Textbook Question
A heat engine uses a heat source at 540°C and has an ideal (Carnot) efficiency of 22%. To increase the ideal efficiency to 42%, what must be the temperature of the heat source?
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23. The Second Law of Thermodynamics
Heat Engines and the Second Law of Thermodynamics
5:45 minutesProblem 68b
Textbook Question
It has been suggested that a heat engine could be developed that made use of the temperature difference between water at the surface of the ocean and water several hundred meters deep. In the tropics, the temperatures may be 27°C and 4°C, respectively. Why might such an engine be feasible in spite of the low efficiency?
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Step 1: Recall the concept of a heat engine. A heat engine operates by transferring heat from a high-temperature reservoir to a low-temperature reservoir, converting part of the heat into useful work. The efficiency of a heat engine is determined by the temperature difference between these reservoirs.
Step 2: Use the Carnot efficiency formula to understand the theoretical maximum efficiency of such an engine. The Carnot efficiency is given by: , where is the temperature of the hot reservoir and is the temperature of the cold reservoir. These temperatures must be in Kelvin.
Step 3: Convert the given temperatures from Celsius to Kelvin. The surface temperature is 27°C, which is . The deep water temperature is 4°C, which is .
Step 4: Substitute the Kelvin temperatures into the Carnot efficiency formula to calculate the theoretical efficiency. This will show that the efficiency is low due to the small temperature difference between the reservoirs.
Step 5: Discuss why such an engine might still be feasible. Despite the low efficiency, the ocean provides an abundant and renewable energy source. The vast amount of water and the continuous temperature gradient in tropical regions could make this engine practical for generating energy on a large scale, even if the efficiency is not high.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Heat Engine
A heat engine is a device that converts thermal energy into mechanical work by exploiting temperature differences. It operates on a thermodynamic cycle, absorbing heat from a high-temperature reservoir, performing work, and releasing waste heat to a low-temperature reservoir. The efficiency of a heat engine is defined as the ratio of work output to heat input, and it is influenced by the temperature difference between the reservoirs.
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Thermodynamic Efficiency
Thermodynamic efficiency measures how well a heat engine converts heat into work. It is often low for engines operating with small temperature differences, as seen in ocean thermal energy conversion (OTEC) systems. Despite low efficiency, such systems can still be viable due to the abundance of thermal energy available and the potential for continuous operation, making them suitable for sustainable energy production.
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Ocean Thermal Energy Conversion (OTEC)
Ocean Thermal Energy Conversion (OTEC) is a technology that harnesses the temperature gradient between warm surface water and cold deep water to generate electricity. This method can operate continuously in tropical regions, where the temperature difference is significant. Although the efficiency of OTEC systems is typically low, the vast and consistent energy source from the ocean makes them a promising option for renewable energy generation.
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