Question

Consider heat conduction through a wall of thickness \(L\) and area \(A\). Under what conditions will the temperature distributions in the wall be a straight line?

Short answer

Answer: The temperature distribution in a wall will be a straight line when the wall is of uniform composition with constant thermal conductivity (k), the heat transfer rate (Q) is constant along the wall's thickness, and the area (A) of the wall remains constant throughout its thickness.

Step-by-step solution

Setting Up Fourier's Law for Heat Conduction

Fourier's law for one-dimensional heat conduction through a solid wall states that the heat transfer rate (Q) is proportional to the temperature gradient and the surface area of the wall. It is given by the formula: \(Q = -kA\frac{dT}{dx}\) Where: - \(Q\) is the heat transfer rate through the wall; - \(k\) is the thermal conductivity of the wall material (constant); - \(A\) is the area of the wall; - \(\frac{dT}{dx}\) is the temperature gradient in the wall (change in temperature per unit length in the x-direction).

Analyzing the Condition for a Linear Temperature Distribution

In order for the temperature distribution to be a straight line, \(\frac{dT}{dx}\) must be constant. Since this quantity represents the temperature gradient, it means that the change in temperature must be the same along the entire thickness of the wall. This would happen when: 1. The thermal conductivity \(k\) is constant throughout the wall, meaning that the wall is of uniform composition; 2. The heat transfer rate \(Q\) remains constant along the length of the wall; 3. The area \(A\) of the wall remains constant along the wall's thickness.

Interpretation of Results

For a straight-line temperature distribution, the following conditions must be met: - The wall must be of uniform composition and constant thermal conductivity \(k\); - The heat transfer rate \(Q\) must be constant along the wall's thickness; - The area \(A\) of the wall must remain constant throughout its thickness. In conclusion, if these conditions are satisfied, the temperature distribution in the wall will be a straight line.