Abstract

A detailed study of rainbow thermometry and its application to droplets in reactive systems is presented. To this end the light-scattering history of a vaporizing droplet under unsteady conditions is discussed. Unlike in previous papers, the reduction of the droplet’s diameter is also taken into account in addition to the variation of the refractive-index profile. A finely stratified sphere model with thousands of layers (i.e., 20,000) is used to compute the scattering patterns of a radially inhomogeneous evaporating droplet at different heating times and therefore with different diameters and refractive-index profiles. In the studied case the temperatures inferred from rainbow thermometry do not represent the actual temperatures inside the droplet. They do not represent an average internal temperature or even the surface or the core temperature. For droplets with a temperature that increases from the core to the surface, the inferred values are always lower than the minimum temperature inside the droplet. Therefore the rainbow technique should be applied with caution in all cases in which droplet inhomogeneities are suspected. In addition, a careful analysis of the scattering in the rainbow region is presented. Because of the physical structure of the rainbow, a marked uncertainty in the inferred temperatures also has to be considered in the case of homogenous droplets. For inhomogeneous spheres this intrinsic uncertainty has to be added to the effects caused by the internal profiles of the refractive index.

© 1998 Optical Society of America

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  23. P. Massoli, F. Beretta, A. D’Alessio, “Pyrolysis in the liquid phase inside single droplets of light oil studied with laser light scattering methods,” Combust. Sci. Technol. 72, 271–282 (1990).
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  29. P. L. C. Lage, R. H. Rangel, “Total thermal radiation absorption by a single spherical droplet,” J. Thermophys. Heat Transfer 7, 101–109 (1993).
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  30. P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837–1843 (1962),errata, 134, AB1 (1964).
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  31. Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
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  32. W. A. Sirignano, C. K. Law, “Transient heating and liquid-phase mass diffusion in fuel droplet vaporization,” in Evaporation–Combustion of Fuels, J. T. Zung, ed., Vol. 166 of Advances in Chemistry Series (American Chemical Society, Washington, D.C., 1978), pp. 3–26.
    [CrossRef]
  33. D. J. Coumou, E. L. Mackor, J. Hijmans, “Isotropic light-scattering in pure liquid,” Trans. Faraday Soc. 60, 1539–1553 (1964).
    [CrossRef]
  34. N. Roth, K. Anders, A. Frohn, “Size insensitive rainbow refractometry: theoretical aspects,” presented at the Eighth International Symposium on Application of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, 8–11 July 1996.
  35. Ru T. Wang, H. C. van de Hulst, “Rainbows: Mie computation and the Airy approximation,” Appl. Opt. 30, 106–117 (1990).
    [CrossRef]
  36. V. Khare, H. M. Nussenzweig, “Theory of the rainbows,” Phys. Rev. Lett. 33, 976–980 (1974).
    [CrossRef]
  37. Shermann L. Min, A. Gomez, “High-resolution size measurement of single spherical particles with a fast Fourier transform of the angular scattering intensity,” Appl. Opt. 35, 4919–4926 (1996).
    [CrossRef] [PubMed]
  38. G. Chen, M. M. Mazumder, Y. R. Chemla, A. Serpenguzel, R. K. Chang, “Wavelength variation of laser emission along the entire rim of slightly deformed microdroplets,” Opt. Lett. 18, 1993–1995 (1993).
    [CrossRef] [PubMed]
  39. R. Kneer, M. Schneider, B. Noll, S. Wittig, “Diffusion controlled evaporation of a multicomponent droplet: theoretical studies on the importance of variable liquid properties,” Int. J. Heat Mass Transfer 36, 2403–2415 (1993).
    [CrossRef]
  40. F. Kreith, Principles of Heat Transfer (International Textbook Co., London, 1972).
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    [CrossRef]
  43. H. Niazmand, B. D. Shaw, H. A. Dwyer, I. Aharon, “Effects of Marangoni convection on transient droplet evaporation,” Combust. Sci. Technol. 103, 219–233 (1994).
    [CrossRef]

1996 (2)

K. Anders, N. Roth, A. Frohn, “Influence of refractive index gradients within droplets on rainbow position and implications for rainbow refractometry,” Part. Part. Syst. Charact. 13, 125–129 (1996).
[CrossRef]

Shermann L. Min, A. Gomez, “High-resolution size measurement of single spherical particles with a fast Fourier transform of the angular scattering intensity,” Appl. Opt. 35, 4919–4926 (1996).
[CrossRef] [PubMed]

1994 (4)

H. Niazmand, B. D. Shaw, H. A. Dwyer, I. Aharon, “Effects of Marangoni convection on transient droplet evaporation,” Combust. Sci. Technol. 103, 219–233 (1994).
[CrossRef]

L. Kai, P. Massoli, “Scattering of electromagnetic-plane waves by radially inhomogeneous spheres: a finely stratified sphere model,” Appl. Opt. 33, 501–511 (1994).
[CrossRef] [PubMed]

M. Schneider, E. D. Hirleman, “Influence of internal refractive index gradients on size measurements of spherically symmetric particles by phase Doppler anemometry,” Appl. Opt. 33, 2379–2388 (1994).
[CrossRef] [PubMed]

L. Kai, P. Massoli, A. D’Alessio, “Some far field scattering characteristics of radially inhomogeneous particles,” Part. Part. Syst. Charact. 11, 385–390 (1994).
[CrossRef]

1993 (4)

P. L. C. Lage, R. H. Rangel, “Total thermal radiation absorption by a single spherical droplet,” J. Thermophys. Heat Transfer 7, 101–109 (1993).
[CrossRef]

