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Braz, Wilson
A Method for the Estimation of IR Emissions from Sooty Turbine
Engine Exhausts
by
Wilson Braz
A Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING
Major Subject: MECHANICAL ENGINEERING
Approved:
_________________________________________
Dr. Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, CT
May 2010
(For Graduation Dec 2010)
i
© Copyright 2010
by
Wilson Braz
All Rights Reserved
ii
CONTENTS
List Of Tables ................................................................................................................... iv
List Of Figures ................................................................................................................... v
Acknowledgements........................................................................................................... vi
Abstract..........................................................................................................................- 1 1. Introduction..............................................................................................................- 2 2. Theory / Methodology .............................................................................................- 4 2.1
IR Theory Fundamentals................................................................................- 4 -
2.2
Exhaust Gas Radiance and Transmittance using MODTRAN ......................- 6 -
2.3
The addition of soot model to MODTRAN results........................................- 7 -
2.4
Chemical Composition of Turbine Engine Exhaust ......................................- 9 -
2.5
Step-by-Step Process Breakdown ................................................................- 10 -
3. Results & Discussion.............................................................................................- 13 4. Conclusions............................................................................................................- 17 References....................................................................................................................- 18 Appendix A..................................................................................................................- 19 Appendix B..................................................................................................................- 24 -
iii
List Of Tables
Table 1 – Values for the composition of standard air – pre-combustion.....................- 10 Table 2 – The composition of exhaust plume – 99% combustion efficiency..............- 10 -
iv
List Of Figures
Figure 1 – The electromagnetic spectrum. ....................................................................- 4 Figure 2 – Blackbody curves. ........................................................................................- 5 Figure 3 –Propane absorption coefficient factor a/C plotted against 5/......................- 8 Figure 4 – Sample MODTRAN input card..................................................................- 11 Figure 5 – Spectral radiance combined MODTRAN output and calculated soot. ......- 13 Figure 6 – 3 to 5m spectral results. ...........................................................................- 14 Figure 7 – Spectral absorption broken out by constituent. ..........................................- 15 -
v
List Of Symbols
 = wavelength in mm
T = temperature in Kelvin
i = radiance in W
m 2  sr
W
i = spectral radiance in
m 2  sr  m
i BB = radiance of a blackbody
2
C1 = first radiation constant = 2hc 2 = 3.7418x10-16 W  m
C2 = second radiation constant = hc k = 1.4388x10-2 m  K
c = speed of light in a vacuum = 2.9979x108 m/s
h = Planck constant = 6.6261x10-34 J s
k = Boltzmann constant = 1.3807x10-23 J
K
K = spectral extinction coefficient
S = path length in meters
a = spectral absorption aoefficient
 = spectral scattering coefficient
 = spectral absorptivity
 = spectral emissivity
 = spectral transmissivity
Q = spectral absorption efficiency factor
D = average soot particle size in meters
3
C = volumetric concentration in m
m3
N = number of particles per unit volume in 1
vi
m3
Acknowledgements
I’d like to thank my dearest wife, Joanne, who with her tireless work in tending to the
affairs of our home and family, allowed me the time to complete this project.
I’d also like to thank Dr. Ernesto Gutierrez-Miravete whose very wise direction, and
well organized materials made the completion of this project a possibility.
vii
Abstract
In this study, the MODTRAN atmospheric model is used to model infrared
emissions spectrally, or as a function of wavelength, from common constituents,
primarily CO2 and H2O. The MODTRAN results are then coupled with a proposed
method to account for the emissions caused by soot, or carbon particulate. This method
includes calculations and results of exhaust gas species concentrations using standard
atmospheric air and Jet A fuel properties as inputs. Other parameters such as carbon
soot particle size and concentration are taken from published values. Results for spectral
plume emissions are presented, and sensitivities of varying parameters are discussed.
Also, this project contains comparison with existing results and a discussion of limitation
of the proposed method.
-1-
1. Introduction
Turbine engines such as those used in helicopters generate power using basic
thermodynamic principals of the Brayton cycle [3]. Fuel is mixed with air, compressed,
and ignited. The combustion reaction releases heat which causes expansion of the gases
which are then utilized to generate power. The constituents of the resultant exhaust
gases are released into the atmosphere at an elevated temperature, and thus emit infrared
(IR) radiation. We can quantize the amount of emitted radiation by using terms of
energy flux per unit solid angle which, in SI units, happens to be
W
. This quantity
m 2  sr
will be referred to as ‘radiance’ in this document.
The IR emissions from hot exhaust plume vary considerably than those from opaque
surfaces. Emissions result from light interactions on a molecular scale. Light at certain
frequencies cause resonances with molecules, thus absorbing and emitting in certain
wavelengths, while being transparent or scattering in others. Therefore, it is of interest
to study radiant emissions at each wavelength. When quantities are given in terms of
wavelength, we refer to them as ‘spectral’ quantities. Therefore, radiance given as a
function of wavelength is termed ‘spectral radiance’ and, in SI can be given in units of
W
where W is energy in watts, m2 is area in square meters, sr is the unit of solid
m  sr  
2
angle in steradians, and  is wavelength in micron.
MODTRAN (MODerate resolution TRANsmission) is a software tool developed by
the US Air Force to model atmospheric transmittance and radiance. Much work has
been put into modeling the molecular interactions of the atmosphere, and therefore
utilization of this work for generating plume IR emission and transmissivity is
advantageous. The code is a console based code written in FORTRAN. Utilization of
the tool requires input parameters to be defined in an input card (text file). When the
code is executed, the program reads the input card, performs the necessary calculations,
and returns an output file which contains, among other data, values of transmissivity and
radiance for the atmospheric path specified at each spectral data point.
The methods proposed in this project also take into consideration the effects of
carbon soot particles in the exhaust plume.
