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Ind. Eng. Chem. Res. 1990,29, 1894-1900
Particle Classification Using Inclined Settlers in Series and with Underflow Recycle Xiaoguang Zhang and Robert H. Davis* Department of Chemical Engineering, University of Colorado, Boulder, Colorado 80309-0424
Continuous classification of particles by size in dilute suspensions, using a single inclined settler with partial underflow recycle and using multiple inclined settlers in series, is examined by both theory and experiment. Partial underflow recycling is advantageous, because it reduces the presence of the smaller particles in the coarse fraction (underflow) and increases their recovery in the fine fraction (overflow). The use of multiple settlers in series is desirable because a broad feed distribution may then be divided into several more narrow product distributions. The theory to predict the particle recovery and size distribution in each product fashion is developed for both operating modes. Several experiments are performed in rectangular inclined settlers with dilute suspensions of polystyrene beads, and the experimental observations are found to be in good agreement with the theoretical predictions. Introduction The classification of a broad distribution of small particles into two or more relative narrow distributions constitutes an important step in a variety of industrial processes. One attractive way to carry out the classification is by the use of inclined settlers, since the particle sedimentation rate in such settlers can be greatly increased (Davis and Acrivos, 1985). A detailed description of the use of a single inclined settler for particle classification was presented in a recent paper by Davis et al. (1989). They described the steady-state classification of a dilute feed suspension into a coarse fraction (underflow) and a fine fraction (overflow)using a single inclined settler, and their experimental results confirmed the predictions of the theoretical model. Unfortunately, both the theoretical predictions and the experimental results show shortcomings in the process. In particular, the coarse fraction contains particles of all sizes, although slightly enriched in the larger particles, whereas the fine fraction is essentially free of the larger particles but has a low recovery of the feed particles. In order to alleviate these deficiencies, a modification employing partial recycling of the underflow stream has been investigated. A schematic of the system is shown in Figure 1. Part of the underflow stream is combined with the feed suspension, resulting in a higher particle volume fraction and a coarser size distribution being fed into the settler. The modified feed suspension is then classified due to differences in the sedimentation velocities of the particles, in the manner described by Davis et al. (1989). The overflow stream and the unrecycled underflow stream, containing fine and coarse fractions, respectively, are withdrawn as products. The addition of the underflow stream t o the feed is predicted to increase the particle recovery in the overflow and cause the particle size distribution in the underflow to become coarser relative to a single settler operating without underflow recycle, under otherwise identical conditions. The reason for these benefits of using the underflow recycle is that, after combining with the partial underflow, the feed suspension has an increased particle volume fraction and a coarsened particle size distribution. Consequently more particles, including those smaller than the cutoff size, enter the settler. With the same overflow rate, more particles smaller than the cutoff size are able to reach the overflow, * T o whom correspondence should be addressed.
causing the particle volume fraction in the overflow to increase. A t the same time, this reduces the presence of the smaller particles in the underflow product. When limited to only a single settler, a feed distribution of particles is divided into only two fractions during steady-state operation. In many practical applications, however, more than two fractions are desirable. This may be accomplished by stepwise transient operation of a single settler, as described by Davis et al. (1989). However, if steady-state operation is desirable, then a possible method to continuously classify the particles into more than two fractions would be to connect several settlers in series, as shown in Figure 2 for two settlers. The overflow from each settler is fed into the subsequent settler, and each successive settler has a lower overflow rate and so removes smaller sized particles than the previous settler in the series. In addition, a portion of the underflow from each settler can be recycled by mixing it with the overflow from the previous settler, and the remainder may be withdrawn as a product. In this fashion, a series of N settlers will divide a feed distribution into N + 1fractions, representing the underflows from the N settlers plus the overflow from the last settler in the series. In this paper, we present theoretical models to predict the particle volume fraction and size distribution in each product fraction for steady-state continuous operation of a single settler with partial underflow recycle and of multiple settlers in series. Experiments for each of these two processes are then described, and the results are compared with the theoretical predictions. Theory Steady-State Operation of a Single Settler with Partial Underflow Recycle. We first develop a theoretical model which describes the continuous operation of an inclined settler with partial underflow recycle. Of particular interest is the prediction of particle volume fractions and size distributions in the overflow and underflow streams and their dependence on the recycle ratio, R, which is defined as the volumetric flow rate of the recycled underflow stream divided by that of the unrecycled underflow stream (see Figure 1). The development closely follows that of Davis et al. (19891, to which the reader is referred for details regarding the nomenclature and assumptions involved. The derivation starts with the appropriate mass balances around the entire system. Steady-state mass balances on
0888-5885/90/2629-l894$02.50/0 0 1990 American Chemical Society
Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990 1895
Figure 1. Schematic diagram of a settler with partial underflow recycle.
