A rod of negligible heat capacity has length 20 cm, area of cross section 1 ·0 cm^2 and therma

step 1 We have given a road of negligible heat capacity and it is told that the temperature of one end

A rod of negligible heat capacity has length 20 cm, area of...

A rod of negligible heat capacity has length $20 \mathrm{~cm}$, area of cross section $1 \cdot 0 \mathrm{~cm}^{2}$ and thermal conductivity $200 \mathrm{~W} \mathrm{~m}^{-1_{0}} \mathrm{C}^{-1}$. The temperature of one end is maintained at $0^{\circ} \mathrm{C}$ and that of the other end is slowly and linearly varied from $0^{\circ} \mathrm{C}$ to $60^{\circ} \mathrm{C}$ in 10 minutes. Assuming no loss of heat through the sides, find the total heat transmitted through the rod in these 10 minutes.

Key Concepts

Time-Dependent Temperature Gradient and Integration

When the temperature difference across a material changes over time, the instantaneous heat flux, calculated using Fourier's law, also varies. To determine the total heat transferred over a period, one must integrate the time-dependent heat flux over the given interval. This approach combines the concepts of a variable temperature boundary condition with the fundamental conduction equation.

Steady-State and Quasi-Steady Assumptions

In problems where a system has negligible heat capacity, it is often assumed that the temperature profile establishes almost immediately, resulting in a linear gradient between the boundaries. This quasi?steady or steady-state condition simplifies the analysis, allowing the use of Fourier's law without solving the full transient heat equation.

Fourier's Law of Heat Conduction

This law states that the rate of heat transfer through a material is proportional to the negative of the temperature gradient and the area perpendicular to that gradient. It is fundamental for analyzing how heat conducts through solids, especially when assuming a linear temperature gradient in a system with steady-state conditions.

Thermal Conductivity

Thermal conductivity is a material property that quantifies its ability to conduct heat. It appears in Fourier's law as a proportionality constant, linking the temperature gradient and the resultant heat flux, and plays a critical role in calculating the amount of heat transmitted in conduction problems.

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