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Thermal diffusivity is a significant property when studying heat transfer, particularly in solids. This property essentially measures how quickly heat diffuses through a material. It is denoted by the symbol \( \alpha \) and is defined by the equation: \[\alpha = \frac{k}{\rho c_p}\]where:

• \( k \) is the thermal conductivity
• \( \rho \) represents the density of the material
• \( c_p \) is the specific heat capacity

The unit of thermal diffusivity is square meters per second \( \mathrm{m}^2/\mathrm{s} \), and it provides insight into how fast a material can adjust its temperature to that of its surroundings when a temperature gradient is present. In the given exercise, a high thermal diffusivity value indicates that the plate can quickly equilibrate temperature changes at its surface through its depth. Understanding thermal diffusivity is crucial for materials application where heat transfer efficiency is essential, such as in reactors or heat exchangers.

The Biot number is a dimensionless parameter that compares the rate of internal thermal resistance within a body to the rate of heat transfer across the boundary layer surrounding the body. Mathematically, the Biot number (\(\text{Bi}\)) is expressed as:\[\text{Bi} = \frac{hL_c}{k}\]where:

• \( h \) is the heat transfer coefficient
• \( L_c \) is the characteristic length, which often is the thickness of the body
• \( k \) is the thermal conductivity

A Biot number much less than 1 indicates that surface resistance is dominant, implying that the temperature distribution within the body is uniform compared to the boundary layer. When the Biot number is significant, it suggests that there is an appreciable temperature gradient within the solid itself. In our example, with a Biot number of 0.075, the internal temperature distribution remains relatively uniform, and the body behaves predictably under cooling conditions.

The Fourier number is another pivotal dimensionless parameter when it comes to unsteady heat transfer problems. It is defined as:\[\text{Fo} = \frac{\alpha t}{L^2}\]This number reflects the ratio of heat conduction rate to the rate of thermal energy storage. It helps in understanding how heat is distributed inside the body over time. A higher Fourier number signifies that the heat spreads more rapidly throughout the material, leading to faster temperature equalization. For transient conduction analysis like in the finite difference method, knowing the Fourier number is crucial for ensuring stability and accuracy. In this specific exercise, with a computed Fourier number of 0.448, the focus is on how swiftly the temperature change within the substrate gets accounted during the prescribed time period.

The heat transfer coefficient, denoted as \( h \), is a measure of a surface's ability to transfer heat between itself and a fluid in contact with it. It is primarily used in convective heat transfer problems. The unit for the heat transfer coefficient is \( \text{W/m}^2\text{K} \), and it plays a critical role in defining how effectively a material can lose or gain heat. In contexts like the sample exercise, the heat transfer coefficient is essential in determining the rate at which the plate cools when exposed to the coolant at a different temperature. Factors influencing the heat transfer coefficient include fluid velocity, fluid properties, and the nature of the surface. An efficient design from a thermal management perspective often involves optimizing the heat transfer coefficient to ensure effective thermal regulation.