Modelling Methane Decomposition.italy

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Some issues in modelling methane catalyticdecomposition in fluidized bed reactors

P. Ammendolaa, R. Chironea, G. Ruoppoloa, G. Russoa,b, R. Solimenea,�

aIstituto di Ricerche sulla Combustione, CNR, Piazzale V. Tecchio, 80, 80125 Napoli, ItalybDipartimento di Ingegneria Chimica, Universita di Napoli Federico II, Piazzale V. Tecchio, 80, 80125 Napoli, Italy

a r t i c l e i n f o

Article history:

Received 29 March 2007

Received in revised form

4 January 2008

Accepted 23 March 2008

Available online 12 May 2008

Keywords:

Methane catalytic decomposition

Fluidized bed reactor

Hydrogen production

Attrition phenomena

nt matter & 2008 Internane.2008.03.033

thor. Tel.: +39 [email protected] (R. So

a b s t r a c t

This paper reports a model of fluidized bed thermo-catalytic decomposition (TCD) of

methane. The novelty of the model consists of taking into account the occurrence of

different competitive phenomena: methane catalytic decomposition, catalyst deactivation

due to carbon deposition on the catalyst particles and their reactivation by means of carbon

attrition. Comparison between theoretical and experimental data shows the capability of

the present model to predict methane conversion and deactivation time during the process.

The model demonstrates to be also a useful tool to investigate the role played by operative

parameters such as fluidizing gas velocity, temperature, size and type of the catalyst.

In particular, the model results have been finalized to characterize the attrition phenomena

as a novel strategy in catalyst regeneration.

& 2008 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights

reserved.

1. Introduction

The thermo-catalytic decomposition (TCD) of methane is

an attractive process towards the production of hydrogen

with reduced CO2 emissions [1–3]. The use of a catalyst is

extremely advantageous since the non-catalytic thermal

decomposition would require elevated process temperatures,

i.e. above 1200 1C [1,2]. Fluidized beds have been indicated in

the last years as an efficient reactor solution for TCD process

compared to fixed bed reactors [1–5]. Consequently, new

criteria have to be taken into account for catalyst design. In

particular, a suitable catalyst should be characterized by a low

propensity to attrition in addition to the requirements of high

thermal stability and conversion efficiency [6].

Catalytic systems containing Ni [7,8] and Fe [9,10] have been

largely tested in the past. The Ni-based catalysts have a

maximum operative temperature of 600 1C. As a consequence,

methane conversion being thermodynamically limited at this

temperature, concentrated hydrogen streams (H2460%) can-

tional Association for Hy

; fax: +39 0815936936.limene).

not be obtained using nickel-based catalysts [6]. On the

contrary, Fe-based catalysts are more stable at higher

temperatures in the order of 700–1000 1C, but deactivation

occurs upon repeated cycles, resulting in a short lifetime [6].

In addition, whatever the catalyst, deposited carbon generally

has a filamentous shape, i.e. nanofibres or nanotubes, with

metal particles on their tips. This is an undesirable feature

with reference to applications in fluidized bed reactors where

attrition is likely to remove the active metal phase away from

the support, resulting in a decrease in hydrogen production

and in an increase in consumption of metals [2].

Muradov et al. [1–3] have investigated the feasibility of

using both activated carbon and carbon black as catalysts for

methane decomposition in fluidized beds. They pointed out

that the process is more advantageous than that involving a

metal catalyst: low cost, high temperature resistance and

tolerance to potentially harmful compounds. On the other

hand, the drawbacks are: (i) the poor catalytic activity of

carbon in comparison with metal catalyst; (ii) the carbon

drogen Energy. Published by Elsevier Ltd. All rights reserved.

dx.doi.org/10.1016/j.ijhydene.2008.03.033

mailto:[email protected]

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Nomenclature

a catalyst specific surface area, m2/g

co inlet methane concentration, mol/m3

cb methane concentration in the bubble phase,

mol/m3

ce methane concentration in the dense phase,

mol/m3

db0 bubble diameter just above the distributor, m

dbm maximum bubble diameter, m

d�b effective bubble diameter at x*, m

dc catalyst particle diameter, m

dp mean pore diameter of a catalyst particle, nm

D mean methane diffusivity, m2/s

DCH4methane diffusivity, m2/s

Deff methane bulk diffusivity, m2/s

Dk methane Knudsen diffusivity, m2/s

Dr bed diameter, m

E1 catalyst activation energy, J/mol

E2 deposited carbon activation energy, J/mol

Ec carbon elutriation rate, g/s

hcd thickness of the carbon deposits, nm

H bubbling fluidized bed height, m

Hmf static bed height, m

k1 surface intrinsic kinetic constant of the catalyst,

m3/(m2s)

k01 catalyst frequency factor, m3/(m2s)

k2 surface intrinsic kinetic constant of the deposited

carbon, m3/(m2s)

k02 deposited carbon frequency factor, m3/(m2s)

ka carbon attrition constant, m�1

kg mass transfer coefficient outside the particle, m/s

kp apparent reaction constant per volume of catalyst

particle, s�1

kep apparent external reaction constant per volume

of catalyst particle, s�1

kip apparent internal reaction constant per volume of

catalyst particle, s�1

Kbe overall coefficient of gas interchange between

bubble and dense phase, s�1

Kbc coefficient of gas interchange between bubble and

cloud-wake region, s�1

Kce coefficient of gas interchange between cloud-

wake region and dense phase, s�1

MC carbon molecular weight, g/mol

MCu copper molecular weight, g/mol

MCH4methane molecular weight, g/mol

MH2hydrogen molecular weight, g/mol

Nc number of catalyst particles, –

PC carbon production rate, g/s

PH2hydrogen production rate, g/s

Scd surface of catalyst particle occupied by deposited

carbon per carbon mass, m2/g

Se total external surface of a catalyst particle, m2

Si total internal surface of a catalyst particle, m2

Se1 external active surface of a catalyst particle, m2

Si1 internal active surface of a catalyst particle, m2

Se2 external catalyst surface occupied by deposited

carbon, m2

Si2 internal catalyst surface occupied by deposited

carbon, m2

t time, s

td deactivation time, s

T temperature, K

U fluidizing gas velocity, m/s

Ub rise velocity of bubbles through the bed, m/s

Ubr rise velocity of isolated bubbles, m/s

Umf minimum fluidization velocity, m/s

v generic input variable, a.u.

Vc volume of a catalyst particle, m3

Wc amount of carbon deposited on the external

surface of all bed particles, g

wec amount of carbon deposited on the external

surface of a catalyst particle, g

wic amount of carbon deposited on the internal

surface of a catalyst particle, g

x distance above the distributor, m

x* distance above the distributor corresponding to

the calculation of d�b, m

yCu copper mass fraction of catalyst particles, g/g

z generic output variable, a.u.

