Magnetic stabilization of fluidized beds: Effect of magnetic field orientation

Magnetic stabilization of fluidized beds: Effect of magnetic field orientation M. J. Espina, M. A. S. Quintanillab, J. M. Valverdeb,∗ aDepartment of Applied Physics II. University of Seville. Avenida Reina Mercedes s/n, 41012 Sevilla, Spain bDepartment of Electronics and Electromagnetism. University of Seville. Avenida Reina Mercedes s/n, 41012 Sevilla, Spain Abstract Fluidized beds of granular materials can be stabilized by interparticle attractive forces which confer the expanded bed an elastic modulus that stabilizes it against flow perturbations. Stabilization in a structure of enduring contacts is seen to occur naturally due to the universal van der Waals forces for ∼50 µm particle size beds albeit in a quite reduced interval of gas velocities over the minimum fluidization velocity vmf . As shown in this work, a magnetic field may induce attractive forces between magnetizable particles thus extending the stable fluidization interval well beyond vmf . The structure of the magnetically stabilized bed is however markedly anisotropic since attractive magnetic forces are maximum along the direction of the externally imposed field which leads to the formation of chain particle aggregates. This paper shows experimental measurements on the magnetic yield stress, gas ∗Corresponding author Email addresses: [email protected] (M. J. Espin), [email protected] (M. A. S. Quintanilla), [email protected] (J. M. Valverde) Preprint submitted to Chemical Engineering Journal November 4, 2016 velocity at the transition to marginal stability and microstructure of magnetostabilized beds as affected by the direction of the magnetic field. Data shows that magnetic stabilization is optimized for co-flow fields as chain aggregates are preferentially orientated parallel to the magnetic field along the direction of minimum drag. As the magnetic field is tilted, particle chains become tilted according to a balance between the magnetic attractive force between the particles and the vertical drag force, which reduces the magnetic yield stress and therefore shortens the interval of magnetic stabilization. Keywords: Fluidization, Magnetofluidization, Magnetic stabilization, Interparticle forces, Rheology, Magnetorheological fluids 1. Introduction Fluidization is the phenomenon by virtue of which a granular material becomes suspended in a upwards directed fluid flow. This takes place when the superficial fluid velocity vgreaches a minimum fluidization velocity, vg=vmf , such that the fluid pressure drop across the particle bed is sufficiently high to support the bed weight per unit area. Gas-solid fluidization technology is widely used in many industrial applications due to the highly efficient contact between solid and gas phases that may be achieved in fluidized bed reactors [1, 2, 3, 4, 5]. The behavior of gas-solid fluidized beds depends critically on a variety of physical properties such as particles’ size and density, interparticle attractive 2 forces and fluid density and viscosity. Geldart [6] proposed a well known four different types of fluidization behavior according to particle size when monodisperse solids are fluidized by dry air at ambient pressure and temperature. Particles of size typically below dp≃20 µm are difficult to fluidize and belong to the Geldart’s C group. Strong interparticle attractive forces as compared with particles’ weight lead to a cohesive behavior that hinders fluidization. Consequently, these fine powders, when fluidized, tend to rise as a slug of solids or to form channels through which the fluid escapes rather than being homogeneously distributed through the bulk. Different types of techniques have been proposed to help these granular materials being uniformly fluidized such as the use of mechanical stirrers to break up stable channels, the addition of nanosilica beads to coat the solids and increase flowability, or fluidization under high gas pressure and/or with high viscosity gases [6, 7, 8, 9]. On the opposite side, Geldart’s D group powders, consisting of large-size beads (typically above dp&500 µm), exhibit a spouting behavior. In an spouted bed, there is a centrally located, upwardly moving diluted core above which the particles drop down to