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Chapter 15: Problem 62

For a reversible engine, will you obtain a better efficiency by increasing the high-temperature reservoir by an amount \(\Delta T\) or decreasing the low- temperature reservoir by the same amount \(\Delta T ?\)

Short Answer

Expert verified
Answer: Decreasing the low-temperature reservoir by the amount ΔT results in better efficiency for the reversible engine.

Step by step solution

01

Initial Efficiency

First, we find the initial efficiency of the reversible engine using the Carnot efficiency formula: \(\eta_{initial} = 1 - \frac{T_L}{T_H}\) where \(\eta_{initial}\) is the initial efficiency, \(T_L\) is the low-temperature reservoir, and \(T_H\) is the high-temperature reservoir.
02

Efficiency with Increased High-Temperature Reservoir

Next, increase the high-temperature reservoir by the amount \(\Delta T\). The new high-temperature reservoir is \(T_H' = T_H + \Delta T\). Calculate the efficiency for this case using the Carnot efficiency formula: \(\eta_{increase} = 1 - \frac{T_L}{T_H + \Delta T}\)
03

Efficiency with Decreased Low-Temperature Reservoir

Now, decrease the low-temperature reservoir by the amount \(\Delta T\). The new low-temperature reservoir is \(T_L' = T_L - \Delta T\). Calculate the efficiency for this case using the Carnot efficiency formula: \(\eta_{decrease} = 1 - \frac{T_L - \Delta T}{T_H}\)
04

Comparing Efficiencies

Compare the two efficiencies, \(\eta_{increase}\) and \(\eta_{decrease}\), to determine which change results in a better efficiency. If \(\eta_{increase} > \eta_{decrease}\), increasing the high-temperature reservoir by the amount \(\Delta T\) results in better efficiency. If \(\eta_{increase} < \eta_{decrease}\), decreasing the low-temperature reservoir by the amount \(\Delta T\) results in better efficiency. It can be shown through mathematical comparison that decreasing the low-temperature reservoir by the amount \(\Delta T\) yields better efficiency.

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Most popular questions from this chapter

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In a certain steam engine, the boiler temperature is \(127^{\circ} \mathrm{C}\) and the cold reservoir temperature is \(27^{\circ} \mathrm{C} .\) While this engine does \(8.34 \mathrm{kJ}\) of work, what minimum amount of heat must be discharged into the cold reservoir?
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