Experimental Design Modelization and Optimization of Pickling Process Parameters for Corrosion Inhibition in Steel Construction

Abstract

The present study attempted to investigate the best conditions to use 2-(4-chlorophenyle)-1,4,5-triphenyle-H-imidazole as a corrosion inhibitor of mild steel in a 7% HCl and 20% H₂SO₄ pickling bath mixture, using chemical, electrochemical, and surface response methodologies in a spherical field. For this, a Doehlert matrix and two principal factors of the Pickling Process were examined. An experimental evaluation was carried out using weight loss, electrochemical impedance spectroscopy, and polarization curve measurements. Impedance diagrams and Bode plots for uninhibited and inhibited systems were analyzed and simulated using the Z-view program, the fitted data obtained closely followed the same pattern as the experimental results. This study demonstrates that the 2-(4-chlorophenyle)-1,4,5-triphenyle-H-imidazole compound is an effective inhibitor for mild steel in pickling bath solutions, and corrosion inhibition efficiency increases with increases in inhibitor concentration to attain 93.2% imidazole at 10⁻³ M. This is due to the absorbability of Cl⁻ and SO₄²⁻ present in the pickling bath solution and the synergistic effect between both elements. The response used in the exploitation of the design was the determination of inhibitor efficiency. This was assessed through weight loss measurements and electrochemical studies on samples in the absence and presence of 2-(4-chlorophenyle)-1,4,5-triphenyle-H-imidazole. It has been shown that the compound under investigation is an effective cathodic-type inhibitor of mild steel corrosion in pickling bath mixtures. Therefore, the inhibition efficiency was improved with the concentration of the inhibitor, which depended on the molecular structure. The optimal corrosion inhibition efficiency as a function of variation in 2-(4-chlorophenyle)-1,4,5-triphenyle-H-imidazole concentration and pickling bath temperature was simulated and demonstrated using canonical analysis; the obtained efficiency at 324 K for 6 h was 81.3% for the coded variable and 83.4% for the real variable. The experimental results are based on a real-time system and provide much more precise results than the simulated results.

1. Introduction

To remove surface scales, metals are submerged in acid pickling baths, which typically contain an appropriate acid solution. The acid reacts with and dissolves these scales. However, once they are removed, it can also begin to degrade the underlying metal. Moreover, corrosion frequently causes large financial losses and possible security issues, which is why different researchers are interested in protection technology [1,2,3]. Consequently, to mitigate this issue, the development of effective corrosion inhibitors has become a key area of research. Chemical substances, typically organic compounds, are employed as corrosion inhibitors form a protective barrier on metal surfaces, reducing the interaction between the metal and the corrosive environment [4,5,6]. The synthetic inhibitors linked to high electron (in charge) density prolong conjugation and exhibit polar functional groups, such as –CN, –OH, and –NH, which provide strong metal inhibitor bondings [7,8,9,10,11]. Heteroatoms (P, O, S, and N) transfer their electrons and/or charge to metallic d-orbitals, forming an efficient metal protective layer by coordinate bonding (chemisorption mechanism) [12,13,14]. Furthermore, the presence of homo-atomic or hetero-atomic multiple bonds enhances the adsorption ability of the inhibitor molecules by increasing their electron-donating ability through extensive conjugation [15,16,17,18]. The compound selected, namely “2-(4-chlorophenyl)-1,4,5-triphenyl-1H-imidazole”, has the following traits:

(i)

Organic compounds with aromatic rings containing electronegative functional groups and π-electrons in conjugated double bonds.

(ii)

Specific interactions between functional groups containing nitrogen heteroatoms with free lone pairs of electrons and the metal surface, which can play an important role in inhibition.

Acid pickling inhibitors are expected to be chemically stable in order to provide high protective efficiency [19,20]. A multitude of studies have been conducted on the impact of temperature on acidic corrosion and the inhibition of corrosion in iron and steel, mostly in acidic environments.

Inhibitors’ performance dependence on temperature and the comparison of thermodynamic data acquired from the corrosion process, with and without inhibitors, led to some conclusions regarding the mechanism of the inhibition effect [21,22,23,24].

