Irreversible processes, Clausius

This completes the heuristic derivation of the Boltzmann transport equation. Now we trim to Boltzmaim s argument that his equation implies the Clausius fonn of the second law of thennodynamics, namely, that the entropy of an isolated system will increase as the result of any irreversible process taking place in the system. This result is referred to as Boltzmann s H-theorem. 

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... 

Irreversible processes correspond to the time evolution in which the past and the future play different roles. In processes such as heat conduction, diffusion, and chemical reaction there is an arrow of time. As we have seen, the second law postulates the existence of entropy 5, whose time change can be written as a sum of two parts One is the flow of entropy deS and the other is the entropy production dtS, what Clausius called uncompensated heat, ... 

A third statement of the second law is based on the entropy. In reversible systems all forces must be opposed by equal and opposite forces. Consequently, in an isolated system any change of state by reversible processes must take place under equilibrium conditions. Changes of state that occur in an isolated system by irreversible processes must of necessity be spontaneous or natural processes. For all such processes in an isolated system, the entropy increases. Clausius expressed the second law as The entropy of the universe is always increasing to a maximum. Planck has given a more general statement of the second law Every physical and chemical process in nature takes place in such a way as to increase the sum of the entropies of all bodies taking any part in the process. In the limit, i.e., for reversible processes, the sum of the entropies remains unchanged. 

For an irreversible cyclic process, the summation of dq / T is less than 0, so the efficiency of an irreversible process is lower than that of a reversible one (Clausius) ... 

The work of Carnot, published in 1824, and later the work of Clausius (1850) and Kelvin (1851), advanced the formulation of the properties of entropy and temperature and the second law. Clausius introduced the word entropy in 1865. The first law expresses the qualitative equivalence of heat and work as well as the conservation of energy. The second law is a qualitative statement on the accessibility of energy and the direction of progress of real processes. For example, the efficiency of a reversible engine is a function of temperature only, and efficiency cannot exceed unity. These statements are the results of the first and second laws, and can be used to define an absolute scale of temperature that is independent of ary material properties used to measure it. A quantitative description of the second law emerges by determining entropy and entropy production in irreversible processes. 

This T, the thermodynamic or absolute temperature, is here a function of S, V and x. But it s easy to show that if T were a function of temperature and entropy, or if it were a function of temperature and anything else, we could violate Kelvin s statement. So T depends only on the empirical temperature, and this dependence must be the same for all systems in order for the entropy of a composite to equal the sum of the entropies of the subsystem. In order for Clausius statement to hold in the case of irreversible processes, the equal sign of rfQ = TdS becomes <, and we have Clausius inequality TdS,rwKere T is the... 

This is known as the Clausius inequality and has important applications in irreversible processes. For example, dS > (dQ/T) for an irreversible chemical reaction or material exchange in a closed heterogeneous system, because of the extra disorder created in the system. In summary, when we consider a closed system and its surroundings together, if the process is reversible and if any entropy decrease takes place in either the system or in its surroundings, this decrease in entropy should be compensated by an entropy increase in the other part, and the total entropy change is thus zero. However, if the process is irreversible and thus spontaneous, we should apply Clausius inequality and can state that there is a net increase in total entropy. Total entropy change approaches zero when the process approaches reversibility. 

Boltzmann connected his ideas with those of Rudolf Clausius, who had introduced the concept of entropy in 1865. Somehow related to heat, entropy was known to increase during irreversible processes, but its exact nature was unknown. From the distribution of gas atoms, Boltzmann described a quantity—later symbolized by the letter H—which is a minimum when atoms assume a Maxwell-Boltzmann distribution. He recognized his H function as the negative of entropy, which is a maximum when the atoms reach thermal equilibrium. Thus Boltzmann offered a kinetic explanation for entropy and, more generally, a connection between the behavior of atoms and thermodynamics. 

Equation (5.31) can be extended to include irreversible processes being known as Clausius inequality ... 

Central to the thermodynamic discussion of irreversible processes is the concept of entropy production. Consider the Clausius inequality, dS > Q/T, which we can rearrange to the form... 

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 relation (3.11), which is equivalent to the classical Clausius-Duhem inequality, implies that the dissipative part of the internal work due to stress is converted into heat. This suggests that the irreversible process of classical thermodynamics gives a conversion of energy from one form to another. The result is evident, because its framework is built on a potential theory. 

This is referred to as the Clausius inequality, and implies that in an irreversible process the entropy increases. The heat supply J Q is therefore not completely converted into the heat component of the internal energy T dS. This is the essential concept of the second part of the Second Law of Thermodynamics. 

