Abstract

Laser triangulation is used to measure the thickness of a liquid film in a test section consisting of a quartz viewing window, a water layer, and a hydrophobic membrane. The triangulation sensor acquires measurements to the bounding surfaces of the film while peering through multiple interfaces. This allows the difference between the two measurements to constitute the local film thickness. A refraction model is developed and applied to the analysis of data collected from the experiment. For verification, an empirical method is also developed and compared to the analytical approach. The measurement technique is intended to assess the stability of liquid films for use as gas–liquid contactors.

© 2006 Optical Society of America

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References

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  1. K. Drost, "Mesoscopic heat-actuated heat pump development," in Proceedings of ASME Advanced Energy Systems Division, S. M. Aceves, S. Garimella, and R. Peterson, eds. (American Society of Mechanical Engineers, 1999), Vol. 39, pp. 9-14.
  2. M. K. Drost, M. Friedrich, C. Martin, J. Martin, and R. Cameron, "Recent developments in microtechnology-based chemical heat pumps," in Microreaction Technology: Industrial Prospects--Proceedings of Third International Conference on Microreaction Technology (IMRET 3), W. Ehrfeld, ed. (Springer, 1999), pp. 394-401.
  3. W. E. TeGrotenhuis, R. J. Cameron, V. V. Viswanathan, and R. S. Wegeng, "Solvent extraction and gas absorption using microchannel contactors," in Microreaction Technology:Industrial Prospects--Proceedings of Third International Conference on Microreaction Technology (IMRET 3), W. Ehrfeld, ed. (Springer, 1999), pp. 541-549.
  4. R. J. Dwulet, "Cutting costs with laser triangulation," Mach. Des. 66, 110-112 (1994).
  5. R. J. Dwulet, "Laser triangulation expands measurement options," Des. News 50, 114 (1995).
  6. W. P. Kennedy, "The basics of triangulation sensors," Sensors 15, 76-83 (1998).
  7. W. Pastorius, "Triangulation sensors: an overview," In Tech 48, 58-59 (2001).
  8. D. R. Wiese, "Laser triangulation sensors: a good choice for high speed inspection," Chilton's I&CS 62, 27-29 (1989).
  9. K. B. Smith and Y. F. Zheng, "Accuracy analysis of point laser triangulation probes using simulation," J. Manuf. Sci. Eng. 120, 736-745 (1998).
    [CrossRef]
  10. F. Murakami, "Accuracy assessment of a laser triangulation sensor," in Proceedings of IEEE Instrumentation and Measurement Technology Conference (IEEE, 1994), pp. 802-805.
  11. R. G. Dorsch, G. Häusler, and J. M. Herrmann, "Laser triangulation: fundamental uncertainty in distance measurement," Appl. Opt. 33, 1306-1314 (1994).
    [CrossRef] [PubMed]
  12. D. I. Driscoll, R. L. Schmitt, and W. H. Stevenson, "Thin flowing liquid film thickness measurement by laser induced fluorescence," J. Fluids Eng. 114, 107-112 (1992).
    [CrossRef]
  13. T. A. Shedd and T. A. Newell, "Automated optical liquid film thickness measurement method," Rev. Sci. Instrum. 69, 4205-4213 (1998).
    [CrossRef]
  14. N. Fukamachi, T. Hazuku, T. Takamasa, T. Hibiki, and M. Ishii, "Measurement on liquid film in microchannels using laser focus displacement meter," in Proceedings of the First International Conference on Microchannels and Minichannels, S. G. Kandlikar, ed. (American Society of Mechanical Engineers, 2003), pp. 543-550.
    [CrossRef]
  15. M. Fukata, T. Yanagisawa, S. Miyamura, and Y. Ogi, "Concentration measurement of refrigerant/refrigeration oil mixture by refractive index," Int. J. Refrig. 27, 346-352 (2004).
    [CrossRef]
  16. W. J. H. Okkerse, S. P. P. Ottengraf, and B. Osinga-Kuipers, "Biofilm thickness variability investigated with a laser triangulation sensor," Biotechnol. Bioeng. 70, 619-629 (2000).
    [CrossRef] [PubMed]
  17. G. G. Láng and M. Seo, "On the electrochemical applications of the bending beam method," J. Electroanal. Chem. 490, 98-101 (2000).
    [CrossRef]
  18. D. Halliday, R. Resnick, and K. S. Krane, Physics, 4th ed. (Wiley, 1992), Vol. 2, pp. 904-905.
  19. R. S. Figliola and D. E. Beasley, Theory and Design for Mechanical Measurements, 3rd ed. (Wiley, 2000), pp. 161-163, 181-183.
  20. L. E. Regalado and D. Malacara, "Isotropic amorphous optical materials," in Handbook of Optical Engineering, D. Malacara and B. J. Thompson, eds. (Dekker, 2001), pp. 827-846.

