Estimation and Modeling of Parameters for Direct Reduction
in Iron Ore/Coal Composites: Part I. Physical Parameters
E. DONSKOI and D.L.S. McELWAIN
This article critically assesses literature estimates of the major physical parameters associated with
direct reduction in iron ore/coal composites, including heats of reactions, specific heats, composition,
porosity, density, shrinkage, swelling, and thermal conductivity. Where estimates are not available,
new formulae are given. In particular, this article focuses on the temperature dependence of certain
parameters, since this is required for the development of mathematical models. The article also
highlights areas of investigation, where the parameter estimates are in need of further experimental
research or where different authors provide conflicting estimates. The results are of more general
applicability than to coal-based direct reduction of iron ore and will be useful for researchers investigating any form of direct reduction of iron ore.
I. INTRODUCTION
Wustite to iron:
DIRECT-REDUCED iron is used as feedstock in iron
and steelmaking processes, including electric-arc and blast
furnaces. While the majority of the worldwide production
uses natural gas as the reductant, there are a significant
number of low-reaction-temperature ironmaking processes
in which iron ore/coal agglomerates are used to produce
iron using the carbon in the coal and the volatiles which
evolve from the coal as the reductant. An advantage over
the traditional blast-furnace ironmaking technology is that
these processes do not require the production of coke, which
is expensive and during which the environmentally harmful
off-gases are hard to contain.
In the direct reduction in iron ore/coal composites (DRIOCC), a mixture consisting of fines of iron-bearing oxide
and carbonaceous material (coal, coke, and char) is heated
to a temperature below the melting temperature of any of
the materials involved. If the process requires the formation
of pellets, a small amount of binder is also used. Volatile
matter from the coal, carbon monoxide from the Boudouard
reaction, and hydrogen and carbon monoxide from the watergas reaction (as detailed subsequently) react with the iron
oxide and reduce the iron oxide to iron.
The main reactions for the coal-based reduction of hematite can be summarized by the following scheme:
Hematite to magnetite:
3Fe2O3 ⫹ CO ⫽ 2Fe3O4 ⫹ CO2
[1]
3Fe2O3 ⫹ H2 ⫽ 2Fe3O4 ⫹ H2O
[2]
Magnetite to wustite:
1.202Fe3O4 ⫹ CO ⫽ 3.807Fe0.947O ⫹ CO2
[3]
1.202Fe3O4 ⫹ H2 ⫽ 3.807Fe0.947O ⫹ H2O
[4]
E. DONSKOI, Postdoctoral Fellow, and D.L.S. McELWAIN, Professor,
are with the Centre in Statistical Science and Industrial Mathematics, School
of Mathematical Sciences, Queensland University of Technology, Brisbane,
Queensland, 4007, Australia. Contact e-mail: s.mcelwain@qut.edu.au
Manuscript submitted May 7, 2001.
METALLURGICAL AND MATERIALS TRANSACTIONS B
Fe0.947O ⫹ CO ⫽ 0.947Fe ⫹ CO2
[5]
Fe0.947O ⫹ H2 ⫽ 0.947Fe ⫹ H2O
[6]
Carbon gasification:
Boudouard reaction: C ⫹ CO2 ⫽ 2CO
[7]
Water gas reaction: C ⫹ H2O ⫽ CO ⫹ H2
[8]
Coal devolatilization:
Coal to carbon: Coal → C ⫹ volatile matter
[9]
Often, instead of Eqs. [3] through [6], the following equations are used:
Magnetite to wustite:
Fe3O4 ⫹ CO ⫽ 3FeO ⫹ CO2
[10]
Fe3O4 ⫹ H2 ⫽ 3FeO ⫹ H2O
[11]
Wustite to iron:
[12]
FeO ⫹ CO ⫽ Fe ⫹ CO2
FeO ⫹ H2 ⫽ Fe ⫹ H2O
[13]
since it is thought that the stoichiometric difference between
FeO and Fe0.947O is small and, so, the difference between
the coefficients is also small. However, in reaction [3], for
example, 1 mole of CO produces 3.807 moles of wustite
instead of the 3 moles predicted by Eq. [10], and this can
be quite significant. In section II, we discuss what difference
this makes for estimating the heats of reaction.
The DRIOCC is a very complex process which includes
nonuniform heating and mass transfer and, sometimes, complex geometry. It may, for example, involve two or more
layers of pellets and nonisothermal heterogeneous reactions
in a porous medium.
One of the most critical requirements for the mathematical
modeling of DRIOCC is the estimation of important parameters. As mentioned previously, the process is nonisothermal
and all parameters are temperature- and, therefore, timedependent functions. Estimated values of the same parameters from different published sources can be quite different.
VOLUME 34B, FEBRUARY 2003—93
This article systematizes and compares information from
different sources to establish new formulae for the important
parameters and to develop some new approaches to the
estimation and modeling of parameters for DRIOCC. For
parameters where the published literature appears to be quite
reliable, references only are given to avoid unnecessary
repetition.
II. ESTIMATION OF HEATS OF REACTION
There are many articles that give the heats of reaction for
coal-based direct reduction of iron ore at room temperature
(298.15 K), but a literature survey has shown that only one
of them (Yagi and Szekely[1]) gives formulae to calculate
the heats of reaction for reactions [2], [4], and [6] as functions
of temperature, and these are different from those which
have been established in the present article (Figure 1).
