Finite difference equation

Introduction to Conduction

Heat equation (Cartesian coordinate)

If the thermal conductivity is constant

𝛼=𝑘𝜌𝑐𝑝

> h𝑡 𝑒𝑟𝑚𝑎𝑙𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑣𝑖𝑡𝑦

Under steady-state conditions

For one-dimensional heat transfer, and no energy generation

Heat equation (Cylindrical coordinate)

Introduction to Conduction

Heat equation (Spherical coordinate)

Boundary Conditions for the heat diffusion equation at the surface (x=0)

Because the heat equation is second order, then two boundary conditions must be expressed for each coordinate to describe the system

The equation is first order in time, that’s why only one condition, termed the initial condition , must be specified

Finite-Difference Equations

The Nodal Network

a) Nodal network b) Finite-Difference approximation

For two-dimensional, steady –state conditions with no generation and constant thermal conductivity ;

𝜕2𝑇𝜕𝑥2

+𝜕2𝑇𝜕 𝑦 2

=0

Finite – Difference Form of the Heat Equation

Finite-Difference Equations

The Energy Balance Method

the actual direction of heat flow (into or out of the node) is often unknown, it is convenient to assume that all the heat flow is into the node

Conduction to an interior node from its adjoining nodes

Finite-Difference Equations

Nodal finite-Difference equations for

Case 1. Interior node

Case 2. Node at an internal corner with convection

Finite-Difference Equations

Nodal finite-Difference equations for

Case 3. Node at a plane surface with convection

Case 4. Node at a external corner with convection

Finite-Difference Equations

Nodal finite-Difference equations for

Case 5. Node at a plane surface with uniform heat flux

Solving the Finite – Difference Equations

The Matrix inversion method

Matrix inversion > direct methods

Gauss –Seidel > iterative methods

The Gauss – Seidel iteration

Solving the Finite – Difference Equations

With heat generation

Without heat generation

Example 1 :

Example 2 :

Using Gauss-seidel iteration method;

For each node

K ; level of iteration

Finite-difference equation for an interior node with heat generation ;

divide the long banr into 9 nodes

All nodes are interior nodes