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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|computational fluid dynamics]] (CFD) calculation.  The MP-PIC method achieves greater stability than its [[Particle-in-cell|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 [[Physics:Continuum mechanics|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.<ref name="Andrews1996" />  This method has proven to be stable in dense particle flows, computationally efficient,<ref name="Snider2001" /> and physically accurate.<ref name="Snider2007" />  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.
==History==
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 ([[Organization:Los Alamos National Laboratory|Los Alamos National Laboratory]]),<ref name="Andrews1996" /> who also coined the term MP-PIC.  Subsequent extension of the method to two-dimensions was performed by D.M. Snider and O'Rourke.<ref name="Snider1997" /> By 2001, D.M. Snider had extended the MP-PIC method to full three-dimensions.<ref name="Snider2001" />  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.

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

===Governing equations===

====Fluid phase====
The multiphase particle-in-cell method assumes an incompressible fluid phase with the corresponding continuity equation,

: <math>\frac{\partial \theta_f}{\partial t} + \nabla \cdot ( \theta_f \mathbf{u}_f ) = 0,</math>

where the <math>\theta_f\;</math> is the fluid volume fraction and <math>\mathbf{u}_f\;</math> is the fluid velocity.  Momentum transport is given by a variation of the [[Physics:Navier–Stokes equations|Navier-Stokes equations]] where <math>\rho_f\;</math> is the fluid density, <math>p\;</math> is the fluid pressure, and <math>\mathbf{g}\;</math> is the body force vector (gravity).

: <math>\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}</math>

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 <math>\mathbf{F}\;</math>, 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|probability distribution]] function (PDF), <math>\phi\left(\mathbf{x}, \mathbf{u}_f, \rho_p, \Omega_p, t \right); </math> which indicates the likelihood of finding a particle  with a velocity <math>\mathbf{u}_f\;</math>, particle density <math>\rho_p\;</math>, particle volume <math>\Omega_p\;</math> at location <math>\mathbf{x}\;</math> and time <math>t\;</math>.  The particle PDF changes in time as described by

: <math>\frac{\partial \phi}{\partial t} + \nabla \cdot ( \phi \mathbf{u}_p) + \nabla_{\mathbf{u}_p} \cdot \left(\phi \mathbf{A} \right) = 0</math>

where <math>\mathbf{A}\;</math> 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 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.<ref name="Sundaresan2010" />

====Identities of the particle probability distribution function====

The following local particle properties are determined from integrating the particle probability distribution function:
*Particle volume fraction: <math>\theta_p = \int\!\!\!\int \!\!\! \int\phi\Omega_p \; d \Omega_p d \rho_p d \mathbf{u}_p</math>
*Average particle density: <math>\overline{\theta_p \rho_p} = \int\!\!\!\int \!\!\! \int\phi\Omega_p \rho_p \; d \Omega_p d \rho_p d \mathbf{u}_p</math>
*Mean particle velocity: <math>\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</math>

====Interphase coupling====
The particle phase is coupled to the fluid phase through the particle acceleration term, <math>\mathbf{A}\;</math>, defined as

: <math>\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}. </math>

In the acceleration term, <math>D_p\;</math> is determined from the particle drag model and <math> \tau\;</math> 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, <math>\mathbf{F}\;</math>.  This is defined from the particle population distribution as

: <math>\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</math>

===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, tri-linear interpolation functions and definitions for the products and gradients of interpolated properties are provided by Snider for three-dimensional models.<ref name="Snider2001" />

===Particle stress model===
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 <math>\tau\;</math>, as

: <math>\tau = \frac{P_s {\theta_P}^\beta}{\max \left[ \theta_{cp} - \theta_p, \epsilon \left(1-\theta_p \right) \right]}</math>

where <math>\theta_{cp}\;</math> is the close-pack volume fraction and <math>\beta\;</math>, <math>P_s\;</math>, and <math>\epsilon\;</math> are constants.

==Limitations of the multiphase particle-in-cell method==
*'''Particle shape''' - In the MP-PIC method, all particles are assumed to be spherical.  Corrections for non-spherical particles can be included in particle drag model but for highly non-spherical particles, the true interactions may not be well represented.
*'''Particle size with respect to grid size''' - The size of particles must be small compared to the Eulerian grid in the MP-PIC approach for accurate interpolation.

