CHAPTER 6 MODELING OF ABSORPTION OF RADIANT...
Transcript of CHAPTER 6 MODELING OF ABSORPTION OF RADIANT...
89
CHAPTER 6
MODELING OF ABSORPTION OF RADIANT
ENERGY
6.1. BACKGROUND
The most dominant obstacle in understanding the exhaust plume is that the
radiative properties, namely, the extinction and absorption coefficients, the scattering
coefficient, the temperature distribution and the scattering phase function of the plume are
generally unknown. The modeling of reduction of intensity of radiant energy while it travels
through a medium due to the absorption is the main theme of this chapter. Scattering is
often accompanied by absorption. Both scattering and absorption remove energy from the
radiant energy traversing the medium; then the energy is said to be attenuated. This
attenuation is called extinction. In this chapter, the process of absorption of radiant energy
by the mixture of gases present in the combustion products of the exhaust plume is
explained. The absorption characteristics of both liquid and solid phase alumina are
discussed, as alumina is one of the major constituents of the combustion products of solid
rockets.
6.2. INTRODUCTION
In the domain of exhaust plume of solid rockets, the combustion gases are the main
component which absorbs the radiant energy. If the radiation properties of gases and
opaque solids are compared, the property variations with wavelength for opaque solids are
continuous whereas gas properties exhibit very irregular wavelength dependences. The
absorption or emission by gases is significant only in certain wavelength regions.
Differences in emission spectra are caused by the type of energy transition that occur with a
particular medium. Since the absorption coefficient is a function of density, gases posses a
highly variable absorption coefficient.
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Many problems of radiative heat transfer involve the exchange of radiant energy
between surfaces separated by a medium that absorbs and emits radiant energy. Most
participating media may have radiative properties that vary pronouncedly with the
frequency characterizing the radiation. One of the important issues for radiative heat
transfer in a combustion system is the description of the radiative properties of particle laid
combustion gases. Beer’s law was originally established as a result of experimental studies
of the attenuation of light in colloidal suspensions. Early studies in this area were made by
Bouguer in 1729 and later by Lambert. The law states that the intensity of radiant energy
traversing a semitransparent medium is decreased by absorption in proportion to the
intensity at that point. In thermodynamic equilibrium, the intensity of radiation does not
change along the path. The website http://Wikipedia reveals the contribution of both Beer
and Lambert as follows :
Lambert law states that absorption is proportional to the length of the path,
whereas, the Beer law states that absorption is proportional to concentration of absorbing
species in the material. If the concentration is expressed as a mole fraction, the molar
absorptivity takes the same as the absorption coefficient, m-1 .
From quantum theory, it is known that that the structure of a single molecule is such
that it may emit and absorb only certain energy quanta corresponding to the allowable
energy configurations of the molecule. Thus a gas composed of identical molecules would
absorb and emit photons having exact frequencies corresponding to these allowable
energies. A radiating gas like the combustion products of a solid motor is composed of
molecules, atoms, ions and free electrons which are associated with different energy levels.
It is convenient to discuss the radiation process by utilizing a photon, which is the basic unit
of radiative energy. Radiative absorption is the capture of photons while emission is the
release of photons.
Most solids and liquids absorb energy in rather broad wave length range. Gases on
the other hand, tend to absorb and emit energy in narrow bands that are only a few microns
in width, called a line. The problem of integrating over frequency must therefore be treated
in a different fashion than in the surface radiation problem. In real gases, these emission
lines must have finite width and are shaped as a probability function. The combination of
several phenomena results into the line broadening effect. The phenomena leads to the line
broadening effect are the following:
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(1) Natural line broadening: This is resulted from the variation of emission levels from
several photons from a large number of molecules.
(2) Collision broadening: This is resulted from the interaction of the force fields between
molecules. Transition energies are affected by the proximity of colliding molecules
and thus the frequency of the photon emitted or absorbed in the transition is
influenced. The collision frequency is influenced by the pressure and temperature of
the gas. As the gas pressure is increased, the emission lines broaden into bands.
(3) Doppler broadening: It is the result of movement of the molecules or atoms. Since
the gas is in a state of thermal agitation, photons will be emitted over a small range of
frequencies as a result of this effect.
(4) Stark broadening: this occurs as a result of the interacting electrical fields of ions and
the free electrons present in high temperature gases or plasmas.
Recently, Guo-Biaco Cai etal [2006] proposed a weighted sum of gray gases
model, which replaces the nongray gas behavior by equivalent finite number of gray gases.
The above discussions reveal the complexities of modeling absorption in a mixture
of gases. Thus in most of the engineering applications, the spectral effects in absorption
phenomena of gases are not taken into account. This is justified to some extent because of
the prevailing lower absorption coefficients at higher temperature levels of radiating gases.
By assuming the condition of optical thin limit and gray gas properties, an appropriate
averaging procedure for the mean absorption coefficient of the mixture of gases is used in
this analysis.
6.3. ABSORPTION
The attenuation of radiant energy inside a gas of infinitesimally thin thickness ds is
proportional to the radiation intensity I and thickness ds and can be expressed as
dI ak I ds= − (6.1)
Here the constant ak is called the gas absorption coefficient. This equation is
valid, exactly, for only a monochromatic ray whose absorptivity is a function of the
wavelength, temperature and pressure. In the case of radiant energy of a broad wavelength
band, the above equation is valid only approximately. Equation (1) is linear and the solution
is
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0
akSI I e= − (6. 2)
Where I0 is the initial intensity and I is the resultant intensity after passing through a
distance of S meters in an absorbing medium with absorption coefficient ak . Here the
attenuation factor of the intensity is akSe−
and the product ak S is known as the
absorptive distance or optical length, which is dimensionless.
The absorbing properties of the gases can change by several orders of magnitude
over a small wavelength interval. Another difficulty also arises from the nature of the
monochromatic absorption coefficient being non-continuum and possessing a strong
functional dependence on the total pressure. An appropriate averaging of absorbing
properties over the wavelength spectrum is required to arrive at meaningful radiant heating
rates. Abu_Romia and Tien [1967] have proposed the planck mean absorption coefficient
for gases that approach the optical thin limit. Gas is optically thick in some parts of the
infrared spectrum and optically thin or almost transparent in the rest.
6.4. PLANCK MEAN ABSORPTION COEFFICIENT
Planck means absorption coefficient is a widely accepted method for estimating the
absorption coefficient of gases. The line intensity spacing ratio is a property of absorbing
gases. Therefore in a mixture of absorbing and inert gases, it could be concluded that the
Planck mean absorption coefficient per unit optical depth is independent of the inert gas
pressure and only varies with temperature. Thus Planck means absorption defined by eqn.
(6.3) is used in the analysis
,
0
,
0
, ( , , ) ( , )
( , )
( , )
a b
b
K T P E T d
Ka T P
E T d
λ λ
λ
λ λ λ
λ λ
∞
∞=∫
∫ (6.3)
Here, the spectral variation of the volumetric absorption coefficient is weighted by
the blackbody function. Thus the plank mean absorption coefficient will be
(1) Independent of wavelength
(2) A function of the partial pressure
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(3) A function of the temperature
Figures 6.1, 6.2 and 6.3 give the plank mean absorption coefficients of CO, CO2 and H2O
respectively, calculated for a Partial pressure of 1atm.
