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Ind. Eng. Chem. Res. 1996, 35, 4810-4812
CORRESPONDENCE Comments on “Specific Cake Resistance Determination with a New Capillary Suction Time Apparatus” G. H. Meeten*,† and J. B. A. F. Smeulders‡ Schlumberger Cambridge Research, Madingley Road, Cambridge CB3 0EL, U.K., and Polymers & Colloids Group, Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, U.K.
1. Introduction Sir: The capillary suction time (CST) method was devised by Gale and Baskerville (1967) as a convenient means of measuring filterability of suspensions. The CST experiment is quick to set up and dismantle, but it has some ambiguity of interpretation. Models of the CST operation have been recently discussed and reviewed (Lee and Hsu, 1992; Meeten and Lebreton, 1993; Lee and Hsu, 1993; Lee and Hsu, 1994; Lee, 1994; Meeten and Smeulders, 1995). Most workers have used the original radial geometry of Gale and Baskerville (1967), where the filtrate flows radially away from a central cylindrical source and in the plane of a filter paper. Unno et al. (1983) and Tiller et al. (1990) have described and modeled a linear CST apparatus, where the filtrate flow is largely unidirectional and which allows an exact correction to be made for the hydraulic resistance of the filter paper. Whether radial or linear, the CST method measures the time for a known volume of filter paper to be permeated by filtrate drawn from the suspension by the capillary suction pressure. Herwijn et al. (1995) have recently described the theory and action of a novel CST apparatus. In this paper we discuss their model and some practical consequences of their apparatus. 2. Theory We use the symbols and terminology of Herwijn et al. (1995), who model the circular wetting front of filtrate as it progresses in the plane of a horizontal filter plate. The filtrate derives from a suspension held in an upright cylindrical reservoir whose lower end rests on the filter and inside which a filter cake grows vertically. Gravitational head within the suspension and capillarity within the filter provide the pressure which drives the filtration. Neglecting inertia, this driving pressure is exactly opposed by the friction pressures of filtrate flow through the filter cake and the filter. Herwijn et al. (1995) relate the wetting front radius r to the time t (their eq 9) by
FsgHr02 + 2rhβγ cos θ )
(
Fs is the density of the suspension, g is the gravitational force per unit mass, H is the height of the suspension in the reservoir, r0 is the internal radius of the reservoir, η is the viscosity of the filtrate, Rav is the average specific cake resistance, Cav is the cake mass deposited per unit filtrate volume, h is the thickness of the filter plate, KF is the hydraulic permeability of the filter plate, and is its porosity. The term βγ cos θ is the capillary suction pressure; see Herwijn et al. (1995) for the meaning of the separate symbols. Numerical solutions to eq 1 were given by Herwijn et al. (1995), but analytical solutions exist for two limiting cases, each of which can be closely approached by experiment. Here we consider these cases, one being where the process is controlled by the filter and the other where it is controlled by the filter cake. The first case corresponds to a filter cake of negligible hydraulic resistance compared with the filter and is attained with a pure liquid in the reservoir or by a suspension of particles small enough to flow through the filter. The second corresponds to a filter cake with a hydraulic resistance which greatly exceeds that of the filter. For simplicity we neglect the effect of the gravitational head of the suspension, which typically contributes only a few percent of the driving pressure. Sedimentation of the suspension in the reservoir, which Tiller (1995) has shown to be important in measurements of filtration and CST, is also neglected. 2.1. Filter Control. Here RavCav f 0, and following integration, eq 1 gives
t(r) )
( ()
η r2 r2 ln 2 - r2 + r02 4KFβγ cos θ r0
)
(2)
where t(r) is the time taken for the wetting front to reach a radius r. Equation 2 is in agreement with the expression of Meeten and Lebreton (1993). 2.2. Filter Cake Control. Here 1/KF f 0, and on integration eq 1 gives
t(r) )
ηRavCav2 h(r - r0)2(r + 2r0) βγ cos θ 3r 2
(3)
0
2r3ηRavCav2h2 r02
Equation 3 should be comparable with
-
)
2 2r2 ln(r)ηh 2r ln(r0)ηh dr (1) 2ηRavCav h r + KF KF dt 2 2
with the initial condition that at t ) 0, r ) r0. In eq 1 * Author to whom correspondence is addressed. † Schlumberger Cambridge Research. ‡ University of Cambridge.
