Local solids concentration measurement in a slurry mixing tank

Local solids concentration measurement in a slurry mixing tank

Pergamon Chemical Engineerin9 Science, Vol. 51, No. 8, pp. 1209 1220, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights...

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Chemical Engineerin9 Science, Vol. 51, No. 8, pp. 1209 1220, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0009-2509/96 $15.00 + 0.00

0009-2509(95)00364-9

LOCAL SOLIDS C O N C E N T R A T I O N MEASUREMENT IN A SLURRY MIXING TANK H. A. N A S R - E L - D I N * * Department of Chemical Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G6 R. S. M A C T A G G A R T Syncrude Canada Ltd, P.O. Bag 4009, M.D. 2050, Fort McMurray, Alberta, Canada T9H 3L1 and J. H. M A S L I Y A H Department of Chemical Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G6

(First received 16 August 1994; revised manuscript received 21 September 1995; accepted 4 October 1995) Abstract--The local solids concentration in a mixing tank was measured using both sample withdrawal and a new conductivity probe. The conductivity probe was used to assess the errors associated with various sample withdrawal techniques and to measure solids concentration profiles in the mixing tank. The effects of sampling tube design (tip shape, face angle and inside diameter), sampling position, bulk solids concentration and particle size on the sampling errors were examined in detail. Solids concentration profiles were also measured as a function of particle size, bulk solids concentration and mixer rotational speed. The experimental results indicated that on the impeller plane, the sample withdrawal techniques (tapered sample tube) gave a lower solids concentration than that in the tank at a sampling velocity ratio of unity. These results suggest that the flow at the impeller plane was three-dimensional. The errors associated with sampling techniques depended on the sample tube shape and location in tank, and were significant for the coarse sand particles of 1000 ktm. When sampling at right angle to the flow, sampling errors increased with particle size, especially when a small diameter sample tube was used. The data provided in the present study can be employed to correct for the errors associated with the sampling withdrawal techniques over a wide range of parameters. Solids concentration profiles in the mixing tank were found to be a function of the particle size, bulk solids concentration and mixer rotational speed. Solids concentration varied with the radial position, except when fine sand particles of 82/am were used. A strong variation in solids concentration with the axial position was observed at the impeller plane for the sand particles examined in the present study. This variation increased with the particle mean size and the mixer rotational speed.

INTRODUCTION

Mixing is a c o m m o n unit operation in the process and mineral industries. The suspension of solid particles in a liquid within a stirred vessel is encountered in crystallization, leaching and reactions utilizing a solid catalyst (Baldi et al., 1981). The stirred vessel is the most c o m m o n reactor for polymerization reactions (Gerstenberg et al., 1983). Mixing vessels are also very c o m m o n in gassed processes such as hydrogenation and oxidation reactors, fermentation, waste-water treatment, evaporative crystallization and froth flotation (Chapman et al., 1983a,b). Information on solids distribution in mechanically stirred vessels is relatively scarce, but is very important in processes such as crystallization (Kipke, 1983). Classical sampling is referred to the removal of a small sample from the system. Due to its simplicity, it has been used as a means of measuring suspension uni-

* Corresponding author. *Present address: Saudi Aramco, Lab R & D Center, P.O. Box 62, Dhahran 31311, Saudi Arabia.

formity or local solids concentration. However, representative samples are extremely difficult to withraw from a mixing tank (Nienow, 1985; MacTaggart et al., 1993a) due to inertia differences between the fluid and particles of different size or density (Smith, 1990). Despite this serious shortcoming, sample withdrawal continues to be used as a method of determining local solids concentration in mixing tanks (Buurman et al., 1986; Barresi and Baldi, 1987a,b; Kuzmanic et al., 1992; Barresi et al., 1994). The objective of sample withdrawal is to obtain a sample that is representative or identical in all properties, to the system being sampled at the point of sampling. There are three factors that can lead to non-representative sampling of solids-liquid systems: particle inertia, flow structure ahead of the sampler, and particle bouncing (Nasr-E1-Din, 1989). Particle inertia refers to the inertia of a particle relative to the inertia of an equal volume of the surrounding fluid. Sampling errors due to particle inertia are a result of disturbances of the fluid flow ahead of the sampling device by the sampler and how the particles respond to these disturbances. The response of a particle to

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H.A. NASR-EL-DINet al.

deflections in the fluid streamlines is a function of the particle inertia parameter (K), defined as pe d 2 Uo K - - -

18 ~tL(~b/2)

(1)

where dp is the particle size, Uo is the local velocity, /~L is the liquid viscosity, pp is the particle density, and ~b is the sample tube inside diameter. Rushton (1965), Rehakova and Novosad (1971a,b) and Sharma and Das (1980) developed empirical or semi-empirical equations to describe the solids concentration obtained by the withdrawal of a sample from the impeller plane of a radial flow impeller in a mixing tank. These equations correlate the sample solids concentration on the impeller plane to equal the bulk solids concentration in the mixing tank at the isokinetic sampling velocity (Us = Uo). No independent measure of the local solids concentration was used by Rushton, Rehakova and Novosad or Sharma and Das to determine if the local solids concentration on the impeller plane was the same as the bulk solids concentration in the tank. Other methods to measure solids concentration profiles in mixing tanks have been developed. Optical methods have been used (Bohnet and Niesmak, 1980; Tojo and Miyanami, 1982; Yamazaki et al., 1986; Ayazi Shamlou and Koutsakos, 1989; Magelli et al., 1990), but they are limited to volumetric solids concentration less than 1 or 2%. Conductivity methods were employed to measure local solids concentration in solids-liquid mixing tanks. Machon et al. (1982), Rieger et al. (1988) and recently MacTaggart et al. (1993b) have developed various conductivity probes for measuring solids concentration over a small volume in space. To obtain reliable results however, the conductivity probe must be used carefully. In some industrial situations, sampling may be the only concentration measurement technique available for use. It would, therefore, be desirable to know the differences between the actual local solids concentration and that measured by sample withdrawal techniques. To the best of our knowledge, these differences have never been considered before. The objectives of this study are (1) to quantify the errors associated with sample withdrawal techniques from a slurry mixing tank; (2) to show under what conditions a representative sample may be obtained; and (3) to measure solids concentration profiles in a mixing tank. EXPERIMENTAL STUDIES

Solids concentration measurements were conducted in a mixing tank with a flat bottom. The tank was constructed of plexiglas and was equipped with an impeller which was driven by a ¼ hp variable speed D.C. motor. The tank had a diameter, T, of 0.292 m and was filled with water such that the normalized liquid height, H / T , was unity. The impeller diameter, D, was 0.097 m, i.e. D / T = 0.33. The impeller normalized height above the tank bottom, h/T, was 0.3.

a. Mixing tank and sampling locations

I~"/r ............... ; Hl b. Sample tubes

Blunt Sample Tube

Tapered Sample Tube

45 - Face Angle Tube

Fig. 1. (a) Schematic of the mixing tank and sampling locations. (b) Sample tubes.

