(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 back face at $x=0$ is perfectly insulated. At time $t=0$, a laser deposits an amount of energy $E$ per unit area on the face at $x=L$. Determine the temperature response. (Hint: Let $\theta(x, t)$ be the temperature response for a unit uniform surface heat flux obtained by setting $q_{s}=1$ in the result of Exercise 3-86, part (i). Then, following part (ii) of Exercise 3-86, show that the required temperature response is $$ T-T_{0}=E \frac{\partial \theta(x, t)}{\partial t} $$ by imposing a heat flux $q_{s}$ for time $\Delta t$ and then letting $\Delta t \rightarrow 0$ such that $q_{s} \Delta t=E$.) (ii) Hence show that the time for the back-face temperature rise to equal half of its maximum rise is $t_{1 / 2}=1.39 L^{2} / \pi^{2} \alpha$.

(i) A slab of thickness $L$ is initially at a uniform temperature $T_{0}$. The back face at $x=0$ is perfectly insulated. At time $t=0$, a laser deposits an amount of energy $E$ per unit area on the face at $x=L$. Determine the temperature response. (Hint: Let $\theta(x, t)$ be the temperature response for a unit uniform surface heat flux obtained by setting $q_{s}=1$ in the result of Exercise 3-86, part (i). Then, following part (ii) of Exercise 3-86, show that the required temperature response is
$$
T-T_{0}=E \frac{\partial \theta(x, t)}{\partial t}
$$
by imposing a heat flux $q_{s}$ for time $\Delta t$ and then letting $\Delta t \rightarrow 0$ such that $q_{s} \Delta t=E$.)
(ii) Hence show that the time for the back-face temperature rise to equal half of its maximum rise is $t_{1 / 2}=1.39 L^{2} / \pi^{2} \alpha$.

Key Concepts

Transient Heat Conduction

This concept involves the study of how temperature in a medium evolves over time due to internal conduction. It is governed by the heat equation, a partial differential equation that describes how heat diffuses through a material. In problems like the slab with a sudden energy input, transient conduction analysis helps predict the temperature distribution as it evolves from the initial state to the steady state over time.

Insulated Boundary Condition

An insulated boundary implies that no heat flows across that surface, which mathematically corresponds to a zero flux (Neumann condition). This boundary condition is crucial in solving heat conduction problems because it modifies the eigenfunctions and influences the overall temperature distribution in the domain. It is typically used to represent physical situations where one face of a body is perfectly insulated.

Impulse Response (Green's Function)

The impulse response in the context of heat conduction refers to the temperature response of a system due to a unit instantaneous energy deposition or uniform surface flux. Represented as a Green's function, it provides a fundamental solution that can be used to construct the response to more general time-dependent heat inputs through convolution. This approach is particularly useful when dealing with arbitrary or short-time heating inputs.

Convolution and Duhamel's Principle

Duhamel’s principle is a method for solving non-homogeneous linear problems by expressing the solution as a superposition (or convolution) of impulse responses. In heat conduction, if the system's response to a unit impulse is known, the overall temperature response to an arbitrary heat input can be obtained by convolving the impulse response with the time history of the applied heat flux. This technique is essential for handling problems with time-dependent boundary conditions or inputs.

Eigenfunction Expansion Method

The eigenfunction expansion method is a mathematical technique used to solve partial differential equations like the heat equation. It involves expressing the solution as a series of eigenfunctions that satisfy the spatial part of the problem, along with corresponding time-dependent coefficients. This method is especially useful in problems with simple geometries and well-defined boundary conditions, as it yields an analytical solution that accurately characterizes the transient temperature distribution.

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