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## THERMAL RADIATION FROM NONISOTHERMAL PARTICLES IN COMBINED HEAT TRANSFER PROBLEMS

In analyzing radiation heat transfer in disperse systems containing small particles, a single particle is usually considered to be isothermal. This assumption greatly simplifies the calculation of the thermal radiation of the particle (Dombrovsky, 1996; Modest, 2003; Viskanta, 2005). At the same time, in the case of intensive heating or cooling, the temperature difference between the center and the surface of the particle may be considerable. This can be realized not only in the case of fast heating of solid or liquid particles by laser or concentrated solar radiation (Pendleton, 1985; Park and Armstrong, 1989; Tuntomo and Tien, 1992; Qiu et al., 1995; Astafieva and Prishivalko, 1998; Dombrovsky and Lipiński, 2007), but also in some other heat transfer problems. One example is the known problem of steam explosion due to thermal interaction of core melt droplets with ambient water during a hypothetical severe accident in light water nuclear reactors (Fletcher, 1995; Theofanous, 1995; Berthoud, 2000). Another example is the problem of oxide particles heating and melting in plasma spraying (Pfender and Chang, 1998; Moreau, 1998; Fincke et al., 2001; Fauchais, 2004; Streibl et al., 2006; Fauchais et al., 2006). In both cases, the temperature difference between the center and the surface of the particle cannot be ignored, and one needs calculations of thermal radiation from nonisothermal particles.

Thermal radiation of a nonisothermal particle is an especially interesting problem for particles of a weakly absorbing material (in particular, for metal oxide particles). The point is that materials that are semitransparent in the infrared spectral range are usually characterized by low thermal conductivity. As a result, one can expect a comparatively large temperature difference between the center and the surface of the particle. On the other hand, in the case of a weak absorption, the solution to the problem is expected to be more complicated because of the possible considerable contribution of radiation emitted from the central core of the particle.

In the case of a large spherical particle with a spherically symmetric temperature field, one can describe the radiation heat transfer by use of an MDP0 approximation, as discussed in the article Thermal radiation from nonisothermal spherical particles. This approach, suggested first by Dombrovsky (1998), was then studied in the greatest detail in other papers by Dombrovsky (1999, 2000a,b, 2002, 2004a,b). Following the recent lecture by Dombrovsky (2007), we present the computational model for calculating radiative-conductive heat transfer in nonisothermal spherical particles, and give examples of the model application to the above-mentioned engineering problems. We do not consider the case of a nonsymmetric temperature field in the particle, which may be important for some applied problems. A novel method for solving this problem has been suggested recently by Liu (2004). The method is based on a Monte Carlo ray-tracing technique.

In the MDP0 approximation, we have the following boundary value problem:

 (1a)
 (1b)

where the variable radiation diffusion coefficient is

 (2)

The spectral radiation flux from the particle and the spectral radiation power inside the particle are calculated as follows:

 (3)

The above boundary value problem is much simpler than the radiative transfer formulation based on a combination of ray tracing and zonal methods developed by Liu et al. (2002) for the same transient radiation-conduction problem in a semitransparent spherical particle.

#### REFERENCES

Astafieva, L. G. and Prishivalko, A. P., Heating of Solid Aerosol Particles Exposed to Intense Optical Radiation, Int. J. Heat Mass Transfer, vol. 41, no. 2, pp. 489–499, 1998.

Berthoud, G., Vapor Explosions, Ann. Rev. Fluid Mech., vol. 32, pp. 573–611, 2000.

Dombrovsky, L. A., Radiation Heat Transfer in Disperse Systems, Begell House, New York and Redding, CT, 1996.

Dombrovsky, L. A., Thermal Radiation From Nonisothermal Particles of Weakly Absorbing Substance, Proc. 2nd Russ. Nat. Conf. Heat Mass Transfer, Moscow, Oct. 26–30, vol. 6, Mosc. Inst. Power Eng. (MIPE) Publ. House, Moscow, pp. 278–281, 1998 (in Russian).

