A Stabilized Finite Volume Element Formulation for Sedimentation-Consolidation Processes

Bürger, Raimund; Ruiz–Baier, Ricardo; Torres, Héctor

doi:10.1137/110836559

Cited by 39 publications

(52 citation statements)

References 52 publications

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“…The FVE method can be summarized as follows [2]. We discretize (1) 1 in time by a semi-implicit method.…”

Section: Finite Volume Element (Fve) Methods and Numerical Examplementioning

confidence: 99%

“…The primal mesh T is composed by 7410 elements and 4206 interior nodes. The boundary conditions for velocity at the suction lifts are given by u = (0, −u z,out /4), where u z,out = νu r,in , with ν = 0.8 and ν = 0.01 corresponding to operating conditions that differ from those studied in [2]. See Figure 3 for numerical results.…”

Section: Finite Volume Element (Fve) Methods and Numerical Examplementioning

confidence: 99%

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A finite volume element method for simulating secondary settling tanks

Bürger

Ruiz–Baier

Torres

2012

Proc Appl Math and Mech

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This contribution summarizes a new finite volume element (FVE) method as a unified scheme for coupled sedimentation-flow problems in secondary settling tanks in wastewater treatment. The model is two-dimensional in an axisymmetric setting. A new numerical example is presented. Model problemModels for the simulation of secondary settling tanks in wastewater treatment are usually of macroscopic type, where the governing PDEs are represent the mass and linear momentum balance equations for the solid and liquid phases. A wellknown challenge for 2D or 3D numerical simulation is the strong coupling between the field of solids concentration φ and the flow field of the mixture, which is defined by the velocity u and the pressure p. (Additional equations of motion for u and p do not arise in 1D models [1].) The governing equations can be written as follows, where φ, u and p are sought:Here F (φ, u) = φu + f bk (φ)k is a flux vector, where f bk is the material specific Kynch batch flux density function and k is the upward-pointing unit vector, the diffusion term ∆A(φ) models sediment compressibility (where A is assumed to be a strictly increasing function of φ), and µ(φ)ε(u) − λpI denotes the Cauchy stress tensor, where ε(u) := 1 2 (∇u + ∇u T ) and µ is concentration-dependent viscosity function. In the forcing term G = λφgk, the constant g is the acceleration of gravity and λ is another constant. We here choose f bk (φ) = u ∞ φ(1 − φ/φ max ) 2 and A(φ) = φ 0 a(s) ds, where a(φ) := f bk (φ)σ e (φ)/[( s − f )gφ] and s and f are the solid and fluid mass densities. Moreover, σ e is the so-called effective solid stress function that is assumed to satisfy σ e (0) = 0 and σ e (φ) > 0 for φ > 0. Thus, (1) 1 is a two-point degenerate parabolic PDE. The viscosity function is chosen as µ(φ) = (1 − φ/φ max ) −β withφ max > φ max .The initial and boundary conditions are as follows (see Figure 1). We consider t ∈ [0, T ], and the spatial domain (r, z) ∈ Ω ⊂ R 2 corresponding to an axisymmetric setting. Initial data are specified for φ and u. The unit is fed through the portion of the boundary Γ in ⊂ ∂Ω, where u = u in and φ = φ in are given. The boundary parts Γ out and Γ c correspond to discharge and overflow outlets, where u = u out and u = u c are specified. On the remainder of ∂Ω we impose no-slip conditions (u = 0) and zero-flux conditions for φ. Finite Volume Element (FVE) method and numerical exampleThe FVE method can be summarized as follows [2]. We discretize (1) 1 in time by a semi-implicit method. Starting from the appropriate weak formulation and employing standard stabilization techniques, we discretize (1) in space by a finite element (FE) scheme on a triangular "primal" mesh T h . Since φ may be discontinuous ((1) 1 may be strongly degenerate), we choose piecewise linear, possibly discontinuous functions for φ, piecewise bilinear, continuous functions for u and piecewise linear, continuous functions for p. By standard discretization techniques we then obtain a Galerkin FE formulation for the coupled problem (1) includi...

