THERMAL RADIATION OF A TWO-PHASE EXHAUST JET
In calculating the thermal radiation of a two-phase jet flowing from the exhaust nozzle of solid-propellant rocket engine, one should also take into account the strong variation of the flow field parameters and optical properties. These parameters vary substantially not only along the jet axis (as in a supersonic nozzle), but also in the cross sections of the jet. Nevertheless, the calculation of the exhaust plume radiation is simpler than solving the radiative transfer problem for the nozzle because the spectral optical thickness of the jet in a cross section is usually small, and there are not any reflecting and emitting boundary surfaces. The latter circumstance makes a solution to the radiative transfer equation (RTE) to be independent of combined heat transfer in a boundary layer and of the wall heating. At the same time, there is one special difficulty in calculating the plume radiation, i.e., it is important to determine the angular distribution of the radiation. As a result, we cannot limit ourselves with the diffusion approximation. In addition, the radiative properties of gases are more important than those in the nozzle, which may result in some computational difficulties.
The thermal radiation of the rocket engine exhaust plume is of interest for the determination of the total heat flux to the elements of a rocket or launcher design, so for the remote disclosure and identification of rockets by analysis of the plume signature. For this reason, one should conduct complete calculations with the detailed angular and spectral structure of the radiation field at an arbitrary distance from the exhaust jet. A solution to the thermal radiation problem for a two-phase jet can be subdivided into the following stages: determination of flow parameters, calculation of the medium radiative properties, approximate calculation of the RTE source function by use of diffusion approximation; and integration of the RTE along the directions of observation and (if needed) determination of the radiation flux to a surface outside the jet. We will not discuss the problem of flow field parameters calculations. A reader can study this problem as applied to exhaust jets by use of other papers (Gilinsky et al., 1979; Dash et al., 1980; Avduevsky et al., 1985; Nelson and Fields, 1995; Burt and Boyd, 2005a). Note that there is a principle difference between physical pictures at a low and very high altitude. In the lower atmosphere, the jet parameters are determined by the dynamic and chemical interactions of the jet gases with cocurrent airflow. At high altitude, the jet can be described as a flow of condensed phase particles without essential dynamic and thermal interactions with the rarefied gas medium. In this case, the temperature of particles varies due to the crystallization and thermal radiation of particles to the space. In the present article, we consider the plume radiation at low altitude only, when thermal radiation does not practically participate in the formation of the jet temperature field.
It was shown in the article, “Radiative transfer in multidimensional problems--a combined computational model,” that a rather accurate solution for the problem under consideration can be obtained by use of a combined computational model. This model is based on a transport approximation for the scattering (phase) function and P_{1} approximation for the spectral radiation energy density as a first step of the iterative solution. To determine the function under the integral and to perform integration at the second step of the solution, one should know the coordinates of intercept of a ray with grid surfaces dividing the region into circular elements of quadrangle cross section. The values of α_{λ}, β_{λ}^{tr}, S_{b}, and I_{λ}^{0} at these points are obtained by use of linear interpolation of the corresponding values at the grid nodes. Using the necessary set of rays, one can calculate the angular distribution of the radiation from a given point of the jet surface as well as the radiation flux to an arbitrary oriented surface.
The thermal radiation of various two-phase jets in the visible and infrared spectral ranges has been studied in detail by many researchers. Except for a few papers concerning the radiation of industrial or volcanic plumes (Stailor, 1978; Zhang et al., 2003), the majority of these studies deal with exhaust jets of rocket engines. In an early paper by Morizumi and Carpenter (1964), the surface of a solid-propellant rocket exhaust jet was assumed to be gray and diffusely emitting, and the integral hemispherical emissivity was determined locally in each cross section by use of a solution for the 1D radiative transfer problem in an isotropically scattering medium. It was also assumed that particles of the condensed phase have constant values of the optical parameters, Q_{a} = 0.5 and Q_{t} = 2. Nevertheless, the important role of radiation scattering by particles was demonstrated. More realistic radiative properties of polydisperse alumina particles were considered by Bartky and Bauer (1966), but their model of a homogeneous plume was based on the solution for an isothermal planar medium. Some additional estimates of the far-field infrared emission of a solid-propellant rocket exhaust at vacuum-expansion altitudes have been presented by Worster (1974). A thermal radiation model for solid rocket booster plumes developed by Watson and Lee (1977) was based on a Monte Carlo simulation to account for axial and radial variations in radiative properties of the plumes. The authors used a gray model for the medium radiative properties. The coefficients of absorption and extinction were averaged according to Plank, and the properties of alumina particles were calculated by using Mie theory at a fixed value of the complex index of refraction, i.e., m = 1.8-0.005i. The difference between the temperatures of particles and the ambient gas was not taken into account.
