Heat engines and the second law

State what is meant by an isothermal change.

Figure 6 shows a p – V diagram in which a fixed mass of an ideal gas is taken through a cycle of three processes.

A → B: adiabatic change B → C: isothermal change C → A: reduction in volume at constant pressure

The maximum temperature of the gas in the cycle is 536 K and the minimum temperature is 291 K.

Calculate, in mol, the amount of gas.

Show that the work done in process C → A is about 100 J.

Complete Table 2 for each process and for the whole cycle.

Two criteria for a heat engine to achieve the maximum theoretical efficiency are:

1. the gas must always be at the same temperature as the hot source when energy is being transferred to the gas by heat transfer

2. the gas must always be at the same temperature as the cold sink when energy is being transferred from the gas by heat transfer.

The temperatures of the source and sink do not change.

Deduce whether the cycle in Figure 6 satisfies either or both of these criteria.

The overall efficiency of a heat engine is defined as

An engineer suggests designing a real engine based on the cycle shown in Figure 6.

Discuss the engineer’s suggestion.

In your answer you should:

• use Table 2 to determine the overall efficiency of a theoretical engine that uses the cycle

• compare this overall efficiency with the efficiency of an ideal engine working between the same temperatures as the cycle

• discuss with reasons the problems that would need to be overcome in making a real engine work on this cycle

• explain whether the engineer’s suggestion would lead to a useful engine.

The first law of thermodynamics can be expressed by the equation Q = Δ U + W.

State the meaning of each term in this equation.

A system consists of a perfectly insulated room containing only an empty refrigerator. The refrigerator is connected to the mains electricity and the refrigerator door is open.

Deduce, by applying the first law of thermodynamics, whether the internal energy of the room increases, decreases or stays the same.

One definition of an ideal heat engine is:

‘a device which provides the maximum possible output of work from a given input of energy by heat transfer.’

Figure 8 shows a heat engine that appears to agree with this definition. The engine takes in energy from a high-temperature source and work is done by the engine.

Figure 8

Complete Figure 8 so that the engine obeys the second law of thermodynamics.

Explain why the maximum theoretical efficiency of an ideal heat engine must be less than 100%.

Figure 7 shows the p – V diagram for a theoretical diesel engine cycle. Figure 8 shows the indicator diagram for a real four-stroke diesel engine working between the same maximum and minimum volumes.

Compare the indicator diagram with the theoretical diesel cycle. In your answer, explain:

• the differences between region A on Figure 7 and region A on Figure 8

• the differences between region B on Figure 7 and region B on Figure 8

• why the features shown in region C appear only on the indicator diagram

• why the efficiency of the real engine will always be less than the efficiency predicted by an analysis of the theoretical cycle.

Figure 9 shows an ideal heat engine.

The engine operates between a source at a temperature of 730 K and a sink at a temperature of 290 K. The input power is 16.0 kW and the power rejected to the sink is 6.4 kW.

Calculate the efficiency of this engine.

Figure 10 shows another system operating between the same overall temperatures and with the same input power as the engine in Figure 9. This system consists of two ideal engines.

The sink for engine 1 forms the source for engine 2. The temperature of the intermediate reservoir is 510 K. All the energy rejected by heat transfer in engine 1 provides the input energy to engine 2. A student suggests that the system in Figure 10 can provide more output power and be more efficient than the engine in Figure 9.

Deduce whether the student’s suggestions are correct. You may annotate Figures 9 and 10.

Which is a correct statement about an ideal heat engine? Tick (✓) one box.

An ideal heat engine has an efficiency of 0.33 The same engine works in reverse as an ideal refrigerator between the same hot and cold spaces.

Determine the coefficient of performance COP_ref of the refrigerator.

Explain how the second law of thermodynamics predicts that a heat engine can never be 100% efficient.

A company plans to build a geothermal power station in a region where there is hot rock deep below the surface. The scheme is shown in Figure 7.

In the heat exchanger, energy from the hot rock is used to produce steam at 175 °C. The steam passes through a turbine that drives an electric generator. The exhaust steam is used to heat nearby greenhouses where it condenses before returning to the heat exchanger.

The lowest temperature in the turbine cycle is 30 °C.

The company claims that when the electrical power output is 2.9 MW, the power station will provide 6.4 MW for heating the greenhouses.

Deduce whether this claim is likely to be true. Treat the power station as an ideal heat engine which obeys the second law of thermodynamics.

In an ideal heat-engine cycle a fixed mass of air is taken through the following four processes.

A → B isothermal compression from an initial pressure of 1.0 × 10⁵ Pa and a volume of 9.0 × 10^–2 m³ to a pressure of 2.2 × 10⁵ Pa. The work done on the air is 7100 J.

B → C increase in volume at constant pressure to a volume of 5.9 × 10⁻² m³.

C → D isothermal expansion to a pressure of 1.0 × 10⁵ Pa and a volume of 13 × 10⁻² m³. The work done by the air is 10 300 J.

D → A reduction in volume at constant pressure to the original volume.

Figure 3 shows the cycle.

Show, by calculation, that the volume at B is 4.1 × 10⁻² m³.

The temperature of the air between A and B is 295 K.

Show that the temperature of the air at C is about 420 K.

An ideal engine based on this cycle uses a device called an economiser. The economiser stores all the energy transferred in the cooling process D → A and gives up all this energy to the air in process B → C. This means that an external source supplies energy to the air by heating only in process C → D.

Complete Table 1 to show the values of work done W and energy transfer Q in each of the four processes. Use the space below Table 1 for any calculations. Use a consistent sign convention.

Explain why W is equal to Q in process A → B and in process C → D.

It is claimed that the efficiency of this engine cycle is the same as the maximum theoretical efficiency of a heat engine operating between the same temperatures.

Deduce whether this claim is true.

Discuss one problem that would be faced by an engineer designing a real engine based on this cycle.

The National Grid is supplied mainly from power stations which have overall efficiencies of up to about 40%

Table 1 shows the average power requirements of a large paper-manufacturing business (a paper mill).

The paper mill can either

• take all of its energy from the National Grid, or

• install an electrical generator of output 60 MW driven by a gas turbine of efficiency 36% as part of a combined heat and power (CHP) scheme. The hot exhaust gases

from the turbine are used to produce steam at high temperature and pressure for heating.

The owners of the paper mill are considering the CHP option.

Explain, with reference to the data above and any other factors, the advice you would give them.

In your answer you should

• explain why the maximum theoretical efficiency of a heat engine is much less

than 100%

• use the information above, including the numerical data, to compare the two

options.

maximum and minimum volumes.

Compare the theoretical and real engine cycles. In your answer you should:

state and explain the differences between the cycles

explain why the work output per cycle of the real engine is less than that predicted by an analysis of the theoretical cycle.