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Effect of magnetic field orientation on fluidized beds of magnetic particles: Theory and experiment
Particuology 12 (2014) 54–63
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Effect of magnetic field orientation on fluidized beds of magnetic particles: Theory and experiment Miguel Ángel Sánchez Quintanilla a,∗ , Manuel Jesús Espin b , José Manuel Valverde a a b
Department of Electronics and Electromagnetism, University of Seville, Sevilla, Spain Department of Applied Physics II, University of Seville, Sevilla, Spain
a r t i c l e
i n f o
Article history: Received 14 December 2012 Received in revised form 13 February 2013 Accepted 27 March 2013 Keywords: Magneto-fluidization Jamming transition Fluidization Fine powders Cohesion Magnetization
a b s t r a c t Geldart-A fluidized beds of fine particles experience a jamming transition between a fluid-like state and a solid-like state at a certain superficial gas velocity, that depends on the relative strength of interparticle attractive forces with respect to particle weight. Interparticle forces provide the bed with a certain tensile strength in the jammed state. In the work presented here we analyze the behavior of a fluidized bed of magnetic particles subjected to an externally applied magnetic field, which contributes to enhance interparticle forces. The importance of the magnetic contribution to interparticle forces is measured by the changes in the tensile strength and the superficial gas velocity at the jamming transition. The link of the field orientation with the microstructure of the bed is discussed. © 2013 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.
1. Introduction Gas fluidized beds are a type of multiphase flow in which a gas is forced to pass through a granular material. The particles of the material are suspended in the gas due to drag forces acting on them, while at the same time they form an interface with the gas leaving the bed, much like the molecules of a liquid. Except for small particles (typically less than a few micrometers in size) and large particles (typically above a few hundreds of micrometer in size) (Gidaspow, 1994), gas fluidized beds resemble a fluid. The bed solid fraction , defined as the fraction of the bed volume occupied by the particles, is a function of the superficial gas velocity U, defined as the velocity of the gas if the vessel containing the particles were empty. A fluidized bed generally expands with increasing superficial gas velocity U up to a point at which gas bubbles appear in the bed (Jackson, 2000). Some fluidized beds (known as GeldartB) (Geldart, 1973) bubble as soon as U is large enough to fluidize the particles (i.e., above the minimum fluidization velocity Um ). For some other beds (known as Geldart-A) (Geldart, 1973), bubbles appear at a superficial gas velocity Ub larger than Um . Between Um and Ub the fluidization is apparently homogeneous in the sense that, in volumes with linear dimensions considerably larger than the particle size, the particles and the gas appear to be perfectly
∗ Corresponding author. Tel.: +34 954557450. E-mail address: [email protected] (M.Á. Sánchez Quintanilla).
mixed and the local solid fraction of the bed is independent of position. However, such a situation is never achieved in practice in a fluid-like state because a bed of uniform solid fraction is hydrodynamically unstable (Glasser, Kevrekidis, & Sundaresan, 1997): if, by virtue of the randomness of the particle movements, a region of smaller than average solid fraction appears in the bed, the gas flow lines tend to bend toward this region because it represents a path of lower resistance to the gas flow. As a consequence, the local gas velocity increases in the region of lower solid fraction, making it harder for particles to fill this region. Whether this disturbance in the solid fraction grows to form a macroscopic bubble or gradually disappears depends on the competition between the drag exerted by the gas on the particles and the tendency of the particles to fill in the voids due to the so called particle pressure arising from collisions between particles as they move randomly inside the fluidized bed. On the other hand, particles in Geldart-A fluidized beds display two particular features:
(1) The gas pressure drop across the consolidated bed p needed to break interparticle contacts must exceed the weight per unit area of the particles W by a certain amount (Espin, Valverde, Quintanilla, & Castellanos, 2011). The yield stress p − W, which we call the tensile strength t of the bed, is required to overcome the adhesion between particles before they can begin to move relatively to each other. Measurements of t can
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http://dx.doi.org/10.1016/j.partic.2013.03.002
M.Á. Sánchez Quintanilla et al. / Particuology 12 (2014) 54–63
Nomenclature d fdrag fm f|| f⊥ f H Hc Hp Mp n k Nc P(, ϕ) p r U Ub UJ Um Uo
vs W p
p ˛ ˇ c t tnat mag t mag o J ϕ
particle diameter (m) drag force on a particle magnetic force (N) coefficient of the magnetic force coefficient of the magnetic force coefficient of the magnetic force total magnetic field created by the Helmholtz coils and the sample (A/m) magnetic field created by the Helmholtz coils (A/m) magnetic field inside a particle (A/m) particle magnetization (A/m) exponent in the Richardson–Zaki equation average number of contacts per particle maximum number of particles in a stable chain angular contact distribution gas pressure drop (Pa) distance between particle centers superficial gas velocity (m/s) bubbling velocity (m/s) superficial gas velocity at the jamming transition (m/s) minimum fluidization velocity (m/s) initial value of the superficial gas velocity (m/s) settling velocity of a single particle particle weight per unit area (Pa) particle magnetic susceptibility powder bulk susceptibility particle density (kg/m3 ) angle between the magnetic field and the line joining the centers of particles in contact maximum stable angle between a chain and the magnetic field consolidation stress (Pa) tensile strength (Pa) natural tensile strength (Pa) magnetic tensile strength (Pa) typical value of magnetic stress (Pa) solid fraction solid fraction at the jamming transition polar angle in spherical coordinates azimuthal angle in spherical coordinates gas kinematic viscosity (m2 /s)
Acronyms magnetic fluidized beds MFBs
particles (Trappe, Prasad, Cipelletti, Segre, & Weitz, 2001). For non-interacting hard spheres, the jamming transition occurs at the so-called random loose packing limit, with J ≈ 0.62 (Makse, Johnson, & Schwartz, 2000), which is slightly smaller than the random close packing limit = 0.636 (Gotoh & Finney, 1974) that represents the maximum attainable solid fraction of a random assembly of hard spheres. For real particles, the solid fraction at the jamming transition depends on the particle size (Yu, Feng, Zou, & Yang, 2003); J decreases with decreasing particle size due to the prevalence of interparticle attractive forces compared with the particle weight. For particles of a given size and nature, interparticle forces can be changed by addition of flow additives (Valverde, Quintanilla, & Castellanos, 2004), a capillary-forming liquid (Feng & Yu, 2000) or, in the case of magnetizable particles, such as the ones used in this work, by application of a magnetic field in the so-called magnetofluidized bed (MFB). In the case of a MFB, the jamming transition corresponds to the transition from a magnetically stabilized bed to a magnetically frozen bed (Siegel, 1988; Hristov, 2003). By using magnetizable particles, interparticle forces can be tuned and can be measured quantitatively as a function of the external applied field. Indeed, measurements of interparticle magnetic forces in MFBs have already been presented by other authors (Lee, 1991; Hristov, 2003; and references therein). In practice, MFBs are used in processes in which it is important to maximize the gas velocity while minimizing particle loss due to elutriation and gas by-pass due to bubble formation, such as, for example, dust collection from a gas stream in a fluidized bed (Fan et al., 2007; Wang, Gui, Shi, & Li, 2008). 2. Methods and materials 2.1. Materials We used synthetic magnetite particles supplied by Xerox Co. with average sizes of 35, 50, and 65 m, a density of
p = 5060 kg/m3 , a nearly spherical shape and consisting on agglomerates of sintered particles (Fig. 1). These particles do not present a permanent magnetization and in the range of fields used in the experiments (<5 kA/m) behave as superparamagnetic particles (Mills, 2004). Therefore, for an isolated particle, the magnetization Mp is proportional to the field Hp inside the particle: Mp = p Hp , where p is the particle susceptibility, which does not coincide with the bulk susceptibility of the powder due to the effect of neighboring particles on the magnetization of each individual particle. An approximate relation between the two susceptibilities is given by (Karkkainen, Sihvola, & Nikoskinen, 2001): = p
be used to estimate the intensity of attractive forces between particles (Castellanos, Valverde, & Quintanilla, 2004). (2) For gas velocities U > Um , the pressure drop across the powder bed equals W as long as U is increased monotonically. If U is then decreased from a value larger than Ub , the gas pressure drop remains equal to W down to a superficial gas velocity UJ between Um and Ub . The superficial gas velocity UJ marks the jamming transition of the bed (Espin et al., 2011). At the jamming transition, the relative positions of the particles in the bed become frozen in a permanent network of contacts with solid fraction J . Thus, the p versus U cycle generally presents a hysteretic behavior. The value of the solid fraction at the jamming transition J depends on the intensity of the attractive forces between the
55
3 + 4 . 3 + p + 3
(1)
By measuring the bulk susceptibility of the magnetite powders at different solid fractions and fitting the results with Eq. (1) (Fig. 2), we obtain p = 11.5 (for details of the measurement procedure for our materials, see (Espin, Valverde, Quintanilla, & Castellanos, 2010)). The magnetic force fm between two isolated magnetizable spheres of diameter d whose centers are separated by a distance r and are aligned forming an angle ˛ with a uniform magnetic field H, is given by (Clercx & Bossis, 1993): o fm = fm
o fm
d 4 r
3 =
o d2 16
(2f|| cos2 ˛ − f⊥ sin2 ˛)n + f sin(2˛)t
p 3 + p
2
(2) 2
H ,
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M.Á. Sánchez Quintanilla et al. / Particuology 12 (2014) 54–63
Fig. 3. Components of the magnetic force acting on a magnetizable sphere with diameter d exerted by a second sphere of the same material in the presence of an external magnetic field H.
