Emission tomography in flame diagnostics

Combustion and Flame 160 (2013) 577–588

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Combustion and Flame j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m b u s t fl a m e

Emission tomography in flame diagnostics Natalya Denisova ⇑, Pavel Tretyakov, Andrey Tupikin Institute of Theoretical and Applied Mechanics, Novosibirsk, Institutskaja str. 4/1, Russia

a r t i c l e

i n f o

Article history: Received 27 June 2012 Received in revised form 4 September 2012 Accepted 8 November 2012 Available online 6 December 2012 Keywords: Emission flame tomography Reconstruction algorithms Chemiluminescent radiation in flames

a b s t r a c t Spatial distributions of chemiluminescence from OH, CH and C2 radicals in flames have been drawing great attention in the literature since the ratios of the local intensities OH/CH/C2 could be used to monitor such characteristics as fuel–air ratio, completeness of combustion, rate of combustion, blowout. With the rapidly growing capability of optical sensor technologies there is an increased interest to develop an accurate tomographic reconstruction technique for combustion control. A specific difficulty of flame tomography is the character of the source function to be reconstructed: the function describing the spatial distribution of chemiluminescence has narrow high-gradient peaks. The goal of the present study is to propose a reliable reconstruction technique applicable to axisymmetric and non-axisymmetric flames. Computing algorithm based on the maximum entropy (MENT) concept in combination with data preprocessing procedure has been developed. This algorithm is optimal in conditions with limited observation directions. A new method based on local regularization has been proposed for data processing to achieve more accurate reconstruction of the peak intensities and thicknesses of chemiluminescent zones in flames. The developed approach is tested in reconstruction of OH, CH and C2 radical chemiluminescences in the axisymmetric propane–air flame of a Bunsen-type burner. Ó 2012 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

1. Introduction Optical emissions from flames are widely used in combustion diagnostics. The emission technique working in the visible and ultraviolet spectra has been extensively applied for detection of OH, CH and C2 chemiluminescent radiation in hydrocarbon flames. Chemiluminescences from OH, CH and C2 radicals have been drawing great attention in the literature as indicators of the fuel-to-air ratio. Chemiluminescence provides information about conditions in the reaction zone and heat release. Knowledge of spatial distributions of OH, CH and C2 emitting radicals is important for better understanding of combustion reactions mechanisms under the external actions (acoustic, electrical, laser, etc.). A flame diagnostics based on the rations of the local chemiluminescence intensities CH/C2/OH is becoming a standard approach in laboratory researches and industrial applications [1,2]. However, it should be noted, that self-chemiluminescence detection gives us the planar images, integrated along the line of sight. Unfortunately, a flame structure is difficult to analyze from the planar images. Therefore, the practically important problem is to obtain spatially resolved local intensities. At the moment, one obtains spatial distributions of chemiluminescence intensities in flames using two different approaches: Cassegrain system [3,4] and tomographic techniques [6–14]. Tomographic approach is well known from medicine appli⇑ Corresponding author. E-mail address: [email protected] (N. Denisova).

cations. Reconstruction algorithms are usually applied to obtain 3D structures from planar images. Generally, tomographic techniques require multiple viewing access and multiple cameras, except axisymmetric cases. Alternatively, Cassegrain system collects data from one viewing direction and uses a single camera. However, Cassegrain optics provides chemiluminescence signal only from a sharply focused small control volume (mm3). Scanning the focal plane through the flame, one can create a family of small size images from different planes located along the line-of-sight. After that, using a deconvolution algorithm, chemiluminescence local parameters can be resolved. In fact, the local parameters can be resolved only in a small area along the line of observation. Comparing these two techniques, we can conclude, that the Cassegrain technique is useful to study combustion processes in a small selected region. However, to obtain the major version of a 3D flame structure one needs to scan the focus not only along the line-ofsight, but also on each plane surface. Potentially, such scanning is possible in the cases of steady-state flames, but it is very hard to realize in practice. In the authors opinion, at the moment, the tomographic technique is the only method to reconstruct 3D spatiotemporal behavior of chemiluminescence intensities in flames. The tomographic and the Cassegrain measurements require complex mathematical procedures to resolve the local parameters: reconstruction algorithms in tomography and deconvolution algorithms in the Cassegrain method. From a rigorous mathematical point of view, both the reconstruction and deconvolution problems

0010-2180/$ - see front matter Ó 2012 The Combustion Institute. Published by Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.combustflame.2012.11.005

