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Experimental and Runge-Kutta Method Simulation to Investigate Corrosion Kinetics of Mild Steel in Sulfuric Acid Solutions.

1. Introduction

Mild steel alloy has been extensively utilized in manufacture as a substance for reaction containers, pipes, etc. [1]. Poor corrosion resistance of mild steel in corrosive electrolytes has largely prevented its implementation. Acid solutions are commonly used in different processes such as acid picking, cleaning, and descaling, which may cause the corrosion of metals [2, 3]. The alloy of mild steel destroyed by corrosive materials has led to significant economic losses and has created huge problems in industrial instrument security [4].

The investigation of mild steel corrosion and iron is a major theoretical issue and has attracted significant attention. Many scientists are conducting research on mild steel corrosion. The corrosion inhibitors of two imidazoline derivatives have been investigated for mild steel and the chloride-substituted was found better as compared with the fluoride-substituted [5]. The effect of 4,6-diamino-2-pyrimidinethiol (4D2P) inhibitor on mild steel oxidation in hydrochloride acid media was studied [6]. The rind, seed, and peel extract of watermelon were studied as corrosion inhibitor for mild steel in hydrochloride acid media [7].

Computational methods were extensively used for the purpose of designing new inhibitors with excellent inhibition characteristics. The simulations investigation was adopted to investigate corrosion-resisting aluminum and stainless steel pipes using 3D finite element model [8]. The sodium phosphate, sodium nitrite, and benzotriazole inhibitors were used to simulate steel metal in concrete pore solutions [9]. Thixoforging and simulation of complex parts of aluminum alloy Al[Si.sub.7]Mg were investigated [10]. A mathematical model was carried out for the sulfuric acid and ferric ion diffusion and the copper sulfide mineral leaching process [11]. Carbon fiber, carbon/carbon, and some modified carbon/carbon blends were exposed to a simulated atomic oxygen ambience to study their attitude in low earth orbit [12]. The reactions between silicon and nitrogen were studied using the shrinking core model [13]. Monte Carlo simulation technique was adopted to study the adsorption behavior of furan derivatives on mild steel face in hydrochloric acid [14]. Corrosion inhibition mechanism of two-mercaptoquinoline Schiff based on mild steel surface is investigated by quantum chemical calculation and molecular dynamics simulation [15].

Runge-Kutta method was extensively used for solving the different model of corrosion. Runge-Kutta, Euler-Maruyama, and Milstein methods were utilized to investigate the relationship between the time and pit corrosion depth in nuclear power plant piping systems [16]. The model is suggested to predict the precipitation allocation of corrosion output by using five-order Runge-Kutta format [17]. The Runge-Kutta method was used to solve the two-phase homogeneous model numerically and the major attitude of activated corrosion outputs [18]. A new partial differential model for monitoring and detecting copper corrosion products by sulfur dioxide (S[O.sub.2]) pollution is proposed using Runge-Kutta method [19].

The present paper explores corrosion kinetics of mild steel in sulfuric acid solutions using weight loss techniques. The effect of temperature (30-50[degrees]C) on corrosion is thoroughly assessed and discussed. Thermodynamic parameters were also calculated and discussed. The Runge-Kutta method was furthermore applied in an endeavor to obtain insights into the mechanism of corrosion of mild steel face at the molecular level.

2. Experimental

Mild steel sheet was mechanically press scissor into pieces of measuring 3 x 2 x 0.1 cm. These pieces were utilized without polishing. However, surface curing of the pieces included cleaning, degreasing in absolute ethanol, and drying in acetone. Solutions of (0.1-0.5 M) [H.sub.2]S[O.sub.4] were provided by dilution of 97% sulfuric acid (weight percentage) utilizing bidistilled water. The chemical structure of this alloy specimen is illustrated in Table 1.

