Question

The heat conduction equation in a medium is given in its simplest form as $$ \frac{1}{r} \frac{d}{d r}\left(r k \frac{d T}{d r}\right)+\dot{e}_{\text {gen }}=0 $$ 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

(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. Answer: There is no incorrect statement among the provided options. All the statements (a), (b), (c), (d), and (e) are correct based on the given heat conduction equation.

Step-by-step solution

Analyze the given heat conduction equation

The given heat conduction equation is: $$ \frac{1}{r} \frac{d}{d r}\left(r k \frac{d T}{d r}\right)+\dot{e}_{\text {gen }}=0 $$ In this equation: - r is the radial distance from the center of the medium - k is the thermal conductivity of the medium - T is the temperature - \(\dot{e}_{\text{gen}}\) is the heat generation rate per unit volume

Checking statement (a)

(a) The medium is of cylindrical shape. Since the equation uses radial distance 'r' and has a term \(\frac{1}{r}\), it implies that the medium is of cylindrical shape. So this statement is correct.

Checking statement (b)

(b) The thermal conductivity of the medium is constant. We see that the thermal conductivity 'k' appears as a constant in the equation. It does not depend on the radial distance or temperature. Therefore, this statement is also correct.

Checking statement (c)

(c) Heat transfer through the medium is steady. There is no time-dependent term in the given heat conduction equation, which implies that the heat transfer through the medium is steady. Thus, this statement is correct.

Checking statement (d)

(d) There is heat generation within the medium. The term \(\dot{e}_{\text {gen }}\) represents the heat generation rate per unit volume within the medium. Since it appears in the equation, this means that there is heat generation within the medium. This statement is also correct.

Checking statement (e)

(e) Heat conduction through the medium is one-dimensional. The given heat conduction equation only contains a radial derivative, \(\frac{dT}{dr}\), which suggests that the heat conduction through the medium occurs only in the radial direction and is one-dimensional. So, this statement is correct as well. Since all the statements (a), (b), (c), (d), and (e) are correct, the exercise appears to have a mistake or typo in the given options. There is no wrong statement among the provided options.

Key concepts

Cylindrical Coordinates

To adequately understand the heat conduction equation provided, we need to talk about cylindrical coordinates. These are quite handy in situations where symmetry around an axis is involved, like pipes or wires. The primary components of cylindrical coordinates are radial distance \( r \), azimuthal angle \( \theta \), and the height \( z \). Here, only \( r \) is of interest.

The radial distance \( r \) is crucial since it represents how far you are moving from the center (or axis). In our heat conduction equation, you'll notice the term \( \frac{1}{r} \), indicating we are indeed discussing a cylindrical shape. Cylindrical coordinates simplify the complexity of equations when dealing with objects having circular cross-sections.

Thermal Conductivity

Thermal conductivity \( k \) represents how well a material can transfer heat. In our specific equation, \( k \) is treated as a constant. This assumption simplifies the math and suggests that the medium uniformly transfers heat, irrespective of position and temperature.

Why is \( k \) important? It is because materials with different thermal conductivities behave very differently under heat transfer scenarios. Metals typically have high \( k \) values, making them good heat conductors. On the other hand, insulators like rubber have low values. In practical design, knowing \( k \) helps ensure that the material chosen will meet the desired thermal performance.

Steady State Heat Transfer

Steady state means that temperatures within the system do not change with time. This can be visualized as a stable scenario where even though heat moves through the medium, the distribution remains constant over time.

Our equation doesn’t include any time-dependent terms. This clearly indicates that the problem assumes a steady state condition. Think of a metal bar being heated at one end continuously. After some time, the temperature gradient along its length doesn’t change, irrespective of the time that has passed.

Heat Generation

Heat generation within a medium occurs when energy is added throughout the material, not just from external sources. This can be due to chemical reactions, electric currents, or nuclear reactions inside the medium.

In our scenario, the term \( \dot{e}_{\text{gen}} \) tells us about the heat generation rate per unit volume. It implies that heat originates not only from the boundaries but also within the medium itself. Understanding this concept is vital, especially in contexts like battery design, where internal heat generation must be managed to prevent overheating.

One-Dimensional Heat Transfer

When we talk about one-dimensional heat transfer, it means that the heat flow is concentrated along a single axis or direction. In our given equation, the heat transfer is one-dimensional along the radial direction \( r \).

This is simplified by considering only the radial derivative \( \frac{dT}{dr} \). The temperature varies with radius but not with angular or axial directions. Having a one-dimensional perspective helps in reducing the complexity of equations, making it easier to solve real-world problems where dimension reduction is applicable.