2004 (1)

M. Fukata, T. Yanagisawa, S. Miyamura, and Y. Ogi, "Concentration measurement of refrigerant/refrigeration oil mixture by refractive index," Int. J. Refrig. 27, 346-352 (2004).
[CrossRef]

2001 (1)

W. Pastorius, "Triangulation sensors: an overview," In Tech 48, 58-59 (2001).

2000 (2)

W. J. H. Okkerse, S. P. P. Ottengraf, and B. Osinga-Kuipers, "Biofilm thickness variability investigated with a laser triangulation sensor," Biotechnol. Bioeng. 70, 619-629 (2000).
[CrossRef] [PubMed]

G. G. Láng and M. Seo, "On the electrochemical applications of the bending beam method," J. Electroanal. Chem. 490, 98-101 (2000).
[CrossRef]

1998 (3)

K. B. Smith and Y. F. Zheng, "Accuracy analysis of point laser triangulation probes using simulation," J. Manuf. Sci. Eng. 120, 736-745 (1998).
[CrossRef]

W. P. Kennedy, "The basics of triangulation sensors," Sensors 15, 76-83 (1998).

T. A. Shedd and T. A. Newell, "Automated optical liquid film thickness measurement method," Rev. Sci. Instrum. 69, 4205-4213 (1998).
[CrossRef]

1995 (1)

R. J. Dwulet, "Laser triangulation expands measurement options," Des. News 50, 114 (1995).

1994 (2)

1992 (1)

D. I. Driscoll, R. L. Schmitt, and W. H. Stevenson, "Thin flowing liquid film thickness measurement by laser induced fluorescence," J. Fluids Eng. 114, 107-112 (1992).
[CrossRef]

1989 (1)

D. R. Wiese, "Laser triangulation sensors: a good choice for high speed inspection," Chilton's I&CS 62, 27-29 (1989).

Beasley, D. E.

R. S. Figliola and D. E. Beasley, Theory and Design for Mechanical Measurements, 3rd ed. (Wiley, 2000), pp. 161-163, 181-183.

Cameron, R.

M. K. Drost, M. Friedrich, C. Martin, J. Martin, and R. Cameron, "Recent developments in microtechnology-based chemical heat pumps," in Microreaction Technology: Industrial Prospects--Proceedings of Third International Conference on Microreaction Technology (IMRET 3), W. Ehrfeld, ed. (Springer, 1999), pp. 394-401.

Cameron, R. J.

W. E. TeGrotenhuis, R. J. Cameron, V. V. Viswanathan, and R. S. Wegeng, "Solvent extraction and gas absorption using microchannel contactors," in Microreaction Technology:Industrial Prospects--Proceedings of Third International Conference on Microreaction Technology (IMRET 3), W. Ehrfeld, ed. (Springer, 1999), pp. 541-549.

Dorsch, R. G.

Driscoll, D. I.

D. I. Driscoll, R. L. Schmitt, and W. H. Stevenson, "Thin flowing liquid film thickness measurement by laser induced fluorescence," J. Fluids Eng. 114, 107-112 (1992).
[CrossRef]

Drost, K.

K. Drost, "Mesoscopic heat-actuated heat pump development," in Proceedings of ASME Advanced Energy Systems Division, S. M. Aceves, S. Garimella, and R. Peterson, eds. (American Society of Mechanical Engineers, 1999), Vol. 39, pp. 9-14.

Drost, M. K.

M. K. Drost, M. Friedrich, C. Martin, J. Martin, and R. Cameron, "Recent developments in microtechnology-based chemical heat pumps," in Microreaction Technology: Industrial Prospects--Proceedings of Third International Conference on Microreaction Technology (IMRET 3), W. Ehrfeld, ed. (Springer, 1999), pp. 394-401.

