Simultaneous measurement of flame temperature and species concentration distribution from nonlinear tomographic absorption spectroscopy

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Simultaneous measurement of flame temperature and species concentration distribution from nonlinear tomographic absorption spectroscopy Jing-Wen Shi , Hong Qi , Jun-You Zhang , Ya-Tao Ren , Li-Ming Ruan , Yong Zhang PII: DOI: Reference:

S0022-4073(19)30344-9 https://doi.org/10.1016/j.jqsrt.2019.106693 JQSRT 106693

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Journal of Quantitative Spectroscopy & Radiative Transfer

Received date: Revised date: Accepted date:

18 May 2019 6 August 2019 13 October 2019

Please cite this article as: Jing-Wen Shi , Hong Qi , Jun-You Zhang , Ya-Tao Ren , Li-Ming Ruan , Yong Zhang , Simultaneous measurement of flame temperature and species concentration distribution from nonlinear tomographic absorption spectroscopy, Journal of Quantitative Spectroscopy & Radiative Transfer (2019), doi: https://doi.org/10.1016/j.jqsrt.2019.106693

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HIGHLIGHTS 

Simultaneous measurement of temperature and concentration is investigated.



The CMA-ES algorithm is introduced to solve the inverse problem.



Tikhonov regularization can improve the reconstruction quality.

Simultaneous measurement of flame temperature and species concentration distribution from nonlinear tomographic absorption spectroscopy Jing-Wen Shi1, 2, Hong Qi1, 2*, Jun-You Zhang1, 2, Ya-Tao Ren1, 2, Li-Ming Ruan1, 2, Yong Zhang1, 2 1. School of Energy Science and Engineering, Harbin Institute of Technology, Harbin, China, 150001 2. Key Laboratory of Aerospace Thermophysics, Ministry of Industry and Information Technology, Harbin, China, 150001 *Corresponding author: Hong Qi 1. School of Energy Science and Engineering, Harbin Institute of Technology 2. Key Laboratory of Aerospace Thermophysics, Ministry of Industry and Information Technology 92, West Dazhi Street, Harbin, P. R. China, 150001 Tel: (86)-0451-86412638 Email: [email protected] (H. Qi) Abstract This research introduces the evolution strategy based on covariance matrix adaption algorithm, which has been proven applicable for solving highly inseparable and nonlinear optimization problems, to resolve the nonlinear tomography absorption spectroscopy inverse optimization problem and simultaneously reconstruct a two-dimensional temperature and species concentration distribution fields in an absorption flame. Severe ill-posedness and crosstalk issues exist during multi-parameter field simultaneous reconstruction. To alleviate the ill-posedness in nonlinear tomography absorption spectroscopy equations, two regularization methods, Tikhonov regularization and the regularization based on the generalized Gaussian Markov random field, are applied to reconstruct the temperature distribution field. The impacts of different regularization factors (the temperature and the concentration regularization factors) are investigated as well, and optimal intervals are suggested for reference. Simulation results show that the evolution strategy based on covariance

matrix adaption algorithm worked well in retrieving multi-parameter distribution fields on both symmetric and asymmetric flame distribution models. Moreover, the evolution strategy based on covariance matrix adaption algorithm alleviated the severe crosstalk issue between temperature and species concentration to improve the accuracy of the reconstructed species concentration distribution field. Keyword: Nonlinear tomography; absorption spectroscopy; CMA-ES algorithm; crosstalk

Nomenclature A

the integrated absorbance

Ac

the computed integrated absorbance

Am

the measured integrated absorbance

b

the weight coefficient

c

light speed [cm/s]

cm

the learning rate for updating the mean value

c1

the learning rate for rank-1 updating the covariance matrix

c

the learning rate for rank-m updating the covariance matrix

c

the learning rate for the step size controlling

C

the covariance matrix

d

the damping parameter for step-size renewing

eT

the average error of temperature

eX

the average error of species concentration

E ''

the lower state energy of the transition [cm-1]

Fobj

the objective function

h

Plank‟s constant [J·s]

I

the identity matrix

I0()

the incident laser intensity [W/cm2]

It()

the transmitted laser intensity [W/cm2]

k

Boltzmann‟s constant [J/K]

L

the total length of sample pool [cm]

L

the projection coefficient matrix [cm]

m

the mean value of the search distribution

N

a normal distribution

p

the image sharpness parameter

pc

the evolution path for updating the covariance matrix

ps

the evolution path for updating the step size

P

the pressure [atm]

Q

the partition function

S

the line strength function [cm-2atm-1]

T

the temperature distribution [K]

wi

the positive weight coefficient for recombination

x

the best individual

x

the search point vector

X

the species concentration distribution

Greek symbols



the coefficient of the integrated absorbance



the integrated absorbance coefficient matrix

C

the maximum deviation of reconstructed species concentration

T

the maximum deviation of reconstructed temperature

T

the temperature regularization factor

 X

the concentration regularization factor



the line shape function [cm]



the wavelength [m]



the step-size

subscripts CGMRF

the CGMRF regularization technique

ini

the initial setting value

poisterior

the measured information

prior

the smooth prior information

rec

the reconstructed value

s-r

the neighboring position between r and s

Tkh

Tikhonov regularization technique

1. Introduction Combustion is a common and important physical phenomenon that widely exists in various engineering fields, such as aero-engine plume exhaust, industrial boiler combustion, chemical metallurgical process, and thermal power generation [1-2]. The essence of combustion is a complex heat and mass transfer problem that involves multi-phase fluid flow, high temperature chemical reaction, radiation heat transfer, and other fields. The combustion process and combustion products are two important aspects of research for the combustion phenomenon, which can directly characterize combustion states, including flame structure, reaction speed, combustion stability, combustion efficiency, and pollution emission [3]. Accurate measurement of temperature and combustion product concentration fields in the combustion process, not only contributes to the mechanism analysis of the heat transfer process, but also provides reliable data for the safe operation of equipment, fuel efficiency, and pollutants control [4]. Therefore, it is crucial to develop accurate measurement method to reconstruct temperature and concentration fields simultaneously in combustion. In the past decades, many research efforts have been devoted to measure temperature, pressure, velocity, concentration, and other parameter fields in combustion area. In general, the measurement techniques can be grouped into two categories: contact and non-contact (or non-intrusive) measurement