S0888-5885(95)00393-9 CCC: $12.00
t)
ηRavCav2 h2(r4 - r04) βγ cos θ 2r 4
(4)
0
obtained by putting P ≡ βγ cos θ, and G/K ≡ RavCav in eqs 1 and 18 of Meeten and Smeulders (1995), q.v. for the meaning of P, G, and K. Equations 3 and 4 can be seen to differ in the terms describing the geometry of the apparatus, i.e., those containing h, r, and r0. This © 1996 American Chemical Society
Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 4811
discrepancy we will show to originate from errors in eqs 5 and 7 of Herwijn et al. (1995). Equation 5 of Herwijn et al. (1995) is given as
2πrh∆PF ) 2πrhβγ cos θ - F′
(5)
where r is the radius of the wetting front, h the filter thickness, ∆PF the pressure drop in the filter between the radii r and r0, and βγ cos θ the capillary pressure. The force F′ results from the pressure difference across the filter cake; see eq 2 of Herwijn et al. (1995). However, the forces in eq 5 act in different directions. The forces represented by 2πrh∆PF (due to viscous flow in the filter) and 2πrhβγ cos θ (due to capillarity) lie in the plane of the filter. That represented by F′ is normal to the plane of the filter. Forces can be equated only if they are equal and opposite in both magnitude and direction. The force equation (5) of Herwijn et al. (1995) is thus vectorially incorrect. A similar error was made by Unno et al. (1983) in modeling the linear CST apparatus, later described correctly by Tiller et al. (1990). Over any closed path the total pressure change must be zero. Take a closed path from the top of the suspension, through the filter cake, through the wetted part of the filter to the wetting front, and return to the top of the suspension. For the radial CST geometry this gives a pressure equation
∆PF ) βγ cos θ - F′/A
(6)
where A is the filter cake area in contact with the filter. Note the dissimilarity to eq 5. The error of Unno et al. (1983) gave an erroneous expression for the time dependence of the wetted distance z in the linear CST device. It did not, however, alter the functionality z ∝ xt found in the corrected linear CST theory by Tiller et al. (1990). This is because z is absent in the erroneous force equation of Unno et al. (1983; their eq 5) and also in the correct pressure equation of Tiller et al. (1990; their eq 5 also). For the radial CST device, however, r is present in the erroneous force equation (5) of Herwijn et al. (1995) but absent in the correct pressure equation (6), above. Hence, different dependences on r and h are found in eqs 3 and 4. In the case of filter control, F′ f 0. Thus eq 2 was obtained correctly by Herwijn et al. (1995) as eqs 5 and 6 become identical when F′ ) 0. Equations 3 and 4 also differ owing to the relation assumed between the volume of filtrate V and the radius r of the wetted filter volume. Herwijn et al. (1995) assume (their eq 7)
V ) hπ(r2 - r02)
(7)
If, however, filtration is taken to begin at the instant that suspension is in contact with the filter, then conservation of filtrate volume gives
V ) hπr2
(8)
Equation 4 is regained if eqs 6 and 8 replace the erroneous equations, (5) and (7), respectively. The numerical conclusions of Herwijn et al. (1995) will be invalidated by the errors described above.
Unno et al. (1983) used discretely positioned electrodes to monitor the progress of the wetting front within the plane of the filter. Herwijn et al. (1995) clamp a ceramic filter sheet between two planar copper electrodes. They obtain the radius r(t) of the radial wetting front in the filter by measuring the electrical resistance R(t) between the electrodes from the time t of placing the suspension. To prevent suspension particles from invading the ceramic, they interpose a thin paper filter between the suspension and the ceramic filter, in the manner of Unno et al. (1983). From the earliest time (Gale and Baskerville, 1967) to the present (Lee, 1994; Meeten and Smeulders, 1995), the effective porosity to filtrate of the filter has been acknowledged to depend on the filtrate saturation sw, with sw < 1 owing to the partial displacement of air by the wetting front of filtrate. For the radial capillary suction time device, Lee and Hsu (1992-1994) used a diffusion model to show that sw varied strongly with radius r. Measurements (Meeten and Smeulders, 1995) of sw(r) in a filter paper showed a different but equally strong dependence, with practically complete filtrate saturation directly under the suspension reservoir. The effect of partial filtrate saturation on the experimental method of Herwijn et al. (1995) is now discussed. An empirical relation which describes the resistivity Fw of a partially saturated porous solid containing an electrolyte and another immiscible fluid is (Tomkiewicz, 1993)
Fw ) F0φ-msw-n
(9)
where F0 is the resistivity of the electrolyte. Equation 9 is Archie’s law in petrophysics, where it applies to a rock of porosity φ containing brine and oil, the brine being the rock-wetting component. Typically, both m and n lie in the range 1.5-2 (Edmundson, 1988a,b). We will suppose this law to be reliable for a filter wetted with filtrate, the nonconducting oil now being the nondisplaced air. To obtain the position of the wetting front from measurements of the resistance R(t), the analysis of Herwijn et al. (1995) assumes Fw to be constant everywhere in the wetted region. Equation 9 shows this assumption to be incorrect if sw varies over the wetted region. The electrical sensing method of Herwijn et al. (1995) is additionally complicated for filter cakes having ionexchange properties. These can act as ion-selective membranes and change the filtrate resistivity F0 over the filtration time. Equation 9 shows an additional time dependence of Fw will result, extra time dependence owing to r(t). This effect may be large, depending on the suspension: filtrate normality changes between 10% and 100% have been reported for clays (Smith, 1977; McKelvey and Milne, 1960). The combined effects of a radially-dependent degree of saturation in the filter and possible time variations of filtrate conductivity make the method and the data analysis of Herwijn et al. (1995) prone to errors whose magnitude is difficult to estimate. Acknowledgment The work in this paper was carried out as part of the DTI Colloid Technology Project.
