An air conditioning unit is used to provide cooling during summer for a house. If the air conditioner provides 450 kW cooling by using 150 kW electrical power, determine the coefficient of performance (COP) of the air conditioner.

1. The outside temperature and inside temperature are 40°C and 20°C, respectively. Using the inequality of Clausius, determine if the cycle is possible.

2. Determine the COP of an air conditioner working based on the Carnot cycle between the same temperature difference.

3. Compare the COPs of the Carnot and actual air conditioners and comment on them based on your answer for the previous part (the inequality of Clausius).

The coefficient of performance (COP) of the air conditioner is 3, according to its cooling output and electrical input. Comparing this with the Carnot cycle's theoretical COP of 14.65 showcases real-world inefficiencies but also confirms the air conditioner's cycle is theoretically possible.

Explanation:

To determine the coefficient of performance (COP) of the air conditioner, we use the formula for the COP of a refrigerator which is given as COPref = Qc / W, where Qc is the cooling provided (in this case 450 kW) and W is the electrical work input (150 kW). Thus, the COP of the air conditioner is 450 kW / 150 kW = 3.

The inequality of Clausius implies that for a cycle to be reversible (and thus, for a Carnot efficiency to be achievable), it cannot involve entropy production. However, real devices always have some inefficiency leading to entropy production, making them irreversible. Nevertheless, to check if a cycle is theoretically possible, you compare its COP to a Carnot cycle. The COP for a Carnot cycle is given by COPCarnot = Tc / (Th - Tc), with temperatures in Kelvin. Substituting Tc = 293 K (20°C) and Th = 313 K (40°C), we get COPCarnot = 293 / (313 - 293) = 14.65.

The actual air conditioner's COP of 3 is significantly lower than the Carnot COP of 14.65, which is expected due to real-world inefficiencies and entropy production. This indicates the cycle is possible since real-world COPs will always be lower than Carnot COPs due to irreversible processes. The comparison between Carnot and actual COPs highlights the limitations and inefficiencies in practical systems.