Further investigations have been accomplished for developing advanced computational methods, so for physical analysis of the problem and engineering method constructing. In connection with the first direction, it is worth remembering the papers by Ludwig et al. (1981, 1982), in which solutions based on two-flux and six-flux approximations for radiative transfer in a cylindrical volume were obtained. It was noted in Ludwig et al. (1982) that an error of the two-flux method increases toward observation angles near the jet axis. An engineering computational method based on a simple analytical model for thermal radiation was realized by Edwards and Bobco (1982). A crude description of the jet emission was considered (totally diffuse radiation depending only on axial coordinate). But the authors presented a detailed study of the effect of particle size distribution on infrared thermal radiation of solid-propellant rocket engine (SPRE) plumes.
Advanced computational models for radiative transfer based on the Monte Carlo method have been developed by Daladova et al. (1990), Nelson (1992), Surzhikov (2002, 2003, 2004), Shuai et al. (2005), Burt and Boyd (2005b, 2007a,b), and Kotov and Surzhikov (2007). One should also remember the recent paper by Cai et al. (2007), in which the finite volume method was suggested to solve RTE as applied to the infrared radiative signatures of liquid and solid rocket plumes. The advanced models enable one to take into account the anisotropic scattering of the radiation and the real fields of flow parameters and radiative properties of the medium. In particular, the “searchlight effect” can be calculated--the scattering of the radiation propagating from the nozzle exit. The searchlight effect was considered for the first time by Stockham and Love (1968). A more detailed study of this effect by Edwards and Babikian (1990) showed that the nozzle wall temperature is an important parameter of this problem. A physical analysis of the role of the condensed phase of combustion products in thermal radiation of exhaust jets has been presented by Lyons et al. (1982, 1983), Nelson (1984ab, 1987), Nelson and Tucker (1986), Victor (1989), Thynell (1992), and Laredo and Netzer (1993). It was shown that the radiation of two-phase jets in the visible, as well as in the infrared range outside the gaseous spectral bands, is determined by the radiative properties of the condensed phase. More essential factors having a great effect on the accuracy of the calculations are the particle temperature and index of absorption of the particle material. A particle size distribution also appears to be important.
Following the early papers by Dombrovsky and Barkova (1986) and Dombrovsky (1996a,b,c), we consider an example of the thermal radiation calculation for an isobaric jet of a large-scale rocket engine. The majority of parameters are omitted. In Fig. 1, only several isotherms for the gaseous phase in the computational region are shown.
Figure 1. Gas temperature field in a two-phase isobaric burning-out exhaust jet in cocurrent airflow (dashed line is the conventional boundary of the jet)
In contrast to the transonic region of the large-scale rocket nozzle, the optical thickness in the jet cross sections is small. At the same time, one should use a finite element mesh with the large number of intervals along the jet axis. Consider first the results of the radiation flux calculations on the jet surface in the first stage of the RTE solution based on the diffusion approximation. The corresponding spectral dependences are shown in Fig. 2. We use the dimensionless variable x = x/x_{0}, where x is the axial coordinate measured from the nozzle exit, and x_{0} = 100 m is the length of the computational region.
Figure 2. Spectral radiation flux near the exhaust jet surface at x = 0.253 (a) and x = 0.8 (b): 1--complete calculation, 2--calculation ignoring the contribution of condensed phase particles, 3--calculation ignoring the gas phase contribution
The spectral dependences q_{λ}(λ) have a number of peaks due to absorption bands of gases. The resonance character of the spectrum is more pronounced in the cross section located far from the nozzle exit. Note that thermal radiation at the spectral band λ = 4.3 μm is absorbed by ambient cold gas. Despite the complexity of the emission spectrum, additional calculations showed that variation of the integration increment Δλ leads to a small effect on the values of integral radiation flux. In the visible range, the exhaust plume radiates due to condensed phase particles. Therefore, the self-integral emission of alumina particles in the high-temperature region at x = 0.253 is twice greater than the corresponding emission of gases. At x = 0.8, the particle radiation is small due to low temperature, and one can suppose that the integral radiation flux is determined only by gaseous radiation. But the calculations ignoring the condensed phase contribution show a very high value of the integral radiation flux. This result is explained by the scattering of the high-temperature region radiation by alumina particles.
