111”1
I
I
50
100
500
200
1000
2000
1
I -
-
diameter
d,, vs. j (m3 aggregate
m-’
solids).
literature with clay-aluminium aggregates [16] and with diatomaceous earth aggregates [5], where it is deduced that the density of the floe decreases as the floe diameter increases.
Solving eqn. (26) from t=O, v=O, Xi =0 to Vi= 0.99V,( 1 - &)‘, Xi =X,,99 we can obtain the ratio between the distance X0., and the aggregate diameter d, Xo.&f, = 3.6151/,2~,(1- $~a)~~l((~a - filgd,) = 3.615(1- +a)1o/N
Analysis of the inertial effects in the noncompression zone
where
A debatable aspect of the sedimentation theory is the assumption concerning the inertial effects and their influence on the data obtained in batch testing. Dixon et al. [17] presented an interesting study of the inertial effects in sedimentation. Using a dimensionless parameter N (ratio between acceleration force and inertial effect) equal to 0.07, they concluded that a continually expanding zone as indicated in Fig. 2 (zone b) cannot be formed in the non-compression zone. Dixon et al. [17] used the following expression, deduced from momentum balance applied to the movement of solids:
.
(25) where V, is the settling rate of a sphere at infinite dilution, Vi is the velocity of the ith sphere, 4, is the concentration of the ith sphere. Although in Dixon’s model, the term corresponding to the liquid acceleration is omitted, this equation has been considered only for studying the order of the influence corresponding to the inertial effects. Equation (25) can be written as p,d(V,)ld=g(p,
--MI
- (I - Q-“vi/K)
(27)
(26)
where vi is the downward velocity of the aggregate (equals -u,). Values ‘n’ in accordance with eqn. (14) are between 4.9 and 5.1. A mean value II equal to 5 is considered. When the inertial effects have a considerable influence on the sedimentation process, the acceleration period of an aggregate to obtain the corresponding settling rate in accordance with the momentum balance is not small in comparison with the time period to change other equilibrium conditions (corresponding to another solids concentration). For the batch testing, the aggregates undergo acceleration or deceleration depending on the real settling rate that they have at this time and the settling rate they would have if the solids concentration did not change. Consider the distance run by an aggregate at any solids concentration when it passes from rest (vi =0) to 99% of the settling rate corresponding to the equilibrium forces (in this case 6 = 0.99V,(l- 4.)‘).
N = (~a - &&l(G
Pa)
(28)
The parameter N is a ratio between acceleration and inertial effects. The corresponding values N are presented in Table 4. We can observe that the values N are very far from the value 0.07 considered by Dixon et al. [17]. The ratios X,,,/d, for the initial solids concentrations (when c$~is minimum) are also presented in Table 4. The small values Xo.99/d, obtained confirm the assumption that the inertial effects are negligible. With the exception of test 1, in all the tests an initial induction period can be observed at the beginning of the test, where the upper interface descends with increasing velocity. This can be explained by the formation of stable aggregates, for a small initial period, after agitating the suspensions for their homogenization. In Fig. 6, the mean values of d, are plotted vs. the parameter N (ratio between acceleration force and inertial effects). Despite the variety of metal hydroxides and conditions of precipitation, all the data are well correlated by the single regression equation log,, d, = - 2.344 - 0.5714 log,, N
(29)
where d, is expressed in meters. The correlation coefficient is 0.9625 and the percent of residuals about the mean explained is 96%. Equations (ll), (28) and (29) yield d~25pa(pa-p,) = 1.2658 X lO4(18~(1+ 0.15Re”.687))2/g (30) which is a quadratic equation in p8. Using a binomial TABLE
4. Variation of da, N and X,.,/d,
Test no.
Range of d, (m) 104
N
Xo.%Jda
2 3 4 5 6 7 8 9 10 11 12
2.06-2.07 12-13 8.9-9.4 12-13 8.1-9.0 lo-11 6.4-7.0 lo-11 8.6-8.9 28-30 7.0-7.3 7.7-8.2
173 7.9 17.1 9.3 18.4 13.4 43.1 9.5 18.6 2.5 31.4 18.5
0.001 0.010 0.008 0.005 0.005 0.025 0.002 0.011 0.034 0.014 0.003 0.003
226 101
correlations indicated by eqns. (29) and (30) have been obtained.
50
Acknowledgements Credits: Support for this work was provided by Environmental Agency, Generalidad Valenciana, Valencia, Spain.
E 10 73 b 0 5
List of symbols 2
AVI f
1 2
5
10
20
50
100
200
C
N
Fig. 6. Aggregate diameter force and inertial effects).
