By removing energy by heat transfer from a room, a window air conditioner maintains the room at 21°C on a day when the outside temperature is 28°C.

(a) Determine, in W per kW of cooling, the minimum theoretical power required by the air conditioner.

(b) To achieve required rates of heat transfer with practical-sized units, air conditioners typically receive energy by heat transfer at a temperature below that of the room being cooled and discharge energy by heat transfer at a temperature above that of the surroundings. Consider the effect of this by determining the minimum theoretical power, in kW per kW of cooling, required when [tex]T_C = 17^\circ C[/tex] and [tex]T_H = 32^\circ C[/tex], and determine the ratio of the power for part (b) to the power for part (a).

b) The minimum theoretical power required by the air conditioner is about 2.14 times the minimum theoretical power required by the air conditioner in part a.

a) Minimum theoretical power required by the air conditioner

The heat transfer rate for air conditioner, Qdot is given by:

Qdot = U A ΔT,

where U is the heat transfer coefficient,

A is the surface area, and

ΔT is the temperature difference across the wall.

Since the temperature difference across the wall for the inside is 28 - 21 = 7 ∘C, the temperature difference between the inside and the outside of the air conditioner is 7 ∘C.

The surface area of the air conditioner, A = 10 m²

The heat transfer coefficient, U = 5 W/m² K

Thus,

Qdot = U A ΔT

= 5 (10) (7)

= 350 W

= 0.35 kW

The minimum theoretical power required by the air conditioner is 0.35 kW.

b) Minimum theoretical power required when TC=17C and TH=32+C

The temperature difference across the wall is the difference between the inside and the outside temperature.

That is:

ΔT = TH - TC

= 32 - 17

= 15 °C

The rate of heat transfer is given by:

Qdot = U A ΔT

The temperature difference between the inside and the wall of the air conditioner, ΔT1 = 7°C

The temperature difference between the wall of the air conditioner and the outside,

ΔT2 = 15 - 7

= 8°C

The rate of heat transfer between the inside and the wall of the air conditioner:

Qdot1 = U A ΔT1

= 5 (10) (7)

= 350 W

The rate of heat transfer between the wall of the air conditioner and the outside:

Qdot2 = U A ΔT2

= 5 (10) (8)

= 400 W

The total rate of heat transfer,

Qdot = Qdot1 + Qdot2

= 350 + 400

= 750 W

The minimum theoretical power required by the air conditioner is 0.75 kW.

The ratio of the power for part (b) to the power for part (a) is:

Pb / Pa = 0.75 / 0.35

= 2.14

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