Extended nonequilibrium thermodynamics

Chemical process rate equations involve the quantity related to concentration fluctuations as a kinetic parameter called chemical relaxation. The stochastic theory of chemical kinetics investigates concentration fluctuations (Malyshev, 2005). For diffusion of polymers, flows through porous media, and the description liquid helium, Fick s and Fourier s laws are generally not applicable, since these laws are based on linear flow-force relations. A general formalism with the aim to go beyond the linear flow-force relations is the extended nonequilibrium thermodynamics. Polymer solutions are highly relevant systems for analyses beyond the local equilibrium theory. 

Extended nonequilibrium thermodynamics is concerned with the nonlinear region and deriving the evolution equations with the dissipative flows as independent variables, besides the usual conserved variables. Typical nonequilib-rium variables such as flows and gradients of intensive properties may contribute to the rate of entropy generation. When the relaxation time of these variables differs from the observation time they act as constant parameters. The phenomenon becomes complex when the observation time and the relaxation time are of the same order, and the description of system requires additional variables. 

To coordinate components, the generalized flows and the thermodynamic forces can be used to define the trajectories of the evolution of nonequilibriun systems in time. A trajectory specifies the curve represented by the flow and force components as a function of time in the flow-force space. A useful trajectory can be found and analyzed by a variation principle. In thermodynamics, the variation principles lead to the least energy dissipation and minimum entropy generation at steady states. According to the most general evolutionary criterion, open chemical reaction systems are dissipative, and evolve toward an asymptotic state in time. 

Extended nonequilibrium thermodynamics is not based on the local equilibrium hypothesis, and uses the conserved variables and nonconserved dissipative fluxes as the independent variables to establish evolution equations for the dissipative fluxes satisfying the second law of thermodynamics. For conservation laws in hydrodynamic systems, the independent variables are the mass density, p, velocity, v, and specific internal energy, u, while the nonconserved variables are the heat flux, shear and bulk viscous pressure, diffusion flux, and electrical flux. For the generalized entropy with the properties of additivity and convex function considered, extended nonequilibrium thermodynamics formulations provide a more complete formulation of transport and rate processes beyond local equilibrium. The formulations can relate microscopic phenomena to a macroscopic thermodynamic interpretation by deriving the generalized transport laws expressed in terms of the generalized frequency and wave-vector-dependent transport coefficients. 

Extended nonequilibrium thermodynamics theory is often applied to flowing polymer solutions. This theory includes relevant fluxes and additional independent variables in describing the flowing polymer solutions. Other contemporary thermodynamic approaches for this problem are GENERIC formalism, matrix method, and internal variables (Jou and Casas-Vazquez, 2001), which are summarized in the following sections. 

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The following summary is from Jou and Casas-Vazquez (2001). In the extended nonequilibrium thermodynamics for a binary liquid mixture, the viscous pressure tensor Pv and the diffusion flux J are considered as additional independent variables. The viscous pressure tensor, Pv, by the simplest Maxwell model, is defined by the following constitutive equation ... 

In extended nonequilibrium thermodynamics of polymer solutions, the generalized extended Gibbs equation for a fluid characterized by internal energy U and viscous pressure Pv is... 

Some processes may have forces operating far away from equilibrium where the linear phenomenological equations are no longer applicable. Such a domain of irreversible phenomena, such as some chemical reactions, periodic oscillations, and bifurcation, is examined by extended nonequilibrium thermodynamics. Extending the methods of thermodynamics to treat the linear and nonlinear phenomena, and such dissipative structures are attracting scientists from various disciplines. 

This book introduces the theory of nonequilibrium thermodynamics and its use in transport and rate processes of physical and biological systems. The first chapter briefly presents the equilibrium thermodynamics. In the second chapter, the transport and rate processes have been summarized. The rest of the book covers the theory of nonequilibrium thermodynamics, dissipation function, and various applications based on linear nonequilibrium thermodynamics. Extended nonequilibrium thermodynamics is briefly covered. All the parts of the book can be used for senior- and graduate-level teaching in engineering and science. 

In the extended nonequilibrium thermodynamics, the fluxes are the essential independent variables of the system. Consider a system subject to a heat flux q in non-equilibrium steady states The characteristic residence time of the energy in the system is of the order of Ul aq), where U is the overall heat transfer coefficient, aq the total heat supplied to the system (and leaving the system) per unit time and for a given total area a. Then, if the heat flux is high enough, the energy residence time will be of the order, or even shorter and the... 

Often extended nonequilibrium thermodynamics with maximum-entropy formalism leads to more general expression for the entropy not limited to second order in fluxes. 

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