R1Qu I
streams entering the overflow: one having the flow rate S(u) and being clarified of particles with settling velocity u and one having the flow rate Q, - S(u) and containing particles with settling velocity u that remain in suspension. Only the latter of these two streams contributes directly to the right-hand side of eq 7, because the former stream is devoid of particles with settling velocity u. Of course, when S(u) IQ,, then all of the particles with settling velocity u have sufficient time in the settler to settle out of suspension before reaching the overflow stream, and so eq 8 applies in this case. Equations 7 and 8 represent the relations between the composition of the overflow stream and that of the modified feed suspension. The composition of the modified feed suspension can be determined from the compositions of the feed suspension and the recycle ratio. Using eqs 1, 3,4, and 6 in eqs 7 and 8 in order to eliminate &, and P&) results in an expression for the relationship between the compositions of the overflow and feed suspension:
S(u) < 8, (10)
RiQu I
Figure 2. Schematic diagram of the system with two settlers in series.
the total suspension, total particles, and those particles having settling velocity u in the settler are, respectively, (1) Qr = Qo + Qu (2) Q d f = Q&J, + QdU (3) Q d P r ( U ) = Q,+J',(u) + QuQu(u) The particle volume fraction, probability density function, and flow rate of the original feed suspension ($f, P&), and Qf, respectively) are assumed to be constant and known. Critical factors for particle classification in this mode include the particle volume fraction, probability density function, and flow rate in the modified feed suspension (&, Pin(u), and Bin,respectively). These factors are affected by the underflow recycle rate and determined by mass balances about the fitting where the recycle line enters the feed line. Assuming that the composition of the recycle stream is the same as that in the underflow stream, the appropriate mass balances are (4) Qin = Qf + R Q u (5) Q i n h = Q d f + RQu& (6) Bin@inPin(u) = Q d P A U ) + RQud#,(u) The final mass balance required is that on particles with setting velocity u entering the overflow stream Qo@J'o(u) = (Qo - s ( U ) ) @ i n p i n ( u ) S(u) < Qo (7) P,(u)= 0 S(u) IQ, (8) where the volumetric rate at which a suspension containing particles with settling velocities less than u only is produced due to the sedimentation of those particles with settling velocities greater than or equal to u is S(u) = uw(L sin '6 + b cos 0) (9) Note that S(u) is equal to the vertical settling velocity, IJ, multiplied by the horizontal projection of the area available for particle settling, as described by Davis and Acrivos (1985). The left-hand side of eq 7 represents the volumetric rate a t which particles with settling velocity u are carried out with the overflow stream at the top of the settler. This stream represents a combination of two
P,(u) = 0 S(u) IQ, (11) Integrating eqs 10 and 11 over all particle settling velocities, and then using the result in eqs 1-3, we determined the particle volume fractions and normalized probability density functions for the overflow and underflow streams:
r
(' + R)Qf Pf(u)du s(u))(l R)Q, - RS(u)
+
1
u C u, (14)
where u, is the cutoff settling velocity defined by S(u,) = 8,. It is predicted that only particles with settling velocities less than u, may reach the overflow line. Example theoretical predictions are shown in Figures 3-5. The original feed suspension is chosen to have a normal bell-shape size distribution with u / D = 0.25, where u is the standard deviation and D is the particle median diameter. The probability density function for the size
1896 Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990 6 0
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Figure 4. Theoretical predictions of particle volume fractions in the overflow for steady-state operation of a settler with different underflow recycle ratios and at different QJQ and Qu/Qr.