2.2. Experimental setup
Fig. 1. SEM micrographs of 35 m (top) and 65 m (bottom) magnetite particles.
where n and t are unit vectors along and perpendicular to, respectively, the line that joins the centers (see Fig. 3). For p = 11.5 and using the calculation procedure explained in (Clercx & Bossis, 1993), we obtain f|| = 9.854, f⊥ = 0.582, and f = 1.833. The force between two magnetizable spheres surrounded by other spheres, as is the case for the particles in our fluidized beds, will differ from that given by Eq. (2), but nevertheless we will use Eq. (2) to estimate the magnetic force between particles, where H in Eq. (2) is the magnetic field inside the fluidized bed.
Our fluidized bed consists of a vertically oriented cylindrical vessel (2.6 cm internal diameter and 15 cm in height) fitted with a non-ferromagnetic 316 L stainless steel filter (5 m pore size) that acts as gas distributor. In all of the experiments, 26.4 g of powder were loaded into the fluidization vessel. A uniform magnetic field Hc is created in the volume of the vessel occupied by the particles using a pair of square Helmholtz coils (50 cm × 50 cm). The coils can be rotated to orientate the magnetic field either perpendicular to (cross-flow configuration) or parallel to (co-flow configuration) the gas flow. The magnetic susceptibility of the filter (approximately 0.02) is much smaller than that of the fluidized particles; thus, any effect of the filter on the total magnetic field (which is the sum of the fields that are created by the coils and the magnetite particles) will be disregarded. The fluidizing gas is dry air whose superficial velocity is controlled by a mass flow controller. The gas pressure drop p across the bed is measured using a differential pressure manometer, and the bed height is monitored using an ultrasound sensor placed at the top of the bed. All of the controls and measurements are conducted automatically by a computer equipped with a data acquisition card and software. In a typical run, a sample of particles is fluidized in a magnetic field H with a superficial gas velocity above the bubbling velocity Ub . Then, the superficial gas velocity is reduced to a value Uo , and the consolidation stress c that acts on the particles is taken as the difference W − p(Uo ), where p(Uo ) is the pressure drop across the bed at the superficial gas velocity Uo . In other words, we assume that a stress p(Uo ) is supported by the gas-particle drag and that a stress W − p(Uo ) is supported by the particles at the bottom layer of the bed without any support from the lateral cell walls. According to the Janssen solution (Nedderman, 1992), the walls do not support a significant fraction of the bed if the height of the bed is less than the bed diameter, as it is our case. Next, the superficial gas velocity is increased in small steps back to a value above Ub while maintaining the applied magnetic field Hc to obtain the tensile strength of the bed t and the superficial gas velocity at the jamming transition UJ from the curve of p versus U, as depicted in Fig. 4. 2.3. The magnetic field inside the MFB
Fig. 2. Bulk magnetic susceptibility of the powders used in the experiments as a function of the solid fraction . The solid line represents the best fit with Eq. (1) for p = 11.5.
The magnetic field H inside the MFB does not coincide with the magnetic field created by the coils Hc because H is the sum of the
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57
Fig. 4. Example of the measured gas pressure drop across a MFB (p) versus the superficial gas velocity U, illustrating the consolidation stress c , tensile strength t , and gas velocity at the jamming transition UJ . The plotted curve was measured for the 65 m particle-size magnetite powder bed operated in a co-flow field strength of 0.43 kA/m. The single-headed arrows indicate the trend of the pressure drop for increasing and decreasing superficial gas velocity.
field created by the coils plus the field created by the material. As a result, even if the magnetic field Hc is uniform, the field H inside the sample is non-uniform. In consequence, the magnetic force between two particles given by Eq. (2) will vary inside the bed. To obtain typical values for the magnetic field inside the bed, we calculated the magnetometric demagnetization factors of a cylinder, Hz /Hc for the co-flow configuration and Hx /Hc for the cross-flow configuration, using a finite element method (FEM) commercial program (COMSOL), where . . . indicates volume average (see Fig. 5). The demagnetization factors depend both on the magnetic susceptibility of the cylinder material and the cylinder aspect ratio (Chen, Pardo, & Sanchez, 2006). For our experiments, both of these parameters are related by Eq. (1) because, for a given powder mass, the aspect ratio of the sample inside the cell depends on its solid fraction . For the mass that was used in our experiments, the relation between the solid fraction and the demagnetization factors is presented in Fig. 6. The results can be fitted in the range 0.25 < < 0.55 by the expressions:
Hz
Hc
= 0.0424 − 0.556 ln().
(3)
= 0.2077 − 0.333 ln()
(4)
Hx
Hc
Fig. 5. Magnetometric demagnetization factors for a cylinder with an aspect ratio of h/R for an external field parallel to the cylinder axis (top) and perpendicular to the cylinder axis (bottom), obtained from FEM numerical calculations using COMSOL. The ticks mark the results of individual FEM calculations.
The tensile strength of the three materials increases when an external magnetic field Hc is applied to the samples: see Fig. 8 for the 35 m particle size magnetite bed operated in the cross-flow configuration. As we can see in this figure, the increase in the tensile
for the co-flow and cross-flow configurations, respectively. 3. Results and discussion 3.1. Magnetic tensile strength The beds of the three different particle sizes show a tensile strength t even in the absence of an external magnetic field, which we call the natural tensile strength tnat of the material (Fig. 7). tnat increases with the previous consolidation stress c = W − p(Uo ) acting on the material and decreases with increasing particle diameter. This dependence is typical of fine powders and can be explained by the growth of the interparticle contact area with larger consolidations, which increases the van der Waals attractive forces that act between neighboring particles (Watson, Valverde, & Castellanos, 2001).
Fig. 6. Magnetometric demagnetization factors for a mass of 26.4 g of our powders as a function of the solid fraction in the bed. The dots represent the points obtained from interpolation of the data in Fig. 4. The lines represent the fits with Eqs. (3) and (4) in the text.
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Fig. 7. Natural tensile strength tnat of the magnetite particles used in the experiments in the absence of a magnetic field as a function of the consolidation stress c acting on the bottom of the bed. mag
strength is roughly independent of the consolidation stress, which allows us to define the magnetic contribution to the tensile strength mag t as: mag
t
(Hc ) = t (Hc , c ) − tnat (Hc = 0, c ).
Fig. 9. Magnetic contribution to the tensile strength (t ) as a function of the external magnetic field created by the coils (Hc ) for cross-flow and co-flow field configurations.
(5)
A similar result has been found in previous experiments with tilted beds of magnetic materials (Hristov, 2003), in which the magnetic contribution to the cohesion was found to be independent of the consolidation stress. The magnetic contribution to the tensile mag strength t is plotted as a function of the magnetic field created by the coils Hc in Fig. 9. It can be seen that the orientation of the mag magnetic field is the most significant factor affecting t , and that its effect is stronger in the co-flow configuration, in agreement with tilted bed experiments by other authors (Hristov, 2003). 3.2. Jamming transition The superficial gas velocity at the jamming transition UJ also increases when a magnetic field is applied (Fig. 10). This increase is more marked in the co-flow configuration, and there is a significant effect of the particle size in both the co-flow and cross-flow configurations. The increase in UJ is accompanied by a decrease in the solid fraction at the jamming transition J (Fig. 11). A more open
Fig. 10. Superficial gas velocity (UJ ) at the jamming transition as a function of the external magnetic field created by the coils (Hc ) for cross-flow and co-flow orientations of the field.
Fig. 8. Dependence of the tensile strength of the 35 m particle-size magnetite powder bed with the magnetic field (cross-flow field configuration) and consolidation stress. The labels indicate the magnetic field Hc created by the coils.
Fig. 11. Solid fraction J at the jamming transition as a function of the external magnetic field Hc created by the coils for cross-flow and co-flow orientations of the field.
