Consider a house that has a 10-m \(\times 20-\mathrm{m}\) base and a 4 -m-high wall. All four walls of the house have an \(R\)-value of \(2.31 \mathrm{~m}^{2} \cdot{ }^{\circ} \mathrm{C} / \mathrm{W}\). The two \(10-\mathrm{m} \times 4-\mathrm{m}\) walls have no windows. The third wall has five windows made of \(0.5-\mathrm{cm}\)-thick glass \((k=0.78 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}), 1.2 \mathrm{~m} \times 1.8 \mathrm{~m}\) in size. The fourth wall has the same size and number of windows, but they are doublepaned with a \(1.5-\mathrm{cm}\)-thick stagnant air space \((k=0.026 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) enclosed between two \(0.5\)-cm-thick glass layers. The thermostat in the house is set at \(24^{\circ} \mathrm{C}\) and the average temperature outside at that location is \(8^{\circ} \mathrm{C}\) during the seven-month-long heating season. Disregarding any direct radiation gain or loss through the windows and taking the heat transfer coefficients at the inner and outer surfaces of the house to be 7 and \(18 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), respectively, determine the average rate of heat transfer through each wall. If the house is electrically heated and the price of electricity is \(\$ 0.08 / \mathrm{kWh}\), determine the amount of money this household will save per heating season by converting the single-pane windows to double-pane windows.

Step by step solution

01

Calculate the heat transfer through the first wall

The first wall is 10m x 4m with no windows and an R-value of 2.31 \(m^2 \cdot{ }^{\circ} C / W\). To calculate the heat transfer through the first wall, use the formula: \(q_{1} = \frac{A_{1}(T_{in} - T_{out})}{R_{1}}\) \(q_{1} = \frac{(10m*4m)(24^{\circ}C-8^{\circ}C)}{2.31 m^2 \cdot{ }^{\circ} C / W}\)

02

Calculate the heat transfer through the second wall

The second wall is the same as the first wall, with no windows, and an R-value of 2.31 \(m^2 \cdot{ }^{\circ} C / W\). The heat transfer through the second wall is the same as the first wall. \(q_{2} = q_{1}\)

03

Calculate the heat transfer through the third wall

The third wall is 10m x 4m with five single-pane windows. Calculate the heat transfer through the wall without the windows, and then through the windows. For the wall without windows: \(A_{withoutWindows} = (10m*4m)-(5*1.2m*1.8m)\) \(q_{3wall} = \frac{(A_{withoutWindows})(24^{\circ }C -8^{\circ }C)}{2.31m^2\cdot{ } ^{\circ }C/W}\) Now calculate the heat transfer through the single-pane windows: \(q_{window} = \frac{k_{glass}\cdot A_{window} \cdot (T_{in} - T_{out})}{d_{glass}}\) where \(A_{window} = 5 * 1.2m * 1.8m\) and \(d_{glass} = 0.5cm\) (convert to meters). Finally, sum the heat transfer through the wall section and the window section: \(q_{3} = q_{3wall} + q_{3window}\)

04

Calculate the heat transfer through the fourth wall

The fourth wall is the same as the third wall, but with double-pane windows. Calculate the heat transfer through double-pane windows: \(q_{doubleWindow} = \frac{k_{glass}\cdot A_{window} \cdot (T_{in} - T_{out})}{d_{glass}} + \frac{k_{air}\cdot A_{window} \cdot (T_{in} - T_{out})}{d_{air}}\) where \(d_{air}=1.5cm\) (convert to meters) and \(k_{air}=0.026 W / m\cdot K\). Again, sum the heat transfer through the wall section and the double-pane window section: \(q_{4} = q_{3wall} + q_{4doubleWindow}\)

05

Calculate the heat transfer rates for single-pane and double-pane windows

Calculate the sum of the heat transfer rates for the house with single-pane and double-pane windows. For single-pane: \(q_{single-pane} = q_{1}+ q_{2}+ q_{3}\) For double-pane: \(q_{double-pane} = q_{1}+ q_{2}+ q_{4}\)

