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Solids Suspension in Three-Phase Stirred Tanks
0263±8762/00/$10.00+0.00 q Institution of Chemical Engineers Trans IChemE, Vol 78, Part A, April 2000
SOLIDS SUSPENSION IN THREE-PHASE STIRRED TANKS G. MICALE, V. CARRARA, F. GRISAFI and A. BRUCATO Dipartimento di Ingegneria Chimica dei Processi e dei Materiali (DICPM), Palermo, Italy
T
he `pressure gauge technique’ recently developed for the quantitative assessment of the mass of solid particles suspended at any agitation speed is extended to the case of three-phase stirred tanks. As a result, curves of the fraction of suspended solids at various gas ¯ow rates, versus agitator speed, are presented. The dif®culties involved in the extension of the technique to three phase systems are addressed and discussed. The experimental results show that the presence of the gas phase causes a signi®cant increase of the agitation speed required to attain complete suspension of solids and lowers the degree of suspension at all agitation speeds below it. The experimental data obtained are shown to be well ®tted by the Weibull functions previously adopted for two-phase systems. Correlations for the estimation of the in¯uence of particle size, particle concentration and gas ¯ow rate on the suspension degree are presented and discussed. Finally, the fractional suspension dependence on particle size is found to be very different when very large particles are dealt with. Keywords: stirred tanks; solids suspension; three-phase systems; pressure gauge technique
INTRODUCTION
relied on the visual assessment of Njs according to the method ®rst introduced by Zwietering7 which, apart from being subject to substantial uncertainties, gives no information on what happens at agitation speeds below Njs. In this respect, a novel technique (`Pressure Gauge Technique’, PGT) has been recently introduced for the quantitative assessment of the mass of solid particles suspended at any agitation speed in solid-liquid systems (Brucato et al.8 ). The technique is based on the measurement of the pressure increase on the tank bottom due to the presence of suspended solid particles. As ¯uids of different densities exert different static heads at the vessel bottom, when solid particles are lifted in the liquid the increase of the ¯uid effective density can be measured at the tank bottom by means of a pressure gauge. Being able to characterize all partial suspension conditions, an objective assessment of complete suspension conditions is also allowed by this technique. In the present work the PGT is extended to the case of three phase systems, resulting in the ®rst experimental data on the way the amount of suspended particles is affected by aeration, at all agitation speeds, in three-phase stirred tanks.
A number of important catalytic multiphase processes involves the suspension of solid particles in gas-liquid dispersions inside mechanically stirred tanks. The suspension of solid particles in a liquid has been extensively studied over the last forty years. Most of the research efforts have been devoted to the determination of the minimum agitator speed, Njs, required to attain the suspension of all particles, while only recently attention has been paid to the analysis of the industrially important partial suspension conditions. As regards three phase stirred systems, there is a quite wide lack of information on the complex interactions among solid particles, gas bubbles and the liquid phase at the various agitation regimes, that causes severe uncertainties in the design and development of these apparatuses. As a matter of fact, such interactions are of paramount importance since the presence of the gas phase largely affects 1±6 the values of Njs and the presence of the solid phase modi®es the values of the minimum agitation speed (Ncd) required for an effective and complete dispersion in the vessel, as pointed out for instance by Frijlink et al.1. For each given system, the limiting factor governing the power requirements of the agitated vessel may be either gas dispersion or solids suspension, as pointed out by Chapman et al.2 who observed that there is a critical particle density (relative to the liquid medium) that discriminates between the two circumstances. A number of other studies on three-phase systems can be found in the literature, some of which, including those by Nienow et al.6, Chapman et al.3, Frijlink et al.1, and Pantula and Ahmed4 address the problem of correlating the agitation speed required to attain complete solids suspension under gassed conditions, but the ®eld appears to be still open to improvements. For instance, all of these works
EXPERIMENTAL In Figure 1, the stirred vessel employed for the experimentation is shown. It consists of a ¯at bottomed, fully baf¯ed, transparent vessel (T = 0.19 m), stirred by a six blade Rushton turbine (D = T /2 = 0.095 m) set at a distance from the vessel bottom equal to T/3. A measured mass of solid particles was ®rst introduced into the vessel and then the liquid phase added up to a height H = T = 0.19 m. The gas phase was fed by means of an open tube sparger placed at T/4 from the vessel bottom. 319
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Figure 1. R19 experimental set-up: (A) power supply; (B) DC motor; (C) optical tachometer; (E) connecting hole; (F) gas ¯owmeter; (P) inclined piezometer.
The impeller shaft was driven by a DC electric motor provided with a speed control loop, while an optical tachometer was used to independently measure the impeller speed. A pressure gauge connected to a speci®c point of the vessel bottom allowed pressure readings to be taken. The point selected for this purpose was placed at a radial location midway between the axis and the side wall and at 458 between subsequent baf¯es. A hole in the vessel bottom transmitted the value of the local pressure to a dead chamber to which the pressure gauge (a simple inclined manometer) was connected. The technique, however, does not require the presence of the dead volume and would have worked in the same way had the pressure gauge been directly connected to the hole in the vessel bottom. The solid phase used for the experimental runs consisted of silica particles with a measured density »s of 2500kg m ±3 in three different cuts (180±212, 250±300, 1680±2000 mm). The concentration of solid particles in the slurry was varied from 3.77 to 33.75 per cent by weight. The gas fed to the vessels was compressed air and its ¯ow rate was measured by a digital mass ¯ow-meter. The gas ¯ow rate was varied from 0.00 to 2.23 vvm. The liquid phase was de-ionized water in all experimental runs.
PRESSURE GAUGE TECHNIQUE The Pressure Gauge Technique (PGT)8 is based on the measurement of the pressure increase on the tank bottom due to the suspension of solid particles. In the following, the relationship between the pressure increase and the mass of suspended particles will be derived by simply performing a macroscopic force balance on the system, which is de®ned as the volume occupied by the liquid and particles introduced in the vessel. With reference to Figure 2, where a stirred tank containing a mass of solids Mt and a mass of liquid Ml is shown, a force balance in the vertical direction performed
Figure 2. Solid particle distribution: (a) at N = 0; (b) at N > 0.
