=Paper=
{{Paper
|id=Vol-2744/paper22
|storemode=property
|title=Optical Diagnostics of Temperature and Structural Parameters of an Axisymmetric Flame
|pdfUrl=https://ceur-ws.org/Vol-2744/paper22.pdf
|volume=Vol-2744
|authors=Vitaly Arbuzov,Edward Arbuzov,Yuri Dubnishchev,Olga Zolotukhina,Vladimir Lukashov
}}
==Optical Diagnostics of Temperature and Structural Parameters of an Axisymmetric Flame==
https://ceur-ws.org/Vol-2744/paper22.pdf
Optical Diagnostics of Temperature and Structural
Parameters of an Axisymmetric Flame*
Vitaly Arbuzov1,3[0000-0003-2404-526X], Edward Arbuzov1,2,3[0000-0001-9488-8650], Yuri Dub-
nishchev1,3[0000-0001-7874-039X], Olga Zolotukhina1,3[0000-0003-3486-4459] and Vladimir
Lukashov1[0000-0001-8178-7607]
1Kutateladze Institute of Thermophysics, Siberian Branch of the Russian Academy of Sciences,
Novosibirsk 630090, Russia
2Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, No-
vosibirsk 630090, Russia
3Novosibirsk State Technical University, Novosibirsk 630073, Russia
arbuzov@itp.nsc.ru|arbuzov@math.nsc.ru|dubnistchev@itp.nsc.ru|me
lexina-olga17@ya.ru|luka@itp.nsc.ru
Abstract. In this paper a method for restoring the parameters of multicompo-
nent media for optical diagnostics of jet using the example of a hydrogen-air
flame study is considered. Hilbert visualization and numerical modeling of
phase perturbations induced by the studied medium in the probing light field are
used. The study of the burning jet was carried out using the methods of Hilbert
optics and Abel transformation in the model of axial symmetry of the torch. A
software package has been developed that implements a direct solution to the
problem: calculation of the spatial optical phase structure of the flame and its
corresponding hilbertograms based on the temperature and molar concentra-
tions of the combustion products of the mixture. The reliability of the obtained
results is confirmed by comparing the Hilbert structures obtained in the experi-
ment and the reconstructed optical density field of the phase using the Abel
transform. The results of the comparison are used as a quality criterion for
modeling the phase structure and temperature field in the study of the combus-
tion process. The developed method can be used to solve the inverse problem of
restoring the temperature field from the Hilbert image of the phase structure of
a hydrogen-air flame in the approximation of axial symmetry. The research is
motivated by the scientific and practical significance of the problem, which
consists in finding methods for controlling the structural and thermodynamic
parameters of reacting jets and flames.
Keywords: Optical Diagnostics of Flames, Hilbert Optics, Optical Information
Technology.
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons Li-
cense Attribution 4.0 International (CC BY 4.0).
* The work was done within the framework of the state assignment of IT SB RAS project
III.18.2.5 (state registration No. АААА-А17-117030310010-9, No. АААА-А17-
117022850021-3), partial financial support from RFBR (project No. 20-38-90195).
2 V. Arbuzov, E. Arbuzov, Y. Dubnishchev, O. Zolotukhina, V. Lukashov
1 Introduction
The optical methods are the most important in unperturbing diagnostics of reacting
jets and flames. They allow one to obtain adequate information about the thermody-
namic and structural parameters of the medium under study without disturbing its
state.
In [1], a method for estimating the temperature distribution in an asymmetric flame
using high-contrast stereoscopic photography is described. Spectral reconstruction of
temperature fields using color ratio pyrometry and interferometric tomography is
reported [2]. An example adapted to the problems of flame research is optical diag-
nostics based on the methods of Hilbert optics and interferometry in combination with
pixel-by-pixel processing of the dynamic structure of visualized phase structures in-
duced by temperature fields [3]. However, the problem is to diagnose the spatial opti-
cal phase structure of the flame.
