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On the determination of spectral emissivity in an optically thick particle cloud
J. Quonf. Spec~osc. Rodinr. Transfir. Vol. 8, pp. 631439.
Pergamon Press, 1968. Printed in Great Britain
ON THE DETERMINATION OF SPECTRAL EMISSIVITY IN AN OPTICALLY THICK PARTICLE CLOUD* J. M.
ADAMS?
Aerojet-General Sacramento,
Corporation, California
(Received 27 May 1967) Abstract-The extinction parameter, X/v, and ratio of optical absorption-total cross section, r,,/$j,, are presented for molten alumina and beryllia particle clouds. These parameters, which are determined for unimodal and bimodal polydispersions, are then related to the spectral emissivity of the cloud. Extrema in the emissivity of an optically thick cloud of beryllia particles are indicated, and are related to pyrometrically determined temperatures. It is shown that a high-intensity laser beam can be used in an extinction measurement to determine the particle cloud emissivity within rather close limits, even if no detectable penetration of the cloud by the light beam occurs.
INTRODUCTION IN ORDER to determine temperature of the condensed phase in a spectrally absorbingemitting two-phase medium (e.g., a cloud of particles in a gas), it is necessary to determine the emissivity and radiance associated with the continuum emission of the condensed-phase cloud ; the detailed method for doing this has been the subject of many experimental and analytical investigations”-3’ most of which have presented results only for an optically thin cloud. The emissivity of an optically thin cloud of particles is usually determined from a measurement of the extinction of a beam of light from a reference light source (e.g., a tungsten ribbon lamp of known brightness temperature). This extinction occurs due to both absorption and scattering of the light beam by the particles, but each contribution can be isolated through the use of the Mie scattering theoryf with a knowledge of the particle size distribution and complex refractive index of the condensed phase relative to the surrounding gas. From this treatment, the absorptivity of the particle cloud can readily be determined in the optically thin case. For a system in thermal equilibrium, the emissivity of the cloud is equated to the absorptivity from Kirchhoff’s Law. The purpose of this paper is to present a method for establishing limits on the value of particle cloud emissivity when the cloud is optically thick, and one is therefore prevented from making an extinction measurement under ordinary conditions. To this end, the use of * The author is pleased to acknowledge the assistance of Donald Carson and Charlotte Bartky of Aeronutronics for their contributions to this paper. Research reported in this paper was sponsored by the Air Force Rocket Propulsion Laboratory, Edwards AFB under Contract No. AF 04(611)-10545. Clearance ref: RA/SA AFSC 26 Ott 1966. y Engineering Specialist, Aerophysics. $ The details of the Mie scattering theory as it applies to a cloud of spherical particles is discussed by VAN DE HULST.‘~
631
J. M.
632
ADAMS
optical properties in the determination of spectral emissivity is shown in the analysis, together with an example of such a calculation for the conditions within a small rocket motor chamber. In order to limit the scope of the presentation, gradients of temperature, density, and particle concentration are assumed not to exist in the region under study. This assumption is not necessary to the solution of the problem, but it does eliminate unwarranted complexities in view of other uncertainties. For a method of handling these gradients, the reader is referred to work by NICHOLLS et dt4) Figure 1 presents the normal and hemispherical emissivity of a particle cloud as a function of optical depth, with the ratio y,/jJ as a parameter. The optical depth, 6, is given by 6 = R% + r,1t,
(11
where fl is the number density of particles in the cloud, cm- 3 ; t is the actual thickness of the cloud, cm ; and T,, 7, are the absorption and scattering cross sections for the polydispersion of particles in the cloud, cm2. The curves in Fig. 1, reproduced by permission of Charlotte Bartky, are the result of work by BARTKY@) and by ROMANOVA on special solutions to the radiative transfer equation.@’ 3D= 3 dimenrionol ID =
I dimensional
i = Isotropic
a 8 Anisotropic N = Normol H = Hemispherical
FIG. 1. Emissivity of a particle cloud determined from the radiative transfer equation. (Reproduced by permission of C. D. Etartky of Aeronutronic Corp.)
An optically thin cloud of scatterers and absorbers corresponds to that region to the left of Fig. 1 where the lines are straight ; values of 6 corresponding to the knee of the curves are associated with optically thick clouds. Finally, when the emissivity reaches its maximum value, a condition of infinite optical depth is said to exist, which corresponds to the region to the right of the knee in Fig. 1.
