Τhe second law of thermodynamics can be expressed in different ways, including the direction in which a process happens and its irreversibility, and in terms of entropy. It states that:
The heat transfer occurs naturally only from higher temperature bodies to lower ones but never in the reverse direction.
The entropy of an isolated system never decreases, as isolated systems tend to reach thermodynamic equilibrium, which is a state of maximum entropy.
An everyday example of the first expression of the second law of thermodynamics is a hot drink cooling down and transferring thermal energy to room temperature due to the lower temperature of its surroundings.
The first law of thermodynamics states that the net energy of an isolated system is constant. Energy can, therefore, not be created or destroyed but can only change forms. This is mathematically expressed in saying that the heat that is supplied to a system is equal to the sum of the system’s change in internal energy and the work done by the system.
Hence, perpetual motion, i.e., motion that continues without the need of any energy input to maintain it, is, according to the first and second laws of thermodynamics, impossible.
Entropy is a quantity that demonstrates the impossibility of the conversion of thermal energy into mechanical work.
The limitations of the first and second laws of thermodynamics:
The first law does not specify the direction of the flow of heat or whether a process is spontaneous or not.
According to the second law, heat flows from a higher temperature body to a lower one. The reverse process is not possible. In actual practice, the heat also doesn’t convert completely into work.
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.
A heat sink and a heat source are required for a thermal energy transfer to occur, as a heat source is hotter than the heat sink, thus allowing thermal energy to transfer from the source to the sink.
This is shown in figure 1, which illustrates a heat transfer occurring away from the hot object (QH) and into the cold object (Qc). The diagram also shows the work done by the engine (W) due to the heat transfer between the source and the sink. TH is the temperature of the hot body or hot reservoir, while TC is the temperature of the lower temperature body or cold reservoir.
Figure 1. Heat engine energy flow diagram.
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 QH and the cold reservoir QC.
\[Q_H \rightarrow W + Q_C \qquad W = Q_H - Q_C\]
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.
A cyclical process is a repetitive process that always returns the system to its initial state.
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 (QH) 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 (QH) and the heat loss (QC), the efficiency can be re-written, as seen below. The efficiency can be between 0% and 100% (only if QC 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.
- A process is considered to be irreversible when energy is lost in the form of heat due to friction.
- A process is considered to be reversible when energy is lost as heat due to the natural phenomenon of heat transfer because of a temperature difference between two bodies. The energy flows from a hot body to a cooler one.
The Carnot cycle is shown in figure 2 below in a p-v diagram where a heat transfer QH occurs during the isothermal path AB, while a heat transfer QC occurs during the isothermal path CD. The total work done (W) can be found using the area inside the shape ABCD.
Figure 2. Carnot cycle p-v diagram.
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.
An isothermal process is a process in which the temperature remains constant.
An adiabatic process is a process that does not transfer any mass or energy to its surroundings.
For ideal engines, the Carnot efficiency or maximum efficiency is given by the formula below, where TH and TC 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.
A power station transfers 5⋅1012 J of heat from coal and 1.8⋅1012 J to the environment. Determine the work done by the power station engine and the efficiency of the power station.
To determine the work output, we must consider the heat transfer from the source and the sink. In this case, the source is the coal, and the sink is the environment. Hence, the work output is given by the difference in the heat transfer between the two reservoirs.
\[W = Q_H - Q_C \quad W = 5 \cdot 10^{12} - 1.8 \cdot 10^{12} = 3.2 \cdot 10^{12} J\]
To determine the efficiency, the fraction of the work output over the heat transfer of the source must be calculated.
\[\eta = \frac{W}{Q_H} = \frac{3.2 \cdot 10^{12}}{5 \cdot 10^{12}} = 0.64 \qquad \eta_{\%} = 0.64 \cdot 100= 64\%\]
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.
\[P = \frac{W}{t}\]
Determine the power output of a heat engine that produces 1500J of work per cycle when the time taken to complete a cycle is 0.45 seconds.
\[P = \frac{W}{t} = \frac{1500}{0.45} = 3333 W\]
Second Law and Engines - Key takeaways
- Heat engines operate based on the second law of thermodynamics.
- The second law of thermodynamics deals with the direction of heat flow.
- The maximum efficiency for a heat engine is the Carnot efficiency.
- The power output of a heat engine is the work done per unit of time.
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