Unsteady State Fugacity Model By A Dynamic Control System

Applied Mathematical Modelling 22 (1998) 485±494

Unsteady state fugacity model by a dynamic control system
Rafael Bru
a

a,*

, Jos Maria Carrasco b, Lourival Costa Para e õba

a,c

c

 Departament de Matem—tica Aplicada, ETSIA, Universitat Politcnica de Valncia, Cami de Vera, s/n, 46022 Valncia, e e e Spain b Departament de Biotecnologia, ETSEA, Universitat Politcnica deValncia, Valncia, Spain e e e Centro Nacional de Pesquisa de Monitoramento e Avaliac—o de Impacto Ambiental, CNPMA, EMBRAPA, Jaguarina, Ë~ u SP, Brazil Received 11 November 1997; received in revised form 16 April 1998; accepted 2 June 1998

Abstract A continuous time dynamic system of an unsteady state fugacity model is presented. Properties of this model as stability are studied. In order toevaluate numerical results a discretization preserving the stability and yielding the positivity property of the model is used. Finally, algorithms to determine the values of the fugacities, the concentrations and the dissipation time are given. The above study is illustrated with numerical results in a three compartmental environmental system. Ó 1998 Elsevier Science Inc. All rights reserved.Keywords: Fugacity model; Multiphase environmental model; Unsteady state model; Dynamic control systems; Nonnegative control systems; Fenitrothion

1. Introduction The thermodynamic concept of fugacity was introduced in 1910 for Lewis [1] in order to explain the behaviour of the real gases with respect to that of ideal gases, in the study of free energy corresponding to an expansion process,isotherm, reversible and in®nitesimal. The fugacity is a thermodynamic magnitude related to the chemical potential and characterized by the leak trend of a substance in a compartment [2,3]. The fugacity express the chemistry activity of a substance and has been applied mainly in thermodynamic problems implicating equilibrium among phases, especially in computations encountered in chemical separationprocesses such as liquid extraction, distillation and adsorption. Mathematical models based on the thermodynamic theory of the fugacity are outlined frequently by a linear system of equations describing the bulk balance of a chemical substance in an ecosystem constituted by compartments. Thus, when all fugacities are equal and constant in all compartments the concentrations are evaluated directly,this case corresponds to the well-known `Level I Fugacity Model' or `Level II Fugacity Model' if in addition there are reactions and advections (see [4]). `Level III Fugacity Model' supposes that the distribution of the substance is not in equilibrium and that each fugacity can have di€erent values, which are determined by a
*

Corresponding author. Tel.: +34 96 387 7660; fax: +34 96 387 7669;e-mail: rbru@mat.upv.es.

0307-904X/98/$19.00 Ó 1998 Elsevier Science Inc. All rights reserved. PII: S 0 3 0 7 - 9 0 4 X ( 9 8 ) 1 0 0 4 7 - 1

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R. Bru et al. / Appl. Math. Modelling 22 (1998) 485±494

linear system of equations when there are reactions, advections, emissions and transfers of the substance among compartments in stationary state [5,6]. In addition, there is another modeldescribing the unsteady state behaviour of a substance in the environment, which permits to observe substances whose emissions vary with the time and to determine the time in which the system reaches the steady state. This last model, known as `Level IV Fugacity Model', usually is described by a system of di€erential equations (3) (see [7]). In this work, we will present a proposal of themultiphase fugacity environmental `Level IV' model for a ecosystem constituted by n compartments where the fugacities change with the time in response to m emissions and they are determined by a continuous time dynamic control system describing the total bulk balance of the substance. Some non-stationary models for the study of the kinetic of a substance in the environment have been proposed for the...