Chemical Engineering Science 59 (2004) 109–118 www.elsevier.com/locate/ces Irreversible thermodynamics—a tool to describe phase transitions far from global equilibrium Dick Bedeaux * , Signe Kjelstrup Department of Chemistry, Norwegian University of Science and Technology, Trondheim 7491, Norway Received 21 May 2003; received in revised form 17 September 2003; accepted 23 September 2003 Abstract In the study of multi-component mass transfer it is common to use the lm model, in which all the resistance to mass transfer across a phase boundary is assumed to be localized in two lms on both sides of the interface. At the phase boundary one assumes that the temperature and chemical potentials are continuous, i.e. there is equilibrium across the interface. The coupling of a possible heat ux to the mass uxes across the interface is neglected. Using irreversible thermodynamics for heterogeneous systems, we show how to write ux equations for heat and mass transfer across these lms, but without these assumptions. The system as a whole can be very far from global equilibrium, but there is local equilibrium everywhere, including in the interface. The interface is introduced as a separate thermodynamic system between other phases. Irreversible thermodynamics then gives a consistent and complete description of phase transitions in systems that are in local equilibrium, but are far from global equilibrium. The interface features as an “additional lm” between the two lms in the phases on both sides of the interface. We present how one can systematically set up the complete description, in which the three “lms” sum to one eective lm. The practical use of the description is indicated. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Irreversible thermodynamics; Multi-component mass transfer; Film model; Gibbs surface; Interface transport 1. Introduction Phase transitions are central in many mechanical and chemical engineering processes, see Taylor and Krishna (1993). Take liquefaction of natural gas and distillation as examples, to mention just two. Phase transitions take place without global equilibrium in the system. RHsjorde et al. (2001) and Olivier et al. (2002) stud- ied phase transitions by nonequilibrium molecular dynam- ics (i.e. NEMD). This method (see Hafskjold, 2002, for a review) can deal with systems that are very far from global equilibrium, in one system the gradient in temperature across the system was 10 8 K= m(RHsjorde et al., 2000). An exam- ple of a liquid and vapor in a temperature gradient, gener- ated by NEMD simulations is shown in Fig. 1. Confronted with the problem of having to describe uxes of mass and heat, in industrial or other systems, one may rightly ask: Do the normal thermodynamic relations for equi- librium apply, and if they apply, how do we dene the * Corresponding author. Tel.: +47-73594178; fax: +47-73591676. E-mail address: bedeaux@phys.chem.ntnu.no (D. Bedeaux). variables? How do we dene the interface for the system in Fig. 1, and how do we integrate across it? What are the proper or most practical coecients of transport to use? Phase transitions in industrial systems have since long been described by the lm model (see Taylor and Krishna, 1993). For NEMD systems, there has been no systematic procedure, until recently (RHsjorde et al., 2001). In this work, we shall present a thermodynamic method that can be regarded as a generalization of the lm model, to be used with industrial as well as simulated systems with large gradi- ents. We shall assume that there is local equilibrium every- where in the system, in agreement with the surprising nding of RHsjorde et al. (2000) and of Johannessen and Bedeaux (2003), that this assumption also applies to the interface. The theory that governs the transport phenomena is ir- reversible thermodynamics. This theory was written for homogeneous systems (de Groot and Mazur 1984; Kuiken, 1994; Taylor and Krishna, 1993; Kjelstrup and Bedeaux, 2001), and was further developed for heterogeneous systems by Bedeaux et al. (1976), Bedeaux (1986) and Albano and Bedeaux (1987). We shall see, that this theory allows one to integrate through the interface, without assuming 0009-2509/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2003.09.028