The efficiency of a Carnot engine

This remarkable result shows that the efficiency of a Carnot engine is simply related to the ratio of the two absolute temperatures used in the cycle. In normal applications in a power plant, the cold temperature is around room temperature T = 300 K while the hot temperature in a power plant is around T = fiOO K, and thus has an efficiency of 0.5, or 50 percent. This is approximately the maximum efficiency of a typical power plant. The heated steam in a power plant is used to drive a turbine and some such arrangement is used in most heat engines. A Carnot engine operating between 600 K and 300 K must be inefficient, only approximately 50 percent of the heat being converted to work, or the second law of thermodynamics would be violated. The actual efficiency of heat engines must be lower than the Carnot efficiency because they use different thermodynamic cycles and the processes are not reversible. 

Carnot s research also made a major contribution to the second law of thermodynamics. Since the maximum efficiency of a Carnot engine is given by 1 -T( H, if the engine is to be 100 percent efficient (i.e., Cma = 1), Tc must equal zero. This led William Thomson (Lord Kelvin) to propose in 1848 that Tf must be the absolute zero of the temperature scale later known as the absolute scale or Kelvin scale. ... 

An estimate of the efficiency of a heat engine working between two temperatures T and T - can be obtained by assuming the Carnot cycle is used. By combining the results from applying the first and second laws to the Carnot cycle, the Carnot efficiency e, may be written ... 

As the magnirnde of the heat exchanged in an isothermal step of a Carnot cycle is proportional to a function of an empirical temperature scale, the magnitude of the heat exchanged can be used as a thermometric property. An important advantage of this approach is that the measurement is independent of the properties of any particular material, because the efficiency of a Carnot cycle is independent of the working substance in the engine. Thus we define a thermodynamic temperature scale (symbol T) such that... 

The T-s diagram and schematic diagram of the Curzon and Ahlborn (endoreversible Carnot) cycle are shown in Figs. 7.5 and 7.6, respectively (Cuzon, F.L. and Ahlborn, B., Efficiency of a Carnot engine at maximum... 

An important measure of the quality of an engine is its efficiency, [the fraction of the energy that it removes from a high temperature reservoir (the heat term in step I) that it converts into work].7 For the Carnot cycle engine to work as efficiently as possible, the heat transfers should be reversible. Thus, the heat transferred to the system in step I should be from a heat reservoir at temperature Th, and the heat transferred from the system in step III should be to a reservoir at Tc. From Table 2, we see that the efficiency of a Carnot cycle engine is... 

Students are reminded of the upper thermodynamic limit set on the efficiency of a heat engine, for example the internal combustion and gas-turbine engines. The ideal and totally unrealistic engine would operate on the so-called Carnot cycle where the working substance (e.g. the gas) is taken in at the high temperature (Th) and pressure and after doing external work is exhausted at the lower temperature (Tc) and lower pressure. The Carnot efficiency, /, is given by... 

Equations (5.7) and (5.8) are known as Carnot s equations. In Eq. (5.7) the smallest possible value of QC is zero the corresponding value of Tc is the absolute zero of temperature on the Kelvin scale. As mentioned in Sec. 1.4, this occurs at -273.15°C. Equation (5.8) shows that the thermal efficiency of a Carnot engine can approach unity only when TH approaches infinity or Tc approaches zero. On earth nature provides heat reservoirs at neither of these conditions all heat engines therefore operate at thermal efficiencies less than unity. The cold reservoirs naturally available are the atmosphere, lakes and rivers, and the oceans, for which Tc = 300 K. Practical hot reservoirs are objects such as furnaces maintained at high temperature by combustion of fossil fuels and nuclear reactors held at high temperature by fission of radioactive elements, for which T = 600 K. With these values,... 

This is a rough practical limit for the thermal efficiency of a Carnot engine actual heat engines are irreversible, and their thermal efficiencies rarely exceed 0.35. 

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. 

The thermal efficiency of a Carnot engine depends only on the temperature levels and not upon the working substance of the engine. 

As shown in Sec. 5.2, the efficiency of a Carnot heat engine is independent of the working medium of the engine. Similarly, the coefficient of perfomianee of a Carnot refrigerator is... 

By definition, the efficiency of a heat engine is equal to the ratio of the total work W done in the cycle to the heat Qi taken in at the upper temperature hence, by equations (18.2) and (18.7), the efficiency of the hypothetical Carnot engine is... 

Now there is a theorem called Carnot s Theorem, the validity of which depends on the trustworthiness of the Second Law This theorem states that a reversible engine has the maximum efficiency, and further, that the efficiency of all reversible engines working between the same temperature limits is the same This holds good whatever the nature of the substance domg work 1 It can be shown that the efficiency of a reversible engine is connected with the temperature limits referred to by the expression— ... 

Since there is nothing special about the temperature of the cold reservoir, except that d > do, Eqs. (8.18) and (8.19) apply to any reversible heat engine operating between any two thermodynamic temperatures d and do - Equation (8.18) shows that the work produced in a reversible heat engine is directly proportional to the difference in temperatures on the thermodynamic scale, while the efficiency is equal to the ratio of the difference in temperature to the temperature of the hot reservoir. The Carnot formula, Eq. (8.19), which relates the efficiency of a reversible engine to the temperatures of the reservoirs is probably the most celebrated formula in all of thermodynamics. 

That is, no engine can work more efficiently than the reversible engine. In the above discussions we can exchange engine A with B, therefore the conclusion of Carnot is that the efficiency of all reversible engines is the same, and it depends only on T and T2. Thus the efficiency of a reversible engine can be defined as... 

Carnot (1824) concluded that in an irreversible process, the efficiency of a heat engine is always less than unity, which gives the second part of the Second law. That is, in the irreversible process the rate of absorption of heat AQi from the high temperature source that is converted to work becomes smaller than in a reversible process, and the heat output increases therefore, from (D.4) we have... 

The plan of the remaining sections of this chapter is as follows. In Sec. 4.3, a h)q)o-thetical device called a Carnot engine is introduced and used to prove that the two physical statements of the second law (the Clausius statement and the Kelvin-Planck statement) are equivalent, in the sense that if one is true, so is the other. An expression is also derived for the efficiency of a Carnol engine for Ihe purpose of defining thermodynamic temperature. Section 4.4 combines Carnot cycles and the Kelvin-Planck statement to derive the existence... 

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