Dombrovsky, L. A., Thermal Radiation of a Spherical Particle of Semitransparent Material, High Temp., vol. 37, no. 2, pp. 260–269, 1999.

Dombrovsky, L. A., Thermal Radiation From Nonisothermal Spherical Particles of a Semitransparent Material, Int. J. Heat Mass Transfer, vol. 43, no. 9, pp. 1661–1672, 2000a.

Dombrovsky, L. A., Approximate Calculation of Thermal Radiation of Nonisothermal Semitransparent Particles, High Temp., vol. 38, no. 4, pp. 663–665, 2000b.

Dombrovsky, L. A., A Modified Differential Approximation for Thermal Radiation of Semitransparent Nonisothermal Particles: Application to Optical Diagnostics of Plasma Spraying, J. Quant. Spectrosc. Radiat. Transfer, vol. 73, no. 2-5, pp. 433–441, 2002.

Dombrovsky, L. A., Nonuniform Absorption of Thermal Radiation in Semitransparent Spherical Particles under Conditions of Arbitrary Illumination of a Disperse System, High Temp., vol. 42, no. 6, pp. 975–986, 2004a.

Dombrovsky, L. A., Absorption of Thermal Radiation in Large Semi-Transparent Particles at Arbitrary Illumination of the Polydisperse System, Int. J. Heat Mass Transfer, vol. 47, no. 25, pp. 5511–5522, 2004b.

Dombrovsky, L. A., Thermal Radiation of Nonisothermal Particles in Combined Heat Transfer Problems (dedication lecture), Proc. of 5th Int. Symp. Radiat. Transfer, Bodrum, Turkey, June 17–22, Begell House, New York and Redding (CT), 2007.

Dombrovsky, L. A. and Lipiński, W., Transient Temperature and Thermal Stress Profiles in Semi-Transparent Particles Under High-Flux Irradiation, Int. J. Heat Mass Transfer, vol. 50, no. 11-12, pp. 2117–2123, 2007.

Fauchais, P., Understanding Plasma Spraying, J. Phys. D, vol. 37, no. 9, pp. R86–R108, 2004.

Fauchais, P., Montavon, G., Vardelle, M., and Cedelle, J., Developments in Direct Current Plasma Spraying, Surf. Coat. Tech., vol. 201, no. 5, pp. 1908–1921, 2006.

Fincke, J. R., Haggard, D. C., and Swank, W. D., Particle Temperature Measurement in the Thermal Spray Process, J. Thermal Spray Tech., vol. 10, no. 2, pp. 255–266, 2001.

Fletcher, D. F., Steam Explosion Triggering: A Review of Theoretical and Experimental Investigations, Nucl. Eng. Des., vol. 155, no. 1-2, pp. 27–36, 1995.

Liu, L. H., Anisotropic Emission Characteristics of Semitransparent Spherical Particle with Spherically Asymmetric Temperature Distribution, J. Quant. Spectrosc. Radiat. Transfer, vol. 85, no. 2, pp. 135–143, 2004.

Liu, L. H., Tan, H. P., and Tong, T. W., Transient Coupled Radiation-Conduction in Semitransparent Spherical Particle, J. Thermophys. Heat Transfer, vol. 16, no. 1, pp. 43–49, 2002.

Modest, M. F., Radiative Heat Transfer, 2nd ed., Academic Press, New York, 2003.

Moreau, C., Towards a Better Control of Thermal Spray Process, Proc. of 15th Int. Therm. Spray Conf., Nice, France, ASM International, Materials Park, OH, pp. 1681–1693, 1998.

Park, B.-S. and Armstrong, R. L., Laser Droplet Heating: Fast and Slow Heating Regimes, Appl. Opt., vol. 28, no. 17, pp. 3671–3680, 1989.

