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Abstract
As an important physical quantity to describe the resistance of fluid to flow, viscosity is an essential property of fluids in fluid mechanics, chemistry, medicine, as well as hydraulic engineering. While traditional measurement methods, including the rotating-cylinder method, capillary tube method and falling sphere method, have significant drawbacks especially in terms of accuracy, response time and the sample must be made to move. In this work, a novel Beer-Lambert law-based method was proposed for the viscosity measurement. Specifically, this work demonstrates that viscosity can be quantitatively reflected by spectral line intensity, and castor oil was selected due to its viscous temperature properties (viscosity has been accurately measured under different temperature), and its transmission spectrum at different temperatures ranging from 10 to 50°C was detected firstly. Then, the principal component analysis (PCA) was employed to obtain the intrinsic features of the transmission spectrum. Finally, the processed data was utilized to train and verify the radial basis function (RBF) neural network. As a result, the accuracy of the predictions conducted by means of the RBF reached 98.45%, which indicates the complicated and non-linear relationships between spectra formation and viscosity can be depicted well by RBF. The results show that the real-time in-situ optical detection method adopted in this work represents a great leap forward in the viscosity measurement, which fundamentally reforms the traditional viscosity measurement methods.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
1.1. Background
The coefficient of viscosity of a liquid is also known as the coefficient of internal friction or viscosity and recent researches suggest that such friction may also exist in the quantum realm [1]. It is an important physical quantity that describes the nature of friction within a liquid. It characterizes the ability of a liquid to resist deformation and exists when there is relative motion within the liquid [2,3]. Additionally, the coefficient of viscosity varies from material to material [4–6], and temperature sensitive [7–9]. The fast and accurate detection for viscosity of liquids has an important role to play in geology [10,11], biological medicine [12], quantum electronics [13] and other fields [14–16].
There are mainly two traditional viscosity measurement methods: the rotational viscosity measurement method created by Brookfield family in the 1940s [17] and the capillary viscosity measurement method invented by Wilhelm Ostwald in the early 20th century [18]. However, these methods always have some defects such as large errors and slow responses. Although the researchers have been endeavoring to improve the performance of these methods, there has been no fundamental changes due to the inherent flaws of traditional experimental methods and the requirement to keep the sample move makes the real-time in-situ detection will never be possible [19,20].
To explore a new method that can make up for the above-mentioned shortcomings, attention was turned to the optical detection method, which is considered to have the advantages of fast response, high accuracy and, most importantly, real-time in-situ detection. This means that an optical parameter is needed to characterize the viscosity. Yunus and his team designed a novel optical fiber viscosity detector that employed the relation between the viscosity and the refractive index [21]. This method chose refractive index as the bridge between optics and viscosity, but still requires the fiber to be placed in the object to work, meaning that real-time in-situ detection is not possible. In previous research [22], it was found that there were links between the viscosity and the properties of the transmission spectrum, which greatly inspired us. However, it is difficult to find a specific formula to describe this relationship, while that's exactly what machine learning is designed to do. In this work, machine learning was adopted as a bridge to link the transmission and the viscosity. Finally, the corresponding prediction model can be established by machine learning, and the optical measurement of viscosity can be realized.
It is clear that the relation between the viscosity and the transmission spectrum is both nonlinear and complex. Hence, the RBF was employed due to its good performance in fitting nonlinear function accurately and handling the system with fuzzy regularity. Besides, it has been successfully applied to the time series analysis [23], data classification [24], pattern recognition [25], as well as fault diagnosis [26], etc.
1.2. Our work
In this work, the castor oil was selected for the experiment due to its classical viscosity-temperature properties, which allows the viscosity of the sample to be precisely regulated by temperature. And because of its traceable viscosity, the accuracy of the machine learning model we developed can be well tested. In the experiment, the temperature was controlled between 10 and 50°C with 0.5°C as a step. And the transmission spectra were examined accordingly. Then, the PCA dimensionality reduction method was utilized to obtain the principal components (PC) of the wide spectrum so as to reduce data redundancy. In this way, the RBF neural network was trained and then was tested to obtain the accuracy.
