Introduction to Finite Elements

Heat Transfer

One-Dimensional Conduction

Ein = energy entering volume (J or kWh) A = area perpendicular to heat

U = change in stored energy (kWh) flow (m2)

qx = heat conducted/heat flux (kW/m2) Kxx = thermal conductivity (kW/m.

oC)

t = time (s) T = temperature (oC)

Q = internal heat source (kW/m3) dT/dx = temperature gradient (oC/m)

One-Dimensional Conduction

U = change in stored energy (kWh)

c = specific heat (kW.h/(kg.oC))

= mass density (kg/m3)

A = area perpendicular to heat flow (m2)

T = temperature (oC)

One-Dimensional Conduction

Heat Transfer with Convection

h = convection coefficient (kW/m2.oC)

T = temperature of solid surface at the

interface (oC)

T = temperature of the fluid (oC)

P = perimeter around the constant

cross-sectional area A

Heat Transfer with Convection

fx

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. f6

f5

f4f3

f2

f1

f

x

Theoretical Model Finite 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 temperature gradient/flux

Select element type

One-Dimensional FEM

One-Dimensional FEM

One-Dimensional FEM

(Analogous to the potential energy method)

One-Dimensional FEM

We cannot specify q* and h on the same surface because they cannot

occur simultaneously on the same surface

One-Dimensional FEM

(Analogous to the structural problem)

One-Dimensional FEM On minimizing with respect to t :

[k]{t} = {f}

One-Dimensional FEM

One-Dimensional FEM

2 1

1 2

6

1 1-

1- 1

1

1

Convection Conduction

0

hPL

L

AK

dxL

x

L

x

L

x

L

x

hPBDBtA

dSNNhdVBDBk

xx

LT

T

V S

T

One-Dimensional FEM

2 1

1 2

6

1 1-

1- 1

1

1

Convection Conduction

0

hPL

L

AK

dxL

x

L

x

L

x

L

x

hPBDBtA

dSNNhdVBDBk

xx

LT

T

V S

T

One-Dimensional FEM

One-Dimensional FEM

2 1

1 2

6

1 1-

1- 1

1

1

Convection Conduction

0

hPL

L

AK

dxL

x

L

x

L

x

L

x

hPBDBtA

dSNNhdVBDBk

xx

LT

T

V S

T

One-Dimensional FEM

One-Dimensional FEM