#### Abstract

3D temperature distribution measurement in a furnace based on acoustic tomography (AT) calculates temperature field through multipath acoustic time-of-flight (TOF) data. In this paper, a new 3D temperature field reconstruction model based on radial basis function approximation with polynomial reproduction (RBF-PR) is proposed for solving the AT inverse problem. In addition, the modified reconstruction method that integrates the advantages of the TSVD and Tikhonov regularization methods is presented to reduce the sensitivity of noise on perturbations with the ill-posed problems and improve the reconstruction quality (RQ). Numerical simulations are implemented to evaluate the effectiveness of the proposed reconstruction method using different 3D temperature distribution models, which include the one-peak symmetry distribution, one-peak asymmetry distribution, and two-peak symmetry distribution. To study the antinoise ability of our method, noises are added to the value of TOF. 3D display of reconstructed temperature fields and reconstruction errors is given. The results indicate that our model can reconstruct the temperature distribution with higher accuracy and better antinoise ability compared with the truncated generalized singular value decomposition (TGSVD). Besides that, the proposed method can determine the hot spot position with higher precision, and the temperature error of the hot spot is lower than the other compared methods.

#### 1. Introduction

In real-world industrial productions, obtaining the high-quality temperature distribution plays an important role. Acoustic tomography (AT) has been widely used and provided an effective means for temperature distribution visualization owing to the advantages of wide measuring range, noncontacting measurement, and large measurement space. In recent years, the AT method has been widely used in various applications, such as boiler furnaces [1, 2], deep-sea hydrothermal vents [3, 4], stored grain [5, 6], and atmospheric surface layer [7, 8]. In this paper, we applied the AT method to reconstruct the temperature distribution in a boiler furnace.

In boiler furnaces, the process of combustion is variable and complex, which has the characteristics of high temperature, dust, turbulence, and so on [4–6]. Acoustic tomography, as an important and efficient method, has been applied to measure the temperature fields in furnaces to improve the combustion efficiency of the boiler and ensure the safe operation of the boiler. At present, the research on temperature field reconstruction by AT is mainly conducted in two dimensions [9–11]. 3D temperature field reconstruction is discussed by only few researchers [12, 13]. Due to the complexity of 3D temperature distribution and computational difficulty, the reconstruction performance is not good. So, we focus on the method to reconstruct the 3D temperature distribution in a boiler furnace.

AT technology reconstructs the temperature distribution from the acoustic time of flight (TOF) [14, 15], and it is an inversion process. Successful application of the AT measurement greatly depends on the reconstruction model. In this paper, a 3D reconstruction model based on radial basis function (RBF) approximation with polynomial reproduction is presented to solve the AT inverse problem. The proposed model can improve the accuracy and stability of RBF model [16, 17] in existence.

Another key issue in the AT measurement is to improve the reconstruction quality (RQ) and the antinoise ability. Many existing methods that were developed to reconstruct the temperature field including the iterative methods [18, 19] and noniterative methods [9, 16, 20]. The reconstruction quality of most iterative methods, such as Landweber iterative method, algebraic reconstruction technique (ART), and the simultaneous algebraic reconstruction technique (SART), is relatively low in the influence of noises. The discretization of temperature reconstruction inverse problems generally gives rise to ill-posed systems of algebraic equations. Typically, the linear systems obtained have to be regularized to make the computation of the approximate solution possible. Truncated singular value decomposition (TSVD) [21, 22] and Tikhonov regularization [23, 24] are two popular regularization methods for solving this discrete ill-posed problem. The TSVD method directly discards the smaller singular values to obtain the approximate solution of the original problem and completely eliminates the influence of the smaller singular values on the regularization solution. As a consequence, the useful information of solution may be lost. The Tikhonov method applies an appropriate filter factor to each singular value of the original problem, which can suppress the diffusion of noise by correcting not only the smaller singular values but also the larger singular values. The approximate solution deviates from the real solution to some extent. Hansen [25] proposed a generalized value decomposition (GSVD) regularization technique for solving the linearized ill-posed problem. Dykes and Reichel [26] adopted a simplified GSVD method and compared the errors in approximate computed solutions of Tikhonov regularization. Chen and Chan [27] researched on truncated GSVD (TGSVD) method and chose a proper regularization parameter *L* and a truncation parameter *k* for ill-posed problems. In this paper, a new modified reconstruction method that integrates the advantages of the TSVD and Tikhonov regularization methods is presented to improve the reconstruction quality (RQ) and the antinoise ability.

