- Email: [email protected]
Measurement of temperature gradient in a heated liquid cylinder using rainbow refractometry assisted with infrared thermometry
Optics Communications 380 (2016) 179–185
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Measurement of temperature gradient in a heated liquid cylinder using rainbow refractometry assisted with infrared thermometry Feihu Song a,n, Chuanlong Xu b, Shimin Wang b, Zhenfeng Li a a Jiangsu Key Laboratory of Advanced Food Manufacturing Equipment & Technology, School of Mechanical Engineering Jiangnan University, Wuxi 214122, China b Key Laboratory of Energy Thermal Conversion and Control of Ministry of Education, School of Energy and Environment, Southeast University, Nanjing 210096, China
art ic l e i nf o
a b s t r a c t
Article history: Received 19 February 2016 Received in revised form 27 May 2016 Accepted 5 June 2016 Available online 13 June 2016
In a heated liquid cylinder, a temperature gradient exists between the hotter surface and the colder core. Measurement of the temperature gradient is very important for the in-depth investigation into the mechanism and optimized control of the heat transfer process. Rainbow refractometry has been attempted to achieve the measurement since several years ago. Yet there is no effective inversion algorithm without predicting refractive index profiles based on experience. In the paper, an improved rainbow refractometry assisted with infrared thermometry was proposed to measure the diameter and the inside temperature gradient of a heated liquid cylinder. The inversion algorithm was designed based on the nonlinear least square method and an optimization process. To evaluate the feasibility of the proposed method, numerical simulations and experiments were carried out. The results of the numerical simulation showed that the relative error of the inversion diameter was about 1%, and the error of the refractive index was less than 6 10 4 at all the radial locations. In the experimental research, t rainbows reconstructed with the reversion parameters were all similar to the corresponding captured ones. & 2016 Published by Elsevier B.V.
Keywords: Rainbow refractometry Temperature gradient measurement Heated cylinder Infrared thermometry
1. Introduction In most industrial processes, the temperature inside a liquid cylinder is inhomogeneous because of heat transfer. When the liquid cylinder is heated with radiation or heat convection from its outside vertical surface, the temperature decreases from the surface to the core, and the temperature gradient varies along with time. Yet there is no measurement method for the temperature gradient inside a cylinder. Contacting thermometry is commonly used at present, such as thermocouple, optical fiber thermometry etc. Yet the response is slow, and the flue field is disturbed by the probes [1]. Infrared thermometry is a non-intrusive technique. But only the surface temperature can be determined, because the infrared radiated by the internal liquid cannot transmit through the surface [2]. Ultrasonic thermometry is also a non-intrusive technique which can be used to detect the mean temperature of the liquid cylinder. But as long as the mean temperature is constant, the measurement signal of the probe remains the same no matter the temperature gradient inside the cylinder changes [3,4]. Rainbow refractometry is a potential technique for the n
Corresponding author. E-mail address: [email protected] (F. Song).
http://dx.doi.org/10.1016/j.optcom.2016.06.011 0030-4018/& 2016 Published by Elsevier B.V.
measurement of the temperature gradient inside a cylinder, because the determination is on the basis of the rainbow pattern affected by refractive index gradient instead of the mean refractive index [5]. Rainbow refractometry was put forward in 1988 to determine the refractive index (related to the temperature) and the diameter of a cylinder simultaneously [6]. However, only the mean refractive index could be determined with the proposed algorithms of standard rainbow refractometry at the beginning [7,8]. Due to the urgent requirement for the analysis of heat transfer inside a cylinder, later a number of researches aimed to achieve the measurement of the refractive index gradient. Li proposed an effective calculation model for the scattering intensity of the rainbow scattered by a multilayered droplet [9]. Vetrano proposed an inversion algorithm for determining the diameter and the refractive index gradient of a droplet, and the burning oil droplets were taken for experimental research [10]. However, the sharp of the refractive index profile should be predicted based on the experience that the temperature profile changed as the burning develops. So the method cannot be used if heat transfer condition changes. Saengkaew attempted to determine the diameter and the refractive index gradient of an inhomogeneous droplet with the algorithm for a homogeneous droplet [11]. Only the inversion diameter was similar to the true value, but the inversion refractive
F. Song et al. / Optics Communications 380 (2016) 179–185
index gradient was completely wrong. To sum up, actually there is no valid measurement method for the refractive index distribution inside a cylinder. To solve the problem, an improved rainbow refractometry assisted with infrared thermometry was proposed to simultaneously determine the diameter and the refractive index gradient of a cylinder without any priori value. Firstly, the surface refractive index (related to the surface temperature) was measured with infrared thermometry to decrease the unknown parameters. Secondly, the cylinder diameter was estimated with the interval of the maxima in the first two Airy fringes. Thirdly, the estimated core refractive index is expressed with the shape coefficient, so that only the shape coefficient should be solved with non-linear least square method. Finally, the two estimated values were optimized with an optimization process. Numerical simulations and experiments were carried out to verify the proposed method.
1.340
Refractive index
180
1.335 1.330 1.325 1.320 0.0
True Reversion 1 Reversion 2
0.2
0.4
0.6
0.8
1.0
Normalized radius Fig. 1. The comparison of the true and the inversion refractive index gradients.
