Thermodynamic Length and Dissipated Availability

Abstract

New expressions for the availability dissipated in a finite-time endoreversible process are found by use of Weinhold's metric on ecluilibrium states of a thermodynamic system. In particular, the dissipated availability is given by the square of the length of the corresponding curve, times a mean relaxation time, divided by the total time of the process. '&he results extend to local thermodynamic equilibrium if instead of length one uses distance (length of the shortest curve) between initial and final states. PACS numbers: 05.70.-aThe results presented below give an important tool for finding limits on the efficiency of' operation of thermodynamic processes in finite time.We have been pursuing various approaches to this problem for several years, ' but the results below give a new inroad for a remarkably general class of processes by providing expressions for the inherent irreversibility quantified by the loss of available work: the availability not transformed into work during the process. This dissiPated availabil. ity is sometimes called "irreversibility. '" The dissipated availability dA" is related to the entropy production dS"by dA"= TdS". We derive expressions for the availability dissipated when a thermodynamic system undergoes a process during which it may be assumed to be in internal equilibrium, though interacting with an environment, which is also in equilibrium. Since the dissipated availability is an extensive quantity, an extension of our expressions to local thermodynamic equilibrium is immediate. These expressions involve the thermodynamic length introduced by Weinhold" and hint at the existence of a temporal element in the classical formalism dealing only with equilibrium.We assume that the time scales for internal relaxation of system and surroundings are much shorter than the time scale on which system and surroundings interact. This implies that we may consider both system and surroundings to be in equilibrium states at each instant of time. This assumption is already among the postulates for local thermodynamic equilibrium. The macroscopic form of this assumption was introduced by Rubin'. a process is e~do~evexsible provided the subsystems participating in the process are in internal equilibrium at each instant. The expression given below for dissipation in a shock wave hints that our expressions for the dissipated availability are valid in a context wider than our derivations based on endoreversibility show. Besides the total time for the process, our expressions for the dissipated availability include only one nonequilibrium parameter: the mean relaxation time. The interpretation of this relaxation time is straightforward for processes during which the system and its environment are close to equilibrium with each other. This automatically holds (except perhaps at the boundary) if the local thermodynamic equilibrium model is appropriate.We now define the notion of thermodynamic length. The second-derivative matrix of the internal energy U with respect to extensive variables X =(X".. . , X")...

Nonholonomic Ricci flows. II. Evolution equations and dynamics

“…Here, we would like to mention that there are alternative approaches to geometric and non-equilibrium thermodynamics, locally anisotropic kinetics and kinetic processes in terms of Riemannian and Finsler like objects on phase and thermodynamic spaces, see reviews of results and bibliography in Refs. [30,31,32,33,34,35] and Chapter 6 from [16]. Those models are not related to Ricci flows of geometric objects and do not seem related to the statistical thermodynamics of metrics and connections which can be derived from the holonomic or anholonomic Perelman's functionals.…”

Principles of control thermodynamics

“…A more general claim along these lines was published by Tondeur and Kvaalen [18] but its exact status remains unclear at this time. 12 Closely related is the result of Salamon and Berry [19] which gives a general bound on the entropy produced.…”

Nonequilibrium Statistical Operator for Systems of Finite Size

“…where P is projection operator [21], ρ q (t 0 ,0) is quasi-equilibrium distribution (9) [20][21], has a form {…} is classical Poisson's bracket. The solution (3) contains non-physical dependence on the initial moment of time t 0 and has physical sense at t>>t 0 , when initial conditions become insignificant.…”

“…But qualitative picture gained in works [6][7] corresponds to representations of the present work. Thermodynamic systems in which the transition from an initial state to final one occurs not infinitely slowly (as in equilibrium thermodynamics), but rather during some finite fixed time were considered in works [8][9].…”