2. Experiment
2.1. Experiment setup
The experimental scheme is illustrated in Fig. 1. A broadband Tungsten halogen light source, with an output wavelength range of 350 to 2500 nm, was injected into the simple mode fiber so as to reduce the influence of the environment (type of the optical fiber interface: SMA905). To be specific, the radiation flux is 10∼100 W/nm, and the color temperature is 3000 K. A linear silicon CCD array with the pixel value of 3648 was employed for the detector. The resolution of the spectrometer is 1 nm, and the signal-to-noise ratio is 250:1 (full signal). The stray light level is <0.05% at 600 nm and the thermal stability is 0.03 pixels/°C with the integration time of 0.1 s.
In the experiment, Water Baths (Lichen HH-2) with accuracy of +/-0.1% is used to help maintain constant temperatures. To ensure the transmission spectra of wavelength was recorded under a stable temperature environment, a two layers sample container was designed, the outer layer is filled with water, and the inner layer is castor oil. During the measurement, the sample container was put into the Water Baths and the spectrum was measured after 10 min for equilibration at different temperature. The light from the halogen tungsten light source was focused by a lens and reflected into the tube by an optical system. Then, the transmitted light was detected by the detector through the lens. Next, the detected signal was transferred into the spectrometer. At last, the spectrum information was passed to the processor combined with viscosity data for machine learning. The main ingredients of the castor oil employed in this work are listed in Table 1.
Table 1. Main ingredients of castor oil.
2.2. Sampling modes
The Sampling modes employed in the experiment were divided into the static sampling and the dynamic sampling. Under the static sampling, 100 spectral data in one group were detected, i.e., each spectral data took 0.1 seconds. And a total of 5 groups were measured, each of them represents the castor oil samples at a certain temperature. In dynamic sampling, 100 spectral data in one group were also measured in the first 10 seconds of each minute, but the number of the groups to be measured mainly depends on the actual needs (time actually spent in the process of temperature change). Besides, the temperature changed continuously during the process (10∼50°C).
3. Static viscosity test
3.1. Analysis of spectrum
Having considered the characteristics and applicability of experimental facility, the spectra exhibited by 5 different temperatures were analyzed. In this section, samples in 10°C, 30°C, and 50°C were selected to conduct the spectral analysis. Then, Beer-Lambert law was adopted to calculate the absorbance in different wavelength. In order to reduce the dimension and redundancy of data, PCA was adopted. In this way, the original spectrum of 350∼1000 nm in the resolution of 1 nm 651-dimensional data were reduced to a 4-dimensional data for further analysis.
Four transmission spectra of the castor oil under different temperatures are illustrated in Fig. 2. The black one is the transmission spectrum under 25°C air spectrum and the rest are transmission spectra of castor oil under 10°C, 30°C, and 50°C. The air spectrum is employed as the standard light intensity used in absorbance calculations to avoid the effect of background noise. There is no doubt that, although the spectral intensity of castor oil at other temperature is less than 25°C Air, the tendency of spectral changes is synchronized. It could be seen clearly that all kinds of the spectral intensity reach the peak at about 700 nm and then show a fluctuating trend of falling and rising, reaching two secondary peaks at about 775 nm and 800 nm. The spectral intensity flattens out with decreasing temperature. When the temperature about 10°C is reached, the overall spectral intensity almost becomes a straight line and it is difficult for us to distinguish the peak. In order to further analyze the viscosity coefficient of castor oil, Beer-Lambert law was used as the calculation method.
3.2. Absorbance calculated by Beer-Lambert law
Beer-Lambert law is the fundamental law of light absorption and is applied to all electromagnetic radiation and all light-absorbing substances, including gases, solids, liquids, molecules, atoms and ions [27]. And the law is the quantitative basis for absorbance photometry, colorimetric analysis, etc. Its Definition is that the proportion of light absorbed by a transparent medium is independent of the intensity of the incident light, and the absorbance can be expressed in Eq. (1).
