Chemiecll .&&ehn# Science, VoL 48, No. 5, pp. 921-931. Ptintid in Great Britain. -r
ooo%2509/93 $6.00 + 0.00 1993 Psrsalrlon Plus Ltd
R. S. MacTAGGART Syncrudd Canada Ltd., PO Bag 4009, M.D. 2050 Fort McMurray,
Alberta, Canada T9H 3L1
H. A. NASR-EL-DIN Petroleum
Recovery Institute, 3512-33 St. N.W., Calgary, Alberta, Canada T2L 2A6
and J. H. MASLIYAH’ Department
of Chemical Engineering, University of Alberta, Edmonton, (First received 20 December
Alberta, Canada T6G 2G6
1991,; accepted in revised form12 June 1992)
local solids concentration from a slurry mixing tank has been measured by the withdrawal of a sample from the vessel. It is shown that the sample tuba design (shape, diameter and tip angle) and sampling technique (withdrawal velocity and location in the mixing tank) can significantly affect the solids concentration and the particle size distribution of the sample withdrawn. The particle inertia relative to an equal volume of fluid results in solids concentration measurements that vary with particle size and bulk concentration. It is practically impossible to obtain reliable measurements of local solids concentration by sample withdrawaI from a mixing tank. Sampling errors can be minimized with the use of fine solids and sampling at as high a velocity as possible. The bulk solids concentration can be accurately measured on the plane da radial flow impeller by sampling at the &kinetic velocity, with a tapered sampling tube and when the solids concentration does not display a strong variation in the vertical direction. The fraction of fine sand in a sample obtained from a bimodal mixture of fine and coarse sands was found to be a function of the sample solids concentration.
Slurry mixing tanks are commonly used in crystallization, leaching and reactions utilizing a solid catalyst (Baldi et al., 1981) and in gaseous processes such as hydrogenation and oxidation reactors, fermentation, waste-water treatment, evaporative crystallization and froth flotation (Chapman et al., 1983a, b). Information on local solids concentration in mechanically stirred vessels is relatively scarce. Local solids concentration has been commonly measured by the removal of a small sample from the system. However, representative samples are extremely difficult to obtain from a mixing tank (Nienow, 1985) due to inertia differences between the fluid and particles of different size or density (Nasr-El-Din, 1987; Smith, 1990). Dcspite these serious shortcomings, sample withdrawal continues to be used as a method of determining local solids concentration in mixing tanks (Baldi et al., 1981; Oldshue, 1981; Buurman et al., 1986 Barresi and Baldi, 1987a, b). Sampling errors and particle collection mechanisms The objective of sample withdrawal is to obtain a sample that is identical in all properties to the system being sampled at the point of sampling. The sampling efficiency, or the separation or aspiration coefficient, is defined as (Nasr-El-Din, 1989) Sampling efficiency = CJCO where Cs is the solids.concentration ‘Author to whom correspondence
of the withdrawn
should be addressed. 921
sample and C,, is the local solids concentration in the mixing tank at the point of sampling. A sampling efficiency of unity implies ideal sampling. Nasr-El-Din (1989) showed that particle inertia relative to an equal volume of surrounding fluid, the flow structure ahead of the sampler and particle bouncing can cause sampling errors. Sampling errors due to the particle inertia effect are observed when the particle trajectories deviate from the fluid streamlines. Greater or fewer particles enter the sample tube than are present locally in the system. The particle inertia is represented by the dimensionless group known variously as the Stokes number, Barth number or particle inertia parameter (K) and is defined as K
where d,. is the particle size, U, is the local velocity, pr. is the liquid viscosity, p,, is the particle density and Q is the sample tube inside diameter. Very small particles, or particles with a density approaching the fluid, have very low inertia relative to an equal volume of fluid. These particles will follow the fluid and enter the sample tube, even if the fluid flow pathway is significantly non-rectilinear. The sampling efficiency of these particles approaches unity. Very coarse or dense particles have very high inertia and do not follow the fluid streamlines. These particles tend to travel in straight lines resulting in serious sampling errors (Nasr-El-Din et al., 1989a, 1991). Particle inertia causes significant sampling errors when the particle and fhdd velocities are not co-linear. A typical situation occurs when the sampling velocity