G
d,vs.N (ratio between acceleration
G,
expansion of the surd and noting that 0
g
H HC 4 H2*
Ho i
Conclusions 1. From the experimental data obtained in batch tests, the mean aggregate diameter and the average volume of water retained on an aggregate can be estimated. For the aggregates of metal hydroxides studied, the values of the diameter are in the range 2x 10m4- 1.3 x 10e4 m and the values of the ratio between total volume and the volume of solids in an aggregate are between 70 and 1400. 2. The influence of the inertial effects on the aggregates is measured, on considering a dimensionless parameter N, ratio between acceleration force and inertial effects. Taking into account the experimental results obtained from metal hydroxide suspensions, the
L M N n OF Re Vi Vo VW v,
aggregate volume index (equal j) parameter in eqn. (14) parameter in eqn. (14) parameter in eqn. (14) solids concentration, kg solids mm3 solids concentration corresponding to the boundary between settling zone and compression zone, kg solids mm3 solids concentration of the sediment at the critical point, kg solids mm3 aggregate diameter, m diameter of a hypothetical particle which has the same amount of solids as the aggregate, m gravitational acceleration, m sb2 supematant-pulp height, m value of H at the critical point, m intercept of the tangent to the upper curve with the H axis, m value of H corresponding to the point T (intercept of the characteristic which arises from the origin tangentially to the sediment, m initial value of H in batch testing, m volume of aggregate per volume of solids (equals AVI) sediment height, m parameter of eqn. (22) dimensionless number, ratio between acceleration force and inertial effect (see (eqn. (29)) exponent of eqn. (1) and (13) objective function (see eqn. (21)) Reynolds number ( = p,d, V,/p) downwards settling velocity of the aggregate in the acceleration period, m s-’ unhindered settling velocity, m s-’ unhindered settling velocity of a solid particle that would have the same amount of solids as an aggregate, m s-l settling velocity of aggregates, m s-l
227
V=P
experimental
value of the settling velocity, m
References
S-l
Vcdc
calculated
value of the settling velocity, m
S-l t
X X 0.99
time, s or h distance, m distance run by a floe from rest to 99% of the settling rate, m
Greek letters E suspension porosity volume fraction of solids corresponding to es1 the boundary between settling zone and compression zone liquid viscosity, kg m-’ s-l P aggregate density, kg rnd3 Pa liquid density, kg mT3 PI solid density, kg mm3 PS volume fraction of aggregates or floes volume fraction of solids 2 initial volume fraction of solids 60
1 B. Fitch, AIChE J., 2.5 (1979) 913. 2 J. F. Richardson and W. N. zaki, Tram. Inst. Chem. Eng., 32 (1954) 35. 3 A. S. Michaels and J. C. Bolger, Znd. Eng. Chem. Fundam., I (1962) 24. 4 W. R. Knocke, J. Water Pollut. ControL Fed., 59 (1986) 784. 5 R. V. Stephenson, J. M. Montgomery and E. R. Baumann, Partic. Sci., Technol., 4 (1986) 237. 6 J. M. Couison and J. M. Richardson, Chemical Engineering, Vol. II, Chapter 3, Pergamon Press, oxford, 1978. 7 J. Gardside and M. R. AI-Dibouni, Znd. Eng. Chem. Process Des. Dev., 16, 2 (1977) 206. 8 G. J. Kynch, Trans. Faraday Sot., 48 (1952) 166. 9 F. M. Tiller, AIChE J., 27 (1981) 823. 10 B. Fitch, AZChE J., 29 (1983) 940. 11 R. Font, AIChE J., 34 (1988) 229. 12 R. Font, AIChE J., 36 (1990) 3. 13 P. T. Shannon, R. D. Dehaas, E. P. Stroupe and E. M. Tory, Ind. Eng. Chem. Fundam., 3 (1964) 250. 14 B. Fitch, AIChE I., 36 (1990) 1545. 15 J. Tering, Plat. Surf: Finish., 11 (1986) 34. 16 N. Tambio and Y. Watanabe, Water. Res., 13 (1979) 409. 17 D. C. Dixon, P. Souter and J. E. Buchanan, Chem. Eng. Sci., 31 (1976) 737.
217
Sedimentation batch test: Application to deduce some parameters aggregates in metal hydroxides suspensions
df
R. Font*, A. F. Marcilla and C. ZofFmann Divi.Sn
de Ingenierla Quimica, Facultad de Ciencias, Universidad de Alicante, Apartado 99, Alicante (Spain)
(Received October 11, 1990; in revised form February 14, 1992)
Abstract The size of aggregates and the amount of water in them were estimated from batch sedimentation tests of metal hydroxides suspensions. This method is suitable for slurries in which the initial solids concentration is less than the critical value (limit between the non-compression range and the compression range of solids concentration). Inertial effects were negligible for these suspensions. The aggregate diameter and the parameter N (ratio between the force of acceleration and the inertial effect) were strongly negative correlated.
Fundamentals
Introduction Sedimentation is a unit operation of liquid-solid separation which allows either the solids concentration in a suspension to be increased or a liquid clarification with nil or very small solids concentration to be obtained. In many industrial processes, wastewaters with a high metal ion content are obtained. Sometimes the recovery of the metals is required and on other occasions it is necessary to eliminate the metals contained in the suspension in order to obtain clear waters. These can either be recirculated in the industrial process (with the corresponding commercial profit) or be discharged into the public irrigation channels (in accordance with the existing regulations). Once the metal hydroxides have been precipitated, it is sometimes necessary to add flocculants to the formation of floes or aggregates with great density and/ or size. These aggregates can then be separated by sedimentation due to the force of gravity. In concentrated metal hydroxide suspensions, large aggregates are formed, which can sediment without addition of flocculants. The main aim of this paper is to characterize the aggregates formed from the data obtained in a sedimentation batch test. Two parameters of the aggregates are considered: a) the amount of water retained in an aggregate, and b) the diameter of the aggregate. The results obtained have been proven logical.