P. Massoli, F. Beretta, A. D’Alessio, M. Lazzaro, “Temperature and size of single droplets by light scattering in the forward and rainbow position,” Appl. Opt. 32, 3295–3301 (1993).
[CrossRef] [PubMed]

G. Chen, M. M. Mazumder, Y. R. Chemla, A. Serpenguzel, R. K. Chang, “Wavelength variation of laser emission along the entire rim of slightly deformed microdroplets,” Opt. Lett. 18, 1993–1995 (1993).
[CrossRef] [PubMed]

R. Kneer, M. Schneider, B. Noll, S. Wittig, “Diffusion controlled evaporation of a multicomponent droplet: theoretical studies on the importance of variable liquid properties,” Int. J. Heat Mass Transfer 36, 2403–2415 (1993).
[CrossRef]

1992 (1)

1991 (3)

Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

A. Naqwi, F. Durst, X. Liu, “Extended phase–Doppler system for characterization of multiphase flows,” Part. Part. Syst. Charact. 8, 16–22 (1991).
[CrossRef]

J. B. A. Card, A. R. Jones, “Measurements of the refractive index of atomized liquid drops by light scattering methods,” Part. Part. Syst. Charact. 8, 267–273 (1991).
[CrossRef]

1990 (2)

P. Massoli, F. Beretta, A. D’Alessio, “Pyrolysis in the liquid phase inside single droplets of light oil studied with laser light scattering methods,” Combust. Sci. Technol. 72, 271–282 (1990).
[CrossRef]

Ru T. Wang, H. C. van de Hulst, “Rainbows: Mie computation and the Airy approximation,” Appl. Opt. 30, 106–117 (1990).
[CrossRef]

1989 (2)

P. Massoli, F. Beretta, A. D’Alessio, “A new experimental technique for the simultaneous determination of single droplets size, velocity and optical properties inside an evaporating hydrocarbon spray,” Chem. Eng. Commun. 75, 171–180 (1989).
[CrossRef]

P. Massoli, F. Beretta, A. D’Alessio, “Single droplets size, velocity, and optical characteristics by the polarization properties of scattered light,” Appl. Opt. 28, 1200–1205 (1989).
[CrossRef] [PubMed]

1984 (2)

C. F. Hess, “Nonintrusive optical single-particle counter for measuring the size and velocity of droplets in spray,” Appl. Opt. 23, 4375–4382 (1984).
[CrossRef] [PubMed]

W. D. Bachalo, M. J. Houser, “Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions,” Opt. Eng. 23, 583–590 (1984).
[CrossRef]

1983 (1)

W. A. Sirignano, “Fuel droplet vaporization and spray combustion theory,” Prog. Energy Combust. Sci. 9, 291–322 (1983).
[CrossRef]

1982 (1)

C. K. Law, “Recent advances in droplet vaporization and combustion,” Prog. Energy Combust. Sci. 8, 171–201 (1982).
[CrossRef]

1980 (1)

S. Prakash, W. A. Sirignano, “Liquid fuel droplet heating with internal circulation,” Int. J. Heat Mass Transfer 23, 253–268 (1980).
[CrossRef]

1979 (1)

1974 (1)

V. Khare, H. M. Nussenzweig, “Theory of the rainbows,” Phys. Rev. Lett. 33, 976–980 (1974).
[CrossRef]

1972 (1)

1964 (1)

D. J. Coumou, E. L. Mackor, J. Hijmans, “Isotropic light-scattering in pure liquid,” Trans. Faraday Soc. 60, 1539–1553 (1964).
[CrossRef]

1962 (1)

P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837–1843 (1962),errata, 134, AB1 (1964).
[CrossRef]

Aharon, I.

H. Niazmand, B. D. Shaw, H. A. Dwyer, I. Aharon, “Effects of Marangoni convection on transient droplet evaporation,” Combust. Sci. Technol. 103, 219–233 (1994).
[CrossRef]

Anders, K.

K. Anders, N. Roth, A. Frohn, “Influence of refractive index gradients within droplets on rainbow position and implications for rainbow refractometry,” Part. Part. Syst. Charact. 13, 125–129 (1996).
[CrossRef]

N. Roth, K. Anders, A. Frohn, “Simultaneous measurements of temperature and size of droplets in the micrometer range,” in Proceedings of the Seventh International Congress on Optical Methods in Flow and Particle Diagnostics, (Laser Institute of America, Toledo, Ohio, 1988) pp. 294–304.

N. Roth, K. Anders, A. Frohn, “Size insensitive rainbow refractometry: theoretical aspects,” presented at the Eighth International Symposium on Application of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, 8–11 July 1996.

Bachalo, W. D.

W. D. Bachalo, M. J. Houser, “Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions,” Opt. Eng. 23, 583–590 (1984).
[CrossRef]

S. V. Sankar, D. H. Buermann, K. M. Ibrahim, W. D. Bachalo, “Application of an integrated phase Doppler interferometer/rainbow thermometer/point-diffraction interferometer for characterizing burning droplets,” in Twenty-fifth Symposium (International) on Combustion, (The Combustion Institute, Pittsburgh, Pa., 1994), pp. 413–421.
[CrossRef]

S. V. Sankar, K. M. Ibrahim, D. H. Buermann, M. J. Fidrich, W. D. Bachalo, “An integrated phase Doppler interferometer/rainbow system for simultaneous measurements of droplet size, velocity, and refractive index” presented at the Third International Congress on Optical Particle Sizing, Yokohama, Japan, 23–26 August 1993.

Barber, P. W.

Bassler, G. C.