-2-
There is always a certain level of
inefficiency in the chemical reaction process of combusting hydrocarbon fuels, such as
kerosene. This inefficiency results in un-burned carbon particles which coalesce into
randomly shaped particles [4]. These particles can be assumed to be spherical for the
purposes of calculation [1]. They behave like small gray bodies. The interaction is
modeled using Mie scattering theory [7].
MODTRAN contains several aerosol models for predicting interactions with
aerosols in the atmosphere, primarily clouds, and ash. Unfortunately, these models are
not necessarily valid for the soot in engine exhaust as the carbon particle sized found by
experiment [2] is smaller than that in cloud simulations used in MODTRAN. Therefore,
the supplemental method for modeling soot, Mie theory, is used.
-3-
2. Theory / Methodology
2.1 IR Theory Fundamentals
All substances emit electromagnetic radiation which varies in wavelength greatly
depending on the material’s internal energy. The infrared (IR) spectrum, which is most
important in heat transfer through radiation, spans from approximately 0.5m
wavelength to 1000m. Figure 1 shows the entire electromagnetic spectrum.
Figure 1 – The electromagnetic spectrum.
The infrared portion of the spectrum spans approximately 0.5m to 1000m
http://commons.wikimedia.org/wiki/File:EM_spectrum.svg
Emissive power radiating at all wavelengths from the surface of a ‘perfect’ emitter
or ‘blackbody’ is given by the Planck equation:
iBB  , T  
2C1
5  e

C2
T
[1]

 1

where  is wavelength, T is temperature and C1 and C2 are constants. C1 is known
as the first radiation constant and is equivalent to 2hc 2  3.7418  10 16 W  m 2 . C2 is
known as the second radiation constant and is equivalent to hc k  1.4388  10 2 m  K .
-4-
Figure 2 – Blackbody curves.
Figure 2 shows plots of blackbodies at various temperatures. In engine exhaust
plumes, radiation is absorbed, emitted, and scattered by the non-opaque gases that
comprise the plume.
The amount of radiation decrease di through an elemental
thickness dS is given to be:
di   K  S i dS
[2]
Where –K is the spectral extinction coefficient. The extinction coefficient can be
broken down into two parts, absorption coefficient a, and scattering coefficient .
K   a   
[3]
Because hot exhaust plumes are at a higher temperature than the surroundings, the
plume self-emissions will dominate the total radiant energy emissions. Therefore it is
valid to neglect scattering effects of the plume. Equation 3 can then be simplified:
K   a
[4]
Kirchoff’s Law states that the directional spectral absorptivity is equal to the
directional spectral emissivity.
   , ,  , T      , ,  , T 
[5]
In addition, Kirchoff’s Law also states that for non-opaque media
-5-
   , S   1     , S 
[6]
Where  is the spectral transmittance, or the fractional amount of energy, at a
certain wavelength, transmitted through the media.  is given by the following equation
which is known as Beer’s law [4]:
   e  a S
[7]
A simplifying assumption of this plume analysis is that the gases are well mixed and
as such, the radiant intensity at any point along a given path S is the same. Therefore we
can write:
i S   i 0 e  a S
[8]
The spectral radiance emitted from the plume gas is calculated by multiplying the
spectral radiant intensity of a blackbody i BB  , T  with the spectral emittance   , T  of
the semitransparent plume gas:
igas  , T   iBB  , T    gas  , T 
[9]
The total, or integrated, radiance within a spectral band spanning from 1 to 2 is
given as:
2
i gas   i gas    d
[ 10 ]
1
2.2 Exhaust Gas Radiance and Transmittance using MODTRAN
MODTRAN (MODerate resolution TRANsmission), developed by the US Air
Force to model atmospheric transmittance and radiance is used to calculate the spectral
contributions of gaseous elements of the turbine exhaust plume in this project.
MODTRAN contains profiles for 13 minor and trace atmospheric gases in addition to
separate band models for absorbing molecules H2O, CO2 and others [2]. Absorption is
calculated in 1 cm-1 spectral resolution, which is also the resolution used to present the
results obtained. Further description of the MODTRAN model is presented in [2].
-6-
MODTRAN inputs require knowledge of constituents of exhaust gas.
The
molecular species that affect the model are:
H20, CO2, O3, N2O, CO, CH4, O2, NO, SO2, NO2, NH3, and HNO3
Methodology for determining the concentration of these gases is presented in
section 2.4 of this project.
To perform calculations, properties of air mass flow, fuel flow, and exhaust gas
temperature were taken from the values presented in the experimental apparatus used by
Rosfjord [6].
2.3 The addition of soot model to MODTRAN results
Soot is modeled independently from MODTRAN. MODTRAN contains built-in
aerosol models used to calculate the effects of water and ash clouds [2]. The methods
used for water and ash are not consistent with methods proposed in Siegel and Howell
[1] for calculating soot effects. In MODTRAN, the aerosol models do not allow for the
appropriate manipulation to confidently predict sooting effects as they are described in
[1]. In addition, since the combustion process of kerosene results in 99.9% fuel burn, the
complexities of the MODTRAN aerosol models are not warranted.
Soot particle sizes in jet engine exhaust using Jet A, or kerosene fuel were found by
experiment to be between 0.18 and 0.22 m [6]. Using the particle size parameter
calculation   D  from [1], where D is the particle diameter, and  is wavelength,
in the infrared spectrum, the particle size parameter    0.3 , therefore Rayleigh
scattering technique can be employed. Gustav Mie [7] originally derived scattering
efficiency equations from electromagnetic theory, and are presented in [1]. From the
Mie equations, the spectral absorption efficiency factor Q is given as :
Q  24
D
n
 n 2   2  22  4n 2 2
[ 11 ]
Since particle diameters do not vary considerably, the volumetric concentration of
particles, C, can be given as:
-7-
C
ND 3
6
[ 12 ]
where N is the number of particles per unit volume and D is the average particle
diameter. Equation 11 then becomes (as shown in reference [1]):
a 36
n

C
 n 2   2  2 2  4 n 2 2


[ 13 ]
Values for n and  of propane and acetylene soot are given in [1]. In this project,
a 5

C 
[ 14 ]
is used, since published values of n and  for soot were not found. A comparison of
this alternate approach is shown in the graph below [1].