06

Calculate the energy cost difference between the two cases

Calculate the energy costs for both cases and find the difference in cost per heating season. Energy cost for single-pane windows: \(Cost_{single-pane} = \frac{q_{single-pane} \cdot t_{heating} \cdot Cost_{elec}}{Efficiency}\) Energy cost for double-pane windows: \(Cost_{double-pane} = \frac{q_{double-pane} \cdot t_{heating} \cdot Cost_{elec}}{Efficiency}\) Where \(t_{heating} = 7\) months (convert to hours), \(Cost_{elec} = \$ 0.08 / kWh\), and \(Efficiency = 3.6 \times 10^6 J / kWh\). Calculate the difference in cost per heating season: \(Cost_{savings} = Cost_{single-pane} - Cost_{double-pane}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Thermal Resistance

Understanding thermal resistance is crucial when analyzing heat transfer in buildings, particularly through elements like walls and windows. Thermal resistance, often denoted as 'R-value,' is a measure of a material's resistance to heat flow. The higher the R-value, the better the material insulates against heat transfer.

In the context of the given exercise, the R-value given for the walls is important for calculating the rate of heat transfer through the walls with no windows. A simple way to think of thermal resistance is to compare it to resistance in electrical circuits; just as electrical resistance opposes current flow, thermal resistance opposes heat flow. The formula to calculate the heat transfer rate is: \(q = \frac{\Delta T}{R}\), where \(q\) is the heat transfer rate, \(\Delta T\) is the temperature difference across the material, and \(R\) is the thermal resistance.

When applying this to a practical scenario, like heating a home, you'll find that materials with high R-values can considerably lower energy consumption and costs, since less heat is lost through them. In the exercise provided, the house's walls and single-pane versus double-pane windows demonstrate distinctly different R-values, leading to varying levels of energy efficiency.

Single-pane vs Double-pane Windows

The difference between single-pane and double-pane windows can significantly impact a home's thermal efficiency. Single-pane windows consist of a single layer of glass, which allows for a relatively higher rate of heat transfer due to its lower resistance to heat flow. In contrast, double-pane windows have two layers of glass with a spacer between them, typically filled with a gas like argon or krypton, which has better insulating properties.

In our exercise, the house has two types of windows: one set of single-pane and one set of double-pane with a stagnant air space providing additional insulation. To compare the heat loss through these windows, one must consider the respective R-values or thermal resistances. For the single-pane windows, only the thermal resistance of the glass is considered, while for the double-pane windows, the resistance of both glass layers and the air space must be taken into account.

Double-pane windows, therefore, offer better insulation due to the additional air space, which reduces the rate of heat transfer and leads to energy savings. The calculations for the rate of heat transfer through the windows (\(q_{window}\)) and (\(q_{doubleWindow}\)) show that double-pane windows can effectively lower heat loss, making them an energy-efficient upgrade from single-pane windows.

Energy Savings in Heating

Energy savings in heating is an important consideration for both environmental and economic reasons. By insulating a home effectively, you can ensure that less heat escapes during cooler months, thereby reducing the amount of energy required to maintain comfortable temperatures indoors.

In the given problem, students are tasked to calculate the potential energy savings that could be realized by switching from single-pane to double-pane windows in a home. To calculate these savings, it is necessary to compute the heat transfer through each wall and window type, converting these rates into energy costs using the price of electricity. The respective costs are then compared to determine the savings:

\(Cost_{savings} = Cost_{single-pane} - Cost_{double-pane}\).

The process is not just a matter of simple subtraction; it requires understanding the relationship between heat transfer, thermal resistance, and energy efficiency. It also serves as a real-world application of how upgrading building components, such as windows, can lead to considerable reductions in energy consumption and costs. The calculations suggest that by lowering the rate of heat transfer with better insulation, the household's heating system would need to cycle on less frequently, thereby conserving energy and reducing utility bills.