The mixer rotational speed was measured with a Cole-Parmer Model 8211 optical tachometer. The mixing experiments were carried out using a sixbladed radial flow, disc-type Rushton impeller. The Rushton impeller was chosen for the present study so that comparisons with previous work could be made. Local solids concentration in the mixing tank was measured using various sample withdrawal techniques and a newly developed conductivity probe. Its construction and calibration are fully described by MacTaggart et al. (1993b). Samples of the slurry in the mixing tank were withdrawn using sampling tubes mounted at different axial positions along the vessel wall, midway between two baffles [Fig. l(a)]. Sample tubes with three different shapes were used in the present study: tapered tube (with a tip angle of 18°), a blunt sample tube (with a tip angle 90 ° and a relative wall thickness, wall thickness/sample tube inside radius, of 0.41) and a 45-face angle sample tube (with a sample face angle of 45°). Unless otherwise indicated, the inside diameter of the sample tubes was 4.55 mm. Figure l(b) depicts schematic diagrams for the three sample tubes. The sampling tubes were located at axial positions from near the bottom of the tank (z/H = 0.1) to near the top free liquid surface (z/H = 0.9). The tubes could be inserted through the wall of the tank (r/R = 1) to the impeller shaft (r/R = 0.04), except on the impeller plane, where the range was 0.33 ~< r/R <<.1. Slurry samples were withdrawn from the mixing tank using a variable speed peristaltic pump. The sample mass, volume and the sampling interval were measured to determine the sample solids concentration

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Local solids concentration measurement in a slurry mixing tank

Detail A SENSOR ELECTRODES

SMALLFIELD / ~ ELECTRODE~

I

\\ ~ ~ LARGEFIELD ~ ,~'-~Jl,I ELECTRODE II~ ~II

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2,4 mm

:~41- ............... ~'

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Fig. 2. Schematic of the conductivity probe.

and the sampling velocity. Sampling velocities ranged from 0.3 to 3 m/s and the impeller rotational speed was varied from 440 to 750 rpm. More details on the mixing tank and sample withdrawal techniques are given by MacTaggart et al. (1993a). The top of the mixing tank was fitted with a mounting system to allow positioning of the conductivity probe at any axial position in the tank and radial positions from r/R = 1. (wall of the tank) to r/R = 0.22 (wall of the probe holder). The conductivity probe, shown in Fig. 2, was constructed of straight stainless steel tubing, 4.8 mm in diameter and 610 mm long (MacTaggart et al., 1993b). With these dimensions, the probe was long enough to reach all levels in the mixing tank while being small in diameter to minimize flow disturbances and to measure solids concentration over a small volume in space. Two field and two sensor electrodes were located on the sensing end of the conductivity probe (Details A and B in Fig. 2). The sensor electrodes consisted of two small gauge stainless steel wires set approximately 1 mm apart. This electrode spacing was adequate to measure the local concentration for particles having diameters up to 1 mm (Nasr-El-Din et al., 1987). To minimize the effect of flow disturbances, the four electrodes were mounted flush with the fiat end of the conductivity probe. The large field electrode was the body of the conductivity probe tubing while the small field electrode was a half-round piece of the 4.8 mm tubing. All of the four electrodes were set in place and electrically insulated from each other with a quick setting resin from L D . Caulk Company. On the

FIELD CIRCUIT AMMETER

FUNCTION GENERATOR

::::::::::::::::::::: CONDUCTIVITY

PROBE

-~

~

BALLAST RESISTOR

SENSOR CIRCUIT

A.C. VOLTMETER

Fig. 3. Conductivity probe circuits. opposite end of the conductivity probe, outside the mixing vessel, the four electrode wires were connected to a multi-pin plug. The conductivity probe circuits are shown in Fig. 3. The field current was generated by an Interstate Electronics Corporation F51 function generator and monitored with a Beckman 3010 multimeter with a + 0.001 mA resolution. A square wave of 1000 Hz frequency and approximately 11 V amplitude, was employed. A 5000 f~ variable ballast resistor was used to maintain a constant field current of 1 mA. This current provided an adequate sensor voltage while minimizing the risk of electrolysis of the working fluid. The sensor voltage was measured with a Fluke 77

1212

H. A. NASR-EL-DINet al.

multimeter with a resolution of ___1 mV. The sensor circuit had such a high impedance that there was essentially no current flowing. This minimized polarization of the sensor electrodes and the associated problems. The operation of the conductivity probe described above relies on the variation of the slurry resistance as the concentration of the non-conducting solids (e.g. sand particles) changes. When the probe is placed in a conducting liquid (e.g. tap water) and a potential is applied across the two field electrodes, a small current flows from one field electrode to the other. The value of this current depends, among other factors, on the resistance of the surrounding medium. If non-conducting particles are added to the conducting liquid, then the resistance of the surrounding medium will increase. As solids concentration is increased, the resistance of the surrounding medium will be higher. Solids concentration can be determined by maintaining a constant field current and measuring the voltage across the two sensor electrodes then using a calibration curve. Various techniques were used to calibrate the conductivity probe, including a sedimentation vessel and a slurry pipeline. The calibration of the probe was conducted by measuring solids concentration in a sedimentation vessel where solids concentration was varied from 0.05 to 0.35. Polystyrene particles of pp = 1050 kg/m 3 were used in these experiments. The particles were sieved several times to produce a closely sized cut (300--355/~m). Figure 4 displays the variation of the slurry specific resistance obtained with the probe with the solids concentration. The slurry specific resistance is defined as [(Rm - R : ) / R : ] , where Rm and R : are the slurry and the fluid resistances, respectively. Predictions of the slurry specific resistance based on the Maxwell's equation for non-conducting particles are shown for comparison. For non-conducting solids (ke = 0), the Maxwell's equation is 2(1 - Co) kM = k L - (2 + Co)

(2)

where ku and kL are the electric conductivities of the mixture (slurry) and the working fluid, respectively.

