Abstract

A novel technique is presented for measuring micro flow rate using the near infrared (NIR) absorption method. The principle of this method is based on the temperature dependency of the NIR absorption band of water (O-H band, ν1 + ν3). We obtained the water temperature in the tube in situ condition using NIR absorption method. A calibration curve between temperature and the NIR absorption intensity in the range of 1500 nm – 1700 nm wavelength was obtained. For measuring flow rate in the tube, the tiny spot of water in the tube was heated using NIR laser (1450 nm) through the lens which was absorbed into the water. The temperature profiles along the tube were obtained using the NIR absorption method via laser heating for different flow rates. The simulation results of the temperature profiles were well matched with the experimental results of it for different flow rates. We found that the conduction affected the temperature more when the flow was low in the upstream and the convection more affected when the flow rate was high in the downstream through the heat transfer analysis. The flow rates were obtained from the temperature difference between the room temperature and the obtained temperature from the NIR method. The calibration curves between the flow rate and temperature obtained from the NIR absorption method was obtained in the two flow rates (1-20 mL/h and 40-100 mL/h). The error and uncertainty of the NIR absorption method for measuring flow rate were approximately 1.2% and 1% at the 1-100 mL/min flow rate, respectively. Thus, we confirmed that the NIR absorption method quantitatively measures the flow rate with respect to the in situ condition for the first time. This method is used for various applications including biomedical and chemical processing without causing any contamination owing to the flow meter installation.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Water film thickness imaging based on time-multiplexed near-infrared absorption

Marc Lubnow, Jay B. Jeffries, Thomas Dreier, and Christof Schulz
Opt. Express 26(16) 20902-20912 (2018)

Temperature measurements of turbid aqueous solutions using near-infrared spectroscopy

Naoto Kakuta, Hidenobu Arimoto, Hideyuki Momoki, Fuguo Li, and Yukio Yamada
Appl. Opt. 47(13) 2227-2233 (2008)

Temporally resolved two dimensional temperature field of acoustically excited swirling flames measured by mid-infrared direct absorption spectroscopy

Xunchen Liu, Guoqing Wang, Jianyi Zheng, Liangliang Xu, Sirui Wang, Lei Li, and Fei Qi
Opt. Express 26(24) 31983-31994 (2018)

References

  • View by:
  • |
  • |
  • |

  1. D. Della Valle, P. A. Tanguy, and P. J. Carreau, “Characterization of the extensional properties of complex fluids using an orifice flowmeter,” J. Non-Newt. Fluid Mech. 94(1), 1–13 (2000).
    [Crossref]
  2. M. Henry, D. Clarke, N. Archer, J. Bowles, M. Leahy, R. Liu, J. Vignos, and F. Zhou, “A self-validating digital Coriolis mass-flow meter: an overview,” Control Eng. Pract. 8(5), 487–506 (2000).
    [Crossref]
  3. S. Chun, B.-R. Yoon, W. Kang, and H.-S. Kwon, “Assessment of combined V/Z clamp-on ultrasonic flow metering,” Journal of Mechanical Science and Technology 28(6), 2169–2177 (2014).
    [Crossref]
  4. T. H. Kim, D.-K. Kim, and S. J. Kim, “Study of the sensitivity of a thermal flow sensor,” Int. J. Heat Mass Transfer 52(7-8), 2140–2144 (2009).
    [Crossref]
  5. N. Nguyen, “Micromachined flow sensors—a review,” Flow Meas. Instrum. 8(1), 7–16 (1997).
    [Crossref]
  6. M. Toloui and J. Hong, “High fidelity digital inline holographic method for 3D flow measurements,” Opt. Express 23(21), 27159–27173 (2015).
    [Crossref] [PubMed]
  7. A. Braeuer, S. R. Engel, S. Dowy, S. Luther, J. Goldlücke, and A. Leipertz, “Raman mixture composition and flow velocity imaging with high repetition rates,” Opt. Express 18(24), 24579–24587 (2010).
    [Crossref] [PubMed]
  8. C. S. Garbe, K. Roetmann, V. Beushausen, and B. Jähne, “An optical flow MTV based technique for measuring microfluidic flow in the presence of diffusion and Taylor dispersion,” Exp. Fluids 44(3), 439–450 (2008).
    [Crossref]
  9. M. Olsen and R. Adrian, “Out-of-focus effects on particle image visibility and correlation in microscopic particle image velocimetry,” Exp. Fluids 29(7), S166–S174 (2000).
    [Crossref]
  10. M. Takahashi, T. Furukawa, Y. Sato, and K. Hishida, “Non-intrusive velocity measurement of millichannel flow by spontaneous Raman imaging,” Journal of Thermal Science and Technology 7(3), 406–413 (2012).
    [Crossref]
  11. M. M. Koochesfahani and D. G. Nocera, “Molecular tagging velocimetry,” Handbook of experimental fluid dynamics, 362–382 (2007).
  12. H. W. Siesler, Y. Ozaki, S. Kawata, and H. M. Heise, Near-infrared spectroscopy: principles, instruments, applications (John Wiley & Sons, 2008).
  13. F. O. Libnau, O. M. Kvalheim, A. A. Christy, and J. Toft, “Spectra of water in the near-and mid-infrared region,” Vib. Spectrosc. 7(3), 243–254 (1994).
    [Crossref]
  14. N. Kakuta, Y. Fukuhara, K. Kondo, H. Arimoto, and Y. Yamada, “Temperature imaging of water in a microchannel using thermal sensitivity of near-infrared absorption,” Lab Chip 11(20), 3479–3486 (2011).
    [Crossref] [PubMed]
  15. S. J. Kim and S. P. Jang, “Experimental and numerical analysis of heat transfer phenomena in a sensor tube of a mass flow controller,” Int. J. Heat Mass Transfer 44(9), 1711–1724 (2001).
    [Crossref]

