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In the context of heat transfer, understanding the temperature distribution within a solid object is crucial for predicting how heat will move through it. In our exercise, we examine a truncated solid cone and aim to derive an expression for temperature distribution, denoted as \( T(x) \). The temperature distribution tells us how temperature varies at each point along the axis of the cone.

Given that the sides of the cone are well-insulated and only the top and bottom surfaces have specified temperatures, this greatly simplifies our calculations. These specific temperature boundaries ensure that any heat transfer happens only along the axis of the cone, not across the diameter. This means that the temperature distribution along the x-axis can be calculated by analyzing how heat flows from the hotter top to the cooler bottom of the cone, a process governed by Fourier's Law. Our goal is to establish a mathematical function, \( T(x) \), that models this temperature change from \( T_1 = 100^{\circ}C \) at the top to \( T_2 = 20^{\circ}C \) at the bottom.

Fourier's Law of heat conduction is fundamental in understanding how heat transfers through materials. It articulates that the rate of heat transfer through a material is directly proportional to the negative of the temperature gradient and the area through which the heat flows.

Mathematically, it is represented as:

• \( Q = -kA(x) \left( \frac{\partial T}{\partial x} \right) \)

Here,

• \( Q \) is the heat transfer rate,
• \( k \) is the thermal conductivity of the material,
• \( A(x) \) is the cross-sectional area normal to the direction of heat transfer,
• \( \frac{\partial T}{\partial x} \) is the temperature gradient along the x-axis.

In this exercise, because the sides of the cone are insulated, we focus solely on the axial temperature gradient. Fourier's Law guides us in determining how to distribute the heat flow across the varying cross-sectional area of the cone. By rearranging this relationship, we can find the spatial derivative of the temperature, which, when integrated, provides the desired temperature distribution along the cone.

Thermal conductivity, represented by the symbol \( k \), is a material-specific property that indicates how well it can conduct heat. In our exercise, the cone is made of pure aluminum, a material known for its high thermal conductivity of \( 205 \, \text{W/(m} \cdot \text{K)} \).

Thermal conductivity plays a pivotal role in the equation derived from Fourier's Law. It affects the rate at which heat flows across the cone. Materials with high thermal conductivity, like aluminum, allow heat to move more readily, which directly influences the temperature distribution \( T(x) \). This constant \( k \) remains integral in each step of solving for both the temperature gradient and the overall heat transfer rate \( Q \). By using aluminum's thermal conductivity in our calculations, we align with real-world conditions, ensuring our model accurately reflects the cone's behaviour under the given temperature differences.

Boundary conditions are essential in solving differential equations related to heat transfer because they provide the necessary constraints to find a unique solution. In the case of our truncated cone, the boundary conditions are given by the fixed temperatures at the top and bottom surfaces.

We know:

• At \( x_1 = 0.075 \, \text{m} \), the temperature \( T_1 = 100^{\circ}C \).
• At \( x_2 = 0.225 \, \text{m} \), the temperature \( T_2 = 20^{\circ}C \).

These conditions are applied when integrating to find the temperature distribution \( T(x) \). By integrating with these specific temperature boundaries, we ensure that the solution for \( T(x) \) adheres to the physical constraints of the problem, accurately representing how the temperature changes along the length of the cone. The boundary conditions effectively anchor the solution, validating that the mathematical model reflects the actual thermal behaviour of the truncated cone, making the findings both feasible and applicable.