Dwulet, R. J.

R. J. Dwulet, "Laser triangulation expands measurement options," Des. News 50, 114 (1995).

R. J. Dwulet, "Cutting costs with laser triangulation," Mach. Des. 66, 110-112 (1994).

Figliola, R. S.

R. S. Figliola and D. E. Beasley, Theory and Design for Mechanical Measurements, 3rd ed. (Wiley, 2000), pp. 161-163, 181-183.

Friedrich, M.

M. K. Drost, M. Friedrich, C. Martin, J. Martin, and R. Cameron, "Recent developments in microtechnology-based chemical heat pumps," in Microreaction Technology: Industrial Prospects--Proceedings of Third International Conference on Microreaction Technology (IMRET 3), W. Ehrfeld, ed. (Springer, 1999), pp. 394-401.

Fukamachi, N.

N. Fukamachi, T. Hazuku, T. Takamasa, T. Hibiki, and M. Ishii, "Measurement on liquid film in microchannels using laser focus displacement meter," in Proceedings of the First International Conference on Microchannels and Minichannels, S. G. Kandlikar, ed. (American Society of Mechanical Engineers, 2003), pp. 543-550.
[CrossRef]

Fukata, M.

M. Fukata, T. Yanagisawa, S. Miyamura, and Y. Ogi, "Concentration measurement of refrigerant/refrigeration oil mixture by refractive index," Int. J. Refrig. 27, 346-352 (2004).
[CrossRef]

Halliday, D.

D. Halliday, R. Resnick, and K. S. Krane, Physics, 4th ed. (Wiley, 1992), Vol. 2, pp. 904-905.

Häusler, G.

Hazuku, T.

N. Fukamachi, T. Hazuku, T. Takamasa, T. Hibiki, and M. Ishii, "Measurement on liquid film in microchannels using laser focus displacement meter," in Proceedings of the First International Conference on Microchannels and Minichannels, S. G. Kandlikar, ed. (American Society of Mechanical Engineers, 2003), pp. 543-550.
[CrossRef]

Herrmann, J. M.

Hibiki, T.

N. Fukamachi, T. Hazuku, T. Takamasa, T. Hibiki, and M. Ishii, "Measurement on liquid film in microchannels using laser focus displacement meter," in Proceedings of the First International Conference on Microchannels and Minichannels, S. G. Kandlikar, ed. (American Society of Mechanical Engineers, 2003), pp. 543-550.
[CrossRef]

Ishii, M.

N. Fukamachi, T. Hazuku, T. Takamasa, T. Hibiki, and M. Ishii, "Measurement on liquid film in microchannels using laser focus displacement meter," in Proceedings of the First International Conference on Microchannels and Minichannels, S. G. Kandlikar, ed. (American Society of Mechanical Engineers, 2003), pp. 543-550.
[CrossRef]

Kennedy, W. P.

W. P. Kennedy, "The basics of triangulation sensors," Sensors 15, 76-83 (1998).

Krane, K. S.

D. Halliday, R. Resnick, and K. S. Krane, Physics, 4th ed. (Wiley, 1992), Vol. 2, pp. 904-905.

Láng, G. G.

G. G. Láng and M. Seo, "On the electrochemical applications of the bending beam method," J. Electroanal. Chem. 490, 98-101 (2000).
[CrossRef]

Malacara, D.

L. E. Regalado and D. Malacara, "Isotropic amorphous optical materials," in Handbook of Optical Engineering, D. Malacara and B. J. Thompson, eds. (Dekker, 2001), pp. 827-846.

Martin, C.

M. K. Drost, M. Friedrich, C. Martin, J. Martin, and R. Cameron, "Recent developments in microtechnology-based chemical heat pumps," in Microreaction Technology: Industrial Prospects--Proceedings of Third International Conference on Microreaction Technology (IMRET 3), W. Ehrfeld, ed. (Springer, 1999), pp. 394-401.

Martin, J.

M. K. Drost, M. Friedrich, C. Martin, J. Martin, and R. Cameron, "Recent developments in microtechnology-based chemical heat pumps," in Microreaction Technology: Industrial Prospects--Proceedings of Third International Conference on Microreaction Technology (IMRET 3), W. Ehrfeld, ed. (Springer, 1999), pp. 394-401.