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belong to the class of ill-posed problems and should be solved by using the approach based on the regularization theory [5]. This paper is focused on the emission tomography method for flame diagnostics. With the rapidly growing capability of optical sensor technologies (including optical fiber), there is an increased interest to develop a reliable tomographic reconstruction technique. Many previous studies sought to recover the local flame parameters using various tomographic algorithms and methods. Some examples are discussed in this Section. In [6], tomographic reconstruction of excited-state CH radicals in methane-air and propane–air flames was performed. This radical is an excellent flame-front indicator. Eight projections, equally spaced over 180°, were sequentially recorded. The approach based on the multiplicative algebraic reconstruction technique (MART) was applied for the tomographic study. The used MART algorithm assumed negligible self-absorption. The algorithm behaved well for reconstruction of smooth objects from as few as two or three projections. However, reconstruction of sharp edges, such as those in the methane–oxygen flame, resulted in strong oscillations on the rim if only two or three projections were used. It was also noted that fluctuations in such flames resulted in noisy reconstructions when the projection data were sequentially recorded. In [7], a study of a 3D temperature distribution in a flame was performed. Optical emission technique was used for registration. A reconstruction algorithm based on algebraic reconstruction technique (ART) was applied for reconstruction of 3D temperature fields in non-axisymmetric flame. An absorption correction term was included into the algorithm. The reconstruction was performed using images from four positions around the flame. To test the developed approach numerical simulations were performed. These tests included the evaluation of the algorithm response to random noise on the projection. The ability of the developed approach to be used as a combustion diagnostic tool was assessed in diffusion flame. In [8], an axisymmetric laminar diffusion flame was studied using optical measurements. Integral emission of excited CH and OH radicals was measured. Emission intensity measurements were lineof-sight-integrated and the two-dimensional, in-plane intensity distribution was recovered with an Abel inversion. It was shown that both CH and OH radicals occurred in a very thin region. 3D reconstruction of instantaneous distribution of chemiluminescence of a turbulent propane–air rich premixed flame was performed in [9]. A ‘slice-by-slice’ reconstruction technique was applied. A multi-lens camera equipped with forty small lenses provided forty views for recording data at each slice. Next four hundred slices were vertically accumulated, resulting in an instantaneous 3D distribution of flame-chemiluminescence. A maximum likelihood expectation maximization (MLEM) algorithm was applied for reconstruction. Although inclusion of absorption term is possible in MLEM algorithm, absorption was not considered because the self-absorption at bands of CH, C2 and HCO was assumed very weak in small flames. In [10], the emission tomography method was applied to relate the hydroxyl OH radical’s chemiluminescent emission from a combustion flame to spatial heat release. It has been shown that the OH radical emission is directly proportional to heat release in premixed flames. A set of forty fiber-optic detectors were used to collect the data. The intensities detected by each detector were used as input to a tomographic reconstruction algorithm based on the multiplicative algebraic reconstruction technique (MART). A ‘slice-by-slice’ reconstruction technique was applied to obtain 3-D images. In [11], the design, implementation and evaluation of a 3-D imaging system for the reconstruction of the luminosity distribution of a combustion flame have been presented. The system comprised three identical sets of red–green–blue CCD cameras and is capable of capturing flame images from six equiangular directions. Computer simulations were performed. The Shepp–Logan head phantom, developed

