Question

Consider a solid cylindrical rod whose ends are maintained at constant but different temperatures while the side surface is perfectly insulated. There is no heat generation. It is claimed that the temperature along the axis of the rod varies linearly during steady heat conduction. Do you agree with this claim? Why?

Short answer

Answer: Yes, the temperature along the axis of the cylindrical rod varies linearly during steady heat conduction. This is due to the constant heat flow rate, thermal conductivity, and cross-sectional area of the rod, resulting in a constant temperature gradient and a linear temperature variation.

Step-by-step solution

Understand the given conditions

We have a solid cylindrical rod with ends maintained at different temperatures and perfectly insulated side surfaces. This means that heat can only flow through the ends of the rod.

Analyze steady heat conduction

During steady heat conduction, the heat flow rate is constant throughout the rod. In this case, we can define Fourier's Law: $$q = -kA \frac{dT}{dx},$$ where \(q\) is the heat flux, \(k\) is the thermal conductivity, \(A\) is the area through which the heat is flowing, and \(\frac{dT}{dx}\) is the temperature gradient.

Apply Fourier's Law for a cylindrical rod

For a cylindrical rod, the cross-sectional area \(A\) remains constant along the length of the rod. Since the heat flow rate (\(q\)) is constant during steady heat conduction, and both \(k\) and \(A\) do not change along the length of the rod, the temperature gradient, \(\frac{dT}{dx}\), must also be constant.

Analyze the temperature variation along the rod's axis

Since the temperature gradient \(\frac{dT}{dx}\) is constant, the temperature \(T\) varies linearly with the distance \(x\) along the axis of the rod. We can represent the temperature variation along the rod's axis as a linear function: $$T(x) = m x + b,$$where \(m\) is the slope of the temperature gradient, and \(b\) is the temperature at \(x = 0\) (one end of the rod).

Conclude whether the claim is correct or not

The analysis shows that the temperature along the axis of the rod varies linearly during steady heat conduction, as claimed. The temperature gradient being constant, which leads to a linear temperature variation, is a direct consequence of the heat flow rate being constant during steady heat conduction and the side surfaces being perfectly insulated.