1. FEM FOR HEAT TRANSFER PROBLEMS
Finite Element Method
for readers of all backgrounds
G. R. Liu and S. S. Quek
CHAPTER 12:
FEM FOR HEAT TRANSFER PROBLEMS

2. CONTENTS FIELD PROBLEMS WEIGHTED RESIDUAL APPROACH FOR FEM
1D HEAT TRANSFER PROBLEMS
2D HEAT TRANSFER PROBLEMS
SUMMARY
CASE STUDY

3. FIELD PROBLEMS
General form of system equations of 2D linear steady state field problems:
(Helmholtz equation)
For 1D problems:

4. FIELD PROBLEMS
Heat transfer in 2D fin
Note:

5. FIELD PROBLEMS Heat transfer in long 2D body Note: Dx = kx, Dy =tky,
g = 0 and Q = q

6. FIELD PROBLEMS
Heat transfer in 1D fin
Note:

7. FIELD PROBLEMS
Heat transfer across composite wall
Note:

8. FIELD PROBLEMS Torsional deformation of bar
Note:
Dx=1/G, Dy=1/G, g=0, Q=2q
( - stress function)
Ideal irrotational fluid flow
Note:
Dx = Dy = 1, g = Q = 0
( - streamline function and  - potential function)

9. FIELD PROBLEMS Accoustic problems
P - the pressure above the ambient pressure ;
w - wave frequency ;
c - wave velocity in the medium
Note:
, Dx = Dy = 1, Q = 0

10. WEIGHTED RESIDUAL APPROACH FOR FEM
Establishing FE equations based on governing equations without knowing the functional.
(Strong form)
Approximate solution:
(Weak form)
Weight function

11. WEIGHTED RESIDUAL APPROACH FOR FEM
Discretize into smaller elements to ensure better approximation
In each element,
Using N as the weight functions
where
Galerkin method
Residuals are then assembled for all elements and enforced to zero.

12. 1D HEAT TRANSFER PROBLEM
1D fin
k : thermal conductivity
h : convection coefficient
A : cross-sectional area of the fin
P : perimeter of the fin
: temperature, and
f : ambient temperature in the fluid
(Specified boundary condition)
(Convective heat loss at free end)

13. 1D fin
Using Galerkin approach,
where
D = kA, g = hP, and Q = hP

14. 1D fin
Integration by parts of first term on right-hand side,
Using

15. 1D fin (Strain matrix) where (Thermal conduction) (Thermal convection)
(External heat supplied)
(Temperature gradient at two ends of element)

16. 1D fin For linear elements, (Recall 1D truss element) Therefore,
for truss element
(Recall stiffness matrix of truss element)

17. 1D fin
for truss element
(Recall mass matrix of truss element)

18. 1D fin or (Left end) (Right end)
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.

19. 1D fin When there is heat convection at boundary, E.g.
Since b is the temperature of the fin at the boundary point,
b = j
Therefore,

20. 1D fin
where ,
For convection on left side,
where ,

21. 1D fin
Therefore,
Residuals are assembled for all elements and enforced to zero: KD = F
Same form for static mechanics problem

22. 1D fin
Direct assembly procedure or
Element 1:

23. 1D fin Direct assembly procedure (Cont’d) Element 2:
Considering all contributions to a node, and enforcing to zero
(Node 1)
(Node 2)
(Node 3)

24. 1D fin Direct assembly procedure (Cont’d) In matrix form:
(Note: same as assembly introduced before)

25. 1D fin Worked example: Heat transfer in 1D fin
Calculate temperature distribution using FEM.
4 linear elements, 5 nodes

26. 1D fin
Element 1, 2, 3: ,
not required
Element 4: ,
required

27. 1D fin
For element 1, 2, 3 ,
For element 4 ,

28. 1D fin Heat source (Still unknown)
1 = 80, four unknowns – eliminate Q*
Solving:

29. Composite wall Convective boundary: at x = 0 at x = H
All equations for 1D fin still applies except
Recall: Only for heat convection
and
vanish.
Therefore, ,

30. Composite wall Worked example: Heat transfer through composite wall
Calculate the temperature distribution across the wall using the FEM.
2 linear elements, 3 nodes

31. Composite wall
For element 1,

32. Composite wall For element 2, Upon assembly,
(Unknown but required to balance equations)

33. Composite wall
Solving:

34. Composite wall Worked example: Heat transfer through thin film layers
Raw material
0.2 mm h
=0.01 W/cm 2 / C f
= 150
0.2mm
Heater k
=0.1 W/cm/
2mm
Glass
Iron
Platinum
=0.5 W/cm/
=0.4 W/cm/
Substrate
Plasma 1 3 4
(1)
(2)
(3)
300C

35. Composite wall
For element 1, 1 2 3 4
(1)
(2)
(3)
For element 2,

36. Composite wall
For element 3,

37. Composite wall
Since, 1 = 300°C,
Solving:

38. 2D HEAT TRANSFER PROBLEM
Element equations
For one element,
Note: W = N : Galerkin approach

39. Element equations Therefore, Gauss’s divergence theorem:
(Need to use Gauss’s divergence theorem to evaluate integral in residual.)
(Product rule of differentiation)
Therefore,
Gauss’s divergence theorem:

40. Element equations
2nd integral:
Therefore,

41. Element equations

42. Element equations
where

43. Element equations
Define ,
(Strain matrix) 

44. Triangular elements
Note: constant strain matrix
(Or Ni = Li)

45. Triangular elements
Note:
(Area coordinates)
E.g.
Therefore,

46. Triangular elements
Similarly,
Note: b(e) will be discussed later

47. Rectangular elements

48. Rectangular elements

49. Rectangular elements
Note: In practice, the integrals are usually evaluated using the Gauss integration scheme

50. Boundary conditions and vector b(e)
Internal
Boundary
bB(e) needs to be evaluated at boundary
Vanishing of bI(e)

51. Boundary conditions and vector b(e)
Need not evaluate
Need to be concern with bB(e)

52. Boundary conditions and vector b(e)
on natural boundary 2
Heat flux across boundary

53. Boundary conditions and vector b(e)
Insulated boundary:
M = S = 0 
Convective boundary condition:

54. Boundary conditions and vector b(e)
Specified heat flux on boundary:

55. Boundary conditions and vector b(e)
For other cases whereby M, S  0

56. Boundary conditions and vector b(e)
where ,
For a rectangular element,
(Equal sharing between nodes 1 and 2)

57. Boundary conditions and vector b(e)
Equal sharing valid for all elements with linear shape functions
Applies to triangular elements too

58. Boundary conditions and vector b(e)
for rectangular element

59. Boundary conditions and vector b(e)
Shared in ratio 2/6, 1/6, 1/6, 2/6

60. Boundary conditions and vector b(e)
Similar for triangular elements

61. Point heat source or sink
Preferably place node at source or sink

62. Point heat source or sink within the element
Point source/sink
(Delta function) 

63. SUMMARY

64. CASE STUDY
Road surface heated by heating cables under road surface

65. CASE STUDY Heat convection M=h=0.0034 S=ff h=-0.017 fQ*
Repetitive boundary no heat flow across
M=0, S=0
Repetitive boundary no heat flow across
M=0, S=0
Insulated
M=0, S=0

66. CASE STUDY Surface temperatures: Node Temperature (C) 1 5.861 2 5.832
5.764 4
5.697 5
5.669