(~) o

2.5

i



i

i

i

1.0

0.5

0.10

0.20

0.30

0.40

3Co 2(1 - Co)

(3)

Figure 4 indicates that the experimental data agree very well with Maxwell's theoretical equation. Similar results were obtained when the probe was used to measure solids concentration in a 2-inch slurry pipeline. More details on the probe calibration are given by MacTaggart et al. (1993b). In all mixing experiments, the working fluid employed was tap water and the particles used were sand (Pc --- 2650 kg/m 3) with mean particle sizes of 82, 255, 410, 500, and 1000 ~m. The mean temperature of the system was 24.9°C. RESULTS AND DISCUSSION Prior to examining various errors associated with sample withdrawal techniques, it is useful to understand the mechanisms of sampling from a mixing tank in order to better interpret the results obtained. The flow structure in a mixing tank stirred with a radial flow impeller is rather complicated. On the plane of the impeller, the flow is generally radial towards the tank wall. This is true except for the region near the tank wall where a stagnation point exists and the flow splits into axial flow upwards and downwards (Rushton, 1965; Laufhutte and Mersmann, 1985a, b). Above the impeller plane, near the tank wall, the flow is generally axial. Depending on the position of the sample tube, the flow in the tank can be parallel, perpendicular or inclined to the sampling tube axis. The angle between the sample tube axis and the flow upstream of the tube will determine the mechanism of particle collection by the sampler and the variation of the sample solids concentration with the sampling velocity. Sampling at the impeller plane Solids concentration in mixing tanks is often measured at the impeller plane. At this plane, Rushton (1965) described the radial velocity of flow on the center line of a radial flow turbine as an expanding jet that starts at the tip of the impeller blade. The radial velocity, Uo, is given by B1ND 2 Uo = - r

/

EXPERIMENTAL DATA

1.5

o

(RM -- Rs)/R f

i

2.0

PJ~

According to eq. (2), the slurry specific resistance is

0.50

0,60

Solids Concentration, Volume Fraction Fig. 4. Probe calibration from sedimentation experiments.

(4)

where B1 is a constant dependent upon the number of impeller blades and the ratio of the impeller to tank diameter, N is the impeller rotational speed, D is the impeller diameter, and r is the radial distance measured from the center of the impeller. Equation (4) is applicable only on the plane of a radial flow impeller between the impeller blade tip (r/R -- 0.33) and near the tank wall (r/R ~- 0.95). Solids concentration in the present study was first measured at the impeller plane (z/H = 0.3) and a normalized radial position, r/R, of 0.9 using the tapered sample tube and the conductivity probe. The sample

1213

Local solids concentration measurement in a slurry mixing tank o o

8

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Particle Size = 410 microns Bulk Solids Concentration = 0.1 I 2

I 3

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o

Sampling Velocity Ratio, Us/Uo

Fig. 5. Sample concentration ratio as a function of the sampling velocity ratio. Tapered sample tube, with an inside diameter 4.55 mm, was used at z/H = 0.3 (impeller plane) and r/R = 0.9. tube had an inside diameter of 4.55 mm. Sand particles with a mean particle size of 410 #m were used at a bulk volumetric concentration of 0.1 and a mixer rotational speed of 545 rpm. Figure 5 shows the variation of the sample concentration ratio, C~/Co, with the sampling velocity ratio, Us/Uo, where Cs is the concentration obtained with the sampling withdrawal technique and Co the solids concentration obtained using the conductivity probe. Us is the sampling velocity and Uo the upstream local velocity. According to eq. (4), the upstream local velocity was 0.74 m/s on the impeller plane (z/H = 0.3 and r/R = 0.9) at a mixer rotational speed of 545 rpm. It should be mentioned that the value of the constant B1 in eq. (4) is 1.13 (Rushton, 1965) for the mixer and the tank geometry used in the present study (a mixing tank with D/T = 0.33 and equipped with six blades). The sampling velocity ratio was varied from 0.25 to 4. Sample withdrawal at velocity ratios less than 0.25 was not possible because of the plugging of the sample tube and the connecting lines at low sample velocities. Also, sampling at velocity ratios greater than 4 was not practical because this led to significant changes in the bulk solids concentration in the mixing tank. At a sampling velocity ratio of 0.3, the sample concentration ratio was unity. Increasing the sampling velocity ratio resulted in lower sampling concentration ratios. The sample solids concentration ratio was significantly less than unity at the isokinetic velocity, i.e. at UJUo = 1. This result confirmed that the flow on the impeller plane was not purely radial, but as well consisted of axial and angular flow components. Laufhutte and Mersmann (1985b) have shown that this is true in a mixing tank filled with liquid only and stirred with the same Rushton-type impeller used in the present study. In the immediate vicinity of the impeller, large angular velocity components exist. These angular flows are lower at greater distances from the impeller, but still exist. To assess the influence of angular and axial flow components at the impeller plane on the sample solids concentration, samples were collected at the same local position in the mixing tank as in Fig. 5 with the 45-face angle sample tube. The sample tube was

I

i

1

L

I

2

3

SamplingVelocity,m/s

0.2

(b)

0.15

i

0.1

~ ~ f

0.05

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1

............................

Particle Size = 410 microns Bulk Solids Concentration = 0.1 i

i

2

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Sampling Velocity, m/s

Fig. 6. Effectof the sampler orientation on the sample solids concentration at z/H = 0.3 (impeller plane) and r/R = 0.9. (a) Lateral sampler orientation. (b) Vertical sampler orientation. oriented to four different positions, 90 ° from each other, such that the opening of the sample tube was oriented upwards, downwards, to the front or to the back of the mixing tank. The results of these experiments are shown in Fig. 6. Figure 6(a) shows the sample solids concentration as a function of the sampling velocity for the cases where the orientation of the sample tube was such that the sample tube faced the front and the back of the mixing tank, respectively. It should be mentioned that the probe cross-sectional area was used to calculate the sampling velocity for the 45-face angle probe. As can be seen, large differences in the sample solids concentration occurred for these two orientations, especially at low sampling velocities. The highest sample solids concentration occurred when the sample tube faced the flow direction (indicated by the impeller rotation). The lowest concentration occurred when the sample tube faced away from the flow. These results confirmed the existence and the importance of the angular flow component at this position in the vessel. If the flow were purely radial, the sample solids concentration should not differ for the two orientations shown in Fig. 6(a). The fluid streamlines into the sample tube were diverted more severely in the second case as compared to the first. This resulted in a greater separation of the particle trajectories from the fluid flow leading to lower sample solid concentrations. Figure 6(b) illustrates the variation of the sample solids concentration with the sampling velocity for the 45-face angle sample tube facing upwards and downwards at the same local position and operating conditions as in Fig. 6(a). Differences in the sample solids

