4-109C

A short cylinder initially at a uniform temperature \( T_i \) is subjected to convection from all of its surfaces to a medium at temperature \( T_{\infty} \). Explain how you can determine the temperature of the midpoint of the cylinder at a specified time \( t \).

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4-110C

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 one- or two-dimensional? Explain.

The temperature of the midpoint of the cylinder at a specified time t can be determined by solving the one-dimensional heat conduction equation, which takes into account factors such as the thermal conductivity of the material, the surface area of the cylinder, and the temperature difference between the cylinder and the surrounding medium. By evaluating the temperature function at the midpoint, we can calculate the temperature at that specific location.

Explanation:

To determine the temperature of the midpoint of the cylinder at a specified time t, we can use the one-dimensional heat conduction equation. This equation describes how heat is transferred within a solid object and takes into account factors such as the thermal conductivity of the material, the surface area of the cylinder, and the temperature difference between the cylinder and the surrounding medium.

First, we need to solve the heat conduction equation for the short cylinder. This involves considering the boundary conditions, which in this case are convection from all surfaces of the cylinder to the surrounding medium at temperature T[infinity]. The heat transfer coefficient h is also given.

Once we have solved the heat conduction equation, we can determine the temperature distribution within the cylinder at any point and at any time. To find the temperature at the midpoint of the cylinder, we simply evaluate the temperature function at that specific location.

It is important to note that the temperature distribution within the cylinder will change over time due to the convection process. Therefore, the temperature at the midpoint of the cylinder will also vary with time. By specifying a specific time t, we can calculate the temperature at the midpoint using the solved heat conduction equation.

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