Another problem is that the heats of reaction given for
room temperature, derived from different sources, are not
in agreement. To estimate the heat of the reaction at room
temperature, the sum of the heats of formation of the
reactants has to be subtracted from the sum of the heats of
formation of the products. Table I gives heats of formation
from different sources and Table II gives the corresponding
heats of reaction at room temperature (298.15 K). Perry[3]
and Lide[6] do not have data for Fe0.947O, so, for the calculation of the heats of reaction in this article, Reactions [10]
through [13] have been used.
The results obtained from Perry[3] and Lide[6] are very
different from the others, not only for reactions [3] and [4]
(or, correspondingly, Reactions [10] and [11]), but also for
some other reactions. This makes the estimates for the heats
of formation from these two sources doubtful. Even though
Ross[7] gives an estimate of the heat of formation of Fe0.947O,
that investigation uses expressions [10] and [11]. The estimated heat of reaction for Reaction [10] is 36,250 J/mol
and that for Reaction [11] is 77,404 J/mol, whereas Reactions
[3] and [4] give estimates of 47,059 and 88,221 J/mol,
respectively.
Temperature-dependent heats of reaction have been calculated using two methods. The first uses tables of heats of
formation for different temperatures and applies the same
algorithm as for room temperature, namely, taking the sum
of the heats of formation of the reactants from the sum
of the heats of formation of the products, and then uses
interpolation. The second method calculates the heats of
reaction at room temperature, adds the change in heat content
of the products (Ht ⫺ H298), and subtracts the change in heat
content of the reactants. The change of heat content can be
taken from the tables, but it can also be calculated as an
integral of the specific heat, and, here, this method is used.
This approach was adopted for two reasons: to check the
concurrence of heats of reaction obtained by two different
methods to local changes in the functions of heats of reaction
(an approximation through points can miss some irregularities) and to check the reliability of the expressions for the
specific heats. Initially, functions for the specific heats given
by Kubaschewski et al.[8] were used, but these appear to be
inaccurate, and, where it was significant, improved models
have been developed (discussed in section III).
To decide on which data source should be used, an average
of the heats of reaction from different sources was obtained,
and the sum of squares of the differences between the average
94—VOLUME 34B, FEBRUARY 2003
(a)
(b)
(c)
Fig. 1—Heats of reaction for Reactions (a) [2], (b) 4, and (c) 6 obtained
from different sources. The solid line shows data from Yagi and Szekely,[1]
the dashed line denotes interpolated data from the JANAF tables,[5] and
the points denote data from Barin.[4]
heats of reaction and the heats of reaction from various
sources was computed. The smallest deviation is obtained
for the estimates by Barin,[4] and the second-lowest is
obtained for the JANAF tables.[5] Based on this and also
because the best agreement between the two different methods of estimating the temperature-dependent heats of reaction is obtained from Barin, all formulae for temperaturedependent heats of reaction given in this article are based
on data from Barin. The estimates of specific heats in the
METALLURGICAL AND MATERIALS TRANSACTIONS B
Table I. Heats of Formation for Compounds Involved in DRIOCC from Different Sources (kJ/mol)
Source
Fe2O3
Fe3O4
FeO
CO
CO2
H2O
Knacke[2]
Perry[3]
Barin[4]
JANAF[5]
Lide[6]
Ross[7]
⫺823.411
⫺830.524
⫺824.248
⫺825.503
⫺824.2
⫺820.901
⫺1115.479
⫺1116.71
⫺1118.383
⫺1120.894
⫺1118.4
⫺1116.291
⫺265.955
⫺270.37
⫺266.270
⫺266.270
⫺272.0
⫺265.684
⫺110.528
⫺110.525
⫺110.541
⫺110.527
⫺110.53
⫺110.525
⫺393.521
⫺393.514
⫺393.505
⫺393.522
⫺393.51
⫺393.514
⫺241.856
⫺241.826
⫺241.826
⫺241.826
⫺241.844
⫺241.827
Table II. Heats of Reaction for Reactions [1] through [8] at 298.15 K Obtained from Different Sources (kJ/mol)*
Source
Reaction
Number
2
3
4
5
6
7
1
2
3
4
5
6
7
8
⫺43.718
⫺2.581
45.047
86.184
⫺17.038
24.099
172.465
131.328
⫺24.837
16.326
22.611
63.774
⫺12.619
28.544
172.464
131.301
⫺46.986
⫺5.848
47.367
88.505
⫺16.694
24.444
172.423
131.285
⫺48.247
⫺7.105
50.354
91.523
⫺16.725
24.444
172.468
131.299
⫺47.180
⫺6.044
19.420
60.556
⫺10.980
30.156
172.450
131.314
⫺52.868
⫺11.706
47.059
88.221
⫺17.305
23.857
172.464
131.302
*For reactions (1), (3), (5) the heat of reaction is given per 1 mole of CO, for reactions (2), (4) and (6) it is given per 1 mole of H2
and for reactions (7) and (8) it is given per 1 mole of carbon.