2.1. Principle of the rainbow refractometry for inhomogeneous cylinder When a homogeneous cylinder is illuminated by a light beam, a monochrome primary rainbow pattern can be observed in the backward area [6]. Based on scattering theories, inversion algorithms have been proposed to determine the diameter and the refractive index of the cylinder from the rainbow pattern [12–14]. If heat transfer happens to the cylinder, temperature inside is inhomogeneous, and hence the refractive index gradient exists. To simplify the refractive index gradient, it can be described with the expression as follows:
n (k ) = nc +
(ns − nc )(e bk − 1) eb − 1
(1)
where b is the shape coefficient of the profile, ns is the refractive index at surface, nc is the refractive index at core, k is the serial number of the layers, n(k) is the refractive index of kth layer. To investigate the relationship between of the scattering intensity distribution and the parameters of an inhomogeneous cylinder, the intensity distribution of the rainbow should be calculated with the refractive index gradient and the diameter at first. Li proposed a model to calculate the scattering intensity distribution by iteration from the core layer to the surface layer [9]. And the intensity distribution of the Airy fringes can be calculated as well. According to the reported research, the intensity distribution is related to the inside temperature gradient as well as the cylinder diameter (denoted by D), which means it is potential to determine the temperature gradient from the scattering intensity distribution with a proper inversion algorithm [14,15]. 2.2. Refractive index gradient inversion with non-linear least square method The refractive index gradient is firstly attempted to be determined with non-linear least square method directly, which is a useful method for solving parameters in a nonlinear problem. The objective function is as follows:
f=
∑ ⎡⎣ Icap, Airy (k) − Ire, Airy (k) ⎤⎦2 k
(2)
where f is objective function value, Icap, Airy is the intensity of the Airy fringes in the captured rainbow, Ire, Airy is the intensity of the Airy fringes in the rainbow reconstructed with ns, nc and b during
the iteration process, k is the serial number of the discrete scattering angle. The initial values of ns, nc and b are set at first. Then in the iteration process, the initial ns, nc and b are continuously optimized with non-linear least square method for the purpose of decreasing the objective function value. And finally a refractive index gradient (described with ns, nc and b) is obtained, with which the final reconstructed rainbow is similar to the captured one. The numerical simulation of the inversion algorithm is then carried out. The parameters of a liquid cylinder are given, so that the intensity distribution of the rainbow pattern can be calculated, which is considered as the captured rainbow. A water cylinder heated with heat convection from its outside vertical surface is taken as the object in the numerical simulation. Firstly, true values of b, ns, nc are set 3, 1.326, 1.336. Fig. 1 shows the true refractive index gradient and the inversion gradients. With the non-linear least square method, different inversion values of ns, nc, and b can be obtained from different initial values. The corresponding rainbow patterns are shown in Fig. 2. It can be seen that the intensity distributions of the three rainbows are similar, especially the Airy fringes (i.e. the low frequency components). But the inversion refractive index gradients are both different from the true one. The main reason of the multi-solution problem is that the intensity distribution of the rainbow is affected by all the three Captured Reconstructed 1 Reconstructed 2
1.0
Rainbows 0.5
Intensity (a.u.)
2. Inversion algorithm for temperature gradient in an inhomogeneous cylinder
0.0 136
140
1.0
144 Captured Reconstructed 1 Reconstructed 2
Airy fringes 0.5
0.0 136
140
144
Scattering angle ( ° ) Fig. 2. The comparison of the captured and the reconstructed rainbows, the low frequency component obtained by the low-pass filter with hamming window.
F. Song et al. / Optics Communications 380 (2016) 179–185
Refractive index
1.335
True Reversion 1 Reversion 2
1.330
1.325 0.0
0.2
0.4 0.6 Normalized radius
0.8
1.0
Fig. 3. The comparison of the true and the inversion refractive index gradients when only b is needed to be solved.
181
are attempted to be obtained with another method and only b is needed to be determined, the inversion refractive index gradients with non-linear least square method are shown in Fig. 3. And the corresponding rainbow patterns are shown in Fig. 4. Before the iteration process, the initial values of b are set as 3 and 8 respectively. The inversion values of b are 2.9946 and 9.7547 respectively. One inversion refractive index gradient is almost the same to the true one, but the other is obviously different from the true one. The captured and the reconstructed rainbows shown in Fig. 4 are still similar, especially the Airy fringes. As b changes, the scattering angles of the maxima in the first Airy fringes are shown in Fig. 5. It can be seen that there is a maximum in the curve, where b is about 6. For an arbitrary value of b which is less than 6, another b larger than 6 corresponds to the same scattering angle. Therefore with the two inversion values of b, the intensity distributions of the two reconstructed Airy fringes are similar. Whereas the ripple structures of two reconstructed rainbows are different, so another objective function is used to judge which one is better:
f=
∑ ⎡⎣ Icap (k) − Ire, Mie (k) ⎤⎦2
(3)
k
Captured Reconstructed 1 Reconstructed 2
1.0
Rainbows
where Ire, Mie is the intensity distribution of the rainbow reconstructed with the inversion parameters. In all, more accurate inversion refractive index gradient can be determined, if only b is needed to be solved.
Intensity (a.u.)
0.5
2.3. Improved rainbow refractometry assisted with infrared thermometry
0.0 136
140
144
1.0
Captured Reconstructed 1 Reconstructed 2
Airy fringes 0.5
0.0 136
140
144
Scattering angle ( °) Fig. 4. The comparison of the captured and the reconstructed rainbows when only b is needed to be solved, the low frequency component obtained by the low-pass filter with hamming window.
Scattering angle of the maximum in the first Airy fringe (°)
140.3 140.2
In the improved method, the cylinder diameter is firstly estimated according to Vetrano's work [10]. The angular interval of the maxima in the first two Airy fringes was reported to be mainly related to the cylinder diameter and slightly related to the refractive index gradient. Therefore the interval can be used to estimate the cylinder diameter. The exponential fitting of the angular interval to the cylinder diameter is shown as follows, which is obtained from the rainbows calculated with nc ¼1.336, ns ¼1.326, b¼3.
D = 565.1⋅Δθ −1.447 , R2 = 0.9987
(4)
where Δθ is the angular interval. Infrared thermometry is used as assistance for the measurement of the surface temperature Ts, and therefore the surface refractive index can be obtained. On the other hand, Vetrano proposed a formula to describe the relationship between nc and ns [10]:
nrg = ns + δ
140.1
n c − ns ns2
(5)
where nrg is the inversion refractive index with the algorithm for homogeneous cylinder (which can be determined from the captured rainbow with the algorithm reported in the previous work [12–14]), δ is mainly related to b. As b changes, the fitting polynomial is shown in Formula (6), which is obtained from the rainbows calculated with nc ¼1.34, ns ¼1.32, D ¼100 μm.
140.0 139.9
δ = 0.001673⋅b3−0.04784⋅b2 + 0.404⋅b + 1.321 ,
139.8 2
4
6
b
8
10
12
Fig. 5. b vs. the scattering angle of the maximum in the first Airy fringe (nc ¼ 1.336, ns ¼ 1.326, D¼ 100 μm).
parameters, and the effect of changing one parameter on the rainbow can be offset by changing the other two parameters properly. So the unknown parameters should be decreased. If ns, nc
R2 = 0.9986
(6)
So the evaluated nc can be expressed with b.
nc =
1 + ns δ⋅ns ⋅ns ⋅(nrg −ns )
(7)
Therefore only b is needed to be solved. While using the algorithm described in Section 2.2, b is solved in the ranges of 2–6 and 6–12 respectively. The two estimated parameters (D and nc) are needed to be
182
F. Song et al. / Optics Communications 380 (2016) 179–185
1.0
Intensity (a.u.)
