CET 96

DOI: 10.3303/CET2296072

Paper Received: 20 December 2021; Revised: 14 July 2022; Accepted: 14 July 2022
Please cite this article as: Borgogna A., Iaquaniello G., Biagioni V., Murmura M.A., Annesini M.C., Cerbelli S., 2022, Estimate of the Height of
Molten Metal Reactors for Methane Cracking, Chemical Engineering Transactions, 96, 427-432 DOI:10.3303/CET2296072

CHEMICAL ENGINEERING TRANSACTIONS

VOL. 96, 2022

A publication of

The Italian Association

of Chemical Engineering
Online at www.cetjournal.it

Guest Editors: David Bogle, Flavio Manenti, Piero Salatino

ISBN 978-88-95608-95-2; ISSN 2283-9216

Estimate of the Height of Molten Metal Reactors for Methane
Cracking

Alessia Borgognac, Gaetano Iaquanielloa, Valentina Biagionib, Maria Anna
Murmurab, Maria Cristina Annesinib, Stefano Cerbellib,*
a NextChem SpA, Via di Vannina 88, 00156 Roma (Italy)
b Dip. di Ingegneria Chimica Materiali Ambiente - Sapienza Università di Roma, Via Eudossiana 18, 00184 Roma (Italy)
c MyRechemical srl, Via di Vannina 88, 00156 Roma (Italy)
Stefano.cerbelli@uniroma1.it

Methane Cracking represents one of the most promising routes to CO2-free hydrogen production.The methane
decomposition reaction is typically carried out in fixed or fluidized catalytic beds, where the metal catalyst is
supported on porous ceramic particles. By proper choice of the metal catalyst, the catalytic reaction environment
allows to obtain sizeable reaction rates at operating temperatures as low as 700°C. Besides, in solid catalytic
beds, the catalyst is swiftly deactivated due to the massive (i.e. stoichiometric) deposition of the solid carbon
product. One way to bypass carbon deposition is to use a molten metal bath (which may or may not contain
catalytic metal components) as a reaction environment, where methane bubbles are introduced at the bottom
of the bath and are progressively converted as they rise through the liquid metal. The key point of this process
is that, owing to a large density difference between the solid carbon phase and the molten metal, the solid
product of the reaction floats on top of the liquid metal and can be thus mechanically skimmed. In this article,
we develop an analytical approach to the estimate of the bath height, which constitutes one of the most critical
design parameters of the process. Specifically, based on the observation that in practical applications the
reacting bubble is in the kinetics-controlled regime, we obtain the conversion vs time solution for a bubble of
given initial size. On the assumption of ideal gaseous mixture behaviour, the knowledge of the conversion curves
allows to estimate the bubble diameter as a function of time during the rise of the bubble through the molten
metal. This piece of information is then post-processed to obtain the bubble motion as a function of time. The
elimination of the time parameter between the two solutions allows to construct a conversion-height map for
different diameters of the bubbles.

1. Introduction

Methane (or, more generally, natural gas) Cracking (MC) represents an attractive route to CO2-free hydrogen
production, the main other alternative being electrolysis. The MC reaction is typically carried out in fixed or
fluidized catalytic beds, where the metal catalyst is supported on porous ceramic particles. This way, appreciable
reaction rates can be obtained at temperatures as low as 700°C. The major shortcoming associated with the
solid catalyst is its swift deactivation, which is caused by the massive (i.e. stoichiometric) deposition of the solid
carbon phase [Parolin et Al, 2021]. Frequent catalysis regeneration is thus required in the process, which
impacts adversely on operational costs. Furthermore, in cases where catalyst regeneration is obtained through
oxidation of the deposited carbon, the CO2-free character of the overall reaction is altogether lost. For this
reason, alternatives that avoid the presence of solid catalyst have been sought. One way to bypass carbon
deposition is to use a molten metal bath (which may or may not contain catalytic components) as a reaction
environment, where methane bubbles are introduced at the bottom of the bath through a sparger and are
progressively converted to hydrogen as they rise through the liquid metal [see, e.g. Pérez et Al, 2013 and therein
cited references]. The key point of this process is that, due to a sizeable density difference between the solid
carbon phase and the molten metal, the solid carbon produced by the reaction migrates to the top of the liquid
metal and can be thus skimmed mechanically.