H. A. NASR-EL-DINet

1214

concentration with the sample tube orientation again exist indicating the importance of the axial flow component at this location in the vessel. Axial flow is expected near the wall of the vessel in the impeller plane where the flow splits into axial flow upwards and downwards. The effect of the axial flow component on the sample solids concentration was less than that observed in Fig. 6(a). Figure 6 shows that even on the plane of a radial impeller, the flow structure is such that the solid particles approach the sample tube obliquely and not parallel to the sample tube. This violates the requirement stated by Lundgren et al. (1978) and Nienow (1985) that the sample withdrawal velocity be in the same direction as the fluid-particle flow. It is therefore reasonable to expect the sample solids concentration ratio to be lower than unity even at a sampling velocity ratio of unity as shown in Fig. 5.

Errors associated with various sampling withdrawal techniques The results shown in Figs 5 and 6 indicated that sample withdrawal techniques can produce significant errors in measuring local solids concentration, even at the impeller plane. To assess sampling errors and to provide guidelines to correct for such errors, the local solids concentration was measured using the conductivity probe and sampling withdrawal techniques. The error associated with sample withdrawal was calculated using the following formula: Sampling error = 100(1 - CJCo).

Effect of the sample tube design on the sampling errors Figure 7 depicts the influence of the sample tube tip angle on the sampling error when sampling at the impeller plane. Figure 7(a) shows the variation of the sampling error with the sampling velocity ratio for the 410/~m sand particles at a bulk solids concentration of 0.1. For the blunt sample tube, the sampling error was positive at sampling velocity ratios less than 0.38. The positive sampling error indicated that the sample solids concentration was higher than the actual local solids concentration in the tank. At a sampling velocity ratio of 0.38, the sampling error was zero which signified that the collected sample was identical to that in the tank; i.e. a representative sample was obtained. At sampling velocity ratios greater than 0.38, the sampling error was negative. In this case, the sample solids concentration was lower than the true concentration in the tank. The effect of the sampling velocity ratio on the sampling error for the tapered tube was similar to that observed for the blunt probe. However, the sample solids concentration was consistently lower for the tapered tube. This lower concentration resulted in higher absolute sampling errors at sampling velocity ratios greater than 0.38. Figure 7(b) shows the effect of the sampling velocity ratio on the sampling error for the 1000/~m sand particles using the same sample tubes examined in Fig. 7(a). Unlike the results obtained with the 410 #m

el.

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ParticleSize = 410 microns Bulk Solids Concentration = 0.1

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_ ""A...,~ -A...,=._.. .... ~

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Particle Size = 1000 microns Bulk Solids Concentration = 0.1

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~ --• . . . . . . . . . . . . . . . . . . . . . . . .

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Fig. 7. Effect of the sample tube shape on the sampling error when sampling at the impeller plane in the mixing tank at z/H = 0.3 and r/R = 0.9. (a) dp = 410/~m. (b) dp = 1000/~m. sand, the effect of the tube tip angle on the sample error was significant. To obtain a representative sample using the blunt probe, a sampling velocity ratio of 2.2 was needed. This velocity ratio was 5.5 times higher than that required for the 410/~m sand. The trends shown in Fig. 7 are due to the particle bouncing effect. Solid particles travelling in-line with the thick sample tube edge will strike this edge, loose inertia and be withdrawn into the sample tube. Consequently, sample solids concentration will be higher than that obtained using a tapered sampling tube. The effect of particle bouncing on the sampling error was more pronounced for the 1000/am sand as shown in Fig. 7(b). These results indicate that the effect of particle bouncing on sample solids concentration is a strong function of particle size. The tapered sample tube minimized the deflection of the fluid streamlines and particle bouncing on the sample tube face. However, sampling errors upto 20% were obtained with the tapered tube even at a sampling velocity ratio of unity. These errors are due to the three-dimensional nature of the flow upstream of the sample tube at the impeller plane as explained earlier. Sample tube diameter can have a large impact on the sampling error when sampling normal to the flow direction in the mixing tank. Figure 8 reveals the influence of the sampling velocity on the sampling error using blunt sample tubes having inside diameters of 2.97, 4.55 and 10.87 mm. Samples were collected at z/H = 0.5 and r/R = 0.9 using the 410/tm sand particles at a bulk solids concentration of 0.1 and a mixer rotational speed of 545 rpm. In this plot, the sampling error is plotted against the sampling velocity U, only and not against the sampling velocity ratio Us/Uo, since the local velocity Uo is not known.

1215

Local solids concentration measurement in a slurry mixing tank 4O

0 Sample Tube Inside Diameter (mm) ~-20

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....~'6 ...... 10.87

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Blunt Tube

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4,5-Face Tube (Downwards Orientation) J i

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45-Face Tube (Upwards Orientstion)



2

a

Sampling Velocity, m/s

S a m p l i n g Velocity, m/s

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40

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Particle Size = 255 microns

(b)

Bulk Solids Concentration = 0.3

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~:: ~ ..............~....,

At a sample tube inside diameter of 2.97 mm, the sampling error was 66% at a sampling velocity of 0.35 m/s. Increasing the sampling velocity reduced the absolute value of the sampling error. However, significant sampling errors were still observed even at sampling velocities greater than 3 m/s. At a given sampling velocity, the sampling error approached zero as the sample tube inside diameter was increased. This trend was due to an increased time available for a particle to change its direction to be withdrawn into the sample tube. Clearly when sampling normal to the flow direction, it is desirable to sample with as large a sample tube as possible. This, however, does not ensure that a representative sample can be obtained. Two sets of experiments were conducted to examine the effect of the sample tube shape and orientation on the sampling error. In the first set, samples of the dp = 255 pm sand particles at bulk solids concentrations of 0.1 and 0.3 were collected using the blunt and the 45-face angle sample tubes at z/H = 0.4 and r/R = 0.5. In the second set, samples were collected for the dp = 82 pm and dp = 410/~m sand particles at a bulk solids concentration of 0.3 using the same sample tubes as for the first set. In each set, the 45-face angle sample tube was oriented both upwards and downwards in the tank. Figure 9(a) depicts the results obtained for the 255 pm sand at a bulk solids concentration of 0.1. Large differences in the sample error were obtained at low sampling velocities, simply by using different sample tube shapes. The mechanism for particle collection was completely different as well. Results for the 45-face angle sample tube, downwards orientation, resembled sampling at right angles to the flow, with negative sampling errors which approached zero as the sampling velocity was increased. On the other hand, the performance of the 45-face angle sample tube with the upwards orientation resembled sampling parallel to the flow, with higher sample errors being obtained as the sampling velocity deviated from the upstream local velocity. The blunt sample tube gave sample errors between those obtained with the two orientations tested with the 45face angle sample tube, and was insensitive to the sampling velocity.

