finite difference equation
TRANSCRIPT
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