R. M. Silverstein, G. C. Bassler, Spectrometric Identification of Organic Compounds, 2nd ed. (Wiley, New York, 1963).

Beretta, F.

P. Massoli, F. Beretta, A. D’Alessio, M. Lazzaro, “Temperature and size of single droplets by light scattering in the forward and rainbow position,” Appl. Opt. 32, 3295–3301 (1993).
[CrossRef] [PubMed]

P. Massoli, F. Beretta, A. D’Alessio, “Pyrolysis in the liquid phase inside single droplets of light oil studied with laser light scattering methods,” Combust. Sci. Technol. 72, 271–282 (1990).
[CrossRef]

P. Massoli, F. Beretta, A. D’Alessio, “A new experimental technique for the simultaneous determination of single droplets size, velocity and optical properties inside an evaporating hydrocarbon spray,” Chem. Eng. Commun. 75, 171–180 (1989).
[CrossRef]

P. Massoli, F. Beretta, A. D’Alessio, “Single droplets size, velocity, and optical characteristics by the polarization properties of scattered light,” Appl. Opt. 28, 1200–1205 (1989).
[CrossRef] [PubMed]

Buermann, D. H.

S. V. Sankar, D. H. Buermann, K. M. Ibrahim, W. D. Bachalo, “Application of an integrated phase Doppler interferometer/rainbow thermometer/point-diffraction interferometer for characterizing burning droplets,” in Twenty-fifth Symposium (International) on Combustion, (The Combustion Institute, Pittsburgh, Pa., 1994), pp. 413–421.
[CrossRef]

S. V. Sankar, K. M. Ibrahim, D. H. Buermann, M. J. Fidrich, W. D. Bachalo, “An integrated phase Doppler interferometer/rainbow system for simultaneous measurements of droplet size, velocity, and refractive index” presented at the Third International Congress on Optical Particle Sizing, Yokohama, Japan, 23–26 August 1993.

Card, J. B. A.

J. B. A. Card, A. R. Jones, “Measurements of the refractive index of atomized liquid drops by light scattering methods,” Part. Part. Syst. Charact. 8, 267–273 (1991).
[CrossRef]

Chang, R. K.

Chemla, Y. R.

Chen, G.

Chowdhury, D. Q.

D. Q. Chowdhury, P. W. Barber, S. C. Hill, “Energy-density distribution inside large nonabsorbing spheres by using Mie theory and geometrical optics,” Appl. Opt. 31, 3518–3523 (1992).
[CrossRef] [PubMed]

M. Schneider, E. D. Hirleman, H. I. Saleheen, D. Q. Chowdhury, S. C. Hill, “Light scattering by radially inhomogeneous fuel droplets in a high temperature environment,” in Laser Application in Combustion and Combustion Diagnostics, L. C. Liou, ed., Proc. SPIE1862, 269–286 (1993).
[CrossRef]

Cooke, D. D.

Coumou, D. J.

D. J. Coumou, E. L. Mackor, J. Hijmans, “Isotropic light-scattering in pure liquid,” Trans. Faraday Soc. 60, 1539–1553 (1964).
[CrossRef]

Cunningham, A. T. S.

C. J. Lawn, A. T. S. Cunningham, P. J. Street, K. J. Matthews, M. Sarjeant, A. M. Godridge, “The combustion of heavy fuel-oils,” in Principles of Combustion Engineering for Boilers, C. J. Lawn, ed. (Academic, New York, 1987), pp. 61–196.

D’Alessio, A.

L. Kai, P. Massoli, A. D’Alessio, “Some far field scattering characteristics of radially inhomogeneous particles,” Part. Part. Syst. Charact. 11, 385–390 (1994).
[CrossRef]

P. Massoli, F. Beretta, A. D’Alessio, M. Lazzaro, “Temperature and size of single droplets by light scattering in the forward and rainbow position,” Appl. Opt. 32, 3295–3301 (1993).
[CrossRef] [PubMed]

P. Massoli, F. Beretta, A. D’Alessio, “Pyrolysis in the liquid phase inside single droplets of light oil studied with laser light scattering methods,” Combust. Sci. Technol. 72, 271–282 (1990).
[CrossRef]

P. Massoli, F. Beretta, A. D’Alessio, “Single droplets size, velocity, and optical characteristics by the polarization properties of scattered light,” Appl. Opt. 28, 1200–1205 (1989).
[CrossRef] [PubMed]

P. Massoli, F. Beretta, A. D’Alessio, “A new experimental technique for the simultaneous determination of single droplets size, velocity and optical properties inside an evaporating hydrocarbon spray,” Chem. Eng. Commun. 75, 171–180 (1989).
[CrossRef]

D’Alessio, D.

L. Kai, P. Massoli, D. D’Alessio, “Studying inhomogeneities of spherical particles by light scattering,” presented at the Third International Congress on Optical Particle Sizing, Yokohama, Japan, 23–26 August 1993.

Durst, F.

A. Naqwi, F. Durst, X. Liu, “Extended phase–Doppler system for characterization of multiphase flows,” Part. Part. Syst. Charact. 8, 16–22 (1991).
[CrossRef]

F. Durst, M. Zareé, “Laser Doppler measurements in two-phase flows,” presented at the Laser Doppler Anemometry Symposium, Copenhagen, Denmark, 25–28 August 1975.

Dusel, P. W.

Dwyer, H. A.

H. Niazmand, B. D. Shaw, H. A. Dwyer, I. Aharon, “Effects of Marangoni convection on transient droplet evaporation,” Combust. Sci. Technol. 103, 219–233 (1994).
[CrossRef]

Farmer, W. M.

