RADIATION MODELING FOR
BIO-MEDICAL APPLICATIONS
A THESIS SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
Bachelor of Technology
In
Mechanical Engineering
By
SAMIR CHOUDHURY
Department of Mechanical Engineering
National Institute of Technology
Rourkela, Orissa
2009
RADIATION MODELING FOR
BIO-MEDICAL APPLICATIONS
A THESIS SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
Bachelor of Technology
In
Mechanical Engineering
By
SAMIR CHOUDHURY
Under the Guidance of
Prof. S. K. Mahapatra
Department of Mechanical Engineering
National Institute of Technology
Rourkela, Orissa
2009
National Institute of Technology
Rourkela
CERTIFICATE
This is to certify that the thesis entitled, “RADATION MODELING FOR BIO-MEDICAL
APPLICATIONS” submitted by Sri Samir Choudhury in partial fulfillment of the requirements for
the award of Bachelor of Technology Degree in Mechanical Engineering at the National Institute of
Technology, Rourkela (Deemed University) is an authentic work carried out by him under my
supervision and guidance.
To the best of my knowledge, the matter embodied in the thesis has not been submitted to any other
University / Institute for the award of any Degree.
Date:…………………
Prof. S. K. Mahapatra
Mechanical Engineering
NIT ROURKELA
ACKNOWLEDGEMENT
The satisfaction and euphoria on the successful completion of any task would be
incomplete without the mention of the people who made it possible whose constant
guidance an encouragement crowned out effort with success.
I am grateful to the Dept. of Mechanical Engineering, NIT ROURKELA, for giving
me the opportunity to execute this project, which is an integral part of the curriculum in
B.Tech programme at the National Institute of Technology, Rourkela.
I would also like to take this opportunity to express heartfelt gratitude for my project
guide Prof. S.K. Mahapatra, who provided me with valuable inputs at the critical stages
of this project execution.
I would like to acknowledge the support of every individual who assisted me in making
this project a success and I would like to thank Mr. B. N. Padhi (a research scholar), for
his help whenever it was required.
Date:…………….
SAMIR CHOUDHURY
B.Tech (10503066)
Dept. of Mechanical Engg.
CONTENTS
1. ABSTRACT……………………………………………………………………….........i
2. LIST OF FIGURES…………………………………………………………………….ii
3. INTRODUCTION……………………………………………………………………...1
4. EQUATION OF RADIATIVE HEAT TRANSFER
IN PARTICIPATING
MEDIUM…………………………………………………………………………….........5
4.1 Absorption
4.2 Out-scattering
4.3 Emission
4.4 In-scattering
5. OVERALL ENERGY CONSERVATION EQUATION………………………………8
6. FINITE VOLUME METHOD…………………………………………………….........9
7. FORMULATION OF THE DISCRETIZATION EQUATIONS
7.1 Steady state RTE…………………………………………………………………..10
7.2 Solution procedure………………………………………………………………...13
7.3 Transient state RTE………………………………………………………………..14
7.4 Solution procedure………………………………………………………………...16
7.5 Linear anisotropic phase function…………………………………………………17
8. RESULTS AND DISCUSSIONS……………………………………………………..19
9. CONCLUSION………………………………………………………………………..35
10. FUTURE SCOPE OF RESEARCH……………………………………………........36
11. REFERENCES…………………………………………………………………........37
ABSTRACT
Thermal radiation is important in many applications, and its analysis is difficult in the
presence of a participating medium. In traditional engineering studies, the transient term
of the radiative transfer equation (RTE) can be neglected. The assumption does not lead
to important errors since the temporal variations of the observables e.g. temperature are
slow as compared to the time of light of a photon. However in many new applications in
different fields the transient effect must be considered in the RTE, like it has a great
usability in the field of Bio-medical (applications like optical tomography, detection of
scar tissues and many more all of which is interaction of LASER with the participating
medium, tissue). In the transient phase, the reflected and the transmitted signals have
temporal signatures that persist for a time period greater than the duration of the source
pulse. This could be a source of information about the properties inside the medium.
Hence sufficiently accurate solution methods are required.
In the last few years, the finite volume method (FVM) and discrete transfer method has
emerged as one of the most attractive methods for modeling steady and transient state
radiative transfer.
The present research work deals with the analysis of steady and transient radiative
transfer in two dimensional square enclosure using FVM and analysis of steady and
transient RTE with one boundary subjected to single short pulse irradiation.
i
LIST OF FIGURES
 Fig. I….A typical control volume
 Fig. II….Control angle discretization in Finite Volume method
 Fig.III….Different orientation of angles in a discretized control angle
 Fig. IV….Step Scheme for a control volume
 Fig.1…. Variation of dimensionless heat flux at the bottom wall for absorbing and
emitting square enclosure.
 Fig.2…. Variation of dimensionless heat flux at the bottom wall for purely scattering
square enclosure.
 Fig.3…. Variation of dimensionless heat flux along centerline in y-direction for
isotropic scattering.
 Fig.4…. Variation of dimensionless heat flux along centerline in y-direction for linear
backward scattering.
 Fig.5….Variation of dimensionless heat flux along centerline in y-direction for linear
forward scattering.
