Volumetric Holdup Research Articles

A phenomenological kinetic flotation model: Distinct Time-Variant floatability distributions for the pulp and froth materials

A simple and easy-to-use phenomenological kinetic flotation model, strongly connected with the physics of the process, is proposed in this paper. The model explicitly contains the cell volume, aeration rate, volumetric hold-up, mean bubble size, and particle density as input variables. It can be employed to characterize the floatability distributions of the particles in the pulp and the froth separately any time during the flotation process. Two new time-dependent kinetic parameters, the bubble loading factor φ(t) and the maximum cell mass transfer capacity Ṁmax(t) also appear in the model expression. φ(t) is a measure of the degree of crowding of the bubble surfaces and accounts for the deviations from the first-order rate equation. Ṁmax(t) describes the maximum amount of mass that can be transported to the froth phase by the bubble population in the cell. Screen fractionation of each froth product collected at different time intervals during a single kinetic flotation test is sufficient to generate the data required by the model for analysis. Application of the model to this data yields directly time-dependent functions for the floatability of the particles reporting to froth Kf(t) or remaining in the cell Kp(t) for each size fraction separately, without the need for any empirical parameters. The test of the model was carried out using published kinetic flotation data from the literature.

Towards a Pool-based Model of Volumetric Slurry Hold-up for Cylindrical Ball Mills

ABSTRACTThis paper discusses the incremental development of an empirical model of volumetric hold-up of slurry inside a cylindrical mill as a function of pool volume.Relevant experimental data from three geometrically similar mills were considered for calibration and testing against the pool-based model. The data included information on ball filling, slurry hold-up, mill speed, mill internal volume, shoulder, and toe angles of the media charge, as well as slurry pool angle.Initial results showed that the proposed empirical model fitted well the data not only from the three mills but also from an independent source. The one-parameter model also proved to be capable of estimating the volumetric slurry hold-up of a cylindrical mill from its load behavior to within 9% average error. Further refinement is to follow for the model to be useful over a wide range of operating conditions.

An EKF observer to estimate semi-autogenous grinding mill hold-ups

A non-linear observer model of a semi-autogenous grinding mill is developed. The observer model distinguishes between the volumetric hold-up of water, solids, and the grinding media in the mill. Solids refer to all ore small enough to discharge through the end-discharge grate, and grinding media refers to the rocks and steel balls. The rocks are all ore too large to discharge from the mill. The observer model uses the accumulation rate of solids and the mill's discharge rate as parameters. It is shown that with mill discharge flow-rate, discharge density, and volumetric hold-up measurements, the model states and parameters are linearly observable. Although instrumentation at the mill discharge is not yet included in industrial circuits because of space restrictions, this study motivates the benefits to be gained from including such instrumentation. An extended Kalman filter is applied in simulation to estimate the model states and parameters from data generated by a semi-autogenous mill simulation model from literature. Results indicate that if sufficiently accurate measurements are available, especially at the discharge of the mill, it is possible to reliably estimate grinding media, solids and water hold-ups within the mill. Such an observer can be used as part of an advanced process control strategy.

Nonlinear observability of grinding mill conditions

Abstract: A nonlinear observer model of a grinding mill is developed. The model distinguishes between the volumetric hold-up of water, solids, rocks and balls in the mill, where solids are all ore small enough to discharge through the end-discharge grate, and rocks are all ore too large to discharge. The model includes the discharge rate, abrasion rate of rocks and the abrasion rate of balls as parameters. It is shown that with mill discharge flow-rate, discharge density and volumetric hold-up measurements the model states and parameters are locally (weakly) nonlinearly observable, but not linearly observable. Although instrumentation at the mill discharge is not yet included in industrial circuits because of space restrictions, this study motivates the benefits to be gained from including instrumentation at the mill discharge. If mills are fitted with adequate measurement instrumentation at the discharge, important mill states and parameters can be estimated.

Granular flow in a rotating drum with gaps in the side wall

This paper presents a numerical investigation of the motion of bi-sized particles in a short rotating drum by using Discrete Element Method (DEM). The side wall of the drum has equally spaced gaps whose width is just between the two particle diameters. One end wall of the drum is fixed while the other rotates with the side wall. Small particles are fed into the drum continuously at the center region in the axial direction. The effect of rotating speed on the volumetric holdup and residence time of small particle is investigated. A critical rotating speed is found, below which the decrease of rotating speed will increase the volumetric holdup and the residence time of the small particles sharply. A jump in the axial distribution of the outflow rate of the small particles is observed at the region adjacent to the fixed end wall. The flow pattern inside the drum is analyzed. In the region between the fixed end wall and the feeding point, all small particles, on average, move towards the fixed end wall. While in the region between the rotating end wall and the feeding point, the small particles curve away the rotating end wall in the upper part of the charge and return to this wall in the lower part. The particle temperature distributions at different rotating speeds are explored to understand the flow behavior observed in these simulations.