Figure 3 –Propane absorption coefficient factor a/C plotted against 5/.
Though comparison of the two curves shown in Figure 3 shows a greater than 45%
difference at the 10 micron wavelength point, the amount of soot participation in the
radiometric values in the infrared portion of the spectrum is small enough that the 5/
method of calculating the absorption coefficient does not lead to significant errors in IR
radiance.
-8-
2.4 Chemical Composition of Turbine Engine Exhaust
To calculate the radiant emissions of an exhaust plume, knowledge of the chemical
composition of the plume is required. In order to broaden the applicability of the
methods proposed in this project, the chemical reactions can be modeled and exhaust gas
composition calculated.
The engine exhaust is modeled by balancing the combustion reaction of the air-fuel
mixture introduced into the combustion system.
Air and fuel mass flow rates are
typically known for engines. For the purposes of calculation and generating results, the
air-mass flow rate and fuel mass flow rate are taken from [6]. The apparatus used in the
experiments of [6] utilized a air mass flow rate of 5.07lbs/sec and a fuel flow rate of
0.08lbs/sec. In addition, combustion efficiencies of 99.9% were always achieved, most
likely due to the high combustion temperatures and volatility of kerosene ( Jet A ) fuel.
A simplified one-formula fuel model for kerosene is utilized by Wang [5] and
Rosfjord [6] and shown to “match reasonably well” with molecular weight of kerosene,
the typical fuel used in turbine engine exhausts. Wang and Rosfjord use the C12H24
molecule to represent kerosene, and hence the complete combustion reaction then
becomes:
C12H24 + 18 O2  12 CO2 + 12 H2O
Using ideal gas law, and flow properties from [6], a table of exhaust gas
concentrations can be generated.
Gas Concentrations of dry atmosphere
Nitrogen
Oxygen
Argon
Carbon Dioxide
Neon
Helium
Methane
Krypton
Hydrogen
*(Calculated) Water
nitrous oxide
xenon
ozone
nitrigen dioxide
N2
O2
Ar
CO2
Ne
He
CH4
Kr
H2
H2O
NO
Xe
O3
NO2
ppmv
780840
207460
9340
381
18.18
5.24
1.745
1.14
0.55
1951.453
0.5
0.09
0.07
0.02
% vol
78.08%
20.75%
0.93%
0.04%
0.00%
0.00%
0.00%
0.00%
0.00%
0.20%
0.00%
0.00%
0.00%
0.00%
Atom Wt
g/mol
28.01
32.00
39.95
44.01
20.18
4.00
16.04
83.80
2.02
18.01
30.01
131.29
48.00
62.00
-9-
mass
density (rho)
g/m^3
1164.53
1330.15
1660.63
1829.44
838.88
166.39
666.89
3483.54
83.80
748.87
1247.34
5457.69
1995.22
2577.49
% mass
75.59%
22.94%
1.29%
0.06%
0.00%
0.00%
0.00%
0.00%
0.00%
0.12%
0.00%
0.00%
0.00%
0.00%
mass
mix ratio
gm/kg
755.933
229.405
12.894
0.579
0.013
0.001
0.001
0.003
0.000
1.215
0.001
0.000
0.000
0.000
iodine
carbon monoxide
ammonia
I
CO
NH3
0.01
0.001
0.001
0.00%
0.00%
0.00%
126.90
28.01
17.03
5275.20
1164.37
707.96
0.00%
0.00%
0.00%
0.000
0.000
0.000
Table 1 – Values for the composition of standard air – pre-combustion.
Exhaust Gas Concentrations
Nitrogen
Oxygen
Argon
Carbon Dioxide
Neon
Helium
Methane
Krypton
Hydrogen
*(Calculated) Water
nitrous oxide
xenon
ozone
nitrigen dioxide
iodine
carbon monoxide
ammonia
N2
O2
Ar
CO2
Ne
He
CH4
Kr
H2
H2O
NO
Xe
O3
NO2
I
CO
NH3
ppmv
767902
155635
9185
32618
17.88
5.15
142.38
1.12
0.54
34162
0.49
0.09
0.07
19.16
0.01
979
0.00
% vol
76.79%
15.56%
0.92%
3.26%
0.00%
0.00%
0.01%
0.00%
0.00%
3.42%
0.00%
0.00%
0.00%
0.00%
0.00%
0.10%
0.00%
Atom Wt
g/mol
28.01
32.00
39.95
44.01
20.18
4.00
16.04
83.80
2.02
18.01
30.01
131.29
48.00
62.00
126.90
28.01
17.03
mass
density (rho)
g/m^3
640.52
731.61
913.38
1006.23
461.40
91.52
366.80
1916.01
46.09
411.89
686.06
3001.83
1097.41
1417.67
2901.46
640.42
389.39
% mass
74.40%
17.22%
1.27%
4.96%
0.00%
0.00%
0.01%
0.00%
0.00%
2.13%
0.00%
0.00%
0.00%
0.00%
0.00%
0.09%
0.00%
mass mix
ratio
gm/kg
743.977
172.230
12.690
49.645
0.012
0.001
0.079
0.003
0.000
21.284
0.001
0.000
0.000
0.041
0.000
0.948
0.000
Table 2 – The composition of exhaust plume – 99% combustion efficiency.
2.5 Step-by-Step Process Breakdown
Step 1 – Define MODTRAN input card.
MODTRAN model parameters are defined in a text file and is read by the code
upon program execution in the computer operating system’s command prompt. An
example of the input card is given below.
A detailed description of the various
parameters is given in the MODTRAN user’s manual.
