.. include:: /images.rst DEM02: Bouncing particle ------------------------ This case provides a comparison between the MFIX-DEM linear spring-dashpot collision model and the *hard sphere* model where collisions are instantaneous. The hard sphere model can be seen as the limiting case where the normal spring coefficient is large, :math:k_{n} \rightarrow \infty. This case was originally reported in :cite:Garg2012. .. _description-dem02: Description ~~~~~~~~~~~ A smooth (frictionless), spherical particle falls freely under gravity from an initial height, :math:h_{0}, and bounces upon collision with a fixed wall :numref:dem01fig1. Assuming that the collision is instantaneous, the maximum height the particle reaches after the first collision (bounce), :math:h_{1}^{\mathrm{\max}}, is given by .. math:: h_{1}^{\mathrm{\max}} = \left( h_{0} - r_{p} \right)e_{n}^{2} :label: dem02eq1 where :math:r_{p} is the particle radius, and :math:e_{n} is the restitution coefficient. A general expression for the maximum height following the :math:k^{\text{th}} bounce is .. math:: h_{k}^{\mathrm{\max}} = \left( h_{0} - r_{p} \right)e_{n}^{2k} + r_{p}. :label: dem02eq2 .. _setup-dem02: Setup ~~~~~ .. _dem01table2: .. csv-table:: DEM-02 Setup, Initial and Boundary Conditions. :widths: auto :header: "Computational/Physical model", " ", " " "1D, Transient", " ", " " "Granular flow (no gas)", " ", " " "Gravity", " ", " " "Thermal energy equation is not solved", " ", " " " ", " ", " " "**Geometry**", " ", " " "Coordinate system", "Cartesian", " ", " " "x-length", "1.0", "$$m$$", " " "y-length", "1.0", "$$m$$", " " "z-length", "1.0", "$$m$$", " " " ", " ", " " "**Solids Properties**", " ", " " "Normal spring coefficient, :math:k_{n}", "*varied*", "(N·m\ :sup:-1)" "Restitution coefficient, :math:e_{n}", "*varied*", "" "Friction coefficient, :math:\mu", "0.0", "" " ", " ", " " "**Solids 1 Type**", "DEM", " " "Diameter, :math:d_{p}", "0.2", "$$m$$" "Density, :math:\rho_{s}", "2600", "(kg·m\ :sup:-3)" " ", " ", " " "**Boundary Conditions**", " ", " ", " " "All boundaries", "Solid walls", " " .. _results-dem02: Results ~~~~~~~ Simulations of a freely-falling particle dropped from an initial height of 0.5m were conducted for three normal spring coefficients, :math:\lbrack 0.5,\ 5.0,\ 50.0\rbrack \times 10^{5} N·m\ :sup:-1, and six restitution coefficients, [0.5, 0.6, 0.7, 0.8, 0.9, 1.0]. All simulations employed the Adams-Bashforth time-stepping method. The maximum height attained after the k\ :sup:th collision for all cases is shown in :numref:dem02fig1. .. _dem02fig1: .. figure:: ../media/image52.png :align: center Comparison between the analytic solution from a hard-sphere model (solid lines) and MFIX-DEM (symbols) of the maximum height reached after the k\ :sup:th wall collision for a freely falling particle. Three values for the normal spring coefficient are used (left to right) with six restitution coefficients. :numref:dem02fig2 illustrates the percent relative difference between the analytical solution for a hard-sphere model and the MFIX-DEM simulation. In the limit of the hard-sphere model (shown left to right by an increasing spring coefficient), the difference between the two collision models decreases. .. _dem02fig2: .. figure:: ../media/image53.png :align: center Percent relative difference between the analytic solution for a hard-sphere model and MFIX-DEM of the maximum height reached after the k\ :sup:th wall collision for a freely falling particle. Three values for the normal spring coefficient are used (left to right) with six restitution coefficients.