Dispersive kinetic model for the non-isothermal reduction of nickel oxide by hydrogen

Article abstract of DOI:10.1016/j.physb.2008.08.020

The kinetics of the non-isothermal reduction process of powder nickel oxide samples using hydrogen was investigated by temperature-programmed experiments at the different constant heating rates. The new procedure for the determination of density distribution function of activation energies (ddfE_a), evaluated from the experimentally obtained non-isothermal conversion curves, was developed. The analytical relationships between the corresponding thermo-kinetic parameters for the investigated reduction process were established. From the influence of heating rate on the basic characteristics of ddfE_a's, it was concluded that the evaluated ddfE_a's are completely independent of the heating rate (v_h). It was found that the value of activation energy at the peak of the distribution curve (E_a,max), at all considered heating rates, is in good agreement with the value of E_a,0 (96.6 kJ mol^-1) calculated from the isoconversional dependence of activation energy, in the conversion range of 0.20≤α≤0.60. From the appearances of the true compensation effect, it was concluded that the factor that produces the changes of kinetic parameter values is a conversion fraction (α). Using the model prediction, the experimentally obtained conversion curves are completely described by the evaluated distribution curves (g(E_a)_vhj) at all considered heating rates. It was concluded that the assumption about the distribution of potential energies of oxygen vacancies presented in NiO samples leads to the distribution of activation energies, which determine the kinetics of non-isothermal reduction processes.

Full text of DOI:10.1016/j.physb.2008.08.020

ARTICLE IN PRESS
Physica B 403 (2008) 4132–4138
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Physica B
journal homepage: www.elsevier.com/locate/physb
Dispersive kinetic model for the non-isothermal reduction
of nickel oxide by hydrogen
Ã
Borivoj AdnaXevi c´ , Bojan Jankovi c´
Faculty of Physical Chemistry, University of Belgrade, Studentski Trg 12-16, P.O. Box 137, 11001 Belgrade, Serbia and Montenegro
a r t i c l e i n f o
a b s t r a c t
Article history:
The kinetics of the non-isothermal reduction process of powder nickel oxide samples using hydrogen
was investigated by temperature-programmed experiments at the different constant heating rates. The
Received 29 July 2008
Accepted 27 August 2008
a
new procedure for the determination of density distribution function of activation energies (ddfE ),
evaluated from the experimentally obtained non-isothermal conversion curves, was developed. The
analytical relationships between the corresponding thermo-kinetic parameters for the investigated
reduction process were established. From the influence of heating rate on the basic characteristics of
PACS:
6
1.43.Gt
1.66.Fn
1.72.Cc
1.72.jd
6
a a h
ddfE ’s, it was concluded that the evaluated ddfE ’s are completely independent of the heating rate (v ).
6
It was found that the value of activation energy at the peak of the distribution curve (Ea,max), at all
6
ꢀ1
considered heating rates, is in good agreement with the value of Ea,0 (96.6 kJ mol ) calculated from the
isoconversional dependence of activation energy, in the conversion range of 0.20p
appearances of the true compensation effect, it was concluded that the factor that produces the changes
of kinetic parameter values is a conversion fraction ( ). Using the model prediction, the experimentally
obtained conversion curves are completely described by the evaluated distribution curves (g(E vhj) at
ap0.60. From the
Keywords:
Nickel oxide
a
Non-isothermal reduction
Kinetics
a
)
Density distribution function
Activation energy
Compensation effect
all considered heating rates. It was concluded that the assumption about the distribution of potential
energies of oxygen vacancies presented in NiO samples leads to the distribution of activation energies,
which determine the kinetics of non-isothermal reduction processes.
&
2008 Elsevier B.V. All rights reserved.
1.
Introduction
size of the continuous reduced phase of the oxide interface, and it
entirely covers the solid reactant.
Metal oxides are used as catalysts in a large variety of
Rodriguez et al. [6] observed the existence of the correlation
between the concentration of oxygen vacancies in the NiO lattice
and the rate of oxide reduction. Namely, they are confirmed that
the reduction mechanism is controlled by the number of oxygen
vacancies in the NiO lattice and by their migration from the bulk
to the surface.
commercial processes for the conversion of hydrocarbons [1,2].
