Project ForgeConsider a laterally insulated rod of length L and constant thermal conductivity. Assuming one-dimensional heat conduction in the rod, which of the following steady-state temperature profile(s) can occur without internal heat generation? (A) linear, increasing left to right. (B) linear, decreasing left to right. (C) a smooth concave-down hump, zero slope at both ends. (D) an oscillating profile with an interior local minimum and maximum.
- Linear profile, increasing from left to right
- Linear profile, decreasing from left to right
- CSmooth concave-down hump (zero at both ends, single interior maximum)
- DOscillating profile (interior local minimum and maximum)
Explanation
With no internal heat generation, steady 1-D conduction requires d²T/dx²=0, whose general solution is the LINEAR profile T=ax+b — this can be increasing (A) or decreasing (B) depending on boundary temperatures, but any curved (nonzero second-derivative) profile such as the concave hump (C) or the oscillating profile (D) would require a non-zero heat-generation term to satisfy the governing equation.
Source: GATE ME 2023 Official Paper (2023)
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