Question

Air \(\left(c_{p}=1007 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) enters a \(17-\mathrm{cm}\)-diameter and 4-m-long tube at $65^{\circ} \mathrm{C}\( at a rate of \)0.08 \mathrm{~kg} / \mathrm{s}$ and leaves at \(15^{\circ} \mathrm{C}\). The tube is observed to be nearly isothermal at \(5^{\circ} \mathrm{C}\). The average convection heat transfer coefficient is (a) \(24.5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) (b) \(46.2 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) (c) \(53.9 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) (d) \(67.6 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) (e) \(90.7 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\)

Short answer

Answer: (b) \(46.2 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\).

Step-by-step solution

Calculate the Rate of Heat Transfer

We first need to determine the rate of heat transfer (Q) from the air to the tube. We can use the given mass flow rate and specific heat, as well as the temperature difference, to compute Q. The formula for calculating the rate of heat transfer is: Q = m_dot * c_p * (T_inlet - T_outlet) We are given: - m_dot = 0.08 kg/s (mass flow rate) - c_p = 1007 J/kg*K (specific heat of air) - T_inlet = 65°C - T_outlet = 15°C Plug in the given values into the formula: Q = 0.08 * 1007 * (65-15) The rate of heat transfer, Q, is found to be 3216.8 W.

Calculate the Tube Surface Area

Now, we need to calculate the tube surface area. This will be used to find the average convection heat transfer coefficient. The formula for calculating the tube surface area (A) is: A = π * d * L We are given: - d = 0.17 m (tube diameter) - L = 4 m (tube length) Plug in the given values into the formula: A = π * 0.17 * 4 The tube surface area, A, is found to be 2.137 m².

Calculate the Temperature Difference between the Air and the Tube Surface

We can compute the temperature difference between the air and the tube surface by averaging the inlet and outlet air temperatures and then subtracting the temperature of the tube surface. The formula for calculating the average air temperature is: T_avg = (T_inlet + T_outlet)/2 We are given: - T_inlet = 65°C - T_outlet = 15°C - T_tube = 5°C Plug in the given values into the formula: T_avg = (65+15)/2 The average air temperature, T_avg, is 40°C. The temperature difference between the air and the tube surface is, therefore, ΔT = T_avg - T_tube = 40 - 5 = 35°C.

Calculate the Average Convection Heat Transfer Coefficient (h)

Finally, we can calculate the average convection heat transfer coefficient (h) using the rate of heat transfer (Q), the tube surface area (A), and the temperature difference (ΔT). The formula for calculating the average convection heat transfer coefficient is: h = Q / (A * ΔT) Plug in the calculated values of Q, A and ΔT into the formula: h = 3216.8 / (2.137 * 35) The convection heat transfer coefficient, h, is found to be 43.3 W/m²·K, which is closest to option (b) 46.2 W/m²·K. The final answer is: (b) \(46.2 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\).