mech4450 introduction to finite element methods
DESCRIPTION
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.TRANSCRIPT
MECH4450 Introduction to Finite Element Methods
Chapter 4
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-DHeat flux q: heat transferred per unit area per unit time (W/m2)
dTq kdx
T TA AQ CAx x t
Q: heat generated per unit volume per unit time C: mass heat capacity
Governing equation:
Steady state equation:
0d dTA AQdx dx
: thermal conductivity
Thermal Convection
( )sq h T T
Newton’s Law of Cooling
2: convective heat transfer coefficient ( )oh W m C
Thermal Conduction in 1-D
Dirichlet BC:
Boundary conditions:
Natural BC:
Mixed BC:
Weak Formulation of 1-D Heat Conduction(Steady State Analysis)
• Governing Equation of 1-D Heat Conduction -----
( )( ) ( ) ( ) 0d dT xx A x AQ xdx dx
0<x<L
• Weighted Integral Formulation -----
• Weak Form from Integration-by-Parts -----
0
( )0 ( ) ( ) ( ) ( )L d dT xw x x A x AQ x dx
dx dx
0 0
0LL dw dT dTA wAQ dx w A
dx dx dx
Formulation for 1-D Linear Element
Let 1 1 2 2(x) (x) (x)T T T
f2
x1 x2
1 2
T1
x
T2
f1
2 11 2( ) , ( )x x x xx x
l l
22
11 )( ,)(
xTAxf
xTAxf
Formulation for 1-D Linear Element
Let w(x)= i (x), i = 1, 2
2 2
1 1
2
2 2 1 11
0 ( ) ( )x x
jij i i i
j x x
ddT A dx AQ dx x f x fdx dx
1122
2
1
)()( fxfxQTK iiij
jij
2
1
2212
1211
2
1
2
1
TT
KKKK
ff
2 2
1 21 1
1 2 , Q , f , fx x
jiij i i
x xx x
dd dT dTwhere K A dx AQ dx A Adx dx dx dx
Element Equations of 1-D Linear Element
2
1
2
1
2
1
1111
TT
LA
ff
2
1 21
1 2 Q , f , fx
i ix x x xx
dT dTwhere AQ dx A Adx dx
f2
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 50oCwith a convection coefficient of h = 10 W(m2.K). Find the temperature alongthe composite wall.
t1 t2 t3
0 200oT C 50oT C
x
1 2 3
1 2 3
1 2 3
70 , 40 , 20
2 , 2.5 , 4
W m K W m K W m K
t cm t cm t cm
Thermal Conduction and Convection- Fin
Objective: to enhance heat transfer
dx
t
x
w 2 ( )2 ( ) 2 ( )
lossc c
h T T w th T T dx w h T T dx tQA dx A
Governing equation for 1-D heat transfer in thin fin
0c cd dTA A Qdx dx
0c cd dTA Ph T T A Qdx dx
2P w t where cA w t
Fin - Weak Formulation(Steady State Analysis)
• Governing Equation of 1-D Heat Conduction -----
( )( ) ( ) 0d dT xx A x Ph T T AQdx dx
0<x<L
• Weighted Integral Formulation -----
• Weak Form from Integration-by-Parts -----
0
( )0 ( ) ( ) ( ) ( ) ( )L d dT xw x x A x Ph T T AQ x dx
dx dx
0 0
0 ( )LL dw dT dTA wPh T T wAQ dx w A
dx dx dx
Formulation for 1-D Linear Element
Let w(x)= i (x), i = 1, 2
2 2
1 1
2
1
2 2 1 1
0
( ) ( )
x xji
j i j ij x x
i i
ddT A Ph dx AQ PhT dxdx dx
x f x f
1122
2
1
)()( fxfxQTK iiij
jij
2
1
2212
1211
2
1
2
1
TT
KKKK
ff
2 2
1 1
1 2
1 2
, Q ,
,
x xji
ij i j i ix x
x x x x
ddwhere K A Ph dx AQ PhT dxdx dx
dT dTf A f Adx dx
Element Equations of 1-D Linear Element
1 1 1
2 2 2
1 1 2 11 1 1 26
f Q TA PhLf Q TL
f2
x=0 x=L1 2
T1
x
T2
f1
2
1 21
1 2 Q , , x
i ix x x xx
dT dTwhere AQ PhT dx f A f Adx dx
1-D Heat Conduction – Example 2
A metallic fin extends from a plane wall whose temperature is 235oC. DetermineThe temperature distribution and amount of heat transferred from the fin to the air At 20oC .
0 235oT C
20oT C
x
2360 , 9
10 , 0.1 , w 1
o oW m C h W m C
l cm t cm m
lt