on the system leads to: Ftot = (Ml + Mt )g
(1)
where Ftot is the total force acting on the vessel bottom. This total force may be split into two contributions. The ®rst one is due to the direct contact of solid particles with the vessel bottom. At no agitation conditions (N = 0, Figure 2a), this is equal to the apparent weight of the total mass of solids introduced in the vessel, as in such conditions all particles lay on the vessel bottom. In formula: Fdirect contact = Mt 1 ê
»l g »s
(for N = 0)
(2)
In still conditions (N = 0), the second contribution is due to the pressure exerted by a height of liquid column equal to the total height H reached by the amounts of liquid and solid phases added into the vessel. Its expression can be obtained by simply subtracting the Fdirect contact contribution from the total force Ftot. Fpressure =
(Ml + Mt )
»l g »s
(for N = 0)
(3)
With reference to Figure 2b, when the agitator speed is greater than zero the total force acting on the vessel bottom does not change, but an amount of solid particles Ms will be suspended: therefore the contribution due to particle direct Trans IChemE, Vol 78, Part A, April 2000
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321
contact with the vessel bottom will be correspondingly reduced: Fdirect contact = (Mt ê
Ms ) 1
»l g »s
ê
(for N > 0) (4)
and the contribution due to pressure becomes: Fpressure = Ftot ê
Fdirect contact = (Ml + Mt )g ê
(Mt ê
Ms ) 1
»l g ê »s
(for N > 0) (5)
Hence the increase of Fpressure at any agitator speed (N > 0) with respect to the still stirrer condition (N = 0), can be related to the mass of suspended solids:
D Fpressure = Fpressure (N>0) ê
Fpressure (N=0) = Ms 1 ê
»l g »s (6)
and the related average DP on the whole vessel bottom is simply given by:
D P = P(N>0)
»l g ê »s Ab
ê
Ms 1
P(N=0) =
(7)
Equation (7) shows that simple pressure measurements on the vessel bottom allow the assessment of the mass of suspended solid particles at any agitation speed. It is important to point out here that the above considerations hold true for the average pressure on the vessel bottom. If the pressure gauge is placed at any given point on the bottom, a local pressure is actually read. As ¯uid motion in the vessel gives rise to pressure gradients on the tank bottom, the local pressure increase actually measured at any single point on the tank bottom includes other contributions due to the ¯uid motion (dynamic head effects). Also, there are other vertical forces that arise in agitation conditions and that have been neglected so far. For instance, a signi®cant axial force may be exchanged between the impeller and the ¯uid, especially in the case of axial impellers. Other vertical forces that are not transmitted to the tank bottom through a pressure increase are the vertical friction forces on the vessel lateral wall and baf¯es. All these forces only arise in agitation conditions and contribute to average bottom pressure variations. Their contribution can, however, be included in the above mentioned dynamic head effects, that clearly have to be compensated for somehow before applying equation (7). Such a compensation can only be done approximately8 and therefore the existence of the dynamic head effects leads to more or less severe uncertainties on the measurements made. The Pressure Gauge Technique for Three-Phase Systems When a gas phase is sparged into the vessel, the volume of the ¯uid mixture increases due to the gas hold-up, while the average medium density decreases. If, however, the tank height is large enough to accommodate the volume expansion, so that no liquid or particles exit the vessel, the total weight on the tank bottom is practically unaffected by the presence of the gas phase, due to its negligible Trans IChemE, Vol 78, Part A, April 2000
Figure 3. Dynamic pressure vs agitation speed (gas-liquid systems).
density with respect to that of the solid and liquid phases. Hence, under such conditions equation (7) applies, with no modi®cations, to both three-phase and two-phase systems. In the case of three-phase systems, the dynamic head effects are lowered by the presence of the gas phase fed into the vessel, as shown in Figure 3 which was obtained in the absence of solid particles in the vessel (gas-liquid system) and where the departure of the curve at Qg = 0 from a parabolic trend marks the onset of surface aeration. This, however, does not result in smaller uncertainties on the estimated mass of suspended solids, due to both the larger agitation speeds required to attain fully suspended conditions, and especially to the complexities introduced by the presence of a gas phase, with the consequent involvement of a number of hydrodynamic regimes. The existence of such regimes and their interaction with the varying presence of solids in the three-phase mixture actually results in a much poorer predictability of the dependence of dynamic head effects on agitation speed, and in turn in larger uncertainties in the prediction of the mass of particles suspended at all agitation speeds. It is worth mentioning here that several attempts were made to overcome the dynamic head effects problem by substituting the solid bottom with a porous one, in the hope that in this way the pressure value read by the gauge connected to the dead volume would always coincide with the average pressure of the liquid phase on the bottom. This result would require, however, a very uniform porosity on the whole bottom, and none of the two porous bottoms tested (a polymeric dish for gas spargers and a sheet of sintered porous stainless steel) was found to ful®l this requirement. As a matter of fact, in the case of uneven porosity the measured pressure tends to equal that on the bottom points which are better connected with the dead chamber. Moreover, porous bottoms gave rise to nonreproducible results, presumably due to the progressive and selective clogging of the pores by attrition-generated ®ne particles. The solid bottom with a single hole previously employed for solid-liquid systems8 was preferred in the end. In Figure 4a, a typical over-pressure plot is shown (empty circles). At the highest agitation speeds, the expected trend of the total over-pressure to increase above the maximum
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worth mentioning here that the compensation procedure is based on the dimensional analysis suggestion that dynamic head effects at any given point in the vessel space should be proportional to the square of stirrer speed, but this would hold true only if the system ¯uid-dynamic properties did not change with agitation speed. In summary, the dynamic effects compensation procedure adopted in the present case consisted of: · ®tting a suitable parabola to carefully chosen points in the high agitation speed region (solid parabola in Figure 4a); · subtracting the so-estimated dynamic effects from the total over-pressures and translating, by means of equation (7), the result into a fractional solids suspension curve (empty squares in Figure 4b); · normalizing the resulting curve in order to avoid its meaningless portions below the abscissa axis in the low agitation speed region (solid circles in Figure 4b). Normalization was carried out in such a way that, while maintaining the 100% suspension data points, the relative minimum on the fractional suspension curve was assigned the value of zero fractional suspension, becoming in this way the starting speed for the suspension phenomenon.
Figure 4. Silica Dp = 180 212 mm; B = 7.66%; Qg = 0.56 vvm. (a) Plots ê of over-pressure vs agitation speed: total over-pressure (empty circles), dynamic head compensation (solid parabola). (b) Plots of suspended solids mass fraction vs agitation speed: not corrected (empty squares), corrected (solid circles).
