2859
Ind. Eng. Chem. Res. 1994,33,2859-2867
Permeability Values from Batch Tests of Sedimentation Rafael Font,*’+ Manuel Pbrez? and Carlos Pasto$ Departamento de Ingenieria Quimica and Departamento de Ingenieria de Sistemas y Comunicaciones, Universidad de Alicante, Apartado 99,Alicante, Spain
Three methods for determining the permeability of suspensions in the compression range are compared, with use of calcium carbonate suspensions. The three methods are as follows: (a) the method proposed by Michaels and Bolger (1962)considering the results obtained from some batch tests of sedimentation, (b) a n approximate method presented elsewhere (Font, R. Chern. Eng. Sei. 1991, 46, 2473-2482) assuming a possible distribution concentration inside the sediment, and (c) a method considered in this paper, based on the determination of the solids concentration in a batch test. With the latter method, values of permeability can be determined a t different solids concentration and times. A discussion of the variation of the permeability values is also presented.
Introduction
SOLIDS CONCENTRATION
This paper discusses the determination of the permeability of suspensions from batch tests, taking into account the analysis of the sedimentation presented in previous papers (Font, 1988, 1990, 1991). The data obtained in batch testing have been used for continuous thickener design by different researchers. Two design parameters have been considered: the cross-sectional area of the thickener and the sediment or sludge height. The calculation of the cross-sectional area per unit of solids volumetric flow has been taken into account by different authors (Coe and Clevenger, 1916;Talmage and Fitch, 1955;Fitch, 1966,1979,1983, 1990;Font, 1988;Tiller and Chen, 1988;Landman et al., 1988). The variation of the solids concentration vs depth in a continuous thickener can be similar to that indicated in Figure 1. Above the sediment, there is a zone with a nearly constant solids concentration in the noncompression zone (or hindered settling range), which limits the loading rate of the thickener. In the sediment or compression zone, there is a gradient of solids concentration between the critical one at the sediment surface (at which the solids structure first shows a compressive yield value) and the underflow solids concentration at the bottom. In a previous paper, it was deduced that from data of a batch test with nonopaque suspensions, an upper limit for the sludge height can be obtained, which is greater than that necessary in the performance of the continuous thickener (Font, 1990). A more exact determination can be made on considering the momentum balance presented by Fitch (1983)and Tiller (1981).The equation of motion neglecting inertial effects states that the effective pressure ‘$8)) on each layer of particles is the result of the downward force of the unbuoyed weight of particles minus the upward force of friction due to Darcian flow of the liquid, in accordance with
LIQUID
INLET
OUTLET
t
L--\
I I
L
\
DEPTH
UNDERFLOW STREAM
SU
Figure 1. Volume fraction of solids vs depth in a continuous thickener.
and settling solids respectively, “$’ is the liquid viscosity, and “k” is the permeability. For batch testing where u and ushave opposite signs UE
+ (Us)(l- E) = 0
(2)
u - us= -u&
(3)
From eq 2
For batch testing, from eqs 1 and 3 d P 8
de,
(
8% &) =
_.--
(e, - @)&I
- E) - $-us)
Note that usis a function of the volume fraction of solids alone in the suspension and a function of the permeability k and the pressure gradient of the liquid in the sediment. For continuous thickeners, it can be written that
+ ECU
u,, = €;(Us) where rc&r and “cS” are the volume fraction of voids (porosity) and the volume fraction of solids in sediment respectively, “u” and “us))the average velocity of liquid Departamento de Ingenieria Quimica. *Departamento de Ingenieria de Sistemas y Comunicaciones.