Miyamura, S.

M. Fukata, T. Yanagisawa, S. Miyamura, and Y. Ogi, "Concentration measurement of refrigerant/refrigeration oil mixture by refractive index," Int. J. Refrig. 27, 346-352 (2004).
[CrossRef]

Murakami, F.

F. Murakami, "Accuracy assessment of a laser triangulation sensor," in Proceedings of IEEE Instrumentation and Measurement Technology Conference (IEEE, 1994), pp. 802-805.

Newell, T. A.

T. A. Shedd and T. A. Newell, "Automated optical liquid film thickness measurement method," Rev. Sci. Instrum. 69, 4205-4213 (1998).
[CrossRef]

Ogi, Y.

M. Fukata, T. Yanagisawa, S. Miyamura, and Y. Ogi, "Concentration measurement of refrigerant/refrigeration oil mixture by refractive index," Int. J. Refrig. 27, 346-352 (2004).
[CrossRef]

Okkerse, W. J. H.

W. J. H. Okkerse, S. P. P. Ottengraf, and B. Osinga-Kuipers, "Biofilm thickness variability investigated with a laser triangulation sensor," Biotechnol. Bioeng. 70, 619-629 (2000).
[CrossRef] [PubMed]

Osinga-Kuipers, B.

W. J. H. Okkerse, S. P. P. Ottengraf, and B. Osinga-Kuipers, "Biofilm thickness variability investigated with a laser triangulation sensor," Biotechnol. Bioeng. 70, 619-629 (2000).
[CrossRef] [PubMed]

Ottengraf, S. P. P.

W. J. H. Okkerse, S. P. P. Ottengraf, and B. Osinga-Kuipers, "Biofilm thickness variability investigated with a laser triangulation sensor," Biotechnol. Bioeng. 70, 619-629 (2000).
[CrossRef] [PubMed]

Pastorius, W.

W. Pastorius, "Triangulation sensors: an overview," In Tech 48, 58-59 (2001).

Regalado, L. E.

L. E. Regalado and D. Malacara, "Isotropic amorphous optical materials," in Handbook of Optical Engineering, D. Malacara and B. J. Thompson, eds. (Dekker, 2001), pp. 827-846.

Resnick, R.

D. Halliday, R. Resnick, and K. S. Krane, Physics, 4th ed. (Wiley, 1992), Vol. 2, pp. 904-905.

Schmitt, R. L.

D. I. Driscoll, R. L. Schmitt, and W. H. Stevenson, "Thin flowing liquid film thickness measurement by laser induced fluorescence," J. Fluids Eng. 114, 107-112 (1992).
[CrossRef]

Seo, M.

G. G. Láng and M. Seo, "On the electrochemical applications of the bending beam method," J. Electroanal. Chem. 490, 98-101 (2000).
[CrossRef]

Shedd, T. A.

T. A. Shedd and T. A. Newell, "Automated optical liquid film thickness measurement method," Rev. Sci. Instrum. 69, 4205-4213 (1998).
[CrossRef]

Smith, K. B.

K. B. Smith and Y. F. Zheng, "Accuracy analysis of point laser triangulation probes using simulation," J. Manuf. Sci. Eng. 120, 736-745 (1998).
[CrossRef]

Stevenson, W. H.

D. I. Driscoll, R. L. Schmitt, and W. H. Stevenson, "Thin flowing liquid film thickness measurement by laser induced fluorescence," J. Fluids Eng. 114, 107-112 (1992).
[CrossRef]

Takamasa, T.

N. Fukamachi, T. Hazuku, T. Takamasa, T. Hibiki, and M. Ishii, "Measurement on liquid film in microchannels using laser focus displacement meter," in Proceedings of the First International Conference on Microchannels and Minichannels, S. G. Kandlikar, ed. (American Society of Mechanical Engineers, 2003), pp. 543-550.
[CrossRef]

TeGrotenhuis, W. E.

W. E. TeGrotenhuis, R. J. Cameron, V. V. Viswanathan, and R. S. Wegeng, "Solvent extraction and gas absorption using microchannel contactors," in Microreaction Technology:Industrial Prospects--Proceedings of Third International Conference on Microreaction Technology (IMRET 3), W. Ehrfeld, ed. (Springer, 1999), pp. 541-549.

Viswanathan, V. V.

