Abstract

To characterize milk through light-scattering measurements, a semianalytical radiative transfer model was used to simulate the backscatter of light in milk having homogenized fat levels from 0.05 to 3.2 wt. %. The input parameters to the model include the incident wavelength, refractive index of particles and medium, and particle number densities. By varying the wavelength, we can obtain a reasonable fit between experimental data and the model for lower fat milks. Results indicate that the model is most sensitive to the particle diameter and size distribution and less sensitive to the number and index of refraction of the particles.

© 2002 Optical Society of America

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References

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  1. G. H. Meeten, P. Wood, “Optical fibre methods for measuring the diffuse reflectance of fluids,” Meas. Sci. Technol. 4, 643–648 (1993).
    [CrossRef]
  2. F. A. Payne, C. L. Hicks, P.-S. Shen, “Predicting optimal cutting time of coagulating milk using diffuse reflectance,” J. Dairy Sci. 76, 48–61 (1993).
    [CrossRef]
  3. F. A. Payne, “Automatic control of coagulum cutting in cheese manufacturing,” Appl. Eng. Agric. 11, 691–697 (1995).
    [CrossRef]
  4. F. A. Payne, Y. Zhou, R. C. Sullivan, S. Nokes, “Radial backscatter profiles in milk in the wavelength range of 400 to 1000 nm,” ASAE paper 97-6097 (American Society of Agricultural Engineers, St. Joseph, Mich., 1997).
  5. C. L. Crofcheck, F. A. Payne, S. E. Nokes, “Predicting the cutting time of cottage cheese using backscatter measurements,” Trans. ASAE 42, 1039–1045 (1999).
    [CrossRef]
  6. F. A. Payne, C. L. Crofcheck, S. E. Nokes, K. C. Kang, “Light backscatter of milk products for transition sensing using optical fibers,” Trans. ASAE 42, 1771–1776 (1999).
    [CrossRef]
  7. C. L. Crofcheck, F. A. Payne, C. L. Hicks, M. P. Mengüç, S. E. Nokes, “Fiber optic sensor response to low levels of fat in skim milk,” J. Food Process Eng. 23, 163–175 (2000).
    [CrossRef]
  8. P. Walstra, R. Jenness, Dairy Chemistry and Physics (Wiley, New York, 1984).
  9. K. W. Ruettiman, M. R. Ladisch, “Casein micelles: structure, properties and enzymatic coagulation,” Enzyme Microbiol. Technol. 9, 578–589 (1987).
    [CrossRef]
  10. H. Mulder, P. Walstra, The Milk Fat Globule. Emulsion as Applied to Milk Products and Comparable Foods (Pudoc, Wageningen, The Netherlands, 1974).
  11. R. Attaie, R. L. Ritcher, “Size distribution of fat globules in goat milk,” J. Dairy Sci. 83, 940–944 (2000).
    [CrossRef] [PubMed]
  12. M. F. Modest, Radiative Heat Transfer (McGraw-Hill, New York, 1992).
  13. R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (Taylor & Francis, Washington, D.C., 1992).
  14. B. M. Agarwal, M. P. Mengüç, “Single and multiple scattering of collimated radiation in an axisymmetric system,” Int. J. Heat Mass Transfer 34, 633–647 (1991).
    [CrossRef]
  15. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  16. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  17. G. M. Hale, M. R. Querry, “Optical constants of water in the 200-nm to 200-mm wavelength region,” Appl. Opt. 12, 555–563 (1973).
    [CrossRef] [PubMed]
  18. P. Walstra, “Effect of homogenization of the fat globule size distribution in milk,” Neth. Milk Dairy J. 29, 279–294 (1975).
  19. J. Miller, Winchester Farms Dairy Lexington, Ky. (personal communication, 1999).
  20. D. G. Schmidt, J. Koops, D. Westerbeek, “Properties of artificial casein micelles,” Neth. Milk Dairy J. 31, 328–341 (1977).
  21. M. P. Mengüç, S. Manickavasagam, “Radiation transfer and polarized light,” Int. J. Eng. Sci. 36, 1569–1593 (1998).
    [CrossRef]
  22. J. M. Schmitt, G. Kumar, “Spectral distortions in near-infrared spectroscopy of turbid materials,” Appl. Spectrosc. 50, 1066–1073 (1996).
    [CrossRef]

