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1
FFinite Element Methodinite Element Method
FEM FOR HEAT TRANSFER PROBLEMS
for readers of all backgroundsfor readers of all backgrounds
G. R. Liu and S. S. Quek
CHAPTER 12:
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2Finite Element Method by G. R. Liu and S. S. Quek
CONTENTSCONTENTS
FIELD PROBLEMSWEIGHTED RESIDUAL APPROACH
FOR FEM1D HEAT TRANSFER PROBLEMS2D HEAT TRANSFER PROBLEMSSUMMARYCASE STUDY
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3Finite Element Method by G. R. Liu and S. S. Quek
FIELD PROBLEMSFIELD PROBLEMS
General form of system equations of 2D linear steady state field problems:
2 2
2 20x yD D g Q
x y
(Helmholtz equation)
For 1D problems:
2
20
dD g Q
dx
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4Finite Element Method by G. R. Liu and S. S. Quek
FIELD PROBLEMSFIELD PROBLEMS
Heat transfer in 2D fin
x z
y
t
f
Heat convection on the surface
f
f
f
f
t
2 2
2 2
Heat supplyHeat convectctionHeat conduction
22( ) ( )f
x y
hhD D q
x y t t
22, fhh
g Q qt t
Note:
![Page 5: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/5.jpg)
5Finite Element Method by G. R. Liu and S. S. Quek
FIELD PROBLEMSFIELD PROBLEMS
Heat transfer in long 2D body
x
z y
Representative plane
t=1
2 2
2 2
Heat supply
Heat conduction
0x yk k qx y
Note:
Dx = kx, Dy =tky,
g = 0 and Q = q
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6Finite Element Method by G. R. Liu and S. S. Quek
FIELD PROBLEMSFIELD PROBLEMS
Heat transfer in 1D fin
0
x
P
A
2
2
Heat supplyHeat convectoinHeat conduction
0f
dkA hP hP q
dx
Note:, , fD kA g hP Q q hP
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7Finite Element Method by G. R. Liu and S. S. Quek
FIELD PROBLEMSFIELD PROBLEMS
Heat transfer across composite wall
x
y Heat convection Heat convection
2
2
Heat supplyHeat conduction
0d
kA qdx
Note:
, 0, D kA g Q q
![Page 8: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/8.jpg)
8Finite Element Method by G. R. Liu and S. S. Quek
FIELD PROBLEMSFIELD PROBLEMS
Torsional deformation of bar2 2
2 2
1 12 0
G x G y
Note:
Dx=1/G, Dy=1/G, g=0, Q=2
Ideal irrotational fluid flow 2 2
2 20
x y
2 2
2 20
x y
Note:
Dx = Dy = 1, g = Q = 0
( - stress function)
( - streamline function and - potential function)
![Page 9: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/9.jpg)
9Finite Element Method by G. R. Liu and S. S. Quek
FIELD PROBLEMSFIELD PROBLEMS
Accoustic problems
2 2 2
2 2 20
P P wP
x y c
Note:
2
2
wg
c , Dx = Dy = 1, Q = 0
P - the pressure above the ambient pressure ;
w - wave frequency ;
c - wave velocity in the medium
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10Finite Element Method by G. R. Liu and S. S. Quek
WEIGHTED RESIDUAL WEIGHTED RESIDUAL APPROACH FOR FEMAPPROACH FOR FEM
Establishing FE equations based on governing equations without knowing the functional.
2 2
2 20x yD D g Q
x y
( ( , )) 0f x y
( ( , ))d d 0AWf x y x y
Approximate solution:
Weight function
(Strong form)
(Weak form)
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11Finite Element Method by G. R. Liu and S. S. Quek
WEIGHTED RESIDUAL WEIGHTED RESIDUAL APPROACH FOR FEMAPPROACH FOR FEM
1 2
3
Discretize into smaller elements to ensure better approximation
In each element,
Using N as the weight functions
( )( , ) ( , )h ex y x y N Φ
where 1 1 dnN N N N
Galerkin method
( ) ( ( , ))d de
e T
Af x y x yR N Residuals are then assembled
for all elements and enforced to zero.
