, Left: Powder microstructure, the crystallites, and aggregates can be identified. Right: Agglomerates, SEM images of the UO 2
, Pellets of mixed oxides (UO 2 and PuO 2 ) obtained after the manufacturing process. The pellets have a diameter and a height equal to 1 cm
, Illustration of the main steps of the pellet manufacture process. a) Raw material (UO 2 and PuO 2 powders), b) powders blending: co-milling inside a ball mill, c) pellet shaping, and d) pellet sintering
, Laboratory ball mill used for the study of nuclear powders grinding
, Images of a ball mill with lifters filled with the product and steel balls of two different sizes [88]
, Three phases of the granular flow behaving like a gas
, Apparent friction coefficient µ and b) packing fraction ? vs the Inertial number I reported in various numerical and experimental studies on granular flows with different boundary conditions. The dashed line in (a) corresponds to a fit form that follows equation 1.1 [14]
, Flowing regimes in a rotating drum: (a) surging, (b) rolling, (c) cascading, (d) cataracting, (e) centrifuging
, Different dynamic angles of repose ? that can be defined in a rotating drum geometry: a) maximum and minimum, b) average
, Apparent friction coefficient and b) shear stress as a function of the flow thickness (H/d) [214]
, Vorticity fields of shear tests with different heights (H/d) and inertial numbers (I)
, Examples of aspherical particles generated by means of different approaches: a) superellipsoids [69], b) superquadrics [144], c) clumps or glued spheres [195], and d) non-convex polyhedra [195]
, Segregation patterns when considering particles of different sizes. a) DEM simulations in a mixed state and after 25 rotations, vol.48
, 18 LIST OF FIGURES 1.15 Fracture modes in 3D: a) Mode I: opening, b) Mode II: In-plane shear, and c) Mode III: Out-of-plane shear [95]
, Types of breakage that a single particle can undergo under different loading conditions
, Fracture modes of plaster particles submitted to double impact tests at different impact energies [255]
, Breakage of a single particle in 2D by a) FEM [29] and b) Peridynamics [23]. c) Fracture of a conglomerate using LEM [72]
, Modeling particle breakage using a) BPM [233], b) Replacing method with spheres [32], c) Replacing method with polyhedra
, Discrete nature and complex particle shape of a granular material, p.27
, Three different protocols for measuring the principal dimensions L, I, S of a particle [16]: a) Krumbein standard protocol (STD) [130], b) minimum bounding box [25], and c) maximum and minimum area projections on a plane, p.28
, Types of mechanical loading that particles can undergo in a rotating drum, p.30
, Particles generated with different numbers of cells which are represented by different colors
, Interface behavior along (a) normal direction and (b) tangential direction. The solution for each pair (? n , f n ) and (? t , f t ) lies on the thick line. See the text for the definition of the variables
, Frictional contact law defined at the contact framework in the (a) normal direction (b) tangential direction
, Generic contact types between polyhedra
, Snapshots of a particle impacting a rigid plane, and the evolution of particle breakage. This test was performed with an impact velocity of 6 m/s, p.44
, Effect of the number of cells on the number of generated fragments for three different values of the impact velocity
, Particle damage D w as a function of impact velocity v for different values of fracture energy G f and C n = C t = 1 MPa. For each test, the error bar represents standard deviation over 10 independent tests
, , p.47
, 48 2.10 Fragmentation efficiency ? as a function of the normalized impact energy ?. The dotted line is the fitting form (2.20). The error bars represent standard deviation for 10 independent events
, The error bars represent standard deviation for 10 independent events
, Particle damage D w as a function of the normal stress threshold C n for three values of impact velocity v for G f = 1 J/m 2 . The inset shows the same data for the values of C n normalized by the impulsion mv/(s T ?t), where mv is the change of momentum during collision and ?t is collision time. The error bars represent standard deviation for 10 independent events
, The dotted line is the fitting form (2.21). The error bars represent standard deviation for 10 independent events
, Particle damage D w as a function of friction coefficient µ between cohesionless cells. The dotted line is the fitting form (2.21). The error bars represent standard deviation for 10 independent events
, Signorini relation between normal force f n and normal contact velocity u n , b) Coulomb friction law as the relation between sliding velocity u t and friction force f t , at a contact between two particles, Contact laws used in the contact dynamics method (CDM): a)
, Geometrical parameters of granular flow in a rotating drum at the initial state (left) and in the steady flow state (right)
, Velocity vector fields in drums of different normalized sizes R/r: a) 18.75, b) 37.5, c) 62.5, d) 100. The Froude number is Fr=0.8 in all cases, p.61
, Velocity profile at the center of the drum for different values of drum size R/r at constant rotation speed ? (a) and at constant Froude number (b). The depth, measured in the z direction, is normalized by the bed depth h b . The velocity component V along the mean free surface direction is normalized by R?, p.61
