Non-Equilibrium Thermodynamics for Industry

SIGNE KJELSTRUP, AUDUN RØSJORDE AND EIVIND JOHANNESSEN

1 What Can Non-Equilibrium Thermodynamics Offer?

Non-equilibrium thermodynamics (NET) offers a systematic way to derive the local entropy production rate, σ 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, JoS, minus the entropy flux into the system, JiS:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

The entropy flux difference and the integral over σ 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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where Ji and Xi are conjugate flux-force pairs. Each flux is a linear combination of all forces:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

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 Lij can, for instance, be functions of the state variables. By including internal variables in the thermodynamic description, one can extend the application of NET to activated processes; see Chapter 2. For this reason, NET appears today as a non-linear and versatile theory that applies to many practical conditions. It is a misunderstanding that flux equations need to be linear on the global level.

The total entropy production rate times the temperature of the environment is equal to the exergy destruction rate in a process. Processes with small losses in exergy have a high second law efficiency. A high second law efficiency, or exergy efficiency, is seldom a specific aim in process design. An increasing worldwide concern with CO2 emission may change this. Multiobjective optimisation, with small entropy production as one target, may then be interesting in chemical engineering design.

2 Developments and Status of NET

Non-equilibrium thermodynamics was founded by Onsager. The theory was further elaborated by de Groot and Mazur and Prigogine. The theory is based on the hypothesis of local equilibrium: a volume element in a non-equilibrium system is in local equilibrium when the normal thermodynamic relations apply to the element. Evidence is emerging that show that many systems of interest in the process industry are in local equilibrium by this criterion. Onsager prescribed that each flux be connected to its conjugate force via the extensive variable that defines the flux.

Onsager assumed that the variables and the rate laws were the same on the macroscopic and the microscopic level; this is the so-called regression hypothesis. Also using the assumption of microscopic reversibility, he proved the reciprocal relations:

Lij = Lji (4)

These assumptions restrict the validity of NET, but as stated above, they have a wide range of validity. It has long been known that the Navier–Stokes equations are contained in NET. More recently, NET has been extended to deal with transport across surfaces, quantum mechanical systems, and mesoscopic systems; see Chapter 2. We have chosen to illustrate NET with cases of transport through surfaces in the following sections.