A long, 4 cm-square bar has opposite faces maintained at 80^∘ C and 0^∘ C, respectively, and th

step 1 Hello students in this question we have one end of the metal bar having length L equals to 0 .25

A long, 4 cm-square bar has opposite faces maintained at...

A long, $4 \mathrm{~cm}$-square bar has opposite faces maintained at $80^{\circ} \mathrm{C}$ and $0^{\circ} \mathrm{C}$, respectively, and the other two lose heat by convection to a fluid at $0^{\circ} \mathrm{C} . \mathrm{An}$ electric current flows in the bar and causes a resistance heating of $3 \times 10^{6} \mathrm{~W} / \mathrm{m}^{3}$. The bar has a thermal conductivity of $15 \mathrm{~W} / \mathrm{m} \mathrm{K}$, and the convective heat transfer coefficient is $1500 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}$. (i) Determine the finite-difference approximation for an interior node, and for a surface node on the convectively cooled faces. (ii) For the given mesh, use Gauss-Seidel iteration to determine the temperatures $T_{1}$ through $T_{9}$ after the first and second iterations.

A long, $4 \mathrm{~cm}$-square bar has opposite faces maintained at $80^{\circ} \mathrm{C}$ and $0^{\circ} \mathrm{C}$, respectively, and the other two lose heat by convection to a fluid at $0^{\circ} \mathrm{C} . \mathrm{An}$ electric current flows in the bar and causes a resistance heating of $3 \times 10^{6} \mathrm{~W} / \mathrm{m}^{3}$. The bar has a thermal conductivity of $15 \mathrm{~W} / \mathrm{m} \mathrm{K}$, and the convective heat transfer coefficient is $1500 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}$.
(i) Determine the finite-difference approximation for an interior node, and for a surface node on the convectively cooled faces.
(ii) For the given mesh, use Gauss-Seidel iteration to determine the temperatures $T_{1}$ through $T_{9}$ after the first and second iterations.

Key Concepts

Gauss-Seidel Iterative Method

The Gauss-Seidel method is an iterative technique for solving systems of algebraic equations, especially those arising from discretization of partial differential equations. In this method, the solution is updated sequentially at each node using the most recent values available, which often leads to faster convergence compared to methods where updates occur concurrently. It is particularly useful for large, sparse systems resulting from finite difference approximations.

Convective Boundary Conditions

Convective boundary conditions, often represented by Robin conditions, model heat loss or gain on a surface via convection. These conditions relate the heat flux at the surface to the difference between the local surface temperature and the temperature of the surrounding fluid, scaled by the convective heat transfer coefficient. Incorporating these into the finite difference scheme allows the model to capture the effect of external convective cooling or heating.

Volumetric Heat Generation

Volumetric heat generation represents internal heat sources distributed throughout the material, such as resistance heating in an electrical conductor. In the finite difference formulation, this term appears as a source term in the energy balance equation, adding complexity to the temperature profile by introducing an additional energy input that must be balanced by conduction and convection.

Discretization and Node Classification

Discretization refers to dividing the continuous physical domain into a set of discrete points (nodes) where the temperature is calculated. In such problems, nodes are classified as interior nodes or boundary nodes. Interior nodes are fully surrounded by the physical domain, whereas boundary nodes lie on surfaces with specified conditions. Each type of node requires a tailored finite-difference equation that accounts for heat generation and specific boundary interactions.

Finite Difference Method

This is a numerical technique used to solve differential equations by approximating derivatives with differences. In the context of heat transfer, it involves discretizing the spatial domain into a grid or mesh and then writing balance equations for the temperature at each node. These equations collectively approximate the behavior of the continuous system, making it possible to simulate temperature distributions in complex geometries and under various boundary conditions.

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