, Maps of local volume-change rates? p in drums of four different size ratios R/r : a) 18.75, b) 37.5, c) 62.5 and d) 100, for Fr = 0
, Maps of local shear rates? q in drums of four different size ratios R/r: a) 18.75, b) 37.5, c) 62.5 and d) 100, for Fr= 0.8
, Wall slip S w as a function of rotation speed ? for two values of Fr and three values of drum width W . The error bars reflect the standard deviations of slip velocities V w
, The thickness h a of the active flowing layer normalized by flow thickness h b as a function of drum size R/r for the tested values of system parameters, p.64
, Average speed V w of the particles at contact with the drum wall normalized by the characteristic velocity ? gd versus the thickness h a of the active flowing layer normalized by particle diameter d for all simulations with different parameter values. Error bars on the data points are smaller than symbol size
, 66 LIST OF FIGURES 3.12 Steepest descent angle ? max (a), secant angle ? m (b) and the ratio ? max /? m (c) as a function of R/r for different parameter values. The error bars represent standard deviation of the values of angles
, as a function of ? max (b), for simulations with different sizes at either constant Fr or ?. The error bars represent standard deviation of the values of ? max
, Contact force network inside drums of different sizes R/r: a) 25, b) 62.5, c) 93.75, d) 125. All drums have constant Froude number Fr=0.8. The thickness of the lines joining the particle centers is proportional to the corresponding normal force, p.70
, Probability density function (pdf) of normal forces inside drums of different sizes (R/r) at constant Froude number Fr = 0.8 (a) and at constant rotation speed ? = 5 rad/s (b)
, Probability density function of the logarithm of normalized forces between colliding particles inside drums of different sizes (R/r) at constant Froude number (a) and at constant rotation speed ? (b)
, in drums of different sizes and values of system parameters. The dashed line is a power-law fit following equation 3
, Fr (a) and drum rotation speed ? (b) for all simulations. The dashed lines are power-law trends with exponents 1/4 and -0.45, respectively. The error bars represent standard deviation of the values of ? max, Slope ratio ? max /? m as a function of the Froude number
, Slope ratio ? max /? m as a function of the scaling parameter ? defined by equation (3.8) with ? = 1/4 and ? = 1/2. The dashed line is the fitting form given by equation (3.9). The error bars represent standard deviation of the values of ? max, p.73
, Normalized active layer thickness (a) and standard deviation of the normal force distribution (b) as a function of the scaling parameter ? defined by (3.8) with ? = 1/4 and ? = 1/2
, Slope ratio as a function of the scaling parameter ? defined by equation (3.8) with a range of values of drum size R, rotation speed ? and filling degree f with ? = 1/4, ? = 1/2, ? = 1. The dashed line is a power-law fitting form. The error bars represent standard deviation of the values of ? max
, Slope ratio as a function of the scaling parameter ? for systems with h 0 < 10d and with h 0 > 10d. The dashed line follows equation 3.9. The error bars represent standard deviation of the values of ? max
, For the sake of comparison between the curves in the same range of values as ?, Q * and ? are multiplied by prefactors. The dashed line follows equation 3.9. The error bars represent standard deviation of the values of ? max
, Voronoï tessellation applied to a) pentagonal particles (n sides = 5), b) hexagonal particles (n sides = 6), c) nonagonal particles (n sides = 9), d) dodecagonal particles (n sides = 12). The cells are represented by different arbitrary colors, p.85
, Side-side double-bond contact, b) vertex-side single-bond contact
, the normal direction, b) in the tangential direction. is the side length, and ? n and ? t denote the local displacements between the two cells in the normal and tangential directions, respectively, vol.87
, Signorini relation in the normal direction, b) Coulomb friction law in the tangential direction. u n and u t denote the contact relative velocities in the normal and tangential directions, respectively, p.88
, Geometrical, mechanical and kinematic parameters of the simulated drums, p.89
, Snapshots of a rotating drum simulation for different numbers of rotations n for ? = 5.24 rad/s. The color is proportional to the damage, defined by the number of cells detached from a particle, represented on color scale from bright green for intact particles to black for highly-damaged particles
, 075 m, f = 0.51, and increasing rotation velocity ?. By increasing ?, the Froude number varies from 0.02 to 1. The color is proportional to particle damage, from bright green for intact particles to black for highly-damaged fragments, Flow regimes displayed after 13.75 rotations, for R = 0
, Free surface profiles for different values of ? considering only unbreakable particles, p.92
, The dashed lines correspond to a tangent hyperbolic form d / d 0 ? tanh(t). b) Evolution of the specific surface S normalized by the initial specific surface S 0 . The dashed lines are linear fits up to a transition point to nonlinear regime
, Characteristic time t * as a function of rotation speed ?