for medical applications, was used for examination of different reconstruction algorithms, including the logical filtered back-projection (LFBP), algebraic reconstruction technique (ART) and the combination of LFBP and ART. Simulation results have shown that satisfactory reconstructions of the head phantom by these algorithms can be obtained with 32 observation directions. In [12], those investigations were continued. Optical fiber imaging based tomographic reconstruction of burner flames was performed. Eight imaging fiber bundles coupled with two RGB charge-coupled cameras were used to acquire flame images simultaneously from eight different directions around the burner. A new algorithm combined the LFBP and the simultaneous algebraic reconstruction technique (SIRT) was tested in numerical simulation. Simulation results have demonstrated that the new LFBR-SIRT algorithm provides consistently better reconstruction of the head phantom if compare to the previous LFBR-ART, used in [11]. An axisymmetric flame tomographic study was performed in [13]. The Abel deconvolutionbased algorithm with Tikhonov’s regularization was developed for reconstruction. Various parameter selection methods were studied. The discrepance principle, L-curve curvature and generalized cross-validation regularization parameter selection methods were tested. It was concluded that Tikhonov regularization could stabilize deconvolution of optical data collected from axisymmetric flames. This approach was developed and the performance of the parameter selection algorithms was evaluated by solving deconvolution test problems derived from a laminar-diffusion flame experiment. In [14], a turbulent opposed jet flame was investigated using emission computed tomography. Instantaneous chemiluminescence emission measurements and phantom studies were performed. Chemiluminescence emission was registered using 10 views. Reconstructions were performed using algebraic reconstruction technique (ART). To examine the relation between the number of views and the achievable resolution with ART, numerical simulations were performed using a mathematical phantom. The phantom study has demonstrated that the spatial resolution achievable with a certain number of views by using ART reconstruction was dependent on the object to be reconstructed. The resolution of the concrete phantom reconstruction depends on the number of views available. Reconstructions of realistic phantom data have shown that good resolution in the turbulent opposed jet flame can be achieved from 20 views. Although the list of references is far from being complete, one can see general tendencies in the flame tomographic researches. In the majority of the listed investigations, reconstructions were performed using the ART (MART) and FBP algorithms. The statistical MLEM algorithm was applied in one work [9]. The question arises: which of many reconstruction techniques should be chosen for tomographic investigations under realistic flame conditions? The conditions are characterized by three basic factors: (1) noisy data (fluctuations), (2) limited data (limited access and complexity of the diagnostic system), (3) a specific source function (high-gradient peaks). It is shown in the literature that the ART (MART) algorithm may be applied successfully in the cases when the measured data are relatively complete and the object to be reconstructed is quite smooth [15]. In those cases, such algorithms as ART or FBP can provide reconstruction of good quality. For instance, in [11], it was shown that satisfactory reconstructions of the head phantom by the ART can be obtained with 32 observation directions. In situations with strongly limited and noisy data, one should expect poor reconstructions by these algorithms. The statistical MLEM algorithm can be the algorithm of choice in the problems with complete, but noisy data. In [9], the MLEM was applied in conditions with forty observation views. The MLEM is a non-regularized algorithm [16], the resulting image can be dominated by noise artifacts. In practice, one obtains good MLEM images by stopping the itera-

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tion process. The stopping rule is considered as a kind of regularization in this approach. The goal of the present paper is to propose a reliable tomographic method which can be applied for quantitative diagnostics of chemiluminescence intensities C2/CH/OH rations. The approach based on the maximum entropy concept is chosen for chemiluminescent flame tomography. The maximum entropy-based (MENT) algorithm is known as a promising approach to tomography problems especially in situations with limited number of views [17,18]. The MENT has also shown good results when applied to objects consisting of isolated peaks on a zero background [19]. But, this algorithm assumes that measured data are noiseless. The preprocessing procedure should be applied to smooth noisy data. Based on our previous experience with the MENT [20], we have concluded that this algorithm is excellent in combination with the data preprocessing procedure. An improvement in image quality obtained by using the MENT over the ART and FBP, was demonstrated in [18,21] for limited view tomography. In this paper, the performance of the MENT-based reconstruction technique is evaluated on the problem inspired by the OH, CH, C2 chemiluminescence from the axisymmetric flame of a Bunsen-type burner. The paper is organized as follows. In the next Section 2, the maximum entropy algorithm is described. In the Section 3, the experimental facility and chemiluminescent emission measurements are presented. A modern approach to the tomography considers preliminary numerical modeling as a necessary component of tomographic researches. In the Section 4, numerical simulations are performed as close as possible to the experimental measurements. A new method based on local regularization is developed for data processing to achieve more accurate reconstruction of the peak intensities and thicknesses of chemiluminescent zones in flames. In the Section 5, the developed approach is applied for reconstruction of OH, CH and C2 radical chemiluminescences from measured data in the flame of the Bunsen-type burner. In this work, we do not discuss the interpretation of the reconstructed images and focus on the reconstruction method.