A comparison of the corrosion inhibition performance of some substituted 1,4,5-triphenyl-1H-imidazole compounds of mild steel in 0.5 M sulfuric acid and 1.0 M chloride acid was investigated using DFT, electrochemical, and weight loss methods [25]. It has been revealed that this compound exhibits mixed-type inhibitory behavior and provides excellent resistance. As the concentration of inhibitors increases, inhibition efficiency increases too. The kinetic and thermodynamic parameters that regulate the adsorption process are calculated and examined. For both neutral and protonated forms, DFT computations are performed at the B3LYP stages of theory using the 6–31G (d,p) basis for the gas and aqueous phases [26]. The correlation between corrosion inhibition and global reactivity descriptors is adequately analyzed and presented in the referenced study’s section on quantum chemical computations.

This paper aims to continue these studies by investigating the corrosion inhibition performance of 2-(4-chlorophenyle)-1,4,5-triphenyle-H-imidazole for mild steel in a new acidic pickling bath formula (a mixture of 7% HCl and 20% H₂SO₄). This paper subsequently explores the optimal response surfaces through a mathematical model adapted to the experimental data. Given the growing industrial demand, and for many other reasons, such as industrial efficiency and productivity, this process involves two principal stages:

(i)

Conducting experiments at different points in this region and collecting data;

(ii)

Predicting the results at other points by analyzing the reliability predictions.

Subsequently, the researcher will be positioned to ascertain whether the results of their system can be improved, as well as to what extent the specific values of the parameters should be utilized. Essentially, the response surface methodology is a framework employed to determine optimal experimental locations within the feasible region to enhance prediction accuracy wherever possible.

Once the researcher has defined the problem, the response surface methodology provides both alternative experimental strategies and evaluation criteria for the experimental domain and the response. The main advantage is that the process of fitting the experiment to the studied problem is completed before the experiment is performed. Adhering to this viewpoint, the most relevant optimality criteria are determined based on variance. Additionally, statistical validation of the empirical model is included.

2. Materials and Methods

2.1. Practical Conditions

The used steel specimens have a rectangular form with dimensions of 2.0 cm × 1.0 cm × 0.20 cm and the following chemical composition (wt%):

C, 0.17; Mn, 0.37; Si, 0.20; S, 0.03; P, 0.01; and balanced Fe. The specimen surfaces were polished using emery paper, then rinsed with distilled water, degreased with ethanol, and dried with hot air. For the weight loss measurements, the immersion period was 24 h at 298 ± 1 K. After immersion, the specimens were cleaned, and their weight was remeasured with an accuracy of 10⁻⁴ g to determine the corrosion rate [27]. The aggressive medium (7% HCl and 20% H₂SO₄ mixture) was prepared by diluting 98% H₂SO₄ and 37% HCl of analytical grade with distilled water. The molecular formula of the analyzed inhibitor, 2-(4-chlorophenyle)-1,4,5-triphenyle-H-imidazole, is shown in Figure 1.

The inhibition efficiency, E_inh(%), was calculated using following equation:

$$E_{\omega }\%={\displaystyle \frac{{\omega }_{corr}^0-{\omega }_{corr}}{{\omega }_{corr}^0}}\times 100$$

(1)

where ω⁰_corr and ω_corr are the corrosion rates without and with inhibitors, respectively.

The electrolysis cell was protected by a cap with five apertures. The working electrode was pressure-fitted into a polytetrafluoroethylene holder exposing only 1 cm² of area to the aggressive solution. Potentiodynamic polarization studies were performed on mild steel specimens by automatically changing the electrode potential from −900 to −100 mV/Ag/AgCl versus OCP at a scan rate of 1 mVs⁻¹. The test solution was thermostatically controlled at 298 ± 1 K in an air atmosphere without bubbling. To evaluate the corrosion kinetic parameters, fitting with the Stern–Geary equation was used [18]. The corrosion inhibition efficiency was evaluated from the corrosion current density values using the following relationship (2):

$$E_{PP}\%={\displaystyle \frac{i_{corr}^0-i_{corr}}{i_{corr}^0}}\times 100$$

(2)

where i⁰_corr and i_corr are, respectively, the corrosion current density values with and without inhibitors.

