Chapter 3, Problem 176P

(a)

To determine

To find:the R and U for air space between studs do not have any reflective surface.

  $\begin{array}{l}U=1.203\frac{W}{m^{2}C}\\ R=0.831\frac{m^{2}C}{W}\end{array}$

Given information:

  $\begin{array}{l}\text{dimension of wood studs = 38mm}\times \text{140mm}\\ \text{center distance = 400mm}\\ \text{dimensions of cavity = 140mm}\\ \text{thickness of gypsum wallboard = 13mm}\\ \text{thickness of rigid foam insulation = 25mm}\\ \text{dimension of wood lapping siding = 13mm}\times \text{200mm}\\ \text{the insulation cavity constitutes 80\%heat transmit area and while studs,headers,plates and still constitute 20\%}\text{.}\end{array}$

Formula used:

The expression fortotal thermal resistance for the entire wall is expressed as follows:

  $\begin{array}{l}$ R_{overall}=\frac{1}{U_{overall}}where{U}_{overall}={(U{f}_{area})}_{insulations}+{(U{f}_{area})}_{stud}$R_{overall}=\text{thermal resistance}\\ U_{overall}=\text{heat transfer coefficient}\\ f_{area}=\text{area fraction}\end{array}$

  $\begin{array}{l}{\epsilon }_{effective}=\frac{1}{\frac{1}{{\epsilon }_1}+\frac{1}{{\epsilon }_2}-1}\\ {\epsilon }_1=\text{effective emissivity of surface 1}\\ {\epsilon }_2=\text{effective emissivityof surface 2}\\ {\epsilon }_{collective}=\text{effective emissivityvof surface combination}\end{array}$

Calculation:

The total average thermal resistance for wall is calculated as follows:

  $R_{overall}=\frac{1}{U_{overall}}where{U}_{overall}={(U{f}_{area})}_{insulations}+{(U{f}_{area})}_{stud}$

And the value of the area fraction factor is 0.82 for air space and 0.18 for stud section. ${\epsilon }_{effective}=\frac{1}{\frac{1}{0.9}+\frac{1}{0.9}-1}=0.82$

Construction	R-value $\frac{{\text{m}}^{\text{2}}\text{C}}{\text{W}}$
	Between studs	At studs
1. Still air above ceiling	0.12	0.044
2. Linoleum	0.009	0.14
3. Felt	0.011	0.23
4.Plywood	0.11	----
5. Wood subfloor	0.166	----
6a. Air space, 90 mm, nonreflective	0.16	----
6b. Stud of wood, 38 mm by 90 mm	-----	0.63
7. Wallboard, 13 mm	0.079	0.079
8. Still air near ceiling	0.12	0.12

Total thermal resistance of each part, R $\frac{{\text{m}}^{\text{2}}\text{C}}{\text{W}}$	0.775	1.243
The U-factor of each part,	1.290	0.805
Area fraction of each part,	0.82	0.18
Total U-factor	1.203 $\frac{\text{W}}{{\text{m}}^{\text{2}}\text{C}}$
Total thermal resistance,	0.831 $\frac{{\text{m}}^{\text{2}}\text{C}}{\text{W}}$

(b)

To determine

To find: The winter R-value and the U-factor of a flat ceiling with an air space has reflective surface with $\epsilon \text{=0}\text{.05}$ on one the side.

(c)

To determine

To find: The winter R-value and the U-factor of a flat ceiling with an air space has reflective surface with $\epsilon \text{=0}\text{.05}$ on one the side.