- Email: [email protected]
A comparative study of slip velocity models for the prediction of performance of floatex density separator
International Journal of Mineral Processing 94 (2010) 20–27
Contents lists available at ScienceDirect
International Journal of Mineral Processing j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i j m i n p r o
A comparative study of slip velocity models for the prediction of performance of floatex density separator Biswajit Sarkar a, Avimanyu Das b,⁎ a b
Department of Chemical and Biological Engineering, University at Buffalo, The State University of New York, Buffalo, NY 14260-4200, USA MNP Division, National Metallurgical Laboratory, Jamshedpur, 831007, India
a r t i c l e
i n f o
Article history: Received 5 June 2009 Received in revised form 23 October 2009 Accepted 1 November 2009 Available online 10 November 2009 Keywords: Teeter bed separator Gravity concentration Slip velocity Modeling Simulation
a b s t r a c t The separation features of the floatex density separator (FDS) are investigated through experimental and computational approaches. It has been shown that the performance of the FDS can be predicted reasonably well using a slip velocity model and steady-state mass balance equations. The approach for the formulation of the slip velocity model makes a difference in the prediction of FDS performance. The computed data from four different slip velocity models have been compared and contrasted with the experimental observations. It has been shown that a slip velocity model based on the modified Richardson and Zaki equation, in which the dissipative pressure gradient is considered to be the primary driving force for separation, predicts the performance more accurately than the other three. A deslimed feed is recommended for better performance of the FDS. © 2009 Elsevier B.V. All rights reserved.
1. Introduction The floatex density separator (FDS) is a gravity concentrator in which liquid fluidization is effectively used to concentrate fine minerals. Hindered settling along with the formation of an autogenous dense medium in the form of a teeter bed are the primary physical phenomena in the FDS. The slurry of mineral fines is fed to the FDS from the top through a feed well. The teeter water (TW) is introduced from the bottom uniformly throughout the cross section. The heavier particles settle through the autogenous dense medium generated while the lighter particles are hydraulically transported to the overflow. Detailed description of the FDS and its operation are discussed elsewhere (Sarkar et al., 2008a,b; Das et al., 2009a). Sarkar et al. (2008a) described the separation with respect to the terminal settling velocity of the particles since this variable combines both size and density of the particle effectively into one parameter. Some initiatives were taken to study the performance of the FDS (Galvin et al., 1999b; Kari et al., 2006; Sarkar et al., 2008a,b). Theoretical studies adopting relative velocity approaches were also taken up to predict the performance of the FDS by a few groups (Das et al., 2009a,b; Kapure et al., 2007; Galvin et al., 1999a; Kim and Klima, 2004). These studies were based on simplified role of autogenous suspension generated inside the FDS. The combined effect of liquid fluidization
⁎ Corresponding author. Tel.: +91 657 2349018; fax: +91 657 2345055. E-mail address: [email protected] (A. Das). 0301-7516/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.minpro.2009.11.001
and autogenous dense medium generation renders the problem extremely complex. The movement of the particles in such a complex system was first systematically studied by Richardson and Zaki (1954) in which they proposed a slip velocity correlation. Subsequently, several other groups proposed different slip velocity correlations to describe the movement of the particles suspended in a liquid (Lockett and AlHabbooby, 1974; Masliyah, 1979; Patwardhan and Tien, 1985; Van der Wielen et al., 1996; Galvin et al., 1999a; Das et al., 2009a,b; Asif, 2002). The separation of particles by liquid fluidization was studied by several researchers (Asif, 1998, 2004; Galvin et al., 1999b; Rasul et al., 2000, 2002; Epstein, 2005). Galvin et al. (1999a) used their own slip velocity model to describe the separation in a teetered bed separator. The slip velocity model proposed by Galvin et al. (1999a) was used to predict the performance of the FDS by a few groups (Kari et al., 2006; Kapure et al., 2007). Kari et al. (2006) compared the applicability of Galvin's slip velocity model with reference to the slip velocity model of Lockett and Al-Habbooby (1974) for predicting the separation performance of the FDS in chromite ore beneficiation. They concluded that both these models are good in predicting the performance at a low teeter water flow rate and a low suspension density. However, in all these studies, the effect of the bed pressure on the separation performance was not considered. Recently, a more realistic slip velocity approach considering the bed pressure and an average bed voidage using Galvin's slip velocity model has been adopted by Das et al. (2009a). Galvin's slip velocity correlation (Galvin et al., 1999a) is compared with other slip velocity models proposed (Masliyah (1979), Patwardhan and Tien (1985), Van der Wielen et al. (1996)) which are used for the prediction of performance of FDS.
B. Sarkar, A. Das / International Journal of Mineral Processing 94 (2010) 20–27
Van Der Wielen et al. (1996) proposed that the two functions F(ε) and G(ρ) may be estimated by the following correlations:
2. Mathematical formulation Details of mathematical formulation are available in a previous paper published by the same group (Das et al., 2009a). Here, some parts are briefly reproduced along with other slip velocity formulations for the convenience of the reader.
0:79nij −1
FðεÞ = ε GðρÞ =
2.1. Particle slip velocity The slip velocity of a particle can be decomposed into three different components, namely, (i) the terminal settling velocity, (ii) the inter-particle distance, and (iii) the suspension density (Patwardhan and Tien, 1985), although the suspension density and the interparticle distance are not mutually independent. The model suggested by them can be used in a modified form for the particle slip velocity in multi-solid systems in which the solids differ in size as well as density: Vslip;ij = Uter;ij FðεÞ GðρÞ
ð1Þ
where, Vslip and Uter are the slip velocity and the terminal settling velocity of the particle, i and j denote the size and density class, respectively. The function F(ε) accounts for the effect of neighboring particles and G(ρ) accounts for the effect of the suspension density. Masliyah (1979) proposed the functional form of F(ε) and G(ρ) for multi-particle system as follows: nij −2
FðεÞ = ε
GðρÞ =
ð2Þ
ρij −ρsus ρij −ρf
ð3Þ
where, nij is the Richardson and Zaki index. Happel (1958) and Kuwabara (1959) proposed a cell model for inter-particle interaction. On similar lines, Patwardhan and Tien (1985) proposed a cell model in which the particle–particle interaction in the suspension is represented by the average distance between two neighboring particles. According to them, a suspension may be represented by a particle surrounded by a liquid envelop of certain thickness. By definition, the bed voidage or the liquid volume fraction for identical particles is represented as: 1 ε = 1− 3 1 + ddε
ð4Þ
where, dε/2 is the liquid envelop thickness and d is the particle diameter. For multi-solid suspensions, the bed voidage can be generalized considering the average diameter of the particles as follows: −1 =3
dε = davg ½ð1−εÞ
−1
21
ð5Þ
ρij −ρsus ρij −ρf
ð7Þ !
nij
=4:8
ð8Þ
Galvin et al. (1999a) proposed a generalized Richardson and Zaki correlation for the estimation of the slip velocity in which the product of F(ε) and G(ρ) is computed as a function of the density differences as follows: FðεÞGðρÞ =
ρij −ρsus ρij −ρf
!n
ij −1
ð9Þ
The above four different empirical slip velocity correlations are chosen for the description of separation in the FDS. A similarity is seen in all four slip velocity correlations though they are based on different physical consideration. The Masliyah (1979) slip velocity correlation is based on Wallis's (1969) steady state momentum balance principle for a particle in suspension. The Patwardhan and Tien (1985) slip velocity correlation is based on the combination of the particle hindrance, particle concentration, and local voidage. The particle hindrance is incorporated using Richardson and Zaki (1954) approach and the influence of the particle concentration is accounted for using the Lockett and Al-Habbooby (1974) approach. In their approach, the local porosity is considered using a cell model based on conservation of volume of each phase. The Van der Wielen et al. (1996) steady state slip velocity correlation is based on the drag force originating from the fluidization of mono-dispersed particles. The slip velocity correlation proposed by Galvin et al. (1999a,b) is based on the Richardson and Zaki (1954) approach in which the dissipative pressure gradient due to the weight of the particle in the liquid (excluding the hydrostatic pressure gradient) is considered to be the main driving force. The use of slip velocity correlations developed by Galvin et al. (1999a,b) is restricted to particles having a higher density than the suspending liquid. The effect of particle size is also not taken into account in formulating the dissipative pressure gradient. The other three slip velocity correlations are analytical in nature and therefore, their use is not restricted to particles having a density higher than that of the liquid. The correlations of Masliyah (1979) and Patwardhan and Tien (1985) are similar for mono-dispersed particles with uniform bed voidage. In all four correlations the wall effect, which can assume a significant role, is not considered. In all the models the particles are assumed to be spherical. In the present work, the particulate system has a top size of 1.0 mm (nominally) in which 70% of the particles are less than 0.5 mm. It is a common practice to consider the particles to be prisms with N faces. The higher the value of N, the closer the particle resembles a sphere. The closure condition ensures that the value of N gets larger as the particle size becomes smaller. Thus, the sphericity assumption of the particles is reasonable.
where, davg = Σcij dij/Σcij, is the average particle size and cij is the volumetric concentration of different particles and by definition Σcij + ε = 1. For accurate estimation of the slip velocity of the particles in a concentrated suspension, the particle–particle interactions may be characterized for each particle species. εij for each particle species is expressed as follows:
2.2. Steady-state mass balance
1 εij = 1− 3 1 + dε d ij
where, F, U and O are the feed, underflow and overflow mass flow rates, respectively and fij, uij and oij are the mass composition of the feed, underflow and overflow, respectively.
