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which have undergone any change are the process materials [86]. To see this, we replace dQ/dt by minus the heat delivered to the environment per unit of time, dQ/dt= −dQ₀/dt:

(2.12)

      The second law gives

(2.13)

      where dS/dt is the rate of entropy change of the process materials and dS₀/dt is the rate of entropy change in the environment. For a completely reversible process, the sum of these entropy rate changes is zero. For a non-equilibrium process, the sum is the total entropy production, dS_irr/dt,

(2.14)

      The entropy change in the surroundings is dS₀/dt = (dQ₀/dt)/T₀. By introducing dQ₀/dt into the expression for the entropy production, and combining the result with the first law, we obtain, after some rearrangement,

(2.15)

      The left-hand side of this equation is the work that is needed to accomplish the process with a particular value of the entropy production, dS_irr/dt. Since dS_irr/dt ≥ 0, the ideal (reversible) work requirement of this process is

(2.16)

      We see that dS/dt changes the ideal work. By comparing Eqs. (2.15) and (2.16), we see that T₀(dS_irr/dt) is the additional quantity of work per unit of time that must be used in the actual process compared to the work in an ideal (reversible) process. Thus, T₀(dS_irr/dt) is the wasted or lost work per unit of time:

(2.17)

      The lost work is the energy dissipated as heat. The relation is called the Gouy–Stodola theorem [87, 88]. The right-hand side of Eq. (2.16) is also called the time rate of change of the availability or the exergy [65, 85]. We shall learn how to derive σ in subsequent chapters. One aim of the process operator or designer should be to minimize dW_lost/dt [89], which is equivalent to entropy production minimization, see [88, p. 227; 32, Chapter 6]. Knowledge about the sources of entropy production is central for our work to improve the energy efficiency of processes. In a recent book Non-equilibrium Thermodynamics for Engineers [2, 3], we discussed how irreversible thermodynamics can be used to map the lost work in an industrial process. The aluminium electrolysis was used as an example to illustrate the importance of such a mapping. Furthermore, the first steps in a systematic method for the minimization of entropy production were described, with its basis in non-equilibrium thermodynamics. Second law optimization [88, 90–95] in mechanical and chemical engineering may become increasingly important in the design of systems that waste less work. Non-equilibrium thermodynamics will play a unique role in this context. For details on how to map the second law efficiency of an industrial system or how to systematically improve this efficiency, we refer to Bejan [87, 88] and to the book Non-equilibrium Thermodynamics for Engineers [2, 3].

2.5Consistent thermodynamic models

      The entropy balance of the system and the expression for the entropy production can be used for control of the internal consistency of the thermodynamic description. The change of the entropy in a volume element in a homogeneous phase is given by the flow of entropy in and out of the volume element and by the entropy production inside the element:

(2.18)

      where we take the entropy flux in the x-direction. In the stationary state, there is no change in the system’s entropy density, s, and

(2.19)

      By integrating over the system’s extension, keeping its cross-sectional area Ω constant, we obtain

(2.20)

      The left-hand side is obtained from σ, which is a function of transport coefficients, fluxes and forces, Eqs. (1.1)–(1.3). The right-hand side of this equation, the difference in the entropy flows J_s^o − J_sⁱ, can be calculated from the knowledge of the heat flux, and of the entropy carried by components into the surroundings. These quantities do not depend on the system’s transport properties. The two calculations must give the same result. Chapter 19 gives an example of such a comparison. Other examples were given by Kjelstrupet al. [32], see also Zvolinschi et al. [95].

      Chapter 3

      Thermodynamic relations for heterogeneous systems

      This chapter gives thermodynamic relations for heterogeneous systems that are in global equilibrium, and discusses the meaning of local equilibrium in homogeneous phases, at surfaces and along three-phase contact lines.

      The heterogeneous systems that are treated in this book exchange heat, mass and charge with their surroundings. The systems have homogeneous phases separated by an interfacial region. They are electroneutral, but polarizable. The words surface and interface will be used interchangeably to indicate the interfacial region. Gibbs [65] calls the interfacial region “the surface of discontinuity”. Thermodynamic relations for homogeneous phases, as well as for surfaces, are mandatory for the chapters to follow. Such equations are therefore presented here [65, 96]. Equations for the threephase contact line are also given, but are less central. The equations are given first for systems that are in global equilibrium, and next for nonequilibrium systems where only local equilibrium applies.

      A thermodynamic description of equilibrium surfaces in terms of excess densities was constructed by Gibbs [65]. This description treats the surface as an autonomous thermodynamic system. We use this description in terms of excess densities as our basis also in non-equilibrium systems. This implies for instance that the surface has its own temperature. All excess densities of a surface depend on this temperature alone and not on the temperatures in the adjacent phases.¹ We present evidence for this assumption using non-equilibrium molecular dynamics simulations (Chapter 22) and the square gradient model of van der Waals (Chapter

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