Fidrich, M. J.

S. V. Sankar, K. M. Ibrahim, D. H. Buermann, M. J. Fidrich, W. D. Bachalo, “An integrated phase Doppler interferometer/rainbow system for simultaneous measurements of droplet size, velocity, and refractive index” presented at the Third International Congress on Optical Particle Sizing, Yokohama, Japan, 23–26 August 1993.

Frohn, A.

K. Anders, N. Roth, A. Frohn, “Influence of refractive index gradients within droplets on rainbow position and implications for rainbow refractometry,” Part. Part. Syst. Charact. 13, 125–129 (1996).
[CrossRef]

N. Roth, K. Anders, A. Frohn, “Simultaneous measurements of temperature and size of droplets in the micrometer range,” in Proceedings of the Seventh International Congress on Optical Methods in Flow and Particle Diagnostics, (Laser Institute of America, Toledo, Ohio, 1988) pp. 294–304.

N. Roth, K. Anders, A. Frohn, “Size insensitive rainbow refractometry: theoretical aspects,” presented at the Eighth International Symposium on Application of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, 8–11 July 1996.

Godridge, A. M.

C. J. Lawn, A. T. S. Cunningham, P. J. Street, K. J. Matthews, M. Sarjeant, A. M. Godridge, “The combustion of heavy fuel-oils,” in Principles of Combustion Engineering for Boilers, C. J. Lawn, ed. (Academic, New York, 1987), pp. 61–196.

Gomez, A.

Hess, C. F.

Hijmans, J.

D. J. Coumou, E. L. Mackor, J. Hijmans, “Isotropic light-scattering in pure liquid,” Trans. Faraday Soc. 60, 1539–1553 (1964).
[CrossRef]

Hill, S. C.

D. Q. Chowdhury, P. W. Barber, S. C. Hill, “Energy-density distribution inside large nonabsorbing spheres by using Mie theory and geometrical optics,” Appl. Opt. 31, 3518–3523 (1992).
[CrossRef] [PubMed]

M. Schneider, E. D. Hirleman, H. I. Saleheen, D. Q. Chowdhury, S. C. Hill, “Light scattering by radially inhomogeneous fuel droplets in a high temperature environment,” in Laser Application in Combustion and Combustion Diagnostics, L. C. Liou, ed., Proc. SPIE1862, 269–286 (1993).
[CrossRef]

Hirleman, E. D.

M. Schneider, E. D. Hirleman, “Influence of internal refractive index gradients on size measurements of spherically symmetric particles by phase Doppler anemometry,” Appl. Opt. 33, 2379–2388 (1994).
[CrossRef] [PubMed]

M. Schneider, E. D. Hirleman, H. I. Saleheen, D. Q. Chowdhury, S. C. Hill, “Light scattering by radially inhomogeneous fuel droplets in a high temperature environment,” in Laser Application in Combustion and Combustion Diagnostics, L. C. Liou, ed., Proc. SPIE1862, 269–286 (1993).
[CrossRef]

M. Schneider, E. D. Hirleman, “Rainbows and radially-inhomogeneous droplets,” presented at the Third International Congress on Optical Particle Sizing, Yokohama, Japan, 23–26 August 1993.

Houser, M. J.

W. D. Bachalo, M. J. Houser, “Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions,” Opt. Eng. 23, 583–590 (1984).
[CrossRef]

Ibrahim, K. M.

S. V. Sankar, D. H. Buermann, K. M. Ibrahim, W. D. Bachalo, “Application of an integrated phase Doppler interferometer/rainbow thermometer/point-diffraction interferometer for characterizing burning droplets,” in Twenty-fifth Symposium (International) on Combustion, (The Combustion Institute, Pittsburgh, Pa., 1994), pp. 413–421.
[CrossRef]

S. V. Sankar, K. M. Ibrahim, D. H. Buermann, M. J. Fidrich, W. D. Bachalo, “An integrated phase Doppler interferometer/rainbow system for simultaneous measurements of droplet size, velocity, and refractive index” presented at the Third International Congress on Optical Particle Sizing, Yokohama, Japan, 23–26 August 1993.

Jones, A. R.

J. B. A. Card, A. R. Jones, “Measurements of the refractive index of atomized liquid drops by light scattering methods,” Part. Part. Syst. Charact. 8, 267–273 (1991).
[CrossRef]

Kai, L.

L. Kai, P. Massoli, “Scattering of electromagnetic-plane waves by radially inhomogeneous spheres: a finely stratified sphere model,” Appl. Opt. 33, 501–511 (1994).
[CrossRef] [PubMed]

L. Kai, P. Massoli, A. D’Alessio, “Some far field scattering characteristics of radially inhomogeneous particles,” Part. Part. Syst. Charact. 11, 385–390 (1994).
[CrossRef]

L. Kai, P. Massoli, D. D’Alessio, “Studying inhomogeneities of spherical particles by light scattering,” presented at the Third International Congress on Optical Particle Sizing, Yokohama, Japan, 23–26 August 1993.

Kerker, M.

Khare, V.

V. Khare, H. M. Nussenzweig, “Theory of the rainbows,” Phys. Rev. Lett. 33, 976–980 (1974).
[CrossRef]

Kneer, R.

R. Kneer, M. Schneider, B. Noll, S. Wittig, “Diffusion controlled evaporation of a multicomponent droplet: theoretical studies on the importance of variable liquid properties,” Int. J. Heat Mass Transfer 36, 2403–2415 (1993).
[CrossRef]

Kreith, F.

