Modeling and E ﬃ ciency Optimization of Steam Boilers by Employing Neural Networks and Response-Surface Method (RSM)

: Boiler e ﬃ ciency is called to some extent of total thermal energy which can be recovered from the fuel. Boiler e ﬃ ciency losses are due to four major factors: Dry gas ﬂux, the latent heat of steam in the ﬂue gas, the combustion loss or the loss of unburned fuel, and radiation and convection losses. In this research, the thermal behavior of boilers in gas reﬁnery facilities is studied and their e ﬃ ciency and their losses are calculated. The main part of this research is comprised of analyzing the e ﬀ ect of various parameters on e ﬃ ciency such as excess air, fuel moisture, air humidity, fuel and air temperature, the temperature of combustion gases


Introduction
Steam boilers are designed to supply steam for heat transfer purposes due to considerable latent heat value [1,2].Steam is the main feed of several industries in case of direct or indirect utilization since the heat transfer coefficient of steam is two times more than water.Therefore, it can be utilized effectively in power production plants in order to generate electricity [3,4].Thus, the boiler considered as the most significant component in power plants, refineries, and so forth [5,6].The working conditions of a boiler should be always monitored.It must be highlighted that boilers are working under high temperature and pressurized status, hence explosion is a serious risk, which is threatening in boilers operation [7,8].In the design procedure of the boilers, several aspects, including financial, fuel cost, and maintenance factors should be covered and noticed.The complexity of steam boiler makes it challenging to perform common measurements since several factors affect the performance of the boiler.Traditional evaluation methods of boiler performance are neither cost-effective nor time-saving [9,10].Besides, to carry out a comprehensive, accurate analysis, the fuel composition should be analyzed before and after the combustion process in detail [11,12].
The earlier studies on heat transfer and boiling were carried out by researchers such as Gongur and Winterton [13] and Kandlikar [14].The researchers collected a large amount of laboratory data by conducting several experiments.They achieved to present some correlations that could be employed to estimate the boiling behavior with little error.In the current time, computational fluid dynamic (CFD) methods have been used extensively to investigate the boiling regime and its associated involved mechanisms.Judd and Hwang [15] proposed a model for the prediction of boiling heat, which includes evaporation and natural heat transfer mechanisms.They found that the evaporation heat transfer is a significant proportion of the total heat transfer.The application of various CFD models for forced boiling simulation studied by several researchers [16][17][18][19][20]. Various two-phase models with different simulation approaches employed.In these studies, the conservation equations for mass, momentum, and energy for each dissolved phase have been used.Also, a series of empirical auxiliary relationships utilized in the simulation procedure.Krepper et al. [21] provided a model for the evaluation of the boiling mechanism.They investigated boiling in critical thermal flux conditions.In their research, essential parameters such as rotation, cross-flow between adjacent channels and bubble concentration regions were determined.By calculating the temperature of the bar surface, critical areas were identified, and different geometries were evaluated through CFD modeling.Rivera and Xicale [22] performed experimental evaluation and analytical assessment on an upstream flow of water-lithium bromide in a uniform vertical heated tube and provided valuable laboratory data for the saturated nucleate boiling heat transfer coefficients.Owhaib et al. [23] experimentally and analytically investigated a flow of R-134a fluid in a quartz vertical circular tube which was uniformly heated by a heater.Their primary purpose was to simulate the saturated and sub-cooled boiling of the rising refrigerant in the vertical pipe through the pressurized steam that was flowed out of the pipe.Stevanovic et al. [24] provided a single-dimensional multi-fluid model to predict two-phase flow patterns in vertical pipes.The presented model was based on the conservation of mass, energy and momentum and was applicable to any fluid flow that has two-phase flow patterns.Yang et al. [25] presented numerical simulations and practical experiments for modeling the behavior of the R-141b refrigerant in a horizontal coil based on the fluid volume method and considering the multiphase flow model.There was a good agreement between the numerical predictions of phase change with their laboratory data.Kouhikamali [26] developed a numerical simulation for condensation in a vertical cylinder under the forced convection regime.Condensation simulations were