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Ind. Eng. Chem. Res. 1998, 37, 4152-4156
Pressure Filtration Technique for Complete Characterization of Consolidating Suspensions Matthew D. Green, Kerry A. Landman, Ross G. de Kretser, and David V. Boger* Advanced Mineral Products Special Research Center, Department of Chemical Engineering, University of Melbourne, Parkville 3052, Victoria, Australia
The behavior of suspensions in settling and compression is fully defined by two parameters: the compressive yield stress function Py(φ) and the hindered settling function r(φ). The compressive yield stress quantifies the strength of the suspension network in compression and determines the maximum concentration achievable for any given applied force. The hindered settling function quantifies the hydrodynamic drag forces experienced by consolidating particles in the suspension and determines the settling velocity and the time scale for the consolidation process. Here, both Py(φ) and r(φ) are determined using a constant pressure filtration device. The technique is explained and results for ZrO2 suspensions are presented. Using this single experiment, any consolidation process such as filtration or continuous thickening can be designed or optimized on the basis of the measured properties of the process suspension. Introduction In a flocculated suspension, the compressive yield stress, Py, defines the compressive force that must be exceeded for the suspension to consolidate at a given concentration, φ (units of volume fraction) [Buscall and White (1987)]. The Py(φ) function depends on the properties of the suspension network structuresthat is, the number, strength, and arrangement of interparticle bonds. In filtration, the Py(φ) of the process suspension defines the pressure required for a given filter cake concentration. In continuous thickening, Py(φ) defines the sediment height required to attain a certain underflow concentration. The hindered settling function, r(φ), accounts for the hydrodynamic interactions between particles in a consolidating suspension and is defined by the following equation for the settling velocity, u [Landman and White (1994)]:
u(φ) )
u0 (1 - φ)2 r(φ)
(1)
where u0 is the settling velocity of a single particle in an infinite media (Stokes law for spherical particles). At the infinite dilution limit, φ f 0, particles are unaffected by neighboring particles; thus, r(φ) f 1. As the concentration increases, hydrodynamic interactions between particles hinder the settling velocity and r(φ) increases. The hindered settling function is always finite and characterizes the consolidation rate of the suspension at all concentrations. The name of this function is thus somewhat misleading since it not only characterizes the consolidation rate in hindered settling but also in compression. In compression, the hindered settling function may alternatively be thought of as an inverse function of the permeability; instead of liquid permeating through a solid matrix, solids move through a stationary liquid phase. * To whom correspondence should be addressed. Tel.: (61) 3-9344 7440. Fax: (61) 3-9344 4153. E-mail: d.boger@chemeng. unimelb.edu.au.
While measurements of the permeabilities of suspensions are quite common in the literature, direct measurements or calculation of hindered settling type functions for suspension systems are scarce. Auzerais et al. (1990) measured the settling velocity function, u(φ), for strongly and weakly flocculated stabilized silica suspensions. Davis and Gecol (1994) have proposed a hindered settling function for polydisperse suspensions of spherical particles. The relatively recent emergence of the general consolidation theory of Buscall, Landman, and White, which incorporates the hindered settling function, is one of the reasons for this lack of experimental data. The usefulness of the hindered settling function, r(φ), is that it has a sound fundamental basis and can be used together with the compressive yield stress, Py(φ), to completely model solid-liquid separation in the design of all types of consolidation equipment. A constant pressure filtration device is used to determine both the compressive yield stress and hindered settling function for a suspension. A schematic of such a device is shown in Figure 1. A sample of initial concentration φ0 is placed in a cylindrical cell, initial suspension height H0. A piston is applied to the suspension at a constant pressure ∆P, expelling liquid from the suspension through a permeable membrane. After sufficient time, the suspension consolidates to an equilibrium state. The transient behavior of the pressure filtration is discussed later in the determination of the hindered settling function. At equilibrium, the pressure exerted by the piston is entirely supported by the suspension network structure. The applied pressure is thus equivalent to the compressive yield stress at that final concentration, such that
∆P ) Py(φf)
(2)
where φf is the final concentration of the suspension which at equilibrium is theoretically constant. Repeating the experiment over a range of pressures will thus generate the complete Py(φ) curve. Pressure filtration has been used for some time in geomechanics to determine the solids stress of soils.
