Shock-Wave Behaviour of Sedimentation in Wastewater Treatment: A Rich Problem

Diehl, Stefan

doi:10.1007/978-3-642-20236-0_7

Cited by 6 publications

(7 citation statements)

References 158 publications

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“…We will in specific cases below show that the equations r = R(t; ), = Φ(t; ), define locally a smooth solution = (r, t) (as long as the characteristics do not intersect each other or meet a discontinuity). That such a solution is indeed a smooth solution of (1a) can easily be verified by differentiating the identity Φ(t) = (R(t), t) with respect to t and utilizing (14); then one arrives at the PDE (13). For convenience, we will also denote the characteristic baselines t  → (t, R(t)) by "characteristics."…”

Section: Characteristics In Regions Of Smooth Solutionmentioning

confidence: 97%

Entropy solutions and flux identification of a scalar conservation law modelling centrifugal sedimentation

Careaga

Diehl

2020

Math Meth Appl Sci

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Centrifugal sedimentation of an ideal suspension in a rotating tube or basket can be modelled by an initial‐boundary‐value problem for a scalar conservation law with a nonconvex flux function. The sought unknown is the volume fraction of solids as function of radial distance and time for constant initial data. The method of characteristics is used to construct entropy solutions. The qualitatively different solutions, which depend on the initial value and the vessel radial coordinates, are presented in detail along with numerical simulations. Based on the entropy solutions, a new method of flux identification, which does not require any prescribed functional expression, is presented and illustrated with synthetic data.

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Section: Characteristics In Regions Of Smooth Solutionmentioning

confidence: 97%

Entropy solutions and flux identification of a scalar conservation law modelling centrifugal sedimentation

Careaga

Diehl

2020

Math Meth Appl Sci

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“…An explicit representation for the inverse ⌣ η −1 i requires some preparation. Note that the convexity conditions (25)…”

Section: 2mentioning

confidence: 99%

“…For the estimated functions corresponding to the concave part of the flux, we use the accent ⌢ instead of ⌣. Inequalities (24) and (25) are replaced by ⌢…”

Section: The Inverse Problem Of Estimating a Concave Part Of The Fluxmentioning

confidence: 99%

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Convexity-preserving flux identification for scalar conservation laws modelling sedimentation

Bürger

Diehl

2013

Inverse Problems

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The 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 a problem of theoretical interest and practical importance for the implementation of 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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“…Theories exist in the literature for design of thickeners to achieve a given target solids flux and/or a given target underflow solids fraction Diehl, 2001Diehl, , 2008Diehl, , 2012. Traditionally (Talmage and Fitch, 1955;Fitch, 1966) these have been based around the Kynch theory (Kynch, 1952).…”

Section: Introductionmentioning

confidence: 99%

Designing thickeners by matching hindered settling and gelled suspension zones in the presence of aggregate densification

Zhang

Grassia

Martin

et al. 2015

Chemical Engineering Science

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A model is presented for design of a thickener for solid–liquid separation where the flocs or aggregates within the solid–liquid suspension undergo an aggregate densification process due to the action of rakes. This aggregate densification facilitates suspension dewatering. The novel feature of the model is that it manages to couple together a hindered settling zone (higher up in the thickener, where the flocs are separated from one another, and the suspension cannot bear weight) and a gelled suspension zone (lower down in the thickener, where the flocs are packed together, and the suspension is able to bear weight). The model determines solids fraction profiles throughout the hindered settling zone and the gelled suspension zone, and also gives zone heights and residence times. Parametric investigations using the model are carried out for different suspension fluxes (which influence the solids fluxes and underflow solids fractions attained), and also for different specified amounts of and rates of aggregate densification

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Shock-Wave Behaviour of Sedimentation in Wastewater Treatment: A Rich Problem

Cited by 6 publications

References 158 publications

Entropy solutions and flux identification of a scalar conservation law modelling centrifugal sedimentation

Entropy solutions and flux identification of a scalar conservation law modelling centrifugal sedimentation

Convexity-preserving flux identification for scalar conservation laws modelling sedimentation

Designing thickeners by matching hindered settling and gelled suspension zones in the presence of aggregate densification

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