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 flux with an approach of numerical percentage propagation for multi-component flows. 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.
Accepté le :
DOI : 10.1051/m2an/2017038
Mots clés : clarifier-thickener, invariant-region property, multi-component flow, percentage propagation, wastewater treatment
@article{M2AN_2018__52_2_365_0,
author = {B\"urger, Raimund and Diehl, Stefan and Mej{\'\i}as, Camilo},
title = {A difference scheme for a degenerating convection-diffusion-reaction system modelling continuous sedimentation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {365--392},
publisher = {EDP-Sciences},
volume = {52},
number = {2},
year = {2018},
doi = {10.1051/m2an/2017038},
zbl = {1412.65068},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2017038/}
}
TY - JOUR AU - Bürger, Raimund AU - Diehl, Stefan AU - Mejías, Camilo TI - A difference scheme for a degenerating convection-diffusion-reaction system modelling continuous sedimentation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 365 EP - 392 VL - 52 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2017038/ DO - 10.1051/m2an/2017038 LA - en ID - M2AN_2018__52_2_365_0 ER -
%0 Journal Article %A Bürger, Raimund %A Diehl, Stefan %A Mejías, Camilo %T A difference scheme for a degenerating convection-diffusion-reaction system modelling continuous sedimentation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 365-392 %V 52 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2017038/ %R 10.1051/m2an/2017038 %G en %F M2AN_2018__52_2_365_0
Bürger, Raimund; Diehl, Stefan; Mejías, Camilo. A difference scheme for a degenerating convection-diffusion-reaction system modelling continuous sedimentation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 2, pp. 365-392. doi : 10.1051/m2an/2017038. http://archive.numdam.org/articles/10.1051/m2an/2017038/
[1] , , and , Monotone (A, B) entropy stable numerical scheme for scalar conservation laws with discontinuous flux. ESAIM: M2AN 48 (2014) 1725–1755 | DOI | MR | Zbl
[2] , , and , A combined hydraulic and biological SBR model. Wat. Sci. Tech. 64 (2011) 1025–1031 | DOI
[3] , and , A locally conservative Eulerian–Lagrangian method for a model two-phase flow problem in a one-dimensional porous medium. SIAM J. Sci. Comput. 34 (2012) A1950–A1974 | DOI | MR | Zbl
[4] and A positivity preserving central scheme for shallow water flows in channels with wet-dry states. ESAIM: M2AN 48 (2014) 665–696 | DOI | Numdam | MR | Zbl
[5] , , and , A 2d model for hydrodynamics and biology coupling applied to algae growth simulations. ESAIM: M2AN 47 (2013) 1387–1412 | DOI | Numdam | MR | Zbl
[6] , and , Splitting in systems of PDEs for two-phase multicomponent flow in porous media. Appl. Math. Letters 53 (2016) 25–32 | DOI | MR | Zbl
[7] , , , , and , Simulations of reactive settling of activated sludge with a reduced biokinetic model. Comput. Chem. Eng. 92 (2016) 216–229 | DOI
[8] , and , A semi-implicit monotone difference scheme for an initial-boundary value problem of a strongly degenerate parabolic equation modeling sedimentation-consolidation processes. Math. Comput. 75 (2006) 91–112 | DOI | MR | Zbl
[9] , , , and , A consistent modelling methodology for secondary settling tanks: A reliable numerical method. Water Sci. Tech. 68 (2013) 192–208 | DOI
[10] , and , A consistent modelling methodology for secondary settling tanks in wastewater treatment. Water Res. 45 (2011) 2247–2260 | DOI
[11] , , and , Well-posedness in BVt and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units. Numer. Math. 97 (2004) 25–65 | DOI | MR | Zbl
[12] , and , A model of continuous sedimentation of flocculated suspensions in clarifier-thickener units. SIAM J. Appl. Math. 65 (2005) 882–940 | DOI | MR | Zbl