=
ð6Þ
Component mass balance and overall steady-state mass balance over the unit is expressed by the two-product relationship: Ffij = Uuij + Ooij
ð10aÞ
F=U+O
ð10bÞ
22
B. Sarkar, A. Das / International Journal of Mineral Processing 94 (2010) 20–27
A simultaneous solution of the slip velocity and steady-state mass balance equations gives an adequate description of the FDS performance. The slip velocity of each particle is estimated first. If the slip velocity of the particle is greater than the interstitial fluid velocity, it will report to the underflow. On the other hand, the particles with slip velocities lower than the interstitial fluid velocity will be carried to the overflow stream. If the particle slip velocity is equal to the interstitial fluid velocity, it has equal probability of reporting to either the overflow or the underflow stream. Knowing the size and density distribution of the particles in the feed and applying the mass balance equations, then, the prediction of the size and density distributions of the FDS products, the partition curves and overall estimation of the performance can be made (Das et al., 2009a). 2.3. Modeling of static pressure, bed voidage and suspension density The suspension density is highest at the bottom and it decreases with height. Consequently, the voidage is the lowest at the bottom and increases along the axial direction. Although, considerations for the suspension density and voidage distribution certainly enhance the accuracy of prediction, these increase the complexity many-fold. Das et al. (2009a) have shown that working with the average values of the two properties offers a great simplification without losing much accuracy. The two properties are related to the bed pressure. At the bottom of the bed the pressure is given by the following equation (Kopko et al., 1975): ΔPT = ΔPs + ΔPf + ΔPsw + ΔPfw + ΔPss
ð11Þ
where, ΔPT is the total pressure drop, ΔPs and ΔPf are the static pressure contributions of the solid phase and the liquid phase, respectively, ΔPsw , ΔPfw and ΔPss are the pressure losses due to solid-wall friction, fluid-wall friction and particle–particle interactions, respectively. In order to account for the pressure losses due to particle-wall and fluid-wall friction, the system is assumed to be equivalent to a Newtonian flow through smooth pipe in which the total pressure loss is due to liquid-wall friction alone (Kopko et al., 1975). By definition, the friction factor is related to the pressure loss due to fluid-wall friction. Friction factor is estimated using empirical correlation and the pressure drop is estimated making use of its definition as follows: 2fu2l ρl
ΔPfw = H DFDS
ð12Þ
where, DFDS is the equivalent diameter (4⁎hydraulic radius) of the FDS unit, f is the friction factor and H is the total height of the bed. The solid–solid frictional loss is assumed to be negligible. Therefore, Eq. (11) reduces to: ΔPstatic = ΔPs + ΔPf = ΔPT −ΔPfw
ð13Þ
where, ΔPstatic is the combined static pressure of the fluid and the solids of a suspension. Das et al. (2009a) estimated the average suspension density from bed pressure. They have also used the volume average density of the particles for voidage determination: εavg =
ρavg;particle −ρsus ρavg;particle −ρwater
ð14Þ
where, ρavg,particle is the particle average density and εavg is the average bed voidage. The interstitial teeter water velocity may, then, be obtained by dividing the superficial teeter water velocity by the bed voidage. It may be noted here that depending upon the solid concentration and particle size distribution, the suspension may exhibit non-Newtonian flow behavior. However, since the bed is fluidized, a Newtonian
behavior is a reasonable approximation at most locations in the bed. Moreover, a relative assessment of the four models in a simplified flow regime is the main focus of the work. Hence, a rigorous nonNewtonian analysis is not attempted. The estimation of the necessary parameters such as Richardson and Zaki index (Rowe, 1987), terminal settling velocity (Hartman et al., 1989) and friction factor for the simulation are discussed elsewhere (Das et al., 2009a,b). 3. Experimental Experimental test campaigns were undertaken using a laboratory model FDS at NML pilot plant with coal as the feed material. Details of the feed characteristics, experimental procedures and test campaigns are discussed elsewhere (Sarkar et al., 2008a). Two different feed size distributions were selected for the comparative study; one widely sized feed, F1, having −1180 μm nominal size with an ash content of 33.3% and the other feed, F2, of nominal size − 1180+150 μm and an ash content of 28.3%. The size distribution and size-wise ash distribution of the two feeds are presented in Table 1. The four tests selected for detailed comparison are listed in Table 2 in which the teeter water flow rate is expressed as superficial liquid velocity. It may be noted that a peristaltic pump is used for feeding the system. The teeter water is introduced through a rotameter in a controlled manner and the underflow valve is operated through a PID controller. Thus, the system operates in a fairly controlled manner and the reproducibility of the results is quite good. A reproducibility test indicated that the deviations in the yield and ash values are within 4%. 4. Comparison of model predictions A computer program was written in VC++ in order to solve the slip velocity and mass balance equations. The details of the feed matrix used and the details of computational procedure are available elsewhere (Das et al., 2009a). Comparisons of the performance of the FDS as predicted by the four different models with the experimental results are shown in Table 2. 4.1. Studies with Feed F1 These studies refer to tests T1 & T2 as shown in Table 2. It may be seen from this table that at a lower bed pressure (T1) all four models over-predict the yield. However, as the bed pressure increases (T2) the model predictions become comparable with the experimental results (±11% error). The Galvin, Masliyah and Van der Wielen models marginally under predict the yield (about 7%) whereas the Patwardhan and Tien model predicts a higher yield (about 11%). At a low bed pressure, the bed formation is not complete. The absence of an effective dense medium (Sarkar et al., 2008b) leads to a size based separation in the FDS resembling an elutriation column. Consequently, the low experimental yield values are not reflected in the model predictions. In the present study, an effective suspension density and average bed voidage are assumed. However, at a low bed pressure the actual suspension density is lower and the voidage is higher due to Table 1 Size and ash distribution of the two different feeds. Feed F1
Feed F2
Particle size
Weight
Ash
Particle size
Weight
Ash
μm
%
%
μm
%
%
+ 500 − 500 + 300 − 300+150 − 150 + 75 − 75 Head
32.11 19.62 23.86 7.27 17.14 100.00
24.51 25.31 36.21 40.69 53.03 33.54
+ 500 − 500 + 300 − 300 + 212 − 212 + 150
26.62 24.39 29.05 19.94
23.34 23.46 27.55 41.37
100.00
28.29
B. Sarkar, A. Das / International Journal of Mineral Processing 94 (2010) 20–27
23
Table 2 Comparison of observed overall performance in terms of clean coal (overflow) properties with various model predictions. Test no.
T1 T2 T3 T4
Feed type
F1 F1 F2 F2
Superficial velocity
Bed pressure
Average suspension density
Experimental
Galvin
mm/s
kPa
kg/m3
Yield (%)
Ash (%)
Yield (%)
Ash (%)
Yield (%)
3.15 3.15 3.15 4.42
4.85 5.27 5.27 5.27
1150 1250 1250 1250
27.4 71.0 52.3 66.3
26.5 26.3 15.9 17.0
41.5 65.8 59.2 69.5
34.7 27.8 16.6 21.1
39.5 66.8 66.3 74.7
inadequate development of the bed. Therefore, the estimated interstitial slip velocities are different from the actual slip velocities. The slip velocities of coal particles of different size and density for feed F1 are calculated using Galvin slip velocity model and are shown in Fig. 1. The dotted horizontal line represents the interstitial teeter water velocity calculated based on an average bed voidage. At a low bed pressure the interstitial teeter water velocity is close to the superficial teeter water velocity as denoted by the horizontal solid line due to insufficient dense medium formation. The difference between the predicted yield value and the experimental yield value at low bed pressure is attributed to the inaccurate estimation of the interstitial teeter water velocity under these conditions. It can also be seen from Fig. 1 that at a low bed pressure all particles below 75 μm size will report to the overflow irrespective of their density. The feed F1 contains a higher amount of − 75 μm material having a high ash content which increases the clean coal ash. The misplacement of ultrafine particles is also not considered in the computation which is another source of error. Sarkar et al. (2008a) have shown that the misplacement of ultrafines to the underflow is proportional to the underflow moisture content. According to Fig. 1, no ultrafine particles should report to the underflow. However, in reality, the ultrafine particles will be misplaced to the underflow with underflow moisture leading to an erroneous estimation of clean coal ash from the model. It may be noted that misplacement is more at low bed pressure due to inadequate development of the bed that results in improper dewatering of the settled mass. In the absence of density based separation at a low bed pressure, the predicted ash values are higher than the experimental value. However, at a higher bed pressure the predictions of the clean coal grade are reasonably close to the observed values. In fact, the Galvin model predictions are within 5% of the observed grade. 4.2. Studies with Feed F2 At a low teeter water rate (T3), all models over-predict the yield. However, the Galvin model offers the closest prediction under these
Fig. 1. Slip velocity distribution of feed F1 having − 1180 μm nominal size at a bed pressure of 4.85 kPa and a teeter water velocity of 3.15 mm/s. The horizontal dotted line represents the interstitial teeter water velocity calculated based on average suspension density and the horizontal solid line represents the superficial teeter water velocity in the FDS.