W. E. TeGrotenhuis, R. J. Cameron, V. V. Viswanathan, and R. S. Wegeng, "Solvent extraction and gas absorption using microchannel contactors," in Microreaction Technology:Industrial Prospects--Proceedings of Third International Conference on Microreaction Technology (IMRET 3), W. Ehrfeld, ed. (Springer, 1999), pp. 541-549.

Wegeng, R. S.

W. E. TeGrotenhuis, R. J. Cameron, V. V. Viswanathan, and R. S. Wegeng, "Solvent extraction and gas absorption using microchannel contactors," in Microreaction Technology:Industrial Prospects--Proceedings of Third International Conference on Microreaction Technology (IMRET 3), W. Ehrfeld, ed. (Springer, 1999), pp. 541-549.

Wiese, D. R.

D. R. Wiese, "Laser triangulation sensors: a good choice for high speed inspection," Chilton's I&CS 62, 27-29 (1989).

Yanagisawa, T.

M. Fukata, T. Yanagisawa, S. Miyamura, and Y. Ogi, "Concentration measurement of refrigerant/refrigeration oil mixture by refractive index," Int. J. Refrig. 27, 346-352 (2004).
[CrossRef]

Zheng, Y. F.

K. B. Smith and Y. F. Zheng, "Accuracy analysis of point laser triangulation probes using simulation," J. Manuf. Sci. Eng. 120, 736-745 (1998).
[CrossRef]

Appl. Opt. (1)

Biotechnol. Bioeng. (1)

W. J. H. Okkerse, S. P. P. Ottengraf, and B. Osinga-Kuipers, "Biofilm thickness variability investigated with a laser triangulation sensor," Biotechnol. Bioeng. 70, 619-629 (2000).
[CrossRef] [PubMed]

Chilton's I&CS (1)

D. R. Wiese, "Laser triangulation sensors: a good choice for high speed inspection," Chilton's I&CS 62, 27-29 (1989).

Des. News (1)

R. J. Dwulet, "Laser triangulation expands measurement options," Des. News 50, 114 (1995).

In Tech (1)

W. Pastorius, "Triangulation sensors: an overview," In Tech 48, 58-59 (2001).

Int. J. Refrig. (1)

M. Fukata, T. Yanagisawa, S. Miyamura, and Y. Ogi, "Concentration measurement of refrigerant/refrigeration oil mixture by refractive index," Int. J. Refrig. 27, 346-352 (2004).
[CrossRef]

J. Electroanal. Chem. (1)

G. G. Láng and M. Seo, "On the electrochemical applications of the bending beam method," J. Electroanal. Chem. 490, 98-101 (2000).
[CrossRef]

J. Fluids Eng. (1)

D. I. Driscoll, R. L. Schmitt, and W. H. Stevenson, "Thin flowing liquid film thickness measurement by laser induced fluorescence," J. Fluids Eng. 114, 107-112 (1992).
[CrossRef]

J. Manuf. Sci. Eng. (1)

K. B. Smith and Y. F. Zheng, "Accuracy analysis of point laser triangulation probes using simulation," J. Manuf. Sci. Eng. 120, 736-745 (1998).
[CrossRef]

Mach. Des. (1)

R. J. Dwulet, "Cutting costs with laser triangulation," Mach. Des. 66, 110-112 (1994).

Rev. Sci. Instrum. (1)

T. A. Shedd and T. A. Newell, "Automated optical liquid film thickness measurement method," Rev. Sci. Instrum. 69, 4205-4213 (1998).
[CrossRef]

Sensors (1)

W. P. Kennedy, "The basics of triangulation sensors," Sensors 15, 76-83 (1998).

Other (8)

K. Drost, "Mesoscopic heat-actuated heat pump development," in Proceedings of ASME Advanced Energy Systems Division, S. M. Aceves, S. Garimella, and R. Peterson, eds. (American Society of Mechanical Engineers, 1999), Vol. 39, pp. 9-14.

M. K. Drost, M. Friedrich, C. Martin, J. Martin, and R. Cameron, "Recent developments in microtechnology-based chemical heat pumps," in Microreaction Technology: Industrial Prospects--Proceedings of Third International Conference on Microreaction Technology (IMRET 3), W. Ehrfeld, ed. (Springer, 1999), pp. 394-401.

