Among all statements of Second Law, the existence and uniqueness of stable equilibrium, for each given value of energy content and composition of constituents of any system, have been adopted to define thermodynamic entropy by means of the impossibility of Perpetual Motion Machine of the Second Kind (PMM2) which is a consequence of the Second Law. Equality of temperature, chemical potential and pressure in many-particle systems are proved to be necessary conditions for the stable equilibrium. The proofs assume the stable equilibrium and derive, by means of the Highest-Entropy Principle, equality of temperature, chemical potential and pressure as a consequence. A first novelty of the present research is to demonstrate that equality is also a sufficient condition, in addition to necessity, for stable equilibrium implying that stable equilibrium is a condition also necessary, in addition to sufficiency, for equality of temperature potential and pressure addressed to as generalized potential. The second novelty is that the proof of sufficiency of equality, or necessity of stable equilibrium, is achieved by means of a generalization of entropy property, derived from a generalized definition of exergy, both being state and additive properties accounting for heat, mass and work interactions of the system underpinning the definition of Highest-Generalized-Entropy Principle adopted in the proof.
Installation layout of wind turbines plays a prominent role in the design of every wind farm. Thus, the wind farm layout optimization problem is proposed to maximize the total power output with the minimum cost. In this research, Kahrizak region in Tehran province of Iran is selected as a windy region and its real wind speed data are gleaned. Three different scenarios are also considered, with various number of generations and populations for GA parameters, effective distances, and longitude and latitude distances of turbines from each other. Among these scenarios, the best result is obtained for the one in which the longitudinal distance between turbines is greater than the latitudinal distance. By observing the wind rose of Kahrizak region, it is observed that the dominant wind direction of the region is toward the east and south-east. Therefore, by increasing the longitudinal distance of the turbines from each other, the efficiency can be improved and the turbine layout becomes more realistic. In this case, the efficiency rate and normalized cost of turbines are 89.5% and 37.4, respectively, and also 56 turbines are needed. The amounts of efficiency and power output are very convenient for real wind speed data of a region.
Abstract:The literature reports the proofs that entropy is an inherent property of any system in any state and governs thermal energy, which depends on temperature and is transferred by heat interactions. A first novelty proposed in the present study is that mechanical energy, determined by pressure and transferred by work interactions, is also characterized by the entropy property. The second novelty is that a generalized definition of entropy relating to temperature, chemical potential and pressure of many-particle systems, is established to calculate the thermal, chemical and mechanical entropy contribution due to heat, mass and work interactions. The expression of generalized entropy is derived from generalized exergy, which in turn depends on temperature, chemical potential and pressure of the system, and by the entropy-exergy relationship constituting the basis of the method adopted to analyze the available energy and its transfer interactions with a reference system which may be external or constitute a subsystem. This method is underpinned by the Second Law statement enunciated in terms of existence and uniqueness of stable equilibrium for each value of energy content of the system. The equality of chemical potential and equality of pressure are assumed, in addition to equality of temperature, to be necessary conditions for stable equilibrium.
The literature reports that equality of temperature, equality of potential and equality of pressure between a system and a reservoir are necessary conditions for the stable equilibrium of the system-reservoir composite or, in the opposite and equivalent logical inference, that stable equilibrium is a sufficient condition for equality. The aim and the first novelty of the present study is to prove that equality of temperature, potential and pressure is also a sufficient condition for stable equilibrium, in addition to necessity, implying that stable equilibrium is a condition also necessary, in addition to sufficiency, for equality. The second novelty is that the proof of the sufficiency of equality (or the necessity of stable equilibrium) is attained by means of the generalization of the entropy property, derived from the generalization of exergy property, which is used to demonstrate that stable equilibrium is a logical consequence of equality of generalized potential. This proof is underpinned by the Second Law statement and the Maximum-Entropy Principle based on generalized entropy which depends on temperature, potential and pressure of the reservoir. The conclusion, based on these two novel concepts, consists of the theorem of necessity and sufficiency of stable equilibrium for equality of generalized potentials within a composite constituted by a system and a reservoir.
First intent of the present research is to prove the rationale behind a generalized definition of thermodynamic entropy as a state and additive property inherent to any system, large or small, in any state, equilibrium or non-equilibrium. The second objective is to extend the canonical equation of state in the perspective of thermal and chemical aspect of microscopic configurations of a system related to inter-particle kinetic energy and inter-particle potential energy determining macroscopic parameters. As a consequence, a generalized state equation is formulated accounting for thermal, chemical and mechanical thermodynamic potentials characterizing any system in any state.
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