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The temperature distribution in real-world industrial environments is often in a three-dimensional space, and developing a reliable method to predict such volumetric information is beneficial for the combustion diagnosis, the understandings of the complicated physical and chemical mechanisms behind the combustion process, the increase of the system efficiency, and the reduction of the pollutant emission. In accordance with the machine learning theory, in this paper, a new methodology is proposed to predict three-dimensional temperature distribution from the limited number of the scattered measurement data. The proposed prediction method includes two key phases. In the first phase, traditional technologies are employed to measure the scattered temperature data in a large-scale three-dimensional area. In the second phase, the Gaussian process regression method, with obvious superiorities, including satisfactory generalization ability, high robustness, and low computational complexity, is developed to predict three-dimensional temperature distributions. Numerical simulations and experimental results from a real-world three-dimensional combustion process indicate that the proposed prediction method is effective and robust, holds a good adaptability to cope with complicated, nonlinear, and high-dimensional problems, and can accurately predict three-dimensional temperature distributions under a relatively low sampling ratio. As a result, a practicable and effective method is introduced for three-dimensional temperature distribution.

Three-dimensional (3D) temperature distribution plays an important role in combustion diagnosis tasks. The acquisition of the rich and accurate combustion temperature distribution details is of a great significance to the combustion adjustment and control. Currently, two kinds of approaches are available for achieving such task, for example, the numerical simulation technique and the experiment method. Owing to the challenges, such as high computational cost and complexity, the inaccurate properties of initial conditions, boundary conditions, geometrical conditions, and physical property parameters, it is hard for the former to achieve combustion diagnosis tasks in real-world applications. With the development of modern measurement technologies, the latter has attracted more and more attentions.

Conventional flame monitoring systems are only used to judge whether there is a flame in a combustion space, and it is difficult to realize the quantitative measurement of the combustion process parameters. A lot of technologies based on the radiation image and radiation energy signal processing had been developed for 3D temperature distribution measurements [

Optical CCD methods are based on the radiation transfer characteristics of the flue gas in a combustion system, which focus on the mathematical relationship between the radiation accumulation image and 3D radiation energy distributions, and thus the temperature distribution profiles are reconstructed via an appropriate algorithm [

The acoustic tomography (AT) method employs an appropriate algorithm to realize the reconstruction of the temperature distribution profiles from the given acoustic wave time-of-flight (TOF) data and has gained extensive acceptances in a variety of areas [

Are there other measurement strategies for 3D temperature distributions? The development of modern measurement instruments makes it possible to measure scattered temperature values. Is it possible to predict 3D temperature distribution from the finite number of the temperature measurement data? If the answer is positive, a natural problem will appear, that is, how to achieve it? With such insight in mind, in this work, we put forward a new methodology to predict 3D temperature distributions from the finite local temperature measurement values, and the main highlights of the study can be summarized as follows.

(1) A new methodology is proposed to predict 3D temperature distribution from the limited number of the scattered temperature measurement data. The proposed method includes two key phases. In the first phase, traditional measurement technologies are employed to measure the local temperature values in a large-scale 3D measurement area. In the second phase, the GPR method is developed to predict 3D temperature distribution details. Numerical simulation results indicate that the execution the GPR method is easy and can accurately predict 3D temperature distributions under a relatively low SA. Furthermore, the GPR method holds satisfactory robustness and good adaptability of coping with complicated, nonlinear, and high-dimensional problems.

(2) A 3D combustion experiment system is constructed, and a series of combustion experiments are conducted. The effectiveness and robustness of the GPR method are further validated by experimental results. As a result, a practicable and efficient method is introduced for 3D temperature distribution measurements.

In accordance with the research target, the rest of this paper is arranged as follows. In Section

The development of modern measurement instruments makes it possible to measure scattered temperature values. In this study, the 3D temperature distribution measurement problem will predict unknown temperature distributions in terms of the provided finite observations.

For convenience, 3D temperature distribution predictions from the limited number of the measurement data can be formulized to be a tensor completion (TC) problem. Mathematically speaking, a 3D measurement area can be represented as a tensor

It is noticeable that the TC problem has attracted the increasing attention over the past several years and has been successfully applied to signal and image processing area. A variety of numerical methods have been developed for solving the problem, and the interested readers are referred to [

We find that if the mapping of the measurement point positions and the corresponding temperature values are abstracted, the temperature distribution at other positions in a 3D measurement area can be predicted. In a mathematical notation, the mapping between the measurement point positions and the corresponding temperature values can be formulized by

According to the analysis presented above, in this work, we put forward an alternative methodology to overcome the drawbacks of the TC technique to predict 3D temperature distributions, which can be divided into the following five steps.

In accordance with practical demands, a 3D measurement area is appropriately determined.

