(i) A slab of thickness L is initially at a uniform...

(i) A slab of thickness $L$ is initially at a uniform temperature $T_{0}$. The face at $x=0$ is perfectly insulated. At time $t=0$, a constant heat flux $q_{s}$ is imposed on the face at $x=L$. Show that the temperature response is $$ \frac{T-T_{0}}{q_{s} L / k}=\frac{3 x^{2}-L^{2}}{6 L^{2}}+\frac{\alpha t}{L^{2}}-\frac{2}{\pi^{2}} \sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{2}} e^{-\left(\alpha n^{2} \pi^{2} / L^{2}\right) t} \cos \frac{n \pi x}{L} $$ (ii) If a negative heat flux equal to $-q_{s}$ is subsequently imposed at time $t=t_{1}$, show that the temperature response for $t>t_{1}$ is $$ \frac{T-T_{0}}{q_{s} L / k}=\frac{\alpha t_{1}}{L^{2}}-\frac{2}{\pi^{2}} \sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{2}} e^{-\left(\alpha n^{2} \pi^{2} / L^{2}\right) t}\left[1-e^{\left(\alpha n^{2} \pi^{2} / L^{2}\right) r_{1}}\right] \cos \frac{n \pi x}{L} $$ (Hint: Use the principle of superposition for a linear differential equation.)

(i) A slab of thickness $L$ is initially at a uniform temperature $T_{0}$. The face at $x=0$ is perfectly insulated. At time $t=0$, a constant heat flux $q_{s}$ is imposed on the face at $x=L$. Show that the temperature response is
$$
\frac{T-T_{0}}{q_{s} L / k}=\frac{3 x^{2}-L^{2}}{6 L^{2}}+\frac{\alpha t}{L^{2}}-\frac{2}{\pi^{2}} \sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{2}} e^{-\left(\alpha n^{2} \pi^{2} / L^{2}\right) t} \cos \frac{n \pi x}{L}
$$
(ii) If a negative heat flux equal to $-q_{s}$ is subsequently imposed at time $t=t_{1}$, show that the temperature response for $t>t_{1}$ is
$$
\frac{T-T_{0}}{q_{s} L / k}=\frac{\alpha t_{1}}{L^{2}}-\frac{2}{\pi^{2}} \sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{2}} e^{-\left(\alpha n^{2} \pi^{2} / L^{2}\right) t}\left[1-e^{\left(\alpha n^{2} \pi^{2} / L^{2}\right) r_{1}}\right] \cos \frac{n \pi x}{L}
$$
(Hint: Use the principle of superposition for a linear differential equation.)

Key Concepts

Superposition Principle

This principle applies to linear differential equations and states that the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. In the problem at hand, this principle is used to combine the effects of a positive heat flux followed by a negative heat flux, thereby obtaining the complete temperature response at later times.

Separation of Variables

A mathematical technique used to reduce partial differential equations into simpler, separable ordinary differential equations by assuming that the solution can be written as the product of functions, each depending on only one of the variables. In the context of heat conduction, this method facilitates obtaining the eigenfunctions and eigenvalues used to construct the Fourier series solution.

Fourier Series Solution

This method involves expressing the temperature distribution as an infinite sum of cosine (or sine) functions that satisfy the spatial part of the differential equation while also conforming to the boundary conditions. The Fourier series approach allows for the solution of the transient heat conduction problem by decomposing it into a series of eigenfunctions, each evolving in time with an associated exponential decay factor.

Heat Conduction Equation

This concept involves the partial differential equation that governs the distribution of temperature in a medium over time due to conduction. In one dimension, the transient heat conduction equation describes how temperature changes within the slab as a function of both spatial position and time, taking into account the material’s thermal diffusivity. It is fundamental in solving problems involving time-dependent temperature variations in a medium.

Boundary Conditions

This refers to the specification of temperature or heat flux values at the boundaries of the domain. In this context, one face of the slab is insulated (implying zero heat flux) while the other face has a prescribed constant heat flux. Accurately incorporating these boundary conditions is essential for obtaining a valid solution to the heat conduction equation using mathematical techniques such as separation of variables and Fourier series.

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