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59
structure (smaller J ) has fewer contacts per particle k (Suzuki, Makino, Yamada, & Iinoya, 1981), which may make the contact network weaker but it also offers less resistance to the gas flow. Therefore, part of the increase in the gas velocity at the jamming transition could be related to the decrease in J. If we assume that, for U ≥ UJ the relation between the superficial gas velocity U and the solid fraction follows the Richardson–Zaki law (Jackson, 2000): n
U = vs (1 − ) ,
(6)
in which the exponent n is a function of the Reynolds number Re = dvs / , where is the gas kinematic viscosity and vs is the terminal settling velocity of an individual particle. Keeping all other factors equal, a change in the solid fraction at the jamming transition J should be accompanied by an increase in the gas velocity at the jamming transition UJ : UJ J = −n UJo 1 − Jo
(7)
where Jo and UJo are the solid fraction and the superficial gas velocity at the jamming transition in the absence of magnetic field, respectively. For example, for magnetite with d = 35 m, n = 4.46 (Jackson, 2000), J = –0.04 and Jo = 0.40, which yields −nJ /(1 − Jo ) ≈ 0.30. However, taking the experimental values of UJ and UJo , we obtain UJ /UJo = 1.14/0.76 = 1.5, which is larger than the value given by Eq. (7). Similar numbers are obtained for the other particle sizes. Therefore, we conclude that the principal contribution to the increase in the superficial gas velocity at the jamming transition UJ is not just related to a hydrodynamic effect due to the decrease in J . The primary reason for the increase in UJ with increasing Hc must be the increase in the magnetic attractive forces between the particles, which depends on the particle size and the field orientation. At the jamming transition, the particle weight is supported by the gas drag on the particles and the interparticle attractive forces provide the network of permanent contacts with some rigidity that is able to sustain infinitesimally small stresses. If, at the point of the jamming transition, the gas velocity is increased or the magnetic field is decreased, the network of permanent contacts disappears along with the tensile strength of the bed (Lee, 1991). Since both the gas drag on each particle and the particle weight depend on the particle diameter d, particles of different diameter give different J values for a given Hc (Fig. 11). However, if we plot the gas velocity UJ divided by the settling velocity of a single particle vs against the magnetic Bond number Bomag , (Fig. 12), the most important factor that determines the jamming transition turns out to be the magnetic field orientation. Of the different possible definitions of the magnetic Bond number in fluidized beds (Hristov, 2006), we chose the ratio of the typical value of the magnetic force between particles (Eq. (2)) to the particle weight: Bog =
o 6fm ,
p d3
(8)
o was calculated from Eq. (2). Dividing U by v takes into where fm s J account the fact that the gas drag force on the particles depends on their diameter d. By using this magnetic Bond number, we take into account most of the differences in the interparticle contact forces induced by the magnetic field due to particle size (the nonmagnetic adhesion between the particles is not included in Bog ). Thus, the only parameter that has a significant effect in Fig. 12 is the field orientation. From Figs. 9 and 12, it is apparent that the microstructure of the contact network inside the powder depends on the orientation of the magnetic field because magnetic forces with the same magnitude produce different macroscopic properties for co-flow and
Fig. 12. Ratio of superficial gas velocity UJ at the jamming transition to the terminal settling velocity of a single particle vs as a function of the magnetic bond number defined in the text.
cross-flow field orientations. This dependence is discussed in the next sections. 3.3. Particle chaining When fluidized, particles are free to rearrange their positions and form chains (interpreted as lines of preferential contact) along the most favorable directions under the combined effect of the total magnetic field and the fluid drag. The relevant role played by the microstructure is supported evidenced by the fact that, in the cross-flow configuration, the magnetic tensile strength depends on whether the magnetic field is applied before or after the initialization of the sample (Valverde, Espin, Quintanilla, & Castellanos, 2009). For the co-flow configuration, it is clear that the most favorable direction is the vertical direction because the particle drag is minimized and the magnetic interaction is attractive. For the cross-flow configuration, there is a competition between the magnetic force and the fluid drag. Adapting a model from Martin and Anderson (1996), based on the mechanical stability between the strain exerted by the fluid on a chain and the interparticle attractive force induced by the field, it is expected that chains can be formed up to an angle ˇ with the field, which is given by (Espin et al., 2011):
tan ˇ =
2f|| , 2f + f⊥
(9)
if other contributions to the interparticle forces are neglected. Since the value of ˇ given by Eq. (9) is independent of the particle size, we can assume that, for all of our samples, ˇ ≈ 65◦ in the cross-flow configuration. The dependence of the gas velocity at the jamming transition UJ on the field orientation, shown in Fig. 12, can thus be explained by the fact that the drag force on a group of spheres depends on the arrangement of the spheres with respect to the gas flow direction (Fillipov, 2000). Therefore, spheres of different sizes that have the same ratio of magnetic force to particle weight form similar structures for the same external field orientation. 3.4. Relation between the tensile strength and microstructure Up to this point, we have explained the existence of a magnetic tensile strength in our beds using the dependence of the interparticle force on the magnetic field, i.e., using a discrete description of the bed. However, the magnetic tensile strength is a stress, i.e.,
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a magnitude that is defined by a continuum description. The aim of this section is to obtain a continuum description in term of the stresses of the magnetic forces between the particles in the bed. The validity of the description will be tested by comparing its predictions for the component of the stress that was experimentally measured: the magnetic tensile strength. Similar results to the ones presented here for the tensile strength and the jamming transition have been found by others authors in MFBs operated in the co-flow (Jovanovic, Sornchamni, Atwater, Akse, & Wheeler, 2004) and cross-flow (Gros, Baup, & Aurousseau, 2008) field configurations. The main difference between this work and previous works is the use of a homogenous external magnetic field in the whole volume of the bed. In a heterogeneous field, in addition to the magnetic forces between particles, there is a force proportional to the field gradient that acts on each particle (Jackson, 1999). The forces due to the field gradient are cumulative for all of the particles in the bed; thus, the arrangement of the external magnetic field has an effect on the mechanical properties of the bed. Although in our experimental setup, as discussed in Section 2.3, the total field inside the sample is not homogeneous due to the effect of the sample magnetization, the directions of the forces due to field gradients vary inside the sample and their net effect vanishes when averaged over the whole sample. For this reason, in the following, we only consider the forces due to particle–particle magnetic interaction and approximate their values using Eq. (2). It can be demonstrated that, for an ensemble of spheres with diameter d that occupy a volume V, the stress tensor ij can be found from the forces fj that act on the contacts between the particles in the form (Cambou, Dubujet, Emeriault, & Sidoroff, 1995): d ij = 2V
(fi nj + fj ni ),
(10)
contacts
where ni gives the normal to the particle surfaces at each contact point. By defining the angular distribution function P(,ϕ)d˝ as the probability of having a contact in the direction given by the solid angle d˝(,ϕ) = sin ddϕ in spherical coordinates, we obtain: ij =
3 k 2 d2
(fi nj + fj ni )P(, ϕ)d˝,
(11)
mag
≈
3 k
d2
(fm )z nz P(, ϕ)d˝.
(12)
Substituting (fm )z from Eq. (2) into the above yields: mag
t
≈
3 o kfm
d2
(2f|| cos2 ˛ − f⊥ sin2 ˛)nz + f sin(2˛)tz
× nz P(, ϕ)d˝.
(13)
In Eq. (13), ˛ is the angle between the external field Hc and the contact normal n, while the vector t is given by t = (n × Hc ) × n/|(n × Hc ) × n| (see Fig. 13). From Eq. (13), we can define a typical value of the magnetic mag contribution to the tensile strength o as: mag
o
=
3 o kfm .
d2
mag
The magnetic contribution to the tensile strength t measured mag in our experiments is plotted against o in Fig. 14. To evaluate the coordination number k, we used the relation (Suzuki, Makino, Yamada, & Iinoya, 1981): k ≈ 1.61(1 − )
where k is the average number of contacts per particle, we used V = d3 /6 and there is an additional ½ factor to take into account the fact that the angular distribution function is normalized to unity mag but each contact belongs to two particles. If we assume that t must be of the same order as the contribution of the magnetic forces to zz , the magnetic contribution to the tensile strength of the bed can be estimated from Eq. (11) as: t
Fig. 13. Geometric representation of the magnetic field and interparticle contact orientation for co-flow (a) and cross-flow (b) field configuration. The circular dot on the surface of the sphere represents the position of an interparticle contact. The vectors n and t are the unit vectors normal and tangential to the contact used in Eq. (13).
(14)
−1.46
.
(15) mag
It can be seen from Fig. 14 that the magnetic tensile strength t mag is approximately 10 times larger than o for the experiments operated in the cross-flow field configuration and approximately 100 times larger for the co-flow field configuration. The value of mag o does not include the strength of the magnetic interaction between particles (which is given by the coefficients f|| , f⊥ , and f ) or the effect of the contact orientation (which is given by the angular distribution P(,ϕ)). To evaluate the strength of these effects, we need to estimate the distribution P(,ϕ), which cannot be measured experimentally. The two most extreme cases that can be considered are an isotropic distribution of contacts and a distribution of contacts in which all of the contacts are aligned along the direction of the chains. In the following, the latter case will be referred to as the anisotropic distribution. The next subsections will be devoted to mag finding a theoretical prediction for t , assuming that the angular distribution P(,ϕ) in our experiments is between these extremes. 3.4.1. Co-flow field configuration mag For the co-flow configuration, in which H = Hc uz , t can be calculated from Eq. (13) by setting ˛ = , n = ur and t = u (the unitary vectors in normal spherical coordinates), as schematically shown in Fig. 13(a). In this case: nz = cos ,
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61
mag
versus the typical stress due to the magnetic forces between the particles, as predicted Fig. 14. Experimental values of the magnetic contribution to the tensile strength t from Eq. (14). The boundaries predicted by the continuum model given by Eqs. (19), (21), (24), and (27), are depicted. The horizontal error bars for the co-flow configuration are of the order of the marker size.
tz = − sin ,
o
(fm )z = fm (2f|| cos2 − f⊥ sin2 ) cos − f sin(2) sin .
In this case, the terms in f⊥ and f are zero because all of the contacts are aligned with the field. (16)
(a) For an isotropic distribution, the contacts have the same probability of pointing in any direction; thus: P(, ϕ) =
1 . 4
(17)
Inserting Eqs. (16) and (17) into Eq. (13) leads to: mag
t
1 mag = o 2
(18)
which finally yields: mag
t
mag
= o
5
f|| −
.
