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 42

Chapter 21, Problem 42

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?

Verified step by step guidance

Convert all temperatures to Kelvin. The inside temperature of the freezer is \(-22 + 273.15 = 251.15\,\text{K}\), and the room temperature is \(25 + 273.15 = 298.15\,\text{K}\).

Calculate the coefficient of performance (COP) of a Carnot refrigerator using the formula \(\text{COP}_{\text{Carnot}} = \frac{T_{\text{cold}}}{T_{\text{hot}} - T_{\text{cold}}}\), where \(T_{\text{cold}}\) is the inside temperature of the freezer and \(T_{\text{hot}}\) is the room temperature.

Determine the actual coefficient of performance of the freezer. Since the freezer's COP is 30% of the Carnot COP, use \(\text{COP}_{\text{freezer}} = 0.3 \times \text{COP}_{\text{Carnot}}\).

Calculate the heat that must be removed to freeze the water. First, find the mass of the water using its density (\(\rho = 1000\,\text{kg/m}^3\)) and volume (\(3.0\,\text{L} = 0.003\,\text{m}^3\)). Then, calculate the heat removed in two steps: (1) cooling the water from \(20\,\degree\text{C}\) to \(0\,\degree\text{C}\) using \(Q_1 = mc\Delta T\), where \(c = 4186\,\text{J/kg·K}\), and (2) freezing the water using \(Q_2 = mL_f\), where \(L_f = 334,000\,\text{J/kg}\). The total heat removed is \(Q = Q_1 + Q_2\).

Determine the time required to freeze the water. The work done by the compressor is related to the heat removed by \(W = \frac{Q}{\text{COP}_{\text{freezer}}}\). Since the compressor works at a rate of \(200\,\text{W}\), the time is given by \(t = \frac{W}{200}\).

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Key Concepts

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

Coefficient of Performance (COP)

The Coefficient of Performance (COP) is a measure of the efficiency of a refrigeration system, defined as the ratio of heat removed from the cold reservoir to the work input. In this context, the COP of the freezer is given as 30% of that of an ideal Carnot refrigerator, which sets a theoretical maximum efficiency based on the temperatures of the hot and cold reservoirs.

Heat Transfer and Latent Heat

When water freezes, it undergoes a phase change from liquid to solid, which requires the removal of heat known as latent heat of fusion. For water, this value is approximately 334 kJ/kg. Understanding how much heat must be removed to freeze the water is crucial for calculating the time required for the freezing process.

Power and Energy Relationship

Power is the rate at which energy is transferred or converted, measured in watts (W). In this scenario, the freezer's compressor operates at 200 W, meaning it can remove 200 joules of energy per second. To determine how long it takes to freeze the water, one must relate the total energy required to freeze the water to the power output of the freezer.

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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 refrigerator operates between energy reservoirs at 0℃ and 250℃. A 2.4-cm-diameter, 50-cm-long copper bar connects the two energy reservoirs. At what rate, in W, must work be done on the refrigerator to remove heat from the cold reservoir at the same rate that it arrives through the copper bar?

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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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Textbook Question

The engine that powers a crane burns fuel at a flame temperature of 2000℃. It is cooled by 20℃ air. The crane lifts a 2000 kg steel girder 30 m upward. How much heat energy is transferred to the engine by burning fuel if the engine is 40% as efficient as a Carnot engine?

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Textbook Question

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?

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