*Author to whom correspondence
should be addressed.
a) Basic relationships and the numerical method proposed for the determination of the aggregate diameter and the amount of water retained in an aggregate
Two zones can be distinguished in the sedimentation of stable aggregates (constant amount of solids in each aggregate): a) non-compression zone (or hindered settling zone or free settling zone) where the aggregates descend separately and the solids concentration controls the sedimentation rate, and b) compression zone, where the increase in solids concentration is due to the unbuoyed weight of the upper solids and taking into account the resistance force due to the liquid flow. The calculation of the thickener area per unit of solids flow sometimes depends only on the relationship between the settling rate and the solids concentration [l]. Some equations have been proposed to correlate the settling rate of aggregates and particles as a function of the solids concentration. The main relationships with physical significance are discussed as follows: a) Using dimensional analysis, Richardson and Zaki [2] developed a general correlation for the hindered settling and fluidization of nonflocculent particles of the form: V, = V&
(1)
where V, is the settling velocity, E is the suspension porosity, V0 is the intercept velocity at E= 1, and n is a function mainly of the particle Reynolds number, Re. For spherical particles, the intercept velocity was equal to the terminal, unhindered settling velocity of a single
0 1992 - Elsevier Sequoia. All rights reserved
218
particle, V,. For conditions of creeping flow (Re < 0.2) and a low ratio of particle diameter to settling column diameter, n was 4.65. b) Michaels and Bolger [3] observed that the settling units of the flocculated kaolinite suspensions examined in their investigation were individual aggregates, and the effective porosity of the suspension would, therefore, be interaggregate porosity. Equation (1) can be expressed as: V, = VU(1 - 4$.65
(2)
where +a is the aggregate volume fraction. Letting +a equal j& where j is the volume of the aggregate per volume of solids inside the aggregate and A is equal to the volume fraction of solids (m3 of solids mm3 of suspension), eqn. (2) becomes: V 1/4.65,V~/4.65_ Vol/4-65j~s s
(3)
The parameter ‘j’ is also known as the Aggregate Volume Index (AVI). The volume of water retained per unit of volume of solids in an aggregate equals ‘j- 1’. The amount of water per unit of any extensive property of the solids can easily be calculated from the value y- 1’ and the densities. Michaels and Bolger [3] extended the utility of eqn. (3) to the estimation of the size and density of the aggregates by expressing V, as Stokes’ Law for the terminal settling velocity of a spherical aggregate in creeping flow: V, =g(p, - pX/(l8~)
(4)
where g is the gravitational acceleration, d, is the aggregate diameter, p is the liquid viscosity, pa is the aggregate density and p, is the liquid density. The volume fraction of solids can be calculated by the equation: A = CJPS
(5)
where C, is the solids concentration, and p. is the solids density. For stable aggregates, the parameter j is equal to: j=m’
l/67@) = (d&,)3 aggregate mm3 solid = 1/67~7.Q( (6)
where d, is the diameter of a hypothetical particle which has the same amount of solids as the aggregate. The aggregate density pa equals Pa = (Ps + o’-
l>PJO
(7)
From eqns. (4), (6) and (7) we deduce that: K,=g(p,-pJdZpj”l(l8pj)
= Vwj-1’3
- ~JdzpIW4
v:/4.65 =(Vwj-l/3)1/4.65
_(V,j-l/3)l/4.65j,+s
(10)
If 7 (m3 of aggregate me3 of solids in an aggregate) is constant and the assumptions are acceptable, plotting V,1’4.65vs. 4, should give a linear variation. Frequently, this variation is not linear which could be due to the following reasons: a) Some of the experimental settling velocity data correspond to the compression zone where the sedimentation velocity depends on many factors, and consequently eqn. (2) cannot be used to correlate these data. b) The parameter ‘j’ is the ratio between the volume of an aggregate and thevolume of solids in this aggregate. As an aggregate is the result of the union of several particles, between these particles in each aggregate there is fluid which occupies the free space. When the aggregate descends, it has also a thin exterior layer of fluid 131, which can be large or small depending on the settling rate and on the proximity of the other aggregates. A variation of j, therefore, can be due to the variation of the thickness of the layer surrounding the aggregate. Some researchers have used plots of Vs1’4.65m. $s to deduce parameters of the aggregates formed during the sedimentation of wastewater sludges [4] and flocculated suspensions [5]. Some aspects must be considered when K1’4.65is plotted ZJS.& - Firstly, all the experimental data of settling rates must correspond to the hindered settling zone or noncompression zone. - Secondly, if the variation of V:‘4.65 vs. 4s is curved, it is not correct to determine the values of VW and ‘j’ from the slope and intercept of the tangent at a point in the curve observed. A mathematical discussion of this aspect can be found in [l]. c) Equations (4), (8) and (9) are valid in creeping flow (when Re < 0.2). For values of aggregate Reynolds number (Re = p,daV,/p) between 0.2 and 500, a correction factor must be taken into account (6). In this case, V,=g(p,-p,)~I(18p(1+0.15Re”~687))
(11)
On considering equations ‘(6) and (8), the Reynolds number equals: Re =p,daVolp = p,dpj”3V,j-1”lp
(8)
From equations obtain:
(9)
J’,, =g(ps - p,)$l((18N
where v,=g(ps
V, is the free settling velocity of a hypothetical solid particle which has the same amount of solids as an aggregate. From eqns. (3) and (8), therefore, we deduce that:
=p,d,,V,/p
(12)
(6), (7), (8), (11) and (12) we can + 0.15(p,d,V,lcL)“.687)))
(13)
219
d) Garside and Al-Dibouni [7] have corrected the exponent in eqn. (1) (which has the value, of 4.65 in eqn. (2)) and have proposed the following equation to calculate the value of 12: (5.09 - n)/(n - 2.73) = 0.104ReoS877
_
-j4$
+b4S+c4,2
INTERFACE
4 L=F(l) , SEDINENT SURFACE
(15)
If the thickness of the fluid layer which surrounds the aggregate decreases when the volume fraction of solids increases, a polynomial variation of j VS. 4. can be assumed according to the equation: j=a
n
H=F(T) , UPPER
“i
(14)
This equation has been obtained for porosities (1 - &J of suspensions between 0.8 and 0.5. Under this assumption, the following equation can be written: v,= V,j_‘“(l
H
(16)
By numerical optimization, it is possible to calculate the best values of V,, a, b and c for the correlation of the experimental values of the settling rates as a function of the volume fraction of solids in the equation combined from eqns. (15) and (16). Value d, can be obtained from eqn. (13). In this way, an interval of values of j (a parameter implying the amount of water retained in the aggregate) can be calculated from eqn. (16) and the aggregate diameter d, is calculated from eqn. (6). b) Zones in a batch test of sedimentation
The settling velocity of aggregates for each solids concentration can be determined in two ways: a) Carrying out many batch tests at different solids concentration and measuring the velocity of the supematant-suspension interface in the period of time when this velocity is constant, and b) Performing a batch test in a long column and applying the Kynch [8] theory for calculating the relation between the settling rate and the solids concentration. With the first method mentioned previously, it is necessary to perform several batch tests with different solids concentration. All the suspensions must, however have been prepared and flocculated in the same way. With the second method, only from a batch test, it is possible to obtain a relation between the settling rate and the solids concentration in an interval of solids concentration (according to the hydrodynamics of the suspension and the value of the initial solids concentration). The batch test has been studied by several researchers [9,10,11,12] taking into consideration the compression zone which builds up on the bottom of the column. Consider the data plotted in Fig. 1, where the variation of the upper discontinuity interface height and the
t Fig. 1. Zones in a batch test.