(19)
1 1 ı() + ı( − ) , 4 sin
(20)
where ı(x) represents the Dirac’s distribution function. Using Eq. (13) again, we obtain: mag t
1 mag = o 2
d (2f|| cos2 − f⊥ sin2 ) cos2
0
− f sin(2) sin cos
ı() + ı( − ) .
The terms in the integrand in which sin appears are zero at = 0 and = , and: mag
t
mag
= o
2f|| .
(21)
1/2
, (22)
sin cos sin ϕ
2
cos2 + sin cos2 ϕ
1/2 =
sin cos sin ϕ , sin ˛
which yield:
The terms in f⊥ and f decrease the magnetic tensile strength because they originate from the repulsive term and the term that is perpendicular to the line of contact in Eq. (2), which are nonzero for contacts that are not aligned with the field. Both f⊥ and f are much smaller than f|| for our particles, and, because their coefficients are also much smaller than that of f|| , which represents the attractive interactions, the net effect is a positive magnetic contribution to the tensile strength. (b) For an anisotropic contact distribution in which all of the contacts were directed along the vertical direction, the angle ˛ can only take the values ˛ = 0 and ˛ = ; thus: P(, ϕ) =
nz = cos , tz =
2 4 f⊥ − f 15 15
sin ˛ = uy × n = cos2 + sin2 cos2 ϕ
sin d (2f|| cos2 − f⊥ sin2 ) cos
−f sin(2) sin cos ,
2
cos ˛ = n · uy = sin sin ϕ,
0
3.4.2. Cross-flow field configuration For the cross-flow configuration, assuming that the external mag field Hc points in the uy direction (see Fig. 13(b)), t can be calculated from Eq. (13) by setting n = ur , t = n × (n × uy )/|n × (n × uy )| and using:
o (fm )z = fm 2f|| sin2 sin2 ϕ − f⊥ (1 − sin2 sin2 ϕ) cos
+ 2f sin3 cos sin2 ϕ.
(23)
(a) For an isotropic distribution (Eq. (17)), using Eq. (13) with (fm )z that is given by Eq. (23) leads to: mag
t
mag
= o
2
15
f|| −
4 2 f⊥ + f 15 15
.
(24)
An alternative way of finding Eq. (24) is to use the fact that the zz component of the stress tensor for an isotropic distribution in the cross-flow configuration is equal to the xx component for an isotropic distribution in the co-flow configuration. As in Eq. (19), f⊥ and f measure the contributions of the repulsive terms along and perpendicular to the line of contact in Eq. (2). However, in contrast with the co-flow result, although the repulsive term in this case decreases the magnetic tensile strength, the forces perpendicmag because, ular to the contacts make a positive contribution to t in most contacts, these forces have a component along the vertical direction. The coefficient of f|| is smaller than in the co-flow case, which fits the experimental fact that cross-flow fields have a smaller effect on the tensile strength than co-flow fields. (b) For an anisotropic contact distribution in which all of the contacts were directed at an angle ˇ = 65◦ (Eq. (9)) from the magnetic field in the plane determined by the magnetic field and
62
M.Á. Sánchez Quintanilla et al. / Particuology 12 (2014) 54–63
the vertical direction (the direction of the gas flow): P(, ϕ) =
1 1
2 sin
2
ı ϕ−
+ı ϕ +
2
ı −
ı −
−ˇ 2
+ˇ 2
between the particles is assumed to be equal to the magnetic interaction between isolated particle pairs and the stress tensor inside the powder is built from an ensemble average of the interparticle forces.
.
(25)
Using Eq. (13) with (fm )z given by Eq. (23) yields, after integration over ϕ: mag
t
=
1 mag 2 o
−
d 2f|| sin2 cos2 − f⊥ (1 − sin2 ) cos2
+2f sin2 cos2 × ı −
+ˇ +ı − −ˇ 2 2
, (26)
thus: mag
t
mag
= o
2f|| sin2 ˇ cos2 ˇ−f⊥ sin4 ˇ +
1 f sin2 (2ˇ) . 2
(27)
This case strongly depends strongly on the orientation of the particles chains. In fact, for ˇ = 0 (the chains are aligned with the magnetic field), there would be no contribution of the magnetic forces to the tensile strength. Eqs. (27) and (24) explain why, in the cross-flow configuration, the magnetic contribution to the tensile strength is so dependent on whether the magnetic field is applied to a bubbling bed or to a static bed (Hristov, 2003; Valverde et al., 2009). When the field is applied to a bubbling bed, particles can form chains at an angle to the external field, and Eq. (27) applies, while, if the field is applied to a static bed, no chains can be formed and the microstructure of the bed is more akin to the structure that was assumed in the derivation of Eq. (24). The large difference between Eqs. (24) and (27) in the coefficient of the term that is proportional to f|| results in a large variation in the magnetic contribution to the tensile strength with the method of application of the field. The pairs of Eqs. (19) and (21) and Eqs. (24) and (27) limit the areas where the experimental values of tmat should fit in Fig. 14 for each magnetic field configuration. It can be seen that, by including the effect of contact orientation and the interaction strength, the trend that is followed by the experiments can be qualitatively reproduced. However, there is still a quantitative disagreement of a factor between 5 and 10 for both field configurations. Two facts may explain this disagreement: first, by using Eq. (2) to calculate the magnetic force between particles, we are assuming that magnetic forces are pairwise additive, which is not the case (Clercx & Bossis, 1993) because the particles act as magnetic circuits for the field lines (Valverde et al., 2009). Second, the data of the tensile mag strength from which t was calculated correspond to consolidated states of the bed and not to states at the jamming transition. When the sample is consolidated, some restructuring of the contact network occurs and consequently, the preferential orientation of the contacts may not be the same as that in the jammed state. 4. Conclusions In this work it was found that, for a fluidized bed of particles magnetized by an externally applied magnetic field, the field orientation plays a major role in the solid fraction and the superficial gas velocity at the jamming transition by modifying the microstructural arrangement of the particles, when they are free to rearrange before assembling into a jammed state as the gas velocity is sufficiently decreased. Different microstructures have different strength when subjected to tensile stresses. The order of magnitude of the resulting magnetic tensile strength has been predicted using a continuum model in which the magnetic interaction
Acknowledgments This work was supported by the Andalusian Government (Junta de Andalucia, contract FQM-5735) and by the Spanish Government (Ministerio de Ciencia y Tecnologia, contract FIS2011-25161). We acknowledge the Microscopy Service of the Innovation, Technology and Research Center (CITIUS, University of Seville) and, in particular, the assistance from Dr. Francisco Varela. References Cambou, B., Dubujet, P., Emeriault, F., & Sidoroff, F. (1995). Homogenization for granular materials. European Journal of Mechanics A/Solids, 14, 255–276. Castellanos, A., Valverde, J. M., & Quintanilla, M. A. S. (2004). The Sevilla powder tester: A tool for characterizing the physical properties of fine cohesive powders at very small consolidations. KONA, 22, 66–81. Chen, D.-X., Pardo, E., & Sanchez, A. (2006). Fluxmetric and magnetometric demagnetizing factors for cylinders. Journal of Magnetism and Magnetic Materials, 306, 135–146. Clercx, H. J. H., & Bossis, G. (1993). Many body electrostatic interactions in electrorheological fluids. Physical Review E, 48, 2721–2738. Espin, M. J., Valverde, J. M., Quintanilla, M. A. S., & Castellanos, A. (2010). Rheology of magnetofluidized fine powders: The role of interparticle contact forces. Journal of Rheology, 54(4), 719–740. Espin, M. J., Valverde, J. M., Quintanilla, M. A. S., & Castellanos, A. (2011). Stabilization of gas-fluidized beds of magnetic powders by a cross-flow magnetic field. Journal of Fluid Mechanics, 680, 80–113. Fan, M., Luo, Z., Zhao, Y., Chen, Q., Tao, D., Tao, X., et al. (2007). Effects of magnetic field on fluidization properties of magnetic pearls. China Particuology, 5, 151–155. Feng, C. L., & Yu, A. B. (2000). Quantification of the relationship between porosity and interparticle forces for the packing of wet uniform spheres. Journal of Colloid and Interface Science, 231, 136–142. Fillipov, A. V. (2000). Drag and torque on clusters of N arbitrary spheres at low Reynolds number. Journal of Colloid and Interface Science, 229, 184–195. Geldart, D. (1973). Types of gas fluidization. Powder Technology, 7, 285–292. Gidaspow, D. (1994). Multiphase flow and fluidization: Continuum and kinetic theory descriptions. London: Academic Press. Glasser, B. J., Kevrekidis, I. G., & Sundaresan, S. (1997). Fully developed travelling wave solutions and bubble formation in fluidized beds. Journal of Fluid Mechanics, 334, 157–188. Gotoh, K., & Finney, J. L. (1974). Statistical geometrical approach to random packing density of equal spheres. Nature, 252, 202–205. Gros, F., Baup, S., & Aurousseau, M. (2008). Hydrodynamic study of a liquid/solid fluidized bed under transverse electromagnetic field. Powder Technology, 183, 152–160. Hristov, J. Y. (2003). Magnetic field assisted fluidization – A unified approach. Part 2. Solids batch gas-fluidized beds: Versions and rheology. Reviews in Chemical Engineering, 19, 1–132. Hristov, J. Y. (2006). Magnetic field assisted fluidization – A unified approach. Part 5. A Hydrodynamic treatise on liquid–solid fluidized beds. Reviews in Chemical Engineering, 22, 195–377. Jackson, J. D. (1999). Classical electrodynamics (3rd ed., pp. ). New York: John Wiley & Sons. Jackson, R. (2000). The dynamics of fluidized particles. Cambridge: Cambridge University Press. Jovanovic, G. N., Sornchamni, T., Atwater, J. E., Akse, J. R., & Wheeler@@Jr., R. R. (2004). Magnetically assisted liquid–solid fluidization in normal and microgravity conditions: Experiment and theory. Powder Technology, 148, 80–91. Karkkainen, K., Sihvola, A., & Nikoskinen, K. (2001). Analysis of a three-dimensional dielectric mixture with finite difference method. IEEE Transactions on Geoscence and Remote Sensing, 39, 1013–1018. Lee, W.-K. (1991). A review of the rheology of magnetically stabilized fluidized beds. Powder Technology, 64, 69–80. Makse, H. A., Johnson, D. L., & Schwartz, L. M. (2000). Packing of compressible granular materials. Physical Review Letters, 84(18), 4160–4163. Martin, J. E., & Anderson, R. A. (1996). Chain model of electrorheology. The Journal of Chemical Physics, 104, 4814–4827. Mills, A. A. (2004). The lodestone: History, physics, and formation. Annals of Science, 61, 273–319. Nedderman, R. M. (1992). Statics and kinematics of granular materials. Cambridge: Cambridge University Press. Siegel, J. H. (1988). Magnetically frozen beds. Powder Technology, 55, 127–132. Suzuki, M., Makino, K., Yamada, M., & Iinoya, K. (1981). Study on the coordination number in a system of randomly packed, uniform-sized spherical particles. International Chemical Engineering, 21, 482–488.
