A room air conditioner is modeled as a heat engine run in reverse: an amount of heat QL is absorbed from the room at a temperature TL into cooling coils containing a working gas; this gas is compressed adiabatically to a temperature TH; the gas is compressed isothermally in a coil outside the house, giving off an amount of heat QH; the gas expands adiabatically back to a temperature TL; and the cycle repeats. An amount of energy W is input into the system every cycle through an electric pump. This model describes the air conditioner with the best possible efficiency.
[Figure: Diagram showing pump, heating coil, room, cooling coil, and valve arrangement]
Assume that the outside air temperature is TH and the inside air temperature is TL. The air-conditioner unit consumes electric power P. Assume that the air is sufficiently dry so that no condensation of water occurs in the cooling coils of the air conditioner. Water boils at 373 K and freezes at 273 K at normal atmospheric pressure.
a. Derive an expression for the maximum rate at which heat is removed from the room in terms of the air temperatures TH, TL, and the power consumed by the air conditioner P. Your derivation must refer to the entropy changes that occur in a Carnot cycle in order to receive full marks for this part.
b. The room is insulated, but heat still passes into the room at a rate R=kΔT, where ΔT is the temperature difference between the inside and the outside of the room and k is a constant. Find the coldest possible temperature of the room in terms of TH, k, and P.
c. A typical room has a value of k = 173 W/°C. If the outside temperature is 40°C, what minimum power should the air conditioner have to get the inside temperature down to 25°C?