User:Chemengr82/CPFD method draft

The Multiphase Particle-in-Cell method (MP-PIC) is a numerical approach for modeling particle-fluid and particle-particle interactions. The MP-PIC method achieves greater stability than its Particle-in-cell predecessor by simultaneously treating particles as computational particles and as a continuum fluid. In the MP-PIC algorithm, the particle properties are mapped from the Lagrangian coordinates to an Eulerian grid through the use of interpolation functions. After evaluating continuum derivative terms, the particle properties are mapped back to the individual particles. This method has proven to be stable in dense particle flows (>5% by volume), computationally efficient, and physically accurate allowing the MP-PIC method to be used as particle-flow solver for the simulation of industrial-scale chemical processes.

History
The Multiphase Particle-in-Cell (MP-PIC) method was originally developed for a one-dimensional case in the mid-1990's by M.J. Andrews (Texas A&M University) and P.J. O'Rourke (Los Alamos National Laboratory). Subsequent extension of the method to two-dimensions was performed by D.M. Snider (SAIC corporation), O'Rourke, and Andrews. By 2001, D.M. Snider had extended the MP-PIC method to full three-dimensions. Currently, the MP-PIC method is used in commercial software for the simulation of particle-fluid systems.

Method
The MP-PIC method is described by the governing equations, interpolation operators, and the particle stress model.

Fluid Phase
The Multiphase Particle-in-Cell method assumes an incompressible fluid phase with the corresponding continuity equation,

$$\frac{\partial \theta_f}{\partial t} + \nabla \cdot ( \theta_f \mathbf{u}_f ) = 0$$,

where the $$\theta_f\;$$ is the fluid volume fraction and $$\mathbf{u}_f\;$$ is the fluid velocity. Momentum transport is given by a variation of the Navier-Stokes equations where $$\rho_f\;$$ is the fluid density, $$p\;$$ is the fluid pressure, and $$\mathbf{g}\;$$ is the body force vector (gravity).

$$\frac{\partial \theta_f \mathbf{u}_f}{\partial t} + \nabla \cdot ( \theta_f \mathbf{u}_f \mathbf{u}_f ) = - \frac{\nabla p}{\rho_f} - \frac{\mathbf{F}}{\rho_f}+\theta_f \mathbf{g}$$

The laminar fluid viscosity terms, not included in the fluid momentum equation, can be included if necessary but will have a negligible effect on dense particle flow. In the MP-PIC method, the fluid motion is coupled with the particle motion through $$\mathbf{F}\;$$, the rate of momentum exchange per volume between the fluid and particle phases. The fluid phase equations are solved using a finite volume approach.

Particle Phase
The particle phase is described by a probability distribution function (PDF), $$\phi\left(\mathbf{x}, \mathbf{u}_f, \rho_p, \Omega_p, t \right); $$ which indicates the likelihood of finding a particle with a velocity $$\mathbf{u}_f\;$$, particle density $$\rho_p\;$$, particle volume $$\Omega_p\;$$ at location $$\mathbf{x}\;$$ and time $$t\;$$. The particle PDF changes in time as described by

$$\frac{\partial \phi}{\partial t} + \nabla \cdot ( \phi \mathbf{u}_p) + \nabla_{\mathbf{u}_p} \cdot \left(\phi \mathbf{A} \right) = 0$$

where $$\mathbf{A}\;$$ is the particle acceleration.

A numerical solution of the particle phase is obtained by dividing the distribution into a finite number of "computational particles" that each contain a prescribed quantity of real particles with identical mass density, volume, velocity and location. At each time step, the velocity and location of each computational particle are updated using a discretized form of the above equations.

Identities of the Particle Probability Distribution Function
The following local particle properties are determined from integrating the particle probability distribution function:
 * Particle volume fraction: $$\theta_p = \int\!\!\!\int \!\!\! \int\phi\Omega_p \; d \Omega_p d \rho_p d \mathbf{u}_p$$
 * Average particle density: $$\overline{\theta_p \rho_p} = \int\!\!\!\int \!\!\! \int\phi\Omega_p \rho_p \; d \Omega_p d \rho_p d \mathbf{u}_p$$
 * Mean particle velocity: $$\overline{\mathbf{u}}_p = \frac{1}{\overline{\theta_p \rho_p}}\int\!\!\!\int \!\!\! \int \phi\Omega_p \rho_p \mathbf{u}_p \; d \Omega_p d \rho_p d \mathbf{u}_p$$

Interphase Coupling
The particle phase is coupled to the fluid phase through the particle acceleration term, $$\mathbf{A}\;$$, defined as

$$\mathbf{A}=D_p \left(\mathbf{u}_f - \mathbf{u}_p\right) - \frac{\nabla p}{\rho_p} + \mathbf{g} - \frac{\nabla \tau}{\theta_p \rho_p} $$.

In the acceleration term, $$D_p\;$$ is determined from the particle drag model and $$ \tau\;$$ is determined from the interparticle stress model.

The momentum of the fluid phase is coupled to the particle phase through the rate of momentum exchange, $$\mathbf{F}\;$$. This is defined from the particle population distribution as

$$\mathbf{F} = \int\!\!\!\int \!\!\! \int \phi\Omega_p \rho_p \left[ D_p \left( \mathbf{u}_f - \mathbf{u}_p \right) - \frac{\nabla p}{\rho_p} \right] \; d \Omega_p d \rho_p d \mathbf{u}_p$$

Interpolation Operators
The transfer of particle properties between the Lagrangian particle space and the Eulerian grid is performed using linear interpolation functions. Assuming a rectilinear grid consisting of rectangular cuboid cells, the scalar particle properties are interpolated to the cell centers while the vector properties are interpolated to cell faces. In three dimensions, interpolation functions and definitions for the products and gradients of interpolated properties are provided by Snider for three dimensional models.

Particle Stress Model
More to be added here.

Extensions

 * Chemical Reactions - Coupling the local Eulerian values for fluid velocity in the MP-PIC method with equations for diffusional mass transfer allows the transport of a chemical species within the fluid-particle system to be modeled. Furthermore, reaction kinetics dependent on particle density, surface area, or volume can be included as well for applications in catalysis, gasification , or solid deposition.
 * Liquid Injection - MP-PIC method was extended by Zhao, O'Rourke, and Snider to model the coating of particle with a liquid.
 * Thermal - Conductive, convective heat transfer can be included by coupling MP-PIC variables with equations for heat transfer. Commercial implementations of MP-PIC method include radiative heat transfer as well.

Applications

 * Chemical Looping Combustion (CLC)
 * Coal Gasifiers
 * Cyclones
 * Fluidized Bed Combustion
 * Fluidized Bed Dryers
 * Fluidized Bed Reactors
 * Particle Jets
 * Polysilicon Deposition
 * Spray Coating

Software

 * Barracuda by CPFD Software, LLC (Albuquerque, NM)