2000 (2)

C. L. Crofcheck, F. A. Payne, C. L. Hicks, M. P. Mengüç, S. E. Nokes, “Fiber optic sensor response to low levels of fat in skim milk,” J. Food Process Eng. 23, 163–175 (2000).
[CrossRef]

R. Attaie, R. L. Ritcher, “Size distribution of fat globules in goat milk,” J. Dairy Sci. 83, 940–944 (2000).
[CrossRef] [PubMed]

1999 (2)

C. L. Crofcheck, F. A. Payne, S. E. Nokes, “Predicting the cutting time of cottage cheese using backscatter measurements,” Trans. ASAE 42, 1039–1045 (1999).
[CrossRef]

F. A. Payne, C. L. Crofcheck, S. E. Nokes, K. C. Kang, “Light backscatter of milk products for transition sensing using optical fibers,” Trans. ASAE 42, 1771–1776 (1999).
[CrossRef]

1998 (1)

M. P. Mengüç, S. Manickavasagam, “Radiation transfer and polarized light,” Int. J. Eng. Sci. 36, 1569–1593 (1998).
[CrossRef]

1996 (1)

J. M. Schmitt, G. Kumar, “Spectral distortions in near-infrared spectroscopy of turbid materials,” Appl. Spectrosc. 50, 1066–1073 (1996).
[CrossRef]

1995 (1)

F. A. Payne, “Automatic control of coagulum cutting in cheese manufacturing,” Appl. Eng. Agric. 11, 691–697 (1995).
[CrossRef]

1993 (2)

G. H. Meeten, P. Wood, “Optical fibre methods for measuring the diffuse reflectance of fluids,” Meas. Sci. Technol. 4, 643–648 (1993).
[CrossRef]

F. A. Payne, C. L. Hicks, P.-S. Shen, “Predicting optimal cutting time of coagulating milk using diffuse reflectance,” J. Dairy Sci. 76, 48–61 (1993).
[CrossRef]

1991 (1)

B. M. Agarwal, M. P. Mengüç, “Single and multiple scattering of collimated radiation in an axisymmetric system,” Int. J. Heat Mass Transfer 34, 633–647 (1991).
[CrossRef]

1987 (1)

K. W. Ruettiman, M. R. Ladisch, “Casein micelles: structure, properties and enzymatic coagulation,” Enzyme Microbiol. Technol. 9, 578–589 (1987).
[CrossRef]

1977 (1)

D. G. Schmidt, J. Koops, D. Westerbeek, “Properties of artificial casein micelles,” Neth. Milk Dairy J. 31, 328–341 (1977).

1975 (1)

P. Walstra, “Effect of homogenization of the fat globule size distribution in milk,” Neth. Milk Dairy J. 29, 279–294 (1975).

1973 (1)

Agarwal, B. M.

B. M. Agarwal, M. P. Mengüç, “Single and multiple scattering of collimated radiation in an axisymmetric system,” Int. J. Heat Mass Transfer 34, 633–647 (1991).
[CrossRef]

Attaie, R.

R. Attaie, R. L. Ritcher, “Size distribution of fat globules in goat milk,” J. Dairy Sci. 83, 940–944 (2000).
[CrossRef] [PubMed]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Crofcheck, C. L.

C. L. Crofcheck, F. A. Payne, C. L. Hicks, M. P. Mengüç, S. E. Nokes, “Fiber optic sensor response to low levels of fat in skim milk,” J. Food Process Eng. 23, 163–175 (2000).
[CrossRef]

C. L. Crofcheck, F. A. Payne, S. E. Nokes, “Predicting the cutting time of cottage cheese using backscatter measurements,” Trans. ASAE 42, 1039–1045 (1999).
[CrossRef]

F. A. Payne, C. L. Crofcheck, S. E. Nokes, K. C. Kang, “Light backscatter of milk products for transition sensing using optical fibers,” Trans. ASAE 42, 1771–1776 (1999).
[CrossRef]

Hale, G. M.

