Regression analysis of experimental data in a study of the performance of a flat flame burner

: The complex nature of the combustion process, which simultaneously obeys the laws of thermodynamics, heat transfer, aerodynamics and the chemical kinetics of oxidation reactions, makes numerical modelling very difficult and the experimental approach is currently playing a crucial role in their investigation. The modern highly developed theory of experimental design combines various analytical procedures that allow, with a minimum number of experiments, the obtaining of maximum information about the physical or technological processes under investigation, the properties of materials and phenomena. The ability to determine the influence of the main mode and design parameters on the geometrical characteristics of the flare is a prerequisite for effectively influencing the combustion process in order to intensify it. The present work is an introduction to the methods of planning and knowledge of multifactorial experiments, including: the preparation, conduct and processing of experimental results; mastering the methodology of experimental research; using the methods of mathematical statistics and regression analysis to plan experiments; developing the ability to analyze the object of study; correctly selecting the optimization parameter and the essential factors of the object of study; building an experiment planning matrix to obtain an adequate mathematical model of the object. The objective of this work is to propose an approach to study the effect of mode and design parameters, on the basic


Introduction
Gas fuel has widely been used as an energy carrier in industry in recent decades.There has been a continuous upward trend in its relative share.This is due to the requirements imposed by both energy efficiency and environmental protection legislation.A primary concern is the ability to actively influence the combustion process in order to manage it optimally.The optimization of combustion devices actually means the appropriate selection of the relevant mode and design parameters, on which both the efficiency of the use of the gaseous fuel and the amount of harmful emissions released into the environment depend.Structural, aerodynamic and physical factors have a determining influence on the combustion completeness and ignition conditions (Kostov et al. 2022;Makzumova et al. 2023).
Flat flame burners are widely used in industry due to their high uniformity of heating and complete combustion with low excess air coefficient.A characteristic feature (Fig. 1) is the possibility of creating a flat torch spreading radially on a refractory plate.
The geometry of the flare is determined by a number of factors, the degree of influence of each of which is difficult to assess by a single-factor experiment.Estimation is further compli- cated by the fact that most of these factors influence each other.From previous experiments and data from the literature (Krystev 2021;Zlateva et al. 2020;Dostiyarov et al. 2022), it boils down to the determination of four main independent parameters that have a significant influence on the formation of a flat flame.These are: ) ) Angle of inclination of the gas nozzles -β.This characterizes the degree of rotation of the air flow and defines the conditions of heat and mass exchange in the flame. ) The diameter of the nozzle holes through which the gas fuel flows - d.This determines the fuel distribution in the air flow and thus the heat exchange characteristics of the flame. ) ) Heat load -Q.This determines the temperature regime. ) ) Length of the mixing chamber -l.This determines the residence time of the reacting components in the mixing section and the ignition intensity of the fuel-air mixture.
The method of the full factorial experiment allows obtaining a mathematical description of the studied process in a certain local region of the factor space.A full factorial experiment is an experiment in which all possible non-repeating combinations of the levels of the independent factors are realized.A full factorial experiment is applied 2 κ , at two levels with four factors (k = 4).Sixteen trials were conducted, each of which was equally duplicated three times.This number of trials allows the assessment of the set of main effects together with their interaction effect (Madani et al. 2015).
The purpose of choosing this plan is to obtain as complete information as possible about the influence of the adopted factors on the diameter of the formed flat flame by minimizing the number of trials and respecting the requirements of rotatability and orthogonality of the corresponding matrix,.
In order to eliminate the influence of external factors, the experiments were conducted randomly over time.The use of this approach allows filtering out the influence of irrelevant factors and avoiding the accumulation of systematic error.The local area for the determination of the factors is established by a priori considerations.Each of the factors is varied at two levels.
Two-stage factorial experiments are excellent tools, providing a means by which the factors included in an experiment can be both assessed and tested for significance (González 1998).
Factorial experiments helps to develop a statistical response model by performing the minimum number of well-chosen experiments and determining the optimal values of the process parameters.Factorial experiments are an empirical modelling technique used to evaluate the relationship between experimental variables and corresponding responses (Krishna Prasad and Srivastava 2009).
Data generated in this way is used to calibrate an empirical model, typically including first- or second-order terms, which is eventually refined to exclude unaffected factors and/or add higher -order terms (Montgomery 2013;Cenci et al. 2023).
Normalized first-order sensitivity coefficients are typically applied to combustion to provide a kinetic view of the model and its strengths and weaknesses relative to experimental data.If little is known about the uncertainty distribution of the input parameters then such first-order methods can provide a useful starting point for key parameter identification and model analysis.Input-output dependencies arising from the application of such sampling approaches are typi-cally estimated using statistical methods.This may include calculating the expected value and variance of selected target outputs (Saltelli et al. 2000;Tomlin 2013).
Regression analysis is a statistical method for analyzing and modelling relationships between mass phenomena.It is a statistical method for investigating cause-effect and phenomenon-response relationships in processes and phenomena (Jamshidnezhad 2015).The main purpose of regression analysis is to represent in analytical form, in the form of a mathematical model, the studied correlation relationship.Therefore, this analysis is also called an analytical method for studying correlation dependencies (Ivanov et al. 2022;Denev et al. 2021;Kostov et al. 2021).

