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 Governing equation: Steady state equation:
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:

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 Elementx1 x2 1 2 T1 x T2 f1
Let x2 x1
f1T1
f2T2

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

9. Element Equations of 1-D Linear Elementx1 x2 1 2 T1 x T2 f1

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. t1 t2 t3 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
x=0
x=L 1 2 T1 x T2 f1