Barin tables and the JANAF tables are almost the same. The
JANAF tables’ results for the heats of reaction for Reactions
[5] through [8] are very close to the estimates from Barin
(for example, Figure 1(c) or Table II), and, for Reactions
[1] through [4], the results would be very similar if the
heats of formation at room temperature were the same. This
implies that formulae for the temperature-dependent heats
of reaction based on the JANAF tables can be obtained by
using the formulae given in the present article, provided a
constant shift is applied. This shift can be obtained from
Table II.
Figure 2 shows graphs of the heats of reaction obtained
by both methods, and Tables III and IV give coefficients to
calculate these heats of reaction using the formula
Hr ⫽ A ⫹ B103T ⫺1 ⫹ C10⫺3T ⫹ D10⫺6T 2
⫹ E10⫺9T 3 kJ/mol
[14]
III. ESTIMATION OF SPECIFIC HEATS
A common method of estimating the specific heat is to
use the approach and data given in Kubaschewski et al.[8]
The discrepancy between the different approaches to the
estimation of the heats of reaction may be due to inaccurate
estimates of specific heats. For example, Figure 3 shows the
discrepancy between the two approaches for Reaction [5]
if, for the specific heat of iron, we take expressions from
Kubaschewski et al. For other compounds in this example,
expressions for the specific heat are taken from the work
described in the present article. The aim is to show only the
effect of not estimating the specific heat of iron accurately
enough. It has been found that the expressions for the specific
heats for other compounds should be corrected also.
In this example, the cause of the discrepancy between
the two different approaches to estimating the temperaturedependent heats of reaction is the inaccurate approximation
METALLURGICAL AND MATERIALS TRANSACTIONS B
of the specific heat of iron in the temperature range from
1000 to 1060 K, where it has a sharp peak at about 1042
K. To improve the approximation of the specific heat for
iron, the results from Ho[9] have been used, where data for the
specific heat of iron from 17 different sources were averaged.
In Tables V and VI, the coefficients to calculate the specific heats of compounds involved in DRIOCC from room
temperature to 1650 K are given. The approximations are
based on data from Barin,[4] and, as stated earlier, these data
are very close to the data from the JANAF tables.[5] These
data, for the specific heat of hematite, are only significantly
different in the temperature range of 950 to 1000 K. Comparison of the heats of formation obtained from Barin and the
JANAF tables and by integration of the specific heat show
that data from Barin are consistent, whereas in the JANAF
tables, there is an unexplained rapid increase. Data reported
in Coughlin et al.[10] also support the estimate of the specific
heat of hematite from 950 to 1000 K, obtained by Barin. In
the present investigation, the specific heats are approximated
using the formula
Cp ⫽ A ⫹ B106T ⫺2 ⫹ C10⫺3T ⫹ D10⫺6T 2
⫹ E10⫺9T 3 J/mol/K
[15]
and the coefficients are given in Tables V and VI.
To estimate the specific heat of coal and its solid conversion products, Donskoi and McElwain[11,12] use the method
proposed by Kirov.[13] Kirov estimates the specific heat of
coal or coal char as a sum of the mass-fraction-weighted
specific heats of the major coal components, that is,
Cp ⫽ Xv1Cv1 ⫹ Xv2Cv2 ⫹ XaCa ⫹ XwCw ⫹ XcCc [16]
where Xv1, Xv2, Xa, Xw, and Xc and Cv1, Cv2, Ca, Cw, and Cc
are the mass fractions and specific heats of the primary
volatile matter (type 1), which is evolved at relatively low
temperatures, secondary volatile matter (type 2), which is
VOLUME 34B, FEBRUARY 2003—95
(a)
(b)
(c)
(d )
(e)
(f)
( g)
(h)
Fig. 2—(a) through (h) Heats of reaction for Reactions [1] through [8] based on data from Barin in kJ/mol. The solid line shows estimates obtained by
way with using integration of specific heats (text). The points are obtained by using tables of heats of formation.
96—VOLUME 34B, FEBRUARY 2003
METALLURGICAL AND MATERIALS TRANSACTIONS B
Table III. Heats of Reaction for Reactions [1] through [4] (kJ/mol)*
Reaction
1
2
3
4
T (K)
298
600
850
970
1050
1300
298
600
850
975
1045
1300
298
850
298
850
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
600
850
970
1050
1300
1600
600
850
975
1045
1300
1600
850
1600
850
1600
A
B
C
D
E
⫺47.17
⫺213.54
⫺202.972
—
⫺104.746
⫺21.224
⫺5.470
⫺165.007
436.0
—
144.471
18.432
53.434
165.113
94.07
212.02
—
—
—
—
—
—
—
—
⫺176.913
—
⫺77.322
—
—
—
—
—
—
793.083
408.893
243.201
116.966
—
—
767.912
⫺265.294
299.536
⫺72.869
—
⫺46.41
⫺324.134
⫺38.045
⫺338.358
—
⫺1265.5
⫺247.092
⫺584.769
⫺52.622
⫺21.994
⫺4.824
⫺1244.6
—
⫺586.123
—
⫺29.13
127.212
236.199
101.311
238.31
7.251
683.491
—
301.851
—
7.676
2.112
673.835
—
281.633
—
9.615
⫺127.936
⫺54.003
⫺116.084
⫺54.057
*For Reactions [1] and [3], these values are per 1 mole of CO; for Reactions [2] and [4], the values are per 1 mole of H2.
