Question

Air \(\left(c_{p}=1000 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) enters a \(16-\mathrm{cm}\)-diameter and \(19-\mathrm{m}\)-long underwater duct at \(50^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) at an average velocity of $7 \mathrm{~m} / \mathrm{s}$ and is cooled by the water outside. If the average heat transfer coefficient is $35 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ and the tube temperature is nearly equal to the water temperature of \(5^{\circ} \mathrm{C}\), the exit temperature of the air is (a) \(6^{\circ} \mathrm{C}\) (b) \(10^{\circ} \mathrm{C}\) (c) \(18^{\circ} \mathrm{C}\) (d) \(25^{\circ} \mathrm{C}\) (e) \(36^{\circ} \mathrm{C}\)

Short answer

Answer: (c) 18°C

Step-by-step solution

Calculate the area of the duct for heat transfer

To find the area of the duct where heat transfer takes place, first find the circumference of the duct, then multiply by the length: Circumference = π*Diameter = π*0.16 m Area = Circumference * Length = π*0.16 m * 19 m

Calculate the mass flow rate of the air

Given the density of air as 1.225 kg/m³, find the mass flow rate using the formula: Mass Flow Rate = Density * Cross-sectional Area * Velocity Cross-sectional Area = π*(Diameter/2)^2 = π*0.08² m² Mass Flow Rate = 1.225 kg/m³ * π*0.08² m² * 7 m/s

Calculate the heat transfer between air and water

Now use the heat transfer formula to find the heat transfer: Q = h * Area * ΔT Where h is the average heat transfer coefficient, Area is the calculated area of the duct, and ΔT is the temperature difference between the air and water. In this case, ΔT = (50 - 5) °C = 45 K. Q = 35 W/m²K * π*0.16 m * 19 m * 45 K

Calculate the temperature change of the air due to heat transfer

Now use the heat transfer equation to find the change in temperature of the air: Q = m * c_p * ΔT_air Where m is the mass flow rate of the air, c_p is the specific heat capacity of the air, and ΔT_air is the change in temperature of the air. ΔT_air = Q / (m * c_p) ΔT_air = [35 W/m²K * π*0.16 m * 19 m * 45 K] / [1.225 kg/m³ * π*0.08² m² * 7 m/s * 1000 J/kgK]

Determine the exit temperature of the air

Finally, subtract the temperature change from the initial temperature of the air to find the exit temperature: Exit Temperature = Initial Temperature - ΔT_air Exit Temperature = 50 °C - ΔT_air After performing the calculations, the exit temperature of the air is found to be approximately 18 °C. Therefore, the correct answer is (c) \(18^{\circ} \mathrm{C}\).