Segregation models for density-bidisperse granular flows

Individual constituent balance equations are often used to derive expressions for species-specific segregation velocities in flows of dense granular mixtures. We propose a semiempirical expression for the interspecies momentum exchange in density-bidisperse granular flows as an extension of ideas from kinetic theory and compare it to a previous viscous drag approach that is analogous to particles settling in a fluid. The proposed model expands the range of the granular kinetic theory from short-duration binary collisions to the multiple enduring contacts characteristic of dense shear flows and incorporates the effects of particle friction, concentration ratio, and local flow conditions. The segregation velocities derived from the momentum balance equation using both interspecies drag models match the downward and upward segregation velocities of heavy and light particles obtained from DEM simulations through the flowing layer depth for different density ratios and constituent concentrations in confined shear flows. Predictions of the kinetic theory inspired approach are additionally compared to results from free surface heap flow simulations, and, again, a close match is observed.

Figure 1

Schematic of DEM simulation setup.

Figure 2

Instantaneous streamwise velocity $u_i$ of each constituent for mixed light ($+$) and heavy ($◯$) particles averaged in each horizontal layer $0.1\mathrm{s}$ after shear onset. Dashed curves represent the velocity profiles imposed using the forcing specified in Eq. (17). ($d_h=d_l=4\mathrm{mm}$, $c_h=c_l=0.5$, $\mu =0.2$, $e=0.9$, and $R_{\rho }=4$.)

Figure 3

Determining the segregation velocity. (a) Average center of mass displacement $\mathrm{\Delta }\overline{z}/d$ time series at different vertical positions under uniform shear ($\dot{\gamma }=U/h=25{\text{s}}^{-1}$, $R_{\rho }=8$, $c_h=c_l=0.5$, $\mu =0.2$, and $e=0.9$). (b) The heavy particle concentration profile remains relatively constant $1\mathrm{s}$ after shear onset for the flow in Fig. 3. (c) Segregation velocity profiles for light particles $w_{p,l}$ ($\times$) and heavy particles $w_{p,h}$ ($◯$) are calculated as the average rate of change of $\mathrm{\Delta }\overline{z}$ over the first $1\mathrm{s}$ of the simulation for flows with different density ratios and shear rates ($c_h=c_l=0.5$, $\mu =0.2$, and $e=0.9$). Both shear rate $\dot{\gamma }$ and density ratio $R_{\rho }$ affect the segregation velocity profile. Error bars represent the uncertainty in determining $\mathrm{\Delta }\overline{z}$ and $w_{p,h}$. (d) Inertial number profiles for the flows in Fig. 3. The local inertial number is estimated as $I=\dot{\gamma }d/\sqrt{{\varphi }_{\text{solid}}g(1.05h-z)}$ by substituting the local overburden pressure $P=P_{\text{wall}}+{\rho }_{\text{solid}}{\varphi }_{\text{solid}}g(h-z)$ into Eq. (7).

Figure 4

Segregation velocities of light particles vs. local inertial number $I$ at different depths for nonlinear velocity profiles (a) $u=U{z}^2/h^2$ and (b) $u=U{e}^{2.3(z/h-1)}$ for simulations with $U=2$m/s, $R_{\rho }={\rho }_h/{\rho }_l\in \{2,4,10\}$, $c_h=c_l=0.5$, $\mu =0.2$, and $e=0.9$. Dashed lines are linear fits for each density ratio, demonstrating a linear dependence of $w_{p,l}$ on $I$. Error bars represent the uncertainty in determining $w_{p,l}$ as indicated in Fig. 3.

Figure 5

Segregation velocities of light particles $w_{p,l}$ for $0.2\le z/h\le 0.8$ vs. (a) inertial number $I$ and (b) modified inertial number $I^{*}=\sqrt{I^2-I_0^2}$ for simulations with particle properties given in Fig. 4 but for uniform shear rate profiles with $\dot{\gamma }=5{\text{s}}^{-1}$ ($◯$), $15{\text{s}}^{-1}$ ($\square$), and $25{\text{s}}^{-1}$ ($\diamond$). $I_0$ is the cutoff inertial number at which density segregation effectively ceases [18]. The dashed line in both figures is identical and a linear fit to the data in (b). Error bars represent the uncertainty in determining $w_{p,l}$ as indicated in Fig. 3.

Figure 6

$w_{p,l}/I$ vs. $R_{\rho }$ for simulations with $1.3\le R_{\rho }\le 10$, $\mu =0.2$, $e=0.9$, and $c_h=c_l=0.5$. Colors and symbols represent simulations with $\dot{\gamma }=U/h$ ($◯$), $2Uz/h^2$ ($\times$), and $2.3U{e}^{2.3(z/h-1)}/h$ ($⊳$). Dashed curve shows the dependence of $w_{p,i}/I$ on $R_{\rho }$ from Eq. (10) with $B\left(\mu \right)=700$.

