A predictive model for bubble-particle collisions in turbulence

TL;DR

This study presents a fully predictive model for bubble–particle collision rates in homogeneous isotropic turbulence by extending a frozen-turbulence framework to include finite-size bubble effects and particle inertia. The approach combines a derived bubble-slip-velocity pdf with a physically motivated decomposition of the collision efficiency into interception, gravity, and inertial contributions, enabling the collision kernel $\Gamma = \pi r_b^2 \int_0^\infty E_c(Re_b,St_p') w_b f(w_b)\,\mathrm{d}w_b$ to be predicted from first principles. DNS validation shows good agreement for $1/Fr \ge 4$ (Fr ≤ 0.25) across $St_b \in [0.5,6.3]$ and $St_p \in [0.01,2]$, with settling effects enhancing collisions at large $St_p$ and smaller bubbles favoring higher rates. The model provides a practical subgrid-scale tool for industrial flotation simulations, highlighting the importance of inertia, gravity, and turbulence intensity in collision dynamics and guiding parameter choices for flotation optimizations.

Abstract

The modelling of bubble-particle collisions is crucial to improving the efficiency of industrial processes such as froth flotation. Although such systems usually have turbulent flows and the bubbles are typically much larger than the particles, there currently exist no predictive models for this case which consistently include finite-size effects in the interaction with the bubbles as well as inertial effects for the particles simultaneously. As a first step, Jiang and Krug (J. Fluid Mech., vol. 1006, 2025, A19) proposed a frozen turbulence approach which captures the collision rate between finite-size bubbles and inertial particles in homogeneous isotropic turbulence using the bubble slip velocity probability density function measured from simulations as an input. In this study, we further develop this approach into a model where the bubble-particle collision rate can be predicted a priori based on the bubble, particle, and turbulence properties. By comparing the predicted collision rate with simulations of bubbles with Stokes numbers of 2.8 and 6.3, and particles with Stokes numbers ranging from 0.01 to 2 in turbulence with a Taylor Reynolds number of 64, good agreement is found between model and simulations for Froude number $Fr \leq 0.25$. Beyond this range of bubble Stokes number, we propose a criterion for using our model and discuss the model's validity. Evaluating our model at typical flotation parameters indicates that particle inertia and settling effects are usually important. Generally, smaller bubbles, larger particles, and stronger turbulence increase the overall collision rate.

Figures (14)

• Figure 1: (a) Sketch of the 'frozen turbulence' collision model along the bubble trajectory where it collides with particles at multiple instances. The dashed line represents streamlines around the bubble and the red solid lines denote particle trajectories under the frozen turbulence approximation. The sizes of the bubble, the particles and the particle trajectories are exaggerated for visibility.(b) Sketch of a particle with $St_p' = 0$ travelling along the grazing trajectory (black solid line) to collide with a bubble in still fluid. The red dashed line is the corresponding streamline. The collision cylinder used to define the collision efficiency is also shown. (c) Sketch of a bubble--particle collision with finite $St_p'$ where the grazing trajectory deviates from the corresponding streamline.
• Figure 2: (a) The actual and predicted mean vertical bubble slip velocities and (b) the inverse large-scale Froude number $1/Fr_L$ as a function of $1/Fr$. The dashed line in (b) shows the critical $Fr_L$ corresponding to a change in the scaling of $\langle w_b \rangle$ in (\ref{['eq:meanSlip']}).
• Figure 3: Plot of the standard deviation of the bubble slip velocity as a function of $St_b$. $\sigma_i^{(BCm)}$ is the model by berk_analytical_2024 where the actual value of $f_b = 1 + 0.169\langle Re_b \rangle^{2/3}$ from simulations is used in (\ref{['eq:BCBubbleRMS']}). $\sigma_i^{(Kos)}$ and $\sigma_i^{(LM)}$ are independent of $1/Fr$.
• Figure 4: Plot of the p.d.f. of the slip velocity magnitude at (a) $St_b = 0.5$, (b) $St_b = 1$, (c) $St_b = 2.8$ and (d) $St_b = 6.3$. The black line shows the actual p.d.f. from the simulations. The mean vertical slip velocities $\langle w_b \rangle$ used to obtain the model predictions (coloured solid lines) are given by (\ref{['eq:meanSlip']}). The dotted line shows the p.d.f. that is reconstructed from the measured $\langle w_b \rangle$ and $\sigma_i$.
• Figure 5: (a) The actual and predicted collision kernels at $St_b = 2.8$ and (b) $St_b = 6.3$. Also shown are the collision kernel according to kostoglou_generalized_2020$\Gamma^{(Kos)}$ and the collision kernel obtained from (\ref{['eq:collKFromEc']}) using the actual p.d.f. $\Gamma^{f(w_b)}$. The colour scheme in panel (b) follows that of panel (a). (c) The ratio of the actual collision kernel $\Gamma^{sim}$ and $\Gamma^{f(w_b)}$ to the model prediction at $St_b = 2.8$ and (d) $St_b = 6.3$. The colour scheme in panel (d) follows that of panel (c). (e) Results for $\Gamma$ for $St_p=0$ and for varying $1/Fr$ relative to the model prediction. In addition to the simulation data (markers), different variants of the Kostoglou model are included as $\Gamma^{(Kos)}$ (full model), $\Gamma^{(Kos)}_{nS}$ (neglecting small-scale turbulence), and $\Gamma^{(Kos)}_{nSnW}$(neglecting small-scale turbulence and collisions in the bubble wake). (f) The radial fluid velocity at collision distance for a stationary bubble exposed to a uniform freestream $u_{\infty}$ such that $Re_b^{\infty} = 2r_bu_{\infty}/\nu = 120$ in the simulations compared to the fit used in $\Gamma^{(Kos)}$. $-u_r(r_c)$ is plotted such that positive values imply flow towards the bubble surface.