Materials and methods
Experimental design methodology refers to the use of practical and powerful statistical tools and techniques to develop efficient, balanced, and economical experimental designs that allow the experimenter to determine how controlled and uncontrolled factors relate to the outcome of the process (Antony 2023).
Developing predictive models to be used in model-based activities requires the identification of the model structure, specifically a set of model equations and model calibration, i.e., the precise estimation of model parameters from experimental data (Mihaluta et al. 2008;de Prada et al. 2019;Cenci et al. 2023).
However, it is imperative to develop a model whose factor interactions can be uniformly distributed over any number of desired levels to ensure the unbiased estimation of main effects.Analysis of variance, a diagnostic tool for regression analysis, is used in some cases to analyze the significance of factor interactions (Sathish Kumar et al. 2022;Hatami et al. 2015).
Direct measurements have played an invaluable role in the development and improvement of kinetic mechanisms over many years with the available techniques discussed in the literature (Pilling 2009;Tomlin 2013).In order to validate the optimized experiments, a desirability check is performed for all studies (Sathish Kumar et al. 2022;Antony 2014;Ansari et al. 2019).
Running a full factorial experiment involves determining the effects of variable controllable input factors (X 1 , X 2 , ..., X n ) on the corresponding varied output responses (Y 1 , Y 2 , ..., Y n ) of a process/system.These experimental design approaches are typically implemented by assigning all factors "k" to two levels, usually denoted by a high (+1 or simply +) and a low (-1 or simply -) level for each factor (Antony 2003;Montgomery 2013;Eriksson et al. 2008).
The experimental design chosen for this study is a 2 4 factorial experiment for four independent variables of gas nozzle inclination angle (X 1 ), gas nozzle diameter (X 2 ), burner heat load (X 3 ) and combustor length (X 4 ).Table 1 gives the ranges of variation of the factors (independent variables) and their values in a natural scale at the basic, upper and lower levels.The coded values of the factors (x i ) are associated with the naturel (X i ) by the dependence: After applying this dependency, we obtain: Table 2 shows the expanded matrix of the full factorial experiment 2 4 in a coded and natural scale after applying the dependence between the factors, and the experimental results, with the dependent variable u y , represents the gas flare diameter D. Here, a column of dummy variable is added x 0 , taking in all experiments a positive value necessary to estimate the free term b 0 of the regression equation.
First-order plans are most often used in designing the experiment.These are those plans that allow an experiment to be conducted to determine the regression equation, which is a first-order polynomial.The minimum number of factor levels required is one greater than the order of the regression equation, for example, for a first-degree polynomial, the minimum number of factor levels is two.
The regression analysis was performed using the methodologies described by (Novik and Arsov, 1981;Degtyarev;Lyubimova and Sisykov 2017) and the sequence of experimental treatments in the particular case of equal duplicated trials is as follows: ) ) To determine the regression coefficients in this case, a model of the type: ) ) After implementing the plan, sixteen equations with sixteen unknowns are obtained and from solving these equations, all sixteen coefficients are found of the regression equation.The coefficients are computed using the formulas: ) The experimental variance is calculated and the uniformity of the variance series is checked by calculating the value of the Cochrane criterion or G-criterion: -average of all n u repetitions of the u-th trial, , 1, ., ; 1, ., ; 1, .; 1, , -the number of degrees of freedom in determining the u-th order variance 2 u y S , G* -calculated value of the Cochrane criterion, G* < G table -homogeneity of the dispersion series.
) ) The statistical significance of the coefficients of the regression equation was checked using the Steudent's criterion.
The dispersion of the coefficient estimates is calculated using the formula: where: -the number of degrees of freedom, α -level of significance, Δb i -confidence interval.Coefficients whose absolute value is equal to or greater than the confidence interval are statistically significant. ) The next stage of processing the experimental data is to test the hypothesis of the model adequacy using the Fisher's criterion or F-criterion: where: 2 in S -dispersion of inadequacy, F* -calculated value of Fisher's criterion.The hypothesis of adequacy is accepted when the calculated value of the