Finally, a transfer function analyzer employing a small amplitude a.c. signal (10 mV rms) was used to perform the electrochemical impedance spectroscopy measurements, covering a frequency range of 100 kHz to 100 mHz, with five points each decade. EC-Lab 10.3 (Orlando, FL, USA) software was then employed to evaluate the results in terms of an equivalent electrical circuit. The following equation below was used to determine the inhibition performance:

$$E_{EIS}\%={\displaystyle \frac{R_p^0-R_p}{R_p^0}}\times 100$$

(3)

where R⁰_p and R_p are the polarization resistance values without and with inhibitors, respectively.

To ensure reproducibility, all experiments were repeated three times, and the evaluated inaccuracy did not exceed 10%.

2.2. Design of Experiment Approach

Regarding the implementation of the experimental design, a matrix of Doehlert with two factors [19] was selected. The experimental points in the space of the coded variables were evenly distributed according to the Doehlert experiment matrices.

These experiment matrices allow for the step-by-step investigation of a second-degree response surface; this means that the field of study can be adjusted by recovering some of the experimental points employed in a previous study, or by adding new factors with new experimental points to the investigated design.

To measure how the variables affect the response, a second-degree polynomial model is created [19]:

Y = b₀ + b₁*X₁ + b₂*X₂ + b_1-1*(X₁*X₁) + b_2-2*(X₂*X₂) + b_1-2*(X₁*X₂)

(4)

where X₁ and X₂ are independent variables representing the inhibitor concentration and temperature of the pickling bath solution; b₀ is constant; b₁ and b₂ are coefficients reflecting the effect of factors X₁ and X₂; b_1-2 are coefficients reflecting the interaction between two factors X₁X₂; and b_1-1 and b_2-2 are coefficients reflecting the influence of quadratic X₁ and X₂.

2.3. Study Vicinity

Corrosion and corrosion inhibition were determined to be contingent upon concentration and temperature. Temperature and concentration significantly influence the rate of metal corrosion, and their variations serve as useful tools for investigating and elucidating the adsorption mechanism of an inhibitor. The variations in both influential elements for corrosion inhibition, such as (1) temperature and (2) inhibitor concentration, were selected to simulate the natural conditions encountered in the experimental field. The concentrations of the inhibitor, namely 2-(4-chlorophenyle)-1,4,5-triphenyle-H-imidazole, were studied in the range 1 to 1000 µM, and a blank mixture solution was prepared for comparison. The influence of pickling bath temperature on the potentiodynamic polarization curves in the absence and presence of the 2-(4-chlorophenyle)-1,4,5-triphenyle-H-imidazole inhibitor was studied for mild steel in mixture solution in the range of 298 ± 2 K to 328 ± 2 K.

Classically, the various levels were expressed in a system of coded variables. Level +1 corresponded to the highest natural value and level −1 to the lowest natural value.

The correspondence between real variables and coded ones was ascertained starting from the following Equation (5):

$$X_j={\displaystyle \frac{U_j-U_j^0}{{∆U}_j}}$$

(5)

where X_j is value of the variable factor in the j-coded variable; U_j is corresponding value of factor j in the natural variable; U⁰_j is the central value in the field of variation; and ∆U_j is variation pace.

$${∆U}_j={\displaystyle \frac{U_j^{max}-U_j^{min}}{2}}$$

(6)

All fields of variation for the 3 studied factors are detailed in

Table 1. Center, variation step, and range of parameters studied.

Factor		Unit	Range	Centre	Variation Step
U1	Inhibitor concentration	µM	1 to 1000	500	500
U2	Pickling bath temperature	K	298 ± 2 to 328 ± 2	318	10

.

