Abstract

We describe a method of designing shaped focal plane detectors for achieving a range of objectives in measurement of particles suspended in a fluid. These detectors can be designed to measure the total concentration in a wide size range (e.g. 200:1) or concentration in a size sub-range (e.g. 63<d<500 μm), and Sauter mean or volume mean diameter. The derivation of these shaped focal plane detectors is rooted in small-angle forward light scattering. The detector shapes are completely general, requiring no assumptions on underlying particle size distribution. We show the theoretical development, numerical simulations and laboratory test results.

© 2009 OSA

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References

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  1. Y. C. Agrawal and H. C. Pottsmith, “Instruments for Particle Size and Settling Velocity Observations in Sediment Transport,” Mar. Geol. 168(1-4), 89–114 (2000).
    [CrossRef]
  2. R. J. Davies-Colley and D. G. Smith, “Turbidity, suspended sediment, and water clarity: a review,” J. Am. Water Resour. Assoc. 37(5), 1085–1101 (2001).
    [CrossRef]
  3. T. F. Sutherland, P. M. Lane, C. L. Amos, and J. Downing, “The calibration of optical backscatter sensors for suspended sediment of varying darkness levels,” Mar. Geol. 162(2-4), 587–597 (2000).
    [CrossRef]
  4. P. D. Thorne and D. M. Hanes, “A review of acoustic measurements of small-scale sediment processes,” Cont. Shelf Res. 22(4), 603 (2002).
    [CrossRef]
  5. E. D. Hirleman, “Optimal scaling of the inverse Frannhofer diffraction particle sizing problem: the linear system produced by quadrature,” Particle Characterization 4(1-4), 128–133 (1987).
    [CrossRef]
  6. Y. C. Agrawal, A. Whitmire, O. A. Mikkelsen, and H. C. Pottsmith, “Light Scattering by Random Shaped Particles and Consequences on Measuring Suspended Sediments by Laser Diffraction,” J. Geophys. Res. 113(C4), C04023 (2008).
    [CrossRef]
  7. H. E. Gerber, “Direct measurement of suspended particulate volume concentration and far-infrared extinction coefficient with a laser-diffraction instrument,” Appl. Opt. 30(33), 4824–4830 (1991).
    [CrossRef] [PubMed]
  8. E. D. Boss, W. Slade, and P. Hill, “Effect of particulate aggregation in aquatic environments on the beam attenuation and its utility as a proxy for particulate mass,” Opt. Express 17(11), 9408–9420 (2009).
    [CrossRef] [PubMed]
  9. H. van de Hulst, Light Scattering by Small Particles. Dover Publications Inc., New York, 470 pp (1981).
  10. Y. C. Agrawal, and H. C. Pottsmith, Laser Sensors for Monitoring Sediments: Capabilities and Limitations, a Survey”, Federal Interagency Sedimentation Project Meeting, Reno, NV.(2001).
  11. D. Topping, S. A. Wright, T. S. Melis, and D. M. Rubin, “High resolution monitoring of suspended sediment concentration and grain size in the Colorado river using laser diffraction instruments and a three-frequency acoustic system, in Proc. of 5th Symposium, Federal Interagency Sedimentation Comm., Reno, NV (2006)

2009 (1)

2008 (1)

Y. C. Agrawal, A. Whitmire, O. A. Mikkelsen, and H. C. Pottsmith, “Light Scattering by Random Shaped Particles and Consequences on Measuring Suspended Sediments by Laser Diffraction,” J. Geophys. Res. 113(C4), C04023 (2008).
[CrossRef]

2002 (1)

P. D. Thorne and D. M. Hanes, “A review of acoustic measurements of small-scale sediment processes,” Cont. Shelf Res. 22(4), 603 (2002).
[CrossRef]

2001 (1)

R. J. Davies-Colley and D. G. Smith, “Turbidity, suspended sediment, and water clarity: a review,” J. Am. Water Resour. Assoc. 37(5), 1085–1101 (2001).
[CrossRef]

2000 (2)

T. F. Sutherland, P. M. Lane, C. L. Amos, and J. Downing, “The calibration of optical backscatter sensors for suspended sediment of varying darkness levels,” Mar. Geol. 162(2-4), 587–597 (2000).
[CrossRef]

Y. C. Agrawal and H. C. Pottsmith, “Instruments for Particle Size and Settling Velocity Observations in Sediment Transport,” Mar. Geol. 168(1-4), 89–114 (2000).
[CrossRef]

1991 (1)

1987 (1)

E. D. Hirleman, “Optimal scaling of the inverse Frannhofer diffraction particle sizing problem: the linear system produced by quadrature,” Particle Characterization 4(1-4), 128–133 (1987).
[CrossRef]

Agrawal, Y. C.