techniques. Contact measurement methods always suffer from issues, such as flow field perturbation caused by physical probes, sample dependence, potential physical and chemical reactions [5], thus lead to unreliable measurement results. Moreover, contact measurement methods are incapable to provide high-precision and spatial resolution and to realize real-time measurement. Conversely, by virtue of their unique advantages such as non-intrusiveness, fast response, and abundant spatial information, non-contact (non-intrusive) measurement methods have attracted great attention and developed rapidly in the measurement field. Rapid development of laser and photoelectron technology in recent decades have greatly promotes the relevant research and application to the non-intrusive measurement methods based on optics theory. For example, Cheng et al. [6] carried out an on-line three-dimensional temperature measurement experiment based on radiation image processing technology in a gas-fired pilot tubular furnace. Our group [7] introduced the light-field technique to simultaneously estimate the temperature distribution of participating media, and Xu et al. [8] applied a single light field camera for optical sectioning tomographic reconstruction of three-dimensional flame temperature distribution successfully. The measurement methods discussed above belong to passive detection technology, which mainly utilizes radiation images of flame to ascertain the parameters distribution field. The measurements based on radiation images of flame can achieve temperature and species concentration simultaneously as well [9-12], while its concentration measurement research mainly focus on soot volume fraction or particle concentration. Moreover, cause the detective signal of passive measurement method is coming from the extinction of flame emissive radiation and which is easily distracted by surroundings, so the signal intensity is insufficient for high-accuracy measuring of typical gaseous combustion products, such as H2O, CO2, CO and so on. By contrast, active detection technology employs laser as measured signal, which can provide higher spatial and temporal resolution, non-intrusive optical access, and higher signal intensity, and avoid interference from the surroundings. In which, laser absorption spectroscopy technology attracts much attention, for which can provide „fingerprint‟ absorption of different gases, and avoid interference from the surroundings. Thus, laser absorption spectroscopy technique has drawn considerable attention in combustion diagnosis and presents a promising research direction [13-15]. For instance, Wang et al. [16] reconstructed gas concentration and temperature distributions based on tunable diode laser absorption spectroscopy (TDLAS). Liu et al. [17] developed a fan-beam TDLAS-based sensor for rapid imaging of temperature and gas concentration. Sur et

al. [18] developed high-sensitivity interference-free diagnostic sensors for the measurement of methane in shock tubes and experimentally proved their effectiveness. Wood et al. [19] experimented on the simultaneous reconstruction of temperature, concentration, and pressure imaging of water vapor in a turbine engine. Wei et al. [20] utilized a high-speed infrared camera to image flow fields backlit with tunable mid-wave infrared laser radiation on the basis of laser absorption spectroscopy techniques. The aforementioned studies indicate that the non-contact (non-intrusive) measurement problem can be converted into an inverse optimization problem. The two keys of an inverse optimization problem are (i) measurement models and (ii) inverse algorithms. Tomography absorption spectroscopy (TAS) technology is a non-contact measurement model and is widely applied to estimate temperature and concentration fields due to its species specificity, relative simplicity of equipment, 2D spatial resolution, and versatility [21-24]. Traditional TAS (also called linear TAS) requires large number of projections to obtain a satisfactory result due to the limited spectral information, and which restricts its applications in harsh industrial environments [25-27]. More recently, nonlinear TAS was proposed and developed significantly in the field of combustion diagnosis [28, 29] because the rich spectral information provided by laser technology can be incorporated into tomographic reconstruction, thereby improving the estimation results dramatically. Therefore, the nonlinear TAS model is selected as the measurement model in this work. The inverse problem of nonlinear TAS is more complex and difficult than other inverse problems because of its extensive calculation cost, nonlinear dependence on the temperature, and linear dependence on species concentration of spectral information. Very recently, a new method for solving hyperspectral absorption tomography has been proposed by Grauer et. al [30] which enhances computational efficiency greatly by adopting two-step approach to attain multiparameter of gas, and this is an enlightenment for our subsequent work on solving the huge computational cost issue. Besides, severe crosstalk problems between temperature and species concentration emerge during the reconstruction of the multi-parameter distribution process of the combustion field and poses another enormous challenge for the algorithms. The widely used conventional inverse methods are gradient-based methods, such as the conjugate gradient (CG), quasi-Newton, and sequential quadratic programming (SQP) methods, which converge rapidly but entail occasionally complicated gradient computation. Moreover, gradient-based algorithms are highly affected by the initial estimate. Without correlative experience, obtaining a reasonable result may be difficult unless a proper initial estimation is

available [31-34]. The global convergence algorithm based on stochastic search mechanism have drawn great attraction in recent years. These stochastic algorithms possess its unique superiorities [35-38] like: (1) easy operation for it merely has few control parameters (2) the initial guess is unnecessary; and (3) they can be parallelized easily. And the evolution strategy based on covariance matrix adaption (CMA-ES) algorithm is an excellent gradient-free stochastic method for the optimization of nonlinear functions. In addition, it has performed excellently in solving some optimization problems that are difficult to deal with, such as ill-posed and highly inseparable problems [39-42]. Zhang et al. [47] first used the CMA-ES algorithm to optimize inverse radiative problem based on point measurement, and made significant progress in inversion precision and computational efficiency. However, to the best of our knowledge, this algorithm has not been used to optimize more complex nonlinear TAS inverse problem based on field measurement. The motivation of this work is to employ the CMA-ES algorithm based on a nonlinear TAS measurement model to estimate the temperature and concentration fields of flame combustion. A new inverse model including the measurement model setting and an optimization algorithm coupled with regularization technique is developed. The remainder of this work is organized as follows: Section 2 introduces the theory of laser tomography absorption spectroscopy and its mathematical formulation. Section 3 explains the CMA-ES algorithm principles in brief. Section 4 provides the results and discussions. The final section summarizes this work. 2. Theory 2.1 Direct model for nonlinear TAS When a laser beam traverses the absorbing gas sample, the laser intensity absorbance     at wavelength  can be calculated by the Beer-Lambert Law [15]:

ln

It   

I0   

L



      P  l   X  l   S   , T  l       , l  dl

(1)

