F inite Element Method

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

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