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Stefan Diehl - One of the best experts on this subject based on the ideXlab platform.
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Estimating the hindered-settling Flux Function from a batch test in a cone
Chemical Engineering Science, 2018Co-Authors: Raimund Bürger, Stefan Diehl, Julio Careaga, R.d. Merckel, Jesús ZambranoAbstract:Abstract The hindered-settling velocity Function for the modelling, simulation and control of secondary settling tanks can be determined from batch tests. The conventional method is to measure the velocity of the descending sludge-supernatant interface (sludge blanket) as the change in height over time in a vessel with constant cross-sectional area. Each such experiment provides one point on the Flux curve since, under idealizing assumptions (monodisperse suspension, no wall-effects), the concentration of sludge remains constant just below the sludge blanket until some wave from the bottom reaches it. A newly developed method of estimation, based on the theory of nonlinear hyperbolic partial differential equations, is applied to both synthetic and experimental data. The method demonstrates that a substantial portion of the Flux Function may be estimated from a single batch test in a conical vessel. The new method takes into consideration that during an ideal settling experiment in a cone, the concentration just below the sludge blanket increases with time since the mass of suspended solids occupy a reduced volume over time.
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advanced methods of Flux identification for clarifier thickener simulation models
Minerals Engineering, 2014Co-Authors: Fernando Betancourt, Raimund Bürger, Stefan Diehl, Camilo MejiasAbstract: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 [Burger, 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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Convexity-preserving Flux identification for scalar conservation laws modelling sedimentation
Inverse Problems, 2013Co-Authors: Raimund Bürger, Stefan DiehlAbstract:Sedimentation of a suspension of small particles dispersed in a viscous fluid can be described by a scalar, nonlinear conservation law, whose Flux Function usually has one inflection point. The identification of the Flux Function is of theoretical interest and practical importance for plant-scale simulators of continuous sedimentation. For a real suspension, the 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 curved (convex or concave, respectively) suspension-supernate interfaces from which it is possible to reconstruct portions of the Flux Function to either side of the inflection point. Several Functional forms can be employed to generate a provably convex or concave, twice differentiable accurate approximation of these data points via the solution of a constrained least-squares minimization problem. The resulting spline-like estimated trajectory can be converted into an explicit formula for the Flux Function. It is proved that the inverse problem of Flux identification solved this way has a unique solution. The problem of gluing together the portions of the Flux Function from the Kynch and Diehl tests is addressed. Examples involving synthetic data are presented.
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Estimation of the batch-settling Flux Function for an ideal suspension from only two experiments
Chemical Engineering Science, 2007Co-Authors: Stefan DiehlAbstract:Modelling the sedimentation of suspensions with partial differential equations requires constitutive relations (material properties) to be known. Restricted to suspensions obeying Kynch's assumption (ideal suspensions), this paper deals with the inverse problem, which is to estimate the batch-settling Flux Function from experimental data. A new batch-settling test is suggested, from which it is theoretically possible to estimate a large part of the Flux Function for lower concentrations containing the extreme point. From a standard batch-settling test, a large part of the Flux Function for higher concentrations can be estimated with the famous method by Kynch. For these two parts, simple general explicit formulae are derived, which contain only the initial concentration and height variables, the interface height and its derivative as a Function of time. The method is demonstrated on synthetic and experimental data. Further experimental development of the new test is required. The aim of the paper is to present a theoretical foundation for the method, including the explicit formulae as a solution of the inverse problem.
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Scalar conservation laws with discontinuous Flux Function: I. The viscous profile condition
Communications in Mathematical Physics, 1996Co-Authors: Stefan DiehlAbstract:The equation $$\frac{{\partial u}}{{\partial t}} + \frac{\partial }{{\partial x}}\left( {H(x)f(u) + \left( {1 - H(x)} \right)g(u)} \right) = 0$$ , where H is Heaviside's step Function, appears for example in continuous sedimentation of solid particles in a liquid, in two-phase flow, in traffic-flow analysis and in ion etching. The discontinuity of the Flux Function at x =0 causes a discontinuity of a solution, which is not uniquely determined by the initial data. The equation can be written as a triangular 2×2 non-strictly hyperbolic system. This augmentation is non-unique and a natural definition is given by means of viscous profiles. By a viscous profile we mean a stationary solution of u _ t +( F ^δ)_ x =ε u _ xx , where F ^δ is a smooth approximation of the discontinuous Flux, i.e., H is smoothed. In terms of the 2×2 system, the discontinuity at x =0 is either a regular Lax, an under-or overcompressive, a marginal under- or overcompressive or a degenerate shock wave. In some cases, depending on f and g , there is a unique viscous profile (e.g. undercompressive and regular Lax waves) and in some cases there are infinitely many (e.g. overcompressive waves). The main purpose of the paper is to show the equivalence between a previously introduced uniqueness condition for the discontinuity of the solution at x =0 and the viscous profile condition.
