Conduction Heat Transfer With Two-Dimensional Steady-State

 Two-Dimensional Steady-State Conduction

equations of general form :

 

•In many problems we need to consider heat transfer in two directions

•Solution involves partial differential equations

•Need analytical, graphical or numerical approaches

•Analytical methods involve mathematical series and functions.

            –Exact solutions

–Limited types of problems can be solved

•Numerical methods provide approximate results at discrete points.

–Can accommodate complex geometries and boundary conditions

–Widely used

            –Many commercially available software packages

* In this chapter we will focus on numerical methods – Finite Difference Method

 

Finite Difference Method is an approximate method for determining temperatures at discrete (nodal) points of the physical system.

Procedure:

•Represent the physical system by a nodal network.

•Use the energy balance method to obtain a finite-difference equation for each node of unknown temperature.

•Solve the resulting set of algebraic equations for the unknown nodal temperatures.

Nodal Network

The nodal network identifies discrete points at which the temperature is to be determined and uses an m,n notation to designate their location.

A finite-difference approximation is used to represent temperature gradients in the domain :