Effect of particle size and interparticle force on the fluidization behavior of gas-fluidized beds

Effect of particle size and interparticle force on the fluidization behavior of gas-fluidized beds J. M. Valverde,1A. Castellanos,1P. Mills,2and M. A. S. Quintanilla1 1Departamento de Electronica y Electromagnetismo, Universidad de Sevilla, Avenida Reina Mercedes s/n, 41012 Sevilla, Spain 2LPMDI, Universite ´de Marne-la-Valle ´e, 5 boulevard DesCartes, Champs-sur-Marne, F-77454 Marne-la-Valle ´e Cedex 2, France 共Received 14 November 2002; published 23 May 2003兲 Gas-fluidized powders of fine particles display a fluidlike regime in which the bed does not have a yield strength, it expands uniformly as the gas velocity is increased and macroscopic bubbles are absent. In this paper we test the extension of this fluidlike regime as a function of particle size and interparticle attractive force. Our results show that for sufficiently large particles, bubbling initiates just after the solidlike fluidized regime as it is obtained experimentally by other workers. A scaling behavior of the solid-phase pressure in the fluidlike regime and a predictive criterion for the onset of macroscopic bubbling are analyzed in the light of these results. DOI: 10.1103/PhysRevE.67.051305 PACS number共s兲: 45.70.⫺n, 47.55.Kf, 47.55.Mh I. INTRODUCTION The behavior of fine powders is a keystone of an increasing number of high-technology industrial processes. For example, fluidized beds, consisting of a vertical column vessel partially filled with a powder resting on a porous filter and a flowing fluid pumped from below, are extensively used in applications that take advantage of their excellent fluid-solid mixing and transport capabilities. Back in 1973 Geldart 关1兴proposed an empirical classification of powders according to their fluidization properties that has been widely used by chemical engineers. Geldart summarized his observations in terms of the particle size versus the relative density difference between the fluid phase and the solid particles. According to Geldart, Group A共aeratable兲powders give a region of ‘‘homogeneous’’ fluidization in which the particles are supported by the drag force of the fluid and the bed expands smoothly as the fluid velocity is increased, taking the appearance of a low viscosity fluid. Above a certain fluid velocity this fluidlike regime is followed by a bubbling regime in which a large part of the gas is trapped from the dense phase by rising macroscopic bubbles and bed expansion is curtailed. More common Group Bpowders consist of coarser and denser particles than Group Apowder particles and give only bubbling fluidization. This group includes sandlike materials. Group Cpowders are very fine, cohesive powders that are incapable of fluidize and tend to rise as a slug of solids or to form channels through which the fluid will escape rather than being distributed through the bulk. The border between Group A and Bpowders is not clear cut. For instance, it has been found that the fluid pressure influences the bubbling behavior. Elevated pressure produce more homogeneous gas-solid flow structure by intensifying particle-fluid interaction and suppressing particle-particle interactions, thereby extending the uniform fluidization regime 关2兴. A fine powder showing an amazing behavior and used in the xerographic industry is toner. In the last decades image quality requirements leaded to a reduction of toner particle size in expenses of loosing the good flowability exhibited by the old-fashion toners, for which the interparticle attractive forces were comparable with the inertial forces. Nowadays xerographic toners are made of polymer particles with a volume average size dp⬍⬃10 ␮ m. For such a small particle size we would predict Group Ctypical sticky behavior since interparticle attractive forces are several orders of magnitude larger than particle weight. The need of reducing cohesiveness was approached by means of the use of additives. The addition of fumed silica nanoparticles, which became dispersed on the surface of the toner particles, decreases the interparticle adhesion and thus causes a shift to well-behaved Group Apowders. Moreover, the small particle size and its low density ( ␳ p⬃1 gr/cm3) allowed for an extended interval of ‘‘homogeneous’’ fluidization in which the bed could expand uniformly up to reach solid volume fractions as low as 0.1 关3兴. This behavior contrasts with the behavior of Group A powders usually