Ch 21: Heat Engines and Refrigerators

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 21: Heat Engines and Refrigerators

Problem 51a

Chapter 21, Problem 51a

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?

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 helps in determining how much energy is needed to operate the plant over a specific time period, 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 allows us to relate the mass of coal to its physical volume. By knowing the total mass of coal required for a day of operation, we can calculate the volume of coal needed and subsequently determine the height of the pile in the specified room dimensions.

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Textbook Question

A car's internal combustion engine can be modeled as a heat engine operating between a combustion temperature of 1500℃ and an air temperature of 20℃ with 30% of the Carnot efficiency. The heat of combustion of gasoline is 47 kJ/g. What mass of gasoline is burned to accelerate a 1500 kg car from rest to a speed of 30 m/s?

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Textbook Question

Home air conditioners in the United States have their power specified in the truly obscure units of tons, where 1 ton is the power needed to melt 1 ton (2000 lb or 910 kg) of ice in 24 hours. A modest-size house typically has a 4.0 ton air conditioner. If a 4.0 ton air conditioner has a coefficient of performance of 2.5, a typical value, at what rate in kW is heat energy removed from the house?

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Textbook Question

FIGURE P21.46 shows a Carnot heat engine driving a Carnot refrigerator. Determine Q₂, Q₃ and Q₄.

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Textbook Question

A nuclear power plant generates 3000 MW of heat energy from nuclear reactions in the reactor's core. This energy is used to boil water and produce high-pressure steam at 300℃. The steam spins a turbine, which produces 1000 MW of electric power, then the steam is condensed and the water is cooled to 25℃ before starting the cycle again. What is the plant's actual efficiency?

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Textbook Question

FIGURE P21.57 shows the cycle for a heat engine that uses a gas having γ = 1.25. The initial temperature is T₁ = 300 K, and this engine operates at 20 cycles per second. What is the power output of the engine?

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Textbook Question

A heat engine using a diatomic gas follows the cycle shown in FIGURE P21.55. Its temperature at point 1 is 20℃. Determine W_s, Q, and ∆E_th for each of the three processes in this cycle. Display your results in a table.

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