ANALYTICAL MODELING OF CARBON TRANSPORT

ROCESSES IN HEAT TREATMENT TECHNOLOGY OF STEELS

Jürgen Gegner
SKF GmbH, Material Physics, Ernst-Sachs-Str. 5, D-97424 Schweinfurt, Germany

ABSTRACT

Thermochemical edge zone processes of steels induced by solid-state diffusion of car-

bon represent one of the economically and scientifically closest points of contact of
materials science to industrial application: carburization is the main step of case
hardening, whereas out-diffusion of carbon, though sporadically used (e.g. de-
carburization of cast iron), is primarily relevant to heat treatment and hot-working
processes, like austenitizing or forging, as undesirable side effect that impairs the
mechanical properties of the rim region. In the present paper, the technological back-
ground of both thermodynamically inverse processes is explained. Realistic modeling
of the carbon transport, which can be controlled by diffusion and/or surface reactions,
for profile prediction is important to reliable carburization control and failure analysis
of decarburized parts or defective plants. Suitable analytical solutions of Fick's law
are summarized and a realistic computer model for the consideration of simultaneous
carbide dissolution during decarburization is developed. The effect of influencing
parameters (e.g. carbon potential, mass transfer coefficient, carbon solubility in
austenite) on the depth profiles is discussed in detail. Selected concrete examples of
multi-step carburization and decarburization processes are simulated and compared
with experimental data.

INTRODUCTION

As the first operation step of case hardening, carburization of low-alloyed steels, i.e.
carbon enrichment in the edge zone to typically 0.6 (maximum martensite hardness)
to 0.85 m.% C, is one of the oldest industrially utilized surface refinement processes,
which is based on controlled inward mass transport by solid state diffusion. This
widely used technique permits the production of mechanically and tribologically high-
ly loadable components with hard rim (58 to 67 HRC) and tough core (30 to 50 HRC).
The costs are considerably lower, if compared, for instance, with carbonitriding, nitro-
carburizing or nitriding. Core hardenability (thick components) can be improved by
alloying elements (e.g. Cr, Mo). With a market share of today more than 30 percent,
case hardening of low-carbon grades (0.07 to 0.3 m.% C) is the most important heat
treatment procedure of steels since almost 60 years. It is preferentially applied to mo-
tor, gear, machine or jet engine parts that require high fatigue, wear and shock resis-
tance, like cogwheels, journals, bolts, shafts (e.g. camshafts), and bearings (e.g. wheel
bearings).

In certain applications like forgeability improvement of tempered casting, the inverse

thermochemical process of case decarburization is also used in engineering technol-
ogy. However, this out-diffusion of carbon caused by surface oxidation reactions,
possibly accompanied by scaling and/or internal oxidation, is more relevant as unde-
sirable side effect to heat treatment and hot-working processes particularly of through
hardenable steels, e.g. austenitizing, soft annealing, forging or upsetting. Since the
martensite start temperature increases locally with the decreasing carbon concentra-

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tion, transformation-induced tensile residual stresses are formed in the affected edge
zone. Apart from the resulting higher crack sensitivity and lower fatigue strength, also
the hardness near the surface and thus the wear resistance is reduced.

Although the essential neglect of the composition dependence of the carbon dif-
fusivity in austenite leads to characteristic deviations from real concentration profile
shapes, mathematical modeling with analytical process simulation permits prediction
of the main operation quantities with the same accuracy as numerical methods (see
e.g. Fig. 4). These parameters are the surface carbon content and the carburization
(generally ranging from 0.05 to more than 10 mm, i.e. few hours to over one week) or
decarburization depth that are defined as surface distance at 0.35 m.% C and 0.92c0,
respectively. Here, c0 denotes the initial carbon concentration. Analytical methods,
which are most suitable for the evaluation of the fundamental effect of (changing)
control parameters on process flow, have often proved superior to numerical tech-
niques for the purpose of solving diffusion problems in metallurgy (1). In the present
paper, appropriate mean values of the carbon diffusivity in austenite between limiting
concentrations c1 and c2 are calculated as follows (2):

1 c2
c 2  c1 c1
Dγ  D γ (c C ) d c C (1)

Data derived from several experimental investigations into the binary Fe–C system is
inserted (3). cC denotes the carbon concentration. For the sake of simplicity, D is used
instead of Dγ in the following.