M.Á. Sánchez Quintanilla et al. / Particuology 12 (2014) 54–63 Trappe, V., Prasad, V., Cipelletti, L., Segre, P. N., & Weitz, D. A. (2001). Jamming phase diagram for attractive particles. Nature, 411, 772–775. Valverde, J. M., Espin, M. J., Quintanilla, M. A. S., & Castellanos, A. (2009). Magnetofluidization of fine magnetite powder. Physical Review E, 79, 031306. Valverde, J. M., Quintanilla, M. A. S., & Castellanos, A. (2004). Jamming threshold of dry fine powders. Physical Review Letters, 92(25), 258303.
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Wang, Y., Gui, K., Shi, M., & Li, C. (2008). Removal of dust from flue gas in magnetically stabilized fluidized bed. Particuology, 6, 116–119. Watson, P. K., Valverde, J. M., & Castellanos, A. (2001). The tensile strength and free volume of cohesive powders compressed by gas flow. Powder Technology, 115, 45–50. Yu, A. B., Feng, C. L., Zou, R. P., & Yang, R. Y. (2003). On the relationship between porosity and interparticle forces. Powder Technology, 130, 70–76.
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Particuology journal homepage: www.elsevier.com/locate/partic
Effect of magnetic field orientation on fluidized beds of magnetic particles: Theory and experiment Miguel Ángel Sánchez Quintanilla a,∗ , Manuel Jesús Espin b , José Manuel Valverde a a b
Department of Electronics and Electromagnetism, University of Seville, Sevilla, Spain Department of Applied Physics II, University of Seville, Sevilla, Spain
a r t i c l e
i n f o
Article history: Received 14 December 2012 Received in revised form 13 February 2013 Accepted 27 March 2013 Keywords: Magneto-fluidization Jamming transition Fluidization Fine powders Cohesion Magnetization
a b s t r a c t Geldart-A fluidized beds of fine particles experience a jamming transition between a fluid-like state and a solid-like state at a certain superficial gas velocity, that depends on the relative strength of interparticle attractive forces with respect to particle weight. Interparticle forces provide the bed with a certain tensile strength in the jammed state. In the work presented here we analyze the behavior of a fluidized bed of magnetic particles subjected to an externally applied magnetic field, which contributes to enhance interparticle forces. The importance of the magnetic contribution to interparticle forces is measured by the changes in the tensile strength and the superficial gas velocity at the jamming transition. The link of the field orientation with the microstructure of the bed is discussed. © 2013 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.
1. Introduction Gas fluidized beds are a type of multiphase flow in which a gas is forced to pass through a granular material. The particles of the material are suspended in the gas due to drag forces acting on them, while at the same time they form an interface with the gas leaving the bed, much like the molecules of a liquid. Except for small particles (typically less than a few micrometers in size) and large particles (typically above a few hundreds of micrometer in size) (Gidaspow, 1994), gas fluidized beds resemble a fluid. The bed solid fraction , defined as the fraction of the bed volume occupied by the particles, is a function of the superficial gas velocity U, defined as the velocity of the gas if the vessel containing the particles were empty. A fluidized bed generally expands with increasing superficial gas velocity U up to a point at which gas bubbles appear in the bed (Jackson, 2000). Some fluidized beds (known as GeldartB) (Geldart, 1973) bubble as soon as U is large enough to fluidize the particles (i.e., above the minimum fluidization velocity Um ). For some other beds (known as Geldart-A) (Geldart, 1973), bubbles appear at a superficial gas velocity Ub larger than Um . Between Um and Ub the fluidization is apparently homogeneous in the sense that, in volumes with linear dimensions considerably larger than the particle size, the particles and the gas appear to be perfectly
∗ Corresponding author. Tel.: +34 954557450. E-mail address: [email protected] (M.Á. Sánchez Quintanilla).
mixed and the local solid fraction of the bed is independent of position. However, such a situation is never achieved in practice in a fluid-like state because a bed of uniform solid fraction is hydrodynamically unstable (Glasser, Kevrekidis, & Sundaresan, 1997): if, by virtue of the randomness of the particle movements, a region of smaller than average solid fraction appears in the bed, the gas flow lines tend to bend toward this region because it represents a path of lower resistance to the gas flow. As a consequence, the local gas velocity increases in the region of lower solid fraction, making it harder for particles to fill this region. Whether this disturbance in the solid fraction grows to form a macroscopic bubble or gradually disappears depends on the competition between the drag exerted by the gas on the particles and the tendency of the particles to fill in the voids due to the so called particle pressure arising from collisions between particles as they move randomly inside the fluidized bed. On the other hand, particles in Geldart-A fluidized beds display two particular features:
(1) The gas pressure drop across the consolidated bed p needed to break interparticle contacts must exceed the weight per unit area of the particles W by a certain amount (Espin, Valverde, Quintanilla, & Castellanos, 2011). The yield stress p − W, which we call the tensile strength t of the bed, is required to overcome the adhesion between particles before they can begin to move relatively to each other. Measurements of t can
1674-2001/$ – see front matter © 2013 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.partic.2013.03.002
M.Á. Sánchez Quintanilla et al. / Particuology 12 (2014) 54–63
Nomenclature d fdrag fm f|| f⊥ f H Hc Hp Mp n k Nc P(, ϕ) p r U Ub UJ Um Uo
vs W p
p ˛ ˇ c t tnat mag t mag o J ϕ
particle diameter (m) drag force on a particle magnetic force (N) coefficient of the magnetic force coefficient of the magnetic force coefficient of the magnetic force total magnetic field created by the Helmholtz coils and the sample (A/m) magnetic field created by the Helmholtz coils (A/m) magnetic field inside a particle (A/m) particle magnetization (A/m) exponent in the Richardson–Zaki equation average number of contacts per particle maximum number of particles in a stable chain angular contact distribution gas pressure drop (Pa) distance between particle centers superficial gas velocity (m/s) bubbling velocity (m/s) superficial gas velocity at the jamming transition (m/s) minimum fluidization velocity (m/s) initial value of the superficial gas velocity (m/s) settling velocity of a single particle particle weight per unit area (Pa) particle magnetic susceptibility powder bulk susceptibility particle density (kg/m3 ) angle between the magnetic field and the line joining the centers of particles in contact maximum stable angle between a chain and the magnetic field consolidation stress (Pa) tensile strength (Pa) natural tensile strength (Pa) magnetic tensile strength (Pa) typical value of magnetic stress (Pa) solid fraction solid fraction at the jamming transition polar angle in spherical coordinates azimuthal angle in spherical coordinates gas kinematic viscosity (m2 /s)
Acronyms magnetic fluidized beds MFBs
particles (Trappe, Prasad, Cipelletti, Segre, & Weitz, 2001). For non-interacting hard spheres, the jamming transition occurs at the so-called random loose packing limit, with J ≈ 0.62 (Makse, Johnson, & Schwartz, 2000), which is slightly smaller than the random close packing limit = 0.636 (Gotoh & Finney, 1974) that represents the maximum attainable solid fraction of a random assembly of hard spheres. For real particles, the solid fraction at the jamming transition depends on the particle size (Yu, Feng, Zou, & Yang, 2003); J decreases with decreasing particle size due to the prevalence of interparticle attractive forces compared with the particle weight. For particles of a given size and nature, interparticle forces can be changed by addition of flow additives (Valverde, Quintanilla, & Castellanos, 2004), a capillary-forming liquid (Feng & Yu, 2000) or, in the case of magnetizable particles, such as the ones used in this work, by application of a magnetic field in the so-called magnetofluidized bed (MFB). In the case of a MFB, the jamming transition corresponds to the transition from a magnetically stabilized bed to a magnetically frozen bed (Siegel, 1988; Hristov, 2003). By using magnetizable particles, interparticle forces can be tuned and can be measured quantitatively as a function of the external applied field. Indeed, measurements of interparticle magnetic forces in MFBs have already been presented by other authors (Lee, 1991; Hristov, 2003; and references therein). In practice, MFBs are used in processes in which it is important to maximize the gas velocity while minimizing particle loss due to elutriation and gas by-pass due to bubble formation, such as, for example, dust collection from a gas stream in a fluidized bed (Fan et al., 2007; Wang, Gui, Shi, & Li, 2008). 2. Methods and materials 2.1. Materials We used synthetic magnetite particles supplied by Xerox Co. with average sizes of 35, 50, and 65 m, a density of