an annular region [6, 10]. For intermediate size particles, fluidization is characterized by the development of gas bubbles. Materials having a mean particle size between 20 and 100 µm appreciably expand beyond vmf and macroscopic bubbles are not seen until a minimum bubbling velocity vg> vb> vmf is reached. These powders belong to the Geldart’s A group. Interparticle attractive forces, which for these powders are similar to particles’ weight, may indeed provide 3 the fluidized bed with an effective elastic modulus that stabilizes it against small disturbances [3, 11]. In the stable state (vmf < vg< vb), interparticle contacts are permanently held by these attractive forces; the bed is jammed and takes the appearance of a weak plastic solid. Jammed bed posses a solid structure consolidated at a low stress and requires a small but non zero yield stress in order to make it flows again (see [3, 11, 12, 13, 14, 15] for a more detailed description). Powders of typical size above dp&100 µm belong to Geldart’s B group [6]. Geldart’s B fluidized beds are commonly unstable, being characterized by the development of large gas bubbles rising across the bed just at the onset of fluidization (vb≈vmf ). Bubbling may limit the performance of Geldart’s B fluidized bed reactors due to the bypassing of gas through bubbles, which reduces gas-solid contact effectiveness and leads to uncertainties in process scale up [16]. Artificial enhancement of interparticle forces may suppress the growth of gas bubbles and shift the behavior of fluidized beds from Geldart B to Geldart A. For example, bubbling beds can be stabilized by incremental addition of a liquid [17], by fluidizing them with highly adsorbing gases [18] or by fluidization at high temperature and pressure conditions [3, 7]. However, fluidization at high temperatures, the addition of liquids or the use of gases distinct to air may lead to material sintering, attrition [19], or adsorption/desorption of chemical species [20], which would modify particle properties and cause undesired fluidization behavior of the powder as well. An alternative noninvasive method for enhancing interparticle forces to stabilize bubbling beds 4 consists of the application of external magnetic fields [21, 22]. For magnetizable particles, sufficiently strong magnetic fields cause mechanical structuring of particles [23, 24] which significantly affect heat and mass transfer properties as well as the rheological behavior of these so-called magnetofluidized beds (MFBs). Several studies have demonstrated that the application of a magnetic field to Geldart’s B magnetic granular materials yields bubbling suppression beyond the minimum fluidization velocity. Channels and bubbles could be also eliminated by imposing magnetic fields to Geldart’s group C magnetic powders [25]. In the case of Geldart’s group A magnetic powders the range of gas velocities at which the fluidized bed is stabilized is extended when external magnetic fields are applied [15, 26, 27]. Magnetically stabilized beds (MSBs) are useful for a variety of potential applications such as the adsorption of contaminants from a gas stream. Magnetic stabilization has also been studied for the development of magnetically controlled valves [28, 29, 30]. In addition, a better understanding of the rheological properties of magnetized particles is of interest in geological flows [31] and magnetorheological fluids (MRFs) [32]. The interested reader may find additional details on the history and current state of the art of MFBs in extended reviews reported elsewhere [33, 34, 35, 36, 37]. Histrov has recently published a series of extended reviewes focused on the application of magnetic fields as assisting fluidization technique [38, 39, 40, 41, 42, 43, 44, 45, 46]. In the present work we have investigated the magnetofluidization behavior of a magnetite powder, which belongs to the class of Geldart A powders. Our 5 main goal is to study the effect of the orientation of the magnetic field on the structure and on the mechanical strength of the MSB. 2. Materials and experimental setup 2.1. Materials The magnetic powder used in our work consists of nearly spherical and monodisperse magnetite particles manufactured and supplied to us by Xerox Co (Fig. 1). A detailed material