0

where I0() and It() are the incident and transmitted laser intensity, respectively; L is the total length of the sample pool; P(l), T(l), and X(l) are the pressure, temperature, and concentration of the absorbing species at position l, respectively; and S(,T(l)) and (l) are the line strength function and the line shape function of the ith spectral line, respectively.

The line strength S(, T) is a function of temperature for a particular transition centered at  and can be described as [15]  hc 2  1  exp    hcE''  1 1   Q (T0 ) T0  kTλ  S ( λ,T ) = S ( λ, T0 )   exp        Q (T ) T k  T T0    hc 2   1  exp     kT0l 

(2)

where S(,T0) denotes the line strength at reference temperature T0; Q is the absorber‟s partition function and can be simulated by using a polynomial fitting; E’’ is the lower-state energy of the transition; h, c, and k are the Plank‟s constant, light speed, and Boltzmann‟s constant, respectively. The line shape function is normalized and expressed as [15] (where  = 1  )







  , l  d  1

(3)

The overlapping spectral lines could be decoupled along a uniform-property path. The integrated absorbance of an isolated spectral line can be inferred from Eqs. (1)-(3) as [15] A







I t ( ) d  I 0 ( )

 ln



L

P  X  S ( , T ) dl

(4)

0

Eq. (4) is a formula of path integration that is suitable only for the situation in which the absorber‟s properties (temperature/concentration) are uniform along the path. The spatially resolved 2D distributions of the parameters in the combustion flame are identified by combining laser absorption spectroscopy and tomography. Fig. 1 Following the tomography theory, the region of interest (ROI) is divided into N×N grids (Fig. 1). In each grid, the gas flame parameters, such as pressure P, temperature T, and gas concentration X, are assumed to be constant. According to Eq. (4), the measured integrated absorbance obtained from the ith laser beam at wavelength k can be expressed as [15] N2

Ak ,i 

a

k , j L i , j 

j 1

N2

  PS T  X  j 1

k , j

Li , j

where i and j are the indices of the laser beam and the grids, respectively; Ak ,i is the integrated absorbance of the ith laser beam at wavelength k;  k , j is the coefficient of the integrated

(5)

absorbance in the jth grid at wavelength k; and Li,j denotes the absorption path length of the ith laser beam within the jth grid. Repeatedly applying Eq. (5) for a total number of K spectral lines and I probing beams, a set of equations can be obtained as follow:

Li, j  αk , j  Ak ,i

(6)

where Li,j denotes the projection coefficient matrix whose rows represent the number of laser beams and columns indicate the number of grids; αk , j is the matrix of the absorbance coefficient, and each column of the matrix represents the corresponding absorbance coefficients at wavelength k; and Ak ,i denotes the integrated absorption matrix whose rows and columns depend on the number of the laser beams and spectral lines, respectively. 2.2 Inverse model for simultaneous measurement of temperature and concentration The direct model in the nonlinear TAS problem can be mathematically expressed by Eq. (6). Unlike the traditional linear TAS method, the nonlinear TAS can maximize the utilization of spectral information while reducing the number of projections used, thereby providing easier optical access for implementation on practical industrial applications. However, additional spectral lines increase computational complexity given their nonlinear dependence on temperature. The kernel of the nonlinear TAS problem is to seek the appropriate values of T and X over the ROI. According to theoretical analysis above, we can see that the temperature reconstruction is nonlinear (according to Eq.(2)), while the concentration distribution is linear (see Eq.(5)). All in all, the simultaneous reconstruction of temperature and concentration is a nonlinear problem. The T and X profiles are discretized over the ROI (see Fig. 1), where Tj and Xj represent the temperature and the concentration of the jth grid, respectively. According to these discretized values of T and X, the measured projections can be obtained by calculating Eqs. (1)-(6) in the direct model. According to Ref. [28, 29], the nonlinear TAS inverse problem can be mathematically expressed as the following minimization problem: D  Trec , X rec  

K

I

 A  , i  A c

k 1 i 1

k

m

 k , i   1

2

(7)

where Am(k,i) means the measured projection of the ith laser beam at wavelength k, which is calculated from assumed temperature and concentration phantom distribution according to Eq. (6). Ac(k,i) denotes the computed projection according to the reconstructed temperature and concentration (designated as Trec and Xrec) of the ith laser beam at wavelength k. K and I represent the total number of spectral transitions and laser beams, respectively. The function D provides a quantitative characterization of the deviation between the reconstructed distribution and the real one. Moreover, the function D is a typically nonlinear optimization problem with multiple solutions [28], specifically, the corresponding result is not the real solution even if the function D approximates zero. Analyzed from strict mathematical theory [31], the tomography problems like chemical species topography are either discrete ill-posed or rank deficient [32-35]. For solving recovering the null-space component, the regular information or prior information are necessary. Given the distribution characteristics of the actual field (smooth and not negative), an additional function Fprior is appended to add a practical constraint condition. We define the deviation function D as the posterior objective function (Fposterior). The master objective function Fobj used in this study thus consists of two parts, Fprior, which is determined by spatial smooth characteristic and Fposterior which is decided by the deviation function. Accordingly, the global minimum points of Fobj approximate a true solution of the nonlinear inverse optimization problem. We employ the regularization technique to add a spatial smooth limitation, and simultaneously mitigate the ill-condition of the problem. Section 3 will explain the employed regularization methods and the relevant mathematical formula in detail. The regularization factor is introduced to scale the weight of the smoothness term, and which can balance the impact of Fprior and Fpoisterior on Fobj and ensure the posterior information (measured knowledge) dominates during the tomographic reconstructing process. Thereby the optimized selection of regularization factor is also crucial. The regularization factor is affected not only by the orders of magnitude of the temperature and concentration, but also by the nonlinearity of the problem, which means the L-curve method is no longer applicable for this nonlinear inverse optimization problem. This is because L-curve curvature assumes that the minimum solution norm is the best one, but for rank-deficient problems the „true‟ solution has a larger norm than the minimum norm solution because the true solution has

