In this problem you will estimate the heat lost by a typical house, assuming that the temperature inside is T(in) = 20 degrees celcius and the temperature outside is T(out) = 0 degrees celcius. The walls have fiberglass insulation, which dominates the heat conduction properties of the wall So we can consider the wall to have a thermal conductivity of k(wall) = 0.048 W/m/K
. We will take the thickness of the walls and ceiling to be L(wall) = 12 cm. Assume that the house is a cube of length L = 9.0 m 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.
a)What is H, the total rate of energy loss due to heat conduction for this house?
b)Let us assume that the winter consists of 150 days in which the outside temperature is 0*C on average. This will give the typical number of "heating degree days" observed in a winter in Vancouver. Given that heating oil has an energy content of 35 MJ per litre when burned, how much oil will be needed to supply the heat lost by conduction from this house over a winter? Assume that the heating system is 75% efficient.
a) To calculate the total rate of energy loss due to heat conduction for this house, we need to find the heat flux (Q) through the walls. The heat flux is given by the equation:
Q = (k * A * (T(in) - T(out))) / L
Where:
- k is the thermal conductivity of the wall material (given as 0.048 W/m/K)
- A is the surface area of the walls (since it is a cube, all sides have the same area)
- T(in) is the temperature inside the house (given as 20 degrees Celsius)
- T(out) is the temperature outside the house (given as 0 degrees Celsius)
- L is the thickness of the walls (given as 12 cm)
First, we need to convert the thickness of the walls to meters:
L = 12 cm = 0.12 m
Next, we can calculate the surface area of the walls:
A = 6 * (L * L)
Substituting the values into the equation, we get:
Q = (0.048 * 6 * (9.0 * 9.0) * (20 - 0)) / 0.12
Simplifying the equation gives us:
Q = 3888 W
Therefore, the total rate of energy loss due to heat conduction (H) is 3888 W.
b) To calculate the amount of oil needed to supply the heat lost by conduction from this house over a winter, we need to calculate the total energy loss over 150 days and then convert it to liters of oil.
First, let's calculate the total energy loss over the winter by multiplying the heat loss rate (H) by the number of seconds in 150 days:
Total energy loss = H * (150 days * 24 hours/day * 60 minutes/hour * 60 seconds/minute)
Next, we need to convert the energy loss from watts to joules:
Energy loss (J) = Total energy loss * 1 watt = Total energy loss
Since heating oil has an energy content of 35 MJ per liter when burned, we can calculate the amount of oil needed by dividing the energy loss by the energy content of 1 liter of oil:
Volume of oil (liters) = Energy loss / (35 MJ/liter * 75% efficiency)
Note: We assume 75% efficiency of the heating system.
Substituting the values into the equation, we get:
Volume of oil (liters) = (Total energy loss) / (35 MJ/liter * 0.75)
Convert the result to liters for the final answer.