, and normalized mean particle size (b) as a function of time normalized by the characteristic time t * for drums rotating at different speeds ?
, Maps of local densities of breakage events during the whole simulation for ? =5.24, 7.85 and 10
, This snapshot corresponds to the instants 0.73t * , 0.9t * and 1.16t * , respectively. The particle gray level is proportional to the number of contacts of the particle, Maps of particle connectivity after 13.75 rotations for ? =5.24, 7.85 and 10.47 rad/s
, particle size measured as a function of time for different values of the filling degree f for the same drum size R = 0.075 m and rotation speed ? = 5, p.24
, Normalized specific surface S/S 0 as a function of time. The dashed lines are linear fits up to the transition point
, Free surface profiles for different filling degrees
, Characteristic time t * as a function of f
, as a function of normalized time for different filling degrees, p.97
, 97 LIST OF FIGURES 4.19 Normalized specific surface S/S 0 for drums of different sizes R/r for a constant value of ? (a) and for a constant value of the Froude number (b). The dashed lines are linear fits below the transition point
, The rate of increase of the normalized specific surface shown in Fig. 4.19. b) Characteristic time as a function of drum size ratio R/r for the two sets of simulations. The dashed lines are power-law fits to the data, p.98
, The normalized mean particle size d /d 0 (a) and normalized specific surface S/S 0 (b) as a function of time in drum flows composed of regular polygons of different numbers of sides for fixed drum size, rotation speed and filling degree, p.99
, Dimensionless grinding rate as a function of the scaling parameter ? in equation (4.5) with ? = 3/4, ? = ?1, ? = 1/4, and ? = 3/2 for all our simulations with different values of system parameters. The symbols refer to different sets of simulations in which every time a single parameter (filling degree f , rotation speed ?, R/r at constant Froude number or constant rotation speed) is varied, p.101
, Comparison between the proposed scaling law in equation (4.5) and the same data plotted as function of ? [223] and Q * [184]. A prefactor was applied in order to bring the data to the same range
, Each cell is presented in a different color; (b) Geometry of a side-side contact between two cells i and j. Two contact points (1 and 2) and their respective projections on the two cells, are defined for this type of contact
, Purely frictional contact interactions: a) Relationship between normal force f n and relative normal velocity u n at a contact point; b) Coulomb friction law as a relationship between the friction force f t and sliding velocity u t, p.111
, The powder particle colors range from bright green (intact) to black (highly damaged)
, Snapshots of several simulations: a) Systems with ball sizes D b of 5 mm, 15 mm and 25 mm from left to right; b) Systems with numbers of balls N b = 10, 25 and 50, with D b = 15mm constant
, Evolution with the number n of revolutions of a) the mean particle size d normalized by the initial mean diameter, b) the specific surface S normalized by its initial value for different values of ball size D b . The filling degree, total balls volume V b , and powder volume V p are constant. Each plot consists of 1000 data points, p.116
, 25 mm from left to right. The dashed red line represents the ball size of each case, Spatial localization of breakage events in drums filled with balls of variable size D b : 5, 10, vol.15
, Probability density function of the normal force f n between the balls and the powder a) for each powder-ball contact, b) the sum of the forces per ball, for different ball sizes D b normalized by the cohesion force C N d cell, p.117
, Evolution of the normalized mean particle size with the number of revolutions for different numbers of balls N b . The ball size D b and powder volume V p are constant. Each plot consists of 1000 data points
, Evolution of the normalized specific surface, b) slope of the linear trend adjusted to the S 0 evolution, for different number of balls
, Force chains in a simulation with N b = 50. Red line thickness is proportional to normal force
