Non-Equilibrium Thermodynamics for Industry

SIGNE KJELSTRUP, AUDUN R0SJORDE AND EIVIND JOHANNESSEN 

Non-equilibrium thermodynamics (NET) offers a systematic way to derive the local entropy production rate, c, of a system. The total entropy production rate is the integral of the local entropy production rate over the volume, V, of the system, but, in a stationary state, it is also equal to the entropy flux out, J, minus the entropy flux into the system, 

The entropy flux difference and the integral over a can be calculated independently, and they must give the same answer. The entropy production rate governs the transport processes that take place in the system. We have 

This means that NET gives flux equations in agreement with the second law of thermodynamics, and that the theory offers a possibility, through Equation (1), to check for consistency in the models that are used. 

The usefulness of NET in describing industrial problems has been questioned, because these problems are frequently non-linear. It is then important to know that the flux-force relations in Equation (3) also describe non-linear phenomena, The phenomenological coefficients fj can, for instance, be functions of the state variables. By 

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A reaction at steady state is not in equilibrium. Nor is it a closed system, as it is continuously fed by fresh reactants, which keep the entropy lower than it would be at equilibrium. In this case the deviation from equilibrium is described by the rate of entropy increase, dS/dt, also referred to as entropy production. It can be shown that a reaction at steady state possesses a minimum rate of entropy production, and, when perturbed, it will return to this state, which is dictated by the rate at which reactants are fed to the system [R.A. van Santen and J.W. Niemantsverdriet, Chemical Kinetics and Catalysis (1995), Plenum, New York]. Hence, steady states settle for the smallest deviation from equilibrium possible under the given conditions. Steady state reactions in industry satisfy these conditions and are operated in a regime where linear non-equilibrium thermodynamics holds. Nonlinear non-equilibrium thermodynamics, however, represents a regime where explosions and uncontrolled oscillations may arise. Obviously, industry wants to avoid such situations ... 

The formalism introduced in the previous subsections is able to incorporate the effect of these influences in the crystallization kinetics, thus providing a more realistic modeling of the process, which is mandatoiy for practical and industrial purposes. Due to the strong foundations of our mesoscopic formalism in the roots of standard non-equilibrium thermodynamics, it is easy to incorporate the influence of other transport processes (like heat conduction or diffusion) into the description of crystallization. In addition, our framework naturally accounts for the couplings between all these different influences. 

This new theory of the non-equilibrium thermodynamics of multiphase polymer systems offers a better explanation of the conductivity breakthrough in polymer blends than the percolation theory, and the mesoscopic metal concept explains conductivity on the molecular level better than the exciton model based on semiconductors. It can also be used to explain other complex phenomena, such as the improvement in the impact strength of polymers due to dispersion of rubber particles, the increase in the viscosity of filled systems, or the formation of gels in colloids or microemulsions. It is thus possible to draw valuable conclusions and make forecasts for the industrial application of such systems. 

Classical thermodynamics deals with equilibrium states. Entropy changes are then calculated via reversible processes. By contrast, non-equilibrium thermodynamics deals with systems that are not in global equilibrium. The entropy production can then be calculated from actual fluxes and forces. Real systems, for instance in biology or in industry, are not in equilibrium and are of course more interesting. Transport phenomena are always irreversible, and we shall see how they are contained in non-equilibrium thermodynamics. The list a to e below gives the main reasons for why non-equilibrium thermodynamics is important. 

The multiphase method provides a practical screening tool for industrial process research and development, even though under many circumstances the nonequilibrium effects such as supersaturation of solutions, retarded mass transfer or reaction kinetics and inhomogeneity of suspensions limit the applicability of the thermodynamic calculations. When the thermodynamic multiphase models are developed towards process simulation tools, one should incorporate such methods that include the effects of these non-equilibrium factors. They must be based on... 

Thermodynamic models are widely used for the calculation of equilibrium and thermophysical properties of fluid mixtures. Two types of such models will be examined cubic equations of state and activity coefficient models. In this chapter cubic equations of state models are used. Volumetric equations of state (EoS) are employed for the calculation of fluid phase equilibrium and thermophysical properties required in the design of processes involving non-ideal fluid mixtures in the oil and gas and chemical industries. It is well known that the introduction of empirical parameters in equation of state mixing rules enhances the ability of a given EoS as a tool for process design although the number of interaction parameters should be as small as possible. In general, the phase equilibrium calculations with an EoS are very sensitive to the values of the binary interaction parameters. 

For non-ideal multicomponent mixtures the multiphase flow calculation can be combined with a more rigorous thermodynamic equilibrium calculation to determine the mixture properties at the interface as discussed by [60, 70, 98]. However, describing the chemical reactor performance under industrial operation conditions the heat balance is normally dominated by the heat of reaction term, the transport terms and the external heating/cooling boundary conditions, hence for chemical processes in which the phase change rates are relatively small the latent heat term is often neglected. 

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