Selective oxidation, ammoxidation, and selective dehydrogenation
probably constitute the most important catalytic applications of
metal oxides [2]. In addition, due to their low cost, metal oxides
are also used as supports of many other catalytic materials
(
metals, sulfides, carbides, nitrides, etc.) [1,2]. One method
In the recent papers of Vand [7], Pitt [8], Lakshmanan and
White [9], Bednarek [10], Burnham and Braun [11] and Skrdla and
Robertson [12], the distributed reactivity kinetic model was
placed and developed, which is based on the assumption about
the existence of the activation energy distribution of reaction
species in the considered reaction system.
frequently employed for the preparation of active oxide catalysts
involves partial reduction under hydrogen atmosphere at elevated
temperatures [3,4].
In former studies, two different kinetic models have been
proposed for the reduction of metal oxides: the ‘nucleation model’
[
3,4] and the ‘interface-controlled model’ [5]. The rate-limiting
In accordance with Rodriguez et al. [6], the presence of oxygen
step of the reduction process in the ‘nucleation model’ is a
generation of small aggregates of clusters of the new phase. On
the other hand, in accordance with the ‘interface-controlled
model’, the rate of metal oxide reduction is proportional to the
2
vacancies leads to increase in the adsorption energy of H and
induces the migration of oxygen vacancies from the bulk to the
surface. From these results, it can be logically concluded that the
present oxygen vacancies have different potential energies, and so
2
they contain the different adsorption energies over H adsorption
and leads to the existence of different activation energies for the
investigated reduction process.
Ã
Bearing in mind the above-introduced facts, in this work, the
new procedure for determination of density distribution function
Corresponding author. Tel./fax: +238111187133.
E-mail address: bojanjan@ffh.bg.ac.yu (B. Jankovi c´ ).
0921-4526/$ - see front matter & 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.physb.2008.08.020

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4133
of activation energy probabilities (ddfE
a
’s), evaluated from the
For direct determination of g(E
the dependence of
following relationship:
a
) using Eq. (3), we need to know
non-isothermal reduction conversion curves, was developed.
Based on the established distribution curve, the non-isothermal
kinetics of nickel oxide reduction under hydrogen atmosphere
was described.
a
h
on T and v , i.e. interpret through the
a
¼ a
ꢁðT; vhÞ
(4)
Based on Eq. (4), it is possible directly (from Eq. (3)) to
determine the density distribution function of activation energies
2.
Experimental section
a
(ddfE ’s) as
ꢀ
ꢁ
ꢀ
ꢁ
d
a
dT
dE
2.1. Materials and methods
gðE Þ h
a
v
¼
ꢂ
(5)
dT
a
E
a
;v
h
a
E ;v
h
The nickel oxide samples were obtained by the gel-combustion
where ðd
mined activation energy (E
ðdT=dE Ea;vhj is the differential dependence T ¼ TðE
determined value of E
For the selected values of
of v values (v
¼ vhj), by solving Eq. (4) with respect to
temperature (T), we can obtain the series of solutions in the form
a
=dTÞEa;vh is the non-isothermal reaction rate at deter-
) and considered heating rate (v ) and
vh at the
method described elsewhere [13]. A green-colored transparent gel
was obtained by drying an aqueous solution of nickel nitrate
hexahydrate (Fluka, 99.5%) and citric acid (Fluka 99.5%) dissolved
in a mole ratio of 1.8:1. This gel was made to undergo a self-
ignition by heating in air up to 300 1C, and an additional heating of
up to 500 1C produced very fine nickel oxide powders. The mean
size of nickel oxide particles determined from XRD data is
a
h
a
Þ
a
Þ
a
.
a
(
a
¼
i
a ¼ const.) and a selected set
h
h
a,i,j
of T ¼ T .
j
k
v
hj
2
v
hj
d
m
¼ 30 nm, and this value overlaps very well with the data
By drawing the dependence of ln v =ðT
Þ
versus 1=T in
a
i
hj
a
i
published by Wu et al. [14].
accordance with the Kissinger–Akahira–Sunose (KAS) method
[15,16], it is possible from the slope and intercept of the above
2.2. Temperature-programmed reduction (TPR) measurements
relationship to obtain the following functional forms:
ꢁ
E
a
¼ E ðai
Þ
(6)
(7)
a
The experiments were carried out in a TA SDT 2960 device,
capable of simultaneous TGA-DTA analysis in the temperature
range from 25 to 1500 1C. The nickel oxide samples were reduced
directly within the thermobalance, in korund pans, under
ln A ¼ ln Aꢁðai
where A is the pre-exponential (frequency) factor.