gðp; hÞ ¼

Z

f ðx; yÞdlp;h

ð3Þ

where g(p, h) denotes the measured projection data and f is the source function to be reconstructed. 2.2. Maximum entropy concept A typical problem encountered in flame tomography is incomplete data. The number of views in combustion experiments is often small because of problems with limited access and the cost and complexity of the diagnostic system. From the mathematical point of view, in that case, the problem of reconstruction formulated by Eq. (3) is a strongly underdetermined problem. This problem can be reformulated as a constrained optimization problem: we want to obtain the most probable solution ~f constrained to the given (limited) data g. Fundamental research into this optimization problem has been conducted by Jaynes [22]. He has developed the theory based on the entropy concept. According to the Jaynes’s theory the entropy functional is used as a priori information about a solution ~f . It was proven that the entropy prior yields the most probable solution that is consistent with the available data. The entropy functional is written in the form:

ln H ¼ b

ZZ

f ðx; yÞ ln f ðx; yÞdxdy

ð4Þ

where b is the proportionality factor. At first, Jaynes’s theory was developed for astrophysical images resolving. Minerbo [17] has applied the maximum entropy principle to the tomography problem. On maximizing ln H (4) on condition of Eq. (3) one obtains the maximum entropy (MENT) algorithm to select a solution. The problem is solved by using the Lagrange multiplier technique:

~f ¼ arg f

(

 max b

ZZ

) Z X f ðx; yÞ lnf ðx;yÞdxdy þ Kk ðg k  f ðx;yÞdlp;hk Þ k

l

ð5Þ 2. Theory A general approach that is suitable for the reconstruction of both symmetrical and nonsymmetrical flames is analyzed in this section. 2.1. Basic equations The basic equation describing the radiation transfer along the lines passing through a flame can be written in a general form as follows:

dIk ðlÞ ¼ jk ðlÞ  kk ðlÞIk ðlÞ dl

ð1Þ

where Ik is the radiation intensity, jk and kk are the emission and absorption coefficients for spectral line k and l is the line of sight. The solution of (1) in the general case is a very difficult problem. For the case of emission tomography one usually uses the spectral lines for which the absorption can be neglected. In this work, chemiluminescent emission of OH, CH and C2 radicals in the hydrocarbon Bunsen flame is analyzed. The self-absorption at bands of OH, CH and C2 radicals is weak under fuel-lean conditions. We assume, that the absorption for the radiation of wavelength k can be neglected (kk ¼ 0) and the Eq. (1) can be presented in the form:

Ik ðp; hÞ ¼

Z

1

jk ðx; yÞdlp;h

ð2Þ

1

where p and h are the normal coordinates of line l. Eq. (2) is known as the Fredholm integral equation of first kind and can be rewritten in the form:

where hk corresponds to the kth viewing angle, Kk are the Lagrange multipliers. Eq. (5) gives the estimation ~f of the unknown source function f. In [21], numerical simulations were performed to compare the ART and the MENT reconstruction techniques under limited data conditions. Figure 1 presents the results of reconstructions. Figure 1a demonstrates the original model. The exact projection data were calculated analytically. The data were calculated for six directions of observations only spaced around the object. A 3% Gaussian noise was added to the exact projection data. Figure 1b shows the reconstructed image using the ART algorithm. For comparison, the reconstruction of the same source function by the MENT algorithm is shown in Fig. 1c. It is seen that the ART reconstruction is completely unrecognizable. The MENT reconstruction is much better but also suffers from the presence of noisy artifacts. Figure 1 demonstrates that the MENT provides an improvement in reconstruction image quality over ART for the case of limited number of views. The reason of the noisy artifacts is that the MENT algorithm assumes that the projection data are noise-free. To regularize the reconstruction problem the projection data should be preprocessed. 2.3. Data processing It is known that the problem of recovering a smooth function from noisy data also belongs to the class of ill-posed problems and should be solved by using a regularization method. In this work, the smoothing procedure was developed by using the Tikhonov’s regularization method [5]. The problem of data processing

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c Fig. 1. Comparison of the ART and MENT reconstruction techniques in the case of limited and noisy data [21]. The reconstructions were performed using projection data calculated for six positions around the original model. (a) the original source; (b) reconstruction by the ART algorithm; (c) reconstruction by the MENT algorithm.

can be formulated as follows. Let g(p, h) are noisy data. We would like to find such smoothed function g~ðp; hÞ that corresponds to the ‘exact’ (noiseless) data. According to the theory [23,24], for correct smoothing of the noisy projection data at each view h = hk, one needs to find the cubic spline function g~ðp; hk ; aÞ which will minimize the following functional:

F¼a

Z

kg~00 ðp; hk ; aÞk2 dp þ jg~ðp; hk ; aÞ  gðp; hk Þj

ð6Þ

where a is the regularization parameter, g~00 is the second derivative of the function g~ðp; h; aÞ; k . . . k is the Euclidean norm. Regularization provides a family of solutions g~ðp; h; aÞ depending on the regularization parameter a. The first term in (6) characterizes the smoothness of the function g~ðp; h; aÞ to be found, and the second term characterizes how close the values g~ðp; h; aÞ are to the measured data g(p, h). The chi-square criterion based on the Pearson’s v2 statistic was used for the choice of the regularization parameter [25]. The parameter a was chosen using the statistical form:

v2 ðaÞ ¼

J K X X ðg~ðpj ; hk ; aÞ  gðpj ; hk ÞÞ2 k¼1 j¼1

r2jk

pffiffiffiffiffiffiffiffiffiffiffi ¼JK 2 JK

ture was prepared in a gas system (1). The flame (2) was stabilized on the cut of the quartz tube with inner diameter of 13.5 mm and the length of 800 mm. The flow rates of the gases were measured by Bronkhorst ElFlow flow meters (with accuracy within 1%). Average velocity is 1 m/s. Reynolds number is about 1000, that corresponds to realization of the Poiseuille velocity profile on a cut of a tube. Such flame is well studied. Spectrozonal detection of the flame front was performed using emissions of the OH, CH and C2 radicals. The spectral flame images were recorded by using CCD camera (3) that is a part of the system FlameMaster produced by LaVision.

ð7Þ

where r is the standard deviation, J is the pixel number at each view, K is the number of views. 3. Experiment The experimental arrangement is schematically shown in Fig. 2. Measurements of the chemiluminescence are performed in a Bunsen-type burner. Propane was taken as a fuel. A combustion mix-

Fig. 2. Arrangement of the experiment. (1) Gas system, (2) flame, (3) CCD camera.

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Fig. 3. Propane–air flame photographs (planar pictures) recorded by using appropriate filters. (a) OH radical emission, (b) CH radical emission, (c) C2 radical emission. The arrows show the selected flame cross-section.

For spectral measurements we used the spectra in the visible and ultra-violet regions emitted by OH, CH and C2 radicals, generated in the reaction zone of the flame. Spectra in the visible and ultra-violet regions correspond to transitions from one to another electron molecular state. The hydroxyl radical OH gives a band system with the strongest head at 306.4 nm. The band of 306.3 nm corresponds to the transition A2 Rþ ! X 2 P from the second excited electron state of OH radical to the ground state. The strongest band of CH radical spectra is around 431.5 nm corresponding to the spectral transition A2 D ! X 2 P. Swan band of C2 radical with the head at 540 nm corresponding to the spectral transition A3 Pg ! X 3 Pu was selected. Figure 3 shows a typical set of the 2D measured pictures of the flame. The pictures were recorded by using a CCD camera with appropriate sets of dichroic filters: OH – ‘‘BP307/10-6, CH – ‘‘BP430/10-5’’ and C2 – ‘‘LP540-3’’. For lean mixtures (equivalence ratio u = 0.8), the Bunsen burner flame is laminar and has a narrow reaction zone.

In this work, the ‘slice-by slice’ technique was applied for 3-D reconstruction. This technique is widely used in the field of medical tomography. The reconstruction area is divided into slices along the axes z and each slice is analyzed separately. After that all reconstructed slices are accumulated resulting in an instantaneous 3-D distribution. The reconstruction technique is shown schematically in Fig. 4. In the present study, the spectral flame images were recorded by using only one view due to axisymmetric flame geometry. In a general case, one needs to record the data from multiple directions.

4. Numerical simulation The formal applying of a reconstruction algorithm to measured data results a reconstructed source function ~f . The main problem is, how close is the estimation ~f to the real source function f?

z

y x 1 2

a

b

c

d

Fig. 4. Tomographic ‘slice-by-slice’ reconstruction technique (the axisymmetric case). The dashed lines indicate 4 arbitrary slices. In total, 301 slices were calculated. (a) the flame, (b) the planar flame image, recorded by using CCD camera and a filter, (c) 2D reconstructions at slices, (d) 3D reconstruction (emitting OH radical density distribution), obtained by accumulation of all 301 slices. The cut in the 3D reconstruction demonstrates the inside flame structure. The selected slice number 2 was used in this paper for a detailed analysis of the reconstruction procedure.

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Source function model (a.u.)

