Temperature Carnot cycle

The Carnot cycle is formulated directly from the second law of thermodynamics. It is a perfectly reversible, adiabatic cycle consisting of two constant entropy processes and two constant temperature processes. It defines the ultimate efficiency for any process operating between two temperatures. The coefficient of performance (COP) of the reverse Carnot cycle (refrigerator) is expressed as... 

If the Carnot cycle is used to calculate the work embedded in the thermal flows with the assumption that the heat-transfer coefficient, U, is constant and the process temperature is much greater than AT, a simple derivation yields the following ... 

The Carnot refrigeratiou cycle is reversible and consists of adiabatic (iseutropic due to reversible character) compression (1-2), isothermal rejection of heat (2-3), adiabatic expansion (3-4) and isothermal addition of heat (4-1). The temperature-entropy diagram is shown in Fig. 11-70. The Carnot cycle is an unattainable ideal which serves as a standard of comparison and it provides a convenient guide to the temperatures that should be maintained to achieve maximum effectiveness. 

For a Carnot cycle (where AQ = TA.s), the COP for the refrigeratiou apphcatiou becomes (note than T is absolute temperature [K]) ... 

The Intercooled Regenerative Reheat Cycle The Carnot cycle is the optimum cycle between two temperatures, and all cycles try to approach this optimum. Maximum thermal efficiency is achieved by approaching the isothermal compression and expansion of the Carnot cycle or by intercoohng in compression and reheating in the expansion process. The intercooled regenerative reheat cycle approaches this optimum cycle in a practical fashion. This cycle achieves the maximum efficiency and work output of any of the cycles described to this point. With the insertion of an intercooler in the compressor, the pressure ratio for maximum efficiency moves to a much higher ratio, as indicated in Fig. 29-36. 

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). 

Figure 2.2 The indicator diagrams for the Carnot and the Otto engines. The Carnot cycle operates between the two temperatures Tj and T2 only, whereas the Otto cycle undergoes a temperature increase as a result of combustion.

It follows that the efficiency of the Carnot engine is entirely determined by the temperatures of the two isothermal processes. The Otto cycle, being a real process, does not have ideal isothermal or adiabatic expansion and contraction of the gas phase due to the finite thermal losses of the combustion chamber and resistance to the movement of the piston, and because the product gases are not at tlrermodynamic equilibrium. Furthermore the heat of combustion is mainly evolved during a short time, after the gas has been compressed by the piston. This gives rise to an additional increase in temperature which is not accompanied by a large change in volume due to the constraint applied by tire piston. The efficiency, QE, expressed as a function of the compression ratio (r) can only be assumed therefore to be an approximation to the ideal gas Carnot cycle. 

Carnot cycle The cycle of a perfect heat engine, in which the heat is and rejected at constant temperature and the whole cycle is perfectly reversible. 

The second law of thermodynamics may be used to show that a cyclic heat power plant (or cyclic heat engine) achieves maximum efficiency by operating on a reversible cycle called the Carnot cycle for a given (maximum) temperature of supply (T ax) and given (minimum) temperature of heat rejection (T jn). Such a Carnot power plant receives all its heat (Qq) at the maximum temperature (i.e. Tq = and rejects all its heat (Q ) at the minimum temperature (i.e. 7 = 7, in) the other processes are reversible and adiabatic and therefore isentropic (see the temperature-entropy diagram of Fig. 1.8). Its thermal efficiency is... 

Fig. 1.8. Temperature-entropy diagram for a Carnot cycle (after Ref. (11).

Two objectives are immediately clear. If the top temperature can be raised and the bottom temperature lowered, then the ratio t= (Tjnin/Tjnax) decreased and, as with a Carnot cycle, thermal efficiency will be increased (for given /a,). The limit on top temperature is likely to be metallurgical while that on the bottom temperature is of the surrounding atmosphere. 

