Inequality of Clausius

This is called the Inequality of Clausius, who, however, established it in a different way. 

Ice type, 195 Ideal gas, 47, 135 Independent variables, 103 Indicator diagram, 45, 127 Inequality of Clausius, 79 Intensity factors, 111 Intrinsic energy, 32, 76, 484 Inversion point, 167 Irreversible processes, 67, 69, 75, 82, 84, 87... 

Clausius proceeded to demonstrate the power of entropy to express the deep consequences of the second law. We begin by introducing the inequality of Clausius, which complements Carnot s theorem (4.25) for the irreversible case. 

The pioneering work in the direction of the second law of thermodynamics is considered to be performed in 1825 by Sadi Carnot investigating the Carnot cycle [51] [40]. Carnot s main theoretical contribution was that he realized that the production of work by a steam engine depends on a flow of heat from a higher temperature to a lower temperature. However, Clausius (1822-1888) was the first that clearly stated the basic ideas of the second law of thermodynamics in 1850 [13] and the mathematical relationship called the inequality of Clausius in 1854 [51]. The word entropy was coined by Clausius in 1854 [51]. 

This is known as the inequality of Clausius, after the German physicist Rudolf Julius Emmanuel Clausius (1822-1888), who suggested the relationship in 1850. The word entropy was coined by Clausius in 1865, and comes from the Greek en, in, and trope, transformation. 

Photosynthetic processes have the main responsibility of energy transfer in biological systems. This is possible because living systems are open systems, otherwise, the free energy F would not be available. In open systems, variations of entropy can be the consequence of different processes dgS, is the entropy exchanged with the environment, and dtS, is the entropy variation due to irreversible processes within the system. The second term is clearly positive, but the first term does not have a definite sign. So the inequality of Clausius-Carnot becomes ... 

The second law of thermod5mamics is stated through the Kelvin-Planck statement and the Clausius statement. The Inequality of Clausius is a consequence of the second law of thermod5mamics, and it is stated for a system undergoing a thermodynamic cycle as... 

The theory of Kelvin (1854), developed in the preceding, section, stands midway between these two hypotheses, in that it assumes the existence of potential differences at the junctions, playing the role postulated by Clausius, and also admits the production of electromotive forces in the interior of the homo-, geneous wires due to inequalities of temperature in the latter, these inequalities giving rise to the flow of heat which is regarded as essential in the theory of Kohlrausch. 

An alternative to analyze the Curzon-Ahlborn cycle, taking into account some effects that are nonideal to the adiabatic processes through the time of these processes, is the model proposed in [5] and in [7]. It allows to find the efficiency of a cycle as a function of the compression ratio, rc = Vmax/Vmin. When rc, Fmax>>Vmin, the Curzon-Ahlborn-Novikov-Chambadal efficiency is recovered. The non-endoreversible Curzon and Ahlbom cycle can be analyzed by means of the so-called non-endoreversibility parameter Is, defined first in [14] and later in [15] and in [16], which can be used to analyze diverse particularities of cycles. Furthermore, this parameter leads to equality instead of Clausius inequality [14]. 

Thus we have proved that if we start with the Clausius statement we obtain the mathematical inequality of the second law introduced at the beginning of this chapter. 

Still another statement of the second law is the Clausius inequality which states that... 

The combination of the Clausius inequality (eq. 1.30) and the first law of thermodynamics for a system at constant volume thus gives... 

Carnot efficiency is one of the cornerstones of thermodynamics. This concept was derived by Carnot from the impossibility of a perpetuum mobile of the second kind [ 1]. It was used by Clausius to define the most basic state function of thermodynamics, namely the entropy [2]. The Carnot cycle deals with the extraction, during one full cycle, of an amount of work W from an amount of heat Q, flowing from a hot reservoir (temperature Ti) into a cold reservoir (temperature T2 < T ). The efficiency r] for doing so obeys the following inequality ... 

The simple inequality (4.10) captures the essence of the second law. Its general consistency with universal inductive experience will be established in Section 4.4, and its further consequences (culminating in the final form of the second law as expressed by Clausius) will be developed in Sections 4.5-4.7. Thus, Carnot s remarkable principle provides virtually complete answers to the questions posed at the beginning of this chapter, although the relationship of (4.10) to these broader issues will certainly not become obvious until the following section. 

The general inequality (4.48) leads to the famous Clausius formulation of the second law ... 

The inequality (5.26) merely says that the entropy function was at a maximum before the variation, which is the counterpart of the Clausius statement [cf. (4.48)]... 

Table 5.1 summarizes the various constraint conditions and the associated thermodynamic potentials and second-law statements for direction of spontaneous change or condition of equilibrium. All of these statements are equivalent to Carnot s theorem ( dq/T < 0) or to Clausius inequality ([Pg.164]

This means that A is independent of the external constraints acting on the system. It must be a state function like energy or entropy. Moreover, in terms of the entropy production per unit time, the Clausius inequality... 

Rational thermodynamics is formulated based on the following hypotheses (i) absolute temperature and entropy are not limited to near-equilibrium situations, (ii) it is assumed that systems have memories, their behavior at a given instant of time is determined by the history of the variables, and (iii) the second law of thermodynamics is expressed in mathematical terms by means of the Clausius-Duhem inequality. The balance equations were combined with the Clausius-Duhem inequality by means of arbitrary source terms, or by an approach based on Lagrange multipliers. 

The Clausius-Duhem equation is the fundamental inequality for a single-component system. The selection of the independent constitutive variables depends on the type of system being considered. A process is then described by solving the balance equations with the constitutive relations and the Clausius-Duhem inequality. 

Studies on thermodynamic restrictions on turbulence modeling show that the kinetic energy equation in a turbulent flow is a direct consequence of the first law of thermodynamics, and the turbulent dissipation rate is a thermodynamic internal variable. The principle of entropy generation, expressed in terms of the Clausius-Duhem and the Clausius-Planck inequalities, imposes restrictions on turbulence modeling. On the other hand, the turbulent dissipation rate as a thermodynamic internal variable ensmes that the mean internal dissipation will be positive and the thermodynamic modeling will be meaningful. 

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