REVIEW ARTICLES Nonequilibrium Thermodynamics in Engineering and Science Yas ¸ ar Demirel* Department of Chemical Engineering, Virginia Polytechnic Institute and State UniVersity, Blacksburg, Virginia 24061 Stanley I. Sandler Center for Molecular and Engineering Thermodynamics, Department of Chemical Engineering, UniVersity of Delaware, Newark, Delaware 19716 ReceiVed: March 31, 2003; In Final Form: August 5, 2003 The field of nonequilibrium thermodynamics has been a popular one outside the United States, especially in Europe, and scientists there from various disciplines have published extensively on the use of nonequilibrium thermodynamics in a large variety of biological, chemical, and mechanical engineering applications. In contrast, the number of publications from North America has been relatively modest. Here, we review the field of nonequilibrium thermodynamics to assess its utility and impact in engineering and science. We find that developments in the fields thermodynamic optimality of processes, dissipative structures, coupled transport and rate processes, and biological systems suggest that in some circumstances nonequilibrium thermodynamics can be quite useful. Introduction The application of thermodynamics to nonequilibrium pro- cesses takes many forms. The simplest is based on the assumption of local equilibrium. There it is assumed that even though the system in which a process is occurring is not at global equilibrium, in any small region the thermodynamic properties are related to the state variables in the same manner as in equilibrium. For this assumption to be valid, the internal relaxation processes in the fluid or material must be much faster than the rate of change imposed upon the state variables. The local equilibrium concept is valid for a wide range of macro- scopic systems 1-7 of usual gases and liquids and for most transport processes and chemical reactions where the reactive collision rates are relatively smaller than overall collision rates. 6 It is not valid in highly rarefied gases where collisions are too infrequent. The extension of equilibrium thermodynamics to nonequilibrium systems with the local equilibrium assumption is a well-accepted form of nonequilibrium thermodynamics (NET). 1-8 Such an extension is possible in terms of entropy density, s[T(x),N k (x)], which is a function of the temperature and the mole number densities at location x, when a well-defined local temperature T(x) exists. Consequently, the total entropy and energy can be obtained from the integrals over the volume of the system: S ) ∫ V s[T(x),N k (x)] dV, and U ) ∫ V u[T(x),N k - (x)] dV. 6 From the internal energy density u(x), we obtain the local variables of (∂s/∂u) Nk ) 1/T(x) and (∂s/∂N k ) u )-µ(x)/ T(x). Another well-known application of thermodynamics to non- equilibrium processes is the so-called second law analysis, or availability or exergy analysis, of real processes. Such an analysis is based on the Gouy-Stodola theorem, which states that the lost available energy is directly proportional to the entropy production due to irreversibility in a process. This provides a quantitative measure of irreversibility through the level of entropy production, which by the second law of thermodynamics is always positive. The level of entropy production can be used as one criterion of the optimality of a process, mainly related to reducing the irreversibilities to control the dissipation of useful power and the depletion of natural resources. 9 However, as an availability or exergy analysis does not involve a financial analysis or consideration of materials of construction or other constraints, the process that has the highest second law efficiency may not be the most economical. Indeed, if the highest second law efficiency was all that needed to be considered, all motive power would be supplied by Carnot engines. Nonetheless, a second law analysis can be useful in efficient energy utilization and engineering design, 10-11 and consequently simulation packages such as recent versions of ASPEN PLUS provide methods of estimating second law efficiency and an exergy analysis for engineering process integration and optimization of, for example, distillation col- umns, heat exchangers, and chemical reactors. The next type of nonequilibrium thermodynamic description, and the one that is of more interest to us here, is based on the observation that the level of irreversibility of any step in a process is, as we shall discuss, related to its distance from global equilibrium; this distance may be treated as a parameter of the process. 6 Small values of this parameter result in processes in which there are linear relations between the driving forces or gradients in the system and the fluxes that result; examples are Fourier’s law in heat conduction and Fick’s law in diffusion. Unfortunately, this is sometimes interpreted as that NET is a completely linearized theory and applies only to systems close * Corresponding author. E-mail: ydemirel@vt.edu. Tel: (540)231-2074. 31 J. Phys. Chem. B 2004, 108, 31-43 10.1021/jp030405g CCC: $27.50 © 2004 American Chemical Society Published on Web 12/03/2003