Ind. Eng. Chem. Res. 1996,34, 3405-3410
3405
Analysis of Iron Oxidation at High Temperatures John C. Slattery,*Kuang-Yao Peng, A. M. Gadalla, and N. Gadalla Department of Chemical Engineering, Texas A&M University, College Station, Texas 77843-3122
A new theory for the high-temperature oxidation of iron is proposed, in which the rate-limiting step is ternary diffusion of ferric, ferrous, and oxygen ions in the iron oxides that are formed. The predictions of this theory are compared with previously published experimental data. The only thermodynamic information required is a phase diagram.
Introduction In the oxidation of a metal, there are several steps: gas absorption, surface reaction, and diffusion through one or more layers of metal oxides. Most prior analyses of high-temperature oxidation of metals are based upon the work of Wagner (1951). Although he considered ionic diffusion through the metal oxide to be the rate-limiting step, he identified the local activity of the oxygen ion with the activity of molecular oxygen at a corresponding partial pressure without explanation. He restricted his theory to simple metal oxides in which the valence of the metal ions has only one value. Himmel et al. (1953, p 840) used this theory together with two correction factors to obtain close agreement with experimental data for the high-temperature oxidation of iron. The method by which these correction factors are to be obtained was not clearly explained. Wagner (1969), Smeltzer (1987), and Coates and Dalvi (1970) have extended this theory to the oxidation of binary alloys. Their result is valid only in the limit of dilute solutions (Bird et al., 1960, p 571; Slattery, 1981, p 483). In what follows, we develop a new theory for the hightemperature oxidation of iron, in which the rate-limiting step is ternary diffusion of ferric, ferrous, and oxygen ions in the iron oxides that are formed. Like Wagner (1951), we assume that electrical neutrality is maintained at each point within each phase. Unlike Wagner (1951), we will assume that local equilibrium is established a t all phase interfaces and that the ions form an ideal solution in oxide phases. Although Wagner (1951) did not directly use the assumption of ideal solutions, in measuring their diffusion coefficients Himmel et al. (1953) followed the analysis of (Steigman et al., 19391, requiring the unstated assumption of ideal, binary solutions of "iron" and oxygen.
Problem Statement In attempting to understand this problem, let us begin with an extreme case: iron exposed to 0 2 a t 1 x lo5 Pa and 1200 "C.From the phase diagram shown in Figure 1, we conclude that with time a corrosion layer will develop consisting of two nonstoichiometric phases, magnetite and wiistite, and a monolayer of hematite as shown in Figure 2. In analyzing this problem, we will make several assumptions.
0 0.2 0.4 94
!a4
28
28
ao
4*'k
Figure 1. Phase diagram for iron and ita oxides. (Reprinted with permission from Borg and Dienes (1988,p 115). Copyright 1988 Academic Press.) oxygen
Figure 2. Corrosion layer consisting of two nonstoichiometric phases, magnetite and wiistite, covered by a monolayer of hematite.
1. The reactions in this system are diffusion limited, and equilibrium is established at the three interfaces shown in Figure 2. 2. Neither Fe nor 0 2 can diffuse through wiistite and magnetite. 3. Wustite and magnetite are nonstoichiometric, and we will assume that these materials are fully dissociated. 4. The ionic radius of 02-is 1.40 A; Fe2+is 0.76 A; Fe3+is 0.64 A (Dean 1979). In a frame of reference such that the wiistite-iron interface is stationary, the ferrous ions Fe2+and ferric ions Fe3+diffise through a lattice
0888-5885i95I2634-34Q5~Q9.QQlQ0 1995 American Chemical Society
3406 Ind. Eng.Chem. Res., Vol. 34, No. 10,1995
