Energy Savings through Increased Insulation
This example deals with a discrete optimization problem of determining the most economical amount of attic insulation for a large single-story home in Virginia. In general, the heat lost through the roof of a single-story home is
\begin{array}{c}\text { Heat loss } \\ \text { in Btu } \\\text { per hour }\end{array}={\begin{array}{c}{\left(\Delta \begin{array}{c}\text { Temperature } \\\text { in }^{\circ} \mathrm{F}\end{array}\right)\left(\begin{array}{c}\text { Area } \\\text { in } \\\mathrm{ft}^{2}\end{array}\right)}\left(\begin{array}{c}\text { Conductance in } \\\frac{\text { Btu/hour }}{\mathrm{ft}^{2}-{ }^{\circ} \mathrm{F}}\end{array}\right),\end{array}} \\or
Q=\left(T_{\text {in }}-T_{\text {out }}\right) \cdot A \cdot UIn southwest Virginia, the number of heating days per year is approximately 230, and the annual heating degree-days equals 230 (65°F−46°F) = 4,370 degree-days per year. Here, 65°F is assumed to be the average inside temperature and 46◦F is the average outside temperature each day.
Consider a 2,400-ft^{2} single-story house in Blacksburg. The typical annual space-heating load for this size of a house is 100 × 10^{6} Btu. That is, with no insulation in the attic, we lose about 100 × 10^{6} Btu per year*. Common sense dictates that the “no insulation” alternative is not attractive and is to be avoided.
With insulation in the attic, the amount of heat lost each year will be reduced. The value of energy savings that results from adding insulation and reducing heat loss is dependent on what type of residential heating furnace is installed. For this example, we assume that an electrical resistance furnace is installed by the builder, and its efficiency is near 100%.
Now we’re in a position to answer the following question: What amount of insulation is most economical? An additional piece of data we need involves the cost of electricity, which is $0.074 per kWh. This can be converted to dollars per 10^{6} Btu as follows (1 kWh = 3,413 Btu):
\frac{\mathrm{kWh}}{3,413 \mathrm{Btu}}=293 \mathrm{kWh} per million Btu
*100 \times 10^{6} \mathrm{Btu} / \mathrm{yr} \cong\left(\frac{4,370^{\circ} \mathrm{F} \text {-days per year }}{1.00 \text { efficiency }}\right)\left(2,400 \mathrm{ft}^{2}\right)\left(24\right. hours / day) \left(\frac{0.397 \mathrm{Btu} / \mathrm{hr}}{\mathrm{ft}^{2}-{ }^{\circ} \mathrm{F}}\right), where 0.397 is the U-factor with no insulation.
\frac{293 \mathrm{kWh}}{10^{6} \mathrm{Btu}}\left(\frac{\$ 0.074}{\mathrm{kWh}}\right) \cong \$ 21.75 / 10^{6} \mathrm{Btu}The cost of several insulation alternatives and associated space-heating loads for this house are given in the following table (an R-value indicates the resistance to heat transfer—the higher the number the less the heat transfer):
| Amount of Insulation | ||||
| R38 | R30 | R19 | R11 | |
| 1600 | 1300 | 900 | 600 | Investment cost ($) |
| 66.2 \times 10^{6} | 67.2 \times 10^{6} | 69.8 \times 10^{6} | 74 \times 10^{6} | Annual heating load (Btu/year) |
In view of these data, which amount of attic insulation is most economical? The life of the insulation is estimated to be 25 years.