encountered in commercial gas-fluidized systems, for which the interval of nonbubbling fluidization is rather short. Experiments reported by Rietema and co-workers 关4兴, Tsinontides and Jackson 关5兴, and Menon and Durian 关6兴on gasfluidized particles of size dp⭓50 ␮ m indicated the absence of grain dynamics in the uniformly fluidized state. These researchers asserted that in the fluidized state the particles are static in direct contact with each other, thus providing an elastic modulus to the bed that stabilizes it against small perturbations. This picture was indeed upheld by theoretical investigations based on the kinetic theory. Koch and Sangani 关7兴derived the particle-phase pressure of a homogeneous suspension of particles in a gas interacting via instantaneous hard-sphere collisions and hydrodynamic interactions. Their linear stability analysis indicated that the homogeneous state of a fluidlike gas-fluidized bed is always unstable, leading them to the conclusion that bubbling can be only restrained by a solidlike behavior of the expanded beds in agreement with Rietema and other’s experiments. However, it contradicted empirical observations 关8兴on some particular gasfluidized systems belonging to Group Ain the Geldart’s classification scheme that exhibit a fluidlike ‘‘homogeneous’’ regime. Our measurements on the yield strength and diffusivity of our gas-fluidized fine powders with reduced cohesiveness clearly showed that both regimes, the solidlike 共below a certain gas velocity vclarger than the minimum fluidization velocity vmf) and the fluidlike 共above vc), might be encountered in the so-called interval of ‘‘homogeneous’’ fluidization 关9兴. PHYSICAL REVIEW E 67, 051305 共2003兲 1063-651X/2003/67共5兲/051305共6兲/$20.00 ©2003 The American Physical Society67 051305-1 A deeper insight into the local dynamics of the fluidlike regime has recently revealed the presence of mesoscale spatiotemporal structures and short-lived voidage instabilities 关3兴whose characteristic temporal frequency gradually increases with gas velocity. Well-organized fluctuations 关10,11兴 and transient voids 关12兴have also been discovered in liquidfluidized beds. Duru and Guazelli showed that in liquid fluidization local voids grew into macroscopic real bubble-like structures for sufficiently high density beads 共particle density/liquid density ␳ p/ ␳ f⬇8) 关12兴. Thus, in close similarity with observations in liquid-fluidized beds, we should conclude that our gas-fluidized beds are actually unstable throughout the fluidlike nonbubbling regime. Even though this instability does not grow into fully developed bubbles, it certainly denies the applicability of hydrodynamic linear or weakly nonlinear stability analyses that have been employed in the past to predict the onset of bubbling 关13兴. Still the question of why large bubbles develop above certain fluid velocities remains unexplained. Undoubtedly bubbling must be inhibited by some complex hydrodynamic mechanism for sufficiently small particles, yet a relative role of interparticle forces should not be unequivocally dismissed. Numerical simulations of bubbling beds have recently shown that when the attractive interparticle force was increased the average size of bubbles decreased and the coalescence rate of bubbles slowed down 关14兴. It is well known that van der Waals attractive force, much larger than particle weight for micrometer-sized particles, causes the aggregation of fluidized fine particles 关15兴. Independent empirical observations suggest that the size of aggregates could influence the transition to the bubbly regime: The homogeneity of fluidization has been substantially improved in some cases when the aggregates could be disrupted by means of external agents such as vibration 关16兴or ultrasonics 关17兴. Changes occurring during fluidization of some fcc catalysts have been observed as the interparticle attractive forces were increased by increasing temperature 关18兴. Bubbling of beds of metallic particles could be suppressed controlling interparticle forces by means of an external magnetic field 关19兴. One of our aims in this paper is to contribute to fill the gap between the observations on one side of a bubble-free fluidlike regime for small particles and on the other side of just bubbling fluidization for larger particles. Following a suggestion by Jackson 关20兴, the extension of the fluidlike regime interval and the onset of bubbling are experimentally investigated using specially designed powders of several particle sizes and different cohesiveness. This work will allow us to elucidate a role of interparticle forces and particle size on fluidization behavior. II. EXPERIMENTAL POWDERS Toner particles made of the random copolymer styrene n-butylmethacrylate ( ␳ p⫽1.14 g/cm3) were produced by an attrition process followed by size classification by means of a cyclone separator apparatus. In this way we had available four sets of powders with volume averaged particle size dp ⫽19.1, 15.4, 11.8, and 7.8 ␮ m. These powders were subsequently blended with nanoparticles of fumed silica, which became dispersed on the toner particle surface. 