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

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Finite Element Method by G. R. Liu and S. S. Quek 2 CONTENTS FIELD PROBLEMS WEIGHTED RESIDUAL APPROACH FOR FEM 1D HEAT TRANSFER PROBLEMS 2D HEAT TRANSFER PROBLEMS SUMMARY CASE STUDY

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Finite Element Method by G. R. Liu and S. S. Quek 3 FIELD PROBLEMS General form of system equations of 2D linear steady state field problems: (Helmholtz equation) For 1D problems:

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Finite Element Method by G. R. Liu and S. S. Quek 4 FIELD PROBLEMS Heat transfer in 2D fin Note:

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Finite Element Method by G. R. Liu and S. S. Quek 5 FIELD PROBLEMS Heat transfer in long 2D body Note: D x = k x, Dy =tk y, g = 0 and Q = q

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Finite Element Method by G. R. Liu and S. S. Quek 6 FIELD PROBLEMS Heat transfer in 1D fin Note:

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Finite Element Method by G. R. Liu and S. S. Quek 7 FIELD PROBLEMS Heat transfer across composite wall Note:

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Finite Element Method by G. R. Liu and S. S. Quek 8 FIELD PROBLEMS Torsional deformation of bar Note: D x =1/G, D y =1/G, g=0, Q=2 Ideal irrotational fluid flow Note: D x = D y = 1, g = Q = 0 ( - stress function) ( - streamline function and - potential function)

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Finite Element Method by G. R. Liu and S. S. Quek 9 FIELD PROBLEMS Accoustic problems Note:, D x = D y = 1, Q = 0 P - the pressure above the ambient pressure ; w - wave frequency ; c - wave velocity in the medium

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Finite Element Method by G. R. Liu and S. S. Quek 10 WEIGHTED RESIDUAL APPROACH FOR FEM Establishing FE equations based on governing equations without knowing the functional. Approximate solution: Weight function (Strong form) (Weak form)

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Finite Element Method by G. R. Liu and S. S. Quek 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.

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Finite Element Method by G. R. Liu and S. S. Quek 12 1D HEAT TRANSFER PROBLEM 1D fin (Specified boundary condition) (Convective heat loss at free end) 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

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Finite Element Method by G. R. Liu and S. S. Quek 13 1D fin Using Galerkin approach, where D = kA, g = hP, and Q = hP

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Finite Element Method by G. R. Liu and S. S. Quek 14 1D fin Integration by parts of first term on right-hand side, Using

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Finite Element Method by G. R. Liu and S. S. Quek 15 1D fin where (Thermal conduction) (Thermal convection) (External heat supplied) (Temperature gradient at two ends of element) (Strain matrix)

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Finite Element Method by G. R. Liu and S. S. Quek 16 1D fin For linear elements, (Recall 1D truss element) Therefore, (Recall stiffness matrix of truss element) for truss element

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Finite Element Method by G. R. Liu and S. S. Quek 17 1D fin (Recall mass matrix of truss element) for truss element

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Finite Element Method by G. R. Liu and S. S. Quek 18 1D fin or (Left end)(Right end) At the internal nodes of the fin, b L (e) and b L (e) vanish upon assembly. At boundaries, where temperature is prescribed, no need to calculate b L (e) or b L (e) first.

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Finite Element Method by G. R. Liu and S. S. Quek 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,

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Finite Element Method by G. R. Liu and S. S. Quek 20 1D fin where, For convection on left side, where,

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Finite Element Method by G. R. Liu and S. S. Quek 21 1D fin Therefore, Residuals are assembled for all elements and enforced to zero: KD = F Same form for static mechanics problem

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Finite Element Method by G. R. Liu and S. S. Quek 22 1D fin Direct assembly procedure or Element 1:

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Finite Element Method by G. R. Liu and S. S. Quek 23 1D fin Direct assembly procedure (Contd) Element 2: Considering all contributions to a node, and enforcing to zero (Node 1) (Node 2) (Node 3)

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Finite Element Method by G. R. Liu and S. S. Quek 24 1D fin Direct assembly procedure (Contd) In matrix form: (Note: same as assembly introduced before)

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Finite Element Method by G. R. Liu and S. S. Quek 25 1D fin Worked example: Heat transfer in 1D fin Calculate temperature distribution using FEM. 4 linear elements, 5 nodes

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Finite Element Method by G. R. Liu and S. S. Quek 26 1D fin Element 1, 2, 3:, not required Element 4:,required

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Finite Element Method by G. R. Liu and S. S. Quek 27 1D fin For element 1, 2, 3 For element 4,,

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Finite Element Method by G. R. Liu and S. S. Quek 28 1D fin Heat source (Still unknown) 1 = 80, four unknowns – eliminate Q * Solving:

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Finite Element Method by G. R. Liu and S. S. Quek 29 Composite wall Convective boundary: at x = 0 at x = H All equations for 1D fin still applies except and Therefore,, vanish. Recall: Only for heat convection

