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Two-photon fluorescence lifetime imaging applied to the mixing of two non-isothermal sprays: temperature and mixing fraction measurements

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Droplets temperature is a key parameter for the study of heat and mass transfers in many spray applications. Time correlated single photons counting (TCSPC) is applied to monitor the fluorescence decay and determine the droplet temperature in the mixing zone of two sprays which are injected with significantly different temperatures. For some well-chosen fluorescent dye, like rhodamine B (RhB), the fluorescence lifetime strongly varies with the temperature. Provided sufficiently different fluorescence lifetimes for the droplets of the two sprays, the fluorescence decay is expected to follow a multiple exponential decay. In this study, different approaches are tested for measuring the temperature of the two sprays as well as their mixing fraction based on the analysis of the fluorescence decay. Firstly, the measurement of the mixture fraction alone is tested by considering a configuration where one spray is seeded with eosin Y (EY) and the other with rhodamine 6G (Rh6G). Given the very different lifetimes of these dyes, which are not temperature dependent, the fluorescence decay is function of the volume fraction of liquid from each spray in these tests. A calibration is necessary to evaluate the mixing fraction. Both sprays are mounted on an automated platform allowing 3D scanning and motions which allows obtaining maps of the fluorescence decay. The out-of-field fluorescence, observed in dense sprays when fluorescence is induced by one-photon absorption, is suppressed by using a two-photon fluorescence excitation. This approach significantly improves the spatial resolution of the measurements. Finally, both the droplet temperature and the mixing fraction are measured simultaneously using a single dye, namely RhB, whose fluorescence lifetime is temperature dependent. Special care must be paid to the fact that RhB does not have a purely monoexponential decay at a given temperature. The fluorescence decay in the mixing zone of the two sprays is considered as a combination of two biexponentials. Results show that the volume fraction of a spray must exceed about 10% to make it possible to determine its temperature with an accuracy of about 2–3 °C. Simultaneous measurements of the sprays temperatures and volume fractions provide a means to calculate the mixing temperature (the average between the temperatures of the two sprays weighted by their volume fractions).

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The authors are grateful to the China Scholarship Council (CSC) for the scholarship attributed to Mr. M. Wang, who was hosted by LEMTA at Université de Lorraine, France. The authors also gratefully acknowledge the financial support of the ENERBATIN CPER program and the European FEDER program.

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WM performed the measurements; SM and CG were involved in planning and supervised the work. CH and BS were involved in the implementation of the optical system and the measurement methods. MW, MS and GC processed the experimental data, performed the analysis, drafted the manuscript, and designed the figures. FL and EB provided critical feedback and helped shape the research. GC conceived the original idea and was in charge of overall direction of the project. All authors discussed the results and commented on the manuscript.

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Correspondence to Guillaume Castanet.

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A Monte Carlo method was implemented to estimate the measurement uncertainties. It consists in generating synthetic histograms of the photon arrival times, while taking into account the photon count rate, the acquisition time, the noise level, the IRF, and the time-channel width. Such method has been already employed by Stiti et al. (2021b) for assessing the precision of temperature measurements in a single spray. In this study, a monoexponential function was used to describe the fluorescence decay. Presently, since two sprays are considered, the fluorescence model has been extended to a two-exponential decay to follow the theoretical model presented in Sect. 5.1a. Results obtained with the method are presented hereafter for \({T}_{1}\;\)= 60 °C and \({T}_{2}\)= 20 °C. Two count rates, 2000 Cps and 15000 Cps, are also considered. The count rate of 2000 cps corresponds to the worst-case scenario (no measurements were performed with a lower count rate in the experiments). The count rate of 15,000 cps is a median value among the measurements. For some given values of the volume fraction \(\psi\) and the count rate, more than 2000 synthetic histograms are generated using the Monte Carlo method. These histograms are analyzed the same manner as their experimental counterparts in order to estimate the temperatures \({T}_{1}\) and \({T}_{2}\), the volume fraction \(\psi\), the mixing temperature \({T}_{\mathrm{m}}\) and the LIF temperature \({T}_{\mathrm{LIF}}\). Based on the thousands of generated data, the average and the standard deviation can be calculated for those quantities, which allows to evaluate the uncertainties of their measurements.


Fig. 18
figure 18

Confidence intervals of the estimations of the volume fraction \(\psi\) and the mixing temperature \({T}_{m}\) when the acquisition time is modified. Each data point is the result of the computation of 2000 synthetic fluorescence decays with the Monte Carlo method. A noise of 28 Cps is considered. The count rate of A and B is 2000 Cps and 15000 Cps for C and D

18 shows the influence of the acquisition time on the estimates of the volume fraction \(\psi\) and the mixing temperature \({T}_{\mathrm{m}}\). The colored areas on the graphs correspond to the confidence intervals of half-width equal to the standard deviations \({\sigma }_{\psi }\) or \({\sigma }_{{T}_{m}}\). It can be observed that an acquisition time of 100 s is a good compromise. It allows having a limited estimation error while maintaining an acceptable acquisition time. This value of the acquisition time was thus retained in the experiments and in the tests performed in the following for evaluating the uncertainties with the Monte Carlo method.

Using the same approach, uncertainties were also quantified for different values of the volume fraction \(\psi\) at a fixed acquisition time of 100 s in Fig. 

Fig. 19
figure 19

Evolution of the standard deviation of the estimated temperature of the hot (\({T}_{1}\)) and cols (\({T}_{2}\)), the mixing fraction \((\psi\)), the mixing temperature (\({T}_{\mathrm{m}}\)) and the LIF temperature (\({T}_{\mathrm{LIF}}\)) as a function of the mixing fraction. Each point is the result of 2000 synthetic fluorescence decays using an acquisition time of 100 s, a noise count rate of 28 Cps and in A 2 000 Cps and B 15 000 Cps count rate

19. Significant differences can be observed between \({\sigma }_{{T}_{1}}\) and \({\sigma }_{{T}_{2}}\). These can be explained by the fact that the hot spray has a much shorter fluorescence lifetime than the cold spray (Fig. 7). When \(\psi\) approaches 1, there is very little of the hot spray passing in the measurement volume in comparison with the cold spray. Therefore, the presence of the hot spray is difficult to detect based on the fluorescence decay. Not only it is hardly visible at the long times but also at the short times. So \({\sigma }_{{T}_{1}}\) is large while \({\sigma }_{{T}_{2}}\) is small. When \(\psi\) gets close to 0, the hot spray has a stronger influence on the fluorescence decay especially at the short times, but the cold spray keeps a significant influence at the long times. The uncertainty on the temperature \({T}_{2}\) of the cold spray does not increase that much. The value of \({\sigma }_{{T}_{2}}\) remains smaller than 1 °C for values of \(\psi\) as low as 0.1. Finally, it is interesting to note that the uncertainties on the parameters \({T}_{\mathrm{m}}\) and \({T}_{Lif}\) remain always lower than 1 °C, while the uncertainty on the temperature of the hot spray \(\left({\sigma }_{{T}_{1}}\right)\) reaches several °C. This is due to the fact that the different parameters are correlated, and errors can compensate each other. The error on the volume fraction \(\psi\) remains lower than 1% when the hot spray is predominant in the measurement volume (more precisely for \(\psi\) < 0.4).

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Wang, M., Stiti, M., Chaynes, H. et al. Two-photon fluorescence lifetime imaging applied to the mixing of two non-isothermal sprays: temperature and mixing fraction measurements. Exp Fluids 63, 172 (2022).

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