0
1x10-5
2x10-5
3x10-5
4x10-5
500 750 1000 1250 1500 1750 2000 2250 2500 2750 3000
Planck mean absorption coefficient of CO
TEMPERATURE, K
PL
AN
CK
ME
AN
AB
SO
RP
TIO
N C
OE
FF
ICIE
NT
(Pa.m
)-1
Figure. 6.1 Planck mean absorption coefficient of CO
0
0.0001
0.0002
0.0003
0.0004
0.0005
500 750 1000 1250 1500 1750 2000 2250 2500 2750 3000
Planck mean absorption coefficient of CO2
TEMPERATURE, K
PL
AN
CK
ME
AN
AB
SO
RP
TIO
N C
OE
FF
ICIE
NT
(Pa
.m)
-1
Figure. 6.2 Planck mean absorption coefficient of CO2
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0
0.00005
0.00010
0.00015
0.00020
500 1000 1500 2000 2500 3000
Planck mean absorption coefficient of H2O
TEMPERATURE, K
PL
AN
CK
ME
AN
AB
SO
RP
TIO
N C
OE
FF
ICIE
NT
(P
a.
m)-1
Figure. 6.3 Planck mean absorption coefficient of H2O
For calculation in thermal radiation, it is necessary to ratio these coefficients from
1atm to the partial pressure of the gas being modeled. For a mixture of gases, the Planck
mean absorption coefficient, ak, is defined as
Ka, mixture = ( P/1atm) ∑=
4
1
.i
ii FMKa (6.4)
Where Kai is the Linear absorption coefficient of the radiating gaseous components like
CO,CO2,HCL and H2O and M.Fi is the Mole fraction. The mole fractions of the principal gas
emitters are influenced by the method with which the plume chemistry is modeled. Values
are provided by using equilibrium chemistry. The predicted values of mole fractions of
combustion products at the exit plane of a typical solid rocket are given below:
Table 6.1: Mole fractions of combustion products
Name of combustion product Mole fraction
H2 0.310 Al2O2 Gas species 0.227
HCl 0.155
H2O 0.119
Al2O3 solid phase 0.089
N2 0.079
H 0.003
CO2 0.016
Cl 0.001
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The Planck mean absorption for the above mixture of combustion gases computed
by the eqn (6.4) is shown in fig. 6.4
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
0 250 500 750 1000 1250 1500 1750 2000 2250 2500
Temperature, K
Ab
so
rpti
on
co
eff
icie
nt,
m-1
Figure. 6.4 : Planck mean absorption coefficient
for mixture of gases given in table 6.1 It may be noted that the Planck mean absorption coefficient of HCl is taken to be
that of CO, due to the unavailability of data. This approximation is based on the following
reasons as reported by Watson and Lee (1976).
(1) Spectral absorption coefficient for HCl is lower than that for CO.
(2) The band for HCl subtends a greater fraction at the blackbody function than
CO.
These two effects may mitigate against each other so that the above approximation
is reasonably valid.
6.5. ABSORPTION CHARACTERISTICS OF Al2O3
In the molten state, Carlson DJ (1967), compiled data for the non-dimensional
absorptive index is represented by the equation
)(
,0T
TT
maa
m
ekk∆
−
= (6.5)
where ka,m=2*10-4 and ∆T0 =260 K and Tm=2320 K.
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Carlson’s review indicates a two order of magnitude drop in ka upon freezing. Thus
a solid particle at the very same elevated temperature (2320 K) as a fully molten particle
has negligible emission in comparison. It may be noted that the absorptive characteristics of
molten alumina up to 2500 K is negligible and hence the absorption of both molten alumina
and solid particles upon freezing is not taken into account in the analysis of plume radiosity.
The absorption efficiency for a size parameter (ψ=2πr/λ) in the range 4<=x<=20 and for a
refractive index 1.78, result gives Qa = 11.7 ka *0.68. Both Ka and Qa are plotted.
Figure 6.6 shows the absorptive characteristics of molten aluminia to show the
difference of absorption characteristics of alumina with phase change. Since, the presence
of molten alumina is not reported in the exhaust plume of bigger motors, its effect is not
taken in the analysis.
-0.00001
0.00004
0.00009
0.00014
00.2x10
-50.4x10
-50.6x10
-50.8x10
-51.0x10
-5
1000 0C
100 oC
Wave Length, λλλλ (m.)
Imag
inary
Part
of
Refr
acti
ve
In
dex
Figure. 6.5 Variation of absorption coefficients of solid Al2O3 particles at elevated temperatures.
0
0.05
0.10
0.15
1750 2000 2250 2500 2750 3000 3250 3500
Absorption EfficiencyAbsorptive Index
Temperature, K
Absorp
tive C
hare
cte
ristics
Figure. 6.6 Absorbptive characteristics of Molten Alumina
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6.6. CONCLUSION
Absorption of solid particles is neglected. Planck mean absorption coefficient presented in figure 6.4 will be used as the absorption coefficient of the mixture of gases in the exhaust plume.
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CHAPTER 7
MODELING OF SCATTERING OF RADIANT
ENERGY
7.1. BACKGROUND
Modeling of thermal radiation in participating media necessitates the modeling of
scattering of radiant energy due to solid particles to account the redistribution of radiant
energy in the domain of the system. Scattering is the parameter which influences mostly the
quantum of radiant energy which escapes the exhaust plume boundary of the rockets and a
part of this energy is incident on the base region of the rocket depending on the view factor
of the region of escape from the boundary to the base region of the rocket. Thus modeling
of scattering of radiant energy emanating from each control volume of the computational
domain of the exhaust plume became essential.
The scattered energy depends on wavelength of incident radiant energy, the size
and shape of the scatterer and refractive indices of the scatterer and the medium in which
the scatterer is located. These parameters are interconnected through complex
mathematical functions, infinite series and recurrence relations. Hence designers usually
resort to make approximations for the particle size parameter and wave length to estimate
the scattering coefficients. Such an approximation becomes erroneous since the scattering
coefficients are highly non-linear and sensitive to the size parameter as will be shown in this
chapter. Further, the scattering efficiency predicted by the approximate formulae cannot
exhibit all minor wiggles and some times these formulae are not available in the required
area of interest. Hence a rigorous approach is required to capture all the wiggles in the
curve of scattering efficiency factors and for larger size parameters for achieving better
accuracy for the results.
Phenomenon of scattering and the built in natural size parameter of the scattering is
explained. Influence of parameters such as refractive indices of scatterer and medium, size
parameter and absorption and scattering characteristics of particle is described. Scattering
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diagrams and phase functions for diffuse reflecting spheres are introduced as applicable
cases in the present study for predicting scattering efficiencies for aluminium oxide particles
contained in the exhaust plume of solid rockets. The directional variation of scattered
radiation intensity as predicted by the Rayleigh phase function is presented. The
scattering patterns of small non absorbing spheres and that of a sphere with a
Lambertonian surface are also shown.
The magnitude of scattering as described by Rayleigh-Debye is presented for
predicting scattering efficiency of a particle which is not too large and when the refractive
index of the particle is sufficiently close to that of the external. Finally Mie theory of
scattering is explained which is valid in a larger domain by defining the scattering
coefficients as functions of Ricatti-Bessel functions and their derivatives. The scattering
efficiency factor is predicted using a series expansion of scattering coefficients. To make the
computation more efficient, the Ricatti-Bessel functions and their derivatives are expressed
as polynomial functions of size parameter and this approach realized the convergence of
the value of scattering efficiency with just seven terms in the series expansion of scattering
coefficients. This mathematical model of predicting the scattering efficiency can thus be
used an input for various analysis in other scientific fields as a subroutine which delivers the
input with minimum computational time.
The developed mathematical model is validated in a number of cases using an
approximate formula derived by Van de Hulst [1981], which is valid over a size parameter
range up to 2 and thus could extend to higher range of the size parameter of Mie scattering.
The scattering efficiency of a control volume with particles of different sizes is then defined
for the solution of radiant transport equation in the computational domain.