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12Finite Element Method by G. R. Liu and S. S. Quek
1D HEAT TRANSFER 1D HEAT TRANSFER PROBLEMPROBLEM
1D fin1D fin
2
2
Heat supplyHeat convectoinHeat conduction
0f
dkA hP hP q
dx
00 (Specified boundary condition)
fbhAdx
dkA
(Convective heat loss at free end)
k : thermal conductivityh : convection coefficientA : cross-sectional area of the finP : perimeter of the fin : temperature, and
f : ambient temperature in the fluid
0 x
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13Finite Element Method by G. R. Liu and S. S. Quek
1D fin1D fin
1 2
(i)
i j
(1)
x
xi xj
(i) 0
0 x
(n)
n+1
2
2
2
2
dj
i
j j
i i
hxe T h
x
hx xT T h
x x
dD g Q x
dx
dD Q dx g dx
dx
R N
N N
Using Galerkin approach,
where
D = kA, g = hP, and Q = hP
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14Finite Element Method by G. R. Liu and S. S. Quek
1D fin1D finIntegration by parts of first term on right-hand side,
d d dj
j j j
i i i
i
xh T hx x xe T T T h
x x xx
d d dD D x Q x g x
dx dx dx
N
R N N N
( )( ) ( )h ex x N ΦUsing
( )( )
( ) ( )
d d d( d )
d d d
( d ) ( d )
j
j
i
i
eeD
j j
i i
e egQ
xh Txe eT
xx
x x eT T
x x
D D xx x x
Q x g x
kb
kf
N NR N Φ
N N N Φ
![Page 15: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/15.jpg)
15Finite Element Method by G. R. Liu and S. S. Quek
1D fin1D fin
( ) ( ) ( ) ( )[ ]e ee e e eD g Q R b k k Φ f
where
( ) d dd d
d d
j j
i i
Tx xe T
D x xD x D x
x x
N Nk B B
( ) dj
i
xe Tg x
g xk N N
( ) dj
i
xe TQ x
Q xf N
( ) d
d
j
i
xhe T
x
Dx
b N
(Thermal conduction)
(Thermal convection)
(External heat supplied)
(Temperature gradient at two ends of element)
d
dx
NB (Strain matrix)
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16Finite Element Method by G. R. Liu and S. S. Quek
1D fin1D fin
For linear elements,
( ) j ii j
x x x xx N N
l l
N (Recall 1D truss element)
d d 1 1
d dj i
x x x x
x x l l l l
NB
Therefore,
( )
11 11 1
d1 1 1
j
i
xeD x
kAl D xl l l
l
k(Recall stiffness matrix of truss element)
AE
lfor truss element
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17Finite Element Method by G. R. Liu and S. S. Quek
1D fin1D fin
( ) 2 1d
1 26
j
i
j
x je ig x
i
x xx x x x hPllg x
l lx x
l
k(Recall mass matrix of truss element)
A l for truss element
( )
Heat supply Heat convection
1 1d ( )
1 12 2
j
i
j
xeQ fx
i
x xQl llQ x q hP
x x
l
f
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18Finite Element Method by G. R. Liu and S. S. Quek
1D fin1D fin
( ) ( )
( )
d0ddd
d dd d
d0d
ji
i
i
j
j e eL R
xx xe T
x x
xx x
x x
kAx kA
xDkAx
xkAx
b b
b N
( ) ( ) ( )e e eL R b b bor
(Left end) (Right end)
i1
(n)
i i+1
(n1)
i1 i (n1)
d
dix x
Dx
1
d
dix x
Dx
i i+1 (n)
1
d
dix x
Dx
d
dix x
Dx
0
At the internal nodes of the fin, bL(e)
and bL(e) vanish upon assembly.
At boundaries, where temperature is prescribed, no need to calculate bL
(e)
or bL(e) first.
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19Finite Element Method by G. R. Liu and S. S. Quek
1D fin1D fin
When there is heat convection at boundary,
fbhAdx
dkA
0 0 0e
Rj fb f hA hAhA
bE.g.