 Fig.(i)….Geometry subjected to normal collimated incidence
 Fig.(ii)….Transmittance and Reflectance of a square enclosure
 Fig.6….Variation of transmittance and reflectance with distance in x-direction
 Fig.7…. Variation of transmittance with distance in x-direction for different wall
emissivity
 Fig.8…. Variation of reflectance with distance in x-direction for different wall
emissivity
 Fig.9…. Variation of transmittance with distance in x-direction for different
anisotropic factor
 Fig.10…. Variation of reflectance with distance in x-direction for different anisotropic
factor
 Fig.11…. Variation of transmittance with distance in x-direction for different angle of
incidence
ii
 Fig.12…. Variation of reflectance with distance in x-direction for different angle of
incidence
 Fig.(a)….2-D square enclosure subjected to pulsed collimated incidence
 Fig.(b)….Unit Step pulse used to irradiate the bottom wall
 Fig.13….Variation of transmittance with non-dimensional time
 Fig.14….Variation of reflectance with non-dimensional time
 Fig.15….Variation of transmittance with non-dimensional time for different optical
thickness
 Fig.16….Variation of reflectance with non-dimensional time for different optical
thickness
 Fig.17….Variation of transmittance with non-dimensional time for different pulsewidth
 Fig.18….Variation of reflectance with non-dimensional time for different pulse-width
 Fig.19….Variation of transmittance with non-dimensional time for different scattering
albedo
 Fig.20….Variation of reflectance with non-dimensional time for different scattering
albedo
 Fig.21….Variation of transmittance with non-dimensional time for different
anisotropic factor
 Fig.22….Variation of reflectance with non-dimensional time for different anisotropic
factor
 Fig.23….Variation of transmittance with non-dimensional time for different angle of
incidence
 Fig.24….Variation of reflectance with non-dimensional time for different angle of
incidence
iii
INTRODUCTION
For the last few decades, there is an exponential growth in the research area of transient
radiative heat transfer in participating media. Traditional analysis of radiation transfer
neglects the transient effect of light propagation due to the large speed of light compared
to the local time and length scales [2].
As the technology advanced and the short pulsed laser applications developed, the steady
state assumption was no longer valid as the temporal width of the input pulse was similar
to the order of Pico and Femto-seconds. Ultra-short pulsed lasers are used in a wide
variety of applications such as thin film property measurements, micro-machining,
removal of contamination particles, ablation of polymers, remote sensing of the
atmosphere, combustion chambers and other environments which involve interaction of
the laser beam with scattering and absorbing particles of different sizes [3]. Another
interesting application of short-pulsed lasers is in optical tomography where their use can
potentially provide physiological and morphological information about the interior of
living tissues and organs in a non-intrusive manner. All these applications need models to
predict transient radiation transport in participating media. In the past, various analytical
studies and numerical models of transient radiative transfer have been reviewed by Mitra
and Kumar [4]. The normal-mode-expansion technique is used in [1] to obtain a semianalytical solution for the angular distribution of radiation at any optical distance within a
linearly anisotropic scattering, absorbing, emitting, non-isothermal, gray medium
between two parallel reflecting boundaries. From the literature it is evident that many
researchers adopted different methods to deal with the problem. The commonly used
methods to solve the transient radiative transfer equation are the Monte Carlo method, the
integral equation solution, the finite volume method (FVM), the radiation element
method (REM), discrete transfer method (DTM) and the discrete ordinates method
(DOM).
The Monte Carlo method is used to simulate problems involving radiative heat
transfer because of its simplicity, the ease by which it can be applied to arbitrary
configurations and its ability to capture actual and often complex physical conditions [5].
The Monte Carlo technique has been used by Guo et al. [5] to simulate short-pulsed laser
1|Page
transport in anisotropically scattering and absorbing media. The authors examined the
effects of pulse width, medium properties, and the effects of Fresnel reflection on the
transmissivity and reflectivity of the medium. However, the method has inherent
statistical errors due to its stochastic nature [2]. It also demands a lot of computational
time and computer memory as the histories of the photons have to be stored at every
instant of time [5]. Thus, the Monte-Carlo method is ruled out in practical utilizations
such as real-time clinical diagnostics where computational efficiency and accuracy are
major concerns [6]. Guo and Maruyama [7] evaluated the isotropic law in three
dimensional inhomogeneous and linear anisotropic scattering media. The discrete
ordinates method has been used by various researchers to solve the transient radiative
transfer equation (RTE). Sakami et al. [8] used the DOM to analyze the ultra-short light
pulse propagation in an anisotropically scattering two dimensional medium. Mitra et al
