Heat engines Knowledge-Based Study

Study notes

1. What is a Heat Engine?

   A heat engine is any physical system or device that converts thermal energy into useful mechanical work by operating in a thermodynamic cycle.

   Every heat engine relies on a working substance (such as steam, air, or a fuel-air mixture) that undergoes a series of cyclic processes, periodically returning to its initial thermodynamic state.

   Key Components

   • Hot Reservoir (Heat Source): A high-temperature thermal reservoir at temperature $T_H$ that supplies heat $Q_H$ to the working substance.
   • Working Substance: The medium that absorbs heat, expands to perform mechanical work $W$, and releases exhaust heat.
   • Cold Reservoir (Heat Sink): A lower-temperature thermal reservoir at temperature $T_C$ that absorbs rejected heat $Q_C$.

   Because the engine operates in a closed cycle, the net change in internal energy over one complete cycle is zero: $$\Delta U = 0$$

   Work Output and Conservation of Energy

   Applying the First Law of Thermodynamics to one complete cycle, where $\Delta U = Q_{\text{net}} - W_{\text{net}} = 0$, yields the net work output:

   $$W = Q_H - Q_C$$

   Here, $Q_H$ represents the heat absorbed from the high-temperature reservoir, and $Q_C$ is the magnitude of heat rejected to the low-temperature reservoir.

   Thermal Efficiency

   The thermal efficiency ($\eta$) of a heat engine measures the fraction of input heat successfully converted into mechanical work:

   $$\eta = \frac{W}{Q_H} = \frac{Q_H - Q_C}{Q_H} = 1 - \frac{Q_C}{Q_H}$$

   • Efficiency is a dimensionless ratio, often expressed as a percentage.
   • Because some heat is always rejected to the surroundings, $Q_C > 0$, meaning that $\eta$ is always strictly less than 1 (or 100%).

   The Kelvin-Planck Statement

   The limitation that no engine can convert absorbed heat entirely into work is a fundamental principle of nature formalised by the Second Law of Thermodynamics.

   The Kelvin-Planck statement states:

   It is impossible for any device that operates on a thermodynamic cycle to receive heat from a single reservoir and produce a net amount of work.

   Physical Consequences

   • A heat engine must operate between at least two different thermal reservoirs to generate continuous work.
   • Thermal energy cannot be completely turned into work in a cyclic process without leaving changes in the surroundings.
   • Rejecting waste heat $Q_C$ to a cold reservoir is not an engineering flaw; it is a mandatory thermodynamic requirement.

   The Ideal Carnot Cycle

   In 1824, Nicolas Léonard Sadi Carnot proposed an idealised, fully reversible cyclic engine operating between two thermal reservoirs. The Carnot cycle represents the upper theoretical limit of thermal efficiency for any heat engine operating between two given temperatures.

   Stages of the Carnot Cycle

   1. Reversible Isothermal Expansion: Heat $Q_H$ is absorbed from the reservoir at constant temperature $T_H$.
   2. Reversible Adiabatic Expansion: The working gas expands without heat exchange, cooling from $T_H$ to $T_C$.
   3. Reversible Isothermal Compression: Heat $Q_C$ is rejected to the cold reservoir at constant temperature $T_C$.
   4. Reversible Adiabatic Compression: The gas is compressed without heat exchange, warming back from $T_C$ to $T_H$.

   Carnot Efficiency

   For a Carnot engine, the ratio of heat transferred equals the ratio of absolute temperatures: $\frac{Q_C}{Q_H} = \frac{T_C}{T_H}$. Thus, the maximum theoretical efficiency is:

   $$\eta_{\text{Carnot}} = 1 - \frac{T_C}{T_H}$$

   Important Note: The temperatures $T_H$ and $T_C$ must always be expressed in Kelvin (K). Real heat engines invariably achieve lower efficiencies due to irreversibilities such as friction, turbulence, and rapid heat conduction.