sediment (compression zone), surface height are plotted TX time. An analysis of the different zones that can be distinguished can be found elsewhere [ll, 121. Kynch [8] developed a theory to calculate the relation between the settling rate of solids observed in the upper interface and the solids concentration. In this theory, it is assumed that the settling rate depends only on the solids concentration (in the settling zone, because there is no compression zone due to the fact that the suspension is considered non compressible). The same theorems are valid for zones ‘a’ and ‘b’ presented in Fig. 1, for compressible suspensions (10). According to these theorems [lo, 111, we can obtain the settling rates V, from the slopes dH/df of the curve H=f(t), and calculate the solids fractions or concentrations corresponding to each point by the equation: 4S= +H,IHi
or C, = CfioIH,
(17)
where Hi is the intercept of the tangent drawn with H-axis (Fig. 1). For zone ‘c’ (Fig. 1) the equation used to calculate A or C, is presented elsewhere [ 111. Nevertheless, when the variation of the slope dLJdt, corresponding to the variation of the sediment surface height zx. time, is not very great, the same eqn. (17) can be applied to calculate the solids concentration [ll]. The limit of solids concentration (critical concentration) between the hindered settling range (or noncompression range) and the compression range cannot be easily determined. This limit can be determined by Font’s method [ll] which considers the variations in the height of the supernatant-suspension discontinuity and the surface of the sediment in a batch test for determining the position of the point T as in Fig. 1 (point T is the intercept between the tangent to the sediment surface curve at the origin and the upper interface). In this way, it is possible to calculate the solids concentration C,, -solids mass/suspension volume - or 4, -volume fraction of solids - , corresponding
220
to the limit between the non-compression range and the compression range of solids concentration, by the equations:
where H,* is the height of the point T. Equation (18) is only a consequence of the material balance. Any relationship between the settling rate and the solids concentration has not been taken into account to obtain this equation. Shannon et al. [13] studied the batch thickening of suspensions of rigid spheres in water. They found that the final volume fractions of solids were between 0.63 and 0.65. Considering that with suspensions with different particle sizes the volume fraction of solids in a fixed bed can be somewhat larger (small particles can fill the holes between larger particles), the upper limit 0.65 has been taken into account as the volume fraction of aggregates at the critical concentration (limit between non-compression range and compression range). Nevertheless, similar conclusions to those stated in this paper could be obtained assuming a value of volume fraction of solids between 0.60-0.62 and 0.65-0.67.
Experimental results and discussion a) Experimental results of a batch test where the sediment surjace is visible. Method for corrobating the procedure proposed for the characterization of the aggregates
A suspension of Zn(OH),, Ni(OH)2 and Cu(OH), was prepared from a solution of ZnSO,, NiSO, and CuS04, where Ca(OH), (70 wt.%) +Mg(OH), (3Owt.%) was added up to pH=11.7. The initial solids concentration was 8.4 kg mm3. With this suspension it was possible to observe the sedimentation surface. The test was carried out at 20 “C. This test is labeled with number 1 and more details can be observed in Tables 1 and 2. The variations of the supernatant-suspension discontinuity height and the sediment surface height are shown vs. time in Fig. 2. We can observe that the curvature of the sediment height vs. time variation is small. The tangent to the sediment height vs. time curve at the origin intercepts at the upper interface at point T. Between the points T and C* (critical point) the variation of the upper discontinuity is nearly linear. Applying the Kynch theory we can obtain the relationship between the solids volume fraction and the solids concentration from equation (17). In zone ‘c’, the slope dH/dz is nearly constant, so its corresponding concentration can also be calculated from eqn. (17), at the limit between zones ‘b’ and ‘c’.
