Exponential scaling in early-stage agglomeration of adhesive particles in turbulence

We carry out direct numerical simulation together with an adhesive discrete element method calculation (DNS-DEM) to investigate agglomeration of particles in homogeneous isotropic turbulence (HIT). We report an exponential-form scaling for the size distribution of early-stage agglomerates, which is valid across a wide range of particle inertia and interparticle adhesion values. Such scaling allows one to quantify the state of agglomeration using a single scale parameter. An agglomeration kernel is then constructed containing the information of agglomerate structures and the sticking probability. An explicit relationship between the sticking probability and microscale particle properties is also proposed based on the scaling analysis of the equation for head-on collisions. Our results extend Smoluchowski's theory to the condition of noncoalescing solid adhesive particles and can reproduce DNS-DEM results with a simple one-dimensional simulation.

Figure 1

Snapshot of the simulated system at $t=20$. The enlarged view from the middle slice $\left(x=0\right)$ shows agglomerates and their size $A$ (defined as the number of primary particles contained in the agglomerate, indicated by the color code).

Figure 2

(a) Temporal evolution of the collision kernel $\mathrm{\Gamma }\left(t\right)/{\mathrm{\Gamma }}_0$ for cases with ${\mathrm{St}}_{\mathrm{k}}=5.8$ and $\mathrm{Ad}=0.013$ (circles), 1.3 (left-pointing triangles), 13 (diamonds), 64 (upward triangles), and 128 (squares). (b) Fraction of particles, $P\left(A\right)$, contained in agglomerates of size $A$ at $t=15$ for $\mathrm{Ad}=1.3$ and 64. (c) Gyration radius of agglomerates $R_g\left(A\right)/r_p$ as a function of agglomerate size $A$ at $t=15$ for the cases with ${\mathrm{St}}_{\mathrm{k}}=5.8$ and $\mathrm{Ad}=13$ (diamonds), 64 (triangles), and 128 (squares). The solid line shows Eq. (13) with $\chi =1.64$ and $D_f=1.64$. (d) An agglomerate produced in the simulation with $\text{St}=5.8$ and $\mathrm{Ad}=64$ with its equivalent sphere with radius of gyration (shaded region).

Figure 3

Left [panels (a), (d), and (g)]: Temporal evolution of the collision kernel $\mathrm{\Gamma }\left(t\right)/{\mathrm{\Gamma }}_0$. Middle [panels (b), (e), and (h)]: Fraction of particles, $P\left(A\right)$, contained in agglomerates of size $A$ at $t=15$ for $\mathrm{Ad}=1.3$ and 64. Right [panels (c), (f), and (i)]: Gyration radius of agglomerates $R_g\left(A\right)/r_p$ as a function of agglomerate size $A$ at $t=15$. The solid lines in panels (c), (f), and (i) are fits to Eq. (13) with (c) $\chi =1.80$ and $D_f=1.54$, (f) $\chi =1.70,D_f=1.60$, and (i) $\chi =1.49,D_f=1.71$. Different rows stand for results for different ${\mathrm{St}}_{\mathrm{k}}$: top, ${\mathrm{St}}_{\mathrm{k}}=0.72$; middle, ${\mathrm{St}}_{\mathrm{k}}=12$; and bottom, ${\mathrm{St}}_{\mathrm{k}}=23$.

Figure 4

(a) Scaled number density $n\left(A\right)/n_0$ of agglomerates of size $A$ for the case with ${\mathrm{St}}_{\mathrm{k}}=5.8$ and $\mathrm{Ad}=64$ at $t=5$ (circles), 10 (squares), 15 (crosses), and 20 (triangles). The solid lines are fits to Eq. (14). Inset: scaled number density $n\left(A\right)/\left(n_0\beta \right)$ as a function of $A/\kappa$ for ${\mathrm{St}}_{\mathrm{k}}=5.8$ (circles), 12 (triangles), and 23 (squares). For each ${\mathrm{St}}_{\mathrm{k}}$, results are shown for $\mathrm{Ad}=1.3$ (black), 13 (blue), and 64 (red), at $t=5$, 10, and 15. The solid line is the exponential scaling $y=exp(-x)$. (b) $n\left(A\right)/n_0$ vs $A$ calculated from population balance equations. Legends are the same as in panel (a). In the inset of panel (b), we show the temporal evolution of the scale parameter $\kappa$ from DNS-DEM result (circles) and from PBE (solid line).

Figure 5

(a) Temporal evolution of the parameter $\kappa$ for DNS-DEM simulation with ${\mathrm{St}}_{\mathrm{k}}=5.8$ and $\mathrm{Ad}=0.013$ (circles), 1.3 (left-pointing triangles), 13 (diamonds), 64 (upward triangles), 128 (squares), and 256 (axes). The solid lines spanning from light to dark color are results from PBE with the sticking probability $\mathrm{\Theta }=0$, 0.2, 0.4, 0.8, and 1. (b) Sticking probability $\mathrm{\Theta }$, determined from Eq. (16), as a function of adhesion parameter $\mathrm{Ad}$ for ${\mathrm{St}}_{\mathrm{k}}=2.9$ (circles), 5.8 (triangles), 12 (diamonds), and 23 (squares). The horizontal dashed line is $\mathrm{\Theta }=1$.

Figure 6

Sticking probability $\mathrm{\Theta }$ as a function of the adhesion parameter ${\mathrm{Ad}}_n$, which is defined based on the averaged normal collision velocity $\langle v_{cn}\rangle$, for ${\mathrm{St}}_{\mathrm{k}}=2.9$ (circles), 5.8 (triangles), 12 (diamonds), and 23 (squares). The solid line is $\mathrm{\Theta }=0.017{\mathrm{Ad}}_n$ and the horizontal dashed line is $\mathrm{\Theta }=1$.