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Figure 1. The two stages of filtration of an initially unnetworked (φ0 < φg) suspension: (a) initial state, (b) compact bed formation, (c) compact bed consolidation, and (d) equilibrium state.
Callaghan and Ottewill (1974), Shirato et al. (1970), Meeten (1994), and Sherwood and Meeten (1997) used a compression cell to examine the compressibility of clays. Bo¨mkes and Wagener (1987) and Holdich (1990) have similarly measured the solids pressure for various suspension systems. Recently, Miller et al. (1996) and Green (1997) used pressure filtration to explicitly determine Py(φ) for Al2O3 and ZrO2 suspensions. The hindered settling function, r(φ), is also determined from the pressure filtration experiment. The technique was developed by Landman et al. (1995) and is summarized in Landman and White (1994). An important advance in the technique is that it takes into account both the compressibility and permeability of the filter cake during formation and consolidation. The technique was used by Eberl et al. (1995) who successfully modeled the high-pressure industrial filtration of flocculated kaolin. In this experiment, the initial concentration of the suspension, φ0, must be less than the gel concentration, φgsthe concentration at which particles or flocs just contact. There are thus two stages in the filtration process as shown in Figure 1. In the first stage, a compact bed is developed that grows away from the membrane with time. The compact bed has a concentration gradient φ(z,t), where φ0 > φg, and a particle stress gradient, where Py(φ(z,t)) > 0. At the top of the bed, height z ) l(t), the concentration is the gel concentration, φ[l(t),t] ) φg and the compressive yield stress is zero. At time t ) tcf, the compact bed reaches the piston and the second stage of filtration begins. Consolidation of the bed continues until the bed is uniform in concentration, φf, satisfying eq 2. Analysis of the compact bed formation stage (t < tcf), using the exact solution of the Landman and White model, predicts a t/V versus V curve that is linear with slope m1. Surprisingly, this linear behavior agrees well with the approximate solution of Shirato et al. (1986) and Wakeman et al. (1991). Note also that while the basis of the Landman and White model is for φ0 < φg, linear behavior in t/V versus V is often observed for φ0 > φg. Small time analysis of the model for φ0 > φg predicts this linearity; however, it seems that the small
time behavior also holds for longer experimental times (Landman et al., 1991; Channell and Zukoski, 1997). From the model the following expression is obtained for the hindered settling function
2∆Pm1(φm/φ0 - 1)(1 - φm)2 λ r(φ ) ) Vp m φm
(3)
where λ is the drag coefficient for a single particle in infinite media and Vp is the particle volume. φm is a mean concentration for the developing compact bed calculated from
3 1 φm ) φr + φg 4 4
(4)
The development of and justification of the use of (4) are presented in Landman et al. (1995) who trialed a number of different forms for φm and found (4) to be the most suitable over the pressure range and data sets evaluated. The complete r(φ) curve is obtained by repeating the experiment over a range of applied pressure, (∆P, measuring the slope of the t/V versus V plots, m1, and measuring the final equilibrium concentrations of the packed beds, φf, to find the mean concentrations, φm. It is the objective of this paper to demonstrate the pressure filtration technique for the determination of both the Py(φ) and r(φ) functions for the complete characterization of a consolidating suspension. Experimental Section The filtration device was designed and built at the University of Melbourne. It was based on the device of Wakeman et al. (1991) and is similar to the devices of Lange and Miller (1987) and Miller et al. (1996). Illustrated in Figure 2, the device consisted of a cylinder holding the suspension on a membrane (0.8 µm Millipore) supported by a sintered stainless steel disc, on a perforated stainless steel plate, in turn supported by a solid steel assembly bolted to the cylinder. Two compression cylinders were used: an aluminum cylinder (280.1-mm high, 40.20-mm inside diameter) and a stainless steel cylinder (306.8-mm high, 28.40-mm
4154 Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998
Figure 3. Typical filtration rate data determined by the constant pressure filtration device (ZrO2, pH 5.7-6.0, φ0 ) 0.047-0.050).
