Finite Element Radiative and Conductive Module for use with PHOENICS

† Department of Materials Engineering, University of Swansea, Swansea , SA2 8PP, UK‡ DERA (Defence and Evaluation Research Agency) Farnborough, Hampshire, UK

Nick Lavery †

S.Brown & J.Spittle †L.Hayward & S. Rooks ‡

Structure of this presentation

1. Introduction

2. Governing equations & finite element solution

3. Numerical examples and validation of model

4. PHOENICS Interfacing

5. Conclusions & future work

The transient heat conduction equation:

The Stefan-Boltzmann equation:

The view factor equation:

2.1 Governing Equations

Qz

TK

zy

TK

yx

TK

xt

Tc zzyyxxp

JI

A A

JI

IIJ dAdA

rAF

I J

2

coscos1

(3)

)( 44JIJIIJI TTFAq (2)

(1)

2.2 Finite Element Solution

dSqNdVQN

dVz

N

z

N

y

N

y

N

x

N

x

N

dVNN

BS

i

V

i

V

jijiji

V

ij

F

K

M

FKTTM

Using 8 noded linear hexahedral elements, and the standard

Galerkin finite element discretisation:

(4)

(5)

3.1 View factor verification

Фa

cb

A2

A1

c

b

A1

a

A2

3.2 Transient heat conduction

3.3 Experimental rig for radiative heat transfer

Thermocouple 1Thermocouple 2

Thermocouple 3

3.3.1 Finite element prediction of experimental rig

Thermocouples

12

3

3.3.2 Experimental and numerical comparison

12

3

Position of fire rod

Temperature contours at t=10 s

3.7 Gnome exhaust nozzle

4. Integration with PHOENICS

• Create a coupled fluid/solid mesh (e.g. with FEMSYS)

• Solve for steady state fluid/gas flow equations PHOENICS

• Read in fluid FE mesh with boundary conditions (b.c.)

• Read in run parameters and material properties

• Calculate view factors for radiative heat transfer

• For each time step– Calculate F from equation (5) – Solve matrix equation (4) *– Update temperature, Tn+1

* The thermal conduction equation (1) is modified to include velocities for the gas/fluid

4. Conclusions and future work

• A finite element solver of radiative and conductive heat

transfer has been created

• The solver is accurate when compared to other numerical

and experimental results

• The solver is suitable for application to geometry of varying

complexity

• Fluid flow will be incorporated by the end of this year in the

form of flow fields from Phoenics

3.3.3 Animation of conducted temperature

Position of fire rod