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 [10] [11] [12] [13] . 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 [10] . It generates a mathematical model; its graphical perspective has led to the term Response Surface Methodology [14] . Graphic drawings of the shape of the surfaces allow a visual explanation of the functional relations between the response and the experimental variables [11] [12] . 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 [15] .

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 [10] [16] [17] [18] .

This RSM procedure includes the following steps [19] : 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.

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 [20] :

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 [14] [21] [22] .

Below is the frequently used equation for coding:

codedvalue = actualvalue − mean halfofrange (2)

Frequently, a low- or second-order polynomial model is proper [11] [12] [20] . 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 [19] :

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 [22] . k is the number of variables, and β 0 , β i , β i i and β i j are the regression constants of intercept, linear, quadratic and interaction terms, respectively [23] [24] . This flexible coded model permits the factor coefficients to become directly analogous to one another, which assists for the scale-free ranking of the relative significance of the factors [25] .

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 [9] .

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 [14] . 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) [26] . To guarantee that the equation fits the data well, the suitability of the “fitted” equations has been assessed through the following indicators [27] [28] [29] [30] .

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 [24] .

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 [19] [31] .

Regression Significance F: A substantial correlation is considered to exist between the independent and dependent variables if this value < α = 0.05 [18] .

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 [15] [32] .

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 [30] [33] .

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 [14] [30] [33] .

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 [19] .

Coefficient of Variation (CV) designates the degree of accuracy with which experiments were implemented. Values below 10% might be considered outstanding [10] [34] .

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 [14] [15] .

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) [24] [33] [35] .

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 [38] . 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 [38] .

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.