Angular difference
Captured Reconstructed 0.5
140
141 Scattering angle ( ° )
142
Fig. 8. Airy fringes of the reconstructed rainbow before nc is optimized (for the captured rainbow nc ¼ 1.34, ns ¼ 1.32, b ¼3.5, D ¼100 μm, for the reconstructed rainbow nc ¼1.3414, ns ¼ 1.3202, b ¼2.53, D ¼ 100.6 μm).
Fig. 6. Flow chart of the improved algorithm.
Intensity (a.u.)
1.0
Captured Reconstructed1, b=3.11 Reconstructed2, b=8
0.5
0.0 138
140
142 144 Scattering angle (°)
Fig. 9. Rainbow reconstructed with the inversion results (the captured rainbow is also attached for comparison).
Fig. 7. Airy fringes of the reconstructed rainbow before diameter is optimized (for the captured rainbow nc ¼ 1.34, ns ¼ 1.32, b ¼3.5, D ¼ 100 μm, for the reconstructed rainbow nc ¼ 1.3424, ns ¼1.3202, b¼ 2.02, D¼ 106.6 μm).
optimized. The flow chart of the improved algorithm is shown in Fig. 6. Then the numerical simulation is carried out to explain the flow chart. The true parameters of the cylinder are set as b¼ 3.5, ns ¼1.32, nc ¼ 1.34, D¼ 100 μm, and the intensity distribution can be calculated. For most commonly used instruments, the measurement error of infrared thermometry is 7 1 °C (corresponding to the refractive index error about 2 10 4 for water [15,16]). So the measurement of ns is assumed as 1.3202. The diameter estimated with Formula (5) is 106.6 μm. With the algorithm mentioned above, nc ¼ 1.3424, b¼2.02 are obtained. And the Airy fringes of the reconstructed rainbow are shown in Fig. 7. It can be seen that the maxima intervals in the two rainbows are different, which is mainly due to the error of the diameter. To solve this problem, the diameter is gradually adjusted to 101.2 μm. The inversion values are therefore updated (nc ¼ 1.3414, b¼ 2.53), and the Airy fringes of the reconstructed rainbow is shown in Fig. 8. The angular difference between the peaks of the two rainbows is mainly due to the
Refractive index
1.340 1.335 1.330 1.325
True Reversion 1
1.320 0.0
0.2
0.4 0.6 0.8 Normalized radius
1.0
Fig. 10. Comparison of the true and the final inversion refractive index gradient.
error of nc. So with the angular difference, the evaluation formula shown in Formula (7) is adjusted as follows:
nc =
1 + ns + Δn δ⋅ns ⋅ns ⋅(nrg −ns )
where
(8)
Δn is an offset, which is adjusted according to the angular
F. Song et al. / Optics Communications 380 (2016) 179–185
Pump Steel pipe
b ea
ms
Po
lar o
Electrothermal film covered with heat insulation material
0.5
La ser
1.0
Valve
id
Water tank
Steel capillary
Conves lens
s er
0.0
La
Relative error (%)
1.5
183
-0.5 Liquid cylinder
-1.0
CCD Camera
Infrared thermometer
-1.5
200
400
600
Diameter
800 (µm)
1000
(a) Relative error of the diameter
Fig. 12. Experimental system.
diameter is about 1%, and the absolute error of the refractive index is less than 6 10 4 at all the radial locations.
Error (10
−4
)
3. Experimental results and discussion
8 6 4 2 0 -2 -4 -6 -8
200
400
600
800
1000
True mean diameter (µm) (b) Error of the refractive index Fig. 11. Inversion errors when the diameter is in the range of 100–1100 μm, (a) Relative error of the diameter, (b) Error of the refractive index.
difference, i.e. at first Δn ¼0, and it is increased or decreased after each comparison. Still the refractive index gradient and the reconstructed rainbow are continuously updated until the angular difference shown in Fig. 8 is less than the preset value. Finally, two inversion results are obtained, nc ¼1.3405, ns ¼1.3202, b¼3.11, D ¼99.2 μm and nc ¼1.3404, ns ¼1.3202, b¼ 8, D ¼101.6 μm. The two reconstructed rainbows are shown in Fig. 9, which are similar to the captured rainbows. The objective function values calculated with Formula (3) are 0.8674 and 1.7123 respectively. As a result, the former inversion parameters are taken as the optimal result. The inversion refractive index gradient is shown in Fig. 10. It can be seen that the refractive index gradients are similar. The relative error of the inversion diameter is 0.8%. At all the radial locations, the error of the refractive index is less than 5 10 4, which corresponds to about 2.5 °C for water. Then the true cylinder diameters are set from 100–1100 μm with a step of 200 μm, the values of b are set randomly in the range of 2–12, nc, ns are set randomly in the range of 1.32–1.34, and nc is larger than ns during a heated cylinder (because the surface temperature is higher). The errors of the results are shown in Fig. 11. It can be seen that the absolute relative error of the
Fig. 12 shows the experimental system, consisting of a fluid system and an optical system. The fluid system consists of a water tank, a pump, a valve, a steel pipe, a steel capillary and an electrothermal film covered with heat resolution material. The water was pumped from the rank to the capillary by a water pump, so that a water cylinder was produced at the exit of the capillary. The internal diameter of the capillary was 1 mm. The height of the capillary is 50 mm. The flow rate of the water is 65 mL/min. The electrothermal film was used to heat the water flowing through. The powers of the electrothermal films were 72 W, 120 W and 180 W respectively. The optical system included an optical platform, a laser (100 mW, 532 nm), a convex lens, a CCD camera (e2v EV50YLM1GE2014, resolution 2048 100 pixels, pixel size 14 μm*14 μm). The water cylinder was illuminated by the laser beam. The scattering rays were focused onto the CCD camera by a convex lens. Fig. 13 shows the row rainbow patterns scattered by liquid cylinder heated with different powers. The grey values of the pattern captured by the CCD camera were extracted. The serial number of transverse pixels could be transformed to the scattering angle of the droplets by calibration. Thus the grey value at one certain pixel represented the scattering intensity at the angle. The surface temperature of the water cylinder was measured by an infrared thermometer (Raytek ST20, measurement error 71 °C). With infrared thermometry, the surface temperatures of the
(a) 72W
(b) 120W
(c) 180W Fig. 13. Row rainbow patterns when the heated power was different.
184
F. Song et al. / Optics Communications 380 (2016) 179–185
1.0
Intensity (a.u.)
