Settling velocities of particulate systems: 9. Phenomenological theory of sedimentation processes: numerical simulation of the transient behaviour of flocculated suspensions in an ideal batch or continuous thickener (original) (raw)
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.
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.
International journal of mineral …, 2000
In one space dimension, the phenomenological theory of sedimentation-consolidation processes predicts the settling behaviour of a flocculated suspension in dependence of two constitutive material-specific functions, the Kynch batch flux density function and the effective solid stress. These functions depend only on the local solids concentration. In this paper, we determine these functions from published data of several experimental studies. The mathematical model is then solved numerically making use of these functions. The numerical results are compared to the respective measurements. General good agreement between numerical and experimental data confirms the validity of the phenomenological theory.
Settling velocities of particulate systems
International Journal of Mineral Processing, 2000
In one space dimension, the phenomenological theory of sedimentation-consolidation processes predicts the settling behaviour of a flocculated suspension in dependence of two constitutive material-specific functions, the Kynch batch flux density function and the effective solid stress. These functions depend only on the local solids concentration. In this paper, we determine these functions from published data of several experimental studies. The mathematical model is then solved numerically making use of these functions. The numerical results are compared to the respective measurements. General good agreement between numerical and experimental data confirms the validity of the phenomenological theory.
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.
Settling velocities of particulate systems: 12
International Journal of Mineral Processing, 2001
In this contribution we show how the phenomenological theory of sedimentation-consolidation processes can be extended to the presence of centrifugal field. The modelling starts from the basic mass and linear momentum balances for the solid and liquid phase, which are referred to a rotating frame of reference. These equations are specified for flocculated suspensions by constitutive assumptions that are similar to that of the pure gravity case. The neglection of the influence of the gravitational relative to the centrifugal field and of Coriolis terms leads to one scalar hyperbolicparabolic partial differential equation for the solids concentration distribution as a function of radius and time. Both cases of a rotating tube and of a rotating axisymmetric vessel are included. A numerical algorithm to solve this equation is presented and employed to calculate numerical examples of the dynamic behaviour of a flocculated suspension in a sedimenting centrifuge. The phenomenological model is appropriately embedded into the existing theories of kinematic Ž. centrifugation processes of ideal non-flocculated suspensions.
Settling velocities of particulate systems: 12: Batch centrifugation of flocculated suspensions
International Journal of Mineral Processing, 2001
In this contribution we show how the phenomenological theory of sedimentation-consolidation processes can be extended to the presence of centrifugal field. The modelling starts from the basic mass and linear momentum balances for the solid and liquid phase, which are referred to a rotating frame of reference. These equations are specified for flocculated suspensions by constitutive assumptions that are similar to that of the pure gravity case. The neglection of the influence of the gravitational relative to the centrifugal field and of Coriolis terms leads to one scalar hyperbolicparabolic partial differential equation for the solids concentration distribution as a function of radius and time. Both cases of a rotating tube and of a rotating axisymmetric vessel are included. A numerical algorithm to solve this equation is presented and employed to calculate numerical examples of the dynamic behaviour of a flocculated suspension in a sedimenting centrifuge. The phenomenological model is appropriately embedded into the existing theories of kinematic Ž. centrifugation processes of ideal non-flocculated suspensions.
Settling velocities of particulate systems, 7. Kynch sedimentation processes: continuous thickening
International Journal of Mineral Processing, 1992
Continuous sedimentation is presented as a logical extension of a batch Kynch sedimentation process. All results are presented in graphical form as settling plots. Transient solutions are obtained to continuous Kynch sedimentation processes for an ideal suspension having a flux density function with one inflection point and two constant states as initial conditions. It is shown that there are only two steady-state solutions to the continuous settling of an ideal suspension in an ideal thickener. The use of Kynch's theory for compressible industrial pulps introduces confusions in the interpretation of unit areas. The traditional equation to calculate unit areas for continuous thickeners based on Kynch's theory is not correctly interpreted in the literature. A different explanation is given here.
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.
Numerical methods for the simulation of continuous sedimentation in ideal clarifier-thickener units
International Journal of Mineral Processing, 2004
We consider a model of continuous sedimentation. Under idealizing assumptions, the settling of the solid particles under the influence of gravity can be described by an initial value problem for a nonlinear hyperbolic partial differential equation with a flux function that depends discontinuously on height. The purpose of this contribution is to present and demonstrate two numerical methods for simulating continuous sedimentation: a front tracking method and a finite difference method. The basic building blocks in the front tracking method are the solutions of a finite number of certain Riemann problems and a procedure for tracking local collisions of shocks. The solutions of the Riemann problems are recalled herein and the front tracking algorithm is described. As an alternative to the front tracking method, a simple scalar finite difference algorithm is proposed. This method is based on discretizing the spatially varying flux parameters on a mesh that is staggered with respect to that of the conserved variable, resulting in a straightforward generalization of the well-known Engquist -Osher upwind finite difference method. The result is an easily implemented upwind shock capturing method. Numerical examples demonstrate that the front tracking and finite difference methods can be used as efficient and accurate simulation tools for continuous sedimentation. The numerical results for the finite difference method indicate that discontinuities in the local solids concentration are resolved sharply and agree with those produced by the front tracking method. The latter is free of numerical dissipation, which leads to sharply resolved concentration discontinuities, but is more complicated to implement than the former. Available mathematical results for the proposed numerical methods are also briefly reviewed.
