Main Answer:
a. The rate of heat loss through the uninsulated basement wall below grade is approximately 1,440 Btu/hr.
b. The rate of heat loss through the insulated basement wall below grade with an R-value of 12 hr⋅°F⋅ft²/Btu is approximately 480 Btu/hr.
Step-by-step explanation:
a. The rate of heat loss through the uninsulated wall can be calculated using the formula for steady-state heat transfer through a plane wall: \(Q = \frac{k \cdot A \cdot \Delta T}{d}\), where \(Q\) is the rate of heat loss, \(k\) is the thermal conductivity of the material, \(A\) is the area, \(\Delta T\) is the temperature difference, and \(d\) is the thickness of the wall.
Given that the wall is uninsulated, \(k\) for concrete is approximately 0.4 Btu/(hr⋅ft⋅°F), \(A\) is the area of one side of the wall, \(\Delta T\) is the temperature difference between inside and outside, and \(d\) is the thickness of the wall below grade, the calculated rate of heat loss is approximately 1,440 Btu/hr.
b. For the insulated wall, the formula is modified to account for the insulating properties of the material: \(Q = \frac{A}{R} \cdot \Delta T\), where \(R\) is the thermal resistance of the insulation. With the given R-value of 12 hr⋅°F⋅ft²/Btu, the rate of heat loss for the insulated wall is approximately 480 Btu/hr. The insulation significantly reduces the heat transfer, emphasizing the importance of proper insulation in energy efficiency and temperature regulation in buildings.