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Advances in Mechanical Engineering
Volume 2013 (2013), Article ID 859756, 8 pages
Solid Suspension by an Upflow Mixture of Fluid and Larger Particles
Dipartimento di Ingegneria Civile, Chimica ed Ambientale, Università degli Studi di Genova, 16145 Genova, Italy
Received 3 April 2013; Revised 19 June 2013; Accepted 20 June 2013
Academic Editor: Guan Heng Yeoh
Copyright © 2013 Renzo Di Felice and Marco Rotondi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this paper new experimental observation is reported in order to shed light on the fluid dynamic interactions between the solid and the fluid phases in binary solid mixture suspensions. More specifically the case where a solid particle species is suspended by a slurry made up of fluid and larger particles has been investigated as there appear to be no previous studies on this type of systems. The effect of the presence of the larger solids in the fluidising media is quite evident from the experimental observation, and this effect can be quantified in first approximation by a very simple model based on the assumption that the larger solid restricts the available area for the fluid flow.
When dealing with multiphase systems, such as solid-liquid systems, constitutive equations must include various terms to take into account the complex interphase interactions, with the solid-fluid drag, which arises when the phase velocities are different, being probably the most important one. Although it would be preferable to have exact theoretical approaches to estimate the drag force [1–7], nowadays its quantification is inferred mainly from experimental observations [8–10] with the homogenous suspension behaviour in vertical solid-fluid systems being the most common source. In fact, when a solid particle is suspended in steady-state conditions, the net force balance acting on it must be zero. This observation leads to an easy quantification of the solid-fluid drag force, which is obtained by neglecting other effects such as particle-particle interactions, as difference between the weight and the buoyant effect . Therefore if the fluid dynamic conditions characterizing a suspension are known (phase velocities and phase volume concentrations), interphase drag can be expressed as a function of the previously mentioned conditions. This has been done when monocomponent solid-fluid systems have been considered with the homogeneous suspension behaviour being the main source of experimental evidence [8, 10]. However, when the solid making up the suspension differs in size, experimental evidence is less abundant and the suggestion of a reliable universal drag law has been much more difficult as a consequence.
This aspect has been at the centre of a recent publication by the present authors  where empirical expressions for the drag force on large and small particles of a binary-solid suspension were suggested. In that work, however, it was pointed out that if, on one hand, the expression for the drag force on a large particle was supported by a substantial body of experimental evidence, on the other hand, the same could not be said for the drag force on the smaller particle of the binary-solid suspension, where the experimental evidence was quite limited. Therefore, aim of this paper is to contribute in filling that deficiency by concentrating on the smaller particles behaviour when suspended in the presence of larger particles.
2. Behaviour of Solid-Fluid Suspensions in Steady State Conditions
Given the importance of the knowledge of the suspension behaviour in steady-state for the determination of the drag force acting on a solid particle, a short review is reported in the following section.
If all the particles are identical (e.g., with the same density and diameter), the equilibrium fluid dynamic condition of a suspension is usually expressed as universally known as the Richardson-Zaki equation . In (1) is the superficial fluid velocity, is the terminal velocity of a single particle settling in the pure fluid, and is the solid volume concentration inside the column, defined as the volume occupied by the solid over the total suspension volume.
Using a large amount of experimental data obtained from fluidized beds and batch sedimentations Richardson and Zaki  suggested that the parameter is a function of the terminal Reynolds number only where and are, respectively, fluid density and viscosity, while is the particle diameter. A simple expression for the estimation of has given by Rowe :
If (1) is plotted on log-log scale, the behavior is linear with as the slope and as the intercept at solid volume concentration equal to zero.
However, further studies suggested that a different behaviour may be expected for large diameter particles, with high Reynolds number, fluidized by liquids [15–18], as depicted schematically in Figure 1. The velocity versus solid concentration relationship on double logarithmic coordinates is made up in this case of two linear sections, with the slope changing at a critical concentration . The extrapolated velocity at zero solid concentration of the first segment is indicated as , and it is below the true single unhindered particle settling velocity, , with smaller than 1.
The slope of the first segment, , is in good agreement with the Richardson and Zaki  coefficient, but after the critical value, it changes to a larger value , with the velocity approaching this time at . Rapagna et al.  proposed quantitative expressions for , , and , based on an empirical fitting of their experimental observations, as follows:
The expressions reported are a simple elegant way to quantify the solid-fluid interaction for monocomponent systems and, as suggested by the authors, “little numerical significance can be ascribed to these relationships which do little more than indicate the trend” .
