All undergraduate physics students are taught the Carnot cycle as an example of a thermodynamic engine. The Carnot cycle is optimized for efficiency but unfortunately yields zero power, and is therefore not very useful in reality. However, it’s analysis is mathematically elegant. How can we modify the Carnot analysis for real engines which are optimized for power without sacrificing mathematical simplicity?
The Carnot Cycle
Carnot’s theorem states that for a heat engine operating with a hot thermal reservoir (temperature TH), and a cold thermal reservoir (temperature TC), none can have an efficiency greater than that of the following cycle:
1. Isothermal, reversible expansion of the working fluid at TH
2. Adiabatic, reversible expansion of the working fluid, which cools it to TC
3. Isothermal, reversible compression of the working fluid at TC
4. Adiabatic, reversible expansion of the working fluid, which heats it back up to TH (with temperatures in Kelvin = oCelsius + 273)
In practice, there are difficulties in creating a Carnot heat engine. In order to achieve the theoretical Carnot efficiency, all of the processes in the cycle must be perfectly reversible, and thus the cycle time, τ, approaches infinity and there is zero power output. In the other extreme case, where τ approaches zero (an instantaneous cycle), eventually we find that the heat flows straight from the hot reservoir to the cold reservoir (Qin = Qout), resulting in no work done and once again, zero power.
Fig 1. Pressure-Volume (PV) diagram for the Carnot cycle, with modifications to maximize power
Maximum Theoretical Efficiency
Disregarding issues of power, the efficiency of a Carnot cycle, ${\eta}_{\epsilon}$, is given in Eq. 1 (Carnot, 1824).
${\eta}_{\epsilon}=1-\frac{T_C}{T_H}\tag{1}$
Efficiency at Maximum Power Output
Somewhere between these two extremes, maximum power output is achieved. In order to estimate the efficiency of a heat engine at maximum power, ${\eta}_{p}$, UBC Physics Professors Frank Curzon and Boye Ahlborn considered the fact that there is a limit to the rate at which heat can be exchanged between the working fluid and the hot and cold reservoirs, and obtained a new expression for the efficiency of a heat engine (Eqn. 2)[note] F. L. Curzon and B. Ahlborn, Efficiency of a Carnot engine at maximum power output, American Journal of Physics 43, 22 (1975); https://doi.org/10.1119/1.10023 [/note].
${\eta}_{p}=1-\sqrt\frac{T_C}{T_H}\tag{2}$
Examples from the real world given in the Curzon-Ahlborn paper are shown in Table 1. Reality is much closer to the Curzon-Ahlborn efficiency than to the Carnot efficiency.
Table 1. Examples of real-world power generation efficiencies.
Fig 2. Examples of electricity generating plants.
The UBC Paper
Curzon & Ahlborn published their result in the American Journal of Physics in 1975. While it had previously been derived in 1957 by Ivan Ivanovich Novikov, in the journal Atomnaya Energiya, and independently in the same year by Paul Chambadal, in a book Les centrales nucléaires, these publication in Russian and French had little impact. Neither the UBC authors nor the editors and referees at the American Journal of Physics were aware of them. The publication of the result in an English-language journal brought the result to a much wider audience. To this day, this paper is still cited in the scientific literature about fifty times a year.
Fig 3. Curzon and Ahlborn’s paper, initially rejected as “correct but of no interest”, still garnering ~50 citations a year, 45 years later.
The Authors
Professors Curzon and Ahlborn joined the UBC Physics Department in the early 1960s. Frank Curzon came to Vancouver from the University of London, and Boye Ahlborn came from Ludwig-Maximilians-Universität Munchën. At the time of their famous 1975 paper on thermodynamics, they were working together in the Plasma Physics group. This research work produced two successful Vancouver companies, Vortek Industries and TIR Systems.
Fig.4. Boye Ahlborn c. 1975.
Fig.5. Frank Curzon c.1975.
Boye was later Director of the UBC’s Engineering Physics Program and was instrumental in setting up the Engineering Physics Project Laboratory in the Hennings Building. Frank’s teaching career at UBC eventually spanned a full half-century.
Model of a Four-Stroke Heat Engine
Fig. 6. Otto cycle engine (Alice Lam, UBC Engineering Physics).
There are many named thermodynamic cycles – Otto (four-stroke gasoline engines in vehicles), Rankine (steam generators in coal-fired and nuclear power stations), Brayton (gas-turbines in aviation) etc. – which all trace locii of slightly differing shapes around the PV diagram. In each case the work done per cycle is the area enclosed by the locus and the efficiency is the work done divided by the heat input by the hot reservoir. The Carnot cycle is just an idealized version that is useful to consider as a limiting case.
Updated by CEW 2019-10-02