A 2 mm-thick, 4 cm-long, straight rectangular composite...

A $2 \mathrm{~mm}$-thick, $4 \mathrm{~cm}$-long, straight rectangular composite material fin has a base temperature of $400 \mathrm{~K}$ and is located in a vacuum system. The fin has an emittance of $0.85$ and sees nearly black vessel walls at $360 \mathrm{~K}$. (i) Develop a finite-difference formulation of this steady one-dimensional conduction problem. Use a radiation heat transfer coefficient to account for the radiation heat transfer. (ii) Using a mesh size of $\Delta x=1 \mathrm{~cm}$, obtain temperatures $T_{2}$ through $T_{5}$ for the first iteration. Take $k=4 \mathrm{~W} / \mathrm{m} \mathrm{K}$ for the composite material. Neglect the tip heat loss.

A $2 \mathrm{~mm}$-thick, $4 \mathrm{~cm}$-long, straight rectangular composite material fin has a base temperature of $400 \mathrm{~K}$ and is located in a vacuum system. The fin has an emittance of $0.85$ and sees nearly black vessel walls at $360 \mathrm{~K}$.
(i) Develop a finite-difference formulation of this steady one-dimensional conduction problem. Use a radiation heat transfer coefficient to account for the radiation heat transfer.
(ii) Using a mesh size of $\Delta x=1 \mathrm{~cm}$, obtain temperatures $T_{2}$ through $T_{5}$ for the first iteration.
Take $k=4 \mathrm{~W} / \mathrm{m} \mathrm{K}$ for the composite material. Neglect the tip heat loss.

Key Concepts

Boundary Conditions

Boundary conditions are essential in defining the behavior of the system at its limits. For a fin, common boundary conditions include a fixed base temperature, specified convective or radiative losses along the surface, and an insulated tip where no heat is lost. These conditions are imposed on the discrete nodes and are crucial for the accurate formulation and solution of the finite-difference equations.

Radiation Heat Transfer Coefficient

In problems where surfaces exchange heat by radiation, it is common to approximate the non-linear radiative heat loss using an effective radiation heat transfer coefficient. This coefficient simplifies the radiative boundary condition into a form analogous to convective heat transfer, allowing for easier integration into the finite-difference formulation while accounting for properties like emissivity and the temperatures of the surroundings.

Finite-Difference Method

This is a numerical technique used to approximate derivatives by using differences between function values at discrete points. In heat transfer problems, it enables the conversion of differential equations governing conduction into algebraic equations that can be solved iteratively or directly, which is particularly useful for steady-state or transient problems when analytical solutions are complex or unavailable.

Steady-State Conduction

Steady-state conduction refers to a condition where the temperature distribution in a material no longer changes with time, meaning that the energy entering and leaving a small volume element is balanced. In this context, the one-dimensional conduction equation is discretized over the fin, modeling how temperature varies along its length under a constant thermal conductivity.

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