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Chemical Engineering Journal 175 (2011) 396– 407

Contents lists available at SciVerse ScienceDirect

Chemical Engineering Journal

jo u r n al hom epage: www.elsev ier .com/ locate /ce j

inetic study of double-walled carbon nanotube synthesis by catalytic chemicalapour deposition over an Fe-Mo/MgO catalyst using methane as the carbonource

igrid Douvena,∗, Sophie L. Pirarda, Georges Heyenb, Dominique Toyea, Jean-Paul Pirarda

Laboratoire de Génie Chimique, B6a, Université de Liège, B-4000 Liège, BelgiumLaboratoire d’Analyse et de Synthèse des Systèmes Chimiques, B6a, Université de Liège, B-4000 Liège, Belgium

r t i c l e i n f o

rticle history:eceived 20 June 2011eceived in revised form 19 August 2011ccepted 24 August 2011

eywords:arbon nanotubes

a b s t r a c t

A kinetic study was performed to describe experimental reaction rates of double-walled carbon nanotube(DWNT) synthesis by catalytic chemical vapour deposition (CCVD) over an Fe-Mo/MgO catalyst usingmethane as the carbon source. Initial reaction rates were determined by mass spectrometry for methanepartial pressures ranging from 0.01 atm to 0.9 atm and for three temperatures: 900 ◦C, 950 ◦C and 1000 ◦C.Mass transfers from the bulk of the fluid to the external surface of the catalytic bed and through thecatalytic bed were negligible as determined experimentally and confirmed by the methane mass balance.

CVD processinetic studyethaneass transfereactivation

A detailed kinetic study was carried out to discriminate between phenomenological kinetic models andto validate the good agreement between the chosen model and the experimental data. The best modelwas found to involve the irreversible dissociative adsorption of methane followed by the irreversibledecomposition of the adsorbed methyl group, which is the rate-determining centre. Activation energyof adsorbed methyl decomposition was found to be equal to 58 kJ mol−1. The catalytic deactivation bycoking was expressed by a decreasing function of coke content, which leads to a sigmoid equation.

. Introduction

Hollow carbon fibres have been observed for many decades [1]ut it was the ground-breaking report by Iijima [2] that made car-on nanotubes one of the most actively investigated materials. Theonsiderable interest in carbon nanotubes is due to their uniqueroperties and potential applications [3,4]. Double-walled carbonanotubes (DWNTs) consist of two concentric graphene sheetsolled together in a nanometric cylinder. These are the simplestxample of multi-walled carbon nanotubes (MWNTs), in terms ofheir structural aspect. Nevertheless, DWNTs have a thin diameter,hich confers to them properties similar to those of single-walled

arbon nanotubes (SWNTs). So, the structural stability of MWNTs5,6] associated with the electronic properties of SWNTs makeWNTs an interesting and promising material. Studies [7–9] have

hown that, in the future, it would be theoretically possible toroduce semiconductor–metallic tubes or metallic–semiconductorubes; the latter represent a molecular wire covered by an insulator

or use as connectors in nanoelectronic systems or as a molecularapacitor in memory devices [10].

∗ Corresponding author. Fax: +32 4366 3545.E-mail address: S.Douven@ulg.ac.be (S. Douven).

385-8947/$ – see front matter © 2011 Elsevier B.V. All rights reserved.oi:10.1016/j.cej.2011.08.066

© 2011 Elsevier B.V. All rights reserved.

Different methods have been used to produce carbon nan-otubes: arc discharge [11], laser ablation [12], solar energy furnace[13] and catalytic chemical vapour deposition (CCVD) [14]. TheCCVD method consists of the decomposition of a hydrocarbonvapour on a catalyst into solid carbon and gaseous hydrogen at hightemperature. The CCVD process appears to be the easiest methodto scale-up production capacity at a viable cost [15,16]. Indeed, theCCVD process can operate continuously and offers the potential forhigh yield production. Although other methods can produce high-quality nanotubes, the yields are limited and these techniques arenot easily transferred to industrial production. Large-scale reactorsusing the CCVD process to produce carbon nanotubes by met-ric tons are already running, using either a fluidized bed reactor[17–19] or an inclined mobile-bed rotating reactor [16,20–22]. Theinclined mobile-bed rotating reactor seems to be one of the mostappropriate technologies because the kinetics of carbon nanotubesynthesis by hydrocarbon decomposition is quite slow, and becausethe ratio between the volume of the product and the volume ofthe catalyst can be very large [16]. Successful development of alarge-scale CCVD reactor requires taking into account all the fac-tors governing its performance: geometric, hydrodynamic, physical

and physico-chemical factors [20,23]. The present study deals withthe physico-chemical factor, involving the kinetics of reaction. Theultimate aim is to use kinetic expression in the modelling of thereactor, so as to be able to simulate reactor performance.

S. Douven et al. / Chemical Engineering Journal 175 (2011) 396– 407 397

Nomenclature

a first parameter of catalytic deactivation (–)b second parameter of catalytic deactivation (–)c third parameter of catalytic deactivation (s)C dimensionless constant (–)Cb methane concentration in the bulk of the fluid

(kmolCH4 m−3)CS methane concentration on the external catalytic bed

surface (kmolCH4 m−3)DCH4–He binary diffusion coefficient (m2 s−1)De effective diffusivity (m2 s−1)Dm molecular diffusivity (m2 s−1)db thickness of the catalytic bed (m)Ea1 activation energy of irreversible dissociative

adsorption of methane (kJ mol−1)Ea2 activation energy of irreversible decomposition of

adsorbed methyl group (kJ mol−1)F0.95,�n,�d

Fisher variable with �n numerator and �d denom-inator degrees of freedom for 95% confidence (–)

k kinetic rate constant (kmolCH4 kg−1catalyst s−1 atm−x)

k1 kinetic rate constant of irreversible dissociative ad-sorption of methane (kmolCH4 kg−1

catalyst s−1 atm−1)k2 kinetic rate constant of irreversible decomposition

of adsorbed methyl group (kmolCH4 kg−1catalyst s−1)

k1ref pre-exponential factor of kinetic rate constant k1(kmolCH4 kg−1

catalyst s−1 atm−1)k2ref pre-exponential factor of kinetic rate constant k2

(kmolCH4 kg−1catalyst s−1)

kM mass transfer coefficient (m s−1)l characteristic length (m)L length of the catalytic bed (m)ma dimensionless mass of active catalyst (–)mcatalyst mass of catalyst (kg)MCH4 molecular weight of CH4 (kg kmol−1)MHe molecular weight of He (kg kmol−1)PCH4 methane partial pressure (atm)PHe helium partial pressure (atm)Pt total pressure (atm)r specific reaction rate (kmolCH4 kg−1

catalyst s−1)

r0 initial specific reaction rate (kmolCH4 kg−1catalyst s−1)

r0 initial specific reaction rate corresponding to thetheoretical model (kmolCH4 kg−1

catalyst s−1)

R perfect gas constant (kJ mol−1 K−1)Re Reynolds number (–)s2

e experimental variance ( kmol2CH4kg−2

catalyst s−2)

s2i residual variance of Model i (kmol2CH4

kg−2catalyst s−2)