Masliyah
Van Der Wielen
Patwardhan & Tien
Ash (%)
Yield (%)
Ash (%)
Yield (%)
Ash (%)
36.2 30.7 24.3 25.9
38.9 66.6 66.2 73.7
36.5 31.6 25.1 26.7
44.2 79.2 66.2 80.1
33.3 29.7 18.8 24.3
conditions (Table 2). At a higher teeter water rate (T4) also the yield values predicted by all models are higher than the observed yield. Under these conditions, the Galvin model predicts a yield value which is very close to the experimental value and the Patwardhan & Tien model predicts a value which is farthest from what is experimentally observed. The clean coal grade is predicted quite accurately by the Galvin model at a low teeter water condition while the other models predict higher values. At a high teeter water rate also the predictions for the grade are on the higher side with the Galvin model predictions being the closest again. It may be summarized from Table 2 that the Galvin model offers good predictions for all conditions studied except when the bed pressure is low (T1). The predictions by the Masliyah and Van der Wielen models are similar for all tests. The Patwardhan & Tien model predicts much higher yield values than what is observed experimentally in all tests. The clean coal ash values predicted by this model are also not close to the experimental values. In the development of the Patwardhan & Tien model it is assumed that the particles are surrounded by a liquid envelope. An effective particle size, larger than the actual particle size, is considered based on an average bed voidage. This results in an increase in the frictional force when the particles settle against the rising teeter water. As a result, the particles experience enhanced upward force that leads to a greater transport of the heavier particles to the overflow stream. This explains why this model always tends to over predict the clean coal yield and ash. At a high bed pressure, the Galvin, Masliyah and Van der Wielen models lead to under prediction of the yield for feed F1 (Test T2). However, they provide an over prediction of the yield for feed F2 (Test T3, T4). These observations may be attributed to the difference in the size distributions of feed F1 and F2. Under any set of conditions, the bed voidage with feed F1 is less than that with feed F2 since feed F1 has a wider size distribution. Moreover, the average density of feed F1 is higher than that of feed F2. A lower bed voidage along with the high density of particles imply a higher effective suspension density. In the present study, the bed voidage and suspension density are incorporated in the averaged form into the model formulation. Thus, for feed F1, a lower suspension density and a higher bed voidage are used, which result in the under prediction of the yield. Whereas, with feed F2, slightly higher values of suspension density and lower bed voidage are used that result in somewhat over prediction of the yield values. In the development of the Galvin model (Galvin et al., 1999a), a dissipative pressure gradient is considered, which arises due to the fluid drag. It is argued that the slip velocity is independent of the origin of the drag force. The complex opposing interaction between the bed pressure and teeter water velocity is incorporated as the dissipative pressure gradient. The ability of providing better predictions of the performance of the FDS by the Galvin model compared to the other three models is likely to be due to the above reason that describes the essence of the separation. The Masliyah model is developed based on the steady-state momentum balance principle, whereas Van der Wielen's slip velocity model is developed from a steady-state force balance. Thus, both these slip velocity models are developed under similar physical
24
B. Sarkar, A. Das / International Journal of Mineral Processing 94 (2010) 20–27
considerations, viz. force and momentum balance. Therefore, the performance predictions using these two slip velocity models are similar under all conditions studied. 4.3. Partition curves The partition curves are generated from the size and density distributions of the two products and the feed. The size based partition curves are presented in Figs. 2 and 3. It can be seen from Fig. 2 that at a low bed pressure (4.85 kPa) the overall size-partition curve is sharp and has a higher positive slope for feed F1. With an increase in the bed pressure from 4.85 to 5.27 kPa the partition curve becomes flatter exhibiting a lower positive slope (Fig. 3) even with a closer sized feed, F2. About 20% of the larger particles (N650 μm) report to the overflow stream at 5.27 kPa bed pressure (feed F2) whereas, almost no larger particles, irrespective of their density, report to the overflow stream at 4.85 kPa bed pressure (F1). This is evident from Fig. 2 which indicates that the partition values are unity above 420 μm. At a low bed pressure, as discussed before, size separation is the primary mode of separation. However, fine particles having a high density may not be carried to the overflow. At a higher bed pressure, the particle loading in the bed increases and the autogenous dense medium forms. At a higher suspension density, larger-lighter particles are also transported to the overflow. Therefore, it can be concluded that at a higher bed pressure, the FDS is more effective as a concentrator. This observation also corroborated our previous findings (Sarkar et al., 2008b). The overall density based partition curves are presented in Figs. 4 and 5. It can be seen from these figures that a significant amount of high density particles report to the overflow stream for both the feeds. The settling of the particles through the suspension is strongly influenced by the size and density of the particles as well as the physical properties of the liquid. At a high bed pressure the dense medium develops fully leading to an increase in the effective viscosity and density of the suspension. The particle mass is determined by its size and density. Those having a lower mass experience a lower gravitational force and are unable to settle against this viscous and dense medium. Therefore, they are likely to be transported to the overflow at higher bed pressures. Thus, the complex interplay between the particle density and size prohibit a very sharp separation in terms of either the size or density. Again, the experimental density partition is described reasonably by all the models with different accuracies. It can also be seen from Figs. 2–5 that the partition curves predicted by the slip velocity models proposed by Masliyah (1979) and Van der Wielen et al. (1996) are similar in nature. On the other hand, the predictions using slip velocity models proposed by Galvin et al. (1999a) and Patwardhan and Tien (1985) are similar in nature. As discussed in
Fig. 2. Prediction of performance in terms of size separation using different slip velocity models at 3.15 mm/s TWFR and 4.85 kPa BP for feed having − 1180 μm nominal size.
Fig. 3. Prediction of performance in terms of size partition using different slip velocity models at 3.15 mm/s TWFR and 5.27 kPa BP for feed having − 1180 + 150 μm feed.
the previous section, the two former models are formulated based on similar fundamental concepts, i.e., steady-state momentum and force balance, respectively. Therefore, the predictions based on these models are similar. The slip velocity equations proposed by Patwardhan and Tien (1985) and Galvin et al. (1999a) are essentially extensions of the Richardson and Zaki (1954) approach. Patwardhan and Tien (1985) extended the Richardson and Zaki model by incorporating a bed voidage based on a cell balance approach. Galvin et al. (1999a) considered a dissipative pressure gradient to extend the Richardson and Zaki model. Hence, the predictions using these two models under all conditions are also somewhat similar. In order to investigate the separation features the density partition curves for narrow size classes are plotted in Figs. 6 and 7. The experimental density partition curves at high bed pressure for a few narrow size classes are shown against the predicted curves in Fig. 6. It can be seen from this figure that these curves are nearly S-shaped (Fig. 6A) for the density partition curves of the particles in these size classes. The hydrodynamic dispersion leads to a distribution of particles along the height of the FDS. However, in the present computation the hydrodynamic dispersion is not considered explicitly. The computations here are for steady-state operation in which the formation of a steady bed (having an average density) is assumed. While there is also a distribution of the suspension density and bed voidage in reality, an average suspension density (as given in Table 2) and a bed voidage is assumed in this work through which the particles settle. The slip velocity is estimated from the terminal settling velocity, the estimation of which takes into account the interstitial
Fig. 4. Prediction of performance in terms of density partition using different slip velocity models at 3.15 mm/s TWFR and 5.27 kPa BP for feed having −1180 μm nominal size.
B. Sarkar, A. Das / International Journal of Mineral Processing 94 (2010) 20–27
25
Fig. 5. Prediction of performance in terms of density partition using different slip velocity models at 3.15 mm/s TWFR and 5.27 kPa BP for feed having − 1180 + 150 μm feed.
fluid velocity. In addition, the functions F(ε) and G(ρ), which are strongly dependent on the hydrodynamics of the system, are used for modifying the slip velocity. Thus, the hydrodynamics are taken into consideration implicitly in the formulation. In view of this, the slip velocity is considered to be adequate to track the movement of the particles towards the product streams. In this manner the size and density distribution of the product streams are evaluated and the partition curves obtained. As depicted in Fig. 6, the models do describe the density partition curves of the particles in various size classes reasonably well. The inclusion of detailed hydrodynamics should improve the accuracy of the predictions and our research group has already taken up this study. The poor separation at a lower bed pressure is investigated further from the density partitions of narrow size classes as shown in Fig. 7. It can be seen from this figure that no density based separation has taken place for the particles larger than 300 μm (Fig. 7A). Good separation is observed only for − 300 μm particles (Fig. 7B and C). It is also seen that the slip velocity models capture the separation for larger particles better. For larger particles, the terminal settling velocities are higher which increase the slip velocity values. As a result, though the average suspension density is higher than the actual suspension density, all larger particles, irrespective of their density, are transported to the underflow stream. For −300 + 150 μm size fraction, only Galvin slip velocity provide a reasonable prediction while other three slip velocity correlations fail to predict the particle separation. For particles smaller than 150 μm size, all slip velocity correlations fail to predict the density separation. The failure of the models in predicting particle separation at smaller size fractions is attributed partly to the assumption of average suspension density and partly to the misplacement phenomenon of fine particles to the underflow which is not accounted for in any of the models. 4.4. Performance prediction The performance of the FDS is computed using all four slip velocity models to understand the effect of the bed pressure and teeter water flow rate on the yield and grade of the product. The predictions using all four models follow a similar trend qualitatively with some variation. The predicted performances of the FDS for the beneficiation of coal fines of feed F1 are presented in Fig. 8. From this figure it may be seen that at a low bed pressure the FDS is unable to produce clean coal with less than 28% ash since the separation is essentially size based. The feed F1 contains a higher amount of the fine particles (−150 μm) with a high ash content. Due to their low mass, these fine particles are transported to the overflow stream hydraulically even at a low teeter water flow rate. Therefore, at a low bed pressure, the
Fig. 6. Comparison of performance prediction in terms of size-wise density partition curves using different slip velocity correlations at 3.15 mm/s teeter rate and 5.27 kPa BP for (A) − 500 + 300 μm size fraction, Feed F2 (B) − 300 + 150 μm size fraction, Feed F1, (C) − 212 + 150 μm size fraction, Feed F2.