Since b is the temperature of the fin at the boundary point,
b = j
Therefore,
( ) ( )
( )00 0
0e e
M s
ieR
j fhAhA
k f
b
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20Finite Element Method by G. R. Liu and S. S. Quek
1D fin1D fin( ) ( ) ( ) ( )e e e eR M s b k Φ f
where ( ) 0 0
0e
M hA
k , ( )0
es
fhA
f
For convection on left side,
( ) ( ) ( ) ( )e e e eL M s b k Φ f
where ( ) 0
0 0e
M
hA
k , ( )
0fe
s
hA
f
![Page 21: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/21.jpg)
21Finite Element Method by G. R. Liu and S. S. Quek
1D fin1D fin
Therefore,
( ) ( )
( ) ( ) ( ) ( ) ( )[ ]e e
e ee e e e eD g M Q S
k f
R k k k Φ f f
( ) ( )e ee e R k Φ f
Residuals are assembled for all elements and enforced to zero: KD = F
Same form for static mechanics problem
![Page 22: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/22.jpg)
22Finite Element Method by G. R. Liu and S. S. Quek
1D fin1D fin
Direct assembly procedure
( ) ( )e ee e R k Φ f
or
1 11 12 1 1
2 21 22 2 2
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
e e e e e
e e e e e
R k k f
R k k f
(1) (2)
1 2 3
Element 1:
1 11 1 12 1
(1) (1) (1) (1)2R k k f
2 21 1 22 2
(1) (1) (1) (1)2R k k f
![Page 23: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/23.jpg)
23Finite Element Method by G. R. Liu and S. S. Quek
1D fin1D fin
Direct assembly procedure (Cont’d)
Element 2:
1 11 12 1
(2) (2) (2) (2)2 3R k k f
2 21 22 2
(2) (2) (2) (2)2 3R k k f
1 11 1 12 1
(1) (1) (1) (1)20 : 0R k k f
Considering all contributions to a node, and enforcing to zero
(Node 1)
2 21 1 22 11 12 2 1
(1) (1) (1) (1) (2) (2) (1) (2)1 2 30 : ( ) ( ) 0R R k k k k f f (Node 2)
2 21 2 22
(2) (2) (2) (2)3 20 : 0R k k f (Node 3)
![Page 24: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/24.jpg)
24Finite Element Method by G. R. Liu and S. S. Quek
1D fin1D fin
Direct assembly procedure (Cont’d)
In matrix form:
11 12 1 1
21 22 11 12 2 1
21 22 2
(1) (1) (1)
(1) (1) (2) (2) (1) (2)2
(2) (2) (2)3
0
0
k k f
k k k k f f
k k f
(Note: same as assembly introduced before)
![Page 25: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/25.jpg)
25Finite Element Method by G. R. Liu and S. S. Quek
1D fin1D fin
Worked example: Heat transfer in 1D fin
1 2 3 4 5
8 cm
(1 ) (2 ) (3 ) (4 )
0 .4 cm
1 cm Ccm
Wk
03
Ccm
Wh
021.0
Cf
020
8 0 C
Calculate temperature distribution using FEM.
4 linear elements, 5 nodes
![Page 26: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/26.jpg)
26Finite Element Method by G. R. Liu and S. S. Quek
1D fin1D fin
1 2 3 4 5
8 cm
(1 ) (2 ) (3 ) (4 )
0 .4 cm
1 cm Ccm
Wk
03
Ccm
Wh
021.0
Cf
020
8 0 C
Element 1, 2, 3: ( )eMk ,
( )eSf not required
Element 4:( )eMk ,
( )eSf required
0
3 0.40.6
2
kA W
l C
0
0.1 2.8 20.093
6 6
hPl W
C
00.1 0.4 0.04
WhA
C
0.1 2.8 20 25.6
2 2fhPl
W
0.1 0.4 20 0.8fhA W
![Page 27: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/27.jpg)
27Finite Element Method by G. R. Liu and S. S. Quek
1D fin1D fin
For element 1, 2, 3
1 1 2 1
1 1 1 26e kA hPl
l
k
1
12e fhPL
f
1,2,3 0.786 0.507
0.507 0.786
k
1,2,3 5.6
5.6
f
For element 4
1 1 2 1 0 0
