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Multiphase particle-in-cell method
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<h2 id="history">History</h2> <p>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 (<a href="/facts/Los_Alamos_National_Laboratory/tlrLkV7r">Los Alamos National Laboratory</a>),<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> who also coined the term MP-PIC. Subsequent extension of the method to two-dimensions was performed by D.M. Snider and O'Rourke.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> By 2001, D.M. Snider had extended the MP-PIC method to full three-dimensions.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> 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. </p> <h2 id="method">Method</h2> <p>The MP-PIC method is described by the governing equations, interpolation operators, and the particle stress model. </p> <h3>Governing equations</h3> <h4>Fluid phase</h4> <p>The multiphase particle-in-cell method assumes an incompressible fluid phase with the corresponding continuity equation, </p> ∂ θ f ∂ t + ∇ ⋅ ( θ f u f ) = 0 , {\displaystyle {\frac {\partial \theta _{f}}{\partial t}}+\nabla \cdot (\theta _{f}\mathbf {u} _{f})=0,} <p>where the θ f {\displaystyle \theta _{f}\;} is the fluid volume fraction and u f {\displaystyle \mathbf {u} _{f}\;} is the fluid velocity. Momentum transport is given by a variation of the <a href="/facts/Navier%25E2%2580%2593Stokes_equations/48pQugEa">Navier-Stokes equations</a> where ρ f {\displaystyle \rho _{f}\;} is the fluid density, p {\displaystyle p\;} is the fluid pressure, and g {\displaystyle \mathbf {g} \;} is the body force vector (gravity). </p> ∂ θ f u f ∂ t + ∇ ⋅ ( θ f u f u f ) = − ∇ p ρ f − F ρ f + θ f g {\displaystyle {\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} } <p>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 {\displaystyle \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. </p> <h4>Particle phase</h4> <p>The particle phase is described by a <a href="/facts/Probability_distribution/EpsKKVRu">probability distribution</a> function (PDF), ϕ ( x , u f , ρ p , Ω p , t ) ; {\displaystyle \phi \left(\mathbf {x} ,\mathbf {u} _{f},\rho _{p},\Omega _{p},t\right);} which indicates the likelihood of finding a particle with a velocity u f {\displaystyle \mathbf {u} _{f}\;} , particle density ρ p {\displaystyle \rho _{p}\;} , particle volume Ω p {\displaystyle \Omega _{p}\;} at location x {\displaystyle \mathbf {x} \;} and time t {\displaystyle t\;} . The particle PDF changes in time as described by </p> ∂ ϕ ∂ t + ∇ ⋅ ( ϕ u p ) + ∇ u p ⋅ ( ϕ A ) = 0 {\displaystyle {\frac {\partial \phi }{\partial t}}+\nabla \cdot (\phi \mathbf {u} _{p})+\nabla _{\mathbf {u} _{p}}\cdot \left(\phi \mathbf {A} \right)=0} <p>where A {\displaystyle \mathbf {A} \;} is the particle acceleration. </p><p>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.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p> <h4>Identities of the particle probability distribution function</h4> <p>The following local particle properties are determined from integrating the particle probability distribution function: </p> <ul><li>Particle volume fraction: θ p = ∫ ∫ ∫ ϕ Ω p d Ω p d ρ p d u p {\displaystyle \theta _{p}=\int \!\!\!\int \!\!\!\int \phi \Omega _{p}\;d\Omega _{p}d\rho _{p}d\mathbf {u} _{p}} </li> <li>Average particle density: θ p ρ p ¯ = ∫ ∫ ∫ ϕ Ω p ρ p d Ω p d ρ p d u p {\displaystyle {\overline {\theta _{p}\rho _{p}}}=\int \!\!\!\int \!\!\!\int \phi \Omega _{p}\rho _{p}\;d\Omega _{p}d\rho _{p}d\mathbf {u} _{p}} </li> <li>Mean particle velocity: u ¯ p = 1 θ p ρ p ¯ ∫ ∫ ∫ ϕ Ω p ρ p u p d Ω p d ρ p d u p {\displaystyle {\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}} </li></ul> <h4>Interphase coupling</h4> <p>The particle phase is coupled to the fluid phase through the particle acceleration term, A {\displaystyle \mathbf {A} \;} , defined as </p> A = D p ( u f − u p ) − ∇ p ρ p + g − ∇ τ θ p ρ p . {\displaystyle \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}}}.} <p>In the acceleration term, D p {\displaystyle D_{p}\;} is determined from the particle drag model and τ {\displaystyle \tau \;} is determined from the interparticle stress model. </p><p>The momentum of the fluid phase is coupled to the particle phase through the rate of momentum exchange, F {\displaystyle \mathbf {F} \;} . This is defined from the particle population distribution as </p> F = ∫ ∫ ∫ ϕ Ω p ρ p [ D p ( u f − u p ) − ∇ p ρ p ] d Ω p d ρ p d u p {\displaystyle \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}} <h3>Interpolation operators</h3> <p>The transfer of particle properties between the Lagrangian particle space and the Eulerian grid is performed using linear interpolation functions. Assuming a <a href="/facts/Rectilinear_grid/8vDdL4UB">rectilinear grid</a> consisting of rectangular <a href="/facts/Cuboids/eLLKX4rh">cuboid</a> 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.