It is important to note that correct quantitative results cannot be obtained in the diffusion (P_{1}) approximation, and the data of Fig. 2 should be considered only as an estimate. In particular, the effect of scattering of the radiation propagating along the jet axis appears to be overestimated because of a fictitious scattering due to a linear (in μ = cosθ) approximation of the angular dependence of the radiation intensity in the P_{1} approximation. In Fig. 3, some results of more accurate calculations are shown, i.e., after integration of the RTE with a source function determined in the diffusion approximation. The angular dependences of the radiation intensity on the exhaust jet surface are very complex, especially far from the high-temperature region of the jet. For this reason, a considerable error of the diffusion approximation and the necessity of the second stage of the solution is quite evident. One can see in Fig. 3 from the book by Dombrovsky (1996c) that scattering always leads to decreasing the radiation intensity in the normal direction to the jet axis and to expanding the radiation angular dependence in the plane perpendicular to the axis. It is important to note that the scattering of radiation by c-phase particles is the strongest factor determining the angular distribution of the radiation near the jet surface.
Figure 3. Integral radiation intensity at x = 0.253 (a) and x = 0.8 (b) in the plane of the jet axis (a) and in the plane perpendicular to the axis (b): 1--complete calculation, 2--without scattering, 3--isotropic scattering model
Angular dependences of the integral radiation force of the jet (in conventional units) for the observation at large distances from the jet are presented in Fig. 4, reproduced from the book by Dombrovsky (1996c). The set of variants enables us to analyze the role of various factors. It is seen that emissions of both gases and condensed phase particles should be taken into account. The radiation flux at angles θ > 60 deg to the jet axis can be obtained without accounting for scattering, and the assumption of isotropic scattering appears to be acceptable for calculating the radiation flux at θ > 40 deg. In both cases, the error is less than 10%. In calculating the radiation at low angles to the jet axis, the scattering should be taken into account in full. For spectral jet radiation, the role of the condensed phase is more significant outside the absorption bands of the gases.
Figure 4. Integral radiation flux from the exhaust jet (in conventional units): 1--complete calculation, 2--calculation ignoring the contribution of condensed phase particles, 3--calculation ignoring the gas phase contribution, 4--calculation without scattering, 5--isotropic scattering model
The effects of radiation scattering by condensed phase particles are more pronounced near the jet. In particular, the scattering is of great importance for the radiation flux to the elements of rocket or launcher design. Some typical computational data for a first-stage SPRE are given in Table 1. One can see that scattering leads to a decrease in the integral radiation flux, especially strong for a surface oriented along the jet axis. It is evident that practical calculations of the total heat flux to the engine bottom must take into account the radiation scattering by alumina particles in the exhaust jet. Some additional data for this problem can be found in an early review by Rochelle (1966), and in a more recent paper by Reardon and Nelson (1994), where the methods used to predict the base heating of rockets caused by exhaust plume radiation were considered. Note that the searchlight effect also gives a small contribution to the integral radiation flux in this particular problem.
Table 1. Effect of scattering on integral radiation flux from exhaust jet to surface element located in nozzle exit plane
q, kW/m^{2} | |||
Surface orientation | Without scattering | With scattering | With searchlight effect |
In plane of the nozzle exit | 150 | 124 | 178 |
Parallel to the jet axis | 236 | 154 | 170 |
The numerical data presented above refer to SPRE exhaust plumes at low altitudes when the plume radiation is strong and it is determined by burning out of the jet gases in cocurrent airflow. In this case, scattering of thermal radiation propagating from the nozzle gives only a small contribution to the total radiation of the jet. At high altitudes, this correlation is essentially changed. An estimate of the searchlight effect has been obtained in a latent form in the radiative transfer calculations for a high-altitude nozzle, when a conventional initial zone of the jet was included in the computational region. As was shown above, the radiation flux from the supersonic nozzle toward the jet is determined by the nozzle wall temperature, and hence increases during the engine running. Some quantitative data for this effect can be found in the books by Dombrovsky (1996c) and Dombrovsky and Baillis (2010).