Table IV. Heats of Reaction for Reactions [5] through [8] (KJ/mol)*
Reaction
T (K)
A
C
D
E
5
298 to 900
900 to 1045
1045 to 1184
1184 to 1650
298 to 900
900 to 1045
1045 to 1184
1184 to 1650
298 to 900
900 to 1650
298 to 900
900 to 1650
⫺10.505
—
—
⫺9.868
30.344
—
—
36.469
165.391
175.564
124.574
128.288
⫺25.712
⫺10.906
⫺127.502
⫺6.666
⫺18.687
119.067
⫺16.039
⫺19.969
37.045
1.134
29.834
16.306
16.489
⫺53.86
179.183
1.036
⫺6.773
⫺199.193
69.254
2.705
⫺50.725
⫺7.629
⫺27.128
⫺10.539
—
45.499
⫺70.717
—
10.223
95.601
⫺37.55
—
19.016
1.504
8.605
1.778
6
7
8
*For Reaction [5], these values are per 1 mole of CO; for Reaction [6], these values are per 1 mole of H2; and for Reactions [7] and
[8], these values are per 1 mole of carbon.
secondary volatile matter, the total volatile-matter mass fraction (Vt) on a moisture- and ash-free coal basis has to be
known. Now,
Xv1 ⫹ Xv2
[17]
Vt ⫽
1 ⫺ Xa ⫺ X w
If Vt is less than 0.1, we assume that all volatile matter
is secondary volatile matter, so that Xv1 ⫽ 0. If Vt is greater
than 0.1, the primary-volatile-matter mass fraction, Xv1, is
assumed to be equal to (Vt ⫺ 0.1) (1 ⫺ Xa ⫺ Xw) and
Xv2 ⫽ 0.1 (1 ⫺ Xa ⫺ Xw). To estimate the specific heats of
primary and secondary volatile matter and ash, correlations
proposed by Kirov (in cal/g ⬚C) can be used:
Ca ⫽ 0.18 ⫹ 1.4 ⫻ 10⫺4T
[18]
Fig. 3—Heat of Reaction [5]. The solid line shows estimates with the heats
of formation for room temperature from Barin;[4] temperature-dependent
specific heats for CO, CO2, and Fe0.947O from Barin;[4] and specific heats
for Fe from Kubaschewski et al.[8] The points denote results estimated
from data for heats of formation for different temperatures from Barin.[4]
evolved at higher temperatures, ash, water, and fixed carbon,
respectively. To find the mass fractions of the primary and
METALLURGICAL AND MATERIALS TRANSACTIONS B
Cv1 ⫽ 0.395 ⫹ 8.1 ⫻ 10⫺4T
⫺4
Cv2 ⫽ 0.71 ⫹ 6.1 ⫻ 10 T
[19]
[20]
IV. ESTIMATION OF COMPOSITION,
POROSITY, AND DENSITY
Different authors have published results related to distinct
iron ore/coal compositions. Reddy et al.’s[14] experiments
VOLUME 34B, FEBRUARY 2003—97
Table V. Coefficients to Estimate Temperature-Dependent Specific Heats of Fe2O3, Fe3O4, Fe0.947O, and Fe
Using Formula [15] (J/K/mol)
Substance
T (K)
Fe2O3
298
960
1000
1100
298
850
298
298
800
1000
1040
1042
1060
1184
Fe3O4
Fe0.947O
Fe
to
to
to
to
to
to
to
to
to
to
to
to
to
to
A
960
1000
1100
1700
850
1870
1650
800
1000
1040
1042
1060
1184
1642
100.884
3525.454
1930.36
132.754
⫺72.613
609.742
48.786
14.379
213.76
8759.86
⫺6415.79
124237.633
542.908
24.036
B
C
D
⫺1.729
—
—
—
—
—
0.280
—
—
—
—
—
—
—
75.165
⫺6352.989
⫺3307.81
7.299
1289.43
⫺797.923
8.368
49.006
⫺460.596
⫺17497.2
6237.5
⫺234458.544
⫺816.002
8.295
E
—
2978.286
1528.2
—
⫺2250.183
494.706
—
⫺57.471
301.135
8773.75
—
110661.111
331.482
0.022
—
—
—
—
1463.478
⫺97.605
—
41.243
—
—
—
—
—
—
Table VI. Coefficients to Estimate Temperature-Dependent Specific Heats of C, CO, CO2, H2, and H2O
Using Formula [15] (J/K/mol)
Substance
T (K)
C
298
800
298
800
298
298
700
298
CO
CO2
H2
H2O
to
to
to
to
to
to
to
to
800
1700
800
1700
1700
700
1700
1700
A
B
C
D
E
⫺5.086
4.129
25.887
24.722
22.078
32.342
31.297
32.171
—
—
0.111
—
—
⫺0.145
—
—
55.527
31.068
6.419
11.278
61.423
⫺8.202
⫺7.781
1.545
⫺34.74
⫺16.903
1.113
⫺2.842
⫺37.404
6.402
8.805
11.285
5.327
3.265
—
—
8.349
—
⫺2.113
⫺3.774
used a composition where, on the basis of fixed carbon in
coal, the C/Fe2O3 molar ratio (M ) varied from 0.9713 to
1.9431. In Dey et al.’s[15] work, when they studied the effect
of temperature and molar ratio on the reduction process, the
molar ratio varied from 3 to 6. In Chakravorty et al.’s[16]
work, the molar ratio changed from 1.82 to 2.19.