F. Kreith, Principles of Heat Transfer (International Textbook Co., London, 1972).

Lage, P. L. C.

P. L. C. Lage, R. H. Rangel, “Total thermal radiation absorption by a single spherical droplet,” J. Thermophys. Heat Transfer 7, 101–109 (1993).
[CrossRef]

Law, C. K.

C. K. Law, “Recent advances in droplet vaporization and combustion,” Prog. Energy Combust. Sci. 8, 171–201 (1982).
[CrossRef]

W. A. Sirignano, C. K. Law, “Transient heating and liquid-phase mass diffusion in fuel droplet vaporization,” in Evaporation–Combustion of Fuels, J. T. Zung, ed., Vol. 166 of Advances in Chemistry Series (American Chemical Society, Washington, D.C., 1978), pp. 3–26.
[CrossRef]

Lawn, C. J.

C. J. Lawn, A. T. S. Cunningham, P. J. Street, K. J. Matthews, M. Sarjeant, A. M. Godridge, “The combustion of heavy fuel-oils,” in Principles of Combustion Engineering for Boilers, C. J. Lawn, ed. (Academic, New York, 1987), pp. 61–196.

Lazzaro, M.

Liu, X.

A. Naqwi, F. Durst, X. Liu, “Extended phase–Doppler system for characterization of multiphase flows,” Part. Part. Syst. Charact. 8, 16–22 (1991).
[CrossRef]

Mackor, E. L.

D. J. Coumou, E. L. Mackor, J. Hijmans, “Isotropic light-scattering in pure liquid,” Trans. Faraday Soc. 60, 1539–1553 (1964).
[CrossRef]

Massoli, P.

L. Kai, P. Massoli, “Scattering of electromagnetic-plane waves by radially inhomogeneous spheres: a finely stratified sphere model,” Appl. Opt. 33, 501–511 (1994).
[CrossRef] [PubMed]

L. Kai, P. Massoli, A. D’Alessio, “Some far field scattering characteristics of radially inhomogeneous particles,” Part. Part. Syst. Charact. 11, 385–390 (1994).
[CrossRef]

P. Massoli, F. Beretta, A. D’Alessio, M. Lazzaro, “Temperature and size of single droplets by light scattering in the forward and rainbow position,” Appl. Opt. 32, 3295–3301 (1993).
[CrossRef] [PubMed]

P. Massoli, F. Beretta, A. D’Alessio, “Pyrolysis in the liquid phase inside single droplets of light oil studied with laser light scattering methods,” Combust. Sci. Technol. 72, 271–282 (1990).
[CrossRef]

P. Massoli, F. Beretta, A. D’Alessio, “Single droplets size, velocity, and optical characteristics by the polarization properties of scattered light,” Appl. Opt. 28, 1200–1205 (1989).
[CrossRef] [PubMed]

P. Massoli, F. Beretta, A. D’Alessio, “A new experimental technique for the simultaneous determination of single droplets size, velocity and optical properties inside an evaporating hydrocarbon spray,” Chem. Eng. Commun. 75, 171–180 (1989).
[CrossRef]

L. Kai, P. Massoli, D. D’Alessio, “Studying inhomogeneities of spherical particles by light scattering,” presented at the Third International Congress on Optical Particle Sizing, Yokohama, Japan, 23–26 August 1993.

P. Massoli, “Temperature and size of droplets inferred by light scattering methods: a theoretical analysis of the influence of internal inhomogeneities,” presented at the Thirteenth Annual Conference on Liquid Atomization and Spray Systems ILASS-Europe, Florence, Italy, 9–11 July 1997.

Matthews, K. J.

C. J. Lawn, A. T. S. Cunningham, P. J. Street, K. J. Matthews, M. Sarjeant, A. M. Godridge, “The combustion of heavy fuel-oils,” in Principles of Combustion Engineering for Boilers, C. J. Lawn, ed. (Academic, New York, 1987), pp. 61–196.

Mazumder, M. M.

Min, Shermann L.

Naqwi, A.

A. Naqwi, F. Durst, X. Liu, “Extended phase–Doppler system for characterization of multiphase flows,” Part. Part. Syst. Charact. 8, 16–22 (1991).
[CrossRef]

Niazmand, H.

H. Niazmand, B. D. Shaw, H. A. Dwyer, I. Aharon, “Effects of Marangoni convection on transient droplet evaporation,” Combust. Sci. Technol. 103, 219–233 (1994).
[CrossRef]

Noll, B.

R. Kneer, M. Schneider, B. Noll, S. Wittig, “Diffusion controlled evaporation of a multicomponent droplet: theoretical studies on the importance of variable liquid properties,” Int. J. Heat Mass Transfer 36, 2403–2415 (1993).
[CrossRef]

Nussenzweig, H. M.

V. Khare, H. M. Nussenzweig, “Theory of the rainbows,” Phys. Rev. Lett. 33, 976–980 (1974).
[CrossRef]

Prakash, S.

S. Prakash, W. A. Sirignano, “Liquid fuel droplet heating with internal circulation,” Int. J. Heat Mass Transfer 23, 253–268 (1980).
[CrossRef]

Rangel, R. H.

P. L. C. Lage, R. H. Rangel, “Total thermal radiation absorption by a single spherical droplet,” J. Thermophys. Heat Transfer 7, 101–109 (1993).
[CrossRef]

Rao, C. N. R.

C. N. R. Rao, Ultra-Violet and Visible Spectroscopy (Butterworth, London, 1961).

Roth, N.

