FEA Course Lecture V

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• FEA Course Lecture V Outline
• UCSD - 10/30/03
• Review of Last Lecture (IV) on Plate and Shell
  Elements.
• Thermal Analysis
• Summary of Concepts of Heat Transfer
• Thermal Loads and Boundary Conditions
• Nonlinear Effects
• Thermal transients and Modeling Considerations
• Example FE

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• Thermal Analysis - Introduction
• Thermal Analysis involves calculating
• Temperature distributions
• Amount of Heat lost or gained
• Thermal gradients
• Thermal fluxes.
• There are two types of thermal analysis
• Steady-state analysis
• Transient thermal analysis

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• Two types of Thermal Analysis
• Steady-state Thermal Analysis. It involves
  determining the temperature distribution and
  other thermal quantities under steady-state
  loading conditions. A steady-state loading
  condition is a situation where heat storage
  effects varying over a period of time can be
  ignored.
• Some examples of thermal loads are
• Convections
• Radiation
• Heat Flow Rates
• Heat Fluxes (Heat Flow/unit area)
• Heat Generation Rates (heat flow/unit volume)
• Constant Temperature Boundaries
• Steady State thermal analysis may be linear or
  nonlinear (due to material properties not
  geometry). Radiation is a nonlinear problem.
• Transient Thermal Analysis. It involves
  determining the temperature distribution and
  other thermal quantities under conditions that
  vary over a period of time.

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Theoretical Basis for Thermal Analysis

• KT T Q where KT f
  (conductivity of material).
• T vector of nodal temperatures
• Q vector of thermal loads.
• KT is nonlinear when radiation heat transfer is
  present. Note that convection and radiation BCs
  contribute terms to both KT and Q.
• Heat is transferred to or from a body by
  convection and radiation.

Heat Flow across boundary due to radiation
(in-out)
Prescribed rate of heat flow across boundary (in
or out)
Heat generated internally (eg., Joule heating)
Prescribed temperature (BC) insulated for
example.
Heat Flow across boundary due to radiation
(in-out)
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Equation of Heat Flow (1D Systems)

• fx -k dT/dx Fourier Heat Conduction
  Equation. Heat flows from high
• temperature region to low temperature region.
• Q -kA dT/dx Q heat flow
• fx Q/A where fx heat flux/unit area, k
  thermal conductivity, A area of cross-section,
  dT/dx temperature gradient
• In general, fx, fy, fz -kdT/dX, dT/dY,
  dT/dZ T
• For an elemental area of length dx, the balance
  of energy is given as
• -KA dT/dx qAdx rca dT/dt dx KA dT/dx
  d/dx(KA dT/dx)dx
• d/dx(KA dT/dx) Aq rca dT/dt
• rate in rate out rate of increase within
• For generally anisotropic material
• -d/dx d/dy d/dz fx fy fzT qv cr
  dT/dt
• where c is specific heat, t is time, r mass
  density and qv rate of internal heat generation
  / unit volume.
• Above equation can be re-written as
• Steady state if dT/dt 0

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Some Notes

• If the body is plane and there is convection and
  or radiation heat transfer across its flat
  lateral surfaces, additional equations for flux
  terms are needed
• Convection BC
• f h(Tf T) (Newtons Law of cooling)
• K f(h)
• Q f(h,Tf)
• where f flux normal to the surface Tf
  temperature of surrounding fluid h heat
  transfer coefficients (which may depend on many
  factors like velocity of fluid,
  roughness/geometry of surface, etc) and T-
  temperature of surface.
• Radiation BC
• f hr(Tr T)
• K f(hr)
• Q f(hr,T)
• where, Tr temperature of the surface hr
  temperature dependent heat transfer coefficients.
  
• hr Fs(Tr2T2)(TrT).
• Where F is a factor that accounts for geometries
  of radiating surfaces.s is Stefan-Boltzmann
  constant.

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FEs in Thermal Analysis

• 1D Bar Element
• Uniform bar whose lateral surface is insulated
• 2D Elements
• PLANE35, PLANE55, PLANE77 etc.
• 3D Elements
• SOLID70, SOLID90 etc.
• Transient Thermal Analysis
• KTT CT Q where Q Q(t)
• T dT/dt,
• C Summation of c
• Solution Use Crank Nicholson Method, etc.