Ann turns on her fan trying to cool her room while she is gone. The room temperature when she comes back in the evening needs to be determined.

Assumptions:

• Consider the air as an ideal gas
• All the doors and windows are tightly closed
• Disregard any heat transfer through the walls and the windows
• The room pressure remains at 1 atm

The system is a closed system since all the doors and windows are tightly closed. It is a stationary system by disregarding the changes of the velocity and elevation of air during the cooling process. The energy balance for this system is:

      Q - W = U₂ - U₁

Also, the assumption indicates that the heat transfer through the walls and windows is disregarded, hence no heat transfer to or from this system.

      Q = 0

The electric energy is transferred into the system by the fan, hence this work is negative.

      W = -_fant

For an ideal gas, the internal energy is a function of the temperature. If the specific heat at the average room temperature is used, the difference of internal energy is:

      U₂ - U₁ = c_v,av(T₂ - T₁)

The energy equation for the air in the room becomes:

      _fant = c_v,avm(T₂ - T₁)

      T₂ = _fant/c_v,avm + T₁

Ann leaves at 8:00 in the morning and returns at 6:00 in the evening. Hence the total time is 10 hours.

The average value of specific heat is used to calculate the temperature. A room temperature is assume first. Then iteration is used till a reliable result obtained. Assume the room temperature is 55 ^oC when she is back in the evening.

      T_av = (15 + 55)/2.0 = 35 ^oC

At 1 atm and 35 ^oC, the density of the air is ρ = 1.166 kg/m^3. The mass of the air in the room is:

      m = (4)(6)(6)(1.166) = 167.9 kg

From table, at 35 ^oC, the constant volume specific heat is
      c_v,av = 0.718 kJ/(kg-K) = 718 J/(kg-K).

With all the data known,

      T₂ = (150)(10)(3600)/((718)(167.9) + 15 = 59.8 ^oC

      T_av = (15 + 59.8)/2.0 = 37.4 ^oC

The calculated average temperature is pretty close to 35 ^oC (the assumed temperature), hence no iteration is needed.

As a result, it looks like Ann's dream of a cooler room in the evening is impossible after all.