427

Notwithstanding its numerous advantages, catalytic cracking of methane in molten liquid baths is still far from
industrial maturity, as only laboratory prototypes or small pilot plants have been constructed to date. This is
mainly due to the fact that the theoretical modelling of the process, which should serve as guidance for the
experimental design, must address a complex intertwined phenomenology, where transport-reaction
phenomena occurring in the gas phase within the bubbles are influenced by the external conditions through
momentum and heat transfer between the two phases. In this respect, the availability of simple design criteria
could provide a useful starting point to select the range of operating conditions of the process. Hinging on some
simplifying assumptions, in this contribution we develop an analytical approach for the estimate of the bath
height, which constitutes one of the most critical design parameters. To this end, we first derive the conversion
vs time solution for a generic gas bubble. On the assumption of ideal gaseous mixture behaviour, the bubble
diameter as a function of time during the rise of the bubble can thus be obtained. In turn, the knowledge of the
bubble size as a function of time allows to determine the vertical motion of the bubble.

2. Statement of the problem and modelling

The physical setting next described is one where pure methane at assigned flowrate is fed through a sparger at
the bottom of a liquid metal bath that is externally heated. Owing to its large thermal conductivity, the liquid metal
is assumed at uniform temperature, independently of the heat duty required by the cracking reaction. Depending
on the type of sparger used, the characteristic bubble diameter ranges from order few micrometers to few
millimeters [Pérez et Al, 2013 ]. The wide range of bubble diameters characterizing the experimental prototypes
discussed in the literature suggests that a preliminary analysis of the transport-reaction regimes governing the
overall (effective) rate of methane conversion inside the bubbles may be in place to orient the
theoretical/numerical modelling of the process. In this analysis, the most critical aspect is to understand where
the elementary reaction steps are taking place, i.e. whether in the bulk of the gas phase or at the gas/liquid
metal interface. Many authors consider the non-catalytic methane cracking reaction as occurring in the bulk of
the gas [Catalan et Al, 2020] when the temperature exceeds 103 ◦C. Besides, in conditions where metal
components possessing catalytic activity towards methane cracking are present in the molten metal bath, the
rate of the surface (heterogeneous) reaction at the gas-metal interface largely overcomes that of the bulk
(homogeneous) reaction [Upham et Al, 2017]. In what follows, we consider this to be the case. As a
consequence, we neglect the effect of the bulk gas reaction on methane conversion. From a thermodynamic
standpoint, the methane cracking reaction is an endothermic reaction

𝐶𝐻4(𝑔) ⇄ 𝐶(𝑠) + 2𝐻2(𝑔), ∆𝐻298.15

0 = 74 𝑘𝐽/𝑚𝑜𝑙 (1)

whose advancement increases (doubles) the total number of moles. As such, methane cracking is favoured at
high temperatures and low pressures. In what follows, we take 𝑇 = 1000 Co and 𝑃 = 1 atm as representative
operating conditions for the temperature and the pressure, respectively.
The reaction kinetics is complex and not yet completely understood. Many authors report that depending on the
conditions and on the catalytic environment, different reaction pathways can be observed, some of which are
associated with ethane, ethylene, and acetylene intermediates. In what follows, we consider a pseudo-first order
expression [Upham et Al, 2017], where a linear first-order reaction kinetics with respect to methane
concentration is enforced up to the equilibrium value. At equilibrium, the reaction rate is assumed to drop to
zero instantaneously, in order to make the kinetic expression consistent with thermodynamics. Thus, the kinetic
expression is given by