In this case, the positions of the points C* and T have also been easily determined by considering the variation of the sediment surface height vs. time curve. At the critical point C* where the two curves intersect, we can observe a change in the slope of dH/dt. This criterion has been used to determine the position of the critical point in the other tests, presented later in this paper, where the sediment surface is not visible. When the data corresponding to the sediment curve are not available, the position of point T (intercept of the tangent to the sediment surface height vs. time curve at origin and the upper discontinuity) can only be estimated and considered to be close to the beginning of the linear variation of the upper discontinuity height vs. time (or when values -dH/dt are nearly constant at the section just before the critical point). This procedure for estimating the position of point T can be tested from the run shown in Fig. 2 and from the batch tests with calcium carbonate suspensions presented elsewhere [ll]. It is possible that with other flocculent suspensions with a behaviour distinct from that shown in Fig. 2, the estimated position of the point T is not close to that deduced by the method proposed in this paper. As the determinations are not exact, intervals for the position of the points C* and T are calculated for the following tests, where the sediment surface is not visible. If eqns. (14)-(16) are considered as representative of the relationship between the settling rate and the solids concentration in the non-compression range, and if we accept the modifications of the Kynch theory proposed in [ll], then the following relations must be fulfilled: 1) The values iA (= 4.) corresponding to the characteristics which intercept the upper interface in the batch test must be less than 0.65 (volume fraction of aggregates less than that when the aggregates are in contact). 2) The concentration C,, (limit between non-compression zone and compression zone) calculated from two independent procedures must take a similar value. These two procedures are the following: a) Estimating the position of the point T and applying eqn. (18). b) By the following expression: 0.65(volume fraction of aggregates when they are in contact)=+*, =jCS,/pS
(19)
At the critical concentration C,,, the aggregates are in contact and the volume fraction of aggregates, therefore, must be close to 0.65 [13]. The value 3’ considered corresponds to the value obtained at the higher solids concentration (and consequently close to that corresponding to E,,). From equation (19):
221 TABLE Test no.
1. Concentrations
of metals
.
Initial concentration of the mixture (ppm)
Precipitating agent
PH supematant
1
850 Ni+’ 850 Cu+’ 850 Zn+’
Ca(OH)a 70%wt. Mg(OH)2 30%wt.
11.67
0 70 Ni+’ 0:75 cu+2 0.82 Zn+2
8.4
2
250 Ni+’ 250 cu+2 250 Zn+2
NaOH
1M
8.08
1 10 Ni+2 0:73 Cu+2 0.82 Zn+’
1.50
3
250 Ni+’ 250 cu+2 250 Zn+2
NaOH
1M
8.18
0.66 Ni+’ <0.5 Cu+2 0.44 Zn+’
1.42
4
250 Ni+’ 250 Cu+’ 250 Zn+’
NaOH
1M
8.79
<0.2 Ni+’ <0.5 Cu+2 <0.2 Zn+’
1.43
5
250 NiC2 250 Cu+2 250 Zn+’
Ca(OH)2 70%wt. Mg(OH)2 30%wt.
8.62
0.27 Ni+’ <0.5 cu+r <0.2 Zn+2
1.99
6
87.5 87.5 87.5 87.5 87.5 87.5
Ni+2 Cu+2 Zn+’ Pb+’ Fe+’ Cr+’
NaOH
1M
6.78
17 0 Nit2 0:50 cu+2 1.32 Znf2 <50 Pb+2 0.24 Fe+’ 1 Cr+’
0.756
7
175 175 175 175 175 175
NiC2 cu+2 Zn+2 Pb+’ Fe+3 Cr+3
NaOH
1M
<0.5 Ni+2 <0.5 cu+2 <0.2 Zn+2 <50 Pb+’
1.40
8
875 87:5 87.5 87.5 87.5 87.5
Ni+2 CU+~ Zn+’ Pb+2 Fe+3 Cr+’
Ca(OH)2 70%wt. Mg(OH)2 30%wt.
8.50
0.5 Ni+’ 0.5 cu+2 0.2 Zn+’ <50 Pb+’ 1 Fe+’ 1 Crc3
1.61
9
87 5 87:5 87.5 87.5 87.5 87.5
Ni+2 Cu+’ Zn+2 Pb+2 Fe+’ Cr+’
CaW-02
6.68
35.5 Ni+’ 0.5 Cu+2 11.3 Zn+’ <50 Pb+’ 5 Fe+3 1 Cr+’
0.822
10
87.5 87.5 87.5 87.5 87.5 87.5
Ni+’ Cu+’ Zn+2 Pb+’ Fe+3 Cr+3
WOW2
7.90
11
334 Al+’ 100 cl640 so;
HCI because initial pH was 11
7.0
1 Al+’ 580 Cl573 SO;
2.24
12
334 Al+3 100 cl640 SO;
HISO, because initial pH was 11
7.0
1 Al+3 100 cl1064 SO;
2.24
10.3
Concentration of metals in supematant (ppm)
Solids concentration of the initial mixture (kg m-‘)
0.910
222 TABLE Test no.
2. Parameters of correlation dp
b
a
C
Cm)
Range
Range
i
d.
O.F.