characterization is described in [26, 27, 47]. Particle density is ρp= 5060 kg/m3, as measured by means of an AccuPyc 1330 pycnometer. A Morphologi G3S (Malvern) characterization device was used to obtain an average particle size dp= 26.2µm (determined from the circular equivalent diameter), elongation ep= 0.098 (one minus the particle aspect ratio), and circularity cp= 0.938 (ratio of the particle’s projected area to the square of its perimeter). A magnetic analysis of the powder was performed by two different procedures reported in detail elsewere [15, 26, 47]. In the range of field intensities applied in our experiments (H < 5 kA/m), a superconducting quantum interference device (SQUID) magnetometer (SQUID Quantum Design MPMS XL) showed that the initially demagnetized particles exhibit a superparamagnetic behavior characterized by a linear and reversible response to the applied magnetic field intensity. A complementary method was used to obtain the magnetic susceptibility of the particles by measuring the change in self-inductance Lof a single layer solenoid as the magnetite powder was 6 poured into it. The material (particle) susceptibility was estimated to be χp= 11.5 from the coherent potential approximation (CPA) mixing rule equation [48]. 2.2. Experimental setup The setup used in our work (Fig. 2) is an adaptation of the Sevilla Powder Tester (SPT) to perform magnetofluidization experiments (see [15] and [47] for a detailed description). In this apparatus, the powder bed sample is held in a vertically oriented cylindrical vessel made of polycarbonate (2R= 2.6 cm internal diameter in the tests reported in this paper) and rests on a nonmagnetizable porous plate (5 µm pore size) which acts as gas distributor. Air is used as fluidizing gas after being passed across a set of filters and a Refrigerated Air Dryer (model SMC IDFA3E) to remove any possible polluting particles and humidity. By means of a series of computer-controlled valves and a mass flow controller (MKS model 1179A, 2000 cm3/min full scale), a flow of the filtered and dried air is pumped through the powder bed while the gas pressure drop ∆pacross it is measured by a differential pressure transducer (MKS model 220CD, 10 Torr full scale). The height hof the bed, which provides an average value of the particle volume fraction φ, is measured by means of an ultrasonic sensor (Senix model Ultra-S) placed on top of the vessel. Magnetite powders have been tested as affected by uniform magnetic fields applied along different directions by placing the bed in the center of 7 a pair of square Helmholtz coils (50 cm ×50 cm) that can be rotated. The orientation of the applied magnetic field was determined from the angle θ between the field and the vertical direction (see Fig. 2). The intensity Hc of the externally imposed magnetic field was controlled by adjusting the electrical current supplied by a N8741A Agilent current source circulating through the coils. Measurements taken by a magnetic probe (Hirst Magnetics Gaussmeter) show that the magnetic field strength is uniform throughout the volume occupied by the powder bed. 3. Experimental results Figure 3 illustrates experimental results on the evolution of the gas pressure drop across the bed (normalized by the powder weight per unit area) ∆p/W as the superficial gas velocity vgis increased in a typical magnetofluidization test. The first step of the experiment consists of applying a magnetic field after the sample is subjected to a sufficiently large gas flow to reach the bubbling regime (vg> vb). Under these conditions, ∆pis roughly equal to Wsince the whole structure of the bed is sustained by the gas flow. The bed remains in this fluid-like bubbling regime until vgis slowly reduced to a critical value vJknown as the jamming velocity. A further decrease of vgbelow vJcauses the bed to jam and it transits to a solid-like stable state. Note that in this manuscript (as well as in our previos works devoted to MFBs) the term jamming means the evolution from a bubbling bed to an expanded but fixed bed in part sustained by the gas flow and in part by the endur- 8 ing interparticle contacts. A detail description of the different order-disorder transitions between fixed beds, MSBs and MFBs and the usual terms used to named them can be found in [38]. This transition can be identified in Fig. 3 by the falling of ∆pbelow W. Part of the weight becomes then sustained by the enduring network of interparticle contacts in the stabilized bed, which gives rise to a consolidation stress σc=W−∆p(vg) at the bottom layer of the powder. For each test, a minimun gas velocity v0was set at which the consolidation stress reaches a value σ0=σcmax =W−∆p(v0). According to the Janssen equation [49], friction at the walls does not support a significant fraction of the powder weight if the height of the bed is below the bed diameter as is the case in our experiments. In order to measure the effect of the applied magnetic field on the mechanical strength of the MSB, the gas velocity was then progressively increased from v0while the external magnetic field is kept fixed. This is called the h-on or magnetization last operation mode [47, 50, 51, 52] in contrast with the h-off-on or magnetization first operation mode where the applied magnetic field is imposed once the bed is settled at a gas velocity v0 (the interested reader may check [53] for further details on the effect of the magnetic field orientation on fixed bed). For small values of vg, the solid-like structure remains unperturbed (bed height does not change) and ∆plinearly increases with vg. The slope sof this linear relationship between ∆pand vg is indicated in Fig. 3. At the value of vgat which ∆pequals W(the minimum fluidization velocity, vmf ), the bed should transit again to the fluid-like 9 from this competition. In this figure, measured values of φ0are plotted as a function of Hc,θand v0. As can be observed, the bed stabilizes at more expanded structures as the intensity of the field is increased and/or the field is more vertically orientated (decreasing θ). Expansion becomes progressively less marked as the gas velocity at which the bed stabilizes v0is increased, particularly for Hcbelow 2.0 kA/m. At a given Hc, as more horizontally oriented magnetic fields are applied to the sample (increasing θ), the shear forces progressively increase the angle αbetween the chained particles and the field direction which reduces the strength of fmaccording to Eq. 2. Thus, the shear force due to the gas flow is able to balance the induced magnetic force at lower vJ(Fig. 4) where the bed is more compacted (larger φ0in Fig. 7). Under these circumstances, the resistance to the passing of the gas flow across the MSB is increased as shown by the increase of sin Fig. 5. If now θis fixed and Hcis increased, stronger magnetic interactions reduce the misalignment produced by the drag force between the chained aggregates and the field direction. This fact promotes the strength of the chain of particles, which requires more intense shear forces to be broken. The transition from the bubbling to the stabilized regime takes then place at larger vJ(as seen in Fig. 4) and smaller φ0(Fig. 7). A larger porosity of these MSBs favors the passing of the gas flow, which is reflected in a reduction of sin Fig. 5. Finally, the mechanical strength of the MSB when an external mag- 16 netic field is applied will be discussed. The measured tensile yield stress σt=σt(Hc, θ, v0) in Fig. 6 stems from two different contributions. On one hand, we have the natural yield stress σt(Hc= 0, v0) arising from the van der Waals interaction and plastic yielding of particle contacts at a given consolidation velocity v0(data at Hc= 0 in Fig. 6). On the other hand, we have the magnetic yield stress σmt caused by the interparticle magnetic forces induced by the external magnetic field. Figure 9 shows the values of this magnetic contribution to the measured overall tensile yield strength, which is obtained as σmt =σt(Hc, θ, v0)−σt(Hc= 0, v0) (see [26] for more details about the procedure to estimate σmt). First of all, it is worth mentioning that, within the experimental accuracy, σmt only depends on Hcand θindicating that the increase of number of contacts per particle as v0is decreased is not significant. Note also that σmt is larger than σt(Hc= 0, v0) even for the lowest intensities of the applied magnetic field. This could be explained by the comparison of the magnitude of van deer Waals and the magnetic forces between particles. Of the several possible definitions of the magnetic Bond number