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1. Introduction to Conduction
Heat equation (Cartesian coordinate)
If the thermal conductivity is constant
𝛼 =
𝑘
𝜌𝑐 𝑝
> 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑣𝑖𝑡𝑦
Under steady-state conditions
For one-dimensional heat transfer, and no energy generation

2. Heat equation (Cylindrical coordinate)
Introduction to Conduction
Heat equation (Spherical coordinate)

3. 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

4. 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 ;
𝜕2
𝑇
𝜕𝑥2
+
𝜕2
𝑇
𝜕𝑦2
= 0

5. Finite – Difference Form of the Heat Equation

6. 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

7. Finite-Difference Equations
Nodal finite-Difference equations for ∆𝑥 = ∆𝑦
Case 1. Interior node
Case 2. Node at an internal corner with convection

8. Finite-Difference Equations
Nodal finite-Difference equations for ∆𝑥 = ∆𝑦
Case 3. Node at a plane surface with convection
Case 4. Node at a external corner with convection

9. Finite-Difference Equations
Nodal finite-Difference equations for ∆𝑥 = ∆𝑦
Case 5. Node at a plane surface with uniform heat flux

10. Solving the Finite – Difference Equations
The Matrix inversion method
Matrix inversion > direct methods
Gauss –Seidel > iterative methods

11. The Gauss – Seidel iteration
Solving the Finite – Difference Equations
With heat generation
Without heat generation

12. Example 1 :

14. Example 2 :

15. Using Gauss-seidel iteration method;
For each node
K ; level of iteration

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

18. ∆𝑥 = 10𝑚𝑚 divide the long banr into 9 nodes
All nodes are interior nodes