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AC unit COP 3 and 4: max and min outside temperature

Short answer

Cooling capacity = 3×1.5 = 4.5 kW, heating capacity = 4×1.5 = 6 kW. Heat exchange rate 0.8 kW/°C. Maximum outside temperature = 22 + 4.5/0.8 = 27.6°C, minimum = 22 − 6/0.8 = 14.5°C.

An air conditioning unit with a power input of 1.5 kW. It has a COP of 3 while working as a cooling unit in summer and 4 while working as heating unit in winter. It maintains a hall at 22°C year around, which exchanges heat at a rate of 0.8 kW per degree temperature difference with the surroundings. Determine the maximum and the minimum outside temperature for which this unit is sufficient.

Given: power W = 1.5 kW. COP = 3 (cooling, summer) and 4 (heating, winter). Hall at T_hall = 22 °C. Heat exchange with the surroundings: 0.8 kW per °C difference.

Maximum outside temperature (summer, cooling)

The unit removes heat Q_L = COP × W = 3 × 1.5 = 4.5 kW.

In steady state this equals the heat gained by the hall from hot outside air:

0.8 (T_out − 22) = 4.5 T_out − 22 = 5.625

T_out,max = 27.6 °C

Above this temperature the heat gain exceeds 4.5 kW and the unit cannot hold 22 °C.

Minimum outside temperature (winter, heating)

The unit supplies heat Q_H = COP × W = 4 × 1.5 = 6.0 kW.

In steady state this equals the heat lost by the hall to the cold outside:

0.8 (22 − T_out) = 6.0 22 − T_out = 7.5

T_out,min = 14.5 °C

Below this temperature the heat loss exceeds 6 kW and the unit cannot hold 22 °C.

Answer: the unit is sufficient for outside temperatures from 14.5 °C (minimum, winter) to 27.6 °C (maximum, summer).

✓ Answer checked by Claude Opus 5.5 (high effort)

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