Finite-volume method standard

The finite volume methods have been used to discretised the partial differential equations of the model using the Simple method for pressure-velocity coupling and the second order upwind scheme to interpolate the variables on the surface of the control volume. The segregated solution algorithm was selected. The Reynolds stress turbulence model was used in this model due to the anisotropic nature of the turbulence in cyclones. Standard fluent wall functions were applied and high order discretisation schemes were also used. 

Coupling of the meshless methods with other standard methods such as the finite element and the finite volume methods can enhance the capabilities of both sides significantly. Such couplings can especially offer advantages in modeling problems with extreme deformations within a Lagrangian framework. This area of research can be expanded much further. 

The current section of the chapter on numerical methods is devoted to an outline of the most frequently used numerical methods for solving the population balance equation either for the particle number distribution function or for a few moments of the number density function. The methods considered are the standard method of moments, the quadrature method of moments (QMOM), the direct quadrature method of moments (DQMOM), the sectional quadrature method of moments (SQMOM), the discrete fixed pivot method, the finite volume method, and the family of spectral weighted residual methods with emphasis on the least squares method. 

Filbet and Laurengot [58] developed a particular finite volume method (FVM) scheme for dicretizing the Smoluchowski equation for purely coalescing systems. For the application of the FVM they established a continuous flux form of the PBE coalescence source terms. The FVM thus ensures that the poly-disperse particle fluxes between the individual sections are conserved in the system. This approach deviates from the conventional sectional methods which are applied to the standard discrete form of the PBE source terms. Kumar et al. [113] adapted the FVM scheme for solving the transformed coalescence source terms to pure breakage and simultaneous breakage and coalescence systems. 

Consequently, numerical solution of the equations of change has been an important research topic for many decades, both in solid mechanics and in fluid mechanics. Solid mechanics is significantly simpler than fluid mechanics because of the absence of the nonlinear convection term, and the finite element method has become the standard method. In fluid mechanics, however, the finite element method is primarily used for laminar flows, and other methods, such as the finite difference and finite volume methods, are used for both laminar and turbulent flows. The recently developed lattice-Boltzmann method is also being used, primarily in academic circles. All of these methods involve the approximation of the field equations defined over a continuous domain by discrete eqnalions associated with a finite set of discrete points within the domain and specified by the user, directly or through an antomated algorithm. Regardless of the method, the numerical solution of the conservation equations for fluid flow is known as computational fluid dynamics (CFD). 

In summary, DQMOM is a numerical method for solving the Eulerian joint PDF transport equation using standard numerical algorithms (e.g., finite-difference or finite-volume codes). The method works by forcing the lower-order moments to agree with the corresponding transport equations. For unbounded joint PDFs, DQMOM can be applied... 

A standard MD computer simulation consists in the computation of the trajectory in the phase space of a system of N interacting bodies. The time evolution is determined by solving Newton s equations of motion of classical mechanics with finite difference methods. Such a model system corresponds to the microcanonical ensemble (NVE) of statistical mechanics with a constant number of particles N, volume V, and total energy E. In MD simulations the collective properties are then determined from the trajectory of all particles, i.e., from the time evolution of positions r = r, and momenta p = p,. The method relies on the assumption that stationary values of every average observable A can be defined as time integrals over the trajectory in the phase space ... 

If a Type I isotherm exhibits a nearly constant adsorption at high relative pressure, the micropore volume is given by the amount adsorbed (converted to a liquid volume) in the plateau region, since the mesopore volume and the external surface are both relatively small. In the more usual case where the Type I isotherm has a finite slope at high relative pressures, both the external area and the micropore volume can be evaluated by the a,-method provided that a standard isotherm on a suitable non-porous reference solid is available. Alternatively, the nonane pre-adsorption method may be used in appropriate cases to separate the processes of micropore filling and surface coverage. At present, however, there is no reliable procedure for the computation of micropore size distribution from a single isotherm but if the size extends down to micropores of molecular dimensions, adsorptive molecules of selected size can be employed as molecular probes. 

The simulation is performed in a grand canonical ensemble (GCE) where all microstates have the same volume (V), temperature and chemical potential under the periodic boundary condition to minimize a finite size effect [30, 31]. For thermal equilibrium at a fixed pu, a standard Metropolis algorithm is repetitively employed with single spin-flip dynamics [30, 31]. When equilibrium has been achieved, the lithium content (1 — 5) in the Li, 3 11204 electrode at a given pu is determined from the fraction of occupied sites. The thermodynamic partial molar quantities oflithium ions are theoretically obtained by fluctuation method [32]. The partial molar internal energy Uu at constant Vand T in the GCE is readily given by [32, 33]... 

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