value predicted by equation (7) (dotted line), is observed. This is due to the above discussed dynamic head effects and is similar to previous ®ndings8 with two-phase (solidliquid) systems. As can be seen, at low agitation speeds, signi®cant underpressures exist. These probably depend on the fact that at low agitation speeds the vessel bottom is `reshaped’ by the particle sediment, hence the near ¯ow ®eld is quite strongly affected and this results in a local pressure decrease (negative dynamic head effect) at the particular location where the measurements were taken. This interpretation is supported by the consideration that no under-pressure effects are observed at the same location when no particles are present in the system (see Figure 3). Similar effects were also observed for two-phase (solidliquid) systems, but their extent was, for unclear reasons, smaller than that observed with three-phase systems. They were therefore neglected in the past, but due to their larger extent have to be dealt with here. As concerns the over-pressures at high agitation speeds, they were found to be less regular than for solid-liquid systems, presumably due to interference by the gasliquid hydrodynamic regimes (gas dispersion features, cavity structures, etc). The same compensation procedure previously adopted8 was extended to the present case by exerting great caution and taking advantage by means of visual observation to ascertain which data points surely involved fully suspended conditions. Details on the method can be found elsewhere8. It is
It is clear that there is some degree of arbitrariness in the compensation procedure adopted. It is likely, for instance, that at the agitation speed adopted as the suspension starting point with a zero fractional suspension, some particles were already suspended. The collocation of the starting point for particle suspension is therefore affected by some uncertainty. On the other hand, it should not be forgotten that the overpressure due to suspension of solids typically accounted for the main portion of the total overpressure, so that the unavoidable uncertainties in the estimation of the dynamic head effects resulted in smaller uncertainties on the amount of suspended particles. RESULTS AND DISCUSSION By applying the above procedure to the raw data, smooth `S’ shaped curves were obtained for the suspended mass of solids versus agitation speed, as shown in Figure 5. In order to ®t each data set, the same Weibull function previously adopted8 was used: 8 for N < Nmin <0 2 X= N Nmin (8) ê : 1 exp for N $ Nmin ê ê Nspan
where N is the impeller speed while Nmin and Nspan are parameters which control the shape of the `S’. In Figure 5, the Weibull function (equation (8)) best ®tted to the experimental data set is reported as solid line and a good match with experimental data can be observed. One of the advantages of using such a Weibull function is that the values of the two parameters have an immediate physical meaning, as Nmin is the value at which the suspension phenomenon starts while Nspan is such that twice its value gives the range of N in which most of the suspension takes place, as can be immediately recognized observing that at N = Nmin + 2Nspan the value of X is 0.982, so that at this agitation speed less than 2% of the particles is still lying on the bottom. These considerations suggested8 the de®nition of a Trans IChemE, Vol 78, Part A, April 2000
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Figure 5. Plot of suspended solids mass fraction vs agitation speed: experimental data (empty squares), Weibull function ®t (solid line); silica Dp = 180 212 m m; B = 7.66%; Qg = 0.56 vvm. ê
`suf®cient suspension’ condition as the state attained at the agitation speed Nss given by Nss = Nmin + 2Nspan
(9)
It is clear that Nss is somewhat equivalent to the Njs de®ned by Zwietering7, though not exactly coinciding with it. As regards the in¯uence of gas ¯ow rates on particle suspension condition, in Figure 6 experimental data on particle fractional suspension versus agitation speed are reported for a given case (Dp = 250 300 mm, B = 15.81%) ê and ®ve gas ¯ow rates. The data shown are correlated by means of Weibull functions (solid lines) and a satisfactory agreement can be observed in all cases. As can be seen, fractional suspension curves move to the right while increasing gas ¯ow rate, implying that for any given agitation speed a smaller and smaller fraction of particles is suspended the higher the gas ¯ow rate. This is an expected result, as all previous studies on particle suspension in stirred vessels found that gassing of the system results into higher values of Njs, i.e. higher agitation speed, to attain an X of 100%1±6. The present data show that this is true for all values of the fractional suspension X. In particular the curves appear to be simply shifted with respect to abscissa axis, implying that the larger effect of gassing is on the starting point of the suspension phenomenon rather than on the span of the suspension range. Best ®tting the Weibull function (equation (8)) to the experimental data sets obtained for each given particle size,
Figure 6. Solid particles mass fraction vs agitation speed at various gas ¯ow rates (silica Dp = 250 300 mm; B = 15.81%). ê
Trans IChemE, Vol 78, Part A, April 2000
Figure 7. Experimental trends of: Nmin (a) and Nspan (b) vs Qg . Œ ê ê Dp = 180 212 mm; B = 3.77%; l Dp = 180 212 m m; B = 7.66%; ê 180 212 m m; ê ê ê 15.81%; 180 212 m m; D = B = D = B= F f p p ê ê ê ê ê ê 33.75%; l Dp = 250 300 m m; B = 7.66%; ! Dp = 250 300 mm; ê ê ê ê ê ê B = 15.81%; Dp = 250 300 m m; B = 33.75% Dp = a n ê ê 1680 2000 mm;ê ê Bê =3.77%; e ê Dp = 1680 2000 mm; B = 7.66%; ê ê ê ê Dp = 1680 2000 m m; B = 15.81%; e Dp = 1680 2000 m m s ê ê ê ê ê ê B = 33.75%.
concentration and gas ¯ow rate, resulted in a couple of values for the two parameters Nmin and Nspan. The results obtained are reported in Figures 7a±b. It can be observed there that Nmin clearly increases when increasing gas ¯ow rate (Figure 7a) while the dependence of Nspan on the same variable is more uncertain and practically negligible (Figure 7b). In other words, Figures 7a,b con®rm that aeration affects only the starting point of the suspension range, but not its extent, as qualitatively suggested by the previous analysis of Figure 6. Also, the dependence of Nmin on Qg seems well represented by a simple power law with an exponent of around 1/3, independently of the values of the other parameters here investigated. On the other hand, such a power law correlation would not converge to the correct limit when Qg tends to zero. As an alternative the ratio (Nmin Nmin0 )/Nmin0 could be approximately correlated to Qg0.5 . ê As regards the dependence of Nmin and Nspan on concentration, it was found to approximately follow the same dependencies previously observed for ungassed systems8. The above considerations indicate that a simple modi®cation of the correlations previously proposed for Nmin and Nspan, could extend their validity to the case of threephase systems: Nmin = 49.1 D0.236 (h/D)0.377 (1 + 0.58 Q0.5 p g )
(10)
Nspan = 3.1 D0.522 B0.265 p
(11)
where the average distance h between impeller and particle
MICALE et al.
324
phenomenological model for partial solids suspension, will be required to effectively address the dependency on particle size over very wide size ranges. The matter is not further addressed here, as this will also require improvements in the precision and reliability of data, which in turn requires further developments of the experimental technique in order to minimize the uncertainties associated with the dynamic head effects compensation. In order to obtain a correlation for the suf®cient suspension agitation speed Nss, the binomial correlation resulting by substituting equations (10) and (11) into equation (9) can be used. In the previous work on two-phase systems it was found, however, that a simpler monomial power law equation could effectively correlate such data. One may wonder whether a simple modi®cation of that correlation may account for the gassing effects in three-phase systems. This is actually the case, as by simply multiplying the previous correlation8 by a suitable factor similar to that introduced in equation (10): Nss = 24.1 D0.428 B0.13 (1 + 0.31 Q0.5 p g )
Figure 8. Plots of: (a) Nminexp vs Nmincomp ; (b) Nspanexp vs Nspancomp . Symbols as shown in Figure 7.
