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Scaling behavior in constant pressure batch dewatering of fine particle suspensions
Int. J. Miner. Process. 83 (2007) 28 – 35 www.elsevier.com/locate/ijminpro
Original article
Scaling behavior in constant pressure batch dewatering of fine particle suspensions Rahul Konnur a,⁎, Sasanka Raha b a b
Engineering and Industries Services, Tata Consultancy Services Limited, 1 Mangaldas Road, Pune, 411 001, India Tata Research Development and Design Centre (TRDDC), 54B Hadapsar Industrial Estate, Pune, 411 013, India Received 28 September 2006; received in revised form 19 March 2007; accepted 24 March 2007 Available online 3 April 2007
Abstract The constant pressure batch dewatering process of fine suspension systems in which dewatering occurs in two stages, viz. cake formation and cake consolidation is considered. Scaling transformations for the dewatering time and extent of dewatering are proposed. Using experimental data obtained by varying the applied pressure and feed solids concentration, we show that the distinct temporal dewatering profiles in the cake consolidation stage collapse on to a unique master curve as a result of scaling. For fixed suspension chemistry, the master curve can be generated using data from a single dewatering test. Application of the existence of the master curve for prediction of key dewatering process parameters is illustrated. In addition, it is shown that the scaling behavior can persist even when the chemistry of the suspension is varied. © 2007 Elsevier B.V. All rights reserved. Keywords: Dewatering; Pressure filtration; Modeling; Simulation; Suspensions; Solid–liquid separations
1. Introduction Use of dewatering is common in many of the chemical, mineral processing, and water industries. The objective of dewatering is to increase recovery of liquid or to improve the recovery of suspended solids. Frequently, these objectives are achieved by imposing a pressure gradient across a semi-permeable medium. In cake filtration, the suspension is fed upstream of the filter medium where permeable layers of particulates form initial layers, which aid subsequent dewatering of the remaining suspension as the cake grows with time. The flow of liquid through the filter cake induces viscous drag forces on the constituent particles due to ⁎ Corresponding author. E-mail address: [email protected] (R. Konnur). 0301-7516/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.minpro.2007.03.008
transfer of momentum from the liquid to the solid particles. In suspensions of fine particles, the resulting time dependent particle rearrangement induces a porosity gradient through the cake height and the cake is considered to be compressible. Numerous experimental studies of constant pressure dewatering of materials which form compressible cakes have been reported in the literature (Wakeman et al., 1991; Sis and Chander, 2000; de Kretser et al., 2001; Kapur et al., 2002; Brown and Zukoski, 2003; Raha et al., 2006, and references therein). As described by Shirato and co-workers, dewatering in a variety of systems occurs in two stages (Shirato et al., 1970, 1971, 1974). A typical two-stage dewatering curve in time versus average solids volume fraction (t–ϕ) coordinates is shown in Fig. 1. The kinetics of dewatering in the cake formation and growth stage (stage 1) is characterized by
R. Konnur, S. Raha / Int. J. Miner. Process. 83 (2007) 28–35
Fig. 1. Typical evolution of a two-stage constant pressure batch dewatering process.
the slope parameter β2. The transition from the cake formation to the cake consolidation stage (stage 2) occurs at a critical value of the average cake solids volume fraction ϕc, which is attained at a filtration time tc (Shirato et al., 1970). The onset of stage 2 is typically followed by an exponential increase in dewatering time. The progress of dewatering is influenced by the applied pressure, and the solids concentration in the bulk suspension. Further, a variety of inter-particle forces influence the structure of the particulate network formed by the compressible cake. The nature and strength of these interactions determine how the bed deforms in the presence of a compressional load. Due to these factors, quantitative modeling and predictive simulation of constant pressure batch dewatering of fine particle suspensions is a very difficult task and continues to be a challenging problem (Stamatakis and Tien, 1991; Landman and White, 1997; Burger et al., 2001). Inputs to these advanced models are constitutive relationships for the compressibility and permeability of materials. Two different laboratory systems that are available for determination of these constitutive relationships are the compressibility–permeability (CP) cell approach (Tien et al., 2001) and the step pressure filtration approach (Murase et al., 1989; de Kretser et al., 2001). In general, dewatering experiments of dispersed suspensions and flocculated suspensions at a relatively low applied pressure can be extremely time consuming. In such instances, methods for predicting the final solids concentration in the cake using partial experimental data, i.e. when a dewatering test is stopped before its completion, can be useful, especially in situations when several constant pressure dewatering tests have to be carried out. However, satisfactory empirical or model based methods applicable for predictive purposes are not available at present.