The minimum concentration of carbon that is needed to
obtain complete reduction can be estimated. It depends upon
the heating regime and the ambient atmosphere. Neglecting
the effect of the ambient atmosphere and assuming that pure
carbon is the reductant, the minimum molar ratio, to reduce
hematite to iron is
M⫽
MW
Keq
⫹1
1 KHM
eq ⫹ 1
⫹
0.555
HM
MW
3 2Keq ⫹ 1
2Keq ⫹ 1
KWI
eq ⫹ 1
⫹ 2.112 WI
2Keq ⫹ 1
[21]
MW
WI
where KHM
eq , Keq , and Keq are equilibrium constants for the
hematite-to-magnetite, magnetite-to-wustite, and wustite-toiron reactions, respectively. (Expressions for the equilibrium
constants can be found in the Appendix of Donskoi and
McElwain.[17]) This estimate (Eq. [21]) is based upon a
consideration of the composition of the gas which evolves
during the reduction. At every stage of the reduction, the
CO2/CO ratio should be less than the equilibrium constant
at this stage of the reduction. For the hematite-to-magnetite
reaction, for example, Reactions [1] and [7] can be rewritten
together as
98—VOLUME 34B, FEBRUARY 2003
Fe2O3 ⫹
⫹
KHM
1
2
eq ⫹ 1
⫻ HM
C ⫽ Fe3O4
3 2Keq ⫹ 1
3
1
3(2KHM
eq ⫹ 1)
(KHM
eq CO2
[22]
⫹ CO)
At 900 ⬚C, Eq. [21] gives M ⫽ 2.089 and, for 1200 ⬚C,
M ⫽ 2.16, approximately. These values can serve only as
reference points and should be corrected, depending on the
reducing or reoxidizing potential of the ambient atmosphere.
If coal is a reductant, this value should be smaller, because
iron ore would be partially reduced by volatiles.
However, in the direct reduction process, the gas composition can be far from equilibrium. For example, if the composition of gases for the reduction of hematite to magnetite,
in the (PCO /(PCO ⫹ PCO2))-temperature diagram, is between
the compositions corresponding to the equilibrium for Reactions [1] and [3], the composition of gas for the reduction
of magnetite to wustite is between the compositions corresponding to the equilibria for Reactions [3] and [5], and the
composition of gases for the reduction of wustite to iron is
between the composition corresponding to the equilibrium
for this reaction and pure CO; M is approximately 2.41 for
1200 ⬚C and 2.363 for 900 ⬚C.
The difference in the apparent densities in the iron ore/
coal composites can be quite significant. Wang et al.[18]
state that, in their experiment, the density of the dried iron
ore–hard coal pellet was 3.86 g cm⫺3 and that of the iron
ore–soft coal pellet was 3.23 g cm⫺3 (the molar ratio of
METALLURGICAL AND MATERIALS TRANSACTIONS B
fixed carbon to oxygen in iron oxides in ore was 1.0, so,
for hard coal, the mixture was 17.3 pct coal, 8 pct binder,
and 74.7 pct ore, and, for soft coal, it was 21.29 pct coal,
8 pct binder, and 70.71 pct ore). In Carvalho et al.’s[19]
experiment, the apparent density of the pellets was 2.85
g cm⫺3 (the molar ratio of fixed carbon to oxygen in iron
oxides in ore was 0.97; the mixture was 70.8 pct iron ore,
23.2 pct coke fines, and 6 pct lime for pellet I and 74.06
pct iron ore, 20.94 pct brown coal, and 5 pct lime for pellet
B-13) and the porosity was 0.34. For the packed bed in
Huang and Lu’s[20] experiment, the apparent density was 2.1
g cm⫺3, and Sun and Lu[21] claim that the porosity in their
experiment was 0.35.
During the reduction, the porosity changes. For example,
if the iron ore in a mixture like that in the Carvalho et al.[19]
experiment is reduced by 95 pct without shrinkage, the final
porosity will be 60 to 65 pct and the apparent density would
then be around 1.9 g cm⫺3. Abnormal swelling occurred in
Seaton et al.’s experiment,[22,23] where, after 45 minutes,
there was a 120 pct volume increase and a 55 pct reduction
at a temperature of 900 ⬚C. Here, the initial mixture was 74
pct hematite concentrate, 18 pct bituminous coal, 7 pct lime,
and 1 pct silica, and the estimate of the porosity after reduction was 0.76 to 0.80; this gives an apparent density of about
0.9 g cm⫺3.