Sc Schmidt number (–)Sh Sherwood number (–)t time (s)t0 parameter of catalytic deactivation (s)T temperature (K or ◦C)Tref reference temperature (K or ◦C)u flow velocity (m s−1)x methane partial order (–)xCH4 methane mole fraction (%)xH2 hydrogen mole fraction (%)xHe helium mole fraction (%)Yij value of initial specific reaction rate of experiment j

at temperature i (kmolCH4 kg−1catalyst s−1)

Yij mean value of initial specific reaction rate of exper-iment j at temperature i (kmolCH4 kg−1

catalyst s−1)

Yij calculated value of initial specific reaction rateby Model 2 of experiment j at temperature i(kmolCH4 kg−1

catalyst s−1)Z axial variable from the boat wall to the external sur-

face of the bed (m)

Greek symbols relative loss of mass of the catalyst due to moisture

and solvent elimination at high temperature (–)�2

(0.95,�) �2 variable corresponding to � degrees of freedomfor 95% confidence (–)

ε porosity of the catalytic bed (–) Weisz modulus (–)

� internal effectiveness factor (–)�G global effectiveness factor (–)� kinematic viscosity (m2 s−1)�i number of degrees of freedom of Model i (–)�m number of degrees of freedom of empirical power

law model (–)�e number of degrees of freedom corresponding to

experimental data (–)� density of the catalytic bed (kgcatalyst m−3)

�CH4−He characteristic length (Lennard–Jones potential) (oA)

� tortuosity of the catalytic bed (–)

˝CH4–He diffusion collision integral (Lennard–Jones poten-

tial) (–)

The CCVD process requires the use of an adequate catalyst forthe reaction. The number of walls, and the quality and purity ofcarbon nanotubes will depend on the catalyst, carbon source andoperating variables such as temperature, flow rates and feedstockpartial pressures [24]. Metals used to catalyse carbon nanotube for-mation are most often transition metals, in particular, iron [25,26],nickel [27,28] and cobalt [29,30]. Mixtures of transition metals areoften more efficient than one metal alone. For example, papershave reported carbon nanotube formation on iron-nickel- [31],iron-cobalt- [31] and nickel-cobalt-based [32] catalysts. Some othermetals have also been used as co-catalysts. Even if not necessar-ily active alone, these metals help to improve the performanceof the usual transition-metal-based catalysts. The most importantco-catalyst is molybdenum. The addition of molybdenum to iron[33,34], cobalt [24,35] and nickel [36,37] catalysts can lead to theformation of thin carbon nanotubes.

A lot of research has been done to understand the growth mech-anism of carbon nanotubes [38,39] but very few studies have beencarried out in order to determine the sequence of elementary stepsinvolved in carbon nanotube synthesis. One kinetic study of MWNTsynthesis over an Fe-Co/Al2O3 catalyst using ethylene as the car-bon source assumed that the rate-determining step could be theelimination of the first hydrogen atom of adsorbed ethylene. Theactivation energy was found to be equal to around 130 kJ mol−1

[40]. A study of SWNT synthesis by ferrocene vapour decomposi-tion in carbon monoxide reports an activation energy of 1.39 eV,i.e. 134 kJ mol−1 [41], and considers that the limiting step is thecarbon diffusion in Fe particles. One study of the growth rate of ver-tically aligned small-diameter carbon nanotubes is reported using

acetylene as the carbon source and an Fe/Al2O3 catalyst. The activa-tion energy was found to be equal to around 1 eV, i.e. 96 kJ mol−1,and the authors interpret this as a diffusion-limited growth rate[42]. Another study of MWNT synthesis with acetylene mentioned

398 S. Douven et al. / Chemical Engineering Journal 175 (2011) 396– 407

Flow controllers

FurnaceHot zo ne Cold zon e

Mass spectrometer

Exhaust gases

He

CH4

H2

Bypa ss

for k

a6pbdbosfivthanofa[t

epspdnmftactattotrrfibtur

2

Fptoib

Bypass va lve Reactor

Fig. 1. Experimental set-up

n activation energy of 26 kJ mol−1 over an Fe/Al2O3 catalyst and6 kJ mol−1 over a Ni/Al2O3 catalyst [43]. A kinetic study of SWNTroduction over Co-Mo/MgO catalysts using methane as the car-on source reported that the rate-determining step would be theissociation of methane [24]. The activation energy was found toe equal to 160 kJ mol−1 over Co-Mo/MgO catalysts and 96 kJ mol−1

ver Co/MgO catalysts. The decomposition of methane into carbonpecies has also been investigated by researchers working in theeld of petrochemicals because of its importance in catalyst deacti-ation. Indeed, the formation of filamentous carbon can even causehe catalyst support structure to disintegrate. Two kinetic studiesave reported carbon filament formation by methane cracking over

Ni catalyst but they each assumed different underlying mecha-isms. The first study assumed a reversible molecular adsorptionf methane followed by the abstraction of the first hydrogen atomrom molecularly adsorbed methane as the rate-determining stepnd evaluated the activation energy as being equal to 59 kJ mol−1

44]. The second study assumed that methane adsorbed dissocia-ively before the stepwise dehydrogenation of carbon species [45].

In the present paper, a kinetic study is presented to describexperimental initial reaction rates of DWNT synthesis by the CCVDrocess over an Fe-Mo/MgO catalyst using methane as the carbonource. The determination of initial specific reaction rates is madeossible by the use of a mass spectrometer placed at the exit of theevice. This experimental device is probably unique in its determi-ation of experimental reaction rates. Several phenomenologicalodels are fitted onto experimental data. A statistical study is per-

ormed in order to discriminate between these kinetic models. Ashe models correspond to different mechanisms, the discriminationllows a better understanding of the elementary steps involved inarbon nanotube production. In the reaction rate expression, par-ial pressures are equal to partial pressures in the fluid if therere no diffusional limitations, i.e. in a chemical regime. Indeed,he catalytic bed is a porous medium, through which the reac-ants must diffuse to react. Some mass transfer limitations couldccur depending on the catalyst structure and experimental condi-ions (temperature, pressure etc.). The assumption of the chemicalegime has been checked experimentally and quantitatively by theesolution of the mass balance of methane. A sigmoid equation istted onto experimental data to describe the catalyst deactivationy coking. The final aim of the present study is the determination ofhe specific reaction rate of carbon nanotube synthesis in order tose it in the modelling of a continuous inclined mobile-bed rotatingeactor.

. Experimental

The experimental set-up for kinetic measurements is shown inig. 1. The reactor is a 1.5-m-long quartz tube (diameter of 0.05 m)laced in a 1-m-long thermolyne tubular furnace. The part of the

ube jutting out from the furnace is a cold zone. The reactor is closedff at its downstream end by a removable cap. This removable caps equipped with a system allowing the introduction of a quartzoat containing the catalyst, under reaction gas or under inert gas,

Removable cap

inetic rate measurements.

from the cold zone to the hot zone without opening the reactor.The cold zone allows: (i) the direct introduction of the catalyst intothe hot zone under a reactive atmosphere and (ii) after reaction,the cooling down of the products under an inert atmosphere. Thecarbon source is methane and the carrier gas is helium. The com-position of the feed gas (CH4, H2 and He) and the total flow areadjusted by three Bronkhorst mass flow controllers. By operating abypass valve, the feed gas can be directed towards either the tubu-lar reactor or the bypass. The composition of the exhaust gas isdetermined by a Balzer mass spectrometer placed at the exit of thedevice. The total pressure in the reactor is equal to atmosphericpressure. The catalyst used is an Fe-Mo/MgO catalyst, as describedelsewhere [46,47]. This experimental set-up has been chosen inorder to measure the reaction rate in a geometric configurationclose to industrial conditions. Indeed, this system is well designedconsidering that the ratio between the volume of product and thevolume of catalyst is very large.

