two-dimensional heat analysis finite element method 20 november 2002 michelle blunt brian coldwell

Two-Dimensional Heat AnalysisFinite Element Method

20 November 2002

Michelle Blunt

Brian Coldwell

Two-Dimensional Heat Transfer

Fundamental Concepts Solution Methods

• Adiabatic

• Heat Flux

• Steady-State

• Finite Differences

• Finite Element Analysis

• Mathematical

• Experimental

• Theoretical

dtAqUdtdxAQdtAq

EUEE

outin

outgenin

time t area sectional-cross A

energy stored Usourceheat internal Q

energy kinetic E conductedheat q

dtAqUdtdxAQdtAq

EUEE

outin

outgenin

time t area sectional-cross A

energy stored Usourceheat internal Q

energy kinetic E conductedheat q

dx

dTKq xxx

change re temperatu dT

tyconductivi thermalK

One-Dimensional Conduction

Two-Dimensional Conduction

dtAqdtAqU

dtdxAQdtAqdtAq

EUEE

outZoutX

inZinX

outgenin

dz

dTK

dx

dTKq zzxxx

Experimental Model

• Two-dimensional heat transfer plate from lab 6.

•Upper and left boundary conditions are set at 0oC; lower and right conditions are constant at 80oC.

Theoretical ModelFinite Difference

CAT CTA

:equations of system Solve

Δx

2TT

x

T

x

T

:order 2nd

Δx

TT

dx

dT

x

T

:order1st

1

2,n1,mn1,m

2

2

n1,mnm,

nmTx

f

x

f1

f2f3 f4

f5 f6

x

f Subdomain We

Domain divided with subdomainswith degrees of freedom

Domain W

x

x

Domain with degrees of freedom

f

f

The fundamental concept of FEM is that a continuous function of a continuum (given domain W) having infinite degrees of freedom is replaced by a discrete model, approximated by a set of piecewise continuous functions having a finite degree of freedom.

f6f5

f4f3

f2

f1

f

x

Theoretical ModelFinite Element

Structural vs Heat Transfer

Structural Analysis Thermal Analysis

•Assume displacement function

•Stress/strain relationships

•Derive element stiffness

•Assemble element equations

•Solve nodal displacements

•Solve element forces

•Select element type

•Assume temperature function

•Temperature relationships

•Derive element conduction

•Assemble element equations

•Solve nodal temperatures

•Solve element gradient/flux

•Select element type

Finite Element 2-D Conduction

• 1-d elements are lines• 2-d elements are either

triangles, quadrilaterals, or a mixture as shown

• Label the nodes so that the difference between two nodes on any element is minimized.

Select Element Type

Finite Element 2-D ConductionAssume (Choose) a Temperature Function

3 Nodes 1 Element

2 DOF: x, y

Assume a linear temperature function for each element as:

3

2

1

321

321

y x 1

),(

a

a

a

yaxaa

yaxaayxt

where u and v describe temperature gradients at (xi,yi).

Finite Element 2-D ConductionAssume (Choose) a Temperature Function

re temperatunodalt

function shapeN

function etemperaturT

m

j

i

mji

mmjjii

t

t

t

NNNT

tNtNtNT

Finite Element 2-D ConductionDefine Temperature Gradient Relationships

mji

mji

m

j

i

mji

mji

xN

xB

t

t

t

y

N

y

N

y

N

x

N

x

N

x

N

y

Tx

T

g

1

Analogous to strain matrix: {g}=[B]{t}

[B] is derivative of [N]

gDgq

q

y

x

yy

xx

K 0

0 K

:Gradient ratureflux/TempeHeat

Finite Element 2-D ConductionDerive Element Conduction Matrix and Equations

2 1

1 2

6

1 1-

1- 1

1

1

Convection Conduction

0

hPL

L

AK

dxL

x

L

x

L

xL

x

hPBDBtA

dSNNhdVBDBk

xx

LT

T

V S

T

Finite Element 2-D ConductionDerive Element Conduction Matrix and Equations

sourceheat constant for

1

1

1

3

QV

dVVQfT

V

Q

elementeach for

tkf

Stiffness matrix is general term for a matrix of known coefficients being multiplied by unknown degrees of freedom, i.e., displacement OR temperature, etc. Thus, the element conduction matrix is often referred to as the stiffness matrix.

Finite Element 2-D ConductionAssemble Element Equations, Apply BC’s

tKF

From here on virtually the same as structural approach. Heat flux boundary conditions already accounted for in derivation. Just substitute into above equation and solve for the following:

Solve for Nodal Temperatures

Solve for Element Temperature Gradient & Heat Flux

Algor: How many elements?

Elements: 9 Time: 6s

Nodes: 16 Memory: 0.239MB


Automatic Mesh



Algor Results Options

• Higher accuracy• More time, memory

• Faster• Less storage space


Smaller Elements Fewer Elements

References

Kreyszig, Erwin. Advanced Engineering Mathematics, 8th ed.(1999)

Chapters: 8, 9

Logan, Daryl L. A First Course in the Finite Element Method Using Algor, 2nd ed.(2001)

Chapters: 13

Questions?

Ha ha ha!!!

Here comes your assignment…

two-dimensional heat analysis finite element method 20 november 2002 michelle blunt brian coldwell

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