1 1 1 2 06e kA hPL
hAL
k
01
12e f
f
hPL
hA
f
4 0.786 0.507
0.507 0.826
k
4 5.6
6.4
f
( ) ( )e e eD g k k k
( ) ( ) ( )e e e eD g M k k k k
( )e eQf f,
, ( ) ( )e e eQ S f f f
![Page 28: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/28.jpg)
28Finite Element Method by G. R. Liu and S. S. Quek
1D fin1D fin
*1
2
3
4
5
0.786 0.507 0 0 0 5.6
0.507 1.572 0.507 0 0 11.2 0
0 0.507 1.572 0.507 0 11.2 0
0 0 0.507 1.572 0.507 11.2 0
0 0 0 0.507 0.826 6.4 0
Q
Heat source(Still unknown)
*1
2
3
4
5
800.786 0.507 0 0 0 5.6
0.507 1.572 0.507 0 0 11.2 80 0.507
0 0.507 1.572 0.507 0 11.2
0 0 0.507 1.572 0.507 11.2
0 0 0 0.507 0.826 6.4
Q
1 = 80, four unknowns – eliminate Q*
Solving: {80.0 42.0 28.2 23.3 22.1}T Φ
![Page 29: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/29.jpg)
29Finite Element Method by G. R. Liu and S. S. Quek
Composite wallComposite wall
(1) (2) (3)
x
y
2
2
Heat supplyHeat conduction
0d
kA qdx
fbhAdx
dkA
fbhAdx
dkA
Convective boundary:
at x = 0
at x = H
All equations for 1D fin still applies except ( )e
Gk and ( )eQf
Therefore, 1 1
1 1e e
M
kA
l
k k , ( ) ( )e eSf f
vanish. Recall: Only for heat convection
![Page 30: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/30.jpg)
30Finite Element Method by G. R. Liu and S. S. Quek
Composite wallComposite wall
Worked example: Heat transfer through composite wall
5 c m 2
Cf05
Ccm
Wk
006.0
C020
Ccm
Wk
02.0
Ccm
Wh
021.0
( 1 ) ( 2 )
1 2 3
Calculate the temperature distribution across the wall using the FEM.
2 linear elements, 3 nodes
![Page 31: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/31.jpg)
31Finite Element Method by G. R. Liu and S. S. Quek
Composite wallComposite wall
5 c m 2
Cf05
Ccm
Wk
006.0
C020
Ccm
Wk
02.0
Ccm
Wh
021.0
( 1 ) ( 2 )
1 2 3
For element 1,
C
W
L
kA0
1.02
12.0
C
WhA
01.011.0
WhA f 5.0511.0
1 0.2 0.1
0.1 0.1
k
1 0.5
0S
f
1 1 0 0
1 1 0e kA
hAL
k
( ) ( )
0fe e
s
hA
f f
![Page 32: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/32.jpg)
32Finite Element Method by G. R. Liu and S. S. Quek
Composite wallComposite wall
5 c m 2
Cf05
Ccm
Wk
006.0
C020
Ccm
Wk
02.0
Ccm
Wh
021.0
( 1 ) ( 2 )
1 2 3
For element 2,
0
0.06 10.012
5
kA W
L C
1 1
1 1e kA
L
k 2 0.012 0.012
0.012 0.012
k
Upon assembly,
1
2
3
0.20 0.10 0 0.5 0
0.10 0.112 0.012 0 0
0 0.012 0.012 ( 20) 0 Q
(Unknown but required to balance equations)
![Page 33: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/33.jpg)
33Finite Element Method by G. R. Liu and S. S. Quek
Composite wallComposite wall
1
2
0.20 0.10 0.5
0.10 0.112 0.24( 20 0.012)
Solving: 2.5806 0.1613 20T Φ
![Page 34: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/34.jpg)
34Finite Element Method by G. R. Liu and S. S. Quek
Composite wallComposite wall
Worked example: Heat transfer through thin film layers
Raw material
0.2 mm
h=0.01 W/cm 2/ C f = 150C
0.2mm
Heater
k=0.1 W/cm/ C 2mm Glass
Iron Platinum
k=0.5 W/cm/C
k=0.4 W/cm/C
Substrate
Plasma
1
2 3 4
(1)
(2) (3)
300C
![Page 35: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/35.jpg)
35Finite Element Method by G. R. Liu and S. S. Quek