Hicks, C. L.

C. L. Crofcheck, F. A. Payne, C. L. Hicks, M. P. Mengüç, S. E. Nokes, “Fiber optic sensor response to low levels of fat in skim milk,” J. Food Process Eng. 23, 163–175 (2000).
[CrossRef]

F. A. Payne, C. L. Hicks, P.-S. Shen, “Predicting optimal cutting time of coagulating milk using diffuse reflectance,” J. Dairy Sci. 76, 48–61 (1993).
[CrossRef]

Howell, J. R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (Taylor & Francis, Washington, D.C., 1992).

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Jenness, R.

P. Walstra, R. Jenness, Dairy Chemistry and Physics (Wiley, New York, 1984).

Kang, K. C.

F. A. Payne, C. L. Crofcheck, S. E. Nokes, K. C. Kang, “Light backscatter of milk products for transition sensing using optical fibers,” Trans. ASAE 42, 1771–1776 (1999).
[CrossRef]

Koops, J.

D. G. Schmidt, J. Koops, D. Westerbeek, “Properties of artificial casein micelles,” Neth. Milk Dairy J. 31, 328–341 (1977).

Kumar, G.

J. M. Schmitt, G. Kumar, “Spectral distortions in near-infrared spectroscopy of turbid materials,” Appl. Spectrosc. 50, 1066–1073 (1996).
[CrossRef]

Ladisch, M. R.

K. W. Ruettiman, M. R. Ladisch, “Casein micelles: structure, properties and enzymatic coagulation,” Enzyme Microbiol. Technol. 9, 578–589 (1987).
[CrossRef]

Manickavasagam, S.

M. P. Mengüç, S. Manickavasagam, “Radiation transfer and polarized light,” Int. J. Eng. Sci. 36, 1569–1593 (1998).
[CrossRef]

Meeten, G. H.

G. H. Meeten, P. Wood, “Optical fibre methods for measuring the diffuse reflectance of fluids,” Meas. Sci. Technol. 4, 643–648 (1993).
[CrossRef]

Mengüç, M. P.

C. L. Crofcheck, F. A. Payne, C. L. Hicks, M. P. Mengüç, S. E. Nokes, “Fiber optic sensor response to low levels of fat in skim milk,” J. Food Process Eng. 23, 163–175 (2000).
[CrossRef]

M. P. Mengüç, S. Manickavasagam, “Radiation transfer and polarized light,” Int. J. Eng. Sci. 36, 1569–1593 (1998).
[CrossRef]

B. M. Agarwal, M. P. Mengüç, “Single and multiple scattering of collimated radiation in an axisymmetric system,” Int. J. Heat Mass Transfer 34, 633–647 (1991).
[CrossRef]

Miller, J.

J. Miller, Winchester Farms Dairy Lexington, Ky. (personal communication, 1999).

Modest, M. F.

M. F. Modest, Radiative Heat Transfer (McGraw-Hill, New York, 1992).

Mulder, H.

H. Mulder, P. Walstra, The Milk Fat Globule. Emulsion as Applied to Milk Products and Comparable Foods (Pudoc, Wageningen, The Netherlands, 1974).

Nokes, S.

F. A. Payne, Y. Zhou, R. C. Sullivan, S. Nokes, “Radial backscatter profiles in milk in the wavelength range of 400 to 1000 nm,” ASAE paper 97-6097 (American Society of Agricultural Engineers, St. Joseph, Mich., 1997).

Nokes, S. E.

