Consider a 15-cm-internal-diameter, 10-m-long circular duct whose interior surface is wet. The duct is to be dried by forcing dry air at \(1 \mathrm{~atm}\) and \(15^{\circ} \mathrm{C}\) through it at an average velocity of \(3 \mathrm{~m} / \mathrm{s}\). The duct passes through a chilled room, and it remains at an average temperature of \(15^{\circ} \mathrm{C}\) at all times. Determine the mass transfer coefficient in the duct.

The Reynolds number is a dimensionless quantity that helps to predict fluid flow patterns. It is the ratio of inertial forces to viscous forces and can be calculated using the formula: $$ Re = \frac{ρVD}{μ} $$ Where \(ρ\) is the fluid density, \(V\) is the average velocity of fluid flow, \(D\) is the hydraulic diameter, and \(μ\) is the dynamic viscosity of the fluid. Given, average velocity \(V = 3~m/s\), duct diameter \(D = 0.15~m\). To find the density and dynamic viscosity of dry air at \(1~atm\) and \(15°C\), we can refer to standard air property tables. For dry air at \(1~atm\) and \(15°C\), the density (\(ρ\)) is approximately \(1.225~kg/m^3\), and the dynamic viscosity (\(μ\)) is around \(1.8 × 10^{-5}~Pa.s\) Now, we can calculate the Reynolds number: $$ Re = \frac{(1.225~kg/m^3)(3~m/s)(0.15~m)}{1.8 \times 10^{-5}~Pa.s} ≈ 30625 $$

Schmidt number is a dimensionless number representing the ratio between molecular momentum transport and mass transport. It can be calculated using the formula: $$ Sc = \frac{μ}{ρD_m} $$ Where \(D_m\) is the mass diffusivity of the fluid. For air-water vapor, the mass diffusivity \(D_m\) at \(15°C\) is approximately \(2.5 × 10^{-5}~m^2/s\). Now, we can calculate the Schmidt number: $$ Sc = \frac{1.8 \times 10^{-5}~Pa.s}{(1.225~kg/m^3)(2.5 × 10^{-5}~m^2/s)} ≈ 0.59 $$

The Chilton-Colburn analogy is a useful tool to determine mass transfer coefficients for turbulent flow while knowing the Reynolds and Schmidt numbers. The analogy suggests that: $$ Sh = \frac{k_mD}{D_m} = 0.023 Re^{0.83} Sc^{0.44} $$ Where \(Sh\) is the Sherwood number and \(k_m\) is the mass transfer coefficient. Now, we can calculate the mass transfer coefficient \(k_m\): $$ k_m = \frac{Sh × D_m}{D} = \frac{0.023 × 30625^{0.83} × 0.59^{0.44} × 2.5 \times 10^{-5}~m^2/s}{0.15 ~m} ≈ 9.3 \times 10^{-6}~ m/s $$ The mass transfer coefficient (\(k_m\)) in the duct is approximately \(9.3 \times 10^{-6}~m/s\).