Composite wallComposite wall
1
2 3 4
(1)
(2) (3)
For element 1,
0
0.1 10.5
0.2
kA W
L C
1 0.5 0.5
0.5 0.5
k 1 1
1 1e kA
L
k
For element 2,
0
0.5 125
0.02
kA W
L C
1 1
1 1e kA
L
k 2 25 25
25 25
k
![Page 36: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/36.jpg)
36Finite Element Method by G. R. Liu and S. S. Quek
Composite wallComposite wall
For element 3,
0
0.4 120
0.02
kA W
L C
00.01 1 0.01
WhA
C
0.01 1 150 1.5fhA W
3 20 20
20 20.01
k
3 0
1.5S
f
1 1 0 0
1 1 0e kA
hAL
k
( ) ( )0
e es
fhA
f f
![Page 37: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/37.jpg)
37Finite Element Method by G. R. Liu and S. S. Quek
Composite wallComposite wall
*1
2
3
4
0.5 0.5 0 0
0.5 25.5 25 0 0
0 25 45 20 0
0 0 20 20.01 1.5
Q
2
3
4
25.5 25 0 150
25 45 20 0
0 20 20.01 1.5
T
T
T
Since, 1 = 300°C,
Solving: 300.0 297.1 297.0 296.9T Φ
* 20.5 300 0.5 297.1 1.45 W/cmQ
![Page 38: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/38.jpg)
38Finite Element Method by G. R. Liu and S. S. Quek
2D HEAT TRANSFER 2D HEAT TRANSFER PROBLEMPROBLEM
Element equationsElement equations2 2
2 20x yD D g Q
x y
1 2
3
For one element,
2 2( )
2 2( )d
e
h he T h
x yAD D g Q A
x y
R N
Note: W = N : Galerkin approach
![Page 39: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/39.jpg)
39Finite Element Method by G. R. Liu and S. S. Quek
Element equationsElement equations(Need to use Gauss’s divergence theorem to evaluate integral in residual.)
2 2( )
2 2( )d
e
h he T h
x yAD D g Q A
x y
R N
2
2
TT T
x x x x x
N
N N
2
2 2d ( )d d
e e e
TT T
x x xA A AD A D A D A
x x x x x
N
N NTherefore,
( )d cos de e
T T
AA
x x x
N NGauss’s divergence theorem:
2
2d cos d d
e
TT T
x x xA AD A D D A
x x xx
N
N N
(Product rule of differentiation)
![Page 40: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/40.jpg)
40Finite Element Method by G. R. Liu and S. S. Quek
Element equationsElement equations
2nd integral:2
2d sin d d
e
TT T
y y yA AD A D D A
y y y y
N
N N
( ) cos sin de
h he T
x yD Dx y
R N
de
T h T h
x yAD D A
x x y y
N N
d de e
T h T
A Ag A Q A N N
Therefore,
![Page 41: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/41.jpg)
41Finite Element Method by G. R. Liu and S. S. Quek
Element equationsElement equations
( )( ) ( )h ex x N Φ
( )
( ) cos sin de
e
h he T
x yD Dx y
b
R N
( )
( )de
eD
T Te
x yAD D A
x x y y
k
N N N NΦ
( ) ( )
( )( d ) de e
e eg Q
T e T
A Ag A Q A
k f
N N Φ N
![Page 42: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/42.jpg)
42Finite Element Method by G. R. Liu and S. S. Quek
Element equationsElement equations
( ) ( ) ( ) ( ) ( ) ( )[ ]e e e e e eD g Q R b k k Φ f
where ( ) cos sin de
h he T
x yD Dx y
b N
( ) de
T Te
D x yAD D A
x x y y
N N N Nk
( ) de
e Tg A
g Ak N N
( ) de
e TQ A
Q Af N
![Page 43: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/43.jpg)
43Finite Element Method by G. R. Liu and S. S. Quek
Element equationsElement equations
( ) de
T Te
D x yAD D A
x x y y
N N N Nk
0
0x
y
D
D
DDefine ( ) ( )e ex x
y y
N
Φ BΦN,
1 2
1 2
d
d
n
n
NN N
x x x xNN N
y y y y
N
BN
(Strain matrix)
T TT
x yD Dx x y y