C. L. Crofcheck, F. A. Payne, C. L. Hicks, M. P. Mengüç, S. E. Nokes, “Fiber optic sensor response to low levels of fat in skim milk,” J. Food Process Eng. 23, 163–175 (2000).
[CrossRef]

C. L. Crofcheck, F. A. Payne, S. E. Nokes, “Predicting the cutting time of cottage cheese using backscatter measurements,” Trans. ASAE 42, 1039–1045 (1999).
[CrossRef]

F. A. Payne, C. L. Crofcheck, S. E. Nokes, K. C. Kang, “Light backscatter of milk products for transition sensing using optical fibers,” Trans. ASAE 42, 1771–1776 (1999).
[CrossRef]

Payne, F. A.

C. L. Crofcheck, F. A. Payne, C. L. Hicks, M. P. Mengüç, S. E. Nokes, “Fiber optic sensor response to low levels of fat in skim milk,” J. Food Process Eng. 23, 163–175 (2000).
[CrossRef]

C. L. Crofcheck, F. A. Payne, S. E. Nokes, “Predicting the cutting time of cottage cheese using backscatter measurements,” Trans. ASAE 42, 1039–1045 (1999).
[CrossRef]

F. A. Payne, C. L. Crofcheck, S. E. Nokes, K. C. Kang, “Light backscatter of milk products for transition sensing using optical fibers,” Trans. ASAE 42, 1771–1776 (1999).
[CrossRef]

F. A. Payne, “Automatic control of coagulum cutting in cheese manufacturing,” Appl. Eng. Agric. 11, 691–697 (1995).
[CrossRef]

F. A. Payne, C. L. Hicks, P.-S. Shen, “Predicting optimal cutting time of coagulating milk using diffuse reflectance,” J. Dairy Sci. 76, 48–61 (1993).
[CrossRef]

F. A. Payne, Y. Zhou, R. C. Sullivan, S. Nokes, “Radial backscatter profiles in milk in the wavelength range of 400 to 1000 nm,” ASAE paper 97-6097 (American Society of Agricultural Engineers, St. Joseph, Mich., 1997).

Querry, M. R.

Ritcher, R. L.

R. Attaie, R. L. Ritcher, “Size distribution of fat globules in goat milk,” J. Dairy Sci. 83, 940–944 (2000).
[CrossRef] [PubMed]

Ruettiman, K. W.

K. W. Ruettiman, M. R. Ladisch, “Casein micelles: structure, properties and enzymatic coagulation,” Enzyme Microbiol. Technol. 9, 578–589 (1987).
[CrossRef]

Schmidt, D. G.

D. G. Schmidt, J. Koops, D. Westerbeek, “Properties of artificial casein micelles,” Neth. Milk Dairy J. 31, 328–341 (1977).

Schmitt, J. M.

J. M. Schmitt, G. Kumar, “Spectral distortions in near-infrared spectroscopy of turbid materials,” Appl. Spectrosc. 50, 1066–1073 (1996).
[CrossRef]

Shen, P.-S.

F. A. Payne, C. L. Hicks, P.-S. Shen, “Predicting optimal cutting time of coagulating milk using diffuse reflectance,” J. Dairy Sci. 76, 48–61 (1993).
[CrossRef]

Siegel, R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (Taylor & Francis, Washington, D.C., 1992).

Sullivan, R. C.

F. A. Payne, Y. Zhou, R. C. Sullivan, S. Nokes, “Radial backscatter profiles in milk in the wavelength range of 400 to 1000 nm,” ASAE paper 97-6097 (American Society of Agricultural Engineers, St. Joseph, Mich., 1997).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Walstra, P.

P. Walstra, “Effect of homogenization of the fat globule size distribution in milk,” Neth. Milk Dairy J. 29, 279–294 (1975).

P. Walstra, R. Jenness, Dairy Chemistry and Physics (Wiley, New York, 1984).

H. Mulder, P. Walstra, The Milk Fat Globule. Emulsion as Applied to Milk Products and Comparable Foods (Pudoc, Wageningen, The Netherlands, 1974).

Westerbeek, D.

D. G. Schmidt, J. Koops, D. Westerbeek, “Properties of artificial casein micelles,” Neth. Milk Dairy J. 31, 328–341 (1977).

