Finite difference equation
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This document discusses heat transfer through conduction and various methods for solving the heat equation using finite difference approximations. It introduces the heat equation in Cartesian, cylindrical and spherical coordinates. It discusses boundary conditions and describes setting up a nodal network to discretize the domain. It then presents the finite difference form of the heat equation and describes different cases for nodal finite difference equations, including for interior nodes, nodes at corners or surfaces with convection, and nodes at surfaces with uniform heat flux. It discusses solving the finite difference equations using matrix inversion, Gauss-Seidel iteration, and provides examples.
Finite difference equation
- 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