Greek symbols

e fractional change in volume of the system be-

tween no conversion and complete conversion of

methane, –

ec porosity factor of a catalyst particle, –

emf void fraction in the bed at minimum fluidization

condition, –

rb bulk density of bed of catalyst particles, g/m3

rc apparent catalyst density, g/m3

t tortuosity factor of a catalyst particle, –

tg mean gas residence time, s

Dimensionless numbers

Da Damkohler number

etdmodel relative error of deactivation time

eXimodel relative error of initial methane conversion

eXr model relative error of residual methane conver-

sion

Remf Reynolds number at minimum fluidization con-

dition

Sc Schimdt number

Sh Sherwood number

Sz sensitivity of a generic output variable

X methane conversion

Xbe index of gas exchange between bubbles and

dense phase

Xi initial methane conversion

Xr residual methane conversion

b fluidization velocity excess with respect to the

fluidizing gas velocity

Y Thiele number

Z effectiveness factor of a catalyst particle

I N T E R N AT I O N A L J O U R N A L O F H Y D R O G E N E N E R G Y 3 3 ( 2 0 0 8 ) 2 6 7 9 – 2 6 9 42680

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catalyst must be periodically unloaded because the carbon

produced by the TCD process has a lower activity than that of

the initial catalyst. However, the authors pointed out the

feasibility of using fluidized bed reactors and carbon as

catalyst on the basis of a scale-up model [3].

Ammendola et al. [11] have proposed a copper-based

catalyst. Advantages of this catalytic system are: (i) relatively

high catalytic activity over that reported for carbon- and

nickel-based catalysts; (ii) high operative temperature up to

1000 1C; (iii) relatively high mechanical resistance to attrition.

This catalyst is a good candidate to be used in fluidized bed

applications, considering that, in addition to a low attrition

rate of the catalyst, it is also characterized by carbon

deposition without formation of carbon fibres with the metal

particles on their tips. This catalytic system has been

extensively studied in a laboratory-scale fluidized bed reactor

[12] in terms of methane to hydrogen conversion, amount of

carbon accumulated on the catalyst, effectiveness of attrition

in removing deposited carbon produced by methane TCD

process and deactivation time.

In the present paper a model of methane TCD carried out in

bubbling fluidized bed reactors is presented. The model has

been formulated taking into account the fate of catalytic

particles subjected to different competitive phenomena:

methanemethane

catalyticcatalytic

decompositiondecomposition

carboncarbon

depositiondeposition

carboncarbon

removalremoval

FB hydrodynamics

temperature CH4 inlet

concentration kinetic of decompos

amoundeposicatalys

amounproduc

external active catalyst surface

internal active catalyst surface

total active catalyst surface

Fig. 1 – Block diagra

methane catalytic decomposition, catalyst deactivation due

to carbon deposition on the catalyst and catalyst reactivation

by means of carbon attrition. A main feature of the model is

related to taking into account the positive effect of attrition

phenomena, commonly considered as a disadvantage of the

catalytic fluidized bed processes, on the regeneration of the

active external surface of catalyst particles. The hypotheses

made in the model formulation have been validated by

comparing theoretical results and experimental data ob-

tained by Ammendola et al. [12]. A sensitivity analysis of

the model has been carried out with reference to operative

parameters (properties of catalyst particles and of carbon

deposits) which have been considered relevant for the

uncertainties on their value. The model has been also used

to investigate the role played by operating conditions such as

fluidizing gas velocity, size and type of the catalyst and to

highlight the operating conditions which yield the attrition

phenomena useful for the catalyst regeneration.

2. Mathematical model

A block diagram of the model is shown in Fig. 1. The model is

structured in three sub-models, which account for the main

deposited carbon properties

deactivation time

carbon elutriation rate

CH4ition

catalyst pore distribution

carbon attrition constant

t of carbon ted on external t surface

t of ed carbon

catalyst properties

hydrogen production

methane conversion

m of the model.

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concurring phenomena: catalytic decomposition of methane,

catalyst deactivation due to carbon deposition on catalyst

active surface and removal of deposited carbon by attrition.

The input variables of the first block are the operating

conditions, the fluidized bed hydrodynamics and the catalyst

and deposited carbon properties. Time by time the first block

needs the values of the actual active surface of the catalyst

particle to calculate the reaction rate and, in turn, the

methane conversion and the hydrogen production as a

function of time. Accordingly, it is also possible to calculate

if the deactivation occurs and, in this case, the deactivation

time of catalyst particles. The second block requires as input

variable the catalyst pore size distribution and, considering

the actual active surface of the catalyst, it calculates the

carbon deposited on the internal and the external active

surfaces of the catalyst particles. The third block determines

the amount of carbon removed by attrition from the external

surface of the catalyst particles which is the elutriated

amount of carbon in the flue gas.

The model has been developed by considering the beha-

viour of a single catalytic particle during the TCD of methane

in a bubbling fluidized bed and by integrating this contribu-

tion to the total number of particles forming the bed to obtain

the performance of the reactor. A schematic representation of

the single particle phenomena is shown in Fig. 2A. At t ¼ 0,

the fresh catalyst is represented as uniformly covered by

metallic active sites. At t40, methane decomposes producing

hydrogen and carbon. The carbon deposits on the particle

active surface, determining a decrease in particle activity and,

in turn, reducing methane decomposition rate. In parallel

with these phenomena, attrition of carbon deposited on the

external surface of particle is responsible for carbon emis-

Fig. 2 – A conceptual representation of thermo-catalytic decomp

attrition phenomena; (B) absence of attrition phenomena.

sions as carbon fines transported in the exit gasses as well as

for renewal of a part of external active surface of the catalyst

particle. According to the scheme it has been considered that

constant values of H2 production and carbon elutriation rate,

Ec, can be obtained. This means that carbon attrition rate is

able to balance carbon deposition rate. Under this condition

there is only a partial carbon recovering of the external

surface of the catalyst particle. Fig. 2A reduces to Fig. 2B when

attrition is not present. In this case a full recovering of the

catalyst is obtained and, due to a deposited carbon activity

significantly lower than metal activity, a relatively low value

of H2 production is expected.

2.1. Model assumptions

The main assumptions concerning hydrodynamics, methane

decomposition, carbon deposition and carbon removal by

attrition are, separately, detailed below.

2.1.1. HydrodynamicsThe well-sound two-phase theory of fluidization [13] has been

considered to model the hydrodynamics of the fluidized bed

reactor. Accordingly, two different phases are distinguished

(Fig. 3): the dense phase which contains all the bed material

and the bubble phase which consists of the swarm of bubbles

effectively present inside the bed. The two-phase theory of

fluidization establishes that a gas flow rate corresponding to

the incipient fluidization condition percolates through the

dense phase whereas the remaining part of the total gas flow

rate rises along the bed as bubbles. The volume changes due

to the methane conversion along the bed axis are considered

proportional to the methane conversion by means of a factor,

osition of methane on a single bed particle. (A) Presence of

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Kbe

UmfU-Umf

ce(t)cb (t,H)

ce (t) cb (t, x)

U, c0

Bubble

phase

Dense

phase

X

x

Fig. 3 – Scheme of the fluidized bed according to the two-

phase theory of fluidization.


e, defined as the fractional change in volume of the system

between no conversion and complete conversion of methane

[14]. Mass transfer between the dense phase and the bubble

phase has been described by an overall mass transfer

coefficient, Kbe. Solids and gases are perfectly mixed in the

dense phase [15]. Bubbles rise in plug flow with a constant

size along the bed axis [15]. The adopted bubble size is an

average value calculated by a semi-empirical correlation

which takes into account the variation of the bubble size

due to bubble coalescence and to volume changes owing to

the occurrence of the methane conversion along the bed axis

as a function of the operating conditions [16]. The tempera-

ture is uniform in the reactor [17] and constant with time.