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An objective of this paper is to demonstrate that temporal evolution in the cake consolidation stage of dewatering is governed by the existence of underlying scaling relationships, which result in a unique dewatering profile in the scaled coordinates. The scaling property implies that distinct cake consolidation part of the dewatering curves in t–/ coordinates (obtained as the pressure and/or feed solids concentration is varied), can be all transformed into a unique master curve. Scaling transformations for the dewatering time and extent of dewatering are proposed and validated using experimental constant pressure batch dewatering data obtained using a variety of materials and conditions. It is also shown that scaling behavior can persist even when the chemistry of the suspension is varied. Another objective of this work is to demonstrate that the existence of the master curve can be utilized to develop methods for two common modeling objectives, stated next. For a fixed suspension chemistry, the master curve can be obtained using data from a single dewatering experiment carried out at some pressure and initial solids volume fraction, ϕ0 N ϕg, where ϕg is the gel point of the suspension. As a result, information captured from a single, complete experimental test (at an applied pressure ▵P1) can be used for (i) prediction of the final solid volume fraction ϕ∞, at a different operating pressure, if experimental data till ϕc is available at the latter pressure, and (ii) for prediction of ϕc when ϕ0 is changed. The problem in (i) above is of relevance when the effect of applied pressure on dewatering is to be investigated, while (ii) is of relevance when the effect of solid concentration of the feed on dewatering is to be investigated. This paper is organized as follows: The experimental procedure is described in Section 2. Results pertaining to existence of scaling behavior under different conditions, a discussion of the implications of scaling behavior, and a model based approach for prediction of process parameters ϕc and ϕ∞, are presented in Section 3. Key conclusions of the work are summarized in Section 4. 2. Experimental procedure In this work, we only present results obtained using fine powders of alumina and kaolin, with mean particle size of 0.35 μm and 1.1 μm, respectively. Particulate suspensions for pressure filtration tests were prepared using a standard procedure (Raha et al., 2006). For preparation of the slurry, the suspension was dispersed on a magnetic stirrer for 2 min. During this conditioning
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R. Konnur, S. Raha / Int. J. Miner. Process. 83 (2007) 28–35
process, the pH was adjusted using 4 N HNO3 or NaOH. After conditioning, the suspension was ultrasonicated using a Branson 450 sonicator for 2 min using 50% duty cycle with 40 W power input. For experiments using polymer additive, two different preparation methods were followed. In the first method, the suspensions were prepared by dispersing the powder in water containing the polymer on a magnetic stirrer. Magnetic stirring was done for 2 min. Following this, the suspensions were ultrasonicated using a Branson 450 sonicator for 2 min using 50% duty cycle with 40 W power input. In the second method, sonication was carried out before adding the additives. Following sonication, the drops of additive were added with fast agitation on a magnetic stirrer for 2 min, followed by slow agitation for 3 min. Constant pressure dewatering experiments were carried out in a highly instrumented and programmable computer driven laboratory scale test rig (de Kretser et al., 2001). Whatman filter paper No. 42 was used as the filter medium. The recorded time versus filtrate weight data was converted to time versus average solids volume fraction in the filtration cell by material balance calculations. Filtration tests were conducted to study the effect of various physical–chemical process parameters. In particular, the effect of pressure, solids concentration of feed, pH, and effect of molecular weight of polymer on dewatering, were investigated. Experiments reported in this paper cover a reasonably wide range of operating conditions: the applied pressure was varied in the range 2 kPa–300 kPa, the solids concentration of the feed was varied in the range of 6% to 30% by volume. In addition, experiments at an applied pressure of 5 kPa were also carried out to investigate the role of polymeric additive on dewatering. Here, dewatering experiments were carried out using 100 ppm of (poly-) acrylic acid (PAA), in a range of molecular weights from 8000 to 4.5 million. 3. Results and discussion For the sake of clarity, the discussion summarized in the first section of this paper is revisited and elaborated upon. It is assumed that dewatering of a feed suspension containing ϕ0 volume fraction solids is completed in two stages. Initial dewatering occurs by filtration and is primarily characterized by the formation and growth of a filter cake (Ruth, 1935). The filtration stage (stage 1) ends when the filter cake which is moving upwards meets the piston which is moving down. At this time instant, tc, the average solids volume fraction in the filter cake is denoted by ϕc. Subsequent dewatering, i.e. in
stage 2 is characterized by compression or consolidation of the filter cake. In theory, this stage lasts from time tc to infinite time, when the final solids volume fraction, ϕ∞, becomes uniform throughout the cake and dewatering ceases. The cake formation–consolidation transition point can be determined using the method proposed by Shirato et al. (1970). We first present the scaling transformations which yield the scaled master curve, and then validate the use of the transformations. Application of occurrence of scaling behavior for prediction of the process parameters ϕc and ϕ∞, is discussed in Sections 3.4 and 3.5. Results showing the existence of scaling behavior even when the chemistry of suspension is changed, are presented in Section 3.6. 3.1. Scaling relationships for solids fraction and dewatering time It is well known that constant pressure dewatering experiments carried out by varying the pressure yield distinct curves in t–ϕ coordinates. We have observed that the distinct t–ϕ curves in the cake consolidation stage (stage 2) part of dewatering, collapse onto a single master curve when represented in a plot of scaled dewatering time versus the extent of dewatering. Although it is possible to have different equivalent definitions of the extent of dewatering, the following definition has been employed in this work Extentof dewatering ¼
ð/ /c Þ : ð/l /c Þ
ð1Þ
The scaled dewatering time is defined as the product of the dewatering time, the rate of dewatering at ϕc, given by β2ϕc3 / 2h02ϕ0(ϕc − ϕ0), and the solids loading, h0ϕ0. It has to be noted that ϕc has a non-linear dependence on both the applied pressure and ϕ0. Two of the parameters which can be easily varied in practice are the applied pressure and the solids concentration of the feed suspension. In the next two sections, we first validate the use of the scaling transformations mentioned above by demonstrating the existence of a master curve when each of these operating parameters is varied. 3.2. Effect of applied pressure We present results of dewatering experiments at different pressures carried out using suspensions prepared from two grades of alumina and kaolin. Figs. 2 and 3 show the temporal evolution of dewatering (for
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Fig. 2. Effect of pressure on dewatering of 10% A16-SG alumina suspension: (a) temporal evolution of dewatering; (b) scaled evolution in the consolidation stage of dewatering.
A16-SG alumina) as well as the evolution in scaled coordinates for the following materials: (i) A16-SG alumina, (ii) A16 alumina, and (iii) kaolin, respectively. From Figs. 2(b), 3(a) and (b), existence of the master curve is apparent. Generally, moderate deviations appear to exist in the latter phase of the cake consolidation stage, and this can be attributed to a combination of small variations in experimental conditions (mainly in pH) as well as differences in the structural characteristics of the consolidating bed. Existence of a master curve for dewatering of other alumina suspensions has also been verified (results not shown). The existence of the master curve for evolution in the cake consolidation stage of dewatering indicates that the process parameters ϕc and ϕ∞ are inter-related. More importantly, given a master curve – which can be determined using a single experiment at any pressure in the range of interest – existence of the scaling relationships implies that the final solids content at any pressure can be estimated if experimental data till ϕc at that pressure, is available. A model and method for estimation of ϕ∞ when an estimate of ϕc is available, is presented in Section 3.4. Next, it is shown that the
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Fig. 3. Scaled evolution in the consolidation stage of dewatering for 10% (a) A16 alumina and, (b) kaolin suspensions.
relationship between parameters ϕc and ϕ∞, is also a function of the solids concentration of the feed. 3.3. Effect of solids concentration of the feed suspension Dewatering experiments at an applied pressure of 100 kPa and six different feed solids concentration, ranging from a low of 0.06 to a high of 0.30 solids volume
Fig. 4. Scaled evolution in the consolidation stage of dewatering for A16-SG alumina suspensions of different feed solids concentration.
