4.2 Heat Engines

Heat engines convert thermal energy into mechanical work by operating between a high-temperature reservoir and a low-temperature reservoir. A working substance, such as gas or steam, moves through a cycle that transfers heat and produces useful work.

Efficiency measures how much input heat becomes work. Carnot efficiency sets the theoretical maximum, while real-world factors like friction, heat loss, and incomplete combustion reduce actual efficiency in engines such as Otto and Diesel cycles.

Heat Engines

Components of heat engines

• Convert thermal energy (heat) into mechanical energy (work) by operating between a high-temperature reservoir (heat source) and a low-temperature reservoir (heat sink)
• Heat flows from the high-temperature reservoir to the low-temperature reservoir, with some of the heat converted into work during this process
• Working substance undergoes the thermodynamic cycle (gas or steam)
• Heat source provides heat to the working substance
• Heat sink absorbs heat from the working substance
• Mechanical components convert the expansion and contraction of the working substance into useful work
  • Pistons
  • Turbines

Factors in heat engine efficiency

• Efficiency is the ratio of work output to heat input, calculated using the formula $\eta = \frac{W}{Q_H}$
  • $\eta$ represents efficiency
  • $W$ represents work output
  • $Q_H$ represents heat input from the high-temperature reservoir
• Carnot efficiency is the maximum theoretical efficiency of a heat engine operating between two temperatures, calculated using the formula $\eta_{Carnot} = 1 - \frac{T_C}{T_H}$
  • $T_C$ represents the temperature of the cold reservoir (heat sink)
  • $T_H$ represents the temperature of the hot reservoir (heat source)
  • A larger temperature difference between the heat source and heat sink leads to higher efficiency
• Irreversibilities and losses reduce the actual efficiency of heat engines below the Carnot efficiency
  • Friction
  • Heat loss
  • Incomplete combustion

Efficiency calculations for ideal gas engines

• Thermodynamic cycles represent the series of processes that the working substance undergoes in a heat engine, returning to its initial state after completing a cycle
• Otto cycle (constant volume heat addition)
  • Efficiency calculated using the formula $\eta = 1 - \frac{1}{r^{\gamma-1}}$
    • $r$ represents the compression ratio
    • $\gamma$ represents the specific heat ratio of the gas
• Diesel cycle (constant pressure heat addition)
  • Efficiency calculated using the formula $\eta = 1 - \frac{1}{r^{\gamma-1}} \left(\frac{r_c^{\gamma}-1}{\gamma(r_c-1)}\right)$
    • $r$ represents the compression ratio
    • $r_c$ represents the cutoff ratio
    • $\gamma$ represents the specific heat ratio of the gas
• Brayton cycle (constant pressure heat addition and rejection)
  • Efficiency calculated using the formula $\eta = 1 - \frac{1}{r_p^{\frac{\gamma-1}{\gamma}}}$
    • $r_p$ represents the pressure ratio
    • $\gamma$ represents the specific heat ratio of the gas

Thermodynamic principles and analysis

• The first law of thermodynamics relates the change in internal energy of a system to heat added and work done
• Pressure-volume diagrams are used to visualize and analyze thermodynamic cycles
• An isentropic process is an idealized thermodynamic process that is adiabatic and reversible
• The Kelvin-Planck statement of the second law of thermodynamics states that it is impossible to construct a heat engine that operates in a cycle and produces no effect other than the extraction of heat from a reservoir and the performance of an equivalent amount of work