Modeling the Batch Sedimentation of Calcium Carbonate Particles in Laboratory Experiments—A Systematic Approach (original) (raw)
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International Journal of Mineral Processing, 2004
Based on a numerical method introduced by Bürger and Karlsen [J. Eng. Math. 41 (2001) 145], a software was developed for the simulation of batch and continuous thickening. The paper recalls the application of the phenomenological theory of sedimentation-consolidation processes to batch settling and continuous thickening of flocculated suspensions. The software presents two alternatives, one for each of these possibilities. For batch thickening, the initial and critical concentration and the height of the initial suspension must be entered together with the parameters of the flux density function and the effective solid stress. The output is a settling plot showing as many lines of constant concentration as requested and a plot of the concentration profile for selected times. For continuous thickening, only the steady state is simulated. The input is the solid feed flux and the required underflow concentration or volume underflow rate. If the thickener area is known, the capacity and the concentration profile in the equipment can be predicted. On the other hand, if the capacity is known, the required settling area and the resulting concentration profiles are predicted. Several examples show the application.
2010
A new continuous one-dimensional sedimentation model incorporating a new continuous flocculation model that considers aggregation and fragmentation processes was derived and tested. Additionally, a new procedure to model sediment particle size distribution (PSD) was derived. Basic to this development were three different parametric models: Jaky, Fredlund and the Gamma probability distribution (GPD) were chosen to fit three different glass micro-spheres PSDs having average particle sizes of 7, 25 and 35 microns. The GPD provided the best fit with the least parameters. The bimodal GPD was used to fit ten sediment samples with excellent results (< 5% average error). A continuous flocculation model was derived using the method of moments for solving the continuous Smoluchowski coagulation equation with fragmentation. The initial sediment PSD was modeled using a bimodal GPD. This new flocculation model resulted in a new general moments' equation that considers aggregation and fragmentation processes, which is represented by a system of ordinary differential equations. The model was calibrated using a genetic algorithm with initial and flocculated PSDs of four sediment samples and four anionic polyacrylamides flocculants. The results show excellent correlation between predicted and observed values (R 2 > 0.9878). A new continuous one-dimensional sedimentation model that resulted in a scalar hyperbolic conservation law was derived from the well-known Kynch kinematic sedimentation model. The model was calibrated using column tests results with glass micro-spheres particles. Two different glass microspheres particle size distributions (PSDs) were used with average diameters of 7 and 37 microns. Excellent values of coefficient of determination (R 2 > 0.89, except for one test replicate) were obtained for both the small and large glass micro-spheres PSDs. These results suggest that the proposed sedimentation model can be expanded to model the sedimentation process inside a sediment pond.
A Model of Continuous Sedimentation of Flocculated Suspensions in Clarifier-Thickener Units
SIAM Journal on Applied Mathematics, 2005
The chief purpose of this paper is to formulate and partly analyze a new mathematical model for continuous sedimentation-consolidation processes of flocculated suspensions in clarifierthickener units. This model appears in two variants for cylindrical and variable cross-sectional area units, respectively (Models 1 and 2). In both cases, the governing equation is a scalar, strongly degenerate parabolic equation in which both the convective and diffusion fluxes depend on parameters that are discontinuous functions of the depth variable. The initial value problem for this equation is analyzed for Model 1. We introduce a simple finite difference scheme and prove its convergence to a weak solution that satisfies an entropy condition. A limited analysis of steady states as desired stationary modes of operation is performed. Numerical examples illustrate that the model realistically describes the dynamics of flocculated suspensions in clarifier-thickeners.
Two-phase modeling of batch sedimentation
Applied Mathematical Modelling, 1999
A two-phase model for the simulation of sedimentation processes is presented. The model solves the continuity and momentum equations for the pure-clear liquid and the sludge phases, and it is veri®ed against a well-known benchmark problem, for which analytical solutions exist. Numerical simulations of a typical 1-D batch sedimentation process for mono-dispersed particles are carried out and results are found to be in satisfactory agreement with experimental data and model predictions of other researchers. A further expansion of the model to two-dimensions leads to predictions of the dynamic behavior of settling tanks and the eect of the inclination angle on the sedimentation process. Ó 1999 Elsevier Science Inc. All rights reserved. 0307-904X/99/$ -see front matter Ó 1999 Elsevier Science Inc. All rights reserved. PII: S 0 3 0 7 -9 0 4 X ( 9 9 ) 0 0 0 1 6 -5
International Journal of Mineral Processing, 2003
This paper presents a unified theory of solid-liquid separation of flocculated suspensions including sedimentationthickening, centrifugation and filtration. After identifying the variables and equations for each of the operations, thickening, centrifugation and filtration, and establishing the compatibility between them, we show that these processes can be described by variants of one scalar hyperbolic-parabolic strongly degenerate partial differential equation with appropriate initial and boundary conditions. To complete the description, constitutive equations should be postulated for the solid-fluid interaction forces in the suspension and for the permeability and the compressibility of the porous medium, which is either a sediment or a filter cake. A particular unit operation can then be simulated by solving these equations numerically. The mathematical analysis of the resulting model confirms the well-posedness of the mathematical model and support the design of robust numerical simulation methods. These methods are employed to calculate a variety of examples from thickening, centrifugation and filtration, which illustrate the theory.
Investigation and modelling of sedimentation of mixed particles
Powder technology, 1997
A model was developed to predict the relative velocity necessary for the calculation of the settling velocity in fluid-particle sedimentation systems, both binary and polydisperse suspensions. The model obtained was tested against the experimental data obtained from a countercurrent flow system and was found to give an excellent fit. The Reynolds number ranges between 4 and 1300. The model predictions were compared with those of other published models. The model predictions have shown an improvement and do not have the limitations present in the other models.
Computational simulation of flocculent sedimentation based on experimental results
Water Science and Technology, 2012
Computerised interpolation algorithms as well as the empirical model for analysing the flocculent settling data were developed. A mechanistic semi-empirical model developed from fundamental physical principles of a falling particle in a viscous fluid was tested against actual flocculation column data. The accuracy of the mechanistic model was evaluated using the sum of the squared errors between the interpolated values (real values) and the model predictions. Its fitting capabilities were compared with Özer's model using nine flocculent data sets of which four were obtained from literature and the rest were actual data from the performed experiments. The developed model consistently simulated the flocculation behaviour of particles in settling columns better than Özer's model in eight of the nine data sets considered. It is recommended that the model's performance be further compared with other models like the Rule based and San's model. The errors due to the use of ...
Particle Size Distribution Analysis for Calcium Carbonate Precipitation
Calcium carbonate precipitation is studied experimentally using gas-liquid and liquid-liquid systems, aiming to obtain small particle sizes and narrow particle size distributions. The gas-liquid reaction is carried on by bubbling pure carbon dioxide into a clear solution of calcium hydroxide containing 2 g/L CaO, in a continuous crystallizer. The liquid-liquid reaction is performed in a semi-continuous mode, by controlled double jet precipitation adding simultaneously, at equal flowrates, 1 M and 0.1 M calcium nitrate tetrahydrate and potassium carbonate solutions. The influence of a non-ionic additive, Tween 20 was also studied. The solid phase composition and particle size distribution are analyzed for different operating conditions. A mathematical model based on the population balance equation is developed assuming as main mechanisms nucleation, size-dependent crystal growth and agglomeration. The corresponding kinetic parameters are estimated by correlating the experimental and calculated particle size distribution data.