2015 (1)

2014 (1)

S. Chun, B.-R. Yoon, W. Kang, and H.-S. Kwon, “Assessment of combined V/Z clamp-on ultrasonic flow metering,” Journal of Mechanical Science and Technology 28(6), 2169–2177 (2014).
[Crossref]

2012 (1)

M. Takahashi, T. Furukawa, Y. Sato, and K. Hishida, “Non-intrusive velocity measurement of millichannel flow by spontaneous Raman imaging,” Journal of Thermal Science and Technology 7(3), 406–413 (2012).
[Crossref]

2011 (1)

N. Kakuta, Y. Fukuhara, K. Kondo, H. Arimoto, and Y. Yamada, “Temperature imaging of water in a microchannel using thermal sensitivity of near-infrared absorption,” Lab Chip 11(20), 3479–3486 (2011).
[Crossref] [PubMed]

2010 (1)

2009 (1)

T. H. Kim, D.-K. Kim, and S. J. Kim, “Study of the sensitivity of a thermal flow sensor,” Int. J. Heat Mass Transfer 52(7-8), 2140–2144 (2009).
[Crossref]

2008 (1)

C. S. Garbe, K. Roetmann, V. Beushausen, and B. Jähne, “An optical flow MTV based technique for measuring microfluidic flow in the presence of diffusion and Taylor dispersion,” Exp. Fluids 44(3), 439–450 (2008).
[Crossref]

2001 (1)

S. J. Kim and S. P. Jang, “Experimental and numerical analysis of heat transfer phenomena in a sensor tube of a mass flow controller,” Int. J. Heat Mass Transfer 44(9), 1711–1724 (2001).
[Crossref]

2000 (3)

M. Olsen and R. Adrian, “Out-of-focus effects on particle image visibility and correlation in microscopic particle image velocimetry,” Exp. Fluids 29(7), S166–S174 (2000).
[Crossref]

D. Della Valle, P. A. Tanguy, and P. J. Carreau, “Characterization of the extensional properties of complex fluids using an orifice flowmeter,” J. Non-Newt. Fluid Mech. 94(1), 1–13 (2000).
[Crossref]

M. Henry, D. Clarke, N. Archer, J. Bowles, M. Leahy, R. Liu, J. Vignos, and F. Zhou, “A self-validating digital Coriolis mass-flow meter: an overview,” Control Eng. Pract. 8(5), 487–506 (2000).
[Crossref]

1997 (1)

N. Nguyen, “Micromachined flow sensors—a review,” Flow Meas. Instrum. 8(1), 7–16 (1997).
[Crossref]

1994 (1)

F. O. Libnau, O. M. Kvalheim, A. A. Christy, and J. Toft, “Spectra of water in the near-and mid-infrared region,” Vib. Spectrosc. 7(3), 243–254 (1994).
[Crossref]

Adrian, R.