overflow product is enriched mostly with high ash fine particles. With an increase in the bed pressure, the effective suspension density and viscosity increase and hence, the total upward force (both buoyancy and drag force) exerted on the particle by the fluid also increases. This enhanced upward force helps in carrying larger and lighter particles with lower ash content to the overflow. The transportation of these particles into the overflow stream at a higher bed pressure reduces the overall ash (Fig. 8B) of the overflow product and increases the clean coal yield (Fig. 8A). Further increase in the bed pressure increases the density and viscosity of the separating medium which helps in transporting larger and heavier particles also to the overflow.
26
B. Sarkar, A. Das / International Journal of Mineral Processing 94 (2010) 20–27
Fig. 8. Performance prediction using different slip velocity models for feed having − 1180 μm nominal sized feed at 3.15 mm/s teeter water flow rate as a function of bed pressure (A) variation of yield (B) variation of grade.
teeter water rate, the clean coal grade expectedly stabilizes close to the feed grade as the hydraulic transport becomes the only dominant phenomenon (Fig. 9B).
Fig. 7. Comparison of performance prediction in terms of size-wise density partition curves using different slip velocity correlations at 3.15 mm/s teeter water rate and 4.85 kPa bed pressure for (A) − 500 + 300 μm size fraction, Feed F1 (B) − 300 + 150 μm size fraction, Feed F1, (C) − 150 + 75 μm size fraction, Feed F1.
This results in an increase in the ash content of the overflow product. Thus, the ash content of the overflow product for feed F1 goes through a minimum (Fig. 8B) when plotted against the bed pressure. An increase in the teeter water flow rate also increases the upward force resulting in an enhanced hydraulic transport. Therefore, for feed F2, the clean coal yield increases monotonically with an increase in the teeter water flow rate (Fig. 9A). The clean coal grade becomes poorer with a higher yield as the teeter water rate increases. At a very high
Fig. 9. Performance prediction using different slip velocity models feed having −1180+ 150 μm feed at 5.27 kPa bed pressure as a function of teeter water flow rate (A) variation of yield (B) variation of grade.
B. Sarkar, A. Das / International Journal of Mineral Processing 94 (2010) 20–27
27
predictions than the other three models. The overall partition curves as well as the density partition of the particles in various size classes are described reasonably well by the models with different degrees of accuracy. It is also established that a deslimed feed is more suitable for FDS operation. Detailed hydrodynamic modeling using such slip velocity models may be useful in providing better understanding and control over the operation of the FDS. Acknowledgement The financial assistance from the Department of Science and Technology, Government of India, for the research is gratefully acknowledged. References
Fig. 10. Performance prediction with Galvin slip velocity model for (A) − 1180 μm feed, and (B) − 1180 + 150 μm feed.
The overall performance prediction against the experimental data for the Galvin model is shown in Fig. 10 along with the feed washability data. It may be seen from this figure that the experimental yield values are below the theoretical maximum (washability data) in the case of feed F1. The model predictions are somewhat close to the experimental observations (Fig. 10A). However, the yield data are close to the theoretical maximum achievable in the case of feed F2 (Fig. 10B). The model predictions are also very close to the experimental observations in this case. This is because of the fact that the presence of the ultrafines in feed F1 adversely affects the partition of the other particles. A feed from which the fines are removed is, therefore, recommended for good performance of the FDS. 5. Conclusion The performance of the FDS is investigated through experimentation and detailed simulation studies. Four different slip velocity models are considered for this study. The salient features of all four models are discussed with reference to their ability to predict the performance of the FDS. The computed data using all four models are compared and contrasted with the experimental observations. The difference between the computed performance and experimental results are discussed in the light of the formulation of the slip velocity models and the assumptions made in this work. It is shown that although all four models are capable of capturing the essence of the separation features in the FDS, their accuracies in predicting the performance of the FDS are different. At a low bed pressure, the relatively poor predictions are also addressed. It is established that the model proposed by Galvin et al. (1999a) in which the dissipative pressure gradient is considered to be the main driving force for the separation, offers better performance
Asif, M., 2004. The complete segregation model for liquid fluidized bed: formulation and related issues. Powder Technology 140, 21–29. Asif, M., 2002. Predicting binary solid fluidized bed behavior using averaging approaches. Powder Technology 127, 226–238. Asif, M., 1998. Segregation velocity model for fluidized suspension of binary mixtures of particles. Chemical Engineering and Processing 37, 279–286. Das, A., Sarkar, B., Mehrotra, S.P., 2009a. Prediction of separation performance of floatex density separator for processing of fine coal particles. International Journal of Mineral Processing 91 (1–2), 41–49. Das, A., Sarkar, B., Biswas, P., Roy, S., 2009b. Performance prediction of floatex density separator in processing iron ore fines — a relative velocity approach. Mineral Processing and Extractive Metallurgy 118 (2), 78–84. Epstein, N., 2005. Teetering. Powder Technology 151, 2–14. Galvin, K.P., Pratten, S., Nguyen Tran Lam, G., 1999a. A generalized empirical description for particle slip velocities in liquid fluidized beds. Chemical Engineering Science 54, 1045–1052. Galvin, K.P., Pratten, S.J., Nicol, S.K., 1999b. Dense medium separation using a teetered bed separator. Minerals Engineering 12 (9), 1069–1081. Hartman, M., Havlin, V., Trnka, O., Carsky, M., 1989. Predicting the free fall velocities of spheres. Chemical Engineering Science 44 (8), 1743–1745. Happel, J., 1958. Viscous flow in multiparticle systems — slow motion of fluids relative to beds of spherical particles. AIChE J. 4, 197–201. Kapure, G., Kari, C., Rao, S.M., Rao, N.D., 2007. The feasibility of a slip velocity model for predicting the enrichment of chromite in a Floatex density separator. International Journal of Mineral Processing 82 (2), 86–95. Kari, C., Kapure, G., Rao, S.M., Rao, N.D., 2006. Predicitng the chromite mineral upgradation in Floatex density separator using hindered settling models. Iron and Steel Institute of Japan International 46 (7), 966–973. Kim, B.H., Klima, M.S., 2004. Development and application of a dynamic model for hindered settling column separations. Minerals Engineering 17, 403–410. Kopko, R.J., Barton, P., McCormick, R.H., 1975. Hydrodynamics of vertical liquid–solids transport, industrial and engineering chemistry. Process Design and Development 14 (3), 264–269. Kuwabara, S., 1959. The forces experienced by randomly distributed parallel circular cylinders or spheres in a viscous flow at small Reynolds numbers. Journal of the Physical Society of Japan 14 (4), 527–532. Lockett, M.J., Al-Habbooby, H.M., 1974. Relative particle velocities in two species settling. Powder Technology 10, 67–71. Masliyah, J.H., 1979. Hindered settling in multi-species particle system. Chemical Engineering Science 34, 1166–1168. Patwardhan, V.S., Tien, C., 1985. Sedimentation and liquid fluidization of solid particles of different sizes and densities. Chemical Engineering Science 40 (7), 1051–1060. Rasul, M.G., Rudolph, V., Wang, F.Y., 2000. Particles separation using fluidization techniques. International Journal of Mineral Processing 60, 163–179. Rasul, M.G., Rudolph, V., Carsky, M., 2002. Segregation in binary and ternary liquid fluidized beds. Powder Technology 126, 116–128. Richardson, J.F., Zaki, W.N., 1954. Sedimentation and fluidization: Part I. Transactions of the Institution of Chemical Engineers 32, 35–53. Rowe, P., 1987. A convenient empirical equation for estimation of the Richardson-Zaki exponent. Chemical Engineering Science 42, 2795–2796. Sarkar, B., Das, A., Roy, S., Rai, S.K., 2008a. In depth analysis of alumina removal from iron ores fines using teetered bed gravity separator. Mineral Processing and Extractive Metallurgy 117 (1), 48–55. Sarkar, B., Das, A., Mehrotra, S.P., 2008b. Study of separation features in floatex density separator for cleaning fine coal. International Journal of Mineral Processing 86, 40–49. Van Der Wielen, L.A.M., Van Dam, M.H.H., Luyben, K.Ch.A.M., 1996. On the relative motion of particle in a swarm of different particles. Chemical Engineering Science 51, 995–1008. Wallis, G.B., 1969. One dimensional two phase flow. McGraw Hill, New York.