==Extensions==
*'''Chemical reactions''' – Coupling the local Eulerian values for fluid velocity in the MP-PIC method with equations for [[Physics:Fick's laws of diffusion|diffusional mass transfer]] allows the transport of a chemical species within the fluid-particle system to be modeled.  Reaction kinetics dependent on particle density, surface area, or volume can be included as well for applications in [[Chemistry:Catalysis|catalysis]],<ref name="Snider2010" /> [[Physics:Gasification|gasification]],<ref name="Snider2011" /> or [[Chemistry:Chemical vapor deposition|solid deposition]].
*'''Liquid Injection''' - MP-PIC method was extended by Zhao, O'Rourke, and Snider to model the coating of particle with a liquid.<ref name="Zhao2009" />
*'''Thermal Modeling''' - Conductive and 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.<ref name="CPFD144release" />

==Applications==
*Biomass gasifiers<ref name="Blaser2009" />
*[[Chemistry:Chemical looping combustion|Chemical looping combustion]] (CLC)<ref name="AIChE2010" /><ref name="SniderGuenther2010" /><ref name="Yeomans2006" /><ref name="Blaser2006" /><ref name="Shleg2003" />
*[[Physics:Fluidized bed combustion#Types|Circulating fluidized bed combustion]]<ref name="Weng2010" />
*[[Chemistry:Coal gasification|Coal gasifiers]]<ref name="Snider2011" /><ref name="Snider2009" />
*[[Physics:Cyclonic separation|Cyclones]]<ref name="Williams2006" />
*[[Chemistry:Fluid catalytic cracking|Fluid catalytic cracking reactors and regenerators]]
*Fluidized bed dryers<ref name="Cocco2004" /><ref name="Parker2013" />
*[[Chemistry:Fluidized bed reactor|Fluidized bed reactor]]s<ref name="Karimipour2009" />
*Liquid-solid settlers<ref name="Sundaresan2010" />
*[[Engineering:Metal casting|Metal casting]]<ref name="Yeomans2006" /><ref name="Lefebvre2005" /><ref name="Winartomo2005" />
*Particle jets<ref name="ORourke2010" />
*Polysilicon deposition<ref name="Parker2011" />
*Spray coating<ref name="Zhao2009" />

==Software==
*''Barracuda Virtual Reactor'' by [http://www.cpfd-software.com CPFD Software]
*''MFiX'' by [https://mfix.netl.doe.gov/ NETL]

