Deconvolution of inclined channel elutriation data to infer particle size distribution

In this paper we investigate the application of optimisation techniques in the deconvolution of mineral fractionation data obtained from a mathematical model for the operation of a fluidised bed with a set of inclined parallel channels mounted above. The model involved the transport equation with a stochastic source function and a linearly increasing fluidisation rate, with the overflow solids being collected in a finite number of increments (bags). Deconvolution of this data is an ill-posed problem and regularisation is required to provide feasible solutions. Deconvolution with regularisation is applied to a synthetic feed consisting of particles of constant density that vary in size only. It was found that the feed size distribution could be successfully deconvolved from the bag weights, with an accuracy that improved as the rate acceleration of the fluidisation rate was decreased. The deconvolution error only grew linearly with error in the measured bag masses. It was also shown that combining data from two different liquids can improve the accuracy.

💡 Research Summary

The paper investigates how to recover the particle size distribution of a granular feed from the time‑resolved mass data collected during an inclined‑channel elutriation experiment. The authors first construct a forward model that describes the motion of particles in a set of parallel inclined channels mounted above a fluidised bed. The fluidisation velocity is assumed to increase linearly with time (v(t)=v₀+λ(t‑t₀)), producing a parabolic velocity profile across the channel gap. For a particle of radius s resting on the lower wall, the local fluid velocity is approximated by v(s)=¯v·6s/z·(1‑s/z). The particle’s own tangential settling velocity is obtained from a standard terminal‑velocity correlation (valid for Reₜ < 10⁵) with an effective gravitational component g sin θ. The net upward velocity is defined as v_net=v(s)‑v_tT; particles only enter the channels when v_net>0.

The mass of particles remaining in the vertical section obeys a simple first‑order decay law ∂M/∂t=‑k·c(s,t)·M, where c(s,t)=max{v_net,0} and k is a constant elutriation rate. This leads to a one‑dimensional transport equation for the particle distribution u(x,s,t) along the channel: ∂u/∂t + c(s,t)∂u/∂x = 0, with appropriate initial and boundary conditions. By integrating this model over the channel length and over size bins, the authors generate synthetic “bag‑mass versus time” data that mimic the experimental collection of overflow solids in discrete increments.

The inverse problem—recovering the original size distribution from the bag data—is ill‑posed. The authors therefore apply Tikhonov regularisation, solving a minimisation problem of the form ‖A·x‑b‖²+α‖L·x‖², where A is the forward‑model matrix, x the vector of size‑bin masses, b the measured bag masses, L a smoothing operator, and α the regularisation parameter. They explore a range of α values and use the L‑curve method to select an optimal balance between data fidelity and smoothness.

Synthetic tests are performed with a mono‑density feed (constant particle density, sizes ranging from 0.038 mm to 2 mm). The authors vary the fluid‑velocity acceleration λ and the regularisation strength. Results show that smaller λ (i.e., slower increase of fluidisation rate) yields markedly lower reconstruction error because the mapping from size to elutriation time becomes less ambiguous, improving the conditioning of A. Adding Gaussian noise to the bag masses demonstrates a linear propagation of measurement error into the reconstructed distribution, confirming the robustness of the regularised solution for realistic noise levels (1–5 %).

A further set of experiments combines data from two liquids of different densities (e.g., water and a lithium‑heteropolytungstate solution). By stacking the two measurement vectors and augmenting the forward matrix, the authors achieve a better‑conditioned system, which reduces the dependence on regularisation and improves the accuracy of the recovered size distribution.

Key conclusions are: (1) With an accurate forward model, regularised optimisation can reliably invert elutriation data to obtain particle size distributions, even though the problem is intrinsically ill‑posed. (2) The rate of increase of the fluidisation velocity should be kept low to enhance inversion stability. (3) Multi‑liquid experiments provide complementary information that strengthens the inverse problem and yields more precise results. The paper acknowledges that many simplifying assumptions were made—neglect of particle‑particle interactions, wall friction, shear‑induced lift, and recirculation—so future work should incorporate these effects, extend the methodology to simultaneous size‑and‑density distributions, and validate the approach with real experimental data. Overall, the study offers a solid mathematical foundation and practical guidelines for using inclined‑channel elutriation as a diagnostic tool in mineral processing and related fields.