# FEM FOR HEAT TRANSFER PROBLEMS

## Presentation on theme: "FEM FOR HEAT TRANSFER PROBLEMS"— Presentation transcript:

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

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

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

FIELD PROBLEMS Heat transfer in 2D fin Note:

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

FIELD PROBLEMS Heat transfer in 1D fin Note:

FIELD PROBLEMS Heat transfer across composite wall Note:

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)

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

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

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.

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)

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

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

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

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

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

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.

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,

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

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

1D fin Direct assembly procedure or Element 1:

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)

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

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

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

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

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

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, ,

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

Composite wall For element 1,

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

Composite wall Solving:

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

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

Composite wall For element 3,

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

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

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:

Element equations 2nd integral: Therefore,

Element equations

Element equations where

Element equations Define , (Strain matrix)

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

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

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

Rectangular elements

Rectangular elements

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

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

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

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

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

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

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

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

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

Boundary conditions and vector b(e)
for rectangular element

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

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

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

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

SUMMARY