GAS CARBURIZATION OF LOW-ALLOYED STEELS

Today's state of the art is the well-established two-step gas carburization process at
customary temperatures between 1120 and 1250 K with continuous control by the C
potential (or C level) cp. It is defined as the carbon content of a pure iron sample (e.g.
thin foil) in equilibrium with the furnace atmosphere and usually expressed in m.%.

Analytical process simulation

The main carburizing reaction in industrially applied, so-called fast mixtures of carrier
and enrichment gas containing CO and H2 is the heterogeneous water gas equilibrium,
COH2CH2O. C stands for carbon dissolved in steel. The C level can be derived
from thermodynamic data by applying the activity-concentration relation (4, 5):

cp cp p CO  p H 2 4800
log  0.15  log   5.286 (2)
m.% m.% p H 2O T /K

T and p denote the temperature and the partial pressure of the gas components, re-
spectively. The influence of the steel composition on the carbon activity-concentration
relation is taken into account by the dimensionless alloy factor ka that typically ranges
from 0.9 to 1.1:

c p(corr)  c p  k a (3)

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 (corr)
The corrected C level c p corresponds to the desired surface content and is usually
also called cp. The mass flux density jC, i.e. the carbon amount transferred from gas to
steel per unit time and area, can be derived as reaction rate from the kinetics law (6):

 as  p  pH2
jC  k
pCO
pH O
1    k  CO
 a 
 a p  as   β c p  cs 
  p (4)
1 KO 2 p H 2 O

pH 2

Here, KO denotes the adsorption constant of oxygen at the surface. Due to the low
carbon content of steels, activities (ap, as) are replaced with corresponding concentra-
tions. The rate constant and its modified value are referred to as k and k′, respectively.
The surface carbon concentration cs varies with time t. For constant temperature, the
mass transfer coefficient  is thus a function of partial pressures and, as the small H2O
content does not change significantly, approximately proportional to the CO-H2 prod-
uct. Based on the predominating heterogeneous water gas equilibrium, effective con-
trol parameters cp and  can be derived that additionally consider the slower carbon
releasing surface processes of the Boudouard reaction and the methane decomposi-
tion. The solution of Fick's law for constant initial carbon concentration c0 under the
third-kind boundary condition of Newton's reaction law in Eq. (4) describes the boost
period of gas carburization (2):

 x  βx  β 2 t   x t 
c C  c 0  (c p  c0 ) erfc  exp  erfc β  (5)
 2 Dγ t  D  2 D t D 
  γ   γ γ


Here, x denotes the distance from surface and erfc is referred to as the error-function
complement. The one-dimensional approach holds in almost all cases of practical in-
terest. Figure 1 shows resulting carbon profiles for cp1.2 m.%, c00.2 m.%, t12 h
and several  values, which
typically range from 1105 to
4105 cm/s in industrially ap-
plied atmospheres. The con-
centration line cC0.35 m.%
corresponding to the carbu-
rization depth dc is drawn in.
The inset reveals the develop-
ment of the surface carbon
content cs with time and thus
illustrates the accepted ex-
pressions fast or slow gas
mixture. Figure 2 demon-
strates the influence of the
steel composition on the
Figure 1. Carburization profiles according to Eq. (5). profile and the carburization
depth. This diagram clarifies
that the C level must be chosen appropriately with respect to the desired cs value.
According to Eq. (3), the alloying elements affect the mass flux and thus the carbu-
rization profile without impact on the diffusivity.

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In order to smooth
the sharp near-surface
concentration
gradient (cf. Figs. 1,
2), subsequent to the
boost period of
duration tb a shorter
diffusion anneal (time
td: 10 to 25 % of tb)
with reduced carbon
b
level from c p (0.8 to
d
1.2 m.% C) to c p
(0.6 to 0.85 m.% C)
is performed (two-
step procedure) by Figure 2. Effect of the alloy factor ka on the carburization profile (ka1: Fe–C).
changing the gas
composition in the
furnace. Usually, the mass transfer coefficient  remains almost constant and an iso-
thermal process is applied. Solving Fick's law for c0const. and ttb then yields (7):

 x  βx  β 2 t   x t 
c C  c 0  (c pb  c 0 ) erfc  exp  erfc β
 2 Dγ t  D  2 D t Dγ 
  γ   γ 
 x  βx  β 2 (t  t b ) 
 (c pb  c pd ) erfc  exp  (6)
 2 Dγ (t  t b )  D 
 γ 
 x (t  t b ) 
 erfc β 
 2 D (t  t ) Dγ 
 γ b 

Application of protective atmosphere or vacuum (uncommon) instead of C potential

control
during
the
diffuse
stage in
order to
avoid re-

Figure 3. Analysis of a two-step boost-diffuse gas carburization process.

