Heat_Conduction_Solver - 1D Steady-State Heat Conduction Solver (FVM)

📌 Project Overview

This project implements a numerical solution for 1D steady-state heat conduction with volumetric heat generation using the Finite Volume Method (FVM). The solver determines the temperature distribution across a plane wall subject to asymmetric boundary conditions: convection on one side and a constant heat flux on the other. https://colab.research.google.com/drive/1iaFjC_rSyr0VtP-PlwFLYGQRzFyWjBMj?usp=sharing

This repository demonstrates:

1. Numerical Analysis: Discretization of differential equations using FVM.
2. Validation: Comparison of numerical results against an exact analytical solution.
3. Mesh Sensitivity: Analysis of solution convergence as node count ($N$) increases.
4. Advanced Post-Processing: Implementation of quadratic interpolation to resolve sub-grid thermal peaks.

⚙️ Problem Description

The physical system consists of a plane wall with the following parameters based on the project specifications:

• Geometry: Thickness, Height, & Width: $L = 0.75 m, H = 1.5 m, W = 1 m$.
• Material: Constant thermal conductivity $k = 40~W/m\cdot K$.
• Heat Source: Uniform volumetric heat generation $\dot{g}''' = 5 \times 10^4~W/m^3$.

Boundary Conditions

1. Left Face ($x=0$): Convection with $h_f = 200~W/m^2\cdot K$ and fluid temperature $T_{f} = 20^\circ C$.
2. Right Face ($x=L$): Uniform heat flux input $\dot{q}'' = 100~W/m^2$ (flowing into the wall).

🧮 Mathematical Formulation

The governing differential equation for 1D steady-state conduction is derived from the energy balance:

$$\frac{d^2T}{dx^2} + \frac{\dot{g}}{k} = 0$$

The domain is discretized into $N$ control volumes. Applying the conservation of energy to each node results in a system of linear algebraic equations in the form $[A]{T} = {b}$, which is solved using matrix inversion.

📊 Results & Analysis

1. Validation (Numerical vs. Analytical)

The numerical solution (FVM) was tested against the analytical solution derived from integrating the heat equation.

• Observation: At $N=101$, the Maximum Absolute Error is negligible (< 0.1°C).

(Fig 1: Overlay of Analytical solution (red) and Numerical approximation (blue).)

2. Mesh Independence Study

The system was solved iteratively for node counts ranging from $N=3$ to $N=101$. The maximum temperature stabilizes rapidly, confirming the consistency of the numerical scheme.

3. Handling Discretization Artifacts (Grid Locking)

When plotting the location of the maximum temperature versus node count, a "stair-step" artifact was observed. This occurs because standard peak detection (max()) is limited to discrete node locations.

Solution: I implemented Quadratic Interpolation. By fitting a parabola to the peak node and its nearest neighbors, the solver estimates the true continuous physical location of the thermal peak.

(Fig 3: The red dashed line shows the discrete "staircase" error. The solid blue line shows the corrected sub-grid location using quadratic interpolation.)