Weight loss tests were conducted using beakers (100 ml) of test solutions maintained at "30[degrees]C" for different concentrations (0.1, 0.2, 0.3, 0.4, and 0.5 M) under total immersion conditions. All tests were made in aerated solutions. Weight loss of the specimens was determined by keeping them in test solutions for a time period range of 1-5 days. After completing a duration of treatment time specimens were scrubbed with a bristle brush under running water in order to remove the corrosion product. Specimens were then dried and reweighed. The weight loss was taken as the difference between the weight at a given time and the initial weight and is determined by using LP 120 digital balance with sensitivity of [+ or -]1 mg. The tests were performed in triplicate to guarantee the reliability of the results and the mean value of the weight loss is reported. Weight loss allowed calculation of the mean corrosion rate in mg/[cm.sup.2] h.

3. Theoretical

3.1. The Runge-Kutta Method [20, 21]. The analytical details of the Runge-Kutta method can be outlined with reference to (1) below and an initial condition (y = [y.sub.o] at x = [x.sub.o]). It is desired to find the value of (y) when (x = [x.sub.o] + h) where (h) is some given constant:

dy/dx = f(x, y) (1)

According to the Runge-Kutta method, it can be shown analytically that the ordinate at x = [x.sub.o] + h to the curve through ([x.sub.o], [y.sub.o]) is given by

y = [y.sub.o] + [1/6](K1 + 4K2 + K3) (2)

where [K.sub.1], [K.sub.2], and [K.sub.3] are given by the equations

K1 = h.f([x.sub.o], [y.sub.o]) (3)

K2 = h.f([x.sub.o] + [1/2]h, [y.sub.o] + [1/2]K1) (4)

K3 = h.f([x.sub.o] + h, [y.sub.o] + 2K2 - K1) (5)

Formula (2) is known as the third-order Runge-Kutta formula because the term corresponding to the third derivative term in the Taylor series for y expanded about ([x.sub.o], [y.sub.o]) is correct [20, 21].

A mathematical model for first-order ordinary differential equation can be found to be used in Rung-Kutta method. The mathematical model represents the relationship between the temperature and the rate of reaction as explained below:

The transition state equation [22, 23] is

r = [RT/Nh] x EXP([DELTA][S.sup.o.sub.a]/R) x EXP(-[DELTA][H.sup.o.sub.a]/RT) (6)

where r is the rate of reaction, [DELTA][H.sup.o.sub.a] is the enthalpy of activation at standard condition, [DELTA][S.sup.o.sub.a] is the entropy of activation at standard condition, h is Planck's constant, and N is the Avogadro number.

Equation (6) is rearranged to obtain

r/T = M x EXP(-[DELTA][H.sup.o.sub.a]/RT) (7)

where M = R/Nh x EXP([DELTA][S.sup.o.sub.a]/R)

Take the (ln) function for both sides of (7) to get

ln(r/T) = [-[DELTA][H.sup.o.sub.a]/[R x T]] + ln(M) (8)

ln(r) = ln(T) - [[DELTA][H.sup.o.sub.a]/[R x T]] + ln(M) (9)

Derivate (9) to find

[1/r][dr/dT] = [1/T] + [[DELTA][H.sup.o.sub.a]/R.[T.sup.2]] (10)

Multiply (10) by (r) and the result is

dr/dT = [r/T] + [r.[DELTA][H.sup.o.sub.a]/R.[T.sup.2]] (11)

Equation (11) is the first-order ordinary differential equation which represents the relationship between the temperature (T) and the rate of reaction (r). This equation can be used in Rung-Kutta method to calculate the rate of reaction at different temperature (30-50[degrees]C) for different concentration.

4. Results and Discussion

The corrosion of mild steel was studied in combination between the theoretical and experimental data to analyse the kinetics of reaction with sulfuric acid ([H.sub.2]S[O.sub.4]) atvarious concentrations (0.1-0.5 M). The theoretical data were obtained by numerical methods, especially by the Runge- Kutta method [20, 21]. The corrosion rate of mild steel experimentally was determined using the relation:

R = [DELTA]w/A.t (12)

where [DELTA]w is the mass loss, A is the area, and t is the immersion of period time.

The enthalpy [DELTA][H.sup.o.sub.a] can be found using (11). The enthalpy [DELTA][H.sup.o.sub.a] was found by plotting ln(r/T) versus 1/T for the experimental data shown in Table 2 according to (8) to get a straight line. The slope of straight line represents [DELTA][H.sup.o.sub.a]/R and the intercept is ln(M), where M is equal to (R/Nh x EXP([DELTA][S.sup.o.sub.a]/R) as shown in Table 2 and Figure 1.

