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 52b

Chapter 21, Problem 52b

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?

Verified step by step guidance

Step 1: Recall the formula for efficiency, which is the ratio of useful output energy (or power) to the input energy (or power). The formula is:

e = P

_outP_in, where

P

_out is the output power and

P

_in is the input power.

Step 2: Identify the given values from the problem. The input power

P

_in is 3000 MW (heat energy generated), and the output power

P

_out is 1000 MW (electric power produced).

Step 3: Substitute the given values into the efficiency formula:

e = \frac{1000}{3000}

Step 4: Simplify the fraction to calculate the efficiency as a decimal. Then, multiply the result by 100 to express the efficiency as a percentage.

Step 5: Interpret the result. The efficiency represents the percentage of input energy that is converted into useful output energy (electric power). The remaining energy is lost, primarily as waste heat.

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Key Concepts

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

Efficiency

Efficiency in a power plant context refers to the ratio of useful output energy to the input energy. It is expressed as a percentage and indicates how well the plant converts the energy from fuel into usable electrical power. In this case, the efficiency can be calculated by comparing the electric power output to the thermal energy input from nuclear reactions.

Thermal Energy

Thermal energy is the total kinetic energy of particles in a substance, which in this scenario is generated from nuclear reactions. The heat energy produced in the reactor core is used to convert water into steam, which is essential for driving the turbine. Understanding the amount of thermal energy produced is crucial for calculating the overall efficiency of the power plant.

Turbine Operation

A turbine is a mechanical device that converts thermal energy from steam into mechanical energy, which is then transformed into electrical energy by a generator. The efficiency of the turbine affects the overall efficiency of the power plant, as it determines how much of the steam's energy is converted into electricity. The performance of the turbine is influenced by factors such as steam pressure and temperature.

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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

A heat engine uses a diatomic gas that follows the pV cycle in FIGURE P21.59. Determine the pressure, volume, and temperature at point 2.

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Textbook Question

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?

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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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