==References==
{{Reflist|refs=
<ref name="Snider2001">Snider, D.M. (2001). An Incompressible Three-Dimensional Multiphase Particle-in-Cell Model for Dense Particle Flows.  ''Journal of Computational Physics'', 170:523–549.</ref><ref name="Andrews1996">Andrews, M.J. and O'Rourke, P.J. (1996). The Multiphase Particle-in-Cell (MP-PIC) Method for Dense Particle Flows.  ''International Journal of Multiphase Flow'', 22(2):379–402.</ref><ref name="Snider1997">Snider, D.M., O'Rourke, P.J., and Andrews, M.J. (1997). An Incompressible Two-Dimensional Multiphase Particle-In-Cell Model for Dense Particle Flows, NM, LA-17280-MS (Los Alamos National Laboratories, Los Alamos, NM)</ref><ref name="Williams2006">Williams, K., Snider, D., Badalassi, V., Reddy Karri, S.B., Knowlton, T.M., and Cocco, R.A. (2006).  Computational Particle Fluid Dynamics Simulations and Validation for Cyclones: High and Low Loadings. ''AIChE 2006 National Meeting'' http://aiche.confex.com/aiche/2006/preliminaryprogram/abstract_76001.htm Retrieved Feb. 19, 2011</ref><ref name="Snider2011">Snider, D.M., Clark, S.M., O'Rourke, P.J. (2011).  Eulerian–Lagrangian method for three-dimensional thermal reacting flow with application to coal gasifiers. ''Chemical Engineering Science'' 66:1285–1295.</ref><ref name="Snider2009">Snider, D., Clark, S.(2009).  CPFD Eulerian-Lagrangian Method for Three Dimensional Thermal Reacting Flow. ''2009 AIChE National Meeting'', http://www.aicheproceedings.org/2009/Fall/data/papers/Paper149130.html Retrieved Feb 19, 2011</ref><ref name="ORourke2010">O'Rourke, P.J., Snider, D.M. (2010). An improved collision damping time for MP-PIC calculations of dense particle flows with applications to polydisperse sedimenting beds and colliding particle jets. ''Chemical Engineering Science'', 65:6014–6028.</ref><ref name="AIChE2010">Williams, K., Snider, D., Guenther, C. (2010) CFD Simulations of the NETL Chemical Looping Experiment, ''AIChE 2010 National Meeting'',  http://www.aicheproceedings.org/2010/Fall/data/papers/Paper202402.html Retrieved Feb 8, 2011</ref><ref name="Snider2010">Snider, D. and Banerjee, S. (2010). Heterogeneous gas chemistry in the CPFD Eulerian–Lagrangian numerical scheme (ozone decomposition). ''Powder Technology'' 199(1):100–106</ref><ref name="Zhao2009">Zhao, P., O'Rourke, P.J., Snider, D. Three-dimensional simulation of liquid injection, film formation and transport, in fluidized beds. ''Particuology'' 7:337-346</ref><ref name="CPFD144release">CPFD Software, LLC. ''Barracuda 14.4 Released''. http://www.cpfd-software.com/news/barracuda_14.4_released Retrieved Feb 8, 2011</ref><ref name="SniderGuenther2010">Snider, D., Guenther, C., Dalton J., Williams, K. (2010) CPFD Eulerian-Lagrangian Numerical Scheme Applied to the NETL Bench-top Chemical Looping Experiment.  ''Proceedings of the 1st International Conference on Chemical Looping''</ref><ref name="Cocco2004">Cocco, R. and Williams, K. (2004).  Optimization of Particle Residence Time Inside Commercial Dryers with Arena-flow.  ''AIChE 2004 National Meeting''</ref><ref name="Weng2010">Weng, M., Nies, M., and Plackmeyer, J. (2010). Comparison between Measurements and Numerical Simulation of Particle Flow and Combustion at the CFBC Plant Duisburg.  ''5. Internationaler VGB-Workshop "Betriebserfahrungen mit Wirbelschichtfeuerungen 2010"''</ref><ref name="Snider2007">Snider, D. (2007). Three fundamental granular flow experiments and CPFD predictions.  ''Powder Technology'' 176: 36-46.</ref><ref name="Shleg2003">Schleg, P. (2003). Technology of Metalcasting, ''American Foundry Society'', Des Plaines, IL, pp. 1 and 39.</ref><ref name="Blaser2006">Blaser, P., and Yeomans, N.  (2006). Sand Core Engineering & Process Modeling, ''Japan Foundry Society'', Vol. 2, No. 2, February 2006, pp. 420–427.</ref><ref name="Yeomans2006">Yeomans, N., and Blaser, P.  (2006).  Predicting the Process, ''Foundry Management & Technology'', January 2006, pp 48–49.</ref><ref name="Winartomo2005">Winartomo, B., Vroomen, U., and Buhrig-Polaczek, A., Pelzer, M. (2005). Multiphase modeling of core shooting processes, ''International Journal of Cast Metals Research'', Vol. 18, No. 1.</ref><ref name="Lefebvre2005">Lefebvre, D., Mackenbrock, A., Vidal, V., and Haigh, P. (2005). Development and use of simulation in the design of blown cores and moulds, ''Foundry Trade Journal'', February 2005.</ref><ref name="Blaser2009">Blaser, P. and Chandran, R.  (2009).  Computational Simulation of Fluidization Dynamics Inside a Commercial Biomass Gasifier.  ''AIChE 2009 Annual Meeting.''</ref><ref name="Karimipour2009">Karimipour, S. and Pugsley, T.  (2009).  Application of the Particle-in-Cell Approach for the Simulation of Bubbling Fluidized Beds of Geldhart A Particles, ''Seventh International Conference on CFD in the Minerals and Process Industries''.</ref><ref name="Sundaresan2010">Sundaresan, S. (2010).  Challenges in the Analysis of High-Velocity Gas-Particle Flows in Large Devices, ''University of Houston Neal Amundson Memorial Lecture Series, 2010''.</ref><ref name="Parker2011">Parker, J. (2011).  Validation of CFD Model for Polysilicon Deposition and Production of Silicon Fines in a Silane Deposition FBR, ''International Journal of Chemical Reactor Engineering'', Vol. 9, A40</ref><ref name="Parker2013">Parker, J., LaMarche, K., Chen, W., Williams, K., Stamato, H., Thibault, S. (2013) CFD simulations for prediction of scaling effects in pharmaceutical fluidized bed processors at three scales, ''Powder Technology'', 235: 115-120.</ref>
}}

[[Category:Numerical differential equations]]
[[Category:Computational fluid dynamics]]

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