A cylindrical aluminum fin with adiabatic tip is attached to a wall with surface temperature of \(300^{\circ} \mathrm{C}\), and is exposed to ambient air condition of \(15^{\circ} \mathrm{C}\) with convection heat transfer coefficient of \(150 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The fin has a uniform cross section with diameter of \(1 \mathrm{~cm}\), length of \(5 \mathrm{~cm}\), and thermal conductivity of \(237 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). Assume steady one-dimensional heat transfer along the fin and the nodal spacing to be uniformly \(10 \mathrm{~mm},(a)\) obtain the finite difference equations for use with the Gauss-Seidel iterative method, and \((b)\) determine the nodal temperatures using the GaussSeidel iterative method and compare the results with the analytical solution.

Finite difference equations are central to solving various types of differential equations that arise in engineering and physics, especially when dealing with heat transfer problems. In essence, these equations transform differential equations into a set of algebraic equations that approximate the original continuous problem into a discrete problem.

In the context of heat transfer, using finite difference equations enables us to predict how temperature changes over a solid object like a fin. By discretizing the fin into nodes and applying thermal balance equations to each node, we obtain our finite difference equations. Each equation relates the temperature at a point to its neighboring points, creating a system that can be solved iteratively.

Developing Finite Difference Equations

To develop finite difference equations, we make use of Taylor's expansion to approximate the temperature at points near our point of interest. For example, the temperature at a node, often labeled as \(T_i\), will be determined by its neighboring nodes' temperatures such as \(T_{i-1}\) and \(T_{i+1}\). The relationship between these temperatures will include terms that account for thermal properties of the fin, such as its thermal conductivity and the heat transfer coefficient.

The convection heat transfer coefficient, symbolized as \(h\), is a parameter that quantifies the convective heat transfer between a solid surface and a fluid in contact with it. It is a measure of how effectively heat is transferred from the solid surface to the fluid or vice versa.

In layman's terms, imagine you're holding a warm cup of coffee. The heat that you feel on your hands is being transferred from the cup to your hands through a process called convection, and the effectiveness of this heat transfer depends on a variety of factors including the fluid's velocity, its properties, and the temperature difference.

Significance in Heat Transfer Problems

The value of \(h\) is critical when calculating the rate of heat loss from an object. A higher heat transfer coefficient means that the object will lose heat more quickly to its surroundings. This is particularly important when designing systems that requirer thermal regulation, such as the fins mentioned in the exercise, since the goal is often to maximize or control the rate at which heat is dissipated.

An adiabatic tip in heat transfer refers to the end condition of an object like a fin where the tip is perfectly insulated. This means that no heat is transferred at that point to the surrounding environment.

To visualize this, think of an adiabatic tip as the end of a rod that is wrapped in an insulation layer so thick that none of the heat from the rod can escape through that end. In practical terms, it's a theoretical construct because perfect insulation is impossible, but it serves as an ideal scenario to simplify calculations.

Implications for Temperature Distribution

In the fin example, an adiabatic tip means that the heat transfer through the tip is zero, which affects the temperature distribution along the fin. Mathematically, this translates to a boundary condition that impacts the finite difference equations. For example, the absence of heat transfer at the tip means that the temperature gradient at the last node is zero, which affects the temperature calculations of the nodes close to it.