On the determination
of spectral
emissivity
SOURCE CONE
REFERENCELIGHTSOURCE
in an optically ACCEPTOR CONE
thick particle
633
cloud
ACCEPTOR ASSEMBLY
CARBONARC f ELECTRODE
STOP FIG.2. Optical
GAS PARTICLECLOUD
arrangement
for extinction
measurement.
A schematic of apparatus which is typical of that used in an extinction measurement is shown in Fig. 2.(I) The radiance and emissivity of the particle cloud are determined for the region defined by the acceptor optics ; this region takes the form of two colinear thin cones having a common apex at the centerline of the cloud. It is the scattering and absorption of the light source beam and the emission of the gas and particles within this region which are considered in the determination of temperature. The depth and particle concentration along the cone centerline defines the optical depth, 6. If temperature and concentration gradients through the plume are to be treated, a set of chordal measurements are taken across the cloud from the edge to the center. (4)The acceptor cone is defined by the optical system to be as small as possible to minimize the forward-scattered and multiply-scattered light which can be received and detected. ANALYSIS
AND
RESULTS
FOR
CLOUDS
OF
ARBITRARY
OPTICAL
THICKNESS
The following relations indicate how the variables affecting cloud emissivity are determined. From an interdependence of these variables for a given cloud geometry, it is shown that the limits on cloud emissivity can be determined from a single extinction measurement. An example is then presented of the use of this analysis to determine the spectral emissivity limits for the chamber region of a small rocket motor which is optically thick. The optical depth of a polydispersed particle cloud is given by 6 = N(Y, + r,)r.
(2)
The number density of particles or droplets in the cloud can be calculated from the relation (3) where L,, the velocity lag = up/us,* m
V, the mean particle volume =
s
f,U$$iR
(4)
0
* For this treatment, L, is assumed to be unity. Where velocity form part of the integrand of equation (4).
lag varies widely with particle
size, L,(D) will
634
J.M.
ADAMS
and&(D), the normalized frequency function describing the particle size distribution satis fies the relation 00 &(D) dD = 1.
I
(5)
0
is the mass fraction of particles (see footnote, bottom of p. 637) and pL, ps are the condensed-phase density and gas density respectively. The optical depth can now be written in the form
x
-”
/
--
physical properties of system
properties of particle size and refractive index
actual thickness of plume
which can be written as (7) where Y, = Y,+?,.
(8)
The second factor on the right side of equation (7), called the extinction parameter, can be determined for a polydispersion by the relation io -
Yr ==
F Y,(D>m)f,(D) dD
J
0
v
m %
s
(9)
D3fN(D) dD
0
where m is the complex refractive index. The ratio of absorption-total cross section, jj,/yt, can be similarly represented in terms of integrals and a cross-plot of these two parameters can be generated, as shown in Fig. 3. This figure was generated by 7094 machine computation for molten alumina particles, near the melting temperature (2320”K), by varying the most probable diameter of AW3,+ a typical unimodal distribution of particle sizes, shown in Fig. 4.* Also shown in the latter figure are two bimodal distributions, such as might result where large particles are burning * The “typical unimodal distribution” (T.U.D.) has been generated which closely approximates measured for a distributions from particle samples taken in rocket exhaust plumes. t9,*‘) From these reported distributions reprewide range of number mean particle diameter, it can be shown that a generalized or “typical” distribution sents the data.” ‘) Therefore, the abscissa in Fig. 4 has been non-dimensionalized by dividing by the most probable diameter for a typical unimodal distribution, D,, (T.U.D.). For each value of D,, (T.U.D.) chosen, a newdistribution results and new values of 7,/V and V./F, are determined. Figure 3 and subsequent cross-plots of $,/V vs. V./ut were generated in this manner.
On the determination
NY ““’
I 2
57103
FIG. 3. Absorption-total
of spectral
/
I
emissivity
,/1,,,
4
68lO'
in an optically
1
I
2
thick particle
,I
,,,,,
4
6
/
8 105
cross section vs. extinction parameter for spherical alumina (particle size distribution from Fig. 4).
