## Shaped focal plane detectors for particle concentration and mean size observations

Optics Express, Vol. 17, Issue 25, pp. 23066-23077 (2009)

http://dx.doi.org/10.1364/OE.17.023066

Acrobat PDF (266 KB)

### 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

## 1. Introduction

## 2. Fundamentals of laser diffraction

*n*forward angles. This multi-angle scattering data is inverted to construct concentration in

*n*size classes

*C*. This is essentially a problem in algebra, solving for

_{n}*n*unknowns from equally as many equations. The multi-component concentration vector

**C**containing as its elements the

*n*solutions

*C*is called the size distribution. Due to noise in measurements, fewer than

_{n}*n*solutions may be obtained in the inversion step. These issues are well understood [5

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]

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]

*n*-element column vector representing the size distribution. Inversion of this equation produces the desired solution for the size distribution

*n*= 32, which is applicable to the instrument described by Agrawal and Pottsmith [1

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]

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]

## 3. Definitions

*μl/l.*Alternate uses include area concentration, number concentration etc. We also define two diameters which appear later in the text. Variables with subscripts v and A will refer, respectively, to volume and area.

### 3.1 Sauter mean diameter (SMD)

_{v}and area concentration as C

_{A}, the SMD is defined as:

### 3.2 Volume mean diameter (VMD)

*d*(diameters) of a laser diffraction set up be represented by:where

_{i}*i*represents the size class, ranging from 1 to

*n*, and ρ is ratio of size in bin

*i*to

*i*-1. If one defines an integer

*u*asthen,

_{i}*u*takes values 1 to

_{i}*n*+ 1 and represents lower size in a size bin. The center of each size bin is √(

*d*

_{i}_{+1}•

*d*). Hereafter, we refer to this size as the bin size.

_{i}*p*(

*u*) then the mean size bin

**is, by definition:where**

*u***represents the bin number containing the center of moment of the size distribution,**

*u**i.e.*the mean logarithmic diameter.

**need not be an integer.**

*u**d*

_{1}is the

*inner*diameter of the first size bin. In other words, the bin number of the mean size appears in the exponent of the parameter ρ. Of course,

**may take any value within the range covered by the size bins.**

*u***.**

*u*## 4. Origin of shaped focal plane detectors

### 4.1 A Detector for sensing total volume concentration

**T**, such that the scalar product:where γ

_{v}_{v}is a calibration factor. In other words, we are seeking 32 weight factors (elements of the row vector T

_{v}) such that the weighted sum of Eq. (7),

_{v}, where C

_{v}itself is the sum of all components of the size distribution - the vector

_{v}is next represented as a scalar product of a unit vector

*volume*concentration of a particular size class. Equation (10) forms the essence of the current idea. Again,

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]

*d*. Thus, while the methods here are applicable to estimating volume concentration generally, conversion to mass concentration should be done with care when working in the marine environment.

### 4.2 A shaped detector for sensing volume concentration in a size sub-range

*m*size classes. If

*m*of 32 elements, and ones elsewhere, one would construct a weight factor

*m*+ 1 to 32. In this way, a coarse fraction sensor can be designed. One can take this idea further in principle, and seek a set of weight factors that produce concentration in a single size bin, effectively looking for particles only of a particular size. However, this is not practical due to the properties of the kernel matrix, specifically its eigenvectors and condition number. The subject is more complex and mathematical than the scope of the present paper. Suffice it to say that within the limitations of the properties of the kernel matrix, concentration in a specific range of sizes can be extracted by the method of weighted sum in Eq. (10). The vector

### 4.3 A detector for sensing area concentration

^{2}/liter). The solution then is, following Eq. (10):where the corresponding kernel matrix

*i.e.,*each row of the kernel matrix represents light scattering per unit area concentration of particles. Although such weight factors are interesting, one need not use this method. The optical obscuration obtains an excellent estimate of the area concentration but for the case of very fine particles (

*a~λ*

_{o}). Only then, the above weight factors provide an improvement over optical obscuration. This is due to the fact that scattering efficiencies depart from being 2 for such small particles. For further details, see van de Hulst [9].