[9] used a P1 approximation to model transient radiative transfer in a rectangular
enclosure.
Hsu [10] considered the Monte Carlo simulations for transient radiative transfer
process within the participating media inside the one-dimensional geometry with the
multiple scattering and reflective boundaries. Various effects, including the scattering
albedo, pulse shape and width, surface reflectivity and optical thickness, are examined
and concluded that if the boundary surface is reflective, then the temporal spread is
influenced by multiple reflections and partial transmissions at the surfaces. The backward
or reverse Monte Carlo method was successfully applied by Lu and Hsu [11] to simulate
transient radiative transport in a non-emitting, absorbing, and anisotropically scattering
one-dimensional slab subjected to ultra-short light pulse irradiation. Wu and Coworkers
[12] and Tan and Hsu [13] have used the integral equation formulation to solve the
transient radiative transfer problem analytically. Tan and Hsu [13] used the integral
equation formulation and the radiation element method by Guo and Kumar [14] to
simulate radiative transport in the same problem with black boundaries exposed to diffuse
or collimated irradiation.
Y.Hasegawa, et al., [15] used Monte Carlo method to simulate the transient light
transmission through the living tissue which was characterized by strong forward
scattering phase function. Brewster and Yamada [16] later conducted the transient study
2|Page
using the same MC algorithm used by Hasegawa, et al. They examined various effects
i.e., albedo, optical thickness, anisotropic scattering, and the detector size, on the
reflected and transmitted temporal signals. Finite volume methods developed by Chai et
al., [17] to solve the steady-state RTE have also been employed to solve the transient
RTE by Chai [18,19]. The finite volume technique is used with the step and curved line
advection method (CLAM) [20] spatial discretization schemes to model transient radiative
transfer in 1-D and 2-D geometries. The author found that the CLAM scheme captures
the penetration depths of radiation more accurately than the step scheme for the same
grid.
Rath et al., [21] extended the DTM, to solve transient radiative transport problems
in a one-dimensional planar absorbing and scattering medium, one boundary of which is
subjected to a short-pulse laser and the other boundary is cold. Effects of optical
thickness, scattering albedo, and anisotropy factor on transmittance and reflectance are
analyzed. Sarma et al., [22] analyzed the radiative heat transfer problem in 1-D planar
absorbing, emitting and anisotropically scattering gray medium in radiative equilibrium
subjected to collimated radiation by the discrete transfer method. The Galerkin method is
extended by T. Okutucu; Y. Yener [23] for the solution of the transient radiative transfer
problem in a one-dimensional participating plane-parallel grey medium with a collimated
short-pulse Gaussian irradiation on one of its boundaries. The transient transmittance and
reflectance of the medium are evaluated for various optical thicknesses, scattering
albedos and pulse durations. Muthukumaran and Mishra [24] used the finite volume
method for solving transient radiative heat transfer problem in a planar participating
medium subjected to a short-pulse diffuse or collimated radiation. For a train of pulses,
effects of the extinction coefficient and the scattering albedo on transmittance and
reflectance signals are studied.
A finite element model, which is based on the discrete ordinates method and leastsquares variational principle, is developed by W.An et al., [25] to simulate the transient
radiative transfer in absorbing and scattering media in one dimensional and twodimensional enclosure and W.An et al., [26] extended the same method to simulate shortpulse light radiative transfer in homogeneous and nonhomogeneous media. Their results
indicated that the reflected signals can imply the break of optical properties profile and
3|Page
their location. Most recently Yilmazer A, Kocar C [27] discussed the radiative transfer
problem in plane-parallel, participating medium with linearly anisotropic scattering using
the ultraspherical-polynomials approximation method. Effects of the order of
approximation, optical thickness, specular reflection, anisotropic scattering, and change
of the source term on results are investigated different order of approximation.
Majority of the findings are based on the most simplified assumption of black wall,
whereas the reflective wall assumption resembles more to the practical application [4].
The multiple scattering and reflective boundaries, influences considerably the radiation
transport in a participating medium. When the boundary surface becomes reflective, then
the temporal spread changes significantly by the multiple reflections and partial
transmissions at the surfaces [10]. Therefore, the present article focuses on the problem
of a participating medium bounded by diffusely emitting boundaries, under the condition
of radiative equilibrium. The total intensity is directly solved using FVM without
splitting into the collimated part and diffusive part as cited in existing formulation [21,
22, 24, 28].
4|Page
EQUATION OF RADIATIVE HEAT
TRANSFER IN A PARTICIPATING MEDIUM
When the medium through which the radiative energy is traveling is participating then
any incident beam will undergo

