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Numerical Methods in Heat Conduction

Numerical Methods in Heat Conduction. Reference: Incropera & DeWitt , Chapter 4, sections 4.4-4.5 Chapter 5, section 5.9. Introduction. Numerical methods are necessary to solve many practical problems in heat conduction that involve: complex 2D and 3D geometries

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Numerical Methods in Heat Conduction

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  1. Numerical Methods in Heat Conduction Reference: Incropera & DeWitt, Chapter 4, sections 4.4-4.5 Chapter 5, section 5.9 ME 259

  2. Introduction • Numerical methods are necessary to solve many practical problems in heat conduction that involve: • complex 2D and 3D geometries • complex boundary conditions • variable properties • An appropriate numerical method can produce a useful approximate solution to the temperature field T(x,y,z,t); the method must be • sufficiently accurate • stable • computationally efficient ME 259

  3. General Features • A numerical method involves a discretization process, where the solution domain is divided into subdomains and nodes • The PDE that describes heat conduction is replaced by a system of algebraic equations, one for each subdomain in terms of nodal temperatures • A solution to the system of algebraic equations almost always requires the use of a computer • As the number of nodes (or subdomains) increase, the numerical solution should approach the exact solution • Numerical methods introduce error and the possibility of solution instability ME 259

  4. Types of Numerical Methods • The Finite Difference Method (FDM) • subdomains are rectangular and nodes form a regular grid network • nodal values of temperature constitute the numerical solution; no interpolation functions are included • discretization equations can be derived from Taylor series expansions or from a control volume approach ME 259

  5. Types of Numerical Methods • The Finite Element Method (FEM) • subdomain may be any polygon shape, even with curved sides; nodes can be placed anywhere in subdomain • numerical solution is written as a finite series sum of interpolation functions, which may be linear, quadratic, cubic, etc. • solution provides nodal temperatures and interpolation functions for each subdomain ME 259

  6. FDM versus FEM Structural Mechanics FEM Conduction Heat Transfer FEM Electromagnetics FEM Fluid Dynamics FDM Convection Heat Transfer FDM ME 259

  7. Commercial Software • General FEM • COSMOS • ANSYS • NASTRAN • ALGOR • FEMLAB • FIDAP • General FDM • FLUENT • FLOW-3D • COMPACT • Electronics Cooling • FLOTHERM • COOLIT • ICEPAK ME 259

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