Catalytic combustion of various hydrocarbon fuels over noble metals addresses the interaction between homogeneous and heterogeneous reactions and it is particularly attractive for the latest power generation technologies aiming at mitigating greenhouse CO2 emissions  . Catalytic combustion methodologies are increasingly explored in the last years due to their enhanced combustion stability at very fuel-lean equivalence ratios and the resulting ultra-low NOx emissions   . Catalytic-rich combustion is of interest not only for natural gas but also for hydrogen and syngas fuels. In this case the catalyst does not only have a prime catalytic partial oxidation function, but also acts as a preheater and stabilizer for subsequent homogeneous combustion zone   . Progress in catalytic combustion depends crucially on advances in catalyst technology and in multi-dimensional modeling for reactor design  .
Recently the catalytic combustion has been dealt with numerically. Lucci et al.  investigated the turbulent catalytic combustion of a fuel-lean hydrogen/air mixture by means of three-dimensional direct numerical simulation (DNS) in a platinum-coated plane channel. Schultze et al.  investigated experimentally and numerically employing two-dimensional model the hetero/homogeneous combustion of hydrogen/air mixture over Pt at stochiometries: 2.0 ≤ Ø ≤ 7.0 at 1.0 bar ≤ p ≤ 5 bar. Moreover Schultze & Mantzaras  employed two-dimensional numerical simulations models to deal with the physicochemical processes during the fuel-lean and fuel rich catalytic combustion of hydrogen/air mixtures in platinum-coated channels. Zheng et al.  investigated experimentally and numerically the homogeneous combustion of fuel-lean syngas mixtures over platinum at elevated pressures and preheats. Yan et al.  studied the numerical of effect of wall parameters on catalytic combustion characteristics of CH4/air in a heat recirculation micro-combustor made of different materials. Arani et al.  carried out three-dimensional direct numerical simulations (DNS) with detailed heterogeneous and homogeneous chemistry and transport to investigate the turbulent combustion of fuel-lean hydrogen/air mixtures in a platinum-coated channel with prescribed wall temperatures.
The present study analyzes mathematically through Response Surface Modeling (RSM) and optimization the previously reported experimental data of thermal structure of catalytic stabilized confined turbulent gaseous diffusion flames over Pt/γAl2O3 and Pd/γAl2O3 catalytic disc burners under fuel-rich and fuel-lean conditions  .
The RSM has been utilized to study the effect of two different operating factors, specifically the radial distance from the center line of the flame (r) and axial distance lengthwise the flame over the disc (x) on the mean radial temperature profiles of the established stabilized flames. Also RSM has been utilized to demonstrate such effects in the three and two dimensions and shows the location of the predicted optimum maximum temperature for the scrutinized catalytic disc burners under fuel-rich and fuel-lean conditions.
Details of the experimental setup and the data investigated in this study have been specified the previous work of  .
2.1. Response Surface Methodology
Response surface methodology (RSM) is an assembly of mathematical and statistical technique used for modeling and analyzing a process in which a response of interest is influenced by several variables and the aim is to optimize this response     . This technique is one of the major quantitative tools in industrial decision making as it gives better understanding of the process; it helps the process engineer to see the effect of the control variables simultaneously and the interactions among all the variables  . It generates a mathematical model; its graphical perspective has led to the term Response Surface Methodology  . Graphic drawings of the shape of the surfaces allow a visual explanation of the functional relations between the response and the experimental variables   . RSM also permits the location of the optimum conditions and sensitivity analyses of the optimum conditions to variations in the settings of the experimental variables. This technique has many advantages such as: cost and time reduction, decreasing the number of tests and valuable in attaining maximum efficiency  .
In the last decade, RSM has been extensively utilized for modeling and optimization of several engineering processes and studies as optimization of media, process conditions, catalyzed reaction conditions, oxidation, production, fermentation, biosorption of metals, thermal cracking of petroleum residue oil, water and waste water treatment, membrane systems, electronics, removal of nickel and lead from petroleum waste water chemical systems like distillation and in modeling and optimizing refinery operations     .