Contents lists available at ScienceDirect
International Journal of Mineral Processing j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i j m i n p r o
A comparative study of slip velocity models for the prediction of performance of floatex density separator Biswajit Sarkar a, Avimanyu Das b,⁎ a b
Department of Chemical and Biological Engineering, University at Buffalo, The State University of New York, Buffalo, NY 14260-4200, USA MNP Division, National Metallurgical Laboratory, Jamshedpur, 831007, India
a r t i c l e
i n f o
Article history: Received 5 June 2009 Received in revised form 23 October 2009 Accepted 1 November 2009 Available online 10 November 2009 Keywords: Teeter bed separator Gravity concentration Slip velocity Modeling Simulation
a b s t r a c t The separation features of the floatex density separator (FDS) are investigated through experimental and computational approaches. It has been shown that the performance of the FDS can be predicted reasonably well using a slip velocity model and steady-state mass balance equations. The approach for the formulation of the slip velocity model makes a difference in the prediction of FDS performance. The computed data from four different slip velocity models have been compared and contrasted with the experimental observations. It has been shown that a slip velocity model based on the modified Richardson and Zaki equation, in which the dissipative pressure gradient is considered to be the primary driving force for separation, predicts the performance more accurately than the other three. A deslimed feed is recommended for better performance of the FDS. © 2009 Elsevier B.V. All rights reserved.
1. Introduction The floatex density separator (FDS) is a gravity concentrator in which liquid fluidization is effectively used to concentrate fine minerals. Hindered settling along with the formation of an autogenous dense medium in the form of a teeter bed are the primary physical phenomena in the FDS. The slurry of mineral fines is fed to the FDS from the top through a feed well. The teeter water (TW) is introduced from the bottom uniformly throughout the cross section. The heavier particles settle through the autogenous dense medium generated while the lighter particles are hydraulically transported to the overflow. Detailed description of the FDS and its operation are discussed elsewhere (Sarkar et al., 2008a,b; Das et al., 2009a). Sarkar et al. (2008a) described the separation with respect to the terminal settling velocity of the particles since this variable combines both size and density of the particle effectively into one parameter. Some initiatives were taken to study the performance of the FDS (Galvin et al., 1999b; Kari et al., 2006; Sarkar et al., 2008a,b). Theoretical studies adopting relative velocity approaches were also taken up to predict the performance of the FDS by a few groups (Das et al., 2009a,b; Kapure et al., 2007; Galvin et al., 1999a; Kim and Klima, 2004). These studies were based on simplified role of autogenous suspension generated inside the FDS. The combined effect of liquid fluidization
⁎ Corresponding author. Tel.: +91 657 2349018; fax: +91 657 2345055. E-mail address: [email protected] (A. Das). 0301-7516/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.minpro.2009.11.001
and autogenous dense medium generation renders the problem extremely complex. The movement of the particles in such a complex system was first systematically studied by Richardson and Zaki (1954) in which they proposed a slip velocity correlation. Subsequently, several other groups proposed different slip velocity correlations to describe the movement of the particles suspended in a liquid (Lockett and AlHabbooby, 1974; Masliyah, 1979; Patwardhan and Tien, 1985; Van der Wielen et al., 1996; Galvin et al., 1999a; Das et al., 2009a,b; Asif, 2002). The separation of particles by liquid fluidization was studied by several researchers (Asif, 1998, 2004; Galvin et al., 1999b; Rasul et al., 2000, 2002; Epstein, 2005). Galvin et al. (1999a) used their own slip velocity model to describe the separation in a teetered bed separator. The slip velocity model proposed by Galvin et al. (1999a) was used to predict the performance of the FDS by a few groups (Kari et al., 2006; Kapure et al., 2007). Kari et al. (2006) compared the applicability of Galvin's slip velocity model with reference to the slip velocity model of Lockett and Al-Habbooby (1974) for predicting the separation performance of the FDS in chromite ore beneficiation. They concluded that both these models are good in predicting the performance at a low teeter water flow rate and a low suspension density. However, in all these studies, the effect of the bed pressure on the separation performance was not considered. Recently, a more realistic slip velocity approach considering the bed pressure and an average bed voidage using Galvin's slip velocity model has been adopted by Das et al. (2009a). Galvin's slip velocity correlation (Galvin et al., 1999a) is compared with other slip velocity models proposed (Masliyah (1979), Patwardhan and Tien (1985), Van der Wielen et al. (1996)) which are used for the prediction of performance of FDS.
B. Sarkar, A. Das / International Journal of Mineral Processing 94 (2010) 20–27
Van Der Wielen et al. (1996) proposed that the two functions F(ε) and G(ρ) may be estimated by the following correlations:
2. Mathematical formulation Details of mathematical formulation are available in a previous paper published by the same group (Das et al., 2009a). Here, some parts are briefly reproduced along with other slip velocity formulations for the convenience of the reader.
0:79nij −1
FðεÞ = ε GðρÞ =
2.1. Particle slip velocity The slip velocity of a particle can be decomposed into three different components, namely, (i) the terminal settling velocity, (ii) the inter-particle distance, and (iii) the suspension density (Patwardhan and Tien, 1985), although the suspension density and the interparticle distance are not mutually independent. The model suggested by them can be used in a modified form for the particle slip velocity in multi-solid systems in which the solids differ in size as well as density: Vslip;ij = Uter;ij FðεÞ GðρÞ
ð1Þ
where, Vslip and Uter are the slip velocity and the terminal settling velocity of the particle, i and j denote the size and density class, respectively. The function F(ε) accounts for the effect of neighboring particles and G(ρ) accounts for the effect of the suspension density. Masliyah (1979) proposed the functional form of F(ε) and G(ρ) for multi-particle system as follows: nij −2
FðεÞ = ε
GðρÞ =
ð2Þ
ρij −ρsus ρij −ρf
ð3Þ
where, nij is the Richardson and Zaki index. Happel (1958) and Kuwabara (1959) proposed a cell model for inter-particle interaction. On similar lines, Patwardhan and Tien (1985) proposed a cell model in which the particle–particle interaction in the suspension is represented by the average distance between two neighboring particles. According to them, a suspension may be represented by a particle surrounded by a liquid envelop of certain thickness. By definition, the bed voidage or the liquid volume fraction for identical particles is represented as: 1 ε = 1− 3 1 + ddε
ð4Þ
where, dε/2 is the liquid envelop thickness and d is the particle diameter. For multi-solid suspensions, the bed voidage can be generalized considering the average diameter of the particles as follows: −1 =3
dε = davg ½ð1−εÞ
−1
21
ð5Þ
ρij −ρsus ρij −ρf
ð7Þ !
nij
=4:8
ð8Þ
Galvin et al. (1999a) proposed a generalized Richardson and Zaki correlation for the estimation of the slip velocity in which the product of F(ε) and G(ρ) is computed as a function of the density differences as follows: FðεÞGðρÞ =
ρij −ρsus ρij −ρf
!n
ij −1
ð9Þ
The above four different empirical slip velocity correlations are chosen for the description of separation in the FDS. A similarity is seen in all four slip velocity correlations though they are based on different physical consideration. The Masliyah (1979) slip velocity correlation is based on Wallis's (1969) steady state momentum balance principle for a particle in suspension. The Patwardhan and Tien (1985) slip velocity correlation is based on the combination of the particle hindrance, particle concentration, and local voidage. The particle hindrance is incorporated using Richardson and Zaki (1954) approach and the influence of the particle concentration is accounted for using the Lockett and Al-Habbooby (1974) approach. In their approach, the local porosity is considered using a cell model based on conservation of volume of each phase. The Van der Wielen et al. (1996) steady state slip velocity correlation is based on the drag force originating from the fluidization of mono-dispersed particles. The slip velocity correlation proposed by Galvin et al. (1999a,b) is based on the Richardson and Zaki (1954) approach in which the dissipative pressure gradient due to the weight of the particle in the liquid (excluding the hydrostatic pressure gradient) is considered to be the main driving force. The use of slip velocity correlations developed by Galvin et al. (1999a,b) is restricted to particles having a higher density than the suspending liquid. The effect of particle size is also not taken into account in formulating the dissipative pressure gradient. The other three slip velocity correlations are analytical in nature and therefore, their use is not restricted to particles having a density higher than that of the liquid. The correlations of Masliyah (1979) and Patwardhan and Tien (1985) are similar for mono-dispersed particles with uniform bed voidage. In all four correlations the wall effect, which can assume a significant role, is not considered. In all the models the particles are assumed to be spherical. In the present work, the particulate system has a top size of 1.0 mm (nominally) in which 70% of the particles are less than 0.5 mm. It is a common practice to consider the particles to be prisms with N faces. The higher the value of N, the closer the particle resembles a sphere. The closure condition ensures that the value of N gets larger as the particle size becomes smaller. Thus, the sphericity assumption of the particles is reasonable.
where, davg = Σcij dij/Σcij, is the average particle size and cij is the volumetric concentration of different particles and by definition Σcij + ε = 1. For accurate estimation of the slip velocity of the particles in a concentrated suspension, the particle–particle interactions may be characterized for each particle species. εij for each particle species is expressed as follows:
2.2. Steady-state mass balance
1 εij = 1− 3 1 + dε d ij
where, F, U and O are the feed, underflow and overflow mass flow rates, respectively and fij, uij and oij are the mass composition of the feed, underflow and overflow, respectively.
