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Chapter 5: Problem 26

An inventor claims to have developed a device that executes a power cycle while operating between reservoirs at 800 and \(350 \mathrm{~K}\) that has a thermal efficiency of (a) \(56 \%\), (b) \(40 \%\). Evaluate the claim for each case.

Short Answer

Expert verified
The claims are possible because both 56% and 40% are less than or equal to the Carnot efficiency of 56.25%.

Step by step solution

01

- Understand the Problem

An inventor's device operates between two reservoirs. The temperatures are given as 800 K for the hot reservoir and 350 K for the cold reservoir. The claim involves evaluating the device's thermal efficiency for both 56% and 40%.
02

- Recall the Formula for Carnot Efficiency

The maximum possible efficiency for any heat engine operating between two temperatures is given by the Carnot efficiency formula: \[ \eta_{carnot} = 1 - \frac{T_c}{T_h} \], where \( T_c \) is the temperature of the cold reservoir and \( T_h \) is the temperature of the hot reservoir.
03

- Calculate Carnot Efficiency

Using the given temperatures, calculate Carnot efficiency: \[ \eta_{carnot} = 1 - \frac{350}{800} = 1 - 0.4375 = 0.5625 \] or 56.25%.
04

- Evaluate the Claim for 56%

Compare the given efficiency of 56% to the Carnot efficiency of 56.25%. Since 56% is less than 56.25%, it is physically possible.
05

- Evaluate the Claim for 40%

Compare the given efficiency of 40% to the Carnot efficiency of 56.25%. Since 40% is significantly less than 56.25%, this claim is also physically possible.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

thermal efficiency
Thermal efficiency is a measure of how well a heat engine converts heat into work. It's important because it tells us how much useful work we can get from a certain amount of heat energy. The higher the thermal efficiency, the better. For a heat engine, thermal efficiency \(\text{畏}\) is given by the formula \( \text{畏} = \frac{W_{net}}{Q_{in}} \), where \( W_{net} \) is the net work output and \( Q_{in} \) is the heat input.
  • The efficiency ranges from 0 to 1 (or 0% to 100%).
  • Perfect efficiency (100%) is not possible due to inherent energy losses such as friction and heat dissipation.

Therefore, thermal efficiency helps us understand the effectiveness of a heat engine, and serves as a benchmark when comparing different engines or cycles.
power cycle
A power cycle is a process where a working fluid (like steam or gas) goes through a series of state changes, transforming heat energy into mechanical work. This cycle is repeated over and over.
For example:
  • The Carnot cycle is a theoretical model of an ideal power cycle.
  • Real-life cycles include the Rankine cycle for steam engines and the Otto cycle for internal combustion engines.

The efficiency and effectiveness of a power cycle depend on factors like the type of working fluid, the temperature range, and the design of the engine. In our example, the power cycle involves reservoirs at 800 K and 350 K.
Carnot efficiency formula
The Carnot efficiency formula allows us to calculate the maximum possible efficiency a heat engine can achieve, operating between two temperature reservoirs. It's given by:
\[ \text{畏}_{carnot} = 1 - \frac{T_c}{T_h} \]
where:
  • \( T_c \) is the temperature of the cold reservoir (in Kelvin).
  • \( T_h \) is the temperature of the hot reservoir (in Kelvin).

The Carnot efficiency sets the upper limit for the efficiency of any real engine. Any actual engine will have a lower efficiency than the Carnot efficiency because real processes have losses such as friction and non-reversible processes.
reservoir temperatures
Reservoir temperatures are crucial in determining the Carnot efficiency and thus the overall thermal efficiency of a heat engine. In thermodynamics, reservoirs are large thermal bodies that maintain a constant temperature even when heat is added or extracted.
In our example, the hot reservoir is at 800 K and the cold reservoir is at 350 K.
Higher temperature differences between the hot and cold reservoirs generally result in higher efficiencies:
  • Higher \( T_h \) relative to \( T_c \) means more potential work output.
  • Lower \( T_c \) relative to \( T_h \) also increases efficiency.

Therefore, to achieve higher efficiencies, heat engines strive to operate between a very high temperature source and a very low temperature sink.
heat engine
A heat engine is a device that converts thermal energy into mechanical work. It does this by exploiting the differences in temperatures between two reservoirs. The main components of heat engines include a working fluid, a source of heat (hot reservoir), and a heat sink (cold reservoir).
In this example:
  • The working fluid absorbs heat from the hot reservoir (800 K).
  • It then does work (like moving a piston or turning a turbine).
  • Finally, it releases unused heat to the cold reservoir (350 K).

The effectiveness of a heat engine is measured by its thermal efficiency. By understanding and applying the Carnot efficiency formula, we can determine how close a given heat engine's performance is to the theoretical maximum efficiency.

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Most popular questions from this chapter

To increase the thermal efficiency of a reversible power cycle operating between thermal reservoirs at temperatures \(T_{\mathrm{H}}\) and \(T_{C}\), would it be better to increase \(T_{\mathrm{H}}\) or decrease \(T_{\mathrm{C}}\) by equal amounts?

Write a paper outlining the contributions of Carnot, Clausius, Kelvin, and Planck to the development of the second law of thermodynamics. In what ways did the now-discredited caloric theory influence the development of the second law as we know it today? What is the historical basis for the idea of a perpetual motion machine of the second kind that is sometimes used to state the second law?

A power cycle operates between a reservoir at temperature \(T\) and a lower- temperature reservoir at \(280 \mathrm{~K}\). At steady state, the cycle develops \(40 \mathrm{~kW}\) of power while rejecting 1000 \(\mathrm{kJ} / \mathrm{min}\) of energy by heat transfer to the cold reservoir. Determine the minimum theoretical value for \(T\), in \(\mathrm{K}\).

What are some of the principal irreversibilities present during operation of (a) an automobile engine, (b) a household refrigerator, (c) a gas-fired water heater, (d) an electric water heater?

The preliminary design of a space station calls for a power cycle that at steady state receives energy by heat transfer at \(T_{\mathrm{H}}=600 \mathrm{~K}\) from a nuclear source and rejects energy to space by thermal radiation according to Eq. 2.33. For the radiative surface, the temperature is \(T_{\mathrm{C}}\), the emissivity is \(0.6\), and the surface receives no radiation from any source. The thermal efficiency of the power cycle is one- half that of a reversible power cycle operating between reservoirs at \(T_{\mathrm{H}}\) and \(T_{\mathrm{C}}\) - (a) For \(T_{\mathrm{C}}=400 \mathrm{~K}\), determine \(\dot{W}_{\text {cycle }} / \mathrm{A}\), the net power developed per unit of radiator surface area, in \(\mathrm{kW} / \mathrm{m}^{2}\), and the thermal efficiency. (b) Plot \(\dot{W}_{\text {cycle }} / \mathrm{A}\) and the thermal efficiency versus \(T_{\mathrm{C}}\), and determine the maximum value of \(\dot{W}_{\text {cycle }} / \mathrm{A}\). (c) Determine the range of temperatures \(T_{\mathrm{C}}\), in \(\mathrm{K}\), for which \(\dot{W}_{\text {cycle }} / \mathrm{A}\) is within 2 percent of the maximum value obtained in part (b). The Stefan-Boltzmann constant is \(5.67 \times 10^{-8} \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}^{4}\).
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