Surface Area Flux Research Articles

Hemodialysis: Techniques and Prescription

Hemodialysis: Techniques and Prescription

CFD modelling of bubble–particle attachments in flotation cells

In recent years, computational fluid dynamic (CFD) modelling of mechanically stirred flotation cells has been used to study the complexity of the flow within the cells. In CFD modelling, the flotation cell is discretized into individual finite volumes where local values of flow properties are calculated. The flotation effect is studied as three sub-processes including collision, attachment and detachment. In the present work, these sub-processes are modelled in a laboratory flotation cell. The flotation kinetics involving a population balance for particles in a semi-batch process has been developed. From turbulent collision models, the local rates of bubble–particle encounters have been estimated from the local turbulent velocities. The probabilities of collision, adhesion and stabilization have been calculated at each location in the flotation cell. The net rate of attachment, after accounting for detachments, has been used in the kinetic model involving transient CFD simulations with removal of bubble–particle aggregates to the froth layer. Comparison of the predicted fraction of particles remaining in the cell and the fraction of free particles to the total number of particles remaining in the cell indicates that the particle recovery rate to the pulp–froth interface is much slower than the net attachment rates. For the case studied, the results indicate that the bubbles are loaded with particles quite quickly, and that the bubble surface area flux is the limiting factor in the recovery rate at the froth interface. This explains why the relationship between flotation rate and bubble surface area flux is generally used as a criterion for designing flotation cells. The predicted flotation rate constants also indicate that fine and large particles do not float as well as intermediate sized particles of 120–240 μm range. This is consistent with the flotation recovery generally observed in flotation practice. The magnitude of the flotation rate constants obtained by CFD modelling indicates that transport rates of the bubble–particle aggregates to the froth layer contribute quite significantly to the overall flotation rate and this is likely to be the case especially in plant-scale equipment.

Rate constant modelling for batch flotation, as a function of gas dispersion properties

Gas dispersion is one of the key factors in the mineral flotation process. It can be expressed by many variables such as bubble size, gas holdup, gas superficial velocity and bubble surface area flux ( S b). In particular, S b has been found to have a strong correlation with the flotation rate constant ( K). Bubble size is important because it is related with the transport capacity and with the flotation probability. It also affects the gas holdup and the bubble surface area flux. In this work bubble sizes are measured using image analysis. An empirical rate constant model is developed to relate K with S b and other dispersion variables. This model is found to be able to predict K for two different batch flotation cells.

Testing the k– Sb relationship at the microscale

The proposed relationship between flotation rate constant k and bubble surface area flux S b for the pulp zone, namely k = PS b, where P is the “floatability factor”, is tested at the microscale. Bubble size D b was measured by an adaptation of the McGill bubble size analyzer. The range in S b, up to 50 s −1, is comparable to full-scale machines. Standard statistical procedures were used to asses the plausibility of the model for the two materials tested, NiO—20 μm and galena—37 μm giving P ≈ 1.73 × 10 −3 and ≈2.82 × 10 −3, respectively, comparable to available values. The data, however, cannot unambiguously distinguish among the various relationships of the form k ∝ 1 / D b m where m may range up to 2.