Discussion
The numerical results shown in the summary in Table 2 after implementation of the full factorial experiment and considering the condition of statistical significance of the coefficients, the following regression equation is obtained: After going from the coded to the natural scale, the equation takes the form: (20) After conversion, the following is obtained: Figure 2 gives a plot of the relative magnitude of the influence of the factors and their interactions, in which the value of each coefficient is denoted by the corresponding height of the bar.
Figures 3, 4, 5 and 6 show the variation of the flare diameter with the change of each of the selected independent factors.Figures 7, 8, 9 and 10 show the response surfaces and their plots obtained from the regression equation due to the influential interaction between the selected independent factors, fixing each of the factors at zero level: d = 0.8; Q = 70; l = 230.x 1 x 2 x 3 x 4 x 1 x 2 x 1 x 3 x 1 x 4 x 2 x 3 x 2 x 4 x 3 x 4 x 1 x 2 x 3 x 1 x 2 x 4 x 1 x 3 x 4 x 2 x 3 x 4 x 1 x 2 x 3 x 4   .63 -41.75 36.63 11.25 -11.38 -12 -8.13 -7.75 -3.75 -3.38 -16.75 -5.38 -4.5

Conclusion
Based on the results of the regression analysis, the values of the regression coefficients and their confidence intervals, an adequate mathematical model of the studied process was obtained.The adequacy of the mathematical model was confirmed as a result of statistical hypothesis testing using Fisher's criterion with a significance level of 0.05.From the analysis, it can be seen that of the independent variables selected as factors, the gas flare geometry is strongly influenced by the inclination angle of the gas nozzles, to a lesser extent by the diameter of the gas nozzles and the interaction between them as well as the interaction between the heat load and the length of the mixing chamber.The influence of the heat load, the mixing chamber length and the interaction between the different factors is only weak.Any considerations about the direction and magnitude of the influence of the studied factors on the gas flare diameter can only be expressed for their intervals of variation chosen in the work.In accordance with the planning matrix and the resulting regression equation, appropriate levels of the factors can be selected at which to obtain the desired gas flare geometry.

Fig. 1 .
Fig. 1.Schematic of a flat flame burner Rys. 1. Schemat palnika z płaskim płomieniem in the series, y ug -result of the g-th iteration of the u-th trial, u y

Table 1 .
Intervals for variation of factors