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the forces of transport conjugate to the fluxes J′_q, J and j are the thermal force, d(1/T)/dx, the chemical force, [−(1/T)(dμ_T/dx)], and the electrical force, [−(1/T)(dϕ/dx)], respectively. The subscript T of the chemical potential μT indicates that the differential should be taken at constant temperature. The electric field is given in terms of the electric potential gradient.

      The L-coefficients are the so-called phenomenological coefficients, or Onsager coefficients for conductivity, as we shall call them. The diagonal Onsager coefficients can be related to λ, D and κ. They are called main coefficients. The off-diagonal L-coefficients describe the coupling between the fluxes. They are called coupling coefficients. Another common name is cross coefficients. According to Eq. (1.3), we have here three reciprocal relations or Onsager relations for these coupling coefficients

(2.8)

      The Onsager relations simplify the system. Here, they reduce the number of independent coefficients from nine to six. Coupling coefficients are small in some cases, but large in others. We shall see in Chapter 7 that large coupling coefficients may lead to a low entropy production. This is why the coupling coefficients are so important in the design of industrial systems, see Sec. 2.4. The relation Lqϕ = Lϕq is Thomson’s second relation, which dates from 1854 (see [10]). Miller [74–77] and Spalleket al. [78] gave experimental evidence for the validity of the Onsager relations in electrolytes. Hafskjold and Kjelstrup Ratkje [68] and Xu and coworkers [70] proved their validity for heat and mass transfer, in homogeneous phases and at surfaces, using non-equilibrium molecular dynamics simulations, respectively. Annunziata et al. [79] took advantage of the Onsager relations to obtain thermodynamic data for an aqueous mixture of enzyme and salt near the solubility limit of the salt. Onsager’s proof of the reciprocal relations shall be discussed in Chapter 7.

2.3Experimental designs and controls

      The importance of equilibrium thermodynamics for the design of experiments is well known. The definition, of say a partial molar property, explains what shall be varied, and what shall be kept constant in the experimental determination of the quantity. Also, there are relations between thermodynamic variables that offer alternative measurements. For instance, the enthalpy of evaporation can be measured in a calorimeter, but it can also be determined by finding the vapor pressure of the evaporating gas as a function of temperature.

      Non-equilibrium thermodynamics is, in a similar way, instrumental for design of experiments that aim to find transport properties, cf. Chapter 20. To see this, consider the following exercise.

      Exercise 2.3.1.Find the electric current in terms of the electric field E = −dϕ/dx using Eq. (2.7) in a system where there is no transport of heat and mass J′q = J = 0.

      •Solution: It follows from Eqs. (2.7a) and (2.7b) that

      Solving these equations, using the Onsager relations, one finds

      Substitution into Eq. (2.7c) then gives

      This exercise shows that the electric conductivity, that one measures as the ratio of measured values of j and E, is not necessarily given by Lϕϕ/T as one might have thought, considering Eq. (2.7c). In the stationary state, the coupling coefficients lead to temperature and chemical potential gradients, which again affect the electric current. Mathematically speaking, the electric conductivity then becomes a combination of the conductivity of a homogeneous conductor, which is found if one could measure with zero chemical potential and temperature gradients and additional terms. The combination of coefficients divided by the temperature is the stationary state conductivity. The stationary state electric conductivity, measured with J′_q = J = 0, is experimentally distinguishable from the Ohmic electric conductivity of the homogeneous conductor, measured with dT/dx = dμT/dx = 0. Nonequilibrium thermodynamics helps define conditions that give well-defined experiments.

      One important practical consequence of the Onsager relations is to offer alternative measurements for the same property. For instance, if it is difficult to measure the coefficient Lqϕ, one may rather measure Lϕq [80–83]. A valuable consistency check for measurements is provided if one measures both coupling coefficients in the relation. So, similar to the situation in equilibrium, the non-equilibrium systems also have possibilities for control of internal consistency.

2.4Entropy production, work and lost work

      Non-equilibrium thermodynamics is probably the only method that can be used to assess how energy resources are exploited within a system. This is because the theory deals with energy conversion on the local level in a system, i.e. at the electrode surface [84] or in a biological membrane [55]. By integration to the system level, we obtain a link to exergy analysis [85]. To see the relation between non-equilibrium thermodynamics and exergy analysis, consider again the thermodynamic fluxes and forces, derived from the local entropy production

(2.9)

      Figure 2.1A schematic illustration of thermodynamic variables that are essential for the lost work in an industrial plant.

      The total entropy production is the integral of σ over the volume V of the system

(2.10)

      The quantity dS_irr/dt makes it possible to calculate the energy dissipated as heat or the lost work in a system (see below).

      In an industrial plant, the materials undergo certain transformations in a time interval dt. Materials are taken in and are leaving the plant at the conditions of the environment. The environment is a natural choice as frame of reference for the analysis. It has constant pressure p₀ (1 bar) and constant temperature T₀ (for instance, 298 K) and some average composition that needs to be defined. The first law of thermodynamics gives the energy change of the process per unit of time:

(2.11)

      Here, dQ/dt is the rate that heat is delivered to the materials, p₀dV/dt is the work that the system does per unit of time by volume expansion against the pressure of the environment, and dW/dt is the work done on the materials per unit of time.

      The minimum work needed to perform a process is the least amount of energy that must be supplied when, at the conclusion of

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