Finite Volume MethodFor Predicting

Radiant Heat Transfer

Rajibul Alam

GUIDE

Mid Semester Presentation 23rd September,2015

Dr. Ratan Joarder Department Of Aerospace Engineering IIT Kharagpur

Why? There is a solver already to solve Navier-Stoke equation. In the

present solver, heat transfer term present on the right hand side consists of conduction and convection heat transfers only.

How ever there are several cases like combustion chamber, reentry vehicle etc. where radiation heat transfer is not negligible.

In such cases it becomes necessary to predict radiant heat transfer correctly to save shielding material.

What? Finite volume method used for other cases gives satisfactory

results.

It’s convenient if the radiation model is based on the same philosophy and even if it uses same computational grid as the solver already existing to solve the Navier-Stoke equation.

As the existing solver uses finite volume method, hence to introduce radiation term, finite volume method is used.

How?

Radiative Transfer Equation

With the above quantities , if we do an energy balance, Radiative Transfer Equation is obtained which is given as

•

Where,

(1)

(2)

)().( IKIIKdsdI S

bS

4')',()',(

41),( dsssrIsrI

Integration of the above equation over any specified volume and applying divergence theorem yields the desired integral equation which is

This equation demands that the net radiant energywithin that leaves through all surfaces of the volume is exactly balanced by the excess, within the volume, of inscattering and emission over outscattering and absorption.

dVdIKIIKddAnsI S

V bS

ASS

S

])([).(

Discrete Equation, Level 1 Approximation :

The objective of the finite volume method is to find ,theintensity at each node P associated with each discrete bundle An algebraic equation for is obtained by introducingapproximations into equation (3).

With such an approximation, right hand side of the equation becomes

(4)

lP

lP

SP

lbPP

lP

SPP

S

V bS VIIKIKddVIKIIK

lP

])([])([

To further simplify, control volume surface is sub divided into f no of elementary surface panels . Denoting radiant heat transfer through each panel as , the integrated equation becomes ,

Level 2 approximation: Relations are required between intensity at integration point f

and intensity at the node P. Here ray is traced back to the point uf where intensity can easily found by the interpolation of nodes.

(5)llP

SPbpP

lP

SPPf

N

fffS

N

f

lf VIIKIKdnsIAQ

f

l

f

])([).(1

,1

Such a determination of intensity at the integration point f and putting it back in the basic equation gives,

Where,

(6)

ljff

j j

l

fSk

f

fSlf

SklffS

lf

SkluffS

lf D

xRSke

kA

NeRANeIAQ fff )())1(1(13

1

,,,

l ldnsseDdnsN

kIKIRKk

fjljf

lf

Sb

S

).().(,).( and

/)( and

For special case of radiative equilibrium and isotropic scattering, the value of R at a node for all solid angle becomes equal to the average intensity at that node which is also equal to the black body radiation at that node.

i.e.

With these basics, an one dimensional radiation problem is solved.

4')',(

41 dsrIIIRR bPaPP

lP

Figure shows two isothermal diffuse plates at temperatures and emissivities respectively enclosing a scattering emitting medium and in radiative equilibrium.

Finding the heat transfer on the two surfaces above and below a Given node P, which are designated as North node and South nodeand putting these values in equations (5) & (6) , gives a relation,

Where the notations have their usual meanings.

Now with an assumed value of intensity at the lower plate, intensities at different nodes are found ,sweeping upward till the second last node.

With that intensity, heat transfer from the upper plate is determined .

(7)lP

lP

lN

lS

lS

lP

lP bIaIaIa

Once heat transfer on the upper surface is obtained, intensity on the upper surface is also calculated with the diffuse surface boundary condition which is given by

A similar treatment but in downward direction is done with the intensity on the upper surface.

This upward and downward sweeping constitutes one cycle. This cycle is repeated till two consecutive values of intensities at any node have difference more than 0.000001. [ 10^(-6) ]

The value of R is then iterated till the two successive values of heat transfer on either plate have difference more than 0.0001 [10^(-4)]

Values of heat transfers on either plate for all the solid angles is then summed up to give total heat transfer on that plate.

0. 3es

lTS

lT

l

IAQ

Current Status Of The Project :

A program is written in FORTRAN95 for the above problem. Though the program runs without any error, but desired accuracy is not achieved yet.

References:

1) A Finite volume Method for Predicting a Radiant Heat Transfer in Enclosures With ParticipatingMedia- G. D. Raithby & E. H. Chui

2) Finite Volume Method for Radiation Heat Transfer -John C. Chai , HaeOk S. Lee & Suhas V. Patankar

3)NUMERICAL MODELING AND ANALYSIS OF HEAT TRANSFER IN SEMITRANSPARENT :JEDIA WITH COMBINED RADIATION AND CONDUCTION -ZIRONG HU

4) RADIATION HEAT TRANSFER IN COMBUSTION SYSTEMS -R. VISKANTA and M. P. MENGD