in fluidized beds [42], we chose the ratio of the typical value of the attractive component of the magnetic force between particles (Eq. 2) to the particle weight Bog= 12fkf0/(πρpgd3 p). For Hc≈0.5 kA/m, this dimensionless number is Bog∼4 which is larger than the typical value of the non-magnetic Bond number (Bog∼2). On the other hand, the rheology of MSBs is affected by the orientation of the applied field for Hcabove 1.5 kA/m. Indeed, the effect of the field in a MFB would be expected to be noticeable when Bog&10 17 [64] which occurs at Hc≈1.0 kA/m for the previous definition of the magnetic bond number. This feature is also in agreement with the changes of the microstructure of MSBs illustrated in Fig. 7. The compaction of MSBs is reduced as fmbecomes more intense. Indeed, at a given Hcand as θis decreased, more expanded MSBs can be stabilized due to the existence of larger σmt. On the other hand, at a fixed orientation of the applied field, the mechanical strength (larger σmt) and the porosity of the microstructure of MSBs are promoted as Hcincreases. 4.3. Tilt angle of chained aggregates To conclude this work, we will look for a relationship between the interparticle magnetic force ~ fmand the structuring of MSBs. It must be pointed out that the magnetic field appearing in Eq. 2 should be the magnetic field in the bulk of the sample Hbwhich is the sum of the externally applied field Hc(created by the Helmholtz coils) plus the field induced by the magnetized powder (also known as the demagnetizing field) Hd. For a linearly magnetizable material (as is the case of the magnetite powder used in our work), Hbcan be obtained from the bulk magnetization Mof the bed and the bulk demagnetization factor κas: Hb=Hc+Hd=Hc−κM =Hc 1 + κχb ,(3) where χb=M/Hbis the bulk magnetic susceptibility of the sample. 18 Using the coherent potential approximation (CPA) mixing rule equation [48, 47, 27], χbcan be derived from the particle volume fraction φand the magnetic susceptibility of the material χp: χb=χpφ3 + 4χb 3 + χp+ 3χb .(4) Let us approximate the magnetite powder to a solid cylinder of the same dimensions (height and diameter) and magnetic susceptibility χbthan the MSB subjected to an external magnetic field tilted an angle θ. According to the procedure detailed in [65, 66], a finite element method (FEM) commercial program (COMSOL) was used to calculate the demagnetization factors in the radial κx,θ and vertical κz,θ directions using the bulk magnetic susceptibility of the cylinder and the cylinder aspect ratio [65]. Since χbdepends on χpand φ, Eq. 4 was employed to obtain both demagnetization factors as functions of the particle volume fraction of the MSB. Figure 10 shows the dependence of κx,θ and κz,θ on φfor different tilt angles θ. Lines corresponds to the best fittings of the results to a logarithmic law κ=a+ ln φwhere the best fitting parameters aand bare reported in Table 1. At given values of Hc,θand φ, the radial Hbx and vertical Hbz components of the magnetic field inside the bulk (and hence the overall Hb=pH2 bx +H2 bz) could be obtained from the corresponding values κx,θ and κz,θ as: Hbx =Hccos θ 1 + κx,θχb (5) 19 Hbz =Hcsin θ 1 + κz,θχb (6) Theoretical models for MSBs and MRFs [39, 66, 67, 68] predict that the magnetic contribution to the yield stress should scale as σmt ∼ξφfmwhere ξ is the average number of contacts per particle (coordination number). Since the strength of the MSB is due to the radial component of fmin Eq. 2 and fkis much larger than f⊥, the attractive magnetic force can be roughly approximated as fmr ∼H2 bcos2α. Then σmt can be estimated as: σmt =KξφH2 bcos2α(7) where Kdepends on the magnetic properties and size of the particles as well as the characteristic constants of each particular theoretical model. Figure 11(b) shows an image of the bulk of the magnetite bed stabilized by a vertically oriented magnetic field (θ= 0◦). As observed, particles aggregate into chains aligned with the external applied field (α= 0◦in Eq. 2 and Fig. 8). Using the equation ξ= 1.61(1 −φ)−1.48 to determine the coordination number [69] and Eq. 7, K=σmt/(ξφH2 bcos2α) where σmt,φ0and Hbshould be now replaced by their corresponding values at θ= 0◦(and α= 0◦). This parameter allows us to obtain αfrom Eq. 7 for other orientations of the