Prior to any measurement, the mass spectrometer is calibratedby using a mixture of methane, hydrogen and helium in known pro-portions. These proportions are adjusted by using the three massflow controllers and flowing through the bypass to the mass spec-trometer. Each kinetic measurement follows this procedure: (i) thereactor is vented with helium until the exhaust gas contains at least99.5% helium; (ii) the reactor is opened; (iii) the catalyst powder(1 × 10−3 kg) is deposited in a length of 0.2 m onto a quartz boat,this quartz boat containing the catalyst powder is placed in thecold zone, the reactor is closed and the venting continues untilthe reactor contains at least 99.5% helium; (iv) the desired feedflow composition, for a total flow rate of 3.72 × 10−6 kmol s−1, isset and introduced into the reactor; (v) once the feed is stabilized,the quartz boat is introduced into the middle of the hot zone under areactive atmosphere; simultaneously, the recording of the exhaustgas composition by the mass spectrometer begins; (vi) after 15 min,the reaction is finished and the reactor is vented with helium; (vii)once the reactor contains 99.5% helium, the quartz boat is cooleddown under an inert atmosphere in the cold zone, before weighingthe nanotubes produced. Associated to each kinetic experiment, ablank measurement is performed by introducing the desired feedflow composition into the reactor without any catalyst in the hotzone. The reactor can be considered to be of the differential plugflow type [48].

In the present study, kinetic rates were determined on anFe-Mo/MgO catalyst at three temperatures (900 ◦C, 950 ◦C and1000 ◦C), and at each temperature the methane mole fraction wasvaried between 1% and 90% balanced by helium.

3. Results

3.1. Use of mass spectrometer

Fig. 2 compares the exhaust gas composition during a blankmeasurement and during a synthesis.

As can be seen in Fig. 2, the helium mole fraction decreases dur-ing the reaction, which means that the ongoing reaction increases

S. Douven et al. / Chemical Engineerin

0

10

20

30

40

50

60

70

80

90

100

9008007006005004003002001000Time (s)

Mol

e fr

actio

n (%

)

Helium

Hydrogen

Methane

Fig. 2. Mass spectrometer curves. Hollow symbols refer to blank measurements andf�v

tiCtcfsbts

r

ftctctncot

FiCa

ull ones refer to reaction curves. For both curves: � and � correspond to helium, and � correspond to methane and, © and � correspond to hydrogen. Operatingariables: xCH4 = 30%, xH2 = 0%, xHe = 70% and T = 950 ◦C.

he total number of gaseous molecules. This qualitative observations in agreement with the stoichiometry of the chemical reactionH4 → C + 2H2; the number of molecules of hydrogen produced iswice the number of molecules of methane decomposed. Methaneonsumption and hydrogen production can be assessed by the dif-erence between the blank curve and the reaction curve. The initialpecific reaction rate r0 is determined from those curves providedy the mass spectrometer. Following the reaction stoichiometry,he relationship between hydrogen production and methane con-umption is given by:

0 = − 1(1 − ˛)mcatalyst

d [CH4]dt

= 12(1 − ˛)mcatalyst

d [H2]dt

(1)

Therefore, the initial specific reaction rate, r0, can be calculatedrom either the slope at the initial time of the hydrogen produc-ion curve (Fig. 3) or the slope at the initial time of the methaneonsumption curve. A comparison of the results confirms that ini-ial specific reaction rates calculated from the hydrogen productionurves are equal to those calculated from the methane consump-ion curves, proving that the mass balance is respected. So, there iso side reaction and no leak in the system. Hydrogen production

urves were used in this study. In Eq. (1), mcatalyst is the initial massf catalyst and is the relative loss of mass of the catalyst dueo moisture and solvent elimination when the catalyst is placed

2 (1-α )m catalyst r

0

0.1

0.2

9008007006005004003002001000Time (s)

103 x

Hyd

roge

n pr

oduc

tion

(km

ol)

0

ig. 3. Example of hydrogen production curve as a function of time with the slope atnitial time corresponding to initial reaction rate, r0, for a feed gas composition of 30%H4, 70% He at 950 ◦C. The grey line and the black line correspond to experimentalnd modelled hydrogen production curves respectively (Section 4.7).

g Journal 175 (2011) 396– 407 399

at high temperature. As experimentally determined, is equal to10.6%. The relative loss of mass was determined by weighing thecatalyst beforehand. Then the catalyst powder is placed onto thequartz boat, introduced in the reactor at the reaction tempera-ture under a helium flow for 15 min. The catalyst is then weightedagain. The mass difference is the loss of mass. In Fig. 3, the delayof around 100 s prior to the reaction beginning is the time neces-sary for the gases to pass through the reactor and to reach the massspectrometer, placed at the exit (Fig. 1).

The first part of this study focuses on the initial specific reac-tion rate, r0. As illustrated in Fig. 3, a large part (0.08 × 10−3 kmol)of the total hydrogen production (0.16 × 10−3 kmol) and there-fore of carbon deposits refers to initial specific reaction rate, r0.After approximately 250 s, the slope of the hydrogen productioncurve decreases whereas the operating conditions do not vary. Thisdecrease in the hydrogen production rate is due to catalytic deac-tivation, which is studied in Section 4.7.

A transmission electron micrograph (TEM) image and a scan-ning electron micrograph (SEM) image of a purified sample, i.e. asample whose catalyst is removed, in Fig. 4 show that carbon nan-otubes are the main product of the reaction. Calculated masses fromspectrometer curves are in agreement with the weighted massesof the nanotubes produced. It can be concluded that methaneis decomposed into carbon species such as carbon nanotubes,graphite, amorphous carbon or carbon black without the formationof gaseous or vaporized liquid hydrocarbons heavier than methane.

3.2. Mass transfer

The catalytic bed is a porous medium where active sites arehomogeneously dispersed. So, reactants must diffuse from the bulkof the gas to the external surface of the bed and then from theexternal surface of the bed to the active sites, where the reactiontakes place. As a consequence, the observed reaction rate coulddiffer from the true reaction rate in chemical regime, i.e. withoutmass transfer limitations. The global effectiveness factor, �G, whichincludes both external and internal resistances, is defined as theratio of the reaction rate with pore diffusion resistance and filmresistance to the reaction rate with concentration in the bulk of thefluid [48].

3.2.1. External mass transferIn order to check that there are no external transfer limitations,

the external mass transfer resistance can be estimated by [23]:

fe = 1 − CS

Cb= r0�db

CbkM(2)

where Cb is the methane concentration in the bulk of the fluid and iscalculated by dividing the methane mole fraction by the tempera-ture and the perfect gas constant R, CS is the methane concentrationon the external catalytic bed surface, r0 is the initial specific reac-tion rate, � is the density of the catalytic bed, db is the thickness ofthe catalytic bed and kM is the mass transfer coefficient. fe variesfrom 1, for a mass transfer limited situation, to 0, for a reactionlimited situation.