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fraction, have been carried out using A16-SG powder. The temporal evolution, i.e. the t–ϕ profile of dewatering for these experiments has been shown in Fig. 3 of Raha et al. (2006) and hence is not shown here. Fig. 4 shows the master curve for these sets of six experiments. From Fig. 4, slight deviations in the master curve can be seen in the latter phase of dewatering. This aspect has been discussed in the previous section. When the objective is to investigate the effect of solids concentration of the feed, the existence of the master curve implies that the transition point ϕc, for any desired ϕ0 N ϕg, can be determined if experimental data at one ϕ0 N ϕg is available, and it is assumed that the final solids concentration does not depend on the solids concentration of the feed.
instantaneous average solids concentration of the cake. The model for evolution of the average solids fraction in the cake consolidation stage of dewatering is given by the following form of Darcy's equation
3.4. Prediction of final solids content when the applied pressure is changed
Pf ¼ DP
We now consider the situation where complete experimental dewatering data at one applied pressure, ▵P1, is available, and it is desired to predict the final solids concentration at a different applied pressure, ▵P2. Existence of the master curve can be employed for determining the value of ϕ∞ at the latter applied pressure, if experimental data till ϕc at that pressure is available. Determination of ϕ∞ is essentially a single variable optimization problem involving minimizing the error between two curves, and this can be easily carried out when a model for the cake consolidation stage of dewatering is available. For a given material, a suitable model for the cake consolidation stage of dewatering can be determined using the complete dewatering data (at an applied pressure, ▵P1). 3.4.1. Model for consolidation stage of dewatering Relatively simple models for the cake consolidation stage of dewatering are available in the literature (Shirato et al., 1971, 1974; Kapur et al., 2002). While the model of Shirato and co-workers is more rigorous, it requires knowledge of a large number of parameters whose dependence on the operating parameters (h0, ϕ0) and relationship with the process parameters (β2, ϕc, ϕ∞), remains unclear. We have found that a generalized model for the cake consolidation stage of dewatering can be easily obtained from the model of Kapur et al. (2002). Development of the model is briefly described next. In the cake consolidation stage, the applied pressure is carried partly as particle network stress, and partly as fluid stress. The distribution of the applied pressure between the liquid and solid phases depends on the
d/ k/3 Pf ¼ ; for /N/c : dt ðh0 /0 Þ2
ð2Þ
The parameters in this model are the permeability correlation k, and the fluid stress Pf, both of which are functions of the instantaneous average solids concentration in the consolidating cake. Following Kapur et al. (2002), it is assumed that the fluid stress decreases linearly with an increase in the solids fraction in the cake, i.e. ð/l /Þ : ð/l /c Þ
ð3Þ
Incorporating this relationship in the model Eq. (2), results in the following model equation d/ DP ¼ k/3 ð/l /Þ; for /N/c dt ðh0 /0 Þ2 ð/l /c Þ or d/ ¼ c1 k/3 ð/l /Þ; for /N/c : dt
ð4Þ
The permeability correlation k, is assumed to be of the form k = kck(ϕ). Here, kc is the term independent of the solids volume fraction, and k(ϕ) is the term dependent on the instantaneous solids volume fraction. For modeling purposes, it is assumed that the functional form of k (ϕ) depends only on the material, but does not depend on the applied pressure and the solids concentration of the feed. This assumption allows the determination of a suitable form of k (ϕ) if experimental dewatering data at a single pressure is available. 3.4.2. Method for prediction of final solids content Existence of the master curve for the cake consolidation stage of dewatering provides the basis for estimation of the final solids concentration at a desired pressure, ▵P2, when experimental data till ϕc at that pressure is available. It is assumed that dewatering data at a ϕ0 N ϕg and a different pressure, ▵P1, is also available. The steps of the estimation procedure are described next. First, the master curve is generated by transforming the consolidation stage part of the experimental data (obtained using an applied pressure, ▵P1) using the scaling relationships mentioned in Section 3.1. In the
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Fig. 5. Comparison of experimentally determined dependence of /∞ on applied pressure and the dependence predicted using experimental dewatering data till /c and Eq. (4), for A16-SG alumina, and kaolin systems.
second step, a value of ϕ∞ for the desired pressure, ▵P2, is guessed and the t–ϕ dependence in stage 2 is obtained using the consolidation stage dewatering model Eq. (4). As mentioned earlier, the form of the ϕ-dependent permeability term, i.e. k(ϕ), in Eq. (4) can be determined using the single set of experimental dewatering data. This can be then used to determine the constant term, c1kc, in Eq. (4) (Raha et al., 2006). In the third step, the simulated t–ϕ dependence is scaled using the transformations in Section 3.1, and the resulting scaled dependence is compared with the master curve obtained using the experimental data. A new value of ϕ∞ is then guessed and the second and third steps are repeated till the two curves match. For the A16-SG alumina system, k(ϕ) was determined to be 1 / ϕ from an analysis of experimental data obtained using a feed suspension containing 10% solids (by volume), and an applied pressure of 100 kPa. The model in Eq. (4) has been used to predict the final solids concentration in the cake, /∞, as the operating pressure is varied over a reasonably wide range. From a comparison of the predicted and model based predictions of /∞, shown in Fig. 5, it is apparent that a fairly good prediction of /∞ is possible for the A16-SG alumina and kaolin systems; the maximum error is about 3%. Similar quality of prediction was obtained for the A16 alumina system. These results are not surprising since the magnitude of prediction error is inversely related to the accuracy of the master curve shown in Figs. 2(b), 3(a) and (b). A capability to predict the final solids concentration using partial experimental data can lead to substantial savings in experimentation time. Fig. 6 shows a dependence of the difference, ▵t = t99 tc, on applied pressure, where t99 is the dewatering time to reach a cake solids
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Fig. 6. Dependence of the time difference, ▵t = t99 tc, on the applied pressure for 10% A16-SG alumina suspension.
fraction equal to 0.99ϕ∞. While the average reduction in experimentation time is about 20%, significant reduction in experimentation time is possible when the applied pressure is lower than 10 kPa. In the case of dispersed systems, it can be expected that significant reduction in experimentation time can be achieved even at higher applied pressure. 3.5. Prediction of ϕc when ϕ0 is changed We next consider the problem of predicting the solids concentration at the cake formation–consolidation transition point, ϕc, when the feed solids concentration is changed and the operating pressure remains unchanged. It is assumed that complete dewatering data with a feed solids concentration not below the gel point of the material is available. This also implies that ϕ∞, which is assumed to be independent of ϕ0, is known. It can be seen that the problem of predicting ϕc when ϕ∞ is known, is the inverse of the problem discussed in Section 3.4.2, where ϕ∞ was determined when ϕc was
Fig. 7. Dependence of error in the prediction of parameter ϕc, on the initial solids concentration, for A16-SG alumina suspensions.