N N N N
B DB ( ) d
e
e TD A
Ak B DB
![Page 44: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/44.jpg)
44Finite Element Method by G. R. Liu and S. S. Quek
Triangular elementsTriangular elements
1( ) ( )
1 2 3 2
3
[ ]e e N N N
NΦ
x,
y,
1 (x1, y1) 1
2 (x2, y2) 2
3 (x3, y3) 3
Ae
i i i iN a b x c y 1
( )2
1( )
2
1( )
2
i j k k je
i j ke
i k je
a x y x yA
b y yA
c x xA
( ) d de e
e T T TD eA A
A A A k B DB B DB B DB
2 2
( ) 2 2
2 24 4
i i j i k i i j i kye x
D i j j j k i j j j k
i k j k k i k j k k
b b b b b c c c c cDD
b b b b b c c c c cA A
b b b b b c c c c c
k
Note: constant strain matrix
(Or Ni = Li)
![Page 45: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/45.jpg)
45Finite Element Method by G. R. Liu and S. S. Quek
Triangular elementsTriangular elements
1( )
2 1 2 3
3
21 1 2 1 3
21 2 2 2 3
21 3 2 3 3
d [ ]d
d
e e
e
e Tg A A
A
N
g A g N N N N A
N
N N N N N
g N N N N N A
N N N N N
k N N
Apnm
pnmALLL pn
A
m 2)!2(
!!!d321
Note:
(Area coordinates)
E.g. 1 1 01 2 1 2 3
1!1!0! 1d d 2 2
(1 1 0 2)! 4 3 2 1 12e eA A
AN N A L L L A A A
( )
2 1 1
1 2 112
1 1 2
eg
gA
kTherefore,
![Page 46: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/46.jpg)
46Finite Element Method by G. R. Liu and S. S. Quek
Triangular elementsTriangular elements
1
( )2
3
1
d 13
1e e e
i
e TQ jA A A
k
N LQA
Q A Q N A Q L A
N L
f N d d
Similarly,
Note: b(e) will be discussed later
![Page 47: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/47.jpg)
47Finite Element Method by G. R. Liu and S. S. Quek
Rectangular elementsRectangular elements
1
2( ) ( )1 2 3 4
3
4
[ ]e e N N N N
NΦ
x
y
1 (x1, y1)
2 (x2, y2)
3 (x3, y3)
2a
4 (x4, y4)
2b
1 ( 1, 1)
2 (1, 1)
3 (1, +1)
2a
4 ( 1, +1)
2b
)1)(1(
)1)(1(
)1)(1(
)1)(1(
41
4
41
3
41
2
41
1
N
N
N
N
, byax
1 1 1 1
1 1 1 1
1 1 1 1 1 1 1 1
0 0 0 0
0 0 0 0a a a a
b b b b
a a a ab b b b
B
![Page 48: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/48.jpg)
48Finite Element Method by G. R. Liu and S. S. Quek
Rectangular elementsRectangular elements1 1( )
1 1 d d d
2 2 1 1 2 1 1 2
2 2 1 1 1 2 2 1
1 1 2 2 1 2 2 16 6
1 1 2 2 1 1 1 2
e
e T TD A
yx
A ab
D aD b
a b
k B DB B DB
1 1( )
1 1
21 1 2 1 3 1 4
21 2 2 2 3 2 3
21 3 2 3 3 3 4
21 4 2 4 3 4 4
d d d
d d
e
e
e T Tg A
A
g A abg
N N N N N N N
N N N N N N Nabg
N N N N N N N
N N N N N N N
k N N N N
( )
4 2 1 2
2 4 2 1
1 2 4 236
2 1 2 4
eg
gA
k
![Page 49: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/49.jpg)
49Finite Element Method by G. R. Liu and S. S. Quek
Rectangular elementsRectangular elements
1
2( )
3
4
1
1d
14
1
e e
e TQ A A
N
N QAQ A Q A
N
N
f N d
Note: In practice, the integrals are usually evaluated using the Gauss integration scheme
![Page 50: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/50.jpg)
50Finite Element Method by G. R. Liu and S. S. Quek
Boundary conditions and vector Boundary conditions and vector bb((e)e)
( ) ( ) ( )e e eI B b b b
Internal Boundary
i
j
k
p
m 1 2
3 n
q
r
(2)Ib
(2)Ib
(3)Bb
(3)Ib
Vanishing of bI(e)bB
(e) needs to be evaluated at boundary
![Page 51: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/51.jpg)
