| 1. |
- Betancourt, Fernando, et al.
(författare)
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A Model of Froth Flotation with Drainage : Simulations and Comparison with Experiments
- 2023
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Ingår i: Minerals. - : MDPI AG. - 2075-163X. ; 13:3
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Tidskriftsartikel (refereegranskat)abstract
- The operation of a froth flotation column can be described by a nonlinear convection–diffusion partial differential equation that incorporates the solids–flux and drift–flux theories as well as a model of foam drainage. The resulting model predicts the bubble and (gangue) particle volume fractions as functions of height and time. The steady-state (time-independent) version of the model defines so-called operating charts that map conditions on the gas and pulp feed rates that allow for operation with a stationary froth layer. Operating charts for a suitably adapted version of the model are compared with experimental results obtained with a laboratory flotation column. Experiments were conducted with a two-phase liquid–bubble flow. The results indicate good agreement between the predicted and measured conditions for steady states. Numerical simulations for transient operation, in part for the addition of solid particles, are presented.
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| 2. |
- Betancourt, Fernando, et al.
(författare)
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A random sampling method for a family of Temple-class systems of conservation laws
- 2018
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Ingår i: Numerische Mathematik. - : Springer Science and Business Media LLC. - 0029-599X .- 0945-3245. ; 138:1, s. 37-73
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Tidskriftsartikel (refereegranskat)abstract
- The Aw–Rascle–Zhang traffic model, a model of sedimentation, and other applications lead to nonlinear (Formula presented.) systems of conservation laws that are governed by a single scalar system velocity. Such systems are of the Temple class since rarefaction wave curves and Hugoniot curves coincide. Moreover, one characteristic field is genuinely nonlinear almost everywhere, and the other is linearly degenerate. Two well-known problems associated with these systems are handled via a random sampling approach. Firstly, Godunov’s and related methods produce spurious oscillations near contact discontinuities since the numerical solution invariably leaves the invariant region of the exact solution. It is shown that alternating between averaging (Av) and remap steps similar to the approach by Chalons and Goatin (Commun Math Sci 5:533–551, 2007) generates numerical solutions that do satisfy an invariant region property. If the remap step is made by random sampling (RS), then combined techniques due to Glimm (Commun Pure Appl Math 18:697–715, 1965), LeVeque and Temple (Trans Am Math Soc 288:115–123, 1985) prove that the resulting Av–RS scheme converges to a weak solution. Numerical examples illustrate that the new scheme is superior to Godunov’s method in accuracy and resolution. Secondly, the vacuum state, which may form even from positive initial data, causes potential problems of non-uniqueness and instability. This is resolved by introducing an alternative Riemann solution concept.
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| 3. |
- Betancourt, Fernando, et al.
(författare)
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Advanced methods of flux identification for clarifier–thickener simulation models
- 2014
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Ingår i: Minerals Engineering. - : Elsevier BV. - 0892-6875. ; 63:August 2014, s. 2-15
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Tidskriftsartikel (refereegranskat)abstract
- Mathematical models for the simulation of batch settling and continuous clarifier-thickeners can usually be expressed as a convection-diffusion partial differential equation (PDE). Reliable numerical methods require that the nonlinear flux function of this PDE has been identified for a given material. This contribution summarizes, and applies to experimental data, a recent approach [Bürger, R., Diehl, S., 2013. Inverse Problems 29, 045008] for the flux identification in the case of a suspension that shows no compressive behavior. The experimental Kynch test and the Diehl test, which are based on an initially homogenous suspension either filling the whole settling column or being initially located above clear liquid, respectively, provide data points that represent a convex and concave, respectively, suspension-supernate interface. A provably convex (concave) smooth approximation of this interface is obtained by solving a constrained least-squares minimization problem. The interface-approximating function can be converted uniquely into an explicit formula for a convex (concave) part of the flux function.
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| 4. |
- Betancourt, Fernando, et al.
(författare)
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Modeling and controlling clarifier–thickeners fed by suspensions with time-dependent properties
- 2014
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Ingår i: Minerals Engineering. - : Elsevier BV. - 0892-6875. ; 62, s. 91-101
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Tidskriftsartikel (refereegranskat)abstract
- A one-dimensional model of the process of continuous sedimentation in a clarifier–thickener unit is presented. The governing model is expressed as a system of two nonlinear partial differential equations for the solids volume fraction and the varying settling velocity of the solids as functions of depth and time. This model extends the well-known model for the dynamics of a flocculated suspension in a clarifier–thickener advanced by Bürger et al. (2005). Operating charts are calculated to be used for the control of steady states, in particular, to keep the sediment level and the underflow volume fraction at desired values. A numerical scheme and a simple regulator are proposed and numerical simulations are performed.