V. SHRINKAGE AND SWELLING
As mentioned previously, while some articles on the
reduction of iron ore in a mixture with carbonaceous materials report swelling,[22,24] authors of other articles[25,26]
observe shrinkage in their experiments. The phenomenon
appears to have a straightforward explanation. Swelling
takes place when there is a significant presence of CaO in
the mixture. In the work by Nascimento et al.,[24] a cement,
which was 10 pct of the mixture, included 59.2 pct CaO,
and in Seaton et al.’s experiment,[22,23] lime was added to
the mixture. Moreover, Seaton et al. showed that for a mixture with a larger content of lime (7 pct), the swelling was
much greater than that for a mixture with a small content
of lime (2 pct). It is worth noting that the largest swelling
occurs at 900 ⬚C. Unfortunately, not enough data are given
to produce a quantitative relationship between swelling, the
temperature, and the amount of CaO.
McAdam et al.[26] give the fractional linear shrinkage
of spherical pellets of an iron-sand concentrate–coal/char
mixture (25 pct of coal by weight) as a function of time
and temperature:
S⫽
r0 ⫺ r
⫽ kt2/5 exp (⫺Q/RT )
r0
冢
冣
[23]
where r0 is the initial pellet radius; k is a reaction-rate constant, taken to be 38.8; r is the instantaneous radius; t is the
time; Q is equal to 105 kJ/mol; R is the gas constant; and
T is the temperature (in Kelvin). McAdam et al. write that
the time exponent is characteristic of sintering by volume
diffusion. Using this expression, after 30 minutes of reduction, the fractional linear shrinkage of pellets with 25 pct
of coal at 1200 ⬚C will be 0.146, corresponding to a relative
volume change of about 37.8 pct. Sharma[25] investigated
the volume shrinkage for a mixture of Bailadila ore with
coal (Fe2O3/C ratio of 80:20) and, at 1200 ⬚C, after 30
METALLURGICAL AND MATERIALS TRANSACTIONS B
minutes, it was about 24 pct. In addition, as can be seen
from Sharma’s graph (Figure 2), the time dependence is
close to linear.
Sharma’s[25] volume-shrinkage data have been fitted by a
formula of the same type as McAdam et al.’s,[26] but with
a linear dependence on time:
Sv ⫽
V0 ⫺ V
⫻ 100 pct ⫽ 0.126998 ⫻ t e⫺4372.02/T
V0
冢
冣
[24]
From this expression for the change in volume (or from
McAdam et al.’s expression, which may be more suitable
for an agglomerated medium), the change in pellet radius
can be easily estimated.
Donskoi and McElwain[12] modeled Seaton et al.’s experiment,[22,23] where an abnormal swelling during the reduction
was observed. The process of pellet-size change during the
reduction has been modeled as an interaction between two
opposite processes, swelling and shrinkage, and the formula
for the local volume change is
dV
⫽ V (Sw ⫹ Sh)
dT
[25]
where Sw is the term associated with swelling and Sh is the
term associated with shrinkage.
The expression for the swelling term is
Sw ⫽ 2.9 ⫻ 10⫺7
⫺(T ⫺ T0)2
WF
exp
␴
2␴ 2
冢
冣
[26]
where the swelling term has been modeled as proportional
to the concentrations of wustite (W ) and iron (F) and having
a Gaussian dependence on temperature, with a mean temperature (T0) of 920 ⬚C and a standard deviation (␴ ) of 60 ⬚C.
A model for the shrinkage term is
Sh ⫽ 0.012␾ 2 exp (⫺4372/T )
[27]
where ␾ is a porosity and T is the absolute temperature. The
activation energy for shrinkage has been taken from a fit to
Sharma’s[25] data.
VI. ESTIMATION OF THERMAL
CONDUCTIVITY
The modeling of thermal conductivity in the reducing
mixture of iron ore and coal is difficult. The thermal conductivity of such composites depends on the properties of the
constituent solids and gases, the temperature, and the porosity structure. A major problem is that the thermal conductivity cannot be directly measured. When a sample is heated,
Reactions [1] through [8] occur, and their thermal interference should be taken into account. Thus, the thermal conductivity can best be obtained from modeling the system. The
thermal conductivity can be measured before the reactions
have started or after they have finished, but, as far as we
are aware, there are no articles on this subject.
Sun[27] and Sun and Lu[28] use a formula given by Bear:[29]
Keffmc ⫽
1
2
冢
兺i fmi Kmi ⫹
1
fmi
兺i
Kmi
冣
[28]
VOLUME 34B, FEBRUARY 2003—99
where Keffmc is the effective thermal conductivity of the
mixture, i is the index of component in the mixture (Fe,
FeO, Fe3O4, coal, and gas), and fmi and Kmi are the volume
fraction and thermal conductivity of component i in a control
volume, respectively. To estimate the thermal conductivity
of coal, Sun and Lu use an equation from Atkinson and
Merrick[30] specifically developed for coals, semicokes, and
cokes during their thermal decomposition:
Km ⫽
␳m
4511
3.5
冢 冣
⫻ T 1/2 (W m⫺1 K⫺1)
兺j yj Kgj M j1/3
兺j yj M j1/3
[30]
where yj is the mole fraction of gaseous component j with
a molecular weight of Mj and thermal conductivity of Kgi .
Temperature-dependent thermal conductivities for iron
and iron oxides are given by Akiyama et al.[32]. Using Sun’s
approach for a mixture of 80 pct ore and 20 pct coal, the
thermal conductivity increases from about 0.7 to 5.0 W/
(m K) as reduction proceeds.
