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Chapter 4 - Engines and Refrigerators
Section 4.1 - Heat Engines
Definition. A heat engine is any machine that absorbs heat and converts part of it into work. We model heat engines as accepting heat from a hot reservoir with temperature T_h and rejecting heat to a cold reservoir with temperature T_c, as well as outputting work.
Definition. A reservoir is any object so large that its temperature does not change as it accepts or rejects heat
For a a heat engine, we will denote Q_h and Q_c to represent the heat absorbed from the hot reservoir and heat rejected to the cold reservoir. Then, the net work done by the engine is W. In this model, all signs are positive.
Definition. The efficiency e is the benefit/cost ratio. Fora heat engine, we see that e = \frac{W}{Q_h}. As conservation of energy applies, we know that Q_h = W + Q_c, so that W = Q_h - Q_c. Then, we can write e = \frac{W}{Q_h} = \frac{Q_h - Q_c}{Q_h} = 1 - \frac{Q_c}{Q_h}. We then see that efficiency is always in the range [0, 1].
By the second law of thermodynamics, S_h \geq S_c. We know that S = \frac{Q}{T} for a reservoir, so \frac{Q_h}{T_h} \geq \frac{Q_c}{T_c}, which can be rewritten as \frac{T_c}{T_h} \geq \frac{Q_c}{Q_h}. Then, substituting into the equation for efficiency, e \geq 1 - \frac{T_c}{T_h}. Note that actual efficiency will be less than this limit as entropy will be produced within the engine as well.
Let us now revisit a classic: the Carnot cycle. This cycle consists of isothermal expansion of a gas at temperature T_h, adiabatic expansion of the gas from T = T_h to T = T_c, isothermal compression at T = T_c, and adiabatic compression from T = T_c to T = T_h. By applying the formula of the ideal gas, we see this cycle reaches the maximum efficiency of e = 1 - \frac{T_c}{T_h}. However, this engine is not very practical.