Figure 1 shows that the equation of the straight line is

ln(r/T) = -[2092/T] - 0.542 (13)

Therefore ([DELTA][H.sup.o]/R) is equal to 2092 K and the ln(M) is equal to 0.542 and [DELTA]S= 196.68 J/mol. The value of [DELTA][H.sup.o]/R can be used in (11) to calculate the rate of reaction at different temperature (30, 35, 40, 45, and 50[degrees]C) and at different concentration (0.1, 0.2, 0.3, 0.4, and 0.5 M) by using Rung-Kutta method as shown in Table 3.

Gibbs free energy at temperature 298 K was calculated by the following Gibbs free energy equation [24]:

[DELTA]G = [DELTA]H - T.[DELTA]S

[DELTA][G.sup.o] = 17392.89 J/mol - 298 K x 196.679 J/mol.K

= -41217.32 J/mol (14)

The minus sign in Gibbs free energy indicates that the reaction of mild steel with sulfuric acid was a spontaneous reaction [24].

To calculate the order of reaction (n) for mild steel in sulfuric acid ln(r) versus ln(C) was plotted according to the following equation [24]:

r = dc/dt = K[C.sup.n] (15)

where (r) is the rate of reaction, (K) is the rate of reaction constant, and (n) is the order of reaction. Take the ln function for both sides of (15) to get

ln(r) = n ln(C) + ln(K) (16)

If ln(r) versus ln(C) is plotted according to (16), a straight line is obtained. The slope of straight line represents the order of reaction (n) and the intercept is the rate of reaction constant ln(K) as shown in Table 3 and Figure 2.

The rate of reaction equations at different temperatures (30-50[degrees]C) was found as shown in Figure 2:

ln(r) = 3.787 ln(C) + 7.935 at T = 50C (17)

ln(r) = 3.787 ln(C) + 7.919 at T = 45C (18)

ln(r) = 3.787 ln(C) + 7.903 at T = 40C (19)

ln(r) = 3.787 ln(C) + 7.886 at T = 35C (20)

ln(r) = 3.787 ln(C) + 7.870 at T = 30C (21)

The order of reaction of sulfuric acid with mild steel is a 4th-order reaction as shown in (17), (18), (19), (20), and (21) at the range of temperature (30-50[degrees]C). The rate of reaction constant increases with an increase in temperature because the kinetic energy of molecules increases with increase in temperature [24, 25] as shown in Table 4 and Figure 3. To find the activation energy (E), ln(K) versus (1/T) was plotted according to the following equations [24] below and the results are shown in Figure 3 and Table 3:

K = A[e.sup.-E/RT] (22)

ln(K) = -[E/RT] + ln(A) (23)

The plot of ln(K) versus (1/T) according to (23) obtains a straight line, the slope of which represents the activation energy of a reaction (E/R) and the intercept is equal to the preexponential factor (A).

The activation energy (E/R) for the reaction of mild steel with sulfuric acid was estimated from the slope as shown in Table 3 and Figure 3 and was found equal to 4.829 K for the temperature range of 30-50[degrees]C and the preexponential factor (A) was 3062.54 as shown in the following equation:

ln(K) = -[4.829/T] + 8.027 (24)

The value of activation energy was very low, indicating that mean of the reaction takes place easily and spontaneously, which is in agreement with the result of minus value of Gibbs free energy.

5. Conclusion

The corrosion kinetics of mild steel with sulfuric acid in combination with the experimental data and theoretical approach using the Runge-Kutta method was investigated. The rates of sulfuric acid reaction with mild steel were increased with increased temperatures. Moreover, the rates of reaction constant (K) at temperature range of 30-50[degrees]C were 2618-2793 [L.sup.3]/[mol.sup.3].h, respectively. The reaction order of mild steel was 4 order in all ranges of temperature (30-50[degrees]C). The enthalpy and entropy of reaction were 17.393 KJ/mol and 196.68 J/mol, respectively. The value of Gibbs free energy was minus value (-41.217 KJ/mol), and therefore it was concluded that the reaction of mild steel with sulfuric acid was spontaneous. The activation energy of the reaction of mild steel with sulfuric acid was calculated and it was very low (E/R = 4.829 K) at different temperatures and at different concentration of sulfuric acid, which leads to concluding that the reaction of a mild steel with sulfuric acid readily occurred. Runge-Kutta simulation technique can be used to simulate the corrosion of mild steel surface in different concentrations of [H.sub.2]S[O.sub.4].

https://doi.org/10.1155/2018/9087101

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The author declares that he has no conflicts of interest.