Figure 2. Schematic of filtration device for measurement of Py(φ) and r(φ).
inside diameter). A piston, with a bleed line, compressed the suspension in the cylinder. To apply pressure to the piston, a pneumatic cylinder was used with a line pressure range of 0-600 kPa. The lower limiting pressure of the system was about 50 kPa, below which the frictional resistance of the O-ring in the piston became significant. A pressure transducer (Philips, P13) mounted on the face of the piston directly measured the solids pressure in the cylinder. A pressure gauge in the bleed line measured the hydrostatic pressure in the cylinder. A rotary encoder on the shaft of the piston measured the piston displacement (0.16 mm/pulse). An electronic balance (A&D Mercury, FA2000) measured the efflux rate from the filter. The filtration data was logged on a PC. A ZrO2 powder (mean volumetric particle diameter, d50 ≈ 0.47 µm, F ≈ 5.72 g/cm3) was supplied by ICI Advanced Ceramics Australia for the experimental work. This metal oxide system was chosen to represent a “model” suspension of a polydisperse industrial particulate system. The preparation of the ZrO2 suspensions was carefully controlled to obtain reproducible suspensions. A procedure similar to that of Leong et al. (1993) was used. Appropriate amounts of the powder were mixed with Milli-Q filtered distilled water with 0.01 M KNO3. All samples were prepared in the dispersed state using concentrated HNO3 to adjust the pH. The sample (up to 400 mL) was dispersed using a high shear mixer for 5 min (Janke and Kunkel, Ultra-Turrax operated at 10 000 rpm). The samples were adjusted to their correct pH using 1-5 M KOH and then were allowed to rest for at least 24 h. Determination of the gel point, φg, of the ZrO2 used in the experiments was performed by settling to equilibrium a small sample of suspension in a 50-mL measuring cylinder, starting from a low initial height to avoid compressive effects from the self-weight of the suspension ( φg exhibited no deviation in the t/V versus V plot. A later paper will discuss this deviation in more detail. The slope utilized in calculations is that of the initial linear portion of the t/V versus V plots. In Figure 4, typical r(φ) data are presented. Here, the effect of using the same sample for each successive pressure by re-slurrying the packed bed with the filtrate is compared with that from using a fresh sample at each applied pressure. Note that the pH and initial suspension concentration varied slightly through the filtration runs due to evaporation and other incidental losses. The r(φ) data from the re-slurried sample agrees with that determined from the fresh samples. The r(φ) data can be fitted to a function of the form
λ r(φ) ) C(1 - φ)n Vp
(5)
which satisfies the criterion r(φ) f 1 at φ ) 0 [Landman and White (1992, 1995); Eberl et al. (1995)]. The data in Figure 4 are fit to this function. Note that r(φ) is a quite weak function of φ and a variety of similar functional forms could equally well be used to fit the data in Figure 4. An estimation of the particle size can be obtained from the extrapolated intercept of the r(φ) curve fit. Here, from eq 5
λ )C Vp
(6)
Assuming spherical particles of radius, ap, the Stokes drag coefficient is λ ) 6πηap, and the particle volume is Vp ) 4/3πap3. The equivalent particle diameter, dp,
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Figure 4. Hindered settling data determined from filtration data in Figure 3. The effect of using the same sample for each successive pressure by re-slurrying the packed bed with the filtrate is compared with using a fresh sample at each applied pressure: b, re-slurried sample (pH 5.7-6.0, φ0 ) 0.047-0.050; O, fresh samples (pH 5.2, φ0 ) 0.052-0.055).
Figure 6. Effect of flocculation state on (a) r(φ) and (b) Py(φ) for ZrO2 suspensions (fresh samples used at each pressure): b, pH 5.2, φ0 ) 0.052-0.055; O, pH 6.5-6.9, φ0 ) 0.047-0.050.