,
2
particles
DISTRIBUTION (T.U.D.1
i’
MODERATELY BIMODALDISTRIBUTION
I
\
/
I I II
I \
\_ J
1.0
2.0
3.0
4.0
5.0
D/Dmar (T.U.D.) FIG. 4. Particle
size distributions
used in generating
63.5
cloud
curves in Figs. 3, 5 and 6.
of molten
636
J. M. ADAMS
to form very small “smoke” particles surrounding these large particles. Since the results must be applicable in the spectral region near the sodium doublet, the intensity of which serves as an indication of gas temperature, a wavelength of 0.589~ was used in the calculations. The effect of the type of distribution on the relationship between rt/V and 7,/y, is shown in Fig. 5 for molten beryllium oxide, BeO(,,. It can be seen from this figure that there
Moderately
blmodal distribution
m=1.730-0.02311
x=o?%9/L
/ 2 x/V,
FIG. 5. Absorption-total
I
I
4
I
I I , I I
I
6
2
8 105
,
cm-’
cross section vs. extinction parameter for spherical beryllia (particle size distributions from Fig. 4).
particles
of molten
is little difference in the resulting curves for the three types of distributions studied, indicating that an error in the assumed distribution will not result in a significant error in the relationship between jj,/jj, and Tt/V. The refractive index for beryllia was determined from measurements of KENDALL’l’) and CARLSON. (13) The dependence of the absorption constant (the imaginary part of the refractive index) on temperature was calculated by equating the temperature coefficient to that already determined for alumina, including the region above the melting point. Although the basis for this extrapolation is tenuous, the calculated values of cloud emissivity which result from these extrapolated values for BeOCo appear to be of reasonable magnitude when compared to measurements taken in the exhaust plumes of small rocket motors.(‘4) Having the relationship expressed by Figs. 3 and 5, the emissivity of a particle cloud can be determined from Fig. 1, i.e. Ep =
4YalYt,4.
(10)
Hence, from equations (7) and (9) ;
(11)
On the determination of spectral emissivity in an optically thick particle cloud
637
where < is determined from the physical properties of the system.* The resulting relationship for the emissivity of a typical particle cloud is shown in Fig. 6.
0,4-
o.3,0'
I 2
I
I 4
I Illll 6 6 10' -7/V,
I 2
I
1111111 4 6 8 105
cm-'
FIG. 6.
Particle cloud emissivity vs. extinction parameter for spherical particles of molten beryllia. (Determined for conditions in chamber of small motor, see text. Laser beam penetration levels are indicated.) 1. Upper level of vi/r for which there will be detectable penetration of cloud by 200 MW laser beam. 2. Example of two possible values for cloud emissivity when laser beam penetrates cloud and 7,/V is determined.
This figure was determined for conditions in the chamber of a small rocket motor, such as that which was actually used in experimental measurements of combustion temperature.(1*14’This was a research motor containing one pound of solid propellant. The average chamber pressure was about 650 psia. For the chamber conditions, 5 = 9 x 10W4. The particle sizes indicated in Fig. 6 refer to the number mean particle diameters corresponding to the values of 7,/V and jj,,/jjr,and resulting from an integral of the form ._
B = [ D&(D) dD.
(12)
The following features of Fig. 6 are of interest : 1. Under certain conditions, a minimum value for cloud emissivity is sign$cant enough to make measurement unnecessary. In other words, even if no reference light source is used to measure the extinction parameter, the emissivity can be determined to within rather narrow limits, thereby fixing limits on the temperature determined from the observed radiance. This condition was used to good advantage in the example shown in Fig. 6 : from the figure, 051 < Ep < 1.0, which condition, together with the measured radiance, (14) fixed the particle temperature between the limits 2880°K < T, < 3035°K. * The mass fraction ofparticulate matter is calculated from the propellant properties with the assumption that the system is at thermodynamic equilibrium, i.e. all reactions have gone to completion. A measurement of particle number density from a light scattering measurement”5) would eliminate this restriction.
638
J.M.
ADAMS
2. The emissivity curve is practically independent of the particle size distribution used. This conclusion is established by observing the differences in Fig. 5, y,/jj, vs. 7,/V, calculated for three size distributions. 3. The use of an extremely intense light source, such as a Q-switched laser, allows narrow limits on emissivity to be established, even if no transmitted light from the beam is detected. As is shown in Fig. 6, if for the conditions within this motor chamber the value of 9,/V is less than 1.5 x lo4 cm- ‘, the transmitted beam will be detected.* If the transmitted beam is detected, a value of 7,/V can be determined? and the limits on emissivity are fixed by the two possible values of 7,/y,. If one the other hand, no transmitted radiation from the laser pulse is observed, the limits on emissivity are :
which in Fig. 6 correspond
to the limits 0.51 < .sp < 0.76.