Y. C. Agrawal, A. Whitmire, O. A. Mikkelsen, and H. C. Pottsmith, “Light Scattering by Random Shaped Particles and Consequences on Measuring Suspended Sediments by Laser Diffraction,” J. Geophys. Res. 113(C4), C04023 (2008).
[CrossRef]

Y. C. Agrawal and H. C. Pottsmith, “Instruments for Particle Size and Settling Velocity Observations in Sediment Transport,” Mar. Geol. 168(1-4), 89–114 (2000).
[CrossRef]

Amos, C. L.

T. F. Sutherland, P. M. Lane, C. L. Amos, and J. Downing, “The calibration of optical backscatter sensors for suspended sediment of varying darkness levels,” Mar. Geol. 162(2-4), 587–597 (2000).
[CrossRef]

Boss, E. D.

Davies-Colley, R. J.

R. J. Davies-Colley and D. G. Smith, “Turbidity, suspended sediment, and water clarity: a review,” J. Am. Water Resour. Assoc. 37(5), 1085–1101 (2001).
[CrossRef]

Downing, J.

T. F. Sutherland, P. M. Lane, C. L. Amos, and J. Downing, “The calibration of optical backscatter sensors for suspended sediment of varying darkness levels,” Mar. Geol. 162(2-4), 587–597 (2000).
[CrossRef]

Gerber, H. E.

Hanes, D. M.

P. D. Thorne and D. M. Hanes, “A review of acoustic measurements of small-scale sediment processes,” Cont. Shelf Res. 22(4), 603 (2002).
[CrossRef]

Hill, P.

Hirleman, E. D.

E. D. Hirleman, “Optimal scaling of the inverse Frannhofer diffraction particle sizing problem: the linear system produced by quadrature,” Particle Characterization 4(1-4), 128–133 (1987).
[CrossRef]

Lane, P. M.

T. F. Sutherland, P. M. Lane, C. L. Amos, and J. Downing, “The calibration of optical backscatter sensors for suspended sediment of varying darkness levels,” Mar. Geol. 162(2-4), 587–597 (2000).
[CrossRef]

Mikkelsen, O. A.

Y. C. Agrawal, A. Whitmire, O. A. Mikkelsen, and H. C. Pottsmith, “Light Scattering by Random Shaped Particles and Consequences on Measuring Suspended Sediments by Laser Diffraction,” J. Geophys. Res. 113(C4), C04023 (2008).
[CrossRef]

Pottsmith, H. C.

Y. C. Agrawal, A. Whitmire, O. A. Mikkelsen, and H. C. Pottsmith, “Light Scattering by Random Shaped Particles and Consequences on Measuring Suspended Sediments by Laser Diffraction,” J. Geophys. Res. 113(C4), C04023 (2008).
[CrossRef]

Y. C. Agrawal and H. C. Pottsmith, “Instruments for Particle Size and Settling Velocity Observations in Sediment Transport,” Mar. Geol. 168(1-4), 89–114 (2000).
[CrossRef]

Slade, W.

Smith, D. G.

R. J. Davies-Colley and D. G. Smith, “Turbidity, suspended sediment, and water clarity: a review,” J. Am. Water Resour. Assoc. 37(5), 1085–1101 (2001).
[CrossRef]

Sutherland, T. F.

T. F. Sutherland, P. M. Lane, C. L. Amos, and J. Downing, “The calibration of optical backscatter sensors for suspended sediment of varying darkness levels,” Mar. Geol. 162(2-4), 587–597 (2000).
[CrossRef]

Thorne, P. D.

P. D. Thorne and D. M. Hanes, “A review of acoustic measurements of small-scale sediment processes,” Cont. Shelf Res. 22(4), 603 (2002).
[CrossRef]

Whitmire, A.

Y. C. Agrawal, A. Whitmire, O. A. Mikkelsen, and H. C. Pottsmith, “Light Scattering by Random Shaped Particles and Consequences on Measuring Suspended Sediments by Laser Diffraction,” J. Geophys. Res. 113(C4), C04023 (2008).
[CrossRef]

Appl. Opt. (1)

Cont. Shelf Res. (1)

P. D. Thorne and D. M. Hanes, “A review of acoustic measurements of small-scale sediment processes,” Cont. Shelf Res. 22(4), 603 (2002).
[CrossRef]

J. Am. Water Resour. Assoc. (1)

R. J. Davies-Colley and D. G. Smith, “Turbidity, suspended sediment, and water clarity: a review,” J. Am. Water Resour. Assoc. 37(5), 1085–1101 (2001).
[CrossRef]

J. Geophys. Res. (1)

Y. C. Agrawal, A. Whitmire, O. A. Mikkelsen, and H. C. Pottsmith, “Light Scattering by Random Shaped Particles and Consequences on Measuring Suspended Sediments by Laser Diffraction,” J. Geophys. Res. 113(C4), C04023 (2008).
[CrossRef]