Refractive index
1.336
1.334 72W 120W 180W
1.332
Captured Reconstructed
0.5
1.330 0.0
0.0
0.2
0.4 0.6 Normalized radius
0.8
1.0
138.0
1.0
60 Intensity (a.u.)
50 72W 120W 180W
40 30
139.5
(a) 72W
Fig. 14. Reversion refractive index gradients.
Temperature ( )
138.5 139.0 Scattering angle (° )
Captured Reconstructed
0.5
20 10 0.0
0.0
0.2
0.4
0.6
0.8
1.0
138.0
Normalized radius
139.5
(b) 120W
Fig. 15. Temperature gradients corresponding to the reversion refractive index gradients.
1.0
Intensity (a.u.)
liquid cylinders were 57.9 °C, 50.1 °C and 39.3 °C as the heating power increased. The reversion diameters were 901.2 μm, 893.6 μm and 905.4 μm, whose relative difference was 13.3%. Because of the viscous force between the capillary and the liquid cylinder, the reversion diameters were less than the internal diameter of the capillary. The refractive index gradients and the corresponding temperature gradients are shown in Fig. 14, and therefore the corresponding temperature gradients obtained from the refractive index gradients are Fig. 15 [15,16]. It can be seen that as the heating power increased, the surface temperature of the liquid cylinder increased as well, and the difference among the core temperatures was less than 2 °C. The captured and the reconstructed rainbows are illustrated in Fig. 16. In each graph, the two rainbows are similar, the objective function values calculated with formula (3) are 0.9675, 1.3524 and 1.1486 respectively in the three graphs.
138.5 139.0 Scattering angle (°)
Captured Reconstructed
0.5
0.0 138.0
138.5 139.0 139.5 Scattering angle (°)
(c) 180W 4. Conclusion Rainbow refractometry was investigated to attempt the measurement of the temperature gradient inside a heated liquid cylinder, without predicting any parameters based on experience. At first nonlinear least square method was directly used to solve the
Fig. 16. The captured and the reconstructed rainbows when the heating power was different.
F. Song et al. / Optics Communications 380 (2016) 179–185
parameters, but a multi-solution situation happened. Therefore an improved rainbow refractometry assisted with infrared thermometry was proposed to avoid the problem. The surface refractive index was obtained by measuring the surface temperature of the cylinder with infrared thermometry. And a rough formula between the core refractive index and the shape coefficient was built according to the previous work. Thus only one unknown parameters was left. An inversion algorithm was designed to achieve the measurement the refractive index gradient and the diameter simultaneously based on the nonlinear least square method and an optimization process. Numerical simulation and experiments were carried out to evaluate the method. The results of the numerical simulation showed that the relative error of the inversion diameter was about 1%, and the error of the refractive index was less than 6 10 4 at all the radial locations. In the experimental research, as the heating power increased, the surface refractive index increased, and the core refractive index almost remained the same. The reversion diameter was less than the internal diameter of the capillary because of the viscous force. The rainbows reconstructed with the reversion parameters were all similar to the corresponding captured ones.
Acknowledgment The authors wish to express their gratitude to the Program for National Natural Science Foundation of China (No. 51406068), National Natural Science Foundation of China (No.51376049), Fundamental Research Funds for the Central Universities (JUSRP51511), Foundation of Jiangsu union innovation (201501916) and Foundation of Jiangsu Key Laboratory of Advanced Food Manufacturing Equipment & Technology (FM-201504).
185
References [1] J. Char, J. Yeh, The measurement of open propane flame temperature using infrared technique, J. Quant. Spectrosc. Radiat. Transf. 1 (1996) 133–144. [2] T.P. Newson, H. Kee, All fiber system for simultaneous interrogation of distributed strain and temperature sensing by spontaneous Brillouin scattering, Opt. Lett. 10 (2000) 620–626. [3] A. Puttmer, P. Hauptmann, B. Henning, Ultrasonic Density Sensor for Liquid, IEEE Trans. Ultrason. Ferroelectr. Freq. Cont. 1 (2000) 85–92. [4] W. Tsai, H. Chen, T. Liao, High accuracy ultrasonic air temperature measurement using multi-frequency continuous wave, Sens. Actuators 132 (2006) 526–532. [5] P. Massoli, Rainbow refractometry applied to radially inhomogeneous spheres: the critical case of evaporating droplets, Appl. Opt. 15 (1998) 3227–3235. [6] N. Roth, K. Anders, A. Frohn, Simultaneous measurement of temperature and size of droplets in the micrometer range, in: Proceedings of the 7th International Congress on Optical Methods in Flow and Particle Diagnostics, ICALEO 88, L.I.A., CA, U.S.A, 67,1988, pp. 294–304. [7] N. Roth, K. Anders, A. Frohn, Simultaneous measurement of temperature and size of droplets in the micrometer range, J. Laser Appl. 2 (1990) 37–42. [8] N. Roth, K. Anders, A. Frohn, Refractive-index measurements for the correction of particle sizing methods, Appl. Opt. 33 (1991) 4960–4965. [9] R. Li, X. Han, H. Jiang, K. Ren, Debye series for light scattering by a multilayered sphere, Appl. Opt. 45 (2006) 1260–1270. [10] M.R. Vetrano, J.P.A.J. Van Beeck, M.L. Riethmuller, Assessment of refractive index gradients by standard rainbow thermometry, Appl. Opt. 34 (2005) 7275–7281. [11] S. Saengkaew, T. Charinpanitkul, H. Vanisri, Rainbow refractometry on particles with radial refractive index gradients, Exp. Fluids 4 (2007) 595–601. [12] J.P.A.J. Van Beeck, M.L. Riethmuller, Nonintrusive measurements of temperature and size of single falling raindrops, Appl. Opt. 34 (1995) 1633–1639. [13] F. Song, C. Xu, S. Wang, Y. Yan, An optimization scheme for the measurement of liquid jet parameters with rainbow refractometry based on Debye theory, Opt. Commun. 305 (2013) 204–211. [14] S. Saengkaew, T. Charinpanikul, C. Laurent, Y. Biscos, G. Lavergne, G. Gouesbet, et al., Processing of individual rainbow signals, Exp. Fluids 48 (2010) 111–119. [15] J. Hom, N. Chigier, Rainbow refractometry: simultaneous measurement of temperature, refractive index, and size of droplets, Appl. Opt. 10 (2002) 1899–1907. [16] I. Thormalen, J. Straub, U. Grigull, Refractive index of water and its dependence on wavelength, temperature, and density, J. Phys. Chem. Ref. Data 14 (1985) 933–945.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Measurement of temperature gradient in a heated liquid cylinder using rainbow refractometry assisted with infrared thermometry Feihu Song a,n, Chuanlong Xu b, Shimin Wang b, Zhenfeng Li a a Jiangsu Key Laboratory of Advanced Food Manufacturing Equipment & Technology, School of Mechanical Engineering Jiangnan University, Wuxi 214122, China b Key Laboratory of Energy Thermal Conversion and Control of Ministry of Education, School of Energy and Environment, Southeast University, Nanjing 210096, China
art ic l e i nf o
a b s t r a c t
Article history: Received 19 February 2016 Received in revised form 27 May 2016 Accepted 5 June 2016 Available online 13 June 2016
In a heated liquid cylinder, a temperature gradient exists between the hotter surface and the colder core. Measurement of the temperature gradient is very important for the in-depth investigation into the mechanism and optimized control of the heat transfer process. Rainbow refractometry has been attempted to achieve the measurement since several years ago. Yet there is no effective inversion algorithm without predicting refractive index profiles based on experience. In the paper, an improved rainbow refractometry assisted with infrared thermometry was proposed to measure the diameter and the inside temperature gradient of a heated liquid cylinder. The inversion algorithm was designed based on the nonlinear least square method and an optimization process. To evaluate the feasibility of the proposed method, numerical simulations and experiments were carried out. The results of the numerical simulation showed that the relative error of the inversion diameter was about 1%, and the error of the refractive index was less than 6 10 4 at all the radial locations. In the experimental research, t rainbows reconstructed with the reversion parameters were all similar to the corresponding captured ones. & 2016 Published by Elsevier B.V.