This RSM procedure includes the following steps  : 1) definition of independent input variables and the desired output responses, and carrying out the required experiments, 2) performing regression analysis with the quadratic model of RSM, 3) calculate analysis of variance (ANOVA) for the independent input variables in order to find significant parameters that affect the responses, 4) determination of the situation of the quadratic model of Response Surface Methodology and decide whether the model of RSM needs screening variables or not and finally, 5) optimization.
2.2. Response Surface Models
Assuming all variables to be to be independent; continuous; measurable and controllable by experiments with negligible errors, the correlation between the response y and independent variables could be represented by the following equation  :
The form of the true response function f is unidentified and perhaps very complex and ε is a term that represents a random experimental error not accounted for in f assumed to have a zero mean. The variables in Equation (1) are usually called the natural variables. The units of the natural independent variables vary from one another. Even if some of the parameters have the same units, not all of these parameters will be tested over the same range. Before performing the regression analysis the variables are codified to eliminate the effect of different units and ranges in the experimental domain and allows parameters of different magnitude to be investigated more evenly in a range between −1 and +1    .
Below is the frequently used equation for coding:
Frequently, a low- or second-order polynomial model is proper    . The relationship between the coded variables and the response is modeled, in this study, by modifying the experimental data to a second-order polynomial, which is suitable for studying the interaction effects of process parameters on the response, according to the following equation  :
where, Y is the response variable;
are the coded input variables that affect the response variable and
In order to estimate the unknown parameters in model (3), a series of experiments have been executed in each of which the response y is measured for identified settings of the control variables that result in a maximum or a minimum response over a certain region of interest  .
Ordinary Least Squares (OLS) method that diminishes the variance of the balanced estimators of the coefficients has been performed to evaluate the coefficients of the equation. It assumes that the random errors are identically distributed with a zero mean and a common unknown variance and they are independent of each other  . Removing unimportant coefficients for terms which did not affect the response, OLS method has been applied for the model without them. The final model contains only substantial parameters (with p-value < 0.05)  . To guarantee that the equation fits the data well, the suitability of the “fitted” equations has been assessed through the following indicators     .
2.3. Regression Statistics
Coefficient of determination R2: It is a measure of the ability of the regression equation to estimate the real response data. It also explains the overall predictive capability of the model and confirms its goodness of fit  .
Adjusted R2: It is an unbiased evaluation of the coefficient of determination. It reprimands the statistic R2 if redundant variables terms are included in the model.
F-value is employed to assess the overall significance of the model. At a specific level of significance (α-value) the calculated value of F should be greater than the corresponding tabulated value   .
Regression Significance F: A substantial correlation is considered to exist between the independent and dependent variables if this value < α = 0.05  .
p-value of each coefficient and the Y-intercept less than 0.05 designates that the corresponding variable has a considerable effect on the response with a fitting level of more than 95%. Coefficient with smaller p-value or greater magnitude of |t-value| signifies more impact into the model equation   .
Confidence Limits are the 95% possibility that the real value of the coefficient lies amid the 95% Lower and Upper values. Thinner range is desirable.
2.4. Prediction Statistics
Predictive Error Sum of Squares (PRESS): It evaluates how the equation model predicts each experimental point in the design and how it is likely to forecast the response in a new experiment within the experimental domain. Small values are wanted   .
Predicted R-Squared ( ): It is a measure of the extent of deviation in prediction of new data clarified by the model and it is calculated from PRESS. and should be within 0.20 of each other    .
Adequate precision statistic (Adeqval) is employed to evaluate the ratio of the signal to noise. A ratio greater than 4 is an indicator of suitable model variation and the model could be used to traverse the design space  .
Coefficient of Variation (CV) designates the degree of accuracy with which experiments were implemented. Values below 10% might be considered outstanding   .
Average absolute deviation (AAD) is a sign of the goodness of fit of the equation and a straight method for describing the deviations. The AAD is estimated by the following equation:
where and are the experimental and calculated responses and n is the number of experimental runs   .