The mass transfer coefficient kM can be correlated to the localflow conditions and is typically obtained from a correlation of theform:

Sh = kMl

De= C Re1/2 Sc1/3 (3)

where Sh is the Sherwood number, Re = ul/� is the Reynolds num-ber, Sc = �/De is the Schmidt number, C is a dimensionless constant,l is a characteristic length, De is the effective diffusivity, u is the flowvelocity and � is the kinematic viscosity.

400 S. Douven et al. / Chemical Engineering Journal 175 (2011) 396– 407

SEM im

uttttm

hrvcai

at

ti

r

i

0wI

3

i

F0

2

Fig. 4. (a) TEM image; (b)

Unfortunately, the reactor geometry here is very complex andnusual, so no suitable correlation can be found in the literature;he reactive surface is very small compared to the total surface ofhe tube and the flow of the fluid in the reactor is perturbed byhe quartz boat and the growing nanotubes which, at the end ofhe reaction, occupy a large part of the reactor cross-section. So the

ass transfer coefficient kM cannot be calculated a priori.Therefore, a diagnostic test for external mass transfer limitations

as been carried out. In this experimental test, the initial specificeaction rate r0 is measured at constant space-time, but the flowelocity is varied by adapting simultaneously the volume of theatalytic bed and the total flow rate [49]. That way, the only oper-ting variable that changes is the gas flow velocity, which directlympacts the mass transfer coefficient, according to Eq. (3).

As the initial specific reaction rate, r0, is not affected by the vari-tion of the total flow rate (Fig. 5), the conclusion is drawn thathere are no external transport limitations.

Furthermore, in an external mass transfer situation, i.e. whenhe methane partial pressure at the external catalytic bed surfaces equal to zero, the kinetics can be written as:

0 = kMPCH4

�LRT(4)

The methane partial order is equal to 1 and the activation energys equal to a few J mol−1.

But in this study, the methane partial order is closed to 0 for.3 < PCH4 < 0.9 and the activation energy is equal to 58 kJ mol−1

hich is a characteristic value for a chemical reaction (Section 4.5).n conclusion, there is no external mass transfer resistance.

.2.2. Internal mass transferIn the absence of external diffusional limitations, the effect of

nternal mass transfer on the reaction rate may be evaluated by

0

1

2

3

4

6420

106 Total fl ow rate (kmol s-1)

104

×r

(km

ol

kg

s

) C

H4

cata

lyst

-1

-1

0 ′

ig. 5. Initial specific reaction rate, r0, as a function of the total flow rate for PCH4 =.6 atm, PH2 = 0 atm, PHe = 0.4 atm at 900 ◦C.

age of a purified sample.

measuring the dependence of the initial specific reaction rate, r0,with the thickness of the catalytic bed, db.

The experimental test consists of measuring the initial specificreaction rate, r0, with various thicknesses of the catalytic bed, underidentical operating conditions. This means that the total flow rateand the volume of catalyst are kept constant. With a constant massof catalyst, the variation of the bed thickness is obtained by varyingthe length of the catalytic bed. In the plot of the observed initialspecific reaction rate r0 vs. the catalyst bed thickness expressed interms of mass per unit length, constant values are expected for thechemical regime as the total mass of catalyst does not vary. If theobserved initial specific reaction rate decreases when the catalyticbed thickness increases, internal mass transfer limitations occur[49].

At first (Fig. 6), the initial specific reaction rate, r0, is independentof the mass of catalyst per unit length until around 5 × 10−3 kg m−1

and then it decreases as the mass per unit length increases. Theexperiments used in this kinetic study were performed with a massper unit length of around 4 × 10−3 kg m−1 in order to avoid internaldiffusion problems.

Besides the experimental test, a quantitative approach was usedto evaluate the internal diffusion, based on the calculation of theWeisz modulus. The Weisz modulus compares the initial spe-cific reaction rate to the diffusion flow of methane at the externalcatalytic bed surface. If « 1, methane diffusion is not the limitingfactor and the operating regime is a chemical regime. If » 1, diffu-sion limitations occur in the catalytic bed and the operating regimeis a diffusional regime. The Weisz modulus is defined by [48]:

= r0�dbDeCS

(5)

0

1

2

3

4

1086420

103×m catalyst per unit length (kg m-1)

104

×r

(km

ol

kg

s

) C

H4

cata

lyst

-1

-1

0

Fig. 6. Initial specific reaction rate, r0, as a function of the mass per unit length forPCH4 = 0.6 atm, PH2 = 0 atm, PHe = 0.4 atm at 900 ◦C.

S. Douven et al. / Chemical Engineering Journal 175 (2011) 396– 407 401

Table 1Initial specific reaction rates r0 and associated Weisz modulus with corresponding operating methane and helium partial pressures at 900 ◦C, 950 ◦C and 1000 ◦C.

PCH4 (atm) PHe (atm) 104 × r0 (kmolCH4 kg−1catalyst

s−1) (–)

900 ◦C 950 ◦C 1000 ◦C 900 ◦C 950 ◦C 1000 ◦C

1 0.01 0.99 0.27 0.28 0.28 0.32 0.31 0.312 0.03 0.97 0.74 0.80 0.81 0.29 0.30 0.303–1 0.05 0.95 1.14 1.22 1.24 0.27 0.28 0.283–2 0.05 0.95 1.42 1.39 1.56 0.33 0.32 0.354–1 0.1 0.9 1.72 2.18 2.34 0.20 0.25 0.264–2 0.1 0.9 1.76 0.215–1 0.2 0.8 2.52 3.12 3.39 0.15 0.18 0.195–2 0.2 0.8 2.39 3.18 0.14 0.186–1 0.3 0.7 3.26 3.63 4.15 0.13 0.14 0.156–2 0.3 0.7 4.23 0.167–1 0.4 0.6 3.49 4.10 4.82 0.10 0.12 0.137–2 0.4 0.6 4.82 0.138–1 0.5 0.5 3.30 4.26 5.18 0.08 0.10 0.118–2 0.5 0.5 3.30 0.089–1 0.6 0.4 3.17 4.27 5.41 0.06 0.08 0.109–2 0.6 0.4 3.22 0.0610 0.7 0.3 3.16 4.01 5.55 0.05 0.07 0.0911 0.8 0.2 3.09 3.98 5.07 0.05 0.06 0.0712–1 0.9 0.1 2.78 3.76 4.94 0.04 0.05 0.0612–2 0.9 0.1 2.82 0.0412–3 0.9 0.1 3.28 0.0412–4 0.9 0.1 3.06 0.04

wsttmsd

D

wosafba

D

war�oa

s9celpe

variances is equal to 14.9. As the ratio is higher than the Fisher vari-able, the empirical model is not able to reproduce the experimentaldata.

Table 2Parameter estimation and associated standard deviations for the empirical powerlaw model r0 = kPx

CH4.