Absorption

Scattering

Emission
Scattering away from the direction under consideration is known as out-scattering and
scattering from the other directions into the direction under consideration is known as inscattering.
Absorption:
The absolute amount of absorption is directly proportional to the magnitude of incident
energy as well as the distance the beam travels through the medium.
I
 k (r ) I (r , sˆ, t )
ds
where k(r) is the linear absorption coefficient
Out-scattering:
It is same as absorption but only difference is that absorbed energy is converted into
internal energy while scattered energy is simply redirected along another direction.
I
  s (r ) I (r , sˆ, t )
s
where
 s (r )
is known as the linear scattering coefficient
5|Page
Emission:
The rate of emission from a volume element is proportional to the magnitude of the
volume. So the emitted intensity is proportional to the length of the path and local energy
content in the medium. At the thermodynamic equilibrium intensity everywhere will be
equal to blackbody intensity.
I
 k (r ) I b
s
where k(r) is the emission constant same as for absorption
In-scattering:
It has contribution from all the directions and hence must be calculated by integration
over all solid angles, considering the radiative heat flux impinging on a volume element
from an infinitesimal pencil of rays in a specified direction.
Scattering phase function:
The scattering phase function  in the RTE describes how radiation energy is scattered
by a participating medium. Scattering can be classified into two categories. These are
isotropic and anisotropic scattering. Isotropic scattering indicates energy scattered equally
into all direction whereas anisotropic scattering can be forward and backward scattering.
Scattering phase function satisfies the following relation:
 (sˆ, sˆ)d  4
4
where
sˆ is the direction from which intensity is scattered into a direction ŝ
.
6|Page
Hence finally we have that energy flux scattered into a direction from all incoming
directions is:
I  s (r )