r¢X

-4O =:::=

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c:=,

o

45,-FaceTube (Upwards O d e ~ )



Blunt Tube

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45-Face Tube (Downwards Orientation)

-100

Sampling Velocity, m/s Fig. 9. Effect of the sample tube shape and orientation on the sampling error at z/H = 0.4 and r/R = 0.5. (a) Bulk solids concentration of 0.1. (b) Bulk solids concentration of 0.3. The results shown in Fig. 9(a) indicate that the sample solids concentration was highest when the orientation of the 45-face angle sample tube was upwards. This trend was observed at all sampling velocities examined. This trend suggests that the sample tube was facing the flow at this position in the mixing tank (z/H = 0.4, r/R = 0.5). Particle bouncing off the protruding tip of the 45-face angle sample tube resulted in a loss of particle inertia and an increase in the sample solids concentration. Barresi and Batdi (1987a) observed the same effect when sampling from a mixing tank, stirred by an axial flow impeller. Just above and below the impeller, the 45-face angle sample tube, opening into the flow, gave consistently higher sample solids concentrations than a blunt sample tube. The sample solids concentrations were consistently lower when the orientation of the 45-face angle sample tube was downwards, especially at low sampling velocities. This was due to the orientation of this sample tube relative to the flow. To enter the sample tube, the fluid must bend at least 90 °. The same is true for the solids, but due to particle inertia, separation between the particle trajectories and the flow streamlines occurs and the sample solids concentration is reduced. Figure 9(b) displays the effect of the sample tube shape and orientation for the same particles used in Fig. 9(a), but at a bulk solids concentration of 0.3. The sampling error of the 45-face angle sample tube for the upwards and downwards orientations showed less variation with the sampling velocity. Unlike the resuits obtained at the lower bulk solids concentration,

H.A. NASR-EL-DINet al.

1216 40

Particle Size = 82 microns

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(a)

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(a)

B u l k S o l i d s Concentration = 0.3

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20

g

Bulk Solids Concentration = 0.3 ~'~Lz~ ~

o

:.:~.:.~ -2o .A'"..~.~............... .........~ ...............

~-~ Q.

..40

¢~

-60 - d' -60 -100

0 . . ' ° .... /' ~ ~

(b)

z/H = 0.5

._ -40

~'~

[i

~

Blunt Tube

o

45-Face Tube (Downwards Orientation

i 1

¢J) -80

45-Face Tube (Upwards OrientatiOn)



i 2

-100

j i 3

(b)

= .

o

I~ -20

o

w

i 3

40

Particle Size = 410 microns

12

i 2

Sampling Velocity, m/s

3

4O

0

i 1

-80

Sampling Velocity, m/s

e.

z/H o 0.1 o 0.3 • 0.4 • 0.5

j,-.

4

Sampling Velocity, m / s

Fig. 10. Effect of the sample tube shape and orientation on the sampling error at z/H =0.4 and r/R =0.5. (a) dv = 82 #m. (b) dv = 410 #m.

..... ~" ........ .i i 1

...•. .................. -&----~,................. Particle Size = 1000 microns B u l k S o l i d s C o n c e n t r a t i o n = 0.1 i i 2 3

Sampling Velocity, m/s Fig. 11. Effectof the sampling velocityon the sampling error at r/R = 0.9 and various axial positions in the mixing tank. (a) dp = 410/~m. (b) dp = 1000/tm.

Effect of the axial position on the sampling error the sampling error was less dependent on the shape of the sample tube. As the solids concentration was increased, the drag forces exerted by the fluid on the solid particles increased and, as a result, more particles reported to the sample tube. The effect of the sample tube shape and orientation on the sampling error is a function of the particle mean size. Figure 10 depicts the influence of the sample tube shape and orientation on the sampling error at z/H = 0.4 and r/R = 0.5 and a bulk solids concentration 0.3. The sampling error was independent of the sample tube shape and orientation for the 82 #m sand particles as shown in Fig. 10(a). Also, increasing the sampling velocity did not influence the sampling error. These results indicate that solids concentration can be measured with a 10% sampling error using any of the sample tubes examined at sampling velocities up to 3 m/s. The effect of the sample tube shape and orientation on the sampling error was significant for the 410 #m sand particles as shown in Fig. 10(b). For such coarse particles, measuring solids concentration using sample withdrawal techniques should be conducted very carefully. Obviously, the design of the sampler has a large impact on the sampling errors, especially for particles having a large diameter. The different sampling mechanisms for the different sample tube design makes it extremely difficult in a mixing tank to determine the sampling velocity required to obtain a representative sample without prior knowledge of the solids and the fluid properties as well as the flow velocity and direction upstream of the sample tube.

Solids concentration profiles are usually measured by withdrawing samples at different axial positions in the tank (Einenkel, 1980; Kuzmanic etal., 1992). As explained earlier, the fluid flow in a mixing tank equipped with a radial impeller is complex and, as a result, sampling error will vary with position in the tank. To examine this point further, samples were collected at a normalized radial position, r/R, of 0.9 and normalized axial positions, z/H, of 0.1, 0.3, 0.4 and 0.5 using a blunt sample tube of 4.55 mm inside diameter. Figure ll(a) shows the sampling error as a function of the sampling velocity for the 410 ym sand particles at a volumetric bulk solids concentration of 0.1. The variation of the sampling error with the sampling velocity depended on the axial position in the tank. At axial positions farther from the impeller plane, e.g. z/H = 0.1 and 0.5, the flow was nearly normal to the probe axis and the absolute value of the sampling error decreased as the sampling velocity was increased. At the impeller plane, z/H = 0.3, the flow was nearly parallel to the probe axis and the sampling error increased as the sampling velocity deviated from the upstream local velocity. At a normalized axial position of 0.4, the sampling error was independent of the sampling velocity. The most important aspect of Fig. 1l(a) is that the sample error was independent of both the sampling axial position and the sampling velocity at high sampling velocities. To obtain a representative solids concentration profile, it is strongly recommended to collect the sample at a sampling velocity of 2-3 m/s. Then, the measured solids concentration can be corrected using the data given in Fig. ll(a).