M. Olsen and R. Adrian, “Out-of-focus effects on particle image visibility and correlation in microscopic particle image velocimetry,” Exp. Fluids 29(7), S166–S174 (2000).
[Crossref]

Archer, N.

M. Henry, D. Clarke, N. Archer, J. Bowles, M. Leahy, R. Liu, J. Vignos, and F. Zhou, “A self-validating digital Coriolis mass-flow meter: an overview,” Control Eng. Pract. 8(5), 487–506 (2000).
[Crossref]

Arimoto, H.

N. Kakuta, Y. Fukuhara, K. Kondo, H. Arimoto, and Y. Yamada, “Temperature imaging of water in a microchannel using thermal sensitivity of near-infrared absorption,” Lab Chip 11(20), 3479–3486 (2011).
[Crossref] [PubMed]

Beushausen, V.

C. S. Garbe, K. Roetmann, V. Beushausen, and B. Jähne, “An optical flow MTV based technique for measuring microfluidic flow in the presence of diffusion and Taylor dispersion,” Exp. Fluids 44(3), 439–450 (2008).
[Crossref]

Bowles, J.

M. Henry, D. Clarke, N. Archer, J. Bowles, M. Leahy, R. Liu, J. Vignos, and F. Zhou, “A self-validating digital Coriolis mass-flow meter: an overview,” Control Eng. Pract. 8(5), 487–506 (2000).
[Crossref]

Braeuer, A.

Carreau, P. J.

D. Della Valle, P. A. Tanguy, and P. J. Carreau, “Characterization of the extensional properties of complex fluids using an orifice flowmeter,” J. Non-Newt. Fluid Mech. 94(1), 1–13 (2000).
[Crossref]

Christy, A. A.

F. O. Libnau, O. M. Kvalheim, A. A. Christy, and J. Toft, “Spectra of water in the near-and mid-infrared region,” Vib. Spectrosc. 7(3), 243–254 (1994).
[Crossref]

Chun, S.

S. Chun, B.-R. Yoon, W. Kang, and H.-S. Kwon, “Assessment of combined V/Z clamp-on ultrasonic flow metering,” Journal of Mechanical Science and Technology 28(6), 2169–2177 (2014).
[Crossref]

Clarke, D.

M. Henry, D. Clarke, N. Archer, J. Bowles, M. Leahy, R. Liu, J. Vignos, and F. Zhou, “A self-validating digital Coriolis mass-flow meter: an overview,” Control Eng. Pract. 8(5), 487–506 (2000).
[Crossref]

Della Valle, D.

D. Della Valle, P. A. Tanguy, and P. J. Carreau, “Characterization of the extensional properties of complex fluids using an orifice flowmeter,” J. Non-Newt. Fluid Mech. 94(1), 1–13 (2000).
[Crossref]

Dowy, S.

Engel, S. R.

Fukuhara, Y.

N. Kakuta, Y. Fukuhara, K. Kondo, H. Arimoto, and Y. Yamada, “Temperature imaging of water in a microchannel using thermal sensitivity of near-infrared absorption,” Lab Chip 11(20), 3479–3486 (2011).
[Crossref] [PubMed]

Furukawa, T.

M. Takahashi, T. Furukawa, Y. Sato, and K. Hishida, “Non-intrusive velocity measurement of millichannel flow by spontaneous Raman imaging,” Journal of Thermal Science and Technology 7(3), 406–413 (2012).
[Crossref]

Garbe, C. S.

C. S. Garbe, K. Roetmann, V. Beushausen, and B. Jähne, “An optical flow MTV based technique for measuring microfluidic flow in the presence of diffusion and Taylor dispersion,” Exp. Fluids 44(3), 439–450 (2008).
[Crossref]

Goldlücke, J.

Henry, M.

M. Henry, D. Clarke, N. Archer, J. Bowles, M. Leahy, R. Liu, J. Vignos, and F. Zhou, “A self-validating digital Coriolis mass-flow meter: an overview,” Control Eng. Pract. 8(5), 487–506 (2000).
[Crossref]

Hishida, K.