K. Anders, N. Roth, A. Frohn, “Influence of refractive index gradients within droplets on rainbow position and implications for rainbow refractometry,” Part. Part. Syst. Charact. 13, 125–129 (1996).
[CrossRef]

N. Roth, K. Anders, A. Frohn, “Simultaneous measurements of temperature and size of droplets in the micrometer range,” in Proceedings of the Seventh International Congress on Optical Methods in Flow and Particle Diagnostics, (Laser Institute of America, Toledo, Ohio, 1988) pp. 294–304.

N. Roth, K. Anders, A. Frohn, “Size insensitive rainbow refractometry: theoretical aspects,” presented at the Eighth International Symposium on Application of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, 8–11 July 1996.

Saleheen, H. I.

M. Schneider, E. D. Hirleman, H. I. Saleheen, D. Q. Chowdhury, S. C. Hill, “Light scattering by radially inhomogeneous fuel droplets in a high temperature environment,” in Laser Application in Combustion and Combustion Diagnostics, L. C. Liou, ed., Proc. SPIE1862, 269–286 (1993).
[CrossRef]

Sankar, S. V.

S. V. Sankar, K. M. Ibrahim, D. H. Buermann, M. J. Fidrich, W. D. Bachalo, “An integrated phase Doppler interferometer/rainbow system for simultaneous measurements of droplet size, velocity, and refractive index” presented at the Third International Congress on Optical Particle Sizing, Yokohama, Japan, 23–26 August 1993.

S. V. Sankar, D. H. Buermann, K. M. Ibrahim, W. D. Bachalo, “Application of an integrated phase Doppler interferometer/rainbow thermometer/point-diffraction interferometer for characterizing burning droplets,” in Twenty-fifth Symposium (International) on Combustion, (The Combustion Institute, Pittsburgh, Pa., 1994), pp. 413–421.
[CrossRef]

Sarjeant, M.

C. J. Lawn, A. T. S. Cunningham, P. J. Street, K. J. Matthews, M. Sarjeant, A. M. Godridge, “The combustion of heavy fuel-oils,” in Principles of Combustion Engineering for Boilers, C. J. Lawn, ed. (Academic, New York, 1987), pp. 61–196.

Schneider, M.

M. Schneider, E. D. Hirleman, “Influence of internal refractive index gradients on size measurements of spherically symmetric particles by phase Doppler anemometry,” Appl. Opt. 33, 2379–2388 (1994).
[CrossRef] [PubMed]

R. Kneer, M. Schneider, B. Noll, S. Wittig, “Diffusion controlled evaporation of a multicomponent droplet: theoretical studies on the importance of variable liquid properties,” Int. J. Heat Mass Transfer 36, 2403–2415 (1993).
[CrossRef]

M. Schneider, E. D. Hirleman, H. I. Saleheen, D. Q. Chowdhury, S. C. Hill, “Light scattering by radially inhomogeneous fuel droplets in a high temperature environment,” in Laser Application in Combustion and Combustion Diagnostics, L. C. Liou, ed., Proc. SPIE1862, 269–286 (1993).
[CrossRef]

M. Schneider, E. D. Hirleman, “Rainbows and radially-inhomogeneous droplets,” presented at the Third International Congress on Optical Particle Sizing, Yokohama, Japan, 23–26 August 1993.

Serpenguzel, A.

Shaw, B. D.

H. Niazmand, B. D. Shaw, H. A. Dwyer, I. Aharon, “Effects of Marangoni convection on transient droplet evaporation,” Combust. Sci. Technol. 103, 219–233 (1994).
[CrossRef]

Silverstein, R. M.

R. M. Silverstein, G. C. Bassler, Spectrometric Identification of Organic Compounds, 2nd ed. (Wiley, New York, 1963).

Sirignano, W. A.

W. A. Sirignano, “Fuel droplet vaporization and spray combustion theory,” Prog. Energy Combust. Sci. 9, 291–322 (1983).
[CrossRef]

S. Prakash, W. A. Sirignano, “Liquid fuel droplet heating with internal circulation,” Int. J. Heat Mass Transfer 23, 253–268 (1980).
[CrossRef]

W. A. Sirignano, C. K. Law, “Transient heating and liquid-phase mass diffusion in fuel droplet vaporization,” in Evaporation–Combustion of Fuels, J. T. Zung, ed., Vol. 166 of Advances in Chemistry Series (American Chemical Society, Washington, D.C., 1978), pp. 3–26.
[CrossRef]

Street, P. J.

C. J. Lawn, A. T. S. Cunningham, P. J. Street, K. J. Matthews, M. Sarjeant, A. M. Godridge, “The combustion of heavy fuel-oils,” in Principles of Combustion Engineering for Boilers, C. J. Lawn, ed. (Academic, New York, 1987), pp. 61–196.

van de Hulst, H. C.

Wang, Ru T.

Wang, Y. P.

Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Wittig, S.

R. Kneer, M. Schneider, B. Noll, S. Wittig, “Diffusion controlled evaporation of a multicomponent droplet: theoretical studies on the importance of variable liquid properties,” Int. J. Heat Mass Transfer 36, 2403–2415 (1993).
[CrossRef]

Wu, Z. S.

Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Wyatt, P. J.

P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837–1843 (1962),errata, 134, AB1 (1964).
[CrossRef]

Zareé, M.

F. Durst, M. Zareé, “Laser Doppler measurements in two-phase flows,” presented at the Laser Doppler Anemometry Symposium, Copenhagen, Denmark, 25–28 August 1975.