distribution, P(D), must be transformed into the probability density function for the settling velocity distribution, P(u),in order to use the theory. This may be accomplished by using Stokes law to relate the particle settling velocity to its diameter
or
where k is a correction to the Stokes settling velocity in order to account for calibration errors, wall effects, and hindered settling (Davis et al., 1989). For nondilute suspensions, in which hindered settling is important, the use of a constant value for k is not strictly correct. In particular, both the size range and concentration of suspended particles will be smaller near the top of the suspension than near the bottom of the suspension. As a result, the hindered settling velocity of a particle of a given size will vary with position in the settler. Consequently, the present analysis with a constant k is strictly valid only for dilute suspensions. The possible effects of increased solids loading are discussed later. Predicted normalized probability density functions for particle size distributions in the overflow and the underflow, Po@) and P J D ) , respectively, whsre the dimensionless diameter is defined as D = D/D, are shown in Figure 3 for R = 0.0, R = 5.0, and R = 20.0. The feed suspension is classified with the relative overflow rate Q,/Q
Figure 5. Theoretical predictions of particle volume fractions in the underflow for steady-state operation of a se-ttler with different underflow recycle ratios and at different Q,/Q and QU/Qc
set equal to 1.0, and the relative underflow rate Qu/Qf set equal to 0.17. Note that Q = S(ij)is the volumetric rate of formation of a suspension containing particles with settling velocities only less than ij, where U is the settling velocity of the median-sized particles. Thus, we expect that the operation with Q,/Q = 1.0 will separate particles with diameters less than the median size, D , from the others. Indeed, Figure 3 shows that the overflow is predicted to contain only particles having D < D , whereas the underflow is enriched with larger particles. Increasing the recycle ratio decreases the fraction of particles having D C D in the underflow, as expected, although even very high recycle ratios do not lead to the complete removal of these smaller particles from the underflow. Predicted dependencies of the particle volume fractions in the overflow and underflow on the recycle ratio are shown in Figures 4 and 5. The different lines in these figures corregpond to operations with different combinations of Q,/Q and QJQP Increasing Q,/Q leads to more of the particles being recovered in the overflow, and less in the underflow, because the hold-up time for sedimentation is reduced when the flow rate through the settler is increased. Increasing Q,/Qf has no effect on the overflow concentration for no recycle, whereas a small decrease is predicted when recycle is used. As expected, when the recycle ratio is increased, the particle volume fraction in the overflow, @, is increased and that in the underflow, @”,is decreased. For large recycle ratios, these product volume fractions approach asymptotic values given by eqs 1 2 and 13 with R m. For example, the overflow asymptote for Q,/Q = 1.0 and Qu/Qf = 0.17 is @o/@f = 0.194, indicating that only 32% of the total volume of particles having D C D in the feed stream is recovered in the overflow when a very large recycle ratio is used. Although this is a distinct improvement over the 15% recovery when R = 0, it indicates that multiple stages should be used if higher recoveries are desired. Steady-State Operation of Multiple Settlers in Series. The steady-state classification of particles using two settlers in series is diagrammed in Figure 2. The feed suspension with a constant particle volume fraction and size distribution is continuously fed into the first settler. The suspension in the first settler is classified with a relatively large overflow rate. From the first settler, the underflow is withdrawn as one of the products, and the overflow is fed into the second settler and classified again with a lower overflow rate. Meanwhile, the feed of each settler is modified by combination with part of its underflow stream. In this manner, the feed size distribution is continuously classified into a coarse fraction as the un-
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Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990 1897
- Initio1 su~pension Fine ( R l = R 2 - 0 . 0 )
Medium ( R 1 *R2=0.0)
Coarse ( R 1 - R Z - 0 0 ) (R1 = R 2 = 2 0 . 0 )
