Diagrama de casos de uso para la gestión de recetas

There are several multiphase models available for simulation of a particulate phase. A summary of the different simulation methods for multiphase flows in CFD can be found in (van Wachem & Almstedt, 2003). The multiphase models can be divided into two groups;

the Euler-Euler and the Euler-Lagrange methods.

The Euler-Euler method considers all phases as continuous ones. For each phase, a momentum conservation equation is solved. In the momentum equation of the dispersed phases, the kinetic theory for granular flows (Lun & Savage, 1987) is used. The intensity of the particle velocity fluctuations determines the stresses, viscosity, and pressure of the solid phase. The interaction between phases is considered by means of momentum exchange coefficients. This numerical method has the advantage that it is suitable for high volume concentrations of the disperse phase. The disadvantage is that the properties of each dispersed particle cannot be simulated and a particle size distribution cannot be considered.

Moreover, the stability of the calculation depends on the choice of model parameters and often convergence problems appear. For the simulation of sedimentation and thickening processes there are also some Euler-Euler models based on the Kynch’s kinematics sedimentation theory (Kynch, 1952), which were reviewed by Bürger in (Bürger &

Wendland, 2001). These models are all based on the measurement of fundamental material properties of the suspension to be separated as explained by (Buscall, 1990; Buscall et al., 1987; Landman & White, 1994). These multiphase models are based on mass balances for the fluid and the disperse phase. The fluid flow is not calculated but they consider the relative velocity between fluid and particles. On this basis, these models are unsuitable for studying the particle behaviour in solid bowl centrifuges where the liquid flow plays a crucial role in particle settling. These models usually describe the sedimentation and consolidation in batch processes in simple-geometry centrifuges. An extension of these models is necessary to describe the sediment build-up in two and three dimensions correctly.

The Euler-Lagrange method solves the continuum conservation equations just for the characterization of the continuous phase. It describes the dispersed phase as mass points and determines the particle trajectories by integrating a force balance for each individual particle. This way, instead of a partial differential equation, only an ordinary differential equation must be solved for the dispersed phase. The weakness of the Euler-Lagrange method is that it is only valid for diluted suspensions if a one-way or a two-way coupling method between continuous and disperse phase is applied. With the one-way coupling, the motion of the discrete phase is calculated based on the velocity field of the continuous phase but the particles have no effect on the pressure and velocity field of the continuous phase. By using a two-way coupling, the equations of continuous and discrete phase are calculated alternatively. The momentum conservation equation of the continuous phase has a source term to consider the momentum exchange with the particulate phase. The maximum value for the volume concentration of the disperse phase is between 5% (Schütz et al., 2007) and 10-12%

(ANSYS, 2009), above this value the interaction between particles should be considered. Thus there is a third method, four-way coupling, which involves the interaction within the disperse phase: impacts between particles, particle-wall interactions, van der Waals forces, electrostatic forces, etc. Thus the restriction for the volume fraction is not valid anymore. However, if the number of particles increases, the computational effort rises significantly. Usually the characteristics of the problem and the information that should be obtained from the simulation decide which multiphase model and coupling methodology should be used.

The Discrete Element Method (DEM) (Cundall & Strack, 1979) is a Lagrangian method originally developed for the simulation of bulk solids and then extended to the entire field of particle technology. This method solves the Newton's equations of motion for translation and rotation of the particles. In these equations, impact forces of particle-particle and particle-wall interactions are considered. Other forces such as electrostatic or Brownian forces can also be implemented if necessary. In order to describe the collisions of particles, the soft-sphere approach was chosen. When two particles collide, they actually deform. This deformation is described by an overlap displacement of the particles. A schematic representation of the forces on the particles in the soft-sphere model is depicted in Figure 17.

This approach uses a spring-dashpot-slider system to calculate the behaviour of the particles during the contact time. The soft-sphere model allows contacts of a particle with more neighbouring particles during a time step. The net contact force acting on a particle is added in the case of multiple overlapping. This model for contact forces was first developed by Cundall (Cundall & Strack, 1979), who proposed a parallel linear spring-dashpot model for the normal direction and a parallel linear spring-dashpot in a series with a slider for the tangential direction. A comparison of the performance of several soft-sphere models can be found in (Di Renzo & Di Maio, 2005; Stevens & Hrenya, 2005). There, also the Hertz-Mindlin (Mindlin, 1949) non-linear soft-sphere model used in the present work is analysed. This model proposes a normal force as a function of the normal overlap δn, the Young’s modulus of the particle material and the particle radii. The damping force in normal direction depends on the restitution coefficient, the normal stiffness, and the normal component of the relative velocity between the particles. The tangential force depends on the tangential overlap δt, on the shear modulus of the particle material, and on the particle radii; while in the damping force in tangential direction the tangential stiffness and the tangential component of the relative velocity between the particles are considered.

