A typical coal-fired power plant burns 300 metric tons of coal every hour to generate 750 MW of electricity. 1 metric ton = 1000 kg. The density of coal is 1500 kg/m³ and its heat of combustion is 28 MJ/kg. Assume that all heat is transferred from the fuel to the boiler and that all the work done in spinning the turbine is transformed into electric energy. Suppose the coal is piled up in a 10 m ✕ 10 m room. How tall must the pile be to operate the plant for one day?

검증된 단계별 안내

Step 1: Calculate the total mass of coal required to operate the plant for one day. Since the plant burns 300 metric tons of coal per hour, multiply this by 24 hours to find the total mass in metric tons. Then convert metric tons to kilograms using the conversion 1 metric ton = 1000 kg.

Step 2: Determine the volume of coal required. Use the formula for density, \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \), and rearrange it to find \( \text{Volume} = \frac{\text{Mass}}{\text{Density}} \). Substitute the total mass of coal (from Step 1) and the given density of coal (1500 kg/m³) to calculate the volume in cubic meters.

Step 3: Calculate the height of the coal pile. The coal is stored in a room with a base area of 10 m ✕ 10 m, which is 100 m². Use the formula for volume, \( \text{Volume} = \text{Base Area} \times \text{Height} \), and rearrange it to find \( \text{Height} = \frac{\text{Volume}}{\text{Base Area}} \). Substitute the volume (from Step 2) and the base area to calculate the height of the pile.

Step 4: Ensure all units are consistent throughout the calculations. Verify that the mass is in kilograms, the density is in kg/m³, the base area is in m², and the resulting height is in meters.

Step 5: Interpret the result. The calculated height represents how tall the coal pile must be to supply the plant with enough coal to operate for one day.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Heat of Combustion

The heat of combustion is the amount of energy released when a substance is burned completely in oxygen. For coal, this value is approximately 28 MJ/kg, meaning that burning one kilogram of coal produces 28 megajoules of energy. This concept is crucial for calculating the total energy produced from a given mass of coal, which directly relates to the power output of the plant.

Power and Energy Relationship

Power is the rate at which energy is produced or consumed, measured in watts (W), where 1 watt equals 1 joule per second. In this context, the power plant generates 750 MW, which means it produces 750 million joules of energy every second. Understanding this relationship allows us to determine how much energy is needed to operate the plant over a specified time, such as one day.

Volume and Density

Density is defined as mass per unit volume, typically expressed in kg/m³. For coal, the density is 1500 kg/m³, which means that one cubic meter of coal weighs 1500 kilograms. This concept is essential for calculating the volume of coal needed to sustain the power plant's operation for a day, as it allows us to convert the total mass of coal required into a physical volume.