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. 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.

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:

Thermal Convection Newton’s Law of Cooling

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

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 -----

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

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

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

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

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

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 -----

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

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