(13)
the correlation obtained (equation (13)) is able to satisfactorily describe the experimental data of this work. In Figure 9, equation (13) is compared with the present experimental data and it can be seen that a very good match is obtained with data dispersion much smaller than that observable in Figures 8a and 8b. This is consistent with previous ®ndings8 and depends on the fact that uncertainties on Nss are smaller than those on Nmin and Nspan separately, as previously discussed8. Clearly equation (13), as well as equations (10) and (11), are also consistent with the two-phase data obtained in that work8, as for Qg = 0 they reduce to the two-phase correlations. It is worth noting that these were found to hold true for the particle sizes ranging up to 850±1000 mm investigated. On the other hand, the data obtained with the largest particles (Dp = 1680 2000 mm), not represented in ê Figure 9 to avoid confusion, are strongly overestimated
sediment, is to be computed as: h=C
ê »s (1
4Mt « s )pT 2
(12)
ê In Figures 8a,b the experimental values here obtained for these parameters are compared with those predicted by equations (10) and (11). It can be observed that, as far as Nmin is concerned, there is a good agreement between correlation and experiment (Figure 8). As regards Nspan, the agreement is ®ne as far as the smaller particles employed in this work are concerned (Dp = 180 212 and 250±300 mm), while it is very poor for the largestê particles used (Dp = 1680 2000 mm) whether under gassed or ungassed conditions.ê As these particles are much larger that the largest particles employed in the previous work8, this ®nding implies that the correlation previously proposed cannot be extrapolated outside the experimental range there investigated. The reasons for such a change of dependence are not clear, though it may be speculated that one of the phenomena involved may be a very fast increase of particle drag factor when increasing particle size and turbulence intensity (Brucato et al., 19989). Clearly an in depth analysis of all possible phenomena involved, that would have to be based on a
Figure 9. Nss experimental vs Nss computed from equation (13). Symbols as in Figure 7.
Trans IChemE, Vol 78, Part A, April 2000
SOLIDS SUSPENSION IN THREE-PHASE STIRRED TANKS
Figure 10. D Nss experimental vs Qg (symbols as in Figure 7; solid line equation (14)).
by equation (13), showing that, not surprisingly, equation (13) cannot be extrapolated to such particle sizes. The effect of the gas ¯ow rate on Nss can ®nally be compared with that predicted by existing correlations, as for instance with the very simple one proposed by Nienow et al.6:
D Njs = 0.94 Qg
unaffected by it. These correlations allow, at least for the investigated tank and physical properties of the three phases, the estimation of the two parameters and therefore the estimation of the amount of suspended (or unsuspended) particles in a three-phase agitated tank operated at partial suspension conditions. It was also found that results obtained with very large particles are consistent with those pertaining to smaller particles as far as Nmin is concerned, while they exhibit a different behaviour in the case of Nspan. This implies that the correlation of this last parameter should not be extrapolated outside the particle size range previously investigated. It was remarked again that a suitable combination of the above mentioned parameters de®nes a `suf®cient suspension’ speed (Nss = Nmin + 2Nspan ) somewhat similar to the `just suspended’ speed (Njs) and it was also shown that, when the interest is limited to assessing complete or suf®cient suspension conditions, a simpler modi®cation of the previously proposed monomial correlation8, can be effectively employed. The comparison of the correlation here obtained for the increase of Nss due to aeration with a previous literature correlation6 for the increase of Njs, showed that such dependence is probably more complex than the simple linear dependence indicated there. NOMENCLATURE
(14)
This is done in Figure 10 by necessarily confusing Njs with Nss. It can be observed that equation (14) tends to underestimate most of the D Nss values here obtained. Other differences mainly lie on the assumptions, in equation (14), of a linear relationship between gas ¯ow rate and DNjs as well as on its independence of other parameters such as particle size and concentration, features that do not appear to be con®rmed by the present data. It is not clear whether these discrepancies may stem from the different de®nitions of `complete’ and `suf®cient’ suspension condition or on the uncertainties involved in both experimental techniques, especially in the case of three-phase systems. CONCLUSIONS A simple technique has been developed for measuring the fraction of suspended solids in three-phase stirred tanks operated at any agitation speed. As the technique is based on simple pressure measurements on the vessel bottom, it does not require expensive instrumentation and is very suitable for application to industrial scale agitated tanks. All the fractional suspension data collected showed a typical `S’ shaped dependence on agitation speed. As previously found for two-phase (solid-liquid) systems, the experimental curves were well ®tted by Weibull functions with an exponent of 2. The two parameters (Nmin and Nspan) appearing there have an immediate physical meaning and could be correlated by simple functions with the physical parameters varied in the experimentation (particle size, concentration and gas ¯ow rate). In particular, aeration was found to increase Nmin while Nspan is practically Trans IChemE, Vol 78, Part A, April 2000
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Ab B C D Ftot g h Ml Ms Mt N Ncd Njs Nmin Nspan Nss Qg T Vtot X
vessel bottom area, m2 total solids concentration, % w/w impeller clearance from tank bottom, m impeller diameter, m total force acting on the vessel bottom, N gravitational acceleration, m s ±2 average distance between the impeller and the surface of ®llets, m mass of liquid phase, kg mass of suspended solid particles, kg total mass of solid particles in the vessel, kg agitation speed, rpm agitation speed for complete gas dispersion, rpm just suspended agitation speed according to Zwietering5, rpm agitation speed at which the suspension phenomenon starts, rpm parameter that determines the width of the `S’, rpm suf®cient suspension agitation speed de®ned by equation (3), rpm gas ¯ow rate, vvm vessel diameter, m total volume (liquid phase + solid phase volume), m3 fractional suspension (Ms/Mt)
Greek symbols DP pressure increase at the tank bottom due to particles suspension, Pa fractional voidage of bed of particles, ± « s densities of solid and liquid phases, kg m ±3 »s , »l
REFERENCES 1. Frijlink, J., Bakker, A. and Smith, J., 1990, Suspension of solid particles with gassed impellers, Chem Eng Sci, 45: 1703±1718. 2. Chapman, M., Nienow, A. and Middleton, J., 1981, Particle suspension in a gas sparged Rushton turbine agitated vessel, TransIChemE, 59: 134±137. 3. Chapman, M., Nienow, A., Cooke, M. and Middleton, J., 1983, Particle-gas-liquid mixing in stirred vessels: Part III, Three-phase mixing, Chem Eng Res Des, 61: 167±181. 4. Pantula, P. and Ahmed, M., 1997, The impeller speed require for
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6. 7. 8. 9.
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complete solids supension in aerated vessels: a simple correlation?, Rec Progr Genie des Proc, 11: 11±18. Rewatkar, V. B., Rao, K. and Joshi, J., 1991, Critical impeller speed for solid suspension in mechanically agitated three-phase reactors. 1. Experimental Part; 2. Mathematical model, Ind Eng Chem Res, 30: 1770±1799. Nienow, A., Konno, M. and Bujalski, W., 1986, Studies on threephase mixing: a review and recent results, Chem Eng Res Des, 64: 35±42. Zwietering Th. N., 1958, Suspending of solid particles in liquids by agitators, Chem Eng Sci, 8: 244±253. Brucato, A., Micale G. and Rizzuti, L., 1997, Determination of the amount of unsuspended solid particles inside stirred tanks by means of pressure measurements, Rec Progr Genie des Proc, 11: 3±10. Brucato, A., Grisa®, F. and G. Montante, 1998, Particle drag coef®cients in turbulent ¯uids, Chem Eng Sci, 53: 3295±3314.
ACKNOWLEDGEMENT The ®nancial support by MURST is gratefully acknowledged.
ADDRESS Correspondence concerning this paper should be addressed to Professor A. Brucato, Dipartimento di Ingegneria Chimica dei Processi e dei Materiali, Viale delle Scienze, 90128 Palermo, Italy. (E-mail: [email protected]). The manuscript was received 20 October 1999 and was accepted for publication after revision 14 March 2000. This work was ®rst presented at the Fluid Mixing 6 Symposium, held 7±8 July 1999 at the University of Bradford, UK.