Mathematical models for the sedimentation of suspensions
Lecture Notes in Applied and Computational Mechanics, 2006
Mathematical models for sedimentation processes are needed in numerous industrial applications for the description, simulation, design and control of solid-liquid separation processes of suspensions. The first simple but complete model describing the settling of a monodisperse suspension of small rigid spheres is the kinematic sedimentation model due to Kynch [93], which leads to a scalar nonlinear conservation law. The extension of this model to flocculated suspensions, pressure filters, polydisperse suspensions and continuously operated clarifier-thickener units give rise to a variety of time-dependent partial differential equations with intriguing non-standard properties. These properties include strongly degenerate parabolic equations, free boundary problems, strongly coupled systems of conservation laws which may fail to be hyperbolic, and conservation laws with a discontinuous flux. This contribution gives an overview of the authors' research that has been devoted to the mathematical modeling of solid-liquid separation, the existence and uniqueness analysis of these equations, the design and convergence analysis of numerical schemes, and the application to engineering problems. Extensions to other applications and general contributions to mathematical analysis are also addressed.
Settling characteristics of particles in a suspension of medium to high solids concentration
1992
During the thickening process of sludges with intermediate to high solid concentrations three settling regimes are typically encountered, namely, zone, transition, and compression regimes. Recent studies have indicated that the validity of Kynch's formulation, which is the most widely used for sizing settling basins, is limited to the zone settling regime. His formulation is based on the solids mass balance equation and does not consider the role of the rising sediment at the bottom of the settling basin. This limitation is ...
International Journal of Mineral Processing, 2004
The phenomenological theory of continuous thickening of flocculated suspensions in an ideal cylindrical thickener is extended to vessels having varying cross-section, including divergent or convergent conical vessels. The purpose of this contribution to draw attention to the corresponding mathematical model, whose key ingredient is a strongly degenerate parabolic partial differential equation. For ideal (non-flocculated) suspensions, which do not form compressible sediments, the mathematical model reduces to the kinematic approach by Anestis, who developed a method of construction of exact solution by the method of characteristics. The difficulty lies in the fact that characteristics and iso-concentration lines, unlike the conventional Kynch model for cylindrical vessels, do not coincide, and one has to resort to numerical methods to simulate the thickening process. A numerical algorithm is presented and employed for simulations of continuous thickening. Implications of the mathematical model are also demonstrated by steady-state calculations, which lead to new possibilities in thickener design.
Numerical methods for the simulation of the settling of flocculated suspensions
Chemical Engineering Journal, 2000
For one space dimension, the phenomenological theory of sedimentation of flocculated suspensions yields a model that consists of an initial-boundary value problem for a second order partial differential equation of mixed hyperbolic-parabolic type. Due to the mixed hyperbolic-parabolic nature of the model, its solutions may be discontinuous and difficulties arise if one tries to construct these solutions by classical numerical methods. In this paper we present and elaborate on numerical methods that can be used to correctly simulate this model, i.e. conservative methods satisfying a discrete entropy principle. Included in our discussion are finite difference methods and methods based on operator splitting. In particular, the operator splitting methods are used to simulate the settling of flocculated suspensions.
Steady-state, control, and capacity calculations for flocculated suspensions in clarifier–thickeners
International Journal of Mineral Processing, 2007
A spatially one-dimensional model for the behaviour of flocculated suspensions in clarifier-thickeners is studied. This model combines a theory of sedimentation-consolidation processes of flocculated suspensions, which leads to a strongly degenerate diffusion equation, with the discontinuous flux appearing in the recently analyzed clarifier-thickener (CT) setup. This setup includes both clarification and thickening zones of clarifier-thickener units, while the earlier ideal continuous thickener (ICT) concept explicitly models the thickening zone only. The construction of steady-state concentration profiles attainable in a continuously operated CT is described. This construction incorporates an entropy principle, which implies that only those steady states are admissible for which the solids concentration is increasing downwards. Numerical examples of steady-state profiles and their applications to comparisons between both modes of operation, to the control of sediment height through selection of the clarification/thickening split ratio of the feed flux, and for capacity calculations are presented. A numerical example illustrates the use of a numerical method for the full (time-dependent) model to compare several fill-up strategies.
Sedimentation and liquid fluidization of solid particles of different sizes and densities
Chemical Engineering Science, 1985
The sedimentation and fluidization of particles of mixed sizes and different densities were studied. A new concept-the apparent porosity of suspensions-was introduced, on the basis of which equations have been developed that are capable of predicting the settling velocities of individual types of particles in a suspension containing mixtures of particles. The validity of these equations was tested against experimental data in both sedimentation and fluidization.
2001
The phenomenological theory of continuous thickening of flocculated suspensions in an ideal cylindrical thickener is extended to vessels håving varying cross-section, including divergent or convergent conical vessels. The purpose of this contribution to draw attention to the corresponding mathematical model, whose key ingredient is a strongly degenerate parabolic partial differential equation. For ideal (non-floccuiated) suspensions, which do not form com pressible sediments, the mathematical model reduces to the kinematic approach by Anestis, who developed a method of construction of exact solution by the method of characteristics. The difficulty lies in the fact that characteristics and iso-concentration lines, unlike the conventional Kynch model for cylindrical vessels, do not coincide, and one has to resort to numerical methods to simulate the thickening process. A numerical algorithm is presented and employed for simu lations of continuous thickening. Implications of the mathematical model are also demonstrated by steady-state calculations, which lead to new possibilities in thickener design.