12–5 0.9 0.1 2.98

here r0 is the observed initial specific reaction rate, � is the den-ity of the catalytic bed (200 kgcatalyst m−3), db is the thickness ofhe catalytic bed (1 × 10−3 m), CS is the methane concentration onhe external catalytic bed surface and is calculated by dividing the

ethane mole fraction by the temperature and by the gas con-tant R, and De is the effective diffusivity through the catalytic bedefined by:

e = ε Dm

�(6)

here ε and � are respectively, the porosity and tortuosityf the catalytic bed. The porosity was estimated from mea-urements by helium pycnometry, mercury porosimetry anddsorption–desorption isotherm and varied from 0.5 and 0.7 as aunction of the compression of the catalyst powder. Tortuosity haseen approximated by 1/ε [50,51]. Dm is the molecular diffusivitynd is calculated by Chapman and Enskog’s relation [52]:

m = DCH4–He = 0.0018583T3/2

(1/MCH4 + 1/MHe

)1/2

Pt(�CH4–He)2˝CH4–He

≈ 1.648 × 10−8 T3/2 (7)

here DCH4–He is the binary diffusion coefficient, T is the temper-ture, MCH4 and MHe are the molecular weights of CH4 and Heespectively, and Pt is the total pressure. The characteristic length,CH4–He, and the diffusion collision integral, ˝CH4−He, are functionsf the Lennard–Jones potential for the CH4–He molecular systemnd are determined from tabulated values [52].

The Weisz modulus is calculated for each experiment and ishown in Table 1. In the operating conditions (PCH4 = 0.6 atm at00 ◦C) of Fig. 6, the value of the Weisz modulus is 0.06, whichonfirms the absence of internal diffusion limitations determined

xperimentally. The maximum value of the Weisz modulus, i.e. theeast favourable value, is equal to 0.35, which lets suspect someossible internal diffusion limitations and so could introduce anrror in the estimation of the reaction rate.

0.04

4. Discussion

4.1. Empirical model

The simplest approach for expressing the rate of reaction is todetermine an empirical model corresponding to a power law:

r0 = kPxCH4

(8)

where r0 is the initial specific reaction rate corresponding to themodel, PCH4 is the methane partial pressure, k is the kinetic rateconstant and x is the methane partial order of reaction. Parame-ter estimation was performed for each temperature. Results of theparameter estimation are presented in Table 2.

A statistical Fisher F-test, with 95% confidence is performedin order to validate the agreement between the empirical modeland the experimental data [53,54]. The value of Fisher variableF0.95,�m,�e is compared to the ratio between the variance of themodel and the experimental variance. The degrees of freedom of themodel (�m) for the Fisher F-test are equal to the number of experi-ments (12 for each temperature) minus the number of parameters(2). For the experimental data, the degrees of freedom (�e) corre-spond to the number of replicates minus the number of means. Theresults of the F-tests lead us to conclude that the empirical modelis not able to reproduce the experimental data. For example, at900 ◦C, �m = 10, �e = 9, so F0.95,�m,�e = 3.1, while the ratio between

Parameters 900 ◦C 950 ◦C 1000 ◦C

104 × k (kmolCH4 kg−1catalyst

s−1 atm−x) 3.45 ± 0.15 4.80 ± 0.30 6.30 ± 0.30x (–) 0.30 ± 0.05 0.35 ± 0.05 0.40 ± 0.05

4 neering Journal 175 (2011) 396– 407

4

tdnampAlaw

lpb

tmmts

etore[t(otmimha[

0.80

0.85

0.90

0.95

1.00

10.80.60.40.20Z /d b

e,CH

CH

P

P = 0.14 = 0.18

= 0.25 = 0.28 = 0.30

= 0.31

Φ = 0.05 = 0.06 = 0.07 = 0.08 = 0.10 = 0.12

Fig. 7. Profiles of methane partial pressure (dimensionless) from the bottom (Z/db =0) to the top (Z/db = 1) of the catalytic bed, for Model 2 at 950 ◦C: PCH4,e = 0.01 atm

02 S. Douven et al. / Chemical Engi

.2. Kinetic models

As a simple empirical model is not suitable to formulate the reac-ion rate, the reaction mechanisms have to be studied in greateretail. Several rate equations are derived from various schemes ofanotube synthesis. In fact, different sequences of elementary stepsnd rate-determining reactions can be envisaged to model experi-ental data. The estimation of the reaction order towards methane

artial pressure allows a first discrimination between these models.ccording to the results of the adjustment of the empirical power

aw model in Section 4.1, the methane order lies between 0 and 1nd increases with temperature. The kinetic constant also increasesith temperature.

Among the envisaged mechanisms, the following phenomeno-ogical models are retained; they are in agreement with therevious conclusion, i.e. leading to a methane partial order ofetween 0 and 1:

Model 1: CH4 + s1 + s2k1−→CH3–s1 + H–s2

CH3–s1 + s2k2−→ . . .

leading to the rate equation:

r0 = k1PCH4

1 + k1/k2PCH4

(9)

Model 2: CH4 + 2sk1−→CH3–s + H–s

CH3–s + sk2−→ . . .

leading to the rate equation:

r0 = k1PCH4(1 + k1/k2PCH4

)2(10)

Model 3: CH4 + 2sk1−→CH3–s + H–s

CH3 − s + CH3 − sk2−→ . . .

leading to the rate equation:

r0 = k1PCH4(1 +

(k1/k2

)0.5P0.5

CH4

)2(11)

Models 1, 2 and 3 assume an irreversible dissociative adsorp-ion of methane followed by the dehydrogenation of the adsorbed

ethyl group, which is the rate-determining active centre, i.e. theost abundant centre. They differ by the presence of one type (s) or

wo types (s1 and s2) of active sites and by the number of adsorbedpecies involved in the kinetically significant step.

Some phenomenological mechanisms leading to similar ratequations can be identified. One can assume the reversible adsorp-ion of hydrocarbon on the active site followed by the eliminationf the first atom of hydrogen from the adsorbed hydrocarbon as theate-determining step. This type of mechanism was proposed by Liut al. [55] using acetylene as the carbon source and by Pirard et al.40] using ethylene on iron-based catalysts. Associated rate equa-ions are mathematically similar to the rate equations of Model 1Eq. (9)) and Model 2 (Eq. (10)) but the physico-chemical meaningf the parameters is different [56]. Snoeck et al. [44] also proposedhis type of mechanism for carbon filament formation by catalytic

ethane cracking and justified their choice by the fact that exper-ments were performed at pressures higher than 0.5 MPa. Those

echanisms were rejected, as the molecular adsorption of methaneas a low probability of occurring compared to the dissociativedsorption of methane. It is for this reason that Alstrup and Tavares45], studying carbon filament formation by methane cracking,

(), 0.03 atm (+), 0.05 atm ( ), 0.1 atm (�), 0.2 atm (�), 0.3 atm (�), 0.4 atm (�),0.5 atm (♦), 0.6 atm (×), 0.7 atm (©), 0.8 atm (�), 0.9 atm (�). Weisz modulus isindicated on the right.

assumed the dissociative reversible adsorption of methane; but thismechanism leads to a methane partial order of reaction equal to 1and not one of between 0 and 1.

One can also assume the irreversible adsorption of hydrocarbonon the active site and the bulk diffusion of carbon to be the rate-determining step. This mechanism was proposed by Lee et al. [57]in a study of carbon nanotube synthesis using acetylene as the car-bon source over an iron-based catalyst. The rate equation obtainedis mathematically equivalent to the rate equation of Model 1 (Eq.(9)). The results of parameter adjustment are the same and a sta-tistical study cannot distinguish between these models, since onlythe meaning of the parameters differs [56].