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| 5. |
- Bürger, Raimund, et al.
(författare)
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A consistent modelling methodology for secondary settling tanks: A reliable numerical method.
- 2013
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Ingår i: Water Science and Technology. - : IWA Publishing. - 0273-1223 .- 1996-9732. ; 68:1, s. 192-208
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Tidskriftsartikel (refereegranskat)abstract
- The consistent modelling methodology for secondary settling tanks (SSTs) leads to a partial differential equation (PDE) of nonlinear convection–diffusion type as a one-dimensional model for the solids concentration as a function of depth and time. This PDE includes a flux that depends discontinuously on spatial position modelling hindered settling and bulk flows, a singular source term describing the feed mechanism, a degenerating term accounting for sediment compressibility, and a dispersion term for turbulence. In addition, the solution itself is discontinuous. A consistent, reliable and robust numerical method that properly handles these difficulties is presented. Many constitutive relations for hindered settling, compression and dispersion can be used within the model, allowing the user to switch on and off effects of interest depending on the modelling goal as well as investigate the suitability of certain constitutive expressions. Simulations show the effect of the dispersion term on effluent suspended solids and total sludge mass in the SST. The focus is on correct implementation whereas calibration and validation are not pursued.
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| 6. |
- Bürger, Raimund, et al.
(författare)
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A consistent modelling methodology for secondary settling tanks in wastewater treatment
- 2011
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Ingår i: Water Research. - : Elsevier BV. - 1879-2448 .- 0043-1354. ; 45:6, s. 2247-2260
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Tidskriftsartikel (refereegranskat)abstract
- The aim of this contribution is partly to build consensus on a consistent modelling methodology (CMM) of complex real processes in wastewater treatment by combining classical concepts with results from applied mathematics, and partly to apply it to the clarification-thickening process in the secondary settling tank. In the CMM, the real process should be approximated by a mathematical model (process model; ordinary or partial differential equation (ODE or PDE)), which in turn is approximated by a simulation model (numerical method) implemented on a computer. These steps have often not been carried out in a correct way. The secondary settling tank was chosen as a case since this is one of the most complex processes in a wastewater treatment plant and simulation models developed decades ago have no guarantee of satisfying fundamental mathematical and physical properties. Nevertheless, such methods are still used in commercial tools to date. This particularly becomes of interest as the state-of-the-art practice is moving towards plant-wide modelling. Then all submodels interact and errors propagate through the model and severely hamper any calibration effort and, hence, the predictive purpose of the model. The CMM is described by applying it first to a simple conversion process in the biological reactor yielding an ODE solver, and then to the solideliquid separation in the secondary settling tank, yielding a PDE solver. Time has come to incorporate established mathematical techniques into environmental engineering, and wastewater treatment modelling in particular, and to use proven reliable and consistent simulation models.
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| 7. |
- Bürger, Raimund, et al.
(författare)
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A difference scheme for a degenerating convection-diffusion-reaction system modelling continuous sedimentation
- 2018
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Ingår i: ESAIM: Mathematical Modelling and Numerical Analysis. - : EDP Sciences. - 0764-583X .- 1290-3841. ; 52:2, s. 365-392
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Tidskriftsartikel (refereegranskat)abstract
- Continuously operated settling tanks are used for the gravity separation of solid-liquid suspensions in several industries. Mathematical models of these units form a topic for well-posedness and numerical analysis even in one space dimension due to the spatially discontinuous coefficients of the underlying strongly degenerate parabolic, nonlinear model partial differential equation (PDE). Such a model is extended to describe the sedimentation of multi-component particles that react with several soluble components of the liquid phase. The fundamental balance equations contain the mass percentages of the components of the solid and liquid phases. The equations are reformulated as a system of nonlinear PDEs that can be solved consecutively in each time step by an explicit numerical scheme. This scheme combines a difference scheme for conservation laws with discontinuous ux with an approach of numerical percentage propagation for multi-component ows. The main result is an invariant-region property, which implies that physically relevant numerical solutions are produced. Simulations of denitrification in secondary settling tanks in wastewater treatment illustrate the model and its discretization.
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| 8. |
- Bürger, Raimund, et al.