Where A is the absorbance; T is transmittance; c is the concentration of light-absorbing substance; b is the thickness of absorption layer; K is a coefficient, related to environmental factors such as temperature. And I1 represents the transmitted light intensity; I0 represents the incident light intensity.Accounting for Beer-Lambert law, the absorbance of castor oil under different temperatures was obtained, and 5 of the most typical cases were selected in Fig. 3, it is obvious that castor oil whose temperature is no matter 10°C or 50°C and others, has the same absorbance variation trend. The absorbance decreases with increasing castor oil temperature between 500 nm with 900 nm.
3.3. PCA dimension reduction analysis
Principal component analysis is a very practical analytical method which called PCA for short. In many scientific experiments, it is often necessary to process data with multiple variables. There may be correlations between results, but only one variable analysis leads to incomplete utilization of the data, ignoring some of the connections. In order to further analyze the data, the PCA dimension reduction method was used. Thus, correlation calculations were used to filter valid bands and cumulative contributions. In recent years, the application of PCA in spectral data analysis has become a trend and has achieved great success [28,29].
In this experiment, after the obtaining of the absorbance of such a wide spectrum in different temperatures, only relevant spectral information including 401 data points was utilized for the next step of the identification (wavelength range: 500∼900 nm) due to the lengthiness of the spectral data. However, performing the following neural network prediction on such data means that it will be a 401-dimensional mathematical prediction, which makes it difficult to guarantee both the accuracy and the responsiveness.
But surprisingly, after the utilization of PCA to perform the dimensionality reduction on the spectral data, the 401-dimensional data was successfully reduced to the 4-dimensional data with 100% cumulative contribution rate. Theoretically speaking, when the cumulative contribution rate reaches 95%, all data can be considered sufficient to be identified by these components [30], which are called PCs. Therefore, in order to achieve the optimal processing of the data with guaranteed contribution rate, the most important three PCs were selected, and the contribution rate of each was shown in Fig. 4 respectively.
In order to confirm the results of PCA, the statistical tests were of great significance, among which the combination test of KMO and Bartlett was always the most persuasive [31,32]. Specifically, the value of KMO sampling suitability variables should exceed 0.5, and the importance of Bartlett's sphericity test needs to be less than 0.05. As shown in Table 2, the result was solid enough.
Table 2. KMO and Bartlett test
Having confirmed the validity, one 3D diagram and three 2D diagrams were contained in Fig. 5 so as to illustrate the relationships among the three PCs. As shown obviously in this figure, PC1, PC2 and PC3 could give a relatively specific boundary of the castor oil in five temperatures, which might be difficult for any two PCs. In consequence, these three PCs could be applied to predict the viscosity of the castor oil in different temperatures.
Each PC is composed of the data at multiple wavelengths, which indicates that the change of a single component symbolizes change of the corresponding wavelength. The matrix of coefficient for each wavelength constituting the PC was shown in Table S1 in Supplement 1.
According to the Table S1, the most dominant characteristic wavelength of PC1 is 682∼695 nm, that of PC2, PC3 and PC4 is 406∼409 nm, 358∼361 nm and 377∼380 nm, respectively. Figure 5(b) shows the changes of characteristic wavelength under different temperatures. In general, the strong absorbance band of PC1 in the 682∼695 nm region and it showed a more uniform and continuous decrease with the temperature. Besides, the absorbance in the 406∼409 nm band of PC2 indicated a trend of decreasing first and then increasing with the rise of the temperature. The fitting results showed that the turning point was around 20°C, and the change was slow at 10∼30°C and 40∼50°C while increasing rapidly at 30∼40°C. Furthermore, the absorbance in the 358∼361 nm band of PC3 was basically unchanged at 10∼20°C and 40∼50°C. But it changed faster at 20∼40°C and showed a trend of increasing first and then decreasing, and the turning point was around 30°C.
In general, the absorbance in different wavelengths displays different variation trends with the change of the temperature. Based on this, the viscosity coefficient was introduced as the fourth dimension for static fitting, which meant that each of the original 3D coordinates would be mapped to a value. Although the accuracy of this fitting reached 95%, the sensitivity and stability of this fitted model couldn’t be guaranteed due to the utilization of only static sampling and the small number of the samples fitted. For this reason, further dynamic probing and machine learning methods will be carried out in the following to perform more stable and accurate predictions on the viscosity coefficient.