4.3. Profiles of methane partial pressure

The mass balance of methane has been established in order todetermine the methane gradient within the catalytic bed and toestimate the possible error. In fact, doubt remains over possibleinternal diffusion limitations at the lowest methane partial pres-sures, and neglecting the diffusion problem could lead to an errorin the values of the calculated kinetic parameters. The mass balanceis detailed in the Appendix.

As an example, Fig. 7 illustrates, for Model 2 at the intermediatetemperature 950 ◦C, methane partial pressure profiles through thecatalytic bed. The corresponding Weisz moduli (Table 1) are alsoindicated. We should mention that the y-axis is enlarged to distin-guish the different profiles. Z is the axial variable on the thicknessof the catalytic bed. According to the y-axis, the profiles of methanepartial pressure are rather flat. However, the higher the Weisz mod-ulus is, the more pronounced the profile. In this case, the chemicalreaction is faster than the diffusion and the concentration gradientof methane is higher.

The internal effectiveness factor, �, is defined as the ratiobetween the experimental reaction rate and the reaction rate if allthe catalytic bed was at the same concentration, equal to the con-centration at the surface of the catalytic bed, which is identical tothe concentration in the bulk of the fluid as there are no externaldiffusion problems.

Experimental values of the internal effectiveness factor, �, aresomewhat higher than the characteristic curve of the first-orderreaction represented by the solid line on Fig. 8. In fact, the reac-tion order is between 0 and 1 (Section 4.1). The minimal value

of the internal effectiveness factor is 0.873 for a methane partialpressure of 0.01 atm. It becomes higher than 0.950 for a methanepartial pressure of 0.1 atm. The kinetic parameters were calcu-lated through the methane mass balance. These values were then

S. Douven et al. / Chemical Engineering Journal 175 (2011) 396– 407 403

0.1

1

1010.10.01Φ

η

Fig. 8. Internal effectiveness factor � as a function of Weisz modulus at 900 ◦C(©), 950 ◦C (�) and 1000 ◦C (�). The solid line is the theoretical equation for a first-oEm

ct

4

tpmwmasl

cvoFwtemw

v2m

TKd

Table 4Statistical Fisher F-tests neglecting the internal diffusion limitations and taking intoaccount these limitations.

T (◦C) Model s2i/s2

2 F0.95,�i,�2

Without internaldiffusion limitations

With internaldiffusion limitations

900 1 4.00 3.73 2.982 – – –3 6.59 6.07 2.98

950 1 7.33 6.62 2.982 – – –3 14.87 13.50 2.98

1000 1 2.09 1.98 2.98

rder reaction and a flat plate catalyst. The dotted line is for a zero-order reaction.xperimental values are represented by hollow symbols as a function of the Weiszodulus given in Table 1.

ompared to those obtained neglecting the internal diffusion limi-ations.

.4. Model discrimination

In order to discriminate between the three different rate equa-ions (Eqs. (9)–(11)) on the basis of experimental data, kineticarameters are estimated for each model at each temperature. Aaximum likelihood formulation is adopted, thus minimizing theeighed sum of squares of the differences between calculated andeasured reaction rates. Parameters are shown in Table 3 with

ssociated standard deviations: (i) neglecting the internal diffu-ion limitations and (ii) taking into account the internal diffusionimitations.

A statistical Fisher F-test, with 95% confidence, allows for dis-rimination between the kinetic models. The value of the Fisherariable F0.95,�1,�2 is compared to the ratio between variances (s2

i )f the different models. If the ratio of variances is higher than theisher variable, then the models are statistically different. Other-ise, they are not distinguishable. The degrees of freedom (�i) for

he Fisher F-test are equal to the number of experiments (12 forach temperature) minus the number of parameters (2 for eachodel). F-tests are presented in Table 4 in both cases, i.e. with andithout diffusion limitations.

As can be seen in Table 4, at 900 ◦C and 950 ◦C, the ratios between

ariances of Models 1 or 3 and the variance corresponding to Model

are higher than the Fisher variable. So discrimination between theodels is possible and it can be seen that Model 2 is the best one.

able 3inetic parameters of Model 1, Model 2 and Model 3: (i) neglecting the internaliffusion limitations, (ii) taking into account the internal diffusion limitations.

T (◦C) Model Without internaldiffusion limitations

With internal diffusionlimitations

104 × k1 104 × k2 104 × k1 104 × k2

900 1 42 ± 5 3.5 ± 0.1 46 ± 6 3.5 ± 0.12 26 ± 1 13.1 ± 0.1 28 ± 1 13.1 ± 0.13 140 ± 40 4.7 ± 0.3 161 ± 46 4.6 ± 0.3

950 1 41 ± 5 4.8 ± 0.2 46 ± 6 4.8 ± 0.22 30 ± 1 16.5 ± 0.1 32 ± 1 16.6 ± 0.13 105 ± 30 7.1 ± 0.7 126 ± 37 6.9 ± 0.7

1000 1 37 ± 3 6.6 ± 0.3 40 ± 4 6.5 ± 0.32 28 ± 1 21.0 ± 0.3 30 ± 1 21.0 ± 0.33 85 ± 15 11 ± 1 101 ± 20 10 ± 1

2 – – –3 3.92 3.62 2.98

At 1000 ◦C, Models 3 and 2 are significantly different but Models 1and 2 are not significantly different, since the ratio is smaller thanthe Fisher variable.

As Model 2 is the only one to satisfy the Fisher test for all tem-peratures, it can be concluded that the models can be discriminatedand Model 2 is the one best able to reproduce the experimentalkinetic data.

4.5. Parameter estimation

In order to estimate kinetic parameters, parameter estimationwas performed for the best model (Model 2) with all the kineticdata at all temperatures together. So, kinetic parameters have to beexpressed as a function of temperature following Arrhenius’ law:

k1 = k1ref exp[−Ea1

R

(1T

− 1Tref

)], (12)

k2 = k2ref exp[−Ea2

R

(1T

− 1Tref

)], (13)

where Ea1 and Ea2 are the activation energies of the first and thesecond elementary steps respectively; k1ref and k2ref are the cor-responding pre-exponential factors. T is the temperature and Trefis a reference temperature fixed to an intermediate value in thetemperature domain studied (927 ◦C). Since the results withoutconsidering internal diffusion limitations are almost identical tothose including internal diffusion limitations, the estimation of theparameters was performed neglecting the diffusion phenomenon.

From parameter estimation, the activation energy Ea2 of irre-versible decomposition of the adsorbed methyl group is found to beequal to 58 kJ mol−1, while the activation energy Ea1 of irreversibledissociative adsorption of methane is not significantly differentfrom zero (8 ± 6 kJ mol−1). Model 2 can thereby be adjusted withthe parameter Ea1 being equal to zero so that only three parametershave to be adjusted. Parameter estimation gives exactly the samevalue of activation energy Ea2 as when the value of Ea1 is allowedto vary. This confirms that the assumption of activation energy Ea1equal to 0 kJ mol−1 is relevant. A comparison of the experimen-tal rates of carbon nanotube synthesis and the rates calculated byModel 2 is presented in Fig. 9, while the parity diagram is shown inFig. 10.