R. S. MACTAGGART et al.
(U,) is different from the local system velocity at the point of sampling (U,). This anisokinetic sampling causes the fluid streamlines ahead of the sample nozzle to either converge or diverge (Nasr-El-Din, 1989). The alignment of the sample tube with the upstream flow can also give rise to changes in the flow direction. Representative samples can be withdrawn parallel to a well-developed flow when sampling at the isokinetic velocity with a properly designed sample tube (Nasr-El-Din et al., 1984). In this situation, there is no change in the direction of the fluid flow ahead of the sample tube and, therefore, no sampling error occurs. When the sample tube is not aligned isoaxially with the fluid flow, the sampling efficiency is always less than unity (Lundgren et al., 1978). In addition, the mechanism of sample withdrawal is different from sampling parallel to the flow. The sample solids concentration increases with sampling velocity when the sample tube is orientated at right angles to the flow. The particle bouncing effect occurs when a particle moving in the fluid strikes a surface and loses some of its inertia. If this surface is near a sample tube opening, then the reduction of particle inertia can be sufficient to allow the particle to be withdrawn into the sample tube, where it otherwise would not have been. Sample solids concentrations are higher when particle bouncing is important than when it is not. The particle bouncing effect has been observed when sampling with blunt sample tubes parallel to the flow and with angled sample tubes projecting at right angles into the flow (Nasr-El-Din, 1989). These two devices cause disturbances in the fluid flow ahead of the sample tube. Particles with high inertia cannot deflect around the “blunt body” and strike the surface (particle bouncing effect). The use of tapered tubes, with tip angles of 18” or less, sampling parallel to the flow at the isokinetic velocity eliminated errors caused by the thick, blunt probe tip (Nasr-El-Din and Shook, 1985). Sample withdrawalfiom mixing tanks The flow fields in a mixing tank are very complex. These flow fields have been measured using laser
Doppler anemometry by several authors (Laufhutte and Mersmann, 1985; Costes and Couderc, 1988; Komori and Murakami, 1988). In addition, computational fluid dynamics have been used to predict the flow patterns in mixing vessels (Middleton et al., 1986; Gosman et al., 1989). The complicated flow patterns make it very difficult to obtain a representative sample from a slurry mixing tank. One of the few regions where the flow field is relatively well-defined is on the plane of a radial flow impeller. The centreline radial velocity (U,) ahead of a radial flow impeller was described by Rushton (1965) as an expanding jet from the impeller and is given by u
Particle density (kg/m’)
Liquid density (kg/m”). Velocity ratio (U&J,)
where B I is a constant dependent upon the number of impeller blades and the ratio of the impeller-to-tank diameter, N is the impeller speed, D is the impeller diameter and r is the distance from the centre of the impeller. Equation (3) is applicable only on the plane of a radial flow impeller between the impeller blade tip (r/R = O/2) and near the tank wall (r/R-0.95). It is not applicable to positions nearer to the mixing-tank wall due to the impinging flow upon the tank wall. Rushton (1965), Rehakova and Novosad (1971a, b) and Sharma and Das (1980) developed empirical or semi-empirical correlations to describe sample solids concentration as a function of sampling velocity. All three relations are valid only for fully suspended solids, sampled on the plane of a radial flow impeller midway between two baflles (0 = 45”). The range of parameters covered in each study is listed in Table I. Rushton (1965) developed the following correlations for a system of glass beads in water: C&a
= k,( U&J,)
where Us and Uo are the sampling velocity and local velocity, respectively; C, and C, are the sample solids concentration and bulk solids concentration in the mixing tank, respectively, and kI is equal to unity when r/R equals 0.95.
