Number Density Calculation Theory
The Number Density Calculation feature computes the number density of particles in each mesh element of the selected domains. Optionally, it also computes the average velocity of particles in each mesh element.
The definition of the number density changes depending on which option is selected from the Particle release specification list in the physics interface Particle Release and Propagation section.
Specify Release Times
If Specify release times (the default) is selected, then the number density of particles Nd,p is defined as
where the sum is taken over all particles in a mesh element. The definition of Ω depends on space dimension:
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In 3D, Ω is the mesh element volume and Nd,p has units of 1/m3.
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In 2D, Ω is the mesh element area and Nd,p has units of 1/m2.
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In a 2D axisymmetric geometry, Ω is the mesh element area multiplied by 2πr, where r is the radial coordinate, so Nd,p has units of 1/m3. That is, Ω is the volume of revolution traced out by the mesh element.
The number multiplication factor of the ith particle nn,i can be specified in the settings for the Number Density Calculation node. Alternatively, if the Enable macroparticles check box is selected in the physics interface Additional Variables section, the number multiplication factor will be initialized using the particle release feature settings.
If the Compute average velocity check box is selected, then additional variables for the average velocity components va (SI unit: m/s) are defined in each mesh element:
where vi (SI unit: m/s) is the particle velocity.
Specify Mass Flow Rate
If Specify mass flow rate is selected from the Particle release specification list, then the particle number density is computed by integrating over time,
If the Compute average velocity check box is selected, then additional variables for the total velocity normalized by mesh element volume vt (SI unit: 1/m2·s or 1/ms depending on space dimension) are defined in each mesh element:
Then the average velocity va is found by dividing by the number density,
The implicit assumption with the Specify mass flow rate option is that the total flux of particles through the system is constant over time, with each model particle representing some number of real particles per unit time. Incidentally, this is why the mesh element area or volume can be included on the right-hand side of the previous equations; if the mesh elements were deforming over time, then this assumption of constant particle flux would be violated.