Carnot

Carnot s cycle A hypothetical scheme for an ideal heat machine. Shows that the maximum efficiency for the conversion of heat into work depends only on the two temperatures between which the heat engine works, and not at all on the nature of the substance employed. 

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. 

Fig. 1. Schematic representation of (a) Carnot heat engine and (b) Carnot refrigerator used as a heat pump.

Because batteries direcdy convert chemical energy to electrical energy ia an isothermal process, they are not limited by the Carnot efficiency. The thermodynamic efficiency S for electrochemical processes is given by ... 

Whenever energy is transformed from one form to another, an iaefficiency of conversion occurs. Electrochemical reactions having efficiencies of 90% or greater are common. In contrast, Carnot heat engine conversions operate at about 40% efficiency. The operation of practical cells always results ia less than theoretical thermodynamic prediction for release of useful energy because of irreversible (polarization) losses of the electrode reactions. The overall electrochemical efficiency is, therefore, defined by ... 

The industrial economy depends heavily on electrochemical processes. Electrochemical systems have inherent advantages such as ambient temperature operation, easily controlled reaction rates, and minimal environmental impact (qv). Electrosynthesis is used in a number of commercial processes. Batteries and fuel cells, used for the interconversion and storage of energy, are not limited by the Carnot efficiency of thermal devices. Corrosion, another electrochemical process, is estimated to cost hundreds of millions of dollars aimuaUy in the United States alone (see Corrosion and CORROSION control). Electrochemical systems can be described using the fundamental principles of thermodynamics, kinetics, and transport phenomena. 

The ideal mechanical power requirement of a thermocompression evaporator is given by the Carnot equation ... 

Tc- This may require Carnot engines or heat pumps internal to the system that provide for the reversible transfer of heat from the temperature of the flowing fluid to that of the surroundings. Since Carnot engines and heat pumps are cychc, they undergo uo net change of state. 

Expansion and Exit Losses For ducts of any cross section, the frictional loss for a sudden enlargement (Fig. 6-13c) with turbulent flow is given by the Borda-Carnot equation ... 

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. 

This eyele illustrates several desirable features of a low-temperature proeess. First, the expander should be applied at the lowest temperature level in the eyele beeause this is where it is the most thermo-dynamieally effeetive, that is, it has the best Carnot or Seeond Law... 

The process illustrates the use of mechanical refrigeration in its high-efficiency temperature range the maximum use of compression energy because of its high efficiency and the use of turboexpansion at a low temperature—its Carnot efficiency is best at low temperatures, especially because it permits large use of the efficient pressure effect. 

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. 

The Carnot eyele is the optimum eyele and all eyeles ineline toward this optimum. Maximum thermal effieieney is aehieved by approaehing the isothermal eompression and expansion of the Carnot eyele, or by inter-eooling in eompression and reheating in the expansion proeess. Figure 2-18 shows the intereooled regenerative reheat eyele, whieh approaehes this optimum eyele in a praetieal fashion. 

For any process converting heat energy to mechanical efficiency, the Carnot efficiency is the theoretical maximum. It is calculated as... 

The second law of thermodynamics was actually postulated by Carnot prior to the development of the first law. The original statements made concerning the second law were negative—they said what would not happen. The second law states that heat will not flow, in itself, from cold to hot. While no mathematical relationships come directly from the second law, a set of equations can be developed by adding a few assumptions for use in compressor analysis. For a reversible process, entropy, s, can be defined in differential form as... 

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