MECh300H Introduction to Finite Element Methods

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MECh300H Introduction to Finite Element Methods
Finite Element Analysis (F.E.A.) of 1-D Problems
Heat Conduction
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Heat Transfer Mechanisms

• Conduction heat transfer by molecular
  agitation within a material without any motion of
  the material as a whole.
• Convection heat transfer by motion of a fluid.
• Radiation the exchange of thermal radiation
  between two or more bodies. Thermal radiation is
  the energy emitted from hot surfaces as
  electromagnetic waves.

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Heat Conduction in 1-D
Heat flux q heat transferred per unit area per
unit time (W/m2)
Governing equation
Q heat generated per unit volume per unit time
C mass heat capacity
k thermal conductivity
Steady state equation
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Thermal Convection
Newtons Law of Cooling
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Thermal Conduction in 1-D
Boundary conditions
Dirichlet BC
Natural BC
Mixed BC
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Weak Formulation of 1-D Heat Conduction(Steady
State Analysis)

• Governing Equation of 1-D Heat Conduction -----

0ltxltL

• Weighted Integral Formulation -----

• Weak Form from Integration-by-Parts -----

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Formulation for 1-D Linear Element
Let
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Formulation for 1-D Linear Element
Let w(x) fi (x), i 1, 2
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Element Equations of 1-D Linear Element
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1-D Heat Conduction - Example
A composite wall consists of three materials, as
shown in the figure below. The inside wall
temperature is 200oC and the outside air
temperature is 50oC with a convection coefficient
of h 10 W(m2.K). Find the temperature along the
composite wall.
t3
t2
t1
x
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Thermal Conduction and Convection- Fin
Objective to enhance heat transfer
Governing equation for 1-D heat transfer in thin
fin
w
t
x
dx
where
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Fin - Weak Formulation(Steady State Analysis)

• Governing Equation of 1-D Heat Conduction -----

0ltxltL

• Weighted Integral Formulation -----

• Weak Form from Integration-by-Parts -----

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Formulation for 1-D Linear Element
Let w(x) fi (x), i 1, 2
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Element Equations of 1-D Linear Element