(författare)
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A difference scheme for a triangular system of conservation laws with discontinuous flux modeling three-phase flows
- 2023
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Ingår i: Networks and Heterogeneous Media. - : American Institute of Mathematical Sciences (AIMS). - 1556-1801. ; 18:1, s. 140-190
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Tidskriftsartikel (refereegranskat)abstract
- A triangular system of conservation laws with discontinuous flux that models the one-dimensional flow of two disperse phases through a continuous one is formulated. The triangularity arises from the distinction between a primary and a secondary disperse phase, where the movement of the primary disperse phase does not depend on the local volume fraction of the secondary one. A particular application is the movement of aggregate bubbles and solid particles in flotation columns under feed and discharge operations. This model is formulated under the assumption of a variable cross-sectional area. A monotone numerical scheme to approximate solutions to this model is presented. The scheme is supported by three partial theoretical arguments. Firstly, it is proved that it satisfies an invariant-region property, i.e., the approximate volume fractions of the three phases, and their sum, stay between zero and one. Secondly, under the assumption of flow in a column with constant cross-sectional area it is shown that the scheme for the primary disperse phase converges to a suitably defined entropy solution. Thirdly, under the additional assumption of absence of flux discontinuities it is further demonstrated, by invoking arguments of compensated compactness, that the scheme for the secondary disperse phase converges to a weak solution of the corresponding conservation law. Numerical examples along with estimations of numerical error and convergence rates are presented for counter-current and co-current flows of the two disperse phases.
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| 9. |
- Bürger, Raimund, et al.
(författare)
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A method-of-lines formulation for a model of reactive settling in tanks with varying cross-sectional area
- 2021
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Ingår i: IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications). - : Oxford University Press (OUP). - 0272-4960 .- 1464-3634. ; 86:3, s. 514-546
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Tidskriftsartikel (refereegranskat)abstract
- Reactive settling denotes the process of sedimentation of small solid particles dispersed in a viscous fluid with simultaneous reactions between the components that constitute the solid and liquid phases. This process is of particular importance for the simulation and control of secondary settling tanks (SSTs) in water resource recovery facilities (WRRFs), formerly known as wastewater treatment plants. A spatially 1D model of reactive settling in an SST is formulated by combining a mechanistic model of sedimentation with compression with a model of biokinetic reactions. In addition, the cross-sectional area of the tank is allowed to vary as a function of height. The final model is a system of strongly degenerate parabolic, nonlinear partial differential equations that include discontinuous coefficients to describe the feed, underflow and overflow mechanisms, as well as singular source terms that model the feed mechanism. A finite difference scheme for the final model is developed by first deriving a method-of-lines formulation (discrete in space, continuous in time) and then passing to a fully discrete scheme by a time discretization. The advantage of this formulation is its compatibility with common practice in development of software for WRRFs. The main mathematical result is an invariant-region property, which implies that physically relevant numerical solutions are produced. Simulations of denitrification in SSTs in WRRFs illustrate the model and its discretization.
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| 10. |
- Bürger, Raimund, et al.
(författare)
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A model of reactive settling of activated sludge : Comparison with experimental data
- 2023
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Ingår i: Chemical Engineering Science. - : Elsevier BV. - 0009-2509. ; 267
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Tidskriftsartikel (refereegranskat)abstract
- A non-negligible part of the biological reactions in the activated sludge process for treatment of wastewater takes place in secondary settling tanks (SST) that follow biological reactors. It is therefore of interest to develop models of so-called reactive settling that describe the spatial variability of reaction rates caused by the variation of local concentration of biomass due to hindered settling and compression. A reactive-settling model of an SST is described by a system of nonlinear partial differential equations and a numerical scheme is introduced for the simulation of hindered settling of flocculated particles, compression at high concentrations, dispersion of the flocculated particles in the suspension, dispersion of the dissolved substrates in the fluid, and the mixing that occurs near the feed inlet. The model is fitted to experiments from a pilot plant where the SST has a varying cross-sectional area. For the reactions, the Activated Sludge Model No. 1 (ASM1) is used with temperature-adjusted standard coefficients. The constitutive functions for hindered settling and compression are calibrated to a series of conventional batch settling experiments after the initial induction period of turbulence and reflocculation has been transformed away. The overall conclusion is thus that a careful treatment of batch-settling data to calibrate the sedimentation-compression model is sufficient to ensure good predictability of the reactive sedimentation process in a pilot SST, while the optional inclusion of hydrodynamic dispersion (modelled with two parameters) or a term modelling the mixing of the suspension near the feed inlet at most marginally improve predictability.
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