These limits on emissivity only f50”K.
would result in a temperature
uncertainty
interval
at 2500°K of
CONCLUSIONS
Some essential radiative properties have been presented for clouds of molten alumina and beryllia particles. The use of these properties in the determination of particle cloud spectral emissivity has been shown, together with an example of such a calculation for the conditions in a small rocket motor chamber. It was indicated that for an optically thick cloud of particles, the emissivity of the cloud can have a rather high minimum. The emissivity limits, when used together with a spectral radiance measurement, result in rather narrow limits on the determined temperature. It was further shown that, when an extremely intense reference light source is used, e.g. a Q-switched laser, the uncertainty in particle cloud emissivity can be further reduced, even if no detectable light from the source beam penetrates the cloud. It was shown that this narrow uncertainty in cloud emissivity, together with a measured radiance from the cloud can be used to establish a temperature for an optically thick cloud to rather close tolerances.
NOMENCLATURE
.; ‘L”. m Iv t u V
Y,
particle diameter frequency function for particle size distribution, normalized frequency function, equation (5) velocity lag, up/up complex refractive index number density of particles, equation (3) thickness of cloud velocity volume of particle absorption cross section for particle
* The lower limit on detectable transmitted radiance condition is indicated by flagnote 1 on the figure. t For example see flagnote 2 on Fig. 6.
see Fig. 4
is -0.1
of the emitted
radiance
from the cloud.
This
On the determination
Ys 6 % P X 5
of spectral
emissivity
in an optically
thick particle
cloud
639
scattering cross section for particle optical depth or thickness emissivity of the particle cloud density mass fraction of particles properties parameter, equations (6) (7)
Subscripts a
absorption
g L N P s
gas liquid normalized particles scattering
REFERENCES 1. J. M. ADAMS and S. E. COLUCCI, “The Spectroscopic Measurement of Gas and Particle Temperature in Metalized Propellant Combustion”. Paper SS66-175 ICRPG-AIAA Solid Propulsion Conference, reprinted in CPIA Bulletin No. III, Vol. III (1966). 2. D. J. CARLSON, in Temperature. Its Measurement and Control in Science and Industry 3, Part 2,535. Reinhold, New York (1962). 3. G. S. SUTHERLAND, The Mechanism of Combustion of an Ammonium Perchlorate-Polyester Resin Composite Solid Propellant, p. 53. PhD Thesis, Princeton University (1956). 4. J. A. NICHOLLS et al., An Experimental and Theoretical Study of Gaseous Detonation Waves. University of Michigan 02874-9-F, AD-268927 (1961). 5. H. C. VAN DE HULST, Light Scattering by Small Particles Ch. II, p. 15. Wiley & Sons Inc., New York (1957). 6. C. D. BARTKY, and E. BAUER, J. Spacecraft 3, 1523 (1966). 7. L. M. ROMANOVA, Optics Spectrosc. 14, 135 (1963). 8. S. CHANDRASEKHAR, Radiative Transfer. Dover Publications Inc., New York (1960). 9. L. A. POVINELLI and R. A. ROSENSTEIN,AIAA J. 3, 1754 (1964). 10. R. SEHGAL, The Collection and Evaluation of Particulate Matter JPL Tech. Report 32-238 (March, 1962). 11. R. A. DOBBINS, Light Scattering and Transmission Properties of Sprays, p. 59. AF 18(600)-1527, Princeton University Aero. Engg. Lab. Rept. No. 530 (1960). 12. E. G. KENDALL and J. D. MCCLELLAND, Materials and Structures, Aerospace Rept. No. TDR-930 (2240-66) TR-1 (March, 1962). 13. D. J. CARLSON, “Radiation from Rocket Exhaust Plumes. Part II: Metalized Solid Propellants” Joint Spec. Conference; Boulder, Colorado; June 1966. 14. S. E. COLUCCI and J. M. ADAMS, Flame Temperature Measurement of Metalized Propellant. AF 04(61 I)10545, Final Rept. AFRPL-TR-66-203, 20 et seq. (Sept., 1966) Confidential. 15. R. A. DOBBINS, lot. cit., p. 69.