51Finite Element Method by G. R. Liu and S. S. Quek
Boundary conditions and vector Boundary conditions and vector bb((e)e)
1
2
Essential boundary is known
Natural boundary: the derivatives of is known
n
x
n
x
Need not evaluate ( )e
Bb
Need to be concern with bB
(e)
![Page 52: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/52.jpg)
52Finite Element Method by G. R. Liu and S. S. Quek
Boundary conditions and vector Boundary conditions and vector bb((e)e)
( ) cos sin de
h he T
x yD Dx y
b N
cos sinh h
x y bD D M Sx y
on natural boundary 2
cos sinh h h
x y bD D k M Sx y n
Heat flux across boundary
1
2
Essential boundary is known
Natural boundary: the derivatives of is known
n
x
n
x
![Page 53: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/53.jpg)
53Finite Element Method by G. R. Liu and S. S. Quek
Boundary conditions and vector Boundary conditions and vector bb((e)e)
Insulated boundary:
M = S = 0 ( ) 0eB b
Convective boundary condition:
qc
n
n
k
h b f
q kn
q h
fb hhn
k
M
S
kq kn
h b fq h
b fM
S
k h hn
, fM h S h
![Page 54: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/54.jpg)
54Finite Element Method by G. R. Liu and S. S. Quek
Boundary conditions and vector Boundary conditions and vector bb((e)e)
Specified heat flux on boundary:
n
n
k
s
q kn
q
sk qn
qk
qs M=0 S= qs
0 b sM S
k qn
0, sM S q
Positive if heat flows into the boundary
Negative if heat flows out off the boundary
0 insulated
S
![Page 55: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/55.jpg)
55Finite Element Method by G. R. Liu and S. S. Quek
Boundary conditions and vector Boundary conditions and vector bb((e)e)
For other cases whereby M, S 0
2
2
( ) cos sin d
( + )d
h he T
B x y
Tb
D Dx y
M S
b N
N
eb NΦ
2
2 2
( ) ( )
( ) ( )
( )
( + )d
( d ) d
e eM B
e T eB
T e T
M S
M S
b N NΦ
N N Φ N
k f
![Page 56: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/56.jpg)
56Finite Element Method by G. R. Liu and S. S. Quek
Boundary conditions and vector Boundary conditions and vector bb((e)e)
( ) ( ) ( ) ( )e e e eB M S b k Φ f
where2
( ) de TM M
k N N ,
2
( ) de TS S
f N
x
y
1 (x1, y1)
2 (x2, y2)
3 (x3, y3)
2a
4 (x4, y4)
2b
For a rectangular element,
1 2
1
1 2( )
13
4
de TS
N
NS d S a
N
N
f N
1
1
11
11d
02 0
00
eS
SaSa
f (Equal sharing between nodes 1 and 2)
![Page 57: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/57.jpg)
57Finite Element Method by G. R. Liu and S. S. Quek
Boundary conditions and vector Boundary conditions and vector bb((e)e)
Equal sharing valid for all elements with linear shape functions
,2 3
0
1
1
0
eS Sb
f ,3 4
0
0
1
1
eS Sa
f ,1 4
1
0
0
1
eS Sb
f
x,
y,
1 (x1, y1) 1
2 (x2, y2) 2
3 (x3, y3) 3
Ae
Applies to triangular elements too
12,1 2
1
12
0
eS
SL
f 23,2 3
0
12
1
eS
SL
f 13,1 3
1
02
1
eS
SL
f
![Page 58: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/58.jpg)
58Finite Element Method by G. R. Liu and S. S. Quek
Boundary conditions and vector Boundary conditions and vector bb((e)e)
( )eMk for rectangular element
2
21 1 2 1 3 1 4
2( ) 1 2 2 2 3 2 4
21 3 2 3 3 3 4
21 4 2 4 3 4 4