Acquire finite temperature measurement data using one of conventional measurement technologies.

The raw measurement data is refined by an appropriate method to ameliorate the data quality.

Abstract the mapping between the temperature value and the measurement position according to the finite measured temperature values.

Use the mapping abstracted in Step

The above discussions result in a fact that, in order to successfully predict 3D temperature distributions, we must seek an effective method to abstract the mapping between the temperature information and the position information in terms of the finite temperature measurement values. Naturally, in the rest of the paper, we will answer the above problems to achieve the goal of predicting 3D temperature distributions.

A variety of methods are available for the estimation of the mapping function,

The GPR method is a kernel based learning machine and holds a good adaptability to deal with complicated, nonlinear, and high-dimensional problems. The execution of the GPR method includes two key phases, the training process and the prediction process, and more details can be found in [

Given a training set

If the observation value,

Similarly, the prior distribution of

Eventually, we can express the joint prior distribution of

Meanwhile, we can write the joint posterior distribution of the prediction as follows:

It is necessary to mention that the covariance matrix is positive definite for the limited data set, which is consistent with the property of the kernel in Mercer’s theorem, and thus the covariance function and kernel function are equivalent. Consequentially, we can rewrite (

The selection of various covariance functions is crucial for practical applications of the GPR method. Currently, a variety of covariance functions are available. The squared exponential function has been successfully applied to different areas, and it has following form:

The optimal hyperparameter

In order to obtain the optimal solution of (

Finally, we can compute the prediction mean and variance corresponding to a new input

The execution details of the GPR technique can be divided into the following three steps.

According to the temperature measurement values and the measurement position coordinates, a set of training samples, (

Solve (

The mean and variance of other positions can be computed via solving (

Owing to the conspicuous superiorities, for example, satisfactory generalization ability, easy numerical execution, low computational complexity and cost, and high robustness, in this study, the GPR method is employed to predict 3D temperature distributions. The prediction details can be divided into the following five steps.

A 3D measurement area is determined according to the requirements of practical measurements.

Measure the finite temperature data via traditional measurement technologies according to practical measurement conditions.

Refine the raw measurement data to improve the data quality.

The GPR model is used to abstract the mapping between the temperature values and the position information.

Use the trained GPR model to predict 3D temperature distributions of other positions at a predetermined measurement area.

In this section, we use the temperature distribution function to generate the temperature distribution of the whole measurement area firstly. The known temperature data is obtained by random sampling with different sampling ratio, and other temperature data is considered unknown. Then, we perform numerical simulations to make a fair assessment for the GPR method, and the results are compared with the MNR method, the generalized regression neural network (GRNN) method, and the TC methods [

The mean relative error (MRE) is used to make a quantitative evaluation for the prediction accuracy of the GPR method, which is defined by

In order to simulate a real-world measurement environment, the simulation data is perturbed with the normal distribution random number with different noise variances (NVs), which can be formulated as

We use the SA to make a quantitative evaluation for the competing methods, which is shown in (

In order to fairly assess the effectiveness and robustness, the following temperature distribution is reconstructed:

True temperature distribution.

For a fair comparison, the measurement data is randomly sampled. In the GPR method, the kernel function is the squared exponential kernel, and the quasi-Newton algorithm is employed to solve the hyperparameters. In the MNR method, the polynomial basis function is used, the highest order is two, and the coefficients are estimated by means of the least squares method. When the SA and the NV are 9.80% and 25, the temperature distributions predicted by the GPR method, the MNR method, the GRNN algorithm, and the TC algorithm are shown in Figures

Temperature distribution predicted by the GPR method when the SA and the NV are 9.80% and 25.

Temperature distribution predicted by the MNR method when the SA and the NV are 9.80% and 25.

Temperature distribution predicted by the GRNN method when the SA and the NV are 9.80% and 25.

Temperature distribution predicted by the TC method when the SA and the NV are 9.80% and 25.

Mean relative errors under different NVs.

Mean relative errors under different SAs.

Figures

Under a specific SA level, that is, 7.84%, the MREs for the compared algorithms under varying NVs, that is, 1, 25, 100, 225, and 400, are shown in Figure

The SA is a good criterion for assessing the effectiveness of a prediction method. Under a specific NV level, that is, 25, Figure

In this section, we use another temperature distribution, which is described in (

True temperature distribution.

The numerical simulation conditions and algorithmic parameters are identical to Section

Temperature distribution predicted by the GPR method when the SA and the NV are 9.80% and 25.

Temperature distribution predicted by the MNR method when the SA and the NV are 9.80% and 25.

Temperature distribution predicted by the GRNN method when the SA and the NV are 9.80% and 25.

Temperature distribution predicted by the TC method when the SA and the NV are 9.80% and 25.

Mean relative errors under different NVs.

Mean relative errors under different SAs.

