Systems & Control Letters 62 (2013) 170–177 Contents lists available at SciVerse ScienceDirect Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle On an evolution criterion of homogeneous multi-component mixtures with chemical transformation N. Ha Hoang a,b,∗ , Denis Dochain b a Faculty of Chemical Engineering, University of Technology, VNU-HCM, 268 Ly Thuong Kiet Str., Dist. 10, HCM City, Viet Nam b CESAME, Université catholique de Louvain, 4-6 avenue G. Lemaitre, B-1348, Louvain-la-Neuve, Belgium article info Article history: Received 20 June 2012 Received in revised form 15 November 2012 Accepted 22 November 2012 Available online 29 December 2012 Keywords: Irreversible thermodynamics Non-isothermal CSTR Chemical reaction network Passivity abstract In this paper, a thermodynamically stable evolution criterion for homogeneous multi-component mixtures with chemical transformation is proposed. The approach is motivated and governed by physical considerations strongly related to the second law of the thermodynamics. More precisely we show that there exists some potential function directly defined on the space of the extensive and/or intensive variables for any transformation, and that meets the evolution criterion without any restriction on the chemical reaction kinetics. As a consequence the irreversibility degree or passivity of the mixture under mass transfer and transport phenomena is explicitly expressed. Some numerical simulations for a homogeneous binary mixture with chemical reaction under multiplicity are given to validate our theoretical developments. © 2012 Elsevier B.V. All rights reserved. 1. Introduction In chemical engineering, reaction systems belong to general thermodynamic systems [1]. This notion covers a large class of non- linear dynamical systems where the matter transformation and transport phenomena play a central role, particularly in chemical systems in which the reactants react to give products [2]. Indeed on the one hand the evolution of the system states (such as tem- perature and concentrations of species) is directly linked to the en- ergy and entropy transformations. As a consequence the first and second laws of thermodynamics allow us to predict the evolution of the system states [3–6]. On the other hand, typical phenom- ena (such as heat and mass transfers, and reaction kinetics) can be explained and modelled by thermodynamics [7]. Fig. 1 repre- sents an open reaction system defined by its physical volume V and surrounding surface Ω. Tubular reactors and reactive distillation columns are indeed typical examples of open reaction systems. Let us note that in contrast with mechanical and electrical systems where connections between energy and dynamical behaviour are today fairly well understood [8,9]. For such systems there exists a potential function connected to the dynamics that should be decreasing along the stable trajectories. Links between physics and process dynamics for stability analysis are ∗ Corresponding author at: CESAME, Université catholique de Louvain, 4-6 avenue G. Lemaitre, B-1348, Louvain-la-Neuve, Belgium. Tel.: +32 10 478041; fax: +32 10 472180. E-mail addresses: ha.hoang@uclouvain.be, ngocha.h@gmail.com (N.H. Hoang), denis.dochain@uclouvain.be (D. Dochain). typically restricted to isothermal conditions [10,11] or adiabatic operations [12] or close to the equilibrium state [4,13,14,5]. These links are quite difficult to exhibit in a geometrical framework [15–17] in general. Indeed chemical reaction systems, and in particular the reference case study well known as the Continuous Stirred Tank Reactor (CSTR), belong to highly nonlinear non- equilibrium thermodynamic systems via reaction kinetics and irreversibilities of the coupling between matter and temperature. Following the law of conservation of energy, the total energy (the energies of the simple system under consideration and its surrounding medium) is conserved. Consequently the internal energy, that is considered from a microscopic point of view, as the sum of kinetic and potential energies of all molecules inside the system, is then not dissipated during chemical reaction but is modified by exchanges (material and heat flows as well as volume expansion for gas phase reactions) affected at the system boundary as stated by the first principle of thermodynamics. As a consequence the variation of the internal energy is an impact only on the spatial order, molecular arrangements and chemical structure of chemical species. Indeed it has been shown in [18] about the geometric aspects on the basic of structure matrices of the simple Hamiltonian formulation, 1 that the internal energy cannot be considered as the storage potential function (as shown in 1 Recently, the formulation of the thermodynamic properties using contact geometry by the so-called Thermodynamic Phase Space which generalizes port Hamiltonian systems to port contact systems has then been proposed to represent simultaneously the energy conservation and the entropy production of irreversible processes [19]. This can also be obtained within the GENERIC framework [20]. 0167-6911/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2012.11.013