Numerical Methods in Heat Conduction by S5n477

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									Numerical Methods in
Heat Conduction

          Reference: Incropera & DeWitt,
              Chapter 4, sections 4.4-4.5
                  Chapter 5, section 5.9




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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

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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

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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

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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

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FDM versus FEM

    Attribute                     FDM      FEM
    Simple formulation            YES       NO
    Memory requirement            LOW      HIGH

    CPU time for given            HIGH     LOW
    accuracy
    Adaptability to complex       FAIR     GOOD
    geometries
   Structural Mechanics              FEM
   Conduction Heat Transfer          FEM
   Electromagnetics                  FEM
   Fluid Dynamics                    FDM
   Convection Heat Transfer          FDM

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Commercial Software

          General FEM
           –   COSMOS
           –   ANSYS
           –   NASTRAN
           –   ALGOR
           –   FEMLAB
           –   FIDAP
          General FDM
           – FLUENT
           – FLOW-3D
           – COMPACT
          Electronics Cooling
           – FLOTHERM
           – COOLIT
           – ICEPAK


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