List of contents
1 General introduction..............................................................................................................1
1.1 World fertilizer production and needs..............................................................................1
1.2 Centrifugal spreading in agriculture.................................................................................1
1.3 Need for spreader modelling ............................................................................................3
1.4 Existing spreader models .................................................................................................3
1.5 Introduction DEM in agriculture......................................................................................4
1.6 Objectives of the research ................................................................................................5
1.7 Outline of the research .....................................................................................................6
2 The Discrete Element Method...............................................................................................7
2.1 Introduction ......................................................................................................................7
2.2 DEM Algorithm flow .......................................................................................................8
2.3 Model parameter settings .................................................................................................9
2.4 Initializing the simulation.................................................................................................9
2.5 Contact resolution ..........................................................................................................11
2.5.1 Introduction ............................................................................................................11
2.5.2 Sphere-sphere contact ............................................................................................11
2.5.3 Sphere-rectangle contact ........................................................................................12
2.5.4 Sphere-cylinder contact..........................................................................................13
2.5.5 Sphere-disc contact ................................................................................................14
2.5.6 Contact with edges .................................................................................................16
2.6 Search algorithms...........................................................................................................18
2.7 Force calculation in DEM simulations...........................................................................19
2.7.1 Introduction ............................................................................................................19
2.8 Normal contact force models .........................................................................................22
2.8.1 Introduction ............................................................................................................22
2.8.2 The Kelvin–Voight model......................................................................................23
2.8.3 The Kono-Kuwabara model ...................................................................................23
2.8.4 Other normal contact models .................................................................................26
2.9 Tangential contact forces ...............................................................................................26
2.9.1 Introduction ............................................................................................................26
2.9.2 Linear model implementation ................................................................................27
2.10 Moments of forces..........................................................................................................29
2.11 Body Forces....................................................................................................................30
2.11.1 Aero dynamical drag force.....................................................................................30
2.11.2 Magnus effect.........................................................................................................31
2.12 Time integration algorithms ...........................................................................................31
2.12.1 Introduction ............................................................................................................31
2.12.2 Integration of Newton’s second law.......................................................................32
2.12.3 Euler method ..........................................................................................................32
2.12.4 Leapfrog method ....................................................................................................33
List of contents viii
2.12.5 Other integrators.....................................................................................................33
2.12.6 Choice of the time step...........................................................................................34
2.12.7 Integration of Euler’s law.......................................................................................35
2.12.8 Leapfrog integration...............................................................................................38
2.12.9 Integration errors ....................................................................................................38
2.13 Program overview ..........................................................................................................40
2.14 Conclusions ....................................................................................................................40
3 Particle measuring methods as input for numerical models............................................42
3.1 Introduction ....................................................................................................................42
3.2 Particle mass ( p m ) ........................................................................................................43
3.3 Particle radius ( p R ) and shape .......................................................................................43
3.3.1 Manual procedure...................................................................................................43
3.3.2 Automatic procedure ..............................................................................................44
3.3.3 Particle shape approximation .................................................................................45
3.3.4 Surface roughness ..................................................................................................50
3.4 Particle density (ρ ) p .....................................................................................................52
3.5 Particle stiffness (k ) ....................................................................................................52
3.5.1 Stiffness for sphere-sphere indenting.....................................................................53
3.5.2 Stiffness for sphere-cylinder indenting ..................................................................53
3.6 Particle damping coefficient (c) and restitution coefficient (e) ..................................54
3.6.1 Sphere-sphere damping ..........................................................................................58
3.7 Particle fracture resistance ( ) max N ................................................................................58
3.7.1 Impact with vanes and vane edges .........................................................................61
3.7.2 Static pressure in the bin ........................................................................................62
3.8 Particle friction coefficient (μ ) .....................................................................................62
3.8.1 Wall-particle friction ( wp μ ) ...................................................................................63
3.8.2 Particle-particle friction ( pp μ )................................................................................64
3.8.3 Static friction ..........................................................................................................65
3.9 Particle Aerodynamical drag coefficient ( d C ) ...............................................................65
3.10 Errors of the measurement methods...............................................................................66
3.11 Conclusions ....................................................................................................................67
4 Single particle simulations...................................................................................................68