Keywords: Rainbow refractometry Temperature gradient measurement Heated cylinder Infrared thermometry
1. Introduction In most industrial processes, the temperature inside a liquid cylinder is inhomogeneous because of heat transfer. When the liquid cylinder is heated with radiation or heat convection from its outside vertical surface, the temperature decreases from the surface to the core, and the temperature gradient varies along with time. Yet there is no measurement method for the temperature gradient inside a cylinder. Contacting thermometry is commonly used at present, such as thermocouple, optical fiber thermometry etc. Yet the response is slow, and the flue field is disturbed by the probes [1]. Infrared thermometry is a non-intrusive technique. But only the surface temperature can be determined, because the infrared radiated by the internal liquid cannot transmit through the surface [2]. Ultrasonic thermometry is also a non-intrusive technique which can be used to detect the mean temperature of the liquid cylinder. But as long as the mean temperature is constant, the measurement signal of the probe remains the same no matter the temperature gradient inside the cylinder changes [3,4]. Rainbow refractometry is a potential technique for the n
Corresponding author. E-mail address: [email protected] (F. Song).
http://dx.doi.org/10.1016/j.optcom.2016.06.011 0030-4018/& 2016 Published by Elsevier B.V.
measurement of the temperature gradient inside a cylinder, because the determination is on the basis of the rainbow pattern affected by refractive index gradient instead of the mean refractive index [5]. Rainbow refractometry was put forward in 1988 to determine the refractive index (related to the temperature) and the diameter of a cylinder simultaneously [6]. However, only the mean refractive index could be determined with the proposed algorithms of standard rainbow refractometry at the beginning [7,8]. Due to the urgent requirement for the analysis of heat transfer inside a cylinder, later a number of researches aimed to achieve the measurement of the refractive index gradient. Li proposed an effective calculation model for the scattering intensity of the rainbow scattered by a multilayered droplet [9]. Vetrano proposed an inversion algorithm for determining the diameter and the refractive index gradient of a droplet, and the burning oil droplets were taken for experimental research [10]. However, the sharp of the refractive index profile should be predicted based on the experience that the temperature profile changed as the burning develops. So the method cannot be used if heat transfer condition changes. Saengkaew attempted to determine the diameter and the refractive index gradient of an inhomogeneous droplet with the algorithm for a homogeneous droplet [11]. Only the inversion diameter was similar to the true value, but the inversion refractive
F. Song et al. / Optics Communications 380 (2016) 179–185
index gradient was completely wrong. To sum up, actually there is no valid measurement method for the refractive index distribution inside a cylinder. To solve the problem, an improved rainbow refractometry assisted with infrared thermometry was proposed to simultaneously determine the diameter and the refractive index gradient of a cylinder without any priori value. Firstly, the surface refractive index (related to the surface temperature) was measured with infrared thermometry to decrease the unknown parameters. Secondly, the cylinder diameter was estimated with the interval of the maxima in the first two Airy fringes. Thirdly, the estimated core refractive index is expressed with the shape coefficient, so that only the shape coefficient should be solved with non-linear least square method. Finally, the two estimated values were optimized with an optimization process. Numerical simulations and experiments were carried out to verify the proposed method.
1.340
Refractive index
180
1.335 1.330 1.325 1.320 0.0
True Reversion 1 Reversion 2
0.2
0.4
0.6
0.8
1.0
Normalized radius Fig. 1. The comparison of the true and the inversion refractive index gradients.
