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
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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