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Fig. 9. Effect of pH on scaled evolution in the consolidation stage of dewatering for 10% A16-SG alumina suspension containing 100 ppm of low molecular weight (Mw 8000) PAA.
ways: (i) change in pH, (ii) change in pH in the presence of 100 ppm of low molecular weight (poly-) acrylic acid (PAA) polymer additive, and (iii) change in molecular weight of PAA, at constant pH. Somewhat surprisingly, we have observed that the master curve can persist even
Fig. 8. Effect of pH on dewatering of 10% A16-SG alumina suspension: (a) temporal evolution of dewatering; (b) scaled evolution in the consolidation stage of dewatering.
known. Therefore, the existence of the master curve can be employed to predict ϕc at a desired ϕ0. In particular, it has to be noted that estimation of ϕc for a desired ϕ0, does not require any knowledge of the kinetics of dewatering in cake formation stage. Dependence of the error in estimation of ϕc on ϕ0 shown in Fig. 7, indicates that the method yields fairly accurate estimates of ϕc. Larger magnitude of prediction error at low ϕ 0 (= 0.06) implies that a different approach is needed for modeling of dewatering when the feed solids concentration is less than or approaches the gel point of the suspension. 3.6. Effect of suspension chemistry It is known that suspension chemistry has a strong impact on the dewaterability of suspensions. In this work, we only summarize results obtained by modifying the chemistry of alumina suspensions in the following
Fig. 10. Effect of molecular weight of PAA on dewatering of 10% A16SG alumina suspension, at a pH of 9.5: (a) temporal evolution of dewatering; (b) scaled evolution in the consolidation stage of dewatering.
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when the pH varies over a reasonably wide range. Fig. 8 (a) shows the evolution of dewatering when the pH of the suspension is varied. The corresponding evolution in scaled coordinates is shown in Fig. 8(b). It can be seen that there appear to be two sets of master curves, one each for dispersed and flocculated systems. The reasons for occurrence of two sets of master curves are not yet understood at present. As shown in Fig. 9, similar results are obtained when the pH is changed in the presence of low molecular weight PAA. In complete contrast to the above, clear evidence of unique master curve is seen when the pH is kept constant, and the molecular weight of polymeric additive is varied over a reasonably wide range. These results are shown in Fig. 10(a) and (b). From Fig. 10(b), it is once again apparent that under certain conditions, there can be a strong inter-relationship between the process parameters /c and /∞. As discussed in Section 3.4, and subject to validation of these results using other materials and conditions, such an inter-relationship would imply that the existence of the master curve could be used to predict the final solids concentration for a desired molecular weight of the polymer additive, if partial experimental data, i.e. till /c, were available. 4. Conclusions Constant pressure batch dewatering of alumina and kaolin fine particle suspension systems has been experimentally investigated. Scaling relationships for the extent of dewatering and dewatering time, which yield a unique master curve in the cake consolidation stage of dewatering, have been proposed. In the systems investigated which show two-stage dewatering behavior, the existence of a master curve appears to be a fairly general phenomenon since it persists even when the pressure, initial solids concentration and the suspension chemistry are varied. Using a simple consolidation stage dewatering model, an application involving use of the master curve for prediction of dewatering process parameters has been demonstrated. Results reported here appear to indicate that an understanding of the scaling behavior can provide useful insights about the characteristic quantities governing evolution of the pressure dewatering process. It is hoped that the results presented here will be helpful for the analysis of laboratory pressure dewatering experiments. Acknowledgements Financial support for this work from the Department of Science and Technology, Government of India, is
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gratefully acknowledged. The authors are grateful to Prof. P. C. Kapur and Dr. Pradip of Tata Research Design and Development Centre (TRDDC), Pune, India, for their comments and suggestions. References Brown, L.A., Zukoski, C.F., 2003. Experimental test of two-phase fluid model of drying consolidation. AIChE Journal 49, 362–372. Burger, R., Concha, F., Karlsen, K.H., 2001. Phenomenological model of filtration processes: 1. Cake formation and expression. Chemical Engineering Science 56, 4537–4553. de Kretser, R.G., Usher, S.P., Scales, P.J., Boger, D.V., Landman, K.A., 2001. Rapid filtration measurement of dewatering design and optimization parameters. AIChE Journal 47, 1758–1769. Kapur, P.C., Raha, S., Usher, S., de Kretser, R.G., Scales, P.J., 2002. Modeling of the consolidation stage in pressure filtration of compressible cakes. Journal of Colloid and Interface Science 256, 216–222. Landman, K.A., White, L.R., 1997. Predicting filtration time and maximising throughput in a pressure filter. AIChE Journal 43, 3147–3160. Murase, T., Iritani, E., Cho, J.H., Shirato, M., 1989. Determination of filtration characteristics based upon filtration tests under step-up filtration conditions. Journal of Chemical Engineering of Japan 22, 373–378. Raha, S., Khilar, K.C., Pradip, P.C., Kapur, 2006. A mean phi model for pressure filtration of fine and colloidal suspensions. Canadian Journal of Chemical Engineering 84, 83–93. Ruth, B.F., 1935. Studies in filtration. III. Derivation of general filtration equations. Industrial and Engineering Chemistry 27, 708–723. Shirato, M., Murase, T., Kato, H., Fukaya, S., 1970. Fundamental analysis for expression under constant pressure. Filtration and Separation 7, 277–282. Shirato, M., Murase, T., Negawa, M., Moridera, H., 1971. Analysis of expression operations. Journal of Chemical Engineering of Japan 4, 263–268. Shirato, M., Murase, T., Tokunaga, A., Yamada, O., 1974. Calculations of consolidation period in expression operations. Journal of Chemical Engineering of Japan 7, 229–231. Sis, H., Chander, S., 2000. Pressure filtration of dispersed and flocculated alumina slurries. Minerals and Metallurgical Processing 17, 41–46. Stamatakis, K., Tien, C., 1991. Cake formation and growth in cake filtration. Chemical Engineering Science 46, 1917–1933. Tien, C., Teoh, S.K., Tan, R.B.H., 2001. Cake filtration analysis — the effect of the relationship between the pore liquid pressure and the cake dependent compressive stress. Chemical Engineering Science 56, 5361–5369. Wakeman, R.J., Sabri, M.N., Tarleton, E.S., 1991. Factors affecting the formation and properties of wet compacts. Powder Technology 65, 283–292.