7.2. INTRODUCTION
Theory of scattering has got wide applications in various fields of science and
technology like space sciences, thermal radiation in participating media, image processing,
atmospheric sciences and remote sensing. Everyone engaged in the study of light or its
industrial applications meets the problem of scattering. Scattering is composed of both
diffraction and reflection. The scattered energy depends on the wave length of incident
energy, the size and shape of the scatterer and refractive indices of scatterer and the
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medium in which the scatterer is located. These parameters are usually interconnected
through complex mathematical functions, infinite series and recurrence relations. Both the
physics and mathematical models of scattering of radiant energy are complex.
Even though the study of scattering of electromagnetic fields was initiated in the last
century, especially in the field of optics, its application in thermal radiation remains as an
area where much detail are not readily available. Hence the modeling of scattering cross
section in the field of thermal radiation became a challenging task. “Except for conference
tutorials and a few isolated projects and classroom examples, very little is formally taught
about the subject” as opined by John C Stover [1990]. Many of the topics presented in the
discussion of Mie theory is taken from the classic reference of Van de Hulst [1981]. Since
considerable time of this study has been spent for modeling scattering of radiant energy
alone, full details of the study is presented for the benefit of future research work in various
fields including optics.
Scattering is often accompanied by absorption. A leaf of a tree looks green
because it scatters green light more effectively than red light. The red light incident on the
leaf is absorbed. Both scattering and absorption remove energy from a beam of light
traversing the medium. The beam is said to be attenuated. This attenuation is called
extinction and thus extinction is the sum of scattering and absorption.
Both absorption and scattering of light energy is applicable in its full meaning to
thermal radiation also as it is part of the electromagnetic radiation. If electromagnetic energy
travels through a perfectly homogeneous medium, it is not scattered. Only inhomogenity
causes scattering. The presence of particles or small scale density fluctuations, make part
of the radiation scatter in all directions. Any material medium has inhomogenities as it
consists of molecules, each of which acts as a scattering center, but it depends on the
arrangement of these molecules whether the scattering will be very effective. In a gas or
fluid, statistical fluctuations in the arrangement of the molecules cause a real scattering,
which sometimes may be appreciable. In this case, whether or not the molecules are
arranged in a regular way, the final result is a co-operative effect of all molecules. Then one
has to investigate in detail the phase relations between the waves scattered by neighboring
molecules. The precise description of the co-operation between the particles is called
dependent scattering and is not addressed in this study.
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In the exhaust plume of solid rockets, Liquid particles of Al2O3 possess relatively
large absorption cross sections while solid particles are almost pure scatterers of radiant
energy. However, the presence of liquid particles of Al2O3 in the exhaust plume is almost
negligible and solid particles of varying sizes are present. The solid particles of Al2O3 are
having both absorption and scattering cross sections. Different methods of computations
exist as can be seen in Edwards and Baikiant [1990] and Gilbert N Plass [1966] in
calculating scattering efficiency factors depending on the nature of refractive index of Al2O3.
7.3. COMPLEX REFRACTIVE INDEX OF Al2O3
When the spherical particles are considered as absorbing, the refractive index is
defined by a complex number as m = n1-in2 where 1−=i . The real and imaginary parts
represent the scattering and absorption coefficients respectively. The influence of n1 and n2
on the Mie scattering and absorption coefficients are discussed in detail by Gilbert N. Plass
[1966]. Table-7.1 gives the complex refractive index of Al2O3 as a function of wavelength, λ
Table 7.1 Complex refractive index of Al2O3
Wavelength,
λ(µm) n1 n2
0.5 1.77 1.0E-06
1.0 1.75 1.0E-06
2.0 1.74 1.0E-06
3.0 1.71 1.0E-06
4.0 1.68 1.0E-05
5.0 1.63 1.0E-04
6.0 1.54 2.2E-04
8.0 1.35 3.3E-04
10.0 1.09 5.0E-04
The sensitivity of imaginary part of complex refractive index, namely the absorption
coefficients of Al2O3 on the wave length as a function of temperature is shown in fig.7.1.
Even though there is appreciable increase in the absorption coefficient at elevated
temperatures in the range of 0.25µ to 0.06µ, the absolute value of absorption coefficient is
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negligible when compared to the scattering coefficients listed in table 7.1. Hence the
refractive index of Al2O3 particles is considered as real in this study.
-0.00001
0.00004
0.00009
0.00014
00.2x10
-50.4x10
-50.6x10
-50.8x10
-51.0x10
-5
1000 0C
100 oC
Wave Length, λλλλ (m.)
Ima
gin
ary
Pa
rt o
f R
efr
ac
tiv
e In
de
x
Fig.7.1 Imaginary part of refractive index of Al2O3 at normal
and elevated temperatures
7.4. SCATTERING ANGLE AND CROSS SECTIONS
The scattering of radiant energy in the exhaust plume is modeled as follows:
(1) Using the phase function, the direction of the scattering is predicted.
(2) Depending on the size parameter, either Rayleigh scattering or Mie scattering
formulations are used to estimate the scattering efficiency of a particle.
(3) The scattering efficiency of particles in a control volume is then estimated by
adding the scattering efficiency of each particle contained in the control volume. The particle
density of each category of particles is given by the Rosin-Rammler distribution.
The scattered wave at any point in the distant field has the character of a spherical
wave in which the energy flows outward from the particle. Scattering angle is defined as the
angle between the forward direction of the incident beam and a straight line connecting the
scattering particle and the detector. A schematic of the scattering angle is shown in fig.7.2
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Fig.7.2 Schematic of scattering angle caused by a scattering particle
To fix the point P in a three dimensional space one needs another angle Φ, called
azimuth angle as described in the figure 7.3 Draw Perpendicular to the YZ plane from the
point P. The Perpendicular will intersect the YZ plane at Q. The angle made by OQ with OY
is the azimuth angleφ . If r is the distance OP, then the intensity, I, of the scattered light at
P is given by equation (7.1)
0II = F ),( φη (7.1)
K2r2
where I0 is the Intensity at origin O and λ
π2=K , the wave number. The Function
F ),( φη is the dimensionless function of the direction. The relative values of I or F may be
plotted in a polar diagram as a function of η in a fixed plane through the direction of
incidence. This diagram is called a scattering diagram of the particle and this presents a
total view of nature of scattering as a function of η. Let the total energy scattered in all
directions be equal to the energy of the incident wave falling on the area Csca of the particle.
The term Csca is known as scattering cross section of the particle. It is shown [1] that the
scattering by any finite particle is fully characterized by its four amplitude functions which
are complex functions of the directions of incidence and of scattering. Knowledge of these
amplitude functions suffices for computing the intensity and polarization of scattered light,
the total cross sections of the particle for scattering, absorption and extinction.
η
o Point P
Incident Light
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Fig.7.3 A schematic of geometry in fixing a point in 3 dimensional spaces.
For homogeneous spheres only the first two amplitude functions are to be
evaluated as functions of scattering angle. When the function F ),( φη is divided by K2Csca,
another function of direction known as phase function is obtained. Directional magnitude of
the scattered intensity is related to the entire scattered intensity times the phase function
Robert Siegel, and John R. Howell (2002)
7.5. SCATTERING PHASE FUNCTIONS
Energy absorbed is converted primarily to thermal energy, whereas scattered
radiant energy is redistributed in the medium. The mapping of redistribution of radiant
energy in the medium is given by the scattering phase function. The scattering function
may be determined analytically for simple geometric shapes, such as spheres, by solving
Maxwell’s equations with the coordinate system and boundary conditions corresponding to
an infinite plane wave incident on the object. This finding was first published by Gustav Mie
[6] in connection with his studies of colloidal suspensions of gold. Scattering Coefficient is
the integral of scattering function over the full solid angle 4π.