The exergy equation (2.26) enables useful information on the irreversibilities and lost work to be obtained, in comparison with a Carnot cycle operating within the same temperature limits (T ,ax = Ey and T in = To). Note first that if the heat supplied is the same to each of the two cycles (Carnot and IJB), then the work output from the Carnot engine (Wcar) is greater than that of the IJB cycle (Wijg), and the heat rejected from the former is less than that rejected by the latter. 

In the ultimate version of the reheated and intercooled reversible cycle [CICICIC- HTHTHT- XJr, both the compression and expansion are divided into a large number of small processes, and a heat exchanger is also used (Fig. 3.6). Then the efficiency approaches that of a Carnot cycle since all the heat is supplied at the maximum temperature Tr = T ax and all the heat is rejected at the minimum temperature = r,nin. 

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... 

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 ... 

Transfer of heat through the walls of the evaporator and condenser requires a temperature difference. This is shown on the modified reversed Carnot cycle (Figure 2.5). For temperature differences of 5 K on hoth the evaporator and condenser, the fluid operating temperatures would he 263.15 K and 313.15 K, and the coefficient of performance falls to 5.26. 

Figure 2.10 (a) A schematic Carnot cycle in which isotherms at empirical temperatures 6 and 62 alternate with adiabatics in a reversible closed path. The enclosed area gives the net work produced in the cycle, (b) The area enclosed by a reversible cyclic process can be approximated by the zig-zag closed path of the isothermal and adiabatic lines of many small Carnot cycles. 

In the next chapter, we will return to the Carnot cycle, describe it quantitatively for an ideal gas with constant heat capacity as the working fluid in the engine, and show that the thermodynamic temperature defined through equation (2.34) or (2.35) is proportional to the absolute temperature, defined through the ideal gas equation pVm = RT. The proportionality constant between the two scales can be set equal to one, so that temperatures on the two scales are the same. That is, 7 °Absolute) = T(Kelvin).r... 

We remember from the earlier discussion of the Carnot cycle that an empirical temperature scale is based on some arbitrary physical property (such as density, electrical resistance, magnetic susceptibility, etc.) that changes in a way that is continuous and single valued. 

To summarize, the Carnot cycle or the Caratheodory principle leads to an integrating denominator that converts the inexact differential 8qrev into an exact differential. This integrating denominator can assume an infinite number of forms, one of which is the thermodynamic (Kelvin) temperature T that is equal to the ideal gas (absolute) temperature. The result is... 

In Chapter 2 (Section 2.2a) we qualitatively described the Carnot cycle, but were not able to quantitatively represent the process on a p— V diagram because we did not know the pressure-volume relationship for a reversible adiabatic process. We now know this relationship (see section 3.3c), and in Figure 3.3, we compare a series of p-V adiabats with different starting temperatures for an... 

In summary, the Carnot cycle can be used to define the thermodynamic temperature (see Section 2.2b), show that this thermodynamic temperature is an integrating denominator that converts the inexact differential bq into an exact differential of the entropy dS, and show that this thermodynamic temperature is the same as the absolute temperature obtained from the ideal gas. This hypothetical engine is indeed a useful one to consider. 

The second law of thermodynamics says that in a Carnot cycle Q/T = constant. This law allows for the definition of a temperature scale if we arbitrarily assign the value of a reference temperature. If we give the value T3 = 273.16K to the triple point (see Gibbs law, Section 8.2) of water, the temperature in kelvin units [K] can be expressed as ... 

The experimental realization of a Carnot cycle to measure the temperature T is unusual. The coincidence of the thermodynamic temperature T with the temperature read by a gas thermometer, for example, allows the use of such thermometer to know T. As we shall see, also other laws of physics relating T with physical parameters other than heat can be used to get an absolute measure of T. 

The modern absolute temperature definition, suggested by W. Thomson, is based on Carnot cycle A scale whose definition does not depend on a specific substance is called absolute ... 

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. 

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