Answer
It all depends on how tall the house is and where it is located. We calculate for a Nashville 1 storey house approximately 2.6m (8.53ft) tall. Using these values, it would take 34 days to justify the decision of paying more money.
Work Step by Step
250m^{2} house 2500 square feet
Overall, with a height of 2.6m (estimated value) or 8.53ft, the volume of the house comes to 21,325 cubic feet.
Current efficiency is 65%
80% efficiency 1500 dollars
95% efficiency 2500 dollars
Paying you 100 dollars (not used)
Nashville
- Gas price currently 9.33 dollars, per thousand cubic feet
- The average temperature is 37 Fahrenheit
Therefore to heat the entire house it would cost 198.96 dollars with 100% efficiency (9.33 • volume of house in thousands of cubic feet).
An 80% efficiency means that 20% of the input energy is not outputted in a useful form. This means that the lower efficiency will drive up the cost of heating. So the cost of a 100% efficient system would only cover 80% of the volume of the house. The extra 20 % would equal 39.79 dollars (9.33 • 20% of volume of house in thousands). Therefore the total cost of an 80% efficient system would equal the total cost of a 100% efficient system plus the cost of the remaining 20% of the house (198.96+39.79 = 238.75 dollars).
Similarly to the previous system, the 95% efficiency will only cover 95% of the home for the same price as the 100% efficient system. The price of the remaining 5% can be calculated as follows: 9.33 • 5% of volume of home in thousands = 9.33 • 1.06625 = 9.95 dollars). Therefore the total cost of a 95% efficient system would equal the total cost of a 100% efficient system plus the cost of the remaining 5% of the house (198.96+9.95 = 208.91 dollars).
This means that if the heaters were used every day for 34 days, the overall cost of the more efficient heater would be justified within 34 days (graph the two systems as linear equations). From that point, it would be more beneficial to have the more expensive heater. So the recommended choice is the 95% efficient system, since it becomes more economically viable after a little over a month.