W. E. TeGrotenhuis, R. J. Cameron, V. V. Viswanathan, and R. S. Wegeng, "Solvent extraction and gas absorption using microchannel contactors," in Microreaction Technology:Industrial Prospects--Proceedings of Third International Conference on Microreaction Technology (IMRET 3), W. Ehrfeld, ed. (Springer, 1999), pp. 541-549.

F. Murakami, "Accuracy assessment of a laser triangulation sensor," in Proceedings of IEEE Instrumentation and Measurement Technology Conference (IEEE, 1994), pp. 802-805.

N. Fukamachi, T. Hazuku, T. Takamasa, T. Hibiki, and M. Ishii, "Measurement on liquid film in microchannels using laser focus displacement meter," in Proceedings of the First International Conference on Microchannels and Minichannels, S. G. Kandlikar, ed. (American Society of Mechanical Engineers, 2003), pp. 543-550.
[CrossRef]

D. Halliday, R. Resnick, and K. S. Krane, Physics, 4th ed. (Wiley, 1992), Vol. 2, pp. 904-905.

R. S. Figliola and D. E. Beasley, Theory and Design for Mechanical Measurements, 3rd ed. (Wiley, 2000), pp. 161-163, 181-183.

L. E. Regalado and D. Malacara, "Isotropic amorphous optical materials," in Handbook of Optical Engineering, D. Malacara and B. J. Thompson, eds. (Dekker, 2001), pp. 827-846.

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of a laser triangulation sensor (Refs. 4–8).

Fig. 2
Fig. 2

Film thickness measurement technique utilizing a laser triangulation sensor and vapor-deposited NiCr.

Fig. 3
Fig. 3

Film thickness measurement setup.

Fig. 4
Fig. 4

Sketch of calibration block. Nominal channel depths are shown in micrometers.

Fig. 5
Fig. 5

Parameters involved in theoretical refraction compensation.

Fig. 6
Fig. 6

Parameters involved in experimental determination of C 1.

Fig. 7
Fig. 7

3D surface plot of example liquid film thickness data.

Fig. 8
Fig. 8

Comparison of surface fit to experimental data.

Tables (2)

Tables Icon

Table 1 Seventeen Unknowns in the Theoretical Refraction Model

Tables Icon

Table 2 Eight Input Parameters in the Theoretical Refraction Model

Equations (29)

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

Δ Z app = Z m Z n .
Δ Z act = C T Δ Z app ,
n 1 sin θ 1 = n 2 sin θ 2 ,
L H = x m A + x m Q + x m L ,
L H = x n A + x n Q ,
sin θ m A = x m A ( x m A             2 + t A       2 ) 1 / 2 ,
sin θ n A = x n A ( x n A           2 + t A       2 ) 1 / 2 ,
sin θ m Q = x m Q ( x m Q             2 + t Q       2 ) 1 / 2 ,
sin θ n Q = x n Q ( x n Q           2 + t Q       2 ) 1 / 2 ,
sin θ m L = x m L ( x m L             2 + Δ Z act           2 ) 1 / 2 .
n Q n A = sin θ m A sin θ m Q ,
n Q n A = sin θ n A sin θ n Q ,
n L n Q = sin θ m Q sin θ m L .
t A = D S + C 1 .
L V = t A + t Q .
t A x n A = L V δ n L H ,
t A x m A = L V + Δ Z act δ m L H .
tan θ m A = x 1 Δ Z app ,
t A x m A = L V δ n L H x 1 .
L H = x A * + x Q * ,
sin θ A * = x A * [ ( x A * ) 2 + ( t A * ) 2 ] 1 / 2 ,
sin θ Q * = x Q * [ ( x Q * ) 2 + t Q       2 ] 1 / 2 ,
n Q n A = sin θ A * sin θ Q * ,
tan θ A * = x Q * t Q δ * ,
C 1 = t A * D S * .
Δ Z = Z bottom Z top ,
C E = Δ Z ¯ dry Δ Z ¯ wetted ,
σ Δ Z = [ ( σ Z bottom ) 2 + ( σ Z top ) 2 ] 1 / 2 .
( u C E C E ) 2 = ( u Δ Z ¯ dry Δ Z ¯ dry ) 2 ( u Δ Z ¯ wetted Δ Z ¯ wetted ) 2 .

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