Technology & Engineering
Clausius Theorem
Clausius Theorem, a fundamental concept in thermodynamics, states that heat cannot spontaneously flow from a colder body to a hotter body. It provides a basis for the second law of thermodynamics, which governs the direction of heat transfer and the efficiency of heat engines. The theorem has wide-ranging applications in engineering, including the design and optimization of energy systems.
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3 Key excerpts on "Clausius Theorem"
eBook - PDF- Wilhelm Jost(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
According to the former, you cannot continuously transform heat into work without something else happening. According to the latter, you cannot transfer heat from a cooler to a warmer body without something else happening. From either of these can be deduced two far-reaching mathematical generalizations: 2A. (Carnot's theorem). For all thermodynamic systems, the absolute temperature is an integrating denominator for the reversible heat differential. This implies the existence of a new function of state: the entropy. 2B. (Law of increasing entropy). Entropy increases monotonically in every irreversible adiabatic process. Principles 2A and 2B comprise, in effect, a mathematical statement of the second law. In either the Kelvin-Planck or the Clausius formulation, the derivation of the above mathematical principles is unreasonably lengthy and tenuous. In its course, certain idealized devices and processes must be evoked and extensive use made of engineering terminology and concepts. Al-though the arguments and conclusions are, without doubt, correct, the line of development is open to criticism on the following logical and aesthetic grounds: 1. T h e line of reasoning leading from the physical to the mathematical principles is excessively long. 2. There is no clear separation between the physical and mathematical content of the theory. 3. T h e laws governing the more commonly encountered physico-chem-ical systems ought to be derivable solely from phenomena occurring in such systems—not from the performance characteristics of steam engines! It is understood, of course, that the historical development of a science will not generally proceed in a sequence of logical orderly steps. But once all the facts are known, the subject ought to be restructured with a view toward aesthetics and logical consistency. Caratheodory (1873-1950) was a mathematician of Greek origin, working in Germany during the early years of this century. In 1909,
- Michael Clifford, Kathy Simmons, Philip Shipway(Authors)
- 2009(Publication Date)
- CRC Press(Publisher)
First, imagine the cycle of the heat engine has only one process during which all the heat supply Q 1 takes place from the hot reservoir at constant temperature T 1 , and one process during which all the heat rejection Q 2 takes place to the cold reservoir at constant temperature T 2 . If no heat transfer takes place during any other process in the cycle, then d T Q Q T 1 1 Q T 2 2 (4.11) If the cycle is reversible, both expressions (4.8) and (4.9) for efficiency are valid. This requires Q Q 2 1 T T 2 1 For any reversible heat engine (or closed system undergoing a reversible cycle), the integral around the cycle of d T Q will be zero; for all irreversible heat engines (or closed systems undergoing an irreversible cycle), this integral will be negative. Thermodynamics 229 T 1 1000 K T 3 800 K T 1 400 K 1000 K 400 K 800 K 400 K A B Figure 4.20 A heat engine operating between more than two reservoirs will always have an efficiency less than the Carnot efficiency and hence Q T 2 2 Q T 1 1 Substituting into equation (4.11) gives the result for the first part of the Clausius inequality, that which applies to the operation of a reversible heat engine : d T Q 0 (4.12) If the cycle contains irreversible processes, the cycle will be irreversible and the efficiency of the heat engine must be lower than the Carnot efficiency, Carnot. In this case, equations (4.8) and (4.9) now require Q T 1 1 Q T 2 2 and so Q T 1 1 Q T 2 2 0 Substituting this into equation (4.11) gives the result for the second part of the Clausius inequality, that which applies to the operation of an irreversible heat engine : d T Q 0 (4.13) Although the Clausius inequality has been derived here through arguments applied to heat engines, the result is true for any closed system undergoing a reversible or irreversible cycle. The result for the reversible cycle has particular significance, as it led Clausius to conclude that he had discovered a new property, which he named entropy .
eBook - PDF- Pierluigi Zotto, Sergio Lo Russo, Paolo Sartori(Authors)
- 2022(Publication Date)
- Società Editrice Esculapio(Publisher)
The relevant point of Car- not’s theorem is the independence of the efficiency from the used thermodynamic fluid, implying that the efficiency of a whatever machine operating with only two heat reservoirs is the same as the one of a Carnot cycle using an ideal gas Second Law of Thermodynamics Chapter 17 324 η ( R) = 1 − T 2 T 1 . The efficiency of any machine M, which operates between two reservoirs, absorbing heat Q1 from the hot reservoir at temperature T1 and releasing heat Q2 to the cold reservoir at temperature T2, is η M ( ) = 1 + Q c Q h = 1 + Q 2 Q 1 thus, by applying Carnot’s theorem, we get η M ( ) ≤ η R ( ) ⇒ 1 + Q 2 Q 1 ≤ 1 − T 2 T 1 providing the relationship Q 1 T 1 + Q 2 T 2 ≤ 0 where the equality holds only if also M is reversible, i.e. Q 1 R ( ) T 1 + Q 2 R ( ) T 2 = 0 and the inequality holds if M is irreversible, i.e. Q 1 I ( ) T 1 + Q 2 I ( ) T 2 < 0. 17.5 Clausius’ Theorem Clausius’ theorem represents a consequence of Carnot’s theorem if several heat reser- voirs are used by a heat engine. It states: Given any cyclic machine M, the sum, performed on all the heat reservoirs used by the machine during a cycle, of the ratio between the heat exchanged during every cycle with a reservoir and its temperature is dQ i T i ≤ 0 i =1 N ∑ where equality holds if M is reversible and inequality holds if M is irreversible. Consider a whatever heat engine M, i.e. reversible or irreversible and which uses a whatever thermodynamic fluid, which exchanges heat with N heat reservoirs at tempera- tures T1, T2, …, TN. Consider other N auxiliary reversible heat engines R1, R2, …, RN such that the i-th ma- chine exchanges heat with the i-th heat reservoir at temperature Ti and another heat reser- voir at temperature T0 shared by all the auxiliary machines.
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