I (r , sˆ' , t ) (r , sˆ' , sˆ, t )d'
s
4 4
Making an energy balance on the radiative energy traveling in the direction we have
change in the intensity found by summing all contribution from emission, absorption,
out-scattering and in-scattering
I ( s  ds, sˆ, t  dt )  I ( s, sˆ, t )  k (r ) I (r , sˆ, t )   s (r ) I (r , sˆ, t )
k (r ) I b 
 s (r )
I (r , sˆ' , t ) (r , sˆ' , sˆ, t )d'

4 4
Expanding the left side of the above equation using Taylor’s Series and truncating after
the first term we would the following equation as
1 I (r , sˆ, t ) I (r , sˆ, t )

 k (r ) I (r , sˆ, t )   s (r ) I (r , sˆ, t )
c
t
s
 (r )
k (r ) I b  s
I (r , sˆ' , t ) (r , sˆ' , sˆ, t )d'
4 4
The above equation is known as the Radiative Heat Transfer equation in a participating
medium.
7|Page
OVERALL ENERGY CONSERVATION
EQUATION
Thermal radiation is one of the modes of hest transfer and must compete with conduction
and convection. So the temperature field depends on all the three modes of heat transfer.
The general form of the energy equation is:

U
 .q  p.v    Q' ' '
t
where u
= internal energy
v
=velocity vector
p
= radiation pressure tensor
q
= total heat flux vector
Q'''
= heat generated within the medium

= dissipation function

= density of the medium
As the medium is radiatively participating through emission, absorption and scattering so
the second and third effect is negligible. Assuming U=CvdT and Fourier’s law of
conduction to hold
q = qr + qc
In the absence of the heat generation
Cv
where
T
 .( kT )  .qr
t

= density of the medium
Cv
= specific heat
K
= thermal conductivity
T
= temperature
qr
= radiative heat flux
8|Page
FINITE VOLUME METHOD
The basic idea of the control volume formulation is dividing the calculation domain into a
number of non-overlapping control volumes such that there is one control volume
surrounding each grid point. The differential equation is integrated over each control
volume. Piecewise profiles expressing the variation between the grid points are used to
evaluate the required integrals. The result is the discretization equation for a group of grid
points.
The discretization equation obtained expresses the conservation principle for the finite
control volume as the differential equation expresses it for an infinitesimal control
volume.
(Fig.I)
There are four basic rules which should be kept in mind during the formulation of the
discretization equations are:
1.Consistency at the control-volume faces
2.All coefficients must always be positive
3.Negative slope linearization of the source term
4.Sum of the neighbour coefficients equal to the coefficients of the grid point under
focus.
9|Page
FORMULATION OF DISCRETIZATION
EQUATION
STEADY STATE RTE:
A scattering, absorbing and emitting medium in a square enclosure with black wall is
considered for the analysis. Discretizing the computational domain in both spatially and
angular direction, then integrating the RTE over a control volume dV and control angle
d
and neglecting the effect of transient term
(Fig.II)
(Fig.III)
10 | P a g e
I (r , sˆ, t )
dVd l     (r ) I (r , sˆ, t )dVd l   S l dVd l
l
s
 v
l v
l v
After applying divergence theorem on LHS and intensity is assumed constant within a
control volume and a control angle the above equation can be written as
4
  I il Ai
i 1
 sˆ.nˆ d


l
l
   ml , P I Pl  S m
, P V
l

l
On further simplification, for a control volume and a control angle the equation becomes
I V  I  I A D
  I  S V
l
P
l
l
l
m, P P
l
e
l
m, P
l
w
x
l
cx


l
 I nl  I sl Ay Dcy

l
Using Step spatial differencing scheme (which sets the downstream boundary intensities
equal to the upstream nodal intensities)
(Fig.IV)
l
l
l
IP
 I el  I nl , I sl  I Sl , I w
 IW
11 | P a g e
The discretized equation can be written in the following form
alp I lp  aWl IWl  aSl I Sl  b
where
l
aWl  yDcx
l
aSl  xDcy

l
l
a lp  xDcy
 yDcx
 (  ml ) p vl

b  ( S ml ) p vl
l 
2  2
  sin  d d
1 1
l
Dcex
l
Dcny
  ( sˆ .nˆ x )d 
l
  ( sˆ .nˆ y )d 
l
 ml   
S ml
l
l
2  2
  sin  cos  sin dd   Dcwx  Dcx
l
l
1  1
2  2
  sin  sin  sin dd   Dcsy  Dcy
l
l
1 1
 s (r ) ll l
 
