Question

A long electrical resistance wire of radius $k_{\text {wirc }}=15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. Heat is generated uniformly in the wire as a result of resistance heating at a constant rate of $1.2 \mathrm{~W} / \mathrm{cm}^{3}$. The wire is covered with polyethylene insulation with a thickness of \(0.5 \mathrm{~cm}\) and thermal conductivity of $k_{\text {ins }}=0.4 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. The outer surface of the insulation is subjected to convection and radiation with the surroundings at \(20^{\circ} \mathrm{C}\). The combined convection and radiation heat transfer coefficients is \(7 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Formulate the temperature profiles for the wire and the polyethylene insulation. Use the temperature profiles to determine the temperature at the interface of the wire and the insulation and the temperature at the center of the wire. The ASTM D1351 standard specifies that thermoplastic polyethylene insulation is suitable for use on electrical wire with operation at temperatures up to \(75^{\circ} \mathrm{C}\). Under these conditions, does the polyethylene insulation for the wire meet the ASTM D1351 standard?

Short answer

Answer 2: The temperature profile for the wire with heat generation is given by the equation \(T(r)= -\frac{q_{gen}}{4k_{wire}}r^2 + Cr + D\), while the temperature profile for the insulation without heat generation is \(T(r)= Er + F\). Question 3: What is the significance of applying the boundary conditions in step 4? Answer 3: Applying the boundary conditions in step 4 allows us to determine the values of the constants \(C\), \(D\), \(E\), and \(F\) in the temperature profiles for the wire and the insulation. These constants are important for calculating the temperature at specific points along the wire and insulation. Question 4: How can we check if the polyethylene insulation meets the ASTM D1351 standard? Answer 4: To check if the insulation meets the ASTM D1351 standard, compare the interface temperature calculated in step 5 with the given allowable maximum temperature of \(75^{\circ}\, \mathrm{C}\). If the interface temperature doesn't exceed the standard, then the polyethylene insulation is suitable for use on the electrical wire.

Step-by-step solution

Calculate the heat generation rate per unit length of wire

To find the heat generation rate, we'll need to multiply the given heat generation rate per unit volume by the volume of the wire per unit length. The volume of the wire per unit length can be found using the formula for the volume of a cylinder: \[ V_{wire} = \pi r^2 \] So, the heat generation rate per unit length of wire is given by: \[ q_{gen} = q_{gen,volume} \times V_{wire} \] where \(q_{gen,volume}=1.2\,\mathrm{W/cm^3}\) and \(r=k_{wire}=1.5\,\mathrm{cm}\).

Obtain temperature profile for the wire (with heat generation)

Now, we can use the heat equation for cylindrical coordinates: \[ \frac{d}{dr}\left( k_{wire} \frac{dT}{dr} \right) + q_{gen} = 0 \] Solve the equation differential for the temperature profile in the wire by integrating it twice: \[ T(r)= -\frac{q_{gen}}{4k_{wire}}r^2 + Cr + D \]

Obtain temperature profile for the insulation (without heat generation)

Since no heat is generated in the insulation, the heat equation simplifies to: \[ \frac{d}{dr}\left( k_{ins} \frac{dT}{dr} \right) = 0 \] Solve the differential equation to obtain the temperature profile for the insulation: \[ T(r)= Er + F \]

Apply boundary conditions to connect the temperature profiles

We need to apply the following boundary conditions: 1. Temperature at the wire-insulation interface: \(T(k_{wire}+t_{ins}) = T_I\) 2. Temperature at the center of wire: \(T(k_{wire}) = T_C\) 3. Heat transfer at the outer surface of the insulation: \(q_r=k_{ins}\frac{dT}{dr}\Big|_{r=k_{wire}+t_{ins}}=-h(T-I_0)\), where \(I_0 = 20^{\circ}\,\mathrm{C}\) and \(h = 7\,\mathrm{W/m^2\cdot K}\) Applying these boundary conditions, we can obtain the constants \(C\), \(D\), \(E\), and \(F\).

Calculate temperature at the center and the interface

Finally, apply the temperature profiles found in steps 2 and 3, along with the known constants, to calculate the temperature at the center of the wire and the interface of wire and insulation: \[ T_C = T(k_{wire}) \] \[ T_I = T(k_{wire} + t_{ins}) \]

Check if insulation meets ASTM D1351 standard

We must check if the maximum operation temperature of the polyethylene insulation is no more than \(75^{\circ} \mathrm{C}\) as per the ASTM D1351 standard. To do that, we need to compare the interface temperature we found in step 5 with the given allowable maximum temperature: \[ T_I \leq 75^{\circ}\, \mathrm{C} \] If the interface temperature doesn't exceed the standard, then the polyethylene insulation is suitable for use on the electrical wire.