1. INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 6340(Print), ISSN 0976 6359(Online), Volume 5, Issue 7, July (2014), pp. 101-112 IAEME AND TECHNOLOGY (IJMET) ISSN 0976 6340 (Print) ISSN 0976 6359 (Online) Volume 5, Issue 7, July (2014), pp. 101-112 IAEME: www.iaeme.com/IJMET.asp Journal Impact Factor (2014): 7.5377 (Calculated by GISI) www.jifactor.com IJMET I A E M E MODELING OF OXYGEN DIFFUSION THROUGH IRON OXIDES LAYERS Ion RIZA1, Marius Constantin POPESCU2 1University Politehnica of Cluj Napoca, Department of Mathematics, Cluj Napoca, Romania 2Vasile Goldis Westerns University Arad, Department of Computer of Science, Arad, Romania 101 ABSTRACT In the present paper we carried out several experiments in oxygen or dry air, at low temperature of some metallic samples. In order to be able to extend or estimate the corrosion phenomenon we made use of the modelling of oxygen diffusion through rust layers (oxides) and of solving the parabolic equations of diffusion, respectively. The diffusion equation is important for modelling the oxygen diffusion within biological systems and for modelling the neutron flux from nuclear reactors. Keywords: Atmospheric Corrosion, Non-Linear Parabolic Equation, Fick Equations, Fokker Equation, Bessel Function. 1. INTRODUCTION Although a part of the metal comes back into the circuit by remelting, the losses, in case of iron, will come to a total of at least 10-15% from the metal got by melting. The corrosion of the metals and alloys is defined as being the process of their spontaneous destruction, as a result of the chemical, electrochemical and biochemical interactions with the resistance environment [10]. In practice, the corrosion phenomena are usually extremely complex and they can appear in several forms; this is why it is not possible to strictly classify all these phenomena. The chemical corrosion of metals or dry corrosion- of alloys takes place by reactions at their surface in contact with dry gases or non-electrolytes [1], [2], [4]. The products that come out under the action of these environments generally remain where the metal interacts with the corrosive environments. They become layers that can have different thicknesses and compositions. Among the most corrosive factors, O2 has an important contribution. The evolution of the corrosion is related, among other things, to the evolution of oxygen concentration in oxides and metals. All types of oxidations start with a law that is proportional or linear with time, followed by another logarithmic or parabolic law. All equations with partial derivatives that describe and influence diffusion are parabolic.

2. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 6340(Print), ISSN 0976 6359(Online), Volume 5, Issue 7, July (2014), pp. 101-112 IAEME 2. EXPERIMENTS IN DRY AIR AT LOW TEMPERATURES At low temperatures the iron oxides Fe3O4 and Fe2O3, are thermically stable and at normal temperatures the ordinary rust Fe2O3*nH2O appears. The OL37 iron sample has been vertically exposed in open atmospheric conditions during different periods of time (about 6, 12, and 24 months), during cold and warm periods. During the cold period the corrosion takes place with values above the average ones. For studying different parts of the sample, a rectangular part, having the length of 3.8 [cm], the width of 3.45 [cm], the surface of 13.11 [cm2], the weight of initial sample 2.5634 [g], the weight without rust 2.2911 [g], the rust weight [g] was taken out. Fig.1: Explication regarding the thickness of oxide layer at low temperature The calculation of the thickness of oxide layer makes also possible the calculation of oxygen diffusion. In order to calculate the thickness of the oxide layer at low temperature we should take into account some experimental or calculated, such as rust weight gr (0.2723[g]), density (5195 [mg/), number of months of exposure or exposure time, t (1.5552x107[s], respectively, 3.1x107[s]), thickness of oxide layer ( 102

3. =0.003997 [cm], for t=1.5552x107[s]). 3. MATHEMATICAL MODELLING OF DIFFUSION The equations that describe the diffusion are parabolic partial derivatives, and the mathematical models are based on three remarkable laws: - the equation of heat or the Fick second law for diffusion , (1) - convection-diffusion equation , (2) - and parabolic-diffusion equation , (3) where w(x,t) represents the practical value of a concentration, expressed in [mg/cm3], x is a distance and t, time. As a particular case, there is the function f(x)=e-x, n order to explain the decrease of concentration in time: this decreases from the air-rust interface (outer air) towards rust-metal interface (towards the interior). The study in one dimension has been imposed by a diffusion named

4. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 6340(Print), ISSN 0976 6359(Online), Volume 5, Issue 7, July (2014), pp. 101-112 IAEME D, expressed in [cm2/s]; due to the fact that we didnt have any data about D, the time dependency (t), or (x,t), we considered D=constant (or K). In calculus we considered D=1.12*10-8 [cm2/s]. The following abbreviations were used (specific to the calculus program): ODE normal differential equation (of variable x or t), PDE differential equation with partial derivatives (with two variables x and t) and SOL solution from an expression or effective solution. In Bessel function, I indicates the type of function and . 3.1. Parabolic Homogenous Equation of Diffusion The second law of Fick, (1) for diffusion phenomena that are variable in time and space, in homogenous and isotropic environments, has been studied with several solving methods: - the method of separation the variables with a real function represented by a Fourier integral with Poisson form and solved with erf Laplace function [7]; - the method of integral transformations, respectively the Fourier transformation [8], [9]. We present five solutions to the heat equation or the second law of Fick about diffusion. a). After changing the function !" and, after solving the derivatives 103 and their replacement, the following differential equation results # # " $ %, &! '( ) * &+!,'( ) , (4) with general solution -&! '( ) * &+!,'( ) . !". (5) b). A solution having the form w(x,t)= u(y(x,t)) will be determined with y(x,t)=ex+t and, after derivation and replacements, the following equation will result # #/ 0 * 1 $" # #/ 0 %, (6) The condition is 1 $" % and the result will be a simpler equation # #/ 0 %, u(y)= C1 y + C2, (7) with the general solution &!"2$" * &+3 (8) c). Let us determine the solution of Ficks equation with the form w(x,t)=u(y(x,t)) and !"2"3 Calculating the derivatives and replacing the parabolic differential equation the result will be: # #/ 0 * $ # #/ 0 %, & * &+!,)456789 ) . (9)

5. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 6340(Print), ISSN 0976 6359(Online), Volume 5, Issue 7, July (2014), pp. 101-112 IAEME d). Likewise we look for a solution with the form w(x,t) = u(y(x,t)) and !:,; # #/ 0 * * ; $: # #/ 0 %, & * &+!,)74?9 H I 3 (17) 104 )< . (10) e). The direct solving of the equation with partial derivatives leads to a solution w(x,t)=1(x) 2(t), (11) where @ &!ABC * &+!,ABC, @+ &! $BC 3 (12) 3.2. Convection-diffusion equation In case of convection-diffusion equation, the phenomenon changes with FokkerPlanck equation (2) having the general form: = - D ** D+ *a(x,t)c(x,t)=f(x,t), (13) where a(x,t) si f(x,t) represents a disturbing factor and a source, respectively; in the most frequent case the form is: B = - D * D+ . (14) In particular, if D=+ is considered to be a diffusion coefficient, (x), becoming a speed, v, by derivation EE , or F E. (15) The term is multiplied with a coefficient R named delaying coefficient. This can have a value higher or lower than one unit and it can delay or accelerate the diffusion process; as a result, the equation with Fokker partial derivatives becomes G . (16) A particular case is represented by the introduction of the source (+) or of the consumption (-), term multiplied with coefficient G For convection-diffusion equation there are two solutions, one with no parameter and another with parameter, apart from the transformation into Fick equation [7].

6. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 6340(Print), ISSN 0976 6359(Online), Volume 5, Issue 7, July (2014), pp. 101-112 IAEME a). The convection-diffusion equation has the solution 105 @ &! 5 -4J8KJ8LMCN.( N * &+! 45 -8J8KJ8LMCN.( N , @+ &! 4MCO P ; (18) b). The solution with - parameter of the convection-diffusion equation is @ &! 5 -4J8KJ8LMCN.( N * &+! 5 -8J8KJ8LMCN.( N , @+ &! 4MC8'O P , @@+ . (19) 3.3. Parabolic Diffusion Equation A). From equation (3) the expression PDE will be obtained, starting from the flux notion (physical [3]), or from Planck-Nernst equation QR RSR *TU;UV WXY RSZ, (20) where, QR is a species flux i, R is the species concentration i, Zis the electrostatic potential, R[]^_`[aab[cdec_af[e[_d] gR is the elementary electric load of the electron (1.60217x%,hC), ij 3kl%mn%,+ o p is Boltzman constant, T is the absolute temperature, expressed in q. The equation is specialized in modeling the oxygen diffusion through oxide layers (or porous environments rust) and it controls the oxygen diffusion through rust layers (oxides). If rW is a source that consumes or give oxygen, then the equation for mass balance is rW s * EQW. (21) Considering rW %, the relation (3) becomes U = t tu vRER TU;UV WXY REZw, (22) or if the term containing temperature is omitted U = t tu DRER. (23) If Di is proportional with D through the function x it results , (24) with the general form [5] x yzW s * {W s * & . (25)

7. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 6340(Print), ISSN 0976 6359(Online), Volume 5, Issue 7, July (2014), pp. 101-112 IAEME The parabolic equation that describes the diffusion phenomenon transforms into: - the second law of Fick (1), applied for homogenous environments, for (x)=1; - for (x), function of x, given by the nature of the modeled process, the equation is a component part of Sturm-Liouville operator, in Neumann problem with non-homogenous limit conditions, in Dirichlet problem for unlimited domains and so on; - if (x)=K, with K=constant, the equation of heat can be obtained, where K(=D) can also be K(w); - if (x) is replaced with w(x,t) or with a function f(w(x,t)), several differential equations with different forms will be obtained, with f(w) at m and/or 106 at n or @ at p and some partial derivatives of w can be added, from (n-1) until one and with a free term w(x,t). a1). Solving the equation by using the method of variables separation PDE1: q !, !, (26) the result was the solution @@+ , of components: @ r|}|~ + + &k! * @ ! +&{!!Q&k! * &{!!&k! @+ r|}|~ # # q &k, @+ &!,$. (27) The constants are determined from a system of initial conditions (x=0, t=0, Ci=1575.745 is the initial concentration) and of final conditions (x=30x10-4, t=1.5552x107, Cf=1279.986 is the final concentration). There are two solutions for the two cases: SOL1(C1=57.58669368I; C2=565.9460188 I; C3=1) SOL2(C1=-57.58669368 I; C2= -565.9460188 I; C3=1), with the graphic representation as shown in Fig.2a. a) b) Fig.2: Graphic representation of the solution for: a) t=1.5552x107[s], b) t=3.1104 x107[s]

8. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 6340(Print), ISSN 0976 6359(Online), Volume 5, Issue 7, July (2014), pp. 101-112 IAEME According to the development functions of the function @ it can result a ODE1 variant, 107 having the form ODE3:@ r|} |~k # # # # * *&k! . (28) The form with F(x), comes from an indefinite derivation. We can find the equivalent solution of ODE3 equation (normal differential equation by turning ODE3 into ODE4) by bringing it to the hermitian form |~k # # # # * &k! %. (29) Any equation having the form p0(x)y + p1(x)y+ p2(x)y = 0, (30) can be transformed into # # # # * g %, (31) where p(x)=!