The thermal conductivity of a material of a cylinder of...

The thermal conductivity of a material of a cylinder of radius $R$ is $K_{1}$, which is surrounded by a cylindrical shell of inner radius $R$ and outer radius $2 R$ made of the material of thermal conductivity $K_{2}$. The two ends of the combined system are maintained at two different temperatures. Assume, that there is no loss of heat across the cylindrical surface and the system is in steady state. The effective thermal conductivity of the system is: (a) $K_{1}+K_{2}$ (b) $\frac{K_{1} K_{2}}{\left(K_{1}+K_{2}\right)}$ (c) $\frac{\left(K_{1}+3 K_{2}\right)}{4}$ (d) $\frac{\left(3 K_{1}+K_{2}\right)}{4}$

The thermal conductivity of a material of a cylinder of radius $R$ is $K_{1}$, which is surrounded by a cylindrical shell of inner radius $R$ and outer radius $2 R$ made of the material of thermal conductivity $K_{2}$. The two ends of the combined system are maintained at two different temperatures. Assume, that there is no loss of heat across the cylindrical surface and the system is in steady state. The effective thermal conductivity of the system is:
(a) $K_{1}+K_{2}$
(b) $\frac{K_{1} K_{2}}{\left(K_{1}+K_{2}\right)}$
(c) $\frac{\left(K_{1}+3 K_{2}\right)}{4}$
(d) $\frac{\left(3 K_{1}+K_{2}\right)}{4}$

Key Concepts

Fourier’s Law of Heat Conduction

Fourier’s law states that the rate of heat transfer through a material is proportional to the negative temperature gradient and the material’s thermal conductivity. This principle forms the basis for calculating heat flow in solids and sets up the relation between material properties, geometry, and temperature difference in thermal conduction problems.

Thermal Resistance

The concept of thermal resistance is used to quantify the opposition that a material offers to heat flow. It is analogous to electrical resistance in circuits and allows for the analysis of heat transfer by treating different materials or layers in a system as resistors, which can be arranged in series or parallel to compute the overall heat transfer characteristics.

Parallel Heat Transfer Paths

When heat flows through multiple paths simultaneously, as in the case of composite systems with different materials arranged side-by-side, the thermal conductances of these paths add. This approach is key in analyzing effective thermal conductivity where components contribute in parallel to the net heat flux.

Effective Thermal Conductivity in Composite Systems

Effective thermal conductivity is a measure that allows a composite system made of different materials to be represented by an equivalent homogeneous material. This concept involves combining the individual thermal conductivities and their associated geometric factors (such as cross-sectional areas) according to how the materials are arranged—whether in series or in parallel—to find a single value that describes the overall heat conduction capability.

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