- 10 -
Model settings
T
7
1
F
2T
5
1
0
0
0
1
0
0
1
1
0Def. by Excel Tool
0
0
0
0
0
0
0
1
1
0.000
0.00
381.000
0
0.000
0.000
0.000
0.000
0.000
0.000 1.043E+03 5.500E+02 3.416E+04 3.262E+04 6.884E-02AAAAAAAAAAAAAA
0.000E+00 9.786E+02 1.424E+02 1.556E+05 0
.000E+00 0.000E+00 1.916E+01 9.834E-04
0.000E+00
0.000
0.000
0.001
500.000 10000.000
0.000
1.000
1.000
0.000
0.000
0
0.000
0
Constituent Concentrations
Wave # (band pass) start-stop
Figure 4 – Sample MODTRAN input card
Step 2 – Run MODTRAN and Read Results
MODTRAN model parameters are first defined in a text file which is then read
automatically. This requires that the input card file be named ‘tape5’ and reside in the
same directory as the executable file. Output is written to the ‘tape6 ‘ file. The output
file contains all of the arguments set in the tape5 file, calculated atmospheric parameters,
and most importantly, spectral radiance and transmissivity. MATLAB scripts were
written to read MODTRAN output and perform subsequent calculations.
Step 3 – Calculate Soot absorption
Calculations of soot absorption are performed utilizing MATLAB using equation
14. An array of absorption values for each wave-number corresponding to the same
spectrum defined in Step 1 is generated and stored in memory.
Step 4 – Calculate Soot Radiance
Soot radiance is calculated in MATLAB from the absorption coefficient using
equation 9.
Step 5 – Combine MODTRAN and Soot Model
- 11 -
With the MODTRAN values read, and soot values calculated in MATLAB, the total
plume spectral radiance can be found by first calculating the spectral transmissivity of
the plume:
 Plume     MODTRAN    Soot  
[ 15 ]
Then by calculating the plume spectral radiance:
iPlume    iBB    1   Plume  .
[ 16 ]
- 12 -
3. Results & Discussion
Figure 5 shows the results of the methods presented in this project plotted against a
signature plot published in [8]. The resolution of the published plot is low, and may
account for some of the error shown in figure 7. The method used in this project has a
finer spectral resolution than that of the published plot which is made evident by the
jagged spectral lines. A moving average is used to present a smoother curve.
Figure 5 – Spectral radiance combined MODTRAN output and calculated soot.
- 13 -
Figure 6 – 3 to 5m spectral results.
In addition to analyzing the error on a spectral line basis, we can also look at
differences in integrated radiance as a merit for accuracy. Integrating the radiance
throughout the entire spectrum analyzed (2 – 20m), the calculated radiance is
approximately 13% higher than the integrated radiance calculated from the published
plot. Narrowing the spectrum to the 3-5m spectrum, where many thermal imagers
operate, yielded larger errors in integrated signature. Calculated thermal radiance is
25% lower than the published plot value.
- 14 -
H2O
CO2
CO2
H2O
H2O
Figure 7 – Spectral absorption broken out by constituent.
A closer look at Figure 6 shows that the emission spike at approximately 4.2 –
4.4m varies between calculated and published.
Figure 7 shows a breakdown of
absorption lines for the constituents that account for the majority of the radiometric
effects. From Figure 7, it is evident that the 4.2 – 4.4m spike is attributed to CO2.
The CO2 spike of the published results has a dip at approximately 4.3m. In
addition, the spectral width of the CO2 spike is wider than that of the calculated radiance.
The dip at 4.3m is indicative of a small amount of atmospheric attenuation, possibly as
a result of the sensor being positioned a small distance away from the hot plume gases.
Therefore, it is difficult to conclude that this particular feature points to a problem in this
method.
The difference in the spectral width of the CO2 spike, however, points to a possible
flaw with the methods proposed in this project. Since MODTRAN uses databases based
off of atmospheric measurements, it is very likely that the databases do not span
temperatures as high as the temperature used for the plume calculations. It is possible
that at elevated temperatures the higher internal energy of CO2 molecules causes the
- 15 -
emission spectrum to get wider, and MODTRAN is not accounting for this. As a result,
caution is advised if attempting to use the method of this project to accurately predict the
spectral emissions from plume.
The procedures presented in this project do include the effect of soot on the engine
exhaust, however the radiance levels are low enough to be neglected. This is because
turbine engines burning kerosene produce combustion efficiencies of >99%, and as such
result in very low volumetric concentrations of soot particles. The calculations were
included, and therefore, if conditions of a non-typical engine are such that soot particle
concentrations increase, participation of soot using this procedure can be analyzed.
- 16 -
4. Conclusions
The purpose of this study project was to investigate the effects of soot on the IR
emissions of a typical turbine engine exhaust.
After generating a soot model and
comparing values of soot emissions compared to the emissions of other molecular
contributors, it became apparent that soot concentrations are typically too low for them
to significantly impact IR emissions. Therefore the focus of the project changed to a
determination of the applicability of using MODTRAN to accurately predict plume
radiance.
The published measured values used for comparison with calculated values were
presented by Mahulikar [8] as a plot. This plot was scanned and converted to numerical
values where inaccuracies were undoubtedly introduced. As such, when the comparison
of the low resolution plot is made along the wide spectral band of 2 to 20m, the
calculated result was only 13% higher than the value derived from the published plot.
This result, in the author’s opinion, is a reasonably good correlation considering the low
resolution of the plot.
It must be stated that MODTRAN was apparently never intended to be used as it is
proposed in this project, therefore, expectations should be set accordingly. MODTRAN
was calibrated to perform detailed and accurate analysis of the earth’s atmosphere at
temperatures which are well below those seen in exhaust plumes. As such, it appears as
though the spectral broadening of CO2 emission in the 4.3m portion of the spectrum is
not accurately predicted. In the 3-5m spectrum, where many imaging sensors operate,
a difference of 25% lower than the measured plot was encountered. Therefore, caution
must be used when using this method for predicting plume spectral emissions.
Furthermore, it is recommended that a more detailed study be pursued where more
detailed measurements of plume are compared against a more robust plume model.