external magnetic field. Figure 12 shows data of the estimated angle β=θ−α between chain aggregates and the vertical direction (Fig. 8) vs. Hband θ. As expected, with the exception of θ6= 0◦(Fig. 11(b)), chains of particles 20 are not aligned with the applied magnetic field (β6=θ) due to the shear force of the fluidizing gas. This result is consistent with direct visualization of the chain aggregates in the bulk of the MSB for θ= 90◦in Fig. 11(c). Snapshots were acquired by a binocular magnifying lens coupled to a CCD camera (the arrangement of the Helmholtz coils in the experimental set up did not allow us to take pictures at the other orientations tested —θ= 30◦or 60◦— without modifying the microstructure of the MSB). The tilt angle directly measured from these images is β≈40◦showing a reasonably agreement with the angle obtained from Eq. 7. Note also that chain aggregates become less tilted (smaller β) as Hbis increased. The same trend can be obtained by imposing more vertically orientated (smaller θ) magnetic fields. Histrov reported a similar trend on the inclination of the free surface of MSBs subjected to magnetic fields with different orientations [39]. These data also agree with the above reported measurements of φ0in Fig. 7. The strength of chain aggregates (as determined by fmin Eq. 2) is promoted as Hcis increased and/or θis decreased. This allows chains to keep their integrity as they become more vertically oriented (smaller β) by increasing gas velocities vg which gives rise to looser (smaller φ0) microstructures of the MSB. Although data in Fig. 12 serve to qualitatively describe the changes in the microstructure of MSBs induced by magnetic fields of different orientations, a more complex analysis would be required in a future work. For instance, Eq. 7 overestimates the attractive component fmr of the magnetic force between particles (the repulsive radial component is neglected). In addition, Eq. 21 2 does not include the effects off the surrounding particles on the magnetic interaction between magnetizable particles [70, 71, 72]. On the other hand, all usual expressions reported in literature [73] to compute ξfrom φ(including that used in this work) are derived for random assemblies of equal sized spheres while the microstructure of MSBs is formed by clusters of chained of particles aggregated in a preferential direction. 5. Conclusions This work has analyzed the effect of an externally imposed magnetic field on the stabilization of a fluidized bed of magnetite particles. Due to naturally occurring van der Waals forces of attraction between these particles (of similar order than particle weight), the fluidized bed is stabilized albeit just for a short interval of gas velocities beyond the minimum fluidization velocity over which part of the bed weight is sustained by enduring network of interparticle contacts. The imposition of a magnetic field serves to further extend the stable fluidization interval depending on the strength and orientation of the field as shown in our work. Induced attractive forces between magnetized particles lead to the formation of chain aggregates preferentially orientated along the field direction. On the other hand, the gas drag force favors orientation of these chains along the vertical direction. Thus, magnetic stabilization is optimized for vertical magnetic fields. Experimental measurements reported in this work show that the stable fluidization interval is shortened whereas the magnetic yield stress and bed expansion are reduced 22 as the magnetic field is tilted. Chain aggregates are seen then inclined at an angle that depends on the interplay of the magnetic attractive force between the particles and the gas drag force. Thus, magnetic stabilization may be fine-tuned non-invasively by means of the intensity and orientation of the externally imposed magnetic field. 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The fitting correlation factor is ρ2. κx,θ κz,θ θ(◦)a b ρ2a b ρ2 0 — — — 0.0381 -0.5494 0.9991 30 0.1155 -0.158 0.9974 0.033 -0.4758 0.9991 60 0.2001 -0.2737 0.9974 0.0191 -0.2747 0.9991 90 0.2311 -0.316 0.9974 — — — 35 Figure 1: Scanning electron microscopy (SEM) photographs of the magnetite particles used in the magnetofluidization experiments reported in this paper. Flow controller Valve Compressed air max Differential manometer max Humidity filter Hall probe Helmholtz coils Ultrasound sensor Vessel (a) Helmholtz coils Vessel (b) Figure 2: Sketch of the experimental setup used in the magnetofluidization experiments reported in this work (a). Tilt angle θof the applied external magnetic field ~ Hcwith respect the vertical direction (b). 