Solids Concentration Measurements of Floating Particles

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Ind. Eng. Chem. Res. 1999, 38, 2794-2802

Solids Concentration Measurements of Floating Particles Suspended in a Stirred Vessel Using Sample Withdrawal Techniques N. Kuzmanic´ * and D. Rusˇ ic´ Faculty of Chemical Technology, Department of Chemical Engineering, University of Split, Teslina 10/V, 21000 Split, Croatia

Axial and radial concentration profiles formed during suspension of floating particles in liquid under turbulent agitation are presented in this paper. Monomodal suspensions of polyethylene in tap water stirred by a pitched blade turbine have been studied. The local solids concentration from a slurry mixing tank have been measured by the withdrawal of a sample from the vessel. The dependence of the distribution of the monosized suspension on the sample tube design and sampling technique has been investigated. Three different probes have been tested. The obtained results have to be related to the direction of the local fluid velocity at the point of sampling in the mixing tank and the orientation of the suction section of the sampling tube. The effects of the particle size, bulk solids concentration, and impeller speed on the solids concentration profiles were also examined in detail. The solids concentration profiles in the mixing tank are described by means of the one-dimensional dispersion model. The applied model includes two parameters, Peb, modified Peclet number, based on solids upward velocity due to the buoyance force effect, and q representing the ratio of Peclet numbers for liquid and solid phases. These parameters were found to be a function of the hydrodynamic conditions in the mixing tank. Introduction The process of solid particles suspension in a liquid is a common unit operation of considerable importance in the chemical industry. In general, the suspension of solid particles in a liquid is used to promote a chemical reaction between the phases (operations where mass transfer is accompanied by chemical reaction), enhancing dissolution or getting some new products, as well as to obtain a uniform distribution of suspended solids in used systems. Crystallization, precipitation, dissolving processes, polymerization, biofermentation, and heterogeneous catalytic processes are only some instances of these operations.1,2 However, the density of solids, which should be dispersed, is sometimes considerably lower than that of the continuous phase (FL > Fp) which makes them float on the surface. In this case, the agitation process should incorporate floating solids into a liquid to produce homogeneous slurries, to acquire adequate reaction speed and particularly acceptable mixing power input which could be extremely high in such cases. Often, not all these operations are compatible, and some degree of optimization is necessary.3 Information on floating solids suspension is relatively scarce, particularly if compared to that on settling particle suspension irrespective of the similarity of these two processes.4-7 Namely, in both cases the particles could be fully suspended by the intensive turbulence, the initial location of particles being markedly different. For settling solids it is the bottom of the tank, a stable and firm surface wherefrom they are suspended by stirring. They are completely immersed in a liquid all the time during the suspension process. The initial location of floating solids is the surface of the liquid * To whom correspondence should be addressed. Tel.: ++38521-385 633. Fax: ++385-21-384 964. E-mail: kuzmanic@ ktf-split.hr.

phase into which they should be dispersed. Their buoyancy may result in the partial immersion of the solids in the dispersing liquid, thus reducing their availability to turbulent eddies. It has been established that in the baffled tanks the intensity of turbulence is primarily responsible for solids dispersion. When alternative baffle configurations are used instead, the liquid swirl in the tanks provides a mechanism for pulling down the floating particles into the vortex. There the liquid velocities are high enough to disperse the particles into the bulk.8 The solid particles usually require that they be suspended completely in order to attain the maximum interfacial area between the solid and liquid phase and avoid the accumulation of solids at any position of the vessel. The determination of the minimum mixing speed required for the complete solid suspension state is an essential goal. A too high impeller speed is usually undesirable since it increases the costs of the process while the additional mixing effect remains almost negligible. In some cases, apart from the minimum impeller speed it is necessary to know the distribution of solid particles within reactors, that means the concentration profiles of solids suspended in a stirred tank.9-11 Concentration profiles are closely related to the rate of solid/liquid chemical reactions as well as the masstransport rate in the system. Physical-chemical characteristics of final products of a lot of processes may greatly depend on the degree of homogenization of suspended particles during the process itself. A typical example is the polymerization process in suspension.12 In addition, the operation of continuous withdrawal of slurry from the mixing vessel also requires the knowledge of the concentration profiles of suspended solids. The relationship between the concentration of the discharge stream and mean bulk solids concentration in the mixing vessel is highly affected by the proper