This paper is organized as follows. In Section 2, the AT measurement model is presented based on RBF-PR. The modified reconstruction method for solving the measurement model is introduced in Section 3. In Section 4, numerical simulations are conducted to evaluate the effectiveness and feasibility of the proposed method. Finally, the main conclusions are drawn in Section 5.

#### 2. Acoustic Temperature Measurement Model

##### 2.1. Principle of AT

The theory of AT is based on the relation between the velocity of acoustic wave and the temperature of fuel gas. The temperature is calculated using the following relation [1, 28]:where *u* is the acoustic velocity in the medium, is the isentropic exponent of gas, *R* is the specific gas constant of an ideal gas, *M* is the molar mass, and *T* is the absolute temperature. For a given gas mixture, , *R*, *M*, are fixed constants. The value of is denoted as Z. Suppose that the flue gas is contained in a boiler and *Z* is 19.98.

In the process of acoustic thermometry, when the distance *d* between the sound receiver and sound transmitter is known and the time of flight (TOF) is measured, the velocity of sound can be calculated as .

Combining equation (1), the temperature can be expressed by the following equation:

##### 2.2. 3D Reconstruction Model Based on RBF-PR

For the current power plant boilers, the boiler size is generally above 10 meters. Assume that the boiler furnace is 10 meters high and has a section of 10 m × 10 m square; thus, a three-dimensional system of is studied as shown in Figure 1. The space in the system is divided into blocks. Twenty acoustic sensors are arranged on the periphery to form a number of effective flying paths, which are represented by the solid lines. When one sensor acts as a transmitter and radiates out sound signal, the other sensors play the roles of receivers and detect the signal. Owing to the fact that there is no significant effective signal between the acoustic sensors on the same side wall, the TOF data on the same wall are omitted. Hence, 58 effective acoustic paths are employed in our temperature reconstruction.

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When all TOF of the flying paths are obtained, the three-dimensional temperature distribution will be reconstructed by using a suitable method.

Let denotes the sound velocity, is the velocity of flue gas in the boiler. Then, TOF of the *k*th effective acoustic flying path can be theoretically expressed as

As we know, the speed of sound in the gas is about 340 m/s in normal temperature and pressure conditions. Under the high-temperature conditions in the furnace, the value is much larger, and the flow rate of the flue gas in the furnace is below 10 m/s, so , and this assumption is considered reasonable in the case of general industrial applications. So, equation (3) can be rewritten aswhere describes the reciprocal of sound velocity.

Radial basis function (RBF) is used to approximate the sound velocity distribution. Then, the reciprocal of sound speed in 3D can be written as the linear combination of a number of RBF:where is a basis function, the basis functions used in the RBF model include cubic, Gaussian, spline, and multiquadric functions, *λ*_{n} is the coefficient for the *n*th basis function, and *N* is the number of blocks in the measured area. In this paper, the radial basis function is expressed aswhere is the geometric center coordinate of the *n*th block as the sample point, *C* is the shape parameter of RBF, which is related to the measured area and the arrangement of sensors. It can be predetermined via a numerical method, and in our model, RBF gives the best result for the shape parameter when *C* = 60.

By substituting equation (4) into equation (3), the in the *k*th effective acoustic flying path is determined as

The linear system of equation (7) can be represented as the matrix equation:where , and , in which *K* denotes the sum of effective paths and *N* is the number of blocks.

In order to improve the accuracy of RBF approximation, the RBF approximant is usually extended by polynomial function *P*_{k} of degree *k*. Now, equation (5) can be expressed as

In this paper, a linear polynomial is used as

When substituting equation (9) into equation (8), equation (8) becomes

Then, TOF of the *k*th effective acoustic flying path is

Letand using the matrix notation, we can write

The equation can also be expressed in the form

In practice, once **t** is obtained by actual measurement, undermined coefficients and can be gained from equation (16). Then, the reciprocal of acoustic velocity *f*(*x*, *y*, *z*) is known. In addition, the reconstructed temperature distribution will be expressed as

#### 3. Reconstruction Method

It can be seen that we have a linear system of equations with (*N*+4) variables in equation (16) compared with equation (7). The vector **t** typically represents data that are contaminated by a measurement error **t**_{noise} in applications in science and engineering. Sometimes, we will refer to the vector **t**_{noise} as “noise.” Thus,

We consider the computation of an approximate solution of the least-squares problem (16):where denotes the Euclidean vector norm and **A** is a matrix that has many singular values of different orders of magnitude close to the origin. The minimization problem in equation (19) is referred to as a discrete ill-posed problem. The presence of singular values close to the origin makes the solution of the problem sensitive to the measurement noise **t**_{noise}.