For axially-symmetrical scattering function, i.e. scattering function that depends only
on the scattering angle, the phase function denoted by λφ with unit of [Steradian
1] is
obtained by normalizing the scattering function by the scattering coefficient, Qsca .
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ie; λφ (θ) = scaQ
)(θβλ (7.2)
The phase function can be regarded as a distribution of probability of a photon being
scattered at an angle. For an axially symmetrical phase function [7] we have
2 θθθφπ λ dSin∫Π
0
)( = 2 ∫Π
0
)(
scaQdSin θθθβπ λ =
sca
sca
Q
Q = 1 (7.3)
where the factor of 2π arises from integration over the angle in its full range: from 0 to 2π.
Thus the directional distribution of scattered energy is expressed by the phase function. In
general it is denoted as ),( ΩΩ′λφ , which represents the fraction of energy incident in
direction ,Ω′ and scattered into direction Ω.
Here the angle of incidence is said to be Ω /( η /, φ/). Similarly the angle of the scattered
beam also can be described by Ω ( η , φ). The angle between the direction of the incident
Angle between beam Ω / and the scattered beam Ω is called the scattering angle Θ. The
scattering angle is related to the azimuthal and zenith angles via the following, John R.
Howell and M. Pinar Menguc [1988], equation.
)cos(sinsincoscoscos φϕηηηη ′−′+′=Θ (7.4)
Fig.7.4 Parameters of solid angle Ω / defined by the azimuth angle η / and polar angle φ/
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The scattering phase function is normalized so that the sum of scattered energy in all
directions add up to 100 percent as shown by equation (7.5)
1sin),,,(4
1),(
4
1
0
2
0
4
0
=′′=ΩΩΩ′ ∫∫∫==Ω
ϕηηϕηϕηφπ
φπ
π
λ
π
ϕ
π
λ ddd (7.5)
The phase function is a measure of the anisotropy of the scattering. It provides a factor for
each direction with which the incoming intensity has to be multiplied to give the outgoing
intensity. Hence for isotropic scattering the phase function is 1 for all direction.
It can be seen that with increasing z, the size parameter, the particles become more
forward scattering. Also for non homogeneous spheres, especially where there is a size
distribution, the small variation on the complex index of refraction does not affect the phase
function significantly. The scattering phase function is purely a function of η for a particle,
such as a sphere, where scattering characteristics are independent of the circumferential
direction φ. If the size is small, say 05.0≤z , then the Rayleigh phase function [5] defined
by equation (6.6) can be used.
( )ηηφ 2cos14
3)( += (7.6)
Figure 7.5 given below shows the directional variation of scattered radiation
intensity as predicted by the Rayleigh phase function. Maximum forward scattering is
occurred at 1800 while the maximum backward scattering occurred at 00. The relative
intensity distribution of the scattered radiant energy is expressed by the ratio of the
scattered radiation intensity to the intensity of isotropic scattering in the same direction.
1=φ is the value for isotropic case for all angles and is plotted in the same figure for
comparison.
On the other hand, for particles with diameter much larger than the wave length of
radiation, ie; for large size parameter, a Legendre polynomial expansion may yield better
accuracy, Milton Kerker [1969], which can account for the strong forward scattering peak.
But this leads to a significant increase in computational effort.
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Fig 7.5 Scattering phase function for Rayleigh and isotropic scattering
If the intensity of the thermal radiation emitted from a surface does not vary with
direction it is referred to as a Lambertonian surface. The scattering pattern of a sphere with
a Lambertonian surface (also called as diffuse sphere) has been computed by E
Schoenberg [1929] the derived phase function is given by equation (7.7)
( )ηηηπ
ηφ cossin3
8)( −= (7.7)
The scattering diagram of the phase function given by the equation (7.7) is shown in fig.7.6.
The validity of this phase function is further discussed in the literature as valid for a gray
sphere with a radius sufficiently greater than the wavelength.
For small non absorbing spheres the phase function is given by equation (7.8) and the
scattering diagram is given in figure 7.7.
−+
−=
22
2
1coscos
2
11
5
3)( βββφ (7.8)
00
0.5
0.5
1.0
1.0
1.5
1.5
00 0.50.5 1.01.0 1.51.5180
o
135o
90o
45o
0o
315o
270o
225o
Isotropic Rayleigh Scattering
108
00
2
2
4
4
6
6
00 22 44 66180
o
135o
90o
45o
0o
315o
270o
225o
Isotropic ScatteringPhase functionfor a diffuse sphere
Fig 7.6 Scattering phase function for a diffuse sphere and isotropic scattering
00
1
1
2
2
3
3
00 11 22 33180
o
135o
90o
45o
0o
315o
270o
225o
Isotropic scattering functionPhase function of small nonabsorbing spheres
Fig 7.7 Scattering phase function for small non absorbing spheres and isotropic
scattering
7.6. MAGNITUDE OF SCATTERING
A built in parameter known as size parameter is defined in the scattering theory
connecting the geometrical size and the wave length of the incident energy. The size
109
parameter is defined as λ
π rz
2= and the magnitude of scattering is modeled in different
forms depending on the value of size parameter. The limits of validity of scattering formula
depend on both the size parameter and the refractive indices of the particle and surrounding
medium. Clearly the size parameter is a scaling parameter relating the particle diameter
and the radiation wave length. For 3.00 ≤≤ z Rayleigh-Debye scattering theory is valid.
7.6.1. Rayleigh Scattering
If the refractive index of the particle is sufficiently close to that of the external
medium and if the particle is not too large, each volume element behaves as a Rayleigh
scatterer. Each of the scattered wavelets in turn mutually interferes. The model was
precisely calculated by Lord Rayleigh as follows. According to this law, the scattering
intensity is proportional to λ-4 and on increasing the size of the particles, the scattering
intensity changes more slowly as a function of wavelength. The scattering cross section,
Csca, of a sphere with a dielectric constant ε2
( )4
112
657
12
32
2
λεε
πεε
+
−=
rCsca (7.9)
where ε1 is the dielectric constant of the medium and λ is the wave length of radiant energy.
If the radius of the sphere is comparable with the wave length, then the quantity Qs
oscillates with the rising λ around a mean value of the same order as the area of the
projection of sphere.
7.6.2. Mie Scattering
Here in this study, rigorous scattering theory for spheres of arbitrary size is carried
out in order to capture all the wiggles exhibited by the scattering cross sections of
aluminium oxide spherical particles contained in the exhaust plume. In the field of scattering
theory, perhaps Mie scattering may be the branch which finds majority of the applications. It
captures all the wiggles in the domain of scattering by making use of the asymptotic
behavior of the spherical Bessel functions. Both the physics and the mathematical models
are complex and hence designers usually resort to make approximations for the size
parameter so that the Mie scattering coefficient is taken for the model value of the size
parameter in the computational domain. The scattering and extinction coefficients are
functions of the size parameter z. The intensity scattered in all directions for an incident
110
wave of unit intensity is independent of the state of polarization of the incident beam and is
known as the scattering cross section Csca. The size parameter is a built in single parameter
connecting the size of the scatterer and the wave length of the incident radiation. The
refractive index of the scatterer and medium and the size parameter are interconnected
through complex mathematical functions, infinite series and recurrence relations in the
computation of scattering cross section. The scattering cross section is given by
∑∞
=
++=1
222
)12(2 n
nnsca banCπ
λ (7.10)
The efficiency factor Qsca is the ratio of scattering cross section to he geometrical cross
section and is given by
∑∞
=
++=1
22
2)12(
2
n
nnsca banz
Q (7.11)
The average geometrical cross section of a convex particle with random orientation
is one-fourth its surface area (PP:110 Van de Hulst).