Þ
ꢀ
1
ꢁ
hydrogen (99.9995 vol%) flowing at a rate of 100 mL min , and
using various values of heating rates, namely 2.5, 5, 10 and
If we know the analytical form of dependence T ¼ T ð
a
i
Þ
vhj
,
Eq. (6) can be easily transformed into the following form:
ꢀ1
2
5
0 1C min , in the temperature range from an ambient one up to
00 1C. The sample mass used for non-isothermal measurements
¼ EꢁaꢁðTÞvhj
E
a
(8)
was about 2570.5 mg. The degree of transformation (
reduction process is expressed as
a
) of the
and also in the form
T ¼ TðE Þvhj
a
(9)
m
0
ꢀ m
ꢀ m
t
a
¼ m
(1)
By differentiating Eq. (9) with respect to E
following dependence:
a
we can obtain the
(10)
0
f
where m
0 t f
, m and m refer to the initial, actual and final mass of
ꢀ
ꢁ
dT
dE
the investigated sample, respectively. The data collected from TPR
measurements can be interpreted on a qualitative basis or by
utilizing simple, approximate kinetic methods. The TPR data can
represent the important source of kinetic information.
¼
hj
f½ðE Þꢃ
a
a
v
In addition, by the direct differentiation of Eq. (3), the following
functional relationship can be obtained:
ꢀ
ꢁ
d
a
¼
hj
d
½ðTÞꢃ
(11)
3.
Theoretical background
dT
v
Because we know the functional relationship represented by
Eq. (9), in that case, Eq. (11) can be easily transformed into the
following relationship:
3
.1. Determination of density distribution function of activation
energies (ddfE
a
’s)
ꢀ
ꢁ
In accordance with the definition of ddfE
the following relation is valid:
a
(designated as g(E
a
)),
da
dT
¼
hj
ꢀ
½ðE
a
Þꢃ
(12)
v
d
a
a
By multiplication of Eqs. (10) and (12) at the same E
a
value, we
gðE
a
Þ ¼
(2)
a
is the calculated
dE
is the degree of transformation and E
activation energy.
can obtain the final expression for gðE Þ
,
vhj
a
ꢀ
ꢁ
ꢀ
ꢁ
where
a
d
a
dT
dE
gðE
a
Þvhj
¼
ꢂ
(13)
dT
a
E
a
;vhj
a
E ;vhj
The non-isothermal (dynamic) measurements involve heating
samples at one or more constant heating rates (usually linear) and
following the course of the reaction. In non-isothermal conditions,
mass loss data are converted to a normalized form called degree of
transformation or conversion fraction (a).
The conversion fraction ranges from 0 to 1 and is a measure of
reaction progress as a function of temperature at a constant value
4
.
Results and discussion
Fig. 1 shows the experimentally determined conversion curves
of nickel oxide (NiO) reduction under hydrogen atmosphere
at the different heating rates (vhj). The same figure also shows
the numerically fitted conversion curves by the following
h
of linear heating rate (v ):
a
¼
a
ðTÞvh
(3)

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Table 1
The influence of heating rate on the numerically evaluated fitting parameters
) (Eq. (14)) for the reduction process of nickel oxide under hydrogen atmosphere
1
0
0
0
0
0
.0
.8
.6
.4
.2
.0
j j
(b , Z
ꢀ
1
2
vhj/1C min
Da
b
j
Z
j
/min
R
Chi-square
2
)/ ꢂ 10ꢀ5
(
w
2
5
.5
2.5270.01
2.6570.02
2.8270.03
3.2070.02
11.0370.05
6.3470.04
3.3870.07
2.5470.03
0.9997
0.9995
0.9991
0.9995
3.4660
5.0631
9.5748
8.0687
0.02–0.99
°
-1
10
20
2
5
1
.5 C min
°
-1
C min
°
-1
0 C min
°
-1
2
0 C min
°
-1
Numerical fit: 2.5 C min
Numerical fit: 5 C min
Numerical fit: 10 C min
Numerical fit: 20 C min
°
-1
°
-1
-9.5
°
-1
-10.0
2
50
300
350
400
450
500
Temperature /°C
-
10.5
Fig. 1. The experimentally determined (full and broken lines) and numerically
fitted (symbols) conversion ( –-T) curves for the nickel oxide (NiO) reduction
a
-
11.0
h
under hydrogen atmosphere, at the different heating rates (v ).