carried out using fluid volume model and the effects of parameters such as hydraulic diameter, fluid velocity, Reynolds number, wall temperature difference, and fluid saturation temperature on heat transfer coefficient were investigated.Ozawa et al. [27]examined a range of thermal behaviors, including boiling patterns, heat transfer, pressure drop, and critical thermal flux during high-pressure carbon dioxide boiling for a horizontal microwave channel.The studied variables were tube diameter changes, mass flux, wall thermal flux, and saturation temperature and pressure of the fluid.Saisorn et al. [28] examined the R-134a refrigerant evaporation flow through horizontal and vertical mini channels in order to obtain heat transfer data and also simulation and determination of fluid flow patterns.Dimensions of the fixed tube were considered to be constant, and the parameters of the thermal flux, mass flux and the saturation pressure of the inlet fluid were analyzed to determine the flow of fluid in two horizontal and upward flow conditions.Shen et al. [29] studied a downward fluid flow in a vertical tube for a wide range of parameters such as pressure, mass flux, and thermal flux.Water considered as the working fluid, and the thermal distribution diagram was plotted near the tube wall.In the following, the effect of the thermal flux of the wall on the heat transfer coefficient and wall temperature investigated, new experimental correlations presented, and the heat transfer coefficient compared in a downstream current with a rising flow.
Chen et al. [30] investigated the engineering applications of constructal theory in China.Xie et al. [31] applied constructal theory to optimize heat transfer performance and pressure drop of an evaporator boiler.
Behbahaninia et al. [32] presented a novel auditing approach to monitoring the performance of steam boilers based on exergy approach.This method was proposed based on ASME ptc 4.1 to calculate the exergy loss and exergy efficiency.Some significant factors such as exergy destruction inside the boiler, exergy loss in the wall of the boiler, exergy destruction in the gas-air heater, exergy loss in the flue gas.It concluded that the major irreversibility of the boiler is due to inside exergy destruction, i.e. more than 38%.In addition, the total exergy efficiency of the boiler was calculated by 53.7%.Li et al. [33] investigated the performance of a boiler in a biomass energy production plant and used the exergy method.It was reported that the most irreversibilities was originated from the combustion process.It was found that the total exergy efficiency of the boiler could be increased by decreasing the excess air and by augmenting the superheater temperature.
Li et al. [34] compared various scenarios in order to prevent the inefficient working of a boiler in a 660 MW coal power plant.The authors recommended that the approach of closing part of the 2 nd air nozzle was the most beneficial approach among other studied methods.In another study, Li et al. [35] analyzed the effect of increasing primary air ratio on the performance of a boiler in a coal-fired power plant.It was monitored that increasing the primary air temperature was more practical and also improved the thermal performance of the boiler instead of rising the air ratio.Javan et al. [36] performed an exergoeconomic optimization of a gas-fired steam power plant by applying a genetic algorithm.Based on the results, the boiler was the most exergy destructive source in the power plant.Therefore, two approaches presented and compared to decrease the amount of exergy destruction in the boiler in order to improve the total efficiency of the power plant: reducing the amount of excess air and decreasing the outlet temperature of the boiler's exhaust gas through heat recovery approaches.It was stated that this optimization technique was able to lower the total cost rate by 20% and decreased the dangerous environmental impact by 88%.Pattanayak et al. [37] optimized the sootblowing frequency of a boiler in a coal-fired power plant in order to lower the hazardous combustion emission and increase the boiler's efficiency.Sobota [38] proposed an on-line monitoring method to improve the efficiency of a steam boiler.Nikula et al. [39] proposed a data-driven model to estimate and monitor the thermal performance of the boiler in order to achieve a higher efficiency boiler with a reduction in emission pollutants.Vandani et al. [40] used a genetic algorithm and partial swarm optimization techniques to improve the exergetic efficiency of boiler blowdown.It was stated that outlet pressure and outlet temperature of the boiler played a significant role in improving the exergy efficiency.
In this research, the modeling of boiler efficiency will be performed based on experimental data and response-surface method (RSM).Steam flow rate and output temperature selected as independent variables.
The dependent variable is considered as the boiler efficiency.