Table 1. Parameter ranges for mixing-tank
Axial location Radial location Angular location Impeller type Sample tube tip Bulk solids concentration (volume fraction) Rarticle size o&j
Rehakova and Novosad (1971a, b)
Impeller plane 0.95R e = 45” Radial flow Tapered 0.01-0.20
Impeller plane 0.85R 0 = 45” Radial flow Blunt 0.0018-0.0025
loo-250 2410 1000 Not available
180-900 1018-2665 780-1000 0.2-5.6
Sharma and Das (1980) Impeller
0.75R-OMR t7 = 45” Radial flow Tapered 0.0815-0.163 250 2600 0.8:y20
from a slurry mixing tank
Equations (4) and (5) show that the sample solids concentration decreases with sampling velocity, similar to sample withdrawal from an L-shaped probe sampler in a slurry pipeline. This is reasonable, since in both situations the sample tube is mounted isoaxially to the flow. Rushton found the sample solids concentration (C,) equalled the bulk solids concentration in the mixing tank (C,) at the isokinetic sampling velocity. The sample solids concentration from L-shaped probe samplers in a slurry pipeline equals the local solids concentration (C,) in a pipeline at the isokinetic velocity. Rushton had no independent method to determine whether C, is equal to C, on the radial impeller plane of a mixing tank. He, therefore, limited his comparisons to the bulk solids concentration (C,) in the tank. Rehakova and Novosad (1971a, b) undertook a theoretical analysis of sample withdrawal from the impeller plane of a mixing tank stirred with a radial flow impeller. The correlations covered only very dilute slurries (0.0018 < C, < 0.0025) and hence limit their applicability. Another complication was that Rehakova and Novosad (197la, b) found no effect of particle size on the sample solids concentration. This is in disagreement with Rushton’s work on mixing tanks and results from slurry pipelines (Nasr-El-Din, 1989). An effect of particle size on sample solids concentration is expected since the particle inertia depends on the square of the particle size [eq. (2)]. In a similar study, Sharma and Das (1980) correlated the sampling efficiency from the impeller plane of a mixing tank stirred with a radial flow impeller as follows:
(6) where A equals 0.028 for a 250 pm glass beads-water system and z I is the sample tube insertion length from the mixing-tank wall. Barresi and Baldi (1987a) showed that the shape of a sample tube can significantly affect the sample solids concentration obtained from a point 20 mm from the mixing-tank wall. It was dependent, however, upon the axial position of the sample tube relative to the flow in the tank. When the flow past the sample tube was at right angles to the sample tube face, a sample tube with an angled face opening into the flow, consistently gave a higher sample solids concentration than a straight-faced sample tube. At other locations, the sample solids concentration was similar for both sample tubes. Beker (1970) showed that the particle size distribution of a sample withdrawn from a mixing tank was a function of the sampling velocity. Nasr-El-Din and Shook (1985) and Nasr-El-Din et al. (1989b, 1991) found similar results when withdrawing samples from the side wall of slurry pipelines. As the sampling velocity decreases, fewer coarse particles are withdrawn into the sample tube due to their high inertia.
The sample particle size distribution, therefore, tends to be finer than the bulk solids particle size distribution. As can be seen, sampling errors associated with sampling withdrawal techniques have been recognized in the literature. However, no systematic study was conducted to examine the reliability of measuring solids concentration and particle size distribution from mixing tanks. The objectives of this study are to (1) determine how the solids concentration of a sample withdrawn from a slurry mixing tank varies with sample withdrawal velocity, sample tube geometry, spatial position in the mixing tank, particle size and bulk concentration; (2) examine if the bulk solids concentration in a mixing tank can be measured reliably, as predicted by Rirshton (1965) and Sharma and Das (1980); and (3) investigate how the particle size distribution of a sample withdrawn from a slurry mixing tank varies with sample withdrawal velocity, geometry and spatial position in the mixing tank. EXPERIMENTAL
