Question

2. In the middle of the summer in Houston, Texas you shut off the A/C of your home during the night to conserve energy. The room temperature (home temperature) reaches 82°F and stays there until the next morning when you reset the thermostat to 70F. Air begins to flow in through a supply duct at 250 ft /min and 60°F with an equal amount of mass of air at room temperature leaving your house thr return duct. Assume that the air is very well mixed in the house, that the air pressure is 1 atm everywhere, and heat transfer with the surroundings is negligible Estimate the amount of time it takes for the room temperature to reach 70°F from the time you reset the thermostat. The house is 2,500 ft ough a

Answer 1

Lets first calculate the mass of air in the house

At 82F

density of air = 1.28 Kg/m^3 = 0.08 lb/ft^3

Mass of air = 0.08*2500 = 200 lb

Now the specific heat capacity of air is

Cp = 1 KJ/Kg.K = 0.24 BTU/lb.F

Now we have a system where the air is entering at 60F and 250 ft^3/min and exiting air is 250 ft^3/min

Carrying out the energy balance we get

$mC_p*\frac{\mathrm{d} T}{\mathrm{d} t} = Q_{in} - Q_{out}$

$200*0.24*\frac{\mathrm{d} T}{\mathrm{d} t} = 20*0.24*(T_s - T)$

$10*\frac{\mathrm{d} T}{\mathrm{d} t} = (T_s - T)$

$\frac{dT}{T-60} = -0.1*dt$

Now

at t = 0, T =82 , Tf = 70

Thus

Ln( 82-60) - Ln(70-60) = 0.1*t

Ln(22) - Ln(10) = 0.1*t

0.788 = 0.1*t

t = 7.88

thus the time required for the room to reach 70F is 7.88 min