| 1. |
- Zambrano, Jesús, et al.
(författare)
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Optimal steady-state design of zone volumes of bioreactors with Monod growth kinetics
- 2015
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Ingår i: Biochemical Engineering Journal. - : Elsevier BV. - 1369-703X .- 1873-295X. ; 100, s. 59-66
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Tidskriftsartikel (refereegranskat)abstract
- This paper deals with steady-state analysis and design of bioreactors consisting of a number of completely stirred tank reactors (CSTRs) in series. The study is confined to one consumed (substrate) and one consuming constituent (biomass). The specific microbial growth rate is assumed to be described by Monod kinetics. The death of biomass is assumed to be negligible. Two optimal design problems for a large number of CSTRs in series are studied: to minimize the effluent substrate concentration for a given total volume, and to minimize the total volume for a given effluent substrate concentration. As an appealing alternative to solve these problems numerically, it is proposed to consider the asymptotic case where the number of CSTRs tends to infinity. This is shown to correspond to one CSTR in series with a plug flow reactor (PFR). A CSTR with a sufficient large volume is needed to avoid wash-out of the biomass. The main result is that both design problems for the CSTR + PFR configuration have the same solution with respect to the optimal volume of the CSTR, which is given as an explicit function of the incoming substrate concentration, the volumetric flow rate and the coefficients of the Monod growth rate function. Numerical results indicate that the plug flow approach may be used as a feasible design procedure even for a reasonably low number of CSTRs in series.
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| 2. |
- 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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| 3. |
- 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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| 4. |
- 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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| 5. |
- 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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| 6. |
- Bürger, Raimund, et al.
(författare)
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A moving-boundary model of reactive settling in wastewater treatment. Part 1 : Governing equations
- 2022
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Ingår i: Applied Mathematical Modelling. - : Elsevier BV. - 0307-904X. ; 106, s. 390-401
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Tidskriftsartikel (refereegranskat)abstract
- Reactive settling is the process of sedimentation of small solid particles in a fluid with simultaneous reactions between the components of the solid and liquid phases. This process is important in sequencing batch reactors (SBRs) in wastewater treatment plants. In that application the particles are biomass (bacteria; activated sludge) and the liquid contains substrates (nitrogen, phosphorus) to be removed through reactions with the biomass. The operation of an SBR in cycles of consecutive fill, react, settle, draw, and idle stages is modelled by a system of spatially one-dimensional, nonlinear, strongly degenerate parabolic convection-diffusion-reaction equations. This system is coupled via conditions of mass conservation to transport equations on a half line, whose origin is located at a moving boundary and that model the effluent pipe. An invariant-region-preserving finite difference scheme is used to simulate operating cycles and the denitrification process within an SBR.
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| 7. |
- Burger, Raimund, et al.
(författare)
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Convexity-preserving flux identification for scalar conservation laws modelling sedimentation
- 2013
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Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 29:4
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Tidskriftsartikel (refereegranskat)abstract
- 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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| 8. |
- Bürger, Raimund, et al.
(författare)
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On reliable and unreliable numerical methods for the simulation of secondary settling tanks in wastewater treatment
- 2012
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Ingår i: Computers & Chemical Engineering. - : Elsevier BV. - 1873-4375 .- 0098-1354. ; 41, s. 93-105
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Tidskriftsartikel (refereegranskat)abstract
- Abstract in UndeterminedA one-dimensional model for the sedimentation-compression-dispersion process in the secondary settling tank can be expressed as a nonlinear strongly degenerate parabolic partial differential equation (PDE), which has coefficients with spatial discontinuities. Reliable numerical methods for simulation produce approximate solutions that converge to the physically relevant solution of the PDE as the discretization is refined. We focus on two such methods and assess their performance via simulations for two scenarios. One method is provably convergent and is used as a reference method. The other method is less efficient in reducing numerical errors, but faster and more easily implemented. Furthermore, we demonstrate some pitfalls when deriving numerical methods for this type of PDE and can thereby rule out certain methods as unsuitable; among others, the wide-spread Takacs method. (c) 2012 Elsevier Ltd. All rights reserved.
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| 9. |
- Bürger, Raimund, et al.
(författare)
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On time discretizations for the batch simulation of the settling-compression process in one dimension
- 2016
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Ingår i: Water Science and Technology. - : IWA Publishing. - 0273-1223 .- 1996-9732. ; 73:5, s. 1010-1017
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Tidskriftsartikel (refereegranskat)abstract
- The main purpose of the recently introduced Bürger-Diehl simulation model for secondary settling tanks was to resolve spatial discretization problems when both hindered settling and the phenomena of compression and dispersion are included. Straightforward time integration unfortunately means long computational times. The next step in the development is to introduce and investigate time-integration methods for more efficient simulations, but where other aspects such as implementation complexity and robustness are equally considered. This is done for batch settling simulations. The key findings are partly a new time-discretization method and partly its comparison with other specially tailored and standard methods. Several advantages and disadvantages for each method are given. One conclusion is that the new linearly implicit method is easier to implement than another one (semi-implicit method), but less efficient based on two types of batch sedimentation tests.
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| 10. |
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