Original article
Scaling behavior in constant pressure batch dewatering of fine particle suspensions Rahul Konnur a,⁎, Sasanka Raha b a b
Engineering and Industries Services, Tata Consultancy Services Limited, 1 Mangaldas Road, Pune, 411 001, India Tata Research Development and Design Centre (TRDDC), 54B Hadapsar Industrial Estate, Pune, 411 013, India Received 28 September 2006; received in revised form 19 March 2007; accepted 24 March 2007 Available online 3 April 2007
Abstract The constant pressure batch dewatering process of fine suspension systems in which dewatering occurs in two stages, viz. cake formation and cake consolidation is considered. Scaling transformations for the dewatering time and extent of dewatering are proposed. Using experimental data obtained by varying the applied pressure and feed solids concentration, we show that the distinct temporal dewatering profiles in the cake consolidation stage collapse on to a unique master curve as a result of scaling. For fixed suspension chemistry, the master curve can be generated using data from a single dewatering test. Application of the existence of the master curve for prediction of key dewatering process parameters is illustrated. In addition, it is shown that the scaling behavior can persist even when the chemistry of the suspension is varied. © 2007 Elsevier B.V. All rights reserved. Keywords: Dewatering; Pressure filtration; Modeling; Simulation; Suspensions; Solid–liquid separations
1. Introduction Use of dewatering is common in many of the chemical, mineral processing, and water industries. The objective of dewatering is to increase recovery of liquid or to improve the recovery of suspended solids. Frequently, these objectives are achieved by imposing a pressure gradient across a semi-permeable medium. In cake filtration, the suspension is fed upstream of the filter medium where permeable layers of particulates form initial layers, which aid subsequent dewatering of the remaining suspension as the cake grows with time. The flow of liquid through the filter cake induces viscous drag forces on the constituent particles due to ⁎ Corresponding author. E-mail address: [email protected] (R. Konnur). 0301-7516/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.minpro.2007.03.008
transfer of momentum from the liquid to the solid particles. In suspensions of fine particles, the resulting time dependent particle rearrangement induces a porosity gradient through the cake height and the cake is considered to be compressible. Numerous experimental studies of constant pressure dewatering of materials which form compressible cakes have been reported in the literature (Wakeman et al., 1991; Sis and Chander, 2000; de Kretser et al., 2001; Kapur et al., 2002; Brown and Zukoski, 2003; Raha et al., 2006, and references therein). As described by Shirato and co-workers, dewatering in a variety of systems occurs in two stages (Shirato et al., 1970, 1971, 1974). A typical two-stage dewatering curve in time versus average solids volume fraction (t–ϕ) coordinates is shown in Fig. 1. The kinetics of dewatering in the cake formation and growth stage (stage 1) is characterized by
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Fig. 1. Typical evolution of a two-stage constant pressure batch dewatering process.
the slope parameter β2. The transition from the cake formation to the cake consolidation stage (stage 2) occurs at a critical value of the average cake solids volume fraction ϕc, which is attained at a filtration time tc (Shirato et al., 1970). The onset of stage 2 is typically followed by an exponential increase in dewatering time. The progress of dewatering is influenced by the applied pressure, and the solids concentration in the bulk suspension. Further, a variety of inter-particle forces influence the structure of the particulate network formed by the compressible cake. The nature and strength of these interactions determine how the bed deforms in the presence of a compressional load. Due to these factors, quantitative modeling and predictive simulation of constant pressure batch dewatering of fine particle suspensions is a very difficult task and continues to be a challenging problem (Stamatakis and Tien, 1991; Landman and White, 1997; Burger et al., 2001). Inputs to these advanced models are constitutive relationships for the compressibility and permeability of materials. Two different laboratory systems that are available for determination of these constitutive relationships are the compressibility–permeability (CP) cell approach (Tien et al., 2001) and the step pressure filtration approach (Murase et al., 1989; de Kretser et al., 2001). In general, dewatering experiments of dispersed suspensions and flocculated suspensions at a relatively low applied pressure can be extremely time consuming. In such instances, methods for predicting the final solids concentration in the cake using partial experimental data, i.e. when a dewatering test is stopped before its completion, can be useful, especially in situations when several constant pressure dewatering tests have to be carried out. However, satisfactory empirical or model based methods applicable for predictive purposes are not available at present.
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An objective of this paper is to demonstrate that temporal evolution in the cake consolidation stage of dewatering is governed by the existence of underlying scaling relationships, which result in a unique dewatering profile in the scaled coordinates. The scaling property implies that distinct cake consolidation part of the dewatering curves in t–/ coordinates (obtained as the pressure and/or feed solids concentration is varied), can be all transformed into a unique master curve. Scaling transformations for the dewatering time and extent of dewatering are proposed and validated using experimental constant pressure batch dewatering data obtained using a variety of materials and conditions. It is also shown that scaling behavior can persist even when the chemistry of the suspension is varied. Another objective of this work is to demonstrate that the existence of the master curve can be utilized to develop methods for two common modeling objectives, stated next. For a fixed suspension chemistry, the master curve can be obtained using data from a single dewatering experiment carried out at some pressure and initial solids volume fraction, ϕ0 N ϕg, where ϕg is the gel point of the suspension. As a result, information captured from a single, complete experimental test (at an applied pressure ▵P1) can be used for (i) prediction of the final solid volume fraction ϕ∞, at a different operating pressure, if experimental data till ϕc is available at the latter pressure, and (ii) for prediction of ϕc when ϕ0 is changed. The problem in (i) above is of relevance when the effect of applied pressure on dewatering is to be investigated, while (ii) is of relevance when the effect of solid concentration of the feed on dewatering is to be investigated. This paper is organized as follows: The experimental procedure is described in Section 2. Results pertaining to existence of scaling behavior under different conditions, a discussion of the implications of scaling behavior, and a model based approach for prediction of process parameters ϕc and ϕ∞, are presented in Section 3. Key conclusions of the work are summarized in Section 4. 2. Experimental procedure In this work, we only present results obtained using fine powders of alumina and kaolin, with mean particle size of 0.35 μm and 1.1 μm, respectively. Particulate suspensions for pressure filtration tests were prepared using a standard procedure (Raha et al., 2006). For preparation of the slurry, the suspension was dispersed on a magnetic stirrer for 2 min. During this conditioning