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decarburization (boundary condition of the second kind: jC0 at x0, employed e.g.
for doping of semiconductors) would result in significantly longer equalization times.
Figure 3 reveals the carbon profile of a typical two-step gas carburization process (
c pb 1.2 m.%, c pd 0.7 m.%, tb10 h, td2 h; c00.15 m.%) calculated in accordance
with Eq. (6), where ttbtd12 h is inserted. The cs-t curve is shown in the inset. As
can be seen, proper process control leads to almost uniform concentration (cs0.05 m.
%) in the edge zone after the diffusion period up to a surface distance of around one
third of the carburization depth as precondition of optimal mechanical properties. For
comparison, both corresponding one-step boost operations are also computed
according to Eq. (5) with tb10 h and tb12 h, respectively. Whereas the profile shape
in the near-surface region is significantly changed by the diffusion anneal that is
obviously associated with some carbon loss (re-decarburization), no noticeable effect
on the carburization depth dc occurs, which is thus, under these customary conditions,
controlled by the total process time t (here 12 h) alone. This result also explains why a
parabolic kinetics law, dct, holds for common two-step boost-diffuse gas
carburization treatments.

Figure 4. Comparison between analytical and numerical solution.

Experimental verification

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Since industry demands for high quality procedures, target quantities must be
observed accurately. Consequently, up-to-date process simulation should be able to
particularly predict the carburization depth as main desired parameter within narrow
scatter bands of about 0.1 mm even for large dc values. For a high temperature gas
carburization (T 1233 K) experiment performed at 1243 K in a modern bulk
 

production plant, Fig. 4 shows the carbon profile calculated numerically (FDM: finite
difference method) by the furnace control software. The mass transfer coefficient and
the C level, measured in both process steps boost (tb56.6 h) and diffuse period
b
(td13.2 h) by means of the foil method, reached the following values: c p 1.20 m.%,
c pd 0.80 m.%, 3.010–5 cm/s (bd). Figure 4 also presents the analytical
solution according to Eq. (6). The transition from the boost to the diffuse period
occurred within less than 5 min by controlled air admission to the atmosphere and can
thus be neglected. Note that the diffusivity expression used by the FDM program is
not known. The calculated carburization depth, as well as the surface content, agrees
well with the prediction of the furnace control unit. The deviations from the typical S
profile shape (see also Fig. 6) stem from the neglect of the composition dependence of
the carbon diffusivity in the austenite phase, which increases with concentration cC.

Figure 5 shows the result of the microchemical cross-section analysis by a novel high-
accuracy SIMS technique (secondary ion mass spectrometry). The measured carbon

Figure 6. Analytical fits with respect to carburization depth.

Figure 5. SIMS measurement of the carburization profile.
profile deviates significantly from the prediction of the furnace control software.

The actual carburization depth of 5.48 mm is around 10 % higher than the desired
value, d cdes 5.02 mm, among other things resulting in application and quality
problems (overstepping of specification). For economical aspects, the corresponding
marked exceedance of the required process time of about 20 % is also unacceptable.
One obvious cause of this huge discrepancy could be found in the used diffusion
coefficient D. For instance, the influence of alloying elements on the carbon
diffusivity in the austenite phase of the applied steel is not considered. Figure 6 pres-

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 des SIMS
ents analytical fits according to Eq. (6): the D values yielding d c and d c
deviate by around 20 %.