=
ð6Þ
Component mass balance and overall steady-state mass balance over the unit is expressed by the two-product relationship: Ffij = Uuij + Ooij
ð10aÞ
F=U+O
ð10bÞ
22
B. Sarkar, A. Das / International Journal of Mineral Processing 94 (2010) 20–27
A simultaneous solution of the slip velocity and steady-state mass balance equations gives an adequate description of the FDS performance. The slip velocity of each particle is estimated first. If the slip velocity of the particle is greater than the interstitial fluid velocity, it will report to the underflow. On the other hand, the particles with slip velocities lower than the interstitial fluid velocity will be carried to the overflow stream. If the particle slip velocity is equal to the interstitial fluid velocity, it has equal probability of reporting to either the overflow or the underflow stream. Knowing the size and density distribution of the particles in the feed and applying the mass balance equations, then, the prediction of the size and density distributions of the FDS products, the partition curves and overall estimation of the performance can be made (Das et al., 2009a). 2.3. Modeling of static pressure, bed voidage and suspension density The suspension density is highest at the bottom and it decreases with height. Consequently, the voidage is the lowest at the bottom and increases along the axial direction. Although, considerations for the suspension density and voidage distribution certainly enhance the accuracy of prediction, these increase the complexity many-fold. Das et al. (2009a) have shown that working with the average values of the two properties offers a great simplification without losing much accuracy. The two properties are related to the bed pressure. At the bottom of the bed the pressure is given by the following equation (Kopko et al., 1975): ΔPT = ΔPs + ΔPf + ΔPsw + ΔPfw + ΔPss
ð11Þ
where, ΔPT is the total pressure drop, ΔPs and ΔPf are the static pressure contributions of the solid phase and the liquid phase, respectively, ΔPsw , ΔPfw and ΔPss are the pressure losses due to solid-wall friction, fluid-wall friction and particle–particle interactions, respectively. In order to account for the pressure losses due to particle-wall and fluid-wall friction, the system is assumed to be equivalent to a Newtonian flow through smooth pipe in which the total pressure loss is due to liquid-wall friction alone (Kopko et al., 1975). By definition, the friction factor is related to the pressure loss due to fluid-wall friction. Friction factor is estimated using empirical correlation and the pressure drop is estimated making use of its definition as follows: 2fu2l ρl
ΔPfw = H DFDS
ð12Þ
where, DFDS is the equivalent diameter (4⁎hydraulic radius) of the FDS unit, f is the friction factor and H is the total height of the bed. The solid–solid frictional loss is assumed to be negligible. Therefore, Eq. (11) reduces to: ΔPstatic = ΔPs + ΔPf = ΔPT −ΔPfw
ð13Þ
where, ΔPstatic is the combined static pressure of the fluid and the solids of a suspension. Das et al. (2009a) estimated the average suspension density from bed pressure. They have also used the volume average density of the particles for voidage determination: εavg =
ρavg;particle −ρsus ρavg;particle −ρwater
ð14Þ
where, ρavg,particle is the particle average density and εavg is the average bed voidage. The interstitial teeter water velocity may, then, be obtained by dividing the superficial teeter water velocity by the bed voidage. It may be noted here that depending upon the solid concentration and particle size distribution, the suspension may exhibit non-Newtonian flow behavior. However, since the bed is fluidized, a Newtonian
behavior is a reasonable approximation at most locations in the bed. Moreover, a relative assessment of the four models in a simplified flow regime is the main focus of the work. Hence, a rigorous nonNewtonian analysis is not attempted. The estimation of the necessary parameters such as Richardson and Zaki index (Rowe, 1987), terminal settling velocity (Hartman et al., 1989) and friction factor for the simulation are discussed elsewhere (Das et al., 2009a,b). 3. Experimental Experimental test campaigns were undertaken using a laboratory model FDS at NML pilot plant with coal as the feed material. Details of the feed characteristics, experimental procedures and test campaigns are discussed elsewhere (Sarkar et al., 2008a). Two different feed size distributions were selected for the comparative study; one widely sized feed, F1, having −1180 μm nominal size with an ash content of 33.3% and the other feed, F2, of nominal size − 1180+150 μm and an ash content of 28.3%. The size distribution and size-wise ash distribution of the two feeds are presented in Table 1. The four tests selected for detailed comparison are listed in Table 2 in which the teeter water flow rate is expressed as superficial liquid velocity. It may be noted that a peristaltic pump is used for feeding the system. The teeter water is introduced through a rotameter in a controlled manner and the underflow valve is operated through a PID controller. Thus, the system operates in a fairly controlled manner and the reproducibility of the results is quite good. A reproducibility test indicated that the deviations in the yield and ash values are within 4%. 4. Comparison of model predictions A computer program was written in VC++ in order to solve the slip velocity and mass balance equations. The details of the feed matrix used and the details of computational procedure are available elsewhere (Das et al., 2009a). Comparisons of the performance of the FDS as predicted by the four different models with the experimental results are shown in Table 2. 4.1. Studies with Feed F1 These studies refer to tests T1 & T2 as shown in Table 2. It may be seen from this table that at a lower bed pressure (T1) all four models over-predict the yield. However, as the bed pressure increases (T2) the model predictions become comparable with the experimental results (±11% error). The Galvin, Masliyah and Van der Wielen models marginally under predict the yield (about 7%) whereas the Patwardhan and Tien model predicts a higher yield (about 11%). At a low bed pressure, the bed formation is not complete. The absence of an effective dense medium (Sarkar et al., 2008b) leads to a size based separation in the FDS resembling an elutriation column. Consequently, the low experimental yield values are not reflected in the model predictions. In the present study, an effective suspension density and average bed voidage are assumed. However, at a low bed pressure the actual suspension density is lower and the voidage is higher due to Table 1 Size and ash distribution of the two different feeds. Feed F1
Feed F2
Particle size
Weight
Ash
Particle size
Weight
Ash
μm
%
%
μm
%
%
+ 500 − 500 + 300 − 300+150 − 150 + 75 − 75 Head
32.11 19.62 23.86 7.27 17.14 100.00
24.51 25.31 36.21 40.69 53.03 33.54
+ 500 − 500 + 300 − 300 + 212 − 212 + 150
26.62 24.39 29.05 19.94
23.34 23.46 27.55 41.37
100.00
28.29
B. Sarkar, A. Das / International Journal of Mineral Processing 94 (2010) 20–27
23
Table 2 Comparison of observed overall performance in terms of clean coal (overflow) properties with various model predictions. Test no.
T1 T2 T3 T4
Feed type
F1 F1 F2 F2
Superficial velocity
Bed pressure
Average suspension density
Experimental
Galvin
mm/s
kPa
kg/m3
Yield (%)
Ash (%)
Yield (%)
Ash (%)
Yield (%)
3.15 3.15 3.15 4.42
4.85 5.27 5.27 5.27
1150 1250 1250 1250
27.4 71.0 52.3 66.3
26.5 26.3 15.9 17.0
41.5 65.8 59.2 69.5
34.7 27.8 16.6 21.1
39.5 66.8 66.3 74.7
inadequate development of the bed. Therefore, the estimated interstitial slip velocities are different from the actual slip velocities. The slip velocities of coal particles of different size and density for feed F1 are calculated using Galvin slip velocity model and are shown in Fig. 1. The dotted horizontal line represents the interstitial teeter water velocity calculated based on an average bed voidage. At a low bed pressure the interstitial teeter water velocity is close to the superficial teeter water velocity as denoted by the horizontal solid line due to insufficient dense medium formation. The difference between the predicted yield value and the experimental yield value at low bed pressure is attributed to the inaccurate estimation of the interstitial teeter water velocity under these conditions. It can also be seen from Fig. 1 that at a low bed pressure all particles below 75 μm size will report to the overflow irrespective of their density. The feed F1 contains a higher amount of − 75 μm material having a high ash content which increases the clean coal ash. The misplacement of ultrafine particles is also not considered in the computation which is another source of error. Sarkar et al. (2008a) have shown that the misplacement of ultrafines to the underflow is proportional to the underflow moisture content. According to Fig. 1, no ultrafine particles should report to the underflow. However, in reality, the ultrafine particles will be misplaced to the underflow with underflow moisture leading to an erroneous estimation of clean coal ash from the model. It may be noted that misplacement is more at low bed pressure due to inadequate development of the bed that results in improper dewatering of the settled mass. In the absence of density based separation at a low bed pressure, the predicted ash values are higher than the experimental value. However, at a higher bed pressure the predictions of the clean coal grade are reasonably close to the observed values. In fact, the Galvin model predictions are within 5% of the observed grade. 4.2. Studies with Feed F2 At a low teeter water rate (T3), all models over-predict the yield. However, the Galvin model offers the closest prediction under these
Fig. 1. Slip velocity distribution of feed F1 having − 1180 μm nominal size at a bed pressure of 4.85 kPa and a teeter water velocity of 3.15 mm/s. The horizontal dotted line represents the interstitial teeter water velocity calculated based on average suspension density and the horizontal solid line represents the superficial teeter water velocity in the FDS.
