Question

A thin electrical heater is inserted between a long circular rod and a concentric tube with inner and outer radii of 20 and 40 mm. The rod

(A) has a thermal conductivity of kₐ =0.15 W/m⋅K, while the tube
(B) has a thermal conductivity of kᵦ =1.5 W/m⋅K and its outer surface is subjected to convection with a fluid of temperature T[infinity]=−15∘C and heat transfer coefficient 50 W/m²⋅K. The thermal contact resistance between the cylinder surfaces and the heater is negligible.
(a) Determine the electrical power per unit length of the cylinders (W/m) that is required to maintain the outer surface of cylinder B at 5∘C.
(b) What is the temperature at the center of cylinder A?

Answer 1

To calculate the required electrical power per unit length to keep the outer surface of the cylinder at 5°C, we use heat conduction equations applicable to radial heat flow and convective heat transfer at the outer surface. The temperature at the center of cylinder A can be found using similar principles, considering the thermal conductivity, geometry, and boundary temperatures.

Step-by-step explanation:

In solving this heat transfer problem, we need to first calculate the electrical power per unit length of the cylinders required to maintain the outer surface of cylinder B at 5°C. We will apply the concept of steady-state heat conduction where the heat generated by the electrical heater in the inner rod must be equal to the heat conducted through the concentric tube and also the heat lost to the surrounding by convection.
To find the electrical power per unit length (W/m), we use the equation Q/t = k * A * (T₂ - T₁) / d, where Q/t is the rate of heat transfer, k is the thermal conductivity, A is the surface area, and d is the distance between two surfaces with temperatures T₂ and T₁. For the convective heat transfer at the outer surface of cylinder B, we use the equation Q/t = h * A * (Tₛ - Tₙ), where h is the heat transfer coefficient, and Tₛ and Tₙ are the temperatures of the surface and surrounding fluid, respectively.
By combining the conduction through the rod and tube and the convection from the outer surface, we can solve for the total electrical power required. To determine the temperature at the center of cylinder A, we will also use the relevant heat transfer equations considering the known thermal conductivity and initial and boundary conditions provided.