Question

Consider a solid cylindrical rod whose side surface is maintained at a constant temperature while the end surfaces are perfectly insulated. The thermal conductivity of the rod material is constant, and there is no heat generation. It is claimed that the temperature in the radial direction within the rod will not vary during steady heat conduction. Do you agree with this claim? Why?

Short answer

Answer: No, the temperature in the radial direction within the rod does not vary during steady heat conduction. This is because the heat flux at the side surface is zero, and as a result, the temperature gradient in the radial direction is also zero.

Step-by-step solution

Understand the given information

We are given a solid cylindrical rod with side surface maintained at a constant temperature and perfectly insulated end surfaces. The thermal conductivity of the rod material is constant, and there is no heat generation within the rod.

Analyze steady heat conduction in a solid cylindrical rod

During steady heat conduction, Fourier's Law of heat conduction tells us that the heat flux is directly proportional to the negative gradient of the temperature. Mathematically, this can be written as: q = -k * (dT/dr) where q is the heat flux, k is the constant thermal conductivity, and dT/dr is the gradient of temperature with respect to the radial position (r).

Apply boundary conditions

For the cylindrical rod, let the side surface be at r=R and the constant temperature at this surface be T_s. Since the end surfaces are perfectly insulated, it means that no heat flows through them; thus, the heat flux at r=0 and r=R should be zero. Therefore, at the side surface: q(R) = -k * (dT/dr)|_(r=R) = 0

Determine the temperature variation in radial direction

Since the heat flux at the side surface is zero, we can infer that the gradient of temperature with respect to the radial position (dT/dr)|_(r=R) is zero as well. This means that the temperature does not vary in the radial direction within the solid cylindrical rod during steady heat conduction.

Conclusion

Based on the analysis, we can agree with the claim that the temperature in the radial direction within the rod will not vary during steady heat conduction. This is because the heat flux at the side surface is zero, and as a result, the temperature gradient in the radial direction is also zero.