expression:
-11.5
-12.0
2
3
5
!
b
j
T
vhj ꢀ T0;vhj
4
avhj ¼ 1 ꢀ exp ꢀ
(14)
v
Zj
hj
0
.0016
0.0017
0.0018
0.0019
where Tvhj is the actual temperature at a considered heating rate
j), T0;vhj is the onset temperature, vhj is the heating rate (where
and represent the fitting parameters ( is
is the scale parameter), at a
(
j
k
v
hj
2
v
hj
j ¼ 1, 2, 3, 4), while
b
j
Z
j
b
j
Fig. 2. The isoconversional dependence ln vhj=ðTai
Þ
versus 1=Tai (KAS method)
for the non-isothermal reduction process of nickel oxide under hydrogen
the shape parameter and
Z
j
atmosphere at four constant heating rates (vhj, j ¼ 1, 2, 3, 4).
considered heating rate (j).
Based on the results given in Fig. 1, it can be concluded that the
experimental change of with temperature (T) can be very well
described using Eq. (14), in wide range of values
¼ 0.02–0.99), at the different heating rates of the considered
system.
Table 1 shows the influence of heating rate (vhj) on the values
of fitting parameters ( ).
Analyzing the results given in Table 1, it is possible to establish
that the values of and represent the analytical functions of
linear heating rate, expressed through the following equations:
a
a
a
1
15
110
05
00
(
a
b
j
,
Z
j
1
b
j
Z
j
1
bj ¼ 2:440 þ 0:038vhj
(15)
9
5
ꢀ
ꢁ
vhj
Zj ¼ 2:541 þ 18:764 exp ꢀ 3
(16)
:147
90
The functional change
and the functional changes of
rate enable us to approach the analytical form of the dependence
(T,v ).
If we know the analytical form of equation
possible to determine the series of solutions of Eq. (14), for the
a ¼ aðTÞ , which is defined by Eq. (14),
v
hj
8
5
b
j
and
j
Z parameters with heating
0
.0
0.2
0.4
0.6
0.8
1.0
a
¼
a
h
a
¼
h
a(T,v ), it is
a
Fig. 3. Dependence of the activation energy (E ) on the conversion fraction (a)
evaluated from the isoconversional KAS method for the non-isothermal reduction
process of nickel oxide under hydrogen atmosphere.
few selected constant values
a
¼
i
a ¼ const., and constant heating
rates (vhj).
Fig. 2 shows the isoconversional (
a
¼
a ¼ const.) dependence
i
j
k
The dependence of activation energy (E
fraction ( ) using the KAS method [15,16], for the reduction
process of nickel oxide by hydrogen is given in Fig. 3.
The obtained Ea, plot shows the practically constant E
values in the range of 0.20p p0.60 [17], except for an increase
from 98.0 to 108.7 kJ mol (see Fig. 3).
By the numerical analysis of results presented in Fig. 3, it is
a
) on the conversion
v
a
hj
2
v
hj
a
i
ln v =ðT
Þ
versus 1=T , for the non-isothermal reduction
a
hj
i
process of nickel oxide under hydrogen atmosphere.
j
k
v
a
i
hj
2
a
–
a
a
From Fig. 2 we can see that dependence ln v =ðT
Þ
versus
hj
a
v
hj
ꢀ1
1
=T is a straight line, and from the slope of isoconversional lines
a
i
a
it is possible to determine the values of Ea, at different constant a
values.
*
possible to evaluate the analytical form of dependence E
a
a
¼ E (a)

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4135
1
15
115
110
105
100
95
1
10
1
1
05
00
9
5
9
0
90
280
2
70
280
290
300
290
300
310
320
Temperature /°C
Temperature /°C
1
15
115
110
105
100
95
1
10
1
1
05
00
9
9
5
0
90
2
90
300
310
320
330
340
320
330
340
350
360
370
Temperature /°C
Temperature /°C
Fig. 4. Dependence of the activation energy (E
a
) on temperature (T) at the selected heating rate (vhj, j ¼ 1, 2, 3, 4) (a–d), for the investigated reduction process of nickel oxide
under hydrogen atmosphere.