h4=2(no~?J100
Cm) 2.07E-4 2.06E-4
1.390E-2
5.27
580 455
1.3E-3 1.2E-3
l.O85E-3
1.47
3.944E6
551 475
9.4E-4 8.9E-4
l.l27E-3
1.50
-2.174E5
2.473E6
625 513
1.3E-3 1,2E-3
2.128E-4
0.65
377.0
- 5.535E4
1.831E6
325 239
9.0E-4 8.1E-4
6.676E-3
3.65
1.229E-4
927.2
-3.865E5
2.091E5
831 570
ME-3 l.OE-3
5.161E-3
3.33
7
8.310E-5
702.5
- 1.983E5
8.759E5
592 452
7.0E-4 6.4E-4
4.461E-3
2.99
8
1.273E-4
822.2
-3.05oE5
1.231E6
692 538
l.lE-3 l.OE-3
l.O22E-3
1.43
9
l.O56E-4
726.0
- 2.327E3
- 6.433E5
603 520
8.9E-4 8.6E-4
3.753E-3
2.74
10
2.721E-4
1171.0
1.56OE5
2.141E5
1361 1129
3.OE-3 2.8E-3
6.607E-5
0.36
11
9.340E-5
574.9
- 1.445E5
3.970E6
468 406
7.3E-4 7.0E-4
1.088E-4
0.47
12
1.048E-4
636.1
- 2.OOOE5
4.356E6
487 400
8.2E-4 7.7E-4
2.263E-4
0.67
220.5
69.7 70.7
1
5.013E-5
69.04
2
1.61OE-4
731.6
-2.068E5
1.897E6
3
1.150E-4
651.2
-1.516E5
4
1.490E-4
749.1
5
1.306E-4
6
Cs,=0.65pslj
33.99
(20)
The two procedures for estimating c, are independent. It the values calculated are similar, it can be concluded that the procedure proposed for correlating the experimental data is correct. In addition, this permits the estimation of the mean aggregate diameter and the average amount of water in the aggregate to be obtained. In addition, at the critical point of a batch test the mean solids concentration C, can be calculated by the equation: c, = H&,/K
(21)
where H, is the sediment height at the critical point. The value C, must be greater than C,, due to the compression of the sediment. Logically the value C, calculated by eqn. (21) must always be greater than the value C,, calculated by eqn. (18) but must also be greater than the value C,, obtained by eqn. (20). b) Other experimental results and dik.kon
In order to test the relationships presented previously, we carried out several batch runs with metal hydroxide suspensions precipitated with different bases. Table 1
shows the composition of the initial mixture (content in cations), the precipitating agent, the pH in the supematant, and the solids concentration of the suspension when the metal ions have been precipitated. The first test shown in Table 1 corresponds to that discussed in the previous section. Tests were carried out at 20 “C. In Figs. 3 and 4, the experimental results corresponding to two tests can be observed (the variations of settling rate US. time can be compared to the Tory plots presented by Fitch [14], where the settling rate data are presented VS. a reduced height. Although the magnitudes of the X-axes are different in both plots, some similarities can easily be deduced). The procedure for estimating the position of point T (intercept between the tangent to the sediment curve at origin and the upper discontinuity curve) and the critical point C* has been the following: a) values of - dH/dt have been calculated by fitting the considered value Hk, the previous one Hk_-l and the following one Hk+ 1 to second order polynomial variation with respect to the time t, and calculating the derivate at the intermediate point, b) calculated values of - dH/dt are plotted VS. time as indicated in Figs. 3 and 4. Point T is located where
dH/dt
0.8
lo4
CM/ s)
-2
H (14
0.8
o.j, , , ‘-1 6000
4000
2000
8000
T IflE t (s) Fig. 4. Batch test number 12.
H=f(t)=a,+a,t+a,tZ+a,t3+ 0 0
12
3
4
5
6
7
f(s)xlo-3 Fig. 2. Batch test number 1.
H CM)
0.2
3
2 0,l 1
0 1000
2000
3000
T I ME t W Fig. 3. Batch test number 4.
there is a marked decrease of -dHldr and point C* is located at the end of the zone where the values of -dH/dt are nearly constant or with small variation (this procedure has only been used for estimating the position of the points T and C*). The computer calculation proceeds as follows: a) Correlate the upper interface height with time by a polynomial expression such as:
8
...
(22)
In this way the oscillations of the settling rate values obtained as explained in the previous section are avoided. b) Calculate the values of the settling rates dH/dt by differentiating H=f(t) with respect to time t, and the corresponding values of solids concentration by eqn. (17). The interval considered is between the end of the initial straight line (of the upper discontinuity) and the critical point. c) Optimize by a Flexible Simplex Program, the values of a, b, c and V,,\ according to eqns. (15) and (16). Calculate, then, the value of d, corresponding to V, using eqn. (13). The Objective Function is: OF = 5 ((Vex, - L,)I(K, n-l
+ L))”
(23)
where each I/e._ is an experimental value of the settling rate, and each I’_, is a calculated value. A magnitude M, which indicates an upper value of the mean relative error is: M= 200((OF/N)1~)
(24)
The results of the correlations are shown in Table 2, where the values of d,, a, b and c, intervals of j and d, are presented. The values of it4, which are also presented in Table 2, are low as a consequence of the good agreement between the experimental and calculated values of the settling rate. We can observe that the aggregate diameter varies slightly around 67%. This’small variation is a consequence of the assumption that 7 (AVI) decreases when the solids concentration increases due to a small variation of the water thickness which surrounds the descending aggregate. This
224 TABLE
3. Comparison
No. ref.