32% and 8% of surface area coverage 共SAC兲were used. The addition of these nanoparticles allowed for a reduction of the interparticle adhesion force because silica is considerably harder than polymer and thus the additive increases the contact hardness; silica also decreases the adhesion force by reducing the typical size of the surface asperities at contact 关21,22兴. In the fluidlike regime toner particles are aggregated. Assuming that in sedimentation aggregates behave like effective spheres with a hydrodynamic radius equal to their radius of gyration R, we derived elsewhere the average number of particles per aggregate 共N兲and the average ratio of aggregate size to particle size (k⬅2R/dp)关15,23兴共see Table I兲. III. EXPERIMENTAL SETUP In the experiments reported in this paper a powder sample of mass mwas held in a r⫽2.21 cm radius polycarbonate cylinder, the base of which is a sintered metal filter of 5 ␮ m pore size. For this bed radius wall effects are negligible 关24兴. A dry nitrogen tank furnished with a mass flow controller supplied adjustable gas flow. All the measurements started by driving the powder into the bubbling regime by imposing a very large gas flow to the bed. Once the powder is in the bubbling regime it had lost memory of its previous history. Then the gas flow was decreased down to a given value. After the bed had reached a stationary state its height hwas read from an ultrasonic sensor and the average solid volume fraction ␾ 关 ␾ ⫽m/( ␳ p ␲ r2h)兴was computed. IV. RESULTS In Fig. 1 the solid volume fraction is plotted as a function of the superficial gas velocity vgfor powders with the same SAC 共32%兲, i.e., same value of interparticle van der Waals force (FvdW⯝1.6 nN 关15兴兲. In Fig. 1 we delineate the transitions between the different fluidization regimes. The gas velocity at the transition to the solidlike regime vcis taken from the point at which the development of enduring contacts cause a slow down of the settling process 关15兴. We must note, however, that the transition solidlike-fluidlike takes place along an interval of gas velocities in which transient TABLE I. dp, particle size; N, average number of particles per aggregate in the fluidized bed; k, average ratio of aggregate size to particle size; ␾ b expt , experimental value of the solid volume fraction at the initiation of bubbling; ␾ b theor , theoretical value of the solid volume fraction at the initiation of bubbling according to Eq. 共7兲; Fr, Froude number at the bubbling point based on aggregate size 共values corresponding to 32% SAC兲. dp共 ␮ m兲Nk ␾ b expt ␾ b theor Fr 7.8 63 5.22 0.089 0.087 0.14 11.8 23.7 3.55 0.140 0.146 0.09 15.4 12.4 2.72 0.177 0.188 0.07 19.1 9.6 2.45 0.228 0.229 0.04 VALVERDE et al. PHYSICAL REVIEW E 67, 051305 共2003兲 051305-2 fluidlike regions alternate with transient solid networks. This issue is extensively discussed elsewhere 关3兴. In the fluidlike regime ␾ decreases steadily as the gas velocity is increased down to the minimum value ␾ bthat the two-phase system is able to sustain. If the gas velocity is further increased above this critical point, macroscopic bubbles rise across the bed and ␾ increases, with oscillations of the free surface due to gas bubbles burst. The gas velocity at incipient macroscopic bubbling vbis well determined and we take it from the point in Fig. 1 at which the solid fraction starts to increase as the gas velocity is increased. V. DISCUSSION A clear outcome from Fig. 1 is that the fluidlike regime shortens as particle size is increased. Let us consider the interval ⌬vg⬅vb⫺vc, which is plotted in Fig. 2 against the ratio of interparticle van der Waals force FvdW to particle weight Wp共granular Bond number Bog⬅FvdW /Wp). It is seen that ⌬vgshows a logarithmic increase with Bog.An extrapolation of the results gives ⌬vg→0 for Bog⬍⬃10, indicating that the fluidlike