### 4.4 A detector for sensing volume mean diameter

## 5. Results

### 5.1 Solutions for weight factors T

*i*) direct total volume concentration estimation; (

*ii*) volume estimation in a size sub-range; and (

*iii*) mean size estimation. To solve for the weight factors according to Eqs. (10), (14) and (21), we have employed the Philips-Twomey method, which permits controlled smoothing of the result. Smoothing is desired to keep the resulting shapes from becoming too complex to fabricate, which would defeat some of the purpose of the focal plane detector design. However, smoothing comes at some expense to the accuracy of Eq. (7). An optimal solution is selected in all cases based on desired smoothing and fidelity of achieving a perfect result.

*d*>63 μm (

### 5.2 Shaped detectors

_{v}, and in combination with extinction of the beam for estimating particle area concentration, this produces the SMD. To get the VMD, one needs two comets to implement the scalar products

### 5.3 Numerical simulations and laboratory tests

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]

*factor*of 200 error that would occur over the 200:1 size range with simple turbidity type sensors. The fidelity of the sub-range sensor shows a less than ideal response. Although fines are ignored, as desired, the edge between fines and coarse particles is not sharp. This is the limitation resulting from the properties of kernel matrix and its eigen-vectors. Thus, the sub-range estimator should be considered approximate. To test for the validity of estimating the volume mean diameter, we treat each row of the kernel matrix

*i.e.*ratio of estimated to true concentrations, for random shaped natural particles. These particles spanned the size range from 3.5 microns to 450 microns. All but the 3 points at the smallest diameters are for narrow size distributions, ¼-ϕ wide. The first 3 points are for dusts with relatively wider size distribution, obtained from Particle Technology Inc. These were in size fractions 2-6, 4-8, and 6-11 microns, at the 10 and 90% points of the cumulative distributions. Also shown is the fidelity of concentration estimated from an area-based optical sensor,

*i.e.*from the transmissometer function of the LISST-25 (o). It shows a 2-order magnitude variability in output. Clearly, the

*d*size dependence, as is well known.

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]

## 6. LISST-25X: A marine instrument

## 7. Summary

## Acknowledgements

## References and links

1. | Y. C. Agrawal and H. C. Pottsmith, “Instruments for Particle Size and Settling Velocity Observations in Sediment Transport,” Mar. Geol. |

2. | R. J. Davies-Colley and D. G. Smith, “Turbidity, suspended sediment, and water clarity: a review,” J. Am. Water Resour. Assoc. |

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. |

4. | P. D. Thorne and D. M. Hanes, “A review of acoustic measurements of small-scale sediment processes,” Cont. Shelf Res. |

5. | E. D. Hirleman, “Optimal scaling of the inverse Frannhofer diffraction particle sizing problem: the linear system produced by quadrature,” Particle Characterization |

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. |

7. | H. E. Gerber, “Direct measurement of suspended particulate volume concentration and far-infrared extinction coefficient with a laser-diffraction instrument,” Appl. Opt. |

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 |

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) |

**OCIS Codes**

(040.6070) Detectors : Solid state detectors

(290.0290) Scattering : Scattering

(290.5820) Scattering : Scattering measurements

(280.4788) Remote sensing and sensors : Optical sensing and sensors

**ToC Category:**

Detectors

**History**

Original Manuscript: September 17, 2009

Revised Manuscript: November 23, 2009

Manuscript Accepted: December 1, 2009

Published: December 2, 2009

**Citation**

Y. C. Agrawal and Ole A. Mikkelsen, "Shaped focal plane detectors for particle concentration and mean size observations," Opt. Express **17**, 23066-23077 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-25-23066

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### References

- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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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