Second Law and Engines

What are heat engines?

In thermodynamics, heat engines are systems that convert thermal energy or heat into mechanical work. Some examples of heat engines are gasoline and diesel engines, jet engines, and steam turbines, all of which convert thermal energy to mechanical work, using part of the heat transfer from combustion.

The basic working principle of a heat engine comprises a gas in a cylinder compressed by a piston. When the gas in the cylinder is heated, it expands, thus increasing the volume, which causes the piston to move and convert heat into work. Once the gas reaches an equilibrium, the piston stops moving. In order to continue producing work, the engine has to use cycles featuring a continuous back and forth motion of the piston. This is achieved by the piston cooling and reducing the volume, which causes the piston to move back down. Hence, a cyclical motion of heating and cooling is required for the continuous production of work in a heat engine.

Heat engines and the second law of thermodynamics

Given the working principle of a heat engine, the possibility of work requires the cooperation of a heat sink and a heat source.

This is shown in figure 1, which illustrates a heat transfer occurring away from the hot object (Q_H) and into the cold object (Q_c). The diagram also shows the work done by the engine (W) due to the heat transfer between the source and the sink. T_H is the temperature of the hot body or hot reservoir, while T_C is the temperature of the lower temperature body or cold reservoir.

The diagram is expressed mathematically in the equation below, where the work done by the heat engine (W) measured in Joules equals the difference between the heat transfer of the hot reservoir Q_H and the cold reservoir Q_C.

\[Q_H \rightarrow W + Q_C \qquad W = Q_H - Q_C\]

Heat engines thus work according to the second law of thermodynamics and cannot be explained solely by the first law, which does not refer to the direction of heat.

The second law is expressed in terms of entropy, which is always increasing. Hence, it is not possible in a cyclical process to fully convert heat into work, as that would mean that the system returns to its initial state, which is ruled out by the second law in its second form.

What is engine efficiency?

The efficiency of an engine is a measure of the amount of input energy that is converted into mechanical work. For maximum engine efficiency, the work done by the engine must be equal to the heat transferred from the sink, which would mean that no heat is lost to the environment. However, this is not practically possible, as there will always be some energy lost to the environment. The efficiency of an engine, therefore, is always less than 100%.

Efficiency (η) can be calculated using the equation below as a fraction of the work (W) over the heat transferred to the heat sink (Q_H) and can be converted into a percentage by multiplying by 100.

\[\eta = \frac{W}{Q_H} \text{ or } \eta_{\%}= \frac{W}{Q_H} \cdot 100\]

As the work is the difference between the heat input (Q_H) and the heat loss (Q_C), the efficiency can be re-written, as seen below. The efficiency can be between 0% and 100% (only if Q_C is equal to zero, which is practically impossible). The formula below can be used for cyclical engines.

\[\eta = \frac{Q_H -Q_C}{Q_H} = 1 - \frac{Q_C}{Q_H}\]

Carnot engine

A Carnot engine operates based on the Carnot cycle discovered by Sadi Carnot. The Carnot cycle is an ideal cycle that provides maximum efficiency. The Carnot principle states that no other type of heat engine operating between a heat source and a heat sink can be more efficient than a reversible Carnot engine operating under the same conditions.

The efficiency of a reversible engine is greater than that of any irreversible engine, as reversible engines operating under the Carnot cycle do not lose any energy if the process is reversed, while irreversible engines lose energy under reverse operation.

The Carnot cycle is shown in figure 2 below in a p-v diagram where a heat transfer Q_H occurs during the isothermal path AB, while a heat transfer Q_C occurs during the isothermal path CD. The total work done (W) can be found using the area inside the shape ABCD.

The Carnot cycle is an ideal theoretical thermodynamic cycle. It is a reversible cycle that includes four consecutive stages before returning to its initial state. The four stages include isothermal expansion, adiabatic expansion, isothermal compression, and adiabatic compression.

For ideal engines, the Carnot efficiency or maximum efficiency is given by the formula below, where T_H and T_C are the temperatures of the source and the sink, respectively, in Kelvin. This efficiency is the maximum efficiency achieved by an ideal reversible heat engine operating by the Carnot cycle. In reality, however, heat engines operate at a much lower efficiency than the Carnot efficiency.

\[\eta = 1- \frac{T_C}{T_H}\]

For an engine to reach maximum efficiency, it must, therefore, operate on a reversible cycle in which no energy is lost due to friction. From the equation, it can be derived that the efficiency is maximised when the engine operates under the greatest temperature difference possible. When the temperature difference is at its highest, more heat is transferred more rapidly, and more work is done by the engine.

The second law of thermodynamics has a wide range of applications, which include steam engines, internal combustion engines (petrol and diesel engines), gas turbine engines, and power-producing stations.

The power of an engine

The output power (P) of a heat engine is defined as the work done by the engine per unit time in seconds, as seen in the equation below. The higher the power output, the higher the work done by the engine. The power is measured in Watts.

\[P = \frac{W}{t}\]