[13] , , and , Fully adaptive multiresolution schemes for strongly degenerate parabolicequations in one space dimension. ESAIM: Math. Modell. Num. Anal. 42 (2008) 535–563 | DOI | Numdam | MR | Zbl
[14] ,On scalar conservation laws with point source and discontinuous flux function. SIAM J. Math. Anal. 26 (1995) 1425–1451 | DOI | MR | Zbl
[15] , A conservation law with point source and discontinuous flux function modelling continuous sedimentation. SIAM J. Appl. Math. 56 (1996) 388–419 | DOI | MR | Zbl
[16] , Continuous sedimentation of multi-component particles. Math. Meth. Appl. Sci. 20 (1997) 1345–1364 | DOI | MR | Zbl
[17] ,A uniqueness condition for nonlinear convection-diffusion equations with discontinuous coefficients. J. Hyperbolic Diff. Equ. 6 (2009) 127–159 | DOI | MR | Zbl
[18] and Theory of Multicomponent Fluids, volume 135, Springer-Verlag, New York (1999) | DOI | MR | Zbl
[19] , and , Benchmarking biological nutrient removal in wastewater treatment plants: Influence of mathematical model assumptions. Water Sci. Tech. 65 (2012) 1496–1505 | DOI
[20] , , and , Impact of reactive settler models on simulated WWTP performance. Water Sci. Tech. 53 (2006) 159–167 | DOI
[21] and , Solution of the Cauchy problem for a conservation law with a discontinuous flux function. SIAM J. Math. Anal. 23 (1992) 635–648 | DOI | MR | Zbl
[22] and , Multi-component particle-size segregation in shallow granular avalanches. J. Fluid Mech. 678 (2011) 535–588 | DOI | Zbl
[23] , , , and , Effect of nitrite, limited reactive settler and plant design configuration on the predicted performance of simultaneous C/N/P removal WWTPs. Bioresource Tech. 136 (2013) 680–688 | DOI
[24] , , , , and , Modeling and pilot-scale experimental verification for predenitrification process. J. Environ. Eng. 118 (1992) 38–55 | DOI
[25] and , Numerical transport of an arbitrary number of components. Computer Methods Appl. Mech. Eng. 196 (2007) 3127–3140 | DOI | MR | Zbl
[26] and , On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Discrete Continuous Dynamical Syst. 9 (2003) 1081–1104 | DOI | MR | Zbl
[27] , and , On the convergence rate of finite difference methods for degenerate convection-diffusion equations in several space dimensions. ESAIM: M2AN 50 (2016) 499–539 | DOI | Numdam | MR | Zbl
[28] , and , Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient. IMA J. Numer. Anal. 22 (2002) 623–664 | DOI | MR | Zbl
[29] , and , L1 stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients. Trans. Royal Norwegian Society Sci. Letters (Skr. K. Nor. Vidensk. Selsk.) 3 (2003) 49 | MR | Zbl
[30] and , Practical identifiability and uncertainty analysis of the one-dimensional hindered-compression continuous settling model. Water Res. 90 (2016) 235–246
[31] , , , , and , Cost-performance analysis of nutrient removal in a full-scale oxidation ditch process based on kinetic modeling. J. Environ. Sci. 25 (2013) 26–32 | DOI
[32] , and , Evaluation of different control strategies of the waste water treatment plant based on a modified activated sludge model no. 3. Environ. Eng. Manag. J. 11 (2012) 147–164 | DOI
[33] , Positivity-preserving high-resolution schemes for systems of conservation laws. J. Comp. Phys. 231 (2012) 173–189 | DOI | MR | Zbl
[34] , Quasistationary sedimentation with adsorption. J. Appl. Mech. Tech. Phys. 46 (2005) 513–522 | DOI | MR | Zbl
[35] , Countercurrent washing of solids, Edited by . In: Solid-Liquid Separation, chap. 15. Butterworth Heinemann, Oxford, 4th edition (2001), 442–475. | DOI
[36] , , , and , Impact on sludge inventory and control strategies using the benchmark simulation model no. 1 with the Bürger-Diehl settler model. Water Sci. Tech. 71 (2015) 1524–1535 | DOI
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