30120140507010

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

101

MODELING OF OXYGEN DIFFUSION THROUGH IRON OXIDES LAYERS

Ion RIZA1, Marius Constantin POPESCU

2

1University Politehnica of Cluj Napoca, Department of Mathematics, Cluj Napoca, Romania

2“Vasile Goldis” Westerns University Arad, Department of Computer of Science, Arad, Romania

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.

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

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102

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 ρ (5195

[mg/��), number of months of exposure or exposure time, t (1.5552x107[s], respectively,

3.1x107[s]), thickness of oxide layer (� � ��

�=0.003997 [cm], for t=1.5552x10

7[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



103

D, expressed in [cm2/s]; due to the fact that we didn’t 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 � � √�1.

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 � ,

� , � ��

and their replacement, the following differential equation results

#�

#�� "$ �� 0, �� &!√'(

√) * &+!,√'(√) ,

(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)=eλx+µt

and, after

derivation and replacements, the following equation will result

#�

#/� 0�� * �1 � 1$"�� #

#/ 0�� 0, (6)

The condition is �1 � 1$"�� 0 and the result will be a simpler equation

#�

#/� 0�� 0, u(y)= C1 y + C2, (7)

with the general solution

��, � � &!"�2$"�� * &+. (8)

c). Let us determine the solution of Fick’s equation with the form w(x,t)=u(y(x,t)) and ��, � �!"�2"��. Calculating the derivatives and replacing the parabolic differential equation the result will

be:

#�#/� 0�� * �1 �

$� ##/ 0�� 0, ��, � � & * &+ !,�)45�6λ78λ�9

) . (9)



104

d). Likewise we look for a solution with the form w(x,t) = u(y(x,t)) and ��, � � !:�,;�

#�#/� 0�� * �1 * ;

$:�� ##/ 0�� 0, ��, � � & * &+ !,�)<8=�6>74?9

)< . (10)

e). The direct solving of the equation with partial derivatives leads to a solution

w(x,t)=φ1(x) φ2(t), (11)

where

@�� &!ABC� * &+!,ABC��, @+� � � &!$B C � . (12)

3.2. Convection-diffusion equation

In case of convection-diffusion equation, the phenomenon changes with Fokker–Planck

equation (2) having the general form:

��

� = -

� D��, ��, �� * * � �� D�+��, ��, �� *a(x,t)c(x,t)=f(x,t), (13)

where a(x,t) şi 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

� � � E� �E� � ��, or

� � � �∆� � �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 λ coefficient

G � � � � ��

�� H I��, �. (17)

For convection-diffusion equation there are two solutions, one with no λ parameter and

another with λ parameter, apart from the transformation into Fick equation [7].



105

a). The convection-diffusion equation has the solution

@�� &!5�

-4J8KJ�8LMCN.(N * &+!45

�-8J8KJ�8LMCN.(

N , @+� � � &!4MCOP ; (18)

b). The solution with -λ parameter of the convection-diffusion equation is

@�� &!5�

-4J8KJ�8LMCN.(N * &+!5

�-8J8KJ�8LMCN.(

N , @+� � � &!�4MC8'�OP , ��, � � @�� @+� �.

(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 � �R S �R * TU;UVWXY �RSZ, (20)

where, QR is a species flux i, �R is the species concentration i, Z is the electrostatic potential, �R is the diffusion coefficient, gR is the elementary electric load of the electron (1.60217x10,hC),

ij � 1.38065 10,+� op 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 � 0, the relation (3) becomes

�U � =

ttu v��R E �R� � TU;UV

WXY �REZw, (22)

or if the term containing temperature is omitted

�U � =

ttu D��R E �R��. (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)



106

The parabolic equation that describes the diffusion phenomenon transforms into:

- the second law of Fick (1), applied for homogenous environments, for φ(x)=1;

- for φ(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 φ(x)=K, with K=constant, the equation of heat can be obtained, where K(=D) can also be K(w);

- if φ(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 � � 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 �1��, � � @��@+� �, of components:

@�� r|}1� |�~1: �+��+ �1 � �&3 !��1�� * �

�� 1��, @�� !�+�&1 {!��!�Q�1,2√&3 !� � * &2 {!��!��1,2√&3 !� ��,

@+� � � r|}2 � |�~2 � ##� �2� � � �q �2� �&3 , @+� � � &1 !, $ �� . (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 x10

7[s]



107

According to the development functions of the function @�� it can result a ODE1 variant,

having the form

ODE3: @�� r|} �|�~3: � #�#�� 1�� #

#� �1�� * *&3 !� �1�� , ��.