Akiyama et al.[32] give experimental results for the thermal
conductivity of porous direct-reduced iron. For iron reduced
by CO with a porosity of 0.62, the thermal conductivity is
approximately 1.25 W/(m K) (at 1000 ⬚C), and, for iron
reduced by H2 with a porosity 0.60, for the same temperature,
the thermal conductivity is 1.5 W/(m K). Bear’s[28] formula
for thermal conductivity, in a vacuum, for the same porosity
(0.60) and density, gives 4.2 W/(m K). Sun and Lu[21] used
another formula in an earlier work, namely,
Keffmc ⫽ (1 ⫺ ␾ ) 兺 fmiKsi
[31]
i
where ␾ is the porosity, and fmi and Ksi are the volume
fraction and thermal conductivity of solid component i,
respectively. Using Eq. [31], the thermal conductivity of the
same porous direct-reduced iron is estimated to be 3.3 W/
(m K). The difference from the measured value may be due
to the effect of impurities or an underestimation of the heat
consumed during the reactions. McAdam et al.[26] studied
the reduction of New Zealand iron-sand concentrate in the
composite pellets by coal or coal char. They give a graph
of the dependence of the thermal conductivity on temperature. The thermal conductivity for a mixture of 75 pct ironsand and 25 pct coal changes from about 1.3 W/(m K) at
20 ⬚C to less than 0.1 W/(m K) at 1200 ⬚C. This estimate
is quite different from that given by other authors. For pure
iron-sand concentrate, the thermal conductivity changes
from 2.2 W/(m K) at 200 ⬚C to about 0.5 W/(m K) at 1200 ⬚C.
Donskoi and McElwain,[12] in their modeling of Seaton
et al.’s experimental work[22,23] on the reduction of iron
ore–coal composite pellets, used Dulnev and Zarichnyak’s[33] approach to the modeling of the thermal conductivity. This gives
ke ⫽ (2/3) (␾ /kg ⫹ (1 ⫺ ␾ )/ks)⫺1
[32]
⫹ (1/3) (␾kg ⫹ (1 ⫺ ␾ )ks)
100—VOLUME 34B, FEBRUARY 2003
T (K)
380
800
1025
1175
to
to
to
to
800
1025
1175
1500
A
B
C
D
31.995
94.7509
⫺439.488
⫺22.6774
11.54
—
—
—
41.322
72.1439
835.878
57.1977
37.6127
—
377.668
16.5732
[29]
where ␳m is the true density of the dry-ash-free material.
This formula gives good agreement with published literature
for the thermal conductivity of the studied carbonaceous
materials and of amorphous carbon as a limiting case.
For the thermal conductivity of a gas mixture, Sun and
Lu use an equation given by Rosner:[31]
Kgas ⫽
Table VII. Coefficients to Estimate TemperatureDependent Thermal Conductivity of a Dense Iron
Using Formula [34]
Table VIII. Formulae from Akiyama et al.[32] (p. 831) to
Estimate Thermal Conductivities of Dense Hematite,
Magnetite, and Wustite
Fe2O3
k ⫽ 1/(1.844
k ⫽ 1/(8.319
Fe3O4
k ⫽ 1/(1.693
k ⫽ 1/(2.697
FeO
k ⫽ 1/(2.335
k ⫽ 1/(8.319
⫻ 10⫺4T )
⫻ 10⫺5T ⫹ 9.243 * 10⫺2)
⫻ 10⫺4T )
⫻ 10⫺6T ⫹ 1.508 * 10⫺1)
⫻ 10⫺4T ⫹ 1.136 * 10⫺1)
⫻ 10⫺5T ⫹ 9.243 * 10⫺2)
Temperature (K)
298 ⬍ T ⬍ 912
912 ⬍ T ⬍ 1500
—
298 ⬍ T ⬍ 906
906 ⬍ T ⬍ 1500
—
298 ⬍ T ⬍ 825
825 ⬍ T ⬍ 1500
where kg is the thermal conductivity of gas inside the pores,
ks is the mean dense thermal conductivity of the mixture,
and ␾ is the porosity. For the mean dense thermal conductivity, the following formula has been used:
ks ⫽ c兺 fi Ksi
[33]
i
where fi is the volume fraction of the solid component i,
and Ksi is the thermal conductivity of the solid component
i. Here, c is a factor reflecting the effects of consolidation,
the porosity matrix, and the mixture of different components
in the solid fraction. In our modeling, in the first stage of
reduction (hematite to magnetite), c ⫽ 0.8; later, during the
second stage of reduction (magnetite to wustite), it linearly
decreased to 0.4; and, in the last stage of reduction, it
remained at 0.4.
Using this approach, for reduced pure iron with a porosity
of 0.60, in a vacuum, the thermal conductivity is estimated
to be 1.1 W/(m K). It is slightly lower than Akiyama et
al.’s measurements. However, in coal-based reduced iron,
roughly half of the volume is occupied by coal, so the number
of connections between iron grains is expected to be lower
than in gaseous–reduced iron ore, so that the thermal conductivity, for the same porosity and composition, is expected
to be lower for coal-based reduced iron.