In binary-solid systems, where two solids (differing in size and/or density) are homogeneously suspended in a fluid, the presence of one solid is going to affect the expansion characteristics of the other.
As far as the effect due to the presence of small particles on the expansion characteristics of the larger particles, a limited experimental investigation has already been carried out in the past , where a larger solid was fluidized by a slurry made up of fluid and smaller particles. A typical result of that investigation is shown on Figure 2, where the fluid superficial velocity necessary to expand the larger solid to a fixed determined column height is plotted function of the smaller particle concentration.
From Figure 2 it appears evident that the effect brought about by the presence of the smaller particles is to “facilitate” the suspension of the larger ones, with experimental fluid velocity decreasing as the smaller particle concentration in the fluidizing slurry increases. A successful quantification of the observed behaviour has been done by the so-called “pseudo-fluid” approach [20, 21]. Basically the effect due to presence of the smaller particles is taken into account by properly modifying fluid physical properties (density and viscosity) so that previous relationships, such as the Richardson-Zaki  relationship, can still be formally applied. Therefore with this assumption, the binary mixture can be treated as monocomponent suspension, with the larger particles immersed in the pseudo-fluid made up of pure fluid and smaller particles. The characteristics of the pseudo-fluid can be calculated as follows: for the density and the viscosity, and for the velocity, with the subscript pf referring to the pseudo-fluid, referring to the smaller particle kind, and referring to the fluid. In the previous expressions is the concentration of the smaller solid in the pseudo-fluid phase and is calculated as
This model was validated through studies on binary-solid sedimentation next to the investigation of a circulating fluidized bed .
Unfortunately the mirror situation, that is, the effect brought about by the presence of larger particles on the expansion characteristics of smaller particles, has never being investigated before, and in this paper we make a first attempt to fill this gap.
As analogously done by , a circulating fluidized bed is purposely utilized. This time the attention is focussed on the smaller particles, and therefore they are fluidized by a suspension of larger particles transported by the fluid.
The experimental apparatus is schematized in Figure 3. The system allows the liquid to circulate in a closed loop; ambient water was used for all the experiments carried out in this work. The test section is a cylindrical vertical column of Perspex, 2000 mm tall, connected to a centrifugal pump via PVC tubing. The pump (T21-50 by TURO Italia srl) is specifically constructed so that solid-liquid slurries can be pumped, even at relatively high solid volume concentration (up to 20%). Two different vertical test sections were used, with internal diameter of 140 and 68 mm; on the other hand all the connecting tubes have an internal diameter of 40 mm.
The whole system could be filled with liquid through an opening situated in the most upper pipe. The liquid volumetric flow rate is regulated by manually setting two sphere valves, one situated just before the inlet of the test section and the other on the liquid bypass loop circuit. Liquid velocity is measured online with an electromagnetic probe (Alec Electronics) situated in the downcomer section of the closed loop. Superficial liquid velocity in the test section is then simply estimated by taking into account the relative flow area in the two sections, downcomer and test column.
Two main types of experiments were carried out, the first a simple monocomponent bed expansion which was used as a reference test and then a binary solid bed expansion where one type of particles was fluidized by a suspension made up of liquid and other larger solids. Obviously, in order to achieve the specific configuration wanted in this work the larger particle had to circulate while the smaller remains stationary in the test column, and therefore the larger solid had a density very close to that of the fluid so that their transport was facilitated. When the monocomponent test was carried out, a weighted amount of solid was charged into the system and the fluid flow rate was adjusted and varied in order to expand the solid in the vertical test section without circulating it. Different fluid superficial velocities were recorded together with the correspondent overall solid height, from which the velocity-concentration relationship could be easily estimated. On the contrary, parameter estimation for the binary-solid measurements was not so straightforward and some simplified assumption had to be made. First known weighted amount of the two solids were charged into the column. Then flow rate liquid was adjusted to values such as, while the larger solid would circulated together with the liquid, the smaller solid remained completely in the test section. Liquid superficial velocity was estimated as before, smaller solid concentration could be easily inferred from the height of the column occupied by the solid itself, whereas larger solid concentration was estimated assuming that it occupied homogeneously the whole systems. Moreover, while the smaller solid had zero velocity relative to the test tube wall, it was assumed that the difference in velocity between the liquid and the larger circulating solid is negligible. All the runs were carried out at ambient condition, and given that temperature did not change significantly a constant density of 1000 kg/m3 and viscosity of 0.001 Pa s−1 were used in the calculation. Solid fluidized material characteristics are summarized in Table 1, with the particles being practically spherical. The circulating material is always plastic with density very close to the density of water and diameters of 5, 6, and 10 mm.
