In a composite rod, when two rods of different lengths and...

In a composite rod, when two rods of different lengths and of the same area of cross-section are joined end to end, then if $K$ is the effective coefficient of thermal conductivity, $\frac{l_{1}+l_{2}}{K}$ is equal to (A) $\frac{l_{1}}{K_{1}}-\frac{l_{2}}{K_{2}}$ (B) $\frac{l_{1}}{K_{2}}-\frac{l_{2}}{K_{1}}$ (C) $\frac{l_{1}}{K_{1}}+\frac{l_{2}}{K_{2}}$ (D) $\frac{l_{1}}{K_{2}}+\frac{l_{2}}{K_{1}}$

In a composite rod, when two rods of different lengths and of the same area of cross-section are joined end to end, then if $K$ is the effective coefficient of thermal conductivity, $\frac{l_{1}+l_{2}}{K}$ is equal to
(A) $\frac{l_{1}}{K_{1}}-\frac{l_{2}}{K_{2}}$
(B) $\frac{l_{1}}{K_{2}}-\frac{l_{2}}{K_{1}}$
(C) $\frac{l_{1}}{K_{1}}+\frac{l_{2}}{K_{2}}$
(D) $\frac{l_{1}}{K_{2}}+\frac{l_{2}}{K_{1}}$

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Key Concepts

Thermal Conductivity

Thermal conductivity is a material property that measures the ability of a material to conduct heat. It is defined as the rate at which heat passes through a material of a given thickness under a constant temperature gradient.

Composite Materials in Heat Transfer

When different materials are joined together, the overall heat conduction behavior of the composite is determined by the properties and geometries of the individual components. In a composite rod made by joining two rods end-to-end, each segment contributes to the overall thermal conduction process based on its own thermal conductivity and length.

Thermal Resistance and Series Addition

Thermal resistance represents the opposition to heat flow across a material and is analogous to electrical resistance in circuits. In systems where components are arranged in series, such as a composite rod, the total thermal resistance is the sum of the individual resistances. This principle is used to derive the effective conductivity of the composite system, leading to the relation where the sum of the individual resistances is equated to (l1/K1 + l2/K2) for the overall effective conductivity.

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