Abstract

Turbidity measurement is important for water quality assessment, food safety, medicine, ocean monitoring, etc. In this paper, a method that accurately estimates the turbidity over a wide range is proposed, where the turbidity of the sample is represented as a weighted ratio of the scattered light intensities at a series of angles. An improvement in the accuracy is achieved by expanding the structure of the ratio function, thus adding more flexibility to the turbidity–intensity fitting. Experiments have been carried out with an 850 nm laser and a power meter fixed on a turntable to measure the light intensity at different angles. The results show that the relative estimation error of the proposed method is 0.58% on average for a four-angle intensity combination for all test samples with a turbidity ranging from 160 NTU to 4000 NTU.

© 2015 Optical Society of America

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References

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  1. ISO, “7027: Water quality–Determination of turbidity,” (1999).
  2. J. R. Gray, G. D. Glysson, J. H. Eychaner, and C. W. Anderson, “Introduction to the proceedings of the federal interagency workshop on turbidity and other sediment surrogates,” in Proceeding of Federal Interagency Workshop on Turbidity and Other Sediment Surrogates, J. R. Gray and G. D. Glysson, eds. (U.S. Geological Survey, 2003), pp. 5–8.
  3. M. A. Uhrich and H. M. Bragg, Monitoring instream turbidity to estimate continuous suspended–sediment loads and yields and clay–water volumes in the upper North Santiam River Basin, Oregon, 1998–2000 (US Department of the Interior, US Geological Survey, 2003).
  4. V. G. Christensen, X. Jian, and A. C. Ziegler, Regression analysis and real–time water-quality monitoring to estimate constituent concentrations, loads, and yields in the Little Arkansas River, south-central Kansas, 1995–99 (US Department of the Interior, US Geological Survey, 2000).
  5. Whipple and Jackson, “A comparative study of the methods used for the measurement of turbidity of water,” MIT Quarterly 13, 274 (1900).
  6. M. J. Sadar, Turbidity Science. Technical Information Series–Booklet No. 11 (Hach Company, 1998).
  7. W. R. McCluney, “Radiometry of water turbidity measurements,” J. Water Pollution Control Federation 47, 252–266 (1975).
  8. Hach Company, 2100 Series Laboratory Turbidimeters Data Sheet LIT2498 Rev 4 (2013).
  9. D. R. Dana and R. A. Maffione, “Determining the backward scattering coefficient with fixed–angle backscattering sensors–Revisited,” in Ocean Optics XVI (2002) pp. 18–22.
  10. S. G. Daraigan, “The development of multispectral algorithms and sensors setup for total suspended solids measurement,” Phd thesis (University Science Malaysia, 2006).
  11. A. F. Omar and M. Z. MatJafri, “Water quality measurement using transmittance and 90° scattering techniques through optical fiber sensor,” in Proceeding of IEEE 2008 6th National Conference on Telecommunication Technologies and IEEE 2008 2nd Malaysia Conference on Photonics (IEEE, 2008), pp. 17–21.
  12. D. Barter and J. Paul, “Comparison of portable nephelometric turbidimeters on natural waters and effluents,” New Zealand J. Marine Freshwater Res. 37, 485–492 (2003).
    [Crossref]
  13. R. J. Cooper, K. A. Keilbach, R. P. Kolman, E. R. Paoli, R. D. Stream, and K. L. Stutzman, “Nephelometer instrument,” United States Patent5,506,679 (April9, 1996).
  14. J. T. O. Kirk, Light and Photosynthesis in Aquatic Ecosystems, 3rd ed. (Cambridge University, 2011).
  15. C. Mobley and E. Boss, Ocean Optics Web Book, http://www.oceanopticsbook.info/.
  16. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, 1994).
  17. M. E. Lee and M. R. Lewis, “A new method for the measurement of the optical volume scattering function in the upper ocean,” J. Atmospheric Oceanic Technol. 20, 563–571 (2003).
    [Crossref]
  18. C. J. McCrowey, S. S. Tinilau, G. Calderon, J. E. Koo, and D. B. Curtis, “A portable high–resolution polar nephelometer for measurement of the angular scattering properties of atmospheric aerosol: design and validation,” Aerosol Sci. Technol. 47, 592–605 (2013).
    [Crossref]
  19. J. F. Gayet, J. F. Fournol, and S. Oshchepkov, “A new airborne polar nephelometer for the measurements of optical and microphysical cloud properties. part I: Theoretical design,” Annales Geophysicae 15, 451–459 (1997).
    [Crossref]
  20. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles, 1st ed. (Cambridge University, 2002).
  21. T. J. Petzold, “Volume scattering functions for selected ocean waters,” Tech. Rep. (Scripps Institution of Oceanography, 1972).
  22. M. Sadar, “Turbidity instrumentation–An overview of today’s available technology,” in Proceeding of Federal Interagency Workshop on Turbidity and Other Sediment Surrogates, J. R. Gray and G. D. Glysson, eds. (U.S. Geological Survey, 2003).