Akiyama et al. give the graph of the thermal conductivity
of a dense iron. Those data have been fitted, and an empirical
equation for different ranges of temperatures can be established, namely,
KFe ⫽ A ⫹ B103T ⫺1 ⫹ C10⫺3T
⫹ D10⫺6T 2 W/(m K)
[34]
and the coefficients are given in Table VII.
The thermal conductivities of dense hematite, magnetite,
and wustite can be calculated from formulae given by Akiyama et al. (Reference 32, p. 831), and these are given in
Table VIII.
METALLURGICAL AND MATERIALS TRANSACTIONS B
Table IX. Coefficients to Estimate Temperature-Dependent Thermal Conductivities of CO, CO2, H2, H2O, O2, CH4, and
N2 Using Formula [38] W/(m K)
Substance
CO
CO2
H2
H2O
O2
CH4
N2
T (K)
A
B
C
D
270-1800
270-1800
270-1800
270-1800
270-1800
270-1300
270-1800
1.7291
⫺0.85817
13.822
⫺6.1163
1.9928
⫺1.7186
3.5487
⫺2.0117
—
⫺12.964
10.77
⫺3.0474
1.7406
⫺5.4318
5.0537
9.0754
27.923
14.419
5.6861
13.766
2.6004
—
⫺1.5068
2.2143
0.5757
⫺0.22493
4.6188
0.63342
Table X. Coefficients to Estimate Temperature-Dependent Thermal Conductivities of C2H6, C2H4, C3H6, C3H8, and C6H6
Using Formula [39] W/(m K) (Reproduced from Reid[38]
Substance
C2H6
C2H4
C3H6
C3H8
C6H6
Name
ethane
ethylene
propylene
propane
benzene
T (K)
273
200
175
273
273
to
to
to
to
to
1020
1270
1270
1270
1270
A
⫺3.174
⫺1.760
⫺7.584
1.858
⫺8.455
⫻
⫻
⫻
⫻
⫻
B
10
10⫺2
10⫺3
10⫺3
10⫺3
Fitting data for the thermal conductivity of carbon from
Rohsenov et al.[34] gives
kC ⫽ 4.0526 ⫺ 553.804/T ⫹ 0.00254608T
⫺ 1.57951 ⫻ 10⫺6 ⫻ T 2 W/(m K)
[35]
The thermal conductivity of SiO2 has been modeled using
averaged data from sources given in the Slag Atlas.[35] The
formulae are, for the temperature range from 300 to 1173 K,
k1Si02 ⫽ ⫺0.2557 ⫹ 216.8/T ⫹ 0.003474T
⫺ 1.383 ⫻ 10⫺6T 2 W/(m K)
⫺2
[36]
2.201
1.200
6.101
⫺4.698
3.618
⫻
⫻
⫻
⫻
⫻
C
⫺4
10
10⫺4
10⫺5
10⫺6
10⫺5
⫺1.923
3.335
9.966
2.177
9.799
⫻
⫻
⫻
⫻
⫻
D
⫺7
10
10⫺8
10⫺8
10⫺7
10⫺8
⫺1.664 ⫻ 10⫺10
⫺1.366 ⫻ 10⫺11
⫺3.840 ⫻ 10⫺11
⫺8.409 ⫻ 10⫺11
⫺4.058 ⫻ 10⫺11
of these will prove useful to researchers studying other categories of direct reduction of iron ore. The article has highlighted deficiencies in the published literature and gives
directions for further experimental work. Since direct reduction of iron ore represents an increasing percentage of world
iron production, decision support in the form of accurate
process-parameter estimates and mathematical modeling
provides a cost-effective way to improve industrial processes. This article supplies support for academic and industrial researchers involved in both the development of new
direct reduction processes and the improvement of existing ones.
and, for the temperature range from 1173 to 1773 K,
k2SiO2 ⫽ 1.970 ⫹ 0.0001074T W/(m K)
[37]
Data for thermal conductivities of different compositions
of admixtures can be found in the Slag Atlas.
Data for the thermal conductivities of gases have been
taken from Touloukian,[36] Gray and Muller,[37] Yaws,[38] and
Rohsenow et al.[34] Those data have been fitted using the
following empirical equation:
Kg ⫽ A10⫺2 ⫹ BT ⫺1 ⫹ C10⫺5T
⫹ D10⫺8T 2 W/(m K)
[39]
and the coefficients are in Table X.
VII. CONCLUSIONS
The development and improvement of industrial processes
requires reliable estimates of process parameters. This article
provides an overview of critical physical parameters relating
to coal-based direct reduction of iron ore, although many
METALLURGICAL AND MATERIALS TRANSACTIONS B
This work was partially supported by the Australian
Research Council under the SPIRT scheme. A QUT Postdoctoral Fellowship to ED is gratefully acknowledged. The
authors thank Dr. Rene Olivares for discussions and valuable
comments regarding this article.
[38]
The coefficients are given in Table IX.
Thermal conductivities of major hydrocarbons (other than
CH4) evolving during coal pyrolysis were taken from Reid
et al.[39] These should be modeled according to the formula
Kg ⫽ A ⫹ BT ⫹ CT 2 ⫹ DT 3 W/(m K)
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METALLURGICAL AND MATERIALS TRANSACTIONS B