Trans IChemE, Vol 78, Part A, April 2000
SOLIDS SUSPENSION IN THREE-PHASE STIRRED TANKS G. MICALE, V. CARRARA, F. GRISAFI and A. BRUCATO Dipartimento di Ingegneria Chimica dei Processi e dei Materiali (DICPM), Palermo, Italy
T
he `pressure gauge technique’ recently developed for the quantitative assessment of the mass of solid particles suspended at any agitation speed is extended to the case of three-phase stirred tanks. As a result, curves of the fraction of suspended solids at various gas ¯ow rates, versus agitator speed, are presented. The dif®culties involved in the extension of the technique to three phase systems are addressed and discussed. The experimental results show that the presence of the gas phase causes a signi®cant increase of the agitation speed required to attain complete suspension of solids and lowers the degree of suspension at all agitation speeds below it. The experimental data obtained are shown to be well ®tted by the Weibull functions previously adopted for two-phase systems. Correlations for the estimation of the in¯uence of particle size, particle concentration and gas ¯ow rate on the suspension degree are presented and discussed. Finally, the fractional suspension dependence on particle size is found to be very different when very large particles are dealt with. Keywords: stirred tanks; solids suspension; three-phase systems; pressure gauge technique
INTRODUCTION
relied on the visual assessment of Njs according to the method ®rst introduced by Zwietering7 which, apart from being subject to substantial uncertainties, gives no information on what happens at agitation speeds below Njs. In this respect, a novel technique (`Pressure Gauge Technique’, PGT) has been recently introduced for the quantitative assessment of the mass of solid particles suspended at any agitation speed in solid-liquid systems (Brucato et al.8 ). The technique is based on the measurement of the pressure increase on the tank bottom due to the presence of suspended solid particles. As ¯uids of different densities exert different static heads at the vessel bottom, when solid particles are lifted in the liquid the increase of the ¯uid effective density can be measured at the tank bottom by means of a pressure gauge. Being able to characterize all partial suspension conditions, an objective assessment of complete suspension conditions is also allowed by this technique. In the present work the PGT is extended to the case of three phase systems, resulting in the ®rst experimental data on the way the amount of suspended particles is affected by aeration, at all agitation speeds, in three-phase stirred tanks.
A number of important catalytic multiphase processes involves the suspension of solid particles in gas-liquid dispersions inside mechanically stirred tanks. The suspension of solid particles in a liquid has been extensively studied over the last forty years. Most of the research efforts have been devoted to the determination of the minimum agitator speed, Njs, required to attain the suspension of all particles, while only recently attention has been paid to the analysis of the industrially important partial suspension conditions. As regards three phase stirred systems, there is a quite wide lack of information on the complex interactions among solid particles, gas bubbles and the liquid phase at the various agitation regimes, that causes severe uncertainties in the design and development of these apparatuses. As a matter of fact, such interactions are of paramount importance since the presence of the gas phase largely affects 1±6 the values of Njs and the presence of the solid phase modi®es the values of the minimum agitation speed (Ncd) required for an effective and complete dispersion in the vessel, as pointed out for instance by Frijlink et al.1. For each given system, the limiting factor governing the power requirements of the agitated vessel may be either gas dispersion or solids suspension, as pointed out by Chapman et al.2 who observed that there is a critical particle density (relative to the liquid medium) that discriminates between the two circumstances. A number of other studies on three-phase systems can be found in the literature, some of which, including those by Nienow et al.6, Chapman et al.3, Frijlink et al.1, and Pantula and Ahmed4 address the problem of correlating the agitation speed required to attain complete solids suspension under gassed conditions, but the ®eld appears to be still open to improvements. For instance, all of these works
EXPERIMENTAL In Figure 1, the stirred vessel employed for the experimentation is shown. It consists of a ¯at bottomed, fully baf¯ed, transparent vessel (T = 0.19 m), stirred by a six blade Rushton turbine (D = T /2 = 0.095 m) set at a distance from the vessel bottom equal to T/3. A measured mass of solid particles was ®rst introduced into the vessel and then the liquid phase added up to a height H = T = 0.19 m. The gas phase was fed by means of an open tube sparger placed at T/4 from the vessel bottom. 319
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Figure 1. R19 experimental set-up: (A) power supply; (B) DC motor; (C) optical tachometer; (E) connecting hole; (F) gas ¯owmeter; (P) inclined piezometer.
The impeller shaft was driven by a DC electric motor provided with a speed control loop, while an optical tachometer was used to independently measure the impeller speed. A pressure gauge connected to a speci®c point of the vessel bottom allowed pressure readings to be taken. The point selected for this purpose was placed at a radial location midway between the axis and the side wall and at 458 between subsequent baf¯es. A hole in the vessel bottom transmitted the value of the local pressure to a dead chamber to which the pressure gauge (a simple inclined manometer) was connected. The technique, however, does not require the presence of the dead volume and would have worked in the same way had the pressure gauge been directly connected to the hole in the vessel bottom. The solid phase used for the experimental runs consisted of silica particles with a measured density »s of 2500kg m ±3 in three different cuts (180±212, 250±300, 1680±2000 mm). The concentration of solid particles in the slurry was varied from 3.77 to 33.75 per cent by weight. The gas fed to the vessels was compressed air and its ¯ow rate was measured by a digital mass ¯ow-meter. The gas ¯ow rate was varied from 0.00 to 2.23 vvm. The liquid phase was de-ionized water in all experimental runs.
PRESSURE GAUGE TECHNIQUE The Pressure Gauge Technique (PGT)8 is based on the measurement of the pressure increase on the tank bottom due to the suspension of solid particles. In the following, the relationship between the pressure increase and the mass of suspended particles will be derived by simply performing a macroscopic force balance on the system, which is de®ned as the volume occupied by the liquid and particles introduced in the vessel. With reference to Figure 2, where a stirred tank containing a mass of solids Mt and a mass of liquid Ml is shown, a force balance in the vertical direction performed
Figure 2. Solid particle distribution: (a) at N = 0; (b) at N > 0.