36 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 v0 σ 0/W=1- ∆ p(v0)/W σ t/W vmf vJ s/W ∆ p/W vg (cm/s) Figure 3: Example of a typical fluidization-defluidization cycle obtained in the magnetofluidization experiments. Data of the measured gas pressure drop across the bed ∆p(normalized by the powder weight per unit area W) as a function of the superficial gas velocity vg. The labels indicate the gas velocity at the jamming transition vJ; the gas velocity v0 at which the bed is consolidated at a maximum stress σ0=W−∆p(v0); the initial slope sbetween ∆pand vg(before the fracture of the bed); the minimum fluidization velocity vfm and the measured tensile yield stress of the bed σt. In this test, the powder sample was consolidated at a gas velocity v0= 0.320 cm/s and subjected to an external magnetic field of intensity Hc= 2.89 kA/m tilted θ= 60◦with respect to gravity. 37 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 vJ (cm/s) Hc (kA/m) v0=0.015 cm/s v0=0.160 cm/s v0=0.320 cm/s v0=0.480 cm/s θ =0° θ =30° θ =60° θ =90° Figure 4: Gas velocity at the jamming transition vJversus the magnetic field intensity Hcfor different values of the tilt angle θof the applied field with respect to the vertical direction. Consolidation velocities v0are indicated. 38 0 200 400 600 800 1000 1200 1400 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 s (Pas/cm) Hc (kA/m) (a) v0=0.015 cm/s θ =0° θ =30° θ =60° θ =90° 0 200 400 600 800 1000 1200 1400 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 s (Pas/cm) Hc (kA/m) (b) v0=0.160 cm/s θ =0° θ =30° θ =60° θ =90° 0 200 400 600 800 1000 1200 1400 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 s (Pas/cm) Hc (kA/m) (c) v0=0.320 cm/s θ =0° θ =30° θ =60° θ =90° 0 200 400 600 800 1000 1200 1400 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 s (Pas/cm) Hc (kA/m) (d) v0=0.480 cm/s θ =0° θ =30° θ =60° θ =90° Figure 5: Slope of the gas pressure drop ∆pvs. the gas velocity vgbefore the breaking point as a function of the intensity Hcand tilt angle θof the externally applied magnetic field. Data obtained for v0= 0.015 cm/s (a), v0= 0.160 cm/s (b), v0= 0.320 cm/s (c) and v0= 0.480 cm/s (d), respectively. 39 0 50 100 150 200 250 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 σ t (Pa) Hc (kA/m) (a) v0=0.015 cm/s θ =0° θ =30° θ =60° θ =90° 0 50 100 150 200 250 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 σ t (Pa) Hc (kA/m) (b) v0=0.160 cm/s θ =0° θ =30° θ =60° θ =90° 0 50 100 150 200 250 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 σ t (Pa) Hc (kA/m) (c) v0=0.320 cm/s θ =0° θ =30° θ =60° θ =90° 0 50 100 150 200 250 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 σ t (Pa) Hc (kA/m) (d) v0=0.480 cm/s θ =0° θ =30° θ =60° θ =90° Figure 6: Tensile yield stress σtof the MSBs as a function of the intensity Hcof the external magnetic field applied at the indicated tilt angles with respect to the vertical direction. Experimental data were obtained at different consolidation velocities v0= 0.015 cm/s (a), v0= 0.160 cm/s (b), v0= 0.320 cm/s (c) and v0= 0.480 cm/s (d). 40 0.20 0.30 0.40 0.50 0.60 0.70 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 φ 0 Hc (kA/m) (a) v0=0.015 cm/s θ =0° θ =30° θ =60° θ =90°0.20 0.30 0.40 0.50 0.60 0.70 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 φ 0 Hc (kA/m) (b) v0=0.160 cm/s θ =0° θ =30° θ =60° θ =90° 0.20 0.30 0.40 0.50 0.60 0.70 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 φ 0 Hc (kA/m) (c) v0=0.320 cm/s θ =0° θ =30° θ =60° θ =90°0.20 0.30 0.40 0.50 0.60 0.70 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 φ 0 Hc (kA/m) (d) v0=0.480 cm/s θ =0° θ =30° θ =60° θ =90° Figure 7: Volume fraction of the powder bed φbefore the breaking point as a function of the intensity Hcand tilted angle θof the externally applied magnetic field. Data obtained for v0= 0.015 cm/s (a), v0= 0.160 cm/s (b), v0= 0.320 cm/s (c) and v0= 0.480 cm/s (d), respectively. 41