Question

The COP of heat pump in comparison with COP of refrigeration cycle for the specified temperature limits is given by

• A. \( \text{COP}_{\text{HP}} + \text{COP}_{\text{R}} = 1 \)
• B. \( \text{COP}_{\text{HP}} - \text{COP}_{\text{R}} = 1 \)
• C. \( \text{COP}_{\text{HP}} \times \text{COP}_{\text{R}} = 1 \)
• D. \( \text{COP}_{\text{HP}} / \text{COP}_{\text{R}} = 1 \)

Hint:

The fundamental relationship between the Coefficient of Performance of a heat pump (\(\text{COP}_{\text{HP}}\)) and a refrigerator (\(\text{COP}_{\text{R}}\)) operating between the same temperature limits is \( \text{COP}_{\text{HP}} = \text{COP}_{\text{R}} + 1 \). This comes directly from the energy balance for the cycle: the heat delivered by a heat pump is the sum of heat extracted from the cold source and the work input.

Answer 1

The Correct Option is B

Step 1: Define COP for a refrigeration cycle and a heat pump.
Coefficient of Performance (COP) of a Refrigeration Cycle (\(\text{COP}_{\text{R}}\)): This measures the efficiency of a refrigerator in removing heat from a cold space. It is defined as the ratio of the heat removed from the cold reservoir (\(Q_L\)) to the work input (\(W_{in}\)). \[ \text{COP}_{\text{R}} = \frac{Q_L}{W_{in}} \] Coefficient of Performance (COP) of a Heat Pump (\(\text{COP}_{\text{HP}}\)): This measures the efficiency of a heat pump in delivering heat to a hot space. It is defined as the ratio of the heat delivered to the hot reservoir (\(Q_H\)) to the work input (\(W_{in}\)). \[ \text{COP}_{\text{HP}} = \frac{Q_H}{W_{in}} \]
Step 2: Relate heat quantities and work input.
According to the First Law of Thermodynamics for a cycle, the heat rejected to the hot reservoir (\(Q_H\)) is the sum of the heat absorbed from the cold reservoir (\(Q_L\)) and the work input (\(W_{in}\)): \[ Q_H = Q_L + W_{in} \]
Step 3: Establish the relationship between \(\text{COP}_{\text{HP}}\) and \(\text{COP}_{\text{R}}\).
From the definitions in Step 1, we can express \(Q_L\) and \(Q_H\) in terms of COP and work input:
\(Q_L = \text{COP}_{\text{R}} \cdot W_{in}\)
\(Q_H = \text{COP}_{\text{HP}} \cdot W_{in}\)
Substitute these into the energy balance equation (\(Q_H = Q_L + W_{in}\)): \[ \text{COP}_{\text{HP}} \cdot W_{in} = \text{COP}_{\text{R}} \cdot W_{in} + W_{in} \] Divide the entire equation by \(W_{in}\) (assuming \(W_{in} \neq 0\)): \[ \text{COP}_{\text{HP}} = \text{COP}_{\text{R}} + 1 \] Rearranging this equation gives: \[ \text{COP}_{\text{HP}} - \text{COP}_{\text{R}} = 1 \]
Step 4: Compare with the given options.
The derived relationship \( \text{COP}_{\text{HP}} - \text{COP}_{\text{R}} = 1 \) directly matches Option (2). The final answer is \( \boxed{\text{2}} \).