Analytical solution for a three-dimensional non-homogeneous bivariate population balance equation---a special case

There has been a dramatic increase in the number of research publications using the population balance equation (PBE). The PBE allows the prediction of the spatial distribution of the dispersed phase size for an accurate estimation of the flow fields in multiphase flows. A few recent studies have proposed new efficient numerical methods to solve non-homogeneous multivariate PBE and implemented the same in computational fluid dynamics (CFD) codes. However, these codes are generally benchmarked against other numerical methods and applied without verification. To address this gap, an analytical solution for a three-dimensional non-homogeneous bivariate PBE is presented here for the first time. The method of manufactured solutions (MMS) has been used to construct a solution of the non-homogeneous PBE containing breakage and coalescence terms, and an additional source term appearing as a result of this method. The analytical solution presented in this work can be used for the rigorous verification of computer codes written to solve the non-homogeneous bivariate PBE. Quantification of the errors due to different numerical schemes will also become possible with the availability of this analytical solution for the PBE

Evaluation of collector performance at the bubble-particle scale

Particle attachment and detachment in froth flotation are complex processes and their measurement presents many challenges. Of particular interest is the effect of collectors at the bubble-particle scale, in order to assess the strength or collecting ability of these reagents. However, studies of the effect of collectors on particle attachment at the bubble-particle scale are scarce. In this work, we propose a methodology to characterise collector strength by measuring the attachment rate of particles to a capillary-pinned bubble. An image processing technique was developed to quantify bubble surface coverage over time, which was then used to determine particle attachment kinetics. The image analysis strategy is based on the sessile drop method and uses curve fitting to determine accurately the particle coverage. The methodology was used to assess the collecting ability of three chalcopyrite collectors. Interestingly, although very similar contact angle measurements were found for two of the collectors, they showed distinctly different particle attachment kinetics. It is proposed that this particle-bubble attachment method can be used to gain additional information not currently available from either contact angle measurements or bulk collector performance tests

A methodology to implement a closed-loop feedback-feedforward level control in a laboratory-scale flotation bank using peristaltic pumps

This paper describes the implementation of a level control strategy in a laboratory-scale flotation system. The laboratory-scale system consists of a bank of three flotation tanks connected in series, which mimics a flotation system found in mineral processing plants. Besides the classical feedback control strategy, we have also included a feedforward strategy to better account for process disturbances. Results revealed that the level control performance significantly improves when a feedforward strategy is considered. This methodology uses peristaltic pumps for level control, which has not been extensively documented even though: (1) peristaltic pumps are commonly used in laboratory-scale systems, and (2) the control implementation is not as straightforward as those control strategies that use valves. Therefore, we believe that this paper, which describes a proven methodology that has been validated in an experimental system, can be a useful reference for many researchers in the field.•Preparation of reagents to ensure that the froth stability of the froth layer is representative of an industrial flotation froth.•Calibration of instruments - convert the electrical signal from PLCs to engineering units.•Tuning PI parameters using SIMC rules by performing step-changes in each flotation cell

The effect of particle size distribution on froth stability in flotation

Separation of particles of different surface properties using froth flotation is a widely-used industrial process, particularly in the minerals industry where it is used to concentrate minerals from ore. One of the key challenges in developing models to predict flotation performance is the interdependent nature of the process variables and operating parameters, which limits the application of optimising process control strategies at industrial scale. Froth stability, which can be quantified using air recovery (the fraction of air entering a flotation cell that overflows in the concentrate as unburst bubbles), has been shown to be linked to flotation separation performance, with stable froths yielding improved mineral recoveries. While it is widely acknowledged that there is an optimum particle size range for collection of particles in the pulp phase, the role of particle size on the measured air recovery and the resulting link to changes in flotation performance is less well understood. This is related to the difficulty in separating particle size and liberation effects. In this work, the effects of particle size distribution on air recovery are studied in a single species (silica) system using a continuous steady-state laboratory flotation cell. This allows an investigation into the effects of particle size distribution only on froth stability, using solids content and solids recovery as indicators of flotation performance. It is shown that, as the cell air rate is increased, the air recovery of the silica system passes through a peak, exhibiting the same froth behaviour as measured industrially. The air recovery profiles of systems with three different particle size distributions (d80 of 89.6, 103.5 and 157.1 μm) are compared. The results show that, at lower air rates, the intermediate particle size distribution (103.5 μm) yields the most stable froth, while at higher air rates, the finest particles (89.6 μm) result in higher air recoveries. This is subsequently linked to changes in flotation performance. The results presented here highlight, for the first time, the link between particle size distribution in flotation feeds, air recovery and flotation performance. The results demonstrate that there is an optimal air rate for each particle size distribution, therefore changes in particle size distribution in the feed to flotation cells require a change in air rate in order to maximise mineral recovery

Development and Validation of a Dynamic Model for Flotation Predictive Control Incorporating Froth Physics

Data Availability Statement: The data that support the findings of this study are available from the corresponding author, P.Q, upon reasonable request.Paulina Quintanilla would like to acknowledge the National Agency for Research and Development (ANID) for funding this research with a scholarship from “Becas Chile”. The Society of Chemical Industry is also greatly acknowledged for the support granted by the SCI Messel Scholarship 2020