2013 (1)

C. J. McCrowey, S. S. Tinilau, G. Calderon, J. E. Koo, and D. B. Curtis, “A portable high–resolution polar nephelometer for measurement of the angular scattering properties of atmospheric aerosol: design and validation,” Aerosol Sci. Technol. 47, 592–605 (2013).
[Crossref]

2003 (2)

D. Barter and J. Paul, “Comparison of portable nephelometric turbidimeters on natural waters and effluents,” New Zealand J. Marine Freshwater Res. 37, 485–492 (2003).
[Crossref]

M. E. Lee and M. R. Lewis, “A new method for the measurement of the optical volume scattering function in the upper ocean,” J. Atmospheric Oceanic Technol. 20, 563–571 (2003).
[Crossref]

1997 (1)

J. F. Gayet, J. F. Fournol, and S. Oshchepkov, “A new airborne polar nephelometer for the measurements of optical and microphysical cloud properties. part I: Theoretical design,” Annales Geophysicae 15, 451–459 (1997).
[Crossref]

1975 (1)

W. R. McCluney, “Radiometry of water turbidity measurements,” J. Water Pollution Control Federation 47, 252–266 (1975).

1900 (1)

Whipple and Jackson, “A comparative study of the methods used for the measurement of turbidity of water,” MIT Quarterly 13, 274 (1900).

Anderson, C. W.

J. R. Gray, G. D. Glysson, J. H. Eychaner, and C. W. Anderson, “Introduction to the proceedings of the federal interagency workshop on turbidity and other sediment surrogates,” in Proceeding of Federal Interagency Workshop on Turbidity and Other Sediment Surrogates, J. R. Gray and G. D. Glysson, eds. (U.S. Geological Survey, 2003), pp. 5–8.

Barter, D.

D. Barter and J. Paul, “Comparison of portable nephelometric turbidimeters on natural waters and effluents,” New Zealand J. Marine Freshwater Res. 37, 485–492 (2003).
[Crossref]

Boss, E.

C. Mobley and E. Boss, Ocean Optics Web Book, http://www.oceanopticsbook.info/.

Bragg, H. M.

M. A. Uhrich and H. M. Bragg, Monitoring instream turbidity to estimate continuous suspended–sediment loads and yields and clay–water volumes in the upper North Santiam River Basin, Oregon, 1998–2000 (US Department of the Interior, US Geological Survey, 2003).

Calderon, G.

C. J. McCrowey, S. S. Tinilau, G. Calderon, J. E. Koo, and D. B. Curtis, “A portable high–resolution polar nephelometer for measurement of the angular scattering properties of atmospheric aerosol: design and validation,” Aerosol Sci. Technol. 47, 592–605 (2013).
[Crossref]

Christensen, V. G.

V. G. Christensen, X. Jian, and A. C. Ziegler, Regression analysis and real–time water-quality monitoring to estimate constituent concentrations, loads, and yields in the Little Arkansas River, south-central Kansas, 1995–99 (US Department of the Interior, US Geological Survey, 2000).

Cooper, R. J.

R. J. Cooper, K. A. Keilbach, R. P. Kolman, E. R. Paoli, R. D. Stream, and K. L. Stutzman, “Nephelometer instrument,” United States Patent5,506,679 (April9, 1996).

Curtis, D. B.

C. J. McCrowey, S. S. Tinilau, G. Calderon, J. E. Koo, and D. B. Curtis, “A portable high–resolution polar nephelometer for measurement of the angular scattering properties of atmospheric aerosol: design and validation,” Aerosol Sci. Technol. 47, 592–605 (2013).
[Crossref]

Dana, D. R.

D. R. Dana and R. A. Maffione, “Determining the backward scattering coefficient with fixed–angle backscattering sensors–Revisited,” in Ocean Optics XVI (2002) pp. 18–22.

Daraigan, S. G.

S. G. Daraigan, “The development of multispectral algorithms and sensors setup for total suspended solids measurement,” Phd thesis (University Science Malaysia, 2006).