2.4. Experimental Response

The current study’s only experimental response was the inhibition efficiency of the mixture (E%); this can be calculated using the following Equation (7):

$$E \%={\displaystyle \frac{{\nu }_0-\nu }{{\nu }_0}}$$

(7)

where ν₀ and ν represent the weight loss of steel in the absence and presence of the inhibitor/mixture, respectively.

2.5. Design of Experiment Matrix

All tests were carried out in the presence or absence of an inhibitor. A point was also created in the center of the domain, which was repeated three times.

Table 2. Experimental matrix design and response vectors.

N° Exp	X1	X2	X1-1	X2-2	X1-2
1	1.00000	0.00000	1.00000	0.00000	0.00000
2	−1.00000	0.00000	1.00000	0.00000	0.00000
3	0.50000	0.86603	0.25000	0.74996	0.43300
4	−0.50000	−0.86603	0.25000	0.74996	0.43300
5	0.50000	−0.86603	0.25000	0.74996	−0.43300
6	−0.50000	0.86603	0.25000	0.74996	−0.43300
7	0.00000	0.00000	0.00000	0.00000	0.00000

illustrates the Doehlert matrix design [20].

The use of coding factor relations permitted the transformation of the experience matrix into predicted experimentation designs. Table 2 presents the results of the inhibitor efficiency measurements for the different mixtures. To ensure reproducibility, all experiments were repeated three times and the evaluated inaccuracy did not exceed 3%.

3. Findings

3.1. Weight Loss Investigation

We began this investigation with a comparative weight loss study examining 0.5 M H₂SO₄ and a novel pickling bath solution, formulated with 7% HCl mixed with 20% H₂SO₄. The weight loss of mild steel corrosion in environments studied with and without different concentrations of the 2-(4-chlorophenyle)-1,4,5-triphenyle-H-imidazole compound was determined after 6 h of immersion at 298 ± 2 K. The inhibition efficiencies represent a similar pattern (Figure 2); thus, at 10⁻³ M of 2-(4-chlorophenyle)-1,4,5-triphenyle-H-imidazole compound, the maximum inhibition efficiency (E_ω = 96.7%) was obtained for the 0.5 M H₂SO₄ solution [7]. For the new pickling bath mixture, a slight variation in inhibition efficiency (E_ω = 93.2%) was observed, without any impact on its inhibition performance.

This result is best explained in terms of the absorbability of Cl⁻ and SO₄²⁻ and the synergistic effect between both elements [7,27,28,29]. In fact, anions with less hydration, for example chloride ions, are expected to have a higher specific adsorption. Being selectively adsorbed, they provide an excess of negative charge in the solution phase, favoring the additional adsorption of imidazole compounds and resulting in higher inhibition [7,30,31,32,33]. It is demonstrated during this study that these compounds are good inhibitors for mild steel corrosion in both the environment of 0.5 M H₂SO₄ and in the novel pickling bath solution formulated with 7% HCl mixed with 20% H₂SO₄.

3.2. Electrochemical Investigation—Current Potential

Polarization curves of the mild steel electrode in the novel pickling bath solution formulated on the basis of 7% HCl mixed with 20% H₂SO₄, without and with the addition of 2-(4-chlorophenyle)-1,4,5-triphenyle-H-imidazole at different concentrations, are presented in Figure 3, and their obtained corrosion parameters are provided in

Table 3. Corrosion parameters obtained from potentiodynamic polarization measurements of mild steel in 7% HCl + 20% H₂SO₄ mixture without and with different concentrations of 2-(4-chlorophenyle)-1,4,5-triphenyle-H-imidazole.

Solution	Conc (10−6 M)	Ecorr (mV/Ag-AgCl)	icorr (µA cm2)	-βc (mV dec−1)	βa (mV dec−1)	Epp (%)
Blank	0	499	2436	125	136	--
Inhibited	1	470	1427	131	136	41.4
Inhibited	10	463	1005	129	121	58.7
Inhibited	100	461	415	126	108	82.9
Inhibited	1000	474	155	127	120	93.6

. As it can be seen, both cathodic and anodic reactions of mild-steel electrode corrosion were inhibited by the increase in imidazole compound concentration (7% HCl + 20% H₂SO₄ mixture). It is clear that the studied imidazole compound suppressed the cathodic reaction to greater extents than the anodic one, especially at low concentration.