- 17 -
References
1. Siegel, Robert and Howel, John: “Thermal Radiation Heat Transfer,” 4th ed., Taylor
& Francis, 2002.
2. Kneisys, F.X., Robertson, D.C., Abreu, L.W. and others: The MODTRAN 2/3 and
LOWTRAN 7 Model, Phillips Laboratory, Geophysics Directorate.
3. Lester C. Lichty, “Combustion Engine Processes,” 1967, McGraw-Hill, Inc., Lib.of
Congress 67-10876
4. Beer, J. M. and Howarth, C. R.: “Radiation From Flames in Furnaces”, Department
of Fuel Technology and Chemical Engineering, University of Sheffield, Sheffield,
England, 1971
5. Ten-See Wang: “Thermophysics Characterization of Kerosene Combustion”, NASA
Marshall Space Flight Center, 2000, American Institute of Aeronautics and
Astronautics Inc.
6. Rosfjord, T.J.: “Aviation-Fuel Effects on Combustion”, NASA Contractor Report
16834, United Technologies Research Center, 1984.
7. Mie, Gustav: “Optics of Turbid Media”, Ann. Phys., col. 25, no. 3, pp 377-445, 1908
8. Mahulikar, Shripad P., et al. “Infrared signature studies of aerospace vehicles”,
Progress in Aerospace Sciences, Volume 43, Issues 7-8, October-November 2007,
Pages 218-245, ISSN 0376-0421, DOI: 10.1016/j.paerosci.2007.06.002.
- 18 -
Appendix A
MATLAB scripts and functions
clear all;
close all;
WaveN1 = 500; %wavenumber cm-1
WaveN2 = 10000; %wavenumber cm-1
WaveN = WaveN1:WaveN2;
%generate wavenumber array... units of cm-1
lambda = 1 ./ WaveN .* 10000; %generate wavelength matrix... units of micron
T = 550; %degrees Kelvin
BBrad = planck(T, lambda); %the blackbody curve in W /(m^2 sr micron)
%-------------------------------------------------------------------------D = 2.1e-7; %particle diameter in m^2 from Rosfjord experiment
%N = 4.24e6; %Particle density cm^-3 from Rosfjord experiment
N = 4.24e6;
N = N * 1e6; %Convert to m^-3
C = N * pi * (D^3) / 6; %soot volume concentration... volume of particles per unit
volume
Xi = pi * D ./ (lambda .* 1e-6); %size parameter length units must agree
Le = 1; %mean beam length for the volume.
a = absorpCoeff(lambda, C, 'Simple');
abyC = absorpCoeff(lambda, 1, 'Simple');
e = 1 - exp(-a .* Le); %EQ 12-89 Siegel & Howell
soot.absorption = e;
soot.tau = 1 - e;
soot.radiance = BBrad .* soot.absorption;
%Read values from MODTRAN
[MODTRAN.WaveN MODTRAN.tau MODTRAN.radiance] = ReadMODOUT4;
MODTRAN.WaveN = MODTRAN.WaveN'; %transpose vector
MODTRAN.tau = MODTRAN.tau';
%transpose vector
MODTRAN.radiance = MODTRAN.radiance'; %read from MODTRAN in W / cm^2 sr cm
MODTRAN.radiance = 10000 .* (WaveN ./ lambda) .* MODTRAN.radiance; %convert to W / m^2
sr micron
MODTRAN.absorption = 1 - MODTRAN.tau;
%Calculate plume values
plume.tau = MODTRAN.tau .* soot.tau;
plume.absorption = 1 - plume.tau;
plume.radiance = BBrad .* plume.absorption;
%Generate plots
smoothSize = 10;
figure;
plot(lambda, filter(ones(1,smoothSize)/smoothSize,1,plume.radiance), ...
'Color', [1 0.5 0], 'LineStyle', '-', 'DisplayName','Plume Radiance');
xlim([1 20]);
xlabel('Wavelength (micron)', 'FontSize', 12, 'FontWeight', 'Bold');
ylabel('W / m^2 sr um', 'FontSize', 12, 'FontWeight', 'Bold');
hold on;
%plot(lambda, soot.radiance, 'Color', [0.5, 0.5, 0.5], 'LineStyle', '--', ...
%
'DisplayName', 'Soot contribution');
%plot(lambda, MODTRAN.radiance, 'Color', 'blue', 'LineStyle', '--', ...
%
'DisplayName','CO2, H2O contribution');
plot(lambda, BBrad, 'Color', 'red', 'LineStyle', ':', 'DisplayName', 'BB Curve');
plume.TotalRadiance = IntegrateModtranData(lambda', plume.radiance'); %(W / m^2 - sr)
%plume.TotalRadiance = plume.TotalRadiance / 1000; %convert to kW / m^2 - sr
soot.TotalRadiance = IntegrateModtranData(lambda', soot.radiance'); %(W / m^2 - sr)
MODTRAN.TotalRadiance = IntegrateModtranData(lambda', MODTRAN.radiance'); %(W / m^2 sr)
- 19 -
text(8.5,174,['T = ' num2str(T) '°K'], 'FontSize', 12);
text(8.5,165,['C = ' num2str(C)],'FontSize', 12);
text(8.5,156,['plume radiance = ' num2str(plume.TotalRadiance)], 'FontSize', 12);
%Create 2 columns of data to export to excel
excelout(:,1) = WaveN';
excelout(:,2) = plume.radiance';
excelout(:,3) = soot.radiance';
Published with MATLAB® 7.5
function [ a ] = absorpCoeff( Lambda, C, FuelType )
%ABSORPCOEFF returns absorption coefficient
%
%========================================
Uses method described by Siegel & Howell
%========================================
%Optical constants of soots
%wavelegths in micron
%ARGUMENTS:
%Lambda = array of wavelengths in micron
%C = soot concentration
%FuelType = Acetylene, Propane, Simple
%
Values of Acetulene and Propane taken from Siegel and Howell
%
page536. Simple method describen in Siegel & Howell page 538.