of values
Range i
0.42 0.62 0.55 0.57 0.66 0.53 0.58 0.50 0.42 0.47 0.49 0.49
69.7-70.7 580-455 551475 625-513 325-239 831-570 592-452 692-538 603-520 1361-1129 468-406 487-400
1 2 3 4 5 6 7 8 9 10 11 12
G, by eqn. (20)
G, by eqn. (18)
C, of the sediment at the critical point
(kg m-‘)
(kg m-‘)
(kg m-‘)
22.7 3.920 3.683 3.422 5.670 3.340 3.700 4.528 3.780 1.930 4.950 5.020
21.0 3.5-3.7 3.2-3.3 3.4-3.6 5.2-5.9 2.3-2.5 3.4-3.6 4.14.2 4.W.2 2.2-2.4 5.0-5.8 6.4-7.1
25.2 3.94.1 3.6-3.7 3.9-4.3 6.1-6.4 2.7-2.9 3.9-4.1 4.7-4.9 4.9-5.8 2.5-2.7 6.1-6.4 7.4-8.0
decrease is cu. 0.05-0.08 mm, which is an admissible small value. Table 3 supplies further information: - The values of @Jmax corresponding to the last point considered in each test in the non-compression zone, are always between 0.42-0.66. This agrees with our earlier comment that the maximum value of (jA),,, at the critical point must be less than 0.65. Note that the only mathematical restriction introduced in the calculation of the data is that j4S must be less than 1. - The values of C,, and C, have been calculated as indicated previously. As the determination of the point T is not exact, an interval for the values of C,, is presented. We observe a good agreement between the two values of C,, obtained in different ways. This confirm thevalidity of the assumption and hypothesis considered. The third value C, corresponds to the mean solids concentration of the sediment at the critical point. This latter value must be greater than the values of C,, obtained by the two procedures explained before as occurs in all the tests. Consequently, the procedure proposed for the characterization of the aggregates (diameter and amount of water retained in each aggregate) does not present any contradictory aspect and the similarity of the values of C,, confirms the physical significance of the model’s parameters obtained. Other interesting conclusions can be deduced from the data corresponding to the metal hydroxide aggregates formed with different hydroxides, NaOH, Ca(OH), and Ca(OH), + Mg(OH),. - The aggregates formed with Ca(OH), are similar to those formed with NaOH, when the Ca(OH), is not introduced in excess. This can be deduced by comparing test numbers 6 and 9. If the Ca(OH), is added in excess, the aggregates formed are greater and contain a larger amount of water (see number 10). This agrees
Ca(OH), used (without metal ions) requires the addition of flocculants or a large initial solids concentration. - When Ca(OH), (70% wt.)+Mg(OH), (30% wt.) is used, the formation of larger crystals is possible because of the slower rate of precipitation due to the Mg(OH), [15]. This effect is more pronounced in test numbers 1 and 5 (hydroxides of Ni+‘, Cu+’ and Zn+‘), where the amount of water retained is less than in other cases. Figure 5 shows the variation of the mean aggregate diameter d, VS. the mean value j. Considering the parameters corresponding to test numbers 1 and 9, we can observe a general tendency of variation. This variation is in accordance with others proposed in the
with
Fig. 5. Aggregate
the
exnerimental
fact
that
the
BreciDitate
I
I
of
looh
1
111”1
I
I
50
100
500
200
1000
2000
1
I -
-
diameter
d,, vs. j (m3 aggregate
m-’
solids).
literature with clay-aluminium aggregates [16] and with diatomaceous earth aggregates [5], where it is deduced that the density of the floe decreases as the floe diameter increases.
Solving eqn. (26) from t=O, v=O, Xi =0 to Vi= 0.99V,( 1 - &)‘, Xi =X,,99 we can obtain the ratio between the distance X0., and the aggregate diameter d, Xo.&f, = 3.6151/,2~,(1- $~a)~~l((~a - filgd,) = 3.615(1- +a)1o/N
Analysis of the inertial effects in the noncompression zone
where
A debatable aspect of the sedimentation theory is the assumption concerning the inertial effects and their influence on the data obtained in batch testing. Dixon et al. [17] presented an interesting study of the inertial effects in sedimentation. Using a dimensionless parameter N (ratio between acceleration force and inertial effect) equal to 0.07, they concluded that a continually expanding zone as indicated in Fig. 2 (zone b) cannot be formed in the non-compression zone. Dixon et al. [17] used the following expression, deduced from momentum balance applied to the movement of solids:
.
(25) where V, is the settling rate of a sphere at infinite dilution, Vi is the velocity of the ith sphere, 4, is the concentration of the ith sphere. Although in Dixon’s model, the term corresponding to the liquid acceleration is omitted, this equation has been considered only for studying the order of the influence corresponding to the inertial effects. Equation (25) can be written as p,d(V,)ld=g(p,
--MI
- (I - Q-“vi/K)
(27)
(26)
where vi is the downward velocity of the aggregate (equals -u,). Values ‘n’ in accordance with eqn. (14) are between 4.9 and 5.1. A mean value II equal to 5 is considered. When the inertial effects have a considerable influence on the sedimentation process, the acceleration period of an aggregate to obtain the corresponding settling rate in accordance with the momentum balance is not small in comparison with the time period to change other equilibrium conditions (corresponding to another solids concentration). For the batch testing, the aggregates undergo acceleration or deceleration depending on the real settling rate that they have at this time and the settling rate they would have if the solids concentration did not change. Consider the distance run by an aggregate at any solids concentration when it passes from rest (vi =0) to 99% of the settling rate corresponding to the equilibrium forces (in this case 6 = 0.99V,(l- 4.)‘).
N = (~a - &&l(G
Pa)
(28)
The parameter N is a ratio between acceleration and inertial effects. The corresponding values N are presented in Table 4. We can observe that the values N are very far from the value 0.07 considered by Dixon et al. [17]. The ratios X,,,/d, for the initial solids concentrations (when c$~is minimum) are also presented in Table 4. The small values Xo.99/d, obtained confirm the assumption that the inertial effects are negligible. With the exception of test 1, in all the tests an initial induction period can be observed at the beginning of the test, where the upper interface descends with increasing velocity. This can be explained by the formation of stable aggregates, for a small initial period, after agitating the suspensions for their homogenization. In Fig. 6, the mean values of d, are plotted vs. the parameter N (ratio between acceleration force and inertial effects). Despite the variety of metal hydroxides and conditions of precipitation, all the data are well correlated by the single regression equation log,, d, = - 2.344 - 0.5714 log,, N
(29)
where d, is expressed in meters. The correlation coefficient is 0.9625 and the percent of residuals about the mean explained is 96%. Equations (ll), (28) and (29) yield d~25pa(pa-p,) = 1.2658 X lO4(18~(1+ 0.15Re”.687))2/g (30) which is a quadratic equation in p8. Using a binomial TABLE
4. Variation of da, N and X,.,/d,
Test no.