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c Fig. 5. Numerical simulation. The models simulate the radial profiles of OH (a), CH (b) and C2 (c) chemiluminescences in the selected cross-section (marked by the arrows in Fig. 3 and shown in Fig. 4 as the slice 2). Solid lines show the original OH, CH and C2 models. Dotted lines show the reconstructed profiles using the MENT algorithm without data preprocessing procedure.

Usually this problem is investigated in a numerical simulation that approaches the situation of a real experiment. Currently, numerical modeling is a necessary component of tomographic researches to examine the effectiveness of the reconstruction algorithm to be used in conditions close to real experiment. It should be noted, that in the literature the effectiveness of the algorithms applicable for flame tomography is often tested by using various models (phantoms). The head phantom, developed for medical applications, was used for examination, for example, in [11,12]. The function describing the spatial distributions of chemiluminescence in real flames has narrow high-gradient peaks. The effectiveness of a reconstruction method for such specific source function should be tested using models close to the real source function. The models used for examination should include the narrow highgradient peaks. In this paper, numerical simulations were performed close to the experimental measurements, presented in the Section 3. 4.1. Models The examination of the MENT algorithm effectiveness was performed for one selected flame slice, marked by the arrows in the Fig. 3 and shown in Fig. 4 as the slice 2. The examination may be limited by analysis of one slice only, because the projection data in different slices are similar. The test models simulating the 2D

distributions of OH, CH and C2 chemiluminescence in the selected slice are shown in Fig. 5a–c. The central profiles of the 3D models are presented. The models were generated by using a composition of simple mathematical functions. We denote them as the OH, CH and C2 models, respectively. The presented profiles clearly show two high-gradient peaks that simulate narrow regions of chemiluminescent radical emission in the flame. 4.2. Projection data One-view projection data were calculated for the OH, CH and C2 models by using analytical formulas. The analytical calculation of the exact projection data is a very important moment because such data are obtained independently on the reconstruction algorithm and can be used as really measured data. A Gaussian noise was added to the exact data to simulate the situation in the experimental measurements. A 3% noise was added to the exact data of the OH model, 5% noise was added to the data of the CH model and 10% noise was added to the data of the C2 model. The comparison of the simulated and measured data is presented in Fig. 6. The first row (a–c) shows the simulated data. The second row (d–f) demonstrates the real measured data from the selected sections in Fig. 3a–c. A good similarity between the measured and modeled data is observed. We can conclude that the numerical simulation is performed close to the real experimental situation.

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Fig. 6. Comparison of the simulated and measured one-view projection data from the selected cross-section of the flame. First row (a–c) simulated projection data using the OH, CH and C2 models, shown in Fig. 5. Second row (d–f) measured projection data in the selected cross-section, marked by the arrow in Fig. 3.

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Fig. 7. Numerical simulation. Reconstruction of the CH model by using the MENT algorithm in combination with standard data processing procedure. Solid lines show simulated noisy data (a) from the original CH model (b), dotted lines demonstrate the processed data (a) and the reconstructed CH model (b) obtained by using the processed data.

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Fig. 8. Numerical simulation. Reconstruction of the C2 model by using the MENT algorithm in combination with standard data processing procedure. Solid lines show simulated noisy data (a) from the original C2 model (b), dotted lines demonstrate the processed data (a) and the reconstructed C2 model (b) obtained by using the processed data.

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Fig. 9. Numerical simulation. Reconstruction of the CH model by using the MENT algorithm in combination with a new data processing procedure based on a local regularization. Solid lines show simulated noisy data (a) and the original CH model (b), dotted lines demonstrate the processed data (a) and the reconstructed CH model (b).

4.3. MENT reconstruction 4.3.1. MENT reconstruction without data preprocessing To clarify the resolution achievable with the MENT algorithm without data preprocessing, the reconstructed images are shown in dotted lines in Fig. 5. It is seen that the OH model is well reconstructed with a detailed structure of the source function. However, reconstructions of the CH and C2 Models are suffered from the presence of noisy artifacts in the central regions. Comparing the reconstructed OH, CH and C2 models, one can make the following conclusions. The obtained results clearly demonstrate the ill-posed nature of the reconstruction problem: a solution dependence on the noise level in the data. The noise level in the projection data of the OH Model (Fig. 6a) is substantially less if compare to the data of the CH and C2 Models (Fig. 6b and c). As the result, the reconstructed profile of the OH Model is very close to the exact model, but the reconstructed CH and C2 profiles suffer from the noisy artifacts. Such reconstructions cannot be used for qualitative and

