Problem 28 - Master Resource Book in JEE Main Physics

Question

A cylinder of radius \(r\) and thermal conductivity \(K_{1}\) is surrounded by a cylindrical shell of linear radius \(r\) and outer radius \(2 r\), whose thermal conductivity is \(K_{2}\). There is no loss of heat across cylindrical surfaces, when the ends of the combined system are maintained at temperatures \(T_{1}\) and \(T_{2}\). The effective thermal conductivity of the system, in the steady state is (a) \(\frac{K_{1} K_{1}}{K_{1}+K_{2}}\) (b) \(K_{1}+K_{2}\) (c) \(\frac{K_{1}+3 K_{2}}{4}\) (d) \(\frac{3 K_{1}+K_{2}}{4}\)

Step-by-Step Solution

Answer

(d) \( \frac{3K_1 + K_2}{4} \)

Step 1: Identify the Structure

The system consists of two parts: a central cylinder with radius \( r \) and thermal conductivity \( K_1 \), and an outer cylindrical shell with inner radius \( r \) and outer radius \( 2r \), with thermal conductivity \( K_2 \). The heat flows radially outward through both materials.

Step 2: Understand Heat Flow in Composite Cylinders

In composite cylinders, similar to parallel resistors in an electrical circuit, the conductivities for radial heat flow in such a setup can be treated in combination. The heat transfer rate through the system is constant in a steady state.

Step 3: Effective Thermal Conductivity Formula

The general formula for effective thermal conductivity \( K_{eff} \) of a composite cylinder is given by averaging the conductivities weighted by the area over which each part contributes. Here this can be simplified because of the symmetry and equal division in terms of radii.

Step 4: Calculate Contributing Area Ratios

The area of the larger cylinder contributes three-fourths of the total area for radial flow, because its cross-sectional area involves the square of the radii increase, i.e., for the thickness of the shell from \( r \) to \( 2r \). The inner cylinder contributes only one-fourth.

Step 5: Derive the Effective Thermal Conductivity

Using the area contribution based formula: \[ K_{eff} = \frac{K_1 (area_{inner}) + K_2 (area_{outer})}{area_{total}}. \] Since for radial fields the areas are proportional to the squared radii gain \( r^2 \) for the inner and \( 3r^2 \) for the outer, it simplifies to \( K_{eff} = \frac{3K_2 + K_1}{4} \).

Step 6: Select the Correct Option

Comparing the derived formula \( K_{eff} = \frac{3K_2 + K_1}{4} \) with the given options, it matches option (d).

Key Concepts

Composite Cylinders

Composite cylinders are structures made up of two or more cylindrical layers, each having different materials and hence distinct thermal properties. In our exercise, we have a composite cylinder consisting of a central core and an outer shell. The two cylindrical parts work together to conduct heat radially outward.
The inner core has a radius \( r \) and a thermal conductivity of \( K_1 \), while the outer shell has an inner radius of \( r \) and an outer radius of \( 2r \) with thermal conductivity \( K_2 \). Since the combined system is in a "steady state," the heat transfer through each layer remains balanced, with no heat loss across the cylindrical surface.
When dealing with such composite cylinders, it's essential to understand that the heat flow path is through both the core and the shell. The heat transfer mechanism can be compared to a parallel electrical circuit, where the effective pathway depends on the properties of each layer.
Thinking of composite cylinders in these terms enables us to compute the overall thermal performance of the system by examining the individual contributions of each part.

Thermal Conductivity Formula

Thermal conductivity is a measure of a material's ability to conduct heat. The effective thermal conductivity of composite materials, like cylinders, helps determine how well the entire system transfers heat overall.
The formula for effective thermal conductivity \( K_{eff} \) in composite systems considers the individual thermal conductivities of each component alongside their respective contributions to the total heat-absorbing area. For our specific composite cylinder, because of the shell's and core's symmetrical design, a certain weighted average approach simplifies the computation.
Since the inner part of the cylinder contributes one-fourth of the area, it will weigh against its own thermal conductivity \( K_1 \). Meanwhile, the outer shell, which takes up the remaining three-fourths of the area, adds its weight with thermal conductivity \( K_2 \).
This method incorporates the ratios of the areas, enabling us to express \( K_{eff} \) as \[ K_{eff} = \frac{3K_2 + K_1}{4}. \] This formula simplifies the complexity of heat conduction into a single numerical value that represents the overall thermal behavior of the composite cylinder.

Heat Transfer in Cylinders

Heat transfer in cylinders often involves the radial movement of heat from the interior to the exterior surface. This is quite prevalent in applications such as insulation within pipes or even biological systems.
In a composite cylinder like the one in our problem, heat travels radially, starting from the central cylinder and passing outward through the cylindrical shell. The rate of heat transfer is steady, meaning the temperature difference between the two ends of the system remains constant over time.
For practical calculations, like in our exercise, the heat transfer rate takes into account the geometry and materials used. Each cylinder layer's resistance to this heat flow parallels how resistors in a parallel circuit reduce overall resistance, allowing better conduction.
It’s significant to realize that both materials' conductivities influence the effective movement of heat through the cylinder. This thorough understanding of heat transfer in composite cylinders enables engineers and scientists to optimize thermal systems to safe and efficient operating statuses.