Problem 77 - Pearson Physics

Question

At a coal-burning power plant a steam turbine operates with a power output of \(550 \mathrm{MW}\). The thermal efficiency of the power plant is \(32 \%\). (a) At what rate is heat discarded to the environment by this power plant? (b) At what rate must heat be supplied to the power plant by burning coal?

Step-by-Step Solution

Answer

(a) 1168.75 MW; (b) 1718.75 MW.

Step 1: Understanding Thermal Efficiency

The thermal efficiency (\(e\)) of a power plant is given by the formula \( e = \frac{W}{Q_H} \), where \(W\) is the work output (power output in this case) and \(Q_H\) is the heat input. Given that the efficiency is 32%, we can write \( 0.32 = \frac{550 \text{ MW}}{Q_H} \).

Step 2: Calculate Heat Input

Rearrange the efficiency equation to solve for \(Q_H\). Thus, \(Q_H = \frac{550}{0.32} \text{ MW}\). Calculate \(Q_H\) to find the amount of heat that needs to be supplied to the plant.

Step 3: Calculation: Heat Input

Calculate using the values: \[ Q_H = \frac{550}{0.32} = 1718.75 \text{ MW}. \] Thus, the rate at which heat must be supplied is \(1718.75 \text{ MW}\).

Step 3: Calculate Heat Discarded

The heat discarded to the environment, \(Q_C\), can be found using the relation \( Q_H = W + Q_C \). By rearranging, \( Q_C = Q_H - W \). Use the calculated \(Q_H\) and given \(W\) (550 MW) to find \(Q_C\).

Step 5: Calculation: Heat Discarded

Substitute the known values into the equation: \[ Q_C = 1718.75 \text{ MW} - 550 \text{ MW} = 1168.75 \text{ MW}. \] Thus, the rate at which heat is discarded to the environment is \(1168.75 \text{ MW}\).

Key Concepts

Heat Input

When we talk about heat input in the context of a power plant, we mean the amount of thermal energy that must be provided to produce electricity. The burning of coal provides this necessary energy in a coal-burning power plant. Let's break it down into simple steps:

• Thermal efficiency is a measurement of how well a power plant converts heat into work, or electricity. In our example, the thermal efficiency is 32%.
• This means that only 32% of the energy provided through heat input is converted into usable power.
• The rest of the energy is not lost; instead, it is expelled in other forms, such as waste heat.

Rearranging the thermal efficiency equation, we determine how much heat (\(Q_H\)) is required. The calculation is straightforward and involves some elementary algebra. The power plant needs a substantial heat input of approximately 1718.75 MW to maintain an output of 550 MW.
Understanding this lays the foundation for grasping other concepts related to energy efficiency and power management.

Heat Discarded

Once we have supplied the heat energy to the power plant, not all of it will be converted into electrical power. The leftover energy is expelled as heat discarded. This is a crucial aspect of power plants that affects their environmental impact. Let's analyze how this works:

• The thermal efficiency indicates how effectively the input heat energy is utilized. Here, with 32% efficiency, the remainder of energy must be discarded as heat.
• Using the formula \(Q_C = Q_H - W\), we determine the amount of heat expelled. It's derived from subtracting the power output from the total heat input.
• The calculation reveals that about 1168.75 MW of heat is discarded into the environment because only 550 MW is converted into electrical power.

By understanding how much heat is discarded, industries can take measures to improve efficiency and mitigate environmental impacts through strategies like waste heat recovery.

Power Output

Power output refers to the usable electrical energy generated by the power plant. It is a direct measure of how well the plant converts heat input into electricity.

• In our example, the power output is given as 550 MW. This is the electricity generated and supplied to the grid.
• The efficiency plays a vital role because it determines the relation between the power output and the heat input. Higher efficiency means more electricity for the same amount of heat.
• Understanding the power output helps in planning and managing energy resource utilization effectively.

It's crucial to ensure that the efficiency is maximized to get the best possible power output from the given heat input. This involves using advanced technology and better design to minimize the loss of energy during the conversion process. As we aim for more sustainable practices, improving the power output ratio becomes essential for future energy solutions.