Transcript Lecture6
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
Heat flux q: heat transferred per unit area per unit time (W/m2)
q k
dT
dx
Governing equation:
T
T
A
AQ
CA
x
x
t
Q: heat generated per unit volume per unit time
C: mass heat capacity
: thermal conductivity
Steady state equation:
d
dT
A
dx
dx
AQ 0
Thermal Convection
Newton’s Law of Cooling
q h(Ts T )
h: convective heat transfer coefficient (W m C )
2
o
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 ----
d
dT ( x )
(
x
)
A
(
x
)
AQ( x ) 0 0<x<L
dx
dx
• Weighted Integral Formulation ----dT ( x )
d
0 w( x ) ( x ) A( x)
AQ( x) dx
dx
dx
0
L
• Weak Form from Integration-by-Parts ----dT
dw
0 A
dx
dx
0
L
L
dT
wAQ
dx
w
A
dx
0
Formulation for 1-D Linear Element
x
f1
T1
T2
1
2
x1
x2
f1 ( x) A
Let
T
,
x 1
x1
f 2 ( x) A
T
x
2
T (x) T11 (x) T22 (x)
x2 x
1 ( x )
,
l
1T1
f2
x x1
2 ( x )
l
2T2
x2
Formulation for 1-D Linear Element
Let w(x)= i (x),
i = 1, 2
x
x2
di d j 2
0 T j A
dx i AQ dx i ( x2 ) f 2 i ( x1 ) f1
j 1
dx dx x1
x1
2
2
KijT j Qi i ( x2 ) f 2 i ( x1 ) f1
j 1
f1 Q1 K11
f 2 Q2 K12
K12 T1
K 22 T2
x2
di d j
dT
where Kij A
dx, Qi i AQ dx, f1 A
dx
dx dx
x1
x1
x2
dT
, f2 A
dx
x1
x2
Element Equations of 1-D Linear Element
x
f1
T1
T2
1
2
x1
f2
x2
f1 Q1 A 1 1 T1
f 2 Q2 L 1 1 T2
x2
dT
where Qi i AQ dx, f1 A
dx
x1
dT
, f2 A
dx
x x1
x x2
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.
1 70W m K , 2 40W m K , 3 20W m K
t1 2cm, t2 2.5cm, t3 4cm
1
2
3
t1
t2
t3
T0 200 C
o
T 50o C
x
Thermal Conduction and
Convection- Fin
Objective: to enhance heat transfer
Governing equation for 1-D heat transfer in thin fin
d
dT
A
c
AcQ 0
dx
dx
w
t
Qloss
x
dx
2h(T T ) dx w 2h(T T ) dx t 2h(T T ) w t
Ac dx
Ac
d
dT
A
c
Ph T T AcQ 0
dx
dx
where
P 2w t
Fin - Weak Formulation
(Steady State Analysis)
• Governing Equation of 1-D Heat Conduction ----
d
dT ( x )
(
x
)
A
(
x
)
Ph T T AQ 0
dx
dx
0<x<L
• Weighted Integral Formulation ----dT ( x )
d
0 w( x ) ( x ) A( x )
Ph(T T ) AQ( x ) dx
dx
dx
0
L
• Weak Form from Integration-by-Parts ----dT
dw
0 A
dx
dx
0
L
L
dT
wPh
(
T
T
)
wAQ
dx
w
A
dx
0
Formulation for 1-D Linear Element
Let w(x)= i (x),
i = 1, 2
x2
x2
d
d
j
0 T j A i
Ph
i j dx i AQ PhT dx
dx dx
j 1
x1
x1
i ( x2 ) f 2 i ( x1 ) f1
2
2
KijT j Qi i ( x2 ) f 2 i ( x1 ) f1
j 1
f1 Q1 K11
f 2 Q2 K12
K12 T1
K 22 T2
x2
di d j
where Kij A
Ph
i j dx, Qi i AQ PhT dx,
dx dx
x1
x1
x2
f1 A
dT
dx
, f2 A
x x1
dT
dx
x x2
Element Equations of 1-D Linear Element
x
f1
T1
T2
1
2
x=0
x=L
f2
f1 Q1 A 1 1 Phl 2 1 T1
f 2 Q2 L 1 1 6 1 2 T2
x2
dT
where Qi i AQ PhT dx, f1 A
dx
x1
dT
, f2 A
dx
x x1
x x2