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Chapter 5: Problem 48

The thermal efficiency of a reversible power cycle operating between hot and cold reservoirs is \(20 \%\). Evaluate the coefficient of performance of (a) a reversible refrigeration cycle operating between the same two reservoirs. (b) a reversible heat pump cycle operating between the same two reservoirs.

Short Answer

Expert verified
(a) COP for refrigeration cycle = 4 (b) COP for heat pump cycle = 5

Step by step solution

01

Identify Given Data

The thermal efficiency of the reversible power cycle is given as \( \eta = 20\% = 0.20 \).
02

Find the Temperature Ratio

Using the efficiency equation for a Carnot cycle, \[ \eta = 1 - \frac{T_C}{T_H} \] we can solve for the temperature ratio: \[ \frac{T_C}{T_H} = 1 - \eta = 1 - 0.20 = 0.80 \].
03

Coefficient of Performance for Refrigeration Cycle

The coefficient of performance (COP) for a reversible refrigeration cycle is given by \[ \text{COP}_{\text{refrigeration}} = \frac{T_C}{T_H - T_C} \] Given that \( \frac{T_C}{T_H} = 0.80 \), so \[ \text{COP}_{\text{refrigeration}} = \frac{0.80T_H}{T_H - 0.80T_H} = \frac{0.80T_H}{0.20T_H} = \frac{0.80}{0.20} = 4 \].
04

Coefficient of Performance for Heat Pump Cycle

The coefficient of performance (COP) for a reversible heat pump cycle is given by \[ \text{COP}_{\text{heat pump}} = \frac{T_H}{T_H - T_C} \] Given that \( \frac{T_C}{T_H} = 0.80 \), so \[ \text{COP}_{\text{heat pump}} = \frac{T_H}{T_H - 0.80T_H} = \frac{T_H}{0.20T_H} = \frac{1}{0.20} = 5 \].

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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 key concept in thermodynamics, indicating how efficiently a heat engine converts heat energy into work. It is denoted by the symbol \( \eta \) and is defined as the ratio of work output to heat input: \[ \eta = \frac{W}{Q_H} \]. For a reversible power cycle, it鈥檚 the maximum efficiency that that cycle can achieve when operating between two temperature reservoirs.
Coefficient of Performance (COP)
The coefficient of performance (COP) is a measure of efficiency for refrigerators and heat pumps. It is the ratio of the heat transfer to the work input. For a refrigeration cycle, COP is defined as: \[ COP_{refrigeration} = \frac{Q_C}{W} = \frac{T_C}{T_H - T_C} \]. For a heat pump, COP is: \[ COP_{heat\ pump} = \frac{Q_H}{W} = \frac{T_H}{T_H - T_C} \].
Carnot Cycle
The Carnot cycle is an idealized thermodynamic cycle proposed by Nicolas L茅onard Sadi Carnot. It provides the maximum possible efficiency for a heat engine operating between two temperatures: \[ \eta = 1 - \frac{T_C}{T_H} \]. This efficiency is considered the upper limit for real engines.
Refrigeration Cycle
A refrigeration cycle is used to transfer heat from a cold region to a warm region. The COP for a reversible refrigeration cycle is higher than that of real systems. The given problem presents: \[ COP_{refrigeration} = \frac{T_C}{T_H - T_C} = \frac{0.80T_H}{0.20T_H} = 4 \].
Heat Pump Cycle
The heat pump cycle works similarly to the refrigeration cycle but transfers heat to a hotter region. The COP for a reversible heat pump cycle is calculated by: \[ COP_{heat\/pump} = \frac{T_H}{T_H - T_C} = \frac{T_H}{0.20T_H} = 5 \]. This indicates that the heat pump is very efficient.

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Most popular questions from this chapter

A power cycle receives 1000 Btu by heat transfer from a reservoir at \(1000^{\circ} \mathrm{F}\) and discharges energy by heat transfer to a reservoir at \(300^{\circ} \mathrm{F}\). The thermal efficiency of the cycle is \(75 \%\) of that for a reversible power cycle operating between the same reservoirs, (a) For the actual cycle, determine the thermal efficiency and the energy discharged to the cold reservoir, in Btu. (b) Repeat for the reversible power cycle.

At steady state, a power cycle develops a power output of \(10 \mathrm{~kW}\) while receiving energy by heat transfer at the rate of \(10 \mathrm{~kJ}\) per cycle of operation from a source at temperature \(T\). The cycle rejects energy by heat transfer to cooling water at a lower temperature of \(300 \mathrm{~K}\). If there are 100 cycles per minute, what is the minimum theoretical value for \(T\), in \(\mathrm{K}\) ?

At steady state, a reversible heat pump cycle discharges energy at the rate \(\dot{Q}_{\mathrm{H}}\) to a hot reservoir at temperature \(T_{\mathrm{H}}\), while receiving energy at the rate \(\dot{Q}_{\mathrm{C}}\) from a cold reservoir at temperature \(T_{\mathrm{C}}\). (a) If \(T_{\mathrm{H}}=13^{\circ} \mathrm{C}\) and \(T_{\mathrm{C}}=2^{\circ} \mathrm{C}\), determine the coefficient of performance. (b) If \(\dot{Q}_{\mathrm{H}}=10.5 \mathrm{~kW}, \dot{Q}_{\mathrm{C}}=8.75 \mathrm{~kW}\), and \(T_{\mathrm{C}}=0^{\circ} \mathrm{C}\), determine \(T_{\mathrm{H}}\), in \({ }^{\circ} \mathrm{C}\). (c) If the coefficient of performance is 10 and \(T_{\mathrm{H}}=27^{\circ} \mathrm{C}\), determine \(T_{\mathrm{C}}\), in \({ }^{\circ} \mathrm{C}\). \(5.46\) A heating system must maintain the interior of a building at \(20^{\circ} \mathrm{C}\) during a period when the outside air temperature is \(5^{\circ} \mathrm{C}\) and the heat transfer from the building through its roof and walls is \(3 \times 10^{6} \mathrm{~kJ}\). For this duty heat pumps are under consideration that would operate between the dwelling and (a) the ground at \(15^{\circ} \mathrm{C}\). (b) a pond at \(10^{\circ} \mathrm{C}\). (c) the outside air at \(5^{\circ} \mathrm{C}\). For each case, evaluate the minimum theopatical not worlk input required by any such heat pump, in \(\mathrm{k}\).

An inventor claims to have developed a power cycle having a thermal efficiency of \(40 \%\), while operating between hot and cold reservoirs at temperature \(T_{\mathrm{H}}\) and \(T_{\mathrm{C}}=300 \mathrm{~K}\), respectively, where \(T_{\mathrm{H}}\) is (a) \(900 \mathrm{~K}\), (b) \(500 \mathrm{~K}\), (c) \(375 \mathrm{~K}\). Evaluate the claim for each case.

A heat pump is under consideration for heating a research station located on an Antarctica ice shelf. The interior of the station is to be kept at \(15^{\circ} \mathrm{C}\). Determine the maximum theoretical rate of heating provided by a heat pump, in \(\mathrm{kW}\) per kW of power input, in each of two cases: The role of the cold reservoir is played by (a) the atmosphere at \(-20^{\circ} \mathrm{C}\), (b) ocean water at \(5^{\circ} \mathrm{C}\).
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