(4)
(5)
then the solid velocity relative to the average is
u; = us - u, = €;[Us- ul
(6)
where ‘6: is the volume fraction of solids and “u? is the settling rate with respect to the mixture. Under steady state
0888-5885/94/2633-2859$04.50/0 0 1994 American Chemical Society
2860 Ind. Eng. Chem. Res., Vol. 33, No. 11, 1994
8 = €;Us= €;,U,
H Im)
(7)
= E;uU;
where “eiU,, is the underflow volume fraction of solids, “0” is the thickener volumetric flux density of solids, and ‘‘v? is the downward velocity of pulp resulting from the underflow withdrawal. Solving for usis eq 7 and substituting into eq 6 yields
+ -
Experimental Points
~
lti
0
=0.0626
so
-e e us - u,, - - - e;
efu
Substitution into eq 1 leads t o
The compression zone height “Huncan be calculated by the equation (Font, 1990) (dp$de,)de;
H, =
S’iU ESl
I
1
1 \
(10)
0
1000
2000
3000
t(s)
Figure 2. Batch test with a calcium carbonate suspension and an initial volume fraction of solids equal to 0.0626.
where “esl”is the critical volume fraction of solids (limit between compression zone and noncompression zone) (Dixon, 1981; Landman et al., 1988; Tiller and Chen, 1988). The relationship between the permeability and the solids concentration must be known t o solve the integration of eq 10. This relation can also be useful for fluidized liquid-solid systems, filtration at low pressure, percolation of liquid, and characterization of the suspension. The main objective of this paper is the comparison of three methods of determining values of permeability from the data obtained in batch tests. Each one of these methods is discussed in the following sections and applied t o the experimental data obtained with calcium carbonate suspensions. Prior to these sections, the results of some batch tests are presented, which lead to the determination of the permeability in the noncompression range, the critical volume fraction of solids “ E ~ I ” , and the relationship between the effective pressure ‘>S,, and the fraction of solids “es”. In addition, an analysis of the hypothesis considered is also presented. (Some researchers consider that the application of the Kynch theorems to the noncompression region in batch tests with compressible suspensions is not correct.) The method presented here is applied to a nonopaque suspension (in which the sediment surface can be observed directly) and can also be applied to batch tests where the location of the sediment surface can be estimated from some batch tests by joining the critical points, as proposed by Fitch (1983).
Materials Calcium carbonate suspensions have been used in this research. The chemical composition of the commercial calcium carbonate was the following: CaC03 = 98.47 wt %; MgC03 = 1.00 wt %; Fez03 = 0.016 wt %; A1203 = 0.20 wt %; Si02 = 0.22 wt %. The volume mean diameter obtained by laser diffraction and X-ray absorption sedimentation techniques was 10.5 pm. The granulometric weight distribution was the following: less then 2 pm, 10%; between 2 pm and 4.6 pm, 20%;
between 4.6 pm and 7.6 pm, 20%; between 7.6 pm and 12 pm, 20%; between 12 pm and 22 pm, 20%; and between 22 pm and 80 pm, 10%. All the results presented in this paper correspond to fresh suspensions, with similar hydrodynamic behavior. The experiments were carried out at 20 “C in graduated cylinders of differing heights, from 0.2 to 1.2 m and with an internal diameter equal to 0.06 m. The particle density at 20 “C was 2554 kg/m3. No colorants were used to observe the sediment surface, as its level could be detected by the different aspect of the suspension at both sides of the sediment surface.
Determination of the Relation between the Settling Rate and the Solids Concentration in the Noncompression Range and the Critical Concentration Seven batch tests at different initial volume fraction of solids (0.0245,0.0332,0.0423,0.0518,0.0625,0.0962, 0.1126) were carried out. Figure 2 shows the results obtained in one test where the variation of the heights of the upper discontinuity (or supernatant-suspension discontinuity) and of the sediment (or compression zone) are plotted vs time. The following zones can be distinguished: the sediment or compression zone a t the bottom and the noncompression zone separated by the sediment interface, which could be observed because there is a great difference of solids concentration at both sides of the sediment. In the noncompression zone, it is considered that the settling rate of solids forming separated aggregates depends only on the solids concentration, and as a consequence of the previous assumption, characteristic lines (loci of points with the same settling rate and solids concentration in the diagram height time) can be drawn. On the other hand, in the sediment or compression zone, there is a matrix of solids with internal channels and the settling rate or subsidence rate depends on the solids concentration as well as on the compression pressure transmitted by the upper layers, as indicated by eq 1. Inside the noncompression zone (Figure 21, three zones can be considered: (a) zone I, where the solids