(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

|�~3: #�#�� #

#� �� * &3 !� �� 0. (29)

Any equation having the form

p0(x)y’’

+ p1(x)y’+ p2(x)y = 0, (30)

can be transformed into

#

#� �� ##� �� * g� � 0, (31)

where p(x)=!� <5<� #� and g�� :�

:� !� <5<� #�.

a2). The hermitian form is

|�~4: #�#�� * &3 !,�� 0. (32)

A new solution �2��, � � @��@+� �, with Bessel functions, results:

@�� r|}3 -|�~4: �+��+ �� * &3 !,�� 0.,

@�� &1 {!��!�Q�0,2√&3 !,� � * &2 {!��!��0,2√&3 !,� ��, φ+�t� � &1 !, $ �� .

(33)

a) b)

Fig.3. Graphic representation of the solution with hermitian, for: a) t=1.5552x107[s], b) t=

6.2208 x107[s]



108

The constants are determined with a system with initial and final conditions, getting two

solutions for the two cases

SOL1(C1=165.4454841 I; C2= -91.23862277I;C3=1) SOL2(C1= -165.4454841I; C2= 91.23862277 I ;

C3=1),

with the cu graphic representation shown in Fig.3a.

Respectively,

SOL1(C1= 296.6675495I; C2= 1548.914183I ; C3=1) SOL2(C1= - 296.6675495 I; C2= 1548.914183I;

C3=1)

with the cu graphic representation shown in Fig.3b.

a3). The variant of the solution ODE4 – hermitian with the special function Eiυ(x), named

exponential integral:

@�� &1 * ~� �1, � !,�� &2, φ+�t� � &1 ! $ �� . (34)

The previous functions are components of the solution

�3��, � � @��@+� �, (35)

SOL1(C1=234.2739606–412.0259131I;C2= - 64.08360910 - 112.7061134I),

SOL2(C1=-234.2739606+412.0259131I;C2= + 64.08360910 + 112.7061134I),

with the graphic representation as shown in Fig.4a.

a) b)

Fig.4: Graphic representation of the hermitian solution with the special function Eiυ(x), for:

a) t=1.5552x107[s], b) t=6.2208 x107[s]

a4). In the case of invariant method with solutions of Bessel functions, we transform ODE3 into

an equivalent from with the invariable method. It is known that two ODE have the same solution if

the invariable is common. The equation in question is

ODE3: #�#�� #

#� �� * &3 !� �� 0, (36)

put under the form



109

y’’+p(x)+q(x)=0 (37)

and made the change of variable y=u(x)z, ODE results as

z’’u+(2u

’+p(x))z

’+(u

’’+p(x)u

’+qu=0. (38)

We cancel the coefficient of zi,

2u’+p(x)=0, (39)

we get

0�� !,(� , (40)

and |�~5 � #�

#�� * �� + �

+ ##� �� * g� � � 0, |�~5: #�

#�� * � � � * &3 !�� 0,

(41)

@�� r|}5�|�~5: #�#�� * ��

� * &3 !�� 0, (42)

With the same component t, as in the first solution, it results that

φ+� � � &1 !,$ �� , @�� C1 Bessel J�1, 2 √C3 eu � * &2 {!��!�� 1, √C3 eu �,

�4��, � � !4(� @��@+� �, (43)

The constants are determined with a system with initial and final conditions, getting two

solutions for the two cases

SOL1(C1= 59.95111043 I; C2= 563.3498783 I; C3=1), SOL2(C1=-59.95111043 I; C2=-563.3498783 I;

C3=1).

The graphic representation is identical in both cases and it is shown in Fig.2.

a5). Solutions with Bessel functions, using the Green function [11]. If in parabolic equation (3),

�� , �= φ(w), this becomes:

� � � ��

� . (44)

Because � � , �, ��

� are components of the zero divergence of a function θ(x,t,w), the Green

function can be applied [6]

¡ � � � �

�� , �� * @��, �� ¢T .

