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The Response Surface Methodology (RSM) has been applied to explore the thermal structure of the experimentally studied catalytic combustion of stabilized confined turbulent gaseous diffusion flames. The Pt/
γ
Al
_{2}
O
_{3}
and Pd/
γ
Al
_{2}
O
_{3}
disc burners were situated in the combustion domain and the experiments were performed under both fuel-rich and fuel-lean conditions at a modified equivalence (fuel/air) ratio (
ø
) of 0.75 and 0.25 respectively. The thermal structure of these catalytic flames developed over the Pt and Pd disc burners were inspected via measuring the mean temperature profiles in the radial direction at different discrete axial locations along the flames.
The RSM considers the effect of the two operating parameters explicitly (r), the radial distance from the center line of the flame, and (x), axial distance along the flame over the disc, on the measured temperature of the flames and finds the predicted maximum temperature and the corresponding process variables. Also the RSM has been employed to elucidate such effects in the three and two dimensions and displays the location of the predicted maximum temperature.

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 CO_{2} emissions [

Recently the catalytic combustion has been dealt with numerically. Lucci et al. [_{4}/air in a heat recirculation micro-combustor made of different materials. Arani et al. [

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/γAl_{2}O_{3} and Pd/γAl_{2}O_{3} 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 [

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 [

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 [

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 ξ 1 , ξ 2 , ... , ξ k could be represented by the following equation [

y = f ( ξ 1 , ξ 2 , ... , ξ k ) + ε (1)

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 ξ 1 , ξ 2 , ... , ξ k 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:

codedvalue = actualvalue − mean halfofrange (2)

Frequently, a low- or second-order polynomial model is proper [

Y = β 0 + ∑ i = 1 k β i X i + ∑ i = 1 k β i i X i 2 + ∑ i = 1 i < j k − 1 ∑ j = 2 k β i j X i X j + ε (3)

where, Y is the response variable; X i and X j are the coded input variables that affect the response variable and denotes the random error or uncertainties between predicted and measured values [

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 [

Coefficient of determination R^{2}: 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 R^{2}: It is an unbiased evaluation of the coefficient of determination. It reprimands the statistic R^{2} 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.

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 ( R pred 2 ): It is a measure of the extent of deviation in prediction of new data clarified by the model and it is calculated from PRESS. R pred 2 and R adj 2 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:

A A D = { [ ∑ i = 1 n ( | y i , exp – y i , c a l | / y i , exp ) ] / n } * 100 (4)

where y i , exp and y i , c a l are the experimental and calculated responses and n is the number of experimental runs [

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) [

The following formulas have been employed to calculate the coded factors of (r) and (x):

R = r / 75 , X = x − 127.5 97.5 (5)

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:

Y = β 0 + β 1 ∗ R + β 2 ∗ X + β 11 ∗ R 2 + β 22 ∗ X 2 + β 12 ∗ R ∗ X (6)

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 [

An appropriate power transformation λ for the data is established on the relation Y * = Y λ .

λ is calculated using the experimental given data such that SS E is minimized (where E is the error between the given experimental response values and the analogous transformed ones Y * ). The following relation (7) has been employed to obtain Y λ :

Y λ = { Y λ − 1 λ 1 g λ − 1 , i f λ ≠ 0 ; ln Y × g , i f λ = 0 (7)

where: g is the geometric mean of the experimental response vector.

For a number of λ values the corresponding Y λ and SS E is calculated for these values to obtain a plot of SS E versus λ. Because the range of SS E values is large, this is performed by plotting the ln ( SS E ) versus λ values. The λ value of corresponding to the minimum ln ( SS E ) 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

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

BOX COX Plot | Flame Condition and Disc Type | |||
---|---|---|---|---|

FR_Pt | FL_Pt | FR_Pd | FL_Pd | |

Risk Level α | 0.05 | 0.05 | 0.05 | 0.05 |

Best Lambda | 0.6836 | 0.5622 | 0.5412 | 0.6152 |

λ for Low Confidence | 0.4417 | 0.5101 | 0.46354 | 0.52878 |

λ for High Confidence | 0.9754 | 0.6188 | 0.62388 | 0.70727 |

Ln(RSSE) for Confidence Interval | 14.373 | 13.273 | 13.494 | 13.493 |

and Prediction Statistics and the Analysis of Variance which checks the significance and fitness of quadratic equation models have been depicted in ^{2} ≥ 0.95) indicates the accuracy of the deduced models. The high and very close agreement between the “ R pred 2 ” values with the corresponding “ R adj 2 ” 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

ANOVA and Regression and Prediction Statistics | Flame Condition and Disc Type | |||||
---|---|---|---|---|---|---|

FR_Pt | FL_Pt | FR_Pd | FL_Pd | |||

Regression Statistics | R^{2} | 0.9453 | 0.9812 | 0.9547 | 0.9588 | |

Adjusted R^{2} | 0.9446 | 0.9809 | 0.9536 | 0.9582 | ||

Standard Error | 1.526 | 0.9015 | 1.330 | 1.098 | ||

Analysis of Variance | Regression | MS | 2818.2 | 2867.6 | 1578.1 | 1872.8 |

df | 3 | 3 | 5 | 3 | ||

Residual | MS | 2.328 | 0.8126 | 1.768 | 1.206 | |

df | 210 | 203 | 212 | 200 | ||

F | 1210.7 | 3528.8 | 892.6 | 1552.6 | ||

Significance F | 3.20E−132 | 8.23E−175 | 3.07E−140 | 3.19E−138 | ||

Prediction Statistics | PRESS | 508.4 | 171.7 | 396.1 | 250.7 | |

R pred 2 | 0.9432 | 0.9804 | 0.9521 | 0.9572 | ||

Adeqval | 118.9 | 193.6 | 106.0 | 132.2 | ||

CV | 6.056 | 4.156 | 6.522 | 4.984 | ||

Average Absolute Deviation % | 12.31 | 7.799 | 12.91 | 8.884 |

the small values of (AAD) support the competence of the employed equations [

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

The quadratic terms in Equation (6) indicate the occurrence of curvatures. The negative signs for R 2 & X 2 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 [

