Chapter 20: Problem 13
A Carnot engine whose high-temperature reservoir is at 620 K takes in 550 J of heat at this temperature in each cycle and gives up 335 J to the low- temperature reservoir. (a) How much mechanical work does the engine perform during each cycle? What is (b) the temperature of the low-temperature reservoir; (c) the thermal efficiency of the cycle?
Short Answer
Step by step solution
Calculate Work Done by the Engine
Calculate the Temperature of the Low-Temperature Reservoir
Calculate the Thermal Efficiency of the Cycle
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Thermodynamics
In our example involving a Carnot engine, thermodynamics principles tell us how energy input as heat from a high-temperature reservoir translates into work done by the engine. This involves understanding energy conservation, where total energy in a closed system (like our engine cycle) remains constant. By comprehending how the energy is taken in and given out, we can better understand the workings of engines and other systems in both practical and theoretical contexts.
Thermal Efficiency
In mathematical terms, thermal efficiency \((\eta)\) is expressed as:
- \( \eta = \frac{W}{Q_H} \)
Higher thermal efficiency means less energy is wasted and more is used to do work, which is why it's a significant performance indicator. In the Carnot engine example, we calculated a thermal efficiency of 39.09%, meaning about 39% of the heat energy taken in by the engine is converted into work.
Engineers strive for higher thermal efficiency to reduce energy loss and improve engine performance. However, due to natural laws and real-world complexities, perfect efficiency (100%) is unattainable, but understanding and improving thermal efficiency helps engineers design better and more sustainable systems.
Temperature Reservoirs
The high-temperature reservoir is the heat source. It provides the energy absorbed by the engine. For example, it might be a hot steam boiler.
- In our exercise, the high-temperature reservoir is at 620 K.
- The temperature of the low-temperature reservoir was calculated to be approximately 377.6 K in our solution.
Understanding temperature reservoirs helps learners grasp why certain processes are more efficient in specific temperature conditions and aids in the design of more effective thermal management systems.