for the range of 0.20p
a
p0.60. The mentioned relationship can be
Table 2
ꢁ
The influence of heating rate (vhj) on the values of Ea0; T
and a (Eqs. (19) and
presented by the following equation:
0;vhj
(20)) for the non-isothermal reduction process of nickel oxide under hydrogen
E
a
¼ 96:6 þ 0:20ð
where E
¼ 96.6 kJ mol represents the constant E
considered range of , while
between the constant and increasing regions of E
a
ꢀ
a
0
Þ
(17)
atmosphere
ꢀ
1
,
a 0
a
value in the
ꢀ1
ꢀ1
ꢁ
v
hj/1C min
E
a0 /kJ mol
T
/1C
;vhj
a
0
a
a
0
represents the boundary value of
a,
a
values (
a
ꢄ0.60).
2
5
.5
96.6
96.6
96.6
96.6
286.35
303.86
317.23
345.26
0.007
0.004
0.007
0.004
The constant b ¼ 0.20 is the numerical fitting coefficient in Eq. (17).
Since, for the selected values of heating rates (vhj), the
following equation is valid (see Eq. (14)):
10
20
1=b
j
T
vhj ¼ T0;vhj þ vhjZj½ꢀ lnð1 ꢀ ai;vhj Þꢃ
(18)
a
increasing regions of E values. The constant a is the numerical
it is possible to find the expression for functional dependence E
on T at the selected heating rate (vhj).
a
fitting coefficient in Eqs. (19) and (20) (Table 2).
Table 2 shows the influence of experimentally obtained
heating rates (vhj, where j ¼ 1, 2, 3, 4) on the values of Ea0, Tꢁ
a
Fig. 4 (a)–(d) shows the dependence of activation energy (E )
0;vhj
on temperature (T) at the selected heating rate (vhj, j ¼ 1, 2, 3, 4),
and a for the non-isothermal reduction of nickel oxide under
hydrogen atmosphere.
for the investigated reduction process of nickel oxide by hydrogen.
In the presented E
shapes of Ea, change with T: (a) the region of constant activation
energy value, E
values with temperature. In both cases, the following relations are
valid:
a
–T curves, we can see the two characteristic
It can be seen from Table 2 that with increase in the heating
rate, the value of Ea0 remains constant, while the numerical fitting
coefficient (a) shows the same value at vh1 ¼ 2.5 1C minꢀ and
a
1
a
¼ Ea0 and (b) the region of linear increasing of E
a
ꢀ1
ꢀ1
vh3 ¼ 10 1C min
(for a ¼ 0.007) and at vh2 ¼ 5 1C min
and
ꢀ
1
v
h4 ¼ 20 1C min (for a ¼ 0.004). On the other hand, the value
ꢁ
of T
increases from 286.35 to 345.26 1C.
ðTÞvhj ¼ Ea0 þ aðT ꢀ Tꢁ
Þ
(19)
0;v
hj
E
a
0;v
hj
From the results presented in Table 2, we can conclude that the
heating rate has no influence on the value of Ea0, which represents
the real activation energy value in the kinetically important region
ðE
a
ðTÞvhj ꢀ Ea0
Þ
T ¼ Tꢁ
þ
(20)
0;vhj
a
of
a values (0.20pap0.60).
where Ea0 represents the constant E
.20p
a
value in the range of
Based on the known dependence in the form T ¼ TðE
a
Þ
, the
v
hj
ꢁ
0
a
p0.60, at a considered heating rate, while T
is the
;vhj
functional dependence introduced in Eq. (10) can be easily
determined. Using the differentiation procedure of Eq. (4) with
0
boundary value of T, between the approximately constant and

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0
0
0
0
0
0
0
.30
.25
.20
.15
.10
.05
.00
0.30
0.25
0.20
0.15
0.10
0.05
0.00
9
6
98
100
102
104
106
108
96
98
100
102
104
106
108
0
0
0
0
0
0
0
.30
.25
.20
.15
.10
.05
.00
0.30
0.25
0.20
0.15
0.10
0.05
0.00
9
6
98
100
102
104
106
108
96
98
100
102
104
106
108
a
Fig. 5. The density distribution function of activation energies (g(E )vhj) at the four different heating rates calculated from Eq. (13), for the investigated non-isothermal
reduction process of nickel oxide under hydrogen atmosphere.
