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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]] (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 [[Continuum mechanics|Eulerian]] grid through the use of [[#Interpolation operators|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 (>5% by volume), 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 [[#Applications|industrial-scale chemical processes]] involving particle-fluid flows.
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| ==History==
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| 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]]),<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 [[#Software|commercial software]] for the simulation of particle-fluid systems.
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| ==Method==
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| The MP-PIC method is described by the [[#Governing equations|governing equations]], [[#Interpolation operators|interpolation operators]], and the [[#Particle stress model|particle stress model]].
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| ===Governing equations===
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| ====Fluid phase====
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| The multiphase particle-in-cell method assumes an incompressible fluid phase with the corresponding continuity equation,
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| : <math>\frac{\partial \theta_f}{\partial t} + \nabla \cdot ( \theta_f \bold{u}_f ) = 0,</math>
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| where the <math>\theta_f\;</math> is the fluid volume fraction and <math>\bold{u}_f\;</math> is the fluid velocity. Momentum transport is given by a variation of the [[Navier–Stokes equations|Navier-Stokes equations]] where <math>\rho_f\;</math> is the fluid density, <math>p\;</math> is the fluid pressure, and <math>\bold{g}\;</math> is the body force vector (gravity).
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| : <math>\frac{\partial \theta_f \bold{u}_f}{\partial t} + \nabla \cdot ( \theta_f \bold{u}_f \bold{u}_f ) = - \frac{\nabla p}{\rho_f} - \frac{\bold{F}}{\rho_f}+\theta_f \bold{g}</math>
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| 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>\bold{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.
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| ====Particle phase====
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| The particle phase is described by a probability distribution function (PDF), <math>\phi\left(\bold{x}, \bold{u}_f, \rho_p, \Omega_p, t \right); </math> which indicates the likelihood of finding a particle with a velocity <math>\bold{u}_f\;</math>, particle density <math>\rho_p\;</math>, particle volume <math>\Omega_p\;</math> at location <math>\bold{x}\;</math> and time <math>t\;</math>. The particle PDF changes in time as described by
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| : <math>\frac{\partial \phi}{\partial t} + \nabla \cdot ( \phi \bold{u}_p) + \nabla_{\bold{u}_p} \cdot \left(\phi \bold{A} \right) = 0</math>
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| where <math>\bold{A}\;</math> is the particle acceleration.
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| 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" />
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| ====Identities of the particle probability distribution function====
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| The following local particle properties are determined from integrating the particle probability distribution function:
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| *Particle volume fraction: <math>\theta_p = \int\!\!\!\int \!\!\! \int\phi\Omega_p \; d \Omega_p d \rho_p d \bold{u}_p</math>
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| *Average particle density: <math>\overline{\theta_p \rho_p} = \int\!\!\!\int \!\!\! \int\phi\Omega_p \rho_p \; d \Omega_p d \rho_p d \bold{u}_p</math>
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| *Mean particle velocity: <math>\overline{\bold{u}}_p = \frac{1}{\overline{\theta_p \rho_p}}\int\!\!\!\int \!\!\! \int \phi\Omega_p \rho_p \bold{u}_p \; d \Omega_p d \rho_p d \bold{u}_p</math>
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| ====Interphase coupling====
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| The particle phase is coupled to the fluid phase through the particle acceleration term, <math>\bold{A}\;</math>, defined as
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| : <math>\bold{A}=D_p \left(\bold{u}_f - \bold{u}_p\right) - \frac{\nabla p}{\rho_p} + \bold{g} - \frac{\nabla \tau}{\theta_p \rho_p}. </math>
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| 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.
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| The momentum of the fluid phase is coupled to the particle phase through the rate of momentum exchange, <math>\bold{F}\;</math>. This is defined from the particle population distribution as
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| : <math>\bold{F} = \int\!\!\!\int \!\!\! \int \phi\Omega_p \rho_p \left[ D_p \left( \bold{u}_f - \bold{u}_p \right) - \frac{\nabla p}{\rho_p} \right] \; d \Omega_p d \rho_p d \bold{u}_p</math>
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| ===Interpolation operators===
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| 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 [[Cuboids|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" />
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| ===Particle stress model===
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| 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
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| : <math>\tau = \frac{P_s {\theta_P}^\beta}{\max \left[ \theta_{cp} - \theta_p, \epsilon \left(1-\theta_p \right) \right]}</math>
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| 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.
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| ==Limitations of the multiphase particle-in-cell method==
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| *'''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.
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| *'''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.
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| ==Extensions==
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| *'''Chemical reactions''' – Coupling the local Eulerian values for fluid velocity in the MP-PIC method with equations for [[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 [[catalysis]],<ref name="Snider2010" /> [[gasification]],<ref name="Snider2011" /> or [[Chemical vapor deposition|solid deposition]].
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| *'''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" />
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| *'''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" />
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| ==Applications==
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| *Biomass gasifiers <ref name="Blaser2009" />
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| *[[Chemical looping combustion]] (CLC)<ref name="AIChE2010" /><ref name="SniderGuenther2010" /><ref name="Yeomans2006" /><ref name="Blaser2006" /><ref name="Shleg2003" />
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| *[[Fluidized bed combustion#Types|Circulating fluidized bed combustion]] <ref name="Weng2010" />
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| *[[Coal gasification|Coal gasifiers]]<ref name="Snider2011" /><ref name="Snider2009" />
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| *[[Cyclonic separation|Cyclones]]<ref name="Williams2006" />
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| *[[Fluid catalytic cracking|Fluid catalytic cracking reactors and regenerators]]
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| *Fluidized bed dryers<ref name="Cocco2004" /><ref name="Parker2013" />
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| *[[Fluidized bed reactor]]s<ref name="Karimipour2009" />
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| *Liquid-solid settlers<ref name="Sundaresan2010" />
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| *[[Metal casting]] <ref name="Yeomans2006" /><ref name="Lefebvre2005" /><ref name="Winartomo2005" />
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| *Particle jets<ref name="ORourke2010" />
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| *Polysilicon deposition <ref name="Parker2011" />
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| *Spray coating<ref name="Zhao2009" />
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| ==Software==
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| *''Barracuda'' by [http://www.cpfd-software.com CPFD Software]
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| ==References==
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| {{Reflist|refs=
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| <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>
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| <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>
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| <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> | |
| }}
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| [[Category:Computational physics]]
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| [[Category:Numerical differential equations]]
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| [[Category:Computational fluid dynamics]]
| |
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There was plenty to ɗо ɑt Stagecoach besides listening to the live ѕhows. You could eνen buy camping chairs fгom some vendors since there were no seats, merely field, fоr general programs. You ϲould eνen buy lottery tickets аt the lottery ticket booth. Boot Barn еven had a young store inside Stagecoach. There was plenty оf vendors to look from lots of which was your typical fair type including unusual jewelry, shirts աith funny sayings, and hats and boots. Αnd alѕo ѕince іt wɑs hot, hɑd Ьeen plenty ߋf vendors selling tɦings to keep cool like misters ɑnd umbrellas.
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