Masliyah
Van Der Wielen
Patwardhan & Tien
Ash (%)
Yield (%)
Ash (%)
Yield (%)
Ash (%)
36.2 30.7 24.3 25.9
38.9 66.6 66.2 73.7
36.5 31.6 25.1 26.7
44.2 79.2 66.2 80.1
33.3 29.7 18.8 24.3
conditions (Table 2). At a higher teeter water rate (T4) also the yield values predicted by all models are higher than the observed yield. Under these conditions, the Galvin model predicts a yield value which is very close to the experimental value and the Patwardhan & Tien model predicts a value which is farthest from what is experimentally observed. The clean coal grade is predicted quite accurately by the Galvin model at a low teeter water condition while the other models predict higher values. At a high teeter water rate also the predictions for the grade are on the higher side with the Galvin model predictions being the closest again. It may be summarized from Table 2 that the Galvin model offers good predictions for all conditions studied except when the bed pressure is low (T1). The predictions by the Masliyah and Van der Wielen models are similar for all tests. The Patwardhan & Tien model predicts much higher yield values than what is observed experimentally in all tests. The clean coal ash values predicted by this model are also not close to the experimental values. In the development of the Patwardhan & Tien model it is assumed that the particles are surrounded by a liquid envelope. An effective particle size, larger than the actual particle size, is considered based on an average bed voidage. This results in an increase in the frictional force when the particles settle against the rising teeter water. As a result, the particles experience enhanced upward force that leads to a greater transport of the heavier particles to the overflow stream. This explains why this model always tends to over predict the clean coal yield and ash. At a high bed pressure, the Galvin, Masliyah and Van der Wielen models lead to under prediction of the yield for feed F1 (Test T2). However, they provide an over prediction of the yield for feed F2 (Test T3, T4). These observations may be attributed to the difference in the size distributions of feed F1 and F2. Under any set of conditions, the bed voidage with feed F1 is less than that with feed F2 since feed F1 has a wider size distribution. Moreover, the average density of feed F1 is higher than that of feed F2. A lower bed voidage along with the high density of particles imply a higher effective suspension density. In the present study, the bed voidage and suspension density are incorporated in the averaged form into the model formulation. Thus, for feed F1, a lower suspension density and a higher bed voidage are used, which result in the under prediction of the yield. Whereas, with feed F2, slightly higher values of suspension density and lower bed voidage are used that result in somewhat over prediction of the yield values. In the development of the Galvin model (Galvin et al., 1999a), a dissipative pressure gradient is considered, which arises due to the fluid drag. It is argued that the slip velocity is independent of the origin of the drag force. The complex opposing interaction between the bed pressure and teeter water velocity is incorporated as the dissipative pressure gradient. The ability of providing better predictions of the performance of the FDS by the Galvin model compared to the other three models is likely to be due to the above reason that describes the essence of the separation. The Masliyah model is developed based on the steady-state momentum balance principle, whereas Van der Wielen's slip velocity model is developed from a steady-state force balance. Thus, both these slip velocity models are developed under similar physical
24
B. Sarkar, A. Das / International Journal of Mineral Processing 94 (2010) 20–27
considerations, viz. force and momentum balance. Therefore, the performance predictions using these two slip velocity models are similar under all conditions studied. 4.3. Partition curves The partition curves are generated from the size and density distributions of the two products and the feed. The size based partition curves are presented in Figs. 2 and 3. It can be seen from Fig. 2 that at a low bed pressure (4.85 kPa) the overall size-partition curve is sharp and has a higher positive slope for feed F1. With an increase in the bed pressure from 4.85 to 5.27 kPa the partition curve becomes flatter exhibiting a lower positive slope (Fig. 3) even with a closer sized feed, F2. About 20% of the larger particles (N650 μm) report to the overflow stream at 5.27 kPa bed pressure (feed F2) whereas, almost no larger particles, irrespective of their density, report to the overflow stream at 4.85 kPa bed pressure (F1). This is evident from Fig. 2 which indicates that the partition values are unity above 420 μm. At a low bed pressure, as discussed before, size separation is the primary mode of separation. However, fine particles having a high density may not be carried to the overflow. At a higher bed pressure, the particle loading in the bed increases and the autogenous dense medium forms. At a higher suspension density, larger-lighter particles are also transported to the overflow. Therefore, it can be concluded that at a higher bed pressure, the FDS is more effective as a concentrator. This observation also corroborated our previous findings (Sarkar et al., 2008b). The overall density based partition curves are presented in Figs. 4 and 5. It can be seen from these figures that a significant amount of high density particles report to the overflow stream for both the feeds. The settling of the particles through the suspension is strongly influenced by the size and density of the particles as well as the physical properties of the liquid. At a high bed pressure the dense medium develops fully leading to an increase in the effective viscosity and density of the suspension. The particle mass is determined by its size and density. Those having a lower mass experience a lower gravitational force and are unable to settle against this viscous and dense medium. Therefore, they are likely to be transported to the overflow at higher bed pressures. Thus, the complex interplay between the particle density and size prohibit a very sharp separation in terms of either the size or density. Again, the experimental density partition is described reasonably by all the models with different accuracies. It can also be seen from Figs. 2–5 that the partition curves predicted by the slip velocity models proposed by Masliyah (1979) and Van der Wielen et al. (1996) are similar in nature. On the other hand, the predictions using slip velocity models proposed by Galvin et al. (1999a) and Patwardhan and Tien (1985) are similar in nature. As discussed in
Fig. 2. Prediction of performance in terms of size separation using different slip velocity models at 3.15 mm/s TWFR and 4.85 kPa BP for feed having − 1180 μm nominal size.
Fig. 3. Prediction of performance in terms of size partition using different slip velocity models at 3.15 mm/s TWFR and 5.27 kPa BP for feed having − 1180 + 150 μm feed.
the previous section, the two former models are formulated based on similar fundamental concepts, i.e., steady-state momentum and force balance, respectively. Therefore, the predictions based on these models are similar. The slip velocity equations proposed by Patwardhan and Tien (1985) and Galvin et al. (1999a) are essentially extensions of the Richardson and Zaki (1954) approach. Patwardhan and Tien (1985) extended the Richardson and Zaki model by incorporating a bed voidage based on a cell balance approach. Galvin et al. (1999a) considered a dissipative pressure gradient to extend the Richardson and Zaki model. Hence, the predictions using these two models under all conditions are also somewhat similar. In order to investigate the separation features the density partition curves for narrow size classes are plotted in Figs. 6 and 7. The experimental density partition curves at high bed pressure for a few narrow size classes are shown against the predicted curves in Fig. 6. It can be seen from this figure that these curves are nearly S-shaped (Fig. 6A) for the density partition curves of the particles in these size classes. The hydrodynamic dispersion leads to a distribution of particles along the height of the FDS. However, in the present computation the hydrodynamic dispersion is not considered explicitly. The computations here are for steady-state operation in which the formation of a steady bed (having an average density) is assumed. While there is also a distribution of the suspension density and bed voidage in reality, an average suspension density (as given in Table 2) and a bed voidage is assumed in this work through which the particles settle. The slip velocity is estimated from the terminal settling velocity, the estimation of which takes into account the interstitial
Fig. 4. Prediction of performance in terms of density partition using different slip velocity models at 3.15 mm/s TWFR and 5.27 kPa BP for feed having −1180 μm nominal size.
B. Sarkar, A. Das / International Journal of Mineral Processing 94 (2010) 20–27
25
Fig. 5. Prediction of performance in terms of density partition using different slip velocity models at 3.15 mm/s TWFR and 5.27 kPa BP for feed having − 1180 + 150 μm feed.
fluid velocity. In addition, the functions F(ε) and G(ρ), which are strongly dependent on the hydrodynamics of the system, are used for modifying the slip velocity. Thus, the hydrodynamics are taken into consideration implicitly in the formulation. In view of this, the slip velocity is considered to be adequate to track the movement of the particles towards the product streams. In this manner the size and density distribution of the product streams are evaluated and the partition curves obtained. As depicted in Fig. 6, the models do describe the density partition curves of the particles in various size classes reasonably well. The inclusion of detailed hydrodynamics should improve the accuracy of the predictions and our research group has already taken up this study. The poor separation at a lower bed pressure is investigated further from the density partitions of narrow size classes as shown in Fig. 7. It can be seen from this figure that no density based separation has taken place for the particles larger than 300 μm (Fig. 7A). Good separation is observed only for − 300 μm particles (Fig. 7B and C). It is also seen that the slip velocity models capture the separation for larger particles better. For larger particles, the terminal settling velocities are higher which increase the slip velocity values. As a result, though the average suspension density is higher than the actual suspension density, all larger particles, irrespective of their density, are transported to the underflow stream. For −300 + 150 μm size fraction, only Galvin slip velocity provide a reasonable prediction while other three slip velocity correlations fail to predict the particle separation. For particles smaller than 150 μm size, all slip velocity correlations fail to predict the density separation. The failure of the models in predicting particle separation at smaller size fractions is attributed partly to the assumption of average suspension density and partly to the misplacement phenomenon of fine particles to the underflow which is not accounted for in any of the models. 4.4. Performance prediction The performance of the FDS is computed using all four slip velocity models to understand the effect of the bed pressure and teeter water flow rate on the yield and grade of the product. The predictions using all four models follow a similar trend qualitatively with some variation. The predicted performances of the FDS for the beneficiation of coal fines of feed F1 are presented in Fig. 8. From this figure it may be seen that at a low bed pressure the FDS is unable to produce clean coal with less than 28% ash since the separation is essentially size based. The feed F1 contains a higher amount of the fine particles (−150 μm) with a high ash content. Due to their low mass, these fine particles are transported to the overflow stream hydraulically even at a low teeter water flow rate. Therefore, at a low bed pressure, the
Fig. 6. Comparison of performance prediction in terms of size-wise density partition curves using different slip velocity correlations at 3.15 mm/s teeter rate and 5.27 kPa BP for (A) − 500 + 300 μm size fraction, Feed F2 (B) − 300 + 150 μm size fraction, Feed F1, (C) − 212 + 150 μm size fraction, Feed F2.