The aim of this work is to adapt the methods of Hilbert optics to the study of ax-
isymmetric hydrogen-air flame with the reconstruction of the spatial phase and tem-
perature structure. The research is motivated by the scientific and practical signifi-
cance of the problem, which consists in finding methods for controlling the structural
and thermodynamic parameters of the torch [4].
2 Research method
The complex of optical diagnostics is based on the IAB–463M device [5] with modi-
fied modules of Hilbert filtration, interferometry, source of probing field, registration
and processing of the optical signal [6]. The probing field is formed by the collimator
system from a slotted source. The investigated hydrogen-air flame is localized in the
vicinity of the front Fourier plane of the lens, which serves as an analyzer of phase
disturbances induced in the probing field by this flame. Visualization of the Fourier
spectrum of phase perturbations is performed by a Hilbert filter placed in the frequen-
cy plane of the Fourier lens. Next, the inverse Fourier transform of the Hilbert-
conjugate Fourier spectrum of phase perturbations is performed. The visualized phase
disturbances are registered by the camera's photo matrix. The registered images are
analyzed by a computer connected to the video camera.
In the frequency plane for the Fourier spectrum of the phase optical density of the
light field perturbed by the medium under study (flame), we have immediately after
the filter:
𝐻(𝐾𝑥 , 𝐾𝑦 )𝑠(𝐾𝑥 , 𝐾𝑦 ) = 𝑠(𝐾𝑥 , 𝐾𝑦 )cosφ+𝑠̂𝑥 (𝐾𝑥 , 𝐾𝑦 )sinφ, (1)
where 𝐻(𝐾𝑥 , 𝐾𝑦 ) is the coherent transfer function of the filter performing the one-
dimensional Foucault–Hilbert transform,
Optical Diagnostics of Temperature and Structural Parameters of an Axisymmetric Flame 3
𝐻(𝐾𝑥 , 𝐾𝑦 ) = cosφ − 𝑖sinφ sgn𝐾𝑥 .
The phase shift φ is a function of the wavelength λ of the probing light field,
φ = φ(λ). At a wavelength satisfying the condition φ(λ0 ) = 𝜋/2, the Hilbert image
of the Fourier spectrum of phase perturbations:
𝑠̂𝑥 (𝐾𝑥 , 𝐾𝑦 ) = −𝑖sgn𝐾𝑥 𝑠(𝐾𝑥 , 𝐾𝑥 ). (2)
The inverse Fourier transform of the Hilbert-conjugate Fourier spectrum of phase
perturbations (1) is performed below:
𝑠(𝐾𝑥 , 𝐾𝑥 )cosφ+𝑠̂𝑥 (𝐾𝑥 , 𝐾𝑥 )sinφ ↔ 𝑠(𝑥, 𝑦)cosφ+𝑠̂𝑥 (𝑥, 𝑦)sinφ. (3)
The intensity of the analytical signal (3) is recorded by a video camera:
𝐼(𝑥, 𝑦) = ρ{|𝑠(𝑥, 𝑦)|2 cos2 φ+|𝑠̂𝑥 (𝑥, 𝑦)|2 sin2 φ}, (4)
where ρ is a coefficient that takes into account the sensitivity of the photo matrix.
The Hilbert transform has the properties of redistributing energy from the region of
low spatial frequencies to the high-frequency region. Extremes and gradients of the
phase optical density of the medium under study are transformed into visualized Hil-
bert-bands structures. The spatial distribution of Hilbert bands provides information
about phase optical density perturbations induced by the temperature field.
An example of the phase structure of a hydrogen-air flame is shown in Fig. 1.a.
The phase structure was visualized using Hilbert optics methods. The experiments
were carried out at atmospheric pressure and initial room temperature.
a b
Fig. 1. (a) – experimental hilbertogram of a hydrogen-air flame; (b) – is a diagram of the burner
arrangement.
