A hot surface at \(100^{\circ} \mathrm{C}\) is to be cooled by attaching 3 -cm- long, \(0.25\)-cm-diameter aluminum pin fins \((k=\) \(237 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) ) with a center-to-center distance of \(0.6 \mathrm{~cm}\). The temperature of the surrounding medium is \(30^{\circ} \mathrm{C}\), and the combined heat transfer coefficient on the surfaces is \(35 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Assuming steady one-dimensional heat transfer along the fin and taking the nodal spacing to be \(0.5 \mathrm{~cm}\), determine \((a)\) the finite difference formulation of this problem, \((b)\) the nodal temperatures along the fin by solving these equations, \((c)\) the rate of heat transfer from a single fin, and \((d)\) the rate of heat transfer from a \(1-\mathrm{m} \times 1-\mathrm{m}\) section of the plate.

When we discuss the heat transfer coefficient, we refer to a measure of the thermal resistance of a material to the flow of heat. It represents the amount of heat that is transferred per unit area of the material per unit temperature difference between the material and the surrounding environment. In the context of our fin heat transfer problem, the heat transfer coefficient, denoted by \( h \), characterizes how easily heat is conveyed from the fin to the surrounding air.
In technical terms, a high heat transfer coefficient suggests that the material is an effective conduit for heat transfer, expelling heat rapidly to the surrounding medium. Conversely, a lower value would imply that the material acts more as an insulator, retaining the heat rather than transferring it. This coefficient is crucial when calculating the heat transfer rate from the fin, represented as \( Q_{fin} = h \times A_{fin} \times (T_1 - T_{\text{air}}) \), where \( A_{fin} \) is the fin's surface area, \( T_1 \) is the temperature at the fin's base, and \( T_{\text{air}} \) is the temperature of the surrounding air.

Thermal conductivity, identified by \( k \) in the problem, is an innate property of a material which quantifies its ability to conduct heat. High thermal conductivity indicates that the material can transfer heat effectively, making it a good heat conductor. On the flip side, materials with low thermal conductivity are poor conductors and hence good insulators. Aluminum, the material used for the fins in our exercise, has a thermal conductivity value of\( 237 \mathrm{W/(m \times K)} \).
Understanding thermal conductivity is essential for solving heat transfer problems using the finite difference method. The thermal conductivity is integrated into the finite difference formulation to determine how the temperature distribution within the fin changes from one nodal point to the next. The value influences the transfer of heat internally within the fin, as it is a measure of the material's ability to conduct heat along the length of the fin.

In the realm of heat transfer analysis, steady-state heat conduction implies that the temperature distribution in the material does not change over time. For our fin problem, this means that once the fin's temperature profile reaches an equilibrium, the temperature at any given point along the fin remains constant, assuming the boundary conditions (such as the temperature of the surrounding air and the hot surface) are also unchanging. The steady-state assumption simplifies the analysis significantly since it eliminates the need to examine time as a variable.
To solve the steady-state heat conduction problem using the finite difference method, we develop a set of equations, with each representing a node along the fin. These equations take into account the heat being conducted through the fin and the heat being convected away by the surrounding air. The solution to this set of linear equations provides us with the steady-state temperature at each node, enabling us to compute the rate of heat transfer from the fin. The heat transfer is directly related to these temperatures and to the heat transfer coefficient and thermal conductivity, solidifying their importance in the construct of the mathematically modeled heat transfer process.