Range of d, (m) 104
N
Xo.%Jda
2 3 4 5 6 7 8 9 10 11 12
2.06-2.07 12-13 8.9-9.4 12-13 8.1-9.0 lo-11 6.4-7.0 lo-11 8.6-8.9 28-30 7.0-7.3 7.7-8.2
173 7.9 17.1 9.3 18.4 13.4 43.1 9.5 18.6 2.5 31.4 18.5
0.001 0.010 0.008 0.005 0.005 0.025 0.002 0.011 0.034 0.014 0.003 0.003
226 101
correlations indicated by eqns. (29) and (30) have been obtained.
50
Acknowledgements Credits: Support for this work was provided by Environmental Agency, Generalidad Valenciana, Valencia, Spain.
E 10 73 b 0 5
List of symbols 2
AVI f
1 2
5
10
20
50
100
200
C
N
Fig. 6. Aggregate diameter force and inertial effects).
G
d,vs.N (ratio between acceleration
G,
expansion of the surd and noting that 0
g
H HC 4 H2*
Ho i
Conclusions 1. From the experimental data obtained in batch tests, the mean aggregate diameter and the average volume of water retained on an aggregate can be estimated. For the aggregates of metal hydroxides studied, the values of the diameter are in the range 2x 10m4- 1.3 x 10e4 m and the values of the ratio between total volume and the volume of solids in an aggregate are between 70 and 1400. 2. The influence of the inertial effects on the aggregates is measured, on considering a dimensionless parameter N, ratio between acceleration force and inertial effects. Taking into account the experimental results obtained from metal hydroxide suspensions, the
L M N n OF Re Vi Vo VW v,
aggregate volume index (equal j) parameter in eqn. (14) parameter in eqn. (14) parameter in eqn. (14) solids concentration, kg solids mm3 solids concentration corresponding to the boundary between settling zone and compression zone, kg solids mm3 solids concentration of the sediment at the critical point, kg solids mm3 aggregate diameter, m diameter of a hypothetical particle which has the same amount of solids as the aggregate, m gravitational acceleration, m sb2 supematant-pulp height, m value of H at the critical point, m intercept of the tangent to the upper curve with the H axis, m value of H corresponding to the point T (intercept of the characteristic which arises from the origin tangentially to the sediment, m initial value of H in batch testing, m volume of aggregate per volume of solids (equals AVI) sediment height, m parameter of eqn. (22) dimensionless number, ratio between acceleration force and inertial effect (see (eqn. (29)) exponent of eqn. (1) and (13) objective function (see eqn. (21)) Reynolds number ( = p,d, V,/p) downwards settling velocity of the aggregate in the acceleration period, m s-’ unhindered settling velocity, m s-’ unhindered settling velocity of a solid particle that would have the same amount of solids as an aggregate, m s-l settling velocity of aggregates, m s-l
227
V=P
experimental
value of the settling velocity, m
References
S-l
Vcdc
calculated
value of the settling velocity, m
S-l t
X X 0.99
time, s or h distance, m distance run by a floe from rest to 99% of the settling rate, m
Greek letters E suspension porosity volume fraction of solids corresponding to es1 the boundary between settling zone and compression zone liquid viscosity, kg m-’ s-l P aggregate density, kg rnd3 Pa liquid density, kg mT3 PI solid density, kg mm3 PS volume fraction of aggregates or floes volume fraction of solids 2 initial volume fraction of solids 60
1 B. Fitch, AIChE J., 2.5 (1979) 913. 2 J. F. Richardson and W. N. zaki, Tram. Inst. Chem. Eng., 32 (1954) 35. 3 A. S. Michaels and J. C. Bolger, Znd. Eng. Chem. Fundam., I (1962) 24. 4 W. R. Knocke, J. Water Pollut. ControL Fed., 59 (1986) 784. 5 R. V. Stephenson, J. M. Montgomery and E. R. Baumann, Partic. Sci., Technol., 4 (1986) 237. 6 J. M. Couison and J. M. Richardson, Chemical Engineering, Vol. II, Chapter 3, Pergamon Press, oxford, 1978. 7 J. Gardside and M. R. AI-Dibouni, Znd. Eng. Chem. Process Des. Dev., 16, 2 (1977) 206. 8 G. J. Kynch, Trans. Faraday Sot., 48 (1952) 166. 9 F. M. Tiller, AIChE J., 27 (1981) 823. 10 B. Fitch, AZChE J., 29 (1983) 940. 11 R. Font, AIChE J., 34 (1988) 229. 12 R. Font, AIChE J., 36 (1990) 3. 13 P. T. Shannon, R. D. Dehaas, E. P. Stroupe and E. M. Tory, Ind. Eng. Chem. Fundam., 3 (1964) 250. 14 B. Fitch, AIChE I., 36 (1990) 1545. 15 J. Tering, Plat. Surf: Finish., 11 (1986) 34. 16 N. Tambio and Y. Watanabe, Water. Res., 13 (1979) 409. 17 D. C. Dixon, P. Souter and J. E. Buchanan, Chem. Eng. Sci., 31 (1976) 737.