When the SA and the NV are 9.80% and 25, Figures

With a specific SA, that is, 9.8%, the MREs for all compared prediction techniques under varying NVs, for example, 1, 25, 100, 225, and 400, are shown in Figure

When the NV level of the measurement data is 25, Figure

In order to verify the practicability and effectiveness of the proposed prediction method, that is, the GPR method, an experiment study was carried out in this section. The experimental procedure is summarized as the following three steps.

The scattered temperature values at different positions are measured by traditional thermocouples.

The measurement data is used to train the GPR model to abstract the mapping between the temperature values and the measurement positions.

The trained GPR model is used to predict 3D temperature distributions.

An experiment system is designed and constructed, which contains three main parts, that is, a combustion system, a temperature measurement system, and a data acquisition module. The combustion system consists of a diesel supply system, burners, a gas/air system, a water cooling system, and a control system. The temperature measurement system automatically achieves the temperature measurement. The data acquisition module achieves the data storage and transmission. The layout of the experiment system is shown in Figure

Layout of the combustion system ((1) air distributor; (2) air heater; (3) combustion chamber; (4) burner; (5) fan; (6) thermocouple; (7) diesel tank).

The test chamber is designed as an adiabatic combustion system. Geometric dimensions of the chamber, whose walls are made of light mullite bricks, are 900 mm (

The diesel supply system consists of a VSC63A5-2 electromagnetic pump, 1.25 Danfoss atomizers (spray angle is 60 degrees), diesel filters, a high pressure ignition bags, ignition needles, and stable flame discs. The combustion air is provided by a CX-75 5.5 medium-pressure fan, and the air is distributed into the header to support the combustion. Different conditions can be accomplished via changing the number of the burners.

HT-116 K thermocouples are above the burners, which are arranged around the front and back walls. The temperature can be measured via changing measurement locations. The arrangement of the measurement point is presented in Figures

Measurement points arranged on the front wall.

Measurement points arranged on the back wall.

Four DAM-3038 acquisition modules are employed to collect the measurement data. Each module can simultaneously collect 8 channel electric signals, and the acquisition frequency is 10 Hz, and the electric signal is transferred into the corresponding temperature signal in the module. Through the automatic temperature acquisition system, the temperature under different positions and time instances can be obtained.

In this section, we use the experiment data to evaluate the practicability and effectiveness of the GPR method. The number of the known temperature measurement data is 33. In the GPR method, the squared exponential kernel is used, and the quasi-Newton algorithm is employed to solve the hyperparameters.

Figure

Temperature distribution predicted by the GPR method.

Measurement point locations.

The predicted temperature values and the measured temperature values.

Relative errors.

Figure

Figure

High-precision temperature distribution information is important for various industrial processes. In this study, a new methodology is put forth to predict 3D temperature distributions from the limited number of the temperature measurement data, and the main research findings can be summarized as follows.

(1) A new methodology is proposed to predict 3D temperature distributions. The method includes two key phases. In this first phase, traditional methods are employed to measure local temperature values. In the second phase, the GPR method is developed to predict 3D temperature distributions in accordance with local temperature measurement values. Numerical simulation results indicate that the execution of the GPR method is easy and can accurately predict 3D temperature distributions under a relatively low SA. Furthermore, the GPR method holds satisfactory robustness and good adaptability of coping with complicated, nonlinear, and high-dimensional problems.

(2) The proposed method can predict 3D temperature distributions in terms of the given local temperature observations and can mitigate one of challenges in traditional tomography-based measurement methods; that is, the increase of the spatial scale of the measurement area leads to the degeneration of the measurement sensitivity and the reduction of the measurement accuracy.

(3) Beyond conventional numerical simulation methods and inverse heat transfer problems, the proposed method does not solve complicated partial differential equations, and the computational complexity and cost are low.

(4) The experimental study shows that, under the combustion condition considered in this study, the maximum relative error is 0.021, which verifies the accuracy and effectiveness of the proposed prediction method. As a result, a practicable and efficient method is introduced for temperature distribution measurements in a 3D area.

Gaussian process regression

Three-dimensional

Acoustic tomography

Time-of-flight

Tensor completion

Multivariate linear regression

Multivariate nonlinear regression

Robust estimation

Regularized multivariate linear regression

Artificial neural network

Least squares based MLR

Generalized regression neural network

Mean relative error

Noise variances

Sampling ratio

Number of the samplings

Number of the unknown variables.

The authors declare that they have no conflicts of interest.

The study is supported by the National Natural Science Foundation of China (nos. 51576196 and 61571189), the Fundamental Research Funds for the Central Universities (no. 2017MS012), the National Key Research and Development Program of China (no. 2017YFB0903601), and the 111 Talent Introduction Projects at North China Electric Power University (no. B13009).