Question

Consider a Carnot refrigerator and a Carnot heat pump operating between the same two thermal energy reservoirs. If the COP of the refrigerator is \(3.4,\) the COP of the heat pump is \((a) 1.7\) (b) 2.4 \((c) 3.4\) \((d) 4.4\) \((e) 5.0\)

Short answer

a) 1.4 b) 2.4 c) 3.4 d) 4.4 Answer: d) 4.4

Step-by-step solution

Recall the formula for COP of a refrigerator and heat pump

For a refrigerator, the Coefficient of Performance (COP) is given by: COP(ref) = \(Q_c/W\) For a heat pump, the Coefficient of Performance (COP) is given by: COP(hp) = \(Q_h/W\) Where: \(Q_c\) = Heat absorbed from the cold reservoir \(Q_h\) = Heat delivered to the hot reservoir \(W\) = Work done on the system

Link the COPs of the refrigerator and the heat pump

As both the refrigerator and heat pump are operating between the same two thermal energy reservoirs, we can relate their COPs based on the energy conservation principle as follows: \(Q_h = Q_c + W\) By dividing both sides of the equation by \(W\), we get: \(Q_h / W = Q_c / W + 1\) Now substitute the COP(ref) and COP(hp) in the equation: COP(hp) = COP(ref) + 1

Calculate the COP of the heat pump

Given the COP of the refrigerator is 3.4, we can find the COP of the heat pump using the formula we derived in Step 2: COP(hp) = COP(ref) + 1 COP(hp) = 3.4 + 1 = 4.4 So, the COP of the heat pump is 4.4. The correct answer is (d) 4.4.