Temperatures between 800 and 1000 1C are considered.

2.1.2. Methane decompositionDecomposition of the methane (CH4-C+2H2) is the only

occurring reaction taking place on the catalyst surface. The

reaction is irreversible, valid assumption at the considered

temperatures, as determined by thermodynamical evalua-

tions [18], and first order with respect to methane concentra-

tion as reported in literature for metallic catalysts [19].

Considering that the model is applied to operative tempera-

tures lower than 1000 1C, the homogenous gas phase methane

decomposition has not been taken into account. This

hypothesis, reasonable on the basis of literature indications

[1,2], has been also confirmed by calculations under the tested

operative conditions.

The carbon produced by methane decomposition covers the

catalyst active sites reducing catalyst activity. However,

the deposited carbon also presents a catalytic activity to the

methane decomposition process even if much lower than

that of metallic catalysts [20]. The reaction order with respect

to methane concentration for this chemical process is

unknown. However, literature data show that activated

carbons and carbon blacks are characterized by a reaction

order of 0.5 [21] and 1 [22], respectively. For the sake of

simplicity, in this work a first reaction order is considered.

The actual methane decomposition rate is calculated by

accounting for intrinsic kinetics of metal catalyst and

deposited carbon and their relative active surfaces.

The methane conversion has been assumed as a quasi-

steady state process with respect to variations of decomposi-

tion kinetics due to the catalyst deactivation.

2.1.3. Carbon depositionThe carbon produced by methane decomposition is deposited

on the internal and external surfaces of the catalyst. Methane

decomposes on an active site whether it is carbon or metallic

without the formation of filamentous carbon. It has been also

assumed that the carbon deposits along the pores of catalyst

particle reducing the pore diameter, without reducing its

specific surface area. This assumption limits the validity

of the model to cases far from conditions of pore occlusion.

The model has been used verifying that the reduction of pore

size was smaller than 1 nm with an initial average pore size of

10 nm. The relative amount of carbon deposited on the

external and internal surfaces of the catalyst is calculated

on the basis of the respective methane decomposition rates,

which, in turn, are proportional to the actual CH4 concentra-

tion. CH4 profile inside particles has been calculated through

Thiele number [14], which, on the basis of the previous

discussion on the limited variation in pore diameter, mainly

varies during the decomposition process due to the reduction

of catalyst activity caused by carbon deposition.

2.1.4. Carbon removalThe mechanism assumed for carbon removal is the attrition

of the carbon accumulated on the external surface of

fluidized particles. In particular, it has been assumed that

the mechanical abrasion generated by the motion of the bed

particles enables one to remove the carbon deposited on the

external surface of the catalyst particles restoring the

metallic active sites and producing carbon fines. These latter

are elutriated according to the correlation, proposed by Miccio

and Massimilla [23], valid for the production of carbon fines

due to the mechanical abrasion of carbon-spotted bed

material:

Ec ¼ kaðU� Umf ÞWc (1)

Carbon elutriation rate, Ec, is proportional to an attrition

constant, ka, to the fluidization excess velocity, (U–Umf), and to

the carbon amount, Wc, actually present on the external

surface of the catalyst particles forming the whole bed. It

must be noted that the choice of Eq. (1) has been done on the

basis of the nature of the interaction between copper-based

catalyst and produced carbon, often described by van der

Waals forces [24,25].

2.2. Model equations

On the basis of model assumptions, the model can be reduced

to the following set of equations:

Methane conversion ¼inlet CH4 moles� outlet CH4 moles

inlet CH4 moles

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XðtÞ ¼�½ð1þ �Þð1� be�Xbe Þ þ DaðtÞð1þ �e�Xbe Þ�

�ð1þ �Þð1� e�Xbe Þ

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½ð1þ �Þð1� be�Xbe Þ þ DaðtÞð1þ �e�Xbe Þ�2 þ 2�DaðtÞð1þ �Þð1� e�Xbe Þð1� be�Xbe Þ

q�ð1þ �Þð1� e�Xbe Þ

(2)


Mass rate of hydrogen production

¼ 2MH2

inlet CH4 molestime

� �XðtÞ

PH2ðtÞ ¼ 2MH2

pD2r

4Uc0XðtÞ (3)

Carbon elutriation rate,

EcðtÞ ¼ kaðUð1þ 0:5�XðtÞÞ � Umf ÞwecðtÞNc (4)

Carbon accumulation

rate on the external

surface of a

catalyst particle

0BBBBB@

1CCCCCA ¼

Carbon deposition


surface of a

catalyst particle

0BBBBB@

1CCCCCA

�

Carbon removal

rate by attrition

from a

catalyst particle

0BBBBB@

1CCCCCA

dwecðtÞ

dt¼ ke

pðtÞceðtÞVcMC �EcðtÞNc

(5)

Carbon accumulation

rate on the internal

surface of a

catalyst particle

0BBBBB@

1CCCCCA ¼

Carbon deposition


surface of a

catalyst particle

0BBBBB@

1CCCCCA

dwicðtÞ

dt¼ ki

pðtÞceðtÞVcMC (6)

Variation rate

of the external

active surface of a

catalyst particle

0BBBBB@

1CCCCCA ¼ �Scd

Carbon accumulation


surface of a

catalyst particle

0BBBBB@

1CCCCCA

dSe1ðtÞ

dt¼ �Scd

dwecðtÞ

dt(7)

Table 1 – Input model variables (base case)

Operating conditions Catalyst proper

U (m/s) 0.058 rc (g/m3)

Hmf (m) 0.12 rb (g/m3)

Dr (m) 0.027 a (m2/g)

c0 (mol/m3) 0.568 dc (m)

T (K) 1073 ec (–)

DCH4(m2/s) 2� 10�4 t (–)

dp (nm)

Umf (m/s)

emf (–)

yCu (g/g)

k10 (m3/(m2s))

E1 (J/mol)

Variation rate

of the internal

surface of a

catalyst particle

0BBBBB@

1CCCCCA ¼ �Scd

Carbon accumulation


surface of a

catalyst particle

0BBBBB@

1CCCCCA

dSi1ðtÞ

dt¼ �Scd

dwicðtÞ

dt(8)

Methane conversion is time dependent through Damkohler

number dependence. In fact, according to its definition,

Da(t) ¼ f(kp(t)) [26], Damkohler number varies with time due

to catalyst deactivation as a result of the reduction of the

apparent kinetic constant of catalyst, kp(t), owing to carbon

deposition on both external and internal surface of the

particles. The apparent kinetics of the catalyst is evaluated

time by time during the deactivation process taking into

account: (i) the actual amount of carbon deposited on the

internal, wicðtÞ, and external, we

cðtÞ, active surfaces of a single

catalyst particle; (ii) the actual active surfaces, either internal

or external, of the catalyst, Si1ðtÞ and Se

1ðtÞ.