This result suggests that the addition of 2-(4-chlorophenyle)-1,4,5-triphenyle-H-imidazole reduces anodic dissolution and retards the hydrogen evolution reaction. Tafel lines of nearly equal slopes were obtained, indicating that the hydrogen evolution reaction was activation-controlled.

The corrosion current density (I_corr) values for both solutions (0.5 M H₂SO₄ and 7% HCl + 20% H₂SO₄ mixture) are presented in Figure 4; the data show that I_corr values decreased significantly in the presence of 2-(4-chlorophenyle)-1,4,5-triphenyle-H-imidazole and further decreased with increasing inhibitor concentration in both solutions. No definite trend was observed in the shift in E_corr values in the presence of various concentrations of the inhibitor in the 7% HCl + 20% H₂SO₄ mixture.

In the anodic domain, we observed that the presence of 2-(4-chlorophenyle)-1,4,5-triphenyle-H-imidazole in the acidic mixture reduced the anodic current density. This result indicates that these inhibitors exhibit both cathodic and anodic inhibition effects.

Therefore, 2-(4-chlorophenyle)-1,4,5-triphenyle-H-imidazole can be classified as an inhibitor of relatively mixed effect (anodic/cathodic inhibition) in the 7% HCl + 20% H₂SO₄ mixture. It is also apparent that the inhibition efficiency (η_PP) followed the same order as that found by the weight loss measurements.

According to the results presented in Table 3, it can be seen that for the studied imidazole inhibitor, η(%) increases with the increase in inhibitor concentration, reaching a maximum value of 10⁻³ M.

This shows that the addition of these compounds does not modify the hydrogen evolution mechanism. The reduction in H⁺ ions at the mild steel surface takes place mainly through a charge transfer mechanism [25,30]. On the other hand, the branches correlated to the anodic reaction changed noticeably, indicating that the iron dissolution mechanism varied after the dissolution of the tested imidazole compound [32,33,34]. The interpretation of these cures shows that both reactions (i.e., anodic and cathodic) affect the corrosion process, but the cathodic reaction is more pronounced than the anodic reaction.

The shifts or displacements in E_corr values when the results in the blank acid mixture solution are compared with the inhibited ones are generally less than 85 mV. This suggests that these studied compounds inhibit both the anodic and cathodic reactions involved in the corrosion of mild steel in acid mixture solutions, and they are therefore mixed-type inhibitors [35,36,37]. Also, we can deduct from current-potential data that the adsorption at higher concentrations of the explored inhibitor is probably extremely effective, covering the steel surface almost entirely, resulting in important inhibitory effects [38,39]. Higher concentrations can result in a multilayer adsorption, leaving no part of the mild steel surface unprotected and blocking all active sites.

3.3. Electrochemical Impedance Spectroscopy

Figure 5 below illustrates the Nyquist plots obtained for mild steel in the blank and inhibitor solutions (7% HCl + 20% H₂SO₄ mixture). The observed spectrum consists of capacitive loops associated with the transferred charges, with the diameter increasing significantly after the addition of the inhibitor, reaching a maximum at 10⁻³ M of 2-(4-chlorophenyle)-1,4,5-triphenyle-H-imidazole.

A depressed semicircle of charge transfer was observed, with the peak of the semicircle plot shifted towards high frequencies; this is associated with a single time constant in the Bode plot [40,41,42,43,44]. The low-frequency inductive loop can be attributed to the relaxation process acquired by adsorption species such as SO₄²⁻_ads and H⁺_ads on the electrode surface [45]. The diameter of the semicircular Nyquist plot rises with increasing inhibitor concentration.

Impedance spectroscopy parameters for mild steel in acidic mixtures with and without inhibitor addition are listed in

Table 4. Impedance spectroscopy parameters for mild steel in 7% HCl + 20% H₂SO₄ mixture in blank solution and with addition of various concentrations of 2-(4-chlorophenyle)-1,4,5-triphenyle-H-imidazole at 298 ± 2 K.