Table 3
The influence of heating rate (vhj) on the basic characteristics of the density distribution function of activation energies (g(E
a
)vhj was calculated from Eq. (13)), for the non-
isothermal reduction process of nickel oxide under hydrogen atmosphere
ꢀ
1
ꢀ1
ꢀ1
ꢀ1
vhj,HW/kJ mol
ꢀ1
v
hj/1C min
Ea,max/kJ mol
g(E
a
)vhj,max/kJ mol
g(E
a
)
SF
HW/kJ mol
2
5
.5
96.6
96.6
96.6
96.6
0.25580
0.24952
0.25062
0.25520
0.12948
0.12650
0.12705
0.12934
0.25
0.23
0.23
0.27
1.5
1.6
1.6
1.4
10
2
0
respect to T, at the constant selected value of heating rate, it can be
From the results in Table 3 and Fig. 5, we can see that the value
possible to obtain the dependence described by Eq. (11).
of Ea,max is the same at all considered heating rates, which
The knowledge of dependencies ðdT=dE
a
Þ
and ðd
a
=dTÞ
indicates that the ddfE
system. The presented ddfE
a
’s is independent of the heating rate of the
’s are highly asymmetric (SF51) and
E
a
;vhj
a
E ;vhj
enables us to directly determine the density distribution function
a
of activation energies (ddfE ) for the investigated reduction process
a
exhibits the long right-tail behavior. On the other hand, shape
factor (SF) and the half-wide (HW) vary in a small range with the
heating rate, 0.23oSFo0.27 and 1.4oHWo1.6 (Table 3), respec-
tively. In addition, the values of Ea,max at all heating rates (Table 3)
are in good agreement with the value of Ea0 established previously
(Ea0 ¼ 96.6 kJ mol ). From these results we can conclude that the
evaluated function represents the eigen characteristic of the
investigated system.
of nickel oxide under hydrogen atmosphere. Using Eq. (13), the
density distribution function of activation energies at different
heating rates was calculated.
The influence of heating rate on the shape of the density
ꢀ1
distribution function of activation energies (gðE
a
Þvhj ) is shown
in Fig. 5.
The basic characteristics of the density distribution function
of activation energies: Ea,max—the value of activation energy at the
It can be pointed out that the calculated distribution functions
(Eq. (13)) at all considered heating rates cannot be fitted by any of
the general probability functions (e.g. Gaussian, log-normal,
Lorentz, logistic, Pearson or Weibull distribution functions).
From these facts, it can be concluded that the calculated
density distribution functions for the non-isothermal reduction
process of NiO by hydrogen represent the unique distribution
maximal value of distribution function, gðE
value of density distribution function, gðE
a
Þ
vhj;max—the maximal
vhj;HW—the value of
a
Þ
density distribution function on the half-wide, SF—shape factor
or factor of asymmetry and HW—the half-wide of the density
distribution function, at the different heating rates, are presented
in Table 3.

ARTICLE IN PRESS
B. AdnaXevi ´c , B. Jankovi ´c / Physica B 403 (2008) 4132–4138
4137
function (the finger-print distribution) in describing the kinetics
of the investigated process.
1.0
0.8
a
It was shown [18–21] that if E and A depend on a, the kinetic
parameters are correlated through the relation of the compensa-
tion effect [22,23]:
ln A ¼ a
c
þ b
c
Ea;
x
(21)
x
0
0
0
0
.6
.4
.2
.0
where a
c
and b
c
are the compensation constants and the subscript
x
refers to a factor that produces a change in kinetic parameters.
Analyzing the mutual inter-dependence of kinetic parameters
for the isoconversional KAS method [15,16], this is ln[AE /Rg( )]
) is the integral reaction model) (included in the
) at the same
degree of conversion, it was found that the following linear
relationship is satisfied for every considered for
(
a
a
where g(
a
intercept of the isoconversional line in Fig. 2 and Ea,
a
a
ꢂ
ꢃ
AE
a
ln
¼ ꢀ9:7526 þ 0:2826Ea;
a
(22)
2
50
300
350
400
450
500
RgðaÞ
where the obtained values of compensation constants for the
reduction process of NiO under the hydrogen atmosphere are
Fig. 6. Comparison between the experimental (symbols) and simulated (model-
prediction) (full lines) conversion ( –T) curves for the investigated reduction
ꢀ
1
a
a
c
¼ ꢀ9.7526 and b
According to the literature [24], this is confirmation of a true
compensation effect in the cases when the dependence of ln A
versus E is linear. A necessary condition for a compensation effect
to be true (or real) is the ability of the factor to affect the
c
¼ 0.2826 mol kJ
.
process of nickel oxide using hydrogen.
a
Table 4
x
The residual sum of squares of deviations of actual and calculated model
temperature dependence of the reduction rate [24]. Obviously, for
the investigated non-isothermal reduction process of NiO under
prediction values, from the temperature-programmed reduction (TPR) data at
different heating rates (vhj ¼ 2.5, 5, 10 and 20 1C minꢀ
1
)
the hydrogen atmosphere, the role of
fraction ( ) (Fig. 3).
