A coal-fired power station burns coal to produce steam, which drives a turbine connected to a generator. The station has an input power of 500 MW from burni...

Marking Scheme and Explanation:

(a) Power output of the boiler [1]

The boiler transfers 90% of the 500 MW input to steam:

\( P_{\text{boiler}} = 0.90 \times 500 = 450 \text{ MW} \)

(b) Power output of the turbine [1]

The turbine converts 50% of the steam energy into mechanical energy:

\( P_{\text{turbine}} = 0.50 \times 450 = 225 \text{ MW} \)

(c) Electrical power output of the generator [1]

The generator converts 95% of the mechanical energy into electrical energy:

\( P_{\text{generator}} = 0.95 \times 225 = 213.75 \text{ MW} \approx 214 \text{ MW} \)

(d) Overall efficiency of the power station [2]

The overall efficiency is the ratio of final electrical output to total fuel input:

\( \text{efficiency} = \frac{213.75}{500} = 0.4275 \text{ (42.75\%)} \)

Alternatively, multiply the individual efficiencies: \( 0.90 \times 0.50 \times 0.95 = 0.4275 \). This shows that the overall efficiency of a multi-stage system is the product of the individual stage efficiencies, so each stage of loss compounds.

(e) Power wasted at each stage [3]

• Boiler waste: \( 500 - 450 = 50 \text{ MW} \) [1] (lost as hot exhaust gases)
• Turbine waste: \( 450 - 225 = 225 \text{ MW} \) [1] (lost as thermal energy in cooling water from the condenser)
• Generator waste: \( 225 - 213.75 = 11.25 \text{ MW} \) [1] (lost as thermal energy due to electrical resistance in the coils and friction in the bearings)

The turbine stage wastes the most energy. This is typical: converting thermal energy into mechanical energy is the least efficient step in a thermal power station.

(f) Total electrical energy produced in 24 hours, in kWh [2]

Convert power to kW: \( 213.75 \text{ MW} = 213{,}750 \text{ kW} \)

\( E = P \times t = 213{,}750 \times 24 = 5{,}130{,}000 \text{ kWh} = 5.13 \times 10^6 \text{ kWh} \)

The kilowatt-hour (kWh) is the commercial unit of electrical energy. 1 kWh = the energy used by a 1 kW appliance running for 1 hour = 3.6 MJ.

(g) One use for the waste heat from the cooling water [1]

The waste heat could be used for district heating, supplying hot water to nearby homes and businesses. This is known as combined heat and power (CHP) and can raise the overall useful energy extraction from the fuel to 70-80%.