Carnot efficiency with thermodynamic temperature

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 ... 

This result, called the Carnot efficiency or the thermodynamic efficiency, places a fundamental limit on the efficiency with which heat can be converted to mechanical work. Only if the high temperature, T, were infinite or the low temperature, T , were zero would it be possible to have a heat engine operate with 100% efficiency. To maximize efficiency, the greatest possible temperature difference should be used. Although we derived this result specifically for the ideal gas, we will show later in this section that it applies to any reversible engine operating between two temperatures. For a real engine, which must operate irreversibly, the actual efficiency must be lower than the thermodynamic efficiency. 

In the classical equilibrium thermodynamics, Stirling and Ericsson cycles have an efficiency that goes to the Carnot efficiency, as it is shown in some textbooks. These three cycles have the common characteristics, including two isothermal processes. The objection to the classical point of view is that reservoirs coupled to the engine modeled by any of these cycles do not have the same temperature as the working fluid because this working fluid is not in direct thermal contact with the reservoir. Thus, an alternative study of these cycles is using finite... 

If instead of a rubberUke deformation a phase transition occurs, the isothermal processes are represented by horizontal lines (since the force is independent of the length), as is shown in Fig. 8.20b. If in each of the two cycles described the same adiabatics are involved, the net work performed is greater for the one with the phase transition. This is analogous to using a condensed vapor in the more conventional Carnot cycle. The thermodynamic efficiency remains the same since it depends only on the two temperatures at which the engine operates. The deliverance... 

Therefore in the case of a reversible Carnot cycle, Clausius integral, the algebraic sum of all the by-temperature-reduced heats [both delivered (directly) to the cycle and drained off (directly) from the cycle], equals 0. According to Thomson-Planck s formulation there must be both heat AQ,=w delivered into the cycle O and heat AQ,=o drained off from the cycle. As a consequence of the existence of these heats (the input AQw and the "residual" AQo). and in accordance with the 1. Principle of Thermod5mamics, it must be valid that rjmax < 1-Another formulation of the It. Principle of Thermodynamics is Carnot s theorem, the fitst part of which states The efficiencies of all reversible Carnot cycles with the working temperatures Tyj and To are equal. 

We first check that this formula for a thermodynamically maximal temperature yields the Carnot efficiency. As heat transfer is included explicitly in the formulae, we simply remove the sink (E = S =0) and associate a temperature T E E /S with ss PP P... 

Entropy has both macroscopic and molecular aspects, as we have seen. The thermodynamic entropy is defined in terms of heat transferred in a reversible process. When heat is transferred from a higher to a lower temperature, the entropy of the universe increases. Heat at a lower temperature is less efficient in driving a Carnot engine, so entropy has some connection with the efficiency with which heat can be turned into work. In a later chapter we will discuss free energy, which is closely associated with the thermodynamic entropy. 

In accordance with (4.9), the thermal efficiency t of a reversible Carnot cycle (4.9) solely depends on the ratio between the thermodynamic temperature in in the two heat reservoirs, in this case T and T3. This property of the Carnot process has made possible the definition of an absolute temperature scale, which is independent of the thermometer substances used (Lord Kelvin, 1854). 

The thermal efficiency of the process (QE) should be compared with a thermodynamically ideal Carnot cycle, which can be done by comparing the respective indicator diagrams. These show the variation of temperamre, volume and pressure in the combustion chamber during the operating cycle. In the Carnot cycle one mole of gas is subjected to alternate isothermal and adiabatic compression or expansion at two temperatures. By die first law of thermodynamics the isothermal work done on (compression) or by the gas (expansion) is accompanied by the absorption or evolution of heat (Figure 2.2). 

It was pointed out in Chapter I that the desire for higher maximum temperature (T nx) in thermodynamic cycles, coupled with low heat rejection temperature (Tmin), is essentially based on attempting to emulate the Carnot cycle, in which the efficiency increases with... 

Fuel cells such as the one shown on Fig. 3.4a convert H2 to H20 and produce electrical power with no intermediate combustion cycle. Thus their thermodynamic efficiency compares favorably with thermal power generation which is limited by Carnot-type constraints. One important advantage of solid electrolyte fuel cells is that, due to their high operating temperature (typically 700° to 1100°C), they offer the possibility of "internal reforming" which permits the use of fuels such as methane without a separate external reformer.33 36... 

A fascinating point, especially to physical chemists, is the potential theoretical efficiency of fuel cells. Conventional combustion machines principally transfer energy from hot parts to cold parts of the machine and, thus, convert some of the energy to mechanical work. The theoretical efficiency is given by the so-called Carnot cycle and depends strongly on the temperature difference, see Fig. 13.3. In fuel cells, the maximum efficiency is given by the relation of the useable free reaction enthalpy G to the enthalpy H (AG = AH - T AS). For hydrogen-fuelled cells the reaction takes place as shown in Eq. (13.1a). With A//R = 241.8 kJ/mol and AGr = 228.5 under standard conditions (0 °C andp = 100 kPa) there is a theoretical efficiency of 95%. If the reaction results in condensed H20, the thermodynamic values are A//R = 285.8 kJ/ mol and AGR = 237.1 and the efficiency can then be calculated as 83%. 

In the conversion of fossil and nuclear energy to electricity, the value of high temperature solution phase thermodynamics in improving plant reliability has been far less obvious than that of classical thermodynamics in predicting Carnot cycle efficiency. Experimental studies under conditions appropriate to modern boiler plant are difficult and with little pressure from designers for such studies this area of thermodynamic study has been seriously neglected until the last decade or two. 

The existence of a finite heat transfer in the isothermal processes is affected with the assumption of a non-endoreversible cycle with ideal gas as working substance. Power output and ecological function have also an issue that shows direct dependence on the temperature of the working substance. Expressions obtained with the changes of variables have the virtue of leading directly to the shape of the efficiency through Z, function. Thus, in classical equilibrium thermodynamics, the Stirling cycle has its efficiency like the Carnot cycle efficiency in finite time thermodynamics, this cycle has an efficiency in their limit cases as the Curzon-Ahlborn cycle efficiency. 

Let ns nse an ideal gas in a Carnot cycle and find the efficiency of the cycle by using ideal-gas properties in ennmerating the changes in the four steps of the cycle. Let us designate the intial state of step 1 with the subscript A, the initial state of step 2 with B, and so on. The high temperature at step 1, which is i on the thermodynamic temperatnre scale, will be Tj on the ideal-gas temperature scale. The low temperature of step 3 will be T2, corresponding to 02. The work and heat terms of a step will be designated with subscripts 1, 2, 3, or 4. 

Thus, Sadi Carnot s analysis of Carnot cycle provided the theory for the formulation of the first and the second law of thermodynamics. His concept is that for a system undergoing a cycle, the net heat transfer is equal to the net work done, which led to the first law of thermodynamics. Similarly, the concept that a heat engine cannot convert all the heat absorbed from a heat source at a single temperature into work even under ideal condition led to the second law of thermodynamics. Carnot cycle efficiency gives the idea about the maximmn theoretical efficiency of an engine. Sadi Carnot was rightly honored with the title Father of Thermodynamics for his invaluable contribution to thermodynamics. 

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