Discrete phases motion equations

The Basset-Boussinesq-Oseen (BBO) equation can be simplified for gas-particle flows with a very small density ratio between the carrier phase and the discrete phase ( 10" ) and with the assumption of one-way coupling such that only the carrier phase has influence on the particle but not vice versa. The particles are governed by these nondimensional motion equations ... 

TFM. And the particle phase motion is solved by tracking discrete parcels, each representing a number of particles with the same properties and following the Newton s law of motion. In the MP-PIC method, the coUisional interaction between particles is replaced by using the normal stress of solids (Snider, 2001), which is calculated on the grid points for gas phase and interpolated to the positions of parcels. The gas continuity equation is the same as Eq. (16), whereas the gas momentum equation reads... 

The discrete phase is considered to be constituted by rigid bodies referred to as particles for now. This assumption is clearly an approximation if the discrete phase is in fact deformable, as is the case for droplets and bubbles. Rigid bodies can only translate and rotate. For the translational motion, the evolution of the center-of-mass velocity follows from Newton s equation, while for the rotations the change of angular velocity can be related to the torque (via the moment of inertia). 

In the SMB operation, the countercurrent motion of fluid and solid is simulated with a discrete jump of injection and collection points in the same direction of the fluid phase. The SMB system is then a set of identical fixed-bed columns, connected in series. The transient SMB model equations are summarized below, with initial and boundary conditions, and the necessary mass balances at the nodes between each column. 

Discrete Fourier transform (DFT), non-adiabatic coupling, Longuet-Higgins phase-based treatment, two-dimensional two-surface system, scattering calculation, 153-155 Discrete variable representation (DVR) direct molecular dynamics, nuclear motion Schrodinger equation, 364-373 non-adiabatic coupling, quantum dressed classical mechanics, 177-183 formulation, 181-183... 

Model. A difference equation for the material balance was obtained from a discrete reactor model which was devised by dividing the annulus into a two dimensional array of cells, each taken to be a well stirred batch reactor. The model supposes that axial motion of the mobile phase and bed rotation occur by instantaneous discontinuous jumps, between cells. Reaction occurs only on the solid surface, and for the reaction type A B + C used in this work, -dn /dt = K n - n n. Linear isotherms, n = BiC, were used, and while dispersion was not explicitly included, it could be simulated by adjusting the number of cells. The balance is given by Eq. 2, where subscript n is the cell index in the axial direction, and subscript m is the index in the circumferential direction. 

Let us consider continuous- or discrete-time dynamical systems in /i-dimensional phase space x = (x1, x2,..., xn), whose equations of motion are, respectively,... 

In this approach, the finite volume methods discussed in the previous chapter can be applied to simulate the continuous fluid (in a Eulerian framework). Various algorithms for treating pressure-velocity coupling, and the discussion on other numerical issues like discretization schemes are applicable. The usual interpolation practices (discussed in the previous chapter) can be used. When solving equations of motion for a continuous fluid in the presence of the dispersed phase, the major differences will be (1) consideration of phase volume fraction in calculation of convective and diffusive terms, and (2) calculation of additional source terms due to the presence of dispersed phase particles. For the calculation of phase volume fraction and additional source terms due to dispersed phase particles, it is necessary to calculate trajectories of the dispersed phase particles, in addition to solving the equations of motion of the continuous phase. 

In the continuum model, the motion of a kink occurs without energy barriers if Q is irrational. For rational fl the distribution of ground state must be discrete, which makes it impossible to transform the phases continuously without extra energy. The dynamic solutions of Eq. (33) exploit the isomorphism with nonlinear relativistic wave equations [107,108] and a moving kink (soliton) can be interpreted as an elementary excitation with energy k(u)... 

Some years later, aided by considerably more rapid computers than available to Wall and co-workers, Karplus, Porter, and Sharma reinvestigated the exchange reaction between H2 and H [24]. As with the earlier work, the twelve classical equations of motion were solved. In addition, discrete quantum-mechanical vibrational and rotation states were included in the total energy so that the trajectories were examined as a function of the initial relative velocity of the atom and molecule and the rotational and vibrational quantum numbers j and v of the molecule. The more sophisticated potential energy surface of Porter and Karplus was used [7], and the impact parameter, orientation and momentum of the reactants, and vibration phase were selected at random from appropriate distribution functions. This Monte Carlo approach was used to examine 200-400 trajectories for each set of VyJ, and v. The reaction probability P can be written as... 

The Langevin equation is discretized temporally by a set of equally spaced time intervals. At predetermined times, the ion dynamics is frozen, and the spatial distribution of the force is calculated from the vector sum of all its components, including both the long-range and the short-range contributions. The components of the force are then kept constant, while the dynamics resumes under the effect of the updated field distribution. Self-consistency between the force field and the ionic motion in the phase space is obtained by iterating this procedure for a desired amount of simulation time. The choice of the spatial and temporal discretization schemes plays a crucial role in computational performance and model accuracy. 

In recent years, DEM has been used in combination with computational fluid dynamics (CFD) aiming at investigating particulate behavior in fluid phase. For a two-phase particle-fluid system, the solid motion and fluid mechanics are solved through the application of Newton s equations of motion for the discrete particles and Navier-Stokes equations for the continuum fluid [2]. 

Discrete bubble model (DBM) In order to study the hydrodynamics in large-scale fluidized beds, the DBM approach is used. In the DBM, the emulsion phase is modeled as a continuum, and the bubbles are regarded as discrete elements (Bokkers et al., 2006). The bubble trajectories are computed by integrating the equations of motion (Newton s second law), accounting for bubble coalescence when two bubbles collide, using closures for the forces acting on the individual bubbles. More detailed and fundamental models could be used in addition to experiments to derive the required closures for the bubble behavior and the emulsion... 

Cell dynamics simulations are based on the time dependence of an order parameter, (i) (Eq. 1.23), which varies continuously with coordinate r. For example, this can be the concentration of one species in a binary blend. An equation is written for the time evolution of the order parameter, dir/dt, in terms of the gradient of a free energy that controls, for example, the tendency for local diffusional motions. The corresponding differential equation is solved on a lattice, i.e. the order parameter V (r) is discretized on a lattice, taking a value at lattice point i. This method is useful for modelling long time-scale dynamics such as those associated with phase separation processes. 

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