Based on the calculated ddfE
conversion curves for the non-isothermal reduction process of
NiO under hydrogen atmosphere can be evaluated from the
following equation [25]:
x
factor has a conversion
Temperature-programmed reduction data
a
a
’s ðgðE Þ, the simulated
a
Þ
v
ꢀ1
hj
vhj/1C min
RSS
2.5
5
2.5310 ꢂ 10ꢀ
6
5
5
ꢀ
6.9511 ꢂ 10
ꢀ
10
8.2184 ꢂ 10
Z
ꢂ
Z
ꢀ
ꢁ
ꢃ
ꢀ4
1
T
20
3.1519 ꢂ 10
A
E
a;a
avhj ¼ 1 ꢀ
exp ꢀ v
exp ꢀ
dT gðE
a
Þvhj dE
a
(23)
RT
0
hj
0
The above model equation assumed that all considered
where
point i,
a
a,i is the actual value of conversion fraction in the data
c,i is the value calculated from Eq. (23) in the
reactions that are included in the complex process share the
same pre-exponential (frequency) factor, so the reactivity dis-
tribution is represented by a continuous distribution of activation
energies [11].
a
corresponding point and n is the number of data points.
Table 4 shows the values of RSS for model prediction of the
investigated reduction process described by Eq. (23).
From Table 4 we can see that the model prediction yields the
best agreement at the heating rate value of 2.5 1C min , but at the
In the above equation, A represents the value of the pre-
exponential factor calculated from our previous paper [17]
ꢀ1
8
ꢀ1
(
A ¼ 1.66 ꢂ10 min
(Eq. (23)), while gðE
a
Þ
represent the
v
hj
other values of heating rates, the obtained predictions give
satisfactory results.
density distribution function of activation energies calculated
a
from Eq. (13), for the apparent activation energy (E ) values
The above-established results confirm the previously exhibited
assumption, about the possibility of explanation of the kinetics of
the non-isothermal reduction process of nickel oxide by hydrogen,
using the distributed activation energy model, as the consequence
of the existence of oxygen vacancies with different potential
energies. In accordance with Rodriguez et al. [6], the obtained
results illustrate the complex role played by oxygen vacancies in
the mechanism of nickel oxide reduction under hydrogen atmo-
sphere. The complex behavior of the investigated process can be
interpreted through the interaction between the hydrogen and
oxygen vacancies with the distribution of potential energies,
which leads to the existence of specific distribution functions of
obtained from the KAS isoconversional method (Fig. 3). The
process has been simulated at four linear heating rates of 2.5, 5, 10
ꢀ1
and 20 1C min in the conversion fraction range of 0.05pap0.95.
A number of mathematical approaches have been pursued to
deal with the double integration of Eq. (23) [26]. In this study,
Simpson’s one-third rule [27] for integration has been used for the
numerical solution of Eq. (23). For all mathematical computations,
s
operation tools from the Mathematica 5 program package were
used.
Fig. 6 shows the comparison between the experimental
(symbols) and simulated (model prediction) (full lines) conver-
sion ( –T) curves for the investigated reduction process of NiO
a
a
E ’s (the finger-print distributions), at all considered heating rates.
using hydrogen.
The residual sum of squares of deviations of actual and
calculated model prediction values (Eq. (23)) is commonly used.
It is defined by the following relation [28]:
5. Conclusions
The kinetics of the non-isothermal reduction process of nickel
oxide under hydrogen atmosphere was accurately determined
from a series of temperature-programmed experiments at the
n
X
2
RSS ¼
ð
aa;i
ꢀ
ac;i
Þ
(24)
i¼1

ARTICLE IN PRESS
4138
B. AdnaXevi ´c , B. Jankovi ´c / Physica B 403 (2008) 4132–4138
different constant heating rates. The new procedure for the
determination of the density distribution function of activation
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