Question

Calculate the efficiency of a fossil-fuel power plant that consumes 382 metric tons of coal each hour to produce useful work at the rate of \(755 \mathrm{MW}\). The heat of combustion of coal is \(28.0 \mathrm{MJ} / \mathrm{kg}\).

Short answer

The efficiency of the coal-fired power plant is 25.4%.

Step-by-step solution

Calculate the Total Energy Input Per Hour

To calculate the total energy input into the power plant per hour, we need to convert the mass of consumed coal into energy. We can do this using the heat of combustion, which is the amount of energy released when a specific amount of coal is burned. Given that the heat of combustion of coal is 28.0 MJ/kg and that the plant consumes 382 metric tons of coal per hour, we calculate the total energy input per hour as follows: \(382,000\,\mathrm{kg/hour} \times 28.0\,\mathrm{MJ/kg} = 10,696,000\,\mathrm{MJ/hour}\). This is equivalent to \(10,696,000\,\mathrm{MJ/hour} \div 3600 \, \mathrm{s/hour} = 2971.11 \, \mathrm{MW}\).

Calculate the Efficiency of the Power Plant

Efficiency is defined as the ratio of useful work output to the total energy input. The data provided states that the useful work produced is 755 MW. Using our calculated total energy input from Step 1, we can now find the efficiency of the power plant: \( \frac{755 \, \mathrm{MW}}{2971.11 \, \mathrm{MW}} = 0.254 = 25.4%\).

Key concepts

Fossil-Fuel Power Plant

A fossil-fuel power plant is a type of power station that burns fossil fuels such as coal, natural gas, or oil to generate electricity. It uses heat from burning these fuels to create steam, which then drives a turbine connected to an electric generator. These plants are a primary source of electricity for much of the world, thanks to their ability to produce large amounts of power. Fossil-fuel power plants operate on the principle of converting chemical energy stored in fossil fuels into mechanical energy and, finally, into electrical energy. However, this process is inherently inefficient, as some of the energy is lost as waste heat. The environmental impact is significant, as burning fossil fuels releases carbon dioxide and other pollutants into the atmosphere. Efficiency is a critical factor, as it determines how effectively the plant can convert fuel into electricity, impacting both costs and environmental footprint.

Heat of Combustion

The heat of combustion is a measure of the energy released when a specific amount of fuel is burned completely. It is typically expressed in megajoules per kilogram (MJ/kg) and indicates how much energy a given type and amount of fuel can produce. In the context of a power plant, knowing the heat of combustion helps determine how much fuel is needed to generate a specific amount of energy. For example, in our exercise, we are given the heat of combustion of coal as 28.0 MJ/kg. This means every kilogram of coal burned releases 28 megajoules of energy. Understanding this concept is vital for calculating the total energy input and helps in assessing the plant's overall efficiency. It highlights the importance of selecting fuels with a high heat of combustion to ensure efficient energy production.

Energy Conversion

Energy conversion is the process of transforming energy from one form to another. In a power plant, this involves converting the chemical energy stored in fossil fuels into mechanical energy through combustion, and subsequently into electrical energy. This process can be broken down into several steps:

• Burning fuel in the boiler to create steam from water.
• The steam spins the turbine, converting thermal energy to mechanical energy.
• The turbine drives a generator, converting mechanical energy to electrical energy.

Efficiency during these conversions is crucial. It is calculated by comparing the useful output energy (electricity) to the total input energy from burning fuel. Inefficiencies typically occur due to energy losses, primarily in the form of heat discharged through the exhaust or cooling systems. These losses explain why the efficiency of a typical power plant is less than 100%, usually around 30% to 40%. By understanding these processes, improvements can be made to enhance efficiency, reduce fuel consumption, and minimize environmental impact.