It should be noted that it is difficult to obtain reliable quantitative data for thermal radiation from a solid rocket exhaust jet due to the great uncertainty in size distribution, chemical composition, morphology, and optical properties of condensed phase particles. One can recommend papers by Pluchino and Masturzo (1981), Laredo and Netzer (1993), Reed and Calia (1993), Plastinin et al. (1996, 2001), and Gossé et al. (2003, 2006) for a detailed study of some particular problems concerning this matter. It is interesting that present-day experimental and theoretical methods enable one to use lidar backscatter properties of a rocket exhaust plume for estimating nonsphericity of alumina particles. This technique was employed by Reed et al. (2005) to determine the distribution of particle shapes for the exhaust plume of a Titan-IVA rocket by observation from a high-altitude research aircraft.
In the present article, the variation of the exhaust jet temperature due to thermal radiation is not taken into account in the problem analysis. As was noted above, this effect may be significant for high-altitude solid-propellant rockets (Fontenot, 1965; Cohen, 1988). A solution for the corresponding combined heat transfer problem depends on a great number of parameters, and is mathematically complicated. For this reason, the transfer to a more general physical problem statement in subsequent sections is accompanied by returning to simple flow models.
REFERENCES
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Bartky, C. D. and Bauer, E., Predicting the emittance of a homogeneous plume containing alumina particles, J. Spacecraft Rockets, vol. 3, no. 10, pp. 1523-1527, 1966.
Burt, J. M. and Boyd, I. D., Monte Carlo simulation of a rarefied multiphase plume flow, AIAA Paper No. 964, 2005a.
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Cai, G., Zhu, D., and Zhang, X., Numerical simulation of the infrared radiative signatures of liquid and solid rocket plumes, Aerospace Sci. Tech., vol. 11, no. 6, pp. 473-480, 2007.
Cohen, D. L., Simple model for particle radiative transfer in vacuum particle plumes, J. Thermophys. Heat Transfer, vol. 2, no. 4, pp. 365-367, 1988.
Daladova, L. A., Makienko, A. I., Pavlova, S. N., Surzhikov, S. T., and Khmelinin, B. A., Computational model of thermal radiation from axisymmetrical volumes of scattering two-phase media, Math. Model., vol. 2, no. 4, pp. 54-66, 1990 (in Russian).
Dash, S. M., Pearce, B. E., Pergament, H. S., and Fishburne, E. S., Prediction of rocket plume flowfields for infrared signature studies, J. Spacecraft Rockets, vol. 17, no. 3, pp. 190-199, 1980.
Dombrovsky, L. A. and Barkova, L. G., Solving the two-dimensional problem of thermal-radiation transfer in an anisotropically scattering medium using the finite element method, High Temp., vol. 24, no. 4, pp. 585-592, 1986.
Dombrovsky, L. A., A theoretical investigation of heat transfer by radiation under conditions of two-phase flow in a supersonic nozzle, High Temp., vol. 34, no. 2, pp. 255-262, 1996a.
Dombrovsky, L. A., Approximate methods for calculating radiation heat transfer in dispersed systems, Therm. Eng., vol. 43, no. 3, pp. 235-243, 1996b.
Dombrovsky, L. A., Radiation Heat Transfer in Disperse Systems, Begell House, New York, and Redding, CT, 1996c.
Dombrovsky, L. A. and Baillis, D., Thermal Radiation in Disperse Systems: An Engineering Approach, Begell House, New York, and Redding, CT, 2010.
Edwards, D. K. and Babikian, D. S. () Radiation from a nongray scattering, emitting, and absorbing solid rocket motor plume, J. Thermophys. Heat Transfer, vol. 4, no. 4, pp. 446-453, 1990.
Edwards, D. K. and Bobco, R. P., Effect of particle size distribution on the radiosity of solid-propellant rocket motor plumes, Prog. Astronaut. Aeronaut., vol. 83, pp. 169-188, 1982.
Fontenot, J. E., Jr., Thermal radiation from solid rocket plumes at high altitude, AIAA J., vol. 3, no. 5, pp. 970-972, 1965.