Ind. Eng. Chem. Res., Vol. 33,No. 11, 1994 2861 concentration is constant, (b) zone 11, where the characteristic lines arise from the bottom of vessel, and (c) zone 111, where the characteristic lines arise from the top surface of the sediment. Zones I and I1 are separated by the characteristic corresponding to the initial solids concentration and intercepts on the upper discontinuity height “H“ at the point where the absolute value of the slope “I-dH/dtl” begins to decrease. Zones I1 and I11 are separated by the characteristicthat arises from the bottom, tangently to the sediment. A wider discussion of the assumptions considered can be found elsewhere (Font, 1988,1990,1991).With the procedure presented in a previous paper (Font, 1988)and taking into account the characteristic lines that arise from the bottom of the cylinder or from the sediment surface, for ” each test the critical volume fraction of solids ‘ ‘ ~ ~ 1and some values of the settling rate are calculated for different volume fraction of solids “4s”. The values of the critical volume fraction of solids have been calculated by the equation (Font, 1988)
-
4 S 3 0
-
H“,
(12)
(-
’-t,) dt, - t,
(for zone 111) (13)
Figure 2 shows the significance of each parameter. Drawing from different characteristics, it is possible to obtain values of the settling rate (determined by “-dH/ dt” at the top of the characteristic lines) and the corresponding volume fraction of solids U+s” by eq 12 (for zone 11) or eq 13 (for zone 111). Some values ofthe settling rate obtained from the different tests are plotted vs the volume fraction of solids in accordance with the Richardson and Zaki (1954)relation in Figure 3. It can be observed that the differences between the values obtained from distinct tests are small, thus corroborating the hypothesis considered. The settling rate values have been correlated with the fraction volume of solids by the following expression (-us)= (5.357
I
\
Initial Volume Fraction
0
0.0245
0
0.0332
A T’
00519
0
00626
0n424
I
00962
?K
01135 I
0.2
I
I
I I
I
(11)
being “4so”the initial volume fraction of solids and “If2n the intercept height of the upper discontinuity with the characteristic that arises from the bottom, tangently to the sediment (separating zones I1 and 111). The values calculated from the different tests are between 0.140and 0.160. A mean value of 0.155 has been considered. In zones I1 and I11 (see Figure 2), the relations between the settling rate calculated as “-dH/dt” and the volume fraction of solids are calculated respectively by the equations
4 8 3 0 exp = H,, - L,
0.3
- a4e)4.66 (dS) (14)
where
for 0 < 4s < 0.0668 a = 5.494 for 0.0668 < 4s < 0.155 a = 7.78033.111r$s 27.0284; A discussion of the significance of eq 14 can be found in Appendix I of a previous paper (Font, 1991). In
+
0.1
O
0,05
0,15
081
&S
Figure 3. Settling rate vs volume fraction of solids in the noncompression range.
accordance with this analysis, the aggregate diameter for the calcium carbonate suspension used in this work is 140pm when q5soequals 0 and 118 pm at the critical volume fraction of solids-0.155. This difference between the values of the aggregate diameter can be due to the fact that the layer of water surrounding the aggregates is thinner, at a high volume fraction of solids when the aggregates are closer, than at a low volume fraction of solids. With the estimated values of aggregate diameter, the Reynolds number is “0.42”, which is close to the limit “0.2”for creeping flow. On the other hand, the value of exponent “n” calculated by the Garside and Al-Dibouni (1977)relation is close to the value 4.65.
Determination of the Relation between the Effective Pressure and the Solids Concentration This relation was obtained from several batch tests, on considering only the height data after a long period of time, 30-40 h, when the sediment height does not vary and consequently corresponds t o the maximum possible compression height at time infinity. These values depend on the distribution of the solids, in accordance with the pressure of the upper layers. The method used for obtaining the relation p s = fies) is explained in a previous paper (Font, 1991). Twentythree different experiments were carried out with different initial solids concentration but less than the critical one. Figure 4 shows the variation of the mean volume fraction of solids as a function of the sediment depth “2’.Note that the value eel (critical volume fraction of solids) corresponds t o the composition of the upper layer of the sediment assuming that the initial solids concentration is less than the critical solids concentration. The 23 experiments correspond to different initial solids concentrations greater than the
Ind. Eng. Chem. Res., Vol. 33, No. 11, 1994
2862
1,1
IH ( m ) ~
I = f(z)
€
Initial Volume Fraction
I
--
0 187
,97
= f(z)
b
0,8-
,
Experimental Points
0,7
I
0
2000
4000
6000
8000
10000
t(s)
O 1 0
0,15
0,1
0,05
02
0,25
z (m) Figure 4. Volume fraction of solids vs distance z to the top of sediment at batch tests (time infinity).