Solution	Conc (10−6 M)	Rs (Ω cm2)	Rct (Ω cm2)	Cdc (µF cm−2)	ndl	Q (µF Sn−1)	ƞEIS (%)
Blank	0	1.2 ± 0.3	4.2 ± 0.1	157.0	0.844 ± 0.03	489.8 ± 0.5	-
Inhibited	1	0.9 ± 0.3	7.2 ± 0.2	94.3	0.863 ± 0.02	255.9 ± 0.4	40.8
Inhibited	10	0.7 ± 0.3	10.0 ± 0.3	93.1	0.845 ± 0.03	250.3 ± 0.3	57.7
Inhibited	100	0.7 ± 0.3	24.5 ± 0.4	53.8	0.863 ± 0.01	133.0 ± 0.4	82.7
Inhibited	1000	0.5 ± 0.2	63.2 ± 0.5	45.0	0.826 ± 0.02	127.4 ± 0.3	93.3

. The electrochemical data obtained from the Nyquist plot were fitted, and the proposed equivalent circuit is shown below. It is made up of solution resistance identified by R_s, the charge-transfer resistance given by R_ct, and a constant phase element. Impedance results were analyzed using the Z-view program, and the obtained Bode plots exhibit three distinctive segments for the mild steel/imidazole/acid mixture (Figure 6 and Figure 7).

From Table 4, it can be observed that R_ct increased with increases in inhibitor concentration. This can be attributed to the formation of an isolating protective film at the metal/solution interface. In addition, it can be observed that an increase in the concentration of the imidazole inhibitor causes a decrease in the double layer capacity (C_dl), which confirms that the studied inhibitor adsorbs on the mild steel surface; this situation is due to an increase in surface coverage by the inhibitor molecules, which leads to an increase in inhibition efficiency [46,47,48]. We can also observe that the inhibition efficiency (ƞ_EIS%) improves with increasing imidazole concentration up to 93.3% at 10⁻³ M.

On the other hand, the inhibition efficiency of the organic compounds mostly depends on their size and active centers. The best performance of the compound 2-(4-chlorophenyle)-1,4,5-triphenyle-H-imidazole may be attributed to the presence of nitrogen atoms based on heterocyclic compounds and the benzene rings attached to the imidazole ring via a methylene group.

Figure 8 shows the electrical equivalent circuit employed to analyze the impedance plots.

3.4. Statistical Treatment

In this investigation, the statistical analysis consisted of assessing the gap between the experimental and mathematical models using chemometric tools such as Nemrodw software. Tests of multiple regressions allowed the mathematical equation below.

Y = 81.0 + 24.0X₁ − 0.1X₂ − 14.0X₁X₁ − 3.3X₂X₂ + 4.6X₁X₂

(8)

3.5. Validity of the Model

The statistical analyses that led to establishing the validity of the model are displayed in the form of variance analysis in

Table 5. Analysis of variance (ANOVA) for the response model.

Source of Variation	Sum of Squares	Degree of Freedom	Mean Square	Ratio	Significance of Regression Coefficients
Regression	1.8937 × 103	5	2.7874 × 102	51.20	<0.001
Residues	20.4	1	20.4	-	-
Validity	112.3	1	31.21	12.25	<0.01
Error	2.13	2	1.24	-	-
Total	1.9177 × 103	9	-	-	-

.

This table provides the percentage of variance explained by the mathematical model in comparison to the variance contained within the experimental results. A probability of ANOVA smaller than 5% confirmed the validity of the suggested model.

3.6. Residual Analysis

Residuals are the deviations of the observed values of the dependent variable from the predicted values for the current model. The residual analysis is displayed in

Table 6. Residual deviations of the observed and predicted values identified using design of experiment.