4
 s ( r ) M l ' l 'l l '
 k (r ) I b 
 I  
4 l ' 1, l '  l
12 | P a g e
SOLUTION PROCEDURE:
Algorithm
l
l
W
l
S
l
p
a , a , a , b are calculated
Calculate
 al I l  al I l  b 

I lp   W W l S S


a
p


l
l
Guess unknown I nb(i.e I S , I W )
Set the current intensities as guessed value
I Pl
I lp  I lp' / I lp  10 6
If Yes
Record
I Pl
If No
Take Updated I
l
and
nb
I Pl
13 | P a g e
TRANSIENT STATE RTE:
A scattering, absorbing and emitting medium in a square enclosure with black wall is
considered for the analysis. Discretizing the computational domain in both spatially and
angular direction, then integrating the RTE over a control volume dV, control angle
d
and a small time interval taking into account the effect of transient term
1 I l
I l
l l
l
dtdvd


dtdvd




I

S
dtdvd
m
m
 l V t c t





s

l Vt
l V t


Applying divergence theorem to the 2nd term and the magnitude of intensity is assumed to
be constant over the control volume and a control angle. Under these assumptions and
using the fully implicit scheme the above equation can be written as




4
1 l
I P  I Pl 0 V l   I il Ai t  sˆ.nˆ d l    ml , P I Pl  S ml , P V l t
c
i 1
 l
where
I Pl 0
I Pl
and
are the nodal intensities at the start and at the end of the time step
respectively. On further simplification, for a control volume and a control angle the
equation becomes






1 l
l
l
I P  I Pl 0 Vl  I el  I wl Ax Dcx
t  I nl  I sl Ay Dcy
t 
c
  ml , P I Pl  Sml , P Vl t


where


 ml     s  ll  l
4




14 | P a g e
l
Sm
 k Ib 
 s M l  l l

 I   l
4 l 1
l l
l
Dcx

 sˆ.nˆ x d
 l 
l
l
Dcy

 sˆ.nˆ y d
l
 d

After applying the STEP scheme the final discretized equation looks like this,
 l
 l  V l
V l
l
l
l
 Dcy Ay  I P  
 Dcx Ax   m, P V 
ct


 ct
 l0
I P 


l
l
Dcx
Ax I Wl  S ml , P V l  Dcy
Ay I Sl
In case the radiation intensity leaving a surface that is not black (i.e. gray) emits and
reflects energy diffusely, then the change in the boundary intensity would be as
following:
ˆ, t    ( r ) I b ( r , t ) 
I r , s
1   (r )

 I (r , sˆ, t ) sˆ.nˆ d
ˆ. n
ˆ0
s
where the 1st term is emissivity and the 2nd term is the reflectance of the surface.
15 | P a g e
SOLUTION PROCEDURE:
The finite volume discretization results in a set of algebraic equations with the radiation
intensities as the unknowns. An iterative method is used to solve the resulting set of
algebraic equations within each time step. The solution process adopts a marching
procedure to solve the set of equations.
The algorithm for the solution procedure is as follows:
1.Start with a suitable intensity distribution for the entire domain.
2.Proceed to the next time step.
3.Set the initial or the most current nodal intensities as the guessed values.
4.Update the upstream boundary intensities.
5.Following the marching order, calculate the nodal intensities for all internal control
volumes.
6.Calculate the radiation arriving and living the opposite walls.
7.Return to step 4 and repeat the calculation until convergence.
8.Stop when the desired time is reached or go to step 2 to advance to a new time step.
16 | P a g e
Linear Anisotropic phase function:
In the previously formulated discretized equations the participating medium was taken to
be isotropically emitting, absorbing and scattering. But in reality it doesn’t happens. The
participating medium has some anisotropicity involved in it. So in the following
formulation the linear anisotropic condition is derived and used in the code to make it
more accurate and precise to have a proper understanding of the behavior of the medium.
As seen earlier the scattering phase function is obeying the following condition
 (sˆ, sˆ)d  4
4
In the finite volume method this is approximated as
l
l 'l
l'


