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)=!