4.1 Introduction ....................................................................................................................68
4.2 Comparison with analytical models ...............................................................................68
4.2.1 The rolling and friction force .................................................................................68
4.2.2 The centrifugal force ..............................................................................................71
4.2.3 The Coriolis force...................................................................................................73
4.3 Experimental methods....................................................................................................74
4.4 Experimental verification of single particle trajectories ................................................75
4.5 Analysis of the rotational motion in single particle trajectories.....................................79
4.6 Analysis of bouncing in single particle trajectories .......................................................83
4.7 Non - spherical particle simulations...............................................................................86
List of contents ix
4.8 Conclusions ....................................................................................................................88
5 Multi particle simulations....................................................................................................89
5.1 Introduction ....................................................................................................................89
5.2 Materials and methods ...................................................................................................90
5.2.1 Introduction ............................................................................................................90
5.2.2 Cylindrical measurements ......................................................................................90
5.2.3 Rectangular measurements (TSP) ..........................................................................91
5.2.4 Fertilizer choice and property measurements.........................................................93
5.2.5 Choice of disc type and bin ....................................................................................96
5.3 Computational setup.......................................................................................................99
5.3.1 Introduction ............................................................................................................99
5.3.2 Initial conditions in the bin...................................................................................100
5.3.3 Spinning disc simulations.....................................................................................103
5.3.4 Ballistic model......................................................................................................104
5.3.5 Model output variables.........................................................................................104
5.3.6 Characterisation time of the simulated pattern.....................................................104
5.3.7 Sensitivity to bin filling........................................................................................105
5.4 Results and discussion..................................................................................................105
5.4.1 Cylindrical spread pattern measurements ............................................................105
5.4.2 TSP measurements ...............................................................................................111
5.5 Conclusions of the experiments ...................................................................................118
6 Model parameter sensitivity..............................................................................................120
6.1 Introduction ..................................................................................................................120
6.2 Definition of sensitivity................................................................................................120
6.3 Influence of friction coefficient....................................................................................123
6.3.1 Particle-wall friction.............................................................................................123
6.3.2 Particle-particle friction........................................................................................128
6.3.3 Conclusion on friction coefficients ......................................................................129
6.4 Influence of elasticity and damping .............................................................................130
6.4.1 Influence of particle stiffness ...............................................................................130
6.5 Influence of bulk flow ..................................................................................................132
6.5.1 Flow rate...............................................................................................................133
6.5.2 Influence of orifice shape.....................................................................................134
6.6 Influence of particle diameter and segregation ............................................................136
6.6.1 Particle diameter...................................................................................................136
6.6.2 Segregation...........................................................................................................140
6.7 Influence of particle shape ...........................................................................................141
6.7.1 Introduction ..........................................................................................................141
6.7.2 Results ..................................................................................................................143
6.8 External forces..............................................................................................................146
6.9 Interaction of parameters..............................................................................................147
6.9.1 Friction - particle shape........................................................................................148
6.9.2 Friction - bulk flow rate .......................................................................................149
6.9.3 Bulk flow segregation ..........................................................................................149
6.9.4 Friction – working width......................................................................................151
6.10 Particle interception......................................................................................................151
6.11 Discussion on sensitivity...........................................................................................152
List of contents x
6.12 Conclusions ..................................................................................................................159
7 Case study: DEM simulations of a new concept for lawn spreaders.............................161
7.1 Introduction ..................................................................................................................161
7.2 Situation and current technology..................................................................................161
7.3 Design of the new concept ...........................................................................................162
7.4 Flow rate regulation of the pocket system ...................................................................163
7.5 DEM simulations..........................................................................................................166
7.5.1 Introduction ..........................................................................................................166
7.5.2 Simulation of ideal conditions, current technology .............................................167
7.5.3 Simulation of modified conditions.......................................................................168
7.6 Conclusions ..................................................................................................................170
8 General conclusions and future outline............................................................................171
8.1 General conclusions .....................................................................................................171
8.2 Suggestions for future work .........................................................................................174
9 Bibliography .......................................................................................................................177
DEM simulation of Centrifugal fertilizer (پایان نامه 200 ص به زبان انگلیسی )