M. Takahashi, T. Furukawa, Y. Sato, and K. Hishida, “Non-intrusive velocity measurement of millichannel flow by spontaneous Raman imaging,” Journal of Thermal Science and Technology 7(3), 406–413 (2012).
[Crossref]

Hong, J.

Jähne, B.

C. S. Garbe, K. Roetmann, V. Beushausen, and B. Jähne, “An optical flow MTV based technique for measuring microfluidic flow in the presence of diffusion and Taylor dispersion,” Exp. Fluids 44(3), 439–450 (2008).
[Crossref]

Jang, S. P.

S. J. Kim and S. P. Jang, “Experimental and numerical analysis of heat transfer phenomena in a sensor tube of a mass flow controller,” Int. J. Heat Mass Transfer 44(9), 1711–1724 (2001).
[Crossref]

Kakuta, N.

N. Kakuta, Y. Fukuhara, K. Kondo, H. Arimoto, and Y. Yamada, “Temperature imaging of water in a microchannel using thermal sensitivity of near-infrared absorption,” Lab Chip 11(20), 3479–3486 (2011).
[Crossref] [PubMed]

Kang, W.

S. Chun, B.-R. Yoon, W. Kang, and H.-S. Kwon, “Assessment of combined V/Z clamp-on ultrasonic flow metering,” Journal of Mechanical Science and Technology 28(6), 2169–2177 (2014).
[Crossref]

Kim, D.-K.

T. H. Kim, D.-K. Kim, and S. J. Kim, “Study of the sensitivity of a thermal flow sensor,” Int. J. Heat Mass Transfer 52(7-8), 2140–2144 (2009).
[Crossref]

Kim, S. J.

T. H. Kim, D.-K. Kim, and S. J. Kim, “Study of the sensitivity of a thermal flow sensor,” Int. J. Heat Mass Transfer 52(7-8), 2140–2144 (2009).
[Crossref]

S. J. Kim and S. P. Jang, “Experimental and numerical analysis of heat transfer phenomena in a sensor tube of a mass flow controller,” Int. J. Heat Mass Transfer 44(9), 1711–1724 (2001).
[Crossref]

Kim, T. H.

T. H. Kim, D.-K. Kim, and S. J. Kim, “Study of the sensitivity of a thermal flow sensor,” Int. J. Heat Mass Transfer 52(7-8), 2140–2144 (2009).
[Crossref]

Kondo, K.

N. Kakuta, Y. Fukuhara, K. Kondo, H. Arimoto, and Y. Yamada, “Temperature imaging of water in a microchannel using thermal sensitivity of near-infrared absorption,” Lab Chip 11(20), 3479–3486 (2011).
[Crossref] [PubMed]

Kvalheim, O. M.

F. O. Libnau, O. M. Kvalheim, A. A. Christy, and J. Toft, “Spectra of water in the near-and mid-infrared region,” Vib. Spectrosc. 7(3), 243–254 (1994).
[Crossref]

Kwon, H.-S.

S. Chun, B.-R. Yoon, W. Kang, and H.-S. Kwon, “Assessment of combined V/Z clamp-on ultrasonic flow metering,” Journal of Mechanical Science and Technology 28(6), 2169–2177 (2014).
[Crossref]

Leahy, M.

M. Henry, D. Clarke, N. Archer, J. Bowles, M. Leahy, R. Liu, J. Vignos, and F. Zhou, “A self-validating digital Coriolis mass-flow meter: an overview,” Control Eng. Pract. 8(5), 487–506 (2000).
[Crossref]

Leipertz, A.

Libnau, F. O.

F. O. Libnau, O. M. Kvalheim, A. A. Christy, and J. Toft, “Spectra of water in the near-and mid-infrared region,” Vib. Spectrosc. 7(3), 243–254 (1994).
[Crossref]

Liu, R.

M. Henry, D. Clarke, N. Archer, J. Bowles, M. Leahy, R. Liu, J. Vignos, and F. Zhou, “A self-validating digital Coriolis mass-flow meter: an overview,” Control Eng. Pract. 8(5), 487–506 (2000).
[Crossref]

Luther, S.

Nguyen, N.

N. Nguyen, “Micromachined flow sensors—a review,” Flow Meas. Instrum. 8(1), 7–16 (1997).
[Crossref]

Olsen, M.