L1 = [0.4358 0.45 0.55 0.65 0.8065 2.5 3.0 4.0 5.0 6.0 7.0 8.5 10.0 ];
if nargin < 2 || nargin > 3
disp('Invalid number of arguments');
- 20 -
return
elseif strcmp(FuelType, 'Acetylene')
n = [1.57 1.56 1.57 1.56 1.57 2.04 2.21 2.38 2.07 2.62 3.05 3.26 3.48];
%Extinction coefficient
kappa =[0.46 0.50 0.53 .52 .49 1.15 1.23 1.44 1.72 1.67 1.91 2.10 2.46];
elseif strcmp(FuelType, 'Simple')
a = 5./Lambda*C; %See page 538 Siegel & Howell.
return
else
%Index of refraction of Propane
n = [1.57 1.56 1.57 1.56 1.57 2.04 2.21 2.38 2.07 2.62 3.05 3.26 3.48];
%Extinction coefficient
kappa =[0.46 0.50 0.53 .52 .49 1.15 1.23 1.44 1.72 1.67 1.91 2.10 2.46];
end
%
%----------------------------%absorption coeff. calculations
%alpha = 0.8; %see table page 534
%a = C*k .* lambda^(-alpha) ; %absorption coefficient method 1 (EQ 12-90)
%a = C * k ./ lambda^0.95;
%absorption coeff. Hottel method.
%- - - - - - - - - %electromagnetic theory
F = n .* kappa ./ ((n.^2 - kappa.^2 + 2).^2 + (4 .* n.^2 .* kappa.^2));
abyC = 36 * pi ./ L1 .* F;
a1 = abyC .* C;
a = interp1(L1, a1, Lambda, 'cubic', 'extrap');
end
Published with MATLAB® 7.5
function [ WaveN, Trans, Radnc ] = ReadMODOUT4( file )
%READMODOUT4 Summary of this function goes here
%
Detailed explanation goes here
if nargin == 0
file = 'C:\PCModWin\Usr\MDOUTex2.txt';
end
fid = fopen(file, 'r');
while 1 %find line containing headers
tline = fgetl(fid);
if strncmp(tline, ' FREQ
TOT TRANS', 18) || ~ischar(tline);
break;
end
end
%read values from file
Data = textscan(fid, '%n %n %n %n %n %n %n %n %n %n %n %n %n %n %n %n');
fclose(fid);
%----------------------------------%convert from cell array to array of values
WaveN = Data{1,1};
Trans = Data{1,2};
%Radnc = Data{1,10};
Radnc = Data{1,3};
%----------------------------------%strip out last data point from list
Npts = size(WaveN,1) - 1;
WaveN = WaveN(1:Npts);
Trans = Trans(1:Npts);
Radnc = Radnc(1:Npts);
%-----------------------------------end
Published with MATLAB® 7.5
- 21 -
function [ R ] = planck( T, lambda )
% Spectral radiance of a blackbody in W /(m^2 sr micron).
%
% SYNTAX:
%
I = planck(T, lambda)
% ARGUMENTS:
%
T = Temperature in Kelvin
%
lambda = Wavelength in micron
if nargin ~= 2
error('Invalid number of arguments');
end
C1 = 59552140 * 2; %radiation constant in W micron^4 / (m^2 sr)
C2 = 14387.752;
%radiation constant in micron * K
R = C1 ./ (lambda.^5 .* (exp(C2 ./ lambda ./ T) - 1));
end
Published with MATLAB® 7.5
function [ E ] = intplanck( T, lambda1, lambda2, Tol )
%Integration of Planck's function between lambda1 and lambda2
%
% SYNTAX:
%
E = intplanck(T, lambda)
% ARGUMENTS:
%
E = band radiance in W /(m^2 sr)
%
T = Temperature in Kelvin
%
lambda1 = lower wavelength limit in micron
%
lambda2 = upper wavelength limit in micron
%
Tol = (optional) Tolerance of integration
if nargin == 3
Tol = 1e-6;
elseif nargin < 3
error('Too few input arguments');
elseif nargin > 4
error('Too many input arguments');
end %if
%create a function handle 'F' to pass the integrand to quad()
F = @(x)planck(T, x);
%evaluate the function
%E = quad(F, lambda1, lambda2, Tol);
E = quadl(F, lambda1, lambda2, Tol);
end %function
Published with MATLAB® 7.5
function [ T ] = plancktemp( Radiance, L1, L2, Tol )
%Find temperature in Kelvin using Newton-Raphson Method to solve intplanck
%
% SYNTAX:
%
T = plancktemp(R, L1, L2)
% ARGUMENTS:
%
T = temperature in K
- 22 -
%
%
%
Radiance = radiance in W / (m^2 sr)
L1 = lambda1, lower wavelength limit in micron
L2 = lambda2, upper wavelength limit in micron
%check input arguments
if nargin == 3
Tol = 1e-2;
elseif nargin < 3
error('Too few input arguments');
elseif nargin > 4
error('Too many input arguments');
end %if
InitialGuess = 100; %Kelvin
MaxTemp = 1000000; %This is to prevent divergence
T1 = InitialGuess-10;
T2 = InitialGuess;
while abs(T2 - T1) > Tol
%------------------------------------------%To prevent divergence,
%
set T2 to MaxTemp if solution gets big
if T2 > MaxTemp, T2 = MaxTemp; end
%------------------------------------------T1 = T2;
E = intplanck(T1, L1, L2) - Radiance;
slope = dintplanck(T1, L1, L2);
T2 = T1 - E / slope;
end
T = T2;
end
function D = dintplanck(Temp, L1, L2)
%Returns the f'(T)... The slope of the planck function
%Required for the Newton-Raphson method
T1 = Temp - .0001;
T2 = Temp + .0001;
E1 = intplanck(T1, L1, L2);
E2 = intplanck(T2, L1, L2);
D = (E2 - E1) / (T2 - T1);
end
Published with MATLAB® 7.5
function [ val ] = IntegrateModtranData( Wave_x, Radiance )
%INTEGRATEMODTRANDATA Integrates data read from MODOUT.txt
%
Integrate numerically using trapezoidal approximation technique.