a nontrivial null component [5, 35]. Section 4 will explore the optimization of the regularization factor in this study. Given the above analysis, the master objective function can be expressed as

Fobj  Trec , Xrec   Fpoisterior  Trec , Xrec    T Fprior  Trec    X Fprior  Xrec 

(8)

where T and X denote the regularization factors, in which subscripts T and X represent the reconstructed temperature and concentration distribution, respectively. During the optimization of the master objective function, numerous local minimum values exist, thereby causing difficulty in selecting an appropriate optimization algorithm. For complicated multimodal functions, many algorithms (e.g., the downhill simplex and gradient-based methods) cannot yield satisfactory results without extra information [36]. Cai et al. [28] have applied the simulated annealing (SA) algorithm to reconstruct the temperature and concentration distributions simultaneously by numerical simulation with nonlinear TAS problem. However, the SA algorithm suffers from extensive calculation cost and exhibits lower reconstruction quality of the concentration [29]. In this study, the CMA-ES algorithm is employed to solve the nonlinear TAS inverse optimization problem and is proven to be excellent in solving the ill-posed and highly inseparable problems. 3. Inversion algorithm of CMA-ES As a randomized optimization method without gradient information, the CMA-ES algorithm is proposed on the basis of the evolution strategy (ES). Hansen and their colleagues [40-43] have made considerable contributions to the development of the CMA-ES algorithm. The CMA-ES algorithm combines the advantages of both the covariance matrix adaptation (CMA) learning mechanism and the evolution strategy, such as the directive role by the CMA, and the reliability and the global searching ability of the ES. Consequently, the CMA-ES algorithm has an outstanding capability for solving nonlinear problems [44-47]. In this study, the nonlinear TAS inverse optimization problem can be converted to solve the master objective function as described in Eq. (8), in which the kernel problem is to seek the appropriate temperature and species concentration distribution fields. Given the superior performance of the CMA-ES algorithm in nonlinear optimization problems, we attempt to apply this algorithm to minimize the master objective function (see Eq. (8)) to simultaneously reconstruct the temperature

distribution field (Trec) and the species concentration distribution field (Xrec) in the absorption flame. A brief introduction of the theoretical fundamentals of the CMA-ES algorithm is presented below. In the CMA-ES algorithm, the dimension of the objective vector depends on the number of the parameters to be optimized in the objective function. In this research, the number of parameters to be optimized is 200 (depend on the trellis partition, we use 10 × 10 grids here). The first 100 parameters represent the reconstructed temperature value, and the latter 100 parameters denote the reconstructed species concentration value. New search points are generated according to the sampling of a multivariate normal distribution. These points (for generation number g=0, 1, 2 ......) can be expressed as [43] 1) x(g k 1) where x(g  k

n

m (g)   (g) N , C(g)  for k  1,..., 

is the kth search point from generation g+1.

(9)

denotes the same distribution on

the left and right sides. m(g) is the mean value of the search distribution at generation g.  (g) 



is

the step size at generation g. N(0,C(g)) is a multivariative normal distribution with zero mean and the covariance matrix C(g). m ( g )   ( g ) N 0, C ( g ) 

N m ( g ) ,( ( g ) )2 C ( g )  . C ( g ) 

nn

is the covariance

matrix at generation g. ≥2 is the population size of the offspring. Then, the keys to the CMA-ES algorithm are the distribution parameters of the new mean m(g), the covariance matrix C(g), and the step size σ(g). (1) The new mean m(g) can be modified as a weighted average of  selected points from samples

x1(g1) , x2(g1) ,……，x (g1) [43]. That is,

m

( g 1)



w x

( g 1) i i:

(10)

w1  w2  …  w  0

(11)



i 1



 w 1, i

i 1

where    is the parent population size, i.e., the number of selected points. wi 1...  the positive weight coefficients for recombination. xi(:g 1) is the ith best individual out of



represent

x1( g 1) , x2( g 1) ,……, x( g 1) and the index i: denotes the index of the ith ranked individual.

f  x1:( g 1)   f  x2:( g1)   …  f  x( g:1)  , where f is the objective function to be minimized.

(2) The covariance matrix can be renewed through the covariance matrix adaption on the basis of rank-μ and rank-one updates. The rank-μ update promotes higher reliability of the estimation of the covariance matrix when the search population is small. The rank-one update ensures that only a single point can be used to renew the covariance matrix at each generation in the limit cases. A credible new covariance matrix can be generated by considering both rank-μ and rank-one updates [43].

( g 1)

C



 (1  c1 (h )  c1  c

w )C

(g)

i

i 1

 c1 p(cg 1)

T

p(cg 1)   c  



w y

( g 1) i i:

i 1

rank-one update

 y i(:g1)   

T

(12)

rank- update

can be close or equal to 0

where c  1  c1 and c1  1  c are the learning rates for the update of the covariance matrix according to the rank- and rank-one updates, respectively. The evolution path can be expressed by a sum of the consecutive steps, a process referred to as cumulation [43] p(cg 1)  1  cc  p(cg )  h cc  2  cc  eff

m ( g 1)  m( g )

(13)

 (g)

(3) To update the step size  ( g ) , the matrix norm of the conjugate evolution path of

p( g 1)

is

compared with its expected length E N (0, I ) , that is [43]



( g 1)



(g)

 c exp    d 

 p( g 1)    1   E N (0, I )   

where d  1 is the damping parameter for step size renewal. I 

(14)

nn

is an identity matrix of n×n.