1217

Local solids concentration measurement in a slurry mixing tank Figure l l(b) demonstrates the variation of the sampling error with the sampling velocity at z/H = 0.3 and 0.5 for the 1000/~m sand particles. The sampling error varied significantly with the axial position for the coarse sand particles even at high sampling velocities. Measuring solids concentration profiles for the coarse sand using the procedure recommended for the 410/tm sand particles will be difficult because a correction factor will be needed for each axial position in the tank. Also, the correction factor will be dependent on the sampling velocity. The results shown in Fig. 11 indicate that under the same operating conditions, the variation of the sampling error with sampling velocity depends on the axial position in the mixing tank. This variation increased for the coarse sand particles. Therefore, to obtain reliable solids concentration measurements from a mixing tank using the sample withdrawal techniques, the flow pattern ahead of the sample tube should be known.

Solids concentration profiles in the mixing tank The flow in a mixing tank equipped with a radial impeller consists of two main circulation loops: one above the impeller plane and the second loop below the impeller plane (see for example Nouri and Whitelaw, 1992). Because of the two loops, the flow is radial in some places and axial in others. This change of flow direction will affect the solids distribution in the tank, especially when highly settling particles, the case of interest in the present study, are used. The conductivity probe was used to measure local solids concentration at r/R of 0.3,0.6 and 0.9. At each normalized radial position, measurements were taken at various normalized axial positions from 0.2 to 0.9. The effects of particle size, bulk solids concentration and mixer rotational speed on solids concentration profiles in the tank were examined in detail. Figure 12 shows two vertical solids concentration profiles obtained using sand particles of 82 #m at a bulk solids concentration of 0.3 and a mixer rotational speed of 545 rpm. The local solids concentration was normalized using the bulk solids concentration in the tank. The concentration profiles indicated that the solids distribution was nearly uniform

'I"

1

particle Size = 82 micron= e~ ~" 0.8 "Bulk Solids Concentration = 0.3.',~.'i" O'J ~ ~- 0.6 Mixer Speed = 545 RPM .

in the vertical direction. Also there was no significant concentration variation in the radial direction. The normalized solids concentration for the two radial positions at the impeller plane was higher than those measured below (z/H = 0.2) or above (z/H = 0.4) the impeller plane. Figure 13 depicts the vertical concentration profiles for the 255/~m sand particles at a bulk solids concentration of 0.3. The settling velocity of these particles at infinite dilution was an order of magnitude higher than those used in Fig. 12. At a normalized radial position of 0.3, a steep increase in the normalized concentration was observed in the upper part of the mixing tank down to z/H = 0.7. The normalized concentration became greater than unity and remained constant from z/H = 0.7 down to 0.4. Because of the impeller, no concentration measurements could be taken at r/R = 0.3 and z/H less than 0.4. At normalized radial positions of 0.6 and 0.9, the variation of the normalized concentration with the axial position was similar to that observed at r/R = 0.3. However, at z/H = 0.3, the vertical concentration profiles exhibited a strong variation with the axial position at both radial positions. This variation was more pronounced than that observed with the 82 #m sand particles shown in Fig. 12. Increasing the particle settling velocity at infinite dilution by a factor of 25 (in comparison with the 82/~m sand particles) resulted in more variation in the solids concentration with the axial and radial positions as shown in Fig. 14. The solid particles in this case (401/~m sand particles) were concentrated in the lower half of the tank. Again, the vertical solids concentration showed a strong dependence with the axial position at the impeller plane and r/R = 0.9. It is interesting to note that a maximum in the vertical concentration profiles for settling slurries having coarse particles was observed by Rieger et al. (1988) using a conductivity probe and Barresi and Baldi (1987a, b) using sample withdrawal techniques. Brucato and Rizzuti (1988) developed a mathematical model to predict vertical solids concentration profiles in a mixing tank equipped with a Rushton impeller. The predicted vertical concentration profile showed similar variations with the axial position to those

.~ o.e r/R • •

O.S

0.9

O.e •

~

0.4 •

~ z



i ~

........ Impeller Plane

0.2 0

~

.

.

,.i ~ ~,.

i i

=,k So,~, Co.=..~on = 3 i

0.4

Z

"

~.

I 05

~

i=~. I 1

0.2

,

I 1.5

i

Normalized Solids Concentration, C/Cs

Fig. 12. Vertical solids concentration profiles in the tank for the 82/~m sand particles at a bulk solids concentration of 0.3.

~"~' ~.

:::::~

=

,

...-

015

1

i

h

o ,, •

0,3 oB 0.9

Impeller Plane

......

11.5

Normalized Solids Concentration, C/C_ B

Fig. 13. Vertical solids concentration profiles in the tank for the 255 pm sand particles at a bulk solids concentration of 0.3.

H. A. NASR-EL-DINet al.

1218

o'; "

"i5 '1- 0.e '1o ,J 0.4 ~ 0 Z

i ~ ....

: :

:

:

:

:

~

~

o

0.3

"

0.6

"1"

1

.~"

0.8

~.

0.8

r/R

Bulk SoI~ ~ncentmtion = 0.3i !

::...~.. . . . . . . 1¢"

"'•

i

0.3 0.6 0.9

~'"----~.. ..... Impeller Plane

Bulk 8Olldl Conctmli-don = 0,1

0.2 0 Z

© • •

....~., ~ :":x, ~ '~'i~ e-~ -r

0.4 Particle S~e = 410 rnlcrocm

0.2 • Mixer Speed - 5 4 5 RPM 0

r/R

i

. . . . . . . ~-_'i;:.A . . . . . . . .

~'-'" i

"'"

0

Normalized Solids Concentration, C/C_ B

Normalized Solids Concentration, C / C

Fig. 14. Vertical solids concentration profiles in the tank for the 410 #m sand particles at a bulk solids concentration of 0.3.