Gilinsky, M. M., Zubkov, A. I., and Sukhanovskaya, L. D., Radiation from two-phase supersonic jet, Jet and Separate Flows, Moscow University Press, Moscow, pp. 106-120, 1979.
Gossé, S., Sarou-Kanian, V., Véron, E., Millot, F., Rifflet, J. C., and Simon, P., Characterization and morphology of alumina particles in solid propellant subscale rocket motor plumes, AIAA Paper No. 3649, 2003.
Gossé, S., Hespel, L., Gossart, P., and Delfour, A., Morphological characterization and particle sizing of alumina particles in solid rocket motor, J. Propulsion Power, vol. 22, no. 1, pp. 127-135, 2006.
Kotov, D. V. and Surzhikov, S. T., Local estimation of directional emissivity of light-scattering volumes using the Monte Carlo method, High Temp., vol. 45, no. 6, pp. 807-817, 2007.
Laredo, D. and Netzer, D. W., The dominant effect of alumina on nearfield plume radiation, J. Quant. Spectr. Radiat. Transfer, vol. 50, no. 5, pp. 511-530, 1993.
Ludwig, C. B., Malkmus, W., Walker, J., Slack, M., and Reed, R., The standard infrared radiation model, AIAA Paper No. 1051, 1981.
Ludwig, C. B., Malkmus, W., Freeman, G. N., Slack, M., and Reed, R., A theoretical model for absorbing, emitting and scattering plume radiation, Prog. Astronaut. Aeronaut., vol. 83, pp. 111-127, 1982.
Lyons, R. B., Wormhoudt, J., and Kolb, C. E., Calculation of visible radiation from missile plumes, Prog. Astronaut. Aeronaut., vol. 83, pp. 128-148, 1982.
Lyons, R. B., Wormhoudt, J., and Gruninger, J., Scattering of radiation by particles in low-altitude plumes, J. Spacecraft Rockets, vol. 20, no. 2, pp. 189-192, 1983.
Morizumi, S. J. and Carpenter, H. J., Thermal radiation from the exhaust plume of an aluminized composite propellant rocket, J. Spacecraft Rockets, vol. 1, no. 5, pp. 501-507, 1964.
Nelson, H. F., Influence of particulates on infrared emission from tactical rocket exhausts, J. Spacecraft Rockets, vol. 21, no. 5, pp. 425-432, 1984a.
Nelson, H. F., Influence of scattering on infrared signatures of rocket plumes, J. Spacecraft Rockets, vol. 21, no. 5, pp. 508-510, 1984b.
Nelson, H. F., Evaluation of rocket plume signature uncertainties, J. Spacecraft Rockets, vol. 24, no. 6, pp. 546-551, 1987.
Nelson, H. F., Backward Monte Carlo modeling for rocket plume base heating, J. Thermophys. Heat Transfer, vol. 6, no. 3, pp. 556-558, 1992.
Nelson, H. F. and Fields, J. C., Particle drag coefficient in solid rocket plumes, J. Thermophys. Heat Transfer, vol. 9, no. 3, pp. 567-569, 1995.
Nelson, H. F. and Tucker, E. O., Boron slurry-fueled jet engine exhaust plume infrared signatures, J. Spacecraft Rockets, vol. 23, no. 5, pp. 527-533, 1986.
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Plastinin, Yu. A., Karabadzhak, G., Khmelinin, B. A., Baula, G., and Rodionov, A., Ultraviolet, visible and infrared spectra modeling for solid and liquid-fuel rocket exhausts, AIAA Paper No. 0660, 2001.
Pluchino, A. B. and Masturzo, D. E., Emissivity of Al_{2}O_{3} particles in a rocket plume, AIAA J., vol. 19, no. 9, pp. 1234-1237, 1981.
Reardon, J. E. and Nelson, H. F., Rocket plume base heating methodology, J. Thermophys. Heat Transfer, vol. 8, no. 2, pp. 216-222, 1994.
Reed, R. A. and Calia, V. S., Review of aluminum oxide rocket exhaust particles, AIAA Paper No. 2819, 1993.