The burner is shown schematically in Fig. 1.b. It consists of two coaxially and verti-
cally positioned quartz tubes (dimensions and relative positions are shown in the fig-
ure). Air is supplied through an inner tube. The fuel mixture H 2/N2 enters through an
4 V. Arbuzov, E. Arbuzov, Y. Dubnishchev, O. Zolotukhina, V. Lukashov
annular channel between the inner and outer pipes (36.51% by volume, 3.95% by
mass). The nitrogen is added to reduce heat transfer to the burner. The diffuse flame
research was performed in sections with axial symmetry. The choice of the area with
axial symmetry is explained by the relatively simple geometry when modeling the
flame.
Measurements of the radial temperature distribution in three sections of the flame
at distances of 0, 30, and 60 mm from the cut of tubes (Fig. 2) were completed in a
mode corresponding to an average flow rate of air on the axis of 1 m/s and an average
flow rate of fuel of 0.8 m/s, using a platinum thermocouple.
According to the dispersion formula Cauchy [7, 8]
𝐵 𝐶
𝑛𝑘 (𝜆) − 1 = 𝐴𝑘 (1 + 2𝑘 + 𝑘4 ), (5)
𝜆 𝜆
the refractive index of the kth component of the burning mixture 𝑛𝑘 depends on the
wavelength of the radiation source 𝜆 and the parameters 𝐴𝑘 , 𝐵𝑘 and 𝐶𝑘 (𝐴𝑘 and 𝐵𝑘 are
dispersion coefficients for the kth component, 𝐶𝑘 is the relative molar concentration
of the kth component).The refractive index of the entire mixture is defined as
𝑝 𝑇0 𝐵
𝑛−1= ∑𝑘 𝐴𝑘 (1 + 2𝑘 ) ∙ 𝐶𝑘 , (6)
𝑝0 𝑇 𝜆
where 𝑝 is the pressure; 𝑝0 - pressure (in the room); 𝑇 is the temperature; 𝑇0 - room
temperature.The reconstruction of the values of molar concentrations of fuel combus-
tion products 𝐶𝑘 and, accordingly, the temperature distribution in the flame is possible
by performing measurements at different wavelengths and using formulas (5) - (6).
Fig. 2. The temperature distribution in three sections along the radius at heights from the cut of
tubes of 0, 30 and 60 mm at an average flow rate of air on the axis of 1 m/s and an average flow rate
of fuel of 0.8 m/s.
Optical Diagnostics of Temperature and Structural Parameters of an Axisymmetric Flame 5
An algorithm for the numerical simulation of Hilbert images by radial values of tem-
perature and mole fraction of fuel combustion products in a hydrogen-air flame was
developed. This algorithm was tested on the experimental data given in [9] (tempera-
ture profiles and molar concentrations in four sections of an axisymmetric flame sec-
tion).
The graphs displaying the radial profiles of temperature, molar concentrations of
fuel combustion products, and refractive index in the section at a height from the cut
of the burner tube 𝑦 = 10 mm are shown in Fig. 3.a - 3.b. The radial temperature field
of an axisymmetric section of a hydrogen-air flame, obtained using piecewise-cubic
Hermitian interpolation by the temperature values in sections at heights 𝑦 = 3, 10, 20,
and 30 mm, is shown in Fig. 4.
The phase structure of the probing light field, disturbed by the medium under study
(flame), in a physical experiment is defined as:
𝑧
∆ψ(𝑥, 𝑦) = 𝑘 ∫𝑧 2[𝑛(𝑥, 𝑦, 𝑧) − 𝑛0 ]𝑑𝑧, (7)
1
where 𝑘 = 2𝜋/𝜆 is the wave number of the probing field; 𝑛(𝑥, 𝑦, 𝑧) is the refractive
index of the medium in the spatial structure of the flame; 𝑛0 is the refractive index of
the medium unperturbing by the flame. The 𝑧 axis is determined by the direction of
the probe light beam, the flame torch cross section is described in 𝑥, 𝑧 coordinates.
