Room air conditioning can be described as a heat engine operating in reverse mode - it extracts heat Q m from a room at temperature T m and transfers heat Q v Q m to the air outside the hous

Question

Room air conditioning can be described as a heat engine operating in reverse mode - it extracts heat $Q_{m}$ from a room at temperature $T_{m}$ and transfers heat $Q_{v} > Q_{m}$ to the air outside the house at temperature $T_{V}$ . An electric compressor must supply work W to achieve this. Assume the device operates as a Carnot engine with maximum possible efficiency. Even though the room is insulated, heat enters from outside at a rate $\frac{Δ Q}{Δ t} = k Δ T$ , where $Δ T = T_{v} - T_{m}$ is the temperature difference between the outside environment and the room, and k is a constant. What is the minimum power P (in W) needed for the air conditioner to maintain a room temperature $T_{m} = 2 0^{\circ} C$ , given an outdoor temperature $T_{v} = 4 0^{\circ} C$ and a typical value of the constant $k = 293 W \cdot K^{- 1}$ ?

Answer:

400

Solution:

At steady room temperature, cooling must remove the incoming heat.

$\dot{Q}_{m} = k Δ T = 293 \times 20 W$

For a Carnot refrigerator:

$\frac{Q _{m}}{T _{m}} = \frac{Q _{v}}{T _{v}}$

$P = \dot{Q}_{m} \frac{T _{v} - T _{m}}{T _{m}}$

$P = 293 \times 20 \times \frac{20}{293} = 400 W$

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