Finite element modelling of the thermal outflow of three power plants in Huelva Estuary
Introduction
Man is accustomed to use the waters of the earth as sinks for waste products and has traditionally relied solely on dispersion of these pollutants in the fluid receptors to reduce the degree of concentration. The quantity of matter added to a fluid medium must be under tolerable levels and certain standards, but the increasing amount of contaminants has brought this practice into dispute and higher standards of water management have to be implemented.
Among the bodies of water, waterways such as rivers are initial receptors of dissolved and suspended materials from their watersheds. Wastewater discharges can have a dramatic impact on a river since they contain organic materials, nutrients and a variety of contaminants harmful to aquatic life. In order to minimize the environmental impact, extensive studies must be undertaken to ensure that pollutant dispersion complies with the acceptable levels.
This paper evaluates the environmental repercussions of the discharges of some power plants which might be constructed in Huelva estuary. The study involves the numerical simulation of the dispersion of the thermal outflow, now considered to be the contaminant, which is governed by the convection–diffusion equation.
The methods most commonly used in simulation models are those of finite differences and finite elements. Weighing on the advantages of finite elements over finite differences, the choice now rests on the finite element method more adapted to the present problem. It is common knowledge that the solutions of convection-dominated transport problems through Galerkin methods may produce node-to-node oscillations [1]. These problems can be resolved by mesh (and time step) refinement, which undermines the practical utility of the method. This has led to the development of alternatives to the standard Galerkin formulation which prevent oscillations without requiring mesh or time-step refinement. These are called stabilization techniques and consist of the addition of a stabilizing term to the original Galerkin formulation of the problem [2], [3], [4], [5], [6], [7].
Due to its accuracy, stability and convergence properties, the simple explicit Characteristic-Galerkin method [8], [9] was chosen to solve the convection–diffusion equation. Its derivation involves a local Taylor expansion of the convection–diffusion equation. This method has the required properties which include the introduction of a stabilizing term that stabilizes the convective oscillations. This methodology results in a smaller computational cost, since through the use of an explicit method it requires only the solution of a system of linear equations in every time step with a mass matrix, which is symmetric and positive definite and constant throughout the integration. Moreover, this matrix can be “lumped” resulting in a fast resolution of the problem. Several difficulties are avoided through the use of this method but at the expense of conditional stability, which has a small effect on the practical application of the model.
Section snippets
Convection–diffusion equation
Upon introduction of the contaminant into the fluid receptor, the dilution of the concentration of the plume is dependent on the basic transport processes such as advection and diffusion. To incorporate the transformation processes into the convection-equation, it is necessary to formulate the basic mass conservation law of the material volume under consideration and its time variation. In considering the variations, it is assumed that the rate of dilution of the contaminants is proportional to
Temporal discretization
An appropriate algorithm for the numerical time integration must be selected since phase accuracy is an important consideration for transient problems. The method starts from a particular time discretization of the differential equation before obtaining the weak form. The main behavior patterns of Eq. (2) can be determined by a change of the independent variable x to x′ such thatThe coordinate system of Eq. (8) describes characteristic directions. The convective terms of Eq. (2)
Application to the Huelva Estuary
The code SEASCAPE was applied to the study of the thermal outflow dispersion of some power plants which might be constructed in Huelva estuary. The water pumped from the river will serve as cooling agent for these power plants and will be discharged later as wastewater. Fig. 1 shows a location map of Huelva estuary with the proposed locations of the power plants, namely: Endesa, Union Fenosa and Energia de Huelva S.L. The estuary is located in the Andalusian region of southwestern Spain which
Numerical results
After the modelling parameters have been determined, program executions of the numerical model are performed to simulate seven cases of thermal outflow discharge. Table 1 shows the cases wherein each case varies according to the number of power plants actively discharging wastewater and the number of working groups for each power plant. Union Fenosa, in its full working capacity, will have three discharge groups. At the time of performing the simulations, only one group of Union Fenosa was
Conclusions
Various finite element methods can be used to solve transport equations. This paper has presented the use of a simple explicit Characteristic-Galerkin method to develop a numerical code named SEASCAPE capable of simulating the dispersion of thermal outflow of power plants. The development of the method starts from a particular explicit time discretization of the differential equation, proceeding then to a quasi-3D spatial discretization which results in a smaller computational cost than a fully
Acknowledgements
This paper is part of the project study “Profundización en el Diagnóstico de la Situación Ambiental del Entorno de la Ria de Huelva” of the Regional Government of Andalucia. The first author is under the auspices of the Agencia Española de Cooperación Internacional.
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