To find the heat flow through a pane of glass over a period of 8.0 hours given the pane's dimensions, thickness, and temperature difference between the inner and outer surfaces, we proceed as follows:
1. Convert Dimensions to Consistent Units:
- The area of the glass in square inches is given as [tex]\(24 \times 24 \text{ inches} = 576 \text{ square inches}\)[/tex]. Since thermal conductivity is often expressed in units involving feet and hours, we need to convert the area from square inches to square feet.
- [tex]\(576 \text{ square inches} \div 144 = 4.0 \text{ square feet}\)[/tex].
2. Convert Thickness from Inches to Feet:
- The thickness of the glass is given as [tex]\(0.200 \text{ inches}\)[/tex]. To maintain consistency with the units used for thermal conductivity, we convert the thickness from inches to feet.
- [tex]\(0.200 \text{ inches} \div 12 = 0.0167 \text{ feet}\)[/tex].
3. Calculate the Temperature Difference:
- The inner surface temperature is [tex]\(80 °\text{F}\)[/tex] and the outer surface temperature is [tex]\(50 °\text{F}\)[/tex].
- The temperature difference, [tex]\(\Delta T\)[/tex], is [tex]\(80 °\text{F} - 50 °\text{F} = 30 °\text{F}\)[/tex].
4. Apply Fourier's Law of Thermal Conduction:
Fourier's law states that heat flow rate [tex]\(Q\)[/tex] through a material is determined by:
[tex]\[
Q = \frac{k \cdot A \cdot \Delta T}{d}
\][/tex]
where:
- [tex]\(k\)[/tex] is the thermal conductivity of the material,
- [tex]\(A\)[/tex] is the area,
- [tex]\(\Delta T\)[/tex] is the temperature difference,
- [tex]\(d\)[/tex] is the thickness of the material.
Given:
- The thermal conductivity of glass, [tex]\(k_{\text{glass}}\)[/tex], is [tex]\(0.49 \text{ BTU/(hr·ft·°F)}\)[/tex].
- Area, [tex]\(A\)[/tex], is [tex]\(4.0 \text{ square feet}\)[/tex].
- Temperature difference, [tex]\(\Delta T\)[/tex], is [tex]\(30 °\text{F}\)[/tex].
- Thickness, [tex]\(d\)[/tex], is [tex]\(0.0167 \text{ feet}\)[/tex].
Substituting these values in Fourier's law:
[tex]\[
Q = \frac{0.49 \text{ BTU/(hr·ft·°F)} \cdot 4.0 \text{ ft}^2 \cdot 30 °\text{F}}{0.0167 \text{ ft}} = 3528 \text{ BTU/hr}
\][/tex]
5. Calculate Total Heat Flow Over 8 Hours:
- The heat flow rate we've calculated is for one hour. To find the total heat flow over 8 hours, we multiply the hourly heat flow rate by 8.
- [tex]\(3528 \text{ BTU/hr} \times 8 \text{ hours} = 28224 \text{ BTU}\)[/tex].
Therefore, the total heat flow through the glass pane over an 8-hour period is [tex]\(28224 \text{ BTU}\)[/tex].