Experiments were conducted in a flat-bottomed Plexiglas mixing tank shown in Fig. 1 and described in Table 2. The six-bladed radial-flow Rushton impeller was driven by a variable-speed motor, with the impeller rotational speed measured with an optical tachometer. The Rushton impeller was chosen for this study so that comparisons with previous work could be made. Nine sampling tubes were vertically positioned along the tank wall from near the bottom (z/H = 0.1) to near the top free liquid surface (z/H = 0.9) and midway between two baffles (0 = 45’). Sample tubes could be inserted flush with the tank wall (r/R = 1.0) to the impeller shaft (r/R = OM), except on the range was the impeller plane, where 0.33 =Gr/R G 1.0. Sample tubes of 2.97, 4.55 and 10.87 mm inside diameter with a relative wall thickness (S = sample tube wall thickness/sample tube inside radius) of 1.14, 0.40 and 0.17, respectively, were used. Both tapered (tip angle of y = 1%‘) and blunt (tip angle of y = 90°) stainless-steel sample tubes were employed. In addition, sample tubes with a flat face (/I = 90”) and faces at an angle (/l = 45”, /3 = 135”) were tested. Slurry samples were withdrawn from the mixing tank using a variable-speed peristaltic pump. The sample mass, volume and the sampling interval were
Table 2. Mixing-tank
Parameter Tank diameter (T) Liquid height (n/T) Number of [email protected]
Be& width (E/T) Radial flow impeller diameter (I)) Impeller
height above tank bottom
Radial impeller blade length (L/D) Radial impeller blade width (w/D)
0.292m 1.0 4 0.10 (h/T)
0.097 m 0.30
R. S. MACTAGGART
measured to determine the sample solids concentration and the sampling velocity (Kao and Kazanskij, 1979). The sample volume was limited to 300-400 ml so as not to significantly alter the bulk solids concentration in the mixing tank. Samples were returned to the tank before additional experiments were conducted. Sampling velocities ranged from 0.3 to 3.0 m/s at impeller speeds from 440 to 750 rpm. Sands with mean particle sizes of 82,410, 500 and 1000 pm and a mean solids density of 263 1 kg/m 3 were used. The fluid in all experiments was tap water. All of the tests were conducted in a batch mode. Samples were withdrawn 5 min after starting the mixer. The sample solids concentration did not change after longer periods of time, therefore, 5 min was a reasonable period for the system to reach steady state. The fluid temperature averaged 24.9”C with a standard deviation of 2.4”C. The reproducibility of measuring solids concentration by sample withdrawal was found to be within 4-5% of the value. The particle size distribution of the sample was determined in several tests by dry sieve analysis. The sand was returned to the mixing tank before further experiments were conducted.
RESULTS AND DISCUSSION
Effect of location and sample tube geometry on sample solids concentration The direction of flow in a mixing tank agitated with a radial flow impeller varies with the location in the mixing tank (Lauthutte and Mersmann, 1985; Middleton et al., 1986; Costes and Couderc, 1988; Gosman et al., 1989). Depending upon the position in the tank, the flow can be parallel to a sample tube, at
right angles to the tube or have both parallel and perpendicular components relative to the sample tube. These variations will affect the solids collection mechanism. Theoretical predictions of sample solids concentration from a mixing tank similar to those conducted with slurry pipelines (Nasr-El-Din et al., 1984) are very difficult. Theoretical models that predict solids concentration from a sampling probe require the velocity field ahead of the probe as an input. This is possible when sampling from slurry pipelines using an L-shaped sampling probe. However, in mixing tanks, predicting the Aow field ahead of a sampling probe is complex. First, fluid dynamics of turbulent flow in mixing tanks are not fully understood. Although various turbulence models have been employed in the literature, some adjustments for the boundary conditions at the impeller edge have been always necessary (Kresta and Wood, 1991). Also, complications to the fluid flow that result from the addition of solids would occur (Hetsroni, 1988). Moreover, dynamics of sample withdrawal together with fluid-particle dynamics within the tank renders the sample withdrawal a challenging problem to solve. In the present study however, comparison of the experimental results obtained from slurry pipelines, where the flow direction is well-defined, will be made. The effect of axial position and sampling velocity on sample solids concentration is shown in Fig. 2. On the impeller plane, where z/H equals 0.3, the sample solids concentration decreased with increasing sampling velocity whereas at z/H = 0.5, the sample solids concentration increased with sampling velocity. At a position between these two (z/H = 0.4), the sample concentration was insensitive to sampling velocity.
Sample withdrawalfrom a slurry mixing tank
Sampling Velocily (Us
Fig. 2. Effect of axial location on sample solids concentration (d,, = 410 pm, C, = 0.10. 4 = 4.6 mm, fi = 909, r/R
Sampling Velocity (U s
Fig. 3.Effect of sample tube shape on sample solids coneentration (d,, = 410 pm, C, = 0.30, 6 = 4.6 mm, z/H = 0.4, r/R = 0.1).