Fig. 15. Vertical solids concentration profiles in the tank for the 410/~m sand particles at a bulk solids concentration of 0.1 and a mixer rotational speed of 545 rpm. -r"

shown in Figs 11-13 at the impeller plane and r/R = 0.54. An important aspect of Figs 11-13 is that the circulation flow in the tank affected the solids distribution in the tank. This effect increased with the particle size. The vertical concentration profile cannot be fitted with the sedimentation-diffusionmodel (Fajner et al., 1985) only. The effect of the circulation flow on the concentration profiles should be also considered. The influence of this type of flow on the vertical solids concentration profiles can be explained as follows. At the impeller plane, the flow is radial towards the wall of the tank. As the fluid approaches the wall it splits into two parts or loops. The volumetric flow rate in each loop depends on the location of the impeller in the tank (Brucato and Rizzuti, 1988). The response of the solids particles to the change in the flow direction depends on the particle inertia parameter. Sand particles will lag the fluid flow with a slip velocity which increases with the particle inertia parameter (Nouri and Whitelaw, 1992). Fine sand particles have low inertia parameter and tend to follow the fluid flow. As a result, the effect of the circulation flow on the vertical solids concentration profile was not significant as was shown in Fig. 12 for the 82 ~m sand particles. On the other hand, the 410 #m particles have high inertia parameter and will lag the fluid flow. As a result, solids concentration above and below the impeller plane will be lower than that at the impeller plane. It should be noted that gravitation will increase the lag between the sand particles and the working fluid above the impeller plane and will decrease the lag below the impeller plane. Solids concentration profiles in a mixing tank are also a function of the bulk solids concentration in the tank. Figure 15 illustrates the vertical concentration profiles for the 410 pm sand particles at a bulk solids concentration of 0.1. Unlike the results obtained the higher bulk solids concentration, the vertical profiles indicated better solids distribution in the tank. Also, significant variations in solids concentration were observed above and below the impeller plane at r/R of 0.6 and 0.9. These changes in the solids concentration were more pronounced at the lower bulk solids concentration.

1

• &...e~ .~ 0.8 - " ~ " ~ '~ ~e 0.6 ~N 0.4

i ParticleSize = 410 microns i Bulk S~ids Concentration = 0.1 Mixer Speed = 440 RPM

• e,.. -'~"~-

~

..... ~7~::"

0 Z

i

'n • Impel Ier -Ha

o

03



0,S

02 0

i Normalized Solids Concentration, C/Ca

Fig. 16. Vertical solids concentration profiles in the tank for the 410/tm sand particles at a bulk solids concentration of 0.1 and a mixer rotational speed of 440 rpm. Solids concentration profiles in a mixing tank varied with the mixer rotational speed. Figure 16 displays the vertical concentration profiles for the 410 pm sand at a rotational speed of 440 rpm. It should be mentioned that at a rotational speed of 492 rpm all the solids were in motion off the tank bottom. This rotational speed was calculated using the Zwietering's (1958) correlation for the solid particles examined in Fig. 16. At a rotational speed of 440 rpm, sand particles settled on the bottom of the tank below z/H = 0.1 and could not be measured by the conductivity probe. The variation of the solids concentration with the axial position was less than that observed with a rotational speed of 545 rpm. A strong variation in the solids concentration at the impeller plane with the axial position was observed at r/R = 0.9 only. CONCLUSIONS

The local solids concentration in a mixing tank was measured using the sample withdrawal technique and a new conductivity probe. The conductivity probe was used to assess the errors associated with the various sampling techniques and to measure solids concentration profiles in the mixing tank. The following results were obtained: 1. On the impeller plane, the sample withdrawal techniques (tapered sample tube) gave lower solids concentration than the true local concentration in the tank at the isokinetic sampling velocity. These results uggest that the flow at the impeller plane was threedimensional.

Local solids concentration measurement in a slurry mixing tank 2. The errors associated with the sampling techniques were found to be dependent on the sample tube shape, especially with coarse sand particles of 1000 #m. 3. When sampling at a right angle to the flow, the sampling errors increased with the particle size. Also, higher sampling errors were observed when sampling tubes having a small diameter were used. 4. Sampling errors depended on the sampling velocity and the axial position of the tank. 5. The data obtained in this study can be used to correct for the errors associated with the sample withdrawal techniques over a wide range of parameters. 6. Solids concentration profiles in the mixing tank were found to be dependent on the particle size, bulk solids concentration and mixer rotational speed. 7. Solids concentration varied with the radial position, except when fine sand particles of 82/~m were used. 8. A strong variation in solids concentration with the axial position was observed at the impeller plane for the sand particles examined in the present study. This variation increased with the particle mean size and the mixer rotational speed. It is hoped that the solids concentration profiles provided here will serve to validate computational models for solids distribution in mixing tanks equipped with a radial flow impeller.

Acknowledgement--JHM wishes to thank the Natural Sciences and Engineering Council of Canada for their financial support.

NOTATION BI

CB Co

C~ 4 D h H k K N r R

RI Rm T Uo Us z

CES 51:8-D

constant in eq. (4), bulk solids concentration, volume fraction local solids concentration obtained with the conductivity probe, volume fraction sample solids concentration, volume fraction particle mean diameter, m impeller diameter, m impeller height above tank bottom, m total liquid height in the mixing tank, m electrical conductivity, f~- 1 m particle inertia parameter, defined in eq. (1) mixer rotational speed, rpm radial distance measured from the center of the mixing tank, m radius of the mixing tank, m fluid resistance, mixture (slurry) resistance, diameter of the mixing tank, m local velocity upstream of the sample tube, defined in eq. (4), m/s sampling velocity, m/s axial distance measured from the bottom of the mixing tank, m

1219

Greek symbols /~ fluid viscosity, P a s p density, kg/m a ~b inside diameter of the sampling tube, m Subscripts L liquid M mixture p particle s sample REFERENCES