Appl. Opt. (9)

W. M. Farmer, “Measurement of particle size, number density, and velocity using a laser interferometer,” Appl. Opt. 11, 2603–2612 (1972).
[CrossRef] [PubMed]

C. F. Hess, “Nonintrusive optical single-particle counter for measuring the size and velocity of droplets in spray,” Appl. Opt. 23, 4375–4382 (1984).
[CrossRef] [PubMed]

P. Massoli, F. Beretta, A. D’Alessio, “Single droplets size, velocity, and optical characteristics by the polarization properties of scattered light,” Appl. Opt. 28, 1200–1205 (1989).
[CrossRef] [PubMed]

P. Massoli, F. Beretta, A. D’Alessio, M. Lazzaro, “Temperature and size of single droplets by light scattering in the forward and rainbow position,” Appl. Opt. 32, 3295–3301 (1993).
[CrossRef] [PubMed]

M. Schneider, E. D. Hirleman, “Influence of internal refractive index gradients on size measurements of spherically symmetric particles by phase Doppler anemometry,” Appl. Opt. 33, 2379–2388 (1994).
[CrossRef] [PubMed]

L. Kai, P. Massoli, “Scattering of electromagnetic-plane waves by radially inhomogeneous spheres: a finely stratified sphere model,” Appl. Opt. 33, 501–511 (1994).
[CrossRef] [PubMed]

D. Q. Chowdhury, P. W. Barber, S. C. Hill, “Energy-density distribution inside large nonabsorbing spheres by using Mie theory and geometrical optics,” Appl. Opt. 31, 3518–3523 (1992).
[CrossRef] [PubMed]

Ru T. Wang, H. C. van de Hulst, “Rainbows: Mie computation and the Airy approximation,” Appl. Opt. 30, 106–117 (1990).
[CrossRef]

Shermann L. Min, A. Gomez, “High-resolution size measurement of single spherical particles with a fast Fourier transform of the angular scattering intensity,” Appl. Opt. 35, 4919–4926 (1996).
[CrossRef] [PubMed]

Chem. Eng. Commun. (1)

P. Massoli, F. Beretta, A. D’Alessio, “A new experimental technique for the simultaneous determination of single droplets size, velocity and optical properties inside an evaporating hydrocarbon spray,” Chem. Eng. Commun. 75, 171–180 (1989).
[CrossRef]

Combust. Sci. Technol. (2)

P. Massoli, F. Beretta, A. D’Alessio, “Pyrolysis in the liquid phase inside single droplets of light oil studied with laser light scattering methods,” Combust. Sci. Technol. 72, 271–282 (1990).
[CrossRef]

H. Niazmand, B. D. Shaw, H. A. Dwyer, I. Aharon, “Effects of Marangoni convection on transient droplet evaporation,” Combust. Sci. Technol. 103, 219–233 (1994).
[CrossRef]

Int. J. Heat Mass Transfer (2)

S. Prakash, W. A. Sirignano, “Liquid fuel droplet heating with internal circulation,” Int. J. Heat Mass Transfer 23, 253–268 (1980).
[CrossRef]

R. Kneer, M. Schneider, B. Noll, S. Wittig, “Diffusion controlled evaporation of a multicomponent droplet: theoretical studies on the importance of variable liquid properties,” Int. J. Heat Mass Transfer 36, 2403–2415 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Thermophys. Heat Transfer (1)

P. L. C. Lage, R. H. Rangel, “Total thermal radiation absorption by a single spherical droplet,” J. Thermophys. Heat Transfer 7, 101–109 (1993).
[CrossRef]

Opt. Eng. (1)

W. D. Bachalo, M. J. Houser, “Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions,” Opt. Eng. 23, 583–590 (1984).
[CrossRef]

Opt. Lett. (1)

Part. Part. Syst. Charact. (4)

A. Naqwi, F. Durst, X. Liu, “Extended phase–Doppler system for characterization of multiphase flows,” Part. Part. Syst. Charact. 8, 16–22 (1991).
[CrossRef]

J. B. A. Card, A. R. Jones, “Measurements of the refractive index of atomized liquid drops by light scattering methods,” Part. Part. Syst. Charact. 8, 267–273 (1991).
[CrossRef]

L. Kai, P. Massoli, A. D’Alessio, “Some far field scattering characteristics of radially inhomogeneous particles,” Part. Part. Syst. Charact. 11, 385–390 (1994).
[CrossRef]

K. Anders, N. Roth, A. Frohn, “Influence of refractive index gradients within droplets on rainbow position and implications for rainbow refractometry,” Part. Part. Syst. Charact. 13, 125–129 (1996).
[CrossRef]

Phys. Rev. (1)

P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837–1843 (1962),errata, 134, AB1 (1964).
[CrossRef]

Phys. Rev. Lett. (1)

V. Khare, H. M. Nussenzweig, “Theory of the rainbows,” Phys. Rev. Lett. 33, 976–980 (1974).
[CrossRef]

Prog. Energy Combust. Sci. (2)

W. A. Sirignano, “Fuel droplet vaporization and spray combustion theory,” Prog. Energy Combust. Sci. 9, 291–322 (1983).
[CrossRef]

C. K. Law, “Recent advances in droplet vaporization and combustion,” Prog. Energy Combust. Sci. 8, 171–201 (1982).
[CrossRef]

Radio Sci. (1)

Z. S. Wu, Y. P. Wang, “Electromagnetic scattering for multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Trans. Faraday Soc. (1)

D. J. Coumou, E. L. Mackor, J. Hijmans, “Isotropic light-scattering in pure liquid,” Trans. Faraday Soc. 60, 1539–1553 (1964).
[CrossRef]

Other (15)

N. Roth, K. Anders, A. Frohn, “Size insensitive rainbow refractometry: theoretical aspects,” presented at the Eighth International Symposium on Application of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, 8–11 July 1996.