(−𝑟𝐶𝐻4 ) = {
𝑘_𝑠 𝐶𝐶𝐻4 𝑓𝑜𝑟 𝑐𝐶𝐻4 < 𝑐𝐶𝐻4

𝑒𝑞

0 𝑓𝑜𝑟 𝑐𝐶𝐻4 ≥ 𝑐𝐶𝐻4
𝑒𝑞 (2)

The conversion of methane in the rising bubble results from a two-step process, namely the diffusion of methane
from the center of the bubble to the gas-liquid interface and the surface reaction, the first associated with a
characteristic time 𝜏𝐷 = 𝑅02 ∕ 𝔇, the second characterized by a characteristic time 𝜏𝑅 = 𝑅0/𝑘𝑠. Here, 𝔇 is the
diffusion coefficient of methane in the methane/hydrogen gaseous mixture, whereas 𝑘𝑠[𝑚/𝑠] is the kinetic
constant of the heterogeneous (surface) reaction. The ratio of these characteristic times can be regarded as a
modified Thiele modulus, henceforth denoted as 𝑇ℎ, which quantifies the relative importance of transport with
respect to kinetics. Depending on the value of this parameter, qualitatively different dynamical behaviors of the
reacting system can be observed (see Figure 1).

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Figure 1: Qualitative evolution of methane concentration profiles for different values of the Thiele modulus.

At low values of Th, the rate-limiting step is the surface reaction: the methane concentration profile is essentially
flat and depends only on time. At the other extreme, i.e. for Th>1, the reaction is totally under transport control
and methane concentration at the reaction interface vanishes at all times t>0. In this case, the apparent
(effective) reaction rate can be predicted by solving a transient pure diffusion problem equipped with zero
boundary conditions. When the Thiele modulus is of order unity the profiles and the boundary concentration
values depend on both space and time. If one excludes very high temperatures (> 1100 °C), for reasonable
values of the bubble size (diameter less than 1 cm) and even in the presence of catalytic components in the
molten metal, one finds that the typical reaction regime is controlled by kinetics in that Th < 10-3. This is the
case, e.g., of a molten metal bath composed of a mixture of 17% Ni (catalyst) in Bi (non-catalytic component)
at T=1000 °C and P= 1 atm [Upham et Al, 2017]. For this case, available experimental data suggest 𝑘𝑠 = 2.3 ∙
10−4 [m/s], a value that is two orders of magnitude larger than that of pure Bi (which does not show catalytic
activity towards the methane cracking reaction. The value associated with pure Bi can be taken as
representative of the combined effect of the homogeneous and the surface reaction in the absence of the
catalyst. Because the Ni catalyst has a negligible vapor pressure, it can be assumed that the sizeable increase
of the reaction rate is only due to the surface reaction. This implies that in catalytic metal baths, the contribution
of the homogeneous reaction can be safely neglected, an assumption that we next enforce. Given the relatively
low value of 𝑘𝑠one finds that in typical conditions 𝑘𝑠 < 10−3 so that the regime is kinetics-controlled and the
methane mole balance can be written as
𝑑𝑛𝐶𝐻4

𝑑𝑡
= −𝑘𝑠 𝑛𝐶𝐻4

𝑅(𝑡)
(3)

where R(t) is the instantaneous bubble radius. Besides, the number of moles of hydrogen generated at a given
time instant, say 𝑛𝐻2 (𝑡) is equal to 𝑛𝐻2 (𝑡) = 2(𝑛𝐶𝐻4

0 − 𝑛𝐶𝐻4 (𝑡)) which implies that the total number of moles at
any time instant is given by 𝑛𝑡𝑜𝑡(𝑡) = 2𝑛𝐶𝐻4

0 + 𝑛𝐶𝐻4 (𝑡). By assuming ideal behavior for the hydrogen/methane
mixture, one can relate the bubble volume (and hence the radius) and the total number of moles, obtaining

