Question

Heat is lost steadily through a \(0.5-\mathrm{cm}\) thick, $2-\mathrm{m} \times 3-\mathrm{m}\( window glass whose thermal conductivity is \)0.7 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. The inner and outer surface temperatures of the glass are measured to be \(12^{\circ} \mathrm{C}\) to \(9^{\circ} \mathrm{C}\). The rate of heat loss by conduction through the glass is (a) \(420 \mathrm{~W}\) (b) \(5040 \mathrm{~W}\) (c) \(17,600 \mathrm{~W}\) (d) \(1256 \mathrm{~W}\) (e) \(2520 \mathrm{~W}\)

Short answer

(e) 2520 W

Step-by-step solution

Convert units

All given values should be in SI units. So, we need to convert the thickness of the glass from cm to m, using the relation: \(1 \mathrm{~cm} = 0.01 \mathrm{~m}\). Thus, $$ 0.5 \mathrm{~cm} = 0.5\times 0.01 \mathrm{~m} = 0.005 \mathrm{~m} $$

Calculate the surface area

The surface area of a rectangle can be calculated by multiplying its length by its width, for this window glass, we have: $$ A = 2 \mathrm{~m} \times 3 \mathrm{~m} = 6 \mathrm{~m^2} $$

Apply Fourier's law of heat conduction

We can now calculate the heat loss rate using the Fourier's law, which is: $$ q = kA \dfrac{T_1 - T_2}{d} $$ Substituting the given values into the formula, we have: $$ q = 0.7 \mathrm{~W/m\cdot K} \times 6 \mathrm{~m^2} \times \dfrac{12^{\circ} \mathrm{C} - 9^{\circ} \mathrm{C}}{0.005 \mathrm{~m} } $$ We can now do the calculation: $$ q = 0.7 \times 6 \times \dfrac{3}{0.005} = 0.7 \times 6 \times 600 = 2520 \mathrm{~W} $$ The rate of heat loss by conduction through the window glass is \(2520 \mathrm{~W}\), so the answer is (e).