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 \(\Delta U = Q_h - Q_c - W\) and \(U\) is a state variable (so \(\Delta U = 0\) as this engines are cyclic), 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.
Section 4.2 - Refrigerators
Definition. A refrigerator is a heat engine operated in reverse.
Definition. The coefficient of performance is a fancy name for efficiency, in which
We can then apply the second law to see that
This textbook does not account for heat pumps.
Section 4.3 - Real Heat Engines
An internal combustion engine is a classic example, in which the working substance is a gas.
Definition. An internal combustion engine follows an Otto cycle, in which gas is compressed adiabatically by a piston. Then, during ignition, the temperature and pressure are raised while volume is constant, followed by a power stroke in which the gas expands and does work. Lastly, the gas is vented as pressure is held constant and volume drops.
We can then show that \(e = 1 - (\frac{V_2}{V_1})^{\gamma - 1}\), where \(V_1 / V_2\) is the compression ratio. Unfortunately, if the compression ratio becomes too high, the gas will preignite spontaneously before compression is finished.
Definition. In a diesel engine, only air is compressed, before fuel is sprayed into the engine when air temperature is high enough for ignition. The efficiency then becomes a function of the cutoff ratio.
Definition. In a steam engine, a gas will follow the Rankine cycle, in which water is pumped to a high pressure, converted to a gas and expanded, sent through a turbine as it expands and pressure drops, and then condensed back to an initial volume. The efficiency then becomes a function of enthalpy (\(H = U + PV\)), where \(e = \frac{H_4 - H_1}{H_3 - H_2} \approx 1 - \frac{H_4 - H_1}{H_3 - H_1}\).
For a steam engine, we see steam tables, and everybody becomes unhappy.
Section 4.4 - Real Refrigerators
A refrigerator is normally the reverse of a Rankine cycle. Notably, refrigerants are used instead of water due to the lower freezing and boiling temperatures. However, many are CFCs. We then see that
Definition. The throttling or Joule-Thomson process is used in refrigerators, and is complex. This class skips it for now.
I've also skipped the Liquefaction of Gasses and Towards Absolute Zero sections.