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Finite Element Method by G. R. Liu and S. S. Quek 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

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Finite Element Method by G. R. Liu and S. S. Quek 31 Composite wall For element 1,

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Finite Element Method by G. R. Liu and S. S. Quek 32 Composite wall For element 2, Upon assembly, (Unknown but required to balance equations)

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Finite Element Method by G. R. Liu and S. S. Quek 33 Composite wall Solving:

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Finite Element Method by G. R. Liu and S. S. Quek 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 C 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) 300 C

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Finite Element Method by G. R. Liu and S. S. Quek 35 Composite wall 1 2 3 4 (1) (2) (3) For element 1, For element 2,

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Finite Element Method by G. R. Liu and S. S. Quek 36 Composite wall For element 3,

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Finite Element Method by G. R. Liu and S. S. Quek 37 Composite wall Since, 1 = 300°C, Solving:

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Finite Element Method by G. R. Liu and S. S. Quek 38 2D HEAT TRANSFER PROBLEM Element equations For one element, Note: W = N : Galerkin approach

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Finite Element Method by G. R. Liu and S. S. Quek 39 Element equations (Need to use Gausss divergence theorem to evaluate integral in residual.) Therefore, Gausss divergence theorem: (Product rule of differentiation)

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Finite Element Method by G. R. Liu and S. S. Quek 40 Element equations 2 nd integral: Therefore,

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Finite Element Method by G. R. Liu and S. S. Quek 41 Element equations

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Finite Element Method by G. R. Liu and S. S. Quek 42 Element equations where

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Finite Element Method by G. R. Liu and S. S. Quek 43 Element equations Define, (Strain matrix)

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Finite Element Method by G. R. Liu and S. S. Quek 44 Triangular elements Note: constant strain matrix (Or N i = L i )

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Finite Element Method by G. R. Liu and S. S. Quek 45 Triangular elements Note: (Area coordinates) E.g. Therefore,

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Finite Element Method by G. R. Liu and S. S. Quek 46 Triangular elements Similarly, Note: b (e) will be discussed later

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Finite Element Method by G. R. Liu and S. S. Quek 47 Rectangular elements

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Finite Element Method by G. R. Liu and S. S. Quek 48 Rectangular elements

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Finite Element Method by G. R. Liu and S. S. Quek 49 Rectangular elements Note: In practice, the integrals are usually evaluated using the Gauss integration scheme

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Finite Element Method by G. R. Liu and S. S. Quek 50 Boundary conditions and vector b (e) InternalBoundary Vanishing of b I (e) b B (e) needs to be evaluated at boundary

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Finite Element Method by G. R. Liu and S. S. Quek 51 Boundary conditions and vector b (e) Need not evaluate Need to be concern with b B (e)

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Finite Element Method by G. R. Liu and S. S. Quek 52 Boundary conditions and vector b (e) on natural boundary 2 Heat flux across boundary

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Finite Element Method by G. R. Liu and S. S. Quek 53 Boundary conditions and vector b (e) Insulated boundary: M = S = 0 Convective boundary condition:

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Finite Element Method by G. R. Liu and S. S. Quek 54 Boundary conditions and vector b (e) Specified heat flux on boundary:

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Finite Element Method by G. R. Liu and S. S. Quek 55 Boundary conditions and vector b (e) For other cases whereby M, S 0

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Finite Element Method by G. R. Liu and S. S. Quek 56 Boundary conditions and vector b (e) where, For a rectangular element, (Equal sharing between nodes 1 and 2)

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Finite Element Method by G. R. Liu and S. S. Quek 57 Boundary conditions and vector b (e) Equal sharing valid for all elements with linear shape functions Applies to triangular elements too

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Finite Element Method by G. R. Liu and S. S. Quek 58 Boundary conditions and vector b (e) for rectangular element

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Finite Element Method by G. R. Liu and S. S. Quek 59 Boundary conditions and vector b (e) Shared in ratio 2/6, 1/6, 1/6, 2/6

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Finite Element Method by G. R. Liu and S. S. Quek 60 Boundary conditions and vector b (e) Similar for triangular elements

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Finite Element Method by G. R. Liu and S. S. Quek 61 Point heat source or sink Preferably place node at source or sink

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Finite Element Method by G. R. Liu and S. S. Quek 62 Point heat source or sink within the element (Delta function) Point source/sink

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Finite Element Method by G. R. Liu and S. S. Quek 63 SUMMARY

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Finite Element Method by G. R. Liu and S. S. Quek 64 CASE STUDY Road surface heated by heating cables under road surface

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Finite Element Method by G. R. Liu and S. S. Quek 65 CASE STUDY Heat convection M=h=0.0034 S= f h=-0.017 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 fQ*fQ*

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Finite Element Method by G. R. Liu and S. S. Quek 66 CASE STUDY NodeTemperature ( C) 15.861 25.832 35.764 45.697 55.669 Surface temperatures:

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