The full details of model equations and their derivation are

reported in Appendices A and B.

3. Results and discussion

The input variables, assumed as the base case, are listed

in Table 1 in terms of operating conditions, properties of

catalytic bed material and properties of deposited carbon. The

operating conditions are those of a typical experiment carried

out by Ammendola et al. [12]. The physical properties of the

catalyst particles and their fluidization behaviour have been

obtained from previous characterization tests reported else-

where [11,12,18]. The evaluation of kinetic parameters of the

catalyst has been experimentally obtained in a fixed bed

micro-reactor at a pre-set methane concentration and varying

ties Deposited carbon properties

20� 105 Scd (m2/g) 5500

8.6�105 hcd (nm) 0.335

100 ka (m�1) 10�2

3.5�10�4 k20 (m3/(m2s)) 7

0.6 E2 (J/mol) 2.0� 105

3

10

0.029

0.57

0.084

2.1

1.55�105

ARTICLE IN PRESS


the reaction temperature in the range 600–800 1C [18]. The

data have been elaborated by means of an Arrhenius plot to

estimate k01 (2.1 m3/(m2s)) and E1 (1.55�105 J/mol). It is worth

noting that the obtained value of the activation energy is

between that of metallic (0.6�105 J/mol) [27] and of carbon-

based catalysts (2.0�105 J/mol) [22] according to the effective

intermediate activity of copper. The value of proportionality

constant of Eqs. (7) and (8), Scd, represents the surface of

catalyst particle occupied by deposited carbon per carbon

mass. Assuming that one mole of carbon deactivates one

mole of copper, this parameter has been estimated by

Scd ¼catalyst particle surface

catalyst particle copper molescarbon molecarbon mass

¼arcVcMCu

rcVcyCuMC(9)

where a, rc, Vc, yCu, MC and MCu are the catalyst specific

surface area, the apparent catalyst density, the volume of

catalyst particle, the copper mass fraction of catalyst, the

carbon and copper molecular weight, respectively. Using the

catalyst data reported in Ammendola et al. [12], the estimated

value is in the order of 5000–6000 m2/g. The thickness of the

carbon deposits on Cu-based catalyst is unknown in litera-

ture. However, NiCuAl catalysts have shown the formation of

high-order deposited carbon structurally close to a perfect

graphite [28]. As a consequence, in the model it has been

assumed equal to the thickness of an elementary cell of

graphite. The adopted value of ka has been determined

working out experimental data [12,18] obtained collecting

the elutriated material during methane decomposition tests

and analysing their carbon content. The kinetic parameters

of the carbon deposits have been chosen from literature data

concerning the activity of carbon blacks [22].

Fig. 4 shows typical model output variables: methane

conversion and carbon production rate (solid line) and carbon

elutriation rate (dashed line) as a function of time. Methane

conversion profile, i.e. carbon production rate profile, is

characterized by a first stage, in which it monotonically

0

met

hane

con

vers

ion,

X, -

0.0

0.2

0.4

0.6

0.8

1.0

carb

on p

rodu

ctio

n ra

te, P

C (1

05 ), g

/s

0

4

8

12

16

20

40002000

Fig. 4 – Model results (base case). Methane conversion X and car

Ec (dashed line) as functions of time during methane TCD in a b

decreases with time from an initial value Xi (ffi0.9) to a residual

value Xr (ffi0.2) at tdffi6500s, followed by a second stage, in

which it is constant. The first stage is that expected considering

that methane conversion decreases with time due to the

carbon deposition and the corresponding decrease in catalyst

activity. The second stage is due to two different and constant

contributions: (i) the internal decomposition rate is constant

because the internal catalyst surface, completely recovered by

carbon, does not vary during the carbon deposition according

to model hypothesis; (ii) the external decomposition rate due to

the activity of the accumulated carbon and to the activity of the

metallic catalyst sites renewed by attrition is constant because

carbon attrition rate is able to balance external carbon

deposition rate yielding constant the external surface occupied

by the carbon and by metallic catalyst.

Carbon elutriation rate (Fig. 4) monotonically increases up

to a stationary value at tffi8000 s. Again this steady state

condition corresponds to the balance between the carbon

deposition rate on the external catalyst surface and the

carbon attrition rate. The model has been used to calculate

the relative contribution of external metal activity renewed by

attrition and activity of total deposited carbon on actual CH4

decomposition rate. Under the tested conditions, attrition

contributes to an amount lower than 1% of the residual value

of methane conversion.

The predicted profiles, presented in Fig. 4, are in agreement

with the conceptual representation of methane TCD in a

bubbling fluidized bed presented in Fig. 2A.

A better understanding of the mechanisms active during

the fluidized bed TCD process can be obtained considering

the ratio between the active and the total external surface of a

catalyst particle, Se1=S

e, the ratio between the active and the

total internal surface of a catalyst particle, Si1=S

i, the carbon

amount deposited on the external, wec, and internal, wi

c,

surface of a single bed particle. Fig. 5 shows these variables as

functions of time.

The active external surface of catalyst particle decreases

with time until, at tffi8000 s, a steady state condition is

time, s

carb

on e

lutr

iatio

n ra

te, E

C (.

109 )

, g/s

0

5

10

15

20

25

1000080006000

bon production rate Pc (solid line) and carbon elutriation rate

ubbling fluidized bed reactor.

ARTICLE IN PRESS


approached corresponding to a value of about 13% of the total

external surface. Correspondingly, the amount of carbon

accumulated on the external surface, wec, increases with time

until a stationary value is reached at tffi8000 s corresponding

to a value of about 6�10�11 g. These results can be analysed

by taking into account that the carbon deposition rate is

initially much larger than the carbon attrition rate, propor-

tional to the amount of the deposited carbon, thus the active

external surface of the catalyst particle decreases and the

amount of deposited carbon increases until a stationary state

is achieved.

The active internal surface of catalyst particle decreases

with time up to a complete covering of the surface at

tffi6500 s, corresponding to td, which can be considered the

‘‘primary’’ deactivation time. Contrarily to the external sur-

face, being in this case absent carbon attrition, the amount of

carbon accumulated on the internal surface, wic, increases

even at t4td, but at a lower rate due to the low catalytic

ratio

bet

wee

n ex

tern

al a

ctiv

ean

d to

tal p

artic

le s

urfa

ce, S

1e /Se ,

-

0.0

0.2

0.4

0.6

0.8

1.0

time,0

ratio

bet

wee

n in

tern

al a

ctiv

ean

d to

tal p

artic

le s

urfa

ce, S

1i /Si ,

-

0.0

0.2

0.4

0.6

0.8

1.0

40002000

Fig. 5 – Model results (base case). (A) Ratio of the active to the t

amount deposited on the external surface of a catalyst particle

internal surface of a catalyst particle Si1=S

i and carbon amount

functions of time.

activity of deposited carbon. It has been verified that in the

considered time interval the accumulated carbon does not

significantly modify the total internal surface of the catalyst

particle.