Pergamon Press, 1968. Printed in Great Britain
ON THE DETERMINATION OF SPECTRAL EMISSIVITY IN AN OPTICALLY THICK PARTICLE CLOUD* J. M.
ADAMS?
Aerojet-General Sacramento,
Corporation, California
(Received 27 May 1967) Abstract-The extinction parameter, X/v, and ratio of optical absorption-total cross section, r,,/$j,, are presented for molten alumina and beryllia particle clouds. These parameters, which are determined for unimodal and bimodal polydispersions, are then related to the spectral emissivity of the cloud. Extrema in the emissivity of an optically thick cloud of beryllia particles are indicated, and are related to pyrometrically determined temperatures. It is shown that a high-intensity laser beam can be used in an extinction measurement to determine the particle cloud emissivity within rather close limits, even if no detectable penetration of the cloud by the light beam occurs.
INTRODUCTION IN ORDER to determine temperature of the condensed phase in a spectrally absorbingemitting two-phase medium (e.g., a cloud of particles in a gas), it is necessary to determine the emissivity and radiance associated with the continuum emission of the condensed-phase cloud ; the detailed method for doing this has been the subject of many experimental and analytical investigations”-3’ most of which have presented results only for an optically thin cloud. The emissivity of an optically thin cloud of particles is usually determined from a measurement of the extinction of a beam of light from a reference light source (e.g., a tungsten ribbon lamp of known brightness temperature). This extinction occurs due to both absorption and scattering of the light beam by the particles, but each contribution can be isolated through the use of the Mie scattering theoryf with a knowledge of the particle size distribution and complex refractive index of the condensed phase relative to the surrounding gas. From this treatment, the absorptivity of the particle cloud can readily be determined in the optically thin case. For a system in thermal equilibrium, the emissivity of the cloud is equated to the absorptivity from Kirchhoff’s Law. The purpose of this paper is to present a method for establishing limits on the value of particle cloud emissivity when the cloud is optically thick, and one is therefore prevented from making an extinction measurement under ordinary conditions. To this end, the use of * The author is pleased to acknowledge the assistance of Donald Carson and Charlotte Bartky of Aeronutronics for their contributions to this paper. Research reported in this paper was sponsored by the Air Force Rocket Propulsion Laboratory, Edwards AFB under Contract No. AF 04(611)-10545. Clearance ref: RA/SA AFSC 26 Ott 1966. y Engineering Specialist, Aerophysics. $ The details of the Mie scattering theory as it applies to a cloud of spherical particles is discussed by VAN DE HULST.‘~
631
J. M.
632
ADAMS
optical properties in the determination of spectral emissivity is shown in the analysis, together with an example of such a calculation for the conditions within a small rocket motor chamber. In order to limit the scope of the presentation, gradients of temperature, density, and particle concentration are assumed not to exist in the region under study. This assumption is not necessary to the solution of the problem, but it does eliminate unwarranted complexities in view of other uncertainties. For a method of handling these gradients, the reader is referred to work by NICHOLLS et dt4) Figure 1 presents the normal and hemispherical emissivity of a particle cloud as a function of optical depth, with the ratio y,/jJ as a parameter. The optical depth, 6, is given by 6 = R% + r,1t,
(11
where fl is the number density of particles in the cloud, cm- 3 ; t is the actual thickness of the cloud, cm ; and T,, 7, are the absorption and scattering cross sections for the polydispersion of particles in the cloud, cm2. The curves in Fig. 1, reproduced by permission of Charlotte Bartky, are the result of work by BARTKY@) and by ROMANOVA on special solutions to the radiative transfer equation.@’ 3D= 3 dimenrionol ID =
I dimensional
i = Isotropic
a 8 Anisotropic N = Normol H = Hemispherical
FIG. 1. Emissivity of a particle cloud determined from the radiative transfer equation. (Reproduced by permission of C. D. Etartky of Aeronutronic Corp.)
An optically thin cloud of scatterers and absorbers corresponds to that region to the left of Fig. 1 where the lines are straight ; values of 6 corresponding to the knee of the curves are associated with optically thick clouds. Finally, when the emissivity reaches its maximum value, a condition of infinite optical depth is said to exist, which corresponds to the region to the right of the knee in Fig. 1.