critical solids concentration and to different total mass. The variation of the mean volume fraction of solids corresponding to the sediment follows the same tendency of variation, without modifination due to the different initial solids concentration used, total mass introduced, and section of the cylinder used. Note also that the volume fraction of solids for a height of 0.01 m is close to the critical value “ E ~ I ” , although somewhat higher, this again corroborates the hypothesis considered in the theory of sedimentation. From the different equations considered for the correlation of the experimental data, the most satisfactory one was the following empirical equation proposed by Tiller and Khatib (1984): where cccs_77is a parameter identified with the volume fraction of solids in the most compact state of the cake in the low-pressure range, “ ~ ~ 1is” the critical volume fraction of solids (equal t o 0.155) and “ p ~ is ” also an empirical parameter. In accordance with the relations deduced by Tiller and Khatib (19841, when the sediment height at time infinity is (‘L”, the mean volume of fraction solids %s)) is equal to E,
-=1+
ln((1
+ (E - l)e-L’G)/E)
(16)
LIG
E,,
where
G=
PB (Qs -
e)g%3-
and
By a modified simplex program, the optimum values of G and E calculated are 0.8527 and 3.056 respectively, ” 61.752 from which it can be deduced that “ p ~ equals
Figure 5. Batch tests in accordance with the Michaels and Bolger method.
N/m2 and “cs-” is equal to 0.4755. Figure 4 shows the variation deduced for the volume fraction of solids of any layer vs the distance to the top of sediment. Taking into account the parameters “ E ~ ~and ” (>B” from eq 15 the following relation can be written
p s = 61.752 In
0.3199 0.4755 - E ,
(5)
(17)
Permeability Values from the Michaels and Bolger Method Michaels and Bolger (1962) presented a method for determining the permeability data of suspensions with an intermediate solids concentration. In this paper, this method is applied to suspensions with an initial solids concentration in the compression range. It was necessary to carry out several batch tests with the same initial solids concentration (at the compression range) and with different initial heights. Using the same procedure, several tests were performed at different initial volume fraction of solids (0.152, 0.187, 0.197, 0.200,0.216). Figure 5 shows the results of three tests corresponding to Merent solids concentrations. Michaels and Bolger (1962) explained the increasing slope of the settling curves at the beginning of the test as a consequence of the fact that the structure of the aggregate network changes with time. Initially, the aggregates are interconnected and the fluid flows up between the tortuous aggregates. As settling proceeds, the aggregates tend to coalesce and line up into vertical rows and the upflow paths straighten out. Following the Michaels and Bolger method, Figure 6 shows the plotting of the settling rates “us))vs the inverses of the initial height (‘2,”. The extrapolations of the straight lines drawn to the y axis lead t o the maximum values of settling rates “u’s))in the absence of compressive force, as a consequence of the expression
On considering the methodology presented by Michaels and Bolger (1962) and Fitch (1979), the following
Ind. Eng. Chem. Res., Vol. 33, No. 11, 1994 2863
-u S(m/s)(1 E-6)
H(m)
80
60 70
1
+
0.152
1
+ 0.187
Initial Volume Fraction
0.197
\.
-e-
of Solids
0.200
0,75
0.5
0,25
0 0
600
1200
1800
2400
3000
3600
t(s) Figure 8. Lines of constant solids concentration in the sediment of a batch test (#w = 0.0962).