Modeling Fundamentals
The neural network model is a data-driven model, so it requires experimental data to build the model.
In this study, with the help of the neural network modeling, the target output is predicted through two input data sets.Figure 1 illustrates the upper and lower limit of input and output data.The utilized input data in the neural network modeling process is listed in Table 1.Where, ̇ represents the mass flow rate (kg/s), T is the temperature (ºC), and η states the boiler efficiency.The neural network is expressed on the basis of a particular topology.The topology is referred to the structure of the network.In this investigation, a triple layer neural network, Figure 3, is used to predict the output variable, the boiler efficiency.A sample 2-5-1 structure is depicted in Figure 4.As can be seen, the number of neurons in the first, second and third layers is 2, 5, and 1, respectively.The number of neurons in the second layer, known as the hidden layer, is arbitrary.In this research, 5 neurons are determined to be placed in the hidden layer.The selection the neurons' number in the first and third layers is not arbitrary and is selected based on the number of input data to the model.According to Figure 4, the neurons in each layer are connected by the edges to the neurons in the adjacent layer.These edges determine the relationship between the input and output variables by the weight coefficients assigned to them.This mathematical relation is discussed in the following.The connection between input data and output data is determined by a network of linked nodes as shown in Figure 4.

Number of
Input and output data are correlated with a series of weighted coefficients.There are 10 (2*5=10) edges between the first layer and the second layer.There are 5 (5*1=5) edges between the second layer and the third layer.Each edge is assigned a weight.The weights of these edges are presented as a 5× 2 matrix and a 5 × 1 matrix.The relationship between output and input data is expressed as follows: ( The employed variables in the Eq. ( 1) are listed and defined in Table 2. [  ] × Bias matrix of the 3 rd layer After multiplying the weight matrix in the input signal, the results are summed with the bias.

Modeling Results of Boiler Efficiency Using Neural Network
In this section, the obtained results from the modeling by means of neural network are compared with the actual results.The total number of data used in modeling is 95.Of these, 70% are for training, 15% for validating, and the rest for testing the network.Figure 5 illustrates the graph error of the modeling results.
The graph error is the difference between the output of the network and the actual output.The network performance in respect to modeling error and regression coefficient R2 are obtained according to Table 3.The performance values (error and correlation coefficient) of the network are graphically depicted in Figure 12 and Figure 13, respectively, as follows:  Most of the stopping training criteria are based on the mean square error control.Therefore, the MSE curve is plotted as a function of the repetition of the training algorithm in Figure 14.
Monitoring MSE for test data and validation data is performed in the same way as training data in various repetitions, and when the validation data error begins to increase, the training should be stopped.
The most generalization is occurred at the 11 th epoch.The point where the validation error reaches its minimum is considered as the output of the model.For instance, in the Figure 14  In Eq. ( 2),   denotes the obtained modeling efficiency,   indicates the real measured boiler efficiency, and Ne represents the total number of samples.

Optimization with the Help of the Response-Surface Method
Response-surface method or RSM is a collection of statistical techniques and applied mathematics for creating empirical models.The purpose of response-surface is to optimize the response (output variable), which is influenced by several independent variables (input variables).In this study, two independent variables of flow rate and output steam temperature of the boiler, and dependent variable of efficiency are discussed.
In Figure 15, the fitted response surface on the boiler experimental data is presented, and as can be seen, this surface is well suited with the experimental data.The constant coefficients obtained from the RSM optimization for Eq. ( 3) are presented in the Table 4.In Figure 16, the response surface along with the experimental data is presented in a 3-D graph.
Figure 16.The fitted surface on the empirical data obtained from the boiler The efficiency surface has an optimal point, which is presented in Table 5 for its optimal value.

Sensitivity Analysis of Effecting Variables
With the aid of sensitivity analysis, the effect of each independent variable can be obtained on dependent variables (boiler efficiency).In the sensitivity analysis, all variables are assumed to be constant and only one variable within its permitted range is changed.In Figure 17, sensitivity analysis is performed for the temperature variable.As can be monitored, the magnitude of the efficiency at 196 ° C results to its maximum value for almost all flow rates.As the flow rate increases and becomes close to the designed flow rate of the boiler changes, the rate of change in efficiency will be slower in terms of temperature.
Figure 17.Temperature sensitivity analysis at four different flow rates Figure 18 shows the sensitivity analysis for the flow rate independent variable.As can be seen, in the flow rate of 15.7 Kg /s, the amount of efficiency yields to its maximum.With increasing temperature, first, the efficiency increases and then decreases.