Fig. 17. Contact forces between two particles following the soft-sphere model

While the DEM model was originally used in environments without fluid, increasing efforts have been made to extend this model to the mixture of fluid and particles. For that, a coupling of the DEM calculations with the fluid flow simulation is necessary to be able to include the hydrodynamic forces in the particle simulation. An important aspect in the calculation of particles in a fluid is the energy dissipation into the surrounding fluid. In viscous fluids, there are, in contrast to air, two additional effects that lead to energy losses.

The drag force leads to lower particle velocities and when a particle is about to hit a surface or another particle, a deceleration occurs. This is due to the drainage of the fluid, which is located in the gap between the two particles or the particle and the surface. In his work

(Gondret et al., 2002), Gondret presents the coefficient of restitution as a function of the Stokes number St (Equation 16), which describes the ratio of the dynamic response time of a particle to flow changes and the characteristic flow time. He determined empirically a critical Stokes number of about 10. Below this value, a particle does not bounce when it hits a wall and instead remains in contact with it.



  vx 9

St 1 ^s (16)

In a centrifugal field, the hydrodynamic forces (drag, lift, torque) play a crucial role and must be included in the calculation of the particle trajectories. Therefore the fluid flow and particles movement have to be computed alternatively. A scheme of the coupling between the CFD software FLUENT and the DEM software EDEM can be seen in Figure 18. At first, the flow will be determined with CFD, then for each position in the computational domain, the velocity, pressure, density and viscosity of the flow are transferred to DEM in order to calculate the hydrodynamic forces and include them in the balance. Afterwards, the fluid flow will be computed again considering the influence of the particles as an additional sink term in the momentum equations. This sink term include the force transmitted to the particles by the fluid.

Fig. 18. Coupling between CFD and DEM

Some authors have already simulated such complex multiphase flows in other separation equipment such as filters and hydrocyclones (Chu et al., 2009; Chu & Yu, 2008; Dong et al., 2003; Ni et al., 2006). Nevertheless due to the high velocity gradients and complex interactions between the phases, the simulation of multiphase flow and in particular the sediment behaviour in centrifugal field is still an unsolved challenge. The advantage of this approach is that an accurate description of the flow, the particle paths and the sediment behaviour can be obtained. However, the amount of simulated particles and its size is limited due to the computational demand.

5.1 Particles simulation parameters

The calculation in CFD must be run as an unsteady case when coupled with DEM. The DEM time steps are typically smaller than the CFD time steps in order to correctly capture the contact behaviour. The choice of time steps in DEM simulation is of great importance in order to achieve stability in the calculation but does not perform very long calculations, although still representing the real particle behaviour. Therefore it is necessary to analyse

the forces acting on the particles to determine the different time scales. In the case presented, the Hertzian contact time - the total time of contact in Hertz theory of elastic collision -, the particle sedimentation time - the time required for a particle to settle one diameter -, and the particle relaxation time - the time for a particle to adapt to the surrounding fluid flow - are considered. On the basis of these, the smallest time step must be chosen so that all physical phenomena are represented. In the simulations presented, the Hertzian contact time is the smallest time scale and, thus a time step tDEM = 10^-6 s was chosen in the DEM software, while in the CFD a time step tCFD = 10^-4 s was sufficient.

The size of the particles was defined as a logarithmic normal distribution with a mean diameter of 200 µm and a standard deviation of σ = 10 µm. Simulations with smaller particles lead to divergence of the calculation despite of the small time steps set. Thus simulations with lower densities of the particles and rotational speeds were performed to obtain similar settling velocities as with small quartz particles (mean diameter of 2.07 µm, s=2.15 µm) which were used in the experiments. In order to reproduce the experimental concentration of approximately 0.5 vol. %, 100,000 particles per second were injected at the inlet of the centrifuge for the simulation. In future work, the comparison of predicted and experimentally determined sediment build-up will show whether the simulation methodology leads to correct predictions.

For the contact models, a parameter setting is needed to represent the real particle behaviour. The information needed is mainly about the physical properties of the particle system because the spring and damper constants are calculated as a function of the shear modulus and the Poisson's ratio of the particle and wall materials. The coefficient of restitution depends on the Stokes particle number. The friction coefficient is an empirically determinable property, which can be found in the literature for various material combinations. Some authors have already determined this friction coefficient for sand and glass particles for DEM (Asaf et al., 2007; Li et al., 2005). Here, shear experiments of a sediment made of quartz particles were performed in a Jenike shear cell under normal pressure in order to calculate the friction coefficient as was done by (Hartl & Ooi, 2008).

Parameter Value Particle radius (µm) 200

Particle density (kg/m³) 1016 Shear modulus particle (Pa) 2.2·10⁸

Shear modulus wall (Pa) 7.0·10¹⁰ Poisson’s ratio particle (-) 0.25

Poisson’s ratio wall (-) 0.30 Restitution coefficient (-) 0.010 Static friction coefficient (-) 0.787 Rolling friction coefficient (-) 0.100 Table 3. Parameters used in the DEM simulation