Regression Parameters | Flame Condition and Disc Type | ||||
---|---|---|---|---|---|

FR_Pt | FL_Pt | FR_Pd | FL_Pd | ||

β 0 | Coeff. | 34.29 | 29.80 | 28.04 | 29.47 |

±C.L. | 0.3694 | 0.2216 | 0.3243 | 0.2734677 | |

Standard Error | 0.1874 | 0.1124 | 0.1645 | 0.1387 | |

t-Stat | 183.0 | 265.1 | 170.4 | 212.5 | |

p-value | 1.55E−233 | 7.73E−260 | 8.20E−229 | 1.59E−237 | |

β 1 | Coeff. | 1.085 | |||

±C.L. | 0.3767 | ||||

Standard Error | 0.1911 | ||||

t Stat | 5.678 | ||||

p-value | 4.44E−08 | ||||

β 2 | Coeff. | 3.999 | 5.176 | 6.106 | 4.153 |

±C.L. | 0.3339 | 0.2146 | 0.2997 | 0.2429157 | |

Standard Error | 0.1694 | 0.1088 | 0.1520 | 0.1232 | |

t-Stat | 23.61 | 47.56 | 40.17 | 33.71 | |

p-value | 5.23E−61 | 5.10E−112 | 4.54E−101 | 1.99E−84 | |

β 1 2 | Coeff. | −12.49 | −14.43 | −10.41 | −9.798 |

±C.L. | 0.6268 | 0.3925 | 0.5677 | 0.4740326 | |

Standard Error | 0.3179 | 0.1991 | 0.2880 | 0.2404 | |

t Stat | −39.30 | −72.48 | −36.14 | −40.76 | |

p-value | 9.41E−99 | 4.65E−147 | 1.41E−92 | 7.97E−99 | |

β 2 2 | Coeff. | −10.89 | −6.548 | −7.393 | −8.382 |

±C.L. | 0.5699 | 0.3558 | 0.4957 | 0.4137794 | |

Standard Error | 0.2891 | 0.1804 | 0.2515 | 0.2098 | |

t-Stat | −37.68 | −36.29 | −29.40 | −39.94 | |

p-value | 2.12E−95 | 1.07E−90 | 9.79E−77 | 2.87E−97 | |

β 12 | Coeff. | 1.018 | |||

±C.L. | 0.5907 | ||||

Standard Error | 0.2997 | ||||

t-Stat | 3.397 | ||||

p-value | 8.15E−04 | ||||

Experimental Max. Temp. | 1275 | 1025 | 960 | 970 | |

Predicted Max. Temp. | 1200.8 | 949.9 | 861.8 | 898.9 | |

% AD for Pred. Max. Temp. | 5.82 | 7.32 | 10.23 | 7.334 | |

r for Max. Pred. Temp. | 1.93E−05 | 1.13E−05 | 5.44 | 2.53E−05 | |

x for Max. Pred. Temp. | 145.4 | 166.04 | 168.3 | 151.7 |

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 (R^{2} > 0.9) exposed in Figures 2(a)-(c) [

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:

For FR_Pt

sqrt ( T ) = 10.428 + 0.3332 ∗ x − 2.221 E − 3 ∗ r 2 − 1.146 E − 3 ∗ x 2 (8)

For FL_Pt

sqrt ( T ) = 11.831 + 0.2287 ∗ x − 2.57 E − 3 ∗ r 2 − 6.89 E − 4 ∗ x 2 (9)

For FR_Pd

sqrt ( T ) = 7.414 − 3.28 E − 3 ∗ r + 0.2609 ∗ x − 1.85 E − 3 ∗ r 2 − 7.78 E − 4 ∗ x 2 + 1.39 E − 4 ∗ r ∗ x (10)

For FL_Pd

sqrt ( T ) = 9.703 + 0.2674 ∗ x − 1.74 E − 3 ∗ r 2 − 8.82 E − 4 ∗ x 2 (11)

Matlab 8.1 has been employed to perform the Response Surface plots for the

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 [

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.

The response surface methodology (RSM) with the aid of Box-Cox method has

been exploited to establish the suitable relations for the effect of radial distance from the center line of the flame (r) and the axial distance above the disc (x) on the thermal structure (T) of the flames in the existence of the various catalytic stabilizing discs. The conventional least squares regression models showed outstanding prediction for the experimental results with high values for R^{2} and R adj 2 > 0.95, high values of calculated F-value; Adeqval, and small values of significance F and AAD. Also good predictability for the response in a new experiment would be anticipated as revealed from the small values of PRESS and high values of R pred 2 . The response surface exemplified in the three dimensions depicted the response of the predicted variable T to the variation of the considered variable parameters r & x with the good agreement between the experimental and predicted results and displayed the location of the predicted maximum temperature. The established equation models can be utilized for the estimation of the temperature profile for any new data for the factors x & r within the explored limits.

The authors declare no conflicts of interest regarding the publication of this paper.

Gendy, T.S., Ghoneim, S.A. and Zakhary, A.S. (2019) Response Surface Modeling of Fuel Rich and Fuel Lean Catalytic Combustion of the Stabilized Confined Turbulent Gaseous Diffusion Flames. World Journal of Engineering and Technology, 7, 1-17. https://doi.org/10.4236/wjet.2019.71001