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process, the pH was adjusted using 4 N HNO3 or NaOH. After conditioning, the suspension was ultrasonicated using a Branson 450 sonicator for 2 min using 50% duty cycle with 40 W power input. For experiments using polymer additive, two different preparation methods were followed. In the first method, the suspensions were prepared by dispersing the powder in water containing the polymer on a magnetic stirrer. Magnetic stirring was done for 2 min. Following this, the suspensions were ultrasonicated using a Branson 450 sonicator for 2 min using 50% duty cycle with 40 W power input. In the second method, sonication was carried out before adding the additives. Following sonication, the drops of additive were added with fast agitation on a magnetic stirrer for 2 min, followed by slow agitation for 3 min. Constant pressure dewatering experiments were carried out in a highly instrumented and programmable computer driven laboratory scale test rig (de Kretser et al., 2001). Whatman filter paper No. 42 was used as the filter medium. The recorded time versus filtrate weight data was converted to time versus average solids volume fraction in the filtration cell by material balance calculations. Filtration tests were conducted to study the effect of various physical–chemical process parameters. In particular, the effect of pressure, solids concentration of feed, pH, and effect of molecular weight of polymer on dewatering, were investigated. Experiments reported in this paper cover a reasonably wide range of operating conditions: the applied pressure was varied in the range 2 kPa–300 kPa, the solids concentration of the feed was varied in the range of 6% to 30% by volume. In addition, experiments at an applied pressure of 5 kPa were also carried out to investigate the role of polymeric additive on dewatering. Here, dewatering experiments were carried out using 100 ppm of (poly-) acrylic acid (PAA), in a range of molecular weights from 8000 to 4.5 million. 3. Results and discussion For the sake of clarity, the discussion summarized in the first section of this paper is revisited and elaborated upon. It is assumed that dewatering of a feed suspension containing ϕ0 volume fraction solids is completed in two stages. Initial dewatering occurs by filtration and is primarily characterized by the formation and growth of a filter cake (Ruth, 1935). The filtration stage (stage 1) ends when the filter cake which is moving upwards meets the piston which is moving down. At this time instant, tc, the average solids volume fraction in the filter cake is denoted by ϕc. Subsequent dewatering, i.e. in
stage 2 is characterized by compression or consolidation of the filter cake. In theory, this stage lasts from time tc to infinite time, when the final solids volume fraction, ϕ∞, becomes uniform throughout the cake and dewatering ceases. The cake formation–consolidation transition point can be determined using the method proposed by Shirato et al. (1970). We first present the scaling transformations which yield the scaled master curve, and then validate the use of the transformations. Application of occurrence of scaling behavior for prediction of the process parameters ϕc and ϕ∞, is discussed in Sections 3.4 and 3.5. Results showing the existence of scaling behavior even when the chemistry of suspension is changed, are presented in Section 3.6. 3.1. Scaling relationships for solids fraction and dewatering time It is well known that constant pressure dewatering experiments carried out by varying the pressure yield distinct curves in t–ϕ coordinates. We have observed that the distinct t–ϕ curves in the cake consolidation stage (stage 2) part of dewatering, collapse onto a single master curve when represented in a plot of scaled dewatering time versus the extent of dewatering. Although it is possible to have different equivalent definitions of the extent of dewatering, the following definition has been employed in this work Extentof dewatering ¼
ð/ /c Þ : ð/l /c Þ
ð1Þ
The scaled dewatering time is defined as the product of the dewatering time, the rate of dewatering at ϕc, given by β2ϕc3 / 2h02ϕ0(ϕc − ϕ0), and the solids loading, h0ϕ0. It has to be noted that ϕc has a non-linear dependence on both the applied pressure and ϕ0. Two of the parameters which can be easily varied in practice are the applied pressure and the solids concentration of the feed suspension. In the next two sections, we first validate the use of the scaling transformations mentioned above by demonstrating the existence of a master curve when each of these operating parameters is varied. 3.2. Effect of applied pressure We present results of dewatering experiments at different pressures carried out using suspensions prepared from two grades of alumina and kaolin. Figs. 2 and 3 show the temporal evolution of dewatering (for
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Fig. 2. Effect of pressure on dewatering of 10% A16-SG alumina suspension: (a) temporal evolution of dewatering; (b) scaled evolution in the consolidation stage of dewatering.
A16-SG alumina) as well as the evolution in scaled coordinates for the following materials: (i) A16-SG alumina, (ii) A16 alumina, and (iii) kaolin, respectively. From Figs. 2(b), 3(a) and (b), existence of the master curve is apparent. Generally, moderate deviations appear to exist in the latter phase of the cake consolidation stage, and this can be attributed to a combination of small variations in experimental conditions (mainly in pH) as well as differences in the structural characteristics of the consolidating bed. Existence of a master curve for dewatering of other alumina suspensions has also been verified (results not shown). The existence of the master curve for evolution in the cake consolidation stage of dewatering indicates that the process parameters ϕc and ϕ∞ are inter-related. More importantly, given a master curve – which can be determined using a single experiment at any pressure in the range of interest – existence of the scaling relationships implies that the final solids content at any pressure can be estimated if experimental data till ϕc at that pressure, is available. A model and method for estimation of ϕ∞ when an estimate of ϕc is available, is presented in Section 3.4. Next, it is shown that the
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Fig. 3. Scaled evolution in the consolidation stage of dewatering for 10% (a) A16 alumina and, (b) kaolin suspensions.
relationship between parameters ϕc and ϕ∞, is also a function of the solids concentration of the feed. 3.3. Effect of solids concentration of the feed suspension Dewatering experiments at an applied pressure of 100 kPa and six different feed solids concentration, ranging from a low of 0.06 to a high of 0.30 solids volume
Fig. 4. Scaled evolution in the consolidation stage of dewatering for A16-SG alumina suspensions of different feed solids concentration.