on the system leads to: Ftot = (Ml + Mt )g
(1)
where Ftot is the total force acting on the vessel bottom. This total force may be split into two contributions. The ®rst one is due to the direct contact of solid particles with the vessel bottom. At no agitation conditions (N = 0, Figure 2a), this is equal to the apparent weight of the total mass of solids introduced in the vessel, as in such conditions all particles lay on the vessel bottom. In formula: Fdirect contact = Mt 1 ê
»l g »s
(for N = 0)
(2)
In still conditions (N = 0), the second contribution is due to the pressure exerted by a height of liquid column equal to the total height H reached by the amounts of liquid and solid phases added into the vessel. Its expression can be obtained by simply subtracting the Fdirect contact contribution from the total force Ftot. Fpressure =
(Ml + Mt )
»l g »s
(for N = 0)
(3)
With reference to Figure 2b, when the agitator speed is greater than zero the total force acting on the vessel bottom does not change, but an amount of solid particles Ms will be suspended: therefore the contribution due to particle direct Trans IChemE, Vol 78, Part A, April 2000
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contact with the vessel bottom will be correspondingly reduced: Fdirect contact = (Mt ê
Ms ) 1
»l g »s
ê
(for N > 0) (4)
and the contribution due to pressure becomes: Fpressure = Ftot ê
Fdirect contact = (Ml + Mt )g ê
(Mt ê
Ms ) 1
»l g ê »s
(for N > 0) (5)
Hence the increase of Fpressure at any agitator speed (N > 0) with respect to the still stirrer condition (N = 0), can be related to the mass of suspended solids:
D Fpressure = Fpressure (N>0) ê
Fpressure (N=0) = Ms 1 ê
»l g »s (6)
and the related average DP on the whole vessel bottom is simply given by:
D P = P(N>0)
»l g ê »s Ab
ê
Ms 1
P(N=0) =
(7)
Equation (7) shows that simple pressure measurements on the vessel bottom allow the assessment of the mass of suspended solid particles at any agitation speed. It is important to point out here that the above considerations hold true for the average pressure on the vessel bottom. If the pressure gauge is placed at any given point on the bottom, a local pressure is actually read. As ¯uid motion in the vessel gives rise to pressure gradients on the tank bottom, the local pressure increase actually measured at any single point on the tank bottom includes other contributions due to the ¯uid motion (dynamic head effects). Also, there are other vertical forces that arise in agitation conditions and that have been neglected so far. For instance, a signi®cant axial force may be exchanged between the impeller and the ¯uid, especially in the case of axial impellers. Other vertical forces that are not transmitted to the tank bottom through a pressure increase are the vertical friction forces on the vessel lateral wall and baf¯es. All these forces only arise in agitation conditions and contribute to average bottom pressure variations. Their contribution can, however, be included in the above mentioned dynamic head effects, that clearly have to be compensated for somehow before applying equation (7). Such a compensation can only be done approximately8 and therefore the existence of the dynamic head effects leads to more or less severe uncertainties on the measurements made. The Pressure Gauge Technique for Three-Phase Systems When a gas phase is sparged into the vessel, the volume of the ¯uid mixture increases due to the gas hold-up, while the average medium density decreases. If, however, the tank height is large enough to accommodate the volume expansion, so that no liquid or particles exit the vessel, the total weight on the tank bottom is practically unaffected by the presence of the gas phase, due to its negligible Trans IChemE, Vol 78, Part A, April 2000
Figure 3. Dynamic pressure vs agitation speed (gas-liquid systems).
density with respect to that of the solid and liquid phases. Hence, under such conditions equation (7) applies, with no modi®cations, to both three-phase and two-phase systems. In the case of three-phase systems, the dynamic head effects are lowered by the presence of the gas phase fed into the vessel, as shown in Figure 3 which was obtained in the absence of solid particles in the vessel (gas-liquid system) and where the departure of the curve at Qg = 0 from a parabolic trend marks the onset of surface aeration. This, however, does not result in smaller uncertainties on the estimated mass of suspended solids, due to both the larger agitation speeds required to attain fully suspended conditions, and especially to the complexities introduced by the presence of a gas phase, with the consequent involvement of a number of hydrodynamic regimes. The existence of such regimes and their interaction with the varying presence of solids in the three-phase mixture actually results in a much poorer predictability of the dependence of dynamic head effects on agitation speed, and in turn in larger uncertainties in the prediction of the mass of particles suspended at all agitation speeds. It is worth mentioning here that several attempts were made to overcome the dynamic head effects problem by substituting the solid bottom with a porous one, in the hope that in this way the pressure value read by the gauge connected to the dead volume would always coincide with the average pressure of the liquid phase on the bottom. This result would require, however, a very uniform porosity on the whole bottom, and none of the two porous bottoms tested (a polymeric dish for gas spargers and a sheet of sintered porous stainless steel) was found to ful®l this requirement. As a matter of fact, in the case of uneven porosity the measured pressure tends to equal that on the bottom points which are better connected with the dead chamber. Moreover, porous bottoms gave rise to nonreproducible results, presumably due to the progressive and selective clogging of the pores by attrition-generated ®ne particles. The solid bottom with a single hole previously employed for solid-liquid systems8 was preferred in the end. In Figure 4a, a typical over-pressure plot is shown (empty circles). At the highest agitation speeds, the expected trend of the total over-pressure to increase above the maximum
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worth mentioning here that the compensation procedure is based on the dimensional analysis suggestion that dynamic head effects at any given point in the vessel space should be proportional to the square of stirrer speed, but this would hold true only if the system ¯uid-dynamic properties did not change with agitation speed. In summary, the dynamic effects compensation procedure adopted in the present case consisted of: · ®tting a suitable parabola to carefully chosen points in the high agitation speed region (solid parabola in Figure 4a); · subtracting the so-estimated dynamic effects from the total over-pressures and translating, by means of equation (7), the result into a fractional solids suspension curve (empty squares in Figure 4b); · normalizing the resulting curve in order to avoid its meaningless portions below the abscissa axis in the low agitation speed region (solid circles in Figure 4b). Normalization was carried out in such a way that, while maintaining the 100% suspension data points, the relative minimum on the fractional suspension curve was assigned the value of zero fractional suspension, becoming in this way the starting speed for the suspension phenomenon.
Figure 4. Silica Dp = 180 212 mm; B = 7.66%; Qg = 0.56 vvm. (a) Plots ê of over-pressure vs agitation speed: total over-pressure (empty circles), dynamic head compensation (solid parabola). (b) Plots of suspended solids mass fraction vs agitation speed: not corrected (empty squares), corrected (solid circles).
