Effects of particle size and density on dust dispersion behind a moving shock

Numerical simulations are performed to study the effect of particle size and density on dust dispersion behind a moving shock. The numerical model used in this paper takes into account multiple particle types with a binning approach, and each bin of particles has its own characteristic uniform particle size and density. The model solves one set of governing equations for the gas phase and $M$ sets of governing equations for the $M$ particle types. Equations for each bin of particles are coupled with the gas flow as well as all other particle types. The specific conditions simulate a Mach-1.4 shock passing over a dust layer containing two uniformly mixed particle types. The background gas condition is at 67 kPa and 295 K. The dust layer is 1.27 cm thick and consists of spherical particles with selected sizes and densities. Preliminary one-dimensional computations of a shock passing through a dilute particle curtain containing different particle types suggest that particles with different sizes or densities behave differently and can separate into different clouds. Particles with larger inertia require a longer relaxation time to the postshock condition. The two-dimensional calculations show that larger particles are lifted to a higher level than smaller particles. In regions near the shock front, larger particles experience a smaller drag force, pushing the particles into the dust layer, and a larger lifting force, pulling the particles into the air. In addition, lighter particles are lifted higher than heavier particles due to a smaller inertia. Particle size plays a significant role on dust dispersion, while particle density has only a minor effect.

Figure 1

Initial conditions for a one-dimensional test of a shock moving through a particle cloud containing two uniformly mixed particle types.

Figure 2

Computed (a) volume fractions for the two particle types $\left(d_{s1}=80\mu \mathrm{m},d_{s2}=10\mu \mathrm{m}\right)$ and (b) gas and particle velocities at $17,100$, and $200\mu \mathrm{s}$. The edge of the partice cloud is defined by a volume fraction of $5\times {10}^{-5}$.

Figure 3

Computed (a) volume fractions for the two particle types $({\rho }_{s1}=1300\text{kg}/{\mathrm{m}}^3,{\rho }_{s2}=2700$ kg/${\mathrm{m}}^3)$ and (b) gas and particle velocities at $17,100$, and $200\mu \mathrm{s}$. The edge of the particle cloud is defined by a volume fraction of $5\times {10}^{-5}$. Dashed lines indicate the locations of the gas shock and the edge of the dust clouds.

Figure 4

Computated (a) coal- and rock-dust volume fraction and (b) gas and particle velocities at $0,50,100$, and $200\mu \mathrm{s}$. The edge of the cloud in panel (b) is defined by a volume fraction of ${10}^{-9}$.

Figure 5

Average velocities for both types of particles in case I $(d_s=10,80\mu \mathrm{m}$; ${\rho }_{s1}={\rho }_{s2}=1300$ kg/${\mathrm{m}}^3)$ as a function of time in the postshock flow.

Figure 6

Initial setups for two-dimensional simulations. The layer contains two particle types that are uniformly mixed with each other.

Figure 7

Computed particle volume fractions for both particle types in cases I and II. (a) Particle I in case I $\left(d_s=80\mu \mathrm{m}\right)$; (b) particle II in case I $\left(d_s=10\mu \mathrm{m}\right)$; (c) particle I in case II $\left(d_s=120\mu \mathrm{m}\right)$; and (d) particle II in case II $\left(d_s=10\mu \mathrm{m}\right)$.

Figure 8

Computed dust-lifting height for different particle diameters in both test cases. The edge of the dust layer is defined as ${\alpha }_s=5\times {10}^{-5}$.

Figure 9

Computed particle volume fractions for both particle types in cases I and II. (a) Particle I in case I $({\rho }_s=1000$ kg/${\mathrm{m}}^3)$; (b) particle II in case I $({\rho }_s=1500$ kg/${\mathrm{m}}^3)$; (c) particle I in case II $({\rho }_s=1000$ kg/${\mathrm{m}}^3)$; and (d) particle II in case II $({\rho }_s=3000$ kg/${\mathrm{m}}^3)$.

Figure 10

Computed dust-lifting height for different particle densities. The edge of the dust layer is defined as ${\alpha }_s=5\times {10}^{-5}$.

Figure 11

Net accelerations and accelerations due to drag, lift, Archimedes force, intergranular stress, and particle-hindrance force for both particle types $(d_{s1}=120\mu \mathrm{m},d_{s2}=10\mu \mathrm{m},{\rho }_{s1}={\rho }_{s2}=1300$ kg/${\mathrm{m}}^3)$ along vertical direction. Forces have been normalized by ${\alpha }_s{\rho }_s$.

Figure 12

Vertical accelerations produced by drag, lift, Archimedes force, intergranular stress, and particle-hindrance force along three vertical lines indicated in the net acceleration color map for (a) $d_{s,1}=120\mu \mathrm{m}$ and (b) $d_{s,2}=10\mu \mathrm{m}$. Forces have been normalized by ${\alpha }_s{\rho }_s$.

Figure 13

Computed dust-lifting height of a shock passing over a layer of dust containing two uniformly mixed particle types with three, four, and five levels of refinement $({\rho }_{s1}={\rho }_{s2}=1300$ kg/${\mathrm{m}}^3$; $d_{s1}=10\mu \mathrm{m}$; $d_{s2}=80\mu \mathrm{m})$. The edge of the dust layer is defined as ${\alpha }_s=5\times {10}^{-5}$.

Figure 14

Comparison of dust-lifting height for cases with zero granular temperatures (solid line) and cases using current model (dashed line). Left: case I $\left(d_{s1}=120\mu \text{m},d_{s2}=10\mu \text{m}\right)$; right: case II $\left(d_{s1}=80\mu \text{m},d_{s2}=10\mu \text{m}\right)$.

Figure 15

Particle volume fraction of a shock passing over a dust layer containing two types of particles $\left(d_{s1}=80\mu \text{m},d_{s2}=10\mu \text{m}\right)$. Top: simulation results assuming zero collisional pressure $\left(P_{c,lm}=0\right)$; bottom: results with current model.

Figure 16

Particle volume fraction of a shock passing over a dust layer containing two types of particles $\left(d_{s1}=80\mu \text{m},d_{s2}=10\mu \text{m}\right)$. Top: simulation results with $2\times P_{\mathrm{fric},l}$; bottom: results with current model $\left(1\times P_{\mathrm{fric},l}\right)$.