The total intensity abstracted from the incident beam of unit intensity both by
scattering and absorption is the extinction cross section and is independent of the state of
polarization of the incident beam. The extinction coefficient is given by
∑∞
=
++=1
2
)Re()12(2 n
nnext banCπ
λ (7.12)
The corresponding extinction efficiency factor is obtained by dividing the extinction
coefficient by geometrical cross sectional area
∑∞
=
+=1
2)12(
2
n
ext nz
Q )Re( nn ba + (7.13)
The coefficients an and bn are[1,9] given by
)15.7()()()()(
)()()()(
)14.7()()()()(
)()()()(
zzm
zzmb
zmz
zmza
nnnn
nnnn
n
nnnn
nnnn
n
′−
′
′−
′=
′−
′
′−
′=
ξβψβψξ
ψβψβψψ
ξβψβψξ
ψβψβψψ
111
Here λ
π rz
2= and
2
1
m
mmandmz ==β where r is the radius of particle and λ is the
wavelength in the medium. Here an and bn are known as scattering coefficients. The
functions )()( zandz nn ξψ are expressed in terms of half integral order Neumann and
Hankel functions as given below:
)(2
)(2
1 zJz
zn
n+
=π
ψ (7.16)
where )(2
1 zJn+
is half integral order Bessel Function.
)(2
)()()(2
1)2( zHz
zizz nnnn +=+=π
χψξ (7.17)
where )(2
1)2(zH n+ is the half integral order Hankel function of the second kind. The
function )(2
)(2
1 zNz
zn
n+
−=π
χ (7.18)
Where )(2
1 zNn+
the half integral is order Neumann function and will be used in the
computation of scattering efficiency for spheres with no absorption. Both the functions
)()( zandz nn ξψ are known as Ricatti functions. More precisely the function )(znψ is
known as Ricatti-Bessel function while the function )(znξ is known as Ricatti-Hankel
function. In the case of refractive index with no component of absorption for a sphere, the
scattering efficiency is defined as
)sin)(sin12(2 2
1
2
2 n
n
next nz
Q βα∑∞
=
++= (7.19)
where the angles nn and βα are given by the equations (7.20) and (7.21) given below
)21.7()()()()(
)()()()(tan
)20.7()()()()(
)()()()(tan
zzm
zzm
zmz
zmz
nnnn
nnnn
n
nnnn
nnnn
n
′−
′
′−
′−=
′−
′
′−
′−=
χβψχβψ
ψβψψβψβ
χβψχβψ
ψβψψβψα
112
Here β and βn are different. The parameter β is already defined as mz=β . In the notation,
βn, n represents the order of the term in eq.(7.21). For example, the values of β1, β2, β3 , β4,
etc. are needed in equation (7.19) which can be evaluated from tan β1, tan β2, tan β3, tan
β4,etc.
7.7. BESSEL FUNCTIONS
Bessel’s function is defined by
0)()1()(1)(
2
2
2
2
=−++ zzz
n
dz
zdz
zdz
zzdn
nn (7.22)
Since Bessel’s function is a second order differential equation, it has two independent
solutions. They are known as Bessel function )(zJ n and Neumann function )(zN n each
of order n (Milton Abramowitz and Irene A. Stegun [1972]) and argument z.
The Bessel function is defined as
mn
m
m
n
z
nmmzJ
2
0 2)1(!
)1()(
+∞
=
∑
++Γ
−= (7.23)
Whenever n is not an integer, the Neumann function is defined as
[ ])()cos()()sin(
1)( zJnzJ
nzN nnn −−= π
π (7.24)
The term Bessel function is frequently used as a generic name for both solutions which are
then called Bessel functions of the first and second kind, respectively. Both the solutions in
general are also called as cylinder function and denoted as )(zzn . There exists recurrence
relations valid for all cylinder functions and for their derivatives and are used in the
computation of both Ricatti functions. The following recurrence relations defined by
equations (7.25) and (7.26) are employed.
)(12
)()( 11 zZz
nzZzZ nnn
+=+ +− (7.25)
)(2
1)(
2
1)(11 zZzZ
dz
zdZnn
n
+− −= (7.26)
113
The most efficient form for computation of Ricatti-Bessel functions are given by the
equations (7.27) and (7.28) as given below.
z
z
zdz
dzz
n
n
n
sin)( 1
−= +ψ (7.27)
z
z
zdz
dzz
n
nn
n
cos)1()( 1
−= +χ (7.28)
The first two terms of order 0 and 1 of each Ricatti- Bessel function [9] are given as
)32.7(sincos
)()31.7(cos)(
)30.7(cossin
)()29.7(sin)(
10
10
zz
zzzz
zz
zzzz
+=−−−=
−=−−−=
χχ
ψψ
7.8. EVALUATION OF RICATTI – BESSEL FUNCTION )(αψ n
Putting n=1 in the recurrence formula of cylindrical function, one gets
)(3
)()( 120 zz
zz ψψψ =+ ⇒ zz
zz
z cos)3
(sin)13
()(22 −+−=ψ (7.33)
Similarly n=2 in the recurrence formula of cylindrical function gives rise to
)(5
)()( 231 zz
zz ψψψ =+ ⇒ zz
zzz
z cos)15
1(sin)615
()(233 −+−=ψ (7.34)
The higher orders of the function can be similarly worked out as follows:
zzz
zzz
z cos)10105
(sin)145105
()(3244 +
−++−=ψ (7.35)
zzz
zzzz
z cos)1945105
(sin)15420945
()(42355 −−++−=ψ (7.36)
zzzz
zzzz
z cos)21103951260
(sin)1210472510395
()(532466 −−+−+−=ψ (7.37)
Thus the Ricatti-Bessel function )(znψ can be expressed as equation (7.38) in terms of
the trigonometric functions sin and cos with polynomial functions of the size parameter As
and Bs as their variable coefficients.
114
znBsznAszn cos)(sin)()( +=ψ (7.38)
The number of terms needed for the evaluation of scattering efficiency can be
determined only after seeing the convergence. It is observed that six to seven terms of the
Ricatti Bessel functions gives sufficient accuracy. Hence the details of derivations of seven
terms of these functions are explained below for provide the computational details of the
scattering efficiency factor.
Since the suffix zero cannot be put as suffix in a FORTRAN program, the order of
Ricatti-Bessel function is incremented by an order 1 in representation. ie; )(znψ where
n=0 to 6 defined in the mathematical form are renamed from n=1 to 7 respectively in the
numerical algorithm as shown in the table 7.1. This is done for the ease of computer
programming.
Table 7.2 Coefficients As(n)& Bs(n) of )(znψ of eqn. 7.38
Term, n Coefficient of sin function, As(n) Coefficient of cosin function, Bs(n)
1 1.0 0.0
2
z
1
-1.0
3 1
32
−z
z
3−
4
zz
6153
− 2
151
z−
5 1
4510524
+−zz
zz
101053
+−
6 +−
35
420945
zz z
15 1
94510542
−−zz
7 1
210472510395246
−+−zzz
zzz
2110395126053
−−
115
7.9. EVALUATION OF RICATTI FUNCTION )(znχ
The following initial values are used to generate the higher orders as explained below:
)40.7(sincos
)(
)39.7(cos)(
1
0
−−−+=
−−−=
zz
zz
andzz
χ
χ
Putting n=1 in the recurrence formula of cylindrical function, one gets
)(3
)()( 120 zz
zz χχχ =+ ⇒ zz
zz
z cos)13
(sin)3
()(22 −+=χ (7.41)
Similarly n=2 in the recurrence formula of cylindrical function gives rise to
)(5
)()( 231 zz
zz χχχ =+ ⇒ zzz
zz
z cos)615
(sin)115
()(323 −+−=χ (7.42)
The higher orders of the function can be similarly worked out as follows:
zzz
zzz
z cos)145105
(sin)10105
()(2434 +−+−=χ (7.43)
zzzz
zzz
z cos)15420945
(sin)1105945
()(35245 +−++−=χ (7.44)
zzzz
zzzz
z cos)1210472510395
(sin)21103951260
()(246536 −+−+++−=χ (7.45)
Thus as in the case of the Ricatti-Bessel function )(znψ , )(znχ also can be expressed
as equation (7.46) in terms of the trigonometric functions sin and cos with polynomial
functions of the size parameter As and Bs as their variable coefficients.
znBkznAkzn sin)(cos)()( +=χ (7.46)
As in the case of first Ricatti-Bessel function, here also, the above order of terms of second
Ricatti Bessel function are incremented by an order 1 for the ease of programming. ie;
)(znχ where the suffix n=0 to 6 are renamed from n=1 to 7 respectively as shown in the
table 7.3.
