Question

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 \(1 \mathrm{~atm}\) pressure. Is this a good assumption?

Short answer

Answer: To find the surface temperature of the cable, we need to: 1. Calculate the power generated in the cable, which is found using the formula P = V × I. With a voltage of 60 V and a current of 1.5 A, the power generated in the cable is 90 W. 2. Estimate the surface temperature of the cable using the convective heat transfer equation and an appropriate correlation for convection heat transfer coefficient (e.g., Dittus-Boelter correlation). 3. Calculate the actual film temperature and compare it to the given assumption of 60°C. If the values are close, the assumption is reasonable; otherwise, re-evaluate the cable's surface temperature using air properties based on the actual film temperature. Note: The precise numerical value of the surface temperature depends on the air properties (thermal conductivity, mass velocity, diameter, specific heat, and dynamic viscosity) used in the calculation. These properties can be obtained from engineering textbooks or online resources.

Step-by-step solution

Calculate the power generated in the cable

We are given the voltage drop (\(V\)) and the electric current (\(I\)) across the cable. To find the power generated in the cable due to its electrical resistance, we use the formula: $$ P = V \times I $$ Plugging in the given values: $$ P = 60 \ V \times 1.5 \ A = 90 \ W $$

Estimate the surface temperature of the cable using convective heat transfer equation

To estimate the surface temperature of the cable, we will use the convective heat transfer equation: $$ Q = hA(T_s - T_\infty) $$ Where \(Q\) is the heat transfer rate, \(h\) is the convection heat transfer coefficient, \(A\) is the surface area, \(T_s\) is the surface temperature, and \(T_\infty\) is the surrounding air temperature. We know that \(Q = P = 90 \ W\), and \(T_\infty = 20^{\circ}C\). To proceed, we'll need an estimate of the convection heat transfer coefficient, \(h\). Assuming a forced convection scenario around the cable (which seems more likely, given the specified problem), we can use empirical correlations to estimate \(h\). One such correlation, the Dittus-Boelter correlation, is given by: $$ h = k\left(\frac{Gd}{k}\right)^{0.8}\left(\frac{k}{C_p\mu}\right)^{0.4} $$ Where \(k\) is the thermal conductivity, \(G\) is the mass velocity, \(d\) is the diameter, \(C_p\) is the specific heat, and \(\mu\) is the dynamic viscosity - all of these properties are of the surrounding air. We are also given that we can evaluate these properties at a film temperature of \(60^{\circ}C\) and \(1\ \mathrm{atm}\) pressure. Consult an engineering textbook or online resources to find the air properties at these conditions. Once we have these values, we can calculate \(h\), plug it into the convective heat transfer equation, and solve for the cable surface temperature (\(T_s\)).

Check the given assumption about film temperature

To check the given assumption about film temperature, let's calculate the actual film temperature based on the surface temperature of the cable (\(T_s\), which we found in Step 2) and the surrounding air temperature (\(T_\infty = 20^{\circ}C\)). The film temperature (\(T_f\)) is calculated as the average of the surface temperature and the surrounding air temperature: $$ T_f = \frac{T_s + T_\infty}{2} $$ Compare the calculated film temperature to the given film temperature (i.e., \(60^{\circ}C\)) and assess if the assumption is reasonable. If the calculated value is close to the given value, the assumption can be considered reasonable. If not, you might need to re-evaluate the cable's surface temperature using more accurate air properties based on the actual film temperature.