p = 5060 kg/m3 , a nearly spherical shape and consisting on agglomerates of sintered particles (Fig. 1). These particles do not present a permanent magnetization and in the range of fields used in the experiments (<5 kA/m) behave as superparamagnetic particles (Mills, 2004). Therefore, for an isolated particle, the magnetization Mp is proportional to the field Hp inside the particle: Mp = p Hp , where p is the particle susceptibility, which does not coincide with the bulk susceptibility of the powder due to the effect of neighboring particles on the magnetization of each individual particle. An approximate relation between the two susceptibilities is given by (Karkkainen, Sihvola, & Nikoskinen, 2001): = p
be used to estimate the intensity of attractive forces between particles (Castellanos, Valverde, & Quintanilla, 2004). (2) For gas velocities U > Um , the pressure drop across the powder bed equals W as long as U is increased monotonically. If U is then decreased from a value larger than Ub , the gas pressure drop remains equal to W down to a superficial gas velocity UJ between Um and Ub . The superficial gas velocity UJ marks the jamming transition of the bed (Espin et al., 2011). At the jamming transition, the relative positions of the particles in the bed become frozen in a permanent network of contacts with solid fraction J . Thus, the p versus U cycle generally presents a hysteretic behavior. The value of the solid fraction at the jamming transition J depends on the intensity of the attractive forces between the
55
3 + 4 . 3 + p + 3
(1)
By measuring the bulk susceptibility of the magnetite powders at different solid fractions and fitting the results with Eq. (1) (Fig. 2), we obtain p = 11.5 (for details of the measurement procedure for our materials, see (Espin, Valverde, Quintanilla, & Castellanos, 2010)). The magnetic force fm between two isolated magnetizable spheres of diameter d whose centers are separated by a distance r and are aligned forming an angle ˛ with a uniform magnetic field H, is given by (Clercx & Bossis, 1993): o fm = fm
o fm
d 4 r
3 =
o d2 16
(2f|| cos2 ˛ − f⊥ sin2 ˛)n + f sin(2˛)t
p 3 + p
2
(2) 2
H ,
56
M.Á. Sánchez Quintanilla et al. / Particuology 12 (2014) 54–63
Fig. 3. Components of the magnetic force acting on a magnetizable sphere with diameter d exerted by a second sphere of the same material in the presence of an external magnetic field H.
2.2. Experimental setup
Fig. 1. SEM micrographs of 35 m (top) and 65 m (bottom) magnetite particles.
where n and t are unit vectors along and perpendicular to, respectively, the line that joins the centers (see Fig. 3). For p = 11.5 and using the calculation procedure explained in (Clercx & Bossis, 1993), we obtain f|| = 9.854, f⊥ = 0.582, and f = 1.833. The force between two magnetizable spheres surrounded by other spheres, as is the case for the particles in our fluidized beds, will differ from that given by Eq. (2), but nevertheless we will use Eq. (2) to estimate the magnetic force between particles, where H in Eq. (2) is the magnetic field inside the fluidized bed.
Our fluidized bed consists of a vertically oriented cylindrical vessel (2.6 cm internal diameter and 15 cm in height) fitted with a non-ferromagnetic 316 L stainless steel filter (5 m pore size) that acts as gas distributor. In all of the experiments, 26.4 g of powder were loaded into the fluidization vessel. A uniform magnetic field Hc is created in the volume of the vessel occupied by the particles using a pair of square Helmholtz coils (50 cm × 50 cm). The coils can be rotated to orientate the magnetic field either perpendicular to (cross-flow configuration) or parallel to (co-flow configuration) the gas flow. The magnetic susceptibility of the filter (approximately 0.02) is much smaller than that of the fluidized particles; thus, any effect of the filter on the total magnetic field (which is the sum of the fields that are created by the coils and the magnetite particles) will be disregarded. The fluidizing gas is dry air whose superficial velocity is controlled by a mass flow controller. The gas pressure drop p across the bed is measured using a differential pressure manometer, and the bed height is monitored using an ultrasound sensor placed at the top of the bed. All of the controls and measurements are conducted automatically by a computer equipped with a data acquisition card and software. In a typical run, a sample of particles is fluidized in a magnetic field H with a superficial gas velocity above the bubbling velocity Ub . Then, the superficial gas velocity is reduced to a value Uo , and the consolidation stress c that acts on the particles is taken as the difference W − p(Uo ), where p(Uo ) is the pressure drop across the bed at the superficial gas velocity Uo . In other words, we assume that a stress p(Uo ) is supported by the gas-particle drag and that a stress W − p(Uo ) is supported by the particles at the bottom layer of the bed without any support from the lateral cell walls. According to the Janssen solution (Nedderman, 1992), the walls do not support a significant fraction of the bed if the height of the bed is less than the bed diameter, as it is our case. Next, the superficial gas velocity is increased in small steps back to a value above Ub while maintaining the applied magnetic field Hc to obtain the tensile strength of the bed t and the superficial gas velocity at the jamming transition UJ from the curve of p versus U, as depicted in Fig. 4. 2.3. The magnetic field inside the MFB
Fig. 2. Bulk magnetic susceptibility of the powders used in the experiments as a function of the solid fraction . The solid line represents the best fit with Eq. (1) for p = 11.5.
The magnetic field H inside the MFB does not coincide with the magnetic field created by the coils Hc because H is the sum of the
M.Á. Sánchez Quintanilla et al. / Particuology 12 (2014) 54–63
57
Fig. 4. Example of the measured gas pressure drop across a MFB (p) versus the superficial gas velocity U, illustrating the consolidation stress c , tensile strength t , and gas velocity at the jamming transition UJ . The plotted curve was measured for the 65 m particle-size magnetite powder bed operated in a co-flow field strength of 0.43 kA/m. The single-headed arrows indicate the trend of the pressure drop for increasing and decreasing superficial gas velocity.
field created by the coils plus the field created by the material. As a result, even if the magnetic field Hc is uniform, the field H inside the sample is non-uniform. In consequence, the magnetic force between two particles given by Eq. (2) will vary inside the bed. To obtain typical values for the magnetic field inside the bed, we calculated the magnetometric demagnetization factors of a cylinder, Hz /Hc for the co-flow configuration and Hx /Hc for the cross-flow configuration, using a finite element method (FEM) commercial program (COMSOL), where . . . indicates volume average (see Fig. 5). The demagnetization factors depend both on the magnetic susceptibility of the cylinder material and the cylinder aspect ratio (Chen, Pardo, & Sanchez, 2006). For our experiments, both of these parameters are related by Eq. (1) because, for a given powder mass, the aspect ratio of the sample inside the cell depends on its solid fraction . For the mass that was used in our experiments, the relation between the solid fraction and the demagnetization factors is presented in Fig. 6. The results can be fitted in the range 0.25 < < 0.55 by the expressions:
Hz
Hc
= 0.0424 − 0.556 ln().
(3)
= 0.2077 − 0.333 ln()
(4)
Hx
Hc
Fig. 5. Magnetometric demagnetization factors for a cylinder with an aspect ratio of h/R for an external field parallel to the cylinder axis (top) and perpendicular to the cylinder axis (bottom), obtained from FEM numerical calculations using COMSOL. The ticks mark the results of individual FEM calculations.
The tensile strength of the three materials increases when an external magnetic field Hc is applied to the samples: see Fig. 8 for the 35 m particle size magnetite bed operated in the cross-flow configuration. As we can see in this figure, the increase in the tensile
for the co-flow and cross-flow configurations, respectively. 3. Results and discussion 3.1. Magnetic tensile strength The beds of the three different particle sizes show a tensile strength t even in the absence of an external magnetic field, which we call the natural tensile strength tnat of the material (Fig. 7). tnat increases with the previous consolidation stress c = W − p(Uo ) acting on the material and decreases with increasing particle diameter. This dependence is typical of fine powders and can be explained by the growth of the interparticle contact area with larger consolidations, which increases the van der Waals attractive forces that act between neighboring particles (Watson, Valverde, & Castellanos, 2001).
Fig. 6. Magnetometric demagnetization factors for a mass of 26.4 g of our powders as a function of the solid fraction in the bed. The dots represent the points obtained from interpolation of the data in Fig. 4. The lines represent the fits with Eqs. (3) and (4) in the text.
58
M.Á. Sánchez Quintanilla et al. / Particuology 12 (2014) 54–63
Fig. 7. Natural tensile strength tnat of the magnetite particles used in the experiments in the absence of a magnetic field as a function of the consolidation stress c acting on the bottom of the bed. mag
strength is roughly independent of the consolidation stress, which allows us to define the magnetic contribution to the tensile strength mag t as: mag
t
(Hc ) = t (Hc , c ) − tnat (Hc = 0, c ).