116
Table 7.3 COEFFICIENTS Ak(n) & Bk(n) of )(znψ of eqn. 7.46
Term, n Coefficient Ak(n) Coefficient Bk(n)
1 1.0 0.0
2
z
1
1
3 1
32
−z
z
3
4
zz
6153
− 115
2−
z
5 1
4510524
+−zz
zz
101053
−
6 +−
35
420945
zz z
15 1
10594524
+−zz
7 1
210472510395246
−+−zzz
zzz
2112601039535
+−
It may be noted from Table7.2 and Table7.3, that both the Ricatti Bessel functions
expressed in the form of equations (7.38) and (7.46) can be expressed by the same
coefficients as shown below:
znBsznAszn cos)(sin)()( +=ψ (7.38)
znBkznAkzn sin)(cos)()( +=χ (7.46)
Where
)48.7()()(
)47.7()()(
nBsnBk
nAsnAk
−=
=
This similarity of expressions obtained for both the Ricatti-Bessel functions considerably
simplifies the coding of the computer program for predicting scattering efficiency factors.
117
7.10. DERIVATIVES OF RICATTI BESSEL FUNCTIONS
7.10.1. DERIVATIVE of )(znψ
The function ( ) ( ) sin ( ) cosn
z As n z Bs n zψ = +
is differentiated with respect to the argument z to obtain the derivatives )(zn
′ψ . Since
As(n) and Bs(n) are functions of the size parameter z ,obviously,
; ( ) ( ( ))sin ( ) cos ( ( ))cos ( ))sinn
d die z As n z As n z Bs n z Bs n z
dz dzψ ′ = + + − (7.49)
[ ] [ ]
)52.7()()()(
)51.7()()()(
)50.7(cos)(sin)(
cos)()(sin)()(
cos)())((sin)())(()(;
nAsnsBnBd
andnBsnsAnAdwhere
znBdznAd
znAsnsBznBsnsA
znAsnBsdz
dznBsnAs
dz
dzie n
+′=
−′=
+=
+′+−′=
++
−=
′ψ
Equations (7.51) and (7.52) are evaluated for n=1,2,…7 and the values of Ad(n) and Bd(n)
are listed in table 7.4 .
Table 7.4 COEFFICIENTS Ad(n) & Bd(n) of )(zn
′ψ of eqn 7.50
Term, n Coefficient of sin function, Ad(n) Coefficient of cosin function, Bd(n)
1 0 1.0
2 1
12
+−
z
z
1
3
ZZ
363
+−
16
2−
Z
4 1
452142
−−zz
zZ
6453
−
5
Zzz
1019542035
−+−
155420
24+−
zz
6 1
12022054725246
+−+−
ZZZ
ZZZ
15630472535
+−
7
ZZZZ
2116802929562370357
+−+−
1231850562370
246−+−
ZZZ
118
7.10.2. DERIVATIVE OF )(znχ
Similarly, the function znBkznAkzn sin)(cos)()( +=χ is
differentiated with respect to z as follows:
[ ] [ ]
)56.7()()()(
)55.7()()()(
)54.7(cos)(sin)(
sin)()(cos)()(
)53.7(cos)(sin))(()sin)((cos))(()()(
nBknkAnBkd
andnAknkBnAkdwhere
znBkdznAkd
znAknkBznBknkA
znBkznBkdz
dznAkznAk
dz
d
dz
dz nn
+′=
−′=
+=
−′++′=
++−+==′ χχ
Equations (7.55) and (7.56) are evaluated for n=1,2,…7 and the values of Akd(n) and
Bkd(n) are listed in table 7.5 .
Table 7.5 COEFFICIENTS Akd(n) & Bkd(n) of )(zn
′ψ of eqn 7.54
Term, n Coefficient Akd(n) Coefficient Bkd(n)
1 -1 0
2 -
z
1 1
12
+−
z
3 )1
6(
2−−
Z
ZZ
363
+−
4 )
645(
3zZ
−− 14521
42−−
zz
5 )1
55420(
24+−−
zz
Zzz
1019542035
−+−
6 (
+−−
ZZZ
15630472535
) 112022054725
246+−+
−
ZZZ
7 -( 1
231850562370246
−+−ZZZ
) ZZZZ
2116802929562370357
+−+−
It may be noted from Table7.4 and Table7.5, that derivatives of both the Ricatti Bessel
functions expressed in the form of equations (7.50) and (7.54) can be expressed by the
same coefficients as shown below:
119
)57.7(cos)(sin)()( znBdznAdzn +=′ψ
)58.7(cos)(sin)()( znBkdznAkdzn +=′χ
Where )60.7()()(
)59.7()()(
nBkdnBd
nAkdndA
−=
=
The similarity of expressions of terms of both the Ricatti Bessel functions is also
seen in the case of their derivatives as shown by equations (7.59) and (7.60). This property
of terms of the derivatives of Ricatti Bessel functions also considerably simplifies the coding
of the computer program for predicting scattering efficiency factors.
7.11. EVALUATION OF SCATTERING EFFICIENCY
The trigonometric functions tan(αn) and tan(βn) can be evaluated by solving the
equations (7.38) and (7.46) for )(znψ and )(znχ respectively and equations (7.50) and
(7.54) for their derivatives. Once tan(αn) and tan(βn) are known, sin2(αn) and sin2(βn) can be
directly calculated to obtain the scattering efficiency given by the equation (7.19).
7.12. SCATTERING IN A CONTROL VOLUME CONTAINING MANY PARTILCES
According to Lamberts law, the monochromatic radiant energy traverses a random,
attenuating medium where the probability of a photon experiencing an absorption/scattering
event, the radiant power flux decays exponentially with the product of the path length in the
medium and the attenuation coefficient of the medium. Lamberts law is a limiting case of the
steady state RTE for a medium with no internal sources in which the path function of the
RTE vanishes. In the present context, the path function is the radiance gained per unit
distance, s, by scattering of the radiance field at s in the direction of the axis. Hence
attenuation due to scattering of solid particles scattered in random fashion in each control
volume of the computational domain should be defined.