Figure 5. Effect of the filtration cylinder diameter on r(φ) for ZrO2 suspensions: b, D ) 28.40 mm, re-slurried sample (pH 6.66.8, φ0 ) 0.050-0.052); O, D ) 40.20 mm, fresh samples (pH 6.56.9, φ0 ) 0.047-0.050).
is thus
9η x2C
dp ) 2ap ) 2
(7)
The accuracy of the calculated particle size is completely dependent on the extrapolated curve fit. For the ZrO2 suspension in Figure 4, the particle size calculated from the average intercept of the two curve fits is dp ) 0.12 µm. This average particle size, when compared with the measured fundamental particle size of ZrO2 (0.47 µm), thus confirms the order of magnitude of the measured r(φ) function. Wall effects were found to be significant in the measurement of Py(φ) [Green and Boger (1997)]. The effect of the filtration cylinder diameter on r(φ) for a strongly flocculated ZrO2 suspension is shown in Figure 5. Two filtration cylinders of diameters 28.40 and 40.20 mm were used. The cross-sectional area of the larger cylinder was 2 times that of the smaller cylinder. The r(φ) data obtained from each cylinder superimpose on each other well. The effect of doubling the crosssectional area for filtration and any wall effects on the r(φ) results thus appear to be negligible for the cylinder diameters used here. The effect of the flocculation state on r(φ) and Py(φ) was investigated. In Figure 6a, the r(φ) curves for
strongly and weakly flocculated ZrO2 suspensions are shown. The corresponding Py(φ) data are shown in Figure 6b, where, as expected, the weakly flocculated data are to the right [Chang et al. (1994); Channell and Zukoski (1997); Green and Boger (1997); Franks and Lange (1997)]. An obvious difference in the r(φ) results is not apparent. The weakly flocculated suspension has a slightly higher r(φ) curve, especially at higher concentrations. This result is inconclusive and more experimental work is required. The r(φ) function, however, has been suggested to be only weakly dependent on structural changes to the suspension [Eberl et al. (1995)]. The effect of the initial suspension concentration on measured r(φ) and Py(φ) is shown in Figure 7. In Figure 7a, the r(φ) curve is shifted downward as φ0 increases and approaches φg. In Figure 7b, the corresponding Py(φ) curve for the data determined in Figure 7a is shown. A φ0 effect is not observed in the Py(φ) data, although errors in the technique could mask a minor effect. A significant effect of φ0 on Py(φ) was previously found by Green and Boger (1997). Conclusions A reproducible technique for determination of the hindered settling function and the compressive yield stress for suspensions has been developed and evaluated. Results indicate that the technique is sound with extrapolated particle sizes agreeing in order with independently determined sizes. Observed changes in the hindered settling factor and compressive yield stress
4156 Ind. Eng. Chem. Res., Vol. 37, No. 10, 1998
Figure 7. Effect of initial concentration, φ0, on (a) r(φ) and (b) Py(φ) for strongly flocculated ZrO2 suspensions (fresh samples used at each pressure): b, pH 6.9, φ0 ) 0.021; O, pH 6.5-6.9, φ0 ) 0.047-0.050; 2, pH 6.7-7.1, φ0 ) 0.082-0.085.
with flocculation strength for the zirconia suspensions tested are in agreement with trends observed elsewhere in the literature. The primary importance of the technique lies in its fundamental basis, linking compressibility and permeability, and the usefulness of the unified theory for the design of all types of consolidation equipment. Further developmental and experimental work are in progress. Literature Cited Auzerais, F. M.; Jackson, R.; Russel, W. B.; Murphy W. F. The transient settling of stable and flocculated dispersions. J. Fluid Mech. 1990, 221, 613. Bo¨mkes, F. J.; Wagener, W. Investigations on pressing filter cakes subjected high pressure of one and two dimensions. Aufbereit.Tech. 1987, 9, 489. Buscall, R.; White, L. R. The consolidation of concentrated suspensions. Part 1: The theory of sedimentation. J. Chem. Soc., Faraday Trans. 1 1987, 83, 873. Callaghan, I. C.; Ottewill, R. H. Interparticle forces in montmorillonite gels. J. Chem. Soc., Faraday Discuss. 1974, 57, 110.
