Thick slabs of stainless steel \((k=14.9 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and \(\left.\alpha=3.95 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)\) and copper \((k=401 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and \(\left.\alpha=117 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)\) are subjected to uniform heat flux of \(8 \mathrm{~kW} / \mathrm{m}^{2}\) at the surface. The two slabs have a uniform initial temperature of \(20^{\circ} \mathrm{C}\). Determine the temperatures of both slabs, at \(1 \mathrm{~cm}\) from the surface, after \(60 \mathrm{~s}\) of exposure to the heat flux.

Understanding the heat diffusion equation is crucial in evaluating how heat travels through materials. This equation is a form of the second law of thermodynamics and describes the spread of heat in space over time. In the provided exercise, it's used to calculate the temperature rise in the slabs of stainless steel and copper when exposed to a heat flux.

The general form of the heat diffusion equation is denoted as \( \frac{\partial T}{\partial t} = \alpha abla^2 T \), where \( \alpha \) is the thermal diffusivity of the material and \( T \) represents the temperature. \( abla^2 \) symbolizes the Laplace operator, which in one-dimensional cases can be simplified to the second derivative of temperature with respect to space. The exercise employs a more specific solution to the heat diffusion equation that integrates the effects of uniform heat flux, initial temperature, and thermal properties of the material.

Implementing the accurate solution involves incorporating the uniform initial condition and the specific boundary conditions presented in the exercise, which is the uniform heat flux applied to the surface of the slabs. The equation utilized in the steps integrates these to calculate the temperature at a specific point inside the material at a certain time, showcasing the transient behavior of temperature due to the imposed heat flux.

Thermal diffusivity, represented by \(\alpha\), is a material-specific property that measures the rate at which heat diffuses through a material. It's defined as the ratio of the thermal conductivity \(k\) to the product of density \(\rho\) and specific heat capacity \(c_p\), or \(\alpha = \frac{k}{\rho c_p}\).

In the given exercise, two materials, stainless steel and copper, have different thermal diffusivities, indicating how quickly they can respond to changes in temperature. A higher thermal diffusivity means the material can reach thermal equilibrium faster because it allows heat to diffuse through it more quickly. This property is crucial in the heat diffusion equation when calculating how temperature changes within a material over time.

The step by step solution shows the utilization of the thermal diffusivity values for stainless steel \(3.95 \times 10^{-6} m^2/s\) and copper \(117 \times 10^{-6} m^2/s\) to determine how the heat flux affects their internal temperatures. These values, combined with the heat flux and the heat diffusion equation, enable calculating the temperatures at different points and times.

Uniform heat flux refers to a constant flow of heat energy across a unit area over time. In the context of the exercise, the uniform heat flux is applied to the surfaces of both the stainless steel and copper slabs, causing the temperatures within to change. This condition is given as \(8 kW/m^2\) or \(8000 W/m^2\).

A uniform heat flux is an idealized boundary condition where the heat input does not vary across the surface or with time. Such an assumption simplifies the problem and makes it feasible to predict the temperature profile within the slabs using the heat diffusion equation. The imposition of a uniform heat flux is essential for the step by step solution, as it drives the change in temperature and is part of the initial conditions that shape the final temperature expression.

The solution articulates that the heat flux introduces energy into the system, and over time, this energy propagates through the slabs, affecting their temperature. Such problems are common in engineering applications, involving heat exchange processes, like heating metals in industrial settings and understanding thermal management.