Fig. 9. Magnetic contribution to the tensile strength (t ) as a function of the external magnetic field created by the coils (Hc ) for cross-flow and co-flow field configurations.
(5)
A similar result has been found in previous experiments with tilted beds of magnetic materials (Hristov, 2003), in which the magnetic contribution to the cohesion was found to be independent of the consolidation stress. The magnetic contribution to the tensile mag strength t is plotted as a function of the magnetic field created by the coils Hc in Fig. 9. It can be seen that the orientation of the mag magnetic field is the most significant factor affecting t , and that its effect is stronger in the co-flow configuration, in agreement with tilted bed experiments by other authors (Hristov, 2003). 3.2. Jamming transition The superficial gas velocity at the jamming transition UJ also increases when a magnetic field is applied (Fig. 10). This increase is more marked in the co-flow configuration, and there is a significant effect of the particle size in both the co-flow and cross-flow configurations. The increase in UJ is accompanied by a decrease in the solid fraction at the jamming transition J (Fig. 11). A more open
Fig. 10. Superficial gas velocity (UJ ) at the jamming transition as a function of the external magnetic field created by the coils (Hc ) for cross-flow and co-flow orientations of the field.
Fig. 8. Dependence of the tensile strength of the 35 m particle-size magnetite powder bed with the magnetic field (cross-flow field configuration) and consolidation stress. The labels indicate the magnetic field Hc created by the coils.
Fig. 11. Solid fraction J at the jamming transition as a function of the external magnetic field Hc created by the coils for cross-flow and co-flow orientations of the field.
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59
structure (smaller J ) has fewer contacts per particle k (Suzuki, Makino, Yamada, & Iinoya, 1981), which may make the contact network weaker but it also offers less resistance to the gas flow. Therefore, part of the increase in the gas velocity at the jamming transition could be related to the decrease in J. If we assume that, for U ≥ UJ the relation between the superficial gas velocity U and the solid fraction follows the Richardson–Zaki law (Jackson, 2000): n
U = vs (1 − ) ,
(6)
in which the exponent n is a function of the Reynolds number Re = dvs / , where is the gas kinematic viscosity and vs is the terminal settling velocity of an individual particle. Keeping all other factors equal, a change in the solid fraction at the jamming transition J should be accompanied by an increase in the gas velocity at the jamming transition UJ : UJ J = −n UJo 1 − Jo
(7)
where Jo and UJo are the solid fraction and the superficial gas velocity at the jamming transition in the absence of magnetic field, respectively. For example, for magnetite with d = 35 m, n = 4.46 (Jackson, 2000), J = –0.04 and Jo = 0.40, which yields −nJ /(1 − Jo ) ≈ 0.30. However, taking the experimental values of UJ and UJo , we obtain UJ /UJo = 1.14/0.76 = 1.5, which is larger than the value given by Eq. (7). Similar numbers are obtained for the other particle sizes. Therefore, we conclude that the principal contribution to the increase in the superficial gas velocity at the jamming transition UJ is not just related to a hydrodynamic effect due to the decrease in J . The primary reason for the increase in UJ with increasing Hc must be the increase in the magnetic attractive forces between the particles, which depends on the particle size and the field orientation. At the jamming transition, the particle weight is supported by the gas drag on the particles and the interparticle attractive forces provide the network of permanent contacts with some rigidity that is able to sustain infinitesimally small stresses. If, at the point of the jamming transition, the gas velocity is increased or the magnetic field is decreased, the network of permanent contacts disappears along with the tensile strength of the bed (Lee, 1991). Since both the gas drag on each particle and the particle weight depend on the particle diameter d, particles of different diameter give different J values for a given Hc (Fig. 11). However, if we plot the gas velocity UJ divided by the settling velocity of a single particle vs against the magnetic Bond number Bomag , (Fig. 12), the most important factor that determines the jamming transition turns out to be the magnetic field orientation. Of the different possible definitions of the magnetic Bond number in fluidized beds (Hristov, 2006), we chose the ratio of the typical value of the magnetic force between particles (Eq. (2)) to the particle weight: Bog =
o 6fm ,
p d3
(8)
o was calculated from Eq. (2). Dividing U by v takes into where fm s J account the fact that the gas drag force on the particles depends on their diameter d. By using this magnetic Bond number, we take into account most of the differences in the interparticle contact forces induced by the magnetic field due to particle size (the nonmagnetic adhesion between the particles is not included in Bog ). Thus, the only parameter that has a significant effect in Fig. 12 is the field orientation. From Figs. 9 and 12, it is apparent that the microstructure of the contact network inside the powder depends on the orientation of the magnetic field because magnetic forces with the same magnitude produce different macroscopic properties for co-flow and
Fig. 12. Ratio of superficial gas velocity UJ at the jamming transition to the terminal settling velocity of a single particle vs as a function of the magnetic bond number defined in the text.
cross-flow field orientations. This dependence is discussed in the next sections. 3.3. Particle chaining When fluidized, particles are free to rearrange their positions and form chains (interpreted as lines of preferential contact) along the most favorable directions under the combined effect of the total magnetic field and the fluid drag. The relevant role played by the microstructure is supported evidenced by the fact that, in the cross-flow configuration, the magnetic tensile strength depends on whether the magnetic field is applied before or after the initialization of the sample (Valverde, Espin, Quintanilla, & Castellanos, 2009). For the co-flow configuration, it is clear that the most favorable direction is the vertical direction because the particle drag is minimized and the magnetic interaction is attractive. For the cross-flow configuration, there is a competition between the magnetic force and the fluid drag. Adapting a model from Martin and Anderson (1996), based on the mechanical stability between the strain exerted by the fluid on a chain and the interparticle attractive force induced by the field, it is expected that chains can be formed up to an angle ˇ with the field, which is given by (Espin et al., 2011):
tan ˇ =
2f|| , 2f + f⊥
(9)
if other contributions to the interparticle forces are neglected. Since the value of ˇ given by Eq. (9) is independent of the particle size, we can assume that, for all of our samples, ˇ ≈ 65◦ in the cross-flow configuration. The dependence of the gas velocity at the jamming transition UJ on the field orientation, shown in Fig. 12, can thus be explained by the fact that the drag force on a group of spheres depends on the arrangement of the spheres with respect to the gas flow direction (Fillipov, 2000). Therefore, spheres of different sizes that have the same ratio of magnetic force to particle weight form similar structures for the same external field orientation. 3.4. Relation between the tensile strength and microstructure Up to this point, we have explained the existence of a magnetic tensile strength in our beds using the dependence of the interparticle force on the magnetic field, i.e., using a discrete description of the bed. However, the magnetic tensile strength is a stress, i.e.,
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M.Á. Sánchez Quintanilla et al. / Particuology 12 (2014) 54–63
a magnitude that is defined by a continuum description. The aim of this section is to obtain a continuum description in term of the stresses of the magnetic forces between the particles in the bed. The validity of the description will be tested by comparing its predictions for the component of the stress that was experimentally measured: the magnetic tensile strength. Similar results to the ones presented here for the tensile strength and the jamming transition have been found by others authors in MFBs operated in the co-flow (Jovanovic, Sornchamni, Atwater, Akse, & Wheeler, 2004) and cross-flow (Gros, Baup, & Aurousseau, 2008) field configurations. The main difference between this work and previous works is the use of a homogenous external magnetic field in the whole volume of the bed. In a heterogeneous field, in addition to the magnetic forces between particles, there is a force proportional to the field gradient that acts on each particle (Jackson, 1999). The forces due to the field gradient are cumulative for all of the particles in the bed; thus, the arrangement of the external magnetic field has an effect on the mechanical properties of the bed. Although in our experimental setup, as discussed in Section 2.3, the total field inside the sample is not homogeneous due to the effect of the sample magnetization, the directions of the forces due to field gradients vary inside the sample and their net effect vanishes when averaged over the whole sample. For this reason, in the following, we only consider the forces due to particle–particle magnetic interaction and approximate their values using Eq. (2). It can be demonstrated that, for an ensemble of spheres with diameter d that occupy a volume V, the stress tensor ij can be found from the forces fj that act on the contacts between the particles in the form (Cambou, Dubujet, Emeriault, & Sidoroff, 1995): d ij = 2V
(fi nj + fj ni ),
(10)
contacts
where ni gives the normal to the particle surfaces at each contact point. By defining the angular distribution function P(,ϕ)d˝ as the probability of having a contact in the direction given by the solid angle d˝(,ϕ) = sin ddϕ in spherical coordinates, we obtain: ij =
3 k 2 d2
(fi nj + fj ni )P(, ϕ)d˝,
(11)
mag
≈
3 k
d2
(fm )z nz P(, ϕ)d˝.
(12)
Substituting (fm )z from Eq. (2) into the above yields: mag
t
≈
3 o kfm
d2
(2f|| cos2 ˛ − f⊥ sin2 ˛)nz + f sin(2˛)tz
× nz P(, ϕ)d˝.
