Finite element analysis of flows in secondary settling tanks

Kleine, D. de; Reddy, B. D.

doi:10.1002/nme.1373

Cited by 29 publications

(10 citation statements)

References 49 publications

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“…For our last example we assess the applicability of the method in approximating the stationary flow field on a half-section of a secondary settling tank (see [13,37]). A sketch of the domain is depicted in Fig.…”

Section: Example 4: Secondary Settling Tankmentioning

confidence: 99%

An augmented mixed finite element method for the vorticity–velocity–pressure formulation of the Stokes equations

Anaya

Mora

Ruiz–Baier

2013

Computer Methods in Applied Mechanics and Engineering

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a b s t r a c tThis paper deals with the numerical approximation of the stationary two-dimensional Stokes equations, formulated in terms of vorticity, velocity and pressure, with non-standard boundary conditions. Here, by introducing a Galerkin least-squares term, we end up with a stabilized variational formulation that can be recast as a twofold saddle point problem. We propose two families of mixed finite elements to solve the discrete problem, in the first family, the unknowns are approximated by piecewise continuous and quadratic elements, Brezzi-Douglas-Marini, and piecewise constant finite elements, respectively, while in the second family, the unknowns are approximated by piecewise linear and continuous, Raviart-Thomas, and piecewise constant finite elements, respectively. The wellposedness of the resulting continuous and discrete variational problems are studied employing an extension of the Babuška-Brezzi theory. We establish a priori error estimates in the natural norms, and we finally report some numerical experiments illustrating the behavior of the proposed schemes and confirming our theoretical findings on structured and unstructured meshes. Additional examples of cases not covered by our theory are also presented.

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Section: Example 4: Secondary Settling Tankmentioning

confidence: 99%

An augmented mixed finite element method for the vorticity–velocity–pressure formulation of the Stokes equations

Anaya

Mora

Ruiz–Baier

2013

Computer Methods in Applied Mechanics and Engineering

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“…In that setting, only the equation for the concentration (1.1a) needs to be solved, while the mixture flow velocity is determined by boundary conditions. Only a limited number of papers, including [11,24,30,37,40,41,49], deal with the numerical study of sedimentation-consolidation models in two or three space dimensions.…”

Section: Related Workmentioning

confidence: 99%

“…Fast and reliable numerical methods for related problems in two and three space dimensions include stable FE methods [37,49], discontinuous Galerkin methods [18], finite volume (FV) formulations [5,11], finite difference schemes [43,47], and hybrid/combined FV-FE methods [16,20,33,46]. The particular concept of FVE meth-ods (see [15]) is intermediate between finite volumes and finite elements.…”

Section: Related Workmentioning

confidence: 99%

“…Kleine and Reddy [37] study different geometries for settling tanks using a FE method, while Calgaro, Creusé, and Goudon [16] couple Taylor-Hood finite elements for the flow equations with finite volumes for a conservation law.…”

Section: Related Workmentioning

confidence: 99%

“…We consider a secondary settling tank studied in [37,55]. (See Figure 11, where the geometry is also described.)…”

Section: Examplementioning

confidence: 99%

See 2 more Smart Citations

A Stabilized Finite Volume Element Formulation for Sedimentation-Consolidation Processes

Bürger¹,

Ruiz–Baier²,

Torres³

2012

SIAM J. Sci. Comput.

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Abstract.A model of sedimentation-consolidation processes in so-called clarifier-thickener units is given by a parabolic equation describing the evolution of the local solids concentration coupled with a version of the Stokes system for an incompressible fluid describing the motion of the mixture. In cylindrical coordinates, and if an axially symmetric solution is assumed, the original problem reduces to two space dimensions. This poses the difficulty that the subspaces for the construction of a numerical scheme involve weighted Sobolev spaces. A novel finite volume element method is introduced for the spatial discretization, where the velocity field and the solids concentration are discretized on two different dual meshes. The method is based on a stabilized discontinuous Galerkin formulation for the concentration field, and a multiscale stabilized pair of P 1 -P 1 elements for velocity and pressure, respectively. Numerical experiments illustrate properties of the model and the satisfactory performance of the proposed method.

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Numerical modeling of chlorine concentration in water storage tanks

Codina

Principe

Muñoz

et al. 2015

Numerical Methods in Fluids

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Summary In this paper, we describe a numerical model to simulate the evolution in time of the hydrodynamics of water storage tanks, with particular emphasis on the time evolution of chlorine concentration. The mathematical model contains several ingredients particularly designed for this problem, namely, a boundary condition to model falling jets on free surfaces, an arbitrary Lagrangian–Eulerian formulation to account for the motion of the free surface because of demand and supply of water, and a coupling of the hydrodynamics with a convection–diffusion–reaction equation modeling the time evolution of chlorine. From the numerical point of view, the equations resulting from the mathematical model are approximated using a finite element formulation, with linear continuous interpolations on tetrahedra for all the unknowns. To make it possible, and also to be able to deal with convection‐dominated flows, a stabilized formulation is used. In order to capture the sharp gradients present in the chlorine concentration, particularly near the injection zone, a discontinuity capturing technique is employed. Copyright © 2015 John Wiley & Sons, Ltd.

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Finite element analysis of flows in secondary settling tanks

Cited by 29 publications

References 49 publications

An augmented mixed finite element method for the vorticity–velocity–pressure formulation of the Stokes equations

An augmented mixed finite element method for the vorticity–velocity–pressure formulation of the Stokes equations

A Stabilized Finite Volume Element Formulation for Sedimentation-Consolidation Processes

Numerical modeling of chlorine concentration in water storage tanks

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