Abstract
This work describes the diagnostic implementation and image processing methods to quantitatively measure diesel spray mixing injected into a high-pressure, high-temperature environment. We used a high-repetition-rate pulse-burst laser developed in-house, a high-speed CMOS camera, and optimized the optical configuration to capture Rayleigh scattering images of the vaporized fuel jets inside a constant volume chamber. The experimental installation was modified to reduce reflections and flare levels to maximize the images’ signal-to-noise ratios by anti-reflection coatings on windows and surfaces, as well as series of optical baffles. Because of the specificities of the high-speed system, several image processing techniques had to be developed and implemented to provide quantitative fuel concentration measurements. These methods involve various correction procedures such as camera linearity, laser intensity fluctuation, dynamic background flare, as well as beam-steering effects. Image inpainting was also applied to correct the Rayleigh scattering signal from large scatterers (e.g. particulates). The experiments demonstrate that applying planar laser Rayleigh scattering at high repetition rate to quantitatively resolve the mixing of fuel and ambient gases in diesel jets is challenging, but possible. The thorough analysis of the experimental uncertainty and comparisons to past data prove that such measurements can be accurate, whilst providing valuable information about the mixing processes of high-pressure diesel jets.
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Acknowledgements
This study was performed at the Combustion Research Facility, Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the US Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.
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Appendix
Appendix
1.1 Rayleigh scattering theory
The theory of Rayleigh scattering, or the propagation of light within a continuous, yet transparent medium was laid out a century and a half ago [23]. Rayleigh later decided to convert his mechanical approximation into the electromagnetic model proposed by Maxwell, describing the light scattered by small particles. This formulation led to the expression of the Rayleigh signal intensity scattered by a dipole included in a sphere [24]:
Equation (13) provides the scattered intensity \(I_{\mathrm{Ray}}\) observed at an angle \(\varphi \) and excited by an incident wave of intensity \(I_0\), and of wavelength \(\lambda \). \(\alpha \) is the polarizability of the molecule, \(\varepsilon _0\) the vacuum permittivity and \(R_{\mathrm{dipole}}\) the radius of a sphere surrounding the dipole. By integrating the scattered intensity over the surface of this sphere, the total power \(W_{\mathrm{T}}\) scattered by the dipole can be calculated:
Where \(\varphi \) and \(\beta \) are the two dimensions of the integration to collect the energy scattered all over the surface of the sphere. After integration, the solution of the total power scattered by a dipole small compared to the wavelength equates to [15]:
As described by Miles [15], the Rayleigh cross-section \(\sigma _{\mathrm{Ray}}\) can be expressed as the ratio of the total scattered power to the incident intensity. In a uniform medium, in which a constant Rayleigh cross-section can be assumed, the total scattered power can be expressed as follows:
with \(\varpi \ V\) the number of particles \(\varpi \) within the considered volume V. This means that the total power emitted by an ensemble of gases molecules can be expressed as:
From this simple expression, the scattered power recorded by an acquisition system positioned orthogonal to the illumination plane can be calculated by introducing the collection angle \(\varOmega \) and the global efficiency of the system \(\eta _{\mathrm{opt}}\). The power of the Rayleigh scattered signal \(W_{\mathrm{Ray}}\) recorded by the imaging system can then be expressed as:
Then, the area probed has to be considered to obtain the intensity of the signal scattered by the molecules. This is achieved by converting the number of particles to the number density N and by introducing the probe volume length \(L_{\mathrm{opt}}\). Equation 19 computes the intensity scattered by Rayleigh scattering for a gas of known cross-section and density, and for a given optical system.
Equation (19) is certainly the most widely used for Rayleigh scattering measurements as it directly relates the recorded intensity to the number density of the molecules under study. Quantification of the number density requires thorough calibration of the optical system as well as accurate knowledge of the species under study regarding their Rayleigh cross-sections.
1.2 Rayleigh cross-section
As described in Eq. (19), the Rayleigh cross-section is needed to compute concentration (number density). While this is not a problem for commonly studied species, information about heavy hydrocarbons is limited or non-existent. As explained in the manuscript, the Rayleigh cross-section of molecules depends upon their polarizability, rather than their size as in the Mie regime, and can be expressed as follows after substitution of the total scattered power (Eq. (15)):
The polarizability \(\alpha \) is a coefficient of proportionality that relates the electric field \(\mathbf {E}\) to the induced moment of the dipole \(\mathbf {p}\) such as: \(\mathbf {p} = \alpha \ \mathbf {E}\). Assuming spherical symmetry of the dipole, its moment is induced in the same direction as the incident electromagnetic radiation [15, 32]. The following expression, known as the Lorenz–Lorentz equation [11], relates the polarizability of the molecules to their refractive index:
in which \(N_0\) is the Loschmidt constant and n is the refractive index of the medium (gas). By substituting the polarizability in the Rayleigh cross-section (Eq. 20), the following expression is obtained [15, 16, 26]:
According to the last equation, the Rayleigh cross-section is finally expressed as a function of the Loschmidt constant \(N_0\), the wavelength of the incident radiation \(\lambda \) and the refractive index of the gases n. Therefore, the refractive index has to be estimated with accuracy to get a correct value for the cross-section as the other parameters are experimental input (\(\lambda \)) or constant (\(N_0\)). One accurate method to estimate the refractive index of gases is to use the molar refractivity A:
The refractive index n is then a function of the Avogadro and Loschmidt constants, respectively \(N_A\) and \(N_0\), and the molar refractivity A.
The assumption that the dipole is spherically symmetrical may not be valid for large hydrocarbons, especially long alkane chains such as n-dodecane. The King correction factor compensates for the anisotropy of the molecule’s polarizability by applying a correction factor to the Rayleigh cross-section [9]:
where \(\varrho _p\) is the depolarization ratio of the molecule for a polarized illumination; note that the depolarization ratio is wavelength dependent. The depolarization ratio can be calculated with the Bottcher’s formula adapted to polarized light [26]:
with \(\varPsi \) the anisotropy of the polarizability and \(\bar{\alpha }\) the trace of the polarizability tensor of the molecule. It should be noted that while this correction is believed to provide more reliable Rayleigh scattering cross-section quantities, its total effect on the Rayleigh scattering cross-section for n-dodecane is less than 1%.
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Manin, J., Pickett, L.M., Skeen, S.A. et al. Image processing methods for Rayleigh scattering measurements of diesel spray mixing at high repetition rate. Appl. Phys. B 127, 79 (2021). https://doi.org/10.1007/s00340-021-07624-7
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DOI: https://doi.org/10.1007/s00340-021-07624-7