REVIEWING THE EXPERIMENTAL PROCEDURE TO DETERMINE THE CARRYING CAPACITY IN FLOTATION COLUMNS

Carrying capacity is a parameter required in the sizing of flotation columns that depends on the characteristics of the generated bubble population and on the mineral density and particle size distribution. Two alternatives were originally proposed by Espinosa-Gomez et al. in the mid-80s for its evaluation: a semi-empirical model based on then available data and an experimental procedure for direct estimation. The model has been improved since, but its calibration still depends on values obtained using the original experimental procedure.To re-evaluate the experimental procedure, tests for different combinations of collection and froth zone heights were performed at three different gas velocities in a laboratory column processing a mixture of hematite/silica particles. The column was instrumented to allow for the continuous calculation of bubble surface area flux which was used to account for changes in the characteristics of the bubble population generated in the column and to explain concentrate particle size variations by the estimation of particle coverage of the bubble surface.The experimental results demonstrate that the collection and froth zone heights as well as the gas rate affect the determination of the carrying capacity. Particle coverage was between 10 and 18% and showed almost no variation in spite of the large increase in the concentrate solids rate. This increase was compensated by the reduction in concentrate particle size. These results lead to the conclusion that both the concept of carrying capacity and its experimental determination method require a deep revision.La capacité de transport, un des paramètres requis pour le dimensionnement des colonnes de flottation, dépend des caractéristiques de la population de bulles générées et de la distribution granulométriques des particules minérales à flotter. Deux alternatives pour estimer sa valeur furent proposées par Espinosa et al. lors de l’introduction de ce paramètre dans les années 80: un model semi-empirique basé sur les données alors disponibles et une procédure expérimentale pour son estimation directe. Le modèle a été amélioré par la suite, mais pour son calibrage, des valeurs encore obtenues par la méthode expérimentale originale sont requises.À objet de re-évaluer cette procédure, des essais réalisés avec différentes combinaisons de hauteur de collecte et de nettoyage à trois taux d’aération différents furent complétés dans une colonne de laboratoire traitant un mélange de particules de silice et d’hématite. La colonne était instrumentée afin de permettre l’estimation en continu du taux surfacique de bulles Sb, variable utilisée pour représenter les caractéristiques de la population de bulles générées dans le diffuseur, ainsi que pour expliquer les variations rencontrées dans la granulométrie du concentré produit, par estimation de la couverture minérale de la surface de bulles.Les résultats expérimentaux obtenus démontrent que la hauteur des zones de collecte et de nettoyage ainsi que le débit de gaz utilisés affectent la détermination de la capacité de transport. Des valeurs de 10 à 18% de couverture furent obtenues, et ce malgré l’augmentation importante du taux de production de concentré observée. Une explication possible serait le fait que cette augmentation ait été compensée par la réduction de la taille des particules récupérées au concentré. Ces résultats indiqueraient que le concept même de capacité de transport ainsi que la procédure actuellement employée pour sa détermination expérimentale requièrent d’une sérieuse révision.

Froth recovery measurement in plant scale flotation cells

Over recent years a number of techniques have been developed as part of the AMIRA 2 Australian Mineral Industries Research Association. 2 P9 project to represent mathematically or measure directly sub-processes occurring in flotation cells. These include the determination of: • a machine parameter (bubble surface area flux, S b); • a froth parameter (froth recovery, R f); • entrainment and drainage parameters ( ξ and δ); and • an ore floatability parameter ( P i). A methodology for estimating froth recovery in large scale plant flotation cells is proposed. The primary aim of the technique is to estimate the froth recovery in a way that is simple, inexpensive and non-intrusive. Plant engineers can use the estimate of froth recovery in day to day trouble-shooting and circuit optimisation. The current definition of froth recovery is given, along with a detailed description of the proposed methodology, equipment required, assumptions and calculations. Advantages and limitations of the methodology are described and possible applications of the froth recovery estimate in current flotation circuit modeling are presented.

Gas dispersion and de-inking in a flotation column

The role of four gas dispersion parameters in ink particle collection was investigated in 4″ and 20″ flotation columns. Gas holdup ( ε g) and superficial gas velocity ( J g) were measured on-line and bubble size ( d b) was estimated using drift flux analysis that enabled bubble surface area flux ( S b) to be calculated. Operating with approximately zero froth depth ink recovery as a function of retention time (controlled by underflow rate) was determined. Using a mixing model, the collection zone flotation rate constant ( k c) was estimated from the recovery––time data. The rate constant was not related to J g or d b but was linearly dependent on ε g and S b, similar to findings in mineral flotation studies.

Predicting gas hold-up in flotation columns from the physical characteristics of rigid spargers

Rigid spargers are attracting renewed interest in some applications of flotation columns, such as deinking of recycled paper, deoiling of water and the recovery of metallic ions from hydrometallurgical solutions. The main factors to be taken into consideration in the selection of a sparger are the characteristics of the sparger (material and geometry), the nature of the pulp to be processed (e.g. per cent solids and presence of surfactants) and the dimensions of the column. Lacking so far has been a method of estimating bubble size or gas hold-up (or bubble surface-area flux) for a given combination of sparger, column and pulp, which must account for the formation of a bubble swarm at the sparger and the rise of the swarm against a countercurrent flow of pulp or water. The method now proposed predicts gas hold-up by combining two models: one for bubble formation at the sparger, which relates bubble size to sparger and pulp characteristics; and the second for bubble flow, which relates bubble size to gas hold-up and phase hydrodynamics. Application of the method is illustrated for a set of porous, stainless-steel spargers installed in a column that was run with water and frother.