Wood, P.

G. H. Meeten, P. Wood, “Optical fibre methods for measuring the diffuse reflectance of fluids,” Meas. Sci. Technol. 4, 643–648 (1993).
[CrossRef]

Zhou, Y.

F. A. Payne, Y. Zhou, R. C. Sullivan, S. Nokes, “Radial backscatter profiles in milk in the wavelength range of 400 to 1000 nm,” ASAE paper 97-6097 (American Society of Agricultural Engineers, St. Joseph, Mich., 1997).

Appl. Spectrosc. (1)

J. M. Schmitt, G. Kumar, “Spectral distortions in near-infrared spectroscopy of turbid materials,” Appl. Spectrosc. 50, 1066–1073 (1996).
[CrossRef]

Appl. Eng. Agric. (1)

F. A. Payne, “Automatic control of coagulum cutting in cheese manufacturing,” Appl. Eng. Agric. 11, 691–697 (1995).
[CrossRef]

Appl. Opt. (1)

Enzyme Microbiol. Technol. (1)

K. W. Ruettiman, M. R. Ladisch, “Casein micelles: structure, properties and enzymatic coagulation,” Enzyme Microbiol. Technol. 9, 578–589 (1987).
[CrossRef]

Int. J. Eng. Sci. (1)

M. P. Mengüç, S. Manickavasagam, “Radiation transfer and polarized light,” Int. J. Eng. Sci. 36, 1569–1593 (1998).
[CrossRef]

Int. J. Heat Mass Transfer (1)

B. M. Agarwal, M. P. Mengüç, “Single and multiple scattering of collimated radiation in an axisymmetric system,” Int. J. Heat Mass Transfer 34, 633–647 (1991).
[CrossRef]

J. Dairy Sci. (1)

F. A. Payne, C. L. Hicks, P.-S. Shen, “Predicting optimal cutting time of coagulating milk using diffuse reflectance,” J. Dairy Sci. 76, 48–61 (1993).
[CrossRef]

J. Dairy Sci. (1)

R. Attaie, R. L. Ritcher, “Size distribution of fat globules in goat milk,” J. Dairy Sci. 83, 940–944 (2000).
[CrossRef] [PubMed]

J. Food Process Eng. (1)

C. L. Crofcheck, F. A. Payne, C. L. Hicks, M. P. Mengüç, S. E. Nokes, “Fiber optic sensor response to low levels of fat in skim milk,” J. Food Process Eng. 23, 163–175 (2000).
[CrossRef]

Meas. Sci. Technol. (1)

G. H. Meeten, P. Wood, “Optical fibre methods for measuring the diffuse reflectance of fluids,” Meas. Sci. Technol. 4, 643–648 (1993).
[CrossRef]

Neth. Milk Dairy J. (1)

P. Walstra, “Effect of homogenization of the fat globule size distribution in milk,” Neth. Milk Dairy J. 29, 279–294 (1975).

Neth. Milk Dairy J. (1)

D. G. Schmidt, J. Koops, D. Westerbeek, “Properties of artificial casein micelles,” Neth. Milk Dairy J. 31, 328–341 (1977).

Trans. ASAE (2)

C. L. Crofcheck, F. A. Payne, S. E. Nokes, “Predicting the cutting time of cottage cheese using backscatter measurements,” Trans. ASAE 42, 1039–1045 (1999).
[CrossRef]

F. A. Payne, C. L. Crofcheck, S. E. Nokes, K. C. Kang, “Light backscatter of milk products for transition sensing using optical fibers,” Trans. ASAE 42, 1771–1776 (1999).
[CrossRef]

Other (8)

P. Walstra, R. Jenness, Dairy Chemistry and Physics (Wiley, New York, 1984).

F. A. Payne, Y. Zhou, R. C. Sullivan, S. Nokes, “Radial backscatter profiles in milk in the wavelength range of 400 to 1000 nm,” ASAE paper 97-6097 (American Society of Agricultural Engineers, St. Joseph, Mich., 1997).