ˆ
ˆ

(
s
,
s
)
d






l '1
4
where
 l 'l = average energy scattered from control angle l’ to the control angle l.
The scattering phase function can be represented in the form of a series
M
 ( sˆ, sˆ)  1   AmPm(sˆ, sˆ)
m 1
For isotropic condition M=0
For linear anisotropic condition M=1
Hence for linear anisotropy with M=1 we have
( sˆ, sˆ)  1  A1P1( sˆ, sˆ)  1  A1( sˆ, sˆ)
17 | P a g e
where Pm is the Legendre’s Polynomial
sˆ = Sinθ’ (cosφ’ iˆ + sinφ’ ĵ) + cosθ’ k̂
ŝ = Sinθ’ (cosφ iˆ + sinφ ĵ) + cosθ k̂
So,
( sˆ, sˆ)  1  A1{sin  ' sin  cos( ' )  cos  ' cos  }
Average scattering phase function can be evaluated by
 l 'l 
 (ŝ, ŝ)dΩ
d '
l '
Hence
2  2
 l 'l  
1

1
{1  a cos cos ' a sin  sin  ' cos(   ' )}sin  ' d ' d '
18 | P a g e
RESULTS AND DISCUSSIONS
1. Isothermal
Absorbing-Emitting Medium
The medium is maintained at a constant temperature T. The black, square enclosure is
having cold walls at 0 Kelvin. The calculation domain is discretized into 20x20 uniform
control volumes in the X and Y directions. Finer angular discretizations is used that of
2x12 control angles with uniform