(45)

The function lower than the parabolic integral can be proportional with the partial derivatives

of the function U(x,t)

�£ � ¤ � �� * ¤

� � , £ � � ¤ � �� * � ¤

� � . (46)

Identifying the previous relations, the following system will result

¤ � � ��, �,

¤ � � @��. (47)



110

Replacing the second equation of the previous system and coming back to the initial notation

w(x,t), it results t¥t¦ � �!,� �¤

�� , (48)

Changing the notation, the result is:

� � � � !,� ��

�� , � � � �!,� ��

�� . (49)

Replacing the integration on [0,∞) with an [a,b] subinterval, it is possible that some solutions

to be lost. The differential equation with partial derivatives of this form can be easily integrated, with

a solution

@�� &{!��!� Q�0,2√�&3 !�� * & {!��!� ��0,2√�&3 !��,

(50)

or

@�� &{!��!� ��0,2√&3 !�� * & {!��!� 2 ��0, √&3 !��, @+� � � &!$ �� ,

(51)

with the solution

�5��, � � @��@+� �, (52)

SOL1( C1=147.6003382 I ; C2=-1550.502962 I ; C3= 1),

SOL2( C1= -147.6003382 I; C2=1550.502962 I ; C3= 1).

He graphic representation is identical in both cases, as shown in Fig.5a.

a) b)

Fig.5: Graphic representation of the solution with Green function: a) t=1.5552x107[s],

b) t= 3.1104 x107[s]

Respectively,

SOL1(C1=150.0922757 I; C2=-1575.0834191I; C3=1), SOL2(C1=-150.0922757I;

C2=1575.0834191I; C3=1),

with the cu graphic representation shown in Fig.5b.



111

B). Solutions with exponential component that are solved with Bessel functions:

b1). ��, � � 0��, �!,"� with the solution

@�� &{!��!� Q ��, A,B5W(§¨�W� � * & {!��!� � ��, A,B5W(

§¨ �W� �, @+� � � &!$B5�,

��, � � �@�� @+� ��!"�, (53)

or ��, � � 0��, �!"� with the solution

@�� &{!��!� Q ��, A,B5W(§¨�W� � * & {!��!� � ��, A,B5W(

§¨ �W� �, @+� � � & !,"�!,$B5�,

��, � � �@�� @+� ��!"�. (54)

b2). ��, � � 0��, �!"�2 "��, with one parameter λ [13], with the solution

@�� &{!��!� Q ��, A,B5W(§¨�W� � * & {!��!� � ��, A,B5W(

§¨ �W� �, @+� � � & !,"��!,$B5� , ��, � � �@��@+©�!"� 2 "��, (55)

or ��, � � 0��, �!" � , "�� with the solution

@�� &{!��!� Q ��, A,B5W(§¨�W� � * & {!��!� � ��, A,B5W(

§¨ �W� �, @+� � � & !2"��!,$B5�,

��, � � �@��@+©�!"� 2 "��. (56)

b3). ��, � � 0��, �!"�2 1�, with two parameters, λ and µ [12], with the solution

@�� &{!��!� Q ��, A,B5W(§¨�W� � * & {!��!� � ��, A,B5W(

§¨ �W� �, @+� � � & !,1�!2$B5�,

��, � � �@��@+©�!"� 2 1�, (57)

or ��, � � 0��, �!"� , 1� with the solution

@�� &{!��!� Q ��, A,B5W(§¨�W� � * & {!��!� � ��, A,B5W(

§¨ �W� �, @+� � � & !21�!,$B5�,

��, � � �@��@+©�!"�, 1�. (58)

4. CONCLUSIONS

The losses of metals and alloys produced by corrosion represent about one third of world’s

production. The change of iron to oxides, more stable – corrosion – is due to the thermodynamic

instability of the iron and to diffusion. The cognition of corrosion development means, among other

things, the cognition of the development of oxygen concentration within oxides and metals. In order

to extend or estimate the development of corrosion phenomenon special mathematics were applied.

Thus, Bessel functions led to precise solutions, using several calculus methods. The solving methods

with Bessel functions of the differential equation with partial derivatives led to identical solutions.



112

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ISSN Print: 0976-6480, ISSN Online: 0976-6499.