𝑅(𝑡) = ((
3ℛ𝑇

4𝜋𝑃
) (2𝑛𝐶𝐻4

0 − 𝑛𝐶𝐻4 (𝑡)))
1/3

, (4)

which, substituted in the methane mole balance equation, yields a first-order differential equation that can be
solved analytically by separation of variables. In turn, the solution of this equation allows to explicitly derive the
bubble size as a function of time and this piece of information can be used to integrate the equation of motion
of the bubble. Owing to the large density difference between the gas and the molten metal (four orders of
magnitude in the conditions considered), bubble inertia can be neglected, and the instantaneous bubble velocity
can be computed by balancing Archimede’s lift and viscous drag by the molten metal. Thus, the bubble velocity
v(t) is computed as the root of
(𝜌𝑚 − 𝜌𝑔 )

3
𝜋𝑅3(𝑡) =

2
𝑓(𝑅𝑒) 𝜌𝑚 𝑣

2(𝑡) 4𝜋𝑅2(𝑡) (5)
Where f(Re) is the friction factor for viscous fluid motion around a sphere. Note that the bubble velocity is also
present implicitly within the friction factor. By solving this equation for any given value of R(t) one obtains the
bubble velocity as a function of time, and thus, upon integration of the latter, the bubble motion z(t), where z
measures the current height of the center of the bubble with respect to the gas sparger. Eliminating the
parameter t between 𝑛𝐶𝐻4 (𝑡) and z(t), a direct relationship between the methane conversion and bath height
can be made explicit.

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3. Results

3.1 Methane conversion and bubble motion

In what follows, the advancement of the cracking reaction is quantified in dimensionless terms by computing the

conversion 𝑋(𝑡) =
𝑛𝐶𝐻4

0 −𝑛𝐶𝐻4 (𝑡)

𝑛𝐶𝐻4
0 . Given that both temperature and pressure are kept constant in this analysis,

the performance of the reaction is evaluated as a function of the initial bubble size 𝑅(𝑡 = 0) = 𝑅0, which is
assumed to range between 50 𝜇𝑚 𝑎𝑛𝑑 5 𝑚𝑚. Figure 2-A depicts the conversion vs time curves for different
values of the initial bubble size (increasing in the direction of the arrow).

Figure 2: Panel A. Conversion versus time for a bubble of pure methane entering a liquid Ni-Bi metal bath at

T=1000 °C and P=1atm. The initial bubble size 𝑅0increases in the direction of the arrow as 𝑅0 =

50 𝜇𝑚; 100 𝜇𝑚; 200 𝜇𝑚; 500 𝜇𝑚; 1 𝑚𝑚; 2 𝑚𝑚; 5 𝑚𝑚 . Panel B: bubble velocity vs time for the same bubble

dimensions of Panel A.

The data show that the characteristic conversion time is strongly dependent on the initial size of the bubble. For
instance, the time to reac 80% conversion ranges from order one second up to 40 seconds when goimng from
the smallest to the largest bubble size considered. Panel B of the same figures shows the instantaneous bubble
velocity. All of the curves show the same trend where the bubble velocity is weakly increasing with time until it
saturates towards a constant value which is about 10% higher than the initial bubble velocity. Here, the bubble
acceleration is due to the increasing bubbles size, which, in turn, is a consequence of the fact that the total
number of moles in the gas phase increases with time. The analysis of the instantaneous Reynolds number
shows that the breakdown of laminar (Stokes) regime occurs when the initial bubble size is equal to 𝑅0 =
100 𝜇𝑚 (not shown for brevity).

3.2 Conversion-height map

Upon integrating the time-dependent velocity, the bubble motion 𝑧(𝑡) can be obtained, and the elimination of
the time parameter between X(t) and 𝑧(𝑡) allows to derive a direct relationship between the overall height of
the reaction bath and the conversion within the bubble when the latter rises from the molten metal.

430

Figure 3: Conversion-height maps for different bubble sizes. The initial bubble radius increases in the direction

of the arrow.

