Question

One-dimensional generalized heat conduction equation representing temperature distribution in a sphere, based on thermal conductivity k, specific heat capacity \(C_p\), density \(\rho\), and energy generation E, can be written as \(\dfrac{1}{r^n}\dfrac{\partial}{\partial r}\left(r^n k\dfrac{\partial T}{\partial r}\right) + E = \rho C_p\dfrac{\partial T}{\partial t}\), where the value of n is

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The Correct Option is B

The given problem involves the one-dimensional generalized heat conduction equation in a spherical coordinate system. The equation is:

\(\frac{1}{r^n}\frac{\partial}{\partial r}\left(r^n k\frac{\partial T}{\partial r}\right) + E = \rho C_p\frac{\partial T}{\partial t}\)

To determine the value of \(n\), we need to consider the physical nature of the problem. The equation given is a form of the heat conduction equation in spherical coordinates. Let's analyze the standard form of the heat conduction equation for a sphere:

For spherical coordinates, the general form of the heat conduction equation without the energy generation term (\(E = 0\)) is:

\(\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 k\frac{\partial T}{\partial r}\right) = \rho C_p\frac{\partial T}{\partial t}\)

Here, the coefficient of the term \(\frac{\partial}{\partial r}(r^2 k \frac{\partial T}{\partial r})\) includes \(r^2\). Thus, \(n = 2\) in spherical coordinates. This is because the area across which the heat is conducted is a function of \(r^2\) due to the spherical symmetry.

Conclusion: By comparing the standard form of the equation in spherical coordinates with the given equation, we conclude that the value of \(n\) should be 2 to represent the sphere.

Therefore, the correct answer is 2.