regime should shrink to zero for fluidized beds of large enough particles. For instance, in the case of glass beads ( ␳ p⯝2.5 g/cm3) we estimate FvdW ⬅Ada/(24z0 2)⯝8nN共Hamaker constant A⯝1.5⫻10⫺19 J, typical asperity size da⯝0.2 ␮ m关25兴, and minimum interparticle distance z0⯝4Å关26兴兲, and thus the condition Bog ⬍⬃10 would be met for dp⬎⬃40 ␮ m. Thus, for particles of size larger than ⬃40 ␮ m, macroscopic bubbling would initiate immediately after the solidlike regime. This is in accordance with experiments reported in the literature 关6兴and with a fluidization test we made in our lab, using a system of glass beads with dp⯝60 ␮ m. Interestingly, a similar threshold is found for the aggregation of particles 关15,27兴, suggesting that the formation of low-density aggregates plays a crucial role in promoting the gas-solid interaction and thus in preventing the growth of large bubbles along an extended pseudoturbulent fluidlike interval. Figure 3 shows the effect of reducing the surface silica coverage from 32% down to 8% 共thus increasing the van der Waals force from 1.6 nN up to 3.4 nN 关15兴兲 for a given particle size 共the same effect is obtained for other particle sizes兲. We observe that, in the fluidlike regime, the bed of more cohesive particles shows larger values of ␾ for a given gas velocity, whereas the opposite happens in the solidlike regime. This behavior may be rationalized as follows: As the additive surface coverage is decreased the interparticle attractive force increases and particles tend to be slightly more aggregated 关15兴. Thus for larger aggregates the gas velocity required to reach a given value of the solid volume fraction FIG. 1. Average solid volume fraction ␾ of the fluidized powder as a function of the superficial gas velocity vg. Particle size is indicated in the inset. Surface additive coverage is held constant 共32%兲. The limits between the solidlike, fluidlike, and bubbling regimes are shown. Error bars represent the typical standard deviation of the time signal of ␾ for a given vgin each regime. The arrows show the predicted values for the solid volume fraction at incipient bubbling, using a modified version of the FG criterion. FIG. 2. Extension of the fluidlike regime as a function of the ratio of interparticle van der Waals force to particle weight. The SAC% of the toner particles is indicated. FIG. 3. Average solid volume fraction ␾ of the fluidized powder as a function of the superficial gas velocity vgfor powders with the same particle size (15.4 ␮ m) but different SAC 共shown兲. The arrows indicate the bubbling point. This figure has been recently reviewed by Sundaresan 关39兴. Please note that the figure shown in that review displays the interstitial gas velocity viinstead of the superficial gas velocity vg关vi⫽vg/共1⫺ ␾ 兲兴 for the powder with 32% SAC. This minor difference does not alter the discussion presented in the text. EFFECT OF PARTICLE SIZE AND INTERPARTICLE... PHYSICAL REVIEW E 67, 051305 共2003兲 051305-3 in the diluted fluidlike regime should be higher as we see in Fig. 3. Accordingly the gas velocity needed to reach the bubbling regime increases with the reduction of surface additive. On the other side when the gas velocity is decreased down to the point at which the solidlike regime is reached aggregates roll or slide over each other until they land at static positions and form permanent networks. The higher the interparticle attractive force the less efficient is the rearrangement of aggregates leaving behind larger void spaces. This leads to smaller values of ␾ for a given gas velocity as we see in Fig. 3. We have plotted in Fig. 2 the extension of the fluidlike regime interval for the beds with increased cohesiveness and it can be seen that vb⫺vcfollows the same law with Bog. Even though our experiments are restricted to toner particles for which the van der Waals force provides the main source of interparticle attraction, recent numerical simulations as well as experiments with magnetic and wet particles also show that the role of interparticle force on fluidization behavior must scale with the particle weight 关28兴. Furthermore, our results agree qualitatively with the results of Rhodes et al. 