N° Experience	Yexp. (%)	Ycal. (%)	Deviation	Standard Deviation	dU
1	93.1	91.2	1.9	−0.103	0.180
2	41.3	43.7	−2.4	−0.120	0.236
3	87.2	89.1	−1.9	−0.283	0.152
4	67.6	65.6	2.0	0.244	0.352
5	83.4	85.4	−2.0	−0.771	0.311
6	63.1	61.3	1.8	−0.010	0.710
7	81.7	81.5	0.2	0.182	0.560

below.

The ANOVA models used to analyze the responses of the dependent variable in most of the programs in the Experimental Design module make certain assumptions about the distribution of the residual (but not predicted) values of the dependent variable. These assumptions can be summarized by saying that the ANOVA model assumes normality, linearity, heteroscedasticity, and the independence of residuals. These kinds of propertiesdof residuals for a dependent variable can be inspected using the options and selections available in the Experimental Design module.

The predicted vs. residual values plot shows that the relation between the residual and predicted values is clearly uniform. Fortunately, the uniformity of the residual vs. deleted residual results provide a serial correlation between these marginal means and standard deviations, confirming that the modelization and experimental data are nearly perfect.

3.7. Graphic Analysis of the Model

The aim of this study was to find an inhibitor solution whose features could have been previously defined from the operative conditions extracted from the mathematical model.

Because the direct exploitation of the equation was delicate, it was convenient to restore it under a graphic representation. While fixing two of the four factors of the survey, it was possible to represent the response surface, which resulted in a surface of regression in a three-dimensional space. It was also possible to project the equation of the design under the isoresponse curves, interpreted at the card curves level.

3.8. Evolution of Efficiency as a Function of Inhibitor Concentration and Temperature

Figure 9 represents the evolution of efficiency as a function of temperature and inhibitor efficiency. This figure shows that the inhibitor efficiency increased when the temperature increased. This evolution was, however, more accentuated for the weakest concentrations of inhibitors. The simulation plot and response surface plot are shown in Figure 10, and the maximum inhibition efficiency as a function of the variation in both factors was studied.

3.9. Canonical Correlation Analysis

Canonical correlation analysis was used to determine and measure the relationships between the two sets of variables studied: corrosion inhibitor concentration and acidic pickling bath temperature. It is especially pertinent in the same situations as multiple regression, when there are multiple intercorrelated outcome variables. Canonical correlation analysis allows us to determine a set of canonical variates, that is, orthogonal linear combinations of the variables within each group, that best explain the variability both within and between groups.

Canonical analysis is a method of rewriting a second-degree equation in a form in which it can be more readily understood. Assume that the estimated response is fitted by a second-order model as follows:

$$\widehat{y}=b_0+{\sum }_{j=1}^k{b}_j{x}_j+{\sum }_{i\ge j}\sum b_{ij}{x}_i{x}_j$$

(9)

Given the matrices

$$x=\left(\begin{array}{c}{x}_1\\ ⋮\\ x_k\end{array}\right), b=\left(\begin{array}{c}{b}_1\\ ⋮\\ b_k\end{array}\right), B=\left(\begin{array}{ccc}\begin{array}{cc}{b}_{11}& {\displaystyle \frac{1}{2}}{b}_{12}\\ {\displaystyle \frac{1}{2}}{b}_{12}& b_{22}\end{array}& \cdots & \begin{array}{c}{\displaystyle \frac{1}{2}}{b}_{1k}\\ {\displaystyle \frac{1}{2}}{b}_{2k}\end{array}\\ ⋮& \ddots & ⋮\\ {\displaystyle \frac{1}{2}}{b}_{1k}{\displaystyle \frac{1}{2}}{b}_{2k}& \cdots & b_{kk}\end{array}\right)$$

(10)

Equation (9) is written as

$$\widehat{y}=b_0+x^{\prime }b+x^{\prime }Bx$$

(11)

Mathematically, a second-degree polynomial (Equation (11)) has a necessarily unique stationary point, S, that can be a maximum, a minimum, or a saddle point. By translation of the origin to the stationary point and rotation of the axes, Equation (11) becomes

$$\widehat{y}={\widehat{y}}_s+{\lambda }_1{z}_1^2+{\lambda }_2{z}_2^2+\dots +{\lambda }_k{z}_k^2$$

(12)

which is known as the canonical equation. In Equation (12), $\widehat{y_s}$ stands for the estimated value at the stationary point S, and λ_i, i = 1,…, k, are the eigenvalues of the symmetric matrix B. The canonical coordinates, z, of point x are obtained as z = M′(x−x_S), where M (by column) is the matrix of normalized eigenvectors of B, and x_S represents the coordinates of S.