It should be noted that the Clausius-Duhem inequality, giving the condition of internal dissipation, is exclusively satisfied for the equality part by the reversible process and for the pure inequality part by the irreversible process. The equality of the non-negative condition is not satisfied for the pure irreversible process . ... 

This gives the Eulerian form of the Second Law of Thermodynamics for the continuum, also referred to as the Clausius-Duhem inequality. Note that the first term of the r.h.s. of (D.76) is a consequence of the internal dissipation in mechanical energy. It should also be noted that the equality is provided for reversible processes whereas the pure inequality is applicable to irreversible processes. 

Thermodynamics can be divided into subjects which deal with 1) equilibrium, (2) nonequilibrium, and (3) irreversible processes. Ail three of these subdivisions are important in hydrocarbon reservoirs and in the interpretation of laboratory experiments for the understanding of hydrocarbon reservoirs. However, equilibrium thermodynamics is by far the most important and the best understood subject. According to Tisza (1966), the subdivision of equilibrium thermodynamics can be carried out further into Gibbsian thermodynamics and the early thermodynamics of Clausius and Kelvin. The latter considered the thermodynamic system as a black box, and all the relevant information was then derived from the energy absorbed and the work done by the system. The concepts of internal energy, U, and entropy, S, from the observable quantities are then established. In Gibbsian thermodynamics, the concepts of internal energy and entropy are assumed to be known and are used to provide a detailed description of the subsystems in equilibrium (we will soon define some of the terms used above). 

Constitutive laws for viscoelastic materials are however more complex than the Hooke s law and consequently the number of parameters to identify increases. Frameworks could also be used to ensure that all parameters of the proposed laws are physically admissible. To do so, it is possible to use the thermodynamic of irreversible processes as the framework. Based on the concepts of continuum mechanics and irreversible thermodynamics, the Clausius-Duhem inequality is obtained for given problems where dissipation mechanisms are of importance, e.g., viscous deformation. Fundamental equations leading to a generic form of the Clausius-Duhem inequality have been well covered by many authors (Bazarov, 1964 Coussy, 2010 Lemaitre and Chaboche, 1990 Mase and Mase, 1999) and thus will be only summarized later in section 3. Based on the generic form of the Qausius-Duhem inequality, models or constitutive laws are further developed considering various assumptions closely related to materials of interest. 

The parameters of this tensor must be identified, as well. Based on the thermodynamics of irreversible processes (TIP) and on the choice of the internal state variables (ISV), the Clausius-Duhem inequality can be written as ... 

Perhaps Clausius hoped to, but did not, provide a way of computing N associated with irreversible processes. Nineteenth-century thermodynamics remained in the restricted domain of idealized reversible transformation and without a theory that related entropy explicitly to irreversible processes. Some expressed the view that entropy is a physical quantity that is spatially distributed and transported (e.g. Bertrand [7] in his 1887 text), but still no theory relating irreversible processes to entropy was formulated in the nineteenth century. 

In his pioneering work on the thermodynamics of chemical processes, Theophile De Bonder (1872-1957) [14-16] incorporated the uncompensated transformation or uncompensated heat of Clausius into the formalism of the Second Law through the concept of affinity, which is presented in the next chapter. This modem approach incorporates irreversibility into the formalism of the Second Law by providing explicit expressions for the computation of entropy produced by irreversible processes [17-19]. We shall follow this more general approach in which, along with thermodynamic states, irreversible processes appear explicitly in the formalism. 

The task of finding a mathematical formulation of the second law of thermodynamics was accomplished by Sadi Carnot and Clausius on the traditional, macroscopic side, and, again, by Boltzmann from a molecular, statistical perspective. What is sought is a state function that quantitatively describes the degree of dispersion in a chemical system. This is entropy, and its symbol is S. It must increase in any irreversible process. 

During the adiabatic process, the two systems did not exchange heat with their surroundings. In the process, the system entropy has been increcised by 11.7 J/K therefore, according to the Clausius inequality(4.12), this is an irreversible process. 

The principle of the constant increase of entropy (Clausius 1865) is that any irreversible process leads to an increase of entropy in the universe at reversible processes, the entropy in the universe remains unchanged. 

Some of you may have heard about the thermodynamic time arrow. Gases escape from open containers and heat flows from a hot body to its colder environment. Never has spontaneous reversal of such processes been observed. We call these irreversible processes. The world is always heading forward in time. Mathematically this is expressed by Clausius theorem. 

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