The choice of the section position is determined by the 𝑦 coordinate. The coordinates
𝑧1 , 𝑧2 specify the size of the flame section in the direction of the probe beam.
a b
Fig. 3. (a) – radial profiles of temperature 𝑇 and molar concentrations 𝐶𝑘 ; (b) – radial profile of the
refractive index 𝑛(𝑟), calculated for 𝜆 = 0.630 μm (𝑦 = 10 mm from the end of the burner tube
cut).
Eq. (7) is turned into the Abel equation in the case of axial flame symmetry (Fig. 5):
𝑅 𝑟𝑑𝑟
∆ψ(𝑟, 𝑦) = 2𝑘 ∫𝑥 [𝑛(𝑟, 𝑦) − 𝑛0 ] , (8)
√𝑟 2 −𝑥 2
6 V. Arbuzov, E. Arbuzov, Y. Dubnishchev, O. Zolotukhina, V. Lukashov
where 𝑟 2 = 𝑥 2 + 𝑧 2 , 𝑅 – the radius of the cross section of the considered zone,
𝑛(𝑟, 𝑦) – refractive index at a distance of 𝑟 from the axis of the torch. Thus, the value
of the phase function ∆ψ can be restored by formula (8), knowing 𝑛0 .
The composition of dry air (relative molar concentrations of dry air components),
as well as the dependence of the saturated vapor pressure on temperature, must be
known to calculate the refractive index of the medium (air) 𝑛0 unperturbed by the
flame.
Atmospheric pressure is equal to the sum of the partial pressures of the dry air
components and the partial pressure of water vapor. The partial pressure of water
vapor in the mixture (air) is determined by the formula for the relative humidity of air:
𝜑𝑊 ∙𝑃0𝑊
𝑃𝑊 = , (9)
100
where 𝑃𝑊 is the partial pressure of water vapor, 𝑃0𝑊 is the saturated vapor pressure at
a given temperature, 𝜑𝑊 is the relative air humidity in the room.
Fig. 4. The recovered radial temperature field of hydrogen-air flame.
Fig. 5. The section of the investigated axisymmetric object in the plane 𝑦 = const.
Optical Diagnostics of Temperature and Structural Parameters of an Axisymmetric Flame 7
The molar concentration of water vapor in the mixture (air) can be determined by the
equation of state of an ideal gas, knowing the value of 𝑃𝑊 :
𝑃𝑊
𝜈𝑊 = , (10)
𝑅𝑓 ∙𝑇0
where 𝑅𝑓 is the universal gas constant, 𝑇0 is the room temperature. Similarly, we cal-
culate the value of the molar concentration of the entire mixture (air):
𝑝0
𝜈= , (11)
𝑅𝑓 ∙𝑇0
where 𝑝0 is atmospheric pressure (in the room).
Then the relative molar concentration of water vapor in the air:
𝜈
𝐶𝑊 = 𝑊 . (12)
𝜈
The refractive index 𝑛0 of the medium unperturbing by the flame, taking into account
(6) and (9) - (12), will be determined as
𝑝0 𝑇𝑛.𝑐. 𝐵 𝐵
𝑛0 = 1 + [(∑𝑘 𝐴𝑘 (1 + 2𝑘 )) (1 − 𝐶𝑊 ) + (𝐴𝑊 (1 + 𝑊
2
)) 𝐶𝑊 ], (13)
𝑝𝑛.𝑐. 𝑇0 𝜆 𝜆
where 𝑝𝑛.𝑐. - atmospheric pressure under normal conditions (101.325 ∙ 103 Pa); 𝑇𝑛.𝑐. -
temperature under normal conditions (0°C); 𝐴𝑘 and 𝐵𝑘 - dispersion coefficients for
the kth component of dry air; 𝐴𝑊 and 𝐵𝑊 - dispersion coefficients for water; 𝜆 is the
wavelength of the radiation source.