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fraction, have been carried out using A16-SG powder. The temporal evolution, i.e. the t–ϕ profile of dewatering for these experiments has been shown in Fig. 3 of Raha et al. (2006) and hence is not shown here. Fig. 4 shows the master curve for these sets of six experiments. From Fig. 4, slight deviations in the master curve can be seen in the latter phase of dewatering. This aspect has been discussed in the previous section. When the objective is to investigate the effect of solids concentration of the feed, the existence of the master curve implies that the transition point ϕc, for any desired ϕ0 N ϕg, can be determined if experimental data at one ϕ0 N ϕg is available, and it is assumed that the final solids concentration does not depend on the solids concentration of the feed.
instantaneous average solids concentration of the cake. The model for evolution of the average solids fraction in the cake consolidation stage of dewatering is given by the following form of Darcy's equation
3.4. Prediction of final solids content when the applied pressure is changed
Pf ¼ DP
We now consider the situation where complete experimental dewatering data at one applied pressure, ▵P1, is available, and it is desired to predict the final solids concentration at a different applied pressure, ▵P2. Existence of the master curve can be employed for determining the value of ϕ∞ at the latter applied pressure, if experimental data till ϕc at that pressure is available. Determination of ϕ∞ is essentially a single variable optimization problem involving minimizing the error between two curves, and this can be easily carried out when a model for the cake consolidation stage of dewatering is available. For a given material, a suitable model for the cake consolidation stage of dewatering can be determined using the complete dewatering data (at an applied pressure, ▵P1). 3.4.1. Model for consolidation stage of dewatering Relatively simple models for the cake consolidation stage of dewatering are available in the literature (Shirato et al., 1971, 1974; Kapur et al., 2002). While the model of Shirato and co-workers is more rigorous, it requires knowledge of a large number of parameters whose dependence on the operating parameters (h0, ϕ0) and relationship with the process parameters (β2, ϕc, ϕ∞), remains unclear. We have found that a generalized model for the cake consolidation stage of dewatering can be easily obtained from the model of Kapur et al. (2002). Development of the model is briefly described next. In the cake consolidation stage, the applied pressure is carried partly as particle network stress, and partly as fluid stress. The distribution of the applied pressure between the liquid and solid phases depends on the
d/ k/3 Pf ¼ ; for /N/c : dt ðh0 /0 Þ2
ð2Þ
The parameters in this model are the permeability correlation k, and the fluid stress Pf, both of which are functions of the instantaneous average solids concentration in the consolidating cake. Following Kapur et al. (2002), it is assumed that the fluid stress decreases linearly with an increase in the solids fraction in the cake, i.e. ð/l /Þ : ð/l /c Þ
ð3Þ
Incorporating this relationship in the model Eq. (2), results in the following model equation d/ DP ¼ k/3 ð/l /Þ; for /N/c dt ðh0 /0 Þ2 ð/l /c Þ or d/ ¼ c1 k/3 ð/l /Þ; for /N/c : dt
ð4Þ
The permeability correlation k, is assumed to be of the form k = kck(ϕ). Here, kc is the term independent of the solids volume fraction, and k(ϕ) is the term dependent on the instantaneous solids volume fraction. For modeling purposes, it is assumed that the functional form of k (ϕ) depends only on the material, but does not depend on the applied pressure and the solids concentration of the feed. This assumption allows the determination of a suitable form of k (ϕ) if experimental dewatering data at a single pressure is available. 3.4.2. Method for prediction of final solids content Existence of the master curve for the cake consolidation stage of dewatering provides the basis for estimation of the final solids concentration at a desired pressure, ▵P2, when experimental data till ϕc at that pressure is available. It is assumed that dewatering data at a ϕ0 N ϕg and a different pressure, ▵P1, is also available. The steps of the estimation procedure are described next. First, the master curve is generated by transforming the consolidation stage part of the experimental data (obtained using an applied pressure, ▵P1) using the scaling relationships mentioned in Section 3.1. In the
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Fig. 5. Comparison of experimentally determined dependence of /∞ on applied pressure and the dependence predicted using experimental dewatering data till /c and Eq. (4), for A16-SG alumina, and kaolin systems.
second step, a value of ϕ∞ for the desired pressure, ▵P2, is guessed and the t–ϕ dependence in stage 2 is obtained using the consolidation stage dewatering model Eq. (4). As mentioned earlier, the form of the ϕ-dependent permeability term, i.e. k(ϕ), in Eq. (4) can be determined using the single set of experimental dewatering data. This can be then used to determine the constant term, c1kc, in Eq. (4) (Raha et al., 2006). In the third step, the simulated t–ϕ dependence is scaled using the transformations in Section 3.1, and the resulting scaled dependence is compared with the master curve obtained using the experimental data. A new value of ϕ∞ is then guessed and the second and third steps are repeated till the two curves match. For the A16-SG alumina system, k(ϕ) was determined to be 1 / ϕ from an analysis of experimental data obtained using a feed suspension containing 10% solids (by volume), and an applied pressure of 100 kPa. The model in Eq. (4) has been used to predict the final solids concentration in the cake, /∞, as the operating pressure is varied over a reasonably wide range. From a comparison of the predicted and model based predictions of /∞, shown in Fig. 5, it is apparent that a fairly good prediction of /∞ is possible for the A16-SG alumina and kaolin systems; the maximum error is about 3%. Similar quality of prediction was obtained for the A16 alumina system. These results are not surprising since the magnitude of prediction error is inversely related to the accuracy of the master curve shown in Figs. 2(b), 3(a) and (b). A capability to predict the final solids concentration using partial experimental data can lead to substantial savings in experimentation time. Fig. 6 shows a dependence of the difference, ▵t = t99 tc, on applied pressure, where t99 is the dewatering time to reach a cake solids
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Fig. 6. Dependence of the time difference, ▵t = t99 tc, on the applied pressure for 10% A16-SG alumina suspension.
fraction equal to 0.99ϕ∞. While the average reduction in experimentation time is about 20%, significant reduction in experimentation time is possible when the applied pressure is lower than 10 kPa. In the case of dispersed systems, it can be expected that significant reduction in experimentation time can be achieved even at higher applied pressure. 3.5. Prediction of ϕc when ϕ0 is changed We next consider the problem of predicting the solids concentration at the cake formation–consolidation transition point, ϕc, when the feed solids concentration is changed and the operating pressure remains unchanged. It is assumed that complete dewatering data with a feed solids concentration not below the gel point of the material is available. This also implies that ϕ∞, which is assumed to be independent of ϕ0, is known. It can be seen that the problem of predicting ϕc when ϕ∞ is known, is the inverse of the problem discussed in Section 3.4.2, where ϕ∞ was determined when ϕc was
Fig. 7. Dependence of error in the prediction of parameter ϕc, on the initial solids concentration, for A16-SG alumina suspensions.