Particle collection within the Jameson cell downcomer

Since the invention of the Jameson cell in 1986 various mechanisms have been postulated for particle collection within the cell's downcomer. Possible mechanisms include thin film migration, instantaneous collection, mixing zone collection, residence time dependence and bubble surface area flux. The limited amount of investigation to date has not provided sufficient evidence to allow one mechanism to be supported over the others. A summary of the postulated mechanisms is presented along with the background to their formulation. They are assessed on the basis of available knowledge and categorized according to specific operating parameters within the Jameson cell downcomer. In addition, relationships between operating variables and various hydrodynamic zones within the downcomer are discussed. A review of how these mechanisms relate to flotation operations is conducted for coal flotation, base-metal ore flotation and the removal of organic matter from solution.

Flowing foam bubble size control

Froth flotation performance can be estimated from the bubble surface area flux. To control such systems, a relationship between the bubble size and the operating parameters has to be found. An experimental system was designed that allowed the foam to flow vertically and to be filmed. The results obtained showed the existence of a linear, non-interacting relationship between the bubble area, and the air flowrate and surfactant concentration. The results show that automatic control of foam bubble surface area is possible using on-line image analysis.

Gas dispersion properties: bubble surface area flux and gas holdup

Bubble surface area flux has been related to flotation performance and advocated as a key “machine variable”. Analysis of available data suggests that bubble surface area flux (S b) and gas holdup ( ε g) are related by S b ∼ 5.5 ε g. This was the case for flotation columns and mechanical cells, both laboratory and plant scale, over the approximate range S b < 130s −1 and ε g < 25%. A relationship is expected given their similar dependence on gas rate and bubble size (both increase as gas rate increases and bubble size decreases). Reasons that a relationship may be obscured, especially in full size units because of the difficulty in defining the overall S b and ε g, are discussed. The possibility to replace S b by ε g as the machine variable has the advantage that gas holdup is easier to measure.

On the relationship between flotation rate and bubble surface area flux

Flotation scale-up has been a major difficulty and is becoming more so as the flotation machines continue to grow in size. It has been proposed by Gorain and his coworkers that the flotation rate constant has a linear relationship with the bubble surface area flux. This paper discusses that claim using the data presented to validate that claim. It can be shown that the measurement and computation of superficial gas velocity, and partially also the bubble size may be biased in some conditions. This makes the bubble surface area flux behave such that the final outcome is in doubt. The validation does not address the different particle sizes at all.

A model to relate the flotation rate constant and the bubble surface area flux in mechanical flotation cells

Two recent flotation models developed by the authors are discussed, viz. the bubble population balance model and the attachment-detachment model. The bubble population balance model describes the history of a bubble population in a flotation cell in terms of the sub-processes of bubble breakage and coalescence. The attachment-detachment model allows for the presence of a gas phase in the flotation cell, both in terms of a gas residence time and the attachment and detachment of mineral particles to/from bubbles. When combined the two models predict a near-linear region about a point of inflexion on the (simulated) response between the flotation rate constant (k) and the flux of bubble surface area through the flotation cell (S b). It is proposed by the authors that this region corresponds to the linear k−S b relationship observed in a recent research project on flotation kinetics in mechanical flotation cells by Gorain et al. (1997).

An example of innovation from the waste management industry: Deinking flotation cells

This is an expanded version of the Keynote lecture entitled “Innovations and Future Trends in Flotation Cell Design” delivered at Minerals Engineering '98. A category of flotation cell is discussed based on devices in the waste management industry. Seven examples of flotation cells utilized in waste paper deinking industry are reviewed to show the variety of approaches used to optimize the characteristics of a flotation system. In attempting to identify the key fundamental parameters, it is noted that comparisons of flotation effectiveness on the basis of bubble surface area flux (bsaf) often fail to reveal differences between flotation devices. This led to the conclusion that perhaps comparisons should be based on cost per delivered BSAF.

The JKMRC high bubble surface area flux flotation cell

The High Bubble Surface Area Flux Flotation Cell (HS bFC) is a 16-litre mechanical flotation cell with a bottom driven impeller, which is operated continuously. Bubble formation is carried out using an in-line mixer which enables the cell to achieve superficial gas velocities (Jg) equivalent to those generated in industrial flotation cells (0.7–1.2 cm/s). At the same time, the cell produces considerably smaller bubble sizes; consequently, a high bubble surface area flux (S6) can be generated.

