Final Answer:
The steady-state temperature distribution in a one-dimensional wall of thermal conductivity is governed by Fourier's Law of Heat Conduction, expressed as q = -kA(dT/dx), where q is the heat flux, k is the thermal conductivity, A is the cross-sectional area, and dT/dx is the temperature gradient. This equation signifies that the heat flux is proportional to the negative temperature gradient, indicating heat transfer from higher to lower temperatures.
Step-by-step explanation:
Fourier's Law of Heat Conduction is fundamental in understanding the steady-state temperature distribution in a one-dimensional wall. It states that the heat flux (q) is proportional to the negative temperature gradient (dT/dx) across the material. Mathematically, this is represented as q = -kA(dT/dx), where k is the thermal conductivity of the material, A is the cross-sectional area perpendicular to the heat flow, and dT/dx is the temperature gradient along the length of the wall. The negative sign indicates that heat flows in the direction of decreasing temperature.
To elaborate, consider a one-dimensional wall with a higher temperature on one side and a lower temperature on the other. The heat flux through the material is determined by the thermal conductivity, the area through which heat is transferred, and the temperature gradient. The temperature gradient represents the rate of change of temperature with respect to distance. The resulting heat flow establishes a steady-state temperature distribution, where the temperature remains constant over time. This mathematical relationship is crucial in analyzing and designing structures to optimize thermal performance in various applications.
In engineering and physics, understanding the steady-state temperature distribution is essential for efficient heat management, whether it's in building materials, electronic devices, or industrial processes. The application of Fourier's Law provides a quantitative basis for predicting and controlling heat transfer in one-dimensional systems, contributing to the advancement of thermal engineering and design.