Assume unit depth, steady state conditions and all heat flows are directed into the nodal region of interest.
where i refers to neighbouring nodes
Applying Fourier’s law for conduction from node (m-1,n) to (m,n):
where (Dy.1) is the heat transfer area and (Tm,n-T m-1,n)/Dx is the finite difference approximation to the temperature gradient.
Substituting all q’s into the energy balance:
* Valid for interior points, within the medium
Consider an external corner, with convection heat transfer
*for summary of nodal finite-difference equations for different configurations
We have transformed the system of differential equations to a system of algebraic equations
Solution methods:
–The Matrix Inversion Method
–Gauss-Seidel Iteration
When completed with a calculator à & time-limited ability Matrix Inversion à little knot, Σ a particular operation. Gauss-Seidel iteration à more nodes, Σ is not a specific operation.
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