value predicted by equation (7) (dotted line), is observed. This is due to the above discussed dynamic head effects and is similar to previous ®ndings8 with two-phase (solidliquid) systems. As can be seen, at low agitation speeds, signi®cant underpressures exist. These probably depend on the fact that at low agitation speeds the vessel bottom is `reshaped’ by the particle sediment, hence the near ¯ow ®eld is quite strongly affected and this results in a local pressure decrease (negative dynamic head effect) at the particular location where the measurements were taken. This interpretation is supported by the consideration that no under-pressure effects are observed at the same location when no particles are present in the system (see Figure 3). Similar effects were also observed for two-phase (solidliquid) systems, but their extent was, for unclear reasons, smaller than that observed with three-phase systems. They were therefore neglected in the past, but due to their larger extent have to be dealt with here. As concerns the over-pressures at high agitation speeds, they were found to be less regular than for solid-liquid systems, presumably due to interference by the gasliquid hydrodynamic regimes (gas dispersion features, cavity structures, etc). The same compensation procedure previously adopted8 was extended to the present case by exerting great caution and taking advantage by means of visual observation to ascertain which data points surely involved fully suspended conditions. Details on the method can be found elsewhere8. It is
It is clear that there is some degree of arbitrariness in the compensation procedure adopted. It is likely, for instance, that at the agitation speed adopted as the suspension starting point with a zero fractional suspension, some particles were already suspended. The collocation of the starting point for particle suspension is therefore affected by some uncertainty. On the other hand, it should not be forgotten that the overpressure due to suspension of solids typically accounted for the main portion of the total overpressure, so that the unavoidable uncertainties in the estimation of the dynamic head effects resulted in smaller uncertainties on the amount of suspended particles. RESULTS AND DISCUSSION By applying the above procedure to the raw data, smooth `S’ shaped curves were obtained for the suspended mass of solids versus agitation speed, as shown in Figure 5. In order to ®t each data set, the same Weibull function previously adopted8 was used: 8 for N < Nmin <0 2 X= N Nmin (8) ê : 1 exp for N $ Nmin ê ê Nspan
where N is the impeller speed while Nmin and Nspan are parameters which control the shape of the `S’. In Figure 5, the Weibull function (equation (8)) best ®tted to the experimental data set is reported as solid line and a good match with experimental data can be observed. One of the advantages of using such a Weibull function is that the values of the two parameters have an immediate physical meaning, as Nmin is the value at which the suspension phenomenon starts while Nspan is such that twice its value gives the range of N in which most of the suspension takes place, as can be immediately recognized observing that at N = Nmin + 2Nspan the value of X is 0.982, so that at this agitation speed less than 2% of the particles is still lying on the bottom. These considerations suggested8 the de®nition of a Trans IChemE, Vol 78, Part A, April 2000
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Figure 5. Plot of suspended solids mass fraction vs agitation speed: experimental data (empty squares), Weibull function ®t (solid line); silica Dp = 180 212 m m; B = 7.66%; Qg = 0.56 vvm. ê
`suf®cient suspension’ condition as the state attained at the agitation speed Nss given by Nss = Nmin + 2Nspan
(9)
It is clear that Nss is somewhat equivalent to the Njs de®ned by Zwietering7, though not exactly coinciding with it. As regards the in¯uence of gas ¯ow rates on particle suspension condition, in Figure 6 experimental data on particle fractional suspension versus agitation speed are reported for a given case (Dp = 250 300 mm, B = 15.81%) ê and ®ve gas ¯ow rates. The data shown are correlated by means of Weibull functions (solid lines) and a satisfactory agreement can be observed in all cases. As can be seen, fractional suspension curves move to the right while increasing gas ¯ow rate, implying that for any given agitation speed a smaller and smaller fraction of particles is suspended the higher the gas ¯ow rate. This is an expected result, as all previous studies on particle suspension in stirred vessels found that gassing of the system results into higher values of Njs, i.e. higher agitation speed, to attain an X of 100%1±6. The present data show that this is true for all values of the fractional suspension X. In particular the curves appear to be simply shifted with respect to abscissa axis, implying that the larger effect of gassing is on the starting point of the suspension phenomenon rather than on the span of the suspension range. Best ®tting the Weibull function (equation (8)) to the experimental data sets obtained for each given particle size,
Figure 6. Solid particles mass fraction vs agitation speed at various gas ¯ow rates (silica Dp = 250 300 mm; B = 15.81%). ê
Trans IChemE, Vol 78, Part A, April 2000
Figure 7. Experimental trends of: Nmin (a) and Nspan (b) vs Qg . Œ ê ê Dp = 180 212 mm; B = 3.77%; l Dp = 180 212 m m; B = 7.66%; ê 180 212 m m; ê ê ê 15.81%; 180 212 m m; D = B = D = B= F f p p ê ê ê ê ê ê 33.75%; l Dp = 250 300 m m; B = 7.66%; ! Dp = 250 300 mm; ê ê ê ê ê ê B = 15.81%; Dp = 250 300 m m; B = 33.75% Dp = a n ê ê 1680 2000 mm;ê ê Bê =3.77%; e ê Dp = 1680 2000 mm; B = 7.66%; ê ê ê ê Dp = 1680 2000 m m; B = 15.81%; e Dp = 1680 2000 m m s ê ê ê ê ê ê B = 33.75%.
concentration and gas ¯ow rate, resulted in a couple of values for the two parameters Nmin and Nspan. The results obtained are reported in Figures 7a±b. It can be observed there that Nmin clearly increases when increasing gas ¯ow rate (Figure 7a) while the dependence of Nspan on the same variable is more uncertain and practically negligible (Figure 7b). In other words, Figures 7a,b con®rm that aeration affects only the starting point of the suspension range, but not its extent, as qualitatively suggested by the previous analysis of Figure 6. Also, the dependence of Nmin on Qg seems well represented by a simple power law with an exponent of around 1/3, independently of the values of the other parameters here investigated. On the other hand, such a power law correlation would not converge to the correct limit when Qg tends to zero. As an alternative the ratio (Nmin Nmin0 )/Nmin0 could be approximately correlated to Qg0.5 . ê As regards the dependence of Nmin and Nspan on concentration, it was found to approximately follow the same dependencies previously observed for ungassed systems8. The above considerations indicate that a simple modi®cation of the correlations previously proposed for Nmin and Nspan, could extend their validity to the case of threephase systems: Nmin = 49.1 D0.236 (h/D)0.377 (1 + 0.58 Q0.5 p g )
(10)
Nspan = 3.1 D0.522 B0.265 p
(11)
where the average distance h between impeller and particle
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phenomenological model for partial solids suspension, will be required to effectively address the dependency on particle size over very wide size ranges. The matter is not further addressed here, as this will also require improvements in the precision and reliability of data, which in turn requires further developments of the experimental technique in order to minimize the uncertainties associated with the dynamic head effects compensation. In order to obtain a correlation for the suf®cient suspension agitation speed Nss, the binomial correlation resulting by substituting equations (10) and (11) into equation (9) can be used. In the previous work on two-phase systems it was found, however, that a simpler monomial power law equation could effectively correlate such data. One may wonder whether a simple modi®cation of that correlation may account for the gassing effects in three-phase systems. This is actually the case, as by simply multiplying the previous correlation8 by a suitable factor similar to that introduced in equation (10): Nss = 24.1 D0.428 B0.13 (1 + 0.31 Q0.5 p g )
Figure 8. Plots of: (a) Nminexp vs Nmincomp ; (b) Nspanexp vs Nspancomp . Symbols as shown in Figure 7.
(13)
the correlation obtained (equation (13)) is able to satisfactorily describe the experimental data of this work. In Figure 9, equation (13) is compared with the present experimental data and it can be seen that a very good match is obtained with data dispersion much smaller than that observable in Figures 8a and 8b. This is consistent with previous ®ndings8 and depends on the fact that uncertainties on Nss are smaller than those on Nmin and Nspan separately, as previously discussed8. Clearly equation (13), as well as equations (10) and (11), are also consistent with the two-phase data obtained in that work8, as for Qg = 0 they reduce to the two-phase correlations. It is worth noting that these were found to hold true for the particle sizes ranging up to 850±1000 mm investigated. On the other hand, the data obtained with the largest particles (Dp = 1680 2000 mm), not represented in ê Figure 9 to avoid confusion, are strongly overestimated
sediment, is to be computed as: h=C
ê »s (1
4Mt « s )pT 2
(12)
ê In Figures 8a,b the experimental values here obtained for these parameters are compared with those predicted by equations (10) and (11). It can be observed that, as far as Nmin is concerned, there is a good agreement between correlation and experiment (Figure 8). As regards Nspan, the agreement is ®ne as far as the smaller particles employed in this work are concerned (Dp = 180 212 and 250±300 mm), while it is very poor for the largestê particles used (Dp = 1680 2000 mm) whether under gassed or ungassed conditions.ê As these particles are much larger that the largest particles employed in the previous work8, this ®nding implies that the correlation previously proposed cannot be extrapolated outside the experimental range there investigated. The reasons for such a change of dependence are not clear, though it may be speculated that one of the phenomena involved may be a very fast increase of particle drag factor when increasing particle size and turbulence intensity (Brucato et al., 19989). Clearly an in depth analysis of all possible phenomena involved, that would have to be based on a
Figure 9. Nss experimental vs Nss computed from equation (13). Symbols as in Figure 7.