When the attenuation coefficient is expressed in terms of the concentration of
particles, the formula for the exponential decay of radiation power is known as the Beer
Law. The scattering efficiency of a volume cell with a number of different sized particles is
120
depending on the concentration of particles. It has been found (Van de Hulst PP: 32) that,
for quite different reasons, the scattering and extinction cross sections of the single particles
must be added to give the corresponding cross sections for the entire cloud. It also states
that the scattering pattern caused by reflection on very large convex particles with random
orientation is identical with the scattering pattern by reflection on a very large sphere of the
same material and surface condition (page:111, Vande Hulst)
Let there be )(rdN particles per unit volume in the radii ranging from r to r+dr and
let Csca be the scattering cross section for a particle of radius r. Then integrating over all the
particles, the effective scattering area in a differential volume dV of the particle cloud can be
found as
)()(0)(
rdNrCdVCrN
scasca ∫∞
==
Equivalently, for particles of sizes of m different categories, the scattering efficiency of the
cell is defined as
( )2
1
isi
m
i
c rNcQQ π∑=
=
where Qsi is the scattering efficiency of the ith category of particles
Nc is the number of particles of the ith categrory present in the cell
ri is the radius of the ith category of particles
Here the number of particles, Nc, in each cell is obtained from the particle density
derived from the particle spectrum as per the Rosin-Rammler distribution.
7.13. EXPERIMENTAL VALUES OF EXTINCTION COEFFICIENTS
Experimental Values of Extinction Coefficients, T.E. Mills, PJ Bishop and A.Minardi
[1994] of Laser-Produced Aluminium Plumes were obtained as a function of radial position
and time at an axial location. The purpose of the study was to measure the extinction
coefficient of a two dimensional, axisymmetric laser produced aluminum cylindrical plume at
one wavelength 0.632 µm. The choice of wavelength was made due to the cost efficiency
and ease of operation provided by He:Ne lasers. A He:Ne probe laser scanning revealed
that the aluminium plume is optically thin and that the radial extinction coefficient profile has
a Gaussian distribution. Extinction coefficients were measured as a function of radial
position and time at one axial location. It was found that the extinction coefficients
121
decreased with radial distance from the laser beam center. The maximum extinction
coefficient measured at the center of the laser beam is 0.6 cm-1 and this corresponds to an
optical thickness of about 0.04. However, the authors give precautions that this technique
for measurement of extinction coefficients cannot be used when liquid droplets are present
in high concentrations.
7.14. COMPUTED SCATTERING EFFICIENCY PARAMETERS
Wavelength of the incident radiation is estimated from the universal form of spectral
distribution of blackbody hemispherical emissive power as discussed in chapter 4.
Refractive index of the alumina particle can be taken from the curve shown in fig.7.16. At a
particular axial location of the domain of computation, variation of size parameter is
observed as ranging from 0.46 to 8.76. Fig.7.17 shows typical values of the scattering
efficiency factors evaluated in the computational domain of the present study.
Even though the observations made in this experiment might not be valid for large
solid rocket motors, these observations could be checked in this theoretical study. Figure
7.8 shows the predicted scattering efficiency factor of control volumes defined in the radial
direction from the centre of plume at a typical axial location. This graph is corresponding to
a case where ten control volumes are defined radially. More continuity will be obtained if
more number of control volumes is defined in the radial direction. Even though all types of
particles are contained in the first few control volumes, the scattering efficiency of these
cells are lower because of their lower volumes and thereby lower number of particles.
Similarly in the case of control volumes defined in the outer region and its periphery, the
particle size has come down causing lower scattering efficiency. Further it may be noted
that, the trend of the curve may be approximated to that of a Gaussian distribution starting
from the centre of the plume.
122
0
2.5
5.0
7.5
10.0
12.5
1.2 1.3 1.4 1.5 1.6
Radial distance(m)
Ne
t s
catt
eri
ng
ex
tin
cti
on
fa
cto
r (m
-1)
Figure7.8 Radial scattering efficiency factor from the centre of the plume
7.15. SCATTERING CHARACTERISTICS OF EXHAUST PLUME
The scattering characteristics of exhaust plume and the aluminium oxide particles
are to be studied for estimating the scattering efficiency of solid particles contained in the
plume. They are
(1) Size and shape of aluminum oxide particles contained in the plume
(2) Refractive indices of different constituents of the exhaust plume
(3) Refractive index of the aluminium oxide particles.
Dewban, Kinslow and Watson [1988] reveals that the dust particles from 0.01 to 0.1
µ are irregularly shaped and in many cases have agglomerated into clusters. Those
particles in the range from 1 to 100 µ are spherical. It is quite common to find some of the
smaller spheres attached to the larger ones. Thus the geometrical shapes of aluminium
particles in the exhaust plume of rockets can be very well approximated to spheres of
different radii. Thus the formulation for scattering efficiency of spherical particles can be
adopted in the study.
Refractive indices of different gaseous components in the exhaust plume are listed
in Table 7.6. This table indicates that the refractive index of the gaseous constituents can be
approximated as 1. A discussion on refractive index of alumina is given by Edwards and
123
Babikian [1990] and the finally converged values are published by the website WIKIPEDIA
due to Boston [1991] and is shown in figure 7.9. Thus the ratio of refractive index of
Alumina particles to that of gaseous constituents of the exhaust plume becomes numerically
equal to the refractive index of alumina. Study on particle spectrum of alumina shows that
percentage of particles greater than 5µm is negligible and hence the diameter of particles
taken in this study is in the range 0.5 to 5µm. Wave length of the incident energy is taken
from the spectrum of radiant energy discussed in chapter 4. Thus particle size parameter
and refractive index are defined for predicting the scattering efficiency of the alumina
particles in the exhaust plume of solid rockets.
Table 7.6 Refractive indices of combustion gases
Species Refractive
index
Water vapor 1.000256
Carbon dioxide 1.000449
Carbon monoxide 1.000338
Oxygen 1.000271
Hydrogen 1.000132
Nitrogen 1.000298
Chlorine 1.000773
Air 1.000292
Nitric Oxide 1.000297
Hydrochloric acid 1.000447
124
1.725
1.750
1.775
1.800
1.825
1.850
0 0.5 1.0 1.5 2.0 2.5
Al2o
3 solid particles
Wavelength(µµµµm)
Re
fra
cti
ve
in
de
x
Fig. 7.9 Spectral variation of refractive index of Alumina particles
7.16. RESULTS AND DISCUSSIONS
Since it is often required to mention the developed mathematical model of Mie
scattering, it will be referred as Mie model in this section. It is essential to validate the
developed Mie model. This is accomplished by comparing the predicted values of scattering
efficiency by the Mie model with standard literature. For non-absorbing spheres, Van de
Hulst has derived a formula to define the salient features of the extinction curve not only for
m close to unity but even for values of m as large as 2. The formula for scattering efficiency
factor for a spherical particle is
12)cos1(4
sin4
22
−=−+−= mzwhereQext ρρρ
ρρ
, (7.61)
z is the size parameter and m is the ratio of the refractive index of the scatterer with that of
the medium in which the scatterer is located. Scattering does not occur for m=1. The
developed Mie model encounters a singularity at m=1 and hence no value is delivered by it
for m=1. For m nearer to 1, scattering is small and this is predicted by the model as can be
seen in fig 7.10.
125
0
0.1
0.2
0.3
0 2 4 6 8
Aproximate formulaMie Theory
Size Parameter
Sca
tte
rin
g E
fic
ien
cy F
acto
r
Fig.7.10: Comparison between Mie model and approximate formula for m=1.05
Even for a higher value of the size parameter of 8, the scattering efficiency factor is
only 0.3. Again it may be seen that both approximate formula and the Mie model almost
coincides at the majority of the size parameters except at the beginning and ending of the
domain of the function. Figure 7.11 is the comparison of scattering efficiency factors
predicted by Mie model and the approximate formula. It may be noted that when the value
of m is increased from 1.05 to 1.1, the scattering efficiency factors have increased almost
three times. The slight differences between the values of efficiency factors predicted by Mie
model and approximate formula starts from the size parameter value of 5.5
126
0
0.25
0.50
0.75
1.00
1.25
0 2 4 6 8
Mie theoryApproximate formula
Size parameter
Sc
att
eri
ng
eff
icie
nc
y f
ac
tor
Fig.7.11 Comparison between Mie model and approximate formula for m=1.1
Figure 7.12 shows the difference in the trends of increase of scattering efficiency factors for
the values of m=1.05 and 1.1. The equation of curve for the efficiency factors as a function
of the size parameter for the case m=1.05 is
00444.000377.000412.0)( 2 ++= zzzQext (7.62)
Whereas, the equation of curve for the efficiency factors as a function of the size parameter
for the case m=1.10 is
00830.00405.00124.0)( 2 −+= zzzQext (7.63)
Equations (7.62) and (7.63) show that the scattering efficiency factors are almost linearly
increasing with the increase of size parameters, while for m=1.1, the increase is of non-
linear nature.
