Numerical Methods for 1D and 2D Heat Conduction Problems

Introduction

• The lecture covers numerical methods to solve 1D and 2D heat conduction problems under steady-state conditions.
• Previously, heat conduction problems were solved using analytical methods in Cartesian, cylindrical, and spherical coordinates.
• Simplified problems included heat conduction through plain walls, fins, pipes, and spherical shapes.
• Analytical methods are limited to simple geometries and thermal conditions.
• Numerical methods are needed for complex geometries and thermal conditions.

Numerical Methods Overview

• A basic introduction to numerical methods for simple geometries.
• Numerical methods allow solving complex heat transfer problems.
• Example: Cylindrical fin with specific boundary conditions (e.g., base temperature, fin length, convection coefficient).
• Heat transfer involves both conduction and convection.

Steps in Numerical Methods

1. Divide the Medium into Volume Elements (Nodes)

• Replace the object with a set of nodes.
• Each node represents the average temperature of each volume element.
• Boundary nodes have half the size.
• Nodes are placed at the center of volume elements.
• Example: Fin divided into several nodes (T1 at the base, T2, T3, T4, etc., with specific positions).

2. Formulate the Problem in Finite Difference Form

• Use energy balance approaches to establish numerical equations.
• Example: Energy balance for a node (T2) involves conduction and convection terms.

Energy Balance Equations

• Balance heat conduction and convection for interior and boundary nodes.
• General nodal equation for interior nodes incorporates terms for conduction and convection.
• Adjust equations based on boundary conditions (e.g., insulated tip, convection at tip, prescribed temperature).

Solving Nodal Equations

• Nodal equations can be solved simultaneously using algebraic methods.
• Matrix representation of nodal equations for easier computation (e.g., inverse matrix method in Excel).
• Example: Solving for 13 nodes involves setting up and solving 13 equations simultaneously.

Advantages and Use Cases of Numerical Methods

• Useful for solving problems with complex geometries and thermal conditions where analytical methods fail.
• Allows easy parameter studies (e.g., change surface or tip temperature and see immediate results).

Comparison: Analytical vs. Numerical Solutions

Analytical Solutions

• Solve differential equations with appropriate boundary conditions.
• Results in continuous functions for temperature.
• Provides exact solutions.

Numerical Solutions

• Solve a series of algebraic equations using numerical methods.
• Results in discrete temperature values at nodes.
• Provides approximate solutions (accuracy improved by increasing number of nodes).