关28兴who obtain by means of a Discrete Element Method simulation that the interval of nonbubbling bed expansion shrinks to zero for Bog→1. VI. A PREDICTIVE CRITERION FOR THE INITIATION OF MACROSCOPIC BUBBLING More than half a century ago Wilhelm and Kwauk delineated the transition to the bubbling regime based on the empirical Froude number criterion, Fr⫽v2/(gdp)⬎1 where vis the fluid velocity and gis the gravity field 关29兴. However, researchers found an order of magnitude of difference between, for instance, air-fluidized glass beads 共Fr⫽1.1兲and air-fluidized lead shot 共Fr⫽85兲关30兴. In addition, the criterion failed to predict the observed instability in liquid-fluidized beds, for which Fr⬃0.1⬍1关12,30兴. Using the effective size of the aggregates for our gas-fluidized powders we estimate Fr⬃10⫺1⫺10⫺2共Table I兲at incipient bubbling, thus denying the applicability of the Wilhelm and Kwauk criterion in close analogy with liquid-fluidized beds. Later in 1984, Foscolo and Gibilaro 共FG兲关31兴formulated the predictive criterion for ␾ b, 共gdp兲1/2⫽0.56nvt ␾ b 1/2 共1⫺ ␾ b兲n⫺1.共1兲 Here vtis the Stokes settling velocity of a single particle and nis the exponent in the phenomenological Richardson- Zaki 共RZ兲law 关32兴,vg⫽vt(1⫺ ␾ )n. By including the prefactor 0.56 in the right hand side of the equation FG admitted n⫽4.8 according to earlier RZ empirical observations on the expansion and settling of fluidized beds at low particle Reynolds numbers 关32兴. We find in the literature a variation in the empirical values of the Richardson-Zaki exponent maybe due to wall effects, residual polydispersity effects and inertial screening effects as pointed out by Snabre and Mills 关33兴.As we are dealing with gas-fluidized beds we give above a simplified version of the FG criterion by neglecting the ratio of gas density to particle density. The original derivation of Eq. 共1兲was based on the general criterion proposed by Wallis to describe bubbles as concentration shocks that originate when the propagation velocity of a voidage disturbance (u ␾ ) rises faster than the elastic response rate of the bed. FG used for u ␾ the well-established relation u ␾ ⫽⫺ ␾ dvg d ␾ ,共2兲 where vgwas related to ␾ by the RZ law. On the other hand, the elastic wave velocity uemust be calculated as ue⫽ 冉 1 ␳ p ⳵ p ⳵ ␾ 冊 1/2 ,共3兲 where pis the particle-phase pressure. As earlier suggested by Verloop and Heertjes 关34兴, Foscolo and Gibilaro neglected particle inertia and related pto the fluid-particle drag force in an ideal homogeneous state, arriving at p⫽1 3n ␳ pgdp ␾ 2⬃ ␳ pgdp ␾ 2.共4兲 This is a wrong derivation, however, since the pressure variation cannot be due only to a dissipative process but, as noted by Batchelor 关13兴and Jackson 关35兴, momentum transport by particle fluctuations and collisions must be necessarily considered. The linear stability analysis of Batchelor 关13兴yields for the particle pressure p⬃ ␳ pvg 2 ␾ F( ␾ ), where F( ␾ )is some function of the solids fraction. Taking into account that when a particle is effectively isolated ( ␾ →0) p→0 and that pmust be zero again when the particles are locked in the packed assembly ( ␾ → ␾ c), Batchelor suggested the simple representation F( ␾ )⬇( ␾ / ␾ c)关1⫺( ␾ / ␾ c)兴. This rough estimate drove him to values of pin the dilute limit comparable to the FG prediction 关36兴. Still there is not a general consensus on a correct formulation of the particle pressure. It must be remarked that, even though the bed manifests a smooth appearance to the naked eye in the fluidlike regime, the existence of mesoscale spatiotemporal structures 关3,12兴 indicates that the fluidlike regime is not truly homogeneous after all. In fact, as pointed out by Jackson 关35兴, the FG criterion fails to predict the onset of primary instability observed in liquid fluidized beds consisting of voidage waves that do not develop into recognizable bubbles. In spite of these severe drawbacks, the survival of the FG criterion to discriminate visible bubbling along years of experiments is fascinating. In their paper Foscolo and Gibilaro reported good agreement with experimental data on liquid and gas fluidized beds at both laminar and turbulent conditions 关31兴. Over the next years the FG criterion was successfully applied under systematic variations of relevant parameters on the bubbling point of gas-fluidized beds such as pressure, temperature, and addition of fines 关37兴. Moreover, measurements of the collisional particle pressure in liquid-solid flows yielded, in the range ␾ ⬍⬃0.3, similar results to those predicted from the FG model, whereas results of collision-based models were in poor correspondence with the experiments 关36兴. Therefore it seems that the scaling p⬃ ␳ pgdp ␾ 2can be taken at least as an empirical fact. VALVERDE et al. PHYSICAL REVIEW E 67, 051305 共2003兲 051305-4 Let us extend Eq. 