The multiphase particle-in-cell method (MP-PIC) is a numerical method for modeling particle-fluid and particle-particle interactions in a computational fluid dynamics (CFD) calculation. The MP-PIC method achieves greater stability than its particle-in-cell predecessor by simultaneously treating the solid particles as computational particles and as a continuum. In the MP-PIC approach, the particle properties are mapped from the Lagrangian coordinates to an Eulerian grid through the use of interpolation functions. After evaluation of the continuum derivative terms, the particle properties are mapped back to the individual particles. This method has proven to be stable in dense particle flows, computationally efficient, and physically accurate. This has allowed the MP-PIC method to be used as particle-flow solver for the simulation of industrial-scale chemical processes involving particle-fluid flows.
The multiphase particle-in-cell (MP-PIC) method was originally developed for a one-dimensional case in the mid-1990s by P.J. O'Rourke (Los Alamos National Laboratory), who also coined the term MP-PIC. Subsequent extension of the method to two-dimensions was performed by D.M. Snider and O'Rourke. 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 and also available in MFiX suite by NETL.
The MP-PIC method is described by the governing equations, interpolation operators, and the particle stress model.
The multiphase particle-in-cell method assumes an incompressible fluid phase with the corresponding continuity equation,
\partial\thetaf | |
\partialt |
+\nabla ⋅ (\thetafuf)=0,
where the
\thetaf
uf
\rhof
p
g
\partial\thetafuf | |
\partialt |
+\nabla ⋅ (\thetafufuf)=-
\nablap | |
\rhof |
-
F | |
\rhof |
+\thetafg
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
F
The particle phase is described by a probability distribution function (PDF),
\phi\left(x,uf,\rhop,\Omegap,t\right);
uf
\rhop
\Omegap
x
t
\partial\phi | |
\partialt |
+\nabla ⋅ (\phiup)+
\nabla | |
up |
⋅ \left(\phiA\right)=0
where
A
A numerical solution of the particle phase is obtained by dividing the distribution into a finite number of "computational particles" that each represent a number 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. The use of computational particles allows for a significant reduction in computational requirements with a negligible impact on accuracy under many conditions. The use of the computational particle in the Multiphase Particle-in-Cell method allows a full particle size distribution (PSD) to be modeled within the system as well as the modeling of polydisperse solids.
The following local particle properties are determined from integrating the particle probability distribution function:
\thetap=\int\int\int\phi\Omegap d\Omegapd\rhopdup
\overline{\thetap\rhop}=\int\int\int\phi\Omegap\rhop d\Omegapd\rhopdup
\overline{u
The particle phase is coupled to the fluid phase through the particle acceleration term,
A
A=Dp\left(uf-up\right)-
\nablap | |
\rhop |
+g-
\nabla\tau | |
\thetap\rhop |
.
In the acceleration term,
Dp
\tau
The momentum of the fluid phase is coupled to the particle phase through the rate of momentum exchange,
F
F=\int\int\int\phi\Omegap\rhop\left[Dp\left(uf-up\right)-
\nablap | |
\rhop |
\right] d\Omegapd\rhopdup
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, tri-linear interpolation functions and definitions for the products and gradients of interpolated properties are provided by Snider for three-dimensional models.
The effects of particle packing are modeled in the MP-PIC method with the use of a function of particle stress. Snider (2001) has suggested calculating the particle stress
\tau
\tau=
Ps{\thetaP | |
\beta}{max |
\left[\thetacp-\thetap,\epsilon\left(1-\thetap\right)\right]}
where
\thetacp
\beta
Ps
\epsilon