3.1. Comparison between experimental and model results

The developed model has been validated by comparing the

time series of methane conversion obtained during TCD tests

in a lab-scale bubbling fluidized bed reactor [12,18] and the

experimental values of the initial, Xi, and the residual, Xr,

methane conversion and of the deactivation time, td, with the

model computations in the same operating conditions.

The experimental set-up, procedure and measurement tech-

niques of the experiments have been described elsewhere

[12,18].

Fig. 6 shows the comparison between the model and

the experimental time-resolved methane conversion [12,18].

carb

on d

epos

ited

on e

xter

nal

part

icle

sur

face

, wC

e (.1

011),

g

0

2

4

6

8

10

s

carb

on d

epos

ited

on in

tern

alpa

rtic

le s

urfa

ce, w

Ci (

.107 )

, g

0

2

4

6

8

10

1000080006000

otal external surface of a catalyst particle Se1=S

e and carbon

wec as functions of time; (B) ratio of the active to the total

deposited on the internal surface of a catalyst particle wic as

ARTICLE IN PRESS

0.0

0.2

0.4

0.6

0.8

1.0

met

hane

con

vers

ion,

X, -

0.0

0.2

0.4

0.6

0.8

1.0

time, s0

0.0

0.2

0.4

0.6

0.8

1.0

experimental datamodel computations

Hmf = 0.10 m

Hmf = 0.07 m

Hmf = 0.12 m

600050004000300020001000

Fig. 6 – Model computations: comparison with experimental

data obtained under different bed heights (U ¼ 0.087 m/s).

Table 2 – Comparison between experimental and model resul

Operating conditions Experimental values

U (m/s) Hmf (m) Xi (–) Xr (–) td (s) Xi

0.058 0.07 0.87 0.11 4780 0.8

0.12 0.91 0.16 6760 0.9

0.087 0.07 0.79 0.08 2980 0.7

0.10 0.82 0.11 4280 0.8

0.12 0.84 0.09 4350 0.8

0.116 0.07 0.75 0.07 2850 0.7


In particular, the conditions are the same as the base case

except for the fluidizing gas velocity which is 0.087 m/s.

The influence of bubbling fluidized bed hydrodynamics is also

investigated considering three different bed heights: 0.07, 0.10

and 0.12 m. The agreement between the model computations

and the experimental data is quite good during both the

initial and the final step of the deactivation process.

Table 2 and Fig. 7 present the comparison between model

predictions and experimental results, obtained with different

superficial gas velocities and bed heights [12,18], in terms

of Xi, Xr, td and their relative errors. The model predictions are

in good agreement with the experimental data whatever

operative conditions are considered. As a matter of fact, the

relative error is lower than 710% in most of the investigated

conditions.

3.2. Sensitivity analysis

The model has been also used to investigate the influence of

properties of catalyst and of deposited carbon on the

performance of the fluidized bed reactor (Fig. 8). In particular,

the specific surface area, a, and the intrinsic reaction

constant, k01, of the catalyst, as well as the surface of catalyst

particle occupied by deposited carbon per carbon mass, Scd,

have been increased and decreased with respect to the

base case. The study has been further pursued in order to

determine the sensitivity of relevant output variables to

changes in these input variables. To this end, the standard

procedure for linearized sensitivity [29] has been used. Each

input variable has been changed by some fraction (720%) of

their values assumed as base case. This variation has been

fixed on the basis of either uncertainties in their experimental

determination or intentional changes with respect to base

case due to operative requirements. The sensitivity of the

generic output variable z has been evaluated as

Sz ¼ðz� � zþÞ=zb

ðv� � vþÞ=vb(10)

where subscript b indicates the base case value. Superscripts

�and + indicate, for the generic input variable v, the left and

the right extremes of the assumed range of variation,

whereas for the output variable z they indicate the values

ts obtained under different operating conditions

Model values Relative errors

(–) Xr (–) td (s) eXi(%) eXr (%) etd

(%)

4 0.11 4364 �3.5 0 �8.7

1 0.18 6424 0 12.5 �5.0

8 0.07 3504 �1.3 �8.7 17.6

4 0.11 4398 2.4 0 2.7

6 0.12 4880 2.4 33.3 12.1

2 0.06 3076 �4.0 �14.3 7.9

ARTICLE IN PRESS

experimental methane conversion, -0.0

mod

el m

etha

ne c

onve

rsio

n, -

0.0

0.2

0.4

0.6

0.8

1.0

td experimental, s0

t d m

odel

, s

0

2000

4000

6000

8000

10000

XiXftd

1.00.80.60.40.2

100008000600040002000

Fig. 7 – Model computations: comparison with experimental

data obtained under different operating conditions (dashed

lines 710%).

0.0

0.2

0.4

0.6

0.8

1.0k1

0 = 1.68 m3/m2sk1

0 = 2.10 m3/m2s*k1

0 = 2.52 m3/m2s

time, s

met

hane

con

vers

ion,

-

0.0

0.2

0.4

0.6

0.8

1.0a = 80 m2/ga = 100 m2/g*a = 120 m2/g

00.0

0.2

0.4

0.6

0.8

1.0Scd = 4400 m2/gScd = 5500 m2/g*Scd = 6600 m2/g

100008000600040002000

Fig. 8 – Model computations: influence of the properties of

catalyst and of deposited carbon. (*) continuous line base

case.


that it assumes for these extremes. The sensitivity values

obtained are reported in Table 3.

The catalyst intrinsic kinetic constant plays a twofold role:

on one hand, increasing the kinetics increases the initial

conversion of the methane; on the other, the deactivation

time is smaller and the steady state condition is achieved in a

step-like manner.

The influence of the specific surface area of the catalyst

particle on the methane conversion is rather trivial: the initial

conversion as well the deactivation time increases with

the total internal surface of the particle. The surface of


mass, Scd, is a characteristic parameter of the physical and

chemical structure of the carbon deposits and of its interac-

tion with metallic catalyst sites. It has a crucial effect on

the deactivation process enabling possibly a large variation

of the deactivation time. These considerations are confirmed

by the parametric analysis reported in Table 3. In particular,

it is evident that: (i) the initial methane conversion is

barely influenced positively by the intrinsic kinetics and

by the specific area of the catalyst, k01 and a, respectively;

(ii) the residual methane conversion depends almost linearly

on the specific area of the catalyst, a; (iii) the deactivation

time is subject to the variations of the three investigated

input variables. On the whole, it can be observed that

the deactivation process is strongly influenced by the

specific area of the catalyst, a, and by the surface of


mass, Scd.

3.3. Relevance of attrition phenomena

The model has been also used to highlight the feasibility that

attrition can operate as a useful process in catalyst regenera-

tion for industrial applications. The model computations

have been obtained disregarding the activity of deposited

carbon in order to emphasize the role of carbon attrition

and choosing the values of the operating conditions on the

basis of results achieved by Muradov et al. [3] for methane

TCD process in a large-scale operation (bed diameter about

4 m). In particular, the present investigation has been

carried out at different catalyst particle sizes 3.5�10�4

and 0.8�10�4 m, fluidizing gas velocities 0.1 and 0.5 m/s,

mean gas residence times tg ¼ 5 and 10 s and temperatures

800 and 1000 1C. It is worth noting that at higher considered

temperature the contribution of the homogeneous gas

phase reaction to methane decomposition calculated by

Chemkins simulations is less than 3% of methane conversion

introducing, as a consequence, a negligible error in the model

results.