of stationary (with respect to the moving boundary) oxygen ions 0,- (Davieset al., 1951, p 892; HaufYe, 1965, p 285). For this reason, the magnetite and wiistite must be regarded as consisting of three components. 5 . Within the wiistite and magnetite phases, qoz-) is a constant, because we assume that the oxygen ions 0,are stationary with respect to the moving boundary (see above). Looking ahead to the jump mass balance for 0,- at the wustite-magnetite interface following (37), we see that qoz-) takes the same value in both phases. 6. The oxidation-reduction reaction a t the ironwustite interface
-
+
Fe 2Fe3+ 3Fe2+ results in no generation of free electrons. 7. The oxidation-reduction reaction at the hematiteoxygen interface
+
4Fe2+ 0,
-
4Fe3'
at zz = 0:
x[TLz+,= 0.433
From (11, we have
which allows us to write
or
+ 20,-
results in no generation of free electrons. Because we are assuming that the hematite is a monolayer and because equilibrium is established at the magnetitehematite interface (assumption 11, this reaction can be regarded as taking place at this latter interface. 8. The Fe2+ and Fe3+ move in such a way as to preserve local electrical neutrality: 2C(FeZ+)+ 3C(Fe3+) - &(OZ-) = 0
(1)
X$Z+)
= 0.303
x::z+)
= 0.112
(h) C(Fe2')
=0
9. The oxides form on a flat sheet of iron. 10. In the one-dimensional problem to be considered here, there is no current flow, and, in view of assumptions 6 and 7, there is no free electron flow: m(Fez+)2
+ ~ N ( F ~= s+)z
(2)
Here we have recognized both that c(02-) is independent of position within an oxide and that the oxygen ions 0,are stationary with respect to the moving boundary. 11. Binary diffusion coefficients are taken to be constants. 12. In order to simplify the analysis, we will assume that both wiistite and magnetite form ideal solutions of the Fe3+,Fe2+,and 0,- ions. We will work in a frame of reference in which the iron-wiistite interface is stationary. In view of the requirement
assumption 8 %(Fez+)
+ &(Fe3+) - k ( O Z - ) = 0
(4)
and assumption 9, it is necessary that we seek a solution only for
within the wiistite and magnetite phases. From Figure 1,we find that, at 1200 "C in equilibrium with iron, wiistite has the composition ~ ( 0 =9 0.513. Recognizing (3) and (4), we have two equations to solve
at 2, = h(m,h):
(14)
Here h(w,m)denotes the position of the wustitethe position magnetite interface shown in Figure 2, h(mlh) of the magnetite-hematite interface, superscript (w) a quantity associated with the wiistite phase, superscript (m) a quantity associated with the magnetite phase, and superscript (h) a quantity associated with the hematite phase. It is helpful to begin by looking at the concentration distributions in each phase separately. Consider first the wiistite. With assumptions 4, 8, 10, and 12, the StefanMaxwell equations (Bird et al., 1960, p 570; Slattery,
Ind. Eng. Chem. Res., Vol. 34,No. 10,1995 3407
1981,p 479)require that
This last implies that
= a constant (23) Equation 23 describes the thickness h(wym) of the layer of wiistite as a function of time. Alternatively, it implies that the speed of displacement of the wiistite-magnetite interface
Equation 20 can be integrated consistent with (21)
to find
or
(W)
(W)
- C(Fe2+.)eq,0= cl erf(q)
(25) where C1 is a constant of integration. The boundary condition (22)requires in view of (23) c(Fe2+)
where
Since there are no homogeneous chemical reactions, the overall differential mass balance (Bird et al., 1960, p 556; Slattery, 1981; p 459) requires in view of assumptions 4,8, and 10 as well as (7)
We can immediately write down the similar results for magnetite:
(27) where
Note that this is identical with the differential mass balance for Fe(2+)and Fe3+ (Bird et al., 1960, p 556; Slattery, 1981,p 456). By assumptions 4 and 5, the differential mass balance for 02-is satisfied identically. Let us look for a solution by first transforming (18) into,an ordinary differential equation. In terms of a new independent variable 22
v=&z&iT3
and, by analogy with (71, we have used = -C(m)2+ 1 (Fe ) + +(mi 5 3 3 (0-1
(29)
Equation 27 is to be solved consistent with the boundary conditions (12)and (13)or
(19)
and using assumptions 5 and 8 as well as (16),(18)may be expressed as h(m,h)
(m) (m) C(Fe2f) = C(Fe2+)eq,c
(31)
atp= From (10) and (ll), the corresponding boundary conditions are
We conclude that h(m,h)