The activation energy of irreversible decomposition of themethyl group is equal to 58 ± 2 kJ mol−1. As a comparison, in thekinetic study of the carbon filament formation by methane crack-ing on a Ni catalyst performed by Snoeck et al. [44], the activationenergy was found to be equal to 59 kJ mol−1, whereas Alstrup and

Tavares [45] found the activation energy to be equal to 120 kJ mol−1.However, as mentioned previously, these two kinetic studies dif-fer in the sequence of elementary steps involved in the filamentsynthesis. A kinetic study of SWNT synthesis over Mo-Co/MgO

404 S. Douven et al. / Chemical Engineering Journal 175 (2011) 396– 407

0

1

2

3

4

5

6

10.80.60.40.20

P (atm)CH4

104

×r

(km

ol

kg

s

) C

H4

cata

lyst

-1

-1

0

Fig. 9. Initial specific reaction rate, r0, as a function of methane partial pressure.Symbols correspond to experimental rates at 900 ◦C (©), 950 ◦C (�) and 1000 ◦C (�).S

cvc

4

4

ftctvptatb

n

-3

-2

-1

0

1

2

3

875 900 92 5 950 97 5 100 0 102 5

Res

idua

ls

olid lines represent rates calculated by Model 2.

atalysts using methane as the carbon source reported that acti-ation energy is equal to around 160 kJ mol−1 for Mo-Co/MgOatalysts and around 95 kJ mol−1 for Co/MgO catalysts [24].

.6. Statistical study

.6.1. Experimental validationA �2 test was carried out in order to validate Model 2 as a

unction of temperature. This test allows for the considerationhat experimental variance can vary with temperature. The �2 testompares the �2 variable value to the ratio between the sum ofhe squares of residuals and experimental variance. Experimentalariance s2

e is estimated from replicated experiments at each tem-erature. No temperature dependence is observed and s2

e is equalo 0.02 × 10−8 kmol2CH4

kg−2catalyst s−2. The �2 variable is tabulated

s a function of the degrees of freedom. If the calculated value ofhe ratio exceeds the tabulated �2 variable, the model is rejectedecause of a lack of fit.

The number of degrees of freedom is obtained by deducing the

umber of parameters (3) from the number of experiments (36).

0

1

2

3

4

5

6

6543210

Cal

cula

ted

104 x

r (k

mol

k

g

)

C

H4

cata

lyst

0′-1

s )

Experimental 104x r (kmol kg s )CH4

-1catalyst-1

0′

-1^

Fig. 10. Parity diagram for Model 2.

T (°C)

Fig. 11. Residuals as a function of temperature for Model 2.

So the �2 variable is determined for 33 degrees of freedom and isequal to 47.4 for 95% confidence.

�2(0.95,33) = 47.4 >

∑i

∑j

(Yij − Yij

)2

s2e

= 45.2 (14)

According to Eq. (14), the ratio is smaller than the �2 variableand therefore Model 2 is validated.

4.6.2. Analysis of residualsThe �2 test establishes whether the overall fit of the model is

satisfactory. Important discrepancies can still exist and can often bedetected through the analysis of residuals, i.e. by examining the setof deviations between the experimental and the predicted values.If the model fits well, the residuals should be randomly distributed.Systematic deviations from randomness indicate that the model isnot totally satisfactory.

Generally, experiments are planned in order to avoid systematicerrors. Partial pressures were chosen randomly but each temper-ature was considered consecutively because the time necessaryto stabilize a given temperature is very long. Therefore, an anal-ysis of residuals is necessary to check that there is no systematicerror as a function of temperature. The residual for experiment j attemperature i is defined as [53]:

Residual = Yij − Yij√s2

2

(15)

where Yij is the observed reaction rate, Yij is the reaction rate cal-culated by Model 2 and s2

2 is the residual variance of Model 2.Analyses of residuals were performed for Model 2 as a function

of temperature and methane partial pressure. Figs. 11 and 12 high-light the random distribution of residuals according to temperatureand methane partial pressure, respectively.

-3

-2

-1

0

1

2

3

10.80.60.40.20

P (atm)

Res

idua

ls

CH4

Fig. 12. Residuals as a function of methane partial pressure for Model 2.

neering Journal 175 (2011) 396– 407 405

4

ena

C[aptcstf[

gpensltaAb

m

wcfb

C

a

r

m

wttp

Iiq

m

(bbotmo

pnt

0

0.5

1

8006004002000Time (s)

ma

Fig. 13. Dimensionless mass of active catalyst curve as a function of time andmethane partial pressure at 950 ◦C; PCH4 = 0.1 atm (�), 0.2 atm (�), 0.3 atm (©),0.4 atm (♦), 0.5 atm (�), 0.6 atm (�), 0.7 atm (�), 0.8 atm ( ), 0.9 atm (�).

00.10.20.30.40.50.60.70.80.9

1

10.80.60.40.20b

illustrated in Fig. 3, where hydrogen production keeps on increas-ing slightly. Parameters t0 and c decrease slightly as methane partialpressure increases (Figs. 15 and 16); so the higher the methane par-tial pressure, the faster the deactivation, as illustrated in Fig. 13,

0

100

200

300

0.32CH0 4

130 −= Pt

t (s

)0

S. Douven et al. / Chemical Engi

.7. Catalyst deactivation

In order to design a reactor and allow its scaling-up, the knowl-dge of the initial specific reaction rate, r0, is not sufficient and it isecessary to model the evolution of the specific reaction rate, r, as

function of time: i.e. to model the catalyst deactivation.As seen in Fig. 3, reaction rate decreases as a function of time.

atalysts frequently lose a fraction of activity while in operation48]. There are many causes of deactivation. Among the mech-nisms that might be responsible for this phenomenon are: (i)oisoning, i.e. the strong chemisorption of species on catalytic sites,hereby blocking sites for the catalytic reaction, (ii) covering of theatalyst with carbon, which hinders the accessibility of the activeites to the reactive species and (iii) thermal degradation leadingo the sintering of active sites or the support or to chemical trans-ormations of active phases to non-active phases of the catalyst48,58].

As deactivation may involve several phenomena whose ori-ins are often not well-known, the determination of a deactivationhenomenological equation is complex. Generally only empiricalquations are determined. In this study, after the burning of theanotubes produced, the catalyst is recovered and used for a newynthesis in the same conditions. The activity of the pristine cata-yst is identical to the activity of the recovered catalyst. Therefore,he deactivation is reversible and probably due to the formation ofmorphous carbon, in agreement with previous literature [59,60].

typical expression of catalyst deactivation by coking is expressedy a decreasing function of coke content:

¯ a = 11 + ˛1CP

(16)

here ma is the dimensionless mass of active catalyst and CP is theoncentration in amorphous carbon. Considering a first-order lawor the deposit of amorphous carbon, the concentration CP is giveny:

P = ˛2 exp (˛3(t − t0)) (17)

The final expression combining both equations corresponds to sigmoid catalytic deactivation equation:

= r0ma (18)

¯ a = a − b tanh(

t − t0

c

)(19)

here r0 is the initial specific reaction rate, r is the specific reac-ion rate and ma is the dimensionless mass of active catalyst. Inhe sigmoid equation of ma, t is the time, whereas a, b, c and t0 arearameters.