ARTICLE IN PRESS

0.0

0.2

0.4

0.6

0.8

1.0

0.0001

0.001

0.01

0.1

1

0.1 m/s0.5 m/s

final

ratio

bet

wee

n ex

tern

al a

ctiv

e an

d to

tal p

artic

le s

urfa

ce, <

S 1e /

Se >ss

, -

0.0

0.2

0.4

0.6

0.8

1.0

resi

dual

met

hane

con

vers

ion,

Xr,

-

0.0001

0.001

0.01

0.1

1

0.1 m/s0.5 m/s

T = 800°Cτg = 10s

T = 800°Cτg = 5s

carbon attrition rate, ka, m-10.0001

0.0

0.2

0.4

0.6

0.8

1.0

0.0001

0.001

0.01

0.1

1

1000°C800°C

U = 0.5m/sτg = 10s

1.00000.10000.01000.0010

Fig. 9 – Model computations: influence of carbon attrition

rate on the final active external surface of catalyst particle

and on the residual methane conversion (dc ¼ 3.5�10�4 m).

Table 3 – Sensitivity of output variablesa to changes of individual input variables

Input variable Base case Variation Xi SXiXr SXr td (s) Std

k01 (m3/(m2 s)) 2.10

1.68 0.8840.096

0.1760

6754�0.1702.52 0.919 0.176 6318

a (m2/g) 10080 0.884

0.0960.146

0.8065496

0.722120 0.919 0.204 7350

Scd (m2/g) 55004400 0.905

00.176

08030

�1.0416600 0.905 0.176 5354

a Values of output variables in the base case are: Xi ¼ 0.91, Xr ¼ 0.18, and td ¼ 6424 s.


Fig. 9A shows the final ratio of the active to the total

external surface of a catalyst particle, hSe1=S

eiss, and the

residual methane conversion, Xr, as function of the carbon

attrition constant, ka, obtained with a catalyst particle size

of 3.5�10�4 m, at temperature of 800 1C and with a mean gas

residence time of 5 s for two different fluidization velocities,

0.1 and 0.5 m/s, continuous and dashed line, respectively.

hSe1=S

eiss presents an increasing monotone sigmoid curve

which highlights a strong dependence of the catalyst

regeneration on carbon attrition constant. When the fluidiz-

ing gas velocity is 0.1 m/s, a completed regeneration of

external surface is possible for values of ka higher than

1 m�1. On the other hand, the residual methane conversion

increases with carbon attrition constant reaching a final very

low value in the order of 0.2% corresponding to the complete

regeneration of the external surface of the catalyst particle.

This evidence highlights that under the investigated condi-

tions the contribution of the external surface of the particles

to the reaction rate is negligible also for completely regener-

ated particles. When the fluidization velocity is increased up

to 0.5 m/s, the curves of hSe1=S

eiss and of Xr as a function of

carbon attrition constant maintain the same shape but they

are simply shifted to lower values of ka. This result confirms

that attrition rate—proportional to the fluidization excess

velocity—is more effective towards the external regeneration

of catalyst particle without any modification of the maximum

residual methane conversion.

Fig. 9B shows the final ratio of the active to the

total external surface of a catalyst particle, hSe1=S

eiss, and

the residual methane conversion, Xr, as a function of

the carbon attrition constant, ka, obtained in the same

operating conditions as those of Fig. 9A but with a larger

mean gas residence time of 10 s. The model results under-

line that: (i) the mean gas residence time does not substan-

tially influence the regeneration of the external surface

of the catalyst particle; (ii) an increase of mean gas residence

time determines a higher residual methane conversion

(E0.4%).

Fig. 9C shows the final ratio of the active to the total

external surface of a catalyst particle, hSe1=S

eiss, and the

residual methane conversion, Xr, as a function of the carbon

attrition constant, ka, obtained with a catalyst particle size of

3.5�10�4 m, at a fluidization velocity of 0.5 m/s, with a mean

gas residence time of 10 s and at two different temperatures,

800 and 1000 1C, continuous and dashed line, respectively. A

higher temperature mainly determines an increase in the

intrinsic kinetic constant of the catalyst k1 by about one order

ARTICLE IN PRESS


of magnitude, from 6�10�8 to 9�10�7 m3/(m2s), producing,

as a consequence, the following effects: (i) the complete

regeneration of external surface and, in turn, the maximum

value of Xr are reached at higher values of ka with respect to

800 1C; (ii) the maximum value of Xr increases by about one

order of magnitude, up to about 6%, with respect to 800 1C. It

can be concluded that for the coarser bed particles and the

operating conditions investigated the role played by carbon

attrition is moderate.

Figs. 10A–C show the final ratio of the active to the

total external surface of a catalyst particle, hSe1=S

eiss, and

the residual methane conversion, Xr, as a function of the

carbon attrition constant, ka, obtained with a lower catalyst

particle size of 0.8�10�4 m, at the same operative conditions

as those of Figs. 9A–C, respectively. The observed phenom-

enology is very similar to that obtained with bed particles

with larger size as it regards both the catalyst regeneration

by carbon attrition and the residual methane conversion.

It is worth noting that a decrease in particle size determines

0.00010.0

0.2

0.4

0.6

0.8

1.0

0.0001

0.001

0.01

0.1

1

0.0001

0.001

0.01

0.1

1

0.0

0.2

0.4

0.6

0.8

1.0

final

ratio

bet

wee

n ex

tern

al a

ctiv

e an

d to

tal p

artic

le s

urfa

ce, <

S 1e /

Se >ss

, -

0.0

0.2

0.4

0.6

0.8

1.0

resi

dual

met

hane

con

vers

ion,

Xr,

-

0.0001

0.001

0.01

0.1

1

T = 800°Cτg = 10s

T = 800°Cτg = 5s

U = 0.5m/sτg = 10s

0.1 m/s0.5 m/s

0.1 m/s0.5 m/s

1000°C800°C

carbon attrition rate, ka, m-11.00000.10000.01000.0010

Fig. 10 – Model computations: influence of carbon attrition

rate on the final active external surface of catalyst particle

and on the residual methane conversion (dc ¼ 0.8�10�4 m).

a more effective catalyst regeneration by attrition (i.e.

at the same ka a slightly higher hSe1=S

eiss is observed)

and, moreover, a higher residual methane conversion.

In particular, it can be observed that the maximum values

of Xr obtained in correspondence of the smaller particles is

higher: about 2% and more than 10% at 800 and 1000 1C,

respectively.

The model results suggest that the attrition could play

an important role in the performance of the reactor when

the contribution of external surface of the bed particles

to the reaction rate is significant, i.e. in the case of completely

regenerated catalyst particles and of high surface intrinsic

kinetics. The comparison of Figs. 9 and 10 underlines that for

the finer bed particles investigated, this contribution is

obviously larger as the total external surface of the bed

particles per catalyst mass is higher.

