Thermodynamic optimization - Equipartition: examples and applications

In [BE 8 017], we outlined the general aspects of entropy analysis as a thermodynamic optimization tool, highlighting the value of equipartition, i.e. the homogeneous distribution of entropy production. The present dossier [BE 8 018] is devoted to extending and illustrating this analysis using various examples.

In many systems, there are several different entropy-producing mechanisms. In a heat exchanger, for example, both viscous dissipation (observed as pressure loss) and heat transfer contribute. Constrained minimization of the total entropy production in this case leads to a certain distribution of the contribution (local and/or global) of these mechanisms, which is not generally equipartition, but a more general relationship that depends on the exponents affecting the control variables of the transfer laws.

Equipartition appears as a special case in this problem (§ 1).

A system can also exchange matter and energy with several sources and sinks. There may then be an optimal allocation of tasks between these sources and sinks. Using an example of heat transfer, and relying on the notion of endoreversible processes, we will show that here again, minimizing irreversibilities for a fixed global task corresponds to equipartition of irreversibilities between sources (§ 2).

The coaxial tube heat exchanger will also be used to illustrate more quantitatively the calculations of entropy production and performance in the vicinity of equipartition operation (§ 3).

The distribution of irreversibilities in space concerns both a "continuous" space, such as the coordinate along the tube exchanger, and a discretized space, made up of a sequence of components or equipment, for example the different stages of a compressor. The relevance of equipartition will be illustrated by several examples of this type. In this analysis, we'll return to a notion familiar from thermal and chemical engineering: the notion of "counter-current". Indeed, the counter-current configuration will appear as a convenient way of approaching (if not achieving) equipartition in exchanges between two material currents (§ 4).

This thermodynamic approach does not directly involve monetary costs or notions of depreciation, and should therefore not be confused with techno-economic optimization. We will, however, use an example to show that a problem of optimal resource allocation, in the financial sense, can sometimes lead to the uniform distribution of a quantity that combines financial parameters and entropy production, and is reduced to the latter in a thermodynamic limit (§ 4.5).