10.1021/ie980592i CCC: $18.00 © 1999 American Chemical Society Published on Web 06/10/1999

Ind. Eng. Chem. Res., Vol. 38, No. 7, 1999 2795

location of the withdrawal point. It is well-known that homogeneous suspensions are not recommended in practice and that most operations limit to the state of complete or intermediate suspension. Logically, it may be assumed that in these states concentrations of suspended solids in a vessel are somewhere higher, somewhere lower, but somewhere also equal to mean bulk concentrations (so-called pseudo-homogeneous zones). These areas of pseudohomogeneity are of particular practical interest since they are most suitable as locations for an outlet stream of continuous systems.13-15 The work with suspended solids postulated the problem of introducing a measure of the suspension quality and make possible the comparison of concentration profiles of solids in geometrically similar systems. Einenkel and Mersmann16 suggested the use of variance as a measure of the suspension quality, whereas Bohnet and Niesmak17 employed the square root of this variance, that is, the standard deviation:

σ)

x ∑( ) 1

i

Cs

i i)1 C

2

-1

(1)

where “i” gives the level considered. Of course, the increase of the degree of homogenization of a suspended system is manifested as the reduction of standard deviation. Recently, a few studies have been reported on the suspension or drawdown of floating solids in liquids by agitation. The works on this subject, in the first place, point out the problems connected with mixing power consumption and with the most suitable geometric configuration of reactor systems for this kind of suspension. However, up to now the concentration profiles of suspended floating solids in the mixing vessel at the moment of their total dispersion in the continuous phase have not yet been systematically studied. With respect to its importance, the objectives of this study are to establish the influence of basic characteristics of suspended solids (particle size, bulk solids concentrations, etc.) as well as the effects of hydrodynamic conditions in the tank on the axial and radial distribution of floating solids in the mixing vessel at a complete suspension state. The experimental data have then been analyzed by a dispersion model. Experimental Studies The apparatus where the experiments were conducted is shown in Figure 1. The process of mixing the solid/liquid system was carried out in a flat-bottomed Plexiglas mixing vessel (T ) 0.32 m) with four baffles of standard size (B ) 0.1T) at 90°. The suspension was stirred by a 45° pitched four-blade turbine, pumping downward (D ) T/3), causing markedly axial flow in the tank. The turbine was driven by a variable-speed motor, with the impeller rotational speed measured with an optical tachometer. For safety, the agitation speed was checked with a stroboscope from time to time. The impeller clearance was always equal to H/3. Monosized suspensions of floating solids (polyethylene of high density; particle size within 135-450 µm; mean solids density of 835 kg‚m-3; bulk solids concentrations of 5-10 kg‚m-3) were tested. Tap water was used as a

Figure 1. Schematic diagram of experimental apparatus: (1) vessel; (2) stirrer; (3) pump; (4) solenoid valve; (5) timer; (6) variable-speed motor; (7) optical tachometer; (8) sieve and measuring cylinder; (9) sampling probe.

fluid in all experiments, and its height in the tank was equal to the tank diameter (H ) T). The local solids concentration in a mixing vessel was measured using sample withdrawal. Nine withdrawal tubes were positioned axially midway between two baffles. An L-shape sample tube, sample tube with a flat face, and the tube with a face at an angle of 45° were tested. They were 32 mm long (except in the case of the determination of radial concentrations profiles); all the probes are thin-walled (wall thickness/probe radius ) 0.125) and with an 8 mm inside diameter. Slurry was withdrawn from the mixing vessel at defined points at a constant flow rate, continuously circulated in the shown system, to be returned to the vessel at a point diametrically opposite the point at which it was removed. To minimize the effect of flow disturbance, the suspension was sent back into the vessel at the top of the mixing tank in the close vicinity of the baffle. Slurry samples were withdrawn from the tank using a variable speed pump. Samples were taken by diverting the flow into the collecting device by means of a solenoid valve controlled by a timer which allowed adaptation and control of the sampling time. The sample was then sieved, dried, and weighed to determine the sample solids concentration. In all runs, with exception of those where specifically required, the adopted stirrer velocity was high enough to ensure the complete suspension state (NJS ) 730 rpm) which was defined according to the Joosten visual criterion.4 In this case the minimum impeller speed to achieve a complete suspension state was defined as the speed at which the stagnant zones of floating solids at the liquid surface had just disappeared. Preliminary examination showed that the sampling time of 10 s could be generally adopted as optimum since the sample solids concentration did not change after longer periods of time. Sampling was started when the system reached steady state. Steady state was determined by the frequent measurement of the local concentration by the wall sampling. Hydrodynamics of the