4. Conclusions

The following conclusions can be made:

1.
A simple and novel model of a bubbling fluidized bed TCD
of methane has been developed taking into account the

fate of catalytic particles subjected to different competitive

phenomena: methane catalytic decomposition, catalyst

deactivation due to carbon deposition on the catalyst and

catalyst reactivation by means of carbon attrition. The

novelty of the model is related to taking into account the

effect of attrition phenomena, commonly considered as a

disadvantage of the catalytic fluidized bed processes, on

the regeneration of the external surface of the catalyst

particles.

2.
The model enables one to predict the performance of the
fluidized bed reactor in terms of methane conversion,

hydrogen production, elutriated carbon and deactivation

time. On the whole, the methane TCD process can be

characterized by an initial, Xi, and residual, Xr, methane

conversion and a deactivation time, td.

3.
The model results and experimental data have been
compared in terms of time-resolved methane conversion

curves, initial, Xi, and residual, Xr, methane conversion

and deactivation time, td, for different operating condi-

tions. Good agreement within 710% has been obtained in

most of the investigated cases.

4.
The sensitivity analysis of the model to changes of the
properties of catalyst and of deposited carbon on the

performance of the fluidized bed reactor has demon-

strated that the deactivation process is strongly influenced

by the specific area of the catalyst, a, and by the surface of


mass, Scd.

5.
The model has shown that carbon attrition plays a key role
in the regeneration of the external catalyst surface in all

the investigated conditions. This phenomenon can also

affect the residual methane conversion when the con-

tribution of the external surface of the particle to the

reaction rate is high enough. This condition is emphasized

by a high intrinsic kinetic rate and by a small size of

catalyst particles.

ARTICLE IN PRESS


Appendix A. Hydrodynamics and methane decomposition

Following a procedure analogous to that developed by Harrison and Davidson [15], the actual methane concentration in the

dense phase can be expressed as

ceðtÞ ¼ co1� be�Xbe þ 0:5�XðtÞð1� e�Xbe Þ

1� be�Xbe þ 0:5�XðtÞð1� e�Xbe Þ þDaðtÞ(A.1)

where b, Xbe, Da(t), ce(t), co and e are the fluidization velocity excess with respect to the fluidizing gas velocity, the index of gas

exchange between bubbles and dense phase, the Damkohler number, the concentration of methane in the dense phase and in

the inlet stream, and the fractional change in volume of the system between no conversion and complete conversion of

methane, respectively. The actual methane concentration in the bubble phase cb(t,x) can be expressed as

cbðt; xÞ ¼ ceðtÞ þ ðco � ceðtÞÞe�Xbex=H (A.2)

where x is the distance above the distributor. As a consequence, the exit conversion of methane, X, taking into account the exit

bubble phase methane concentration and the dense phase methane concentration, can be easily obtained [15]

XðtÞ ¼�½ð1þ �Þð1� be�Xbe Þ þ DaðtÞð1þ �e�Xbe Þ�

�ð1þ �Þð1� e�Xbe Þ

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½ð1þ �Þð1� be�Xbe Þ þ DaðtÞð1þ �e�Xbe Þ�2 þ 2�DaðtÞð1þ �Þð1� e�Xbe Þð1� be�Xbe Þ

q�ð1þ �Þð1� e�Xbe Þ

. (A.3)

The fluidization velocity excess with respect to the fluidizing gas velocity, b, is calculated by [15]

b ¼U� Umf

U(A.4)

where U and Umf are the fluidizing gas velocity and the minimum fluidization velocity.

The index of gas exchange between bubbles and dense phase, Xbe, may be expressed as a dimensionless cross-flow ratio,

defined as [15]

Xbe ¼Kbe � H

Ub(A.5)

where Kbe is the overall coefficient of gas interchange between bubble phase and dense phase, H is the bubbling fluidized bed

height and Ub is the rise velocity of bubbles through the bed.

H and Ub can be calculated by the following equation system [16]:

H ¼ HmfUbr þ ðUð1þ 0:5�XðtÞÞ � Umf Þ

Ubr

x� ¼ ð0:4C0:5Þ � H

d�b ¼ dbm � ðdbm � db0Þ � exp�0:3 � x�

Dr

� �

Ubr ¼ 0:711 � ðg � d�bÞ0:5� 1:2 � exp �1:45

d�bDr

� �Ub ¼ Uð1þ 0:5�XðtÞÞ � Umf þUbr

8>>>>>>>>>>>><>>>>>>>>>>>>:

(A.6)

where Hmf, Ubr, x*, d�b; dbm, db0 and Dr are the static bed height, the rise velocity of isolated bubbles, a pre-set distance above the

distributor, the effective bubble diameter at x*, the maximum bubble diameter, the bubble diameter just above the distributor

and the bed diameter, respectively. d�b is the mean value of the bubble size along the bed axis and it is assumed as the bubble

diameter in all the fluidized bed.

Kbe can be calculated from the following relation [17]:

1Kbe¼

1Kbcþ

1Kce

(A.7)

where Kbc and Kce are the coefficient of gas interchange between bubble and cloud-wake region and the coefficient of gas

interchange between cloud-wake region and dense phase, respectively.

Kbc can be evaluated from the expression derived by Davidson and Harrison [15]

Kbc ¼ 4:5Umf

dn

b

!þ 5:85

D1=2CH4� g1=4

dn5=4b

0@

1A (A.8)

where DCH4is the methane diffusion.

ARTICLE IN PRESS


Kce can be calculated from the following equation [17]:

Kce ¼ 6:77DCH4

� �mf � Ub

dn3b

!1=2

(A.9)

where emf is the void fraction in the bed at minimum fluidization condition.

The Damkohler number is calculated by [26]

DaðtÞ ¼kpðtÞrbHmf

rcU(A.10)

where kp(t) is the apparent reaction constant per volume of catalyst particle, rc is the apparent catalyst density and rb is the bulk

density of catalyst bed.

The apparent reaction constant kp(t) varies with time as the active surface of the catalyst varies with time as a consequence of

the deposition of carbon produced by methane decomposition. Accounting for the simultaneous action of the different

resistances, i.e. gas diffusion in the external boundary layer, intra-particle diffusion, intrinsic kinetics on the surface of catalyst

and intrinsic kinetics on the surface of deposited carbon, kp(t) can be expressed as

kpðtÞ ¼1

ðVc=kgSeÞ þ ðVc=ðk1ðS

i1ðtÞZðS

i1ðtÞ; S

i2ðtÞÞ þ Se

1ðtÞÞ þ k2ðSi2ðtÞZðS

i1ðtÞ; S

i2ðtÞÞ þ Se

2ðtÞÞÞÞ(A.11)

where Vc, kg, Se, k1, k2, Si1ðtÞ; S

e1ðtÞ; S

i2ðtÞ;S

e2ðtÞ and ZðSi

1ðtÞ;Si2ðtÞÞ are the volume of a single particle of catalyst, the mass transfer

coefficient outside the particle, the total external surface of the catalyst particle, the surface intrinsic kinetic constant of the

catalyst, the surface intrinsic kinetic constant of the deposited carbon, the internal and external active surfaces of a single

catalyst particle, the internal and external catalyst surfaces occupied by the deposited carbon and the efficiency of a catalyst

particle, respectively. The surface intrinsic kinetic constants, k1 and k2, can be expressed by Arrhenius’s law as first order of

approximation as

k1 ¼ k01 e�E1=RT

k2 ¼ k02 e�E2=RT (A.12)

where k01, k0

2, E1 and E2 are the catalyst and carbon frequency factors and the activation energies of the catalyst and of the carbon

deposits, respectively. The efficiency of the catalytic particle is a function of time accounting for the effective surface active

inside the particle and the reduction of pore dimension due to the carbon deposition. The relationship between the efficiency

and the Thiele number [14], YðSi1ðtÞ;S

i2ðtÞÞ; in this case, is

ZðSi1ðtÞ;S

i2ðtÞÞ ¼

3YðSi1ðtÞ; S

i2ðtÞÞ cothð3YðSi

1ðtÞ; Si2ðtÞÞÞ � 1

3YðSi1ðtÞ; S

i2ðtÞÞ

2(A.13)