Bolger method is based on a hypothesis that needs clarification, which will be the subject of a future paper. Non compression
r
Permeability Values from the Variation of the Discontinuity Heights in a Batch Test The fundamentals of this method are presented elsewhere (Font, 1991). In this paper, only the application of the method and the limitations are presented. Consider the lines of constant solids concentration in the batch test shown in Figure 8. In the sediment two kinds of lines are plotted: the continuous lines correspond to the real ones, which are unknown, and the dashed lines, which correspond to a hypothetical situation, are equidistant lines to the sediment surface and correspond to a volume fraction of solids calculated by
i-
1,000E-11
I \
1,000E-12
t t
I
No compression
0
Compression
(20)
I 1 I
1,000E-13
1 I
0,OS
I1
0.1
0,15
1
0,2
0,25
03
0,35
Figure 7. Permeability vs volume fraction of solids.
equivalence is obtained:
(19) From the data of ordinate at the origin for each straight line drawn and considering the relations 18 and 19, the values of the permeability “k” have been deduced. The values of permeability “k” are plotted vs volume fraction of solids in Figure 7. As can be expected, it can be observed that the permeability decreases when the volume fraction of solids increases. In spite of the agreement between the values of permeability obtained by this method and by the other two methods discussed in this paper, the Michaels and
being “€&” the volume fraction of solids a t the bottom when the hypothetical lines arise from the bottom a t time “tl”, “q&2” the volume fraction of solids of the characteristic that arises from the top of sediment when the hypothetical line of constant concentration arises from the bottom at time “tl)),‘‘(-ue2”’ the settling rate corresponding to a characteristic line (coinciding with -dHddtZ when the characteristic intercepts with the upper discontinuity), and “dLlldt1” the slope of the sediment surface at time “tl”. Two aspects can be considered regarding the dashed lines: (a) they are imaginary and do not correspond to the real lines, and (b) although they are imaginary, the material balance is fulfilled with the distribution of solids concentration calculated. When the variation of the sediment becomes nearly lineal, a region with the maximum solids concentration in the range of low effective pressure is formed on the bottom and arises with the lines of constant concentration. The volume fraction of solids can be estimated by eq 20 using the value “dLlldtl” corresponding to the constant slope of the sediment surface (Font, 1990). A
2864 Ind. Eng. Chem. Res., Vol. 33, No. 11, 1994
normal variation of the sediment surface and the lines of constant solids concentration can be similar to those shown in Figure 8. When “dLlldt1” becomes nearly constant, the real lines of the greatest constant solids concentration can be close to the hypothetical lines. In this way, it is possible to estimate the location of the lines of constant solids concentration and to obtain therefore, a variation of solids concentration when the variation of the sediment surface becomes nearly lineal. In the real situation there can be lines above and below the dashed ones, and consequently the range of the values of permeability deduced with the assumption considered must be close to the real one. The assumption of the method is based on the fact that when “a11 dtl” is nearly constant, the real lines of constant solids concentration coincide with the hypothetical ones. On the other hand, when “dLlldt1”is nearly constant, the settling rate corresponding to the different lines of constant solids concentration can be calculated by
-
zs ( m3 sol./m3 sus.)
0,35
1=300 Sac
t=600sec
0
where “&)” is the volume fraction of solids of the line where the distance to the sediment is “z”, “dLlldt1”is the nearly constant value, and “cHb” can be obtained from
(22) where “q5~z” is the volume fraction of solids of the characteristic that arises from the sediment, when “(dL11 dtl?” is nearly constant, (i.e. the characteristic that coincides with the sediment interface and intercepts with the upper discontinuity at the critical point), “(us&” the value of the settling rate of the previous characteristic, (i.e. the value -dHddtz just before the critical point), and “(dLlldt1)”’the value constant of settling rate of sediment. Once the distribution of solids concentration and the corresponding settling rates are estimated in the sediment when “dLlldtl”becomes constant, the calculation of the permeability can be done by the momentum balance-eqs 4 and 21-written as
k=
(23)
Figure 8 shows the variation of the sediment surface of the batch test selected for the application of the method (+so = 0.0962). For the value (dLddtl? = 7.248 x d s , just before the critical point (which is nearly constant from 2400 to 3600 s), the following values have been obtained: & = 0.138; (-uSz)’ = 1.020 x 10-3 d s ; cib = 0.332. Values “k” of permeability deduced from the data of Figure 9, using eq 23 are also plotted in Figure 7.
