Control Volume Analysis, Entropy Balance and the Entropy Production in Flow Systems

Abstract

This chapter concerns "control volume analysis", the standard engineering tool for the analysis of flow systems, and its application to entropy balance calculations. Firstly, the principles of control volume analysis are enunciated and applied to flows of conserved quantities (e.g. mass, momentum, energy) through a control volume, giving integral (Reynolds transport theorem) and differential forms of the conservation equations. Several definitions of steady state are discussed. The concept of "entropy" is then established using Jaynes' maximum entropy method, both in general and in equilibrium thermodynamics. The thermodynamic entropy then gives the "entropy production" concept. Equations for the entropy production are then derived for simple, integral and infinitesimal flow systems. Some technical aspects are examined, including discrete and continuum representations of volume elements, the effect of radiation, and the analysis of systems subdivided into compartments. A Reynolds decomposition of the entropy production equation then reveals an "entropy production closure problem" in fluctuating dissipative systems: even at steady state, the entropy production based on mean flow rates and gradients is not necessarily in balance with the outward entropy fluxes based on mean quantities. Finally, a direct analysis of an infinitesimal element by Jaynes' maximum entropy method yields a theoretical framework with which to predict the steady state of a flow system. This is cast in terms of a "minimum flux potential" principle, which reduces, in different circumstances, to maximum or minimum entropy production (MaxEP or MinEP) principles. It is hoped that this chapter inspires others to attain a deeper understanding and higher technical rigour in the calculation and extremisation of the entropy production in flow systems of all types.

Maximum Entropy Analysis of Flow Networks: Theoretical Foundation and Applications

“…In typical thermodynamic systems involving isolated systems or canonical ensembles—for which the probabilities and constraints are defined over the contents of the system or the ensemble—the stationary state is interpreted as the equilibrium position of the system, e.g., [ 57 , 58 , 59 , 60 ]. However, in forced flow systems such as flow networks—in which the probabilities and constraints are defined over constant flow rates or fluxes into or through part of the system—the stationary state must instead be interpreted as the steady state of the system [ 61 , 62 , 63 ]. (We note that the term “steady-state” is somewhat misleading, since it refers only to the mean flow and not its fluctuations; a steady-state flow need not be steady in time, only in the mean.)…”

Depletion, Chemical Reaction and Transport in High Burnup Nuclear Fuel

“…The two forms of the discretized processes are shown in Figure 2 (Niven and Noack, 2014). The discretization in Figure 2a represents processes where each localized volume is at equilibrium for the given composition and temperature.…”

Discussion of Di Vita A on Information Thermodynamics and Scale Invariance in Fluid Dynamics

“…Comments on the theoretical basis of the maximum entropy production (MaxEP) hypothesis for prediction of the steady state of a flow system, advocated by a number of authors [3][4][5][6][7]. The latter is connected to a direct MaxEnt analysis of flow systems given by us, based on an entropy defined on the set of instantaneous flux states [8][9][10][11].…”

“…Di Vita's criticism could also be leveled at the various formulations of upper bound theory in turbulent uid mechanics [37][38][39], in which the postulated extrema may appear rather ad hoc. However, other theoretical treatments have been proposed to explain entropy production extrema, including our above mentioned MaxEnt analysis of an infinitesima low system [8][9][10][11]. The analysis is based on maximization of an entropy depend on the set of instantaneous flux states and reaction rates, giving a potential function (negative Massieu function) which is minimized at steadystate low.…”

“…These comments are so vague that we do not see how to interpret them as a criticism. In the first study [8] we do adopt a uniform prior probability, but in later works [9][10][11] we consider a prior probability in its full generality, which must be selectedfor the problem at hand. This is a feature of all MaxEnt analyses [40].…”