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recycled portion of the underflow of the first settler, a medium fraction as the unrecycled portion of the underflow of the second settler, and a fine fraction as the overflow of the second settler. The derivation of the model to predict the operation of two or more settlers in series is a relatively straightforward extension of the previous model, since the operation of each settler is similar to that of one settler operating by itself. The formulas derived previously to predict the particle volume fractions and normalized probability density functions in the overflow and underflow of a single settler can be used directly for the first settler. Thus
L
where u ~ ,defined ~, by S ( U , = ~ )Qo,i,is the cutoff particle settling velocity for the overflow from the ith settler. Figure 6 shows theoretical predictions for the normalized probability density functions of the product fractions which are produced from a suspension with a bell-shaped size distribution having c/D = 0.25 and using a system of two settlers in series with Q0,,/Q = 1.5, QU,,/Qf= 0.17, Q0,2/Q = 0.5, and Qu,2/Qo,l = 0.67. Operating conditions both with recycle (R, = R2 = 20.0) and without recycle (R, = R2 = 0.0) are considered. The predicted size distributions in Figure 6 show a successful separation among three product fractions, especially for the case when recycle is used. As described in the last section, the use of underflow recycle for each settler causes the particle size distribution in each underflow to become coarser and leads to less overlap of the three product fractions. Underflow recycling does not, however, affect the predicted cutoff settling velocity.
Further, these same equations may be applied to any settler in series (say, the ith settler) by simply replacing the feed particle volume fraction $*,normalized probability density function Pf(u), and flow rate Qf by the corresponding values from the overflow of the previous settler: 4qi-1, Z'o,i-l(U), and Qo,i-l. These are, respectively:
Experiments Steady-State Operation of a Single Settler with Partial Underflow Recycle. Several experiments were carried out in order to establish the operating characteristics of a single settler with partial underflow recycle operating at steady state. The objectives of the experiments reported here were to test the theoretical model and to investigate the benefits of underflow recycle for improving particle classification. Of particular interest is the effect of the recycle ratio on the particle volume fraction in the overflow and on the size distribution in the underflow.
1898 Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990 Table I. Summarv of Classification of Polvstvrene Beads in a Sinale Settler with Different Underflow Recvcle Ratios median diameter quartile deviation (theorylexptl), pm (theory/exptl), pm df, % d", % (theory/exptl) fine coarse fine coarse R $Jf, % (theorylexptl) 0.0 0.97 0.053/0.051 8.55/9.08 44.6/46.1 63.3/64.5 4.50/5.10 8.2118.37 1.0 0.97 0.014j0.073 7.87j8.50 45.6j46.7 63.9j64.3 4.3oj5.03 8.21ja.48 46.2149.4 0.11/0.13 7.661734 64.4165.4 4.0915.32 5.0 0.84 8.2718.37
The experiments were performed in a rectangular settler with L = 60 cm, w = 3.5 cm, and b = 2.5 cm. Its angle of inclination was set equal to 6 = 45' for all experiments. The suspension was composed of polystyrene beads with median diameter 62.6 pm, density 1.049 g/cm3, and quartile deviation 8.8 pm. The latter is a measure of the spread of the size distribution and is defined as (DT5D2,)/2, where 25% of the particles (by volume) is smaller thrn the first quartile diameter, D25, and 75% of the particles is smaller than the third quartile diameter, DT5. The suspending fluid was distilled water at room temperature, with 3 drops/L of Triton X-100 added to aid in dispersing the particles and 1 drop/L of water bed preservative added to prevent microbial growth. Dry particles were weighed and added to the fluid so that the particle volume fraction was approximately 1'70 (measured values of & were slightly lower, from 0.8% to 1.0%, because some of the particles were retained as sediment in the settler). In steady-state operation, the constant-feed flow rate Qr and underflow rate Q, were controlled with Masterflex peristaltic tubing pumps and set to be 1.38 and 0.15 mL/s, respectively. The overflow rate was subject to level control and was the difference between the flow rates of the feed and the underflow. Its value of 1.23 mL/s corresponds to a cutoff size of Do = 55 pm in the overflow from the settler. The correction factor k was determined by a series of experiments for which the overflow rate was set such that not