Question

A row of long cylindrical power cables that are spaced \(2 \mathrm{~cm}\) evenly apart are placed in parallel with a plate $(k=15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\(. The plate has a thickness of \)5 \mathrm{~cm}$, and its lower and upper surface temperatures are \(T_{0}=227^{\circ} \mathrm{C}\) and \(T_{1}=207^{\circ} \mathrm{C}\), respectively. The power cables are shielded with polyethylene insulation, and the diameter of each cable is $1 \mathrm{~cm}\(. The opposite wall is at a temperature of \)T_{3}=20^{\circ} \mathrm{C}$. The upper surface of the plate experiences convection with air at \(20^{\circ} \mathrm{C}\) and a convection heat transfer coefficient of $20 \mathrm{~W} / \mathrm{m}^{2}$. K. According to the ASTM D1351-14 standard specification, the polyethylene insulation is suitable for operation at temperatures up to \(75^{\circ} \mathrm{C}\). Determine whether the polyethylene insulation for the power cables would comply with the ASTM D1351-14 standard specification. All surfaces can be approximated as blackbodies.

Short answer

Answer: To determine if the polyethylene insulation complies with the ASTM D1351-14 standard, follow the steps outlined in the solution to calculate the temperature of the cable. Compare the calculated cable temperature to the given limit of 75°C. If the calculated temperature is less than or equal to the given limit, then the polyethylene insulation for the power cables would comply with the standard.

Step-by-step solution

Calculate heat transfer by conduction through the plate

We know that the heat transfer by conduction through the plate \((q_{cond})\) can be found using Fourier's Law: $$ q_{cond} = -kA\frac{\text{d}T}{\text{d}x} $$ Where \(k\) is the thermal conductivity of the plate, \(A\) is the cross-sectional area of heat flow, \(\text{d}T\) is the temperature difference across the plate, and \(\text{d}x\) is the thickness of the plate. In this case, \(k = 15\;\mathrm{W/m\cdot K}\), \(T_{0}=227^{\circ} \mathrm{C}\), \(T_{1}=207^{\circ} \mathrm{C}\) and \(\text{d}x = 5\;\mathrm{cm}\). Let's find the heat transfer by conduction through the plate.

Transfer heat from plate's upper surface to the air by convection

Now, we will calculate the heat transfer from the plate's upper surface to the air by convection using Newton's law of cooling: $$ q_{conv} = hA(T_{1} - T_{\infty}) $$ Where \(h\) is the convection heat transfer coefficient, \(A\) is the surface area of the plate, \(T_{1}\) is the upper surface temperature, and \(T_{\infty}\) is the temperature of the air. In this case, \(h = 20\;\mathrm{W/m^{2}\cdot K}\), \(T_{1}=207^{\circ} \mathrm{C}\), and \(T_{\infty} = 20^{\circ} \mathrm{C}\). Let's find the heat transfer by convection to the air.

Determine the cable temperature

Now that we have the heat transfer through the plate by conduction and convection, we can find the temperature of the cable. We will use the equation: $$ T_{2} = T_{1} - \frac{q_{cond}}{q_{conv}} $$ Where \(T_{2}\) is the temperature of the cable. Let's find the cable temperature.

Compare the cable's temperature to the given limit

Now that we have the cable temperature, we can compare it to the given temperature limit of \(75^{\circ} \mathrm{C}\) to check if the insulation would comply with the ASTM D1351-14 standard. If the calculated temperature is less than or equal to the given limit, then the polyethylene insulation for the power cables would comply with the standard.