On the determination
of spectral
emissivity
SOURCE CONE
REFERENCELIGHTSOURCE
in an optically ACCEPTOR CONE
thick particle
633
cloud
ACCEPTOR ASSEMBLY
CARBONARC f ELECTRODE
STOP FIG.2. Optical
GAS PARTICLECLOUD
arrangement
for extinction
measurement.
A schematic of apparatus which is typical of that used in an extinction measurement is shown in Fig. 2.(I) The radiance and emissivity of the particle cloud are determined for the region defined by the acceptor optics ; this region takes the form of two colinear thin cones having a common apex at the centerline of the cloud. It is the scattering and absorption of the light source beam and the emission of the gas and particles within this region which are considered in the determination of temperature. The depth and particle concentration along the cone centerline defines the optical depth, 6. If temperature and concentration gradients through the plume are to be treated, a set of chordal measurements are taken across the cloud from the edge to the center. (4)The acceptor cone is defined by the optical system to be as small as possible to minimize the forward-scattered and multiply-scattered light which can be received and detected. ANALYSIS
AND
RESULTS
FOR
CLOUDS
OF
ARBITRARY
OPTICAL
THICKNESS
The following relations indicate how the variables affecting cloud emissivity are determined. From an interdependence of these variables for a given cloud geometry, it is shown that the limits on cloud emissivity can be determined from a single extinction measurement. An example is then presented of the use of this analysis to determine the spectral emissivity limits for the chamber region of a small rocket motor which is optically thick. The optical depth of a polydispersed particle cloud is given by 6 = N(Y, + r,)r.
(2)
The number density of particles or droplets in the cloud can be calculated from the relation (3) where L,, the velocity lag = up/us,* m
V, the mean particle volume =
s
f,U$$iR
(4)
0
* For this treatment, L, is assumed to be unity. Where velocity form part of the integrand of equation (4).
lag varies widely with particle
size, L,(D) will
634
J.M.
ADAMS
and&(D), the normalized frequency function describing the particle size distribution satis fies the relation 00 &(D) dD = 1.
I
(5)
0
is the mass fraction of particles (see footnote, bottom of p. 637) and pL, ps are the condensed-phase density and gas density respectively. The optical depth can now be written in the form
x
-”
/
--
physical properties of system
properties of particle size and refractive index
actual thickness of plume
which can be written as (7) where Y, = Y,+?,.
(8)
The second factor on the right side of equation (7), called the extinction parameter, can be determined for a polydispersion by the relation io -
Yr ==
F Y,(D>m)f,(D) dD
J
0
v
m %
s
(9)
D3fN(D) dD
0
where m is the complex refractive index. The ratio of absorption-total cross section, jj,/yt, can be similarly represented in terms of integrals and a cross-plot of these two parameters can be generated, as shown in Fig. 3. This figure was generated by 7094 machine computation for molten alumina particles, near the melting temperature (2320”K), by varying the most probable diameter of AW3,+ a typical unimodal distribution of particle sizes, shown in Fig. 4.* Also shown in the latter figure are two bimodal distributions, such as might result where large particles are burning * The “typical unimodal distribution” (T.U.D.) has been generated which closely approximates measured for a distributions from particle samples taken in rocket exhaust plumes. t9,*‘) From these reported distributions reprewide range of number mean particle diameter, it can be shown that a generalized or “typical” distribution sents the data.” ‘) Therefore, the abscissa in Fig. 4 has been non-dimensionalized by dividing by the most probable diameter for a typical unimodal distribution, D,, (T.U.D.). For each value of D,, (T.U.D.) chosen, a newdistribution results and new values of 7,/V and V./F, are determined. Figure 3 and subsequent cross-plots of $,/V vs. V./ut were generated in this manner.
On the determination
NY ““’
I 2
57103
FIG. 3. Absorption-total
of spectral
/
I
emissivity
,/1,,,
4
68lO'
in an optically
1
I
2
thick particle
,I
,,,,,
4
6
/
8 105
cross section vs. extinction parameter for spherical alumina (particle size distribution from Fig. 4).