more than 10% of the particles (by volume) in the overflow had diameters larger than the cutoff size. From these experiments, k was set to be 0.8. Seven experiments were performed, having recycle ratios R = 0.0,0.5, 1.0, 3.0, 5.0, 10.0, and 20.0. A peristaltic tubing pump was used for the underflow recycle, and the recycle ratio was varied by varying the speed of this pump. For each experiment, the settler was initially run for 2 h at a fixed recycle ratio with underflow and overflow being returned to the feed reservoir so that a steady state was reached. The underflow and overflow were then separated from the feed reservoir and collected as the products. The samples were analyzed by an Elzone 180XY particle size analyzer produced by Particle Data, Inc. This machine uses electric conductivity measurements to yield the probability density function of the size distribution, P(D), as well as the particle volume fraction, 4. The results for the particle volume fraction in the overflow as a function of the recycle ratio for all seven experiments are shown in Figure 7. As predicted, the particle volume fraction in the overflow stream increases as the recycle ratio increases. A t low recycle ratios, the experimental results are in good agreement with the model predictions. However, at higher recycle ratios ( R > 5.01, the measured particle volume fractions in the overflow increase much faster than predicted, and it was found that more particles larger than the cutoff size were present in the overflow. The most likely explanation for this is that flow instabilities beneath the upper wall of the settler were induced due to the high flow rates and pulsating nature (due to the peristaltic pumps) of the entering and exiting streams. Such instabilities manifested in the form of waves along the suspension-clear fluid interface were observed in experiments. When these waves broke near the top of
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Figure 7. Particle volume fractions in the overflow for steady-state operation of a settler with different underflow recycle ratios.
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Figure 8. Normalized probability density functions of the initial suspension and of the product fractions classified by steady-state operation of a settler at the underflow recycle ratio R = 0.0.
the settler, they caused particles larger than the cutoff size to remix into the overflow and contaminate it. This phenomenon has been observed and studied by Davis (1982) and Borhan (1989). They found that the flow in a inclined settler should be unstable when the feed rate exceeded a critical value, and the occurrence of such flow instabilities contaminated the overflow and therefore limited the efficiency of the inclined settling process. Davis (1982) and Borhan (1989) also observed that the propagation and breakup of waves along the suspension-clear fluid interface beneath the upper wall of the inclined settler greatly increased the particle volume fraction in the overflow product. The effects of underflow recycle on particle classification can be seen more clearly in Table I and Figures 8-10, where examples of the measured final size distributions are compared with those predicted by the theory for R = 0.0, 1.0, and 5.0. The theoretical predictions were made by using the measured feed distribution in eqs 12-18. The experimental measurements of the particle concentrations, size distributions, and median and quartile diameters were made directly by the Elzone 180 particle size analyzer. From these results, it is evident that, at lower recycle ratios, there is a good agreement between the model predictions
Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990 1899 0.08
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Figure 11. Normalized probability density functions of the initial suspension and of the product fractions classified by steady-state operation of two settlers in series with no underflow recycle. Table 11. Summary of Classification of Polystyrene Beads in the System of Two Settlers in Series without Underflow Recycle $, % ' median diameter quartile deviation (theory/ (theory/exptl), (theory/exptl), exptl) wn w initial suspension 0.90 62.2 0.02 fine fraction 0.031/0.035 43.5146.7 4.8015.23 52.5153.6 4.1015.59 medium fraction 0.41/0.31 7.3317.72 64.4164.7 0.21/8.51 coarse fraction
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Figure 9. Normalized probability density functions of the initial suspension and of the product fractions classified by steady-state operation of a settler at the underflow recycle ratio R = 1.0.