J. Miller, Winchester Farms Dairy Lexington, Ky. (personal communication, 1999).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

M. F. Modest, Radiative Heat Transfer (McGraw-Hill, New York, 1992).

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (Taylor & Francis, Washington, D.C., 1992).

H. Mulder, P. Walstra, The Milk Fat Globule. Emulsion as Applied to Milk Products and Comparable Foods (Pudoc, Wageningen, The Netherlands, 1974).

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Figures (12)

Fig. 1
Fig. 1

Positioning of the light-emitting and light-detecting optical fibers to measure backscatter.

Fig. 2
Fig. 2

First-order scattering; the finite solid angle of the detector is shown by the dotted lines.

Fig. 3
Fig. 3

Second-order scattering. Example of light ray tracing used in the semianalytical radiative transport model.

Fig. 4
Fig. 4

Observed (symbols) and predicted (curves) normalized intensities (NI) at three different wavelengths (710, 940, and 1010 nm, respectively) as a function of radial distance for different milkfat levels [0.05 (◆), 0.5 (□), 1 (△), 2 (●), and 3.2 wt. % (*)].

Fig. 5
Fig. 5

Observed (symbols) and predicted (curves) normalized intensities (NI) for the three lower milkfat milks [0.05 (◆), 0.5 (□), and 1.0 wt. % (△)] at the appropriate wavelengths (710, 940, and 1010 nm, respectively) as a function of radial distance.

Fig. 6
Fig. 6

Observed (symbols) and predicted (curves) normalized intensities (NI) based on various size distributions [where the diameters are 90%, 110%, and 120% of the diameters used in the base case (BC)] as a function of radial distance for three different milkfat levels (0.05, 0.5, and 1.0 wt. %).

Fig. 7
Fig. 7

Observed (symbols) and predicted (curves and curves with symbols) normalized intensities (NI) based on various single particle diameters (indicated on the plots in nanometers) and including the prediction based on the size distribution considered in the base case (BC) as a function of radial distance for three different milkfat levels (0.05, 0.5, and 1.0 wt. %).

Fig. 8
Fig. 8

Observed (symbols) and predicted (curves) normalized intensities (NI) at seven different volume fractions (±10%, ±0.01, ±0.006 of the size distribution of the base case) and as a function of radial distance for three different milkfat levels (0.05, 0.5, and 1.0 wt. %).

Fig. 9
Fig. 9

Observed (symbols) and predicted (curves) normalized intensities (NI) at three different indices of refraction of fat (1.454, 1.47, and the base case 1.462) as a function of radial distance for three different milkfat levels (0.05, 0.5, and 1.0 wt. %).

Fig. 10
Fig. 10

Predicted intensity ratios (IRs) as a function of radial distance for different milkfat levels (0.05, 0.5, and 1.0 wt. %). The labels to the left and the right indicate the order of milkfat levels at 0 and 6 mm, respectively. An inversion of order is shown from short to long radial distances.

Fig. 11
Fig. 11

Phase function versus the scattering angle for different milkfat levels (0.05, 0.5, and 1.0 wt. %). The labels to the right and the left of the curves indicate the order of milkfat levels, showing an inversion in order from the forward to the backward-scattering regimes.

Fig. 12
Fig. 12

Scattering into milk with respect to the number of milkfat particles and the distance between the emitter and the detector, where the scattering into 0.05-wt. % milkfat (left) and 1-wt. % milkfat (right) are both shown.

Equations (5)

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

I P = I 0   exp - β ¯ l 1 ,
I Q = I P σ   Φ α 1 4 π exp - β ¯ l 2 .
I R = I Q σ   Φ α 2 Δ Ω 2 4 π exp - β ¯ l 3 .
I R r I 0 = 00 L 1   L 3   σ 2 Φ α 1 4 π Φ α 2 Δ Ω 2 4 π exp - β ¯ L d l 3 d l 1 ,
d ¯ mn p = i = 1 n   N i d i m i = 1 n   N i d i n ,

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