The curved surface of uniform rod is thermally isolated from...

The curved surface of uniform rod is thermally isolated from surrounding. Its ends are maintained at temperature $T_{1}$ and $T_{2}\left(T_{1}>T_{2}\right) .$ If in steady state, temperature gradient at a distance $x$ from hot end is equal to $\frac{d T}{d x}$, then which one of the following graphs is correct? (A) (B) (C) (D)

Key Concepts

Boundary Conditions

Boundary conditions in heat conduction problems define the temperatures or heat fluxes at the boundaries of the system. In the context of a rod with its ends maintained at fixed temperatures, these conditions determine the linear temperature profile and constant gradient along the rod in the steady state.

Thermal Isolation

Thermal isolation refers to the condition where a body is insulated from its surroundings except at specific boundaries. This ensures that heat flow occurs only along the intended path (e.g., along the length of the rod) and that no additional heat is lost or gained along other surfaces, thereby simplifying the analysis to one-dimensional conduction.

Fourier’s Law

Fourier’s Law of heat conduction relates the heat flux to the negative of the temperature gradient multiplied by the material’s thermal conductivity. This principle underpins the analysis of heat transfer in a medium under conduction, suggesting that a uniform medium will have a constant temperature gradient if the heat flux is uniform.

Steady State Heat Conduction

In steady state heat conduction, the temperature at any point in the material does not change with time. This condition implies that the rate of heat input equals the rate of heat output at every point, resulting in a constant heat flux along the medium when the physical properties and cross-sectional area remain constant.

Uniform Material and Geometry

In a uniform rod, where the material properties, cross-sectional area, and geometric dimensions do not vary along the length, the temperature distribution in a steady state between two fixed temperatures is linear. This directly implies that the temperature gradient remains constant along the rod.

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