The evolution path utilized for step size controlling p(g) differs from Eq. (13), but a conjugate evolution path p(g) 

n

can be built through a similar process, and expressed as follow [43]

p( g 1)  1  c  p( g )  c  2  c  eff C( g ) 

0.5

m( g 1)  m( g )

c  2  c  eff

where c  1 denotes the cumulative learning rate for the step size controlling. normalized constant. C( g ) 

0.5

1

2

 B( g ) D( g )  B( g )  , where C( g )  B( g ) D( g )  B( g ) 

decomposition of C( g ) . B( g )  of the diagonal matrix D( g ) 

T

(15)

 (g)

n

n

T

is a

is an eigen

is an orthonormal basis of eigenvectors, and the diagonal elements

are square roots of the corresponding positive eigenvalues.

Detailed setting of the other parameters used in the following simulation is shown here: the step size representing coordinate wise standard deviation is set as 0.3 (which is labeled as „sigma‟ in program code). Besides, the stopping criterion used in the CMA-ES algorithm are requested to satisfy number of function evaluations (maximum number of iterations is set as 103×N2, N denotes the number of searching vector and which is 200 in this research) or stop fitness (stop fitness is set as 8×10-6). Other relevant parameters setting is available in Ref. [43] and are omitted here for simplicity. 4. Results and discussion 4.1 Physical model and parameter settings In this work, we use two kinds of distribution models created by the superposition of Gaussian functions to imitate the temperature distributions and concentration fields of flame and calculate the measured projections Am. The distributions of the temperature after bicubic interpolation are depicted in Figs. 2(a) and 2(b). The temperature in the symmetrical temperature distribution model is between 800 K and 1800 K, and its peak is located at (5, 5) in the coordinates. As for the asymmetrical temperature distribution, the temperature value varies from 650 K to 1850 K, and its peaks are positioned at (2, 5) and (7, 5). Similar to the temperature distributions, the distribution models of the concentration field are divided into symmetrical and asymmetrical distributions (Figs. 2(c) and 2(d)). The concentration value of the symmetrical distribution model varies from 0.02 to 0.1. The asymmetrical concentration distribution model has two peak positions located at (7, 5) and (2, 5), and its value ranges from 0.04 to 0.085. The ROI is divided into 10×10 grids with a 1 cm interval. In this study, the pressure P is assumed to be uniform at 1 atm. The simulative research is

performed with 20 laser beam configurations (10 horizontal beams and 10 vertical beams according to the coordinate axis) except special explanation. Fig. 2 H2O is employed as the target species in this study given its abundant presence in air and as a major product of hydrocarbon combustion combustors. Thousands of non-negligible H2O spectral lines occur in near-infrared spectroscopy [48]. The variation of spectral parameters in the line strength function influences the solution of the nonlinear TAS equations as well. Thus, the spectral lines should be properly selected to maintain high sensitivity in different temperature ranges and guarantee the quality of the reconstructed results. Given the temperature sensitive feature in various states, the transitions employed in this work are taken from Ref. [48]. Table 1 lists the detailed parameters for the selected spectral lines. As shown, ten wavelengths were chosen [48]: 4 for monitoring peak absorbance of temperature-sensitive features, 4 for monitoring feature broadening (2 are best when the broadening is high and 2 are best when the broadening is low), and 2 for tracking baseline changes. Table 1 As discussed in Section 2.2, addition of regularization effectively alleviates the ill-condition of the nonlinear TAS inverse optimization problem. For the execution of the smooth constraint, we attempt two kinds of regularization methods, Tikhonov regularization [49] and regularization on the basis of the generalized Gaussian Markov random field (CGMRF) [50]. Smoothness is characterized by means of the discrepancy between the value of each discrete grid and the nearest surrounding grids. In the Tikhonov regularization method, the effect of each grid surrounding the central point is considered equal and can be expressed as [28]

Fprior-Tkh  T  

N

N

 m 1

1   Tm,n  8 Tm 1, n 1  Tm 1, n  Tm 1, n 1  Tm, n 1  Tm, n 1  Tm 1, n 1  Tm 1, n  Tm 1, n 1    n 1





2

(16)

where Fprior-Tkh(T) denotes the prior objective function based on the smooth confine on the temperature distribution under the Tikhonov regularization technique. Tm, n is the temperature value of the discrete grid

whose coordinate is (m, n). Similarly, Fprior-Tkh(X) represents the prior objective function based on the smooth confine on the concentration distribution under the Tikhonov regularization technique. The CGMRF regularization method provides good edge preservation in the reconstructed image and measures the impact of grids surrounding the central point by the distance between the central point and the surrounding points, which can be expressed as [50] Fprior-CGMRF  T   

1 p

p

b   

s r

Ts  Tr

p

(17)

s ,r N

where Fprior-CGMRF (T) presents the prior objective function based on the smooth confine on the temperature distribution when using the CGMRF regularization technique. p is the image sharpness parameter. σ is the scale parameter. r is the neighboring position of s, and bs-r the weight coefficient. In this study, we use p=1.1,



σ=50, bs r  4  2 2



bs r  4  4 2



1



1

in the case where r is the nearest neighboring position of s, and

when r is the diagonal neighboring position of s. Similarly, Fprior-CGMRF (X) denotes the

prior objective function based on the smooth confine on the concentration distribution under the CGMRF regularization technique. The average relative deviation provides quantitative characterizations of the quality of the reconstructed results, which can be defined as N

N

 T

rec

eT 

 m, n   Tini  m, n 

m 1 n 1

N

(18)