Permeability from Batch Tests with Determination of Solids Concentration at Different Heights This method is based on the same fundamentals presented previously, but determining the solids con-
02
0,1
t=1200sec
*
t=18oosec
~
L
0,1
+
_ 03
I
_
x(m) Figure 9. Solids concentration at different heights and times from side extracting samples
=
0.0962).
centration in the sediment, instead of estimating it. A cylinder with side tubes for extracting samples at different heights was used. The determination of the solids concentration at different heights was obtained at distinct times. The procedure was the following: (a) First of all, a batch test without extracting samples was carried out at an initial volume fraction of solids equal to 0.0962. This test was the same as that used in the previous section (Figure 8). (b) Seven batch tests with the same initial solids concentration and initial height as the previous one were carried out with a different final time, when samples were extracted to determine the solids concentration in the sediment region. It was tested that the evolution of the interfaces was similar in all the tests. Figure 9 shows the experimental data of the volume fraction of solids vs distance to the bottom obtained at different times. At the top of sediment, the volume fraction of solids has been considered equal to “eSl = 0.155”. By the curves obtained from the correlation of the data shown in Figure 9, lines of constant solids concentration have been deduced and plotted in Figure 10. It can be observed that the lines of constant solids concentration are divergent, as commented on in other papers (Font, 1990, 1991). The determination of the values “(-us)” at difference distances “z” and times “tl” has been carried out described in the following. The material balance corresponding to solids is the following
where “cS” is the volume fraction of solids which is a function of the distance “x” to the bottom of cylinder at time “t”. For a layer of distance P” to the sediment interface, which arises with a velocity “dL,lldtl”,it can be written that
0,35
x(m)
0,25
0,15 1,000E-12 1
8 0.22
* ++
024
0.26 0.28
0,05
1,000E-13
0 0
600
2400
1800
1200
0
3000
t(s) Figure 10. Lines of constant volume fraction of solids
’
=
0.0962).
x
0.30
~
1
1000
2000
3000
4000
5000
t(s) Figure 11. Permeability vs time in batch testing.
that the following relation must be fulfilled: aE,(z,t) - ac,(x,t) at at
a+,t) aX
+--
d ~ , dt,
(25)
From eqs 24 and 25, taking into account that ‘‘x = L
- z”, it can be deduced that a[~,(~,t)(-u,)i_ az
- - a+,t) ~ - dt,
az
ag,(z,t) at
(26)
Integrating eq 26 from the top of the sediment (z = 0, E , = ~ ~ (-us) 1 , = (-usl)),the following can be written
hz(--)
dz (27)
On the other hand, from a material balance around the sediment surface, it can be deduced that
From eqs 27 and 28, the following expression is deduced
Using eq 29 and the data presented in Figure 10 the values of (-us)can be calculated for different distances ‘Y7and distinct times 21”. The permeability “K” can be calculated by the equation I-..
1
Taking into account that at the bottom of the cylinder the settling rate is zero, from eq 29 it can be deduced
The determination of the permeability by eq 30 for different values of time has been carried out with the following parameters deduced at each time: (a) value “dLl/dtl”obtained from Figure 8, (b)variation of “8cS(z,t)/ at” vs distance z a t different values of time from polynomial correlations obtained from data presented in Figure 10, (c) relationship between the effective pressure and the volume fraction of solids indicated by eq 17, (d) variation of “a&,t)/&” vs the distance “z” to the top of the sediment (from the polynomial correlations obtained from data presented in Figure 9), and (e) values g,, e, g, and p . The values of permeability determined for some volume fractions of solids are plotted vs time in Figure 11. It can be observed that the permeability values for each solids concentration increase with time to a constant value, showing that there is a variation in the particle packing arrangement, as indicated in other papers (Dixon, 1980; Font, 1990, 1991). The values of permeability are also plotted vs volume fraction of solids in Figure 7, together with those obtained from the previous methods. Note that with a single apparatus-a cylinder with many side exit tubes-the determination of permeabilities can be obtained from some batch tests that end at different times, when samples are extracted.
Discussion of Results Figure 7 shows the values of permeability determined by different methods. For the noncompression range, the values of permeability are obtained from the corresponding data of the settling rate (applying eq 19 substituting 4, by cs). Note that there is a logical variation of permeability from the noncompression range to the compression range. In the compression range, the three methods lead t o similar results. There are nevertheless some interesting aspects that merit consideration.
