A comprehensive non-equilibrium thermodynamic analysis applied to a vapor–liquid two-phase flow of a pure fluid Rémi Revellin a,b,⇑ , Stéphane Lips a,b , Pierre Neveu c , Jocelyn Bonjour a,b a Université de Lyon, CNRS, France b INSA-Lyon, CETHIL, CNRS, UMR5008, F-69621 Villeurbanne, France c Processes, Materials, and Solar Energy Laboratory (PROMES–CNRS, UPR 8521), 7 Rue du Four Solaire, 66120 Font-Romeu, France article info Article history: Received 16 September 2011 Received in revised form 7 February 2012 Accepted 18 February 2012 Available online 27 February 2012 Keywords: Two-phase flow Non-equilibrium thermodynamics Vapor quality Entropy Exergy abstract In this study, a comprehensive thermodynamic 1-D analysis applied to a steady state two-phase flow of a pure fluid is proposed taking into account the non-equilibrium state between the phases and the capillary work. Different equations explicitly expressing the first and the second law of thermodynamics are pre- sented. The role of the specific heat capacities at constant vapor quality is emphasized. In addition, it is shown that the capillary work can always be neglected in the calculation of the variation of the vapor quality whereas the flashing effect must be taken into account under certain conditions. Furthermore, the role of the vapor enthalpy variation is highlighted in the definition of the elementary variation of the vapor quality at a non-equilibrium state between the phases (dryout conditions). Finally, from an entropy analysis, the mean two-phase flow temperature under non-equilibrium conditions applicable in the Newton law has been proposed. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction In this paper, a comprehensive thermodynamic 1-D analysis ap- plied to a vapor–liquid two-phase flow of a pure fluid is proposed taking into account the non-equilibrium state between the phases. Different equations explicitly expressing the first and the second law of thermodynamics are presented. We particularly focus on the elementary variation of the vapor flow quality, the entropy generation and the exergy variation. These equations are devel- oped with a minimum of assumptions (which will be listed later in the paper) in order to remain as general as possible. Especially, the capillary work per unit time is taken into account. The motiva- tions for proposing such equations are the following:  Most of the papers in the literature on vapor–liquid two-phase flow base their reasoning on energy balance and on the well- known vapor flow quality (see any two-phase heat transfer or pressure drop models). Most of the time, whatever the condi- tions (geometry, heat flux, mass velocity...), equations are based on the same assumptions (among which negligible accel- eration of the phases, flashing effect or capillary work). Histor- ically, first flow boiling or condensation tests were performed in macrochannel tubes (around 10 mm or more). Capillary work, flashing effect and phase acceleration are negligible in such con- ditions. Nevertheless, in microchannels flashing effect is not necessarily negligible (Revellin et al., 2009b) and capillary work may be significant. Many other examples may be found in the literature where the energy balance (first law) is not written with justifiable assumptions. This paper aims at proposing more general equations that could also be used for data reduction when performing experiments. These equations must be easy to apply. This is the reason why a 1D analysis is considered here.  Working at non-equilibrium state between the phases offers the possibility of treating the mist flow regime or the dryout condi- tions for which the vapor is superheated and the liquid at satu- ration. Various relations have been proposed so far to define the variation of the vapor flow quality for non-equilibrium state between the phases (Chen et al., 1979) but we demonstrate in this paper that these empirical or intuitive definitions were not thermodynamically accurate. Bilicki et al. (2002) proposed a theoretical model for two-phase flow characterization by the method of irreversible thermodynamics in both classical (CIT: each phase is at equilibrium but exhibits a temperature difference, as a result there is a non-equilibrium state between the phases) and extended (EIT: equations include dissipative fluxes and reduce to the classical constitutive laws in the limit of slow phenomena) formulations. The work by Bilicki et al. is probably the most comprehensive available in the literature but it was restricted to a 1-D homogeneous model in the absence of capillary work. We see the importance of developing 1-D simple equations with a minimum of assumptions 0301-9322/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmultiphaseflow.2012.02.008 ⇑ Corresponding author at: CETHIL, UMR5008, INSA-Lyon, Bât Sadi Carnot, 9 Rue de la Physique, 69621 Villeurbanne cedex, France. Tel.: +33 4 72 43 72 31; fax: +33 4 72 43 88 10. E-mail address: remi.revellin@insa-lyon.fr (R. Revellin). International Journal of Multiphase Flow 42 (2012) 184–193 Contents lists available at SciVerse ScienceDirect International Journal of Multiphase Flow journal homepage: www.elsevier.com/locate/ijmulflow