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“…The FVE method can be summarized as follows [2]. We discretize (1) 1 in time by a semi-implicit method.…”

Section: Finite Volume Element (Fve) Methods and Numerical Examplementioning

confidence: 99%

Section: Finite Volume Element (Fve) Methods and Numerical Examplementioning

confidence: 99%

A finite volume element method for simulating secondary settling tanks

Bürger

Ruiz–Baier

Torres

2012

Proc Appl Math and Mech

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“…The CT model provides an idealized description of secondary settling tanks in wastewater treatment or of thickeners in mineral processing [7]. For so-called ideal suspensions of small solid, non-flocculent particles that do not exhibit the effect of sediment compressibility, the complete set of governing partial differential equations is given by a first-order scalar conservation law for the local solids concentration that involves as a coefficient the local volumeaveraged flow velocity of the mixture (in short, bulk velocity); this velocity must satisfy a divergence-free condition plus possibly an additional equation of motion [13,14]. It is frequently assumed, however, that all variables are horizontally constant such that a spatially one-dimensional description is sufficient.…”

Section: Scopementioning

confidence: 99%

“…Obviously, only in d = 1 dimensions are the two scalar equations (5) solvable for u and q = q. In d ≥ 2 dimensions an additional equation of motion must be solved; for example, in [12,13], (5) is solved along with the equation…”

Section: Deterministic Versionsmentioning

confidence: 99%

“…In the multi-dimensional setting, the right-hand side of (6) describes the coupling between the concentration and velocity fields. The coupled system (5), (6) is solved (in slight variants) in [12,13] by finite volume and finite volume element techniques, respectively. In the present work we wish to partially address the multi-dimensional case, but prefer not to solve the full coupled system (5), (6) for reasons of the considerable additional computational effort, and to avoid that the uncertainty in the concentration field be introduced into the flow field.…”

Section: Deterministic Versionsmentioning

confidence: 99%

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A hybrid stochastic Galerkin method for uncertainty quantification applied to a conservation law modelling a clarifier‐thickener unit

2013

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The continuous sedimentation process in a clarifier-thickener can be described by a scalar nonlinear conservation law for the local solids volume fraction. The flux density function is discontinuous with respect to spatial position due to feed and discharge mechanisms. Typically, the feed flow cannot be given deterministically and efficient numerical simulation requires a concept for quantifying uncertainty. In this paper uncertainty quantification is expressed by a new hybrid stochastic Galerkin (HSG) method that extends the classical polynomial chaos approximation by multiresolution discretization in the stochastic space. The new approach leads to a deterministic hyperbolic system for a finite number of stochastic moments which is however partially decoupled and thus allows efficient parallelisation. The complexity of the problem is further reduced by stochastic adaptivity. For the approximate solution of the resulting high-dimensional system a finite volume scheme is introduced. Numerical experiments cover one-and two-dimensional situations.

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Finite element and finite volume-element simulation of pseudo-ECGs and cardiac alternans

Dupraz

Filippi

Gizzi

et al. 2014

Math. Meth. Appl. Sci.

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In this paper, we are interested in the spatio-temporal dynamics of the transmembrane potential in paced isotropic and anisotropic cardiac tissues. In particular, we observe a specific precursor of cardiac arrhythmias that is the presence of alternans in the action potential duration. The underlying mathematical model consists of a reaction-diffusion system describing the propagation of the electric potential and the nonlinear interaction with ionic gating variables. Either conforming piecewise continuous finite elements or a finite volume-element scheme are employed for the spatial discretization of all fields, whereas operator splitting strategies of first and second order are used for the time integration. We also describe an efficient mechanism to compute pseudo-ECG signals, and we analyze restitution curves and alternans patterns for physiological and pathological cardiac rhythms.

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A Stabilized Finite Volume Element Formulation for Sedimentation-Consolidation Processes

Cited by 39 publications

References 52 publications

A finite volume element method for simulating secondary settling tanks

A finite volume element method for simulating secondary settling tanks

A hybrid stochastic Galerkin method for uncertainty quantification applied to a conservation law modelling a clarifier‐thickener unit

Finite element and finite volume-element simulation of pseudo-ECGs and cardiac alternans

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