Question

Cold air at \(5^{\circ} \mathrm{C}\) enters a 12 -cm-diameter, 20 -m-long isothermal pipe at a velocity of \(2.5 \mathrm{~m} / \mathrm{s}\) and leaves at \(19^{\circ} \mathrm{C}\). Estimate the surface temperature of the pipe.

Short answer

Based on the given information and calculated heat transfer rate, we can estimate the surface temperature of the pipe to be approximately equal to the final temperature of the air, which is \(19^{\circ} \mathrm{C}\).

Step-by-step solution

Calculate the area of the pipe

We can find the area of the pipe using the formula for the surface area of a cylinder (ignoring the top and bottom caps) as follows: Area \(A = 2 \pi r L\) where \(r\) is the radius of the pipe (half of the diameter) and \(L\) is the length of the pipe. Given the diameter of the pipe as \(12\,\text{cm}\), its radius is \(r = 6\,\text{cm}\). Also, the length is given as \(L = 20\,\text{m}\). Remember to convert \(r\) from centimeters to meters: \(r = 0.06\,\text{m}\). Now we can calculate the area: \(A = 2 \pi (0.06\,\text{m})(20\,\text{m}) \approx 7.54\,\text{m}^2\)

Assume a value for the heat transfer coefficient (h)

Since we do not have information about the air's properties or the pipe's material, we can assume a value for the heat transfer coefficient (h) for gases: \(h = 25\,\text{W} / \text{m}^2 \text{K}\)

Calculate the heat transfer rate (Q)

We can use the formula for the heat transfer rate (Q) by convection: \(Q = h A \Delta T\) Where \(\Delta T\) is the temperature difference between the air and the pipe surface. Given the initial and final temperatures of the air, we can determine the average air temperature as follows: \(T_{avg} = \frac{5 + 19}{2} = 12^{\circ} \mathrm{C}\) Assuming that the pipe surface temperature (\(T_s\)) is approximately equal to the final temperature of the air (\(T_f\)), we can estimate the temperature difference between the air and the pipe surface: \(\Delta T = T_s - T_{avg}\) Now we can calculate the heat transfer rate: \(Q = h A \Delta T\) \(Q = (25\,\text{W} / \text{m}^2 \text{K})(7.54\,\text{m}^2)(19 - 12)\) \(Q = (25\,\text{W} / \text{m}^2 \text{K})(7.54\,\text{m}^2)(7\,K)\) \(Q \approx 1310.64\,\text{W}\)

Estimate the surface temperature of the pipe

We have assumed that the pipe surface temperature (\(T_s\)) is approximately equal to the final temperature of the air (\(T_f\)). Therefore, we can estimate the surface temperature of the pipe as: \(T_s \approx 19^{\circ} \mathrm{C}\) Considering the assumptions made, such as heat transfer coefficients and temperature approximation, it's important to notice that this value may not be very precise, but it is a reasonable estimate for the surface temperature of the pipe.