The truncated singular value decomposition (TSVD) is a common method for solving least-squares problems as equation (19). Now, we introduce SVD of the matrix *A* ∈ *R*^{m×n} in brief:where *U* and *V* are orthogonal matrices andwhere *r* is the rank of *A*.

Then, it can give the truncated SVD:where is a best rank-*r* approximation of *A* in the spectral norm. Thus, we can get the TSVD solution, which is given by

From equation (23), it can be found that some small singular values σ_{i}(*k* ≤ *i* ≤ *r*) are ignored. As a consequence, useful information of the exact solution **x** may be lost.

The above discrete ill-posed problem is often solved by Tikhonov regularization, which replaces equation (19) by a penalized least-squares problem of the form:where *μ* > 0 is a regularization parameter and *L* is referred to as the regularization matrix. The minimization problem (24) is called to be in standard form when *L* = *I* (identity operator) and in general form otherwise. For any fixed *μ* > 0, the above problem (20) has the unique solution.

The matrix *L* is assumed to satisfywhere *N*(·) denotes the null space of (·).

The generalized singular value decomposition (GSVD) of the matrix pair {*A*, *L*} is factorized aswhere are orthogonal, *X* is nonsingular, and , and are *p* × *p* nonnegative matrixes with entries and satisfy

Please refer to [25] for details. The quotients (1 ≤ *j* ≤ *p*) are referred to as generalized singular values of {*A*, *L*}.

Define the *A*-weighted generalized inverse of *L* and vector **x**_{0} as

Then, the standard form quantities of can be written as

The least-squares problem of TSVD can be transformed into of truncated GSVD (TGSVD), and the approximate solution of the TGSVD method can be expressed as

The regularization matrix *L* and the truncation parameter *k* affect the computational accuracy. When *L* = *I* (identity matrix), the TGSVD method is similar to the TSVD method. Many examples of regularization matrices can be found in [25, 29, 30]. Common regularization matrices *L* are finite difference matrices, such as

In our paper, we develop a new method to construct the regularization matrix. Tikhonov regularization problem (20) can be rewritten as

Then, the corresponding approximate solution of equation (19) can be given as

By substituting equation (15), *L*_{μ} = *μI*, into equation (34), we have

In addition, the solution of equation (28) is given by ; it satisfies .

From equation (35), it follows that Tikhonov regularization with *L*_{μ} = *μI* and *μ* > 0 dampens all solution components of . On the contrary, TSVD does not dampen any solution component that is not set to 0. When the regularization parameter is well determined, Tikhonov regularization may improve the accuracy of computed solution.

The regularization matrix can be given as

Then, it can be obtained as

Due to , then

In order to avoid severe propagation of the error **t**_{noise} in **t** into the computed approximate solution, the smallest eigenvalue of the matrix , which is now max {}, has to be sufficiently large. In addition, the matrix should be of small norm.

By substituting equations (20) and (38) into equation (37), we have the solutionwhere is the filter factor and

According to equation (40), it is important to select the values of truncation index *s* and regularization parameter . There are a variety of ways to choose the suitable values, including the threshold setting method, discrepancy principle, generalized cross validation, and *L*-curve criterion (see [21, 31]). In this paper, we apply the *L*-curve criterion to determine the truncation index *k* and the regularization parameter simply.

#### 4. Numerical Simulation and Discussion

In this section, in order to validate the performance of our proposed RBF-PR model and TT method, simulation experiments are carried out, and the results’ analysis is given. All computations are simulated using MATLAB R2016a software.

As an improved model on the basis of the RBF model, reconstruction results of the presented model based on RBF-PR are compared with those of the model based on RBF. For convenience, the two models are called RBF model and RBF-PR model, respectively.

The internal temperature field of the furnace is complex and variable. When the combustion is stable and homogeneous, the furnace center temperature can be as high as 2000 K and the temperature near the furnace wall is about 500 K. Combustion imbalance, off-center firing, may be appeared by some complicated factors. Three typical temperature models with different complexity levels, i.e., one-peak symmetry model, one-peak asymmetry model, and two-peak model, are chosen in this paper.