Figure 3 shows the results of this approach as a function of the initial bubble radius. As can be gathered by
inspection, the height values range in a wider range than the characteristic conversion times (five orders of
magnitude are here involved whereas conversion times are distributed in less than three decades, see Figure
2-A). This is ultimately due to a different scaling of Archimedean’s lift (proportional to 𝑅3(𝑡)) with respect to
viscous drag forces, whose dependence on the radius shifts from a proportionality relationship, 𝐹~𝑅(𝑡), in the
Stokes regime, to a square dependence 𝐹~𝑅2(𝑡) when the bubble enters the turbulent regime.
By considering a fixed conversion target, say 𝑋%, a different representation of the data depicted in Fig. 3 can be
obtained in terms of height, 𝐻(𝑋%), needed to obtain the conversion target as a function of the initial bubble
size (see Figure 4). This representation makes it evident the occurrence of a shift between two regimes when
the bubble size increases. For small bubble diameters, 𝐻(𝑋%) increases approximately as the square root of
the bubble size, whereas for large bubbles the increase obeys a 3/2 type power-law. It is worth highlighting that
the results depicted in Fig. 4 are quantitatively consistent with the experimental data reported in [Upham et at.,
2017] which find a reactor height of the order of one meter to obtain conversion higher than 90% with an initial
bubble radius of 1 mm.

Figure 4: Height to obtain a target conversion value as a function of the initial bubble radius (see main text for

details).

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4. Conclusions

Molten metal baths represent a promising reaction environment to carry out methane cracking due to the natural
separation of the solid carbon phase which prevents catalyst deactivation. We derive an analytical model for
predicting methane conversion rate as a function of the residence time of the bubble. The model is based on
the assumption that the prevalent contribution to methane conversion is due to the heterogeneous surface
reaction at the gas/liquid-metal interface. This assumption is supported by experimental data in the literature
that estimate that in favourable conditions (T=1000 °C, P=1atm, presence of metal catalyst) the rate of reaction
in the bulk gas phase two orders of magnitude lower than the heterogeneous reaction). Furthermore, an order-
of-magnitude analysis shows that in these conditions the reacting system is in the kinetics-controlled regime,
which allows to assume that spatial concentration gradients inside the bubble can be neglected. Based on these
assumptions and enforcing the validity of ideal gas behavior inside for the hydrogen/methane gas phase, we
derive an analytical solution for the molar concentration of the species and the bubble size, wich increases with
time due to the increasing number of moles cause by the advancement of the reaction. The influence of the
initial bubble size on the conversion time and the bath height is analyzed at fixed temperature and pressure.
Two regimes are identified, where the bath height scale with different power laws depending on the initial bubble
size. Future work on the subject will address the influence of temperature and pressure on the reactor
performance.

References

Catalan, L. J., & Rezaei, E., 2020, Coupled hydrodynamic and kinetic model of liquid metal bubble reactor for
hydrogen production by noncatalytic thermal decomposition of methane, International Journal of Hydrogen
Energy, 45(4), 2486-2503.

Murmura, M. A., Cerbelli, S., & Annesini, M. C., 2017, Progress on modeling and design of membrane reactors
for hydrogen production, Chemical Engineering Transactions, 57, 919-924.

Parolin, G., Borgogna, A., Iaquaniello, G., Salladini, A., & Cerbelli, S., 2021, Deactivation-induced dynamics of
the reaction front in a fixed-bed catalytic membrane reactor: Methane cracking as a case study, International
Journal of Hydrogen Energy, 46(38), 20159-20170.

Pérez, B. J. L., Jiménez, J. A. M., Bhardwaj, R., Goetheer, E., van Sint Annaland, M., & Gallucci, F., 2013,
Methane pyrolysis in a molten gallium bubble column reactor for sustainable hydrogen production: Proof of
concept & techno-economic assessment, International Journal of Hydrogen Energy, 46(7), 4917-4935.

Upham, D. C., Agarwal, V., Khechfe, A., Snodgrass, Z. R., Gordon, M. J., Metiu, H., & McFarland, E. W., 2017,
Catalytic molten metals for the direct conversion of methane to hydrogen and separable
carbon. Science, 358(6365), 917-921.

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Estimate of the Height of Molten Metal Reactors for Methane Cracking