2.1. Principle of the rainbow refractometry for inhomogeneous cylinder When a homogeneous cylinder is illuminated by a light beam, a monochrome primary rainbow pattern can be observed in the backward area [6]. Based on scattering theories, inversion algorithms have been proposed to determine the diameter and the refractive index of the cylinder from the rainbow pattern [12–14]. If heat transfer happens to the cylinder, temperature inside is inhomogeneous, and hence the refractive index gradient exists. To simplify the refractive index gradient, it can be described with the expression as follows:
n (k ) = nc +
(ns − nc )(e bk − 1) eb − 1
(1)
where b is the shape coefficient of the profile, ns is the refractive index at surface, nc is the refractive index at core, k is the serial number of the layers, n(k) is the refractive index of kth layer. To investigate the relationship between of the scattering intensity distribution and the parameters of an inhomogeneous cylinder, the intensity distribution of the rainbow should be calculated with the refractive index gradient and the diameter at first. Li proposed a model to calculate the scattering intensity distribution by iteration from the core layer to the surface layer [9]. And the intensity distribution of the Airy fringes can be calculated as well. According to the reported research, the intensity distribution is related to the inside temperature gradient as well as the cylinder diameter (denoted by D), which means it is potential to determine the temperature gradient from the scattering intensity distribution with a proper inversion algorithm [14,15]. 2.2. Refractive index gradient inversion with non-linear least square method The refractive index gradient is firstly attempted to be determined with non-linear least square method directly, which is a useful method for solving parameters in a nonlinear problem. The objective function is as follows:
f=
∑ ⎡⎣ Icap, Airy (k) − Ire, Airy (k) ⎤⎦2 k
(2)
where f is objective function value, Icap, Airy is the intensity of the Airy fringes in the captured rainbow, Ire, Airy is the intensity of the Airy fringes in the rainbow reconstructed with ns, nc and b during
the iteration process, k is the serial number of the discrete scattering angle. The initial values of ns, nc and b are set at first. Then in the iteration process, the initial ns, nc and b are continuously optimized with non-linear least square method for the purpose of decreasing the objective function value. And finally a refractive index gradient (described with ns, nc and b) is obtained, with which the final reconstructed rainbow is similar to the captured one. The numerical simulation of the inversion algorithm is then carried out. The parameters of a liquid cylinder are given, so that the intensity distribution of the rainbow pattern can be calculated, which is considered as the captured rainbow. A water cylinder heated with heat convection from its outside vertical surface is taken as the object in the numerical simulation. Firstly, true values of b, ns, nc are set 3, 1.326, 1.336. Fig. 1 shows the true refractive index gradient and the inversion gradients. With the non-linear least square method, different inversion values of ns, nc, and b can be obtained from different initial values. The corresponding rainbow patterns are shown in Fig. 2. It can be seen that the intensity distributions of the three rainbows are similar, especially the Airy fringes (i.e. the low frequency components). But the inversion refractive index gradients are both different from the true one. The main reason of the multi-solution problem is that the intensity distribution of the rainbow is affected by all the three Captured Reconstructed 1 Reconstructed 2
1.0
Rainbows 0.5
Intensity (a.u.)
2. Inversion algorithm for temperature gradient in an inhomogeneous cylinder
0.0 136
140
1.0
144 Captured Reconstructed 1 Reconstructed 2
Airy fringes 0.5
0.0 136
140
144
Scattering angle ( ° ) Fig. 2. The comparison of the captured and the reconstructed rainbows, the low frequency component obtained by the low-pass filter with hamming window.
F. Song et al. / Optics Communications 380 (2016) 179–185
Refractive index
1.335
True Reversion 1 Reversion 2
1.330
1.325 0.0
0.2
0.4 0.6 Normalized radius
0.8
1.0
Fig. 3. The comparison of the true and the inversion refractive index gradients when only b is needed to be solved.
181
are attempted to be obtained with another method and only b is needed to be determined, the inversion refractive index gradients with non-linear least square method are shown in Fig. 3. And the corresponding rainbow patterns are shown in Fig. 4. Before the iteration process, the initial values of b are set as 3 and 8 respectively. The inversion values of b are 2.9946 and 9.7547 respectively. One inversion refractive index gradient is almost the same to the true one, but the other is obviously different from the true one. The captured and the reconstructed rainbows shown in Fig. 4 are still similar, especially the Airy fringes. As b changes, the scattering angles of the maxima in the first Airy fringes are shown in Fig. 5. It can be seen that there is a maximum in the curve, where b is about 6. For an arbitrary value of b which is less than 6, another b larger than 6 corresponds to the same scattering angle. Therefore with the two inversion values of b, the intensity distributions of the two reconstructed Airy fringes are similar. Whereas the ripple structures of two reconstructed rainbows are different, so another objective function is used to judge which one is better:
f=
∑ ⎡⎣ Icap (k) − Ire, Mie (k) ⎤⎦2
(3)
k
Captured Reconstructed 1 Reconstructed 2
1.0
Rainbows
where Ire, Mie is the intensity distribution of the rainbow reconstructed with the inversion parameters. In all, more accurate inversion refractive index gradient can be determined, if only b is needed to be solved.
Intensity (a.u.)
0.5
2.3. Improved rainbow refractometry assisted with infrared thermometry
0.0 136
140
144
1.0
Captured Reconstructed 1 Reconstructed 2
Airy fringes 0.5
0.0 136
140
144
Scattering angle ( °) Fig. 4. The comparison of the captured and the reconstructed rainbows when only b is needed to be solved, the low frequency component obtained by the low-pass filter with hamming window.
Scattering angle of the maximum in the first Airy fringe (°)
140.3 140.2
In the improved method, the cylinder diameter is firstly estimated according to Vetrano's work [10]. The angular interval of the maxima in the first two Airy fringes was reported to be mainly related to the cylinder diameter and slightly related to the refractive index gradient. Therefore the interval can be used to estimate the cylinder diameter. The exponential fitting of the angular interval to the cylinder diameter is shown as follows, which is obtained from the rainbows calculated with nc ¼1.336, ns ¼1.326, b¼3.
D = 565.1⋅Δθ −1.447 , R2 = 0.9987
(4)
where Δθ is the angular interval. Infrared thermometry is used as assistance for the measurement of the surface temperature Ts, and therefore the surface refractive index can be obtained. On the other hand, Vetrano proposed a formula to describe the relationship between nc and ns [10]:
nrg = ns + δ
140.1
n c − ns ns2
(5)
where nrg is the inversion refractive index with the algorithm for homogeneous cylinder (which can be determined from the captured rainbow with the algorithm reported in the previous work [12–14]), δ is mainly related to b. As b changes, the fitting polynomial is shown in Formula (6), which is obtained from the rainbows calculated with nc ¼1.34, ns ¼1.32, D ¼100 μm.
140.0 139.9
δ = 0.001673⋅b3−0.04784⋅b2 + 0.404⋅b + 1.321 ,
139.8 2
4
6
b
8
10
12
Fig. 5. b vs. the scattering angle of the maximum in the first Airy fringe (nc ¼ 1.336, ns ¼ 1.326, D¼ 100 μm).
parameters, and the effect of changing one parameter on the rainbow can be offset by changing the other two parameters properly. So the unknown parameters should be decreased. If ns, nc
R2 = 0.9986
(6)
So the evaluated nc can be expressed with b.
nc =
1 + ns δ⋅ns ⋅ns ⋅(nrg −ns )
(7)
Therefore only b is needed to be solved. While using the algorithm described in Section 2.2, b is solved in the ranges of 2–6 and 6–12 respectively. The two estimated parameters (D and nc) are needed to be
182
F. Song et al. / Optics Communications 380 (2016) 179–185
1.0
Intensity (a.u.)