Reed, R. A., Nolen, M. K., Zolensky, M. E., Bankman, I., Giles, J., Lang, P., Mishchenko, M. I., and Ross, M. R., Lidar backscatter properties of Al_{2}O_{3} rocket exhaust particles, J. Spacecraft Rockets, vol. 42, no. 4, pp. 711-715, 2005.
Rochelle, W. C., Theoretical and experimental investigation of heating from saturn solid propellant rocket exhausts, AIAA Paper No. 653, 1966.
Shuai, Y., Dong, S. K., and Tan, H. P., Simulation of the infrared radiation characteristics of high-altitude exhaust plume including particles using the backward Monte Carlo method, J. Quant. Spectrosc. Radiat. Transfer, vol. 95, no. 2, pp. 231-240, 2005.
Stailor, W. F., Determination of stack plume properties from satellite imagery, J. Spacecraft Rockets, vol. 15, no. 2, pp. 92-99, 1978.
Stockham, L. W. and Love, T. J., Radiative heat transfer from a cylindrical cloud of particles, AIAA J., vol. 6, no. 10, pp. 1935-1940, 1968.
Surzhikov, S. T., Direct simulation Monte Carlo algorithms for the rocket exhaust plumes emissivity prediction, AIAA Paper No. 0795, 2002.
Surzhikov, S. T., Monte Carlo simulation of plumes spectral emission, AIAA Paper No. 3895, 2003.
Surzhikov, S. T., Three-dimensional model of the spectral emissivity of light-scattering exhaust plumes, High Temp., vol. 42, no. 5, pp. 763-775, 2004.
Thynell, S. T., Effect of linear-anisotropic scattering on spectral emission from cylindrical plumes, J. Thermophys. Heat Transfer, vol. 6, no. 2, pp. 224-231, 1992.
Victor, A. C., Effects of multiple scattering on rocket exhaust plume smoke visibility, J. Spacecraft Rockets, vol. 26, no. 4, pp. 274-278, 1989.
Watson, G. N. and Lee, A. L., Thermal radiation model for solid motor booster plumes, J. Spacecraft Rockets, vol. 14, no. 11, pp. 641-647, 1977.
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Zhang, J., Goldstein, D. B., Varghese, P. L., Gimelshine, N. E., Gimelshein, S. F., and Levin, D. A., Simulation of gas dynamics and radiation in volcanic plumes on Io, Icarus, vol. 163, no. 1, pp. 182-197, 2003.
Verweise
- Avduevsky, V. S., Ashratov, E. A., Ivanov, A. V., and Pirumov, U. G., Supersonic Nonisobaric Jets of Gases, Mech. Eng., Moscow, 1985 (in Russian).
- Bartky, C. D. and Bauer, E., Predicting the emittance of a homogeneous plume containing alumina particles, J. Spacecraft Rockets, vol. 3, no. 10, pp. 1523-1527, 1966.
- Burt, J. M. and Boyd, I. D., Monte Carlo simulation of a rarefied multiphase plume flow, AIAA Paper No. 964, 2005a.
- Burt, J. M. and Boyd, I. D., A Monte carlo radiation model for simulating rarefied multiphase plume flows, AIAA Paper No. 4691, 2005b.
- Burt, J. M. and Boyd, I. D., High-altitude plume simulations for a solid propellant rocket, AIAA J., vol. 45, no. 12, pp. 2872-2884, 2007a.
- Burt, J. M. and Boyd, I. D., Monte Carlo simulation of particle radiation in high altitude solid rocket plumes, AIAA Paper No. 5703, 2007b.
- Cai, G., Zhu, D., and Zhang, X., Numerical simulation of the infrared radiative signatures of liquid and solid rocket plumes, Aerospace Sci. Tech., vol. 11, no. 6, pp. 473-480, 2007.
- Cohen, D. L., Simple model for particle radiative transfer in vacuum particle plumes, J. Thermophys. Heat Transfer, vol. 2, no. 4, pp. 365-367, 1988.
- Daladova, L. A., Makienko, A. I., Pavlova, S. N., Surzhikov, S. T., and Khmelinin, B. A., Computational model of thermal radiation from axisymmetrical volumes of scattering two-phase media, Math. Model., vol. 2, no. 4, pp. 54-66, 1990 (in Russian).
- Dash, S. M., Pearce, B. E., Pergament, H. S., and Fishburne, E. S., Prediction of rocket plume flowfields for infrared signature studies, J. Spacecraft Rockets, vol. 17, no. 3, pp. 190-199, 1980.
