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 45

Chapter 21, Problem 45

A Carnot engine operates between temperatures of 5℃ and 500℃. The output is used to run a Carnot refrigerator operating between -5℃ and 25℃. How many joules of heat energy does the refrigerator exhaust into the room for each joule of heat energy used by the heat engine?

Verified step by step guidance

Convert all temperatures from Celsius to Kelvin by adding 273.15 to each value. For the Carnot engine: T_hot = 500 + 273.15 K and T_cold = 5 + 273.15 K. For the Carnot refrigerator: T_high = 25 + 273.15 K and T_low = -5 + 273.15 K.

Calculate the efficiency of the Carnot engine using the formula: η = 1 - (T_cold / T_hot), where T_cold and T_hot are the temperatures of the Carnot engine in Kelvin.

Determine the coefficient of performance (COP) of the Carnot refrigerator using the formula: COP = T_low / (T_high - T_low), where T_low and T_high are the temperatures of the refrigerator in Kelvin.

Relate the work output of the Carnot engine to the work input of the Carnot refrigerator. For each joule of work produced by the engine, it is used as input work for the refrigerator. Use the relationship: Q_exhaust = COP × Work_input, where Q_exhaust is the heat energy exhausted by the refrigerator into the room.

Combine the efficiency of the Carnot engine and the COP of the refrigerator to find the total heat energy exhausted into the room for each joule of heat energy used by the heat engine. Use the relationship: Q_exhaust = COP × (1 / η).

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Key Concepts

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

Carnot Engine

A Carnot engine is an idealized heat engine that operates on the Carnot cycle, which is the most efficient cycle possible between two temperature reservoirs. It converts heat energy from a hot reservoir into work while rejecting some heat to a cold reservoir. The efficiency of a Carnot engine depends solely on the temperatures of the hot and cold reservoirs, given by the formula η = 1 - (T_c/T_h), where T_c and T_h are the absolute temperatures of the cold and hot reservoirs, respectively.

Carnot Refrigerator

A Carnot refrigerator is an idealized refrigeration cycle that operates between two temperature reservoirs, absorbing heat from a cold reservoir and expelling it to a hot reservoir. Its efficiency is defined by the coefficient of performance (COP), which is the ratio of heat removed from the cold reservoir to the work input. The COP for a Carnot refrigerator is given by COP = T_c / (T_h - T_c), where T_c and T_h are the absolute temperatures of the cold and hot reservoirs, respectively.

Heat Transfer and Energy Conservation

Heat transfer refers to the movement of thermal energy from one object or system to another due to a temperature difference. In thermodynamic systems, the principle of energy conservation states that energy cannot be created or destroyed, only transformed. In the context of the Carnot engine and refrigerator, the heat energy output from the engine becomes the input for the refrigerator, illustrating the conservation of energy as it moves through the two systems.

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Related Practice

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

A freezer with a coefficient of performance 30% that of a Carnot refrigerator keeps the inside temperature at -22℃ in a 25℃ room. 3.0 L of water at 20℃ are placed in the freezer. How long does it take for the water to freeze if the freezer's compressor does work at the rate of 200 W while the water is freezing?

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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

An ideal refrigerator utilizes a Carnot cycle operating between 0℃ and 25℃. To turn 10 kg of liquid water at 0℃ into 10 kg of ice at 0℃, (a) how much heat is exhausted into the room and (b) how much energy must be supplied to the refrigerator?

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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 Carnot heat engine operates between reservoirs at 182℃ and 0℃. If the engine extracts 25 J of energy from the hot reservoir per cycle, how many cycles will it take to lift a 10 kg mass a height of 10 m?

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