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")...
Until the 19th century, technology was essentially the domain of skilled artisans and constructors who relied on practical experience to design and build their machines. One of the first efforts to use physical theory to study the functioning of machines was undertaken by the French engineer Sadi Carnot. Motivated by the concern of the French about the superiority of British steam engines, he undertook a systematic study of the physical processes governing steam engines, resulting in his remarkable paper Reflexions sur la puissance motrice du feu (On the Motive Power of Heat) published in 1826. Among the earliest successes of this new science, thermodynamics, was the formulation of criteria describing how well real processes perform in comparison with an ideal model. Carnot showed that any engine, using heat from a hot reservoir at temperature Th to do work, has to transfer some heat to a reservoir at lower temperature T1, and that no engine could convert into work more of the heat taken in at Th than the fraction ηC = 1−(T1/Th) known as the Carnot efficiency.
We apply the method of optimal control theory to determine the optimal piston trajectory for successively less idealized models of the Otto cycle. The optimal path has significantly smaller losses from friction and heat leaks than the path with conventional piston motion and the same loss parameters. The resulting increases in efficiency are of the order of 10%.
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