2866 Ind. Eng. Chem. Res., Vol. 33, No. 11, 1994
(1)With the Michaels and Bolger method, values of permeability are determined in the range 0.155-0.216. The determination at greater values of solids concentration could not be carried out, because it was difficult to observe the straight line in the corresponding batch test. In the other two methods, however, the range of determinate values is much greater, from 0.155 to 0.330. (2) The two latter methods based on the estimation or determination of a solids concentration distribution lead to similar results at the greater solids concentration when the divergence between the lines of constant solids concentration is not great. In any case, as commented previously, by the method of estimation of a solids concentration distribution, a range of permeability “k” is obtained, but it is possible that some values of “k” were greater and others smaller than the real ones. ( 3 )It can be observed that the decay in the values of permeability, plotted in logarithmic scale, corresponds to a solids concentration at the bottom of the cylinder in a batch test when “dLl/dtl”becomes nearly constant, as expected. Much greater sediment heights would be necessary to significantly increase the solids concentration a t the bottom of the cylinder in batch testing and also in continuous thickeners. (4)The procedure explained in the methods based on the estimation or determination of the solids concentration distribution inside the sediment has been applied to nonopaque suspensions (where the sediment surface is visible). For opaque suspensions, the variation of the sediment surface can be determined from some batch tests, detecting the critical points (where there is a significant change in the slope of the supernatantsuspension interface), as indicated by Fitch (1983). On the other hand, it has been tested that the hypothesis, assumed from the model of sedimentation with different treatments for the noncompression zone (where Kynch theorems can be applied) and the compression zone taken into account, leads to logical results and conclusions in all the parameters and relationships obtained.
Conclusions Data of permeability can be obtained from the distribution of solids concentration inside a sediment in a batch test. This distribution can be measured by extracting samples at different heights or can be estimated, on considering that, when the variation of the sediment surface height becomes lineal with time, the real lines of constant solids concentration coincide or are close to the hypothetical lines parallel to the sediment surface curve, where volume fractions of solids can easily be calculated. On considering the data of calcium carbonate suspensions, the values of permeability obtained with the determination of distribution of solids concentration are close to those found with the estimation of solids concentration (in the case of greatest values of solids concentration) and with the values deduced by the Michaels and Bolger method. In addition, the methods proposed in this paper have been useful for determining the permeability in a range of solids concentration wider than the range possible by the Michaels and Bolger method.
Nomenclature g = gravity acceleration, d s 2
H , = compression zone height in a continuous thickener, m H , = intercept height of tangent to curve H(t),m H , = initial height of suspension at t = 0, m Hz = height of descending interface at t = tz, m H i = intercept height of tangent to curve L ( t )at t = 0 with the descending interface, m Hlz = intercept height between parallel line to x axis that passes through (t1,Ll)and the tangent to curve H ( t ) k = permeability, m2 L1 = sediment height at time tl, m p s = effective pressure, N/m2 tl = time when the characteristic line arises from the sediment, s tz = time when the characteristic line intercepts to the upper discontinuity height, m us = settling velocity of solids, d s u’,= settling velocity of solids in absence of compressive force, m / s us2 = settling velocity of the solids corresponding to the characteristic lines that arise from the sediment at (L1,tl) and intercept the supernatant suspension discontinuity at (H2,td, mk x = distance down column, m 2, = initial height, m Z,1 = parameter of eq 18, m z = distance to sediment surface, m Greek Letters es = volume fraction of solids in sediment es1 = value of cs at cake surface (limit value between noncompression and compression zone) F S = mean value of E , = value of eS at the bottom of cylinder e,, = empirical parameter of eq 15 0 = volumetric flux density of the thickener, m3 solids s-l m-2 cSu = underflow volume fraction of solids Q = fluid density, Kg/m3 es = solid density, Kg/m3 p = viscosity of fluid, Kg/(m s) r$s = volume fraction of solids in suspension r $ s ~ = volume fraction of solids, corresponding to the characteristic line that arises tangently from the sediment Symbols
(y
= referring
to continuous thickener
0 = referring to when the rising rate of sediment is constant
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Received for review March 17,1994 Revised manuscript received August 3 , 1994 Accepted August 15,1994@
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@
Abstract published in Advance ACS Abstracts, October 15,
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