2796 Ind. Eng. Chem. Res., Vol. 38, No. 7, 1999

system and the sampling conditions were kept the same (the same probe, sampling location, and withdrawal velocity). For the studied system, the changes of the local concentrations were almost negligible, 6 min after the beginning of agitation. Discussion Experiments were carried out in a reactor system usually recommended for settling solids suspension, even though some authors for suspension of floating solids recommend a mixing tank with partial baffles and an axial stirrer of nonstandard dimensions (T/D > 3).4 However, it should be kept in mind that in practice both settling and floating particles are frequently present in the system, and both should be suspended under the same conditions. Therefore, this study would be taken as preliminary in this respect. In addition, if a tank with partial baffles was used, a developed vortex would produce surface aeration, which significantly disturbs the process of continuous withdrawal of suspension from the vessel. Sampling of suspensions from a slurry-handling apparatus is of great interest in the chemical industry. Because of its simplicity and versatility, sampling is widely employed in order to measure the particle size distribution and to obtain information on local solids concentrations in mechanically stirred vessels.18,19 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. The sampling efficiency is usually defined as the ratio Cs/C0, where Cs is the solids concentration of the withdrawn sample and C0 is the local solids concentration in the mixing tank at the point of sampling. A sampling efficiency of unity implies ideal sampling. In this work effectiveness of sampling devices is defined h where C h is the mean concentration in the vessel. as Cs/C Of course, this parameter is affected by the variation of the solid concentration in the vessel as a function of the spatial coordinates; however, at constant C h and stirrer speed C/C h can be used to analyze the results of the wall-sampling procedure. As already pointed out, in the present paper local concentrations of suspended solids at different axial and radial positions were measured by the method of wall sampling. However, some earlier studies connected with the problem of wall sampling of settling and floating suspended particles from mixing tanks showed that representative samples are extremely difficult to obtain.20-24 There is a series of factors affecting sampling efficiency, Cs/C h , taking us apart from the sampling efficiency of unity. The samh is primarily the function of inertia pling efficiency Cs/C differences between the fluid and particles, sampling device geometry, disturbances of the fluid flow ahead of the sampler, and the particles response to these disturbances, as well as the sampling technique (withdrawal velocity, position of sampling in the mixing tank). Therefore, the first part of the present paper deals with the effects of the sample tube design and sample withdrawal technique on the distribution of floating suspended solids in the mixing tank. To establish axial solids distribution, local concentrations were measured at nine vertical positions along the tank wall, but the measurements in the immediate vicinity to the liquid surface were not included in the analysis because of the results’ nonreproducibility. With

Figure 2. Effect of the sampling tube shape and sampling velocity on the axial solids concentration profiles in the tank (C h ) 5 kg‚m-3; d ) 135 µm; N ) NJS ) 730 rpm).

respect to the mixing vessel size the number of sampling positions gives quite satisfactory insight into the formed axial concentration profiles in the tank. It should be mentioned that literature points out that a representative sample of suspended solids may be obtained at isokinetic sampling conditions (vs/v0 ) 1). Therefore, the fluid velocity field in the tank was determined in the previous paper24 as well as the effect of the withdrawal velocity, vs/v0, on sample concentrah . The results showed that the sample contions, Cs/C centration increased with the increase of sampling velocity, but already at vs/v0 ) 7 the sample concentration reached a constant value. Therefore, this paper presents axial concentration profiles obtained by all three sample tube types at two markedly different sampling velocities (Figure 2). This means the samplings were performed at very low sampling velocities (e.g., vs/v0 ratios were considerably lower than 1 for all the sampling points), and also at sampling velocities when vs/v0 ratios markedly exceeded the value of 1 for all the sampling points. At a lower sampling velocity, that is, at vs ) 0.08 m‚s-1 (Figure 2a), the effect of the sample tube shape


