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### 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

- 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

(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

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

Therefore,

Residuals are assembled for all elements and enforced to zero: KD = F

Same form for static mechanics problem

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

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

Solving:

300C

·

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

Composite wall

For element 3,

Element equations

(Need to use Gauss’s divergence theorem to evaluate integral in residual.)

(Product rule of differentiation)

Therefore,

Gauss’s divergence theorem:

Element equations

where

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)

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

CASE STUDY

Road surface heated by heating cables under road surface

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

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