3. Experimental Section Herwijn et al. (1995) describe a novel radial capillary suction time apparatus. The original radial design of Baskerville and Gale (1968) and the linear variant of
Nomenclature A ) area of the filter and cake in contact (m2) Cav ) cake mass deposited per unit filtrate volume (kg m-3)
4812 Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 F′ ) suction force exerted by the filter medium on the filter cake (N) g ) gravitational force per unit mass (N m-1) h ) thickness of the filter (m) H ) height of the suspension in the reservoir (m) KF ) hydraulic permeability of the filter (m2) m ) cementation exponent n ) saturation exponent P ) capillary suction pressure (Pa) ∆PF ) pressure drop in the filter (Pa) r ) radius (m) r0 ) internal radius of the reservoir (m) R(t) ) electrical resistance (Ω) sw ) filtrate saturation t ) time (s) V ) volume of the filtrate (m3) Greek Symbols Rav ) average specific cake resistance (m kg-1) ) porosity of the filter η ) viscosity of the filtrate (Pa s) Fw ) wetted filter electrical resistivity (Ω m) Fs ) suspension density (kg m-3) F0 ) filtrate electrical resistivity (Ω m) βγ cos θ ) capillary suction pressure (Pa); see Herwijn et al. (1995) for the meaning of the separate symbols
Literature Cited Baskerville, R. C.; Gale, R. S. A simple automatic instrument for determining the filtrability of sewage sludges. Water Pollut. Control 1968, 2, 3-11. Edmundson, H. N. Archie’s law: Electrical conduction in clean, water-bearing rock. Tech. Rev. 1988a, 36 (3), 4-13. Edmundson, H. N. Archie II: Electrical conduction in hydrocarbonbearing rock. Tech. Rev. 1988b, 36 (4), 12-21.
Gale, R. S.; Baskerville, R. C. Capillary suction method for determination of the filtration properties of a solid/liquid suspension. Chem. Ind. 1967, 355-356. Herwijn, A. J. M.; La Heij, E. J.; IJzermans, J. J.; Coumans, W. J.; Kerkhof, P. J. A. M. Determination of specific cake resistance with a new capillary suction time apparatus. Ind. Eng. Chem. Res. 1995, 34, 1310-1319. Lee, D. J. A dynamic model of capillary suction apparatus. J. Chem. Eng. Jpn. 1994, 27, 216-221. Lee, D. J.; Hsu, Y. H. Fluid flow in capillary suction apparatus. Ind. Eng. Chem. Res. 1992, 31, 2379-2385. Lee, D. J.; Hsu, Y. H. Cake formation in capillary suction apparatus. Ind. Eng. Chem. Res. 1993, 32, 1180-1185. Lee, D. J.; Hsu, Y. H. Use of capillary suction apparatus for estimating the averaged specific resistance of filtration cake. J. Chem. Technol. Biotechnol. 1994, 59, 45-51. McKelvey, J. G.; Milne, I. H. The flow of salt solutions through compacted clay. Clays Clay Miner. 1962, 9, 248-259. Meeten, G. H.; Lebreton, C. A filtration model of the capillary suction time method. J. Pet. Sci. Eng. 1993, 9, 155-162. Meeten, G. H.; Smeulders, J. B. A. F. Interpretation of filterability measured by the capillary suction time method. Chem. Eng. Sci. 1995, 50, 1273-1279. Smith, J. E. Thermodynamics of salinity changes accompanying compaction of shaly rocks. SPE 6329. SPE J. 1977, 377-386. Tiller, F. M. Role of porosity in filtration: XII. Filtration with sedimentation. AIChE J. 1995, 41, 1153-1164. Tiller, F. M.; Shen, Y. L.; Adin, A. Capillary suction theory for rectangular cells. Res. J. Water Pollut. Control Fed. 1990, 62 (2), 130-136. Tomkiewicz, M. Impedance of composite media. Electrochim. Acta 1993, 38, 1923-1928. Unno, H.; Muraiso, H.; Akehata, T. Theoretical and experimental study of factors affecting capillary suction time (CST). Water Res. 1983, 17, 149-156.
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