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 $T_{\mathrm{\infty }}$, with a heat transfer coefficient of $h$. Is the heat transfer in this short cylinder one- or twodimensional? Explain.

Short answer

Explain your answer. Answer: The heat transfer in the given short cylinder is one-dimensional. This is because it only involves radial heat transfer due to convection at the side surface of the cylinder, while the insulated top and bottom surfaces prevent heat transfer in the axial direction.

Step-by-step solution

Insulated top and bottom surfaces

Since the top and bottom surfaces of the cylinder are insulated, there is no heat transfer through these surfaces. Therefore, we only need to consider the heat transfer along the side surface of the cylinder.

Heat transfer through the side surface

Now let's analyze the heat transfer through the side surface of the cylinder. With the given information, we know that the entire side surface is subjected to convection with the surrounding medium at temperature $T_{\mathrm{\infty }}$ and a heat transfer coefficient of $h$. Convection causes heat transfer in the radial direction (from the inside to the outside of the cylinder or vice versa). Also, note that the insulated top and bottom surfaces of the cylinder won't affect the radial heat transfer.

One-dimensional or two-dimensional heat transfer

As we have concluded that there is no heat transfer along the axis of the cylinder due to the insulated top and bottom surfaces, it leaves us with the possibility of heat transfer in the radial direction only. Thus, the heat transfer in this short cylinder is one-dimensional.

Explanation of the result

In conclusion, the heat transfer in this short cylinder is one-dimensional because it only involves radial heat transfer due to convection at the side surface of the cylinder. The insulated top and bottom surfaces prevent heat transfer in the axial direction, leading to a one-dimensional heat transfer scenario.

Key concepts

Convection Heat Transfer

Convection is one of the primary mechanisms of heat transfer and plays a pivotal role in everyday applications such as heating a room, cooking, and cooling electronic devices. It involves the movement of heat from one place to another through the motion of fluids, which could be either liquids or gases. For instance, when a pot of water is heated on a stove, the water at the bottom becomes hotter, less dense, and rises, while the cooler, more dense water descends, setting up a cyclic motion known as convection currents. This motion transports heat from the bottom of the pot to the top, evenly distributing the temperature.

In our exercise, convection heat transfer is occurring at the side surface of the cylinder, where heat is transferred from the cylinder to the surrounding medium. This process is driven by the difference in temperature between the cylinder's surface, initially at a temperature of $T_i$, and the surrounding medium at temperature $T_{\text{infinity}}$. The heat transfer coefficient, $h$, determines the efficiency of this convection process. Simplifying for educational purposes, one can envision the cylinder's side as a 'heat emitter', getting rid of excess heat energy via the medium around it, much like skin radiates heat into the air.

Thermal Insulation

When we talk about thermal insulation, we're referring to the practice or materials that are used to reduce heat transfer between objects in thermal contact or within the range of radiative influence. Materials that are good insulators have low thermal conductivity and are used extensively in daily life in a variety of ways – from insulating our homes to keeping our drinks hot or cold in thermoses.

In terms of the present exercise, the cylinder has insulated top and bottom surfaces. These surfaces have been modified or designed in such a way that they significantly impede the flow of heat. This minimizes energy loss and ensures that for the purposes of our exercise, the heat transfer from these areas can be ignored. The thermal insulation effectively restricts the pathway for heat flow, leading to the analysis of heat transfer in one-dimensional terms only. This constrained path simplifies the analysis and underscores the importance of insulation in controlling heat flow within a system.

Heat Transfer Coefficient

The heat transfer coefficient, $h$, is a critical value in the study of heat transfer as it quantifies the convective heat transfer from a solid to a fluid or vice versa. It's defined as the amount of heat transferred per unit area, per unit temperature difference between the solid surface and the fluid. In the context of our exercise, the heat transfer coefficient would inform us how readily the cylinder’s side surface is going to lose its heat to the surrounding medium.

To put it into perspective, a high value of $h$ would suggest that heat is briskly being transferred, possibly because of a strong breeze or a fluid with great thermal properties flowing past the cylinder. A low $h$ could mean that the medium is relatively still or a poor conductor of heat, leading to a slow rate of heat loss. The heat transfer coefficient is fundamental in calculating the rate of heat transfer and plays a central role in the design of thermal systems.