2.5. Residuals Analysis
Graphical residual analysis should be performed to legalize the assumptions involved in the ANOVA of a normal residuals distribution (the normal probability plot vs studentzed residuals will be like a straight line) and homogeneity of the variance (structure less plot of studentized residuals vs. run time or the predicted response)    .
2.5.1. Application of RSM to the Present Work
The following formulas have been employed to calculate the coded factors of (r) and (x):
where: r: radial distance from the center line of the flame (mm); x: Axial distance along the flame over the disc (mm).
In the present work the following equation has been applied employing the above formulas (5) for coding the factors:
To establish the correlation between dependent response and independent variables numerous mathematical models have been suggested. A suitable power transformation to the response data could be recognized using the Box-Cox method for normalizing the data or equalizing its variance   .
2.5.2. Box-Cox Method (BC)
An appropriate power transformation λ for the data is established on the relation .
λ is calculated using the experimental given data such that is minimized (where E is the error between the given experimental response values and the analogous transformed ones ). The following relation (7) has been employed to obtain :
where: g is the geometric mean of the experimental response vector.
For a number of λ values the corresponding and is calculated for these values to obtain a plot of versus λ. Because the range of values is large, this is performed by plotting the versus λ values. The λ value of corresponding to the minimum is selected and its 100(1 − α) percent confidence interval is calculated. If the interval for λ do not include the value of one, then the conversion is applicable for the given response data. For details of this method refer to   .
This method has been exploited and the results are portrayed in Figures 1(a)-(d) and Table 1. The values of the sqrt (T) of the mean experimental temperature T dependent variable cited in  have been employed to represent the response Y in Equation (6).
3. Results and Discussions
The regression has been accomplished by means of Microsoft Excel 2010 and Matlab 8.1 to estimate the coefficients of Equation (6) for the coded factors along side with the statistical parameters which validate the results. Regression
Table 1. Values of BOX COX Plot.
Figure 1. (a) BOX COX Plot for FR_Pt; (b) BOX COX Plot for FL_Pt; (c) BOX COX Plot for FR_Pd; (d) BOX COX Plot for FL_Pd.
and Prediction Statistics and the Analysis of Variance which checks the significance and fitness of quadratic equation models have been depicted in Table 2. The very large F-values and a very small Significance F values implies that the equation models are highly significant for all terms in the polynomial equation within 95% confidence interval and are adequate to predict the responses. The high determination coefficient (R2 ≥ 0.95) indicates the accuracy of the deduced models. The high and very close agreement between the “ ” values with the corresponding “ ” ones imply the real and good relation between the independent and dependent variables together with the high degree of correlation between the measured and predicted data from the regression models. The adequate precision values of (106 - 193) show good models discrimination and each of these models could be used to traverse the design space. The low values of PRESS and CV < 10 display the high accuracy; reliability and good consistency of the accomplished experiments and that the models were reproducible. Finally
Table 2. Regression and prediction statistics and analysis of variance for the catalyticflame stabilizing disc burners.
the small values of (AAD) support the competence of the employed equations       . These results indicate that the prediction of the temperature profile for any new data for the factors x & r within the investigated limits could be achieved employing the developed equation models.
The values of only the important coefficients of Equation (6) beside the corresponding low values of p < α = 0.05, small Standard Error and large t-Stat are depicted in Table 3. The very small coefficients limits in comparison with their corresponding ones which mean that they do not span the zero as a value for the parameter ensures the reliability of recognized equations for prediction for new data  . The empty cells belong to the excluded non-significant coefficients. The positive sign in front of the model terms indicates synergistic effect while the negative sign indicates antagonistic effect of the independent variables  .