Different sampling mechanisms dominate depending upon the flow direction relative to the sample tube. On the impeller plane (z/H = 0.3) the flow is parallel to the sample tube, similar to sampling from a slurry pipeline with an L-shaped probe sampler. In both cases, the sample solids concentration decreases with sampling velocity. Above the impeller plane (z/H = OS), the flow is generally perpendicular to the sample tube, similar to wall sampling from a vertical slurry pipeline. In both situations, the sample solids concentration increases with sampling velocity. The largest differences in the sample solids concentration shown in Fig. 2 occur at low sampling velocities. The particle inertia causes the solids to travel generally in straight lines. At low sampling velocities, the straight particle trajectories do not correspond to the curved liquid flow. When sampling parallel to the flow (z/H = 0.3), the liquid flow paths diverge around the sample tube. The huge solids follow a straight trajectory, however, and a high sample solids concentration results. When sampling at a right angle to the flow, the liquid bends at a 90” angle into the sample tube. The solids do not completely follow the fluid streamlines and, hence, result in low sample solids concentration. At high sampling velocities, the drag force exerted by the fluid on the particles overcomes the particle inertia to some extent, and the sample concentration becomes less dependent on the sampling velocity. Different sample solids concentrations can be obtained from the same location in the mixing tank, depending upon the shape of the sample withdrawal probe (Barresi and Baldi, 1987a). This is confirmed in Fig. 3. Differences in sample solids concentration of more than 100% were obtained depending upon the sample tube shape and sampling velocity_ This occurred even though the local solids concentration in the tank was the same. The different sampling results of Fig. 3 are again due to the flow structure ahead of the sample tube, particle inertia and particle bouncing. The highest sample solids concentrations are obtained from the
0.24 0.18 -
Sempling Velcdty (Us
Fig. 4. Effect of sample tube diameter on sample solids concentration (d,, = 410 pm, CB = 0.30, p = 90”, z/H = 0.4, r/R = 1.0).
45” sampling tube, opening into the flow. The fluid streamlines into this sampling tube are distorted much less than are the streamlines into the 135” sampling tube, opening away from the flow. For the 45” tube, the particle trajectories correspond more closely to the liquid streamlines, resulting in more solids entering this sample tube and, hence, higher sample solids concentrations. In addition, solids impinge upon the tip of the 45” sample tube, lose inertia and are more easily withdrawn into the sample tube (particle bouncing effect). This occurs even at sampling velocities less than 0.5 m/s. Increasing the sampling velocity minimizes the effect of sample tube shape on sample solids concentration. The effect of sample tube diameter on sample solids concentration is shown in Fig. 4 for flow at a right angle to the sample tube. The sample solids concentration is over 20% lower for the 3.0 mm sampling tube than for the 10.9 mm tube. Similar variations were observed by Nasr-El-Din and Shook (1986) when sampling from the wall of a vertical slurry pipeline with different port diameters. Lower sample solids concentrations are obtained with small sampling tubes since at a given velocity past the tube, there is less time available for a particle to change direction and enter the small sample tube than to
R. S. MACTAGGART et al.
. . . . . . . . . . _. . . . . . ..*....._.
.... .. ..
and Daa (ls3q
I I L 1
WA .; El* 0.D 1 . I 3
SamplingVelocHyRatio (U s / U g 1
SamplingVelocityRatio &Is I U, )
enter the larger sample tube. Fewer particles, therefore, enter the small tube and lower sample solids concentrations are obtained with the small sample tube than with a larger tube. The effect of sample tube tip angle is shown in Fig. 5 for sample withdrawal on the impeiler plane of the mixing tank (flow parallel to the sample tube). The local velocity in the tank (V,) was determined with eq. (3). The tapered sample tube (y = 18”) gives slightly lower sample solids concentrations at all sampling velocities than does the blunt sample tube (y = 900). Also, the deviation of sample solids concentration from the bulk value is significantly larger at low sampling velocities than at higher sampling velocities. This indicates lower sampling errors when sampling at high velocities. The results shown so far, indicate that it is possible to obtain a wide variety of sample solids concentrations, depending upon the sample tube geometry, velocity and its position in the tank. It would be very difficult to determine a priori how to obtain the desired solids concentration by sample withdrawal. It is possible, however, to measure the bulk solids concentration (C,) in the mixing tank by sample withdrawal. In agreement with Rushton (1965) and Sharma and Das (1980), a tapered sample tube used on the plane of the radial flow impeller, at the isokinetic sampling velocity (Us/U, = 1) gives Cs = Ca for the 410 pm sand. The correlations of Rushton (1965) and Sharma and Das (1980) [eqs (4)-(6)] are shown for comparison in Fig. 5. These correlations adequately predict the sample solids concentration as a function of sampling velocity for samples obtained with the tapered sample tube. Data for the blunt sample tube were consistently higher than the predicted values due to the particle bouncing effect. This phenomenon was not considered by Rushton nor Sharma and Das. The effect of sample tube tip angle was also examined for the 1000 pm sand on the impeller plane as shown in Fig. 6. In this case, the solids were not fully
.. .... shannamd
Fig. 5. Effect of sample tube tip angle on sample solids concentration (medium sand, low concentration, d,., = 410 pm, C, = 0.10, qb= 4.6 mm, B = 90”, z/H = 0.3, r/R = 0.9).