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Ismaeel M. Alwaan (ID)

Department of Materials Engineering, College of Engineering, University of Kufa, Najaf, Iraq

Correspondence should be addressed to Ismaeel M. Alwaan; ism10alw@yahoo.com

Received 19 January 2018; Revised 14 May 2018; Accepted 12 June 2018; Published 8 July 2018

Academic Editor: Ramana M. Pidaparti

Caption: Figure 1: Log(r/T) versus (1/T) of mild steel in sulfuric acid at (0.1 M) concentration and at different temperature (30-50[degrees]C).

Caption: Figure 2: ln(r) versus ln(C) of mild steel in sulfuric acid at different concentration (0.1-0.5 M) and at different temperature (30-50[degrees]C).

Caption: Figure 3: ln(K) versus (1/T) of mild steel in sulfuric acid at different temperature (30-50[degrees]C).
Table 1: The chemical structure of mild steel specimen (weight
percentage).

Elements              C      Si     Mn     P      S      Cr

Weight percentage    0.19   0.26   0.64   0.06   0.05   0.08
  (wt %)

Elements              Mo     Cu     Fe

Weight percentage    0.02   0.27   Bal.
  (wt %)

Table 2: Rate of reaction (mg/[cm.sup.2].h) of mild steel
in sulfuric acid at 0.1 M concentration and at different
temperature (30-50[degrees]C).

Temperature      1/T ([K.sup.?1])
([degrees]C)

30               0.003299
35               0.003245
40               0.003193
45               0.003143
50               0.003095

Temperature          Rate(r) (mg/[cm.sup.2].h)
([degrees]C)       AT C=0.1 M [H.sub.2]S[O.sub.4]

30                             0.175
35                            0.20275
40                             0.2305
45                            0.25825
50                             0.286

Temperature       r/T(mg/[cm.sup.2].h.K)
([degrees]C)

30                       0.000577
35                       0.000658
40                       0.000736
45                       0.000812
50                       0.000885

Table 3: The rate of reaction (mg/[cm.sup.2].h) of mild
steel in sulfuric acid at different concentration
(0.1-0.5 M) and at different temperature (30-50[degrees]C).

                              Rate of reaction
                            (mg/[cm.sup.2].h) at
Concentration
[H.sub.2]S[O.sub.4]     30[degrees]C *    35[degrees]C
(M)

0.1                          0.175          0.198961
0.2                        22.48125         25.55936
0.3                         44.7875         50.91976
0.4                        67.09375         76.28016
0.5                          89.94          102.2545

                                      Rate of reaction
                                    (mg/[cm.sup.2].h) at
Concentration
[H.sub.2]S[O.sub.4]     40[degrees]C    45[degrees] C    50[degrees]C
(M)

0.1                       0.225336        0.25427462       0.285929
0.2                       28.94761        32.6652073       36.73163
0.3                       57.66988        65.0761399       73.17734
0.4                       86.39215        97.4870725        109.623
0.5                       115.8097        130.682624        146.951

* denotes experimental data and other data are theoretical data
that were found by Runge-Kutta method.

Table 4: The rate of reaction constant of mild steel in
sulfuric acid at different temperature (30-50[degrees]C).

Temperature       1/T (K)      Rate of reaction
([degrees]C)                      constant K

30               0.0032987        2617.56559
35              0.00324517        2659.78348
40              0.00319336        2705.38633
45              0.00314317        2749.02065
50              0.00309454        2793.35874
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Title Annotation:Research Article
Author:Alwaan, Ismaeel M.
Publication:International Journal of Corrosion
Date:Jan 1, 2018
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