Pendleton, J. D., Water Droplets Irradiated by a Pulsed CO2 Laser: Comparison of Computed Temperature Contours with Explosive Vaporization Patterns, Appl. Opt., vol. 24, no. 11, 1631–1637, 1985.

Pfender, E. and Chang, C. H., Plasma Spray Jets and Plasma-Particulate Interaction: Modeling and Experiments, Proc. of 15th Int. Therm. Spray Conf., Nice, France, pp. 315–327, 1998.

Qiu, T. Q., Longtin, J. P., and Tien, C. L., Characteristics of Radiation Absorption in Metallic Particles, ASME J. Heat Transfer, vol. 117, no. 2, pp. 340–345, 1995.

Streibl, T., Vaidya, A., Friis, M., Srinivasan, V., and Sampath, S., A Critical Assessment of Particle Temperature Distributions During Plasma Spraying: Experimental Results for YSZ, Plasma Chem. Plasma Process., vol. 26, no. 1, pp. 73–102, 2006.

Theofanous, T. G., The Study of Steam Explosions in Nuclear Systems, Nucl. Eng. Des., vol. 155, no. 1-2, pp. 1–26, 1995.

Tuntomo, A. and Tien, C. L., Transient Heat Transfer in a Conducting Particle with Internal Radiant Absorption, ASME J. Heat Transfer, vol. 114, no. 2, pp. 304–309, 1992.

Viskanta, R., Radiative Transfer in Combustion Systems: Fundamentals and Applications, Begell House, New York and Redding, CT, 2005.