on the solids distribution is evident, particularly above the impeller zone. Since samplings were carried out under identical experimental conditions, sample tube geometry was concluded to affect the floating solids distribution at lower sampling velocities. The sampling efficiency above the impeller plane is highest when using an L-shaped sampling probe and lowest when using a straight probe. To account for obtained results, it is necessary to relate the direction of the fluid flow at the point of sampling in the mixing tank to the sample tube shape. When using an L-shaped sampling probe, the local fluid velocity vector is perpendicular to the suction section of the probe, causing negligible distortion of the fluid streamlines ahead of the probe. The particle trajectories, characterized otherwise by very low inertia, correspond more closely to the fluid streamlines, resulting in higher sample solids concentrations. When sampling using a straight probe, the particles should bend at a 90° angle into the sample tube since in this case the velocity vector direction is parallel to the suction section of the withdrawal probe. However, the shape of this sampling probe obviously causes considerable deviations of fluid streamlines in its close vicinity. Particles respond to these deviations by themselves reorienting and hence, fewer particles enter the tube. Also, this is the case for low withdrawal velocity so that the sampling flow effect on particle flow is considerably lower relative to high-speed axial flow in the mixing vessel. Strong fluid flow will enable a solid particle to continue to follow the fluid flow, and hence, a lower sample solids concentrations result. The third type of the probe (e.g., the 45° tube) is a kind of combination of the above two mechanisms of sampling, proved with the obtained results. However, on the plane below the impeller the sampling efficiency is almost insensitive to sample tube geometry at the same sampling velocity. This may be related to the circulating fluid flow in the mixing tank and also to suspended particles inertia. The used axial impeller pumps the fluid downward. Fluid streamlines strike the bottom and suddenly change direction, ascending along the tank wall to the surface. That is, close to the bottom fluid velocity direction changes are rather pronounced and considerably affect particles movement. Obviously, a pronounced degree of turbulence in this zone diminishes the sample tube shape effect. The effect of the sample tube shape on the sample solids concentration at a higher withdrawal velocity, vs ) 0.38 m‚s-1, that is, sampling velocity significantly higher than local fluid velocities in the sampling points, was also examined as shown in Figure 2b. The differences in the concentrations ratio Cs/C h obtained with different types of sampling devices all along the axial axis are so small that it may be stated that at higher vs the sample probe shape does not affect the sampling efficiency. The concentration profiles indicated that the solids distribution was nearly uniform in the vertical direction. There were no significant concentration variations in the radial direction as is frequently the case with profiles of settling suspended particles at the state of complete suspension.10,25-27 The given results show that in no case does the sampling efficiency, Cs/C h , exceed the value of unity. This is clearly indicative of the fact that it is impossible to