According to Van de Hulst, If the absolute value of (m-1) is very small, ie;
,1 smallveryiswherem δδ≈− it makes no difference in the scattering pattern
whether 0101 ≺ −− morm . Figures 7.13 and 7.14 are provided to study the
127
range of the domain of size parameter for validity of this theory. It can be seen from these
figures that up to the value of z=4.5, the values of scattering
0
0.4
0.8
1.2
0 2 4 6 8
Mie theory (m=1.1)Aproximate formula (m=1.1)Mie theory (m=1.05)Approximate formula (m=1.05)
Size parameter
Sc
att
eri
ng
eff
icie
nc
y f
ac
tor
Figure 7.12 Comparison of scattering efficiency factors for the values of
m=1.05 and 1.1.
0
0.1
0.2
0.3
0 2 4 6 8
m=0.95m=1.05
Size parameter
Sc
att
eri
ng
eff
icie
nc
y f
ac
tor
Figure 7.13 Comparison of scattering efficiency factors for the values of m=0.95 and
m=1.05
128
Efficiencies are matching for the case 05.01 =−m and then onwards start deviating.
Again the values of scattering matches up to the value of z=3.0 and then starts deviating for
the case 10.01 =−m . This indicates that as the value of refractive index starts deviating
more from the value 1, the domain of size parameter where the same scattering pattern
exists is reduced. This validates the view of Van de Hulst.
0
0.2
0.4
0.6
0.8
1.0
0 2 4 6 8
m=1.1m=0.9
Size parameter
Scatt
eri
ng
eff
icie
ncy
Figure 7.14 Comparison of scattering efficiency factors for the values of
m=1.1 and m=0.9
Figures 7.15 and 7.16 give the predicted scattering efficiency factors by Mie theory
for m=1.5 and m=2.0 respectively and their comparison with the approximate formula. It
may be seen from figures 7.15 and 7.16 that up to the value of particle size of 0.7, the trend
of Mie theory values are just opposite to that of approximate formula. Then afterwards, both
are in the same trend and Mie theory captures more wiggles in the values of scattering
efficiency. Figure 7.17 gives the comparison of scattering efficiency factors with the
approximate formula.
129
0
2
4
6
0 2 4 6 8
Approximate formulaMie theory
Particle size
Sca
tteri
ng
Eff
icie
nc
y f
ac
tor
Fig. 7.15 Comparison of values of scattering efficiency factor for m=1.5
0
2
4
6
8
10
0 2 4 6 8
Mie theory (m=2.0)Approximate formula (m=2.0)
Particle size
Sca
tteri
ng
eff
icie
ncy
fac
tor
Fig 7.16 Comparison of values of scattering efficiency factor for m=2.0
130
0
1
2
3
4
0 2 4 6 8
Approximate formula (m=3.0)Approximate formula (m=2.0) Approximate formula (m=1.5)
Particle size
sc
att
eri
ng
eff
icie
nc
y f
ac
tor
Fig. 7.17 scattering efficiency factor predicted by approximate formula
0
4
8
12
16
0 2 4 6 8
Mie theory (m=3.0)Mie theory (m=2.0)Mie theory (m=1.5)
Particle size
Sc
att
eri
ng
eff
icie
nc
y f
ac
tor
Fig. 7.18 scattering efficiency factor predicted by Mie theory
7.16.1. Sensitivity of Scattering Efficiency on Wavelength
Figures 7.19 and 7.20 shows the variation of scattering efficiency of particles of 5
µm and 2 µm radii with a refractive index of 1.80. Both curves show differences in scattering
efficiency for lower values of wavelength, but show the same trend of approximate theory.
131
Mie theory could capture more number of wiggles as seen in these curves. For the higher
size particle, scattering efficiency exhibits an increasing trend with the increase of
wavelength as seen in figure 7.19. Whereas, for the lower size particle, scattering efficiency
exhibits an increasing followed by a decreasing trend with the increase of wavelength as
seen in figure 7.20
0
2
4
6
8
10
0.2x10-5
0.4x10-5
0.6x10-5
0.8x10-5
1.0x10-5
0
2
4
6
0
3
6
9
12
15
size parameterMie theory (5µ)Approximate formula
Wavelength,m
Sc
att
eri
ng
eff
icie
nc
y
Fig.7.19 Sensitivity of scattering efficiency of a particle of radius 5 µ on wavelength
0
2
4
6
8
10
0.35x10-5
0.70x10-5
1.05x10-5
0
5
10
15
Mie theory (radius=2µ)Approximate formulaSize parameter
Wavelength (m.)
Sc
att
eri
ng
eff
icie
nc
y
Fig.7.20 Sensitivity of scattering efficiency of a particle of radius 2 µ on wavelength
7.17. CLOSURE
The required fundamental parameters for modeling the scattering of radiant energy
are described in this chapter. Scattering laws apply with equal validity to all wavelengths.
Interestingly, these depend upon the size parameter which is the ratio of the circumference
132
of particle to the wavelength of incident radiation. Depending on the range of values of this
natural size parameter, different theories exist to define the scattering efficiencies. Both
Rayleigh and Mie theory of scattering are described in this context. The scattering cross
sections and efficiencies of spherical particles of Al2O3 are estimated as a function of their
size parameter and their dependence on wavelength of the incident energy is separately
addressed. Rayleigh formula is a single formula showing its inverse dependence to the
fourth power of the wavelength. The Mie scattering is valid over a wide range of the size
parameter. Theory of Mie scattering makes use of Ricatti-Bessel functions because of their
asymptotic behavior for capturing the wiggles in the curves of scattering efficiencies.
Approximate formula exist for scattering efficiencies in certain range of the size parameter
and is used for comparing the scattering efficiencies predicted by Mie theory in the limited
range where the formula is valid . It is observed that the trends of the curves generated by
the two methods are the same and show reasonably good comparison in certain region.
However the approximate formulae are not able to capture the wiggles for higher particle
sizes. The minor wiggles appearing in the two curves infer that scattering efficiency vary
drastically and highly non-linear with the wavelength and size parameter so that prediction
of scattering efficiency by interpolating even in the close intervals becomes erroneous.
In general, the scattering efficiency is used as an input for various analysis and
hence simple and fast computing model is preferred. For this, the Ricatti Bessel functions
and their derivatives appearing in the Mie coefficients are expressed in terms of polynomials
of the size parameter. These functions are derived up to the 7th term using their recurrence
relations to obtain good convergence. In general, the scattering efficiency is used as an
input for various analysis and hence simple model is preferred. Ideally, the subroutines
made from these models should take less computer time without reducing the accuracy.
This objective could be met in this study by expressing the Mie coefficients with the
polynomials of size parameter. Major findings of this chapter is published as a paper titled
“Modelling of Scattering in Thermal Radiation from Aluminium Oxide Particles in the
Exhaust Plume of a Solid Motor ” in the proceedings of 19th National and 8th ISHMT-ASME
Heat and Mass Transfer Conference , 03-05,Jan,2008, JNTU, Hyderabad.