YðSi1ðtÞ; S

i2ðtÞÞ ¼

dc

6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik1Si

1ðtÞ þ k2Si2ðtÞ

VcDðSi2ðtÞÞ

s(A.14)

DðSi2ðtÞÞ ¼

11

Deffþ 1

DkðSi2ðtÞÞ

Deff ¼DCH4

�Ct

DkðSi2ðtÞÞ ¼ 485

�Ct dp � 2

wicðtÞScd

Si hcd

� � ffiffiffiffiffiffiffiffiffiffiffiT

MCH4

q8>><>>: (A.15)

where dc, DðSi2ðtÞÞ; Deff, DkðS

i2ðtÞÞ; ec, t, dp, wi

cðtÞ; Scd, Si, hcd, T and MCH4are the catalyst particle diameter, the mean, bulk and

Knudsen diffusivity inside the particle, the porosity, the tortuosity factor and the mean pore diameter of a catalyst particle, the

carbon deposited on the internal surface of a single bed particle, the surface of catalyst particle occupied by deposited carbon per

carbon mass, the internal surface of a catalyst particle, the thickness of the deposited carbon, temperature and molecular weight

of methane, respectively. It is noteworthy that the Thiele number varies during reaction time as a result of changes in pore

diameter and in catalyst particle activity due to carbon deposition. The actual average pore size is calculated by considering that

the carbon deposits uniformly along the pores reducing, in turn, the diameter. The temporal evolution of the deactivation

process has been taken into account considering the variation of the active internal and external surfaces of the catalyst particle

with time.

The mass transfer coefficient outside the particle kg can be evaluated from the expression derived by Hayhurst and Parmar for

fluidized beds [30]

Sh ¼dckg

DCH4

¼ 2�mf þ 0:68Remf

�mf

� �1=2

� Sc1=3 (A.16)

where Sh, Remf and Sc are the Sherwood number, the Reynolds number at minimum fluidizing conditions and the Schimdt

number [31], respectively.

ARTICLE IN PRESS


Appendix B. Carbon deposition and removal

The deactivation process occurring on the internal and external surfaces of the catalytic particles can be modelled taking into

account that:

(I)
The transient mass balance on the carbon deposited on the external surface of a single bed particle, wecðtÞ; is determined
by both the effective external reaction rate of carbon accumulation proportional to the methane decomposition rate and

the removal rate of carbon by the mechanical abrasion of bed particles:

dwecðtÞ

dt¼ ke

pðtÞceðtÞVcMC � kawecðtÞðUð1þ 0:5�XðtÞÞ � UmfÞ (B.1)

where kepðtÞ is the external surface reaction constant, MC is carbon molecular weight, ka is the carbon attrition constant.

i
(II) The transient mass balance on the carbon deposited on the internal surface of a single bed particle, wcðtÞ; is determined
by the effective internal reaction rate of carbon accumulation proportional to the methane decomposition rate:

dwicðtÞ

dt¼ ki

pðtÞceðtÞVcMC (B.2)

where kipðtÞ is the internal surface reaction constant.

e
(III) The transient balance on the external active surface of a single catalyst particle, S1ðtÞ; is similar to the mass balance on
the deposited carbon, but it is referred to the external active catalyst surface:

dSe1ðtÞ

dt¼ �Scdðk

epðtÞceðtÞVcMC � kawe

cðtÞðUð1þ 0:5�XðtÞÞ � Umf ÞÞ. (B.3)

i
(IV) The transient balance on the internal active surface of a single catalyst particle, S1ðtÞ; only accounts for the
accumulation of carbon deposits on the internal active surface of catalyst:

dSi1ðtÞ

dt¼ �Scdki

pðtÞceðtÞVcMC (B.4)

e i
The external and internal surfaces’ reaction constants, kpðtÞ and kpðtÞ; can be expressed as
kepðtÞ ¼

1

ðVc=kgSeÞ þ ðVc=ðk1Se

1ðtÞ þ k2Se2ðtÞÞÞ þ ððk1Si

1ðtÞ þ k2Si2ðtÞÞZðS

i1ðtÞ; S

i2ðtÞÞVcÞ=ððk1Se

1ðtÞ þ k2Se2ðtÞÞkgSe

ÞÞ(B.5)

kipðtÞ ¼

1

ðVc=kgSeÞ þ ðVc=ððk1Si


i1ðtÞ;S

i2ðtÞÞÞÞ þ ðððk1Se

1ðtÞ þ k2Se2ðtÞÞVcÞ=ððk1Si


i1ðtÞ; S

i2ðtÞÞkgSe

ÞÞ(B.6)

and the active and the deactivated surfaces are related to each other by

Se1ðtÞ þ Se

2ðtÞ ¼ Se

Si1ðtÞ þ Si

2ðtÞ ¼ Si

((B.7)

The carbon elutriation rate, Ec(t), can be calculated by

EcðtÞ ¼ kaðUð1þ 0:5�XðtÞÞ �Umf ÞWcðtÞ ¼ kaðUð1þ 0:5�XðtÞÞ � Umf ÞwecðtÞNc (B.8)

where Wc(t) and Nc are the carbon deposited on the external surface of bed particles and the number of catalyst particles,

respectively. Nc can be evaluated as

Nc ¼pD2

r=4Hmfrb

Vcrc(B.9)

The production rates of carbon, PC, and of hydrogen, PH2; can be calculated by

PCðtÞ ¼MCpD2

r

4Uc0XðtÞ (B.10)

PH2ðtÞ ¼ 2MH2

pD2r

4Uc0XðtÞ (B.11)

Eqs. (A.1), (A.2), (A.4)–(A.16) and (B.5)–(B.7) have been rear-

ranged and, then, substituted in Eqs. (B.1)–(B.4) in order to

obtain four independent first order differential equations.

This differential equations system has been solved in

Mathcad environment using a fourth order Runge–Kutta

algorithm with adaptive integration step size. The computed

results of the differential equations system are used to

evaluate the exit methane conversion, the carbon elutriation

rate and the carbon and hydrogen production rates by means

of Eqs. (A.3) and (B.8)–(B.11), respectively. All the variables

ARTICLE IN PRESS


involved were evaluated as functions of time during the TCD

of methane.

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