obtain representative samples by sample withdrawal all along the axial axis, irrespective of the sampling velocity. Working with suspended solids sampling Naser-ElDin18 found that three main factors can cause this ratio to deviate from unity (i.e., ideal sampling) for liquidsolid systems, including particle inertia, particle bouncing, and flow structure ahead of the sampler. Particle inertia is a major source of sampling errors when the densities of the two phases are significantly different. In such a case, because particle inertia is different from that of an equivalent volume of fluid, particle motion does not follow the distorted fluid streamlines. Consequently, sample solids concentration and composition will be significantly different from those in the tank. However, in the case of floating solids, characterized by very low inertia, they may be assumed to correspond more closely to the fluid streamlines, even in cases of rather pronounced liquid flow deviation. Thus, lower sample solids concentrations cannot be due to particle inertia. The bouncing effect is the next possible cause of disturbances during solid suspension. Because of specific hydrodynamic conditions during sampling, a part of the solids hit the probe wall, lose some of their inertia, and are more easily withdrawn into the sample tube. However, Naser-El-Din18 and Kuzmanic´ and Kessler24 found that the wall thickness could affect exclusively an increase of sample solids concentration and in no case its decrease. It is quite obvious that lower sample solids concentrations result from complex hydrodynamics in the mixing tank. In the first place, sample tube geometry and sample withdrawal velocity cause considerable deviations of liquid flow in the close vicinity of the sample tube. Further, it must be kept in mind that in an agitated tank the local velocity of the fluid changes with time quickly and noticeably in direction and modulus, because of marked turbulence, and periodical flow induced by stirrer blades. Probably, fluctuating velocities are of the same order of magnitude of the mean velocity. As a consequence the orientation of solid trajectories is less precise and quite complex. Therefore, it may be concluded that lower sample solids concentrations than the expected result from the complex hydrodynamics in the mixing tank. In addition, the analysis of the profiles given in Figure 2 shows an L-shaped probe sampler to be least sensitive to withdrawal velocity changes. Obviously, this probe shape is best suitable for axial fluid flow since the trajectories of suspended solids and fluid streamlines are distorted much less ahead of the suction section of the sampling probe. In fact, the local hydrodynamical changes due to the withdrawal flow and probe shape are least pronounced in this case. To confirm the reports on the effects of the sampling velocity on the axial solids concentration profiles, the local solids concentrations were determined using a simple sampling probe at four different withdrawal velocities. Figure 3 shows that the sampling efficiency increases with increasing vs all along the axial axis, but only up to a defined value of this velocity. For the vs > 0.38 m‚s-1 this impact is very low, almost negligible. Applying eq 1, the values of standard deviation σ were determined for obtained axial concentration profiles. It gives an insight into the degree of homogenization of a suspended system (Table 1). The given results show the differences in the quality


Figure 3. Influence of the withdrawal velocity on the axial concentration profiles (C h ) 5 kg‚m-3; d ) 135 µm; N ) NJS ) 730 rpm).

Figure 4. Axial concentration profiles as a function of the bulk solids concentrations (d ) 135 µm; N ) NJS ) 730 rpm; vs ) 0.38 m‚s-1).

Table 1. Comparison of Standard Deviation for Obtained Axial Concentration Profiles (C h ) 5 kg‚m-3; d ) 135 µm)

function of the mean bulk solids concentration. The examined bulk concentrations were in the range of 5, 7.5, and 10 kg‚m-3. The concentration profiles were always analyzed at the same impeller speed since the preliminary studies have shown quite a negligible effect of bulk concentrations on the minimum stirrer speed required for complete suspension of the floating solids.4,29 It is evident that the degree of homogenization increases if the bulk solids concentration increases. The particle distribution is almost uniform vertically with no significant concentration variations in the radial direction. Thus, the sample solids concentrations increase all along the axial axis with increasing solids concentration in a slurry tank. In fact, because of an increase in the drag force exerted by the fluid on the particles, the tendency for particles to follow the fluid flow into the sample device increased. In the case of C h h ratio value approaches unity ) 10 kg‚m-3, the Cs/C which points to the fact that at higher solids concentrations in the system it is possible to obtain a representative sample or to determine the so-called pseudohomogeneous areas. Further, the effects of the floating particle size on the axial concentration profiles in the mixing tank were examined. As shown by Figure 5, within the studied size range, the particle diameter does not affect the particles distribution. Some differences in the sampling efficiency were recorded only in the close vicinity of the liquid surface. However, the illogical results obtained do not allow any definite conclusion. It may be eventually said that because of the decrease of the axial fluid velocity and change of its direction in this zone, the buoyance force is more pronounced relative to the other parts of the mixing tank. More reliable conclusions call for further studies of solids in the size range of which is considerably wider. It was also obvious that the local concentrations were relatively lower in the upper part of the tank. It was not to be expected with respect to the initial position of the floating particles before the suspension process. This may be due to the fluid flow in the tank generated by the impeller type used. Axialflow impeller discharge flows toward the base of the vessel. As the discharge flow impinges on the vessel base, it is redirected in the outward radial direction

standard deviation, σ sampling probe

vs ) 0.08 m‚s-1

vs ) 0.38 m‚s-1

straight tube 45° edged L-shaped

0.141 0.165 0.100

0.107 0.095 0.083

of distribution for different probes and different sampling velocities, directly confirming the effects of these two parameters on solids concentrations. The σ values point to a considerably high degree of system homogenization at NJS, which was not the case with settling suspended particles in the mixing vessel. In fact, working with settling solids Penaz et al.28 found out that solids distribution in the mixing tank depended on the operating conditions, that is, on the geometry of the system, and slurry hydrodynamics. Bohnet et al.,17 in particular, observed the effects of particle size, their settling velocity, and density differences between the solid and liquid phases on the degree of homogenization. A higher degree of homogenization requires higher power consumption, these authors concluded. In their studies the degree of homogenization of a suspended system increases with increasing impeller speed, but only up to a defined extent. Any further velocity increase may impair the distribution of solids in the tank Because of the centrifugal force generated by the impeller. Machon et al.,9 who dealt with the same problems, reported mean bulk solids concentration affects on the degree of homogenization. This conclusion is partly in disagreement with that of Baldi et al.13 who stated that the concentration profiles are dependent on the settling particle sizes but not on their concentration. However, it should be pointed out that Baldi et al. worked with suspensions of relatively low bulk concentrations. With respect to the mentioned results connected with settling solids concentration profiles in the mixing tank, the effects of mean solids concentrations, suspended particle sizes, and impeller speeds on floating particles distribution in the mixing vessel was also examined. The results shown in Figure 4 point to the fact that axial concentration profiles of floating solids are a


Figure 5. Vertical solids concentration profiles in the tank as a function of the particle size (C h ) 5 kg‚m-3; vs ) 0.38 m‚s-1; N ) NJS ) 730 rpm).

which leads to suspension of the solid particles at the periphery of the vessel. However, if an axial-flow impeller is sufficiently far from the vessel base, its discharge flow will impinge on the vessel wall rather than on the base. This leads to two flow loops in the vessel. The primary flow loop heads up the wall and then returns to the impeller at or above the impeller plane. The secondary flow loop is characterized by low velocity, radially inward flow at the base of the vessel which returns to the impeller via upflow at the center of the vessel. This flow pattern is referred to as reversed flow.30 Such a flow causes a higher quantity of suspended particles in the lower part of the tank, which directly affects the sample concentration. Obviously, further studies are called for using more sophisticated techniques, such as laser Doppler anemometry (LDA) or digital particle image velocimetry (DPIV). This would confirm the presence of the reversed flow, and the impact of the impeller off-bottom clearance on suspended particles distribution in the mixing vessel would be studied in detail. The concentration profiles of floating solids vary, as expected, with the variations of mixing speed. The results shown in Figure 6 represent axial profiles of solids of 135 µm at four different impeller speeds. The N/NJS were as follows: 0.8, 1, 1.1, and 1.2. The sampling efficiency along the axial axis is considerably increased with the increase of the rotational speed. It may be observed that the increase of mixing speed affects the degree of suspension homogenization, but only up to a defined value. No further increase in rpm to more than N > 803 rpm stirrer speed will affect the degree of homogenization. Identical results were obtained with other probe types as well. However, it should be kept in mind that the sampling efficiency values obtained at different impeller speeds are not comparable with respect to the different local solids concentrations in the mixing vessel at different impeller speeds. The effect of mentioned parameters on radial solids distribution profiles in the slurry was also observed (Figures 7 and 8). The given diagrams show some differences in local solids concentrations along the radial axis. The local concentrations increase with the distance of the sampling location from the vessel wall. With

Figure 6. Effect of the impeller speed on the vertical solids concentration profiles (C h ) 5 kg‚m-3; d ) 135 µm; vs ) 0.38 m‚s-1).

Figure 7. Radial concentration profiles of the suspended solids in the tank as a function of the withdrawal velocity (L-sampling device; C h ) 5 kg‚m-3; d ) 135 µm; N ) NJS ) 730 rpm; O, vs ) 0.08 m‚s-1; 9, vs ) 0.38 m‚s-1).

respect to probe shapes and fluid flow oriented downward in the vicinity of the impeller shaft, these results are not expected. This is probably due to the lower local velocity of the downward fluid flow. Thus, irrespective of the fact that the sampling velocity remains constant, the vs/v0 ratio changes. This change may be the cause of an increase of local concentration in the vicinity of the impeller shaft.