DECARBURIZATION OF LOW-ALLOYED STEELS

In reactive atmospheres, e.g. contaminated protective gas or (wet) air, the following
decarburizing processes may occur: 2CO22CO, CO2CO2, CCO22CO
(Boudouard reaction), and CH2OCOH2 (heterogeneous water gas equilibrium).
The resulting carbon removal from the surface generates the concentration (activity)
gradient as driving force of (outward) diffusion. Mathematical examination and pre-
diction of decarburization profiles is of special importance to quality assurance (e.g.
grinding allowance) of heat treatment processes, like austenitizing, and failure analy-
ses of components and industrial plants.

Figure 7. Phase diagram of bearing steel 100Cr6.

Decarburization of through hardenable bearing steel

In order to estimate potential edge zone damage during austenitizing (typical

temperatures between 1100 and 1170 K for around 30 min, high-speed steels up to
1500 K for only few minutes) of standard bearing steel 100Cr6 (SAE 52100,
measured initial carbon content: c01.02 m.% C), two isothermal experiments at 1163
K are performed. The course of decarburization in the phase diagram (intersection
through the ternary Fe–Cr–C system at 1.5 m.% Cr) is shown in Fig. 7 (8).

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 The first experiment describes the
worst case, i.e. annealing in air for
30 min. Figure 8 represents the
result of the SIMS analysis that
yields a decarburization depth,
ddx(cC0.94 m.% C), of 350 µm.
The inset reveals the decarburized
microstructure with scaling. As
the atmosphere offers oxygen in
excess, it can be assumed that the
surface carbon concentration cs re-
mains constant at 0 m.% during
the whole anneal (9). Under the
simple initial and boundary con-
Figure 8. Decarburization in ambient air. ditions c0const. and csconst., the
depth profile is given by the Van
Ostrand-Dewey equation. Figure
8 confirms that the decarburization induced austenite-ferrite () phase trans-
formation in the edge zone (cf. Fig. 7), which leads to the formation of a penetrating
 interlayer and thus to the development of a near-surface inflexion point in the
measured concentration-distance curve as for the diffusivities D100D(cC0) is
valid, can be considered by shifting the origin in order to model the inward moving
outset of the  region (10):

xξ
c C  cs  (c 0  cs )  erf (7)
2 Dγ t

No blocking effect of the porous, poorly adhering scale (see Fig. 8, inset) on carbon
out-diffusion is observed.

Figure 9. Controlled admission of air for 10 min.

In the second experiment, after 20 min annealing in protective gas, dosed increasing
admission of air occurs in order to simulate furnace leakage. Figure 9 demonstrates

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that the highly accurate SIMS technique reveals the preserved carbide segregation,
which stem from steel production, in a depth of around 60 µm (see microstructure,
arrow). The decarburization depth reaches 100 µm. No appreciable scaling is
observed. As shown in the inset of Fig. 9, the controlled air contamination of the
neutral gas atmosphere for 10 min is realistically modeled by linearly decreasing
surface concentration:

c s  c 0  mt (8)

The positive slope parameter m reaches 6. 6 10–4 m.%/s. The solution of Fick's law
under the first-kind boundary condition of Eq. (8) for c0const. can be expressed as
follows (2):

 x 2 1 x x  x 2 
cC  c0  4mt    erfc  exp   (9)
 8 Dγ t 4  2 Dγ t 4 π Dγ t  4 D t 
 γ 

Figure 9 supports that the measured decarburization profile quantitatively agrees with
the result of this analytical simulation.

Computer model for decarburization with simultaneous carbide dissolution

M C
If the initial content c0 exceeds the solubility of carbon in austenite, c γ 3 , the char-
M C
acteristic phase diagram in Fig. 7 shows that decarburization, i.e. cs c γ 3 , is accom-
panied by the dissolution of M3C carbide particles (metal fraction M: Fe, Cr; general
stoichiometry MC, e.g. 3, 7/3). To consider this background process, which is
neglected in the computations presented in the previous section, an existing Fortran
code is upgraded that can also be used to calculate carburization or decarburization
profiles of complex multi-step processes and solves the plane-sheet diffusion problem
(here: thickness l2√Dt, i.e. no center effects) for an arbitrary, discretely defined
(evaluation points xi) initial concentration distribution fC(x) under the boundary condi-
tion csconst. (2):

4 cs 
1 (2n  1) π x  D t
cC  cs 
π
 2n  1 sin l
exp  ( 2n  1) 2 π 2 2γ 
l 
n0 
(10)
2  n πx  Dγ t  l n π x
  sin exp  n 2 π 2 2    f C ( x) sin d x
l n 1 l  l  0 l