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Figure 10. Normalized probability density functions of the initial suspension and of the product fractions classified by steady-state operation of a settler a t the underflow recycle ratio R = 5.0.
and experimental measurements of particle size distributions in the product fractions. Moreover, increasing the recycle ratio in a limited range reduced the presence of the smaller particles in the coarse fraction, as predicted by the model. However, increasing the recycle ratio also caused a widening of the size distribution in the overflow over that predicted by the model due to flow instabilities along the upper wall of the settler, as described above. From the predictions of the linear stability analyses by Herbolzheimer (1983), Davis et al. (1983), and Borhan and Acrivos (1988), it is expected that this problem may be overcome by reducing the feed flow rate, by increasing the angle of inclination from the vertical, by decreasing the spacing between the inclined plates, or by using a more viscous suspending fluid. It is also expected that introducing the feed in a more uniform manner would reduce the source of the instabilities. Steady-State Operation of Two Settlers in Series. Steady-state operation of two settlers in series without underflow recycle was performed for classifying the feed suspension described above into three fractions of different size ranges. Two settlers with the same dimensions as the one used in the previous experiments were used, and both were angled at 0 = 4 5 O . Feed suspension was drawn up into the first settler at a flow rate Qf = 2.07 mL/s, which was controlled by the combination of the overflow rate Qo,l = 1.87 mL/s and the underflow rate Qu,l = 0.20 mL/s. The overflow of the first settler was fed into the second settler, in which the overflow rate Qo,z = 1.17 mL/s was controlled by the difference between the overflow rate of the first settler and the underflow rate of the second settler (Qu,2
= 0.70 mL/s). The correction factors for both settlers were set to be kl = kz = 0.8, by the procedures described previously. Corresponding to the given overflow rates, the predicted cutoff sizes for the first and second settlers were 66 and 56 pm, respectively. The system was first run for 2 h to reach a steady state, during which time the overflow of the second settler and the underflows of the two settlers were returned to the feed reservoir. The overflow from the second settler and the underflows from the two settlers were then separated and collected as products. The final size distributions of the products are shown in Figure 11; the corresponding theoretical predictions are also presented in this figure for comparison. A summary of the final classification of particles is also tabulated in Table 11. The experimental results of the product size distributions are in good agreement with the model predictions. However, as is the case when operating a single settler without recycle, there is considerable overlap in the particle size distributions between the product fractions. This deficiency may be improved by partial underflow recycle, as discussed earlier and as shown in Figure 6.
Concluding Remarks Two schemes, one using a single settler with partial underflow recycle to classify a dilute suspension into a fine and a coarse fraction in steady-state operation and a second using multiple settlers in series to classify a dilute suspension into several narrow distributions in steady-state operation, have been proposed and tested. The theoretical models to predict the operations of these schemes are also developed, and the experimental results for low recycle ratios are in good agreement with the theoretical predictions. The application of the partial underflow recycle in steady-state operation of a single inclined settler improves the particle classification; that is, it increases the particle recovery in the overflow stream (fine fraction) and causes the particle size distribution in the underflow stream
1900 Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990
