Question

A 1-mm-diameter nickel wire with electrical resistance of $0.129 \Omega / \mathrm{m}$ is submerged horizontally in water at atmospheric pressure. Determine the electrical current at which the wire would be in danger of burnout in nucleate boiling.

Short answer

Answer: To find the electrical current at which the wire would be in danger of burnout in nucleate boiling, you need the specific values of the heat transfer coefficient (h), the wall temperature (T_w), and the water temperature (T_∞) from an experiment with the given dimensions and material of the wire. Once you have these values, you can use the equation 0.129I^2 = hA(T_w - T_∞) to solve for the electrical current (I).

Step-by-step solution

Calculate Power Dissipation Per Unit Length

The power dissipated \(P\) in the wire per unit length due to electrical current can be calculated using the formula: \(P = RI^2\) Where \(R\) is the electrical resistance per unit length and \(I\) is the electrical current. Given that \(R=0.129\, \Omega/m\), we have: \(P = 0.129I^2\)

Determine Burnout Point

Burnout occurs when the heat transfer rate through nucleate boiling equals the power dissipation in the wire. The formula for heat transfer rate \(q\) through nucleate boiling can be calculated using the formula: \(q = hA(T_w - T_{\infty})\) Where \(h\) is the heat transfer coefficient, \(A\) is the area of the wire, \(T_w\) is the wall temperature, and \(T_{\infty}\) is the water temperature.

Calculate Area of the Wire

The cross-sectional area \(A\) of a cylindrical wire with diameter \(d=1 \,\mathrm{mm}\) can be calculated using the formula: \(A = \pi(\frac{d}{2})^2\) Substituting the value for \(d\) gives: \(A = \pi(\frac{1}{2})^2=0.785\, \mathrm{mm^2}\)

Relate Power Dissipation and Heat Transfer Rate

Since at burnout, the power dissipation in the wire equals the heat transfer rate, we can write the equation: \(P = q\) Substitute the expressions for \(P\) and \(q\) obtained in Steps 1 and 2: \(0.129I^2 = hA(T_w - T_{\infty})\)

Solve for Electrical Current

From the experiment, the values of \(h\), \(T_w\), and \(T_{\infty}\) can be obtained for nucleate boiling. Then, the equation from Step 4 can be solved for the electrical current \(I\). Nucleate boiling might occur under a wide range of heat transfer coefficients and water temperatures, so you need specific values for \(h\), \(T_w\), and \(T_{\infty}\) from an experiment with the given dimensions and material of the wire. Once you have these values, you can plug them into the equation and solve for \(I\) algebraically. Finally, you'll have the electrical current at which the wire would be in danger of burnout in nucleate boiling.