Raimund Bürger, Julio Careaga, Stefan Diehl, Romel Pineda
{"title":"序批式反应器移动边界对流扩散反应模型的数值格式","authors":"Raimund Bürger, Julio Careaga, Stefan Diehl, Romel Pineda","doi":"10.1051/m2an/2023068","DOIUrl":null,"url":null,"abstract":"Sequencing batch reactors (SBRs) are devices widely used in wastewater treatment, chemical engineering, and other areas. They allow for the sedimentation and compression of solid particles of biomass simultaneously with biochemical reactions with nutrients dissolved in the liquid. The kinetics of these reactions may be given by one of the established activated sludge models (ASMx). An SBR is operated in various stages and is equipped with a movable extraction and fill device and a discharge opening. A one-dimensional model of this unit can be formulated as a moving-boundary problem for a degenerating system of convection-diffusion-reaction equations whose unknowns are the concentrations of the components forming the solid and liquid phases, respectively. This model is transformed to a fixed computational domain and is discretized by an explicit monotone scheme along with an alternative semi-implicit variant. The semi-implicit variant is based on solving, during each time step, a system of nonlinear equations for the total solids concentration followed by the solution of linear systems for the solid component percentages and liquid component concentrations. It is demonstrated that the semi-implicit scheme is well posed and that both variants produce approximations that satisfy an invariant region principle: solids concentrations are nonnegative and less or equal to a set maximal one, percentages are nonnegative and sum up to one, and substrate concentrations are nonnegative. These properties are achieved under a Courant-Friedrichs-Lewy (CFL) condition that is less restrictive for the semi-implicit than for the explicit variant. Numerical examples with realistic parameters illustrate that as a consequence, the semi-implicit variant is more efficient than the explicit one.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"41 1","pages":"0"},"PeriodicalIF":0.6000,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Numerical schemes for a moving-boundary convection-diffusion-reaction model of sequencing batch reactors\",\"authors\":\"Raimund Bürger, Julio Careaga, Stefan Diehl, Romel Pineda\",\"doi\":\"10.1051/m2an/2023068\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Sequencing batch reactors (SBRs) are devices widely used in wastewater treatment, chemical engineering, and other areas. They allow for the sedimentation and compression of solid particles of biomass simultaneously with biochemical reactions with nutrients dissolved in the liquid. The kinetics of these reactions may be given by one of the established activated sludge models (ASMx). An SBR is operated in various stages and is equipped with a movable extraction and fill device and a discharge opening. A one-dimensional model of this unit can be formulated as a moving-boundary problem for a degenerating system of convection-diffusion-reaction equations whose unknowns are the concentrations of the components forming the solid and liquid phases, respectively. This model is transformed to a fixed computational domain and is discretized by an explicit monotone scheme along with an alternative semi-implicit variant. The semi-implicit variant is based on solving, during each time step, a system of nonlinear equations for the total solids concentration followed by the solution of linear systems for the solid component percentages and liquid component concentrations. It is demonstrated that the semi-implicit scheme is well posed and that both variants produce approximations that satisfy an invariant region principle: solids concentrations are nonnegative and less or equal to a set maximal one, percentages are nonnegative and sum up to one, and substrate concentrations are nonnegative. These properties are achieved under a Courant-Friedrichs-Lewy (CFL) condition that is less restrictive for the semi-implicit than for the explicit variant. 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Numerical schemes for a moving-boundary convection-diffusion-reaction model of sequencing batch reactors
Sequencing batch reactors (SBRs) are devices widely used in wastewater treatment, chemical engineering, and other areas. They allow for the sedimentation and compression of solid particles of biomass simultaneously with biochemical reactions with nutrients dissolved in the liquid. The kinetics of these reactions may be given by one of the established activated sludge models (ASMx). An SBR is operated in various stages and is equipped with a movable extraction and fill device and a discharge opening. A one-dimensional model of this unit can be formulated as a moving-boundary problem for a degenerating system of convection-diffusion-reaction equations whose unknowns are the concentrations of the components forming the solid and liquid phases, respectively. This model is transformed to a fixed computational domain and is discretized by an explicit monotone scheme along with an alternative semi-implicit variant. The semi-implicit variant is based on solving, during each time step, a system of nonlinear equations for the total solids concentration followed by the solution of linear systems for the solid component percentages and liquid component concentrations. It is demonstrated that the semi-implicit scheme is well posed and that both variants produce approximations that satisfy an invariant region principle: solids concentrations are nonnegative and less or equal to a set maximal one, percentages are nonnegative and sum up to one, and substrate concentrations are nonnegative. These properties are achieved under a Courant-Friedrichs-Lewy (CFL) condition that is less restrictive for the semi-implicit than for the explicit variant. Numerical examples with realistic parameters illustrate that as a consequence, the semi-implicit variant is more efficient than the explicit one.
期刊介绍:
The journal publishes original research and survey papers in the area of Probability and Statistics. It covers theoretical and practical aspects, in any field of these domains.
Of particular interest are methodological developments with application in other scientific areas, for example Biology and Genetics, Information Theory, Finance, Bioinformatics, Random structures and Random graphs, Econometrics, Physics.
Long papers are very welcome.
Indeed, we intend to develop the journal in the direction of applications and to open it to various fields where random mathematical modelling is important. In particular we will call (survey) papers in these areas, in order to make the random community aware of important problems of both theoretical and practical interest. We all know that many recent fascinating developments in Probability and Statistics are coming from "the outside" and we think that ESAIM: P&S should be a good entry point for such exchanges. Of course this does not mean that the journal will be only devoted to practical aspects.