Fig. 6. Effect of sample tube tip angle on sample solids d,, concentration (coarse sand, low concentration, = 1000 w, C, = 0.10,4 = 4.6 mm, p = 90’, z/H = 0.3, r/R - 0.9).
suspended (fully suspended being that no particle remains on the bottom of the tank longer than one second) and formed a sediment on the tank bottom. Consequently, the sample solids concentration measured was significantly lower than the bulk solids concentration at all sampling velocities. The Rushton (1965) and Sharma and Das (1980) correlations, therefore, cannot be used in situations where the solids in the mixing tank are not fully suspended. The bulk solids concentration in a mixing tank can be reliably measured by sample withdrawal on the plane of a radial flow impeller only when the solids are completely suspended. A similar situation is shown in Fig. 7 for the 410 pm sand at a bulk solids concentration of 0.30. The solids were not fully suspended in the vessel, with a significant portion of the solids being settled on the tank bottom. In this situation, the sample solids concentrations were significantly higher than the bulk solids concentration at most sampling velocities.
SamplingVelocity Ratlo (U s I U g )
Fig, 7. Effect of sample tube tip angle on sample solids concentration (medium sand, high concentration, d,o = 410 pm, C, = 030.4 = 4.6 mm, p = 90”, z/H = 0.3, t/R = 0.9).
Sample withdrawal from a slurry mixing tank Once again, the Rushton and Sharma and Das correlations do not predict the sample solids concentrations in a situation where the solids concentration displayed a strong variation in the vertical direction. More work needs to be conducted to ascertain whether in all systems the bulk solids concentration in a mixing tank can be measured by sample withdrawal on the plane of a radial flow impeller. For a suspension of alumina-type cracking catalyst (pp = 1600 kg/m ‘). Rushton (1965) found the sample solids concentration to be 90% of the bulk solids concentration at the isokinetic velocity. The data of Fig. 5 confirm that C, equals C, when U, equals U,, as shown by Rushton (1965) and Sharma and Das (1980). Figure 8 shows the effect of the mixer speed on the solids concentration obtained by sample withdrawal. Varying the mixer speed alters the velocity of the liquid and solids in the mixing tank. In addition, the distribution of the solids in the tank and, therefore, the local solids concentration changes with the mixer speed. The just suspended mixer speed (N,s, the point where all the solids are in motion off the tank bottom with some velocity), calculated from the Zweitering correlation (Zwietering, 1958), is 492 rpm for the suspension used in Fig. g. The difference in sample solids concentration obtained at 440 rpm and at 545 or 680 rpm is most likely due to the difference in the degree of solids suspension and, thus, the local solids concentration. The variation of sample solids concentration with sampling velocity was similar at the three impeller rotational speeds examined. The particle inertia has been shown to significantly affect the sampling efficiency from a slurry mixing tank. Particle size and bulk solids concentration strongly affect the particle inertia and should, therefore, affect the sampling efficiency. The effect of particle size on the sample solids concentration is shown in Fig. 9 for a single location (z/H = 0.5, r/R = 0.9) in the mixing tank. The difference between the sample solids concentration and the bulk solids concentration is progressively larger as the particle size increases. It is difficult, however, to determine whether
[ 103,@-j L!= 4???‘%
. . . . . .. . . . . . .. . . . . .. . . . . .
. . . . . . . . . . . _._... _. . . . .