- Dombrovsky, L. A. and Barkova, L. G., Solving the two-dimensional problem of thermal-radiation transfer in an anisotropically scattering medium using the finite element method, High Temp., vol. 24, no. 4, pp. 585-592, 1986.
- Dombrovsky, L. A., A theoretical investigation of heat transfer by radiation under conditions of two-phase flow in a supersonic nozzle, High Temp., vol. 34, no. 2, pp. 255-262, 1996a.
- Dombrovsky, L. A., Approximate methods for calculating radiation heat transfer in dispersed systems, Therm. Eng., vol. 43, no. 3, pp. 235-243, 1996b.
- Dombrovsky, L. A., Radiation Heat Transfer in Disperse Systems, Begell House, New York, and Redding, CT, 1996c.
- Dombrovsky, L. A. and Baillis, D., Thermal Radiation in Disperse Systems: An Engineering Approach, Begell House, New York, and Redding, CT, 2010.
- Edwards, D. K. and Babikian, D. S. () Radiation from a nongray scattering, emitting, and absorbing solid rocket motor plume, J. Thermophys. Heat Transfer, vol. 4, no. 4, pp. 446-453, 1990.
- Edwards, D. K. and Bobco, R. P., Effect of particle size distribution on the radiosity of solid-propellant rocket motor plumes, Prog. Astronaut. Aeronaut., vol. 83, pp. 169-188, 1982.
- Fontenot, J. E., Jr., Thermal radiation from solid rocket plumes at high altitude, AIAA J., vol. 3, no. 5, pp. 970-972, 1965.
- Gilinsky, M. M., Zubkov, A. I., and Sukhanovskaya, L. D., Radiation from two-phase supersonic jet, Jet and Separate Flows, Moscow University Press, Moscow, pp. 106-120, 1979.
- GossÃ©, S., Sarou-Kanian, V., VÃ©ron, E., Millot, F., Rifflet, J. C., and Simon, P., Characterization and morphology of alumina particles in solid propellant subscale rocket motor plumes, AIAA Paper No. 3649, 2003.
- GossÃ©, S., Hespel, L., Gossart, P., and Delfour, A., Morphological characterization and particle sizing of alumina particles in solid rocket motor, J. Propulsion Power, vol. 22, no. 1, pp. 127-135, 2006.
- Kotov, D. V. and Surzhikov, S. T., Local estimation of directional emissivity of light-scattering volumes using the Monte Carlo method, High Temp., vol. 45, no. 6, pp. 807-817, 2007.
- Laredo, D. and Netzer, D. W., The dominant effect of alumina on nearfield plume radiation, J. Quant. Spectr. Radiat. Transfer, vol. 50, no. 5, pp. 511-530, 1993.
- Ludwig, C. B., Malkmus, W., Walker, J., Slack, M., and Reed, R., The standard infrared radiation model, AIAA Paper No. 1051, 1981.
- Ludwig, C. B., Malkmus, W., Freeman, G. N., Slack, M., and Reed, R., A theoretical model for absorbing, emitting and scattering plume radiation, Prog. Astronaut. Aeronaut., vol. 83, pp. 111-127, 1982.
- Lyons, R. B., Wormhoudt, J., and Kolb, C. E., Calculation of visible radiation from missile plumes, Prog. Astronaut. Aeronaut., vol. 83, pp. 128-148, 1982.
- Lyons, R. B., Wormhoudt, J., and Gruninger, J., Scattering of radiation by particles in low-altitude plumes, J. Spacecraft Rockets, vol. 20, no. 2, pp. 189-192, 1983.
- Morizumi, S. J. and Carpenter, H. J., Thermal radiation from the exhaust plume of an aluminized composite propellant rocket, J. Spacecraft Rockets, vol. 1, no. 5, pp. 501-507, 1964.
- Nelson, H. F., Influence of particulates on infrared emission from tactical rocket exhausts, J. Spacecraft Rockets, vol. 21, no. 5, pp. 425-432, 1984a.
- Nelson, H. F., Influence of scattering on infrared signatures of rocket plumes, J. Spacecraft Rockets, vol. 21, no. 5, pp. 508-510, 1984b.
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