Abstract

Retrieving the distribution of droplet size in a polydispersed system from spectral turbidity data is essentially a process of finding a solution to the integral equation involved. Three well-established methods are available: modeling the unknown distribution with an empirical function, an analytical method where the light-scattering kernel is approximated by a simple formula, and the matrix inversion method with smoothing constraint. These methods are reviewed by application to the measured spectral turbidities of polydispersed fogs of water droplets formed by spontaneous condensation in supersonic steam flows. On the basis of the examples considered, which include highly skewed monomodal and bimodal distributions, it was concluded that the matrix inversion method offered the best chance of obtaining solutions in the absence of any a priori information.

© 1980 Optical Society of America

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References

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  1. E. J. Durbin, NACA Tech. Note 2441 (Aug.1951).
  2. S. Twomey, J. Franklin Inst. 279, 95 (1965).
    [CrossRef]
  3. R. A. Dobbins, G. S. Jizmagian, J. Opt. Soc. Am. 56, 1345 (1966).
    [CrossRef]
  4. A. L. Fymat, Appl. Math. Comp. 1, 131 (1975).
    [CrossRef]
  5. R. A. Mugele, H. D. Evans, Ind. Eng. Chem. 43, 1317 (1951).
    [CrossRef]
  6. J. H. Roberts, M. J. Webb, AIAA J. 2, 3583 (1964).
  7. D. L. Phillips, J. Assoc. Comput. Mach. 9, 84 (1962).
    [CrossRef]
  8. S. Twomey, J. Assoc. Comput. Mach. 10, 97 (1963).
    [CrossRef]
  9. K. S. Shifrin, A. Ya. Perelman, in Proceedings, Second Interdisciplinary Conference on Electromagnetic Scattering, U. Massachusetts, June 1965, R. L. Rowell, R. S. Stein, Eds. (Gordon and Breach, New York, 1965).
  10. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1956).
  11. K. S. Shifrin, A. Ya. Perelman, Opt. Spectrosc. 15, 285 (1963).
  12. Ref. 11, p. 362.
  13. Ref. 11, p. 434.
  14. M. J. Moore, P. T. Walters, R. I. Crane, B. J. Davidson, in Conference Proceedings (Institution of Mechanical Engineers, London, 1973), Vol. 3.
  15. P. T. Walters, in Conference Proceedings (Institution of Mechanical Engineers, London, 1973), Vol. 3.
  16. R. B. Penndorf, J. Opt. Soc. Am. 47, 101 (1957).
    [CrossRef]
  17. A correction for dispersion may be incorporated by using phase shift parameter p = 2α(m − 1) as the reference parameter for E (see Ref. 10). This was found to be a very small effect.
  18. R. B. Penndorf, Geophys. Res. Pap. No. 45, Pt. 6 (1956).
  19. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  20. A. Arnulf, J. Bricard, E. Cure, C. Veret, J. Opt. Soc. Am. 47, 491 (1957).
    [CrossRef]
  21. G. Yamamoto, M. Tanaka, Appl. Opt. 8, 447 (1969).
    [CrossRef] [PubMed]

1975 (1)

A. L. Fymat, Appl. Math. Comp. 1, 131 (1975).
[CrossRef]

1969 (1)

1966 (1)

1965 (1)

S. Twomey, J. Franklin Inst. 279, 95 (1965).
[CrossRef]

1964 (1)

J. H. Roberts, M. J. Webb, AIAA J. 2, 3583 (1964).

1963 (2)

S. Twomey, J. Assoc. Comput. Mach. 10, 97 (1963).
[CrossRef]

K. S. Shifrin, A. Ya. Perelman, Opt. Spectrosc. 15, 285 (1963).

1962 (1)

D. L. Phillips, J. Assoc. Comput. Mach. 9, 84 (1962).
[CrossRef]

1957 (2)

1956 (1)

R. B. Penndorf, Geophys. Res. Pap. No. 45, Pt. 6 (1956).

1951 (1)

R. A. Mugele, H. D. Evans, Ind. Eng. Chem. 43, 1317 (1951).
[CrossRef]

Arnulf, A.

