Question

In \(2001,\) the United States produced 51 percent of its electricity in the amount of \(1.878 \times 10^{12} \mathrm{kWh}\) from coalfired power plants. Taking the average thermal efficiency to be 34 percent, determine the amount of thermal energy rejected by the coal-fired power plants in the United States that year.

Short answer

Answer: The thermal energy rejected by the coal-fired power plants in the United States in 2001 was approximately \(3.645 \times 10^{12} \mathrm{kWh}\).

Step-by-step solution

Identify the given values

The exercise provides us with the following information: - Total electricity generated in 2001: \(1.878 \times 10^{12} \mathrm{kWh}\) - Average thermal efficiency of coal-fired power plants: 34%

Calculate the energy input

To calculate the energy input, we will use the formula for thermal efficiency: Thermal Efficiency = (Output Energy) / (Input Energy) We can rearrange the formula to solve for Input Energy: Input Energy = (Output Energy) / (Thermal Efficiency) Plugging in the given values: Input Energy = \((1.878 \times 10^{12} \mathrm{kWh}) / 0.34\)

Evaluate the expression to find the energy input

Calculating the energy input: Input Energy = \((1.878 \times 10^{12} \mathrm{kWh}) / 0.34 \approx 5.523 \times 10^{12} \mathrm{kWh}\)

Calculate the energy rejected

To find the energy rejected, we subtract the output energy (electricity generated) from the input energy: Energy Rejected = Input Energy - Output Energy Then, plugging in the values: Energy Rejected = \(5.523 \times 10^{12} \mathrm{kWh} - 1.878 \times 10^{12} \mathrm{kWh}\)

Evaluate the expression to find the energy rejected

Calculating the energy rejected: Energy Rejected = \(5.523 \times 10^{12} \mathrm{kWh} - 1.878 \times 10^{12} \mathrm{kWh} \approx 3.645 \times 10^{12} \mathrm{kWh}\) Therefore, the amount of thermal energy rejected by the coal-fired power plants in the United States in 2001 was approximately \(3.645 \times 10^{12} \mathrm{kWh}\).

Key concepts

Energy Input Calculation

When dealing with the efficiency of a power plant, one key aspect to consider is the energy input calculation. The energy input is the total amount of energy needed to produce a certain output of electricity. To find this critical value, you start with the output – the amount of electricity generated by the plant – and consider the plant's thermal efficiency.

The thermal efficiency of a power plant tells us what fraction of the input energy gets converted to electrical energy. It's calculated by taking the electrical energy output and dividing it by the total energy input. To find the energy input when we know the plant’s output and its thermal efficiency, we simply rearrange the formula:
Input Energy = Output Energy / Thermal Efficiency

For instance, if a power plant has an efficiency of 34% and produces a given output, the energy input is the output divided by 0.34. Remember, to express thermal efficiency as a decimal in calculations, you convert the percentage into a decimal by dividing by 100. Thus, 34% becomes 0.34. It's crucial to convert your units consistently to avoid any miscalculations, typically energy is measured in kilowatt-hours (kWh) for such large-scale engineering problems.

Energy Rejected by Power Plants

A significant amount of the energy generated by power plants is not converted into electricity, but is instead rejected or wasted. This is an inevitable consequence of the Second Law of Thermodynamics, which dictates that not all energy input can be transformed into useful work.

Understanding the Rejected Energy

The energy that is not used in electricity production typically takes the form of heat and is discharged into the environment. This is a loss for the plant since it represents energy that was produced (often through the burning of fuel) but did not convert to electricity. It is calculated as the difference between the total energy input and the energy that was actually converted to electricity:
Energy Rejected = Input Energy - Output Energy

To optimize the performance of power plants, engineers work on minimizing this energy rejection, enhancing the plant's thermal efficiency, and reducing the environmental impact. For example, improvements in materials and the integration of systems like combined cycle power plants, where waste heat is used to drive another cycle to produce more electricity are ways to achieve this.

Thermal Energy Conversion

Thermal energy conversion refers to the process of converting heat energy into electricity, which is a central function of thermal power plants. Coal-fired power plants, for instance, burn coal to generate heat. This heat is then used to produce steam, which drives a turbine connected to an electric generator, ultimately producing electricity.

The Efficiency of Conversion

The efficiency of this conversion process is a vital metric and is quantitatively expressed by the thermal efficiency. This implies that higher thermal efficiency equates to more electricity per unit of fuel consumed.

The efficiency is not just a measure of performance, but also of cost-effectiveness and environmental impact. A plant with higher thermal efficiency utilizes less fuel to produce the same amount of electricity, which also means it emits lower quantities of greenhouse gases if the fuel is a fossil fuel. Students and engineers alike are interested in finding ways to improve thermal efficiency, such as using higher temperature materials, waste heat recovery systems, and advanced turbine designs, making the conversion process more economical and environmentally friendly.