∂C ∂C ∂2C ∂C )-v + vb + Dp 2 ∂t ∂z ∂z ∂z

Figure 8. Influence of the impeller speed on the radial concentration profiles (45° angle sampling probe; C h ) 5 kg‚m-3; d ) 135 µm; N ) NJS ) 730 rpm; vs ) 0.38 m‚s-1).

(2)

where v is the liquid superficial velocity, vb is the solids upward velocity due to buoyance force, and Dp is the dispersion coefficient for the solid phase. The z-axis is directed upward, that is, z ) 0 and z ) H correspond to the bottom and the top of the vessel, respectively. The axial stirrer used in the present study pumps the liquid flow downward. After impinging on the tank bottom, the liquid flow changes direction, ascending along the tank wall to the liquid surface, that is, to the initial position of the floating solids. Fluid flow then returns to the impeller plane along the impeller shift. And just this part of the flow, from the liquid surface to the impeller plane, is essential for the suspension of these particles. This means that in the applied coordinate system fluid velocity will have the negative sign. Although the solids upward velocity may depend on the solids concentration and impeller speed, vb was assumed to be constant in the present model. It was also assumed that the solids phase dispersion coefficient is equal to the liquid phase dispersion coefficient, believing that under mentioned conditions suspended particles, much smaller than the macroscale of turbulence, do follow the fluid very closely and consequently the particle eddy diffusivity tends to be equal to the liquid eddy diffusivity.36 The boundary conditions are derived from the massbalance equation for the volume elements at the top and bottom of the vessel:

Dp

dC ) v(CT - C) + vbC at z ) H (top of the tank) dz (3) Dp

dC ) vbCB at z ) 0 (bottom of the tank) (4) dz

Under steady-state conditions subject to the boundary conditions, eqs 3 and 4, the dimensionless solution of eq 2 is

Θ)

q q-1

(5)

Figure 9. Schematic diagram of the dispersion model.

In addition, Figure 8 also shows that the sampling velocity effect and impeller speed effect on radial solids h is distribution are rather low. In this case also, Cs/C lower than unity and already mentioned explanations could be given. Obviously, the method of continuous sampling gives no reliable measures of local solids concentrations because of the complex hydrodynamics, but it reveals the actual situation in the mixing tank. Still, to obtain reliable measure of the local solids concentrations in a mixing tank with suspended floating particles, it is necessery to use a technique other than sample withdrawal, which would be the subject of further studies. To interpret mathematically the axial solids distribution profiles in the mixing vessel, the one-dimensional dispersion model was adopted (Figure 9). Up to now this model has been applied successfully to interpret the behavior of the solid phase in the mixing vessels and bubble columns.31-35 According to this model, the continuity equation for the solid phase in a differential element of the tank volume can be written as

where Peb is the modified Peclet number with the velocity of the particles orientated upward by the buoyance force, Peb ) vbH/Dp; q is the ratio of Peclet numbers based on fluid velocity and particle upward velocity. This means that q ) Pef/Peb ) v/vb. Concentration profiles calculated by eq 5 are plotted for comparison. The lines in these figures were computed by adjusting the model parameters Peb and q, so as to agree with the experimental profiles. As can be seen, the calculated profiles agree fairly well with the experimental data (Figure 10). Therefore, the experimental obtained concentration profiles can be described satisfactorily by the present dispersion model. Axial concentration profiles in a slurry mixing vessel largely depend on the operating variables, in the first place on the number of impeller speeds. The effect of this variable on the concentration profiles can be interpreted approximately by the present model (Figure 11). Figure 11 shows the effects of the impeller speed on the model parameters Peb and q. With an increase of the impeller speed, Peb decreases and assumes a defined