The quadratic terms in Equation (6) indicate the occurrence of curvatures. The negative signs for reveal that the quadratic curves are concave. This means that the temperature increases with an increase in X up to a maximum value beyond which the temperature decreases with further increase of the distance above the disk  . The temperature always decreases with the increase of radial distance from the center line of the flame. This is more pronounced for the case of FL Pt as it has the largest value of and ǀt-statǀ and the smallest p-value. The magnitude of the coefficients of the quadratic terms indicates the steepness of the curvature (Figures 7(a)-(d)) FL_Pt > FR_Pt > FR_Pd > FL_Pd  .
Table 3. Estimated Regression Parameters for the Catalytic Flame Stabilizing Disc Burners. .
The evidence for the validity of the regression models and their high capability to forecast the response has been affirmed in the high correlation between the experimental data and predicted values (R2 > 0.9) exposed in Figures 2(a)-(c)   . The straight line of the normal probability plot of the studentized residuals in Figures 3(a)-(d) implies the normal distribution of errors  . The structureless pattern in the plots of the studentized residuals versus run number or predicted response (Figure 4 & Figures 5(a)-(d)) indicate the satisfaction of the equation models and the absence of any violation of the independence or constant variance involved in the ANOVA assumptions    .
3.1. Equation Models in Terms of Natural Factors
OLS method also has been utilized to explore the relation between the response variable and the natural independent variables. The following equations have been attained for the fuel rich (FR) and fuel lean (FL) cases:
Figure 2. (a) Predicted Temp. vs Experimental Temp. for FR_Pt; (b) Predicted Temp. vs Experimental Temp. for FL_Pt; (c) Predicted Temp. vs Experimental Temp. for FR_Pd; (d) Predicted Temp. vs Experimental Temp. for FL_Pd.
Figure 3. (a) A polt of Studentized residuals for FR-Pt; (b) Normal polt of Studentized residuals for FL-Pt; (b) Normal polt of Studentized residuals for FR-Pd; (b) Normal polt of Studentized residuals for FL-Pd.
Figure 4. (a) Studentized residuals vs Experiment Number for FR_Pt; (b) Studentized residuals vs Experiment Number for FL_Pt; (c) Studentized residuals vs Experiment Number for FR_Pd; (d) Studentized residuals vs Experiment Number for FL_Pd.
3.2. Response Surface Plots
Matlab 8.1 has been employed to perform the Response Surface plots for the
Figure 5. (a) Studentized residuals vs Predicted Temperature for FR_Pt; (b) Studentized residuals vs Predicted Temperature for FL_Pt; (c) Studentized residuals vs Predicted Temperature for FR_Pd; (d) Studentized residuals vs Predicted Temperature for FL_Pd.
predicted temperatures versus the actual natural variables of (r) and (x) for the above developed equation models (8)-(11) as exemplified in Figures 6(a)-(d) together with the corresponding experimental values. This graphical interpretation displays the interactions among the input variables and shows the best operating conditions of a process along with the maximum response  . Most of the experimental values fall on the predicted Response Surface which supports the accuracy of the developed model equations. The two dimensional contour plots have been exposed in Figures 7(a)-(d) along with the corresponding maximum predicted responses. The curvature natures of these plots indicate the interaction of their independent variables  .
An optimization process has been performed for the above presented Equations (8)-(11) to estimate the maximum predicted temperature and the corresponding r and (x) values. This has been achieved with the aid of the provisional maximum values obtained from the RSM plots for initial guessing, and utilizing Matlab 8.1. The Matlab implements a multidimensional unconstrained nonlinear optimization employing the Nelder-Mead simplex (direct search) method.
Table 3 depicts the maximum predicted temperature together with the analogous maximum experimental one. The small values of the % absolute deviation (AD) ≤ 10 for the forecasted maximum temperature from the conforming maximum experimental values supports the capability of the employed predictive equations.
The response surface methodology (RSM) with the aid of Box-Cox method has
Figure 6. (a) Surface plot for FR_Pt; (b) Surface plot for FL_Pt; (c) Surface plot for FR_Pd; (d): Surface plot for FL_Pd.
Figure 7. (a): Contour plot for FR_Pt; (b) Contour plot for FL_Pt; (c) Contour plot for FR_Pd; (d) Contour plot for FL_Pd.
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