Conclusion
The plant's steam production and distribution unit is designed and installed by Iranian Energy Engineering Company "Garmagostar" in order to provide steam and heating loads of reboilers, heaters, flares, sweetening packages, seawater treatment plant and other steam utilities.The total number of data used in the modeling process is 95 which are categorized as 70% for training, 15% for validation, and the rest for test procedure of the network.Modeling by means of neural network yields to the prediction error in the range of ∓1.The associated error of the training stage, validation stage, and test stage were obtained less than 0.6%, 0.5%, and 1%, respectively.As expected, the data test error is higher than training and validation due to the lack of participation of the test data at the training and learning stage of the neural network.In the ideal mode, when the model error is zero, all points are placed on the line of Y = T (Bisection of 1 st and 3 rd coordination sections).In practice there is a small amount of error that causes the points to be scattered up and down of the line.The best-passing equation of the points along with the correlation coefficient as well as the mean square error are calculated for validation and test phases of the total data.Response-surface method (RSM) is a set of statistical techniques and applied mathematics for constructing empirical models.
The target goal of the RSM is to optimize the response (output variable), which is influenced by several independent variables (inputs).In this study, two independent variables of steam flow rate and temperature of the generated steam and dependent variable of efficiency are discussed.The amount of efficiency at temperatures of 196 ° C results in maximum efficiency for almost all flow rates.As the flow rate increases and approaches to the specific boiler design flow rate, the trend of variations of efficiency in term of temperature is getting slower.Also, sensitivity analysis for the flow rate variable was performed and it is monitored that the maximum amount of efficiency is yielded at flow rate of 15.7 Kg/s.With increasing temperature, first, the efficiency increases and then decreases.

Figure 1 .
Figure 1.The upper and lower limit of input and output data for construction of a neural network model

Figure 2 .Figure 3 .
Figure 2. Randomized classification of empirical data in order to construct a neural network model

Figure 4 .
Figure 4. Sample schematic structure of a 2-5-1 neural network model: 2 neurons in the input layer, 5 neurons in the hidden layer, and a neuron in the output layer

Figure 5 .
Figure 5.Comparison of model estimations with actual efficiency measurements (95 Data)

Figure 6 .Figure 7 .
Figure 6.Comparison of experimental data and modeling for training stage (75 data)

Figure 8 .
Figure 8.Comparison of experimental data and modeling for test stage (14 data)

Figure 12 .
Figure 12.Values of mean squared error (MSE) of training, validation, test, and all.

Figure 13 .
Figure 13.Correlation coefficient of training, validation, test, and all.

Figure 14 .
Figure 14.Mean Squared Error Diagram in Different epochs of Training Process for estimating the Boiler Efficiency.

Table 1 .
The experimental data used to form the neural network model Training data is used to create a model.The validation data is used to check the quality and correctness of the training stage.Test data is not used in the training phase and is used only for performance evaluation and examining the neural network modeling.

Table 2 .
Definition of required variables in modeling using neural network × Bias matrix of the 2 nd layerAfter multiplying the weight matrix in the input signal, the results are summed with the bias.

Table 3 .
The performance of the neural network in predicting efficiency before the 11 th epoch, the network error trend for the training, validation, and testing data is decreasing and from repetition 11, the verification error is increasing, while the training data error continues to decline.From Repeat 11 to Repeat 17 (6 consecutive repetitions), the validation error has an increasing trend, so the training algorithm ends and Repeat No. 11 is considered as an output.In other words, the training is stopped if the evaluation set error is raised in 6 consecutive repetitions.This stop occurred at repeat No. 11.It should be noted that the algorithm calculates the mean squared error as follows:

Table 4 .
The optimal values of the constants which are obtained by RSM optimization

Table 5 .
The optimal boiler efficiency (obtained from the RSM)