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MECH4450 Introduction to Finite Element Methods

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Presentation on theme: "MECH4450 Introduction to Finite Element Methods"— Presentation transcript:

1 MECH4450 Introduction to Finite Element Methods
Chapter 4 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 Element
x1 x2 1 2 T1 x T2 f1 Let

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

9 Element Equations of 1-D Linear Element
x1 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

15 1-D Heat Conduction – Example 2
A metallic fin extends from a plane wall whose temperature is 235oC. Determine The temperature distribution and amount of heat transferred from the fin to the air At 20oC . t l x


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