Question

The inner and outer surfaces of a 0.5 -cm-thick \(2-\mathrm{m} \times\) \(2-m\) window glass in winter are \(15^{\circ} \mathrm{C}\) and \(6^{\circ} \mathrm{C},\) respectively. If the thermal conductivity of the glass is \(0.78 \mathrm{W} / \mathrm{m} \cdot^{\circ} \mathrm{C}\), determine the amount of heat loss, in \(\mathrm{kJ}\), through the glass over a period of \(10 \mathrm{h}\). What would your answer be if the glass were \(1-\mathrm{cm}\) thick?

Short answer

Answer: For a 0.5 cm thick glass, 202,176 kJ of heat is lost in 10 hours. For a 1 cm thick glass, 101,088 kJ of heat is lost in 10 hours.

Step-by-step solution

Identify the Given Information

We're given the following information: - Inner surface temperature: \(15^{\circ} C\) - Outer surface temperature: \(6^{\circ} C\) - Glass thickness: \(0.5 cm\) - Glass dimensions (Area): \(2 m \times 2 m\) - Thermal conductivity (k): \(0.78 W/m \cdot ^{\circ} C\) - Time period: \(10 h\)

Convert Units

Convert the thickness and area to the same unit of length (meters) to use in the calculation. Thickness in meters: \(0.5 cm = 0.005 m\) Area: \(2 m \times 2 m = 4 m^2\)

Calculate Temperature Difference

The temperature difference, denoted by \(\Delta T\), between the inner and outer surface is: \(\Delta T = T_{\text{inner}} - T_{\text{outer}} = 15^{\circ} C - 6^{\circ} C = 9^{\circ} C\)

Calculate Heat Loss Rate

Use the formula for heat transfer by conduction: \(q = kA \frac{\Delta T}{d}\), where \(q\) is the heat loss rate, and \(d\) is the thickness. Plug in the given values: \(q = 0.78 \cdot 4 \cdot \frac{9}{0.005} = 5616 W\)

Calculate Total Heat Loss for 10 hours

To find the total heat loss for 10 hours, we need to multiply heat loss rate by time in seconds (convert 10 hours to seconds): Total heat loss = \(q \times time = 5616 \cdot (10 \cdot 3600) = 202176000 J\) Convert the answer to kJ: \(202176000 J = 202176 kJ\) So, the total heat loss with \(0.5\) cm thick glass is \(202176 kJ\)

Calculate Heat Loss for 1 cm Thick Glass

Recalculate the heat loss using the same formula but with a new thickness of \(1 cm = 0.01 m\): \(q_2 = 0.78 \cdot 4 \cdot \frac{9}{0.01} = 2808 W\) Calculate the total heat loss for 10 hours with the new heat loss rate: Total heat loss = \(q_2 \times time = 2808 \cdot (10 \cdot 3600) = 101088000 J\) Convert the answer to kJ: \(101088000 J = 101088 kJ\) So, the total heat loss with \(1 cm\) thick glass is \(101088 kJ\) The answer to the problem is: - For \(0.5 cm\) thick glass: \(202176 kJ\) of heat is lost in 10 hours - For \(1 cm\) thick glass: \(101088 kJ\) of heat is lost in 10 hours