#### 参考文献

1. Astafieva, L. G. and Prishivalko, A. P., Heating of Solid Aerosol Particles Exposed to Intense Optical Radiation, Int. J. Heat Mass Transfer, vol. 41, no. 2, pp. 489â€“499, 1998.
2. Berthoud, G., Vapor Explosions, Ann. Rev. Fluid Mech., vol. 32, pp. 573â€“611, 2000.
3. Dombrovsky, L. A., Radiation Heat Transfer in Disperse Systems, Begell House, New York and Redding, CT, 1996.
4. Dombrovsky, L. A., Thermal Radiation From Nonisothermal Particles of Weakly Absorbing Substance, Proc. 2nd Russ. Nat. Conf. Heat Mass Transfer, Moscow, Oct. 26â€“30, vol. 6, Mosc. Inst. Power Eng. (MIPE) Publ. House, Moscow, pp. 278â€“281, 1998 (in Russian).
5. Dombrovsky, L. A., Thermal Radiation of a Spherical Particle of Semitransparent Material, High Temp., vol. 37, no. 2, pp. 260â€“269, 1999.
6. Dombrovsky, L. A., Thermal Radiation From Nonisothermal Spherical Particles of a Semitransparent Material, Int. J. Heat Mass Transfer, vol. 43, no. 9, pp. 1661â€“1672, 2000a.
7. Dombrovsky, L. A., Approximate Calculation of Thermal Radiation of Nonisothermal Semitransparent Particles, High Temp., vol. 38, no. 4, pp. 663â€“665, 2000b.
8. Dombrovsky, L. A., A Modified Differential Approximation for Thermal Radiation of Semitransparent Nonisothermal Particles: Application to Optical Diagnostics of Plasma Spraying, J. Quant. Spectrosc. Radiat. Transfer, vol. 73, no. 2-5, pp. 433â€“441, 2002.
9. Dombrovsky, L. A., Nonuniform Absorption of Thermal Radiation in Semitransparent Spherical Particles under Conditions of Arbitrary Illumination of a Disperse System, High Temp., vol. 42, no. 6, pp. 975â€“986, 2004a.
10. Dombrovsky, L. A., Absorption of Thermal Radiation in Large Semi-Transparent Particles at Arbitrary Illumination of the Polydisperse System, Int. J. Heat Mass Transfer, vol. 47, no. 25, pp. 5511â€“5522, 2004b.
11. Dombrovsky, L. A., Thermal Radiation of Nonisothermal Particles in Combined Heat Transfer Problems (dedication lecture), Proc. of 5th Int. Symp. Radiat. Transfer, Bodrum, Turkey, June 17â€“22, Begell House, New York and Redding (CT), 2007.
12. Dombrovsky, L. A. and LipiÅ„ski, W., Transient Temperature and Thermal Stress Profiles in Semi-Transparent Particles Under High-Flux Irradiation, Int. J. Heat Mass Transfer, vol. 50, no. 11-12, pp. 2117â€“2123, 2007.
13. Fauchais, P., Understanding Plasma Spraying, J. Phys. D, vol. 37, no. 9, pp. R86â€“R108, 2004.
14. Fauchais, P., Montavon, G., Vardelle, M., and Cedelle, J., Developments in Direct Current Plasma Spraying, Surf. Coat. Tech., vol. 201, no. 5, pp. 1908â€“1921, 2006.
15. Fincke, J. R., Haggard, D. C., and Swank, W. D., Particle Temperature Measurement in the Thermal Spray Process, J. Thermal Spray Tech., vol. 10, no. 2, pp. 255â€“266, 2001.
16. Fletcher, D. F., Steam Explosion Triggering: A Review of Theoretical and Experimental Investigations, Nucl. Eng. Des., vol. 155, no. 1-2, pp. 27â€“36, 1995.
17. Liu, L. H., Anisotropic Emission Characteristics of Semitransparent Spherical Particle with Spherically Asymmetric Temperature Distribution, J. Quant. Spectrosc. Radiat. Transfer, vol. 85, no. 2, pp. 135â€“143, 2004.
18. Liu, L. H., Tan, H. P., and Tong, T. W., Transient Coupled Radiation-Conduction in Semitransparent Spherical Particle, J. Thermophys. Heat Transfer, vol. 16, no. 1, pp. 43â€“49, 2002.
19. Modest, M. F., Radiative Heat Transfer, 2nd ed., Academic Press, New York, 2003.
20. Moreau, C., Towards a Better Control of Thermal Spray Process, Proc. of 15th Int. Therm. Spray Conf., Nice, France, ASM International, Materials Park, OH, pp. 1681â€“1693, 1998.
21. Park, B.-S. and Armstrong, R. L., Laser Droplet Heating: Fast and Slow Heating Regimes, Appl. Opt., vol. 28, no. 17, pp. 3671â€“3680, 1989.
22. Pendleton, J. D., Water Droplets Irradiated by a Pulsed CO2 Laser: Comparison of Computed Temperature Contours with Explosive Vaporization Patterns, Appl. Opt., vol. 24, no. 11, 1631â€“1637, 1985.
23. Pfender, E. and Chang, C. H., Plasma Spray Jets and Plasma-Particulate Interaction: Modeling and Experiments, Proc. of 15th Int. Therm. Spray Conf., Nice, France, pp. 315â€“327, 1998.
24. Qiu, T. Q., Longtin, J. P., and Tien, C. L., Characteristics of Radiation Absorption in Metallic Particles, ASME J. Heat Transfer, vol. 117, no. 2, pp. 340â€“345, 1995.
25. Streibl, T., Vaidya, A., Friis, M., Srinivasan, V., and Sampath, S., A Critical Assessment of Particle Temperature Distributions During Plasma Spraying: Experimental Results for YSZ, Plasma Chem. Plasma Process., vol. 26, no. 1, pp. 73â€“102, 2006.
26. Theofanous, T. G., The Study of Steam Explosions in Nuclear Systems, Nucl. Eng. Des., vol. 155, no. 1-2, pp. 1â€“26, 1995.
27. Tuntomo, A. and Tien, C. L., Transient Heat Transfer in a Conducting Particle with Internal Radiant Absorption, ASME J. Heat Transfer, vol. 114, no. 2, pp. 304â€“309, 1992.
28. Viskanta, R., Radiative Transfer in Combustion Systems: Fundamentals and Applications, Begell House, New York and Redding, CT, 2005.