Question

Air (1 atm) enters into a \(5-\mathrm{cm}\)-diameter circular tube at \(20^{\circ} \mathrm{C}\) with an average velocity of $5 \mathrm{~m} / \mathrm{s}$. The tube wall is maintained at a constant surface temperature of \(160^{\circ} \mathrm{C}\), and the outlet mean temperature is $80^{\circ} \mathrm{C}$. Estimate the length of the tube. Is the flow fully developed?

Short answer

Short Answer: First, calculate the heat transfer coefficient using the Dittus-Boelter equation, which requires determining the values of the Reynolds number and the Prandtl number. Next, calculate the mass flow rate using the air's density, cross-sectional area of the tube, and average velocity. Then, find the specific heat capacity of air and the temperature difference to calculate the heat transfer. Afterwards, calculate the heat transfer area and find the length of the tube. Lastly, check if the flow is fully developed by comparing the calculated tube length with the entrance length for flow development.

Step-by-step solution

Determine the heat transfer coefficient

The heat transfer coefficient can be estimated using the Dittus-Boelter equation for turbulent flow: \(h = k\frac{Nu}{D}\) We know the diameter of the tube, \(D = 5 cm = 0.05 m\). But first, we need to calculate the Nusselt number (Nu). For air at 20°C, the thermal conductivity, \(k\), can be found in standard tables and is approximately \(k = 0.02587 \mathrm{W/mK}\). The Nusselt number is calculated as: \(Nu = 0.023\,Re^{0.8}Pr^{0.4}\) where \(Re\) is the Reynolds number, calculated as: \(Re = \frac{ρvD}{μ}\) and \(Pr\) is the Prandtl number, calculated as: \(Pr = \frac{μcp}{k}\) Next, we will find the values of the Reynolds number and the Prandtl number.

Calculate the mass flow rate

The mass flow rate of the air can be determined using the following formula: \(\dot{m} = ρAv\) Where \(ρ\) is the air density at 20°C, which can be found in standard tables and is approximately \(ρ = 1.204 \mathrm{kg/m^3}\), \(A\) is the cross-sectional area of the tube, and \(v\) is the average velocity of the air. The area of the tube can be calculated as: \(A = π(\frac{D}{2})^2\) Now we can calculate the mass flow rate.

Determine the specific heat capacity and temperature difference

We need to determine the specific heat capacity of air at constant pressure, \(c_p\), and the temperature difference to calculate the heat transfer. The specific heat capacity of air at constant pressure can be found in standard tables and is approximately \(c_p = 1005 \mathrm{J/kgK}\). The temperature difference between the inlet and outlet is given as: ΔT = \(80^{\circ}\mathrm{C} - 20^{\circ}\mathrm{C} = 60^{\circ}\mathrm{C}\) Now we can calculate the heat transfer as: \(Q = \dot{m}c_pΔT\)

Calculate the heat transfer area

We can calculate the heat transfer area using the following equation: \(A_h = \frac{Q}{hΔT_s}\) Where \(A_h\) is the heat transfer area, and \(ΔT_s\) is the temperature difference between the surface and the outlet, which is: \(ΔT_s = 160^{\circ}\mathrm{C} - 80^{\circ}\mathrm{C} = 80^{\circ}\mathrm{C}\) Now, we can calculate the length of the tube (\(L\)) using the formula: \(L = \frac{A_h}{πD}\)

Determine if the flow is fully developed

Now, we will check if the flow is fully developed. The entrance length for flow development is given by: \(L_e = 0.05\,Re\,D\) If \(L > L_e\), the flow is fully developed. Otherwise, it is developing flow. Finally, compare the calculated tube length with the entrance length for flow development to determine whether the flow is fully developed or not.