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SOLIDS SUSPENSION IN THREE-PHASE STIRRED TANKS
Figure 10. D Nss experimental vs Qg (symbols as in Figure 7; solid line equation (14)).
by equation (13), showing that, not surprisingly, equation (13) cannot be extrapolated to such particle sizes. The effect of the gas ¯ow rate on Nss can ®nally be compared with that predicted by existing correlations, as for instance with the very simple one proposed by Nienow et al.6:
D Njs = 0.94 Qg
unaffected by it. These correlations allow, at least for the investigated tank and physical properties of the three phases, the estimation of the two parameters and therefore the estimation of the amount of suspended (or unsuspended) particles in a three-phase agitated tank operated at partial suspension conditions. It was also found that results obtained with very large particles are consistent with those pertaining to smaller particles as far as Nmin is concerned, while they exhibit a different behaviour in the case of Nspan. This implies that the correlation of this last parameter should not be extrapolated outside the particle size range previously investigated. It was remarked again that a suitable combination of the above mentioned parameters de®nes a `suf®cient suspension’ speed (Nss = Nmin + 2Nspan ) somewhat similar to the `just suspended’ speed (Njs) and it was also shown that, when the interest is limited to assessing complete or suf®cient suspension conditions, a simpler modi®cation of the previously proposed monomial correlation8, can be effectively employed. The comparison of the correlation here obtained for the increase of Nss due to aeration with a previous literature correlation6 for the increase of Njs, showed that such dependence is probably more complex than the simple linear dependence indicated there. NOMENCLATURE
(14)
This is done in Figure 10 by necessarily confusing Njs with Nss. It can be observed that equation (14) tends to underestimate most of the D Nss values here obtained. Other differences mainly lie on the assumptions, in equation (14), of a linear relationship between gas ¯ow rate and DNjs as well as on its independence of other parameters such as particle size and concentration, features that do not appear to be con®rmed by the present data. It is not clear whether these discrepancies may stem from the different de®nitions of `complete’ and `suf®cient’ suspension condition or on the uncertainties involved in both experimental techniques, especially in the case of three-phase systems. CONCLUSIONS A simple technique has been developed for measuring the fraction of suspended solids in three-phase stirred tanks operated at any agitation speed. As the technique is based on simple pressure measurements on the vessel bottom, it does not require expensive instrumentation and is very suitable for application to industrial scale agitated tanks. All the fractional suspension data collected showed a typical `S’ shaped dependence on agitation speed. As previously found for two-phase (solid-liquid) systems, the experimental curves were well ®tted by Weibull functions with an exponent of 2. The two parameters (Nmin and Nspan) appearing there have an immediate physical meaning and could be correlated by simple functions with the physical parameters varied in the experimentation (particle size, concentration and gas ¯ow rate). In particular, aeration was found to increase Nmin while Nspan is practically Trans IChemE, Vol 78, Part A, April 2000
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Ab B C D Ftot g h Ml Ms Mt N Ncd Njs Nmin Nspan Nss Qg T Vtot X
vessel bottom area, m2 total solids concentration, % w/w impeller clearance from tank bottom, m impeller diameter, m total force acting on the vessel bottom, N gravitational acceleration, m s ±2 average distance between the impeller and the surface of ®llets, m mass of liquid phase, kg mass of suspended solid particles, kg total mass of solid particles in the vessel, kg agitation speed, rpm agitation speed for complete gas dispersion, rpm just suspended agitation speed according to Zwietering5, rpm agitation speed at which the suspension phenomenon starts, rpm parameter that determines the width of the `S’, rpm suf®cient suspension agitation speed de®ned by equation (3), rpm gas ¯ow rate, vvm vessel diameter, m total volume (liquid phase + solid phase volume), m3 fractional suspension (Ms/Mt)
Greek symbols DP pressure increase at the tank bottom due to particles suspension, Pa fractional voidage of bed of particles, ± « s densities of solid and liquid phases, kg m ±3 »s , »l
REFERENCES 1. Frijlink, J., Bakker, A. and Smith, J., 1990, Suspension of solid particles with gassed impellers, Chem Eng Sci, 45: 1703±1718. 2. Chapman, M., Nienow, A. and Middleton, J., 1981, Particle suspension in a gas sparged Rushton turbine agitated vessel, TransIChemE, 59: 134±137. 3. Chapman, M., Nienow, A., Cooke, M. and Middleton, J., 1983, Particle-gas-liquid mixing in stirred vessels: Part III, Three-phase mixing, Chem Eng Res Des, 61: 167±181. 4. Pantula, P. and Ahmed, M., 1997, The impeller speed require for
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6. 7. 8. 9.
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complete solids supension in aerated vessels: a simple correlation?, Rec Progr Genie des Proc, 11: 11±18. Rewatkar, V. B., Rao, K. and Joshi, J., 1991, Critical impeller speed for solid suspension in mechanically agitated three-phase reactors. 1. Experimental Part; 2. Mathematical model, Ind Eng Chem Res, 30: 1770±1799. Nienow, A., Konno, M. and Bujalski, W., 1986, Studies on threephase mixing: a review and recent results, Chem Eng Res Des, 64: 35±42. Zwietering Th. N., 1958, Suspending of solid particles in liquids by agitators, Chem Eng Sci, 8: 244±253. Brucato, A., Micale G. and Rizzuti, L., 1997, Determination of the amount of unsuspended solid particles inside stirred tanks by means of pressure measurements, Rec Progr Genie des Proc, 11: 3±10. Brucato, A., Grisa®, F. and G. Montante, 1998, Particle drag coef®cients in turbulent ¯uids, Chem Eng Sci, 53: 3295±3314.
ACKNOWLEDGEMENT The ®nancial support by MURST is gratefully acknowledged.
ADDRESS Correspondence concerning this paper should be addressed to Professor A. Brucato, Dipartimento di Ingegneria Chimica dei Processi e dei Materiali, Viale delle Scienze, 90128 Palermo, Italy. (E-mail: [email protected]). The manuscript was received 20 October 1999 and was accepted for publication after revision 14 March 2000. This work was ®rst presented at the Fluid Mixing 6 Symposium, held 7±8 July 1999 at the University of Bradford, UK.
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