Eychaner, J. H.

J. R. Gray, G. D. Glysson, J. H. Eychaner, and C. W. Anderson, “Introduction to the proceedings of the federal interagency workshop on turbidity and other sediment surrogates,” in Proceeding of Federal Interagency Workshop on Turbidity and Other Sediment Surrogates, J. R. Gray and G. D. Glysson, eds. (U.S. Geological Survey, 2003), pp. 5–8.

Fournol, J. F.

J. F. Gayet, J. F. Fournol, and S. Oshchepkov, “A new airborne polar nephelometer for the measurements of optical and microphysical cloud properties. part I: Theoretical design,” Annales Geophysicae 15, 451–459 (1997).
[Crossref]

Gayet, J. F.

J. F. Gayet, J. F. Fournol, and S. Oshchepkov, “A new airborne polar nephelometer for the measurements of optical and microphysical cloud properties. part I: Theoretical design,” Annales Geophysicae 15, 451–459 (1997).
[Crossref]

Glysson, G. D.

J. R. Gray, G. D. Glysson, J. H. Eychaner, and C. W. Anderson, “Introduction to the proceedings of the federal interagency workshop on turbidity and other sediment surrogates,” in Proceeding of Federal Interagency Workshop on Turbidity and Other Sediment Surrogates, J. R. Gray and G. D. Glysson, eds. (U.S. Geological Survey, 2003), pp. 5–8.

Gray, J. R.

J. R. Gray, G. D. Glysson, J. H. Eychaner, and C. W. Anderson, “Introduction to the proceedings of the federal interagency workshop on turbidity and other sediment surrogates,” in Proceeding of Federal Interagency Workshop on Turbidity and Other Sediment Surrogates, J. R. Gray and G. D. Glysson, eds. (U.S. Geological Survey, 2003), pp. 5–8.

Jackson,

Whipple and Jackson, “A comparative study of the methods used for the measurement of turbidity of water,” MIT Quarterly 13, 274 (1900).

Jian, X.

V. G. Christensen, X. Jian, and A. C. Ziegler, Regression analysis and real–time water-quality monitoring to estimate constituent concentrations, loads, and yields in the Little Arkansas River, south-central Kansas, 1995–99 (US Department of the Interior, US Geological Survey, 2000).

Keilbach, K. A.

R. J. Cooper, K. A. Keilbach, R. P. Kolman, E. R. Paoli, R. D. Stream, and K. L. Stutzman, “Nephelometer instrument,” United States Patent5,506,679 (April9, 1996).

Kirk, J. T. O.

J. T. O. Kirk, Light and Photosynthesis in Aquatic Ecosystems, 3rd ed. (Cambridge University, 2011).

Kolman, R. P.

R. J. Cooper, K. A. Keilbach, R. P. Kolman, E. R. Paoli, R. D. Stream, and K. L. Stutzman, “Nephelometer instrument,” United States Patent5,506,679 (April9, 1996).

Koo, J. E.

C. J. McCrowey, S. S. Tinilau, G. Calderon, J. E. Koo, and D. B. Curtis, “A portable high–resolution polar nephelometer for measurement of the angular scattering properties of atmospheric aerosol: design and validation,” Aerosol Sci. Technol. 47, 592–605 (2013).
[Crossref]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles, 1st ed. (Cambridge University, 2002).

Lee, M. E.

M. E. Lee and M. R. Lewis, “A new method for the measurement of the optical volume scattering function in the upper ocean,” J. Atmospheric Oceanic Technol. 20, 563–571 (2003).
[Crossref]

Lewis, M. R.

M. E. Lee and M. R. Lewis, “A new method for the measurement of the optical volume scattering function in the upper ocean,” J. Atmospheric Oceanic Technol. 20, 563–571 (2003).
[Crossref]

Maffione, R. A.

D. R. Dana and R. A. Maffione, “Determining the backward scattering coefficient with fixed–angle backscattering sensors–Revisited,” in Ocean Optics XVI (2002) pp. 18–22.

MatJafri, M. Z.

A. F. Omar and M. Z. MatJafri, “Water quality measurement using transmittance and 90° scattering techniques through optical fiber sensor,” in Proceeding of IEEE 2008 6th National Conference on Telecommunication Technologies and IEEE 2008 2nd Malaysia Conference on Photonics (IEEE, 2008), pp. 17–21.

McCluney, W. R.

