Question

Consider a short cylinder whose top and bottom surfaces are insulated. The cylinder is initially at a uniform temperature \(T_{i}\) and is subjected to convection from its side surface to a medium at temperature \(T_{\infty}\) with a heat transfer coefficient of \(h\). Is the heat transfer in this short cylinder oneor two-dimensional? Explain.

Short answer

Provide an explanation for your answer.

Step-by-step solution

Define the problem and the geometry

We have a short cylinder with insulated top and bottom surfaces, meaning that there is no heat transfer through these surfaces. The cylinder is subjected to convection from its side surface, which is in contact with the medium at a temperature \(T_\infty\) and a heat transfer coefficient \(h\). We aim to find if the heat transfer in the cylinder is one-dimensional or two-dimensional.

Visualize the heat transfer process

Since the top and bottom surfaces of the cylinder are insulated, the only direction in which heat transfer can take place is through the side surface, along the radial direction (perpendicular to the axis of the cylinder). Therefore, heat transfer must occur outward from the center of the cylinder to the side surface, and then from the side surface of the cylinder to the external medium by convection.

Analyze the heat conduction in the radial direction

The heat conduction in the radial direction occurs due to the temperature difference between the center of the cylinder and the side surface. As the heat is transferred from the center of the cylinder to the side surface, the temperature distribution within the cylinder will change, and the temperature profile will develop along the radial direction.

Analyze the influence of the axial dimension

Since the top and bottom surfaces of the cylinder are insulated, there is no heat transfer in the axial direction (parallel to the axis of the cylinder). Thus, the temperature profile along the axial direction remains uniform. Because the temperature within the cylinder doesn't depend on the axial position, we can conclude that the heat transfer in the cylinder is one-dimensional.

Conclusion

The heat transfer in the short cylinder is one-dimensional since heat is only transferred in the radial direction, perpendicularly to the axis of the cylinder. The axial dimension does not affect the temperature distribution within the cylinder, as the top and bottom surfaces are insulated, which means no heat is transferred along the axial direction.