Question

A vertical \(0.9\)-m-high and \(1.5\)-m-wide double-pane window consists of two sheets of glass separated by a \(2.0-\mathrm{cm}\) air gap at atmospheric pressure. If the glass surface temperatures across the air gap are measured to be \(20^{\circ} \mathrm{C}\) and \(30^{\circ} \mathrm{C}\), the rate of heat transfer through the window is (a) \(16.3 \mathrm{~W}\) (b) \(21.7 \mathrm{~W}\) (c) \(24.0 \mathrm{~W}\) $\begin{array}{ll}\text { (d) } 31.3 \mathrm{~W} & \text { (e) } 44.6 \mathrm{~W}\end{array}$ (For air, use $k=0.02551 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \quad \operatorname{Pr}=0.7296\(, \)\nu=1.562 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\(. Also, the applicable correlation is \)\left.\mathrm{Nu}=0.42 \mathrm{Ra}^{1 / 4} \operatorname{Pr}^{0.012}(H / L)^{-0.3} .\right)$

Short answer

Based on the given information and calculations, the rate of heat transfer through the double-pane window is approximately 24.0 W.

Step-by-step solution

Find relevant parameters

To find the rate of heat transfer, first, we need to calculate the temperature difference, \(\Delta T\), the height of the window \(H\), the width of the window \(L\), and the air gap \(d\). Given: - \(T_1 = 30^\circ C\) (outer glass surface temperature) - \(T_2 = 20^\circ C\) (inner glass surface temperature) - \(H = 0.9\,\text{m}\) - \(L = 1.5\,\text{m}\) - \(d = 0.02\,\text{m}\) (air gap)

Calculate the temperature difference

The temperature difference, \(\Delta T\), between the two glass surfaces is given by: $$\Delta T = T_1 - T_2$$ So, the temperature difference is: $$\Delta T = 30 - 20 = 10\,K$$

Calculate Grashof number

The Grashof number (Gr) is a dimensionless number that helps to describe natural convection. It can be calculated as follows: $$\text{Gr} = \frac{g \beta \Delta T L^3}{\nu^2}$$ Where: - \(g = 9.81\,\text{m/s}^2\) (gravitational acceleration), - \(\beta = \frac{1}{T_\text{avg}} \approx \frac{1}{293\,\text{K}}\) (coefficient of thermal expansion for air, approximated as the inverse of the average temperature) - \(\Delta T\) is the temperature difference, - \(L\) is the characteristic length (in this case, window height) - \(\nu = 1.562 \times 10^{-5}\,\text{m}^2/\text{s}\) (kinematic viscosity of air) Now plug in the values: $$\text{Gr} = \frac{9.81\frac{1}{293}\cdot10\cdot(0.9)^3}{(1.562\times10^{-5})^2}$$ Compute the result to get the Grashof number: $$\text{Gr} \approx 1.962 \times 10^7$$

Calculate the Rayleigh number

The Rayleigh number (Ra) is another dimensionless number that describes natural convection. It is the product of the Grashof number and the Prandtl number (Pr): $$\text{Ra} = \text{Gr} \cdot \text{Pr}$$ - \(\text{Pr} = 0.7296\) (Prandtl number of air) Now calculate the Rayleigh number: $$\text{Ra} = 1.962 \times 10^7 \cdot 0.7296$$ Compute the result to get the Rayleigh number: $$\text{Ra} \approx 1.43 \times 10^7$$

Calculate the Nusselt number

The Nusselt number (Nu) can be calculated using the given correlation: $$\text{Nu} = 0.42 \text{Ra}^{1/4} \text{Pr}^{0.012} (H/L)^{-0.3}$$ Now plug in the values: $$\text{Nu} = 0.42 (1.43 \times 10^7)^{1/4} (0.7296)^{0.012} (0.9/1.5)^{-0.3}$$ Compute the result to get the Nusselt number: $$\text{Nu} \approx 18.63$$

Calculate the heat transfer coefficient

From the Nusselt number, we can determine the heat transfer coefficient (h) using the formula: $$h = \frac{k}{L} \cdot \text{Nu}$$ Where: - \(k = 0.02551\,\text{W/m}\cdot\text{K}\) (thermal conductivity of air) Now plug in the values: $$h = \frac{0.02551}{0.9} \cdot 18.63$$ Compute the result to get the heat transfer coefficient: $$h \approx 0.535\,\text{W/m}^2\cdot\text{K}$$

Calculate the rate of heat transfer

Finally, compute the rate of heat transfer (Q) using the formula: $$Q = h \cdot A \cdot \Delta T$$ Where: - \(A = H \cdot L\) Now plug in the values: $$Q = 0.535 \cdot (0.9 \cdot 1.5) \cdot 10$$ Compute the result to get the rate of heat transfer: $$Q \approx 24.0\,\text{W}$$ Therefore, the rate of heat transfer through the window is approximately 24.0 W, which corresponds to the answer choice (c).