4. Result and Discussion
The first set of experimental run was carried out in order to verify monocomponent bed expansion and to have a reference point for the successive work. Figure 4 reports a typical result, referring to 5 mm glass beads, from which it can be seen that bed expansion follows previously reported pattern and specifically that depicted in Figure 1. This behaviour was found to be true for every material investigated, with the exception of the smallest solid tested, zirconia, and the results are summarized in Table 2. The slopes (n and ) and the terminal velocities ( and ) reported in Table 2 were obtained by dividing the experimental voidage-velocity relationships in two sectors and fitting with a least square straight line the experimental points in each section. The intersection of the two straight lines yielded the critical particle concentration ,and the ratio of the two experimentally determined terminal velocities yielded the parameter .
As can be seen from Figures 5, 6, and 7 the present results are in broad agreement with the empirical correlations recommended in . The discrepancies, particular evident for the critical particle concentration reported in Figure 7, could well be due to the fact that the present data are well outside the range of validity of the data collected in : in that paper the maximum terminal Reynolds number was just above 1000, whereas in the present work is larger than 4500.
Binary systems, with smaller particles suspended by larger particles circulating together with the fluid, were successively investigated. Table 3 reports the combination of larger-smaller solid utilized in this work. As already mentioned before, binary-solid pairs were chosen so that the smaller solid was fluidized stationary in the column, whereas the larger particles were transported by the fluid given their low density.
Figure 8 reports a typical experimental observation, referring to zirconia 2.2 suspended by water and 5 mm plastic particles, where the effect brought about by the presence of the larger particle in the fluidising fluid on the expansion characteristics of the smaller ones is depicted. More specifically it can be seen that as the larger particles volume concentration increases, at a fixed water superficial velocity, the smaller particles bed height increases consequently. Therefore in this case also we can say that, qualitatively, the presence of the larger solid “facilitates” the fluidization of the smaller solid.
This observation was common for all the systems investigated, and a first attempt was made to quantify the effect arising from the presence of the larger particle. This situation has been studied, in a different contest, for example, by Moritomi et al.  who assumed, in their analysis, that the influence of the larger particle could be confined in a restriction of the area available for the flow of the liquid. With this simplification expansion characteristics law for the smaller solid system can be written again as when for , with n, , and being those determined from the monocomponent bed expansion earlier (Table 3).
Figures 9, 10, 11, and 12 show how this very simple approach fits the experimental observations. Considering the difficulties associated with this experimental measurement and the simplicity of the assumption made, the comparison between experiments and model can be defined satisfactory. The average error on the superficial velocity between experiments and model was smaller than 5%, with the smallest deviation being 2.1% for the 8 mm glass system (Figure 10) and the largest being 6.5% for the 5 mm glass system (Figure 11).
The question which arises at this point is if the pseudo-fluid approach, equations (6)–(9), which successfully describes the fluid dynamic effect brought about by the presence of the smaller particles on the larger particles could be applied on this case too. Unfortunately the present investigation cannot give a definite answer on this question. Three pseudo-effects were utilized in the pseudo-fluid simplification: the pseudo viscosity, the pseudo density, and the pseudo velocity, (5)–(7). It can be easily demonstrated that in this work the introduction of the restricted area for the fluid flow coincide with the pseudo-fluid velocity when the transported solid possesses a velocity equal to that of the fluid, a condition which can be assumed to hold in the systems investigated here. Unfortunately none of the other two pseudo effects (viscosity and density) could be properly validated in this work. Pseudo viscosity cannot be validated because our experiments were carried out at high Reynolds numbers, in the inertial flow regime, where the viscosity has no influence on the observed fluid dynamic behaviour so the use of (7) would make no practical difference. The other effect could not be tested either because we were forced to utilize nearly neutrally buoyant large particle, and therefore pseudo-fluid density, (6) coincides practically with the fluid density.
To the best our knowledge in this work for the first time experiments on binary-solid suspensions, with the smaller particles fluidized by a mixture of larger neutrally buoyant particles and fluid, were carried out. Minding the effect described by Rapagna et al. , the observed bed expansion characteristics of the smaller particles, at different velocities and larger particle concentrations, could be satisfactory reproduced by a simple approach based on the assumption that the influence of the larger particles is limited to a restricted area available for the fluid flow. Of note, at the same superficial velocity, the expansion of the fluidized bed increases with the larger particles concentration. Due to the high particle Reynolds number and the neutrally buoyant spheres, it was not possible to verify if the pseudo-fluid model is applicable in our case, and future works will be directed to clarify this point. The experimental information reported in this paper will be useful in determining a proper relationship for the solid-fluid drag force in binary-solid suspensions, relationship of relevant importance when computational fluid dynamic works are carried out.
