Carnots Cycle

The Carnot cycle is an ideal thermodynamic cycle that represents the most efficient cycle for a heat engine and refrigeration machine operating between two temperature limits. It consists of four reversible processes (1) reversible 

Thermal efficiency of this Carnot cycle is given as 

With the application of the second law of thermodynamics, it can be shown that for reversible heat addition at high temperature and heat rejection at low 

Substituting Equation 3.40 into Equation 3.39, the Carnot cycle efficiency 

An automobile engine burns a fuel to the combustion product at a temperature of 1100°C and rejects heat in a radiator with exhaust temperature of 200°C. What is the maximum possible efficiency of this 

Sadi Carnot s most valuable contribution to thermodynamics is Carnot s ideal heat engine operating with Carnot cycle. His works on ideal heat engine provided the foundation for quantitative mathematical formulation of Carnot efficiency based on Carnot s theorem. However, Carnot s research findings were not well known until another scientist Benoit Pierre Emile Clapeyron followed in his footsteps and experimented with the change in pressure and volume of the processes of a cycle and its effect on work done. Clapeyron developed Carnot s idea of the efficiency of 

From Fig. 4.1, it can be seen that maximum Carnot efficiency of a reversible engine is given by 

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. 

The ideal engine, that is, without loss due to friction etc., is referred to as the reversible engine. This reversible engine can be operated under AQ2 = 0, therefore the efficiency is = 1. Note that the working material of this reversible engine is not necessarily a perfect gas. 

The Carnot cycle for this heat engine is shown in Fig. D.2a where the processes are represented as follows  

Isothermal expansion process Keeping the same temperature Ti with a high temperature thermal bath, the piston is slowly pulled up to absorb heat AQoo- 

Adiabatic expansion process The engine is moved out of the bath, and the piston continues to be pulled up until the temperature reaches T2. Note that Ti Ti. 

Isothermal compression process Keeping the same temperature T2 with a low temperature thermal bath the piston is slowly pushed down to exhaust heat A Qg. 

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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 ... 

Rankine Cycle Thermodynamics. Carnot cycles provide the highest theoretical efficiency possible, but these are entirely gas phase. A drawback to a Carnot cycle is the need for gas compression. Producing efficient, large-volume compressors has been such a problem that combustion turbines and jet engines were not practical until the late 1940s. 

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 COP in real refrigeratiou cycles is always less than for the ideal (Carnot) cycle and there is constant effort to achieve this ideal value. 

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. 

The thermal efficiency of an ideal simple cycle is decreased by the addition of an intercooler. Figure 2-7 shows the schematic of such a cycle. The ideal simple gas turbine cycle is 1-2-3-4-1, and the cycle with the intercooling added is -a-b-c-2- i-A-. Both cycles in their ideal form are reversible and can be simulated by a number of Carnot cycles. Thus, if the simple gas turbine cycle 1-2-3-4-1 is divided into a number of cycles like m-n-o-p-m,... 

All the Carnot cycles making up the simple gas turbine cycle have the same efficiency. Likewise, all of the Carnot cycles into which the cycle a-b-c-2-a might similarly be divided have a common value of efficiency lower than the Carnot cycles which comprise cycle 1-2-3-4-1. Thus, the addition of an intercooler, which adds a-b-c-2-a to the simple cycle, lowers the efficiency of the 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).

In his search for high efficiency, the designer of a gas turbine power plant will attempt to emulate these features of the Carnot cycle. 

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. 

In Chapter 1, the gas turbine plant was considered briefly in relation to an ideal plant based on the Carnot cycle. From the simple analysis in Section 1.4, it was explained that the closed cycle gas turbine failed to match the Carnot plant in thermal efficiency because of... 

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... 

Stirling engines also have the maximum theoretical possible efficiency because their power cycle (their theoretical pressure volume diagram) matches the Carnot cycle. The Carnot cycle, first described by the French physicist Sadi Carnot, determines the maximum theoretical efficiency of any heat engine operating between a hot and a cold reservoir. The Carnot efficiency formula is... 

Applying the first law of thermodynamics to the Carnot cycle gives... 

Because the gas in the Carnot cycle starts and ends at the same state, the system s entropy does not change during a cycle. Now apply the second law to the universe for the case of the Carnot cycle. Because the processes are reversible, the entropy of the universe does not change by Equation 2b. This can be written ... 

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