Estimate the percentage savings in the fuel bill if the thermostat is set back from 68 degrees F to 50 degrees F for 12 hours of the day, and left at 68 degrees F for the other 12 hours, assuming an outside temperature of 15 degrees F all day.
The amount of heat per unit time that leaks out of the house is approximately proportionate to difference between the inside and outside temperature. The thermostat will make sure that as much heat is produced by the heating as is leaking away, so that thee temperature stays constant on average.
So, if the thermostat is left at 68 F all day, the total amount of heat used per day will be some constant c times (68-15), while if it is set at 50 F for half a day and 68 for the other half, you'll use
1/2 c (50-15) + 1/2 c (68-15)
per day.
c (68 - 15) = 53 c
1/2 c (50-15) + 1/2 c (68-15) =
44 c
44c/(53 c) = 0.83
So, you save 17%
To estimate the percentage savings in the fuel bill, we need to calculate the fuel usage for both scenarios and compare them.
First, let's calculate the fuel usage when the thermostat is set to 68 degrees F all day. We assume that the fuel usage is directly proportional to the temperature difference between the indoor and outdoor temperatures.
Fuel usage for 24 hours at 68 degrees F:
Temperature difference = 68 - 15 = 53 degrees F
Fuel usage = 53 * 24 = 1,272 heating degree hours (HDH)
Next, let's calculate the fuel usage when the thermostat is set to 50 degrees F for 12 hours and left at 68 degrees F for the other 12 hours.
Fuel usage for 12 hours at 50 degrees F:
Temperature difference = 50 - 15 = 35 degrees F
Fuel usage = 35 * 12 = 420 HDH
Fuel usage for 12 hours at 68 degrees F:
Temperature difference = 68 - 15 = 53 degrees F
Fuel usage = 53 * 12 = 636 HDH
Total fuel usage = 420 + 636 = 1,056 HDH
To calculate the percentage savings, we compare the total fuel usage for both scenarios:
Percentage savings = ((1,272 - 1,056) / 1,272) * 100
Percentage savings = 216 / 1,272 * 100
Percentage savings ≈ 16.98%
Therefore, by setting the thermostat back from 68 degrees F to 50 degrees F for 12 hours of the day and leaving it at 68 degrees F for the other 12 hours, the estimated percentage savings in the fuel bill would be approximately 16.98%.
To estimate the percentage savings in the fuel bill, we can start by calculating the amount of fuel used for both scenarios: when the thermostat is set back and when it is kept at a constant temperature.
1. Calculate the fuel used when the thermostat is set back:
Since the outside temperature is 15 degrees F all day and the thermostat is set back to 50 degrees F for 12 hours, the temperature difference is 50 - 15 = 35 degrees F.
Assuming that the fuel consumption is directly proportional to the temperature difference, we can calculate the fuel used during the 12 hours of setback as a fraction of the fuel used for a 24-hour day at a constant 68 degrees F.
2. Calculate the fuel used when the thermostat is constant:
In this scenario, the temperature is maintained at 68 degrees F for 24 hours, regardless of the outside temperature.
We assume that the fuel consumption is directly proportional to the temperature difference, so the fuel used is the same as the total fuel used for a 24-hour day at 68 degrees F.
3. Calculate the percentage savings:
The percentage savings can be calculated by finding the difference between the fuel used in the two scenarios and expressing it as a percentage of the fuel used when the thermostat is kept at 68 degrees F for the entire day.
Note: To get an accurate estimate, it is important to consider the efficiency of the heating system and other factors, such as insulation of the house, type of fuel, and local climate conditions. This calculation assumes a simplified model for illustrative purposes.
For a more accurate estimation, it may be best to consult a qualified energy professional or refer to specific energy consumption data for your heating system.