Question

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?

Short answer

Answer: The rate of heat loss from the uninsulated square duct to the attic space is approximately 5915 W, and the pressure difference between the inlet and outlet sections of the duct is 278 Pa.

Step-by-step solution

Calculate the Reynolds number for the air flow

To calculate the Reynolds number, we first need to find the velocity of the air flow and the hydraulic diameter of the duct. The hydraulic diameter is given by the formula \(D_h = \frac{4 \cdot A}{P}\), where \(A\) is the cross-sectional area and \(P\) is the wetted perimeter of the duct. The cross-sectional area is: \(A = 0.2 \mathrm{~m} \times 0.2 \mathrm{~m} = 0.04 \mathrm{~m}^2\) The wetted perimeter is: \(P = 4 \cdot 0.2 \mathrm{~m} = 0.8 \mathrm{~m}\) The hydraulic diameter is therefore: \(D_h = \frac{4 \cdot 0.04 \mathrm{~m}^2}{0.8 \mathrm{~m}} = 0.2 \mathrm{~m}\) The velocity of the air flow can be found using the volume flow rate: \(v = \frac{0.15 \mathrm{~m}^3 / \mathrm{s}}{0.04 \mathrm{~m}^2} = 3.75 \mathrm{~m} / \mathrm{s}\) To calculate the Reynolds number, we use the formula \(Re = \frac{\rho v D_h}{\mu}\), where \(\rho\) is the air density, \(v\) is the velocity, \(D_h\) is the hydraulic diameter, and \(\mu\) is the dynamic viscosity. We're required to evaluate the air properties at a bulk mean temperature of \(80^{\circ} \mathrm{C}\): \(\rho = 1.027 \mathrm{~kg} / \mathrm{m}^3\) (from air tables) \(\mu = 2.073 \times 10^{-5} \mathrm{~kg} / (\mathrm{m} \cdot \mathrm{s})\) (from air tables) Now, we can calculate the Reynolds number: \(Re = \frac{1.027 \mathrm{~kg} / \mathrm{m}^3 \cdot 3.75 \mathrm{~m} / \mathrm{s} \cdot 0.2 \mathrm{~m}}{2.073 \times 10^{-5} \mathrm{~kg} / (\mathrm{m} \cdot \mathrm{s})} = 37412\)

Calculate the friction factor for the duct

Given the Reynolds number and the relative roughness \(\epsilon = 10^{-3}\), we can use the Moody chart or an approximation formula to find the friction factor (\(f\)) for the duct. In this case, let's use the approximation formula by Haaland: \(f = \frac{1}{(1.8 \log_{10} (Re \cdot \sqrt{f}) - 1.64)^2}\) Solving this equation for \(f\) yields a friction factor of \(f \approx 0.0249\).

Calculate the pressure loss in the duct

To calculate the pressure loss in the duct, we can use the Darcy-Weisbach equation: \(\Delta P = f \frac{L}{D_h} \frac{1}{2} \rho v^2\) Substituting the values for friction factor, duct length, hydraulic diameter, density, and velocity: \(\Delta P = 0.0249 \cdot \frac{8 \mathrm{~m}}{0.2 \mathrm{~m}} \cdot \frac{1}{2} \cdot 1.027 \mathrm{~kg} / \mathrm{m}^3 \cdot (3.75 \mathrm{~m} / \mathrm{s})^2\) \(\Delta P \approx 278 \mathrm{~Pa}\) The pressure difference between the inlet and the outlet sections of the duct is \(278 \mathrm{~Pa}\).

Determine the rate of heat loss from the duct to the attic space

To calculate the rate of heat loss, we use the formula: \(Q = h A_s (T_{surface} - T_{ambient})\) where \(h\) is the convective heat transfer coefficient, \(A_s\) is the surface area of the duct, \(T_{surface}\) is the surface temperature of the duct, and \(T_{ambient}\) is the ambient temperature in the attic space. Given that the duct surface is nearly isothermal at \(60^{\circ} \mathrm{C}\) and assuming the ambient temperature in the attic space to be \(25^{\circ} \mathrm{C}\), we can substitute these values into the formula: \(Q = h A_s (60^{\circ} \mathrm{C} - 25^{\circ} \mathrm{C})\) The surface area of the duct is: \(A_s = 4 \times 0.2 \mathrm{~m} \times 8 \mathrm{~m} = 6.4 \mathrm{~m}^2\) We can use the simplified formula for the convective heat transfer coefficient (\(h\)) for horizontal flows in ducts, given by (Re, Pr are Reynolds and Prandtl numbers, respectively): \(h = \frac{k}{D_h} \cdot 0.023 \cdot Re^{0.8} \cdot Pr^{0.33}\) Using the air property values at \(80^{\circ} \mathrm{C}\) and given a Prandtl number \(\mathrm{Pr} = 0.7\): \(k = 0.0327 \mathrm{~W} / (\mathrm{m} \cdot \mathrm{K})\) \(h = \frac{0.0327 \mathrm{~W} / (\mathrm{m} \cdot \mathrm{K})}{0.2 \mathrm{~m}} \cdot 0.023 \cdot 37412^{0.8} \cdot 0.7^{0.33} \approx 26.18 \mathrm{~W} / (\mathrm{m}^2 \cdot \mathrm{K})\) Finally, the rate of heat loss is: \(Q = 26.18 \mathrm{~W} / (\mathrm{m}^2 \cdot \mathrm{K}) \cdot 6.4 \mathrm{~m}^2 \cdot 35 \mathrm{~K} \approx 5915 \mathrm{~W}\) The rate of heat loss from the duct to the attic space is approximately \(5915 \mathrm{~W}\).

Evaluate the assumption of air properties at \(80^{\circ} \mathrm{C}\)

Since there is a significant temperature difference between the duct surface and the ambient temperature in the attic space, the assumption of air properties at a bulk mean temperature of \(80^{\circ} \mathrm{C}\) may not be completely accurate. However, the variation in air properties due to temperature difference is relatively small. Therefore, the assumption can be considered reasonable for this particular problem.