,
2
particles
DISTRIBUTION (T.U.D.1
i’
MODERATELY BIMODALDISTRIBUTION
I
\
/
I I II
I \
\_ J
1.0
2.0
3.0
4.0
5.0
D/Dmar (T.U.D.) FIG. 4. Particle
size distributions
used in generating
63.5
cloud
curves in Figs. 3, 5 and 6.
of molten
636
J. M. ADAMS
to form very small “smoke” particles surrounding these large particles. Since the results must be applicable in the spectral region near the sodium doublet, the intensity of which serves as an indication of gas temperature, a wavelength of 0.589~ was used in the calculations. The effect of the type of distribution on the relationship between rt/V and 7,/y, is shown in Fig. 5 for molten beryllium oxide, BeO(,,. It can be seen from this figure that there
Moderately
blmodal distribution
m=1.730-0.02311
x=o?%9/L
/ 2 x/V,
FIG. 5. Absorption-total
I
I
4
I
I I , I I
I
6
2
8 105
,
cm-’
cross section vs. extinction parameter for spherical beryllia (particle size distributions from Fig. 4).
particles
of molten
is little difference in the resulting curves for the three types of distributions studied, indicating that an error in the assumed distribution will not result in a significant error in the relationship between jj,/jj, and Tt/V. The refractive index for beryllia was determined from measurements of KENDALL’l’) and CARLSON. (13) The dependence of the absorption constant (the imaginary part of the refractive index) on temperature was calculated by equating the temperature coefficient to that already determined for alumina, including the region above the melting point. Although the basis for this extrapolation is tenuous, the calculated values of cloud emissivity which result from these extrapolated values for BeOCo appear to be of reasonable magnitude when compared to measurements taken in the exhaust plumes of small rocket motors.(‘4) Having the relationship expressed by Figs. 3 and 5, the emissivity of a particle cloud can be determined from Fig. 1, i.e. Ep =
4YalYt,4.
(10)
Hence, from equations (7) and (9) ;
(11)
On the determination of spectral emissivity in an optically thick particle cloud
637
where < is determined from the physical properties of the system.* The resulting relationship for the emissivity of a typical particle cloud is shown in Fig. 6.
0,4-
o.3,0'
I 2
I
I 4
I Illll 6 6 10' -7/V,
I 2
I
1111111 4 6 8 105
cm-'
FIG. 6.
Particle cloud emissivity vs. extinction parameter for spherical particles of molten beryllia. (Determined for conditions in chamber of small motor, see text. Laser beam penetration levels are indicated.) 1. Upper level of vi/r for which there will be detectable penetration of cloud by 200 MW laser beam. 2. Example of two possible values for cloud emissivity when laser beam penetrates cloud and 7,/V is determined.
This figure was determined for conditions in the chamber of a small rocket motor, such as that which was actually used in experimental measurements of combustion temperature.(1*14’This was a research motor containing one pound of solid propellant. The average chamber pressure was about 650 psia. For the chamber conditions, 5 = 9 x 10W4. The particle sizes indicated in Fig. 6 refer to the number mean particle diameters corresponding to the values of 7,/V and jj,,/jjr,and resulting from an integral of the form ._
B = [ D&(D) dD.
(12)
The following features of Fig. 6 are of interest : 1. Under certain conditions, a minimum value for cloud emissivity is sign$cant enough to make measurement unnecessary. In other words, even if no reference light source is used to measure the extinction parameter, the emissivity can be determined to within rather narrow limits, thereby fixing limits on the temperature determined from the observed radiance. This condition was used to good advantage in the example shown in Fig. 6 : from the figure, 051 < Ep < 1.0, which condition, together with the measured radiance, (14) fixed the particle temperature between the limits 2880°K < T, < 3035°K. * The mass fraction ofparticulate matter is calculated from the propellant properties with the assumption that the system is at thermodynamic equilibrium, i.e. all reactions have gone to completion. A measurement of particle number density from a light scattering measurement”5) would eliminate this restriction.
638
J.M.
ADAMS
2. The emissivity curve is practically independent of the particle size distribution used. This conclusion is established by observing the differences in Fig. 5, y,/jj, vs. 7,/V, calculated for three size distributions. 3. The use of an extremely intense light source, such as a Q-switched laser, allows narrow limits on emissivity to be established, even if no transmitted light from the beam is detected. As is shown in Fig. 6, if for the conditions within this motor chamber the value of 9,/V is less than 1.5 x lo4 cm- ‘, the transmitted beam will be detected.* If the transmitted beam is detected, a value of 7,/V can be determined? and the limits on emissivity are fixed by the two possible values of 7,/y,. If one the other hand, no transmitted radiation from the laser pulse is observed, the limits on emissivity are :
which in Fig. 6 correspond
to the limits 0.51 < .sp < 0.76.