overflow product is enriched mostly with high ash fine particles. With an increase in the bed pressure, the effective suspension density and viscosity increase and hence, the total upward force (both buoyancy and drag force) exerted on the particle by the fluid also increases. This enhanced upward force helps in carrying larger and lighter particles with lower ash content to the overflow. The transportation of these particles into the overflow stream at a higher bed pressure reduces the overall ash (Fig. 8B) of the overflow product and increases the clean coal yield (Fig. 8A). Further increase in the bed pressure increases the density and viscosity of the separating medium which helps in transporting larger and heavier particles also to the overflow.
26
B. Sarkar, A. Das / International Journal of Mineral Processing 94 (2010) 20–27
Fig. 8. Performance prediction using different slip velocity models for feed having − 1180 μm nominal sized feed at 3.15 mm/s teeter water flow rate as a function of bed pressure (A) variation of yield (B) variation of grade.
teeter water rate, the clean coal grade expectedly stabilizes close to the feed grade as the hydraulic transport becomes the only dominant phenomenon (Fig. 9B).
Fig. 7. Comparison of performance prediction in terms of size-wise density partition curves using different slip velocity correlations at 3.15 mm/s teeter water rate and 4.85 kPa bed pressure for (A) − 500 + 300 μm size fraction, Feed F1 (B) − 300 + 150 μm size fraction, Feed F1, (C) − 150 + 75 μm size fraction, Feed F1.
This results in an increase in the ash content of the overflow product. Thus, the ash content of the overflow product for feed F1 goes through a minimum (Fig. 8B) when plotted against the bed pressure. An increase in the teeter water flow rate also increases the upward force resulting in an enhanced hydraulic transport. Therefore, for feed F2, the clean coal yield increases monotonically with an increase in the teeter water flow rate (Fig. 9A). The clean coal grade becomes poorer with a higher yield as the teeter water rate increases. At a very high
Fig. 9. Performance prediction using different slip velocity models feed having −1180+ 150 μm feed at 5.27 kPa bed pressure as a function of teeter water flow rate (A) variation of yield (B) variation of grade.
B. Sarkar, A. Das / International Journal of Mineral Processing 94 (2010) 20–27
27
predictions than the other three models. The overall partition curves as well as the density partition of the particles in various size classes are described reasonably well by the models with different degrees of accuracy. It is also established that a deslimed feed is more suitable for FDS operation. Detailed hydrodynamic modeling using such slip velocity models may be useful in providing better understanding and control over the operation of the FDS. Acknowledgement The financial assistance from the Department of Science and Technology, Government of India, for the research is gratefully acknowledged. References
Fig. 10. Performance prediction with Galvin slip velocity model for (A) − 1180 μm feed, and (B) − 1180 + 150 μm feed.
The overall performance prediction against the experimental data for the Galvin model is shown in Fig. 10 along with the feed washability data. It may be seen from this figure that the experimental yield values are below the theoretical maximum (washability data) in the case of feed F1. The model predictions are somewhat close to the experimental observations (Fig. 10A). However, the yield data are close to the theoretical maximum achievable in the case of feed F2 (Fig. 10B). The model predictions are also very close to the experimental observations in this case. This is because of the fact that the presence of the ultrafines in feed F1 adversely affects the partition of the other particles. A feed from which the fines are removed is, therefore, recommended for good performance of the FDS. 5. Conclusion The performance of the FDS is investigated through experimentation and detailed simulation studies. Four different slip velocity models are considered for this study. The salient features of all four models are discussed with reference to their ability to predict the performance of the FDS. The computed data using all four models are compared and contrasted with the experimental observations. The difference between the computed performance and experimental results are discussed in the light of the formulation of the slip velocity models and the assumptions made in this work. It is shown that although all four models are capable of capturing the essence of the separation features in the FDS, their accuracies in predicting the performance of the FDS are different. At a low bed pressure, the relatively poor predictions are also addressed. It is established that the model proposed by Galvin et al. (1999a) in which the dissipative pressure gradient is considered to be the main driving force for the separation, offers better performance
Asif, M., 2004. The complete segregation model for liquid fluidized bed: formulation and related issues. Powder Technology 140, 21–29. Asif, M., 2002. Predicting binary solid fluidized bed behavior using averaging approaches. Powder Technology 127, 226–238. Asif, M., 1998. Segregation velocity model for fluidized suspension of binary mixtures of particles. Chemical Engineering and Processing 37, 279–286. Das, A., Sarkar, B., Mehrotra, S.P., 2009a. Prediction of separation performance of floatex density separator for processing of fine coal particles. International Journal of Mineral Processing 91 (1–2), 41–49. Das, A., Sarkar, B., Biswas, P., Roy, S., 2009b. Performance prediction of floatex density separator in processing iron ore fines — a relative velocity approach. Mineral Processing and Extractive Metallurgy 118 (2), 78–84. Epstein, N., 2005. Teetering. Powder Technology 151, 2–14. Galvin, K.P., Pratten, S., Nguyen Tran Lam, G., 1999a. A generalized empirical description for particle slip velocities in liquid fluidized beds. Chemical Engineering Science 54, 1045–1052. Galvin, K.P., Pratten, S.J., Nicol, S.K., 1999b. Dense medium separation using a teetered bed separator. Minerals Engineering 12 (9), 1069–1081. Hartman, M., Havlin, V., Trnka, O., Carsky, M., 1989. Predicting the free fall velocities of spheres. Chemical Engineering Science 44 (8), 1743–1745. Happel, J., 1958. Viscous flow in multiparticle systems — slow motion of fluids relative to beds of spherical particles. AIChE J. 4, 197–201. Kapure, G., Kari, C., Rao, S.M., Rao, N.D., 2007. The feasibility of a slip velocity model for predicting the enrichment of chromite in a Floatex density separator. International Journal of Mineral Processing 82 (2), 86–95. Kari, C., Kapure, G., Rao, S.M., Rao, N.D., 2006. Predicitng the chromite mineral upgradation in Floatex density separator using hindered settling models. Iron and Steel Institute of Japan International 46 (7), 966–973. Kim, B.H., Klima, M.S., 2004. Development and application of a dynamic model for hindered settling column separations. Minerals Engineering 17, 403–410. Kopko, R.J., Barton, P., McCormick, R.H., 1975. Hydrodynamics of vertical liquid–solids transport, industrial and engineering chemistry. Process Design and Development 14 (3), 264–269. Kuwabara, S., 1959. The forces experienced by randomly distributed parallel circular cylinders or spheres in a viscous flow at small Reynolds numbers. Journal of the Physical Society of Japan 14 (4), 527–532. Lockett, M.J., Al-Habbooby, H.M., 1974. Relative particle velocities in two species settling. Powder Technology 10, 67–71. Masliyah, J.H., 1979. Hindered settling in multi-species particle system. Chemical Engineering Science 34, 1166–1168. Patwardhan, V.S., Tien, C., 1985. Sedimentation and liquid fluidization of solid particles of different sizes and densities. Chemical Engineering Science 40 (7), 1051–1060. Rasul, M.G., Rudolph, V., Wang, F.Y., 2000. Particles separation using fluidization techniques. International Journal of Mineral Processing 60, 163–179. Rasul, M.G., Rudolph, V., Carsky, M., 2002. Segregation in binary and ternary liquid fluidized beds. Powder Technology 126, 116–128. Richardson, J.F., Zaki, W.N., 1954. Sedimentation and fluidization: Part I. Transactions of the Institution of Chemical Engineers 32, 35–53. Rowe, P., 1987. A convenient empirical equation for estimation of the Richardson-Zaki exponent. Chemical Engineering Science 42, 2795–2796. Sarkar, B., Das, A., Roy, S., Rai, S.K., 2008a. In depth analysis of alumina removal from iron ores fines using teetered bed gravity separator. Mineral Processing and Extractive Metallurgy 117 (1), 48–55. Sarkar, B., Das, A., Mehrotra, S.P., 2008b. Study of separation features in floatex density separator for cleaning fine coal. International Journal of Mineral Processing 86, 40–49. Van Der Wielen, L.A.M., Van Dam, M.H.H., Luyben, K.Ch.A.M., 1996. On the relative motion of particle in a swarm of different particles. Chemical Engineering Science 51, 995–1008. Wallis, G.B., 1969. One dimensional two phase flow. McGraw Hill, New York.