N

 T

ini

 m, n 

m 1 n 1

N

eX 

N

 X

rec

 m, n   X ini  m, n 

m 1 n 1

N

(19)

N

 X

ini

 m, n 

m 1 n 1

where the subscripts rec and ini represent the reconstructed and actual distributions, respectively. 4.2 Temperature reconstruction

Both Tikhonov and CGMRF regularization methods are applied on the symmetric temperature distribution model to compare their respective performances on the reconstructed results. The numerical simulation is conducted under the same conditions except for the regularization method. For the Tikhonov method, a value of 4×10-8 is set as the temperature regularization factor. For the CGMRF regularization, the temperature regularization factor is set as 6×10-5. Fig. 3 The reconstructed images restricted by the two regularization methods are shown in Fig. 3. Fig. 3(a) presents the reconstructed temperature field image from the Tikhonov method, and Fig. 3(b) shows the reconstructed temperature under CGMRF regularization. A comparison of Fig. 3 shows that the Tikhonov method is more effective than the CGMRF method in recovering the geometrical characteristic feature of the Gaussian distribution model. Fig. 3(c) illustrates that the reconstructed central high-temperature field by using the CGMRF method is roughly square in shape and differs considerably from the original distribution. Analysis from the error distributions shown in Fig. 3 (b) and (d), the larger errors mainly occur in the centric high-temperature zone for CGMRF method, and for Tikhonov the larger errors emerge in the edge of distribution. Table 2 Table 2 exhibit reconstructed quality data comparison between Tikhonov and CGMRF regularization methods. From the Table 2, it can be found that the average relative error of the reconstructed temperature distribution constrained by the Tihkonov technique is only 0.29%, and the deviation between the reconstructed and the original model ranges within ±14K. With the restraint of the CGMRF method, the average relative error is 1.18%, the margin of error is ±78K. By comparing computational efficiency, Tikhonov consume time of 2234.1s while CGMRF consume more time (>1000s) than Tikhonov. All simulation work is implemented based on the same computer with an Inter® Core™ i7-7700 CPU of Pentium 3.60GHz and 15.9 GB RAM. The above comparative analysis indicates that the Tikhonov method outperforms the CGMRF method on computational accuracy and efficiency. Hence, we adopt the Tikhonov regularization technique as the optimal regularization method in the following numerical simulation research. Fig. 4

The temperature reconstruction based on nonlinear TAS theory belongs to the inverse problem, which may lead to „inverse crime‟. To avoid this problem, a coarser grid should be arranged to the inverse problem than the direct problem. For verification on the independence of grid and avoidance of falling into the „inverse crime‟, we adopted 10×10 grids for the direct model, and coarse grid 6×6 grids for inverse model to search 72 values in total for temperature and species concentration. The reconstructed result with 6×6 grid arrangement is shown in Fig. 4. Fig. 4(a) exhibits spatial distribution of reconstructed temperature field, and Fig. 4(b) shows the section distribution (section is chosen at the edge of the distribution) of the reconstructed temperature field. From all the figures in Fig. 4, it can be seen that temperature distribution can be retrieved well even with coarse grid arrangement, reconstructed temperature have a good fit with the original temperature model. Fig. 5 We also examine the validity of the Tikhonov regularization method in reconstructing the asymmetrical temperature distribution model. The reconstructed results (see Fig. 5(a)) indicate that reconstruction successfully restores the bimodal distribution characteristics of the asymmetrical distribution model. The average relative error is 0.5%, and the maximum temperature deviation is 35 K, which is a completely acceptable error tolerance for a high-temperature (about 1500 K) flame. The reconstructed error of the asymmetric model increases mainly due to the aggravation of the ill-posedness problem. According to Fig. 5(b), the maximum temperature error appeared on the left central peak area of the distribution. This phenomenon is supposedly caused by the smoothing constraint on the whole distribution that results in excessive smoothness for the peak area whose gradient is larger in practice. Fig. 6 The regularization factor γT, a normalized quantity that can be utilized to scale the proportion of prior information and measurement knowledge, directly affects the reconstructed quality as well. Therefore, determination of the optimal regularization factor is indispensable. Fig. 6(a) depicts the trend chart of the average relative error eT of the reconstructed field as they vary with the temperature regularization factor γT. It can be observed that the reconstruction error varies irregularly with variation of temperature regularization factor. Viewing from the whole trend, we can find that if the temperature regularization factor γT is beyond

the optimal order range, the reconstruction error will rise significantly and rapidly. When the temperature regularization factor is too small, the restriction effect of regularization is insufficient and causes the solution to the master objective function to fall into the local minimum. If the temperature regularization factor is too big (γT>10-8, as shown in Fig. 6(a)), the smoothing constraints are so excessive that the reconstructed distribution approximates the „plane‟ field. According to Fig. 6, the order of 10-9~10-8 is the optimal regularization interval of the temperature regularization factor, in which the average relative error is maintained relatively low. Specifically, the optimal temperature regularization factor is set as 2.4×10-8 in the following simulative research. Furthermore, we‟ve investigated the influence of transitions‟ number on reconstructed quality of temperature reconstruction based on the nonlinear TAS theory. The result is shown in Fig. 6(b). Clearly, with the number of transitions increasing, reconstructed relative error of temperature is reducing, which mean reconstructed quality is improving effectively. In actual measurement, all kinds of measured noise are inevitable. Besides, the noise has obvious influence on availability of algorithm as well. Thus, we take the noise effect into the consideration to the reconstructed quality during temperature reconstructing process, because the temperature reconstruction is a typically nonlinear optimization problem and is quite sensitive to the noise. Here, the SNR (signal to noise) is adopted to describe the noise effect. Fig. 6(c) exhibits the reconstructed quality vary with SNR between 34dB to 80dB. The reconstructed quality is described by relative error of temperature field eT. It can be seen from Fig. 6(c) that the reconstructed quality is improved with higher SNR. At the situation that SNR>60dB, the relative error achieves less than 0.5%. When SNR>55dB, the maximum deviation temperature is only nearly 20K. At the lowest SNR=34dB, the relative error is still less than 4%, the corresponding maximum deviation temperature is nearly 100K, which is completely acceptable for engineering application. 4.3 Simultaneous temperature and concentration reconstruction In addition to temperature, the chemical species concentration of the product is another focal point in the combustion diagnosis field. The SA algorithm has been proven capable of solving the nonlinear TAS inverse optimization problem to simultaneously reconstruct temperature and concentration [28]. Unfortunately, our attempt to use SA algorithm is failed, so we make a comparison on the reconstructed