1 2 Sampltng Velocity (Us
Sampling Velocky (U s
Fig. 9. Effect of particle size on sample concentration (C, = 0.10, $ = 4.6 mm-b = 90”, z/H = 0.5, r/R = 0.9).
Fig. 10. Effect of bulk solids concentration on sample concentration (d*,, = 410 m, $ = 4.6 mm, fi = 90’, z/H = 0.5, r/R = 0.9).
this difference is due to increasing sampling error with particle size or simply due to the fact that the concentration of the coarser sand particles will most likely display a strong variation in the vertical direction in the tank. This question can only be answered with an accurate, independent measure of the local solids concentration. The effect of the bulk solids concentration on the sample concentration is shown in Fig. 10 for z/H equal to 0.5 and r/R equal to 0.9. In this figure, the sample solids concentration (C,) is normalized by the bulk solids concentration in the mixing tank (C,). Nasr-El-Din (1989) found that sampling errors decrease with increasing solids concentration in a slurry pipeline due to an increase in the drag force exerted by the fluid on the particles, thus increasing the tendency for particles to follow the fluid flow into the sample tube. This appears to be the case in Fig. 10, as the sensitivity of the sample solids concentration to velocity decreases with increasing bulk solids concentration.
Fig. 8. E&et of mixer speed on sample concentration (d,e = 410 m c, = 0.10, 4 = 4.6 mm, /3 = W”, z/H = 0.3, rfR = 0.9).
Eflect of sampling velocity, geometry and location on sample particle size distribution The results presented thus far show that the sampling efficiency varies with the size of particles being
R. S. MACWGGART et al.
Table 3. Effect of sampling velocity, geometry and locatkn ok sample parttile
800 801 802 803
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.1 0.1 0.5 0.5 0.5 0.5
0.9 0.5 0.8 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4
:; 806 807 808 810 811 813 814 815 816
:I E E” 90” 0
90” 90” 90” 90” 135” 45” 135’ 45”
sampled. It is, therefore, reasonable to expect that the particle size distribution (PSD) of a sample withdrawn from a slurry with a range of particle sizes will not be the same as the local PSD in the tank. In addition, the PSD of a sample may vary with sampling technique and position in the tank. This was tested with a mixture of 10 ~01% fine 82 pm sand and 10 ~01% coarse 500 p sand. A summary of the various runs is given in Table 3. Inspection of runs 800 and 803 indicates that the reproducibility of the data was good for both the sample solids concentration and for the mean particle size (d,,). The effect of the sample tube shape on sample PSD is shown in Fig. 11 for runs 813 and 814. The samples were obtained at the same location and sampling velocity, but using different sample tubes (fi = 45”, 135”). The mean sample particle size (d,,)from the 45” sampler was more than double the mean particle size from the 135” sampler (95 pm vs 210 pm). This was again due to particle inertia, the flow field relative
Fig. 11. Effect of sample tube shape on sample particle size distribution (C, = 0.20, 4 = 4.6 mm, U, = 0.30 m/s, z/H = 0.4, r/R = 0.5).
1.61 1.58 1.63 1.62 ,1.59 1.60 1.58 0.66 2.79 0.32 2.52 0.29 0.30 2.85 2.76
(vo!un$actioa) 0.222 0.200 0.229 0.222 0.137 0.218 0.169 0.231 0.217 0.149 0.157 0.138 0.204 0.178 01184
(2) 280 160 355 290 90 220 110 330 330 100 110 95 210 125 135
to the sample tube and the particle bouncing effect. This effect was less significant at a higher sampling velocity as shown by runs 815 and 816 of Table 3. The variation of the sample particle size distribution with radial position is shown in Fig. 12 for runs 808 and 811. It appears that the coarse solids are less abundant near the centre of the mixing tank (r/R = 0.1)than at the wall of the tank (r/R = 1.O). Similar results are shown by runs 807 and 810 for a much lower sampling velocity. This seems reasonable as the centrifugal force generated by the impeller tends to concentrate the coarse particles near the wall of the tank. Previous sample withdrawal work has indicated that fine particles are more easily suspended than coarse particles in a mixing tank (McLaren et al., 1932; Oldshue, 1981). This appears to be confirmed in Fig. 13 where progressively finer samples are obtained above the impeller plane in the mixing tank. The coarsest sample was obtained at the plane of the
Fig. 12. Variation of particle size distribution with radial position (C, = 0.20, 4 = 4.6 mm, B = 90”, Usa2.6 m/s. zfH = 0.4).
