Question

Consider a house with a flat roof whose outer dimensions are $12 \mathrm{~m} \times 12 \mathrm{~m}\(. The outer walls of the house are \)6 \mathrm{~m}$ high. The walls and the roof of the house are made of 20 -cm-thick concrete $(k=0.75 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$. The temperatures of the inner and outer surfaces of the house are \(15^{\circ} \mathrm{C}\) and $3^{\circ} \mathrm{C}$, respectively. Accounting for the effects of the edges of adjoining surfaces, determine the rate of heat loss from the house through its walls and the roof. What is the error involved in ignoring the effects of the edges and corners and treating the roof as a $12-\mathrm{m} \times 12-\mathrm{m}\( surface and the walls as \)6-\mathrm{m} \times 12-\mathrm{m}$ surfaces for simplicity?

Short answer

Answer: We can conclude that the total heat loss through the walls and roof of the house, without considering the effects of edges and corners, is estimated to be 19440 W. The actual heat loss would be higher due to the increased complexity and surface area around edges and corners. The error in our calculations is positive, indicating that our estimate is an underestimate of the true heat loss. To more accurately estimate the error, we would need additional information on the material properties and the structure's geometrical complexity, or utilize engineering software and numerical simulations to better account for the effects of edges and corners.

Step-by-step solution

Find the area of the walls and roof

Let us first calculate the area of the walls and the roof. Since the house has a flat roof and outer dimensions of \(12 \mathrm{~m} \times 12 \mathrm{~m}\), the area of the roof will be \(A_{roof} = 12 m \times 12 m = 144 m^2\). The house has four walls, each with dimensions \(12 \mathrm{~m} \times 6 \mathrm{~m}\), so the total area of the walls will be \(A_{walls} = 4(12 m \times 6 m) = 288 m^2\).

Calculate the heat loss without considering the effects of edges and corners

To calculate the heat loss through the roof and walls, we'll use the formula \(Q = kA\frac{\Delta T}{d}\), 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 material. We are given: \(k = 0.75 \mathrm{~W/(m \cdot K)}\), \(d = 0.2 \mathrm{~m}\), \(\Delta T = 15^{\circ} \mathrm{C} - 3^{\circ} \mathrm{C} = 12^\circ \mathrm{C}\). First, let's calculate the heat loss for the roof without considering the effects of edges and corners: \(Q_{roof} = k \times A_{roof} \times \frac{\Delta T}{d} = 0.75 \mathrm{W/(m \cdot K)} \times 144 \mathrm{m^2} \times \frac{12^\circ \mathrm{C}}{0.2 \mathrm{m}} = 6480 \mathrm{W}\). Next, calculate the heat loss for the walls without considering the edges and corners: \(Q_{walls} = k \times A_{walls} \times \frac{\Delta T}{d} = 0.75 \mathrm{W/(m \cdot K)} \times 288 \mathrm{m^2} \times \frac{12^\circ \mathrm{C}}{0.2 \mathrm{m}} = 12960 \mathrm{W}\). The total heat loss for the house without considering the edges and corners is \(Q_{total} = Q_{roof} + Q_{walls} = 6480 \mathrm{W} + 12960 \mathrm{W} = 19440 \mathrm{W}\).

Calculate the heat loss with considering the effects of edges and corners

Accounting for the effects of edges and adjoining surfaces is a more complex process and would require integration or numerical analysis, which is beyond the scope of a high school exercise. However, it is possible to estimate that the actual heat loss would be higher than the calculated value without considering the edges and corners due to the increased complexity in the interactions of the materials and the increased surface area around edges and corners.

Estimate the error involved in ignoring the effects of edges and corners

It is difficult to determine the error in our calculations without performing a more complex analysis. However, we can be certain about one thing: the error is positive, meaning that our calculated heat loss without considering the edges and corners is an underestimate. To estimate this error, we would need more in-depth knowledge of the material properties and the geometrical complexity of the structure. In practice, engineering software or numerical simulation could be used to more accurately determine the effects of edges and corners on the heat loss through the roof and walls. Our goal in this exercise was to determine the rate of heat loss from the house through its walls and the roof, and to see what the error was if we ignore the effects of the edges and corners and treat the roof as a simple geometric shape. Although we couldn't quantify the error precisely, we can be certain that our simplistic approach provides an underestimate of the true heat loss.