W. R. McCluney, “Radiometry of water turbidity measurements,” J. Water Pollution Control Federation 47, 252–266 (1975).

McCrowey, C. J.

C. J. McCrowey, S. S. Tinilau, G. Calderon, J. E. Koo, and D. B. Curtis, “A portable high–resolution polar nephelometer for measurement of the angular scattering properties of atmospheric aerosol: design and validation,” Aerosol Sci. Technol. 47, 592–605 (2013).
[Crossref]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles, 1st ed. (Cambridge University, 2002).

Mobley, C.

C. Mobley and E. Boss, Ocean Optics Web Book, http://www.oceanopticsbook.info/.

Mobley, C. D.

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, 1994).

Omar, A. F.

A. F. Omar and M. Z. MatJafri, “Water quality measurement using transmittance and 90° scattering techniques through optical fiber sensor,” in Proceeding of IEEE 2008 6th National Conference on Telecommunication Technologies and IEEE 2008 2nd Malaysia Conference on Photonics (IEEE, 2008), pp. 17–21.

Oshchepkov, S.

J. F. Gayet, J. F. Fournol, and S. Oshchepkov, “A new airborne polar nephelometer for the measurements of optical and microphysical cloud properties. part I: Theoretical design,” Annales Geophysicae 15, 451–459 (1997).
[Crossref]

Paoli, E. R.

R. J. Cooper, K. A. Keilbach, R. P. Kolman, E. R. Paoli, R. D. Stream, and K. L. Stutzman, “Nephelometer instrument,” United States Patent5,506,679 (April9, 1996).

Paul, J.

D. Barter and J. Paul, “Comparison of portable nephelometric turbidimeters on natural waters and effluents,” New Zealand J. Marine Freshwater Res. 37, 485–492 (2003).
[Crossref]

Petzold, T. J.

T. J. Petzold, “Volume scattering functions for selected ocean waters,” Tech. Rep. (Scripps Institution of Oceanography, 1972).

Sadar, M.

M. Sadar, “Turbidity instrumentation–An overview of today’s available technology,” in Proceeding of Federal Interagency Workshop on Turbidity and Other Sediment Surrogates, J. R. Gray and G. D. Glysson, eds. (U.S. Geological Survey, 2003).

Sadar, M. J.

M. J. Sadar, Turbidity Science. Technical Information Series–Booklet No. 11 (Hach Company, 1998).

Stream, R. D.

R. J. Cooper, K. A. Keilbach, R. P. Kolman, E. R. Paoli, R. D. Stream, and K. L. Stutzman, “Nephelometer instrument,” United States Patent5,506,679 (April9, 1996).

Stutzman, K. L.

R. J. Cooper, K. A. Keilbach, R. P. Kolman, E. R. Paoli, R. D. Stream, and K. L. Stutzman, “Nephelometer instrument,” United States Patent5,506,679 (April9, 1996).

Tinilau, S. S.

C. J. McCrowey, S. S. Tinilau, G. Calderon, J. E. Koo, and D. B. Curtis, “A portable high–resolution polar nephelometer for measurement of the angular scattering properties of atmospheric aerosol: design and validation,” Aerosol Sci. Technol. 47, 592–605 (2013).
[Crossref]

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles, 1st ed. (Cambridge University, 2002).

Uhrich, M. A.

M. A. Uhrich and H. M. Bragg, Monitoring instream turbidity to estimate continuous suspended–sediment loads and yields and clay–water volumes in the upper North Santiam River Basin, Oregon, 1998–2000 (US Department of the Interior, US Geological Survey, 2003).

Whipple,

Whipple and Jackson, “A comparative study of the methods used for the measurement of turbidity of water,” MIT Quarterly 13, 274 (1900).

Ziegler, A. C.

V. G. Christensen, X. Jian, and A. C. Ziegler, Regression analysis and real–time water-quality monitoring to estimate constituent concentrations, loads, and yields in the Little Arkansas River, south-central Kansas, 1995–99 (US Department of the Interior, US Geological Survey, 2000).

