finite volume method for predicting radiant heat transfer
DESCRIPTION
Mid semester presentation at IIT Kharagpur, Department Of Aerospace EngineeringTRANSCRIPT
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