B. Saramito - One of the best experts on this subject based on the ideXlab platform.
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Steady tearing mode instabilities with a resistivity depending on a Flux Function
ESAIM: Mathematical Modelling and Numerical Analysis, 1999Co-Authors: Atanda Boussari, Erich K. Maschke, B. SaramitoAbstract:We consider plasma tearing mode instabilities when the resistivity depends on a Flux Function (ψ ), for the plane slab model. This problem, represented by the MHD equations, is studied as a bifurcation problem. For so doing, it is written in the form (I(.)-T(S,.)) = 0 , where T(S,.) is a compact operator in a suitable space and S is the bifurcation parameter. In this work, the resistivity is not assumed to be a given quantity (as usually done in previous papers, see [1,2,5,7,8,9,10], but it depends non linearly of the unknowns of the problem; this is the main difficulty, with new mathematical results. We also develop in this paper a 1D code to compute bifurcation points from the trivial branch (equilibrium state).
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Existence of regular solution and attractor for tearing mode instabilities with a resistivity depending on a Flux Function
Applied Mathematics Letters, 1999Co-Authors: Atanda Boussari, B. SaramitoAbstract:Abstract We consider tearing mode instabilities when the resistivity depends on a Flux Function (ψ) for a bidimensional layer of plasma. This problem modelized by M.H.D. equations is written in terms of Flux Functions, and in this work, we first show, using a fixed-point method, existence of a local regular solution of the considered problem. Next we show existence of a global solution, and we end this paper with existence of a global attractor.
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Bifurcations of stationary tearing modes with a resistivity depending on a Flux Function
Applied Mathematics Letters, 1998Co-Authors: Atanda Boussari, Erich K. Maschke, B. SaramitoAbstract:Abstract We consider plasma tearing mode instabilities when the resistivity depends on a Flux Function (ψ), for the plane slab model. This problem, represented by MHD equations, is studied as a bifurcation problem. For so doing, it is written in the form (I − T(S, ·)), where T(S, ·) is a compact operator in a suitable space and S is the bifurcation parameter. In this work, the resistivity is not assumed to be a given quantity (as usually done in previous papers), but it depends nonlinearly of the unknowns of the problem; this is the main difficulty, with new mathematical results. We also develop in this paper a 1-D code to compute bifurcation points from the trivial branch (equilibrium state).
Sebastian Franz - One of the best experts on this subject based on the ideXlab platform.
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OPTIMIZATION OF AN IMPLICIT SUBGRID-SCALE MODEL FOR LES
2004Co-Authors: Stefan Hickel, Sebastian Franz, P. KoumoutsakosAbstract:Summary We give a summary of the derivation of an implicit subgrid-scale model for LES which is obtained from a new approach for the approximation of hyperbolic conservation laws. Adaptive local deconvolution is performed using a quasi-linear solution-adaptive combination of local interpolation polynomials. The physical Flux Function is substituted by a suitable numerical Flux Function. The truncation error has physical significance and effectively acts as subgrid-scale model. It can be determined by a modified-differentialequation analysis and is adjustable through free parameters. Computational results for Burgers equation show that the model with parameters identified by evolutionary optimization give significantly better results than other models.
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Implicit subgrid-scale modeling by adaptive deconvolution
Journal of Computational Physics, 2004Co-Authors: Nikolaus A. Adams, Stefan Hickel, Sebastian FranzAbstract:A new approach for the construction of implicit subgrid-scale models for large-eddy simulation based on adaptive local deconvolution is proposed. An approximation of the unfiltered solution is obtained from a quasi-linear combination of local interpolation polynomials. The physical Flux Function is modeled by a suitable numerical Flux Function. The effective subgrid-scale model can be determined by a modified-differential equation analysis. Discretization parameters which determine the behavior of the implicit model in regions of developed turbulence can be adjusted so that a given explicit subgrid-scale model is recovered to leading order in filter width. Alternatively, improved discretization parameters can be found directly by evolutionary optimization. Computational results for stochastically forced and decaying Burgers turbulence are provided. An assessment of the computational experiments shows that results for a given explicit subgrid-scale model can be matched by computations with an implicit representation. A considerable improvement can be achieved if instead of the parameters matching an explicit model discretization parameters determined by evolutionary optimization are used.
Raimund Bürger - One of the best experts on this subject based on the ideXlab platform.
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Estimating the hindered-settling Flux Function from a batch test in a cone
Chemical Engineering Science, 2018Co-Authors: Raimund Bürger, Stefan Diehl, Julio Careaga, R.d. Merckel, Jesús ZambranoAbstract:Abstract The hindered-settling velocity Function for the modelling, simulation and control of secondary settling tanks can be determined from batch tests. The conventional method is to measure the velocity of the descending sludge-supernatant interface (sludge blanket) as the change in height over time in a vessel with constant cross-sectional area. Each such experiment provides one point on the Flux curve since, under idealizing assumptions (monodisperse suspension, no wall-effects), the concentration of sludge remains constant just below the sludge blanket until some wave from the bottom reaches it. A newly developed method of estimation, based on the theory of nonlinear hyperbolic partial differential equations, is applied to both synthetic and experimental data. The method demonstrates that a substantial portion of the Flux Function may be estimated from a single batch test in a conical vessel. The new method takes into consideration that during an ideal settling experiment in a cone, the concentration just below the sludge blanket increases with time since the mass of suspended solids occupy a reduced volume over time.