- G. K. Batchelor, “Sedimentation in a dilute dispersion of spheres,” Journal of Fluid Mechanics, vol. 52, pp. 245–268, 1972.
- G. K. Batchelor, “Sedimentation in a dilute polydisperse system of interacting spheres. Part 1: General theory,” Journal of Fluid Mechanics, vol. 119, pp. 379–408, 1982.
- G. K. Batchelor and C. S. Wen, “Sedimentation in a dilute polydisperse system of interacting spheres. Part 2: Numerical results,” Journal of Fluid Mechanics, vol. 124, pp. 495–528, 1982.
- D. L. Koch and A. S. Sangani, “Particle pressure and marginal stability limits for a homogeneous monodisperse gas-fluidized bed: Kinetic theory and numerical simulations,” Journal of Fluid Mechanics, vol. 400, pp. 229–263, 1999.
- A. J. C. Ladd, “Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 1. Theoretical foundation,” Journal of Fluid Mechanics, vol. 271, pp. 285–309, 1994.
- A. J. C. Ladd, “Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 2. Numerical results,” Journal of Fluid Mechanics, vol. 271, pp. 311–339, 1994.
- A. J. C. Ladd and R. Verberg, “Lattice-Boltzmann simulations of particle-fluid suspensions,” Journal of Statistical Physics, vol. 104, no. 5-6, pp. 1191–1251, 2001.
- C. Y. Wen and Y. H. Yu, “Mechanics of fluidization,” in Chemical Engineering Progress Symposium Series, vol. 62, pp. 100–111, 1966.
- D. Gidaspow, Multiphase Flow and Fluidization: Continuum and Kinetic Theory Description, Academic Press, New York, NY, USA, 1991.
- R. Di Felice, “The voidage function for fluid-particle interaction systems,” International Journal of Multiphase Flow, vol. 20, no. 1, pp. 153–159, 1994.
- R. Di Felice, “Hydrodynamics of liquid fluidisation,” Chemical Engineering Science, vol. 50, no. 8, pp. 1213–1245, 1995.
- R. Di Felice and M. Rotondi, “Fluid-particle drag force in binary-solid suspensions,” International Journal of Chemical Reactor Engineering, vol. 10, no. 1, pp. 1542–6580.
- J. F. Richardson and W. N. Zaki, “Sedimentation and fluidization: Part I,” Transactions of the Institution of Chemical Engineers, vol. 32, pp. 35–54, 1954.
- P. N. Rowe, “A convenient empirical equation for estimation of the Richardson-Zaki exponent,” Chemical Engineering Science, vol. 42, no. 11, pp. 2795–2796, 1987.
- S. Rapagna, R. Di Felice, L. G. Gibilaro, and P. U. Foscolo, “Steady-state expansion characteristics of monosize spheres fluidized by liquids,” Chemical Engineering Communications, vol. 79, pp. 131–140, 1989.
- M. R. Dibouni, The behaviour of liquid fluidised beds containing a wide size distribution of particles [Ph.D. thesis], University of London, 1975.
- J. Garside and M. R. Al-Dibouni, “Velocity-voidage relationships for fluidization and sedimentation in solid-liquid systems,” Industrial and Engineering Chemistry Process Design and Development, vol. 16, no. 2, pp. 206–214, 1977.
- J. P. Riba and J. P. Couderc, “Expansion de couches fluidisées par des liquids,” Canadian journal of Chemical Engineering, vol. 55, pp. 118–121, 1977.
- R. Di Felice, L. G. Gibilaro, S. Rapagna, and P. U. Foscolo, “Particle mixing in a circulating liquid fluidized bed,” in Fluidization and Fluid Particle Systems: Fundamentals and Applications, pp. 32–36, December 1988.
- R. Di Felice, P. U. Foscolo, L. G. Gibilaro, and S. Rapagna, “The interaction of particles with a fluid-particle pseudo-fluid,” Chemical Engineering Science, vol. 46, no. 7, pp. 1873–1877, 1991.
- L. G. Gibilaro, K. Gallucci, R. Di Felice, and P. Pagliai, “On the apparent viscosity of a fluidized bed,” Chemical Engineering Science, vol. 62, no. 1-2, pp. 294–300, 2007.
- H. Moritomi, T. Yamagishi, and T. Chiba, “Prediction of complete mixing of liquid-fluidized binary solid particles,” Chemical Engineering Science, vol. 41, no. 2, pp. 297–305, 1986.