4. Dynamic monitoring of viscosity-temperature characteristics
4.1. Analysis of dynamic monitoring of viscosity
After the static measurement, the dynamic measurement was carried out in temperature range 10∼50°C. To be specific, the data under several hours measuring were selected for analysis firstly. Then, a 3D curved surface was created with the X-axis representing temperature, Y-axis meaning wavelength, and Z-axis referring to absorbance. In order to highlight the pattern of the change in the image, the curved surface was projected onto the 2D plane of X and Y, and the change in absorbance was shown via the color change and the attached graph in Fig. 6.
The results shown in Fig. 6 are very similar to those analyzed in the previous section. The most significant changes occurred when the temperature reached about 20∼40°C, and the corresponding wavelengths also varied significantly from 500 to 900 nm. While, there was a large data redundancy phenomenon due to the same variation trend of many wavelengths. Thus, the attention should be focused on the bands analyzed in the previous section. The next step of the neural network prediction can be performed by normalizing the corresponding dynamic detection data.
4.2. Prediction of castor oil viscosity by RBF neutral network
RBF neutral network is essentially a multivariable interpolation radial basis function method [33]. And RBF is a real-valued function whose value depends only on the distance from the origin shown in Eq. (2).
After obtaining the trained neural network, the predicted results can be expressed in Eq. (3) by inputting the spectral data at the corresponding combustion level.
In this study, 70% of the data were used to train the neural network and 30% were used for testing. In order to obtain the most desirable prediction results, for each PC, a weighted K nearest-neighbor algorithm was employed to calculate its central wavelength, using which as a basis to adjust the wavelength range of the input data for scanning. The results of neural network training with different range of wavelengths are shown in Fig. 7. It could be found that the highest accuracy of 98.45% was achieved when the sampling radius is 15.74 nm. Therefore, it can be concluded that the viscosity coefficient of castor oil at different temperatures can be predicted by trained RBF neutral network using the spectral information, and the wavelength range used is centered on each PC with a radius of 15.74 nm. Also, this method can be applied to the viscosity of other fluid measurement by simply adjusting the composition of PCs and sampling range of wavelength.
5. Conclusion
In this work, a novel detection system was developed for the viscosity detection of the castor oil by means of two modes: the static model and dynamic model. In the static mode, the spectra and absorbance of the castor oil at several specific temperatures were detected and calculated for analysis. Based on this type of detection, the PCA algorithm was applied and successfully reduced the ultra-high-dimensional data to the 4-dimensional data, which greatly reduced the data redundancy. Besides, a 3D coordinate system was established based on the top three PCs to analyze the variation pattern of each PC, and the wavelengths in the main components of each PC were calculated. As a result, the accuracy of the prediction of castor oil viscosity via static detection mode reached 95%, but the sensitivity and stability of the fitted model could not be guaranteed due to the shortcomings of this mode. In view of this, the dynamic detection and machine learning method RBF were applied to perform more accurate predictions and finally achieved an accuracy of 98.45%. In summary, this real-time in-situ optical detection method adopted in this work symbolizes a great leap forward in the fluid viscosity measurement, which fundamentally reforms the traditional viscosity measurement methods. Meanwhile, this method can also provide theoretical references for the researchers in related fields and shows far-reaching application potential in the medicine, materials science and other fields.
Funding
National Key Research and Development Program of China (2018YFB1800901); National Natural Science Foundation of China (61822507, 61835005, 61935005, 61975084, 62075038, 62075097, 62175113); NUIST Students’ Platform for Innovation and Entrepreneurship Training Program (202210300134Y).
Acknowledgments
This work was supported in part by the National Key Research and Development Program of China (No. 2018YFB1800901), the National Natural Science Foundation of China (NSFC) (No. 62075097, 62075038, 61975084, 61935005, 61835005, 61822507, 62175113), NUIST Students’ Platform for Innovation and Entrepreneurship Training Program (202210300134Y).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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