Question

Consider a plane wall of thickness \(L\) whose thermal conductivity varies in a specified temperature range as \(k(T)=k_{0}\left(1+\beta T^{2}\right)\) where \(k_{0}\) and \(\beta\) are two specified constants.

Short answer

Answer: The importance of considering temperature-dependent thermal conductivity in a plane wall is that it accounts for the changes in a material's ability to conduct heat as its temperature changes. This leads to a more accurate representation of heat transfer within the wall, considering the actual behavior of the material. Temperature-dependent thermal conductivity can affect the temperature distribution and heat transfer rate, which is crucial for designing and analyzing materials used in various applications, such as insulation or heat exchangers.

Step-by-step solution

Identify the given information and constants

We are given the following information: - The thickness of the plane wall, \(L\). - The thermal conductivity formula: \(k(T) = k_{0}(1 + \beta T^{2})\). - The constants \(k_{0}\) and \(\beta\).

Understand the thermal conductivity formula

The thermal conductivity formula, \(k(T) = k_{0}(1 + \beta T^{2})\), helps us determine the temperature-dependent thermal conductivity of the material in the plane wall. As the temperature increases, the material's thermal conductivity changes to follow this formula. This relationship may be relevant for solving heat transfer problems involving temperature-dependent materials.

Recall the general heat conduction equation

In a steady-state heat conduction problem, the heat transfer rate through the wall can be found using Fourier's law: \(Q = -k(T)A\frac{dT}{dx}\) Where \(Q\) is the heat transfer rate, \(A\) is the surface area, \(k(T)\) is the temperature-dependent thermal conductivity, and \(\frac{dT}{dx}\) is the temperature gradient in the x-direction. Due to the temperature-dependent thermal conductivity, the heat conduction equation becomes more complex, and solving for the temperature distribution may involve using differential equations.

Determine the temperature distribution

To determine the temperature distribution within the wall, we can consider the heat conduction equation along with the given thermal conductivity formula. Since the problem does not provide more specific information like boundary conditions or the heat generation rate within the material, we cannot proceed further in finding the temperature distribution or the heat transfer rate at this stage. If more information was given, we would have solved the problem by using an appropriate mathematical method, such as the separation of variables, finite difference, or finite element analysis, to determine the temperature distribution and heat transfer rate through the wall.

Summary

In the given problem, the plane wall of thickness \(L\) has a temperature-dependent thermal conductivity, defined by \(k(T) = k_{0}(1 + \beta T^{2})\), with constants \(k_{0}\) and \(\beta\) given. The goal of solving problems involving temperature-dependent materials usually includes finding the temperature distribution and heat transfer rate. However, without more information like boundary conditions or heat generation rate, we cannot provide a more specific solution to this exercise.