Aerosol Sci. Technol. (1)

C. J. McCrowey, S. S. Tinilau, G. Calderon, J. E. Koo, and D. B. Curtis, “A portable high–resolution polar nephelometer for measurement of the angular scattering properties of atmospheric aerosol: design and validation,” Aerosol Sci. Technol. 47, 592–605 (2013).
[Crossref]

Annales Geophysicae (1)

J. F. Gayet, J. F. Fournol, and S. Oshchepkov, “A new airborne polar nephelometer for the measurements of optical and microphysical cloud properties. part I: Theoretical design,” Annales Geophysicae 15, 451–459 (1997).
[Crossref]

J. Atmospheric Oceanic Technol. (1)

M. E. Lee and M. R. Lewis, “A new method for the measurement of the optical volume scattering function in the upper ocean,” J. Atmospheric Oceanic Technol. 20, 563–571 (2003).
[Crossref]

J. Water Pollution Control Federation (1)

W. R. McCluney, “Radiometry of water turbidity measurements,” J. Water Pollution Control Federation 47, 252–266 (1975).

MIT Quarterly (1)

Whipple and Jackson, “A comparative study of the methods used for the measurement of turbidity of water,” MIT Quarterly 13, 274 (1900).

New Zealand J. Marine Freshwater Res. (1)

D. Barter and J. Paul, “Comparison of portable nephelometric turbidimeters on natural waters and effluents,” New Zealand J. Marine Freshwater Res. 37, 485–492 (2003).
[Crossref]

Other (16)

R. J. Cooper, K. A. Keilbach, R. P. Kolman, E. R. Paoli, R. D. Stream, and K. L. Stutzman, “Nephelometer instrument,” United States Patent5,506,679 (April9, 1996).

J. T. O. Kirk, Light and Photosynthesis in Aquatic Ecosystems, 3rd ed. (Cambridge University, 2011).

C. Mobley and E. Boss, Ocean Optics Web Book, http://www.oceanopticsbook.info/.

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, 1994).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles, 1st ed. (Cambridge University, 2002).

T. J. Petzold, “Volume scattering functions for selected ocean waters,” Tech. Rep. (Scripps Institution of Oceanography, 1972).

M. Sadar, “Turbidity instrumentation–An overview of today’s available technology,” in Proceeding of Federal Interagency Workshop on Turbidity and Other Sediment Surrogates, J. R. Gray and G. D. Glysson, eds. (U.S. Geological Survey, 2003).

Hach Company, 2100 Series Laboratory Turbidimeters Data Sheet LIT2498 Rev 4 (2013).

D. R. Dana and R. A. Maffione, “Determining the backward scattering coefficient with fixed–angle backscattering sensors–Revisited,” in Ocean Optics XVI (2002) pp. 18–22.

S. G. Daraigan, “The development of multispectral algorithms and sensors setup for total suspended solids measurement,” Phd thesis (University Science Malaysia, 2006).

A. F. Omar and M. Z. MatJafri, “Water quality measurement using transmittance and 90° scattering techniques through optical fiber sensor,” in Proceeding of IEEE 2008 6th National Conference on Telecommunication Technologies and IEEE 2008 2nd Malaysia Conference on Photonics (IEEE, 2008), pp. 17–21.

ISO, “7027: Water quality–Determination of turbidity,” (1999).

J. R. Gray, G. D. Glysson, J. H. Eychaner, and C. W. Anderson, “Introduction to the proceedings of the federal interagency workshop on turbidity and other sediment surrogates,” in Proceeding of Federal Interagency Workshop on Turbidity and Other Sediment Surrogates, J. R. Gray and G. D. Glysson, eds. (U.S. Geological Survey, 2003), pp. 5–8.

M. A. Uhrich and H. M. Bragg, Monitoring instream turbidity to estimate continuous suspended–sediment loads and yields and clay–water volumes in the upper North Santiam River Basin, Oregon, 1998–2000 (US Department of the Interior, US Geological Survey, 2003).

V. G. Christensen, X. Jian, and A. C. Ziegler, Regression analysis and real–time water-quality monitoring to estimate constituent concentrations, loads, and yields in the Little Arkansas River, south-central Kansas, 1995–99 (US Department of the Interior, US Geological Survey, 2000).

M. J. Sadar, Turbidity Science. Technical Information Series–Booklet No. 11 (Hach Company, 1998).

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Figures (8)