In Eq. (10), n is a non-negative integer. All the following exemplarily simulations

refer to rolling bearing steel 100Cr6 at 1123 K with cs0.1 m.% C (no  trans-
formation, cf. Fig. 7). c Cl and c Ccarb denote the carbon content in the austenite lattice,
(tot)
where (volume) diffusion only occurs, and the carbides, respectively: c C  c Cl 
c Ccarb . Firstly, a homogeneous distribution of the M 3C particles is assumed. The initial
M C
values are taken from Fig. 7: c0l  c γ 3 0.70 m.% C, c 0carb 0.32 m.% C ( c0l  c 0carb
c01.02 m.% C). Carbide dissolution is described by a first-order kinetics law with
the positive rate constant k (11):

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 d cCcarb
  k  cCcarb (11)
dt
In the proposed extended computer-aided iterative process simulation, Eq. (11) is
computed after any time step t at each evaluation point (e.g., xi1xi1 µm). The
calculated amount,  c Ccarb (xi)k c Ccarb (xi)t, is then totally (if possible: c Ccarb 0,
c Cl  c 0l ) or partly (maximum release from M C carbon store) added to c Cl (xi) and
correspondingly (mass conservation) subtracted from c Ccarb (xi). Note that therefore in
this material model, between two diffusion steps of same duration t in the  lattice
that are respectively evaluated according to Eq. (10) with c Cl fC, the described local
carbon redistribution from
the carbides (source) to the
austenite matrix phase (sink)
occurs up to their complete
dissolution, i.e. c Ccarb (xi)0,
changing the concentration
l
profile c C (x) continuously. Figure 10. Carbide dissolution during decarburization.

Figure 11. Decarburization with fast carbide dissolution.

Figure 10 represents the microstructure of decarburized through hardened steel

90MnCrV8 after faulty austenitizing. The sharp boundary of carbide dissolution that
has been formed, points to a large k value in Eq. (11).

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This metallographic finding is taken into account in the first simulation shown in Fig.
11: it illustrates the effect of including carbide dissolution in the decarburization
model even if diffusion remains the rate-controlling reaction step. For comparison, the
one-phase calculation for pure austenitic material () according to the Van Ostrand-
Dewey equation, cCcs(c0cs)erfx/(2Dt), is drawn in. Applying the same process
time of t1 h, Fig. 12 demonstrates the other limiting case, k0. This supposition cor-
responds to the unrealistic assumption of stable carbides during decarburization of the
 matrix. An intermediate value of the control parameter k is considered in the simu-
lation of Fig. 13. The gradual car-
bide dissolution leads to a broad
inward moving transition region.

Figure 12. Decarburization with stable carbides.

In Fig. 13, the

resulting total
carbon
(tot)
profile, c C
l carb
 cC  cC ,
is
exemplarily
drawn in.
From these
c C(tot) –x
curves, the
de-
carburization
Figure 13. Decarburization with medium carbide dissolution rate. depth dd can
be
determined.

Figure 14 reveals the time development of dd as the result of this evaluation. For
comparison, the one-phase computation is also included. Validity of the parabolic
kinetics law for the decarburization depth, ddt, points to a diffusion controlled
process.

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 Figure 14. Influence of carbide dissolution rate on decarburization depth.

According to Fig. 14, consideration of simultaneous carbide dissolution yields lower

dd values that decrease further with increasing rate constant k. Note that the effect on
the mechanical properties, e.g. hardness loss in the edge zone, mainly depends on the
decarburization level of the austenite matrix that can be characterized, for instance, by
the surface distance x( c Cl 0.92 c0l ).

In the above computations (cf. Figs. 11 to 14), a uniform initial c Ccarb profile is as-
sumed, i.e. c Ccarb (x0,t0) c 0carb const. However, with the presented simulation tool,
also real heterogeneous carbide distributions (e.g. segregations) within the austenitic
matrix can be included in the extended decarburization model. An example is given in
Fig. 15. The calculation involves fast carbide dissolution according to Eq. (11), i.e.
k. In order to emphasize the effect on the decarburization depth-time curve, dd(t),
the longer process duration of t10 h is considered.