quantitative flame diagnostics. As noted, the reason of the noisy artifacts is that the MENT algorithm assumes that the projection data are noise free. To regularize the solution the projection data should be correctly preprocessed. 4.3.2. MENT reconstruction with standard data preprocessing procedure A processing procedure described in the Section 2.3 was applied to smooth the simulated noisy data. The regularization parameter was chosen by using the chi-square v2 criterion (7). The standard approach supposes that one regularization parameter is selected for the whole set of data. Figures 7 and 8 demonstrate the results of data processing by using the standard approach and the corresponding reconstructed CH and C2 profiles. The comparison between the original and the reconstructed profiles demonstrates that the peaks in the reconstructed profiles are over-smoothed. Such reconstructed images are useful for qualitative illustrations, but cannot be used for quantitative flame diagnostics. The correct

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Fig. 10. Numerical simulation. Reconstruction of the C2 model by using the MENT algorithm in combination with a new data processing procedure based on a local regularization. Solid lines show simulated noisy data (a) and the original model (b), dotted lines show the processed data (a) and the reconstructed model (b).

Fig. 11. Experiment. The reconstructed OH, CH and C2 chemiluminescence spatial distributions in the flame of the Bunsen burner. (a) 3D reconstruction of CH chemiluminescence. The inside structure is shown in a cut. The longitudinal sections of 3D reconstructions of CH (b), OH (c) and C2 (d) chemiluminescences are shown. 3D reconstructions were obtained from the data, shown in Fig. 3.

reconstruction of the peak intensity is an important moment in flame tomography because the ratio of peak intensities CH/C2/ OH has been shown to be a robust measure of local equivalence ratio in lean burning combusters [26].

4.3.3. MENT reconstruction with data preprocessing using local regularization Projection data in flames have a specific character: the second derivative g~00 is relatively small in the center and it is discontinuous

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at the peaks. The second derivative g~00 characterizes the smooth~ðp; h; aÞ to be found. In accordance with Eq. ness of the function g (7), the parameter a controls the degree of smoothing. Numerical tests have shown that as the parameter a was relatively large value, g~ðp; h; aÞ was well smoothed in the center and over-smoothed in the peaks. As the parameter was decreased, g~ðp; h; aÞ approaches the peaks, but became too noisy in the center. By analyzing the numerical results one can conclude that the regularization parameter a should be preferably different in that different areas. To avoid over-smoothing of the peaks and to ensure a feasible smoothing, a new approach based on the spatially adaptive – ‘local’ regularization has been proposed in this work for flame tomography. The idea of local regularization was introduced first in the work [27] for solving integral equations of the convolution type. The data set was divided into two areas: the central zone and the peak zone. The chi-square v2 criterion (7) was used for choosing of the optimal regularization parameter with the exception that it was applied separately to each of two selected areas. Figures 9 and 10 demonstrate the results of data preprocessing procedure using the local regularization method and the corresponding reconstructed profiles of the CH and C2 models. New peak intensity values were obtained after correction of the preprocessing pro-

cedure. The excellent agreement on detailed profiles between the original and reconstructed models is achieved. For diagnostic purposes peak intensities of OH, CH and C2 profiles are representative quantities. As seen data preprocessing with local regularization allowed us to obtain correct reconstructions of the intensity peaks and thicknesses of chemiluminescent zones. These results show that chemiluminescence can be effectively used to detect a thickness of a flame-front zone.

5. Experimental reconstruction results In this Section, the developed approach is applied for reconstruction of emitting radical distributions in the flame of a Bunsen burner. The OH, CH and C2 chemiluminescence spatial distributions in the flame reaction zone were reconstructed using measured data (planar pictures), shown in Fig. 3. The 2D digital matrixes of these pictures are projection data for tomographic reconstruction. Since the studied flame is an axisymmetric object, one-view projection data were used for reconstruction. In a Cartesian coordinate system (x, y, z) a 3D flame area is normalized as follows: {1 < x < 1; 1 < y < 1; 2 < z < 2}. It is assumed that the