(coarse fraction) to be coarser, thus leading to less overlap in particle size distributions between product fractions. Both theory and experiments verify that this scheme allows for effective classification of small particles into two different size fractions, with the exception of operation using high recycle ratios for which flow instabilities in the settler were observed. In order to avoid the contamination of the overflow that occurs due to these flow instabilities beneath the upper wall of the settler, we recommend that only modest recycle ratios be used and that the inclination angle, 8, be set as large as possible and the spacing, b, and underflow rate, Q,, be set as low as possible, so long as the sediment layer on the lower wall can be withdrawn continuously. Of particular interest is the potential use of several inclined settlers in series, which would allow for the continuous classification of a broad size distribution of dilute suspension into several narrow size distributions simultaneously. As confirmed by the experimental results for two settlers in series, the theoretical model predicts effective and relatively sharp classification. The extension of this mode to more settlers in series is relatively straightforward. Additionally, a new observation has been noted concerning particle classification by inclined settling, which seems to bear further study. With respect to flow instabilities, a more complete understanding of the wave formation and growth at the interface between the suspension and the sediment on the upward-facing wall is needed. We observed flow instabilities along both the suspension-clear fluid interface and the suspensionsediment interface. The former contaminated the overflow (fine fraction) as described earlier, whereas the latter made the particle size distribution in the underflow stream (coarse fraction) shift toward a larger size range because these instabilities resuspended the sediment. Some of the resuspended particles were smaller than the cutoff size and were carried to the overflow instead of being withdrawn from the underflow. It is possible that more effective classification will be obtained in processes designed to take advantage of this phenomenon. Finally, it is of practical interest to extend the present work to higher solids loadings. In general, hindered settling will reduce the volumetric rate a t which suspension may be processed. Moreover, hindered settling leads to a self-sharpening of the diffuse region in the upper portion of the settler (Davis et al., 1982). This may reduce the efficiency of the classification process because it would make it difficult to collect fine particles in the overflow without contaminating it with coarse particles. On the other hand, hindered settling effects coupled with particle flux continuity requirements dictate that the volume fraction of suspended fine particles near the top of the settler is higher than that in the feed stream (Davis et al., 1982),which may increase the efficiency of portioning these particles to the overflow stream. Clearly, ramifications of higher solids loading on particle classification by sedimentation warrant further study by theory and experiments.
Acknowledgment This work was supported by Grant CBT 8451014 from the National Science Foundation and by the Dow Chemical Co. Nomenclature b = spacing between inclined plates (cm) D = particle diameter (pm) D = average diameter (pm) D = dimensionless diameter (pm) Do = cutoff diameter in overflow (pm) g = gravity constant (980 cm/s2) 12 = correction factor for settling velocity L = settler length (cm) P(D) = normalized probability size density function (volume basis) (l/pm) 8 = volumetric flow rate (cm3/s) Q = flow rate of suspension with only particles smaller than D R = underflow recycle ratio S = volumetric inclined settling rate (cm3/s) L’ = settling velocity (cm/s) u, = settling velocity of particle with cutoff size (cm/s) w = settler width (cm) = particle volume fraction u = standard deviation of particle size distribution of suspension 0 = angle of inclination of settler from the vertical Subscripts 1, i = order of the settlers f = feed in = modified feed o = overflow R = recycle flow u = underflow
Literature Cited Borhan, A. An experimental Study of the Effect of Suspension Concentration on the Stability and Efficiency of Inclined Settlers. Phys. Fluids 1989, AI (l), 108-123. Borhan, A.; Acrivos, A. The Sedimentation of Nondilute Suspensions in Inclined Settlers. Phys. Fluids 1988,31 (12), 3488-3501. Davis, R. H. The Operation and Stability of Continuous Inclined Supersettlers. Ph.D. Thesis, Stanford University, 1982. Davis, R. H.; Acrivos, A. Sedimentation of Noncolloidal Particles at Low Reynolds Numbers. Annu. Rev. Fluid Mech. 1985, 17, 91-118. Davis, R. H.; Herbolzheimer, E.; Acrivos, A. The Sedimentation of Polydisperse Suspensions in Vessels Having Inclined Walls. Znt. J . Multiple Flow 1982, 8 (6), 571-585. Davis, R. H.; Herbolzheimer, E.; Acrivos, A. Wave Formation and Growth During Sedimentation in Narrow Tilted Channels. Phys. Fluids 1983,26, 2055-2064. Davis, R. H.; Zhang, X.; Agarwala, J. P. Particle Classification for Dilute Suspensions Using an Inclined Settler. Znd. Eng. Chem. Res. 1989, 28, 785-793. Herbolzheimer, E. The Stability of the Flow During Sedimentation Beneath Inclined Surface. Phys. Fluids 1983,26, 2043-2054.
Received for review January 30, 1990 Revised manuscript received May 11, 1990 Accepted May 24, 1990