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advanced methods of Flux identification for clarifier thickener simulation models
Minerals Engineering, 2014Co-Authors: Fernando Betancourt, Raimund Bürger, Stefan Diehl, Camilo MejiasAbstract: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 [Burger, 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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Convexity-preserving Flux identification for scalar conservation laws modelling sedimentation
Inverse Problems, 2013Co-Authors: Raimund Bürger, Stefan DiehlAbstract:Sedimentation of a suspension of small particles dispersed in a viscous fluid can be described by a scalar, nonlinear conservation law, whose Flux Function usually has one inflection point. The identification of the Flux Function is of theoretical interest and practical importance for plant-scale simulators of continuous sedimentation. For a real suspension, the 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 curved (convex or concave, respectively) suspension-supernate interfaces from which it is possible to reconstruct portions of the Flux Function to either side of the inflection point. Several Functional forms can be employed to generate a provably convex or concave, twice differentiable accurate approximation of these data points via the solution of a constrained least-squares minimization problem. The resulting spline-like estimated trajectory can be converted into an explicit formula for the Flux Function. It is proved that the inverse problem of Flux identification solved this way has a unique solution. The problem of gluing together the portions of the Flux Function from the Kynch and Diehl tests is addressed. Examples involving synthetic data are presented.
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a front tracking approach to a model of continuous sedimentation in ideal clarifier thickener units
Nonlinear Analysis-real World Applications, 2003Co-Authors: Raimund Bürger, Kenneth H. Karlsen, Christian Klingenberg, Nils Henrik RisebroAbstract:We study a model of continuous sedimentation. Under idealizing assumptions, the settling of the solid particles under the influence of gravity can be described by the initial value problem for a one-dimensional scalar conservation law with a Flux Function that depends discontinuously on the spatial position. We construct a weak solution to the sedimentation model by proving the convergence of a front tracking method. The basic building block in this method is the solution of the Riemann problem, which is complicated by the fact that the Flux Function is discontinuous. A feature of the convergence analysis is the difficulty of bounding the total variation of the conserved variable. To overcome this obstacle, we rely on a certain non-linear Temple Functional under which the total variation can be bounded. The total variation bound on the transformed variable also implies that the front tracking construction is well defined. Finally, via some numerical examples, we demonstrate that the front tracking method can be used as a highly efficient and accurate simulation tool for continuous sedimentation.
Atanda Boussari - One of the best experts on this subject based on the ideXlab platform.
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Steady tearing mode instabilities with a resistivity depending on a Flux Function
ESAIM: Mathematical Modelling and Numerical Analysis, 1999Co-Authors: Atanda Boussari, Erich K. Maschke, B. SaramitoAbstract:We consider plasma tearing mode instabilities when the resistivity depends on a Flux Function (ψ ), for the plane slab model. This problem, represented by the MHD equations, is studied as a bifurcation problem. For so doing, it is written in the form (I(.)-T(S,.)) = 0 , where T(S,.) is a compact operator in a suitable space and S is the bifurcation parameter. In this work, the resistivity is not assumed to be a given quantity (as usually done in previous papers, see [1,2,5,7,8,9,10], but it depends non linearly of the unknowns of the problem; this is the main difficulty, with new mathematical results. We also develop in this paper a 1D code to compute bifurcation points from the trivial branch (equilibrium state).
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Existence of regular solution and attractor for tearing mode instabilities with a resistivity depending on a Flux Function
Applied Mathematics Letters, 1999Co-Authors: Atanda Boussari, B. SaramitoAbstract:Abstract We consider tearing mode instabilities when the resistivity depends on a Flux Function (ψ) for a bidimensional layer of plasma. This problem modelized by M.H.D. equations is written in terms of Flux Functions, and in this work, we first show, using a fixed-point method, existence of a local regular solution of the considered problem. Next we show existence of a global solution, and we end this paper with existence of a global attractor.
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Bifurcations of stationary tearing modes with a resistivity depending on a Flux Function
Applied Mathematics Letters, 1998Co-Authors: Atanda Boussari, Erich K. Maschke, B. SaramitoAbstract:Abstract We consider plasma tearing mode instabilities when the resistivity depends on a Flux Function (ψ), for the plane slab model. This problem, represented by MHD equations, is studied as a bifurcation problem. For so doing, it is written in the form (I − T(S, ·)), where T(S, ·) is a compact operator in a suitable space and S is the bifurcation parameter. In this work, the resistivity is not assumed to be a given quantity (as usually done in previous papers), but it depends nonlinearly of the unknowns of the problem; this is the main difficulty, with new mathematical results. We also develop in this paper a 1-D code to compute bifurcation points from the trivial branch (equilibrium state).