Question

The heat conduction equation in a medium is given in its simplest form as $$ \frac{1}{r} \frac{d}{d r}\left(r k \frac{d T}{d r}\right)+\hat{e}_{\mathrm{gen}}=0 $$ Select the wrong statement below. (a) The medium is of cylindrical shape. (b) The thermal conductivity of the medium is constant. (c) Heat transfer through the medium is steady. (d) There is heat generation within the medium. (e) Heat conduction through the medium is one-dimensional.

Short answer

Question: Identify the incorrect statement in the given heat conduction equation in a medium: $$\frac{1}{r} \frac{d}{d r}\left(r k \frac{d T}{d r}\right)+\hat{e}_{\mathrm{gen}}=0$$ Answer: There is a mistake in the problem statement as all the given statements (a) to (e) are consistent with the heat conduction equation.

Step-by-step solution

(Understanding the heat conduction equation)

First, let's analyze the given heat conduction equation and understand each of the terms in it: $$ \frac{1}{r} \frac{d}{d r}\left(r k \frac{d T}{d r}\right)+\hat{e}_{\mathrm{gen}}=0 $$ In this equation, - \(r\) is the radial distance from the axis of the cylinder - \(k\) is the thermal conductivity - \(T\) is the temperature - \(\frac{dT}{dr}\) is the temperature gradient with respect to the radial distance \(r\) - \(\hat{e}_{\mathrm{gen}}\) is the heat generation rate per unit volume Now, let's examine each statement and determine if they are consistent with the given equation:

(a) The medium is of cylindrical shape.

Since the equation contains terms related to the radial distance \(r\), it suggests that the heat conduction is happening in a medium with a cylindrical shape.

(b) The thermal conductivity of the medium is constant.

The thermal conductivity \(k\) is not a function of \(r\) in the given equation, which implies that the thermal conductivity of the medium is constant.

(c) Heat transfer through the medium is steady.

The given equation does not have any terms related to time, which means that heat transfer through the medium is steady.

(d) There is heat generation within the medium.

The term \(\hat{e}_{\mathrm{gen}}\) in the equation represents heat generation rate per unit volume. Since the term is not equal to zero, there is heat generation within the medium.

(e) Heat conduction through the medium is one-dimensional.

The given heat conduction equation only has terms related to the radial distance r (with no terms representing the heat conduction in other directions). This means that heat conduction through the medium is one-dimensional. Based on our analysis, all the statements (a) to (e) are consistent with the heat conduction equation, implying they are true. However, the problem is to identify the wrong statement, which means there is a mistake in the problem statement provided.