An 8-m-long, uninsulated square duct of cross section \(0.2 \mathrm{~m} \times 0.2 \mathrm{~m}\) and relative roughness \(10^{-3}\) passes through the attic space of a house. Hot air enters the duct at \(1 \mathrm{~atm}\) and \(80^{\circ} \mathrm{C}\) at a volume flow rate of \(0.15 \mathrm{~m}^{3} / \mathrm{s}\). The duct surface is nearly isothermal at \(60^{\circ} \mathrm{C}\). Determine the rate of heat loss from the duct to the attic space and the pressure difference between the inlet and outlet sections of the duct. Evaluate air properties at a bulk mean temperature of \(80^{\circ} \mathrm{C}\). Is this a good assumption?

At a bulk mean temperature of \(80^{\circ}\mathrm{C}\) and \(1\,\text{atm}\), look up the properties of air in a thermodynamic table or use an online calculator. You should find the following values: - \(\rho = 0.964\,\text{kg/m}^3\) - \(\mu = 2.5\times10^{-5}\,\text{kg/(m s)}\) - \(c_p = 1005\,\text{J/(kg K)}\) - \(k = 0.0287\,\text{W/(m K)}\) These will be used in the subsequent calculations.

The heat loss per unit length (Q' [W/m]) can be calculated using the following equation: $$ Q' = hD(T_{\text{air}} - T_{\text{surface}}) $$ Where \(h\) is the convective heat transfer coefficient, \(D\) is the hydraulic diameter, and \(T_{\text{air}}\) and \(T_{\text{surface}}\) are the air and surface temperatures, respectively. First, calculate the hydraulic diameter: $$ D = \frac{4A}{P} = \frac{4 \times (0.2\,\text{m} \times 0.2\,\text{m})}{4 \times 0.2\,\text{m}} = 0.2\,\text{m} $$ Where \(A\) is the cross-sectional area of the duct and \(P\) is its perimeter. Now, find the convective heat transfer coefficient: $$ h = 0.026\,\text{k}\left(\frac{\text{Re}\cdot\text{Pr}}{D}\right)^{\frac{1}{3}} = 0.026 \times 0.0287\,\text{W/(m K)}\left(\frac{\text{Re}\cdot\text{Pr}}{0.2\,\text{m}}\right)^{\frac{1}{3}} $$ However, before we can compute \(h\), we need to determine the Reynolds number (Re) and Prandtl number (Pr).

The Reynolds number can be calculated using the following equation: $$ \text{Re} = \frac{\rho VD}{\mu} = \frac{0.964\,\text{kg/m}^3 \times 0.15\,\text{m/s} \times 0.2\,\text{m}}{2.5\times10^{-5}\,\text{kg/(m s)}} = 1152 $$ The Prandtl number can be calculated as follows: $$ \text{Pr} = \frac{\mu c_p}{k} = \frac{2.5\times 10^{-5}\,\text{kg/(m s)} \times 1005\,\text{J/(kg K)}}{0.0287\,\text{W/(m K)}} = 0.871 $$ Now, we can calculate the convective heat transfer coefficient, \(h\): $$ h = 0.026 \times 0.0287\,\text{W/(m K)}\left(\frac{1152 \times 0.871}{0.2\,\text{m}}\right)^{\frac{1}{3}} = 5.29\,\text{W/(m K)} $$ Finally, calculate the heat loss per unit length, \(Q'\): $$ Q' = hD(T_{\text{air}} - T_{\text{surface}}) = 5.29\,\text{W/(m K)} \times 0.2\,\text{m} \times (80\,\text{C} - 60\,\text{C}) = 21.16\,\text{W/m} $$ To find the total heat loss, multiply \(Q'\) by the length of the duct (8 m): $$ Q_{\text{total}} = Q' \times \text{length} = 21.16\,\text{W/m} \times 8\,\text{m} = 169.28\,\text{W} $$

Here, we need to calculate the friction factor (\(f\)) and Darcy friction factor (\(f_D\)) first: $$ f = 0.079\,\text{Re}^{-0.25} = 0.079 \times (1152)^{-0.25} = 0.150 $$ $$ f_D = \frac{4f}{\text{Re}} = \frac{4 \times 0.150}{1152} = 5.21\times 10^{-4} $$ Now, calculate the pressure difference using the following equation: $$ \Delta P = f_D \times \frac{\rho V^2 L}{2D} = 5.21\times 10^{-4} \times \frac{0.964\,\text{kg/m}^3 \times (0.15\,\text{m/s})^2 \times 8\,\text{m}}{2 \times 0.2\,\text{m}} = 18.30\,\text{Pa} $$ #Summary# The rate of heat loss from the duct to the attic space is \(169.28\,\text{W}\). The pressure difference between the inlet and outlet sections of the duct is \(18.30\,\text{Pa}\). The analysis above indicates that evaluating the air properties at a bulk mean temperature of \(80^{\circ}\mathrm{C}\) was a reasonable assumption.