Written as in Equation (13), the analysis of the quadratic equation is very simple. If all the coefficients (λ_i, i = 1,…, k) are positive, the stationary point S is a minimum; if all the coefficients are negative, S is a maximum; and if some coefficients are positive and other negatives, S is a saddle point. We illustrate this analysis with Example 1. The second-order fitted model is as follows:

$$\widehat{y}=82.34+5.27x_1+21.53x_2-2.87x_1^2-13.54x_2^2-3.465x_1{x}_2$$

(13)

The contour lines of this surface are shown in Figure 11, and the canonical equation is

$$\widehat{y}=81.00+4.75 z_1+23.44{ z}_2-3.16{ z}_1^2-15.01 z_2^2$$

(14)

Thus, the graphical representation of the optimum corrosion inhibition of 2-(4-chlorophenyle)-1,4,5-triphenyle-H-imidazole, determined on the basis of canonical analysis, is displayed below (Figure 11).

Therefore, the stationary point S is the maximum, as we also observe in Figure 12.

The z = M′(x−x_S) transformation is

$$Z_1=0.207 x_1-0.978 x_2$$

(15)

$$Z_2=0.978 x_1-0.207 x_2$$

(16)

and its inverse is

$$x_1=0.21 z_1+0.98{ z}_2$$

(17)

$$x_2=-0.98{ z}_1-0.21 z_2$$

(18)

If z₁ = z₂ = 0 in the system of Equations (17) and (18), the coordinates of the stationary point x_S are obtained. Again, this information is also seen in Figure 11. The estimated response in S, ${\widehat{y}}_s$, is 81% (independent term in Equation (14)).

The eigenvectors associated with the eigenvalues are (0.207, −0.978)′ and (−0.978, −0.207)′, which are the coefficients of x₁ and x₂, respectively, in Equations (15) and (16).

The boxplots displayed below in Figure 13 also confirm the canonical correlation obtained.

4. Conclusions

The inhibiting properties of this film generally remain independent of variations in the concentration of hydrochloric acid pickling baths. It is demonstrated during this study that the 2-(4-chlorophenyle)-1,4,5-triphenyle-H-imidazole compound is a good inhibitor for mild steel in pickling bath solutions, and corrosion inhibition efficiency increases with the increase in inhibitor concentration to attain 93.2% imidazole at 10⁻³ M. This is due to the absorbability of Cl⁻ and SO₄²⁻, present in the pickling bath solution, and the synergistic effect between these elements.

In fact, anions with less hydration of chloride ions are expected to have a higher specific adsorption. Being selectively adsorbed, they provide an excess of negative charge in the pickling bath solution phase, favoring the additional adsorption of 2-(4-chlorophenyle)-1,4,5-triphenyle-H-imidazole compounds and resulting in higher inhibition.

The surface response methodology was used to find out the best conditions for 2-(4-chlorophenyle)-1,4,5-triphenyle-H-imidazole as a corrosion inhibitor of mild steel, considering low and high levels of pickling hydrochloric acidic solution. Impedance diagrams and Bode plots for uninhibited and inhibited systems were analyzed and simulated using the Z-view program. The fitted data obtained trends with nearly the same patterns as the experimental results.

The experiment design methodology has enabled us, through a limited number of carefully selected tests, to obtain a description of the studied response behavior within the experimental area, and thus determine the experimental conditions that maximize the efficiency of inhibition. In this regard, the obtained efficiency at 324 K for 6 h for the coded variable was 81.3%, and 83.4% for the real variable.