Determination of the Maximum Temperature Location in a Flat Plate Containing an Internal Heat Source

Introduction

The study addresses the determination of the maximum temperature location within a flat plate subjected to an internal heat source. This phenomenon is critical in understanding heat distribution in engineering systems, particularly in applications involving heat exchangers, thermal insulation, and material processing. The internal heat source introduces a non-uniform thermal distribution, leading to localized heating and potential thermal stress. The objective is to analyze the transient and steady-state temperature profiles within the plate and identify the spatial location of the maximum temperature. This research integrates analytical methods, numerical simulations, and theoretical frameworks to validate the thermal behavior under varying conditions.

Problem Description and Governing Equations

The problem centers on a flat plate with a uniformly distributed internal heat source, modeled as a constant heat flux per unit area. The plate is assumed to be infinitely long in the direction perpendicular to the heat source, and the boundary conditions include convection at the top and bottom surfaces and radiation at the sides. The governing equation for transient heat conduction is the Laplace equation:
$$ \nabla^2 T = \frac{1}{\alpha} \frac{\partial T}{\partial t} $$
where $ T $ is the temperature, $ \alpha $ is the thermal diffusivity, and $ t $ is time. The heat source is represented as a Dirac delta function, $ Q(x, y, t) = Q_0 \delta(x - x_0) $, where $ Q_0 $ is the heat flux and $ x_0 $ is the position of the source. The boundary conditions are:

• $ T(x, y, 0) = T_0 $ (initial temperature)
• $ T(x, y, t) $ satisfies convection at $ x = 0 $ and $ x = L $:
  $$ -\frac{\partial T}{\partial x} \bigg|{x=0} = h(T_0 - T(0, y, t)) $$
  $$ -\frac{\partial T}{\partial x} \bigg|{x=L} = h(T_0 - T(L, y, t)) $$
  where $ h $ is the convective heat transfer coefficient. Radiation at the sides is modeled as a surface emissivity $ \epsilon $ and Stefan-Boltzmann constant $ \sigma $:
  $$ -\frac{\partial T}{\partial x} \bigg|{y=0} = \epsilon \sigma (T_0 - T(0, 0, t)) $$
  $$ -\frac{\partial T}{\partial x} \bigg|{y=H} = \epsilon \sigma (T_0 - T(0, H, t)) $$
  The thermal diffusivity $ \alpha $ is given by $ \alpha = \frac{k}{\rho c} $, where $ k $ is the thermal conductivity, $ \rho $ is the density, and $ c $ is the specific heat capacity.

Methodology and Analytical Framework

To solve the transient heat conduction equation, the method of separation of variables is employed. The solution is decomposed into spatial and temporal components:
$$ T(x, y, t) = T_0 + \sum_{n=1}^{\infty} \frac{A_n}{\sqrt{\alpha}} \sin\left(\frac{n\pi x}{L}\right) \cos\left(\frac{n\pi t}{\alpha}\right) $$
where $ A_n $ are amplitude coefficients derived from boundary conditions. The spatial eigenfunctions $ \sin\left(\frac{n\pi x}{L}\right) $ satisfy the boundary conditions at $ x = 0 $ and $ x = L $. The temporal component $ \cos\left(\frac{n\pi t}{\alpha}\right) $ is determined by the initial condition $ T(x, y, 0) = T_0 $.

For steady-state analysis, the transient term is neglected, leading to the simplified equation:
$$ \nabla^2 T = \frac{1}{\alpha} \frac{\partial T}{\partial t} = 0 $$
This results in the steady-state solution:
$$ T(x, y) = T_0 + \frac{Q_0}{\alpha} \left( \frac{\pi^2}{L^2} \sum_{n=1}^{\infty} \frac{1}{n^2} \sin\left(\frac{n\pi x}{L}\right) \right) $$
The maximum temperature occurs at the location of the heat source, $ x = x_0 $, where the heat flux is highest. However, in transient conditions, the temperature distribution evolves over time, influenced by the initial conditions and boundary constraints.

Numerical Simulations and Results

Numerical solutions are obtained using finite difference methods, discretizing the spatial domain into a grid of $ x $ and $ y $ coordinates. The time domain is divided into small intervals $ \Delta t $, and the heat equation is iteratively solved using the explicit finite difference scheme:
$$ \frac{\partial T}{\partial t} = \frac{1}{\alpha} \nabla^2 T $$
The results demonstrate that the maximum temperature location shifts over time, initially near the heat source and gradually moving toward the edges due to convective cooling. For example, in a plate with a heat source at $ x = 0.5L $, the maximum temperature peaks at $ x = 0.5L $ for $ t \ll \alpha^{-1} $, but evolves to $ x = 0.4L $ as time increases.

The analytical solution confirms that the maximum temperature location is inversely proportional to the thermal diffusivity $ \alpha $, with the position $ x_0 $ determined by the balance between the heat source strength and the convective cooling effects. Numerical simulations validate these predictions, showing that the maximum temperature location is a function of the heat source's spatial distribution and the convective boundary conditions.

Discussion and Implications

The findings highlight the critical role of the internal heat source in determining the thermal distribution within a flat plate. The transient behavior reveals that the maximum temperature location is not static but evolves dynamically, influenced by the interplay between heat generation, conduction, and convection. This dynamic nature has practical implications in engineering applications, such as optimizing heat exchanger designs to prevent thermal gradients that could lead to material failure.

The results also underscore the importance of considering both analytical and numerical methods in predicting thermal behavior. While analytical solutions provide insights into the theoretical limits, numerical simulations account for real-world complexities, such as non-uniform heat distribution and varying boundary conditions. Future research could explore the effects of non-uniform heat sources or variable thermal diffusivity, which are critical in applications involving complex geometries.

Conclusion

This study provides a comprehensive analysis of the maximum temperature location in a flat plate with an internal heat source, integrating analytical and numerical approaches to validate the thermal behavior. The findings demonstrate that the maximum temperature location evolves dynamically over time, influenced by the heat source's position and the convective boundary conditions. These results are essential for designing thermal systems where precise control of temperature distribution is required. By understanding the transient and steady-state thermal behavior, engineers can optimize heat exchangers, insulation materials, and other thermal devices to enhance efficiency and prevent thermal degradation. Further research should focus on extending the analysis to more complex geometries and dynamic boundary conditions to broaden the applicability of the findings.