Log out Go to App
Go to App

Learning Materials

Features

Discover

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A heat engine takes in \(125 \mathrm{kJ}\) of heat from a reservoir at $815 \mathrm{K}\( and exhausts \)82 \mathrm{kJ}\( to a reservoir at \)293 \mathrm{K}$ (a) What is the efficiency of the engine? (b) What is the efficiency of an ideal engine operating between the same two reservoirs?
For a more realistic estimate of the maximum coefficient of performance of a heat pump, assume that a heat pump takes in heat from outdoors at $10^{\circ} \mathrm{C}$ below the ambient outdoor temperature, to account for the temperature difference across its heat exchanger. Similarly, assume that the output must be \(10^{\circ} \mathrm{C}\) hotter than the house (which itself might be kept at \(20^{\circ} \mathrm{C}\) ) to make the heat flow into the house. Make a graph of the coefficient of performance of a reversible heat pump under these conditions as a function of outdoor temperature (from $\left.-15^{\circ} \mathrm{C} \text { to }+15^{\circ} \mathrm{C} \text { in } 5^{\circ} \mathrm{C} \text { increments }\right)$
A balloon contains \(160 \mathrm{L}\) of nitrogen gas at \(25^{\circ} \mathrm{C}\) and 1.0 atm. How much energy must be added to raise the temperature of the nitrogen to \(45^{\circ} \mathrm{C}\) while allowing the balloon to expand at atmospheric pressure?
Suppose you mix 4.0 mol of a monatomic gas at \(20.0^{\circ} \mathrm{C}\) and 3.0 mol of another monatomic gas at \(30.0^{\circ} \mathrm{C} .\) If the mixture is allowed to reach equilibrium, what is the final temperature of the mixture? [Hint: Use energy conservation.]
A container holding \(1.20 \mathrm{kg}\) of water at \(20.0^{\circ} \mathrm{C}\) is placed in a freezer that is kept at \(-20.0^{\circ} \mathrm{C} .\) The water freezes and comes to thermal equilibrium with the interior of the freezer. What is the minimum amount of electrical energy required by the freezer to do this if it operates between reservoirs at temperatures of $20.0^{\circ} \mathrm{C}\( and \)-20.0^{\circ} \mathrm{C} ?$
See all solutions

Recommended explanations on Physics Textbooks

Space Physics

Read Explanation

Waves Physics

Read Explanation

Fundamentals of Physics

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Wave Optics

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free