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1

A multi-model approach to Saint-Venant equations: A stability study by LMIs

Dos Santos Martins V., Rodrigues M., Diagne M.

International Journal of Applied Mathematics and Computer Science

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2012

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Vol. 22, no. 3

539-550

EN

This paper deals with the stability study of the nonlinear Saint-Venant Partial Differential Equation (PDE). The proposed approach is based on the multi-model concept which takes into account some Linear Time Invariant (LTI) models defined around a set of operating points. This method allows describing the dynamics of this nonlinear system in an infinite dimensional space over a wide operating range. A stability analysis of the nonlinear Saint-Venant PDE is proposed both by using Linear Matrix Inequalities (LMIs) and an Internal Model Boundary Control (IMBC) structure. The method is applied both in simulations and real experiments through a microchannel, illustrating thus the theoretical results developed in the paper.

2

Modelowanie wpływu napełnienia kolektora kanalizacji grawitacyjnej na przebieg tlenowych procesów biodegradacji ścieków

Łagód G., Sobczuk H.

Proceedings of ECOpole

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2007

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Vol. 1, No. 1/2

181--186

PL

W grawitacyjnych systemach kanalizacyjnych zachodzą procesy zarówno fizyczne, chemiczne, jak i biologiczne. Biodegradacja ścieków, prowadząca do rzeczywistego ubytku ładunku zanieczyszczeń podczas ich przepływu w kanalizacji, jest ważnym procesem zmieniającym ilość i jakość niesionych zanieczyszczeń. Stąd też kolektor grawitacyjny powinien być traktowany zarówno jako reaktor biologiczny, jak i urządzenie do zbierania i transportu ścieków. W prezentowanej pracy proces biodegradacji ścieków opisano za pomocą modelu matematycznego wzrostu i rozwoju populacji mikroorganizmów, który stanowi człon źródłowy w równaniu adwekcji-dyspersji. Parametry hydrodynamiczne systemu kanalizacyjnego wykorzystywane w symulacjach są obliczane za pomocą równania Saint-Venanta. Prezentowany model może być pomocny podczas określania dynamiki zmian ładunków zanieczyszczeń dopływających do oczyszczalni poprzez kolektory systemu kanalizacyjnego oraz prognozowania oddziaływania przelewów burzowych na wody odbiornika.

EN

The interceptor of urban wastewater should be treated as a collector and transporter of sewage and also as a bioreactor. with a continuous inflow, growth and washing out of biomass. Specific sewage biodégradation processes were described by suitable mathematical models of biomass growth and decay. For presented system it is possible to compose the matrix of integrated process of organic substance transformation in the gravity sewer system. Numerical model based on described processes contains stoichiometric and kinetic parameters of sewage biodégradation appropriate to living microfauna of saprobionts as a biological processing factor in sewer pipe and a precursor of activated sewage sludge in WWTP. Complete numerical implementation of a model includes also a module of sewer channel hydrodynamic calculation based on Saint-Venant equation.As a last part of necessary modules adveclion-dispersion equation is used. This kind of model, makes it possible to demonstrate the dynamics of pollutants load change delivered to the wastewater treatment plant through interceptor of a sewage system. Also it can be used to predict influence of combined sewer overflows on receiving waters. This paper, basing on the previous achievements is a case study to create a model describing the process of biodegradation of urban sewage running in gravity sewer in the presence of saprobiontic microfauna.

3

Finite-volume solvers for a multilayer Saint-Venant system

Audusse E., Bristeau M. O.

International Journal of Applied Mathematics and Computer Science

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2007

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Vol. 17, no 3

311-320

EN

We consider the numerical investigation of two hyperbolic shallow water models. We focus on the treatment of the hyperbolic part. We first recall some efficient finite volume solvers for the classical Saint-Venant system. Then we study their extensions to a new multilayer Saint-Venant system. Finally, we use a kinetic solver to perform some numerical tests which prove that the 2D multilayer Saint-Venant system is a relevant alternative to 3D hydrostatic Navier-Stokes equations.

4

The Stability of an Irrigation Canal System

Bounit H.

International Journal of Applied Mathematics and Computer Science

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2003

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Vol. 13, no 4

453-468

EN

In this paper we examine the stability of an irrigation canal system. The system considered is a single reach of an irrigation canal which is derived from Saint-Venant's equations. It is modelled as a system of nonlinear partial differential equations which is then linearized. The linearized system consists of hyperbolic partial differential equations. Both the control and observation operators are unbounded but admissible. From the theory of symmetric hyperbolic systems, we derive the exponential (or internal) stability of the semigroup underlying the system. Next, we compute explicitly the transfer functions of the system and we show that the input-output (or external) stability holds. Finally, we prove that the system is regular in the sense of (Weiss, 1994) and give various properties related to its transfer functions.