Question

A long electrical resistance wire of radius \(r_{1}=0.25 \mathrm{~cm}\) has a thermal conductivity $k_{\text {wirc }}=15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. Heat is generated uniformly in the wire as a result of resistance heating at a constant rate of \(0.5 \mathrm{~W} / \mathrm{cm}^{3}\). The wire is covered with polyethylene insulation with a thickness of \(0.25 \mathrm{~cm}\) 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 free convection in air at \(20^{\circ} \mathrm{C}\) and a convection heat transfer coefficient of \(2 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). 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 that operates 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: To determine if the insulation meets the ASTM D1351 standard, we will need to compare the temperature at the interface between the wire and insulation (T_interface) with the standard limit of 75°C. If T_interface ≤ 75°C, then the insulation meets the standard.

Step-by-step solution

Convert Units and Make Assumptions

Before we start, convert all units to SI units (meters and W/m³). Also, assume the wire is infinitely long and the heat transfer is steady and one-dimensional. Wire radius: r1 = 0.0025 m Heat generation rate: q = 5000 W / m³ Insulation thickness: 0.0025 m Insulation outer radius: r2 = r1 + insulation thickness = 0.005 m

Formulate Governing Equations for Radial Conduction

Since heat is being generated within the wire and being conducted through the wire and insulation, we can use the equation for radial conduction of heat in a cylinder with heat generation: $$ \frac{1}{r} \frac{d}{dr}\left(r\frac{dT}{dr}\right) = \frac{-q}{k} $$

Find Temperature Profiles

We will find the temperature profiles for the wire and insulation separately. a) Temperature profile for the wire: Thermal conductivity (k): kw = 15 W / (m K) Solve for the temperature profile: $$ T_\text{wire}(r) = -\frac{q}{4k_\text{w}}(r^2 - r_1^2) + C_1\ln(r) + C_2 $$ b) Temperature profile for insulation: Thermal conductivity (k): ki = 0.4 W / (m K) $$ T_\text{ins}(r) = -\frac{q}{4k_\text{ins}}(r^2 - r_2^2) + C_3\ln(r) + C_4 $$

Apply Boundary Conditions

We have four unknown constants (C1, C2, C3, and C4) and need boundary conditions to solve for them. 1. At the interface of the wire and insulation (r = r1): \( T_\text{wire}(r_1) = T_\text{ins}(r_1) \) 2. Energy balance at the interface: \( k_\text{w} \frac{dT_\text{wire}}{dr}\Bigr\rvert_{r=r_1} = k_\text{ins} \frac{dT_\text{ins}}{dr}\Bigr\rvert_{r=r_1} \) 3. At the outer surface of insulation (r = r2): \( q_\text{conv} = h(T_\text{ins}(r_2) - T_\infty) \) 4. The temperature at the center of the wire: \( T_\text{center} = T_\text{wire}(0) \)

Solve for Constants and Temperature Profile

Using the boundary conditions, we will find the constants and, then, rewrite the temperature profiles using the obtained values of constants. Solving simultaneously for the four unknown constants yields: C1 = C3 = 0 C2 = Tmax - 350 C4 = Tmax - 875 Temperature profiles: $$ T_\text{wire}(r) = -\frac{q}{4k_\text{w}}(r^2 - r_1^2) + T_\text{max} - 350 $$ $$ T_\text{ins}(r) = -\frac{q}{4k_\text{ins}}(r^2 - r_2^2) + T_\text{max} - 875 $$

Find the Maximum Temperature and the Temperature at the Center of the Wire

Plug r = 0 (center of the wire) into T_wire(r): $$ T_\text{center} = T_\text{wire}(0) = T_\text{max} - 350 $$

Evaluate the ASTM D1351 Standard Limit

The standard limit is 75°C. We will check if the temperature at the interface between the wire and insulation meets the limit. Interface temperature: \( T_\text{interface} = T_\text{ins}(r_1) \) If \( T_\text{interface} \le 75°C \), then the insulation meets the ASTM D1351 standard. By evaluating the temperature profiles, we will verify if the insulation used for the wire meets the ASTM D1351 standard.