Carnot heat engine maximum power

Chen, J. (1994). The Maximum Power Output and Maximum Efficiency of an Irreversible Carnot Heat Engine, /, Phys. D Appl. Phys., Vol. 27, pp. 1144-1149 Chen, J. (1996). The Efficiency of an Irreversible Combined Cycle at Maximum Specific Power Output, /, Phys. D Appl. Phys. Vol. 29, pp. 2818-2822 Chen, L. . Zhang, W. Sun, F. (2007). Power, efficiency, entropy generation rate and ecological optimization for a class of generalized universal heat engine cycles. Applied Energy, Vol. 84, pp. 512-525... 

Consider an example for a heat engine where the high temperature reservoir is at Ti, say 1,000°K and the low temperature reservoir at To, say 500°K. The efficiency of the reversible Carnot engine, (13.11), is 0.5, whereas the efficiency of an heat engine with power output, at maximum power, (13.44), is about 0.3. Due to the requirement of power output there is a 40% drop in the efficiency of the heat engine This is the main lesson of this chapter. The analysis is idealized but the makes the point, that achieving power output necessitates losses in efficieny. 

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... 

For a heat engine like a steam turbine in an electric power plant the low temperature is determined by the outdoor environment. This temperature is about 300 K. Engineering considerations limit the high temperature to about 800 K. The maximum efficiency according to Carnot is 0.63 or 63 percent. No matter how skilled the builders of a steam turbine, if the temperatures are 300 K and 800 K, the efficiency will never exceed 63 percent. When you realize that the efficiency can never be larger than about 63 percent, a realizable efficiency of 50 percent looks quite good. 

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... 

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. 

The conventional generation of electrical energy from a fuel requires the use of a heat engine which converts thermal energy to mechanical energy. All heat engines operate by the Carnot cycle, and their maximum efficiency is about 40-50% (for the modern gas-fired power stations, the efficiency is about 55%). 

An internal combustion engine, as well as a major electrical power station are both "heat engines" in the thermodynamic sense, and their theoretical maximum efficiency is that of the Carnot cycle. 

Carnot did not derive a mathematical expression for the maximum efficiency attained by a reversible heat engine in terms of the temperatures between which it operated. This was done later by others who realized the importance of his conclusion. Carnot did, however, find a way of calculating the maximum work that can be generated. (For example, he concluded that 1000 units of heat passing from a body maintained at the temperature of 1 degree to another body maintained at zero would produce, in acting upon the air, 1.395 units of motive power [1, p. 42].)... 

Fig. 14.2. Schematic portrayal of the time history for a Carnot-like engine with friction and finite heat conductance to its reservoirs, optimized to produce maximum average power. The upper figure shows the temperature variation and the lower, the volume changes, as the system goes through its isothermal and adiabatic branches...

On the second, Carnot himself states "In accordance with the principles we have now established, we can reasonably compare the motive power of heat with that of a head of water for both of them there is a maximum which cannot be exceeded, whatever the type of hydraulic machine and whatever the type of heat engine employed. The motive power of a head of water depends upon its height and the quantity of water the motive power of heat depends also on the quantity of caloric and on what may be called - on what we shall call - the height of its fall, that is on the temperature difference of the bodies between which the caloric flows."... 

These delicate engines provide value as educational tools, but they immediately inspire curiosity into the possibility of generating power from one of the many sources of low temperature waste heat (less than 100°C) that are available. A quick look at the Carnot formula shows that an engine operating with a hot side at 100°C and a cold side at 23°C will have a maximum Carnot efficiency of [((373 K—296 K)/373 K) X 100] about 21 percent. If an engine could be built that achieved 25 percent of the possible 21 percent Carnot efficiency it would have about 5 percent overall Carnot efficiency. 

Figure 13.6 schematically gives W, as a function of Qin and at the same time, the corresponding thermodynamic efficiency n = (Wout/Qm) has its highest value, the Carnot value, for an infinitely slow operation of the engine at a zero heat input rate Q but also at zero power output. Note that then T2 —> and T3 T0. The thermodynamic efficiency T is zero when Wout is zero, but now at the maximum possible heat input rate that the engine can absorb, namely, when T2 = T3. Somewhere between these extremes the power output... 

The Carnot efficiency has little practical value. It is a maximum theoretical efficiency of a hypothetical engine. Even if such an engine could be constructed, it would have to be operated at infinitesimally low velocities to allow the heat transfer to occur. It would be very efficient, but it would generate no power (Figure 2-8), thus it would be useless. The same applies to the theoretical fuel cell efficiency. The fuel cell operating at theoretical efficiency would generate no current, and therefore it would be of no practical value. 

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