Ayazi Shamlou, P. and Koutsakos, E., 1989, Solids suspension and distribution in liquids under turbulent agitation. Chem. Engng Sci. 44, 529-542. Baldi, G., Conti, R. and Gianetto, A., 1981, Concentration profiles for solids suspended in a continuous agitated reactor. A.I.Ch.E.J. 27, 1017-1020. Barresi, A. and Baldi, G., 1987a, Solid dispersion in an agitated vessel. Chem. Engng Sci. 42, 2949-2956. Barresi, A. and Baldi, G., 1987b, Solid dispersion in an agitated vessel:effect of particle shape and density. Chem. Engng Sci. 42, 2969-2972. Barresi, A., Kuzmanic, N. and Baldi, G., 1994, Continuous sampling of a slurry from a stirred vessel: analysis of the sampling efficiencyand affectingparameters. Proceedings of the 8th European Conference on Mixing, BHRA, Cambridge, UK, 21-23 September, p. 1724. Bohnet, M. and Niesmak, G., 1980, Distribution of solids in stirred suspensions. Ger. chem. Engng 3, 57~5 Brucato, A. and Rizzuti, L., 1988, The application of the network-of-zonesmodel to solid-liquid suspensions. Proceedings of the 6th European Conference On Mixing, BHRA, Pavia, Italy, 24-26 May, pp. 273-280. Buurman, C., Resoort, G. and Plaschkes, A., 1986, Scalingup rules for solids suspension in stirred vessels. Chem. Engng Sci. 41, 2865-2871. Chapman, C. M., Nienow, A. W., Cooke, M. and Middleton, J. C., 1983a, Particle-gas-liquid mixing in stirred vessels. Part I: Particle-liquid mixing. Chem. Engng Res. Des. 61, 71-81. Chapman, C. M., Nienow, A. W., Cooke, M. and Middleton, J. C., 1983b, Particle-gas-liquid mixing in stirred vessels. Part II: Gasqiquid mixing.Chem. Eng. Res. Des. 61, 82-95. Einenkel, W. -D., 1980, Influence of physical properties and equipment design on the homogeneity of suspensions in agitated vessels. Get. chem. Engng 3, 118-124. Fajner, D., Magelli,F., Nocentini, M. and Pasquali, G., 1985, Solids concentration profilesin a mechanicallystirred and staged column slurry reactor. Chem. Engng Res. Des. 63, 235 240. Gerstenberg, H., Sckuhr, P. and Steiner, R., 1983, Stirred tank reactors for polymerization. Ger. chem. Engng 6, 129-141. Kipke, K., 1983, Problems in mixing technology. Get. chem. Engng 6, 119-128. Kuzmanic, N., Barresi, A. A and Baldi, G., 1992, Wall sampling of suspensions from stirred vessels. Proceedings of 3rd International Symposium Multiphase Hydrodynamics in Industrial Plants, Castelnuovo di Garfagnana, Italy, 10-11 September, pp. 437-452. Laufhutte, H. D. and Mersmann, A. B., 1985a, Dissipation of power in stirred vessels. Proceedings of the 5th European Conference on Mixing, BHRA, pp. 331-340. Laufhutte, H. D. and Mersmann, A. B., 1985b, Laser-Doppler velocimetry as a suitable measuring technique for the determination of flow behaviour in stirred fluids. Get. chem. Engng 8, 371-379. Lundgren, D. A., Durhan, M. D. and Mason, K. W., 1978, Sampling from tangential flow streams. Am. Ind. Hyg. Assoc. 39, 64ff644.

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H. A. NASR-EL-DINet al.

Machon, V., Fort, I. and Skrivanek, J., 1982, Local solids distribution in the space of a stirred vessel. Proceedings of the 4th European Conference on Mixing, Leeuwenhorst, The Netherlands, 27-29 April, BHRA, Cranfield, pp. 289-302. MacTaggart, R. S., Nasr-EI-Din, H. A. and Masliyah, J. H., 1993b, Sample withdrawal from a slurry mixing tank. Chem. Engng Sci. 48, 921-931. MacTaggart, R. S., Nasr-E1-Din, H. A. and Masliyah, J. H., 1993b, A conductivity probe for measuring local solids concentration in a slurry mixing tank. Separat. Technol. 3, 151-160. Magelli, R., Fajner, D., Nocentini, M. and Pasquali, G., 1990, Solid distribution in vessels stirred with multiple impellers. Chem. Engng Sci. 45, 615-625. Maxwell, J.C., 1881, A Treatise on Electricity and Magnetism. Calender Press, Oxford. Nasr-El-Din, H. A., 1989, Sampling from slurry pipelines, in Wastewater Treatment Technology, (Edited by P. N. Cheremisinoff), Vol. 3, Chap. 14. Gulf Publishing Company, Houston. Nasr-E1-Din, H., Shook, C. A. and Colwell, J., 1987, A conductivity probe for measuring local concentrations in slurry systems. Int. J. Multiphase Flow 13, 365-378. Nienow, A. W., 1985, The suspension of solid particles, in Mixing in the Process Industries (Edited by N. Harnby, M. F. Edwards and A. W. Nienow). Butterworths, London. Nouri, J. M. and Whitelaw, J. H., 1992, Particle velocity characteristics of dilute to moderately dense suspension

flows in stirred reactions. Int. J. Multiphase Flow 18, 21-33. Rehakova, M. and Novosad, Z., 1971a, The separation effect at the outlet of a vessel with a mechanically agitated suspension. Coll. Czech. Chem. Commun. 36, 3004-3007. Rehakova, M. and Novosad, Z., 1971b, Determination of the separation effect in a stirred vessel with a suspension by a dynamic method. Coll. Czech. Chem. Commun. 36, 3008-3012. Rieger, F., Ditl, P. and Havelkova, O., 1988, Suspension of solid particles-concentration profiles and particle layer on the vessel bottom. Proceedings of the 6th European Conference on Mixing, Pavia, Italy, 24-26 May, pp. 251-258. Rushton, J. H., 1965, The continuous removal of mixed phases from a mixing tank. AIChE L Chem. E. Syrup. Series 10, 3-10. Sharma, R. N. and Das, H. C. L., 1980, Effect of withdrawal flow velocity on the composition of a two phase system in a mixing tank. Coll. Czech. Chem. Commum. 45, 3293-3301. Smith, J. M., 1990, Industrial needs for mixing research. Trans. I. Chem. E. 68 A, 3-6. Tojo, K. and Miyanami, K., 1982, Solids suspension in mixing tanks. Ind. Eng. Chem. Fund. 21, 214-220. Yamazaki, H., Tojo, K. and Miyanami, K., 1986, Concentration profiles of solids suspended in a stirred tank. Powder Technol. 48, 205-216. Zwietering, Th. N., 1958, Suspending of solids particles in liquid by agitators. Chem. Engng Sci. 8, 244-253.