end
Npts = size(Wave_x,1);
val = 0;
for ci = 1:Npts-1
val = val + ((Wave_x(ci)-Wave_x(ci+1)) * ((Radiance(ci)+Radiance(ci+1))/2));
end
Published with MATLAB® 7.5
- 23 -
Appendix B
Sample MODTRAN output file
1
*****
CARD 1
*****T
CARD 1A *****F
MODTRAN3.7
7
1
1
0
2T
5 381.00000
Version 1.1
0
0
0
*****
1
CARD 2C *****
1
0
1
MODEL ATMOSPHERE NO.
7
0
*****
0
0
0
1
1
0.000
0.00
0.000
MOLECULAR BAND MODEL PARAMETERS FILE:
CARD 2
MAR98
0
0
DATA/BMP97_01.BIN
0
0
0.00000
0.00000
0.00000
0.00000
0.00000
0Def. by Excel Tool
ICLD =
0
MODEL 0 / 7 USER INPUT DATA
0.00000 1.013E+03 5.330E+02 3.416E+04 3.262E+04 6.884E-02
AAAAAAAAAAAAAA
0.000E+00 9.786E+02 1.424E+02 1.556E+05 0.000E+00 0.000E+00 1.916E+01 9.834E-04
0.000E+00
1
Z
(KM)
0.000
P
(MB)
1013.000
T
(K)
533.00
REL H
H2O
CLD AMT
RAIN RATE
(%) (GM / M3) (GM / M3) (MM / HR) TYPE
0.06 1.407E+01 0.000E+00 0.000E+00 RURAL
CARD 3
*****
0.00000
0.00000
0.00000
0.00100
CARD 4
*****
500
10000
1
1
0.00000
AEROSOL
PROFILE
RURAL
0.00000
0
0.00000
PROGRAM WILL COMPUTE RADIANCE
ATMOSPHERIC MODEL
TEMPERATURE =
7
WATER VAPOR =
7
OZONE
=
7
M4 =
HORIZONTAL PATH
ALTITUDE =
RANGE
=
FREQUENCY RANGE
IV1 =
IV2 =
IDV =
IFWHM =
Def. by Excel Tool
Def. by Excel Tool
Def. by Excel Tool
0 M5 =
0 M6 =
0 MDEF =
0
0.00000 KM
0.00100 KM
500
10000
1
1
CM-1
CM-1
CM-1
CM-1
HORIZONTAL PATH AT ALTITUDE =
(
(
20.00 MICRONS)
1.00 MICRONS)
0.000 KM WITH RANGE =
0.001 KM, MODEL =
7
TOTAL COLUMN ABSORBER AMOUNTS FOR THE LINE-OF-SIGHT PATH:
HNO3
O3 UV
(ATM CM)
(ATM CM)
0.0000E+00 3.5246E-06
AER 1
1
0
0.000000
H2O
(
1.7502E+00
3.5246E-06
O2
(
7.9667E+00
NH3
5.0350E-08
CFC-11
(
0.0000E+00
0.0000E+00
CFC-115
(
0.0000E+00
0.0000E+00
FREQ
WAVLEN
SURF
(CM-1) (MICRN) ALBEDO
500.
AER 2
0.000158
20.000
0.000
O3
CFC-12
CLONO2
CNTMSLF1
CNTMSLF2
CNTMFRN
N2 CONT
(MOL CM-2) (MOL CM-2) (MOL CM-2)
8.9188E+17 -5.8716E+18 2.5217E+19 2.4075E-04
AER 3
0.000000
AER 4
CIRRUS
WAT DROP
(KM GM/M3)
0.000000
0.000000
0.000000
CO
ATM CM
5.0104E-02
CH4
N2O
1.6701E+00
7.2908E-03
0.0000E+00
NO
ATM CM
0.0000E+00
9.8099E-04
SO2
)
0.0000E+00
CFC-14
ATM CM
0.0000E+00
0.0000E+00
CO2
CFC-13
0.0000E+00
NO2
MOL SCAT
5.1235E-04
ICE PART
(KM GM/M3)
0.000000
)
CFC-22
CFC-113
0.0000E+00
CFC-114
)
0.0000E+00
CHCL2F
CCL4
N2O5
ATM CM
)
0.0000E+00 0.0000E+00 0.0000E+00
RADIANCE(WATTS/CM2-STER-XXX)
PATH THERMAL
SCAT PART
SURFACE EMISSION
SURFACE REFLECTED
(CM-1)
(MICRN)
(CM-1)
(CM-1)
(MICRN)
(CM-1)
(MICRN)
3.17E-07
MEAN AER RH
(PRCNT)
0.06
HNO4
0.0000E+00
7.92E-06
0.00E+00
0.00E+00
0.00E+00
- 24 -
0.00E+00
0.00E+00
TOTAL RADIANCE
(CM-1)
(MICRN)
3.17E-07
7.92E-06
INTEGRAL
TOTAL
TRANS
1.58E-07
0.99392
501.
502.
503.
504.
505.
506.
507.
508.
509.
510.
511.
512.
513.
514.
515.
516.
517.
518.
519.
520.
521.
522.
523.
524.
525.
526.
527.
528.
529.
530.
531.
532.
533.
534.
535.
536.
537.
538.
539.
540.
541.
542.
543.
544.
545.
546.
547.
548.
549.