Figure 10. Axial concentration profiles of solid particles at different impeller speeds; 45° angle sampling probe; C h ) 5 kg‚m-3; d ) 135 µm; vs ) 0.38 m‚s-1; (points, experimental data; solid line, theoretical data).

tank are measured by the wall-sampling technique, the concentration profiles are dependent on both the sampling probe geometry and the sampling velocity. The sampling tube shape effect is particularly pronounced at lower sampling velocities. By an increase of the sampling velocity, this effect is diminished so that at vs > 0.38 m‚s-1 it may be concluded that sampling probe geometry has practically no effect on the sampling efficiency. Such a behavior is closely related to low particle inertia and hydrodynamic conditions in the vessel. Irrespective of the sampling velocity, it may be concluded that the studied system has a high degree of homogenization. The axial concentration profiles do not display significant variations of local solids concentrations in the radial direction. Axial concentration profiles of suspended solids are also a function of mean bulk solids concentration and impeller speed, whereas the particle size within the examined size range does not affect the distribution of the suspended solids. No significant dependence of radial profiles on operating conditions were recorded. To interpret mathematically the axial concentration profiles of the suspended floating solids in the mixing vessel, the one-dimensional dispersion model adopted to floating solids was used. It is characterized by two parameterssthe modified Peclet number with upward particle velocity and q which represents the ratio of fluid velocity to the mentioned particle velocity. These two parameters render possible a very good mathematical description of the suspension, experimentally determined by the sampling technique. The values of the model parameters are a function of the established hydrodynamic conditions in the system. Irrespective of the mentioned difficulties in floating suspended solids sampling, the results give an insight and a series of important information on the distribution of these kinds of particles in the mixing vessel. The results may be particularly useful for resolving the problems of continuous withdrawal and wall sampling from mixing tanks, very frequent in industrial practice. To get a more precise measure of the local concentrations of floating solids in a slurry, mixing requires the use of some other nonintrusive techniques, if possible, which will be the subject of our further research. Nomenclature

Figure 11. Correlation of model parameters, Peb and Pef/Peb.

asymptotic value. Namely, with an increase of the impeller speed the fluid velocity in the mixing vessel is increased, diminishing the buoyance force effect and consequently the Peb value. The other models’ parameter q increases if the impeller speed increases. This can be easily understood since liquid velocity, which is the numerator of the mentioned parameter, is increased with the impeller speed. Conclusion The axial and radial distribution of floating solids in a mixing tank in a complete suspension state was measured. If the local solids concentrations in a mixing

B ) baffle width, m C ) solids concentration, kg‚m-3 CB ) solids concentration at bottom of tank, kg‚m-3 CT ) solids concentration at top of tank, kg‚m-3 Cs ) sample solids concentration, kg‚m-3 C h ) mean bulk solids concentration, kg‚m-3 d ) particle diameter, m D ) agitator diameter, m Dp ) dispersion coefficient for solid particles, m2‚s-1 FrJS ) Froude number to just-suspended particles off the surface vessel H ) height of liquid from the bottom of the vessel, m N ) impeller speed, rpm NJS ) impeller speed to just-suspended particles, rpm Peb ) Peclet number based on upward velocity of solid particles ()vbH/Dp) Pef ) Peclet number based on fluid velocity ()vH/Dp) r ) length of the sampling probe, m R ) diameter of the tank, m q ) Pef/Peb ) v/vb t ) time, s T ) vessel diameter, m v ) superfical velocity of liquid phase, m‚s-1

2802 Ind. Eng. Chem. Res., Vol. 38, No. 7, 1999 vb ) hindered upward velocity of solid particles, m‚s-1 vs ) sampling velocity, m‚s-1 v0 ) local fluid velocity, m‚s-1 z ) axial sampling position from the bottom, m ) dimensionless height ()z/H) µL ) viscosity of liquid, N‚s‚m-2 Θ ) dimensionless solids concentration FL ) density of liquid, kg‚m-3 Fp ) density of particle, kg‚m-3 ∆F ) density difference between liquid and solid, kg‚m-3 σ ) standard deviation

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Received for review September 14, 1998 Revised manuscript received January 26, 1999 Accepted March 3, 1999 IE980592I

Solids Concentration Measurements of Floating Particles

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