Control of continuous sedimentation of ideal suspensions as an initial and boundary value problem

Bustos, María Cristina; Paiva, Freddy; Wendland, Wolfgang L.

doi:10.1002/mma.1670120607

Cited by 39 publications

(32 citation statements)

References 17 publications

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“…An approximation of t 3 is obtained by (22). Numerical values corresponding to Figure 26 and obtained by the formulae are A numerical simulation is shown in Figure 28.…”

Section: Step Response Asmentioning

confidence: 99%

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Operating Charts for Continuous Sedimentation II: Step Responses

Diehl

2005

J Eng Math

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Abstract. The process of continuous sedimentation of particles in a liquid has often been predicted by means of operating charts and mass-balance considerations, where the underlying constitutive assumption is the one by Kynch. Much more complex operating charts (concentration-flux diagrams) can be obtained from a onedimensional model of an ideal continuous clarifier-thickener unit. The engineering concept of 'optimal operation' is defined generally as a special type of solution of the model equation, which is a conservation law with a source term and a space-discontinuous flux function. All qualitatively different step responses (with the unit initially in optimal operation in steady state) are presented and classified in terms of operating charts. Quantitative information relating several interesting variables are also presented concerning, for example, the time until overflow occurs as a function of the feed concentration and flux.

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“…An approximation of t 3 is obtained by (22). Numerical values corresponding to Figure 26 and obtained by the formulae are A numerical simulation is shown in Figure 28.…”

Section: Step Response Asmentioning

confidence: 99%

“…8.5], [18], [19,Ch. 6.7], [20][21][22][23]. The problem of giving a satisfactory treatment of the behaviour around the inlet and outlets, respectively, was not solved until the 1990's, see [24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

Operating Charts for Continuous Sedimentation II: Step Responses

Diehl

2005

J Eng Math

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“…Repeating the above procedure, by a finite number of steps (because, from (A 4 ), it follows that a(v (2) (x, t 2 )) has a finite number of inflection points), we can construct the global weak entropy solution of (1.1). Case (II) A shock wave x = X 0 (t) emanates at the point (0, t 0 ) in the x-t plane, with non-negative original speed for the problem (2.3).…”

Section: Lemma 22 If U(x T) Is a Weak Entropy Solution Ofmentioning

confidence: 99%

“…The initial boundary value problem of scalar conservation laws plays an important roles in the mathematical modelling and simulation of the practical problem of the one-dimensional sedimentation processes and traffic flow on highways [1][2][3][4][5]. Bardos et al [6] established the existence and uniqueness of the weak entropy solution in the BV-setting for the initial boundary value problems of scalar conservation laws, respectively, by the vanishing viscosity method and by Kruzkov's method [7] (about proving the existence of the solution of the initial value problem for conservation laws by the vanishing viscosity method, see also [8][9][10] etc.).…”

Section: Introductionmentioning

confidence: 99%

Construction of Solutions and L 1–error Estimates of Viscous Methods for Scalar Conservation Laws with Boundary

Liu

Pan

2005

Acta Math Sinica

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This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. By the structure of the weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux-Nedelec, we give a construction method to the weak entropy solution of the initial boundary value problem. Compared with the initial value problem, the weak entropy solution of the initial boundary value problem includes the following new interaction type: an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary. According to the structure and some global estimates of the weak entropy solution, we derive the global L 1 -error estimate for viscous methods to this initial boundary value problem by using the matching travelling wave solutions method. If the inviscid solution includes the interaction that an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary, or the inviscid solution includes some shock wave which is tangent to the boundary, then the error of the viscosity solution to the inviscid solution is bounded by O(ε 1/2 ) in L 1 -norm; otherwise, as in the initial value problem, the L 1 -error bound is O(ε| ln ε|).

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“…We refer to the previous papers in this series [1,2] for references and discussions of previous works. For the particular problem of controlling the process, different angles of approach can be found in [3][4][5][6][7][8][9].…”

Section: Introductionmentioning

confidence: 99%

Operating Charts for Continuous Sedimentation III: Control of Step Inputs

Diehl

2006

J Eng Math

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Abstract. The main purposes of a clarifier-thickener unit is that it should produce a high underflow concentration and a zero effluent concentration. The main difficulty in the control of the clarification-thickening process (by adjusting a volume flow) is that it is non-linear with complex relations between concentrations and volume flows via the solution of a PDE -a conservation law with a source term and a space-discontinuous flux function. In order to approach this problem, control objectives for dynamic operation and strategies on how to meet these objectives are presented in the case when the clarifier-thickener unit initially is in steady state in optimal operation and is subjected to step input data. A complete classification of such solutions is given by means of an operating chart (concentration-flux diagram).

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Control of continuous sedimentation of ideal suspensions as an initial and boundary value problem

Abstract: We present the control of continuous sedimentation in an ideal thickener as an initial and boundary value problem and construct the entropy solution. ,

Cited by 39 publications

References 17 publications

Operating Charts for Continuous Sedimentation II: Step Responses

Operating Charts for Continuous Sedimentation II: Step Responses

Construction of Solutions and L 1–error Estimates of Viscous Methods for Scalar Conservation Laws with Boundary

Operating Charts for Continuous Sedimentation III: Control of Step Inputs

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