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> <h3>Particle stress model</h3> <p>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 τ {\displaystyle \tau \;} , as </p> τ = P s θ P β max [ θ c p − θ p , ϵ ( 1 − θ p ) ] {\displaystyle \tau ={\frac {P_{s}{\theta _{P}}^{\beta }}{\max \left[\theta _{cp}-\theta _{p},\epsilon \left(1-\theta _{p}\right)\right]}}} <p>where θ c p {\displaystyle \theta _{cp}\;} is the close-pack volume fraction and β {\displaystyle \beta \;} , P s {\displaystyle P_{s}\;} , and ϵ {\displaystyle \epsilon \;} are constants. </p> <h2 id="limitations-of-the-multiphase-particle-in-cell-method">Limitations of the multiphase particle-in-cell method</h2> <ul><li>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.</li> <li>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.</li></ul> <h2 id="extensions">Extensions</h2> <ul><li>Chemical reactions – Coupling the local Eulerian values for fluid velocity in the MP-PIC method with equations for <a href="/facts/Fick%2527s_laws_of_diffusion/2guvKD4T">diffusional mass transfer</a> 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 <a href="/facts/Catalysis/LPBS7TQC">catalysis</a>,<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> <a href="/facts/Gasification/kN2Ng4hL">gasification</a>,<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> or <a href="/facts/Chemical_vapor_deposition/7khLxeU5">solid deposition</a>.</li> <li>Liquid Injection - MP-PIC method was extended by Zhao, O'Rourke, and Snider to model the coating of particle with a liquid.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a></li> <li>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.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a></li></ul> <h2 id="applications">Applications</h2> <ul><li>Biomass gasifiers<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a></li> <li><a href="/facts/Chemical_looping_combustion/1MBgSfss">Chemical looping combustion</a> (CLC)<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a><a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a><a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a><a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a><a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a></li> <li><a href="/facts/Fluidized_bed_combustion/JUoVwnIX">Circulating fluidized bed combustion</a><a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a></li> <li><a href="/facts/Coal_gasification/hzn6l8dF">Coal gasifiers</a><a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a><a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a></li> <li><a href="/facts/Cyclonic_separation/CrWt47Yy">Cyclones</a><a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a></li> <li><a href="/facts/Fluid_catalytic_cracking/F8cpnstx">Fluid catalytic cracking reactors and regenerators</a></li> <li>Fluidized bed dryers<a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a><a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a></li> <li><a href="/facts/Fluidized_bed_reactor/63oVuJdQ">Fluidized bed reactors</a><a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a></li> <li>Liquid-solid settlers<a class="footnote-ref" id="fnref:26" href="#fn:26"><sup>26</sup></a></li> <li><a href="/facts/Metal_casting/7mMSBWTs">Metal casting</a><a class="footnote-ref" id="fnref:27" href="#fn:27"><sup>27</sup></a><a class="footnote-ref" id="fnref:28" href="#fn:28"><sup>28</sup></a><a class="footnote-ref" id="fnref:29" href="#fn:29"><sup>29</sup></a></li> <li>Particle jets<a class="footnote-ref" id="fnref:30" href="#fn:30"><sup>30</sup></a></li> <li>Polysilicon deposition<a class="footnote-ref" id="fnref:31" href="#fn:31"><sup>31</sup></a></li> <li>Spray coating<a class="footnote-ref" id="fnref:32" href="#fn:32"><sup>32</sup></a></li></ul> <h2 id="software">Software</h2> <ul><li><i>Barracuda Virtual Reactor</i> by <a href="http://www.cpfd-software.com">CPFD Software</a></li> <li><i>MFiX</i> by <a href="https://mfix.netl.doe.gov/">NETL</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>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. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Snider, D.M. (2001). An Incompressible Three-Dimensional Multiphase Particle-in-Cell Model for Dense Particle Flows. Journal of Computational Physics, 170:523–549. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Snider, D. (2007). Three fundamental granular flow experiments and CPFD predictions. Powder Technology 176: 36-46. <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>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. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>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) <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Snider, D.M. (2001). An Incompressible Three-Dimensional Multiphase Particle-in-Cell Model for Dense Particle Flows. Journal of Computational Physics, 170:523–549. <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>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. <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Snider, D.M. (2001). An Incompressible Three-Dimensional Multiphase Particle-in-Cell Model for Dense Particle Flows. Journal of Computational Physics, 170:523–549. <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Snider, D. and Banerjee, S. (2010). Heterogeneous gas chemistry in the CPFD Eulerian–Lagrangian numerical scheme (ozone decomposition). Powder Technology 199(1):100–106 <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>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. <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>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 <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>CPFD Software, LLC. Barracuda 14.4 Released. http://www.cpfd-software.com/news/barracuda_14.4_released Retrieved Feb 8, 2011 <a href="http://www.cpfd-software.com/news/barracuda_14.4_released" target="_blank">http://www.cpfd-software.com/news/barracuda_14.4_released</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Blaser, P. and Chandran, R. (2009). Computational Simulation of Fluidization Dynamics Inside a Commercial Biomass Gasifier. AIChE 2009 Annual Meeting. <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>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 <a href="http://www.aicheproceedings.org/2010/Fall/data/papers/Paper202402.html" target="_blank">http://www.aicheproceedings.org/2010/Fall/data/papers/Paper202402.html</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>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 <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Yeomans, N., and Blaser, P. (2006). Predicting the Process, Foundry Management & Technology, January 2006, pp 48–49. <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Blaser, P., and Yeomans, N. (2006). Sand Core Engineering & Process Modeling, Japan Foundry Society, Vol. 2, No. 2, February 2006, pp. 420–427. <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Schleg, P. (2003). Technology of Metalcasting, American Foundry Society, Des Plaines, IL, pp. 1 and 39. <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>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" <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>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. <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><p>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 <a href="http://www.aicheproceedings.org/2009/Fall/data/papers/Paper149130.html" target="_blank">http://www.aicheproceedings.org/2009/Fall/data/papers/Paper149130.html</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li> <li id="fn:22"><p>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 <a href="http://aiche.confex.com/aiche/2006/preliminaryprogram/abstract_76001.htm" target="_blank">http://aiche.confex.com/aiche/2006/preliminaryprogram/abstract_76001.htm</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></p></li> <li id="fn:23"><p>Cocco, R. and Williams, K. (2004). Optimization of Particle Residence Time Inside Commercial Dryers with Arena-flow. AIChE 2004 National Meeting <a href="#fnref:23" class="footnote-back-ref">↩</a></p></li> <li id="fn:24"><p>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. <a href="#fnref:24" class="footnote-back-ref">↩</a></p></li> <li id="fn:25"><p>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. <a href="#fnref:25" class="footnote-back-ref">↩</a></p></li> <li id="fn:26"><p>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. <a href="#fnref:26" class="footnote-back-ref">↩</a></p></li> <li id="fn:27"><p>Yeomans, N., and Blaser, P. (2006). Predicting the Process, Foundry Management & Technology, January 2006, pp 48–49. <a href="#fnref:27" class="footnote-back-ref">↩</a></p></li> <li id="fn:28"><p>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. <a href="#fnref:28" class="footnote-back-ref">↩</a></p></li> <li id="fn:29"><p>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. <a href="#fnref:29" class="footnote-back-ref">↩</a></p></li> <li id="fn:30"><p>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. <a href="#fnref:30" class="footnote-back-ref">↩</a></p></li> <li id="fn:31"><p>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 <a href="#fnref:31" class="footnote-back-ref">↩</a></p></li> <li id="fn:32"><p>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 <a href="#fnref:32" class="footnote-back-ref">↩</a></p></li> </ol>