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MECh300H Introduction to Finite Element Methods

MECh300H Introduction to Finite Element Methods. Finite Element Analysis (F.E.A.) of 1-D Problems – Heat Conduction. Heat Transfer Mechanisms. Conduction – heat transfer by molecular agitation within a material without any motion of the material as a whole.

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MECh300H Introduction to Finite Element Methods

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

  2. 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.

  3. Heat Conduction in 1-D Heat flux q: heat transferred per unit areaper unit time(W/m2) Governing equation: Q: heat generated per unit volumeper unit time C: mass heat capacity k: thermal conductivity Steady state equation:

  4. Thermal Convection Newton’s Law of Cooling

  5. Thermal Conduction in 1-D Boundary conditions: Dirichlet BC: Natural BC: Mixed BC:

  6. Weak Formulation of 1-D Heat Conduction(Steady State Analysis) • Governing Equation of 1-D Heat Conduction ----- 0<x<L • Weighted Integral Formulation ----- • Weak Form from Integration-by-Parts -----

  7. Formulation for 1-D Linear Element T1 T2 f1 f2 x 1 2 x1 x2 f2T2 f1T1 x2 x1 Let

  8. Formulation for 1-D Linear Element Let w(x)= fi (x), i = 1, 2

  9. Element Equations of 1-D Linear Element T1 T2 f1 f2 x 1 2 x1 x2

  10. 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

  11. 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

  12. Fin - Weak Formulation(Steady State Analysis) • Governing Equation of 1-D Heat Conduction ----- 0<x<L • Weighted Integral Formulation ----- • Weak Form from Integration-by-Parts -----

  13. Formulation for 1-D Linear Element Let w(x)= fi (x), i = 1, 2

  14. Element Equations of 1-D Linear Element T1 T2 f1 f2 x 1 2 x=0 x=L

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