EPIOPTICS-10

The following sections are included:

• PREFACE

• Contents

The aim of this book is to present the essence of non-equilibrium thermodynamics for engineers. The field was established in 1931 and developed during the forties and fifties for transport in homogeneous phases. Applications of the theory are now increasing. Some perspectives on the applications are given after a brief introduction.

This chapter explains in more detail what non-equilibrium thermodynamics is, and how it adds to engineering fields.

We derive the entropy production for a volume element of a homogeneous phase where diffusion, conduction, and chemical reaction can take place along the x-axis. The system is in mechanical equilibrium. Equivalent pairs of conjugate fluxes and forces are derived.

We present examples of flux equations that follow from the entropy production in Chapter 3. The determination of Onsager coefficients from experimental data, for instance, conductivities, diffusion coefficients and transport numbers, is discussed for simple homogeneous systems. We show that the coupling coefficients can be related to stored energy or work.

Maxwell-Stefan equations were derived before Onsager established non-equilibrium thermodynamics. The equations are important, because they place all components on the same footing, and contain a diffusion coefficient which for homogeneous fluids is relatively constant. We show how the equations are compatible with non-equilibrium thermodynamics and give rules of transformations between alternatives frames of reference for transport.

We derive the entropy production for a system with chemical reactions, temperature gradients and shear flow. The momentum balance contributes to a change in the internal energy. The Navier-Stokes equation and other relations are given. The Navier-Stokes equation contains a mechanical force due to the gradient of the reaction Gibbs energy. The rate of the chemical reaction obtains for symmetry reasons a term due to expansion or contraction. We discuss stationary pipe- and plug flow.

Chemical reactions have rates which normally are non-linear in their driving force. In order to describe this and derive the law of mass action using non-equilibrium thermodynamics, we shall introduce as internal variable, the probability that the reaction is in a state γ on the path between the reactant and product states. The processes along this path occur on an intermediate (mesoscopic) level, and the extended theory is called mesoscopic non-equilibrium thermodynamics. A linear force-flux relation along the mesoscopic reaction path leads to a non-linear macroscopic law of an Arrhenius-type. This theory captures the nature of the chemical reaction, and gives it a thermodynamic basis.

The purpose of this chapter is to map the entropy production in an industrial process and to discuss the information in such a map. The lost work in a 230 kA aluminum electrolysis cell is calculated from the entropy production in the various parts of the cell. The thermodynamic efficiency is calculated. The entropy production due to charge transfer and heat exchange are both large. The first can be changed by changing materials and electrode distances. The second can be changed by increasing the cell dimensions.

We present a procedure to minimize the entropy production in a process where defined output conditions are maintained. The procedure is demonstrated for ideal gas expansion and single plate heat exchange. We show that a process with minimum total entropy production can be characterized by uniform local entropy production.

We shall see how control theory can be used in order to find paths of operation with minimum entropy production in two typical process units, the chemical reactor and the distillation tower. Some typical results are presented and concluded with practical guidelines for second law efficient design.

The following sections are included:

• Appendix A

• Bibliography

• List of Symbols

• Index

• About the authors