- A(m,h)
$I-= a constant
(32)
describes the thickness h(mph) of the layer of magnetite as a function of time. This is turn implies that the speed
3408 Ind. Eng.Chem. Res., Vol. 34,No. 10,1995
of displacement of the magnetite-hematite interface is
Equation 27 can be integrated consistent with (30) through (32)to find (m) '(Fez+)
as assumptions 8 and 10,the jump mass balance for Fe3+ a t the wiistite-magnetite interface also reduces to (37). At the magnetite-hematite interface, the jump mass balance for Fe2+can be simplified using the jump mass balance for Fe3+as well as assumptions 7,8,and 10 to obtain
(m)
- '(Fez+)eq,b =
Here CZis a constant of integration. At the iron-wiistite interface, the jump mass balance for Fe2+is satisfied automatically using the jump mass balance for Fe3+as well as assumptions 6 and 10. The jump mass balance for Fe3+ is satisfied automatically in a similar manner. The jump mass balance for Ozis automatically satisfied. From the jump mass balance for Fez+at the wiistitemagnetite interface (Slattery, 1981,p 451)together with (7),(16),(24),and (291,we find a t z2 = h(w'm):
=O
(39)
We can estimate that at 1200 "C(Chen and Peterson, 1975)
!2$$+) = 3.80 x 10-l' m2/s
(40)
and (Himmel et al., 1953) Q$$+)
= 3.59 x 10-l' m2/s
(41)
and (Touloukian, 1966,p 481)
e(w)= 5.36 x
lo3kg/m3
(42)
[We will show in a subsequent manuscript that, when we analyze the experiments of Himmel et al. (1953)and of Chen and Peterson (1975)using our theory, we find that their "self-diffusion coefficients for iron" can be = interpreted as our q i e 2 + ) in the limit @$e2+) where i = w or m.1 If we assume that the thermal expansion coefficient is the same for magnetite and wiistite, given the densities at room temperature Weast, 1982,p B-109),we find
@ie3+),
In view of assumption 4, the jump mass balance for Ozrequires that c(02-) be continuous across the interface, as indicated in assumption 5. With this result as well
e'"'
= 4.82 x lo3 kg/m3
(43) In view of assumption 5, we can compute (at the iron-
Ind. Eng. Chem. Res., Vol. 34,No. 10,1995 3409 Table 1. Comparison of the Predictions of (47) with the Experimental Observations of Davies et ul. (1951) T,"C Kexp,kg m-2 s-u2 K, kg m-2 s-1'2 700 0.00077 0.000633 800 900 1035 1090 1200 50
150
100
200
0.002388 0.005035 0.0116 0.0142 0.024
0.00254 0.00532 0.0135 0.0171 0.026
We estimate that
ZZ
Figure 3. Mole fractions of the three ions (02-,Fez+, and Fe3+) in the wiistite phase as functions of z2 h m ) at 1000 s.
0.026 kg --
&
I
200
400
600
800
1000
t
Figure 4. Position h(w*m) of the wiistite-magnetite interface (upper curve) and position h(m,h)of the magnetite-hematite interface (lower curve) in pm as functions of time t in seconds for iron exposed to 0 2 at 1 x lo6 Pa and 1200 "C.
wiistite phase interface)
io3 35.5
5.36
= 0.513
= 77.8 kg-mol/m3
(44)
Under these circumstances, we can solve (26),(351,(36), and (38).
A(wpm)= 0.600
(47)
m2s Further comparisons between their measurements and our predictions for a broad range of temperatures are shown in Table 1. Much of the difference between theory and experiment may be attributable to our rough estimates for the physical properties, particularly (40).
Conclusion The comparison between the calculated and observed values for the rate constant K shown in Table 1 was obtained without the use of adjustable parameters as needed by Himmel et al. (1953,p 840). In contrast with prior theories (Wagner, 1951, 1969; Smeltzer, 1987; Coates and Dalvi, 1970), neither do we assume dilute solutions in treating this problem of ternary diffusion, nor do we require any thermodynamic data other than the phase diagram.
Acknowledgment The authors appreciate the help given them by M. W. Vaughn in using Mathematica (Wolfram Research 1994).
Nomenclature c = total molar concentration cti)- molar
C, = -36.1
c, = -1.84 io3
(45)
By way of illustrating these results, Figure 3 shows the mole fractions of the three ions in the wiistite phase a t a particular time, 1000 s. Equations 23 and 31 permit us to plot h(w,m) and h(mlB) as functions of time t in Figure 4.
