Equilibrium theory of bidensity particle-laden suspensions in thin-film flow down a spiral separator

Abstract

Spiral gravity separators are designed to separate multi-species slurry components based on differences in density and size. Previous studies have investigated steady-state solutions for mixtures of liquids and single particle species in thin-film flows. However, these models are constrained to single-species systems and cannot describe the dynamics of multi-species separation. In contrast, our analysis extends to mixtures containing two particle species of differing densities, revealing that they undergo radial separation, which is an essential mechanism for practical applications in separating particles of varying densities. This work models gravity-driven bidensity slurries in a spiral trough by incorporating particle interactions, using empirically derived formulas for particle fluxes from previous bidensity studies on inclined planes. Specifically, we study a thin-film bidensity slurry flowing down a rectangular channel helically wound around a vertical axis. Through a thin-film approximation, we derive equilibrium profiles for the concentration of each particle species and the fluid depth. Additionally, we analyze the influence of key design parameters, such as spiral radius and channel width, on particle concentration profiles. Our findings provide valuable insights into optimizing spiral separator designs for enhanced applicability and adaptability.

Figures (6)

• Figure 1: A schematic of a helical channel with rectangular cross-section. The spiral inner radius is $R_{i}$, the outer radius is $R_{o}$, while the vertical spacing between each turn is the pitch $2\pi \alpha$. The channel width is $R=R_{o}-R_{i}$.
• Figure 2: The stable (a) and unstable (b) equilibrium configuration. Heavy particles are shown in dark blue and lighter particles are red. The rows increase in time from top to bottom, demonstrating the behavior of different starting configurations when perturbed by the secondary flow.
• Figure 3: The vertical equilibrium single species particle concentration profile for different $\bar{\phi}=\int_{0}^{h (r)}\phi\mathrm{d}z$ by numerically solving equations \ref{['eq:equilibrium flux region 23']}-\ref{['eq:equilibrium flux region 23d']}. The parameters are the $\rho_{2}=2.5$, $\alpha=1$, $R=1$, $r=1$, and $h=1$. Equation \ref{['eq:critical volume fraction']} yields $\phi_{c}= 0.3481$ which is marked with a dashed line. Panel (a) shows two dilute scenarios, when $\bar{\phi}$ is near 0, in red and blue. The particles concentrate near the channel bottom. Panel (b) shows that when $\bar{\phi} > \phi_{c}$ (blue), the extra particles are near the surface, and when $\bar{\phi} > \phi_{c}$ (red), the fluid is near the free surface.
• Figure 4: The vertical equilibrium particle concentration profile for different $\bar{\phi}_{i}=\int_{0}^{h (r)}\phi_{i}\mathrm{d}z$, $i=1,2$ by numerically solving equations \ref{['eq:equilibrium flux region 12']}-\ref{['eq:equilibrium flux region 12e']}. The parameters are $\rho_{1}=2.5$, $\rho_{2}=3.8$, $r=1$, $R=1$, $\alpha=1$, and $h=1$. Equation \ref{['eq:critical volume fraction']} yields $\phi_{c,1}=0.3481$ and $\phi_{c,2}=0.4131$ shown as dashed lines. The red and blue dashed line represents the critical volume fraction for the light and heavy particle species respectively. Panel (a) shows a case when small amount heavy particles ($\bar{\phi}_{2}=0.05$) is added into region (ii) which mainly consists of the light particles ($\bar{\phi}_{1}=0.30$). Panel (b) shows the case when small amount of light particles ($\bar{\phi}_{1}=0.05$) is added into heavy particle concentrated region ($\bar{\phi}_{2}=0.4$).
• Figure 5: Equilibrium profiles and geometries for two different spiral concentrator setups (see parameters in Table \ref{['table: parameter 1']}). Row (a,b) show the fluid depth $h$ as functions of $r$ and the stream function $\psi$ defined in equation \ref{['eq:stream function']}. The vertical black dotted lines indicate the location of the interface of each region (i.e. left is $R_p$ and right is $R_f$). Row (c,d) show the particle volume fractions $\phi$ as functions of $r$. The red solid line represents the light particles $\phi_1$ and the blue dash line represents the heavy particles $\phi_2$. The particle volume fraction $\phi_{c,i}$ is computed by equation \ref{['eq:critical volume fraction']}. The plots (e) and (f) show their respective geometries with respect to dimensionless variables.