Fig. 1
Fig. 1 (a) Schematic of the turbidity measurement. A transmissometer only collects the intensity of the transmitted light (i.e., θ = 0°) to measure the turbidity, whereas a nephelometer collects the scattered light at scattering angles θ other than 0°. (b) Qualitative illustration of the relationship between the light intensity and the turbidity for θ = 0°,30°,90°, and 135°. Each curve is normalized to its maximum value.
Fig. 2
Fig. 2 Definition of the volume scattering function (VSF; from [15]), where Φ i a s , and Φ t denote the radiant fluxes of the incident light, absorbed light, scattered light, and transmitted light, respectively. ΔA is the area onto which the incident light is projected. ΔV is the volume of water that is illuminated by the incident light. The medium is assumed to be isotropic, and the light is assumed to be unpolarized; therefore, the scattering process is azimuthally symmetric so that the shape of the VSF depends only on the scattering angle.
Fig. 3
Fig. 3 Top view (left) and side view (right) of the experimental setup. An 850 nm laser is used as the light source. A power meter is fixed on a turntable to collect the scattered light intensity at a series of angles. The intensity is collected as the turntable is rotated in intervals of 5°.
Fig. 4
Fig. 4 Light power distribution for various turbidities and measurement angles: (a) 3D view and (b) contour plot. The highest light power appears at a low turbidity at small angles, and the smallest light power appears at a low turbidity at large angles.
Fig. 5
Fig. 5 Light intensity changes with respect to different angles and turbidities. (a) Angular distribution of the scattered light intensity for different turbidities. (b) Relationship between the light intensity and the turbidity. It can be seen that the light scattered from different angles is sensitive to the variation in the turbidity in different regions.
Fig. 6
Fig. 6 Comparison between the ratio and GWR methods. (a) Mean relative errors for different methods. (b) Relative error of each sample in the test set. The marked plus sign and circle in (a) indicate the average values of the dashed lines with inverted triangles and plus signs in (b), respectively. When three or more angles are used, the minimum value of εmean of the GWR method is less than that of the ratio method.
Fig. 7
Fig. 7 Frequency of angle appearance in the first 100 combinations with the minimum mean relative error. The frequency is counted every 10°, e.g., 0° on the abscissa represents 0° and 5°, and 10° on the abscissa represents 10° and 15°.
Fig. 8
Fig. 8 Geometrical configuration of the sensor and the water sample in the experimental setup.

Tables (3)

Tables Icon

Table 1 Turbidities of the standard formazin solutions used as samples. The boldfaced and underlined values (e.g., 320 ) are the turbidities of the samples for tests. The others (e.g., 160) are for the calibration.

Tables Icon

Table 2 Algorithm 1 Procedure for determining the optimal angle combinations for the GWR method

Tables Icon

Table 2 Best angle combinations and εmean of the three best combinations with two to six angles. The boldfaced column represents the minimum value of εmean .

Equations (20)

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

Φ t = Φ i e c r ,
β ( ψ ) = lim Δ r 0 lim Δ Ω 0 Φ s ( ψ ) Φ i Δ r Δ Ω ( m 1 s r 1 ) ,
b = 2 π ψ = 0 π β ( ψ ) s i n ( ψ ) d ψ .
I 0 = m e n T ,
T = 1 n ( l n I 0 l n m ) .
T = k 1 I 90 + k 2 ,
T = I 90 I 0 k 1 + I 30 k 2 + I 90 k 3 + I 138 k 4 ,
T = f ( I , K ) ,
T = k p + 1 I 1 + k p + 2 I 2 + + k 2 p I p k 1 I 1 + k 2 I 2 + + k p I p ,
T = k p + 1 I 1 + k p + 2 I 2 + + k 2 p I p I 1 + k 2 I 2 + + k p I p ,
T j = 2 p k j I j I 1 + j = 1 p k p + j I j I 1 = T .
[ T I 2 T I 3 T I p 1 I 2 I p ] [ k 2 k 3 k 2 p ] = T .
[ T 1 I 2 , 1 T 1 I 3 , 1 T 1 I p , 1 1 I 2 , 1 I p , 1 T 2 I 2 , 2 T 2 I 3 , 2 T 2 I p , 2 1 I 2 , 2 I p , 2 T M I 2 , M T M I 3 , M T M I p , M 1 I 2 , M I p , M ] A [ k 2 k 3 k 2 p ] K = [ T 1 T 2 T M ] B
A K = B ,
K ^ = ( A T A ) 1 A T B ,
T ^ = k ^ p + 1 I 1 + k ^ p + 2 I 2 + + k ^ 2 p I p I 1 + k ^ 2 I 2 + + k ^ p I p ,
ε m e a n = 1 n i = 1 n | T ^ i T i | T i ,
T d = T 0 V 0 V d ,
Δ T d | T d V 0 | Δ V 0 + | T d V d | Δ V d = T 0 V d Δ V 0 + T 0 V d 2 Δ V d ,
Δ T d T d 4000 / 25 × 0.005 s + 4000 / 25 2 × 0.03 160 s = 0.005 + 0.0012 / s .

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