Question

Using the Kelvin-Planck statement of the second law of thermodynamics, demonstrate the following corollaries: (a) The coefficient of performance of an irreversible refrigeration cycle is always less than the coefficient of performance of a reversible refrigeration cycle when both exchange energy by heat transfer with the same two reservoirs. (b) All reversible refrigeration cycles operating between the same two reservoirs have the same coefficient of performance. (c) The coefficient of performance of an irreversible heat pump cycle is always less than the coefficient of performance of a reversible heat pump cycle when both exchange energy by heat transfer with the same two reservoirs. (d) All reversible heat pump cycles operating between the same two reservoirs have the same coefficient of performance.

Short answer

Reversible cycles have higher COPs than irreversible ones. For both refrigeration and heat pump cycles, all reversible cycles between the same reservoirs have the same COP.

Step-by-step solution

- Understand the given statement of the second law of thermodynamics

The Kelvin-Planck statement of the second law of thermodynamics states that it is impossible to construct a device that operates in a cycle and produces no effect other than the transfer of heat from a single reservoir and the performance of an equal amount of work. In simple terms, no heat engine can be 100% efficient.

- Define the coefficient of performance (COP) for refrigeration and heat pump cycles

The coefficient of performance (COP) for a refrigeration cycle is given by COP_{ref} = \frac{Q_L}{W_{cycle}}where \(Q_L\) is the heat removed from the low-temperature reservoir and \(W_{cycle}\) is the work input. For a heat pump cycle, the COP is given by COP_{hp} = \frac{Q_H}{W_{cycle}}where \(Q_H\) is the heat delivered to the high-temperature reservoir.

- Corollary (a): Irreversible refrigeration vs. reversible refrigeration

For any irreversible process, there are losses due to irreversibilities such as friction, unrestrained expansion, and heat transfer across finite temperature differences. These losses reduce the work input-effectiveness in an irreversible refrigeration cycle. Since a reversible cycle is an idealization with no irreversibilities, it will always have a higher COP compared to an irreversible cycle when dealing with the same two reservoirs. Hence, COP_{ref,irrev} < COP_{ref,rev}.

- Corollary (b): Reversible refrigeration cycles have the same COP

All reversible refrigeration cycles operating between the same two temperature reservoirs must comply with the Carnot principle, which states that the COP of such cycles depends only on the temperatures of the reservoirs:COP_{rev} = \frac{T_L}{T_H - T_L}. Since this relationship depends only on the temperatures \(T_L\) and \(T_H\), and not on the specifics of the cycle, all reversible refrigeration cycles between the same reservoirs share the same COP.

- Corollary (c): Irreversible heat pump vs. reversible heat pump

Similar to refrigeration cycles, irreversible heat pump cycles suffer from irreversibilities leading to efficiencies lower than the corresponding reversible cycles. Therefore, when both operate between the same high and low-temperature reservoirs, the COP of the irreversible heat pump cycle is always less than that of the reversible heat pump cycle: COP_{hp,irrev} < COP_{hp,rev}.

- Corollary (d): Reversible heat pump cycles have the same COP

For reversible heat pump cycles operating between the same two reservoirs, the COP depends only on the temperatures of those reservoirs. Using the Carnot principle, we have: COP_{hp,rev} = \frac{T_H}{T_H - T_L}. As this formula solely relies on the temperatures \(T_L\) and \(T_H\), any reversible heat pump cycle operating between the same two reservoirs will have the identical COP.

Key concepts

Coefficient of Performance (COP)

The Coefficient of Performance (COP) is a measure of the efficiency of refrigeration and heat pump cycles. For a refrigeration cycle, COP is defined as the ratio of the heat removed from the low-temperature reservoir (\(Q_L\)) to the work input (\(W_{cycle}\)). Mathematically, it is given by:
\[ COP_{ref} = \frac{Q_L}{W_{cycle}} \]
For a heat pump cycle, COP is defined as the ratio of the heat delivered to the high-temperature reservoir (\(Q_H\)) to the work input (\(W_{cycle}\)). It is given by:
\[ COP_{hp} = \frac{Q_H}{W_{cycle}} \]
Higher COP values indicate more efficient cycles. Remember, the efficiencies vary for reversible and irreversible cycles.

Reversible vs Irreversible Cycles

Reversible cycles are theoretical models that assume no losses or inefficiencies during the process. These cycles are idealized and represent maximum efficiency scenarios. Irreversible cycles, on the other hand, are realistic and account for actual inefficiencies such as friction, unrestrained expansion, and finite temperature differences in heat transfer.
Because reversible cycles are perfect and have no losses, their COP is higher compared to irreversible ones. This holds true for both refrigeration and heat pump cycles. For example:

• Refrigeration: \[ COP_{ref,irrev} < COP_{ref,rev} \]
• Heat Pump: \[ COP_{hp,irrev} < COP_{hp,rev} \]

Second Law of Thermodynamics

The Second Law of Thermodynamics is a fundamental principle that governs the direction of heat transfer and the efficiency of energy conversion processes. The Kelvin-Planck statement of this law says it's impossible to build a heat engine that converts all absorbed heat into work without any losses.
In simpler terms, no machine can be perfectly efficient. This law also implies that the COP of any irreversible cycle will be lower than that of a reversible cycle operating between the same two thermal reservoirs.
This principle helps in understanding why identical temperatures don't guarantee identical efficiencies if irreversibilities are present.

Heat Pump Cycles

Heat pumps move heat from a colder area to a warmer one using mechanical work. They are essentially the reverse of refrigeration cycles and are commonly used for heating purposes. The performance, or COP, of a heat pump cycle is higher in a reversible process due to the absence of irreversibilities. Mathematically:
\[ COP_{hp,rev} = \frac{T_H}{T_H - T_L} \]
Where \( T_L \) is the temperature of the low-temperature reservoir and \( T_H \) is the temperature of the high-temperature reservoir. In any practical setup, expect some loss, leading to a lower COP for irreversible cycles.

Refrigeration Cycles

Refrigeration cycles are used to remove heat from a low-temperature space and release it to a higher-temperature environment. The most common cycles used are vapor-compression, absorption, and gas cycles. For an ideal, reversible refrigeration cycle, the COP can be represented by:
\[ COP_{ref,rev} = \frac{T_L}{T_H - T_L} \]
Here, \( T_L \) is the temperature at which heat is absorbed, and \( T_H \) is the temperature at which heat is released. Since all reversible cycles between the same reservoirs have the same efficiency, the COP remains constant for those conditions, making them a benchmark for comparing real-life (irreversible) refrigeration systems.