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      Chapter 2

      Why non-equilibrium thermodynamics?

      This chapter explains what the field of non-equilibrium thermodynamics adds to the analysis of common scientific and engineering problems. Accurate and reliable flux equations can be obtained. Experiments can be well defined. Knowledge of transport properties gives information on the system’s ability to convert energy.

      The most common industrial and living systems have transport of heat, mass and charge, alone or in combination with a chemical reaction. The process industry, the electrochemical industry, biological systems, as well as laboratory experiments, all concern heterogeneous systems, which are not in global equilibrium. There are four major reasons why non-equilibrium thermodynamics is important for such systems. In the first place, the theory gives an accurate description of coupled transport processes. In the second place, a framework is obtained for definition of experiments. In the third place, the theory quantifies not only the entropy that is produced during transport, but also the work that is done and the lost work. Last but not least, this theory allows us to check that the thermodynamic equations we use to model our system are in agreement with the second law.

      The aim of this book is to give a systematic description of transport in heterogeneous systems. Non-equilibrium thermodynamics has so far mostly been used in science [23, 24, 30]. Accurate expressions for the fluxes are now required also in engineering [32, 45, 72]. In order to see immediately what non-equilibrium thermodynamics can add to the description of real systems, we compare simple flux equations to flux equations given by non-equilibrium thermodynamics in the following four sections.

2.1Simple flux equations

      The simplest descriptions of heat, mass and charge transports are the equations of Fourier, Fick and Ohm. Fourier’s law expresses the measurable heat flux in terms of the temperature gradient by

(2.1)

      where λ is the thermal conductivity, T is the absolute temperature and the direction of transport is along the x-axis. Fick’s law gives the flux of one of the components in terms of the gradient of its concentration c:

(2.2)

      where D is the diffusion coefficient. Similarly, Ohm’s law gives the electric current in terms of the gradient of the electric potential:

(2.3)

      where κ is the electrical conductivity and ϕ is the electric potential.

      In a stationary state, there is no accumulation of internal energy, mass or charge. This means that the heat, molar and electric fluxes are independent of position. The derivatives of the above equations with respect to x are then zero:

	(2.4)
	(2.5)
	(2.6)

      These equations can be used to calculate the temperature, concentration and electric potential as a function of the position, when their values on the boundaries of the system and λ, D and κ are known. Such a calculation is illustrated by the following exercise.

      Exercise 2.1.1.Calculate the temperature as a function of position between two walls at a distance of 10 cm which are kept at constant temperatures of 5°C and 25°C, assuming that the thermal conductivity is constant.

      •Solution: According to Eq. (2.4), d²T/dx² = 0. The general solution of this equation is T(x) = a + bx. The constants a and b follow from the boundary condition. We have T(0) = 5°C and T(10) = 25°C. It follows that T(x) = (5 + 2x)°C.

      Equations (2.1)–(2.3) describe a pure degradation of thermal, chemical and electrical energy. In reality, there is also conversion between the energy forms. For an efficient exploitation of energy resources, such a conversion is essential. This is captured in non-equilibrium thermodynamics by the so-called coupling coefficients. Mass transport occurs, for instance, not only because dc/dx ≠ 0, but also becaused T/dx ≠ 0 or dϕ/dx ≠ 0.

      Chemical and mechanical engineers need theories of transport for increasingly complex systems with gradients in pressure, concentration and temperature. Simple vectorial transport laws have long worked well in engineering, but there is now an increasing effort to be more precise. The books by Taylor and Krishna [72], Kuiken [34] and Cussler [73], which use Maxwell–Stefan’s formulation of the flux equations, are important books in this context. A need for more accurate flux equations in modelling [45, 72] makes non-equilibrium thermodynamics a necessary tool.

2.2Flux equations with coupling terms

      Many natural and man-made processes are not adequately described by the simple flux equations given above. There are, for instance, always large fluxes of mass and heat that accompany charge transport in batteries and electrolysis cells. The resulting local cooling in electrolysis cells may lead to unwanted freezing of electrolyte. Electrical energy is frequently used to transport mass in biological systems. Large temperature gradients across space ships have been used to supply electric power to the ships. Salt concentration differences between river water and sea water can be used to generate electric power. Pure water can be generated from salt water by application of pressure gradients. In all these more or less randomly chosen examples, one needs transport equations that describe coupling between various fluxes. The flux equations given above become too simple.

      Non-equilibrium thermodynamics introduces coupling among fluxes. Coupling means that transport of mass will take place in a system not only when the gradient in the chemical potential is different from zero, but also when there are gradients in temperature or electric potential. Coupling between fluxes can describe the phenomena mentioned above.

      For example, in a bulk system with transport of heat, mass, and electric charge in the x-direction, we shall find that the linear relations (1.2) take the form

	(2.7a)
	(2.7b)
	(2.7b)

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