These limits on emissivity only f50”K.
would result in a temperature
uncertainty
interval
at 2500°K of
CONCLUSIONS
Some essential radiative properties have been presented for clouds of molten alumina and beryllia particles. The use of these properties in the determination of particle cloud spectral emissivity has been shown, together with an example of such a calculation for the conditions in a small rocket motor chamber. It was indicated that for an optically thick cloud of particles, the emissivity of the cloud can have a rather high minimum. The emissivity limits, when used together with a spectral radiance measurement, result in rather narrow limits on the determined temperature. It was further shown that, when an extremely intense reference light source is used, e.g. a Q-switched laser, the uncertainty in particle cloud emissivity can be further reduced, even if no detectable light from the source beam penetrates the cloud. It was shown that this narrow uncertainty in cloud emissivity, together with a measured radiance from the cloud can be used to establish a temperature for an optically thick cloud to rather close tolerances.
NOMENCLATURE
.; ‘L”. m Iv t u V
Y,
particle diameter frequency function for particle size distribution, normalized frequency function, equation (5) velocity lag, up/up complex refractive index number density of particles, equation (3) thickness of cloud velocity volume of particle absorption cross section for particle
* The lower limit on detectable transmitted radiance condition is indicated by flagnote 1 on the figure. t For example see flagnote 2 on Fig. 6.
see Fig. 4
is -0.1
of the emitted
radiance
from the cloud.
This
On the determination
Ys 6 % P X 5
of spectral
emissivity
in an optically
thick particle
cloud
639
scattering cross section for particle optical depth or thickness emissivity of the particle cloud density mass fraction of particles properties parameter, equations (6) (7)
Subscripts a
absorption
g L N P s
gas liquid normalized particles scattering
REFERENCES 1. J. M. ADAMS and S. E. COLUCCI, “The Spectroscopic Measurement of Gas and Particle Temperature in Metalized Propellant Combustion”. Paper SS66-175 ICRPG-AIAA Solid Propulsion Conference, reprinted in CPIA Bulletin No. III, Vol. III (1966). 2. D. J. CARLSON, in Temperature. Its Measurement and Control in Science and Industry 3, Part 2,535. Reinhold, New York (1962). 3. G. S. SUTHERLAND, The Mechanism of Combustion of an Ammonium Perchlorate-Polyester Resin Composite Solid Propellant, p. 53. PhD Thesis, Princeton University (1956). 4. J. A. NICHOLLS et al., An Experimental and Theoretical Study of Gaseous Detonation Waves. University of Michigan 02874-9-F, AD-268927 (1961). 5. H. C. VAN DE HULST, Light Scattering by Small Particles Ch. II, p. 15. Wiley & Sons Inc., New York (1957). 6. C. D. BARTKY, and E. BAUER, J. Spacecraft 3, 1523 (1966). 7. L. M. ROMANOVA, Optics Spectrosc. 14, 135 (1963). 8. S. CHANDRASEKHAR, Radiative Transfer. Dover Publications Inc., New York (1960). 9. L. A. POVINELLI and R. A. ROSENSTEIN,AIAA J. 3, 1754 (1964). 10. R. SEHGAL, The Collection and Evaluation of Particulate Matter JPL Tech. Report 32-238 (March, 1962). 11. R. A. DOBBINS, Light Scattering and Transmission Properties of Sprays, p. 59. AF 18(600)-1527, Princeton University Aero. Engg. Lab. Rept. No. 530 (1960). 12. E. G. KENDALL and J. D. MCCLELLAND, Materials and Structures, Aerospace Rept. No. TDR-930 (2240-66) TR-1 (March, 1962). 13. D. J. CARLSON, “Radiation from Rocket Exhaust Plumes. Part II: Metalized Solid Propellants” Joint Spec. Conference; Boulder, Colorado; June 1966. 14. S. E. COLUCCI and J. M. ADAMS, Flame Temperature Measurement of Metalized Propellant. AF 04(61 I)10545, Final Rept. AFRPL-TR-66-203, 20 et seq. (Sept., 1966) Confidential. 15. R. A. DOBBINS, lot. cit., p. 69.