Mar. Geol. (2)

T. F. Sutherland, P. M. Lane, C. L. Amos, and J. Downing, “The calibration of optical backscatter sensors for suspended sediment of varying darkness levels,” Mar. Geol. 162(2-4), 587–597 (2000).
[CrossRef]

Y. C. Agrawal and H. C. Pottsmith, “Instruments for Particle Size and Settling Velocity Observations in Sediment Transport,” Mar. Geol. 168(1-4), 89–114 (2000).
[CrossRef]

Opt. Express (1)

Particle Characterization (1)

E. D. Hirleman, “Optimal scaling of the inverse Frannhofer diffraction particle sizing problem: the linear system produced by quadrature,” Particle Characterization 4(1-4), 128–133 (1987).
[CrossRef]

Other (3)

H. van de Hulst, Light Scattering by Small Particles. Dover Publications Inc., New York, 470 pp (1981).

Y. C. Agrawal, and H. C. Pottsmith, Laser Sensors for Monitoring Sediments: Capabilities and Limitations, a Survey”, Federal Interagency Sedimentation Project Meeting, Reno, NV.(2001).

D. Topping, S. A. Wright, T. S. Melis, and D. M. Rubin, “High resolution monitoring of suspended sediment concentration and grain size in the Colorado river using laser diffraction instruments and a three-frequency acoustic system, in Proc. of 5th Symposium, Federal Interagency Sedimentation Comm., Reno, NV (2006)

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Figures (7)

Fig. 1
Fig. 1

Schematic of a laser diffraction set up, showing left to right, a collimated beam, a receiving lens, and a set of ring detectors in the focal plane of a receiving lens.

Fig. 2
Fig. 2

Normalized weight factors for T D , T v , and T v2 . These magnitudes scale with the magnitude of corresponding matrix K .

Fig. 3
Fig. 3

The formation of the comet shape by modulating azimuth or arc-width of each detector in proportion to the weight factors. The inner-most rings cover a 120-degree arc-width. The detectors with positive weights are connected together, and likewise, negatives are also connected together. The darker parts are the active areas. This is the conceptual view. In reality, all positive elements would be contiguous, and negative elements would be similarly contiguous, thus forming comet-shaped detectors. The thin comet on right measures volume concentration, while the fat one on left finds the constant X.

Fig. 4
Fig. 4

Numerical simulations: Fidelity testing of weight functions for estimating mean diameter, total, and sub-range concentrations. (The ordinate is fidelity).

Fig. 5
Fig. 5

Fidelity of measurement of concentration of random shaped natural grains. The weighted sum (*) shows a near constant value. In contrast, an optical transmission-based estimate (o) shows essentially 1/d dependence, producing a 2-order of magnitude variation in calibration.

Fig. 6
Fig. 6

Fidelity of estimation of mean diameter (*). The straight line is 1:1.

Fig. 7
Fig. 7

Effectiveness of the coarse-particle sensor weight factors T v2. This relative sensitivity to particles of different sizes closely mirrors the theoretical response of Fig. 4.

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

E ¯ = K ¯ ¯ C ¯
d S M D = 1.5 C v / C A
d i = d 1 ρ i 1
u i = log ( d i / d 1 ) / log ( ρ )
u = i u i p ( u i ) / p ( u i )
V M D = d 1 ρ u 1
T ¯ v E = γ v C v
U ¯ C ¯ v = i C ¯ v i             = C v
T ¯ v ( K ¯ ¯ v C ¯ v ) = γ v U ¯ C ¯ v
T ¯ v = γ v U ¯ K ¯ ¯ v - 1
a m i n = 2 / k θ m a x ;
a m a x = 2 / k θ m i n ;
θ m i n , m a x = arctan ( r m i n , m a x / f )
T ¯ v 2 = γ v U ¯ 2 K ¯ ¯ v - 1 , where U ¯ 2             = [ 0 0 0 m times .... 1 1 1 ... 32 - m times . ] ;
T ¯ A = γ A K ¯ ¯ A - 1 U ¯
T ¯ D E ¯ = γ R ¯ C ¯ v
R ¯ = [ 1 : 32 ]
R m = R ¯ C ¯ v / U ¯ C ¯ v
X ¯ = 6 . 6 - 0 . 2 R ¯
X ¯ = [ 6 . 4 : - 0 . 2 : 0 . 2 ]
T ¯ D = K ¯ ¯ v - 1 X ¯
X m = T ¯ D E ¯ / T ¯ v E ¯
R m = 5 ( 6 . 6 - X m )
d m = d 1 ρ [ R m - 1 ]

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