Angular difference
Captured Reconstructed 0.5
140
141 Scattering angle ( ° )
142
Fig. 8. Airy fringes of the reconstructed rainbow before nc is optimized (for the captured rainbow nc ¼ 1.34, ns ¼ 1.32, b ¼3.5, D ¼100 μm, for the reconstructed rainbow nc ¼1.3414, ns ¼ 1.3202, b ¼2.53, D ¼ 100.6 μm).
Fig. 6. Flow chart of the improved algorithm.
Intensity (a.u.)
1.0
Captured Reconstructed1, b=3.11 Reconstructed2, b=8
0.5
0.0 138
140
142 144 Scattering angle (°)
Fig. 9. Rainbow reconstructed with the inversion results (the captured rainbow is also attached for comparison).
Fig. 7. Airy fringes of the reconstructed rainbow before diameter is optimized (for the captured rainbow nc ¼ 1.34, ns ¼ 1.32, b ¼3.5, D ¼ 100 μm, for the reconstructed rainbow nc ¼ 1.3424, ns ¼1.3202, b¼ 2.02, D¼ 106.6 μm).
optimized. The flow chart of the improved algorithm is shown in Fig. 6. Then the numerical simulation is carried out to explain the flow chart. The true parameters of the cylinder are set as b¼ 3.5, ns ¼1.32, nc ¼ 1.34, D¼ 100 μm, and the intensity distribution can be calculated. For most commonly used instruments, the measurement error of infrared thermometry is 7 1 °C (corresponding to the refractive index error about 2 10 4 for water [15,16]). So the measurement of ns is assumed as 1.3202. The diameter estimated with Formula (5) is 106.6 μm. With the algorithm mentioned above, nc ¼ 1.3424, b¼2.02 are obtained. And the Airy fringes of the reconstructed rainbow are shown in Fig. 7. It can be seen that the maxima intervals in the two rainbows are different, which is mainly due to the error of the diameter. To solve this problem, the diameter is gradually adjusted to 101.2 μm. The inversion values are therefore updated (nc ¼ 1.3414, b¼ 2.53), and the Airy fringes of the reconstructed rainbow is shown in Fig. 8. The angular difference between the peaks of the two rainbows is mainly due to the
Refractive index
1.340 1.335 1.330 1.325
True Reversion 1
1.320 0.0
0.2
0.4 0.6 0.8 Normalized radius
1.0
Fig. 10. Comparison of the true and the final inversion refractive index gradient.
error of nc. So with the angular difference, the evaluation formula shown in Formula (7) is adjusted as follows:
nc =
1 + ns + Δn δ⋅ns ⋅ns ⋅(nrg −ns )
where
(8)
Δn is an offset, which is adjusted according to the angular
F. Song et al. / Optics Communications 380 (2016) 179–185
Pump Steel pipe
b ea
ms
Po
lar o
Electrothermal film covered with heat insulation material
0.5
La ser
1.0
Valve
id
Water tank
Steel capillary
Conves lens
s er
0.0
La
Relative error (%)
1.5
183
-0.5 Liquid cylinder
-1.0
CCD Camera
Infrared thermometer
-1.5
200
400
600
Diameter
800 (µm)
1000
(a) Relative error of the diameter
Fig. 12. Experimental system.
diameter is about 1%, and the absolute error of the refractive index is less than 6 10 4 at all the radial locations.
Error (10
−4
)
3. Experimental results and discussion
8 6 4 2 0 -2 -4 -6 -8
200
400
600
800
1000
True mean diameter (µm) (b) Error of the refractive index Fig. 11. Inversion errors when the diameter is in the range of 100–1100 μm, (a) Relative error of the diameter, (b) Error of the refractive index.
difference, i.e. at first Δn ¼0, and it is increased or decreased after each comparison. Still the refractive index gradient and the reconstructed rainbow are continuously updated until the angular difference shown in Fig. 8 is less than the preset value. Finally, two inversion results are obtained, nc ¼1.3405, ns ¼1.3202, b¼3.11, D ¼99.2 μm and nc ¼1.3404, ns ¼1.3202, b¼ 8, D ¼101.6 μm. The two reconstructed rainbows are shown in Fig. 9, which are similar to the captured rainbows. The objective function values calculated with Formula (3) are 0.8674 and 1.7123 respectively. As a result, the former inversion parameters are taken as the optimal result. The inversion refractive index gradient is shown in Fig. 10. It can be seen that the refractive index gradients are similar. The relative error of the inversion diameter is 0.8%. At all the radial locations, the error of the refractive index is less than 5 10 4, which corresponds to about 2.5 °C for water. Then the true cylinder diameters are set from 100–1100 μm with a step of 200 μm, the values of b are set randomly in the range of 2–12, nc, ns are set randomly in the range of 1.32–1.34, and nc is larger than ns during a heated cylinder (because the surface temperature is higher). The errors of the results are shown in Fig. 11. It can be seen that the absolute relative error of the
Fig. 12 shows the experimental system, consisting of a fluid system and an optical system. The fluid system consists of a water tank, a pump, a valve, a steel pipe, a steel capillary and an electrothermal film covered with heat resolution material. The water was pumped from the rank to the capillary by a water pump, so that a water cylinder was produced at the exit of the capillary. The internal diameter of the capillary was 1 mm. The height of the capillary is 50 mm. The flow rate of the water is 65 mL/min. The electrothermal film was used to heat the water flowing through. The powers of the electrothermal films were 72 W, 120 W and 180 W respectively. The optical system included an optical platform, a laser (100 mW, 532 nm), a convex lens, a CCD camera (e2v EV50YLM1GE2014, resolution 2048 100 pixels, pixel size 14 μm*14 μm). The water cylinder was illuminated by the laser beam. The scattering rays were focused onto the CCD camera by a convex lens. Fig. 13 shows the row rainbow patterns scattered by liquid cylinder heated with different powers. The grey values of the pattern captured by the CCD camera were extracted. The serial number of transverse pixels could be transformed to the scattering angle of the droplets by calibration. Thus the grey value at one certain pixel represented the scattering intensity at the angle. The surface temperature of the water cylinder was measured by an infrared thermometer (Raytek ST20, measurement error 71 °C). With infrared thermometry, the surface temperatures of the
(a) 72W
(b) 120W
(c) 180W Fig. 13. Row rainbow patterns when the heated power was different.
184
F. Song et al. / Optics Communications 380 (2016) 179–185
1.0
Intensity (a.u.)