Holdup and Residence Time Distributions in Inclined Dishes

Inclined rotating dishes are commonly used for particle processing, but the flow behavior of the bed has received little attention. Experiments using nongranulating solids showed that the fractional volumetric holdup was related to the material flow pattern, being dependent upon the rotational Froude number, the dish aspect ratio (rim height/diameter), and the angle of dish inclination. A magnetic tracer method was used to determine the particle exit age distributions. For a monosized feed, the dimensionless variance of the exit age distribution and the rotational Froude number were linearly related and independent of the dish diameter. For two-component feeds, the exit age distributions of the components were not the same and considerable differences were found when one component was present in a small proportion. If the components were of equal proportions, there was a marked and fairly consistent extent of segregation, whatever the operating conditions.

Modeling and simulation of semi-batch photo reactive distillation

A reaction model for batch reactive distillation is developed considering two-phase (vapor and liquid) reactions, and assuming equilibrium stages/compartments, constant pressure in each stage/compartment, and constant volumetric holdup. This reaction model was used to simulate the chlorination of toluene in a photo reactive distillation column operated in a semi-batch mode. A downward computational approach was employed to determine the initial values in solving the differential and algebraic equations. Simulation predictions were compared with experimental results.

How to predict the metastable zone width

Industrial crystallizers have to be operated at the optimum supersaturation which is less than the metastable zone width to obtain a desired product quality, with respect to median particle size, crystal size distribution (CSD),shape and purity. Due to the time-consuming experimental determination of metastable zone widths, a theoretical model based on the birth of nuclei on the surface of crystals present in the solution and their outgrowth has been developed. In a solution cooled down at a constant cooling rate (polythermal method), the surface-born nuclei have to grow to a certain size to be detected by means of a particle analysing system or the human eye. Calculations were carried out for 28 inorganic systems. A comparison with results from the literature showed that the visible shower of nuclei occurred at a volumetric hold-up of crystals between 10 −4 and 10 −3 and a corresponding size between 10 and 100 μm. A simplified model is presented which takes only into account the growth step.

Hold-up and flooding in a vibrating plate extractor

The mean volumetric hold-up of the dispersed phase and the limiting flow rates of phases in five sizes of experimental vibrating plate extractors VPE were measured and the results compared with the relations of Richardson and Zaki, Míšek and Ishii and Zuber for the relative phase velocity. The experiments were performed with the system toluene-water, toluene dispersed. All the relations were found to describe the dependence of hold-up on flow rates very well. They also allowed a satisfactory estimation of the limiting phase velocities. Within the range of the experimental conditions the characteristic velocity as well as the limiting phase velocities could be approximated by linear functions of the frequency of vibrations. For the columns studied no influence of column size on limiting velocities was detected. The deviations of individual columns from geometrical similarity, however, brought about an appreciable effect both on the hold-up and on the flooding rates. This phenomenon has been explained in terms of longitudinal hold-up profiles, the shape of which depends on the boundary conditions in the particular column. This explanation was corroborated by case studies using a polydisperse mathematical model which show that even in the presence of hold-up profiles the effect of phase velocities on the mean hold-up can be correlated by means of the investigated relations with constant parameters and that the parameters depend on the hold-up profiles form.

Residence Time Distribution and Mass Transport Characteristics of Large Industrial Grinding Mills

In the mathematical modelling of grinding circuit dynamics, the distribution of the residence time is a parameter of prime importance to describe the flow characteristics of the material through the mill. This paper will present a simple mathematical model which has been developped from experimental studies. Residence time distribution determination have been performed on different industrial scale grinding circuits; one of which was processing a Cu-Pb-Zn ore while another one a niobium mineral. The residence time distributions have been measured with an impulse of tritiated water as the tracer. From these results, a model has been developped which allows to obtain directly from the tracer measurements the residence time distribution. From a dynamic simulation and control point of view a relationship between the pulp volumetric hold-up and the mill pulp feed rate is also presented and discussed.