The radial field of the refractive index calculated by formula (6) is shown in
Fig. 6.a. The conditions were accepted in the calculation: 𝑇0 = 27°С, 𝜆 = 0.630 μm,
𝑝 = 𝑝0 = 101.325 ∙ 103 Pa (normal conditions), 𝐴𝑘 and 𝐵𝑘 are the values taken from
the reference data.
The phase function must be reconstructed from the hilbertogram according to the
algorithm proposed in [2], and the refractive index of the medium must be determined
by determing the Abel equation to solve the inverse problem (reconstruction of the
flame temperature from the hilbertograms). The calculated inverse field of the phase
function (for the wavelength 𝜆 = 0.630 μm), obtained by calculating the Abel integral
(8), is shown in Fig. 6.b.
The refractive index depends on temperature and 4-x values of mole fraction of
combustion products, which follows from formula (6). Measurements at three differ-
ent wavelengths of the probing field are sufficient to determine all unknown parame-
ters, since the combustion area is divided into zones consisting of 3 components of
combustion products.
The radial profiles of temperature, molar concentrations of combustion products,
refractive index and the corresponding values of the phase function, hilbertogram and
interferogram are shown in Fig. 7 for the section 𝑦 = 10 mm and 𝜆 = 0.630 μm.
The numerically simulated hilbertograms obtained from the calculated phase func-
tion ∆ψ for 𝜆 = 0.630, 0.520, 0.405 μm, respectively, are shown in Fig. 8. The disper-
sion coefficients were taken from the reference data, and the relative air humidity in
the room was taken 𝜑𝑊 = 60% in the numerical simulation of the hilbertograms.
8 V. Arbuzov, E. Arbuzov, Y. Dubnishchev, O. Zolotukhina, V. Lukashov
a b
Fig. 6. (a) – radial field of the refractive index of a hydrogen-air flame (𝜆 = 0.630 μm); (b) – field
of the phase function for a radius 𝑟 from 0 to 15 mm 𝜆 = 0.630 μm.
a
b
Fig. 7. (a) – radial profiles of temperature 𝑇 and of molar concentrations 𝐶𝑘 ; (b) – radial profile of
the refractive index 𝑛(𝑟, 𝑦); (c) – phase function, interferogram and hilbertogram (𝑦 = 10 mm,
𝜆 = 0.630 μm). c
The results obtained confirm the feasibility of multi-wave optical diagnostics in direct
and inverse problems of studying phase structure and temperature fields in an ax-
isymmetric flame.
Optical Diagnostics of Temperature and Structural Parameters of an Axisymmetric Flame 9
a b
c
Fig. 8. Hilbert images numerically modeled by the field of the phase function ∆ψ for 𝑟 from -15 to
15 mm: (a) - 𝑦 = 0.630 μm; (b) - 𝑦 = 0.520 μm; (c) - 𝑦 = 0.405 μm.
3 Conclusion
The study of a hydrogen-air flame was done in the presented work using the methods
of Hilbert optics and numerical modeling in the approximation of axial symmetry of
the flame using the Abel integral. The reliability of the results is confirmed by com-
paring the hilbertograms obtained in the experiment (Fig. 1.a) and reconstructed from
the phase structure according to the Abel equation (Fig. 8). The comparison results
are used as a quality criterion of the modeling the phase structure and temperature
field using the Abel transform in the study of the combustion process. The application
of the developed method for solving the inverse problem - restoration of the tempera-
ture field from the Hilbert image of the phase structure of a hydrogen-air flame will
become the next stage of work.
The work was done within the framework of the state assignment of IT SB RAS
project III.18.2.5 (state registration No. АААА-А17-117030310010-9, No. АААА-
А17-117022850021-3), partial financial support from RFBR (project No. 20-38-
90195).
10 V. Arbuzov, E. Arbuzov, Y. Dubnishchev, O. Zolotukhina, V. Lukashov
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