共1兲to our case of aggregated fine particles. According to our previous model 关23兴, aggregates are viewed like effective spheres with a hydrodynamic radius equal to their radius of gyration and undergoing a collisional dynamics. The average number of aggregated particles Nand the ratio of aggregate size to particle size kare derived fitting a generalized RZ law vg *⫽vt *(1⫺ ␾ *)n, where vt *⫽vtN/k 关23兴is the Stokes velocity of a single aggregate and ␾ * ⫽ ␾ k3/N关23兴is the volume fraction filled by the aggregates, to the results of settling experiments 关15,23兴. We admit that the voidage wave velocity can be obtained as u ␾ *⫽⫺ ␾ *dvg * d ␾ *⫽k2 ␾ vtn 冉 1⫺ ␾ k3 N 冊 n⫺1 .共5兲 Following the same approach a straightforward generalization of the particle pressure is given by p* ⬃ ␳ *gkdp( ␾ *)2, where ␳ *⫽ ␳ pN/k3关23兴is the density of the aggregate. Thus, for our effective system of aggregates, the elastic wave velocity can be calculated as ue *⫽ 冉 1 ␳ * ⳵ p* ⳵ ␾ * 冊 1/2 ⬃ 冉 gdp ␾ k4 N 冊 1/2 .共6兲 If we admit that visible bubbling is a consequence of a nonlinear process leading to the development of concentration shocks we can recover the Wallis criterion relating bubbling to the crisis that results from the balance between the voidage and elastic waves velocities. Thus, equating u ␾ * ⬃ue *we obtain a predictive criterion for the solid volume fraction at the onset of macroscopic bubbling 共gdp兲1/2⬃nvtN1/2 ␾ b 1/2 冉 1⫺ ␾ b k3 N 冊 n⫺1 .共7兲 As can be seen in Table I this modified criterion yields values of ␾ bquite close to the experimental ones. It must be stressed, however, that we propose an order of magnitude for ue *based on the scaling of the particle pressure p* ⬃ ␳ *gkdp( ␾ *)2. Since aggregates of our cohesive particles have a fractal dimension (D⫽lnN/lnk) close to 2.5 共diffusion-limited aggregation limit兲关15兴we can write p ⬃ ␳ pgdpk3/2 ␾ 2where the influence of interparticle forces is explicit in the ratio of aggregate size to particle size k, showing that the aggregation of particles contributes to rise particle pressure and thus has a stabilizing action. VII. CONCLUSIONS By extending the range of particle size downwards we have illustrated the connection between the two mechanisms that have been proposed in the literature as responsible for suppressing bubbling in fluidized beds, on one hand hydrodynamic forces and on the other interparticle forces. Although both mechanisms seemed to be contradictory the present work shows that they are rather complementary. The extent of the fluidlike regime and thus the relative role of hydrodynamic stabilization shortens as particle size is increased, indicating that large bubbles are just restrained by yield stresses for sufficiently large particles. Nevertheless the inherent instability of the fluidlike regime confirms that bubble macrostructures cannot be the result of a linear departure from a homogeneous state but a consequence of a complex nonlinear hydrodynamic interaction in which interparticle forces play a relative role by inducing particle aggregation. Application of the Wallis criterion to predict the onset of bubbling suggests that the particle pressure in the pseudoturbulent fluidlike regime scales as p⬃ ␳ pgdpk3/2 ␾ 2. This law is in contradiction with the sublinear dependence on ␾ theoretically predicted for a homogeneous state 关7兴but agrees with direct measurements made on a liquid-fluidized bed 关36兴. Presumably mesoscale structures affect the flow characteristics profoundly 共like is the case in riser flows 关38兴兲, and this would provide a stabilizing action against the growth of macroscopic bubbles. To our knowledge we are the first workers able to succeed in fluidizing particles small enough to see an extended interval of uniform expansion with hydrodynamic stabilization. The complexities of hydrodynamic interactions of the pseudoturbulent two-phase flow make a theoretical model capable of predicting the scaling of the solid phase pressure a rather complicated challenge that remains to be addressed. ACKNOWLEDGMENTS We are glad to acknowledge R. Jackson for suggesting to us the reported experiments. This research has been supported by the Xerox Foundation, Spanish Government Agency Ministerio de Ciencia y Tecnologia 共DGES兲under Contract No. BMF2000-1056, and NATO Grant No. LINKAGE PST.CLG.976575. 关1兴D. Geldart, Powder Technol. 7, 285 共1973兲. 关2兴J. Li and J.A.M. 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