quality which can be acquired from Ref. [29]. The error of the reconstructed temperature distribution retrieved by the SA algorithm is minimal (1.50%), whereas the error of the reconstructed concentration field is considerable (7.23%) according to Ref. [29]. Such results are mainly due to the crosstalk between the temperature and concentration fields. To validate the simultaneous reconstruction of the multi-parameter fields, the CMA-ES algorithm, which is effective in solving the inseparable problems, is applied to simultaneously rebuild the temperature and concentration distributions. Fig. 2 presents the temperature and concentration distribution models for the numerical simulation. Figs. 2 depicts the symmetric distribution models and asymmetric distribution models, respectively. Fig. 7 A comparison of Figs. 2 and 7 shows excellent consistency between the reconstruction results and the distribution models. Figs. 7(c) and 7(d) depict the reconstructed error distributions of the reconstructed temperature and concentration fields, respectively. The overall distribution of the reconstructed error is relatively uniform, and the value of the error is within the acceptable range (see Figs. 7(c) and 7(d)). Specifically, the maximum temperature error of the reconstruction is only 25 K, which is a satisfactory result for a combustion field with an average temperature over 1000 K. Moreover, the average relative error is only 0.6%. Besides, according to Fig. 7(a) it can be seen that the whole reconstructed temperature distribution has a good fit on the original model distribution. The error of the concentration reconstruction is within ±0.0032 with an average relative error of 0.76%. The error of the reconstructed concentration distribution and the error of temperature reconstruction are both at the same accuracy level, proving the CMA-ES algorithm can effectively alleviate the crosstalk problem between the temperature and concentration. Thus, the CMA-ES algorithm proves its availability on reconstructing temperature and concentration distribution simultaneously. When the temperature and concentration are reconstructed simultaneously, the influence on the reconstructed quality is due to the temperature and concentration regularization factors. To explore the respective impacts on the reconstructed quality, comparative simulation analysis between the two kinds of regularization factor is conducted. Fig. 8

Fig. 8 illustrates the effect of the two regularization factors on the reconstructed results. Fig. 8(a) shows the reconstruction results without temperature regularization (γT=0), in which the left side represents the reconstructed temperature field and the right side is the reconstructed concentration field. By comparison, Fig. 8(b) shows the reconstruction results without concentration regularization (γC=0), in which the left side represents the reconstructed temperature field and the right side is the reconstructed concentration field. Obviously, from these two figures we can see that, in the absence of concentration regularization, the regularization of temperature seems to be invalid on smoothing for both temperature and concentration reconstructions, the reconstructed results have already lost characteristic feature of field (non-negative and smooth). Conversely, the quality of reconstructed results only restrained by concentration regularization factor as shown in Fig. 8(a) is better, for which still recovering the most characteristics of original model distribution. Further analysis from Fig. 8(b), the temperature reconstructed field exhibit fluctuations on its edge area while the reconstructed concentration is smoother as a contrast, which clearly can attribute to the absence of temperature regularization factor. From all the analysis above, a basic conclusion can be drawn that both regularization factors are necessary for high-accuracy simultaneous reconstruction of temperature and concentration and the concentration regularization works better than temperature regularization on smoothing influence. On the basis of discussion above, it‟s necessary to make it clear the choosing principle of different regularization factors in reconstruction process. Because the influence research on temperature regularization factor has been done in Section 4.2, we keep the focus on the concentration regularization influence in this section. To figure out the effect of concentration regularization and to further have an optimal choice for concentration regularization factor, some simulative reconstructions are performed with different regularization settings, the result is shown in Fig. 9. Fig. 9 Fig. 9 depicts the impact of the concentration regularization factor on the reconstructed result, which shows the trend charts to specify the influence of the concentration regularization factor on the reconstructed errors of temperature and concentration, respectively. Like the temperature regularization factor, the concentration regularization factor has an effect on reconstructed quality of temperature and concentration reconstructions, respectively. Fig. 9 indicates that the reconstructed error of both temperature and concentration remains low during the interval of the concentration regularization factor γC between 10-1 and

10-3. To facilitate a follow-up investigation, the optimal concentration regularization factor γC is selected as 2.2×10-1. Fig. 10 In practice, parameter distributions in engine combustion flame are usually asymmetrical. We also test the CMA-ES algorithm for reconstructing temperature and concentration fields for the asymmetrical model, with the same settings following the optimal selection discussed above. The reconstruction results on the asymmetric distribution are shown in Fig. 10, and the reconstructions successfully reflect the two-peak characteristics of the asymmetric phantom of temperature and concentration, with the peaks of the two parameter field positions located precisely. The average relative errors of the asymmetric temperature distribution and asymmetric concentration distribution are 2.04% and 1.11%, respectively. The maximum deviations of asymmetric temperature distribution and concentration distribution appear in the peak position, which is mainly introduced by the over-smoothness. The maximum deviation between the reconstructions and distribution models is 67K and 0.0021, respectively.

