In this problem you will estimate the heat lost by a typical house, assuming that the temperature inside is T_in= 20degrees C and the temperature outside is T_out = 0degrees C. The walls and uppermost ceiling of a typical house are supported by 2 x 6-inch wooden beams (k_wood = 0.12 W/m /K) with fiberglass insulation (k_ins = 0.04 (W/m /K) in between. The true depth of the beams is actually 5 and 5/8 inches, but we will take the thickness of the walls and ceiling to be L_wall = 18 cm to allow for the interior and exterior covering. Assume that the house is a cube of length L = 9.0m on a side. Assume that the roof has very high conductivity, so that the air in the attic is at the same temperature as the outside air. Ignore heat loss through the ground.
The first step is to calculate k_eff, the effective thermal conductivity of the wall (or ceiling), allowing for the fact that the 2 x 6 beams are actually only 1and 5/8 wide and are spaced 16 inches center to center.
The Attempt at a Solution
I know that k_eff = f(k_wood-k_ins)+k_ins, so I have everything except f, which is the fraction of the total wall/ceiling area in which the heat is conducted by wood. I'm having trouble finding this...