M. Olsen and R. Adrian, “Out-of-focus effects on particle image visibility and correlation in microscopic particle image velocimetry,” Exp. Fluids 29(7), S166–S174 (2000).
[Crossref]

Roetmann, K.

C. S. Garbe, K. Roetmann, V. Beushausen, and B. Jähne, “An optical flow MTV based technique for measuring microfluidic flow in the presence of diffusion and Taylor dispersion,” Exp. Fluids 44(3), 439–450 (2008).
[Crossref]

Sato, Y.

M. Takahashi, T. Furukawa, Y. Sato, and K. Hishida, “Non-intrusive velocity measurement of millichannel flow by spontaneous Raman imaging,” Journal of Thermal Science and Technology 7(3), 406–413 (2012).
[Crossref]

Takahashi, M.

M. Takahashi, T. Furukawa, Y. Sato, and K. Hishida, “Non-intrusive velocity measurement of millichannel flow by spontaneous Raman imaging,” Journal of Thermal Science and Technology 7(3), 406–413 (2012).
[Crossref]

Tanguy, P. A.

D. Della Valle, P. A. Tanguy, and P. J. Carreau, “Characterization of the extensional properties of complex fluids using an orifice flowmeter,” J. Non-Newt. Fluid Mech. 94(1), 1–13 (2000).
[Crossref]

Toft, J.

F. O. Libnau, O. M. Kvalheim, A. A. Christy, and J. Toft, “Spectra of water in the near-and mid-infrared region,” Vib. Spectrosc. 7(3), 243–254 (1994).
[Crossref]

Toloui, M.

Vignos, J.

M. Henry, D. Clarke, N. Archer, J. Bowles, M. Leahy, R. Liu, J. Vignos, and F. Zhou, “A self-validating digital Coriolis mass-flow meter: an overview,” Control Eng. Pract. 8(5), 487–506 (2000).
[Crossref]

Yamada, Y.

N. Kakuta, Y. Fukuhara, K. Kondo, H. Arimoto, and Y. Yamada, “Temperature imaging of water in a microchannel using thermal sensitivity of near-infrared absorption,” Lab Chip 11(20), 3479–3486 (2011).
[Crossref] [PubMed]

Yoon, B.-R.

S. Chun, B.-R. Yoon, W. Kang, and H.-S. Kwon, “Assessment of combined V/Z clamp-on ultrasonic flow metering,” Journal of Mechanical Science and Technology 28(6), 2169–2177 (2014).
[Crossref]

Zhou, F.

M. Henry, D. Clarke, N. Archer, J. Bowles, M. Leahy, R. Liu, J. Vignos, and F. Zhou, “A self-validating digital Coriolis mass-flow meter: an overview,” Control Eng. Pract. 8(5), 487–506 (2000).
[Crossref]

Control Eng. Pract. (1)

M. Henry, D. Clarke, N. Archer, J. Bowles, M. Leahy, R. Liu, J. Vignos, and F. Zhou, “A self-validating digital Coriolis mass-flow meter: an overview,” Control Eng. Pract. 8(5), 487–506 (2000).
[Crossref]

Exp. Fluids (2)

C. S. Garbe, K. Roetmann, V. Beushausen, and B. Jähne, “An optical flow MTV based technique for measuring microfluidic flow in the presence of diffusion and Taylor dispersion,” Exp. Fluids 44(3), 439–450 (2008).
[Crossref]

M. Olsen and R. Adrian, “Out-of-focus effects on particle image visibility and correlation in microscopic particle image velocimetry,” Exp. Fluids 29(7), S166–S174 (2000).
[Crossref]

Flow Meas. Instrum. (1)

N. Nguyen, “Micromachined flow sensors—a review,” Flow Meas. Instrum. 8(1), 7–16 (1997).
[Crossref]

Int. J. Heat Mass Transfer (2)

T. H. Kim, D.-K. Kim, and S. J. Kim, “Study of the sensitivity of a thermal flow sensor,” Int. J. Heat Mass Transfer 52(7-8), 2140–2144 (2009).
[Crossref]

S. J. Kim and S. P. Jang, “Experimental and numerical analysis of heat transfer phenomena in a sensor tube of a mass flow controller,” Int. J. Heat Mass Transfer 44(9), 1711–1724 (2001).
[Crossref]