5. Conclusions In this research, the integral absorbance of different spectral transitions is employed as an input measuring signal for the nonlinear TAS inverse optimization problem in the absorption flame. Severe ill-posedness and crosstalk issues exist during the reconstruction of the multi-physical parameter field in flame. The CMA-ES algorithm is employed to solve the nonlinear TAS inverse optimization problem to simultaneously reconstruct 2D temperature and concentration distributions in absorption flame. The concluding remarks are summarized as follows: (1) The temperature distribution can be reconstructed accurately by using the CMA-ES algorithm in the nonlinear TAS model. The algorithm is accurate for both symmetric and asymmetric temperature distribution reconstructions. (2) The CMA-ES algorithm is available in solving the nonlinear inverse optimization problem and successfully reconstructs the temperature and species concentration fields simultaneously. Moreover, the

CMA-ES algorithm alleviates the severe crosstalk during multi-physical parameter reconstruction and improves the reconstructed accuracy on species concentration distribution. (3) The Tikhonov regularization method is more suitable than the CGMRF regularization method for reconstructing a Gaussian-shaped distribution model. The optimal regularization intervals of the temperature and concentration regularization factors are specified for the Tikhonov regularization method, and the reconstructed results are satisfactory. Further study will focus on simultaneously reconstructing the 3D temperature field and other physical parameters, e.g., pressure, species concentration, absorption coefficient field and scattering field of the flame. Acknowledgements The supports of this work by the National Natural Science Foundation of China (No. 51976044, 51706053) and the National Science and Technology Major Project (2017-V-0016) are gratefully acknowledged. A very special acknowledgement is also made to the editors and referees who make important comments to improve this paper. Conflict of Interest Statement

Authors do not have any conflict on interest in publishing the manuscript entitled “Simultaneous measurement of flame temperature and species concentration distribution from nonlinear tomographic absorption spectroscopy” by Jing-Wen Shi, Hong Qi, Jun-You Zhang, Ya-Tao Ren, Li-Ming Ruan, Yong Zhang in Journal of Quantitative Spectroscopy & Radiative Transfer.

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Authors do not have any conflict on interest in publishing the manuscript entitled “Simultaneous measurement of flame temperature and species concentration distribution from nonlinear tomographic absorption spectroscopy” by Jing-Wen Shi, Hong Qi, Jun-You Zhang, Ya-Tao Ren, Li-Ming Ruan, Yong Zhang in Journal of Quantitative Spectroscopy & Radiative Transfer. Figure captions Fig. 1

Schematic diagram of discretization of LAS tomographic problem

Fig. 2

Distribution models for simulation

Fig. 3

Reconstructed results of temperature

Fig. 4

Error distribution of reconstructed temperature

Fig. 5

Reconstructed results of asymmetrical temperature distribution

Fig. 6

Effect of different factors on reconstruction quality

Fig. 7 Simultaneous reconstructed results of symmetric distribution models

Fig. 8

Effect of regularization factor on reconstruction quality

Fig. 9

Effect of concentration regularization factor on reconstruction quality

Fig. 10

Simultaneous reconstruction results of asymmetric distribution models

y

ROI

k-th spectral line

αν, j Li,j

i-th laser beam

αν,1

...

αν,N x

Fig. 1 Schematic diagram of discretization of LAS tomographic problem

T(K)

(a)

(a) Symmetric distribution model for temperature T(K)

(b)

(b) Asymmetric distribution model for temperature concentration

(c)

X

(c) Symmetric distribution model for concentration

concentration

(d)

X

(d) Asymmetric distribution model for concentration Fig. 2 Distribution models for simulation

(a) Tikhonov result

(b) Error distribution of Tikhonov result

(c) CGMRF result

(d) Error distribution of CGMRF result Fig. 3 Reconstruction results of temperature

(a) reconstructed temperature distribution

(b) reconstructed temperature section distribution Fig. 4 Verification of grid independence

(a) Reconstructed result of temperature

(b) Reconstructed error distribution Fig. 5 Reconstruction results of asymmetrical temperature distribution

(a) Temperature regularization factor

(b) Transitions’ number

(c) Noise Fig. 6 Effect of different factors on reconstruction quality

(a) Reconstructed result of temperature distribution

(b) Reconstructed result of concentration distribution

(c) Error distribution of temperature

(d) Error distribution of concentration Fig. 7 Simultaneous reconstruction results of symmetric distribution models

(a) Reconstruction results without temperature regularization

(b) Reconstruction results without concentration regularization Fig. 8 Effect of regularization factor on reconstruction quality

(a)

X

(a) Effect on the reconstructed temperature

(b) X

X

X

X

(b) Effect on the reconstructed concentration Fig. 9 Effect of concentration regularization factor on reconstruction quality

(a) Reconstruction result of temperature distribution

(b) Reconstruction result of concentration distribution

(c) Error distribution of temperature

(d) Error distribution of concentration Fig. 10 Simultaneous reconstruction results of asymmetric distribution models Table caption

Table 1 Spectral parameters of selected H2O transitions [48]

Table 2 Reconstructed quality comparison between regularization methods

Table 1 Spectral parameters of selected H2O transitions [48] Line strength S0 [cm-2atm-1]

Lower state energy E‫״‬ [cm-1]

1328.604

1.313×10-3

300.4

Bandhead baseline

1334.839

5.777×10-1

4174

High temperature Temperature sensitive feature (TSF)

1341.587

1.600×10-3

1050

High broadening on medium temperature TSF

1341.842

5.146×10-3

2042

Medium temperature TSF

1341.884

2.453×10-3

2434

Low broadening on medium temperature TSF

1350.410

2.535×10-3

920.2

Dual temperature TSF

1364.682

8.628×10-2

206.3

Cold TSF

1364.724

1.435×10-1

142.3

Low broadening on cold TSF

1365.076

2.138×10-1

134.9

High broadening on cold TSF

1373.749

4.361×10-3

504.0

Bandcenter baseline

Wavelength [nm]

Description

Table 2 Reconstructed quality comparison between regularization methods Regularization method

eT

eTmax

Computational time

Iterations

Tikhonov

0.29

14K

2234.1 s

588236

CGMRF

1.18

78K

3435.6 s

588236