, Time evolution of the volume of each size population normalized by the total volume V for a) D b = 15 and b) N b = 32. The dashed lines are analytical fits obtained from the system of equations 5.8
, Cumulative volume transfers: from big to small (? s b ), from big to medium (? m b ), and from medium to small (? s m ), for a) D b = 15 and b) N b = 32, p.121
, Snapshot of a simulation of crushable particles inside a rotating drum in 3D, p.128
, Images séquentielles de la fragmentation d'un grain lors de l'impact avec un plan rigide
, de l'écoulement stationnaire dans des systèmes avec des rapports entre la taille du tambour et la taille du grain R/r= a) 18.5, b) 37.5, c) 62.5, d) 93.75, et e) 125. La taille de particule est gardée constante. Les grains sont colorés en fonction de leur vitesse projetée sur la direction d'écoulement et normalisée par la vitesse de la, p.134
, Simulation 2D d'un tambour tournant rempli avec des particules sécables. Le niveau d'endommagement des particules est représenté du vert au noir; les particules noires étant celles ayant subit le plus fort endomagement, p.135
, Simulation d'un broyeur à boulets en 2D. Les zooms illustrent les modes de comminution dans différentes zones du broyeur
, Cone crack present at the contact point, view from the impact plane. b) Fissure propagation from the cone crack
, A.2 a) Front view of meridional cracks developed but no fragments generated, only debris from the cone crack at the impact point. b) Front view of a single meridional crack that generates two fragments. c) Top view of a fragmentation in 5 slices generated from the meridional crack development. d) Secondary level of cracking, p.143
, Snapshots taken at the end of the impact of particles at 6 m/s, 8 m/s, and 10 m/s from left to right
, Mean particle size distribution of the fragments for tests performed at different impact speeds with the respective values of ?/? * , a) particle size normalized by d 0 (initial particle size), b) particle size normalized by the maximum fragment size d max , specific for each velocity. The dashed lines correspond to a cummulative weibull fucntion (equation (A.2)). The legend in (b) corresponds also to (a), p.144
, Multiple values of ?/? * representing various impact velocities are included, as well as the black dashed line that represents the power law fitted to the intermediate and small fragments, A.5 Probability density function of the fragments volume
, Evolution of a) coefficient of uniformity (C u ) and b) coefficient of curvature (C c ) regarding the impact velocity. The red dashed lines show the values of C u and C c for a uniform in volume particle size distribution
, Fragments with different sizes (d/d 0 ) generated at two different impact energies (?/? * )
, Schematic illustration of the measurement of: a) form dimensions and b) ? = ?R/R147
, 10 Evolution of the elongation calculated as the ratio between the longer and intermediate dimension of the oriented bounding box a) mean value and error bars issue of 10 simulations b)
, Evolution of a) sphericity (?) and b) specific surface (SSA) normalized by the specific surface of the initial particle (SSA 0 ), with impact energy ?/? * for different sets of fragment size d/d 0 . The points represent mean values of the fragments generated in 10 independent tests and the blue line correspond to the mean value for each speed. The legend in (a) corresponds also to (b)
, Evolution of (?) mean value and error bars issue of 10 simulations for the different values of ?/? *
, Illustration of a rotating drum with two plans closing the end walls of the cylinder, vol.152
, Some granular material properties evolution regarding the friction coefficient µ: a) average free surface angle, b) packing fraction, c) coordination Number, d) ratio between the particles velocity at the bottom and the drum wall velocity, for test performed under different rotation speeds. The legend in (d) correspond to the other subfigures
, Parameters of the breakage model and material properties of powder particles and balls for all simulations
, 2 Geometrical characteristics of the two case studies
, Values of the model (eq. 5.8) parameters found for the two studied cases in this section
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