RECUPERACIÓN DE PERCLORETILENO EN INDUCALSA

A recovery system for perchlorine ethylene from hazardous liquid waste (Ministry Agreement 026 MAE) which stems from shoe making polyurethane sole washing has been designed (Brito H. / Rodríguez B., 2014). Such waste is directly disposed into the environment, causing losses in flora and fauna (GUARIN, O. / RUEDA, G. / PEREZ, H. 2010). Besides contributing with environmental liabilities, the recovered perchlorine ethylene complies the requirements of the DIRSA safety card, because of what it is reutilized in the manufacture process, which contributes to the decrease of production costs and the conservation of the ecosystem. This hazardous waste should be delivered to an environmental manager before paying for its transportation and final disposal (90 USD per kg of waste). For this reason, INDUCALSA along with ESPOCH researchers carried out a scaling of the recovery system by working with 20 samples of perchlorine ethylene – silicone during a lapse of four weeks. An initial characterization was followed by in – lab recovery to measure the variables (temperature, time, input and output flow) with whom the process was scaled for 45,73 Kg/h of waste. The pH, density, viscosity and IR spectrum of the obtained product were analyzed. They showed the purity of the solvent (Disolventes Reunidas, S.A., 2003), which is acceptable because of its high economic value in the market as well as the contribution to the change of the productive matrix in Ecuador (SENPLADES, 2012)

Economic model predictive control for a rougher froth flotation cell using physics-based models

The development of an economic model predictive control (E-MPC) strategy is presented. The strategy uses a novel dynamic flotation model that incorporates the physics of the froth phase in a flotation cell. The dynamic model was previously calibrated and validated using experimental data. Sensitivity analyses were conducted to select a suitable objective function that accounted for both process economics and control variable sensitivities. While the ultimate goal of a rougher flotation cell is to maximise the metallurgical recovery at a steady state for a specified minimum grade, it was evident that the incorporation of air recovery dynamics (which can be measured in real-time) and concentrate grade dynamics (calculated through first-principle models) led to the best results. The addition of a dynamic variable that can be easily measured online, i.e. air recovery, offers great potential to improve plant performance in existing froth flotation systems. Furthermore, a minimum concentrate grade was imposed in the E-MPC strategy. This acts as an economic constraint as it allows the metallurgical recovery to be optimised while ensuring that concentrate grade requirements are met. The dynamic optimisation problem for the E-MPC strategy was discretised using orthogonal collocations, and was implemented in Matlab using automatic differentiation via CasADi. Two typical manipulated variables were considered: air flowrate and pulp height setpoints. Based on laboratory-scale data, the implementation of the E-MPC strategy resulted in improvements ranging from +8 to +22 % in metallurgical recovery, while maintaining the specified grade. This is therefore an encouraging control strategy to explore in larger flotation systems

Economic model predictive control for a rougher froth flotation cell using physics-based models

Data availability: Data will be made available on request.Supplementary data are available online at: https://www.sciencedirect.com/science/article/pii/S089268752300064X#appSC .The development of an economic model predictive control (E-MPC) strategy is presented. The strategy uses a novel dynamic flotation model that incorporates the physics of the froth phase in a flotation cell. The dynamic model was previously calibrated and validated using experimental data. Sensitivity analyses were conducted to select a suitable objective function that accounted for both process economics and control variable sensitivities. While the ultimate goal of a rougher flotation cell is to maximise the metallurgical recovery at a steady state for a specified minimum grade, it was evident that the incorporation of air recovery dynamics (which can be measured in real-time) and concentrate grade dynamics (calculated through first-principle models) led to the best results. The addition of a dynamic variable that can be easily measured online, i.e. air recovery, offers great potential to improve plant performance in existing froth flotation systems. Furthermore, a minimum concentrate grade was imposed in the E-MPC strategy. This acts as an economic constraint as it allows the metallurgical recovery to be optimised while ensuring that concentrate grade requirements are met. The dynamic optimisation problem for the E-MPC strategy was discretised using orthogonal collocations, and was implemented in Matlab using automatic differentiation via CasADi. Two typical manipulated variables were considered: air flowrate and pulp height setpoints. Based on laboratory-scale data, the implementation of the E-MPC strategy resulted in improvements ranging from +8 to +22 % in metallurgical recovery, while maintaining the specified grade. This is therefore an encouraging control strategy to explore in larger flotation systems.Paulina Quintanilla would like to acknowledge the National Agency for Research and Development (ANID) for funding this research with a scholarship from “Becas Chile”

Evaluation of Changes in Feed Particle Size within an Economic Model Predictive Control Strategy for Froth Flotation

This study presents the evaluation of the impact of feed particle size on an Economic Model Predictive Control (E-MPC) strategy for a flotation bank. The effect of particle size was assessed under two scenarios: (1) assuming constant floatability with no dependency on particle size, and (2) assuming variable floatability as a function of particle size. The E-MPC strategy uses a dynamic model that includes froth physics, which was previously calibrated and validated using experimental data. Two typical control variables were considered: air flowrate and pulp height setpoints. The proposed objective function depends on three flotation variables: (1) dynamic air recovery, a measurable variable used to quantify froth stability and is directly related to flotation performance, (2) metallurgical recovery at steady-state, and (3) dynamic concentrate grade. A moving horizon estimator (MHE) was implemented to estimate the model states in both scenarios. Simulation results showed that the estimation of metallurgical indicators (concentrate grade and recovery) was significantly affected by changes in the floatability parameter. A poor estimate of floatability is likely to lead to very different results for the control strategy. Future research should focus on estimating and updating the most significant parameters of the dynamic model, such as floatability, with an appropriate sampling time