Bricard, J.

Crane, R. I.

M. J. Moore, P. T. Walters, R. I. Crane, B. J. Davidson, in Conference Proceedings (Institution of Mechanical Engineers, London, 1973), Vol. 3.

Cure, E.

Davidson, B. J.

M. J. Moore, P. T. Walters, R. I. Crane, B. J. Davidson, in Conference Proceedings (Institution of Mechanical Engineers, London, 1973), Vol. 3.

Dobbins, R. A.

Durbin, E. J.

E. J. Durbin, NACA Tech. Note 2441 (Aug.1951).

Evans, H. D.

R. A. Mugele, H. D. Evans, Ind. Eng. Chem. 43, 1317 (1951).
[CrossRef]

Fymat, A. L.

A. L. Fymat, Appl. Math. Comp. 1, 131 (1975).
[CrossRef]

Jizmagian, G. S.

Kerker, M.

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

Moore, M. J.

M. J. Moore, P. T. Walters, R. I. Crane, B. J. Davidson, in Conference Proceedings (Institution of Mechanical Engineers, London, 1973), Vol. 3.

Mugele, R. A.

R. A. Mugele, H. D. Evans, Ind. Eng. Chem. 43, 1317 (1951).
[CrossRef]

Penndorf, R. B.

R. B. Penndorf, J. Opt. Soc. Am. 47, 101 (1957).
[CrossRef]

R. B. Penndorf, Geophys. Res. Pap. No. 45, Pt. 6 (1956).

Perelman, A. Ya.

K. S. Shifrin, A. Ya. Perelman, Opt. Spectrosc. 15, 285 (1963).

K. S. Shifrin, A. Ya. Perelman, in Proceedings, Second Interdisciplinary Conference on Electromagnetic Scattering, U. Massachusetts, June 1965, R. L. Rowell, R. S. Stein, Eds. (Gordon and Breach, New York, 1965).

Phillips, D. L.

D. L. Phillips, J. Assoc. Comput. Mach. 9, 84 (1962).
[CrossRef]

Roberts, J. H.

J. H. Roberts, M. J. Webb, AIAA J. 2, 3583 (1964).

Shifrin, K. S.

K. S. Shifrin, A. Ya. Perelman, Opt. Spectrosc. 15, 285 (1963).

K. S. Shifrin, A. Ya. Perelman, in Proceedings, Second Interdisciplinary Conference on Electromagnetic Scattering, U. Massachusetts, June 1965, R. L. Rowell, R. S. Stein, Eds. (Gordon and Breach, New York, 1965).

Tanaka, M.

Twomey, S.

S. Twomey, J. Franklin Inst. 279, 95 (1965).
[CrossRef]

S. Twomey, J. Assoc. Comput. Mach. 10, 97 (1963).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1956).

Veret, C.

Walters, P. T.

P. T. Walters, in Conference Proceedings (Institution of Mechanical Engineers, London, 1973), Vol. 3.

M. J. Moore, P. T. Walters, R. I. Crane, B. J. Davidson, in Conference Proceedings (Institution of Mechanical Engineers, London, 1973), Vol. 3.

Webb, M. J.

J. H. Roberts, M. J. Webb, AIAA J. 2, 3583 (1964).

Yamamoto, G.

AIAA J. (1)

J. H. Roberts, M. J. Webb, AIAA J. 2, 3583 (1964).

Appl. Math. Comp. (1)

A. L. Fymat, Appl. Math. Comp. 1, 131 (1975).
[CrossRef]

Appl. Opt. (1)

Geophys. Res. Pap. No. 45 (1)

R. B. Penndorf, Geophys. Res. Pap. No. 45, Pt. 6 (1956).

Ind. Eng. Chem. (1)

R. A. Mugele, H. D. Evans, Ind. Eng. Chem. 43, 1317 (1951).
[CrossRef]

J. Assoc. Comput. Mach. (2)

D. L. Phillips, J. Assoc. Comput. Mach. 9, 84 (1962).
[CrossRef]

S. Twomey, J. Assoc. Comput. Mach. 10, 97 (1963).
[CrossRef]

J. Franklin Inst. (1)

S. Twomey, J. Franklin Inst. 279, 95 (1965).
[CrossRef]

J. Opt. Soc. Am. (3)

Opt. Spectrosc. (1)

K. S. Shifrin, A. Ya. Perelman, Opt. Spectrosc. 15, 285 (1963).

Other (9)

Ref. 11, p. 362.