Davies et al. (1951)observe experimentally that for iron exposed to 02 a t 1 x lo5 Pa and 1200 "C
concentration of species i in phase j c ( ~ ~=~molar + ~ concentration , ~ of Fez+in wiistite at the iron-wiistite phase interface, introduced in (10) ciTizi), ,? = molar concentration of Fez+in wiistite at the wtis&te-magnetite phase interface, introduced in (11) ~ i F : 2 = ~ ~ molar , ~ concentration of Fez+in magnetite at the wustite-magnetite phase interface, introduced in (12) c $ & ~ , , = molar concentration of Fez+in magnetite at the magnetite-hematite phase interface, introduced in (13) Ci = constants of integration i = 1, 2 @i k,jk - binary diffusion coefficient for species i and j in p ase k = ternary diffusion coefficient for species i in phase k h(w,m) = position of the wtistite-magnetite phase interface h(m,h) = position of the magnetite-hematite phase interface Kexp= experimental value of rate constant introduced in
4;)) (47)
=-0.0241kg (46)
K = rate constant introduced in (48)
Z$g.$+,2 = z2 component of the molar flux of species Fez+in wiistite
3410 Ind. Eng. Chem. Res., Vol. 34, No. 10,1995
component of the speed of displacement of the wiistite-magnetite phase interface uLmsh)= z2 component of the speed of displacement of the magnetite-hematite phase interface u ; ( ~= ) molar averaged velocity in wiistite xo’) (i) - mole fraction of species i in phasej zi = rectangular Cartesian coordinates, i = 1, 2, 3 U ( w , m )= 2 2
Greek Letters
7 = defined by (19) p = defined by (28) L(w,m)= defined by (23)
L(m,h)= defined by (32) = density of phase k Superscripts
(w)= indicates wiistite phase (m) = indicates magnetite phase (h) = indicates hematite phase
Literature Cited Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; New York, John Wiley: 1960. Borg, R. J.; Dienes, G. J. An Introduction to Solid State D i m i o n ; New York, Academic Press: 1988. Chen, W. K.; Peterson, N. L. Effect of the deviation from stoichiometry on cation self-diffusion and isotope effect in wiistite, Fel-,O. J.Phys. Chem. Solids 1976,36,1097-1103. Coates, D.; Dalvi, A. An extention of the Wagner theory of alloy oxidation and sulfidation. Oxid. Met. 1970,2, 331-347. Davies, M. H.; Simnad, M. T.; Birchenall, C. E. On the mechanism and kinetics of the scaling of iron. J.Met. 1951, Oct, 889-896. Dean, J. A. Lunge’s Handbook of Chemistry, 12th ed.; McGrawHill: New York, 1979.
Hauffe, K. Oxidation of Metals; Plenum Press: New York, 1965. Himmel, L.; Mehl, R. F.; Birchenall, C. E. Self-diffision of iron oxides and the Wagner theory of oxidation. J.Met. 1963, June, 827-843. Slattery, J. C. Momentum, Energy, and Mass Transfer in Continua, 2nd ed.; Robert E. Krieger: Malabar, FL, 1981; 1st ed.; McGraw-Hill: New York, 1972. Smeltzer, W. Diffisional growth of multiphase scales and subscales on binary alloys. Mater. Sci. Eng. 1987, 87, 35-43. Steigman, J.; Shockly, J.; Nix, F. The self-diffision of copper. Phys. Rev. 1939,56, 13-21. Touloukian, Y. S. Recommended values of the thermophysical properties of eight alloys, mojor constituents and their oxides; Thermophysical Properties Research Center Purdue University: Lafayette, IN, 1966. Wagner, C. Diffusion and high temperature oxidation of metal. InAtom Movement; American Society for metals: 1951; pp 153171. Wagner, C. The distribution of cations in metal oxide and metal sulphide solid solutions formed during the oxidation of alloys. Corrosion Sci. 1969, 9, 91-109. Weast, R. C. CRC Handbook of Chemistry and Physics, 63rd ed.; CRC Press: Boca Raton, FL, 1982. Wolfram Research, I. Mathematica, Version 2.2 ed.; Wolfram Research Inc.: Champaign, IL, 1994.
Received for review January 4, 1995 Revised manuscript received April 21, 1995 Accepted May 17, 1995” I39500174
Abstract published in Advance ACS Abstracts, August 15, 1995.