The expressions of these 3D temperature distributions created are shown as follows:

One-peak symmetrical temperature distribution:

One-peak asymmetrical temperature distribution:

Two-peak temperature distribution:

The RQ is given by calculating the maximum relative error , the mean relative error , and the root-mean-square percent error :where Num is the number of calculated points, are temperatures of point for the modeled one and reconstructed one, respectively, and is the mean temperature of the model.

In addition, to evaluate the reconstruction accuracy of hot spot location and hot spot temperature, other RQ of the reconstructed hot spot positioning error and the temperature error of the hot spot are defined aswhere is the reconstructed hot spot coordinates, while is the original hot spot coordinates; and are the reconstructed temperature of the hot spot and original temperature of the hot spot, respectively.

##### 4.1. Simulation Results without Measurement Noise

In this subsection, three kinds of 3D temperature distribution are employed to evaluate the feasibility of the RBF-PR model without measurement noise. Because SVD is noise-free, it is chosen as the reconstruction method. Figures 2–4 show the reconstructed results for one-peak symmetrical temperature distribution, one-peak asymmetrical one, and two-peak distribution. The original temperature distributions shown in Figures 2(a), 4(a), 4(b), and 4(d) show the reconstructed temperature distributions, gained using the RBF model and our proposed RBF-PR model. Figures 4(d) and 4(e) show the absolute errors of the temperature distributions reconstructed using the RBF and RBF-PR model, respectively. The RQ of reconstructed models for these three kinds of temperature distribution is shown in Table 1.

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In Figure 2, for simple one-peak symmetrical temperature distribution, the reconstruction based on the proposed RBF-PR model conforms better to its original field, comparing with the one based on the RBF model. The absolute errors of the RBF-PR model are small, almost all of which are below 8 K. While the errors of RBF are obviously larger, especially somewhere error is nearly 120 K. Particularly, from Table 1, we find that the maximum relative error of the RBF model is up to 29.45% and the mean relative error and root-mean-square percent error are 3.93% and 4.59%, respectively, while of the proposed RBF-PR model is only 0.47% and and are within 0.38%. The reconstructed hot spot positions of these two models are the same as the original position. Besides that, the temperature error of the hot spot of the RBF model is 1.02%, while the one of RBF-PB model is nearly to 0%.

In industrial production, the asymmetrical temperature distribution may be appeared by some complicated factors, and the reconstruction of that is more difficult than reconstruction of a symmetrical one. Figure 3 shows the reconstructed temperature distribution for one-peak asymmetrical distribution. It can be seen that the accuracy of the results based on the RBF-PR model is improved remarkably compared with that of the results of the RBF model. The error of the RBF model is around 100 K in most area, while it will dramatically increase nearly 200 K. However, in the proposed RBF-PR model, the most area’s error and the maximum error are reduced to about 5 K and 25 K, respectively. In addition, it can be observed from Table 1 that the RQ of the RBF-PR model is high and , , , and are 0.33%, 0.26%, 0.38%, and 0.11%, which are lower than that of the RBF model. There are some little deviations in hot spot positioning of the RBF model, where is 0.1 m, whereas the RBF-PR model can achieve precise positioning.

For two-peak distribution, the temperature distribution gradually becomes much more complex. Therefore, it is important to demonstrate the possibility of reconstructing the temperature field. From the displays of reconstructed temperature distribution and absolute errors of two-peak asymmetrical distribution in Figure 4, it indicates that the results of the RBF-PR model have more excellent reconstruction performance than one of the RBF model. In addition, we can also observe from Table 1 that the most errors of the RBF model are lower, only is a little higher than ones of the RBF-PR model. Besides that, the RBF-PR model can also give the accurate hot spot location, while the RBF model still has some location biases, and is 0.57 m. In the following section, the RBF-PR model is used to reconstruct the temperature distribution.

##### 4.2. Simulation Results with Measurement Noise

In practical AT application, the TOF data obtained via actual measurement may be influenced by various factors, such as accuracy limit of devices and interference of environment. In this section, we generate a noise vector corresponding to the noise level *α*, and the vector **t** can be rewritten aswhere *N*_{noise} is a standard normal distribution vector and the noise level *α* is chosen between 0 and 0.1 (or 10%).

For convenience, the proposed method is called the TT (*L*_{μ}) method, which will be compared with other reconstructed methods, TGSVD (*L*_{1}) and TGSVD (*L*_{2}), in the subsequent sections.