Chang, J. C.; Lange, F. F.; Pearson, D. S. Pressure sensitivity for particle packing of aqueous Al2O3 slurries vs interparticle potential. J. Am. Ceram. Soc. 1994, 77 (5), 1357. Channell, G. M.; Zukoski, C. F. Shear and compressive rheology of aggregated alumina suspensions. AIChE J. 1997, 43, 1700. Davis, R. H.; Gecol, H. Hindered settling function with no empirical parameters for polydisperse suspensions. AIChE J. 1994, 40(3), 570. Eberl, M.; Landman, K. A.; Scales, P. J. Scale up procedures and test methods in filtration: a test case on kaolin plant data. Colloids Surf. A 1995, 103, 1. Franks, G. V.; Lange, F. F. Consolidation and mechanical behaviour of saturated alumina powder compacts. Submitted for publication in J. Am. Ceram. Soc. Green, M. D. Characterisation of suspensions in settling and compression. Ph.D. Dissertation, University of Melbourne, Parkville, Victoria, Australia, 1997. Green, M. D.; Boger, D. V. The yielding of suspensions in compression. Ind. Eng. Chem. Res. 1997, 36, 4984. Holdich, R. G. Solids concentration and pressure profiles during compressible cake filtration. Chem. Eng. Commun. 1990, 91, 255. Landman, K. A.; White, L. R. Determination of the hindered settling factor for flocculated suspensions. AIChE J. 1992, 38 (2), 184. Landman, K. A.; White, L. R. Solid/liquid separation of flocculated suspensions. Adv. Colloid Interface Sci. 1994, 51, 175. Landman, K. A.; Sirakoff, C.; White, L. R. Dewatering of flocculated suspensions by pressure filtration. Phys. Fluids A 1991, 3(6), 1495-1509. Landman, K. A.; White, L. R.; Eberl, M. Pressure filtration of flocculated suspensions. AIChE J. 1995, 41(7), 1687. Lange, F. F.; Miller, K. T. Pressure filtration: Consolidation kinetics and mechanics. Am. Ceram. Soc. Bull. 1987, 66 (10), 1498. Leong, Y. K.; Katiforis, N.; Harding, D. B. O.; Healy, T. W.; Boger, D. V. Role of rheology in colloidal processing of ZrO2. J. Mater. Process. Manuf. Sci. 1993, 1, 445. Meeten, G. H. Shear and compressive yield in the filtration of a bentonite suspension. Colloids Surf. A 1994, 82, 77. Miller, K. T.; Melant, R. M.; Zukoski, C. F. Comparison of the compressive response of aggregated suspensions: pressure filtration, centrifugation, and osmotic consolidation. J. Am. Ceram. Soc. 1996, 79 (10), 2545. Sherwood, J. D.; Meeten, G. H. The filtration properties of compressible mud filtercakes. J. Pet. Sci. Eng. 1997, 18, 73. Shirato, M.; Kato, H.; Kobayashi, K.; Sakazaki, H. Analysis of settling of thick slurries due to consolidation. J. Chem. Eng. Jpn. 1970, 3 (1), 98. Shirato, M.; Murase, T.; Iwata, M.; Nakatsuka, S. The TerzaghiVoigt combined model for constant-pressure consolidation of filter cakes and homogeneous semi-solid materials. Chem. Eng. Sci. 1986, 41 (12), 3213. Wakeman, R. J.; Sabri, M. N.; Tarleton, E. S. Factors affecting the formation and properties wet compacts. Powder Technol. 1991, 65, 283.
Resubmitted for review July 2, 1998 Accepted July 2, 1998 IE970544I