19.960
19.920
19.881
19.841
19.802
19.763
19.724
19.685
19.646
19.608
19.569
19.531
19.493
19.455
19.417
19.380
19.342
19.305
19.268
19.231
19.194
19.157
19.120
19.084
19.048
19.011
18.975
18.939
18.904
18.868
18.832
18.797
18.762
18.727
18.692
18.657
18.622
18.587
18.553
18.519
18.484
18.450
18.416
18.382
18.349
18.315
18.282
18.248
18.215
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
8.19E-07
1.62E-05
2.18E-06
2.15E-06
4.58E-07
8.78E-07
1.10E-05
9.19E-07
2.28E-07
2.00E-06
2.29E-06
3.79E-07
1.44E-07
1.22E-06
1.51E-06
1.47E-06
1.22E-05
8.29E-06
1.96E-06
1.10E-05
4.02E-07
1.80E-07
2.07E-07
3.18E-07
8.52E-07
1.47E-05
7.45E-07
1.84E-07
1.90E-07
9.83E-08
5.98E-08
7.30E-08
1.40E-07
6.29E-07
1.51E-07
2.99E-06
5.17E-07
1.60E-07
6.45E-08
4.28E-07
1.15E-06
9.97E-08
8.97E-08
3.31E-07
3.80E-06
5.12E-06
7.70E-07
3.57E-06
2.19E-07
2.05E-05
4.09E-04
5.52E-05
5.45E-05
1.17E-05
2.25E-05
2.83E-04
2.37E-05
5.92E-06
5.19E-05
5.97E-05
9.93E-06
3.79E-06
3.23E-05
4.02E-05
3.92E-05
3.26E-04
2.22E-04
5.29E-05
2.96E-04
1.09E-05
4.90E-06
5.66E-06
8.73E-06
2.35E-05
4.07E-04
2.07E-05
5.14E-06
5.31E-06
2.76E-06
1.69E-06
2.07E-06
3.96E-06
1.80E-05
4.32E-06
8.59E-05
1.49E-05
4.62E-06
1.87E-06
1.25E-05
3.36E-05
2.93E-06
2.65E-06
9.78E-06
1.13E-04
1.53E-04
2.30E-05
1.07E-04
6.61E-06
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
INTEGRATED ABSORPTION FROM
500 TO 10000 CM-1 =
AVERAGE TRANSMITTANCE = 0.9436
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
535.6866 CM-1
INTEGRATED TOTAL RADIANCE = 1.740401E-02 WATTS CM-2 STER-1 (FROM
MINIMUM SPECTRAL RADIANCE = 1.621898E-16 WATTS CM-2 STER-1 / CM-1
MAXIMUM SPECTRAL RADIANCE = 7.011691E-05 WATTS CM-2 STER-1 / CM-1
BOUNDARY TEMPERATURE =
0.000
BOUNDARY EMISSIVITY =
1.000
CARD 5 *****
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
500 TO 10000 CM-1 )
AT 10000 CM-1
AT
669 CM-1
0
- 25 -
8.19E-07
1.62E-05
2.18E-06
2.15E-06
4.58E-07
8.78E-07
1.10E-05
9.19E-07
2.28E-07
2.00E-06
2.29E-06
3.79E-07
1.44E-07
1.22E-06
1.51E-06
1.47E-06
1.22E-05
8.29E-06
1.96E-06
1.10E-05
4.02E-07
1.80E-07
2.07E-07
3.18E-07
8.52E-07
1.47E-05
7.45E-07
1.84E-07
1.90E-07
9.83E-08
5.98E-08
7.30E-08
1.40E-07
6.29E-07
1.51E-07
2.99E-06
5.17E-07
1.60E-07
6.45E-08
4.28E-07
1.15E-06
9.97E-08
8.97E-08
3.31E-07
3.80E-06
5.12E-06
7.70E-07
3.57E-06
2.19E-07
2.05E-05
4.09E-04
5.52E-05
5.45E-05
1.17E-05
2.25E-05
2.83E-04
2.37E-05
5.92E-06
5.19E-05
5.97E-05
9.93E-06
3.79E-06
3.23E-05
4.02E-05
3.92E-05
3.26E-04
2.22E-04
5.29E-05
2.96E-04
1.09E-05
4.90E-06
5.66E-06
8.73E-06
2.35E-05
4.07E-04
2.07E-05
5.14E-06
5.31E-06
2.76E-06
1.69E-06
2.07E-06
3.96E-06
1.80E-05
4.32E-06
8.59E-05
1.49E-05
4.62E-06
1.87E-06
1.25E-05
3.36E-05
2.93E-06
2.65E-06
9.78E-06
1.13E-04
1.53E-04
2.30E-05
1.07E-04
6.61E-06
9.77E-07
1.72E-05
1.94E-05
2.15E-05
2.20E-05
2.29E-05
3.39E-05
3.48E-05
3.50E-05
3.70E-05
3.93E-05
3.97E-05
3.98E-05
4.11E-05
4.26E-05
4.40E-05
5.62E-05
6.45E-05
6.65E-05
7.74E-05
7.78E-05
7.80E-05
7.82E-05
7.86E-05
7.94E-05
9.41E-05
9.48E-05
9.50E-05
9.52E-05
9.53E-05
9.54E-05
9.55E-05
9.56E-05
9.62E-05
9.64E-05
9.94E-05
9.99E-05
1.00E-04
1.00E-04
1.01E-04
1.02E-04
1.02E-04
1.02E-04
1.02E-04
1.06E-04
1.11E-04
1.12E-04
1.15E-04
1.16E-04
0.98433
0.69008
0.95841
0.95921
0.99132
0.98340
0.79199
0.98269
0.99571
0.96259
0.95723
0.99293
0.99732
0.97730
0.97193
0.97279
0.77514
0.84738
0.96392
0.79899
0.99265
0.99672
0.99623
0.99422
0.98455
0.73408
0.98654
0.99668
0.99659
0.99824
0.99893
0.99870
0.99751
0.98880
0.99732
0.94704
0.99086
0.99718
0.99886
0.99247
0.97989
0.99826
0.99843
0.99424
0.93384
0.91107
0.98665
0.93827
0.99622
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