The original one-dimensional distance distribution of the carbides, taken from a light-
optical micrograph of rolling bearing steel 100Cr6 by digital image analysis, covers a
range of 0.530 mm. As usual for engineering materials, the detected arrangement
reveals pronounced accumulation and depletion regions around the average value. It is
recurrently repeated (period p) to provide the complete initial carbide distribution. At
1123 K, c 0carb amounts to 0.32 m.% C.

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 Figure 16 finally
presents the
resultant
development of
the
decarburization
depth with time.
The drawn
comparison with
the curve
describing the
corresponding
uniform particle
arrangement,
Figure 15. Decarburization with real initial carbide distribution. c Ccarb (t0)0.32
m.% C,
evidently illustrates the influence of the considered heterogeneous carbide distribution
on the program output.

Figure 16. Effect of heterogeneous carbide distribution on decarburization depth.

SUMMARY AND CONCLUSIONS

Analytical simulations of industrially highly relevant carburization and

decarburization processes of low-alloyed steels are presented and discussed in detail.
Comparisons with experimental data confirm their applicability.

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Consideration of steel dependent carbon diffusivities is particularly recommended to
optimize the process control of state-of-the-art two-step boost-diffuse gas carbu-
rization heat treatments. The new SIMS technique (secondary ion mass spectrometry)
for measuring carbon profiles with high accuracy should be used.

The presented computer model of decarburization with simultaneous carbide dissol-

ution illustrates the potential of analytical simulation methods, as realization of this
complex process scheme with less flexible standard numerical software systems is dif-
ficult.

ACKNOWLEDGEMENT

The SIMS measurements were performed in cooperation with the Institute of Material
Physics, University of Göttingen (Germany). The author is grateful to Dr. Peter J.
Wilbrandt and Prof. Dr. Reiner Kirchheim for support and helpful discussions.

REFERENCES

1. Fredman T P: 'An Analytical Solution Method for Composite Layer Diffusion

Problems with an Application in Metallurgy'. Heat and Mass Transfer 2003 39 (4)
285-95.
2. Crank J: The Mathematics of Diffusion. Oxford, Clarendon Press, 1990.
3. Ågren J: 'A Revised Expression for the Diffusivity of Carbon in Binary Fe–C Aus-
tenite'. Scripta Met. 1986 20 (11) 1507-10.
4. Neumann F, Person B: 'Beitrag zur Metallurgie der Gasaufkohlung  Zusammen-
hang zwischen dem Kohlenstoffpotential der Gasphase und des Werkstückes unter
Berücksichtigung der Legierungselemente'. HTM 1968 23 (4) 296-310.
5. Brünner P, Weissohn K K: 'Computer Simulation of Carburization and Nitriding
Processes'. Mater. Sci. Forum 1994 163-165 699-706.
6. Grabke H J: 'Aufkohlung, Carbidbildung und Metal Dusting'. HTM 2002 57 (1)
11-16.
7. Pavlossoglou J: Mathematical Modelling, Computer Simulation and Optimisation
of Multi-Stage Gas Carburising Processes. Thesis, University of Birmingham,
1975.
8. Bungardt K, Kunze E, Horn E: 'Untersuchungen über den Aufbau des Systems Ei-
sen-Chrom-Kohlenstoff'. Arch. Eisenhüttenwes. 1958 29 (3) 193-203.
9. Birks N, Meier G H: Introduction to High Temperature Oxidation of Metals.
London, Edward Arnold Publishers Ltd., 1983.
10. Adamaszek K, Bro P, Kučera J: 'Decarburization and Hardness Changes in Carbon
Steels Caused by High-Temperature Surface Oxidation in Ambient Air'. Defect
and Diffusion Forum 2001 194-199 1701-06.
11. Esser F, Eckstein H-J: Über die Kinetik der Auflösungsvorgänge von Karbiden bei
der Wärmebehandlung von unlegierten Kohlenstoffstählen  Beitrag zur mathe-
matischen Beschreibung der Karbidauflösung und Entwicklung eines Systemmo-
dells, das für einen Computer programmiert wird. Leipzig, VEB Deutscher Verlag
für Grundstoffindustrie, Freiberger Forschungshefte, 1971.

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