(13)
In Eq. (13), ˛ is the angle between the external field Hc and the contact normal n, while the vector t is given by t = (n × Hc ) × n/|(n × Hc ) × n| (see Fig. 13). From Eq. (13), we can define a typical value of the magnetic mag contribution to the tensile strength o as: mag
o
=
3 o kfm .
d2
mag
The magnetic contribution to the tensile strength t measured mag in our experiments is plotted against o in Fig. 14. To evaluate the coordination number k, we used the relation (Suzuki, Makino, Yamada, & Iinoya, 1981): k ≈ 1.61(1 − )
where k is the average number of contacts per particle, we used V = d3 /6 and there is an additional ½ factor to take into account the fact that the angular distribution function is normalized to unity mag but each contact belongs to two particles. If we assume that t must be of the same order as the contribution of the magnetic forces to zz , the magnetic contribution to the tensile strength of the bed can be estimated from Eq. (11) as: t
Fig. 13. Geometric representation of the magnetic field and interparticle contact orientation for co-flow (a) and cross-flow (b) field configuration. The circular dot on the surface of the sphere represents the position of an interparticle contact. The vectors n and t are the unit vectors normal and tangential to the contact used in Eq. (13).
(14)
−1.46
.
(15) mag
It can be seen from Fig. 14 that the magnetic tensile strength t mag is approximately 10 times larger than o for the experiments operated in the cross-flow field configuration and approximately 100 times larger for the co-flow field configuration. The value of mag o does not include the strength of the magnetic interaction between particles (which is given by the coefficients f|| , f⊥ , and f ) or the effect of the contact orientation (which is given by the angular distribution P(,ϕ)). To evaluate the strength of these effects, we need to estimate the distribution P(,ϕ), which cannot be measured experimentally. The two most extreme cases that can be considered are an isotropic distribution of contacts and a distribution of contacts in which all of the contacts are aligned along the direction of the chains. In the following, the latter case will be referred to as the anisotropic distribution. The next subsections will be devoted to mag finding a theoretical prediction for t , assuming that the angular distribution P(,ϕ) in our experiments is between these extremes. 3.4.1. Co-flow field configuration mag For the co-flow configuration, in which H = Hc uz , t can be calculated from Eq. (13) by setting ˛ = , n = ur and t = u (the unitary vectors in normal spherical coordinates), as schematically shown in Fig. 13(a). In this case: nz = cos ,
M.Á. Sánchez Quintanilla et al. / Particuology 12 (2014) 54–63
61
mag
versus the typical stress due to the magnetic forces between the particles, as predicted Fig. 14. Experimental values of the magnetic contribution to the tensile strength t from Eq. (14). The boundaries predicted by the continuum model given by Eqs. (19), (21), (24), and (27), are depicted. The horizontal error bars for the co-flow configuration are of the order of the marker size.
tz = − sin ,
o
(fm )z = fm (2f|| cos2 − f⊥ sin2 ) cos − f sin(2) sin .
In this case, the terms in f⊥ and f are zero because all of the contacts are aligned with the field. (16)
(a) For an isotropic distribution, the contacts have the same probability of pointing in any direction; thus: P(, ϕ) =
1 . 4
(17)
Inserting Eqs. (16) and (17) into Eq. (13) leads to: mag
t
1 mag = o 2
(18)
which finally yields: mag
t
mag
= o
5
f|| −
.
(19)
1 1 ı() + ı( − ) , 4 sin
(20)
where ı(x) represents the Dirac’s distribution function. Using Eq. (13) again, we obtain: mag t
1 mag = o 2
d (2f|| cos2 − f⊥ sin2 ) cos2
0
− f sin(2) sin cos
ı() + ı( − ) .
The terms in the integrand in which sin appears are zero at = 0 and = , and: mag
t
mag
= o
2f|| .
(21)
1/2
, (22)
sin cos sin ϕ
2
cos2 + sin cos2 ϕ
1/2 =
sin cos sin ϕ , sin ˛
which yield:
The terms in f⊥ and f decrease the magnetic tensile strength because they originate from the repulsive term and the term that is perpendicular to the line of contact in Eq. (2), which are nonzero for contacts that are not aligned with the field. Both f⊥ and f are much smaller than f|| for our particles, and, because their coefficients are also much smaller than that of f|| , which represents the attractive interactions, the net effect is a positive magnetic contribution to the tensile strength. (b) For an anisotropic contact distribution in which all of the contacts were directed along the vertical direction, the angle ˛ can only take the values ˛ = 0 and ˛ = ; thus: P(, ϕ) =
nz = cos , tz =
2 4 f⊥ − f 15 15
sin ˛ = uy × n = cos2 + sin2 cos2 ϕ
sin d (2f|| cos2 − f⊥ sin2 ) cos
−f sin(2) sin cos ,
2
cos ˛ = n · uy = sin sin ϕ,
0
3.4.2. Cross-flow field configuration For the cross-flow configuration, assuming that the external mag field Hc points in the uy direction (see Fig. 13(b)), t can be calculated from Eq. (13) by setting n = ur , t = n × (n × uy )/|n × (n × uy )| and using:
o (fm )z = fm 2f|| sin2 sin2 ϕ − f⊥ (1 − sin2 sin2 ϕ) cos
+ 2f sin3 cos sin2 ϕ.
(23)
(a) For an isotropic distribution (Eq. (17)), using Eq. (13) with (fm )z that is given by Eq. (23) leads to: mag
t
mag
= o
2
15
f|| −
4 2 f⊥ + f 15 15
.
(24)
An alternative way of finding Eq. (24) is to use the fact that the zz component of the stress tensor for an isotropic distribution in the cross-flow configuration is equal to the xx component for an isotropic distribution in the co-flow configuration. As in Eq. (19), f⊥ and f measure the contributions of the repulsive terms along and perpendicular to the line of contact in Eq. (2). However, in contrast with the co-flow result, although the repulsive term in this case decreases the magnetic tensile strength, the forces perpendicmag because, ular to the contacts make a positive contribution to t in most contacts, these forces have a component along the vertical direction. The coefficient of f|| is smaller than in the co-flow case, which fits the experimental fact that cross-flow fields have a smaller effect on the tensile strength than co-flow fields. (b) For an anisotropic contact distribution in which all of the contacts were directed at an angle ˇ = 65◦ (Eq. (9)) from the magnetic field in the plane determined by the magnetic field and
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M.Á. Sánchez Quintanilla et al. / Particuology 12 (2014) 54–63
the vertical direction (the direction of the gas flow): P(, ϕ) =
1 1
2 sin
2
ı ϕ−
+ı ϕ +
2
ı −
ı −
−ˇ 2
+ˇ 2
between the particles is assumed to be equal to the magnetic interaction between isolated particle pairs and the stress tensor inside the powder is built from an ensemble average of the interparticle forces.
.
(25)
Using Eq. (13) with (fm )z given by Eq. (23) yields, after integration over ϕ: mag
t
=
1 mag 2 o
−
d 2f|| sin2 cos2 − f⊥ (1 − sin2 ) cos2
+2f sin2 cos2 × ı −
+ˇ +ı − −ˇ 2 2
, (26)
thus: mag
t
mag
= o
2f|| sin2 ˇ cos2 ˇ−f⊥ sin4 ˇ +
1 f sin2 (2ˇ) . 2
(27)
This case strongly depends strongly on the orientation of the particles chains. In fact, for ˇ = 0 (the chains are aligned with the magnetic field), there would be no contribution of the magnetic forces to the tensile strength. Eqs. (27) and (24) explain why, in the cross-flow configuration, the magnetic contribution to the tensile strength is so dependent on whether the magnetic field is applied to a bubbling bed or to a static bed (Hristov, 2003; Valverde et al., 2009). When the field is applied to a bubbling bed, particles can form chains at an angle to the external field, and Eq. (27) applies, while, if the field is applied to a static bed, no chains can be formed and the microstructure of the bed is more akin to the structure that was assumed in the derivation of Eq. (24). The large difference between Eqs. (24) and (27) in the coefficient of the term that is proportional to f|| results in a large variation in the magnetic contribution to the tensile strength with the method of application of the field. The pairs of Eqs. (19) and (21) and Eqs. (24) and (27) limit the areas where the experimental values of tmat should fit in Fig. 14 for each magnetic field configuration. It can be seen that, by including the effect of contact orientation and the interaction strength, the trend that is followed by the experiments can be qualitatively reproduced. However, there is still a quantitative disagreement of a factor between 5 and 10 for both field configurations. Two facts may explain this disagreement: first, by using Eq. (2) to calculate the magnetic force between particles, we are assuming that magnetic forces are pairwise additive, which is not the case (Clercx & Bossis, 1993) because the particles act as magnetic circuits for the field lines (Valverde et al., 2009). Second, the data of the tensile mag strength from which t was calculated correspond to consolidated states of the bed and not to states at the jamming transition. When the sample is consolidated, some restructuring of the contact network occurs and consequently, the preferential orientation of the contacts may not be the same as that in the jammed state. 4. Conclusions In this work it was found that, for a fluidized bed of particles magnetized by an externally applied magnetic field, the field orientation plays a major role in the solid fraction and the superficial gas velocity at the jamming transition by modifying the microstructural arrangement of the particles, when they are free to rearrange before assembling into a jammed state as the gas velocity is sufficiently decreased. Different microstructures have different strength when subjected to tensile stresses. The order of magnitude of the resulting magnetic tensile strength has been predicted using a continuum model in which the magnetic interaction
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