Question

If the thermal efficiency of a reversible power cycle operating between two reservoirs is denoted by \(\eta_{\max }\), develop an expression in terms of \(\eta_{\max }\) for the coefficient of performance of (a) a reversible refrigeration cycle operating between the same two reservoirs. (b) a reversible heat pump operating between the same two reservoirs.

Short answer

(a) \text{COP}_{\text{refrigerator}} = \frac{1 - \eta_{\text{max}}}{\text{\eta_{\text{max}}}}\(b) \text{COP}_{\text{heat pump}} = \frac{1}{\text{\eta_{\text{max}}}}

Step-by-step solution

Identify the Thermal Efficiency Formula

Given that the thermal efficiency of a reversible power cycle is denoted by \eta_{\text{max}}, the efficiency can be expressed as: \[ \eta_{\text{max}} = 1 - \frac{T_L}{T_H} \]where \(T_L\) is the temperature of the low-temperature reservoir and \(T_H\) is the temperature of the high-temperature reservoir.

Coefficient of Performance for Refrigeration Cycle

For a reversible refrigeration cycle, the coefficient of performance (COP) in terms of the temperatures \(T_L\) and \(T_H\) is given by: \[ \text{COP}_{\text{refrigerator}} = \frac{T_L}{T_H - T_L} \]Rewrite this using \eta_{\text{max}}: \[ \text{COP}_{\text{refrigerator}} = \frac{T_L}{T_H(1 - \frac{T_L}{T_H})} = \frac{T_L}{T_H - T_L} = \frac{T_L}{T_H} \frac{1}{1 - \frac{T_L}{T_H}} \]Use \eta_{\text{max}} = 1 - \frac{T_L}{T_H}: \[ \text{COP}_{\text{refrigerator}} = \frac{1 - \eta_{\text{max}}}{\text{\eta_{\text{max}}}} \]

Coefficient of Performance for Heat Pump

For a reversible heat pump cycle, the coefficient of performance (COP) in terms of \(T_L\) and \(T_H\) is given by: \[ \text{COP}_{\text{heat pump}} = \frac{T_H}{T_H - T_L} \]Rewrite using the known efficiency relation: \[ \text{COP}_{\text{heat pump}} = \frac{T_H}{T_H(1 - \frac{T_L}{T_H})} = \frac{1}{1 - \frac{T_L}{T_H}} \]Use \eta_{\text{max}} = 1 - \frac{T_L}{T_H}: \[ \text{COP}_{\text{heat pump}} = \frac{1}{\text{\eta_{\text{max}}}} \]

Key concepts

Reversible Power Cycle

A reversible power cycle is a thermodynamic process where a system returns to its initial state at the end of the cycle, making the whole process reversible. This means no energy is lost or dissipated, and the cycle can operate in a forward or reverse manner with identical performance. Systems operating on such cycles often involve heat engines or other machinery that work with heat exchange between two reservoirs. The thermal efficiency of a reversible power cycle is denoted by \( \eta_{\text{max}} \), representing the maximum efficiency achievable under ideal conditions.

Coefficient of Performance

The coefficient of performance (COP) is a measure of efficiency for refrigeration cycles and heat pumps. Unlike thermal efficiency, which is often less than 1, the COP can be greater than 1 as it measures the ratio of useful heating or cooling provided versus the work input. For refrigeration cycles, COP determines how well the system keeps a space cool. For heat pumps, it measures the heating efficiency. For instance, the COP for a refrigeration cycle can be expressed in terms of temperature of the reservoirs, using the relationship: \[ \text{COP}_{\text{refrigerator}} = \frac{T_L}{T_H - T_L} \]

Refrigeration Cycle

A refrigeration cycle is a process that removes heat from a low-temperature reservoir and transfers it to a high-temperature reservoir. This is achieved by using a refrigerant, which absorbs the heat from the desired space and dissipates it outside. The performance of such a system is quantified by the COP, showing how much cooling effect is delivered for each unit of work input. Using the maximum thermal efficiency \( \eta_{\text{max}} \):

• \[ \text{COP}_{\text{refrigerator}} = \frac{1 - \eta_{\text{max}}}{\text{ \eta_{\text{max}} }} \]

Heat Pump

A heat pump is a device that transfers heat from a low-temperature area to a high-temperature one, opposite to natural heat flow, using work input. This is similar to a refrigeration cycle but focused on heating instead of cooling. The COP for a heat pump is expressed as the heat delivered to the high-temperature reservoir per unit of work input. By using \( \eta_{\text{max}} \):

• \[ \text{COP}_{\text{heat pump}} = \frac{1}{\text{ \eta_{\text{max}}}} \]

Thermodynamic Cycles

Thermodynamic cycles are processes where a system undergoes a series of state changes, eventually returning to its initial state. Examples include the Carnot cycle, Rankine cycle, and refrigeration cycle. These cycles are crucial for understanding energy conversion systems. Cycles can either be power cycles (like in engines) that convert heat into work or refrigeration cycles that transfer heat for cooling processes. The efficiency and performance of these cycles hinge on concepts like reversible processes and the effectiveness of heat transfer. The laws of thermodynamics govern these cycles, dictating possible efficiencies and constraints.