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)  T11 (x)  T22 (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
 di 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
 di 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  2w  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


di 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