Sample withdrawal from a slurry mixing tank
(1) The sample sohds concentration
from a slurry mixing tank can vary widely. Some salient features are outlined below:
(a) The sampling velocity significantly affects the sample solids concentration.
(W The shape of the sample tube face also
influences the sample solids concentration obtained. (4 The sample tube diameter significantly affects the sample solids concentration. Cd)When sampling parallel to the fluid flow, a sample tube with a blunt tip provides a higher sample solids concentration than a tapered sample tube. (4 The sample solids concentration is a strong function of the particle size and bulk solids concentration in the tank.
Fig. 13. Variation of particle size distribution with axial position (C, = 0.20, q5 = 4.6 mm, B = 90°, Us ~1.6 m/s, r/R = 1.0).
(2) The bulk solids concentration in a slurry mix-
Sample Solids Concentration (C, , Volume Fraction) Fig. 14. Sample fines fraction vs sample solids concentration.
impeller (z/H = 0.3) and a slightly finer sample was obtained below the impeller. As mentioned previously, these and earlier results from the literature, must be viewed with caution since the absolute particle size distribution measured by sample withdrawal is inaccurate due to sample withdrawal errors. However, the trend of generally decreasing solids size distribution with increasing height in the mixing tank is correct. Within the set of data examined in this study, it appears that the sample particle size distribution is a function of the sample solids concentration. This is shown in Fig. 14, where the mass fraction of fine sand in a sample is plotted against the sample solids concentration. The mean particle size of the bulk sand mixture was 150 pm, with the 150 pm fraction consisting of one-half coarse sand and one-half fine sand. The fraction of fine sand ( Y) is, therefore, defined as Y=
0.5M150 + MT
ing tank, stirred by a radial flow impeller, can be measured on the plane of the impeller, with a sample tube tapered to 18” or less operating at the isokinetic velocity, when the solids concentration does not display a strong dependence in the vertical direction. (3) The sample particle size distribution varies with the sample tube shape, sampling velocity and position in the mixing tank. With the bimodal sand used in this study, the fraction of fine sand in a sample is a function of the sample solids concentration.
It is practically impossible to obtain a representative measure of the local solids concentration in a slurry mixing tank by sample withdrawal. The variation of sample solids concentration with operating conditions (sampling velocity, tube geometry, particle size and concentration) can bc minimized, however, if fine solids are used and if the samples are withdrawn at as high a velocity as possible using a tapered sampling probe. To obtain a reliable measure of the local solids concentrations in a slurry mixing tank, it is necessary to use a technique other than sample withdrawal. This will be the subject of a future study. Acknowledgement-JHM Science and Engineering cial support.
The trend of Fig. 14 simply states that at low sample solids concentrations, the fraction of fine sand is high.
wishes to thank the Natural Council of Canada for their finan-
G d D
constant in eq. (6), dimensionless baffle width, m constant in eq. (3). dimensionless bulk particle concentration, volume fraction local particle concentration, volume fraction sample particle volume concentration, volume fraction particle diameter, m impeller diameter, m
R. S. MACTACSGART et al.
impeller height above tank bottom, m total liquid height in mixing tank, m constant in eqs (4) and (5), dimensionless particle inertia parameter, dimensionless impeller blade lengjh, m mass, kg
h H kt K L M N r
radial distance from the centre of mixing tank, m mixing-tank miting-tank
radius, m diameter, m
pipeline bulk velocity, m/s local upstream velocity, m/s sampling velocity, m/s impeller blade width, m fraction of fine sand in a sample, dimensionless axial height in mixing tank, m
sample tube insertion length from the mixing-tank wall [eq. (6)], m Greek
B : e p P 4
letters angle of sample tube face, degrees angle of sample tube tip, degrees relative sample tube wall thickness (wall thickness/sample tube inside radius) angular position in mixing tank, degrees viscosity, Pa s density, kg/m3 sample tube inside diameter, m
Subscripts JS L M P
total mean particle retained
150 pm sand fraction
liquid mixture particle size,
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