J. Non-Newt. Fluid Mech. (1)

D. Della Valle, P. A. Tanguy, and P. J. Carreau, “Characterization of the extensional properties of complex fluids using an orifice flowmeter,” J. Non-Newt. Fluid Mech. 94(1), 1–13 (2000).
[Crossref]

Journal of Mechanical Science and Technology (1)

S. Chun, B.-R. Yoon, W. Kang, and H.-S. Kwon, “Assessment of combined V/Z clamp-on ultrasonic flow metering,” Journal of Mechanical Science and Technology 28(6), 2169–2177 (2014).
[Crossref]

Journal of Thermal Science and Technology (1)

M. Takahashi, T. Furukawa, Y. Sato, and K. Hishida, “Non-intrusive velocity measurement of millichannel flow by spontaneous Raman imaging,” Journal of Thermal Science and Technology 7(3), 406–413 (2012).
[Crossref]

Lab Chip (1)

N. Kakuta, Y. Fukuhara, K. Kondo, H. Arimoto, and Y. Yamada, “Temperature imaging of water in a microchannel using thermal sensitivity of near-infrared absorption,” Lab Chip 11(20), 3479–3486 (2011).
[Crossref] [PubMed]

Opt. Express (2)

Vib. Spectrosc. (1)

F. O. Libnau, O. M. Kvalheim, A. A. Christy, and J. Toft, “Spectra of water in the near-and mid-infrared region,” Vib. Spectrosc. 7(3), 243–254 (1994).
[Crossref]

Other (2)

M. M. Koochesfahani and D. G. Nocera, “Molecular tagging velocimetry,” Handbook of experimental fluid dynamics, 362–382 (2007).

H. W. Siesler, Y. Ozaki, S. Kawata, and H. M. Heise, Near-infrared spectroscopy: principles, instruments, applications (John Wiley & Sons, 2008).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1
Fig. 1 (a) Experimental setup (b) Magnification of measurement position shown in Fig. 1(a).
Fig. 2
Fig. 2 Absorption spectra for different temperatures obtained by FTNIR.
Fig. 3
Fig. 3 Absorption intensity obtained from FTNIR, according to the DI water temperature.
Fig. 4
Fig. 4 Absorption spectra at different (a) upstream and (b) downstream positions according to the laser-focused position (0 mm) for 1 mL/h flow rate.
Fig. 5
Fig. 5 Absorption spectra for different flow rates at the 1 mm position with IR laser heating.
Fig. 6
Fig. 6 Temperature profiles according to the position of tube for different flow rates (1–100 mL/h) with IR laser heating at the 0 mm position.
Fig. 7
Fig. 7 Fluid cell considering laser induced heat generation, conductive and convective heat transfer and heat loss to the atmosphere.
Fig. 8
Fig. 8 Comparison of temperature profiles between the experimental and simulation results of 1mL/h, 10mL/h and 60mL/h flow rate case.
Fig. 9
Fig. 9 Calculated percentage of heat transfer (conduction and convection) for different flow rates (1, 10, and 60 mL/h) at 0 mm position.
Fig. 10
Fig. 10 Difference between room temperature (Troom, 22.5 °C) and temperature (T), according to the flow rate for different positions (a) upstream and (b) downstream with IR laser heating at the 0 mm position.
Fig. 11
Fig. 11 Flow rate according to the difference of room temperature (Troom, 22.5 °C) and temperature(T) at 1 mm position with exponential curve fitting for different flow rate ranges (1-20 mL/h, 40-100 mL/h).
Fig. 12
Fig. 12 Error of NIR absorption method according to the flow rate with measurement uncertainty.

Tables (1)

Tables Icon

Table 1 Error (%) and measurement uncertainty for syringe pump with respect to the flow rate of a micro flow gravimetric system

Equations (5)

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

q= ε( W 2 W 1 +δW) ρt
q ˙ laser + q ˙ conv,i1i = q ˙ cond,ii1 + q ˙ cond,ii+1 + q ˙ loss
q ˙ laser + m ˙ C p ( T i T i1 )=KA T i T i1 Δx +KA T i T i+1 Δx +hPΔx( T i T )
T i=1 = T , T t=N =0
Error= ( q NIR q REF ) q REF ×100

Metrics