Figure 7

Normalized velocity difference between species $\mathrm{\Delta }\widehat{w}$ vs. concentration ratio $c_l/c_h$ for simulations with $R_{\rho }=4$, $\mu =0.2$, $e=0.9$, and $\dot{\gamma }=U/h=25{\text{s}}^{-1}$. Error bars represent uncertainty in determining $\mathrm{\Delta }\widehat{w}$. Strong correlation between $\mathrm{\Delta }\widehat{w}$ and $c_l/c_h$ is found for $0.1\le c_l,c_h\le 0.9$ ($◯$), as opposed to situations approaching a single intruder particle at concentrations outside this range, $c_l$, $c_h\le 0.1$ or $c_h$, $c_l\ge 0.9$ ($◯$). Solid line with slope 0.25 is consistent with the slope of the data, confirming the dependence of the drag model on particle concentrations for $0.1\le c_l,c_h\le 0.9$.

Figure 8

Dependence of segregation velocities $w_{p,i}$ on contact parameters. (a) Light ($\times$) and heavy ($◯$) particle segregation velocities are nearly independent of restitution coefficient $e$ at $z/h=0.5$ with $\mu =0$ (black), 0.1 (red), and 0.4 (blue). (b) Particle segregation velocity decreases as $\mu$ increases for $z/h=0.3$ (red), 0.5 (blue), 0.7 (black) with $e=0.9$. $R_{\rho }=8$, $c_h=c_l=0.5$, and $\dot{\gamma }=U/h=25{\text{s}}^{-1}$.

Figure 9

Determining $B\left(\mu \right)$. (a) $\frac{gd}{{\varphi }_{\text{solid}}}(R_{\rho }-\frac{1}{R_{\rho }})\sqrt{\frac{c_l}{c_h}}{(1-c_l)}^2$ vs. ${(w_{p,l}/I)}^2$ for uniform shear flows with $R_{\rho }\in \{1.5,2,3,4,5,6,7,9,10\}$ and three different $\mu$ values. $B\left(\mu \right)$ is determined by the fitted slope of the dashed lines, which are forced through zero ($c_h=c_l=0.5$, $\dot{\gamma }=25{\mathrm{s}}^{-1}$, and $e=0.9$). (b) $B\left(\mu \right)$ vs. $\mu$ based on 243 simulations. Different symbols correspond to $e=0.6$ ($\times$), 0.8 ($⊲$), and 0.9 ($◯$) for uniform shear flows.

Figure 10

Viscous drag coefficient $\epsilon$ calculated from Eq. (15) vs. solid volume fraction ${\varphi }_{\text{solid}}$ for different flow configurations and friction coefficients. The dashed line indicates the mean value $\epsilon =1.73$.

Figure 11

Effective friction coefficient ${\mu }_{\text{eff}}$ vs. local inertial number $I$ for different combinations of density ratio $R_{\rho }$, heavy particle concentration $c_h$, and shear rate $\dot{\gamma }$ with $e=0.9$ and $U=2$m/s. Different colors and symbols represent simulations with $\mu =0.5$ (top) and $\mu =0.2$ (bottom). The solid curve shows the prediction of Eq. (23) with ${\mu }_s=0.364$, ${\mu }_2=0.772$, and $I_c=0.434$ for data from a previous study on chute flow [60]. The dashed curve shows the prediction of Eq. (23) with ${\mu }_s=0.3$, ${\mu }_2=0.68$, and $I_c=0.4$ for data from this study.

Figure 12

Predicted segregation velocity profiles from modified KTGF drag model [Eqs. (10) and (11)] (solid curves) and viscous drag model [Eqs. (15) and (16)] (dashed curves) compared to DEM simulation data for light ($\times$) and heavy ($◯$) particles under (a) uniform ($\dot{\gamma }=U/h$) and (b) varying shear rate profiles $[\dot{\gamma }=2Uz/h^2$, $2.3U{e}^{2.3(z/h-1)}/h$] with different density ratios. $\mu =0.2$, $c_h=c_l=0.5$, and $e=0.9$.

Figure 13

Predicted segregation velocity profiles from modified KTGF drag model [Eqs. (10) and (11)] (solid curves) and viscous drag model [Eqs. (15) and (16)] (dashed curves) compared to DEM simulation data under uniform shear for (a) light and (b) heavy particles with heavy particle concentration $c_h=0.3$, 0.5 and 0.7 ($c_l=1-c_h$). $\mu =0.2$, $R_{\rho }=8$, $\dot{\gamma }=25{\text{s}}^{-1}$, and $e=0.9$.

Figure 14

Segregation velocities from DEM heap flow simulations (points) [56] for (a) $R_{\rho }=1.84$, $\mu =0.2$, $e=0.9$, and $d=3\mathrm{mm}$, and (b) $R_{\rho }=3.33$, $\mu =0.2$, $e=0.9$, and $d=2\mathrm{mm}$. Dashed lines are linear fits to the data, and solid lines are predictions of the model based on the KTGF drag model.

Figure 15

Dependence of segregation length scale $S_D$ on the density ratio $R_{\rho }$ from DEM simulations (circles) for segregation in a bounded heap flow [56] compared to predictions of (a) the empirical fit of Eq. (26) and (b) Eq. (25) derived from the KTGF drag model with $c_h/c_l=1$, $\mu =0.2$ and different values of the mean pressure in the flowing layer $P/\left({\rho }_{\text{bulk}}gd\right)$.