 in
and
the

and

directions
respectively.
Present
John Chai
1.1
1
q*
0.9
0.8
0.7
0.6
0
0.1
0.2
0.3
0.4
0.5
X
Lx  Ly  1, ka  10, q* 
q
T 4
(Fig.1)
The above figure shows dimensionless heat flux at the bottom wall which is in good
agreement with the published results.
19 | P a g e
2. Purely
scattering medium
A square enclosure is considered with black walls and the medium scatters energy
isotropically with the scattering albedo ( = /) as unity. The bottom wall is hot
maintained at a temperature Th with the remaining walls maintained at 0 Kelvin.
Lx  Ly  1, s  1, q*** 
q
 (Th4  Tc4 )
(Fig.2)
The above figure shows dimensionless heat flux at the bottom wall which is in good
agreement with the published results.
20 | P a g e
3. Linear
Anisotropically Scattering Medium
A square enclosure is considered with black walls and the medium scatters energy
anisotropically with the scattering albedo ( = /) as unity. The bottom wall is hot
maintained at a temperature Th with the remaining walls maintained at 0 Kelvin. The
phase functions are studied using 25x25 control volumes and 6x24 control angles.
Lx  Ly  1, s  1, a  0, q *** 
q
 (Th4  Tc4 )
(Fig.3)
The above figure shows dimensionless heat flux at the centerline in y-direction which
is in good agreement with the published results.
The figure below shows dimensionless heat flux at the centerline in y-direction. It is in
good agreement with the published results for backward scattering B2 series as both have
same number of terms in its expansion.
21 | P a g e
Lx  Ly  1, s  1, a  1, q *** 
q
 (T  Tc4 )
(Fig.4)
Lx  Ly  1, s  1, a  1, q *** 
q
 (T  Tc4 )
(Fig.5)
4
h
4
h
The above figure shows dimensionless heat flux at the centerline in y-direction. The
F1 series has more number of terms than linear forward scattering hence they do not
agree but the trend is same which assures the correctness of simulation.
22 | P a g e
4. Collimated
Incidence
The top wall of the black, square enclosure is subjected to a normal collimated incidence
(as shown in figure). The other walls are maintained at 0 Kelvin and the medium scatters
energy isotropically with a scattering albedo of unity. The domain is divided into 25 x 25
control volumes and 3 x 24 control angles in the
 and 
directions. Step scheme is
used in the present problem. The control angles are adjusted to capture the collimated
incidence.
Reflectance
Collimated
beam
Fig. (i) Square Enclosure with
collimated irradiation
Transmittance
Fig. (ii) Reflectance & Transmittance for
an enclosure with collimated beam
23 | P a g e
Transmittance
Reflectance
Lx  Ly  1, s  1, k  0, q** 
q
qc
(Fig.6)
Variation of wall emissivity:
ε=1
ε=0.8
ε=0.4
(Fig.7)
24 | P a g e
ε=1
ε=0.8
ε=0.4
(Fig.8)
Variation of anisotropic factor:
a=1
a=0
a=-1
(Fig.9)
25 | P a g e
a=-1
a=0
a=1
(Fig.10)
Variation of angle of incidence:
𝜭=90°
𝜭=45°
𝜭=30°
(Fig.11)
26 | P a g e
𝜭=90°
𝜭=45°
𝜭=30°
(Fig.12)
27 | P a g e
5.
Temporal variations of the optical signals
The bottom wall of the black, square enclosure is subjected to a normal collimated
incidence (as shown in figure). The other walls are maintained at 0 Kelvin and the
medium scatters energy isotropically with a scattering albedo of unity. A unit step pulse
*
as shown of a non-dimensional pulse width t p (ct p )  1 is used to irradiate the wall and
the different flux variation with non-dimensional time t ( ct ) is observed.
*
Fig.(a) collimated irradiation at the bottom wall.
Fig.(b) Time of arrival of the collimated square pulse at different locations.
28 | P a g e
(Fig.13)
(Fig.14)
a  0,   1, (L)  1, t *p (ct p )  1
29 | P a g e
Variation of Optical thickness:
=0.5
=1
=2
(Fig.15)
=0.5
=1
=2
(Fig.16)
a  0,   1, t *p  1
30 | P a g e
Variation of pulse-width:
tp*=1
tp*=3
tp*=2
(Fig.17)
tp*=3
tp*=1
tp*=2
a  0,   1,  1
(Fig.18)
31 | P a g e
Variation of scattering albedo:
=1
=0.75
=0
=0.5
(Fig.19)
=1
=0.75
=0.5
=0
a  0,  1, t  1
(Fig.20)
*
p
32 | P a g e
Variation of the anisotropic factor:
a=1
a=0
a=-1
(Fig.21)
a=-1
a=0
a=1
(Fig.22)
t *p  1,   1,  1
33 | P a g e
Variation of angle of incidence:
=90
=45
=30
(Fig.23)
=30
=45
=90
(Fig.24)
a  0,   1,  1, t *p  1
34 | P a g e
CONCLUSION
 Finite volume method found to be a suitable method for the simulation of
collimated incidence problem with Steady and Transient RTE.
 Transmitted flux decreases considerably with the increase in optical thickness
whereas reflectance increases.
 Reflected flux increases with the pulse width (nearly twice) whereas there is not much
considerable change in transmitted flux.
 In the fully scattering medium both transmitted flux and reflected flux increases and
reach their maximum.
 Linear forward scattering has more transmittance and less reflectance than the
isotropic condition and vice-versa for the linear backward scattering.
 Transmittance decreases with the increase in the collimated angle whereas the reverse
happens in the case of reflectance.
 Present study helps us to create a database for the healthy tissues.
 By comparing with the available information from the experiments, it will help
to indentify the turbid tissues (damaged) if present.
 This created database will help to simulate through the inverse radiation to
obtain the unknown tissue properties.
35 | P a g e
FUTURE SCOPE OF RESEARCH
Study of the effect during the interaction of short-pulse collimated radiation in
participating medium on the transmittance and the reflectance signals for:
 The different type of scheme like the CLAM scheme
 The different type of pulse profile like time varying Gaussian profile
 Combined conduction and radiation
 Multi-dimensional enclosures
 Cylindrical co-ordinates
36 | P a g e
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