deM
N N N N N N N
N N N N N N NM
N N N N N N N
N N N N N N N
k
21 1 2
21( ) 2 1 2
,1 2 1
0 0
0 0d
0 0 0 0
0 0 0 0
eM
N N N
N N NaM
k
![Page 59: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/59.jpg)
59Finite Element Method by G. R. Liu and S. S. Quek
Boundary conditions and vector Boundary conditions and vector bb((e)e)
21 12
11 1
1 2d d
4 3N
1 1
1 21 1
1 1 2d d
4 6N N
21 12
21 1
1 2d d
4 3N
( ),1 2
2 16 6
1 26 6
2 1 0 0
1 2 0 02
0 0 0 06
0 0 0 0
0 0
0 0(2 )
0 0 0 0
0 0 0 0
eM
aM
aM
k
Shared in ratio 2/6, 1/6, 1/6, 2/6( ),2 3
0 0 0 0
0 2 1 02
0 1 2 06
0 0 0 0
eM
M b
k
( ),3 4
0 0 0 0
0 0 0 02
0 0 2 16
0 0 1 2
eM
M a
k ( ),1 4
2 0 0 1
0 0 0 02
0 0 0 06
1 0 0 2
eM
M b
k
![Page 60: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/60.jpg)
60Finite Element Method by G. R. Liu and S. S. Quek
Boundary conditions and vector Boundary conditions and vector bb((e)e)
Similar for triangular elements
( ),
2 1 0
1 2 06
0 0 0
ijeM i j
ML
k ( ),
0 0 0
0 2 16
0 1 2
jkeM j k
ML
k
( ),
2 0 1
0 0 06
1 0 2
e ikM i k
ML
k
![Page 61: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/61.jpg)
61Finite Element Method by G. R. Liu and S. S. Quek
Point heat source or sinkPoint heat source or sink
Preferably place node at source or sink
Q* i j
k
1
2
*
F
f
f
j Q
![Page 62: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/62.jpg)
62Finite Element Method by G. R. Liu and S. S. Quek
Point heat source or sink within the elementPoint heat source or sink within the element
( ) de
e TQ A
Q Af N
(X0, Y0)
Q*
i
j
k
x
y
00 YyXxQQ (Delta function)
( )0 0 d d
e
i
eQ jA
k
N
Q N x X y Y x y
N
f
Point source/sink
0 0
0 0
0 0
,
,
,
ie
Q j
k
N X Y
Q N X Y
N X Y
f
![Page 63: 1 Finite Element Method FEM FOR HEAT TRANSFER PROBLEMS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 12:](https://reader033.vdocument.in/reader033/viewer/2022061306/5514793d550346d36e8b45de/html5/thumbnails/63.jpg)
63Finite Element Method by G. R. Liu and S. S. Quek
SUMMARYSUMMARY
2 2
2 2
2 2
0fx y
h hk t k
y
t
xD D g Qx y
02
2
2
2
QGy
Dx
D yx
*
( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )e e
e e e e e e e e eD g M Q SQ
k f
R b k k k Φ f f f
( ) c o s s i n d
h fh
bM S
e Tx yD D
x y
b N
y
n
x
S o u r c e a n d s i n k
O n l y f o r e l e m e n t s o n t h e d e r i v a t i v e b o u n d a r y
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64Finite Element Method by G. R. Liu and S. S. Quek
CASE STUDYCASE STUDY
6 cm
2 cm
4 cm
Heating cables
Representative section to be analyzed
Insulation
f= 0C
Road surface heated by heating cables under road surface
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65Finite Element Method by G. R. Liu and S. S. Quek
CASE STUDYCASE STUDYHeat convectionM=h=0.0034S=f h=-0.017
Repetitive boundary no heat flow acrossM=0, S=0
Repetitive boundary no heat flow acrossM=0, S=0
InsulatedM=0, S=0
fQ*
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66Finite Element Method by G. R. Liu and S. S. Quek
CASE STUDYCASE STUDY
Node Temperature (C)
1 5.861
2 5.832
3 5.764
4 5.697
5 5.669
Surface temperatures: