Thermal energy generated by the electrical resistance of a \(5-\mathrm{mm}\)-diameter and 4-m-long bare cable is dissipated to the surrounding air at \(20^{\circ} \mathrm{C}\). The voltage drop and the electric current across the cable in steady operation are measured to be \(60 \mathrm{~V}\) and \(1.5 \mathrm{~A}\), respectively. Disregarding radiation, estimate the surface temperature of the cable. Evaluate air properties at a film temperature of \(60^{\circ} \mathrm{C}\) and 1 atm pressure. Is this a good assumption?

To determine the total power loss in the cable due to electrical resistance, we can use the equation \(P = IV\), where ‘P’ is the power loss, ‘I’ is the electric current, and ‘V’ is the voltage drop across the cable. Here, ‘I’ = 1.5 A, and ‘V’ = 60 V. Calculate the power loss: $$ P = IV $$

To calculate the heat transfer coefficient, we need to consider the air properties at the film temperature of \(60^{\circ} \mathrm{C}\) and 1 atm pressure. You can refer to standard air properties tables and find the following values: - Thermal conductivity (\(k\)) = 0.028 W/mK - Dynamic viscosity (\(\mu\)) = 2.0 × 10^{-5} kg/m.s - Prandtl number (\(Pr\)) = 0.7 Next, calculate the heat transfer coefficient using the following formula: $$ h = \frac{k}{D} \times Nusselt \ number $$ Where, 'D' is the diameter of the cable and Nusselt number is given by the following formula: $$ Nu = 0.3 + \frac{0.62 * (Re^{0.5}) * (Pr^{1/3})}{[1+(0.4/Pr)^{2/3}]^{1/4}*[1+(7.601/Re)^{0.85}]^{1.378}} $$ Here, ‘Re’ is the Reynolds number, and it can be calculated using the formula: $$ Re = \frac{4 * Q}{\pi * D * \mu} $$ Where ‘Q’ is the volumetric flow rate. We will first calculate the Reynolds number, and using the Nusselt number calculation, we will determine the heat transfer coefficient.

Now that we have the heat transfer coefficient (h), we can determine the surface temperature of the cable using the following equation: $$ T_s = T_\infty + \frac{P}{h * A} $$ Where, 'T_s' is the surface temperature of the cable, 'T_∞' is the temperature of the surrounding air, 'P' is the power loss calculated in step 1, 'h' is the heat transfer coefficient calculated in step 2, and 'A' is the total surface area of the cable. Calculate the surface temperature of the cable using the above equation.

Now that we have calculated the surface temperature of the cable, we can check if our initial assumption of evaluating air properties at a film temperature of \(60^{\circ} \mathrm{C}\) and 1 atm pressure was accurate. Calculate the actual film temperature using the following formula: $$ T_f = \frac{T_s + T_\infty}{2} $$ Compare the actual film temperature with the initial assumption of \(60^{\circ} \mathrm{C}\). If the difference is significant, revise the air properties using the actual film temperature and repeat the process until the calculated surface temperature converges to a constant value.