Ref. 11, p. 434.

M. J. Moore, P. T. Walters, R. I. Crane, B. J. Davidson, in Conference Proceedings (Institution of Mechanical Engineers, London, 1973), Vol. 3.

P. T. Walters, in Conference Proceedings (Institution of Mechanical Engineers, London, 1973), Vol. 3.

A correction for dispersion may be incorporated by using phase shift parameter p = 2α(m − 1) as the reference parameter for E (see Ref. 10). This was found to be a very small effect.

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

K. S. Shifrin, A. Ya. Perelman, in Proceedings, Second Interdisciplinary Conference on Electromagnetic Scattering, U. Massachusetts, June 1965, R. L. Rowell, R. S. Stein, Eds. (Gordon and Breach, New York, 1965).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1956).

E. J. Durbin, NACA Tech. Note 2441 (Aug.1951).

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Figures (8)

Fig. 1
Fig. 1

Spectral turbidity data.

Fig. 2
Fig. 2

E ¯ curves for upper limit functions with negative skews.

Fig. 3
Fig. 3

Fitting E ¯ curves to case 2 data.

Fig. 4
Fig. 4

Smoothing solutions for case 2.

Fig. 5
Fig. 5

Comparison between solution found by trial and error and numerical solution.

Fig. 6
Fig. 6

Inversion of case 2 by the method of Shifrin and Perelman.

Fig. 7
Fig. 7

Numerical inversion of case 3.

Fig. 8
Fig. 8

Inversion of case 3 by the method of Shifrin and Perelman.

Equations (20)

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f / f 0 = exp [ π 4 C n l 0 E ( α ) N r ( D ) D 2 dD ] .
C n D 1 D 2 N r ( D ) dD
g ( ν = π λ ) = 1 l log e f 0 / f = 0 E ( ν , D ) f ( D ) dD ,
E ¯ = 0 D E ( ν , D ) N r ( D ) D 2 dD / 0 D N r ( D ) D 2 dD .
N r ( D ) α exp { δ ln [ aD / ( D D ) ] } 2 D 4 ( D D ) ,
a b K ( y , x ) f ( y ) dy = g ( x ) + e ( x ) ,
A f = g + e ,
i ( f i 1 2 f i + f i + 1 ) 2 + γ 1 i ϵ i 2 ,
f = ( A T A + γ H ) 1 A T g ,
[ 1 2 1 0 0 · · · 2 5 4 1 0 · · · 1 4 6 4 1 0 · · 0 1 4 6 4 1 0 · · · · · · · · · ] .
f = ( A T A + γ I ) 1 ( A T g + γ f t ) ,
E = 2 K ( xa / 2 ) = 1 2 sin ( xa / 2 ) xa + 2 [ 1 cos ( xa ) ] ( xa ) 2
g ( x / 2 ) = 0 K ( xa 2 ) m ( a ) da ,
m ( a ) = 2 a 2 r 0 4 f ( r ) .
m ( a ) = 1 π j = 1 m g ( x j / 2 ) ω ( ax j ) Δ x j + C 0 τ ω 0 ( a τ ) + C 2 ω 2 ( a τ ) τ ,
ω ( y ) = y sin y + cos y 1 ω 0 ( y ) = cos y 2 sin y / y + 1 ω 2 ( y ) = cos y 1 } .
g ( x / 2 ) = C 0 + C 2 / x 2 .
1 n n | E ¯ g |
av | ϵ j | = 1 / n | A f i g j | ( j = 0 , 1 , . . . , n ) ,
C 0 K + C 2 i = 1 k ( 1 / x i ) 2 = i = 1 k g ( x i / 2 ) .

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