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Fig. 9. Effect of pH on scaled evolution in the consolidation stage of dewatering for 10% A16-SG alumina suspension containing 100 ppm of low molecular weight (Mw 8000) PAA.
ways: (i) change in pH, (ii) change in pH in the presence of 100 ppm of low molecular weight (poly-) acrylic acid (PAA) polymer additive, and (iii) change in molecular weight of PAA, at constant pH. Somewhat surprisingly, we have observed that the master curve can persist even
Fig. 8. Effect of pH on dewatering of 10% A16-SG alumina suspension: (a) temporal evolution of dewatering; (b) scaled evolution in the consolidation stage of dewatering.
known. Therefore, the existence of the master curve can be employed to predict ϕc at a desired ϕ0. In particular, it has to be noted that estimation of ϕc for a desired ϕ0, does not require any knowledge of the kinetics of dewatering in cake formation stage. Dependence of the error in estimation of ϕc on ϕ0 shown in Fig. 7, indicates that the method yields fairly accurate estimates of ϕc. Larger magnitude of prediction error at low ϕ 0 (= 0.06) implies that a different approach is needed for modeling of dewatering when the feed solids concentration is less than or approaches the gel point of the suspension. 3.6. Effect of suspension chemistry It is known that suspension chemistry has a strong impact on the dewaterability of suspensions. In this work, we only summarize results obtained by modifying the chemistry of alumina suspensions in the following
Fig. 10. Effect of molecular weight of PAA on dewatering of 10% A16SG alumina suspension, at a pH of 9.5: (a) temporal evolution of dewatering; (b) scaled evolution in the consolidation stage of dewatering.
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when the pH varies over a reasonably wide range. Fig. 8 (a) shows the evolution of dewatering when the pH of the suspension is varied. The corresponding evolution in scaled coordinates is shown in Fig. 8(b). It can be seen that there appear to be two sets of master curves, one each for dispersed and flocculated systems. The reasons for occurrence of two sets of master curves are not yet understood at present. As shown in Fig. 9, similar results are obtained when the pH is changed in the presence of low molecular weight PAA. In complete contrast to the above, clear evidence of unique master curve is seen when the pH is kept constant, and the molecular weight of polymeric additive is varied over a reasonably wide range. These results are shown in Fig. 10(a) and (b). From Fig. 10(b), it is once again apparent that under certain conditions, there can be a strong inter-relationship between the process parameters /c and /∞. As discussed in Section 3.4, and subject to validation of these results using other materials and conditions, such an inter-relationship would imply that the existence of the master curve could be used to predict the final solids concentration for a desired molecular weight of the polymer additive, if partial experimental data, i.e. till /c, were available. 4. Conclusions Constant pressure batch dewatering of alumina and kaolin fine particle suspension systems has been experimentally investigated. Scaling relationships for the extent of dewatering and dewatering time, which yield a unique master curve in the cake consolidation stage of dewatering, have been proposed. In the systems investigated which show two-stage dewatering behavior, the existence of a master curve appears to be a fairly general phenomenon since it persists even when the pressure, initial solids concentration and the suspension chemistry are varied. Using a simple consolidation stage dewatering model, an application involving use of the master curve for prediction of dewatering process parameters has been demonstrated. Results reported here appear to indicate that an understanding of the scaling behavior can provide useful insights about the characteristic quantities governing evolution of the pressure dewatering process. It is hoped that the results presented here will be helpful for the analysis of laboratory pressure dewatering experiments. Acknowledgements Financial support for this work from the Department of Science and Technology, Government of India, is
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gratefully acknowledged. The authors are grateful to Prof. P. C. Kapur and Dr. Pradip of Tata Research Design and Development Centre (TRDDC), Pune, India, for their comments and suggestions. References Brown, L.A., Zukoski, C.F., 2003. Experimental test of two-phase fluid model of drying consolidation. AIChE Journal 49, 362–372. Burger, R., Concha, F., Karlsen, K.H., 2001. Phenomenological model of filtration processes: 1. Cake formation and expression. Chemical Engineering Science 56, 4537–4553. de Kretser, R.G., Usher, S.P., Scales, P.J., Boger, D.V., Landman, K.A., 2001. Rapid filtration measurement of dewatering design and optimization parameters. AIChE Journal 47, 1758–1769. Kapur, P.C., Raha, S., Usher, S., de Kretser, R.G., Scales, P.J., 2002. Modeling of the consolidation stage in pressure filtration of compressible cakes. Journal of Colloid and Interface Science 256, 216–222. Landman, K.A., White, L.R., 1997. Predicting filtration time and maximising throughput in a pressure filter. AIChE Journal 43, 3147–3160. Murase, T., Iritani, E., Cho, J.H., Shirato, M., 1989. Determination of filtration characteristics based upon filtration tests under step-up filtration conditions. Journal of Chemical Engineering of Japan 22, 373–378. Raha, S., Khilar, K.C., Pradip, P.C., Kapur, 2006. A mean phi model for pressure filtration of fine and colloidal suspensions. Canadian Journal of Chemical Engineering 84, 83–93. Ruth, B.F., 1935. Studies in filtration. III. Derivation of general filtration equations. Industrial and Engineering Chemistry 27, 708–723. Shirato, M., Murase, T., Kato, H., Fukaya, S., 1970. Fundamental analysis for expression under constant pressure. Filtration and Separation 7, 277–282. Shirato, M., Murase, T., Negawa, M., Moridera, H., 1971. Analysis of expression operations. Journal of Chemical Engineering of Japan 4, 263–268. Shirato, M., Murase, T., Tokunaga, A., Yamada, O., 1974. Calculations of consolidation period in expression operations. Journal of Chemical Engineering of Japan 7, 229–231. Sis, H., Chander, S., 2000. Pressure filtration of dispersed and flocculated alumina slurries. Minerals and Metallurgical Processing 17, 41–46. Stamatakis, K., Tien, C., 1991. Cake formation and growth in cake filtration. Chemical Engineering Science 46, 1917–1933. Tien, C., Teoh, S.K., Tan, R.B.H., 2001. Cake filtration analysis — the effect of the relationship between the pore liquid pressure and the cake dependent compressive stress. Chemical Engineering Science 56, 5361–5369. Wakeman, R.J., Sabri, M.N., Tarleton, E.S., 1991. Factors affecting the formation and properties of wet compacts. Powder Technology 65, 283–292.