The empirical prediction of bubble surface area flux in mechanical flotation cells from cell design and operating data

In the operation of mechanical flotation cells, the dispersion of gas into fine bubbles may be expressed by three indicators : bubble size, gas holdup and superficial gas velocity. Taken together, these properties determine the bubble surface area flux (S b) in the cell, which has been found to have a strong correlation with the flotation rate constant (k). Previous work by the authors has indicated that it is possible to predict the value of k for a known ore in a cell from a knowledge of the bubble surface area flux generated in that cell. In order to make good use of this finding, an empirical model has been developed to predict S b in mechanical flotation cells, using data from extensive pilot industrial scale test programs. The model is able to predict Sb from cell operating conditions, impeller design and feed properties. The model has been validated for different types and cell sizes, impeller types and ore types, in different independent investigations carried out at several concentrators in Australia and South Africa. This paper outlines the development of the model, the parameter estimation, and the validation using a number of additional data sets.

Studies on impeller type, impeller speed and air flow rate in an industrial scale flotation cell. Part 5: validation of k-S b relationship and effect of froth depth

A previous investigation carried out by the authors at the Hellyer concentrator, using a 3 m 3 cell fitted with four different impellers treating plant zinc cleaner feed ore, suggested a linear correlation between flotation rate constant k and bubble surface area flux S b. The relationship between k and Sb was found to be independent of impeller type. This paper describes an investigation at the Scuddles concentrator in Western Australia to validate the findings at Hellyer for a different ore. Unlike the Hellyer work for which only one froth depth was used, the Scuddles work was carried out at different froth depths using the same 3 m 3 cell fitted in turn with three different impellers viz. Batequip, Dorr-Oliver and Outokumpu. The results confirmed the strong correlation between k and S b at three different froth depths used for the study. Moreover, this relationship was found to be practically independent of impeller type. However, at shallow froth depth the kS b relationship was found to be linear, whereas at intermediate and deep froth depths the relationship was found to be non-linear with the froth playing an important role in the overall kinetics.

The effect of froth residence time on the kinetics of flotation

Previous studies by the authors have shown that there is a strong correlation between the overall flotation rate constant k and the bubble surface area flux, S b, in a flotation cell, and that this relationship is independent of impeller type. More recently it was found that results obtained from a 250 litre cell and a 3 m 3 cell treating the same feed could be represented by the same k-S b relationship, as long as the ratio of froth height to superficial gas rate in the cells was the same. This ratio is termed the froth residence time, τ fg . This paper examines the effect of τ fg on the kinetics of flotation, by re-evaluating and re-interpreting these results. It is found that k decreases exponentially with τ fg but the relationship is dependent on cell size and therefore not directly useful for scale-up. When τ fg is divided by a typical cell dimension to take into account the effect of froth transportation distance in cells of different sizes, the relationship between k and the “specific froth residence time” fits the data better and and is found to be independent of cell size. Finally, when the results are evaluated in terms of τ fg , the froth residence time calculated on the basis of the concentrate slurry flow rate, the fitting od the data is found to be very much better again. The implications of these relationships for flotation modelling and scale-up are discussed.

Carrying capacity in a pilot flotation column

Concentrate flow rates were measured for the flotation of −100-mesh Lower Kittaning coal in a computer-controlled 0.076 x 3.5-m pilot flotation column. The variables studied were recovery zone height, airflow and solid feed rates. The specific projected surface areas of the feed, concentrate and tailing streams were determined from the measured particle size distributions. The results obtained show that the concentrate flow rate was independent of recovery zone height but was dependent on the hydrostatic pressure setting (measure of froth depth), gas flow rate, solid loading and particle size. New correlations for column carrying capacity and yield arc provided as a function of bubble surface area flux and the specific particle projected surface area.

Studies on impeller type, impeller speed and air flow rate in an industrial scale flotation cell. Part 4: Effect of bubble surface area flux on flotation performance

The metallurgical performance of a 2.8m 3 portable industrial scale flotation cell was measured when treating zinc cleaner feed at Hellyer concentrator in Tasmania, Australia. The cell was fitted in turn with four different impeller-stator systems and operated over a wide range of air flow rates and impeller speeds. Bubble size, gas holdup and superficial gas velocity were measured at each of 64 different operating conditions along with the metallurgical performance of the cell. When metallurgical performance was expressed in terms of a kinetic constant, it was found that neither bubble size nor gas holdup nor superficial gas velocity could be related to flotation rate individually; but when taken together, they determine the bubble surface area flux in the cell, which could be related to flotation rate extremely well. A linear relationship between flotation rate and bubble surface area flux was found for all four impellers investigated: the slope of the line was independent of the type of impeller used. The linear relationship was verified for different size fractions of the ore: the slope of the straight line was different for different size fractions, values being greater for the smaller size particles. The relationship was also independently confirmed at another zinc cleaner operation. This finding has potential practical application in flotation plant modelling, design and optimisation.