Figures 5–7 show us the displays of reconstructed temperature distribution and absolute errors of temperature distribution reconstructed with the noise level *α* = 5% for one-peak symmetrical, one-peak asymmetrical, and two-peak temperature distribution, respectively. (b), (c), and (d) of Figures 5–7 are the reconstructed results by using the TGSVD (*L*_{1}), TGSVD (*L*_{2}), and proposed TT (*L*_{μ}) methods, respectively. Table 2 lists the reconstruction errors of different methods with noise for these three temperature distributions. From the figures and Tables 1 and 2, we can observe that the RQ of reconstructed temperature decreases when the noise is added to TOF data compared with the ones without noise.

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For one-peak symmetrical temperature distribution in Figure 5 and one-peak asymmetrical temperature distribution in Figure 6, the reconstruction temperature fields using three different methods can be accepted with noise. Furthermore, the results of the proposed TT (*L*_{μ}) methods have more excellent reconstruction performance than the ones of TGSVD (*L*_{1}) and TGSVD (*L*_{2}) methods. The absolute errors of them are small, almost of which are below 80 K. In addition, from Table 2 for one-peak symmetrical distribution, the RQ of the TT (*L*_{μ}) method is high and , , , and are 4.87%, 3.07%, 3.56%, and 3.28%, which are lower than those of other methods. Besides that, the TGSVD (*L*_{1}) and TT (*L*_{μ}) methods can gain precise positioning of the hot spot, while there are some little deviations of the TGSVD (*L*_{2}) method, and is 0.6 m. For one-peak asymmetrical distribution, , , and of the TT (*L*_{μ}) method are lower than the ones of the TGSVD (*L*_{1}) and TGSVD (*L*_{2}) methods, but *E*_{hot} is a litter higher.

For two-peak temperature distribution in Figure 7, the reconstructed results of the TGSVD (*L*_{1}), TGSVD (*L*_{2}), and TT (L_{μ}) methods all get worse with noise. Especially, the reconstruction temperature fields have been distorted, and the absolute errors are very large by using the TGSVD (*L*_{2}) method. By contrast, the reconstructed temperature fields by the proposed TT (*L*_{μ}) methods are more close to the original fields compared with other methods. From Table 2, , , , and of the TT (*L*_{μ}) method are smaller than the other compared reconstruction methods. The results indicate that the TT (*L*_{μ}) method gives the best reflection of temperature distribution and improves the RQ in the AT measurement. In addition, for the complex temperature distribution, these methods can not achieve accurate positioning of hot spots. For two-peak temperature distribution, due to two hot spots, *E*_{p} are 0.28 m and 0.28 m in the TT (*L*_{μ}) method, while E_{p} are 1.23 m and 1.14 m in the TGSVD (*L*_{1}) method and 1.33 m and 1.45 m in the TGSVD (*L*_{2}) method, respectively. The results show that the TT (L_{μ}) method can improve the accuracy of hot spot positioning.

#### 5. Conclusion

To gain the high-quality 3D reconstruction temperature field of a furnace in AT measurement, we proposed a new 3D temperature field reconstruction model based on RBF-PR in this paper. In addition, the modified reconstruction method was presented to improve RQ and antinoise ability. Numerical simulations were implemented to evaluate the feasibility and effectiveness of the proposed reconstruction model and method using 3D different temperature distributions. On the basis of the simulation results, we can have the following conclusions: (1) the proposed reconstructed model based on RBF-PR can improve the reconstruction performance compared with the model based on RBF approximation; (2) the modified reconstructed method can reconstruct temperature distribution with higher accuracy and better antinoise ability compared with TGSVD methods; and (3) in the three temperature distribution models with different complexity, the results show that the modified method can improve the accuracy of hot spot positioning. In addition, the temperature error of the hot spot is lower. The RQ of our reconstructed method is superior to the other compared methods.

Due to the advantages of the proposed reconstructed method, the algorithm can be extended to many applications in other areas, such as atmospheric temperature distribution measurement, temperature field measurement in the storage grain, and deep-sea hydrothermal temperature field measurement. In the process of reconstructed temperature, most researchers assumed that sound wave propagates along the straight line, but the temperature distribution in a furnace is inhomogeneous; according to the acoustic principle, sound waves are no longer propagating in a straight line in a furnace. The errors because of refraction will influence the temperature field reconstruction. The refraction effect on the reconstructed temperature field is about to be performed in further work.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This research is supported by the National Science Foundation of China (nos. 11674093 and 11474091), the Fundamental Research Funds for the Central Universities of China (No. 2018 MS131), and the State Key Laboratory of Acoustics, Institute of Acoustics, Chinese Academy of Sciences, China (SKLA201808).