Parameter a can be expressed as a function of other parameters.n fact, at the initial time, the dimensionless mass of active catalysts equal to the initial dimensionless mass of catalyst placed in theuartz boat introduced in the reactor. Therefore,

¯ a = 1 + b tanh(−t0

c

)− b tanh

(t − t0

c

)(20)

So three parameters remain to be fitted to experimental dataFig. 13): (i) parameter b corresponds to half of the differenceetween the maximum dimensionless mass of active catalyst, at theeginning of the reaction, and the minimum dimensionless massf active catalyst, at the end of the reaction; (ii) parameter t0 ishe time corresponding to the inflection point of the dimensionless

ass of active catalyst curve; (iii) parameter c represents the slopef the dimensionless mass of active catalyst curve at time t0.

In order to determine b, c and t0, parameter estimation waserformed for each experiment previously used for the determi-ation of initial reaction rates. The approach is identical whateverhe temperature considered. Results are qualitatively identical at

P (atm)CH4

Fig. 14. Evolution of parameter b as a function of methane partial pressure at 950 ◦C.

each temperature. The results at 950 ◦C are reported here. The fit-ting between the experimental hydrogen production curve and themodelled hydrogen production curve is excellent as illustrated inFig. 3. The associated sigmoid catalytic deactivation curve can beseen in Fig. 13. It is important to note that the model is fitted ontodata from the initial time of hydrogen production curve, i.e. around100 s.

Parameters b, c and t0 of the sigmoid equation (Eq. (20)) haveto be determined as a function of methane partial pressure. Resultsof the parameter estimation at 950 ◦C are presented in Figs. 14–16respectively.

Parameter b is independent of methane partial pressure for agiven temperature (Fig. 14) and is equal to 0.46. It is important tonote that b is not exactly equal to 0.5 because the mass of activecatalyst is not equal to zero after a reaction time of 900 s. This is

10.80.60.40.20P (atm)CH 4

Fig. 15. Evolution of parameter t0 as a function of methane partial pressure at 950 ◦C.

406 S. Douven et al. / Chemical Engineerin

0

50

100

150

10.80.60.40.20P (atm)

c (s

)

0.24CH4

79 −= Pc

CH4

Fig. 16. Evolution of parameter c as a function of methane partial pressure at 950 ◦C.

wcpSo

5

toa

daeop

tttteTro

adm

A

G

Fig. A.1. Differential volume element of the catalytic bed.

hich represents the evolution of the dimensionless mass of activeatalyst as a function of methane partial pressure. The rate of amor-hous carbon deposit is proportional to the methane concentration.o the higher the methane partial pressure is, the higher the ratef amorphous carbon deposit.

. Conclusion

The kinetic study performed allows us to determine the equa-ion of the initial specific reaction rate of DWNT synthesis by CCVDver an Fe-Mo/MgO catalyst using methane as the carbon source,s well as the equation of catalytic deactivation.

The methane mass balance on the catalytic bed allows us toetermine the profiles of methane partial pressure through the cat-lytic bed taking into account the internal diffusion. The internalffectiveness factor has been calculated and the kinetic parametersbtained are in agreement with those calculated when the methaneartial pressure is kept equal to that in the bulk of the fluid.

From a statistical point of view, the kinetic equation best ableo reproduce the whole set of experimental data correspondso a mechanism involving the irreversible dissociative adsorp-ion of methane followed by the irreversible decomposition ofhe adsorbed methyl group as the significant step. The activationnergy of methyl decomposition is found to be equal to 58 kJ mol−1.he catalyst deactivation by coking during the reaction can be cor-ectly fitted by a sigmoid equation whose parameters are a functionf methane partial pressure and temperature.

The initial specific reaction rate equation and associated cat-lytic deactivation equation are the essential physico-chemicalata required to model the performance of the continuous inclinedobile-bed rotating reactor and to allow its scaling-up.

cknowledgements

S. Douven is grateful to the Région Wallonne – Directionénérale des Technologies, de la Recherche et de l’Énergie for

g Journal 175 (2011) 396– 407

a Ph.D. grant and also for a Postdoctoral Researcher position inthe framework of the Programme de Formation et d’Impulsionà la Recherche Scientifique et Technologique (FIRST PINSYNACand FIRST AGATHA). S.L. Pirard is grateful to the National Fundfor Scientific Research, Belgium (F.R.S.-FNRS) for a PostdoctoralResearcher position. The authors also thank the Belgian Fonds pourla Recherche Fondamentale Collective (FRFC), the Ministère de laCommunauté Franc aise – Direction de la Recherche Scientifique,the Fonds de Bay and the Interuniversity Attraction Pole (IAP 6/17INANOMAT) for their financial support. The involvement of the Uni-versity of Liège in the Project SOMABAT of the European Unionseventh framework program is also acknowledged.

Appendix A.

Consider the catalytic bed as a flat plate of thickness db, in con-tact with reactant on one side but sealed on the other side as shownin Fig. A.1.

The mass balance of methane on the differential volume elementof the catalytic bed dZ is:

−DedCCH4

dZ= � r0 dZ − De

(dCCH4

dZ+ d2CCH4

dZ2dZ

)(A.1)

Leading to:

Ded2CCH4

dZ2= � r0 (A.2)

The dimensionless expression is:

1

d2bR�

De

T

d2PCH4

dz2= r0 (A.3)

In this equation, 1/d2bR� is a constant and De/T is only function

of temperature, r0 corresponds to the models presented in Section4.2.

For each model, the corresponding equation is then:

1

d2bR�

De

T

d2PCH4

dz2= k1PCH4

1 + k1/k2PCH4

for Model 1; (A.4)

1

d2bR�

De

T

d2PCH4

dz2= k1PCH4(

1 + k1/k2PCH4

)2for Model 2; (A.5)

1

d2bR�

De

T

d2PCH4

dz2= k1PCH4(

1 +(

k1/k2)0.5

P0.5CH4

)2for Model 3; (A.6)

This is to be solved with the boundary conditions: (i) at z = 0,dPCH4 /dz = 0, i.e. the gradient of methane partial pressure is zeroat the wall of the quartz boat and (ii) at z = 1, PCH4 = PCH4,b, that is,the methane partial pressure at the external surface of the catalyticbed is equal to the methane partial pressure in the bulk of the fluid,as there is no external diffusion limitation.

For initial values of k1 and k2, the resolution of equations(A.4)–(A.6) leads to PCH4 and dPCH4 /dz profiles within the catalyticbed. The integration, on the thickness of the bed, of these equationscorresponds to the measured initial specific reaction rate r0. New

values of kinetic parameters are obtained. The iterative resolutionleads to the values of kinetic parameters k1 and k2 presented inTable 3.

In these equations:

neerin

R

[

[

[

[

[[[[[

[

[[

[[

[

[

[

[[[

[

[

[

[

[

[

[

[

[

[

[

[

[

[[[[[

[

[[

[

[

[

[[[

S. Douven et al. / Chemical Engi

CCH4 Methane concentration (kmolCH4 m−3)db Catalytic bed thickness (m)De Effective diffusivity (m2 s−1)PCH4 Methane partial pressure (atm)PCH4,b Methane partial pressure in the bulk of the fluid (atm)r0 Initial specific reaction rate corresponding to the

theoretical model (kmolCH4 kg−1catalyst

s−1)R Perfect gas constant (kJ mol−1 K−1)T Temperature (K)z = Z/db Dimensionless axial variable (–)Z Axial variable from the boat wall to the external surface of

the bed (m)� Density of the catalytic bed (kg m−3)

eferences

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