10.9: Calculating Heat Loss of Windows
- Page ID
- 50060
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)As you may recall from Chapter 7, heat loss is calculated using the formula
\[ Heat \, Loss = \dfrac{Area * HDD * 24}{R-value} \]
Using equation 10.9.1, you can calculate the heat loss for windows.
Example 1
A house in State College, PA has 380 ft 2 of windows (R-value = 1.1), 2750 ft 2 of walls and 1920 ft 2 of roof (R = 30). The composite R-value of the walls is 19. Calculate the heating requirement for the house for the heating season. What is the percentage of heat that is lost through the windows? Let HDD be 6000 °F days.
- Answer
-
Heat loss through the windows is
\[ Heat \, Loss = \dfrac{380 * 6000 * 24}{1.1} = 49,745,455 \, BTUs = 49.74 \, MMBTUs \nonumber\]
Heat loss through the walls is
\[ Heat \, Loss = \dfrac{2750 * 6000 * 24}{19} = 20,842,105 \, BTUs \nonumber\]
Heat loss through the roof is
\[ Heat \, Loss = \dfrac{1920 * 6000 * 24}{30} = 9,216,000 \, BTUs \nonumber\]
The total heat loss is then the sum of the heat losses, which is 79,803,560 BTUs = 79.80 MMBTUs.
The percentage of heat loss through the windows is
\[ \% \, Heat \, Loss = \dfrac{49.74}{79.80} * 100 = 62.3\% \nonumber\]
Example 2
Windows in the house described in Example 1 are upgraded at a cost of $1,550. The upgraded windows have an R-value of 4.0.
(a) What is the percent savings in the energy and the heating bill if the energy cost is $11.15/MMBTUs?
(b) What is the payback period for this modification?
- Answer
-
Part (a):
The new heat loss for the same window size with the new R-value is
\[ New \, Heat \, Loss = \dfrac{380 * 6000 * 24}{4.0} = 13,680,000 \, BTUs = 13.68 \, MMBTUs \nonumber\]
The annual energy savings is
\[ Annual \, Savings = 49.74 - 13.68 = 36.06 \, MMBTUs \nonumber\]
The percent savings is
\[ \% \, Savings = \dfrac{36.06}{79.84} * 100 = 45.1 \% \nonumber\]
Part (b):
The old heating bill is
\[ Old \, Bill = 79.80 \, MMBTUs * \dfrac{\$11.15}{MMBTU} = \$889.90 \nonumber\]
The new heating bill is
\[ New \, Bill = 43.74 \, MMBTUs * \dfrac{\$11.15}{MMBTU} = \$487.73 \nonumber\]
The monetary savings is thus
\[ 889.90 - 487.73 = \$402.06 \, per \, year \nonumber\]
The payback period is
\[ Payback \, Period = \dfrac{Additional \, Investment}{Annual \, Savings} = \dfrac{1550}{402.06} = 3.85 \, years \nonumber\]
Table 10.9.1 shows the cost effectiveness of replacing old windows with new and improved windows. The costs are calculated using a computer program called RESFEN developed by US Department of Energy.
Table 10.9.1. Cost effectiveness of using improved windows
| Performance | Base Model | Recommended Level | Best Available |
|---|---|---|---|
| Window description | Double-paned, clear glass, aluminum frame | Double-paned, low-e coating, wood or vinyl frame | Triple-paned, tinted, two spectrally selective low-e coatings, krypton-filled, wood or vinyl frame |
| SHGCa | 0.61 | 0.55 | 0.20 |
| U-factorb | 0.87 | 0.40 | 0.15 |
| Annual heating energy use | 547 therms | 429 therms | 426 therms |
| Annual cooling energy use | 1,134 kWh | 1,103 kWh | 588 kWh |
| Annual energy cost | $290 | $240 | $210 |
| Lifetime energy cost | $4,700 | $3,900 | $3,400 |
| Lifetime energy cost savings | $800 | $1,300 |
a SHGC, or Solar Heat Gain Coefficient, is a measure of the solar radiation admitted through a window. SHGC ranges between 0 and 1; the lower the number, the lower the transmission of solar heat. SHGC has replaced shading coefficient (SC) as the standard indicator of a window's shading ability. SHGC is approximately equal to the SC multiplied by 0.87.
b U-factor is a measure of the rate of heat flow through a window. The U-factor is the inverse of the R-value, or resistance, the common measure of insulation.
c Lifetime energy cost savings is the sum of the discounted value of annual energy cost savings, based on average usage and an assumed window life of 25 years. Future energy price trends and a discount rate of 3.4 percent are based on Federal guidelines (effective from April 2000 to March 2001). Assumed electricity price: $0.06/kWh, the Federal average electricity price in the U.S.Assumed gas price: $0.40/therm, the Federal average gas price in the U.S.
Cost-Effectiveness Assumptions: The model shown above is the result of a simulation using a residential windows modeling program called RESFEN. Calculations are based on a prototype house: 1,540 ft2, two stories, a standard efficiency gas furnace and central air conditioner, and window area covering 15 percent of the exterior wall surface area.