Determine the rate of flow of heat through a spherical shell, whose inner wall of radius r1 is maintained at temperature t1 and outer wall of radius r2 is maintained at temperature t2 (greater than t1)?

To determine the rate of heat flow through a spherical shell, we can apply the principles of thermal conduction, specifically using Fourier's law of heat conduction. This law states that the heat transfer rate is proportional to the temperature difference and the area through which the heat is being conducted. Let's break this down step by step.

Understanding the Setup

We have a spherical shell with an inner radius \( r_1 \) and an outer radius \( r_2 \). The inner wall is kept at a temperature \( t_1 \), while the outer wall is at a higher temperature \( t_2 \). The goal is to find the rate of heat transfer, denoted as \( Q \), through this shell.

Applying Fourier's Law

Fourier's law in three dimensions can be expressed as:

Q = -k A \frac{dT}{dr}

Where:

• Q = rate of heat transfer (W, watts)
• k = thermal conductivity of the material (W/m·K)
• A = surface area through which heat is flowing (m²)
• dT/dr = temperature gradient (K/m)

Calculating the Surface Area

The surface area \( A \) of a spherical shell at radius \( r \) is given by:

A = 4 \pi r^2

Since the heat transfer occurs radially, we will consider a thin spherical shell at radius \( r \) where \( r_1 < r < r_2 \).

Finding the Temperature Gradient

The temperature gradient \( \frac{dT}{dr} \) can be approximated for a steady-state condition (where the temperature does not change with time) as:

\(\frac{dT}{dr} = \frac{t_2 - t_1}{r_2 - r_1}\)

This assumes a linear temperature distribution between the two radii.

Combining the Equations

Substituting the expression for \( A \) and \( \frac{dT}{dr} \) into Fourier's law gives us:

Q = -k (4 \pi r^2) \left(\frac{t_2 - t_1}{r_2 - r_1}\right)

Since \( t_2 > t_1 \), the negative sign indicates the direction of heat flow from the higher temperature to the lower temperature. Therefore, we can express the rate of heat transfer as:

Q = \frac{4 \pi k (t_2 - t_1) r^2}{r_2 - r_1}

Final Considerations

To find the total heat transfer rate through the entire shell, we can integrate this expression from \( r_1 \) to \( r_2 \). However, in many practical applications, we often consider the average radius \( r = \frac{r_1 + r_2}{2} \) for simplification.

In summary, the rate of flow of heat through a spherical shell is influenced by the thermal conductivity of the material, the temperature difference between the inner and outer walls, and the geometry of the shell itself. This understanding is crucial in fields such as thermal engineering and materials science, where managing heat transfer is essential for efficiency and safety.