W. A. Sirignano, C. K. Law, “Transient heating and liquid-phase mass diffusion in fuel droplet vaporization,” in Evaporation–Combustion of Fuels, J. T. Zung, ed., Vol. 166 of Advances in Chemistry Series (American Chemical Society, Washington, D.C., 1978), pp. 3–26.
[CrossRef]

R. M. Silverstein, G. C. Bassler, Spectrometric Identification of Organic Compounds, 2nd ed. (Wiley, New York, 1963).

C. N. R. Rao, Ultra-Violet and Visible Spectroscopy (Butterworth, London, 1961).

C. J. Lawn, A. T. S. Cunningham, P. J. Street, K. J. Matthews, M. Sarjeant, A. M. Godridge, “The combustion of heavy fuel-oils,” in Principles of Combustion Engineering for Boilers, C. J. Lawn, ed. (Academic, New York, 1987), pp. 61–196.

M. Schneider, E. D. Hirleman, H. I. Saleheen, D. Q. Chowdhury, S. C. Hill, “Light scattering by radially inhomogeneous fuel droplets in a high temperature environment,” in Laser Application in Combustion and Combustion Diagnostics, L. C. Liou, ed., Proc. SPIE1862, 269–286 (1993).
[CrossRef]

L. Kai, P. Massoli, D. D’Alessio, “Studying inhomogeneities of spherical particles by light scattering,” presented at the Third International Congress on Optical Particle Sizing, Yokohama, Japan, 23–26 August 1993.

S. V. Sankar, D. H. Buermann, K. M. Ibrahim, W. D. Bachalo, “Application of an integrated phase Doppler interferometer/rainbow thermometer/point-diffraction interferometer for characterizing burning droplets,” in Twenty-fifth Symposium (International) on Combustion, (The Combustion Institute, Pittsburgh, Pa., 1994), pp. 413–421.
[CrossRef]

S. V. Sankar, K. M. Ibrahim, D. H. Buermann, M. J. Fidrich, W. D. Bachalo, “An integrated phase Doppler interferometer/rainbow system for simultaneous measurements of droplet size, velocity, and refractive index” presented at the Third International Congress on Optical Particle Sizing, Yokohama, Japan, 23–26 August 1993.

M. Schneider, E. D. Hirleman, “Rainbows and radially-inhomogeneous droplets,” presented at the Third International Congress on Optical Particle Sizing, Yokohama, Japan, 23–26 August 1993.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

N. Roth, K. Anders, A. Frohn, “Simultaneous measurements of temperature and size of droplets in the micrometer range,” in Proceedings of the Seventh International Congress on Optical Methods in Flow and Particle Diagnostics, (Laser Institute of America, Toledo, Ohio, 1988) pp. 294–304.

F. Durst, M. Zareé, “Laser Doppler measurements in two-phase flows,” presented at the Laser Doppler Anemometry Symposium, Copenhagen, Denmark, 25–28 August 1975.

F. Kreith, Principles of Heat Transfer (International Textbook Co., London, 1972).

P. Massoli, “Temperature and size of droplets inferred by light scattering methods: a theoretical analysis of the influence of internal inhomogeneities,” presented at the Thirteenth Annual Conference on Liquid Atomization and Spray Systems ILASS-Europe, Florence, Italy, 9–11 July 1997.

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Figures (6)

Fig. 1
Fig. 1

Temperature (top) (from Ref. 32) and refractive-index (bottom) profiles established inside the droplet at different heating times during the vaporization of a single n-octane droplet. For each profile the normalized heating time τ = t/ t 0 and diameter ξ = D/ D 0 are also reported. t 0 and D 0 represent the lifetime and the initial diameter of the droplet, respectively.

Fig. 2
Fig. 2

Angular patterns computed by the FSSM at various heating times for vertically (solid curve) and horizontally (dashed curve) polarized incident light. The radial profiles of the refractive index used in input at the FSSM are shown in Fig. 1 (bottom). The initial droplet diameter was 40 μm.

Fig. 3
Fig. 3

Angular pattern of the C VV cross section in the primary rainbow region (solid curve) and FFT fit (dashed curve) for a 40-μm homogeneous droplet with n = 1.392.

Fig. 4
Fig. 4

Primary rainbow angles that were determined by the application of the FFT procedure to the angular patterns of Fig. 2 and that are therefore relative at the various heating times (triangles). For comparison, the rainbow positions of droplets that have the same refractive-index profiles as in Fig. 2 but a fixed size of 40 μm are also reported (circles).

Fig. 5
Fig. 5

Primary rainbow angles computed for a D 0 = 31.48 μm homogeneous sphere as a function of the refractive index. The rhombi represent the values obtained by the application of the FFT procedure; for comparison, the rainbow angles defined as the relative maximum of the angularly averaged C VV cross section are also reported (solid curve). The dotted and the dashed curves indicate the variation bands of the θ R - n correlation when D 0 is varied within ±5% and ±10%, respectively.

Fig. 6
Fig. 6

Temperature of a 40-μm droplet of n-octane at various heating times, as determined by the rainbow light-scattering technique. At each time the corresponding real temperatures (i.e., the temperature profiles input into the FSSM calculus code) are shown in Fig. 1 (top). The vertical bar represents the uncertainty, on the inferred temperature, that is intrinsic to the rainbow thermometry when a ±5% size indetermination is assumed.

Equations (1)

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n λ T = n λ T 0 + δ n / δ T T - T 0 ,

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