Question

A cylindrical rod is placed in a crossflow of air at $20^{\circ} \mathrm{C}(1 \mathrm{~atm})\( with velocity of \)10 \mathrm{~m} / \mathrm{s}$. The rod has a diameter of \(5 \mathrm{~mm}\) and a constant surface temperature of \(120^{\circ} \mathrm{C}\). Determine \((a)\) the average drag coefficient, \((b)\) the convection heat transfer coefficient using the Churchill and Bernstein relation, and (c) the convection heat transfer coefficient using Table 7-1.

Short answer

Answer: The values are as follows: (a) average drag coefficient (\(C_D\)) (b) convection heat transfer coefficient using Churchill and Bernstein relation (\(h_{CB}\)) (c) convection heat transfer coefficient using Table 7-1 (\(h_{T}\)).

Step-by-step solution

Calculate Reynolds Number

In order to determine the drag coefficient and convection heat transfer coefficient, we first need to calculate the Reynolds number using the following formula: \[ Re = \frac{\rho VD}{\mu} \] where: - \(Re\) = Reynolds number - \(\rho\) = air density at \(20^{\circ} \mathrm{C}\) (approximately \(1.2 \ kg/m^3\)) - \(V\) = air velocity (\(10 \ m/s\)) - \(D\) = rod diameter (\(5 \times 10^{-3} \ m\)) - \(\mu\) = air dynamic viscosity at \(20^{\circ} \mathrm{C}\) (approximately \(1.81 \times 10^{-5} \ kg/(m \cdot s)\)) Now, we can plug the values into the formula to find \(Re\).

Calculate the Drag Coefficient

The average drag coefficient \((C_D)\) can be calculated from the Reynolds number using the following correlation (valid for a smooth cylinder): \[C_D = 1.328 Re^{-0.5}\] Using the Reynolds number calculated in step 1, we can determine the average drag coefficient \((C_D)\).

Calculate the Convection Heat Transfer Coefficient Using Churchill and Bernstein Relation

Using the Churchill and Bernstein correlation, we can calculate the Nusselt number \((Nu_D)\) as: \[Nu_D = 0.3 + \frac{0.62 Re^{0.5} Pr^{1/3}}{\left[1 + (0.4 / Pr)^{2/3}\right]^{1/4}}\left[1 + \left(\frac{Re}{282000}\right)^{5/8}\right]^{4/5}\] Where: - \(Nu_D\) = Nusselt number - \(Re\) = Reynolds number - \(Pr\) = Prandtl number of the air at \(20^{\circ} \mathrm{C}\)(approximately 0.707) Using the values we already have, we can determine \(Nu_D\) and subsequently the convection heat transfer coefficient \((h)\) using the formula: \[h = \frac{k Nu_D}{D}\] The thermal conductivity of the air at \(20^{\circ} \mathrm{C}\) (approximately \(0.0262 \ W/(m \cdot K)\)).

Calculate the Convection Heat Transfer Coefficient Using Table 7-1

To calculate the convection heat transfer coefficient \((h)\) using Table 7-1, we first need to find the corresponding row in the table with the data for our given Reynolds number. Once we have found the row, we can read off the Nusselt number (\(Nu_D\)) and then use the same formula as in step 3 to calculate the convection heat transfer coefficient \((h)\): \[h = \frac{k Nu_D}{D}\] Now, we have calculated the average drag coefficient (a), the convection heat transfer coefficient using the Churchill and Bernstein relation (b), and the convection heat transfer coefficient using Table 7-1 (c).