Refractive index
1.336
1.334 72W 120W 180W
1.332
Captured Reconstructed
0.5
1.330 0.0
0.0
0.2
0.4 0.6 Normalized radius
0.8
1.0
138.0
1.0
60 Intensity (a.u.)
50 72W 120W 180W
40 30
139.5
(a) 72W
Fig. 14. Reversion refractive index gradients.
Temperature ( )
138.5 139.0 Scattering angle (° )
Captured Reconstructed
0.5
20 10 0.0
0.0
0.2
0.4
0.6
0.8
1.0
138.0
Normalized radius
139.5
(b) 120W
Fig. 15. Temperature gradients corresponding to the reversion refractive index gradients.
1.0
Intensity (a.u.)
liquid cylinders were 57.9 °C, 50.1 °C and 39.3 °C as the heating power increased. The reversion diameters were 901.2 μm, 893.6 μm and 905.4 μm, whose relative difference was 13.3%. Because of the viscous force between the capillary and the liquid cylinder, the reversion diameters were less than the internal diameter of the capillary. The refractive index gradients and the corresponding temperature gradients are shown in Fig. 14, and therefore the corresponding temperature gradients obtained from the refractive index gradients are Fig. 15 [15,16]. It can be seen that as the heating power increased, the surface temperature of the liquid cylinder increased as well, and the difference among the core temperatures was less than 2 °C. The captured and the reconstructed rainbows are illustrated in Fig. 16. In each graph, the two rainbows are similar, the objective function values calculated with formula (3) are 0.9675, 1.3524 and 1.1486 respectively in the three graphs.
138.5 139.0 Scattering angle (°)
Captured Reconstructed
0.5
0.0 138.0
138.5 139.0 139.5 Scattering angle (°)
(c) 180W 4. Conclusion Rainbow refractometry was investigated to attempt the measurement of the temperature gradient inside a heated liquid cylinder, without predicting any parameters based on experience. At first nonlinear least square method was directly used to solve the
Fig. 16. The captured and the reconstructed rainbows when the heating power was different.
F. Song et al. / Optics Communications 380 (2016) 179–185
parameters, but a multi-solution situation happened. Therefore an improved rainbow refractometry assisted with infrared thermometry was proposed to avoid the problem. The surface refractive index was obtained by measuring the surface temperature of the cylinder with infrared thermometry. And a rough formula between the core refractive index and the shape coefficient was built according to the previous work. Thus only one unknown parameters was left. An inversion algorithm was designed to achieve the measurement the refractive index gradient and the diameter simultaneously based on the nonlinear least square method and an optimization process. Numerical simulation and experiments were carried out to evaluate the method. The results of the numerical simulation showed that the relative error of the inversion diameter was about 1%, and the error of the refractive index was less than 6 10 4 at all the radial locations. In the experimental research, as the heating power increased, the surface refractive index increased, and the core refractive index almost remained the same. The reversion diameter was less than the internal diameter of the capillary because of the viscous force. The rainbows reconstructed with the reversion parameters were all similar to the corresponding captured ones.
Acknowledgment The authors wish to express their gratitude to the Program for National Natural Science Foundation of China (No. 51406068), National Natural Science Foundation of China (No.51376049), Fundamental Research Funds for the Central Universities (JUSRP51511), Foundation of Jiangsu union innovation (201501916) and Foundation of Jiangsu Key Laboratory of Advanced Food Manufacturing Equipment & Technology (FM-201504).
185
References [1] J. Char, J. Yeh, The measurement of open propane flame temperature using infrared technique, J. Quant. Spectrosc. Radiat. Transf. 1 (1996) 133–144. [2] T.P. Newson, H. Kee, All fiber system for simultaneous interrogation of distributed strain and temperature sensing by spontaneous Brillouin scattering, Opt. Lett. 10 (2000) 620–626. [3] A. Puttmer, P. Hauptmann, B. Henning, Ultrasonic Density Sensor for Liquid, IEEE Trans. Ultrason. Ferroelectr. Freq. Cont. 1 (2000) 85–92. [4] W. Tsai, H. Chen, T. Liao, High accuracy ultrasonic air temperature measurement using multi-frequency continuous wave, Sens. Actuators 132 (2006) 526–532. [5] P. Massoli, Rainbow refractometry applied to radially inhomogeneous spheres: the critical case of evaporating droplets, Appl. Opt. 15 (1998) 3227–3235. [6] N. Roth, K. Anders, A. Frohn, Simultaneous measurement of temperature and size of droplets in the micrometer range, in: Proceedings of the 7th International Congress on Optical Methods in Flow and Particle Diagnostics, ICALEO 88, L.I.A., CA, U.S.A, 67,1988, pp. 294–304. [7] N. Roth, K. Anders, A. Frohn, Simultaneous measurement of temperature and size of droplets in the micrometer range, J. Laser Appl. 2 (1990) 37–42. [8] N. Roth, K. Anders, A. Frohn, Refractive-index measurements for the correction of particle sizing methods, Appl. Opt. 33 (1991) 4960–4965. [9] R. Li, X. Han, H. Jiang, K. Ren, Debye series for light scattering by a multilayered sphere, Appl. Opt. 45 (2006) 1260–1270. [10] M.R. Vetrano, J.P.A.J. Van Beeck, M.L. Riethmuller, Assessment of refractive index gradients by standard rainbow thermometry, Appl. Opt. 34 (2005) 7275–7281. [11] S. Saengkaew, T. Charinpanitkul, H. Vanisri, Rainbow refractometry on particles with radial refractive index gradients, Exp. Fluids 4 (2007) 595–601. [12] J.P.A.J. Van Beeck, M.L. Riethmuller, Nonintrusive measurements of temperature and size of single falling raindrops, Appl. Opt. 34 (1995) 1633–1639. [13] F. Song, C. Xu, S. Wang, Y. Yan, An optimization scheme for the measurement of liquid jet parameters with rainbow refractometry based on Debye theory, Opt. Commun. 305 (2013) 204–211. [14] S. Saengkaew, T. Charinpanikul, C. Laurent, Y. Biscos, G. Lavergne, G. Gouesbet, et al., Processing of individual rainbow signals, Exp. Fluids 48 (2010) 111–119. [15] J. Hom, N. Chigier, Rainbow refractometry: simultaneous measurement of temperature, refractive index, and size of droplets, Appl. Opt. 10 (2002) 1899–1907. [16] I. Thormalen, J. Straub, U. Grigull, Refractive index of water and its dependence on wavelength, temperature, and density, J. Phys. Chem. Ref. Data 14 (1985) 933–945.