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with local regularization are also shown in these figures. These smoothed data were used as input data for the MENT reconstruction algorithm. Similar processing was performed with data in the slice 1. The reconstructed OH, CH and C2 radical chemiluminescence profiles in these two selected slices are presented in Fig. 13(a-b). The ratio of peak intensities in the slice 1 is estimated (Fig. 13a) as follows: CH:C2:OH = 1.3:0.8:1. The similar ratio in the slice 2 gives (Fig. 13b) CH:C2:OH = 1.1:0.7:1. The profiles in the slice 1 show higher intensities, however, the rations CH/C2/OH in the slices 1 and 2 are close to each other. This result is correct for this type of flame. The OH distribution has the narrowest emission zone in both slices. The C2 intensity shows a wider profile than OH and CH ones. Similar distributions were obtained for all of 301 slices that allowed us to analyze the relations CH/C2/OH along the height of the flame. The developed reconstruction technique could be applied for reconstructions of both axisymmetric and non-axisymmetric flames. Further study is necessary for the non-axisymmetric flame tomography. In that case, the question arises: how many observation views should be used in measurements to obtain good results of reconstruction? This question could be answered in preliminary numerical simulations. As it was noted above, the MENT provides good reconstruction using minimal necessary number of views. Potentially, this technique could be extended for reconstruction of spatiotemporal behavior of flames. Previously [28], the MENT algorithm was successfully applied for reconstruction of fast plasmachemical processes with a time step of 0.2 ls.

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flame is oriented vertically along the axis z. The 3D area was divided into 301 slices along z and each slice {x, y, zk = const, k = 1, 2, 3 . . . 301} is analyzed separately. The reconstruction technique is shown in Fig. 4. The measured data digital matrix was also divided into 301 rows. Therefore, the 3D reconstruction problem was reduced to the 2D problem. 2D reconstruction method was tested in numerical simulation in the Section 4. The 201  201 grid was used for reconstruction at each slice. The reconstruction was computed repeatedly for every of 301 slices. The reconstruction procedure was organized in that way that the data in each slice were automatically processed using proper regularization. 301 horizontal images were reconstructed and accumulated vertically to obtain 3D image. Figure 11 shows the reconstructed images which correspond to the recorded data. The images demonstrate the OH, CH and C2 chemiluminescence spatial distributions in the flame of the Bunsen burner. Figure 11a shows 3D distribution of CH chemiluminescence. Chemiluminescence profiles of CH (b), OH (c) and C2 (d) chemiluminescence spatial distributions in the longitudinal sections of 3D reconstructions are shown in Fig. 11. More detailed analysis was performed for two slices. These slices positions are shown approximately in Fig. 4 (slices 1 and 2). The measured data corresponding to the slice 2 (marked by the arrows in Fig. 3) are presented in Fig. 12a–c for OH, CH and C2 radical emission. The preprocessed data using the smoothing procedure

Spatial distributions of OH, CH and C2 chemiluminescence have been drawing great attention for monitoring such characteristics as fuel–air ratio, completeness of combustion, rate of combustion, blowout and for investigation of various external actions (acoustic, electrical, laser, etc.) on combustion processes. There is an increased interest in the application of tomographic algorithms and methods to combustion control. In this work, the maximum entropy-based method is applied for quantitative tomographic diagnostics of chemiluminescence intensities C2/CH/OH rations. The MENT is chosen because this approach yields the most probable solution consistent with the limited data and has previously shown good results in reconstruction objects consisting of isolated peaks. The performance of the MENT technique was tested on the problem inspired by the OH, CH, C2 chemiluminescence from the axisymmetric flame of a Bunsen-type burner. A new approach based on local regularization has provided accurate reconstruction of the chemiluminescence peak intensities and thicknesses of reaction zones in flames. The obtained results show that tomography can be effectively used to detect the ratio of peak intensities CH/C2/OH for monitoring fuel–air ratio, completeness of combustion, rate of combustion, etc. Potentially, this technique could be used to study spatiotemporal behavior of both axisymmetric and non-axisymmetric flames. Acknowledgment This work was partly supported by Russian Foundation for Basic Research (Grant 11-01-00158-a). References [1] N. Docquier, S. Candel, Prog. Energy Combust. Sci. 28 (2002) 107–150. [2] T.M. Muruganandam, B. Kim, R. Olsen, M. Patel, B. Romig, J.M. Seitzman, in: AIAA-paper 2003–4490 (2003) 39th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, Huntsville, AL, July 20–23, 2003. [3] J. Kojima, Y. Ikeda, T. Nakajima, Combust. Flame 140 (2005) 34–45. [4] K. Gosselin, M. Renfro, Appl. Opt. 51 (2012) 1671–1680.

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